Properties

Label 2304.2.f
Level $2304$
Weight $2$
Character orbit 2304.f
Rep. character $\chi_{2304}(1151,\cdot)$
Character field $\Q$
Dimension $32$
Newform subspaces $8$
Sturm bound $768$
Trace bound $43$

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Defining parameters

Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2304.f (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 24 \)
Character field: \(\Q\)
Newform subspaces: \( 8 \)
Sturm bound: \(768\)
Trace bound: \(43\)
Distinguishing \(T_p\): \(5\), \(7\), \(23\), \(43\)

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

\( 32q + O(q^{10}) \) \( 32q + 32q^{25} - 96q^{49} - 64q^{73} + 64q^{97} + O(q^{100}) \)

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.f.a \(4\) \(18.398\) \(\Q(\zeta_{8})\) None \(0\) \(0\) \(-8\) \(0\) \(q+(-2+\zeta_{8}^{3})q^{5}+(\zeta_{8}-2\zeta_{8}^{2})q^{7}+\cdots\)
2304.2.f.b \(4\) \(18.398\) \(\Q(\zeta_{8})\) None \(0\) \(0\) \(-8\) \(0\) \(q+(-2+\zeta_{8}^{3})q^{5}+(-\zeta_{8}+2\zeta_{8}^{2})q^{7}+\cdots\)
2304.2.f.c \(4\) \(18.398\) \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(0\) \(q+\zeta_{8}^{3}q^{5}+2\zeta_{8}q^{7}-4\zeta_{8}^{2}q^{11}-2\zeta_{8}q^{13}+\cdots\)
2304.2.f.d \(4\) \(18.398\) \(\Q(\zeta_{8})\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(0\) \(0\) \(q+\zeta_{8}^{3}q^{5}+2\zeta_{8}q^{13}+5\zeta_{8}^{2}q^{17}+\cdots\)
2304.2.f.e \(4\) \(18.398\) \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(0\) \(q+\zeta_{8}^{3}q^{5}+2\zeta_{8}q^{7}-4\zeta_{8}^{2}q^{11}+2\zeta_{8}q^{13}+\cdots\)
2304.2.f.f \(4\) \(18.398\) \(\Q(\zeta_{8})\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{8}^{3}q^{5}-2\zeta_{8}q^{13}-\zeta_{8}^{2}q^{17}+13q^{25}+\cdots\)
2304.2.f.g \(4\) \(18.398\) \(\Q(\zeta_{8})\) None \(0\) \(0\) \(8\) \(0\) \(q+(2+\zeta_{8}^{3})q^{5}+(\zeta_{8}+2\zeta_{8}^{2})q^{7}+2\zeta_{8}^{2}q^{11}+\cdots\)
2304.2.f.h \(4\) \(18.398\) \(\Q(\zeta_{8})\) None \(0\) \(0\) \(8\) \(0\) \(q+(2+\zeta_{8}^{3})q^{5}+(\zeta_{8}+2\zeta_{8}^{2})q^{7}+2\zeta_{8}^{2}q^{11}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(2304, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(24, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(72, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(96, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(192, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(288, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(384, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(576, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(768, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1152, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ (\( ( 1 + 4 T + 12 T^{2} + 20 T^{3} + 25 T^{4} )^{2} \))(\( ( 1 + 4 T + 12 T^{2} + 20 T^{3} + 25 T^{4} )^{2} \))(\( ( 1 + 8 T^{2} + 25 T^{4} )^{2} \))(\( ( 1 + 8 T^{2} + 25 T^{4} )^{2} \))(\( ( 1 + 8 T^{2} + 25 T^{4} )^{2} \))(\( ( 1 - 8 T^{2} + 25 T^{4} )^{2} \))(\( ( 1 - 4 T + 12 T^{2} - 20 T^{3} + 25 T^{4} )^{2} \))(\( ( 1 - 4 T + 12 T^{2} - 20 T^{3} + 25 T^{4} )^{2} \))
