# Properties

 Label 684.2.k.g Level $684$ Weight $2$ Character orbit 684.k Analytic conductor $5.462$ Analytic rank $0$ Dimension $4$ CM no 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{7})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 7x^{2} + 49$$ x^4 + 7*x^2 + 49 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 228) 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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} + \beta_1 + 1) q^{5} + ( - \beta_{3} + 2) q^{7}+O(q^{10})$$ q + (b2 + b1 + 1) * q^5 + (-b3 + 2) * q^7 $$q + (\beta_{2} + \beta_1 + 1) q^{5} + ( - \beta_{3} + 2) q^{7} + ( - \beta_{3} - 3) q^{11} + ( - 2 \beta_{3} + \beta_{2} - 2 \beta_1) q^{13} + (2 \beta_{2} - 2 \beta_1 + 2) q^{17} + (\beta_{3} + 4 \beta_{2} + 2) q^{19} + (\beta_{3} + 3 \beta_{2} + \beta_1) q^{23} + (2 \beta_{3} + 3 \beta_{2} + 2 \beta_1) q^{25} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{29} + (\beta_{3} + 2) q^{31} + (9 \beta_{2} + 3 \beta_1 + 9) q^{35} - 5 q^{37} + (4 \beta_{2} + 4) q^{41} + ( - 8 \beta_{2} + \beta_1 - 8) q^{43} + ( - 2 \beta_{3} - 2 \beta_1) q^{47} + ( - 4 \beta_{3} + 4) q^{49} + ( - 3 \beta_{3} + \beta_{2} - 3 \beta_1) q^{53} + (4 \beta_{2} - 2 \beta_1 + 4) q^{55} + (5 \beta_{2} + \beta_1 + 5) q^{59} + 5 \beta_{2} q^{61} + ( - \beta_{3} + 13) q^{65} + (3 \beta_{3} + 6 \beta_{2} + 3 \beta_1) q^{67} + ( - 10 \beta_{2} - 10) q^{71} + (\beta_{2} + 2 \beta_1 + 1) q^{73} + (\beta_{3} + 1) q^{77} + ( - 6 \beta_{2} + \beta_1 - 6) q^{79} + (2 \beta_{3} - 10) q^{83} - 12 \beta_{2} q^{85} + ( - \beta_{3} - 11 \beta_{2} - \beta_1) q^{89} + ( - 3 \beta_{3} - 12 \beta_{2} - 3 \beta_1) q^{91} + (4 \beta_{3} - 5 \beta_{2} + \beta_1 - 9) q^{95} - 4 \beta_1 q^{97}+O(q^{100})$$ q + (b2 + b1 + 1) * q^5 + (-b3 + 2) * q^7 + (-b3 - 3) * q^11 + (-2*b3 + b2 - 2*b1) * q^13 + (2*b2 - 2*b1 + 2) * q^17 + (b3 + 4*b2 + 2) * q^19 + (b3 + 3*b2 + b1) * q^23 + (2*b3 + 3*b2 + 2*b1) * q^25 + (-2*b3 - 2*b2 - 2*b1) * q^29 + (b3 + 2) * q^31 + (9*b2 + 3*b1 + 9) * q^35 - 5 * q^37 + (4*b2 + 4) * q^41 + (-8*b2 + b1 - 8) * q^43 + (-2*b3 - 2*b1) * q^47 + (-4*b3 + 4) * q^49 + (-3*b3 + b2 - 3*b1) * q^53 + (4*b2 - 2*b1 + 4) * q^55 + (5*b2 + b1 + 5) * q^59 + 5*b2 * q^61 + (-b3 + 13) * q^65 + (3*b3 + 6*b2 + 3*b1) * q^67 + (-10*b2 - 10) * q^71 + (b2 + 2*b1 + 1) * q^73 + (b3 + 1) * q^77 + (-6*b2 + b1 - 6) * q^79 + (2*b3 - 10) * q^83 - 12*b2 * q^85 + (-b3 - 11*b2 - b1) * q^89 + (-3*b3 - 12*b2 - 3*b1) * q^91 + (4*b3 - 5*b2 + b1 - 9) * q^95 - 4*b1 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{5} + 8 q^{7}+O(q^{10})$$ 4 * q + 2 * q^5 + 8 * q^7 $$4 q + 2 q^{5} + 8 q^{7} - 12 q^{11} - 2 q^{13} + 4 q^{17} - 6 q^{23} - 6 q^{25} + 4 q^{29} + 8 q^{31} + 18 q^{35} - 20 q^{37} + 8 q^{41} - 16 q^{43} + 16 q^{49} - 2 q^{53} + 8 q^{55} + 10 q^{59} - 10 q^{61} + 52 q^{65} - 12 q^{67} - 20 q^{71} + 2 q^{73} + 4 q^{77} - 12 q^{79} - 40 q^{83} + 24 q^{85} + 22 q^{89} + 24 q^{91} - 26 q^{95}+O(q^{100})$$ 4 * q + 2 * q^5 + 8 * q^7 - 12 * q^11 - 2 * q^13 + 4 * q^17 - 6 * q^23 - 6 * q^25 + 4 * q^29 + 8 * q^31 + 18 * q^35 - 20 * q^37 + 8 * q^41 - 16 * q^43 + 16 * q^49 - 2 * q^53 + 8 * q^55 + 10 * q^59 - 10 * q^61 + 52 * q^65 - 12 * q^67 - 20 * q^71 + 2 * q^73 + 4 * q^77 - 12 * q^79 - 40 * q^83 + 24 * q^85 + 22 * q^89 + 24 * q^91 - 26 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 7x^{2} + 49$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 7$$ (v^2) / 7 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 7$$ (v^3) / 7
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$7\beta_{2}$$ 7*b2 $$\nu^{3}$$ $$=$$ $$7\beta_{3}$$ 7*b3

