Properties

Label 1575.2.g
Level 1575
Weight 2
Character orbit g
Rep. character \(\chi_{1575}(1574,\cdot)\)
Character field \(\Q\)
Dimension 48
Newform subspaces 5
Sturm bound 480
Trace bound 14

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Defining parameters

Level: \( N \) = \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1575.g (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 105 \)
Character field: \(\Q\)
Newform subspaces: \( 5 \)
Sturm bound: \(480\)
Trace bound: \(14\)
Distinguishing \(T_p\): \(2\), \(59\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1575, [\chi])\).

Total New Old
Modular forms 264 48 216
Cusp forms 216 48 168
Eisenstein series 48 0 48

Trace form

\( 48q + 48q^{4} + O(q^{10}) \) \( 48q + 48q^{4} + 48q^{16} + 96q^{46} - 36q^{49} + 192q^{64} - 48q^{79} - 12q^{91} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1575, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1575.2.g.a \(8\) \(12.576\) \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{24}^{6}q^{2}+\zeta_{24}^{3}q^{4}+(-\zeta_{24}-\zeta_{24}^{5}+\cdots)q^{7}+\cdots\)
1575.2.g.b \(8\) \(12.576\) \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(0\) \(q+\zeta_{24}^{6}q^{2}+(\zeta_{24}-\zeta_{24}^{4})q^{7}-2\zeta_{24}^{6}q^{8}+\cdots\)
1575.2.g.c \(8\) \(12.576\) \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{24}^{6}q^{2}+\zeta_{24}^{3}q^{4}+(\zeta_{24}+\zeta_{24}^{5}+\cdots)q^{7}+\cdots\)
1575.2.g.d \(8\) \(12.576\) 8.0.157351936.1 \(\Q(\sqrt{-7}) \) \(0\) \(0\) \(0\) \(0\) \(q+\beta _{6}q^{2}+(2+\beta _{5})q^{4}-\beta _{1}q^{7}+(\beta _{3}+\cdots)q^{8}+\cdots\)
1575.2.g.e \(16\) \(12.576\) 16.0.\(\cdots\).1 \(\Q(\sqrt{-7}) \) \(0\) \(0\) \(0\) \(0\) \(q-\beta _{10}q^{2}+(2+\beta _{4})q^{4}-\beta _{9}q^{7}+(\beta _{3}+\cdots)q^{8}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1575, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1575, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(315, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(525, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 + 4 T^{2} + 9 T^{4} + 16 T^{6} + 16 T^{8} )^{2} \))(\( ( 1 + 2 T^{2} + 4 T^{4} )^{4} \))(\( ( 1 + 4 T^{2} + 9 T^{4} + 16 T^{6} + 16 T^{8} )^{2} \))(\( ( 1 + T^{4} + 16 T^{8} )^{2} \))(\( ( 1 - T^{4} - 15 T^{8} - 16 T^{12} + 256 T^{16} )^{2} \))
$3$ 1
$5$ 1
$7$ (\( ( 1 - 10 T^{2} + 49 T^{4} )^{2} \))(\( ( 1 - 13 T^{2} + 49 T^{4} )^{2} \))(\( ( 1 - 10 T^{2} + 49 T^{4} )^{2} \))(\( ( 1 + 7 T^{2} )^{4} \))(\( ( 1 + 7 T^{2} )^{8} \))
$11$ (\( ( 1 - 16 T^{2} + 114 T^{4} - 1936 T^{6} + 14641 T^{8} )^{2} \))(\( ( 1 - 20 T^{2} + 121 T^{4} )^{4} \))(\( ( 1 - 16 T^{2} + 114 T^{4} - 1936 T^{6} + 14641 T^{8} )^{2} \))(\( ( 1 - 206 T^{4} + 14641 T^{8} )^{2} \))(\( ( 1 + 206 T^{4} + 27795 T^{8} + 3016046 T^{12} + 214358881 T^{16} )^{2} \))
