# Properties

 Label 847.2.b.b Level $847$ Weight $2$ Character orbit 847.b Analytic conductor $6.763$ Analytic rank $0$ Dimension $8$ CM discriminant -7 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$847 = 7 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 847.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.76332905120$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.37515625.1 Defining polynomial: $$x^{8} - x^{7} - x^{6} + 3 x^{5} - x^{4} + 6 x^{3} - 4 x^{2} - 8 x + 16$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2^{4}\cdot 11^{2}$$ Twist minimal: no (minimal twist has level 77) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( -1 + \beta_{2} ) q^{4} + ( -\beta_{1} + \beta_{4} ) q^{7} + ( -\beta_{1} + \beta_{3} ) q^{8} + 3 q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( -1 + \beta_{2} ) q^{4} + ( -\beta_{1} + \beta_{4} ) q^{7} + ( -\beta_{1} + \beta_{3} ) q^{8} + 3 q^{9} + ( 1 - \beta_{2} + \beta_{7} ) q^{14} + ( 1 - 2 \beta_{2} - \beta_{6} + \beta_{7} ) q^{16} + 3 \beta_{1} q^{18} + ( 2 - \beta_{2} + \beta_{6} ) q^{23} + 5 q^{25} + ( 2 \beta_{1} - \beta_{3} - \beta_{5} ) q^{28} + ( \beta_{1} + 2 \beta_{4} - \beta_{5} ) q^{29} + ( 3 \beta_{1} - 2 \beta_{3} - \beta_{4} ) q^{32} + ( -3 + 3 \beta_{2} ) q^{36} + ( -\beta_{2} - \beta_{6} + 2 \beta_{7} ) q^{37} + ( \beta_{1} + 2 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{43} + ( 5 \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{46} -7 q^{49} + 5 \beta_{1} q^{50} + ( -3 + \beta_{2} + \beta_{6} + 2 \beta_{7} ) q^{53} + ( -4 + 4 \beta_{2} + \beta_{6} ) q^{56} + ( -7 + 2 \beta_{2} + \beta_{7} ) q^{58} + ( -3 \beta_{1} + 3 \beta_{4} ) q^{63} + ( -5 + 5 \beta_{2} - \beta_{7} ) q^{64} + ( 1 - 3 \beta_{2} - \beta_{6} - 2 \beta_{7} ) q^{67} + ( 5 + \beta_{2} - \beta_{6} - 2 \beta_{7} ) q^{71} + ( -3 \beta_{1} + 3 \beta_{3} ) q^{72} + ( 3 \beta_{1} - 3 \beta_{3} - 3 \beta_{4} - \beta_{5} ) q^{74} + ( -3 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{79} + 9 q^{81} + ( 1 - 6 \beta_{2} - 2 \beta_{6} + \beta_{7} ) q^{86} + ( -9 + \beta_{2} + \beta_{6} - \beta_{7} ) q^{92} -7 \beta_{1} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 10q^{4} + 24q^{9} + O(q^{10})$$ $$8q - 10q^{4} + 24q^{9} + 14q^{14} + 18q^{16} + 16q^{23} + 40q^{25} - 30q^{36} + 12q^{37} - 56q^{49} - 20q^{53} - 42q^{56} - 56q^{58} - 54q^{64} + 8q^{67} + 32q^{71} + 72q^{81} + 28q^{86} - 80q^{92} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{7} - x^{6} + 3 x^{5} - x^{4} + 6 x^{3} - 4 x^{2} - 8 x + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{7} - \nu^{6} - \nu^{5} + 