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Properties

Label 1340.2.a
Level $1340$
Weight $2$
Character orbit 1340.a
Rep. character $\chi_{1340}(1,\cdot)$
Character field $\Q$
Dimension $22$
Newform subspaces $9$
Sturm bound $408$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 1340 = 2^{2} \cdot 5 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1340.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 9 \)
Sturm bound: \(408\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1340))\).

Total New Old
Modular forms 210 22 188
Cusp forms 199 22 177
Eisenstein series 11 0 11

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

\(2\)\(5\)\(67\)FrickeDim
\(-\)\(+\)\(+\)$-$\(5\)
\(-\)\(+\)\(-\)$+$\(6\)
\(-\)\(-\)\(+\)$+$\(5\)
\(-\)\(-\)\(-\)$-$\(6\)
Plus space\(+\)\(11\)
Minus space\(-\)\(11\)

Trace form

\( 22 q + 18 q^{9} + O(q^{10}) \) \( 22 q + 18 q^{9} - 4 q^{15} + 6 q^{17} + 2 q^{19} - 6 q^{23} + 22 q^{25} - 24 q^{27} - 18 q^{29} - 8 q^{33} + 8 q^{35} + 10 q^{37} - 16 q^{39} - 12 q^{41} - 12 q^{43} - 8 q^{45} - 10 q^{47} + 42 q^{49} - 24 q^{51} + 20 q^{53} - 32 q^{57} - 26 q^{59} - 12 q^{61} + 8 q^{65} + 2 q^{67} + 8 q^{69} + 36 q^{71} - 6 q^{73} - 8 q^{77} - 24 q^{79} + 46 q^{81} - 24 q^{83} - 12 q^{85} + 12 q^{87} + 14 q^{89} + 16 q^{91} + 28 q^{93} - 8 q^{95} + 24 q^{97} - 12 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1340))\) 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 67
1340.2.a.a 1340.a 1.a $1$ $10.700$ \(\Q\) None \(0\) \(-2\) \(-1\) \(2\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{3}-q^{5}+2q^{7}+q^{9}+2q^{13}+\cdots\)
1340.2.a.b 1340.a 1.a $1$ $10.700$ \(\Q\) None \(0\) \(1\) \(-1\) \(1\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{5}+q^{7}-2q^{9}-6q^{11}+2q^{13}+\cdots\)
1340.2.a.c 1340.a 1.a $1$ $10.700$ \(\Q\) None \(0\) \(1\) \(-1\) \(5\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{5}+5q^{7}-2q^{9}+2q^{13}+\cdots\)
1340.2.a.d 1340.a 1.a $2$ $10.700$ \(\Q(\sqrt{17}) \) None \(0\) \(-1\) \(2\) \(-3\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta q^{3}+q^{5}+(-2+\beta )q^{7}+(1+\beta )q^{9}+\cdots\)
1340.2.a.e 1340.a 1.a $2$ $10.700$ \(\Q(\sqrt{3}) \) None \(0\) \(2\) \(2\) \(4\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{3}+q^{5}+2q^{7}+(1+2\beta )q^{9}+\cdots\)
1340.2.a.f 1340.a 1.a $3$ $10.700$ \(\Q(\zeta_{14})^+\) None \(0\) \(-2\) \(3\) \(-2\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{2})q^{3}+q^{5}+(1-3\beta _{1}+2\beta _{2})q^{7}+\cdots\)
1340.2.a.g 1340.a 1.a $3$ $10.700$ 3.3.257.1 None \(0\) \(4\) \(-3\) \(-2\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta _{1})q^{3}-q^{5}+(-1+\beta _{1}+\beta _{2})q^{7}+\cdots\)
1340.2.a.h 1340.a 1.a $4$ $10.700$ 4.4.60513.1 None \(0\) \(-1\) \(4\) \(5\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}+q^{5}+(1+\beta _{3})q^{7}+(1+\beta _{1}+\cdots)q^{9}+\cdots\)
1340.2.a.i 1340.a 1.a $5$ $10.700$ 5.5.1465016.1 None \(0\) \(-2\) \(-5\) \(-10\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}-q^{5}+(-2+\beta _{2})q^{7}+(2+\beta _{1}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1340))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1340)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(67))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(134))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(268))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(335))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(670))\)\(^{\oplus 2}\)