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Properties

Label 688.2.bg
Level $688$
Weight $2$
Character orbit 688.bg
Rep. character $\chi_{688}(17,\cdot)$
Character field $\Q(\zeta_{21})$
Dimension $252$
Newform subspaces $6$
Sturm bound $176$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 688 = 2^{4} \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 688.bg (of order \(21\) and degree \(12\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 43 \)
Character field: \(\Q(\zeta_{21})\)
Newform subspaces: \( 6 \)
Sturm bound: \(176\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(688, [\chi])\).

Total New Old
Modular forms 1128 276 852
Cusp forms 984 252 732
Eisenstein series 144 24 120

Trace form

\( 252 q + 11 q^{3} - 13 q^{5} - 4 q^{7} + 10 q^{9} + O(q^{10}) \) \( 252 q + 11 q^{3} - 13 q^{5} - 4 q^{7} + 10 q^{9} + 2 q^{11} - 13 q^{13} + 19 q^{15} - 15 q^{17} + 3 q^{19} + 13 q^{21} + 17 q^{23} + 4 q^{25} + 32 q^{27} - 13 q^{29} + 77 q^{31} - 24 q^{33} - 27 q^{35} - 14 q^{37} + 10 q^{39} - 14 q^{41} + 14 q^{43} - 18 q^{45} + 22 q^{47} - 92 q^{49} + 40 q^{51} + 7 q^{53} - 66 q^{55} + q^{57} + 38 q^{59} - 13 q^{61} - 7 q^{63} - 20 q^{65} + 65 q^{67} - 23 q^{69} + 17 q^{71} - 15 q^{73} + 50 q^{75} + 35 q^{77} + 4 q^{79} + q^{81} + 61 q^{83} - 34 q^{85} - 50 q^{87} - 3 q^{89} - 62 q^{91} - 12 q^{93} + 119 q^{95} + 26 q^{97} + 30 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(688, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
688.2.bg.a 688.bg 43.g $12$ $5.494$ \(\Q(\zeta_{21})\) None \(0\) \(-8\) \(3\) \(1\) $\mathrm{SU}(2)[C_{21}]$ \(q+(-1-\zeta_{21}^{7}+\zeta_{21}^{9})q^{3}+(1-\zeta_{21}+\cdots)q^{5}+\cdots\)
688.2.bg.b 688.bg 43.g $24$ $5.494$ None \(0\) \(6\) \(3\) \(-3\) $\mathrm{SU}(2)[C_{21}]$
688.2.bg.c 688.bg 43.g $36$ $5.494$ None \(0\) \(16\) \(-17\) \(-6\) $\mathrm{SU}(2)[C_{21}]$
688.2.bg.d 688.bg 43.g $48$ $5.494$ None \(0\) \(-3\) \(0\) \(2\) $\mathrm{SU}(2)[C_{21}]$
688.2.bg.e 688.bg 43.g $60$ $5.494$ None \(0\) \(9\) \(-1\) \(-1\) $\mathrm{SU}(2)[C_{21}]$
688.2.bg.f 688.bg 43.g $72$ $5.494$ None \(0\) \(-9\) \(-1\) \(3\) $\mathrm{SU}(2)[C_{21}]$

Decomposition of \(S_{2}^{\mathrm{old}}(688, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(688, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(43, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(86, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(172, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(344, [\chi])\)\(^{\oplus 2}\)