# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$171 = 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 171.p (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.65941252056$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ 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: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{6} - 1) q^{2} - \zeta_{6} q^{4} + (4 \zeta_{6} - 4) q^{5} + 5 q^{7} + (10 \zeta_{6} - 5) q^{8} +O(q^{10})$$ q + (-z - 1) * q^2 - z * q^4 + (4*z - 4) * q^5 + 5 * q^7 + (10*z - 5) * q^8 $$q + ( - \zeta_{6} - 1) q^{2} - \zeta_{6} q^{4} + (4 \zeta_{6} - 4) q^{5} + 5 q^{7} + (10 \zeta_{6} - 5) q^{8} + ( - 4 \zeta_{6} + 8) q^{10} + 4 q^{11} + ( - 7 \zeta_{6} + 14) q^{13} + ( - 5 \zeta_{6} - 5) q^{14} + ( - 11 \zeta_{6} + 11) q^{16} + ( - 8 \zeta_{6} + 8) q^{17} + 19 q^{19} + 4 q^{20} + ( - 4 \zeta_{6} - 4) q^{22} - 4 \zeta_{6} q^{23} + 9 \zeta_{6} q^{25} - 21 q^{26} - 5 \zeta_{6} q^{28} + ( - 32 \zeta_{6} + 64) q^{29} + (6 \zeta_{6} - 3) q^{31} + ( - 9 \zeta_{6} + 18) q^{32} + (8 \zeta_{6} - 16) q^{34} + (20 \zeta_{6} - 20) q^{35} + (18 \zeta_{6} - 9) q^{37} + ( - 19 \zeta_{6} - 19) q^{38} + ( - 20 \zeta_{6} - 20) q^{40} + (8 \zeta_{6} + 8) q^{41} + (11 \zeta_{6} - 11) q^{43} - 4 \zeta_{6} q^{44} + (8 \zeta_{6} - 4) q^{46} + 56 \zeta_{6} q^{47} - 24 q^{49} + ( - 18 \zeta_{6} + 9) q^{50} + ( - 7 \zeta_{6} - 7) q^{52} + (28 \zeta_{6} - 56) q^{53} + (16 \zeta_{6} - 16) q^{55} + (50 \zeta_{6} - 25) q^{56} - 96 q^{58} + ( - 28 \zeta_{6} - 28) q^{59} + 79 \zeta_{6} q^{61} + ( - 9 \zeta_{6} + 9) q^{62} - 71 q^{64} + (56 \zeta_{6} - 28) q^{65} + (11 \zeta_{6} - 22) q^{67} - 8 q^{68} + ( - 20 \zeta_{6} + 40) q^{70} + ( - 64 \zeta_{6} - 64) q^{71} + ( - \zeta_{6} + 1) q^{73} + ( - 27 \zeta_{6} + 27) q^{74} - 19 \zeta_{6} q^{76} + 20 q^{77} + (19 \zeta_{6} + 19) q^{79} + 44 \zeta_{6} q^{80} - 24 \zeta_{6} q^{82} + 112 q^{83} + 32 \zeta_{6} q^{85} + ( - 11 \zeta_{6} + 22) q^{86} + (40 \zeta_{6} - 20) q^{88} + ( - 44 \zeta_{6} + 88) q^{89} + ( - 35 \zeta_{6} + 70) q^{91} + (4 \zeta_{6} - 4) q^{92} + ( - 112 \zeta_{6} + 56) q^{94} + (76 \zeta_{6} - 76) q^{95} + (64 \zeta_{6} + 64) q^{97} + (24 \zeta_{6} + 24) q^{98} +O(q^{100})$$ q + (-z - 1) * q^2 - z * q^4 + (4*z - 4) * q^5 + 5 * q^7 + (10*z - 5) * q^8 + (-4*z + 8) * q^10 + 4 * q^11 + (-7*z + 14) * q^13 + (-5*z - 5) * q^14 + (-11*z + 11) * q^16 + (-8*z + 8) * q^17 + 19 * q^19 + 4 * q^20 + (-4*z - 4) * q^22 - 4*z * q^23 + 9*z * q^25 - 21 * q^26 - 5*z * q^28 + (-32*z + 64) * q^29 + (6*z - 3) * q^31 + (-9*z + 18) * q^32 + (8*z - 16) * q^34 + (20*z - 20) * q^35 + (18*z - 9) * q^37 + (-19*z - 19) * q^38 + (-20*z - 20) * q^40 + (8*z + 8) * q^41 + (11*z - 11) * q^43 - 4*z * q^44 + (8*z - 4) * q^46 + 56*z * q^47 - 24 * q^49 + (-18*z + 9) * q^50 + (-7*z - 7) * q^52 + (28*z - 56) * q^53 + (16*z - 16) * q^55 + (50*z - 25) * q^56 - 96 * q^58 + (-28*z - 28) * q^59 + 79*z * q^61 + (-9*z + 9) * q^62 - 71 * q^64 + (56*z - 28) * q^65 + (11*z - 22) * q^67 - 8 * q^68 + (-20*z + 40) * q^70 + (-64*z - 64) * q^71 + (-z + 1) * q^73 + (-27*z + 27) * q^74 - 19*z * q^76 + 20 * q^77 + (19*z + 19) * q^79 + 44*z * q^80 - 24*z * q^82 + 112 * q^83 + 32*z * q^85 + (-11*z + 22) * q^86 + (40*z - 20) * q^88 + (-44*z + 88) * q^89 + (-35*z + 70) * q^91 + (4*z - 4) * q^92 + (-112*z + 56) * q^94 + (76*z - 76) * q^95 + (64*z + 64) * q^97 + (24*z + 24) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{2} - q^{4} - 4 q^{5} + 10 q^{7}+O(q^{10})$$ 2 * q - 3 * q^2 - q^4 - 4 * q^5 + 10 * q^7 $$2 q - 3 q^{2} - q^{4} - 4 q^{5} + 10 q^{7} + 12 q^{10} + 8 q^{11} + 21 q^{13} - 15 q^{14} + 11 q^{16} + 8 q^{17} + 38 q^{19} + 8 q^{20} - 12 q^{22} - 4 q^{23} + 9 q^{25} - 42 q^{26} - 5 q^{28} + 96 q^{29} + 27 q^{32} - 24 q^{34} - 20 q^{35} - 57 q^{38} - 60 q^{40} + 24 q^{41} - 11 q^{43} - 4 q^{44} + 56 q^{47} - 48 q^{49} - 21 q^{52} - 84 q^{53} - 16 q^{55} - 192 q^{58} - 84 q^{59} + 79 q^{61} + 9 q^{62} - 142 q^{64} - 33 q^{67} - 16 q^{68} + 60 q^{70} - 192 q^{71} + q^{73} + 27 q^{74} - 19 q^{76} + 40 q^{77} + 57 q^{79} + 44 q^{80} - 24 q^{82} + 224 q^{83} + 32 q^{85} + 33 q^{86} + 132 q^{89} + 105 q^{91} - 4 q^{92} - 76 q^{95} + 192 q^{97} + 72 q^{98}+O(q^{100})$$ 2 * q - 3 * q^2 - q^4 - 4 * q^5 + 10 * q^7 + 12 * q^10 + 8 * q^11 + 21 * q^13 - 15 * q^14 + 11 * q^16 + 8 * q^17 + 38 * q^19 + 8 * q^20 - 12 * q^22 - 4 * q^23 + 9 * q^25 - 42 * q^26 - 5 * q^28 + 96 * q^29 + 27 * q^32 - 24 * q^34 - 20 * q^35 - 57 * q^38 - 60 * q^40 + 24 * q^41 - 11 * q^43 - 4 * q^44 + 56 * q^47 - 48 * q^49 - 21 * q^52 - 84 * q^53 - 16 * q^55 - 192 * q^58 - 84 * q^59 + 79 * q^61 + 9 * q^62 - 142 * q^64 - 33 * q^67 - 16 * q^68 + 60 * q^70 - 192 * q^71 + q^73 + 27 * q^74 - 19 * q^76 + 40 * q^77 + 57 * q^79 + 44 * q^80 - 24 * q^82 + 224 * q^83 + 32 * q^85 + 33 * q^86 + 132 * q^89 + 105 * q^91 - 4 * q^92 - 76 * q^95 + 192 * q^97 + 72 * 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
46.1
 0.5 + 0.866025i 0.5 − 0.866025i
−1.50000 0.866025i 0 −0.500000 0.866025i −2.00000 + 3.46410i 0 5.00000 8.66025i 0 6.00000 3.46410i
145.1 −1.50000 + 0.866025i 0 −0.500000 + 0.866025i −2.00000 3.46410i 0 5.00000 8.66025i 0 6.00000 + 3.46410i
 $$n$$: 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.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 171.3.p.a 2
3.b odd 2 1 171.3.p.b yes 2
19.d odd 6 1 inner 171.3.p.a 2
57.f even 6 1 171.3.p.b yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
171.3.p.a 2 1.a even 1 1 trivial
171.3.p.a 2 19.d odd 6 1 inner
171.3.p.b yes 2 3.b odd 2 1
171.3.p.b yes 2 57.f even 6 1