3 \nu^{4} - \nu^{3} + 6 \nu^{2} + 4 \nu - 8$$$$)/8$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{7} - \nu^{6} + 3 \nu^{5} - \nu^{4} - 5 \nu^{3} + 4 \nu^{2} - 8 \nu + 8$$$$)/8$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{7} + \nu^{6} + \nu^{5} - 3 \nu^{4} + 9 \nu^{3} + 16 \nu^{2} - 4 \nu + 8$$$$)/8$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{7} - \nu^{6} - 17 \nu^{5} + 3 \nu^{4} - \nu^{3} + 6 \nu^{2} + 4 \nu - 96$$$$)/8$$ $$\beta_{5}$$ $$=$$ $$($$$$3 \nu^{7} - 7 \nu^{6} - \nu^{5} - \nu^{4} - \nu^{3} + 20 \nu^{2} - 38 \nu - 12$$$$)/4$$ $$\beta_{6}$$ $$=$$ $$($$$$-6 \nu^{7} - \nu^{6} + 7 \nu^{5} - 5 \nu^{4} + 3 \nu^{3} - 31 \nu^{2} - 4 \nu + 40$$$$)/4$$ $$\beta_{7}$$ $$=$$ $$($$$$-6 \nu^{7} - 3 \nu^{6} + 7 \nu^{5} - \nu^{4} + 3 \nu^{3} - 31 \nu^{2} - 26 \nu + 44$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-3 \beta_{7} + 4 \beta_{6} - \beta_{2} + 11 \beta_{1} + 5$$$$)/22$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{5} + 2 \beta_{4} + 7 \beta_{3} + 11 \beta_{2} + 8 \beta_{1} + 11$$$$)/22$$ $$\nu^{3}$$ $$=$$ $$($$$$2 \beta_{7} + \beta_{6} + 11 \beta_{3} - 14 \beta_{2} + 11 \beta_{1} - 18$$$$)/22$$ $$\nu^{4}$$ $$=$$ $$($$$$11 \beta_{7} - 11 \beta_{6} - 5 \beta_{5} - \beta_{4} + 2 \beta_{3} + 29 \beta_{1} - 11$$$$)/22$$ $$\nu^{5}$$ $$=$$ $$($$$$-\beta_{4} + \beta_{1} - 11$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$11 \beta_{7} - 22 \beta_{6} - 10 \beta_{5} - 2 \beta_{4} + 4 \beta_{3} + 11 \beta_{2} - 63 \beta_{1} - 33$$$$)/22$$ $$\nu^{7}$$ $$=$$ $$($$$$-8 \beta_{7} - 4 \beta_{6} + 11 \beta_{5} - 22 \beta_{4} - 33 \beta_{3} - 65 \beta_{2} - 44 \beta_{1} - 49$$$$)/22$$

## Character values

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

 $$n$$ $$122$$ $$365$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
846.1
 −0.373058 − 1.36412i 1.10362 − 0.884319i 1.18208 − 0.776336i −1.41264 − 0.0667372i −1.41264 + 0.0667372i 1.18208 + 0.776336i 1.10362 + 0.884319i −0.373058 + 1.36412i
2.72824i 0 −5.44331 0 0 2.64575i 9.39419i 3.00000 0
846.2 1.76864i 0 −1.12808 0 0 2.64575i 1.54211i 3.00000 0
846.3 1.55267i 0 −0.410792 0 0 2.64575i 2.46752i 3.00000 0
846.4 0.133474i 0 1.98218 0 0 2.64575i 0.531520i 3.00000 0
846.5 0.133474i 0 1.98218 0 0 2.64575i 0.531520i 3.00000 0
846.6 1.55267i 0 −0.410792 0 0 2.64575i 2.46752i 3.00000 0
846.7 1.76864i 0 −1.12808 0 0 2.64575i 1.54211i 3.00000 0
846.8 2.72824i 0 −5.44331 0 0 2.64575i 9.39419i 3.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 846.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by $$\Q(\sqrt{-7})$$
11.b odd 2 1 inner
