# Properties

 Label 171.3.c.c Level $171$ Weight $3$ Character orbit 171.c Analytic conductor $4.659$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [171,3,Mod(37,171)]

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, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("171.37");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$171 = 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 171.c (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: $$4.65941252056$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 57) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$\beta = \sqrt{-3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{2} + q^{4} - 4 q^{5} - 10 q^{7} - 5 \beta q^{8} +O(q^{10})$$ q - b * q^2 + q^4 - 4 * q^5 - 10 * q^7 - 5*b * q^8 $$q - \beta q^{2} + q^{4} - 4 q^{5} - 10 q^{7} - 5 \beta q^{8} + 4 \beta q^{10} - 10 q^{11} - 14 \beta q^{13} + 10 \beta q^{14} - 11 q^{16} - 10 q^{17} + 19 q^{19} - 4 q^{20} + 10 \beta q^{22} + 20 q^{23} - 9 q^{25} - 42 q^{26} - 10 q^{28} + 20 \beta q^{29} + 10 \beta q^{31} - 9 \beta q^{32} + 10 \beta q^{34} + 40 q^{35} - 6 \beta q^{37} - 19 \beta q^{38} + 20 \beta q^{40} - 20 \beta q^{41} - 10 q^{43} - 10 q^{44} - 20 \beta q^{46} + 80 q^{47} + 51 q^{49} + 9 \beta q^{50} - 14 \beta q^{52} - 24 \beta q^{53} + 40 q^{55} + 50 \beta q^{56} + 60 q^{58} - 20 \beta q^{59} - 10 q^{61} + 30 q^{62} - 71 q^{64} + 56 \beta q^{65} - 44 \beta q^{67} - 10 q^{68} - 40 \beta q^{70} - 60 \beta q^{71} - 10 q^{73} - 18 q^{74} + 19 q^{76} + 100 q^{77} - 10 \beta q^{79} + 44 q^{80} - 60 q^{82} - 70 q^{83} + 40 q^{85} + 10 \beta q^{86} + 50 \beta q^{88} + 60 \beta q^{89} + 140 \beta q^{91} + 20 q^{92} - 80 \beta q^{94} - 76 q^{95} + 44 \beta q^{97} - 51 \beta q^{98} +O(q^{100})$$ q - b * q^2 + q^4 - 4 * q^5 - 10 * q^7 - 5*b * q^8 + 4*b * q^10 - 10 * q^11 - 14*b * q^13 + 10*b * q^14 - 11 * q^16 - 10 * q^17 + 19 * q^19 - 4 * q^20 + 10*b * q^22 + 20 * q^23 - 9 * q^25 - 42 * q^26 - 10 * q^28 + 20*b * q^29 + 10*b * q^31 - 9*b * q^32 + 10*b * q^34 + 40 * q^35 - 6*b * q^37 - 19*b * q^38 + 20*b * q^40 - 20*b * q^41 - 10 * q^43 - 10 * q^44 - 20*b * q^46 + 80 * q^47 + 51 * q^49 + 9*b * q^50 - 14*b * q^52 - 24*b * q^53 + 40 * q^55 + 50*b * q^56 + 60 * q^58 - 20*b * q^59 - 10 * q^61 + 30 * q^62 - 71 * q^64 + 56*b * q^65 - 44*b * q^67 - 10 * q^68 - 40*b * q^70 - 60*b * q^71 - 10 * q^73 - 18 * q^74 + 19 * q^76 + 100 * q^77 - 10*b * q^79 + 44 * q^80 - 60 * q^82 - 70 * q^83 + 40 * q^85 + 10*b * q^86 + 50*b * q^88 + 60*b * q^89 + 140*b * q^91 + 20 * q^92 - 80*b * q^94 - 76 * q^95 + 44*b * q^97 - 51*b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{4} - 8 q^{5} - 20 q^{7}+O(q^{10})$$ 2 * q + 2 * q^4 - 8 * q^5 - 20 * q^7 $$2 q + 2 q^{4} - 8 q^{5} - 20 q^{7} - 20 q^{11} - 22 q^{16} - 20 q^{17} + 38 q^{19} - 8 q^{20} + 40 q^{23} - 18 q^{25} - 84 q^{26} - 20 q^{28} + 80 q^{35} - 20 q^{43} - 20 q^{44} + 160 q^{47} + 102 q^{49} + 80 q^{55} + 120 q^{58} - 20 q^{61} + 60 q^{62} - 142 q^{64} - 20 q^{68} - 20 q^{73} - 36 q^{74} + 38 q^{76} + 200 q^{77} + 88 q^{80} - 120 q^{82} - 140 q^{83} + 80 q^{85} + 40 q^{92} - 152 q^{95}+O(q^{100})$$ 2 * q + 2 * q^4 - 8 * q^5 - 20 * q^7 - 20 * q^11 - 22 * q^16 - 20 * q^17 + 38 * q^19 - 8 * q^20 + 40 * q^23 - 18 * q^25 - 84 * q^26 - 20 * q^28 + 80 * q^35 - 20 * q^43 - 20 * q^44 + 160 * q^47 + 102 * q^49 + 80 * q^55 + 120 * q^58 - 20 * q^61 + 60 * q^62 - 142 * q^64 - 20 * q^68 - 20 * q^73 - 36 * q^74 + 38 * q^76 + 200 * q^77 + 88 * q^80 - 120 * q^82 - 140 * q^83 + 80 * q^85 + 40 * q^92 - 152 * q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/171\mathbb{Z}\right)^\times$$.

