LMFDB - Embedded newform 3087.2.c.c.3086.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3087,2,Mod(3086,3087)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3087, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3087.3086");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3087 = 3^{2} \cdot 7^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3087.c (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:	$24.6498191040$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3086.16
Character	$\chi$	$=$	3087.3086
Dual form			3087.2.c.c.3086.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.223929i q^{2} +1.94986 q^{4} +3.02565 q^{5} +0.884487i q^{8} +O(q^{10})$	$q+0.223929i q^{2} +1.94986 q^{4} +3.02565 q^{5} +0.884487i q^{8} +0.677530i q^{10} +2.32440i q^{11} +6.38925i q^{13} +3.70165 q^{16} +1.14736 q^{17} -3.16976i q^{19} +5.89958 q^{20} -0.520500 q^{22} +4.24909i q^{23} +4.15455 q^{25} -1.43074 q^{26} -2.88568i q^{29} -5.49978i q^{31} +2.59788i q^{32} +0.256928i q^{34} +5.09447 q^{37} +0.709802 q^{38} +2.67615i q^{40} +3.05199 q^{41} -2.57547 q^{43} +4.53224i q^{44} -0.951493 q^{46} -11.9772 q^{47} +0.930324i q^{50} +12.4581i q^{52} +10.8964i q^{53} +7.03280i q^{55} +0.646188 q^{58} -10.5715 q^{59} +7.99201i q^{61} +1.23156 q^{62} +6.82156 q^{64} +19.3316i q^{65} +1.22716 q^{67} +2.23719 q^{68} -9.56431i q^{71} -10.2895i q^{73} +1.14080i q^{74} -6.18058i q^{76} +9.12059 q^{79} +11.1999 q^{80} +0.683428i q^{82} -12.9254 q^{83} +3.47152 q^{85} -0.576723i q^{86} -2.05590 q^{88} +13.9772 q^{89} +8.28511i q^{92} -2.68204i q^{94} -9.59059i q^{95} -11.0998i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q - 40 q^{16} + 64 q^{22} + 136 q^{25} + 32 q^{37} + 32 q^{43} + 32 q^{46} - 128 q^{58} + 32 q^{64} + 40 q^{67} + 8 q^{79} + 64 q^{85} + 16 q^{88}+O(q^{100}) $ 24 * q - 40 * q^16 + 64 * q^22 + 136 * q^25 + 32 * q^37 + 32 * q^43 + 32 * q^46 - 128 * q^58 + 32 * q^64 + 40 * q^67 + 8 * q^79 + 64 * q^85 + 16 * q^88

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3087\mathbb{Z}\right)^\times$.

$n$	$344$	$2404$
$\chi(n)$	$-1$	$-1$

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 $

$2$	0.223929i	0.158342i	0.996861	+	0.0791708i	$0.0252273\pi$
$2$	0.223929i	0.158342i	−0.996861	+	0.0791708i	$0.974773\pi$
$3$	0	0
$3$	0	0
$4$	1.94986	0.974928
$5$	3.02565	1.35311	0.676556	−	0.736392i	$-0.263472\pi$
$5$	3.02565	1.35311	0.676556	+	0.736392i	$0.263472\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0.884487i	0.312713i
$9$	0	0
$10$	0.677530i	0.214254i
$11$	2.32440i	0.700832i	0.936594	+	0.350416i	$0.113960\pi$
$11$	2.32440i	0.700832i	−0.936594	+	0.350416i	$0.886040\pi$
$12$	0	0
$13$	6.38925i	1.77206i	0.463629	+	0.886029i	$0.346547\pi$
$13$	6.38925i	1.77206i	−0.463629	+	0.886029i	$0.653453\pi$
$14$	0	0
$15$	0	0
$16$	3.70165	0.925412
$17$	1.14736	0.278276	0.139138	−	0.990273i	$-0.455567\pi$
$17$	1.14736	0.278276	0.139138	+	0.990273i	$0.455567\pi$
$18$	0	0
$19$	− 3.16976i	− 0.727194i	−0.931556	−	0.363597i	$-0.881549\pi$
$19$	− 3.16976i	− 0.727194i	0.931556	−	0.363597i	$-0.118451\pi$
$20$	5.89958	1.31919
$21$	0	0
$22$	−0.520500	−0.110971
$23$	4.24909i	0.885996i	0.896523	+	0.442998i	$0.146085\pi$
$23$	4.24909i	0.885996i	−0.896523	+	0.442998i	$0.853915\pi$
$24$	0	0
$25$	4.15455	0.830910
$26$	−1.43074	−0.280591
$27$	0	0
$28$	0	0
$29$	− 2.88568i	− 0.535858i	−0.963439	−	0.267929i	$-0.913661\pi$
$29$	− 2.88568i	− 0.535858i	0.963439	−	0.267929i	$-0.0863392\pi$
$30$	0	0
$31$	− 5.49978i	− 0.987790i	−0.869522	−	0.493895i	$-0.835573\pi$
$31$	− 5.49978i	− 0.987790i	0.869522	−	0.493895i	$-0.164427\pi$
$32$	2.59788i	0.459245i
$33$	0	0
$34$	0.256928i	0.0440627i
$35$	0	0
$36$	0	0
$37$	5.09447	0.837526	0.418763	−	0.908096i	$-0.362464\pi$
$37$	5.09447	0.837526	0.418763	+	0.908096i	$0.362464\pi$
$38$	0.709802	0.115145
$39$	0	0
$40$	2.67615i	0.423136i
$41$	3.05199	0.476640	0.238320	−	0.971187i	$-0.423403\pi$
$41$	3.05199	0.476640	0.238320	+	0.971187i	$0.423403\pi$
$42$	0	0
$43$	−2.57547	−0.392756	−0.196378	−	0.980528i	$-0.562918\pi$
$43$	−2.57547	−0.392756	−0.196378	+	0.980528i	$0.562918\pi$
$44$	4.53224i	0.683260i
$45$	0	0
$46$	−0.951493	−0.140290
$47$	−11.9772	−1.74706	−0.873528	−	0.486774i	$-0.838174\pi$
$47$	−11.9772	−1.74706	−0.873528	+	0.486774i	$0.838174\pi$
$48$	0	0
$49$	0	0
$50$	0.930324i	0.131568i
$51$	0	0
$52$	12.4581i	1.72763i
$53$	10.8964i	1.49673i	0.663285	+	0.748367i	$0.269162\pi$
$53$	10.8964i	1.49673i	−0.663285	+	0.748367i	$0.730838\pi$
$54$	0	0
$55$	7.03280i	0.948303i
$56$	0	0
$57$	0	0
$58$	0.646188	0.0848486
$59$	−10.5715	−1.37629	−0.688146	−	0.725573i	$-0.741575\pi$
$59$	−10.5715	−1.37629	−0.688146	+	0.725573i	$0.741575\pi$
$60$	0	0
$61$	7.99201i	1.02327i	0.859202	+	0.511636i	$0.170960\pi$
$61$	7.99201i	1.02327i	−0.859202	+	0.511636i	$0.829040\pi$
$62$	1.23156	0.156408
$63$	0	0
$64$	6.82156	0.852695
$65$	19.3316i	2.39779i
$66$	0	0
$67$	1.22716	0.149922	0.0749610	−	0.997186i	$-0.476117\pi$
