LMFDB - Embedded newform 9065.2.a.bc.1.13

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9065,2,Mod(1,9065)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9065.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9065, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 9065 = 5 \cdot 7^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9065.a (trivial)

Newform invariants

comment:select newform

sage:traces = [34,-2,10,30,-34,8,0,-6,28,2,-30] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$72.3843894323$
Analytic rank:	$0$
Dimension:	$34$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Character	$\chi$	$=$	9065.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.909782 q^{2} +0.118964 q^{3} -1.17230 q^{4} -1.00000 q^{5} -0.108231 q^{6} +2.88610 q^{8} -2.98585 q^{9} +0.909782 q^{10} -3.42513 q^{11} -0.139461 q^{12} -5.76743 q^{13} -0.118964 q^{15} -0.281123 q^{16} +2.77627 q^{17} +2.71647 q^{18} +1.36891 q^{19} +1.17230 q^{20} +3.11612 q^{22} +6.90159 q^{23} +0.343342 q^{24} +1.00000 q^{25} +5.24710 q^{26} -0.712100 q^{27} +0.231883 q^{29} +0.108231 q^{30} +5.09069 q^{31} -5.51643 q^{32} -0.407467 q^{33} -2.52580 q^{34} +3.50030 q^{36} +1.00000 q^{37} -1.24541 q^{38} -0.686116 q^{39} -2.88610 q^{40} -6.46459 q^{41} -7.93371 q^{43} +4.01527 q^{44} +2.98585 q^{45} -6.27894 q^{46} +5.19617 q^{47} -0.0334435 q^{48} -0.909782 q^{50} +0.330276 q^{51} +6.76114 q^{52} -14.3523 q^{53} +0.647856 q^{54} +3.42513 q^{55} +0.162851 q^{57} -0.210963 q^{58} -12.5786 q^{59} +0.139461 q^{60} -8.43298 q^{61} -4.63141 q^{62} +5.58100 q^{64} +5.76743 q^{65} +0.370706 q^{66} -4.29168 q^{67} -3.25461 q^{68} +0.821040 q^{69} -12.1484 q^{71} -8.61745 q^{72} -10.1017 q^{73} -0.909782 q^{74} +0.118964 q^{75} -1.60477 q^{76} +0.624216 q^{78} +5.38429 q^{79} +0.281123 q^{80} +8.87283 q^{81} +5.88136 q^{82} -5.36151 q^{83} -2.77627 q^{85} +7.21794 q^{86} +0.0275857 q^{87} -9.88526 q^{88} +13.1555 q^{89} -2.71647 q^{90} -8.09072 q^{92} +0.605608 q^{93} -4.72738 q^{94} -1.36891 q^{95} -0.656257 q^{96} -9.66100 q^{97} +10.2269 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 34 q - 2 q^{2} + 10 q^{3} + 30 q^{4} - 34 q^{5} + 8 q^{6} - 6 q^{8} + 28 q^{9} + 2 q^{10} - 30 q^{11} + 20 q^{12} + 18 q^{13} - 10 q^{15} + 18 q^{16} + 10 q^{17} + 40 q^{19} - 30 q^{20} - 4 q^{22} - 16 q^{23}+ \cdots - 100 q^{99}+O(q^{100}) $ 34 * q - 2 * q^2 + 10 * q^3 + 30 * q^4 - 34 * q^5 + 8 * q^6 - 6 * q^8 + 28 * q^9 + 2 * q^10 - 30 * q^11 + 20 * q^12 + 18 * q^13 - 10 * q^15 + 18 * q^16 + 10 * q^17 + 40 * q^19 - 30 * q^20 - 4 * q^22 - 16 * q^23 + 24 * q^24 + 34 * q^25 + 12 * q^26 + 34 * q^27 - 22 * q^29 - 8 * q^30 + 52 * q^31 + 6 * q^32 + 10 * q^33 + 36 * q^34 + 14 * q^36 + 34 * q^37 + 16 * q^38 - 2 * q^39 + 6 * q^40 + 16 * q^41 - 56 * q^44 - 28 * q^45 - 28 * q^46 + 38 * q^47 + 18 * q^48 - 2 * q^50 - 2 * q^51 + 52 * q^52 + 4 * q^53 + 32 * q^54 + 30 * q^55 - 12 * q^57 - 8 * q^58 + 16 * q^59 - 20 * q^60 + 60 * q^61 - 14 * q^62 - 38 * q^64 - 18 * q^65 + 14 * q^66 + 12 * q^67 + 36 * q^68 + 8 * q^69 - 44 * q^71 - 6 * q^72 + 48 * q^73 - 2 * q^74 + 10 * q^75 + 50 * q^76 + 8 * q^78 - 26 * q^79 - 18 * q^80 - 10 * q^81 + 60 * q^82 + 28 * q^83 - 10 * q^85 - 28 * q^86 + 86 * q^87 + 42 * q^88 - 4 * q^89 - 22 * q^92 + 48 * q^93 + 40 * q^94 - 40 * q^95 + 86 * q^96 + 18 * q^97 - 100 * q^99

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 $

$2$	−0.909782	−0.643313	−0.321656	−	0.946856i	$-0.604240\pi$
$2$	−0.909782	−0.643313	−0.321656	+	0.946856i	$0.604240\pi$
$3$	0.118964	0.0686839	0.0343419	−	0.999410i	$-0.489066\pi$
$3$	0.118964	0.0686839	0.0343419	+	0.999410i	$0.489066\pi$
$4$	−1.17230	−0.586149
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−0.108231	−0.0441852
$7$	0	0
$7$	0	0
$8$	2.88610	1.02039
$9$	−2.98585	−0.995283
$10$	0.909782	0.287698
$11$	−3.42513	−1.03272	−0.516358	−	0.856373i	$-0.672713\pi$
$11$	−3.42513	−1.03272	−0.516358	+	0.856373i	$0.672713\pi$
$12$	−0.139461	−0.0402590
$13$	−5.76743	−1.59960	−0.799799	−	0.600268i	$-0.795060\pi$
$13$	−5.76743	−1.59960	−0.799799	+	0.600268i	$0.795060\pi$
$14$	0	0
$15$	−0.118964	−0.0307164
$16$	−0.281123	−0.0702808
$17$	2.77627	0.673344	0.336672	−	0.941622i	$-0.390699\pi$
$17$	2.77627	0.673344	0.336672	+	0.941622i	$0.390699\pi$
$18$	2.71647	0.640278
$19$	1.36891	0.314050	0.157025	−	0.987595i	$-0.449810\pi$
$19$	1.36891	0.314050	0.157025	+	0.987595i	$0.449810\pi$
$20$	1.17230	0.262134
$21$	0	0
$22$	3.11612	0.664359
$23$	6.90159	1.43908	0.719540	−	0.694451i	$-0.244353\pi$
$23$	6.90159	1.43908	0.719540	+	0.694451i	$0.244353\pi$
$24$	0.343342	0.0700843
$25$	1.00000	0.200000
$26$	5.24710	1.02904
$27$	−0.712100	−0.137044
$28$	0	0
$29$	0.231883	0.0430595	0.0215298	−	0.999768i	$-0.493146\pi$
$29$	0.231883	0.0430595	0.0215298	+	0.999768i	$0.493146\pi$
$30$	0.108231	0.0197602
$31$	5.09069	0.914314	0.457157	−	0.889386i	$-0.348868\pi$
$31$	5.09069	0.914314	0.457157	+	0.889386i	$0.348868\pi$
$32$	−5.51643	−0.975177
$33$	−0.407467	−0.0709309
$34$	−2.52580	−0.433171
$35$	0	0
$36$	3.50030	0.583384
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	−1.24541	−0.202032
$39$	−0.686116	−0.109867
$40$	−2.88610	−0.456332
$41$	−6.46459	−1.00960	−0.504799	−	0.863237i	$-0.668434\pi$
$41$	−6.46459	−1.00960	−0.504799	+	0.863237i	$0.668434\pi$
$42$	0	0
$43$	−7.93371	−1.20988	−0.604940	−	0.796271i	$-0.706803\pi$
$43$	−7.93371	−1.20988	−0.604940	+	0.796271i	$0.706803\pi$
$44$	4.01527	0.605325
$45$	2.98585	0.445104
$46$	−6.27894	−0.925779
$47$	5.19617	0.757939	0.378969	−	0.925409i	$-0.376279\pi$
$47$	5.19617	0.757939	0.378969	+	0.925409i	$0.376279\pi$
$48$	−0.0334435	−0.00482716
