Properties

Label 3630.2.a
Level $3630$
Weight $2$
Character orbit 3630.a
Rep. character $\chi_{3630}(1,\cdot)$
Character field $\Q$
Dimension $74$
Newform subspaces $46$
Sturm bound $1584$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 3630 = 2 \cdot 3 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3630.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 46 \)
Sturm bound: \(1584\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(7\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3630))\).

Total New Old
Modular forms 840 74 766
Cusp forms 745 74 671
Eisenstein series 95 0 95

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

\(2\)\(3\)\(5\)\(11\)FrickeDim
\(+\)\(+\)\(+\)\(+\)$+$\(4\)
\(+\)\(+\)\(+\)\(-\)$-$\(5\)
\(+\)\(+\)\(-\)\(+\)$-$\(4\)
\(+\)\(+\)\(-\)\(-\)$+$\(5\)
\(+\)\(-\)\(+\)\(+\)$-$\(7\)
\(+\)\(-\)\(+\)\(-\)$+$\(2\)
\(+\)\(-\)\(-\)\(+\)$+$\(3\)
\(+\)\(-\)\(-\)\(-\)$-$\(7\)
\(-\)\(+\)\(+\)\(+\)$-$\(4\)
\(-\)\(+\)\(+\)\(-\)$+$\(5\)
\(-\)\(+\)\(-\)\(+\)$+$\(4\)
\(-\)\(+\)\(-\)\(-\)$-$\(5\)
\(-\)\(-\)\(+\)\(+\)$+$\(3\)
\(-\)\(-\)\(+\)\(-\)$-$\(7\)
\(-\)\(-\)\(-\)\(+\)$-$\(7\)
\(-\)\(-\)\(-\)\(-\)$+$\(2\)
Plus space\(+\)\(28\)
Minus space\(-\)\(46\)

