# Properties

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

# Learn more

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: $$2$$ 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: $$1$$ 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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{6} + 1) q^{2} + \zeta_{6} q^{4} + q^{7} + 3 q^{8}+O(q^{10})$$ q + (-z + 1) * q^2 + z * q^4 + q^7 + 3 * q^8 $$q + ( - \zeta_{6} + 1) q^{2} + \zeta_{6} q^{4} + q^{7} + 3 q^{8} + 2 q^{11} - 5 \zeta_{6} q^{13} + ( - \zeta_{6} + 1) q^{14} + ( - \zeta_{6} + 1) q^{16} + (4 \zeta_{6} - 4) q^{17} + (2 \zeta_{6} - 5) q^{19} + ( - 2 \zeta_{6} + 2) q^{22} - 4 \zeta_{6} q^{23} + 5 \zeta_{6} q^{25} - 5 q^{26} + \zeta_{6} q^{28} - 8 \zeta_{6} q^{29} - 3 q^{31} + 5 \zeta_{6} q^{32} + 4 \zeta_{6} q^{34} + 3 q^{37} + (5 \zeta_{6} - 3) q^{38} + (12 \zeta_{6} - 12) q^{41} + ( - \zeta_{6} + 1) q^{43} + 2 \zeta_{6} q^{44} - 4 q^{46} - 6 \zeta_{6} q^{47} - 6 q^{49} + 5 q^{50} + ( - 5 \zeta_{6} + 5) q^{52} + 4 \zeta_{6} q^{53} + 3 q^{56} - 8 q^{58} + ( - 10 \zeta_{6} + 10) q^{59} + 13 \zeta_{6} q^{61} + (3 \zeta_{6} - 3) q^{62} + 7 q^{64} - 11 \zeta_{6} q^{67} - 4 q^{68} + ( - 6 \zeta_{6} + 6) q^{71} + ( - 11 \zeta_{6} + 11) q^{73} + ( - 3 \zeta_{6} + 3) q^{74} + ( - 3 \zeta_{6} - 2) q^{76} + 2 q^{77} + (\zeta_{6} - 1) q^{79} + 12 \zeta_{6} q^{82} - \zeta_{6} q^{86} + 6 q^{88} - 6 \zeta_{6} q^{89} - 5 \zeta_{6} q^{91} + ( - 4 \zeta_{6} + 4) q^{92} - 6 q^{94} + (2 \zeta_{6} - 2) q^{97} + (6 \zeta_{6} - 6) q^{98} +O(q^{100})$$ q + (-z + 1) * q^2 + z * q^4 + q^7 + 3 * q^8 + 2 * q^11 - 5*z * q^13 + (-z + 1) * q^14 + (-z + 1) * q^16 + (4*z - 4) * q^17 + (2*z - 5) * q^19 + (-2*z + 2) * q^22 - 4*z * q^23 + 5*z * q^25 - 5 * q^26 + z * q^28 - 8*z * q^29 - 3 * q^31 + 5*z * q^32 + 4*z * q^34 + 3 * q^37 + (5*z - 3) * q^38 + (12*z - 12) * q^41 + (-z + 1) * q^43 + 2*z * q^44 - 4 * q^46 - 6*z * q^47 - 6 * q^49 + 5 * q^50 + (-5*z + 5) * q^52 + 4*z * q^53 + 3 * q^56 - 8 * q^58 + (-10*z + 10) * q^59 + 13*z * q^61 + (3*z - 3) * q^62 + 7 * q^64 - 11*z * q^67 - 4 * q^68 + (-6*z + 6) * q^71 + (-11*z + 11) * q^73 + (-3*z + 3) * q^74 + (-3*z - 2) * q^76 + 2 * q^77 + (z - 1) * q^79 + 12*z * q^82 - z * q^86 + 6 * q^88 - 6*z * q^89 - 5*z * q^91 + (-4*z + 4) * q^92 - 6 * q^94 + (2*z - 2) * q^97 + (6*z - 6) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} + q^{4} + 2 q^{7} + 6 q^{8}+O(q^{10})$$ 2 * q + q^2 + q^4 + 2 * q^7 + 6 * q^8 $$2 q + q^{2} + q^{4} + 2 q^{7} + 6 q^{8} + 4 q^{11} - 5 q^{13} + q^{14} + q^{16} - 4 q^{17} - 8 q^{19} + 2 q^{22} - 4 q^{23} + 5 q^{25} - 10 q^{26} + q^{28} - 8 q^{29} - 6 q^{31} + 5 q^{32} + 4 q^{34} + 6 q^{37} - q^{38} - 12 q^{41} + q^{43} + 2 q^{44} - 8 q^{46} - 6 q^{47} - 12 q^{49} + 10 q^{50} + 5 q^{52} + 4 q^{53} + 6 q^{56} - 16 q^{58} + 10 q^{59} + 13 q^{61} - 3 q^{62} + 14 q^{64} - 11 q^{67} - 8 q^{68} + 6 q^{71} + 11 q^{73} + 3 q^{74} - 7 q^{76} + 4 q^{77} - q^{79} + 12 q^{82} - q^{86} + 12 q^{88} - 6 q^{89} - 5 q^{91} + 4 q^{92} - 12 q^{94} - 2 q^{97} - 6 q^{98}+O(q^{100})$$ 2 * q + q^2 + q^4 + 2 * q^7 + 6 * q^8 + 4 * q^11 - 5 * q^13 + q^14 + q^16 - 4 * q^17 - 8 * q^19 + 2 * q^22 - 4 * q^23 + 5 * q^25 - 10 * q^26 + q^28 - 8 * q^29 - 6 * q^31 + 5 * q^32 + 4 * q^34 + 6 * q^37 - q^38 - 12 * q^41 + q^43 + 2 * q^44 - 8 * q^46 - 6 * q^47 - 12 * q^49 + 10 * q^50 + 5 * q^52 + 4 * q^53 + 6 * q^56 - 16 * q^58 + 10 * q^59 + 13 * q^61 - 3 * q^62 + 14 * q^64 - 11 * q^67 - 8 * q^68 + 6 * q^71 + 11 * q^73 + 3 * q^74 - 7 * q^76 + 4 * q^77 - q^79 + 12 * q^82 - q^86 + 12 * q^88 - 6 * q^89 - 5 * q^91 + 4 * q^92 - 12 * q^94 - 2 * q^97 - 6 * q^98

