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Properties

Label 510.2.a
Level $510$
Weight $2$
Character orbit 510.a
Rep. character $\chi_{510}(1,\cdot)$
Character field $\Q$
Dimension $9$
Newform subspaces $8$
Sturm bound $216$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 510 = 2 \cdot 3 \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 510.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 8 \)
Sturm bound: \(216\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(7\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(510))\).

Total New Old
Modular forms 116 9 107
Cusp forms 101 9 92
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.

\(2\)\(3\)\(5\)\(17\)FrickeDim
\(+\)\(+\)\(+\)\(-\)\(-\)\(1\)
\(+\)\(+\)\(-\)\(+\)\(-\)\(2\)
\(+\)\(-\)\(-\)\(-\)\(-\)\(1\)
\(-\)\(+\)\(+\)\(+\)\(-\)\(1\)
\(-\)\(+\)\(+\)\(-\)\(+\)\(1\)
\(-\)\(+\)\(-\)\(-\)\(-\)\(1\)
\(-\)\(-\)\(+\)\(-\)\(-\)\(1\)
\(-\)\(-\)\(-\)\(+\)\(-\)\(1\)
Plus space\(+\)\(1\)
Minus space\(-\)\(8\)

Trace form

\( 9 q + q^{2} - 3 q^{3} + 9 q^{4} + q^{5} + q^{6} + q^{8} + 9 q^{9} + O(q^{10}) \) \( 9 q + q^{2} - 3 q^{3} + 9 q^{4} + q^{5} + q^{6} + q^{8} + 9 q^{9} - 3 q^{10} + 4 q^{11} - 3 q^{12} + 6 q^{13} + q^{15} + 9 q^{16} + q^{17} + q^{18} + 4 q^{19} + q^{20} + 4 q^{22} + 8 q^{23} + q^{24} + 9 q^{25} - 10 q^{26} - 3 q^{27} + 30 q^{29} + q^{30} + 8 q^{31} + q^{32} + 12 q^{33} + q^{34} + 9 q^{36} + 6 q^{37} - 12 q^{38} - 10 q^{39} - 3 q^{40} + 2 q^{41} + 8 q^{42} + 4 q^{43} + 4 q^{44} + q^{45} - 3 q^{48} + 17 q^{49} + q^{50} + q^{51} + 6 q^{52} + 6 q^{53} + q^{54} + 12 q^{55} - 12 q^{57} - 10 q^{58} + 4 q^{59} + q^{60} + 6 q^{61} + 9 q^{64} - 10 q^{65} - 4 q^{66} - 4 q^{67} + q^{68} + 8 q^{69} + 8 q^{70} - 24 q^{71} + q^{72} - 46 q^{73} - 18 q^{74} - 3 q^{75} + 4 q^{76} + 6 q^{78} + 8 q^{79} + q^{80} + 9 q^{81} - 38 q^{82} - 28 q^{83} - 3 q^{85} - 28 q^{86} - 2 q^{87} + 4 q^{88} - 6 q^{89} - 3 q^{90} - 32 q^{91} + 8 q^{92} - 40 q^{93} + 20 q^{95} + q^{96} - 6 q^{97} - 39 q^{98} + 4 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(510))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 5 17
510.2.a.a 510.a 1.a $1$ $4.072$ \(\Q\) None 510.2.a.a \(-1\) \(-1\) \(-1\) \(2\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}+2q^{7}+\cdots\)
510.2.a.b 510.a 1.a $1$ $4.072$ \(\Q\) None 510.2.a.b \(-1\) \(1\) \(1\) \(-2\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}-2q^{7}+\cdots\)
510.2.a.c 510.a 1.a $1$ $4.072$ \(\Q\) None 510.2.a.c \(1\) \(-1\) \(-1\) \(-4\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{5}-q^{6}-4q^{7}+\cdots\)
510.2.a.d 510.a 1.a $1$ $4.072$ \(\Q\) None 510.2.a.d \(1\) \(-1\) \(-1\) \(2\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{5}-q^{6}+2q^{7}+\cdots\)
510.2.a.e 510.a 1.a $1$ $4.072$ \(\Q\) None 510.2.a.e \(1\) \(-1\) \(1\) \(0\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}+q^{8}+\cdots\)
510.2.a.f 510.a 1.a $1$ $4.072$ \(\Q\) None 510.2.a.f \(1\) \(1\) \(-1\) \(0\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}+q^{8}+\cdots\)
510.2.a.g 510.a 1.a $1$ $4.072$ \(\Q\) None 510.2.a.g \(1\) \(1\) \(1\) \(2\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{5}+q^{6}+2q^{7}+\cdots\)
510.2.a.h 510.a 1.a $2$ $4.072$ \(\Q(\sqrt{6}) \) None 510.2.a.h \(-2\) \(-2\) \(2\) \(0\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}+\beta q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(510))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(510)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(34))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(51))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(85))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(102))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(170))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(255))\)\(^{\oplus 2}\)