Properties

 Label 441.2.c.a Level $441$ Weight $2$ Character orbit 441.c 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(440,441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("441.440");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 441.c (of order $$2$$, degree $$1$$, 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: $$4$$ 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: $$2^{2}$$ Twist minimal: no (minimal twist has level 63) 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.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + \beta_{3} q^{5} - 2 \beta_1 q^{8}+O(q^{10})$$ q - b1 * q^2 + b3 * q^5 - 2*b1 * q^8 $$q - \beta_1 q^{2} + \beta_{3} q^{5} - 2 \beta_1 q^{8} - 2 \beta_{2} q^{10} - \beta_1 q^{11} + 3 \beta_{2} q^{13} - 4 q^{16} + 2 \beta_{3} q^{17} - \beta_{2} q^{19} - 2 q^{22} - 4 \beta_1 q^{23} + q^{25} + 3 \beta_{3} q^{26} + 2 \beta_1 q^{29} + \beta_{2} q^{31} - 4 \beta_{2} q^{34} - q^{37} - \beta_{3} q^{38} - 4 \beta_{2} q^{40} - 3 \beta_{3} q^{41} - q^{43} - 8 q^{46} - 5 \beta_{3} q^{47} - \beta_1 q^{50} + 2 \beta_1 q^{53} - 2 \beta_{2} q^{55} + 4 q^{58} + 2 \beta_{3} q^{59} + 2 \beta_{2} q^{61} + \beta_{3} q^{62} - 8 q^{64} + 9 \beta_1 q^{65} + 11 q^{67} + 5 \beta_1 q^{71} + \beta_{2} q^{73} + \beta_1 q^{74} + 5 q^{79} - 4 \beta_{3} q^{80} + 6 \beta_{2} q^{82} + 3 \beta_{3} q^{83} + 12 q^{85} + \beta_1 q^{86} - 4 q^{88} - 2 \beta_{3} q^{89} + 10 \beta_{2} q^{94} - 3 \beta_1 q^{95} + 6 \beta_{2} q^{97}+O(q^{100})$$ q - b1 * q^2 + b3 * q^5 - 2*b1 * q^8 - 2*b2 * q^10 - b1 * q^11 + 3*b2 * q^13 - 4 * q^16 + 2*b3 * q^17 - b2 * q^19 - 2 * q^22 - 4*b1 * q^23 + q^25 + 3*b3 * q^26 + 2*b1 * q^29 + b2 * q^31 - 4*b2 * q^34 - q^37 - b3 * q^38 - 4*b2 * q^40 - 3*b3 * q^41 - q^43 - 8 * q^46 - 5*b3 * q^47 - b1 * q^50 + 2*b1 * q^53 - 2*b2 * q^55 + 4 * q^58 + 2*b3 * q^59 + 2*b2 * q^61 + b3 * q^62 - 8 * q^64 + 9*b1 * q^65 + 11 * q^67 + 5*b1 * q^71 + b2 * q^73 + b1 * q^74 + 5 * q^79 - 4*b3 * q^80 + 6*b2 * q^82 + 3*b3 * q^83 + 12 * q^85 + b1 * q^86 - 4 * q^88 - 2*b3 * q^89 + 10*b2 * q^94 - 3*b1 * q^95 + 6*b2 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q - 16 q^{16} - 8 q^{22} + 4 q^{25} - 4 q^{37} - 4 q^{43} - 32 q^{46} + 16 q^{58} - 32 q^{64} + 44 q^{67} + 20 q^{79} + 48 q^{85} - 16 q^{88}+O(q^{100})$$ 4 * q - 16 * q^16 - 8 * q^22 + 4 * q^25 - 4 * q^37 - 4 * q^43 - 32 * q^46 + 16 * q^58 - 32 * q^64 + 44 * q^67 + 20 * q^79 + 48 * q^85 - 16 * q^88

