# Properties

 Label 847.2.f.r Level $847$ Weight $2$ Character orbit 847.f Analytic conductor $6.763$ Analytic rank $0$ Dimension $8$ 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: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{5})$$ Coefficient field: 8.0.446265625.1 Defining polynomial: $$x^{8} - x^{7} + 4x^{6} - 7x^{5} + 19x^{4} + 21x^{3} + 36x^{2} + 27x + 81$$ x^8 - x^7 + 4*x^6 - 7*x^5 + 19*x^4 + 21*x^3 + 36*x^2 + 27*x + 81 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ 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 basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{5} q^{2} + (\beta_{6} + \beta_{5} + \beta_{2} - \beta_1) q^{3} + (\beta_{3} - \beta_1) q^{4} + (2 \beta_{6} + \beta_{4}) q^{5} + (\beta_{6} - 3 \beta_{4}) q^{6} + \beta_{3} q^{7} + (3 \beta_{7} - 3 \beta_{4} + 3 \beta_{3} + 3) q^{8} + \beta_{5} q^{9}+O(q^{10})$$ q - b5 * q^2 + (b6 + b5 + b2 - b1) * q^3 + (b3 - b1) * q^4 + (2*b6 + b4) * q^5 + (b6 - 3*b4) * q^6 + b3 * q^7 + (3*b7 - 3*b4 + 3*b3 + 3) * q^8 + b5 * q^9 $$q - \beta_{5} q^{2} + (\beta_{6} + \beta_{5} + \beta_{2} - \beta_1) q^{3} + (\beta_{3} - \beta_1) q^{4} + (2 \beta_{6} + \beta_{4}) q^{5} + (\beta_{6} - 3 \beta_{4}) q^{6} + \beta_{3} q^{7} + (3 \beta_{7} - 3 \beta_{4} + 3 \beta_{3} + 3) q^{8} + \beta_{5} q^{9} + ( - \beta_{2} - 6) q^{10} + ( - 2 \beta_{2} - 3) q^{12} + ( - 2 \beta_{7} - 2 \beta_{5}) q^{13} + (\beta_{6} + \beta_{5} + \beta_{2} - \beta_1) q^{14} + ( - 6 \beta_{3} + \beta_1) q^{15} + (\beta_{6} + 2 \beta_{4}) q^{16} + (\beta_{6} + 5 \beta_{4}) q^{17} + ( - 3 \beta_{3} + \beta_1) q^{18} + ( - 3 \beta_{7} + 3 \beta_{4} - 3 \beta_{3} - 3) q^{19} + (5 \beta_{7} + 3 \beta_{5}) q^{20} - \beta_{2} q^{21} + (\beta_{2} - 5) q^{23} + 3 \beta_{5} q^{24} + ( - 8 \beta_{7} + 8 \beta_{4} - 8 \beta_{3} - 8) q^{25} + (6 \beta_{3} - 4 \beta_1) q^{26} + (2 \beta_{6} + 3 \beta_{4}) q^{27} + (\beta_{6} - \beta_{4}) q^{28} + ( - 7 \beta_{3} - \beta_1) q^{29} + ( - 3 \beta_{7} - 7 \beta_{6} - 7 \beta_{5} + 3 \beta_{4} - 3 \beta_{3} - 7 \beta_{2} + \cdots - 3) q^{30}+ \cdots - \beta_{2} q^{98}+O(q^{100})$$ q - b5 * q^2 + (b6 + b5 + b2 - b1) * q^3 + (b3 - b1) * q^4 + (2*b6 + b4) * q^5 + (b6 - 3*b4) * q^6 + b3 * q^7 + (3*b7 - 3*b4 + 3*b3 + 3) * q^8 + b5 * q^9 + (-b2 - 6) * q^10 + (-2*b2 - 3) * q^12 + (-2*b7 - 2*b5) * q^13 + (b6 + b5 + b2 - b1) * q^14 + (-6*b3 + b1) * q^15 + (b6 + 2*b4) * q^16 + (b6 + 5*b4) * q^17 + (-3*b3 + b1) * q^18 + (-3*b7 + 3*b4 - 3*b3 - 3) * q^19 + (5*b7 + 3*b5) * q^20 - b2 * q^21 + (b2 - 5) * q^23 + 3*b5 * q^24 + (-8*b7 + 8*b4 - 8*b3 - 8) * q^25 + (6*b3 - 4*b1) * q^26 + (2*b6 + 3*b4) * q^27 + (b6 - b4) * q^28 + (-7*b3 - b1) * q^29 + (-3*b7 - 7*b6 - 7*b5 + 3*b4 - 3*b3 - 7*b2 + 7*b1 - 3) * q^30 + b7 * q^31 + (b2 + 3) * q^32 + (4*b2 - 3) * q^34 + (-b7 + 2*b5) * q^35 + (-3*b7 - 2*b6 - 2*b5 + 3*b4 - 3*b3 - 2*b2 + 2*b1 - 3) * q^36 + (4*b3 + 4*b1) * q^37 - 3*b6 * q^38 + (4*b6 - 6*b4) * q^39 + (3*b3 + 6*b1) * q^40 + (7*b7 - 7*b4 + 7*b3 + 7) * q^41 + (3*b7 + b5) * q^42 + (b2 - 4) * q^43 + (b2 + 6) * q^45 + (-3*b7 + 4*b5) * q^46 + (-5*b7 + 3*b6 + 3*b5 + 5*b4 - 5*b3 + 3*b2 - 3*b1 - 5) * q^47 + (-3*b3 - b1) * q^48 - b4 * q^49 - 8*b6 * q^50 + (-3*b3 - 4*b1) * q^51 + (8*b7 + 6*b6 + 6*b5 - 8*b4 + 8*b3 + 6*b2 - 6*b1 + 8) * q^52 + (-6*b7 - 3*b5) * q^53 + (b2 - 6) * q^54 - 3 * q^56 - 3*b5 * q^57 + (3*b7 - 6*b6 - 6*b5 - 3*b4 + 3*b3 - 6*b2 + 6*b1 + 3) * q^58 + (9*b3 + b1) * q^59 + (-8*b6 + 9*b4) * q^60 + (b6 - 2*b4) * q^61 + b1 * q^62 + (-b6 - b5 - b2 + b1) * q^63 + (b7 - 6*b5) * q^64 + (-6*b2 - 10) * q^65 + (5*b2 - 3) * q^67 + (-2*b7 - 3*b5) * q^68 + (3*b7 - 4*b6 - 4*b5 - 3*b4 + 3*b3 - 4*b2 + 4*b1 + 3) * q^69 + (-6*b3 + b1) * q^70 + (-b6 + 2*b4) * q^71 - 3*b6 * q^72 + 5*b3 * q^73 + (-12*b7 + 12*b4 - 12*b3 - 12) * q^74 - 8*b5 * q^75 + (3*b2 + 3) * q^76 + (-10*b2 - 12) * q^78 + (6*b7 + b5) * q^79 + (-8*b7 + 3*b6 + 3*b5 + 8*b4 - 8*b3 + 3*b2 - 3*b1 - 8) * q^80 + (-6*b3 + 2*b1) * q^81 + 7*b6 * q^82 + 3*b4 * q^83 + (-3*b3 + 2*b1) * q^84 + (-11*b7 + 9*b6 + 9*b5 + 11*b4 - 11*b3 + 9*b2 - 9*b1 - 11) * q^85 + (-3*b7 + 3*b5) * q^86 + (6*b2 - 3) * q^87 + (-9*b2 + 6) * q^89 + (-3*b7 - 7*b5) * q^90 + (2*b7 + 2*b6 + 2*b5 - 2*b4 + 2*b3 + 2*b2 - 2*b1 + 2) * q^91 + (-2*b3 + 3*b1) * q^92 - b6 * q^93 + (-2*b6 - 9*b4) * q^94 + (-3*b3 - 6*b1) * q^95 + (3*b7 + 4*b6 + 4*b5 - 3*b4 + 3*b3 + 4*b2 - 4*b1 + 3) * q^96 + (b7 - 2*b5) * q^97 - b2 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + q^{2} + q^{3} - 3 q^{4} - 7 q^{6} - 2 q^{7} + 6 q^{8} - q^{9}+O(q^{10})$$ 8 * q + q^2 + q^3 - 3 * q^4 - 7 * q^6 - 2 * q^7 + 6 * q^8 - q^9 $$8 q + q^{2} + q^{3} - 3 q^{4} - 7 q^{6} - 2 q^{7} + 6 q^{8} - q^{9} - 52 q^{10} - 32 q^{12} + 6 q^{13} + q^{14} + 13 q^{15} + 3 q^{16} + 9 q^{17} + 7 q^{18} - 6 q^{19} - 13 q^{20} - 4 q^{21} - 36 q^{23} - 3 q^{24} - 16 q^{25} - 16 q^{26} + 4 q^{27} - 3 q^{28} + 13 q^{29} - 