Properties

Label 1710.2.a
Level $1710$
Weight $2$
Character orbit 1710.a
Rep. character $\chi_{1710}(1,\cdot)$
Character field $\Q$
Dimension $30$
Newform subspaces $25$
Sturm bound $720$
Trace bound $13$

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Defining parameters

Level: \( N \) \(=\) \( 1710 = 2 \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1710.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 25 \)
Sturm bound: \(720\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(7\), \(11\), \(13\), \(53\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1710))\).

Total New Old
Modular forms 376 30 346
Cusp forms 345 30 315
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.

\(2\)\(3\)\(5\)\(19\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(+\)\(1\)
\(+\)\(+\)\(+\)\(-\)\(-\)\(2\)
\(+\)\(+\)\(-\)\(+\)\(-\)\(2\)
\(+\)\(+\)\(-\)\(-\)\(+\)\(1\)
\(+\)\(-\)\(+\)\(+\)\(-\)\(2\)
\(+\)\(-\)\(+\)\(-\)\(+\)\(3\)
\(+\)\(-\)\(-\)\(+\)\(+\)\(1\)
\(+\)\(-\)\(-\)\(-\)\(-\)\(3\)
\(-\)\(+\)\(+\)\(+\)\(-\)\(2\)
\(-\)\(+\)\(+\)\(-\)\(+\)\(1\)
\(-\)\(+\)\(-\)\(+\)\(+\)\(1\)
\(-\)\(+\)\(-\)\(-\)\(-\)\(2\)
\(-\)\(-\)\(+\)\(+\)\(+\)\(3\)
\(-\)\(-\)\(+\)\(-\)\(-\)\(2\)
\(-\)\(-\)\(-\)\(+\)\(-\)\(3\)
\(-\)\(-\)\(-\)\(-\)\(+\)\(1\)
Plus space\(+\)\(12\)
Minus space\(-\)\(18\)

