Properties

Label 1575.2.b
Level 1575
Weight 2
Character orbit b
Rep. character \(\chi_{1575}(251,\cdot)\)
Character field \(\Q\)
Dimension 52
Newform subspaces 8
Sturm bound 480
Trace bound 22

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Defining parameters

Level: \( N \) = \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1575.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 21 \)
Character field: \(\Q\)
Newform subspaces: \( 8 \)
Sturm bound: \(480\)
Trace bound: \(22\)
Distinguishing \(T_p\): \(2\), \(37\), \(47\), \(67\)

Dimensions

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

Total New Old
Modular forms 264 52 212
Cusp forms 216 52 164
Eisenstein series 48 0 48

Trace form

\( 52q - 56q^{4} - 8q^{7} + O(q^{10}) \) \( 52q - 56q^{4} - 8q^{7} + 60q^{16} + 20q^{22} + 28q^{28} - 16q^{37} - 16q^{43} - 52q^{46} + 32q^{49} + 44q^{58} - 120q^{64} + 48q^{67} + 112q^{79} - 20q^{88} - 28q^{91} + O(q^{100}) \)

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.b.a \(4\) \(12.576\) \(\Q(\sqrt{-2}, \sqrt{7})\) \(\Q(\sqrt{-7}) \) \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{2}+(-2+\beta _{2})q^{4}+\beta _{2}q^{7}+(-2\beta _{1}+\cdots)q^{8}+\cdots\)
1575.2.b.b \(4\) \(12.576\) \(\Q(\sqrt{-2}, \sqrt{3})\) None \(0\) \(0\) \(0\) \(-4\) \(q+\beta _{1}q^{2}+\beta _{2}q^{4}+(-1+\beta _{1}+\beta _{3})q^{7}+\cdots\)
1575.2.b.c \(4\) \(12.576\) \(\Q(\sqrt{-2}, \sqrt{3})\) None \(0\) \(0\) \(0\) \(-4\) \(q+\beta _{1}q^{2}+\beta _{2}q^{4}+(-1-\beta _{1}-\beta _{3})q^{7}+\cdots\)
1575.2.b.d \(4\) \(12.576\) \(\Q(\sqrt{-2}, \sqrt{-3})\) None \(0\) \(0\) \(0\) \(-2\) \(q-\beta _{1}q^{2}-\beta _{2}q^{7}-2\beta _{1}q^{8}-\beta _{1}q^{11}+\cdots\)
1575.2.b.e \(4\) \(12.576\) \(\Q(\sqrt{-2}, \sqrt{-3})\) None \(0\) \(0\) \(0\) \(2\) \(q+\beta _{1}q^{2}+\beta _{2}q^{7}+2\beta _{1}q^{8}-\beta _{1}q^{11}+\cdots\)
1575.2.b.f \(8\) \(12.576\) 8.0.\(\cdots\).3 \(\Q(\sqrt{-7}) \) \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{2}+(-2+\beta _{2})q^{4}-\beta _{4}q^{7}+(-2\beta _{1}+\cdots)q^{8}+\cdots\)
1575.2.b.g \(8\) \(12.576\) 8.0.\(\cdots\).3 \(\Q(\sqrt{-7}) \) \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{2}+(-2+\beta _{2})q^{4}+\beta _{4}q^{7}+(-2\beta _{1}+\cdots)q^{8}+\cdots\)
1575.2.b.h \(16\) \(12.576\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{3}q^{2}+(-1-\beta _{1})q^{4}-\beta _{8}q^{7}+(-\beta _{2}+\cdots)q^{8}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(1575, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(315, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(525, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T^{4} + 16 T^{8} \))(\( 1 - 4 T^{2} + 9 T^{4} - 16 T^{6} + 16 T^{8} \))(\( 1 - 4 T^{2} + 9 T^{4} - 16 T^{6} + 16 T^{8} \))(\( ( 1 - 2 T^{2} + 4 T^{4} )^{2} \))(\( ( 1 - 2 T^{2} + 4 T^{4} )^{2} \))(\( 1 - T^{4} - 15 T^{8} - 16 T^{12} + 256 T^{16} \))(\( 1 - T^{4} - 15 T^{8} - 16 T^{12} + 256 T^{16} \))(\( ( 1 - 2 T^{2} + 3 T^{4} - 8 T^{6} + 16 T^{8} )^{4} \))
$3$ 1
$5$ 1
