Properties

Label 4009.2
Level 4009
Weight 2
Dimension 660505
Nonzero newspaces 64
Sturm bound 2671200

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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

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