LMFDB - Embedded newform 5312.2.a.bn.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5312,2,Mod(1,5312)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5312.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5312, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5312 = 2^{6} \cdot 83 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5312.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,-1,0,-2,0,3,0,3,0,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$42.4165335537$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.9059636.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 8x^{4} + 11x^{3} + 4x^{2} - 4x - 1 $ x^6 - x^5 - 8x^4 + 11x^3 + 4x^2 - 4x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 83)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$2.14357$ of defining polynomial
Character	$\chi$	$=$	5312.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.04941 q^{3} -2.79494 q^{5} +3.61008 q^{7} +1.20006 q^{9} +3.94898 q^{11} -5.70400 q^{13} -5.72796 q^{15} +1.07186 q^{17} -6.03605 q^{19} +7.39852 q^{21} -4.88050 q^{23} +2.81168 q^{25} -3.68880 q^{27} -0.653487 q^{29} -4.43696 q^{31} +8.09306 q^{33} -10.0899 q^{35} +2.58601 q^{37} -11.6898 q^{39} -0.360608 q^{41} +1.95509 q^{43} -3.35411 q^{45} -5.32783 q^{47} +6.03266 q^{49} +2.19668 q^{51} -9.17524 q^{53} -11.0372 q^{55} -12.3703 q^{57} +3.18599 q^{59} +13.3223 q^{61} +4.33233 q^{63} +15.9423 q^{65} -5.74581 q^{67} -10.0021 q^{69} -10.9440 q^{71} -13.2048 q^{73} +5.76227 q^{75} +14.2561 q^{77} +4.12809 q^{79} -11.1600 q^{81} -1.00000 q^{83} -2.99579 q^{85} -1.33926 q^{87} +7.81114 q^{89} -20.5919 q^{91} -9.09314 q^{93} +16.8704 q^{95} -14.4189 q^{97} +4.73903 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - q^{3} - 2 q^{5} + 3 q^{7} + 3 q^{9} + 3 q^{11} - 14 q^{13} - 6 q^{15} - 5 q^{17} + 4 q^{19} + 2 q^{21} - 5 q^{23} + 14 q^{25} - 10 q^{27} + q^{29} + 3 q^{31} + 2 q^{33} + 10 q^{35} - 39 q^{37} - 8 q^{39}+ \cdots + 18 q^{99}+O(q^{100}) $ 6 * q - q^3 - 2 * q^5 + 3 * q^7 + 3 * q^9 + 3 * q^11 - 14 * q^13 - 6 * q^15 - 5 * q^17 + 4 * q^19 + 2 * q^21 - 5 * q^23 + 14 * q^25 - 10 * q^27 + q^29 + 3 * q^31 + 2 * q^33 + 10 * q^35 - 39 * q^37 - 8 * q^39 - q^41 + 8 * q^43 - 10 * q^45 - 12 * q^47 + 11 * q^49 + 24 * q^51 - 14 * q^53 - 24 * q^55 - 10 * q^57 + 17 * q^59 + 5 * q^61 - 11 * q^63 - 8 * q^65 - 16 * q^67 - 8 * q^69 - 26 * q^71 - 6 * q^73 + 45 * q^75 - 6 * q^77 - 12 * q^79 - 34 * q^81 - 6 * q^83 - 18 * q^85 - 3 * q^87 - 22 * q^89 + 2 * q^91 - 34 * q^93 - 4 * q^95 + 6 * q^97 + 18 * 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
$3$	2.04941	1.18323	0.591613	−	0.806222i	$-0.298492\pi$
$3$	2.04941	1.18323	0.591613	+	0.806222i	$0.298492\pi$
$4$	0	0
$5$	−2.79494	−1.24993	−0.624967	−	0.780651i	$-0.714888\pi$
$5$	−2.79494	−1.24993	−0.624967	+	0.780651i	$0.714888\pi$
$6$	0	0
$7$	3.61008	1.36448	0.682241	−	0.731128i	$-0.261006\pi$
$7$	3.61008	1.36448	0.682241	+	0.731128i	$0.261006\pi$
$8$	0	0
$9$	1.20006	0.400022
$10$	0	0
$11$	3.94898	1.19066	0.595331	−	0.803480i	$-0.297021\pi$
$11$	3.94898	1.19066	0.595331	+	0.803480i	$0.297021\pi$
$12$	0	0
$13$	−5.70400	−1.58201	−0.791003	−	0.611812i	$-0.790441\pi$
$13$	−5.70400	−1.58201	−0.791003	+	0.611812i	$0.790441\pi$
$14$	0	0
$15$	−5.72796	−1.47895
$16$	0	0
$17$	1.07186	0.259965	0.129982	−	0.991516i	$-0.458508\pi$
$17$	1.07186	0.259965	0.129982	+	0.991516i	$0.458508\pi$
$18$	0	0
$19$	−6.03605	−1.38476	−0.692382	−	0.721531i	$-0.743439\pi$
$19$	−6.03605	−1.38476	−0.692382	+	0.721531i	$0.743439\pi$
$20$	0	0
$21$	7.39852	1.61449
$22$	0	0
$23$	−4.88050	−1.01765	−0.508827	−	0.860869i	$-0.669921\pi$
$23$	−4.88050	−1.01765	−0.508827	+	0.860869i	$0.669921\pi$
$24$	0	0
$25$	2.81168	0.562336
$26$	0	0
$27$	−3.68880	−0.709910
$28$	0	0
$29$	−0.653487	−0.121350	−0.0606748	−	0.998158i	$-0.519325\pi$
$29$	−0.653487	−0.121350	−0.0606748	+	0.998158i	$0.519325\pi$
$30$	0	0
$31$	−4.43696	−0.796902	−0.398451	−	0.917190i	$-0.630452\pi$
$31$	−4.43696	−0.796902	−0.398451	+	0.917190i	$0.630452\pi$
$32$	0	0
$33$	8.09306	1.40882
$34$	0	0
$35$	−10.0899	−1.70551
$36$	0	0
$37$	2.58601	0.425137	0.212569	−	0.977146i	$-0.431817\pi$
$37$	2.58601	0.425137	0.212569	+	0.977146i	$0.431817\pi$
$38$	0	0
$39$	−11.6898	−1.87187
$40$	0	0
$41$	−0.360608	−0.0563174	−0.0281587	−	0.999603i	$-0.508964\pi$
$41$	−0.360608	−0.0563174	−0.0281587	+	0.999603i	$0.508964\pi$
$42$	0	0
$43$	1.95509	0.298148	0.149074	−	0.988826i	$-0.452371\pi$
$43$	1.95509	0.298148	0.149074	+	0.988826i	$0.452371\pi$
$44$	0	0
$45$	−3.35411	−0.500001
$46$	0	0
$47$	−5.32783	−0.777144	−0.388572	−	0.921418i	$-0.627032\pi$
$47$	−5.32783	−0.777144	−0.388572	+	0.921418i	$0.627032\pi$
$48$	0	0
$49$	6.03266	0.861809
$50$	0	0
$51$	2.19668	0.307597
$52$	0	0
