# Properties

 Label 441.2.p.b Level $441$ Weight $2$ Character orbit 441.p Analytic conductor $3.521$ Analytic rank $0$ Dimension $8$ CM discriminant -7 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,2,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: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.12745506816.5 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 8x^{6} + 55x^{4} - 72x^{2} + 81$$ x^8 - 8*x^6 + 55*x^4 - 72*x^2 + 81 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: no (minimal twist has level 63) Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

## $q$-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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{6} - 2 \beta_{4} + \beta_{3} + 2) q^{4} + (\beta_{7} - 2 \beta_{5} + \cdots + 2 \beta_1) q^{8}+O(q^{10})$$ q + b1 * q^2 + (b6 - 2*b4 + b3 + 2) * q^4 + (b7 - 2*b5 + b2 + 2*b1) * q^8 $$q + \beta_1 q^{2} + (\beta_{6} - 2 \beta_{4} + \beta_{3} + 2) q^{4} + (\beta_{7} - 2 \beta_{5} + \cdots + 2 \beta_1) q^{8}+ \cdots + ( - 3 \beta_{7} + 15 \beta_{5} + \cdots - 15 \beta_1) q^{92}+O(q^{100})$$ q + b1 * q^2 + (b6 - 2*b4 + b3 + 2) * q^4 + (b7 - 2*b5 + b2 + 2*b1) * q^8 + (b7 + b5) * q^11 + (2*b6 - 3*b4) * q^16 + (-b3 + 5) * q^22 + (-b2 - 3*b1) * q^23 + (-5*b4 + 5) * q^25 + (b7 + 3*b5 + b2 - 3*b1) * q^29 - 3*b5 * q^32 - 4*b6 * q^37 + 2*b3 * q^43 + (b2 + b1) * q^44 + (-5*b6 + 11*b4 - 5*b3 - 11) * q^46 + (-5*b5 + 5*b1) * q^50 + (b7 - 5*b5) * q^53 + (-b6 + 13*b4) * q^58 + (b3 - 6) * q^64 + (-4*b4 + 4) * q^67 + (-3*b7 + b5 - 3*b2 - b1) * q^71 + (-4*b7 + 8*b5) * q^74 - 8*b4 * q^79 + (2*b2 + 4*b1) * q^86 + (5*b6 + 7*b4 + 5*b3 - 7) * q^88 + (-3*b7 + 15*b5 - 3*b2 - 15*b1) * q^92 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 8 q^{4}+O(q^{10})$$ 8 * q + 8 * q^4 $$8 q + 8 q^{4} - 12 q^{16} + 40 q^{22} + 20 q^{25} - 44 q^{46} + 52 q^{58} - 48 q^{64} + 16 q^{67} - 32 q^{79} - 28 q^{88}+O(q^{100})$$ 8 * q + 8 * q^4 - 12 * q^16 + 40 * q^22 + 20 * q^25 - 44 * q^46 + 52 * q^58 - 48 * q^64 + 16 * q^67 - 32 * q^79 - 28 * q^88

