# Properties

 Label 8624.2.a.bf Level $8624$ Weight $2$ Character orbit 8624.a Self dual yes Analytic conductor $68.863$ 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] = [8624,2,Mod(1,8624)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8624, 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("8624.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$8624 = 2^{4} \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8624.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: $$68.8629867032$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 154) 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 $$\beta = \sqrt{5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta - 1) q^{3} + ( - \beta - 1) q^{5} + (2 \beta + 3) q^{9}+O(q^{10})$$ q + (-b - 1) * q^3 + (-b - 1) * q^5 + (2*b + 3) * q^9 $$q + ( - \beta - 1) q^{3} + ( - \beta - 1) q^{5} + (2 \beta + 3) q^{9} - q^{11} + ( - \beta + 1) q^{13} + (2 \beta + 6) q^{15} + (2 \beta + 2) q^{17} + (\beta - 5) q^{19} - 4 q^{23} + (2 \beta + 1) q^{25} + ( - 2 \beta - 10) q^{27} - 2 \beta q^{29} + 2 q^{31} + (\beta + 1) q^{33} + ( - 4 \beta - 2) q^{37} + 4 q^{39} + ( - 2 \beta - 2) q^{41} + ( - 2 \beta + 6) q^{43} + ( - 5 \beta - 13) q^{45} - 2 q^{47} + ( - 4 \beta - 12) q^{51} + ( - 2 \beta + 4) q^{53} + (\beta + 1) q^{55} + 4 \beta q^{57} + (\beta + 5) q^{59} + (\beta + 3) q^{61} + 4 q^{65} + (6 \beta + 2) q^{67} + (4 \beta + 4) q^{69} + (2 \beta - 2) q^{71} + (4 \beta - 4) q^{73} + ( - 3 \beta - 11) q^{75} + (6 \beta + 11) q^{81} + (5 \beta - 1) q^{83} + ( - 4 \beta - 12) q^{85} + (2 \beta + 10) q^{87} - 10 q^{89} + ( - 2 \beta - 2) q^{93} + 4 \beta q^{95} + (2 \beta - 8) q^{97} + ( - 2 \beta - 3) q^{99} +O(q^{100})$$ q + (-b - 1) * q^3 + (-b - 1) * q^5 + (2*b + 3) * q^9 - q^11 + (-b + 1) * q^13 + (2*b + 6) * q^15 + (2*b + 2) * q^17 + (b - 5) * q^19 - 4 * q^23 + (2*b + 1) * q^25 + (-2*b - 10) * q^27 - 2*b * q^29 + 2 * q^31 + (b + 1) * q^33 + (-4*b - 2) * q^37 + 4 * q^39 + (-2*b - 2) * q^41 + (-2*b + 6) * q^43 + (-5*b - 13) * q^45 - 2 * q^47 + (-4*b - 12) * q^51 + (-2*b + 4) * q^53 + (b + 1) * q^55 + 4*b * q^57 + (b + 5) * q^59 + (b + 3) * q^61 + 4 * q^65 + (6*b + 2) * q^67 + (4*b + 4) * q^69 + (2*b - 2) * q^71 + (4*b - 4) * q^73 + (-3*b - 11) * q^75 + (6*b + 11) * q^81 + (5*b - 1) * q^83 + (-4*b - 12) * q^85 + (2*b + 10) * q^87 - 10 * q^89 + (-2*b - 2) * q^93 + 4*b * q^95 + (2*b - 8) * q^97 + (-2*b - 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} - 2 q^{5} + 6 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 - 2 * q^5 + 6 * q^9 $$2 q - 2 q^{3} - 2 q^{5} + 6 q^{9} - 2 q^{11} + 2 q^{13} + 12 q^{15} + 4 q^{17} - 10 q^{19} - 8 q^{23} + 2 q^{25} - 20 q^{27} + 4 q^{31} + 2 q^{33} - 4 q^{37} + 8 q^{39} - 4 q^{41} + 12 q^{43} - 26 q^{45} - 4 q^{47} - 24 q^{51} + 8 q^{53} + 2 q^{55} + 10 q^{59} + 6 q^{61} + 8 q^{65} + 4 q^{67} + 8 q^{69} - 4 q^{71} - 8 q^{73} - 22 q^{75} + 22 q^{81} - 2 q^{83} - 24 q^{85} + 20 q^{87} - 20 q^{89} - 4 q^{93} - 16 q^{97} - 6 q^{99}+O(q^{100})$$ 2 * q - 2 * q^3 - 2 * q^5 + 6 * q^9 - 2 * q^11 + 2 * q^13 + 12 * q^15 + 4 * q^17 - 10 * q^19 - 8 * q^23 + 2 * q^25 - 20 * q^27 + 4 * q^31 + 2 * q^33 - 4 * q^37 + 8 * q^39 - 4 * q^41 + 12 * q^43 - 26 * q^45 - 4 * q^47 - 24 * q^51 + 8 * q^53 + 2 * q^55 + 10 * q^59 + 6 * q^61 + 8 * q^65 + 4 * q^67 + 8 * q^69 - 4 * q^71 - 8 * q^73 - 22 * q^75 + 22 * q^81 - 2 * q^83 - 24 * q^85 + 20 * q^87 - 20 * q^89 - 4 * q^93 - 16 * 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
 1.61803 −0.618034
0 −3.23607 0 −3.23607 0 0 0 7.47214 0
1.2 0 1.23607 0 1.23607 0 0 0 −1.47214 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
$$2$$ $$-1$$
$$7$$ $$-1$$
$$11$$ $$1$$

