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: $$x^{8} - x^{7} + 4 x^{6} - 7 x^{5} + 19 x^{4} + 21 x^{3} + 36 x^{2} + 27 x + 81$$ 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.

 $$f(q)$$ $$=$$ $$q -\beta_{5} q^{2} + ( -\beta_{1} + \beta_{2} + \beta_{5} + \beta_{6} ) q^{3} + ( -\beta_{1} + \beta_{3} ) q^{4} + ( \beta_{4} + 2 \beta_{6} ) q^{5} + ( -3 \beta_{4} + \beta_{6} ) q^{6} + \beta_{3} q^{7} + ( 3 + 3 \beta_{3} - 3 \beta_{4} + 3 \beta_{7} ) q^{8} + \beta_{5} q^{9} +O(q^{10})$$ $$q -\beta_{5} q^{2} + ( -\beta_{1} + \beta_{2} + \beta_{5} + \beta_{6} ) q^{3} + ( -\beta_{1} + \beta_{3} ) q^{4} + ( \beta_{4} + 2 \beta_{6} ) q^{5} + ( -3 \beta_{4} + \beta_{6} ) q^{6} + \beta_{3} q^{7} + ( 3 + 3 \beta_{3} - 3 \beta_{4} + 3 \beta_{7} ) q^{8} + \beta_{5} q^{9} + ( -6 - \beta_{2} ) q^{10} + ( -3 - 2 \beta_{2} ) q^{12} + ( -2 \beta_{5} - 2 \beta_{7} ) q^{13} + ( -\beta_{1} + \beta_{2} + \beta_{5} + \beta_{6} ) q^{14} + ( \beta_{1} - 6 \beta_{3} ) q^{15} + ( 2 \beta_{4} + \beta_{6} ) q^{16} + ( 5 \beta_{4} + \beta_{6} ) q^{17} + ( \beta_{1} - 3 \beta_{3} ) q^{18} + ( -3 - 3 \beta_{3} + 3 \beta_{4} - 3 \beta_{7} ) q^{19} + ( 3 \beta_{5} + 5 \beta_{7} ) q^{20} -\beta_{2} q^{21} + ( -5 + \beta_{2} ) q^{23} + 3 \beta_{5} q^{24} + ( -8 - 8 \beta_{3} + 8 \beta_{4} - 8 \beta_{7} ) q^{25} + ( -4 \beta_{1} + 6 \beta_{3} ) q^{26} + ( 3 \beta_{4} + 2 \beta_{6} ) q^{27} + ( -\beta_{4} + \beta_{6} ) q^{28} + ( -\beta_{1} - 7 \beta_{3} ) q^{29} + ( -3 + 7 \beta_{1} - 7 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - 7 \beta_{5} - 7 \beta_{6} - 3 \beta_{7} ) q^{30} + \beta_{7} q^{31} + ( 3 + \beta_{2} ) q^{32} + ( -3 + 4 \beta_{2} ) q^{34} + ( 2 \beta_{5} - \beta_{7} ) q^{35} + ( -3 + 2 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 3 \beta_{7} ) q^{36} + ( 4 \beta_{1} + 4 \beta_{3} ) q^{37} -3 \beta_{6} q^{38} + ( -6 \beta_{4} + 4 \beta_{6} ) q^{39} + ( 6 \beta_{1} + 3 \beta_{3} ) q^{40} + ( 7 + 7 \beta_{3} - 7 \beta_{4} + 7 \beta_{7} ) q^{41} + ( \beta_{5} + 3 \beta_{7} ) q^{42} + ( -4 + \beta_{2} ) q^{43} + ( 6 + \beta_{2} ) q^{45} + ( 4 \beta_{5} - 3 \beta_{7} ) q^{46} + ( -5 - 3 \beta_{1} + 3 \beta_{2} - 5 \beta_{3} + 5 \beta_{4} + 3 \beta_{5} + 3 \beta_{6} - 5 \beta_{7} ) q^{47} + ( -\beta_{1} - 3 \beta_{3} ) q^{48} -\beta_{4} q^{49} -8 \beta_{6} q^{50} + ( -4 \beta_{1} - 3 \beta_{3} ) q^{51} + ( 8 - 6 \beta_{1} + 6 \beta_{2} + 8 \beta_{3} - 8 \beta_{4} + 6 \beta_{5} + 6 \beta_{6} + 8 \beta_{7} ) q^{52} + ( -3 \beta_{5} - 6 \beta_{7} ) q^{53} + ( -6 + \beta_{2} ) q^{54} -3 q^{56} -3 \beta_{5} q^{57} + ( 3 + 6 \beta_{1} - 6 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} - 6 \beta_{5} - 6 \beta_{6} + 3 \beta_{7} ) q^{58} + ( \beta_{1} + 9 \beta_{3} ) q^{59} + ( 9 \beta_{4} - 8 \beta_{6} ) q^{60} + ( -2 \beta_{4} + \beta_{6} ) q^{61} + \beta_{1} q^{62} + ( \beta_{1} - \beta_{2} - \beta_{5} - \beta_{6} ) q^{63} + ( -6 \beta_{5} + \beta_{7} ) q^{64} + ( -10 - 6 \beta_{2} ) q^{65} + ( -3 + 5 \beta_{2} ) q^{67} + ( -3 \beta_{5} - 2 \beta_{7} ) q^{68} + ( 3 + 4 \beta_{1} - 4 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} - 4 \beta_{5} - 4 \beta_{6} + 3 \beta_{7} ) q^{69} + ( \beta_{1} - 6 \beta_{3} ) q^{70} + ( 2 \beta_{4} - \beta_{6} ) q^{71} -3 \beta_{6} q^{72} + 5 \beta_{3} q^{73} + ( -12 - 12 \beta_{3} + 12 \beta_{4} - 12 \beta_{7} ) q^{74} -8 \beta_{5} q^{75} + ( 3 + 3 \beta_{2} ) q^{76} + ( -12 - 10 \beta_{2} ) q^{78} + ( \beta_{5} + 6 \beta_{7} ) q^{79} + ( -8 - 3 \beta_{1} + 3 \beta_{2} - 8 \beta_{3} + 8 \beta_{4} + 3 \beta_{5} + 3 \beta_{6} - 8 \beta_{7} ) q^{80} + ( 2 \beta_{1} - 6 \beta_{3} ) q^{81} + 7 \beta_{6} q^{82} + 3 \beta_{4} q^{83} + ( 2 \beta_{1} - 3 \beta_{3} ) q^{84} + ( -11 - 9 \beta_{1} + 9 \beta_{2} - 11 \beta_{3} + 11 \beta_{4} + 9 \beta_{5} + 9 \beta_{6} - 11 \beta_{7} ) q^{85} + ( 3 \beta_{5} - 3 \beta_{7} ) q^{86} + ( -3 + 6 \beta_{2} ) q^{87} + ( 6 - 9 \beta_{2} ) q^{89} + ( -7 \beta_{5} - 3 \beta_{7} ) q^{90} + ( 2 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{91} + ( 3 \beta_{1} - 2 \beta_{3} ) q^{92} -\beta_{6} q^{93} + ( -9 \beta_{4} - 2 \beta_{6} ) q^{94} + ( -6 \beta_{1} - 3 \beta_{3} ) q^{95} + ( 3 - 4 \beta_{1} + 4 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} + 4 \beta_{5} + 4 \beta_{6} + 3 \beta_{7} ) q^{96} + ( -2 \beta_{5} + \beta_{7} ) q^{97} -\beta_{2} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + q^{2} + q^{3} - 3 q^{4} - 7 q^{6} - 2 q^{7} + 6 q^{8} - q^{9} + O(q^{10})$$ $$8 q + q^{2} + q^{3} - 3 q^{4} - 7 q^{6} - 2 q^{7} + 6 q^{8} - q^{9} - 52 q^{10} - 32 q^{12} + 6 q^{13} + q^{14} + 13 q^{15} + 3 q^{16} + 9 q^{17} + 7 q^{18} - 6 q^{19} - 13 q^{20} - 4 q^{21} - 36 q^{23} - 3 q^{24} - 16 q^{25} - 16 q^{26} + 4 q^{27} - 3 q^{28} + 13 q^{29} - 13 q^{30} - 2 q^{31} + 28 q^{32} - 8 q^{34} - 8 q^{36} - 4 q^{37} + 3 q^{38} - 16 q^{39} + 14 q^{41} - 7 q^{42} - 28 q^{43} + 52 q^{45} + 2 q^{46} - 7 q^{47} + 5 q^{48} - 2 q^{49} + 8 q^{50} + 2 q^{51} + 22 q^{52} + 15 q^{53} - 44 q^{54} - 24 q^{56} + 3 q^{57} - 17 q^{59} + 26 q^{60} - 5 q^{61} + q^{62} - q^{63} + 4 q^{64} - 104 q^{65} - 4 q^{67} + 7 q^{68} + 2 q^{69} + 13 q^{70} + 5 q^{71} + 3 q^{72} - 10 q^{73} - 24 q^{74} + 8 q^{75} + 36 q^{76} - 136 q^{78} - 13 q^{79} - 13 q^{80} + 14 q^{81} - 7 q^{82} + 6 q^{83} + 8 q^{84} - 13 q^{85} + 3 q^{86} + 12 q^{89} + 13 q^{90} + 6 q^{91} + 7 q^{92} + q^{93} - 16 q^{94} + 10 q^{96} - 4 q^{98} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{5} - 21$$$$)/19$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{6} + 40 \nu$$$$)/57$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{7} - 97 \nu^{2}$$$$)/171$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{7} + 4 \nu^{6} - 7 \nu^{5} + 19 \nu^{4} - 76 \nu^{3} + 36 \nu^{2} + 27 \nu + 81$$$$)/171$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{7} - 40 \nu^{2}$$$$)/57$$ $$\beta_{7}$$ $$=$$ $$($$$$4 \nu^{7} - 16 \nu^{6} + 28 \nu^{5} - 76 \nu^{4} + 133 \nu^{3} - 144 \nu^{2} - 108 \nu - 324$$$$)/513$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{6} - 3 \beta_{4}$$ $$\nu^{3}$$ $$=$$ $$-3 \beta_{7} - 4 \beta_{5}$$ $$\nu^{4}$$ $$=$$ $$-12 \beta_{7} - 7 \beta_{6} - 7 \beta_{5} + 12 \beta_{4} - 12 \beta_{3} - 7 \beta_{2} + 7 \beta_{1} - 12$$ $$\nu^{5}$$ $$=$$ $$-19 \beta_{2} - 21$$ $$\nu^{6}$$ $$=$$ $$57 \beta_{3} - 40 \beta_{1}$$ $$\nu^{7}$$ $$=$$ $$-97 \beta_{6} + 120 \beta_{4}$$

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 $$S_{2}^{\mathrm{new}}(847, [\chi])$$:

 $$T_{2}^{8} - \cdots$$ $$T_{3}^{8} - \cdots$$ $$T_{13}^{8} - \cdots$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$81 + 27 T + 36 T^{2} + 21 T^{3} + 19 T^{4} - 7 T^{5} + 4 T^{6} - T^{7} + T^{8}$$
$3$ $$81 + 27 T + 36 T^{2} + 21 T^{3} + 19 T^{4} - 7 T^{5} + 4 T^{6} - T^{7} + T^{8}$$
$5$ $$28561 + 2197 T^{2} + 169 T^{4} + 13 T^{6} + T^{8}$$
$7$ $$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$
$11$ $$T^{8}$$
$13$ $$256 + 384 T + 640 T^{2} + 1056 T^{3} + 1744 T^{4} - 264 T^{5} + 40 T^{6} - 6 T^{7} + T^{8}$$
$17$ $$83521 - 44217 T + 18496 T^{2} - 7191 T^{3} + 2719 T^{4} - 423 T^{5} + 64 T^{6} - 9 T^{7} + T^{8}$$
$19$ $$( 81 + 27 T + 9 T^{2} + 3 T^{3} + T^{4} )^{2}$$
$23$ $$( 17 + 9 T + T^{2} )^{4}$$
$29$ $$2313441 - 771147 T + 197730 T^{2} - 46137 T^{3} + 10309 T^{4} - 1183 T^{5} + 130 T^{6} - 13 T^{7} + T^{8}$$
$31$ $$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$
$37$ $$5308416 - 442368 T + 147456 T^{2} - 21504 T^{3} + 4864 T^{4} + 448 T^{5} + 64 T^{6} + 4 T^{7} + T^{8}$$
$41$ $$( 2401 - 343 T + 49 T^{2} - 7 T^{3} + T^{4} )^{2}$$
$43$ $$( 9 + 7 T + T^{2} )^{4}$$
$47$ $$83521 - 34391 T + 19074 T^{2} - 9877 T^{3} + 5189 T^{4} + 581 T^{5} + 66 T^{6} + 7 T^{7} + T^{8}$$
$53$ $$531441 - 295245 T + 144342 T^{2} - 69255 T^{3} + 33129 T^{4} - 2565 T^{5} + 198 T^{6} - 15 T^{7} + T^{8}$$
$59$ $$22667121 + 5584653 T + 1047420 T^{2} + 177123 T^{3} + 28459 T^{4} + 2567 T^{5} + 220 T^{6} + 17 T^{7} + T^{8}$$
$61$ $$81 + 135 T + 198 T^{2} + 285 T^{3} + 409 T^{4} + 95 T^{5} + 22 T^{6} + 5 T^{7} + T^{8}$$
$67$ $$( -81 + T + T^{2} )^{4}$$
$71$ $$81 - 135 T + 198 T^{2} - 285 T^{3} + 409 T^{4} - 95 T^{5} + 22 T^{6} - 5 T^{7} + T^{8}$$
$73$ $$( 625 + 125 T + 25 T^{2} + 5 T^{3} + T^{4} )^{2}$$
$79$ $$2313441 + 771147 T + 197730 T^{2} + 46137 T^{3} + 10309 T^{4} + 1183 T^{5} + 130 T^{6} + 13 T^{7} + T^{8}$$
$83$ $$( 81 - 27 T + 9 T^{2} - 3 T^{3} + T^{4} )^{2}$$
$89$ $$( -261 - 3 T + T^{2} )^{4}$$
$97$ $$28561 + 2197 T^{2} + 169 T^{4} + 13 T^{6} + T^{8}$$