Properties

Label 448.3.s
Level $448$
Weight $3$
Character orbit 448.s
Rep. character $\chi_{448}(129,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $60$
Newform subspaces $8$
Sturm bound $192$
Trace bound $9$

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Defining parameters

Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 448.s (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 8 \)
Sturm bound: \(192\)
Trace bound: \(9\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(448, [\chi])\).

Total New Old
Modular forms 280 68 212
Cusp forms 232 60 172
Eisenstein series 48 8 40

Trace form

\( 60 q + 6 q^{5} + 76 q^{9} + O(q^{10}) \) \( 60 q + 6 q^{5} + 76 q^{9} - 6 q^{17} + 118 q^{21} + 108 q^{25} + 40 q^{29} - 6 q^{33} - 30 q^{37} + 60 q^{45} - 4 q^{49} + 82 q^{53} - 44 q^{57} + 6 q^{61} + 48 q^{65} - 6 q^{73} - 46 q^{77} - 142 q^{81} + 396 q^{85} + 282 q^{89} + 182 q^{93} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(448, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
448.3.s.a 448.s 7.d $2$ $12.207$ \(\Q(\sqrt{-3}) \) None \(0\) \(-3\) \(-3\) \(-14\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-1-\zeta_{6})q^{3}+(-2+\zeta_{6})q^{5}-7q^{7}+\cdots\)
448.3.s.b 448.s 7.d $2$ $12.207$ \(\Q(\sqrt{-3}) \) None \(0\) \(3\) \(-3\) \(14\) $\mathrm{SU}(2)[C_{6}]$ \(q+(1+\zeta_{6})q^{3}+(-2+\zeta_{6})q^{5}+7q^{7}+\cdots\)
448.3.s.c 448.s 7.d $4$ $12.207$ \(\Q(\sqrt{2}, \sqrt{-3})\) None \(0\) \(-6\) \(6\) \(-8\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-2-\beta _{1}+\beta _{3})q^{3}+(1-\beta _{1}-2\beta _{2}+\cdots)q^{5}+\cdots\)
448.3.s.d 448.s 7.d $4$ $12.207$ \(\Q(\sqrt{2}, \sqrt{-3})\) None \(0\) \(6\) \(6\) \(8\) $\mathrm{SU}(2)[C_{6}]$ \(q+(2+\beta _{1}+\beta _{3})q^{3}+(1-\beta _{1}+2\beta _{2}+\cdots)q^{5}+\cdots\)
448.3.s.e 448.s 7.d $8$ $12.207$ 8.0.\(\cdots\).2 None \(0\) \(0\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{6}]$ \(q-\beta _{2}q^{3}+\beta _{7}q^{5}+(-2+3\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)
448.3.s.f 448.s 7.d $8$ $12.207$ 8.0.\(\cdots\).2 None \(0\) \(0\) \(0\) \(4\) $\mathrm{SU}(2)[C_{6}]$ \(q-\beta _{5}q^{3}-\beta _{4}q^{5}+(-1+3\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)
448.3.s.g 448.s 7.d $16$ $12.207$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q-\beta _{2}q^{3}-\beta _{10}q^{5}+\beta _{13}q^{7}+(1+\beta _{1}+\cdots)q^{9}+\cdots\)
448.3.s.h 448.s 7.d $16$ $12.207$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q-\beta _{2}q^{3}+\beta _{5}q^{5}+(-\beta _{2}-\beta _{12})q^{7}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(448, [\chi])\) into lower level spaces

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