Properties

Label 112.7.c
Level $112$
Weight $7$
Character orbit 112.c
Rep. character $\chi_{112}(97,\cdot)$
Character field $\Q$
Dimension $23$
Newform subspaces $5$
Sturm bound $112$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 112.c (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q\)
Newform subspaces: \( 5 \)
Sturm bound: \(112\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{7}(112, [\chi])\).

Total New Old
Modular forms 102 25 77
Cusp forms 90 23 67
Eisenstein series 12 2 10

Trace form

\( 23 q + 361 q^{7} - 5105 q^{9} + O(q^{10}) \) \( 23 q + 361 q^{7} - 5105 q^{9} - 678 q^{11} - 1456 q^{15} + 8752 q^{21} + 14530 q^{23} - 83233 q^{25} - 33738 q^{29} + 82032 q^{35} + 1798 q^{37} - 249040 q^{39} - 211078 q^{43} + 9767 q^{49} - 273984 q^{51} - 232442 q^{53} + 6992 q^{57} - 828431 q^{63} - 456080 q^{65} + 410666 q^{67} - 604270 q^{71} - 380890 q^{77} + 1766978 q^{79} + 1398999 q^{81} + 356544 q^{85} + 1190928 q^{91} - 863872 q^{93} + 2829840 q^{95} + 5472170 q^{99} + O(q^{100}) \)

Decomposition of \(S_{7}^{\mathrm{new}}(112, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
112.7.c.a 112.c 7.b $1$ $25.766$ \(\Q\) \(\Q(\sqrt{-7}) \) 7.7.b.a \(0\) \(0\) \(0\) \(343\) $\mathrm{U}(1)[D_{2}]$ \(q+7^{3}q^{7}+3^{6}q^{9}-1962q^{11}+22734q^{23}+\cdots\)
112.7.c.b 112.c 7.b $2$ $25.766$ \(\Q(\sqrt{-510}) \) None 7.7.b.b \(0\) \(0\) \(0\) \(-266\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta q^{3}+\beta q^{5}+(-133+7\beta )q^{7}-1311q^{9}+\cdots\)
112.7.c.c 112.c 7.b $4$ $25.766$ 4.0.211968.1 None 14.7.b.a \(0\) \(0\) \(0\) \(-308\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{1}q^{3}+5\beta _{2}q^{5}+(-77+14\beta _{1}+\cdots)q^{7}+\cdots\)
112.7.c.d 112.c 7.b $4$ $25.766$ 4.0.903168.1 None 28.7.b.a \(0\) \(0\) \(0\) \(28\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{2}q^{3}-\beta _{1}q^{5}+(7-\beta _{1}+2\beta _{2}+\beta _{3})q^{7}+\cdots\)
112.7.c.e 112.c 7.b $12$ $25.766$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None 56.7.c.a \(0\) \(0\) \(0\) \(564\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{4}q^{3}-\beta _{8}q^{5}+(47+\beta _{4}-\beta _{5})q^{7}+\cdots\)

Decomposition of \(S_{7}^{\mathrm{old}}(112, [\chi])\) into lower level spaces

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