## Hecke kernels

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

 $$T_{2}^{2} + 3T_{2} + 3$$ T2^2 + 3*T2 + 3 $$T_{5}^{2} + 4T_{5} + 16$$ T5^2 + 4*T5 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 3T + 3$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 4T + 16$$
$7$ $$(T - 5)^{2}$$
$11$ $$(T - 4)^{2}$$
$13$ $$T^{2} - 21T + 147$$
$17$ $$T^{2} - 8T + 64$$
$19$ $$(T - 19)^{2}$$
$23$ $$T^{2} + 4T + 16$$
$29$ $$T^{2} - 96T + 3072$$
$31$ $$T^{2} + 27$$
$37$ $$T^{2} + 243$$
$41$ $$T^{2} - 24T + 192$$
$43$ $$T^{2} + 11T + 121$$
$47$ $$T^{2} - 56T + 3136$$
$53$ $$T^{2} + 84T + 2352$$
$59$ $$T^{2} + 84T + 2352$$
$61$ $$T^{2} - 79T + 6241$$
$67$ $$T^{2} + 33T + 363$$
$71$ $$T^{2} + 192T + 12288$$
$73$ $$T^{2} - T + 1$$
$79$ $$T^{2} - 57T + 1083$$
$83$ $$(T - 112)^{2}$$
$89$ $$T^{2} - 132T + 5808$$
$97$ $$T^{2} - 192T + 12288$$