LMFDB - Embedded newform 3344.2.a.q.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3344,2,Mod(1,3344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3344, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3344.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3344 = 2^{4} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3344.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:	$26.7019744359$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.621.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 3 $ x^3 - 6*x - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 418)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.14510$ of defining polynomial
Character	$\chi$	$=$	3344.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.14510 q^{3} -1.60147 q^{5} -2.89167 q^{7} +1.60147 q^{9} +1.00000 q^{11} +4.89167 q^{13} -3.43531 q^{15} -5.74657 q^{17} +1.00000 q^{19} -6.20293 q^{21} +7.74657 q^{23} -2.43531 q^{25} -3.00000 q^{27} +5.34803 q^{29} +7.43531 q^{31} +2.14510 q^{33} +4.63091 q^{35} -7.49314 q^{37} +10.4931 q^{39} -2.05783 q^{41} +10.6887 q^{43} -2.56469 q^{45} +7.08727 q^{47} +1.36176 q^{49} -12.3270 q^{51} +8.32698 q^{53} -1.60147 q^{55} +2.14510 q^{57} +2.54364 q^{59} +13.4931 q^{61} -4.63091 q^{63} -7.83384 q^{65} +10.3848 q^{67} +16.6172 q^{69} -14.9789 q^{71} -3.74657 q^{73} -5.22399 q^{75} -2.89167 q^{77} +8.69607 q^{79} -11.2397 q^{81} +6.10833 q^{83} +9.20293 q^{85} +11.4721 q^{87} +3.20293 q^{89} -14.1451 q^{91} +15.9495 q^{93} -1.60147 q^{95} -6.98627 q^{97} +1.60147 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{5} + 6 q^{7} + 3 q^{9} + 3 q^{11} + 9 q^{15} - 9 q^{17} + 3 q^{19} - 15 q^{21} + 15 q^{23} + 12 q^{25} - 9 q^{27} + 6 q^{29} + 3 q^{31} - 6 q^{37} + 15 q^{39} - 9 q^{41} + 21 q^{43} - 27 q^{45}+ \cdots + 3 q^{99}+O(q^{100}) $ 3 * q - 3 * q^5 + 6 * q^7 + 3 * q^9 + 3 * q^11 + 9 * q^15 - 9 * q^17 + 3 * q^19 - 15 * q^21 + 15 * q^23 + 12 * q^25 - 9 * q^27 + 6 * q^29 + 3 * q^31 - 6 * q^37 + 15 * q^39 - 9 * q^41 + 21 * q^43 - 27 * q^45 + 12 * q^47 + 27 * q^49 - 3 * q^51 - 9 * q^53 - 3 * q^55 + 3 * q^59 + 24 * q^61 - 6 * q^65 + 3 * q^69 - 21 * q^71 - 3 * q^73 - 36 * q^75 + 6 * q^77 + 6 * q^79 - 9 * q^81 + 33 * q^83 + 24 * q^85 - 6 * q^87 + 6 * q^89 - 36 * q^91 + 36 * q^93 - 3 * q^95 + 12 * q^97 + 3 * 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	0
$2$	0	0
$3$	2.14510	1.23848	0.619238	−	0.785204i	$-0.287442\pi$
$3$	2.14510	1.23848	0.619238	+	0.785204i	$0.287442\pi$
$4$	0	0
$5$	−1.60147	−0.716197	−0.358099	−	0.933684i	$-0.616575\pi$
$5$	−1.60147	−0.716197	−0.358099	+	0.933684i	$0.616575\pi$
$6$	0	0
$7$	−2.89167	−1.09295	−0.546474	−	0.837476i	$-0.684030\pi$
$7$	−2.89167	−1.09295	−0.546474	+	0.837476i	$0.684030\pi$
$8$	0	0
$9$	1.60147	0.533822
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	4.89167	1.35671	0.678353	−	0.734736i	$-0.262694\pi$
$13$	4.89167	1.35671	0.678353	+	0.734736i	$0.262694\pi$
$14$	0	0
$15$	−3.43531	−0.886993
$16$	0	0
$17$	−5.74657	−1.39375	−0.696874	−	0.717194i	$-0.745426\pi$
$17$	−5.74657	−1.39375	−0.696874	+	0.717194i	$0.745426\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−6.20293	−1.35359
$22$	0	0
$23$	7.74657	1.61527	0.807636	−	0.589682i	$-0.200747\pi$
$23$	7.74657	1.61527	0.807636	+	0.589682i	$0.200747\pi$
$24$	0	0
$25$	−2.43531	−0.487062
$26$	0	0
$27$	−3.00000	−0.577350
$28$	0	0
$29$	5.34803	0.993105	0.496552	−	0.868007i	$-0.334599\pi$
$29$	5.34803	0.993105	0.496552	+	0.868007i	$0.334599\pi$
$30$	0	0
$31$	7.43531	1.33542	0.667710	−	0.744421i	$-0.267274\pi$
$31$	7.43531	1.33542	0.667710	+	0.744421i	$0.267274\pi$
$32$	0	0
$33$	2.14510	0.373414
$34$	0	0
$35$	4.63091	0.782767
$36$	0	0
$37$	−7.49314	−1.23186	−0.615932	−	0.787799i	$-0.711220\pi$
$37$	−7.49314	−1.23186	−0.615932	+	0.787799i	$0.711220\pi$
$38$	0	0
$39$	10.4931	1.68025
$40$	0	0
$41$	−2.05783	−0.321379	−0.160689	−	0.987005i	$-0.551372\pi$
$41$	−2.05783	−0.321379	−0.160689	+	0.987005i	$0.551372\pi$
$42$	0	0
$43$	10.6887	1.63002	0.815009	−	0.579449i	$-0.196732\pi$
$43$	10.6887	1.63002	0.815009	+	0.579449i	$0.196732\pi$
$44$	0	0
$45$	−2.56469	−0.382322
$46$	0	0
$47$	7.08727	1.03379	0.516893	−	0.856050i	$-0.327089\pi$
$47$	7.08727	1.03379	0.516893	+	0.856050i	$0.327089\pi$
$48$	0	0
$49$	1.36176	0.194537
$50$	0	0
$51$	−12.3270	−1.72612
$52$	0	0
$53$	8.32698	1.14380	0.571899	−	0.820324i	$-0.306207\pi$
$53$	8.32698	1.14380	0.571899	+	0.820324i	$0.306207\pi$
$54$	0	0
$55$	−1.60147	−0.215942
$56$	0	0
$57$	2.14510	0.284126
$58$	0	0
$59$	2.54364	0.331153	0.165577	−	0.986197i	$-0.447051\pi$
$59$	2.54364	0.331153	0.165577	+	0.986197i	$0.447051\pi$
$60$	0	0
$61$	13.4931	1.72762	0.863810	−	0.503818i	$-0.168072\pi$
