Properties

Label 980.2.a
Level $980$
Weight $2$
Character orbit 980.a
Rep. character $\chi_{980}(1,\cdot)$
Character field $\Q$
Dimension $13$
Newform subspaces $11$
Sturm bound $336$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 980.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 11 \)
Sturm bound: \(336\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(3\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(980))\).

Total New Old
Modular forms 192 13 179
Cusp forms 145 13 132
Eisenstein series 47 0 47

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.
\(-\)\(+\)\(+\)\(-\)\(4\)
\(-\)\(+\)\(-\)\(+\)\(2\)
\(-\)\(-\)\(+\)\(+\)\(2\)
\(-\)\(-\)\(-\)\(-\)\(5\)
Plus space\(+\)\(4\)
Minus space\(-\)\(9\)

Trace form

\( 13 q - 2 q^{3} + q^{5} + 9 q^{9} + O(q^{10}) \) \( 13 q - 2 q^{3} + q^{5} + 9 q^{9} + 2 q^{13} - 2 q^{15} + 10 q^{17} - 4 q^{19} - 6 q^{23} + 13 q^{25} - 8 q^{27} + 10 q^{29} + 12 q^{33} + 34 q^{37} + 44 q^{39} - 2 q^{41} + 34 q^{43} + 9 q^{45} + 10 q^{47} - 4 q^{51} - 6 q^{53} - 8 q^{55} - 8 q^{57} - 16 q^{59} - 2 q^{61} - 2 q^{65} + 14 q^{67} + 4 q^{71} - 18 q^{73} - 2 q^{75} + 17 q^{81} + 10 q^{83} - 10 q^{85} + 48 q^{87} - 10 q^{89} + 32 q^{93} + 4 q^{95} - 6 q^{97} - 48 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(980))\) 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
980.2.a.a 980.a 1.a $1$ $7.825$ \(\Q\) None \(0\) \(-3\) \(-1\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}-q^{5}+6q^{9}-2q^{11}-6q^{13}+\cdots\)
980.2.a.b 980.a 1.a $1$ $7.825$ \(\Q\) None \(0\) \(-3\) \(1\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}+q^{5}+6q^{9}-5q^{11}+3q^{13}+\cdots\)
980.2.a.c 980.a 1.a $1$ $7.825$ \(\Q\) None \(0\) \(-1\) \(-1\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{5}-2q^{9}+3q^{11}+q^{13}+\cdots\)
980.2.a.d 980.a 1.a $1$ $7.825$ \(\Q\) None \(0\) \(-1\) \(1\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{5}-2q^{9}-q^{11}+5q^{13}+\cdots\)
980.2.a.e 980.a 1.a $1$ $7.825$ \(\Q\) None \(0\) \(-1\) \(1\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{5}-2q^{9}+6q^{11}-2q^{13}+\cdots\)
980.2.a.f 980.a 1.a $1$ $7.825$ \(\Q\) None \(0\) \(1\) \(-1\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{5}-2q^{9}-q^{11}-5q^{13}+\cdots\)
980.2.a.g 980.a 1.a $1$ $7.825$ \(\Q\) None \(0\) \(1\) \(-1\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{5}-2q^{9}+6q^{11}+2q^{13}+\cdots\)
980.2.a.h 980.a 1.a $1$ $7.825$ \(\Q\) None \(0\) \(2\) \(1\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{3}+q^{5}+q^{9}-2q^{13}+2q^{15}+\cdots\)
980.2.a.i 980.a 1.a $1$ $7.825$ \(\Q\) None \(0\) \(3\) \(1\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+q^{5}+6q^{9}-2q^{11}+6q^{13}+\cdots\)
980.2.a.j 980.a 1.a $2$ $7.825$ \(\Q(\sqrt{2}) \) None \(0\) \(-2\) \(2\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{3}+q^{5}-2\beta q^{9}+(-1+\cdots)q^{11}+\cdots\)
980.2.a.k 980.a 1.a $2$ $7.825$ \(\Q(\sqrt{2}) \) None \(0\) \(2\) \(-2\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{3}-q^{5}+2\beta q^{9}+(-1+2\beta )q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(980))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(980)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(140))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(196))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(245))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(490))\)\(^{\oplus 2}\)