Properties

Label 315.2.bz
Level 315
Weight 2
Character orbit bz
Rep. character \(\chi_{315}(73,\cdot)\)
Character field \(\Q(\zeta_{12})\)
Dimension 72
Newform subspaces 4
Sturm bound 96
Trace bound 5

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Defining parameters

Level: \( N \) = \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 315.bz (of order \(12\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 35 \)
Character field: \(\Q(\zeta_{12})\)
Newform subspaces: \( 4 \)
Sturm bound: \(96\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(315, [\chi])\).

Total New Old
Modular forms 224 88 136
Cusp forms 160 72 88
Eisenstein series 64 16 48

Trace form

\( 72q + 2q^{2} + 12q^{5} - 2q^{7} + 28q^{8} + O(q^{10}) \) \( 72q + 2q^{2} + 12q^{5} - 2q^{7} + 28q^{8} - 18q^{10} + 4q^{11} + 28q^{16} - 12q^{17} - 8q^{22} - 2q^{23} + 4q^{25} - 70q^{28} - 24q^{31} - 6q^{32} - 36q^{35} - 16q^{37} - 102q^{40} - 28q^{43} + 28q^{46} + 36q^{47} - 100q^{50} + 24q^{52} + 4q^{53} + 32q^{56} + 66q^{58} - 72q^{61} + 28q^{65} + 18q^{67} - 120q^{68} - 8q^{70} - 8q^{71} + 48q^{73} - 40q^{77} + 36q^{80} + 18q^{82} + 88q^{85} - 20q^{86} + 16q^{88} - 56q^{91} + 92q^{92} + 4q^{95} + 54q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(315, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
315.2.bz.a \(4\) \(2.515\) \(\Q(\zeta_{12})\) None \(-2\) \(0\) \(-4\) \(-10\) \(q+(-1+\zeta_{12}^{2}+\zeta_{12}^{3})q^{2}+(-1-\zeta_{12}^{2}+\cdots)q^{4}+\cdots\)
315.2.bz.b \(4\) \(2.515\) \(\Q(\zeta_{12})\) None \(4\) \(0\) \(4\) \(0\) \(q+(1-\zeta_{12})q^{2}+(1+\zeta_{12}^{2})q^{4}+(\zeta_{12}+\cdots)q^{5}+\cdots\)
315.2.bz.c \(32\) \(2.515\) None \(0\) \(0\) \(0\) \(0\)
315.2.bz.d \(32\) \(2.515\) None \(0\) \(0\) \(12\) \(8\)

