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Properties

Label 578.4.a
Level $578$
Weight $4$
Character orbit 578.a
Rep. character $\chi_{578}(1,\cdot)$
Character field $\Q$
Dimension $68$
Newform subspaces $18$
Sturm bound $306$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 578 = 2 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 578.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 18 \)
Sturm bound: \(306\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(578))\).

Total New Old
Modular forms 248 68 180
Cusp forms 212 68 144
Eisenstein series 36 0 36

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

\(2\)\(17\)FrickeDim
\(+\)\(+\)$+$\(18\)
\(+\)\(-\)$-$\(16\)
\(-\)\(+\)$-$\(14\)
\(-\)\(-\)$+$\(20\)
Plus space\(+\)\(38\)
Minus space\(-\)\(30\)

Trace form

\( 68 q - 2 q^{3} + 272 q^{4} + 6 q^{5} - 20 q^{6} - 8 q^{7} + 704 q^{9} + O(q^{10}) \) \( 68 q - 2 q^{3} + 272 q^{4} + 6 q^{5} - 20 q^{6} - 8 q^{7} + 704 q^{9} + 4 q^{10} - 50 q^{11} - 8 q^{12} + 100 q^{13} + 40 q^{14} + 136 q^{15} + 1088 q^{16} - 16 q^{18} + 120 q^{19} + 24 q^{20} + 92 q^{21} + 124 q^{22} + 84 q^{23} - 80 q^{24} + 1200 q^{25} + 112 q^{26} - 164 q^{27} - 32 q^{28} - 418 q^{29} + 240 q^{30} + 432 q^{31} - 576 q^{33} - 556 q^{35} + 2816 q^{36} - 70 q^{37} - 56 q^{38} + 580 q^{39} + 16 q^{40} - 516 q^{41} - 200 q^{42} + 280 q^{43} - 200 q^{44} + 558 q^{45} - 336 q^{46} + 600 q^{47} - 32 q^{48} + 4340 q^{49} + 256 q^{50} + 400 q^{52} + 440 q^{53} + 472 q^{54} + 148 q^{55} + 160 q^{56} - 472 q^{57} - 876 q^{58} - 880 q^{59} + 544 q^{60} - 878 q^{61} - 1208 q^{62} + 448 q^{63} + 4352 q^{64} + 740 q^{65} - 1488 q^{66} + 600 q^{67} - 508 q^{69} + 600 q^{70} - 1884 q^{71} - 64 q^{72} - 736 q^{73} - 1220 q^{74} - 278 q^{75} + 480 q^{76} - 1188 q^{77} + 1064 q^{78} + 2052 q^{79} + 96 q^{80} + 5132 q^{81} - 584 q^{82} + 592 q^{83} + 368 q^{84} + 1680 q^{86} - 1556 q^{87} + 496 q^{88} + 892 q^{89} + 1300 q^{90} + 1672 q^{91} + 336 q^{92} - 372 q^{93} - 376 q^{94} + 120 q^{95} - 320 q^{96} + 1804 q^{97} - 272 q^{98} - 1082 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(578))\) 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 17
578.4.a.a 578.a 1.a $1$ $34.103$ \(\Q\) None \(-2\) \(2\) \(-16\) \(-24\) $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+2q^{3}+4q^{4}-2^{4}q^{5}-4q^{6}+\cdots\)
578.4.a.b 578.a 1.a $1$ $34.103$ \(\Q\) None \(-2\) \(2\) \(18\) \(10\) $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+2q^{3}+4q^{4}+18q^{5}-4q^{6}+\cdots\)
578.4.a.c 578.a 1.a $1$ $34.103$ \(\Q\) None \(2\) \(-2\) \(-8\) \(34\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}-2q^{3}+4q^{4}-8q^{5}-4q^{6}+\cdots\)
578.4.a.d 578.a 1.a $1$ $34.103$ \(\Q\) None \(2\) \(2\) \(8\) \(-34\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+2q^{3}+4q^{4}+8q^{5}+4q^{6}+\cdots\)
578.4.a.e 578.a 1.a $2$ $34.103$ \(\Q(\sqrt{13}) \) None \(-4\) \(-7\) \(1\) \(-19\) $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+(-2-3\beta )q^{3}+4q^{4}+(1-\beta )q^{5}+\cdots\)
578.4.a.f 578.a 1.a $2$ $34.103$ \(\Q(\sqrt{2}) \) None \(-4\) \(0\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+2\beta q^{3}+4q^{4}+\beta q^{5}-4\beta q^{6}+\cdots\)
578.4.a.g 578.a 1.a $2$ $34.103$ \(\Q(\sqrt{13}) \) None \(-4\) \(7\) \(-1\) \(19\) $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+(2+3\beta )q^{3}+4q^{4}+(-1+\beta )q^{5}+\cdots\)
578.4.a.h 578.a 1.a $2$ $34.103$ \(\Q(\sqrt{13}) \) None \(4\) \(-6\) \(4\) \(6\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+(-3-\beta )q^{3}+4q^{4}+(2+4\beta )q^{5}+\cdots\)
578.4.a.i 578.a 1.a $2$ $34.103$ \(\Q(\sqrt{205}) \) None \(4\) \(-3\) \(17\) \(33\) $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+(-1-\beta )q^{3}+4q^{4}+(8+\beta )q^{5}+\cdots\)
578.4.a.j 578.a 1.a $2$ $34.103$ \(\Q(\sqrt{2}) \) None \(4\) \(0\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+3\beta q^{3}+4q^{4}-7\beta q^{5}+6\beta q^{6}+\cdots\)
578.4.a.k 578.a 1.a $2$ $34.103$ \(\Q(\sqrt{205}) \) None \(4\) \(3\) \(-17\) \(-33\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+(1+\beta )q^{3}+4q^{4}+(-8-\beta )q^{5}+\cdots\)
578.4.a.l 578.a 1.a $6$ $34.103$ 6.6.525778857.1 None \(-12\) \(0\) \(0\) \(-51\) $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+(-\beta _{2}-\beta _{3}+\beta _{5})q^{3}+4q^{4}+\cdots\)
578.4.a.m 578.a 1.a $6$ $34.103$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(-12\) \(0\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+(-\beta _{1}-\beta _{3})q^{3}+4q^{4}+(-2\beta _{3}+\cdots)q^{5}+\cdots\)
578.4.a.n 578.a 1.a $6$ $34.103$ 6.6.525778857.1 None \(-12\) \(0\) \(0\) \(51\) $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+(\beta _{2}+\beta _{5})q^{3}+4q^{4}+(-1+\cdots)q^{5}+\cdots\)
578.4.a.o 578.a 1.a $6$ $34.103$ 6.6.\(\cdots\).1 None \(12\) \(-18\) \(-30\) \(-33\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+(-3+\beta _{1}-\beta _{2})q^{3}+4q^{4}+\cdots\)
578.4.a.p 578.a 1.a $6$ $34.103$ 6.6.\(\cdots\).1 None \(12\) \(18\) \(30\) \(33\) $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+(3-\beta _{1}+\beta _{2})q^{3}+4q^{4}+(5+\cdots)q^{5}+\cdots\)
578.4.a.q 578.a 1.a $8$ $34.103$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-16\) \(0\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+\beta _{4}q^{3}+4q^{4}+(-\beta _{3}-\beta _{6}+\cdots)q^{5}+\cdots\)
578.4.a.r 578.a 1.a $12$ $34.103$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(24\) \(0\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+\beta _{2}q^{3}+4q^{4}+\beta _{5}q^{5}+2\beta _{2}q^{6}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(578))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(578)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(34))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(289))\)\(^{\oplus 2}\)