# Properties

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

# Related objects

## 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: $$\Q(\zeta_{10})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + ( -2 + \zeta_{10} - \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{2} + ( \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{3} + ( 3 - 3 \zeta_{10} - 3 \zeta_{10}^{3} ) q^{4} -\zeta_{10} q^{5} + ( -1 - \zeta_{10}^{2} ) q^{6} + \zeta_{10}^{3} q^{7} + ( 4 \zeta_{10} + \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{8} + ( 2 - \zeta_{10} + \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{9} +O(q^{10})$$ $$q + ( -2 + \zeta_{10} - \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{2} + ( \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{3} + ( 3 - 3 \zeta_{10} - 3 \zeta_{10}^{3} ) q^{4} -\zeta_{10} q^{5} + ( -1 - \zeta_{10}^{2} ) q^{6} + \zeta_{10}^{3} q^{7} + ( 4 \zeta_{10} + \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{8} + ( 2 - \zeta_{10} + \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{9} + ( 2 + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{10} + 3 q^{12} + ( 2 - 2 \zeta_{10}^{3} ) q^{13} + ( -\zeta_{10} - \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{14} + ( 1 - \zeta_{10} ) q^{15} + ( -3 - 5 \zeta_{10} - 3 \zeta_{10}^{2} ) q^{16} + ( -5 - 5 \zeta_{10}^{2} ) q^{17} + ( -3 + 3 \zeta_{10} + 5 \zeta_{10}^{3} ) q^{18} + ( 2 \zeta_{10} + 3 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{19} + ( -3 + 3 \zeta_{10}^{3} ) q^{20} + ( -\zeta_{10}^{2} + \zeta_{10}^{3} ) q^{21} + ( -3 - 5 \zeta_{10}^{2} + 5 \zeta_{10}^{3} ) q^{23} + ( -4 + 3 \zeta_{10} - 3 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{24} -4 \zeta_{10}^{2} q^{25} + ( -4 + 4 \zeta_{10} + 6 \zeta_{10}^{3} ) q^{26} + ( 4 - 3 \zeta_{10} + 4 \zeta_{10}^{2} ) q^{27} + ( 3 + 3 \zeta_{10}^{2} ) q^{28} + ( -1 + \zeta_{10} - 3 \zeta_{10}^{3} ) q^{29} + ( \zeta_{10} + \zeta_{10}^{3} ) q^{30} + ( 1 - 3 \zeta_{10} + 3 \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{31} + ( 9 + 3 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{32} + ( 15 + 10 \zeta_{10}^{2} - 10 \zeta_{10}^{3} ) q^{34} + ( 1 - \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{35} + ( -6 \zeta_{10} - 3 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{36} + ( -4 + 4 \zeta_{10} ) q^{37} + ( -7 - 5 \zeta_{10} - 7 \zeta_{10}^{2} ) q^{38} + 2 \zeta_{10} q^{39} + ( 4 - 4 \zeta_{10} - 5 \zeta_{10}^{3} ) q^{40} + ( -10 \zeta_{10} + 5 \zeta_{10}^{2} - 10 \zeta_{10}^{3} ) q^{41} + ( 1 - \zeta_{10}^{3} ) q^{42} + ( -2 - 9 \zeta_{10}^{2} + 9 \zeta_{10}^{3} ) q^{43} + ( -2 - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{45} + ( 11 - 3 \zeta_{10} + 3 \zeta_{10}^{2} - 11 \zeta_{10}^{3} ) q^{46} + ( \zeta_{10} - 6 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{47} + ( 5 - 5 \zeta_{10} - 3 \zeta_{10}^{3} ) q^{48} -\zeta_{10} q^{49} + ( 4 + 4 \zeta_{10} + 4 \zeta_{10}^{2} ) q^{50} -5 \zeta_{10}^{3} q^{51} + ( -6 \zeta_{10} - 6 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{52} + ( 4 - 3 \zeta_{10} + 3 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{53} + ( -6 - 5 \zeta_{10}^{2} + 5 \zeta_{10}^{3} ) q^{54} + ( -5 - 4 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{56} + ( -2 - \zeta_{10} + \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{57} + ( 2 \zeta_{10} + 3 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{58} + ( -5 + 5 \zeta_{10} - 3 \zeta_{10}^{3} ) q^{59} -3 \zeta_{10} q^{60} + ( 1 - 7 \zeta_{10} + \zeta_{10}^{2} ) q^{61} + ( 1 - \zeta_{10} ) q^{62} + ( \zeta_{10} + \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{63} + ( -5 - \zeta_{10} + \zeta_{10}^{2} + 5 \zeta_{10}^{3} ) q^{64} + ( -2 - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{65} + ( 3 + 7 \zeta_{10}^{2} - 7 \zeta_{10}^{3} ) q^{67} + ( -30 + 15 \zeta_{10} - 15 \zeta_{10}^{2} + 30 \zeta_{10}^{3} ) q^{68} + ( 2 \zeta_{10} - 7 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{69} + ( -1 + \zeta_{10} + 2 \zeta_{10}^{3} ) q^{70} + ( -5 + 13 \zeta_{10} - 5 \zeta_{10}^{2} ) q^{71} + ( 9 + 5 \zeta_{10} + 9 \zeta_{10}^{2} ) q^{72} + ( -2 + 2 \zeta_{10} - 11 \zeta_{10}^{3} ) q^{73} + ( -4 \zeta_{10} - 4 \zeta_{10}^{3} ) q^{74} + ( 4 \zeta_{10} - 4 \zeta_{10}^{2} ) q^{75} + ( 21 + 15 \zeta_{10}^{2} - 15 \zeta_{10}^{3} ) q^{76} + ( -4 - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{78} + ( 8 - 7 \zeta_{10} + 7 \zeta_{10}^{2} - 8 \zeta_{10}^{3} ) q^{79} + ( 3 \zeta_{10} + 5 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{80} + ( 6 - 6 \zeta_{10} - 2 \zeta_{10}^{3} ) q^{81} + ( 15 + 5 \zeta_{10} + 15 \zeta_{10}^{2} ) q^{82} + ( -6 - \zeta_{10} - 6 \zeta_{10}^{2} ) q^{83} + 3 \zeta_{10}^{3} q^{84} + ( 5 \zeta_{10} + 5 \zeta_{10}^{3} ) q^{85} + ( 13 - 2 \zeta_{10} + 2 \zeta_{10}^{2} - 13 \zeta_{10}^{3} ) q^{86} + ( -1 + 4 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{87} + ( 2 - 3 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{89} + ( 5 - 2 \zeta_{10} + 2 \zeta_{10}^{2} - 5 \zeta_{10}^{3} ) q^{90} + ( 2 \zeta_{10} + 2 \zeta_{10}^{3} ) q^{91} + ( -9 + 9 \zeta_{10} + 24 \zeta_{10}^{3} ) q^{92} + ( 3 - 5 \zeta_{10} + 3 \zeta_{10}^{2} ) q^{93} + ( 4 + 5 \zeta_{10} + 4 \zeta_{10}^{2} ) q^{94} + ( 2 - 2 \zeta_{10} - 5 \zeta_{10}^{3} ) q^{95} + ( 6 \zeta_{10} - 3 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{96} + ( -7 + 7 \zeta_{10} - 7 \zeta_{10}^{2} + 7 \zeta_{10}^{3} ) q^{97} + ( 2 + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{2} + 3q^{3} + 6q^{4} - q^{5} - 3q^{6} + q^{7} + 7q^{8} + 4q^{9} + O(q^{10})$$ $$4q - 4q^{2} + 3q^{3} + 6q^{4} - q^{5} - 3q^{6} + q^{7} + 7q^{8} + 4q^{9} + 6q^{10} + 12q^{12} + 6q^{13} - q^{14} + 3q^{15} - 14q^{16} - 15q^{17} - 4q^{18} + q^{19} - 9q^{20} + 2q^{21} - 2q^{23} - 6q^{24} + 4q^{25} - 6q^{26} + 9q^{27} + 9q^{28} - 6q^{29} + 2q^{30} - 3q^{31} + 30q^{32} + 40q^{34} + q^{35} - 9q^{36} - 12q^{37} - 26q^{38} + 2q^{39} + 7q^{40} - 25q^{41} + 3q^{42} + 10q^{43} - 6q^{45} + 27q^{46} + 8q^{47} + 12q^{48} - q^{49} + 16q^{50} - 5q^{51} - 6q^{52} + 6q^{53} - 14q^{54} - 12q^{56} - 8q^{57} + q^{58} - 18q^{59} - 3q^{60} - 4q^{61} + 3q^{62} + q^{63} - 17q^{64} - 4q^{65} - 2q^{67} - 60q^{68} + 11q^{69} - q^{70} - 2q^{71} + 32q^{72} - 17q^{73} - 8q^{74} + 8q^{75} + 54q^{76} - 12q^{78} + 10q^{79} + q^{80} + 16q^{81} + 50q^{82} - 19q^{83} + 3q^{84} + 10q^{85} + 35q^{86} - 12q^{87} + 14q^{89} + 11q^{90} + 4q^{91} - 3q^{92} + 4q^{93} + 17q^{94} + q^{95} + 15q^{96} - 7q^{97} + 6q^{98} + O(q^{100})$$

