# Properties

 Label 3024.2.k.e Level $3024$ Weight $2$ Character orbit 3024.k Analytic conductor $24.147$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3024,2,Mod(1889,3024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3024, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))

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

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

chi := DirichletCharacter("3024.1889");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3024 = 2^{4} \cdot 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3024.k (of order $$2$$, degree $$1$$, not 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: $$24.1467615712$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}\cdot 3$$ Twist minimal: no (minimal twist has level 378) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{5} + (\beta_1 - 2) q^{7}+O(q^{10})$$ q - b2 * q^5 + (b1 - 2) * q^7 $$q - \beta_{2} q^{5} + (\beta_1 - 2) q^{7} + \beta_{3} q^{11} - 2 \beta_1 q^{13} - 4 \beta_{2} q^{17} + \beta_1 q^{19} + \beta_{3} q^{23} - 2 q^{25} - 2 \beta_{3} q^{29} - 3 \beta_1 q^{31} + ( - \beta_{3} + 2 \beta_{2}) q^{35} + 7 q^{37} + 7 \beta_{2} q^{41} + 2 q^{43} - 2 \beta_{2} q^{47} + ( - 4 \beta_1 + 1) q^{49} + 4 \beta_{3} q^{53} - 3 \beta_1 q^{55} + 2 \beta_{2} q^{59} - 4 \beta_1 q^{61} + 2 \beta_{3} q^{65} - 2 q^{67} - \beta_{3} q^{71} + 2 \beta_1 q^{73} + ( - 2 \beta_{3} - 3 \beta_{2}) q^{77} + 10 q^{79} - 10 \beta_{2} q^{83} + 12 q^{85} + 3 \beta_{2} q^{89} + (4 \beta_1 + 6) q^{91} - \beta_{3} q^{95} - 8 \beta_1 q^{97}+O(q^{100})$$ q - b2 * q^5 + (b1 - 2) * q^7 + b3 * q^11 - 2*b1 * q^13 - 4*b2 * q^17 + b1 * q^19 + b3 * q^23 - 2 * q^25 - 2*b3 * q^29 - 3*b1 * q^31 + (-b3 + 2*b2) * q^35 + 7 * q^37 + 7*b2 * q^41 + 2 * q^43 - 2*b2 * q^47 + (-4*b1 + 1) * q^49 + 4*b3 * q^53 - 3*b1 * q^55 + 2*b2 * q^59 - 4*b1 * q^61 + 2*b3 * q^65 - 2 * q^67 - b3 * q^71 + 2*b1 * q^73 + (-2*b3 - 3*b2) * q^77 + 10 * q^79 - 10*b2 * q^83 + 12 * q^85 + 3*b2 * q^89 + (4*b1 + 6) * q^91 - b3 * q^95 - 8*b1 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 8 q^{7}+O(q^{10})$$ 4 * q - 8 * q^7 $$4 q - 8 q^{7} - 8 q^{25} + 28 q^{37} + 8 q^{43} + 4 q^{49} - 8 q^{67} + 40 q^{79} + 48 q^{85} + 24 q^{91}+O(q^{100})$$ 4 * q - 8 * q^7 - 8 * q^25 + 28 * q^37 + 8 * q^43 + 4 * q^49 - 8 * q^67 + 40 * q^79 + 48 * q^85 + 24 * q^91

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$2\zeta_{12}^{2} - 1$$ 2*v^2 - 1 $$\beta_{2}$$ $$=$$ $$-\zeta_{12}^{3} + 2\zeta_{12}$$ -v^3 + 2*v $$\beta_{3}$$ $$=$$ $$3\zeta_{12}^{3}$$ 3*v^3
 $$\zeta_{12}$$ $$=$$ $$( \beta_{3} + 3\beta_{2} ) / 6$$ (b3 + 3*b2) / 6 $$\zeta_{12}^{2}$$ $$=$$ $$( \beta _1 + 1 ) / 2$$ (b1 + 1) / 2 $$\zeta_{12}^{3}$$ $$=$$ $$( \beta_{3} ) / 3$$ (b3) / 3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3024\mathbb{Z}\right)^\times$$.