$53$	−9.17524	−1.26032	−0.630158	−	0.776467i	$-0.717010\pi$
$53$	−9.17524	−1.26032	−0.630158	+	0.776467i	$0.717010\pi$
$54$	0	0
$55$	−11.0372	−1.48825
$56$	0	0
$57$	−12.3703	−1.63849
$58$	0	0
$59$	3.18599	0.414780	0.207390	−	0.978258i	$-0.433503\pi$
$59$	3.18599	0.414780	0.207390	+	0.978258i	$0.433503\pi$
$60$	0	0
$61$	13.3223	1.70575	0.852875	−	0.522115i	$-0.174857\pi$
$61$	13.3223	1.70575	0.852875	+	0.522115i	$0.174857\pi$
$62$	0	0
$63$	4.33233	0.545822
$64$	0	0
$65$	15.9423	1.97740
$66$	0	0
$67$	−5.74581	−0.701963	−0.350981	−	0.936382i	$-0.614152\pi$
$67$	−5.74581	−0.701963	−0.350981	+	0.936382i	$0.614152\pi$
$68$	0	0
$69$	−10.0021	−1.20411
$70$	0	0
$71$	−10.9440	−1.29881	−0.649406	−	0.760442i	$-0.724982\pi$
$71$	−10.9440	−1.29881	−0.649406	+	0.760442i	$0.724982\pi$
$72$	0	0
$73$	−13.2048	−1.54551	−0.772755	−	0.634705i	$-0.781122\pi$
$73$	−13.2048	−1.54551	−0.772755	+	0.634705i	$0.781122\pi$
$74$	0	0
$75$	5.76227	0.665370
$76$	0	0
$77$	14.2561	1.62464
$78$	0	0
$79$	4.12809	0.464447	0.232223	−	0.972662i	$-0.425400\pi$
$79$	4.12809	0.464447	0.232223	+	0.972662i	$0.425400\pi$
$80$	0	0
$81$	−11.1600	−1.24000
$82$	0	0
$83$	−1.00000	−0.109764
$83$	−1.00000	−0.109764
$84$	0	0
$85$	−2.99579	−0.324939
$86$	0	0
$87$	−1.33926	−0.143584
$88$	0	0
$89$	7.81114	0.827979	0.413990	−	0.910282i	$-0.364135\pi$
$89$	7.81114	0.827979	0.413990	+	0.910282i	$0.364135\pi$
$90$	0	0
$91$	−20.5919	−2.15862
$92$	0	0
$93$	−9.09314	−0.942914
$94$	0	0
$95$	16.8704	1.73086
$96$	0	0
$97$	−14.4189	−1.46402	−0.732008	−	0.681296i	$-0.761417\pi$
$97$	−14.4189	−1.46402	−0.732008	+	0.681296i	$0.761417\pi$
$98$	0	0
$99$	4.73903	0.476291
$100$	0	0
$101$	−8.15271	−0.811225	−0.405613	−	0.914045i	$-0.632942\pi$
$101$	−8.15271	−0.811225	−0.405613	+	0.914045i	$0.632942\pi$
$102$	0	0
$103$	18.4508	1.81801	0.909006	−	0.416784i	$-0.136843\pi$
$103$	18.4508	1.81801	0.909006	+	0.416784i	$0.136843\pi$
$104$	0	0
$105$	−20.6784	−2.01800
$106$	0	0
$107$	−4.24254	−0.410142	−0.205071	−	0.978747i	$-0.565742\pi$
$107$	−4.24254	−0.410142	−0.205071	+	0.978747i	$0.565742\pi$
$108$	0	0
$109$	−3.71776	−0.356096	−0.178048	−	0.984022i	$-0.556978\pi$
$109$	−3.71776	−0.356096	−0.178048	+	0.984022i	$0.556978\pi$
$110$	0	0
$111$	5.29978	0.503033
$112$	0	0
$113$	−4.06149	−0.382073	−0.191037	−	0.981583i	$-0.561185\pi$
$113$	−4.06149	−0.382073	−0.191037	+	0.981583i	$0.561185\pi$
$114$	0	0
$115$	13.6407	1.27200
$116$	0	0
$117$	−6.84517	−0.632836
$118$	0	0
$119$	3.86951	0.354717
$120$	0	0
$121$	4.59445	0.417677
$122$	0	0
$123$	−0.739031	−0.0666362
$124$	0	0
$125$	6.11622	0.547051
$126$	0	0
$127$	−10.9066	−0.967802	−0.483901	−	0.875123i	$-0.660780\pi$
$127$	−10.9066	−0.967802	−0.483901	+	0.875123i	$0.660780\pi$
$128$	0	0
$129$	4.00677	0.352776
$130$	0	0
$131$	−3.29062	−0.287503	−0.143752	−	0.989614i	$-0.545917\pi$
$131$	−3.29062	−0.287503	−0.143752	+	0.989614i	$0.545917\pi$
$132$	0	0
$133$	−21.7906	−1.88948
$134$	0	0
$135$	10.3100	0.887340
$136$	0	0
$137$	3.52824	0.301438	0.150719	−	0.988577i	$-0.451841\pi$
$137$	3.52824	0.301438	0.150719	+	0.988577i	$0.451841\pi$
$138$	0	0
$139$	9.84970	0.835441	0.417720	−	0.908576i	$-0.362829\pi$
$139$	9.84970	0.835441	0.417720	+	0.908576i	$0.362829\pi$
$140$	0	0
$141$	−10.9189	−0.919537
$142$	0	0
$143$	−22.5250	−1.88363
$144$	0	0
$145$	1.82646	0.151679
$146$	0	0
$147$	12.3634	1.01971
$148$	0	0
$149$	11.0899	0.908522	0.454261	−	0.890869i	$-0.349903\pi$
$149$	11.0899	0.908522	0.454261	+	0.890869i	$0.349903\pi$
$150$	0	0
$151$	−11.1368	−0.906300	−0.453150	−	0.891434i	$-0.649700\pi$
$151$	−11.1368	−0.906300	−0.453150	+	0.891434i	$0.649700\pi$
$152$	0	0
$153$	1.28630	0.103992
$154$	0	0
$155$	12.4010	0.996075
$156$	0	0
$157$	−2.29600	−0.183241	−0.0916203	−	0.995794i	$-0.529205\pi$
$157$	−2.29600	−0.183241	−0.0916203	+	0.995794i	$0.529205\pi$
$158$	0	0
$159$	−18.8038	−1.49124
$160$	0	0
$161$	−17.6190	−1.38857
$162$	0	0
$163$	−4.72606	−0.370174	−0.185087	−	0.982722i	$-0.559257\pi$
$163$	−4.72606	−0.370174	−0.185087	+	0.982722i	$0.559257\pi$
$164$	0	0
$165$	−22.6196	−1.76093
$166$	0	0
$167$	20.6780	1.60011	0.800057	−	0.599924i	$-0.204802\pi$
$167$	20.6780	1.60011	0.800057	+	0.599924i	$0.204802\pi$
$168$	0	0
$169$	19.5356	1.50274
$170$	0	0
$171$	−7.24365	−0.553935
$172$	0	0
$173$	16.1382	1.22697	0.613483	−	0.789708i	$-0.289768\pi$
$173$	16.1382	1.22697	0.613483	+	0.789708i	$0.289768\pi$
$174$	0	0
$175$	10.1504	0.767297
$176$	0	0
$177$	6.52938	0.490779
$178$	0	0
$179$	12.1449	0.907749	0.453874	−	0.891066i	$-0.350041\pi$
$179$	12.1449	0.907749	0.453874	+	0.891066i	$0.350041\pi$
$180$	0	0
$181$	17.7008	1.31569	0.657844	−	0.753154i	$-0.271469\pi$
$181$	17.7008	1.31569	0.657844	+	0.753154i	$0.271469\pi$
$182$	0	0
$183$	27.3029	2.01829
$184$	0	0
$185$	−7.22773	−0.531393
$186$	0	0
