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Properties

Label 1290.2.a
Level $1290$
Weight $2$
Character orbit 1290.a
Rep. character $\chi_{1290}(1,\cdot)$
Character field $\Q$
Dimension $29$
Newform subspaces $21$
Sturm bound $528$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 1290 = 2 \cdot 3 \cdot 5 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1290.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 21 \)
Sturm bound: \(528\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1290))\).

Total New Old
Modular forms 272 29 243
Cusp forms 257 29 228
Eisenstein series 15 0 15

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

\(2\)\(3\)\(5\)\(43\)FrickeDim
\(+\)\(+\)\(+\)\(+\)$+$\(3\)
\(+\)\(+\)\(-\)\(+\)$-$\(2\)
\(+\)\(+\)\(-\)\(-\)$+$\(2\)
\(+\)\(-\)\(+\)\(+\)$-$\(2\)
\(+\)\(-\)\(+\)\(-\)$+$\(1\)
\(+\)\(-\)\(-\)\(+\)$+$\(1\)
\(+\)\(-\)\(-\)\(-\)$-$\(3\)
\(-\)\(+\)\(+\)\(+\)$-$\(2\)
\(-\)\(+\)\(+\)\(-\)$+$\(2\)
\(-\)\(+\)\(-\)\(+\)$+$\(1\)
\(-\)\(+\)\(-\)\(-\)$-$\(2\)
\(-\)\(-\)\(+\)\(+\)$+$\(1\)
\(-\)\(-\)\(+\)\(-\)$-$\(3\)
\(-\)\(-\)\(-\)\(+\)$-$\(4\)
Plus space\(+\)\(11\)
Minus space\(-\)\(18\)

