# Properties

 Label 847.2.f.j 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: $$\Q(\zeta_{10})$$ Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ 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.

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

## 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
0.809017 + 2.48990i −0.500000 0.363271i −3.92705 + 2.85317i 0.309017 0.951057i 0.500000 1.53884i 0.809017 0.587785i −6.04508 4.39201i −0.809017 2.48990i 2.61803
323.1 −0.309017 0.224514i −0.500000 + 1.53884i −0.572949 1.76336i −0.809017 + 0.587785i 0.500000 0.363271i −0.309017 0.951057i −0.454915 + 1.40008i 0.309017 + 0.224514i 0.381966
372.1 0.809017 2.48990i −0.500000 + 0.363271i −3.92705 2.85317i 0.309017 + 0.951057i 0.500000 + 1.53884i 0.809017 + 0.587785i −6.04508 + 4.39201i −0.809017 + 2.48990i 2.61803
729.1 −0.309017 + 0.224514i −0.500000 1.53884i −0.572949 + 1.76336i −0.809017 0.587785i 0.500000 + 0.363271i −0.309017 + 0.951057i −0.454915 1.40008i 0.309017 0.224514i 0.381966
 $$n$$: 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.j 4
11.b odd 2 1 847.2.f.d 4
11.c even 5 1 847.2.a.h yes 2
11.c even 5 2 847.2.f.c 4
11.c even 5 1 inner 847.2.f.j 4
11.d odd 10 1 847.2.a.d 2
11.d odd 10 1 847.2.f.d 4
11.d odd 10 2 847.2.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 11.c even 5 2
847.2.f.d 4 11.b odd 2 1
847.2.f.d 4 11.d odd 10 1
847.2.f.j 4 1.a even 1 1 trivial
847.2.f.j 4 11.c even 5 1 inner
847.2.f.l 4 11.d odd 10 2
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 $$S_{2}^{\mathrm{new}}(847, [\chi])$$:

 $$T_{2}^{4} - T_{2}^{3} + 6 T_{2}^{2} + 4 T_{2} + 1$$ $$T_{3}^{4} + 2 T_{3}^{3} + 4 T_{3}^{2} + 3 T_{3} + 1$$ $$T_{13}^{4} + 4 T_{13}^{3} + 16 T_{13}^{2} + 24 T_{13} + 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 4 T + 6 T^{2} - T^{3} + T^{4}$$
$3$ $$1 + 3 T + 4 T^{2} + 2 T^{3} + T^{4}$$
$5$ $$1 + T + T^{2} + T^{3} + T^{4}$$
$7$ $$1 - T + T^{2} - T^{3} + T^{4}$$
$11$ $$T^{4}$$
$13$ $$16 + 24 T + 16 T^{2} + 4 T^{3} + T^{4}$$
$17$ $$625 - 375 T + 100 T^{2} - 10 T^{3} + T^{4}$$
$19$ $$121 - 11 T + 31 T^{2} + 9 T^{3} + T^{4}$$
$23$ $$( -31 + T + T^{2} )^{2}$$
$29$ $$121 + 66 T + 16 T^{2} + T^{3} + T^{4}$$
$31$ $$1 - 3 T + 19 T^{2} - 7 T^{3} + T^{4}$$
$37$ $$256 - 192 T + 64 T^{2} - 8 T^{3} + T^{4}$$
$41$ $$15625 - 3125 T + 375 T^{2} - 25 T^{3} + T^{4}$$
$43$ $$( -95 - 5 T + T^{2} )^{2}$$
$47$ $$841 - 232 T + 34 T^{2} - 3 T^{3} + T^{4}$$
$53$ $$121 - 66 T + 16 T^{2} - T^{3} + T^{4}$$
$59$ $$1 - 18 T + 124 T^{2} - 7 T^{3} + T^{4}$$
$61$ $$1681 + 164 T + 46 T^{2} + 9 T^{3} + T^{4}$$
$67$ $$( -61 + T + T^{2} )^{2}$$
$71$ $$6241 + 158 T + 204 T^{2} - 23 T^{3} + T^{4}$$
$73$ $$19321 + 2363 T + 159 T^{2} + 7 T^{3} + T^{4}$$
$79$ $$3025 - 550 T + 60 T^{2} - 5 T^{3} + T^{4}$$
$83$ $$841 - 551 T + 151 T^{2} - 11 T^{3} + T^{4}$$
$89$ $$( 1 - 7 T + T^{2} )^{2}$$
$97$ $$2401 + 343 T + 49 T^{2} + 7 T^{3} + T^{4}$$