LMFDB - Embedded newform 3969.2.a.y.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3969,2,Mod(1,3969)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3969, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

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

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

chi := DirichletCharacter("3969.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3969 = 3^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3969.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$31.6926245622$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} + 4 $ x^4 - 6*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.874032$ of defining polynomial
Character	$\chi$	$=$	3969.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.61803 q^{2} +0.618034 q^{4} +0.874032 q^{5} -2.23607 q^{8} +O(q^{10})$	$q+1.61803 q^{2} +0.618034 q^{4} +0.874032 q^{5} -2.23607 q^{8} +1.41421 q^{10} +1.00000 q^{11} +1.74806 q^{13} -4.85410 q^{16} -3.16228 q^{17} -5.45052 q^{19} +0.540182 q^{20} +1.61803 q^{22} -2.76393 q^{23} -4.23607 q^{25} +2.82843 q^{26} -3.23607 q^{29} -0.333851 q^{31} -3.38197 q^{32} -5.11667 q^{34} +2.70820 q^{37} -8.81913 q^{38} -1.95440 q^{40} -10.0270 q^{41} -11.9443 q^{43} +0.618034 q^{44} -4.47214 q^{46} +12.5216 q^{47} -6.85410 q^{50} +1.08036 q^{52} -2.70820 q^{53} +0.874032 q^{55} -5.23607 q^{58} +8.15143 q^{59} +13.1893 q^{61} -0.540182 q^{62} +4.23607 q^{64} +1.52786 q^{65} -7.47214 q^{67} -1.95440 q^{68} -3.47214 q^{71} +13.0618 q^{73} +4.38197 q^{74} -3.36861 q^{76} -12.2361 q^{79} -4.24264 q^{80} -16.2241 q^{82} -7.19859 q^{83} -2.76393 q^{85} -19.3262 q^{86} -2.23607 q^{88} +6.32456 q^{89} -1.70820 q^{92} +20.2604 q^{94} -4.76393 q^{95} -9.35931 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} - 2 q^{4}+O(q^{10}) $ 4 * q + 2 * q^2 - 2 * q^4	$ 4 q + 2 q^{2} - 2 q^{4} + 4 q^{11} - 6 q^{16} + 2 q^{22} - 20 q^{23} - 8 q^{25} - 4 q^{29} - 18 q^{32} - 16 q^{37} - 12 q^{43} - 2 q^{44} - 14 q^{50} + 16 q^{53} - 12 q^{58} + 8 q^{64} + 24 q^{65} - 12 q^{67} + 4 q^{71} + 22 q^{74} - 40 q^{79} - 20 q^{85} - 46 q^{86} + 20 q^{92} - 28 q^{95}+O(q^{100}) $ 4 * q + 2 * q^2 - 2 * q^4 + 4 * q^11 - 6 * q^16 + 2 * q^22 - 20 * q^23 - 8 * q^25 - 4 * q^29 - 18 * q^32 - 16 * q^37 - 12 * q^43 - 2 * q^44 - 14 * q^50 + 16 * q^53 - 12 * q^58 + 8 * q^64 + 24 * q^65 - 12 * q^67 + 4 * q^71 + 22 * q^74 - 40 * q^79 - 20 * q^85 - 46 * q^86 + 20 * q^92 - 28 * q^95

Display 100 coefficients 10 coefficients

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 $

$2$	1.61803	1.14412	0.572061	−	0.820211i	$-0.306144\pi$
$2$	1.61803	1.14412	0.572061	+	0.820211i	$0.306144\pi$
$3$	0	0
$3$	0	0
$4$	0.618034	0.309017
$5$	0.874032	0.390879	0.195440	−	0.980716i	$-0.437387\pi$
$5$	0.874032	0.390879	0.195440	+	0.980716i	$0.437387\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−2.23607	−0.790569
$9$	0	0
$10$	1.41421	0.447214
$11$	1.00000	0.301511	0.150756	−	0.988571i	$-0.451829\pi$
$11$	1.00000	0.301511	0.150756	+	0.988571i	$0.451829\pi$
$12$	0	0
$13$	1.74806	0.484826	0.242413	−	0.970173i	$-0.422061\pi$
$13$	1.74806	0.484826	0.242413	+	0.970173i	$0.422061\pi$
$14$	0	0
$15$	0	0
$16$	−4.85410	−1.21353
$17$	−3.16228	−0.766965	−0.383482	−	0.923548i	$-0.625275\pi$
$17$	−3.16228	−0.766965	−0.383482	+	0.923548i	$0.625275\pi$
$18$	0	0
$19$	−5.45052	−1.25044	−0.625218	−	0.780450i	$-0.714990\pi$
$19$	−5.45052	−1.25044	−0.625218	+	0.780450i	$0.714990\pi$
$20$	0.540182	0.120788
$21$	0	0
$22$	1.61803	0.344966
$23$	−2.76393	−0.576320	−0.288160	−	0.957582i	$-0.593043\pi$
$23$	−2.76393	−0.576320	−0.288160	+	0.957582i	$0.593043\pi$
$24$	0	0
$25$	−4.23607	−0.847214
$26$	2.82843	0.554700
$27$	0	0
$28$	0	0
$29$	−3.23607	−0.600923	−0.300461	−	0.953794i	$-0.597141\pi$
$29$	−3.23607	−0.600923	−0.300461	+	0.953794i	$0.597141\pi$
$30$	0	0
$31$	−0.333851	−0.0599613	−0.0299807	−	0.999550i	$-0.509545\pi$
$31$	−0.333851	−0.0599613	−0.0299807	+	0.999550i	$0.509545\pi$
$32$	−3.38197	−0.597853
$33$	0	0
$34$	−5.11667	−0.877502
$35$	0	0
$36$	0	0
$37$	2.70820	0.445226	0.222613	−	0.974907i	$-0.428541\pi$
$37$	2.70820	0.445226	0.222613	+	0.974907i	$0.428541\pi$
$38$	−8.81913	−1.43065
$39$	0	0
$40$	−1.95440	−0.309017
$41$	−10.0270	−1.56596	−0.782978	−	0.622049i	$-0.786300\pi$
$41$	−10.0270	−1.56596	−0.782978	+	0.622049i	$0.786300\pi$
$42$	0	0
$43$	−11.9443	−1.82148	−0.910742	−	0.412975i	$-0.864490\pi$
$43$	−11.9443	−1.82148	−0.910742	+	0.412975i	$0.864490\pi$
$44$	0.618034	0.0931721
$45$	0	0
$46$	−4.47214	−0.659380
$47$	12.5216	1.82646	0.913231	−	0.407442i	$-0.133579\pi$
$47$	12.5216	1.82646	0.913231	+	0.407442i	$0.133579\pi$
$48$	0	0
$49$	0	0
$50$	−6.85410	−0.969316
$51$	0	0
$52$	1.08036	0.149819
$53$	−2.70820	−0.372000	−0.186000	−	0.982550i	$-0.559553\pi$
$53$	−2.70820	−0.372000	−0.186000	+	0.982550i	$0.559553\pi$
$54$	0	0
$55$	0.874032	0.117854
