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Properties

Label 3610.2.a
Level $3610$
Weight $2$
Character orbit 3610.a
Rep. character $\chi_{3610}(1,\cdot)$
Character field $\Q$
Dimension $115$
Newform subspaces $36$
Sturm bound $1140$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 3610 = 2 \cdot 5 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3610.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 36 \)
Sturm bound: \(1140\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(3\), \(7\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3610))\).

Total New Old
Modular forms 610 115 495
Cusp forms 531 115 416
Eisenstein series 79 0 79

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

\(2\)\(5\)\(19\)FrickeDim
\(+\)\(+\)\(+\)$+$\(17\)
\(+\)\(+\)\(-\)$-$\(12\)
\(+\)\(-\)\(+\)$-$\(16\)
\(+\)\(-\)\(-\)$+$\(12\)
\(-\)\(+\)\(+\)$-$\(17\)
\(-\)\(+\)\(-\)$+$\(12\)
\(-\)\(-\)\(+\)$+$\(8\)
\(-\)\(-\)\(-\)$-$\(21\)
Plus space\(+\)\(49\)
Minus space\(-\)\(66\)

Trace form

\( 115 q + q^{2} + 4 q^{3} + 115 q^{4} - q^{5} + 8 q^{7} + q^{8} + 115 q^{9} + O(q^{10}) \) \( 115 q + q^{2} + 4 q^{3} + 115 q^{4} - q^{5} + 8 q^{7} + q^{8} + 115 q^{9} + q^{10} - 4 q^{11} + 4 q^{12} + 6 q^{13} + 4 q^{14} - 4 q^{15} + 115 q^{16} + 2 q^{17} - 3 q^{18} - q^{20} - 24 q^{21} + 12 q^{22} - 8 q^{23} + 115 q^{25} - 2 q^{26} + 16 q^{27} + 8 q^{28} + 10 q^{29} - 4 q^{30} - 8 q^{31} + q^{32} - 8 q^{33} + 10 q^{34} - 4 q^{35} + 115 q^{36} + 22 q^{37} + 32 q^{39} + q^{40} - 10 q^{41} + 8 q^{42} - 4 q^{44} + 3 q^{45} - 12 q^{46} + 16 q^{47} + 4 q^{48} + 119 q^{49} + q^{50} - 16 q^{51} + 6 q^{52} + 6 q^{53} + 24 q^{54} - 12 q^{55} + 4 q^{56} + 10 q^{58} + 12 q^{59} - 4 q^{60} + 2 q^{61} + 8 q^{62} + 48 q^{63} + 115 q^{64} - 2 q^{65} - 8 q^{66} + 4 q^{67} + 2 q^{68} + 40 q^{69} - 4 q^{70} - 16 q^{71} - 3 q^{72} + 10 q^{73} + 14 q^{74} + 4 q^{75} - 16 q^{77} - 24 q^{78} - q^{80} + 139 q^{81} + 18 q^{82} + 8 q^{83} - 24 q^{84} - 18 q^{85} - 16 q^{86} - 8 q^{87} + 12 q^{88} - 2 q^{89} + 13 q^{90} + 16 q^{91} - 8 q^{92} + 8 q^{93} - 4 q^{94} + 18 q^{97} - 23 q^{98} + 12 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3610))\) 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 19
3610.2.a.a 3610.a 1.a $1$ $28.826$ \(\Q\) None \(-1\) \(-1\) \(-1\) \(2\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}+2q^{7}+\cdots\)
3610.2.a.b 3610.a 1.a $1$ $28.826$ \(\Q\) None \(-1\) \(-1\) \(1\) \(-1\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}-q^{7}+\cdots\)
3610.2.a.c 3610.a 1.a $1$ $28.826$ \(\Q\) None \(-1\) \(0\) \(-1\) \(1\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-q^{5}+q^{7}-q^{8}-3q^{9}+\cdots\)
3610.2.a.d 3610.a 1.a $1$ $28.826$ \(\Q\) None \(-1\) \(1\) \(1\) \(-3\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}-3q^{7}+\cdots\)
3610.2.a.e 3610.a 1.a $1$ $28.826$ \(\Q\) None \(-1\) \(3\) \(-1\) \(-5\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+3q^{3}+q^{4}-q^{5}-3q^{6}-5q^{7}+\cdots\)
