# Properties

 Label 847.2.a.a Level $847$ Weight $2$ Character orbit 847.a Self dual yes Analytic conductor $6.763$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [847,2,Mod(1,847)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(847, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("847.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$847 = 7 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 847.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$6.76332905120$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 77) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
−1.00000 2.00000 −1.00000 −2.00000 −2.00000 1.00000 3.00000 1.00000 2.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$7$$ $$-1$$
$$11$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 847.2.a.a 1
3.b odd 2 1 7623.2.a.n 1
7.b odd 2 1 5929.2.a.b 1
11.b odd 2 1 77.2.a.c 1
11.c even 5 4 847.2.f.k 4
11.d odd 10 4 847.2.f.e 4
33.d even 2 1 693.2.a.a 1
44.c even 2 1 1232.2.a.a 1
55.d odd 2 1 1925.2.a.c 1
55.e even 4 2 1925.2.b.d 2
77.b even 2 1 539.2.a.d 1
77.h odd 6 2 539.2.e.a 2
77.i even 6 2 539.2.e.b 2
88.b odd 2 1 4928.2.a.g 1
88.g even 2 1 4928.2.a.bi 1
231.h odd 2 1 4851.2.a.a 1
308.g odd 2 1 8624.2.a.bc 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.a.c 1 11.b odd 2 1
539.2.a.d 1 77.b even 2 1
539.2.e.a 2 77.h odd 6 2
539.2.e.b 2 77.i even 6 2
693.2.a.a 1 33.d even 2 1
847.2.a.a 1 1.a even 1 1 trivial
847.2.f.e 4 11.d odd 10 4
847.2.f.k 4 11.c even 5 4
1232.2.a.a 1 44.c even 2 1
1925.2.a.c 1 55.d odd 2 1
1925.2.b.d 2 55.e even 4 2
4851.2.a.a 1 231.h odd 2 1
4928.2.a.g 1 88.b odd 2 1
4928.2.a.bi 1 88.g even 2 1
5929.2.a.b 1 7.b odd 2 1
7623.2.a.n 1 3.b odd 2 1
8624.2.a.bc 1 308.g odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(847))$$:

 $$T_{2} + 1$$ T2 + 1 $$T_{3} - 2$$ T3 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T - 2$$
$5$ $$T + 2$$
$7$ $$T - 1$$
$11$ $$T$$
$13$ $$T + 4$$
$17$ $$T + 4$$
$19$ $$T$$
$23$ $$T + 4$$
$29$ $$T - 6$$
$31$ $$T - 10$$
$37$ $$T + 6$$
$41$ $$T + 4$$
$43$ $$T + 12$$
$47$ $$T + 10$$
$53$ $$T + 6$$
$59$ $$T - 2$$
$61$ $$T$$
$67$ $$T - 8$$
$71$ $$T + 12$$
$73$ $$T - 8$$
$79$ $$T + 8$$
$83$ $$T$$
$89$ $$T + 6$$
$97$ $$T + 10$$