$56$	0	0
$57$	0	0
$58$	−5.23607	−0.687529
$59$	8.15143	1.06123	0.530613	−	0.847614i	$-0.321962\pi$
$59$	8.15143	1.06123	0.530613	+	0.847614i	$0.321962\pi$
$60$	0	0
$61$	13.1893	1.68872	0.844358	−	0.535780i	$-0.179982\pi$
$61$	13.1893	1.68872	0.844358	+	0.535780i	$0.179982\pi$
$62$	−0.540182	−0.0686031
$63$	0	0
$64$	4.23607	0.529508
$65$	1.52786	0.189508
$66$	0	0
$67$	−7.47214	−0.912867	−0.456433	−	0.889758i	$-0.650873\pi$
$67$	−7.47214	−0.912867	−0.456433	+	0.889758i	$0.650873\pi$
$68$	−1.95440	−0.237005
$69$	0	0
$70$	0	0
$71$	−3.47214	−0.412067	−0.206033	−	0.978545i	$-0.566056\pi$
$71$	−3.47214	−0.412067	−0.206033	+	0.978545i	$0.566056\pi$
$72$	0	0
$73$	13.0618	1.52876	0.764382	−	0.644763i	$-0.223044\pi$
$73$	13.0618	1.52876	0.764382	+	0.644763i	$0.223044\pi$
$74$	4.38197	0.509393
$75$	0	0
$76$	−3.36861	−0.386406
$77$	0	0
$78$	0	0
$79$	−12.2361	−1.37667	−0.688333	−	0.725395i	$-0.741657\pi$
$79$	−12.2361	−1.37667	−0.688333	+	0.725395i	$0.741657\pi$
$80$	−4.24264	−0.474342
$81$	0	0
$82$	−16.2241	−1.79165
$83$	−7.19859	−0.790148	−0.395074	−	0.918649i	$-0.629281\pi$
$83$	−7.19859	−0.790148	−0.395074	+	0.918649i	$0.629281\pi$
$84$	0	0
$85$	−2.76393	−0.299791
$86$	−19.3262	−2.08400
$87$	0	0
$88$	−2.23607	−0.238366
$89$	6.32456	0.670402	0.335201	−	0.942147i	$-0.391196\pi$
$89$	6.32456	0.670402	0.335201	+	0.942147i	$0.391196\pi$
$90$	0	0
$91$	0	0
$92$	−1.70820	−0.178093
$93$	0	0
$94$	20.2604	2.08970
$95$	−4.76393	−0.488769
$96$	0	0
$97$	−9.35931	−0.950294	−0.475147	−	0.879906i	$-0.657605\pi$
$97$	−9.35931	−0.950294	−0.475147	+	0.879906i	$0.657605\pi$
$98$	0	0
$99$	0	0
$100$	−2.61803	−0.261803
$101$	−12.3941	−1.23326	−0.616628	−	0.787255i	$-0.711502\pi$
$101$	−12.3941	−1.23326	−0.616628	+	0.787255i	$0.711502\pi$
$102$	0	0
$103$	−10.0270	−0.987991	−0.493996	−	0.869464i	$-0.664464\pi$
$103$	−10.0270	−0.987991	−0.493996	+	0.869464i	$0.664464\pi$
$104$	−3.90879	−0.383288
$105$	0	0
$106$	−4.38197	−0.425614
$107$	−5.94427	−0.574654	−0.287327	−	0.957832i	$-0.592767\pi$
$107$	−5.94427	−0.574654	−0.287327	+	0.957832i	$0.592767\pi$
$108$	0	0
$109$	2.94427	0.282010	0.141005	−	0.990009i	$-0.454967\pi$
$109$	2.94427	0.282010	0.141005	+	0.990009i	$0.454967\pi$
$110$	1.41421	0.134840
$111$	0	0
$112$	0	0
$113$	−8.23607	−0.774784	−0.387392	−	0.921915i	$-0.626624\pi$
$113$	−8.23607	−0.774784	−0.387392	+	0.921915i	$0.626624\pi$
$114$	0	0
$115$	−2.41577	−0.225271
$116$	−2.00000	−0.185695
$117$	0	0
$118$	13.1893	1.21417
$119$	0	0
$120$	0	0
$121$	−10.0000	−0.909091
$122$	21.3407	1.93210
$123$	0	0
$124$	−0.206331	−0.0185291
$125$	−8.07262	−0.722037
$126$	0	0
$127$	13.6525	1.21146	0.605731	−	0.795670i	$-0.292881\pi$
$127$	13.6525	1.21146	0.605731	+	0.795670i	$0.292881\pi$
$128$	13.6180	1.20368
$129$	0	0
$130$	2.47214	0.216821
$131$	1.87558	0.163871	0.0819353	−	0.996638i	$-0.473890\pi$
$131$	1.87558	0.163871	0.0819353	+	0.996638i	$0.473890\pi$
$132$	0	0
$133$	0	0
$134$	−12.0902	−1.04443
$135$	0	0
$136$	7.07107	0.606339
$137$	3.76393	0.321574	0.160787	−	0.986989i	$-0.448597\pi$
$137$	3.76393	0.321574	0.160787	+	0.986989i	$0.448597\pi$
$138$	0	0
$139$	0.206331	0.0175008	0.00875038	−	0.999962i	$-0.497215\pi$
$139$	0.206331	0.0175008	0.00875038	+	0.999962i	$0.497215\pi$
$140$	0	0
$141$	0	0
$142$	−5.61803	−0.471455
$143$	1.74806	0.146180
$144$	0	0
$145$	−2.82843	−0.234888
$146$	21.1344	1.74909
$147$	0	0
$148$	1.67376	0.137582
$149$	−19.4721	−1.59522	−0.797610	−	0.603174i	$-0.793903\pi$
$149$	−19.4721	−1.59522	−0.797610	+	0.603174i	$0.793903\pi$
$150$	0	0
$151$	−5.29180	−0.430640	−0.215320	−	0.976544i	$-0.569079\pi$
$151$	−5.29180	−0.430640	−0.215320	+	0.976544i	$0.569079\pi$
$152$	12.1877	0.988556
$153$	0	0
$154$	0	0
$155$	−0.291796	−0.0234376
$156$	0	0
$157$	2.16073	0.172445	0.0862224	−	0.996276i	$-0.472520\pi$
$157$	2.16073	0.172445	0.0862224	+	0.996276i	$0.472520\pi$
$158$	−19.7984	−1.57507
$159$	0	0
$160$	−2.95595	−0.233688
$161$	0	0
$162$	0	0
$163$	−8.23607	−0.645099	−0.322549	−	0.946553i	$-0.604540\pi$
$163$	−8.23607	−0.645099	−0.322549	+	0.946553i	$0.604540\pi$
$164$	−6.19704	−0.483907
$165$	0	0
$166$	−11.6476	−0.904026
$167$	−8.69161	−0.672577	−0.336289	−	0.941759i	$-0.609172\pi$
$167$	−8.69161	−0.672577	−0.336289	+	0.941759i	$0.609172\pi$
$168$	0	0
$169$	−9.94427	−0.764944
$170$	−4.47214	−0.342997
$171$	0	0
$172$	−7.38197	−0.562870
$173$	19.2588	1.46422	0.732110	−	0.681186i	$-0.238536\pi$
$173$	19.2588	1.46422	0.732110	+	0.681186i	$0.238536\pi$
$174$	0	0
$175$	0	0
$176$	−4.85410	−0.365892
$177$	0	0
$178$	10.2333	0.767022
$179$	20.3607	1.52183	0.760914	−	0.648852i	$-0.224751\pi$
$179$	20.3607	1.52183	0.760914	+	0.648852i	$0.224751\pi$
$180$	0	0
$181$	−22.9613	−1.70670	−0.853349	−	0.521340i	$-0.825432\pi$
$181$	−22.9613	−1.70670	−0.853349	+	0.521340i	$0.825432\pi$
$182$	0	0
$183$	0	0
$184$	6.18034	0.455621
$185$	2.36706	0.174029
$186$	0	0
$187$	−3.16228	−0.231249
