Properties

Label 525.2.g
Level 525
Weight 2
Character orbit g
Rep. character \(\chi_{525}(524,\cdot)\)
Character field \(\Q\)
Dimension 44
Newform subspaces 6
Sturm bound 160
Trace bound 6

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Defining parameters

Level: \( N \) = \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 525.g (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 105 \)
Character field: \(\Q\)
Newform subspaces: \( 6 \)
Sturm bound: \(160\)
Trace bound: \(6\)
Distinguishing \(T_p\): \(2\), \(41\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(525, [\chi])\).

Total New Old
Modular forms 92 52 40
Cusp forms 68 44 24
Eisenstein series 24 8 16

Trace form

\( 44q + 48q^{4} + 8q^{9} + O(q^{10}) \) \( 44q + 48q^{4} + 8q^{9} + 40q^{16} - 18q^{21} - 24q^{36} - 48q^{39} - 88q^{46} + 34q^{49} - 60q^{51} - 32q^{64} + 8q^{79} + 112q^{81} - 36q^{84} + 54q^{91} + 4q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(525, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
525.2.g.a \(4\) \(4.192\) \(\Q(\zeta_{12})\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{12}q^{3}-2q^{4}+(\zeta_{12}-\zeta_{12}^{2})q^{7}+\cdots\)
525.2.g.b \(4\) \(4.192\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) \(q+\zeta_{12}q^{2}-\zeta_{12}q^{3}+q^{4}-3q^{6}+(-\zeta_{12}+\cdots)q^{7}+\cdots\)
525.2.g.c \(4\) \(4.192\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{12}q^{2}-\zeta_{12}q^{3}+q^{4}+3q^{6}+(-\zeta_{12}+\cdots)q^{7}+\cdots\)
525.2.g.d \(8\) \(4.192\) 8.0.303595776.1 None \(0\) \(0\) \(0\) \(0\) \(q+(\beta _{1}-\beta _{4}-\beta _{5})q^{2}-\beta _{1}q^{3}+(1-\beta _{2}+\cdots)q^{4}+\cdots\)
525.2.g.e \(8\) \(4.192\) 8.0.303595776.1 None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{6}q^{2}+(-\beta _{2}-\beta _{4}-\beta _{6})q^{3}+(1+\cdots)q^{4}+\cdots\)
525.2.g.f \(16\) \(4.192\) \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{2}q^{2}+\beta _{4}q^{3}+(1-\beta _{5})q^{4}-\beta _{8}q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(525, [\chi])\) into lower level spaces

