LMFDB - Embedded newform 441.3.b.e.197.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,3,Mod(197,441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("441.197");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 441 = 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	441.b (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$12.0163796583$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.959512576.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 7x^{4} + 81 $ x^8 + 7*x^4 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6}\cdot 7^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			197.8
Root			$-0.819051 + 1.52616i$ of defining polynomial
Character	$\chi$	$=$	441.197
Dual form			441.3.b.e.197.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.05231i q^{2} -5.31662 q^{4} +8.31662i q^{5} -4.01875i q^{8} +O(q^{10})$	$q+3.05231i q^{2} -5.31662 q^{4} +8.31662i q^{5} -4.01875i q^{8} -25.3850 q^{10} +19.7281i q^{11} +17.3474 q^{13} -9.00000 q^{16} -5.68338i q^{17} +4.31353 q^{19} -44.2164i q^{20} -60.2164 q^{22} -15.1086i q^{23} -44.1662 q^{25} +52.9499i q^{26} +0.824662i q^{29} +39.0084 q^{31} -43.5458i q^{32} +17.3474 q^{34} -35.8997 q^{37} +13.1662i q^{38} +33.4225 q^{40} -47.6834i q^{41} -11.3668 q^{43} -104.887i q^{44} +46.1161 q^{46} -1.79950i q^{47} -134.809i q^{50} -92.2299 q^{52} +35.3553i q^{53} -164.071 q^{55} -2.51713 q^{58} -68.3325i q^{59} +37.7360 q^{61} +119.066i q^{62} +96.9156 q^{64} +144.272i q^{65} +33.1662 q^{67} +30.2164i q^{68} -0.824662i q^{71} +94.7748 q^{73} -109.577i q^{74} -22.9334 q^{76} +12.7335 q^{79} -74.8496i q^{80} +145.545 q^{82} +86.7652i q^{83} +47.2665 q^{85} -34.6949i q^{86} +79.2824 q^{88} +102.016i q^{89} +80.3266i q^{92} +5.49263 q^{94} +35.8740i q^{95} +41.4600 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 16 q^{4}+O(q^{10}) $ 8 * q - 16 * q^4	$ 8 q - 16 q^{4} - 72 q^{16} - 296 q^{22} - 88 q^{25} - 128 q^{37} - 144 q^{43} + 24 q^{46} - 312 q^{58} + 112 q^{64} + 208 q^{79} + 272 q^{85} + 24 q^{88}+O(q^{100}) $ 8 * q - 16 * q^4 - 72 * q^16 - 296 * q^22 - 88 * q^25 - 128 * q^37 - 144 * q^43 + 24 * q^46 - 312 * q^58 + 112 * q^64 + 208 * q^79 + 272 * q^85 + 24 * q^88

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/441\mathbb{Z}\right)^\times$.

$n$	$199$	$344$
$\chi(n)$	$1$	$-1$

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 $

$2$	3.05231i	1.52616i	0.646306	+	0.763079i	$0.276313\pi$
$2$	3.05231i	1.52616i	−0.646306	+	0.763079i	$0.723687\pi$
$3$	0	0
$3$	0	0
$4$	−5.31662	−1.32916
$5$	8.31662i	1.66332i	0.555281	+	0.831662i	$0.312611\pi$
$5$	8.31662i	1.66332i	−0.555281	+	0.831662i	$0.687389\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	− 4.01875i	− 0.502344i
$9$	0	0
$10$	−25.3850	−2.53850
$11$	19.7281i	1.79346i	0.442574	+	0.896732i	$0.354065\pi$
$11$	19.7281i	1.79346i	−0.442574	+	0.896732i	$0.645935\pi$
$12$	0	0
$13$	17.3474	1.33442	0.667210	−	0.744870i	$-0.267489\pi$
$13$	17.3474	1.33442	0.667210	+	0.744870i	$0.267489\pi$
$14$	0	0
$15$	0	0
$16$	−9.00000	−0.562500
$17$	− 5.68338i	− 0.334316i	−0.985930	−	0.167158i	$-0.946541\pi$
$17$	− 5.68338i	− 0.334316i	0.985930	−	0.167158i	$-0.0534590\pi$
$18$	0	0
$19$	4.31353	0.227028	0.113514	−	0.993536i	$-0.463789\pi$
$19$	4.31353	0.227028	0.113514	+	0.993536i	$0.463789\pi$
$20$	− 44.2164i	− 2.21082i
$21$	0	0
$22$	−60.2164	−2.73711
$23$	− 15.1086i	− 0.656895i	−0.944522	−	0.328447i	$-0.893475\pi$
$23$	− 15.1086i	− 0.656895i	0.944522	−	0.328447i	$-0.106525\pi$
$24$	0	0
$25$	−44.1662	−1.76665
$26$	52.9499i	2.03653i
$27$	0	0
$28$	0	0
$29$	0.824662i	0.0284366i	0.999899	+	0.0142183i	$0.00452598\pi$
$29$	0.824662i	0.0284366i	−0.999899	+	0.0142183i	$0.995474\pi$
$30$	0	0
$31$	39.0084	1.25834	0.629168	−	0.777269i	$-0.283396\pi$
$31$	39.0084	1.25834	0.629168	+	0.777269i	$0.283396\pi$
$32$	− 43.5458i	− 1.36081i
$33$	0	0
$34$	17.3474	0.510219
$35$	0	0
$36$	0	0
$37$	−35.8997	−0.970263	−0.485132	−	0.874441i	$-0.661228\pi$
$37$	−35.8997	−0.970263	−0.485132	+	0.874441i	$0.661228\pi$
$38$	13.1662i	0.346480i
$39$	0	0
$40$	33.4225	0.835562
$41$	− 47.6834i	− 1.16301i	−0.813543	−	0.581505i	$-0.802464\pi$
$41$	− 47.6834i	− 1.16301i	0.813543	−	0.581505i	$-0.197536\pi$
$42$	0	0
$43$	−11.3668	−0.264343	−0.132172	−	0.991227i	$-0.542195\pi$
$43$	−11.3668	−0.264343	−0.132172	+	0.991227i	$0.542195\pi$
$44$	− 104.887i	− 2.38379i
$45$	0	0
$46$	46.1161	1.00252
$47$	− 1.79950i	− 0.0382872i	−0.999817	−	0.0191436i	$-0.993906\pi$
$47$	− 1.79950i	− 0.0382872i	0.999817	−	0.0191436i	$-0.00609397\pi$
$48$	0	0
$49$	0	0
$50$	− 134.809i	− 2.69619i
$51$	0	0
$52$	−92.2299	−1.77365
$53$	35.3553i	0.667082i	0.942736	+	0.333541i	$0.108244\pi$
$53$	35.3553i	0.667082i	−0.942736	+	0.333541i	$0.891756\pi$
$54$	0	0
$55$	−164.071	−2.98311
$56$	0	0
$57$	0	0
$58$	−2.51713	−0.0433987
$59$	− 68.3325i	− 1.15818i	−0.815264	−	0.579089i	$-0.803408\pi$
$59$	− 68.3325i	− 1.15818i	0.815264	−	0.579089i	$-0.196592\pi$
$60$	0	0
$61$	37.7360	0.618623	0.309311	−	0.950961i	$-0.399901\pi$