13 q^{30} - 2 q^{31} + 28 q^{32} - 8 q^{34} - 8 q^{36} - 4 q^{37} + 3 q^{38} - 16 q^{39} + 14 q^{41} - 7 q^{42} - 28 q^{43} + 52 q^{45} + 2 q^{46} - 7 q^{47} + 5 q^{48} - 2 q^{49} + 8 q^{50} + 2 q^{51} + 22 q^{52} + 15 q^{53} - 44 q^{54} - 24 q^{56} + 3 q^{57} - 17 q^{59} + 26 q^{60} - 5 q^{61} + q^{62} - q^{63} + 4 q^{64} - 104 q^{65} - 4 q^{67} + 7 q^{68} + 2 q^{69} + 13 q^{70} + 5 q^{71} + 3 q^{72} - 10 q^{73} - 24 q^{74} + 8 q^{75} + 36 q^{76} - 136 q^{78} - 13 q^{79} - 13 q^{80} + 14 q^{81} - 7 q^{82} + 6 q^{83} + 8 q^{84} - 13 q^{85} + 3 q^{86} + 12 q^{89} + 13 q^{90} + 6 q^{91} + 7 q^{92} + q^{93} - 16 q^{94} + 10 q^{96} - 4 q^{98}+O(q^{100})$$ 8 * q + q^2 + q^3 - 3 * q^4 - 7 * q^6 - 2 * q^7 + 6 * q^8 - q^9 - 52 * q^10 - 32 * q^12 + 6 * q^13 + q^14 + 13 * q^15 + 3 * q^16 + 9 * q^17 + 7 * q^18 - 6 * q^19 - 13 * q^20 - 4 * q^21 - 36 * q^23 - 3 * q^24 - 16 * q^25 - 16 * q^26 + 4 * q^27 - 3 * q^28 + 13 * q^29 - 13 * q^30 - 2 * q^31 + 28 * q^32 - 8 * q^34 - 8 * q^36 - 4 * q^37 + 3 * q^38 - 16 * q^39 + 14 * q^41 - 7 * q^42 - 28 * q^43 + 52 * q^45 + 2 * q^46 - 7 * q^47 + 5 * q^48 - 2 * q^49 + 8 * q^50 + 2 * q^51 + 22 * q^52 + 15 * q^53 - 44 * q^54 - 24 * q^56 + 3 * q^57 - 17 * q^59 + 26 * q^60 - 5 * q^61 + q^62 - q^63 + 4 * q^64 - 104 * q^65 - 4 * q^67 + 7 * q^68 + 2 * q^69 + 13 * q^70 + 5 * q^71 + 3 * q^72 - 10 * q^73 - 24 * q^74 + 8 * q^75 + 36 * q^76 - 136 * q^78 - 13 * q^79 - 13 * q^80 + 14 * q^81 - 7 * q^82 + 6 * q^83 + 8 * q^84 - 13 * q^85 + 3 * q^86 + 12 * q^89 + 13 * q^90 + 6 * q^91 + 7 * q^92 + q^93 - 16 * q^94 + 10 * q^96 - 4 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{7} + 4x^{6} - 7x^{5} + 19x^{4} + 21x^{3} + 36x^{2} + 27x + 81$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -\nu^{5} - 21 ) / 19$$ (-v^5 - 21) / 19 $$\beta_{3}$$ $$=$$ $$( \nu^{6} + 40\nu ) / 57$$ (v^6 + 40*v) / 57 $$\beta_{4}$$ $$=$$ $$( -\nu^{7} - 97\nu^{2} ) / 171$$ (-v^7 - 97*v^2) / 171 $$\beta_{5}$$ $$=$$ $$( -\nu^{7} + 4\nu^{6} - 7\nu^{5} + 19\nu^{4} - 76\nu^{3} + 36\nu^{2} + 27\nu + 81 ) / 171$$ (-v^7 + 4*v^6 - 7*v^5 + 19*v^4 - 76*v^3 + 36*v^2 + 27*v + 81) / 171 $$\beta_{6}$$ $$=$$ $$( -\nu^{7} - 40\nu^{2} ) / 57$$ (-v^7 - 40*v^2) / 57 $$\beta_{7}$$ $$=$$ $$( 4\nu^{7} - 16\nu^{6} + 28\nu^{5} - 76\nu^{4} + 133\nu^{3} - 144\nu^{2} - 108\nu - 324 ) / 513$$ (4*v^7 - 16*v^6 + 28*v^5 - 76*v^4 + 133*v^3 - 144*v^2 - 108*v - 324) / 513
