LMFDB - Embedded newform 171.2.a.d.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [171,2,Mod(1,171)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("171.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(171, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

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

Newform invariants

comment:select newform

sage:traces = [1,2,0,2,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$1.36544187456$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 57)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	171.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q+2.00000 q^{2} +2.00000 q^{4} +3.00000 q^{5} -5.00000 q^{7} +6.00000 q^{10} -1.00000 q^{11} +2.00000 q^{13} -10.0000 q^{14} -4.00000 q^{16} +1.00000 q^{17} -1.00000 q^{19} +6.00000 q^{20} -2.00000 q^{22} +4.00000 q^{23} +4.00000 q^{25} +4.00000 q^{26} -10.0000 q^{28} +2.00000 q^{29} -6.00000 q^{31} -8.00000 q^{32} +2.00000 q^{34} -15.0000 q^{35} -2.00000 q^{38} -1.00000 q^{43} -2.00000 q^{44} +8.00000 q^{46} +9.00000 q^{47} +18.0000 q^{49} +8.00000 q^{50} +4.00000 q^{52} -10.0000 q^{53} -3.00000 q^{55} +4.00000 q^{58} +8.00000 q^{59} -1.00000 q^{61} -12.0000 q^{62} -8.00000 q^{64} +6.00000 q^{65} +8.00000 q^{67} +2.00000 q^{68} -30.0000 q^{70} +12.0000 q^{71} -11.0000 q^{73} -2.00000 q^{76} +5.00000 q^{77} +16.0000 q^{79} -12.0000 q^{80} -12.0000 q^{83} +3.00000 q^{85} -2.00000 q^{86} +6.00000 q^{89} -10.0000 q^{91} +8.00000 q^{92} +18.0000 q^{94} -3.00000 q^{95} -10.0000 q^{97} +36.0000 q^{98} +O(q^{100})$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.00000	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$2$	2.00000	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$3$	0	0
$3$	0	0
$4$	2.00000	1.00000
$5$	3.00000	1.34164	0.670820	−	0.741620i	$-0.265942\pi$
$5$	3.00000	1.34164	0.670820	+	0.741620i	$0.265942\pi$
$6$	0	0
$7$	−5.00000	−1.88982	−0.944911	−	0.327327i	$-0.893852\pi$
$7$	−5.00000	−1.88982	−0.944911	+	0.327327i	$0.893852\pi$
$8$	0	0
$9$	0	0
$10$	6.00000	1.89737
$11$	−1.00000	−0.301511	−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000	−0.301511	−0.150756	+	0.988571i	$0.548171\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	−10.0000	−2.67261
$15$	0	0
$16$	−4.00000	−1.00000
$17$	1.00000	0.242536	0.121268	−	0.992620i	$-0.461304\pi$
$17$	1.00000	0.242536	0.121268	+	0.992620i	$0.461304\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	6.00000	1.34164
$21$	0	0
$22$	−2.00000	−0.426401
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	4.00000	0.784465
$27$	0	0
$28$	−10.0000	−1.88982
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	−6.00000	−1.07763	−0.538816	−	0.842424i	$-0.681128\pi$
$31$	−6.00000	−1.07763	−0.538816	+	0.842424i	$0.681128\pi$
$32$	−8.00000	−1.41421
$33$	0	0
$34$	2.00000	0.342997
$35$	−15.0000	−2.53546
$36$	0	0
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	−2.00000	−0.324443
$39$	0	0
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	−1.00000	−0.152499	−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000	−0.152499	−0.0762493	+	0.997089i	$0.524294\pi$
$44$	−2.00000	−0.301511
$45$	0	0
$46$	8.00000	1.17954
$47$	9.00000	1.31278	0.656392	−	0.754420i	$-0.272082\pi$
$47$	9.00000	1.31278	0.656392	+	0.754420i	$0.272082\pi$
$48$	0	0
$49$	18.0000	2.57143
$50$	8.00000	1.13137
$51$	0	0
$52$	4.00000	0.554700
$53$	−10.0000	−1.37361	−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000	−1.37361	−0.686803	+	0.726844i	$0.740986\pi$
$54$	0	0
$55$	−3.00000	−0.404520
$56$	0	0
$57$	0	0
$58$	4.00000	0.525226
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	0	0
$61$	−1.00000	−0.128037	−0.0640184	−	0.997949i	$-0.520392\pi$
$61$	−1.00000	−0.128037	−0.0640184	+	0.997949i	$0.520392\pi$
$62$	−12.0000	−1.52400
$63$	0	0
$64$	−8.00000	−1.00000
$65$	6.00000	0.744208
$66$	0	0
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	2.00000	0.242536
$69$	0	0
$70$	−30.0000	−3.58569
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	0	0
$73$	−11.0000	−1.28745	−0.643726	−	0.765256i	$-0.722612\pi$
$73$	−11.0000	−1.28745	−0.643726	+	0.765256i	$0.722612\pi$
$74$	0	0
$75$	0	0
$76$	−2.00000	−0.229416
$77$	5.00000	0.569803
$78$	0	0
$79$	16.0000	1.80014	0.900070	−	0.435745i	$-0.143515\pi$
$79$	16.0000	1.80014	0.900070	+	0.435745i	$0.143515\pi$
$80$	−12.0000	−1.34164
$81$	0	0
$82$	0	0
$83$	−12.0000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	0	0