$7$ (\( 1 - 4 T^{2} - 26 T^{4} - 196 T^{6} + 2401 T^{8} \))(\( 1 - 4 T^{2} - 26 T^{4} - 196 T^{6} + 2401 T^{8} \))(\( ( 1 + 2 T^{2} + 49 T^{4} )^{2} \))(\( ( 1 - 7 T^{2} )^{4} \))(\( ( 1 + 2 T^{2} + 49 T^{4} )^{2} \))(\( ( 1 - 7 T^{2} )^{4} \))(\( 1 - 4 T^{2} - 26 T^{4} - 196 T^{6} + 2401 T^{8} \))(\( 1 - 4 T^{2} - 26 T^{4} - 196 T^{6} + 2401 T^{8} \))
$11$ (\( ( 1 - 6 T + 11 T^{2} )^{2}( 1 + 6 T + 11 T^{2} )^{2} \))(\( ( 1 - 6 T + 11 T^{2} )^{2}( 1 + 6 T + 11 T^{2} )^{2} \))(\( ( 1 + 10 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 - 11 T^{2} )^{4} \))(\( ( 1 + 10 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 - 11 T^{2} )^{4} \))(\( ( 1 - 6 T + 11 T^{2} )^{2}( 1 + 6 T + 11 T^{2} )^{2} \))(\( ( 1 - 6 T + 11 T^{2} )^{2}( 1 + 6 T + 11 T^{2} )^{2} \))
$13$ (\( ( 1 - 18 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 - 18 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 - 6 T + 13 T^{2} )^{2}( 1 + 6 T + 13 T^{2} )^{2} \))(\( ( 1 - 6 T + 13 T^{2} )^{2}( 1 + 6 T + 13 T^{2} )^{2} \))(\( ( 1 - 6 T + 13 T^{2} )^{2}( 1 + 6 T + 13 T^{2} )^{2} \))(\( ( 1 - 6 T + 13 T^{2} )^{2}( 1 + 6 T + 13 T^{2} )^{2} \))(\( ( 1 - 18 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 - 18 T^{2} + 169 T^{4} )^{2} \))
$17$ (\( 1 - 32 T^{2} + 706 T^{4} - 9248 T^{6} + 83521 T^{8} \))(\( 1 - 32 T^{2} + 706 T^{4} - 9248 T^{6} + 83521 T^{8} \))(\( ( 1 - 16 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 + 16 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 - 16 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 - 16 T^{2} + 289 T^{4} )^{2} \))(\( 1 - 32 T^{2} + 706 T^{4} - 9248 T^{6} + 83521 T^{8} \))(\( 1 - 32 T^{2} + 706 T^{4} - 9248 T^{6} + 83521 T^{8} \))
$19$ (\( ( 1 + 6 T^{2} + 361 T^{4} )^{2} \))(\( ( 1 + 6 T^{2} + 361 T^{4} )^{2} \))(\( ( 1 + 19 T^{2} )^{4} \))(\( ( 1 + 19 T^{2} )^{4} \))(\( ( 1 + 19 T^{2} )^{4} \))(\( ( 1 + 19 T^{2} )^{4} \))(\( ( 1 + 6 T^{2} + 361 T^{4} )^{2} \))(\( ( 1 + 6 T^{2} + 361 T^{4} )^{2} \))
$23$ (\( ( 1 + 8 T + 54 T^{2} + 184 T^{3} + 529 T^{4} )^{2} \))(\( ( 1 - 8 T + 54 T^{2} - 184 T^{3} + 529 T^{4} )^{2} \))(\( ( 1 + 14 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 + 23 T^{2} )^{4} \))(\( ( 1 + 14 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 + 23 T^{2} )^{4} \))(\( ( 1 + 8 T + 54 T^{2} + 184 T^{3} + 529 T^{4} )^{2} \))(\( ( 1 - 8 T + 54 T^{2} - 184 T^{3} + 529 T^{4} )^{2} \))
$29$ (\( ( 1 - 4 T + 60 T^{2} - 116 T^{3} + 841 T^{4} )^{2} \))(\( ( 1 - 4 T + 60 T^{2} - 116 T^{3} + 841 T^{4} )^{2} \))(\( ( 1 + 56 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 40 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 + 56 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 + 40 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 + 4 T + 60 T^{2} + 116 T^{3} + 841 T^{4} )^{2} \))(\( ( 1 + 4 T + 60 T^{2} + 116 T^{3} + 841 T^{4} )^{2} \))
$31$ (\( 1 - 36 T^{2} + 1094 T^{4} - 34596 T^{6} + 923521 T^{8} \))(\( 1 - 36 T^{2} + 1094 T^{4} - 34596 T^{6} + 923521 T^{8} \))(\( ( 1 - 46 T^{2} + 961 T^{4} )^{2} \))(\( ( 1 - 31 T^{2} )^{4} \))(\( ( 1 - 46 T^{2} + 961 T^{4} )^{2} \))(\( ( 1 - 31 T^{2} )^{4} \))(\( 1 - 36 T^{2} + 1094 T^{4} - 34596 T^{6} + 923521 T^{8} \))(\( 1 - 36 T^{2} + 1094 T^{4} - 34596 T^{6} + 923521 T^{8} \))
