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Properties

Label 760.2.a
Level $760$
Weight $2$
Character orbit 760.a
Rep. character $\chi_{760}(1,\cdot)$
Character field $\Q$
Dimension $18$
Newform subspaces $10$
Sturm bound $240$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 760 = 2^{3} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 760.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 10 \)
Sturm bound: \(240\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(760))\).

Total New Old
Modular forms 128 18 110
Cusp forms 113 18 95
Eisenstein series 15 0 15

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

\(2\)\(5\)\(19\)FrickeDim
\(+\)\(+\)\(+\)$+$\(3\)
\(+\)\(+\)\(-\)$-$\(2\)
\(+\)\(-\)\(+\)$-$\(3\)
\(+\)\(-\)\(-\)$+$\(1\)
\(-\)\(+\)\(+\)$-$\(2\)
\(-\)\(+\)\(-\)$+$\(3\)
\(-\)\(-\)\(+\)$+$\(1\)
\(-\)\(-\)\(-\)$-$\(3\)
Plus space\(+\)\(8\)
Minus space\(-\)\(10\)

Trace form

\( 18 q - 2 q^{5} + 14 q^{9} + O(q^{10}) \) \( 18 q - 2 q^{5} + 14 q^{9} - 8 q^{17} - 8 q^{21} + 18 q^{25} + 12 q^{29} - 8 q^{31} - 16 q^{33} + 12 q^{35} + 4 q^{39} - 4 q^{41} - 4 q^{43} + 6 q^{45} + 12 q^{47} + 6 q^{49} + 8 q^{51} - 16 q^{53} - 8 q^{55} - 4 q^{57} + 16 q^{59} - 20 q^{61} - 44 q^{63} - 24 q^{67} + 56 q^{69} - 8 q^{71} - 16 q^{73} + 16 q^{77} + 16 q^{79} - 30 q^{81} + 12 q^{83} - 4 q^{85} + 28 q^{87} - 12 q^{89} - 32 q^{91} - 16 q^{97} + 96 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(760))\) 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 5 19
760.2.a.a 760.a 1.a $1$ $6.069$ \(\Q\) None \(0\) \(-2\) \(1\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{3}+q^{5}+q^{9}-4q^{11}+4q^{13}+\cdots\)
760.2.a.b 760.a 1.a $1$ $6.069$ \(\Q\) None \(0\) \(-2\) \(1\) \(4\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{3}+q^{5}+4q^{7}+q^{9}+4q^{11}+\cdots\)
760.2.a.c 760.a 1.a $1$ $6.069$ \(\Q\) None \(0\) \(0\) \(1\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{5}-3q^{9}-4q^{11}-6q^{13}-6q^{17}+\cdots\)
760.2.a.d 760.a 1.a $1$ $6.069$ \(\Q\) None \(0\) \(2\) \(1\) \(4\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{3}+q^{5}+4q^{7}+q^{9}-4q^{11}+\cdots\)
760.2.a.e 760.a 1.a $1$ $6.069$ \(\Q\) None \(0\) \(3\) \(1\) \(-1\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}+q^{5}-q^{7}+6q^{9}+4q^{11}+\cdots\)
760.2.a.f 760.a 1.a $2$ $6.069$ \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(-2\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{3}-q^{5}+2\beta q^{7}-q^{9}+(2+2\beta )q^{11}+\cdots\)
760.2.a.g 760.a 1.a $2$ $6.069$ \(\Q(\sqrt{3}) \) None \(0\) \(2\) \(-2\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{3}-q^{5}+(1+2\beta )q^{9}+2q^{11}+\cdots\)
760.2.a.h 760.a 1.a $3$ $6.069$ 3.3.568.1 None \(0\) \(-1\) \(-3\) \(-5\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}-q^{5}+(-2-\beta _{2})q^{7}+(1+2\beta _{1}+\cdots)q^{9}+\cdots\)
760.2.a.i 760.a 1.a $3$ $6.069$ 3.3.316.1 None \(0\) \(-1\) \(-3\) \(-1\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}-q^{5}+(-1+2\beta _{1}-\beta _{2})q^{7}+\cdots\)
760.2.a.j 760.a 1.a $3$ $6.069$ 3.3.229.1 None \(0\) \(-1\) \(3\) \(-1\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{2}q^{3}+q^{5}+\beta _{1}q^{7}+(1-\beta _{1})q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(760))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(760)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(76))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(95))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(152))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(190))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(380))\)\(^{\oplus 2}\)