LMFDB - Embedded newform 171.2.a.c.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,-1] 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)

$f(q)$

$=$

$q+2.00000 q^{2} +2.00000 q^{4} -1.00000 q^{5} +3.00000 q^{7} -2.00000 q^{10} +3.00000 q^{11} -6.00000 q^{13} +6.00000 q^{14} -4.00000 q^{16} -3.00000 q^{17} -1.00000 q^{19} -2.00000 q^{20} +6.00000 q^{22} -4.00000 q^{23} -4.00000 q^{25} -12.0000 q^{26} +6.00000 q^{28} +10.0000 q^{29} +2.00000 q^{31} -8.00000 q^{32} -6.00000 q^{34} -3.00000 q^{35} +8.00000 q^{37} -2.00000 q^{38} +8.00000 q^{41} -1.00000 q^{43} +6.00000 q^{44} -8.00000 q^{46} -3.00000 q^{47} +2.00000 q^{49} -8.00000 q^{50} -12.0000 q^{52} +6.00000 q^{53} -3.00000 q^{55} +20.0000 q^{58} +7.00000 q^{61} +4.00000 q^{62} -8.00000 q^{64} +6.00000 q^{65} +8.00000 q^{67} -6.00000 q^{68} -6.00000 q^{70} -12.0000 q^{71} -11.0000 q^{73} +16.0000 q^{74} -2.00000 q^{76} +9.00000 q^{77} +4.00000 q^{80} +16.0000 q^{82} -4.00000 q^{83} +3.00000 q^{85} -2.00000 q^{86} -10.0000 q^{89} -18.0000 q^{91} -8.00000 q^{92} -6.00000 q^{94} +1.00000 q^{95} -2.00000 q^{97} +4.00000 q^{98} +O(q^{100})$

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
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	0	0
$7$	3.00000	1.13389	0.566947	−	0.823754i	$-0.308125\pi$
$7$	3.00000	1.13389	0.566947	+	0.823754i	$0.308125\pi$
$8$	0	0
$9$	0	0
$10$	−2.00000	−0.632456
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	0	0
$13$	−6.00000	−1.66410	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	−6.00000	−1.66410	−0.832050	+	0.554700i	$0.812833\pi$
$14$	6.00000	1.60357
$15$	0	0
$16$	−4.00000	−1.00000
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−2.00000	−0.447214
$21$	0	0
$22$	6.00000	1.27920
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	−12.0000	−2.35339
$27$	0	0
$28$	6.00000	1.13389
$29$	10.0000	1.85695	0.928477	−	0.371391i	$-0.121119\pi$
$29$	10.0000	1.85695	0.928477	+	0.371391i	$0.121119\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	−8.00000	−1.41421
$33$	0	0
$34$	−6.00000	−1.02899
$35$	−3.00000	−0.507093
$36$	0	0
$37$	8.00000	1.31519	0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000	1.31519	0.657596	+	0.753371i	$0.271573\pi$
$38$	−2.00000	−0.324443
$39$	0	0
$40$	0	0
$41$	8.00000	1.24939	0.624695	−	0.780869i	$-0.285223\pi$
$41$	8.00000	1.24939	0.624695	+	0.780869i	$0.285223\pi$
$42$	0	0
$43$	−1.00000	−0.152499	−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000	−0.152499	−0.0762493	+	0.997089i	$0.524294\pi$
$44$	6.00000	0.904534
$45$	0	0
$46$	−8.00000	−1.17954
$47$	−3.00000	−0.437595	−0.218797	−	0.975770i	$-0.570213\pi$
$47$	−3.00000	−0.437595	−0.218797	+	0.975770i	$0.570213\pi$
$48$	0	0
$49$	2.00000	0.285714
$50$	−8.00000	−1.13137
$51$	0	0
$52$	−12.0000	−1.66410
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	−3.00000	−0.404520
$56$	0	0
$57$	0	0
$58$	20.0000	2.62613
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	7.00000	0.896258	0.448129	−	0.893969i	$-0.352090\pi$
$61$	7.00000	0.896258	0.448129	+	0.893969i	$0.352090\pi$
$62$	4.00000	0.508001
$63$	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	$0.337479\pi$
$68$	−6.00000	−0.727607
$69$	0	0
$70$	−6.00000	−0.717137
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	0	0
$73$	−11.0000	−1.28745	−0.643726	−	0.765256i	$-0.722612\pi$
$73$	−11.0000	−1.28745	−0.643726	+	0.765256i	$0.722612\pi$
$74$	16.0000	1.85996
$75$	0	0
$76$	−2.00000	−0.229416
$77$	9.00000	1.02565
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	4.00000	0.447214
$81$	0	0
$82$	16.0000	1.76690
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	0	0
$85$	3.00000	0.325396
$86$	−2.00000	−0.215666
$87$	0	0
$88$	0	0
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	−18.0000	−1.88691
$92$	−8.00000	−0.834058
$93$	0	0
$94$	−6.00000	−0.618853
$95$	1.00000	0.102598
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	4.00000	0.404061
$99$	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
$103$	14.0000	1.37946	0.689730	−	0.724066i	$-0.257729\pi$
$103$	14.0000	1.37946	0.689730	+	0.724066i	$0.257729\pi$
$104$	0	0
$105$	0	0
$106$	12.0000	1.16554