$61$	37.7360	0.618623	0.309311	+	0.950961i	$0.399901\pi$
$62$	119.066i	1.92042i
$63$	0	0
$64$	96.9156	1.51431
$65$	144.272i	2.21957i
$66$	0	0
$67$	33.1662	0.495019	0.247509	−	0.968886i	$-0.420388\pi$
$67$	33.1662	0.495019	0.247509	+	0.968886i	$0.420388\pi$
$68$	30.2164i	0.444358i
$69$	0	0
$70$	0	0
$71$	− 0.824662i	− 0.0116150i	−0.999983	−	0.00580748i	$-0.998151\pi$
$71$	− 0.824662i	− 0.0116150i	0.999983	−	0.00580748i	$-0.00184859\pi$
$72$	0	0
$73$	94.7748	1.29828	0.649142	−	0.760667i	$-0.275128\pi$
$73$	94.7748	1.29828	0.649142	+	0.760667i	$0.275128\pi$
$74$	− 109.577i	− 1.48077i
$75$	0	0
$76$	−22.9334	−0.301755
$77$	0	0
$78$	0	0
$79$	12.7335	0.161184	0.0805918	−	0.996747i	$-0.474319\pi$
$79$	12.7335	0.161184	0.0805918	+	0.996747i	$0.474319\pi$
$80$	− 74.8496i	− 0.935620i
$81$	0	0
$82$	145.545	1.77493
$83$	86.7652i	1.04536i	0.852528	+	0.522682i	$0.175069\pi$
$83$	86.7652i	1.04536i	−0.852528	+	0.522682i	$0.824931\pi$
$84$	0	0
$85$	47.2665	0.556076
$86$	− 34.6949i	− 0.403429i
$87$	0	0
$88$	79.2824	0.900936
$89$	102.016i	1.14625i	0.819469	+	0.573123i	$0.194268\pi$
$89$	102.016i	1.14625i	−0.819469	+	0.573123i	$0.805732\pi$
$90$	0	0
$91$	0	0
$92$	80.3266i	0.873115i
$93$	0	0
$94$	5.49263	0.0584323
$95$	35.8740i	0.377621i
$96$	0	0
$97$	41.4600	0.427422	0.213711	−	0.976897i	$-0.431445\pi$
$97$	41.4600	0.427422	0.213711	+	0.976897i	$0.431445\pi$
$98$	0	0
$99$	0	0
$100$	234.815	2.34815
$101$	11.7836i	0.116670i	0.998297	+	0.0583348i	$0.0185791\pi$
$101$	11.7836i	0.116670i	−0.998297	+	0.0583348i	$0.981421\pi$
$102$	0	0
$103$	−192.497	−1.86891	−0.934453	−	0.356087i	$-0.884111\pi$
$103$	−192.497	−1.86891	−0.934453	+	0.356087i	$0.884111\pi$
$104$	− 69.7151i	− 0.670338i
$105$	0	0
$106$	−107.916	−1.01807
$107$	− 86.4312i	− 0.807769i	−0.914810	−	0.403884i	$-0.867660\pi$
$107$	− 86.4312i	− 0.807769i	0.914810	−	0.403884i	$-0.132340\pi$
$108$	0	0
$109$	187.266	1.71804	0.859021	−	0.511941i	$-0.171073\pi$
$109$	187.266	1.71804	0.859021	+	0.511941i	$0.171073\pi$
$110$	− 500.797i	− 4.55270i
$111$	0	0
$112$	0	0
$113$	101.588i	0.899011i	0.893278	+	0.449506i	$0.148400\pi$
$113$	101.588i	0.899011i	−0.893278	+	0.449506i	$0.851600\pi$
$114$	0	0
$115$	125.652	1.09263
$116$	− 4.38442i	− 0.0377967i
$117$	0	0
$118$	208.572	1.76756
$119$	0	0
$120$	0	0
$121$	−268.198	−2.21651
$122$	115.182i	0.944116i
$123$	0	0
$124$	−207.393	−1.67253
$125$	− 159.398i	− 1.27519i
$126$	0	0
$127$	−22.2322	−0.175057	−0.0875285	−	0.996162i	$-0.527897\pi$
$127$	−22.2322	−0.175057	−0.0875285	+	0.996162i	$0.527897\pi$
$128$	121.634i	0.950262i
$129$	0	0
$130$	−440.364	−3.38742
$131$	196.264i	1.49820i	0.662458	+	0.749099i	$0.269513\pi$
$131$	196.264i	1.49820i	−0.662458	+	0.749099i	$0.730487\pi$
$132$	0	0
$133$	0	0
$134$	101.234i	0.755476i
$135$	0	0
$136$	−22.8401	−0.167942
$137$	144.933i	1.05790i	0.848652	+	0.528951i	$0.177415\pi$
$137$	144.933i	1.05790i	−0.848652	+	0.528951i	$0.822585\pi$
$138$	0	0
$139$	−65.4792	−0.471073	−0.235537	−	0.971865i	$-0.575685\pi$
$139$	−65.4792	−0.471073	−0.235537	+	0.971865i	$0.575685\pi$
$140$	0	0
$141$	0	0
$142$	2.51713	0.0177262
$143$	342.232i	2.39323i
$144$	0	0
$145$	−6.85840	−0.0472993
$146$	289.282i	1.98139i
$147$	0	0
$148$	190.865	1.28963
$149$	99.7972i	0.669780i	0.942257	+	0.334890i	$0.108699\pi$
$149$	99.7972i	0.669780i	−0.942257	+	0.334890i	$0.891301\pi$
$150$	0	0
$151$	−212.897	−1.40992	−0.704958	−	0.709249i	$-0.749034\pi$
$151$	−212.897	−1.40992	−0.704958	+	0.709249i	$0.749034\pi$
$152$	− 17.3350i	− 0.114046i
$153$	0	0
$154$	0	0
$155$	324.418i	2.09302i
$156$	0	0
$157$	−115.940	−0.738468	−0.369234	−	0.929336i	$-0.620380\pi$
$157$	−115.940	−0.738468	−0.369234	+	0.929336i	$0.620380\pi$
$158$	38.8667i	0.245991i
$159$	0	0
$160$	362.154	2.26347
$161$	0	0
$162$	0	0
$163$	138.633	0.850511	0.425255	−	0.905073i	$-0.360184\pi$
$163$	138.633	0.850511	0.425255	+	0.905073i	$0.360184\pi$
$164$	253.515i	1.54582i
$165$	0	0
$166$	−264.835	−1.59539
$167$	− 217.198i	− 1.30059i	−0.759683	−	0.650293i	$-0.774646\pi$
$167$	− 217.198i	− 1.30059i	0.759683	−	0.650293i	$-0.225354\pi$
$168$	0	0
$169$	131.934	0.780675
$170$	144.272i	0.848660i
$171$	0	0
$172$	60.4327	0.351353
$173$	− 224.681i	− 1.29873i	−0.760475	−	0.649367i	$-0.775034\pi$
$173$	− 224.681i	− 1.29873i	0.760475	−	0.649367i	$-0.224966\pi$
$174$	0	0
$175$	0	0
$176$	− 177.553i	− 1.00882i
$177$	0	0
$178$	−311.385	−1.74935
$179$	176.941i	0.988497i	0.869321	+	0.494248i	$0.164557\pi$
$179$	176.941i	0.988497i	−0.869321	+	0.494248i	$0.835443\pi$
$180$	0	0
$181$	53.2215	0.294041	0.147021	−	0.989133i	$-0.453032\pi$
$181$	53.2215	0.294041	0.147021	+	0.989133i	$0.453032\pi$
$182$	0	0
$183$	0	0
$184$	−60.7176	−0.329987
$185$	− 298.565i	− 1.61386i
$186$	0	0
$187$	112.122	0.599584
$188$	9.56725i	0.0508896i
$189$	0	0
$190$	−109.499	−0.576309
$191$	103.872i	0.543833i	0.962321	+	0.271916i	$0.0876574\pi$
$191$	103.872i	0.543833i	−0.962321	+	0.271916i	$0.912343\pi$
$192$	0	0
$193$	192.665	0.998264	0.499132	−	0.866526i	$-0.333652\pi$
