Properties

Label 630.2.g
Level 630
Weight 2
Character orbit g
Rep. character \(\chi_{630}(379,\cdot)\)
Character field \(\Q\)
Dimension 16
Newform subspaces 7
Sturm bound 288
Trace bound 11

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Defining parameters

Level: \( N \) = \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 630.g (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 5 \)
Character field: \(\Q\)
Newform subspaces: \( 7 \)
Sturm bound: \(288\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(11\), \(29\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(630, [\chi])\).

Total New Old
Modular forms 160 16 144
Cusp forms 128 16 112
Eisenstein series 32 0 32

Trace form

\( 16q - 16q^{4} - 4q^{5} + O(q^{10}) \) \( 16q - 16q^{4} - 4q^{5} + 4q^{14} + 16q^{16} + 16q^{19} + 4q^{20} - 12q^{25} - 8q^{26} + 8q^{29} + 32q^{31} + 4q^{35} + 8q^{41} + 8q^{46} - 16q^{49} - 12q^{50} + 8q^{55} - 4q^{56} - 48q^{59} - 48q^{61} - 16q^{64} + 20q^{65} + 4q^{70} + 64q^{71} + 16q^{74} - 16q^{76} + 40q^{79} - 4q^{80} - 16q^{85} - 8q^{86} - 40q^{89} - 8q^{91} - 4q^{95} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(630, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
630.2.g.a \(2\) \(5.031\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(-4\) \(0\) \(q+iq^{2}-q^{4}+(-2-i)q^{5}+iq^{7}+\cdots\)
630.2.g.b \(2\) \(5.031\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(-2\) \(0\) \(q+iq^{2}-q^{4}+(-1+2i)q^{5}-iq^{7}+\cdots\)
630.2.g.c \(2\) \(5.031\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(-2\) \(0\) \(q+iq^{2}-q^{4}+(-1-2i)q^{5}+iq^{7}+\cdots\)
630.2.g.d \(2\) \(5.031\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(2\) \(0\) \(q+iq^{2}-q^{4}+(1+2i)q^{5}-iq^{7}-iq^{8}+\cdots\)
630.2.g.e \(2\) \(5.031\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(2\) \(0\) \(q+iq^{2}-q^{4}+(1+2i)q^{5}+iq^{7}-iq^{8}+\cdots\)
630.2.g.f \(2\) \(5.031\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(4\) \(0\) \(q+iq^{2}-q^{4}+(2-i)q^{5}-iq^{7}-iq^{8}+\cdots\)
630.2.g.g \(4\) \(5.031\) \(\Q(i, \sqrt{6})\) None \(0\) \(0\) \(-4\) \(0\) \(q-\beta _{2}q^{2}-q^{4}+(-1-\beta _{1}+\beta _{2})q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(630, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(630, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(70, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(90, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(210, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(315, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( ( 1 + T^{2} )^{2} \))
$3$ 1
$5$ (\( 1 + 4 T + 5 T^{2} \))(\( 1 + 2 T + 5 T^{2} \))(\( 1 + 2 T + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))(\( 1 - 4 T + 5 T^{2} \))(\( 1 + 4 T + 8 T^{2} + 20 T^{3} + 25 T^{4} \))
$7$ (\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( ( 1 + T^{2} )^{2} \))
