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Properties

Label 3234.2.a
Level $3234$
Weight $2$
Character orbit 3234.a
Rep. character $\chi_{3234}(1,\cdot)$
Character field $\Q$
Dimension $68$
Newform subspaces $39$
Sturm bound $1344$
Trace bound $17$

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Defining parameters

Level: \( N \) \(=\) \( 3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3234.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 39 \)
Sturm bound: \(1344\)
Trace bound: \(17\)
Distinguishing \(T_p\): \(5\), \(13\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3234))\).

Total New Old
Modular forms 704 68 636
Cusp forms 641 68 573
Eisenstein series 63 0 63

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

\(2\)\(3\)\(7\)\(11\)FrickeDim
\(+\)\(+\)\(+\)\(+\)$+$\(5\)
\(+\)\(+\)\(+\)\(-\)$-$\(4\)
\(+\)\(+\)\(-\)\(+\)$-$\(4\)
\(+\)\(+\)\(-\)\(-\)$+$\(4\)
\(+\)\(-\)\(+\)\(+\)$-$\(6\)
\(+\)\(-\)\(+\)\(-\)$+$\(3\)
\(+\)\(-\)\(-\)\(+\)$+$\(2\)
\(+\)\(-\)\(-\)\(-\)$-$\(5\)
\(-\)\(+\)\(+\)\(+\)$-$\(5\)
\(-\)\(+\)\(+\)\(-\)$+$\(4\)
\(-\)\(+\)\(-\)\(+\)$+$\(3\)
\(-\)\(+\)\(-\)\(-\)$-$\(6\)
\(-\)\(-\)\(+\)\(+\)$+$\(2\)
\(-\)\(-\)\(+\)\(-\)$-$\(7\)
\(-\)\(-\)\(-\)\(+\)$-$\(7\)
\(-\)\(-\)\(-\)\(-\)$+$\(1\)
Plus space\(+\)\(24\)
Minus space\(-\)\(44\)

