# Properties

 Label 8470.2.a.j Level $8470$ Weight $2$ Character orbit 8470.a Self dual yes Analytic conductor $67.633$ 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] = [8470,2,Mod(1,8470)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8470, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8470.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8470.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: $$67.6332905120$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 70) 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)

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} - q^{5} + q^{7} - q^{8} - 3 q^{9}+O(q^{10})$$ q - q^2 + q^4 - q^5 + q^7 - q^8 - 3 * q^9 $$q - q^{2} + q^{4} - q^{5} + q^{7} - q^{8} - 3 q^{9} + q^{10} + 6 q^{13} - q^{14} + q^{16} - 2 q^{17} + 3 q^{18} - q^{20} + q^{25} - 6 q^{26} + q^{28} - 6 q^{29} + 8 q^{31} - q^{32} + 2 q^{34} - q^{35} - 3 q^{36} - 10 q^{37} + q^{40} - 2 q^{41} - 4 q^{43} + 3 q^{45} + 8 q^{47} + q^{49} - q^{50} + 6 q^{52} - 2 q^{53} - q^{56} + 6 q^{58} - 8 q^{59} + 14 q^{61} - 8 q^{62} - 3 q^{63} + q^{64} - 6 q^{65} - 12 q^{67} - 2 q^{68} + q^{70} - 16 q^{71} + 3 q^{72} - 2 q^{73} + 10 q^{74} + 8 q^{79} - q^{80} + 9 q^{81} + 2 q^{82} - 8 q^{83} + 2 q^{85} + 4 q^{86} + 10 q^{89} - 3 q^{90} + 6 q^{91} - 8 q^{94} + 2 q^{97} - q^{98}+O(q^{100})$$ q - q^2 + q^4 - q^5 + q^7 - q^8 - 3 * q^9 + q^10 + 6 * q^13 - q^14 + q^16 - 2 * q^17 + 3 * q^18 - q^20 + q^25 - 6 * q^26 + q^28 - 6 * q^29 + 8 * q^31 - q^32 + 2 * q^34 - q^35 - 3 * q^36 - 10 * q^37 + q^40 - 2 * q^41 - 4 * q^43 + 3 * q^45 + 8 * q^47 + q^49 - q^50 + 6 * q^52 - 2 * q^53 - q^56 + 6 * q^58 - 8 * q^59 + 14 * q^61 - 8 * q^62 - 3 * q^63 + q^64 - 6 * q^65 - 12 * q^67 - 2 * q^68 + q^70 - 16 * q^71 + 3 * q^72 - 2 * q^73 + 10 * q^74 + 8 * q^79 - q^80 + 9 * q^81 + 2 * q^82 - 8 * q^83 + 2 * q^85 + 4 * q^86 + 10 * q^89 - 3 * q^90 + 6 * q^91 - 8 * q^94 + 2 * 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 0 1.00000 −1.00000 0 1.00000 −1.00000 −3.00000 1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$5$$ $$1$$
$$7$$ $$-1$$
$$11$$ $$-1$$

## 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 8470.2.a.j 1
11.b odd 2 1 70.2.a.a 1
33.d even 2 1 630.2.a.d 1
44.c even 2 1 560.2.a.d 1
55.d odd 2 1 350.2.a.b 1
55.e even 4 2 350.2.c.b 2
77.b even 2 1 490.2.a.h 1
77.h odd 6 2 490.2.e.d 2
77.i even 6 2 490.2.e.c 2
88.b odd 2 1 2240.2.a.n 1
88.g even 2 1 2240.2.a.q 1
132.d odd 2 1 5040.2.a.bm 1
165.d even 2 1 3150.2.a.bj 1
165.l odd 4 2 3150.2.g.c 2
220.g even 2 1 2800.2.a.m 1
220.i odd 4 2 2800.2.g.n 2
231.h odd 2 1 4410.2.a.b 1
308.g odd 2 1 3920.2.a.t 1
385.h even 2 1 2450.2.a.l 1
385.l odd 4 2 2450.2.c.k 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.a.a 1 11.b odd 2 1
350.2.a.b 1 55.d odd 2 1
350.2.c.b 2 55.e even 4 2
490.2.a.h 1 77.b even 2 1
490.2.e.c 2 77.i even 6 2
490.2.e.d 2 77.h odd 6 2
560.2.a.d 1 44.c even 2 1
630.2.a.d 1 33.d even 2 1
2240.2.a.n 1 88.b odd 2 1
2240.2.a.q 1 88.g even 2 1
2450.2.a.l 1 385.h even 2 1
2450.2.c.k 2 385.l odd 4 2
2800.2.a.m 1 220.g even 2 1
2800.2.g.n 2 220.i odd 4 2
3150.2.a.bj 1 165.d even 2 1
3150.2.g.c 2 165.l odd 4 2
3920.2.a.t 1 308.g odd 2 1
4410.2.a.b 1 231.h odd 2 1
5040.2.a.bm 1 132.d odd 2 1
8470.2.a.j 1 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(8470))$$:

 $$T_{3}$$ T3 $$T_{13} - 6$$ T13 - 6 $$T_{17} + 2$$ T17 + 2 $$T_{19}$$ T19

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T$$
$5$ $$T + 1$$
$7$ $$T - 1$$
$11$ $$T$$
$13$ $$T - 6$$
$17$ $$T + 2$$
$19$ $$T$$
$23$ $$T$$
$29$ $$T + 6$$
$31$ $$T - 8$$
$37$ $$T + 10$$
$41$ $$T + 2$$
$43$ $$T + 4$$
$47$ $$T - 8$$
$53$ $$T + 2$$
$59$ $$T + 8$$
$61$ $$T - 14$$
$67$ $$T + 12$$
$71$ $$T + 16$$
$73$ $$T + 2$$
$79$ $$T - 8$$
$83$ $$T + 8$$
$89$ $$T - 10$$
$97$ $$T - 2$$