$193$	192.665	0.998264	0.499132	+	0.866526i	$0.333652\pi$
$194$	126.549i	0.652314i
$195$	0	0
$196$	0	0
$197$	− 176.068i	− 0.893745i	−0.894598	−	0.446873i	$-0.852538\pi$
$197$	− 176.068i	− 0.893745i	0.894598	−	0.446873i	$-0.147462\pi$
$198$	0	0
$199$	−167.206	−0.840229	−0.420115	−	0.907471i	$-0.638010\pi$
$199$	−167.206	−0.840229	−0.420115	+	0.907471i	$0.638010\pi$
$200$	177.493i	0.887466i
$201$	0	0
$202$	−35.9673	−0.178056
$203$	0	0
$204$	0	0
$205$	396.565	1.93446
$206$	− 587.562i	− 2.85224i
$207$	0	0
$208$	−156.127	−0.750611
$209$	85.0977i	0.407166i
$210$	0	0
$211$	−211.799	−1.00379	−0.501895	−	0.864929i	$-0.667363\pi$
$211$	−211.799	−1.00379	−0.501895	+	0.864929i	$0.667363\pi$
$212$	− 187.971i	− 0.886656i
$213$	0	0
$214$	263.815	1.23278
$215$	− 94.5330i	− 0.439688i
$216$	0	0
$217$	0	0
$218$	571.596i	2.62200i
$219$	0	0
$220$	872.305	3.96502
$221$	− 98.5921i	− 0.446118i
$222$	0	0
$223$	−168.385	−0.755089	−0.377544	−	0.925992i	$-0.623231\pi$
$223$	−168.385	−0.755089	−0.377544	+	0.925992i	$0.623231\pi$
$224$	0	0
$225$	0	0
$226$	−310.079	−1.37203
$227$	239.931i	1.05697i	0.848944	+	0.528483i	$0.177239\pi$
$227$	239.931i	1.05697i	−0.848944	+	0.528483i	$0.822761\pi$
$228$	0	0
$229$	443.306	1.93583	0.967916	−	0.251272i	$-0.0808490\pi$
$229$	443.306	1.93583	0.967916	+	0.251272i	$0.0808490\pi$
$230$	383.530i	1.66752i
$231$	0	0
$232$	3.31411	0.0142850
$233$	95.4836i	0.409801i	0.978783	+	0.204901i	$0.0656871\pi$
$233$	95.4836i	0.409801i	−0.978783	+	0.204901i	$0.934313\pi$
$234$	0	0
$235$	14.9657	0.0636840
$236$	363.298i	1.53940i
$237$	0	0
$238$	0	0
$239$	− 206.449i	− 0.863804i	−0.901921	−	0.431902i	$-0.857843\pi$
$239$	− 206.449i	− 0.863804i	0.901921	−	0.431902i	$-0.142157\pi$
$240$	0	0
$241$	154.358	0.640491	0.320246	−	0.947335i	$-0.396235\pi$
$241$	154.358	0.640491	0.320246	+	0.947335i	$0.396235\pi$
$242$	− 818.625i	− 3.38275i
$243$	0	0
$244$	−200.628	−0.822246
$245$	0	0
$246$	0	0
$247$	74.8287	0.302950
$248$	− 156.765i	− 0.632118i
$249$	0	0
$250$	486.534	1.94614
$251$	− 382.433i	− 1.52364i	−0.647791	−	0.761818i	$-0.724307\pi$
$251$	− 382.433i	− 1.52364i	0.647791	−	0.761818i	$-0.275693\pi$
$252$	0	0
$253$	298.063	1.17812
$254$	− 67.8598i	− 0.267165i
$255$	0	0
$256$	16.3985	0.0640566
$257$	23.0184i	0.0895657i	0.998997	+	0.0447828i	$0.0142596\pi$
$257$	23.0184i	0.0895657i	−0.998997	+	0.0447828i	$0.985740\pi$
$258$	0	0
$259$	0	0
$260$	− 767.041i	− 2.95016i
$261$	0	0
$262$	−599.059	−2.28649
$263$	− 127.537i	− 0.484931i	−0.970160	−	0.242465i	$-0.922044\pi$
$263$	− 127.537i	− 0.484931i	0.970160	−	0.242465i	$-0.0779561\pi$
$264$	0	0
$265$	−294.037	−1.10957
$266$	0	0
$267$	0	0
$268$	−176.332	−0.657957
$269$	192.380i	0.715168i	0.933881	+	0.357584i	$0.116399\pi$
$269$	192.380i	0.715168i	−0.933881	+	0.357584i	$0.883601\pi$
$270$	0	0
$271$	108.212	0.399305	0.199652	−	0.979867i	$-0.436019\pi$
$271$	108.212	0.399305	0.199652	+	0.979867i	$0.436019\pi$
$272$	51.1504i	0.188053i
$273$	0	0
$274$	−442.380	−1.61453
$275$	− 871.316i	− 3.16842i
$276$	0	0
$277$	367.129	1.32538	0.662689	−	0.748895i	$-0.269415\pi$
$277$	367.129	1.32538	0.662689	+	0.748895i	$0.269415\pi$
$278$	− 199.863i	− 0.718932i
$279$	0	0
$280$	0	0
$281$	− 172.747i	− 0.614757i	−0.951587	−	0.307379i	$-0.900548\pi$
$281$	− 172.747i	− 0.614757i	0.951587	−	0.307379i	$-0.0994518\pi$
$282$	0	0
$283$	369.322	1.30503	0.652513	−	0.757777i	$-0.273715\pi$
$283$	369.322	1.30503	0.652513	+	0.757777i	$0.273715\pi$
$284$	4.38442i	0.0154381i
$285$	0	0
$286$	−1044.60	−3.65245
$287$	0	0
$288$	0	0
$289$	256.699	0.888233
$290$	− 20.9340i	− 0.0721862i
$291$	0	0
$292$	−503.882	−1.72562
$293$	92.4486i	0.315524i	0.987477	+	0.157762i	$0.0504279\pi$
$293$	92.4486i	0.315524i	−0.987477	+	0.157762i	$0.949572\pi$
$294$	0	0
$295$	568.296	1.92643
$296$	144.272i	0.487406i
$297$	0	0
$298$	−304.612	−1.02219
$299$	− 262.095i	− 0.876573i
$300$	0	0
$301$	0	0
$302$	− 649.829i	− 2.15175i
$303$	0	0
$304$	−38.8218	−0.127703
$305$	313.836i	1.02897i
$306$	0	0
$307$	499.662	1.62756	0.813781	−	0.581171i	$-0.197405\pi$
$307$	499.662	1.62756	0.813781	+	0.581171i	$0.197405\pi$
$308$	0	0
$309$	0	0
$310$	−990.227	−3.19428
$311$	89.1345i	0.286606i	0.989679	+	0.143303i	$0.0457724\pi$
$311$	89.1345i	0.286606i	−0.989679	+	0.143303i	$0.954228\pi$
$312$	0	0
$313$	63.0276	0.201366	0.100683	−	0.994919i	$-0.467897\pi$
$313$	63.0276	0.201366	0.100683	+	0.994919i	$0.467897\pi$
$314$	− 353.884i	− 1.12702i
$315$	0	0
$316$	−67.6992	−0.214238
$317$	− 263.465i	− 0.831121i	−0.909565	−	0.415561i	$-0.863585\pi$
$317$	− 263.465i	− 0.831121i	0.909565	−	0.415561i	$-0.136415\pi$
$318$	0	0
$319$	−16.2690	−0.0510000
$320$	806.011i	2.51878i
$321$	0	0
$322$	0	0
$323$	− 24.5154i	− 0.0758991i
$324$	0	0
$325$	−766.172	−2.35745
$326$	423.152i	1.29801i
$327$	0	0
$328$	−191.628	−0.584231
$329$	0	0
$330$	0	0
$331$	−358.997	−1.08458	−0.542292	−	0.840190i	$-0.682443\pi$
$331$	−358.997	−1.08458	−0.542292	+	0.840190i	$0.682443\pi$
$332$	− 461.298i	− 1.38945i
$333$	0	0
$334$	662.957	1.98490
$335$	275.831i	0.823377i
