Properties

Label 585.2.a
Level $585$
Weight $2$
Character orbit 585.a
Rep. character $\chi_{585}(1,\cdot)$
Character field $\Q$
Dimension $20$
Newform subspaces $14$
Sturm bound $168$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 585.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 14 \)
Sturm bound: \(168\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(2\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(585))\).

Total New Old
Modular forms 92 20 72
Cusp forms 77 20 57
Eisenstein series 15 0 15

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

\(3\)\(5\)\(13\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(3\)
\(+\)\(+\)\(-\)\(-\)\(1\)
\(+\)\(-\)\(+\)\(-\)\(3\)
\(+\)\(-\)\(-\)\(+\)\(1\)
\(-\)\(+\)\(+\)\(-\)\(3\)
\(-\)\(+\)\(-\)\(+\)\(2\)
\(-\)\(-\)\(+\)\(+\)\(2\)
\(-\)\(-\)\(-\)\(-\)\(5\)
Plus space\(+\)\(8\)
Minus space\(-\)\(12\)

Trace form

\( 20 q - 2 q^{2} + 20 q^{4} + 2 q^{5} - 4 q^{7} - 6 q^{8} + O(q^{10}) \) \( 20 q - 2 q^{2} + 20 q^{4} + 2 q^{5} - 4 q^{7} - 6 q^{8} + 4 q^{10} - 4 q^{11} - 2 q^{13} + 16 q^{14} + 20 q^{16} + 4 q^{17} - 8 q^{19} + 6 q^{20} - 24 q^{22} + 20 q^{25} - 12 q^{28} - 16 q^{29} + 12 q^{31} + 6 q^{32} - 4 q^{35} + 12 q^{38} + 28 q^{41} - 16 q^{43} + 32 q^{44} - 32 q^{46} + 4 q^{47} + 12 q^{49} - 2 q^{50} - 6 q^{52} - 32 q^{53} + 4 q^{55} + 48 q^{56} - 8 q^{58} + 16 q^{61} - 16 q^{62} + 20 q^{64} + 4 q^{65} - 36 q^{67} - 20 q^{68} + 8 q^{70} - 24 q^{71} - 24 q^{73} - 16 q^{74} - 44 q^{76} - 36 q^{77} + 14 q^{80} + 48 q^{82} + 44 q^{83} + 8 q^{85} - 80 q^{86} - 52 q^{88} - 8 q^{89} + 4 q^{91} - 12 q^{92} + 72 q^{94} - 8 q^{95} - 20 q^{97} + 38 q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(585))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 5 13
585.2.a.a 585.a 1.a $1$ $4.671$ \(\Q\) None \(-2\) \(0\) \(-1\) \(-3\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+2q^{4}-q^{5}-3q^{7}+2q^{10}+\cdots\)
585.2.a.b 585.a 1.a $1$ $4.671$ \(\Q\) None \(-2\) \(0\) \(-1\) \(3\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+2q^{4}-q^{5}+3q^{7}+2q^{10}+\cdots\)
585.2.a.c 585.a 1.a $1$ $4.671$ \(\Q\) None \(-2\) \(0\) \(1\) \(-1\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+2q^{4}+q^{5}-q^{7}-2q^{10}+\cdots\)
585.2.a.d 585.a 1.a $1$ $4.671$ \(\Q\) None \(-1\) \(0\) \(-1\) \(2\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{4}-q^{5}+2q^{7}+3q^{8}+q^{10}+\cdots\)
585.2.a.e 585.a 1.a $1$ $4.671$ \(\Q\) None \(0\) \(0\) \(-1\) \(-1\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{4}-q^{5}-q^{7}+3q^{11}+q^{13}+\cdots\)
585.2.a.f 585.a 1.a $1$ $4.671$ \(\Q\) None \(0\) \(0\) \(1\) \(-1\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{4}+q^{5}-q^{7}-3q^{11}+q^{13}+\cdots\)
585.2.a.g 585.a 1.a $1$ $4.671$ \(\Q\) None \(1\) \(0\) \(-1\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{4}-q^{5}-3q^{8}-q^{10}-4q^{11}+\cdots\)
585.2.a.h 585.a 1.a $1$ $4.671$ \(\Q\) None \(1\) \(0\) \(1\) \(-4\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{4}+q^{5}-4q^{7}-3q^{8}+q^{10}+\cdots\)
585.2.a.i 585.a 1.a $1$ $4.671$ \(\Q\) None \(1\) \(0\) \(1\) \(2\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{4}+q^{5}+2q^{7}-3q^{8}+q^{10}+\cdots\)
585.2.a.j 585.a 1.a $2$ $4.671$ \(\Q(\sqrt{17}) \) None \(-1\) \(0\) \(-2\) \(-5\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+(2+\beta )q^{4}-q^{5}+(-3+\beta )q^{7}+\cdots\)
585.2.a.k 585.a 1.a $2$ $4.671$ \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(2\) \(4\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+q^{4}+q^{5}+2q^{7}-\beta q^{8}+\beta q^{10}+\cdots\)
585.2.a.l 585.a 1.a $2$ $4.671$ \(\Q(\sqrt{17}) \) None \(1\) \(0\) \(2\) \(-5\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+(2+\beta )q^{4}+q^{5}+(-3+\beta )q^{7}+\cdots\)
585.2.a.m 585.a 1.a $2$ $4.671$ \(\Q(\sqrt{2}) \) None \(2\) \(0\) \(-2\) \(4\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}+(1+2\beta )q^{4}-q^{5}+(2+2\beta )q^{7}+\cdots\)
585.2.a.n 585.a 1.a $3$ $4.671$ 3.3.316.1 None \(0\) \(0\) \(3\) \(1\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(3+\beta _{2})q^{4}+q^{5}-\beta _{2}q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(585))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(585)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(65))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(117))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(195))\)\(^{\oplus 2}\)