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Properties

Label 784.2.i
Level $784$
Weight $2$
Character orbit 784.i
Rep. character $\chi_{784}(177,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $36$
Newform subspaces $14$
Sturm bound $224$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 784.i (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 14 \)
Sturm bound: \(224\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(3\), \(5\), \(11\)

Dimensions

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

Total New Old
Modular forms 272 44 228
Cusp forms 176 36 140
Eisenstein series 96 8 88

Trace form

\( 36 q - 3 q^{3} + q^{5} - 13 q^{9} + O(q^{10}) \) \( 36 q - 3 q^{3} + q^{5} - 13 q^{9} + 3 q^{11} + 4 q^{13} + 14 q^{15} + q^{17} - 5 q^{19} + 9 q^{23} - 13 q^{25} + 18 q^{27} + 16 q^{29} + 13 q^{31} - 9 q^{33} - 7 q^{37} - 26 q^{39} + 12 q^{41} - 16 q^{43} - 10 q^{45} + 3 q^{47} - 47 q^{51} - 11 q^{53} - 22 q^{55} - 54 q^{57} - 27 q^{59} - 3 q^{61} + 18 q^{65} + 15 q^{67} - 2 q^{69} + 64 q^{71} + 13 q^{73} + 4 q^{75} + 33 q^{79} + 22 q^{81} + 40 q^{83} - 50 q^{85} + 26 q^{87} + 13 q^{89} + 21 q^{93} - 33 q^{95} + 12 q^{97} + 4 q^{99} + O(q^{100}) \)

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.i.a 784.i 7.c $2$ $6.260$ \(\Q(\sqrt{-3}) \) None \(0\) \(-3\) \(-1\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-3+3\zeta_{6})q^{3}-\zeta_{6}q^{5}-6\zeta_{6}q^{9}+\cdots\)
784.2.i.b 784.i 7.c $2$ $6.260$ \(\Q(\sqrt{-3}) \) None \(0\) \(-2\) \(-4\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-2+2\zeta_{6})q^{3}-4\zeta_{6}q^{5}-\zeta_{6}q^{9}+\cdots\)
784.2.i.c 784.i 7.c $2$ $6.260$ \(\Q(\sqrt{-3}) \) None \(0\) \(-2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-2+2\zeta_{6})q^{3}-\zeta_{6}q^{9}-4q^{13}+\cdots\)
784.2.i.d 784.i 7.c $2$ $6.260$ \(\Q(\sqrt{-3}) \) None \(0\) \(-1\) \(3\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{3}+3\zeta_{6}q^{5}+2\zeta_{6}q^{9}+\cdots\)
784.2.i.e 784.i 7.c $2$ $6.260$ \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-2\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-2\zeta_{6}q^{5}+3\zeta_{6}q^{9}+(-4+4\zeta_{6})q^{11}+\cdots\)
784.2.i.f 784.i 7.c $2$ $6.260$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-7}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{3}]$ \(q+3\zeta_{6}q^{9}+(4-4\zeta_{6})q^{11}+8\zeta_{6}q^{23}+\cdots\)
784.2.i.g 784.i 7.c $2$ $6.260$ \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(2\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+2\zeta_{6}q^{5}+3\zeta_{6}q^{9}+(-4+4\zeta_{6})q^{11}+\cdots\)
784.2.i.h 784.i 7.c $2$ $6.260$ \(\Q(\sqrt{-3}) \) None \(0\) \(1\) \(-1\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{3}-\zeta_{6}q^{5}+2\zeta_{6}q^{9}+(3+\cdots)q^{11}+\cdots\)
784.2.i.i 784.i 7.c $2$ $6.260$ \(\Q(\sqrt{-3}) \) None \(0\) \(2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(2-2\zeta_{6})q^{3}-\zeta_{6}q^{9}+4q^{13}+(6+\cdots)q^{17}+\cdots\)
784.2.i.j 784.i 7.c $2$ $6.260$ \(\Q(\sqrt{-3}) \) None \(0\) \(2\) \(4\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(2-2\zeta_{6})q^{3}+4\zeta_{6}q^{5}-\zeta_{6}q^{9}+8q^{15}+\cdots\)
784.2.i.k 784.i 7.c $4$ $6.260$ \(\Q(\sqrt{2}, \sqrt{-3})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{1}q^{3}+(\beta _{1}+\beta _{3})q^{5}+5\beta _{2}q^{9}+(-4+\cdots)q^{11}+\cdots\)
784.2.i.l 784.i 7.c $4$ $6.260$ \(\Q(\sqrt{2}, \sqrt{-3})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+2\beta _{1}q^{3}+(-\beta _{1}-\beta _{3})q^{5}+5\beta _{2}q^{9}+\cdots\)
784.2.i.m 784.i 7.c $4$ $6.260$ \(\Q(\sqrt{2}, \sqrt{-3})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{1}q^{3}+(-2\beta _{1}-2\beta _{3})q^{5}-\beta _{2}q^{9}+\cdots\)
784.2.i.n 784.i 7.c $4$ $6.260$ \(\Q(\sqrt{2}, \sqrt{-3})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{1}q^{3}+(2\beta _{1}+2\beta _{3})q^{5}-\beta _{2}q^{9}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(784, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(49, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(56, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(98, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(112, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(196, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(392, [\chi])\)\(^{\oplus 2}\)