$61$	13.4931	1.72762	0.863810	+	0.503818i	$0.168072\pi$
$62$	0	0
$63$	−4.63091	−0.583440
$64$	0	0
$65$	−7.83384	−0.971669
$66$	0	0
$67$	10.3848	1.26871	0.634353	−	0.773043i	$-0.281267\pi$
$67$	10.3848	1.26871	0.634353	+	0.773043i	$0.281267\pi$
$68$	0	0
$69$	16.6172	2.00047
$70$	0	0
$71$	−14.9789	−1.77767	−0.888837	−	0.458224i	$-0.848486\pi$
$71$	−14.9789	−1.77767	−0.888837	+	0.458224i	$0.848486\pi$
$72$	0	0
$73$	−3.74657	−0.438503	−0.219251	−	0.975668i	$-0.570361\pi$
$73$	−3.74657	−0.438503	−0.219251	+	0.975668i	$0.570361\pi$
$74$	0	0
$75$	−5.22399	−0.603214
$76$	0	0
$77$	−2.89167	−0.329536
$78$	0	0
$79$	8.69607	0.978384	0.489192	−	0.872176i	$-0.337292\pi$
$79$	8.69607	0.978384	0.489192	+	0.872176i	$0.337292\pi$
$80$	0	0
$81$	−11.2397	−1.24886
$82$	0	0
$83$	6.10833	0.670476	0.335238	−	0.942133i	$-0.391183\pi$
$83$	6.10833	0.670476	0.335238	+	0.942133i	$0.391183\pi$
$84$	0	0
$85$	9.20293	0.998198
$86$	0	0
$87$	11.4721	1.22994
$88$	0	0
$89$	3.20293	0.339510	0.169755	−	0.985486i	$-0.445702\pi$
$89$	3.20293	0.339510	0.169755	+	0.985486i	$0.445702\pi$
$90$	0	0
$91$	−14.1451	−1.48281
$92$	0	0
$93$	15.9495	1.65389
$94$	0	0
$95$	−1.60147	−0.164307
$96$	0	0
$97$	−6.98627	−0.709349	−0.354674	−	0.934990i	$-0.615408\pi$
$97$	−6.98627	−0.709349	−0.354674	+	0.934990i	$0.615408\pi$
$98$	0	0
$99$	1.60147	0.160953
$100$	0	0
$101$	−1.20293	−0.119696	−0.0598481	−	0.998207i	$-0.519062\pi$
$101$	−1.20293	−0.119696	−0.0598481	+	0.998207i	$0.519062\pi$
$102$	0	0
$103$	−16.8779	−1.66303	−0.831517	−	0.555500i	$-0.812527\pi$
$103$	−16.8779	−1.66303	−0.831517	+	0.555500i	$0.812527\pi$
$104$	0	0
$105$	9.93378	0.969438
$106$	0	0
$107$	8.03677	0.776944	0.388472	−	0.921460i	$-0.373003\pi$
$107$	8.03677	0.776944	0.388472	+	0.921460i	$0.373003\pi$
$108$	0	0
$109$	11.5299	1.10437	0.552183	−	0.833723i	$-0.313795\pi$
$109$	11.5299	1.10437	0.552183	+	0.833723i	$0.313795\pi$
$110$	0	0
$111$	−16.0735	−1.52563
$112$	0	0
$113$	−8.00000	−0.752577	−0.376288	−	0.926503i	$-0.622800\pi$
$113$	−8.00000	−0.752577	−0.376288	+	0.926503i	$0.622800\pi$
$114$	0	0
$115$	−12.4059	−1.15685
$116$	0	0
$117$	7.83384	0.724239
$118$	0	0
$119$	16.6172	1.52329
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−4.41425	−0.398020
$124$	0	0
$125$	11.9074	1.06503
$126$	0	0
$127$	21.2765	1.88798	0.943991	−	0.329971i	$-0.107039\pi$
$127$	21.2765	1.88798	0.943991	+	0.329971i	$0.107039\pi$
$128$	0	0
$129$	22.9284	2.01874
$130$	0	0
$131$	3.43531	0.300144	0.150072	−	0.988675i	$-0.452049\pi$
$131$	3.43531	0.300144	0.150072	+	0.988675i	$0.452049\pi$
$132$	0	0
$133$	−2.89167	−0.250740
$134$	0	0
$135$	4.80440	0.413497
$136$	0	0
$137$	13.5877	1.16088	0.580439	−	0.814303i	$-0.302881\pi$
$137$	13.5877	1.16088	0.580439	+	0.814303i	$0.302881\pi$
$138$	0	0
$139$	−3.19560	−0.271048	−0.135524	−	0.990774i	$-0.543272\pi$
$139$	−3.19560	−0.271048	−0.135524	+	0.990774i	$0.543272\pi$
$140$	0	0
$141$	15.2029	1.28032
$142$	0	0
$143$	4.89167	0.409062
$144$	0	0
$145$	−8.56469	−0.711259
$146$	0	0
$147$	2.92112	0.240930
$148$	0	0
$149$	−12.0735	−0.989104	−0.494552	−	0.869148i	$-0.664668\pi$
$149$	−12.0735	−0.989104	−0.494552	+	0.869148i	$0.664668\pi$
$150$	0	0
$151$	4.91273	0.399792	0.199896	−	0.979817i	$-0.435940\pi$
$151$	4.91273	0.399792	0.199896	+	0.979817i	$0.435940\pi$
$152$	0	0
$153$	−9.20293	−0.744013
$154$	0	0
$155$	−11.9074	−0.956425
$156$	0	0
$157$	8.00733	0.639054	0.319527	−	0.947577i	$-0.396476\pi$
$157$	8.00733	0.639054	0.319527	+	0.947577i	$0.396476\pi$
$158$	0	0
$159$	17.8622	1.41657
$160$	0	0
$161$	−22.4005	−1.76541
$162$	0	0
$163$	−1.88434	−0.147593	−0.0737966	−	0.997273i	$-0.523512\pi$
$163$	−1.88434	−0.147593	−0.0737966	+	0.997273i	$0.523512\pi$
$164$	0	0
$165$	−3.43531	−0.267438
$166$	0	0
$167$	4.98627	0.385849	0.192925	−	0.981214i	$-0.438203\pi$
$167$	4.98627	0.385849	0.192925	+	0.981214i	$0.438203\pi$
$168$	0	0
$169$	10.9284	0.840650
$170$	0	0
$171$	1.60147	0.122467
$172$	0	0
$173$	6.51419	0.495265	0.247632	−	0.968854i	$-0.420347\pi$
$173$	6.51419	0.495265	0.247632	+	0.968854i	$0.420347\pi$
$174$	0	0
$175$	7.04211	0.532333
$176$	0	0
$177$	5.45636	0.410125
$178$	0	0
$179$	5.55096	0.414899	0.207449	−	0.978246i	$-0.433484\pi$
$179$	5.55096	0.414899	0.207449	+	0.978246i	$0.433484\pi$
$180$	0	0
$181$	1.88434	0.140062	0.0700311	−	0.997545i	$-0.477690\pi$
$181$	1.88434	0.140062	0.0700311	+	0.997545i	$0.477690\pi$
$182$	0	0
$183$	28.9442	2.13961
$184$	0	0
$185$	12.0000	0.882258
$186$	0	0
$187$	−5.74657	−0.420231
$188$	0	0
$189$	8.67501	0.631014
$190$	0	0
$191$	−2.44264	−0.176743	−0.0883715	−	0.996088i	$-0.528166\pi$
$191$	−2.44264	−0.176743	−0.0883715	+	0.996088i	$0.528166\pi$
$192$	0	0
$193$	−9.67501	−0.696423	−0.348211	−	0.937416i	$-0.613211\pi$
