Properties

Label 847.2.f
Level $847$
Weight $2$
Character orbit 847.f
Rep. character $\chi_{847}(148,\cdot)$
Character field $\Q(\zeta_{5})$
Dimension $216$
Newform subspaces $26$
Sturm bound $176$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 847 = 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 847.f (of order \(5\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 11 \)
Character field: \(\Q(\zeta_{5})\)
Newform subspaces: \( 26 \)
Sturm bound: \(176\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(2\), \(3\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(847, [\chi])\).

Total New Old
Modular forms 400 216 184
Cusp forms 304 216 88
Eisenstein series 96 0 96

Trace form

\( 216 q + 4 q^{2} + 8 q^{3} - 52 q^{4} + 4 q^{5} - 6 q^{6} + 2 q^{7} + 2 q^{8} - 44 q^{9} + O(q^{10}) \) \( 216 q + 4 q^{2} + 8 q^{3} - 52 q^{4} + 4 q^{5} - 6 q^{6} + 2 q^{7} + 2 q^{8} - 44 q^{9} + 16 q^{10} - 36 q^{12} + 2 q^{13} - 3 q^{14} + 22 q^{15} - 74 q^{16} + 16 q^{17} - 15 q^{18} - 10 q^{19} - 14 q^{20} - 12 q^{21} - 40 q^{23} + 14 q^{24} - 66 q^{25} + 10 q^{26} + 26 q^{27} - 11 q^{28} + 6 q^{29} - 12 q^{30} + 28 q^{31} - 12 q^{32} + 48 q^{34} + 8 q^{35} - 102 q^{36} - 4 q^{37} - 26 q^{38} - 6 q^{39} + 26 q^{40} + 32 q^{41} - 8 q^{43} - 200 q^{45} + 13 q^{46} + 28 q^{47} + 52 q^{48} - 54 q^{49} - 18 q^{50} + 26 q^{51} - 54 q^{52} + 14 q^{53} - 64 q^{54} + 18 q^{56} + 14 q^{57} + 35 q^{58} - 40 q^{59} + 108 q^{60} - 74 q^{62} + 10 q^{63} - 56 q^{64} - 16 q^{65} + 44 q^{67} - 6 q^{68} + 12 q^{69} + 32 q^{70} + 42 q^{71} - 3 q^{72} + 32 q^{73} + 38 q^{74} + 6 q^{75} - 4 q^{78} - 6 q^{79} + 160 q^{80} + 6 q^{81} - 30 q^{82} - 28 q^{83} - 12 q^{84} + 4 q^{85} + 101 q^{86} + 60 q^{87} - 128 q^{89} + 12 q^{90} - 18 q^{91} + 129 q^{92} - 10 q^{93} + 54 q^{94} - 56 q^{95} + 32 q^{96} - 32 q^{97} - 6 q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(847, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
847.2.f.a 847.f 11.c $4$ $6.763$ \(\Q(\zeta_{10})\) None \(-5\) \(-6\) \(2\) \(-1\) $\mathrm{SU}(2)[C_{5}]$ \(q+(-1-\zeta_{10}+\zeta_{10}^{2}+\zeta_{10}^{3})q^{2}+\cdots\)
847.2.f.b 847.f 11.c $4$ $6.763$ \(\Q(\zeta_{10})\) None \(-5\) \(4\) \(2\) \(1\) $\mathrm{SU}(2)[C_{5}]$ \(q+(-1-\zeta_{10}+\zeta_{10}^{2}+\zeta_{10}^{3})q^{2}+\cdots\)
