Properties

Label 578.2.a
Level $578$
Weight $2$
Character orbit 578.a
Rep. character $\chi_{578}(1,\cdot)$
Character field $\Q$
Dimension $23$
Newform subspaces $9$
Sturm bound $153$
Trace bound $9$

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Defining parameters

Level: \( N \) \(=\) \( 578 = 2 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 578.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 9 \)
Sturm bound: \(153\)
Trace bound: \(9\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(578))\).

Total New Old
Modular forms 94 23 71
Cusp forms 59 23 36
Eisenstein series 35 0 35

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
\(+\)\(+\)$+$\(5\)
\(+\)\(-\)$-$\(7\)
\(-\)\(+\)$-$\(8\)
\(-\)\(-\)$+$\(3\)
Plus space\(+\)\(8\)
Minus space\(-\)\(15\)

Trace form

\( 23 q - q^{2} + 2 q^{3} + 23 q^{4} + 2 q^{6} + 4 q^{7} - q^{8} + 23 q^{9} + O(q^{10}) \) \( 23 q - q^{2} + 2 q^{3} + 23 q^{4} + 2 q^{6} + 4 q^{7} - q^{8} + 23 q^{9} - 6 q^{11} + 2 q^{12} - 6 q^{13} + 4 q^{14} - 4 q^{15} + 23 q^{16} - 5 q^{18} - 12 q^{21} - 6 q^{22} + 2 q^{24} + 25 q^{25} - 6 q^{26} - 4 q^{27} + 4 q^{28} + 4 q^{31} - q^{32} + 4 q^{33} - 4 q^{35} + 23 q^{36} + 4 q^{37} + 4 q^{39} - 6 q^{41} - 12 q^{42} - 16 q^{43} - 6 q^{44} - 4 q^{47} + 2 q^{48} + 11 q^{49} + q^{50} - 6 q^{52} - 2 q^{53} - 4 q^{54} - 8 q^{55} + 4 q^{56} - 8 q^{57} - 8 q^{59} - 4 q^{60} + 4 q^{61} + 4 q^{62} + 4 q^{63} + 23 q^{64} + 8 q^{66} - 16 q^{67} - 8 q^{69} - 4 q^{70} - 5 q^{72} - 2 q^{73} + 4 q^{74} - 10 q^{75} + 16 q^{77} + 4 q^{78} - 8 q^{79} + 23 q^{81} - 6 q^{82} - 12 q^{83} - 12 q^{84} - 12 q^{86} - 12 q^{87} - 6 q^{88} - 2 q^{89} + 8 q^{91} - 20 q^{93} - 4 q^{94} + 2 q^{96} - 14 q^{97} - 17 q^{98} - 6 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(578))\) 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
578.2.a.a 578.a 1.a $1$ $4.615$ \(\Q\) None \(1\) \(2\) \(0\) \(4\) $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+2q^{3}+q^{4}+2q^{6}+4q^{7}+q^{8}+\cdots\)
578.2.a.b 578.a 1.a $2$ $4.615$ \(\Q(\sqrt{2}) \) None \(-2\) \(0\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+\beta q^{5}-q^{8}-3q^{9}-\beta q^{10}+\cdots\)
578.2.a.c 578.a 1.a $2$ $4.615$ \(\Q(\sqrt{2}) \) None \(2\) \(0\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta q^{3}+q^{4}+2\beta q^{5}+\beta q^{6}+\cdots\)
578.2.a.d 578.a 1.a $2$ $4.615$ \(\Q(\sqrt{2}) \) None \(2\) \(0\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta q^{3}+q^{4}-\beta q^{5}+\beta q^{6}+q^{8}+\cdots\)
578.2.a.e 578.a 1.a $3$ $4.615$ \(\Q(\zeta_{18})^+\) None \(-3\) \(-3\) \(0\) \(-9\) $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+(-1-2\beta _{1}+\beta _{2})q^{3}+q^{4}+\cdots\)
578.2.a.f 578.a 1.a $3$ $4.615$ \(\Q(\zeta_{18})^+\) None \(-3\) \(3\) \(0\) \(9\) $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+(1-\beta _{1}-\beta _{2})q^{3}+q^{4}+(2\beta _{1}+\cdots)q^{5}+\cdots\)
578.2.a.g 578.a 1.a $3$ $4.615$ \(\Q(\zeta_{18})^+\) None \(3\) \(-3\) \(-6\) \(-3\) $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+(-1-\beta _{1})q^{3}+q^{4}+(-2+\beta _{1}+\cdots)q^{5}+\cdots\)
578.2.a.h 578.a 1.a $3$ $4.615$ \(\Q(\zeta_{18})^+\) None \(3\) \(3\) \(6\) \(3\) $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+(1+\beta _{1})q^{3}+q^{4}+(2-\beta _{1}+\beta _{2})q^{5}+\cdots\)
578.2.a.i 578.a 1.a $4$ $4.615$ \(\Q(\zeta_{16})^+\) None \(-4\) \(0\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta _{1}q^{3}+q^{4}+2\beta _{1}q^{5}-\beta _{1}q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(578))\) into lower level spaces

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