# Properties

 Label 441.4.e.t Level $441$ Weight $4$ Character orbit 441.e Analytic conductor $26.020$ Analytic rank $0$ Dimension $4$ CM discriminant -7 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,4,Mod(226,441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("441.226");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 441.e (of order $$3$$, degree $$2$$, 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: $$26.0198423125$$ 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: $$\Z[a_1, \ldots, a_{4}]$$ 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 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} - \beta_{2} q^{4} - 9 \beta_{3} q^{8}+O(q^{10})$$ q + b1 * q^2 - b2 * q^4 - 9*b3 * q^8 $$q + \beta_1 q^{2} - \beta_{2} q^{4} - 9 \beta_{3} q^{8} + ( - 10 \beta_{3} - 10 \beta_1) q^{11} + (55 \beta_{2} + 55) q^{16} + 70 q^{22} - 82 \beta_1 q^{23} - 125 \beta_{2} q^{25} - 100 \beta_{3} q^{29} + ( - 17 \beta_{3} - 17 \beta_1) q^{32} + (450 \beta_{2} + 450) q^{37} + 180 q^{43} - 10 \beta_1 q^{44} - 574 \beta_{2} q^{46} - 125 \beta_{3} q^{50} + ( - 188 \beta_{3} - 188 \beta_1) q^{53} + (700 \beta_{2} + 700) q^{58} + 559 q^{64} - 740 \beta_{2} q^{67} + 370 \beta_{3} q^{71} + (450 \beta_{3} + 450 \beta_1) q^{74} + (1384 \beta_{2} + 1384) q^{79} + 180 \beta_1 q^{86} - 630 \beta_{2} q^{88} + 82 \beta_{3} q^{92}+O(q^{100})$$ q + b1 * q^2 - b2 * q^4 - 9*b3 * q^8 + (-10*b3 - 10*b1) * q^11 + (55*b2 + 55) * q^16 + 70 * q^22 - 82*b1 * q^23 - 125*b2 * q^25 - 100*b3 * q^29 + (-17*b3 - 17*b1) * q^32 + (450*b2 + 450) * q^37 + 180 * q^43 - 10*b1 * q^44 - 574*b2 * q^46 - 125*b3 * q^50 + (-188*b3 - 188*b1) * q^53 + (700*b2 + 700) * q^58 + 559 * q^64 - 740*b2 * q^67 + 370*b3 * q^71 + (450*b3 + 450*b1) * q^74 + (1384*b2 + 1384) * q^79 + 180*b1 * q^86 - 630*b2 * q^88 + 82*b3 * q^92 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{4}+O(q^{10})$$ 4 * q + 2 * q^4 $$4 q + 2 q^{4} + 110 q^{16} + 280 q^{22} + 250 q^{25} + 900 q^{37} + 720 q^{43} + 1148 q^{46} + 1400 q^{58} + 2236 q^{64} + 1480 q^{67} + 2768 q^{79} + 1260 q^{88}+O(q^{100})$$ 4 * q + 2 * q^4 + 110 * q^16 + 280 * q^22 + 250 * q^25 + 900 * q^37 + 720 * q^43 + 1148 * q^46 + 1400 * q^58 + 2236 * q^64 + 1480 * q^67 + 2768 * q^79 + 1260 * q^88

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 $$\left(\mathbb{Z}/441\mathbb{Z}\right)^\times$$.

 $$n$$ $$199$$ $$344$$ $$\chi(n)$$ $$-1 - \beta_{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
226.1
 −1.32288 + 2.29129i 1.32288 − 2.29129i −1.32288 − 2.29129i 1.32288 + 2.29129i
−1.32288 + 2.29129i 0 0.500000 + 0.866025i 0 0 0 −23.8118 0 0
226.2 1.32288 2.29129i 0 0.500000 + 0.866025i 0 0 0 23.8118 0 0
361.1 −1.32288 2.29129i 0 0.500000 0.866025i 0 0 0 −23.8118 0 0
361.2 1.32288 + 2.29129i 0 0.500000 0.866025i 0 0 0 23.8118 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
7.b odd 2 1 CM by $$\Q(\sqrt{-7})$$
3.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
21.c even 2 1 inner
21.g even 6 1 inner
21.h odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.4.e.t 4
3.b odd 2 1 inner 441.4.e.t 4
7.b odd 2 1 CM 441.4.e.t 4
7.c even 3 1 441.4.a.p 2
7.c even 3 1 inner 441.4.e.t 4
7.d odd 6 1 441.4.a.p 2
7.d odd 6 1 inner 441.4.e.t 4
21.c even 2 1 inner 441.4.e.t 4
21.g even 6 1 441.4.a.p 2
21.g even 6 1 inner 441.4.e.t 4
21.h odd 6 1 441.4.a.p 2
21.h odd 6 1 inner 441.4.e.t 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.4.a.p 2 7.c even 3 1
441.4.a.p 2 7.d odd 6 1
441.4.a.p 2 21.g even 6 1
441.4.a.p 2 21.h odd 6 1
441.4.e.t 4 1.a even 1 1 trivial
441.4.e.t 4 3.b odd 2 1 inner
441.4.e.t 4 7.b odd 2 1 CM
441.4.e.t 4 7.c even 3 1 inner
441.4.e.t 4 7.d odd 6 1 inner
441.4.e.t 4 21.c even 2 1 inner
441.4.e.t 4 21.g even 6 1 inner
441.4.e.t 4 21.h odd 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(441, [\chi])$$:

 $$T_{2}^{4} + 7T_{2}^{2} + 49$$ T2^4 + 7*T2^2 + 49 $$T_{5}$$ T5 $$T_{13}$$ T13

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 7T^{2} + 49$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$T^{4} + 700 T^{2} + 490000$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$T^{4} + 47068 T^{2} + \cdots + 2215396624$$
$29$ $$(T^{2} - 70000)^{2}$$
$31$ $$T^{4}$$
$37$ $$(T^{2} - 450 T + 202500)^{2}$$
$41$ $$T^{4}$$
$43$ $$(T - 180)^{4}$$
$47$ $$T^{4}$$
$53$ $$T^{4} + 247408 T^{2} + \cdots + 61210718464$$
$59$ $$T^{4}$$
$61$ $$T^{4}$$
$67$ $$(T^{2} - 740 T + 547600)^{2}$$
$71$ $$(T^{2} - 958300)^{2}$$
$73$ $$T^{4}$$
$79$ $$(T^{2} - 1384 T + 1915456)^{2}$$
$83$ $$T^{4}$$
$89$ $$T^{4}$$
$97$ $$T^{4}$$