LMFDB - Embedded newform 6008.2.a.e.1.28

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6008,2,Mod(1,6008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6008, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6008 = 2^{3} \cdot 751 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6008.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$47.9741215344$
Analytic rank:	$0$
Dimension:	$50$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.28
Character	$\chi$	$=$	6008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.363113 q^{3} +1.56553 q^{5} +3.79058 q^{7} -2.86815 q^{9} +O(q^{10})$	$q+0.363113 q^{3} +1.56553 q^{5} +3.79058 q^{7} -2.86815 q^{9} -0.247976 q^{11} +4.99282 q^{13} +0.568466 q^{15} +5.01347 q^{17} -8.47134 q^{19} +1.37641 q^{21} +5.06856 q^{23} -2.54911 q^{25} -2.13080 q^{27} +0.226828 q^{29} -3.67858 q^{31} -0.0900433 q^{33} +5.93428 q^{35} +2.82049 q^{37} +1.81296 q^{39} +4.03334 q^{41} +3.40433 q^{43} -4.49018 q^{45} +7.95479 q^{47} +7.36851 q^{49} +1.82046 q^{51} +4.00370 q^{53} -0.388214 q^{55} -3.07606 q^{57} +8.61749 q^{59} -1.44468 q^{61} -10.8720 q^{63} +7.81642 q^{65} +13.6298 q^{67} +1.84046 q^{69} +0.864128 q^{71} -2.13913 q^{73} -0.925615 q^{75} -0.939972 q^{77} -12.9958 q^{79} +7.83072 q^{81} -12.6165 q^{83} +7.84876 q^{85} +0.0823642 q^{87} +11.5379 q^{89} +18.9257 q^{91} -1.33574 q^{93} -13.2622 q^{95} +1.16300 q^{97} +0.711231 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9}+O(q^{10}) $ 50 * q + 6 * q^3 + 23 * q^5 + 12 * q^7 + 56 * q^9	$ 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9} - 5 q^{11} + 36 q^{13} + 5 q^{15} + 14 q^{17} + 9 q^{19} + 30 q^{21} + 3 q^{23} + 71 q^{25} + 24 q^{27} + 61 q^{29} + 27 q^{31} + 24 q^{33} - 7 q^{35} + 56 q^{37} - 2 q^{39} + 10 q^{41} + 19 q^{43} + 76 q^{45} + 3 q^{47} + 82 q^{49} - q^{51} + 56 q^{53} + 7 q^{55} + 35 q^{57} - q^{59} + 67 q^{61} + 25 q^{63} + 27 q^{65} + 46 q^{67} + 68 q^{69} + 4 q^{71} + 62 q^{73} + 27 q^{75} + 71 q^{77} + 7 q^{79} + 74 q^{81} - q^{83} + 72 q^{85} + 25 q^{87} + 19 q^{89} + 45 q^{91} + 72 q^{93} - 24 q^{95} + 81 q^{97} + 16 q^{99}+O(q^{100}) $ 50 * q + 6 * q^3 + 23 * q^5 + 12 * q^7 + 56 * q^9 - 5 * q^11 + 36 * q^13 + 5 * q^15 + 14 * q^17 + 9 * q^19 + 30 * q^21 + 3 * q^23 + 71 * q^25 + 24 * q^27 + 61 * q^29 + 27 * q^31 + 24 * q^33 - 7 * q^35 + 56 * q^37 - 2 * q^39 + 10 * q^41 + 19 * q^43 + 76 * q^45 + 3 * q^47 + 82 * q^49 - q^51 + 56 * q^53 + 7 * q^55 + 35 * q^57 - q^59 + 67 * q^61 + 25 * q^63 + 27 * q^65 + 46 * q^67 + 68 * q^69 + 4 * q^71 + 62 * q^73 + 27 * q^75 + 71 * q^77 + 7 * q^79 + 74 * q^81 - q^83 + 72 * q^85 + 25 * q^87 + 19 * q^89 + 45 * q^91 + 72 * q^93 - 24 * q^95 + 81 * q^97 + 16 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0.363113	0.209644	0.104822	−	0.994491i	$-0.466573\pi$
$3$	0.363113	0.209644	0.104822	+	0.994491i	$0.466573\pi$
$4$	0	0
$5$	1.56553	0.700128	0.350064	−	0.936726i	$-0.386160\pi$
$5$	1.56553	0.700128	0.350064	+	0.936726i	$0.386160\pi$
$6$	0	0
$7$	3.79058	1.43271	0.716353	−	0.697738i	$-0.245810\pi$
$7$	3.79058	1.43271	0.716353	+	0.697738i	$0.245810\pi$
$8$	0	0
$9$	−2.86815	−0.956050
$10$	0	0
$11$	−0.247976	−0.0747675	−0.0373838	−	0.999301i	$-0.511902\pi$
$11$	−0.247976	−0.0747675	−0.0373838	+	0.999301i	$0.511902\pi$
$12$	0	0
$13$	4.99282	1.38476	0.692380	−	0.721533i	$-0.256562\pi$
$13$	4.99282	1.38476	0.692380	+	0.721533i	$0.256562\pi$
$14$	0	0
$15$	0.568466	0.146777
$16$	0	0
$17$	5.01347	1.21595	0.607973	−	0.793958i	$-0.291983\pi$
$17$	5.01347	1.21595	0.607973	+	0.793958i	$0.291983\pi$
$18$	0	0
$19$	−8.47134	−1.94346	−0.971729	−	0.236097i	$-0.924132\pi$
$19$	−8.47134	−1.94346	−0.971729	+	0.236097i	$0.924132\pi$
$20$	0	0
$21$	1.37641	0.300358
$22$	0	0
$23$	5.06856	1.05687	0.528434	−	0.848974i	$-0.322779\pi$
$23$	5.06856	1.05687	0.528434	+	0.848974i	$0.322779\pi$
$24$	0	0
$25$	−2.54911	−0.509821
$26$	0	0
$27$	−2.13080	−0.410073
$28$	0	0
$29$	0.226828	0.0421209	0.0210604	−	0.999778i	$-0.493296\pi$
$29$	0.226828	0.0421209	0.0210604	+	0.999778i	$0.493296\pi$
$30$	0	0
$31$	−3.67858	−0.660693	−0.330347	−	0.943860i	$-0.607166\pi$
$31$	−3.67858	−0.660693	−0.330347	+	0.943860i	$0.607166\pi$
$32$	0	0
$33$	−0.0900433	−0.0156745
$34$	0	0
$35$	5.93428	1.00308
$36$	0	0
$37$	2.82049	0.463686	0.231843	−	0.972753i	$-0.425524\pi$
$37$	2.82049	0.463686	0.231843	+	0.972753i	$0.425524\pi$
$38$	0	0
$39$	1.81296	0.290306
$40$	0	0
$41$	4.03334	0.629902	0.314951	−	0.949108i	$-0.398012\pi$
$41$	4.03334	0.629902	0.314951	+	0.949108i	$0.398012\pi$
$42$	0	0
$43$	3.40433	0.519155	0.259578	−	0.965722i	$-0.416417\pi$
$43$	3.40433	0.519155	0.259578	+	0.965722i	$0.416417\pi$
$44$	0	0
$45$	−4.49018	−0.669357
$46$	0	0
$47$	7.95479	1.16033	0.580163	−	0.814501i	$-0.302989\pi$
$47$	7.95479	1.16033	0.580163	+	0.814501i	$0.302989\pi$
$48$	0	0