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 + 2 T^{2} )^{4} \))(\( ( 1 + T^{2} + 4 T^{4} )^{2} \))(\( ( 1 + T^{2} + 4 T^{4} )^{2} \))(\( ( 1 + T^{2} + 4 T^{6} + 16 T^{8} )^{2} \))(\( ( 1 + T^{2} + 4 T^{6} + 16 T^{8} )^{2} \))(\( ( 1 + T^{2} + 3 T^{4} + 4 T^{6} + 16 T^{8} )^{4} \))
$3$ (\( ( 1 - 3 T^{2} )^{2} \))(\( ( 1 - 3 T^{2} )^{2} \))(\( ( 1 - 3 T^{2} )^{2} \))(\( 1 + 5 T^{2} + 16 T^{4} + 45 T^{6} + 81 T^{8} \))(\( 1 + 5 T^{2} + 16 T^{4} + 45 T^{6} + 81 T^{8} \))(\( ( 1 + 2 T^{2} - 2 T^{4} + 18 T^{6} + 81 T^{8} )^{2} \))
$5$ (\( \))(\( \))(\( \))(\( \))(\( \))(\( \))
$7$ (\( 1 + 11 T^{2} + 49 T^{4} \))(\( 1 + 2 T^{2} + 49 T^{4} \))(\( 1 + 2 T^{2} + 49 T^{4} \))(\( ( 1 + 2 T^{2} + 49 T^{4} )^{2} \))(\( ( 1 + 2 T^{2} + 49 T^{4} )^{2} \))(\( ( 1 - 10 T^{2} + 49 T^{4} )^{4} \))
$11$ (\( ( 1 - 11 T^{2} )^{4} \))(\( ( 1 - 10 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 - 10 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 - 37 T^{2} + 576 T^{4} - 4477 T^{6} + 14641 T^{8} )^{2} \))(\( ( 1 - 37 T^{2} + 576 T^{4} - 4477 T^{6} + 14641 T^{8} )^{2} \))(\( ( 1 - 16 T^{2} + 285 T^{4} - 1936 T^{6} + 14641 T^{8} )^{4} \))
$13$ (\( ( 1 - T^{2} + 169 T^{4} )^{2} \))(\( ( 1 + 13 T^{2} )^{4} \))(\( ( 1 + 13 T^{2} )^{4} \))(\( ( 1 + T^{2} + 264 T^{4} + 169 T^{6} + 28561 T^{8} )^{2} \))(\( ( 1 + T^{2} + 264 T^{4} + 169 T^{6} + 28561 T^{8} )^{2} \))(\( ( 1 + 20 T^{2} + 169 T^{4} )^{8} \))
$17$ (\( ( 1 - 17 T^{2} )^{4} \))(\( ( 1 + 2 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 + 2 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 - 47 T^{2} + 1056 T^{4} - 13583 T^{6} + 83521 T^{8} )^{2} \))(\( ( 1 - 47 T^{2} + 1056 T^{4} - 13583 T^{6} + 83521 T^{8} )^{2} \))(\( ( 1 - 26 T^{2} + 558 T^{4} - 7514 T^{6} + 83521 T^{8} )^{4} \))
$19$ (\( ( 1 - T + 19 T^{2} )^{2}( 1 + T + 19 T^{2} )^{2} \))(\( ( 1 - 8 T + 19 T^{2} )^{2}( 1 + 8 T + 19 T^{2} )^{2} \))(\( ( 1 - 8 T + 19 T^{2} )^{2}( 1 + 8 T + 19 T^{2} )^{2} \))(\( ( 1 - 8 T + 19 T^{2} )^{4}( 1 + 8 T + 19 T^{2} )^{4} \))(\( ( 1 - 8 T + 19 T^{2} )^{4}( 1 + 8 T + 19 T^{2} )^{4} \))(\( ( 1 - 10 T^{2} + 558 T^{4} - 3610 T^{6} + 130321 T^{8} )^{4} \))
$23$ (\( ( 1 + 23 T^{2} )^{4} \))(\( ( 1 + 34 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 + 34 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 + 16 T^{2} - 66 T^{4} + 8464 T^{6} + 279841 T^{8} )^{2} \))(\( ( 1 + 16 T^{2} - 66 T^{4} + 8464 T^{6} + 279841 T^{8} )^{2} \))(\( ( 1 + 64 T^{2} + 1893 T^{4} + 33856 T^{6} + 279841 T^{8} )^{4} \))