$336$	0	0
$337$	−266.834	−0.791792	−0.395896	−	0.918295i	$-0.629566\pi$
$337$	−266.834	−0.791792	−0.395896	+	0.918295i	$0.629566\pi$
$338$	402.704i	1.19143i
$339$	0	0
$340$	−251.298	−0.739112
$341$	769.562i	2.25678i
$342$	0	0
$343$	0	0
$344$	45.6802i	0.132791i
$345$	0	0
$346$	685.797	1.98207
$347$	− 639.907i	− 1.84411i	−0.387055	−	0.922057i	$-0.626508\pi$
$347$	− 639.907i	− 1.84411i	0.387055	−	0.922057i	$-0.373492\pi$
$348$	0	0
$349$	363.520	1.04161	0.520803	−	0.853677i	$-0.325633\pi$
$349$	363.520	1.04161	0.520803	+	0.853677i	$0.325633\pi$
$350$	0	0
$351$	0	0
$352$	859.077	2.44056
$353$	− 198.918i	− 0.563507i	−0.959487	−	0.281754i	$-0.909084\pi$
$353$	− 198.918i	− 0.563507i	0.959487	−	0.281754i	$-0.0909161\pi$
$354$	0	0
$355$	6.85840	0.0193194
$356$	− 542.380i	− 1.52354i
$357$	0	0
$358$	−540.079	−1.50860
$359$	288.332i	0.803153i	0.915826	+	0.401576i	$0.131538\pi$
$359$	288.332i	0.803153i	−0.915826	+	0.401576i	$0.868462\pi$
$360$	0	0
$361$	−342.393	−0.948458
$362$	162.449i	0.448753i
$363$	0	0
$364$	0	0
$365$	788.206i	2.15947i
$366$	0	0
$367$	−582.179	−1.58632	−0.793159	−	0.609015i	$-0.791565\pi$
$367$	−582.179	−1.58632	−0.793159	+	0.609015i	$0.791565\pi$
$368$	135.977i	0.369503i
$369$	0	0
$370$	911.314	2.46301
$371$	0	0
$372$	0	0
$373$	425.261	1.14011	0.570056	−	0.821606i	$-0.306922\pi$
$373$	425.261	1.14011	0.570056	+	0.821606i	$0.306922\pi$
$374$	342.232i	0.915059i
$375$	0	0
$376$	−7.23174	−0.0192333
$377$	14.3058i	0.0379464i
$378$	0	0
$379$	−735.462	−1.94053	−0.970266	−	0.242039i	$-0.922184\pi$
$379$	−735.462	−1.94053	−0.970266	+	0.242039i	$0.922184\pi$
$380$	− 190.729i	− 0.501917i
$381$	0	0
$382$	−317.050	−0.829974
$383$	− 495.398i	− 1.29347i	−0.762715	−	0.646734i	$-0.776134\pi$
$383$	− 495.398i	− 1.29347i	0.762715	−	0.646734i	$-0.223866\pi$
$384$	0	0
$385$	0	0
$386$	588.074i	1.52351i
$387$	0	0
$388$	−220.427	−0.568111
$389$	30.7134i	0.0789547i	0.999220	+	0.0394773i	$0.0125693\pi$
$389$	30.7134i	0.0789547i	−0.999220	+	0.0394773i	$0.987431\pi$
$390$	0	0
$391$	−85.8677	−0.219610
$392$	0	0
$393$	0	0
$394$	537.414	1.36400
$395$	105.900i	0.268101i
$396$	0	0
$397$	−110.850	−0.279219	−0.139609	−	0.990207i	$-0.544585\pi$
$397$	−110.850	−0.279219	−0.139609	+	0.990207i	$0.544585\pi$
$398$	− 510.364i	− 1.28232i
$399$	0	0
$400$	397.496	0.993741
$401$	563.159i	1.40439i	0.711986	+	0.702194i	$0.247796\pi$
$401$	563.159i	1.40439i	−0.711986	+	0.702194i	$0.752204\pi$
$402$	0	0
$403$	676.697	1.67915
$404$	− 62.6491i	− 0.155072i
$405$	0	0
$406$	0	0
$407$	− 708.234i	− 1.74013i
$408$	0	0
$409$	−504.845	−1.23434	−0.617170	−	0.786830i	$-0.711721\pi$
$409$	−504.845	−1.23434	−0.617170	+	0.786830i	$0.711721\pi$
$410$	1210.44i	2.95229i
$411$	0	0
$412$	1023.44	2.48407
$413$	0	0
$414$	0	0
$415$	−721.594	−1.73878
$416$	− 755.409i	− 1.81589i
$417$	0	0
$418$	−259.745	−0.621400
$419$	− 603.098i	− 1.43937i	−0.694299	−	0.719687i	$-0.744285\pi$
$419$	− 603.098i	− 1.43937i	0.694299	−	0.719687i	$-0.255715\pi$
$420$	0	0
$421$	−664.259	−1.57781	−0.788906	−	0.614514i	$-0.789352\pi$
$421$	−664.259	−1.57781	−0.788906	+	0.614514i	$0.789352\pi$
$422$	− 646.479i	− 1.53194i
$423$	0	0
$424$	142.084	0.335105
$425$	251.013i	0.590620i
$426$	0	0
$427$	0	0
$428$	459.523i	1.07365i
$429$	0	0
$430$	288.544	0.671034
$431$	251.936i	0.584537i	0.956336	+	0.292269i	$0.0944101\pi$
$431$	251.936i	0.584537i	−0.956336	+	0.292269i	$0.905590\pi$
$432$	0	0
$433$	132.417	0.305814	0.152907	−	0.988241i	$-0.451137\pi$
$433$	132.417	0.305814	0.152907	+	0.988241i	$0.451137\pi$
$434$	0	0
$435$	0	0
$436$	−995.626	−2.28355
$437$	− 65.1713i	− 0.149133i
$438$	0	0
$439$	−624.941	−1.42355	−0.711777	−	0.702405i	$-0.752110\pi$
$439$	−624.941	−1.42355	−0.711777	+	0.702405i	$0.752110\pi$
$440$	659.362i	1.49855i
$441$	0	0
$442$	300.934	0.680846
$443$	81.9535i	0.184997i	0.995713	+	0.0924983i	$0.0294853\pi$
$443$	81.9535i	0.184997i	−0.995713	+	0.0924983i	$0.970515\pi$
$444$	0	0
$445$	−848.428	−1.90658
$446$	− 513.963i	− 1.15238i
$447$	0	0
$448$	0	0
$449$	335.639i	0.747525i	0.927524	+	0.373763i	$0.121933\pi$
$449$	335.639i	0.747525i	−0.927524	+	0.373763i	$0.878067\pi$
$450$	0	0
$451$	940.702	2.08581
$452$	− 540.107i	− 1.19493i
$453$	0	0
$454$	−732.346	−1.61310
$455$	0	0
$456$	0	0
$457$	488.201	1.06827	0.534136	−	0.845398i	$-0.320637\pi$
$457$	488.201	1.06827	0.534136	+	0.845398i	$0.320637\pi$
$458$	1353.11i	2.95439i
$459$	0	0
$460$	−668.046	−1.45227
$461$	508.850i	1.10380i	0.833912	+	0.551898i	$0.186096\pi$
$461$	508.850i	1.10380i	−0.833912	+	0.551898i	$0.813904\pi$
$462$	0	0
$463$	680.364	1.46947	0.734735	−	0.678355i	$-0.237307\pi$
$463$	680.364	1.46947	0.734735	+	0.678355i	$0.237307\pi$
$464$	− 7.42196i	− 0.0159956i
$465$	0	0
$466$	−291.446	−0.625421
$467$	− 546.834i	− 1.17095i	−0.810690	−	0.585475i	$-0.800908\pi$
$467$	− 546.834i	− 1.17095i	0.810690	−	0.585475i	$-0.199092\pi$
$468$	0	0
$469$	0	0
$470$	45.6802i	0.0971918i
$471$	0	0
$472$	−274.611	−0.581804
$473$	− 224.244i	− 0.474090i
$474$	0	0
$475$	−190.512	−0.401079