Trace form

\( 29 q + q^{2} + q^{3} + 29 q^{4} + q^{5} + q^{6} + 8 q^{7} + q^{8} + 29 q^{9} + O(q^{10}) \) \( 29 q + q^{2} + q^{3} + 29 q^{4} + q^{5} + q^{6} + 8 q^{7} + q^{8} + 29 q^{9} - 3 q^{10} - 4 q^{11} + q^{12} - 2 q^{13} - 16 q^{14} + q^{15} + 29 q^{16} - 22 q^{17} + q^{18} - 4 q^{19} + q^{20} + 8 q^{21} - 4 q^{22} + q^{24} + 29 q^{25} + 6 q^{26} + q^{27} + 8 q^{28} - 2 q^{29} + q^{30} + 24 q^{31} + q^{32} + 12 q^{33} + 18 q^{34} + 29 q^{36} + 22 q^{37} + 4 q^{38} + 14 q^{39} - 3 q^{40} + 26 q^{41} + 8 q^{42} - 3 q^{43} - 4 q^{44} + q^{45} + 16 q^{46} - 8 q^{47} + q^{48} + 13 q^{49} + q^{50} + 2 q^{51} - 2 q^{52} - 2 q^{53} + q^{54} - 4 q^{55} - 16 q^{56} + 12 q^{57} + 6 q^{58} - 20 q^{59} + q^{60} + 14 q^{61} + 8 q^{63} + 29 q^{64} + 6 q^{65} - 4 q^{66} - 12 q^{67} - 22 q^{68} - 8 q^{69} + 8 q^{70} + 8 q^{71} + q^{72} - 22 q^{73} + 14 q^{74} + q^{75} - 4 q^{76} - 64 q^{77} + 6 q^{78} - 24 q^{79} + q^{80} + 29 q^{81} + 10 q^{82} - 52 q^{83} + 8 q^{84} + 18 q^{85} + q^{86} - 2 q^{87} - 4 q^{88} - 46 q^{89} - 3 q^{90} + 16 q^{93} - 8 q^{94} - 12 q^{95} + q^{96} + 10 q^{97} + 25 q^{98} - 4 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1290))\) 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 3 5 43
1290.2.a.a 1290.a 1.a $1$ $10.301$ \(\Q\) None \(-1\) \(-1\) \(1\) \(0\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}-q^{8}+\cdots\)
1290.2.a.b 1290.a 1.a $1$ $10.301$ \(\Q\) None \(-1\) \(-1\) \(1\) \(0\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}-q^{8}+\cdots\)
1290.2.a.c 1290.a 1.a $1$ $10.301$ \(\Q\) None \(-1\) \(1\) \(-1\) \(0\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{5}-q^{6}-q^{8}+\cdots\)
1290.2.a.d 1290.a 1.a $1$ $10.301$ \(\Q\) None \(-1\) \(1\) \(-1\) \(1\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{5}-q^{6}+q^{7}+\cdots\)
1290.2.a.e 1290.a 1.a $1$ $10.301$ \(\Q\) None \(-1\) \(1\) \(-1\) \(4\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{5}-q^{6}+4q^{7}+\cdots\)
1290.2.a.f 1290.a 1.a $1$ $10.301$ \(\Q\) None \(-1\) \(1\) \(1\) \(-3\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}-3q^{7}+\cdots\)
1290.2.a.g 1290.a 1.a $1$ $10.301$ \(\Q\) None \(-1\) \(1\) \(1\) \(-2\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}-2q^{7}+\cdots\)
1290.2.a.h 1290.a 1.a $1$ $10.301$ \(\Q\) None \(-1\) \(1\) \(1\) \(2\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}+2q^{7}+\cdots\)
1290.2.a.i 1290.a 1.a $1$ $10.301$ \(\Q\) None \(-1\) \(1\) \(1\) \(4\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}+4q^{7}+\cdots\)
1290.2.a.j 1290.a 1.a $1$ $10.301$ \(\Q\) None \(1\) \(-1\) \(-1\) \(-4\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{5}-q^{6}-4q^{7}+\cdots\)
1290.2.a.k 1290.a 1.a $1$ $10.301$ \(\Q\) None \(1\) \(-1\) \(-1\) \(2\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{5}-q^{6}+2q^{7}+\cdots\)
1290.2.a.l 1290.a 1.a $1$ $10.301$ \(\Q\) None \(1\) \(-1\) \(1\) \(-1\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}-q^{7}+\cdots\)
1290.2.a.m 1290.a 1.a $1$ $10.301$ \(\Q\) None \(1\) \(1\) \(-1\) \(-4\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}-4q^{7}+\cdots\)
1290.2.a.n 1290.a 1.a $1$ $10.301$ \(\Q\) None \(1\) \(1\) \(-1\) \(2\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}+2q^{7}+\cdots\)
1290.2.a.o 1290.a 1.a $2$ $10.301$ \(\Q(\sqrt{17}) \) None \(-2\) \(-2\) \(2\) \(3\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}+(1+\beta )q^{7}+\cdots\)
1290.2.a.p 1290.a 1.a $2$ $10.301$ \(\Q(\sqrt{17}) \) None \(2\) \(-2\) \(-2\) \(-3\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{5}-q^{6}+(-1+\cdots)q^{7}+\cdots\)
1290.2.a.q 1290.a 1.a $2$ $10.301$ \(\Q(\sqrt{2}) \) None \(2\) \(-2\) \(2\) \(0\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}+\beta q^{7}+\cdots\)
1290.2.a.r 1290.a 1.a $2$ $10.301$ \(\Q(\sqrt{41}) \) None \(2\) \(2\) \(-2\) \(3\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}+(1+\beta )q^{7}+\cdots\)
1290.2.a.s 1290.a 1.a $2$ $10.301$ \(\Q(\sqrt{33}) \) None \(2\) \(2\) \(2\) \(-1\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{5}+q^{6}-\beta q^{7}+\cdots\)
1290.2.a.t 1290.a 1.a $2$ $10.301$ \(\Q(\sqrt{5}) \) None \(2\) \(2\) \(2\) \(2\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{5}+q^{6}+(1+\beta )q^{7}+\cdots\)
1290.2.a.u 1290.a 1.a $3$ $10.301$ 3.3.469.1 None \(-3\) \(-3\) \(-3\) \(3\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}+(1+\beta _{1}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1290))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1290)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(43))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(86))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(129))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(215))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(258))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(430))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(645))\)\(^{\oplus 2}\)