# Properties

 Label 8624.2.a.bh 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: $$\Q(\sqrt{2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ 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 $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta - 1) q^{3} + (\beta + 2) q^{5} - 2 \beta q^{9} +O(q^{10})$$ q + (b - 1) * q^3 + (b + 2) * q^5 - 2*b * q^9 $$q + (\beta - 1) q^{3} + (\beta + 2) q^{5} - 2 \beta q^{9} - q^{11} + ( - 2 \beta + 1) q^{13} + \beta q^{15} + (4 \beta + 2) q^{17} + ( - \beta - 2) q^{19} + ( - 3 \beta + 2) q^{23} + (4 \beta + 1) q^{25} + ( - \beta - 1) q^{27} + ( - 4 \beta - 3) q^{29} - 4 q^{31} + ( - \beta + 1) q^{33} + (\beta - 8) q^{37} + (3 \beta - 5) q^{39} + ( - \beta + 4) q^{41} - 4 \beta q^{43} + ( - 4 \beta - 4) q^{45} + (6 \beta - 2) q^{47} + ( - 2 \beta + 6) q^{51} + ( - 7 \beta - 2) q^{53} + ( - \beta - 2) q^{55} - \beta q^{57} + ( - \beta - 7) q^{59} + (2 \beta - 9) q^{61} + ( - 3 \beta - 2) q^{65} + ( - 3 \beta - 7) q^{67} + (5 \beta - 8) q^{69} + ( - 5 \beta + 4) q^{71} + ( - \beta + 8) q^{73} + ( - 3 \beta + 7) q^{75} + ( - 3 \beta + 9) q^{79} + (6 \beta - 1) q^{81} + (10 \beta + 2) q^{83} + (10 \beta + 12) q^{85} + (\beta - 5) q^{87} + (6 \beta - 4) q^{89} + ( - 4 \beta + 4) q^{93} + ( - 4 \beta - 6) q^{95} + ( - 2 \beta + 1) q^{97} + 2 \beta q^{99} +O(q^{100})$$ q + (b - 1) * q^3 + (b + 2) * q^5 - 2*b * q^9 - q^11 + (-2*b + 1) * q^13 + b * q^15 + (4*b + 2) * q^17 + (-b - 2) * q^19 + (-3*b + 2) * q^23 + (4*b + 1) * q^25 + (-b - 1) * q^27 + (-4*b - 3) * q^29 - 4 * q^31 + (-b + 1) * q^33 + (b - 8) * q^37 + (3*b - 5) * q^39 + (-b + 4) * q^41 - 4*b * q^43 + (-4*b - 4) * q^45 + (6*b - 2) * q^47 + (-2*b + 6) * q^51 + (-7*b - 2) * q^53 + (-b - 2) * q^55 - b * q^57 + (-b - 7) * q^59 + (2*b - 9) * q^61 + (-3*b - 2) * q^65 + (-3*b - 7) * q^67 + (5*b - 8) * q^69 + (-5*b + 4) * q^71 + (-b + 8) * q^73 + (-3*b + 7) * q^75 + (-3*b + 9) * q^79 + (6*b - 1) * q^81 + (10*b + 2) * q^83 + (10*b + 12) * q^85 + (b - 5) * q^87 + (6*b - 4) * q^89 + (-4*b + 4) * q^93 + (-4*b - 6) * q^95 + (-2*b + 1) * q^97 + 2*b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + 4 q^{5}+O(q^{10})$$ 2 * q - 2 * q^3 + 4 * q^5 $$2 q - 2 q^{3} + 4 q^{5} - 2 q^{11} + 2 q^{13} + 4 q^{17} - 4 q^{19} + 4 q^{23} + 2 q^{25} - 2 q^{27} - 6 q^{29} - 8 q^{31} + 2 q^{33} - 16 q^{37} - 10 q^{39} + 8 q^{41} - 8 q^{45} - 4 q^{47} + 12 q^{51} - 4 q^{53} - 4 q^{55} - 14 q^{59} - 18 q^{61} - 4 q^{65} - 14 q^{67} - 16 q^{69} + 8 q^{71} + 16 q^{73} + 14 q^{75} + 18 q^{79} - 2 q^{81} + 4 q^{83} + 24 q^{85} - 10 q^{87} - 8 q^{89} + 8 q^{93} - 12 q^{95} + 2 q^{97}+O(q^{100})$$ 2 * q - 2 * q^3 + 4 * q^5 - 2 * q^11 + 2 * q^13 + 4 * q^17 - 4 * q^19 + 4 * q^23 + 2 * q^25 - 2 * q^27 - 6 * q^29 - 8 * q^31 + 2 * q^33 - 16 * q^37 - 10 * q^39 + 8 * q^41 - 8 * q^45 - 4 * q^47 + 12 * q^51 - 4 * q^53 - 4 * q^55 - 14 * q^59 - 18 * q^61 - 4 * q^65 - 14 * q^67 - 16 * q^69 + 8 * q^71 + 16 * q^73 + 14 * q^75 + 18 * q^79 - 2 * q^81 + 4 * q^83 + 24 * q^85 - 10 * q^87 - 8 * q^89 + 8 * q^93 - 12 * q^95 + 2 * q^97

