# Properties

 Label 847.2.f.f 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: $$x^{4} - x^{3} + x^{2} - x + 1$$ x^4 - x^3 + x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{10}^{2} q^{3} + 2 \zeta_{10}^{3} q^{4} - 3 \zeta_{10} q^{5} - \zeta_{10}^{3} q^{7} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} + 2) q^{9} +O(q^{10})$$ q + z^2 * q^3 + 2*z^3 * q^4 - 3*z * q^5 - z^3 * q^7 + (-2*z^3 + 2*z^2 - 2*z + 2) * q^9 $$q + \zeta_{10}^{2} q^{3} + 2 \zeta_{10}^{3} q^{4} - 3 \zeta_{10} q^{5} - \zeta_{10}^{3} q^{7} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} + 2) q^{9} - 2 q^{12} + ( - 4 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 4 \zeta_{10} + 4) q^{13} - 3 \zeta_{10}^{3} q^{15} - 4 \zeta_{10} q^{16} + 6 \zeta_{10} q^{17} + 2 \zeta_{10}^{2} q^{19} + ( - 6 \zeta_{10}^{3} + 6 \zeta_{10}^{2} - 6 \zeta_{10} + 6) q^{20} + q^{21} + 3 q^{23} + 4 \zeta_{10}^{2} q^{25} + 5 \zeta_{10} q^{27} + 2 \zeta_{10} q^{28} + 6 \zeta_{10}^{3} q^{29} + (5 \zeta_{10}^{3} - 5 \zeta_{10}^{2} + 5 \zeta_{10} - 5) q^{31} + (3 \zeta_{10}^{3} - 3 \zeta_{10}^{2} + 3 \zeta_{10} - 3) q^{35} + 4 \zeta_{10}^{2} q^{36} - 11 \zeta_{10}^{3} q^{37} + 4 \zeta_{10} q^{39} + 6 \zeta_{10}^{2} q^{41} + 8 q^{43} - 6 q^{45} - 4 \zeta_{10}^{3} q^{48} - \zeta_{10} q^{49} + 6 \zeta_{10}^{3} q^{51} + 8 \zeta_{10}^{2} q^{52} + ( - 6 \zeta_{10}^{3} + 6 \zeta_{10}^{2} - 6 \zeta_{10} + 6) q^{53} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} - 2) q^{57} + 9 \zeta_{10}^{3} q^{59} + 6 \zeta_{10} q^{60} + 10 \zeta_{10} q^{61} - 2 \zeta_{10}^{2} q^{63} + ( - 8 \zeta_{10}^{3} + 8 \zeta_{10}^{2} - 8 \zeta_{10} + 8) q^{64} - 12 q^{65} + 5 q^{67} + (12 \zeta_{10}^{3} - 12 \zeta_{10}^{2} + 12 \zeta_{10} - 12) q^{68} + 3 \zeta_{10}^{2} q^{69} - 9 \zeta_{10} q^{71} - 2 \zeta_{10}^{3} q^{73} + (4 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 4 \zeta_{10} - 4) q^{75} - 4 q^{76} + ( - 10 \zeta_{10}^{3} + 10 \zeta_{10}^{2} - 10 \zeta_{10} + 10) q^{79} + 12 \zeta_{10}^{2} q^{80} - \zeta_{10}^{3} q^{81} - 12 \zeta_{10} q^{83} + 2 \zeta_{10}^{3} q^{84} - 18 \zeta_{10}^{2} q^{85} - 6 q^{87} - 3 q^{89} - 4 \zeta_{10}^{2} q^{91} + 6 \zeta_{10}^{3} q^{92} - 5 \zeta_{10} q^{93} - 6 \zeta_{10}^{3} q^{95} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{97} +O(q^{100})$$ q + z^2 * q^3 + 2*z^3 * q^4 - 3*z * q^5 - z^3 * q^7 + (-2*z^3 + 2*z^2 - 2*z + 2) * q^9 - 2 * q^12 + (-4*z^3 + 4*z^2 - 4*z + 4) * q^13 - 3*z^3 * q^15 - 4*z * q^16 + 6*z * q^17 + 2*z^2 * q^19 + (-6*z^3 + 6*z^2 - 6*z + 6) * q^20 + q^21 + 3 * q^23 + 4*z^2 * q^25 + 5*z * q^27 + 2*z * q^28 + 6*z^3 * q^29 + (5*z^3 - 5*z^2 + 5*z - 5) * q^31 + (3*z^3 - 3*z^2 + 3*z - 3) * q^35 + 4*z^2 * q^36 - 11*z^3 * q^37 + 4*z * q^39 + 6*z^2 * q^41 + 8 * q^43 - 6 * q^45 - 4*z^3 * q^48 - z * q^49 + 6*z^3 * q^51 + 8*z^2 * q^52 + (-6*z^3 + 6*z^2 - 6*z + 6) * q^53 + (2*z^3 - 2*z^2 + 2*z - 2) * q^57 + 9*z^3 * q^59 + 6*z * q^60 + 10*z * q^61 - 2*z^2 * q^63 + (-8*z^3 + 8*z^2 - 8*z + 8) * q^64 - 12 * q^65 + 5 * q^67 + (12*z^3 - 12*z^2 + 12*z - 12) * q^68 + 3*z^2 * q^69 - 9*z * q^71 - 2*z^3 * q^73 + (4*z^3 - 4*z^2 + 4*z - 4) * q^75 - 4 * q^76 + (-10*z^3 + 10*z^2 - 10*z + 10) * q^79 + 12*z^2 * q^80 - z^3 * q^81 - 12*z * q^83 + 2*z^3 * q^84 - 18*z^2 * q^85 - 6 * q^87 - 3 * q^89 - 4*z^2 * q^91 + 6*z^3 * q^92 - 5*z * q^93 - 6*z^3 * q^95 + (-z^3 + z^2 - z + 1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - q^{3} + 2 q^{4} - 3 q^{5} - q^{7} + 2 q^{9}+O(q^{10})$$ 4 * q - q^3 + 2 * q^4 - 3 * q^5 - q^7 + 2 * q^9 $$4 q - q^{3} + 2 q^{4} - 3 q^{5} - q^{7} + 2 q^{9} - 8 q^{12} + 4 q^{13} - 3 q^{15} - 4 q^{16} + 6 q^{17} - 2 q^{19} + 6 q^{20} + 4 q^{21} + 12 q^{23} - 4 q^{25} + 5 q^{27} + 2 q^{28} + 6 q^{29} - 5 q^{31} - 3 q^{35} - 4 q^{36} - 11 q^{37} + 4 q^{39} - 6 q^{41} + 32 q^{43} - 24 q^{45} - 4 q^{48} - q^{49} + 6 q^{51} - 8 q^{52} + 6 q^{53} - 2 q^{57} + 9 q^{59} + 6 q^{60} + 10 q^{61} + 2 q^{63} + 8 q^{64} - 48 q^{65} + 20 q^{67} - 12 q^{68} - 3 q^{69} - 9 q^{71} - 2 q^{73} - 4 q^{75} - 16 q^{76} + 10 q^{79} - 12 q^{80} - q^{81} - 12 q^{83} + 2 q^{84} + 18 q^{85} - 24 q^{87} - 12 q^{89} + 4 q^{91} + 6 q^{92} - 5 q^{93} - 6 q^{95} + q^{97}+O(q^{100})$$ 4 * q - q^3 + 2 * q^4 - 3 * q^5 - q^7 + 2 * q^9 - 8 * q^12 + 4 * q^13 - 3 * q^15 - 4 * q^16 + 6 * q^17 - 2 * q^19 + 6 * q^20 + 4 * q^21 + 12 * q^23 - 4 * q^25 + 5 * q^27 + 2 * q^28 + 6 * q^29 - 5 * q^31 - 3 * q^35 - 4 * q^36 - 11 * q^37 + 4 * q^39 - 6 * q^41 + 32 * q^43 - 24 * q^45 - 4 * q^48 - q^49 + 6 * q^51 - 8 * q^52 + 6 * q^53 - 2 * q^57 + 9 * q^59 + 6 * q^60 + 10 * q^61 + 2 * q^63 + 8 * q^64 - 48 * q^65 + 20 * q^67 - 12 * q^68 - 3 * q^69 - 9 * q^71 - 2 * q^73 - 4 * q^75 - 16 * q^76 + 10 * q^79 - 12 * q^80 - q^81 - 12 * q^83 + 2 * q^84 + 18 * q^85 - 24 * q^87 - 12 * q^89 + 4 * q^91 + 6 * q^92 - 5 * q^93 - 6 * q^95 + q^97