$193$	−9.67501	−0.696423	−0.348211	+	0.937416i	$0.613211\pi$
$194$	0	0
$195$	−16.8044	−1.20339
$196$	0	0
$197$	−17.4931	−1.24633	−0.623167	−	0.782089i	$-0.714154\pi$
$197$	−17.4931	−1.24633	−0.623167	+	0.782089i	$0.714154\pi$
$198$	0	0
$199$	25.2397	1.78920	0.894598	−	0.446873i	$-0.147462\pi$
$199$	25.2397	1.78920	0.894598	+	0.446873i	$0.147462\pi$
$200$	0	0
$201$	22.2765	1.57126
$202$	0	0
$203$	−15.4648	−1.08541
$204$	0	0
$205$	3.29554	0.230171
$206$	0	0
$207$	12.4059	0.862267
$208$	0	0
$209$	1.00000	0.0691714
$210$	0	0
$211$	14.8338	1.02120	0.510602	−	0.859817i	$-0.329423\pi$
$211$	14.8338	1.02120	0.510602	+	0.859817i	$0.329423\pi$
$212$	0	0
$213$	−32.1314	−2.20161
$214$	0	0
$215$	−17.1176	−1.16741
$216$	0	0
$217$	−21.5005	−1.45955
$218$	0	0
$219$	−8.03677	−0.543075
$220$	0	0
$221$	−28.1103	−1.89090
$222$	0	0
$223$	−11.4931	−0.769637	−0.384819	−	0.922992i	$-0.625736\pi$
$223$	−11.4931	−0.769637	−0.384819	+	0.922992i	$0.625736\pi$
$224$	0	0
$225$	−3.90006	−0.260004
$226$	0	0
$227$	−8.73284	−0.579619	−0.289810	−	0.957084i	$-0.593592\pi$
$227$	−8.73284	−0.579619	−0.289810	+	0.957084i	$0.593592\pi$
$228$	0	0
$229$	7.84117	0.518159	0.259080	−	0.965856i	$-0.416581\pi$
$229$	7.84117	0.518159	0.259080	+	0.965856i	$0.416581\pi$
$230$	0	0
$231$	−6.20293	−0.408123
$232$	0	0
$233$	−12.5069	−0.819352	−0.409676	−	0.912231i	$-0.634358\pi$
$233$	−12.5069	−0.819352	−0.409676	+	0.912231i	$0.634358\pi$
$234$	0	0
$235$	−11.3500	−0.740394
$236$	0	0
$237$	18.6540	1.21170
$238$	0	0
$239$	−24.6245	−1.59283	−0.796414	−	0.604752i	$-0.793272\pi$
$239$	−24.6245	−1.59283	−0.796414	+	0.604752i	$0.793272\pi$
$240$	0	0
$241$	16.0809	1.03586	0.517930	−	0.855423i	$-0.326703\pi$
$241$	16.0809	1.03586	0.517930	+	0.855423i	$0.326703\pi$
$242$	0	0
$243$	−15.1103	−0.969328
$244$	0	0
$245$	−2.18081	−0.139327
$246$	0	0
$247$	4.89167	0.311250
$248$	0	0
$249$	13.1030	0.830368
$250$	0	0
$251$	−13.0873	−0.826061	−0.413031	−	0.910717i	$-0.635530\pi$
$251$	−13.0873	−0.826061	−0.413031	+	0.910717i	$0.635530\pi$
$252$	0	0
$253$	7.74657	0.487023
$254$	0	0
$255$	19.7412	1.23624
$256$	0	0
$257$	0.723522	0.0451320	0.0225660	−	0.999745i	$-0.492816\pi$
$257$	0.723522	0.0451320	0.0225660	+	0.999745i	$0.492816\pi$
$258$	0	0
$259$	21.6677	1.34636
$260$	0	0
$261$	8.56469	0.530141
$262$	0	0
$263$	−13.7760	−0.849465	−0.424733	−	0.905319i	$-0.639632\pi$
$263$	−13.7760	−0.849465	−0.424733	+	0.905319i	$0.639632\pi$
$264$	0	0
$265$	−13.3354	−0.819185
$266$	0	0
$267$	6.87062	0.420475
$268$	0	0
$269$	−19.9579	−1.21685	−0.608427	−	0.793610i	$-0.708199\pi$
$269$	−19.9579	−1.21685	−0.608427	+	0.793610i	$0.708199\pi$
$270$	0	0
$271$	−28.7486	−1.74635	−0.873175	−	0.487406i	$-0.837943\pi$
$271$	−28.7486	−1.74635	−0.873175	+	0.487406i	$0.837943\pi$
$272$	0	0
$273$	−30.3427	−1.83642
$274$	0	0
$275$	−2.43531	−0.146855
$276$	0	0
$277$	3.08727	0.185496	0.0927482	−	0.995690i	$-0.470435\pi$
$277$	3.08727	0.185496	0.0927482	+	0.995690i	$0.470435\pi$
$278$	0	0
$279$	11.9074	0.712877
$280$	0	0
$281$	9.09460	0.542538	0.271269	−	0.962504i	$-0.412557\pi$
$281$	9.09460	0.542538	0.271269	+	0.962504i	$0.412557\pi$
$282$	0	0
$283$	−13.1451	−0.781395	−0.390698	−	0.920519i	$-0.627766\pi$
$283$	−13.1451	−0.781395	−0.390698	+	0.920519i	$0.627766\pi$
$284$	0	0
$285$	−3.43531	−0.203490
$286$	0	0
$287$	5.95056	0.351251
$288$	0	0
$289$	16.0230	0.942532
$290$	0	0
$291$	−14.9863	−0.878511
$292$	0	0
$293$	3.10833	0.181591	0.0907953	−	0.995870i	$-0.471059\pi$
$293$	3.10833	0.181591	0.0907953	+	0.995870i	$0.471059\pi$
$294$	0	0
$295$	−4.07355	−0.237171
$296$	0	0
$297$	−3.00000	−0.174078
$298$	0	0
$299$	37.8937	2.19145
$300$	0	0
$301$	−30.9083	−1.78153
$302$	0	0
$303$	−2.58041	−0.148241
$304$	0	0
$305$	−21.6088	−1.23732
$306$	0	0
$307$	−2.18920	−0.124944	−0.0624722	−	0.998047i	$-0.519898\pi$
$307$	−2.18920	−0.124944	−0.0624722	+	0.998047i	$0.519898\pi$
$308$	0	0
$309$	−36.2049	−2.05963
$310$	0	0
$311$	−3.92112	−0.222346	−0.111173	−	0.993801i	$-0.535461\pi$
$311$	−3.92112	−0.222346	−0.111173	+	0.993801i	$0.535461\pi$
$312$	0	0
$313$	−21.7118	−1.22722	−0.613611	−	0.789608i	$-0.710284\pi$
$313$	−21.7118	−1.22722	−0.613611	+	0.789608i	$0.710284\pi$
$314$	0	0
$315$	7.41625	0.417858
$316$	0	0
$317$	6.90739	0.387958	0.193979	−	0.981006i	$-0.437861\pi$
$317$	6.90739	0.387958	0.193979	+	0.981006i	$0.437861\pi$
$318$	0	0
$319$	5.34803	0.299432
$320$	0	0
$321$	17.2397	0.962226
$322$	0	0
$323$	−5.74657	−0.319748
$324$	0	0
$325$	−11.9127	−0.660799
$326$	0	0
$327$	24.7328	1.36773
$328$	0	0
$329$	−20.4941	−1.12987
$330$	0	0
$331$	−32.1030	−1.76454	−0.882270	−	0.470744i	$-0.843986\pi$
$331$	−32.1030	−1.76454	−0.882270	+	0.470744i	$0.843986\pi$