847.2.f.c 847.f 11.c $4$ $6.763$ \(\Q(\zeta_{10})\) None \(-4\) \(3\) \(-1\) \(1\) $\mathrm{SU}(2)[C_{5}]$ \(q+(-2+\zeta_{10}-\zeta_{10}^{2}+2\zeta_{10}^{3})q^{2}+\cdots\)
847.2.f.d 847.f 11.c $4$ $6.763$ \(\Q(\zeta_{10})\) None \(-1\) \(-2\) \(-1\) \(-1\) $\mathrm{SU}(2)[C_{5}]$ \(q+(1-2\zeta_{10}+2\zeta_{10}^{2}-\zeta_{10}^{3})q^{2}+\cdots\)
847.2.f.e 847.f 11.c $4$ $6.763$ \(\Q(\zeta_{10})\) None \(-1\) \(-2\) \(2\) \(1\) $\mathrm{SU}(2)[C_{5}]$ \(q+(-1+\zeta_{10}-\zeta_{10}^{2}+\zeta_{10}^{3})q^{2}+\cdots\)
847.2.f.f 847.f 11.c $4$ $6.763$ \(\Q(\zeta_{10})\) None \(0\) \(-1\) \(-3\) \(-1\) $\mathrm{SU}(2)[C_{5}]$ \(q+\zeta_{10}^{2}q^{3}+2\zeta_{10}^{3}q^{4}-3\zeta_{10}q^{5}+\cdots\)
847.2.f.g 847.f 11.c $4$ $6.763$ \(\Q(\zeta_{10})\) None \(0\) \(-1\) \(-3\) \(1\) $\mathrm{SU}(2)[C_{5}]$ \(q+\zeta_{10}^{2}q^{3}+2\zeta_{10}^{3}q^{4}-3\zeta_{10}q^{5}+\cdots\)
847.2.f.h 847.f 11.c $4$ $6.763$ \(\Q(\zeta_{10})\) None \(0\) \(3\) \(1\) \(-1\) $\mathrm{SU}(2)[C_{5}]$ \(q-3\zeta_{10}^{2}q^{3}+2\zeta_{10}^{3}q^{4}+\zeta_{10}q^{5}+\cdots\)
847.2.f.i 847.f 11.c $4$ $6.763$ \(\Q(\zeta_{10})\) None \(0\) \(3\) \(1\) \(1\) $\mathrm{SU}(2)[C_{5}]$ \(q-3\zeta_{10}^{2}q^{3}+2\zeta_{10}^{3}q^{4}+\zeta_{10}q^{5}+\cdots\)
847.2.f.j 847.f 11.c $4$ $6.763$ \(\Q(\zeta_{10})\) None \(1\) \(-2\) \(-1\) \(1\) $\mathrm{SU}(2)[C_{5}]$ \(q+(-1+2\zeta_{10}-2\zeta_{10}^{2}+\zeta_{10}^{3})q^{2}+\cdots\)
847.2.f.k 847.f 11.c $4$ $6.763$ \(\Q(\zeta_{10})\) None \(1\) \(-2\) \(2\) \(-1\) $\mathrm{SU}(2)[C_{5}]$ \(q+(1-\zeta_{10}+\zeta_{10}^{2}-\zeta_{10}^{3})q^{2}+2\zeta_{10}^{2}q^{3}+\cdots\)
847.2.f.l 847.f 11.c $4$ $6.763$ \(\Q(\zeta_{10})\) None \(4\) \(3\) \(-1\) \(-1\) $\mathrm{SU}(2)[C_{5}]$ \(q+(2-\zeta_{10}+\zeta_{10}^{2}-2\zeta_{10}^{3})q^{2}+(\zeta_{10}+\cdots)q^{3}+\cdots\)
847.2.f.m 847.f 11.c $4$ $6.763$ \(\Q(\zeta_{10})\) None \(5\) \(-6\) \(2\) \(1\) $\mathrm{SU}(2)[C_{5}]$ \(q+(1+\zeta_{10}-\zeta_{10}^{2}-\zeta_{10}^{3})q^{2}+(-2\zeta_{10}+\cdots)q^{3}+\cdots\)
847.2.f.n 847.f 11.c $4$ $6.763$ \(\Q(\zeta_{10})\) None \(5\) \(4\) \(2\) \(-1\) $\mathrm{SU}(2)[C_{5}]$ \(q+(1+\zeta_{10}-\zeta_{10}^{2}-\zeta_{10}^{3})q^{2}+(2\zeta_{10}+\cdots)q^{3}+\cdots\)
847.2.f.o 847.f 11.c $8$ $6.763$ 8.0.446265625.1 None \(-1\) \(1\) \(0\) \(2\) $\mathrm{SU}(2)[C_{5}]$ \(q+\beta _{5}q^{2}+(-\beta _{1}+\beta _{2}+\beta _{5}+\beta _{6})q^{3}+\cdots\)
