Properties

Label 847.6.a
Level $847$
Weight $6$
Character orbit 847.a
Rep. character $\chi_{847}(1,\cdot)$
Character field $\Q$
Dimension $273$
Newform subspaces $19$
Sturm bound $528$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 847 = 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 847.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 19 \)
Sturm bound: \(528\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(847))\).

Total New Old
Modular forms 452 273 179
Cusp forms 428 273 155
Eisenstein series 24 0 24

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(7\)\(11\)FrickeDim.
\(+\)\(+\)\(+\)\(67\)
\(+\)\(-\)\(-\)\(70\)
\(-\)\(+\)\(-\)\(71\)
\(-\)\(-\)\(+\)\(65\)
Plus space\(+\)\(132\)
Minus space\(-\)\(141\)

Trace form

\( 273 q - 7 q^{2} + 20 q^{3} + 4415 q^{4} + 42 q^{5} - 74 q^{6} - 49 q^{7} - 867 q^{8} + 22169 q^{9} + O(q^{10}) \) \( 273 q - 7 q^{2} + 20 q^{3} + 4415 q^{4} + 42 q^{5} - 74 q^{6} - 49 q^{7} - 867 q^{8} + 22169 q^{9} - 372 q^{10} + 934 q^{12} - 1134 q^{13} - 147 q^{14} + 3160 q^{15} + 74375 q^{16} + 4534 q^{17} - 983 q^{18} - 1640 q^{19} - 684 q^{20} - 392 q^{21} - 3276 q^{23} - 13742 q^{24} + 166655 q^{25} + 17164 q^{26} - 5152 q^{27} + 3087 q^{28} - 7138 q^{29} - 6336 q^{30} - 6656 q^{31} - 27931 q^{32} + 35606 q^{34} - 7938 q^{35} + 400215 q^{36} + 45870 q^{37} - 12150 q^{38} - 5320 q^{39} - 26160 q^{40} + 10862 q^{41} - 16562 q^{42} - 2148 q^{43} + 5390 q^{45} - 31480 q^{46} - 4212 q^{47} + 105726 q^{48} + 655473 q^{49} + 36995 q^{50} - 56020 q^{51} - 12016 q^{52} + 51410 q^{53} + 45620 q^{54} + 20433 q^{56} + 8364 q^{57} + 41470 q^{58} + 4712 q^{59} - 33124 q^{60} - 21978 q^{61} + 81104 q^{62} - 10045 q^{63} + 1206171 q^{64} + 13156 q^{65} + 37348 q^{67} + 75318 q^{68} - 87584 q^{69} + 37044 q^{70} + 72624 q^{71} - 140203 q^{72} + 93262 q^{73} + 191306 q^{74} + 7112 q^{75} + 24990 q^{76} - 102340 q^{78} + 125916 q^{79} + 130976 q^{80} + 1776945 q^{81} - 274702 q^{82} - 111492 q^{83} - 51450 q^{84} - 163712 q^{85} + 236940 q^{86} + 43964 q^{87} - 479390 q^{89} + 188212 q^{90} + 22050 q^{91} - 805276 q^{92} + 177120 q^{93} - 10944 q^{94} + 189260 q^{95} + 116890 q^{96} + 196642 q^{97} - 16807 q^{98} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(847))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 7 11
847.6.a.a 847.a 1.a $1$ $135.845$ \(\Q\) None \(2\) \(-6\) \(-74\) \(49\) $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}-6q^{3}-28q^{4}-74q^{5}-12q^{6}+\cdots\)
847.6.a.b 847.a 1.a $1$ $135.845$ \(\Q\) None \(10\) \(-14\) \(-56\) \(49\) $-$ $-$ $\mathrm{SU}(2)$ \(q+10q^{2}-14q^{3}+68q^{4}-56q^{5}+\cdots\)
847.6.a.c 847.a 1.a $2$ $135.845$ \(\Q(\sqrt{57}) \) None \(-9\) \(-6\) \(-18\) \(-98\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-4-\beta )q^{2}-6\beta q^{3}+(-2+9\beta )q^{4}+\cdots\)
847.6.a.d 847.a 1.a $4$ $135.845$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(8\) \(-43\) \(57\) \(-196\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(2-\beta _{1})q^{2}+(-11+\beta _{1}+\beta _{3})q^{3}+\cdots\)
847.6.a.e 847.a 1.a $5$ $135.845$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(-10\) \(17\) \(119\) \(245\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-2+\beta _{1})q^{2}+(3+\beta _{1}-\beta _{4})q^{3}+\cdots\)
847.6.a.f 847.a 1.a $6$ $135.845$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(-4\) \(-12\) \(-62\) \(294\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+(-2+\beta _{1}+\beta _{3})q^{3}+\cdots\)
847.6.a.g 847.a 1.a $8$ $135.845$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-4\) \(42\) \(50\) \(-392\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+(6-\beta _{1}+\beta _{3})q^{3}+\cdots\)
847.6.a.h 847.a 1.a $12$ $135.845$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(-12\) \(-1\) \(52\) \(588\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+\beta _{3}q^{3}+(2^{4}-\beta _{1}+\cdots)q^{4}+\cdots\)
847.6.a.i 847.a 1.a $12$ $135.845$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(-4\) \(-1\) \(-10\) \(588\) $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(\beta _{1}+\beta _{3})q^{3}+(15+\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
847.6.a.j 847.a 1.a $12$ $135.845$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(4\) \(-1\) \(-10\) \(-588\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(\beta _{1}+\beta _{3})q^{3}+(15+\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
847.6.a.k 847.a 1.a $12$ $135.845$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(12\) \(-1\) \(52\) \(-588\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+\beta _{3}q^{3}+(2^{4}-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
847.6.a.l 847.a 1.a $13$ $135.845$ \(\mathbb{Q}[x]/(x^{13} - \cdots)\) None \(-8\) \(-1\) \(-69\) \(-637\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+(\beta _{1}+\beta _{5})q^{3}+(19+\cdots)q^{4}+\cdots\)
847.6.a.m 847.a 1.a $13$ $135.845$ \(\mathbb{Q}[x]/(x^{13} - \cdots)\) None \(8\) \(-1\) \(-69\) \(637\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+(\beta _{1}+\beta _{5})q^{3}+(19-\beta _{1}+\cdots)q^{4}+\cdots\)
847.6.a.n 847.a 1.a $26$ $135.845$ None \(-16\) \(-2\) \(156\) \(-1274\) $+$ $+$ $\mathrm{SU}(2)$
847.6.a.o 847.a 1.a $26$ $135.845$ None \(16\) \(-2\) \(156\) \(1274\) $-$ $+$ $\mathrm{SU}(2)$
847.6.a.p 847.a 1.a $28$ $135.845$ None \(-14\) \(-28\) \(-258\) \(1372\) $-$ $-$ $\mathrm{SU}(2)$
847.6.a.q 847.a 1.a $28$ $135.845$ None \(14\) \(-28\) \(-258\) \(-1372\) $+$ $+$ $\mathrm{SU}(2)$
847.6.a.r 847.a 1.a $32$ $135.845$ None \(-3\) \(54\) \(142\) \(-1568\) $+$ $-$ $\mathrm{SU}(2)$
847.6.a.s 847.a 1.a $32$ $135.845$ None \(3\) \(54\) \(142\) \(1568\) $-$ $+$ $\mathrm{SU}(2)$

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(847))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(847)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(77))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(121))\)\(^{\oplus 2}\)