Trace form

\( 30 q + 30 q^{4} - 2 q^{5} + O(q^{10}) \) \( 30 q + 30 q^{4} - 2 q^{5} + 8 q^{13} - 4 q^{14} + 30 q^{16} - 2 q^{20} - 16 q^{23} + 30 q^{25} - 16 q^{26} - 4 q^{29} - 8 q^{31} - 8 q^{34} + 4 q^{35} - 16 q^{37} - 6 q^{38} - 4 q^{41} - 20 q^{43} - 12 q^{46} + 20 q^{47} + 50 q^{49} + 8 q^{52} + 16 q^{53} - 24 q^{55} - 4 q^{56} - 20 q^{58} + 48 q^{59} - 12 q^{61} - 8 q^{62} + 30 q^{64} + 12 q^{70} + 8 q^{71} - 8 q^{73} + 12 q^{74} - 2 q^{80} + 16 q^{82} + 36 q^{83} + 36 q^{85} - 12 q^{86} + 4 q^{89} + 32 q^{91} - 16 q^{92} + 4 q^{94} - 16 q^{97} - 16 q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1710))\) 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 19
1710.2.a.a 1710.a 1.a $1$ $13.654$ \(\Q\) None \(-1\) \(0\) \(-1\) \(-4\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-q^{5}-4q^{7}-q^{8}+q^{10}+\cdots\)
1710.2.a.b 1710.a 1.a $1$ $13.654$ \(\Q\) None \(-1\) \(0\) \(-1\) \(-2\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-q^{5}-2q^{7}-q^{8}+q^{10}+\cdots\)
1710.2.a.c 1710.a 1.a $1$ $13.654$ \(\Q\) None \(-1\) \(0\) \(-1\) \(-2\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-q^{5}-2q^{7}-q^{8}+q^{10}+\cdots\)
1710.2.a.d 1710.a 1.a $1$ $13.654$ \(\Q\) None \(-1\) \(0\) \(-1\) \(-1\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-q^{5}-q^{7}-q^{8}+q^{10}+\cdots\)
1710.2.a.e 1710.a 1.a $1$ $13.654$ \(\Q\) None \(-1\) \(0\) \(-1\) \(4\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-q^{5}+4q^{7}-q^{8}+q^{10}+\cdots\)
1710.2.a.f 1710.a 1.a $1$ $13.654$ \(\Q\) None \(-1\) \(0\) \(-1\) \(4\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-q^{5}+4q^{7}-q^{8}+q^{10}+\cdots\)
1710.2.a.g 1710.a 1.a $1$ $13.654$ \(\Q\) None \(-1\) \(0\) \(1\) \(-5\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+q^{5}-5q^{7}-q^{8}-q^{10}+\cdots\)
1710.2.a.h 1710.a 1.a $1$ $13.654$ \(\Q\) None \(-1\) \(0\) \(1\) \(-2\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+q^{5}-2q^{7}-q^{8}-q^{10}+\cdots\)
1710.2.a.i 1710.a 1.a $1$ $13.654$ \(\Q\) None \(-1\) \(0\) \(1\) \(0\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+q^{5}-q^{8}-q^{10}-4q^{11}+\cdots\)
1710.2.a.j 1710.a 1.a $1$ $13.654$ \(\Q\) None \(-1\) \(0\) \(1\) \(2\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+q^{5}+2q^{7}-q^{8}-q^{10}+\cdots\)
1710.2.a.k 1710.a 1.a $1$ $13.654$ \(\Q\) None \(-1\) \(0\) \(1\) \(2\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+q^{5}+2q^{7}-q^{8}-q^{10}+\cdots\)
1710.2.a.l 1710.a 1.a $1$ $13.654$ \(\Q\) None \(1\) \(0\) \(-1\) \(-4\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-q^{5}-4q^{7}+q^{8}-q^{10}+\cdots\)
1710.2.a.m 1710.a 1.a $1$ $13.654$ \(\Q\) None \(1\) \(0\) \(-1\) \(-2\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-q^{5}-2q^{7}+q^{8}-q^{10}+\cdots\)
1710.2.a.n 1710.a 1.a $1$ $13.654$ \(\Q\) None \(1\) \(0\) \(-1\) \(-2\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-q^{5}-2q^{7}+q^{8}-q^{10}+\cdots\)
1710.2.a.o 1710.a 1.a $1$ $13.654$ \(\Q\) None \(1\) \(0\) \(-1\) \(2\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-q^{5}+2q^{7}+q^{8}-q^{10}+\cdots\)
1710.2.a.p 1710.a 1.a $1$ $13.654$ \(\Q\) None \(1\) \(0\) \(1\) \(-4\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+q^{5}-4q^{7}+q^{8}+q^{10}+\cdots\)
1710.2.a.q 1710.a 1.a $1$ $13.654$ \(\Q\) None \(1\) \(0\) \(1\) \(-2\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+q^{5}-2q^{7}+q^{8}+q^{10}+\cdots\)
1710.2.a.r 1710.a 1.a $1$ $13.654$ \(\Q\) None \(1\) \(0\) \(1\) \(-1\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+q^{5}-q^{7}+q^{8}+q^{10}+\cdots\)
1710.2.a.s 1710.a 1.a $1$ $13.654$ \(\Q\) None \(1\) \(0\) \(1\) \(2\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+q^{5}+2q^{7}+q^{8}+q^{10}+\cdots\)
1710.2.a.t 1710.a 1.a $1$ $13.654$ \(\Q\) None \(1\) \(0\) \(1\) \(4\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+q^{5}+4q^{7}+q^{8}+q^{10}+\cdots\)
1710.2.a.u 1710.a 1.a $2$ $13.654$ \(\Q(\sqrt{3}) \) None \(-2\) \(0\) \(-2\) \(4\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-q^{5}+2q^{7}-q^{8}+q^{10}+\cdots\)
1710.2.a.v 1710.a 1.a $2$ $13.654$ \(\Q(\sqrt{17}) \) None \(-2\) \(0\) \(2\) \(2\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+q^{5}+(1+\beta )q^{7}-q^{8}+\cdots\)
1710.2.a.w 1710.a 1.a $2$ $13.654$ \(\Q(\sqrt{17}) \) None \(2\) \(0\) \(-2\) \(-1\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-q^{5}-\beta q^{7}+q^{8}-q^{10}+\cdots\)
1710.2.a.x 1710.a 1.a $2$ $13.654$ \(\Q(\sqrt{17}) \) None \(2\) \(0\) \(-2\) \(2\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-q^{5}+(1+\beta )q^{7}+q^{8}+\cdots\)
1710.2.a.y 1710.a 1.a $2$ $13.654$ \(\Q(\sqrt{3}) \) None \(2\) \(0\) \(2\) \(4\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+q^{5}+2q^{7}+q^{8}+q^{10}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1710))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1710)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(57))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(95))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(114))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(171))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(190))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(285))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(342))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(570))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(855))\)\(^{\oplus 2}\)