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Properties

Label 722.2.e
Level $722$
Weight $2$
Character orbit 722.e
Rep. character $\chi_{722}(99,\cdot)$
Character field $\Q(\zeta_{9})$
Dimension $174$
Newform subspaces $19$
Sturm bound $190$
Trace bound $13$

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Defining parameters

Level: \( N \) \(=\) \( 722 = 2 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 722.e (of order \(9\) and degree \(6\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 19 \)
Character field: \(\Q(\zeta_{9})\)
Newform subspaces: \( 19 \)
Sturm bound: \(190\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(3\), \(5\), \(7\), \(13\)

Dimensions

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

Total New Old
Modular forms 690 174 516
Cusp forms 450 174 276
Eisenstein series 240 0 240

Trace form

\( 174 q + 3 q^{3} + 3 q^{6} + 12 q^{7} + 3 q^{8} + 3 q^{9} + O(q^{10}) \) \( 174 q + 3 q^{3} + 3 q^{6} + 12 q^{7} + 3 q^{8} + 3 q^{9} + 12 q^{11} - 12 q^{13} - 12 q^{14} + 6 q^{15} + 12 q^{17} + 6 q^{18} - 24 q^{20} - 24 q^{21} + 12 q^{23} + 3 q^{24} - 6 q^{26} - 3 q^{27} + 6 q^{28} + 18 q^{29} - 6 q^{31} - 3 q^{33} + 12 q^{34} + 12 q^{35} + 3 q^{36} + 12 q^{37} - 24 q^{39} - 3 q^{41} + 12 q^{42} + 6 q^{43} + 24 q^{45} - 30 q^{47} - 6 q^{48} - 45 q^{49} - 3 q^{50} - 21 q^{51} + 6 q^{52} - 24 q^{53} - 9 q^{54} - 18 q^{55} - 12 q^{56} - 48 q^{58} + 3 q^{59} + 6 q^{60} - 6 q^{61} - 18 q^{62} - 12 q^{63} - 87 q^{64} - 12 q^{65} - 3 q^{66} + 9 q^{67} + 9 q^{68} + 6 q^{69} + 12 q^{70} + 18 q^{71} - 6 q^{72} + 30 q^{73} + 18 q^{74} - 12 q^{77} + 18 q^{78} - 6 q^{79} + 33 q^{81} - 3 q^{82} + 36 q^{83} - 6 q^{84} + 24 q^{85} - 12 q^{86} + 24 q^{87} + 6 q^{88} - 12 q^{90} + 12 q^{91} - 6 q^{92} - 6 q^{93} + 12 q^{94} - 3 q^{97} - 21 q^{99} + O(q^{100}) \)