$332$	0	0
$333$	−12.0000	−0.657596
$334$	0	0
$335$	−16.6309	−0.908644
$336$	0	0
$337$	−1.55096	−0.0844864	−0.0422432	−	0.999107i	$-0.513450\pi$
$337$	−1.55096	−0.0844864	−0.0422432	+	0.999107i	$0.513450\pi$
$338$	0	0
$339$	−17.1608	−0.932048
$340$	0	0
$341$	7.43531	0.402645
$342$	0	0
$343$	16.3039	0.880330
$344$	0	0
$345$	−26.6118	−1.43273
$346$	0	0
$347$	0.506864	0.0272099	0.0136049	−	0.999907i	$-0.495669\pi$
$347$	0.506864	0.0272099	0.0136049	+	0.999907i	$0.495669\pi$
$348$	0	0
$349$	10.8117	0.578738	0.289369	−	0.957218i	$-0.406554\pi$
$349$	10.8117	0.578738	0.289369	+	0.957218i	$0.406554\pi$
$350$	0	0
$351$	−14.6750	−0.783294
$352$	0	0
$353$	27.0966	1.44221	0.721103	−	0.692828i	$-0.243635\pi$
$353$	27.0966	1.44221	0.721103	+	0.692828i	$0.243635\pi$
$354$	0	0
$355$	23.9883	1.27316
$356$	0	0
$357$	35.6456	1.88656
$358$	0	0
$359$	20.7486	1.09507	0.547534	−	0.836784i	$-0.315567\pi$
$359$	20.7486	1.09507	0.547534	+	0.836784i	$0.315567\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	2.14510	0.112589
$364$	0	0
$365$	6.00000	0.314054
$366$	0	0
$367$	−25.6402	−1.33841	−0.669205	−	0.743078i	$-0.733365\pi$
$367$	−25.6402	−1.33841	−0.669205	+	0.743078i	$0.733365\pi$
$368$	0	0
$369$	−3.29554	−0.171559
$370$	0	0
$371$	−24.0789	−1.25011
$372$	0	0
$373$	25.2049	1.30506	0.652531	−	0.757762i	$-0.273707\pi$
$373$	25.2049	1.30506	0.652531	+	0.757762i	$0.273707\pi$
$374$	0	0
$375$	25.5426	1.31901
$376$	0	0
$377$	26.1608	1.34735
$378$	0	0
$379$	−3.47208	−0.178349	−0.0891744	−	0.996016i	$-0.528423\pi$
$379$	−3.47208	−0.178349	−0.0891744	+	0.996016i	$0.528423\pi$
$380$	0	0
$381$	45.6402	2.33822
$382$	0	0
$383$	−23.1176	−1.18126	−0.590628	−	0.806944i	$-0.701120\pi$
$383$	−23.1176	−1.18126	−0.590628	+	0.806944i	$0.701120\pi$
$384$	0	0
$385$	4.63091	0.236013
$386$	0	0
$387$	17.1176	0.870139
$388$	0	0
$389$	−26.7623	−1.35690	−0.678451	−	0.734646i	$-0.737348\pi$
$389$	−26.7623	−1.35690	−0.678451	+	0.734646i	$0.737348\pi$
$390$	0	0
$391$	−44.5162	−2.25128
$392$	0	0
$393$	7.36909	0.371721
$394$	0	0
$395$	−13.9265	−0.700716
$396$	0	0
$397$	−31.1030	−1.56101	−0.780507	−	0.625147i	$-0.785039\pi$
$397$	−31.1030	−1.56101	−0.780507	+	0.625147i	$0.785039\pi$
$398$	0	0
$399$	−6.20293	−0.310535
$400$	0	0
$401$	29.3500	1.46567	0.732835	−	0.680406i	$-0.238197\pi$
$401$	29.3500	1.46567	0.732835	+	0.680406i	$0.238197\pi$
$402$	0	0
$403$	36.3711	1.81177
$404$	0	0
$405$	18.0000	0.894427
$406$	0	0
$407$	−7.49314	−0.371421
$408$	0	0
$409$	8.58774	0.424636	0.212318	−	0.977201i	$-0.431899\pi$
$409$	8.58774	0.424636	0.212318	+	0.977201i	$0.431899\pi$
$410$	0	0
$411$	29.1471	1.43772
$412$	0	0
$413$	−7.35536	−0.361934
$414$	0	0
$415$	−9.78228	−0.480193
$416$	0	0
$417$	−6.85490	−0.335686
$418$	0	0
$419$	−7.59414	−0.370998	−0.185499	−	0.982644i	$-0.559390\pi$
$419$	−7.59414	−0.370998	−0.185499	+	0.982644i	$0.559390\pi$
$420$	0	0
$421$	8.76030	0.426951	0.213475	−	0.976948i	$-0.431522\pi$
$421$	8.76030	0.426951	0.213475	+	0.976948i	$0.431522\pi$
$422$	0	0
$423$	11.3500	0.551857
$424$	0	0
$425$	13.9947	0.678841
$426$	0	0
$427$	−39.0177	−1.88820
$428$	0	0
$429$	10.4931	0.506613
$430$	0	0
$431$	33.6677	1.62172	0.810858	−	0.585243i	$-0.199001\pi$
$431$	33.6677	1.62172	0.810858	+	0.585243i	$0.199001\pi$
$432$	0	0
$433$	30.5383	1.46758	0.733789	−	0.679378i	$-0.237750\pi$
$433$	30.5383	1.46758	0.733789	+	0.679378i	$0.237750\pi$
$434$	0	0
$435$	−18.3721	−0.880877
$436$	0	0
$437$	7.74657	0.370569
$438$	0	0
$439$	12.5069	0.596920	0.298460	−	0.954422i	$-0.403527\pi$
$439$	12.5069	0.596920	0.298460	+	0.954422i	$0.403527\pi$
$440$	0	0
$441$	2.18081	0.103848
$442$	0	0
$443$	17.6677	0.839417	0.419709	−	0.907659i	$-0.362132\pi$
$443$	17.6677	0.839417	0.419709	+	0.907659i	$0.362132\pi$
$444$	0	0
$445$	−5.12938	−0.243156
$446$	0	0
$447$	−25.8990	−1.22498
$448$	0	0
$449$	−17.7098	−0.835777	−0.417888	−	0.908498i	$-0.637230\pi$
$449$	−17.7098	−0.835777	−0.417888	+	0.908498i	$0.637230\pi$
$450$	0	0
$451$	−2.05783	−0.0968994
$452$	0	0
$453$	10.5383	0.495133
$454$	0	0
$455$	22.6529	1.06198
$456$	0	0
$457$	−24.8064	−1.16039	−0.580197	−	0.814476i	$-0.697024\pi$
$457$	−24.8064	−1.16039	−0.580197	+	0.814476i	$0.697024\pi$
$458$	0	0
$459$	17.2397	0.804681
$460$	0	0
$461$	−2.98627	−0.139085	−0.0695423	−	0.997579i	$-0.522154\pi$
$461$	−2.98627	−0.139085	−0.0695423	+	0.997579i	$0.522154\pi$
$462$	0	0
$463$	39.5667	1.83882	0.919410	−	0.393301i	$-0.128667\pi$
$463$	39.5667	1.83882	0.919410	+	0.393301i	$0.128667\pi$
$464$	0	0
$465$	−25.5426	−1.18451
$466$	0	0
$467$	−4.46475	−0.206604	−0.103302	−	0.994650i	$-0.532941\pi$
$467$	−4.46475	−0.206604	−0.103302	+	0.994650i	$0.532941\pi$
$468$	0	0
$469$	−30.0294	−1.38663
$470$	0	0
