Properties

Label 630.2.a
Level $630$
Weight $2$
Character orbit 630.a
Rep. character $\chi_{630}(1,\cdot)$
Character field $\Q$
Dimension $10$
Newform subspaces $10$
Sturm bound $288$
Trace bound $13$

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Defining parameters

Level: \( N \) \(=\) \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 630.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 10 \)
Sturm bound: \(288\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(11\), \(13\), \(17\), \(19\), \(29\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(630))\).

Total New Old
Modular forms 160 10 150
Cusp forms 129 10 119
Eisenstein series 31 0 31

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)\(3\)\(5\)\(7\)FrickeDim.
\(+\)\(+\)\(+\)\(-\)\(-\)\(1\)
\(+\)\(+\)\(-\)\(+\)\(-\)\(1\)
\(+\)\(-\)\(+\)\(+\)\(-\)\(1\)
\(+\)\(-\)\(+\)\(-\)\(+\)\(1\)
\(+\)\(-\)\(-\)\(+\)\(+\)\(1\)
\(+\)\(-\)\(-\)\(-\)\(-\)\(1\)
\(-\)\(+\)\(+\)\(+\)\(-\)\(1\)
\(-\)\(+\)\(-\)\(-\)\(-\)\(1\)
\(-\)\(-\)\(+\)\(-\)\(-\)\(1\)
\(-\)\(-\)\(-\)\(+\)\(-\)\(1\)
Plus space\(+\)\(2\)
Minus space\(-\)\(8\)

Trace form

\( 10 q - 2 q^{2} + 10 q^{4} - 2 q^{8} + O(q^{10}) \) \( 10 q - 2 q^{2} + 10 q^{4} - 2 q^{8} + 8 q^{13} + 10 q^{16} + 12 q^{17} + 20 q^{19} + 16 q^{22} + 24 q^{23} + 10 q^{25} + 8 q^{26} - 20 q^{29} - 8 q^{31} - 2 q^{32} + 4 q^{34} - 2 q^{35} - 4 q^{37} + 12 q^{38} + 4 q^{41} - 8 q^{46} + 8 q^{47} + 10 q^{49} - 2 q^{50} + 8 q^{52} - 4 q^{53} - 8 q^{55} - 12 q^{58} + 12 q^{59} - 16 q^{61} - 8 q^{62} + 10 q^{64} - 4 q^{65} + 8 q^{67} + 12 q^{68} + 2 q^{70} + 16 q^{71} - 20 q^{73} - 4 q^{74} + 20 q^{76} - 8 q^{77} - 8 q^{79} - 20 q^{82} - 28 q^{83} + 16 q^{85} + 8 q^{86} + 16 q^{88} - 36 q^{89} + 4 q^{91} + 24 q^{92} - 24 q^{94} - 12 q^{95} + 4 q^{97} - 2 q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(630))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 5 7
630.2.a.a $1$ $5.031$ \(\Q\) None \(-1\) \(0\) \(-1\) \(-1\) $+$ $-$ $+$ $+$ \(q-q^{2}+q^{4}-q^{5}-q^{7}-q^{8}+q^{10}+\cdots\)
630.2.a.b $1$ $5.031$ \(\Q\) None \(-1\) \(0\) \(-1\) \(1\) $+$ $-$ $+$ $-$ \(q-q^{2}+q^{4}-q^{5}+q^{7}-q^{8}+q^{10}+\cdots\)
630.2.a.c $1$ $5.031$ \(\Q\) None \(-1\) \(0\) \(-1\) \(1\) $+$ $+$ $+$ $-$ \(q-q^{2}+q^{4}-q^{5}+q^{7}-q^{8}+q^{10}+\cdots\)
630.2.a.d $1$ $5.031$ \(\Q\) None \(-1\) \(0\) \(1\) \(-1\) $+$ $-$ $-$ $+$ \(q-q^{2}+q^{4}+q^{5}-q^{7}-q^{8}-q^{10}+\cdots\)
630.2.a.e $1$ $5.031$ \(\Q\) None \(-1\) \(0\) \(1\) \(-1\) $+$ $+$ $-$ $+$ \(q-q^{2}+q^{4}+q^{5}-q^{7}-q^{8}-q^{10}+\cdots\)
630.2.a.f $1$ $5.031$ \(\Q\) None \(-1\) \(0\) \(1\) \(1\) $+$ $-$ $-$ $-$ \(q-q^{2}+q^{4}+q^{5}+q^{7}-q^{8}-q^{10}+\cdots\)
630.2.a.g $1$ $5.031$ \(\Q\) None \(1\) \(0\) \(-1\) \(-1\) $-$ $+$ $+$ $+$ \(q+q^{2}+q^{4}-q^{5}-q^{7}+q^{8}-q^{10}+\cdots\)
630.2.a.h $1$ $5.031$ \(\Q\) None \(1\) \(0\) \(-1\) \(1\) $-$ $-$ $+$ $-$ \(q+q^{2}+q^{4}-q^{5}+q^{7}+q^{8}-q^{10}+\cdots\)
630.2.a.i $1$ $5.031$ \(\Q\) None \(1\) \(0\) \(1\) \(-1\) $-$ $-$ $-$ $+$ \(q+q^{2}+q^{4}+q^{5}-q^{7}+q^{8}+q^{10}+\cdots\)
630.2.a.j $1$ $5.031$ \(\Q\) None \(1\) \(0\) \(1\) \(1\) $-$ $+$ $-$ $-$ \(q+q^{2}+q^{4}+q^{5}+q^{7}+q^{8}+q^{10}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(630))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(630)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(105))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(126))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(210))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(315))\)\(^{\oplus 2}\)