# Properties

 Label 176.2.e.b Level $176$ Weight $2$ Character orbit 176.e Analytic conductor $1.405$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-11})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 2x^{2} - 3x + 9$$ x^4 - x^3 - 2*x^2 - 3*x + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ 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 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} + (\beta_{3} - 2) q^{5} + (\beta_{3} - 1) q^{9}+O(q^{10})$$ q + b1 * q^3 + (b3 - 2) * q^5 + (b3 - 1) * q^9 $$q + \beta_1 q^{3} + (\beta_{3} - 2) q^{5} + (\beta_{3} - 1) q^{9} + ( - \beta_{2} + 2 \beta_1) q^{11} + ( - 2 \beta_{2} - 3 \beta_1) q^{15} + (4 \beta_{2} + \beta_1) q^{23} + ( - 3 \beta_{3} + 7) q^{25} + ( - 2 \beta_{2} + \beta_1) q^{27} + (4 \beta_{2} - 3 \beta_1) q^{31} + (\beta_{3} - 6) q^{33} + ( - 3 \beta_{3} - 2) q^{37} + ( - 2 \beta_{3} + 10) q^{45} + (2 \beta_{2} - 4 \beta_1) q^{47} - 7 q^{49} + 6 q^{53} + ( - 4 \beta_{2} - 3 \beta_1) q^{55} + ( - 8 \beta_{2} + \beta_1) q^{59} + (8 \beta_{2} - 3 \beta_1) q^{67} + (5 \beta_{3} - 12) q^{69} + ( - 4 \beta_{2} + 5 \beta_1) q^{71} + (6 \beta_{2} + 10 \beta_1) q^{75} + (2 \beta_{3} - 3) q^{81} + (5 \beta_{3} + 2) q^{89} + (\beta_{3} + 4) q^{93} + ( - 3 \beta_{3} + 10) q^{97} + ( - 5 \beta_{2} - \beta_1) q^{99}+O(q^{100})$$ q + b1 * q^3 + (b3 - 2) * q^5 + (b3 - 1) * q^9 + (-b2 + 2*b1) * q^11 + (-2*b2 - 3*b1) * q^15 + (4*b2 + b1) * q^23 + (-3*b3 + 7) * q^25 + (-2*b2 + b1) * q^27 + (4*b2 - 3*b1) * q^31 + (b3 - 6) * q^33 + (-3*b3 - 2) * q^37 + (-2*b3 + 10) * q^45 + (2*b2 - 4*b1) * q^47 - 7 * q^49 + 6 * q^53 + (-4*b2 - 3*b1) * q^55 + (-8*b2 + b1) * q^59 + (8*b2 - 3*b1) * q^67 + (5*b3 - 12) * q^69 + (-4*b2 + 5*b1) * q^71 + (6*b2 + 10*b1) * q^75 + (2*b3 - 3) * q^81 + (5*b3 + 2) * q^89 + (b3 + 4) * q^93 + (-3*b3 + 10) * q^97 + (-5*b2 - b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 6 q^{5} - 2 q^{9}+O(q^{10})$$ 4 * q - 6 * q^5 - 2 * q^9 $$4 q - 6 q^{5} - 2 q^{9} + 22 q^{25} - 22 q^{33} - 14 q^{37} + 36 q^{45} - 28 q^{49} + 24 q^{53} - 38 q^{69} - 8 q^{81} + 18 q^{89} + 18 q^{93} + 34 q^{97}+O(q^{100})$$ 4 * q - 6 * q^5 - 2 * q^9 + 22 * q^25 - 22 * q^33 - 14 * q^37 + 36 * q^45 - 28 * q^49 + 24 * q^53 - 38 * q^69 - 8 * q^81 + 18 * q^89 + 18 * q^93 + 34 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 2x^{2} - 3x + 9$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{3} + \nu^{2} - \nu + 3 ) / 3$$ (-v^3 + v^2 - v + 3) / 3 $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 2\nu^{2} - 2\nu - 6 ) / 3$$ (v^3 + 2*v^2 - 2*v - 6) / 3 $$\beta_{3}$$ $$=$$ $$( -\nu^{3} + \nu^{2} + 5\nu + 3 ) / 3$$ (-v^3 + v^2 + 5*v + 3) / 3
 $$\nu$$ $$=$$ $$( \beta_{3} - \beta_1 ) / 2$$ (b3 - b1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + 2\beta_{2} + \beta _1 + 2 ) / 2$$ (b3 + 2*b2 + b1 + 2) / 2 $$\nu^{3}$$ $$=$$ $$\beta_{2} - 2\beta _1 + 4$$ b2 - 2*b1 + 4

## 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
 −1.18614 + 1.26217i 1.68614 + 0.396143i 1.68614 − 0.396143i −1.18614 − 1.26217i
0 2.52434i 0 −4.37228 0 0 0 −3.37228 0
175.2 0 0.792287i 0 1.37228 0 0 0 2.37228 0
175.3 0 0.792287i 0 1.37228 0 0 0 2.37228 0
175.4 0 2.52434i 0 −4.37228 0 0 0 −3.37228 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.b 4
3.b odd 2 1 1584.2.o.e 4
4.b odd 2 1 inner 176.2.e.b 4
8.b even 2 1 704.2.e.c 4
8.d odd 2 1 704.2.e.c 4
11.b odd 2 1 CM 176.2.e.b 4
12.b even 2 1 1584.2.o.e 4
16.e even 4 2 2816.2.g.c 8
16.f odd 4 2 2816.2.g.c 8
33.d even 2 1 1584.2.o.e 4
44.c even 2 1 inner 176.2.e.b 4
88.b odd 2 1 704.2.e.c 4
88.g even 2 1 704.2.e.c 4
132.d odd 2 1 1584.2.o.e 4
176.i even 4 2 2816.2.g.c 8
176.l odd 4 2 2816.2.g.c 8

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 7T_{3}^{2} + 4$$ acting on $$S_{2}^{\mathrm{new}}(176, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 7T^{2} + 4$$
$5$ $$(T^{2} + 3 T - 6)^{2}$$
$7$ $$T^{4}$$
$11$ $$(T^{2} + 11)^{2}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$T^{4} + 127T^{2} + 3364$$
$29$ $$T^{4}$$
$31$ $$T^{4} + 87T^{2} + 36$$
$37$ $$(T^{2} + 7 T - 62)^{2}$$
$41$ $$T^{4}$$
$43$ $$T^{4}$$
$47$ $$(T^{2} + 44)^{2}$$
$53$ $$(T - 6)^{4}$$
$59$ $$T^{4} + 343 T^{2} + 27556$$
$61$ $$T^{4}$$
$67$ $$T^{4} + 303 T^{2} + 10404$$
$71$ $$T^{4} + 151T^{2} + 3844$$
$73$ $$T^{4}$$
$79$ $$T^{4}$$
$83$ $$T^{4}$$
$89$ $$(T^{2} - 9 T - 186)^{2}$$
$97$ $$(T^{2} - 17 T - 2)^{2}$$