$471$	17.1765	0.791453
$472$	0	0
$473$	10.6887	0.491469
$474$	0	0
$475$	−2.43531	−0.111740
$476$	0	0
$477$	13.3354	0.610585
$478$	0	0
$479$	20.8863	0.954321	0.477161	−	0.878816i	$-0.341666\pi$
$479$	20.8863	0.954321	0.477161	+	0.878816i	$0.341666\pi$
$480$	0	0
$481$	−36.6540	−1.67128
$482$	0	0
$483$	−48.0514	−2.18642
$484$	0	0
$485$	11.1883	0.508033
$486$	0	0
$487$	−15.2838	−0.692575	−0.346288	−	0.938128i	$-0.612558\pi$
$487$	−15.2838	−0.692575	−0.346288	+	0.938128i	$0.612558\pi$
$488$	0	0
$489$	−4.04211	−0.182791
$490$	0	0
$491$	−3.31965	−0.149814	−0.0749069	−	0.997191i	$-0.523866\pi$
$491$	−3.31965	−0.149814	−0.0749069	+	0.997191i	$0.523866\pi$
$492$	0	0
$493$	−30.7328	−1.38414
$494$	0	0
$495$	−2.56469	−0.115274
$496$	0	0
$497$	43.3142	1.94291
$498$	0	0
$499$	2.07355	0.0928247	0.0464124	−	0.998922i	$-0.485221\pi$
$499$	2.07355	0.0928247	0.0464124	+	0.998922i	$0.485221\pi$
$500$	0	0
$501$	10.6961	0.477865
$502$	0	0
$503$	1.11672	0.0497921	0.0248960	−	0.999690i	$-0.492075\pi$
$503$	1.11672	0.0497921	0.0248960	+	0.999690i	$0.492075\pi$
$504$	0	0
$505$	1.92645	0.0857260
$506$	0	0
$507$	23.4426	1.04112
$508$	0	0
$509$	−38.2481	−1.69532	−0.847659	−	0.530542i	$-0.821988\pi$
$509$	−38.2481	−1.69532	−0.847659	+	0.530542i	$0.821988\pi$
$510$	0	0
$511$	10.8338	0.479261
$512$	0	0
$513$	−3.00000	−0.132453
$514$	0	0
$515$	27.0294	1.19106
$516$	0	0
$517$	7.08727	0.311698
$518$	0	0
$519$	13.9736	0.613373
$520$	0	0
$521$	10.8853	0.476892	0.238446	−	0.971156i	$-0.423362\pi$
$521$	10.8853	0.476892	0.238446	+	0.971156i	$0.423362\pi$
$522$	0	0
$523$	15.7191	0.687349	0.343674	−	0.939089i	$-0.388328\pi$
$523$	15.7191	0.687349	0.343674	+	0.939089i	$0.388328\pi$
$524$	0	0
$525$	15.1060	0.659282
$526$	0	0
$527$	−42.7275	−1.86124
$528$	0	0
$529$	37.0093	1.60910
$530$	0	0
$531$	4.07355	0.176777
$532$	0	0
$533$	−10.0662	−0.436016
$534$	0	0
$535$	−12.8706	−0.556445
$536$	0	0
$537$	11.9074	0.513842
$538$	0	0
$539$	1.36176	0.0586552
$540$	0	0
$541$	−26.4373	−1.13663	−0.568314	−	0.822812i	$-0.692404\pi$
$541$	−26.4373	−1.13663	−0.568314	+	0.822812i	$0.692404\pi$
$542$	0	0
$543$	4.04211	0.173464
$544$	0	0
$545$	−18.4648	−0.790943
$546$	0	0
$547$	24.9588	1.06716	0.533581	−	0.845749i	$-0.320846\pi$
$547$	24.9588	1.06716	0.533581	+	0.845749i	$0.320846\pi$
$548$	0	0
$549$	21.6088	0.922241
$550$	0	0
$551$	5.34803	0.227834
$552$	0	0
$553$	−25.1462	−1.06932
$554$	0	0
$555$	25.7412	1.09265
$556$	0	0
$557$	8.30486	0.351888	0.175944	−	0.984400i	$-0.443702\pi$
$557$	8.30486	0.351888	0.175944	+	0.984400i	$0.443702\pi$
$558$	0	0
$559$	52.2858	2.21145
$560$	0	0
$561$	−12.3270	−0.520445
$562$	0	0
$563$	−41.6402	−1.75493	−0.877463	−	0.479644i	$-0.840766\pi$
$563$	−41.6402	−1.75493	−0.877463	+	0.479644i	$0.840766\pi$
$564$	0	0
$565$	12.8117	0.538993
$566$	0	0
$567$	32.5015	1.36494
$568$	0	0
$569$	−14.9138	−0.625219	−0.312609	−	0.949882i	$-0.601203\pi$
$569$	−14.9138	−0.625219	−0.312609	+	0.949882i	$0.601203\pi$
$570$	0	0
$571$	23.0020	0.962603	0.481302	−	0.876555i	$-0.340164\pi$
$571$	23.0020	0.962603	0.481302	+	0.876555i	$0.340164\pi$
$572$	0	0
$573$	−5.23970	−0.218892
$574$	0	0
$575$	−18.8653	−0.786737
$576$	0	0
$577$	−33.6613	−1.40134	−0.700669	−	0.713487i	$-0.747115\pi$
$577$	−33.6613	−1.40134	−0.700669	+	0.713487i	$0.747115\pi$
$578$	0	0
$579$	−20.7539	−0.862502
$580$	0	0
$581$	−17.6633	−0.732796
$582$	0	0
$583$	8.32698	0.344868
$584$	0	0
$585$	−12.5456	−0.518698
$586$	0	0
$587$	−4.00000	−0.165098	−0.0825488	−	0.996587i	$-0.526306\pi$
$587$	−4.00000	−0.165098	−0.0825488	+	0.996587i	$0.526306\pi$
$588$	0	0
$589$	7.43531	0.306367
$590$	0	0
$591$	−37.5246	−1.54355
$592$	0	0
$593$	−30.9588	−1.27133	−0.635663	−	0.771967i	$-0.719273\pi$
$593$	−30.9588	−1.27133	−0.635663	+	0.771967i	$0.719273\pi$
$594$	0	0
$595$	−26.6118	−1.09098
$596$	0	0
$597$	54.1418	2.21587
$598$	0	0
$599$	−3.93378	−0.160730	−0.0803650	−	0.996766i	$-0.525609\pi$
$599$	−3.93378	−0.160730	−0.0803650	+	0.996766i	$0.525609\pi$
$600$	0	0
$601$	23.1451	0.944108	0.472054	−	0.881570i	$-0.343513\pi$
$601$	23.1451	0.944108	0.472054	+	0.881570i	$0.343513\pi$
$602$	0	0
$603$	16.6309	0.677263
$604$	0	0
$605$	−1.60147	−0.0651088
$606$	0	0
$607$	1.78334	0.0723836	0.0361918	−	0.999345i	$-0.488477\pi$
$607$	1.78334	0.0723836	0.0361918	+	0.999345i	$0.488477\pi$
$608$	0	0
$609$	−33.1735	−1.34426
$610$	0	0
$611$	34.6686	1.40254
$612$	0	0
$613$	−33.0598	−1.33527	−0.667637	−	0.744487i	$-0.732694\pi$
$613$	−33.0598	−1.33527	−0.667637	+	0.744487i	$0.732694\pi$
$614$	0	0
$615$	7.06927	0.285061
$616$	0	0
$617$	−21.8412	−0.879292	−0.439646	−	0.898171i	$-0.644896\pi$
$617$	−21.8412	−0.879292	−0.439646	+	0.898171i	$0.644896\pi$
$618$	0	0
$619$	46.2206	1.85776	0.928882	−	0.370375i	$-0.120771\pi$
