Properties

Label 722.6.a
Level $722$
Weight $6$
Character orbit 722.a
Rep. character $\chi_{722}(1,\cdot)$
Character field $\Q$
Dimension $143$
Newform subspaces $20$
Sturm bound $570$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 722 = 2 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 722.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 20 \)
Sturm bound: \(570\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(722))\).

Total New Old
Modular forms 495 143 352
Cusp forms 455 143 312
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.
\(+\)\(+\)\(+\)\(35\)
\(+\)\(-\)\(-\)\(37\)
\(-\)\(+\)\(-\)\(39\)
\(-\)\(-\)\(+\)\(32\)
Plus space\(+\)\(67\)
Minus space\(-\)\(76\)

Trace form

\( 143q - 4q^{2} + 4q^{3} + 2288q^{4} - 44q^{5} - 8q^{6} - 76q^{7} - 64q^{8} + 11929q^{9} + O(q^{10}) \) \( 143q - 4q^{2} + 4q^{3} + 2288q^{4} - 44q^{5} - 8q^{6} - 76q^{7} - 64q^{8} + 11929q^{9} - 200q^{10} - 162q^{11} + 64q^{12} - 1338q^{13} - 80q^{14} - 1868q^{15} + 36608q^{16} + 934q^{17} - 628q^{18} - 704q^{20} + 2940q^{21} + 1424q^{22} + 4068q^{23} - 128q^{24} + 97427q^{25} + 2144q^{26} - 3464q^{27} - 1216q^{28} + 5566q^{29} + 7792q^{30} + 11164q^{31} - 1024q^{32} + 6800q^{33} - 3080q^{34} + 4264q^{35} + 190864q^{36} - 10038q^{37} - 11460q^{39} - 3200q^{40} - 25722q^{41} - 8656q^{42} + 998q^{43} - 2592q^{44} - 19720q^{45} - 688q^{46} - 2400q^{47} + 1024q^{48} + 369203q^{49} + 16260q^{50} - 47816q^{51} - 21408q^{52} - 53370q^{53} + 42208q^{54} - 33836q^{55} - 1280q^{56} + 14640q^{58} + 31028q^{59} - 29888q^{60} - 133476q^{61} + 46192q^{62} + 54796q^{63} + 585728q^{64} + 108904q^{65} + 62304q^{66} + 29516q^{67} + 14944q^{68} + 121196q^{69} - 65232q^{70} - 5892q^{71} - 10048q^{72} + 199254q^{73} + 14432q^{74} + 136620q^{75} - 47832q^{77} + 2800q^{78} - 202780q^{79} - 11264q^{80} + 1283759q^{81} + 71272q^{82} - 134810q^{83} + 47040q^{84} + 176372q^{85} + 83968q^{86} + 232284q^{87} + 22784q^{88} - 202750q^{89} - 85720q^{90} - 481736q^{91} + 65088q^{92} - 225568q^{93} - 61312q^{94} - 2048q^{96} - 16390q^{97} - 243076q^{98} - 260246q^{99} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(722))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 19
722.6.a.a \(1\) \(115.797\) \(\Q\) None \(-4\) \(14\) \(-45\) \(-121\) \(+\) \(-\) \(q-4q^{2}+14q^{3}+2^{4}q^{4}-45q^{5}-56q^{6}+\cdots\)
722.6.a.b \(1\) \(115.797\) \(\Q\) None \(4\) \(6\) \(31\) \(-27\) \(-\) \(-\) \(q+4q^{2}+6q^{3}+2^{4}q^{4}+31q^{5}+24q^{6}+\cdots\)
722.6.a.c \(2\) \(115.797\) \(\Q(\sqrt{1441}) \) None \(8\) \(-3\) \(-45\) \(114\) \(-\) \(-\) \(q+4q^{2}+(-1-\beta )q^{3}+2^{4}q^{4}+(-24+\cdots)q^{5}+\cdots\)
722.6.a.d \(3\) \(115.797\) \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(-12\) \(-13\) \(81\) \(228\) \(+\) \(-\) \(q-4q^{2}+(-4-\beta _{1})q^{3}+2^{4}q^{4}+(3^{3}+\cdots)q^{5}+\cdots\)
722.6.a.e \(3\) \(115.797\) \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(-12\) \(15\) \(-14\) \(-312\) \(+\) \(+\) \(q-4q^{2}+(5-\beta _{1})q^{3}+2^{4}q^{4}+(-5+\cdots)q^{5}+\cdots\)
