# Properties

 Label 171.2.f.b Level $171$ Weight $2$ Character orbit 171.f Analytic conductor $1.365$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [171,2,Mod(64,171)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(171, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2]))

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

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

chi := DirichletCharacter("171.64");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$171 = 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 171.f (of order $$3$$, degree $$2$$, 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: $$1.36544187456$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.954288.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27$$ x^6 - x^5 - 2*x^4 + 3*x^3 - 6*x^2 - 9*x + 27 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 57) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{5} + \beta_{4}) q^{2} + (2 \beta_{3} - \beta_{2} - \beta_1 - 2) q^{4} + (\beta_{3} - \beta_1) q^{5} + \beta_{2} q^{7} + ( - \beta_{4} + \beta_{2} + 1) q^{8}+O(q^{10})$$ q + (-b5 + b4) * q^2 + (2*b3 - b2 - b1 - 2) * q^4 + (b3 - b1) * q^5 + b2 * q^7 + (-b4 + b2 + 1) * q^8 $$q + ( - \beta_{5} + \beta_{4}) q^{2} + (2 \beta_{3} - \beta_{2} - \beta_1 - 2) q^{4} + (\beta_{3} - \beta_1) q^{5} + \beta_{2} q^{7} + ( - \beta_{4} + \beta_{2} + 1) q^{8} + ( - 2 \beta_{5} - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{10} + ( - 2 \beta_{4} - \beta_{2} - 1) q^{11} + (2 \beta_{5} - \beta_{3} + 1) q^{13} + ( - \beta_{5} + \beta_{4} - \beta_{3} + \beta_1) q^{14} + ( - 2 \beta_{5} + 2 \beta_{4} - \beta_{3}) q^{16} + ( - 2 \beta_{5} - \beta_{2}) q^{19} + (2 \beta_{4} - \beta_{2} - 7) q^{20} + (2 \beta_{5} - 2 \beta_{4} - 7 \beta_{3} + \beta_1) q^{22} + (2 \beta_{5} - 5 \beta_{3} - \beta_{2} - \beta_1 + 5) q^{23} + (2 \beta_{5} + \beta_{3} - 1) q^{25} + (\beta_{4} + 2 \beta_{2} + 8) q^{26} + (2 \beta_{5} + 5 \beta_{3} - 5) q^{28} + ( - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 2) q^{29} + ( - 2 \beta_{4} - \beta_{2} + 4) q^{31} + ( - \beta_{5} + 6 \beta_{3} - 6) q^{32} + (2 \beta_{5} - 2 \beta_{4} + 5 \beta_{3} + \beta_1) q^{35} - q^{37} + (\beta_{5} - \beta_{4} + \beta_{3} - 2 \beta_{2} - \beta_1 - 8) q^{38} + (4 \beta_{5} - 4 \beta_{4} + 7 \beta_{3} - \beta_1) q^{40} + ( - 2 \beta_{3} + 2 \beta_1) q^{41} + ( - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - \beta_1) q^{43} + (4 \beta_{5} - 5 \beta_{3} - \beta_{2} - \beta_1 + 5) q^{44} + (4 \beta_{4} + 3 \beta_{2} + 9) q^{46} + (6 \beta_{3} - 6) q^{47} + ( - 2 \beta_{4} - 2 \beta_{2} - 2) q^{49} + ( - \beta_{4} + 2 \beta_{2} + 8) q^{50} + ( - 6 \beta_{5} + 6 \beta_{4} + \beta_1) q^{52} + (3 \beta_{3} - 3 \beta_{2} - 3 \beta_1 - 3) q^{53} + (2 \beta_{5} - 2 \beta_{4} - 4 \beta_{3} - 2 \beta_1) q^{55} + ( - 3 \beta_{4} + 6) q^{56} + (4 \beta_{4} - 2 \beta_{2} - 2) q^{58} + (2 \beta_{5} - 2 \beta_{4} - \beta_{3} + \beta_1) q^{59} + (2 \beta_{5} + 3 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 3) q^{61} + ( - 3 \beta_{5} + 3 \beta_{4} - 7 \beta_{3} + \beta_1) q^{62} + ( - 2 \beta_{4} - \beta_{2} - 6) q^{64} + (4 \beta_{4} - \beta_{2} - 1) q^{65} + ( - 4 \beta_{5} + 4 \beta_{3} + \beta_{2} + \beta_1 - 4) q^{67} + ( - 4 \beta_{5} - 7 \beta_{3} + \beta_{2} + \beta_1 + 7) q^{70} + ( - 4 \beta_{5} + 4 \beta_{4} - 4 \beta_{3} - 2 \beta_1) q^{71} + ( - 7 \beta_{3} + 2 \beta_1) q^{73} + (\beta_{5} - \beta_{4}) q^{74} + (4 \beta_{5} - 6 \beta_{4} - 3 \beta_{3} - 2 \beta_1 + 5) q^{76} + (3 \beta_{2} - 3) q^{77} + (2 \beta_{5} - 2 \beta_{4} - 4 \beta_{3} - \beta_1) q^{79} + ( - 4 \beta_{5} - 3 \beta_{3} + 3 \beta_{2} + 3 \beta_1 + 3) q^{80} + (4 \beta_{5} + 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 2) q^{82} - 4 \beta_{4} q^{83} + ( - 3 \beta_{5} + 7 \beta_{3} - \beta_{2} - \beta_1 - 7) q^{86} + (3 \beta_{2} + 3) q^{88} + (5 \beta_{3} + \beta_{2} + \beta_1 - 5) q^{89} + (2 \beta_{5} + 2 \beta_{3} - \beta_{2} - \beta_1 - 2) q^{91} + ( - 8 \beta_{5} + 8 \beta_{4} + 3 \beta_{3} - 3 \beta_1) q^{92} - 6 \beta_{4} q^{94} + ( - 2 \beta_{5} - 2 \beta_{4} - 5 \beta_{3} + 2 \beta_{2} - \beta_1 + 2) q^{95} + ( - 6 \beta_{5} + 6 \beta_{4} + 2 \beta_{3} + 2 \beta_1) q^{97} + (4 \beta_{5} - 4 \beta_{4} - 6 \beta_{3}) q^{98}+O(q^{100})$$ q + (-b5 + b4) * q^2 + (2*b3 - b2 - b1 - 2) * q^4 + (b3 - b1) * q^5 + b2 * q^7 + (-b4 + b2 + 1) * q^8 + (-2*b5 - b3 + b2 + b1 + 1) * q^10 + (-2*b4 - b2 - 1) * q^11 + (2*b5 - b3 + 1) * q^13 + (-b5 + b4 - b3 + b1) * q^14 + (-2*b5 + 2*b4 - b3) * q^16 + (-2*b5 - b2) * q^19 + (2*b4 - b2 - 7) * q^20 + (2*b5 - 2*b4 - 7*b3 + b1) * q^22 + (2*b5 - 5*b3 - b2 - b1 + 5) * q^23 + (2*b5 + b3 - 1) * q^25 + (b4 + 2*b2 + 8) * q^26 + (2*b5 + 5*b3 - 5) * q^28 + (-2*b3 + 2*b2 + 2*b1 + 2) * q^29 + (-2*b4 - b2 + 4) * q^31 + (-b5 + 6*b3 - 6) * q^32 + (2*b5 - 2*b4 + 5*b3 + b1) * q^35 - q^37 + (b5 - b4 + b3 - 2*b2 - b1 - 8) * q^38 + (4*b5 - 4*b4 + 7*b3 - b1) * q^40 + (-2*b3 + 2*b1) * q^41 + (-2*b5 + 2*b4 + 2*b3 - b1) * q^43 + (4*b5 - 5*b3 - b2 - b1 + 5) * q^44 + (4*b4 + 3*b2 + 9) * q^46 + (6*b3 - 6) * q^47 + (-2*b4 - 2*b2 - 2) * q^49 + (-b4 + 2*b2 + 8) * q^50 + (-6*b5 + 6*b4 + b1) * q^52 + (3*b3 - 3*b2 - 3*b1 - 3) * q^53 + (2*b5 - 2*b4 - 4*b3 - 2*b1) * q^55 + (-3*b4 + 6) * q^56 + (4*b4 - 2*b2 - 2) * q^58 + (2*b5 - 2*b4 - b3 + b1) * q^59 + (2*b5 + 3*b3 + 2*b2 + 2*b1 - 3) * q^61 + (-3*b5 + 3*b4 - 7*b3 + b1) * q^62 + (-2*b4 - b2 - 6) * q^64 + (4*b4 - b2 - 1) * q^65 + (-4*b5 + 4*b3 + b2 + b1 - 4) * q^67 + (-4*b5 - 7*b3 + b2 + b1 + 7) * q^70 + (-4*b5 + 4*b4 - 4*b3 - 2*b1) * q^71 + (-7*b3 + 2*b1) * q^73 + (b5 - b4) * q^74 + (4*b5 - 6*b4 - 3*b3 - 2*b1 + 5) * q^76 + (3*b2 - 3) * q^77 + (2*b5 - 2*b4 - 4*b3 - b1) * q^79 + (-4*b5 - 3*b3 + 3*b2 + 3*b1 + 3) * q^80 + (4*b5 + 2*b3 - 2*b2 - 2*b1 - 2) * q^82 - 4*b4 * q^83 + (-3*b5 + 7*b3 - b2 - b1 - 7) * q^86 + (3*b2 + 3) * q^88 + (5*b3 + b2 + b1 - 5) * q^89 + (2*b5 + 2*b3 - b2 - b1 - 2) * q^91 + (-8*b5 + 8*b4 + 3*b3 - 3*b1) * q^92 - 6*b4 * q^94 + (-2*b5 - 2*b4 - 5*b3 + 2*b2 - b1 + 2) * q^95 + (-6*b5 + 6*b4 + 2*b3 + 2*b1) * q^97 + (4*b5 - 4*b4 - 6*b3) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - q^{2} - 5 q^{4} + 2 q^{5} - 2 q^{7} + 6 q^{8}+O(q^{10})$$ 6 * q - q^2 - 5 * q^4 + 2 * q^5 - 2 * q^7 + 6 * q^8 $$6 q - q^{2} - 5 q^{4} + 2 q^{5} - 2 q^{7} + 6 q^{8} + 4 q^{10} + q^{13} - 3 q^{14} - 5 q^{16} + 4 q^{19} - 44 q^{20} - 18 q^{22} + 14 q^{23} - 5 q^{25} + 42 q^{26} - 17 q^{28} + 4 q^{29} + 30 q^{31} - 17 q^{32} + 18 q^{35} - 6 q^{37} - 41 q^{38} + 24 q^{40} - 4 q^{41} + 3 q^{43} + 12 q^{44} + 40 q^{46} - 18 q^{47} - 4 q^{49} + 46 q^{50} - 5 q^{52} - 6 q^{53} - 12 q^{55} + 42 q^{56} - 16 q^{58} - 13 q^{61} - 23 q^{62} - 30 q^{64} - 12 q^{65} - 9 q^{67} + 24 q^{70} - 18 q^{71} - 19 q^{73} + q^{74} + 27 q^{76} - 24 q^{77} - 11 q^{79} + 10 q^{80} - 8 q^{82} + 8 q^{83} - 17 q^{86} + 12 q^{88} - 16 q^{89} - 7 q^{91} - 2 q^{92} + 12 q^{94} - 2 q^{95} + 2 q^{97} - 14 q^{98}+O(q^{100})$$ 6 * q - q^2 - 5 * q^4 + 2 * q^5 - 2 * q^7 + 6 * q^8 + 4 * q^10 + q^13 - 3 * q^14 - 5 * q^16 + 4 * q^19 - 44 * q^20 - 18 * q^22 + 14 * q^23 - 5 * q^25 + 42 * q^26 - 17 * q^28 + 4 * q^29 + 30 * q^31 - 17 * q^32 + 18 * q^35 - 6 * q^37 - 41 * q^38 + 24 * q^40 - 4 * q^41 + 3 * q^43 + 12 * q^44 + 40 * q^46 - 18 * q^47 - 4 * q^49 + 46 * q^50 - 5 * q^52 - 6 * q^53 - 12 * q^55 + 42 * q^56 - 16 * q^58 - 13 * q^61 - 23 * q^62 - 30 * q^64 - 12 * q^65 - 9 * q^67 + 24 * q^70 - 18 * q^71 - 19 * q^73 + q^74 + 27 * q^76 - 24 * q^77 - 11 * q^79 + 10 * q^80 - 8 * q^82 + 8 * q^83 - 17 * q^86 + 12 * q^88 - 16 * q^89 - 7 * q^91 - 2 * q^92 + 12 * q^94 - 2 * q^95 + 2 * q^97 - 14 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{5} + 4\nu^{4} - \nu^{3} + 9\nu^{2} - 21\nu - 9 ) / 27$$ (-v^5 + 4*v^4 - v^3 + 9*v^2 - 21*v - 9) / 27 $$\beta_{2}$$ $$=$$ $$( -\nu^{5} + 4\nu^{4} - \nu^{3} - 18\nu^{2} + 33\nu - 9 ) / 27$$ (-v^5 + 4*v^4 - v^3 - 18*v^2 + 33*v - 9) / 27 $$\beta_{3}$$ $$=$$ $$( -2\nu^{5} - \nu^{4} - 2\nu^{3} + 12\nu + 36 ) / 27$$ (-2*v^5 - v^4 - 2*v^3 + 12*v + 36) / 27 $$\beta_{4}$$ $$=$$ $$( -2\nu^{5} - \nu^{4} + 7\nu^{3} + 9\nu^{2} + 12\nu + 9 ) / 27$$ (-2*v^5 - v^4 + 7*v^3 + 9*v^2 + 12*v + 9) / 27 $$\beta_{5}$$ $$=$$ $$( 4\nu^{5} + 2\nu^{4} - 5\nu^{3} + 18\nu^{2} + 3\nu - 72 ) / 27$$ (4*v^5 + 2*v^4 - 5*v^3 + 18*v^2 + 3*v - 72) / 27
 $$\nu$$ $$=$$ $$( \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 3$$ (b5 + b4 + b3 + b2 - b1 + 1) / 3 $$\nu^{2}$$ $$=$$ $$( 2\beta_{5} + 2\beta_{4} + 2\beta_{3} - \beta_{2} + \beta _1 + 2 ) / 3$$ (2*b5 + 2*b4 + 2*b3 - b2 + b1 + 2) / 3 $$\nu^{3}$$ $$=$$ $$( -2\beta_{5} + 7\beta_{4} - 11\beta_{3} + \beta_{2} - \beta _1 + 7 ) / 3$$ (-2*b5 + 7*b4 - 11*b3 + b2 - b1 + 7) / 3 $$\nu^{4}$$ $$=$$ $$( 2\beta_{5} + 2\beta_{4} - 7\beta_{3} + 8\beta_{2} + 10\beta _1 + 20 ) / 3$$ (2*b5 + 2*b4 - 7*b3 + 8*b2 + 10*b1 + 20) / 3 $$\nu^{5}$$ $$=$$ $$( 7\beta_{5} - 2\beta_{4} - 20\beta_{3} + \beta_{2} - 10\beta _1 + 43 ) / 3$$ (7*b5 - 2*b4 - 20*b3 + b2 - 10*b1 + 43) / 3

## Character values

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

 $$n$$ $$20$$ $$154$$ $$\chi(n)$$ $$1$$ $$-1 + \beta_{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
64.1
 0.403374 + 1.68443i −1.62241 − 0.606458i 1.71903 − 0.211943i 0.403374 − 1.68443i −1.62241 + 0.606458i 1.71903 + 0.211943i
−1.25707 2.17731i 0 −2.16044 + 3.74200i 1.66044 + 2.87597i 0 2.32088 5.83502 0 4.17458 7.23058i
64.2 −0.285997 0.495361i 0 0.836412 1.44871i −1.33641 2.31473i 0 −3.67282 −2.10083 0 −0.764419 + 1.32401i
64.3 1.04307 + 1.80664i 0 −1.17597 + 2.03684i 0.675970 + 1.17081i 0 0.351939 −0.734191 0 −1.41016 + 2.44247i
163.1 −1.25707 + 2.17731i 0 −2.16044 3.74200i 1.66044 2.87597i 0 2.32088 5.83502 0 4.17458 + 7.23058i