Trace form

\( 74 q + 2 q^{3} + 74 q^{4} + 4 q^{7} + 74 q^{9} + O(q^{10}) \) \( 74 q + 2 q^{3} + 74 q^{4} + 4 q^{7} + 74 q^{9} - 2 q^{10} + 2 q^{12} - 8 q^{13} - 12 q^{14} + 74 q^{16} - 8 q^{17} + 8 q^{19} + 4 q^{21} - 8 q^{23} + 74 q^{25} - 12 q^{26} + 2 q^{27} + 4 q^{28} + 8 q^{29} - 2 q^{30} - 16 q^{31} - 12 q^{34} + 4 q^{35} + 74 q^{36} + 8 q^{37} - 8 q^{38} + 8 q^{39} - 2 q^{40} + 12 q^{41} - 4 q^{42} + 16 q^{43} + 8 q^{46} - 16 q^{47} + 2 q^{48} + 50 q^{49} - 8 q^{51} - 8 q^{52} - 16 q^{53} - 12 q^{56} + 8 q^{57} - 12 q^{58} + 4 q^{59} - 12 q^{61} + 8 q^{62} + 4 q^{63} + 74 q^{64} + 4 q^{65} + 16 q^{67} - 8 q^{68} - 8 q^{69} + 4 q^{70} + 8 q^{71} - 20 q^{73} - 4 q^{74} + 2 q^{75} + 8 q^{76} + 12 q^{78} + 74 q^{81} - 16 q^{82} - 16 q^{83} + 4 q^{84} + 4 q^{85} + 16 q^{86} + 8 q^{87} - 44 q^{89} - 2 q^{90} + 136 q^{91} - 8 q^{92} + 136 q^{93} + 24 q^{95} + 116 q^{97} + 16 q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3630))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 5 11
3630.2.a.a 3630.a 1.a $1$ $28.986$ \(\Q\) None \(-1\) \(-1\) \(-1\) \(-4\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}-4q^{7}+\cdots\)
3630.2.a.b 3630.a 1.a $1$ $28.986$ \(\Q\) None \(-1\) \(-1\) \(-1\) \(-3\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}-3q^{7}+\cdots\)
3630.2.a.c 3630.a 1.a $1$ $28.986$ \(\Q\) None \(-1\) \(-1\) \(-1\) \(-1\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}-q^{7}+\cdots\)
3630.2.a.d 3630.a 1.a $1$ $28.986$ \(\Q\) None \(-1\) \(-1\) \(1\) \(-3\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}-3q^{7}+\cdots\)
3630.2.a.e 3630.a 1.a $1$ $28.986$ \(\Q\) None \(-1\) \(-1\) \(1\) \(-1\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}-q^{7}+\cdots\)
3630.2.a.f 3630.a 1.a $1$ $28.986$ \(\Q\) None \(-1\) \(-1\) \(1\) \(0\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}-q^{8}+\cdots\)
3630.2.a.g 3630.a 1.a $1$ $28.986$ \(\Q\) None \(-1\) \(1\) \(-1\) \(-1\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{5}-q^{6}-q^{7}+\cdots\)
3630.2.a.h 3630.a 1.a $1$ $28.986$ \(\Q\) None \(-1\) \(1\) \(-1\) \(1\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{5}-q^{6}+q^{7}+\cdots\)
3630.2.a.i 3630.a 1.a $1$ $28.986$ \(\Q\) None \(-1\) \(1\) \(-1\) \(2\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{5}-q^{6}+2q^{7}+\cdots\)
3630.2.a.j 3630.a 1.a $1$ $28.986$ \(\Q\) None \(-1\) \(1\) \(1\) \(-2\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}-2q^{7}+\cdots\)
3630.2.a.k 3630.a 1.a $1$ $28.986$ \(\Q\) None \(-1\) \(1\) \(1\) \(0\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}-q^{8}+\cdots\)
3630.2.a.l 3630.a 1.a $1$ $28.986$ \(\Q\) None \(-1\) \(1\) \(1\) \(3\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}+3q^{7}+\cdots\)
3630.2.a.m 3630.a 1.a $1$ $28.986$ \(\Q\) None \(-1\) \(1\) \(1\) \(5\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}+5q^{7}+\cdots\)
3630.2.a.n 3630.a 1.a $1$ $28.986$ \(\Q\) None \(1\) \(-1\) \(-1\) \(0\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{5}-q^{6}+q^{8}+\cdots\)
3630.2.a.o 3630.a 1.a $1$ $28.986$ \(\Q\) None \(1\) \(-1\) \(-1\) \(1\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{5}-q^{6}+q^{7}+\cdots\)
3630.2.a.p 3630.a 1.a $1$ $28.986$ \(\Q\) None \(1\) \(-1\) \(-1\) \(3\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{5}-q^{6}+3q^{7}+\cdots\)
3630.2.a.q 3630.a 1.a $1$ $28.986$ \(\Q\) None \(1\) \(-1\) \(1\) \(1\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}+q^{7}+\cdots\)
3630.2.a.r 3630.a 1.a $1$ $28.986$ \(\Q\) None \(1\) \(-1\) \(1\) \(3\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}+3q^{7}+\cdots\)
3630.2.a.s 3630.a 1.a $1$ $28.986$ \(\Q\) None \(1\) \(-1\) \(1\) \(4\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}+4q^{7}+\cdots\)
3630.2.a.t 3630.a 1.a $1$ $28.986$ \(\Q\) None \(1\) \(1\) \(-1\) \(-2\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}-2q^{7}+\cdots\)
3630.2.a.u 3630.a 1.a $1$ $28.986$ \(\Q\) None \(1\) \(1\) \(-1\) \(-1\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}-q^{7}+\cdots\)
3630.2.a.v 3630.a 1.a $1$ $28.986$ \(\Q\) None \(1\) \(1\) \(-1\) \(1\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}+q^{7}+\cdots\)
3630.2.a.w 3630.a 1.a $1$ $28.986$ \(\Q\) None \(1\) \(1\) \(-1\) \(4\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}+4q^{7}+\cdots\)
3630.2.a.x 3630.a 1.a $1$ $28.986$ \(\Q\) None \(1\) \(1\) \(1\) \(-5\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{5}+q^{6}-5q^{7}+\cdots\)
