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Properties

Label 3762.2.a
Level $3762$
Weight $2$
Character orbit 3762.a
Rep. character $\chi_{3762}(1,\cdot)$
Character field $\Q$
Dimension $76$
Newform subspaces $38$
Sturm bound $1440$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 3762 = 2 \cdot 3^{2} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3762.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 38 \)
Sturm bound: \(1440\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(5\), \(7\), \(13\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3762))\).

Total New Old
Modular forms 736 76 660
Cusp forms 705 76 629
Eisenstein series 31 0 31

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

\(2\)\(3\)\(11\)\(19\)FrickeDim
\(+\)\(+\)\(+\)\(+\)\(+\)\(3\)
\(+\)\(+\)\(+\)\(-\)\(-\)\(5\)
\(+\)\(+\)\(-\)\(+\)\(-\)\(5\)
\(+\)\(+\)\(-\)\(-\)\(+\)\(3\)
\(+\)\(-\)\(+\)\(+\)\(-\)\(4\)
\(+\)\(-\)\(+\)\(-\)\(+\)\(7\)
\(+\)\(-\)\(-\)\(+\)\(+\)\(5\)
\(+\)\(-\)\(-\)\(-\)\(-\)\(5\)
\(-\)\(+\)\(+\)\(+\)\(-\)\(5\)
\(-\)\(+\)\(+\)\(-\)\(+\)\(3\)
\(-\)\(+\)\(-\)\(+\)\(+\)\(3\)
\(-\)\(+\)\(-\)\(-\)\(-\)\(5\)
\(-\)\(-\)\(+\)\(+\)\(+\)\(5\)
\(-\)\(-\)\(+\)\(-\)\(-\)\(7\)
\(-\)\(-\)\(-\)\(+\)\(-\)\(8\)
\(-\)\(-\)\(-\)\(-\)\(+\)\(3\)
Plus space\(+\)\(32\)
Minus space\(-\)\(44\)

