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Properties

Label 722.4.a
Level $722$
Weight $4$
Character orbit 722.a
Rep. character $\chi_{722}(1,\cdot)$
Character field $\Q$
Dimension $85$
Newform subspaces $21$
Sturm bound $380$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 722 = 2 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 722.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 21 \)
Sturm bound: \(380\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(722))\).

Total New Old
Modular forms 305 85 220
Cusp forms 265 85 180
Eisenstein series 40 0 40

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)\(19\)FrickeDim
\(+\)\(+\)$+$\(22\)
\(+\)\(-\)$-$\(20\)
\(-\)\(+\)$-$\(18\)
\(-\)\(-\)$+$\(25\)
Plus space\(+\)\(47\)
Minus space\(-\)\(38\)

Trace form

\( 85 q + 2 q^{2} - 8 q^{3} + 340 q^{4} + 16 q^{5} - 20 q^{6} - 12 q^{7} + 8 q^{8} + 775 q^{9} + O(q^{10}) \) \( 85 q + 2 q^{2} - 8 q^{3} + 340 q^{4} + 16 q^{5} - 20 q^{6} - 12 q^{7} + 8 q^{8} + 775 q^{9} + 20 q^{10} - 78 q^{11} - 32 q^{12} + 22 q^{13} + 88 q^{14} + 304 q^{15} + 1360 q^{16} - 2 q^{17} - 22 q^{18} + 64 q^{20} + 48 q^{21} + 168 q^{22} + 168 q^{23} - 80 q^{24} + 1909 q^{25} - 112 q^{26} - 368 q^{27} - 48 q^{28} + 226 q^{29} - 8 q^{30} - 228 q^{31} + 32 q^{32} - 28 q^{33} + 196 q^{34} - 1136 q^{35} + 3100 q^{36} - 346 q^{37} + 168 q^{39} + 80 q^{40} + 1098 q^{41} + 968 q^{42} - 486 q^{43} - 312 q^{44} + 1244 q^{45} - 600 q^{46} - 168 q^{47} - 128 q^{48} + 2945 q^{49} - 18 q^{50} + 1924 q^{51} + 88 q^{52} + 1254 q^{53} - 80 q^{54} + 732 q^{55} + 352 q^{56} - 248 q^{58} + 416 q^{59} + 1216 q^{60} + 232 q^{61} - 392 q^{62} - 680 q^{63} + 5440 q^{64} - 272 q^{65} + 336 q^{66} + 572 q^{67} - 8 q^{68} - 2020 q^{69} - 264 q^{70} - 1248 q^{71} - 88 q^{72} - 458 q^{73} - 496 q^{74} + 84 q^{75} + 1020 q^{77} - 2120 q^{78} - 756 q^{79} + 256 q^{80} + 8069 q^{81} - 2068 q^{82} - 1298 q^{83} + 192 q^{84} - 3100 q^{85} - 896 q^{86} - 2232 q^{87} + 672 q^{88} + 1706 q^{89} + 2588 q^{90} + 688 q^{91} + 672 q^{92} + 2288 q^{93} + 1536 q^{94} - 320 q^{96} - 854 q^{97} + 3170 q^{98} - 434 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(722))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 19
722.4.a.a 722.a 1.a $1$ $42.599$ \(\Q\) None \(-2\) \(-5\) \(3\) \(-32\) $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}-5q^{3}+4q^{4}+3q^{5}+10q^{6}+\cdots\)
722.4.a.b 722.a 1.a $1$ $42.599$ \(\Q\) None \(-2\) \(5\) \(-12\) \(8\) $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+5q^{3}+4q^{4}-12q^{5}-10q^{6}+\cdots\)
722.4.a.c 722.a 1.a $1$ $42.599$ \(\Q\) None \(2\) \(-5\) \(-12\) \(8\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}-5q^{3}+4q^{4}-12q^{5}-10q^{6}+\cdots\)
722.4.a.d 722.a 1.a $1$ $42.599$ \(\Q\) None \(2\) \(2\) \(-9\) \(-31\) $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+2q^{3}+4q^{4}-9q^{5}+4q^{6}+\cdots\)
722.4.a.e 722.a 1.a $1$ $42.599$ \(\Q\) None \(2\) \(5\) \(3\) \(-32\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+5q^{3}+4q^{4}+3q^{5}+10q^{6}+\cdots\)
722.4.a.f 722.a 1.a $2$ $42.599$ \(\Q(\sqrt{73}) \) None \(-4\) \(-9\) \(-9\) \(-18\) $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+(-4-\beta )q^{3}+4q^{4}+(-3+\cdots)q^{5}+\cdots\)