Trace form

\( 68 q + 2 q^{2} - 2 q^{3} + 68 q^{4} + 4 q^{5} + 2 q^{8} + 68 q^{9} + O(q^{10}) \) \( 68 q + 2 q^{2} - 2 q^{3} + 68 q^{4} + 4 q^{5} + 2 q^{8} + 68 q^{9} + 4 q^{10} - 2 q^{12} + 68 q^{16} + 12 q^{17} + 2 q^{18} - 4 q^{19} + 4 q^{20} + 2 q^{22} - 4 q^{23} + 52 q^{25} + 16 q^{26} - 2 q^{27} - 12 q^{29} - 24 q^{31} + 2 q^{32} - 2 q^{33} + 68 q^{36} + 56 q^{37} - 4 q^{38} + 36 q^{39} + 4 q^{40} + 12 q^{41} + 12 q^{43} + 4 q^{45} + 12 q^{47} - 2 q^{48} + 62 q^{50} - 8 q^{51} + 60 q^{53} + 44 q^{57} + 64 q^{58} - 8 q^{59} + 24 q^{61} + 16 q^{62} + 68 q^{64} + 72 q^{65} - 4 q^{66} + 40 q^{67} + 12 q^{68} - 4 q^{69} + 4 q^{71} + 2 q^{72} + 24 q^{73} + 68 q^{74} - 6 q^{75} - 4 q^{76} - 20 q^{78} + 52 q^{79} + 4 q^{80} + 68 q^{81} - 8 q^{82} + 16 q^{83} + 56 q^{85} - 20 q^{86} - 8 q^{87} + 2 q^{88} + 24 q^{89} + 4 q^{90} - 4 q^{92} + 64 q^{93} + 8 q^{94} - 32 q^{95} + 16 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3234))\) 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 7 11
3234.2.a.a 3234.a 1.a $1$ $25.824$ \(\Q\) None \(-1\) \(-1\) \(-3\) \(0\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}-3q^{5}+q^{6}-q^{8}+\cdots\)
3234.2.a.b 3234.a 1.a $1$ $25.824$ \(\Q\) None \(-1\) \(-1\) \(0\) \(0\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{6}-q^{8}+q^{9}+\cdots\)
3234.2.a.c 3234.a 1.a $1$ $25.824$ \(\Q\) None \(-1\) \(-1\) \(0\) \(0\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{6}-q^{8}+q^{9}+\cdots\)
3234.2.a.d 3234.a 1.a $1$ $25.824$ \(\Q\) None \(-1\) \(-1\) \(0\) \(0\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{6}-q^{8}+q^{9}+\cdots\)
3234.2.a.e 3234.a 1.a $1$ $25.824$ \(\Q\) None \(-1\) \(-1\) \(0\) \(0\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{6}-q^{8}+q^{9}+\cdots\)
3234.2.a.f 3234.a 1.a $1$ $25.824$ \(\Q\) None \(-1\) \(-1\) \(0\) \(0\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{6}-q^{8}+q^{9}+\cdots\)
3234.2.a.g 3234.a 1.a $1$ $25.824$ \(\Q\) None \(-1\) \(-1\) \(3\) \(0\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+3q^{5}+q^{6}-q^{8}+\cdots\)
3234.2.a.h 3234.a 1.a $1$ $25.824$ \(\Q\) None \(-1\) \(1\) \(-3\) \(0\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-3q^{5}-q^{6}-q^{8}+\cdots\)
3234.2.a.i 3234.a 1.a $1$ $25.824$ \(\Q\) None \(-1\) \(1\) \(-2\) \(0\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-2q^{5}-q^{6}-q^{8}+\cdots\)
3234.2.a.j 3234.a 1.a $1$ $25.824$ \(\Q\) None \(-1\) \(1\) \(0\) \(0\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{8}+q^{9}+\cdots\)
3234.2.a.k 3234.a 1.a $1$ $25.824$ \(\Q\) None \(-1\) \(1\) \(0\) \(0\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{8}+q^{9}+\cdots\)
3234.2.a.l 3234.a 1.a $1$ $25.824$ \(\Q\) None \(-1\) \(1\) \(0\) \(0\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{8}+q^{9}+\cdots\)
3234.2.a.m 3234.a 1.a $1$ $25.824$ \(\Q\) None \(-1\) \(1\) \(0\) \(0\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{8}+q^{9}+\cdots\)
3234.2.a.n 3234.a 1.a $1$ $25.824$ \(\Q\) None \(-1\) \(1\) \(2\) \(0\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}+2q^{5}-q^{6}-q^{8}+\cdots\)
3234.2.a.o 3234.a 1.a $1$ $25.824$ \(\Q\) None \(-1\) \(1\) \(3\) \(0\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}+3q^{5}-q^{6}-q^{8}+\cdots\)
3234.2.a.p 3234.a 1.a $1$ $25.824$ \(\Q\) None \(1\) \(-1\) \(-2\) \(0\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-2q^{5}-q^{6}+q^{8}+\cdots\)
3234.2.a.q 3234.a 1.a $1$ $25.824$ \(\Q\) None \(1\) \(-1\) \(0\) \(0\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{6}+q^{8}+q^{9}+\cdots\)
3234.2.a.r 3234.a 1.a $1$ $25.824$ \(\Q\) None \(1\) \(-1\) \(2\) \(0\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+2q^{5}-q^{6}+q^{8}+\cdots\)
3234.2.a.s 3234.a 1.a $1$ $25.824$ \(\Q\) None \(1\) \(-1\) \(4\) \(0\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+4q^{5}-q^{6}+q^{8}+\cdots\)
3234.2.a.t 3234.a 1.a $1$ $25.824$ \(\Q\) None \(1\) \(1\) \(-2\) \(0\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}-2q^{5}+q^{6}+q^{8}+\cdots\)