Trace form

\( 76 q + 2 q^{2} + 76 q^{4} - 4 q^{5} + 2 q^{8} + O(q^{10}) \) \( 76 q + 2 q^{2} + 76 q^{4} - 4 q^{5} + 2 q^{8} + 4 q^{10} - 2 q^{11} + 4 q^{13} - 8 q^{14} + 76 q^{16} + 4 q^{17} - 4 q^{20} - 4 q^{23} + 96 q^{25} + 20 q^{29} + 28 q^{31} + 2 q^{32} + 12 q^{34} - 8 q^{35} + 16 q^{37} - 6 q^{38} + 4 q^{40} + 28 q^{41} - 8 q^{43} - 2 q^{44} + 8 q^{46} + 132 q^{49} + 30 q^{50} + 4 q^{52} + 32 q^{53} + 12 q^{55} - 8 q^{56} + 8 q^{58} + 44 q^{59} + 60 q^{61} - 16 q^{62} + 76 q^{64} - 16 q^{65} - 20 q^{67} + 4 q^{68} + 8 q^{70} + 20 q^{71} - 44 q^{73} + 24 q^{74} - 32 q^{79} - 4 q^{80} + 20 q^{82} - 40 q^{83} - 16 q^{85} - 8 q^{86} + 8 q^{89} - 64 q^{91} - 4 q^{92} + 16 q^{94} - 32 q^{97} + 34 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3762))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 11 19
3762.2.a.a 3762.a 1.a $1$ $30.040$ \(\Q\) None 3762.2.a.a \(-1\) \(0\) \(-3\) \(-4\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-3q^{5}-4q^{7}-q^{8}+3q^{10}+\cdots\)
3762.2.a.b 3762.a 1.a $1$ $30.040$ \(\Q\) None 1254.2.a.k \(-1\) \(0\) \(-3\) \(2\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-3q^{5}+2q^{7}-q^{8}+3q^{10}+\cdots\)
3762.2.a.c 3762.a 1.a $1$ $30.040$ \(\Q\) None 418.2.a.b \(-1\) \(0\) \(-2\) \(2\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-2q^{5}+2q^{7}-q^{8}+2q^{10}+\cdots\)
3762.2.a.d 3762.a 1.a $1$ $30.040$ \(\Q\) None 1254.2.a.j \(-1\) \(0\) \(-1\) \(-2\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-q^{5}-2q^{7}-q^{8}+q^{10}+\cdots\)
3762.2.a.e 3762.a 1.a $1$ $30.040$ \(\Q\) None 1254.2.a.h \(-1\) \(0\) \(1\) \(2\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+q^{5}+2q^{7}-q^{8}-q^{10}+\cdots\)
3762.2.a.f 3762.a 1.a $1$ $30.040$ \(\Q\) None 1254.2.a.i \(-1\) \(0\) \(2\) \(-4\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+2q^{5}-4q^{7}-q^{8}-2q^{10}+\cdots\)
3762.2.a.g 3762.a 1.a $1$ $30.040$ \(\Q\) None 418.2.a.a \(-1\) \(0\) \(2\) \(-3\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+2q^{5}-3q^{7}-q^{8}-2q^{10}+\cdots\)
3762.2.a.h 3762.a 1.a $1$ $30.040$ \(\Q\) None 1254.2.a.g \(-1\) \(0\) \(2\) \(0\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+2q^{5}-q^{8}-2q^{10}-q^{11}+\cdots\)
3762.2.a.i 3762.a 1.a $1$ $30.040$ \(\Q\) None 1254.2.a.f \(-1\) \(0\) \(2\) \(0\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+2q^{5}-q^{8}-2q^{10}+q^{11}+\cdots\)
3762.2.a.j 3762.a 1.a $1$ $30.040$ \(\Q\) None 418.2.a.c \(-1\) \(0\) \(2\) \(1\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+2q^{5}+q^{7}-q^{8}-2q^{10}+\cdots\)
3762.2.a.k 3762.a 1.a $1$ $30.040$ \(\Q\) None 3762.2.a.k \(-1\) \(0\) \(3\) \(0\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+3q^{5}-q^{8}-3q^{10}-q^{11}+\cdots\)
3762.2.a.l 3762.a 1.a $1$ $30.040$ \(\Q\) None 3762.2.a.k \(1\) \(0\) \(-3\) \(0\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-3q^{5}+q^{8}-3q^{10}+q^{11}+\cdots\)
3762.2.a.m 3762.a 1.a $1$ $30.040$ \(\Q\) None 1254.2.a.c \(1\) \(0\) \(-3\) \(2\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-3q^{5}+2q^{7}+q^{8}-3q^{10}+\cdots\)
3762.2.a.n 3762.a 1.a $1$ $30.040$ \(\Q\) None 1254.2.a.b \(1\) \(0\) \(0\) \(-4\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-4q^{7}+q^{8}+q^{11}-4q^{14}+\cdots\)
3762.2.a.o 3762.a 1.a $1$ $30.040$ \(\Q\) None 1254.2.a.e \(1\) \(0\) \(0\) \(-2\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-2q^{7}+q^{8}+q^{11}-4q^{13}+\cdots\)
3762.2.a.p 3762.a 1.a $1$ $30.040$ \(\Q\) None 1254.2.a.d \(1\) \(0\) \(2\) \(-2\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+2q^{5}-2q^{7}+q^{8}+2q^{10}+\cdots\)
3762.2.a.q 3762.a 1.a $1$ $30.040$ \(\Q\) None 3762.2.a.a \(1\) \(0\) \(3\) \(-4\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+3q^{5}-4q^{7}+q^{8}+3q^{10}+\cdots\)
3762.2.a.r 3762.a 1.a $1$ $30.040$ \(\Q\) None 1254.2.a.a \(1\) \(0\) \(4\) \(0\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+4q^{5}+q^{8}+4q^{10}-q^{11}+\cdots\)
3762.2.a.s 3762.a 1.a $2$ $30.040$ \(\Q(\sqrt{21}) \) None 418.2.a.f \(-2\) \(0\) \(-3\) \(-5\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+(-1-\beta )q^{5}+(-2-\beta )q^{7}+\cdots\)
3762.2.a.t 3762.a 1.a $2$ $30.040$ \(\Q(\sqrt{5}) \) None 1254.2.a.o \(-2\) \(0\) \(-2\) \(6\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+(-1-\beta )q^{5}+(3-\beta )q^{7}+\cdots\)