$49$	7.36851	1.05264
$50$	0	0
$51$	1.82046	0.254915
$52$	0	0
$53$	4.00370	0.549950	0.274975	−	0.961451i	$-0.411330\pi$
$53$	4.00370	0.549950	0.274975	+	0.961451i	$0.411330\pi$
$54$	0	0
$55$	−0.388214	−0.0523468
$56$	0	0
$57$	−3.07606	−0.407434
$58$	0	0
$59$	8.61749	1.12190	0.560951	−	0.827849i	$-0.310436\pi$
$59$	8.61749	1.12190	0.560951	+	0.827849i	$0.310436\pi$
$60$	0	0
$61$	−1.44468	−0.184972	−0.0924859	−	0.995714i	$-0.529481\pi$
$61$	−1.44468	−0.184972	−0.0924859	+	0.995714i	$0.529481\pi$
$62$	0	0
$63$	−10.8720	−1.36974
$64$	0	0
$65$	7.81642	0.969508
$66$	0	0
$67$	13.6298	1.66514	0.832570	−	0.553920i	$-0.186869\pi$
$67$	13.6298	1.66514	0.832570	+	0.553920i	$0.186869\pi$
$68$	0	0
$69$	1.84046	0.221566
$70$	0	0
$71$	0.864128	0.102553	0.0512766	−	0.998684i	$-0.483671\pi$
$71$	0.864128	0.102553	0.0512766	+	0.998684i	$0.483671\pi$
$72$	0	0
$73$	−2.13913	−0.250366	−0.125183	−	0.992134i	$-0.539952\pi$
$73$	−2.13913	−0.250366	−0.125183	+	0.992134i	$0.539952\pi$
$74$	0	0
$75$	−0.925615	−0.106881
$76$	0	0
$77$	−0.939972	−0.107120
$78$	0	0
$79$	−12.9958	−1.46214	−0.731071	−	0.682302i	$-0.760979\pi$
$79$	−12.9958	−1.46214	−0.731071	+	0.682302i	$0.760979\pi$
$80$	0	0
$81$	7.83072	0.870080
$82$	0	0
$83$	−12.6165	−1.38485	−0.692423	−	0.721492i	$-0.743457\pi$
$83$	−12.6165	−1.38485	−0.692423	+	0.721492i	$0.743457\pi$
$84$	0	0
$85$	7.84876	0.851317
$86$	0	0
$87$	0.0823642	0.00883037
$88$	0	0
$89$	11.5379	1.22301	0.611506	−	0.791240i	$-0.290564\pi$
$89$	11.5379	1.22301	0.611506	+	0.791240i	$0.290564\pi$
$90$	0	0
$91$	18.9257	1.98395
$92$	0	0
$93$	−1.33574	−0.138510
$94$	0	0
$95$	−13.2622	−1.36067
$96$	0	0
$97$	1.16300	0.118084	0.0590422	−	0.998255i	$-0.481195\pi$
$97$	1.16300	0.118084	0.0590422	+	0.998255i	$0.481195\pi$
$98$	0	0
$99$	0.711231	0.0714814
$100$	0	0
$101$	7.04166	0.700671	0.350336	−	0.936624i	$-0.386068\pi$
$101$	7.04166	0.700671	0.350336	+	0.936624i	$0.386068\pi$
$102$	0	0
$103$	−19.0516	−1.87721	−0.938603	−	0.345000i	$-0.887879\pi$
$103$	−19.0516	−1.87721	−0.938603	+	0.345000i	$0.887879\pi$
$104$	0	0
$105$	2.15482	0.210289
$106$	0	0
$107$	15.7332	1.52099	0.760494	−	0.649344i	$-0.224957\pi$
$107$	15.7332	1.52099	0.760494	+	0.649344i	$0.224957\pi$
$108$	0	0
$109$	−7.16256	−0.686049	−0.343025	−	0.939326i	$-0.611451\pi$
$109$	−7.16256	−0.686049	−0.343025	+	0.939326i	$0.611451\pi$
$110$	0	0
$111$	1.02416	0.0972088
$112$	0	0
$113$	−2.68481	−0.252565	−0.126283	−	0.991994i	$-0.540305\pi$
$113$	−2.68481	−0.252565	−0.126283	+	0.991994i	$0.540305\pi$
$114$	0	0
$115$	7.93500	0.739943
$116$	0	0
$117$	−14.3202	−1.32390
$118$	0	0
$119$	19.0040	1.74209
$120$	0	0
$121$	−10.9385	−0.994410
$122$	0	0
$123$	1.46456	0.132055
$124$	0	0
$125$	−11.8184	−1.05707
$126$	0	0
$127$	17.7521	1.57524	0.787622	−	0.616159i	$-0.211312\pi$
$127$	17.7521	1.57524	0.787622	+	0.616159i	$0.211312\pi$
$128$	0	0
$129$	1.23616	0.108838
$130$	0	0
$131$	−9.35653	−0.817483	−0.408742	−	0.912650i	$-0.634032\pi$
$131$	−9.35653	−0.817483	−0.408742	+	0.912650i	$0.634032\pi$
$132$	0	0
$133$	−32.1113	−2.78440
$134$	0	0
$135$	−3.33584	−0.287104
$136$	0	0
$137$	−2.29891	−0.196409	−0.0982046	−	0.995166i	$-0.531310\pi$
$137$	−2.29891	−0.196409	−0.0982046	+	0.995166i	$0.531310\pi$
$138$	0	0
$139$	16.2411	1.37755	0.688775	−	0.724976i	$-0.258149\pi$
$139$	16.2411	1.37755	0.688775	+	0.724976i	$0.258149\pi$
$140$	0	0
$141$	2.88849	0.243255
$142$	0	0
$143$	−1.23810	−0.103535
$144$	0	0
$145$	0.355106	0.0294900
$146$	0	0
$147$	2.67561	0.220680
$148$	0	0
$149$	−9.65723	−0.791151	−0.395576	−	0.918433i	$-0.629455\pi$
$149$	−9.65723	−0.791151	−0.395576	+	0.918433i	$0.629455\pi$
$150$	0	0
$151$	−11.5971	−0.943761	−0.471880	−	0.881663i	$-0.656425\pi$
$151$	−11.5971	−0.943761	−0.471880	+	0.881663i	$0.656425\pi$
$152$	0	0
$153$	−14.3794	−1.16250
$154$	0	0
$155$	−5.75894	−0.462570
$156$	0	0
$157$	15.1889	1.21221	0.606103	−	0.795386i	$-0.292732\pi$
$157$	15.1889	1.21221	0.606103	+	0.795386i	$0.292732\pi$
$158$	0	0
$159$	1.45380	0.115293
$160$	0	0
$161$	19.2128	1.51418
$162$	0	0
$163$	14.2886	1.11917	0.559586	−	0.828772i	$-0.310960\pi$
$163$	14.2886	1.11917	0.559586	+	0.828772i	$0.310960\pi$
$164$	0	0
$165$	−0.140966	−0.0109742
$166$	0	0
$167$	−3.06504	−0.237180	−0.118590	−	0.992943i	$-0.537837\pi$
$167$	−3.06504	−0.237180	−0.118590	+	0.992943i	$0.537837\pi$
$168$	0	0
$169$	11.9283	0.917558
$170$	0	0
$171$	24.2971	1.85804
$172$	0	0
$173$	18.5299	1.40880	0.704400	−	0.709803i	$-0.251216\pi$
$173$	18.5299	1.40880	0.704400	+	0.709803i	$0.251216\pi$
$174$	0	0
$175$	−9.66260	−0.730424
$176$	0	0
$177$	3.12913	0.235199
$178$	0	0
$179$	6.54681	0.489331	0.244666	−	0.969608i	$-0.421322\pi$
$179$	6.54681	0.489331	0.244666	+	0.969608i	$0.421322\pi$
$180$	0	0
$181$	−0.0903538	−0.00671595	−0.00335797	−	0.999994i	$-0.501069\pi$
$181$	−0.0903538	−0.00671595	−0.00335797	+	0.999994i	$0.501069\pi$
