Properties

 Label 4009.2 Level 4009 Weight 2 Dimension 660505 Nonzero newspaces 64 Sturm bound 2.6712e+06

Defining parameters

 Level: $$N$$ = $$4009\( 4009 = 19 \cdot 211$$ \) Weight: $$k$$ = $$2$$ Nonzero newspaces: $$64$$ Sturm bound: $$2671200$$

Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(4009))$$.

Total New Old
Modular forms 671580 667613 3967
Cusp forms 664021 660505 3516
Eisenstein series 7559 7108 451

Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(4009))$$

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
4009.2.a $$\chi_{4009}(1, \cdot)$$ 4009.2.a.a 1 1
4009.2.a.b 3
4009.2.a.c 71
4009.2.a.d 75
4009.2.a.e 82
4009.2.a.f 83
4009.2.d $$\chi_{4009}(4008, \cdot)$$ n/a 350 1
4009.2.e $$\chi_{4009}(1702, \cdot)$$ n/a 704 2
4009.2.f $$\chi_{4009}(634, \cdot)$$ n/a 700 2
4009.2.g $$\chi_{4009}(647, \cdot)$$ n/a 636 2
4009.2.h $$\chi_{4009}(1280, \cdot)$$ n/a 704 2
4009.2.i $$\chi_{4009}(704, \cdot)$$ n/a 1272 4
4009.2.l $$\chi_{4009}(2953, \cdot)$$ n/a 700 2
4009.2.m $$\chi_{4009}(1703, \cdot)$$ n/a 704 2
4009.2.n $$\chi_{4009}(2336, \cdot)$$ n/a 704 2
4009.2.u $$\chi_{4009}(1281, \cdot)$$ n/a 704 2
4009.2.v $$\chi_{4009}(58, \cdot)$$ n/a 1908 6
4009.2.w $$\chi_{4009}(1251, \cdot)$$ n/a 2106 6
4009.2.x $$\chi_{4009}(423, \cdot)$$ n/a 2100 6
4009.2.y $$\chi_{4009}(196, \cdot)$$ n/a 2106 6
4009.2.z $$\chi_{4009}(1500, \cdot)$$ n/a 1400 4
4009.2.bc $$\chi_{4009}(645, \cdot)$$ n/a 2100 6
4009.2.bf $$\chi_{4009}(201, \cdot)$$ n/a 2816 8
4009.2.bg $$\chi_{4009}(134, \cdot)$$ n/a 2544 8
4009.2.bh $$\chi_{4009}(1337, \cdot)$$ n/a 2800 8
4009.2.bi $$\chi_{4009}(83, \cdot)$$ n/a 2816 8
4009.2.bj $$\chi_{4009}(15, \cdot)$$ n/a 2106 6
4009.2.bn $$\chi_{4009}(648, \cdot)$$ n/a 2106 6
4009.2.bo $$\chi_{4009}(421, \cdot)$$ n/a 2112 6
4009.2.bs $$\chi_{4009}(254, \cdot)$$ n/a 4224 12
4009.2.bt $$\chi_{4009}(476, \cdot)$$ n/a 3816 12
