# Properties

 Label 176.2.a.a Level $176$ Weight $2$ Character orbit 176.a Self dual yes Analytic conductor $1.405$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [176,2,Mod(1,176)]

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, 0]))

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

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

chi := DirichletCharacter("176.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$176 = 2^{4} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 176.a (trivial)

## 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: $$1.40536707557$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 44) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - q^{3} - 3 q^{5} - 2 q^{7} - 2 q^{9}+O(q^{10})$$ q - q^3 - 3 * q^5 - 2 * q^7 - 2 * q^9 $$q - q^{3} - 3 q^{5} - 2 q^{7} - 2 q^{9} + q^{11} - 4 q^{13} + 3 q^{15} + 6 q^{17} - 8 q^{19} + 2 q^{21} + 3 q^{23} + 4 q^{25} + 5 q^{27} - 5 q^{31} - q^{33} + 6 q^{35} - q^{37} + 4 q^{39} + 10 q^{43} + 6 q^{45} - 3 q^{49} - 6 q^{51} - 6 q^{53} - 3 q^{55} + 8 q^{57} - 3 q^{59} - 4 q^{61} + 4 q^{63} + 12 q^{65} + q^{67} - 3 q^{69} - 15 q^{71} - 4 q^{73} - 4 q^{75} - 2 q^{77} - 2 q^{79} + q^{81} - 6 q^{83} - 18 q^{85} - 9 q^{89} + 8 q^{91} + 5 q^{93} + 24 q^{95} - 7 q^{97} - 2 q^{99}+O(q^{100})$$ q - q^3 - 3 * q^5 - 2 * q^7 - 2 * q^9 + q^11 - 4 * q^13 + 3 * q^15 + 6 * q^17 - 8 * q^19 + 2 * q^21 + 3 * q^23 + 4 * q^25 + 5 * q^27 - 5 * q^31 - q^33 + 6 * q^35 - q^37 + 4 * q^39 + 10 * q^43 + 6 * q^45 - 3 * q^49 - 6 * q^51 - 6 * q^53 - 3 * q^55 + 8 * q^57 - 3 * q^59 - 4 * q^61 + 4 * q^63 + 12 * q^65 + q^67 - 3 * q^69 - 15 * q^71 - 4 * q^73 - 4 * q^75 - 2 * q^77 - 2 * q^79 + q^81 - 6 * q^83 - 18 * q^85 - 9 * q^89 + 8 * q^91 + 5 * q^93 + 24 * q^95 - 7 * q^97 - 2 * q^99

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 −1.00000 0 −3.00000 0 −2.00000 0 −2.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$11$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 176.2.a.a 1
3.b odd 2 1 1584.2.a.p 1
4.b odd 2 1 44.2.a.a 1
5.b even 2 1 4400.2.a.v 1
5.c odd 4 2 4400.2.b.k 2
7.b odd 2 1 8624.2.a.w 1
8.b even 2 1 704.2.a.i 1
8.d odd 2 1 704.2.a.f 1
11.b odd 2 1 1936.2.a.c 1
12.b even 2 1 396.2.a.c 1
16.e even 4 2 2816.2.c.k 2
16.f odd 4 2 2816.2.c.e 2
20.d odd 2 1 1100.2.a.b 1
20.e even 4 2 1100.2.b.c 2
24.f even 2 1 6336.2.a.j 1
24.h odd 2 1 6336.2.a.i 1
28.d even 2 1 2156.2.a.a 1
28.f even 6 2 2156.2.i.c 2
28.g odd 6 2 2156.2.i.b 2
36.f odd 6 2 3564.2.i.j 2
36.h even 6 2 3564.2.i.a 2
44.c even 2 1 484.2.a.a 1
44.g even 10 4 484.2.e.b 4
44.h odd 10 4 484.2.e.a 4
52.b odd 2 1 7436.2.a.d 1
60.h even 2 1 9900.2.a.h 1
60.l odd 4 2 9900.2.c.g 2
88.b odd 2 1 7744.2.a.bc 1
88.g even 2 1 7744.2.a.m 1
132.d odd 2 1 4356.2.a.j 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
44.2.a.a 1 4.b odd 2 1
176.2.a.a 1 1.a even 1 1 trivial
396.2.a.c 1 12.b even 2 1
484.2.a.a 1 44.c even 2 1
484.2.e.a 4 44.h odd 10 4
484.2.e.b 4 44.g even 10 4
704.2.a.f 1 8.d odd 2 1
704.2.a.i 1 8.b even 2 1
1100.2.a.b 1 20.d odd 2 1
1100.2.b.c 2 20.e even 4 2
1584.2.a.p 1 3.b odd 2 1
1936.2.a.c 1 11.b odd 2 1
2156.2.a.a 1 28.d even 2 1
2156.2.i.b 2 28.g odd 6 2
2156.2.i.c 2 28.f even 6 2
2816.2.c.e 2 16.f odd 4 2
2816.2.c.k 2 16.e even 4 2
3564.2.i.a 2 36.h even 6 2
3564.2.i.j 2 36.f odd 6 2
4356.2.a.j 1 132.d odd 2 1
4400.2.a.v 1 5.b even 2 1
4400.2.b.k 2 5.c odd 4 2
6336.2.a.i 1 24.h odd 2 1
6336.2.a.j 1 24.f even 2 1
7436.2.a.d 1 52.b odd 2 1
7744.2.a.m 1 88.g even 2 1
7744.2.a.bc 1 88.b odd 2 1
8624.2.a.w 1 7.b odd 2 1
9900.2.a.h 1 60.h even 2 1
9900.2.c.g 2 60.l odd 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} + 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(176))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 1$$
$5$ $$T + 3$$
$7$ $$T + 2$$
$11$ $$T - 1$$
$13$ $$T + 4$$
$17$ $$T - 6$$
$19$ $$T + 8$$
$23$ $$T - 3$$
$29$ $$T$$
$31$ $$T + 5$$
$37$ $$T + 1$$
$41$ $$T$$
$43$ $$T - 10$$
$47$ $$T$$
$53$ $$T + 6$$
$59$ $$T + 3$$
$61$ $$T + 4$$
$67$ $$T - 1$$
$71$ $$T + 15$$
$73$ $$T + 4$$
$79$ $$T + 2$$
$83$ $$T + 6$$
$89$ $$T + 9$$
$97$ $$T + 7$$