# Properties

 Label 684.2.k.a Level $684$ Weight $2$ Character orbit 684.k Analytic conductor $5.462$ Analytic rank $1$ 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: $$1$$ 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: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + (4 \zeta_{6} - 4) q^{5} - 3 q^{7}+O(q^{10})$$ q + (4*z - 4) * q^5 - 3 * q^7 $$q + (4 \zeta_{6} - 4) q^{5} - 3 q^{7} + 4 q^{11} - 5 \zeta_{6} q^{13} + (2 \zeta_{6} - 5) q^{19} - 4 \zeta_{6} q^{23} - 11 \zeta_{6} q^{25} - 8 \zeta_{6} q^{29} + q^{31} + ( - 12 \zeta_{6} + 12) q^{35} - 5 q^{37} + (8 \zeta_{6} - 8) q^{41} + ( - 5 \zeta_{6} + 5) q^{43} + 8 \zeta_{6} q^{47} + 2 q^{49} - 4 \zeta_{6} q^{53} + (16 \zeta_{6} - 16) q^{55} + (12 \zeta_{6} - 12) q^{59} + \zeta_{6} q^{61} + 20 q^{65} - 3 \zeta_{6} q^{67} + (16 \zeta_{6} - 16) q^{71} + ( - 15 \zeta_{6} + 15) q^{73} - 12 q^{77} + ( - 7 \zeta_{6} + 7) q^{79} + 12 \zeta_{6} q^{89} + 15 \zeta_{6} q^{91} + ( - 20 \zeta_{6} + 12) q^{95} + ( - 2 \zeta_{6} + 2) q^{97} +O(q^{100})$$ q + (4*z - 4) * q^5 - 3 * q^7 + 4 * q^11 - 5*z * q^13 + (2*z - 5) * q^19 - 4*z * q^23 - 11*z * q^25 - 8*z * q^29 + q^31 + (-12*z + 12) * q^35 - 5 * q^37 + (8*z - 8) * q^41 + (-5*z + 5) * q^43 + 8*z * q^47 + 2 * q^49 - 4*z * q^53 + (16*z - 16) * q^55 + (12*z - 12) * q^59 + z * q^61 + 20 * q^65 - 3*z * q^67 + (16*z - 16) * q^71 + (-15*z + 15) * q^73 - 12 * q^77 + (-7*z + 7) * q^79 + 12*z * q^89 + 15*z * q^91 + (-20*z + 12) * q^95 + (-2*z + 2) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{5} - 6 q^{7}+O(q^{10})$$ 2 * q - 4 * q^5 - 6 * q^7 $$2 q - 4 q^{5} - 6 q^{7} + 8 q^{11} - 5 q^{13} - 8 q^{19} - 4 q^{23} - 11 q^{25} - 8 q^{29} + 2 q^{31} + 12 q^{35} - 10 q^{37} - 8 q^{41} + 5 q^{43} + 8 q^{47} + 4 q^{49} - 4 q^{53} - 16 q^{55} - 12 q^{59} + q^{61} + 40 q^{65} - 3 q^{67} - 16 q^{71} + 15 q^{73} - 24 q^{77} + 7 q^{79} + 12 q^{89} + 15 q^{91} + 4 q^{95} + 2 q^{97}+O(q^{100})$$ 2 * q - 4 * q^5 - 6 * q^7 + 8 * q^11 - 5 * q^13 - 8 * q^19 - 4 * q^23 - 11 * q^25 - 8 * q^29 + 2 * q^31 + 12 * q^35 - 10 * q^37 - 8 * q^41 + 5 * q^43 + 8 * q^47 + 4 * q^49 - 4 * q^53 - 16 * q^55 - 12 * q^59 + q^61 + 40 * q^65 - 3 * q^67 - 16 * q^71 + 15 * q^73 - 24 * q^77 + 7 * q^79 + 12 * q^89 + 15 * q^91 + 4 * q^95 + 2 * 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 −2.00000 + 3.46410i 0 −3.00000 0 0 0
577.1 0 0 0 −2.00000 3.46410i 0 −3.00000 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
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.a 2
3.b odd 2 1 684.2.k.e yes 2
4.b odd 2 1 2736.2.s.b 2
12.b even 2 1 2736.2.s.r 2
19.c even 3 1 inner 684.2.k.a 2
57.h odd 6 1 684.2.k.e yes 2
76.g odd 6 1 2736.2.s.b 2
228.m even 6 1 2736.2.s.r 2

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

 $$T_{5}^{2} + 4T_{5} + 16$$ T5^2 + 4*T5 + 16 $$T_{7} + 3$$ T7 + 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 4T + 16$$
$7$ $$(T + 3)^{2}$$
$11$ $$(T - 4)^{2}$$
$13$ $$T^{2} + 5T + 25$$
$17$ $$T^{2}$$
$19$ $$T^{2} + 8T + 19$$
$23$ $$T^{2} + 4T + 16$$
$29$ $$T^{2} + 8T + 64$$
$31$ $$(T - 1)^{2}$$
$37$ $$(T + 5)^{2}$$
$41$ $$T^{2} + 8T + 64$$
$43$ $$T^{2} - 5T + 25$$
$47$ $$T^{2} - 8T + 64$$
$53$ $$T^{2} + 4T + 16$$
$59$ $$T^{2} + 12T + 144$$
$61$ $$T^{2} - T + 1$$
$67$ $$T^{2} + 3T + 9$$
$71$ $$T^{2} + 16T + 256$$
$73$ $$T^{2} - 15T + 225$$
$79$ $$T^{2} - 7T + 49$$
$83$ $$T^{2}$$
$89$ $$T^{2} - 12T + 144$$
$97$ $$T^{2} - 2T + 4$$