$187$	4.23276	0.309530
$188$	0	0
$189$	−13.3168	−0.968658
$190$	0	0
$191$	−6.65255	−0.481362	−0.240681	−	0.970604i	$-0.577371\pi$
$191$	−6.65255	−0.481362	−0.240681	+	0.970604i	$0.577371\pi$
$192$	0	0
$193$	5.52687	0.397832	0.198916	−	0.980016i	$-0.436258\pi$
$193$	5.52687	0.397832	0.198916	+	0.980016i	$0.436258\pi$
$194$	0	0
$195$	32.6723	2.33971
$196$	0	0
$197$	−1.91450	−0.136402	−0.0682011	−	0.997672i	$-0.521726\pi$
$197$	−1.91450	−0.136402	−0.0682011	+	0.997672i	$0.521726\pi$
$198$	0	0
$199$	9.19381	0.651732	0.325866	−	0.945416i	$-0.394344\pi$
$199$	9.19381	0.651732	0.325866	+	0.945416i	$0.394344\pi$
$200$	0	0
$201$	−11.7755	−0.830580
$202$	0	0
$203$	−2.35914	−0.165579
$204$	0	0
$205$	1.00788	0.0703931
$206$	0	0
$207$	−5.85692	−0.407084
$208$	0	0
$209$	−23.8362	−1.64879
$210$	0	0
$211$	−17.4617	−1.20211	−0.601057	−	0.799206i	$-0.705254\pi$
$211$	−17.4617	−1.20211	−0.601057	+	0.799206i	$0.705254\pi$
$212$	0	0
$213$	−22.4287	−1.53679
$214$	0	0
$215$	−5.46435	−0.372665
$216$	0	0
$217$	−16.0178	−1.08736
$218$	0	0
$219$	−27.0621	−1.82869
$220$	0	0
$221$	−6.11391	−0.411266
$222$	0	0
$223$	−22.5554	−1.51042	−0.755211	−	0.655482i	$-0.772465\pi$
$223$	−22.5554	−1.51042	−0.755211	+	0.655482i	$0.772465\pi$
$224$	0	0
$225$	3.37420	0.224947
$226$	0	0
$227$	−23.1778	−1.53836	−0.769182	−	0.639030i	$-0.779336\pi$
$227$	−23.1778	−1.53836	−0.769182	+	0.639030i	$0.779336\pi$
$228$	0	0
$229$	9.23153	0.610036	0.305018	−	0.952347i	$-0.401337\pi$
$229$	9.23153	0.610036	0.305018	+	0.952347i	$0.401337\pi$
$230$	0	0
$231$	29.2166	1.92231
$232$	0	0
$233$	2.66475	0.174573	0.0872867	−	0.996183i	$-0.472180\pi$
$233$	2.66475	0.174573	0.0872867	+	0.996183i	$0.472180\pi$
$234$	0	0
$235$	14.8910	0.971380
$236$	0	0
$237$	8.46014	0.549545
$238$	0	0
$239$	22.8068	1.47525	0.737625	−	0.675211i	$-0.235947\pi$
$239$	22.8068	1.47525	0.737625	+	0.675211i	$0.235947\pi$
$240$	0	0
$241$	−0.358446	−0.0230895	−0.0115448	−	0.999933i	$-0.503675\pi$
$241$	−0.358446	−0.0230895	−0.0115448	+	0.999933i	$0.503675\pi$
$242$	0	0
$243$	−11.8051	−0.757295
$244$	0	0
$245$	−16.8609	−1.07720
$246$	0	0
$247$	34.4296	2.19070
$248$	0	0
$249$	−2.04941	−0.129876
$250$	0	0
$251$	−20.1333	−1.27080	−0.635401	−	0.772182i	$-0.719165\pi$
$251$	−20.1333	−1.27080	−0.635401	+	0.772182i	$0.719165\pi$
$252$	0	0
$253$	−19.2730	−1.21168
$254$	0	0
$255$	−6.13959	−0.384476
$256$	0	0
$257$	15.6673	0.977299	0.488649	−	0.872480i	$-0.337490\pi$
$257$	15.6673	0.977299	0.488649	+	0.872480i	$0.337490\pi$
$258$	0	0
$259$	9.33569	0.580092
$260$	0	0
$261$	−0.784227	−0.0485424
$262$	0	0
$263$	−21.9193	−1.35160	−0.675800	−	0.737085i	$-0.736202\pi$
$263$	−21.9193	−1.35160	−0.675800	+	0.737085i	$0.736202\pi$
$264$	0	0
$265$	25.6442	1.57531
$266$	0	0
$267$	16.0082	0.979686
$268$	0	0
$269$	17.2756	1.05331	0.526655	−	0.850079i	$-0.323446\pi$
$269$	17.2756	1.05331	0.526655	+	0.850079i	$0.323446\pi$
$270$	0	0
$271$	−15.3874	−0.934716	−0.467358	−	0.884068i	$-0.654794\pi$
$271$	−15.3874	−0.934716	−0.467358	+	0.884068i	$0.654794\pi$
$272$	0	0
$273$	−42.2012	−2.55413
$274$	0	0
$275$	11.1033	0.669552
$276$	0	0
$277$	−13.6895	−0.822524	−0.411262	−	0.911517i	$-0.634912\pi$
$277$	−13.6895	−0.822524	−0.411262	+	0.911517i	$0.634912\pi$
$278$	0	0
$279$	−5.32464	−0.318778
$280$	0	0
$281$	−5.78233	−0.344945	−0.172473	−	0.985014i	$-0.555176\pi$
$281$	−5.78233	−0.344945	−0.172473	+	0.985014i	$0.555176\pi$
$282$	0	0
$283$	−24.8977	−1.48002	−0.740008	−	0.672598i	$-0.765178\pi$
$283$	−24.8977	−1.48002	−0.740008	+	0.672598i	$0.765178\pi$
$284$	0	0
$285$	34.5743	2.04800
$286$	0	0
$287$	−1.30182	−0.0768441
$288$	0	0
$289$	−15.8511	−0.932418
$290$	0	0
$291$	−29.5502	−1.73226
$292$	0	0
$293$	−18.0431	−1.05409	−0.527043	−	0.849838i	$-0.676699\pi$
$293$	−18.0431	−1.05409	−0.527043	+	0.849838i	$0.676699\pi$
$294$	0	0
$295$	−8.90464	−0.518448
$296$	0	0
$297$	−14.5670	−0.845263
$298$	0	0
$299$	27.8384	1.60994
$300$	0	0
$301$	7.05802	0.406817
$302$	0	0
$303$	−16.7082	−0.959862
$304$	0	0
$305$	−37.2351	−2.13208
$306$	0	0
$307$	8.48030	0.483996	0.241998	−	0.970277i	$-0.422197\pi$
$307$	8.48030	0.483996	0.241998	+	0.970277i	$0.422197\pi$
$308$	0	0
$309$	37.8132	2.15112
$310$	0	0
$311$	7.75724	0.439873	0.219936	−	0.975514i	$-0.429415\pi$
$311$	7.75724	0.439873	0.219936	+	0.975514i	$0.429415\pi$
$312$	0	0
$313$	−1.33779	−0.0756164	−0.0378082	−	0.999285i	$-0.512038\pi$
$313$	−1.33779	−0.0756164	−0.0378082	+	0.999285i	$0.512038\pi$
$314$	0	0
$315$	−12.1086	−0.682242
$316$	0	0
$317$	−17.8320	−1.00154	−0.500772	−	0.865579i	$-0.666951\pi$
$317$	−17.8320	−1.00154	−0.500772	+	0.865579i	$0.666951\pi$
$318$	0	0
$319$	−2.58061	−0.144486
$320$	0	0
$321$	−8.69468	−0.485290
$322$	0	0
$323$	−6.46981	−0.359990
$324$	0	0
$325$	−16.0378	−0.889619
$326$	0	0
