Properties

Label 3330.2.a
Level $3330$
Weight $2$
Character orbit 3330.a
Rep. character $\chi_{3330}(1,\cdot)$
Character field $\Q$
Dimension $60$
Newform subspaces $38$
Sturm bound $1368$
Trace bound $13$

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Defining parameters

Level: \( N \) \(=\) \( 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3330.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 38 \)
Sturm bound: \(1368\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(7\), \(11\), \(13\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3330))\).

Total New Old
Modular forms 700 60 640
Cusp forms 669 60 609
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.

\(2\)\(3\)\(5\)\(37\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(+\)\(1\)
\(+\)\(+\)\(+\)\(-\)\(-\)\(5\)
\(+\)\(+\)\(-\)\(+\)\(-\)\(4\)
\(+\)\(+\)\(-\)\(-\)\(+\)\(2\)
\(+\)\(-\)\(+\)\(+\)\(-\)\(5\)
\(+\)\(-\)\(+\)\(-\)\(+\)\(5\)
\(+\)\(-\)\(-\)\(+\)\(+\)\(5\)
\(+\)\(-\)\(-\)\(-\)\(-\)\(4\)
\(-\)\(+\)\(+\)\(+\)\(-\)\(4\)
\(-\)\(+\)\(+\)\(-\)\(+\)\(2\)
\(-\)\(+\)\(-\)\(+\)\(+\)\(1\)
\(-\)\(+\)\(-\)\(-\)\(-\)\(5\)
\(-\)\(-\)\(+\)\(+\)\(+\)\(3\)
\(-\)\(-\)\(+\)\(-\)\(-\)\(6\)
\(-\)\(-\)\(-\)\(+\)\(-\)\(6\)
\(-\)\(-\)\(-\)\(-\)\(+\)\(2\)
Plus space\(+\)\(21\)
Minus space\(-\)\(39\)