4009.2.bu $$\chi_{4009}(144, \cdot)$$ n/a 4200 12
4009.2.bv $$\chi_{4009}(178, \cdot)$$ n/a 4224 12
4009.2.bw $$\chi_{4009}(825, \cdot)$$ n/a 2816 8
4009.2.cd $$\chi_{4009}(322, \cdot)$$ n/a 2816 8
4009.2.ce $$\chi_{4009}(221, \cdot)$$ n/a 2816 8
4009.2.cf $$\chi_{4009}(445, \cdot)$$ n/a 2800 8
4009.2.ci $$\chi_{4009}(96, \cdot)$$ n/a 7632 24
4009.2.cj $$\chi_{4009}(810, \cdot)$$ n/a 4224 12
4009.2.cq $$\chi_{4009}(94, \cdot)$$ n/a 4224 12
4009.2.cr $$\chi_{4009}(31, \cdot)$$ n/a 4224 12
4009.2.cs $$\chi_{4009}(12, \cdot)$$ n/a 4200 12
4009.2.cv $$\chi_{4009}(100, \cdot)$$ n/a 8424 24
4009.2.cw $$\chi_{4009}(55, \cdot)$$ n/a 8448 24
4009.2.cx $$\chi_{4009}(137, \cdot)$$ n/a 8424 24
4009.2.cy $$\chi_{4009}(43, \cdot)$$ n/a 12636 36
4009.2.cz $$\chi_{4009}(123, \cdot)$$ n/a 12672 36
4009.2.da $$\chi_{4009}(54, \cdot)$$ n/a 12636 36
4009.2.dd $$\chi_{4009}(18, \cdot)$$ n/a 8400 24
4009.2.dh $$\chi_{4009}(526, \cdot)$$ n/a 8448 24
4009.2.di $$\chi_{4009}(128, \cdot)$$ n/a 8424 24
4009.2.dm $$\chi_{4009}(10, \cdot)$$ n/a 8424 24
4009.2.dn $$\chi_{4009}(30, \cdot)$$ n/a 16896 48
4009.2.do $$\chi_{4009}(11, \cdot)$$ n/a 16800 48
4009.2.dp $$\chi_{4009}(20, \cdot)$$ n/a 15264 48
4009.2.dq $$\chi_{4009}(45, \cdot)$$ n/a 16896 48
4009.2.du $$\chi_{4009}(40, \cdot)$$ n/a 12672 36
4009.2.dv $$\chi_{4009}(32, \cdot)$$ n/a 12636 36
4009.2.dz $$\chi_{4009}(110, \cdot)$$ n/a 12636 36
4009.2.ec $$\chi_{4009}(8, \cdot)$$ n/a 16800 48
4009.2.ed $$\chi_{4009}(141, \cdot)$$ n/a 16896 48
4009.2.ee $$\chi_{4009}(75, \cdot)$$ n/a 16896 48
4009.2.el $$\chi_{4009}(145, \cdot)$$ n/a 16896 48
4009.2.em $$\chi_{4009}(4, \cdot)$$ n/a 50544 144
4009.2.en $$\chi_{4009}(5, \cdot)$$ n/a 50688 144
4009.2.eo $$\chi_{4009}(47, \cdot)$$ n/a 50544 144
4009.2.ep $$\chi_{4009}(41, \cdot)$$ n/a 50544 144
4009.2.et $$\chi_{4009}(2, \cdot)$$ n/a 50544 144
4009.2.eu $$\chi_{4009}(60, \cdot)$$ n/a 50688 144