$188$	7.73877	0.564408
$189$	0	0
$190$	−7.70820	−0.559212
$191$	−4.41641	−0.319560	−0.159780	−	0.987153i	$-0.551078\pi$
$191$	−4.41641	−0.319560	−0.159780	+	0.987153i	$0.551078\pi$
$192$	0	0
$193$	−16.4721	−1.18569	−0.592845	−	0.805316i	$-0.701995\pi$
$193$	−16.4721	−1.18569	−0.592845	+	0.805316i	$0.701995\pi$
$194$	−15.1437	−1.08725
$195$	0	0
$196$	0	0
$197$	18.4164	1.31211	0.656057	−	0.754711i	$-0.272223\pi$
$197$	18.4164	1.31211	0.656057	+	0.754711i	$0.272223\pi$
$198$	0	0
$199$	11.4412	0.811047	0.405524	−	0.914085i	$-0.367089\pi$
$199$	11.4412	0.811047	0.405524	+	0.914085i	$0.367089\pi$
$200$	9.47214	0.669781
$201$	0	0
$202$	−20.0540	−1.41100
$203$	0	0
$204$	0	0
$205$	−8.76393	−0.612100
$206$	−16.2241	−1.13038
$207$	0	0
$208$	−8.48528	−0.588348
$209$	−5.45052	−0.377021
$210$	0	0
$211$	−3.29180	−0.226617	−0.113308	−	0.993560i	$-0.536145\pi$
$211$	−3.29180	−0.226617	−0.113308	+	0.993560i	$0.536145\pi$
$212$	−1.67376	−0.114954
$213$	0	0
$214$	−9.61803	−0.657475
$215$	−10.4397	−0.711980
$216$	0	0
$217$	0	0
$218$	4.76393	0.322654
$219$	0	0
$220$	0.540182	0.0364190
$221$	−5.52786	−0.371844
$222$	0	0
$223$	−11.6476	−0.779978	−0.389989	−	0.920819i	$-0.627521\pi$
$223$	−11.6476	−0.779978	−0.389989	+	0.920819i	$0.627521\pi$
$224$	0	0
$225$	0	0
$226$	−13.3262	−0.886448
$227$	−23.0888	−1.53246	−0.766228	−	0.642568i	$-0.777869\pi$
$227$	−23.0888	−1.53246	−0.766228	+	0.642568i	$0.777869\pi$
$228$	0	0
$229$	10.0270	0.662604	0.331302	−	0.943525i	$-0.392512\pi$
$229$	10.0270	0.662604	0.331302	+	0.943525i	$0.392512\pi$
$230$	−3.90879	−0.257738
$231$	0	0
$232$	7.23607	0.475071
$233$	22.9443	1.50313	0.751565	−	0.659659i	$-0.229299\pi$
$233$	22.9443	1.50313	0.751565	+	0.659659i	$0.229299\pi$
$234$	0	0
$235$	10.9443	0.713926
$236$	5.03786	0.327937
$237$	0	0
$238$	0	0
$239$	13.9443	0.901980	0.450990	−	0.892529i	$-0.351071\pi$
$239$	13.9443	0.901980	0.450990	+	0.892529i	$0.351071\pi$
$240$	0	0
$241$	12.1089	0.780005	0.390002	−	0.920814i	$-0.372474\pi$
$241$	12.1089	0.780005	0.390002	+	0.920814i	$0.372474\pi$
$242$	−16.1803	−1.04011
$243$	0	0
$244$	8.15143	0.521842
$245$	0	0
$246$	0	0
$247$	−9.52786	−0.606243
$248$	0.746512	0.0474036
$249$	0	0
$250$	−13.0618	−0.826099
$251$	2.70091	0.170480	0.0852399	−	0.996360i	$-0.472834\pi$
$251$	2.70091	0.170480	0.0852399	+	0.996360i	$0.472834\pi$
$252$	0	0
$253$	−2.76393	−0.173767
$254$	22.0902	1.38606
$255$	0	0
$256$	13.5623	0.847644
$257$	−15.8902	−0.991203	−0.495602	−	0.868550i	$-0.665052\pi$
$257$	−15.8902	−0.991203	−0.495602	+	0.868550i	$0.665052\pi$
$258$	0	0
$259$	0	0
$260$	0.944272	0.0585613
$261$	0	0
$262$	3.03476	0.187488
$263$	14.2361	0.877834	0.438917	−	0.898528i	$-0.355362\pi$
$263$	14.2361	0.877834	0.438917	+	0.898528i	$0.355362\pi$
$264$	0	0
$265$	−2.36706	−0.145407
$266$	0	0
$267$	0	0
$268$	−4.61803	−0.282091
$269$	21.1344	1.28859	0.644293	−	0.764778i	$-0.277152\pi$
$269$	21.1344	1.28859	0.644293	+	0.764778i	$0.277152\pi$
$270$	0	0
$271$	27.5865	1.67576	0.837879	−	0.545856i	$-0.183795\pi$
$271$	27.5865	1.67576	0.837879	+	0.545856i	$0.183795\pi$
$272$	15.3500	0.930732
$273$	0	0
$274$	6.09017	0.367921
$275$	−4.23607	−0.255445
$276$	0	0
$277$	5.76393	0.346321	0.173161	−	0.984894i	$-0.444602\pi$
$277$	5.76393	0.346321	0.173161	+	0.984894i	$0.444602\pi$
$278$	0.333851	0.0200230
$279$	0	0
$280$	0	0
$281$	−5.00000	−0.298275	−0.149137	−	0.988816i	$-0.547650\pi$
$281$	−5.00000	−0.298275	−0.149137	+	0.988816i	$0.547650\pi$
$282$	0	0
$283$	−12.1089	−0.719801	−0.359901	−	0.932991i	$-0.617189\pi$
$283$	−12.1089	−0.719801	−0.359901	+	0.932991i	$0.617189\pi$
$284$	−2.14590	−0.127336
$285$	0	0
$286$	2.82843	0.167248
$287$	0	0
$288$	0	0
$289$	−7.00000	−0.411765
$290$	−4.57649	−0.268741
$291$	0	0
$292$	8.07262	0.472414
$293$	14.9374	0.872650	0.436325	−	0.899789i	$-0.356280\pi$
$293$	14.9374	0.872650	0.436325	+	0.899789i	$0.356280\pi$
$294$	0	0
$295$	7.12461	0.414811
$296$	−6.05573	−0.351982
$297$	0	0
$298$	−31.5066	−1.82513
$299$	−4.83153	−0.279415
$300$	0	0
$301$	0	0
$302$	−8.56231	−0.492705
$303$	0	0
$304$	26.4574	1.51744
$305$	11.5279	0.660084
$306$	0	0
$307$	−18.6398	−1.06383	−0.531915	−	0.846798i	$-0.678528\pi$
$307$	−18.6398	−1.06383	−0.531915	+	0.846798i	$0.678528\pi$
$308$	0	0
$309$	0	0
$310$	−0.472136	−0.0268155
$311$	28.4118	1.61108	0.805542	−	0.592538i	$-0.201874\pi$
$311$	28.4118	1.61108	0.805542	+	0.592538i	$0.201874\pi$
$312$	0	0
$313$	−13.3956	−0.757165	−0.378583	−	0.925567i	$-0.623588\pi$
$313$	−13.3956	−0.757165	−0.378583	+	0.925567i	$0.623588\pi$
$314$	3.49613	0.197298
$315$	0	0
$316$	−7.56231	−0.425413
$317$	18.8328	1.05776	0.528878	−	0.848698i	$-0.322613\pi$
$317$	18.8328	1.05776	0.528878	+	0.848698i	$0.322613\pi$
$318$	0	0
$319$	−3.23607	−0.181185
$320$	3.70246	0.206974
$321$	0	0
$322$	0	0
$323$	17.2361	0.959040
$324$	0	0
$325$	−7.40492	−0.410751
$326$	−13.3262	−0.738072
$327$	0	0
$328$	22.4211	1.23800
