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Properties

Label 1110.2.a
Level $1110$
Weight $2$
Character orbit 1110.a
Rep. character $\chi_{1110}(1,\cdot)$
Character field $\Q$
Dimension $25$
Newform subspaces $19$
Sturm bound $456$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1110.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 19 \)
Sturm bound: \(456\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(7\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1110))\).

Total New Old
Modular forms 236 25 211
Cusp forms 221 25 196
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.

\(2\)\(3\)\(5\)\(37\)FrickeDim
\(+\)\(+\)\(+\)\(+\)$+$\(3\)
\(+\)\(+\)\(-\)\(+\)$-$\(2\)
\(+\)\(+\)\(-\)\(-\)$+$\(1\)
\(+\)\(-\)\(+\)\(+\)$-$\(2\)
\(+\)\(-\)\(+\)\(-\)$+$\(1\)
\(+\)\(-\)\(-\)\(-\)$-$\(3\)
\(-\)\(+\)\(+\)\(+\)$-$\(1\)
\(-\)\(+\)\(+\)\(-\)$+$\(2\)
\(-\)\(+\)\(-\)\(+\)$+$\(1\)
\(-\)\(+\)\(-\)\(-\)$-$\(2\)
\(-\)\(-\)\(+\)\(+\)$+$\(1\)
\(-\)\(-\)\(+\)\(-\)$-$\(2\)
\(-\)\(-\)\(-\)\(+\)$-$\(4\)
Plus space\(+\)\(9\)
Minus space\(-\)\(16\)

Trace form

\( 25 q + q^{2} + q^{3} + 25 q^{4} + q^{5} + q^{6} + 8 q^{7} + q^{8} + 25 q^{9} + O(q^{10}) \) \( 25 q + q^{2} + q^{3} + 25 q^{4} + q^{5} + q^{6} + 8 q^{7} + q^{8} + 25 q^{9} + q^{10} - 4 q^{11} + q^{12} - 10 q^{13} - 8 q^{14} + q^{15} + 25 q^{16} - 14 q^{17} + q^{18} + 20 q^{19} + q^{20} + 8 q^{21} - 12 q^{22} - 8 q^{23} + q^{24} + 25 q^{25} - 2 q^{26} + q^{27} + 8 q^{28} + 14 q^{29} + q^{30} + q^{32} + 12 q^{33} + 18 q^{34} + 8 q^{35} + 25 q^{36} - 3 q^{37} - 20 q^{38} + 14 q^{39} + q^{40} + 2 q^{41} + 4 q^{43} - 4 q^{44} + q^{45} - 32 q^{47} + q^{48} + 33 q^{49} + q^{50} + 2 q^{51} - 10 q^{52} - 34 q^{53} + q^{54} + 12 q^{55} - 8 q^{56} + 12 q^{57} - 18 q^{58} - 4 q^{59} + q^{60} - 2 q^{61} + 8 q^{62} + 8 q^{63} + 25 q^{64} - 2 q^{65} - 4 q^{66} - 12 q^{67} - 14 q^{68} + 8 q^{69} + 8 q^{70} - 24 q^{71} + q^{72} - 14 q^{73} + q^{74} + q^{75} + 20 q^{76} - 16 q^{77} - 2 q^{78} + 32 q^{79} + q^{80} + 25 q^{81} + 10 q^{82} - 28 q^{83} + 8 q^{84} + 18 q^{85} - 12 q^{86} + 14 q^{87} - 12 q^{88} - 6 q^{89} + q^{90} - 16 q^{91} - 8 q^{92} + 8 q^{93} + 16 q^{94} - 12 q^{95} + q^{96} + 2 q^{97} - 7 q^{98} - 4 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1110))\) 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 37
1110.2.a.a 1110.a 1.a $1$ $8.863$ \(\Q\) None \(-1\) \(-1\) \(-1\) \(-4\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}-4q^{7}+\cdots\)
1110.2.a.b 1110.a 1.a $1$ $8.863$ \(\Q\) None \(-1\) \(-1\) \(-1\) \(1\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}+q^{7}+\cdots\)
1110.2.a.c 1110.a 1.a $1$ $8.863$ \(\Q\) None \(-1\) \(-1\) \(-1\) \(3\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}+3q^{7}+\cdots\)
1110.2.a.d 1110.a 1.a $1$ $8.863$ \(\Q\) None \(-1\) \(-1\) \(1\) \(3\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}+3q^{7}+\cdots\)
1110.2.a.e 1110.a 1.a $1$ $8.863$ \(\Q\) None \(-1\) \(1\) \(-1\) \(-1\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{5}-q^{6}-q^{7}+\cdots\)
1110.2.a.f 1110.a 1.a $1$ $8.863$ \(\Q\) None \(-1\) \(1\) \(-1\) \(1\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{5}-q^{6}+q^{7}+\cdots\)
1110.2.a.g 1110.a 1.a $1$ $8.863$ \(\Q\) None \(-1\) \(1\) \(-1\) \(4\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{5}-q^{6}+4q^{7}+\cdots\)
1110.2.a.h 1110.a 1.a $1$ $8.863$ \(\Q\) None \(-1\) \(1\) \(1\) \(-4\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}-4q^{7}+\cdots\)
1110.2.a.i 1110.a 1.a $1$ $8.863$ \(\Q\) None \(1\) \(-1\) \(-1\) \(0\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{5}-q^{6}+q^{8}+\cdots\)
1110.2.a.j 1110.a 1.a $1$ $8.863$ \(\Q\) None \(1\) \(-1\) \(1\) \(-3\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}-3q^{7}+\cdots\)
1110.2.a.k 1110.a 1.a $1$ $8.863$ \(\Q\) None \(1\) \(-1\) \(1\) \(0\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}+q^{8}+\cdots\)
1110.2.a.l 1110.a 1.a $1$ $8.863$ \(\Q\) None \(1\) \(-1\) \(1\) \(3\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}+3q^{7}+\cdots\)
1110.2.a.m 1110.a 1.a $1$ $8.863$ \(\Q\) None \(1\) \(1\) \(-1\) \(-5\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}-5q^{7}+\cdots\)
1110.2.a.n 1110.a 1.a $1$ $8.863$ \(\Q\) None \(1\) \(1\) \(-1\) \(-1\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}-q^{7}+\cdots\)
1110.2.a.o 1110.a 1.a $1$ $8.863$ \(\Q\) None \(1\) \(1\) \(-1\) \(4\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}+4q^{7}+\cdots\)
1110.2.a.p 1110.a 1.a $2$ $8.863$ \(\Q(\sqrt{33}) \) None \(-2\) \(-2\) \(2\) \(-1\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}-\beta q^{7}+\cdots\)
1110.2.a.q 1110.a 1.a $2$ $8.863$ \(\Q(\sqrt{113}) \) None \(-2\) \(2\) \(2\) \(6\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}+3q^{7}+\cdots\)
1110.2.a.r 1110.a 1.a $2$ $8.863$ \(\Q(\sqrt{17}) \) None \(2\) \(-2\) \(-2\) \(-2\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{5}-q^{6}-q^{7}+\cdots\)
1110.2.a.s 1110.a 1.a $4$ $8.863$ 4.4.54764.1 None \(4\) \(4\) \(4\) \(4\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{5}+q^{6}+(1+\beta _{2}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1110))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1110)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(37))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(74))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(111))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(185))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(222))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(370))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(555))\)\(^{\oplus 2}\)