Properties

Label 1296.2.a
Level $1296$
Weight $2$
Character orbit 1296.a
Rep. character $\chi_{1296}(1,\cdot)$
Character field $\Q$
Dimension $22$
Newform subspaces $17$
Sturm bound $432$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 1296 = 2^{4} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1296.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 17 \)
Sturm bound: \(432\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(5\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1296))\).

Total New Old
Modular forms 252 26 226
Cusp forms 181 22 159
Eisenstein series 71 4 67

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

\(2\)\(3\)FrickeDim.
\(+\)\(+\)\(+\)\(5\)
\(+\)\(-\)\(-\)\(7\)
\(-\)\(+\)\(-\)\(6\)
\(-\)\(-\)\(+\)\(4\)
Plus space\(+\)\(9\)
Minus space\(-\)\(13\)

Trace form

\( 22 q - 2 q^{7} + O(q^{10}) \) \( 22 q - 2 q^{7} + 2 q^{13} + 4 q^{19} + 16 q^{25} + 10 q^{31} + 8 q^{37} + 10 q^{43} + 12 q^{49} - 6 q^{55} + 14 q^{61} + 22 q^{67} - 16 q^{73} + 22 q^{79} + 24 q^{85} + 2 q^{91} - 10 q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1296))\) 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
1296.2.a.a 1296.a 1.a $1$ $10.349$ \(\Q\) None \(0\) \(0\) \(-3\) \(-2\) $-$ $+$ $\mathrm{SU}(2)$ \(q-3q^{5}-2q^{7}-6q^{11}+5q^{13}+3q^{17}+\cdots\)
1296.2.a.b 1296.a 1.a $1$ $10.349$ \(\Q\) None \(0\) \(0\) \(-3\) \(1\) $-$ $-$ $\mathrm{SU}(2)$ \(q-3q^{5}+q^{7}+3q^{11}-q^{13}-6q^{17}+\cdots\)
1296.2.a.c 1296.a 1.a $1$ $10.349$ \(\Q\) None \(0\) \(0\) \(-3\) \(4\) $-$ $+$ $\mathrm{SU}(2)$ \(q-3q^{5}+4q^{7}-q^{13}-3q^{17}+4q^{19}+\cdots\)
1296.2.a.d 1296.a 1.a $1$ $10.349$ \(\Q\) None \(0\) \(0\) \(-1\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}+4q^{11}-5q^{13}-5q^{17}-8q^{19}+\cdots\)
1296.2.a.e 1296.a 1.a $1$ $10.349$ \(\Q\) None \(0\) \(0\) \(-1\) \(3\) $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}+3q^{7}-5q^{11}-5q^{13}-2q^{17}+\cdots\)
1296.2.a.f 1296.a 1.a $1$ $10.349$ \(\Q\) None \(0\) \(0\) \(0\) \(-2\) $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{7}-3q^{11}+2q^{13}+3q^{17}+q^{19}+\cdots\)
1296.2.a.g 1296.a 1.a $1$ $10.349$ \(\Q\) None \(0\) \(0\) \(0\) \(-2\) $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{7}+3q^{11}+2q^{13}-3q^{17}+q^{19}+\cdots\)
1296.2.a.h 1296.a 1.a $1$ $10.349$ \(\Q\) None \(0\) \(0\) \(1\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{5}-4q^{11}-5q^{13}+5q^{17}-8q^{19}+\cdots\)
1296.2.a.i 1296.a 1.a $1$ $10.349$ \(\Q\) None \(0\) \(0\) \(1\) \(3\) $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}+3q^{7}+5q^{11}-5q^{13}+2q^{17}+\cdots\)
1296.2.a.j 1296.a 1.a $1$ $10.349$ \(\Q\) None \(0\) \(0\) \(3\) \(-2\) $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{5}-2q^{7}+6q^{11}+5q^{13}-3q^{17}+\cdots\)
1296.2.a.k 1296.a 1.a $1$ $10.349$ \(\Q\) None \(0\) \(0\) \(3\) \(1\) $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{5}+q^{7}-3q^{11}-q^{13}+6q^{17}+\cdots\)
1296.2.a.l 1296.a 1.a $1$ $10.349$ \(\Q\) None \(0\) \(0\) \(3\) \(4\) $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{5}+4q^{7}-q^{13}+3q^{17}+4q^{19}+\cdots\)
1296.2.a.m 1296.a 1.a $2$ $10.349$ \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(-4\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-2+\beta )q^{5}+2\beta q^{7}+2q^{11}+(1+\cdots)q^{13}+\cdots\)
1296.2.a.n 1296.a 1.a $2$ $10.349$ \(\Q(\sqrt{33}) \) None \(0\) \(0\) \(-1\) \(-3\) $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta q^{5}+(-2+\beta )q^{7}-q^{11}+(2+\beta )q^{13}+\cdots\)
1296.2.a.o 1296.a 1.a $2$ $10.349$ \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(0\) \(-4\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{5}-2q^{7}-2\beta q^{11}-q^{13}-3\beta q^{17}+\cdots\)
1296.2.a.p 1296.a 1.a $2$ $10.349$ \(\Q(\sqrt{33}) \) None \(0\) \(0\) \(1\) \(-3\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{5}+(-2+\beta )q^{7}+q^{11}+(2+\beta )q^{13}+\cdots\)
1296.2.a.q 1296.a 1.a $2$ $10.349$ \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(4\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(2+\beta )q^{5}-2\beta q^{7}-2q^{11}+(1-2\beta )q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1296))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1296)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(81))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(108))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(144))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(162))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(216))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(324))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(432))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(648))\)\(^{\oplus 2}\)