163.2 −0.285997 + 0.495361i 0 0.836412 + 1.44871i −1.33641 + 2.31473i 0 −3.67282 −2.10083 0 −0.764419 1.32401i
163.3 1.04307 1.80664i 0 −1.17597 2.03684i 0.675970 1.17081i 0 0.351939 −0.734191 0 −1.41016 2.44247i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 64.3 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.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 171.2.f.b 6
3.b odd 2 1 57.2.e.b 6
4.b odd 2 1 2736.2.s.z 6
12.b even 2 1 912.2.q.l 6
19.c even 3 1 inner 171.2.f.b 6
19.c even 3 1 3249.2.a.y 3
19.d odd 6 1 3249.2.a.t 3
57.f even 6 1 1083.2.a.o 3
57.h odd 6 1 57.2.e.b 6
57.h odd 6 1 1083.2.a.l 3
76.g odd 6 1 2736.2.s.z 6
228.m even 6 1 912.2.q.l 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.2.e.b 6 3.b odd 2 1
57.2.e.b 6 57.h odd 6 1
171.2.f.b 6 1.a even 1 1 trivial
171.2.f.b 6 19.c even 3 1 inner
912.2.q.l 6 12.b even 2 1
912.2.q.l 6 228.m even 6 1
1083.2.a.l 3 57.h odd 6 1
1083.2.a.o 3 57.f even 6 1
2736.2.s.z 6 4.b odd 2 1
2736.2.s.z 6 76.g odd 6 1
3249.2.a.t 3 19.d odd 6 1
3249.2.a.y 3 19.c even 3 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} + T_{2}^{5} + 6T_{2}^{4} + T_{2}^{3} + 28T_{2}^{2} + 15T_{2} + 9$$ acting on $$S_{2}^{\mathrm{new}}(171, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + T^{5} + 6 T^{4} + T^{3} + 28 T^{2} + \cdots + 9$$
$3$ $$T^{6}$$
$5$ $$T^{6} - 2 T^{5} + 12 T^{4} - 8 T^{3} + \cdots + 144$$
$7$ $$(T^{3} + T^{2} - 9 T + 3)^{2}$$
$11$ $$(T^{3} - 24 T + 36)^{2}$$
$13$ $$T^{6} - T^{5} + 22 T^{4} + 27 T^{3} + \cdots + 9$$
$17$ $$T^{6}$$
$19$ $$T^{6} - 4 T^{5} + 17 T^{4} + \cdots + 6859$$
$23$ $$T^{6} - 14 T^{5} + 168 T^{4} + \cdots + 24336$$
$29$ $$T^{6} - 4 T^{5} + 48 T^{4} + \cdots + 9216$$
$31$ $$(T^{3} - 15 T^{2} + 51 T + 31)^{2}$$
$37$ $$(T + 1)^{6}$$
$41$ $$T^{6} + 4 T^{5} + 48 T^{4} + \cdots + 9216$$
$43$ $$T^{6} - 3 T^{5} + 30 T^{4} + 89 T^{3} + \cdots + 169$$
$47$ $$(T^{2} + 6 T + 36)^{3}$$
$53$ $$T^{6} + 6 T^{5} + 108 T^{4} + \cdots + 104976$$
$59$ $$T^{6} + 24 T^{4} - 72 T^{3} + \cdots + 1296$$
$61$ $$T^{6} + 13 T^{5} + 158 T^{4} + \cdots + 5329$$
$67$ $$T^{6} + 9 T^{5} + 162 T^{4} + \cdots + 292681$$
$71$ $$T^{6} + 18 T^{5} + 312 T^{4} + \cdots + 419904$$
$73$ $$T^{6} + 19 T^{5} + 278 T^{4} + \cdots + 961$$
$79$ $$T^{6} + 11 T^{5} + 118 T^{4} + \cdots + 29241$$
$83$ $$(T^{3} - 4 T^{2} - 80 T + 192)^{2}$$
$89$ $$T^{6} + 16 T^{5} + 180 T^{4} + \cdots + 11664$$
$97$ $$T^{6} - 2 T^{5} + 272 T^{4} + \cdots + 2096704$$