$85$	3.00000	0.325396
$86$	−2.00000	−0.215666
$87$	0	0
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	−10.0000	−1.04828
$92$	8.00000	0.834058
$93$	0	0
$94$	18.0000	1.85656
$95$	−3.00000	−0.307794
$96$	0	0
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	36.0000	3.63655
$99$	0	0
$100$	8.00000	0.800000
$101$	−2.00000	−0.199007	−0.0995037	−	0.995037i	$-0.531726\pi$
$101$	−2.00000	−0.199007	−0.0995037	+	0.995037i	$0.531726\pi$
$102$	0	0
$103$	−2.00000	−0.197066	−0.0985329	−	0.995134i	$-0.531415\pi$
$103$	−2.00000	−0.197066	−0.0985329	+	0.995134i	$0.531415\pi$
$104$	0	0
$105$	0	0
$106$	−20.0000	−1.94257
$107$	−6.00000	−0.580042	−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000	−0.580042	−0.290021	+	0.957020i	$0.593662\pi$
$108$	0	0
$109$	4.00000	0.383131	0.191565	−	0.981480i	$-0.438644\pi$
$109$	4.00000	0.383131	0.191565	+	0.981480i	$0.438644\pi$
$110$	−6.00000	−0.572078
$111$	0	0
$112$	20.0000	1.88982
$113$	−2.00000	−0.188144	−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000	−0.188144	−0.0940721	+	0.995565i	$0.529988\pi$
$114$	0	0
$115$	12.0000	1.11901
$116$	4.00000	0.371391
$117$	0	0
$118$	16.0000	1.47292
$119$	−5.00000	−0.458349
$120$	0	0
$121$	−10.0000	−0.909091
$122$	−2.00000	−0.181071
$123$	0	0
$124$	−12.0000	−1.07763
$125$	−3.00000	−0.268328
$126$	0	0
$127$	−2.00000	−0.177471	−0.0887357	−	0.996055i	$-0.528283\pi$
$127$	−2.00000	−0.177471	−0.0887357	+	0.996055i	$0.528283\pi$
$128$	0	0
$129$	0	0
$130$	12.0000	1.05247
$131$	−7.00000	−0.611593	−0.305796	−	0.952097i	$-0.598923\pi$
$131$	−7.00000	−0.611593	−0.305796	+	0.952097i	$0.598923\pi$
$132$	0	0
$133$	5.00000	0.433555
$134$	16.0000	1.38219
$135$	0	0
$136$	0	0
$137$	9.00000	0.768922	0.384461	−	0.923141i	$-0.374387\pi$
$137$	9.00000	0.768922	0.384461	+	0.923141i	$0.374387\pi$
$138$	0	0
$139$	−13.0000	−1.10265	−0.551323	−	0.834292i	$-0.685877\pi$
$139$	−13.0000	−1.10265	−0.551323	+	0.834292i	$0.685877\pi$
$140$	−30.0000	−2.53546
$141$	0	0
$142$	24.0000	2.01404
$143$	−2.00000	−0.167248
$144$	0	0
$145$	6.00000	0.498273
$146$	−22.0000	−1.82073
$147$	0	0
$148$	0	0
$149$	21.0000	1.72039	0.860194	−	0.509968i	$-0.170343\pi$
$149$	21.0000	1.72039	0.860194	+	0.509968i	$0.170343\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	0	0
$154$	10.0000	0.805823
$155$	−18.0000	−1.44579
$156$	0	0
$157$	−18.0000	−1.43656	−0.718278	−	0.695756i	$-0.755069\pi$
$157$	−18.0000	−1.43656	−0.718278	+	0.695756i	$0.755069\pi$
$158$	32.0000	2.54578
$159$	0	0
$160$	−24.0000	−1.89737
$161$	−20.0000	−1.57622
$162$	0	0
$163$	0	0		−	1.00000i	$-0.5\pi$
$163$	0	0			1.00000i	$0.5\pi$
$164$	0	0
$165$	0	0
$166$	−24.0000	−1.86276
$167$	−10.0000	−0.773823	−0.386912	−	0.922117i	$-0.626458\pi$
$167$	−10.0000	−0.773823	−0.386912	+	0.922117i	$0.626458\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	6.00000	0.460179
$171$	0	0
$172$	−2.00000	−0.152499
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	0	0
$175$	−20.0000	−1.51186
$176$	4.00000	0.301511
$177$	0	0
$178$	12.0000	0.899438
$179$	18.0000	1.34538	0.672692	−	0.739923i	$-0.265138\pi$
$179$	18.0000	1.34538	0.672692	+	0.739923i	$0.265138\pi$
$180$	0	0
$181$	−14.0000	−1.04061	−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000	−1.04061	−0.520306	+	0.853980i	$0.674182\pi$
$182$	−20.0000	−1.48250
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−1.00000	−0.0731272
$188$	18.0000	1.31278
$189$	0	0
$190$	−6.00000	−0.435286
$191$	−9.00000	−0.651217	−0.325609	−	0.945505i	$-0.605569\pi$
$191$	−9.00000	−0.651217	−0.325609	+	0.945505i	$0.605569\pi$
$192$	0	0
$193$	4.00000	0.287926	0.143963	−	0.989583i	$-0.454015\pi$
$193$	4.00000	0.287926	0.143963	+	0.989583i	$0.454015\pi$
$194$	−20.0000	−1.43592
$195$	0	0
$196$	36.0000	2.57143
$197$	2.00000	0.142494	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000	0.142494	0.0712470	+	0.997459i	$0.477302\pi$
$198$	0	0
$199$	−21.0000	−1.48865	−0.744325	−	0.667817i	$-0.767229\pi$
$199$	−21.0000	−1.48865	−0.744325	+	0.667817i	$0.767229\pi$
$200$	0	0
$201$	0	0
$202$	−4.00000	−0.281439
$203$	−10.0000	−0.701862
$204$	0	0
$205$	0	0
$206$	−4.00000	−0.278693
$207$	0	0
$208$	−8.00000	−0.554700
$209$	1.00000	0.0691714
$210$	0	0
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	−20.0000	−1.37361
$213$	0	0
$214$	−12.0000	−0.820303
$215$	−3.00000	−0.204598
$216$	0	0
$217$	30.0000	2.03653
$218$	8.00000	0.541828
$219$	0	0