$329$	0	0
$330$	0	0
$331$	2.00000	0.109930	0.0549650	−	0.998488i	$-0.482495\pi$
$331$	2.00000	0.109930	0.0549650	+	0.998488i	$0.482495\pi$
$332$	−4.44897	−0.244169
$333$	0	0
$334$	−14.0633	−0.769511
$335$	−6.53089	−0.356820
$336$	0	0
$337$	−17.1803	−0.935873	−0.467936	−	0.883762i	$-0.655002\pi$
$337$	−17.1803	−0.935873	−0.467936	+	0.883762i	$0.655002\pi$
$338$	−16.0902	−0.875190
$339$	0	0
$340$	−1.70820	−0.0926404
$341$	−0.333851	−0.0180790
$342$	0	0
$343$	0	0
$344$	26.7082	1.44001
$345$	0	0
$346$	31.1614	1.67525
$347$	21.9443	1.17803	0.589015	−	0.808122i	$-0.299516\pi$
$347$	21.9443	1.17803	0.589015	+	0.808122i	$0.299516\pi$
$348$	0	0
$349$	28.6668	1.53450	0.767250	−	0.641348i	$-0.221625\pi$
$349$	28.6668	1.53450	0.767250	+	0.641348i	$0.221625\pi$
$350$	0	0
$351$	0	0
$352$	−3.38197	−0.180259
$353$	20.5154	1.09192	0.545962	−	0.837810i	$-0.316164\pi$
$353$	20.5154	1.09192	0.545962	+	0.837810i	$0.316164\pi$
$354$	0	0
$355$	−3.03476	−0.161068
$356$	3.90879	0.207165
$357$	0	0
$358$	32.9443	1.74116
$359$	17.1803	0.906744	0.453372	−	0.891321i	$-0.350221\pi$
$359$	17.1803	0.906744	0.453372	+	0.891321i	$0.350221\pi$
$360$	0	0
$361$	10.7082	0.563590
$362$	−37.1521	−1.95267
$363$	0	0
$364$	0	0
$365$	11.4164	0.597562
$366$	0	0
$367$	−1.74806	−0.0912482	−0.0456241	−	0.998959i	$-0.514528\pi$
$367$	−1.74806	−0.0912482	−0.0456241	+	0.998959i	$0.514528\pi$
$368$	13.4164	0.699379
$369$	0	0
$370$	3.82998	0.199111
$371$	0	0
$372$	0	0
$373$	−9.18034	−0.475340	−0.237670	−	0.971346i	$-0.576384\pi$
$373$	−9.18034	−0.475340	−0.237670	+	0.971346i	$0.576384\pi$
$374$	−5.11667	−0.264577
$375$	0	0
$376$	−27.9991	−1.44394
$377$	−5.65685	−0.291343
$378$	0	0
$379$	−31.6525	−1.62588	−0.812939	−	0.582349i	$-0.802134\pi$
$379$	−31.6525	−1.62588	−0.812939	+	0.582349i	$0.802134\pi$
$380$	−2.94427	−0.151038
$381$	0	0
$382$	−7.14590	−0.365616
$383$	−13.3168	−0.680457	−0.340229	−	0.940343i	$-0.610504\pi$
$383$	−13.3168	−0.680457	−0.340229	+	0.940343i	$0.610504\pi$
$384$	0	0
$385$	0	0
$386$	−26.6525	−1.35658
$387$	0	0
$388$	−5.78437	−0.293657
$389$	10.9443	0.554897	0.277448	−	0.960741i	$-0.410511\pi$
$389$	10.9443	0.554897	0.277448	+	0.960741i	$0.410511\pi$
$390$	0	0
$391$	8.74032	0.442017
$392$	0	0
$393$	0	0
$394$	29.7984	1.50122
$395$	−10.6947	−0.538110
$396$	0	0
$397$	28.8245	1.44666	0.723329	−	0.690504i	$-0.242611\pi$
$397$	28.8245	1.44666	0.723329	+	0.690504i	$0.242611\pi$
$398$	18.5123	0.927938
$399$	0	0
$400$	20.5623	1.02812
$401$	37.0689	1.85113	0.925566	−	0.378587i	$-0.123590\pi$
$401$	37.0689	1.85113	0.925566	+	0.378587i	$0.123590\pi$
$402$	0	0
$403$	−0.583592	−0.0290708
$404$	−7.65996	−0.381097
$405$	0	0
$406$	0	0
$407$	2.70820	0.134241
$408$	0	0
$409$	−4.11512	−0.203480	−0.101740	−	0.994811i	$-0.532441\pi$
$409$	−4.11512	−0.203480	−0.101740	+	0.994811i	$0.532441\pi$
$410$	−14.1803	−0.700317
$411$	0	0
$412$	−6.19704	−0.305306
$413$	0	0
$414$	0	0
$415$	−6.29180	−0.308852
$416$	−5.91189	−0.289854
$417$	0	0
$418$	−8.81913	−0.431358
$419$	−24.4242	−1.19320	−0.596600	−	0.802539i	$-0.703482\pi$
$419$	−24.4242	−1.19320	−0.596600	+	0.802539i	$0.703482\pi$
$420$	0	0
$421$	14.8885	0.725623	0.362812	−	0.931863i	$-0.381817\pi$
$421$	14.8885	0.725623	0.362812	+	0.931863i	$0.381817\pi$
$422$	−5.32624	−0.259277
$423$	0	0
$424$	6.05573	0.294092
$425$	13.3956	0.649783
$426$	0	0
$427$	0	0
$428$	−3.67376	−0.177578
$429$	0	0
$430$	−16.8918	−0.814593
$431$	−23.4164	−1.12793	−0.563964	−	0.825799i	$-0.690724\pi$
$431$	−23.4164	−1.12793	−0.563964	+	0.825799i	$0.690724\pi$
$432$	0	0
$433$	−20.5154	−0.985907	−0.492954	−	0.870056i	$-0.664083\pi$
$433$	−20.5154	−0.985907	−0.492954	+	0.870056i	$0.664083\pi$
$434$	0	0
$435$	0	0
$436$	1.81966	0.0871459
$437$	15.0649	0.720651
$438$	0	0
$439$	−11.6963	−0.558232	−0.279116	−	0.960257i	$-0.590041\pi$
$439$	−11.6963	−0.558232	−0.279116	+	0.960257i	$0.590041\pi$
$440$	−1.95440	−0.0931721
$441$	0	0
$442$	−8.94427	−0.425436
$443$	−2.29180	−0.108887	−0.0544433	−	0.998517i	$-0.517338\pi$
$443$	−2.29180	−0.108887	−0.0544433	+	0.998517i	$0.517338\pi$
$444$	0	0
$445$	5.52786	0.262046
$446$	−18.8461	−0.892391
$447$	0	0
$448$	0	0
$449$	31.4721	1.48526	0.742631	−	0.669701i	$-0.233578\pi$
$449$	31.4721	1.48526	0.742631	+	0.669701i	$0.233578\pi$
$450$	0	0
$451$	−10.0270	−0.472154
$452$	−5.09017	−0.239421
$453$	0	0
$454$	−37.3584	−1.75332
$455$	0	0
$456$	0	0
$457$	23.8328	1.11485	0.557426	−	0.830227i	$-0.311789\pi$
$457$	23.8328	1.11485	0.557426	+	0.830227i	$0.311789\pi$
$458$	16.2241	0.758100
$459$	0	0
$460$	−1.49302	−0.0696126
$461$	−18.1784	−0.846655	−0.423327	−	0.905977i	$-0.639138\pi$
$461$	−18.1784	−0.846655	−0.423327	+	0.905977i	$0.639138\pi$
$462$	0	0
$463$	−36.8885	−1.71436	−0.857178	−	0.515020i	$-0.827784\pi$
$463$	−36.8885	−1.71436	−0.857178	+	0.515020i	$0.827784\pi$
$464$	15.7082	0.729235
$465$	0	0
$466$	37.1246	1.71976
