Properties

Label 2304.2.c
Level $2304$
Weight $2$
Character orbit 2304.c
Rep. character $\chi_{2304}(2303,\cdot)$
Character field $\Q$
Dimension $32$
Newform subspaces $10$
Sturm bound $768$
Trace bound $49$

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Defining parameters

Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2304.c (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 12 \)
Character field: \(\Q\)
Newform subspaces: \( 10 \)
Sturm bound: \(768\)
Trace bound: \(49\)
Distinguishing \(T_p\): \(5\), \(7\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(2304, [\chi])\).

Total New Old
Modular forms 432 32 400
Cusp forms 336 32 304
Eisenstein series 96 0 96

Trace form

\( 32q + O(q^{10}) \) \( 32q - 32q^{25} + 32q^{49} - 64q^{73} - 64q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(2304, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
2304.2.c.a \(2\) \(18.398\) \(\Q(\sqrt{-2}) \) None \(0\) \(0\) \(0\) \(0\) \(q+\beta q^{5}-2\beta q^{7}-4q^{11}-2q^{13}+\beta q^{17}+\cdots\)
2304.2.c.b \(2\) \(18.398\) \(\Q(\sqrt{-2}) \) None \(0\) \(0\) \(0\) \(0\) \(q+\beta q^{5}-2\beta q^{7}-4q^{11}+2q^{13}-\beta q^{17}+\cdots\)
2304.2.c.c \(2\) \(18.398\) \(\Q(\sqrt{-2}) \) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(0\) \(0\) \(q+3\beta q^{5}-6q^{13}-5\beta q^{17}-13q^{25}+\cdots\)
2304.2.c.d \(2\) \(18.398\) \(\Q(\sqrt{-2}) \) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(0\) \(0\) \(q+\beta q^{5}-6q^{13}-3\beta q^{17}+3q^{25}+\cdots\)
2304.2.c.e \(2\) \(18.398\) \(\Q(\sqrt{-2}) \) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(0\) \(0\) \(q+3\beta q^{5}+6q^{13}+5\beta q^{17}-13q^{25}+\cdots\)
2304.2.c.f \(2\) \(18.398\) \(\Q(\sqrt{-2}) \) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(0\) \(0\) \(q+\beta q^{5}+6q^{13}+3\beta q^{17}+3q^{25}+\cdots\)
2304.2.c.g \(2\) \(18.398\) \(\Q(\sqrt{-2}) \) None \(0\) \(0\) \(0\) \(0\) \(q+\beta q^{5}+2\beta q^{7}+4q^{11}-2q^{13}+\beta q^{17}+\cdots\)
2304.2.c.h \(2\) \(18.398\) \(\Q(\sqrt{-2}) \) None \(0\) \(0\) \(0\) \(0\) \(q+\beta q^{5}+2\beta q^{7}+4q^{11}+2q^{13}-\beta q^{17}+\cdots\)
2304.2.c.i \(8\) \(18.398\) \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{24}^{5}q^{5}+\zeta_{24}^{3}q^{7}+\zeta_{24}^{6}q^{11}+\cdots\)
2304.2.c.j \(8\) \(18.398\) \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{24}^{3}q^{5}-\zeta_{24}q^{7}+\zeta_{24}^{6}q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(2304, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(2304, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 