$11$ (\( ( 1 + 11 T^{2} )^{2} \))(\( ( 1 + 6 T + 11 T^{2} )^{2} \))(\( ( 1 + 2 T + 11 T^{2} )^{2} \))(\( ( 1 - 2 T + 11 T^{2} )^{2} \))(\( ( 1 - 6 T + 11 T^{2} )^{2} \))(\( ( 1 + 11 T^{2} )^{2} \))(\( ( 1 - 2 T^{2} + 121 T^{4} )^{2} \))
$13$ (\( ( 1 - 6 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} ) \))(\( 1 - 22 T^{2} + 169 T^{4} \))(\( ( 1 - 4 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} ) \))(\( 1 - 22 T^{2} + 169 T^{4} \))(\( 1 - 22 T^{2} + 169 T^{4} \))(\( ( 1 - 6 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} ) \))(\( 1 - 32 T^{2} + 498 T^{4} - 5408 T^{6} + 28561 T^{8} \))
$17$ (\( ( 1 - 8 T + 17 T^{2} )( 1 + 8 T + 17 T^{2} ) \))(\( ( 1 - 8 T + 17 T^{2} )( 1 + 8 T + 17 T^{2} ) \))(\( 1 - 18 T^{2} + 289 T^{4} \))(\( ( 1 - 2 T + 17 T^{2} )( 1 + 2 T + 17 T^{2} ) \))(\( ( 1 - 8 T + 17 T^{2} )( 1 + 8 T + 17 T^{2} ) \))(\( ( 1 - 8 T + 17 T^{2} )( 1 + 8 T + 17 T^{2} ) \))(\( ( 1 - 8 T + 17 T^{2} )^{2}( 1 + 8 T + 17 T^{2} )^{2} \))
$19$ (\( ( 1 - 8 T + 19 T^{2} )^{2} \))(\( ( 1 + 4 T + 19 T^{2} )^{2} \))(\( ( 1 - 6 T + 19 T^{2} )^{2} \))(\( ( 1 - 2 T + 19 T^{2} )^{2} \))(\( ( 1 + 4 T + 19 T^{2} )^{2} \))(\( ( 1 - 8 T + 19 T^{2} )^{2} \))(\( ( 1 + 8 T + 48 T^{2} + 152 T^{3} + 361 T^{4} )^{2} \))
$23$ (\( 1 + 18 T^{2} + 529 T^{4} \))(\( 1 - 30 T^{2} + 529 T^{4} \))(\( 1 + 18 T^{2} + 529 T^{4} \))(\( ( 1 - 23 T^{2} )^{2} \))(\( 1 - 30 T^{2} + 529 T^{4} \))(\( 1 + 18 T^{2} + 529 T^{4} \))(\( 1 - 36 T^{2} + 998 T^{4} - 19044 T^{6} + 279841 T^{8} \))
$29$ (\( ( 1 + 8 T + 29 T^{2} )^{2} \))(\( ( 1 - 2 T + 29 T^{2} )^{2} \))(\( ( 1 - 6 T + 29 T^{2} )^{2} \))(\( ( 1 + 6 T + 29 T^{2} )^{2} \))(\( ( 1 + 2 T + 29 T^{2} )^{2} \))(\( ( 1 - 8 T + 29 T^{2} )^{2} \))(\( ( 1 - 4 T + 38 T^{2} - 116 T^{3} + 841 T^{4} )^{2} \))
$31$ (\( ( 1 - 4 T + 31 T^{2} )^{2} \))(\( ( 1 + 2 T + 31 T^{2} )^{2} \))(\( ( 1 + 2 T + 31 T^{2} )^{2} \))(\( ( 1 - 6 T + 31 T^{2} )^{2} \))(\( ( 1 + 2 T + 31 T^{2} )^{2} \))(\( ( 1 - 4 T + 31 T^{2} )^{2} \))(\( ( 1 - 8 T + 54 T^{2} - 248 T^{3} + 961 T^{4} )^{2} \))
$37$ (\( 1 - 10 T^{2} + 1369 T^{4} \))(\( 1 + 26 T^{2} + 1369 T^{4} \))(\( 1 - 58 T^{2} + 1369 T^{4} \))(\( 1 - 10 T^{2} + 1369 T^{4} \))(\( 1 + 26 T^{2} + 1369 T^{4} \))(\( 1 - 10 T^{2} + 1369 T^{4} \))(\( ( 1 - 12 T + 37 T^{2} )^{2}( 1 + 12 T + 37 T^{2} )^{2} \))
$41$ (\( ( 1 - 12 T + 41 T^{2} )^{2} \))(\( ( 1 + 6 T + 41 T^{2} )^{2} \))(\( ( 1 + 2 T + 41 T^{2} )^{2} \))(\( ( 1 + 6 T + 41 T^{2} )^{2} \))(\( ( 1 - 6 T + 41 T^{2} )^{2} \))(\( ( 1 + 12 T + 41 T^{2} )^{2} \))(\( ( 1 - 12 T + 94 T^{2} - 492 T^{3} + 1681 T^{4} )^{2} \))
$43$ (\( 1 - 22 T^{2} + 1849 T^{4} \))(\( 1 - 82 T^{2} + 1849 T^{4} \))(\( 1 - 70 T^{2} + 1849 T^{4} \))(\( 1 - 22 T^{2} + 1849 T^{4} \))(\( 1 - 82 T^{2} + 1849 T^{4} \))(\( 1 - 22 T^{2} + 1849 T^{4} \))(\( 1 - 92 T^{2} + 4278 T^{4} - 170108 T^{6} + 3418801 T^{8} \))