$619$	46.2206	1.85776	0.928882	+	0.370375i	$0.120771\pi$
$620$	0	0
$621$	−23.2397	−0.932577
$622$	0	0
$623$	−9.26182	−0.371067
$624$	0	0
$625$	−6.89273	−0.275709
$626$	0	0
$627$	2.14510	0.0856671
$628$	0	0
$629$	43.0598	1.71691
$630$	0	0
$631$	−0.317659	−0.0126458	−0.00632291	−	0.999980i	$-0.502013\pi$
$631$	−0.317659	−0.0126458	−0.00632291	+	0.999980i	$0.502013\pi$
$632$	0	0
$633$	31.8201	1.26474
$634$	0	0
$635$	−34.0735	−1.35217
$636$	0	0
$637$	6.66129	0.263930
$638$	0	0
$639$	−23.9883	−0.948961
$640$	0	0
$641$	−41.5246	−1.64012	−0.820061	−	0.572276i	$-0.806061\pi$
$641$	−41.5246	−1.64012	−0.820061	+	0.572276i	$0.806061\pi$
$642$	0	0
$643$	−7.26182	−0.286378	−0.143189	−	0.989695i	$-0.545736\pi$
$643$	−7.26182	−0.286378	−0.143189	+	0.989695i	$0.545736\pi$
$644$	0	0
$645$	−36.7191	−1.44581
$646$	0	0
$647$	−29.0819	−1.14333	−0.571664	−	0.820487i	$-0.693702\pi$
$647$	−29.0819	−1.14333	−0.571664	+	0.820487i	$0.693702\pi$
$648$	0	0
$649$	2.54364	0.0998465
$650$	0	0
$651$	−46.1207	−1.80761
$652$	0	0
$653$	12.6529	0.495146	0.247573	−	0.968869i	$-0.420367\pi$
$653$	12.6529	0.495146	0.247573	+	0.968869i	$0.420367\pi$
$654$	0	0
$655$	−5.50153	−0.214962
$656$	0	0
$657$	−6.00000	−0.234082
$658$	0	0
$659$	−49.4878	−1.92777	−0.963886	−	0.266317i	$-0.914193\pi$
$659$	−49.4878	−1.92777	−0.963886	+	0.266317i	$0.914193\pi$
$660$	0	0
$661$	4.94950	0.192513	0.0962566	−	0.995357i	$-0.469313\pi$
$661$	4.94950	0.192513	0.0962566	+	0.995357i	$0.469313\pi$
$662$	0	0
$663$	−60.2995	−2.34184
$664$	0	0
$665$	4.63091	0.179579
$666$	0	0
$667$	41.4289	1.60413
$668$	0	0
$669$	−24.6540	−0.953177
$670$	0	0
$671$	13.4931	0.520897
$672$	0	0
$673$	21.9001	0.844185	0.422093	−	0.906553i	$-0.361296\pi$
$673$	21.9001	0.844185	0.422093	+	0.906553i	$0.361296\pi$
$674$	0	0
$675$	7.30592	0.281205
$676$	0	0
$677$	26.4353	1.01599	0.507996	−	0.861360i	$-0.330387\pi$
$677$	26.4353	1.01599	0.507996	+	0.861360i	$0.330387\pi$
$678$	0	0
$679$	20.2020	0.775282
$680$	0	0
$681$	−18.7328	−0.717844
$682$	0	0
$683$	−9.82545	−0.375960	−0.187980	−	0.982173i	$-0.560194\pi$
$683$	−9.82545	−0.375960	−0.187980	+	0.982173i	$0.560194\pi$
$684$	0	0
$685$	−21.7603	−0.831418
$686$	0	0
$687$	16.8201	0.641727
$688$	0	0
$689$	40.7328	1.55180
$690$	0	0
$691$	19.8255	0.754196	0.377098	−	0.926173i	$-0.376922\pi$
$691$	19.8255	0.754196	0.377098	+	0.926173i	$0.376922\pi$
$692$	0	0
$693$	−4.63091	−0.175914
$694$	0	0
$695$	5.11765	0.194123
$696$	0	0
$697$	11.8255	0.447921
$698$	0	0
$699$	−26.8285	−1.01475
$700$	0	0
$701$	−10.7550	−0.406209	−0.203105	−	0.979157i	$-0.565103\pi$
$701$	−10.7550	−0.406209	−0.203105	+	0.979157i	$0.565103\pi$
$702$	0	0
$703$	−7.49314	−0.282609
$704$	0	0
$705$	−24.3470	−0.916960
$706$	0	0
$707$	3.47848	0.130822
$708$	0	0
$709$	−1.82651	−0.0685962	−0.0342981	−	0.999412i	$-0.510920\pi$
$709$	−1.82651	−0.0685962	−0.0342981	+	0.999412i	$0.510920\pi$
$710$	0	0
$711$	13.9265	0.522283
$712$	0	0
$713$	57.5981	2.15707
$714$	0	0
$715$	−7.83384	−0.292969
$716$	0	0
$717$	−52.8221	−1.97268
$718$	0	0
$719$	4.65929	0.173762	0.0868812	−	0.996219i	$-0.472310\pi$
$719$	4.65929	0.173762	0.0868812	+	0.996219i	$0.472310\pi$
$720$	0	0
$721$	48.8055	1.81761
$722$	0	0
$723$	34.4951	1.28289
$724$	0	0
$725$	−13.0241	−0.483703
$726$	0	0
$727$	−19.9358	−0.739377	−0.369688	−	0.929156i	$-0.620536\pi$
$727$	−19.9358	−0.739377	−0.369688	+	0.929156i	$0.620536\pi$
$728$	0	0
$729$	1.30592	0.0483676
$730$	0	0
$731$	−61.4236	−2.27183
$732$	0	0
$733$	13.7687	0.508558	0.254279	−	0.967131i	$-0.418162\pi$
$733$	13.7687	0.508558	0.254279	+	0.967131i	$0.418162\pi$
$734$	0	0
$735$	−4.67807	−0.172553
$736$	0	0
$737$	10.3848	0.382529
$738$	0	0
$739$	−22.4867	−0.827188	−0.413594	−	0.910461i	$-0.635727\pi$
$739$	−22.4867	−0.827188	−0.413594	+	0.910461i	$0.635727\pi$
$740$	0	0
$741$	10.4931	0.385475
$742$	0	0
$743$	50.6540	1.85831	0.929157	−	0.369686i	$-0.120535\pi$
$743$	50.6540	1.85831	0.929157	+	0.369686i	$0.120535\pi$
$744$	0	0
$745$	19.3354	0.708393
$746$	0	0
$747$	9.78228	0.357915
$748$	0	0
$749$	−23.2397	−0.849160
$750$	0	0
$751$	38.9127	1.41995	0.709973	−	0.704229i	$-0.248707\pi$
$751$	38.9127	1.41995	0.709973	+	0.704229i	$0.248707\pi$
$752$	0	0
$753$	−28.0735	−1.02306
$754$	0	0
$755$	−7.86756	−0.286330
$756$	0	0
$757$	−20.5877	−0.748274	−0.374137	−	0.927373i	$-0.622061\pi$
$757$	−20.5877	−0.748274	−0.374137	+	0.927373i	$0.622061\pi$
$758$	0	0
$759$	16.6172	0.603166
$760$	0	0
$761$	10.6025	0.384341	0.192171	−	0.981362i	$-0.438447\pi$
$761$	10.6025	0.384341	0.192171	+	0.981362i	$0.438447\pi$
$762$	0	0
$763$	−33.3407	−1.20701
$764$	0	0
$765$	14.7382	0.532860
$766$	0	0
$767$	12.4426	0.449278
$768$	0	0
