Properties

Label 688.6.a
Level $688$
Weight $6$
Character orbit 688.a
Rep. character $\chi_{688}(1,\cdot)$
Character field $\Q$
Dimension $105$
Newform subspaces $12$
Sturm bound $528$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 688 = 2^{4} \cdot 43 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 688.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 12 \)
Sturm bound: \(528\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(688))\).

Total New Old
Modular forms 446 105 341
Cusp forms 434 105 329
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.
\(+\)\(+\)\(+\)\(25\)
\(+\)\(-\)\(-\)\(28\)
\(-\)\(+\)\(-\)\(26\)
\(-\)\(-\)\(+\)\(26\)
Plus space\(+\)\(51\)
Minus space\(-\)\(54\)

Trace form

\( 105 q + 38 q^{5} + 8505 q^{9} + O(q^{10}) \) \( 105 q + 38 q^{5} + 8505 q^{9} + 726 q^{11} - 122 q^{13} - 3768 q^{15} + 202 q^{17} + 2360 q^{19} - 3174 q^{23} + 61339 q^{25} - 576 q^{27} - 4282 q^{29} + 1394 q^{31} + 9432 q^{33} + 17088 q^{35} - 9410 q^{37} + 20172 q^{39} - 10878 q^{41} + 5547 q^{43} + 15390 q^{45} + 43484 q^{47} + 283257 q^{49} - 92900 q^{51} - 37634 q^{53} - 41296 q^{57} + 55404 q^{59} + 106438 q^{61} + 123856 q^{63} + 109636 q^{65} + 71562 q^{67} - 85504 q^{69} - 94812 q^{71} - 19158 q^{73} - 5712 q^{75} + 74824 q^{77} - 194156 q^{79} + 812729 q^{81} + 165990 q^{83} - 247708 q^{85} - 34932 q^{87} - 39830 q^{89} - 100056 q^{91} + 330776 q^{93} - 467276 q^{95} + 106290 q^{97} - 194018 q^{99} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(688))\) 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
688.6.a.a 688.a 1.a $3$ $110.344$ 3.3.159992.1 None \(0\) \(-8\) \(-14\) \(74\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-3-\beta _{2})q^{3}+(-5+2\beta _{1}-\beta _{2})q^{5}+\cdots\)
688.6.a.b 688.a 1.a $3$ $110.344$ 3.3.146508.1 None \(0\) \(28\) \(-14\) \(182\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(9-\beta _{2})q^{3}+(-5+2\beta _{1}-3\beta _{2})q^{5}+\cdots\)
688.6.a.c 688.a 1.a $5$ $110.344$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(1\) \(61\) \(-122\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}+(12-2\beta _{1}-\beta _{2}-\beta _{3})q^{5}+\cdots\)
688.6.a.d 688.a 1.a $6$ $110.344$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(0\) \(-17\) \(61\) \(-210\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-3+\beta _{1})q^{3}+(10-\beta _{1}+\beta _{4}-\beta _{5})q^{5}+\cdots\)
688.6.a.e 688.a 1.a $8$ $110.344$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(26\) \(-212\) \(136\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(3+\beta _{4})q^{3}+(-3^{3}+\beta _{4}-\beta _{6}+\beta _{7})q^{5}+\cdots\)
688.6.a.f 688.a 1.a $8$ $110.344$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(26\) \(-40\) \(60\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(3+\beta _{1})q^{3}+(-5+\beta _{7})q^{5}+(8-\beta _{1}+\cdots)q^{7}+\cdots\)
688.6.a.g 688.a 1.a $9$ $110.344$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(0\) \(-28\) \(-40\) \(-136\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-3-\beta _{1})q^{3}+(-4+\beta _{4})q^{5}+(-15+\cdots)q^{7}+\cdots\)
688.6.a.h 688.a 1.a $10$ $110.344$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(0\) \(-28\) \(138\) \(-60\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-3-\beta _{4})q^{3}+(14+\beta _{7})q^{5}+(-6+\cdots)q^{7}+\cdots\)
688.6.a.i 688.a 1.a $11$ $110.344$ \(\mathbb{Q}[x]/(x^{11} - \cdots)\) None \(0\) \(0\) \(24\) \(100\) $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}+(2+\beta _{2})q^{5}+(9+\beta _{1}-\beta _{3}+\cdots)q^{7}+\cdots\)
688.6.a.j 688.a 1.a $13$ $110.344$ \(\mathbb{Q}[x]/(x^{13} - \cdots)\) None \(0\) \(18\) \(-50\) \(232\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta _{1})q^{3}+(-4+\beta _{3})q^{5}+(18-\beta _{1}+\cdots)q^{7}+\cdots\)
688.6.a.k 688.a 1.a $14$ $110.344$ \(\mathbb{Q}[x]/(x^{14} - \cdots)\) None \(0\) \(-27\) \(25\) \(-160\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-2+\beta _{1})q^{3}+(2+\beta _{4})q^{5}+(-11+\cdots)q^{7}+\cdots\)
688.6.a.l 688.a 1.a $15$ $110.344$ \(\mathbb{Q}[x]/(x^{15} - \cdots)\) None \(0\) \(9\) \(99\) \(-96\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{3}+(7-\beta _{4})q^{5}+(-6-\beta _{2}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(688))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(688)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(43))\)\(^{\oplus 5}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(86))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(172))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(344))\)\(^{\oplus 2}\)