LMFDB - Embedded newform 4624.2.a.bi.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4624,2,Mod(1,4624)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4624, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4624.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 4624 = 2^{4} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4624.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,3,0,-6,0,6,0,12,0,6,0,6,0,-15] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)

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

Self dual:	yes
Analytic conductor:	$36.9228258946$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2312)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.87939$ of defining polynomial
Character	$\chi$	$=$	4624.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.22668 q^{3} -3.87939 q^{5} +0.467911 q^{7} +7.41147 q^{9} +1.30541 q^{11} -2.94356 q^{13} -12.5175 q^{15} +3.18479 q^{19} +1.50980 q^{21} -3.17024 q^{23} +10.0496 q^{25} +14.2344 q^{27} +8.80066 q^{29} -0.857097 q^{31} +4.21213 q^{33} -1.81521 q^{35} +3.49020 q^{37} -9.49794 q^{39} +8.12061 q^{41} -7.00774 q^{43} -28.7520 q^{45} +2.41147 q^{47} -6.78106 q^{49} +1.30541 q^{53} -5.06418 q^{55} +10.2763 q^{57} -9.96585 q^{59} +3.35504 q^{61} +3.46791 q^{63} +11.4192 q^{65} +12.7811 q^{67} -10.2294 q^{69} +12.6236 q^{71} +5.10607 q^{73} +32.4270 q^{75} +0.610815 q^{77} -4.06418 q^{79} +23.6955 q^{81} +8.51754 q^{83} +28.3969 q^{87} -4.31315 q^{89} -1.37733 q^{91} -2.76558 q^{93} -12.3550 q^{95} -6.93582 q^{97} +9.67499 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} - 6 q^{5} + 6 q^{7} + 12 q^{9} + 6 q^{11} + 6 q^{13} - 15 q^{15} + 6 q^{19} + 6 q^{21} + 12 q^{23} + 3 q^{25} + 12 q^{27} + 12 q^{29} - 3 q^{31} - 12 q^{33} - 9 q^{35} + 9 q^{37} - 3 q^{39}+ \cdots + 24 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 - 6 * q^5 + 6 * q^7 + 12 * q^9 + 6 * q^11 + 6 * q^13 - 15 * q^15 + 6 * q^19 + 6 * q^21 + 12 * q^23 + 3 * q^25 + 12 * q^27 + 12 * q^29 - 3 * q^31 - 12 * q^33 - 9 * q^35 + 9 * q^37 - 3 * q^39 + 30 * q^41 + 3 * q^43 - 33 * q^45 - 3 * q^47 - 3 * q^49 + 6 * q^53 - 6 * q^55 - 3 * q^57 - 9 * q^59 - 15 * q^61 + 15 * q^63 + 21 * q^67 - 15 * q^69 + 3 * q^71 + 3 * q^73 + 39 * q^75 + 6 * q^77 - 3 * q^79 + 3 * q^81 + 3 * q^83 + 57 * q^87 + 9 * q^89 + 27 * q^91 - 39 * q^93 - 12 * q^95 - 30 * q^97 + 24 * 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
$3$	3.22668	1.86293	0.931463	−	0.363837i	$-0.118533\pi$
$3$	3.22668	1.86293	0.931463	+	0.363837i	$0.118533\pi$
$4$	0	0
$5$	−3.87939	−1.73491	−0.867457	−	0.497512i	$-0.834247\pi$
$5$	−3.87939	−1.73491	−0.867457	+	0.497512i	$0.834247\pi$
$6$	0	0
$7$	0.467911	0.176854	0.0884269	−	0.996083i	$-0.471816\pi$
$7$	0.467911	0.176854	0.0884269	+	0.996083i	$0.471816\pi$
$8$	0	0
$9$	7.41147	2.47049
$10$	0	0
$11$	1.30541	0.393595	0.196798	−	0.980444i	$-0.436946\pi$
$11$	1.30541	0.393595	0.196798	+	0.980444i	$0.436946\pi$
$12$	0	0
$13$	−2.94356	−0.816397	−0.408199	−	0.912893i	$-0.633843\pi$
$13$	−2.94356	−0.816397	−0.408199	+	0.912893i	$0.633843\pi$
$14$	0	0
$15$	−12.5175	−3.23202
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	3.18479	0.730642	0.365321	−	0.930882i	$-0.380959\pi$
$19$	3.18479	0.730642	0.365321	+	0.930882i	$0.380959\pi$
$20$	0	0
$21$	1.50980	0.329465
$22$	0	0
$23$	−3.17024	−0.661042	−0.330521	−	0.943799i	$-0.607224\pi$
$23$	−3.17024	−0.661042	−0.330521	+	0.943799i	$0.607224\pi$
$24$	0	0
$25$	10.0496	2.00993
$26$	0	0
$27$	14.2344	2.73942
$28$	0	0
$29$	8.80066	1.63424	0.817121	−	0.576467i	$-0.195569\pi$
$29$	8.80066	1.63424	0.817121	+	0.576467i	$0.195569\pi$
$30$	0	0
$31$	−0.857097	−0.153939	−0.0769695	−	0.997033i	$-0.524524\pi$
$31$	−0.857097	−0.153939	−0.0769695	+	0.997033i	$0.524524\pi$
$32$	0	0
$33$	4.21213	0.733238
$34$	0	0
$35$	−1.81521	−0.306826
$36$	0	0
$37$	3.49020	0.573785	0.286893	−	0.957963i	$-0.407378\pi$
$37$	3.49020	0.573785	0.286893	+	0.957963i	$0.407378\pi$
$38$	0	0
$39$	−9.49794	−1.52089
$40$	0	0
$41$	8.12061	1.26823	0.634113	−	0.773240i	$-0.281365\pi$
$41$	8.12061	1.26823	0.634113	+	0.773240i	$0.281365\pi$
$42$	0	0
$43$	−7.00774	−1.06867	−0.534335	−	0.845273i	$-0.679438\pi$
$43$	−7.00774	−1.06867	−0.534335	+	0.845273i	$0.679438\pi$
$44$	0	0
$45$	−28.7520	−4.28609
$46$	0	0
$47$	2.41147	0.351750	0.175875	−	0.984413i	$-0.443725\pi$
$47$	2.41147	0.351750	0.175875	+	0.984413i	$0.443725\pi$
$48$	0	0
$49$	−6.78106	−0.968723
$50$	0	0
$51$	0	0
$52$	0	0
$53$	1.30541	0.179311	0.0896557	−	0.995973i	$-0.471423\pi$
$53$	1.30541	0.179311	0.0896557	+	0.995973i	$0.471423\pi$
$54$	0	0
$55$	−5.06418	−0.682854
$56$	0	0
$57$	10.2763	1.36113
$58$	0	0
$59$	−9.96585	−1.29744	−0.648722	−	0.761026i	$-0.724696\pi$