$467$	−40.3144	−1.86553	−0.932764	−	0.360488i	$-0.882610\pi$
$467$	−40.3144	−1.86553	−0.932764	+	0.360488i	$0.882610\pi$
$468$	0	0
$469$	0	0
$470$	17.7082	0.816819
$471$	0	0
$472$	−18.2272	−0.838973
$473$	−11.9443	−0.549198
$474$	0	0
$475$	23.0888	1.05939
$476$	0	0
$477$	0	0
$478$	22.5623	1.03198
$479$	11.2349	0.513336	0.256668	−	0.966500i	$-0.417375\pi$
$479$	11.2349	0.513336	0.256668	+	0.966500i	$0.417375\pi$
$480$	0	0
$481$	4.73411	0.215857
$482$	19.5927	0.892421
$483$	0	0
$484$	−6.18034	−0.280925
$485$	−8.18034	−0.371450
$486$	0	0
$487$	−13.9443	−0.631875	−0.315938	−	0.948780i	$-0.602319\pi$
$487$	−13.9443	−0.631875	−0.315938	+	0.948780i	$0.602319\pi$
$488$	−29.4922	−1.33505
$489$	0	0
$490$	0	0
$491$	22.0000	0.992846	0.496423	−	0.868081i	$-0.334646\pi$
$491$	22.0000	0.992846	0.496423	+	0.868081i	$0.334646\pi$
$492$	0	0
$493$	10.2333	0.460887
$494$	−15.4164	−0.693617
$495$	0	0
$496$	1.62054	0.0727646
$497$	0	0
$498$	0	0
$499$	−13.8885	−0.621737	−0.310868	−	0.950453i	$-0.600620\pi$
$499$	−13.8885	−0.621737	−0.310868	+	0.950453i	$0.600620\pi$
$500$	−4.98915	−0.223122
$501$	0	0
$502$	4.37016	0.195050
$503$	44.4295	1.98101	0.990507	−	0.137463i	$-0.0438947\pi$
$503$	44.4295	1.98101	0.990507	+	0.137463i	$0.0438947\pi$
$504$	0	0
$505$	−10.8328	−0.482054
$506$	−4.47214	−0.198811
$507$	0	0
$508$	8.43769	0.374362
$509$	10.1545	0.450092	0.225046	−	0.974348i	$-0.427747\pi$
$509$	10.1545	0.450092	0.225046	+	0.974348i	$0.427747\pi$
$510$	0	0
$511$	0	0
$512$	−5.29180	−0.233867
$513$	0	0
$514$	−25.7109	−1.13406
$515$	−8.76393	−0.386185
$516$	0	0
$517$	12.5216	0.550699
$518$	0	0
$519$	0	0
$520$	−3.41641	−0.149819
$521$	21.8809	0.958620	0.479310	−	0.877646i	$-0.340887\pi$
$521$	21.8809	0.958620	0.479310	+	0.877646i	$0.340887\pi$
$522$	0	0
$523$	17.0981	0.747647	0.373823	−	0.927500i	$-0.378047\pi$
$523$	17.0981	0.747647	0.373823	+	0.927500i	$0.378047\pi$
$524$	1.15917	0.0506388
$525$	0	0
$526$	23.0344	1.00435
$527$	1.05573	0.0459882
$528$	0	0
$529$	−15.3607	−0.667856
$530$	−3.82998	−0.166364
$531$	0	0
$532$	0	0
$533$	−17.5279	−0.759216
$534$	0	0
$535$	−5.19548	−0.224620
$536$	16.7082	0.721684
$537$	0	0
$538$	34.1962	1.47430
$539$	0	0
$540$	0	0
$541$	−18.5967	−0.799537	−0.399768	−	0.916616i	$-0.630909\pi$
$541$	−18.5967	−0.799537	−0.399768	+	0.916616i	$0.630909\pi$
$542$	44.6358	1.91727
$543$	0	0
$544$	10.6947	0.458532
$545$	2.57339	0.110232
$546$	0	0
$547$	−2.70820	−0.115794	−0.0578972	−	0.998323i	$-0.518440\pi$
$547$	−2.70820	−0.115794	−0.0578972	+	0.998323i	$0.518440\pi$
$548$	2.32624	0.0993720
$549$	0	0
$550$	−6.85410	−0.292260
$551$	17.6383	0.751415
$552$	0	0
$553$	0	0
$554$	9.32624	0.396234
$555$	0	0
$556$	0.127520	0.00540803
$557$	20.7082	0.877435	0.438717	−	0.898625i	$-0.355433\pi$
$557$	20.7082	0.877435	0.438717	+	0.898625i	$0.355433\pi$
$558$	0	0
$559$	−20.8794	−0.883103
$560$	0	0
$561$	0	0
$562$	−8.09017	−0.341263
$563$	−34.2750	−1.44452	−0.722259	−	0.691623i	$-0.756896\pi$
$563$	−34.2750	−1.44452	−0.722259	+	0.691623i	$0.756896\pi$
$564$	0	0
$565$	−7.19859	−0.302847
$566$	−19.5927	−0.823541
$567$	0	0
$568$	7.76393	0.325767
$569$	−42.4164	−1.77819	−0.889094	−	0.457724i	$-0.848665\pi$
$569$	−42.4164	−1.77819	−0.889094	+	0.457724i	$0.848665\pi$
$570$	0	0
$571$	−40.1803	−1.68149	−0.840747	−	0.541427i	$-0.817884\pi$
$571$	−40.1803	−1.68149	−0.840747	+	0.541427i	$0.817884\pi$
$572$	1.08036	0.0451722
$573$	0	0
$574$	0	0
$575$	11.7082	0.488266
$576$	0	0
$577$	−12.5216	−0.521281	−0.260640	−	0.965436i	$-0.583934\pi$
$577$	−12.5216	−0.521281	−0.260640	+	0.965436i	$0.583934\pi$
$578$	−11.3262	−0.471109
$579$	0	0
$580$	−1.74806	−0.0725844
$581$	0	0
$582$	0	0
$583$	−2.70820	−0.112162
$584$	−29.2070	−1.20859
$585$	0	0
$586$	24.1692	0.998418
$587$	−2.90724	−0.119995	−0.0599973	−	0.998199i	$-0.519109\pi$
$587$	−2.90724	−0.119995	−0.0599973	+	0.998199i	$0.519109\pi$
$588$	0	0
$589$	1.81966	0.0749778
$590$	11.5279	0.474595
$591$	0	0
$592$	−13.1459	−0.540293
$593$	−27.9504	−1.14779	−0.573893	−	0.818930i	$-0.694568\pi$
$593$	−27.9504	−1.14779	−0.573893	+	0.818930i	$0.694568\pi$
$594$	0	0
$595$	0	0
$596$	−12.0344	−0.492950
$597$	0	0
$598$	−7.81758	−0.319685
$599$	39.8328	1.62752	0.813762	−	0.581198i	$-0.197416\pi$
$599$	39.8328	1.62752	0.813762	+	0.581198i	$0.197416\pi$
$600$	0	0
$601$	−31.9867	−1.30477	−0.652383	−	0.757889i	$-0.726231\pi$
$601$	−31.9867	−1.30477	−0.652383	+	0.757889i	$0.726231\pi$
$602$	0	0
$603$	0	0
$604$	−3.27051	−0.133075
$605$	−8.74032	−0.355345
$606$	0	0
$607$	26.2511	1.06550	0.532749	−	0.846273i	$-0.321159\pi$
$607$	26.2511	1.06550	0.532749	+	0.846273i	$0.321159\pi$
$608$	18.4335	0.747577
$609$	0	0
$610$	18.6525	0.755217
$611$	21.8885	0.885516
$612$	0	0
$613$	39.1803	1.58248	0.791240	−	0.611506i	$-0.209436\pi$
$613$	39.1803	1.58248	0.791240	+	0.611506i	$0.209436\pi$
$614$	−30.1599	−1.21715
$615$	0	0
$616$	0	0
$617$	21.4164	0.862192	0.431096	−	0.902306i	$-0.358127\pi$