$7$ (\( ( 1 - 7 T^{2} )^{2} \))(\( ( 1 + 2 T + 7 T^{2} )^{2} \))(\( ( 1 + 2 T + 7 T^{2} )^{2} \))(\( ( 1 + T + 7 T^{2} )^{2} \))(\( ( 1 - T + 7 T^{2} )^{2} \))(\( ( 1 - 7 T^{2} )^{4} \))(\( ( 1 - 7 T^{2} )^{4} \))(\( ( 1 + 4 T^{2} + 6 T^{4} + 196 T^{6} + 2401 T^{8} )^{2} \))
$11$ (\( 1 - 206 T^{4} + 14641 T^{8} \))(\( 1 - 16 T^{2} + 114 T^{4} - 1936 T^{6} + 14641 T^{8} \))(\( 1 - 16 T^{2} + 114 T^{4} - 1936 T^{6} + 14641 T^{8} \))(\( ( 1 - 20 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 - 20 T^{2} + 121 T^{4} )^{2} \))(\( 1 + 206 T^{4} + 27795 T^{8} + 3016046 T^{12} + 214358881 T^{16} \))(\( 1 + 206 T^{4} + 27795 T^{8} + 3016046 T^{12} + 214358881 T^{16} \))(\( ( 1 - 20 T^{2} + 121 T^{4} )^{8} \))
$13$ (\( ( 1 - 13 T^{2} )^{4} \))(\( 1 - 4 T^{2} - 90 T^{4} - 676 T^{6} + 28561 T^{8} \))(\( 1 - 4 T^{2} - 90 T^{4} - 676 T^{6} + 28561 T^{8} \))(\( ( 1 - 5 T + 13 T^{2} )^{2}( 1 + 5 T + 13 T^{2} )^{2} \))(\( ( 1 - 5 T + 13 T^{2} )^{2}( 1 + 5 T + 13 T^{2} )^{2} \))(\( ( 1 - 13 T^{2} )^{8} \))(\( ( 1 - 13 T^{2} )^{8} \))(\( ( 1 - 20 T^{2} + 222 T^{4} - 3380 T^{6} + 28561 T^{8} )^{4} \))
$17$ (\( ( 1 + 17 T^{2} )^{4} \))(\( ( 1 + 22 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 + 22 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 - 20 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 - 20 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 + 17 T^{2} )^{8} \))(\( ( 1 + 17 T^{2} )^{8} \))(\( ( 1 + 4 T^{2} + 558 T^{4} + 1156 T^{6} + 83521 T^{8} )^{4} \))
$19$ (\( ( 1 - 19 T^{2} )^{4} \))(\( 1 - 28 T^{2} + 486 T^{4} - 10108 T^{6} + 130321 T^{8} \))(\( 1 - 28 T^{2} + 486 T^{4} - 10108 T^{6} + 130321 T^{8} \))(\( ( 1 - 7 T + 19 T^{2} )^{2}( 1 + 7 T + 19 T^{2} )^{2} \))(\( ( 1 - 7 T + 19 T^{2} )^{2}( 1 + 7 T + 19 T^{2} )^{2} \))(\( ( 1 - 19 T^{2} )^{8} \))(\( ( 1 - 19 T^{2} )^{8} \))(\( ( 1 - 4 T^{2} + 510 T^{4} - 1444 T^{6} + 130321 T^{8} )^{4} \))
$23$ (\( 1 - 734 T^{4} + 279841 T^{8} \))(\( ( 1 - 44 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 - 44 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 + 4 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 + 4 T^{2} + 529 T^{4} )^{2} \))(\( 1 + 734 T^{4} + 258915 T^{8} + 205403294 T^{12} + 78310985281 T^{16} \))(\( 1 + 734 T^{4} + 258915 T^{8} + 205403294 T^{12} + 78310985281 T^{16} \))(\( ( 1 - 56 T^{2} + 1818 T^{4} - 29624 T^{6} + 279841 T^{8} )^{4} \))
$29$ (\( 1 + 1234 T^{4} + 707281 T^{8} \))(\( ( 1 - 56 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 56 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 56 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 56 T^{2} + 841 T^{4} )^{2} \))(\( 1 - 1234 T^{4} + 815475 T^{8} - 872784754 T^{12} + 500246412961 T^{16} \))(\( 1 - 1234 T^{4} + 815475 T^{8} - 872784754 T^{12} + 500246412961 T^{16} \))(\( ( 1 - 16 T^{2} + 1362 T^{4} - 13456 T^{6} + 707281 T^{8} )^{4} \))
$31$ (\( ( 1 - 31 T^{2} )^{4} \))(\( 1 - 76 T^{2} + 2934 T^{4} - 73036 T^{6} + 923521 T^{8} \))(\( 1 - 76 T^{2} + 2934 T^{4} - 73036 T^{6} + 923521 T^{8} \))(\( ( 1 - 35 T^{2} + 961 T^{4} )^{2} \))(\( ( 1 - 35 T^{2} + 961 T^{4} )^{2} \))(\( ( 1 - 31 T^{2} )^{8} \))(\( ( 1 - 31 T^{2} )^{8} \))(\( ( 1 - 52 T^{2} + 1422 T^{4} - 49972 T^{6} + 923521 T^{8} )^{4} \))
