Properties

Label 448.7.c
Level $448$
Weight $7$
Character orbit 448.c
Rep. character $\chi_{448}(321,\cdot)$
Character field $\Q$
Dimension $94$
Newform subspaces $12$
Sturm bound $448$
Trace bound $9$

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Defining parameters

Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 448.c (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q\)
Newform subspaces: \( 12 \)
Sturm bound: \(448\)
Trace bound: \(9\)
Distinguishing \(T_p\): \(3\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{7}(448, [\chi])\).

Total New Old
Modular forms 396 98 298
Cusp forms 372 94 278
Eisenstein series 24 4 20

Trace form

\( 94 q - 21874 q^{9} + O(q^{10}) \) \( 94 q - 21874 q^{9} - 20896 q^{21} - 268754 q^{25} + 33204 q^{29} + 107284 q^{37} - 2 q^{49} + 221844 q^{53} + 2912 q^{57} + 62496 q^{65} - 560108 q^{77} + 3430942 q^{81} - 3153024 q^{85} + 2990848 q^{93} + O(q^{100}) \)

Decomposition of \(S_{7}^{\mathrm{new}}(448, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
448.7.c.a 448.c 7.b $1$ $103.064$ \(\Q\) \(\Q(\sqrt{-7}) \) 7.7.b.a \(0\) \(0\) \(0\) \(-343\) $\mathrm{U}(1)[D_{2}]$ \(q-7^{3}q^{7}+3^{6}q^{9}-1962q^{11}-22734q^{23}+\cdots\)
448.7.c.b 448.c 7.b $1$ $103.064$ \(\Q\) \(\Q(\sqrt{-7}) \) 7.7.b.a \(0\) \(0\) \(0\) \(343\) $\mathrm{U}(1)[D_{2}]$ \(q+7^{3}q^{7}+3^{6}q^{9}+1962q^{11}+22734q^{23}+\cdots\)
448.7.c.c 448.c 7.b $2$ $103.064$ \(\Q(\sqrt{-510}) \) None 7.7.b.b \(0\) \(0\) \(0\) \(-266\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta q^{3}+\beta q^{5}+(-133-7\beta )q^{7}-1311q^{9}+\cdots\)
448.7.c.d 448.c 7.b $2$ $103.064$ \(\Q(\sqrt{-510}) \) None 7.7.b.b \(0\) \(0\) \(0\) \(266\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta q^{3}-\beta q^{5}+(133-7\beta )q^{7}-1311q^{9}+\cdots\)
448.7.c.e 448.c 7.b $4$ $103.064$ 4.0.211968.1 None 14.7.b.a \(0\) \(0\) \(0\) \(-308\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{1}q^{3}+5\beta _{2}q^{5}+(-77-14\beta _{1}+\cdots)q^{7}+\cdots\)
448.7.c.f 448.c 7.b $4$ $103.064$ 4.0.903168.1 None 28.7.b.a \(0\) \(0\) \(0\) \(-28\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{2}q^{3}+\beta _{1}q^{5}+(-7+\beta _{1}-2\beta _{2}+\cdots)q^{7}+\cdots\)
448.7.c.g 448.c 7.b $4$ $103.064$ 4.0.903168.1 None 28.7.b.a \(0\) \(0\) \(0\) \(28\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{2}q^{3}-\beta _{1}q^{5}+(7+\beta _{1}-2\beta _{2}+\beta _{3})q^{7}+\cdots\)
448.7.c.h 448.c 7.b $4$ $103.064$ 4.0.211968.1 None 14.7.b.a \(0\) \(0\) \(0\) \(308\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{1}q^{3}-5\beta _{2}q^{5}+(77-14\beta _{1}+7\beta _{2}+\cdots)q^{7}+\cdots\)
448.7.c.i 448.c 7.b $12$ $103.064$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None 56.7.c.a \(0\) \(0\) \(0\) \(-564\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{4}q^{3}+\beta _{8}q^{5}+(-47-\beta _{4}+\beta _{5}+\cdots)q^{7}+\cdots\)
448.7.c.j 448.c 7.b $12$ $103.064$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None 56.7.c.a \(0\) \(0\) \(0\) \(564\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{4}q^{3}-\beta _{8}q^{5}+(47-\beta _{4}-\beta _{5})q^{7}+\cdots\)
448.7.c.k 448.c 7.b $24$ $103.064$ None 224.7.c.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$
448.7.c.l 448.c 7.b $24$ $103.064$ None 224.7.c.b \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$

Decomposition of \(S_{7}^{\mathrm{old}}(448, [\chi])\) into lower level spaces

\( S_{7}^{\mathrm{old}}(448, [\chi]) \simeq \) \(S_{7}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 7}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(56, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(112, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(224, [\chi])\)\(^{\oplus 2}\)