$617$	21.4164	0.862192	0.431096	+	0.902306i	$0.358127\pi$
$618$	0	0
$619$	−35.5617	−1.42934	−0.714672	−	0.699460i	$-0.753424\pi$
$619$	−35.5617	−1.42934	−0.714672	+	0.699460i	$0.753424\pi$
$620$	−0.180340	−0.00724262
$621$	0	0
$622$	45.9712	1.84328
$623$	0	0
$624$	0	0
$625$	14.1246	0.564984
$626$	−21.6746	−0.866290
$627$	0	0
$628$	1.33540	0.0532883
$629$	−8.56409	−0.341473
$630$	0	0
$631$	−11.3475	−0.451738	−0.225869	−	0.974158i	$-0.572522\pi$
$631$	−11.3475	−0.451738	−0.225869	+	0.974158i	$0.572522\pi$
$632$	27.3607	1.08835
$633$	0	0
$634$	30.4721	1.21020
$635$	11.9327	0.473535
$636$	0	0
$637$	0	0
$638$	−5.23607	−0.207298
$639$	0	0
$640$	11.9026	0.470492
$641$	−12.5279	−0.494821	−0.247410	−	0.968911i	$-0.579580\pi$
$641$	−12.5279	−0.494821	−0.247410	+	0.968911i	$0.579580\pi$
$642$	0	0
$643$	28.5393	1.12548	0.562740	−	0.826634i	$-0.309747\pi$
$643$	28.5393	1.12548	0.562740	+	0.826634i	$0.309747\pi$
$644$	0	0
$645$	0	0
$646$	27.8885	1.09726
$647$	29.6684	1.16638	0.583192	−	0.812334i	$-0.301803\pi$
$647$	29.6684	1.16638	0.583192	+	0.812334i	$0.301803\pi$
$648$	0	0
$649$	8.15143	0.319972
$650$	−11.9814	−0.469950
$651$	0	0
$652$	−5.09017	−0.199346
$653$	3.29180	0.128818	0.0644090	−	0.997924i	$-0.479484\pi$
$653$	3.29180	0.128818	0.0644090	+	0.997924i	$0.479484\pi$
$654$	0	0
$655$	1.63932	0.0640535
$656$	48.6722	1.90033
$657$	0	0
$658$	0	0
$659$	−11.7639	−0.458258	−0.229129	−	0.973396i	$-0.573588\pi$
$659$	−11.7639	−0.458258	−0.229129	+	0.973396i	$0.573588\pi$
$660$	0	0
$661$	−17.6383	−0.686049	−0.343024	−	0.939326i	$-0.611451\pi$
$661$	−17.6383	−0.686049	−0.343024	+	0.939326i	$0.611451\pi$
$662$	3.23607	0.125773
$663$	0	0
$664$	16.0965	0.624667
$665$	0	0
$666$	0	0
$667$	8.94427	0.346324
$668$	−5.37171	−0.207838
$669$	0	0
$670$	−10.5672	−0.408246
$671$	13.1893	0.509167
$672$	0	0
$673$	9.65248	0.372076	0.186038	−	0.982543i	$-0.440435\pi$
$673$	9.65248	0.372076	0.186038	+	0.982543i	$0.440435\pi$
$674$	−27.7984	−1.07075
$675$	0	0
$676$	−6.14590	−0.236381
$677$	−7.07107	−0.271763	−0.135882	−	0.990725i	$-0.543387\pi$
$677$	−7.07107	−0.271763	−0.135882	+	0.990725i	$0.543387\pi$
$678$	0	0
$679$	0	0
$680$	6.18034	0.237005
$681$	0	0
$682$	−0.540182	−0.0206846
$683$	−17.7639	−0.679718	−0.339859	−	0.940476i	$-0.610379\pi$
$683$	−17.7639	−0.679718	−0.339859	+	0.940476i	$0.610379\pi$
$684$	0	0
$685$	3.28980	0.125697
$686$	0	0
$687$	0	0
$688$	57.9787	2.21042
$689$	−4.73411	−0.180355
$690$	0	0
$691$	18.7974	0.715088	0.357544	−	0.933896i	$-0.383614\pi$
$691$	18.7974	0.715088	0.357544	+	0.933896i	$0.383614\pi$
$692$	11.9026	0.452469
$693$	0	0
$694$	35.5066	1.34781
$695$	0.180340	0.00684068
$696$	0	0
$697$	31.7082	1.20103
$698$	46.3839	1.75566
$699$	0	0
$700$	0	0
$701$	−13.1803	−0.497815	−0.248907	−	0.968527i	$-0.580071\pi$
$701$	−13.1803	−0.497815	−0.248907	+	0.968527i	$0.580071\pi$
$702$	0	0
$703$	−14.7611	−0.556727
$704$	4.23607	0.159653
$705$	0	0
$706$	33.1946	1.24930
$707$	0	0
$708$	0	0
$709$	−5.65248	−0.212283	−0.106142	−	0.994351i	$-0.533850\pi$
$709$	−5.65248	−0.212283	−0.106142	+	0.994351i	$0.533850\pi$
$710$	−4.91034	−0.184282
$711$	0	0
$712$	−14.1421	−0.529999
$713$	0.922740	0.0345569
$714$	0	0
$715$	1.52786	0.0571389
$716$	12.5836	0.470271
$717$	0	0
$718$	27.7984	1.03743
$719$	17.8145	0.664368	0.332184	−	0.943215i	$-0.392214\pi$
$719$	17.8145	0.664368	0.332184	+	0.943215i	$0.392214\pi$
$720$	0	0
$721$	0	0
$722$	17.3262	0.644816
$723$	0	0
$724$	−14.1908	−0.527399
$725$	13.7082	0.509110
$726$	0	0
$727$	7.02236	0.260445	0.130222	−	0.991485i	$-0.458431\pi$
$727$	7.02236	0.260445	0.130222	+	0.991485i	$0.458431\pi$
$728$	0	0
$729$	0	0
$730$	18.4721	0.683684
$731$	37.7711	1.39701
$732$	0	0
$733$	−10.0270	−0.370356	−0.185178	−	0.982705i	$-0.559286\pi$
$733$	−10.0270	−0.370356	−0.185178	+	0.982705i	$0.559286\pi$
$734$	−2.82843	−0.104399
$735$	0	0
$736$	9.34752	0.344554
$737$	−7.47214	−0.275240
$738$	0	0
$739$	−33.3607	−1.22719	−0.613596	−	0.789620i	$-0.710278\pi$
$739$	−33.3607	−1.22719	−0.613596	+	0.789620i	$0.710278\pi$
$740$	1.46292	0.0537781
$741$	0	0
$742$	0	0
$743$	−41.2492	−1.51329	−0.756644	−	0.653828i	$-0.773162\pi$
$743$	−41.2492	−1.51329	−0.756644	+	0.653828i	$0.773162\pi$
$744$	0	0
$745$	−17.0193	−0.623538
$746$	−14.8541	−0.543847
$747$	0	0
$748$	−1.95440	−0.0714598
$749$	0	0
$750$	0	0
$751$	30.3050	1.10584	0.552922	−	0.833233i	$-0.313513\pi$
$751$	30.3050	1.10584	0.552922	+	0.833233i	$0.313513\pi$
$752$	−60.7811	−2.21646
$753$	0	0
$754$	−9.15298	−0.333332
$755$	−4.62520	−0.168328
$756$	0	0
$757$	16.0689	0.584034	0.292017	−	0.956413i	$-0.405674\pi$
$757$	16.0689	0.584034	0.292017	+	0.956413i	$0.405674\pi$
$758$	−51.2148	−1.86020
$759$	0	0
$760$	10.6525	0.386406
$761$	1.87558	0.0679899	0.0339949	−	0.999422i	$-0.489177\pi$
$761$	1.87558	0.0679899	0.0339949	+	0.999422i	$0.489177\pi$
$762$	0	0
$763$	0	0
$764$	−2.72949	−0.0987495