$107$	2.00000	0.193347	0.0966736	−	0.995316i	$-0.469180\pi$
$107$	2.00000	0.193347	0.0966736	+	0.995316i	$0.469180\pi$
$108$	0	0
$109$	20.0000	1.91565	0.957826	−	0.287348i	$-0.0927736\pi$
$109$	20.0000	1.91565	0.957826	+	0.287348i	$0.0927736\pi$
$110$	−6.00000	−0.572078
$111$	0	0
$112$	−12.0000	−1.13389
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	4.00000	0.373002
$116$	20.0000	1.85695
$117$	0	0
$118$	0	0
$119$	−9.00000	−0.825029
$120$	0	0
$121$	−2.00000	−0.181818
$122$	14.0000	1.26750
$123$	0	0
$124$	4.00000	0.359211
$125$	9.00000	0.804984
$126$	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
$131$	13.0000	1.13582	0.567908	−	0.823092i	$-0.307753\pi$
$131$	13.0000	1.13582	0.567908	+	0.823092i	$0.307753\pi$
$132$	0	0
$133$	−3.00000	−0.260133
$134$	16.0000	1.38219
$135$	0	0
$136$	0	0
$137$	−3.00000	−0.256307	−0.128154	−	0.991754i	$-0.540905\pi$
$137$	−3.00000	−0.256307	−0.128154	+	0.991754i	$0.540905\pi$
$138$	0	0
$139$	−5.00000	−0.424094	−0.212047	−	0.977259i	$-0.568013\pi$
$139$	−5.00000	−0.424094	−0.212047	+	0.977259i	$0.568013\pi$
$140$	−6.00000	−0.507093
$141$	0	0
$142$	−24.0000	−2.01404
$143$	−18.0000	−1.50524
$144$	0	0
$145$	−10.0000	−0.830455
$146$	−22.0000	−1.82073
$147$	0	0
$148$	16.0000	1.31519
$149$	−15.0000	−1.22885	−0.614424	−	0.788976i	$-0.710612\pi$
$149$	−15.0000	−1.22885	−0.614424	+	0.788976i	$0.710612\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	0	0
$153$	0	0
$154$	18.0000	1.45048
$155$	−2.00000	−0.160644
$156$	0	0
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	0	0
$159$	0	0
$160$	8.00000	0.632456
$161$	−12.0000	−0.945732
$162$	0	0
$163$	−16.0000	−1.25322	−0.626608	−	0.779334i	$-0.715557\pi$
$163$	−16.0000	−1.25322	−0.626608	+	0.779334i	$0.715557\pi$
$164$	16.0000	1.24939
$165$	0	0
$166$	−8.00000	−0.620920
$167$	−18.0000	−1.39288	−0.696441	−	0.717614i	$-0.745234\pi$
$167$	−18.0000	−1.39288	−0.696441	+	0.717614i	$0.745234\pi$
$168$	0	0
$169$	23.0000	1.76923
$170$	6.00000	0.460179
$171$	0	0
$172$	−2.00000	−0.152499
$173$	−14.0000	−1.06440	−0.532200	−	0.846619i	$-0.678635\pi$
$173$	−14.0000	−1.06440	−0.532200	+	0.846619i	$0.678635\pi$
$174$	0	0
$175$	−12.0000	−0.907115
$176$	−12.0000	−0.904534
$177$	0	0
$178$	−20.0000	−1.49906
$179$	10.0000	0.747435	0.373718	−	0.927543i	$-0.378083\pi$
$179$	10.0000	0.747435	0.373718	+	0.927543i	$0.378083\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	−36.0000	−2.66850
$183$	0	0
$184$	0	0
$185$	−8.00000	−0.588172
$186$	0	0
$187$	−9.00000	−0.658145
$188$	−6.00000	−0.437595
$189$	0	0
$190$	2.00000	0.145095
$191$	3.00000	0.217072	0.108536	−	0.994092i	$-0.465384\pi$
$191$	3.00000	0.217072	0.108536	+	0.994092i	$0.465384\pi$
$192$	0	0
$193$	4.00000	0.287926	0.143963	−	0.989583i	$-0.454015\pi$
$193$	4.00000	0.287926	0.143963	+	0.989583i	$0.454015\pi$
$194$	−4.00000	−0.287183
$195$	0	0
$196$	4.00000	0.285714
$197$	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
$199$	−5.00000	−0.354441	−0.177220	−	0.984171i	$-0.556711\pi$
$199$	−5.00000	−0.354441	−0.177220	+	0.984171i	$0.556711\pi$
$200$	0	0
$201$	0	0
$202$	−4.00000	−0.281439
$203$	30.0000	2.10559
$204$	0	0
$205$	−8.00000	−0.558744
$206$	28.0000	1.95085
$207$	0	0
$208$	24.0000	1.66410
$209$	−3.00000	−0.207514
$210$	0	0
$211$	−28.0000	−1.92760	−0.963800	−	0.266627i	$-0.914091\pi$
$211$	−28.0000	−1.92760	−0.963800	+	0.266627i	$0.914091\pi$
$212$	12.0000	0.824163
$213$	0	0
$214$	4.00000	0.273434
$215$	1.00000	0.0681994
$216$	0	0
$217$	6.00000	0.407307
$218$	40.0000	2.70914
$219$	0	0
$220$	−6.00000	−0.404520
$221$	18.0000	1.21081
$222$	0	0
$223$	4.00000	0.267860	0.133930	−	0.990991i	$-0.457240\pi$
$223$	4.00000	0.267860	0.133930	+	0.990991i	$0.457240\pi$
$224$	−24.0000	−1.60357
$225$	0	0
$226$	12.0000	0.798228
$227$	−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
$229$	−15.0000	−0.991228	−0.495614	−	0.868543i	$-0.665057\pi$
$229$	−15.0000	−0.991228	−0.495614	+	0.868543i	$0.665057\pi$
$230$	8.00000	0.527504
$231$	0	0
$232$	0	0
$233$	11.0000	0.720634	0.360317	−	0.932830i	$-0.382669\pi$
$233$	11.0000	0.720634	0.360317	+	0.932830i	$0.382669\pi$
$234$	0	0
$235$	3.00000	0.195698
$236$	0	0
$237$	0	0
$238$	−18.0000	−1.16677
