# Properties

 Label 171.2.g.b Level $171$ Weight $2$ Character orbit 171.g Analytic conductor $1.365$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [171,2,Mod(106,171)]

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([4, 4]))

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

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

chi := DirichletCharacter("171.106");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$171 = 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 171.g (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: yes 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{6} q^{2} + ( - \zeta_{6} + 2) q^{3} + ( - \zeta_{6} + 1) q^{4} - q^{5} + (\zeta_{6} + 1) q^{6} + (3 \zeta_{6} - 3) q^{7} + 3 q^{8} + ( - 3 \zeta_{6} + 3) q^{9}+O(q^{10})$$ q + z * q^2 + (-z + 2) * q^3 + (-z + 1) * q^4 - q^5 + (z + 1) * q^6 + (3*z - 3) * q^7 + 3 * q^8 + (-3*z + 3) * q^9 $$q + \zeta_{6} q^{2} + ( - \zeta_{6} + 2) q^{3} + ( - \zeta_{6} + 1) q^{4} - q^{5} + (\zeta_{6} + 1) q^{6} + (3 \zeta_{6} - 3) q^{7} + 3 q^{8} + ( - 3 \zeta_{6} + 3) q^{9} - \zeta_{6} q^{10} + (3 \zeta_{6} - 3) q^{11} + ( - 2 \zeta_{6} + 1) q^{12} + ( - 6 \zeta_{6} + 6) q^{13} - 3 q^{14} + (\zeta_{6} - 2) q^{15} + \zeta_{6} q^{16} + (3 \zeta_{6} - 3) q^{17} + 3 q^{18} + ( - 2 \zeta_{6} - 3) q^{19} + (\zeta_{6} - 1) q^{20} + (6 \zeta_{6} - 3) q^{21} - 3 q^{22} + (8 \zeta_{6} - 8) q^{23} + ( - 3 \zeta_{6} + 6) q^{24} - 4 q^{25} + 6 q^{26} + ( - 6 \zeta_{6} + 3) q^{27} + 3 \zeta_{6} q^{28} - 5 q^{29} + ( - \zeta_{6} - 1) q^{30} + 7 \zeta_{6} q^{31} + ( - 5 \zeta_{6} + 5) q^{32} + (6 \zeta_{6} - 3) q^{33} - 3 q^{34} + ( - 3 \zeta_{6} + 3) q^{35} - 3 \zeta_{6} q^{36} + 2 q^{37} + ( - 5 \zeta_{6} + 2) q^{38} + ( - 12 \zeta_{6} + 6) q^{39} - 3 q^{40} - q^{41} + (3 \zeta_{6} - 6) q^{42} - 8 \zeta_{6} q^{43} + 3 \zeta_{6} q^{44} + (3 \zeta_{6} - 3) q^{45} - 8 q^{46} + 9 q^{47} + (\zeta_{6} + 1) q^{48} - 2 \zeta_{6} q^{49} - 4 \zeta_{6} q^{50} + (6 \zeta_{6} - 3) q^{51} - 6 \zeta_{6} q^{52} - 3 \zeta_{6} q^{53} + ( - 3 \zeta_{6} + 6) q^{54} + ( - 3 \zeta_{6} + 3) q^{55} + (9 \zeta_{6} - 9) q^{56} + (\zeta_{6} - 8) q^{57} - 5 \zeta_{6} q^{58} + 3 q^{59} + (2 \zeta_{6} - 1) q^{60} + 7 q^{61} + (7 \zeta_{6} - 7) q^{62} + 9 \zeta_{6} q^{63} + 7 q^{64} + (6 \zeta_{6} - 6) q^{65} + (3 \zeta_{6} - 6) q^{66} + ( - 4 \zeta_{6} + 4) q^{67} + 3 \zeta_{6} q^{68} + (16 \zeta_{6} - 8) q^{69} + 3 q^{70} + ( - 15 \zeta_{6} + 15) q^{71} + ( - 9 \zeta_{6} + 9) q^{72} + ( - 5 \zeta_{6} + 5) q^{73} + 2 \zeta_{6} q^{74} + (4 \zeta_{6} - 8) q^{75} + (3 \zeta_{6} - 5) q^{76} - 9 \zeta_{6} q^{77} + ( - 6 \zeta_{6} + 12) q^{78} + 12 \zeta_{6} q^{79} - \zeta_{6} q^{80} - 9 \zeta_{6} q^{81} - \zeta_{6} q^{82} + ( - \zeta_{6} + 1) q^{83} + (3 \zeta_{6} + 3) q^{84} + ( - 3 \zeta_{6} + 3) q^{85} + ( - 8 \zeta_{6} + 8) q^{86} + (5 \zeta_{6} - 10) q^{87} + (9 \zeta_{6} - 9) q^{88} + \zeta_{6} q^{89} - 3 q^{90} + 18 \zeta_{6} q^{91} + 8 \zeta_{6} q^{92} + (7 \zeta_{6} + 7) q^{93} + 9 \zeta_{6} q^{94} + (2 \zeta_{6} + 3) q^{95} + ( - 10 \zeta_{6} + 5) q^{96} + 2 \zeta_{6} q^{97} + ( - 2 \zeta_{6} + 2) q^{98} + 9 \zeta_{6} q^{99} +O(q^{100})$$ q + z * q^2 + (-z + 2) * q^3 + (-z + 1) * q^4 - q^5 + (z + 1) * q^6 + (3*z - 3) * q^7 + 3 * q^8 + (-3*z + 3) * q^9 - z * q^10 + (3*z - 3) * q^11 + (-2*z + 1) * q^12 + (-6*z + 6) * q^13 - 3 * q^14 + (z - 2) * q^15 + z * q^16 + (3*z - 3) * q^17 + 3 * q^18 + (-2*z - 