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 8x^{6} + 55x^{4} - 72x^{2} + 81$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -\nu^{7} - 258\nu ) / 55$$ (-v^7 - 258*v) / 55 $$\beta_{3}$$ $$=$$ $$( -\nu^{6} - 148 ) / 55$$ (-v^6 - 148) / 55 $$\beta_{4}$$ $$=$$ $$( -8\nu^{6} + 55\nu^{4} - 440\nu^{2} + 576 ) / 495$$ (-8*v^6 + 55*v^4 - 440*v^2 + 576) / 495 $$\beta_{5}$$ $$=$$ $$( -8\nu^{7} + 55\nu^{5} - 440\nu^{3} + 576\nu ) / 495$$ (-8*v^7 + 55*v^5 - 440*v^3 + 576*v) / 495 $$\beta_{6}$$ $$=$$ $$( -23\nu^{6} + 220\nu^{4} - 1265\nu^{2} + 1656 ) / 495$$ (-23*v^6 + 220*v^4 - 1265*v^2 + 1656) / 495 $$\beta_{7}$$ $$=$$ $$( -13\nu^{7} + 110\nu^{5} - 715\nu^{3} + 936\nu ) / 165$$ (-13*v^7 + 110*v^5 - 715*v^3 + 936*v) / 165
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{6} - 4\beta_{4} + \beta_{3} + 4$$ b6 - 4*b4 + b3 + 4 $$\nu^{3}$$ $$=$$ $$\beta_{7} - 6\beta_{5} + \beta_{2} + 6\beta_1$$ b7 - 6*b5 + b2 + 6*b1 $$\nu^{4}$$ $$=$$ $$8\beta_{6} - 23\beta_{4}$$ 8*b6 - 23*b4 $$\nu^{5}$$ $$=$$ $$8\beta_{7} - 39\beta_{5}$$ 8*b7 - 39*b5 $$\nu^{6}$$ $$=$$ $$-55\beta_{3} - 148$$ -55*b3 - 148 $$\nu^{7}$$ $$=$$ $$-55\beta_{2} - 258\beta_1$$ -55*b2 - 258*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/441\mathbb{Z}\right)^\times$$.