3762.2.a.u 3762.a 1.a $2$ $30.040$ \(\Q(\sqrt{3}) \) None 3762.2.a.u \(-2\) \(0\) \(0\) \(-6\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+(-3-\beta )q^{7}-q^{8}-q^{11}+\cdots\)
3762.2.a.v 3762.a 1.a $2$ $30.040$ \(\Q(\sqrt{5}) \) None 1254.2.a.n \(-2\) \(0\) \(2\) \(4\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+(1+\beta )q^{5}+2q^{7}-q^{8}+\cdots\)
3762.2.a.w 3762.a 1.a $2$ $30.040$ \(\Q(\sqrt{6}) \) None 3762.2.a.w \(-2\) \(0\) \(4\) \(0\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+2q^{5}-\beta q^{7}-q^{8}-2q^{10}+\cdots\)
3762.2.a.x 3762.a 1.a $2$ $30.040$ \(\Q(\sqrt{6}) \) None 3762.2.a.w \(2\) \(0\) \(-4\) \(0\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-2q^{5}+\beta q^{7}+q^{8}-2q^{10}+\cdots\)
3762.2.a.y 3762.a 1.a $2$ $30.040$ \(\Q(\sqrt{17}) \) None 418.2.a.e \(2\) \(0\) \(-4\) \(3\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-2q^{5}+(1+\beta )q^{7}+q^{8}+\cdots\)
3762.2.a.z 3762.a 1.a $2$ $30.040$ \(\Q(\sqrt{33}) \) None 1254.2.a.l \(2\) \(0\) \(-3\) \(-4\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+(-1-\beta )q^{5}-2q^{7}+q^{8}+\cdots\)
3762.2.a.ba 3762.a 1.a $2$ $30.040$ \(\Q(\sqrt{3}) \) None 3762.2.a.u \(2\) \(0\) \(0\) \(-6\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+(-3+\beta )q^{7}+q^{8}+q^{11}+\cdots\)
3762.2.a.bb 3762.a 1.a $2$ $30.040$ \(\Q(\sqrt{13}) \) None 418.2.a.d \(2\) \(0\) \(1\) \(-1\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+\beta q^{5}+(-1+\beta )q^{7}+q^{8}+\cdots\)
3762.2.a.bc 3762.a 1.a $2$ $30.040$ \(\Q(\sqrt{33}) \) None 1254.2.a.m \(2\) \(0\) \(1\) \(4\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+\beta q^{5}+2q^{7}+q^{8}+\beta q^{10}+\cdots\)
3762.2.a.bd 3762.a 1.a $3$ $30.040$ 3.3.469.1 None 418.2.a.h \(-3\) \(0\) \(-5\) \(1\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+(-2+\beta _{1})q^{5}-\beta _{2}q^{7}+\cdots\)
3762.2.a.be 3762.a 1.a $3$ $30.040$ 3.3.229.1 None 1254.2.a.q \(-3\) \(0\) \(-1\) \(0\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+\beta _{1}q^{5}+(-\beta _{1}+\beta _{2})q^{7}+\cdots\)
3762.2.a.bf 3762.a 1.a $3$ $30.040$ \(\Q(\zeta_{14})^+\) None 1254.2.a.p \(3\) \(0\) \(2\) \(4\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+(1+\beta _{1})q^{5}+(1+\beta _{2})q^{7}+\cdots\)
3762.2.a.bg 3762.a 1.a $3$ $30.040$ 3.3.621.1 None 418.2.a.g \(3\) \(0\) \(3\) \(-6\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+(1+\beta _{1}+\beta _{2})q^{5}+(-2+\cdots)q^{7}+\cdots\)
3762.2.a.bh 3762.a 1.a $4$ $30.040$ 4.4.23377.1 None 1254.2.a.r \(4\) \(0\) \(-3\) \(2\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+(-1-\beta _{2})q^{5}+\beta _{1}q^{7}+\cdots\)
3762.2.a.bi 3762.a 1.a $5$ $30.040$ 5.5.7578576.1 None 3762.2.a.bi \(-5\) \(0\) \(-3\) \(6\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+(-1+\beta _{1})q^{5}+(1-\beta _{3}+\cdots)q^{7}+\cdots\)
3762.2.a.bj 3762.a 1.a $5$ $30.040$ 5.5.7318736.1 None 3762.2.a.bj \(-5\) \(0\) \(-1\) \(4\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-\beta _{4}q^{5}+(1-\beta _{3})q^{7}-q^{8}+\cdots\)
3762.2.a.bk 3762.a 1.a $5$ $30.040$ 5.5.7318736.1 None 3762.2.a.bj \(5\) \(0\) \(1\) \(4\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+\beta _{4}q^{5}+(1-\beta _{3})q^{7}+q^{8}+\cdots\)
3762.2.a.bl 3762.a 1.a $5$ $30.040$ 5.5.7578576.1 None 3762.2.a.bi \(5\) \(0\) \(3\) \(6\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+(1-\beta _{1})q^{5}+(1-\beta _{3})q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3762))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3762)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(22))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(57))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(66))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(99))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(114))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(171))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(198))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(209))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(342))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(418))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(627))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1254))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1881))\)\(^{\oplus 2}\)