Properties

Label 616.2.a
Level $616$
Weight $2$
Character orbit 616.a
Rep. character $\chi_{616}(1,\cdot)$
Character field $\Q$
Dimension $14$
Newform subspaces $8$
Sturm bound $192$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 616 = 2^{3} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 616.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 8 \)
Sturm bound: \(192\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(616))\).

Total New Old
Modular forms 104 14 90
Cusp forms 89 14 75
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.

\(2\)\(7\)\(11\)FrickeDim
\(+\)\(+\)\(+\)$+$\(1\)
\(+\)\(+\)\(-\)$-$\(4\)
\(+\)\(-\)\(+\)$-$\(2\)
\(+\)\(-\)\(-\)$+$\(1\)
\(-\)\(+\)\(-\)$+$\(2\)
\(-\)\(-\)\(+\)$+$\(1\)
\(-\)\(-\)\(-\)$-$\(3\)
Plus space\(+\)\(5\)
Minus space\(-\)\(9\)

Trace form

\( 14 q + 4 q^{5} + 10 q^{9} + O(q^{10}) \) \( 14 q + 4 q^{5} + 10 q^{9} + 6 q^{11} - 4 q^{13} + 4 q^{15} + 4 q^{17} - 12 q^{23} + 6 q^{25} + 12 q^{27} - 4 q^{29} - 4 q^{31} + 24 q^{37} - 8 q^{39} - 12 q^{41} - 8 q^{43} + 32 q^{45} + 8 q^{47} + 14 q^{49} - 8 q^{51} - 4 q^{53} - 8 q^{59} + 20 q^{61} - 8 q^{63} + 32 q^{65} - 4 q^{67} - 12 q^{69} + 12 q^{71} - 12 q^{73} + 12 q^{75} - 4 q^{77} - 24 q^{79} + 14 q^{81} + 16 q^{83} - 56 q^{87} - 16 q^{89} + 4 q^{91} - 4 q^{93} - 32 q^{95} - 32 q^{97} + 18 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(616))\) 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 7 11
616.2.a.a 616.a 1.a $1$ $4.919$ \(\Q\) None \(0\) \(-2\) \(2\) \(1\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{3}+2q^{5}+q^{7}+q^{9}-q^{11}+4q^{13}+\cdots\)
616.2.a.b 616.a 1.a $1$ $4.919$ \(\Q\) None \(0\) \(-1\) \(-1\) \(1\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{5}+q^{7}-2q^{9}+q^{11}+q^{15}+\cdots\)
616.2.a.c 616.a 1.a $1$ $4.919$ \(\Q\) None \(0\) \(0\) \(-2\) \(1\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{5}+q^{7}-3q^{9}-q^{11}+2q^{13}+\cdots\)
616.2.a.d 616.a 1.a $1$ $4.919$ \(\Q\) None \(0\) \(0\) \(0\) \(-1\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{7}-3q^{9}-q^{11}-6q^{13}-2q^{19}+\cdots\)
616.2.a.e 616.a 1.a $1$ $4.919$ \(\Q\) None \(0\) \(2\) \(2\) \(1\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{3}+2q^{5}+q^{7}+q^{9}-q^{11}+4q^{15}+\cdots\)
616.2.a.f 616.a 1.a $2$ $4.919$ \(\Q(\sqrt{17}) \) None \(0\) \(-1\) \(-3\) \(-2\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{3}+(-2+\beta )q^{5}-q^{7}+(1+\beta )q^{9}+\cdots\)
616.2.a.g 616.a 1.a $3$ $4.919$ 3.3.229.1 None \(0\) \(1\) \(1\) \(3\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{2}q^{3}-\beta _{2}q^{5}+q^{7}+(1-\beta _{1})q^{9}+\cdots\)
616.2.a.h 616.a 1.a $4$ $4.919$ 4.4.11348.1 None \(0\) \(1\) \(5\) \(-4\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{2}q^{3}+(1-\beta _{1})q^{5}-q^{7}+(2+\beta _{2}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(616))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(616)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(44))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(77))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(88))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(154))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(308))\)\(^{\oplus 2}\)