3234.2.a.u 3234.a 1.a $1$ $25.824$ \(\Q\) None \(1\) \(1\) \(-2\) \(0\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}-2q^{5}+q^{6}+q^{8}+\cdots\)
3234.2.a.v 3234.a 1.a $1$ $25.824$ \(\Q\) None \(1\) \(1\) \(4\) \(0\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+4q^{5}+q^{6}+q^{8}+\cdots\)
3234.2.a.w 3234.a 1.a $2$ $25.824$ \(\Q(\sqrt{7}) \) None \(-2\) \(-2\) \(0\) \(0\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+\beta q^{5}+q^{6}-q^{8}+\cdots\)
3234.2.a.x 3234.a 1.a $2$ $25.824$ \(\Q(\sqrt{3}) \) None \(-2\) \(-2\) \(0\) \(0\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+\beta q^{5}+q^{6}-q^{8}+\cdots\)
3234.2.a.y 3234.a 1.a $2$ $25.824$ \(\Q(\sqrt{2}) \) None \(-2\) \(-2\) \(0\) \(0\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{6}-q^{8}+q^{9}+\cdots\)
3234.2.a.z 3234.a 1.a $2$ $25.824$ \(\Q(\sqrt{2}) \) None \(-2\) \(2\) \(0\) \(0\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{8}+q^{9}+\cdots\)
3234.2.a.ba 3234.a 1.a $2$ $25.824$ \(\Q(\sqrt{7}) \) None \(-2\) \(2\) \(0\) \(0\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}+\beta q^{5}-q^{6}-q^{8}+\cdots\)
3234.2.a.bb 3234.a 1.a $2$ $25.824$ \(\Q(\sqrt{5}) \) None \(2\) \(-2\) \(-2\) \(0\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+(-1-\beta )q^{5}-q^{6}+\cdots\)
3234.2.a.bc 3234.a 1.a $2$ $25.824$ \(\Q(\sqrt{2}) \) None \(2\) \(-2\) \(4\) \(0\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+2q^{5}-q^{6}+q^{8}+\cdots\)
3234.2.a.bd 3234.a 1.a $2$ $25.824$ \(\Q(\sqrt{2}) \) None \(2\) \(2\) \(-4\) \(0\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}-2q^{5}+q^{6}+q^{8}+\cdots\)
3234.2.a.be 3234.a 1.a $2$ $25.824$ \(\Q(\sqrt{5}) \) None \(2\) \(2\) \(2\) \(0\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+(1+\beta )q^{5}+q^{6}+\cdots\)
3234.2.a.bf 3234.a 1.a $3$ $25.824$ 3.3.2700.1 None \(3\) \(-3\) \(0\) \(0\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+\beta _{1}q^{5}-q^{6}+q^{8}+\cdots\)
3234.2.a.bg 3234.a 1.a $3$ $25.824$ 3.3.621.1 None \(3\) \(-3\) \(0\) \(0\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-\beta _{2}q^{5}-q^{6}+q^{8}+\cdots\)
3234.2.a.bh 3234.a 1.a $3$ $25.824$ 3.3.2700.1 None \(3\) \(3\) \(0\) \(0\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}-\beta _{1}q^{5}+q^{6}+q^{8}+\cdots\)
3234.2.a.bi 3234.a 1.a $3$ $25.824$ 3.3.621.1 None \(3\) \(3\) \(0\) \(0\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+\beta _{2}q^{5}+q^{6}+q^{8}+\cdots\)
3234.2.a.bj 3234.a 1.a $4$ $25.824$ 4.4.14336.1 None \(-4\) \(-4\) \(0\) \(0\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+\beta _{2}q^{5}+q^{6}-q^{8}+\cdots\)
3234.2.a.bk 3234.a 1.a $4$ $25.824$ 4.4.14336.1 None \(-4\) \(4\) \(0\) \(0\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}+\beta _{2}q^{5}-q^{6}-q^{8}+\cdots\)
3234.2.a.bl 3234.a 1.a $4$ $25.824$ 4.4.4352.1 None \(4\) \(-4\) \(-4\) \(0\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+(-1-\beta _{3})q^{5}-q^{6}+\cdots\)
3234.2.a.bm 3234.a 1.a $4$ $25.824$ 4.4.4352.1 None \(4\) \(4\) \(4\) \(0\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+(1+\beta _{3})q^{5}+q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3234))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3234)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(22))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(66))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(77))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(147))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(154))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(231))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(294))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(462))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(539))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1078))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1617))\)\(^{\oplus 2}\)