$182$	0	0
$183$	−0.524581	−0.0387781
$184$	0	0
$185$	4.41557	0.324639
$186$	0	0
$187$	−1.24322	−0.0909132
$188$	0	0
$189$	−8.07698	−0.587514
$190$	0	0
$191$	−21.8064	−1.57786	−0.788930	−	0.614484i	$-0.789364\pi$
$191$	−21.8064	−1.57786	−0.788930	+	0.614484i	$0.789364\pi$
$192$	0	0
$193$	10.0685	0.724747	0.362373	−	0.932033i	$-0.381967\pi$
$193$	10.0685	0.724747	0.362373	+	0.932033i	$0.381967\pi$
$194$	0	0
$195$	2.83825	0.203251
$196$	0	0
$197$	−2.03221	−0.144789	−0.0723944	−	0.997376i	$-0.523064\pi$
$197$	−2.03221	−0.144789	−0.0723944	+	0.997376i	$0.523064\pi$
$198$	0	0
$199$	13.7343	0.973601	0.486800	−	0.873513i	$-0.338164\pi$
$199$	13.7343	0.973601	0.486800	+	0.873513i	$0.338164\pi$
$200$	0	0
$201$	4.94915	0.349086
$202$	0	0
$203$	0.859809	0.0603468
$204$	0	0
$205$	6.31433	0.441012
$206$	0	0
$207$	−14.5374	−1.01042
$208$	0	0
$209$	2.10069	0.145308
$210$	0	0
$211$	−20.7374	−1.42762	−0.713810	−	0.700339i	$-0.753032\pi$
$211$	−20.7374	−1.42762	−0.713810	+	0.700339i	$0.753032\pi$
$212$	0	0
$213$	0.313776	0.0214996
$214$	0	0
$215$	5.32959	0.363475
$216$	0	0
$217$	−13.9440	−0.946579
$218$	0	0
$219$	−0.776747	−0.0524877
$220$	0	0
$221$	25.0314	1.68379
$222$	0	0
$223$	15.2223	1.01936	0.509681	−	0.860363i	$-0.329763\pi$
$223$	15.2223	1.01936	0.509681	+	0.860363i	$0.329763\pi$
$224$	0	0
$225$	7.31122	0.487415
$226$	0	0
$227$	−20.0482	−1.33064	−0.665322	−	0.746557i	$-0.731706\pi$
$227$	−20.0482	−1.33064	−0.665322	+	0.746557i	$0.731706\pi$
$228$	0	0
$229$	24.2437	1.60207	0.801033	−	0.598620i	$-0.204284\pi$
$229$	24.2437	1.60207	0.801033	+	0.598620i	$0.204284\pi$
$230$	0	0
$231$	−0.341317	−0.0224570
$232$	0	0
$233$	0.653982	0.0428438	0.0214219	−	0.999771i	$-0.493181\pi$
$233$	0.653982	0.0428438	0.0214219	+	0.999771i	$0.493181\pi$
$234$	0	0
$235$	12.4535	0.812376
$236$	0	0
$237$	−4.71895	−0.306529
$238$	0	0
$239$	−5.16420	−0.334044	−0.167022	−	0.985953i	$-0.553415\pi$
$239$	−5.16420	−0.334044	−0.167022	+	0.985953i	$0.553415\pi$
$240$	0	0
$241$	0.716483	0.0461527	0.0230764	−	0.999734i	$-0.492654\pi$
$241$	0.716483	0.0461527	0.0230764	+	0.999734i	$0.492654\pi$
$242$	0	0
$243$	9.23585	0.592480
$244$	0	0
$245$	11.5356	0.736985
$246$	0	0
$247$	−42.2959	−2.69122
$248$	0	0
$249$	−4.58123	−0.290324
$250$	0	0
$251$	−19.9941	−1.26202	−0.631010	−	0.775775i	$-0.717359\pi$
$251$	−19.9941	−1.26202	−0.631010	+	0.775775i	$0.717359\pi$
$252$	0	0
$253$	−1.25688	−0.0790194
$254$	0	0
$255$	2.84999	0.178473
$256$	0	0
$257$	14.8853	0.928520	0.464260	−	0.885699i	$-0.346320\pi$
$257$	14.8853	0.928520	0.464260	+	0.885699i	$0.346320\pi$
$258$	0	0
$259$	10.6913	0.664325
$260$	0	0
$261$	−0.650576	−0.0402696
$262$	0	0
$263$	−18.6297	−1.14876	−0.574379	−	0.818589i	$-0.694756\pi$
$263$	−18.6297	−1.14876	−0.574379	+	0.818589i	$0.694756\pi$
$264$	0	0
$265$	6.26792	0.385035
$266$	0	0
$267$	4.18956	0.256397
$268$	0	0
$269$	7.10569	0.433241	0.216621	−	0.976256i	$-0.430497\pi$
$269$	7.10569	0.433241	0.216621	+	0.976256i	$0.430497\pi$
$270$	0	0
$271$	14.8249	0.900550	0.450275	−	0.892890i	$-0.351326\pi$
$271$	14.8249	0.900550	0.450275	+	0.892890i	$0.351326\pi$
$272$	0	0
$273$	6.87217	0.415923
$274$	0	0
$275$	0.632117	0.0381181
$276$	0	0
$277$	−2.15211	−0.129308	−0.0646539	−	0.997908i	$-0.520594\pi$
$277$	−2.15211	−0.129308	−0.0646539	+	0.997908i	$0.520594\pi$
$278$	0	0
$279$	10.5507	0.631655
$280$	0	0
$281$	−15.9058	−0.948859	−0.474429	−	0.880294i	$-0.657346\pi$
$281$	−15.9058	−0.948859	−0.474429	+	0.880294i	$0.657346\pi$
$282$	0	0
$283$	15.3382	0.911759	0.455880	−	0.890041i	$-0.349325\pi$
$283$	15.3382	0.911759	0.455880	+	0.890041i	$0.349325\pi$
$284$	0	0
$285$	−4.81567	−0.285256
$286$	0	0
$287$	15.2887	0.902464
$288$	0	0
$289$	8.13491	0.478524
$290$	0	0
$291$	0.422299	0.0247556
$292$	0	0
$293$	−32.0061	−1.86981	−0.934907	−	0.354892i	$-0.884517\pi$
$293$	−32.0061	−1.86981	−0.934907	+	0.354892i	$0.884517\pi$
$294$	0	0
$295$	13.4910	0.785474
$296$	0	0
$297$	0.528388	0.0306602
$298$	0	0
$299$	25.3064	1.46351
$300$	0	0
$301$	12.9044	0.743796
$302$	0	0
$303$	2.55692	0.146891
$304$	0	0
$305$	−2.26169	−0.129504
$306$	0	0
$307$	−22.8601	−1.30469	−0.652347	−	0.757920i	$-0.726215\pi$
$307$	−22.8601	−1.30469	−0.652347	+	0.757920i	$0.726215\pi$
$308$	0	0
$309$	−6.91787	−0.393544
$310$	0	0
$311$	−16.4603	−0.933379	−0.466690	−	0.884421i	$-0.654554\pi$
$311$	−16.4603	−0.933379	−0.466690	+	0.884421i	$0.654554\pi$
$312$	0	0
$313$	18.3332	1.03625	0.518127	−	0.855304i	$-0.326630\pi$
$313$	18.3332	1.03625	0.518127	+	0.855304i	$0.326630\pi$
$314$	0	0
$315$	−17.0204	−0.958991
$316$	0	0
$317$	−27.6693	−1.55406	−0.777031	−	0.629463i	$-0.783275\pi$
$317$	−27.6693	−1.55406	−0.777031	+	0.629463i	$0.783275\pi$
$318$	0	0
$319$	−0.0562478	−0.00314927
$320$	0	0
$321$	5.71295	0.318866
$322$	0	0
$323$	−42.4708	−2.36314
$324$	0	0