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{6} - 3\beta_{4}$$ b6 - 3*b4 $$\nu^{3}$$ $$=$$ $$-3\beta_{7} - 4\beta_{5}$$ -3*b7 - 4*b5 $$\nu^{4}$$ $$=$$ $$-12\beta_{7} - 7\beta_{6} - 7\beta_{5} + 12\beta_{4} - 12\beta_{3} - 7\beta_{2} + 7\beta _1 - 12$$ -12*b7 - 7*b6 - 7*b5 + 12*b4 - 12*b3 - 7*b2 + 7*b1 - 12 $$\nu^{5}$$ $$=$$ $$-19\beta_{2} - 21$$ -19*b2 - 21 $$\nu^{6}$$ $$=$$ $$57\beta_{3} - 40\beta_1$$ 57*b3 - 40*b1 $$\nu^{7}$$ $$=$$ $$-97\beta_{6} + 120\beta_{4}$$ -97*b6 + 120*b4

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/847\mathbb{Z}\right)^\times$$.

 $$n$$ $$122$$ $$365$$ $$\chi(n)$$ $$1$$ $$\beta_{3}$$

## 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
 1.86298 − 1.35354i −1.05397 + 0.765752i 0.402580 + 1.23901i −0.711597 − 2.19007i 1.86298 + 1.35354i −1.05397 − 0.765752i 0.402580 − 1.23901i −0.711597 + 2.19007i
−0.711597 2.19007i 1.86298 + 1.35354i −2.67200 + 1.94132i 1.11418 3.42908i 1.63865 5.04324i −0.809017 + 0.587785i 2.42705 + 1.76336i 0.711597 + 2.19007i −8.30278
148.2 0.402580 + 1.23901i −1.05397 0.765752i 0.244951 0.177967i −1.11418 + 3.42908i 0.524471 1.61416i −0.809017 + 0.587785i 2.42705 + 1.76336i −0.402580 1.23901i −4.69722
323.1 −1.05397 0.765752i 0.402580 1.23901i −0.0935628 0.287957i 2.91695 2.11929i −1.37308 + 0.997603i 0.309017 + 0.951057i −0.927051 + 2.85317i 1.05397 + 0.765752i −4.69722
323.2 1.86298 + 1.35354i −0.711597 + 2.19007i 1.02061 + 3.14113i −2.91695 + 2.11929i −4.29004 + 3.11689i 0.309017 + 0.951057i −0.927051 + 2.85317i −1.86298 1.35354i −8.30278
372.1 −0.711597 + 2.19007i 1.86298 1.35354i −2.67200 1.94132i 1.11418 + 3.42908i 1.63865 + 5.04324i −0.809017 0.587785i 2.42705 1.76336i 0.711597 2.19007i −8.30278
372.2 0.402580 1.23901i −1.05397 + 0.765752i 0.244951 + 0.177967i −1.11418 3.42908i 0.524471 + 1.61416i −0.809017 0.587785i 2.42705 1.76336i −0.402580 + 1.23901i −4.69722
729.1 −1.05397 + 0.765752i 0.402580 + 1.23901i −0.0935628 + 0.287957i 2.91695 + 2.11929i −1.37308 0.997603i 0.309017 0.951057i −0.927051 2.85317i 1.05397 0.765752i −4.69722
729.2 1.86298 1.35354i −0.711597 2.19007i 1.02061 3.14113i −2.91695 2.11929i −4.29004 3.11689i 0.309017 0.951057i −0.927051 2.85317i −1.86298 + 1.35354i −8.30278
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 729.2 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.r 8
11.b odd 2 1 847.2.f.o 8
11.c even 5 1 847.2.a.e 2
11.c even 5 3 inner 847.2.f.r 8
11.d odd 10 1 847.2.a.g yes 2