$220$	−6.00000	−0.404520
$221$	2.00000	0.134535
$222$	0	0
$223$	12.0000	0.803579	0.401790	−	0.915732i	$-0.368388\pi$
$223$	12.0000	0.803579	0.401790	+	0.915732i	$0.368388\pi$
$224$	40.0000	2.67261
$225$	0	0
$226$	−4.00000	−0.266076
$227$	−18.0000	−1.19470	−0.597351	−	0.801980i	$-0.703780\pi$
$227$	−18.0000	−1.19470	−0.597351	+	0.801980i	$0.703780\pi$
$228$	0	0
$229$	25.0000	1.65205	0.826023	−	0.563636i	$-0.190598\pi$
$229$	25.0000	1.65205	0.826023	+	0.563636i	$0.190598\pi$
$230$	24.0000	1.58251
$231$	0	0
$232$	0	0
$233$	−9.00000	−0.589610	−0.294805	−	0.955557i	$-0.595255\pi$
$233$	−9.00000	−0.589610	−0.294805	+	0.955557i	$0.595255\pi$
$234$	0	0
$235$	27.0000	1.76129
$236$	16.0000	1.04151
$237$	0	0
$238$	−10.0000	−0.648204
$239$	3.00000	0.194054	0.0970269	−	0.995282i	$-0.469067\pi$
$239$	3.00000	0.194054	0.0970269	+	0.995282i	$0.469067\pi$
$240$	0	0
$241$	20.0000	1.28831	0.644157	−	0.764894i	$-0.277208\pi$
$241$	20.0000	1.28831	0.644157	+	0.764894i	$0.277208\pi$
$242$	−20.0000	−1.28565
$243$	0	0
$244$	−2.00000	−0.128037
$245$	54.0000	3.44993
$246$	0	0
$247$	−2.00000	−0.127257
$248$	0	0
$249$	0	0
$250$	−6.00000	−0.379473
$251$	−7.00000	−0.441836	−0.220918	−	0.975292i	$-0.570905\pi$
$251$	−7.00000	−0.441836	−0.220918	+	0.975292i	$0.570905\pi$
$252$	0	0
$253$	−4.00000	−0.251478
$254$	−4.00000	−0.250982
$255$	0	0
$256$	16.0000	1.00000
$257$	8.00000	0.499026	0.249513	−	0.968371i	$-0.419729\pi$
$257$	8.00000	0.499026	0.249513	+	0.968371i	$0.419729\pi$
$258$	0	0
$259$	0	0
$260$	12.0000	0.744208
$261$	0	0
$262$	−14.0000	−0.864923
$263$	−23.0000	−1.41824	−0.709120	−	0.705087i	$-0.750908\pi$
$263$	−23.0000	−1.41824	−0.709120	+	0.705087i	$0.750908\pi$
$264$	0	0
$265$	−30.0000	−1.84289
$266$	10.0000	0.613139
$267$	0	0
$268$	16.0000	0.977356
$269$	14.0000	0.853595	0.426798	−	0.904347i	$-0.359642\pi$
$269$	14.0000	0.853595	0.426798	+	0.904347i	$0.359642\pi$
$270$	0	0
$271$	12.0000	0.728948	0.364474	−	0.931214i	$-0.381249\pi$
$271$	12.0000	0.728948	0.364474	+	0.931214i	$0.381249\pi$
$272$	−4.00000	−0.242536
$273$	0	0
$274$	18.0000	1.08742
$275$	−4.00000	−0.241209
$276$	0	0
$277$	−11.0000	−0.660926	−0.330463	−	0.943819i	$-0.607205\pi$
$277$	−11.0000	−0.660926	−0.330463	+	0.943819i	$0.607205\pi$
$278$	−26.0000	−1.55938
$279$	0	0
$280$	0	0
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	0	0
$283$	−13.0000	−0.772770	−0.386385	−	0.922338i	$-0.626276\pi$
$283$	−13.0000	−0.772770	−0.386385	+	0.922338i	$0.626276\pi$
$284$	24.0000	1.42414
$285$	0	0
$286$	−4.00000	−0.236525
$287$	0	0
$288$	0	0
$289$	−16.0000	−0.941176
$290$	12.0000	0.704664
$291$	0	0
$292$	−22.0000	−1.28745
$293$	28.0000	1.63578	0.817889	−	0.575376i	$-0.195144\pi$
$293$	28.0000	1.63578	0.817889	+	0.575376i	$0.195144\pi$
$294$	0	0
$295$	24.0000	1.39733
$296$	0	0
$297$	0	0
$298$	42.0000	2.43299
$299$	8.00000	0.462652
$300$	0	0
$301$	5.00000	0.288195
$302$	0	0
$303$	0	0
$304$	4.00000	0.229416
$305$	−3.00000	−0.171780
$306$	0	0
$307$	−12.0000	−0.684876	−0.342438	−	0.939540i	$-0.611253\pi$
$307$	−12.0000	−0.684876	−0.342438	+	0.939540i	$0.611253\pi$
$308$	10.0000	0.569803
$309$	0	0
$310$	−36.0000	−2.04466
$311$	21.0000	1.19080	0.595400	−	0.803429i	$-0.296993\pi$
$311$	21.0000	1.19080	0.595400	+	0.803429i	$0.296993\pi$
$312$	0	0
$313$	−2.00000	−0.113047	−0.0565233	−	0.998401i	$-0.518002\pi$
$313$	−2.00000	−0.113047	−0.0565233	+	0.998401i	$0.518002\pi$
$314$	−36.0000	−2.03160
$315$	0	0
$316$	32.0000	1.80014
$317$	4.00000	0.224662	0.112331	−	0.993671i	$-0.464168\pi$
$317$	4.00000	0.224662	0.112331	+	0.993671i	$0.464168\pi$
$318$	0	0
$319$	−2.00000	−0.111979
$320$	−24.0000	−1.34164
$321$	0	0
$322$	−40.0000	−2.22911
$323$	−1.00000	−0.0556415
$324$	0	0
$325$	8.00000	0.443760
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−45.0000	−2.48093
$330$	0	0
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	−24.0000	−1.31717
$333$	0	0
$334$	−20.0000	−1.09435
$335$	24.0000	1.31126
$336$	0	0
$337$	−14.0000	−0.762629	−0.381314	−	0.924445i	$-0.624528\pi$
$337$	−14.0000	−0.762629	−0.381314	+	0.924445i	$0.624528\pi$
$338$	−18.0000	−0.979071
$339$	0	0
$340$	6.00000	0.325396
$341$	6.00000	0.324918
$342$	0	0
$343$	−55.0000	−2.96972
$344$	0	0
$345$	0	0
$346$	−12.0000	−0.645124
$347$	25.0000	1.34207	0.671035	−	0.741426i	$-0.265850\pi$
$347$	25.0000	1.34207	0.671035	+	0.741426i	$0.265850\pi$
$348$	0	0
$349$	9.00000	0.481759	0.240879	−	0.970555i	$-0.422564\pi$