## Character values

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

 $$n$$ $$325$$ $$343$$ $$533$$ $$\chi(n)$$ $$\beta_{2}$$ $$1$$ $$1$$

## 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
 −1.32288 + 2.29129i 1.32288 − 2.29129i −1.32288 − 2.29129i 1.32288 + 2.29129i
0 0 0 −0.822876 + 1.42526i 0 −0.645751 0 0 0
505.2 0 0 0 1.82288 3.15731i 0 4.64575 0 0 0
577.1 0 0 0 −0.822876 1.42526i 0 −0.645751 0 0 0
577.2 0 0 0 1.82288 + 3.15731i 0 4.64575 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.g 4
3.b odd 2 1 228.2.i.b 4
4.b odd 2 1 2736.2.s.u 4
12.b even 2 1 912.2.q.g 4
19.c even 3 1 inner 684.2.k.g 4
57.f even 6 1 4332.2.a.m 2
57.h odd 6 1 228.2.i.b 4
57.h odd 6 1 4332.2.a.h 2
76.g odd 6 1 2736.2.s.u 4
228.m even 6 1 912.2.q.g 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
228.2.i.b 4 3.b odd 2 1
228.2.i.b 4 57.h odd 6 1
684.2.k.g 4 1.a even 1 1 trivial
684.2.k.g 4 19.c even 3 1 inner
912.2.q.g 4 12.b even 2 1
912.2.q.g 4 228.m even 6 1
2736.2.s.u 4 4.b odd 2 1
2736.2.s.u 4 76.g odd 6 1
4332.2.a.h 2 57.h odd 6 1
4332.2.a.m 2 57.f even 6 1

## Hecke kernels

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

 $$T_{5}^{4} - 2T_{5}^{3} + 10T_{5}^{2} + 12T_{5} + 36$$ T5^4 - 2*T5^3 + 10*T5^2 + 12*T5 + 36 $$T_{7}^{2} - 4T_{7} - 3$$ T7^2 - 4*T7 - 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4} - 2 T^{3} + 10 T^{2} + 12 T + 36$$
$7$ $$(T^{2} - 4 T - 3)^{2}$$
$11$ $$(T^{2} + 6 T + 2)^{2}$$
$13$ $$T^{4} + 2 T^{3} + 31 T^{2} - 54 T + 729$$
$17$ $$T^{4} - 4 T^{3} + 40 T^{2} + 96 T + 576$$
$19$ $$T^{4} + 10T^{2} + 361$$
$23$ $$T^{4} + 6 T^{3} + 34 T^{2} + 12 T + 4$$
$29$ $$T^{4} - 4 T^{3} + 40 T^{2} + 96 T + 576$$
$31$ $$(T^{2} - 4 T - 3)^{2}$$
$37$ $$(T + 5)^{4}$$
$41$ $$(T^{2} - 4 T + 16)^{2}$$
$43$ $$T^{4} + 16 T^{3} + 199 T^{2} + \cdots + 3249$$
$47$ $$T^{4} + 28T^{2} + 784$$
$53$ $$T^{4} + 2 T^{3} + 66 T^{2} + \cdots + 3844$$
$59$ $$T^{4} - 10 T^{3} + 82 T^{2} + \cdots + 324$$
$61$ $$(T^{2} + 5 T + 25)^{2}$$
$67$ $$T^{4} + 12 T^{3} + 171 T^{2} + \cdots + 729$$
$71$ $$(T^{2} + 10 T + 100)^{2}$$
$73$ $$T^{4} - 2 T^{3} + 31 T^{2} + 54 T + 729$$
$79$ $$T^{4} + 12 T^{3} + 115 T^{2} + \cdots + 841$$
$83$ $$(T^{2} + 20 T + 72)^{2}$$
$89$ $$T^{4} - 22 T^{3} + 370 T^{2} + \cdots + 12996$$
$97$ $$T^{4} + 112 T^{2} + 12544$$