$37$ (\( ( 1 - 38 T^{2} + 1369 T^{4} )^{2} \))(\( ( 1 + 4 T + 66 T^{2} + 148 T^{3} + 1369 T^{4} )^{2} \))(\( ( 1 + 4 T + 66 T^{2} + 148 T^{3} + 1369 T^{4} )^{2} \))(\( ( 1 - 8 T + 37 T^{2} )^{4} \))(\( ( 1 + 8 T + 37 T^{2} )^{4} \))(\( ( 1 + 38 T^{2} + 75 T^{4} + 52022 T^{6} + 1874161 T^{8} )^{2} \))(\( ( 1 + 38 T^{2} + 75 T^{4} + 52022 T^{6} + 1874161 T^{8} )^{2} \))(\( ( 1 + 76 T^{2} + 3318 T^{4} + 104044 T^{6} + 1874161 T^{8} )^{4} \))
$41$ (\( ( 1 + 41 T^{2} )^{4} \))(\( ( 1 - 26 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 - 26 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 + 41 T^{2} )^{4} \))(\( ( 1 + 41 T^{2} )^{4} \))(\( ( 1 + 41 T^{2} )^{8} \))(\( ( 1 + 41 T^{2} )^{8} \))(\( ( 1 + 44 T^{2} + 3246 T^{4} + 73964 T^{6} + 2825761 T^{8} )^{4} \))
$43$ (\( ( 1 + 58 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 + 4 T + 42 T^{2} + 172 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 + 4 T + 42 T^{2} + 172 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 - 5 T + 43 T^{2} )^{4} \))(\( ( 1 + 5 T + 43 T^{2} )^{4} \))(\( ( 1 - 58 T^{2} + 1515 T^{4} - 107242 T^{6} + 3418801 T^{8} )^{2} \))(\( ( 1 - 58 T^{2} + 1515 T^{4} - 107242 T^{6} + 3418801 T^{8} )^{2} \))(\( ( 1 + 100 T^{2} + 6102 T^{4} + 184900 T^{6} + 3418801 T^{8} )^{4} \))
$47$ (\( ( 1 + 47 T^{2} )^{4} \))(\( ( 1 + 12 T + 118 T^{2} + 564 T^{3} + 2209 T^{4} )^{2} \))(\( ( 1 - 12 T + 118 T^{2} - 564 T^{3} + 2209 T^{4} )^{2} \))(\( ( 1 + 40 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 + 40 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 + 47 T^{2} )^{8} \))(\( ( 1 + 47 T^{2} )^{8} \))(\( ( 1 + 124 T^{2} + 7398 T^{4} + 273916 T^{6} + 4879681 T^{8} )^{4} \))
$53$ (\( 1 - 5582 T^{4} + 7890481 T^{8} \))(\( 1 - 88 T^{2} + 5826 T^{4} - 247192 T^{6} + 7890481 T^{8} \))(\( 1 - 88 T^{2} + 5826 T^{4} - 247192 T^{6} + 7890481 T^{8} \))(\( ( 1 - 74 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 - 74 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 - 5582 T^{4} + 7890481 T^{8} )^{2} \))(\( ( 1 - 5582 T^{4} + 7890481 T^{8} )^{2} \))(\( ( 1 - 104 T^{2} + 6378 T^{4} - 292136 T^{6} + 7890481 T^{8} )^{4} \))
$59$ (\( ( 1 + 59 T^{2} )^{4} \))(\( ( 1 - 12 T + 142 T^{2} - 708 T^{3} + 3481 T^{4} )^{2} \))(\( ( 1 + 12 T + 142 T^{2} + 708 T^{3} + 3481 T^{4} )^{2} \))(\( ( 1 + 64 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 + 64 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 + 59 T^{2} )^{8} \))(\( ( 1 + 59 T^{2} )^{8} \))(\( ( 1 + 92 T^{2} + 4374 T^{4} + 320252 T^{6} + 12117361 T^{8} )^{4} \))
$61$ (\( ( 1 - 61 T^{2} )^{4} \))(\( 1 - 52 T^{2} + 1206 T^{4} - 193492 T^{6} + 13845841 T^{8} \))(\( 1 - 52 T^{2} + 1206 T^{4} - 193492 T^{6} + 13845841 T^{8} \))(\( ( 1 - 95 T^{2} + 3721 T^{4} )^{2} \))(\( ( 1 - 95 T^{2} + 3721 T^{4} )^{2} \))(\( ( 1 - 61 T^{2} )^{8} \))(\( ( 1 - 61 T^{2} )^{8} \))(\( ( 1 - 52 T^{2} + 6582 T^{4} - 193492 T^{6} + 13845841 T^{8} )^{4} \))
