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Properties

Label 1170.2.a
Level $1170$
Weight $2$
Character orbit 1170.a
Rep. character $\chi_{1170}(1,\cdot)$
Character field $\Q$
Dimension $20$
Newform subspaces $17$
Sturm bound $504$
Trace bound $17$

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Defining parameters

Level: \( N \) \(=\) \( 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1170.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 17 \)
Sturm bound: \(504\)
Trace bound: \(17\)
Distinguishing \(T_p\): \(7\), \(11\), \(17\), \(31\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1170))\).

Total New Old
Modular forms 268 20 248
Cusp forms 237 20 217
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.

\(2\)\(3\)\(5\)\(13\)FrickeDim
\(+\)\(+\)\(+\)\(-\)$-$\(2\)
\(+\)\(+\)\(-\)\(+\)$-$\(1\)
\(+\)\(+\)\(-\)\(-\)$+$\(1\)
\(+\)\(-\)\(+\)\(+\)$-$\(2\)
\(+\)\(-\)\(+\)\(-\)$+$\(2\)
\(+\)\(-\)\(-\)\(+\)$+$\(2\)
\(+\)\(-\)\(-\)\(-\)$-$\(1\)
\(-\)\(+\)\(+\)\(+\)$-$\(1\)
\(-\)\(+\)\(+\)\(-\)$+$\(1\)
\(-\)\(+\)\(-\)\(-\)$-$\(2\)
\(-\)\(-\)\(+\)\(+\)$+$\(1\)
\(-\)\(-\)\(+\)\(-\)$-$\(2\)
\(-\)\(-\)\(-\)\(+\)$-$\(2\)
Plus space\(+\)\(7\)
Minus space\(-\)\(13\)

Trace form

\( 20 q - 2 q^{2} + 20 q^{4} - 2 q^{5} - 2 q^{8} + O(q^{10}) \) \( 20 q - 2 q^{2} + 20 q^{4} - 2 q^{5} - 2 q^{8} - 4 q^{11} + 2 q^{13} + 20 q^{16} + 8 q^{17} + 20 q^{19} - 2 q^{20} + 16 q^{22} - 16 q^{23} + 20 q^{25} + 16 q^{29} - 8 q^{31} - 2 q^{32} - 4 q^{34} + 8 q^{35} - 20 q^{37} + 8 q^{38} - 8 q^{41} + 4 q^{43} - 4 q^{44} + 20 q^{46} + 32 q^{47} + 36 q^{49} - 2 q^{50} + 2 q^{52} + 4 q^{53} - 8 q^{55} - 28 q^{58} - 4 q^{59} - 16 q^{61} + 20 q^{62} + 20 q^{64} - 4 q^{65} + 24 q^{67} + 8 q^{68} + 16 q^{70} - 12 q^{73} + 24 q^{74} + 20 q^{76} - 2 q^{80} - 12 q^{82} + 12 q^{85} + 16 q^{86} + 16 q^{88} + 64 q^{89} - 8 q^{91} - 16 q^{92} + 8 q^{94} + 8 q^{95} + 4 q^{97} + 14 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1170))\) 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 13
1170.2.a.a 1170.a 1.a $1$ $9.342$ \(\Q\) None \(-1\) \(0\) \(-1\) \(0\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-q^{5}-q^{8}+q^{10}-4q^{11}+\cdots\)
1170.2.a.b 1170.a 1.a $1$ $9.342$ \(\Q\) None \(-1\) \(0\) \(-1\) \(0\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-q^{5}-q^{8}+q^{10}+q^{13}+\cdots\)
1170.2.a.c 1170.a 1.a $1$ $9.342$ \(\Q\) None \(-1\) \(0\) \(1\) \(-4\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+q^{5}-4q^{7}-q^{8}-q^{10}+\cdots\)
1170.2.a.d 1170.a 1.a $1$ $9.342$ \(\Q\) None \(-1\) \(0\) \(1\) \(-4\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+q^{5}-4q^{7}-q^{8}-q^{10}+\cdots\)
1170.2.a.e 1170.a 1.a $1$ $9.342$ \(\Q\) None \(-1\) \(0\) \(1\) \(2\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+q^{5}+2q^{7}-q^{8}-q^{10}+\cdots\)
1170.2.a.f 1170.a 1.a $1$ $9.342$ \(\Q\) None \(-1\) \(0\) \(1\) \(2\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+q^{5}+2q^{7}-q^{8}-q^{10}+\cdots\)
1170.2.a.g 1170.a 1.a $1$ $9.342$ \(\Q\) None \(-1\) \(0\) \(1\) \(2\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+q^{5}+2q^{7}-q^{8}-q^{10}+\cdots\)
1170.2.a.h 1170.a 1.a $1$ $9.342$ \(\Q\) None \(1\) \(0\) \(-1\) \(-4\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-q^{5}-4q^{7}+q^{8}-q^{10}+\cdots\)
1170.2.a.i 1170.a 1.a $1$ $9.342$ \(\Q\) None \(1\) \(0\) \(-1\) \(-4\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-q^{5}-4q^{7}+q^{8}-q^{10}+\cdots\)
1170.2.a.j 1170.a 1.a $1$ $9.342$ \(\Q\) None \(1\) \(0\) \(-1\) \(-2\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-q^{5}-2q^{7}+q^{8}-q^{10}+\cdots\)
1170.2.a.k 1170.a 1.a $1$ $9.342$ \(\Q\) None \(1\) \(0\) \(-1\) \(2\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-q^{5}+2q^{7}+q^{8}-q^{10}+\cdots\)
1170.2.a.l 1170.a 1.a $1$ $9.342$ \(\Q\) None \(1\) \(0\) \(-1\) \(2\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-q^{5}+2q^{7}+q^{8}-q^{10}+\cdots\)
1170.2.a.m 1170.a 1.a $1$ $9.342$ \(\Q\) None \(1\) \(0\) \(1\) \(0\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+q^{5}+q^{8}+q^{10}-q^{13}+\cdots\)
1170.2.a.n 1170.a 1.a $1$ $9.342$ \(\Q\) None \(1\) \(0\) \(1\) \(4\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+q^{5}+4q^{7}+q^{8}+q^{10}+\cdots\)
1170.2.a.o 1170.a 1.a $2$ $9.342$ \(\Q(\sqrt{2}) \) None \(-2\) \(0\) \(-2\) \(0\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-q^{5}+\beta q^{7}-q^{8}+q^{10}+\cdots\)
1170.2.a.p 1170.a 1.a $2$ $9.342$ \(\Q(\sqrt{13}) \) None \(-2\) \(0\) \(-2\) \(2\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-q^{5}+(1+\beta )q^{7}-q^{8}+\cdots\)
1170.2.a.q 1170.a 1.a $2$ $9.342$ \(\Q(\sqrt{13}) \) None \(2\) \(0\) \(2\) \(2\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+q^{5}+(1+\beta )q^{7}+q^{8}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1170))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1170)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(65))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(78))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(117))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(130))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(195))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(234))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(390))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(585))\)\(^{\oplus 2}\)