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Properties

Label 726.4.a
Level $726$
Weight $4$
Character orbit 726.a
Rep. character $\chi_{726}(1,\cdot)$
Character field $\Q$
Dimension $55$
Newform subspaces $26$
Sturm bound $528$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 726 = 2 \cdot 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 726.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 26 \)
Sturm bound: \(528\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(726))\).

Total New Old
Modular forms 420 55 365
Cusp forms 372 55 317
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
\(+\)\(+\)\(+\)$+$\(8\)
\(+\)\(+\)\(-\)$-$\(6\)
\(+\)\(-\)\(+\)$-$\(7\)
\(+\)\(-\)\(-\)$+$\(7\)
\(-\)\(+\)\(+\)$-$\(6\)
\(-\)\(+\)\(-\)$+$\(8\)
\(-\)\(-\)\(+\)$+$\(9\)
\(-\)\(-\)\(-\)$-$\(4\)
Plus space\(+\)\(32\)
Minus space\(-\)\(23\)

Trace form

\( 55 q - 2 q^{2} - 3 q^{3} + 220 q^{4} - 26 q^{5} - 6 q^{6} - 12 q^{7} - 8 q^{8} + 495 q^{9} + O(q^{10}) \) \( 55 q - 2 q^{2} - 3 q^{3} + 220 q^{4} - 26 q^{5} - 6 q^{6} - 12 q^{7} - 8 q^{8} + 495 q^{9} - 28 q^{10} - 12 q^{12} - 142 q^{13} - 32 q^{14} + 18 q^{15} + 880 q^{16} + 186 q^{17} - 18 q^{18} + 128 q^{19} - 104 q^{20} - 36 q^{21} + 168 q^{23} - 24 q^{24} + 1745 q^{25} + 148 q^{26} - 27 q^{27} - 48 q^{28} + 310 q^{29} - 36 q^{30} + 172 q^{31} - 32 q^{32} - 132 q^{34} + 528 q^{35} + 1980 q^{36} - 466 q^{37} + 664 q^{38} - 138 q^{39} - 112 q^{40} + 2 q^{41} + 264 q^{42} + 112 q^{43} - 234 q^{45} + 656 q^{46} - 320 q^{47} - 48 q^{48} + 1939 q^{49} - 366 q^{50} - 258 q^{51} - 568 q^{52} - 962 q^{53} - 54 q^{54} - 128 q^{56} - 360 q^{57} + 364 q^{58} - 500 q^{59} + 72 q^{60} - 806 q^{61} - 112 q^{62} - 108 q^{63} + 3520 q^{64} - 1756 q^{65} - 804 q^{67} + 744 q^{68} + 1536 q^{69} + 240 q^{70} - 896 q^{71} - 72 q^{72} - 138 q^{73} - 172 q^{74} - 261 q^{75} + 512 q^{76} + 228 q^{78} + 12 q^{79} - 416 q^{80} + 4455 q^{81} + 940 q^{82} + 1244 q^{83} - 144 q^{84} + 1644 q^{85} + 488 q^{86} + 2418 q^{87} - 262 q^{89} - 252 q^{90} + 9968 q^{91} + 672 q^{92} + 4920 q^{93} + 272 q^{94} + 3560 q^{95} - 96 q^{96} + 6726 q^{97} - 2418 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(726))\) 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
726.4.a.a 726.a 1.a $1$ $42.835$ \(\Q\) None \(-2\) \(-3\) \(0\) \(11\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}-3q^{3}+4q^{4}+6q^{6}+11q^{7}+\cdots\)
726.4.a.b 726.a 1.a $1$ $42.835$ \(\Q\) None \(-2\) \(-3\) \(10\) \(-16\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}-3q^{3}+4q^{4}+10q^{5}+6q^{6}+\cdots\)
726.4.a.c 726.a 1.a $1$ $42.835$ \(\Q\) None \(-2\) \(3\) \(-5\) \(16\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+3q^{3}+4q^{4}-5q^{5}-6q^{6}+\cdots\)
726.4.a.d 726.a 1.a $1$ $42.835$ \(\Q\) None \(-2\) \(3\) \(6\) \(-6\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+3q^{3}+4q^{4}+6q^{5}-6q^{6}+\cdots\)
726.4.a.e 726.a 1.a $1$ $42.835$ \(\Q\) None \(2\) \(-3\) \(0\) \(-11\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}-3q^{3}+4q^{4}-6q^{6}-11q^{7}+\cdots\)
726.4.a.f 726.a 1.a $1$ $42.835$ \(\Q\) None \(2\) \(-3\) \(6\) \(16\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}-3q^{3}+4q^{4}+6q^{5}-6q^{6}+\cdots\)
726.4.a.g 726.a 1.a $1$ $42.835$ \(\Q\) None \(2\) \(3\) \(-5\) \(-16\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+3q^{3}+4q^{4}-5q^{5}+6q^{6}+\cdots\)
