Properties

Label 3276.2.a
Level $3276$
Weight $2$
Character orbit 3276.a
Rep. character $\chi_{3276}(1,\cdot)$
Character field $\Q$
Dimension $30$
Newform subspaces $19$
Sturm bound $1344$
Trace bound $19$

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Defining parameters

Level: \( N \) \(=\) \( 3276 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3276.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 19 \)
Sturm bound: \(1344\)
Trace bound: \(19\)
Distinguishing \(T_p\): \(5\), \(11\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3276))\).

Total New Old
Modular forms 696 30 666
Cusp forms 649 30 619
Eisenstein series 47 0 47

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

\(2\)\(3\)\(7\)\(13\)FrickeDim
\(-\)\(+\)\(+\)\(+\)\(-\)\(4\)
\(-\)\(+\)\(+\)\(-\)\(+\)\(2\)
\(-\)\(+\)\(-\)\(+\)\(+\)\(2\)
\(-\)\(+\)\(-\)\(-\)\(-\)\(4\)
\(-\)\(-\)\(+\)\(+\)\(+\)\(4\)
\(-\)\(-\)\(+\)\(-\)\(-\)\(5\)
\(-\)\(-\)\(-\)\(+\)\(-\)\(5\)
\(-\)\(-\)\(-\)\(-\)\(+\)\(4\)
Plus space\(+\)\(12\)
Minus space\(-\)\(18\)

Trace form

\( 30 q + 4 q^{5} + O(q^{10}) \) \( 30 q + 4 q^{5} - 8 q^{11} - 8 q^{17} + 8 q^{19} - 2 q^{23} + 8 q^{25} - 6 q^{29} - 4 q^{31} - 2 q^{35} - 12 q^{37} + 14 q^{43} + 30 q^{49} + 2 q^{53} + 20 q^{55} - 20 q^{59} + 4 q^{61} - 10 q^{65} + 36 q^{67} + 56 q^{71} + 32 q^{73} + 22 q^{79} + 12 q^{83} + 12 q^{85} + 16 q^{89} + 2 q^{91} - 22 q^{95} - 8 q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3276))\) 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 13
3276.2.a.a 3276.a 1.a $1$ $26.159$ \(\Q\) None 3276.2.a.a \(0\) \(0\) \(-2\) \(-1\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{5}-q^{7}+2q^{11}+q^{13}+4q^{17}+\cdots\)
3276.2.a.b 3276.a 1.a $1$ $26.159$ \(\Q\) None 364.2.a.a \(0\) \(0\) \(-1\) \(-1\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{5}-q^{7}+4q^{11}+q^{13}+2q^{17}+\cdots\)
3276.2.a.c 3276.a 1.a $1$ $26.159$ \(\Q\) None 1092.2.a.d \(0\) \(0\) \(-1\) \(1\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}+q^{7}+6q^{11}-q^{13}+8q^{17}+\cdots\)
3276.2.a.d 3276.a 1.a $1$ $26.159$ \(\Q\) None 1092.2.a.c \(0\) \(0\) \(0\) \(-1\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{7}+2q^{11}-q^{13}-4q^{17}+4q^{19}+\cdots\)
3276.2.a.e 3276.a 1.a $1$ $26.159$ \(\Q\) None 3276.2.a.e \(0\) \(0\) \(0\) \(1\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{7}-4q^{11}-q^{13}+4q^{17}-4q^{19}+\cdots\)
3276.2.a.f 3276.a 1.a $1$ $26.159$ \(\Q\) None 3276.2.a.e \(0\) \(0\) \(0\) \(1\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{7}+4q^{11}-q^{13}-4q^{17}-4q^{19}+\cdots\)
3276.2.a.g 3276.a 1.a $1$ $26.159$ \(\Q\) None 1092.2.a.b \(0\) \(0\) \(1\) \(-1\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}-q^{7}-2q^{11}+q^{13}+4q^{17}+\cdots\)
3276.2.a.h 3276.a 1.a $1$ $26.159$ \(\Q\) None 3276.2.a.a \(0\) \(0\) \(2\) \(-1\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{5}-q^{7}-2q^{11}+q^{13}-4q^{17}+\cdots\)
3276.2.a.i 3276.a 1.a $1$ $26.159$ \(\Q\) None 1092.2.a.e \(0\) \(0\) \(2\) \(-1\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{5}-q^{7}-q^{13}+4q^{17}-8q^{19}+\cdots\)
3276.2.a.j 3276.a 1.a $1$ $26.159$ \(\Q\) None 1092.2.a.a \(0\) \(0\) \(2\) \(1\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{5}+q^{7}-q^{13}-4q^{17}+6q^{23}+\cdots\)
3276.2.a.k 3276.a 1.a $1$ $26.159$ \(\Q\) None 364.2.a.b \(0\) \(0\) \(3\) \(1\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{5}+q^{7}+2q^{11}-q^{13}+4q^{17}+\cdots\)
3276.2.a.l 3276.a 1.a $2$ $26.159$ \(\Q(\sqrt{5}) \) None 1092.2.a.f \(0\) \(0\) \(-2\) \(2\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta )q^{5}+q^{7}+(-1+\beta )q^{11}+\cdots\)
3276.2.a.m 3276.a 1.a $2$ $26.159$ \(\Q(\sqrt{33}) \) None 1092.2.a.g \(0\) \(0\) \(-1\) \(2\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta q^{5}+q^{7}-2q^{11}-q^{13}+(4-\beta )q^{19}+\cdots\)
3276.2.a.n 3276.a 1.a $2$ $26.159$ \(\Q(\sqrt{3}) \) None 364.2.a.d \(0\) \(0\) \(0\) \(2\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{5}+q^{7}+(-3+\beta )q^{11}+q^{13}+\cdots\)
3276.2.a.o 3276.a 1.a $2$ $26.159$ \(\Q(\sqrt{3}) \) None 3276.2.a.o \(0\) \(0\) \(0\) \(2\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{5}+q^{7}+\beta q^{11}+q^{13}-4q^{19}+\cdots\)
3276.2.a.p 3276.a 1.a $2$ $26.159$ \(\Q(\sqrt{3}) \) None 3276.2.a.p \(0\) \(0\) \(0\) \(2\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{5}+q^{7}-2\beta q^{11}+q^{13}+5q^{19}+\cdots\)
3276.2.a.q 3276.a 1.a $2$ $26.159$ \(\Q(\sqrt{6}) \) None 364.2.a.c \(0\) \(0\) \(2\) \(-2\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{5}-q^{7}+(-4-\beta )q^{11}-q^{13}+\cdots\)
3276.2.a.r 3276.a 1.a $3$ $26.159$ 3.3.1373.1 None 1092.2.a.h \(0\) \(0\) \(-1\) \(-3\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{2}q^{5}-q^{7}-\beta _{1}q^{11}+q^{13}-4q^{17}+\cdots\)
3276.2.a.s 3276.a 1.a $4$ $26.159$ \(\Q(\sqrt{3}, \sqrt{19})\) None 3276.2.a.s \(0\) \(0\) \(0\) \(-4\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{5}-q^{7}+2\beta _{1}q^{11}-q^{13}+(\beta _{1}+\cdots)q^{17}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3276))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3276)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(52))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(78))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(91))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(117))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(126))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(156))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(182))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(234))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(252))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(273))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(364))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(468))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(546))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(819))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1092))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1638))\)\(^{\oplus 2}\)