# Properties

 Label 8512.2.a.x Level $8512$ Weight $2$ Character orbit 8512.a Self dual yes Analytic conductor $67.969$ Analytic rank $1$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 3$$ x^2 - x - 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1064) 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)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{13})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} + q^{5} - q^{7} + \beta q^{9}+O(q^{10})$$ q + b * q^3 + q^5 - q^7 + b * q^9 $$q + \beta q^{3} + q^{5} - q^{7} + \beta q^{9} + ( - \beta + 1) q^{11} + ( - 2 \beta + 1) q^{13} + \beta q^{15} + (\beta + 3) q^{17} - q^{19} - \beta q^{21} + ( - 2 \beta - 3) q^{23} - 4 q^{25} + ( - 2 \beta + 3) q^{27} + ( - \beta + 3) q^{29} + (\beta - 7) q^{31} - 3 q^{33} - q^{35} + (2 \beta - 1) q^{37} + ( - \beta - 6) q^{39} + ( - \beta - 1) q^{41} + (4 \beta - 2) q^{43} + \beta q^{45} + (2 \beta - 5) q^{47} + q^{49} + (4 \beta + 3) q^{51} - 3 \beta q^{53} + ( - \beta + 1) q^{55} - \beta q^{57} + (2 \beta - 3) q^{59} + (4 \beta - 3) q^{61} - \beta q^{63} + ( - 2 \beta + 1) q^{65} + ( - 3 \beta + 7) q^{67} + ( - 5 \beta - 6) q^{69} + ( - 6 \beta + 1) q^{71} + (3 \beta - 2) q^{73} - 4 \beta q^{75} + (\beta - 1) q^{77} - 14 q^{79} + ( - 2 \beta - 6) q^{81} + ( - 5 \beta + 4) q^{83} + (\beta + 3) q^{85} + (2 \beta - 3) q^{87} + ( - 6 \beta + 6) q^{89} + (2 \beta - 1) q^{91} + ( - 6 \beta + 3) q^{93} - q^{95} + ( - 6 \beta + 9) q^{97} - 3 q^{99} +O(q^{100})$$ q + b * q^3 + q^5 - q^7 + b * q^9 + (-b + 1) * q^11 + (-2*b + 1) * q^13 + b * q^15 + (b + 3) * q^17 - q^19 - b * q^21 + (-2*b - 3) * q^23 - 4 * q^25 + (-2*b + 3) * q^27 + (-b + 3) * q^29 + (b - 7) * q^31 - 3 * q^33 - q^35 + (2*b - 1) * q^37 + (-b - 6) * q^39 + (-b - 1) * q^41 + (4*b - 2) * q^43 + b * q^45 + (2*b - 5) * q^47 + q^49 + (4*b + 3) * q^51 - 3*b * q^53 + (-b + 1) * q^55 - b * q^57 + (2*b - 3) * q^59 + (4*b - 3) * q^61 - b * q^63 + (-2*b + 1) * q^65 + (-3*b + 7) * q^67 + (-5*b - 6) * q^69 + (-6*b + 1) * q^71 + (3*b - 2) * q^73 - 4*b * q^75 + (b - 1) * q^77 - 14 * q^79 + (-2*b - 6) * q^81 + (-5*b + 4) * q^83 + (b + 3) * q^85 + (2*b - 3) * q^87 + (-6*b + 6) * q^89 + (2*b - 1) * q^91 + (-6*b + 3) * q^93 - q^95 + (-6*b + 9) * q^97 - 3 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} + 2 q^{5} - 2 q^{7} + q^{9}+O(q^{10})$$ 2 * q + q^3 + 2 * q^5 - 2 * q^7 + q^9 $$2 q + q^{3} + 2 q^{5} - 2 q^{7} + q^{9} + q^{11} + q^{15} + 7 q^{17} - 2 q^{19} - q^{21} - 8 q^{23} - 8 q^{25} + 4 q^{27} + 5 q^{29} - 13 q^{31} - 6 q^{33} - 2 q^{35} - 13 q^{39} - 3 q^{41} + q^{45} - 8 q^{47} + 2 q^{49} + 10 q^{51} - 3 q^{53} + q^{55} - q^{57} - 4 q^{59} - 2 q^{61} - q^{63} + 11 q^{67} - 17 q^{69} - 4 q^{71} - q^{73} - 4 q^{75} - q^{77} - 28 q^{79} - 14 q^{81} + 3 q^{83} + 7 q^{85} - 4 q^{87} + 6 q^{89} - 2 q^{95} + 12 q^{97} - 6 q^{99}+O(q^{100})$$ 2 * q + q^3 + 2 * q^5 - 2 * q^7 + q^9 + q^11 + q^15 + 7 * q^17 - 2 * q^19 - q^21 - 8 * q^23 - 8 * q^25 + 4 * q^27 + 5 * q^29 - 13 * q^31 - 6 * q^33 - 2 * q^35 - 13 * q^39 - 3 * q^41 + q^45 - 8 * q^47 + 2 * q^49 + 10 * q^51 - 3 * q^53 + q^55 - q^57 - 4 * q^59 - 2 * q^61 - q^63 + 11 * q^67 - 17 * q^69 - 4 * q^71 - q^73 - 4 * q^75 - q^77 - 28 * q^79 - 14 * q^81 + 3 * q^83 + 7 * q^85 - 4 * q^87 + 6 * q^89 - 2 * q^95 + 12 * q^97 - 6 * 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
 −1.30278 2.30278
0 −1.30278 0 1.00000 0 −1.00000 0 −1.30278 0
1.2 0 2.30278 0 1.00000 0 −1.00000 0 2.30278 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.x 2
4.b odd 2 1 8512.2.a.r 2
8.b even 2 1 1064.2.a.b 2
8.d odd 2 1 2128.2.a.i 2
24.h odd 2 1 9576.2.a.bs 2
56.h odd 2 1 7448.2.a.bb 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1064.2.a.b 2 8.b even 2 1
2128.2.a.i 2 8.d odd 2 1
7448.2.a.bb 2 56.h odd 2 1
8512.2.a.r 2 4.b odd 2 1
8512.2.a.x 2 1.a even 1 1 trivial
9576.2.a.bs 2 24.h odd 2 1