$325$	−12.7272	−0.705980
$326$	0	0
$327$	−2.60082	−0.143826
$328$	0	0
$329$	30.1533	1.66240
$330$	0	0
$331$	−24.6163	−1.35303	−0.676516	−	0.736427i	$-0.736511\pi$
$331$	−24.6163	−1.35303	−0.676516	+	0.736427i	$0.736511\pi$
$332$	0	0
$333$	−8.08959	−0.443307
$334$	0	0
$335$	21.3378	1.16581
$336$	0	0
$337$	−0.399526	−0.0217636	−0.0108818	−	0.999941i	$-0.503464\pi$
$337$	−0.399526	−0.0217636	−0.0108818	+	0.999941i	$0.503464\pi$
$338$	0	0
$339$	−0.974889	−0.0529487
$340$	0	0
$341$	0.912200	0.0493984
$342$	0	0
$343$	1.39687	0.0754239
$344$	0	0
$345$	2.88130	0.155124
$346$	0	0
$347$	−25.9293	−1.39196	−0.695979	−	0.718062i	$-0.745030\pi$
$347$	−25.9293	−1.39196	−0.695979	+	0.718062i	$0.745030\pi$
$348$	0	0
$349$	30.5889	1.63738	0.818692	−	0.574233i	$-0.194700\pi$
$349$	30.5889	1.63738	0.818692	+	0.574233i	$0.194700\pi$
$350$	0	0
$351$	−10.6387	−0.567853
$352$	0	0
$353$	5.03394	0.267930	0.133965	−	0.990986i	$-0.457229\pi$
$353$	5.03394	0.267930	0.133965	+	0.990986i	$0.457229\pi$
$354$	0	0
$355$	1.35282	0.0718003
$356$	0	0
$357$	6.90060	0.365218
$358$	0	0
$359$	6.39274	0.337396	0.168698	−	0.985668i	$-0.446044\pi$
$359$	6.39274	0.337396	0.168698	+	0.985668i	$0.446044\pi$
$360$	0	0
$361$	52.7636	2.77703
$362$	0	0
$363$	−3.97192	−0.208472
$364$	0	0
$365$	−3.34888	−0.175288
$366$	0	0
$367$	−20.7256	−1.08187	−0.540933	−	0.841066i	$-0.681929\pi$
$367$	−20.7256	−1.08187	−0.540933	+	0.841066i	$0.681929\pi$
$368$	0	0
$369$	−11.5682	−0.602218
$370$	0	0
$371$	15.1763	0.787916
$372$	0	0
$373$	28.7957	1.49099	0.745493	−	0.666513i	$-0.232214\pi$
$373$	28.7957	1.49099	0.745493	+	0.666513i	$0.232214\pi$
$374$	0	0
$375$	−4.29141	−0.221607
$376$	0	0
$377$	1.13251	0.0583273
$378$	0	0
$379$	6.36089	0.326737	0.163368	−	0.986565i	$-0.447764\pi$
$379$	6.36089	0.326737	0.163368	+	0.986565i	$0.447764\pi$
$380$	0	0
$381$	6.44602	0.330240
$382$	0	0
$383$	−22.1432	−1.13146	−0.565731	−	0.824590i	$-0.691406\pi$
$383$	−22.1432	−1.13146	−0.565731	+	0.824590i	$0.691406\pi$
$384$	0	0
$385$	−1.47156	−0.0749975
$386$	0	0
$387$	−9.76412	−0.496338
$388$	0	0
$389$	20.1533	1.02181	0.510906	−	0.859637i	$-0.329310\pi$
$389$	20.1533	1.02181	0.510906	+	0.859637i	$0.329310\pi$
$390$	0	0
$391$	25.4111	1.28509
$392$	0	0
$393$	−3.39748	−0.171380
$394$	0	0
$395$	−20.3453	−1.02369
$396$	0	0
$397$	−7.28166	−0.365456	−0.182728	−	0.983163i	$-0.558493\pi$
$397$	−7.28166	−0.365456	−0.182728	+	0.983163i	$0.558493\pi$
$398$	0	0
$399$	−11.6600	−0.583732
$400$	0	0
$401$	12.1771	0.608096	0.304048	−	0.952657i	$-0.401662\pi$
$401$	12.1771	0.608096	0.304048	+	0.952657i	$0.401662\pi$
$402$	0	0
$403$	−18.3665	−0.914901
$404$	0	0
$405$	12.2593	0.609167
$406$	0	0
$407$	−0.699414	−0.0346686
$408$	0	0
$409$	5.80267	0.286924	0.143462	−	0.989656i	$-0.454177\pi$
$409$	5.80267	0.286924	0.143462	+	0.989656i	$0.454177\pi$
$410$	0	0
$411$	−0.834765	−0.0411759
$412$	0	0
$413$	32.6653	1.60735
$414$	0	0
$415$	−19.7516	−0.969568
$416$	0	0
$417$	5.89735	0.288794
$418$	0	0
$419$	−14.6649	−0.716428	−0.358214	−	0.933639i	$-0.616614\pi$
$419$	−14.6649	−0.716428	−0.358214	+	0.933639i	$0.616614\pi$
$420$	0	0
$421$	19.3955	0.945281	0.472640	−	0.881255i	$-0.343301\pi$
$421$	19.3955	0.945281	0.472640	+	0.881255i	$0.343301\pi$
$422$	0	0
$423$	−22.8155	−1.10933
$424$	0	0
$425$	−12.7799	−0.619915
$426$	0	0
$427$	−5.47616	−0.265010
$428$	0	0
$429$	−0.449570	−0.0217055
$430$	0	0
$431$	−9.54357	−0.459698	−0.229849	−	0.973226i	$-0.573823\pi$
$431$	−9.54357	−0.459698	−0.229849	+	0.973226i	$0.573823\pi$
$432$	0	0
$433$	−30.9789	−1.48875	−0.744374	−	0.667762i	$-0.767252\pi$
$433$	−30.9789	−1.48875	−0.744374	+	0.667762i	$0.767252\pi$
$434$	0	0
$435$	0.128944	0.00618239
$436$	0	0
$437$	−42.9375	−2.05398
$438$	0	0
$439$	33.0681	1.57825	0.789126	−	0.614231i	$-0.210534\pi$
$439$	33.0681	1.57825	0.789126	+	0.614231i	$0.210534\pi$
$440$	0	0
$441$	−21.1340	−1.00638
$442$	0	0
$443$	36.0251	1.71160	0.855802	−	0.517304i	$-0.173064\pi$
$443$	36.0251	1.71160	0.855802	+	0.517304i	$0.173064\pi$
$444$	0	0
$445$	18.0629	0.856265
$446$	0	0
$447$	−3.50667	−0.165860
$448$	0	0
$449$	13.0293	0.614889	0.307445	−	0.951566i	$-0.400526\pi$
$449$	13.0293	0.614889	0.307445	+	0.951566i	$0.400526\pi$
$450$	0	0
$451$	−1.00017	−0.0470962
$452$	0	0
$453$	−4.21107	−0.197853
$454$	0	0
$455$	29.6288	1.38902
$456$	0	0
$457$	−2.29899	−0.107542	−0.0537712	−	0.998553i	$-0.517124\pi$
$457$	−2.29899	−0.107542	−0.0537712	+	0.998553i	$0.517124\pi$
$458$	0	0
$459$	−10.6827	−0.498627
$460$	0	0
$461$	7.38626	0.344012	0.172006	−	0.985096i	$-0.444975\pi$
$461$	7.38626	0.344012	0.172006	+	0.985096i	$0.444975\pi$
$462$	0	0
$463$	−5.79595	−0.269361	−0.134680	−	0.990889i	$-0.543001\pi$
$463$	−5.79595	−0.269361	−0.134680	+	0.990889i	$0.543001\pi$
$464$	0	0
$465$	−2.09115	−0.0969748
$466$	0	0
$467$	1.52157	0.0704100	0.0352050	−	0.999380i	$-0.488792\pi$