722.6.a.f \(3\) \(115.797\) \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(12\) \(-15\) \(-14\) \(-312\) \(-\) \(-\) \(q+4q^{2}+(-5+\beta _{1})q^{3}+2^{4}q^{4}+(-5+\cdots)q^{5}+\cdots\)
722.6.a.g \(4\) \(115.797\) \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-16\) \(-14\) \(36\) \(38\) \(+\) \(-\) \(q-4q^{2}+(-3-\beta _{1})q^{3}+2^{4}q^{4}+(9+\cdots)q^{5}+\cdots\)
722.6.a.h \(4\) \(115.797\) \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-16\) \(1\) \(-14\) \(97\) \(+\) \(+\) \(q-4q^{2}+\beta _{1}q^{3}+2^{4}q^{4}+(-4+\beta _{1}+\cdots)q^{5}+\cdots\)
722.6.a.i \(4\) \(115.797\) \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(16\) \(-1\) \(-14\) \(97\) \(-\) \(+\) \(q+4q^{2}-\beta _{1}q^{3}+2^{4}q^{4}+(-4+\beta _{1}+\cdots)q^{5}+\cdots\)
722.6.a.j \(4\) \(115.797\) \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(16\) \(14\) \(36\) \(38\) \(-\) \(+\) \(q+4q^{2}+(3+\beta _{1})q^{3}+2^{4}q^{4}+(9+\beta _{2}+\cdots)q^{5}+\cdots\)
722.6.a.k \(6\) \(115.797\) \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(-24\) \(24\) \(-83\) \(104\) \(+\) \(-\) \(q-4q^{2}+(5-\beta _{1}-\beta _{2})q^{3}+2^{4}q^{4}+\cdots\)
722.6.a.l \(6\) \(115.797\) \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(24\) \(-24\) \(-83\) \(104\) \(-\) \(-\) \(q+4q^{2}+(-5+\beta _{1}+\beta _{2})q^{3}+2^{4}q^{4}+\cdots\)
722.6.a.m \(8\) \(115.797\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-32\) \(22\) \(-108\) \(94\) \(+\) \(-\) \(q-4q^{2}+(3-\beta _{1})q^{3}+2^{4}q^{4}+(-13+\cdots)q^{5}+\cdots\)
722.6.a.n \(8\) \(115.797\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(32\) \(-22\) \(-108\) \(94\) \(-\) \(-\) \(q+4q^{2}+(-3+\beta _{1})q^{3}+2^{4}q^{4}+(-13+\cdots)q^{5}+\cdots\)
722.6.a.o \(12\) \(115.797\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(-48\) \(0\) \(-42\) \(-438\) \(+\) \(+\) \(q-4q^{2}+\beta _{1}q^{3}+2^{4}q^{4}+(-3+\beta _{7}+\cdots)q^{5}+\cdots\)
722.6.a.p \(12\) \(115.797\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(48\) \(0\) \(-42\) \(-438\) \(-\) \(-\) \(q+4q^{2}-\beta _{1}q^{3}+2^{4}q^{4}+(-3+\beta _{7}+\cdots)q^{5}+\cdots\)
722.6.a.q \(15\) \(115.797\) \(\mathbb{Q}[x]/(x^{15} - \cdots)\) None \(-60\) \(0\) \(108\) \(84\) \(+\) \(-\) \(q-4q^{2}+\beta _{1}q^{3}+2^{4}q^{4}+(7+\beta _{5})q^{5}+\cdots\)
722.6.a.r \(15\) \(115.797\) \(\mathbb{Q}[x]/(x^{15} - \cdots)\) None \(60\) \(0\) \(108\) \(84\) \(-\) \(+\) \(q+4q^{2}-\beta _{1}q^{3}+2^{4}q^{4}+(7+\beta _{5})q^{5}+\cdots\)
722.6.a.s \(16\) \(115.797\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(-64\) \(-46\) \(84\) \(198\) \(+\) \(+\) \(q-4q^{2}+(-3+\beta _{1})q^{3}+2^{4}q^{4}+(5+\cdots)q^{5}+\cdots\)
722.6.a.t \(16\) \(115.797\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(64\) \(46\) \(84\) \(198\) \(-\) \(+\) \(q+4q^{2}+(3-\beta _{1})q^{3}+2^{4}q^{4}+(5-\beta _{1}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(722))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(722)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(361))\)\(^{\oplus 2}\)