Properties

Label 648.2.a
Level $648$
Weight $2$
Character orbit 648.a
Rep. character $\chi_{648}(1,\cdot)$
Character field $\Q$
Dimension $12$
Newform subspaces $8$
Sturm bound $216$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 648 = 2^{3} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 648.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 8 \)
Sturm bound: \(216\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(648))\).

Total New Old
Modular forms 132 12 120
Cusp forms 85 12 73
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.

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

Trace form

\( 12 q + O(q^{10}) \) \( 12 q - 6 q^{13} + 6 q^{19} + 6 q^{25} - 12 q^{31} + 18 q^{37} - 6 q^{43} + 24 q^{49} + 12 q^{55} + 30 q^{61} + 6 q^{67} - 24 q^{73} + 12 q^{79} + 6 q^{85} - 36 q^{91} - 30 q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(648))\) 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
648.2.a.a 648.a 1.a $1$ $5.174$ \(\Q\) None \(0\) \(0\) \(-1\) \(-3\) $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}-3q^{7}+5q^{11}-5q^{13}-2q^{17}+\cdots\)
648.2.a.b 648.a 1.a $1$ $5.174$ \(\Q\) None \(0\) \(0\) \(-1\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}-4q^{11}-5q^{13}-5q^{17}+8q^{19}+\cdots\)
648.2.a.c 648.a 1.a $1$ $5.174$ \(\Q\) None \(0\) \(0\) \(1\) \(-3\) $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}-3q^{7}-5q^{11}-5q^{13}+2q^{17}+\cdots\)
648.2.a.d 648.a 1.a $1$ $5.174$ \(\Q\) None \(0\) \(0\) \(1\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{5}+4q^{11}-5q^{13}+5q^{17}+8q^{19}+\cdots\)
648.2.a.e 648.a 1.a $2$ $5.174$ \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(-4\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-2+\beta )q^{5}-2\beta q^{7}-2q^{11}+(1+\cdots)q^{13}+\cdots\)
648.2.a.f 648.a 1.a $2$ $5.174$ \(\Q(\sqrt{33}) \) None \(0\) \(0\) \(-1\) \(3\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta q^{5}+(2-\beta )q^{7}+q^{11}+(2+\beta )q^{13}+\cdots\)
648.2.a.g 648.a 1.a $2$ $5.174$ \(\Q(\sqrt{33}) \) None \(0\) \(0\) \(1\) \(3\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{5}+(2-\beta )q^{7}-q^{11}+(2+\beta )q^{13}+\cdots\)
648.2.a.h 648.a 1.a $2$ $5.174$ \(\Q(\sqrt{3}) \) 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 \(S_{2}^{\mathrm{old}}(\Gamma_0(648))\) into lower level spaces

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