Properties

Label 630.2.d
Level 630
Weight 2
Character orbit d
Rep. character \(\chi_{630}(629,\cdot)\)
Character field \(\Q\)
Dimension 16
Newform subspaces 4
Sturm bound 288
Trace bound 23

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Defining parameters

Level: \( N \) = \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 630.d (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 105 \)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(288\)
Trace bound: \(23\)
Distinguishing \(T_p\): \(11\), \(23\)

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

\( 16q + 16q^{4} + O(q^{10}) \) \( 16q + 16q^{4} + 16q^{16} + 16q^{46} + 8q^{49} + 16q^{64} + 40q^{70} + 16q^{79} - 80q^{85} - 40q^{91} + O(q^{100}) \)

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.d.a \(4\) \(5.031\) \(\Q(\sqrt{-2}, \sqrt{-5})\) None \(-4\) \(0\) \(0\) \(0\) \(q-q^{2}+q^{4}+\beta _{3}q^{5}+(-\beta _{1}-\beta _{2})q^{7}+\cdots\)
630.2.d.b \(4\) \(5.031\) \(\Q(\sqrt{-2}, \sqrt{5})\) None \(-4\) \(0\) \(0\) \(0\) \(q-q^{2}+q^{4}+\beta _{3}q^{5}+(-\beta _{1}-\beta _{3})q^{7}+\cdots\)
630.2.d.c \(4\) \(5.031\) \(\Q(\sqrt{-2}, \sqrt{5})\) None \(4\) \(0\) \(0\) \(0\) \(q+q^{2}+q^{4}-\beta _{3}q^{5}+(\beta _{1}-\beta _{3})q^{7}+\cdots\)
630.2.d.d \(4\) \(5.031\) \(\Q(\sqrt{-2}, \sqrt{-5})\) None \(4\) \(0\) \(0\) \(0\) \(q+q^{2}+q^{4}-\beta _{3}q^{5}+(-\beta _{1}-\beta _{2})q^{7}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(630, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(210, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(315, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 + T )^{4} \))(\( ( 1 + T )^{4} \))(\( ( 1 - T )^{4} \))(\( ( 1 - T )^{4} \))
$3$ 1
$5$ (\( ( 1 + 5 T^{2} )^{2} \))(\( ( 1 - 5 T^{2} )^{2} \))(\( ( 1 - 5 T^{2} )^{2} \))(\( ( 1 + 5 T^{2} )^{2} \))
$7$ (\( 1 + 4 T^{2} + 49 T^{4} \))(\( 1 - 6 T^{2} + 49 T^{4} \))(\( 1 - 6 T^{2} + 49 T^{4} \))(\( 1 + 4 T^{2} + 49 T^{4} \))
$11$ (\( ( 1 - 20 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 + 10 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 + 10 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 - 20 T^{2} + 121 T^{4} )^{2} \))
$13$ (\( ( 1 + 16 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 + 6 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 + 6 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 + 16 T^{2} + 169 T^{4} )^{2} \))
$17$ (\( ( 1 - 14 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 - 24 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 - 24 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 - 14 T^{2} + 289 T^{4} )^{2} \))
$19$ (\( ( 1 - 19 T^{2} )^{4} \))(\( ( 1 - 28 T^{2} + 361 T^{4} )^{2} \))(\( ( 1 - 28 T^{2} + 361 T^{4} )^{2} \))(\( ( 1 - 19 T^{2} )^{4} \))
$23$ (\( ( 1 + 6 T + 23 T^{2} )^{4} \))(\( ( 1 - 4 T + 23 T^{2} )^{4} \))(\( ( 1 + 4 T + 23 T^{2} )^{4} \))(\( ( 1 - 6 T + 23 T^{2} )^{4} \))
$29$ (\( ( 1 - 50 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 50 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 50 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 50 T^{2} + 841 T^{4} )^{2} \))
$31$ (\( ( 1 - 31 T^{2} )^{4} \))(\( ( 1 - 22 T^{2} + 961 T^{4} )^{2} \))(\( ( 1 - 22 T^{2} + 961 T^{4} )^{2} \))(\( ( 1 - 31 T^{2} )^{4} \))
$37$ (\( ( 1 - 56 T^{2} + 1369 T^{4} )^{2} \))(\( ( 1 + 24 T^{2} + 1369 T^{4} )^{2} \))(\( ( 1 + 24 T^{2} + 1369 T^{4} )^{2} \))(\( ( 1 - 56 T^{2} + 1369 T^{4} )^{2} \))
$41$ (\( ( 1 - 8 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 + 62 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 + 62 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 - 8 T^{2} + 1681 T^{4} )^{2} \))
$43$ (\( ( 1 - 10 T + 43 T^{2} )^{2}( 1 + 10 T + 43 T^{2} )^{2} \))(\( ( 1 - 84 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 84 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 10 T + 43 T^{2} )^{2}( 1 + 10 T + 43 T^{2} )^{2} \))
$47$ (\( ( 1 - 74 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 - 4 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 - 4 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 - 74 T^{2} + 2209 T^{4} )^{2} \))
$53$ (\( ( 1 + 6 T + 53 T^{2} )^{4} \))(\( ( 1 - 4 T + 53 T^{2} )^{4} \))(\( ( 1 + 4 T + 53 T^{2} )^{4} \))(\( ( 1 - 6 T + 53 T^{2} )^{4} \))
$59$ (\( ( 1 + 28 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 + 98 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 + 98 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 + 28 T^{2} + 3481 T^{4} )^{2} \))
$61$ (\( ( 1 - 8 T + 61 T^{2} )^{2}( 1 + 8 T + 61 T^{2} )^{2} \))(\( ( 1 - 32 T^{2} + 3721 T^{4} )^{2} \))(\( ( 1 - 32 T^{2} + 3721 T^{4} )^{2} \))(\( ( 1 - 8 T + 61 T^{2} )^{2}( 1 + 8 T + 61 T^{2} )^{2} \))
$67$ (\( ( 1 - 67 T^{2} )^{4} \))(\( ( 1 - 84 T^{2} + 4489 T^{4} )^{2} \))(\( ( 1 - 84 T^{2} + 4489 T^{4} )^{2} \))(\( ( 1 - 67 T^{2} )^{4} \))
$71$ (\( ( 1 - 110 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 - 140 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 - 140 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 - 110 T^{2} + 5041 T^{4} )^{2} \))
$73$ (\( ( 1 + 106 T^{2} + 5329 T^{4} )^{2} \))(\( ( 1 - 34 T^{2} + 5329 T^{4} )^{2} \))(\( ( 1 - 34 T^{2} + 5329 T^{4} )^{2} \))(\( ( 1 + 106 T^{2} + 5329 T^{4} )^{2} \))
$79$ (\( ( 1 + 4 T + 79 T^{2} )^{4} \))(\( ( 1 - 6 T + 79 T^{2} )^{4} \))(\( ( 1 - 6 T + 79 T^{2} )^{4} \))(\( ( 1 + 4 T + 79 T^{2} )^{4} \))
$83$ (\( ( 1 - 86 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 - 6 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 - 6 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 - 86 T^{2} + 6889 T^{4} )^{2} \))
$89$ (\( ( 1 + 88 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 + 158 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 + 158 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 + 88 T^{2} + 7921 T^{4} )^{2} \))
$97$ (\( ( 1 + 34 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 + 97 T^{2} )^{4} \))(\( ( 1 + 97 T^{2} )^{4} \))(\( ( 1 + 34 T^{2} + 9409 T^{4} )^{2} \))
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