$239$	15.0000	0.970269	0.485135	−	0.874439i	$-0.338771\pi$
$239$	15.0000	0.970269	0.485135	+	0.874439i	$0.338771\pi$
$240$	0	0
$241$	12.0000	0.772988	0.386494	−	0.922292i	$-0.373686\pi$
$241$	12.0000	0.772988	0.386494	+	0.922292i	$0.373686\pi$
$242$	−4.00000	−0.257130
$243$	0	0
$244$	14.0000	0.896258
$245$	−2.00000	−0.127775
$246$	0	0
$247$	6.00000	0.381771
$248$	0	0
$249$	0	0
$250$	18.0000	1.13842
$251$	−27.0000	−1.70422	−0.852112	−	0.523359i	$-0.824679\pi$
$251$	−27.0000	−1.70422	−0.852112	+	0.523359i	$0.824679\pi$
$252$	0	0
$253$	−12.0000	−0.754434
$254$	−4.00000	−0.250982
$255$	0	0
$256$	16.0000	1.00000
$257$	−8.00000	−0.499026	−0.249513	−	0.968371i	$-0.580271\pi$
$257$	−8.00000	−0.499026	−0.249513	+	0.968371i	$0.580271\pi$
$258$	0	0
$259$	24.0000	1.49129
$260$	12.0000	0.744208
$261$	0	0
$262$	26.0000	1.60629
$263$	21.0000	1.29492	0.647458	−	0.762101i	$-0.275832\pi$
$263$	21.0000	1.29492	0.647458	+	0.762101i	$0.275832\pi$
$264$	0	0
$265$	−6.00000	−0.368577
$266$	−6.00000	−0.367884
$267$	0	0
$268$	16.0000	0.977356
$269$	30.0000	1.82913	0.914566	−	0.404436i	$-0.132532\pi$
$269$	30.0000	1.82913	0.914566	+	0.404436i	$0.132532\pi$
$270$	0	0
$271$	12.0000	0.728948	0.364474	−	0.931214i	$-0.381249\pi$
$271$	12.0000	0.728948	0.364474	+	0.931214i	$0.381249\pi$
$272$	12.0000	0.727607
$273$	0	0
$274$	−6.00000	−0.362473
$275$	−12.0000	−0.723627
$276$	0	0
$277$	13.0000	0.781094	0.390547	−	0.920583i	$-0.372286\pi$
$277$	13.0000	0.781094	0.390547	+	0.920583i	$0.372286\pi$
$278$	−10.0000	−0.599760
$279$	0	0
$280$	0	0
$281$	−2.00000	−0.119310	−0.0596550	−	0.998219i	$-0.519000\pi$
$281$	−2.00000	−0.119310	−0.0596550	+	0.998219i	$0.519000\pi$
$282$	0	0
$283$	19.0000	1.12943	0.564716	−	0.825285i	$-0.308986\pi$
$283$	19.0000	1.12943	0.564716	+	0.825285i	$0.308986\pi$
$284$	−24.0000	−1.42414
$285$	0	0
$286$	−36.0000	−2.12872
$287$	24.0000	1.41668
$288$	0	0
$289$	−8.00000	−0.470588
$290$	−20.0000	−1.17444
$291$	0	0
$292$	−22.0000	−1.28745
$293$	−4.00000	−0.233682	−0.116841	−	0.993151i	$-0.537277\pi$
$293$	−4.00000	−0.233682	−0.116841	+	0.993151i	$0.537277\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	−30.0000	−1.73785
$299$	24.0000	1.38796
$300$	0	0
$301$	−3.00000	−0.172917
$302$	−16.0000	−0.920697
$303$	0	0
$304$	4.00000	0.229416
$305$	−7.00000	−0.400819
$306$	0	0
$307$	−12.0000	−0.684876	−0.342438	−	0.939540i	$-0.611253\pi$
$307$	−12.0000	−0.684876	−0.342438	+	0.939540i	$0.611253\pi$
$308$	18.0000	1.02565
$309$	0	0
$310$	−4.00000	−0.227185
$311$	−7.00000	−0.396934	−0.198467	−	0.980108i	$-0.563596\pi$
$311$	−7.00000	−0.396934	−0.198467	+	0.980108i	$0.563596\pi$
$312$	0	0
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	−4.00000	−0.225733
$315$	0	0
$316$	0	0
$317$	12.0000	0.673987	0.336994	−	0.941507i	$-0.390590\pi$
$317$	12.0000	0.673987	0.336994	+	0.941507i	$0.390590\pi$
$318$	0	0
$319$	30.0000	1.67968
$320$	8.00000	0.447214
$321$	0	0
$322$	−24.0000	−1.33747
$323$	3.00000	0.166924
$324$	0	0
$325$	24.0000	1.33128
$326$	−32.0000	−1.77232
$327$	0	0
$328$	0	0
$329$	−9.00000	−0.496186
$330$	0	0
$331$	12.0000	0.659580	0.329790	−	0.944054i	$-0.393022\pi$
$331$	12.0000	0.659580	0.329790	+	0.944054i	$0.393022\pi$
$332$	−8.00000	−0.439057
$333$	0	0
$334$	−36.0000	−1.96983
$335$	−8.00000	−0.437087
$336$	0	0
$337$	−22.0000	−1.19842	−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000	−1.19842	−0.599208	+	0.800593i	$0.704518\pi$
$338$	46.0000	2.50207
$339$	0	0
$340$	6.00000	0.325396
$341$	6.00000	0.324918
$342$	0	0
$343$	−15.0000	−0.809924
$344$	0	0
$345$	0	0
$346$	−28.0000	−1.50529
$347$	−3.00000	−0.161048	−0.0805242	−	0.996753i	$-0.525659\pi$
$347$	−3.00000	−0.161048	−0.0805242	+	0.996753i	$0.525659\pi$
$348$	0	0
$349$	25.0000	1.33822	0.669110	−	0.743164i	$-0.266676\pi$
$349$	25.0000	1.33822	0.669110	+	0.743164i	$0.266676\pi$
$350$	−24.0000	−1.28285
$351$	0	0
$352$	−24.0000	−1.27920
$353$	−14.0000	−0.745145	−0.372572	−	0.928003i	$-0.621524\pi$
$353$	−14.0000	−0.745145	−0.372572	+	0.928003i	$0.621524\pi$
$354$	0	0
$355$	12.0000	0.636894
$356$	−20.0000	−1.06000
$357$	0	0
$358$	20.0000	1.05703
$359$	−25.0000	−1.31945	−0.659725	−	0.751507i	$-0.729327\pi$