$59$	−9.96585	−1.29744	−0.648722	+	0.761026i	$0.724696\pi$
$60$	0	0
$61$	3.35504	0.429568	0.214784	−	0.976662i	$-0.431095\pi$
$61$	3.35504	0.429568	0.214784	+	0.976662i	$0.431095\pi$
$62$	0	0
$63$	3.46791	0.436916
$64$	0	0
$65$	11.4192	1.41638
$66$	0	0
$67$	12.7811	1.56145	0.780727	−	0.624872i	$-0.214849\pi$
$67$	12.7811	1.56145	0.780727	+	0.624872i	$0.214849\pi$
$68$	0	0
$69$	−10.2294	−1.23147
$70$	0	0
$71$	12.6236	1.49815	0.749073	−	0.662487i	$-0.230499\pi$
$71$	12.6236	1.49815	0.749073	+	0.662487i	$0.230499\pi$
$72$	0	0
$73$	5.10607	0.597620	0.298810	−	0.954313i	$-0.403410\pi$
$73$	5.10607	0.597620	0.298810	+	0.954313i	$0.403410\pi$
$74$	0	0
$75$	32.4270	3.74434
$76$	0	0
$77$	0.610815	0.0696088
$78$	0	0
$79$	−4.06418	−0.457256	−0.228628	−	0.973514i	$-0.573424\pi$
$79$	−4.06418	−0.457256	−0.228628	+	0.973514i	$0.573424\pi$
$80$	0	0
$81$	23.6955	2.63284
$82$	0	0
$83$	8.51754	0.934922	0.467461	−	0.884014i	$-0.345169\pi$
$83$	8.51754	0.934922	0.467461	+	0.884014i	$0.345169\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	28.3969	3.04447
$88$	0	0
$89$	−4.31315	−0.457193	−0.228596	−	0.973521i	$-0.573414\pi$
$89$	−4.31315	−0.457193	−0.228596	+	0.973521i	$0.573414\pi$
$90$	0	0
$91$	−1.37733	−0.144383
$92$	0	0
$93$	−2.76558	−0.286777
$94$	0	0
$95$	−12.3550	−1.26760
$96$	0	0
$97$	−6.93582	−0.704226	−0.352113	−	0.935957i	$-0.614537\pi$
$97$	−6.93582	−0.704226	−0.352113	+	0.935957i	$0.614537\pi$
$98$	0	0
$99$	9.67499	0.972373
$100$	0	0
$101$	13.9094	1.38404	0.692019	−	0.721879i	$-0.256721\pi$
$101$	13.9094	1.38404	0.692019	+	0.721879i	$0.256721\pi$
$102$	0	0
$103$	13.7219	1.35206	0.676031	−	0.736873i	$-0.263698\pi$
$103$	13.7219	1.35206	0.676031	+	0.736873i	$0.263698\pi$
$104$	0	0
$105$	−5.85710	−0.571594
$106$	0	0
$107$	15.9290	1.53992	0.769958	−	0.638095i	$-0.220277\pi$
$107$	15.9290	1.53992	0.769958	+	0.638095i	$0.220277\pi$
$108$	0	0
$109$	−1.22163	−0.117011	−0.0585054	−	0.998287i	$-0.518633\pi$
$109$	−1.22163	−0.117011	−0.0585054	+	0.998287i	$0.518633\pi$
$110$	0	0
$111$	11.2618	1.06892
$112$	0	0
$113$	−2.47565	−0.232890	−0.116445	−	0.993197i	$-0.537150\pi$
$113$	−2.47565	−0.232890	−0.116445	+	0.993197i	$0.537150\pi$
$114$	0	0
$115$	12.2986	1.14685
$116$	0	0
$117$	−21.8161	−2.01690
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−9.29591	−0.845083
$122$	0	0
$123$	26.2026	2.36261
$124$	0	0
$125$	−19.5895	−1.75213
$126$	0	0
$127$	−2.36184	−0.209580	−0.104790	−	0.994494i	$-0.533417\pi$
$127$	−2.36184	−0.209580	−0.104790	+	0.994494i	$0.533417\pi$
$128$	0	0
$129$	−22.6117	−1.99085
$130$	0	0
$131$	1.07098	0.0935724	0.0467862	−	0.998905i	$-0.485102\pi$
$131$	1.07098	0.0935724	0.0467862	+	0.998905i	$0.485102\pi$
$132$	0	0
$133$	1.49020	0.129217
$134$	0	0
$135$	−55.2208	−4.75265
$136$	0	0
$137$	19.5749	1.67240	0.836199	−	0.548426i	$-0.184773\pi$
$137$	19.5749	1.67240	0.836199	+	0.548426i	$0.184773\pi$
$138$	0	0
$139$	7.95811	0.674998	0.337499	−	0.941326i	$-0.390419\pi$
$139$	7.95811	0.674998	0.337499	+	0.941326i	$0.390419\pi$
$140$	0	0
$141$	7.78106	0.655283
$142$	0	0
$143$	−3.84255	−0.321330
$144$	0	0
$145$	−34.1411	−2.83527
$146$	0	0
$147$	−21.8803	−1.80466
$148$	0	0
$149$	1.65270	0.135395	0.0676974	−	0.997706i	$-0.478435\pi$
$149$	1.65270	0.135395	0.0676974	+	0.997706i	$0.478435\pi$
$150$	0	0
$151$	−15.8280	−1.28806	−0.644032	−	0.764998i	$-0.722740\pi$
$151$	−15.8280	−1.28806	−0.644032	+	0.764998i	$0.722740\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	3.32501	0.267071
$156$	0	0
$157$	−6.80840	−0.543370	−0.271685	−	0.962386i	$-0.587581\pi$
$157$	−6.80840	−0.543370	−0.271685	+	0.962386i	$0.587581\pi$
$158$	0	0
$159$	4.21213	0.334044
$160$	0	0
$161$	−1.48339	−0.116908
$162$	0	0
$163$	3.81521	0.298830	0.149415	−	0.988775i	$-0.452261\pi$
$163$	3.81521	0.298830	0.149415	+	0.988775i	$0.452261\pi$
$164$	0	0
$165$	−16.3405	−1.27211
$166$	0	0
$167$	−11.7023	−0.905554	−0.452777	−	0.891624i	$-0.649567\pi$
$167$	−11.7023	−0.905554	−0.452777	+	0.891624i	$0.649567\pi$
$168$	0	0
$169$	−4.33544	−0.333495
$170$	0	0
$171$	23.6040	1.80504
$172$	0	0
$173$	15.1361	1.15078	0.575388	−	0.817881i	$-0.304851\pi$
$173$	15.1361	1.15078	0.575388	+	0.817881i	$0.304851\pi$
$174$	0	0
$175$	4.70233	0.355463
$176$	0	0
$177$	−32.1566	−2.41704
$178$	0	0
$179$	−6.10607	−0.456389	−0.228194	−	0.973616i	$-0.573282\pi$
$179$	−6.10607	−0.456389	−0.228194	+	0.973616i	$0.573282\pi$
$180$	0	0
$181$	7.28581	0.541550	0.270775	−	0.962643i	$-0.412720\pi$
$181$	7.28581	0.541550	0.270775	+	0.962643i	$0.412720\pi$
$182$	0	0
$183$	10.8256	0.800254
$184$	0	0
$185$	−13.5398	−0.995468
$186$	0	0
$187$	0	0
$188$	0	0
$189$	6.66044	0.484476
$190$	0	0
$191$	16.5253	1.19573	0.597864	−	0.801598i	$-0.296016\pi$