$769$	−8.77495	−0.316433	−0.158216	−	0.987404i	$-0.550574\pi$
$769$	−8.77495	−0.316433	−0.158216	+	0.987404i	$0.550574\pi$
$770$	0	0
$771$	1.55203	0.0558949
$772$	0	0
$773$	−46.8653	−1.68563	−0.842813	−	0.538206i	$-0.819102\pi$
$773$	−46.8653	−1.68563	−0.842813	+	0.538206i	$0.819102\pi$
$774$	0	0
$775$	−18.1073	−0.650432
$776$	0	0
$777$	46.4794	1.66744
$778$	0	0
$779$	−2.05783	−0.0737294
$780$	0	0
$781$	−14.9789	−0.535989
$782$	0	0
$783$	−16.0441	−0.573369
$784$	0	0
$785$	−12.8235	−0.457689
$786$	0	0
$787$	27.4143	0.977213	0.488606	−	0.872504i	$-0.337505\pi$
$787$	27.4143	0.977213	0.488606	+	0.872504i	$0.337505\pi$
$788$	0	0
$789$	−29.5510	−1.05204
$790$	0	0
$791$	23.1334	0.822528
$792$	0	0
$793$	66.0040	2.34387
$794$	0	0
$795$	−28.6057	−1.01454
$796$	0	0
$797$	−9.16616	−0.324682	−0.162341	−	0.986735i	$-0.551904\pi$
$797$	−9.16616	−0.324682	−0.162341	+	0.986735i	$0.551904\pi$
$798$	0	0
$799$	−40.7275	−1.44084
$800$	0	0
$801$	5.12938	0.181238
$802$	0	0
$803$	−3.74657	−0.132214
$804$	0	0
$805$	35.8737	1.26438
$806$	0	0
$807$	−42.8117	−1.50704
$808$	0	0
$809$	48.0829	1.69050	0.845252	−	0.534368i	$-0.179450\pi$
$809$	48.0829	1.69050	0.845252	+	0.534368i	$0.179450\pi$
$810$	0	0
$811$	2.67395	0.0938951	0.0469475	−	0.998897i	$-0.485051\pi$
$811$	2.67395	0.0938951	0.0469475	+	0.998897i	$0.485051\pi$
$812$	0	0
$813$	−61.6686	−2.16281
$814$	0	0
$815$	3.01771	0.105706
$816$	0	0
$817$	10.6887	0.373952
$818$	0	0
$819$	−22.6529	−0.791556
$820$	0	0
$821$	29.5078	1.02983	0.514915	−	0.857242i	$-0.327824\pi$
$821$	29.5078	1.02983	0.514915	+	0.857242i	$0.327824\pi$
$822$	0	0
$823$	2.39654	0.0835382	0.0417691	−	0.999127i	$-0.486701\pi$
$823$	2.39654	0.0835382	0.0417691	+	0.999127i	$0.486701\pi$
$824$	0	0
$825$	−5.22399	−0.181876
$826$	0	0
$827$	−7.78868	−0.270839	−0.135419	−	0.990788i	$-0.543238\pi$
$827$	−7.78868	−0.270839	−0.135419	+	0.990788i	$0.543238\pi$
$828$	0	0
$829$	13.2818	0.461296	0.230648	−	0.973037i	$-0.425915\pi$
$829$	13.2818	0.461296	0.230648	+	0.973037i	$0.425915\pi$
$830$	0	0
$831$	6.62252	0.229733
$832$	0	0
$833$	−7.82545	−0.271136
$834$	0	0
$835$	−7.98534	−0.276344
$836$	0	0
$837$	−22.3059	−0.771006
$838$	0	0
$839$	33.9063	1.17058	0.585288	−	0.810825i	$-0.300981\pi$
$839$	33.9063	1.17058	0.585288	+	0.810825i	$0.300981\pi$
$840$	0	0
$841$	−0.398534	−0.0137426
$842$	0	0
$843$	19.5089	0.671921
$844$	0	0
$845$	−17.5015	−0.602071
$846$	0	0
$847$	−2.89167	−0.0993590
$848$	0	0
$849$	−28.1976	−0.967739
$850$	0	0
$851$	−58.0461	−1.98979
$852$	0	0
$853$	40.0735	1.37209	0.686046	−	0.727558i	$-0.259345\pi$
$853$	40.0735	1.37209	0.686046	+	0.727558i	$0.259345\pi$
$854$	0	0
$855$	−2.56469	−0.0877106
$856$	0	0
$857$	0.665693	0.0227396	0.0113698	−	0.999935i	$-0.496381\pi$
$857$	0.665693	0.0227396	0.0113698	+	0.999935i	$0.496381\pi$
$858$	0	0
$859$	−13.8990	−0.474228	−0.237114	−	0.971482i	$-0.576201\pi$
$859$	−13.8990	−0.474228	−0.237114	+	0.971482i	$0.576201\pi$
$860$	0	0
$861$	12.7646	0.435015
$862$	0	0
$863$	40.3941	1.37503	0.687516	−	0.726169i	$-0.258701\pi$
$863$	40.3941	1.37503	0.687516	+	0.726169i	$0.258701\pi$
$864$	0	0
$865$	−10.4323	−0.354707
$866$	0	0
$867$	34.3711	1.16730
$868$	0	0
$869$	8.69607	0.294994
$870$	0	0
$871$	50.7991	1.72126
$872$	0	0
$873$	−11.1883	−0.378666
$874$	0	0
$875$	−34.4323	−1.16402
$876$	0	0
$877$	15.6108	0.527139	0.263569	−	0.964640i	$-0.415100\pi$
$877$	15.6108	0.527139	0.263569	+	0.964640i	$0.415100\pi$
$878$	0	0
$879$	6.66769	0.224895
$880$	0	0
$881$	19.9843	0.673288	0.336644	−	0.941632i	$-0.390708\pi$
$881$	19.9843	0.673288	0.336644	+	0.941632i	$0.390708\pi$
$882$	0	0
$883$	−20.2167	−0.680345	−0.340172	−	0.940363i	$-0.610485\pi$
$883$	−20.2167	−0.680345	−0.340172	+	0.940363i	$0.610485\pi$
$884$	0	0
$885$	−8.73818	−0.293731
$886$	0	0
$887$	−20.1471	−0.676473	−0.338237	−	0.941061i	$-0.609830\pi$
$887$	−20.1471	−0.676473	−0.338237	+	0.941061i	$0.609830\pi$
$888$	0	0
$889$	−61.5246	−2.06347
$890$	0	0
$891$	−11.2397	−0.376544
$892$	0	0
$893$	7.08727	0.237167
$894$	0	0
$895$	−8.88968	−0.297149
$896$	0	0
$897$	81.2858	2.71405
$898$	0	0
$899$	39.7643	1.32621
$900$	0	0
$901$	−47.8516	−1.59417
$902$	0	0
$903$	−66.3015	−2.20638
$904$	0	0
$905$	−3.01771	−0.100312
$906$	0	0
$907$	−4.57109	−0.151781	−0.0758903	−	0.997116i	$-0.524180\pi$
$907$	−4.57109	−0.151781	−0.0758903	+	0.997116i	$0.524180\pi$
$908$	0	0
$909$	−1.92645	−0.0638964
$910$	0	0
$911$	−51.6402	−1.71092	−0.855459	−	0.517871i	$-0.826725\pi$
$911$	−51.6402	−1.71092	−0.855459	+	0.517871i	$0.826725\pi$
$912$	0	0
$913$	6.10833	0.202156
$914$	0	0
$915$	−46.3531	−1.53239
$916$	0	0
$917$	−9.93378	−0.328042
$918$	0	0
$919$	48.8412	1.61112	0.805561	−	0.592513i	$-0.201864\pi$
$919$	48.8412	1.61112	0.805561	+	0.592513i	$0.201864\pi$