$467$	1.52157	0.0704100	0.0352050	+	0.999380i	$0.488792\pi$
$468$	0	0
$469$	51.6647	2.38565
$470$	0	0
$471$	5.51529	0.254131
$472$	0	0
$473$	−0.844191	−0.0388159
$474$	0	0
$475$	21.5944	0.990817
$476$	0	0
$477$	−11.4832	−0.525779
$478$	0	0
$479$	−4.17058	−0.190559	−0.0952794	−	0.995451i	$-0.530374\pi$
$479$	−4.17058	−0.190559	−0.0952794	+	0.995451i	$0.530374\pi$
$480$	0	0
$481$	14.0822	0.642094
$482$	0	0
$483$	6.97642	0.317438
$484$	0	0
$485$	1.82071	0.0826741
$486$	0	0
$487$	36.3937	1.64916	0.824579	−	0.565747i	$-0.191412\pi$
$487$	36.3937	1.64916	0.824579	+	0.565747i	$0.191412\pi$
$488$	0	0
$489$	5.18840	0.234627
$490$	0	0
$491$	−15.5397	−0.701295	−0.350648	−	0.936508i	$-0.614038\pi$
$491$	−15.5397	−0.701295	−0.350648	+	0.936508i	$0.614038\pi$
$492$	0	0
$493$	1.13719	0.0512167
$494$	0	0
$495$	1.11346	0.0500461
$496$	0	0
$497$	3.27555	0.146928
$498$	0	0
$499$	38.9248	1.74251	0.871256	−	0.490829i	$-0.163306\pi$
$499$	38.9248	1.74251	0.871256	+	0.490829i	$0.163306\pi$
$500$	0	0
$501$	−1.11296	−0.0497232
$502$	0	0
$503$	−20.1298	−0.897544	−0.448772	−	0.893646i	$-0.648139\pi$
$503$	−20.1298	−0.897544	−0.448772	+	0.893646i	$0.648139\pi$
$504$	0	0
$505$	11.0239	0.490559
$506$	0	0
$507$	4.33131	0.192360
$508$	0	0
$509$	−0.0456568	−0.00202370	−0.00101185	−	0.999999i	$-0.500322\pi$
$509$	−0.0456568	−0.00202370	−0.00101185	+	0.999999i	$0.500322\pi$
$510$	0	0
$511$	−8.10855	−0.358701
$512$	0	0
$513$	18.0508	0.796961
$514$	0	0
$515$	−29.8258	−1.31428
$516$	0	0
$517$	−1.97259	−0.0867546
$518$	0	0
$519$	6.72844	0.295346
$520$	0	0
$521$	−11.2088	−0.491066	−0.245533	−	0.969388i	$-0.578963\pi$
$521$	−11.2088	−0.491066	−0.245533	+	0.969388i	$0.578963\pi$
$522$	0	0
$523$	0.116293	0.00508516	0.00254258	−	0.999997i	$-0.499191\pi$
$523$	0.116293	0.00508516	0.00254258	+	0.999997i	$0.499191\pi$
$524$	0	0
$525$	−3.50862	−0.153129
$526$	0	0
$527$	−18.4425	−0.803367
$528$	0	0
$529$	2.69032	0.116970
$530$	0	0
$531$	−24.7162	−1.07259
$532$	0	0
$533$	20.1378	0.872263
$534$	0	0
$535$	24.6309	1.06489
$536$	0	0
$537$	2.37723	0.102585
$538$	0	0
$539$	−1.82721	−0.0787036
$540$	0	0
$541$	0.0606811	0.00260889	0.00130444	−	0.999999i	$-0.499585\pi$
$541$	0.0606811	0.00260889	0.00130444	+	0.999999i	$0.499585\pi$
$542$	0	0
$543$	−0.0328087	−0.00140796
$544$	0	0
$545$	−11.2132	−0.480322
$546$	0	0
$547$	−27.6883	−1.18387	−0.591933	−	0.805987i	$-0.701635\pi$
$547$	−27.6883	−1.18387	−0.591933	+	0.805987i	$0.701635\pi$
$548$	0	0
$549$	4.14354	0.176842
$550$	0	0
$551$	−1.92154	−0.0818602
$552$	0	0
$553$	−49.2616	−2.09482
$554$	0	0
$555$	1.60335	0.0680586
$556$	0	0
$557$	42.7445	1.81114	0.905571	−	0.424194i	$-0.139442\pi$
$557$	42.7445	1.81114	0.905571	+	0.424194i	$0.139442\pi$
$558$	0	0
$559$	16.9972	0.718905
$560$	0	0
$561$	−0.451430	−0.0190594
$562$	0	0
$563$	18.5974	0.783787	0.391893	−	0.920011i	$-0.371820\pi$
$563$	18.5974	0.783787	0.391893	+	0.920011i	$0.371820\pi$
$564$	0	0
$565$	−4.20315	−0.176828
$566$	0	0
$567$	29.6830	1.24657
$568$	0	0
$569$	−14.4580	−0.606111	−0.303055	−	0.952973i	$-0.598007\pi$
$569$	−14.4580	−0.606111	−0.303055	+	0.952973i	$0.598007\pi$
$570$	0	0
$571$	18.7204	0.783424	0.391712	−	0.920088i	$-0.371883\pi$
$571$	18.7204	0.783424	0.391712	+	0.920088i	$0.371883\pi$
$572$	0	0
$573$	−7.91821	−0.330788
$574$	0	0
$575$	−12.9203	−0.538814
$576$	0	0
$577$	41.3729	1.72238	0.861189	−	0.508285i	$-0.169720\pi$
$577$	41.3729	1.72238	0.861189	+	0.508285i	$0.169720\pi$
$578$	0	0
$579$	3.65601	0.151939
$580$	0	0
$581$	−47.8240	−1.98407
$582$	0	0
$583$	−0.992820	−0.0411184
$584$	0	0
$585$	−22.4187	−0.926898
$586$	0	0
$587$	28.3558	1.17037	0.585186	−	0.810899i	$-0.301022\pi$
$587$	28.3558	1.17037	0.585186	+	0.810899i	$0.301022\pi$
$588$	0	0
$589$	31.1625	1.28403
$590$	0	0
$591$	−0.737921	−0.0303540
$592$	0	0
$593$	−31.9921	−1.31376	−0.656879	−	0.753996i	$-0.728124\pi$
$593$	−31.9921	−1.31376	−0.656879	+	0.753996i	$0.728124\pi$
$594$	0	0
$595$	29.7514	1.21969
$596$	0	0
$597$	4.98712	0.204109
$598$	0	0
$599$	−10.0853	−0.412076	−0.206038	−	0.978544i	$-0.566057\pi$
$599$	−10.0853	−0.412076	−0.206038	+	0.978544i	$0.566057\pi$
$600$	0	0
$601$	45.6405	1.86171	0.930856	−	0.365385i	$-0.119063\pi$
$601$	45.6405	1.86171	0.930856	+	0.365385i	$0.119063\pi$
$602$	0	0
$603$	−39.0922	−1.59196
$604$	0	0
$605$	−17.1246	−0.696214
$606$	0	0
$607$	−44.3535	−1.80025	−0.900126	−	0.435629i	$-0.856526\pi$
$607$	−44.3535	−1.80025	−0.900126	+	0.435629i	$0.856526\pi$
$608$	0	0
$609$	0.312208	0.0126513
$610$	0	0
$611$	39.7168	1.60677
$612$	0	0
$613$	−34.5779	−1.39659	−0.698295	−	0.715811i	$-0.746057\pi$
$613$	−34.5779	−1.39659	−0.698295	+	0.715811i	$0.746057\pi$
$614$	0	0
$615$	2.29282	0.0924553
$616$	0	0
$617$	27.6199	1.11193	0.555967	−	0.831204i	$-0.312348\pi$
$617$	27.6199	1.11193	0.555967	+	0.831204i	$0.312348\pi$