Decomposition of \(S_{2}^{\mathrm{old}}(315, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(315, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 2 T + 5 T^{2} + 8 T^{3} + 13 T^{4} + 16 T^{5} + 20 T^{6} + 16 T^{7} + 16 T^{8} \))(\( 1 - 4 T + 5 T^{2} + 2 T^{3} - 11 T^{4} + 4 T^{5} + 20 T^{6} - 32 T^{7} + 16 T^{8} \))
$3$ 1
$5$ (\( 1 + 4 T + 11 T^{2} + 20 T^{3} + 25 T^{4} \))(\( 1 - 4 T + 11 T^{2} - 20 T^{3} + 25 T^{4} \))
$7$ (\( ( 1 + 5 T + 7 T^{2} )^{2} \))(\( 1 + 11 T^{2} + 49 T^{4} \))
$11$ (\( 1 + 2 T - 16 T^{2} - 4 T^{3} + 235 T^{4} - 44 T^{5} - 1936 T^{6} + 2662 T^{7} + 14641 T^{8} \))(\( 1 + 2 T - 16 T^{2} - 4 T^{3} + 235 T^{4} - 44 T^{5} - 1936 T^{6} + 2662 T^{7} + 14641 T^{8} \))
$13$ (\( ( 1 + 4 T + 8 T^{2} + 52 T^{3} + 169 T^{4} )^{2} \))(\( ( 1 - 4 T + 8 T^{2} - 52 T^{3} + 169 T^{4} )^{2} \))
$17$ (\( 1 + 8 T + 20 T^{2} - 52 T^{3} - 545 T^{4} - 884 T^{5} + 5780 T^{6} + 39304 T^{7} + 83521 T^{8} \))(\( 1 + 4 T + 20 T^{2} + 100 T^{3} + 271 T^{4} + 1700 T^{5} + 5780 T^{6} + 19652 T^{7} + 83521 T^{8} \))
$19$ (\( 1 + 2 T - 32 T^{2} - 4 T^{3} + 859 T^{4} - 76 T^{5} - 11552 T^{6} + 13718 T^{7} + 130321 T^{8} \))(\( 1 - 2 T - 32 T^{2} + 4 T^{3} + 859 T^{4} + 76 T^{5} - 11552 T^{6} - 13718 T^{7} + 130321 T^{8} \))
$23$ (\( 1 + 14 T + 53 T^{2} - 226 T^{3} - 2552 T^{4} - 5198 T^{5} + 28037 T^{6} + 170338 T^{7} + 279841 T^{8} \))(\( 1 - 4 T + 53 T^{2} - 244 T^{3} + 1588 T^{4} - 5612 T^{5} + 28037 T^{6} - 48668 T^{7} + 279841 T^{8} \))
$29$ (\( ( 1 - 49 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 49 T^{2} + 841 T^{4} )^{2} \))
$31$ (\( 1 + 12 T + 106 T^{2} + 696 T^{3} + 3891 T^{4} + 21576 T^{5} + 101866 T^{6} + 357492 T^{7} + 923521 T^{8} \))(\( 1 + 12 T + 106 T^{2} + 696 T^{3} + 3891 T^{4} + 21576 T^{5} + 101866 T^{6} + 357492 T^{7} + 923521 T^{8} \))
$37$ (\( 1 + 12 T + 72 T^{2} + 288 T^{3} + 983 T^{4} + 10656 T^{5} + 98568 T^{6} + 607836 T^{7} + 1874161 T^{8} \))(\( 1 - 12 T + 72 T^{2} - 288 T^{3} + 983 T^{4} - 10656 T^{5} + 98568 T^{6} - 607836 T^{7} + 1874161 T^{8} \))
$41$ (\( 1 - 122 T^{2} + 6651 T^{4} - 205082 T^{6} + 2825761 T^{8} \))(\( 1 - 122 T^{2} + 6651 T^{4} - 205082 T^{6} + 2825761 T^{8} \))
$43$ (\( 1 + 6 T + 18 T^{2} + 60 T^{3} - 889 T^{4} + 2580 T^{5} + 33282 T^{6} + 477042 T^{7} + 3418801 T^{8} \))(\( 1 + 6 T + 18 T^{2} + 60 T^{3} - 889 T^{4} + 2580 T^{5} + 33282 T^{6} + 477042 T^{7} + 3418801 T^{8} \))
$47$ (\( 1 + 6 T + 90 T^{2} + 672 T^{3} + 5159 T^{4} + 31584 T^{5} + 198810 T^{6} + 622938 T^{7} + 4879681 T^{8} \))(\( 1 + 18 T + 90 T^{2} - 528 T^{3} - 8377 T^{4} - 24816 T^{5} + 198810 T^{6} + 1868814 T^{7} + 4879681 T^{8} \))
$53$ (\( ( 1 - 4 T - 37 T^{2} - 212 T^{3} + 2809 T^{4} )( 1 + 14 T + 143 T^{2} + 742 T^{3} + 2809 T^{4} ) \))(\( ( 1 - 4 T - 37 T^{2} - 212 T^{3} + 2809 T^{4} )( 1 + 14 T + 143 T^{2} + 742 T^{3} + 2809 T^{4} ) \))
$59$ (\( 1 - 6 T - 64 T^{2} + 108 T^{3} + 4395 T^{4} + 6372 T^{5} - 222784 T^{6} - 1232274 T^{7} + 12117361 T^{8} \))(\( 1 + 6 T - 64 T^{2} - 108 T^{3} + 4395 T^{4} - 6372 T^{5} - 222784 T^{6} + 1232274 T^{7} + 12117361 T^{8} \))
$61$ (\( 1 + 12 T + 157 T^{2} + 1308 T^{3} + 11088 T^{4} + 79788 T^{5} + 584197 T^{6} + 2723772 T^{7} + 13845841 T^{8} \))(\( 1 + 12 T + 157 T^{2} + 1308 T^{3} + 11088 T^{4} + 79788 T^{5} + 584197 T^{6} + 2723772 T^{7} + 13845841 T^{8} \))
$67$ (\( 1 - 8 T + 137 T^{2} - 1224 T^{3} + 11492 T^{4} - 82008 T^{5} + 614993 T^{6} - 2406104 T^{7} + 20151121 T^{8} \))(\( 1 + 22 T + 137 T^{2} - 834 T^{3} - 16648 T^{4} - 55878 T^{5} + 614993 T^{6} + 6616786 T^{7} + 20151121 T^{8} \))
$71$ (\( ( 1 + 6 T + 148 T^{2} + 426 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 + 6 T + 148 T^{2} + 426 T^{3} + 5041 T^{4} )^{2} \))
$73$ (\( 1 + 144 T^{2} + 600 T^{3} + 10991 T^{4} + 43800 T^{5} + 767376 T^{6} + 28398241 T^{8} \))(\( 1 + 24 T + 144 T^{2} - 1752 T^{3} - 31057 T^{4} - 127896 T^{5} + 767376 T^{6} + 9336408 T^{7} + 28398241 T^{8} \))
$79$ (\( 1 + 6 T + 148 T^{2} + 816 T^{3} + 13203 T^{4} + 64464 T^{5} + 923668 T^{6} + 2958234 T^{7} + 38950081 T^{8} \))(\( 1 - 6 T + 148 T^{2} - 816 T^{3} + 13203 T^{4} - 64464 T^{5} + 923668 T^{6} - 2958234 T^{7} + 38950081 T^{8} \))
$83$ (\( 1 + 2 T + 2 T^{2} + 140 T^{3} + 9631 T^{4} + 11620 T^{5} + 13778 T^{6} + 1143574 T^{7} + 47458321 T^{8} \))(\( 1 - 2 T + 2 T^{2} - 140 T^{3} + 9631 T^{4} - 11620 T^{5} + 13778 T^{6} - 1143574 T^{7} + 47458321 T^{8} \))
$89$ (\( ( 1 + 16 T + 89 T^{2} )^{2}( 1 - 16 T + 167 T^{2} - 1424 T^{3} + 7921 T^{4} ) \))(\( ( 1 - 16 T + 89 T^{2} )^{2}( 1 + 16 T + 167 T^{2} + 1424 T^{3} + 7921 T^{4} ) \))
$97$ (\( 1 - 4 T + 8 T^{2} - 12 T^{3} - 8818 T^{4} - 1164 T^{5} + 75272 T^{6} - 3650692 T^{7} + 88529281 T^{8} \))(\( 1 + 4 T + 8 T^{2} + 12 T^{3} - 8818 T^{4} + 1164 T^{5} + 75272 T^{6} + 3650692 T^{7} + 88529281 T^{8} \))
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