Properties

Label 418.2.a
Level $418$
Weight $2$
Character orbit 418.a
Rep. character $\chi_{418}(1,\cdot)$
Character field $\Q$
Dimension $15$
Newform subspaces $8$
Sturm bound $120$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 418 = 2 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 418.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 8 \)
Sturm bound: \(120\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(418))\).

Total New Old
Modular forms 64 15 49
Cusp forms 57 15 42
Eisenstein series 7 0 7

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

\(2\)\(11\)\(19\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(3\)
\(+\)\(-\)\(+\)\(-\)\(2\)
\(+\)\(-\)\(-\)\(+\)\(2\)
\(-\)\(+\)\(+\)\(-\)\(3\)
\(-\)\(+\)\(-\)\(+\)\(1\)
\(-\)\(-\)\(-\)\(-\)\(4\)
Plus space\(+\)\(6\)
Minus space\(-\)\(9\)

Trace form

\( 15 q + q^{2} + 15 q^{4} + 6 q^{5} + 4 q^{6} - 8 q^{7} + q^{8} + 19 q^{9} + O(q^{10}) \) \( 15 q + q^{2} + 15 q^{4} + 6 q^{5} + 4 q^{6} - 8 q^{7} + q^{8} + 19 q^{9} + 6 q^{10} + q^{11} - 2 q^{13} - 16 q^{15} + 15 q^{16} - 14 q^{17} - 3 q^{18} - q^{19} + 6 q^{20} - 16 q^{21} - q^{22} - 20 q^{23} + 4 q^{24} + 13 q^{25} + 10 q^{26} - 8 q^{28} + 6 q^{29} + 8 q^{30} - 4 q^{31} + q^{32} + 2 q^{34} - 8 q^{35} + 19 q^{36} - 14 q^{37} + 5 q^{38} - 44 q^{39} + 6 q^{40} + 2 q^{41} + 12 q^{42} - 20 q^{43} + q^{44} - 18 q^{45} + 16 q^{46} - 16 q^{47} + 15 q^{49} - q^{50} - 32 q^{51} - 2 q^{52} - 14 q^{53} + 4 q^{54} + 6 q^{55} - 4 q^{57} + 18 q^{58} - 8 q^{59} - 16 q^{60} + 6 q^{61} - 36 q^{63} + 15 q^{64} + 12 q^{65} + 4 q^{66} + 8 q^{67} - 14 q^{68} + 16 q^{69} - 8 q^{70} - 12 q^{71} - 3 q^{72} - 14 q^{73} + 10 q^{74} + 32 q^{75} - q^{76} + 8 q^{77} - 24 q^{78} - 16 q^{79} + 6 q^{80} + 7 q^{81} + 18 q^{82} + 20 q^{83} - 16 q^{84} + 52 q^{85} - 4 q^{86} - 28 q^{87} - q^{88} - 26 q^{89} - 10 q^{90} - 16 q^{91} - 20 q^{92} + 8 q^{93} - 6 q^{95} + 4 q^{96} + 38 q^{97} - 23 q^{98} + 13 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(418))\) 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 11 19
418.2.a.a 418.a 1.a $1$ $3.338$ \(\Q\) None \(1\) \(-1\) \(-2\) \(-3\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-2q^{5}-q^{6}-3q^{7}+\cdots\)
418.2.a.b 418.a 1.a $1$ $3.338$ \(\Q\) None \(1\) \(0\) \(2\) \(2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+2q^{5}+2q^{7}+q^{8}-3q^{9}+\cdots\)
418.2.a.c 418.a 1.a $1$ $3.338$ \(\Q\) None \(1\) \(3\) \(-2\) \(1\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+3q^{3}+q^{4}-2q^{5}+3q^{6}+q^{7}+\cdots\)
418.2.a.d 418.a 1.a $2$ $3.338$ \(\Q(\sqrt{13}) \) None \(-2\) \(-3\) \(-1\) \(-1\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+(-1-\beta )q^{3}+q^{4}+(-1+\beta )q^{5}+\cdots\)
418.2.a.e 418.a 1.a $2$ $3.338$ \(\Q(\sqrt{17}) \) None \(-2\) \(1\) \(4\) \(3\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+(1-\beta )q^{3}+q^{4}+2q^{5}+(-1+\cdots)q^{6}+\cdots\)
418.2.a.f 418.a 1.a $2$ $3.338$ \(\Q(\sqrt{21}) \) None \(2\) \(-1\) \(3\) \(-5\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-\beta q^{3}+q^{4}+(2-\beta )q^{5}-\beta q^{6}+\cdots\)
418.2.a.g 418.a 1.a $3$ $3.338$ 3.3.621.1 None \(-3\) \(0\) \(-3\) \(-6\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta _{1}q^{3}+q^{4}+(-1-\beta _{1}-\beta _{2})q^{5}+\cdots\)
418.2.a.h 418.a 1.a $3$ $3.338$ 3.3.469.1 None \(3\) \(1\) \(5\) \(1\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta _{1}q^{3}+q^{4}+(2-\beta _{1})q^{5}+\beta _{1}q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(418))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(418)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(209))\)\(^{\oplus 2}\)