3610.2.a.f 3610.a 1.a $1$ $28.826$ \(\Q\) None \(1\) \(-1\) \(1\) \(-3\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}-3q^{7}+\cdots\)
3610.2.a.g 3610.a 1.a $1$ $28.826$ \(\Q\) None \(1\) \(0\) \(-1\) \(1\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-q^{5}+q^{7}+q^{8}-3q^{9}+\cdots\)
3610.2.a.h 3610.a 1.a $1$ $28.826$ \(\Q\) None \(1\) \(1\) \(-1\) \(-1\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}-q^{7}+\cdots\)
3610.2.a.i 3610.a 1.a $1$ $28.826$ \(\Q\) None \(1\) \(1\) \(-1\) \(2\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}+2q^{7}+\cdots\)
3610.2.a.j 3610.a 1.a $2$ $28.826$ \(\Q(\sqrt{5}) \) None \(-2\) \(-3\) \(2\) \(1\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+(-1-\beta )q^{3}+q^{4}+q^{5}+(1+\cdots)q^{6}+\cdots\)
3610.2.a.k 3610.a 1.a $2$ $28.826$ \(\Q(\sqrt{17}) \) None \(-2\) \(-1\) \(2\) \(5\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-\beta q^{3}+q^{4}+q^{5}+\beta q^{6}+(3+\cdots)q^{7}+\cdots\)
3610.2.a.l 3610.a 1.a $2$ $28.826$ \(\Q(\sqrt{5}) \) None \(-2\) \(1\) \(-2\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta q^{3}+q^{4}-q^{5}-\beta q^{6}-q^{8}+\cdots\)
3610.2.a.m 3610.a 1.a $2$ $28.826$ \(\Q(\sqrt{17}) \) None \(-2\) \(1\) \(-2\) \(3\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta q^{3}+q^{4}-q^{5}-\beta q^{6}+(2+\cdots)q^{7}+\cdots\)
3610.2.a.n 3610.a 1.a $2$ $28.826$ \(\Q(\sqrt{5}) \) None \(-2\) \(3\) \(-2\) \(-2\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+(1+\beta )q^{3}+q^{4}-q^{5}+(-1+\cdots)q^{6}+\cdots\)
3610.2.a.o 3610.a 1.a $2$ $28.826$ \(\Q(\sqrt{5}) \) None \(-2\) \(3\) \(2\) \(-2\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+(1+\beta )q^{3}+q^{4}+q^{5}+(-1+\cdots)q^{6}+\cdots\)
3610.2.a.p 3610.a 1.a $2$ $28.826$ \(\Q(\sqrt{5}) \) None \(2\) \(-3\) \(-2\) \(-2\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+(-1-\beta )q^{3}+q^{4}-q^{5}+(-1+\cdots)q^{6}+\cdots\)
3610.2.a.q 3610.a 1.a $2$ $28.826$ \(\Q(\sqrt{5}) \) None \(2\) \(-3\) \(2\) \(-2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+(-1-\beta )q^{3}+q^{4}+q^{5}+(-1+\cdots)q^{6}+\cdots\)
3610.2.a.r 3610.a 1.a $2$ $28.826$ \(\Q(\sqrt{5}) \) None \(2\) \(-1\) \(-2\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-\beta q^{3}+q^{4}-q^{5}-\beta q^{6}+q^{8}+\cdots\)
3610.2.a.s 3610.a 1.a $2$ $28.826$ \(\Q(\sqrt{17}) \) None \(2\) \(-1\) \(-2\) \(3\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-\beta q^{3}+q^{4}-q^{5}-\beta q^{6}+(2+\cdots)q^{7}+\cdots\)
3610.2.a.t 3610.a 1.a $2$ $28.826$ \(\Q(\sqrt{17}) \) None \(2\) \(1\) \(2\) \(-1\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta q^{3}+q^{4}+q^{5}+\beta q^{6}-\beta q^{7}+\cdots\)
3610.2.a.u 3610.a 1.a $2$ $28.826$ \(\Q(\sqrt{17}) \) None \(2\) \(1\) \(2\) \(5\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta q^{3}+q^{4}+q^{5}+\beta q^{6}+(3+\cdots)q^{7}+\cdots\)
3610.2.a.v 3610.a 1.a $2$ $28.826$ \(\Q(\sqrt{5}) \) None \(2\) \(3\) \(2\) \(1\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+(1+\beta )q^{3}+q^{4}+q^{5}+(1+\beta )q^{6}+\cdots\)