$359$	−25.0000	−1.31945	−0.659725	+	0.751507i	$0.729327\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	4.00000	0.210235
$363$	0	0
$364$	−36.0000	−1.88691
$365$	11.0000	0.575766
$366$	0	0
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	16.0000	0.834058
$369$	0	0
$370$	−16.0000	−0.831800
$371$	18.0000	0.934513
$372$	0	0
$373$	−16.0000	−0.828449	−0.414224	−	0.910175i	$-0.635947\pi$
$373$	−16.0000	−0.828449	−0.414224	+	0.910175i	$0.635947\pi$
$374$	−18.0000	−0.930758
$375$	0	0
$376$	0	0
$377$	−60.0000	−3.09016
$378$	0	0
$379$	−30.0000	−1.54100	−0.770498	−	0.637442i	$-0.779993\pi$
$379$	−30.0000	−1.54100	−0.770498	+	0.637442i	$0.779993\pi$
$380$	2.00000	0.102598
$381$	0	0
$382$	6.00000	0.306987
$383$	−14.0000	−0.715367	−0.357683	−	0.933843i	$-0.616433\pi$
$383$	−14.0000	−0.715367	−0.357683	+	0.933843i	$0.616433\pi$
$384$	0	0
$385$	−9.00000	−0.458682
$386$	8.00000	0.407189
$387$	0	0
$388$	−4.00000	−0.203069
$389$	15.0000	0.760530	0.380265	−	0.924878i	$-0.375833\pi$
$389$	15.0000	0.760530	0.380265	+	0.924878i	$0.375833\pi$
$390$	0	0
$391$	12.0000	0.606866
$392$	0	0
$393$	0	0
$394$	4.00000	0.201517
$395$	0	0
$396$	0	0
$397$	−7.00000	−0.351320	−0.175660	−	0.984451i	$-0.556206\pi$
$397$	−7.00000	−0.351320	−0.175660	+	0.984451i	$0.556206\pi$
$398$	−10.0000	−0.501255
$399$	0	0
$400$	16.0000	0.800000
$401$	28.0000	1.39825	0.699127	−	0.714998i	$-0.253572\pi$
$401$	28.0000	1.39825	0.699127	+	0.714998i	$0.253572\pi$
$402$	0	0
$403$	−12.0000	−0.597763
$404$	−4.00000	−0.199007
$405$	0	0
$406$	60.0000	2.97775
$407$	24.0000	1.18964
$408$	0	0
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	−16.0000	−0.790184
$411$	0	0
$412$	28.0000	1.37946
$413$	0	0
$414$	0	0
$415$	4.00000	0.196352
$416$	48.0000	2.35339
$417$	0	0
$418$	−6.00000	−0.293470
$419$	−20.0000	−0.977064	−0.488532	−	0.872546i	$-0.662467\pi$
$419$	−20.0000	−0.977064	−0.488532	+	0.872546i	$0.662467\pi$
$420$	0	0
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	−56.0000	−2.72604
$423$	0	0
$424$	0	0
$425$	12.0000	0.582086
$426$	0	0
$427$	21.0000	1.01626
$428$	4.00000	0.193347
$429$	0	0
$430$	2.00000	0.0964486
$431$	18.0000	0.867029	0.433515	−	0.901146i	$-0.357273\pi$
$431$	18.0000	0.867029	0.433515	+	0.901146i	$0.357273\pi$
$432$	0	0
$433$	−26.0000	−1.24948	−0.624740	−	0.780833i	$-0.714795\pi$
$433$	−26.0000	−1.24948	−0.624740	+	0.780833i	$0.714795\pi$
$434$	12.0000	0.576018
$435$	0	0
$436$	40.0000	1.91565
$437$	4.00000	0.191346
$438$	0	0
$439$	10.0000	0.477274	0.238637	−	0.971109i	$-0.423299\pi$
$439$	10.0000	0.477274	0.238637	+	0.971109i	$0.423299\pi$
$440$	0	0
$441$	0	0
$442$	36.0000	1.71235
$443$	−39.0000	−1.85295	−0.926473	−	0.376361i	$-0.877175\pi$
$443$	−39.0000	−1.85295	−0.926473	+	0.376361i	$0.877175\pi$
$444$	0	0
$445$	10.0000	0.474045
$446$	8.00000	0.378811
$447$	0	0
$448$	−24.0000	−1.13389
$449$	20.0000	0.943858	0.471929	−	0.881636i	$-0.343558\pi$
$449$	20.0000	0.943858	0.471929	+	0.881636i	$0.343558\pi$
$450$	0	0
$451$	24.0000	1.13012
$452$	12.0000	0.564433
$453$	0	0
$454$	−36.0000	−1.68956
$455$	18.0000	0.843853
$456$	0	0
$457$	3.00000	0.140334	0.0701670	−	0.997535i	$-0.477647\pi$
$457$	3.00000	0.140334	0.0701670	+	0.997535i	$0.477647\pi$
$458$	−30.0000	−1.40181
$459$	0	0
$460$	8.00000	0.373002
$461$	33.0000	1.53696	0.768482	−	0.639872i	$-0.221013\pi$
$461$	33.0000	1.53696	0.768482	+	0.639872i	$0.221013\pi$
$462$	0	0
$463$	−31.0000	−1.44069	−0.720346	−	0.693615i	$-0.756017\pi$
$463$	−31.0000	−1.44069	−0.720346	+	0.693615i	$0.756017\pi$
$464$	−40.0000	−1.85695
$465$	0	0
$466$	22.0000	1.01913
$467$	17.0000	0.786666	0.393333	−	0.919396i	$-0.371322\pi$
$467$	17.0000	0.786666	0.393333	+	0.919396i	$0.371322\pi$
$468$	0	0
$469$	24.0000	1.10822
$470$	6.00000	0.276759
$471$	0	0
$472$	0	0
$473$	−3.00000	−0.137940
$474$	0	0
$475$	4.00000	0.183533
$476$	−18.0000	−0.825029
$477$	0	0
$478$	30.0000	1.37217
$479$	40.0000	1.82765	0.913823	−	0.406112i	$-0.133116\pi$
$479$	40.0000	1.82765	0.913823	+	0.406112i	$0.133116\pi$
$480$	0	0
$481$	−48.0000	−2.18861
$482$	24.0000	1.09317
$483$	0	0
$484$	−4.00000	−0.181818
$485$	2.00000	0.0908153
$486$	0	0
$487$	8.00000	0.362515	0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000	0.362515	0.181257	+	0.983436i	$0.441983\pi$