$618$	0	0
$619$	−45.1522	−1.81482	−0.907410	−	0.420247i	$-0.861943\pi$
$619$	−45.1522	−1.81482	−0.907410	+	0.420247i	$0.861943\pi$
$620$	0	0
$621$	−10.8001	−0.433393
$622$	0	0
$623$	43.7353	1.75222
$624$	0	0
$625$	−5.75652	−0.230261
$626$	0	0
$627$	0.762788	0.0304628
$628$	0	0
$629$	14.1405	0.563817
$630$	0	0
$631$	22.6526	0.901787	0.450893	−	0.892578i	$-0.351105\pi$
$631$	22.6526	0.901787	0.450893	+	0.892578i	$0.351105\pi$
$632$	0	0
$633$	−7.53002	−0.299291
$634$	0	0
$635$	27.7915	1.10287
$636$	0	0
$637$	36.7897	1.45766
$638$	0	0
$639$	−2.47845	−0.0980459
$640$	0	0
$641$	33.3532	1.31737	0.658687	−	0.752417i	$-0.271112\pi$
$641$	33.3532	1.31737	0.658687	+	0.752417i	$0.271112\pi$
$642$	0	0
$643$	−29.3107	−1.15590	−0.577951	−	0.816072i	$-0.696147\pi$
$643$	−29.3107	−1.15590	−0.577951	+	0.816072i	$0.696147\pi$
$644$	0	0
$645$	1.93524	0.0762002
$646$	0	0
$647$	−1.83563	−0.0721662	−0.0360831	−	0.999349i	$-0.511488\pi$
$647$	−1.83563	−0.0721662	−0.0360831	+	0.999349i	$0.511488\pi$
$648$	0	0
$649$	−2.13693	−0.0838818
$650$	0	0
$651$	−5.06324	−0.198444
$652$	0	0
$653$	10.9497	0.428493	0.214247	−	0.976780i	$-0.431270\pi$
$653$	10.9497	0.428493	0.214247	+	0.976780i	$0.431270\pi$
$654$	0	0
$655$	−14.6479	−0.572343
$656$	0	0
$657$	6.13534	0.239363
$658$	0	0
$659$	10.1057	0.393662	0.196831	−	0.980437i	$-0.436935\pi$
$659$	10.1057	0.393662	0.196831	+	0.980437i	$0.436935\pi$
$660$	0	0
$661$	22.6395	0.880574	0.440287	−	0.897857i	$-0.354877\pi$
$661$	22.6395	0.880574	0.440287	+	0.897857i	$0.354877\pi$
$662$	0	0
$663$	9.08923	0.352996
$664$	0	0
$665$	−50.2713	−1.94944
$666$	0	0
$667$	1.14969	0.0445162
$668$	0	0
$669$	5.52743	0.213703
$670$	0	0
$671$	0.358244	0.0138299
$672$	0	0
$673$	31.2335	1.20396	0.601982	−	0.798510i	$-0.294378\pi$
$673$	31.2335	1.20396	0.601982	+	0.798510i	$0.294378\pi$
$674$	0	0
$675$	5.43165	0.209064
$676$	0	0
$677$	−32.3766	−1.24433	−0.622166	−	0.782885i	$-0.713747\pi$
$677$	−32.3766	−1.24433	−0.622166	+	0.782885i	$0.713747\pi$
$678$	0	0
$679$	4.40843	0.169180
$680$	0	0
$681$	−7.27976	−0.278961
$682$	0	0
$683$	19.7237	0.754708	0.377354	−	0.926069i	$-0.376834\pi$
$683$	19.7237	0.754708	0.377354	+	0.926069i	$0.376834\pi$
$684$	0	0
$685$	−3.59902	−0.137512
$686$	0	0
$687$	8.80320	0.335863
$688$	0	0
$689$	19.9897	0.761548
$690$	0	0
$691$	2.90500	0.110511	0.0552557	−	0.998472i	$-0.482403\pi$
$691$	2.90500	0.110511	0.0552557	+	0.998472i	$0.482403\pi$
$692$	0	0
$693$	2.69598	0.102412
$694$	0	0
$695$	25.4259	0.964460
$696$	0	0
$697$	20.2210	0.765927
$698$	0	0
$699$	0.237469	0.00898192
$700$	0	0
$701$	−41.3415	−1.56145	−0.780723	−	0.624878i	$-0.785149\pi$
$701$	−41.3415	−1.56145	−0.780723	+	0.624878i	$0.785149\pi$
$702$	0	0
$703$	−23.8934	−0.901155
$704$	0	0
$705$	4.52203	0.170309
$706$	0	0
$707$	26.6920	1.00386
$708$	0	0
$709$	25.8126	0.969412	0.484706	−	0.874677i	$-0.338927\pi$
$709$	25.8126	0.969412	0.484706	+	0.874677i	$0.338927\pi$
$710$	0	0
$711$	37.2739	1.39788
$712$	0	0
$713$	−18.6451	−0.698266
$714$	0	0
$715$	−1.93828	−0.0724877
$716$	0	0
$717$	−1.87519	−0.0700302
$718$	0	0
$719$	−4.71324	−0.175774	−0.0878872	−	0.996130i	$-0.528011\pi$
$719$	−4.71324	−0.175774	−0.0878872	+	0.996130i	$0.528011\pi$
$720$	0	0
$721$	−72.2165	−2.68948
$722$	0	0
$723$	0.260165	0.00967563
$724$	0	0
$725$	−0.578208	−0.0214741
$726$	0	0
$727$	15.1640	0.562403	0.281202	−	0.959649i	$-0.409267\pi$
$727$	15.1640	0.562403	0.281202	+	0.959649i	$0.409267\pi$
$728$	0	0
$729$	−20.1385	−0.745871
$730$	0	0
$731$	17.0675	0.631264
$732$	0	0
$733$	5.96043	0.220153	0.110077	−	0.993923i	$-0.464890\pi$
$733$	5.96043	0.220153	0.110077	+	0.993923i	$0.464890\pi$
$734$	0	0
$735$	4.18875	0.154504
$736$	0	0
$737$	−3.37985	−0.124498
$738$	0	0
$739$	9.23179	0.339597	0.169798	−	0.985479i	$-0.445688\pi$
$739$	9.23179	0.339597	0.169798	+	0.985479i	$0.445688\pi$
$740$	0	0
$741$	−15.3582	−0.564198
$742$	0	0
$743$	−51.0084	−1.87132	−0.935658	−	0.352909i	$-0.885193\pi$
$743$	−51.0084	−1.87132	−0.935658	+	0.352909i	$0.885193\pi$
$744$	0	0
$745$	−15.1187	−0.553907
$746$	0	0
$747$	36.1861	1.32398
$748$	0	0
$749$	59.6381	2.17913
$750$	0	0
$751$	−1.00000	−0.0364905
$751$	−1.00000	−0.0364905
$752$	0	0
$753$	−7.26014	−0.264574
$754$	0	0
$755$	−18.1557	−0.660753
$756$	0	0
$757$	40.4183	1.46903	0.734514	−	0.678594i	$-0.237410\pi$
$757$	40.4183	1.46903	0.734514	+	0.678594i	$0.237410\pi$
$758$	0	0
$759$	−0.456390	−0.0165659
$760$	0	0
$761$	−30.5931	−1.10900	−0.554500	−	0.832184i	$-0.687090\pi$
$761$	−30.5931	−1.10900	−0.554500	+	0.832184i	$0.687090\pi$
$762$	0	0
$763$	−27.1503	−0.982906
$764$	0	0
$765$	−22.5114	−0.813901
$766$	0	0
$767$	43.0256	1.55356
$768$	0	0
$769$	−40.7009	−1.46771	−0.733855	−	0.679306i	$-0.762281\pi$
$769$	−40.7009	−1.46771	−0.733855	+	0.679306i	$0.762281\pi$
$770$	0	0
$771$	5.40506	0.194658
$772$	0	0