$49$	0	0
$50$	−0.909782	−0.128663
$51$	0.330276	0.0462479
$52$	6.76114	0.937602
$53$	−14.3523	−1.97144	−0.985722	−	0.168381i	$-0.946146\pi$
$53$	−14.3523	−1.97144	−0.985722	+	0.168381i	$0.946146\pi$
$54$	0.647856	0.0881620
$55$	3.42513	0.461844
$56$	0	0
$57$	0.162851	0.0215701
$58$	−0.210963	−0.0277007
$59$	−12.5786	−1.63759	−0.818795	−	0.574087i	$-0.805357\pi$
$59$	−12.5786	−1.63759	−0.818795	+	0.574087i	$0.805357\pi$
$60$	0.139461	0.0180044
$61$	−8.43298	−1.07973	−0.539866	−	0.841751i	$-0.681525\pi$
$61$	−8.43298	−1.07973	−0.539866	+	0.841751i	$0.681525\pi$
$62$	−4.63141	−0.588190
$63$	0	0
$64$	5.58100	0.697625
$65$	5.76743	0.715362
$66$	0.370706	0.0456307
$67$	−4.29168	−0.524313	−0.262156	−	0.965025i	$-0.584434\pi$
$67$	−4.29168	−0.524313	−0.262156	+	0.965025i	$0.584434\pi$
$68$	−3.25461	−0.394680
$69$	0.821040	0.0988416
$70$	0	0
$71$	−12.1484	−1.44175	−0.720875	−	0.693065i	$-0.756260\pi$
$71$	−12.1484	−1.44175	−0.720875	+	0.693065i	$0.756260\pi$
$72$	−8.61745	−1.01558
$73$	−10.1017	−1.18232	−0.591160	−	0.806555i	$-0.701330\pi$
$73$	−10.1017	−1.18232	−0.591160	+	0.806555i	$0.701330\pi$
$74$	−0.909782	−0.105760
$75$	0.118964	0.0137368
$76$	−1.60477	−0.184080
$77$	0	0
$78$	0.624216	0.0706785
$79$	5.38429	0.605780	0.302890	−	0.953025i	$-0.402048\pi$
$79$	5.38429	0.605780	0.302890	+	0.953025i	$0.402048\pi$
$80$	0.281123	0.0314305
$81$	8.87283	0.985870
$82$	5.88136	0.649488
$83$	−5.36151	−0.588502	−0.294251	−	0.955728i	$-0.595070\pi$
$83$	−5.36151	−0.588502	−0.294251	+	0.955728i	$0.595070\pi$
$84$	0	0
$85$	−2.77627	−0.301129
$86$	7.21794	0.778331
$87$	0.0275857	0.00295750
$88$	−9.88526	−1.05377
$89$	13.1555	1.39448	0.697240	−	0.716837i	$-0.254411\pi$
$89$	13.1555	1.39448	0.697240	+	0.716837i	$0.254411\pi$
$90$	−2.71647	−0.286341
$91$	0	0
$92$	−8.09072	−0.843515
$93$	0.605608	0.0627986
$94$	−4.72738	−0.487592
$95$	−1.36891	−0.140447
$96$	−0.656257	−0.0669789
$97$	−9.66100	−0.980926	−0.490463	−	0.871462i	$-0.663172\pi$
$97$	−9.66100	−0.980926	−0.490463	+	0.871462i	$0.663172\pi$
$98$	0	0
$99$	10.2269	1.02784
$100$	−1.17230	−0.117230
$101$	−13.5023	−1.34353	−0.671763	−	0.740766i	$-0.734463\pi$
$101$	−13.5023	−1.34353	−0.671763	+	0.740766i	$0.734463\pi$
$102$	−0.300479	−0.0297519
$103$	1.09413	0.107808	0.0539038	−	0.998546i	$-0.482834\pi$
$103$	1.09413	0.107808	0.0539038	+	0.998546i	$0.482834\pi$
$104$	−16.6454	−1.63221
$105$	0	0
$106$	13.0575	1.26825
$107$	−5.92325	−0.572622	−0.286311	−	0.958137i	$-0.592429\pi$
$107$	−5.92325	−0.572622	−0.286311	+	0.958137i	$0.592429\pi$
$108$	0.834793	0.0803280
$109$	−7.48184	−0.716631	−0.358315	−	0.933601i	$-0.616649\pi$
$109$	−7.48184	−0.716631	−0.358315	+	0.933601i	$0.616649\pi$
$110$	−3.11612	−0.297110
$111$	0.118964	0.0112916
$112$	0	0
$113$	18.0278	1.69591	0.847957	−	0.530064i	$-0.177832\pi$
$113$	18.0278	1.69591	0.847957	+	0.530064i	$0.177832\pi$
$114$	−0.148159	−0.0138763
$115$	−6.90159	−0.643576
$116$	−0.271835	−0.0252393
$117$	17.2207	1.59205
$118$	11.4437	1.05348
$119$	0	0
$120$	−0.343342	−0.0313427
$121$	0.731514	0.0665013
$122$	7.67217	0.694605
$123$	−0.769053	−0.0693432
$124$	−5.96780	−0.535924
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−19.5894	−1.73827	−0.869137	−	0.494571i	$-0.835325\pi$
$127$	−19.5894	−1.73827	−0.869137	+	0.494571i	$0.835325\pi$
$128$	5.95538	0.526386
$129$	−0.943825	−0.0830992
$130$	−5.24710	−0.460201
$131$	−13.2963	−1.16170	−0.580852	−	0.814009i	$-0.697281\pi$
$131$	−13.2963	−1.16170	−0.580852	+	0.814009i	$0.697281\pi$
$132$	0.477673	0.0415761
$133$	0	0
$134$	3.90450	0.337297
$135$	0.712100	0.0612878
$136$	8.01258	0.687073
$137$	−7.05082	−0.602392	−0.301196	−	0.953562i	$-0.597386\pi$
$137$	−7.05082	−0.602392	−0.301196	+	0.953562i	$0.597386\pi$
$138$	−0.746967	−0.0635861
$139$	8.61941	0.731089	0.365545	−	0.930794i	$-0.380883\pi$
$139$	8.61941	0.731089	0.365545	+	0.930794i	$0.380883\pi$
$140$	0	0
$141$	0.618157	0.0520582
$142$	11.0524	0.927496
$143$	19.7542	1.65193
$144$	0.839391	0.0699493
$145$	−0.231883	−0.0192568
$146$	9.19038	0.760601
$147$	0	0
$148$	−1.17230	−0.0963623
$149$	14.6470	1.19993	0.599963	−	0.800028i	$-0.295182\pi$
$149$	14.6470	1.19993	0.599963	+	0.800028i	$0.295182\pi$
$150$	−0.108231	−0.00883704
$151$	−4.06092	−0.330473	−0.165236	−	0.986254i	$-0.552839\pi$
$151$	−4.06092	−0.330473	−0.165236	+	0.986254i	$0.552839\pi$
$152$	3.95081	0.320453
$153$	−8.28952	−0.670168
$154$	0	0
$155$	−5.09069	−0.408894
$156$	0.804333	0.0643981
$157$	−2.15028	−0.171611	−0.0858055	−	0.996312i	$-0.527346\pi$
$157$	−2.15028	−0.171611	−0.0858055	+	0.996312i	$0.527346\pi$
$158$	−4.89853	−0.389706
$159$	−1.70741	−0.135406
$160$	5.51643	0.436112
$161$	0	0
$162$	−8.07234	−0.634223
$163$	−16.4404	−1.28771	−0.643857	−	0.765146i	$-0.722667\pi$
$163$	−16.4404	−1.28771	−0.643857	+	0.765146i	$0.722667\pi$
$164$	7.57842	0.591775
$165$	0.407467	0.0317213
$166$	4.87780	0.378591
$167$	9.45862	0.731930	0.365965	−	0.930629i	$-0.380739\pi$
$167$	9.45862	0.731930	0.365965	+	0.930629i	$0.380739\pi$
$168$	0	0
$169$	20.2633	1.55871
$170$	2.52580	0.193720
$171$	−4.08736	−0.312568
$172$	9.30067	0.709169
$173$	21.6879	1.64890	0.824451	−	0.565934i	$-0.191484\pi$
$173$	21.6879	1.64890	0.824451	+	0.565934i	$0.191484\pi$
$174$	−0.0250969	−0.00190259
$175$	0	0
$176$	0.962884	0.0725801
$177$	−1.49640	−0.112476
$178$	−11.9686	−0.897087
$179$	21.7559	1.62611	0.813055	−	0.582187i	$-0.197803\pi$
$179$	21.7559	1.62611	0.813055	+	0.582187i	$0.197803\pi$
$180$	−3.50030	−0.260897
$181$	15.8771	1.18014	0.590069	−	0.807353i	$-0.299101\pi$
$181$	15.8771	1.18014	0.590069	+	0.807353i	$0.299101\pi$
$182$	0	0
$183$	−1.00322	−0.0741602
