# Properties

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

# 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: $$\Z[a_1, \ldots, a_{4}]$$ 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.

 $$f(q)$$ $$=$$ $$q -3 \zeta_{10}^{2} q^{3} + 2 \zeta_{10}^{3} q^{4} + \zeta_{10} q^{5} + \zeta_{10}^{3} q^{7} + ( -6 + 6 \zeta_{10} - 6 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{9} +O(q^{10})$$ $$q -3 \zeta_{10}^{2} q^{3} + 2 \zeta_{10}^{3} q^{4} + \zeta_{10} q^{5} + \zeta_{10}^{3} q^{7} + ( -6 + 6 \zeta_{10} - 6 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{9} + 6 q^{12} + ( 4 - 4 \zeta_{10} + 4 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{13} -3 \zeta_{10}^{3} q^{15} -4 \zeta_{10} q^{16} -2 \zeta_{10} q^{17} -6 \zeta_{10}^{2} q^{19} + ( -2 + 2 \zeta_{10} - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{20} + 3 q^{21} -5 q^{23} -4 \zeta_{10}^{2} q^{25} + 9 \zeta_{10} q^{27} -2 \zeta_{10} q^{28} -10 \zeta_{10}^{3} q^{29} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{31} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{35} -12 \zeta_{10}^{2} q^{36} + 5 \zeta_{10}^{3} q^{37} -12 \zeta_{10} q^{39} -2 \zeta_{10}^{2} q^{41} -8 q^{43} -6 q^{45} + 8 \zeta_{10}^{2} q^{47} + 12 \zeta_{10}^{3} q^{48} -\zeta_{10} q^{49} + 6 \zeta_{10}^{3} q^{51} + 8 \zeta_{10}^{2} q^{52} + ( 6 - 6 \zeta_{10} + 6 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{53} + ( -18 + 18 \zeta_{10} - 18 \zeta_{10}^{2} + 18 \zeta_{10}^{3} ) q^{57} -3 \zeta_{10}^{3} q^{59} + 6 \zeta_{10} q^{60} + 2 \zeta_{10} q^{61} -6 \zeta_{10}^{2} q^{63} + ( 8 - 8 \zeta_{10} + 8 \zeta_{10}^{2} - 8 \zeta_{10}^{3} ) q^{64} + 4 q^{65} -3 q^{67} + ( 4 - 4 \zeta_{10} + 4 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{68} + 15 \zeta_{10}^{2} q^{69} -\zeta_{10} q^{71} -10 \zeta_{10}^{3} q^{73} + ( -12 + 12 \zeta_{10} - 12 \zeta_{10}^{2} + 12 \zeta_{10}^{3} ) q^{75} + 12 q^{76} + ( -6 + 6 \zeta_{10} - 6 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{79} -4 \zeta_{10}^{2} q^{80} -9 \zeta_{10}^{3} q^{81} -12 \zeta_{10} q^{83} + 6 \zeta_{10}^{3} q^{84} -2 \zeta_{10}^{2} q^{85} -30 q^{87} -15 q^{89} + 4 \zeta_{10}^{2} q^{91} -10 \zeta_{10}^{3} q^{92} + 3 \zeta_{10} q^{93} -6 \zeta_{10}^{3} q^{95} + ( 5 - 5 \zeta_{10} + 5 \zeta_{10}^{2} - 5 \zeta_{10}^{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 3 q^{3} + 2 q^{4} + q^{5} + q^{7} - 6 q^{9} + O(q^{10})$$ $$4 q + 3 q^{3} + 2 q^{4} + q^{5} + q^{7} - 6 q^{9} + 24 q^{12} + 4 q^{13} - 3 q^{15} - 4 q^{16} - 2 q^{17} + 6 q^{19} - 2 q^{20} + 12 q^{21} - 20 q^{23} + 4 q^{25} + 9 q^{27} - 2 q^{28} - 10 q^{29} - q^{31} - q^{35} + 12 q^{36} + 5 q^{37} - 12 q^{39} + 2 q^{41} - 32 q^{43} - 24 q^{45} - 8 q^{47} + 12 q^{48} - q^{49} + 6 q^{51} - 8 q^{52} + 6 q^{53} - 18 q^{57} - 3 q^{59} + 6 q^{60} + 2 q^{61} + 6 q^{63} + 8 q^{64} + 16 q^{65} - 12 q^{67} + 4 q^{68} - 15 q^{69} - q^{71} - 10 q^{73} - 12 q^{75} + 48 q^{76} - 6 q^{79} + 4 q^{80} - 9 q^{81} - 12 q^{83} + 6 q^{84} + 2 q^{85} - 120 q^{87} - 60 q^{89} - 4 q^{91} - 10 q^{92} + 3 q^{93} - 6 q^{95} + 5 q^{97} + O(q^{100})$$