3) * q^19 + (z - 1) * q^20 + (6*z - 3) * q^21 - 3 * q^22 + (8*z - 8) * q^23 + (-3*z + 6) * q^24 - 4 * q^25 + 6 * q^26 + (-6*z + 3) * q^27 + 3*z * q^28 - 5 * q^29 + (-z - 1) * q^30 + 7*z * q^31 + (-5*z + 5) * q^32 + (6*z - 3) * q^33 - 3 * q^34 + (-3*z + 3) * q^35 - 3*z * q^36 + 2 * q^37 + (-5*z + 2) * q^38 + (-12*z + 6) * q^39 - 3 * q^40 - q^41 + (3*z - 6) * q^42 - 8*z * q^43 + 3*z * q^44 + (3*z - 3) * q^45 - 8 * q^46 + 9 * q^47 + (z + 1) * q^48 - 2*z * q^49 - 4*z * q^50 + (6*z - 3) * q^51 - 6*z * q^52 - 3*z * q^53 + (-3*z + 6) * q^54 + (-3*z + 3) * q^55 + (9*z - 9) * q^56 + (z - 8) * q^57 - 5*z * q^58 + 3 * q^59 + (2*z - 1) * q^60 + 7 * q^61 + (7*z - 7) * q^62 + 9*z * q^63 + 7 * q^64 + (6*z - 6) * q^65 + (3*z - 6) * q^66 + (-4*z + 4) * q^67 + 3*z * q^68 + (16*z - 8) * q^69 + 3 * q^70 + (-15*z + 15) * q^71 + (-9*z + 9) * q^72 + (-5*z + 5) * q^73 + 2*z * q^74 + (4*z - 8) * q^75 + (3*z - 5) * q^76 - 9*z * q^77 + (-6*z + 12) * q^78 + 12*z * q^79 - z * q^80 - 9*z * q^81 - z * q^82 + (-z + 1) * q^83 + (3*z + 3) * q^84 + (-3*z + 3) * q^85 + (-8*z + 8) * q^86 + (5*z - 10) * q^87 + (9*z - 9) * q^88 + z * q^89 - 3 * q^90 + 18*z * q^91 + 8*z * q^92 + (7*z + 7) * q^93 + 9*z * q^94 + (2*z + 3) * q^95 + (-10*z + 5) * q^96 + 2*z * q^97 + (-2*z + 2) * q^98 + 9*z * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} + 3 q^{3} + q^{4} - 2 q^{5} + 3 q^{6} - 3 q^{7} + 6 q^{8} + 3 q^{9}+O(q^{10})$$ 2 * q + q^2 + 3 * q^3 + q^4 - 2 * q^5 + 3 * q^6 - 3 * q^7 + 6 * q^8 + 3 * q^9 $$2 q + q^{2} + 3 q^{3} + q^{4} - 2 q^{5} + 3 q^{6} - 3 q^{7} + 6 q^{8} + 3 q^{9} - q^{10} - 3 q^{11} + 6 q^{13} - 6 q^{14} - 3 q^{15} + q^{16} - 3 q^{17} + 6 q^{18} - 8 q^{19} - q^{20} - 6 q^{22} - 8 q^{23} + 9 q^{24} - 8 q^{25} + 12 q^{26} + 3 q^{28} - 10 q^{29} - 3 q^{30} + 7 q^{31} + 5 q^{32} - 6 q^{34} + 3 q^{35} - 3 q^{36} + 4 q^{37} - q^{38} - 6 q^{40} - 2 q^{41} - 9 q^{42} - 8 q^{43} + 3 q^{44} - 3 q^{45} - 16 q^{46} + 18 q^{47} + 3 q^{48} - 2 q^{49} - 4 q^{50} - 6 q^{52} - 3 q^{53} + 9 q^{54} + 3 q^{55} - 9 q^{56} - 15 q^{57} - 5 q^{58} + 6 q^{59} + 14 q^{61} - 7 q^{62} + 9 q^{63} + 14 q^{64} - 6 q^{65} - 9 q^{66} + 4 q^{67} + 3 q^{68} + 6 q^{70} + 15 q^{71} + 9 q^{72} + 5 q^{73} + 2 q^{74} - 12 q^{75} - 7 q^{76} - 9 q^{77} + 18 q^{78} + 12 q^{79} - q^{80} - 9 q^{81} - q^{82} + q^{83} + 9 q^{84} + 3 q^{85} + 8 q^{86} - 15 q^{87} - 9 q^{88} + q^{89} - 6 q^{90} + 18 q^{91} + 8 q^{92} + 21 q^{93} + 9 q^{94} + 8 q^{95} + 2 q^{97} + 2 q^{98} + 9 q^{99}+O(q^{100})$$ 2 * q + q^2 + 3 * q^3 + q^4 - 2 * q^5 + 3 * q^6 - 3 * q^7 + 6 * q^8 + 3 * q^9 - q^10 - 3 * q^11 + 6 * q^13 - 6 * q^14 - 3 * q^15 + q^16 - 3 * q^17 + 6 * q^18 - 8 * q^19 - q^20 - 6 * q^22 - 8 * q^23 + 9 * q^24 - 8 * q^25 + 12 * q^26 + 3 * q^28 - 10 * q^29 - 3 * q^30 + 7 * q^31 + 5 * q^32 - 6 * q^34 + 3 * q^35 - 3 * q^36 + 4 * q^37 - q^38 - 6 * q^40 - 2 * q^41 - 9 * q^42 - 8 * q^43 + 3 * q^44 - 3 * q^45 - 16 * q^46 + 18 * q^47 + 3 * q^48 - 2 * q^49 - 4 * q^50 - 6 * q^52 - 3 * q^53 + 9 * q^54 + 3 * q^55 - 9 * q^56 - 15 * q^57 - 5 * q^58 + 6 * q^59 + 14 * q^61 - 7 * q^62 + 9 * q^63 + 14 * q^64 - 6 * q^65 - 9 * q^66 + 4 * q^67 + 3 * q^68 + 6 * q^70 + 15 * q^71 + 9 * q^72 + 5 * q^73 + 2 * q^74 - 12 * q^75 - 7 * q^76 - 9 * q^77 + 18 * q^78 + 12 * q^79 - q^80 - 9 * q^81 - q^82 + q^83 + 9 * q^84 + 3 * q^85 + 8 * q^86 - 15 * q^87 - 9 * q^88 + q^89 - 6 * q^90 + 18 * q^91 + 8 * q^92 + 21 * q^93 + 9 * q^94 + 8 * q^95 + 2 * q^97 + 2 * q^98 + 9 * q^99