$349$	9.00000	0.481759	0.240879	+	0.970555i	$0.422564\pi$
$350$	−40.0000	−2.13809
$351$	0	0
$352$	8.00000	0.426401
$353$	2.00000	0.106449	0.0532246	−	0.998583i	$-0.483050\pi$
$353$	2.00000	0.106449	0.0532246	+	0.998583i	$0.483050\pi$
$354$	0	0
$355$	36.0000	1.91068
$356$	12.0000	0.635999
$357$	0	0
$358$	36.0000	1.90266
$359$	−37.0000	−1.95279	−0.976393	−	0.216003i	$-0.930698\pi$
$359$	−37.0000	−1.95279	−0.976393	+	0.216003i	$0.930698\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−28.0000	−1.47165
$363$	0	0
$364$	−20.0000	−1.04828
$365$	−33.0000	−1.72730
$366$	0	0
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	−16.0000	−0.834058
$369$	0	0
$370$	0	0
$371$	50.0000	2.59587
$372$	0	0
$373$	16.0000	0.828449	0.414224	−	0.910175i	$-0.364053\pi$
$373$	16.0000	0.828449	0.414224	+	0.910175i	$0.364053\pi$
$374$	−2.00000	−0.103418
$375$	0	0
$376$	0	0
$377$	4.00000	0.206010
$378$	0	0
$379$	34.0000	1.74646	0.873231	−	0.487306i	$-0.162020\pi$
$379$	34.0000	1.74646	0.873231	+	0.487306i	$0.162020\pi$
$380$	−6.00000	−0.307794
$381$	0	0
$382$	−18.0000	−0.920960
$383$	34.0000	1.73732	0.868659	−	0.495410i	$-0.164982\pi$
$383$	34.0000	1.73732	0.868659	+	0.495410i	$0.164982\pi$
$384$	0	0
$385$	15.0000	0.764471
$386$	8.00000	0.407189
$387$	0	0
$388$	−20.0000	−1.01535
$389$	27.0000	1.36895	0.684477	−	0.729034i	$-0.260031\pi$
$389$	27.0000	1.36895	0.684477	+	0.729034i	$0.260031\pi$
$390$	0	0
$391$	4.00000	0.202289
$392$	0	0
$393$	0	0
$394$	4.00000	0.201517
$395$	48.0000	2.41514
$396$	0	0
$397$	25.0000	1.25471	0.627357	−	0.778732i	$-0.284137\pi$
$397$	25.0000	1.25471	0.627357	+	0.778732i	$0.284137\pi$
$398$	−42.0000	−2.10527
$399$	0	0
$400$	−16.0000	−0.800000
$401$	−36.0000	−1.79775	−0.898877	−	0.438201i	$-0.855616\pi$
$401$	−36.0000	−1.79775	−0.898877	+	0.438201i	$0.855616\pi$
$402$	0	0
$403$	−12.0000	−0.597763
$404$	−4.00000	−0.199007
$405$	0	0
$406$	−20.0000	−0.992583
$407$	0	0
$408$	0	0
$409$	−14.0000	−0.692255	−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000	−0.692255	−0.346128	+	0.938187i	$0.612504\pi$
$410$	0	0
$411$	0	0
$412$	−4.00000	−0.197066
$413$	−40.0000	−1.96827
$414$	0	0
$415$	−36.0000	−1.76717
$416$	−16.0000	−0.784465
$417$	0	0
$418$	2.00000	0.0978232
$419$	−28.0000	−1.36789	−0.683945	−	0.729534i	$-0.739737\pi$
$419$	−28.0000	−1.36789	−0.683945	+	0.729534i	$0.739737\pi$
$420$	0	0
$421$	26.0000	1.26716	0.633581	−	0.773676i	$-0.281584\pi$
$421$	26.0000	1.26716	0.633581	+	0.773676i	$0.281584\pi$
$422$	24.0000	1.16830
$423$	0	0
$424$	0	0
$425$	4.00000	0.194029
$426$	0	0
$427$	5.00000	0.241967
$428$	−12.0000	−0.580042
$429$	0	0
$430$	−6.00000	−0.289346
$431$	34.0000	1.63772	0.818861	−	0.573992i	$-0.194606\pi$
$431$	34.0000	1.63772	0.818861	+	0.573992i	$0.194606\pi$
$432$	0	0
$433$	6.00000	0.288342	0.144171	−	0.989553i	$-0.453949\pi$
$433$	6.00000	0.288342	0.144171	+	0.989553i	$0.453949\pi$
$434$	60.0000	2.88009
$435$	0	0
$436$	8.00000	0.383131
$437$	−4.00000	−0.191346
$438$	0	0
$439$	26.0000	1.24091	0.620456	−	0.784241i	$-0.286947\pi$
$439$	26.0000	1.24091	0.620456	+	0.784241i	$0.286947\pi$
$440$	0	0
$441$	0	0
$442$	4.00000	0.190261
$443$	5.00000	0.237557	0.118779	−	0.992921i	$-0.462102\pi$
$443$	5.00000	0.237557	0.118779	+	0.992921i	$0.462102\pi$
$444$	0	0
$445$	18.0000	0.853282
$446$	24.0000	1.13643
$447$	0	0
$448$	40.0000	1.88982
$449$	36.0000	1.69895	0.849473	−	0.527633i	$-0.176920\pi$
$449$	36.0000	1.69895	0.849473	+	0.527633i	$0.176920\pi$
$450$	0	0
$451$	0	0
$452$	−4.00000	−0.188144
$453$	0	0
$454$	−36.0000	−1.68956
$455$	−30.0000	−1.40642
$456$	0	0
$457$	−29.0000	−1.35656	−0.678281	−	0.734802i	$-0.737275\pi$
$457$	−29.0000	−1.35656	−0.678281	+	0.734802i	$0.737275\pi$
$458$	50.0000	2.33635
$459$	0	0
$460$	24.0000	1.11901
$461$	−27.0000	−1.25752	−0.628758	−	0.777601i	$-0.716436\pi$
$461$	−27.0000	−1.25752	−0.628758	+	0.777601i	$0.716436\pi$
$462$	0	0
$463$	17.0000	0.790057	0.395029	−	0.918669i	$-0.370735\pi$
$463$	17.0000	0.790057	0.395029	+	0.918669i	$0.370735\pi$
$464$	−8.00000	−0.371391
$465$	0	0
$466$	−18.0000	−0.833834
$467$	5.00000	0.231372	0.115686	−	0.993286i	$-0.463093\pi$
$467$	5.00000	0.231372	0.115686	+	0.993286i	$0.463093\pi$
$468$	0	0
$469$	−40.0000	−1.84703
$470$	54.0000	2.49083
$471$	0	0
$472$	0	0
$473$	1.00000	0.0459800
$474$	0	0
$475$	−4.00000	−0.183533
$476$	−10.0000	−0.458349
$477$	0	0
$478$	6.00000	0.274434
$479$	16.0000	0.731059	0.365529	−	0.930800i	$-0.380888\pi$