$920$	0	0
$921$	−4.69607	−0.154741
$922$	0	0
$923$	−73.2721	−2.41178
$924$	0	0
$925$	18.2481	0.599994
$926$	0	0
$927$	−27.0294	−0.887763
$928$	0	0
$929$	−57.8819	−1.89904	−0.949522	−	0.313700i	$-0.898431\pi$
$929$	−57.8819	−1.89904	−0.949522	+	0.313700i	$0.898431\pi$
$930$	0	0
$931$	1.36176	0.0446299
$932$	0	0
$933$	−8.41120	−0.275370
$934$	0	0
$935$	9.20293	0.300968
$936$	0	0
$937$	−57.6496	−1.88333	−0.941664	−	0.336553i	$-0.890739\pi$
$937$	−57.6496	−1.88333	−0.941664	+	0.336553i	$0.890739\pi$
$938$	0	0
$939$	−46.5740	−1.51989
$940$	0	0
$941$	−15.0966	−0.492135	−0.246067	−	0.969253i	$-0.579138\pi$
$941$	−15.0966	−0.492135	−0.246067	+	0.969253i	$0.579138\pi$
$942$	0	0
$943$	−15.9411	−0.519114
$944$	0	0
$945$	−13.8927	−0.451931
$946$	0	0
$947$	−33.3775	−1.08462	−0.542311	−	0.840178i	$-0.682451\pi$
$947$	−33.3775	−1.08462	−0.542311	+	0.840178i	$0.682451\pi$
$948$	0	0
$949$	−18.3270	−0.594919
$950$	0	0
$951$	14.8171	0.480476
$952$	0	0
$953$	−12.4794	−0.404248	−0.202124	−	0.979360i	$-0.564784\pi$
$953$	−12.4794	−0.404248	−0.202124	+	0.979360i	$0.564784\pi$
$954$	0	0
$955$	3.91180	0.126583
$956$	0	0
$957$	11.4721	0.370840
$958$	0	0
$959$	−39.2913	−1.26878
$960$	0	0
$961$	24.2838	0.783349
$962$	0	0
$963$	12.8706	0.414750
$964$	0	0
$965$	15.4942	0.498776
$966$	0	0
$967$	12.5804	0.404559	0.202279	−	0.979328i	$-0.435165\pi$
$967$	12.5804	0.404559	0.202279	+	0.979328i	$0.435165\pi$
$968$	0	0
$969$	−12.3270	−0.396000
$970$	0	0
$971$	1.79067	0.0574653	0.0287327	−	0.999587i	$-0.490853\pi$
$971$	1.79067	0.0574653	0.0287327	+	0.999587i	$0.490853\pi$
$972$	0	0
$973$	9.24063	0.296241
$974$	0	0
$975$	−25.5540	−0.818384
$976$	0	0
$977$	6.40586	0.204942	0.102471	−	0.994736i	$-0.467325\pi$
$977$	6.40586	0.204942	0.102471	+	0.994736i	$0.467325\pi$
$978$	0	0
$979$	3.20293	0.102366
$980$	0	0
$981$	18.4648	0.589534
$982$	0	0
$983$	20.5647	0.655912	0.327956	−	0.944693i	$-0.393640\pi$
$983$	20.5647	0.655912	0.327956	+	0.944693i	$0.393640\pi$
$984$	0	0
$985$	28.0147	0.892621
$986$	0	0
$987$	−43.9619	−1.39932
$988$	0	0
$989$	82.8011	2.63292
$990$	0	0
$991$	29.0166	0.921744	0.460872	−	0.887467i	$-0.347537\pi$
$991$	29.0166	0.921744	0.460872	+	0.887467i	$0.347537\pi$
$992$	0	0
$993$	−68.8642	−2.18534
$994$	0	0
$995$	−40.4205	−1.28142
$996$	0	0
$997$	42.3638	1.34167	0.670837	−	0.741605i	$-0.265935\pi$
$997$	42.3638	1.34167	0.670837	+	0.741605i	$0.265935\pi$
$998$	0	0
$999$	22.4794	0.711217

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.a.q.1.3		3
4.3	odd	2		418.2.a.g.1.1	✓	3
12.11	even	2		3762.2.a.bg.1.2		3
44.43	even	2		4598.2.a.bo.1.1		3
76.75	even	2		7942.2.a.bi.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
418.2.a.g.1.1	✓	3	4.3	odd	2
3344.2.a.q.1.3		3	1.1	even	1	trivial
3762.2.a.bg.1.2		3	12.11	even	2
4598.2.a.bo.1.1		3	44.43	even	2
7942.2.a.bi.1.3		3	76.75	even	2

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:	\( N \)	\(=\)	\( 3344 = 2^{4} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3344.a (trivial)

\(f(q)\)	\(=\)	\(q+2.14510 q^{3} -1.60147 q^{5} -2.89167 q^{7} +1.60147 q^{9} +1.00000 q^{11} +4.89167 q^{13} -3.43531 q^{15} -5.74657 q^{17} +1.00000 q^{19} -6.20293 q^{21} +7.74657 q^{23} -2.43531 q^{25} -3.00000 q^{27} +5.34803 q^{29} +7.43531 q^{31} +2.14510 q^{33} +4.63091 q^{35} -7.49314 q^{37} +10.4931 q^{39} -2.05783 q^{41} +10.6887 q^{43} -2.56469 q^{45} +7.08727 q^{47} +1.36176 q^{49} -12.3270 q^{51} +8.32698 q^{53} -1.60147 q^{55} +2.14510 q^{57} +2.54364 q^{59} +13.4931 q^{61} -4.63091 q^{63} -7.83384 q^{65} +10.3848 q^{67} +16.6172 q^{69} -14.9789 q^{71} -3.74657 q^{73} -5.22399 q^{75} -2.89167 q^{77} +8.69607 q^{79} -11.2397 q^{81} +6.10833 q^{83} +9.20293 q^{85} +11.4721 q^{87} +3.20293 q^{89} -14.1451 q^{91} +15.9495 q^{93} -1.60147 q^{95} -6.98627 q^{97} +1.60147 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{5} + 6 q^{7} + 3 q^{9} + 3 q^{11} + 9 q^{15} - 9 q^{17} + 3 q^{19} - 15 q^{21} + 15 q^{23} + 12 q^{25} - 9 q^{27} + 6 q^{29} + 3 q^{31} - 6 q^{37} + 15 q^{39} - 9 q^{41} + 21 q^{43} - 27 q^{45}+ \cdots + 3 q^{99}+O(q^{100}) \) 3 * q - 3 * q^5 + 6 * q^7 + 3 * q^9 + 3 * q^11 + 9 * q^15 - 9 * q^17 + 3 * q^19 - 15 * q^21 + 15 * q^23 + 12 * q^25 - 9 * q^27 + 6 * q^29 + 3 * q^31 - 6 * q^37 + 15 * q^39 - 9 * q^41 + 21 * q^43 - 27 * q^45 + 12 * q^47 + 27 * q^49 - 3 * q^51 - 9 * q^53 - 3 * q^55 + 3 * q^59 + 24 * q^61 - 6 * q^65 + 3 * q^69 - 21 * q^71 - 3 * q^73 - 36 * q^75 + 6 * q^77 + 6 * q^79 - 9 * q^81 + 33 * q^83 + 24 * q^85 - 6 * q^87 + 6 * q^89 - 36 * q^91 + 36 * q^93 - 3 * q^95 + 12 * q^97 + 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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