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Properties

Label 726.6.a
Level $726$
Weight $6$
Character orbit 726.a
Rep. character $\chi_{726}(1,\cdot)$
Character field $\Q$
Dimension $91$
Newform subspaces $35$
Sturm bound $792$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 726 = 2 \cdot 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 726.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 35 \)
Sturm bound: \(792\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(726))\).

Total New Old
Modular forms 684 91 593
Cusp forms 636 91 545
Eisenstein series 48 0 48

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

\(2\)\(3\)\(11\)FrickeDim
\(+\)\(+\)\(+\)$+$\(10\)
\(+\)\(+\)\(-\)$-$\(13\)
\(+\)\(-\)\(+\)$-$\(11\)
\(+\)\(-\)\(-\)$+$\(11\)
\(-\)\(+\)\(+\)$-$\(12\)
\(-\)\(+\)\(-\)$+$\(11\)
\(-\)\(-\)\(+\)$+$\(9\)
\(-\)\(-\)\(-\)$-$\(14\)
Plus space\(+\)\(41\)
Minus space\(-\)\(50\)

Trace form

\( 91 q + 4 q^{2} - 9 q^{3} + 1456 q^{4} + 22 q^{5} + 36 q^{6} - 372 q^{7} + 64 q^{8} + 7371 q^{9} + O(q^{10}) \) \( 91 q + 4 q^{2} - 9 q^{3} + 1456 q^{4} + 22 q^{5} + 36 q^{6} - 372 q^{7} + 64 q^{8} + 7371 q^{9} - 360 q^{10} - 144 q^{12} + 258 q^{13} - 704 q^{14} - 594 q^{15} + 23296 q^{16} - 3942 q^{17} + 324 q^{18} - 4272 q^{19} + 352 q^{20} + 180 q^{21} - 3408 q^{23} + 576 q^{24} + 59989 q^{25} - 6920 q^{26} - 729 q^{27} - 5952 q^{28} + 526 q^{29} - 2376 q^{30} + 6028 q^{31} + 1024 q^{32} + 2696 q^{34} + 35184 q^{35} + 117936 q^{36} - 18202 q^{37} - 9872 q^{38} - 5238 q^{39} - 5760 q^{40} + 21890 q^{41} - 720 q^{42} + 24960 q^{43} + 1782 q^{45} + 8736 q^{46} + 26896 q^{47} - 2304 q^{48} + 250199 q^{49} + 10620 q^{50} + 9162 q^{51} + 4128 q^{52} + 77206 q^{53} + 2916 q^{54} - 11264 q^{56} + 27936 q^{57} + 7176 q^{58} + 1660 q^{59} - 9504 q^{60} + 35322 q^{61} + 43424 q^{62} - 30132 q^{63} + 372736 q^{64} - 20908 q^{65} + 44604 q^{67} - 63072 q^{68} - 56520 q^{69} + 15712 q^{70} + 46456 q^{71} + 5184 q^{72} - 151410 q^{73} + 153272 q^{74} - 126567 q^{75} - 68352 q^{76} + 23688 q^{78} - 51180 q^{79} + 5632 q^{80} + 597051 q^{81} - 230424 q^{82} - 34732 q^{83} + 2880 q^{84} - 96228 q^{85} + 8720 q^{86} - 16722 q^{87} - 248926 q^{89} - 29160 q^{90} - 836216 q^{91} - 54528 q^{92} - 403056 q^{93} + 295968 q^{94} - 253384 q^{95} + 9216 q^{96} - 520986 q^{97} + 64932 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(726))\) 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 3 11
726.6.a.a 726.a 1.a $1$ $116.439$ \(\Q\) None \(-4\) \(-9\) \(-66\) \(-176\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}-9q^{3}+2^{4}q^{4}-66q^{5}+6^{2}q^{6}+\cdots\)
726.6.a.b 726.a 1.a $1$ $116.439$ \(\Q\) None \(-4\) \(-9\) \(-36\) \(108\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}-9q^{3}+2^{4}q^{4}-6^{2}q^{5}+6^{2}q^{6}+\cdots\)
726.6.a.c 726.a 1.a $1$ $116.439$ \(\Q\) None \(-4\) \(-9\) \(50\) \(-2\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}-9q^{3}+2^{4}q^{4}+50q^{5}+6^{2}q^{6}+\cdots\)
726.6.a.d 726.a 1.a $1$ $116.439$ \(\Q\) None \(-4\) \(-9\) \(66\) \(-198\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}-9q^{3}+2^{4}q^{4}+66q^{5}+6^{2}q^{6}+\cdots\)