$191$	16.5253	1.19573	0.597864	+	0.801598i	$0.296016\pi$
$192$	0	0
$193$	−5.65270	−0.406891	−0.203445	−	0.979086i	$-0.565214\pi$
$193$	−5.65270	−0.406891	−0.203445	+	0.979086i	$0.565214\pi$
$194$	0	0
$195$	36.8462	2.63861
$196$	0	0
$197$	0.834808	0.0594776	0.0297388	−	0.999558i	$-0.490532\pi$
$197$	0.834808	0.0594776	0.0297388	+	0.999558i	$0.490532\pi$
$198$	0	0
$199$	23.0719	1.63552	0.817762	−	0.575556i	$-0.195214\pi$
$199$	23.0719	1.63552	0.817762	+	0.575556i	$0.195214\pi$
$200$	0	0
$201$	41.2404	2.90887
$202$	0	0
$203$	4.11793	0.289022
$204$	0	0
$205$	−31.5030	−2.20026
$206$	0	0
$207$	−23.4962	−1.63310
$208$	0	0
$209$	4.15745	0.287577
$210$	0	0
$211$	−17.8530	−1.22905	−0.614525	−	0.788897i	$-0.710652\pi$
$211$	−17.8530	−1.22905	−0.614525	+	0.788897i	$0.710652\pi$
$212$	0	0
$213$	40.7324	2.79094
$214$	0	0
$215$	27.1857	1.85405
$216$	0	0
$217$	−0.401045	−0.0272247
$218$	0	0
$219$	16.4757	1.11332
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−14.6750	−0.982710	−0.491355	−	0.870959i	$-0.663498\pi$
$223$	−14.6750	−0.982710	−0.491355	+	0.870959i	$0.663498\pi$
$224$	0	0
$225$	74.4826	4.96550
$226$	0	0
$227$	18.6013	1.23461	0.617306	−	0.786723i	$-0.288224\pi$
$227$	18.6013	1.23461	0.617306	+	0.786723i	$0.288224\pi$
$228$	0	0
$229$	7.78880	0.514698	0.257349	−	0.966318i	$-0.417151\pi$
$229$	7.78880	0.514698	0.257349	+	0.966318i	$0.417151\pi$
$230$	0	0
$231$	1.97090	0.129676
$232$	0	0
$233$	−15.4466	−1.01194	−0.505969	−	0.862552i	$-0.668865\pi$
$233$	−15.4466	−1.01194	−0.505969	+	0.862552i	$0.668865\pi$
$234$	0	0
$235$	−9.35504	−0.610255
$236$	0	0
$237$	−13.1138	−0.851833
$238$	0	0
$239$	−26.9368	−1.74239	−0.871197	−	0.490934i	$-0.836656\pi$
$239$	−26.9368	−1.74239	−0.871197	+	0.490934i	$0.836656\pi$
$240$	0	0
$241$	−19.7246	−1.27057	−0.635287	−	0.772276i	$-0.719118\pi$
$241$	−19.7246	−1.27057	−0.635287	+	0.772276i	$0.719118\pi$
$242$	0	0
$243$	33.7547	2.16536
$244$	0	0
$245$	26.3063	1.68065
$246$	0	0
$247$	−9.37464	−0.596494
$248$	0	0
$249$	27.4834	1.74169
$250$	0	0
$251$	−26.9864	−1.70337	−0.851683	−	0.524058i	$-0.824418\pi$
$251$	−26.9864	−1.70337	−0.851683	+	0.524058i	$0.824418\pi$
$252$	0	0
$253$	−4.13846	−0.260183
$254$	0	0
$255$	0	0
$256$	0	0
$257$	28.2422	1.76170	0.880849	−	0.473398i	$-0.156973\pi$
$257$	28.2422	1.76170	0.880849	+	0.473398i	$0.156973\pi$
$258$	0	0
$259$	1.63310	0.101476
$260$	0	0
$261$	65.2259	4.03738
$262$	0	0
$263$	−5.96585	−0.367870	−0.183935	−	0.982938i	$-0.558884\pi$
$263$	−5.96585	−0.367870	−0.183935	+	0.982938i	$0.558884\pi$
$264$	0	0
$265$	−5.06418	−0.311090
$266$	0	0
$267$	−13.9172	−0.851716
$268$	0	0
$269$	23.7392	1.44740	0.723701	−	0.690113i	$-0.242439\pi$
$269$	23.7392	1.44740	0.723701	+	0.690113i	$0.242439\pi$
$270$	0	0
$271$	6.56212	0.398620	0.199310	−	0.979936i	$-0.436130\pi$
$271$	6.56212	0.398620	0.199310	+	0.979936i	$0.436130\pi$
$272$	0	0
$273$	−4.44419	−0.268975
$274$	0	0
$275$	13.1189	0.791097
$276$	0	0
$277$	13.7638	0.826988	0.413494	−	0.910507i	$-0.364308\pi$
$277$	13.7638	0.826988	0.413494	+	0.910507i	$0.364308\pi$
$278$	0	0
$279$	−6.35235	−0.380305
$280$	0	0
$281$	−23.0446	−1.37472	−0.687362	−	0.726315i	$-0.741231\pi$
$281$	−23.0446	−1.37472	−0.687362	+	0.726315i	$0.741231\pi$
$282$	0	0
$283$	24.0077	1.42711	0.713556	−	0.700598i	$-0.247083\pi$
$283$	24.0077	1.42711	0.713556	+	0.700598i	$0.247083\pi$
$284$	0	0
$285$	−39.8658	−2.36144
$286$	0	0
$287$	3.79973	0.224291
$288$	0	0
$289$	0	0
$290$	0	0
$291$	−22.3797	−1.31192
$292$	0	0
$293$	−21.2472	−1.24128	−0.620638	−	0.784097i	$-0.713126\pi$
$293$	−21.2472	−1.24128	−0.620638	+	0.784097i	$0.713126\pi$
$294$	0	0
$295$	38.6614	2.25095
$296$	0	0
$297$	18.5817	1.07822
$298$	0	0
$299$	9.33181	0.539673
$300$	0	0
$301$	−3.27900	−0.188998
$302$	0	0
$303$	44.8813	2.57836
$304$	0	0
$305$	−13.0155	−0.745264
$306$	0	0
$307$	−13.3277	−0.760652	−0.380326	−	0.924853i	$-0.624188\pi$
$307$	−13.3277	−0.760652	−0.380326	+	0.924853i	$0.624188\pi$
$308$	0	0
$309$	44.2763	2.51879
$310$	0	0
$311$	−10.5895	−0.600473	−0.300237	−	0.953865i	$-0.597066\pi$
$311$	−10.5895	−0.600473	−0.300237	+	0.953865i	$0.597066\pi$
$312$	0	0
$313$	−4.38919	−0.248091	−0.124046	−	0.992277i	$-0.539587\pi$
$313$	−4.38919	−0.248091	−0.124046	+	0.992277i	$0.539587\pi$
$314$	0	0
$315$	−13.4534	−0.758011
$316$	0	0
$317$	−1.36184	−0.0764888	−0.0382444	−	0.999268i	$-0.512177\pi$
$317$	−1.36184	−0.0764888	−0.0382444	+	0.999268i	$0.512177\pi$
$318$	0	0
$319$	11.4884	0.643229
$320$	0	0
$321$	51.3979	2.86875
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−29.5817	−1.64090
$326$	0	0
$327$	−3.94181	−0.217982
$328$	0	0
$329$	1.12836	0.0622083
$330$	0	0
$331$	−17.9418	−0.986171	−0.493085	−	0.869981i	$-0.664131\pi$