$476$	0	0
$477$	0	0
$478$	630.148	1.31830
$479$	683.235i	1.42638i	0.700972	+	0.713189i	$0.252750\pi$
$479$	683.235i	1.42638i	−0.700972	+	0.713189i	$0.747250\pi$
$480$	0	0
$481$	−622.769	−1.29474
$482$	471.150i	0.977490i
$483$	0	0
$484$	1425.91	2.94609
$485$	344.807i	0.710942i
$486$	0	0
$487$	−208.770	−0.428686	−0.214343	−	0.976758i	$-0.568761\pi$
$487$	−208.770	−0.428686	−0.214343	+	0.976758i	$0.568761\pi$
$488$	− 151.652i	− 0.310762i
$489$	0	0
$490$	0	0
$491$	− 383.177i	− 0.780402i	−0.920730	−	0.390201i	$-0.872405\pi$
$491$	− 383.177i	− 0.780402i	0.920730	−	0.390201i	$-0.127595\pi$
$492$	0	0
$493$	4.68686	0.00950682
$494$	228.401i	0.462350i
$495$	0	0
$496$	−351.076	−0.707814
$497$	0	0
$498$	0	0
$499$	−569.435	−1.14115	−0.570576	−	0.821245i	$-0.693280\pi$
$499$	−569.435	−1.14115	−0.570576	+	0.821245i	$0.693280\pi$
$500$	847.462i	1.69492i
$501$	0	0
$502$	1167.31	2.32531
$503$	317.699i	0.631609i	0.948824	+	0.315804i	$0.102274\pi$
$503$	317.699i	0.631609i	−0.948824	+	0.315804i	$0.897726\pi$
$504$	0	0
$505$	−98.0000	−0.194059
$506$	909.784i	1.79799i
$507$	0	0
$508$	118.201	0.232678
$509$	− 298.280i	− 0.586012i	−0.956111	−	0.293006i	$-0.905344\pi$
$509$	− 298.280i	− 0.586012i	0.956111	−	0.293006i	$-0.0946555\pi$
$510$	0	0
$511$	0	0
$512$	536.588i	1.04802i
$513$	0	0
$514$	−70.2593	−0.136691
$515$	− 1600.93i	− 3.10860i
$516$	0	0
$517$	35.5007	0.0686667
$518$	0	0
$519$	0	0
$520$	579.794	1.11499
$521$	− 689.578i	− 1.32357i	−0.749695	−	0.661783i	$-0.769800\pi$
$521$	− 689.578i	− 1.32357i	0.749695	−	0.661783i	$-0.230200\pi$
$522$	0	0
$523$	422.234	0.807331	0.403666	−	0.914907i	$-0.367736\pi$
$523$	422.234	0.807331	0.403666	+	0.914907i	$0.367736\pi$
$524$	− 1043.46i	− 1.99134i
$525$	0	0
$526$	389.282	0.740081
$527$	− 221.700i	− 0.420682i
$528$	0	0
$529$	300.731	0.568490
$530$	− 897.494i	− 1.69338i
$531$	0	0
$532$	0	0
$533$	− 827.185i	− 1.55194i
$534$	0	0
$535$	718.816	1.34358
$536$	− 133.287i	− 0.248670i
$537$	0	0
$538$	−587.205	−1.09146
$539$	0	0
$540$	0	0
$541$	157.472	0.291076	0.145538	−	0.989353i	$-0.453509\pi$
$541$	157.472	0.291076	0.145538	+	0.989353i	$0.453509\pi$
$542$	330.296i	0.609402i
$543$	0	0
$544$	−247.487	−0.454940
$545$	1557.43i	2.85766i
$546$	0	0
$547$	116.195	0.212423	0.106212	−	0.994344i	$-0.466128\pi$
$547$	116.195	0.212423	0.106212	+	0.994344i	$0.466128\pi$
$548$	− 770.553i	− 1.40612i
$549$	0	0
$550$	2659.53	4.83551
$551$	3.55720i	0.00645590i
$552$	0	0
$553$	0	0
$554$	1120.59i	2.02273i
$555$	0	0
$556$	348.128	0.626130
$557$	886.138i	1.59091i	0.606012	+	0.795456i	$0.292768\pi$
$557$	886.138i	1.59091i	−0.606012	+	0.795456i	$0.707232\pi$
$558$	0	0
$559$	−197.184	−0.352744
$560$	0	0
$561$	0	0
$562$	527.277	0.938216
$563$	− 1108.72i	− 1.96931i	−0.174503	−	0.984657i	$-0.555832\pi$
$563$	− 1108.72i	− 1.96931i	0.174503	−	0.984657i	$-0.444168\pi$
$564$	0	0
$565$	−844.871	−1.49535
$566$	1127.29i	1.99168i
$567$	0	0
$568$	−3.31411	−0.00583470
$569$	− 618.213i	− 1.08649i	−0.839574	−	0.543245i	$-0.817195\pi$
$569$	− 618.213i	− 1.08649i	0.839574	−	0.543245i	$-0.182805\pi$
$570$	0	0
$571$	−540.132	−0.945940	−0.472970	−	0.881078i	$-0.656818\pi$
$571$	−540.132	−0.945940	−0.472970	+	0.881078i	$0.656818\pi$
$572$	− 1819.52i	− 3.18098i
$573$	0	0
$574$	0	0
$575$	667.289i	1.16050i
$576$	0	0
$577$	−87.5430	−0.151721	−0.0758605	−	0.997118i	$-0.524170\pi$
$577$	−87.5430	−0.151721	−0.0758605	+	0.997118i	$0.524170\pi$
$578$	783.527i	1.35558i
$579$	0	0
$580$	36.4636	0.0628682
$581$	0	0
$582$	0	0
$583$	−697.494	−1.19639
$584$	− 380.876i	− 0.652186i
$585$	0	0
$586$	−282.182	−0.481540
$587$	157.335i	0.268032i	0.990979	+	0.134016i	$0.0427874\pi$
$587$	157.335i	0.268032i	−0.990979	+	0.134016i	$0.957213\pi$
$588$	0	0
$589$	168.264	0.285677
$590$	1734.62i	2.94003i
$591$	0	0
$592$	323.098	0.545773
$593$	682.380i	1.15073i	0.817898	+	0.575363i	$0.195139\pi$
$593$	682.380i	1.15073i	−0.817898	+	0.575363i	$0.804861\pi$
$594$	0	0
$595$	0	0
$596$	− 530.584i	− 0.890242i
$597$	0	0
$598$	799.997	1.33779
$599$	230.024i	0.384014i	0.981394	+	0.192007i	$0.0614996\pi$
$599$	230.024i	0.384014i	−0.981394	+	0.192007i	$0.938500\pi$
$600$	0	0
$601$	471.172	0.783980	0.391990	−	0.919970i	$-0.371787\pi$
$601$	471.172	0.783980	0.391990	+	0.919970i	$0.371787\pi$
$602$	0	0
$603$	0	0
$604$	1131.89	1.87400
$605$	− 2230.50i	− 3.68678i
$606$	0	0
$607$	522.968	0.861562	0.430781	−	0.902456i	$-0.358238\pi$
$607$	522.968	0.861562	0.430781	+	0.902456i	$0.358238\pi$
$608$	− 187.836i	− 0.308941i
$609$	0	0
$610$	−957.926	−1.57037
$611$	− 31.2167i	− 0.0510911i
$612$	0	0
$613$	295.831	0.482596	0.241298	−	0.970451i	$-0.422427\pi$
$613$	295.831	0.482596	0.241298	+	0.970451i	$0.422427\pi$
$614$	1525.12i	2.48392i
$615$	0	0
$616$	0	0
$617$	175.568i	0.284551i	0.989827	+	0.142276i	$0.0454419\pi$
$617$	175.568i	0.284551i	−0.989827	+	0.142276i	$0.954558\pi$
$618$	0	0
$619$	723.936	1.16952	0.584762	−	0.811205i	$-0.301188\pi$
$619$	723.936	1.16952	0.584762	+	0.811205i	$0.301188\pi$
$620$	− 1724.81i	− 2.78195i
$621$	0	0