$327$	−7.61919	−0.421342
$328$	0	0
$329$	−19.2339	−1.06040
$330$	0	0
$331$	3.00788	0.165328	0.0826639	−	0.996577i	$-0.473657\pi$
$331$	3.00788	0.165328	0.0826639	+	0.996577i	$0.473657\pi$
$332$	0	0
$333$	3.10338	0.170064
$334$	0	0
$335$	16.0592	0.877408
$336$	0	0
$337$	−29.3225	−1.59730	−0.798649	−	0.601797i	$-0.794452\pi$
$337$	−29.3225	−1.59730	−0.798649	+	0.601797i	$0.794452\pi$
$338$	0	0
$339$	−8.32365	−0.452078
$340$	0	0
$341$	−17.5215	−0.948841
$342$	0	0
$343$	−3.49216	−0.188559
$344$	0	0
$345$	27.9553	1.50506
$346$	0	0
$347$	−4.64488	−0.249350	−0.124675	−	0.992198i	$-0.539789\pi$
$347$	−4.64488	−0.249350	−0.124675	+	0.992198i	$0.539789\pi$
$348$	0	0
$349$	−8.71573	−0.466542	−0.233271	−	0.972412i	$-0.574943\pi$
$349$	−8.71573	−0.466542	−0.233271	+	0.972412i	$0.574943\pi$
$350$	0	0
$351$	21.0409	1.12308
$352$	0	0
$353$	10.1205	0.538659	0.269329	−	0.963048i	$-0.413198\pi$
$353$	10.1205	0.538659	0.269329	+	0.963048i	$0.413198\pi$
$354$	0	0
$355$	30.5878	1.62343
$356$	0	0
$357$	7.93019	0.419710
$358$	0	0
$359$	−2.86374	−0.151143	−0.0755713	−	0.997140i	$-0.524078\pi$
$359$	−2.86374	−0.151143	−0.0755713	+	0.997140i	$0.524078\pi$
$360$	0	0
$361$	17.4339	0.917572
$362$	0	0
$363$	9.41588	0.494206
$364$	0	0
$365$	36.9067	1.93179
$366$	0	0
$367$	1.49317	0.0779428	0.0389714	−	0.999240i	$-0.487592\pi$
$367$	1.49317	0.0779428	0.0389714	+	0.999240i	$0.487592\pi$
$368$	0	0
$369$	−0.432752	−0.0225282
$370$	0	0
$371$	−33.1233	−1.71968
$372$	0	0
$373$	−13.7121	−0.709985	−0.354992	−	0.934869i	$-0.615517\pi$
$373$	−13.7121	−0.709985	−0.354992	+	0.934869i	$0.615517\pi$
$374$	0	0
$375$	12.5346	0.647285
$376$	0	0
$377$	3.72749	0.191976
$378$	0	0
$379$	−20.0582	−1.03032	−0.515159	−	0.857095i	$-0.672267\pi$
$379$	−20.0582	−1.03032	−0.515159	+	0.857095i	$0.672267\pi$
$380$	0	0
$381$	−22.3520	−1.14513
$382$	0	0
$383$	4.44760	0.227262	0.113631	−	0.993523i	$-0.463752\pi$
$383$	4.44760	0.227262	0.113631	+	0.993523i	$0.463752\pi$
$384$	0	0
$385$	−39.8450	−2.03069
$386$	0	0
$387$	2.34623	0.119266
$388$	0	0
$389$	−16.8921	−0.856463	−0.428232	−	0.903669i	$-0.640863\pi$
$389$	−16.8921	−0.856463	−0.428232	+	0.903669i	$0.640863\pi$
$390$	0	0
$391$	−5.23123	−0.264554
$392$	0	0
$393$	−6.74383	−0.340181
$394$	0	0
$395$	−11.5378	−0.580528
$396$	0	0
$397$	−17.8421	−0.895472	−0.447736	−	0.894166i	$-0.647770\pi$
$397$	−17.8421	−0.895472	−0.447736	+	0.894166i	$0.647770\pi$
$398$	0	0
$399$	−44.6578	−2.23569
$400$	0	0
$401$	−33.5384	−1.67483	−0.837413	−	0.546570i	$-0.815933\pi$
$401$	−33.5384	−1.67483	−0.837413	+	0.546570i	$0.815933\pi$
$402$	0	0
$403$	25.3084	1.26070
$404$	0	0
$405$	31.1916	1.54992
$406$	0	0
$407$	10.2121	0.506195
$408$	0	0
$409$	−13.6062	−0.672783	−0.336391	−	0.941722i	$-0.609207\pi$
$409$	−13.6062	−0.672783	−0.336391	+	0.941722i	$0.609207\pi$
$410$	0	0
$411$	7.23080	0.356669
$412$	0	0
$413$	11.5017	0.565960
$414$	0	0
$415$	2.79494	0.137198
$416$	0	0
$417$	20.1860	0.988514
$418$	0	0
$419$	35.3138	1.72519	0.862595	−	0.505895i	$-0.168838\pi$
$419$	35.3138	1.72519	0.862595	+	0.505895i	$0.168838\pi$
$420$	0	0
$421$	−36.2195	−1.76523	−0.882615	−	0.470097i	$-0.844219\pi$
$421$	−36.2195	−1.76523	−0.882615	+	0.470097i	$0.844219\pi$
$422$	0	0
$423$	−6.39374	−0.310875
$424$	0	0
$425$	3.01373	0.146188
$426$	0	0
$427$	48.0947	2.32746
$428$	0	0
$429$	−46.1629	−2.22876
$430$	0	0
$431$	12.8505	0.618989	0.309495	−	0.950901i	$-0.399840\pi$
$431$	12.8505	0.618989	0.309495	+	0.950901i	$0.399840\pi$
$432$	0	0
$433$	3.63930	0.174894	0.0874468	−	0.996169i	$-0.472129\pi$
$433$	3.63930	0.174894	0.0874468	+	0.996169i	$0.472129\pi$
$434$	0	0
$435$	3.74315	0.179470
$436$	0	0
$437$	29.4589	1.40921
$438$	0	0
$439$	19.2550	0.918993	0.459497	−	0.888180i	$-0.348030\pi$
$439$	19.2550	0.918993	0.459497	+	0.888180i	$0.348030\pi$
$440$	0	0
$441$	7.23959	0.344742
$442$	0	0
$443$	14.0770	0.668819	0.334410	−	0.942428i	$-0.391463\pi$
$443$	14.0770	0.668819	0.334410	+	0.942428i	$0.391463\pi$
$444$	0	0
$445$	−21.8317	−1.03492
$446$	0	0
$447$	22.7278	1.07499
$448$	0	0
$449$	33.7090	1.59083	0.795414	−	0.606066i	$-0.207253\pi$
$449$	33.7090	1.59083	0.795414	+	0.606066i	$0.207253\pi$
$450$	0	0
$451$	−1.42403	−0.0670551
$452$	0	0
$453$	−22.8238	−1.07236
$454$	0	0
$455$	57.5531	2.69813
$456$	0	0
$457$	12.8003	0.598773	0.299386	−	0.954132i	$-0.403218\pi$
$457$	12.8003	0.598773	0.299386	+	0.954132i	$0.403218\pi$
$458$	0	0
$459$	−3.95388	−0.184552
$460$	0	0
$461$	18.2924	0.851963	0.425981	−	0.904732i	$-0.359929\pi$
$461$	18.2924	0.851963	0.425981	+	0.904732i	$0.359929\pi$
$462$	0	0
$463$	−7.72298	−0.358918	−0.179459	−	0.983765i	$-0.557435\pi$
$463$	−7.72298	−0.358918	−0.179459	+	0.983765i	$0.557435\pi$
$464$	0	0
$465$	25.4148	1.17858
$466$	0	0
$467$	−12.2488	−0.566808	−0.283404	−	0.959001i	$-0.591464\pi$