$331$	−17.9418	−0.986171	−0.493085	+	0.869981i	$0.664131\pi$
$332$	0	0
$333$	25.8675	1.41753
$334$	0	0
$335$	−49.5827	−2.70899
$336$	0	0
$337$	−32.6091	−1.77633	−0.888164	−	0.459526i	$-0.848019\pi$
$337$	−32.6091	−1.77633	−0.888164	+	0.459526i	$0.848019\pi$
$338$	0	0
$339$	−7.98814	−0.433856
$340$	0	0
$341$	−1.11886	−0.0605897
$342$	0	0
$343$	−6.44831	−0.348176
$344$	0	0
$345$	39.6837	2.13650
$346$	0	0
$347$	−11.3696	−0.610351	−0.305176	−	0.952296i	$-0.598715\pi$
$347$	−11.3696	−0.610351	−0.305176	+	0.952296i	$0.598715\pi$
$348$	0	0
$349$	−14.0000	−0.749403	−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000	−0.749403	−0.374701	+	0.927146i	$0.622255\pi$
$350$	0	0
$351$	−41.8999	−2.23645
$352$	0	0
$353$	−11.1506	−0.593489	−0.296744	−	0.954957i	$-0.595901\pi$
$353$	−11.1506	−0.593489	−0.296744	+	0.954957i	$0.595901\pi$
$354$	0	0
$355$	−48.9718	−2.59916
$356$	0	0
$357$	0	0
$358$	0	0
$359$	19.4442	1.02623	0.513113	−	0.858321i	$-0.328492\pi$
$359$	19.4442	1.02623	0.513113	+	0.858321i	$0.328492\pi$
$360$	0	0
$361$	−8.85710	−0.466163
$362$	0	0
$363$	−29.9949	−1.57433
$364$	0	0
$365$	−19.8084	−1.03682
$366$	0	0
$367$	29.4662	1.53812	0.769060	−	0.639176i	$-0.220724\pi$
$367$	29.4662	1.53812	0.769060	+	0.639176i	$0.220724\pi$
$368$	0	0
$369$	60.1857	3.13314
$370$	0	0
$371$	0.610815	0.0317119
$372$	0	0
$373$	−31.2891	−1.62009	−0.810044	−	0.586369i	$-0.800557\pi$
$373$	−31.2891	−1.62009	−0.810044	+	0.586369i	$0.800557\pi$
$374$	0	0
$375$	−63.2089	−3.26410
$376$	0	0
$377$	−25.9053	−1.33419
$378$	0	0
$379$	31.9932	1.64338	0.821690	−	0.569935i	$-0.193032\pi$
$379$	31.9932	1.64338	0.821690	+	0.569935i	$0.193032\pi$
$380$	0	0
$381$	−7.62092	−0.390432
$382$	0	0
$383$	4.00505	0.204649	0.102324	−	0.994751i	$-0.467372\pi$
$383$	4.00505	0.204649	0.102324	+	0.994751i	$0.467372\pi$
$384$	0	0
$385$	−2.36959	−0.120765
$386$	0	0
$387$	−51.9377	−2.64014
$388$	0	0
$389$	13.0155	0.659911	0.329956	−	0.943996i	$-0.392966\pi$
$389$	13.0155	0.659911	0.329956	+	0.943996i	$0.392966\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	3.45573	0.174318
$394$	0	0
$395$	15.7665	0.793299
$396$	0	0
$397$	−37.0523	−1.85960	−0.929801	−	0.368062i	$-0.880021\pi$
$397$	−37.0523	−1.85960	−0.929801	+	0.368062i	$0.880021\pi$
$398$	0	0
$399$	4.80840	0.240721
$400$	0	0
$401$	8.84255	0.441576	0.220788	−	0.975322i	$-0.429137\pi$
$401$	8.84255	0.441576	0.220788	+	0.975322i	$0.429137\pi$
$402$	0	0
$403$	2.52292	0.125675
$404$	0	0
$405$	−91.9241	−4.56774
$406$	0	0
$407$	4.55613	0.225839
$408$	0	0
$409$	10.0942	0.499126	0.249563	−	0.968359i	$-0.419713\pi$
$409$	10.0942	0.499126	0.249563	+	0.968359i	$0.419713\pi$
$410$	0	0
$411$	63.1620	3.11555
$412$	0	0
$413$	−4.66313	−0.229458
$414$	0	0
$415$	−33.0428	−1.62201
$416$	0	0
$417$	25.6783	1.25747
$418$	0	0
$419$	19.2226	0.939084	0.469542	−	0.882910i	$-0.344419\pi$
$419$	19.2226	0.939084	0.469542	+	0.882910i	$0.344419\pi$
$420$	0	0
$421$	13.1925	0.642965	0.321482	−	0.946916i	$-0.395819\pi$
$421$	13.1925	0.642965	0.321482	+	0.946916i	$0.395819\pi$
$422$	0	0
$423$	17.8726	0.868994
$424$	0	0
$425$	0	0
$426$	0	0
$427$	1.56986	0.0759708
$428$	0	0
$429$	−12.3987	−0.598614
$430$	0	0
$431$	−18.8307	−0.907042	−0.453521	−	0.891245i	$-0.649832\pi$
$431$	−18.8307	−0.907042	−0.453521	+	0.891245i	$0.649832\pi$
$432$	0	0
$433$	−4.19665	−0.201678	−0.100839	−	0.994903i	$-0.532153\pi$
$433$	−4.19665	−0.201678	−0.100839	+	0.994903i	$0.532153\pi$
$434$	0	0
$435$	−110.163	−5.28189
$436$	0	0
$437$	−10.0966	−0.482985
$438$	0	0
$439$	32.8708	1.56884	0.784419	−	0.620231i	$-0.212961\pi$
$439$	32.8708	1.56884	0.784419	+	0.620231i	$0.212961\pi$
$440$	0	0
$441$	−50.2576	−2.39322
$442$	0	0
$443$	13.3979	0.636552	0.318276	−	0.947998i	$-0.396896\pi$
$443$	13.3979	0.636552	0.318276	+	0.947998i	$0.396896\pi$
$444$	0	0
$445$	16.7324	0.793190
$446$	0	0
$447$	5.33275	0.252230
$448$	0	0
$449$	−10.3446	−0.488192	−0.244096	−	0.969751i	$-0.578491\pi$
$449$	−10.3446	−0.488192	−0.244096	+	0.969751i	$0.578491\pi$
$450$	0	0
$451$	10.6007	0.499168
$452$	0	0
$453$	−51.0719	−2.39957
$454$	0	0
$455$	5.34318	0.250492
$456$	0	0
$457$	−20.4929	−0.958617	−0.479308	−	0.877647i	$-0.659112\pi$
$457$	−20.4929	−0.958617	−0.479308	+	0.877647i	$0.659112\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	12.9017	0.600891	0.300445	−	0.953799i	$-0.402865\pi$
$461$	12.9017	0.600891	0.300445	+	0.953799i	$0.402865\pi$
$462$	0	0
$463$	−9.54488	−0.443588	−0.221794	−	0.975094i	$-0.571191\pi$
$463$	−9.54488	−0.443588	−0.221794	+	0.975094i	$0.571191\pi$
$464$	0	0
$465$	10.7287	0.497533
$466$	0	0
$467$	−16.5790	−0.767186	−0.383593	−	0.923502i	$-0.625313\pi$
$467$	−16.5790	−0.767186	−0.383593	+	0.923502i	$0.625313\pi$