$67$	1.22716	0.149922	0.0749610	+	0.997186i	$0.476117\pi$
$68$	2.23719	0.271299
$69$	0	0
$70$	0	0
$71$	− 9.56431i	− 1.13508i	−0.823347	−	0.567538i	$-0.807896\pi$
$71$	− 9.56431i	− 1.13508i	0.823347	−	0.567538i	$-0.192104\pi$
$72$	0	0
$73$	− 10.2895i	− 1.20430i	−0.798383	−	0.602150i	$-0.794311\pi$
$73$	− 10.2895i	− 1.20430i	0.798383	−	0.602150i	$-0.205689\pi$
$74$	1.14080i	0.132615i
$75$	0	0
$76$	− 6.18058i	− 0.708962i
$77$	0	0
$78$	0	0
$79$	9.12059	1.02615	0.513073	−	0.858345i	$-0.328507\pi$
$79$	9.12059	1.02615	0.513073	+	0.858345i	$0.328507\pi$
$80$	11.1999	1.25219
$81$	0	0
$82$	0.683428i	0.0754720i
$83$	−12.9254	−1.41875	−0.709373	−	0.704833i	$-0.751022\pi$
$83$	−12.9254	−1.41875	−0.709373	+	0.704833i	$0.751022\pi$
$84$	0	0
$85$	3.47152	0.376539
$86$	− 0.576723i	− 0.0621897i
$87$	0	0
$88$	−2.05590	−0.219159
$89$	13.9772	1.48158	0.740788	−	0.671738i	$-0.234452\pi$
$89$	13.9772	1.48158	0.740788	+	0.671738i	$0.234452\pi$
$90$	0	0
$91$	0	0
$92$	8.28511i	0.863782i
$93$	0	0
$94$	− 2.68204i	− 0.276632i
$95$	− 9.59059i	− 0.983974i
$96$	0	0
$97$	− 11.0998i	− 1.12701i	−0.826112	−	0.563506i	$-0.809452\pi$
$97$	− 11.0998i	− 1.12701i	0.826112	−	0.563506i	$-0.190548\pi$
$98$	0	0
$99$	0	0
$100$	8.10077	0.810077
$101$	3.01393	0.299897	0.149948	−	0.988694i	$-0.452089\pi$
$101$	3.01393	0.299897	0.149948	+	0.988694i	$0.452089\pi$
$102$	0	0
$103$	− 5.05076i	− 0.497667i	−0.968546	−	0.248833i	$-0.919953\pi$
$103$	− 5.05076i	− 0.497667i	0.968546	−	0.248833i	$-0.0800471\pi$
$104$	−5.65121	−0.554147
$105$	0	0
$106$	−2.44002	−0.236995
$107$	16.3005i	1.57583i	0.615784	+	0.787915i	$0.288840\pi$
$107$	16.3005i	1.57583i	−0.615784	+	0.787915i	$0.711160\pi$
$108$	0	0
$109$	−6.91055	−0.661911	−0.330955	−	0.943646i	$-0.607371\pi$
$109$	−6.91055	−0.661911	−0.330955	+	0.943646i	$0.607371\pi$
$110$	−1.57485	−0.150156
$111$	0	0
$112$	0	0
$113$	9.54328i	0.897756i	0.893593	+	0.448878i	$0.148176\pi$
$113$	9.54328i	0.897756i	−0.893593	+	0.448878i	$0.851824\pi$
$114$	0	0
$115$	12.8562i	1.19885i
$116$	− 5.62666i	− 0.522423i
$117$	0	0
$118$	− 2.36726i	− 0.217924i
$119$	0	0
$120$	0	0
$121$	5.59718	0.508835
$122$	−1.78964	−0.162027
$123$	0	0
$124$	− 10.7238i	− 0.963024i
$125$	−2.55804	−0.228798
$126$	0	0
$127$	0.0393502	0.00349176	0.00174588	−	0.999998i	$-0.499444\pi$
$127$	0.0393502	0.00349176	0.00174588	+	0.999998i	$0.499444\pi$
$128$	6.72331i	0.594262i
$129$	0	0
$130$	−4.32891	−0.379670
$131$	7.66082	0.669329	0.334665	−	0.942337i	$-0.391377\pi$
$131$	7.66082	0.669329	0.334665	+	0.942337i	$0.391377\pi$
$132$	0	0
$133$	0	0
$134$	0.274798i	0.0237389i
$135$	0	0
$136$	1.01483i	0.0870207i
$137$	− 18.8659i	− 1.61182i	−0.592038	−	0.805910i	$-0.701677\pi$
$137$	− 18.8659i	− 1.61182i	0.592038	−	0.805910i	$-0.298323\pi$
$138$	0	0
$139$	− 19.9952i	− 1.69597i	−0.530019	−	0.847986i	$-0.677815\pi$
$139$	− 19.9952i	− 1.69597i	0.530019	−	0.847986i	$-0.322185\pi$
$140$	0	0
$141$	0	0
$142$	2.14173	0.179730
$143$	−14.8511	−1.24191
$144$	0	0
$145$	− 8.73106i	− 0.725075i
$146$	2.30413	0.190691
$147$	0	0
$148$	9.93349	0.816528
$149$	22.1466i	1.81432i	0.420783	+	0.907161i	$0.361755\pi$
$149$	22.1466i	1.81432i	−0.420783	+	0.907161i	$0.638245\pi$
$150$	0	0
$151$	8.53160	0.694291	0.347146	−	0.937811i	$-0.387151\pi$
$151$	8.53160	0.694291	0.347146	+	0.937811i	$0.387151\pi$
$152$	2.80362	0.227403
$153$	0	0
$154$	0	0
$155$	− 16.6404i	− 1.33659i
$156$	0	0
$157$	− 17.5709i	− 1.40231i	−0.713010	−	0.701154i	$-0.752668\pi$
$157$	− 17.5709i	− 1.40231i	0.713010	−	0.701154i	$-0.247332\pi$
$158$	2.04236i	0.162482i
$159$	0	0
$160$	7.86027i	0.621409i
$161$	0	0
$162$	0	0
$163$	22.0627	1.72809	0.864043	−	0.503418i	$-0.167924\pi$
$163$	22.0627	1.72809	0.864043	+	0.503418i	$0.167924\pi$
$164$	5.95093	0.464690
$165$	0	0
$166$	− 2.89437i	− 0.224647i
$167$	2.91067	0.225234	0.112617	−	0.993638i	$-0.464077\pi$
$167$	2.91067	0.225234	0.112617	+	0.993638i	$0.464077\pi$
$168$	0	0
$169$	−27.8225	−2.14019
$170$	0.777373i	0.0596218i
$171$	0	0
$172$	−5.02180	−0.382909
$173$	−16.6720	−1.26755	−0.633774	−	0.773519i	$-0.718495\pi$
$173$	−16.6720	−1.26755	−0.633774	+	0.773519i	$0.718495\pi$
$174$	0	0
$175$	0	0
$176$	8.60410i	0.648558i
$177$	0	0
$178$	3.12989i	0.234595i
$179$	− 16.0778i	− 1.20171i	−0.799357	−	0.600856i	$-0.794827\pi$
$179$	− 16.0778i	− 1.20171i	0.799357	−	0.600856i	$-0.205173\pi$
$180$	0	0
$181$	− 19.0644i	− 1.41705i	−0.705687	−	0.708524i	$-0.749362\pi$
$181$	− 19.0644i	− 1.41705i	0.705687	−	0.708524i	$-0.250638\pi$
$182$	0	0
$183$	0	0
$184$	−3.75826	−0.277063
$185$	15.4141	1.13327
$186$	0	0
$187$	2.66692i	0.195025i
$188$	−23.3538	−1.70325
$189$	0	0
$190$	2.14761	0.155804
$191$	− 5.45997i	− 0.395070i	−0.980296	−	0.197535i	$-0.936706\pi$
$191$	− 5.45997i	− 0.395070i	0.980296	−	0.197535i	$-0.0632935\pi$
$192$	0	0
$193$	0.198662	0.0143000	0.00715001	−	0.999974i	$-0.497724\pi$
$193$	0.198662	0.0143000	0.00715001	+	0.999974i	$0.497724\pi$
$194$	2.48556	0.178453
$195$	0	0
$196$	0	0
