# Properties

 Label 847.2.f.k Level 847 Weight 2 Character orbit 847.f Analytic conductor 6.763 Analytic rank 0 Dimension 4 CM no Inner twists 4

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## Newspace parameters

 Level: $$N$$ $$=$$ $$847 = 7 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 847.f (of order $$5$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.76332905120$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 77) Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{10}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{2} + 2 \zeta_{10}^{2} q^{3} + \zeta_{10}^{3} q^{4} + 2 \zeta_{10} q^{5} + 2 \zeta_{10} q^{6} -\zeta_{10}^{3} q^{7} + 3 \zeta_{10}^{2} q^{8} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{9} +O(q^{10})$$ $$q + ( 1 - \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{2} + 2 \zeta_{10}^{2} q^{3} + \zeta_{10}^{3} q^{4} + 2 \zeta_{10} q^{5} + 2 \zeta_{10} q^{6} -\zeta_{10}^{3} q^{7} + 3 \zeta_{10}^{2} q^{8} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{9} + 2 q^{10} -2 q^{12} + ( 4 - 4 \zeta_{10} + 4 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{13} -\zeta_{10}^{2} q^{14} + 4 \zeta_{10}^{3} q^{15} + \zeta_{10} q^{16} + 4 \zeta_{10} q^{17} + \zeta_{10}^{3} q^{18} + ( -2 + 2 \zeta_{10} - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{20} + 2 q^{21} -4 q^{23} + ( -6 + 6 \zeta_{10} - 6 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{24} -\zeta_{10}^{2} q^{25} -4 \zeta_{10}^{3} q^{26} + 4 \zeta_{10} q^{27} + \zeta_{10} q^{28} -6 \zeta_{10}^{3} q^{29} + 4 \zeta_{10}^{2} q^{30} + ( -10 + 10 \zeta_{10} - 10 \zeta_{10}^{2} + 10 \zeta_{10}^{3} ) q^{31} -5 q^{32} + 4 q^{34} + ( 2 - 2 \zeta_{10} + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{35} -\zeta_{10}^{2} q^{36} + 6 \zeta_{10}^{3} q^{37} + 8 \zeta_{10} q^{39} + 6 \zeta_{10}^{3} q^{40} -4 \zeta_{10}^{2} q^{41} + ( 2 - 2 \zeta_{10} + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{42} -12 q^{43} -2 q^{45} + ( -4 + 4 \zeta_{10} - 4 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{46} -10 \zeta_{10}^{2} q^{47} + 2 \zeta_{10}^{3} q^{48} -\zeta_{10} q^{49} -\zeta_{10} q^{50} + 8 \zeta_{10}^{3} q^{51} + 4 \zeta_{10}^{2} q^{52} + ( 6 - 6 \zeta_{10} + 6 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{53} + 4 q^{54} + 3 q^{56} -6 \zeta_{10}^{2} q^{58} -2 \zeta_{10}^{3} q^{59} -4 \zeta_{10} q^{60} + 10 \zeta_{10}^{3} q^{62} + \zeta_{10}^{2} q^{63} + ( -7 + 7 \zeta_{10} - 7 \zeta_{10}^{2} + 7 \zeta_{10}^{3} ) q^{64} + 8 q^{65} + 8 q^{67} + ( -4 + 4 \zeta_{10} - 4 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{68} -8 \zeta_{10}^{2} q^{69} -2 \zeta_{10}^{3} q^{70} + 12 \zeta_{10} q^{71} -3 \zeta_{10} q^{72} -8 \zeta_{10}^{3} q^{73} + 6 \zeta_{10}^{2} q^{74} + ( 2 - 2 \zeta_{10} + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{75} + 8 q^{78} + ( 8 - 8 \zeta_{10} + 8 \zeta_{10}^{2} - 8 \zeta_{10}^{3} ) q^{79} + 2 \zeta_{10}^{2} q^{80} + 11 \zeta_{10}^{3} q^{81} -4 \zeta_{10} q^{82} + 2 \zeta_{10}^{3} q^{84} + 8 \zeta_{10}^{2} q^{85} + ( -12 + 12 \zeta_{10} - 12 \zeta_{10}^{2} + 12 \zeta_{10}^{3} ) q^{86} + 12 q^{87} -6 q^{89} + ( -2 + 2 \zeta_{10} - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{90} -4 \zeta_{10}^{2} q^{91} -4 \zeta_{10}^{3} q^{92} -20 \zeta_{10} q^{93} -10 \zeta_{10} q^{94} -10 \zeta_{10}^{2} q^{96} + ( 10 - 10 \zeta_{10} + 10 \zeta_{10}^{2} - 10 \zeta_{10}^{3} ) q^{97} - q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + q^{2} - 2q^{3} + q^{4} + 2q^{5} + 2q^{6} - q^{7} - 3q^{8} - q^{9} + O(q^{10})$$ $$4q + q^{2} - 2q^{3} + q^{4} + 2q^{5} + 2q^{6} - q^{7} - 3q^{8} - q^{9} + 8q^{10} - 8q^{12} + 4q^{13} + q^{14} + 4q^{15} + q^{16} + 4q^{17} + q^{18} - 2q^{20} + 8q^{21} - 16q^{23} - 6q^{24} + q^{25} - 4q^{26} + 4q^{27} + q^{28} - 6q^{29} - 4q^{30} - 10q^{31} - 20q^{32} + 16q^{34} + 2q^{35} + q^{36} + 6q^{37} + 8q^{39} + 6q^{40} + 4q^{41} + 2q^{42} - 48q^{43} - 8q^{45} - 4q^{46} + 10q^{47} + 2q^{48} - q^{49} - q^{50} + 8q^{51} - 4q^{52} + 6q^{53} + 16q^{54} + 12q^{56} + 6q^{58} - 2q^{59} - 4q^{60} + 10q^{62} - q^{63} - 7q^{64} + 32q^{65} + 32q^{67} - 4q^{68} + 8q^{69} - 2q^{70} + 12q^{71} - 3q^{72} - 8q^{73} - 6q^{74} + 2q^{75} + 32q^{78} + 8q^{79} - 2q^{80} + 11q^{81} - 4q^{82} + 2q^{84} - 8q^{85} - 12q^{86} + 48q^{87} - 24q^{89} - 2q^{90} + 4q^{91} - 4q^{92} - 20q^{93} - 10q^{94} + 10q^{96} + 10q^{97} - 4q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/847\mathbb{Z}\right)^\times$$.

