# Properties

 Label 441.2.p.a Level $441$ Weight $2$ Character orbit 441.p Analytic conductor $3.521$ Analytic rank $0$ Dimension $4$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,2,Mod(80,441)]

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([3, 1]))

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

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

chi := DirichletCharacter("441.80");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 441.p (of order $$6$$, 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: $$3.52140272914$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ 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: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 63) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + ( - 2 \beta_{3} + \beta_1) q^{5} - 2 \beta_{3} q^{8}+O(q^{10})$$ q + b1 * q^2 + (-2*b3 + b1) * q^5 - 2*b3 * q^8 $$q + \beta_1 q^{2} + ( - 2 \beta_{3} + \beta_1) q^{5} - 2 \beta_{3} q^{8} + ( - 2 \beta_{2} + 4) q^{10} + (\beta_{3} - \beta_1) q^{11} + ( - 6 \beta_{2} + 3) q^{13} + ( - 4 \beta_{2} + 4) q^{16} + (2 \beta_{3} + 2 \beta_1) q^{17} + ( - \beta_{2} - 1) q^{19} - 2 q^{22} + 4 \beta_1 q^{23} - \beta_{2} q^{25} + ( - 6 \beta_{3} + 3 \beta_1) q^{26} + 2 \beta_{3} q^{29} + (\beta_{2} - 2) q^{31} + (8 \beta_{2} - 4) q^{34} + ( - \beta_{2} + 1) q^{37} + ( - \beta_{3} - \beta_1) q^{38} + ( - 4 \beta_{2} - 4) q^{40} + ( - 3 \beta_{3} + 6 \beta_1) q^{41} - q^{43} + 8 \beta_{2} q^{46} + (10 \beta_{3} - 5 \beta_1) q^{47} - \beta_{3} q^{50} + ( - 2 \beta_{3} + 2 \beta_1) q^{53} + (4 \beta_{2} - 2) q^{55} + (4 \beta_{2} - 4) q^{58} + (2 \beta_{3} + 2 \beta_1) q^{59} + (2 \beta_{2} + 2) q^{61} + (\beta_{3} - 2 \beta_1) q^{62} - 8 q^{64} - 9 \beta_1 q^{65} - 11 \beta_{2} q^{67} + 5 \beta_{3} q^{71} + (\beta_{2} - 2) q^{73} + ( - \beta_{3} + \beta_1) q^{74} + (5 \beta_{2} - 5) q^{79} + ( - 4 \beta_{3} - 4 \beta_1) q^{80} + (6 \beta_{2} + 6) q^{82} + (3 \beta_{3} - 6 \beta_1) q^{83} + 12 q^{85} - \beta_1 q^{86} + 4 \beta_{2} q^{88} + (4 \beta_{3} - 2 \beta_1) q^{89} + (10 \beta_{2} - 20) q^{94} + (3 \beta_{3} - 3 \beta_1) q^{95} + ( - 12 \beta_{2} + 6) q^{97}+O(q^{100})$$ q + b1 * q^2 + (-2*b3 + b1) * q^5 - 2*b3 * q^8 + (-2*b2 + 4) * q^10 + (b3 - b1) * q^11 + (-6*b2 + 3) * q^13 + (-4*b2 + 4) * q^16 + (2*b3 + 2*b1) * q^17 + (-b2 - 1) * q^19 - 2 * q^22 + 4*b1 * q^23 - b2 * q^25 + (-6*b3 + 3*b1) * q^26 + 2*b3 * q^29 + (b2 - 2) * q^31 + (8*b2 - 4) * q^34 + (-b2 + 1) * q^37 + (-b3 - b1) * q^38 + (-4*b2 - 4) * q^40 + (-3*b3 + 6*b1) * q^41 - q^43 + 8*b2 * q^46 + (10*b3 - 5*b1) * q^47 - b3 * q^50 + (-2*b3 + 2*b1) * q^53 + (4*b2 - 2) * q^55 + (4*b2 - 4) * q^58 + (2*b3 + 2*b1) * q^59 + (2*b2 + 2) * q^61 + (b3 - 2*b1) * q^62 - 8 * q^64 - 9*b1 * q^65 - 11*b2 * q^67 + 5*b3 * q^71 + (b2 - 2) * q^73 + (-b3 + b1) * q^74 + (5*b2 - 5) * q^79 + (-4*b3 - 4*b1) * q^80 + (6*b2 + 6) * q^82 + (3*b3 - 6*b1) * q^83 + 12 * q^85 - b1 * q^86 + 4*b2 * q^88 + (4*b3 - 2*b1) * q^89 + (10*b2 - 20) * q^94 + (3*b3 - 3*b1) * q^95 + (-12*b2 + 6) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q + 12 q^{10} + 8 q^{16} - 6 q^{19} - 8 q^{22} - 2 q^{25} - 6 q^{31} + 2 q^{37} - 24 q^{40} - 4 q^{43} + 16 q^{46} - 8 q^{58} + 12 q^{61} - 32 q^{64} - 22 q^{67} - 6 q^{73} - 10 q^{79} + 36 q^{82} + 48 q^{85} + 8 q^{88} - 60 q^{94}+O(q^{100})$$ 4 * q + 12 * q^10 + 8 * q^16 - 6 * q^19 - 8 * q^22 - 2 * q^25 - 6 * q^31 + 2 * q^37 - 24 * q^40 - 4 * q^43 + 16 * q^46 - 8 * q^58 + 12 * q^61 - 32 * q^64 - 22 * q^67 - 6 * q^73 - 10 * q^79 + 36 * q^82 + 48 * q^85 + 8 * q^88 - 60 * q^94