$622$	−272.067	−0.437406
$623$	0	0
$624$	0	0
$625$	221.501	0.354402
$626$	192.380i	0.307316i
$627$	0	0
$628$	616.407	0.981540
$629$	204.032i	0.324375i
$630$	0	0
$631$	911.161	1.44400	0.721998	−	0.691895i	$-0.243224\pi$
$631$	911.161	1.44400	0.721998	+	0.691895i	$0.243224\pi$
$632$	− 51.1728i	− 0.0809696i
$633$	0	0
$634$	804.180	1.26842
$635$	− 184.897i	− 0.291177i
$636$	0	0
$637$	0	0
$638$	− 49.6581i	− 0.0778341i
$639$	0	0
$640$	−1011.58	−1.58060
$641$	508.807i	0.793771i	0.917868	+	0.396886i	$0.129909\pi$
$641$	508.807i	0.793771i	−0.917868	+	0.396886i	$0.870091\pi$
$642$	0	0
$643$	576.686	0.896868	0.448434	−	0.893816i	$-0.351982\pi$
$643$	576.686	0.896868	0.448434	+	0.893816i	$0.351982\pi$
$644$	0	0
$645$	0	0
$646$	74.8287	0.115834
$647$	1157.79i	1.78947i	0.446594	+	0.894737i	$0.352637\pi$
$647$	1157.79i	1.78947i	−0.446594	+	0.894737i	$0.647363\pi$
$648$	0	0
$649$	1348.07	2.07715
$650$	− 2338.60i	− 3.59784i
$651$	0	0
$652$	−737.061	−1.13046
$653$	923.265i	1.41388i	0.707272	+	0.706941i	$0.249925\pi$
$653$	923.265i	1.41388i	−0.707272	+	0.706941i	$0.750075\pi$
$654$	0	0
$655$	−1632.25	−2.49199
$656$	429.150i	0.654193i
$657$	0	0
$658$	0	0
$659$	− 121.880i	− 0.184947i	−0.995715	−	0.0924734i	$-0.970523\pi$
$659$	− 121.880i	− 0.184947i	0.995715	−	0.0924734i	$-0.0294773\pi$
$660$	0	0
$661$	437.007	0.661131	0.330565	−	0.943783i	$-0.392761\pi$
$661$	437.007	0.661131	0.330565	+	0.943783i	$0.392761\pi$
$662$	− 1095.77i	− 1.65525i
$663$	0	0
$664$	348.688	0.525133
$665$	0	0
$666$	0	0
$667$	12.4595	0.0186799
$668$	1154.76i	1.72868i
$669$	0	0
$670$	−841.924	−1.25660
$671$	744.459i	1.10948i
$672$	0	0
$673$	108.269	0.160875	0.0804376	−	0.996760i	$-0.474368\pi$
$673$	108.269	0.160875	0.0804376	+	0.996760i	$0.474368\pi$
$674$	− 814.461i	− 1.20840i
$675$	0	0
$676$	−701.444	−1.03764
$677$	− 262.950i	− 0.388405i	−0.980962	−	0.194202i	$-0.937788\pi$
$677$	− 262.950i	− 0.388405i	0.980962	−	0.194202i	$-0.0622118\pi$
$678$	0	0
$679$	0	0
$680$	− 189.952i	− 0.279342i
$681$	0	0
$682$	−2348.95	−3.44420
$683$	347.207i	0.508355i	0.967158	+	0.254178i	$0.0818048\pi$
$683$	347.207i	0.508355i	−0.967158	+	0.254178i	$0.918195\pi$
$684$	0	0
$685$	−1205.35	−1.75964
$686$	0	0
$687$	0	0
$688$	102.301	0.148693
$689$	613.325i	0.890167i
$690$	0	0
$691$	−1146.92	−1.65979	−0.829896	−	0.557918i	$-0.811600\pi$
$691$	−1146.92	−1.65979	−0.829896	+	0.557918i	$0.811600\pi$
$692$	1194.54i	1.72622i
$693$	0	0
$694$	1953.20	2.81441
$695$	− 544.566i	− 0.783548i
$696$	0	0
$697$	−271.003	−0.388813
$698$	1109.58i	1.58965i
$699$	0	0
$700$	0	0
$701$	205.643i	0.293357i	0.989184	+	0.146679i	$0.0468583\pi$
$701$	205.643i	0.293357i	−0.989184	+	0.146679i	$0.953142\pi$
$702$	0	0
$703$	−154.855	−0.220277
$704$	1911.96i	2.71585i
$705$	0	0
$706$	607.161	0.860001
$707$	0	0
$708$	0	0
$709$	−732.259	−1.03281	−0.516403	−	0.856346i	$-0.672729\pi$
$709$	−732.259	−1.03281	−0.516403	+	0.856346i	$0.672729\pi$
$710$	20.9340i	0.0294845i
$711$	0	0
$712$	409.977	0.575810
$713$	− 589.362i	− 0.826594i
$714$	0	0
$715$	−2846.22	−3.98072
$716$	− 940.728i	− 1.31387i
$717$	0	0
$718$	−880.079	−1.22574
$719$	− 768.728i	− 1.06916i	−0.845117	−	0.534582i	$-0.820469\pi$
$719$	− 768.728i	− 1.06916i	0.845117	−	0.534582i	$-0.179531\pi$
$720$	0	0
$721$	0	0
$722$	− 1045.09i	− 1.44750i
$723$	0	0
$724$	−282.958	−0.390827
$725$	− 36.4222i	− 0.0502375i
$726$	0	0
$727$	961.681	1.32281	0.661403	−	0.750030i	$-0.269961\pi$
$727$	961.681	1.32281	0.661403	+	0.750030i	$0.269961\pi$
$728$	0	0
$729$	0	0
$730$	−2405.85	−3.29569
$731$	64.6015i	0.0883742i
$732$	0	0
$733$	406.498	0.554568	0.277284	−	0.960788i	$-0.410566\pi$
$733$	406.498	0.554568	0.277284	+	0.960788i	$0.410566\pi$
$734$	− 1776.99i	− 2.42097i
$735$	0	0
$736$	−657.916	−0.893907
$737$	654.307i	0.887798i
$738$	0	0
$739$	−1024.76	−1.38668	−0.693339	−	0.720611i	$-0.743861\pi$
$739$	−1024.76	−1.38668	−0.693339	+	0.720611i	$0.743861\pi$
$740$	1587.36i	2.14508i
$741$	0	0
$742$	0	0
$743$	− 952.258i	− 1.28164i	−0.767691	−	0.640820i	$-0.778595\pi$
$743$	− 952.258i	− 1.28164i	0.767691	−	0.640820i	$-0.221405\pi$
$744$	0	0
$745$	−829.976	−1.11406
$746$	1298.03i	1.73999i
$747$	0	0
$748$	−596.112	−0.796941
$749$	0	0
$750$	0	0
$751$	−1061.46	−1.41339	−0.706696	−	0.707518i	$-0.749815\pi$
$751$	−1061.46	−1.41339	−0.706696	+	0.707518i	$0.749815\pi$
$752$	16.1955i	0.0215365i
$753$	0	0
$754$	−43.6657	−0.0579121
$755$	− 1770.59i	− 2.34515i
$756$	0	0
$757$	−308.132	−0.407044	−0.203522	−	0.979070i	$-0.565239\pi$
$757$	−308.132	−0.407044	−0.203522	+	0.979070i	$0.565239\pi$
$758$	− 2244.86i	− 2.96156i
$759$	0	0
$760$	144.169	0.189696
$761$	− 675.272i	− 0.887349i	−0.896188	−	0.443674i	$-0.853675\pi$
$761$	− 675.272i	− 0.887349i	0.896188	−	0.443674i	$-0.146325\pi$
$762$	0	0
$763$	0	0
$764$	− 552.249i	− 0.722839i
$765$	0	0
$766$	1512.11	1.97404
$767$	− 1185.39i	− 1.54549i
$768$	0	0
$769$	193.740	0.251938	0.125969	−	0.992034i	$-0.459796\pi$
$769$	193.740	0.251938	0.125969	+	0.992034i	$0.459796\pi$
$770$	0	0
$771$	0	0
$772$	−1024.33	−1.32685