3630.2.a.y 3630.a 1.a $1$ $28.986$ \(\Q\) None \(1\) \(1\) \(1\) \(-3\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{5}+q^{6}-3q^{7}+\cdots\)
3630.2.a.z 3630.a 1.a $1$ $28.986$ \(\Q\) None \(1\) \(1\) \(1\) \(2\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{5}+q^{6}+2q^{7}+\cdots\)
3630.2.a.ba 3630.a 1.a $2$ $28.986$ \(\Q(\sqrt{3}) \) None \(-2\) \(-2\) \(-2\) \(0\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}+\beta q^{7}+\cdots\)
3630.2.a.bb 3630.a 1.a $2$ $28.986$ \(\Q(\sqrt{5}) \) None \(-2\) \(-2\) \(-2\) \(2\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}+2\beta q^{7}+\cdots\)
3630.2.a.bc 3630.a 1.a $2$ $28.986$ \(\Q(\sqrt{5}) \) None \(-2\) \(-2\) \(-2\) \(5\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}+(3-\beta )q^{7}+\cdots\)
3630.2.a.bd 3630.a 1.a $2$ $28.986$ \(\Q(\sqrt{3}) \) None \(-2\) \(-2\) \(2\) \(0\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}+\beta q^{7}+\cdots\)
3630.2.a.be 3630.a 1.a $2$ $28.986$ \(\Q(\sqrt{5}) \) None \(-2\) \(-2\) \(2\) \(2\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}+2\beta q^{7}+\cdots\)
3630.2.a.bf 3630.a 1.a $2$ $28.986$ \(\Q(\sqrt{5}) \) None \(-2\) \(-2\) \(2\) \(5\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}+(3-\beta )q^{7}+\cdots\)
3630.2.a.bg 3630.a 1.a $2$ $28.986$ \(\Q(\sqrt{3}) \) None \(-2\) \(2\) \(-2\) \(4\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{5}-q^{6}+(2+\beta )q^{7}+\cdots\)
3630.2.a.bh 3630.a 1.a $2$ $28.986$ \(\Q(\sqrt{3}) \) None \(-2\) \(2\) \(2\) \(-4\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}+(-2+\cdots)q^{7}+\cdots\)
3630.2.a.bi 3630.a 1.a $2$ $28.986$ \(\Q(\sqrt{5}) \) None \(2\) \(-2\) \(-2\) \(-5\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{5}-q^{6}+(-2+\cdots)q^{7}+\cdots\)
3630.2.a.bj 3630.a 1.a $2$ $28.986$ \(\Q(\sqrt{5}) \) None \(2\) \(-2\) \(-2\) \(-2\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{5}-q^{6}-2\beta q^{7}+\cdots\)
3630.2.a.bk 3630.a 1.a $2$ $28.986$ \(\Q(\sqrt{3}) \) None \(2\) \(-2\) \(-2\) \(0\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{5}-q^{6}+\beta q^{7}+\cdots\)
3630.2.a.bl 3630.a 1.a $2$ $28.986$ \(\Q(\sqrt{5}) \) None \(2\) \(-2\) \(2\) \(-5\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}+(-2+\cdots)q^{7}+\cdots\)
3630.2.a.bm 3630.a 1.a $2$ $28.986$ \(\Q(\sqrt{5}) \) None \(2\) \(-2\) \(2\) \(-2\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}-2\beta q^{7}+\cdots\)
3630.2.a.bn 3630.a 1.a $2$ $28.986$ \(\Q(\sqrt{3}) \) None \(2\) \(-2\) \(2\) \(0\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}+\beta q^{7}+\cdots\)
3630.2.a.bo 3630.a 1.a $2$ $28.986$ \(\Q(\sqrt{3}) \) None \(2\) \(2\) \(-2\) \(-4\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}+(-2+\cdots)q^{7}+\cdots\)
3630.2.a.bp 3630.a 1.a $2$ $28.986$ \(\Q(\sqrt{3}) \) None \(2\) \(2\) \(2\) \(4\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{5}+q^{6}+(2+\beta )q^{7}+\cdots\)
3630.2.a.bq 3630.a 1.a $4$ $28.986$ 4.4.52625.1 None \(-4\) \(4\) \(-4\) \(-1\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{5}-q^{6}+\beta _{3}q^{7}+\cdots\)
3630.2.a.br 3630.a 1.a $4$ $28.986$ 4.4.43025.1 None \(-4\) \(4\) \(4\) \(-1\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}-\beta _{1}q^{7}+\cdots\)
3630.2.a.bs 3630.a 1.a $4$ $28.986$ 4.4.52625.1 None \(4\) \(4\) \(-4\) \(1\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}+(-\beta _{2}+\cdots)q^{7}+\cdots\)
3630.2.a.bt 3630.a 1.a $4$ $28.986$ 4.4.43025.1 None \(4\) \(4\) \(4\) \(1\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{5}+q^{6}+\beta _{1}q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3630))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3630)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(165))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(55))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(66))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(110))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(121))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(242))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(330))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(363))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(605))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(726))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1210))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1815))\)\(^{\oplus 2}\)