## 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 8624.2.a.bf 2
4.b odd 2 1 1078.2.a.w 2
7.b odd 2 1 1232.2.a.p 2
12.b even 2 1 9702.2.a.cu 2
28.d even 2 1 154.2.a.d 2
28.f even 6 2 1078.2.e.q 4
28.g odd 6 2 1078.2.e.n 4
56.e even 2 1 4928.2.a.bt 2
56.h odd 2 1 4928.2.a.bk 2
84.h odd 2 1 1386.2.a.m 2
140.c even 2 1 3850.2.a.bj 2
140.j odd 4 2 3850.2.c.q 4
308.g odd 2 1 1694.2.a.l 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.a.d 2 28.d even 2 1
1078.2.a.w 2 4.b odd 2 1
1078.2.e.n 4 28.g odd 6 2
1078.2.e.q 4 28.f even 6 2
1232.2.a.p 2 7.b odd 2 1
1386.2.a.m 2 84.h odd 2 1
1694.2.a.l 2 308.g odd 2 1
3850.2.a.bj 2 140.c even 2 1
3850.2.c.q 4 140.j odd 4 2
4928.2.a.bk 2 56.h odd 2 1
4928.2.a.bt 2 56.e even 2 1
8624.2.a.bf 2 1.a even 1 1 trivial
9702.2.a.cu 2 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(8624))$$:

 $$T_{3}^{2} + 2T_{3} - 4$$ T3^2 + 2*T3 - 4 $$T_{5}^{2} + 2T_{5} - 4$$ T5^2 + 2*T5 - 4 $$T_{13}^{2} - 2T_{13} - 4$$ T13^2 - 2*T13 - 4 $$T_{17}^{2} - 4T_{17} - 16$$ T17^2 - 4*T17 - 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 2T - 4$$
$5$ $$T^{2} + 2T - 4$$
$7$ $$T^{2}$$
$11$ $$(T + 1)^{2}$$
$13$ $$T^{2} - 2T - 4$$
$17$ $$T^{2} - 4T - 16$$
$19$ $$T^{2} + 10T + 20$$
$23$ $$(T + 4)^{2}$$
$29$ $$T^{2} - 20$$
$31$ $$(T - 2)^{2}$$
$37$ $$T^{2} + 4T - 76$$
$41$ $$T^{2} + 4T - 16$$
$43$ $$T^{2} - 12T + 16$$
$47$ $$(T + 2)^{2}$$
$53$ $$T^{2} - 8T - 4$$
$59$ $$T^{2} - 10T + 20$$
$61$ $$T^{2} - 6T + 4$$
$67$ $$T^{2} - 4T - 176$$
$71$ $$T^{2} + 4T - 16$$
$73$ $$T^{2} + 8T - 64$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 2T - 124$$
$89$ $$(T + 10)^{2}$$
$97$ $$T^{2} + 16T + 44$$