 $$n$$ $$20$$ $$154$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1
 0.5 + 0.866025i 0.5 − 0.866025i
1.73205i 0 1.00000 −4.00000 0 −10.0000 8.66025i 0 6.92820i
37.2 1.73205i 0 1.00000 −4.00000 0 −10.0000 8.66025i 0 6.92820i
 $$n$$: 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
19.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 171.3.c.c 2
3.b odd 2 1 57.3.c.a 2
4.b odd 2 1 2736.3.o.e 2
12.b even 2 1 912.3.o.a 2
19.b odd 2 1 inner 171.3.c.c 2
57.d even 2 1 57.3.c.a 2
76.d even 2 1 2736.3.o.e 2
228.b odd 2 1 912.3.o.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.3.c.a 2 3.b odd 2 1
57.3.c.a 2 57.d even 2 1
171.3.c.c 2 1.a even 1 1 trivial
171.3.c.c 2 19.b odd 2 1 inner
912.3.o.a 2 12.b even 2 1
912.3.o.a 2 228.b odd 2 1
2736.3.o.e 2 4.b odd 2 1
2736.3.o.e 2 76.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(171, [\chi])$$:

 $$T_{2}^{2} + 3$$ T2^2 + 3 $$T_{5} + 4$$ T5 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 3$$
$3$ $$T^{2}$$
$5$ $$(T + 4)^{2}$$
$7$ $$(T + 10)^{2}$$
$11$ $$(T + 10)^{2}$$
$13$ $$T^{2} + 588$$
$17$ $$(T + 10)^{2}$$
$19$ $$(T - 19)^{2}$$
$23$ $$(T - 20)^{2}$$
$29$ $$T^{2} + 1200$$
$31$ $$T^{2} + 300$$
$37$ $$T^{2} + 108$$
$41$ $$T^{2} + 1200$$
$43$ $$(T + 10)^{2}$$
$47$ $$(T - 80)^{2}$$
$53$ $$T^{2} + 1728$$
$59$ $$T^{2} + 1200$$
$61$ $$(T + 10)^{2}$$
$67$ $$T^{2} + 5808$$
$71$ $$T^{2} + 10800$$
$73$ $$(T + 10)^{2}$$
$79$ $$T^{2} + 300$$
$83$ $$(T + 70)^{2}$$
$89$ $$T^{2} + 10800$$
$97$ $$T^{2} + 5808$$