# Properties

 Label 684.2.k.f 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: $$\Q(\sqrt{-2}, \sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{2} + 4$$ x^4 - 2*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$3$$ 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.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{5} + (\beta_{3} - 1) q^{7}+O(q^{10})$$ q - b2 * q^5 + (b3 - 1) * q^7 $$q - \beta_{2} q^{5} + (\beta_{3} - 1) q^{7} - \beta_{3} q^{11} + ( - \beta_1 + 1) q^{13} - 2 \beta_{2} q^{17} + (\beta_{3} - 2 \beta_{2} + 1) q^{19} + (\beta_{3} - \beta_{2} + 6 \beta_1 - 6) q^{23} + (\beta_1 - 1) q^{25} + (2 \beta_{3} - 2 \beta_{2}) q^{29} + (\beta_{3} - 7) q^{31} + (\beta_{2} - 6 \beta_1) q^{35} + ( - 4 \beta_{3} - 1) q^{37} + ( - \beta_{2} + \beta_1) q^{43} + (2 \beta_{3} - 2 \beta_{2} - 6 \beta_1 + 6) q^{47} - 2 \beta_{3} q^{49} + (\beta_{3} - \beta_{2}) q^{53} + 6 \beta_1 q^{55} + 3 \beta_{2} q^{59} + (2 \beta_{3} - 2 \beta_{2} - 7 \beta_1 + 7) q^{61} - \beta_{3} q^{65} + (\beta_{3} - \beta_{2} - 7 \beta_1 + 7) q^{67} + 6 \beta_1 q^{71} + ( - 2 \beta_{2} + 7 \beta_1) q^{73} + (\beta_{3} - 6) q^{77} + (3 \beta_{2} - 5 \beta_1) q^{79} + ( - 2 \beta_{3} + 12) q^{83} + (12 \beta_1 - 12) q^{85} + ( - \beta_{3} + \beta_{2} + 6 \beta_1 - 6) q^{89} + (\beta_{3} - \beta_{2} + \beta_1 - 1) q^{91} + ( - \beta_{2} + 6 \beta_1 - 12) q^{95} + (6 \beta_{2} + 4 \beta_1) q^{97}+O(q^{100})$$ q - b2 * q^5 + (b3 - 1) * q^7 - b3 * q^11 + (-b1 + 1) * q^13 - 2*b2 * q^17 + (b3 - 2*b2 + 1) * q^19 + (b3 - b2 + 6*b1 - 6) * q^23 + (b1 - 1) * q^25 + (2*b3 - 2*b2) * q^29 + (b3 - 7) * q^31 + (b2 - 6*b1) * q^35 + (-4*b3 - 1) * q^37 + (-b2 + b1) * q^43 + (2*b3 - 2*b2 - 6*b1 + 6) * q^47 - 2*b3 * q^49 + (b3 - b2) * q^53 + 6*b1 * q^55 + 3*b2 * q^59 + (2*b3 - 2*b2 - 7*b1 + 7) * q^61 - b3 * q^65 + (b3 - b2 - 7*b1 + 7) * q^67 + 6*b1 * q^71 + (-2*b2 + 7*b1) * q^73 + (b3 - 6) * q^77 + (3*b2 - 5*b1) * q^79 + (-2*b3 + 12) * q^83 + (12*b1 - 12) * q^85 + (-b3 + b2 + 6*b1 - 6) * q^89 + (b3 - b2 + b1 - 1) * q^91 + (-b2 + 6*b1 - 12) * q^95 + (6*b2 + 4*b1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{7}+O(q^{10})$$ 4 * q - 4 * q^7 $$4 q - 4 q^{7} + 2 q^{13} + 4 q^{19} - 12 q^{23} - 2 q^{25} - 28 q^{31} - 12 q^{35} - 4 q^{37} + 2 q^{43} + 12 q^{47} + 12 q^{55} + 14 q^{61} + 14 q^{67} + 12 q^{71} + 14 q^{73} - 24 q^{77} - 10 q^{79} + 48 q^{83} - 24 q^{85} - 12 q^{89} - 2 q^{91} - 36 q^{95} + 8 q^{97}+O(q^{100})$$ 4 * q - 4 * q^7 + 2 * q^13 + 4 * q^19 - 12 * q^23 - 2 * q^25 - 28 * q^31 - 12 * q^35 - 4 * q^37 + 2 * q^43 + 12 * q^47 + 12 * q^55 + 14 * q^61 + 14 * q^67 + 12 * q^71 + 14 * q^73 - 24 * q^77 - 10 * q^79 + 48 * q^83 - 24 * q^85 - 12 * q^89 - 2 * q^91 - 36 * q^95 + 8 * q^97

