Properties

Label 525.2.bc
Level 525
Weight 2
Character orbit bc
Rep. character \(\chi_{525}(82,\cdot)\)
Character field \(\Q(\zeta_{12})\)
Dimension 96
Newform subspaces 5
Sturm bound 160
Trace bound 11

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Defining parameters

Level: \( N \) = \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 525.bc (of order \(12\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 35 \)
Character field: \(\Q(\zeta_{12})\)
Newform subspaces: \( 5 \)
Sturm bound: \(160\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(2\)

Dimensions

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

Total New Old
Modular forms 368 96 272
Cusp forms 272 96 176
Eisenstein series 96 0 96

Trace form

\( 96q - 8q^{7} + 24q^{8} + O(q^{10}) \) \( 96q - 8q^{7} + 24q^{8} + 16q^{11} + 80q^{16} - 4q^{21} + 8q^{22} + 8q^{23} - 48q^{26} + 24q^{28} - 48q^{31} - 24q^{32} + 36q^{33} - 96q^{36} - 4q^{37} - 12q^{38} - 16q^{42} - 40q^{43} - 40q^{46} + 60q^{47} + 16q^{51} + 108q^{52} + 24q^{53} + 96q^{56} - 16q^{57} - 4q^{58} - 12q^{61} - 4q^{63} - 144q^{66} - 8q^{67} - 132q^{68} + 32q^{71} - 12q^{72} - 36q^{73} - 60q^{77} - 80q^{78} + 48q^{81} - 12q^{82} - 88q^{86} + 24q^{87} + 32q^{88} - 92q^{91} + 56q^{92} + 24q^{93} + 72q^{98} + O(q^{100}) \)

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.bc.a \(8\) \(4.192\) \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(0\) \(q+\zeta_{24}^{7}q^{3}+(-2\zeta_{24}^{2}+2\zeta_{24}^{6})q^{4}+\cdots\)
525.2.bc.b \(8\) \(4.192\) \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(0\) \(q+(2\zeta_{24}-\zeta_{24}^{5})q^{2}+\zeta_{24}^{7}q^{3}+(\zeta_{24}^{2}+\cdots)q^{4}+\cdots\)
525.2.bc.c \(24\) \(4.192\) None \(0\) \(0\) \(0\) \(0\)
525.2.bc.d \(24\) \(4.192\) None \(0\) \(0\) \(0\) \(0\)
525.2.bc.e \(32\) \(4.192\) None \(0\) \(0\) \(0\) \(-8\)