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

Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(4009))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(4009)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(19))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(211))$$$$^{\oplus 2}$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ ($$1 + T + 2 T^{2}$$)($$1 - 2 T + 5 T^{2} - 7 T^{3} + 10 T^{4} - 8 T^{5} + 8 T^{6}$$)
$3$ ($$1 - 2 T + 3 T^{2}$$)($$1 + 2 T + 8 T^{2} + 11 T^{3} + 24 T^{4} + 18 T^{5} + 27 T^{6}$$)
$5$ ($$1 - 3 T + 5 T^{2}$$)($$1 + 3 T + 11 T^{2} + 17 T^{3} + 55 T^{4} + 75 T^{5} + 125 T^{6}$$)
$7$ ($$1 + 7 T^{2}$$)($$1 - 3 T + 17 T^{2} - 29 T^{3} + 119 T^{4} - 147 T^{5} + 343 T^{6}$$)
$11$ ($$1 + 11 T^{2}$$)($$1 - T + 24 T^{2} - 21 T^{3} + 264 T^{4} - 121 T^{5} + 1331 T^{6}$$)
$13$ ($$1 - 4 T + 13 T^{2}$$)($$( 1 + 13 T^{2} )^{3}$$)
$17$ ($$1 + 17 T^{2}$$)($$1 + 2 T + 50 T^{2} + 67 T^{3} + 850 T^{4} + 578 T^{5} + 4913 T^{6}$$)
$19$ ($$1 + T$$)($$( 1 - T )^{3}$$)
$23$ ($$1 + T + 23 T^{2}$$)($$1 - 2 T + 68 T^{2} - 91 T^{3} + 1564 T^{4} - 1058 T^{5} + 12167 T^{6}$$)
$29$ ($$1 - 3 T + 29 T^{2}$$)($$1 - 6 T + 92 T^{2} - 335 T^{3} + 2668 T^{4} - 5046 T^{5} + 24389 T^{6}$$)
$31$ ($$1 - 4 T + 31 T^{2}$$)($$1 - 5 T + 57 T^{2} - 143 T^{3} + 1767 T^{4} - 4805 T^{5} + 29791 T^{6}$$)
$37$ ($$1 + 37 T^{2}$$)($$1 - 9 T + 47 T^{2} - 217 T^{3} + 1739 T^{4} - 12321 T^{5} + 50653 T^{6}$$)
$41$ ($$1 - 10 T + 41 T^{2}$$)($$1 + 7 T + 95 T^{2} + 581 T^{3} + 3895 T^{4} + 11767 T^{5} + 68921 T^{6}$$)
$43$ ($$1 - 6 T + 43 T^{2}$$)($$1 + T + 120 T^{2} + 85 T^{3} + 5160 T^{4} + 1849 T^{5} + 79507 T^{6}$$)
$47$ ($$1 + 47 T^{2}$$)($$1 + 3 T + 53 T^{2} + 479 T^{3} + 2491 T^{4} + 6627 T^{5} + 103823 T^{6}$$)
$53$ ($$1 - 6 T + 53 T^{2}$$)($$1 + 27 T + 395 T^{2} + 3535 T^{3} + 20935 T^{4} + 75843 T^{5} + 148877 T^{6}$$)
$59$ ($$1 - 11 T + 59 T^{2}$$)($$1 + 2 T + 148 T^{2} + 249 T^{3} + 8732 T^{4} + 6962 T^{5} + 205379 T^{6}$$)
$61$ ($$1 + 61 T^{2}$$)($$1 - 4 T + 179 T^{2} - 480 T^{3} + 10919 T^{4} - 14884 T^{5} + 226981 T^{6}$$)
$67$ ($$1 - 2 T + 67 T^{2}$$)($$1 - 8 T + 45 T^{2} + 280 T^{3} + 3015 T^{4} - 35912 T^{5} + 300763 T^{6}$$)
$71$ ($$1 + 9 T + 71 T^{2}$$)($$1 + T + 127 T^{2} - 195 T^{3} + 9017 T^{4} + 5041 T^{5} + 357911 T^{6}$$)
$73$ ($$1 + 15 T + 73 T^{2}$$)($$1 - 12 T + 246 T^{2} - 1739 T^{3} + 17958 T^{4} - 63948 T^{5} + 389017 T^{6}$$)
$79$ ($$1 - 5 T + 79 T^{2}$$)($$1 + 33 T + 593 T^{2} + 6475 T^{3} + 46847 T^{4} + 205953 T^{5} + 493039 T^{6}$$)
$83$ ($$1 + 83 T^{2}$$)($$1 + 7 T + 221 T^{2} + 959 T^{3} + 18343 T^{4} + 48223 T^{5} + 571787 T^{6}$$)
$89$ ($$1 - 9 T + 89 T^{2}$$)($$1 - 4 T - 24 T^{2} + 1641 T^{3} - 2136 T^{4} - 31684 T^{5} + 704969 T^{6}$$)
$97$ ($$1 - 11 T + 97 T^{2}$$)($$1 + 7 T + 291 T^{2} + 1309 T^{3} + 28227 T^{4} + 65863 T^{5} + 912673 T^{6}$$)