Properties

 Label 171.3.p.f Level $171$ Weight $3$ Character orbit 171.p Analytic conductor $4.659$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

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: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.19163381760000.1 Defining polynomial: $$x^{8} - 14x^{6} + 177x^{4} - 266x^{2} + 361$$ x^8 - 14*x^6 + 177*x^4 - 266*x^2 + 361 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (2 \beta_{6} + \beta_{4} + 3 \beta_{2}) q^{4} + ( - \beta_{7} + \beta_{5} - 2 \beta_{3} + \beta_1) q^{5} + (\beta_{6} - \beta_{4} - 5) q^{7} + (2 \beta_{7} - \beta_{5} - 2 \beta_{3} + 3 \beta_1) q^{8}+O(q^{10})$$ q + b1 * q^2 + (2*b6 + b4 + 3*b2) * q^4 + (-b7 + b5 - 2*b3 + b1) * q^5 + (b6 - b4 - 5) * q^7 + (2*b7 - b5 - 2*b3 + 3*b1) * q^8 $$q + \beta_1 q^{2} + (2 \beta_{6} + \beta_{4} + 3 \beta_{2}) q^{4} + ( - \beta_{7} + \beta_{5} - 2 \beta_{3} + \beta_1) q^{5} + (\beta_{6} - \beta_{4} - 5) q^{7} + (2 \beta_{7} - \beta_{5} - 2 \beta_{3} + 3 \beta_1) q^{8} + ( - \beta_{4} + 4 \beta_{2} - 8) q^{10} + (\beta_{5} + \beta_{3} + 2 \beta_1) q^{11} + (2 \beta_{4} + 5 \beta_{2} - 10) q^{13} + (\beta_{7} + \beta_{5}) q^{14} + (2 \beta_{6} + 4 \beta_{4} + 11 \beta_{2} - 11) q^{16} + (4 \beta_{3} - 2 \beta_1) q^{17} + ( - 3 \beta_{6} - 5 \beta_{4} - 6 \beta_{2} - 9) q^{19} + ( - 3 \beta_{5} + \beta_{3} - 2 \beta_1) q^{20} + (7 \beta_{6} + 10 \beta_{2} + 10) q^{22} + ( - 7 \beta_{7} + \beta_{3} - 9 \beta_1) q^{23} + (12 \beta_{6} + 6 \beta_{4} - 9 \beta_{2}) q^{25} + ( - 2 \beta_{5} - 7 \beta_{3} - 9 \beta_1) q^{26} + ( - 4 \beta_{6} - 2 \beta_{4} + 15 \beta_{2}) q^{28} + 10 \beta_{3} q^{29} + ( - \beta_{6} - \beta_{4} + 22 \beta_{2} - 11) q^{31} + ( - 2 \beta_{7} + 4 \beta_{5} - 9 \beta_{3} + 2 \beta_1) q^{32} + ( - 6 \beta_{4} - 14 \beta_{2} + 28) q^{34} + (11 \beta_{7} - 11 \beta_{5} + 18 \beta_{3} - 9 \beta_1) q^{35} + (6 \beta_{6} + 6 \beta_{4} - 50 \beta_{2} + 25) q^{37} + ( - 3 \beta_{7} + 5 \beta_{5} + 14 \beta_{3} - 14 \beta_1) q^{38} + ( - 13 \beta_{6} + 14 \beta_{2} + 14) q^{40} + (4 \beta_{7} + 4 \beta_{5}) q^{41} + (7 \beta_{6} + 14 \beta_{4} + 25 \beta_{2} - 