Trace form

\( 60 q - 2 q^{2} + 60 q^{4} - 2 q^{5} - 2 q^{8} + O(q^{10}) \) \( 60 q - 2 q^{2} + 60 q^{4} - 2 q^{5} - 2 q^{8} - 4 q^{11} + 12 q^{13} + 4 q^{14} + 60 q^{16} + 4 q^{17} + 12 q^{19} - 2 q^{20} + 24 q^{22} + 16 q^{23} + 60 q^{25} - 8 q^{26} - 4 q^{29} + 16 q^{31} - 2 q^{32} + 12 q^{34} - 4 q^{35} + 2 q^{37} + 16 q^{38} + 12 q^{41} + 8 q^{43} - 4 q^{44} + 40 q^{47} + 96 q^{49} - 2 q^{50} + 12 q^{52} - 4 q^{53} + 4 q^{56} - 24 q^{58} - 12 q^{59} + 20 q^{61} + 12 q^{62} + 60 q^{64} + 4 q^{67} + 4 q^{68} + 12 q^{70} + 8 q^{71} + 8 q^{73} + 12 q^{76} + 24 q^{79} - 2 q^{80} + 4 q^{82} + 52 q^{83} + 24 q^{85} + 20 q^{86} + 24 q^{88} + 64 q^{91} + 16 q^{92} + 4 q^{94} + 16 q^{95} + 12 q^{97} - 2 q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3330))\) 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 37
3330.2.a.a 3330.a 1.a $1$ $26.590$ \(\Q\) None \(-1\) \(0\) \(-1\) \(-3\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-q^{5}-3q^{7}-q^{8}+q^{10}+\cdots\)
3330.2.a.b 3330.a 1.a $1$ $26.590$ \(\Q\) None \(-1\) \(0\) \(-1\) \(-3\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-q^{5}-3q^{7}-q^{8}+q^{10}+\cdots\)
3330.2.a.c 3330.a 1.a $1$ $26.590$ \(\Q\) None \(-1\) \(0\) \(-1\) \(0\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-q^{5}-q^{8}+q^{10}-4q^{11}+\cdots\)
3330.2.a.d 3330.a 1.a $1$ $26.590$ \(\Q\) None \(-1\) \(0\) \(-1\) \(2\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-q^{5}+2q^{7}-q^{8}+q^{10}+\cdots\)
3330.2.a.e 3330.a 1.a $1$ $26.590$ \(\Q\) None \(-1\) \(0\) \(-1\) \(3\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-q^{5}+3q^{7}-q^{8}+q^{10}+\cdots\)
3330.2.a.f 3330.a 1.a $1$ $26.590$ \(\Q\) None \(-1\) \(0\) \(1\) \(-5\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+q^{5}-5q^{7}-q^{8}-q^{10}+\cdots\)
3330.2.a.g 3330.a 1.a $1$ $26.590$ \(\Q\) None \(-1\) \(0\) \(1\) \(-3\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+q^{5}-3q^{7}-q^{8}-q^{10}+\cdots\)
3330.2.a.h 3330.a 1.a $1$ $26.590$ \(\Q\) None \(-1\) \(0\) \(1\) \(-1\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+q^{5}-q^{7}-q^{8}-q^{10}+\cdots\)
3330.2.a.i 3330.a 1.a $1$ $26.590$ \(\Q\) None \(-1\) \(0\) \(1\) \(0\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+q^{5}-q^{8}-q^{10}-2q^{11}+\cdots\)
3330.2.a.j 3330.a 1.a $1$ $26.590$ \(\Q\) None \(-1\) \(0\) \(1\) \(0\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+q^{5}-q^{8}-q^{10}+2q^{11}+\cdots\)
3330.2.a.k 3330.a 1.a $1$ $26.590$ \(\Q\) None \(-1\) \(0\) \(1\) \(4\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+q^{5}+4q^{7}-q^{8}-q^{10}+\cdots\)
3330.2.a.l 3330.a 1.a $1$ $26.590$ \(\Q\) None \(-1\) \(0\) \(1\) \(4\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+q^{5}+4q^{7}-q^{8}-q^{10}+\cdots\)
3330.2.a.m 3330.a 1.a $1$ $26.590$ \(\Q\) None \(1\) \(0\) \(-1\) \(-4\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-q^{5}-4q^{7}+q^{8}-q^{10}+\cdots\)
3330.2.a.n 3330.a 1.a $1$ $26.590$ \(\Q\) None \(1\) \(0\) \(-1\) \(-3\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-q^{5}-3q^{7}+q^{8}-q^{10}+\cdots\)
3330.2.a.o 3330.a 1.a $1$ $26.590$ \(\Q\) None \(1\) \(0\) \(-1\) \(0\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-q^{5}+q^{8}-q^{10}+2q^{11}+\cdots\)
3330.2.a.p 3330.a 1.a $1$ $26.590$ \(\Q\) None \(1\) \(0\) \(-1\) \(1\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-q^{5}+q^{7}+q^{8}-q^{10}+\cdots\)
3330.2.a.q 3330.a 1.a $1$ $26.590$ \(\Q\) None \(1\) \(0\) \(-1\) \(3\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-q^{5}+3q^{7}+q^{8}-q^{10}+\cdots\)
3330.2.a.r 3330.a 1.a $1$ $26.590$ \(\Q\) None \(1\) \(0\) \(-1\) \(4\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-q^{5}+4q^{7}+q^{8}-q^{10}+\cdots\)
3330.2.a.s 3330.a 1.a $1$ $26.590$ \(\Q\) None \(1\) \(0\) \(1\) \(-4\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+q^{5}-4q^{7}+q^{8}+q^{10}+\cdots\)