77.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 847.2.b.b 8
7.b odd 2 1 CM 847.2.b.b 8
11.b odd 2 1 inner 847.2.b.b 8
11.c even 5 1 77.2.l.a 8
11.c even 5 1 847.2.l.a 8
11.c even 5 1 847.2.l.c 8
11.c even 5 1 847.2.l.d 8
11.d odd 10 1 77.2.l.a 8
11.d odd 10 1 847.2.l.a 8
11.d odd 10 1 847.2.l.c 8
11.d odd 10 1 847.2.l.d 8
33.f even 10 1 693.2.bu.a 8
33.h odd 10 1 693.2.bu.a 8
77.b even 2 1 inner 847.2.b.b 8
77.j odd 10 1 77.2.l.a 8
77.j odd 10 1 847.2.l.a 8
77.j odd 10 1 847.2.l.c 8
77.j odd 10 1 847.2.l.d 8
77.l even 10 1 77.2.l.a 8
77.l even 10 1 847.2.l.a 8
77.l even 10 1 847.2.l.c 8
77.l even 10 1 847.2.l.d 8
77.m even 15 2 539.2.s.a 16
77.n even 30 2 539.2.s.a 16
77.o odd 30 2 539.2.s.a 16
77.p odd 30 2 539.2.s.a 16
231.r odd 10 1 693.2.bu.a 8
231.u even 10 1 693.2.bu.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.l.a 8 11.c even 5 1
77.2.l.a 8 11.d odd 10 1
77.2.l.a 8 77.j odd 10 1
77.2.l.a 8 77.l even 10 1
539.2.s.a 16 77.m even 15 2
539.2.s.a 16 77.n even 30 2
539.2.s.a 16 77.o odd 30 2
539.2.s.a 16 77.p odd 30 2
693.2.bu.a 8 33.f even 10 1
693.2.bu.a 8 33.h odd 10 1
693.2.bu.a 8 231.r odd 10 1
693.2.bu.a 8 231.u even 10 1
847.2.b.b 8 1.a even 1 1 trivial
847.2.b.b 8 7.b odd 2 1 CM
847.2.b.b 8 11.b odd 2 1 inner
847.2.b.b 8 77.b even 2 1 inner
847.2.l.a 8 11.c even 5 1
847.2.l.a 8 11.d odd 10 1
847.2.l.a 8 77.j odd 10 1
847.2.l.a 8 77.l even 10 1
847.2.l.c 8 11.c even 5 1
847.2.l.c 8 11.d odd 10 1
847.2.l.c 8 77.j odd 10 1
847.2.l.c 8 77.l even 10 1
847.2.l.d 8 11.c even 5 1
847.2.l.d 8 11.d odd 10 1
847.2.l.d 8 77.j odd 10 1
847.2.l.d 8 77.l even 10 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} + 13 T_{2}^{6} + 49 T_{2}^{4} + 57 T_{2}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(847, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 57 T^{2} + 49 T^{4} + 13 T^{6} + T^{8}$$
$3$ $$T^{8}$$
$5$ $$T^{8}$$
$7$ $$( 7 + T^{2} )^{4}$$
$11$ $$T^{8}$$
$13$ $$T^{8}$$
$17$ $$T^{8}$$
$19$ $$T^{8}$$
$23$ $$( -619 + 408 T - 51 T^{2} - 8 T^{3} + T^{4} )^{2}$$
$29$ $$259081 + 155562 T^{2} + 9499 T^{4} + 178 T^{6} + T^{8}$$
$31$ $$T^{8}$$
$37$ $$( 1481 + 894 T - 149 T^{2} - 6 T^{3} + T^{4} )^{2}$$
$41$ $$T^{8}$$
$43$ $$16072081 + 2478498 T^{2} + 53459 T^{4} + 402 T^{6} + T^{8}$$
$47$ $$T^{8}$$
$53$ $$( -2455 - 1650 T - 165 T^{2} + 10 T^{3} + T^{4} )^{2}$$
$59$ $$T^{8}$$
$61$ $$T^{8}$$
$67$ $$( 17341 + 1276 T - 319 T^{2} - 4 T^{3} + T^{4} )^{2}$$
$71$ $$( -139 + 1584 T - 99 T^{2} - 16 T^{3} + T^{4} )^{2}$$
$73$ $$T^{8}$$
$79$ $$23338561 + 3771482 T^{2} + 82859 T^{4} + 538 T^{6} + T^{8}$$
$83$ $$T^{8}$$
$89$ $$T^{8}$$
$97$ $$T^{8}$$