$488$	0	0
$489$	0	0
$490$	−4.00000	−0.180702
$491$	8.00000	0.361035	0.180517	−	0.983572i	$-0.442223\pi$
$491$	8.00000	0.361035	0.180517	+	0.983572i	$0.442223\pi$
$492$	0	0
$493$	−30.0000	−1.35113
$494$	12.0000	0.539906
$495$	0	0
$496$	−8.00000	−0.359211
$497$	−36.0000	−1.61482
$498$	0	0
$499$	−35.0000	−1.56682	−0.783408	−	0.621508i	$-0.786520\pi$
$499$	−35.0000	−1.56682	−0.783408	+	0.621508i	$0.786520\pi$
$500$	18.0000	0.804984
$501$	0	0
$502$	−54.0000	−2.41014
$503$	−24.0000	−1.07011	−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000	−1.07011	−0.535054	+	0.844818i	$0.679709\pi$
$504$	0	0
$505$	2.00000	0.0889988
$506$	−24.0000	−1.06693
$507$	0	0
$508$	−4.00000	−0.177471
$509$	10.0000	0.443242	0.221621	−	0.975133i	$-0.428865\pi$
$509$	10.0000	0.443242	0.221621	+	0.975133i	$0.428865\pi$
$510$	0	0
$511$	−33.0000	−1.45983
$512$	32.0000	1.41421
$513$	0	0
$514$	−16.0000	−0.705730
$515$	−14.0000	−0.616914
$516$	0	0
$517$	−9.00000	−0.395820
$518$	48.0000	2.10900
$519$	0	0
$520$	0	0
$521$	28.0000	1.22670	0.613351	−	0.789810i	$-0.289821\pi$
$521$	28.0000	1.22670	0.613351	+	0.789810i	$0.289821\pi$
$522$	0	0
$523$	14.0000	0.612177	0.306089	−	0.952003i	$-0.400980\pi$
$523$	14.0000	0.612177	0.306089	+	0.952003i	$0.400980\pi$
$524$	26.0000	1.13582
$525$	0	0
$526$	42.0000	1.83129
$527$	−6.00000	−0.261364
$528$	0	0
$529$	−7.00000	−0.304348
$530$	−12.0000	−0.521247
$531$	0	0
$532$	−6.00000	−0.260133
$533$	−48.0000	−2.07911
$534$	0	0
$535$	−2.00000	−0.0864675
$536$	0	0
$537$	0	0
$538$	60.0000	2.58678
$539$	6.00000	0.258438
$540$	0	0
$541$	−13.0000	−0.558914	−0.279457	−	0.960158i	$-0.590154\pi$
$541$	−13.0000	−0.558914	−0.279457	+	0.960158i	$0.590154\pi$
$542$	24.0000	1.03089
$543$	0	0
$544$	24.0000	1.02899
$545$	−20.0000	−0.856706
$546$	0	0
$547$	−2.00000	−0.0855138	−0.0427569	−	0.999086i	$-0.513614\pi$
$547$	−2.00000	−0.0855138	−0.0427569	+	0.999086i	$0.513614\pi$
$548$	−6.00000	−0.256307
$549$	0	0
$550$	−24.0000	−1.02336
$551$	−10.0000	−0.426014
$552$	0	0
$553$	0	0
$554$	26.0000	1.10463
$555$	0	0
$556$	−10.0000	−0.424094
$557$	−3.00000	−0.127114	−0.0635570	−	0.997978i	$-0.520244\pi$
$557$	−3.00000	−0.127114	−0.0635570	+	0.997978i	$0.520244\pi$
$558$	0	0
$559$	6.00000	0.253773
$560$	12.0000	0.507093
$561$	0	0
$562$	−4.00000	−0.168730
$563$	−44.0000	−1.85438	−0.927189	−	0.374593i	$-0.877783\pi$
$563$	−44.0000	−1.85438	−0.927189	+	0.374593i	$0.877783\pi$
$564$	0	0
$565$	−6.00000	−0.252422
$566$	38.0000	1.59726
$567$	0	0
$568$	0	0
$569$	−30.0000	−1.25767	−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000	−1.25767	−0.628833	+	0.777541i	$0.716467\pi$
$570$	0	0
$571$	32.0000	1.33916	0.669579	−	0.742741i	$-0.266474\pi$
$571$	32.0000	1.33916	0.669579	+	0.742741i	$0.266474\pi$
$572$	−36.0000	−1.50524
$573$	0	0
$574$	48.0000	2.00348
$575$	16.0000	0.667246
$576$	0	0
$577$	3.00000	0.124892	0.0624458	−	0.998048i	$-0.480110\pi$
$577$	3.00000	0.124892	0.0624458	+	0.998048i	$0.480110\pi$
$578$	−16.0000	−0.665512
$579$	0	0
$580$	−20.0000	−0.830455
$581$	−12.0000	−0.497844
$582$	0	0
$583$	18.0000	0.745484
$584$	0	0
$585$	0	0
$586$	−8.00000	−0.330477
$587$	37.0000	1.52715	0.763577	−	0.645717i	$-0.223441\pi$
$587$	37.0000	1.52715	0.763577	+	0.645717i	$0.223441\pi$
$588$	0	0
$589$	−2.00000	−0.0824086
$590$	0	0
$591$	0	0
$592$	−32.0000	−1.31519
$593$	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
$595$	9.00000	0.368964
$596$	−30.0000	−1.22885
$597$	0	0
$598$	48.0000	1.96287
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	−8.00000	−0.326327	−0.163163	−	0.986599i	$-0.552170\pi$
$601$	−8.00000	−0.326327	−0.163163	+	0.986599i	$0.552170\pi$
$602$	−6.00000	−0.244542
$603$	0	0
$604$	−16.0000	−0.651031
$605$	2.00000	0.0813116
$606$	0	0
$607$	18.0000	0.730597	0.365299	−	0.930890i	$-0.380967\pi$
$607$	18.0000	0.730597	0.365299	+	0.930890i	$0.380967\pi$
$608$	8.00000	0.324443
$609$	0	0
$610$	−14.0000	−0.566843
$611$	18.0000	0.728202
$612$	0	0
$613$	9.00000	0.363507	0.181753	−	0.983344i	$-0.441823\pi$
$613$	9.00000	0.363507	0.181753	+	0.983344i	$0.441823\pi$
$614$	−24.0000	−0.968561
$615$	0	0
$616$	0	0