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

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.a.b 1 11.c even 5 1
539.2.a.b 1 77.j odd 10 1
539.2.e.d 2 77.m even 15 2
539.2.e.e 2 77.p odd 30 2
693.2.a.b 1 33.h odd 10 1
847.2.a.c 1 11.d odd 10 1
847.2.f.f 4 1.a even 1 1 trivial
847.2.f.f 4 11.c even 5 3 inner
847.2.f.g 4 11.b odd 2 1
847.2.f.g 4 11.d odd 10 3
1232.2.a.d 1 44.h odd 10 1
1925.2.a.f 1 55.j even 10 1
1925.2.b.g 2 55.k odd 20 2
4851.2.a.k 1 231.u even 10 1
4928.2.a.i 1 88.o even 10 1
4928.2.a.x 1 88.l odd 10 1
5929.2.a.d 1 77.l even 10 1
7623.2.a.i 1 33.f even 10 1
8624.2.a.s 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 $$S_{2}^{\mathrm{new}}(847, [\chi])$$:

 $$T_{2}$$ T2 $$T_{3}^{4} + T_{3}^{3} + T_{3}^{2} + T_{3} + 1$$ T3^4 + T3^3 + T3^2 + T3 + 1 $$T_{13}^{4} - 4T_{13}^{3} + 16T_{13}^{2} - 64T_{13} + 256$$ T13^4 - 4*T13^3 + 16*T13^2 - 64*T13 + 256

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + T^{3} + T^{2} + T + 1$$
$5$ $$T^{4} + 3 T^{3} + 9 T^{2} + 27 T + 81$$
$7$ $$T^{4} + T^{3} + T^{2} + T + 1$$
$11$ $$T^{4}$$
$13$ $$T^{4} - 4 T^{3} + 16 T^{2} - 64 T + 256$$
$17$ $$T^{4} - 6 T^{3} + 36 T^{2} + \cdots + 1296$$
$19$ $$T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16$$
$23$ $$(T - 3)^{4}$$
$29$ $$T^{4} - 6 T^{3} + 36 T^{2} + \cdots + 1296$$
$31$ $$T^{4} + 5 T^{3} + 25 T^{2} + 125 T + 625$$
$37$ $$T^{4} + 11 T^{3} + 121 T^{2} + \cdots + 14641$$
$41$ $$T^{4} + 6 T^{3} + 36 T^{2} + \cdots + 1296$$
$43$ $$(T - 8)^{4}$$
$47$ $$T^{4}$$
$53$ $$T^{4} - 6 T^{3} + 36 T^{2} + \cdots + 1296$$
$59$ $$T^{4} - 9 T^{3} + 81 T^{2} + \cdots + 6561$$
$61$ $$T^{4} - 10 T^{3} + 100 T^{2} + \cdots + 10000$$
$67$ $$(T - 5)^{4}$$
$71$ $$T^{4} + 9 T^{3} + 81 T^{2} + \cdots + 6561$$
$73$ $$T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16$$
$79$ $$T^{4} - 10 T^{3} + 100 T^{2} + \cdots + 10000$$
$83$ $$T^{4} + 12 T^{3} + 144 T^{2} + \cdots + 20736$$
$89$ $$(T + 3)^{4}$$
$97$ $$T^{4} - T^{3} + T^{2} - T + 1$$