$197$	− 15.0110i	− 1.06949i	−0.845014	−	0.534743i	$-0.820408\pi$
$197$	− 15.0110i	− 1.06949i	0.845014	−	0.534743i	$-0.179592\pi$
$198$	0	0
$199$	− 0.0497196i	− 0.00352453i	−0.999998	−	0.00176226i	$-0.999439\pi$
$199$	− 0.0497196i	− 0.00352453i	0.999998	−	0.00176226i	$-0.000560946\pi$
$200$	3.67464i	0.259837i
$201$	0	0
$202$	0.674906i	0.0474862i
$203$	0	0
$204$	0	0
$205$	9.23424	0.644947
$206$	1.13101	0.0788014
$207$	0	0
$208$	23.6508i	1.63989i
$209$	7.36779	0.509641
$210$	0	0
$211$	19.8681	1.36777	0.683887	−	0.729588i	$-0.260288\pi$
$211$	19.8681	1.36777	0.683887	+	0.729588i	$0.260288\pi$
$212$	21.2464i	1.45921i
$213$	0	0
$214$	−3.65016	−0.249520
$215$	−7.79248	−0.531443
$216$	0	0
$217$	0	0
$218$	− 1.54747i	− 0.104808i
$219$	0	0
$220$	13.7130i	0.924527i
$221$	7.33078i	0.493122i
$222$	0	0
$223$	11.1172i	0.744461i	0.928140	+	0.372231i	$0.121407\pi$
$223$	11.1172i	0.744461i	−0.928140	+	0.372231i	$0.878593\pi$
$224$	0	0
$225$	0	0
$226$	−2.13702	−0.142152
$227$	−17.0090	−1.12893	−0.564463	−	0.825458i	$-0.690917\pi$
$227$	−17.0090	−1.12893	−0.564463	+	0.825458i	$0.690917\pi$
$228$	0	0
$229$	28.1557i	1.86058i	0.366821	+	0.930292i	$0.380446\pi$
$229$	28.1557i	1.86058i	−0.366821	+	0.930292i	$0.619554\pi$
$230$	−2.87888	−0.189828
$231$	0	0
$232$	2.55235	0.167570
$233$	− 6.83120i	− 0.447527i	−0.974643	−	0.223763i	$-0.928166\pi$
$233$	− 6.83120i	− 0.447527i	0.974643	−	0.223763i	$-0.0718343\pi$
$234$	0	0
$235$	−36.2388	−2.36396
$236$	−20.6129	−1.34178
$237$	0	0
$238$	0	0
$239$	− 7.63188i	− 0.493666i	−0.969058	−	0.246833i	$-0.920610\pi$
$239$	− 7.63188i	− 0.493666i	0.969058	−	0.246833i	$-0.0793898\pi$
$240$	0	0
$241$	− 15.0422i	− 0.968954i	−0.874804	−	0.484477i	$-0.839010\pi$
$241$	− 15.0422i	− 0.968954i	0.874804	−	0.484477i	$-0.160990\pi$
$242$	1.25337i	0.0805698i
$243$	0	0
$244$	15.5833i	0.997617i
$245$	0	0
$246$	0	0
$247$	20.2524	1.28863
$248$	4.86448	0.308895
$249$	0	0
$250$	− 0.572819i	− 0.0362282i
$251$	30.6693	1.93583	0.967915	−	0.251279i	$-0.0808510\pi$
$251$	30.6693	1.93583	0.967915	+	0.251279i	$0.0808510\pi$
$252$	0	0
$253$	−9.87656	−0.620934
$254$	0.00881165i	0 0.000552892i
$255$	0	0
$256$	12.1376	0.758598
$257$	24.3456	1.51864	0.759318	−	0.650720i	$-0.225533\pi$
$257$	24.3456	1.51864	0.759318	+	0.650720i	$0.225533\pi$
$258$	0	0
$259$	0	0
$260$	37.6939i	2.33767i
$261$	0	0
$262$	1.71548i	0.105983i
$263$	1.00517i	0.0619814i	0.999520	+	0.0309907i	$0.00986623\pi$
$263$	1.00517i	0.0619814i	−0.999520	+	0.0309907i	$0.990134\pi$
$264$	0	0
$265$	32.9686i	2.02525i
$266$	0	0
$267$	0	0
$268$	2.39279	0.146163
$269$	0.0217090	0.00132362	0.000661811	−	1.00000i	$-0.499789\pi$
$269$	0.0217090	0.00132362	0.000661811		1.00000i	$0.499789\pi$
$270$	0	0
$271$	− 14.5128i	− 0.881592i	−0.897607	−	0.440796i	$-0.854696\pi$
$271$	− 14.5128i	− 0.881592i	0.897607	−	0.440796i	$-0.145304\pi$
$272$	4.24713	0.257520
$273$	0	0
$274$	4.22461	0.255218
$275$	9.65682i	0.582328i
$276$	0	0
$277$	15.0680	0.905346	0.452673	−	0.891677i	$-0.350470\pi$
$277$	15.0680	0.905346	0.452673	+	0.891677i	$0.350470\pi$
$278$	4.47751	0.268543
$279$	0	0
$280$	0	0
$281$	9.94171i	0.593073i	0.955022	+	0.296536i	$0.0958316\pi$
$281$	9.94171i	0.593073i	−0.955022	+	0.296536i	$0.904168\pi$
$282$	0	0
$283$	13.5621i	0.806186i	0.915159	+	0.403093i	$0.132065\pi$
$283$	13.5621i	0.806186i	−0.915159	+	0.403093i	$0.867935\pi$
$284$	− 18.6490i	− 1.10662i
$285$	0	0
$286$	− 3.32560i	− 0.196647i
$287$	0	0
$288$	0	0
$289$	−15.6836	−0.922562
$290$	1.95514	0.114810
$291$	0	0
$292$	− 20.0631i	− 1.17410i
$293$	−8.24489	−0.481671	−0.240836	−	0.970566i	$-0.577422\pi$
$293$	−8.24489	−0.481671	−0.240836	+	0.970566i	$0.577422\pi$
$294$	0	0
$295$	−31.9856	−1.86227
$296$	4.50600i	0.261906i
$297$	0	0
$298$	−4.95927	−0.287283
$299$	−27.1485	−1.57004
$300$	0	0
$301$	0	0
$302$	1.91047i	0.109935i
$303$	0	0
$304$	− 11.7334i	− 0.672954i
$305$	24.1810i	1.38460i
$306$	0	0
$307$	3.21948i	0.183746i	0.995771	+	0.0918728i	$0.0292853\pi$
$307$	3.21948i	0.183746i	−0.995771	+	0.0918728i	$0.970715\pi$
$308$	0	0
$309$	0	0
$310$	3.72627	0.211638
$311$	8.35230	0.473615	0.236808	−	0.971557i	$-0.423899\pi$
$311$	8.35230	0.473615	0.236808	+	0.971557i	$0.423899\pi$
$312$	0	0
$313$	− 2.60985i	− 0.147517i	−0.997276	−	0.0737587i	$-0.976501\pi$
$313$	− 2.60985i	− 0.147517i	0.997276	−	0.0737587i	$-0.0234995\pi$
$314$	3.93463	0.222044
$315$	0	0
$316$	17.7838	1.00042
$317$	− 6.43056i	− 0.361176i	−0.983559	−	0.180588i	$-0.942200\pi$
$317$	− 6.43056i	− 0.361176i	0.983559	−	0.180588i	$-0.0578001\pi$
$318$	0	0
$319$	6.70747	0.375546
$320$	20.6396	1.15379
$321$	0	0
$322$	0	0
$323$	− 3.63687i	− 0.202361i
$324$	0	0
$325$	26.5444i	1.47242i
$326$	4.94048i	0.273628i
$327$	0	0
$328$	2.69944i	0.149052i
$329$	0	0
$330$	0	0
$331$	−24.6831	−1.35671	−0.678353	−	0.734736i	$-0.737306\pi$
$331$	−24.6831	−1.35671	−0.678353	+	0.734736i	$0.737306\pi$
$332$	−25.2027	−1.38318
$333$	0	0
$334$	0.651783i	0.0356640i
$335$	3.71297	0.202861
$336$	0	0