## 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
0.118034 + 0.363271i 1.30902 + 0.951057i 1.50000 1.08981i 0.309017 0.951057i −0.190983 + 0.587785i 0.809017 0.587785i 1.19098 + 0.865300i −0.118034 0.363271i 0.381966
323.1 −2.11803 1.53884i 0.190983 0.587785i 1.50000 + 4.61653i −0.809017 + 0.587785i −1.30902 + 0.951057i −0.309017 0.951057i 2.30902 7.10642i 2.11803 + 1.53884i 2.61803
372.1 0.118034 0.363271i 1.30902 0.951057i 1.50000 + 1.08981i 0.309017 + 0.951057i −0.190983 0.587785i 0.809017 + 0.587785i 1.19098 0.865300i −0.118034 + 0.363271i 0.381966
729.1 −2.11803 + 1.53884i 0.190983 + 0.587785i 1.50000 4.61653i −0.809017 0.587785i −1.30902 0.951057i −0.309017 + 0.951057i 2.30902 + 7.10642i 2.11803 1.53884i 2.61803
 $$n$$: 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 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 847.2.f.c 4
11.b odd 2 1 847.2.f.l 4
11.c even 5 1 847.2.a.h yes 2
11.c even 5 1 inner 847.2.f.c 4
11.c even 5 2 847.2.f.j 4
11.d odd 10 1 847.2.a.d 2
11.d odd 10 2 847.2.f.d 4
11.d odd 10 1 847.2.f.l 4
33.f even 10 1 7623.2.a.bx 2
33.h odd 10 1 7623.2.a.t 2
77.j odd 10 1 5929.2.a.s 2
77.l even 10 1 5929.2.a.i 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
847.2.a.d 2 11.d odd 10 1
847.2.a.h yes 2 11.c even 5 1
847.2.f.c 4 1.a even 1 1 trivial
847.2.f.c 4 11.c even 5 1 inner
847.2.f.d 4 11.d odd 10 2
847.2.f.j 4 11.c even 5 2
847.2.f.l 4 11.b odd 2 1
847.2.f.l 4 11.d odd 10 1
5929.2.a.i 2 77.l even 10 1
5929.2.a.s 2 77.j odd 10 1
7623.2.a.t 2 33.h odd 10 1
7623.2.a.bx 2 33.f even 10 1

