Properties

Label 630.2.z
Level 630
Weight 2
Character orbit z
Rep. character \(\chi_{630}(169,\cdot)\)
Character field \(\Q(\zeta_{6})\)
Dimension 72
Newform subspaces 3
Sturm bound 288
Trace bound 1

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Defining parameters

Level: \( N \) = \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 630.z (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 45 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 3 \)
Sturm bound: \(288\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(11\)

Dimensions

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

Total New Old
Modular forms 304 72 232
Cusp forms 272 72 200
Eisenstein series 32 0 32

Trace form

\( 72q + 36q^{4} + 4q^{5} + 16q^{9} + O(q^{10}) \) \( 72q + 36q^{4} + 4q^{5} + 16q^{9} - 12q^{11} + 8q^{14} + 20q^{15} - 36q^{16} - 4q^{20} + 4q^{21} - 12q^{25} + 16q^{29} - 16q^{30} + 24q^{31} + 20q^{36} - 4q^{39} - 16q^{41} - 24q^{44} - 24q^{45} + 36q^{49} + 4q^{50} - 112q^{51} - 72q^{54} + 24q^{55} - 8q^{56} - 16q^{59} + 4q^{60} - 72q^{64} + 48q^{65} + 40q^{66} - 16q^{69} + 88q^{71} + 56q^{74} + 12q^{79} - 8q^{80} + 32q^{81} + 8q^{84} + 24q^{85} - 64q^{86} + 160q^{89} + 60q^{90} + 24q^{91} - 12q^{95} - 108q^{99} + O(q^{100}) \)

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.z.a \(4\) \(5.031\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(4\) \(0\) \(q+\zeta_{12}q^{2}+(-2\zeta_{12}+\zeta_{12}^{3})q^{3}+\zeta_{12}^{2}q^{4}+\cdots\)
630.2.z.b \(24\) \(5.031\) None \(0\) \(0\) \(-2\) \(0\)
630.2.z.c \(44\) \(5.031\) None \(0\) \(0\) \(2\) \(0\)

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

\( S_{2}^{\mathrm{old}}(630, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(90, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(315, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 - T^{2} + T^{4} \))
$3$ (\( ( 1 - 3 T^{2} )^{2} \))
$5$ (\( 1 - 4 T + 11 T^{2} - 20 T^{3} + 25 T^{4} \))
$7$ (\( 1 - T^{2} + T^{4} \))
$11$ (\( ( 1 + 5 T + 14 T^{2} + 55 T^{3} + 121 T^{4} )^{2} \))
$13$ (\( 1 + 25 T^{2} + 456 T^{4} + 4225 T^{6} + 28561 T^{8} \))
$17$ (\( ( 1 - 33 T^{2} + 289 T^{4} )^{2} \))
$19$ (\( ( 1 - 2 T + 19 T^{2} )^{4} \))
$23$ (\( ( 1 + 23 T^{2} + 529 T^{4} )^{2} \))
$29$ (\( ( 1 + 6 T + 7 T^{2} + 174 T^{3} + 841 T^{4} )^{2} \))
$31$ (\( ( 1 - 31 T^{2} + 961 T^{4} )^{2} \))
$37$ (\( ( 1 - 10 T^{2} + 1369 T^{4} )^{2} \))
$41$ (\( ( 1 - 6 T - 5 T^{2} - 246 T^{3} + 1681 T^{4} )^{2} \))
$43$ (\( 1 + 70 T^{2} + 3051 T^{4} + 129430 T^{6} + 3418801 T^{8} \))
$47$ (\( 1 + 93 T^{2} + 6440 T^{4} + 205437 T^{6} + 4879681 T^{8} \))
$53$ (\( ( 1 - 102 T^{2} + 2809 T^{4} )^{2} \))
$59$ (\( ( 1 + 4 T - 43 T^{2} + 236 T^{3} + 3481 T^{4} )^{2} \))
$61$ (\( ( 1 - 13 T + 61 T^{2} )^{2}( 1 - T + 61 T^{2} )^{2} \))
$67$ (\( 1 - 10 T^{2} - 4389 T^{4} - 44890 T^{6} + 20151121 T^{8} \))
$71$ (\( ( 1 - 11 T + 71 T^{2} )^{4} \))
$73$ (\( ( 1 - 25 T^{2} + 5329 T^{4} )^{2} \))
$79$ (\( ( 1 - 7 T - 30 T^{2} - 553 T^{3} + 6241 T^{4} )^{2} \))
$83$ (\( 1 + 45 T^{2} - 4864 T^{4} + 310005 T^{6} + 47458321 T^{8} \))
$89$ (\( ( 1 - 10 T + 89 T^{2} )^{4} \))
$97$ (\( 1 + 25 T^{2} - 8784 T^{4} + 235225 T^{6} + 88529281 T^{8} \))
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