$773$	−29.2149	−1.05079	−0.525393	−	0.850859i	$-0.676082\pi$
$773$	−29.2149	−1.05079	−0.525393	+	0.850859i	$0.676082\pi$
$774$	0	0
$775$	9.37710	0.336836
$776$	0	0
$777$	3.88216	0.139272
$778$	0	0
$779$	−34.1678	−1.22419
$780$	0	0
$781$	−0.214283	−0.00766764
$782$	0	0
$783$	−0.483325	−0.0172726
$784$	0	0
$785$	23.7787	0.848699
$786$	0	0
$787$	−16.2828	−0.580420	−0.290210	−	0.956963i	$-0.593725\pi$
$787$	−16.2828	−0.580420	−0.290210	+	0.956963i	$0.593725\pi$
$788$	0	0
$789$	−6.76470	−0.240830
$790$	0	0
$791$	−10.1770	−0.361852
$792$	0	0
$793$	−7.21300	−0.256141
$794$	0	0
$795$	2.27596	0.0807202
$796$	0	0
$797$	−26.7775	−0.948509	−0.474254	−	0.880388i	$-0.657282\pi$
$797$	−26.7775	−0.948509	−0.474254	+	0.880388i	$0.657282\pi$
$798$	0	0
$799$	39.8811	1.41089
$800$	0	0
$801$	−33.0923	−1.16926
$802$	0	0
$803$	0.530452	0.0187193
$804$	0	0
$805$	30.0783	1.06012
$806$	0	0
$807$	2.58017	0.0908263
$808$	0	0
$809$	−14.0495	−0.493955	−0.246978	−	0.969021i	$-0.579437\pi$
$809$	−14.0495	−0.493955	−0.246978	+	0.969021i	$0.579437\pi$
$810$	0	0
$811$	10.3335	0.362859	0.181430	−	0.983404i	$-0.441928\pi$
$811$	10.3335	0.362859	0.181430	+	0.983404i	$0.441928\pi$
$812$	0	0
$813$	5.38313	0.188795
$814$	0	0
$815$	22.3693	0.783563
$816$	0	0
$817$	−28.8392	−1.00896
$818$	0	0
$819$	−54.2817	−1.89676
$820$	0	0
$821$	19.9187	0.695169	0.347584	−	0.937649i	$-0.387002\pi$
$821$	19.9187	0.695169	0.347584	+	0.937649i	$0.387002\pi$
$822$	0	0
$823$	−15.8346	−0.551961	−0.275980	−	0.961163i	$-0.589002\pi$
$823$	−15.8346	−0.551961	−0.275980	+	0.961163i	$0.589002\pi$
$824$	0	0
$825$	0.229530	0.00799121
$826$	0	0
$827$	−40.6275	−1.41275	−0.706377	−	0.707835i	$-0.749672\pi$
$827$	−40.6275	−1.41275	−0.706377	+	0.707835i	$0.749672\pi$
$828$	0	0
$829$	−17.3777	−0.603552	−0.301776	−	0.953379i	$-0.597580\pi$
$829$	−17.3777	−0.603552	−0.301776	+	0.953379i	$0.597580\pi$
$830$	0	0
$831$	−0.781460	−0.0271086
$832$	0	0
$833$	36.9418	1.27996
$834$	0	0
$835$	−4.79842	−0.166056
$836$	0	0
$837$	7.83834	0.270933
$838$	0	0
$839$	−42.7478	−1.47582	−0.737910	−	0.674900i	$-0.764187\pi$
$839$	−42.7478	−1.47582	−0.737910	+	0.674900i	$0.764187\pi$
$840$	0	0
$841$	−28.9485	−0.998226
$842$	0	0
$843$	−5.77560	−0.198922
$844$	0	0
$845$	18.6741	0.642408
$846$	0	0
$847$	−41.4633	−1.42470
$848$	0	0
$849$	5.56949	0.191145
$850$	0	0
$851$	14.2958	0.490055
$852$	0	0
$853$	−36.7516	−1.25835	−0.629176	−	0.777263i	$-0.716608\pi$
$853$	−36.7516	−1.25835	−0.629176	+	0.777263i	$0.716608\pi$
$854$	0	0
$855$	38.0379	1.30087
$856$	0	0
$857$	2.70706	0.0924714	0.0462357	−	0.998931i	$-0.485277\pi$
$857$	2.70706	0.0924714	0.0462357	+	0.998931i	$0.485277\pi$
$858$	0	0
$859$	−45.4054	−1.54921	−0.774605	−	0.632445i	$-0.782051\pi$
$859$	−45.4054	−1.54921	−0.774605	+	0.632445i	$0.782051\pi$
$860$	0	0
$861$	5.55154	0.189196
$862$	0	0
$863$	−47.1118	−1.60370	−0.801852	−	0.597522i	$-0.796152\pi$
$863$	−47.1118	−1.60370	−0.801852	+	0.597522i	$0.796152\pi$
$864$	0	0
$865$	29.0091	0.986339
$866$	0	0
$867$	2.95389	0.100319
$868$	0	0
$869$	3.22264	0.109321
$870$	0	0
$871$	68.0509	2.30582
$872$	0	0
$873$	−3.33565	−0.112894
$874$	0	0
$875$	−44.7985	−1.51447
$876$	0	0
$877$	−45.6039	−1.53993	−0.769967	−	0.638084i	$-0.779727\pi$
$877$	−45.6039	−1.53993	−0.769967	+	0.638084i	$0.779727\pi$
$878$	0	0
$879$	−11.6218	−0.391995
$880$	0	0
$881$	−48.3132	−1.62771	−0.813857	−	0.581066i	$-0.802636\pi$
$881$	−48.3132	−1.62771	−0.813857	+	0.581066i	$0.802636\pi$
$882$	0	0
$883$	39.0276	1.31338	0.656691	−	0.754160i	$-0.271956\pi$
$883$	39.0276	1.31338	0.656691	+	0.754160i	$0.271956\pi$
$884$	0	0
$885$	4.89875	0.164670
$886$	0	0
$887$	53.8234	1.80721	0.903607	−	0.428362i	$-0.140909\pi$
$887$	53.8234	1.80721	0.903607	+	0.428362i	$0.140909\pi$
$888$	0	0
$889$	67.2908	2.25686
$890$	0	0
$891$	−1.94183	−0.0650537
$892$	0	0
$893$	−67.3877	−2.25504
$894$	0	0
$895$	10.2492	0.342594
$896$	0	0
$897$	9.18910	0.306815
$898$	0	0
$899$	−0.834405	−0.0278290
$900$	0	0
$901$	20.0724	0.668709
$902$	0	0
$903$	4.68575	0.155932
$904$	0	0
$905$	−0.141452	−0.00470202
$906$	0	0
$907$	−25.7716	−0.855731	−0.427866	−	0.903842i	$-0.640734\pi$
$907$	−25.7716	−0.855731	−0.427866	+	0.903842i	$0.640734\pi$
$908$	0	0
$909$	−20.1965	−0.669876
$910$	0	0
$911$	−19.1198	−0.633468	−0.316734	−	0.948514i	$-0.602586\pi$
$911$	−19.1198	−0.633468	−0.316734	+	0.948514i	$0.602586\pi$
$912$	0	0
$913$	3.12860	0.103541
$914$	0	0
$915$	−0.821249	−0.0271496
$916$	0	0
$917$	−35.4667	−1.17121
$918$	0	0
$919$	−40.5991	−1.33924	−0.669620	−	0.742704i	$-0.733543\pi$
$919$	−40.5991	−1.33924	−0.669620	+	0.742704i	$0.733543\pi$
$920$	0	0
$921$	−8.30080	−0.273521
$922$	0	0
$923$	4.31444	0.142011
$924$	0	0
$925$	−7.18974	−0.236397
$926$	0	0
$927$	54.6427	1.79470
$928$	0	0
$929$	44.6533	1.46503	0.732514	−	0.680752i	$-0.238347\pi$