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 2$$ (v^3) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$2\beta_{2}$$ 2*b2 $$\nu^{3}$$ $$=$$ $$2\beta_{3}$$ 2*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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
80.1
 −1.22474 + 0.707107i 1.22474 − 0.707107i −1.22474 − 0.707107i 1.22474 + 0.707107i
−1.22474 + 0.707107i 0 0 −1.22474 2.12132i 0 0 2.82843i 0 3.00000 + 1.73205i
80.2 1.22474 0.707107i 0 0 1.22474 + 2.12132i 0 0 2.82843i 0 3.00000 + 1.73205i
215.1 −1.22474 0.707107i 0 0 −1.22474 + 2.12132i 0 0 2.82843i 0 3.00000 1.73205i
215.2 1.22474 + 0.707107i 0 0 1.22474 2.12132i 0 0 2.82843i 0 3.00000 1.73205i
 $$n$$: 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.d odd 6 1 inner
21.g even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.2.p.a 4
3.b odd 2 1 inner 441.2.p.a 4
7.b odd 2 1 63.2.p.a 4
7.c even 3 1 63.2.p.a 4
7.c even 3 1 441.2.c.a 4
7.d odd 6 1 441.2.c.a 4
7.d odd 6 1 inner 441.2.p.a 4
21.c even 2 1 63.2.p.a 4
21.g even 6 1 441.2.c.a 4
21.g even 6 1 inner 441.2.p.a 4
21.h odd 6 1 63.2.p.a 4
21.h odd 6 1 441.2.c.a 4
28.d even 2 1 1008.2.bt.b 4
28.f even 6 1 7056.2.k.b 4
28.g odd 6 1 1008.2.bt.b 4
28.g odd 6 1 7056.2.k.b 4
35.c odd 2 1 1575.2.bk.c 4
35.f even 4 2 1575.2.bc.a 8
35.j even 6 1 1575.2.bk.c 4
35.l odd 12 2 1575.2.bc.a 8
63.g even 3 1 567.2.i.d 4
63.h even 3 1 567.2.s.d 4
63.j odd 6 1 567.2.s.d 4
63.l odd 6 1 567.2.i.d 4
63.l odd 6 1 567.2.s.d 4
63.n odd 6 1 567.2.i.d 4
63.o even 6 1 567.2.i.d 4
63.o even 6 1 567.2.s.d 4
84.h odd 2 1 1008.2.bt.b 4
84.j odd 6 1 7056.2.k.b 4
84.n even 6 1 1008.2.bt.b 4
84.n even 6 1 7056.2.k.b 4
105.g even 2 1 1575.2.bk.c 4
105.k odd 4 2 1575.2.bc.a 8
105.o odd 6 1 1575.2.bk.c 4
105.x even 12 2 1575.2.bc.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.p.a 4 7.b odd 2 1