$765$	0	0
$766$	−21.5471	−0.778527
$767$	14.2492	0.514510
$768$	0	0
$769$	−52.2958	−1.88583	−0.942917	−	0.333027i	$-0.891930\pi$
$769$	−52.2958	−1.88583	−0.942917	+	0.333027i	$0.891930\pi$
$770$	0	0
$771$	0	0
$772$	−10.1803	−0.366398
$773$	−14.3184	−0.514996	−0.257498	−	0.966279i	$-0.582898\pi$
$773$	−14.3184	−0.514996	−0.257498	+	0.966279i	$0.582898\pi$
$774$	0	0
$775$	1.41421	0.0508001
$776$	20.9281	0.751274
$777$	0	0
$778$	17.7082	0.634870
$779$	54.6525	1.95813
$780$	0	0
$781$	−3.47214	−0.124243
$782$	14.1421	0.505722
$783$	0	0
$784$	0	0
$785$	1.88854	0.0674050
$786$	0	0
$787$	−34.1475	−1.21723	−0.608613	−	0.793467i	$-0.708274\pi$
$787$	−34.1475	−1.21723	−0.608613	+	0.793467i	$0.708274\pi$
$788$	11.3820	0.405466
$789$	0	0
$790$	−17.3044	−0.615663
$791$	0	0
$792$	0	0
$793$	23.0557	0.818733
$794$	46.6389	1.65515
$795$	0	0
$796$	7.07107	0.250627
$797$	27.2039	0.963612	0.481806	−	0.876278i	$-0.339981\pi$
$797$	27.2039	0.963612	0.481806	+	0.876278i	$0.339981\pi$
$798$	0	0
$799$	−39.5967	−1.40083
$800$	14.3262	0.506509
$801$	0	0
$802$	59.9787	2.11792
$803$	13.0618	0.460940
$804$	0	0
$805$	0	0
$806$	−0.944272	−0.0332606
$807$	0	0
$808$	27.7140	0.974975
$809$	43.5410	1.53082	0.765410	−	0.643543i	$-0.222536\pi$
$809$	43.5410	1.53082	0.765410	+	0.643543i	$0.222536\pi$
$810$	0	0
$811$	31.5254	1.10701	0.553503	−	0.832847i	$-0.313291\pi$
$811$	31.5254	1.10701	0.553503	+	0.832847i	$0.313291\pi$
$812$	0	0
$813$	0	0
$814$	4.38197	0.153588
$815$	−7.19859	−0.252156
$816$	0	0
$817$	65.1025	2.27765
$818$	−6.65841	−0.232806
$819$	0	0
$820$	−5.41641	−0.189149
$821$	−33.1803	−1.15800	−0.579001	−	0.815327i	$-0.696557\pi$
$821$	−33.1803	−1.15800	−0.579001	+	0.815327i	$0.696557\pi$
$822$	0	0
$823$	−9.41641	−0.328235	−0.164118	−	0.986441i	$-0.552478\pi$
$823$	−9.41641	−0.328235	−0.164118	+	0.986441i	$0.552478\pi$
$824$	22.4211	0.781076
$825$	0	0
$826$	0	0
$827$	−21.6525	−0.752930	−0.376465	−	0.926431i	$-0.622861\pi$
$827$	−21.6525	−0.752930	−0.376465	+	0.926431i	$0.622861\pi$
$828$	0	0
$829$	22.2148	0.771550	0.385775	−	0.922593i	$-0.373934\pi$
$829$	22.2148	0.771550	0.385775	+	0.922593i	$0.373934\pi$
$830$	−10.1803	−0.353365
$831$	0	0
$832$	7.40492	0.256719
$833$	0	0
$834$	0	0
$835$	−7.59675	−0.262896
$836$	−3.36861	−0.116506
$837$	0	0
$838$	−39.5192	−1.36517
$839$	−47.4643	−1.63865	−0.819324	−	0.573330i	$-0.805651\pi$
$839$	−47.4643	−1.63865	−0.819324	+	0.573330i	$0.805651\pi$
$840$	0	0
$841$	−18.5279	−0.638892
$842$	24.0902	0.830202
$843$	0	0
$844$	−2.03444	−0.0700284
$845$	−8.69161	−0.299001
$846$	0	0
$847$	0	0
$848$	13.1459	0.451432
$849$	0	0
$850$	21.6746	0.743432
$851$	−7.48529	−0.256592
$852$	0	0
$853$	−47.6405	−1.63118	−0.815590	−	0.578631i	$-0.803587\pi$
$853$	−47.6405	−1.63118	−0.815590	+	0.578631i	$0.803587\pi$
$854$	0	0
$855$	0	0
$856$	13.2918	0.454304
$857$	−26.9675	−0.921191	−0.460596	−	0.887610i	$-0.652364\pi$
$857$	−26.9675	−0.921191	−0.460596	+	0.887610i	$0.652364\pi$
$858$	0	0
$859$	35.0215	1.19492	0.597459	−	0.801900i	$-0.296177\pi$
$859$	35.0215	1.19492	0.597459	+	0.801900i	$0.296177\pi$
$860$	−6.45207	−0.220014
$861$	0	0
$862$	−37.8885	−1.29049
$863$	−22.3050	−0.759269	−0.379635	−	0.925136i	$-0.623950\pi$
$863$	−22.3050	−0.759269	−0.379635	+	0.925136i	$0.623950\pi$
$864$	0	0
$865$	16.8328	0.572333
$866$	−33.1946	−1.12800
$867$	0	0
$868$	0	0
$869$	−12.2361	−0.415080
$870$	0	0
$871$	−13.0618	−0.442581
$872$	−6.58359	−0.222949
$873$	0	0
$874$	24.3755	0.824513
$875$	0	0
$876$	0	0
$877$	9.76393	0.329705	0.164852	−	0.986318i	$-0.447285\pi$
$877$	9.76393	0.329705	0.164852	+	0.986318i	$0.447285\pi$
$878$	−18.9250	−0.638686
$879$	0	0
$880$	−4.24264	−0.143019
$881$	−21.7534	−0.732890	−0.366445	−	0.930440i	$-0.619425\pi$
$881$	−21.7534	−0.732890	−0.366445	+	0.930440i	$0.619425\pi$
$882$	0	0
$883$	41.6525	1.40172	0.700859	−	0.713300i	$-0.252800\pi$
$883$	41.6525	1.40172	0.700859	+	0.713300i	$0.252800\pi$
$884$	−3.41641	−0.114906
$885$	0	0
$886$	−3.70820	−0.124580
$887$	14.1421	0.474846	0.237423	−	0.971406i	$-0.423697\pi$
$887$	14.1421	0.474846	0.237423	+	0.971406i	$0.423697\pi$
$888$	0	0
$889$	0	0
$890$	8.94427	0.299813
$891$	0	0
$892$	−7.19859	−0.241027
$893$	−68.2492	−2.28387
$894$	0	0
$895$	17.7959	0.594851
$896$	0	0
$897$	0	0
$898$	50.9230	1.69932
$899$	1.08036	0.0360321
$900$	0	0
$901$	8.56409	0.285311
$902$	−16.2241	−0.540202
$903$	0	0
$904$	18.4164	0.612521
$905$	−20.0689	−0.667112
$906$	0	0
$907$	11.5410	0.383213	0.191607	−	0.981472i	$-0.438630\pi$
$907$	11.5410	0.383213	0.191607	+	0.981472i	$0.438630\pi$
$908$	−14.2697	−0.473555
$909$	0	0
$910$	0	0
$911$	−40.2492	−1.33352	−0.666758	−	0.745274i	$-0.732319\pi$
$911$	−40.2492	−1.33352	−0.666758	+	0.745274i	$0.732319\pi$
$912$	0	0
$913$	−7.19859	−0.238238
$914$	38.5623	1.27553
$915$	0	0
$916$	6.19704	0.204756
$917$	0	0
$918$	0	0
$919$	19.1803	0.632701	0.316351	−	0.948642i	$-0.397542\pi$