722.4.a.g 722.a 1.a $2$ $42.599$ \(\Q(\sqrt{73}) \) None \(-4\) \(5\) \(13\) \(4\) $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+(3-\beta )q^{3}+4q^{4}+(8-3\beta )q^{5}+\cdots\)
722.4.a.h 722.a 1.a $2$ $42.599$ \(\Q(\sqrt{73}) \) None \(4\) \(-5\) \(13\) \(4\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+(-2-\beta )q^{3}+4q^{4}+(5+3\beta )q^{5}+\cdots\)
722.4.a.i 722.a 1.a $2$ $42.599$ \(\Q(\sqrt{177}) \) None \(4\) \(-1\) \(10\) \(57\) $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}-\beta q^{3}+4q^{4}+(6-2\beta )q^{5}+\cdots\)
722.4.a.j 722.a 1.a $3$ $42.599$ 3.3.253788.1 None \(-6\) \(5\) \(1\) \(26\) $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+(2-\beta _{1})q^{3}+4q^{4}+(1-\beta _{1}+\cdots)q^{5}+\cdots\)
722.4.a.k 722.a 1.a $3$ $42.599$ 3.3.253788.1 None \(6\) \(-5\) \(1\) \(26\) $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+(-2+\beta _{1})q^{3}+4q^{4}+(1-\beta _{1}+\cdots)q^{5}+\cdots\)
722.4.a.l 722.a 1.a $4$ $42.599$ 4.4.322225.1 None \(-8\) \(-3\) \(27\) \(4\) $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+(-1+\beta _{1})q^{3}+4q^{4}+(8+2\beta _{2}+\cdots)q^{5}+\cdots\)
722.4.a.m 722.a 1.a $4$ $42.599$ 4.4.322225.1 None \(8\) \(3\) \(27\) \(4\) $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+(1-\beta _{1})q^{3}+4q^{4}+(8+2\beta _{2}+\cdots)q^{5}+\cdots\)
722.4.a.n 722.a 1.a $6$ $42.599$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(-12\) \(-7\) \(22\) \(-18\) $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+(-2-\beta _{1}-\beta _{2})q^{3}+4q^{4}+\cdots\)
722.4.a.o 722.a 1.a $6$ $42.599$ 6.6.6719782761.1 None \(-12\) \(9\) \(-27\) \(-21\) $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+(1+\beta _{1})q^{3}+4q^{4}+(-4-\beta _{3}+\cdots)q^{5}+\cdots\)
722.4.a.p 722.a 1.a $6$ $42.599$ 6.6.6719782761.1 None \(12\) \(-9\) \(-27\) \(-21\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+(-1+\beta _{3})q^{3}+4q^{4}+(-5+\cdots)q^{5}+\cdots\)
722.4.a.q 722.a 1.a $6$ $42.599$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(12\) \(7\) \(22\) \(-18\) $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+(2+\beta _{1}+\beta _{2})q^{3}+4q^{4}+(4+\cdots)q^{5}+\cdots\)
722.4.a.r 722.a 1.a $8$ $42.599$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-16\) \(10\) \(-18\) \(-14\) $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+(1-\beta _{3})q^{3}+4q^{4}+(-1+\beta _{2}+\cdots)q^{5}+\cdots\)
722.4.a.s 722.a 1.a $8$ $42.599$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(16\) \(-10\) \(-18\) \(-14\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+(-1+\beta _{3})q^{3}+4q^{4}+(-1+\cdots)q^{5}+\cdots\)
722.4.a.t 722.a 1.a $9$ $42.599$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(-18\) \(-9\) \(3\) \(33\) $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+(-1+\beta _{3})q^{3}+4q^{4}-\beta _{6}q^{5}+\cdots\)
722.4.a.u 722.a 1.a $9$ $42.599$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(18\) \(9\) \(3\) \(33\) $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+(1-\beta _{3})q^{3}+4q^{4}-\beta _{6}q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(722))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(722)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(361))\)\(^{\oplus 2}\)