Properties

Label 350.2.a
Level $350$
Weight $2$
Character orbit 350.a
Rep. character $\chi_{350}(1,\cdot)$
Character field $\Q$
Dimension $10$
Newform subspaces $8$
Sturm bound $120$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 350.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 8 \)
Sturm bound: \(120\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(350))\).

Total New Old
Modular forms 72 10 62
Cusp forms 49 10 39
Eisenstein series 23 0 23

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

\(2\)\(5\)\(7\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(1\)
\(+\)\(+\)\(-\)\(-\)\(2\)
\(+\)\(-\)\(+\)\(-\)\(2\)
\(-\)\(+\)\(+\)\(-\)\(1\)
\(-\)\(-\)\(+\)\(+\)\(1\)
\(-\)\(-\)\(-\)\(-\)\(3\)
Plus space\(+\)\(2\)
Minus space\(-\)\(8\)

Trace form

\( 10 q + 2 q^{3} + 10 q^{4} - 2 q^{6} + 18 q^{9} + O(q^{10}) \) \( 10 q + 2 q^{3} + 10 q^{4} - 2 q^{6} + 18 q^{9} + 2 q^{12} + 10 q^{13} + 2 q^{14} + 10 q^{16} - 8 q^{17} + 4 q^{18} - 2 q^{19} + 6 q^{21} - 4 q^{22} - 2 q^{24} - 18 q^{26} - 4 q^{27} - 16 q^{29} + 4 q^{31} - 8 q^{34} + 18 q^{36} + 8 q^{37} + 2 q^{38} + 24 q^{39} - 12 q^{41} - 2 q^{42} - 12 q^{43} + 8 q^{46} + 4 q^{47} + 2 q^{48} + 10 q^{49} + 10 q^{52} - 4 q^{53} - 32 q^{54} + 2 q^{56} + 4 q^{57} - 12 q^{58} + 2 q^{59} - 6 q^{61} - 12 q^{62} - 4 q^{63} + 10 q^{64} - 12 q^{66} + 16 q^{67} - 8 q^{68} - 48 q^{69} - 48 q^{71} + 4 q^{72} - 4 q^{73} + 12 q^{74} - 2 q^{76} + 4 q^{77} + 8 q^{78} + 32 q^{79} - 18 q^{81} + 4 q^{82} - 2 q^{83} + 6 q^{84} - 20 q^{86} - 12 q^{87} - 4 q^{88} - 88 q^{89} + 10 q^{91} - 8 q^{93} - 4 q^{94} - 2 q^{96} + 8 q^{97} - 84 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(350))\) 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 5 7
350.2.a.a 350.a 1.a $1$ $2.795$ \(\Q\) None \(-1\) \(-1\) \(0\) \(-1\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{6}-q^{7}-q^{8}+\cdots\)
350.2.a.b 350.a 1.a $1$ $2.795$ \(\Q\) None \(-1\) \(0\) \(0\) \(1\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+q^{7}-q^{8}-3q^{9}+4q^{11}+\cdots\)
350.2.a.c 350.a 1.a $1$ $2.795$ \(\Q\) None \(-1\) \(3\) \(0\) \(1\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+3q^{3}+q^{4}-3q^{6}+q^{7}-q^{8}+\cdots\)
350.2.a.d 350.a 1.a $1$ $2.795$ \(\Q\) None \(1\) \(-3\) \(0\) \(-1\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-3q^{3}+q^{4}-3q^{6}-q^{7}+q^{8}+\cdots\)
350.2.a.e 350.a 1.a $1$ $2.795$ \(\Q\) None \(1\) \(1\) \(0\) \(1\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{6}+q^{7}+q^{8}+\cdots\)
350.2.a.f 350.a 1.a $1$ $2.795$ \(\Q\) None \(1\) \(2\) \(0\) \(-1\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+2q^{3}+q^{4}+2q^{6}-q^{7}+q^{8}+\cdots\)
350.2.a.g 350.a 1.a $2$ $2.795$ \(\Q(\sqrt{6}) \) None \(-2\) \(0\) \(0\) \(-2\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta q^{3}+q^{4}-\beta q^{6}-q^{7}-q^{8}+\cdots\)
350.2.a.h 350.a 1.a $2$ $2.795$ \(\Q(\sqrt{6}) \) None \(2\) \(0\) \(0\) \(2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta q^{3}+q^{4}+\beta q^{6}+q^{7}+q^{8}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(350))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(350)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(175))\)\(^{\oplus 2}\)