$617$	−23.0000	−0.925945	−0.462973	−	0.886373i	$-0.653217\pi$
$617$	−23.0000	−0.925945	−0.462973	+	0.886373i	$0.653217\pi$
$618$	0	0
$619$	20.0000	0.803868	0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000	0.803868	0.401934	+	0.915669i	$0.368338\pi$
$620$	−4.00000	−0.160644
$621$	0	0
$622$	−14.0000	−0.561349
$623$	−30.0000	−1.20192
$624$	0	0
$625$	11.0000	0.440000
$626$	28.0000	1.11911
$627$	0	0
$628$	−4.00000	−0.159617
$629$	−24.0000	−0.956943
$630$	0	0
$631$	7.00000	0.278666	0.139333	−	0.990246i	$-0.455504\pi$
$631$	7.00000	0.278666	0.139333	+	0.990246i	$0.455504\pi$
$632$	0	0
$633$	0	0
$634$	24.0000	0.953162
$635$	2.00000	0.0793676
$636$	0	0
$637$	−12.0000	−0.475457
$638$	60.0000	2.37542
$639$	0	0
$640$	0	0
$641$	−2.00000	−0.0789953	−0.0394976	−	0.999220i	$-0.512576\pi$
$641$	−2.00000	−0.0789953	−0.0394976	+	0.999220i	$0.512576\pi$
$642$	0	0
$643$	−1.00000	−0.0394362	−0.0197181	−	0.999806i	$-0.506277\pi$
$643$	−1.00000	−0.0394362	−0.0197181	+	0.999806i	$0.506277\pi$
$644$	−24.0000	−0.945732
$645$	0	0
$646$	6.00000	0.236067
$647$	27.0000	1.06148	0.530740	−	0.847535i	$-0.321914\pi$
$647$	27.0000	1.06148	0.530740	+	0.847535i	$0.321914\pi$
$648$	0	0
$649$	0	0
$650$	48.0000	1.88271
$651$	0	0
$652$	−32.0000	−1.25322
$653$	1.00000	0.0391330	0.0195665	−	0.999809i	$-0.493771\pi$
$653$	1.00000	0.0391330	0.0195665	+	0.999809i	$0.493771\pi$
$654$	0	0
$655$	−13.0000	−0.507952
$656$	−32.0000	−1.24939
$657$	0	0
$658$	−18.0000	−0.701713
$659$	−10.0000	−0.389545	−0.194772	−	0.980848i	$-0.562397\pi$
$659$	−10.0000	−0.389545	−0.194772	+	0.980848i	$0.562397\pi$
$660$	0	0
$661$	12.0000	0.466746	0.233373	−	0.972387i	$-0.425024\pi$
$661$	12.0000	0.466746	0.233373	+	0.972387i	$0.425024\pi$
$662$	24.0000	0.932786
$663$	0	0
$664$	0	0
$665$	3.00000	0.116335
$666$	0	0
$667$	−40.0000	−1.54881
$668$	−36.0000	−1.39288
$669$	0	0
$670$	−16.0000	−0.618134
$671$	21.0000	0.810696
$672$	0	0
$673$	−16.0000	−0.616755	−0.308377	−	0.951264i	$-0.599786\pi$
$673$	−16.0000	−0.616755	−0.308377	+	0.951264i	$0.599786\pi$
$674$	−44.0000	−1.69482
$675$	0	0
$676$	46.0000	1.76923
$677$	22.0000	0.845529	0.422764	−	0.906240i	$-0.361060\pi$
$677$	22.0000	0.845529	0.422764	+	0.906240i	$0.361060\pi$
$678$	0	0
$679$	−6.00000	−0.230259
$680$	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
$685$	3.00000	0.114624
$686$	−30.0000	−1.14541
$687$	0	0
$688$	4.00000	0.152499
$689$	−36.0000	−1.37149
$690$	0	0
$691$	17.0000	0.646710	0.323355	−	0.946278i	$-0.395189\pi$
$691$	17.0000	0.646710	0.323355	+	0.946278i	$0.395189\pi$
$692$	−28.0000	−1.06440
$693$	0	0
$694$	−6.00000	−0.227757
$695$	5.00000	0.189661
$696$	0	0
$697$	−24.0000	−0.909065
$698$	50.0000	1.89253
$699$	0	0
$700$	−24.0000	−0.907115
$701$	−42.0000	−1.58632	−0.793159	−	0.609015i	$-0.791565\pi$
$701$	−42.0000	−1.58632	−0.793159	+	0.609015i	$0.791565\pi$
$702$	0	0
$703$	−8.00000	−0.301726
$704$	−24.0000	−0.904534
$705$	0	0
$706$	−28.0000	−1.05379
$707$	−6.00000	−0.225653
$708$	0	0
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	24.0000	0.900704
$711$	0	0
$712$	0	0
$713$	−8.00000	−0.299602
$714$	0	0
$715$	18.0000	0.673162
$716$	20.0000	0.747435
$717$	0	0
$718$	−50.0000	−1.86598
$719$	35.0000	1.30528	0.652640	−	0.757668i	$-0.273661\pi$
$719$	35.0000	1.30528	0.652640	+	0.757668i	$0.273661\pi$
$720$	0	0
$721$	42.0000	1.56416
$722$	2.00000	0.0744323
$723$	0	0
$724$	4.00000	0.148659
$725$	−40.0000	−1.48556
$726$	0	0
$727$	−7.00000	−0.259616	−0.129808	−	0.991539i	$-0.541436\pi$
$727$	−7.00000	−0.259616	−0.129808	+	0.991539i	$0.541436\pi$
$728$	0	0
$729$	0	0
$730$	22.0000	0.814257
$731$	3.00000	0.110959
$732$	0	0
$733$	34.0000	1.25582	0.627909	−	0.778287i	$-0.283911\pi$
$733$	34.0000	1.25582	0.627909	+	0.778287i	$0.283911\pi$
$734$	16.0000	0.590571
$735$	0	0
$736$	32.0000	1.17954
$737$	24.0000	0.884051
$738$	0	0
$739$	−45.0000	−1.65535	−0.827676	−	0.561206i	$-0.810337\pi$
$739$	−45.0000	−1.65535	−0.827676	+	0.561206i	$0.810337\pi$
$740$	−16.0000	−0.588172
$741$	0	0
$742$	36.0000	1.32160
$743$	−24.0000	−0.880475	−0.440237	−	0.897881i	$-0.645106\pi$
$743$	−24.0000	−0.880475	−0.440237	+	0.897881i	$0.645106\pi$
$744$	0	0
$745$	15.0000	0.549557