$468$	0	0
$469$	5.98040	0.276149
$470$	0	0
$471$	−21.9685	−1.01226
$472$	0	0
$473$	−9.14796	−0.420623
$474$	0	0
$475$	32.0060	1.46854
$476$	0	0
$477$	9.67499	0.442987
$478$	0	0
$479$	−1.13247	−0.0517441	−0.0258720	−	0.999665i	$-0.508236\pi$
$479$	−1.13247	−0.0517441	−0.0258720	+	0.999665i	$0.508236\pi$
$480$	0	0
$481$	−10.2736	−0.468437
$482$	0	0
$483$	−4.78644	−0.217790
$484$	0	0
$485$	26.9067	1.22177
$486$	0	0
$487$	−26.2267	−1.18844	−0.594222	−	0.804301i	$-0.702540\pi$
$487$	−26.2267	−1.18844	−0.594222	+	0.804301i	$0.702540\pi$
$488$	0	0
$489$	12.3105	0.556698
$490$	0	0
$491$	−3.40104	−0.153487	−0.0767435	−	0.997051i	$-0.524452\pi$
$491$	−3.40104	−0.153487	−0.0767435	+	0.997051i	$0.524452\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−37.5330	−1.68698
$496$	0	0
$497$	5.90673	0.264953
$498$	0	0
$499$	−29.6587	−1.32771	−0.663853	−	0.747863i	$-0.731080\pi$
$499$	−29.6587	−1.32771	−0.663853	+	0.747863i	$0.731080\pi$
$500$	0	0
$501$	−37.7597	−1.68698
$502$	0	0
$503$	−12.7861	−0.570105	−0.285052	−	0.958512i	$-0.592011\pi$
$503$	−12.7861	−0.570105	−0.285052	+	0.958512i	$0.592011\pi$
$504$	0	0
$505$	−53.9600	−2.40119
$506$	0	0
$507$	−13.9891	−0.621277
$508$	0	0
$509$	−29.3337	−1.30019	−0.650096	−	0.759852i	$-0.725271\pi$
$509$	−29.3337	−1.30019	−0.650096	+	0.759852i	$0.725271\pi$
$510$	0	0
$511$	2.38919	0.105691
$512$	0	0
$513$	45.3337	2.00153
$514$	0	0
$515$	−53.2327	−2.34571
$516$	0	0
$517$	3.14796	0.138447
$518$	0	0
$519$	48.8394	2.14381
$520$	0	0
$521$	22.8075	0.999213	0.499607	−	0.866252i	$-0.333478\pi$
$521$	22.8075	0.999213	0.499607	+	0.866252i	$0.333478\pi$
$522$	0	0
$523$	14.5422	0.635886	0.317943	−	0.948110i	$-0.397008\pi$
$523$	14.5422	0.635886	0.317943	+	0.948110i	$0.397008\pi$
$524$	0	0
$525$	15.1729	0.662201
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−12.9495	−0.563024
$530$	0	0
$531$	−73.8617	−3.20532
$532$	0	0
$533$	−23.9035	−1.03538
$534$	0	0
$535$	−61.7948	−2.67162
$536$	0	0
$537$	−19.7023	−0.850218
$538$	0	0
$539$	−8.85204	−0.381285
$540$	0	0
$541$	20.0651	0.862667	0.431333	−	0.902193i	$-0.358043\pi$
$541$	20.0651	0.862667	0.431333	+	0.902193i	$0.358043\pi$
$542$	0	0
$543$	23.5090	1.00887
$544$	0	0
$545$	4.73917	0.203004
$546$	0	0
$547$	−14.1019	−0.602956	−0.301478	−	0.953473i	$-0.597480\pi$
$547$	−14.1019	−0.602956	−0.301478	+	0.953473i	$0.597480\pi$
$548$	0	0
$549$	24.8658	1.06125
$550$	0	0
$551$	28.0283	1.19404
$552$	0	0
$553$	−1.90167	−0.0808674
$554$	0	0
$555$	−43.6887	−1.85448
$556$	0	0
$557$	−27.2344	−1.15396	−0.576980	−	0.816758i	$-0.695769\pi$
$557$	−27.2344	−1.15396	−0.576980	+	0.816758i	$0.695769\pi$
$558$	0	0
$559$	20.6277	0.872460
$560$	0	0
$561$	0	0
$562$	0	0
$563$	11.6869	0.492542	0.246271	−	0.969201i	$-0.420795\pi$
$563$	11.6869	0.492542	0.246271	+	0.969201i	$0.420795\pi$
$564$	0	0
$565$	9.60401	0.404044
$566$	0	0
$567$	11.0874	0.465627
$568$	0	0
$569$	−10.7297	−0.449811	−0.224906	−	0.974381i	$-0.572207\pi$
$569$	−10.7297	−0.449811	−0.224906	+	0.974381i	$0.572207\pi$
$570$	0	0
$571$	8.86484	0.370982	0.185491	−	0.982646i	$-0.440612\pi$
$571$	8.86484	0.370982	0.185491	+	0.982646i	$0.440612\pi$
$572$	0	0
$573$	53.3218	2.22755
$574$	0	0
$575$	−31.8598	−1.32864
$576$	0	0
$577$	28.8307	1.20024	0.600119	−	0.799911i	$-0.295120\pi$
$577$	28.8307	1.20024	0.600119	+	0.799911i	$0.295120\pi$
$578$	0	0
$579$	−18.2395	−0.758007
$580$	0	0
$581$	3.98545	0.165344
$582$	0	0
$583$	1.70409	0.0705761
$584$	0	0
$585$	84.6332	3.49915
$586$	0	0
$587$	4.58584	0.189278	0.0946389	−	0.995512i	$-0.469830\pi$
$587$	4.58584	0.189278	0.0946389	+	0.995512i	$0.469830\pi$
$588$	0	0
$589$	−2.72967	−0.112474
$590$	0	0
$591$	2.69366	0.110802
$592$	0	0
$593$	−21.8108	−0.895661	−0.447830	−	0.894119i	$-0.647803\pi$
$593$	−21.8108	−0.895661	−0.447830	+	0.894119i	$0.647803\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	74.4457	3.04686
$598$	0	0
$599$	5.33544	0.218000	0.109000	−	0.994042i	$-0.465235\pi$
$599$	5.33544	0.218000	0.109000	+	0.994042i	$0.465235\pi$
$600$	0	0
$601$	16.7273	0.682321	0.341161	−	0.940005i	$-0.389180\pi$
$601$	16.7273	0.682321	0.341161	+	0.940005i	$0.389180\pi$
$602$	0	0
$603$	94.7265	3.85756
$604$	0	0
$605$	36.0624	1.46615
$606$	0	0
$607$	−30.8794	−1.25336	−0.626678	−	0.779278i	$-0.715586\pi$
$607$	−30.8794	−1.25336	−0.626678	+	0.779278i	$0.715586\pi$
$608$	0	0
$609$	13.2872	0.538426
$610$	0	0
$611$	−7.09833	−0.287168
$612$	0	0
$613$	−46.6432	−1.88390	−0.941951	−	0.335751i	$-0.891010\pi$
$613$	−46.6432	−1.88390	−0.941951	+	0.335751i	$0.891010\pi$
$614$	0	0
$615$	−101.650	−4.09893
$616$	0	0
$617$	2.08109	0.0837815	0.0418908	−	0.999122i	$-0.486662\pi$
$617$	2.08109	0.0837815	0.0418908	+	0.999122i	$0.486662\pi$
$618$	0	0