$184$	19.9187	1.46842
$185$	−1.00000	−0.0735215
$186$	−0.550971	−0.0403992
$187$	−9.50908	−0.695373
$188$	−6.09145	−0.444265
$189$	0	0
$190$	1.24541	0.0903515
$191$	−5.39945	−0.390690	−0.195345	−	0.980735i	$-0.562583\pi$
$191$	−5.39945	−0.390690	−0.195345	+	0.980735i	$0.562583\pi$
$192$	0.663938	0.0479156
$193$	17.6972	1.27387	0.636936	−	0.770917i	$-0.280202\pi$
$193$	17.6972	1.27387	0.636936	+	0.770917i	$0.280202\pi$
$194$	8.78940	0.631042
$195$	0.686116	0.0491338
$196$	0	0
$197$	−13.8079	−0.983771	−0.491885	−	0.870660i	$-0.663692\pi$
$197$	−13.8079	−0.983771	−0.491885	+	0.870660i	$0.663692\pi$
$198$	−9.30426	−0.661225
$199$	6.38067	0.452314	0.226157	−	0.974091i	$-0.427384\pi$
$199$	6.38067	0.452314	0.226157	+	0.974091i	$0.427384\pi$
$200$	2.88610	0.204078
$201$	−0.510556	−0.0360118
$202$	12.2841	0.864308
$203$	0	0
$204$	−0.387182	−0.0271081
$205$	6.46459	0.451506
$206$	−0.995417	−0.0693540
$207$	−20.6071	−1.43229
$208$	1.62136	0.112421
$209$	−4.68869	−0.324324
$210$	0	0
$211$	1.27993	0.0881141	0.0440571	−	0.999029i	$-0.485972\pi$
$211$	1.27993	0.0881141	0.0440571	+	0.999029i	$0.485972\pi$
$212$	16.8252	1.15556
$213$	−1.44522	−0.0990249
$214$	5.38886	0.368375
$215$	7.93371	0.541074
$216$	−2.05519	−0.139838
$217$	0	0
$218$	6.80684	0.461018
$219$	−1.20174	−0.0812063
$220$	−4.01527	−0.270710
$221$	−16.0119	−1.07708
$222$	−0.108231	−0.00726400
$223$	−1.30538	−0.0874150	−0.0437075	−	0.999044i	$-0.513917\pi$
$223$	−1.30538	−0.0874150	−0.0437075	+	0.999044i	$0.513917\pi$
$224$	0	0
$225$	−2.98585	−0.199057
$226$	−16.4014	−1.09100
$227$	−4.51518	−0.299683	−0.149842	−	0.988710i	$-0.547876\pi$
$227$	−4.51518	−0.299683	−0.149842	+	0.988710i	$0.547876\pi$
$228$	−0.190910	−0.0126433
$229$	14.9064	0.985045	0.492522	−	0.870300i	$-0.336075\pi$
$229$	14.9064	0.985045	0.492522	+	0.870300i	$0.336075\pi$
$230$	6.27894	0.414021
$231$	0	0
$232$	0.669236	0.0439375
$233$	24.9005	1.63129	0.815644	−	0.578554i	$-0.196383\pi$
$233$	24.9005	1.63129	0.815644	+	0.578554i	$0.196383\pi$
$234$	−15.6670	−1.02419
$235$	−5.19617	−0.338961
$236$	14.7458	0.959871
$237$	0.640537	0.0416073
$238$	0	0
$239$	13.2821	0.859149	0.429574	−	0.903031i	$-0.358664\pi$
$239$	13.2821	0.859149	0.429574	+	0.903031i	$0.358664\pi$
$240$	0.0334435	0.00215877
$241$	23.5113	1.51450	0.757249	−	0.653126i	$-0.226543\pi$
$241$	23.5113	1.51450	0.757249	+	0.653126i	$0.226543\pi$
$242$	−0.665518	−0.0427811
$243$	3.19185	0.204757
$244$	9.88596	0.632884
$245$	0	0
$246$	0.699670	0.0446093
$247$	−7.89509	−0.502353
$248$	14.6922	0.932957
$249$	−0.637826	−0.0404206
$250$	0.909782	0.0575396
$251$	4.74538	0.299526	0.149763	−	0.988722i	$-0.452149\pi$
$251$	4.74538	0.299526	0.149763	+	0.988722i	$0.452149\pi$
$252$	0	0
$253$	−23.6388	−1.48616
$254$	17.8220	1.11825
$255$	−0.330276	−0.0206827
$256$	−16.5801	−1.03626
$257$	−10.6488	−0.664253	−0.332127	−	0.943235i	$-0.607766\pi$
$257$	−10.6488	−0.664253	−0.332127	+	0.943235i	$0.607766\pi$
$258$	0.858675	0.0534588
$259$	0	0
$260$	−6.76114	−0.419308
$261$	−0.692366	−0.0428564
$262$	12.0967	0.747339
$263$	4.01470	0.247557	0.123778	−	0.992310i	$-0.460499\pi$
$263$	4.01470	0.247557	0.123778	+	0.992310i	$0.460499\pi$
$264$	−1.17599	−0.0723772
$265$	14.3523	0.881657
$266$	0	0
$267$	1.56503	0.0957783
$268$	5.03113	0.307325
$269$	8.20712	0.500397	0.250198	−	0.968195i	$-0.419504\pi$
$269$	8.20712	0.500397	0.250198	+	0.968195i	$0.419504\pi$
$270$	−0.647856	−0.0394272
$271$	−18.1888	−1.10489	−0.552447	−	0.833548i	$-0.686306\pi$
$271$	−18.1888	−1.10489	−0.552447	+	0.833548i	$0.686306\pi$
$272$	−0.780474	−0.0473232
$273$	0	0
$274$	6.41470	0.387526
$275$	−3.42513	−0.206543
$276$	−0.962504	−0.0579359
$277$	−6.56435	−0.394414	−0.197207	−	0.980362i	$-0.563187\pi$
$277$	−6.56435	−0.394414	−0.197207	+	0.980362i	$0.563187\pi$
$278$	−7.84178	−0.470319
$279$	−15.2000	−0.910001
$280$	0	0
$281$	14.9195	0.890024	0.445012	−	0.895525i	$-0.353199\pi$
$281$	14.9195	0.890024	0.445012	+	0.895525i	$0.353199\pi$
$282$	−0.562388	−0.0334897
$283$	−10.6099	−0.630690	−0.315345	−	0.948977i	$-0.602120\pi$
$283$	−10.6099	−0.630690	−0.315345	+	0.948977i	$0.602120\pi$
$284$	14.2415	0.845080
$285$	−0.162851	−0.00964646
$286$	−17.9720	−1.06271
$287$	0	0
$288$	16.4712	0.970577
$289$	−9.29233	−0.546608
$290$	0.210963	0.0123881
$291$	−1.14931	−0.0673738
$292$	11.8422	0.693015
$293$	−7.77341	−0.454127	−0.227064	−	0.973880i	$-0.572913\pi$
$293$	−7.77341	−0.454127	−0.227064	+	0.973880i	$0.572913\pi$
$294$	0	0
$295$	12.5786	0.732352
$296$	2.88610	0.167751
$297$	2.43904	0.141527
$298$	−13.3255	−0.771928
$299$	−39.8044	−2.30195
$300$	−0.139461	−0.00805179
$301$	0	0
$302$	3.69455	0.212597
$303$	−1.60628	−0.0922786
$304$	−0.384833	−0.0220717
$305$	8.43298	0.482871
$306$	7.54165	0.431127
$307$	7.02522	0.400950	0.200475	−	0.979699i	$-0.435751\pi$
$307$	7.02522	0.400950	0.200475	+	0.979699i	$0.435751\pi$
$308$	0	0
$309$	0.130162	0.00740464
$310$	4.63141	0.263047
$311$	32.0804	1.81911	0.909555	−	0.415583i	$-0.136422\pi$
$311$	32.0804	1.81911	0.909555	+	0.415583i	$0.136422\pi$
$312$	−1.98020	−0.112107
$313$	29.4689	1.66568	0.832840	−	0.553513i	$-0.186713\pi$
$313$	29.4689	1.66568	0.832840	+	0.553513i	$0.186713\pi$
$314$	1.95628	0.110400
$315$	0	0
$316$	−6.31199	−0.355077
$317$	−12.3292	−0.692475	−0.346237	−	0.938147i	$-0.612541\pi$
$317$	−12.3292	−0.692475	−0.346237	+	0.938147i	$0.612541\pi$
$318$	1.55337	0.0871087
$319$	−0.794228	−0.0444682
$320$	−5.58100	−0.311987
$321$	−0.704653	−0.0393299
$322$	0	0
$323$	3.80046	0.211463
$324$	−10.4016	−0.577866
$325$	−5.76743	−0.319919
$326$	14.9572	0.828402
$327$	−0.890070	−0.0492210
$328$	−18.6574	−1.03018