25) q^{43} + (3 \beta_{7} - 13 \beta_{3} + 29 \beta_1) q^{44} + ( - 31 \beta_{6} - 31 \beta_{4} - 56 \beta_{2} + 28) q^{46} + ( - 8 \beta_{7} - 8 \beta_1) q^{47} + ( - 10 \beta_{6} + 10 \beta_{4} + 6) q^{49} + (12 \beta_{7} - 6 \beta_{5} - 9 \beta_{3} + 15 \beta_1) q^{50} + ( - 21 \beta_{6} - 35 \beta_{2} - 35) q^{52} + ( - 5 \beta_{7} + 10 \beta_{5} - 21 \beta_{3} + 5 \beta_1) q^{53} + (4 \beta_{6} + 8 \beta_{4} - 10 \beta_{2} + 10) q^{55} + ( - 12 \beta_{7} + 6 \beta_{5} - 17 \beta_{3} + 11 \beta_1) q^{56} + (10 \beta_{6} - 10 \beta_{4} + 70) q^{58} + ( - 5 \beta_{7} - 5 \beta_{5} + 25 \beta_1) q^{59} + ( - 20 \beta_{6} - 10 \beta_{4} - 17 \beta_{2}) q^{61} + ( - \beta_{7} + \beta_{5} - 20 \beta_{3} + 10 \beta_1) q^{62} + (7 \beta_{6} - 7 \beta_{4} - 1) q^{64} + (2 \beta_{7} - \beta_{5} + 7 \beta_{3} - 6 \beta_1) q^{65} + ( - 11 \beta_{4} + 15 \beta_{2} - 30) q^{67} + (6 \beta_{5} + 12 \beta_{3} + 18 \beta_1) q^{68} + (17 \beta_{4} - 30 \beta_{2} + 60) q^{70} + (8 \beta_{7} + 8 \beta_{5} + 10 \beta_1) q^{71} + (18 \beta_{6} + 36 \beta_{4} + 15 \beta_{2} - 15) q^{73} + (6 \beta_{7} - 6 \beta_{5} + 38 \beta_{3} - 19 \beta_1) q^{74} + ( - 18 \beta_{6} - 30 \beta_{4} - 55 \beta_{2} + 98) q^{76} + ( - 5 \beta_{5} + 5 \beta_{3}) q^{77} + (19 \beta_{6} + 35 \beta_{2} + 35) q^{79} + ( - \beta_{7} + 3 \beta_{3} - 7 \beta_1) q^{80} + (16 \beta_{6} + 8 \beta_{4} - 20 \beta_{2}) q^{82} + (4 \beta_{5} - 14 \beta_{3} - 10 \beta_1) q^{83} + ( - 4 \beta_{6} - 2 \beta_{4} + 24 \beta_{2}) q^{85} + (7 \beta_{7} - 14 \beta_{5} - 46 \beta_{3} - 7 \beta_1) q^{86} + (23 \beta_{6} + 23 \beta_{4} + 120 \beta_{2} - 60) q^{88} + ( - 3 \beta_{7} + 6 \beta_{5} - 51 \beta_{3} + 3 \beta_1) q^{89} + (5 \beta_{4} - 5 \beta_{2} + 10) q^{91} + ( - 3 \beta_{7} + 3 \beta_{5} + 110 \beta_{3} - 55 \beta_1) q^{92} + ( - 32 \beta_{6} - 32 \beta_{4} - 48 \beta_{2} + 24) q^{94} + (23 \beta_{7} - 13 \beta_{5} + 32 \beta_{3} - 13 \beta_1) q^{95} + (8 \beta_{6} - 70 \beta_{2} - 70) q^{97} + ( - 10 \beta_{7} - 10 \beta_{5} - 44 \beta_1) q^{98}+O(q^{100})$$ q + b1 * q^2 + (2*b6 + b4 + 3*b2) * q^4 + (-b7 + b5 - 2*b3 + b1) * q^5 + (b6 - b4 - 5) * q^7 + (2*b7 - b5 - 2*b3 + 3*b1) * q^8 + (-b4 + 4*b2 - 8) * q^10 + (b5 + b3 + 2*b1) * q^11 + (2*b4 + 5*b2 - 10) * q^13 + (b7 + b5) * q^14 + (2*b6 + 4*b4 + 11*b2 - 11) * q^16 + (4*b3 - 2*b1) * q^17 + (-3*b6 - 5*b4 - 6*b2 - 9) * q^19 + (-3*b5 + b3 - 2*b1) * q^20 + (7*b6 + 10*b2 + 10) * q^22 + (-7*b7 + b3 - 9*b1) * q^23 + (12*b6 + 6*b4 - 9*b2) * q^25 + (-2*b5 - 7*b3 - 9*b1) * q^26 + (-4*b6 - 2*b4 + 15*b2) * q^28 + 10*b3 * q^29 + (-b6 - b4 + 22*b2 - 11) * q^31 + (-2*b7 + 4*b5 - 9*b3 + 2*b1) * q^32 + (-6*b4 - 14*b2 + 28) * q^34 + (11*b7 - 11*b5 + 18*b3 - 9*b1) * q^35 + (6*b6 + 6*b4 - 50*b2 + 25) * q^37 + (-3*b7 + 5*b5 + 14*b3 - 14*b1) * q^38 + (-13*b6 + 14*b2 + 14) * q^40 + (4*b7 + 4*b5) * q^41 + (7*b6 + 14*b4 + 25*b2 - 25) * q^43 + (3*b7 - 13*b3 + 29*b1) * q^44 + (-31*b6 - 31*b4 - 56*b2 + 28) * q^46 + (-8*b7 - 8*b1) * q^47 + (-10*b6 + 10*b4 + 6) * q^49 + (12*b7 - 6*b5 - 9*b3 + 15*b1) * q^50 + (-21*b6 - 35*b2 - 35) * q^52 + (-5*b7 + 10*b5 - 21*b3 + 5*b1) * q^53 + (4*b6 + 8*b4 - 10*b2 + 10) * q^55 + (-12*b7 + 6*b5 - 17*b3 + 11*b1) * q^56 + (10*b6 - 10*b4 + 70) * q^58 + (-5*b7 - 5*b5 + 25*b1) * q^59 + (-20*b6 - 10*b4 - 17*b2) * q^61 + (-b7 + b5 - 20*b3 + 10*b1) * q^62 + (7*b6 - 7*b4 - 1) * q^64 + (2*b7 - b5 + 7*b3 - 6*b1) * q^65 + (-11*b4 + 15*b2 - 30) * q^67 + (6*b5 + 12*b3 + 18*b1) * q^68 + (17*b4 - 30*b2 + 60) * q^70 + (8*b7 + 8*b5 + 10*b1) * q^71 + (18*b6 + 36*b4 + 15*b2 - 15) * q^73 + (6*b7 - 6*b5 + 38*b3 - 19*b1) * q^74 + (-18*b6 - 30*b4 - 55*b2 + 98) * q^76 + (-5*b5 + 5*b3) * q^77 + (19*b6 + 35*b2 + 35) * q^79 + (-b7 + 3*b3 - 7*b1) * q^80 + (16*b6 + 8*b4 - 20*b2) * q^82 + (4*b5 - 14*b3 - 10*b1) * q^83 + (-4*b6 - 2*b4 + 24*b2) * q^85 + (7*b7 - 14*b5 - 46*b3 - 7*b1) * q^86 + (23*b6 + 23*b4 + 120*b2 - 60) * q^88 + (-3*b7 + 6*b5 - 51*b3 + 3*b1) * q^89 + (5*b4 - 5*b2 + 10) * q^91 + (-3*b7 + 3*b5 + 110*b3 - 55*b1) * q^92 + (-32*b6 - 32*b4 - 48*b2 + 24) * q^94 + (23*b7 - 13*b5 + 32*b3 - 13*b1) * q^95 + (8*b6 - 70*b2 - 70) * q^97 + (-10*b7 - 10*b5 - 44*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 12 q^{4} - 40 q^{7}+O(q^{10})$$ 8 * q + 12 * q^4 - 40 * q^7 $$8 q + 12 q^{4} - 