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Properties

Label 130.2.e
Level $130$
Weight $2$
Character orbit 130.e
Rep. character $\chi_{130}(61,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $12$
Newform subspaces $4$
Sturm bound $42$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 130 = 2 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 130.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 13 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 4 \)
Sturm bound: \(42\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(130, [\chi])\).

Total New Old
Modular forms 48 12 36
Cusp forms 32 12 20
Eisenstein series 16 0 16

Trace form

\( 12 q - 6 q^{4} + 8 q^{7} - 14 q^{9} + O(q^{10}) \) \( 12 q - 6 q^{4} + 8 q^{7} - 14 q^{9} + 2 q^{10} + 2 q^{11} - 4 q^{13} - 4 q^{14} - 4 q^{15} - 6 q^{16} + 8 q^{17} + 16 q^{18} - 6 q^{19} - 24 q^{21} + 8 q^{22} - 8 q^{23} + 12 q^{25} + 8 q^{26} + 24 q^{27} + 8 q^{28} - 12 q^{29} + 4 q^{30} + 8 q^{31} - 20 q^{33} - 16 q^{34} + 2 q^{35} - 14 q^{36} - 12 q^{37} - 24 q^{38} + 28 q^{39} - 4 q^{40} + 4 q^{41} - 12 q^{42} - 16 q^{43} - 4 q^{44} + 16 q^{45} - 8 q^{46} - 32 q^{47} + 4 q^{49} + 8 q^{51} + 8 q^{52} - 32 q^{53} + 12 q^{54} + 4 q^{55} + 2 q^{56} + 64 q^{57} + 12 q^{58} + 8 q^{60} - 8 q^{61} + 8 q^{62} + 32 q^{63} + 12 q^{64} - 2 q^{65} + 40 q^{66} + 8 q^{68} + 12 q^{69} - 40 q^{71} - 8 q^{72} + 56 q^{73} - 6 q^{74} - 6 q^{76} - 8 q^{77} - 20 q^{78} - 8 q^{79} - 38 q^{81} + 16 q^{82} + 12 q^{84} - 12 q^{85} + 8 q^{86} - 40 q^{87} + 8 q^{88} + 10 q^{89} - 36 q^{90} + 18 q^{91} + 16 q^{92} - 44 q^{93} - 2 q^{94} - 16 q^{95} + 4 q^{97} + 16 q^{98} + 20 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(130, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
130.2.e.a 130.e 13.c $2$ $1.038$ \(\Q(\sqrt{-3}) \) None \(-1\) \(0\) \(2\) \(3\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{2}-\zeta_{6}q^{4}+q^{5}+3\zeta_{6}q^{7}+\cdots\)
130.2.e.b 130.e 13.c $2$ $1.038$ \(\Q(\sqrt{-3}) \) None \(1\) \(2\) \(-2\) \(1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{2}+(2-2\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)
130.2.e.c 130.e 13.c $4$ $1.038$ \(\Q(\sqrt{-3}, \sqrt{10})\) None \(-2\) \(0\) \(-4\) \(2\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{2}q^{2}+(\beta _{1}+\beta _{3})q^{3}+(-1-\beta _{2}+\cdots)q^{4}+\cdots\)
130.2.e.d 130.e 13.c $4$ $1.038$ \(\Q(\zeta_{12})\) None \(2\) \(-2\) \(4\) \(2\) $\mathrm{SU}(2)[C_{3}]$ \(q+\zeta_{12}q^{2}+(-\zeta_{12}+\zeta_{12}^{2})q^{3}+(-1+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(130, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(130, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(26, [\chi])\)\(^{\oplus 2}\)