$773$	− 388.554i	− 0.502657i	−0.967902	−	0.251329i	$-0.919133\pi$
$773$	− 388.554i	− 0.502657i	0.967902	−	0.251329i	$-0.0808675\pi$
$774$	0	0
$775$	−1722.86	−2.22304
$776$	− 166.617i	− 0.214713i
$777$	0	0
$778$	−93.7469	−0.120497
$779$	− 205.684i	− 0.264035i
$780$	0	0
$781$	16.2690	0.0208310
$782$	− 262.095i	− 0.335160i
$783$	0	0
$784$	0	0
$785$	− 964.225i	− 1.22831i
$786$	0	0
$787$	−606.694	−0.770895	−0.385447	−	0.922730i	$-0.625953\pi$
$787$	−606.694	−0.770895	−0.385447	+	0.922730i	$0.625953\pi$
$788$	936.086i	1.18793i
$789$	0	0
$790$	−323.239	−0.409164
$791$	0	0
$792$	0	0
$793$	654.623	0.825502
$794$	− 338.348i	− 0.426131i
$795$	0	0
$796$	888.970	1.11680
$797$	637.483i	0.799853i	0.916547	+	0.399927i	$0.130964\pi$
$797$	637.483i	0.799853i	−0.916547	+	0.399927i	$0.869036\pi$
$798$	0	0
$799$	−10.2272	−0.0128000
$800$	1923.26i	2.40407i
$801$	0	0
$802$	−1718.94	−2.14332
$803$	1869.73i	2.32843i
$804$	0	0
$805$	0	0
$806$	2065.49i	2.56264i
$807$	0	0
$808$	47.3555	0.0586083
$809$	− 628.893i	− 0.777370i	−0.921371	−	0.388685i	$-0.872929\pi$
$809$	− 628.893i	− 0.777370i	0.921371	−	0.388685i	$-0.127071\pi$
$810$	0	0
$811$	−410.689	−0.506398	−0.253199	−	0.967414i	$-0.581483\pi$
$811$	−410.689	−0.506398	−0.253199	+	0.967414i	$0.581483\pi$
$812$	0	0
$813$	0	0
$814$	2161.75	2.65572
$815$	1152.96i	1.41468i
$816$	0	0
$817$	−49.0308	−0.0600132
$818$	− 1540.94i	− 1.88380i
$819$	0	0
$820$	−2108.39	−2.57120
$821$	751.179i	0.914956i	0.889221	+	0.457478i	$0.151247\pi$
$821$	751.179i	0.914956i	−0.889221	+	0.457478i	$0.848753\pi$
$822$	0	0
$823$	121.799	0.147995	0.0739973	−	0.997258i	$-0.476424\pi$
$823$	121.799	0.147995	0.0739973	+	0.997258i	$0.476424\pi$
$824$	773.599i	0.938834i
$825$	0	0
$826$	0	0
$827$	460.859i	0.557266i	0.960398	+	0.278633i	$0.0898813\pi$
$827$	460.859i	0.557266i	−0.960398	+	0.278633i	$0.910119\pi$
$828$	0	0
$829$	211.427	0.255038	0.127519	−	0.991836i	$-0.459299\pi$
$829$	211.427	0.255038	0.127519	+	0.991836i	$0.459299\pi$
$830$	− 2202.53i	− 2.65365i
$831$	0	0
$832$	1681.24	2.02072
$833$	0	0
$834$	0	0
$835$	1806.35	2.16330
$836$	− 452.433i	− 0.541187i
$837$	0	0
$838$	1840.84	2.19671
$839$	372.692i	0.444209i	0.975023	+	0.222105i	$0.0712927\pi$
$839$	372.692i	0.444209i	−0.975023	+	0.222105i	$0.928707\pi$
$840$	0	0
$841$	840.320	0.999191
$842$	− 2027.53i	− 2.40799i
$843$	0	0
$844$	1126.06	1.33419
$845$	1097.25i	1.29852i
$846$	0	0
$847$	0	0
$848$	− 318.198i	− 0.375234i
$849$	0	0
$850$	−766.172	−0.901379
$851$	542.394i	0.637361i
$852$	0	0
$853$	929.064	1.08917	0.544586	−	0.838705i	$-0.316687\pi$
$853$	929.064	1.08917	0.544586	+	0.838705i	$0.316687\pi$
$854$	0	0
$855$	0	0
$856$	−347.346	−0.405778
$857$	− 1346.48i	− 1.57116i	−0.618763	−	0.785578i	$-0.712366\pi$
$857$	− 1346.48i	− 1.57116i	0.618763	−	0.785578i	$-0.287634\pi$
$858$	0	0
$859$	980.733	1.14171	0.570857	−	0.821049i	$-0.306611\pi$
$859$	980.733	1.14171	0.570857	+	0.821049i	$0.306611\pi$
$860$	502.596i	0.584415i
$861$	0	0
$862$	−768.987	−0.892096
$863$	− 1002.64i	− 1.16181i	−0.813972	−	0.580904i	$-0.802699\pi$
$863$	− 1002.64i	− 1.16181i	0.813972	−	0.580904i	$-0.197301\pi$
$864$	0	0
$865$	1868.59	2.16022
$866$	404.180i	0.466720i
$867$	0	0
$868$	0	0
$869$	251.208i	0.289077i
$870$	0	0
$871$	575.350	0.660562
$872$	− 752.578i	− 0.863048i
$873$	0	0
$874$	198.923	0.227601
$875$	0	0
$876$	0	0
$877$	−733.963	−0.836902	−0.418451	−	0.908239i	$-0.637427\pi$
$877$	−733.963	−0.836902	−0.418451	+	0.908239i	$0.637427\pi$
$878$	− 1907.52i	− 2.17257i
$879$	0	0
$880$	1476.64	1.67800
$881$	− 1171.68i	− 1.32994i	−0.746869	−	0.664971i	$-0.768444\pi$
$881$	− 1171.68i	− 1.32994i	0.746869	−	0.664971i	$-0.231556\pi$
$882$	0	0
$883$	−168.042	−0.190308	−0.0951539	−	0.995463i	$-0.530334\pi$
$883$	−168.042	−0.190308	−0.0951539	+	0.995463i	$0.530334\pi$
$884$	524.177i	0.592960i
$885$	0	0
$886$	−250.148	−0.282334
$887$	− 411.003i	− 0.463362i	−0.972792	−	0.231681i	$-0.925577\pi$
$887$	− 411.003i	− 0.463362i	0.972792	−	0.231681i	$-0.0744226\pi$
$888$	0	0
$889$	0	0
$890$	− 2589.67i	− 2.90974i
$891$	0	0
$892$	895.239	1.00363
$893$	− 7.76218i	− 0.00869226i
$894$	0	0
$895$	−1471.55	−1.64419
$896$	0	0
$897$	0	0
$898$	−1024.48	−1.14084
$899$	32.1688i	0.0357828i
$900$	0	0
$901$	200.938	0.223016
$902$	2871.32i	3.18328i
$903$	0	0
$904$	408.258	0.451613
$905$	442.623i	0.489086i
$906$	0	0
$907$	1147.08	1.26470	0.632350	−	0.774683i	$-0.282091\pi$
$907$	1147.08	1.26470	0.632350	+	0.774683i	$0.282091\pi$
$908$	− 1275.63i	− 1.40487i
$909$	0	0
$910$	0	0
$911$	1384.30i	1.51954i	0.650193	+	0.759769i	$0.274688\pi$
$911$	1384.30i	1.51954i	−0.650193	+	0.759769i	$0.725312\pi$
$912$	0	0
$913$	−1711.71	−1.87482
$914$	1490.14i	1.63035i
$915$	0	0
$916$	−2356.89	−2.57302
$917$	0	0
$918$	0	0
$919$	139.430	0.151720	0.0758598	−	0.997118i	$-0.475830\pi$
$919$	139.430	0.151720	0.0758598	+	0.997118i	$0.475830\pi$
$920$	− 504.966i	− 0.548876i
$921$	0	0
$922$	−1553.17	−1.68457
$923$	− 14.3058i	− 0.0154992i
$924$	0	0
$925$	1585.56	1.71412
$926$	2076.69i	2.24264i