$467$	−12.2488	−0.566808	−0.283404	+	0.959001i	$0.591464\pi$
$468$	0	0
$469$	−20.7428	−0.957815
$470$	0	0
$471$	−4.70543	−0.216815
$472$	0	0
$473$	7.72060	0.354994
$474$	0	0
$475$	−16.9714	−0.778703
$476$	0	0
$477$	−11.0109	−0.504154
$478$	0	0
$479$	9.72827	0.444496	0.222248	−	0.974990i	$-0.428661\pi$
$479$	9.72827	0.444496	0.222248	+	0.974990i	$0.428661\pi$
$480$	0	0
$481$	−14.7506	−0.672569
$482$	0	0
$483$	−36.1085	−1.64299
$484$	0	0
$485$	40.2999	1.82992
$486$	0	0
$487$	33.3554	1.51148	0.755740	−	0.654872i	$-0.227278\pi$
$487$	33.3554	1.51148	0.755740	+	0.654872i	$0.227278\pi$
$488$	0	0
$489$	−9.68562	−0.437999
$490$	0	0
$491$	−0.879784	−0.0397041	−0.0198520	−	0.999803i	$-0.506320\pi$
$491$	−0.879784	−0.0397041	−0.0198520	+	0.999803i	$0.506320\pi$
$492$	0	0
$493$	−0.700448	−0.0315466
$494$	0	0
$495$	−13.2453	−0.595332
$496$	0	0
$497$	−39.5086	−1.77220
$498$	0	0
$499$	−11.5141	−0.515441	−0.257720	−	0.966220i	$-0.582971\pi$
$499$	−11.5141	−0.515441	−0.257720	+	0.966220i	$0.582971\pi$
$500$	0	0
$501$	42.3777	1.89330
$502$	0	0
$503$	7.82934	0.349093	0.174546	−	0.984649i	$-0.444154\pi$
$503$	7.82934	0.349093	0.174546	+	0.984649i	$0.444154\pi$
$504$	0	0
$505$	22.7863	1.01398
$506$	0	0
$507$	40.0365	1.77808
$508$	0	0
$509$	34.4776	1.52819	0.764096	−	0.645102i	$-0.223185\pi$
$509$	34.4776	1.52819	0.764096	+	0.645102i	$0.223185\pi$
$510$	0	0
$511$	−47.6705	−2.10882
$512$	0	0
$513$	22.2658	0.983057
$514$	0	0
$515$	−51.5688	−2.27240
$516$	0	0
$517$	−21.0395	−0.925317
$518$	0	0
$519$	33.0737	1.45178
$520$	0	0
$521$	26.3564	1.15469	0.577347	−	0.816499i	$-0.304088\pi$
$521$	26.3564	1.15469	0.577347	+	0.816499i	$0.304088\pi$
$522$	0	0
$523$	−4.92628	−0.215411	−0.107706	−	0.994183i	$-0.534350\pi$
$523$	−4.92628	−0.215411	−0.107706	+	0.994183i	$0.534350\pi$
$524$	0	0
$525$	20.8023	0.907885
$526$	0	0
$527$	−4.75581	−0.207167
$528$	0	0
$529$	0.819291	0.0356214
$530$	0	0
$531$	3.82339	0.165921
$532$	0	0
$533$	2.05691	0.0890945
$534$	0	0
$535$	11.8576	0.512650
$536$	0	0
$537$	24.8897	1.07407
$538$	0	0
$539$	23.8229	1.02612
$540$	0	0
$541$	21.1500	0.909311	0.454655	−	0.890667i	$-0.349762\pi$
$541$	21.1500	0.909311	0.454655	+	0.890667i	$0.349762\pi$
$542$	0	0
$543$	36.2761	1.55676
$544$	0	0
$545$	10.3909	0.445097
$546$	0	0
$547$	3.19334	0.136537	0.0682687	−	0.997667i	$-0.478252\pi$
$547$	3.19334	0.136537	0.0682687	+	0.997667i	$0.478252\pi$
$548$	0	0
$549$	15.9877	0.682337
$550$	0	0
$551$	3.94448	0.168040
$552$	0	0
$553$	14.9027	0.633729
$554$	0	0
$555$	−14.8126	−0.628758
$556$	0	0
$557$	25.7327	1.09033	0.545165	−	0.838329i	$-0.316467\pi$
$557$	25.7327	1.09033	0.545165	+	0.838329i	$0.316467\pi$
$558$	0	0
$559$	−11.1518	−0.471672
$560$	0	0
$561$	8.67465	0.366244
$562$	0	0
$563$	22.7318	0.958030	0.479015	−	0.877807i	$-0.340994\pi$
$563$	22.7318	0.958030	0.479015	+	0.877807i	$0.340994\pi$
$564$	0	0
$565$	11.3516	0.477566
$566$	0	0
$567$	−40.2886	−1.69196
$568$	0	0
$569$	−12.5423	−0.525799	−0.262899	−	0.964823i	$-0.584679\pi$
$569$	−12.5423	−0.525799	−0.262899	+	0.964823i	$0.584679\pi$
$570$	0	0
$571$	9.13574	0.382319	0.191159	−	0.981559i	$-0.438775\pi$
$571$	9.13574	0.382319	0.191159	+	0.981559i	$0.438775\pi$
$572$	0	0
$573$	−13.6338	−0.569560
$574$	0	0
$575$	−13.7224	−0.572264
$576$	0	0
$577$	−7.90995	−0.329296	−0.164648	−	0.986352i	$-0.552649\pi$
$577$	−7.90995	−0.329296	−0.164648	+	0.986352i	$0.552649\pi$
$578$	0	0
$579$	11.3268	0.470725
$580$	0	0
$581$	−3.61008	−0.149771
$582$	0	0
$583$	−36.2329	−1.50061
$584$	0	0
$585$	19.1318	0.791004
$586$	0	0
$587$	35.3383	1.45857	0.729283	−	0.684212i	$-0.239854\pi$
$587$	35.3383	1.45857	0.729283	+	0.684212i	$0.239854\pi$
$588$	0	0
$589$	26.7817	1.10352
$590$	0	0
$591$	−3.92358	−0.161395
$592$	0	0
$593$	19.9190	0.817977	0.408989	−	0.912539i	$-0.365882\pi$
$593$	19.9190	0.817977	0.408989	+	0.912539i	$0.365882\pi$
$594$	0	0
$595$	−10.8150	−0.443373
$596$	0	0
$597$	18.8419	0.771146
$598$	0	0
$599$	4.92162	0.201092	0.100546	−	0.994932i	$-0.467941\pi$
$599$	4.92162	0.201092	0.100546	+	0.994932i	$0.467941\pi$
$600$	0	0
$601$	22.9308	0.935366	0.467683	−	0.883896i	$-0.345089\pi$
$601$	22.9308	0.935366	0.467683	+	0.883896i	$0.345089\pi$
$602$	0	0
$603$	−6.89535	−0.280800
$604$	0	0
$605$	−12.8412	−0.522069
$606$	0	0
$607$	−31.2888	−1.26997	−0.634986	−	0.772524i	$-0.718994\pi$
$607$	−31.2888	−1.26997	−0.634986	+	0.772524i	$0.718994\pi$
$608$	0	0
$609$	−4.83484	−0.195917
$610$	0	0
$611$	30.3900	1.22945
$612$	0	0
$613$	−37.5402	−1.51623	−0.758117	−	0.652119i	$-0.773880\pi$
$613$	−37.5402	−1.51623	−0.758117	+	0.652119i	$0.773880\pi$
$614$	0	0
$615$	2.06555	0.0832909
$616$	0	0
$617$	27.0766	1.09006	0.545032	−	0.838415i	$-0.316517\pi$
$617$	27.0766	1.09006	0.545032	+	0.838415i	$0.316517\pi$
$618$	0	0