40 q^{7} - 48 q^{10} - 60 q^{13} - 44 q^{16} - 96 q^{19} + 120 q^{22} - 36 q^{25} + 60 q^{28} + 168 q^{34} + 168 q^{40} - 100 q^{43} + 48 q^{49} - 420 q^{52} + 40 q^{55} + 560 q^{58} - 68 q^{61} - 8 q^{64} - 180 q^{67} + 360 q^{70} - 60 q^{73} + 564 q^{76} + 420 q^{79} - 80 q^{82} + 96 q^{85} + 60 q^{91} - 840 q^{97}+O(q^{100})$$ 8 * q + 12 * q^4 - 40 * q^7 - 48 * q^10 - 60 * q^13 - 44 * q^16 - 96 * q^19 + 120 * q^22 - 36 * q^25 + 60 * q^28 + 168 * q^34 + 168 * q^40 - 100 * q^43 + 48 * q^49 - 420 * q^52 + 40 * q^55 + 560 * q^58 - 68 * q^61 - 8 * q^64 - 180 * q^67 + 360 * q^70 - 60 * q^73 + 564 * q^76 + 420 * q^79 - 80 * q^82 + 96 * q^85 + 60 * q^91 - 840 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 14x^{6} + 177x^{4} - 266x^{2} + 361$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( 14\nu^{6} - 177\nu^{4} + 2478\nu^{2} - 361 ) / 3363$$ (14*v^6 - 177*v^4 + 2478*v^2 - 361) / 3363 $$\beta_{3}$$ $$=$$ $$( -14\nu^{7} + 177\nu^{5} - 2478\nu^{3} + 3724\nu ) / 3363$$ (-14*v^7 + 177*v^5 - 2478*v^3 + 3724*v) / 3363 $$\beta_{4}$$ $$=$$ $$( -20\nu^{6} + 413\nu^{4} - 4661\nu^{2} + 13167 ) / 3363$$ (-20*v^6 + 413*v^4 - 4661*v^2 + 13167) / 3363 $$\beta_{5}$$ $$=$$ $$( 34\nu^{7} - 590\nu^{5} + 7139\nu^{3} - 23617\nu ) / 3363$$ (34*v^7 - 590*v^5 + 7139*v^3 - 23617*v) / 3363 $$\beta_{6}$$ $$=$$ $$( -39\nu^{6} + 413\nu^{4} - 4661\nu^{2} - 5320 ) / 3363$$ (-39*v^6 + 413*v^4 - 4661*v^2 - 5320) / 3363 $$\beta_{7}$$ $$=$$ $$( -53\nu^{7} + 590\nu^{5} - 7139\nu^{3} - 11685\nu ) / 3363$$ (-53*v^7 + 590*v^5 - 7139*v^3 - 11685*v) / 3363
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$2\beta_{6} + \beta_{4} + 7\beta_{2}$$ 2*b6 + b4 + 7*b2 $$\nu^{3}$$ $$=$$ $$2\beta_{7} - \beta_{5} - 10\beta_{3} + 11\beta_1$$ 2*b7 - b5 - 10*b3 + 11*b1 $$\nu^{4}$$ $$=$$ $$14\beta_{6} + 28\beta_{4} + 79\beta_{2} - 79$$ 14*b6 + 28*b4 + 79*b2 - 79 $$\nu^{5}$$ $$=$$ $$14\beta_{7} - 28\beta_{5} - 121\beta_{3} - 14\beta_1$$ 14*b7 - 28*b5 - 121*b3 - 14*b1 $$\nu^{6}$$ $$=$$ $$-177\beta_{6} + 177\beta_{4} - 973$$ -177*b6 + 177*b4 - 973 $$\nu^{7}$$ $$=$$ $$-177\beta_{7} - 177\beta_{5} - 1858\beta_1$$ -177*b7 - 177*b5 - 1858*b1

