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Properties

Label 784.2.bg
Level $784$
Weight $2$
Character orbit 784.bg
Rep. character $\chi_{784}(65,\cdot)$
Character field $\Q(\zeta_{21})$
Dimension $324$
Newform subspaces $6$
Sturm bound $224$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 784.bg (of order \(21\) and degree \(12\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 49 \)
Character field: \(\Q(\zeta_{21})\)
Newform subspaces: \( 6 \)
Sturm bound: \(224\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(784, [\chi])\).

Total New Old
Modular forms 1416 348 1068
Cusp forms 1272 324 948
Eisenstein series 144 24 120

Trace form

\( 324 q + 11 q^{3} - 13 q^{5} + 10 q^{7} + 12 q^{9} + O(q^{10}) \) \( 324 q + 11 q^{3} - 13 q^{5} + 10 q^{7} + 12 q^{9} + 13 q^{11} - 10 q^{13} + 16 q^{15} - 13 q^{17} + 44 q^{19} - 11 q^{21} + 19 q^{23} + 6 q^{25} + 32 q^{27} - 30 q^{29} - 22 q^{31} - 23 q^{33} + 35 q^{35} - 13 q^{37} + 26 q^{39} - 2 q^{41} - 6 q^{43} - 24 q^{45} + 17 q^{47} - 20 q^{49} + 9 q^{51} - 17 q^{53} - 8 q^{55} - 3 q^{57} - 13 q^{59} - 17 q^{61} + 32 q^{63} - 12 q^{65} + 2 q^{67} - 72 q^{69} - 30 q^{71} - q^{73} + 144 q^{75} - 31 q^{77} + 20 q^{79} + 41 q^{81} + 54 q^{83} - 20 q^{85} + 40 q^{87} - q^{89} - 46 q^{91} - 6 q^{93} - 16 q^{97} + 12 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(784, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
784.2.bg.a 784.bg 49.g $24$ $6.260$ None \(0\) \(-7\) \(0\) \(0\) $\mathrm{SU}(2)[C_{21}]$
784.2.bg.b 784.bg 49.g $24$ $6.260$ None \(0\) \(7\) \(0\) \(0\) $\mathrm{SU}(2)[C_{21}]$
784.2.bg.c 784.bg 49.g $48$ $6.260$ None \(0\) \(14\) \(-14\) \(14\) $\mathrm{SU}(2)[C_{21}]$
784.2.bg.d 784.bg 49.g $60$ $6.260$ None \(0\) \(-1\) \(3\) \(-4\) $\mathrm{SU}(2)[C_{21}]$
784.2.bg.e 784.bg 49.g $84$ $6.260$ None \(0\) \(-10\) \(-1\) \(4\) $\mathrm{SU}(2)[C_{21}]$
784.2.bg.f 784.bg 49.g $84$ $6.260$ None \(0\) \(8\) \(-1\) \(-4\) $\mathrm{SU}(2)[C_{21}]$

Decomposition of \(S_{2}^{\mathrm{old}}(784, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(784, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(49, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(98, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(196, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(392, [\chi])\)\(^{\oplus 2}\)