3610.2.a.w 3610.a 1.a $3$ $28.826$ \(\Q(\zeta_{18})^+\) None \(-3\) \(0\) \(3\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-\beta _{1}q^{3}+q^{4}+q^{5}+\beta _{1}q^{6}+\cdots\)
3610.2.a.x 3610.a 1.a $3$ $28.826$ \(\Q(\zeta_{18})^+\) None \(3\) \(0\) \(3\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta _{1}q^{3}+q^{4}+q^{5}+\beta _{1}q^{6}+\cdots\)
3610.2.a.y 3610.a 1.a $4$ $28.826$ 4.4.5225.1 None \(-4\) \(-2\) \(4\) \(3\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-\beta _{3}q^{3}+q^{4}+q^{5}+\beta _{3}q^{6}+\cdots\)
3610.2.a.z 3610.a 1.a $4$ $28.826$ \(\Q(\zeta_{20})^+\) None \(-4\) \(4\) \(4\) \(-2\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+(1+\beta _{1})q^{3}+q^{4}+q^{5}+(-1+\cdots)q^{6}+\cdots\)
3610.2.a.ba 3610.a 1.a $4$ $28.826$ \(\Q(\zeta_{20})^+\) None \(4\) \(-4\) \(4\) \(-2\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+(-1+\beta _{1})q^{3}+q^{4}+q^{5}+(-1+\cdots)q^{6}+\cdots\)
3610.2.a.bb 3610.a 1.a $4$ $28.826$ 4.4.5225.1 None \(4\) \(2\) \(4\) \(3\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta _{3}q^{3}+q^{4}+q^{5}+\beta _{3}q^{6}+\cdots\)
3610.2.a.bc 3610.a 1.a $6$ $28.826$ 6.6.15536448.1 None \(-6\) \(0\) \(-6\) \(-6\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta _{1}q^{3}+q^{4}-q^{5}-\beta _{1}q^{6}+\cdots\)
3610.2.a.bd 3610.a 1.a $6$ $28.826$ 6.6.23153769.1 None \(-6\) \(0\) \(-6\) \(6\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta _{1}q^{3}+q^{4}-q^{5}-\beta _{1}q^{6}+\cdots\)
3610.2.a.be 3610.a 1.a $6$ $28.826$ 6.6.15536448.1 None \(6\) \(0\) \(-6\) \(-6\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta _{1}q^{3}+q^{4}-q^{5}+\beta _{1}q^{6}+\cdots\)
3610.2.a.bf 3610.a 1.a $6$ $28.826$ 6.6.23153769.1 None \(6\) \(0\) \(-6\) \(6\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-\beta _{1}q^{3}+q^{4}-q^{5}-\beta _{1}q^{6}+\cdots\)
3610.2.a.bg 3610.a 1.a $8$ $28.826$ 8.8.\(\cdots\).1 None \(-8\) \(-6\) \(-8\) \(2\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+(-1+\beta _{1})q^{3}+q^{4}-q^{5}+(1+\cdots)q^{6}+\cdots\)
3610.2.a.bh 3610.a 1.a $8$ $28.826$ 8.8.\(\cdots\).1 None \(8\) \(6\) \(-8\) \(2\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+(1-\beta _{1})q^{3}+q^{4}-q^{5}+(1-\beta _{1}+\cdots)q^{6}+\cdots\)
3610.2.a.bi 3610.a 1.a $9$ $28.826$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(-9\) \(0\) \(9\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta _{1}q^{3}+q^{4}+q^{5}-\beta _{1}q^{6}+\cdots\)
3610.2.a.bj 3610.a 1.a $9$ $28.826$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(9\) \(0\) \(9\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-\beta _{1}q^{3}+q^{4}+q^{5}-\beta _{1}q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3610))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3610)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(95))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(190))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(361))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(722))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1805))\)\(^{\oplus 2}\)