 $$n$$ $$757$$ $$785$$ $$1135$$ $$2593$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1889.1
 0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 − 0.500000i
0 0 0 −1.73205 0 −2.00000 1.73205i 0 0 0
1889.2 0 0 0 −1.73205 0 −2.00000 + 1.73205i 0 0 0
1889.3 0 0 0 1.73205 0 −2.00000 1.73205i 0 0 0
1889.4 0 0 0 1.73205 0 −2.00000 + 1.73205i 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 inner
7.b odd 2 1 inner
21.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3024.2.k.e 4
3.b odd 2 1 inner 3024.2.k.e 4
4.b odd 2 1 378.2.d.b 4
7.b odd 2 1 inner 3024.2.k.e 4
12.b even 2 1 378.2.d.b 4
21.c even 2 1 inner 3024.2.k.e 4
28.d even 2 1 378.2.d.b 4
36.f odd 6 1 1134.2.m.b 4
36.f odd 6 1 1134.2.m.c 4
36.h even 6 1 1134.2.m.b 4
36.h even 6 1 1134.2.m.c 4
84.h odd 2 1 378.2.d.b 4
252.s odd 6 1 1134.2.m.b 4
252.s odd 6 1 1134.2.m.c 4
252.bi even 6 1 1134.2.m.b 4
252.bi even 6 1 1134.2.m.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.d.b 4 4.b odd 2 1
378.2.d.b 4 12.b even 2 1
378.2.d.b 4 28.d even 2 1
378.2.d.b 4 84.h odd 2 1
1134.2.m.b 4 36.f odd 6 1
1134.2.m.b 4 36.h even 6 1
1134.2.m.b 4 252.s odd 6 1
1134.2.m.b 4 252.bi even 6 1
1134.2.m.c 4 36.f odd 6 1
1134.2.m.c 4 36.h even 6 1
1134.2.m.c 4 252.s odd 6 1
1134.2.m.c 4 252.bi even 6 1
3024.2.k.e 4 1.a even 1 1 trivial
3024.2.k.e 4 3.b odd 2 1 inner
3024.2.k.e 4 7.b odd 2 1 inner
3024.2.k.e 4 21.c even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(3024, [\chi])$$:

 $$T_{5}^{2} - 3$$ T5^2 - 3 $$T_{11}^{2} + 9$$ T11^2 + 9 $$T_{13}^{2} + 12$$ T13^2 + 12

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$(T^{2} - 3)^{2}$$
$7$ $$(T^{2} + 4 T + 7)^{2}$$
$11$ $$(T^{2} + 9)^{2}$$
$13$ $$(T^{2} + 12)^{2}$$
$17$ $$(T^{2} - 48)^{2}$$
$19$ $$(T^{2} + 3)^{2}$$
$23$ $$(T^{2} + 9)^{2}$$
$29$ $$(T^{2} + 36)^{2}$$
$31$ $$(T^{2} + 27)^{2}$$
$37$ $$(T - 7)^{4}$$
$41$ $$(T^{2} - 147)^{2}$$
$43$ $$(T - 2)^{4}$$
$47$ $$(T^{2} - 12)^{2}$$
$53$ $$(T^{2} + 144)^{2}$$
$59$ $$(T^{2} - 12)^{2}$$
$61$ $$(T^{2} + 48)^{2}$$
$67$ $$(T + 2)^{4}$$
$71$ $$(T^{2} + 9)^{2}$$
$73$ $$(T^{2} + 12)^{2}$$
$79$ $$(T - 10)^{4}$$
$83$ $$(T^{2} - 300)^{2}$$
$89$ $$(T^{2} - 27)^{2}$$
$97$ $$(T^{2} + 192)^{2}$$