11.d odd 10 3 847.2.f.o 8
33.f even 10 1 7623.2.a.bc 2
33.h odd 10 1 7623.2.a.bs 2
77.j odd 10 1 5929.2.a.k 2
77.l even 10 1 5929.2.a.p 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
847.2.a.e 2 11.c even 5 1
847.2.a.g yes 2 11.d odd 10 1
847.2.f.o 8 11.b odd 2 1
847.2.f.o 8 11.d odd 10 3
847.2.f.r 8 1.a even 1 1 trivial
847.2.f.r 8 11.c even 5 3 inner
5929.2.a.k 2 77.j odd 10 1
5929.2.a.p 2 77.l even 10 1
7623.2.a.bc 2 33.f even 10 1
7623.2.a.bs 2 33.h odd 10 1

## Hecke kernels

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

 $$T_{2}^{8} - T_{2}^{7} + 4T_{2}^{6} - 7T_{2}^{5} + 19T_{2}^{4} + 21T_{2}^{3} + 36T_{2}^{2} + 27T_{2} + 81$$ T2^8 - T2^7 + 4*T2^6 - 7*T2^5 + 19*T2^4 + 21*T2^3 + 36*T2^2 + 27*T2 + 81 $$T_{3}^{8} - T_{3}^{7} + 4T_{3}^{6} - 7T_{3}^{5} + 19T_{3}^{4} + 21T_{3}^{3} + 36T_{3}^{2} + 27T_{3} + 81$$ T3^8 - T3^7 + 4*T3^6 - 7*T3^5 + 19*T3^4 + 21*T3^3 + 36*T3^2 + 27*T3 + 81 $$T_{13}^{8} - 6T_{13}^{7} + 40T_{13}^{6} - 264T_{13}^{5} + 1744T_{13}^{4} + 1056T_{13}^{3} + 640T_{13}^{2} + 384T_{13} + 256$$ T13^8 - 6*T13^7 + 40*T13^6 - 264*T13^5 + 1744*T13^4 + 1056*T13^3 + 640*T13^2 + 384*T13 + 256

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} - T^{7} + 4 T^{6} - 7 T^{5} + \cdots + 81$$
$3$ $$T^{8} - T^{7} + 4 T^{6} - 7 T^{5} + \cdots + 81$$
$5$ $$T^{8} + 13 T^{6} + 169 T^{4} + \cdots + 28561$$
$7$ $$(T^{4} + T^{3} + T^{2} + T + 1)^{2}$$
$11$ $$T^{8}$$
$13$ $$T^{8} - 6 T^{7} + 40 T^{6} - 264 T^{5} + \cdots + 256$$
$17$ $$T^{8} - 9 T^{7} + 64 T^{6} + \cdots + 83521$$
$19$ $$(T^{4} + 3 T^{3} + 9 T^{2} + 27 T + 81)^{2}$$
$23$ $$(T^{2} + 9 T + 17)^{4}$$
$29$ $$T^{8} - 13 T^{7} + 130 T^{6} + \cdots + 2313441$$
$31$ $$(T^{4} + T^{3} + T^{2} + T + 1)^{2}$$
$37$ $$T^{8} + 4 T^{7} + 64 T^{6} + \cdots + 5308416$$
$41$ $$(T^{4} - 7 T^{3} + 49 T^{2} - 343 T + 2401)^{2}$$
$43$ $$(T^{2} + 7 T + 9)^{4}$$
$47$ $$T^{8} + 7 T^{7} + 66 T^{6} + \cdots + 83521$$
$53$ $$T^{8} - 15 T^{7} + 198 T^{6} + \cdots + 531441$$
$59$ $$T^{8} + 17 T^{7} + 220 T^{6} + \cdots + 22667121$$
$61$ $$T^{8} + 5 T^{7} + 22 T^{6} + 95 T^{5} + \cdots + 81$$
$67$ $$(T^{2} + T - 81)^{4}$$
$71$ $$T^{8} - 5 T^{7} + 22 T^{6} - 95 T^{5} + \cdots + 81$$
$73$ $$(T^{4} + 5 T^{3} + 25 T^{2} + 125 T + 625)^{2}$$
$79$ $$T^{8} + 13 T^{7} + 130 T^{6} + \cdots + 2313441$$
$83$ $$(T^{4} - 3 T^{3} + 9 T^{2} - 27 T + 81)^{2}$$
$89$ $$(T^{2} - 3 T - 261)^{4}$$
$97$ $$T^{8} + 13 T^{6} + 169 T^{4} + \cdots + 28561$$