$13$ (\( ( 1 + 4 T^{2} - 90 T^{4} + 676 T^{6} + 28561 T^{8} )^{2} \))(\( ( 1 - T^{2} + 169 T^{4} )^{4} \))(\( ( 1 + 4 T^{2} - 90 T^{4} + 676 T^{6} + 28561 T^{8} )^{2} \))(\( ( 1 + 13 T^{2} )^{8} \))(\( ( 1 + 13 T^{2} )^{16} \))
$17$ (\( ( 1 - 22 T^{2} + 289 T^{4} )^{4} \))(\( ( 1 + 20 T^{2} + 289 T^{4} )^{4} \))(\( ( 1 - 22 T^{2} + 289 T^{4} )^{4} \))(\( ( 1 - 17 T^{2} )^{8} \))(\( ( 1 - 17 T^{2} )^{16} \))
$19$ (\( ( 1 - 28 T^{2} + 486 T^{4} - 10108 T^{6} + 130321 T^{8} )^{2} \))(\( ( 1 - 7 T + 19 T^{2} )^{4}( 1 + 7 T + 19 T^{2} )^{4} \))(\( ( 1 - 28 T^{2} + 486 T^{4} - 10108 T^{6} + 130321 T^{8} )^{2} \))(\( ( 1 - 19 T^{2} )^{8} \))(\( ( 1 - 19 T^{2} )^{16} \))
$23$ (\( ( 1 + 44 T^{2} + 529 T^{4} )^{4} \))(\( ( 1 - 4 T^{2} + 529 T^{4} )^{4} \))(\( ( 1 + 44 T^{2} + 529 T^{4} )^{4} \))(\( ( 1 - 734 T^{4} + 279841 T^{8} )^{2} \))(\( ( 1 + 734 T^{4} + 258915 T^{8} + 205403294 T^{12} + 78310985281 T^{16} )^{2} \))
$29$ (\( ( 1 - 56 T^{2} + 841 T^{4} )^{4} \))(\( ( 1 - 56 T^{2} + 841 T^{4} )^{4} \))(\( ( 1 - 56 T^{2} + 841 T^{4} )^{4} \))(\( ( 1 + 1234 T^{4} + 707281 T^{8} )^{2} \))(\( ( 1 - 1234 T^{4} + 815475 T^{8} - 872784754 T^{12} + 500246412961 T^{16} )^{2} \))
$31$ (\( ( 1 - 76 T^{2} + 2934 T^{4} - 73036 T^{6} + 923521 T^{8} )^{2} \))(\( ( 1 - 35 T^{2} + 961 T^{4} )^{4} \))(\( ( 1 - 76 T^{2} + 2934 T^{4} - 73036 T^{6} + 923521 T^{8} )^{2} \))(\( ( 1 - 31 T^{2} )^{8} \))(\( ( 1 - 31 T^{2} )^{16} \))
$37$ (\( ( 1 - 116 T^{2} + 5910 T^{4} - 158804 T^{6} + 1874161 T^{8} )^{2} \))(\( ( 1 - 10 T^{2} + 1369 T^{4} )^{4} \))(\( ( 1 - 116 T^{2} + 5910 T^{4} - 158804 T^{6} + 1874161 T^{8} )^{2} \))(\( ( 1 - 6 T + 37 T^{2} )^{4}( 1 + 6 T + 37 T^{2} )^{4} \))(\( ( 1 - 6 T - T^{2} - 222 T^{3} + 1369 T^{4} )^{4}( 1 + 6 T - T^{2} + 222 T^{3} + 1369 T^{4} )^{4} \))
$41$ (\( ( 1 - 26 T^{2} + 1681 T^{4} )^{4} \))(\( ( 1 + 41 T^{2} )^{8} \))(\( ( 1 - 26 T^{2} + 1681 T^{4} )^{4} \))(\( ( 1 + 41 T^{2} )^{8} \))(\( ( 1 + 41 T^{2} )^{16} \))
$43$ (\( ( 1 - 68 T^{2} + 4086 T^{4} - 125732 T^{6} + 3418801 T^{8} )^{2} \))(\( ( 1 - 61 T^{2} + 1849 T^{4} )^{4} \))(\( ( 1 - 68 T^{2} + 4086 T^{4} - 125732 T^{6} + 3418801 T^{8} )^{2} \))(\( ( 1 - 12 T + 43 T^{2} )^{4}( 1 + 12 T + 43 T^{2} )^{4} \))(\( ( 1 - 12 T + 101 T^{2} - 516 T^{3} + 1849 T^{4} )^{4}( 1 + 12 T + 101 T^{2} + 516 T^{3} + 1849 T^{4} )^{4} \))
$47$ (\( ( 1 - 92 T^{2} + 4806 T^{4} - 203228 T^{6} + 4879681 T^{8} )^{2} \))(\( ( 1 - 40 T^{2} + 2209 T^{4} )^{4} \))(\( ( 1 - 92 T^{2} + 4806 T^{4} - 203228 T^{6} + 4879681 T^{8} )^{2} \))(\( ( 1 - 47 T^{2} )^{8} \))(\( ( 1 - 47 T^{2} )^{16} \))