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

 $$\beta_{1}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 2\nu ) / 2$$ (v^3 + 2*v) / 2 $$\beta_{3}$$ $$=$$ $$( -\nu^{3} + 4\nu ) / 2$$ (-v^3 + 4*v) / 2
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 3$$ (b3 + b2) / 3 $$\nu^{2}$$ $$=$$ $$2\beta_1$$ 2*b1 $$\nu^{3}$$ $$=$$ $$( -2\beta_{3} + 4\beta_{2} ) / 3$$ (-2*b3 + 4*b2) / 3

## Character values

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

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

## 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
 1.22474 − 0.707107i −1.22474 + 0.707107i 1.22474 + 0.707107i −1.22474 − 0.707107i
0 0 0 −1.22474 + 2.12132i 0 1.44949 0 0 0
505.2 0 0 0 1.22474 2.12132i 0 −3.44949 0 0 0
577.1 0 0 0 −1.22474 2.12132i 0 1.44949 0 0 0
577.2 0 0 0 1.22474 + 2.12132i 0 −3.44949 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.f 4
3.b odd 2 1 228.2.i.a 4
4.b odd 2 1 2736.2.s.t 4
12.b even 2 1 912.2.q.i 4
19.c even 3 1 inner 684.2.k.f 4
57.f even 6 1 4332.2.a.g 2
57.h odd 6 1 228.2.i.a 4
57.h odd 6 1 4332.2.a.l 2
76.g odd 6 1 2736.2.s.t 4
228.m even 6 1 912.2.q.i 4

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

## Hecke kernels

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

 $$T_{5}^{4} + 6T_{5}^{2} + 36$$ T5^4 + 6*T5^2 + 36 $$T_{7}^{2} + 2T_{7} - 5$$ T7^2 + 2*T7 - 5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 6T^{2} + 36$$
$7$ $$(T^{2} + 2 T - 5)^{2}$$
$11$ $$(T^{2} - 6)^{2}$$
$13$ $$(T^{2} - T + 1)^{2}$$
$17$ $$T^{4} + 24T^{2} + 576$$
$19$ $$(T^{2} - 2 T + 19)^{2}$$
$23$ $$T^{4} + 12 T^{3} + 114 T^{2} + \cdots + 900$$
$29$ $$T^{4} + 24T^{2} + 576$$
$31$ $$(T^{2} + 14 T + 43)^{2}$$
$37$ $$(T^{2} + 2 T - 95)^{2}$$
$41$ $$T^{4}$$
$43$ $$T^{4} - 2 T^{3} + 9 T^{2} + 10 T + 25$$
$47$ $$T^{4} - 12 T^{3} + 132 T^{2} + \cdots + 144$$
$53$ $$T^{4} + 6T^{2} + 36$$
$59$ $$T^{4} + 54T^{2} + 2916$$
$61$ $$T^{4} - 14 T^{3} + 171 T^{2} + \cdots + 625$$
$67$ $$T^{4} - 14 T^{3} + 153 T^{2} + \cdots + 1849$$
$71$ $$(T^{2} - 6 T + 36)^{2}$$
$73$ $$T^{4} - 14 T^{3} + 171 T^{2} + \cdots + 625$$
$79$ $$T^{4} + 10 T^{3} + 129 T^{2} + \cdots + 841$$
$83$ $$(T^{2} - 24 T + 120)^{2}$$
$89$ $$T^{4} + 12 T^{3} + 114 T^{2} + \cdots + 900$$
$97$ $$T^{4} - 8 T^{3} + 264 T^{2} + \cdots + 40000$$