# Properties

 Label 847.2.f.e 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: $$\Z[a_1, a_2]$$ 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.

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

## 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.e 4
11.b odd 2 1 847.2.f.k 4
11.c even 5 1 77.2.a.c 1
11.c even 5 3 inner 847.2.f.e 4
11.d odd 10 1 847.2.a.a 1
11.d odd 10 3 847.2.f.k 4
33.f even 10 1 7623.2.a.n 1
33.h odd 10 1 693.2.a.a 1
44.h odd 10 1 1232.2.a.a 1
55.j even 10 1 1925.2.a.c 1
55.k odd 20 2 1925.2.b.d 2
77.j odd 10 1 539.2.a.d 1
77.l even 10 1 5929.2.a.b 1
77.m even 15 2 539.2.e.a 2
77.p odd 30 2 539.2.e.b 2
88.l odd 10 1 4928.2.a.bi 1
88.o even 10 1 4928.2.a.g 1
231.u even 10 1 4851.2.a.a 1
308.t even 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.c even 5 1
539.2.a.d 1 77.j odd 10 1
539.2.e.a 2 77.m even 15 2
539.2.e.b 2 77.p odd 30 2
693.2.a.a 1 33.h odd 10 1
847.2.a.a 1 11.d odd 10 1
847.2.f.e 4 1.a even 1 1 trivial
847.2.f.e 4 11.c even 5 3 inner
847.2.f.k 4 11.b odd 2 1
847.2.f.k 4 11.d odd 10 3
1232.2.a.a 1 44.h odd 10 1
1925.2.a.c 1 55.j even 10 1
1925.2.b.d 2 55.k odd 20 2
4851.2.a.a 1 231.u even 10 1
4928.2.a.g 1 88.o even 10 1
4928.2.a.bi 1 88.l odd 10 1
5929.2.a.b 1 77.l even 10 1
7623.2.a.n 1 33.f even 10 1
8624.2.a.bc 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 $$S_{2}^{\mathrm{new}}(847, [\chi])$$:

 $$T_{2}^{4} + T_{2}^{3} + T_{2}^{2} + T_{2} + 1$$ T2^4 + T2^3 + T2^2 + T2 + 1 $$T_{3}^{4} + 2T_{3}^{3} + 4T_{3}^{2} + 8T_{3} + 16$$ T3^4 + 2*T3^3 + 4*T3^2 + 8*T3 + 16 $$T_{13}^{4} + 4T_{13}^{3} + 16T_{13}^{2} + 64T_{13} + 256$$ T13^4 + 4*T13^3 + 16*T13^2 + 64*T13 + 256

## Hecke characteristic polynomials

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