$619$	−35.6168	−1.43156	−0.715780	−	0.698326i	$-0.753929\pi$
$619$	−35.6168	−1.43156	−0.715780	+	0.698326i	$0.753929\pi$
$620$	0	0
$621$	−45.1266	−1.81087
$622$	0	0
$623$	−2.01817	−0.0808563
$624$	0	0
$625$	25.7469	1.02988
$626$	0	0
$627$	13.4148	0.535734
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−23.7939	−0.947218	−0.473609	−	0.880735i	$-0.657049\pi$
$631$	−23.7939	−0.947218	−0.473609	+	0.880735i	$0.657049\pi$
$632$	0	0
$633$	−57.6059	−2.28963
$634$	0	0
$635$	9.16250	0.363603
$636$	0	0
$637$	19.9605	0.790863
$638$	0	0
$639$	93.5595	3.70116
$640$	0	0
$641$	−9.01455	−0.356053	−0.178027	−	0.984026i	$-0.556971\pi$
$641$	−9.01455	−0.356053	−0.178027	+	0.984026i	$0.556971\pi$
$642$	0	0
$643$	3.71419	0.146473	0.0732367	−	0.997315i	$-0.476667\pi$
$643$	3.71419	0.146473	0.0732367	+	0.997315i	$0.476667\pi$
$644$	0	0
$645$	87.7197	3.45396
$646$	0	0
$647$	−4.28405	−0.168423	−0.0842117	−	0.996448i	$-0.526837\pi$
$647$	−4.28405	−0.168423	−0.0842117	+	0.996448i	$0.526837\pi$
$648$	0	0
$649$	−13.0095	−0.510667
$650$	0	0
$651$	−1.29404	−0.0507176
$652$	0	0
$653$	−4.64765	−0.181877	−0.0909383	−	0.995857i	$-0.528987\pi$
$653$	−4.64765	−0.181877	−0.0909383	+	0.995857i	$0.528987\pi$
$654$	0	0
$655$	−4.15476	−0.162340
$656$	0	0
$657$	37.8435	1.47641
$658$	0	0
$659$	45.0259	1.75396	0.876980	−	0.480526i	$-0.159554\pi$
$659$	45.0259	1.75396	0.876980	+	0.480526i	$0.159554\pi$
$660$	0	0
$661$	−17.7537	−0.690540	−0.345270	−	0.938503i	$-0.612213\pi$
$661$	−17.7537	−0.690540	−0.345270	+	0.938503i	$0.612213\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−5.78106	−0.224180
$666$	0	0
$667$	−27.9002	−1.08030
$668$	0	0
$669$	−47.3515	−1.83072
$670$	0	0
$671$	4.37969	0.169076
$672$	0	0
$673$	34.5357	1.33125	0.665627	−	0.746285i	$-0.268164\pi$
$673$	34.5357	1.33125	0.665627	+	0.746285i	$0.268164\pi$
$674$	0	0
$675$	143.051	5.50602
$676$	0	0
$677$	29.1138	1.11893	0.559467	−	0.828852i	$-0.311006\pi$
$677$	29.1138	1.11893	0.559467	+	0.828852i	$0.311006\pi$
$678$	0	0
$679$	−3.24535	−0.124545
$680$	0	0
$681$	60.0205	2.29999
$682$	0	0
$683$	5.03508	0.192662	0.0963310	−	0.995349i	$-0.469289\pi$
$683$	5.03508	0.192662	0.0963310	+	0.995349i	$0.469289\pi$
$684$	0	0
$685$	−75.9386	−2.90147
$686$	0	0
$687$	25.1320	0.958845
$688$	0	0
$689$	−3.84255	−0.146389
$690$	0	0
$691$	−11.4270	−0.434702	−0.217351	−	0.976094i	$-0.569742\pi$
$691$	−11.4270	−0.434702	−0.217351	+	0.976094i	$0.569742\pi$
$692$	0	0
$693$	4.52704	0.171968
$694$	0	0
$695$	−30.8726	−1.17106
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−49.8411	−1.88516
$700$	0	0
$701$	38.0806	1.43828	0.719142	−	0.694863i	$-0.244535\pi$
$701$	38.0806	1.43828	0.719142	+	0.694863i	$0.244535\pi$
$702$	0	0
$703$	11.1156	0.419231
$704$	0	0
$705$	−30.1857	−1.13686
$706$	0	0
$707$	6.50837	0.244772
$708$	0	0
$709$	−13.7382	−0.515950	−0.257975	−	0.966152i	$-0.583055\pi$
$709$	−13.7382	−0.515950	−0.257975	+	0.966152i	$0.583055\pi$
$710$	0	0
$711$	−30.1215	−1.12965
$712$	0	0
$713$	2.71721	0.101760
$714$	0	0
$715$	14.9067	0.557480
$716$	0	0
$717$	−86.9163	−3.24595
$718$	0	0
$719$	−8.36722	−0.312045	−0.156022	−	0.987754i	$-0.549867\pi$
$719$	−8.36722	−0.312045	−0.156022	+	0.987754i	$0.549867\pi$
$720$	0	0
$721$	6.42065	0.239117
$722$	0	0
$723$	−63.6451	−2.36699
$724$	0	0
$725$	88.4434	3.28470
$726$	0	0
$727$	2.78518	0.103297	0.0516483	−	0.998665i	$-0.483553\pi$
$727$	2.78518	0.103297	0.0516483	+	0.998665i	$0.483553\pi$
$728$	0	0
$729$	37.8289	1.40107
$730$	0	0
$731$	0	0
$732$	0	0
$733$	5.34730	0.197507	0.0987534	−	0.995112i	$-0.468514\pi$
$733$	5.34730	0.197507	0.0987534	+	0.995112i	$0.468514\pi$
$734$	0	0
$735$	84.8822	3.13093
$736$	0	0
$737$	16.6845	0.614581
$738$	0	0
$739$	21.7374	0.799624	0.399812	−	0.916597i	$-0.369075\pi$
$739$	21.7374	0.799624	0.399812	+	0.916597i	$0.369075\pi$
$740$	0	0
$741$	−30.2490	−1.11122
$742$	0	0
$743$	31.2371	1.14598	0.572989	−	0.819563i	$-0.305784\pi$
$743$	31.2371	1.14598	0.572989	+	0.819563i	$0.305784\pi$
$744$	0	0
$745$	−6.41147	−0.234898
$746$	0	0
$747$	63.1275	2.30972
$748$	0	0
$749$	7.45336	0.272340
$750$	0	0
$751$	22.9608	0.837851	0.418926	−	0.908021i	$-0.362407\pi$
$751$	22.9608	0.837851	0.418926	+	0.908021i	$0.362407\pi$
$752$	0	0
$753$	−87.0765	−3.17324
$754$	0	0
$755$	61.4029	2.23468
$756$	0	0
$757$	−32.1756	−1.16944	−0.584721	−	0.811234i	$-0.698796\pi$
$757$	−32.1756	−1.16944	−0.584721	+	0.811234i	$0.698796\pi$
$758$	0	0
$759$	−13.3535	−0.484701
$760$	0	0
$761$	−24.8348	−0.900261	−0.450131	−	0.892963i	$-0.648623\pi$
$761$	−24.8348	−0.900261	−0.450131	+	0.892963i	$0.648623\pi$
$762$	0	0
$763$	−0.571614	−0.0206938
$764$	0	0
$765$	0	0
$766$	0	0
$767$	29.3351	1.05923
$768$	0	0