726.4.a.h 726.a 1.a $1$ $42.835$ \(\Q\) None \(2\) \(3\) \(0\) \(-14\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+3q^{3}+4q^{4}+6q^{6}-14q^{7}+\cdots\)
726.4.a.i 726.a 1.a $1$ $42.835$ \(\Q\) None \(2\) \(3\) \(6\) \(6\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+3q^{3}+4q^{4}+6q^{5}+6q^{6}+\cdots\)
726.4.a.j 726.a 1.a $2$ $42.835$ \(\Q(\sqrt{3}) \) None \(-4\) \(-6\) \(-6\) \(12\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}-3q^{3}+4q^{4}+(-3+4\beta )q^{5}+\cdots\)
726.4.a.k 726.a 1.a $2$ $42.835$ \(\Q(\sqrt{1081}) \) None \(-4\) \(-6\) \(-5\) \(-27\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}-3q^{3}+4q^{4}+(-2-\beta )q^{5}+\cdots\)
726.4.a.l 726.a 1.a $2$ $42.835$ \(\Q(\sqrt{5}) \) None \(-4\) \(-6\) \(-5\) \(28\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}-3q^{3}+4q^{4}+(1-7\beta )q^{5}+\cdots\)
726.4.a.m 726.a 1.a $2$ $42.835$ \(\Q(\sqrt{273}) \) None \(-4\) \(-6\) \(6\) \(6\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}-3q^{3}+4q^{4}+(3+\beta )q^{5}+6q^{6}+\cdots\)
726.4.a.n 726.a 1.a $2$ $42.835$ \(\Q(\sqrt{5}) \) None \(-4\) \(6\) \(-25\) \(8\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+3q^{3}+4q^{4}+(-12-\beta )q^{5}+\cdots\)
726.4.a.o 726.a 1.a $2$ $42.835$ \(\Q(\sqrt{97}) \) None \(-4\) \(6\) \(10\) \(2\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+3q^{3}+4q^{4}+(5-\beta )q^{5}-6q^{6}+\cdots\)
726.4.a.p 726.a 1.a $2$ $42.835$ \(\Q(\sqrt{3}) \) None \(4\) \(-6\) \(-6\) \(-12\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}-3q^{3}+4q^{4}+(-3+4\beta )q^{5}+\cdots\)
726.4.a.q 726.a 1.a $2$ $42.835$ \(\Q(\sqrt{5}) \) None \(4\) \(-6\) \(-5\) \(-28\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}-3q^{3}+4q^{4}+(1-7\beta )q^{5}+\cdots\)
726.4.a.r 726.a 1.a $2$ $42.835$ \(\Q(\sqrt{1081}) \) None \(4\) \(-6\) \(-5\) \(27\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}-3q^{3}+4q^{4}+(-2-\beta )q^{5}+\cdots\)
726.4.a.s 726.a 1.a $2$ $42.835$ \(\Q(\sqrt{273}) \) None \(4\) \(-6\) \(6\) \(-6\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}-3q^{3}+4q^{4}+(3+\beta )q^{5}-6q^{6}+\cdots\)
726.4.a.t 726.a 1.a $2$ $42.835$ \(\Q(\sqrt{5}) \) None \(4\) \(6\) \(-25\) \(-8\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+3q^{3}+4q^{4}+(-12-\beta )q^{5}+\cdots\)
726.4.a.u 726.a 1.a $4$ $42.835$ 4.4.244225.1 None \(-8\) \(-12\) \(-6\) \(1\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}-3q^{3}+4q^{4}+(-2-\beta _{1}+\beta _{3})q^{5}+\cdots\)
726.4.a.v 726.a 1.a $4$ $42.835$ 4.4.5157648.2 None \(-8\) \(12\) \(-6\) \(-12\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+3q^{3}+4q^{4}+(-2+2\beta _{1}+\cdots)q^{5}+\cdots\)
726.4.a.w 726.a 1.a $4$ $42.835$ 4.4.12421225.1 None \(-8\) \(12\) \(20\) \(-21\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+3q^{3}+4q^{4}+(5-\beta _{2})q^{5}+\cdots\)
726.4.a.x 726.a 1.a $4$ $42.835$ 4.4.244225.1 None \(8\) \(-12\) \(-6\) \(-1\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}-3q^{3}+4q^{4}+(-2-\beta _{1}+\beta _{3})q^{5}+\cdots\)
726.4.a.y 726.a 1.a $4$ $42.835$ 4.4.5157648.2 None \(8\) \(12\) \(-6\) \(12\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+3q^{3}+4q^{4}+(-2+2\beta _{1}+\cdots)q^{5}+\cdots\)
726.4.a.z 726.a 1.a $4$ $42.835$ 4.4.12421225.1 None \(8\) \(12\) \(20\) \(21\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+3q^{3}+4q^{4}+(5-\beta _{2})q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(726))\) into lower level spaces

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