$479$	16.0000	0.731059	0.365529	+	0.930800i	$0.380888\pi$
$480$	0	0
$481$	0	0
$482$	40.0000	1.82195
$483$	0	0
$484$	−20.0000	−0.909091
$485$	−30.0000	−1.36223
$486$	0	0
$487$	−16.0000	−0.725029	−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000	−0.725029	−0.362515	+	0.931978i	$0.618082\pi$
$488$	0	0
$489$	0	0
$490$	108.000	4.87894
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	2.00000	0.0900755
$494$	−4.00000	−0.179969
$495$	0	0
$496$	24.0000	1.07763
$497$	−60.0000	−2.69137
$498$	0	0
$499$	5.00000	0.223831	0.111915	−	0.993718i	$-0.464301\pi$
$499$	5.00000	0.223831	0.111915	+	0.993718i	$0.464301\pi$
$500$	−6.00000	−0.268328
$501$	0	0
$502$	−14.0000	−0.624851
$503$	−16.0000	−0.713405	−0.356702	−	0.934218i	$-0.616099\pi$
$503$	−16.0000	−0.713405	−0.356702	+	0.934218i	$0.616099\pi$
$504$	0	0
$505$	−6.00000	−0.266996
$506$	−8.00000	−0.355643
$507$	0	0
$508$	−4.00000	−0.177471
$509$	−6.00000	−0.265945	−0.132973	−	0.991120i	$-0.542452\pi$
$509$	−6.00000	−0.265945	−0.132973	+	0.991120i	$0.542452\pi$
$510$	0	0
$511$	55.0000	2.43306
$512$	32.0000	1.41421
$513$	0	0
$514$	16.0000	0.705730
$515$	−6.00000	−0.264392
$516$	0	0
$517$	−9.00000	−0.395820
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−36.0000	−1.57719	−0.788594	−	0.614914i	$-0.789191\pi$
$521$	−36.0000	−1.57719	−0.788594	+	0.614914i	$0.789191\pi$
$522$	0	0
$523$	−34.0000	−1.48672	−0.743358	−	0.668894i	$-0.766768\pi$
$523$	−34.0000	−1.48672	−0.743358	+	0.668894i	$0.766768\pi$
$524$	−14.0000	−0.611593
$525$	0	0
$526$	−46.0000	−2.00570
$527$	−6.00000	−0.261364
$528$	0	0
$529$	−7.00000	−0.304348
$530$	−60.0000	−2.60623
$531$	0	0
$532$	10.0000	0.433555
$533$	0	0
$534$	0	0
$535$	−18.0000	−0.778208
$536$	0	0
$537$	0	0
$538$	28.0000	1.20717
$539$	−18.0000	−0.775315
$540$	0	0
$541$	3.00000	0.128980	0.0644900	−	0.997918i	$-0.479458\pi$
$541$	3.00000	0.128980	0.0644900	+	0.997918i	$0.479458\pi$
$542$	24.0000	1.03089
$543$	0	0
$544$	−8.00000	−0.342997
$545$	12.0000	0.514024
$546$	0	0
$547$	−26.0000	−1.11168	−0.555840	−	0.831289i	$-0.687603\pi$
$547$	−26.0000	−1.11168	−0.555840	+	0.831289i	$0.687603\pi$
$548$	18.0000	0.768922
$549$	0	0
$550$	−8.00000	−0.341121
$551$	−2.00000	−0.0852029
$552$	0	0
$553$	−80.0000	−3.40195
$554$	−22.0000	−0.934690
$555$	0	0
$556$	−26.0000	−1.10265
$557$	41.0000	1.73723	0.868613	−	0.495491i	$-0.165012\pi$
$557$	41.0000	1.73723	0.868613	+	0.495491i	$0.165012\pi$
$558$	0	0
$559$	−2.00000	−0.0845910
$560$	60.0000	2.53546
$561$	0	0
$562$	−20.0000	−0.843649
$563$	12.0000	0.505740	0.252870	−	0.967500i	$-0.418626\pi$
$563$	12.0000	0.505740	0.252870	+	0.967500i	$0.418626\pi$
$564$	0	0
$565$	−6.00000	−0.252422
$566$	−26.0000	−1.09286
$567$	0	0
$568$	0	0
$569$	18.0000	0.754599	0.377300	−	0.926091i	$-0.376853\pi$
$569$	18.0000	0.754599	0.377300	+	0.926091i	$0.376853\pi$
$570$	0	0
$571$	0	0		−	1.00000i	$-0.5\pi$
$571$	0	0			1.00000i	$0.5\pi$
$572$	−4.00000	−0.167248
$573$	0	0
$574$	0	0
$575$	16.0000	0.667246
$576$	0	0
$577$	27.0000	1.12402	0.562012	−	0.827129i	$-0.310027\pi$
$577$	27.0000	1.12402	0.562012	+	0.827129i	$0.310027\pi$
$578$	−32.0000	−1.33102
$579$	0	0
$580$	12.0000	0.498273
$581$	60.0000	2.48922
$582$	0	0
$583$	10.0000	0.414158
$584$	0	0
$585$	0	0
$586$	56.0000	2.31334
$587$	−7.00000	−0.288921	−0.144460	−	0.989511i	$-0.546145\pi$
$587$	−7.00000	−0.288921	−0.144460	+	0.989511i	$0.546145\pi$
$588$	0	0
$589$	6.00000	0.247226
$590$	48.0000	1.97613
$591$	0	0
$592$	0	0
$593$	6.00000	0.246390	0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000	0.246390	0.123195	+	0.992382i	$0.460686\pi$
$594$	0	0
$595$	−15.0000	−0.614940
$596$	42.0000	1.72039
$597$	0	0
$598$	16.0000	0.654289
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	0	0		−	1.00000i	$-0.5\pi$
$601$	0	0			1.00000i	$0.5\pi$
$602$	10.0000	0.407570
$603$	0	0
$604$	0	0
$605$	−30.0000	−1.21967
$606$	0	0
$607$	26.0000	1.05531	0.527654	−	0.849460i	$-0.323072\pi$
$607$	26.0000	1.05531	0.527654	+	0.849460i	$0.323072\pi$
$608$	8.00000	0.324443
$609$	0	0
$610$	−6.00000	−0.242933
$611$	18.0000	0.728202
$612$	0	0
$613$	33.0000	1.33286	0.666429	−	0.745569i	$-0.267822\pi$
$613$	33.0000	1.33286	0.666429	+	0.745569i	$0.267822\pi$
$614$	−24.0000	−0.968561
$615$	0	0
$616$	0	0
$617$	−27.0000	−1.08698	−0.543490	−	0.839416i	$-0.682897\pi$
$617$	−27.0000	−1.08698	−0.543490	+	0.839416i	$0.682897\pi$
$618$	0	0