$769$	−2.52671	−0.0911156	−0.0455578	−	0.998962i	$-0.514507\pi$
$769$	−2.52671	−0.0911156	−0.0455578	+	0.998962i	$0.514507\pi$
$770$	0	0
$771$	91.1285	3.28191
$772$	0	0
$773$	−35.9968	−1.29472	−0.647358	−	0.762186i	$-0.724126\pi$
$773$	−35.9968	−1.29472	−0.647358	+	0.762186i	$0.724126\pi$
$774$	0	0
$775$	−8.61350	−0.309406
$776$	0	0
$777$	5.26950	0.189042
$778$	0	0
$779$	25.8625	0.926619
$780$	0	0
$781$	16.4789	0.589663
$782$	0	0
$783$	125.272	4.47687
$784$	0	0
$785$	26.4124	0.942699
$786$	0	0
$787$	24.4338	0.870970	0.435485	−	0.900196i	$-0.356577\pi$
$787$	24.4338	0.870970	0.435485	+	0.900196i	$0.356577\pi$
$788$	0	0
$789$	−19.2499	−0.685315
$790$	0	0
$791$	−1.15839	−0.0411874
$792$	0	0
$793$	−9.87576	−0.350699
$794$	0	0
$795$	−16.3405	−0.579537
$796$	0	0
$797$	6.45161	0.228528	0.114264	−	0.993450i	$-0.463549\pi$
$797$	6.45161	0.228528	0.114264	+	0.993450i	$0.463549\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−31.9668	−1.12949
$802$	0	0
$803$	6.66550	0.235220
$804$	0	0
$805$	5.75465	0.202825
$806$	0	0
$807$	76.5987	2.69640
$808$	0	0
$809$	22.6777	0.797305	0.398652	−	0.917102i	$-0.369478\pi$
$809$	22.6777	0.797305	0.398652	+	0.917102i	$0.369478\pi$
$810$	0	0
$811$	20.7989	0.730348	0.365174	−	0.930939i	$-0.381009\pi$
$811$	20.7989	0.730348	0.365174	+	0.930939i	$0.381009\pi$
$812$	0	0
$813$	21.1739	0.742600
$814$	0	0
$815$	−14.8007	−0.518444
$816$	0	0
$817$	−22.3182	−0.780815
$818$	0	0
$819$	−10.2080	−0.356697
$820$	0	0
$821$	−42.5381	−1.48459	−0.742295	−	0.670074i	$-0.766262\pi$
$821$	−42.5381	−1.48459	−0.742295	+	0.670074i	$0.766262\pi$
$822$	0	0
$823$	−49.3474	−1.72014	−0.860071	−	0.510174i	$-0.829581\pi$
$823$	−49.3474	−1.72014	−0.860071	+	0.510174i	$0.829581\pi$
$824$	0	0
$825$	42.3304	1.47375
$826$	0	0
$827$	20.3577	0.707907	0.353954	−	0.935263i	$-0.384837\pi$
$827$	20.3577	0.707907	0.353954	+	0.935263i	$0.384837\pi$
$828$	0	0
$829$	24.0820	0.836403	0.418202	−	0.908354i	$-0.362661\pi$
$829$	24.0820	0.836403	0.418202	+	0.908354i	$0.362661\pi$
$830$	0	0
$831$	44.4115	1.54062
$832$	0	0
$833$	0	0
$834$	0	0
$835$	45.3979	1.57106
$836$	0	0
$837$	−12.2003	−0.421703
$838$	0	0
$839$	−51.1513	−1.76594	−0.882969	−	0.469432i	$-0.844459\pi$
$839$	−51.1513	−1.76594	−0.882969	+	0.469432i	$0.844459\pi$
$840$	0	0
$841$	48.4516	1.67075
$842$	0	0
$843$	−74.3575	−2.56101
$844$	0	0
$845$	16.8188	0.578585
$846$	0	0
$847$	−4.34966	−0.149456
$848$	0	0
$849$	77.4653	2.65860
$850$	0	0
$851$	−11.0648	−0.379296
$852$	0	0
$853$	0.737415	0.0252486	0.0126243	−	0.999920i	$-0.495981\pi$
$853$	0.737415	0.0252486	0.0126243	+	0.999920i	$0.495981\pi$
$854$	0	0
$855$	−91.5690	−3.13160
$856$	0	0
$857$	−18.2932	−0.624885	−0.312442	−	0.949937i	$-0.601147\pi$
$857$	−18.2932	−0.624885	−0.312442	+	0.949937i	$0.601147\pi$
$858$	0	0
$859$	37.9077	1.29339	0.646696	−	0.762748i	$-0.276150\pi$
$859$	37.9077	1.29339	0.646696	+	0.762748i	$0.276150\pi$
$860$	0	0
$861$	12.2605	0.417837
$862$	0	0
$863$	38.2719	1.30279	0.651395	−	0.758739i	$-0.274184\pi$
$863$	38.2719	1.30279	0.651395	+	0.758739i	$0.274184\pi$
$864$	0	0
$865$	−58.7187	−1.99650
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−5.30541	−0.179974
$870$	0	0
$871$	−37.6219	−1.27477
$872$	0	0
$873$	−51.4047	−1.73978
$874$	0	0
$875$	−9.16613	−0.309872
$876$	0	0
$877$	−38.2249	−1.29076	−0.645382	−	0.763860i	$-0.723302\pi$
$877$	−38.2249	−1.29076	−0.645382	+	0.763860i	$0.723302\pi$
$878$	0	0
$879$	−68.5580	−2.31240
$880$	0	0
$881$	−6.38144	−0.214996	−0.107498	−	0.994205i	$-0.534284\pi$
$881$	−6.38144	−0.214996	−0.107498	+	0.994205i	$0.534284\pi$
$882$	0	0
$883$	−26.8266	−0.902786	−0.451393	−	0.892325i	$-0.649073\pi$
$883$	−26.8266	−0.902786	−0.451393	+	0.892325i	$0.649073\pi$
$884$	0	0
$885$	124.748	4.19336
$886$	0	0
$887$	−8.87433	−0.297971	−0.148985	−	0.988839i	$-0.547601\pi$
$887$	−8.87433	−0.297971	−0.148985	+	0.988839i	$0.547601\pi$
$888$	0	0
$889$	−1.10513	−0.0370650
$890$	0	0
$891$	30.9323	1.03627
$892$	0	0
$893$	7.68004	0.257003
$894$	0	0
$895$	23.6878	0.791795
$896$	0	0
$897$	30.1108	1.00537
$898$	0	0
$899$	−7.54301	−0.251574
$900$	0	0
$901$	0	0
$902$	0	0
$903$	−10.5803	−0.352090
$904$	0	0
$905$	−28.2645	−0.939542
$906$	0	0
$907$	−37.5827	−1.24791	−0.623956	−	0.781460i	$-0.714475\pi$
$907$	−37.5827	−1.24791	−0.623956	+	0.781460i	$0.714475\pi$
$908$	0	0
$909$	103.089	3.41926
$910$	0	0
$911$	36.5289	1.21026	0.605128	−	0.796128i	$-0.293122\pi$
$911$	36.5289	1.21026	0.605128	+	0.796128i	$0.293122\pi$
$912$	0	0
$913$	11.1189	0.367981
$914$	0	0
$915$	−41.9968	−1.38837
$916$	0	0
$917$	0.501126	0.0165486
$918$	0	0
$919$	37.8108	1.24726	0.623631	−	0.781719i	$-0.285657\pi$
$919$	37.8108	1.24726	0.623631	+	0.781719i	$0.285657\pi$
$920$	0	0
$921$	−43.0042	−1.41704
