Properties

Label 688.4.a
Level $688$
Weight $4$
Character orbit 688.a
Rep. character $\chi_{688}(1,\cdot)$
Character field $\Q$
Dimension $63$
Newform subspaces $13$
Sturm bound $352$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 688 = 2^{4} \cdot 43 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 688.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 13 \)
Sturm bound: \(352\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(688))\).

Total New Old
Modular forms 270 63 207
Cusp forms 258 63 195
Eisenstein series 12 0 12

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

\(2\)\(43\)FrickeDim
\(+\)\(+\)$+$\(17\)
\(+\)\(-\)$-$\(14\)
\(-\)\(+\)$-$\(16\)
\(-\)\(-\)$+$\(16\)
Plus space\(+\)\(33\)
Minus space\(-\)\(30\)

Trace form

\( 63 q + 2 q^{5} + 567 q^{9} + O(q^{10}) \) \( 63 q + 2 q^{5} + 567 q^{9} - 66 q^{11} - 46 q^{13} + 72 q^{15} - 26 q^{17} + 24 q^{19} + 138 q^{23} + 1765 q^{25} + 576 q^{27} - 142 q^{29} - 78 q^{31} - 24 q^{33} - 192 q^{35} + 330 q^{37} - 852 q^{39} + 414 q^{41} - 129 q^{43} + 90 q^{45} + 1244 q^{47} + 2783 q^{49} + 268 q^{51} + 106 q^{53} + 848 q^{57} - 2052 q^{59} - 1102 q^{61} - 1712 q^{63} - 164 q^{65} - 1326 q^{67} + 512 q^{69} + 1092 q^{71} - 1354 q^{73} + 1776 q^{75} + 472 q^{77} - 236 q^{79} + 4799 q^{81} - 1602 q^{83} - 52 q^{85} - 3540 q^{87} - 746 q^{89} + 1800 q^{91} - 1336 q^{93} + 1972 q^{95} - 290 q^{97} + 3382 q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(688))\) 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 43
688.4.a.a 688.a 1.a $1$ $40.593$ \(\Q\) None \(0\) \(-8\) \(6\) \(-14\) $-$ $-$ $\mathrm{SU}(2)$ \(q-8q^{3}+6q^{5}-14q^{7}+37q^{9}+43q^{11}+\cdots\)
688.4.a.b 688.a 1.a $1$ $40.593$ \(\Q\) None \(0\) \(4\) \(-14\) \(14\) $-$ $-$ $\mathrm{SU}(2)$ \(q+4q^{3}-14q^{5}+14q^{7}-11q^{9}+11q^{11}+\cdots\)
688.4.a.c 688.a 1.a $2$ $40.593$ \(\Q(\sqrt{3}) \) None \(0\) \(-2\) \(-14\) \(22\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+3\beta )q^{3}+(-7+7\beta )q^{5}+(11+\cdots)q^{7}+\cdots\)
688.4.a.d 688.a 1.a $3$ $40.593$ 3.3.85685.1 None \(0\) \(9\) \(-5\) \(8\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(3+\beta _{1})q^{3}+(-2-2\beta _{1}+\beta _{2})q^{5}+\cdots\)
688.4.a.e 688.a 1.a $4$ $40.593$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(-11\) \(1\) \(-42\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-3+\beta _{1})q^{3}+(\beta _{1}-\beta _{2}-\beta _{3})q^{5}+\cdots\)
688.4.a.f 688.a 1.a $4$ $40.593$ 4.4.45868.1 None \(0\) \(11\) \(-27\) \(20\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(3-\beta _{1}-\beta _{3})q^{3}+(-6+2\beta _{1}-2\beta _{2}+\cdots)q^{5}+\cdots\)
688.4.a.g 688.a 1.a $5$ $40.593$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(11\) \(-1\) \(8\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(2-\beta _{4})q^{3}+(-\beta _{3}-\beta _{4})q^{5}+(3-\beta _{1}+\cdots)q^{7}+\cdots\)
688.4.a.h 688.a 1.a $6$ $40.593$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(0\) \(-7\) \(-1\) \(-20\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{1})q^{3}+(\beta _{1}+\beta _{5})q^{5}+(-3+\cdots)q^{7}+\cdots\)
688.4.a.i 688.a 1.a $6$ $40.593$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(0\) \(-7\) \(43\) \(-8\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{3}+(7+\beta _{1}-\beta _{4})q^{5}+\cdots\)
688.4.a.j 688.a 1.a $6$ $40.593$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(0\) \(-3\) \(-5\) \(16\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{3}+(-1-\beta _{3})q^{5}+(3+\cdots)q^{7}+\cdots\)
688.4.a.k 688.a 1.a $7$ $40.593$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(0\) \(9\) \(-3\) \(32\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{3})q^{3}-\beta _{2}q^{5}+(5-\beta _{2}+\beta _{3}+\cdots)q^{7}+\cdots\)
688.4.a.l 688.a 1.a $8$ $40.593$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(-6\) \(12\) \(-24\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{3}+(2-\beta _{6})q^{5}+(-3+\cdots)q^{7}+\cdots\)
688.4.a.m 688.a 1.a $10$ $40.593$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(0\) \(0\) \(10\) \(-12\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+(1+\beta _{4})q^{5}+(-1+\beta _{5})q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(688))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(688)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(43))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(86))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(172))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(344))\)\(^{\oplus 2}\)