# Properties

 Label 171.2.a.a Level $171$ Weight $2$ Character orbit 171.a Self dual yes Analytic conductor $1.365$ Analytic rank $0$ 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] = [171,2,Mod(1,171)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(171, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("171.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$171 = 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 171.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: $$1.36544187456$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 57) 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)

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{4} + 2 q^{5} + 3 q^{8}+O(q^{10})$$ q - q^2 - q^4 + 2 * q^5 + 3 * q^8 $$q - q^{2} - q^{4} + 2 q^{5} + 3 q^{8} - 2 q^{10} + 6 q^{13} - q^{16} + 6 q^{17} - q^{19} - 2 q^{20} - 4 q^{23} - q^{25} - 6 q^{26} - 2 q^{29} + 8 q^{31} - 5 q^{32} - 6 q^{34} - 10 q^{37} + q^{38} + 6 q^{40} + 2 q^{41} - 4 q^{43} + 4 q^{46} - 12 q^{47} - 7 q^{49} + q^{50} - 6 q^{52} + 6 q^{53} + 2 q^{58} + 12 q^{59} - 2 q^{61} - 8 q^{62} + 7 q^{64} + 12 q^{65} - 4 q^{67} - 6 q^{68} + 10 q^{73} + 10 q^{74} + q^{76} - 2 q^{80} - 2 q^{82} - 16 q^{83} + 12 q^{85} + 4 q^{86} + 2 q^{89} + 4 q^{92} + 12 q^{94} - 2 q^{95} + 10 q^{97} + 7 q^{98}+O(q^{100})$$ q - q^2 - q^4 + 2 * q^5 + 3 * q^8 - 2 * q^10 + 6 * q^13 - q^16 + 6 * q^17 - q^19 - 2 * q^20 - 4 * q^23 - q^25 - 6 * q^26 - 2 * q^29 + 8 * q^31 - 5 * q^32 - 6 * q^34 - 10 * q^37 + q^38 + 6 * q^40 + 2 * q^41 - 4 * q^43 + 4 * q^46 - 12 * q^47 - 7 * q^49 + q^50 - 6 * q^52 + 6 * q^53 + 2 * q^58 + 12 * q^59 - 2 * q^61 - 8 * q^62 + 7 * q^64 + 12 * q^65 - 4 * q^67 - 6 * q^68 + 10 * q^73 + 10 * q^74 + q^76 - 2 * q^80 - 2 * q^82 - 16 * q^83 + 12 * q^85 + 4 * q^86 + 2 * q^89 + 4 * q^92 + 12 * q^94 - 2 * q^95 + 10 * q^97 + 7 * q^98

## 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
−1.00000 0 −1.00000 2.00000 0 0 3.00000 0 −2.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$19$$ $$1$$

## 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 171.2.a.a 1
3.b odd 2 1 57.2.a.c 1
4.b odd 2 1 2736.2.a.s 1
5.b even 2 1 4275.2.a.m 1
7.b odd 2 1 8379.2.a.e 1
12.b even 2 1 912.2.a.b 1
15.d odd 2 1 1425.2.a.a 1
15.e even 4 2 1425.2.c.g 2
19.b odd 2 1 3249.2.a.g 1
21.c even 2 1 2793.2.a.i 1
24.f even 2 1 3648.2.a.bf 1
24.h odd 2 1 3648.2.a.o 1
33.d even 2 1 6897.2.a.a 1
39.d odd 2 1 9633.2.a.h 1
57.d even 2 1 1083.2.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.2.a.c 1 3.b odd 2 1
171.2.a.a 1 1.a even 1 1 trivial
912.2.a.b 1 12.b even 2 1
1083.2.a.a 1 57.d even 2 1
1425.2.a.a 1 15.d odd 2 1
1425.2.c.g 2 15.e even 4 2
2736.2.a.s 1 4.b odd 2 1
2793.2.a.i 1 21.c even 2 1
3249.2.a.g 1 19.b odd 2 1
3648.2.a.o 1 24.h odd 2 1
3648.2.a.bf 1 24.f even 2 1
4275.2.a.m 1 5.b even 2 1
6897.2.a.a 1 33.d even 2 1
8379.2.a.e 1 7.b odd 2 1
9633.2.a.h 1 39.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(171))$$:

 $$T_{2} + 1$$ T2 + 1 $$T_{5} - 2$$ T5 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T$$
$5$ $$T - 2$$
$7$ $$T$$
$11$ $$T$$
$13$ $$T - 6$$
$17$ $$T - 6$$
$19$ $$T + 1$$
$23$ $$T + 4$$
$29$ $$T + 2$$
$31$ $$T - 8$$
$37$ $$T + 10$$
$41$ $$T - 2$$
$43$ $$T + 4$$
$47$ $$T + 12$$
$53$ $$T - 6$$
$59$ $$T - 12$$
$61$ $$T + 2$$
$67$ $$T + 4$$
$71$ $$T$$
$73$ $$T - 10$$
$79$ $$T$$
$83$ $$T + 16$$
$89$ $$T - 2$$
$97$ $$T - 10$$