$922$	0	0
$923$	−37.1584	−1.22308
$924$	0	0
$925$	35.0752	1.15327
$926$	0	0
$927$	101.700	3.34026
$928$	0	0
$929$	10.5645	0.346609	0.173305	−	0.984868i	$-0.444555\pi$
$929$	10.5645	0.346609	0.173305	+	0.984868i	$0.444555\pi$
$930$	0	0
$931$	−21.5963	−0.707789
$932$	0	0
$933$	−34.1688	−1.11864
$934$	0	0
$935$	0	0
$936$	0	0
$937$	28.0378	0.915954	0.457977	−	0.888964i	$-0.348574\pi$
$937$	28.0378	0.915954	0.457977	+	0.888964i	$0.348574\pi$
$938$	0	0
$939$	−14.1625	−0.462176
$940$	0	0
$941$	26.5871	0.866715	0.433357	−	0.901222i	$-0.357329\pi$
$941$	26.5871	0.866715	0.433357	+	0.901222i	$0.357329\pi$
$942$	0	0
$943$	−25.7443	−0.838351
$944$	0	0
$945$	−25.8384	−0.840524
$946$	0	0
$947$	17.7888	0.578058	0.289029	−	0.957320i	$-0.406668\pi$
$947$	17.7888	0.578058	0.289029	+	0.957320i	$0.406668\pi$
$948$	0	0
$949$	−15.0300	−0.487895
$950$	0	0
$951$	−4.39424	−0.142493
$952$	0	0
$953$	−20.1061	−0.651299	−0.325650	−	0.945490i	$-0.605583\pi$
$953$	−20.1061	−0.651299	−0.325650	+	0.945490i	$0.605583\pi$
$954$	0	0
$955$	−64.1079	−2.07448
$956$	0	0
$957$	37.0696	1.19829
$958$	0	0
$959$	9.15932	0.295770
$960$	0	0
$961$	−30.2654	−0.976303
$962$	0	0
$963$	118.057	3.80435
$964$	0	0
$965$	21.9290	0.705920
$966$	0	0
$967$	−37.0068	−1.19006	−0.595029	−	0.803704i	$-0.702860\pi$
$967$	−37.0068	−1.19006	−0.595029	+	0.803704i	$0.702860\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−13.3618	−0.428802	−0.214401	−	0.976746i	$-0.568780\pi$
$971$	−13.3618	−0.428802	−0.214401	+	0.976746i	$0.568780\pi$
$972$	0	0
$973$	3.72369	0.119376
$974$	0	0
$975$	−95.4508	−3.05687
$976$	0	0
$977$	−20.6304	−0.660025	−0.330013	−	0.943976i	$-0.607053\pi$
$977$	−20.6304	−0.660025	−0.330013	+	0.943976i	$0.607053\pi$
$978$	0	0
$979$	−5.63041	−0.179949
$980$	0	0
$981$	−9.05407	−0.289074
$982$	0	0
$983$	16.8452	0.537280	0.268640	−	0.963241i	$-0.413426\pi$
$983$	16.8452	0.537280	0.268640	+	0.963241i	$0.413426\pi$
$984$	0	0
$985$	−3.23854	−0.103189
$986$	0	0
$987$	3.64084	0.115889
$988$	0	0
$989$	22.2163	0.706436
$990$	0	0
$991$	−42.3432	−1.34508	−0.672538	−	0.740063i	$-0.734796\pi$
$991$	−42.3432	−1.34508	−0.672538	+	0.740063i	$0.734796\pi$
$992$	0	0
$993$	−57.8925	−1.83716
$994$	0	0
$995$	−89.5049	−2.83749
$996$	0	0
$997$	−24.4097	−0.773064	−0.386532	−	0.922276i	$-0.626327\pi$
$997$	−24.4097	−0.773064	−0.386532	+	0.922276i	$0.626327\pi$
$998$	0	0
$999$	49.6810	1.57184

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4624.2.a.bi.1.3		3
4.3	odd	2		2312.2.a.o.1.1	✓	3
17.16	even	2		4624.2.a.bb.1.1		3
68.47	odd	4		2312.2.b.l.577.1		6
68.55	odd	4		2312.2.b.l.577.6		6
68.67	odd	2		2312.2.a.r.1.3	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2312.2.a.o.1.1	✓	3	4.3	odd	2
2312.2.a.r.1.3	yes	3	68.67	odd	2
2312.2.b.l.577.1		6	68.47	odd	4
2312.2.b.l.577.6		6	68.55	odd	4
4624.2.a.bb.1.1		3	17.16	even	2
4624.2.a.bi.1.3		3	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:	\( N \)	\(=\)	\( 4624 = 2^{4} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4624.a (trivial)

\(f(q)\)	\(=\)	\(q+3.22668 q^{3} -3.87939 q^{5} +0.467911 q^{7} +7.41147 q^{9} +1.30541 q^{11} -2.94356 q^{13} -12.5175 q^{15} +3.18479 q^{19} +1.50980 q^{21} -3.17024 q^{23} +10.0496 q^{25} +14.2344 q^{27} +8.80066 q^{29} -0.857097 q^{31} +4.21213 q^{33} -1.81521 q^{35} +3.49020 q^{37} -9.49794 q^{39} +8.12061 q^{41} -7.00774 q^{43} -28.7520 q^{45} +2.41147 q^{47} -6.78106 q^{49} +1.30541 q^{53} -5.06418 q^{55} +10.2763 q^{57} -9.96585 q^{59} +3.35504 q^{61} +3.46791 q^{63} +11.4192 q^{65} +12.7811 q^{67} -10.2294 q^{69} +12.6236 q^{71} +5.10607 q^{73} +32.4270 q^{75} +0.610815 q^{77} -4.06418 q^{79} +23.6955 q^{81} +8.51754 q^{83} +28.3969 q^{87} -4.31315 q^{89} -1.37733 q^{91} -2.76558 q^{93} -12.3550 q^{95} -6.93582 q^{97} +9.67499 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} - 6 q^{5} + 6 q^{7} + 12 q^{9} + 6 q^{11} + 6 q^{13} - 15 q^{15} + 6 q^{19} + 6 q^{21} + 12 q^{23} + 3 q^{25} + 12 q^{27} + 12 q^{29} - 3 q^{31} - 12 q^{33} - 9 q^{35} + 9 q^{37} - 3 q^{39}+ \cdots + 24 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 - 6 * q^5 + 6 * q^7 + 12 * q^9 + 6 * q^11 + 6 * q^13 - 15 * q^15 + 6 * q^19 + 6 * q^21 + 12 * q^23 + 3 * q^25 + 12 * q^27 + 12 * q^29 - 3 * q^31 - 12 * q^33 - 9 * q^35 + 9 * q^37 - 3 * q^39 + 30 * q^41 + 3 * q^43 - 33 * q^45 - 3 * q^47 - 3 * q^49 + 6 * q^53 - 6 * q^55 - 3 * q^57 - 9 * q^59 - 15 * q^61 + 15 * q^63 + 21 * q^67 - 15 * q^69 + 3 * q^71 + 3 * q^73 + 39 * q^75 + 6 * q^77 - 3 * q^79 + 3 * q^81 + 3 * q^83 + 57 * q^87 + 9 * q^89 + 27 * q^91 - 39 * q^93 - 12 * q^95 - 30 * q^97 + 24 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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