Properties

Label 644.2.a
Level $644$
Weight $2$
Character orbit 644.a
Rep. character $\chi_{644}(1,\cdot)$
Character field $\Q$
Dimension $12$
Newform subspaces $4$
Sturm bound $192$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 644 = 2^{2} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 644.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(192\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(644))\).

Total New Old
Modular forms 102 12 90
Cusp forms 91 12 79
Eisenstein series 11 0 11

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

\(2\)\(7\)\(23\)FrickeDim.
\(-\)\(+\)\(+\)\(-\)\(5\)
\(-\)\(+\)\(-\)\(+\)\(1\)
\(-\)\(-\)\(+\)\(+\)\(1\)
\(-\)\(-\)\(-\)\(-\)\(5\)
Plus space\(+\)\(2\)
Minus space\(-\)\(10\)

Trace form

\( 12 q + 4 q^{3} + 4 q^{5} + 16 q^{9} + O(q^{10}) \) \( 12 q + 4 q^{3} + 4 q^{5} + 16 q^{9} + 12 q^{13} - 4 q^{15} + 8 q^{17} + 12 q^{19} - 4 q^{21} + 28 q^{25} + 16 q^{27} + 20 q^{29} - 8 q^{31} + 24 q^{33} + 4 q^{35} + 4 q^{37} + 4 q^{39} + 12 q^{41} + 24 q^{45} - 16 q^{47} + 12 q^{49} + 8 q^{51} - 12 q^{53} - 20 q^{55} - 28 q^{57} + 24 q^{59} - 8 q^{61} - 12 q^{65} - 4 q^{67} + 12 q^{71} + 4 q^{73} - 8 q^{75} + 12 q^{79} + 28 q^{81} - 32 q^{83} + 8 q^{85} - 48 q^{87} - 12 q^{91} - 4 q^{93} - 32 q^{95} + 44 q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(644))\) 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 7 23
644.2.a.a 644.a 1.a $1$ $5.142$ \(\Q\) None \(0\) \(-1\) \(0\) \(1\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{7}-2q^{9}-2q^{11}-3q^{13}+\cdots\)
644.2.a.b 644.a 1.a $1$ $5.142$ \(\Q\) None \(0\) \(1\) \(-2\) \(-1\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-2q^{5}-q^{7}-2q^{9}-2q^{11}+\cdots\)
644.2.a.c 644.a 1.a $5$ $5.142$ 5.5.8580816.1 None \(0\) \(1\) \(4\) \(5\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+(1+\beta _{4})q^{5}+q^{7}+(2+\beta _{2}+\cdots)q^{9}+\cdots\)
644.2.a.d 644.a 1.a $5$ $5.142$ 5.5.6963152.1 None \(0\) \(3\) \(2\) \(-5\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{3}+(1-\beta _{2}+\beta _{3})q^{5}-q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(644))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(644)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(46))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(92))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(161))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(322))\)\(^{\oplus 2}\)