7}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(48, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(96, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(144, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(192, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(288, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(384, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(576, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(768, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1152, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ (\( 1 - 8 T^{2} + 25 T^{4} \))(\( 1 - 8 T^{2} + 25 T^{4} \))(\( 1 + 8 T^{2} + 25 T^{4} \))(\( 1 - 8 T^{2} + 25 T^{4} \))(\( 1 + 8 T^{2} + 25 T^{4} \))(\( 1 - 8 T^{2} + 25 T^{4} \))(\( 1 - 8 T^{2} + 25 T^{4} \))(\( 1 - 8 T^{2} + 25 T^{4} \))(\( ( 1 - 4 T^{2} + 25 T^{4} )^{4} \))(\( ( 1 - 4 T^{2} + 25 T^{4} )^{4} \))
$7$ (\( 1 - 6 T^{2} + 49 T^{4} \))(\( 1 - 6 T^{2} + 49 T^{4} \))(\( ( 1 - 7 T^{2} )^{2} \))(\( ( 1 - 7 T^{2} )^{2} \))(\( ( 1 - 7 T^{2} )^{2} \))(\( ( 1 - 7 T^{2} )^{2} \))(\( 1 - 6 T^{2} + 49 T^{4} \))(\( 1 - 6 T^{2} + 49 T^{4} \))(\( ( 1 - 4 T + 7 T^{2} )^{4}( 1 + 4 T + 7 T^{2} )^{4} \))(\( ( 1 - 10 T^{2} + 49 T^{4} )^{4} \))
$11$ (\( ( 1 + 4 T + 11 T^{2} )^{2} \))(\( ( 1 + 4 T + 11 T^{2} )^{2} \))(\( ( 1 + 11 T^{2} )^{2} \))(\( ( 1 + 11 T^{2} )^{2} \))(\( ( 1 + 11 T^{2} )^{2} \))(\( ( 1 + 11 T^{2} )^{2} \))(\( ( 1 - 4 T + 11 T^{2} )^{2} \))(\( ( 1 - 4 T + 11 T^{2} )^{2} \))(\( ( 1 + 14 T^{2} + 121 T^{4} )^{4} \))(\( ( 1 - 2 T^{2} + 121 T^{4} )^{4} \))
$13$ (\( ( 1 + 2 T + 13 T^{2} )^{2} \))(\( ( 1 - 2 T + 13 T^{2} )^{2} \))(\( ( 1 + 6 T + 13 T^{2} )^{2} \))(\( ( 1 + 6 T + 13 T^{2} )^{2} \))(\( ( 1 - 6 T + 13 T^{2} )^{2} \))(\( ( 1 - 6 T + 13 T^{2} )^{2} \))(\( ( 1 + 2 T + 13 T^{2} )^{2} \))(\( ( 1 - 2 T + 13 T^{2} )^{2} \))(\( ( 1 + 14 T^{2} + 169 T^{4} )^{4} \))(\( ( 1 + 14 T^{2} + 169 T^{4} )^{4} \))
$17$ (\( 1 - 32 T^{2} + 289 T^{4} \))(\( 1 - 32 T^{2} + 289 T^{4} \))(\( 1 + 16 T^{2} + 289 T^{4} \))(\( 1 - 16 T^{2} + 289 T^{4} \))(\( 1 + 16 T^{2} + 289 T^{4} \))(\( 1 - 16 T^{2} + 289 T^{4} \))(\( 1 - 32 T^{2} + 289 T^{4} \))(\( 1 - 32 T^{2} + 289 T^{4} \))(\( ( 1 - 32 T^{2} + 289 T^{4} )^{4} \))(\( ( 1 - 16 T^{2} + 289 T^{4} )^{4} \))
$19$ (\( 1 - 6 T^{2} + 361 T^{4} \))(\( 1 - 6 T^{2} + 361 T^{4} \))(\( ( 1 - 19 T^{2} )^{2} \))(\( ( 1 - 19 T^{2} )^{2} \))(\( ( 1 - 19 T^{2} )^{2} \))(\( ( 1 - 19 T^{2} )^{2} \))(\( 1 - 6 T^{2} + 361 T^{4} \))(\( 1 - 6 T^{2} + 361 T^{4} \))(\( ( 1 - 22 T^{2} + 361 T^{4} )^{4} \))(\( ( 1 + 10 T^{2} + 361 T^{4} )^{4} \))
$23$ (\( ( 1 - 4 T + 23 T^{2} )^{2} \))(\( ( 1 + 4 T + 23 T^{2} )^{2} \))(\( ( 1 + 23 T^{2} )^{2} \))(\( ( 1 + 23 T^{2} )^{2} \))(\( ( 1 + 23 T^{2} )^{2} \))(\( ( 1 + 23 T^{2} )^{2} \))(\( ( 1 + 4 T + 23 T^{2} )^{2} \))(\( ( 1 - 4 T + 23 T^{2} )^{2} \))(\( ( 1 + 22 T^{2} + 529 T^{4} )^{4} \))(\( ( 1 - 26 T^{2} + 529 T^{4} )^{4} \))