 $$n$$ $$122$$ $$365$$ $$\chi(n)$$ $$1$$ $$-\zeta_{10}^{3}$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
148.1
 −0.309017 + 0.951057i 0.809017 − 0.587785i −0.309017 − 0.951057i 0.809017 + 0.587785i
−0.309017 0.951057i −1.61803 1.17557i 0.809017 0.587785i −0.618034 + 1.90211i −0.618034 + 1.90211i −0.809017 + 0.587785i −2.42705 1.76336i 0.309017 + 0.951057i 2.00000
323.1 0.809017 + 0.587785i 0.618034 1.90211i −0.309017 0.951057i 1.61803 1.17557i 1.61803 1.17557i 0.309017 + 0.951057i 0.927051 2.85317i −0.809017 0.587785i 2.00000
372.1 −0.309017 + 0.951057i −1.61803 + 1.17557i 0.809017 + 0.587785i −0.618034 1.90211i −0.618034 1.90211i −0.809017 0.587785i −2.42705 + 1.76336i 0.309017 0.951057i 2.00000
729.1 0.809017 0.587785i 0.618034 + 1.90211i −0.309017 + 0.951057i 1.61803 + 1.17557i 1.61803 + 1.17557i 0.309017 0.951057i 0.927051 + 2.85317i −0.809017 + 0.587785i 2.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 3 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 847.2.f.k 4
11.b odd 2 1 847.2.f.e 4
11.c even 5 1 847.2.a.a 1
11.c even 5 3 inner 847.2.f.k 4
11.d odd 10 1 77.2.a.c 1
11.d odd 10 3 847.2.f.e 4
33.f even 10 1 693.2.a.a 1
33.h odd 10 1 7623.2.a.n 1
44.g even 10 1 1232.2.a.a 1
55.h odd 10 1 1925.2.a.c 1
55.l even 20 2 1925.2.b.d 2
77.j odd 10 1 5929.2.a.b 1
77.l even 10 1 539.2.a.d 1
77.n even 30 2 539.2.e.b 2
77.o odd 30 2 539.2.e.a 2
88.k even 10 1 4928.2.a.bi 1
88.p odd 10 1 4928.2.a.g 1
231.r odd 10 1 4851.2.a.a 1
308.s odd 10 1 8624.2.a.bc 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.a.c 1 11.d odd 10 1
539.2.a.d 1 77.l even 10 1
539.2.e.a 2 77.o odd 30 2
539.2.e.b 2 77.n even 30 2
693.2.a.a 1 33.f even 10 1
847.2.a.a 1 11.c even 5 1
847.2.f.e 4 11.b odd 2 1
847.2.f.e 4 11.d odd 10 3
847.2.f.k 4 1.a even 1 1 trivial
847.2.f.k 4 11.c even 5 3 inner
1232.2.a.a 1 44.g even 10 1
1925.2.a.c 1 55.h odd 10 1
1925.2.b.d 2 55.l even 20 2
4851.2.a.a 1 231.r odd 10 1
4928.2.a.g 1 88.p odd 10 1
4928.2.a.bi 1 88.k even 10 1
5929.2.a.b 1 77.j odd 10 1
7623.2.a.n 1 33.h odd 10 1
8624.2.a.bc 1 308.s odd 10 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(847, [\chi])$$:

 $$T_{2}^{4} - T_{2}^{3} + T_{2}^{2} - T_{2} + 1$$ $$T_{3}^{4} + 2 T_{3}^{3} + 4 T_{3}^{2} + 8 T_{3} + 16$$ $$T_{13}^{4} - 4 T_{13}^{3} + 16 T_{13}^{2} - 64 T_{13} + 256$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T - T^{2} + 3 T^{3} - T^{4} + 6 T^{5} - 4 T^{6} - 8 T^{7} + 16 T^{8}$$
$3$ $$1 + 2 T + T^{2} - 4 T^{3} - 11 T^{4} - 12 T^{5} + 9 T^{6} + 54 T^{7} + 81 T^{8}$$
$5$ $$1 - 2 T - T^{2} + 12 T^{3} - 19 T^{4} + 60 T^{5} - 25 T^{6} - 250 T^{7} + 625 T^{8}$$
$7$ $$1 + T + T^{2} + T^{3} + T^{4}$$
$11$ 1
$13$ $$1 - 4 T + 3 T^{2} + 40 T^{3} - 199 T^{4} + 520 T^{5} + 507 T^{6} - 8788 T^{7} + 28561 T^{8}$$
$17$ $$1 - 4 T - T^{2} + 72 T^{3} - 271 T^{4} + 1224 T^{5} - 289 T^{6} - 19652 T^{7} + 83521 T^{8}$$
$19$ $$1 - 19 T^{2} + 361 T^{4} - 6859 T^{6} + 130321 T^{8}$$
$23$ $$( 1 + 4 T + 23 T^{2} )^{4}$$
$29$ $$1 + 6 T + 7 T^{2} - 132 T^{3} - 995 T^{4} - 3828 T^{5} + 5887 T^{6} + 146334 T^{7} + 707281 T^{8}$$
$31$ $$1 + 10 T + 69 T^{2} + 380 T^{3} + 1661 T^{4} + 11780 T^{5} + 66309 T^{6} + 297910 T^{7} + 923521 T^{8}$$
$37$ $$1 - 6 T - T^{2} + 228 T^{3} - 1331 T^{4} + 8436 T^{5} - 1369 T^{6} - 303918 T^{7} + 1874161 T^{8}$$
$41$ $$1 - 4 T - 25 T^{2} + 264 T^{3} - 31 T^{4} + 10824 T^{5} - 42025 T^{6} - 275684 T^{7} + 2825761 T^{8}$$
$43$ $$( 1 + 12 T + 43 T^{2} )^{4}$$
$47$ $$1 - 10 T + 53 T^{2} - 60 T^{3} - 1891 T^{4} - 2820 T^{5} + 117077 T^{6} - 1038230 T^{7} + 4879681 T^{8}$$
$53$ $$1 - 6 T - 17 T^{2} + 420 T^{3} - 1619 T^{4} + 22260 T^{5} - 47753 T^{6} - 893262 T^{7} + 7890481 T^{8}$$
$59$ $$1 + 2 T - 55 T^{2} - 228 T^{3} + 2789 T^{4} - 13452 T^{5} - 191455 T^{6} + 410758 T^{7} + 12117361 T^{8}$$
$61$ $$1 - 61 T^{2} + 3721 T^{4} - 226981 T^{6} + 13845841 T^{8}$$
$67$ $$( 1 - 8 T + 67 T^{2} )^{4}$$
$71$ $$1 - 12 T + 73 T^{2} - 24 T^{3} - 4895 T^{4} - 1704 T^{5} + 367993 T^{6} - 4294932 T^{7} + 25411681 T^{8}$$
$73$ $$1 + 8 T - 9 T^{2} - 656 T^{3} - 4591 T^{4} - 47888 T^{5} - 47961 T^{6} + 3112136 T^{7} + 28398241 T^{8}$$
$79$ $$1 - 8 T - 15 T^{2} + 752 T^{3} - 4831 T^{4} + 59408 T^{5} - 93615 T^{6} - 3944312 T^{7} + 38950081 T^{8}$$
$83$ $$1 - 83 T^{2} + 6889 T^{4} - 571787 T^{6} + 47458321 T^{8}$$
$89$ $$( 1 + 6 T + 89 T^{2} )^{4}$$
$97$ $$1 - 10 T + 3 T^{2} + 940 T^{3} - 9691 T^{4} + 91180 T^{5} + 28227 T^{6} - 9126730 T^{7} + 88529281 T^{8}$$
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