$37$ (\( 1 - 76 T^{2} + 3670 T^{4} - 104044 T^{6} + 1874161 T^{8} \))(\( 1 - 76 T^{2} + 3670 T^{4} - 104044 T^{6} + 1874161 T^{8} \))(\( ( 1 - 38 T^{2} + 1369 T^{4} )^{2} \))(\( ( 1 - 12 T + 37 T^{2} )^{2}( 1 + 12 T + 37 T^{2} )^{2} \))(\( ( 1 - 38 T^{2} + 1369 T^{4} )^{2} \))(\( ( 1 - 12 T + 37 T^{2} )^{2}( 1 + 12 T + 37 T^{2} )^{2} \))(\( 1 - 76 T^{2} + 3670 T^{4} - 104044 T^{6} + 1874161 T^{8} \))(\( 1 - 76 T^{2} + 3670 T^{4} - 104044 T^{6} + 1874161 T^{8} \))
$41$ (\( 1 - 128 T^{2} + 7330 T^{4} - 215168 T^{6} + 2825761 T^{8} \))(\( 1 - 128 T^{2} + 7330 T^{4} - 215168 T^{6} + 2825761 T^{8} \))(\( ( 1 + 16 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 - 80 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 + 16 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 + 80 T^{2} + 1681 T^{4} )^{2} \))(\( 1 - 128 T^{2} + 7330 T^{4} - 215168 T^{6} + 2825761 T^{8} \))(\( 1 - 128 T^{2} + 7330 T^{4} - 215168 T^{6} + 2825761 T^{8} \))
$43$ (\( ( 1 - 8 T + 70 T^{2} - 344 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 + 8 T + 70 T^{2} + 344 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 + 8 T + 43 T^{2} )^{4} \))(\( ( 1 + 43 T^{2} )^{4} \))(\( ( 1 - 8 T + 43 T^{2} )^{4} \))(\( ( 1 + 43 T^{2} )^{4} \))(\( ( 1 + 8 T + 70 T^{2} + 344 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 - 8 T + 70 T^{2} - 344 T^{3} + 1849 T^{4} )^{2} \))
$47$ (\( ( 1 - 8 T + 38 T^{2} - 376 T^{3} + 2209 T^{4} )^{2} \))(\( ( 1 + 8 T + 38 T^{2} + 376 T^{3} + 2209 T^{4} )^{2} \))(\( ( 1 + 62 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 + 47 T^{2} )^{4} \))(\( ( 1 + 62 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 + 47 T^{2} )^{4} \))(\( ( 1 - 8 T + 38 T^{2} - 376 T^{3} + 2209 T^{4} )^{2} \))(\( ( 1 + 8 T + 38 T^{2} + 376 T^{3} + 2209 T^{4} )^{2} \))
$53$ (\( ( 1 + 4 T + 60 T^{2} + 212 T^{3} + 2809 T^{4} )^{2} \))(\( ( 1 + 4 T + 60 T^{2} + 212 T^{3} + 2809 T^{4} )^{2} \))(\( ( 1 + 88 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 + 56 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 + 88 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 - 56 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 - 4 T + 60 T^{2} - 212 T^{3} + 2809 T^{4} )^{2} \))(\( ( 1 - 4 T + 60 T^{2} - 212 T^{3} + 2809 T^{4} )^{2} \))
$59$ (\( 1 - 44 T^{2} - 746 T^{4} - 153164 T^{6} + 12117361 T^{8} \))(\( 1 - 44 T^{2} - 746 T^{4} - 153164 T^{6} + 12117361 T^{8} \))(\( ( 1 + 10 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 - 59 T^{2} )^{4} \))(\( ( 1 + 10 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 - 59 T^{2} )^{4} \))(\( 1 - 44 T^{2} - 746 T^{4} - 153164 T^{6} + 12117361 T^{8} \))(\( 1 - 44 T^{2} - 746 T^{4} - 153164 T^{6} + 12117361 T^{8} \))
$61$ (\( 1 - 172 T^{2} + 14326 T^{4} - 640012 T^{6} + 13845841 T^{8} \))(\( 1 - 172 T^{2} + 14326 T^{4} - 640012 T^{6} + 13845841 T^{8} \))(\( ( 1 - 118 T^{2} + 3721 T^{4} )^{2} \))(\( ( 1 - 12 T + 61 T^{2} )^{2}( 1 + 12 T + 61 T^{2} )^{2} \))(\( ( 1 - 118 T^{2} + 3721 T^{4} )^{2} \))(\( ( 1 - 12 T + 61 T^{2} )^{2}( 1 + 12 T + 61 T^{2} )^{2} \))(\( 1 - 172 T^{2} + 14326 T^{4} - 640012 T^{6} + 13845841 T^{8} \))(\( 1 - 172 T^{2} + 14326 T^{4} - 640012 T^{6} + 13845841 T^{8} \))