Character values

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

 $$n$$ $$20$$ $$154$$ $$\chi(n)$$ $$1$$ $$\beta_{2}$$

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
 −3.05907 − 1.76616i −1.06868 − 0.617004i 1.06868 + 0.617004i 3.05907 + 1.76616i −3.05907 + 1.76616i −1.06868 + 0.617004i 1.06868 − 0.617004i 3.05907 − 1.76616i
−3.05907 1.76616i 0 4.23861 + 7.34149i 0.533068 0.923301i 0 0.477226 15.8150i 0 −3.26139 + 1.88296i
46.2 −1.06868 0.617004i 0 −1.23861 2.14534i 4.08850 7.08149i 0 −10.4772 7.99294i 0 −8.73861 + 5.04524i
46.3 1.06868 + 0.617004i 0 −1.23861 2.14534i −4.08850 + 7.08149i 0 −10.4772 7.99294i 0 −8.73861 + 5.04524i
46.4 3.05907 + 1.76616i 0 4.23861 + 7.34149i −0.533068 + 0.923301i 0 0.477226 15.8150i 0 −3.26139 + 1.88296i
145.1 −3.05907 + 1.76616i 0 4.23861 7.34149i 0.533068 + 0.923301i 0 0.477226 15.8150i 0 −3.26139 1.88296i
145.2 −1.06868 + 0.617004i 0 −1.23861 + 2.14534i 4.08850 + 7.08149i 0 −10.4772 7.99294i 0 −8.73861 5.04524i
145.3 1.06868 0.617004i 0 −1.23861 + 2.14534i −4.08850 7.08149i 0 −10.4772 7.99294i 0 −8.73861 5.04524i
145.4 3.05907 1.76616i 0 4.23861 7.34149i −0.533068 0.923301i 0 0.477226 15.8150i 0 −3.26139 1.88296i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 145.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
19.d odd 6 1 inner
57.f even 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.f 8
3.b odd 2 1 inner 171.3.p.f 8
19.d odd 6 1 inner 171.3.p.f 8
57.f even 6 1 inner 171.3.p.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
171.3.p.f 8 1.a even 1 1 trivial
171.3.p.f 8 3.b odd 2 1 inner
171.3.p.f 8 19.d odd 6 1 inner
171.3.p.f 8 57.f even 6 1 inner

Hecke kernels

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

 $$T_{2}^{8} - 14T_{2}^{6} + 177T_{2}^{4} - 266T_{2}^{2} + 361$$ T2^8 - 14*T2^6 + 177*T2^4 - 266*T2^2 + 361 $$T_{5}^{8} + 68T_{5}^{6} + 4548T_{5}^{4} + 5168T_{5}^{2} + 5776$$ T5^8 + 68*T5^6 + 4548*T5^4 + 5168*T5^2 + 5776

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} - 14 T^{6} + 177 T^{4} + \cdots + 361$$
$3$ $$T^{8}$$
$5$ $$T^{8} + 68 T^{6} + 4548 T^{4} + \cdots + 5776$$
$7$ $$(T^{2} + 10 T - 5)^{4}$$
$11$ $$(T^{4} - 140 T^{2} + 1900)^{2}$$
$13$ $$(T^{4} + 30 T^{3} + 335 T^{2} + 1050 T + 1225)^{2}$$
$17$ $$T^{8} + 168 T^{6} + 25488 T^{4} + \cdots + 7485696$$
$19$ $$(T^{4} + 48 T^{3} + 1178 T^{2} + \cdots + 130321)^{2}$$
$23$ $$T^{8} + 3332 T^{6} + \cdots + 7687066773136$$
$29$ $$T^{8} - 1400 T^{6} + \cdots + 36100000000$$
$31$ $$(T^{4} + 746 T^{2} + 124609)^{2}$$
$37$ $$(T^{4} + 4470 T^{2} + 2295225)^{2}$$
$41$ $$T^{8} - 2720 T^{6} + \cdots + 14786560000$$
$43$ $$(T^{4} + 50 T^{3} + 3345 T^{2} + \cdots + 714025)^{2}$$
$47$ $$T^{8} + 3968 T^{6} + \cdots + 14541836517376$$
$53$ $$T^{8} - 7044 T^{6} + \cdots + 32083206953616$$
$59$ $$T^{8} - 15500 T^{6} + \cdots + 33\!\cdots\!00$$
$61$ $$(T^{4} + 34 T^{3} + 3867 T^{2} + \cdots + 7349521)^{2}$$
$67$ $$(T^{4} + 90 T^{3} + 2165 T^{2} + \cdots + 286225)^{2}$$
$71$ $$T^{8} + \cdots + 390758009760000$$
$73$ $$(T^{4} + 30 T^{3} + 10395 T^{2} + \cdots + 90155025)^{2}$$
$79$ $$(T^{4} - 210 T^{3} + 14765 T^{2} + \cdots + 4225)^{2}$$
$83$ $$(T^{4} - 7208 T^{2} + 12527536)^{2}$$
$89$ $$T^{8} - 32580 T^{6} + \cdots + 44\!\cdots\!00$$
$97$ $$(T^{4} + 420 T^{3} + 72860 T^{2} + \cdots + 197683600)^{2}$$