## Hecke kernels

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

 $$T_{2}^{4} + 4 T_{2}^{3} + 6 T_{2}^{2} - T_{2} + 1$$ $$T_{3}^{4} - 3 T_{3}^{3} + 4 T_{3}^{2} - 2 T_{3} + 1$$ $$T_{13}^{4} - 6 T_{13}^{3} + 16 T_{13}^{2} - 16 T_{13} + 16$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + 4 T + 4 T^{2} - 7 T^{3} - 21 T^{4} - 14 T^{5} + 16 T^{6} + 32 T^{7} + 16 T^{8}$$
$3$ $$1 - 3 T + T^{2} + T^{3} + 4 T^{4} + 3 T^{5} + 9 T^{6} - 81 T^{7} + 81 T^{8}$$
$5$ $$1 + T - 4 T^{2} - 9 T^{3} + 11 T^{4} - 45 T^{5} - 100 T^{6} + 125 T^{7} + 625 T^{8}$$
$7$ $$1 - T + T^{2} - T^{3} + T^{4}$$
$11$ 
$13$ $$1 - 6 T + 3 T^{2} + 10 T^{3} + 81 T^{4} + 130 T^{5} + 507 T^{6} - 13182 T^{7} + 28561 T^{8}$$
$17$ $$1 + 15 T + 83 T^{2} + 165 T^{3} + 64 T^{4} + 2805 T^{5} + 23987 T^{6} + 73695 T^{7} + 83521 T^{8}$$
$19$ $$1 - T + 12 T^{2} - 53 T^{3} + 425 T^{4} - 1007 T^{5} + 4332 T^{6} - 6859 T^{7} + 130321 T^{8}$$
$23$ $$( 1 + T + 15 T^{2} + 23 T^{3} + 529 T^{4} )^{2}$$
$29$ $$1 + 6 T - 13 T^{2} - 192 T^{3} - 575 T^{4} - 5568 T^{5} - 10933 T^{6} + 146334 T^{7} + 707281 T^{8}$$
$31$ $$1 + 3 T - 12 T^{2} + 131 T^{3} + 1365 T^{4} + 4061 T^{5} - 11532 T^{6} + 89373 T^{7} + 923521 T^{8}$$
$37$ $$1 + 12 T + 27 T^{2} - 20 T^{3} + 441 T^{4} - 740 T^{5} + 36963 T^{6} + 607836 T^{7} + 1874161 T^{8}$$
$41$ $$1 + 25 T + 334 T^{2} + 3125 T^{3} + 22431 T^{4} + 128125 T^{5} + 561454 T^{6} + 1723025 T^{7} + 2825761 T^{8}$$
$43$ $$( 1 - 5 T - 9 T^{2} - 215 T^{3} + 1849 T^{4} )^{2}$$
$47$ $$1 - 8 T - 13 T^{2} + 430 T^{3} - 2449 T^{4} + 20210 T^{5} - 28717 T^{6} - 830584 T^{7} + 4879681 T^{8}$$
$53$ $$1 - 6 T - 37 T^{2} + 360 T^{3} + 121 T^{4} + 19080 T^{5} - 103933 T^{6} - 893262 T^{7} + 7890481 T^{8}$$
$59$ $$1 + 18 T + 65 T^{2} - 642 T^{3} - 8141 T^{4} - 37878 T^{5} + 226265 T^{6} + 3696822 T^{7} + 12117361 T^{8}$$
$61$ $$1 + 4 T - 15 T^{2} - 424 T^{3} - 271 T^{4} - 25864 T^{5} - 55815 T^{6} + 907924 T^{7} + 13845841 T^{8}$$
$67$ $$( 1 + T + 73 T^{2} + 67 T^{3} + 4489 T^{4} )^{2}$$
$71$ $$1 + 2 T + 133 T^{2} - 326 T^{3} + 8655 T^{4} - 23146 T^{5} + 670453 T^{6} + 715822 T^{7} + 25411681 T^{8}$$
$73$ $$1 + 17 T + 86 T^{2} - 779 T^{3} - 15281 T^{4} - 56867 T^{5} + 458294 T^{6} + 6613289 T^{7} + 28398241 T^{8}$$
$79$ $$1 - 10 T - 19 T^{2} + 910 T^{3} - 6929 T^{4} + 71890 T^{5} - 118579 T^{6} - 4930390 T^{7} + 38950081 T^{8}$$
$83$ $$1 + 19 T + 68 T^{2} - 345 T^{3} - 2479 T^{4} - 28635 T^{5} + 468452 T^{6} + 10863953 T^{7} + 47458321 T^{8}$$
$89$ $$( 1 - 7 T + 179 T^{2} - 623 T^{3} + 7921 T^{4} )^{2}$$
$97$ $$1 + 7 T - 48 T^{2} - 1015 T^{3} - 2449 T^{4} - 98455 T^{5} - 451632 T^{6} + 6388711 T^{7} + 88529281 T^{8}$$