$337$	1.66492	0.0906942	0.0453471	−	0.998971i	$-0.485561\pi$
$337$	1.66492	0.0906942	0.0453471	+	0.998971i	$0.485561\pi$
$338$	− 6.23026i	− 0.338882i
$339$	0	0
$340$	6.76896	0.367098
$341$	12.7837	0.692274
$342$	0	0
$343$	0	0
$344$	− 2.27797i	− 0.122820i
$345$	0	0
$346$	− 3.73334i	− 0.200706i
$347$	24.7331i	1.32774i	0.747848	+	0.663870i	$0.231087\pi$
$347$	24.7331i	1.32774i	−0.747848	+	0.663870i	$0.768913\pi$
$348$	0	0
$349$	8.05909i	0.431393i	0.976460	+	0.215697i	$0.0692022\pi$
$349$	8.05909i	0.431393i	−0.976460	+	0.215697i	$0.930798\pi$
$350$	0	0
$351$	0	0
$352$	−6.03850	−0.321853
$353$	2.70943	0.144208	0.0721042	−	0.997397i	$-0.477029\pi$
$353$	2.70943	0.144208	0.0721042	+	0.997397i	$0.477029\pi$
$354$	0	0
$355$	− 28.9383i	− 1.53588i
$356$	27.2535	1.44443
$357$	0	0
$358$	3.60029	0.190281
$359$	18.0536i	0.952833i	0.879220	+	0.476416i	$0.158064\pi$
$359$	18.0536i	0.952833i	−0.879220	+	0.476416i	$0.841936\pi$
$360$	0	0
$361$	8.95259	0.471189
$362$	4.26908	0.224378
$363$	0	0
$364$	0	0
$365$	− 31.1325i	− 1.62955i
$366$	0	0
$367$	− 19.8669i	− 1.03704i	−0.855064	−	0.518522i	$-0.826482\pi$
$367$	− 19.8669i	− 1.03704i	0.855064	−	0.518522i	$-0.173518\pi$
$368$	15.7286i	0.819911i
$369$	0	0
$370$	3.45166i	0.179443i
$371$	0	0
$372$	0	0
$373$	18.3014	0.947612	0.473806	−	0.880629i	$-0.342880\pi$
$373$	18.3014	0.947612	0.473806	+	0.880629i	$0.342880\pi$
$374$	−0.597202	−0.0308806
$375$	0	0
$376$	− 10.5937i	− 0.546328i
$377$	18.4373	0.949571
$378$	0	0
$379$	−2.26172	−0.116177	−0.0580883	−	0.998311i	$-0.518500\pi$
$379$	−2.26172	−0.116177	−0.0580883	+	0.998311i	$0.518500\pi$
$380$	− 18.7003i	− 0.959304i
$381$	0	0
$382$	1.22265	0.0625560
$383$	9.19920	0.470057	0.235029	−	0.971988i	$-0.424482\pi$
$383$	9.19920	0.470057	0.235029	+	0.971988i	$0.424482\pi$
$384$	0	0
$385$	0	0
$386$	0.0444863i	0.00226429i
$387$	0	0
$388$	− 21.6430i	− 1.09876i
$389$	6.06660i	0.307589i	0.988103	+	0.153794i	$0.0491493\pi$
$389$	6.06660i	0.307589i	−0.988103	+	0.153794i	$0.950851\pi$
$390$	0	0
$391$	4.87524i	0.246552i
$392$	0	0
$393$	0	0
$394$	3.36139	0.169344
$395$	27.5957	1.38849
$396$	0	0
$397$	14.6875i	0.737146i	0.929599	+	0.368573i	$0.120153\pi$
$397$	14.6875i	0.737146i	−0.929599	+	0.368573i	$0.879847\pi$
$398$	0.0111336	0.000558079 0
$399$	0	0
$400$	15.3787	0.768934
$401$	− 35.9779i	− 1.79665i	−0.439330	−	0.898326i	$-0.644784\pi$
$401$	− 35.9779i	− 1.79665i	0.439330	−	0.898326i	$-0.355216\pi$
$402$	0	0
$403$	35.1395	1.75042
$404$	5.87672	0.292378
$405$	0	0
$406$	0	0
$407$	11.8416i	0.586965i
$408$	0	0
$409$	17.0212i	0.841642i	0.907144	+	0.420821i	$0.138258\pi$
$409$	17.0212i	0.841642i	−0.907144	+	0.420821i	$0.861742\pi$
$410$	2.06781i	0.102122i
$411$	0	0
$412$	− 9.84826i	− 0.485189i
$413$	0	0
$414$	0	0
$415$	−39.1077	−1.91972
$416$	−16.5985	−0.813809
$417$	0	0
$418$	1.64986i	0.0806973i
$419$	−9.93475	−0.485345	−0.242672	−	0.970108i	$-0.578024\pi$
$419$	−9.93475	−0.485345	−0.242672	+	0.970108i	$0.578024\pi$
$420$	0	0
$421$	−29.7851	−1.45164	−0.725819	−	0.687886i	$-0.758539\pi$
$421$	−29.7851	−1.45164	−0.725819	+	0.687886i	$0.758539\pi$
$422$	4.44904i	0.216576i
$423$	0	0
$424$	−9.63771	−0.468049
$425$	4.76677	0.231222
$426$	0	0
$427$	0	0
$428$	31.7837i	1.53632i
$429$	0	0
$430$	− 1.74496i	− 0.0841495i
$431$	− 23.3144i	− 1.12301i	−0.827472	−	0.561507i	$-0.810222\pi$
$431$	− 23.3144i	− 1.12301i	0.827472	−	0.561507i	$-0.189778\pi$
$432$	0	0
$433$	23.4403i	1.12647i	0.826297	+	0.563234i	$0.190443\pi$
$433$	23.4403i	1.12647i	−0.826297	+	0.563234i	$0.809557\pi$
$434$	0	0
$435$	0	0
$436$	−13.4746	−0.645315
$437$	13.4686	0.644291
$438$	0	0
$439$	4.21244i	0.201049i	0.994935	+	0.100524i	$0.0320521\pi$
$439$	4.21244i	0.201049i	−0.994935	+	0.100524i	$0.967948\pi$
$440$	−6.22042	−0.296547
$441$	0	0
$442$	−1.64157	−0.0780818
$443$	− 2.38333i	− 0.113235i	−0.998396	−	0.0566177i	$-0.981968\pi$
$443$	− 2.38333i	− 0.113235i	0.998396	−	0.0566177i	$-0.0180316\pi$
$444$	0	0
$445$	42.2900	2.00474
$446$	−2.48946	−0.117879
$447$	0	0
$448$	0	0
$449$	− 29.8102i	− 1.40683i	−0.710778	−	0.703416i	$-0.751657\pi$
$449$	− 29.8102i	− 1.40683i	0.710778	−	0.703416i	$-0.248343\pi$
$450$	0	0
$451$	7.09402i	0.334045i
$452$	18.6080i	0.875248i
$453$	0	0
$454$	− 3.80881i	− 0.178756i
$455$	0	0
$456$	0	0
$457$	11.1343	0.520842	0.260421	−	0.965495i	$-0.416139\pi$
$457$	11.1343	0.520842	0.260421	+	0.965495i	$0.416139\pi$
$458$	−6.30488	−0.294608
$459$	0	0
$460$	25.0678i	1.16879i
$461$	−34.6787	−1.61515	−0.807574	−	0.589766i	$-0.799220\pi$
$461$	−34.6787	−1.61515	−0.807574	+	0.589766i	$0.799220\pi$
$462$	0	0
$463$	−11.8060	−0.548673	−0.274337	−	0.961634i	$-0.588458\pi$
$463$	−11.8060	−0.548673	−0.274337	+	0.961634i	$0.588458\pi$
$464$	− 10.6818i	− 0.495889i
$465$	0	0
$466$	1.52970	0.0708622
$467$	−15.5098	−0.717707	−0.358853	−	0.933394i	$-0.616832\pi$
$467$	−15.5098	−0.717707	−0.358853	+	0.933394i	$0.616832\pi$
$468$	0	0
$469$	0	0
$470$	− 8.11492i	− 0.374314i
$471$	0	0
