# Properties

 Label 171.1.o.a Level $171$ Weight $1$ Character orbit 171.o Analytic conductor $0.085$ Analytic rank $0$ Dimension $4$ Projective image $A_{4}$ CM/RM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [171,1,Mod(94,171)]

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([2, 3]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("171.94");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$171 = 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 171.o (of order $$6$$, 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: $$0.0853401171602$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$A_{4}$$ Projective field: Galois closure of 4.0.29241.1 Artin image: $\SL(2,3):C_2$ Artin field: Galois closure of $$\mathbb{Q}[x]/(x^{16} - \cdots)$$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + \zeta_{12}^{5} q^{2} + \zeta_{12}^{3} q^{3} + \zeta_{12}^{4} q^{5} - \zeta_{12}^{2} q^{6} - \zeta_{12}^{2} q^{7} - \zeta_{12}^{3} q^{8} - q^{9} +O(q^{10})$$ q + z^5 * q^2 + z^3 * q^3 + z^4 * q^5 - z^2 * q^6 - z^2 * q^7 - z^3 * q^8 - q^9 $$q + \zeta_{12}^{5} q^{2} + \zeta_{12}^{3} q^{3} + \zeta_{12}^{4} q^{5} - \zeta_{12}^{2} q^{6} - \zeta_{12}^{2} q^{7} - \zeta_{12}^{3} q^{8} - q^{9} - \zeta_{12}^{3} q^{10} + \zeta_{12}^{2} q^{11} + \zeta_{12} q^{13} + \zeta_{12} q^{14} - \zeta_{12} q^{15} + \zeta_{12}^{2} q^{16} - \zeta_{12}^{5} q^{18} + \zeta_{12}^{3} q^{19} - \zeta_{12}^{5} q^{21} - \zeta_{12} q^{22} - \zeta_{12}^{4} q^{23} + q^{24} - q^{26} - \zeta_{12}^{3} q^{27} - \zeta_{12}^{5} q^{29} + q^{30} - \zeta_{12} q^{31} + \zeta_{12}^{5} q^{33} + q^{35} - \zeta_{12}^{2} q^{38} + \zeta_{12}^{4} q^{39} + \zeta_{12} q^{40} + \zeta_{12} q^{41} + \zeta_{12}^{4} q^{42} - \zeta_{12}^{2} q^{43} - \zeta_{12}^{4} q^{45} + \zeta_{12}^{3} q^{46} - \zeta_{12}^{2} q^{47} + \zeta_{12}^{5} q^{48} + \zeta_{12}^{2} q^{54} - q^{55} + \zeta_{12}^{5} q^{56} - q^{57} + \zeta_{12}^{4} q^{58} - \zeta_{12} q^{59} + \zeta_{12}^{2} q^{61} + q^{62} + \zeta_{12}^{2} q^{63} - q^{64} + \zeta_{12}^{5} q^{65} - \zeta_{12}^{4} q^{66} - \zeta_{12} q^{67} + \zeta_{12} q^{69} + \zeta_{12}^{5} q^{70} + \zeta_{12}^{3} q^{72} - \zeta_{12}^{4} q^{77} - \zeta_{12}^{3} q^{78} - \zeta_{12}^{5} q^{79} - q^{80} + q^{81} - q^{82} + \zeta_{12}^{2} q^{83} + \zeta_{12} q^{86} + \zeta_{12}^{2} q^{87} - \zeta_{12}^{5} q^{88} + \zeta_{12}^{3} q^{90} - \zeta_{12}^{3} q^{91} - \zeta_{12}^{4} q^{93} + \zeta_{12} q^{94} - \zeta_{12} q^{95} + \zeta_{12}^{5} q^{97} - \zeta_{12}^{2} q^{99} +O(q^{100})$$ q + z^5 * q^2 + z^3 * q^3 + z^4 * q^5 - z^2 * q^6 - z^2 * q^7 - z^3 * q^8 - q^9 - z^3 * q^10 + z^2 * q^11 + z * q^13 + z * q^14 - z * q^15 + z^2 * q^16 - z^5 * q^18 + z^3 * q^19 - z^5 * q^21 - z * q^22 - z^4 * q^23 + q^24 - q^26 - z^3 * q^27 - z^5 * q^29 + q^30 - z * q^31 + z^5 * q^33 + q^35 - z^2 * q^38 + z^4 * q^39 + z * q^40 + z * q^41 + z^4 * q^42 - z^2 * q^43 - z^4 * q^45 + z^3 * q^46 - z^2 * q^47 + z^5 * q^48 + z^2 * q^54 - q^55 + z^5 * q^56 - q^57 + z^4 * q^58 - z * q^59 + z^2 * q^61 + q^62 + z^2 * q^63 - q^64 + z^5 * q^65 - z^4 * q^66 - z * q^67 + z * q^69 + z^5 * q^70 + z^3 * q^72 - z^4 * q^77 - z^3 * q^78 - z^5 * q^79 - q^80 + q^81 - q^82 + z^2 * q^83 + z * q^86 + z^2 * q^87 - z^5 * q^88 + z^3 * q^90 - z^3 * q^91 - z^4 * q^93 + z * q^94 - z * q^95 + z^5 * q^97 - z^2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{5} - 2 q^{6} - 2 q^{7} - 4 q^{9}+O(q^{10})$$ 4 * q - 2 * q^5 - 2 * q^6 - 2 * q^7 - 4 * q^9 $$4 q - 2 q^{5} - 2 q^{6} - 2 q^{7} - 4 q^{9} + 2 q^{11} + 2 q^{16} + 2 q^{23} + 4 q^{24} - 4 q^{26} + 4 q^{30} + 4 q^{35} - 2 q^{38} - 2 q^{39} - 2 q^{42} - 2 q^{43} + 2 q^{45} - 2 q^{47} + 2 q^{54} - 4 q^{55} - 4 q^{57} - 2 q^{58} + 2 q^{61} + 4 q^{62} + 2 q^{63} - 4 q^{64} + 2 q^{66} + 2 q^{77} - 4 q^{80} + 4 q^{81} - 4 q^{82} + 2 q^{83} + 2 q^{87} + 2 q^{93} - 2 q^{99}+O(q^{100})$$ 4 * q - 2 * q^5 - 2 * q^6 - 2 * q^7 - 4 * q^9 + 2 * q^11 + 2 * q^16 + 2 * q^23 + 4 * q^24 - 4 * q^26 + 4 * q^30 + 4 * q^35 - 2 * q^38 - 2 * q^39 - 2 * q^42 - 2 * q^43 + 2 * q^45 - 2 * q^47 + 2 * q^54 - 4 * q^55 - 4 * q^57 - 2 * q^58 + 2 * q^61 + 4 * q^62 + 2 * q^63 - 4 * q^64 + 2 * q^66 + 2 * q^77 - 4 * q^80 + 4 * q^81 - 4 * q^82 + 2 * q^83 + 2 * q^87 + 2 * q^93 - 2 * q^99