$746$	−32.0000	−1.17160
$747$	0	0
$748$	−18.0000	−0.658145
$749$	6.00000	0.219235
$750$	0	0
$751$	−8.00000	−0.291924	−0.145962	−	0.989290i	$-0.546628\pi$
$751$	−8.00000	−0.291924	−0.145962	+	0.989290i	$0.546628\pi$
$752$	12.0000	0.437595
$753$	0	0
$754$	−120.000	−4.37014
$755$	8.00000	0.291150
$756$	0	0
$757$	23.0000	0.835949	0.417975	−	0.908459i	$-0.362740\pi$
$757$	23.0000	0.835949	0.417975	+	0.908459i	$0.362740\pi$
$758$	−60.0000	−2.17930
$759$	0	0
$760$	0	0
$761$	13.0000	0.471250	0.235625	−	0.971844i	$-0.424286\pi$
$761$	13.0000	0.471250	0.235625	+	0.971844i	$0.424286\pi$
$762$	0	0
$763$	60.0000	2.17215
$764$	6.00000	0.217072
$765$	0	0
$766$	−28.0000	−1.01168
$767$	0	0
$768$	0	0
$769$	−45.0000	−1.62274	−0.811371	−	0.584532i	$-0.801278\pi$
$769$	−45.0000	−1.62274	−0.811371	+	0.584532i	$0.801278\pi$
$770$	−18.0000	−0.648675
$771$	0	0
$772$	8.00000	0.287926
$773$	36.0000	1.29483	0.647415	−	0.762138i	$-0.275850\pi$
$773$	36.0000	1.29483	0.647415	+	0.762138i	$0.275850\pi$
$774$	0	0
$775$	−8.00000	−0.287368
$776$	0	0
$777$	0	0
$778$	30.0000	1.07555
$779$	−8.00000	−0.286630
$780$	0	0
$781$	−36.0000	−1.28818
$782$	24.0000	0.858238
$783$	0	0
$784$	−8.00000	−0.285714
$785$	2.00000	0.0713831
$786$	0	0
$787$	8.00000	0.285169	0.142585	−	0.989783i	$-0.454459\pi$
$787$	8.00000	0.285169	0.142585	+	0.989783i	$0.454459\pi$
$788$	4.00000	0.142494
$789$	0	0
$790$	0	0
$791$	18.0000	0.640006
$792$	0	0
$793$	−42.0000	−1.49146
$794$	−14.0000	−0.496841
$795$	0	0
$796$	−10.0000	−0.354441
$797$	12.0000	0.425062	0.212531	−	0.977154i	$-0.431829\pi$
$797$	12.0000	0.425062	0.212531	+	0.977154i	$0.431829\pi$
$798$	0	0
$799$	9.00000	0.318397
$800$	32.0000	1.13137
$801$	0	0
$802$	56.0000	1.97743
$803$	−33.0000	−1.16454
$804$	0	0
$805$	12.0000	0.422944
$806$	−24.0000	−0.845364
$807$	0	0
$808$	0	0
$809$	−5.00000	−0.175791	−0.0878953	−	0.996130i	$-0.528014\pi$
$809$	−5.00000	−0.175791	−0.0878953	+	0.996130i	$0.528014\pi$
$810$	0	0
$811$	2.00000	0.0702295	0.0351147	−	0.999383i	$-0.488820\pi$
$811$	2.00000	0.0702295	0.0351147	+	0.999383i	$0.488820\pi$
$812$	60.0000	2.10559
$813$	0	0
$814$	48.0000	1.68240
$815$	16.0000	0.560456
$816$	0	0
$817$	1.00000	0.0349856
$818$	20.0000	0.699284
$819$	0	0
$820$	−16.0000	−0.558744
$821$	33.0000	1.15171	0.575854	−	0.817553i	$-0.304670\pi$
$821$	33.0000	1.15171	0.575854	+	0.817553i	$0.304670\pi$
$822$	0	0
$823$	19.0000	0.662298	0.331149	−	0.943578i	$-0.392564\pi$
$823$	19.0000	0.662298	0.331149	+	0.943578i	$0.392564\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	52.0000	1.80822	0.904109	−	0.427303i	$-0.140536\pi$
$827$	52.0000	1.80822	0.904109	+	0.427303i	$0.140536\pi$
$828$	0	0
$829$	20.0000	0.694629	0.347314	−	0.937749i	$-0.387094\pi$
$829$	20.0000	0.694629	0.347314	+	0.937749i	$0.387094\pi$
$830$	8.00000	0.277684
$831$	0	0
$832$	48.0000	1.66410
$833$	−6.00000	−0.207888
$834$	0	0
$835$	18.0000	0.622916
$836$	−6.00000	−0.207514
$837$	0	0
$838$	−40.0000	−1.38178
$839$	−30.0000	−1.03572	−0.517858	−	0.855467i	$-0.673270\pi$
$839$	−30.0000	−1.03572	−0.517858	+	0.855467i	$0.673270\pi$
$840$	0	0
$841$	71.0000	2.44828
$842$	4.00000	0.137849
$843$	0	0
$844$	−56.0000	−1.92760
$845$	−23.0000	−0.791224
$846$	0	0
$847$	−6.00000	−0.206162
$848$	−24.0000	−0.824163
$849$	0	0
$850$	24.0000	0.823193
$851$	−32.0000	−1.09695
$852$	0	0
$853$	34.0000	1.16414	0.582069	−	0.813139i	$-0.302243\pi$
$853$	34.0000	1.16414	0.582069	+	0.813139i	$0.302243\pi$
$854$	42.0000	1.43721
$855$	0	0
$856$	0	0
$857$	−48.0000	−1.63965	−0.819824	−	0.572615i	$-0.805929\pi$
$857$	−48.0000	−1.63965	−0.819824	+	0.572615i	$0.805929\pi$
$858$	0	0
$859$	35.0000	1.19418	0.597092	−	0.802173i	$-0.296323\pi$
$859$	35.0000	1.19418	0.597092	+	0.802173i	$0.296323\pi$
$860$	2.00000	0.0681994
$861$	0	0
$862$	36.0000	1.22616
$863$	−44.0000	−1.49778	−0.748889	−	0.662696i	$-0.769412\pi$
$863$	−44.0000	−1.49778	−0.748889	+	0.662696i	$0.769412\pi$
$864$	0	0
$865$	14.0000	0.476014
$866$	−52.0000	−1.76703
$867$	0	0
$868$	12.0000	0.407307
$869$	0	0
$870$	0	0
$871$	−48.0000	−1.62642
$872$	0	0
$873$	0	0
$874$	8.00000	0.270604
$875$	27.0000	0.912767
$876$	0	0
$877$	−22.0000	−0.742887	−0.371444	−	0.928456i	$-0.621137\pi$