$472$	− 9.35035i	− 0.430385i
$473$	− 5.98642i	− 0.275256i
$474$	0	0
$475$	− 13.1689i	− 0.604232i
$476$	0	0
$477$	0	0
$478$	1.70900	0.0781678
$479$	7.36161	0.336360	0.168180	−	0.985756i	$-0.446211\pi$
$479$	7.36161	0.336360	0.168180	+	0.985756i	$0.446211\pi$
$480$	0	0
$481$	32.5499i	1.48415i
$482$	3.36839	0.153426
$483$	0	0
$484$	10.9137	0.496077
$485$	− 33.5840i	− 1.52497i
$486$	0	0
$487$	−41.3340	−1.87302	−0.936510	−	0.350640i	$-0.885964\pi$
$487$	−41.3340	−1.87302	−0.936510	+	0.350640i	$0.885964\pi$
$488$	−7.06883	−0.319991
$489$	0	0
$490$	0	0
$491$	− 22.2997i	− 1.00637i	−0.864179	−	0.503185i	$-0.832162\pi$
$491$	− 22.2997i	− 1.00637i	0.864179	−	0.503185i	$-0.167838\pi$
$492$	0	0
$493$	− 3.31092i	− 0.149117i
$494$	4.53510i	0.204044i
$495$	0	0
$496$	− 20.3583i	− 0.914113i
$497$	0	0
$498$	0	0
$499$	−15.5901	−0.697907	−0.348953	−	0.937140i	$-0.613463\pi$
$499$	−15.5901	−0.697907	−0.348953	+	0.937140i	$0.613463\pi$
$500$	−4.98781	−0.223061
$501$	0	0
$502$	6.86775i	0.306523i
$503$	−28.0258	−1.24961	−0.624803	−	0.780782i	$-0.714821\pi$
$503$	−28.0258	−1.24961	−0.624803	+	0.780782i	$0.714821\pi$
$504$	0	0
$505$	9.11908	0.405794
$506$	− 2.21165i	− 0.0983197i
$507$	0	0
$508$	0.0767272	0.00340422
$509$	−23.4522	−1.03950	−0.519751	−	0.854318i	$-0.673975\pi$
$509$	−23.4522	−1.03950	−0.519751	+	0.854318i	$0.673975\pi$
$510$	0	0
$511$	0	0
$512$	16.1646i	0.714380i
$513$	0	0
$514$	5.45168i	0.240463i
$515$	− 15.2818i	− 0.673398i
$516$	0	0
$517$	− 27.8398i	− 1.22439i
$518$	0	0
$519$	0	0
$520$	−17.0986	−0.749822
$521$	−19.6897	−0.862622	−0.431311	−	0.902203i	$-0.641949\pi$
$521$	−19.6897	−0.862622	−0.431311	+	0.902203i	$0.641949\pi$
$522$	0	0
$523$	− 29.3345i	− 1.28271i	−0.767246	−	0.641353i	$-0.778373\pi$
$523$	− 29.3345i	− 1.28271i	0.767246	−	0.641353i	$-0.221627\pi$
$524$	14.9375	0.652548
$525$	0	0
$526$	−0.225087	−0.00981424
$527$	− 6.31024i	− 0.274878i
$528$	0	0
$529$	4.94526	0.215012
$530$	−7.38263	−0.320681
$531$	0	0
$532$	0	0
$533$	19.4999i	0.844634i
$534$	0	0
$535$	49.3196i	2.13227i
$536$	1.08541i	0.0468826i
$537$	0	0
$538$	0.00486128i	0 0.000209584i
$539$	0	0
$540$	0	0
$541$	−14.1811	−0.609693	−0.304847	−	0.952401i	$-0.598605\pi$
$541$	−14.1811	−0.609693	−0.304847	+	0.952401i	$0.598605\pi$
$542$	3.24984	0.139593
$543$	0	0
$544$	2.98071i	0.127797i
$545$	−20.9089	−0.895639
$546$	0	0
$547$	−2.29433	−0.0980983	−0.0490492	−	0.998796i	$-0.515619\pi$
$547$	−2.29433	−0.0980983	−0.0490492	+	0.998796i	$0.515619\pi$
$548$	− 36.7857i	− 1.57141i
$549$	0	0
$550$	−2.16244	−0.0922068
$551$	−9.14693	−0.389672
$552$	0	0
$553$	0	0
$554$	3.37415i	0.143354i
$555$	0	0
$556$	− 38.9878i	− 1.65345i
$557$	29.7218i	1.25935i	0.776858	+	0.629676i	$0.216812\pi$
$557$	29.7218i	1.25935i	−0.776858	+	0.629676i	$0.783188\pi$
$558$	0	0
$559$	− 16.4553i	− 0.695987i
$560$	0	0
$561$	0	0
$562$	−2.22624	−0.0939081
$563$	−4.47480	−0.188590	−0.0942951	−	0.995544i	$-0.530060\pi$
$563$	−4.47480	−0.188590	−0.0942951	+	0.995544i	$0.530060\pi$
$564$	0	0
$565$	28.8746i	1.21476i
$566$	−3.03696	−0.127653
$567$	0	0
$568$	8.45951	0.354953
$569$	− 1.12610i	− 0.0472086i	−0.999721	−	0.0236043i	$-0.992486\pi$
$569$	− 1.12610i	− 0.0472086i	0.999721	−	0.0236043i	$-0.00751418\pi$
$570$	0	0
$571$	−26.6881	−1.11686	−0.558431	−	0.829551i	$-0.688596\pi$
$571$	−26.6881	−1.11686	−0.558431	+	0.829551i	$0.688596\pi$
$572$	−28.9576	−1.21078
$573$	0	0
$574$	0	0
$575$	17.6530i	0.736183i
$576$	0	0
$577$	2.17852i	0.0906930i	0.998971	+	0.0453465i	$0.0144392\pi$
$577$	2.17852i	0.0906930i	−0.998971	+	0.0453465i	$0.985561\pi$
$578$	− 3.51200i	− 0.146080i
$579$	0	0
$580$	− 17.0243i	− 0.706896i
$581$	0	0
$582$	0	0
$583$	−25.3275	−1.04896
$584$	9.10096	0.376600
$585$	0	0
$586$	− 1.84627i	− 0.0762687i
$587$	−5.94259	−0.245277	−0.122639	−	0.992451i	$-0.539136\pi$
$587$	−5.94259	−0.245277	−0.122639	+	0.992451i	$0.539136\pi$
$588$	0	0
$589$	−17.4330	−0.718315
$590$	− 7.16251i	− 0.294876i
$591$	0	0
$592$	18.8580	0.775057
$593$	33.7933	1.38773	0.693863	−	0.720107i	$-0.255907\pi$
$593$	33.7933	1.38773	0.693863	+	0.720107i	$0.255907\pi$
$594$	0	0
$595$	0	0
$596$	43.1827i	1.76883i
$597$	0	0
$598$	− 6.07933i	− 0.248602i
$599$	− 0.576506i	− 0.0235554i	−0.999931	−	0.0117777i	$-0.996251\pi$
$599$	− 0.576506i	− 0.0235554i	0.999931	−	0.0117777i	$-0.00374905\pi$
$600$	0	0
$601$	− 9.20844i	− 0.375620i	−0.982205	−	0.187810i	$-0.939861\pi$
$601$	− 9.20844i	− 0.375620i	0.982205	−	0.187810i	$-0.0601389\pi$
$602$	0	0
$603$	0	0
$604$	16.6354	0.676884
$605$	16.9351	0.688510
$606$	0	0
$607$	− 24.9707i	− 1.01353i	−0.862085	−	0.506764i	$-0.830841\pi$
$607$	− 24.9707i	− 1.01353i	0.862085	−	0.506764i	$-0.169159\pi$
$608$	8.23467	0.333960
$609$	0	0
$610$	−5.41483	−0.219240
$611$	− 76.5254i	− 3.09589i
$612$	0	0
$613$	−32.2739	−1.30353	−0.651765	−	0.758421i	$-0.725971\pi$
$613$	−32.2739	−1.30353	−0.651765	+	0.758421i	$0.725971\pi$
$614$	−0.720936	−0.0290946
$615$	0	0
$616$	0	0
$617$	− 38.0815i	− 1.53310i	−0.642183	−	0.766551i	$-0.721971\pi$