 $$n$$ $$199$$ $$344$$ $$\chi(n)$$ $$\beta_{4}$$ $$-1$$

## 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
 −2.23256 + 1.28897i −1.00781 + 0.581861i 1.00781 − 0.581861i 2.23256 − 1.28897i −2.23256 − 1.28897i −1.00781 − 0.581861i 1.00781 + 0.581861i 2.23256 + 1.28897i
−2.23256 + 1.28897i 0 2.32288 4.02334i 0 0 0 6.82058i 0 0
80.2 −1.00781 + 0.581861i 0 −0.322876 + 0.559237i 0 0 0 3.07892i 0 0
80.3 1.00781 0.581861i 0 −0.322876 + 0.559237i 0 0 0 3.07892i 0 0
80.4 2.23256 1.28897i 0 2.32288 4.02334i 0 0 0 6.82058i 0 0
215.1 −2.23256 1.28897i 0 2.32288 + 4.02334i 0 0 0 6.82058i 0 0
215.2 −1.00781 0.581861i 0 −0.322876 0.559237i 0 0 0 3.07892i 0 0
215.3 1.00781 + 0.581861i 0 −0.322876 0.559237i 0 0 0 3.07892i 0 0
215.4 2.23256 + 1.28897i 0 2.32288 + 4.02334i 0 0 0 6.82058i 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 80.4 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 441.2.p.b 8
3.b odd 2 1 inner 441.2.p.b 8
7.b odd 2 1 CM 441.2.p.b 8
7.c even 3 1 63.2.c.a 4
7.c even 3 1 inner 441.2.p.b 8
7.d odd 6 1 63.2.c.a 4
7.d odd 6 1 inner 441.2.p.b 8
21.c even 2 1 inner 441.2.p.b 8
21.g even 6 1 63.2.c.a 4
21.g even 6 1 inner 441.2.p.b 8
21.h odd 6 1 63.2.c.a 4
21.h odd 6 1 inner 441.2.p.b 8
28.f even 6 1 1008.2.k.a 4
28.g odd 6 1 1008.2.k.a 4
35.i odd 6 1 1575.2.b.a 4
35.j even 6 1 1575.2.b.a 4
35.k even 12 2 1575.2.g.d 8
35.l odd 12 2 1575.2.g.d 8
56.j odd 6 1 4032.2.k.c 4
56.k odd 6 1 4032.2.k.b 4
56.m even 6 1 4032.2.k.b 4
56.p even 6 1 4032.2.k.c 4
63.g even 3 1 567.2.o.f 8
63.h even 3 1 567.2.o.f 8
63.i even 6 1 567.2.o.f 8
63.j odd 6 1 567.2.o.f 8
63.k odd 6 1 567.2.o.f 8
63.n odd 6 1 567.2.o.f 8
63.s even 6 1 567.2.o.f 8
63.t odd 6 1 567.2.o.f 8
84.j odd 6 1 1008.2.k.a 4
84.n even 6 1 1008.2.k.a 4
105.o odd 6 1 1575.2.b.a 4
105.p even 6 1 1575.2.b.a 4
105.w odd 12 2 1575.2.g.d 8
105.x even 12 2 1575.2.g.d 8
168.s odd 6 1 4032.2.k.c 4
168.v even 6 1 4032.2.k.b 4
168.ba even 6 1 4032.2.k.c 4
168.be odd 6 1 4032.2.k.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.c.a 4 7.c even 3 1
63.2.c.a 4 7.d odd 6 1
63.2.c.a 4 21.g even 6 1
63.2.c.a 4 21.h odd 6 1
441.2.p.b 8 1.a even 1 1 trivial
441.2.p.b 8 3.b odd 2 1 inner
441.2.p.b 8 7.b odd 2 1 CM
441.2.p.b 8 7.c even 3 1 inner
441.2.p.b 8 7.d odd 6 1 inner
441.2.p.b 8 21.c even 2 1 inner
441.2.p.b 8 21.g even 6 1 inner
441.2.p.b 8 21.h odd 6 1 inner
567.2.o.f 8 63.g even 3 1
567.2.o.f 8 63.h even 3 1
567.2.o.f 8 63.i even 6 1
567.2.o.f 8 63.j odd 6 1
567.2.o.f 8 63.k odd 6 1
567.2.o.f 8 63.n odd 6 1
567.2.o.f 8 63.s even 6 1
567.2.o.f 8 63.t odd 6 1
1008.2.k.a 4 28.f even 6 1
1008.2.k.a 4 28.g odd 6 1
1008.2.k.a 4 84.j odd 6 1
1008.2.k.a 4 84.n even 6 1
1575.2.b.a 4 35.i odd 6 1
1575.2.b.a 4 35.j even 6 1
1575.2.b.a 4 105.o odd 6 1
1575.2.b.a 4 105.p even 6 1
1575.2.g.d 8 35.k even 12 2
1575.2.g.d 8 35.l odd 12 2
1575.2.g.d 8 105.w odd 12 2
1575.2.g.d 8 105.x even 12 2
4032.2.k.b 4 56.k odd 6 1
4032.2.k.b 4 56.m even 6 1
4032.2.k.b 4 168.v even 6 1
4032.2.k.b 4 168.be odd 6 1
4032.2.k.c 4 56.j odd 6 1
4032.2.k.c 4 56.p even 6 1
4032.2.k.c 4 168.s odd 6 1
4032.2.k.c 4 168.ba even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} - 8T_{2}^{6} + 55T_{2}^{4} - 72T_{2}^{2} + 81$$ acting on $$S_{2}^{\mathrm{new}}(441, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} - 8 T^{6} + \cdots + 81$$
$3$ $$T^{8}$$
$5$ $$T^{8}$$
$7$ $$T^{8}$$
$11$ $$T^{8} - 44 T^{6} + \cdots + 1296$$
$13$ $$T^{8}$$
$17$ $$T^{8}$$
$19$ $$T^{8}$$
$23$ $$T^{8} - 92 T^{6} + \cdots + 104976$$
$29$ $$(T^{4} + 116 T^{2} + 2916)^{2}$$
$31$ $$T^{8}$$
$37$ $$(T^{4} + 112 T^{2} + 12544)^{2}$$
$41$ $$T^{8}$$
$43$ $$(T^{2} - 28)^{4}$$
$47$ $$T^{8}$$
$53$ $$T^{8} - 212 T^{6} + \cdots + 1296$$
$59$ $$T^{8}$$
$61$ $$T^{8}$$
$67$ $$(T^{2} - 4 T + 16)^{4}$$
$71$ $$(T^{4} + 284 T^{2} + 12996)^{2}$$
$73$ $$T^{8}$$
$79$ $$(T^{2} + 8 T + 64)^{4}$$
$83$ $$T^{8}$$
$89$ $$T^{8}$$
$97$ $$T^{8}$$