$47$ (\( 1 - 78 T^{2} + 2209 T^{4} \))(\( 1 - 90 T^{2} + 2209 T^{4} \))(\( 1 - 30 T^{2} + 2209 T^{4} \))(\( 1 - 78 T^{2} + 2209 T^{4} \))(\( 1 - 90 T^{2} + 2209 T^{4} \))(\( 1 - 78 T^{2} + 2209 T^{4} \))(\( 1 - 108 T^{2} + 5798 T^{4} - 238572 T^{6} + 4879681 T^{8} \))
$53$ (\( 1 - 70 T^{2} + 2809 T^{4} \))(\( 1 - 70 T^{2} + 2809 T^{4} \))(\( 1 - 70 T^{2} + 2809 T^{4} \))(\( 1 - 102 T^{2} + 2809 T^{4} \))(\( 1 - 70 T^{2} + 2809 T^{4} \))(\( 1 - 70 T^{2} + 2809 T^{4} \))(\( 1 - 92 T^{2} + 4278 T^{4} - 258428 T^{6} + 7890481 T^{8} \))
$59$ (\( ( 1 + 8 T + 59 T^{2} )^{2} \))(\( ( 1 + 4 T + 59 T^{2} )^{2} \))(\( ( 1 + 8 T + 59 T^{2} )^{2} \))(\( ( 1 + 8 T + 59 T^{2} )^{2} \))(\( ( 1 - 4 T + 59 T^{2} )^{2} \))(\( ( 1 - 8 T + 59 T^{2} )^{2} \))(\( ( 1 + 8 T + 128 T^{2} + 472 T^{3} + 3481 T^{4} )^{2} \))
$61$ (\( ( 1 + 6 T + 61 T^{2} )^{2} \))(\( ( 1 + 12 T + 61 T^{2} )^{2} \))(\( ( 1 + 10 T + 61 T^{2} )^{2} \))(\( ( 1 - 10 T + 61 T^{2} )^{2} \))(\( ( 1 + 12 T + 61 T^{2} )^{2} \))(\( ( 1 + 6 T + 61 T^{2} )^{2} \))(\( ( 1 - 12 T + 152 T^{2} - 732 T^{3} + 3721 T^{4} )^{2} \))
$67$ (\( 1 - 70 T^{2} + 4489 T^{4} \))(\( 1 - 34 T^{2} + 4489 T^{4} \))(\( 1 - 70 T^{2} + 4489 T^{4} \))(\( 1 + 10 T^{2} + 4489 T^{4} \))(\( 1 - 34 T^{2} + 4489 T^{4} \))(\( 1 - 70 T^{2} + 4489 T^{4} \))(\( ( 1 - 70 T^{2} + 4489 T^{4} )^{2} \))
$71$ (\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 12 T + 71 T^{2} )^{2} \))(\( ( 1 - 6 T + 71 T^{2} )^{2} \))(\( ( 1 - 14 T + 71 T^{2} )^{2} \))(\( ( 1 - 12 T + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 - 12 T + 154 T^{2} - 852 T^{3} + 5041 T^{4} )^{2} \))
$73$ (\( 1 - 130 T^{2} + 5329 T^{4} \))(\( 1 - 142 T^{2} + 5329 T^{4} \))(\( 1 + 50 T^{2} + 5329 T^{4} \))(\( 1 - 46 T^{2} + 5329 T^{4} \))(\( 1 - 142 T^{2} + 5329 T^{4} \))(\( 1 - 130 T^{2} + 5329 T^{4} \))(\( 1 - 236 T^{2} + 24198 T^{4} - 1257644 T^{6} + 28398241 T^{8} \))
$79$ (\( ( 1 + 8 T + 79 T^{2} )^{2} \))(\( ( 1 - 16 T + 79 T^{2} )^{2} \))(\( ( 1 - 12 T + 79 T^{2} )^{2} \))(\( ( 1 + 4 T + 79 T^{2} )^{2} \))(\( ( 1 - 16 T + 79 T^{2} )^{2} \))(\( ( 1 + 8 T + 79 T^{2} )^{2} \))(\( ( 1 + 4 T + 138 T^{2} + 316 T^{3} + 6241 T^{4} )^{2} \))
$83$ (\( ( 1 - 83 T^{2} )^{2} \))(\( 1 - 22 T^{2} + 6889 T^{4} \))(\( 1 - 102 T^{2} + 6889 T^{4} \))(\( 1 + 90 T^{2} + 6889 T^{4} \))(\( 1 - 22 T^{2} + 6889 T^{4} \))(\( ( 1 - 83 T^{2} )^{2} \))(\( ( 1 - 160 T^{2} + 6889 T^{4} )^{2} \))
$89$ (\( ( 1 - 4 T + 89 T^{2} )^{2} \))(\( ( 1 - 14 T + 89 T^{2} )^{2} \))(\( ( 1 + 10 T + 89 T^{2} )^{2} \))(\( ( 1 - 10 T + 89 T^{2} )^{2} \))(\( ( 1 + 14 T + 89 T^{2} )^{2} \))(\( ( 1 + 4 T + 89 T^{2} )^{2} \))(\( ( 1 + 10 T + 89 T^{2} )^{4} \))
$97$ (\( 1 - 50 T^{2} + 9409 T^{4} \))(\( ( 1 - 8 T + 97 T^{2} )( 1 + 8 T + 97 T^{2} ) \))(\( 1 - 94 T^{2} + 9409 T^{4} \))(\( 1 - 94 T^{2} + 9409 T^{4} \))(\( ( 1 - 8 T + 97 T^{2} )( 1 + 8 T + 97 T^{2} ) \))(\( 1 - 50 T^{2} + 9409 T^{4} \))(\( 1 - 124 T^{2} + 8838 T^{4} - 1166716 T^{6} + 88529281 T^{8} \))
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