## Hecke kernels

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

 $$T_{3}^{2} - T_{3} - 3$$ T3^2 - T3 - 3 $$T_{5} - 1$$ T5 - 1 $$T_{11}^{2} - T_{11} - 3$$ T11^2 - T11 - 3 $$T_{23}^{2} + 8T_{23} + 3$$ T23^2 + 8*T23 + 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - T - 3$$
$5$ $$(T - 1)^{2}$$
$7$ $$(T + 1)^{2}$$
$11$ $$T^{2} - T - 3$$
$13$ $$T^{2} - 13$$
$17$ $$T^{2} - 7T + 9$$
$19$ $$(T + 1)^{2}$$
$23$ $$T^{2} + 8T + 3$$
$29$ $$T^{2} - 5T + 3$$
$31$ $$T^{2} + 13T + 39$$
$37$ $$T^{2} - 13$$
$41$ $$T^{2} + 3T - 1$$
$43$ $$T^{2} - 52$$
$47$ $$T^{2} + 8T + 3$$
$53$ $$T^{2} + 3T - 27$$
$59$ $$T^{2} + 4T - 9$$
$61$ $$T^{2} + 2T - 51$$
$67$ $$T^{2} - 11T + 1$$
$71$ $$T^{2} + 4T - 113$$
$73$ $$T^{2} + T - 29$$
$79$ $$(T + 14)^{2}$$
$83$ $$T^{2} - 3T - 79$$
$89$ $$T^{2} - 6T - 108$$
$97$ $$T^{2} - 12T - 81$$