726.6.a.e 726.a 1.a $1$ $116.439$ \(\Q\) None \(-4\) \(9\) \(30\) \(90\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}+9q^{3}+2^{4}q^{4}+30q^{5}-6^{2}q^{6}+\cdots\)
726.6.a.f 726.a 1.a $1$ $116.439$ \(\Q\) None \(4\) \(-9\) \(-36\) \(-108\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}-9q^{3}+2^{4}q^{4}-6^{2}q^{5}-6^{2}q^{6}+\cdots\)
726.6.a.g 726.a 1.a $1$ $116.439$ \(\Q\) None \(4\) \(-9\) \(-14\) \(-130\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}-9q^{3}+2^{4}q^{4}-14q^{5}-6^{2}q^{6}+\cdots\)
726.6.a.h 726.a 1.a $1$ $116.439$ \(\Q\) None \(4\) \(-9\) \(-14\) \(112\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}-9q^{3}+2^{4}q^{4}-14q^{5}-6^{2}q^{6}+\cdots\)
726.6.a.i 726.a 1.a $1$ $116.439$ \(\Q\) None \(4\) \(-9\) \(66\) \(198\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}-9q^{3}+2^{4}q^{4}+66q^{5}-6^{2}q^{6}+\cdots\)
726.6.a.j 726.a 1.a $1$ $116.439$ \(\Q\) None \(4\) \(9\) \(-14\) \(130\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+9q^{3}+2^{4}q^{4}-14q^{5}+6^{2}q^{6}+\cdots\)
726.6.a.k 726.a 1.a $1$ $116.439$ \(\Q\) None \(4\) \(9\) \(30\) \(-90\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}+9q^{3}+2^{4}q^{4}+30q^{5}+6^{2}q^{6}+\cdots\)
726.6.a.l 726.a 1.a $2$ $116.439$ \(\Q(\sqrt{10129}) \) None \(-8\) \(-18\) \(-25\) \(-79\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}-9q^{3}+2^{4}q^{4}+(-12-\beta )q^{5}+\cdots\)
726.6.a.m 726.a 1.a $2$ $116.439$ \(\Q(\sqrt{3}) \) None \(-8\) \(18\) \(-48\) \(-120\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}+9q^{3}+2^{4}q^{4}+(-24+20\beta )q^{5}+\cdots\)
726.6.a.n 726.a 1.a $2$ $116.439$ \(\Q(\sqrt{18049}) \) None \(-8\) \(18\) \(-25\) \(-21\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+9q^{3}+2^{4}q^{4}+(-12-\beta )q^{5}+\cdots\)
726.6.a.o 726.a 1.a $2$ $116.439$ \(\Q(\sqrt{3}) \) None \(-8\) \(18\) \(18\) \(300\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}+9q^{3}+2^{4}q^{4}+(9+40\beta )q^{5}+\cdots\)
726.6.a.p 726.a 1.a $2$ $116.439$ \(\Q(\sqrt{2161}) \) None \(-8\) \(18\) \(50\) \(-96\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+9q^{3}+2^{4}q^{4}+(5^{2}-\beta )q^{5}+\cdots\)
726.6.a.q 726.a 1.a $2$ $116.439$ \(\Q(\sqrt{10129}) \) None \(8\) \(-18\) \(-25\) \(79\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}-9q^{3}+2^{4}q^{4}+(-12-\beta )q^{5}+\cdots\)
726.6.a.r 726.a 1.a $2$ $116.439$ \(\Q(\sqrt{3}) \) None \(8\) \(18\) \(-48\) \(120\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}+9q^{3}+2^{4}q^{4}+(-24+20\beta )q^{5}+\cdots\)
726.6.a.s 726.a 1.a $2$ $116.439$ \(\Q(\sqrt{18049}) \) None \(8\) \(18\) \(-25\) \(21\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+9q^{3}+2^{4}q^{4}+(-12-\beta )q^{5}+\cdots\)
726.6.a.t 726.a 1.a $2$ $116.439$ \(\Q(\sqrt{8761}) \) None \(8\) \(18\) \(-14\) \(-210\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+9q^{3}+2^{4}q^{4}+(-7-\beta )q^{5}+\cdots\)
726.6.a.u 726.a 1.a $2$ $116.439$ \(\Q(\sqrt{3}) \) None \(8\) \(18\) \(18\) \(-300\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}+9q^{3}+2^{4}q^{4}+(9+40\beta )q^{5}+\cdots\)
726.6.a.v 726.a 1.a $3$ $116.439$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(-12\) \(-27\) \(14\) \(95\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}-9q^{3}+2^{4}q^{4}+(5+\beta _{2})q^{5}+\cdots\)