## Character values

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

 $$n$$ $$20$$ $$154$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$

## 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
 0.5 − 0.866025i 0.5 + 0.866025i
0.500000 + 0.866025i 0 0.500000 0.866025i 0 0 1.00000 3.00000 0 0
163.1 0.500000 0.866025i 0 0.500000 + 0.866025i 0 0 1.00000 3.00000 0 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
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.a 2
3.b odd 2 1 57.2.e.a 2
4.b odd 2 1 2736.2.s.j 2
12.b even 2 1 912.2.q.a 2
19.c even 3 1 inner 171.2.f.a 2
19.c even 3 1 3249.2.a.c 1
19.d odd 6 1 3249.2.a.f 1
57.f even 6 1 1083.2.a.b 1
57.h odd 6 1 57.2.e.a 2
57.h odd 6 1 1083.2.a.c 1
76.g odd 6 1 2736.2.s.j 2
228.m even 6 1 912.2.q.a 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(171, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$(T - 1)^{2}$$
$11$ $$(T - 2)^{2}$$
$13$ $$T^{2} + 5T + 25$$
$17$ $$T^{2} + 4T + 16$$
$19$ $$T^{2} + 8T + 19$$
$23$ $$T^{2} + 4T + 16$$
$29$ $$T^{2} + 8T + 64$$
$31$ $$(T + 3)^{2}$$
$37$ $$(T - 3)^{2}$$
$41$ $$T^{2} + 12T + 144$$
$43$ $$T^{2} - T + 1$$
$47$ $$T^{2} + 6T + 36$$
$53$ $$T^{2} - 4T + 16$$
$59$ $$T^{2} - 10T + 100$$
$61$ $$T^{2} - 13T + 169$$
$67$ $$T^{2} + 11T + 121$$
$71$ $$T^{2} - 6T + 36$$
$73$ $$T^{2} - 11T + 121$$
$79$ $$T^{2} + T + 1$$
$83$ $$T^{2}$$
$89$ $$T^{2} + 6T + 36$$
$97$ $$T^{2} + 2T + 4$$
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