847.2.f.p 847.f 11.c $8$ $6.763$ 8.0.159390625.1 None \(-1\) \(6\) \(3\) \(2\) $\mathrm{SU}(2)[C_{5}]$ \(q-\beta _{1}q^{2}+(1-\beta _{3})q^{3}+(\beta _{1}-\beta _{2}-\beta _{3}+\cdots)q^{4}+\cdots\)
847.2.f.q 847.f 11.c $8$ $6.763$ 8.0.159390625.1 None \(1\) \(-4\) \(3\) \(-2\) $\mathrm{SU}(2)[C_{5}]$ \(q-\beta _{4}q^{2}+(-\beta _{2}-\beta _{3})q^{3}+(1-\beta _{1}+\cdots)q^{4}+\cdots\)
847.2.f.r 847.f 11.c $8$ $6.763$ 8.0.446265625.1 None \(1\) \(1\) \(0\) \(-2\) $\mathrm{SU}(2)[C_{5}]$ \(q-\beta _{5}q^{2}+(-\beta _{1}+\beta _{2}+\beta _{5}+\beta _{6})q^{3}+\cdots\)
847.2.f.s 847.f 11.c $8$ $6.763$ 8.0.159390625.1 None \(1\) \(6\) \(3\) \(-2\) $\mathrm{SU}(2)[C_{5}]$ \(q+\beta _{1}q^{2}+(1-\beta _{3})q^{3}+(\beta _{1}-\beta _{2}-\beta _{3}+\cdots)q^{4}+\cdots\)
847.2.f.t 847.f 11.c $12$ $6.763$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(-2\) \(1\) \(-1\) \(-3\) $\mathrm{SU}(2)[C_{5}]$ \(q+(-1-\beta _{1}-\beta _{2}-\beta _{3}+\beta _{6}+\beta _{7}+\cdots)q^{2}+\cdots\)
847.2.f.u 847.f 11.c $12$ $6.763$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(2\) \(1\) \(-1\) \(3\) $\mathrm{SU}(2)[C_{5}]$ \(q+(1+\beta _{1}+\beta _{2}+\beta _{3}-\beta _{6}-\beta _{7}-\beta _{8}+\cdots)q^{2}+\cdots\)
847.2.f.v 847.f 11.c $16$ $6.763$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(-2\) \(-2\) \(-5\) \(4\) $\mathrm{SU}(2)[C_{5}]$ \(q-\beta _{6}q^{2}+(-\beta _{10}+\beta _{11}-\beta _{14}-\beta _{15})q^{3}+\cdots\)
847.2.f.w 847.f 11.c $16$ $6.763$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(2\) \(-2\) \(-5\) \(-4\) $\mathrm{SU}(2)[C_{5}]$ \(q+\beta _{6}q^{2}+(-\beta _{10}+\beta _{11}-\beta _{14}-\beta _{15})q^{3}+\cdots\)
847.2.f.x 847.f 11.c $16$ $6.763$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(3\) \(-2\) \(-5\) \(4\) $\mathrm{SU}(2)[C_{5}]$ \(q+(\beta _{5}+\beta _{6})q^{2}+(\beta _{9}-\beta _{11}+\beta _{13})q^{3}+\cdots\)
847.2.f.y 847.f 11.c $24$ $6.763$ None \(-4\) \(2\) \(4\) \(-6\) $\mathrm{SU}(2)[C_{5}]$
847.2.f.z 847.f 11.c $24$ $6.763$ None \(4\) \(2\) \(4\) \(6\) $\mathrm{SU}(2)[C_{5}]$

Decomposition of \(S_{2}^{\mathrm{old}}(847, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(847, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(77, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(121, [\chi])\)\(^{\oplus 2}\)