$919$	19.1803	0.632701	0.316351	+	0.948642i	$0.397542\pi$
$920$	5.40182	0.178093
$921$	0	0
$922$	−29.4133	−0.968677
$923$	−6.06952	−0.199781
$924$	0	0
$925$	−11.4721	−0.377202
$926$	−59.6869	−1.96143
$927$	0	0
$928$	10.9443	0.359263
$929$	53.2974	1.74863	0.874315	−	0.485360i	$-0.161311\pi$
$929$	53.2974	1.74863	0.874315	+	0.485360i	$0.161311\pi$
$930$	0	0
$931$	0	0
$932$	14.1803	0.464492
$933$	0	0
$934$	−65.2301	−2.13439
$935$	−2.76393	−0.0903902
$936$	0	0
$937$	−19.5440	−0.638473	−0.319237	−	0.947675i	$-0.603426\pi$
$937$	−19.5440	−0.638473	−0.319237	+	0.947675i	$0.603426\pi$
$938$	0	0
$939$	0	0
$940$	6.76393	0.220615
$941$	−46.5902	−1.51880	−0.759399	−	0.650625i	$-0.774507\pi$
$941$	−46.5902	−1.51880	−0.759399	+	0.650625i	$0.774507\pi$
$942$	0	0
$943$	27.7140	0.902492
$944$	−39.5679	−1.28782
$945$	0	0
$946$	−19.3262	−0.628350
$947$	−47.2361	−1.53497	−0.767483	−	0.641069i	$-0.778491\pi$
$947$	−47.2361	−1.53497	−0.767483	+	0.641069i	$0.778491\pi$
$948$	0	0
$949$	22.8328	0.741185
$950$	37.3584	1.21207
$951$	0	0
$952$	0	0
$953$	30.5836	0.990700	0.495350	−	0.868694i	$-0.335040\pi$
$953$	30.5836	0.990700	0.495350	+	0.868694i	$0.335040\pi$
$954$	0	0
$955$	−3.86008	−0.124909
$956$	8.61803	0.278727
$957$	0	0
$958$	18.1784	0.587319
$959$	0	0
$960$	0	0
$961$	−30.8885	−0.996405
$962$	7.65996	0.246967
$963$	0	0
$964$	7.48373	0.241035
$965$	−14.3972	−0.463461
$966$	0	0
$967$	8.47214	0.272446	0.136223	−	0.990678i	$-0.456504\pi$
$967$	8.47214	0.272446	0.136223	+	0.990678i	$0.456504\pi$
$968$	22.3607	0.718699
$969$	0	0
$970$	−13.2361	−0.424985
$971$	−0.333851	−0.0107138	−0.00535689	−	0.999986i	$-0.501705\pi$
$971$	−0.333851	−0.0107138	−0.00535689	+	0.999986i	$0.501705\pi$
$972$	0	0
$973$	0	0
$974$	−22.5623	−0.722943
$975$	0	0
$976$	−64.0222	−2.04930
$977$	−31.3050	−1.00153	−0.500767	−	0.865582i	$-0.666949\pi$
$977$	−31.3050	−1.00153	−0.500767	+	0.865582i	$0.666949\pi$
$978$	0	0
$979$	6.32456	0.202134
$980$	0	0
$981$	0	0
$982$	35.5967	1.13594
$983$	−24.5030	−0.781524	−0.390762	−	0.920492i	$-0.627789\pi$
$983$	−24.5030	−0.781524	−0.390762	+	0.920492i	$0.627789\pi$
$984$	0	0
$985$	16.0965	0.512878
$986$	16.5579	0.527311
$987$	0	0
$988$	−5.88854	−0.187340
$989$	33.0132	1.04976
$990$	0	0
$991$	−6.41641	−0.203824	−0.101912	−	0.994793i	$-0.532496\pi$
$991$	−6.41641	−0.203824	−0.101912	+	0.994793i	$0.532496\pi$
$992$	1.12907	0.0358480
$993$	0	0
$994$	0	0
$995$	10.0000	0.317021
$996$	0	0
$997$	−45.1760	−1.43074	−0.715369	−	0.698746i	$-0.753742\pi$
$997$	−45.1760	−1.43074	−0.715369	+	0.698746i	$0.753742\pi$
$998$	−22.4721	−0.711343
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3969.2.a.y.1.4	yes	4
3.2	odd	2		3969.2.a.r.1.1	✓	4
7.6	odd	2	inner	3969.2.a.y.1.3	yes	4
21.20	even	2		3969.2.a.r.1.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3969.2.a.r.1.1	✓	4	3.2	odd	2
3969.2.a.r.1.2	yes	4	21.20	even	2
3969.2.a.y.1.3	yes	4	7.6	odd	2	inner
3969.2.a.y.1.4	yes	4	1.1	even	1	trivial

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 3969 = 3^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3969.a (trivial)

\(f(q)\)	\(=\)	\(q+1.61803 q^{2} +0.618034 q^{4} +0.874032 q^{5} -2.23607 q^{8} +O(q^{10})\)	\(q+1.61803 q^{2} +0.618034 q^{4} +0.874032 q^{5} -2.23607 q^{8} +1.41421 q^{10} +1.00000 q^{11} +1.74806 q^{13} -4.85410 q^{16} -3.16228 q^{17} -5.45052 q^{19} +0.540182 q^{20} +1.61803 q^{22} -2.76393 q^{23} -4.23607 q^{25} +2.82843 q^{26} -3.23607 q^{29} -0.333851 q^{31} -3.38197 q^{32} -5.11667 q^{34} +2.70820 q^{37} -8.81913 q^{38} -1.95440 q^{40} -10.0270 q^{41} -11.9443 q^{43} +0.618034 q^{44} -4.47214 q^{46} +12.5216 q^{47} -6.85410 q^{50} +1.08036 q^{52} -2.70820 q^{53} +0.874032 q^{55} -5.23607 q^{58} +8.15143 q^{59} +13.1893 q^{61} -0.540182 q^{62} +4.23607 q^{64} +1.52786 q^{65} -7.47214 q^{67} -1.95440 q^{68} -3.47214 q^{71} +13.0618 q^{73} +4.38197 q^{74} -3.36861 q^{76} -12.2361 q^{79} -4.24264 q^{80} -16.2241 q^{82} -7.19859 q^{83} -2.76393 q^{85} -19.3262 q^{86} -2.23607 q^{88} +6.32456 q^{89} -1.70820 q^{92} +20.2604 q^{94} -4.76393 q^{95} -9.35931 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} - 2 q^{4}+O(q^{10}) \) 4 * q + 2 * q^2 - 2 * q^4	\( 4 q + 2 q^{2} - 2 q^{4} + 4 q^{11} - 6 q^{16} + 2 q^{22} - 20 q^{23} - 8 q^{25} - 4 q^{29} - 18 q^{32} - 16 q^{37} - 12 q^{43} - 2 q^{44} - 14 q^{50} + 16 q^{53} - 12 q^{58} + 8 q^{64} + 24 q^{65} - 12 q^{67} + 4 q^{71} + 22 q^{74} - 40 q^{79} - 20 q^{85} - 46 q^{86} + 20 q^{92} - 28 q^{95}+O(q^{100}) \) 4 * q + 2 * q^2 - 2 * q^4 + 4 * q^11 - 6 * q^16 + 2 * q^22 - 20 * q^23 - 8 * q^25 - 4 * q^29 - 18 * q^32 - 16 * q^37 - 12 * q^43 - 2 * q^44 - 14 * q^50 + 16 * q^53 - 12 * q^58 + 8 * q^64 + 24 * q^65 - 12 * q^67 + 4 * q^71 + 22 * q^74 - 40 * q^79 - 20 * q^85 - 46 * q^86 + 20 * q^92 - 28 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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