3330.2.a.t 3330.a 1.a $1$ $26.590$ \(\Q\) None \(1\) \(0\) \(1\) \(-3\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+q^{5}-3q^{7}+q^{8}+q^{10}+\cdots\)
3330.2.a.u 3330.a 1.a $1$ $26.590$ \(\Q\) None \(1\) \(0\) \(1\) \(-1\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+q^{5}-q^{7}+q^{8}+q^{10}+\cdots\)
3330.2.a.v 3330.a 1.a $1$ $26.590$ \(\Q\) None \(1\) \(0\) \(1\) \(-1\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+q^{5}-q^{7}+q^{8}+q^{10}+\cdots\)
3330.2.a.w 3330.a 1.a $1$ $26.590$ \(\Q\) None \(1\) \(0\) \(1\) \(0\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+q^{5}+q^{8}+q^{10}+4q^{11}+\cdots\)
3330.2.a.x 3330.a 1.a $1$ $26.590$ \(\Q\) None \(1\) \(0\) \(1\) \(1\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+q^{5}+q^{7}+q^{8}+q^{10}+\cdots\)
3330.2.a.y 3330.a 1.a $1$ $26.590$ \(\Q\) None \(1\) \(0\) \(1\) \(1\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+q^{5}+q^{7}+q^{8}+q^{10}+\cdots\)
3330.2.a.z 3330.a 1.a $1$ $26.590$ \(\Q\) None \(1\) \(0\) \(1\) \(3\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+q^{5}+3q^{7}+q^{8}+q^{10}+\cdots\)
3330.2.a.ba 3330.a 1.a $1$ $26.590$ \(\Q\) None \(1\) \(0\) \(1\) \(4\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+q^{5}+4q^{7}+q^{8}+q^{10}+\cdots\)
3330.2.a.bb 3330.a 1.a $2$ $26.590$ \(\Q(\sqrt{33}) \) None \(-2\) \(0\) \(-2\) \(-3\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-q^{5}+(-1-\beta )q^{7}-q^{8}+\cdots\)
3330.2.a.bc 3330.a 1.a $2$ $26.590$ \(\Q(\sqrt{17}) \) None \(-2\) \(0\) \(2\) \(-2\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+q^{5}-q^{7}-q^{8}-q^{10}+\cdots\)
3330.2.a.bd 3330.a 1.a $2$ $26.590$ \(\Q(\sqrt{3}) \) None \(2\) \(0\) \(-2\) \(-6\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-q^{5}+(-3+\beta )q^{7}+q^{8}+\cdots\)
3330.2.a.be 3330.a 1.a $2$ $26.590$ \(\Q(\sqrt{33}) \) None \(2\) \(0\) \(-2\) \(-1\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-q^{5}-\beta q^{7}+q^{8}-q^{10}+\cdots\)
3330.2.a.bf 3330.a 1.a $2$ $26.590$ \(\Q(\sqrt{113}) \) None \(2\) \(0\) \(-2\) \(6\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-q^{5}+3q^{7}+q^{8}-q^{10}+\cdots\)
3330.2.a.bg 3330.a 1.a $3$ $26.590$ 3.3.892.1 None \(-3\) \(0\) \(3\) \(-1\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+q^{5}-\beta _{1}q^{7}-q^{8}-q^{10}+\cdots\)
3330.2.a.bh 3330.a 1.a $3$ $26.590$ 3.3.316.1 None \(-3\) \(0\) \(3\) \(-1\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+q^{5}+(-1+2\beta _{1})q^{7}+\cdots\)
3330.2.a.bi 3330.a 1.a $3$ $26.590$ 3.3.316.1 None \(3\) \(0\) \(-3\) \(-1\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-q^{5}+(-1+2\beta _{1})q^{7}+\cdots\)
3330.2.a.bj 3330.a 1.a $4$ $26.590$ 4.4.54764.1 None \(-4\) \(0\) \(-4\) \(4\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-q^{5}+(1+\beta _{2})q^{7}-q^{8}+\cdots\)
3330.2.a.bk 3330.a 1.a $5$ $26.590$ 5.5.23544108.1 None \(-5\) \(0\) \(-5\) \(3\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-q^{5}+(1+\beta _{4})q^{7}-q^{8}+\cdots\)
3330.2.a.bl 3330.a 1.a $5$ $26.590$ 5.5.23544108.1 None \(5\) \(0\) \(5\) \(3\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+q^{5}+(1+\beta _{4})q^{7}+q^{8}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3330))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3330)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(37))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(74))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(111))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(185))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(222))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(333))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(370))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(555))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(666))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1110))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1665))\)\(^{\oplus 2}\)