Comer Coppie is a modest, unassuming fellow who sits in a modest, unassuming office on Lower Broadway playing with numbers. Big numbers. The numbers that will determine the future of New York City - if it has one.

Coppie does not exactly work for Gov. Hugh L. Carey, and he does not exactly work against Mayor Edward I. Koch. He is the financial "bridge" between them - but in terms of raw fiscal power, he may have more than either one.

Who else in the Big Applle can cancel any proposed city expenditure of $100,000 or more, simply on his say-so? Who else can invalidate a negotiated labor agreement just because he suspects it will be financially risky for the city? Who else has final authority over every penny of the next four New York City budgets?

No one but Coppie. He is executive director of the Financial Control Board, the agency set up by the state legislature in 1975 to oversee the financial recovery of its wayward child New York City.

Coppie assumed his $57,500-a-year job last November. For the previous nine years, he was the $47,500-a-year budget director of Washington, D.C., and a confidant of former Mayor Walter E. Washington.

"I had planned to stay on (in and with Washington)," Coppie says, But then came what he wryly calls "the events of last fall" - Washington's defeat in the D.C. Democratic primary by Marion Barry, who went on to be elected mayor last November.

In the same week that Barry was winning the general election, Coppie was becoming the first top-ranked staffer from Washington's first home rule government to find a new job.

Barry offered Coppie a job in the new city administration - although Barry specified that it would not be the same one Coppie held under Walter Washington. But Coppie found the New York offer "a challenge to my expertise and commitment I couldn't turn down." His D.C. job went to Gladys Mack, formerly his deputy.

Coppie, who is 47, still is in the process of selling his Northwest Washington home. He spends the week with friends on the East Side here, then rides a Friday night train back to Washington to spend the weekend with his wife and three children. On Sunday nights, he reverses the process. The Coppie family plans to buy a house in Queens County this month.

The New York financial situation that Coppie inherited may be turning the corner, he thinks.

In 1975, New York City's bond rating was in minus numbers, private industry was beginning to desert what looked like a sinking ship, and many people on Capitol Hill wanted to let the city twist slowly in the wind to teach it a lesson.

As Coppie was arriving last fall, however, New York was preparing to offer its first notes in three years (it has since done so, to a solid reception on Wall Street). And under the Koch administration, the city has balanced its budget by the state's legal definition of the term (although not yet "according to generally accepted accounting principles," Coppie says).

One symbol of a changed situation was the new name the state legislature gave to the agency Coppie heads. It used to be called the Emergency Financial Control Board. The first word was officially dropped by the legislature in 1978, at the same time that the agency was given life until 2008.

The name change was "both cosmetic and substantive," says Coppie, a man so thoughtful and careful that he will sometimes spend 20 silent seconds groping for the right adjective.

"I think at the very least it means we're (pause . . . pause) institutionalized."

Coppie's role as executive director of the control board is one of persuasion.

He is not the governor's "heavy," peering over the shoulders of City Hall's bookkeepers. For example, he has not yet vetoed anything Koch has done or proposed. Nor is he an errandrunner when funds have to be gotten from Washington. He is a man whose job is "to strike a balance, to move toward a point constructively."

Coppie says Koch and his staff have exhibited "a great attitude" so far, even though "I suspect they fantasize that one day I will simply go away." The suspicion is born of his experience in Washington, which "made me very sensitive to question of home rule authority," Coppie says.

He emphasizes, however, that "if we are doing our job properly, there won't be any resentment. We want to get the city back into the bond market by 1982 (as state law requires), and there is a direct correlation between our responsibility and their doing that."

Although Washington is hardly without fiscal difficulties, Coppie said his old employer "will never have the problems New York City has because it's not in the credit markets. (Washington) has been showing a successful capacity to balance its budget for some time."

But Mayor Barry could stumble on the same "15-5 rule" that bedeviled Mayor Washington, Coppie warned.

"What it means is that, in planning for a new budget year, you know that there will be a 15 percent growth in mandatory expenses and only a 5 percent growth in revenues. So you start planning with a 10 percent deficit. That makes it tough."

Another festering sore, Coppie said, is that "60 percent of all personal income is earned by non-residents of Washington." The long-debated but never-passed commuter tax would be a remedy for that, but can it ever get through the House of Representatives? I'm hopeful that it will," Coppie said. Without it, "the budget problem is a serious one."

Washington could have avoided any budget squeezes "If Mayor Washington had not made a decision in 1969 essentially to rebuild the physical plant," Coppie said.

"That meant we went from $8 million in debt service in 1968 to more than $100 million in the early 1980s. But the benefit is very real indeed. That's one of the reasons you're seeing the renaissance in Washington.

"You go into most neighborhoods in D.C., you see the traditional fire and police stations. But you also see a library, a swimming pool, a recreation center, not to mention the public school rebuilding. That's all been invaluable."

So has his experience on the executive side, Coppie said. "It makes you sensitive to the politics and the pressures," he said. "I got a lot out of my years in Washington, but one of the biggest things I got was the sensitivity to how much of a difference a spirit of cooperation makes.

"The real point here is attitude and commitment. You need them both. We've got them both in New York. I'm very optimistic." CAPTION: Picture, Comer Coppie in New York City. By Donal Holway for The Washington Post