Properties

Label 672.4.q
Level $672$
Weight $4$
Character orbit 672.q
Rep. character $\chi_{672}(193,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $96$
Newform subspaces $12$
Sturm bound $512$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 672.q (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 12 \)
Sturm bound: \(512\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(672, [\chi])\).

Total New Old
Modular forms 800 96 704
Cusp forms 736 96 640
Eisenstein series 64 0 64

Trace form

\( 96 q - 432 q^{9} + O(q^{10}) \) \( 96 q - 432 q^{9} - 144 q^{13} - 240 q^{21} - 1368 q^{25} + 24 q^{33} - 504 q^{37} + 1184 q^{41} - 1184 q^{49} - 784 q^{53} - 336 q^{57} + 1760 q^{61} + 560 q^{65} - 2184 q^{73} - 3872 q^{77} - 3888 q^{81} - 7296 q^{85} - 3424 q^{89} - 936 q^{93} - 7632 q^{97} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(672, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
672.4.q.a 672.q 7.c $2$ $39.649$ \(\Q(\sqrt{-3}) \) None \(0\) \(-3\) \(-15\) \(7\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-3+3\zeta_{6})q^{3}-15\zeta_{6}q^{5}+(-7+\cdots)q^{7}+\cdots\)
672.4.q.b 672.q 7.c $2$ $39.649$ \(\Q(\sqrt{-3}) \) None \(0\) \(3\) \(-15\) \(-7\) $\mathrm{SU}(2)[C_{3}]$ \(q+(3-3\zeta_{6})q^{3}-15\zeta_{6}q^{5}+(7-21\zeta_{6})q^{7}+\cdots\)
672.4.q.c 672.q 7.c $4$ $39.649$ \(\Q(\sqrt{-3}, \sqrt{37})\) None \(0\) \(-6\) \(12\) \(-36\) $\mathrm{SU}(2)[C_{3}]$ \(q-3\beta _{1}q^{3}+(6-6\beta _{1}-\beta _{2}-\beta _{3})q^{5}+\cdots\)
672.4.q.d 672.q 7.c $4$ $39.649$ \(\Q(\sqrt{-3}, \sqrt{37})\) None \(0\) \(6\) \(12\) \(36\) $\mathrm{SU}(2)[C_{3}]$ \(q+3\beta _{1}q^{3}+(6-6\beta _{1}-\beta _{2}-\beta _{3})q^{5}+\cdots\)
672.4.q.e 672.q 7.c $6$ $39.649$ 6.0.2972215728.1 None \(0\) \(-9\) \(-4\) \(17\) $\mathrm{SU}(2)[C_{3}]$ \(q-3\beta _{3}q^{3}+(-1+\beta _{3}+\beta _{5})q^{5}+(3+\cdots)q^{7}+\cdots\)
672.4.q.f 672.q 7.c $6$ $39.649$ 6.0.2972215728.1 None \(0\) \(9\) \(-4\) \(-17\) $\mathrm{SU}(2)[C_{3}]$ \(q+3\beta _{3}q^{3}+(-1+\beta _{3}+\beta _{5})q^{5}+(-3+\cdots)q^{7}+\cdots\)
672.4.q.g 672.q 7.c $10$ $39.649$ \(\mathbb{Q}[x]/(x^{10} + \cdots)\) None \(0\) \(-15\) \(0\) \(1\) $\mathrm{SU}(2)[C_{3}]$ \(q-3\beta _{3}q^{3}+\beta _{6}q^{5}+(1-\beta _{3}+\beta _{5}-\beta _{7}+\cdots)q^{7}+\cdots\)
672.4.q.h 672.q 7.c $10$ $39.649$ \(\mathbb{Q}[x]/(x^{10} + \cdots)\) None \(0\) \(15\) \(0\) \(-1\) $\mathrm{SU}(2)[C_{3}]$ \(q+3\beta _{3}q^{3}+\beta _{6}q^{5}+(-1+\beta _{3}-\beta _{5}+\cdots)q^{7}+\cdots\)
672.4.q.i 672.q 7.c $12$ $39.649$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(-18\) \(7\) \(22\) $\mathrm{SU}(2)[C_{3}]$ \(q+3\beta _{1}q^{3}+(1+\beta _{1}+\beta _{5})q^{5}+(1-2\beta _{1}+\cdots)q^{7}+\cdots\)
672.4.q.j 672.q 7.c $12$ $39.649$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(18\) \(7\) \(-22\) $\mathrm{SU}(2)[C_{3}]$ \(q-3\beta _{1}q^{3}+(1+\beta _{1}+\beta _{5})q^{5}+(-1+\cdots)q^{7}+\cdots\)
672.4.q.k 672.q 7.c $14$ $39.649$ \(\mathbb{Q}[x]/(x^{14} - \cdots)\) None \(0\) \(-21\) \(0\) \(9\) $\mathrm{SU}(2)[C_{3}]$ \(q+3\beta _{3}q^{3}-\beta _{9}q^{5}+(1+\beta _{3}+\beta _{5})q^{7}+\cdots\)
672.4.q.l 672.q 7.c $14$ $39.649$ \(\mathbb{Q}[x]/(x^{14} - \cdots)\) None \(0\) \(21\) \(0\) \(-9\) $\mathrm{SU}(2)[C_{3}]$ \(q-3\beta _{3}q^{3}-\beta _{9}q^{5}+(-1-\beta _{3}-\beta _{5}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(672, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(672, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(56, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(112, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(168, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(224, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(336, [\chi])\)\(^{\oplus 2}\)