$29$ (\( 1 - 8 T^{2} + 841 T^{4} \))(\( 1 - 8 T^{2} + 841 T^{4} \))(\( 1 - 40 T^{2} + 841 T^{4} \))(\( 1 + 40 T^{2} + 841 T^{4} \))(\( 1 - 40 T^{2} + 841 T^{4} \))(\( 1 + 40 T^{2} + 841 T^{4} \))(\( 1 - 8 T^{2} + 841 T^{4} \))(\( 1 - 8 T^{2} + 841 T^{4} \))(\( ( 1 - 52 T^{2} + 841 T^{4} )^{4} \))(\( ( 1 - 4 T^{2} + 841 T^{4} )^{4} \))
$31$ (\( 1 + 10 T^{2} + 961 T^{4} \))(\( 1 + 10 T^{2} + 961 T^{4} \))(\( ( 1 - 31 T^{2} )^{2} \))(\( ( 1 - 31 T^{2} )^{2} \))(\( ( 1 - 31 T^{2} )^{2} \))(\( ( 1 - 31 T^{2} )^{2} \))(\( 1 + 10 T^{2} + 961 T^{4} \))(\( 1 + 10 T^{2} + 961 T^{4} \))(\( ( 1 - 50 T^{2} + 961 T^{4} )^{4} \))(\( ( 1 - 58 T^{2} + 961 T^{4} )^{4} \))
$37$ (\( ( 1 + 8 T + 37 T^{2} )^{2} \))(\( ( 1 - 8 T + 37 T^{2} )^{2} \))(\( ( 1 + 12 T + 37 T^{2} )^{2} \))(\( ( 1 - 12 T + 37 T^{2} )^{2} \))(\( ( 1 - 12 T + 37 T^{2} )^{2} \))(\( ( 1 + 12 T + 37 T^{2} )^{2} \))(\( ( 1 + 8 T + 37 T^{2} )^{2} \))(\( ( 1 - 8 T + 37 T^{2} )^{2} \))(\( ( 1 + 37 T^{2} )^{8} \))(\( ( 1 + 26 T^{2} + 1369 T^{4} )^{4} \))
$41$ (\( 1 - 64 T^{2} + 1681 T^{4} \))(\( 1 - 64 T^{2} + 1681 T^{4} \))(\( 1 - 80 T^{2} + 1681 T^{4} \))(\( 1 + 80 T^{2} + 1681 T^{4} \))(\( 1 - 80 T^{2} + 1681 T^{4} \))(\( 1 + 80 T^{2} + 1681 T^{4} \))(\( 1 - 64 T^{2} + 1681 T^{4} \))(\( 1 - 64 T^{2} + 1681 T^{4} \))(\( ( 1 - 80 T^{2} + 1681 T^{4} )^{4} \))(\( ( 1 - 64 T^{2} + 1681 T^{4} )^{4} \))
$43$ (\( 1 + 42 T^{2} + 1849 T^{4} \))(\( 1 + 42 T^{2} + 1849 T^{4} \))(\( ( 1 - 43 T^{2} )^{2} \))(\( ( 1 - 43 T^{2} )^{2} \))(\( ( 1 - 43 T^{2} )^{2} \))(\( ( 1 - 43 T^{2} )^{2} \))(\( 1 + 42 T^{2} + 1849 T^{4} \))(\( 1 + 42 T^{2} + 1849 T^{4} \))(\( ( 1 - 22 T^{2} + 1849 T^{4} )^{4} \))(\( ( 1 - 43 T^{2} )^{8} \))
$47$ (\( ( 1 + 12 T + 47 T^{2} )^{2} \))(\( ( 1 - 12 T + 47 T^{2} )^{2} \))(\( ( 1 + 47 T^{2} )^{2} \))(\( ( 1 + 47 T^{2} )^{2} \))(\( ( 1 + 47 T^{2} )^{2} \))(\( ( 1 + 47 T^{2} )^{2} \))(\( ( 1 - 12 T + 47 T^{2} )^{2} \))(\( ( 1 + 12 T + 47 T^{2} )^{2} \))(\( ( 1 + 70 T^{2} + 2209 T^{4} )^{4} \))(\( ( 1 + 22 T^{2} + 2209 T^{4} )^{4} \))
$53$ (\( 1 + 56 T^{2} + 2809 T^{4} \))(\( 1 + 56 T^{2} + 2809 T^{4} \))(\( 1 + 56 T^{2} + 2809 T^{4} \))(\( 1 - 56 T^{2} + 2809 T^{4} \))(\( 1 + 56 T^{2} + 2809 T^{4} \))(\( 1 - 56 T^{2} + 2809 T^{4} \))(\( 1 + 56 T^{2} + 2809 T^{4} \))(\( 1 + 56 T^{2} + 2809 T^{4} \))(\( ( 1 - 52 T^{2} + 2809 T^{4} )^{4} \))(\( ( 1 - 100 T^{2} + 2809 T^{4} )^{4} \))
$59$ (\( ( 1 + 59 T^{2} )^{2} \))(\( ( 1 + 59 T^{2} )^{2} \))(\( ( 1 + 59 T^{2} )^{2} \))(\( ( 1 + 59 T^{2} )^{2} \))(\( ( 1 + 59 T^{2} )^{2} \))(\( ( 1 + 59 T^{2} )^{2} \))(\( ( 1 + 59 T^{2} )^{2} \))(\( ( 1 + 59 T^{2} )^{2} \))(\( ( 1 - 10 T^{2} + 3481 T^{4} )^{4} \))(\( ( 1 + 59 T^{2} )^{8} \))