63.2.p.a 4 7.c even 3 1
63.2.p.a 4 21.c even 2 1
63.2.p.a 4 21.h odd 6 1
441.2.c.a 4 7.c even 3 1
441.2.c.a 4 7.d odd 6 1
441.2.c.a 4 21.g even 6 1
441.2.c.a 4 21.h odd 6 1
441.2.p.a 4 1.a even 1 1 trivial
441.2.p.a 4 3.b odd 2 1 inner
441.2.p.a 4 7.d odd 6 1 inner
441.2.p.a 4 21.g even 6 1 inner
567.2.i.d 4 63.g even 3 1
567.2.i.d 4 63.l odd 6 1
567.2.i.d 4 63.n odd 6 1
567.2.i.d 4 63.o even 6 1
567.2.s.d 4 63.h even 3 1
567.2.s.d 4 63.j odd 6 1
567.2.s.d 4 63.l odd 6 1
567.2.s.d 4 63.o even 6 1
1008.2.bt.b 4 28.d even 2 1
1008.2.bt.b 4 28.g odd 6 1
1008.2.bt.b 4 84.h odd 2 1
1008.2.bt.b 4 84.n even 6 1
1575.2.bc.a 8 35.f even 4 2
1575.2.bc.a 8 35.l odd 12 2
1575.2.bc.a 8 105.k odd 4 2
1575.2.bc.a 8 105.x even 12 2
1575.2.bk.c 4 35.c odd 2 1
1575.2.bk.c 4 35.j even 6 1
1575.2.bk.c 4 105.g even 2 1
1575.2.bk.c 4 105.o odd 6 1
7056.2.k.b 4 28.f even 6 1
7056.2.k.b 4 28.g odd 6 1
7056.2.k.b 4 84.j odd 6 1
7056.2.k.b 4 84.n even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 2T_{2}^{2} + 4$$ acting on $$S_{2}^{\mathrm{new}}(441, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 2T^{2} + 4$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 6T^{2} + 36$$
$7$ $$T^{4}$$
$11$ $$T^{4} - 2T^{2} + 4$$
$13$ $$(T^{2} + 27)^{2}$$
$17$ $$T^{4} + 24T^{2} + 576$$
$19$ $$(T^{2} + 3 T + 3)^{2}$$
$23$ $$T^{4} - 32T^{2} + 1024$$
$29$ $$(T^{2} + 8)^{2}$$
$31$ $$(T^{2} + 3 T + 3)^{2}$$
$37$ $$(T^{2} - T + 1)^{2}$$
$41$ $$(T^{2} - 54)^{2}$$
$43$ $$(T + 1)^{4}$$
$47$ $$T^{4} + 150 T^{2} + 22500$$
$53$ $$T^{4} - 8T^{2} + 64$$
$59$ $$T^{4} + 24T^{2} + 576$$
$61$ $$(T^{2} - 6 T + 12)^{2}$$
$67$ $$(T^{2} + 11 T + 121)^{2}$$
$71$ $$(T^{2} + 50)^{2}$$
$73$ $$(T^{2} + 3 T + 3)^{2}$$
$79$ $$(T^{2} + 5 T + 25)^{2}$$
$83$ $$(T^{2} - 54)^{2}$$
$89$ $$T^{4} + 24T^{2} + 576$$
$97$ $$(T^{2} + 108)^{2}$$