$619$	18.5116	0.744046	0.372023	−	0.928224i	$-0.378664\pi$
$619$	18.5116	0.744046	0.372023	+	0.928224i	$0.378664\pi$
$620$	0	0
$621$	18.0032	0.722443
$622$	0	0
$623$	28.1988	1.12976
$624$	0	0
$625$	−31.1529	−1.24611
$626$	0	0
$627$	−48.8501	−1.95089
$628$	0	0
$629$	2.77184	0.110521
$630$	0	0
$631$	−17.2785	−0.687848	−0.343924	−	0.938998i	$-0.611756\pi$
$631$	−17.2785	−0.687848	−0.343924	+	0.938998i	$0.611756\pi$
$632$	0	0
$633$	−35.7861	−1.42237
$634$	0	0
$635$	30.4832	1.20969
$636$	0	0
$637$	−34.4103	−1.36339
$638$	0	0
$639$	−13.1335	−0.519553
$640$	0	0
$641$	45.4417	1.79484	0.897419	−	0.441179i	$-0.145440\pi$
$641$	45.4417	1.79484	0.897419	+	0.441179i	$0.145440\pi$
$642$	0	0
$643$	6.23775	0.245993	0.122997	−	0.992407i	$-0.460750\pi$
$643$	6.23775	0.245993	0.122997	+	0.992407i	$0.460750\pi$
$644$	0	0
$645$	−11.1987	−0.440947
$646$	0	0
$647$	16.8823	0.663712	0.331856	−	0.943330i	$-0.392325\pi$
$647$	16.8823	0.663712	0.331856	+	0.943330i	$0.392325\pi$
$648$	0	0
$649$	12.5814	0.493863
$650$	0	0
$651$	−32.8269	−1.28659
$652$	0	0
$653$	25.9556	1.01572	0.507860	−	0.861440i	$-0.330437\pi$
$653$	25.9556	1.01572	0.507860	+	0.861440i	$0.330437\pi$
$654$	0	0
$655$	9.19709	0.359360
$656$	0	0
$657$	−15.8467	−0.618237
$658$	0	0
$659$	−34.8250	−1.35659	−0.678295	−	0.734789i	$-0.737281\pi$
$659$	−34.8250	−1.35659	−0.678295	+	0.734789i	$0.737281\pi$
$660$	0	0
$661$	−10.2220	−0.397590	−0.198795	−	0.980041i	$-0.563703\pi$
$661$	−10.2220	−0.397590	−0.198795	+	0.980041i	$0.563703\pi$
$662$	0	0
$663$	−12.5299	−0.486620
$664$	0	0
$665$	60.9034	2.36173
$666$	0	0
$667$	3.18934	0.123492
$668$	0	0
$669$	−46.2252	−1.78717
$670$	0	0
$671$	52.6096	2.03097
$672$	0	0
$673$	−32.8882	−1.26775	−0.633874	−	0.773437i	$-0.718536\pi$
$673$	−32.8882	−1.26775	−0.633874	+	0.773437i	$0.718536\pi$
$674$	0	0
$675$	−10.3717	−0.399208
$676$	0	0
$677$	21.8948	0.841487	0.420743	−	0.907180i	$-0.361769\pi$
$677$	21.8948	0.841487	0.420743	+	0.907180i	$0.361769\pi$
$678$	0	0
$679$	−52.0533	−1.99762
$680$	0	0
$681$	−47.5007	−1.82023
$682$	0	0
$683$	38.8541	1.48671	0.743355	−	0.668898i	$-0.233234\pi$
$683$	38.8541	1.48671	0.743355	+	0.668898i	$0.233234\pi$
$684$	0	0
$685$	−9.86121	−0.376778
$686$	0	0
$687$	18.9191	0.721810
$688$	0	0
$689$	52.3356	1.99383
$690$	0	0
$691$	−5.18133	−0.197107	−0.0985535	−	0.995132i	$-0.531422\pi$
$691$	−5.18133	−0.197107	−0.0985535	+	0.995132i	$0.531422\pi$
$692$	0	0
$693$	17.1083	0.649890
$694$	0	0
$695$	−27.5293	−1.04425
$696$	0	0
$697$	−0.386522	−0.0146406
$698$	0	0
$699$	5.46115	0.206560
$700$	0	0
$701$	−3.06591	−0.115798	−0.0578988	−	0.998322i	$-0.518440\pi$
$701$	−3.06591	−0.115798	−0.0578988	+	0.998322i	$0.518440\pi$
$702$	0	0
$703$	−15.6093	−0.588715
$704$	0	0
$705$	30.5176	1.14936
$706$	0	0
$707$	−29.4319	−1.10690
$708$	0	0
$709$	8.03122	0.301619	0.150809	−	0.988563i	$-0.451812\pi$
$709$	8.03122	0.301619	0.150809	+	0.988563i	$0.451812\pi$
$710$	0	0
$711$	4.95398	0.185789
$712$	0	0
$713$	21.6546	0.810971
$714$	0	0
$715$	62.9560	2.35442
$716$	0	0
$717$	46.7404	1.74555
$718$	0	0
$719$	−24.7233	−0.922023	−0.461012	−	0.887394i	$-0.652513\pi$
$719$	−24.7233	−0.922023	−0.461012	+	0.887394i	$0.652513\pi$
$720$	0	0
$721$	66.6088	2.48064
$722$	0	0
$723$	−0.734601	−0.0273201
$724$	0	0
$725$	−1.83740	−0.0682392
$726$	0	0
$727$	−37.3913	−1.38677	−0.693383	−	0.720569i	$-0.743881\pi$
$727$	−37.3913	−1.38677	−0.693383	+	0.720569i	$0.743881\pi$
$728$	0	0
$729$	9.28677	0.343954
$730$	0	0
$731$	2.09558	0.0775080
$732$	0	0
$733$	−7.30120	−0.269676	−0.134838	−	0.990868i	$-0.543051\pi$
$733$	−7.30120	−0.269676	−0.134838	+	0.990868i	$0.543051\pi$
$734$	0	0
$735$	−34.5549	−1.27458
$736$	0	0
$737$	−22.6901	−0.835801
$738$	0	0
$739$	25.9837	0.955826	0.477913	−	0.878407i	$-0.341393\pi$
$739$	25.9837	0.955826	0.477913	+	0.878407i	$0.341393\pi$
$740$	0	0
$741$	70.5603	2.59210
$742$	0	0
$743$	−16.7818	−0.615664	−0.307832	−	0.951441i	$-0.599603\pi$
$743$	−16.7818	−0.615664	−0.307832	+	0.951441i	$0.599603\pi$
$744$	0	0
$745$	−30.9957	−1.13559
$746$	0	0
$747$	−1.20006	−0.0439081
$748$	0	0
$749$	−15.3159	−0.559630
$750$	0	0
$751$	18.6067	0.678967	0.339484	−	0.940612i	$-0.389748\pi$
$751$	18.6067	0.678967	0.339484	+	0.940612i	$0.389748\pi$
$752$	0	0
$753$	−41.2613	−1.50364
$754$	0	0
$755$	31.1267	1.13282
$756$	0	0
$757$	−42.7062	−1.55218	−0.776091	−	0.630621i	$-0.782800\pi$
$757$	−42.7062	−1.55218	−0.776091	+	0.630621i	$0.782800\pi$
$758$	0	0
$759$	−39.4982	−1.43369
$760$	0	0
$761$	−15.4171	−0.558869	−0.279435	−	0.960165i	$-0.590147\pi$
$761$	−15.4171	−0.558869	−0.279435	+	0.960165i	$0.590147\pi$
$762$	0	0
$763$	−13.4214	−0.485887
$764$	0	0
$765$	−3.59514	−0.129983
$766$	0	0
$767$	−18.1729	−0.656185
$768$	0	0
$769$	40.4747	1.45955	0.729777	−	0.683685i	$-0.239624\pi$