$29$ (\( ( 1 - 29 T^{2} )^{4} \))(\( ( 1 - 10 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 10 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 97 T^{2} + 3960 T^{4} - 81577 T^{6} + 707281 T^{8} )^{2} \))(\( ( 1 - 97 T^{2} + 3960 T^{4} - 81577 T^{6} + 707281 T^{8} )^{2} \))(\( ( 1 - 88 T^{2} + 3429 T^{4} - 74008 T^{6} + 707281 T^{8} )^{4} \))
$31$ (\( ( 1 - 7 T + 31 T^{2} )^{2}( 1 + 7 T + 31 T^{2} )^{2} \))(\( ( 1 - 50 T^{2} + 961 T^{4} )^{2} \))(\( ( 1 - 50 T^{2} + 961 T^{4} )^{2} \))(\( ( 1 - 50 T^{2} + 961 T^{4} )^{4} \))(\( ( 1 - 50 T^{2} + 961 T^{4} )^{4} \))(\( ( 1 - 58 T^{2} + 2574 T^{4} - 55738 T^{6} + 923521 T^{8} )^{4} \))
$37$ (\( ( 1 + 26 T^{2} + 1369 T^{4} )^{2} \))(\( ( 1 - 12 T + 37 T^{2} )^{2}( 1 + 12 T + 37 T^{2} )^{2} \))(\( ( 1 - 12 T + 37 T^{2} )^{2}( 1 + 12 T + 37 T^{2} )^{2} \))(\( ( 1 - 80 T^{2} + 4206 T^{4} - 109520 T^{6} + 1874161 T^{8} )^{2} \))(\( ( 1 - 80 T^{2} + 4206 T^{4} - 109520 T^{6} + 1874161 T^{8} )^{2} \))(\( ( 1 - 74 T^{2} + 2763 T^{4} - 101306 T^{6} + 1874161 T^{8} )^{4} \))
$41$ (\( ( 1 + 41 T^{2} )^{4} \))(\( ( 1 - 6 T + 41 T^{2} )^{4} \))(\( ( 1 + 6 T + 41 T^{2} )^{4} \))(\( ( 1 + 6 T + 41 T^{2} )^{8} \))(\( ( 1 - 6 T + 41 T^{2} )^{8} \))(\( ( 1 + 38 T^{2} + 2022 T^{4} + 63878 T^{6} + 2825761 T^{8} )^{4} \))
$43$ (\( ( 1 - 61 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 22 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 22 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 104 T^{2} + 6270 T^{4} - 192296 T^{6} + 3418801 T^{8} )^{2} \))(\( ( 1 - 104 T^{2} + 6270 T^{4} - 192296 T^{6} + 3418801 T^{8} )^{2} \))(\( ( 1 - 122 T^{2} + 7083 T^{4} - 225578 T^{6} + 3418801 T^{8} )^{4} \))
$47$ (\( ( 1 - 47 T^{2} )^{4} \))(\( ( 1 + 50 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 + 50 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 - 131 T^{2} + 8040 T^{4} - 289379 T^{6} + 4879681 T^{8} )^{2} \))(\( ( 1 - 131 T^{2} + 8040 T^{4} - 289379 T^{6} + 4879681 T^{8} )^{2} \))(\( ( 1 - 146 T^{2} + 9558 T^{4} - 322514 T^{6} + 4879681 T^{8} )^{4} \))
$53$ (\( ( 1 + 53 T^{2} )^{4} \))(\( ( 1 + 53 T^{2} )^{4} \))(\( ( 1 + 53 T^{2} )^{4} \))(\( ( 1 + 136 T^{2} + 9054 T^{4} + 382024 T^{6} + 7890481 T^{8} )^{2} \))(\( ( 1 + 136 T^{2} + 9054 T^{4} + 382024 T^{6} + 7890481 T^{8} )^{2} \))(\( ( 1 + 16 T^{2} - 1122 T^{4} + 44944 T^{6} + 7890481 T^{8} )^{4} \))
$59$ (\( ( 1 + 59 T^{2} )^{4} \))(\( ( 1 - 12 T + 59 T^{2} )^{4} \))(\( ( 1 + 12 T + 59 T^{2} )^{4} \))(\( ( 1 - 6 T + 94 T^{2} - 354 T^{3} + 3481 T^{4} )^{4} \))(\( ( 1 + 6 T + 94 T^{2} + 354 T^{3} + 3481 T^{4} )^{4} \))(\( ( 1 + 68 T^{2} + 7362 T^{4} + 236708 T^{6} + 12117361 T^{8} )^{4} \))