$329$	0	0
$330$	−0.370706	−0.0204067
$331$	0.635861	0.0349501	0.0174750	−	0.999847i	$-0.494437\pi$
$331$	0.635861	0.0349501	0.0174750	+	0.999847i	$0.494437\pi$
$332$	6.28528	0.344950
$333$	−2.98585	−0.163623
$334$	−8.60528	−0.470860
$335$	4.29168	0.234480
$336$	0	0
$337$	−16.8590	−0.918369	−0.459184	−	0.888341i	$-0.651858\pi$
$337$	−16.8590	−0.918369	−0.459184	+	0.888341i	$0.651858\pi$
$338$	−18.4351	−1.00274
$339$	2.14466	0.116482
$340$	3.25461	0.176506
$341$	−17.4363	−0.944226
$342$	3.71860	0.201079
$343$	0	0
$344$	−22.8975	−1.23455
$345$	−0.821040	−0.0442033
$346$	−19.7313	−1.06076
$347$	13.9412	0.748402	0.374201	−	0.927348i	$-0.377917\pi$
$347$	13.9412	0.748402	0.374201	+	0.927348i	$0.377917\pi$
$348$	−0.0323386	−0.00173353
$349$	−18.9357	−1.01361	−0.506803	−	0.862062i	$-0.669173\pi$
$349$	−18.9357	−1.01361	−0.506803	+	0.862062i	$0.669173\pi$
$350$	0	0
$351$	4.10699	0.219215
$352$	18.8945	1.00708
$353$	15.4050	0.819928	0.409964	−	0.912102i	$-0.365541\pi$
$353$	15.4050	0.819928	0.409964	+	0.912102i	$0.365541\pi$
$354$	1.36139	0.0723572
$355$	12.1484	0.644770
$356$	−15.4222	−0.817373
$357$	0	0
$358$	−19.7931	−1.04610
$359$	−23.0521	−1.21664	−0.608322	−	0.793690i	$-0.708157\pi$
$359$	−23.0521	−1.21664	−0.608322	+	0.793690i	$0.708157\pi$
$360$	8.61745	0.454179
$361$	−17.1261	−0.901373
$362$	−14.4447	−0.759198
$363$	0.0870238	0.00456756
$364$	0	0
$365$	10.1017	0.528749
$366$	0.912711	0.0477082
$367$	−9.61073	−0.501676	−0.250838	−	0.968029i	$-0.580706\pi$
$367$	−9.61073	−0.501676	−0.250838	+	0.968029i	$0.580706\pi$
$368$	−1.94020	−0.101140
$369$	19.3023	1.00484
$370$	0.909782	0.0472973
$371$	0	0
$372$	−0.709953	−0.0368093
$373$	35.2555	1.82546	0.912731	−	0.408561i	$-0.133969\pi$
$373$	35.2555	1.82546	0.912731	+	0.408561i	$0.133969\pi$
$374$	8.65119	0.447342
$375$	−0.118964	−0.00614327
$376$	14.9966	0.773393
$377$	−1.33737	−0.0688779
$378$	0	0
$379$	−3.05407	−0.156877	−0.0784384	−	0.996919i	$-0.524993\pi$
$379$	−3.05407	−0.156877	−0.0784384	+	0.996919i	$0.524993\pi$
$380$	1.60477	0.0823230
$381$	−2.33043	−0.119391
$382$	4.91232	0.251336
$383$	−1.57600	−0.0805298	−0.0402649	−	0.999189i	$-0.512820\pi$
$383$	−1.57600	−0.0805298	−0.0402649	+	0.999189i	$0.512820\pi$
$384$	0.708476	0.0361543
$385$	0	0
$386$	−16.1006	−0.819498
$387$	23.6888	1.20417
$388$	11.3256	0.574968
$389$	−36.1055	−1.83062	−0.915310	−	0.402749i	$-0.868055\pi$
$389$	−36.1055	−1.83062	−0.915310	+	0.402749i	$0.868055\pi$
$390$	−0.624216	−0.0316084
$391$	19.1607	0.968997
$392$	0	0
$393$	−1.58178	−0.0797904
$394$	12.5622	0.632872
$395$	−5.38429	−0.270913
$396$	−11.9890	−0.602469
$397$	−27.3834	−1.37433	−0.687167	−	0.726500i	$-0.741146\pi$
$397$	−27.3834	−1.37433	−0.687167	+	0.726500i	$0.741146\pi$
$398$	−5.80502	−0.290979
$399$	0	0
$400$	−0.281123	−0.0140562
$401$	−3.26403	−0.162998	−0.0814990	−	0.996673i	$-0.525971\pi$
$401$	−3.26403	−0.162998	−0.0814990	+	0.996673i	$0.525971\pi$
$402$	0.464494	0.0231669
$403$	−29.3602	−1.46253
$404$	15.8287	0.787507
$405$	−8.87283	−0.440894
$406$	0	0
$407$	−3.42513	−0.169777
$408$	0.953209	0.0471909
$409$	4.83761	0.239204	0.119602	−	0.992822i	$-0.461838\pi$
$409$	4.83761	0.239204	0.119602	+	0.992822i	$0.461838\pi$
$410$	−5.88136	−0.290460
$411$	−0.838793	−0.0413746
$412$	−1.28264	−0.0631913
$413$	0	0
$414$	18.7480	0.921412
$415$	5.36151	0.263186
$416$	31.8157	1.55989
$417$	1.02540	0.0502140
$418$	4.26569	0.208642
$419$	−19.0158	−0.928982	−0.464491	−	0.885578i	$-0.653763\pi$
$419$	−19.0158	−0.928982	−0.464491	+	0.885578i	$0.653763\pi$
$420$	0	0
$421$	5.11929	0.249499	0.124749	−	0.992188i	$-0.460187\pi$
$421$	5.11929	0.249499	0.124749	+	0.992188i	$0.460187\pi$
$422$	−1.16446	−0.0566849
$423$	−15.5150	−0.754363
$424$	−41.4222	−2.01164
$425$	2.77627	0.134669
$426$	1.31484	0.0637040
$427$	0	0
$428$	6.94381	0.335642
$429$	2.35004	0.113461
$430$	−7.21794	−0.348080
$431$	−35.6342	−1.71644	−0.858219	−	0.513284i	$-0.828429\pi$
$431$	−35.6342	−1.71644	−0.858219	+	0.513284i	$0.828429\pi$
$432$	0.200188	0.00963155
$433$	−6.22688	−0.299245	−0.149622	−	0.988743i	$-0.547806\pi$
$433$	−6.22688	−0.299245	−0.149622	+	0.988743i	$0.547806\pi$
$434$	0	0
$435$	−0.0275857	−0.00132263
$436$	8.77095	0.420052
$437$	9.44765	0.451943
$438$	1.09332	0.0522410
$439$	3.22731	0.154031	0.0770156	−	0.997030i	$-0.475461\pi$
$439$	3.22731	0.154031	0.0770156	+	0.997030i	$0.475461\pi$
$440$	9.88526	0.471261
$441$	0	0
$442$	14.5674	0.692899
$443$	−15.9417	−0.757415	−0.378707	−	0.925517i	$-0.623631\pi$
$443$	−15.9417	−0.757415	−0.378707	+	0.925517i	$0.623631\pi$
$444$	−0.139461	−0.00661853
$445$	−13.1555	−0.623631
$446$	1.18761	0.0562352
$447$	1.74246	0.0824156
$448$	0	0
$449$	8.88574	0.419344	0.209672	−	0.977772i	$-0.432760\pi$
$449$	8.88574	0.419344	0.209672	+	0.977772i	$0.432760\pi$
$450$	2.71647	0.128056
$451$	22.1420	1.04263
$452$	−21.1340	−0.994058
$453$	−0.483103	−0.0226981
$454$	4.10783	0.192790
$455$	0	0
$456$	0.470004	0.0220099
$457$	2.59196	0.121247	0.0606235	−	0.998161i	$-0.480691\pi$
$457$	2.59196	0.121247	0.0606235	+	0.998161i	$0.480691\pi$
$458$	−13.5616	−0.633692
$459$	−1.97698	−0.0922776
$460$	8.09072	0.377232
$461$	−0.947535	−0.0441311	−0.0220655	−	0.999757i	$-0.507024\pi$
$461$	−0.947535	−0.0441311	−0.0220655	+	0.999757i	$0.507024\pi$
$462$	0	0
$463$	−27.6639	−1.28565	−0.642825	−	0.766013i	$-0.722238\pi$
$463$	−27.6639	−1.28565	−0.642825	+	0.766013i	$0.722238\pi$
$464$	−0.0651876	−0.00302626
$465$	−0.605608	−0.0280844
$466$	−22.6541	−1.04943
$467$	41.0432	1.89925	0.949627	−	0.313381i	$-0.101462\pi$
$467$	41.0432	1.89925	0.949627	+	0.313381i	$0.101462\pi$
$468$	−20.1877	−0.933179
