# Properties

 Label 684.2.k.b Level $684$ Weight $2$ Character orbit 684.k Analytic conductor $5.462$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [684,2,Mod(505,684)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(684, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 4]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("684.505");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$684 = 2^{2} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 684.k (of order $$3$$, degree $$2$$, minimal)

## 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: no Analytic conductor: $$5.46176749826$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 76) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\zeta_{6} - 1) q^{5}+O(q^{10})$$ q + (z - 1) * q^5 $$q + (\zeta_{6} - 1) q^{5} + 4 q^{11} + \zeta_{6} q^{13} + ( - 3 \zeta_{6} + 3) q^{17} + (2 \zeta_{6} - 5) q^{19} + 5 \zeta_{6} q^{23} + 4 \zeta_{6} q^{25} + 7 \zeta_{6} q^{29} + 4 q^{31} + 10 q^{37} + (5 \zeta_{6} - 5) q^{41} + ( - 5 \zeta_{6} + 5) q^{43} - 7 \zeta_{6} q^{47} - 7 q^{49} + 11 \zeta_{6} q^{53} + (4 \zeta_{6} - 4) q^{55} + ( - 3 \zeta_{6} + 3) q^{59} - 11 \zeta_{6} q^{61} - q^{65} + 3 \zeta_{6} q^{67} + ( - 11 \zeta_{6} + 11) q^{71} + (15 \zeta_{6} - 15) q^{73} + ( - 13 \zeta_{6} + 13) q^{79} + 3 \zeta_{6} q^{85} + 3 \zeta_{6} q^{89} + ( - 5 \zeta_{6} + 3) q^{95} + ( - 5 \zeta_{6} + 5) q^{97} +O(q^{100})$$ q + (z - 1) * q^5 + 4 * q^11 + z * q^13 + (-3*z + 3) * q^17 + (2*z - 5) * q^19 + 5*z * q^23 + 4*z * q^25 + 7*z * q^29 + 4 * q^31 + 10 * q^37 + (5*z - 5) * q^41 + (-5*z + 5) * q^43 - 7*z * q^47 - 7 * q^49 + 11*z * q^53 + (4*z - 4) * q^55 + (-3*z + 3) * q^59 - 11*z * q^61 - q^65 + 3*z * q^67 + (-11*z + 11) * q^71 + (15*z - 15) * q^73 + (-13*z + 13) * q^79 + 3*z * q^85 + 3*z * q^89 + (-5*z + 3) * q^95 + (-5*z + 5) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{5}+O(q^{10})$$ 2 * q - q^5 $$2 q - q^{5} + 8 q^{11} + q^{13} + 3 q^{17} - 8 q^{19} + 5 q^{23} + 4 q^{25} + 7 q^{29} + 8 q^{31} + 20 q^{37} - 5 q^{41} + 5 q^{43} - 7 q^{47} - 14 q^{49} + 11 q^{53} - 4 q^{55} + 3 q^{59} - 11 q^{61} - 2 q^{65} + 3 q^{67} + 11 q^{71} - 15 q^{73} + 13 q^{79} + 3 q^{85} + 3 q^{89} + q^{95} + 5 q^{97}+O(q^{100})$$ 2 * q - q^5 + 8 * q^11 + q^13 + 3 * q^17 - 8 * q^19 + 5 * q^23 + 4 * q^25 + 7 * q^29 + 8 * q^31 + 20 * q^37 - 5 * q^41 + 5 * q^43 - 7 * q^47 - 14 * q^49 + 11 * q^53 - 4 * q^55 + 3 * q^59 - 11 * q^61 - 2 * q^65 + 3 * q^67 + 11 * q^71 - 15 * q^73 + 13 * q^79 + 3 * q^85 + 3 * q^89 + q^95 + 5 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/684\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$343$$ $$533$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
505.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 0 −0.500000 + 0.866025i 0 0 0 0 0
577.1 0 0 0 −0.500000 0.866025i 0 0 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 684.2.k.b 2
3.b odd 2 1 76.2.e.a 2
4.b odd 2 1 2736.2.s.g 2
12.b even 2 1 304.2.i.a 2
15.d odd 2 1 1900.2.i.a 2
15.e even 4 2 1900.2.s.a 4
19.c even 3 1 inner 684.2.k.b 2
24.f even 2 1 1216.2.i.g 2
24.h odd 2 1 1216.2.i.c 2
57.d even 2 1 1444.2.e.b 2
57.f even 6 1 1444.2.a.c 1
57.f even 6 1 1444.2.e.b 2
57.h odd 6 1 76.2.e.a 2
57.h odd 6 1 1444.2.a.b 1
76.g odd 6 1 2736.2.s.g 2
228.m even 6 1 304.2.i.a 2
228.m even 6 1 5776.2.a.k 1
228.n odd 6 1 5776.2.a.f 1
285.n odd 6 1 1900.2.i.a 2
285.v even 12 2 1900.2.s.a 4
456.u even 6 1 1216.2.i.g 2
456.x odd 6 1 1216.2.i.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.2.e.a 2 3.b odd 2 1
76.2.e.a 2 57.h odd 6 1
304.2.i.a 2 12.b even 2 1
304.2.i.a 2 228.m even 6 1
684.2.k.b 2 1.a even 1 1 trivial
684.2.k.b 2 19.c even 3 1 inner
1216.2.i.c 2 24.h odd 2 1
1216.2.i.c 2 456.x odd 6 1
1216.2.i.g 2 24.f even 2 1
1216.2.i.g 2 456.u even 6 1
1444.2.a.b 1 57.h odd 6 1
1444.2.a.c 1 57.f even 6 1
1444.2.e.b 2 57.d even 2 1
1444.2.e.b 2 57.f even 6 1
1900.2.i.a 2 15.d odd 2 1
1900.2.i.a 2 285.n odd 6 1
1900.2.s.a 4 15.e even 4 2
1900.2.s.a 4 285.v even 12 2
2736.2.s.g 2 4.b odd 2 1
2736.2.s.g 2 76.g odd 6 1
5776.2.a.f 1 228.n odd 6 1
5776.2.a.k 1 228.m even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(684, [\chi])$$:

 $$T_{5}^{2} + T_{5} + 1$$ T5^2 + T5 + 1 $$T_{7}$$ T7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + T + 1$$
$7$ $$T^{2}$$
$11$ $$(T - 4)^{2}$$
$13$ $$T^{2} - T + 1$$
$17$ $$T^{2} - 3T + 9$$
$19$ $$T^{2} + 8T + 19$$
$23$ $$T^{2} - 5T + 25$$
$29$ $$T^{2} - 7T + 49$$
$31$ $$(T - 4)^{2}$$
$37$ $$(T - 10)^{2}$$
$41$ $$T^{2} + 5T + 25$$
$43$ $$T^{2} - 5T + 25$$
$47$ $$T^{2} + 7T + 49$$
$53$ $$T^{2} - 11T + 121$$
$59$ $$T^{2} - 3T + 9$$
$61$ $$T^{2} + 11T + 121$$
$67$ $$T^{2} - 3T + 9$$
$71$ $$T^{2} - 11T + 121$$
$73$ $$T^{2} + 15T + 225$$
$79$ $$T^{2} - 13T + 169$$
$83$ $$T^{2}$$
$89$ $$T^{2} - 3T + 9$$
$97$ $$T^{2} - 5T + 25$$