$617$	− 38.0815i	− 1.53310i	0.642183	−	0.766551i	$-0.278029\pi$
$618$	0	0
$619$	− 30.4687i	− 1.22464i	−0.790610	−	0.612319i	$-0.790237\pi$
$619$	− 30.4687i	− 1.22464i	0.790610	−	0.612319i	$-0.209763\pi$
$620$	− 32.4464i	− 1.30308i
$621$	0	0
$622$	1.87032i	0.0749931i
$623$	0	0
$624$	0	0
$625$	−28.5125	−1.14050
$626$	0.584421	0.0233582
$627$	0	0
$628$	− 34.2607i	− 1.36715i
$629$	5.84521	0.233064
$630$	0	0
$631$	15.9820	0.636232	0.318116	−	0.948052i	$-0.396950\pi$
$631$	15.9820	0.636232	0.318116	+	0.948052i	$0.396950\pi$
$632$	8.06704i	0.320890i
$633$	0	0
$634$	1.43999	0.0571893
$635$	0.119060	0.00472475
$636$	0	0
$637$	0	0
$638$	1.50200i	0.0594646i
$639$	0	0
$640$	20.3424i	0.804102i
$641$	22.0916i	0.872565i	0.899810	+	0.436283i	$0.143705\pi$
$641$	22.0916i	0.872565i	−0.899810	+	0.436283i	$0.856295\pi$
$642$	0	0
$643$	− 2.61541i	− 0.103142i	−0.998669	−	0.0515709i	$-0.983577\pi$
$643$	− 2.61541i	− 0.103142i	0.998669	−	0.0515709i	$-0.0164228\pi$
$644$	0	0
$645$	0	0
$646$	0.814400	0.0320422
$647$	31.5745	1.24132	0.620660	−	0.784079i	$-0.286865\pi$
$647$	31.5745	1.24132	0.620660	+	0.784079i	$0.286865\pi$
$648$	0	0
$649$	− 24.5723i	− 0.964548i
$650$	−5.94407	−0.233146
$651$	0	0
$652$	43.0191	1.68476
$653$	18.8178i	0.736397i	0.929747	+	0.368199i	$0.120025\pi$
$653$	18.8178i	0.736397i	−0.929747	+	0.368199i	$0.879975\pi$
$654$	0	0
$655$	23.1790	0.905677
$656$	11.2974	0.441089
$657$	0	0
$658$	0	0
$659$	− 47.7090i	− 1.85848i	−0.369480	−	0.929239i	$-0.620464\pi$
$659$	− 47.7090i	− 1.85848i	0.369480	−	0.929239i	$-0.379536\pi$
$660$	0	0
$661$	24.3935i	0.948799i	0.880310	+	0.474399i	$0.157335\pi$
$661$	24.3935i	0.948799i	−0.880310	+	0.474399i	$0.842665\pi$
$662$	− 5.52727i	− 0.214823i
$663$	0	0
$664$	− 11.4323i	− 0.443661i
$665$	0	0
$666$	0	0
$667$	12.2615	0.474768
$668$	5.67538	0.219587
$669$	0	0
$670$	0.831441i	0.0321214i
$671$	−18.5766	−0.717142
$672$	0	0
$673$	31.5844	1.21749	0.608745	−	0.793366i	$-0.291673\pi$
$673$	31.5844	1.21749	0.608745	+	0.793366i	$0.291673\pi$
$674$	0.372825i	0.0143607i
$675$	0	0
$676$	−54.2499	−2.08653
$677$	17.7436	0.681941	0.340971	−	0.940074i	$-0.389244\pi$
$677$	17.7436	0.681941	0.340971	+	0.940074i	$0.389244\pi$
$678$	0	0
$679$	0	0
$680$	3.07051i	0.117749i
$681$	0	0
$682$	2.86263i	0.109616i
$683$	− 51.3592i	− 1.96521i	−0.185720	−	0.982603i	$-0.559462\pi$
$683$	− 51.3592i	− 1.96521i	0.185720	−	0.982603i	$-0.440538\pi$
$684$	0	0
$685$	− 57.0815i	− 2.18097i
$686$	0	0
$687$	0	0
$688$	−9.53350	−0.363461
$689$	−69.6197	−2.65230
$690$	0	0
$691$	− 8.59175i	− 0.326846i	−0.986556	−	0.163423i	$-0.947747\pi$
$691$	− 8.59175i	− 0.326846i	0.986556	−	0.163423i	$-0.0522535\pi$
$692$	−32.5080	−1.23577
$693$	0	0
$694$	−5.53845	−0.210237
$695$	− 60.4985i	− 2.29484i
$696$	0	0
$697$	3.50173	0.132638
$698$	−1.80466	−0.0683075
$699$	0	0
$700$	0	0
$701$	1.97794i	0.0747059i	0.999302	+	0.0373529i	$0.0118926\pi$
$701$	1.97794i	0.0747059i	−0.999302	+	0.0373529i	$0.988107\pi$
$702$	0	0
$703$	− 16.1483i	− 0.609044i
$704$	15.8560i	0.597596i
$705$	0	0
$706$	0.606720i	0.0228342i
$707$	0	0
$708$	0	0
$709$	−39.8892	−1.49807	−0.749035	−	0.662530i	$-0.769483\pi$
$709$	−39.8892	−1.49807	−0.749035	+	0.662530i	$0.769483\pi$
$710$	6.48011	0.243194
$711$	0	0
$712$	12.3626i	0.463309i
$713$	23.3690	0.875178
$714$	0	0
$715$	−44.9343	−1.68045
$716$	− 31.3494i	− 1.17158i
$717$	0	0
$718$	−4.04272	−0.150873
$719$	14.8851	0.555122	0.277561	−	0.960708i	$-0.410474\pi$
$719$	14.8851	0.555122	0.277561	+	0.960708i	$0.410474\pi$
$720$	0	0
$721$	0	0
$722$	2.00474i	0.0746089i
$723$	0	0
$724$	− 37.1729i	− 1.38152i
$725$	− 11.9887i	− 0.445249i
$726$	0	0
$727$	38.0550i	1.41138i	0.708521	+	0.705690i	$0.249363\pi$
$727$	38.0550i	1.41138i	−0.708521	+	0.705690i	$0.750637\pi$
$728$	0	0
$729$	0	0
$730$	6.97147	0.258026
$731$	−2.95500	−0.109295
$732$	0	0
$733$	36.2727i	1.33976i	0.742468	+	0.669882i	$0.233655\pi$
$733$	36.2727i	1.33976i	−0.742468	+	0.669882i	$0.766345\pi$
$734$	4.44878	0.164207
$735$	0	0
$736$	−11.0386	−0.406889
$737$	2.85242i	0.105070i
$738$	0	0
$739$	−42.6658	−1.56949	−0.784744	−	0.619820i	$-0.787206\pi$
$739$	−42.6658	−1.56949	−0.784744	+	0.619820i	$0.787206\pi$
$740$	30.0552	1.10485
$741$	0	0
$742$	0	0
$743$	− 1.00817i	− 0.0369862i	−0.999829	−	0.0184931i	$-0.994113\pi$
$743$	− 1.00817i	− 0.0369862i	0.999829	−	0.0184931i	$-0.00588688\pi$
$744$	0	0
$745$	67.0079i	2.45498i
$746$	4.09822i	0.150046i
$747$	0	0
$748$	5.20012i	0.190135i
$749$	0	0
$750$	0	0
$751$	−32.2217	−1.17579	−0.587893	−	0.808938i	$-0.700043\pi$
$751$	−32.2217	−1.17579	−0.587893	+	0.808938i	$0.700043\pi$
$752$	−44.3354	−1.61675
$753$	0	0
$754$	4.12865i	0.150357i
$755$	25.8136	0.939454
$756$	0	0
$757$	−18.4357	−0.670057	−0.335028	−	0.942208i	$-0.608746\pi$
$757$	−18.4357	−0.670057	−0.335028	+	0.942208i	$0.608746\pi$
$758$	− 0.506464i	− 0.0183956i
$759$	0	0
$760$	8.48276	0.307702
$761$	32.2294	1.16832	0.584158	−	0.811640i	$-0.301425\pi$
