Properties

Label 490.2.a
Level $490$
Weight $2$
Character orbit 490.a
Rep. character $\chi_{490}(1,\cdot)$
Character field $\Q$
Dimension $15$
Newform subspaces $13$
Sturm bound $168$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 490 = 2 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 490.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 13 \)
Sturm bound: \(168\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(3\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(490))\).

Total New Old
Modular forms 100 15 85
Cusp forms 69 15 54
Eisenstein series 31 0 31

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.
\(+\)\(+\)\(+\)\(+\)\(3\)
\(+\)\(+\)\(-\)\(-\)\(1\)
\(+\)\(-\)\(+\)\(-\)\(3\)
\(+\)\(-\)\(-\)\(+\)\(1\)
\(-\)\(+\)\(+\)\(-\)\(2\)
\(-\)\(+\)\(-\)\(+\)\(1\)
\(-\)\(-\)\(-\)\(-\)\(4\)
Plus space\(+\)\(5\)
Minus space\(-\)\(10\)

Trace form

\( 15 q - q^{2} + 15 q^{4} + q^{5} - q^{8} + 23 q^{9} + O(q^{10}) \) \( 15 q - q^{2} + 15 q^{4} + q^{5} - q^{8} + 23 q^{9} + q^{10} + 6 q^{13} + 4 q^{15} + 15 q^{16} - 2 q^{17} + 3 q^{18} + q^{20} - 4 q^{22} + 15 q^{25} + 6 q^{26} - 2 q^{29} - 8 q^{31} - q^{32} - 2 q^{34} + 23 q^{36} - 46 q^{37} - 48 q^{39} + q^{40} - 2 q^{41} - 52 q^{43} - 3 q^{45} + 8 q^{46} - 8 q^{47} - q^{50} + 32 q^{51} + 6 q^{52} + 18 q^{53} + 4 q^{55} - 40 q^{57} + 2 q^{58} + 8 q^{59} + 4 q^{60} + 14 q^{61} - 8 q^{62} + 15 q^{64} - 2 q^{65} - 36 q^{67} - 2 q^{68} + 40 q^{71} + 3 q^{72} - 2 q^{73} + 14 q^{74} + 24 q^{78} - 32 q^{79} + q^{80} + 55 q^{81} - 2 q^{82} - 8 q^{83} + 14 q^{85} + 16 q^{86} - 4 q^{88} - 10 q^{89} - 3 q^{90} - 24 q^{93} - 8 q^{94} + 4 q^{95} - 2 q^{97} - 48 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(490))\) 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
490.2.a.a 490.a 1.a $1$ $3.913$ \(\Q\) None \(-1\) \(-2\) \(1\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-2q^{3}+q^{4}+q^{5}+2q^{6}-q^{8}+\cdots\)
490.2.a.b 490.a 1.a $1$ $3.913$ \(\Q\) None \(-1\) \(-1\) \(1\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}-q^{8}+\cdots\)
490.2.a.c 490.a 1.a $1$ $3.913$ \(\Q\) None \(-1\) \(1\) \(-1\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{5}-q^{6}-q^{8}+\cdots\)
490.2.a.d 490.a 1.a $1$ $3.913$ \(\Q\) None \(-1\) \(2\) \(-1\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+2q^{3}+q^{4}-q^{5}-2q^{6}-q^{8}+\cdots\)
490.2.a.e 490.a 1.a $1$ $3.913$ \(\Q\) None \(1\) \(-3\) \(1\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-3q^{3}+q^{4}+q^{5}-3q^{6}+q^{8}+\cdots\)
490.2.a.f 490.a 1.a $1$ $3.913$ \(\Q\) None \(1\) \(-2\) \(-1\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-2q^{3}+q^{4}-q^{5}-2q^{6}+q^{8}+\cdots\)
490.2.a.g 490.a 1.a $1$ $3.913$ \(\Q\) None \(1\) \(-2\) \(-1\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-2q^{3}+q^{4}-q^{5}-2q^{6}+q^{8}+\cdots\)
490.2.a.h 490.a 1.a $1$ $3.913$ \(\Q\) None \(1\) \(0\) \(1\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+q^{5}+q^{8}-3q^{9}+q^{10}+\cdots\)
490.2.a.i 490.a 1.a $1$ $3.913$ \(\Q\) None \(1\) \(2\) \(1\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+2q^{3}+q^{4}+q^{5}+2q^{6}+q^{8}+\cdots\)
490.2.a.j 490.a 1.a $1$ $3.913$ \(\Q\) None \(1\) \(2\) \(1\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+2q^{3}+q^{4}+q^{5}+2q^{6}+q^{8}+\cdots\)
490.2.a.k 490.a 1.a $1$ $3.913$ \(\Q\) None \(1\) \(3\) \(-1\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+3q^{3}+q^{4}-q^{5}+3q^{6}+q^{8}+\cdots\)
490.2.a.l 490.a 1.a $2$ $3.913$ \(\Q(\sqrt{2}) \) None \(-2\) \(-4\) \(-2\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+(-2+\beta )q^{3}+q^{4}-q^{5}+(2+\cdots)q^{6}+\cdots\)
490.2.a.m 490.a 1.a $2$ $3.913$ \(\Q(\sqrt{2}) \) None \(-2\) \(4\) \(2\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+(2+\beta )q^{3}+q^{4}+q^{5}+(-2+\cdots)q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(490))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(490)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(245))\)\(^{\oplus 2}\)