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	−36.0000	−1.44579
$621$	0	0
$622$	42.0000	1.68405
$623$	−30.0000	−1.20192
$624$	0	0
$625$	−29.0000	−1.16000
$626$	−4.00000	−0.159872
$627$	0	0
$628$	−36.0000	−1.43656
$629$	0	0
$630$	0	0
$631$	15.0000	0.597141	0.298570	−	0.954388i	$-0.403490\pi$
$631$	15.0000	0.597141	0.298570	+	0.954388i	$0.403490\pi$
$632$	0	0
$633$	0	0
$634$	8.00000	0.317721
$635$	−6.00000	−0.238103
$636$	0	0
$637$	36.0000	1.42637
$638$	−4.00000	−0.158362
$639$	0	0
$640$	0	0
$641$	−18.0000	−0.710957	−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000	−0.710957	−0.355479	+	0.934684i	$0.615682\pi$
$642$	0	0
$643$	−1.00000	−0.0394362	−0.0197181	−	0.999806i	$-0.506277\pi$
$643$	−1.00000	−0.0394362	−0.0197181	+	0.999806i	$0.506277\pi$
$644$	−40.0000	−1.57622
$645$	0	0
$646$	−2.00000	−0.0786889
$647$	39.0000	1.53325	0.766624	−	0.642096i	$-0.221935\pi$
$647$	39.0000	1.53325	0.766624	+	0.642096i	$0.221935\pi$
$648$	0	0
$649$	−8.00000	−0.314027
$650$	16.0000	0.627572
$651$	0	0
$652$	0	0
$653$	−3.00000	−0.117399	−0.0586995	−	0.998276i	$-0.518695\pi$
$653$	−3.00000	−0.117399	−0.0586995	+	0.998276i	$0.518695\pi$
$654$	0	0
$655$	−21.0000	−0.820538
$656$	0	0
$657$	0	0
$658$	−90.0000	−3.50857
$659$	14.0000	0.545363	0.272681	−	0.962104i	$-0.412090\pi$
$659$	14.0000	0.545363	0.272681	+	0.962104i	$0.412090\pi$
$660$	0	0
$661$	12.0000	0.466746	0.233373	−	0.972387i	$-0.425024\pi$
$661$	12.0000	0.466746	0.233373	+	0.972387i	$0.425024\pi$
$662$	−8.00000	−0.310929
$663$	0	0
$664$	0	0
$665$	15.0000	0.581675
$666$	0	0
$667$	8.00000	0.309761
$668$	−20.0000	−0.773823
$669$	0	0
$670$	48.0000	1.85440
$671$	1.00000	0.0386046
$672$	0	0
$673$	−24.0000	−0.925132	−0.462566	−	0.886585i	$-0.653071\pi$
$673$	−24.0000	−0.925132	−0.462566	+	0.886585i	$0.653071\pi$
$674$	−28.0000	−1.07852
$675$	0	0
$676$	−18.0000	−0.692308
$677$	−34.0000	−1.30673	−0.653363	−	0.757045i	$-0.726642\pi$
$677$	−34.0000	−1.30673	−0.653363	+	0.757045i	$0.726642\pi$
$678$	0	0
$679$	50.0000	1.91882
$680$	0	0
$681$	0	0
$682$	12.0000	0.459504
$683$	6.00000	0.229584	0.114792	−	0.993390i	$-0.463380\pi$
$683$	6.00000	0.229584	0.114792	+	0.993390i	$0.463380\pi$
$684$	0	0
$685$	27.0000	1.03162
$686$	−110.000	−4.19982
$687$	0	0
$688$	4.00000	0.152499
$689$	−20.0000	−0.761939
$690$	0	0
$691$	−31.0000	−1.17930	−0.589648	−	0.807661i	$-0.700733\pi$
$691$	−31.0000	−1.17930	−0.589648	+	0.807661i	$0.700733\pi$
$692$	−12.0000	−0.456172
$693$	0	0
$694$	50.0000	1.89797
$695$	−39.0000	−1.47935
$696$	0	0
$697$	0	0
$698$	18.0000	0.681310
$699$	0	0
$700$	−40.0000	−1.51186
$701$	22.0000	0.830929	0.415464	−	0.909610i	$-0.363619\pi$
$701$	22.0000	0.830929	0.415464	+	0.909610i	$0.363619\pi$
$702$	0	0
$703$	0	0
$704$	8.00000	0.301511
$705$	0	0
$706$	4.00000	0.150542
$707$	10.0000	0.376089
$708$	0	0
$709$	−42.0000	−1.57734	−0.788672	−	0.614815i	$-0.789231\pi$
$709$	−42.0000	−1.57734	−0.788672	+	0.614815i	$0.789231\pi$
$710$	72.0000	2.70211
$711$	0	0
$712$	0	0
$713$	−24.0000	−0.898807
$714$	0	0
$715$	−6.00000	−0.224387
$716$	36.0000	1.34538
$717$	0	0
$718$	−74.0000	−2.76166
$719$	−33.0000	−1.23069	−0.615346	−	0.788257i	$-0.710984\pi$
$719$	−33.0000	−1.23069	−0.615346	+	0.788257i	$0.710984\pi$
$720$	0	0
$721$	10.0000	0.372419
$722$	2.00000	0.0744323
$723$	0	0
$724$	−28.0000	−1.04061
$725$	8.00000	0.297113
$726$	0	0
$727$	−23.0000	−0.853023	−0.426511	−	0.904482i	$-0.640258\pi$
$727$	−23.0000	−0.853023	−0.426511	+	0.904482i	$0.640258\pi$
$728$	0	0
$729$	0	0
$730$	−66.0000	−2.44277
$731$	−1.00000	−0.0369863
$732$	0	0
$733$	−14.0000	−0.517102	−0.258551	−	0.965998i	$-0.583245\pi$
$733$	−14.0000	−0.517102	−0.258551	+	0.965998i	$0.583245\pi$
$734$	−16.0000	−0.590571
$735$	0	0
$736$	−32.0000	−1.17954
$737$	−8.00000	−0.294684
$738$	0	0
$739$	−5.00000	−0.183928	−0.0919640	−	0.995762i	$-0.529314\pi$
$739$	−5.00000	−0.183928	−0.0919640	+	0.995762i	$0.529314\pi$
$740$	0	0
$741$	0	0
$742$	100.000	3.67112
$743$	8.00000	0.293492	0.146746	−	0.989174i	$-0.453120\pi$
$743$	8.00000	0.293492	0.146746	+	0.989174i	$0.453120\pi$
$744$	0	0
$745$	63.0000	2.30814
$746$	32.0000	1.17160
$747$	0	0
$748$	−2.00000	−0.0731272
$749$	30.0000	1.09618
$750$	0	0
$751$	24.0000	0.875772	0.437886	−	0.899030i	$-0.355727\pi$
$751$	24.0000	0.875772	0.437886	+	0.899030i	$0.355727\pi$
$752$	−36.0000	−1.31278
$753$	0	0
$754$	8.00000	0.291343
$755$	0	0
$756$	0	0
$757$	−17.0000	−0.617876	−0.308938	−	0.951082i	$-0.599973\pi$
