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Properties

Label 616.4.a
Level $616$
Weight $4$
Character orbit 616.a
Rep. character $\chi_{616}(1,\cdot)$
Character field $\Q$
Dimension $46$
Newform subspaces $11$
Sturm bound $384$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 616 = 2^{3} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 616.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 11 \)
Sturm bound: \(384\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(616))\).

Total New Old
Modular forms 296 46 250
Cusp forms 280 46 234
Eisenstein series 16 0 16

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
\(+\)\(+\)\(+\)$+$\(7\)
\(+\)\(+\)\(-\)$-$\(4\)
\(+\)\(-\)\(+\)$-$\(6\)
\(+\)\(-\)\(-\)$+$\(7\)
\(-\)\(+\)\(+\)$-$\(7\)
\(-\)\(+\)\(-\)$+$\(5\)
\(-\)\(-\)\(+\)$+$\(6\)
\(-\)\(-\)\(-\)$-$\(4\)
Plus space\(+\)\(25\)
Minus space\(-\)\(21\)

Trace form

\( 46 q - 28 q^{5} + 450 q^{9} + O(q^{10}) \) \( 46 q - 28 q^{5} + 450 q^{9} - 66 q^{11} + 108 q^{13} + 100 q^{15} - 172 q^{17} - 28 q^{23} + 926 q^{25} + 444 q^{27} + 556 q^{29} - 20 q^{31} - 1056 q^{37} - 488 q^{39} + 324 q^{41} + 312 q^{43} - 1752 q^{45} + 952 q^{47} + 2254 q^{49} + 120 q^{51} + 1276 q^{53} - 288 q^{57} + 440 q^{59} + 340 q^{61} - 1064 q^{63} - 656 q^{65} - 820 q^{67} + 492 q^{69} - 1284 q^{71} - 2620 q^{73} + 2172 q^{75} + 308 q^{77} + 1544 q^{79} + 5998 q^{81} - 592 q^{83} + 6992 q^{85} + 4136 q^{87} - 3944 q^{89} - 1932 q^{91} + 2212 q^{93} - 1408 q^{95} + 3048 q^{97} - 1782 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(616))\) 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
616.4.a.a 616.a 1.a $1$ $36.345$ \(\Q\) None \(0\) \(9\) \(15\) \(-7\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+9q^{3}+15q^{5}-7q^{7}+54q^{9}+11q^{11}+\cdots\)
616.4.a.b 616.a 1.a $2$ $36.345$ \(\Q(\sqrt{37}) \) None \(0\) \(-4\) \(0\) \(-14\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{3}+\beta q^{5}-7q^{7}-23q^{9}+11q^{11}+\cdots\)
616.4.a.c 616.a 1.a $2$ $36.345$ \(\Q(\sqrt{5}) \) None \(0\) \(-2\) \(-8\) \(-14\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1-3\beta )q^{3}+(-4+2\beta )q^{5}-7q^{7}+\cdots\)
616.4.a.d 616.a 1.a $2$ $36.345$ \(\Q(\sqrt{5}) \) None \(0\) \(-2\) \(6\) \(-14\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1-3\beta )q^{3}+(3+\beta )q^{5}-7q^{7}+\cdots\)
616.4.a.e 616.a 1.a $4$ $36.345$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(-3\) \(-19\) \(-28\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{3}+(-5-\beta _{2})q^{5}-7q^{7}+\cdots\)
616.4.a.f 616.a 1.a $4$ $36.345$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(-3\) \(-13\) \(28\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{3}+(-4+\beta _{1}-\beta _{2})q^{5}+\cdots\)
616.4.a.g 616.a 1.a $5$ $36.345$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(2\) \(-14\) \(-35\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+(-2-\beta _{1}-\beta _{2})q^{5}-7q^{7}+\cdots\)
616.4.a.h 616.a 1.a $6$ $36.345$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(0\) \(0\) \(-14\) \(42\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{2}q^{3}+(-2-\beta _{3})q^{5}+7q^{7}+(4+\cdots)q^{9}+\cdots\)
616.4.a.i 616.a 1.a $6$ $36.345$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(0\) \(0\) \(2\) \(42\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{4}q^{3}+(-\beta _{2}+\beta _{4})q^{5}+7q^{7}+(11+\cdots)q^{9}+\cdots\)
616.4.a.j 616.a 1.a $7$ $36.345$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(0\) \(0\) \(6\) \(-49\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}+(1-\beta _{2})q^{5}-7q^{7}+(15-\beta _{2}+\cdots)q^{9}+\cdots\)
616.4.a.k 616.a 1.a $7$ $36.345$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(0\) \(3\) \(11\) \(49\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+(2-\beta _{2})q^{5}+7q^{7}+(15+\beta _{3}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(616))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(616)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(22))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(44))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(77))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(88))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(154))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(308))\)\(^{\oplus 2}\)