Properties

Label 640.2.f
Level $640$
Weight $2$
Character orbit 640.f
Rep. character $\chi_{640}(449,\cdot)$
Character field $\Q$
Dimension $24$
Newform subspaces $7$
Sturm bound $192$
Trace bound $9$

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Defining parameters

Level: \( N \) \(=\) \( 640 = 2^{7} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 640.f (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 40 \)
Character field: \(\Q\)
Newform subspaces: \( 7 \)
Sturm bound: \(192\)
Trace bound: \(9\)
Distinguishing \(T_p\): \(3\), \(7\), \(13\), \(37\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(640, [\chi])\).

Total New Old
Modular forms 112 24 88
Cusp forms 80 24 56
Eisenstein series 32 0 32

Trace form

\( 24 q + 24 q^{9} + O(q^{10}) \) \( 24 q + 24 q^{9} - 8 q^{25} + 16 q^{41} - 24 q^{49} - 16 q^{65} + 56 q^{81} - 16 q^{89} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(640, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
640.2.f.a 640.f 40.f $2$ $5.110$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(-2\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+(-1+i)q^{5}-3q^{9}-6q^{13}-4iq^{17}+\cdots\)
640.2.f.b 640.f 40.f $2$ $5.110$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(2\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+(1+i)q^{5}-3q^{9}+6q^{13}+4iq^{17}+\cdots\)
640.2.f.c 640.f 40.f $4$ $5.110$ \(\Q(i, \sqrt{6})\) None \(0\) \(0\) \(-8\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{3}q^{3}+(-2-\beta _{1})q^{5}+\beta _{2}q^{7}+3q^{9}+\cdots\)
640.2.f.d 640.f 40.f $4$ $5.110$ \(\Q(\sqrt{-2}, \sqrt{5})\) \(\Q(\sqrt{-10}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+\beta _{3}q^{5}-\beta _{1}q^{7}-3q^{9}-\beta _{2}q^{11}+\cdots\)
640.2.f.e 640.f 40.f $4$ $5.110$ \(\Q(\sqrt{2}, \sqrt{-5})\) \(\Q(\sqrt{-5}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+\beta _{1}q^{3}-\beta _{3}q^{5}+\beta _{2}q^{7}-q^{9}+\beta _{2}q^{15}+\cdots\)
640.2.f.f 640.f 40.f $4$ $5.110$ \(\Q(\sqrt{-2}, \sqrt{-5})\) \(\Q(\sqrt{-5}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-\beta _{3}q^{3}-\beta _{2}q^{5}-3\beta _{1}q^{7}+7q^{9}+\cdots\)
640.2.f.g 640.f 40.f $4$ $5.110$ \(\Q(i, \sqrt{6})\) None \(0\) \(0\) \(8\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{3}q^{3}+(2+\beta _{1})q^{5}+\beta _{2}q^{7}+3q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(640, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(640, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(160, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(320, [\chi])\)\(^{\oplus 2}\)