$757$	−17.0000	−0.617876	−0.308938	+	0.951082i	$0.599973\pi$
$758$	68.0000	2.46987
$759$	0	0
$760$	0	0
$761$	−15.0000	−0.543750	−0.271875	−	0.962333i	$-0.587644\pi$
$761$	−15.0000	−0.543750	−0.271875	+	0.962333i	$0.587644\pi$
$762$	0	0
$763$	−20.0000	−0.724049
$764$	−18.0000	−0.651217
$765$	0	0
$766$	68.0000	2.45694
$767$	16.0000	0.577727
$768$	0	0
$769$	11.0000	0.396670	0.198335	−	0.980134i	$-0.436447\pi$
$769$	11.0000	0.396670	0.198335	+	0.980134i	$0.436447\pi$
$770$	30.0000	1.08112
$771$	0	0
$772$	8.00000	0.287926
$773$	−20.0000	−0.719350	−0.359675	−	0.933078i	$-0.617112\pi$
$773$	−20.0000	−0.719350	−0.359675	+	0.933078i	$0.617112\pi$
$774$	0	0
$775$	−24.0000	−0.862105
$776$	0	0
$777$	0	0
$778$	54.0000	1.93599
$779$	0	0
$780$	0	0
$781$	−12.0000	−0.429394
$782$	8.00000	0.286079
$783$	0	0
$784$	−72.0000	−2.57143
$785$	−54.0000	−1.92734
$786$	0	0
$787$	40.0000	1.42585	0.712923	−	0.701242i	$-0.247371\pi$
$787$	40.0000	1.42585	0.712923	+	0.701242i	$0.247371\pi$
$788$	4.00000	0.142494
$789$	0	0
$790$	96.0000	3.41553
$791$	10.0000	0.355559
$792$	0	0
$793$	−2.00000	−0.0710221
$794$	50.0000	1.77443
$795$	0	0
$796$	−42.0000	−1.48865
$797$	44.0000	1.55856	0.779280	−	0.626676i	$-0.215585\pi$
$797$	44.0000	1.55856	0.779280	+	0.626676i	$0.215585\pi$
$798$	0	0
$799$	9.00000	0.318397
$800$	−32.0000	−1.13137
$801$	0	0
$802$	−72.0000	−2.54241
$803$	11.0000	0.388182
$804$	0	0
$805$	−60.0000	−2.11472
$806$	−24.0000	−0.845364
$807$	0	0
$808$	0	0
$809$	55.0000	1.93370	0.966849	−	0.255351i	$-0.0821909\pi$
$809$	55.0000	1.93370	0.966849	+	0.255351i	$0.0821909\pi$
$810$	0	0
$811$	−38.0000	−1.33436	−0.667180	−	0.744896i	$-0.732499\pi$
$811$	−38.0000	−1.33436	−0.667180	+	0.744896i	$0.732499\pi$
$812$	−20.0000	−0.701862
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	1.00000	0.0349856
$818$	−28.0000	−0.978997
$819$	0	0
$820$	0	0
$821$	45.0000	1.57051	0.785255	−	0.619172i	$-0.212532\pi$
$821$	45.0000	1.57051	0.785255	+	0.619172i	$0.212532\pi$
$822$	0	0
$823$	43.0000	1.49889	0.749443	−	0.662069i	$-0.230321\pi$
$823$	43.0000	1.49889	0.749443	+	0.662069i	$0.230321\pi$
$824$	0	0
$825$	0	0
$826$	−80.0000	−2.78356
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	0	0
$829$	52.0000	1.80603	0.903017	−	0.429604i	$-0.141347\pi$
$829$	52.0000	1.80603	0.903017	+	0.429604i	$0.141347\pi$
$830$	−72.0000	−2.49916
$831$	0	0
$832$	−16.0000	−0.554700
$833$	18.0000	0.623663
$834$	0	0
$835$	−30.0000	−1.03819
$836$	2.00000	0.0691714
$837$	0	0
$838$	−56.0000	−1.93449
$839$	−54.0000	−1.86429	−0.932144	−	0.362089i	$-0.882064\pi$
$839$	−54.0000	−1.86429	−0.932144	+	0.362089i	$0.882064\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	52.0000	1.79204
$843$	0	0
$844$	24.0000	0.826114
$845$	−27.0000	−0.928828
$846$	0	0
$847$	50.0000	1.71802
$848$	40.0000	1.37361
$849$	0	0
$850$	8.00000	0.274398
$851$	0	0
$852$	0	0
$853$	−14.0000	−0.479351	−0.239675	−	0.970853i	$-0.577041\pi$
$853$	−14.0000	−0.479351	−0.239675	+	0.970853i	$0.577041\pi$
$854$	10.0000	0.342193
$855$	0	0
$856$	0	0
$857$	8.00000	0.273275	0.136637	−	0.990621i	$-0.456370\pi$
$857$	8.00000	0.273275	0.136637	+	0.990621i	$0.456370\pi$
$858$	0	0
$859$	27.0000	0.921228	0.460614	−	0.887601i	$-0.347629\pi$
$859$	27.0000	0.921228	0.460614	+	0.887601i	$0.347629\pi$
$860$	−6.00000	−0.204598
$861$	0	0
$862$	68.0000	2.31609
$863$	44.0000	1.49778	0.748889	−	0.662696i	$-0.230588\pi$
$863$	44.0000	1.49778	0.748889	+	0.662696i	$0.230588\pi$
$864$	0	0
$865$	−18.0000	−0.612018
$866$	12.0000	0.407777
$867$	0	0
$868$	60.0000	2.03653
$869$	−16.0000	−0.542763
$870$	0	0
$871$	16.0000	0.542139
$872$	0	0
$873$	0	0
$874$	−8.00000	−0.270604
$875$	15.0000	0.507093
$876$	0	0
$877$	−6.00000	−0.202606	−0.101303	−	0.994856i	$-0.532301\pi$
$877$	−6.00000	−0.202606	−0.101303	+	0.994856i	$0.532301\pi$
$878$	52.0000	1.75491
$879$	0	0
$880$	12.0000	0.404520
$881$	37.0000	1.24656	0.623281	−	0.781998i	$-0.285799\pi$
$881$	37.0000	1.24656	0.623281	+	0.781998i	$0.285799\pi$
$882$	0	0
$883$	35.0000	1.17784	0.588922	−	0.808190i	$-0.299553\pi$
$883$	35.0000	1.17784	0.588922	+	0.808190i	$0.299553\pi$
$884$	4.00000	0.134535
$885$	0	0
$886$	10.0000	0.335957
$887$	12.0000	0.402921	0.201460	−	0.979497i	$-0.435431\pi$
$887$	12.0000	0.402921	0.201460	+	0.979497i	$0.435431\pi$
$888$	0	0
$889$	10.0000	0.335389
$890$	36.0000	1.20672
$891$	0	0
$892$	24.0000	0.803579
$893$	−9.00000	−0.301174
$894$	0	0
