# Properties

 Label 4680.2.a.w Level $4680$ Weight $2$ Character orbit 4680.a Self dual yes Analytic conductor $37.370$ Analytic rank $0$ 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] = [4680,2,Mod(1,4680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4680, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4680.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4680 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4680.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: $$37.3699881460$$ Analytic rank: $$0$$ 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: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 520) 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.

 $$f(q)$$ $$=$$ $$q - q^{5} - 2 q^{7}+O(q^{10})$$ q - q^5 - 2 * q^7 $$q - q^{5} - 2 q^{7} - 3 \beta q^{11} - q^{13} + (2 \beta + 2) q^{17} + ( - 3 \beta - 4) q^{19} + ( - 5 \beta + 2) q^{23} + q^{25} + (4 \beta + 4) q^{29} - \beta q^{31} + 2 q^{35} + (2 \beta - 4) q^{37} + (2 \beta - 2) q^{41} + (3 \beta - 6) q^{43} - 2 q^{47} - 3 q^{49} + ( - 2 \beta + 6) q^{53} + 3 \beta q^{55} + (\beta + 4) q^{59} + ( - 8 \beta + 4) q^{61} + q^{65} + (2 \beta - 2) q^{67} + 7 \beta q^{71} + (2 \beta + 4) q^{73} + 6 \beta q^{77} + (2 \beta + 4) q^{79} + 2 q^{83} + ( - 2 \beta - 2) q^{85} + 10 q^{89} + 2 q^{91} + (3 \beta + 4) q^{95} + ( - 4 \beta + 6) q^{97} +O(q^{100})$$ q - q^5 - 2 * q^7 - 3*b * q^11 - q^13 + (2*b + 2) * q^17 + (-3*b - 4) * q^19 + (-5*b + 2) * q^23 + q^25 + (4*b + 4) * q^29 - b * q^31 + 2 * q^35 + (2*b - 4) * q^37 + (2*b - 2) * q^41 + (3*b - 6) * q^43 - 2 * q^47 - 3 * q^49 + (-2*b + 6) * q^53 + 3*b * q^55 + (b + 4) * q^59 + (-8*b + 4) * q^61 + q^65 + (2*b - 2) * q^67 + 7*b * q^71 + (2*b + 4) * q^73 + 6*b * q^77 + (2*b + 4) * q^79 + 2 * q^83 + (-2*b - 2) * q^85 + 10 * q^89 + 2 * q^91 + (3*b + 4) * q^95 + (-4*b + 6) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{5} - 4 q^{7}+O(q^{10})$$ 2 * q - 2 * q^5 - 4 * q^7 $$2 q - 2 q^{5} - 4 q^{7} - 2 q^{13} + 4 q^{17} - 8 q^{19} + 4 q^{23} + 2 q^{25} + 8 q^{29} + 4 q^{35} - 8 q^{37} - 4 q^{41} - 12 q^{43} - 4 q^{47} - 6 q^{49} + 12 q^{53} + 8 q^{59} + 8 q^{61} + 2 q^{65} - 4 q^{67} + 8 q^{73} + 8 q^{79} + 4 q^{83} - 4 q^{85} + 20 q^{89} + 4 q^{91} + 8 q^{95} + 12 q^{97}+O(q^{100})$$ 2 * q - 2 * q^5 - 4 * q^7 - 2 * q^13 + 4 * q^17 - 8 * q^19 + 4 * q^23 + 2 * q^25 + 8 * q^29 + 4 * q^35 - 8 * q^37 - 4 * q^41 - 12 * q^43 - 4 * q^47 - 6 * q^49 + 12 * q^53 + 8 * q^59 + 8 * q^61 + 2 * q^65 - 4 * q^67 + 8 * q^73 + 8 * q^79 + 4 * q^83 - 4 * q^85 + 20 * q^89 + 4 * q^91 + 8 * q^95 + 12 * 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 0 0 −1.00000 0 −2.00000 0 0 0
1.2 0 0 0 −1.00000 0 −2.00000 0 0 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
$$2$$ $$1$$
$$3$$ $$-1$$
$$5$$ $$1$$
$$13$$ $$1$$

## 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 4680.2.a.w 2
3.b odd 2 1 520.2.a.c 2
4.b odd 2 1 9360.2.a.ck 2
12.b even 2 1 1040.2.a.n 2
15.d odd 2 1 2600.2.a.w 2
15.e even 4 2 2600.2.d.i 4
24.f even 2 1 4160.2.a.u 2
24.h odd 2 1 4160.2.a.bn 2
39.d odd 2 1 6760.2.a.n 2
60.h even 2 1 5200.2.a.bl 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
520.2.a.c 2 3.b odd 2 1
1040.2.a.n 2 12.b even 2 1
2600.2.a.w 2 15.d odd 2 1
2600.2.d.i 4 15.e even 4 2
4160.2.a.u 2 24.f even 2 1
4160.2.a.bn 2 24.h odd 2 1
4680.2.a.w 2 1.a even 1 1 trivial
5200.2.a.bl 2 60.h even 2 1
6760.2.a.n 2 39.d odd 2 1
9360.2.a.ck 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4680))$$:

 $$T_{7} + 2$$ T7 + 2 $$T_{11}^{2} - 18$$ T11^2 - 18 $$T_{17}^{2} - 4T_{17} - 4$$ T17^2 - 4*T17 - 4 $$T_{19}^{2} + 8T_{19} - 2$$ T19^2 + 8*T19 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$(T + 1)^{2}$$
$7$ $$(T + 2)^{2}$$
$11$ $$T^{2} - 18$$
$13$ $$(T + 1)^{2}$$
$17$ $$T^{2} - 4T - 4$$
$19$ $$T^{2} + 8T - 2$$
$23$ $$T^{2} - 4T - 46$$
$29$ $$T^{2} - 8T - 16$$
$31$ $$T^{2} - 2$$
$37$ $$T^{2} + 8T + 8$$
$41$ $$T^{2} + 4T - 4$$
$43$ $$T^{2} + 12T + 18$$
$47$ $$(T + 2)^{2}$$
$53$ $$T^{2} - 12T + 28$$
$59$ $$T^{2} - 8T + 14$$
$61$ $$T^{2} - 8T - 112$$
$67$ $$T^{2} + 4T - 4$$
$71$ $$T^{2} - 98$$
$73$ $$T^{2} - 8T + 8$$
$79$ $$T^{2} - 8T + 8$$
$83$ $$(T - 2)^{2}$$
$89$ $$(T - 10)^{2}$$
$97$ $$T^{2} - 12T + 4$$