Properties

Label 3864.2.a
Level $3864$
Weight $2$
Character orbit 3864.a
Rep. character $\chi_{3864}(1,\cdot)$
Character field $\Q$
Dimension $68$
Newform subspaces $24$
Sturm bound $1536$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 3864 = 2^{3} \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3864.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 24 \)
Sturm bound: \(1536\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3864))\).

Total New Old
Modular forms 784 68 716
Cusp forms 753 68 685
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.

\(2\)\(3\)\(7\)\(23\)FrickeDim
\(+\)\(+\)\(+\)\(+\)\(+\)\(4\)
\(+\)\(+\)\(+\)\(-\)\(-\)\(4\)
\(+\)\(+\)\(-\)\(+\)\(-\)\(3\)
\(+\)\(+\)\(-\)\(-\)\(+\)\(5\)
\(+\)\(-\)\(+\)\(+\)\(-\)\(4\)
\(+\)\(-\)\(+\)\(-\)\(+\)\(4\)
\(+\)\(-\)\(-\)\(+\)\(+\)\(3\)
\(+\)\(-\)\(-\)\(-\)\(-\)\(5\)
\(-\)\(+\)\(+\)\(+\)\(-\)\(6\)
\(-\)\(+\)\(+\)\(-\)\(+\)\(3\)
\(-\)\(+\)\(-\)\(+\)\(+\)\(4\)
\(-\)\(+\)\(-\)\(-\)\(-\)\(5\)
\(-\)\(-\)\(+\)\(+\)\(+\)\(3\)
\(-\)\(-\)\(+\)\(-\)\(-\)\(6\)
\(-\)\(-\)\(-\)\(+\)\(-\)\(7\)
\(-\)\(-\)\(-\)\(-\)\(+\)\(2\)
Plus space\(+\)\(28\)
Minus space\(-\)\(40\)