$877$	−22.0000	−0.742887	−0.371444	+	0.928456i	$0.621137\pi$
$878$	20.0000	0.674967
$879$	0	0
$880$	12.0000	0.404520
$881$	−7.00000	−0.235836	−0.117918	−	0.993023i	$-0.537622\pi$
$881$	−7.00000	−0.235836	−0.117918	+	0.993023i	$0.537622\pi$
$882$	0	0
$883$	−21.0000	−0.706706	−0.353353	−	0.935490i	$-0.614959\pi$
$883$	−21.0000	−0.706706	−0.353353	+	0.935490i	$0.614959\pi$
$884$	36.0000	1.21081
$885$	0	0
$886$	−78.0000	−2.62046
$887$	52.0000	1.74599	0.872995	−	0.487730i	$-0.162175\pi$
$887$	52.0000	1.74599	0.872995	+	0.487730i	$0.162175\pi$
$888$	0	0
$889$	−6.00000	−0.201234
$890$	20.0000	0.670402
$891$	0	0
$892$	8.00000	0.267860
$893$	3.00000	0.100391
$894$	0	0
$895$	−10.0000	−0.334263
$896$	0	0
$897$	0	0
$898$	40.0000	1.33482
$899$	20.0000	0.667037
$900$	0	0
$901$	−18.0000	−0.599667
$902$	48.0000	1.59823
$903$	0	0
$904$	0	0
$905$	−2.00000	−0.0664822
$906$	0	0
$907$	−2.00000	−0.0664089	−0.0332045	−	0.999449i	$-0.510571\pi$
$907$	−2.00000	−0.0664089	−0.0332045	+	0.999449i	$0.510571\pi$
$908$	−36.0000	−1.19470
$909$	0	0
$910$	36.0000	1.19339
$911$	−2.00000	−0.0662630	−0.0331315	−	0.999451i	$-0.510548\pi$
$911$	−2.00000	−0.0662630	−0.0331315	+	0.999451i	$0.510548\pi$
$912$	0	0
$913$	−12.0000	−0.397142
$914$	6.00000	0.198462
$915$	0	0
$916$	−30.0000	−0.991228
$917$	39.0000	1.28789
$918$	0	0
$919$	−40.0000	−1.31948	−0.659739	−	0.751495i	$-0.729333\pi$
$919$	−40.0000	−1.31948	−0.659739	+	0.751495i	$0.729333\pi$
$920$	0	0
$921$	0	0
$922$	66.0000	2.17359
$923$	72.0000	2.36991
$924$	0	0
$925$	−32.0000	−1.05215
$926$	−62.0000	−2.03745
$927$	0	0
$928$	−80.0000	−2.62613
$929$	50.0000	1.64045	0.820223	−	0.572043i	$-0.193849\pi$
$929$	50.0000	1.64045	0.820223	+	0.572043i	$0.193849\pi$
$930$	0	0
$931$	−2.00000	−0.0655474
$932$	22.0000	0.720634
$933$	0	0
$934$	34.0000	1.11251
$935$	9.00000	0.294331
$936$	0	0
$937$	53.0000	1.73143	0.865717	−	0.500533i	$-0.166863\pi$
$937$	53.0000	1.73143	0.865717	+	0.500533i	$0.166863\pi$
$938$	48.0000	1.56726
$939$	0	0
$940$	6.00000	0.195698
$941$	−2.00000	−0.0651981	−0.0325991	−	0.999469i	$-0.510378\pi$
$941$	−2.00000	−0.0651981	−0.0325991	+	0.999469i	$0.510378\pi$
$942$	0	0
$943$	−32.0000	−1.04206
$944$	0	0
$945$	0	0
$946$	−6.00000	−0.195077
$947$	−28.0000	−0.909878	−0.454939	−	0.890523i	$-0.650339\pi$
$947$	−28.0000	−0.909878	−0.454939	+	0.890523i	$0.650339\pi$
$948$	0	0
$949$	66.0000	2.14245
$950$	8.00000	0.259554
$951$	0	0
$952$	0	0
$953$	16.0000	0.518291	0.259145	−	0.965838i	$-0.416559\pi$
$953$	16.0000	0.518291	0.259145	+	0.965838i	$0.416559\pi$
$954$	0	0
$955$	−3.00000	−0.0970777
$956$	30.0000	0.970269
$957$	0	0
$958$	80.0000	2.58468
$959$	−9.00000	−0.290625
$960$	0	0
$961$	−27.0000	−0.870968
$962$	−96.0000	−3.09516
$963$	0	0
$964$	24.0000	0.772988
$965$	−4.00000	−0.128765
$966$	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
$970$	4.00000	0.128432
$971$	−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
$973$	−15.0000	−0.480878
$974$	16.0000	0.512673
$975$	0	0
$976$	−28.0000	−0.896258
$977$	42.0000	1.34370	0.671850	−	0.740688i	$-0.265500\pi$
$977$	42.0000	1.34370	0.671850	+	0.740688i	$0.265500\pi$
$978$	0	0
$979$	−30.0000	−0.958804
$980$	−4.00000	−0.127775
$981$	0	0
$982$	16.0000	0.510581
$983$	−4.00000	−0.127580	−0.0637901	−	0.997963i	$-0.520319\pi$
$983$	−4.00000	−0.127580	−0.0637901	+	0.997963i	$0.520319\pi$
$984$	0	0
$985$	−2.00000	−0.0637253
$986$	−60.0000	−1.91079
$987$	0	0
$988$	12.0000	0.381771
$989$	4.00000	0.127193
$990$	0	0
$991$	−8.00000	−0.254128	−0.127064	−	0.991894i	$-0.540555\pi$
$991$	−8.00000	−0.254128	−0.127064	+	0.991894i	$0.540555\pi$
$992$	−16.0000	−0.508001
$993$	0	0
$994$	−72.0000	−2.28370
$995$	5.00000	0.158511
$996$	0	0
$997$	−7.00000	−0.221692	−0.110846	−	0.993838i	$-0.535356\pi$
$997$	−7.00000	−0.221692	−0.110846	+	0.993838i	$0.535356\pi$
$998$	−70.0000	−2.21581
$999$	0	0

Twists

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

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

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Representations

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Database

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