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

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

Character values

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

 $$n$$ $$199$$ $$344$$ $$\chi(n)$$ $$-1$$ $$-1$$

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
440.1
 −1.22474 + 0.707107i 1.22474 + 0.707107i −1.22474 − 0.707107i 1.22474 − 0.707107i
1.41421i 0 0 −2.44949 0 0 2.82843i 0 3.46410i
440.2 1.41421i 0 0 2.44949 0 0 2.82843i 0 3.46410i
440.3 1.41421i 0 0 −2.44949 0 0 2.82843i 0 3.46410i
440.4 1.41421i 0 0 2.44949 0 0 2.82843i 0 3.46410i
 $$n$$: 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 441.2.c.a 4
3.b odd 2 1 inner 441.2.c.a 4
4.b odd 2 1 7056.2.k.b 4
7.b odd 2 1 inner 441.2.c.a 4
7.c even 3 1 63.2.p.a 4
7.c even 3 1 441.2.p.a 4
7.d odd 6 1 63.2.p.a 4
7.d odd 6 1 441.2.p.a 4
12.b even 2 1 7056.2.k.b 4
21.c even 2 1 inner 441.2.c.a 4
21.g even 6 1 63.2.p.a 4
21.g even 6 1 441.2.p.a 4
21.h odd 6 1 63.2.p.a 4
21.h odd 6 1 441.2.p.a 4
28.d even 2 1 7056.2.k.b 4
28.f even 6 1 1008.2.bt.b 4
28.g odd 6 1 1008.2.bt.b 4
35.i odd 6 1 1575.2.bk.c 4
35.j even 6 1 1575.2.bk.c 4
35.k even 12 2 1575.2.bc.a 8
35.l odd 12 2 1575.2.bc.a 8
63.g even 3 1 567.2.s.d 4
63.h even 3 1 567.2.i.d 4
63.i even 6 1 567.2.i.d 4
63.j odd 6 1 567.2.i.d 4
63.k odd 6 1 567.2.s.d 4
63.n odd 6 1 567.2.s.d 4
63.s even 6 1 567.2.s.d 4
63.t odd 6 1 567.2.i.d 4
84.h odd 2 1 7056.2.k.b 4
84.j odd 6 1 1008.2.bt.b 4
84.n even 6 1 1008.2.bt.b 4
105.o odd 6 1 1575.2.bk.c 4
105.p even 6 1 1575.2.bk.c 4
105.w odd 12 2 1575.2.bc.a 8
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.c even 3 1
63.2.p.a 4 7.d odd 6 1
63.2.p.a 4 21.g even 6 1
63.2.p.a 4 21.h odd 6 1
441.2.c.a 4 1.a even 1 1 trivial
441.2.c.a 4 3.b odd 2 1 inner
441.2.c.a 4 7.b odd 2 1 inner
441.2.c.a 4 21.c even 2 1 inner
441.2.p.a 4 7.c even 3 1
441.2.p.a 4 7.d odd 6 1
441.2.p.a 4 21.g even 6 1
441.2.p.a 4 21.h odd 6 1
567.2.i.d 4 63.h even 3 1
567.2.i.d 4 63.i even 6 1
567.2.i.d 4 63.j odd 6 1
567.2.i.d 4 63.t odd 6 1
567.2.s.d 4 63.g even 3 1
567.2.s.d 4 63.k odd 6 1
567.2.s.d 4 63.n odd 6 1
567.2.s.d 4 63.s even 6 1
1008.2.bt.b 4 28.f even 6 1
1008.2.bt.b 4 28.g odd 6 1
1008.2.bt.b 4 84.j odd 6 1
1008.2.bt.b 4 84.n even 6 1
1575.2.bc.a 8 35.k even 12 2
1575.2.bc.a 8 35.l odd 12 2
1575.2.bc.a 8 105.w odd 12 2
1575.2.bc.a 8 105.x even 12 2
1575.2.bk.c 4 35.i odd 6 1
1575.2.bk.c 4 35.j even 6 1
1575.2.bk.c 4 105.o odd 6 1
1575.2.bk.c 4 105.p even 6 1
7056.2.k.b 4 4.b odd 2 1
7056.2.k.b 4 12.b even 2 1
7056.2.k.b 4 28.d even 2 1
7056.2.k.b 4 84.h odd 2 1

Hecke kernels

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

Hecke characteristic polynomials

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