# Properties

 Label 684.2.k.c Level $684$ Weight $2$ Character orbit 684.k Analytic conductor $5.462$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

# 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: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - q^{7}+O(q^{10})$$ q - q^7 $$q - q^{7} + 7 \zeta_{6} q^{13} + ( - 2 \zeta_{6} + 5) q^{19} + 5 \zeta_{6} q^{25} + 11 q^{31} - q^{37} + (5 \zeta_{6} - 5) q^{43} - 6 q^{49} + 13 \zeta_{6} q^{61} - 5 \zeta_{6} q^{67} + ( - 7 \zeta_{6} + 7) q^{73} + ( - 13 \zeta_{6} + 13) q^{79} - 7 \zeta_{6} q^{91} + (14 \zeta_{6} - 14) q^{97} +O(q^{100})$$ q - q^7 + 7*z * q^13 + (-2*z + 5) * q^19 + 5*z * q^25 + 11 * q^31 - q^37 + (5*z - 5) * q^43 - 6 * q^49 + 13*z * q^61 - 5*z * q^67 + (-7*z + 7) * q^73 + (-13*z + 13) * q^79 - 7*z * q^91 + (14*z - 14) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{7}+O(q^{10})$$ 2 * q - 2 * q^7 $$2 q - 2 q^{7} + 7 q^{13} + 8 q^{19} + 5 q^{25} + 22 q^{31} - 2 q^{37} - 5 q^{43} - 12 q^{49} + 13 q^{61} - 5 q^{67} + 7 q^{73} + 13 q^{79} - 7 q^{91} - 14 q^{97}+O(q^{100})$$ 2 * q - 2 * q^7 + 7 * q^13 + 8 * q^19 + 5 * q^25 + 22 * q^31 - 2 * q^37 - 5 * q^43 - 12 * q^49 + 13 * q^61 - 5 * q^67 + 7 * q^73 + 13 * q^79 - 7 * q^91 - 14 * 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 0 −1.00000 0 0 0
577.1 0 0 0 0 0 −1.00000 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
19.c even 3 1 inner
57.h odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 684.2.k.c 2
3.b odd 2 1 CM 684.2.k.c 2
4.b odd 2 1 2736.2.s.l 2
12.b even 2 1 2736.2.s.l 2
19.c even 3 1 inner 684.2.k.c 2
57.h odd 6 1 inner 684.2.k.c 2
76.g odd 6 1 2736.2.s.l 2
228.m even 6 1 2736.2.s.l 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
684.2.k.c 2 1.a even 1 1 trivial
684.2.k.c 2 3.b odd 2 1 CM
684.2.k.c 2 19.c even 3 1 inner
684.2.k.c 2 57.h odd 6 1 inner
2736.2.s.l 2 4.b odd 2 1
2736.2.s.l 2 12.b even 2 1
2736.2.s.l 2 76.g odd 6 1
2736.2.s.l 2 228.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 $$S_{2}^{\mathrm{new}}(684, [\chi])$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$(T + 1)^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2} - 7T + 49$$
$17$ $$T^{2}$$
$19$ $$T^{2} - 8T + 19$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$(T - 11)^{2}$$
$37$ $$(T + 1)^{2}$$
$41$ $$T^{2}$$
$43$ $$T^{2} + 5T + 25$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} - 13T + 169$$
$67$ $$T^{2} + 5T + 25$$
$71$ $$T^{2}$$
$73$ $$T^{2} - 7T + 49$$
$79$ $$T^{2} - 13T + 169$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2} + 14T + 196$$