$469$	0	0
$470$	4.72738	0.218058
$471$	−0.255806	−0.0117869
$472$	−36.3030	−1.67098
$473$	27.1740	1.24946
$474$	−0.582748	−0.0267665
$475$	1.36891	0.0628099
$476$	0	0
$477$	42.8539	1.96214
$478$	−12.0838	−0.552701
$479$	26.7087	1.22035	0.610176	−	0.792266i	$-0.291099\pi$
$479$	26.7087	1.22035	0.610176	+	0.792266i	$0.291099\pi$
$480$	0.656257	0.0299539
$481$	−5.76743	−0.262972
$482$	−21.3902	−0.974296
$483$	0	0
$484$	−0.857552	−0.0389796
$485$	9.66100	0.438683
$486$	−2.90388	−0.131723
$487$	26.3740	1.19512	0.597561	−	0.801824i	$-0.296137\pi$
$487$	26.3740	1.19512	0.597561	+	0.801824i	$0.296137\pi$
$488$	−24.3384	−1.10175
$489$	−1.95582	−0.0884451
$490$	0	0
$491$	−5.68407	−0.256518	−0.128259	−	0.991741i	$-0.540939\pi$
$491$	−5.68407	−0.256518	−0.128259	+	0.991741i	$0.540939\pi$
$492$	0.901559	0.0406454
$493$	0.643769	0.0289939
$494$	7.18281	0.323170
$495$	−10.2269	−0.459666
$496$	−1.43111	−0.0642588
$497$	0	0
$498$	0.580282	0.0260031
$499$	−16.8859	−0.755917	−0.377959	−	0.925822i	$-0.623374\pi$
$499$	−16.8859	−0.755917	−0.377959	+	0.925822i	$0.623374\pi$
$500$	1.17230	0.0524267
$501$	1.12523	0.0502718
$502$	−4.31726	−0.192689
$503$	34.0097	1.51642	0.758210	−	0.652011i	$-0.226074\pi$
$503$	34.0097	1.51642	0.758210	+	0.652011i	$0.226074\pi$
$504$	0	0
$505$	13.5023	0.600844
$506$	21.5062	0.956066
$507$	2.41060	0.107058
$508$	22.9646	1.01889
$509$	31.1895	1.38245	0.691225	−	0.722639i	$-0.257071\pi$
$509$	31.1895	1.38245	0.691225	+	0.722639i	$0.257071\pi$
$510$	0.300479	0.0133054
$511$	0	0
$512$	3.17350	0.140250
$513$	−0.974801	−0.0430385
$514$	9.68807	0.427323
$515$	−1.09413	−0.0482130
$516$	1.10644	0.0487085
$517$	−17.7975	−0.782735
$518$	0	0
$519$	2.58008	0.113253
$520$	16.6454	0.729948
$521$	−23.7071	−1.03863	−0.519313	−	0.854584i	$-0.673812\pi$
$521$	−23.7071	−1.03863	−0.519313	+	0.854584i	$0.673812\pi$
$522$	0.629902	0.0275701
$523$	28.7392	1.25668	0.628339	−	0.777939i	$-0.283735\pi$
$523$	28.7392	1.25668	0.628339	+	0.777939i	$0.283735\pi$
$524$	15.5872	0.680932
$525$	0	0
$526$	−3.65250	−0.159256
$527$	14.1331	0.615648
$528$	0.114548	0.00498508
$529$	24.6319	1.07095
$530$	−13.0575	−0.567181
$531$	37.5577	1.62986
$532$	0	0
$533$	37.2841	1.61495
$534$	−1.42384	−0.0616154
$535$	5.92325	0.256084
$536$	−12.3862	−0.535003
$537$	2.58816	0.111688
$538$	−7.46668	−0.321912
$539$	0	0
$540$	−0.834793	−0.0359238
$541$	39.4701	1.69695	0.848476	−	0.529234i	$-0.177520\pi$
$541$	39.4701	1.69695	0.848476	+	0.529234i	$0.177520\pi$
$542$	16.5479	0.710792
$543$	1.88881	0.0810564
$544$	−15.3151	−0.656630
$545$	7.48184	0.320487
$546$	0	0
$547$	44.0951	1.88537	0.942686	−	0.333682i	$-0.108291\pi$
$547$	44.0951	1.88537	0.942686	+	0.333682i	$0.108291\pi$
$548$	8.26565	0.353091
$549$	25.1796	1.07464
$550$	3.11612	0.132872
$551$	0.317426	0.0135228
$552$	2.36960	0.100857
$553$	0	0
$554$	5.97213	0.253731
$555$	−0.118964	−0.00504974
$556$	−10.1045	−0.428527
$557$	−30.0805	−1.27455	−0.637275	−	0.770637i	$-0.719938\pi$
$557$	−30.0805	−1.27455	−0.637275	+	0.770637i	$0.719938\pi$
$558$	13.8287	0.585415
$559$	45.7571	1.93532
$560$	0	0
$561$	−1.13124	−0.0477609
$562$	−13.5735	−0.572564
$563$	−18.7236	−0.789106	−0.394553	−	0.918873i	$-0.629100\pi$
$563$	−18.7236	−0.789106	−0.394553	+	0.918873i	$0.629100\pi$
$564$	−0.724664	−0.0305138
$565$	−18.0278	−0.758436
$566$	9.65265	0.405731
$567$	0	0
$568$	−35.0615	−1.47115
$569$	27.5561	1.15521	0.577605	−	0.816316i	$-0.303987\pi$
$569$	27.5561	1.15521	0.577605	+	0.816316i	$0.303987\pi$
$570$	0.148159	0.00620569
$571$	−14.5930	−0.610698	−0.305349	−	0.952240i	$-0.598773\pi$
$571$	−14.5930	−0.610698	−0.305349	+	0.952240i	$0.598773\pi$
$572$	−23.1578	−0.968276
$573$	−0.642340	−0.0268341
$574$	0	0
$575$	6.90159	0.287816
$576$	−16.6640	−0.694334
$577$	23.9266	0.996077	0.498039	−	0.867155i	$-0.334054\pi$
$577$	23.9266	0.996077	0.498039	+	0.867155i	$0.334054\pi$
$578$	8.45399	0.351640
$579$	2.10533	0.0874945
$580$	0.271835	0.0112874
$581$	0	0
$582$	1.04562	0.0433424
$583$	49.1586	2.03594
$584$	−29.1546	−1.20643
$585$	−17.2207	−0.711987
$586$	7.07211	0.292146
$587$	−35.2937	−1.45673	−0.728364	−	0.685190i	$-0.759719\pi$
$587$	−35.2937	−1.45673	−0.728364	+	0.685190i	$0.759719\pi$
$588$	0	0
$589$	6.96869	0.287140
$590$	−11.4437	−0.471131
$591$	−1.64264	−0.0675692
$592$	−0.281123	−0.0115541
$593$	26.8803	1.10384	0.551921	−	0.833897i	$-0.313895\pi$
$593$	26.8803	1.10384	0.551921	+	0.833897i	$0.313895\pi$
$594$	−2.21899	−0.0910462
$595$	0	0
$596$	−17.1706	−0.703335
$597$	0.759070	0.0310667
$598$	36.2133	1.48087
$599$	35.4368	1.44791	0.723953	−	0.689849i	$-0.242323\pi$
$599$	35.4368	1.44791	0.723953	+	0.689849i	$0.242323\pi$
$600$	0.343342	0.0140169
$601$	38.2283	1.55936	0.779682	−	0.626175i	$-0.215380\pi$
$601$	38.2283	1.55936	0.779682	+	0.626175i	$0.215380\pi$
$602$	0	0
$603$	12.8143	0.521839
$604$	4.76060	0.193706
$605$	−0.731514	−0.0297403
$606$	1.46137	0.0593640
$607$	26.6899	1.08331	0.541656	−	0.840601i	$-0.317798\pi$
$607$	26.6899	1.08331	0.541656	+	0.840601i	$0.317798\pi$
$608$	−7.55150	−0.306254
$609$	0	0
$610$	−7.67217	−0.310637
$611$	−29.9685	−1.21240
$612$	9.71778	0.392818
$613$	−38.3787	−1.55010	−0.775050	−	0.631900i	$-0.782275\pi$
$613$	−38.3787	−1.55010	−0.775050	+	0.631900i	$0.782275\pi$
$614$	−6.39142	−0.257937
$615$	0.769053	0.0310112
$616$	0	0
$617$	−6.74256	−0.271445	−0.135723	−	0.990747i	$-0.543336\pi$
$617$	−6.74256	−0.271445	−0.135723	+	0.990747i	$0.543336\pi$
$618$	−0.118419	−0.00476350
$619$	31.6835	1.27347	0.636733	−	0.771084i	$-0.280285\pi$
$619$	31.6835	1.27347	0.636733	+	0.771084i	$0.280285\pi$
$620$	5.96780	0.239673
$621$	−4.91462	−0.197217
