# Properties

 Label 176.9.h.a Level $176$ Weight $9$ Character orbit 176.h Self dual yes Analytic conductor $71.699$ Analytic rank $0$ Dimension $1$ CM discriminant -11 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [176,9,Mod(65,176)]

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([0, 0, 1]))

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("176.65");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$176 = 2^{4} \cdot 11$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 176.h (of order $$2$$, degree $$1$$, 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: yes Analytic conductor: $$71.6986353708$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 11) 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)

 $$f(q)$$ $$=$$ $$q + 113 q^{3} + 1151 q^{5} + 6208 q^{9}+O(q^{10})$$ q + 113 * q^3 + 1151 * q^5 + 6208 * q^9 $$q + 113 q^{3} + 1151 q^{5} + 6208 q^{9} - 14641 q^{11} + 130063 q^{15} + 531793 q^{23} + 934176 q^{25} - 39889 q^{27} + 1541233 q^{31} - 1654433 q^{33} + 716447 q^{37} + 7145408 q^{45} + 6080638 q^{47} + 5764801 q^{49} - 15265438 q^{53} - 16851791 q^{55} + 4101553 q^{59} - 19806767 q^{67} + 60092609 q^{69} - 7043087 q^{71} + 105561888 q^{75} - 45238145 q^{81} - 84100993 q^{89} + 174159329 q^{93} - 81155713 q^{97} - 90891328 q^{99}+O(q^{100})$$ q + 113 * q^3 + 1151 * q^5 + 6208 * q^9 - 14641 * q^11 + 130063 * q^15 + 531793 * q^23 + 934176 * q^25 - 39889 * q^27 + 1541233 * q^31 - 1654433 * q^33 + 716447 * q^37 + 7145408 * q^45 + 6080638 * q^47 + 5764801 * q^49 - 15265438 * q^53 - 16851791 * q^55 + 4101553 * q^59 - 19806767 * q^67 + 60092609 * q^69 - 7043087 * q^71 + 105561888 * q^75 - 45238145 * q^81 - 84100993 * q^89 + 174159329 * q^93 - 81155713 * q^97 - 90891328 * q^99

## Character values

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

 $$n$$ $$111$$ $$133$$ $$145$$ $$\chi(n)$$ $$0$$ $$0$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
65.1
 0
0 113.000 0 1151.00 0 0 0 6208.00 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 $$\Q(\sqrt{-11})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 176.9.h.a 1
4.b odd 2 1 11.9.b.a 1
11.b odd 2 1 CM 176.9.h.a 1
12.b even 2 1 99.9.c.a 1
44.c even 2 1 11.9.b.a 1
132.d odd 2 1 99.9.c.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.9.b.a 1 4.b odd 2 1
11.9.b.a 1 44.c even 2 1
99.9.c.a 1 12.b even 2 1
99.9.c.a 1 132.d odd 2 1
176.9.h.a 1 1.a even 1 1 trivial
176.9.h.a 1 11.b odd 2 1 CM

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} - 113$$ acting on $$S_{9}^{\mathrm{new}}(176, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 113$$
$5$ $$T - 1151$$
$7$ $$T$$
$11$ $$T + 14641$$
$13$ $$T$$
$17$ $$T$$
$19$ $$T$$
$23$ $$T - 531793$$
$29$ $$T$$
$31$ $$T - 1541233$$
$37$ $$T - 716447$$
$41$ $$T$$
$43$ $$T$$
$47$ $$T - 6080638$$
$53$ $$T + 15265438$$
$59$ $$T - 4101553$$
$61$ $$T$$
$67$ $$T + 19806767$$
$71$ $$T + 7043087$$
$73$ $$T$$
$79$ $$T$$
$83$ $$T$$
$89$ $$T + 84100993$$
$97$ $$T + 81155713$$