$895$	54.0000	1.80502
$896$	0	0
$897$	0	0
$898$	72.0000	2.40267
$899$	−12.0000	−0.400222
$900$	0	0
$901$	−10.0000	−0.333148
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−42.0000	−1.39613
$906$	0	0
$907$	−26.0000	−0.863316	−0.431658	−	0.902037i	$-0.642071\pi$
$907$	−26.0000	−0.863316	−0.431658	+	0.902037i	$0.642071\pi$
$908$	−36.0000	−1.19470
$909$	0	0
$910$	−60.0000	−1.98898
$911$	6.00000	0.198789	0.0993944	−	0.995048i	$-0.468309\pi$
$911$	6.00000	0.198789	0.0993944	+	0.995048i	$0.468309\pi$
$912$	0	0
$913$	12.0000	0.397142
$914$	−58.0000	−1.91847
$915$	0	0
$916$	50.0000	1.65205
$917$	35.0000	1.15580
$918$	0	0
$919$	8.00000	0.263896	0.131948	−	0.991257i	$-0.457877\pi$
$919$	8.00000	0.263896	0.131948	+	0.991257i	$0.457877\pi$
$920$	0	0
$921$	0	0
$922$	−54.0000	−1.77840
$923$	24.0000	0.789970
$924$	0	0
$925$	0	0
$926$	34.0000	1.11731
$927$	0	0
$928$	−16.0000	−0.525226
$929$	2.00000	0.0656179	0.0328089	−	0.999462i	$-0.489555\pi$
$929$	2.00000	0.0656179	0.0328089	+	0.999462i	$0.489555\pi$
$930$	0	0
$931$	−18.0000	−0.589926
$932$	−18.0000	−0.589610
$933$	0	0
$934$	10.0000	0.327210
$935$	−3.00000	−0.0981105
$936$	0	0
$937$	21.0000	0.686040	0.343020	−	0.939328i	$-0.388550\pi$
$937$	21.0000	0.686040	0.343020	+	0.939328i	$0.388550\pi$
$938$	−80.0000	−2.61209
$939$	0	0
$940$	54.0000	1.76129
$941$	−42.0000	−1.36916	−0.684580	−	0.728937i	$-0.740015\pi$
$941$	−42.0000	−1.36916	−0.684580	+	0.728937i	$0.740015\pi$
$942$	0	0
$943$	0	0
$944$	−32.0000	−1.04151
$945$	0	0
$946$	2.00000	0.0650256
$947$	28.0000	0.909878	0.454939	−	0.890523i	$-0.349661\pi$
$947$	28.0000	0.909878	0.454939	+	0.890523i	$0.349661\pi$
$948$	0	0
$949$	−22.0000	−0.714150
$950$	−8.00000	−0.259554
$951$	0	0
$952$	0	0
$953$	32.0000	1.03658	0.518291	−	0.855204i	$-0.326568\pi$
$953$	32.0000	1.03658	0.518291	+	0.855204i	$0.326568\pi$
$954$	0	0
$955$	−27.0000	−0.873699
$956$	6.00000	0.194054
$957$	0	0
$958$	32.0000	1.03387
$959$	−45.0000	−1.45313
$960$	0	0
$961$	5.00000	0.161290
$962$	0	0
$963$	0	0
$964$	40.0000	1.28831
$965$	12.0000	0.386294
$966$	0	0
$967$	−32.0000	−1.02905	−0.514525	−	0.857475i	$-0.672032\pi$
$967$	−32.0000	−1.02905	−0.514525	+	0.857475i	$0.672032\pi$
$968$	0	0
$969$	0	0
$970$	−60.0000	−1.92648
$971$	−12.0000	−0.385098	−0.192549	−	0.981287i	$-0.561675\pi$
$971$	−12.0000	−0.385098	−0.192549	+	0.981287i	$0.561675\pi$
$972$	0	0
$973$	65.0000	2.08380
$974$	−32.0000	−1.02535
$975$	0	0
$976$	4.00000	0.128037
$977$	−54.0000	−1.72761	−0.863807	−	0.503824i	$-0.831926\pi$
$977$	−54.0000	−1.72761	−0.863807	+	0.503824i	$0.831926\pi$
$978$	0	0
$979$	−6.00000	−0.191761
$980$	108.000	3.44993
$981$	0	0
$982$	0	0
$983$	44.0000	1.40338	0.701691	−	0.712481i	$-0.252429\pi$
$983$	44.0000	1.40338	0.701691	+	0.712481i	$0.252429\pi$
$984$	0	0
$985$	6.00000	0.191176
$986$	4.00000	0.127386
$987$	0	0
$988$	−4.00000	−0.127257
$989$	−4.00000	−0.127193
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	48.0000	1.52400
$993$	0	0
$994$	−120.000	−3.80617
$995$	−63.0000	−1.99723
$996$	0	0
$997$	−47.0000	−1.48850	−0.744252	−	0.667898i	$-0.767194\pi$
$997$	−47.0000	−1.48850	−0.744252	+	0.667898i	$0.767194\pi$
$998$	10.0000	0.316544
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	171.2.a.d.1.1		1
3.2	odd	2		57.2.a.a.1.1	✓	1
4.3	odd	2		2736.2.a.v.1.1		1
5.4	even	2		4275.2.a.b.1.1		1
7.6	odd	2		8379.2.a.p.1.1		1
12.11	even	2		912.2.a.g.1.1		1
15.2	even	4		1425.2.c.b.799.1		2
15.8	even	4		1425.2.c.b.799.2		2
15.14	odd	2		1425.2.a.j.1.1		1
19.18	odd	2		3249.2.a.b.1.1		1
21.20	even	2		2793.2.a.b.1.1		1
24.5	odd	2		3648.2.a.bh.1.1		1
24.11	even	2		3648.2.a.r.1.1		1
33.32	even	2		6897.2.a.f.1.1		1
39.38	odd	2		9633.2.a.o.1.1		1
57.56	even	2		1083.2.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
57.2.a.a.1.1	✓	1	3.2	odd	2
171.2.a.d.1.1		1	1.1	even	1	trivial
912.2.a.g.1.1		1	12.11	even	2
1083.2.a.e.1.1		1	57.56	even	2
1425.2.a.j.1.1		1	15.14	odd	2
1425.2.c.b.799.1		2	15.2	even	4
1425.2.c.b.799.2		2	15.8	even	4
2736.2.a.v.1.1		1	4.3	odd	2
2793.2.a.b.1.1		1	21.20	even	2
3249.2.a.b.1.1		1	19.18	odd	2
3648.2.a.r.1.1		1	24.11	even	2
3648.2.a.bh.1.1		1	24.5	odd	2
4275.2.a.b.1.1		1	5.4	even	2
6897.2.a.f.1.1		1	33.32	even	2
8379.2.a.p.1.1		1	7.6	odd	2
9633.2.a.o.1.1		1	39.38	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 171 = 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	171.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more