## 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 2.42705 + 1.76336i 1.61803 1.17557i −0.309017 + 0.951057i 0 0.809017 0.587785i 0 1.85410 + 5.70634i 0
323.1 0 −0.927051 + 2.85317i −0.618034 1.90211i 0.809017 0.587785i 0 −0.309017 0.951057i 0 −4.85410 3.52671i 0
372.1 0 2.42705 1.76336i 1.61803 + 1.17557i −0.309017 0.951057i 0 0.809017 + 0.587785i 0 1.85410 5.70634i 0
729.1 0 −0.927051 2.85317i −0.618034 + 1.90211i 0.809017 + 0.587785i 0 −0.309017 + 0.951057i 0 −4.85410 + 3.52671i 0
 $$n$$: 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.i 4
11.b odd 2 1 847.2.f.h 4
11.c even 5 1 77.2.a.a 1
11.c even 5 3 inner 847.2.f.i 4
11.d odd 10 1 847.2.a.b 1
11.d odd 10 3 847.2.f.h 4
33.f even 10 1 7623.2.a.j 1
33.h odd 10 1 693.2.a.c 1
44.h odd 10 1 1232.2.a.l 1
55.j even 10 1 1925.2.a.h 1
55.k odd 20 2 1925.2.b.e 2
77.j odd 10 1 539.2.a.c 1
77.l even 10 1 5929.2.a.f 1
77.m even 15 2 539.2.e.f 2
77.p odd 30 2 539.2.e.c 2
88.l odd 10 1 4928.2.a.a 1
88.o even 10 1 4928.2.a.bj 1
231.u even 10 1 4851.2.a.j 1
308.t even 10 1 8624.2.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.a.a 1 11.c even 5 1
539.2.a.c 1 77.j odd 10 1
539.2.e.c 2 77.p odd 30 2
539.2.e.f 2 77.m even 15 2
693.2.a.c 1 33.h odd 10 1
847.2.a.b 1 11.d odd 10 1
847.2.f.h 4 11.b odd 2 1
847.2.f.h 4 11.d odd 10 3
847.2.f.i 4 1.a even 1 1 trivial
847.2.f.i 4 11.c even 5 3 inner
1232.2.a.l 1 44.h odd 10 1
1925.2.a.h 1 55.j even 10 1
1925.2.b.e 2 55.k odd 20 2
4851.2.a.j 1 231.u even 10 1
4928.2.a.a 1 88.l odd 10 1
4928.2.a.bj 1 88.o even 10 1
5929.2.a.f 1 77.l even 10 1
7623.2.a.j 1 33.f even 10 1
8624.2.a.a 1 308.t even 10 1

## Hecke kernels

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

 $$T_{2}$$ $$T_{3}^{4} - 3 T_{3}^{3} + 9 T_{3}^{2} - 27 T_{3} + 81$$ $$T_{13}^{4} - 4 T_{13}^{3} + 16 T_{13}^{2} - 64 T_{13} + 256$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$81 - 27 T + 9 T^{2} - 3 T^{3} + T^{4}$$
$5$ $$1 - T + T^{2} - T^{3} + T^{4}$$
$7$ $$1 - T + T^{2} - T^{3} + T^{4}$$
$11$ $$T^{4}$$
$13$ $$256 - 64 T + 16 T^{2} - 4 T^{3} + T^{4}$$
$17$ $$16 + 8 T + 4 T^{2} + 2 T^{3} + T^{4}$$
$19$ $$1296 - 216 T + 36 T^{2} - 6 T^{3} + T^{4}$$
$23$ $$( 5 + T )^{4}$$
$29$ $$10000 + 1000 T + 100 T^{2} + 10 T^{3} + T^{4}$$
$31$ $$1 + T + T^{2} + T^{3} + T^{4}$$
$37$ $$625 - 125 T + 25 T^{2} - 5 T^{3} + T^{4}$$
$41$ $$16 - 8 T + 4 T^{2} - 2 T^{3} + T^{4}$$
$43$ $$( 8 + T )^{4}$$
$47$ $$4096 + 512 T + 64 T^{2} + 8 T^{3} + T^{4}$$
$53$ $$1296 - 216 T + 36 T^{2} - 6 T^{3} + T^{4}$$
$59$ $$81 + 27 T + 9 T^{2} + 3 T^{3} + T^{4}$$
$61$ $$16 - 8 T + 4 T^{2} - 2 T^{3} + T^{4}$$
$67$ $$( 3 + T )^{4}$$
$71$ $$1 + T + T^{2} + T^{3} + T^{4}$$
$73$ $$10000 + 1000 T + 100 T^{2} + 10 T^{3} + T^{4}$$
$79$ $$1296 + 216 T + 36 T^{2} + 6 T^{3} + T^{4}$$
$83$ $$20736 + 1728 T + 144 T^{2} + 12 T^{3} + T^{4}$$
$89$ $$( 15 + T )^{4}$$
$97$ $$625 - 125 T + 25 T^{2} - 5 T^{3} + T^{4}$$