Properties

 Label 176.2.e.a Level $176$ Weight $2$ Character orbit 176.e Analytic conductor $1.405$ Analytic rank $0$ Dimension $2$ CM discriminant -11 Inner twists $4$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [176,2,Mod(175,176)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(176, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("176.175");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$176 = 2^{4} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 176.e (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: $$1.40536707557$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-11})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 3$$ x^2 - x + 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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 $$\beta = \sqrt{-11}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{3} + 3 q^{5} - 8 q^{9} +O(q^{10})$$ q - b * q^3 + 3 * q^5 - 8 * q^9 $$q - \beta q^{3} + 3 q^{5} - 8 q^{9} + \beta q^{11} - 3 \beta q^{15} - \beta q^{23} + 4 q^{25} + 5 \beta q^{27} + 3 \beta q^{31} + 11 q^{33} + 7 q^{37} - 24 q^{45} - 2 \beta q^{47} - 7 q^{49} + 6 q^{53} + 3 \beta q^{55} - \beta q^{59} + 3 \beta q^{67} - 11 q^{69} - 5 \beta q^{71} - 4 \beta q^{75} + 31 q^{81} - 9 q^{89} + 33 q^{93} - 17 q^{97} - 8 \beta q^{99} +O(q^{100})$$ q - b * q^3 + 3 * q^5 - 8 * q^9 + b * q^11 - 3*b * q^15 - b * q^23 + 4 * q^25 + 5*b * q^27 + 3*b * q^31 + 11 * q^33 + 7 * q^37 - 24 * q^45 - 2*b * q^47 - 7 * q^49 + 6 * q^53 + 3*b * q^55 - b * q^59 + 3*b * q^67 - 11 * q^69 - 5*b * q^71 - 4*b * q^75 + 31 * q^81 - 9 * q^89 + 33 * q^93 - 17 * q^97 - 8*b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{5} - 16 q^{9}+O(q^{10})$$ 2 * q + 6 * q^5 - 16 * q^9 $$2 q + 6 q^{5} - 16 q^{9} + 8 q^{25} + 22 q^{33} + 14 q^{37} - 48 q^{45} - 14 q^{49} + 12 q^{53} - 22 q^{69} + 62 q^{81} - 18 q^{89} + 66 q^{93} - 34 q^{97}+O(q^{100})$$ 2 * q + 6 * q^5 - 16 * q^9 + 8 * q^25 + 22 * q^33 + 14 * q^37 - 48 * q^45 - 14 * q^49 + 12 * q^53 - 22 * q^69 + 62 * q^81 - 18 * q^89 + 66 * q^93 - 34 * q^97

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/176\mathbb{Z}\right)^\times$$.

 $$n$$ $$111$$ $$133$$ $$145$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
175.1
 0.5 + 1.65831i 0.5 − 1.65831i
0 3.31662i 0 3.00000 0 0 0 −8.00000 0
175.2 0 3.31662i 0 3.00000 0 0 0 −8.00000 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
11.b odd 2 1 CM by $$\Q(\sqrt{-11})$$
4.b odd 2 1 inner
44.c even 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 176.2.e.a 2
3.b odd 2 1 1584.2.o.a 2
4.b odd 2 1 inner 176.2.e.a 2
8.b even 2 1 704.2.e.a 2
8.d odd 2 1 704.2.e.a 2
11.b odd 2 1 CM 176.2.e.a 2
12.b even 2 1 1584.2.o.a 2
16.e even 4 2 2816.2.g.a 4
16.f odd 4 2 2816.2.g.a 4
33.d even 2 1 1584.2.o.a 2
44.c even 2 1 inner 176.2.e.a 2
88.b odd 2 1 704.2.e.a 2
88.g even 2 1 704.2.e.a 2
132.d odd 2 1 1584.2.o.a 2
176.i even 4 2 2816.2.g.a 4
176.l odd 4 2 2816.2.g.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
176.2.e.a 2 1.a even 1 1 trivial
176.2.e.a 2 4.b odd 2 1 inner
176.2.e.a 2 11.b odd 2 1 CM
176.2.e.a 2 44.c even 2 1 inner
704.2.e.a 2 8.b even 2 1
704.2.e.a 2 8.d odd 2 1
704.2.e.a 2 88.b odd 2 1
704.2.e.a 2 88.g even 2 1
1584.2.o.a 2 3.b odd 2 1
1584.2.o.a 2 12.b even 2 1
1584.2.o.a 2 33.d even 2 1
1584.2.o.a 2 132.d odd 2 1
2816.2.g.a 4 16.e even 4 2
2816.2.g.a 4 16.f odd 4 2
2816.2.g.a 4 176.i even 4 2
2816.2.g.a 4 176.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 11$$ acting on $$S_{2}^{\mathrm{new}}(176, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 11$$
$5$ $$(T - 3)^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 11$$
$13$ $$T^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2} + 11$$
$29$ $$T^{2}$$
$31$ $$T^{2} + 99$$
$37$ $$(T - 7)^{2}$$
$41$ $$T^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2} + 44$$
$53$ $$(T - 6)^{2}$$
$59$ $$T^{2} + 11$$
$61$ $$T^{2}$$
$67$ $$T^{2} + 99$$
$71$ $$T^{2} + 275$$
$73$ $$T^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$(T + 9)^{2}$$
$97$ $$(T + 17)^{2}$$