$622$	−29.1861	−1.17026
$623$	0	0
$624$	0.192883	0.00772151
$625$	1.00000	0.0400000
$626$	−26.8103	−1.07155
$627$	−0.557786	−0.0222758
$628$	2.52077	0.100590
$629$	2.77627	0.110697
$630$	0	0
$631$	−32.2900	−1.28544	−0.642722	−	0.766099i	$-0.722195\pi$
$631$	−32.2900	−1.28544	−0.642722	+	0.766099i	$0.722195\pi$
$632$	15.5396	0.618132
$633$	0.152266	0.00605202
$634$	11.2168	0.445478
$635$	19.5894	0.777380
$636$	2.00159	0.0793683
$637$	0	0
$638$	0.722574	0.0286070
$639$	36.2733	1.43495
$640$	−5.95538	−0.235407
$641$	−4.80164	−0.189653	−0.0948267	−	0.995494i	$-0.530230\pi$
$641$	−4.80164	−0.189653	−0.0948267	+	0.995494i	$0.530230\pi$
$642$	0.641080	0.0253014
$643$	40.8091	1.60935	0.804677	−	0.593713i	$-0.202338\pi$
$643$	40.8091	1.60935	0.804677	+	0.593713i	$0.202338\pi$
$644$	0	0
$645$	0.943825	0.0371631
$646$	−3.45759	−0.136037
$647$	−44.2759	−1.74067	−0.870333	−	0.492464i	$-0.836096\pi$
$647$	−44.2759	−1.74067	−0.870333	+	0.492464i	$0.836096\pi$
$648$	25.6079	1.00597
$649$	43.0832	1.69116
$650$	5.24710	0.205808
$651$	0	0
$652$	19.2731	0.754792
$653$	−48.5592	−1.90027	−0.950134	−	0.311843i	$-0.899054\pi$
$653$	−48.5592	−1.90027	−0.950134	+	0.311843i	$0.899054\pi$
$654$	0.809769	0.0316645
$655$	13.2963	0.519530
$656$	1.81735	0.0709554
$657$	30.1623	1.17674
$658$	0	0
$659$	−24.9277	−0.971044	−0.485522	−	0.874224i	$-0.661370\pi$
$659$	−24.9277	−0.971044	−0.485522	+	0.874224i	$0.661370\pi$
$660$	−0.477673	−0.0185934
$661$	31.0666	1.20835	0.604175	−	0.796851i	$-0.293503\pi$
$661$	31.0666	1.20835	0.604175	+	0.796851i	$0.293503\pi$
$662$	−0.578495	−0.0224838
$663$	−1.90484	−0.0739780
$664$	−15.4738	−0.600501
$665$	0	0
$666$	2.71647	0.105261
$667$	1.60036	0.0619661
$668$	−11.0883	−0.429020
$669$	−0.155294	−0.00600400
$670$	−3.90450	−0.150844
$671$	28.8840	1.11506
$672$	0	0
$673$	14.7397	0.568175	0.284088	−	0.958798i	$-0.408309\pi$
$673$	14.7397	0.568175	0.284088	+	0.958798i	$0.408309\pi$
$674$	15.3380	0.590798
$675$	−0.712100	−0.0274087
$676$	−23.7546	−0.913637
$677$	18.5304	0.712180	0.356090	−	0.934452i	$-0.384110\pi$
$677$	18.5304	0.712180	0.356090	+	0.934452i	$0.384110\pi$
$678$	−1.95117	−0.0749343
$679$	0	0
$680$	−8.01258	−0.307269
$681$	−0.537144	−0.0205834
$682$	15.8632	0.607433
$683$	−11.1961	−0.428407	−0.214204	−	0.976789i	$-0.568716\pi$
$683$	−11.1961	−0.428407	−0.214204	+	0.976789i	$0.568716\pi$
$684$	4.79160	0.183211
$685$	7.05082	0.269398
$686$	0	0
$687$	1.77333	0.0676567
$688$	2.23035	0.0850313
$689$	82.7761	3.15352
$690$	0.746967	0.0284366
$691$	21.3612	0.812617	0.406309	−	0.913736i	$-0.366816\pi$
$691$	21.3612	0.812617	0.406309	+	0.913736i	$0.366816\pi$
$692$	−25.4247	−0.966502
$693$	0	0
$694$	−12.6834	−0.481456
$695$	−8.61941	−0.326953
$696$	0.0796150	0.00301780
$697$	−17.9474	−0.679807
$698$	17.2274	0.652066
$699$	2.96227	0.112043
$700$	0	0
$701$	−46.1611	−1.74348	−0.871739	−	0.489970i	$-0.837008\pi$
$701$	−46.1611	−1.74348	−0.871739	+	0.489970i	$0.837008\pi$
$702$	−3.73646	−0.141024
$703$	1.36891	0.0516294
$704$	−19.1156	−0.720448
$705$	−0.618157	−0.0232811
$706$	−14.0152	−0.527470
$707$	0	0
$708$	1.75422	0.0659277
$709$	8.31765	0.312376	0.156188	−	0.987727i	$-0.450079\pi$
$709$	8.31765	0.312376	0.156188	+	0.987727i	$0.450079\pi$
$710$	−11.0524	−0.414789
$711$	−16.0767	−0.602922
$712$	37.9681	1.42291
$713$	35.1338	1.31577
$714$	0	0
$715$	−19.7542	−0.738765
$716$	−25.5043	−0.953142
$717$	1.58009	0.0590097
$718$	20.9724	0.782682
$719$	−41.1217	−1.53358	−0.766790	−	0.641898i	$-0.778147\pi$
$719$	−41.1217	−1.53358	−0.766790	+	0.641898i	$0.778147\pi$
$720$	−0.839391	−0.0312823
$721$	0	0
$722$	15.5810	0.579865
$723$	2.79700	0.104022
$724$	−18.6127	−0.691736
$725$	0.231883	0.00861190
$726$	−0.0791726	−0.00293837
$727$	26.2779	0.974592	0.487296	−	0.873237i	$-0.337983\pi$
$727$	26.2779	0.974592	0.487296	+	0.873237i	$0.337983\pi$
$728$	0	0
$729$	−26.2388	−0.971806
$730$	−9.19038	−0.340151
$731$	−22.0261	−0.814665
$732$	1.17607	0.0434689
$733$	18.0969	0.668424	0.334212	−	0.942498i	$-0.391530\pi$
$733$	18.0969	0.668424	0.334212	+	0.942498i	$0.391530\pi$
$734$	8.74367	0.322735
$735$	0	0
$736$	−38.0722	−1.40336
$737$	14.6996	0.541466
$738$	−17.5608	−0.646424
$739$	−16.3375	−0.600986	−0.300493	−	0.953784i	$-0.597151\pi$
$739$	−16.3375	−0.600986	−0.300493	+	0.953784i	$0.597151\pi$
$740$	1.17230	0.0430945
$741$	−0.939232	−0.0345035
$742$	0	0
$743$	4.16469	0.152788	0.0763938	−	0.997078i	$-0.475659\pi$
$743$	4.16469	0.152788	0.0763938	+	0.997078i	$0.475659\pi$
$744$	1.74784	0.0640791
$745$	−14.6470	−0.536623
$746$	−32.0748	−1.17434
$747$	16.0086	0.585725
$748$	11.1475	0.407592
$749$	0	0
$750$	0.108231	0.00395205
$751$	−5.35719	−0.195487	−0.0977433	−	0.995212i	$-0.531162\pi$
$751$	−5.35719	−0.195487	−0.0977433	+	0.995212i	$0.531162\pi$
$752$	−1.46076	−0.0532686
$753$	0.564530	0.0205726
$754$	1.21671	0.0443100
$755$	4.06092	0.147792
$756$	0	0
$757$	14.4682	0.525857	0.262928	−	0.964815i	$-0.415312\pi$
$757$	14.4682	0.525857	0.262928	+	0.964815i	$0.415312\pi$
$758$	2.77853	0.100921
$759$	−2.81217	−0.102075
$760$	−3.95081	−0.143311
$761$	−0.0352578	−0.00127809	−0.000639047	−	1.00000i	$-0.500203\pi$
$761$	−0.0352578	−0.00127809	−0.000639047		1.00000i	$0.500203\pi$
$762$	2.12018	0.0768060
$763$	0	0
$764$	6.32976	0.229003
$765$	8.28952	0.299708
$766$	1.43382	0.0518058
$767$	72.5460	2.61948
$768$	−1.97243	−0.0711741
$769$	−17.0129	−0.613502	−0.306751	−	0.951790i	$-0.599242\pi$
$769$	−17.0129	−0.613502	−0.306751	+	0.951790i	$0.599242\pi$
$770$	0	0
$771$	−1.26682	−0.0456235
$772$	−20.7464	−0.746679
$773$	8.57388	0.308381	0.154191	−	0.988041i	$-0.450723\pi$