Trace form

\( 68 q - 8 q^{5} + 68 q^{9} + O(q^{10}) \) \( 68 q - 8 q^{5} + 68 q^{9} - 8 q^{13} - 8 q^{17} + 68 q^{25} + 8 q^{29} - 8 q^{37} + 8 q^{41} - 8 q^{45} + 16 q^{47} + 68 q^{49} - 8 q^{53} + 16 q^{55} + 16 q^{59} + 8 q^{61} - 16 q^{65} + 16 q^{67} + 32 q^{71} - 8 q^{73} + 32 q^{79} + 68 q^{81} + 16 q^{83} + 16 q^{87} + 40 q^{89} + 8 q^{93} + 32 q^{95} + 8 q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3864))\) 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 7 23
3864.2.a.a 3864.a 1.a $1$ $30.854$ \(\Q\) None 3864.2.a.a \(0\) \(-1\) \(-3\) \(-1\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-3q^{5}-q^{7}+q^{9}-q^{13}+3q^{15}+\cdots\)
3864.2.a.b 3864.a 1.a $1$ $30.854$ \(\Q\) None 3864.2.a.b \(0\) \(-1\) \(0\) \(-1\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{7}+q^{9}-2q^{11}-6q^{13}+\cdots\)
3864.2.a.c 3864.a 1.a $1$ $30.854$ \(\Q\) None 3864.2.a.c \(0\) \(1\) \(0\) \(-1\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{7}+q^{9}+q^{11}-6q^{13}+7q^{19}+\cdots\)
3864.2.a.d 3864.a 1.a $1$ $30.854$ \(\Q\) None 3864.2.a.d \(0\) \(1\) \(2\) \(1\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+2q^{5}+q^{7}+q^{9}+6q^{13}+2q^{15}+\cdots\)
3864.2.a.e 3864.a 1.a $1$ $30.854$ \(\Q\) None 3864.2.a.e \(0\) \(1\) \(2\) \(1\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+2q^{5}+q^{7}+q^{9}+4q^{11}+4q^{13}+\cdots\)
3864.2.a.f 3864.a 1.a $1$ $30.854$ \(\Q\) None 3864.2.a.f \(0\) \(1\) \(4\) \(1\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+4q^{5}+q^{7}+q^{9}+3q^{11}-2q^{13}+\cdots\)
3864.2.a.g 3864.a 1.a $2$ $30.854$ \(\Q(\sqrt{41}) \) None 3864.2.a.g \(0\) \(-2\) \(0\) \(2\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{7}+q^{9}-\beta q^{11}+2q^{13}+\cdots\)
3864.2.a.h 3864.a 1.a $2$ $30.854$ \(\Q(\sqrt{29}) \) None 3864.2.a.h \(0\) \(-2\) \(1\) \(-2\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+\beta q^{5}-q^{7}+q^{9}+5q^{11}+(1+\cdots)q^{13}+\cdots\)
3864.2.a.i 3864.a 1.a $2$ $30.854$ \(\Q(\sqrt{33}) \) None 3864.2.a.i \(0\) \(-2\) \(1\) \(2\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+\beta q^{5}+q^{7}+q^{9}+(1-\beta )q^{11}+\cdots\)
3864.2.a.j 3864.a 1.a $2$ $30.854$ \(\Q(\sqrt{5}) \) None 3864.2.a.j \(0\) \(2\) \(-1\) \(2\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-\beta q^{5}+q^{7}+q^{9}+(-3+2\beta )q^{11}+\cdots\)
3864.2.a.k 3864.a 1.a $3$ $30.854$ 3.3.621.1 None 3864.2.a.k \(0\) \(-3\) \(-3\) \(3\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+(-1-\beta _{1}+\beta _{2})q^{5}+q^{7}+q^{9}+\cdots\)
3864.2.a.l 3864.a 1.a $3$ $30.854$ 3.3.229.1 None 3864.2.a.l \(0\) \(-3\) \(-1\) \(-3\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+\beta _{2}q^{5}-q^{7}+q^{9}+(1+\beta _{1}+\cdots)q^{11}+\cdots\)
3864.2.a.m 3864.a 1.a $3$ $30.854$ 3.3.837.1 None 3864.2.a.m \(0\) \(-3\) \(0\) \(3\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+\beta _{1}q^{5}+q^{7}+q^{9}+(2-\beta _{1}+\cdots)q^{11}+\cdots\)
3864.2.a.n 3864.a 1.a $3$ $30.854$ 3.3.1229.1 None 3864.2.a.n \(0\) \(-3\) \(1\) \(3\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+\beta _{1}q^{5}+q^{7}+q^{9}+(-1+\beta _{1}+\cdots)q^{11}+\cdots\)
3864.2.a.o 3864.a 1.a $3$ $30.854$ 3.3.733.1 None 3864.2.a.o \(0\) \(3\) \(-3\) \(-3\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+(-1-\beta _{2})q^{5}-q^{7}+q^{9}+(-1+\cdots)q^{11}+\cdots\)
3864.2.a.p 3864.a 1.a $3$ $30.854$ 3.3.229.1 None 3864.2.a.p \(0\) \(3\) \(-3\) \(3\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+(-1+\beta _{1})q^{5}+q^{7}+q^{9}+(-1+\cdots)q^{11}+\cdots\)
3864.2.a.q 3864.a 1.a $3$ $30.854$ 3.3.229.1 None 3864.2.a.q \(0\) \(3\) \(1\) \(-3\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-\beta _{2}q^{5}-q^{7}+q^{9}+(-1-\beta _{1}+\cdots)q^{11}+\cdots\)
3864.2.a.r 3864.a 1.a $4$ $30.854$ 4.4.2225.1 None 3864.2.a.r \(0\) \(-4\) \(-3\) \(4\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+(-1-\beta _{3})q^{5}+q^{7}+q^{9}+(1+\cdots)q^{11}+\cdots\)
3864.2.a.s 3864.a 1.a $4$ $30.854$ 4.4.75645.1 None 3864.2.a.s \(0\) \(-4\) \(1\) \(-4\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+\beta _{1}q^{5}-q^{7}+q^{9}+(-1-\beta _{1}+\cdots)q^{11}+\cdots\)
3864.2.a.t 3864.a 1.a $4$ $30.854$ 4.4.39605.1 None 3864.2.a.t \(0\) \(4\) \(-2\) \(-4\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+(-1+\beta _{1})q^{5}-q^{7}+q^{9}+(-1+\cdots)q^{11}+\cdots\)
3864.2.a.u 3864.a 1.a $4$ $30.854$ 4.4.256549.1 None 3864.2.a.u \(0\) \(4\) \(2\) \(4\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+\beta _{1}q^{5}+q^{7}+q^{9}+(-1-\beta _{2}+\cdots)q^{11}+\cdots\)
3864.2.a.v 3864.a 1.a $5$ $30.854$ 5.5.17679757.1 None 3864.2.a.v \(0\) \(5\) \(-6\) \(5\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+(-1-\beta _{1})q^{5}+q^{7}+q^{9}+(-\beta _{1}+\cdots)q^{11}+\cdots\)
3864.2.a.w 3864.a 1.a $6$ $30.854$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None 3864.2.a.w \(0\) \(-6\) \(2\) \(-6\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-\beta _{3}q^{5}-q^{7}+q^{9}+\beta _{4}q^{11}+\cdots\)
3864.2.a.x 3864.a 1.a $6$ $30.854$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None 3864.2.a.x \(0\) \(6\) \(0\) \(-6\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+\beta _{5}q^{5}-q^{7}+q^{9}+(1+\beta _{4}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3864))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3864)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(46))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(69))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(92))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(138))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(161))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(168))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(184))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(276))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(322))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(483))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(552))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(644))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(966))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1288))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1932))\)\(^{\oplus 2}\)