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Properties

Label 570.2.a
Level $570$
Weight $2$
Character orbit 570.a
Rep. character $\chi_{570}(1,\cdot)$
Character field $\Q$
Dimension $13$
Newform subspaces $13$
Sturm bound $240$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 570.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 13 \)
Sturm bound: \(240\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(7\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(570))\).

Total New Old
Modular forms 128 13 115
Cusp forms 113 13 100
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.

\(2\)\(3\)\(5\)\(19\)FrickeDim
\(+\)\(+\)\(+\)\(+\)$+$\(1\)
\(+\)\(+\)\(+\)\(-\)$-$\(1\)
\(+\)\(+\)\(-\)\(-\)$+$\(1\)
\(+\)\(-\)\(+\)\(+\)$-$\(1\)
\(+\)\(-\)\(-\)\(+\)$+$\(1\)
\(+\)\(-\)\(-\)\(-\)$-$\(1\)
\(-\)\(+\)\(+\)\(+\)$-$\(1\)
\(-\)\(+\)\(-\)\(-\)$-$\(2\)
\(-\)\(-\)\(+\)\(-\)$-$\(2\)
\(-\)\(-\)\(-\)\(+\)$-$\(2\)
Plus space\(+\)\(3\)
Minus space\(-\)\(10\)

Trace form

\( 13 q + q^{2} + q^{3} + 13 q^{4} + q^{5} + q^{6} + 8 q^{7} + q^{8} + 13 q^{9} + O(q^{10}) \) \( 13 q + q^{2} + q^{3} + 13 q^{4} + q^{5} + q^{6} + 8 q^{7} + q^{8} + 13 q^{9} + q^{10} - 4 q^{11} + q^{12} - 2 q^{13} + 8 q^{14} + q^{15} + 13 q^{16} + 2 q^{17} + q^{18} + q^{19} + q^{20} + 8 q^{21} + 12 q^{22} + 8 q^{23} + q^{24} + 13 q^{25} + 14 q^{26} + q^{27} + 8 q^{28} + 14 q^{29} - 3 q^{30} - 24 q^{31} + q^{32} - 4 q^{33} + 10 q^{34} - 8 q^{35} + 13 q^{36} + 22 q^{37} + q^{38} + 6 q^{39} + q^{40} - 6 q^{41} - 12 q^{43} - 4 q^{44} + q^{45} - 16 q^{47} + q^{48} + 5 q^{49} + q^{50} - 30 q^{51} - 2 q^{52} - 10 q^{53} + q^{54} - 12 q^{55} + 8 q^{56} - 3 q^{57} - 2 q^{58} - 36 q^{59} + q^{60} - 10 q^{61} + 16 q^{62} + 8 q^{63} + 13 q^{64} - 2 q^{65} - 4 q^{66} - 28 q^{67} + 2 q^{68} - 8 q^{69} + 8 q^{70} - 24 q^{71} + q^{72} - 30 q^{73} - 10 q^{74} + q^{75} + q^{76} - 16 q^{77} - 2 q^{78} + 8 q^{79} + q^{80} + 13 q^{81} - 14 q^{82} - 44 q^{83} + 8 q^{84} - 6 q^{85} - 4 q^{86} - 2 q^{87} + 12 q^{88} - 6 q^{89} + q^{90} + 8 q^{92} + 16 q^{93} - 8 q^{94} + q^{95} + q^{96} + 34 q^{97} - 7 q^{98} - 4 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(570))\) 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 5 19
570.2.a.a 570.a 1.a $1$ $4.551$ \(\Q\) None \(-1\) \(-1\) \(-1\) \(-2\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}-2q^{7}+\cdots\)
570.2.a.b 570.a 1.a $1$ $4.551$ \(\Q\) None \(-1\) \(-1\) \(-1\) \(2\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}+2q^{7}+\cdots\)
570.2.a.c 570.a 1.a $1$ $4.551$ \(\Q\) None \(-1\) \(-1\) \(1\) \(-2\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}-2q^{7}+\cdots\)
570.2.a.d 570.a 1.a $1$ $4.551$ \(\Q\) None \(-1\) \(1\) \(-1\) \(4\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{5}-q^{6}+4q^{7}+\cdots\)
570.2.a.e 570.a 1.a $1$ $4.551$ \(\Q\) None \(-1\) \(1\) \(1\) \(-4\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}-4q^{7}+\cdots\)
570.2.a.f 570.a 1.a $1$ $4.551$ \(\Q\) None \(-1\) \(1\) \(1\) \(2\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}+2q^{7}+\cdots\)
570.2.a.g 570.a 1.a $1$ $4.551$ \(\Q\) None \(1\) \(-1\) \(-1\) \(0\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{5}-q^{6}+q^{8}+\cdots\)
570.2.a.h 570.a 1.a $1$ $4.551$ \(\Q\) None \(1\) \(-1\) \(1\) \(-2\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}-2q^{7}+\cdots\)
570.2.a.i 570.a 1.a $1$ $4.551$ \(\Q\) None \(1\) \(-1\) \(1\) \(4\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}+4q^{7}+\cdots\)
570.2.a.j 570.a 1.a $1$ $4.551$ \(\Q\) None \(1\) \(1\) \(-1\) \(2\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}+2q^{7}+\cdots\)
570.2.a.k 570.a 1.a $1$ $4.551$ \(\Q\) None \(1\) \(1\) \(-1\) \(2\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}+2q^{7}+\cdots\)
570.2.a.l 570.a 1.a $1$ $4.551$ \(\Q\) None \(1\) \(1\) \(1\) \(-2\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{5}+q^{6}-2q^{7}+\cdots\)
570.2.a.m 570.a 1.a $1$ $4.551$ \(\Q\) None \(1\) \(1\) \(1\) \(4\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{5}+q^{6}+4q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(570))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(570)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(285))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(57))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(95))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(114))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(190))\)\(^{\oplus 2}\)