$761$	32.2294	1.16832	0.584158	+	0.811640i	$0.301425\pi$
$762$	0	0
$763$	0	0
$764$	− 10.6462i	− 0.385164i
$765$	0	0
$766$	2.05997i	0.0744297i
$767$	− 67.5439i	− 2.43887i
$768$	0	0
$769$	43.1786i	1.55706i	0.627608	+	0.778530i	$0.284034\pi$
$769$	43.1786i	1.55706i	−0.627608	+	0.778530i	$0.715966\pi$
$770$	0	0
$771$	0	0
$772$	0.387363	0.0139415
$773$	−0.0127856	−0.000459867 0	−0.000229934	−	1.00000i	$-0.500073\pi$
$773$	−0.0127856	−0.000459867 0	−0.000229934		1.00000i	$0.500073\pi$
$774$	0	0
$775$	− 22.8491i	− 0.820764i
$776$	9.81761	0.352432
$777$	0	0
$778$	−1.35849	−0.0487041
$779$	− 9.67408i	− 0.346610i
$780$	0	0
$781$	22.2313	0.795497
$782$	−1.09171	−0.0390394
$783$	0	0
$784$	0	0
$785$	− 53.1633i	− 1.89748i
$786$	0	0
$787$	− 34.8954i	− 1.24389i	−0.783062	−	0.621943i	$-0.786343\pi$
$787$	− 34.8954i	− 1.24389i	0.783062	−	0.621943i	$-0.213657\pi$
$788$	− 29.2692i	− 1.04267i
$789$	0	0
$790$	6.17947i	0.219856i
$791$	0	0
$792$	0	0
$793$	−51.0630	−1.81330
$794$	−3.28896	−0.116721
$795$	0	0
$796$	− 0.0969460i	− 0.00343616i
$797$	2.12544	0.0752869	0.0376434	−	0.999291i	$-0.488015\pi$
$797$	2.12544	0.0752869	0.0376434	+	0.999291i	$0.488015\pi$
$798$	0	0
$799$	−13.7422	−0.486164
$800$	10.7930i	0.381591i
$801$	0	0
$802$	8.05650	0.284485
$803$	23.9170	0.844011
$804$	0	0
$805$	0	0
$806$	7.86874i	0.277165i
$807$	0	0
$808$	2.66578i	0.0937818i
$809$	5.67156i	0.199401i	0.995017	+	0.0997007i	$0.0317885\pi$
$809$	5.67156i	0.199401i	−0.995017	+	0.0997007i	$0.968211\pi$
$810$	0	0
$811$	47.7024i	1.67506i	0.546394	+	0.837528i	$0.316000\pi$
$811$	47.7024i	1.67506i	−0.546394	+	0.837528i	$0.684000\pi$
$812$	0	0
$813$	0	0
$814$	−2.65167	−0.0929410
$815$	66.7541	2.33829
$816$	0	0
$817$	8.16365i	0.285610i
$818$	−3.81153	−0.133267
$819$	0	0
$820$	18.0054	0.628777
$821$	− 18.4310i	− 0.643247i	−0.946868	−	0.321624i	$-0.895771\pi$
$821$	− 18.4310i	− 0.643247i	0.946868	−	0.321624i	$-0.104229\pi$
$822$	0	0
$823$	13.5774	0.473278	0.236639	−	0.971598i	$-0.423954\pi$
$823$	13.5774	0.473278	0.236639	+	0.971598i	$0.423954\pi$
$824$	4.46734	0.155627
$825$	0	0
$826$	0	0
$827$	19.9807i	0.694799i	0.937717	+	0.347399i	$0.112935\pi$
$827$	19.9807i	0.694799i	−0.937717	+	0.347399i	$0.887065\pi$
$828$	0	0
$829$	− 9.53405i	− 0.331131i	−0.986199	−	0.165566i	$-0.947055\pi$
$829$	− 9.53405i	− 0.331131i	0.986199	−	0.165566i	$-0.0529449\pi$
$830$	− 8.75735i	− 0.303972i
$831$	0	0
$832$	43.5846i	1.51103i
$833$	0	0
$834$	0	0
$835$	8.80666	0.304767
$836$	14.3661	0.496863
$837$	0	0
$838$	− 2.22468i	− 0.0768503i
$839$	−1.12969	−0.0390013	−0.0195006	−	0.999810i	$-0.506208\pi$
$839$	−1.12969	−0.0390013	−0.0195006	+	0.999810i	$0.506208\pi$
$840$	0	0
$841$	20.6728	0.712856
$842$	− 6.66975i	− 0.229855i
$843$	0	0
$844$	38.7399	1.33348
$845$	−84.1811	−2.89592
$846$	0	0
$847$	0	0
$848$	40.3346i	1.38510i
$849$	0	0
$850$	1.06742i	0.0366122i
$851$	21.6469i	0.742045i
$852$	0	0
$853$	− 38.2466i	− 1.30954i	−0.755828	−	0.654770i	$-0.772766\pi$
$853$	− 38.2466i	− 1.30954i	0.755828	−	0.654770i	$-0.227234\pi$
$854$	0	0
$855$	0	0
$856$	−14.4176	−0.492783
$857$	39.8391	1.36088	0.680439	−	0.732805i	$-0.261789\pi$
$857$	39.8391	1.36088	0.680439	+	0.732805i	$0.261789\pi$
$858$	0	0
$859$	10.5008i	0.358283i	0.983823	+	0.179141i	$0.0573320\pi$
$859$	10.5008i	0.358283i	−0.983823	+	0.179141i	$0.942668\pi$
$860$	−15.1942	−0.518118
$861$	0	0
$862$	5.22076	0.177820
$863$	18.1541i	0.617971i	0.951067	+	0.308986i	$0.0999895\pi$
$863$	18.1541i	0.617971i	−0.951067	+	0.308986i	$0.900010\pi$
$864$	0	0
$865$	−50.4436	−1.71513
$866$	−5.24896	−0.178367
$867$	0	0
$868$	0	0
$869$	21.1999i	0.719156i
$870$	0	0
$871$	7.84066i	0.265671i
$872$	− 6.11229i	− 0.206988i
$873$	0	0
$874$	3.01601i	0.102018i
$875$	0	0
$876$	0	0
$877$	25.4339	0.858843	0.429421	−	0.903104i	$-0.358718\pi$
$877$	25.4339	0.858843	0.429421	+	0.903104i	$0.358718\pi$
$878$	−0.943288	−0.0318344
$879$	0	0
$880$	26.0330i	0.877571i
$881$	41.4113	1.39518	0.697591	−	0.716496i	$-0.254255\pi$
$881$	41.4113	1.39518	0.697591	+	0.716496i	$0.254255\pi$
$882$	0	0
$883$	−0.807894	−0.0271878	−0.0135939	−	0.999908i	$-0.504327\pi$
$883$	−0.807894	−0.0271878	−0.0135939	+	0.999908i	$0.504327\pi$
$884$	14.2940i	0.480758i
$885$	0	0
$886$	0.533697	0.0179299
$887$	−42.6601	−1.43239	−0.716193	−	0.697902i	$-0.754117\pi$
$887$	−42.6601	−1.43239	−0.716193	+	0.697902i	$0.754117\pi$
$888$	0	0
$889$	0	0
$890$	9.46995i	0.317434i
$891$	0	0
$892$	21.6769i	0.725796i
$893$	37.9649i	1.27045i
$894$	0	0
$895$	− 48.6458i	− 1.62605i
$896$	0	0
$897$	0	0
$898$	6.67537	0.222760
$899$	−15.8706	−0.529315
$900$	0	0
$901$	12.5021i	0.416506i
$902$	−1.58856	−0.0528932
$903$	0	0
$904$	−8.44091	−0.280740
$905$	− 57.6822i	− 1.91742i
$906$	0	0
$907$	−26.8200	−0.890545	−0.445272	−	0.895395i	$-0.646893\pi$
$907$	−26.8200	−0.890545	−0.445272	+	0.895395i	$0.646893\pi$
$908$	−33.1651	−1.10062
$909$	0	0
$910$	0	0
$911$	− 28.3322i	− 0.938689i	−0.883015	−	0.469345i	$-0.844490\pi$
$911$	− 28.3322i	− 0.938689i	0.883015	−	0.469345i	$-0.155510\pi$