$61$ (\( ( 1 - 13 T + 61 T^{2} )^{2}( 1 + 13 T + 61 T^{2} )^{2} \))(\( ( 1 - 14 T + 61 T^{2} )^{2}( 1 + 14 T + 61 T^{2} )^{2} \))(\( ( 1 - 14 T + 61 T^{2} )^{2}( 1 + 14 T + 61 T^{2} )^{2} \))(\( ( 1 - 14 T + 61 T^{2} )^{4}( 1 + 14 T + 61 T^{2} )^{4} \))(\( ( 1 - 14 T + 61 T^{2} )^{4}( 1 + 14 T + 61 T^{2} )^{4} \))(\( ( 1 - 178 T^{2} + 15174 T^{4} - 662338 T^{6} + 13845841 T^{8} )^{4} \))
$67$ (\( ( 1 - 109 T^{2} + 4489 T^{4} )^{2} \))(\( ( 1 - 70 T^{2} + 4489 T^{4} )^{2} \))(\( ( 1 - 70 T^{2} + 4489 T^{4} )^{2} \))(\( ( 1 - 200 T^{2} + 18846 T^{4} - 897800 T^{6} + 20151121 T^{8} )^{2} \))(\( ( 1 - 200 T^{2} + 18846 T^{4} - 897800 T^{6} + 20151121 T^{8} )^{2} \))(\( ( 1 - 50 T^{2} + 1203 T^{4} - 224450 T^{6} + 20151121 T^{8} )^{4} \))
$71$ (\( ( 1 - 71 T^{2} )^{4} \))(\( ( 1 - 130 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 - 130 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 - 100 T^{2} + 4134 T^{4} - 504100 T^{6} + 25411681 T^{8} )^{2} \))(\( ( 1 - 100 T^{2} + 4134 T^{4} - 504100 T^{6} + 25411681 T^{8} )^{2} \))(\( ( 1 - 256 T^{2} + 26445 T^{4} - 1290496 T^{6} + 25411681 T^{8} )^{4} \))
$73$ (\( ( 1 - 46 T^{2} + 5329 T^{4} )^{2} \))(\( ( 1 + 98 T^{2} + 5329 T^{4} )^{2} \))(\( ( 1 + 98 T^{2} + 5329 T^{4} )^{2} \))(\( ( 1 + 98 T^{2} + 5329 T^{4} )^{4} \))(\( ( 1 + 98 T^{2} + 5329 T^{4} )^{4} \))(\( ( 1 - 8 T^{2} - 1422 T^{4} - 42632 T^{6} + 28398241 T^{8} )^{4} \))
$79$ (\( ( 1 - 4 T + 79 T^{2} )^{4} \))(\( ( 1 + 8 T + 79 T^{2} )^{4} \))(\( ( 1 + 8 T + 79 T^{2} )^{4} \))(\( ( 1 + T + 150 T^{2} + 79 T^{3} + 6241 T^{4} )^{4} \))(\( ( 1 + T + 150 T^{2} + 79 T^{3} + 6241 T^{4} )^{4} \))(\( ( 1 - 8 T + 153 T^{2} - 632 T^{3} + 6241 T^{4} )^{8} \))
$83$ (\( ( 1 - 83 T^{2} )^{4} \))(\( ( 1 - 83 T^{2} )^{4} \))(\( ( 1 - 83 T^{2} )^{4} \))(\( ( 1 + 4 T^{2} - 5226 T^{4} + 27556 T^{6} + 47458321 T^{8} )^{2} \))(\( ( 1 + 4 T^{2} - 5226 T^{4} + 27556 T^{6} + 47458321 T^{8} )^{2} \))(\( ( 1 - 38 T^{2} + 4878 T^{4} - 261782 T^{6} + 47458321 T^{8} )^{4} \))
$89$ (\( ( 1 + 89 T^{2} )^{4} \))(\( ( 1 + 6 T + 89 T^{2} )^{4} \))(\( ( 1 - 6 T + 89 T^{2} )^{4} \))(\( ( 1 + 18 T + 226 T^{2} + 1602 T^{3} + 7921 T^{4} )^{4} \))(\( ( 1 - 18 T + 226 T^{2} - 1602 T^{3} + 7921 T^{4} )^{4} \))(\( ( 1 + 188 T^{2} + 23922 T^{4} + 1489148 T^{6} + 62742241 T^{8} )^{4} \))
$97$ (\( ( 1 - 169 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 + 146 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 + 146 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 + 265 T^{2} + 32736 T^{4} + 2493385 T^{6} + 88529281 T^{8} )^{2} \))(\( ( 1 + 265 T^{2} + 32736 T^{4} + 2493385 T^{6} + 88529281 T^{8} )^{2} \))(\( ( 1 + 250 T^{2} + 29718 T^{4} + 2352250 T^{6} + 88529281 T^{8} )^{4} \))
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