# Properties

 Label 8512.2.a.c Level $8512$ Weight $2$ Character orbit 8512.a Self dual yes Analytic conductor $67.969$ 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] = [8512,2,Mod(1,8512)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8512, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8512.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$8512 = 2^{6} \cdot 7 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8512.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: $$67.9686622005$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 532) 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)

 $$f(q)$$ $$=$$ $$q + 2 q^{5} - q^{7} - 3 q^{9}+O(q^{10})$$ q + 2 * q^5 - q^7 - 3 * q^9 $$q + 2 q^{5} - q^{7} - 3 q^{9} + 4 q^{11} - 4 q^{13} + 6 q^{17} + q^{19} - 4 q^{23} - q^{25} - 6 q^{29} - 4 q^{31} - 2 q^{35} + 10 q^{37} + 4 q^{41} - 8 q^{43} - 6 q^{45} + q^{49} - 10 q^{53} + 8 q^{55} - 4 q^{59} - 14 q^{61} + 3 q^{63} - 8 q^{65} - 6 q^{67} + 6 q^{71} - 2 q^{73} - 4 q^{77} + 10 q^{79} + 9 q^{81} - 4 q^{83} + 12 q^{85} + 12 q^{89} + 4 q^{91} + 2 q^{95} - 12 q^{97} - 12 q^{99}+O(q^{100})$$ q + 2 * q^5 - q^7 - 3 * q^9 + 4 * q^11 - 4 * q^13 + 6 * q^17 + q^19 - 4 * q^23 - q^25 - 6 * q^29 - 4 * q^31 - 2 * q^35 + 10 * q^37 + 4 * q^41 - 8 * q^43 - 6 * q^45 + q^49 - 10 * q^53 + 8 * q^55 - 4 * q^59 - 14 * q^61 + 3 * q^63 - 8 * q^65 - 6 * q^67 + 6 * q^71 - 2 * q^73 - 4 * q^77 + 10 * q^79 + 9 * q^81 - 4 * q^83 + 12 * q^85 + 12 * q^89 + 4 * q^91 + 2 * q^95 - 12 * q^97 - 12 * q^99

## 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
0 0 0 2.00000 0 −1.00000 0 −3.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$7$$ $$+1$$
$$19$$ $$-1$$

## 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 8512.2.a.c 1
4.b odd 2 1 8512.2.a.d 1
8.b even 2 1 2128.2.a.a 1
8.d odd 2 1 532.2.a.a 1
24.f even 2 1 4788.2.a.e 1
56.e even 2 1 3724.2.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
532.2.a.a 1 8.d odd 2 1
2128.2.a.a 1 8.b even 2 1
3724.2.a.b 1 56.e even 2 1
4788.2.a.e 1 24.f even 2 1
8512.2.a.c 1 1.a even 1 1 trivial
8512.2.a.d 1 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(8512))$$:

 $$T_{3}$$ T3 $$T_{5} - 2$$ T5 - 2 $$T_{11} - 4$$ T11 - 4 $$T_{23} + 4$$ T23 + 4

## Hecke characteristic polynomials

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