## 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
 −1.41421 1.41421
0 −2.41421 0 0.585786 0 0 0 2.82843 0
1.2 0 0.414214 0 3.41421 0 0 0 −2.82843 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.bh 2
4.b odd 2 1 1078.2.a.x 2
7.b odd 2 1 8624.2.a.cc 2
7.d odd 6 2 1232.2.q.f 4
12.b even 2 1 9702.2.a.ch 2
28.d even 2 1 1078.2.a.t 2
28.f even 6 2 154.2.e.e 4
28.g odd 6 2 1078.2.e.m 4
84.h odd 2 1 9702.2.a.cx 2
84.j odd 6 2 1386.2.k.t 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.e.e 4 28.f even 6 2
1078.2.a.t 2 28.d even 2 1
1078.2.a.x 2 4.b odd 2 1
1078.2.e.m 4 28.g odd 6 2
1232.2.q.f 4 7.d odd 6 2
1386.2.k.t 4 84.j odd 6 2
8624.2.a.bh 2 1.a even 1 1 trivial
8624.2.a.cc 2 7.b odd 2 1
9702.2.a.ch 2 12.b even 2 1
9702.2.a.cx 2 84.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8624))$$:

 $$T_{3}^{2} + 2T_{3} - 1$$ T3^2 + 2*T3 - 1 $$T_{5}^{2} - 4T_{5} + 2$$ T5^2 - 4*T5 + 2 $$T_{13}^{2} - 2T_{13} - 7$$ T13^2 - 2*T13 - 7 $$T_{17}^{2} - 4T_{17} - 28$$ T17^2 - 4*T17 - 28

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 2T - 1$$
$5$ $$T^{2} - 4T + 2$$
$7$ $$T^{2}$$
$11$ $$(T + 1)^{2}$$
$13$ $$T^{2} - 2T - 7$$
$17$ $$T^{2} - 4T - 28$$
$19$ $$T^{2} + 4T + 2$$
$23$ $$T^{2} - 4T - 14$$
$29$ $$T^{2} + 6T - 23$$
$31$ $$(T + 4)^{2}$$
$37$ $$T^{2} + 16T + 62$$
$41$ $$T^{2} - 8T + 14$$
$43$ $$T^{2} - 32$$
$47$ $$T^{2} + 4T - 68$$
$53$ $$T^{2} + 4T - 94$$
$59$ $$T^{2} + 14T + 47$$
$61$ $$T^{2} + 18T + 73$$
$67$ $$T^{2} + 14T + 31$$
$71$ $$T^{2} - 8T - 34$$
$73$ $$T^{2} - 16T + 62$$
$79$ $$T^{2} - 18T + 63$$
$83$ $$T^{2} - 4T - 196$$
$89$ $$T^{2} + 8T - 56$$
$97$ $$T^{2} - 2T - 7$$