$929$	44.6533	1.46503	0.732514	+	0.680752i	$0.238347\pi$
$930$	0	0
$931$	−62.4212	−2.04577
$932$	0	0
$933$	−5.97696	−0.195677
$934$	0	0
$935$	−1.94630	−0.0636509
$936$	0	0
$937$	−43.8615	−1.43289	−0.716447	−	0.697642i	$-0.754233\pi$
$937$	−43.8615	−1.43289	−0.716447	+	0.697642i	$0.754233\pi$
$938$	0	0
$939$	6.65703	0.217244
$940$	0	0
$941$	26.7665	0.872564	0.436282	−	0.899810i	$-0.356295\pi$
$941$	26.7665	0.872564	0.436282	+	0.899810i	$0.356295\pi$
$942$	0	0
$943$	20.4432	0.665724
$944$	0	0
$945$	−12.6448	−0.411335
$946$	0	0
$947$	52.4011	1.70281	0.851404	−	0.524510i	$-0.175752\pi$
$947$	52.4011	1.70281	0.851404	+	0.524510i	$0.175752\pi$
$948$	0	0
$949$	−10.6803	−0.346697
$950$	0	0
$951$	−10.0471	−0.325799
$952$	0	0
$953$	−27.5861	−0.893600	−0.446800	−	0.894634i	$-0.647436\pi$
$953$	−27.5861	−0.893600	−0.446800	+	0.894634i	$0.647436\pi$
$954$	0	0
$955$	−34.1387	−1.10470
$956$	0	0
$957$	−0.0204243	−0.000660225 0
$958$	0	0
$959$	−8.71421	−0.281397
$960$	0	0
$961$	−17.4680	−0.563484
$962$	0	0
$963$	−45.1252	−1.45414
$964$	0	0
$965$	15.7626	0.507415
$966$	0	0
$967$	−25.3095	−0.813899	−0.406949	−	0.913451i	$-0.633407\pi$
$967$	−25.3095	−0.813899	−0.406949	+	0.913451i	$0.633407\pi$
$968$	0	0
$969$	−15.4217	−0.495417
$970$	0	0
$971$	−12.5560	−0.402941	−0.201471	−	0.979495i	$-0.564572\pi$
$971$	−12.5560	−0.402941	−0.201471	+	0.979495i	$0.564572\pi$
$972$	0	0
$973$	61.5631	1.97362
$974$	0	0
$975$	−4.62143	−0.148004
$976$	0	0
$977$	−37.4838	−1.19921	−0.599606	−	0.800295i	$-0.704676\pi$
$977$	−37.4838	−1.19921	−0.599606	+	0.800295i	$0.704676\pi$
$978$	0	0
$979$	−2.86111	−0.0914416
$980$	0	0
$981$	20.5433	0.655897
$982$	0	0
$983$	−37.2078	−1.18674	−0.593372	−	0.804929i	$-0.702204\pi$
$983$	−37.2078	−1.18674	−0.593372	+	0.804929i	$0.702204\pi$
$984$	0	0
$985$	−3.18149	−0.101371
$986$	0	0
$987$	10.9491	0.348512
$988$	0	0
$989$	17.2550	0.548678
$990$	0	0
$991$	−41.9488	−1.33255	−0.666274	−	0.745707i	$-0.732112\pi$
$991$	−41.9488	−1.33255	−0.666274	+	0.745707i	$0.732112\pi$
$992$	0	0
$993$	−8.93850	−0.283655
$994$	0	0
$995$	21.5015	0.681645
$996$	0	0
$997$	19.4934	0.617361	0.308681	−	0.951166i	$-0.400113\pi$
$997$	19.4934	0.617361	0.308681	+	0.951166i	$0.400113\pi$
$998$	0	0
$999$	−6.00991	−0.190145

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.e.1.28	✓	50

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.e.1.28	✓	50	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:	\( N \)	\(=\)	\( 6008 = 2^{3} \cdot 751 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6008.a (trivial)

\(f(q)\)	\(=\)	\(q+0.363113 q^{3} +1.56553 q^{5} +3.79058 q^{7} -2.86815 q^{9} +O(q^{10})\)	\(q+0.363113 q^{3} +1.56553 q^{5} +3.79058 q^{7} -2.86815 q^{9} -0.247976 q^{11} +4.99282 q^{13} +0.568466 q^{15} +5.01347 q^{17} -8.47134 q^{19} +1.37641 q^{21} +5.06856 q^{23} -2.54911 q^{25} -2.13080 q^{27} +0.226828 q^{29} -3.67858 q^{31} -0.0900433 q^{33} +5.93428 q^{35} +2.82049 q^{37} +1.81296 q^{39} +4.03334 q^{41} +3.40433 q^{43} -4.49018 q^{45} +7.95479 q^{47} +7.36851 q^{49} +1.82046 q^{51} +4.00370 q^{53} -0.388214 q^{55} -3.07606 q^{57} +8.61749 q^{59} -1.44468 q^{61} -10.8720 q^{63} +7.81642 q^{65} +13.6298 q^{67} +1.84046 q^{69} +0.864128 q^{71} -2.13913 q^{73} -0.925615 q^{75} -0.939972 q^{77} -12.9958 q^{79} +7.83072 q^{81} -12.6165 q^{83} +7.84876 q^{85} +0.0823642 q^{87} +11.5379 q^{89} +18.9257 q^{91} -1.33574 q^{93} -13.2622 q^{95} +1.16300 q^{97} +0.711231 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9}+O(q^{10}) \) 50 * q + 6 * q^3 + 23 * q^5 + 12 * q^7 + 56 * q^9	\( 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9} - 5 q^{11} + 36 q^{13} + 5 q^{15} + 14 q^{17} + 9 q^{19} + 30 q^{21} + 3 q^{23} + 71 q^{25} + 24 q^{27} + 61 q^{29} + 27 q^{31} + 24 q^{33} - 7 q^{35} + 56 q^{37} - 2 q^{39} + 10 q^{41} + 19 q^{43} + 76 q^{45} + 3 q^{47} + 82 q^{49} - q^{51} + 56 q^{53} + 7 q^{55} + 35 q^{57} - q^{59} + 67 q^{61} + 25 q^{63} + 27 q^{65} + 46 q^{67} + 68 q^{69} + 4 q^{71} + 62 q^{73} + 27 q^{75} + 71 q^{77} + 7 q^{79} + 74 q^{81} - q^{83} + 72 q^{85} + 25 q^{87} + 19 q^{89} + 45 q^{91} + 72 q^{93} - 24 q^{95} + 81 q^{97} + 16 q^{99}+O(q^{100}) \) 50 * q + 6 * q^3 + 23 * q^5 + 12 * q^7 + 56 * q^9 - 5 * q^11 + 36 * q^13 + 5 * q^15 + 14 * q^17 + 9 * q^19 + 30 * q^21 + 3 * q^23 + 71 * q^25 + 24 * q^27 + 61 * q^29 + 27 * q^31 + 24 * q^33 - 7 * q^35 + 56 * q^37 - 2 * q^39 + 10 * q^41 + 19 * q^43 + 76 * q^45 + 3 * q^47 + 82 * q^49 - q^51 + 56 * q^53 + 7 * q^55 + 35 * q^57 - q^59 + 67 * q^61 + 25 * q^63 + 27 * q^65 + 46 * q^67 + 68 * q^69 + 4 * q^71 + 62 * q^73 + 27 * q^75 + 71 * q^77 + 7 * q^79 + 74 * q^81 - q^83 + 72 * q^85 + 25 * q^87 + 19 * q^89 + 45 * q^91 + 72 * q^93 - 24 * q^95 + 81 * q^97 + 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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