$67$ (\( ( 1 - 12 T + 67 T^{2} )^{4} \))(\( ( 1 + 12 T + 67 T^{2} )^{4} \))(\( ( 1 - 8 T + 67 T^{2} )^{4} \))(\( ( 1 + 67 T^{2} )^{4} \))(\( ( 1 + 8 T + 67 T^{2} )^{4} \))(\( ( 1 + 67 T^{2} )^{4} \))(\( ( 1 + 12 T + 67 T^{2} )^{4} \))(\( ( 1 - 12 T + 67 T^{2} )^{4} \))
$71$ (\( ( 1 + 8 T + 86 T^{2} + 568 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 - 8 T + 86 T^{2} - 568 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 + 110 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 + 71 T^{2} )^{4} \))(\( ( 1 + 110 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 + 71 T^{2} )^{4} \))(\( ( 1 + 8 T + 86 T^{2} + 568 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 - 8 T + 86 T^{2} - 568 T^{3} + 5041 T^{4} )^{2} \))
$73$ (\( ( 1 + 4 T + 73 T^{2} )^{4} \))(\( ( 1 + 4 T + 73 T^{2} )^{4} \))(\( ( 1 + 73 T^{2} )^{4} \))(\( ( 1 - 16 T + 73 T^{2} )^{4} \))(\( ( 1 + 73 T^{2} )^{4} \))(\( ( 1 + 16 T + 73 T^{2} )^{4} \))(\( ( 1 + 4 T + 73 T^{2} )^{4} \))(\( ( 1 + 4 T + 73 T^{2} )^{4} \))
$79$ (\( 1 - 164 T^{2} + 18054 T^{4} - 1023524 T^{6} + 38950081 T^{8} \))(\( 1 - 164 T^{2} + 18054 T^{4} - 1023524 T^{6} + 38950081 T^{8} \))(\( ( 1 - 142 T^{2} + 6241 T^{4} )^{2} \))(\( ( 1 - 79 T^{2} )^{4} \))(\( ( 1 - 142 T^{2} + 6241 T^{4} )^{2} \))(\( ( 1 - 79 T^{2} )^{4} \))(\( 1 - 164 T^{2} + 18054 T^{4} - 1023524 T^{6} + 38950081 T^{8} \))(\( 1 - 164 T^{2} + 18054 T^{4} - 1023524 T^{6} + 38950081 T^{8} \))
$83$ (\( 1 - 188 T^{2} + 20566 T^{4} - 1295132 T^{6} + 47458321 T^{8} \))(\( 1 - 188 T^{2} + 20566 T^{4} - 1295132 T^{6} + 47458321 T^{8} \))(\( ( 1 - 134 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 - 83 T^{2} )^{4} \))(\( ( 1 - 134 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 - 83 T^{2} )^{4} \))(\( 1 - 188 T^{2} + 20566 T^{4} - 1295132 T^{6} + 47458321 T^{8} \))(\( 1 - 188 T^{2} + 20566 T^{4} - 1295132 T^{6} + 47458321 T^{8} \))
$89$ (\( 1 - 192 T^{2} + 20450 T^{4} - 1520832 T^{6} + 62742241 T^{8} \))(\( 1 - 192 T^{2} + 20450 T^{4} - 1520832 T^{6} + 62742241 T^{8} \))(\( ( 1 - 160 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 + 160 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 - 160 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 - 160 T^{2} + 7921 T^{4} )^{2} \))(\( 1 - 192 T^{2} + 20450 T^{4} - 1520832 T^{6} + 62742241 T^{8} \))(\( 1 - 192 T^{2} + 20450 T^{4} - 1520832 T^{6} + 62742241 T^{8} \))
$97$ (\( ( 1 - 16 T + 226 T^{2} - 1552 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 - 16 T + 226 T^{2} - 1552 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 + 8 T + 97 T^{2} )^{4} \))(\( ( 1 - 8 T + 97 T^{2} )^{4} \))(\( ( 1 + 8 T + 97 T^{2} )^{4} \))(\( ( 1 + 8 T + 97 T^{2} )^{4} \))(\( ( 1 - 16 T + 226 T^{2} - 1552 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 - 16 T + 226 T^{2} - 1552 T^{3} + 9409 T^{4} )^{2} \))
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