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Properties

Label 722.2.c
Level $722$
Weight $2$
Character orbit 722.c
Rep. character $\chi_{722}(429,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $54$
Newform subspaces $14$
Sturm bound $190$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 722 = 2 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 722.c (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 19 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 14 \)
Sturm bound: \(190\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(3\), \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(722, [\chi])\).

Total New Old
Modular forms 230 54 176
Cusp forms 150 54 96
Eisenstein series 80 0 80

Trace form

\( 54 q + q^{2} + q^{3} - 27 q^{4} + 4 q^{5} - q^{6} - 2 q^{8} - 24 q^{9} + O(q^{10}) \) \( 54 q + q^{2} + q^{3} - 27 q^{4} + 4 q^{5} - q^{6} - 2 q^{8} - 24 q^{9} + 2 q^{10} - 2 q^{11} - 2 q^{12} + 6 q^{13} + 6 q^{14} - 14 q^{15} - 27 q^{16} - 4 q^{17} - 20 q^{18} - 8 q^{20} + 10 q^{21} - 7 q^{22} - 2 q^{23} - q^{24} - 27 q^{25} - 4 q^{26} + 10 q^{27} - 2 q^{29} + 28 q^{30} + 8 q^{31} + q^{32} - 11 q^{33} + 6 q^{34} - 8 q^{35} - 24 q^{36} + 8 q^{37} - 16 q^{39} + 2 q^{40} + 19 q^{41} + 24 q^{42} + 18 q^{43} + q^{44} - 36 q^{45} - 16 q^{46} + 22 q^{47} + q^{48} + 18 q^{49} - 22 q^{50} - 6 q^{51} + 6 q^{52} + 2 q^{53} + 11 q^{54} + 22 q^{55} - 12 q^{56} - 4 q^{58} - 9 q^{59} - 14 q^{60} - 12 q^{61} - 8 q^{62} + 32 q^{63} + 54 q^{64} - 8 q^{65} - 13 q^{66} - 11 q^{67} + 8 q^{68} + 40 q^{69} - 12 q^{70} - 22 q^{71} + 10 q^{72} + 27 q^{73} + 22 q^{74} + 66 q^{75} + 52 q^{77} - 2 q^{78} - 12 q^{79} + 4 q^{80} - 31 q^{81} + 5 q^{82} - 26 q^{83} - 20 q^{84} + 12 q^{85} + 16 q^{86} - 56 q^{87} + 14 q^{88} + 6 q^{89} + 8 q^{90} - 4 q^{91} - 2 q^{92} - 28 q^{94} + 2 q^{96} - q^{97} - 7 q^{98} - 12 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(722, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
722.2.c.a 722.c 19.c $2$ $5.765$ \(\Q(\sqrt{-3}) \) None \(-1\) \(-3\) \(-2\) \(-6\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{2}+(-3+3\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)
722.2.c.b 722.c 19.c $2$ $5.765$ \(\Q(\sqrt{-3}) \) None \(-1\) \(1\) \(0\) \(-8\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{2}+(1-\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)
722.2.c.c 722.c 19.c $2$ $5.765$ \(\Q(\sqrt{-3}) \) None \(-1\) \(1\) \(0\) \(-2\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{2}+(1-\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)
722.2.c.d 722.c 19.c $2$ $5.765$ \(\Q(\sqrt{-3}) \) None \(-1\) \(1\) \(4\) \(6\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{2}+(1-\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)
722.2.c.e 722.c 19.c $2$ $5.765$ \(\Q(\sqrt{-3}) \) None \(1\) \(-1\) \(0\) \(-2\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{2}+(-1+\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)
722.2.c.f 722.c 19.c $2$ $5.765$ \(\Q(\sqrt{-3}) \) None \(1\) \(-1\) \(4\) \(6\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{2}+(-1+\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)
722.2.c.g 722.c 19.c $2$ $5.765$ \(\Q(\sqrt{-3}) \) None \(1\) \(3\) \(-2\) \(-6\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{2}+(3-3\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)
722.2.c.h 722.c 19.c $4$ $5.765$ \(\Q(\sqrt{-3}, \sqrt{5})\) None \(-2\) \(2\) \(5\) \(-4\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1-\beta _{3})q^{2}+2\beta _{1}q^{3}+\beta _{3}q^{4}+\cdots\)
722.2.c.i 722.c 19.c $4$ $5.765$ \(\Q(\sqrt{-3}, \sqrt{5})\) None \(2\) \(-2\) \(5\) \(-4\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1+\beta _{3})q^{2}-2\beta _{1}q^{3}+\beta _{3}q^{4}+(3+\cdots)q^{5}+\cdots\)
722.2.c.j 722.c 19.c $4$ $5.765$ \(\Q(\sqrt{-3}, \sqrt{7})\) None \(2\) \(0\) \(-2\) \(4\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1+\beta _{2})q^{2}-\beta _{1}q^{3}+\beta _{2}q^{4}+(-1+\cdots)q^{5}+\cdots\)
722.2.c.k 722.c 19.c $6$ $5.765$ \(\Q(\zeta_{18})\) None \(-3\) \(0\) \(-6\) \(12\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{18})q^{2}+(\zeta_{18}^{2}-\zeta_{18}^{3})q^{3}+\cdots\)
722.2.c.l 722.c 19.c $6$ $5.765$ \(\Q(\zeta_{18})\) None \(3\) \(0\) \(-6\) \(12\) $\mathrm{SU}(2)[C_{3}]$ \(q+\zeta_{18}q^{2}+(-\zeta_{18}^{2}+\zeta_{18}^{3}+\zeta_{18}^{4}+\cdots)q^{3}+\cdots\)
722.2.c.m 722.c 19.c $8$ $5.765$ 8.0.324000000.2 None \(-4\) \(-2\) \(2\) \(-4\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1-\beta _{4})q^{2}+(\beta _{3}+\beta _{6})q^{3}+\beta _{4}q^{4}+\cdots\)
722.2.c.n 722.c 19.c $8$ $5.765$ 8.0.324000000.2 None \(4\) \(2\) \(2\) \(-4\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1+\beta _{4})q^{2}+(-\beta _{3}-\beta _{6})q^{3}+\beta _{4}q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(722, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(722, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(38, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(361, [\chi])\)\(^{\oplus 2}\)