$61$ (\( ( 1 + 8 T + 61 T^{2} )^{2} \))(\( ( 1 - 8 T + 61 T^{2} )^{2} \))(\( ( 1 - 12 T + 61 T^{2} )^{2} \))(\( ( 1 + 12 T + 61 T^{2} )^{2} \))(\( ( 1 + 12 T + 61 T^{2} )^{2} \))(\( ( 1 - 12 T + 61 T^{2} )^{2} \))(\( ( 1 + 8 T + 61 T^{2} )^{2} \))(\( ( 1 - 8 T + 61 T^{2} )^{2} \))(\( ( 1 - 70 T^{2} + 3721 T^{4} )^{4} \))(\( ( 1 + 74 T^{2} + 3721 T^{4} )^{4} \))
$67$ (\( 1 - 102 T^{2} + 4489 T^{4} \))(\( 1 - 102 T^{2} + 4489 T^{4} \))(\( ( 1 - 67 T^{2} )^{2} \))(\( ( 1 - 67 T^{2} )^{2} \))(\( ( 1 - 67 T^{2} )^{2} \))(\( ( 1 - 67 T^{2} )^{2} \))(\( 1 - 102 T^{2} + 4489 T^{4} \))(\( 1 - 102 T^{2} + 4489 T^{4} \))(\( ( 1 - 118 T^{2} + 4489 T^{4} )^{4} \))(\( ( 1 - 86 T^{2} + 4489 T^{4} )^{4} \))
$71$ (\( ( 1 - 4 T + 71 T^{2} )^{2} \))(\( ( 1 + 4 T + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 4 T + 71 T^{2} )^{2} \))(\( ( 1 - 4 T + 71 T^{2} )^{2} \))(\( ( 1 - 74 T^{2} + 5041 T^{4} )^{4} \))(\( ( 1 + 70 T^{2} + 5041 T^{4} )^{4} \))
$73$ (\( ( 1 + 8 T + 73 T^{2} )^{2} \))(\( ( 1 + 8 T + 73 T^{2} )^{2} \))(\( ( 1 + 16 T + 73 T^{2} )^{2} \))(\( ( 1 - 16 T + 73 T^{2} )^{2} \))(\( ( 1 + 16 T + 73 T^{2} )^{2} \))(\( ( 1 - 16 T + 73 T^{2} )^{2} \))(\( ( 1 + 8 T + 73 T^{2} )^{2} \))(\( ( 1 + 8 T + 73 T^{2} )^{2} \))(\( ( 1 - 4 T + 73 T^{2} )^{8} \))(\( ( 1 + 4 T + 73 T^{2} )^{8} \))
$79$ (\( 1 - 150 T^{2} + 6241 T^{4} \))(\( 1 - 150 T^{2} + 6241 T^{4} \))(\( ( 1 - 79 T^{2} )^{2} \))(\( ( 1 - 79 T^{2} )^{2} \))(\( ( 1 - 79 T^{2} )^{2} \))(\( ( 1 - 79 T^{2} )^{2} \))(\( 1 - 150 T^{2} + 6241 T^{4} \))(\( 1 - 150 T^{2} + 6241 T^{4} \))(\( ( 1 - 146 T^{2} + 6241 T^{4} )^{4} \))(\( ( 1 + 38 T^{2} + 6241 T^{4} )^{4} \))
$83$ (\( ( 1 - 12 T + 83 T^{2} )^{2} \))(\( ( 1 - 12 T + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 12 T + 83 T^{2} )^{2} \))(\( ( 1 + 12 T + 83 T^{2} )^{2} \))(\( ( 1 - 34 T^{2} + 6889 T^{4} )^{4} \))(\( ( 1 + 142 T^{2} + 6889 T^{4} )^{4} \))
$89$ (\( 1 + 64 T^{2} + 7921 T^{4} \))(\( 1 + 64 T^{2} + 7921 T^{4} \))(\( 1 + 160 T^{2} + 7921 T^{4} \))(\( 1 - 160 T^{2} + 7921 T^{4} \))(\( 1 + 160 T^{2} + 7921 T^{4} \))(\( 1 - 160 T^{2} + 7921 T^{4} \))(\( 1 + 64 T^{2} + 7921 T^{4} \))(\( 1 + 64 T^{2} + 7921 T^{4} \))(\( ( 1 - 128 T^{2} + 7921 T^{4} )^{4} \))(\( ( 1 - 160 T^{2} + 7921 T^{4} )^{4} \))
$97$ (\( ( 1 + 97 T^{2} )^{2} \))(\( ( 1 + 97 T^{2} )^{2} \))(\( ( 1 + 8 T + 97 T^{2} )^{2} \))(\( ( 1 - 8 T + 97 T^{2} )^{2} \))(\( ( 1 + 8 T + 97 T^{2} )^{2} \))(\( ( 1 - 8 T + 97 T^{2} )^{2} \))(\( ( 1 + 97 T^{2} )^{2} \))(\( ( 1 + 97 T^{2} )^{2} \))(\( ( 1 - 8 T + 97 T^{2} )^{8} \))(\( ( 1 + 16 T + 97 T^{2} )^{8} \))
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