## Character values

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

 $$n$$ $$20$$ $$154$$ $$\chi(n)$$ $$\zeta_{12}^{4}$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
94.1
 0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i
−0.866025 + 0.500000i 1.00000i 0 −0.500000 + 0.866025i −0.500000 0.866025i −0.500000 0.866025i 1.00000i −1.00000 1.00000i
94.2 0.866025 0.500000i 1.00000i 0 −0.500000 + 0.866025i −0.500000 0.866025i −0.500000 0.866025i 1.00000i −1.00000 1.00000i
151.1 −0.866025 0.500000i 1.00000i 0 −0.500000 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 1.00000i −1.00000 1.00000i
151.2 0.866025 + 0.500000i 1.00000i 0 −0.500000 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 1.00000i −1.00000 1.00000i
 $$n$$: 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
9.c even 3 1 inner
19.b odd 2 1 inner
171.o odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 171.1.o.a 4
3.b odd 2 1 513.1.o.a 4
4.b odd 2 1 2736.1.bs.a 4
9.c even 3 1 inner 171.1.o.a 4
9.c even 3 1 1539.1.c.d 2
9.d odd 6 1 513.1.o.a 4
9.d odd 6 1 1539.1.c.c 2
19.b odd 2 1 inner 171.1.o.a 4
19.c even 3 1 3249.1.i.a 4
19.c even 3 1 3249.1.s.a 4
19.d odd 6 1 3249.1.i.a 4
19.d odd 6 1 3249.1.s.a 4
19.e even 9 3 3249.1.bc.a 12
19.e even 9 3 3249.1.be.a 12
19.f odd 18 3 3249.1.bc.a 12
19.f odd 18 3 3249.1.be.a 12
36.f odd 6 1 2736.1.bs.a 4
57.d even 2 1 513.1.o.a 4
76.d even 2 1 2736.1.bs.a 4
171.g even 3 1 3249.1.i.a 4
171.h even 3 1 3249.1.s.a 4
171.i odd 6 1 3249.1.s.a 4
171.l even 6 1 513.1.o.a 4
171.l even 6 1 1539.1.c.c 2
171.o odd 6 1 inner 171.1.o.a 4
171.o odd 6 1 1539.1.c.d 2
171.s odd 6 1 3249.1.i.a 4
171.v even 9 3 3249.1.be.a 12
171.w even 9 3 3249.1.bc.a 12
171.bc odd 18 3 3249.1.be.a 12
171.be odd 18 3 3249.1.bc.a 12
684.w even 6 1 2736.1.bs.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
171.1.o.a 4 1.a even 1 1 trivial
171.1.o.a 4 9.c even 3 1 inner
171.1.o.a 4 19.b odd 2 1 inner
171.1.o.a 4 171.o odd 6 1 inner
513.1.o.a 4 3.b odd 2 1
513.1.o.a 4 9.d odd 6 1
513.1.o.a 4 57.d even 2 1
513.1.o.a 4 171.l even 6 1
1539.1.c.c 2 9.d odd 6 1
1539.1.c.c 2 171.l even 6 1
1539.1.c.d 2 9.c even 3 1
1539.1.c.d 2 171.o odd 6 1
2736.1.bs.a 4 4.b odd 2 1
2736.1.bs.a 4 36.f odd 6 1
2736.1.bs.a 4 76.d even 2 1
2736.1.bs.a 4 684.w even 6 1
3249.1.i.a 4 19.c even 3 1
3249.1.i.a 4 19.d odd 6 1
3249.1.i.a 4 171.g even 3 1
3249.1.i.a 4 171.s odd 6 1
3249.1.s.a 4 19.c even 3 1
3249.1.s.a 4 19.d odd 6 1
3249.1.s.a 4 171.h even 3 1
3249.1.s.a 4 171.i odd 6 1
3249.1.bc.a 12 19.e even 9 3
3249.1.bc.a 12 19.f odd 18 3
3249.1.bc.a 12 171.w even 9 3
3249.1.bc.a 12 171.be odd 18 3
3249.1.be.a 12 19.e even 9 3
3249.1.be.a 12 19.f odd 18 3
3249.1.be.a 12 171.v even 9 3
3249.1.be.a 12 171.bc odd 18 3

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(171, [\chi])$$.

## Hecke characteristic polynomials

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