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

\( S_{2}^{\mathrm{old}}(525, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(175, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 - 2 T + 2 T^{2} - 4 T^{3} + 4 T^{4} )^{2}( 1 + 2 T + 2 T^{2} + 4 T^{3} + 4 T^{4} )^{2} \))(\( 1 + 7 T^{4} + 33 T^{8} + 112 T^{12} + 256 T^{16} \))
$3$ (\( 1 - T^{4} + T^{8} \))(\( 1 - T^{4} + T^{8} \))
$5$ 1
$7$ (\( 1 - 94 T^{4} + 2401 T^{8} \))(\( 1 + 23 T^{4} + 2401 T^{8} \))
$11$ (\( ( 1 + 6 T + 25 T^{2} + 66 T^{3} + 121 T^{4} )^{4} \))(\( ( 1 - 6 T + 25 T^{2} - 66 T^{3} + 121 T^{4} )^{4} \))
$13$ (\( ( 1 + 146 T^{4} + 28561 T^{8} )^{2} \))(\( ( 1 - 337 T^{4} + 28561 T^{8} )^{2} \))
$17$ (\( ( 1 - 8 T + 32 T^{2} + 16 T^{3} - 353 T^{4} + 272 T^{5} + 9248 T^{6} - 39304 T^{7} + 83521 T^{8} )( 1 + 8 T + 32 T^{2} - 16 T^{3} - 353 T^{4} - 272 T^{5} + 9248 T^{6} + 39304 T^{7} + 83521 T^{8} ) \))(\( ( 1 - 289 T^{4} + 83521 T^{8} )^{2} \))
$19$ (\( ( 1 - 37 T^{2} + 361 T^{4} )^{2}( 1 + 11 T^{2} + 361 T^{4} )^{2} \))(\( ( 1 - 37 T^{2} + 361 T^{4} )^{2}( 1 + 26 T^{2} + 361 T^{4} )^{2} \))
$23$ (\( ( 1 - 529 T^{4} + 279841 T^{8} )^{2} \))(\( 1 - 98 T^{4} - 270237 T^{8} - 27424418 T^{12} + 78310985281 T^{16} \))
$29$ (\( ( 1 - 29 T^{2} )^{8} \))(\( ( 1 - 29 T^{2} )^{8} \))
$31$ (\( ( 1 + 9 T + 58 T^{2} + 279 T^{3} + 961 T^{4} )^{4} \))(\( ( 1 - 6 T + 43 T^{2} - 186 T^{3} + 961 T^{4} )^{4} \))
$37$ (\( 1 + 2737 T^{4} + 5617008 T^{8} + 5129578657 T^{12} + 3512479453921 T^{16} \))(\( 1 + 2737 T^{4} + 5617008 T^{8} + 5129578657 T^{12} + 3512479453921 T^{16} \))
$41$ (\( ( 1 + 26 T^{2} + 1681 T^{4} )^{4} \))(\( ( 1 + 26 T^{2} + 1681 T^{4} )^{4} \))
$43$ (\( ( 1 - 217 T^{4} + 3418801 T^{8} )^{2} \))(\( ( 1 + 1778 T^{4} + 3418801 T^{8} )^{2} \))
$47$ (\( 1 + 1054 T^{4} - 3768765 T^{8} + 5143183774 T^{12} + 23811286661761 T^{16} \))(\( ( 1 - 2209 T^{4} + 4879681 T^{8} )^{2} \))
$53$ (\( 1 + 5614 T^{4} + 23626515 T^{8} + 44297160334 T^{12} + 62259690411361 T^{16} \))(\( ( 1 - 2809 T^{4} + 7890481 T^{8} )^{2} \))
$59$ (\( ( 1 - 10 T^{2} - 3381 T^{4} - 34810 T^{6} + 12117361 T^{8} )^{2} \))(\( ( 1 - 70 T^{2} + 1419 T^{4} - 243670 T^{6} + 12117361 T^{8} )^{2} \))
$61$ (\( ( 1 - 14 T + 61 T^{2} )^{4}( 1 - T + 61 T^{2} )^{4} \))(\( ( 1 + 3 T + 64 T^{2} + 183 T^{3} + 3721 T^{4} )^{4} \))
$67$ (\( ( 1 - 8809 T^{4} + 20151121 T^{8} )( 1 + 2903 T^{4} + 20151121 T^{8} ) \))(\( 1 + 5497 T^{4} + 10065888 T^{8} + 110770712137 T^{12} + 406067677556641 T^{16} \))
$71$ (\( ( 1 + 6 T + 71 T^{2} )^{8} \))(\( ( 1 - 12 T + 71 T^{2} )^{8} \))
$73$ (\( 1 - 10367 T^{4} + 79076448 T^{8} - 294404564447 T^{12} + 806460091894081 T^{16} \))(\( ( 1 - 8542 T^{4} + 28398241 T^{8} )( 1 + 9791 T^{4} + 28398241 T^{8} ) \))
$79$ (\( ( 1 + 133 T^{2} + 11448 T^{4} + 830053 T^{6} + 38950081 T^{8} )^{2} \))(\( ( 1 + 37 T^{2} - 4872 T^{4} + 230917 T^{6} + 38950081 T^{8} )^{2} \))
$83$ (\( ( 1 + 3122 T^{4} + 47458321 T^{8} )^{2} \))(\( ( 1 - 13294 T^{4} + 47458321 T^{8} )^{2} \))
$89$ (\( ( 1 - 70 T^{2} - 3021 T^{4} - 554470 T^{6} + 62742241 T^{8} )^{2} \))(\( ( 1 + 14 T^{2} - 7725 T^{4} + 110894 T^{6} + 62742241 T^{8} )^{2} \))
$97$ (\( ( 1 - 18193 T^{4} + 88529281 T^{8} )^{2} \))(\( ( 1 + 9743 T^{4} + 88529281 T^{8} )^{2} \))
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