726.6.a.w 726.a 1.a $3$ $116.439$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(-12\) \(27\) \(14\) \(5\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+9q^{3}+2^{4}q^{4}+(5-\beta _{1})q^{5}+\cdots\)
726.6.a.x 726.a 1.a $3$ $116.439$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(12\) \(-27\) \(14\) \(-95\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}-9q^{3}+2^{4}q^{4}+(5+\beta _{2})q^{5}+\cdots\)
726.6.a.y 726.a 1.a $3$ $116.439$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(12\) \(27\) \(14\) \(-5\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+9q^{3}+2^{4}q^{4}+(5-\beta _{1})q^{5}+\cdots\)
726.6.a.z 726.a 1.a $4$ $116.439$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-16\) \(-36\) \(-30\) \(-180\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}-9q^{3}+2^{4}q^{4}+(-7+\beta _{1}+\cdots)q^{5}+\cdots\)
726.6.a.ba 726.a 1.a $4$ $116.439$ 4.4.264400.1 None \(-16\) \(-36\) \(14\) \(216\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}-9q^{3}+2^{4}q^{4}+(1-5\beta _{2})q^{5}+\cdots\)
726.6.a.bb 726.a 1.a $4$ $116.439$ 4.4.8052400.2 None \(-16\) \(36\) \(-150\) \(68\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+9q^{3}+2^{4}q^{4}+(-38+\beta _{1}+\cdots)q^{5}+\cdots\)
726.6.a.bc 726.a 1.a $4$ $116.439$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(16\) \(-36\) \(-30\) \(180\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}-9q^{3}+2^{4}q^{4}+(-7+\beta _{1}+\cdots)q^{5}+\cdots\)
726.6.a.bd 726.a 1.a $4$ $116.439$ 4.4.264400.1 None \(16\) \(-36\) \(14\) \(-216\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}-9q^{3}+2^{4}q^{4}+(1-5\beta _{2})q^{5}+\cdots\)
726.6.a.be 726.a 1.a $4$ $116.439$ 4.4.8052400.2 None \(16\) \(36\) \(-150\) \(-68\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}+9q^{3}+2^{4}q^{4}+(-38+\beta _{1}+\cdots)q^{5}+\cdots\)
726.6.a.bf 726.a 1.a $6$ $116.439$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(-24\) \(-54\) \(41\) \(157\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}-9q^{3}+2^{4}q^{4}+(6+\beta _{1}-\beta _{3}+\cdots)q^{5}+\cdots\)
726.6.a.bg 726.a 1.a $6$ $116.439$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(-24\) \(54\) \(139\) \(-265\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}+9q^{3}+2^{4}q^{4}+(23+\beta _{1})q^{5}+\cdots\)
726.6.a.bh 726.a 1.a $6$ $116.439$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(24\) \(-54\) \(41\) \(-157\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}-9q^{3}+2^{4}q^{4}+(6+\beta _{1}-\beta _{3}+\cdots)q^{5}+\cdots\)
726.6.a.bi 726.a 1.a $6$ $116.439$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(24\) \(54\) \(139\) \(265\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+9q^{3}+2^{4}q^{4}+(23+\beta _{1})q^{5}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(726))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(726)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(22))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(66))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(121))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(242))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(363))\)\(^{\oplus 2}\)