## Character values

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

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
106.1
 0.5 + 0.866025i 0.5 − 0.866025i
0.500000 + 0.866025i 1.50000 0.866025i 0.500000 0.866025i −1.00000 1.50000 + 0.866025i −1.50000 + 2.59808i 3.00000 1.50000 2.59808i −0.500000 0.866025i
121.1 0.500000 0.866025i 1.50000 + 0.866025i 0.500000 + 0.866025i −1.00000 1.50000 0.866025i −1.50000 2.59808i 3.00000 1.50000 + 2.59808i −0.500000 + 0.866025i
 $$n$$: 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
171.g 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.g.b 2
3.b odd 2 1 513.2.g.b 2
9.c even 3 1 171.2.h.b yes 2
9.d odd 6 1 513.2.h.a 2
19.c even 3 1 171.2.h.b yes 2
57.h odd 6 1 513.2.h.a 2
171.g even 3 1 inner 171.2.g.b 2
171.n odd 6 1 513.2.g.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
171.2.g.b 2 1.a even 1 1 trivial
171.2.g.b 2 171.g even 3 1 inner
171.2.h.b yes 2 9.c even 3 1
171.2.h.b yes 2 19.c even 3 1
513.2.g.b 2 3.b odd 2 1
513.2.g.b 2 171.n odd 6 1
513.2.h.a 2 9.d odd 6 1
513.2.h.a 2 57.h odd 6 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T + 1$$
$3$ $$T^{2} - 3T + 3$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2} + 3T + 9$$
$11$ $$T^{2} + 3T + 9$$
$13$ $$T^{2} - 6T + 36$$
$17$ $$T^{2} + 3T + 9$$
$19$ $$T^{2} + 8T + 19$$
$23$ $$T^{2} + 8T + 64$$
$29$ $$(T + 5)^{2}$$
$31$ $$T^{2} - 7T + 49$$
$37$ $$(T - 2)^{2}$$
$41$ $$(T + 1)^{2}$$
$43$ $$T^{2} + 8T + 64$$
$47$ $$(T - 9)^{2}$$
$53$ $$T^{2} + 3T + 9$$
$59$ $$(T - 3)^{2}$$
$61$ $$(T - 7)^{2}$$
$67$ $$T^{2} - 4T + 16$$
$71$ $$T^{2} - 15T + 225$$
$73$ $$T^{2} - 5T + 25$$
$79$ $$T^{2} - 12T + 144$$
$83$ $$T^{2} - T + 1$$
$89$ $$T^{2} - T + 1$$
$97$ $$T^{2} - 2T + 4$$