$912$	0	0
$913$	− 30.0437i	− 0.994303i
$914$	2.49330i	0.0824709i
$915$	0	0
$916$	54.8996i	1.81393i
$917$	0	0
$918$	0	0
$919$	33.2895	1.09812	0.549060	−	0.835783i	$-0.314986\pi$
$919$	33.2895	1.09812	0.549060	+	0.835783i	$0.314986\pi$
$920$	−11.3712	−0.374897
$921$	0	0
$922$	− 7.76556i	− 0.255745i
$923$	61.1088	2.01142
$924$	0	0
$925$	21.1652	0.695909
$926$	− 2.64371i	− 0.0868778i
$927$	0	0
$928$	7.49666	0.246090
$929$	15.3510	0.503650	0.251825	−	0.967773i	$-0.418969\pi$
$929$	15.3510	0.503650	0.251825	+	0.967773i	$0.418969\pi$
$930$	0	0
$931$	0	0
$932$	− 13.3199i	− 0.436307i
$933$	0	0
$934$	− 3.47309i	− 0.113643i
$935$	8.06918i	0.263890i
$936$	0	0
$937$	19.4414i	0.635124i	0.948238	+	0.317562i	$0.102864\pi$
$937$	19.4414i	0.635124i	−0.948238	+	0.317562i	$0.897136\pi$
$938$	0	0
$939$	0	0
$940$	−70.6605	−2.30469
$941$	−40.5112	−1.32063	−0.660314	−	0.750989i	$-0.729577\pi$
$941$	−40.5112	−1.32063	−0.660314	+	0.750989i	$0.729577\pi$
$942$	0	0
$943$	12.9682i	0.422301i
$944$	−39.1320	−1.27364
$945$	0	0
$946$	1.34053	0.0435845
$947$	7.07798i	0.230004i	0.993365	+	0.115002i	$0.0366874\pi$
$947$	7.07798i	0.230004i	−0.993365	+	0.115002i	$0.963313\pi$
$948$	0	0
$949$	65.7424	2.13409
$950$	2.94891	0.0956752
$951$	0	0
$952$	0	0
$953$	20.9587i	0.678919i	0.940621	+	0.339459i	$0.110244\pi$
$953$	20.9587i	0.678919i	−0.940621	+	0.339459i	$0.889756\pi$
$954$	0	0
$955$	− 16.5199i	− 0.534573i
$956$	− 14.8811i	− 0.481288i
$957$	0	0
$958$	1.64848i	0.0532599i
$959$	0	0
$960$	0	0
$961$	0.752414	0.0242714
$962$	−7.28885	−0.235002
$963$	0	0
$964$	− 29.3302i	− 0.944661i
$965$	0.601082	0.0193495
$966$	0	0
$967$	7.53992	0.242468	0.121234	−	0.992624i	$-0.461315\pi$
$967$	7.53992	0.242468	0.121234	+	0.992624i	$0.461315\pi$
$968$	4.95064i	0.159119i
$969$	0	0
$970$	7.52044	0.241467
$971$	−33.6134	−1.07871	−0.539353	−	0.842080i	$-0.681331\pi$
$971$	−33.6134	−1.07871	−0.539353	+	0.842080i	$0.681331\pi$
$972$	0	0
$973$	0	0
$974$	− 9.25587i	− 0.296577i
$975$	0	0
$976$	29.5836i	0.946949i
$977$	19.7436i	0.631655i	0.948817	+	0.315827i	$0.102282\pi$
$977$	19.7436i	0.631655i	−0.948817	+	0.315827i	$0.897718\pi$
$978$	0	0
$979$	32.4885i	1.03834i
$980$	0	0
$981$	0	0
$982$	4.99354	0.159350
$983$	−42.1779	−1.34526	−0.672632	−	0.739977i	$-0.734836\pi$
$983$	−42.1779	−1.34526	−0.672632	+	0.739977i	$0.734836\pi$
$984$	0	0
$985$	− 45.4179i	− 1.44713i
$986$	0.741412	0.0236114
$987$	0	0
$988$	39.4893	1.25632
$989$	− 10.9434i	− 0.347980i
$990$	0	0
$991$	−21.6276	−0.687023	−0.343511	−	0.939149i	$-0.611616\pi$
$991$	−21.6276	−0.687023	−0.343511	+	0.939149i	$0.611616\pi$
$992$	14.2878	0.453637
$993$	0	0
$994$	0	0
$995$	− 0.150434i	− 0.00476907i
$996$	0	0
$997$	10.6195i	0.336322i	0.985760	+	0.168161i	$0.0537828\pi$
$997$	10.6195i	0.336322i	−0.985760	+	0.168161i	$0.946217\pi$
$998$	− 3.49107i	− 0.110508i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3087.2.c.c.3086.16	yes	24
3.2	odd	2	inner	3087.2.c.c.3086.9	✓	24
7.6	odd	2	inner	3087.2.c.c.3086.10	yes	24
21.20	even	2	inner	3087.2.c.c.3086.15	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3087.2.c.c.3086.9	✓	24	3.2	odd	2	inner
3087.2.c.c.3086.10	yes	24	7.6	odd	2	inner
3087.2.c.c.3086.15	yes	24	21.20	even	2	inner
3087.2.c.c.3086.16	yes	24	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:	\( N \)	\(=\)	\( 3087 = 3^{2} \cdot 7^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3087.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.223929i q^{2} +1.94986 q^{4} +3.02565 q^{5} +0.884487i q^{8} +O(q^{10})\)	\(q+0.223929i q^{2} +1.94986 q^{4} +3.02565 q^{5} +0.884487i q^{8} +0.677530i q^{10} +2.32440i q^{11} +6.38925i q^{13} +3.70165 q^{16} +1.14736 q^{17} -3.16976i q^{19} +5.89958 q^{20} -0.520500 q^{22} +4.24909i q^{23} +4.15455 q^{25} -1.43074 q^{26} -2.88568i q^{29} -5.49978i q^{31} +2.59788i q^{32} +0.256928i q^{34} +5.09447 q^{37} +0.709802 q^{38} +2.67615i q^{40} +3.05199 q^{41} -2.57547 q^{43} +4.53224i q^{44} -0.951493 q^{46} -11.9772 q^{47} +0.930324i q^{50} +12.4581i q^{52} +10.8964i q^{53} +7.03280i q^{55} +0.646188 q^{58} -10.5715 q^{59} +7.99201i q^{61} +1.23156 q^{62} +6.82156 q^{64} +19.3316i q^{65} +1.22716 q^{67} +2.23719 q^{68} -9.56431i q^{71} -10.2895i q^{73} +1.14080i q^{74} -6.18058i q^{76} +9.12059 q^{79} +11.1999 q^{80} +0.683428i q^{82} -12.9254 q^{83} +3.47152 q^{85} -0.576723i q^{86} -2.05590 q^{88} +13.9772 q^{89} +8.28511i q^{92} -2.68204i q^{94} -9.59059i q^{95} -11.0998i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q - 40 q^{16} + 64 q^{22} + 136 q^{25} + 32 q^{37} + 32 q^{43} + 32 q^{46} - 128 q^{58} + 32 q^{64} + 40 q^{67} + 8 q^{79} + 64 q^{85} + 16 q^{88}+O(q^{100}) \) 24 * q - 40 * q^16 + 64 * q^22 + 136 * q^25 + 32 * q^37 + 32 * q^43 + 32 * q^46 - 128 * q^58 + 32 * q^64 + 40 * q^67 + 8 * q^79 + 64 * q^85 + 16 * q^88

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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