# Properties

 Label 4031.2 Level 4031 Weight 2 Dimension 670819 Nonzero newspaces 24 Sturm bound 2.7048e+06

## Defining parameters

 Level: $$N$$ = $$4031 = 29 \cdot 139$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$24$$ Sturm bound: $$2704800$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(4031))$$.

Total New Old
Modular forms 680064 678219 1845
Cusp forms 672337 670819 1518
Eisenstein series 7727 7400 327

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(4031))$$

We only show spaces with even parity, since no modular forms exist when this condition is not satisfied. Within each space $$S_k^{\mathrm{new}}(N, \chi)$$ we list the newforms together with their dimension.

Label $$\chi$$ Newforms Dimension $$\chi$$ degree
4031.2.a $$\chi_{4031}(1, \cdot)$$ 4031.2.a.a 2 1
4031.2.a.b 59
4031.2.a.c 61
4031.2.a.d 98
4031.2.a.e 103
4031.2.b $$\chi_{4031}(1391, \cdot)$$ n/a 344 1
4031.2.e $$\chi_{4031}(320, \cdot)$$ n/a 652 2
4031.2.g $$\chi_{4031}(3057, \cdot)$$ n/a 696 2
4031.2.j $$\chi_{4031}(1710, \cdot)$$ n/a 696 2
4031.2.k $$\chi_{4031}(140, \cdot)$$ n/a 2076 6
4031.2.l $$\chi_{4031}(1433, \cdot)$$ n/a 1392 4
4031.2.p $$\chi_{4031}(557, \cdot)$$ n/a 2064 6
4031.2.q $$\chi_{4031}(181, \cdot)$$ n/a 4176 12
4031.2.r $$\chi_{4031}(175, \cdot)$$ n/a 7216 22
4031.2.s $$\chi_{4031}(416, \cdot)$$ n/a 4176 12
4031.2.u $$\chi_{4031}(42, \cdot)$$ n/a 4176 12
4031.2.z $$\chi_{4031}(57, \cdot)$$ n/a 7656 22
4031.2.ba $$\chi_{4031}(30, \cdot)$$ n/a 14344 44
4031.2.bc $$\chi_{4031}(43, \cdot)$$ n/a 8352 24
4031.2.bd $$\chi_{4031}(75, \cdot)$$ n/a 15312 44
4031.2.bf $$\chi_{4031}(28, \cdot)$$ n/a 15312 44
4031.2.bi $$\chi_{4031}(36, \cdot)$$ n/a 45936 132
4031.2.bk $$\chi_{4031}(12, \cdot)$$ n/a 30624 88
4031.2.bl $$\chi_{4031}(6, \cdot)$$ n/a 45936 132
4031.2.bo $$\chi_{4031}(7, \cdot)$$ n/a 91872 264
4031.2.bq $$\chi_{4031}(8, \cdot)$$ n/a 91872 264
4031.2.bt $$\chi_{4031}(4, \cdot)$$ n/a 91872 264
4031.2.bu $$\chi_{4031}(2, \cdot)$$ n/a 183744 528

"n/a" means that newforms for that character have not been added to the database yet

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(4031))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(4031)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(29))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(139))$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$1 - 2 T + 3 T^{2} - 4 T^{3} + 4 T^{4}$$)
$3$ ($$1 - 2 T + 5 T^{2} - 6 T^{3} + 9 T^{4}$$)
$5$ ($$( 1 - 2 T + 5 T^{2} )^{2}$$)
$7$ ($$1 + 2 T + 7 T^{2} + 14 T^{3} + 49 T^{4}$$)
$11$ ($$( 1 + 11 T^{2} )^{2}$$)
$13$ ($$( 1 - 5 T + 13 T^{2} )^{2}$$)
$17$ ($$1 - 6 T + 41 T^{2} - 102 T^{3} + 289 T^{4}$$)
$19$ ($$1 + 2 T + 21 T^{2} + 38 T^{3} + 361 T^{4}$$)
$23$ ($$1 - 12 T + 74 T^{2} - 276 T^{3} + 529 T^{4}$$)
$29$ ($$( 1 - T )^{2}$$)
$31$ ($$1 - 4 T + 58 T^{2} - 124 T^{3} + 961 T^{4}$$)
$37$ ($$1 + 2 T^{2} + 1369 T^{4}$$)
$41$ ($$( 1 - 2 T + 41 T^{2} )^{2}$$)
$43$ ($$1 + 2 T + 85 T^{2} + 86 T^{3} + 1849 T^{4}$$)
$47$ ($$1 - 4 T + 90 T^{2} - 188 T^{3} + 2209 T^{4}$$)
$53$ ($$1 - 8 T + 114 T^{2} - 424 T^{3} + 2809 T^{4}$$)
$59$ ($$1 + 12 T + 122 T^{2} + 708 T^{3} + 3481 T^{4}$$)
$61$ ($$1 - 2 T + 25 T^{2} - 122 T^{3} + 3721 T^{4}$$)
$67$ ($$1 + 2 T + 7 T^{2} + 134 T^{3} + 4489 T^{4}$$)
$71$ ($$1 + 6 T + 23 T^{2} + 426 T^{3} + 5041 T^{4}$$)
$73$ ($$1 - 6 T + 105 T^{2} - 438 T^{3} + 5329 T^{4}$$)
$79$ ($$1 - 12 T + 122 T^{2} - 948 T^{3} + 6241 T^{4}$$)
$83$ ($$( 1 + 7 T + 83 T^{2} )^{2}$$)
$89$ ($$1 + 28 T + 366 T^{2} + 2492 T^{3} + 7921 T^{4}$$)
$97$ ($$1 - 10 T + 201 T^{2} - 970 T^{3} + 9409 T^{4}$$)