$67$ (\( ( 1 - 4 T + 67 T^{2} )^{4} \))(\( ( 1 - 8 T + 102 T^{2} - 536 T^{3} + 4489 T^{4} )^{2} \))(\( ( 1 - 8 T + 102 T^{2} - 536 T^{3} + 4489 T^{4} )^{2} \))(\( ( 1 + 7 T + 67 T^{2} )^{4} \))(\( ( 1 - 7 T + 67 T^{2} )^{4} \))(\( ( 1 + 4 T - 51 T^{2} + 268 T^{3} + 4489 T^{4} )^{4} \))(\( ( 1 - 4 T - 51 T^{2} - 268 T^{3} + 4489 T^{4} )^{4} \))(\( ( 1 + 124 T^{2} + 12438 T^{4} + 556636 T^{6} + 20151121 T^{8} )^{4} \))
$71$ (\( 1 + 2914 T^{4} + 25411681 T^{8} \))(\( 1 + 32 T^{2} + 5538 T^{4} + 161312 T^{6} + 25411681 T^{8} \))(\( 1 + 32 T^{2} + 5538 T^{4} + 161312 T^{6} + 25411681 T^{8} \))(\( ( 1 - 134 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 - 134 T^{2} + 5041 T^{4} )^{2} \))(\( 1 - 2914 T^{4} - 16920285 T^{8} - 74049638434 T^{12} + 645753531245761 T^{16} \))(\( 1 - 2914 T^{4} - 16920285 T^{8} - 74049638434 T^{12} + 645753531245761 T^{16} \))(\( ( 1 - 184 T^{2} + 18162 T^{4} - 927544 T^{6} + 25411681 T^{8} )^{4} \))
$73$ (\( ( 1 - 73 T^{2} )^{4} \))(\( 1 - 244 T^{2} + 25110 T^{4} - 1300276 T^{6} + 28398241 T^{8} \))(\( 1 - 244 T^{2} + 25110 T^{4} - 1300276 T^{6} + 28398241 T^{8} \))(\( ( 1 - 38 T^{2} + 5329 T^{4} )^{2} \))(\( ( 1 - 38 T^{2} + 5329 T^{4} )^{2} \))(\( ( 1 - 73 T^{2} )^{8} \))(\( ( 1 - 73 T^{2} )^{8} \))(\( ( 1 - 164 T^{2} + 13326 T^{4} - 873956 T^{6} + 28398241 T^{8} )^{4} \))
$79$ (\( ( 1 - 8 T + 79 T^{2} )^{4} \))(\( ( 1 + 8 T + 126 T^{2} + 632 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 + 8 T + 126 T^{2} + 632 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 - 14 T + 79 T^{2} )^{4} \))(\( ( 1 - 14 T + 79 T^{2} )^{4} \))(\( ( 1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4} )^{4} \))(\( ( 1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4} )^{4} \))(\( ( 1 - 4 T + 79 T^{2} )^{16} \))
$83$ (\( ( 1 + 83 T^{2} )^{4} \))(\( ( 1 - 12 T + 190 T^{2} - 996 T^{3} + 6889 T^{4} )^{2} \))(\( ( 1 + 12 T + 190 T^{2} + 996 T^{3} + 6889 T^{4} )^{2} \))(\( ( 1 + 112 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 + 112 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 + 83 T^{2} )^{8} \))(\( ( 1 + 83 T^{2} )^{8} \))(\( ( 1 + 268 T^{2} + 30870 T^{4} + 1846252 T^{6} + 47458321 T^{8} )^{4} \))
$89$ (\( ( 1 + 89 T^{2} )^{4} \))(\( ( 1 - 12 T + 166 T^{2} - 1068 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 + 12 T + 166 T^{2} + 1068 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 - 38 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 - 38 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 + 89 T^{2} )^{8} \))(\( ( 1 + 89 T^{2} )^{8} \))(\( ( 1 + 140 T^{2} + 18798 T^{4} + 1108940 T^{6} + 62742241 T^{8} )^{4} \))
$97$ (\( ( 1 - 97 T^{2} )^{4} \))(\( 1 - 52 T^{2} + 15606 T^{4} - 489268 T^{6} + 88529281 T^{8} \))(\( 1 - 52 T^{2} + 15606 T^{4} - 489268 T^{6} + 88529281 T^{8} \))(\( ( 1 - 19 T + 97 T^{2} )^{2}( 1 + 19 T + 97 T^{2} )^{2} \))(\( ( 1 - 19 T + 97 T^{2} )^{2}( 1 + 19 T + 97 T^{2} )^{2} \))(\( ( 1 - 97 T^{2} )^{8} \))(\( ( 1 - 97 T^{2} )^{8} \))(\( ( 1 - 356 T^{2} + 50286 T^{4} - 3349604 T^{6} + 88529281 T^{8} )^{4} \))
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