$773$	8.57388	0.308381	0.154191	+	0.988041i	$0.450723\pi$
$774$	−21.5517	−0.774659
$775$	5.09069	0.182863
$776$	−27.8826	−1.00093
$777$	0	0
$778$	32.8481	1.17766
$779$	−8.84944	−0.317064
$780$	−0.804333	−0.0287997
$781$	41.6098	1.48892
$782$	−17.4320	−0.623368
$783$	−0.165124	−0.00590104
$784$	0	0
$785$	2.15028	0.0767468
$786$	1.43908	0.0513302
$787$	38.3303	1.36633	0.683164	−	0.730265i	$-0.260604\pi$
$787$	38.3303	1.36633	0.683164	+	0.730265i	$0.260604\pi$
$788$	16.1869	0.576636
$789$	0.477604	0.0170032
$790$	4.89853	0.174282
$791$	0	0
$792$	29.5159	1.04880
$793$	48.6366	1.72714
$794$	24.9129	0.884126
$795$	1.70741	0.0605556
$796$	−7.48005	−0.265123
$797$	5.49628	0.194688	0.0973441	−	0.995251i	$-0.468965\pi$
$797$	5.49628	0.194688	0.0973441	+	0.995251i	$0.468965\pi$
$798$	0	0
$799$	14.4260	0.510354
$800$	−5.51643	−0.195035
$801$	−39.2803	−1.38790
$802$	2.96956	0.104859
$803$	34.5998	1.22100
$804$	0.598523	0.0211083
$805$	0	0
$806$	26.7113	0.940867
$807$	0.976351	0.0343692
$808$	−38.9689	−1.37092
$809$	10.7887	0.379310	0.189655	−	0.981851i	$-0.439263\pi$
$809$	10.7887	0.379310	0.189655	+	0.981851i	$0.439263\pi$
$810$	8.07234	0.283633
$811$	−10.2753	−0.360813	−0.180407	−	0.983592i	$-0.557741\pi$
$811$	−10.2753	−0.360813	−0.180407	+	0.983592i	$0.557741\pi$
$812$	0	0
$813$	−2.16382	−0.0758883
$814$	3.11612	0.109220
$815$	16.4404	0.575883
$816$	−0.0928483	−0.00325034
$817$	−10.8605	−0.379962
$818$	−4.40116	−0.153883
$819$	0	0
$820$	−7.57842	−0.264650
$821$	−16.7294	−0.583859	−0.291929	−	0.956440i	$-0.594297\pi$
$821$	−16.7294	−0.583859	−0.291929	+	0.956440i	$0.594297\pi$
$822$	0.763118	0.0266168
$823$	−1.86444	−0.0649903	−0.0324952	−	0.999472i	$-0.510345\pi$
$823$	−1.86444	−0.0649903	−0.0324952	+	0.999472i	$0.510345\pi$
$824$	3.15776	0.110006
$825$	−0.407467	−0.0141862
$826$	0	0
$827$	−9.92220	−0.345029	−0.172514	−	0.985007i	$-0.555189\pi$
$827$	−9.92220	−0.345029	−0.172514	+	0.985007i	$0.555189\pi$
$828$	24.1576	0.839536
$829$	−15.6383	−0.543139	−0.271569	−	0.962419i	$-0.587543\pi$
$829$	−15.6383	−0.543139	−0.271569	+	0.962419i	$0.587543\pi$
$830$	−4.87780	−0.169311
$831$	−0.780921	−0.0270899
$832$	−32.1880	−1.11592
$833$	0	0
$834$	−0.932890	−0.0323033
$835$	−9.45862	−0.327329
$836$	5.49655	0.190102
$837$	−3.62508	−0.125301
$838$	17.3002	0.597626
$839$	23.0045	0.794204	0.397102	−	0.917774i	$-0.370016\pi$
$839$	23.0045	0.794204	0.397102	+	0.917774i	$0.370016\pi$
$840$	0	0
$841$	−28.9462	−0.998146
$842$	−4.65743	−0.160506
$843$	1.77489	0.0611303
$844$	−1.50046	−0.0516480
$845$	−20.2633	−0.697077
$846$	14.1152	0.485292
$847$	0	0
$848$	4.03477	0.138555
$849$	−1.26219	−0.0433183
$850$	−2.52580	−0.0866342
$851$	6.90159	0.236583
$852$	1.69423	0.0580434
$853$	31.4272	1.07605	0.538024	−	0.842930i	$-0.319171\pi$
$853$	31.4272	1.07605	0.538024	+	0.842930i	$0.319171\pi$
$854$	0	0
$855$	4.08736	0.139785
$856$	−17.0951	−0.584298
$857$	40.0780	1.36904	0.684519	−	0.728995i	$-0.260012\pi$
$857$	40.0780	1.36904	0.684519	+	0.728995i	$0.260012\pi$
$858$	−2.13802	−0.0729908
$859$	−9.91020	−0.338132	−0.169066	−	0.985605i	$-0.554075\pi$
$859$	−9.91020	−0.338132	−0.169066	+	0.985605i	$0.554075\pi$
$860$	−9.30067	−0.317150
$861$	0	0
$862$	32.4193	1.10421
$863$	−8.09458	−0.275543	−0.137771	−	0.990464i	$-0.543994\pi$
$863$	−8.09458	−0.275543	−0.137771	+	0.990464i	$0.543994\pi$
$864$	3.92825	0.133642
$865$	−21.6879	−0.737411
$866$	5.66510	0.192508
$867$	−1.10545	−0.0375431
$868$	0	0
$869$	−18.4419	−0.625599
$870$	0.0250969	0.000850866 0
$871$	24.7520	0.838689
$872$	−21.5933	−0.731242
$873$	28.8463	0.976298
$874$	−8.59530	−0.290740
$875$	0	0
$876$	1.40880	0.0475990
$877$	12.4206	0.419415	0.209707	−	0.977764i	$-0.432749\pi$
$877$	12.4206	0.419415	0.209707	+	0.977764i	$0.432749\pi$
$878$	−2.93615	−0.0990902
$879$	−0.924756	−0.0311912
$880$	−0.962884	−0.0324588
$881$	−44.0888	−1.48539	−0.742695	−	0.669630i	$-0.766453\pi$
$881$	−44.0888	−1.48539	−0.742695	+	0.669630i	$0.766453\pi$
$882$	0	0
$883$	−15.2448	−0.513030	−0.256515	−	0.966540i	$-0.582574\pi$
$883$	−15.2448	−0.513030	−0.256515	+	0.966540i	$0.582574\pi$
$884$	18.7708	0.631329
$885$	1.49640	0.0503008
$886$	14.5035	0.487254
$887$	14.0153	0.470588	0.235294	−	0.971924i	$-0.424395\pi$
$887$	14.0153	0.470588	0.235294	+	0.971924i	$0.424395\pi$
$888$	0.343342	0.0115218
$889$	0	0
$890$	11.9686	0.401190
$891$	−30.3906	−1.01812
$892$	1.53030	0.0512382
$893$	7.11309	0.238030
$894$	−1.58526	−0.0530190
$895$	−21.7559	−0.727218
$896$	0	0
$897$	−4.73529	−0.158107
$898$	−8.08408	−0.269769
$899$	1.18044	0.0393699
$900$	3.50030	0.116677
$901$	−39.8459	−1.32746
$902$	−20.1444	−0.670736
$903$	0	0
$904$	52.0301	1.73049
$905$	−15.8771	−0.527774
$906$	0.439518	0.0146020
$907$	0.378018	0.0125519	0.00627594	−	0.999980i	$-0.498002\pi$
$907$	0.378018	0.0125519	0.00627594	+	0.999980i	$0.498002\pi$
$908$	5.29314	0.175659
$909$	40.3157	1.33719
$910$	0	0
$911$	−6.47689	−0.214589	−0.107295	−	0.994227i	$-0.534219\pi$
$911$	−6.47689	−0.214589	−0.107295	+	0.994227i	$0.534219\pi$
$912$	−0.0457812	−0.00151597
$913$	18.3639	0.607755
$914$	−2.35812	−0.0779997
$915$	1.00322	0.0331654
$916$	−17.4748	−0.577383
$917$	0	0
$918$	1.79862	0.0593633
$919$	−25.6946	−0.847588	−0.423794	−	0.905759i	$-0.639302\pi$
$919$	−25.6946	−0.847588	−0.423794	+	0.905759i	$0.639302\pi$
$920$	−19.9187	−0.656699
$921$	0.835748	0.0275388
$922$	0.862050	0.0283901
$923$	70.0650	2.30622
$924$	0	0
$925$	1.00000	0.0328798
$926$	25.1681	0.827075
$927$	−3.26690	−0.107299
$928$	−1.27917	−0.0419907
$929$	−41.0136	−1.34561	−0.672806	−	0.739819i	$-0.734911\pi$
$929$	−41.0136	−1.34561	−0.672806	+	0.739819i	$0.734911\pi$
$930$	0.550971	0.0180671