$927$	0	0
$928$	35.9106	0.0386968
$929$	− 282.744i	− 0.304353i	−0.988353	−	0.152177i	$-0.951372\pi$
$929$	− 282.744i	− 0.304353i	0.988353	−	0.152177i	$-0.0486283\pi$
$930$	0	0
$931$	0	0
$932$	− 507.651i	− 0.544690i
$933$	0	0
$934$	1669.11	1.78705
$935$	932.478i	0.997303i
$936$	0	0
$937$	1076.69	1.14908	0.574539	−	0.818477i	$-0.305181\pi$
$937$	1076.69	1.14908	0.574539	+	0.818477i	$0.305181\pi$
$938$	0	0
$939$	0	0
$940$	−79.5673	−0.0846460
$941$	316.622i	0.336474i	0.985747	+	0.168237i	$0.0538075\pi$
$941$	316.622i	0.336474i	−0.985747	+	0.168237i	$0.946193\pi$
$942$	0	0
$943$	−720.428	−0.763974
$944$	614.992i	0.651475i
$945$	0	0
$946$	684.464	0.723535
$947$	− 1353.13i	− 1.42886i	−0.699705	−	0.714432i	$-0.746685\pi$
$947$	− 1353.13i	− 1.42886i	0.699705	−	0.714432i	$-0.253315\pi$
$948$	0	0
$949$	1644.10	1.73246
$950$	− 581.504i	− 0.612109i
$951$	0	0
$952$	0	0
$953$	506.717i	0.531708i	0.964013	+	0.265854i	$0.0856538\pi$
$953$	506.717i	0.531708i	−0.964013	+	0.265854i	$0.914346\pi$
$954$	0	0
$955$	−863.865	−0.904570
$956$	1097.61i	1.14813i
$957$	0	0
$958$	−2085.45	−2.17688
$959$	0	0
$960$	0	0
$961$	560.657	0.583410
$962$	− 1900.89i	− 1.97597i
$963$	0	0
$964$	−820.666	−0.851313
$965$	1602.32i	1.66044i
$966$	0	0
$967$	966.423	0.999403	0.499701	−	0.866198i	$-0.333443\pi$
$967$	966.423	0.999403	0.499701	+	0.866198i	$0.333443\pi$
$968$	1077.82i	1.11345i
$969$	0	0
$970$	−1052.46	−1.08501
$971$	− 549.425i	− 0.565834i	−0.959144	−	0.282917i	$-0.908698\pi$
$971$	− 549.425i	− 0.565834i	0.959144	−	0.282917i	$-0.0913022\pi$
$972$	0	0
$973$	0	0
$974$	− 637.233i	− 0.654243i
$975$	0	0
$976$	−339.624	−0.347975
$977$	− 1571.58i	− 1.60858i	−0.594239	−	0.804289i	$-0.702547\pi$
$977$	− 1571.58i	− 1.60858i	0.594239	−	0.804289i	$-0.297453\pi$
$978$	0	0
$979$	−2012.58	−2.05575
$980$	0	0
$981$	0	0
$982$	1169.58	1.19102
$983$	810.638i	0.824657i	0.911035	+	0.412329i	$0.135284\pi$
$983$	810.638i	0.824657i	−0.911035	+	0.412329i	$0.864716\pi$
$984$	0	0
$985$	1464.29	1.48659
$986$	14.3058i	0.0145089i
$987$	0	0
$988$	−397.836	−0.402668
$989$	171.735i	0.173645i
$990$	0	0
$991$	1071.68	1.08142	0.540708	−	0.841210i	$-0.318156\pi$
$991$	1071.68	1.08142	0.540708	+	0.841210i	$0.318156\pi$
$992$	− 1698.65i	− 1.71235i
$993$	0	0
$994$	0	0
$995$	− 1390.59i	− 1.39757i
$996$	0	0
$997$	1065.30	1.06850	0.534252	−	0.845325i	$-0.320593\pi$
$997$	1065.30	1.06850	0.534252	+	0.845325i	$0.320593\pi$
$998$	− 1738.10i	− 1.74158i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.3.b.e.197.8	yes	8
3.2	odd	2	inner	441.3.b.e.197.1	✓	8
7.2	even	3		441.3.q.e.116.8		16
7.3	odd	6		441.3.q.e.422.2		16
7.4	even	3		441.3.q.e.422.1		16
7.5	odd	6		441.3.q.e.116.7		16
7.6	odd	2	inner	441.3.b.e.197.7	yes	8
21.2	odd	6		441.3.q.e.116.1		16
21.5	even	6		441.3.q.e.116.2		16
21.11	odd	6		441.3.q.e.422.8		16
21.17	even	6		441.3.q.e.422.7		16
21.20	even	2	inner	441.3.b.e.197.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.3.b.e.197.1	✓	8	3.2	odd	2	inner
441.3.b.e.197.2	yes	8	21.20	even	2	inner
441.3.b.e.197.7	yes	8	7.6	odd	2	inner
441.3.b.e.197.8	yes	8	1.1	even	1	trivial
441.3.q.e.116.1		16	21.2	odd	6
441.3.q.e.116.2		16	21.5	even	6
441.3.q.e.116.7		16	7.5	odd	6
441.3.q.e.116.8		16	7.2	even	3
441.3.q.e.422.1		16	7.4	even	3
441.3.q.e.422.2		16	7.3	odd	6
441.3.q.e.422.7		16	21.17	even	6
441.3.q.e.422.8		16	21.11	odd	6

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:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	441.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.05231i q^{2} -5.31662 q^{4} +8.31662i q^{5} -4.01875i q^{8} +O(q^{10})\)	\(q+3.05231i q^{2} -5.31662 q^{4} +8.31662i q^{5} -4.01875i q^{8} -25.3850 q^{10} +19.7281i q^{11} +17.3474 q^{13} -9.00000 q^{16} -5.68338i q^{17} +4.31353 q^{19} -44.2164i q^{20} -60.2164 q^{22} -15.1086i q^{23} -44.1662 q^{25} +52.9499i q^{26} +0.824662i q^{29} +39.0084 q^{31} -43.5458i q^{32} +17.3474 q^{34} -35.8997 q^{37} +13.1662i q^{38} +33.4225 q^{40} -47.6834i q^{41} -11.3668 q^{43} -104.887i q^{44} +46.1161 q^{46} -1.79950i q^{47} -134.809i q^{50} -92.2299 q^{52} +35.3553i q^{53} -164.071 q^{55} -2.51713 q^{58} -68.3325i q^{59} +37.7360 q^{61} +119.066i q^{62} +96.9156 q^{64} +144.272i q^{65} +33.1662 q^{67} +30.2164i q^{68} -0.824662i q^{71} +94.7748 q^{73} -109.577i q^{74} -22.9334 q^{76} +12.7335 q^{79} -74.8496i q^{80} +145.545 q^{82} +86.7652i q^{83} +47.2665 q^{85} -34.6949i q^{86} +79.2824 q^{88} +102.016i q^{89} +80.3266i q^{92} +5.49263 q^{94} +35.8740i q^{95} +41.4600 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{4}+O(q^{10}) \) 8 * q - 16 * q^4	\( 8 q - 16 q^{4} - 72 q^{16} - 296 q^{22} - 88 q^{25} - 128 q^{37} - 144 q^{43} + 24 q^{46} - 312 q^{58} + 112 q^{64} + 208 q^{79} + 272 q^{85} + 24 q^{88}+O(q^{100}) \) 8 * q - 16 * q^4 - 72 * q^16 - 296 * q^22 - 88 * q^25 - 128 * q^37 - 144 * q^43 + 24 * q^46 - 312 * q^58 + 112 * q^64 + 208 * q^79 + 272 * q^85 + 24 * q^88

Introduction

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