Properties

Label 640.4.d
Level $640$
Weight $4$
Character orbit 640.d
Rep. character $\chi_{640}(321,\cdot)$
Character field $\Q$
Dimension $48$
Newform subspaces $9$
Sturm bound $384$
Trace bound $17$

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Defining parameters

Level: \( N \) \(=\) \( 640 = 2^{7} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 640.d (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 8 \)
Character field: \(\Q\)
Newform subspaces: \( 9 \)
Sturm bound: \(384\)
Trace bound: \(17\)
Distinguishing \(T_p\): \(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(640, [\chi])\).

Total New Old
Modular forms 304 48 256
Cusp forms 272 48 224
Eisenstein series 32 0 32

Trace form

\( 48 q - 432 q^{9} + O(q^{10}) \) \( 48 q - 432 q^{9} - 1200 q^{25} - 928 q^{33} + 160 q^{41} - 528 q^{49} - 32 q^{57} - 1728 q^{73} + 10032 q^{81} - 1696 q^{89} + 6336 q^{97} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(640, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
640.4.d.a 640.d 8.b $4$ $37.761$ \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+7\zeta_{8}^{2}q^{3}-5\zeta_{8}q^{5}+13\zeta_{8}^{3}q^{7}+\cdots\)
640.4.d.b 640.d 8.b $4$ $37.761$ \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\zeta_{8}^{2}q^{3}+5\zeta_{8}q^{5}+3\zeta_{8}^{3}q^{7}-23q^{9}+\cdots\)
640.4.d.c 640.d 8.b $4$ $37.761$ \(\Q(i, \sqrt{34})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{2}q^{3}-5\beta _{1}q^{5}+\beta _{3}q^{7}-7q^{9}+\cdots\)
640.4.d.d 640.d 8.b $4$ $37.761$ \(\Q(i, \sqrt{10})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{2}q^{3}+5\beta _{1}q^{5}-\beta _{3}q^{7}+17q^{9}+\cdots\)
640.4.d.e 640.d 8.b $4$ $37.761$ \(\Q(i, \sqrt{10})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{2}q^{3}-5\beta _{1}q^{5}+5\beta _{3}q^{7}+17q^{9}+\cdots\)
640.4.d.f 640.d 8.b $4$ $37.761$ \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\zeta_{8}^{2}q^{3}-\zeta_{8}q^{5}+3\zeta_{8}^{3}q^{7}+5^{2}q^{9}+\cdots\)
640.4.d.g 640.d 8.b $8$ $37.761$ 8.0.\(\cdots\).1 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{3}q^{3}-\beta _{1}q^{5}+(\beta _{2}-\beta _{6})q^{7}-31q^{9}+\cdots\)
640.4.d.h 640.d 8.b $8$ $37.761$ 8.0.\(\cdots\).2 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{6}q^{3}+5\beta _{1}q^{5}+(-\beta _{3}-\beta _{5})q^{7}+\cdots\)
640.4.d.i 640.d 8.b $8$ $37.761$ 8.0.40960000.1 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{3}+\beta _{2}q^{5}+(-\beta _{3}+\beta _{4})q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(640, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(640, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(8, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(32, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(64, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(128, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(160, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(320, [\chi])\)\(^{\oplus 2}\)