$931$	0	0
$932$	−29.1908	−0.956178
$933$	3.81641	0.124944
$934$	−37.3404	−1.22181
$935$	9.50908	0.310980
$936$	49.7005	1.62451
$937$	−26.9690	−0.881038	−0.440519	−	0.897743i	$-0.645205\pi$
$937$	−26.9690	−0.881038	−0.440519	+	0.897743i	$0.645205\pi$
$938$	0	0
$939$	3.50574	0.114405
$940$	6.09145	0.198681
$941$	−46.2711	−1.50839	−0.754197	−	0.656648i	$-0.771974\pi$
$941$	−46.2711	−1.50839	−0.754197	+	0.656648i	$0.771974\pi$
$942$	0.232727	0.00758267
$943$	−44.6159	−1.45289
$944$	3.53613	0.115091
$945$	0	0
$946$	−24.7224	−0.803794
$947$	−49.6807	−1.61440	−0.807202	−	0.590275i	$-0.799019\pi$
$947$	−49.6807	−1.61440	−0.807202	+	0.590275i	$0.799019\pi$
$948$	−0.750900	−0.0243881
$949$	58.2611	1.89123
$950$	−1.24541	−0.0404064
$951$	−1.46673	−0.0475619
$952$	0	0
$953$	−9.65313	−0.312696	−0.156348	−	0.987702i	$-0.549972\pi$
$953$	−9.65313	−0.312696	−0.156348	+	0.987702i	$0.549972\pi$
$954$	−38.9877	−1.26227
$955$	5.39945	0.174722
$956$	−15.5706	−0.503589
$957$	−0.0944845	−0.00305425
$958$	−24.2991	−0.785068
$959$	0	0
$960$	−0.663938	−0.0214285
$961$	−5.08492	−0.164030
$962$	5.24710	0.169173
$963$	17.6859	0.569921
$964$	−27.5623	−0.887722
$965$	−17.6972	−0.569693
$966$	0	0
$967$	54.6499	1.75742	0.878711	−	0.477354i	$-0.158404\pi$
$967$	54.6499	1.75742	0.878711	+	0.477354i	$0.158404\pi$
$968$	2.11122	0.0678572
$969$	0.452118	0.0145241
$970$	−8.78940	−0.282210
$971$	−4.72360	−0.151588	−0.0757938	−	0.997124i	$-0.524149\pi$
$971$	−4.72360	−0.151588	−0.0757938	+	0.997124i	$0.524149\pi$
$972$	−3.74179	−0.120018
$973$	0	0
$974$	−23.9946	−0.768837
$975$	−0.686116	−0.0219733
$976$	2.37071	0.0758845
$977$	−7.11372	−0.227588	−0.113794	−	0.993504i	$-0.536300\pi$
$977$	−7.11372	−0.227588	−0.113794	+	0.993504i	$0.536300\pi$
$978$	1.77937	0.0568979
$979$	−45.0593	−1.44010
$980$	0	0
$981$	22.3396	0.713250
$982$	5.17126	0.165022
$983$	6.02200	0.192072	0.0960359	−	0.995378i	$-0.469384\pi$
$983$	6.02200	0.192072	0.0960359	+	0.995378i	$0.469384\pi$
$984$	−2.21956	−0.0707570
$985$	13.8079	0.439956
$986$	−0.585689	−0.0186521
$987$	0	0
$988$	9.25540	0.294453
$989$	−54.7552	−1.74111
$990$	9.30426	0.295709
$991$	−10.1242	−0.321607	−0.160804	−	0.986986i	$-0.551409\pi$
$991$	−10.1242	−0.321607	−0.160804	+	0.986986i	$0.551409\pi$
$992$	−28.0824	−0.891618
$993$	0.0756446	0.00240051
$994$	0	0
$995$	−6.38067	−0.202281
$996$	0.747722	0.0236925
$997$	17.8669	0.565851	0.282925	−	0.959142i	$-0.408695\pi$
$997$	17.8669	0.565851	0.282925	+	0.959142i	$0.408695\pi$
$998$	15.3625	0.486291
$999$	−0.712100	−0.0225299

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9065.2.a.bc.1.13	yes	34
7.6	odd	2		9065.2.a.bb.1.13	✓	34

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9065.2.a.bb.1.13	✓	34	7.6	odd	2
9065.2.a.bc.1.13	yes	34	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 9065 = 5 \cdot 7^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9065.a (trivial)

\(f(q)\)	\(=\)	\(q-0.909782 q^{2} +0.118964 q^{3} -1.17230 q^{4} -1.00000 q^{5} -0.108231 q^{6} +2.88610 q^{8} -2.98585 q^{9} +0.909782 q^{10} -3.42513 q^{11} -0.139461 q^{12} -5.76743 q^{13} -0.118964 q^{15} -0.281123 q^{16} +2.77627 q^{17} +2.71647 q^{18} +1.36891 q^{19} +1.17230 q^{20} +3.11612 q^{22} +6.90159 q^{23} +0.343342 q^{24} +1.00000 q^{25} +5.24710 q^{26} -0.712100 q^{27} +0.231883 q^{29} +0.108231 q^{30} +5.09069 q^{31} -5.51643 q^{32} -0.407467 q^{33} -2.52580 q^{34} +3.50030 q^{36} +1.00000 q^{37} -1.24541 q^{38} -0.686116 q^{39} -2.88610 q^{40} -6.46459 q^{41} -7.93371 q^{43} +4.01527 q^{44} +2.98585 q^{45} -6.27894 q^{46} +5.19617 q^{47} -0.0334435 q^{48} -0.909782 q^{50} +0.330276 q^{51} +6.76114 q^{52} -14.3523 q^{53} +0.647856 q^{54} +3.42513 q^{55} +0.162851 q^{57} -0.210963 q^{58} -12.5786 q^{59} +0.139461 q^{60} -8.43298 q^{61} -4.63141 q^{62} +5.58100 q^{64} +5.76743 q^{65} +0.370706 q^{66} -4.29168 q^{67} -3.25461 q^{68} +0.821040 q^{69} -12.1484 q^{71} -8.61745 q^{72} -10.1017 q^{73} -0.909782 q^{74} +0.118964 q^{75} -1.60477 q^{76} +0.624216 q^{78} +5.38429 q^{79} +0.281123 q^{80} +8.87283 q^{81} +5.88136 q^{82} -5.36151 q^{83} -2.77627 q^{85} +7.21794 q^{86} +0.0275857 q^{87} -9.88526 q^{88} +13.1555 q^{89} -2.71647 q^{90} -8.09072 q^{92} +0.605608 q^{93} -4.72738 q^{94} -1.36891 q^{95} -0.656257 q^{96} -9.66100 q^{97} +10.2269 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 34 q - 2 q^{2} + 10 q^{3} + 30 q^{4} - 34 q^{5} + 8 q^{6} - 6 q^{8} + 28 q^{9} + 2 q^{10} - 30 q^{11} + 20 q^{12} + 18 q^{13} - 10 q^{15} + 18 q^{16} + 10 q^{17} + 40 q^{19} - 30 q^{20} - 4 q^{22} - 16 q^{23}+ \cdots - 100 q^{99}+O(q^{100}) \) 34 * q - 2 * q^2 + 10 * q^3 + 30 * q^4 - 34 * q^5 + 8 * q^6 - 6 * q^8 + 28 * q^9 + 2 * q^10 - 30 * q^11 + 20 * q^12 + 18 * q^13 - 10 * q^15 + 18 * q^16 + 10 * q^17 + 40 * q^19 - 30 * q^20 - 4 * q^22 - 16 * q^23 + 24 * q^24 + 34 * q^25 + 12 * q^26 + 34 * q^27 - 22 * q^29 - 8 * q^30 + 52 * q^31 + 6 * q^32 + 10 * q^33 + 36 * q^34 + 14 * q^36 + 34 * q^37 + 16 * q^38 - 2 * q^39 + 6 * q^40 + 16 * q^41 - 56 * q^44 - 28 * q^45 - 28 * q^46 + 38 * q^47 + 18 * q^48 - 2 * q^50 - 2 * q^51 + 52 * q^52 + 4 * q^53 + 32 * q^54 + 30 * q^55 - 12 * q^57 - 8 * q^58 + 16 * q^59 - 20 * q^60 + 60 * q^61 - 14 * q^62 - 38 * q^64 - 18 * q^65 + 14 * q^66 + 12 * q^67 + 36 * q^68 + 8 * q^69 - 44 * q^71 - 6 * q^72 + 48 * q^73 - 2 * q^74 + 10 * q^75 + 50 * q^76 + 8 * q^78 - 26 * q^79 - 18 * q^80 - 10 * q^81 + 60 * q^82 + 28 * q^83 - 10 * q^85 - 28 * q^86 + 86 * q^87 + 42 * q^88 - 4 * q^89 - 22 * q^92 + 48 * q^93 + 40 * q^94 - 40 * q^95 + 86 * q^96 + 18 * q^97 - 100 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more