$769$	40.4747	1.45955	0.729777	+	0.683685i	$0.239624\pi$
$770$	0	0
$771$	32.1086	1.15636
$772$	0	0
$773$	−52.2403	−1.87895	−0.939476	−	0.342614i	$-0.888688\pi$
$773$	−52.2403	−1.87895	−0.939476	+	0.342614i	$0.888688\pi$
$774$	0	0
$775$	−12.4753	−0.448127
$776$	0	0
$777$	19.1326	0.686379
$778$	0	0
$779$	2.17664	0.0779864
$780$	0	0
$781$	−43.2176	−1.54645
$782$	0	0
$783$	2.41058	0.0861472
$784$	0	0
$785$	6.41717	0.229039
$786$	0	0
$787$	34.4654	1.22856	0.614279	−	0.789089i	$-0.289447\pi$
$787$	34.4654	1.22856	0.614279	+	0.789089i	$0.289447\pi$
$788$	0	0
$789$	−44.9215	−1.59925
$790$	0	0
$791$	−14.6623	−0.521332
$792$	0	0
$793$	−75.9906	−2.69851
$794$	0	0
$795$	52.5555	1.86395
$796$	0	0
$797$	−8.32223	−0.294788	−0.147394	−	0.989078i	$-0.547089\pi$
$797$	−8.32223	−0.294788	−0.147394	+	0.989078i	$0.547089\pi$
$798$	0	0
$799$	−5.71071	−0.202030
$800$	0	0
$801$	9.37388	0.331210
$802$	0	0
$803$	−52.1457	−1.84018
$804$	0	0
$805$	49.2440	1.73562
$806$	0	0
$807$	35.4046	1.24630
$808$	0	0
$809$	40.1161	1.41041	0.705203	−	0.709005i	$-0.250856\pi$
$809$	40.1161	1.41041	0.705203	+	0.709005i	$0.250856\pi$
$810$	0	0
$811$	30.1648	1.05923	0.529614	−	0.848239i	$-0.322337\pi$
$811$	30.1648	1.05923	0.529614	+	0.848239i	$0.322337\pi$
$812$	0	0
$813$	−31.5350	−1.10598
$814$	0	0
$815$	13.2091	0.462693
$816$	0	0
$817$	−11.8010	−0.412865
$818$	0	0
$819$	−24.7116	−0.863493
$820$	0	0
$821$	−24.7839	−0.864963	−0.432481	−	0.901643i	$-0.642362\pi$
$821$	−24.7839	−0.864963	−0.432481	+	0.901643i	$0.642362\pi$
$822$	0	0
$823$	26.7616	0.932849	0.466425	−	0.884561i	$-0.345542\pi$
$823$	26.7616	0.932849	0.466425	+	0.884561i	$0.345542\pi$
$824$	0	0
$825$	22.7551	0.792231
$826$	0	0
$827$	−20.2138	−0.702904	−0.351452	−	0.936206i	$-0.614312\pi$
$827$	−20.2138	−0.702904	−0.351452	+	0.936206i	$0.614312\pi$
$828$	0	0
$829$	22.2180	0.771662	0.385831	−	0.922569i	$-0.373915\pi$
$829$	22.2180	0.771662	0.385831	+	0.922569i	$0.373915\pi$
$830$	0	0
$831$	−28.0554	−0.973231
$832$	0	0
$833$	6.46619	0.224040
$834$	0	0
$835$	−57.7938	−2.00004
$836$	0	0
$837$	16.3671	0.565728
$838$	0	0
$839$	4.83237	0.166832	0.0834159	−	0.996515i	$-0.473417\pi$
$839$	4.83237	0.166832	0.0834159	+	0.996515i	$0.473417\pi$
$840$	0	0
$841$	−28.5730	−0.985274
$842$	0	0
$843$	−11.8503	−0.408148
$844$	0	0
$845$	−54.6009	−1.87833
$846$	0	0
$847$	16.5863	0.569912
$848$	0	0
$849$	−51.0255	−1.75119
$850$	0	0
$851$	−12.6210	−0.432643
$852$	0	0
$853$	28.7951	0.985925	0.492963	−	0.870050i	$-0.335914\pi$
$853$	28.7951	0.985925	0.492963	+	0.870050i	$0.335914\pi$
$854$	0	0
$855$	20.2455	0.692383
$856$	0	0
$857$	16.2165	0.553944	0.276972	−	0.960878i	$-0.410669\pi$
$857$	16.2165	0.553944	0.276972	+	0.960878i	$0.410669\pi$
$858$	0	0
$859$	−35.7031	−1.21818	−0.609088	−	0.793103i	$-0.708464\pi$
$859$	−35.7031	−1.21818	−0.609088	+	0.793103i	$0.708464\pi$
$860$	0	0
$861$	−2.66796	−0.0909239
$862$	0	0
$863$	6.63919	0.226001	0.113000	−	0.993595i	$-0.463954\pi$
$863$	6.63919	0.226001	0.113000	+	0.993595i	$0.463954\pi$
$864$	0	0
$865$	−45.1053	−1.53363
$866$	0	0
$867$	−32.4854	−1.10326
$868$	0	0
$869$	16.3018	0.552999
$870$	0	0
$871$	32.7741	1.11051
$872$	0	0
$873$	−17.3036	−0.585638
$874$	0	0
$875$	22.0800	0.746441
$876$	0	0
$877$	−37.0604	−1.25144	−0.625720	−	0.780048i	$-0.715195\pi$
$877$	−37.0604	−1.25144	−0.625720	+	0.780048i	$0.715195\pi$
$878$	0	0
$879$	−36.9776	−1.24722
$880$	0	0
$881$	47.1178	1.58744	0.793719	−	0.608285i	$-0.208142\pi$
$881$	47.1178	1.58744	0.793719	+	0.608285i	$0.208142\pi$
$882$	0	0
$883$	−13.6661	−0.459899	−0.229950	−	0.973203i	$-0.573856\pi$
$883$	−13.6661	−0.459899	−0.229950	+	0.973203i	$0.573856\pi$
$884$	0	0
$885$	−18.2492	−0.613441
$886$	0	0
$887$	−38.4600	−1.29136	−0.645680	−	0.763608i	$-0.723426\pi$
$887$	−38.4600	−1.29136	−0.645680	+	0.763608i	$0.723426\pi$
$888$	0	0
$889$	−39.3736	−1.32055
$890$	0	0
$891$	−44.0708	−1.47643
$892$	0	0
$893$	32.1591	1.07616
$894$	0	0
$895$	−33.9441	−1.13463
$896$	0	0
$897$	57.0522	1.90492
$898$	0	0
$899$	2.89950	0.0967037
$900$	0	0
$901$	−9.83460	−0.327638
$902$	0	0
$903$	14.4647	0.481356
$904$	0	0
$905$	−49.4726	−1.64452
$906$	0	0
$907$	−14.6814	−0.487487	−0.243743	−	0.969840i	$-0.578375\pi$
$907$	−14.6814	−0.487487	−0.243743	+	0.969840i	$0.578375\pi$
$908$	0	0
$909$	−9.78378	−0.324507
$910$	0	0
$911$	−1.43324	−0.0474853	−0.0237426	−	0.999718i	$-0.507558\pi$
$911$	−1.43324	−0.0474853	−0.0237426	+	0.999718i	$0.507558\pi$
$912$	0	0
$913$	−3.94898	−0.130692
$914$	0	0
$915$	−76.3098	−2.52273
$916$	0	0
$917$	−11.8794	−0.392293
$918$	0	0
$919$	30.7396	1.01401	0.507003	−	0.861944i	$-0.330753\pi$
$919$	30.7396	1.01401	0.507003	+	0.861944i	$0.330753\pi$
$920$	0	0
$921$	17.3796	0.572677
$922$	0	0
$923$	62.4245	2.05473
$924$	0	0
$925$	7.27103	0.239070
$926$	0	0