$53$ (\( ( 1 + 88 T^{2} + 5826 T^{4} + 247192 T^{6} + 7890481 T^{8} )^{2} \))(\( ( 1 + 74 T^{2} + 2809 T^{4} )^{4} \))(\( ( 1 + 88 T^{2} + 5826 T^{4} + 247192 T^{6} + 7890481 T^{8} )^{2} \))(\( ( 1 - 5582 T^{4} + 7890481 T^{8} )^{2} \))(\( ( 1 - 5582 T^{4} + 7890481 T^{8} )^{4} \))
$59$ (\( ( 1 - 12 T + 142 T^{2} - 708 T^{3} + 3481 T^{4} )^{4} \))(\( ( 1 + 64 T^{2} + 3481 T^{4} )^{4} \))(\( ( 1 + 12 T + 142 T^{2} + 708 T^{3} + 3481 T^{4} )^{4} \))(\( ( 1 + 59 T^{2} )^{8} \))(\( ( 1 + 59 T^{2} )^{16} \))
$61$ (\( ( 1 - 52 T^{2} + 1206 T^{4} - 193492 T^{6} + 13845841 T^{8} )^{2} \))(\( ( 1 - 95 T^{2} + 3721 T^{4} )^{4} \))(\( ( 1 - 52 T^{2} + 1206 T^{4} - 193492 T^{6} + 13845841 T^{8} )^{2} \))(\( ( 1 - 61 T^{2} )^{8} \))(\( ( 1 - 61 T^{2} )^{16} \))
$67$ (\( ( 1 - 140 T^{2} + 10806 T^{4} - 628460 T^{6} + 20151121 T^{8} )^{2} \))(\( ( 1 - 85 T^{2} + 4489 T^{4} )^{4} \))(\( ( 1 - 140 T^{2} + 10806 T^{4} - 628460 T^{6} + 20151121 T^{8} )^{2} \))(\( ( 1 - 118 T^{2} + 4489 T^{4} )^{4} \))(\( ( 1 + 118 T^{2} + 9435 T^{4} + 529702 T^{6} + 20151121 T^{8} )^{4} \))
$71$ (\( ( 1 + 32 T^{2} + 5538 T^{4} + 161312 T^{6} + 25411681 T^{8} )^{2} \))(\( ( 1 - 134 T^{2} + 5041 T^{4} )^{4} \))(\( ( 1 + 32 T^{2} + 5538 T^{4} + 161312 T^{6} + 25411681 T^{8} )^{2} \))(\( ( 1 + 2914 T^{4} + 25411681 T^{8} )^{2} \))(\( ( 1 - 2914 T^{4} - 16920285 T^{8} - 74049638434 T^{12} + 645753531245761 T^{16} )^{2} \))
$73$ (\( ( 1 + 244 T^{2} + 25110 T^{4} + 1300276 T^{6} + 28398241 T^{8} )^{2} \))(\( ( 1 + 38 T^{2} + 5329 T^{4} )^{4} \))(\( ( 1 + 244 T^{2} + 25110 T^{4} + 1300276 T^{6} + 28398241 T^{8} )^{2} \))(\( ( 1 + 73 T^{2} )^{8} \))(\( ( 1 + 73 T^{2} )^{16} \))
$79$ (\( ( 1 - 8 T + 126 T^{2} - 632 T^{3} + 6241 T^{4} )^{4} \))(\( ( 1 + 14 T + 79 T^{2} )^{8} \))(\( ( 1 - 8 T + 126 T^{2} - 632 T^{3} + 6241 T^{4} )^{4} \))(\( ( 1 + 8 T + 79 T^{2} )^{8} \))(\( ( 1 - 8 T - 15 T^{2} - 632 T^{3} + 6241 T^{4} )^{8} \))
$83$ (\( ( 1 - 236 T^{2} + 25974 T^{4} - 1625804 T^{6} + 47458321 T^{8} )^{2} \))(\( ( 1 - 112 T^{2} + 6889 T^{4} )^{4} \))(\( ( 1 - 236 T^{2} + 25974 T^{4} - 1625804 T^{6} + 47458321 T^{8} )^{2} \))(\( ( 1 - 83 T^{2} )^{8} \))(\( ( 1 - 83 T^{2} )^{16} \))
$89$ (\( ( 1 - 12 T + 166 T^{2} - 1068 T^{3} + 7921 T^{4} )^{4} \))(\( ( 1 - 38 T^{2} + 7921 T^{4} )^{4} \))(\( ( 1 + 12 T + 166 T^{2} + 1068 T^{3} + 7921 T^{4} )^{4} \))(\( ( 1 + 89 T^{2} )^{8} \))(\( ( 1 + 89 T^{2} )^{16} \))
$97$ (\( ( 1 + 52 T^{2} + 15606 T^{4} + 489268 T^{6} + 88529281 T^{8} )^{2} \))(\( ( 1 + 167 T^{2} + 9409 T^{4} )^{4} \))(\( ( 1 + 52 T^{2} + 15606 T^{4} + 489268 T^{6} + 88529281 T^{8} )^{2} \))(\( ( 1 + 97 T^{2} )^{8} \))(\( ( 1 + 97 T^{2} )^{16} \))
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