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.e.a 722.e 19.e $6$ $5.765$ \(\Q(\zeta_{18})\) None \(0\) \(-6\) \(0\) \(-6\) $\mathrm{SU}(2)[C_{9}]$ \(q+(-\zeta_{18}+\zeta_{18}^{4})q^{2}+(-1-\zeta_{18}^{2}+\cdots)q^{3}+\cdots\)
722.2.e.b 722.e 19.e $6$ $5.765$ \(\Q(\zeta_{18})\) None \(0\) \(-3\) \(0\) \(-6\) $\mathrm{SU}(2)[C_{9}]$ \(q+(\zeta_{18}-\zeta_{18}^{4})q^{2}+(-1+\zeta_{18}^{3}-\zeta_{18}^{5})q^{3}+\cdots\)
722.2.e.c 722.e 19.e $6$ $5.765$ \(\Q(\zeta_{18})\) None \(0\) \(0\) \(0\) \(-9\) $\mathrm{SU}(2)[C_{9}]$ \(q+(\zeta_{18}-\zeta_{18}^{4})q^{2}+\zeta_{18}q^{3}-\zeta_{18}^{5}q^{4}+\cdots\)
722.2.e.d 722.e 19.e $6$ $5.765$ \(\Q(\zeta_{18})\) None \(0\) \(0\) \(0\) \(-9\) $\mathrm{SU}(2)[C_{9}]$ \(q+(-\zeta_{18}+\zeta_{18}^{4})q^{2}-\zeta_{18}q^{3}-\zeta_{18}^{5}q^{4}+\cdots\)
722.2.e.e 722.e 19.e $6$ $5.765$ \(\Q(\zeta_{18})\) None \(0\) \(0\) \(0\) \(3\) $\mathrm{SU}(2)[C_{9}]$ \(q+(\zeta_{18}-\zeta_{18}^{4})q^{2}+\zeta_{18}q^{3}-\zeta_{18}^{5}q^{4}+\cdots\)
722.2.e.f 722.e 19.e $6$ $5.765$ \(\Q(\zeta_{18})\) None \(0\) \(0\) \(0\) \(3\) $\mathrm{SU}(2)[C_{9}]$ \(q+(-\zeta_{18}+\zeta_{18}^{4})q^{2}-\zeta_{18}q^{3}-\zeta_{18}^{5}q^{4}+\cdots\)
722.2.e.g 722.e 19.e $6$ $5.765$ \(\Q(\zeta_{18})\) None \(0\) \(0\) \(0\) \(9\) $\mathrm{SU}(2)[C_{9}]$ \(q+(\zeta_{18}-\zeta_{18}^{4})q^{2}-3\zeta_{18}q^{3}-\zeta_{18}^{5}q^{4}+\cdots\)
722.2.e.h 722.e 19.e $6$ $5.765$ \(\Q(\zeta_{18})\) None \(0\) \(0\) \(0\) \(9\) $\mathrm{SU}(2)[C_{9}]$ \(q+(-\zeta_{18}+\zeta_{18}^{4})q^{2}+3\zeta_{18}q^{3}-\zeta_{18}^{5}q^{4}+\cdots\)
722.2.e.i 722.e 19.e $6$ $5.765$ \(\Q(\zeta_{18})\) None \(0\) \(0\) \(0\) \(12\) $\mathrm{SU}(2)[C_{9}]$ \(q+(\zeta_{18}-\zeta_{18}^{4})q^{2}+\zeta_{18}q^{3}-\zeta_{18}^{5}q^{4}+\cdots\)
722.2.e.j 722.e 19.e $6$ $5.765$ \(\Q(\zeta_{18})\) None \(0\) \(0\) \(0\) \(12\) $\mathrm{SU}(2)[C_{9}]$ \(q+(-\zeta_{18}+\zeta_{18}^{4})q^{2}-\zeta_{18}q^{3}-\zeta_{18}^{5}q^{4}+\cdots\)
722.2.e.k 722.e 19.e $6$ $5.765$ \(\Q(\zeta_{18})\) None \(0\) \(3\) \(0\) \(-6\) $\mathrm{SU}(2)[C_{9}]$ \(q+(-\zeta_{18}+\zeta_{18}^{4})q^{2}+(\zeta_{18}^{2}+\zeta_{18}^{3}+\cdots)q^{3}+\cdots\)
722.2.e.l 722.e 19.e $6$ $5.765$ \(\Q(\zeta_{18})\) None \(0\) \(3\) \(0\) \(-6\) $\mathrm{SU}(2)[C_{9}]$ \(q+(-\zeta_{18}+\zeta_{18}^{4})q^{2}+(1-\zeta_{18}^{3}+\zeta_{18}^{5})q^{3}+\cdots\)
722.2.e.m 722.e 19.e $6$ $5.765$ \(\Q(\zeta_{18})\) None \(0\) \(6\) \(0\) \(-6\) $\mathrm{SU}(2)[C_{9}]$ \(q+(\zeta_{18}-\zeta_{18}^{4})q^{2}+(1+\zeta_{18}^{2})q^{3}+\cdots\)
722.2.e.n 722.e 19.e $12$ $5.765$ 12.0.\(\cdots\).3 None \(0\) \(0\) \(0\) \(-6\) $\mathrm{SU}(2)[C_{9}]$ \(q+\beta _{10}q^{2}+(-\beta _{1}-\beta _{7})q^{3}+\beta _{2}q^{4}+\cdots\)
722.2.e.o 722.e 19.e $12$ $5.765$ 12.0.\(\cdots\).3 None \(0\) \(0\) \(0\) \(-6\) $\mathrm{SU}(2)[C_{9}]$ \(q-\beta _{10}q^{2}+(-\beta _{1}-\beta _{7})q^{3}+\beta _{2}q^{4}+\cdots\)
722.2.e.p 722.e 19.e $12$ $5.765$ 12.0.\(\cdots\).1 None \(0\) \(0\) \(0\) \(6\) $\mathrm{SU}(2)[C_{9}]$ \(q-\beta _{9}q^{2}-2\beta _{7}q^{3}-\beta _{11}q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
722.2.e.q 722.e 19.e $12$ $5.765$ 12.0.\(\cdots\).1 None \(0\) \(0\) \(0\) \(6\) $\mathrm{SU}(2)[C_{9}]$ \(q+(\beta _{3}-\beta _{9})q^{2}-2\beta _{1}q^{3}+(-\beta _{5}+\beta _{11})q^{4}+\cdots\)
722.2.e.r 722.e 19.e $24$ $5.765$ None \(0\) \(0\) \(0\) \(6\) $\mathrm{SU}(2)[C_{9}]$
722.2.e.s 722.e 19.e $24$ $5.765$ None \(0\) \(0\) \(0\) \(6\) $\mathrm{SU}(2)[C_{9}]$

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

\( S_{2}^{\mathrm{old}}(722, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(19, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(38, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(361, [\chi])\)\(^{\oplus 2}\)