$927$	22.1422	0.727244
$928$	0	0
$929$	−3.38196	−0.110959	−0.0554793	−	0.998460i	$-0.517669\pi$
$929$	−3.38196	−0.110959	−0.0554793	+	0.998460i	$0.517669\pi$
$930$	0	0
$931$	−36.4134	−1.19340
$932$	0	0
$933$	15.8977	0.520469
$934$	0	0
$935$	−11.8303	−0.386893
$936$	0	0
$937$	43.8412	1.43223	0.716115	−	0.697982i	$-0.245918\pi$
$937$	43.8412	1.43223	0.716115	+	0.697982i	$0.245918\pi$
$938$	0	0
$939$	−2.74168	−0.0894712
$940$	0	0
$941$	29.3566	0.956998	0.478499	−	0.878088i	$-0.341181\pi$
$941$	29.3566	0.956998	0.478499	+	0.878088i	$0.341181\pi$
$942$	0	0
$943$	1.75995	0.0573117
$944$	0	0
$945$	37.2198	1.21076
$946$	0	0
$947$	18.6458	0.605906	0.302953	−	0.953006i	$-0.402028\pi$
$947$	18.6458	0.605906	0.302953	+	0.953006i	$0.402028\pi$
$948$	0	0
$949$	75.3205	2.44501
$950$	0	0
$951$	−36.5450	−1.18505
$952$	0	0
$953$	−47.4761	−1.53790	−0.768950	−	0.639308i	$-0.779221\pi$
$953$	−47.4761	−1.53790	−0.768950	+	0.639308i	$0.779221\pi$
$954$	0	0
$955$	18.5935	0.601671
$956$	0	0
$957$	−5.28871	−0.170960
$958$	0	0
$959$	12.7372	0.411306
$960$	0	0
$961$	−11.3134	−0.364947
$962$	0	0
$963$	−5.09132	−0.164065
$964$	0	0
$965$	−15.4472	−0.497264
$966$	0	0
$967$	40.2215	1.29344	0.646718	−	0.762729i	$-0.276141\pi$
$967$	40.2215	1.29344	0.646718	+	0.762729i	$0.276141\pi$
$968$	0	0
$969$	−13.2593	−0.425949
$970$	0	0
$971$	7.39815	0.237418	0.118709	−	0.992929i	$-0.462124\pi$
$971$	7.39815	0.237418	0.118709	+	0.992929i	$0.462124\pi$
$972$	0	0
$973$	35.5582	1.13994
$974$	0	0
$975$	−32.8680	−1.05262
$976$	0	0
$977$	24.8952	0.796469	0.398234	−	0.917284i	$-0.369623\pi$
$977$	24.8952	0.796469	0.398234	+	0.917284i	$0.369623\pi$
$978$	0	0
$979$	30.8460	0.985844
$980$	0	0
$981$	−4.46155	−0.142446
$982$	0	0
$983$	53.8004	1.71597	0.857983	−	0.513678i	$-0.171717\pi$
$983$	53.8004	1.71597	0.857983	+	0.513678i	$0.171717\pi$
$984$	0	0
$985$	5.35090	0.170494
$986$	0	0
$987$	−39.4181	−1.25469
$988$	0	0
$989$	−9.54180	−0.303412
$990$	0	0
$991$	−35.5111	−1.12805	−0.564024	−	0.825758i	$-0.690748\pi$
$991$	−35.5111	−1.12805	−0.564024	+	0.825758i	$0.690748\pi$
$992$	0	0
$993$	6.16436	0.195620
$994$	0	0
$995$	−25.6961	−0.814622
$996$	0	0
$997$	61.1251	1.93585	0.967926	−	0.251234i	$-0.0808363\pi$
$997$	61.1251	1.93585	0.967926	+	0.251234i	$0.0808363\pi$
$998$	0	0
$999$	−9.53926	−0.301809

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5312.2.a.bn.1.6		6
4.3	odd	2		5312.2.a.bo.1.1		6
8.3	odd	2		1328.2.a.l.1.6		6
8.5	even	2		83.2.a.b.1.2	✓	6
24.5	odd	2		747.2.a.j.1.5		6
40.29	even	2		2075.2.a.g.1.5		6
56.13	odd	2		4067.2.a.d.1.2		6
664.165	odd	2		6889.2.a.e.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
83.2.a.b.1.2	✓	6	8.5	even	2
747.2.a.j.1.5		6	24.5	odd	2
1328.2.a.l.1.6		6	8.3	odd	2
2075.2.a.g.1.5		6	40.29	even	2
4067.2.a.d.1.2		6	56.13	odd	2
5312.2.a.bn.1.6		6	1.1	even	1	trivial
5312.2.a.bo.1.1		6	4.3	odd	2
6889.2.a.e.1.5		6	664.165	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 5312 = 2^{6} \cdot 83 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5312.a (trivial)

\(f(q)\)	\(=\)	\(q+2.04941 q^{3} -2.79494 q^{5} +3.61008 q^{7} +1.20006 q^{9} +3.94898 q^{11} -5.70400 q^{13} -5.72796 q^{15} +1.07186 q^{17} -6.03605 q^{19} +7.39852 q^{21} -4.88050 q^{23} +2.81168 q^{25} -3.68880 q^{27} -0.653487 q^{29} -4.43696 q^{31} +8.09306 q^{33} -10.0899 q^{35} +2.58601 q^{37} -11.6898 q^{39} -0.360608 q^{41} +1.95509 q^{43} -3.35411 q^{45} -5.32783 q^{47} +6.03266 q^{49} +2.19668 q^{51} -9.17524 q^{53} -11.0372 q^{55} -12.3703 q^{57} +3.18599 q^{59} +13.3223 q^{61} +4.33233 q^{63} +15.9423 q^{65} -5.74581 q^{67} -10.0021 q^{69} -10.9440 q^{71} -13.2048 q^{73} +5.76227 q^{75} +14.2561 q^{77} +4.12809 q^{79} -11.1600 q^{81} -1.00000 q^{83} -2.99579 q^{85} -1.33926 q^{87} +7.81114 q^{89} -20.5919 q^{91} -9.09314 q^{93} +16.8704 q^{95} -14.4189 q^{97} +4.73903 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - q^{3} - 2 q^{5} + 3 q^{7} + 3 q^{9} + 3 q^{11} - 14 q^{13} - 6 q^{15} - 5 q^{17} + 4 q^{19} + 2 q^{21} - 5 q^{23} + 14 q^{25} - 10 q^{27} + q^{29} + 3 q^{31} + 2 q^{33} + 10 q^{35} - 39 q^{37} - 8 q^{39}+ \cdots + 18 q^{99}+O(q^{100}) \) 6 * q - q^3 - 2 * q^5 + 3 * q^7 + 3 * q^9 + 3 * q^11 - 14 * q^13 - 6 * q^15 - 5 * q^17 + 4 * q^19 + 2 * q^21 - 5 * q^23 + 14 * q^25 - 10 * q^27 + q^29 + 3 * q^31 + 2 * q^33 + 10 * q^35 - 39 * q^37 - 8 * q^39 - q^41 + 8 * q^43 - 10 * q^45 - 12 * q^47 + 11 * q^49 + 24 * q^51 - 14 * q^53 - 24 * q^55 - 10 * q^57 + 17 * q^59 + 5 * q^61 - 11 * q^63 - 8 * q^65 - 16 * q^67 - 8 * q^69 - 26 * q^71 - 6 * q^73 + 45 * q^75 - 6 * q^77 - 12 * q^79 - 34 * q^81 - 6 * q^83 - 18 * q^85 - 3 * q^87 - 22 * q^89 + 2 * q^91 - 34 * q^93 - 4 * q^95 + 6 * q^97 + 18 * q^99

Introduction

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