LMFDB - Embedded newform 4352.2.a.bd.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4352,2,Mod(1,4352)] 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("4352.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4352, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4352 = 2^{8} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4352.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,-8,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)

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

Self dual:	yes
Analytic conductor:	$34.7508949597$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.8.16845963264.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 10x^{6} + 23x^{4} - 16x^{2} + 1 $ x^8 - 10x^6 + 23x^4 - 16*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 2176)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.24296$ of defining polynomial
Character	$\chi$	$=$	4352.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.84240 q^{3} +1.67000 q^{5} -5.08101 q^{7} +0.394449 q^{9} -4.67083 q^{11} -1.67000 q^{13} -3.07681 q^{15} -1.00000 q^{17} -6.51323 q^{19} +9.36127 q^{21} -2.71927 q^{23} -2.21110 q^{25} +4.80048 q^{27} +1.67000 q^{29} -7.44274 q^{31} +8.60555 q^{33} -8.48528 q^{35} -9.36127 q^{37} +3.07681 q^{39} -6.00000 q^{41} +0.658729 q^{45} +13.2388 q^{47} +18.8167 q^{49} +1.84240 q^{51} -7.69127 q^{53} -7.80028 q^{55} +12.0000 q^{57} +5.65685 q^{59} -5.01000 q^{61} -2.00420 q^{63} -2.78890 q^{65} -8.48528 q^{67} +5.01000 q^{69} +10.5196 q^{71} -7.21110 q^{73} +4.07374 q^{75} +23.7325 q^{77} -7.44274 q^{79} -10.0278 q^{81} -7.36961 q^{83} -1.67000 q^{85} -3.07681 q^{87} +10.6056 q^{89} +8.48528 q^{91} +13.7125 q^{93} -10.8771 q^{95} +2.00000 q^{97} -1.84240 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 32 q^{9} - 8 q^{17} + 40 q^{25} + 40 q^{33} - 48 q^{41} + 64 q^{49} + 96 q^{57} - 80 q^{65} + 64 q^{81} + 56 q^{89} + 16 q^{97}+O(q^{100}) $ 8 * q + 32 * q^9 - 8 * q^17 + 40 * q^25 + 40 * q^33 - 48 * q^41 + 64 * q^49 + 96 * q^57 - 80 * q^65 + 64 * q^81 + 56 * q^89 + 16 * q^97

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
$3$	−1.84240	−1.06371	−0.531856	−	0.846835i	$-0.678505\pi$
$3$	−1.84240	−1.06371	−0.531856	+	0.846835i	$0.678505\pi$
$4$	0	0
$5$	1.67000	0.746846	0.373423	−	0.927661i	$-0.378184\pi$
$5$	1.67000	0.746846	0.373423	+	0.927661i	$0.378184\pi$
$6$	0	0
$7$	−5.08101	−1.92044	−0.960220	−	0.279243i	$-0.909917\pi$
$7$	−5.08101	−1.92044	−0.960220	+	0.279243i	$0.909917\pi$
$8$	0	0
$9$	0.394449	0.131483
$10$	0	0
$11$	−4.67083	−1.40831	−0.704154	−	0.710047i	$-0.748674\pi$
$11$	−4.67083	−1.40831	−0.704154	+	0.710047i	$0.748674\pi$
$12$	0	0
$13$	−1.67000	−0.463174	−0.231587	−	0.972814i	$-0.574392\pi$
$13$	−1.67000	−0.463174	−0.231587	+	0.972814i	$0.574392\pi$
$14$	0	0
$15$	−3.07681	−0.794429
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−6.51323	−1.49424	−0.747119	−	0.664690i	$-0.768564\pi$
$19$	−6.51323	−1.49424	−0.747119	+	0.664690i	$0.768564\pi$
$20$	0	0
$21$	9.36127	2.04280
$22$	0	0
$23$	−2.71927	−0.567008	−0.283504	−	0.958971i	$-0.591497\pi$
$23$	−2.71927	−0.567008	−0.283504	+	0.958971i	$0.591497\pi$
$24$	0	0
$25$	−2.21110	−0.442221
$26$	0	0
$27$	4.80048	0.923852
$28$	0	0
$29$	1.67000	0.310111	0.155056	−	0.987906i	$-0.450444\pi$
$29$	1.67000	0.310111	0.155056	+	0.987906i	$0.450444\pi$
$30$	0	0
$31$	−7.44274	−1.33676	−0.668378	−	0.743822i	$-0.733011\pi$
$31$	−7.44274	−1.33676	−0.668378	+	0.743822i	$0.733011\pi$
$32$	0	0
$33$	8.60555	1.49803
$34$	0	0
$35$	−8.48528	−1.43427
$36$	0	0
$37$	−9.36127	−1.53898	−0.769491	−	0.638657i	$-0.779490\pi$
$37$	−9.36127	−1.53898	−0.769491	+	0.638657i	$0.779490\pi$
$38$	0	0
$39$	3.07681	0.492684
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	0.658729	0.0981975
$46$	0	0
$47$	13.2388	1.93108	0.965541	−	0.260251i	$-0.0838054\pi$
$47$	13.2388	1.93108	0.965541	+	0.260251i	$0.0838054\pi$
$48$	0	0
$49$	18.8167	2.68809
$50$	0	0
$51$	1.84240	0.257988
$52$	0	0
$53$	−7.69127	−1.05648	−0.528238	−	0.849096i	$-0.677147\pi$
$53$	−7.69127	−1.05648	−0.528238	+	0.849096i	$0.677147\pi$
$54$	0	0
$55$	−7.80028	−1.05179
$56$	0	0
$57$	12.0000	1.58944
$58$	0	0
$59$	5.65685	0.736460	0.368230	−	0.929735i	$-0.379964\pi$
$59$	5.65685	0.736460	0.368230	+	0.929735i	$0.379964\pi$
$60$	0	0
$61$	−5.01000	−0.641464	−0.320732	−	0.947170i	$-0.603929\pi$
$61$	−5.01000	−0.641464	−0.320732	+	0.947170i	$0.603929\pi$
$62$	0	0
$63$	−2.00420	−0.252505
$64$	0	0
$65$	−2.78890	−0.345920
$66$	0	0
$67$	−8.48528	−1.03664	−0.518321	−	0.855186i	$-0.673443\pi$
$67$	−8.48528	−1.03664	−0.518321	+	0.855186i	$0.673443\pi$
$68$	0	0
$69$	5.01000	0.603133
$70$	0	0
$71$	10.5196	1.24844	0.624221	−	0.781248i	$-0.285417\pi$
$71$	10.5196	1.24844	0.624221	+	0.781248i	$0.285417\pi$
$72$	0	0
$73$	−7.21110	−0.843996	−0.421998	−	0.906597i	$-0.638671\pi$
$73$	−7.21110	−0.843996	−0.421998	+	0.906597i	$0.638671\pi$
$74$	0	0
$75$	4.07374	0.470395
$76$	0	0
$77$	23.7325	2.70457
$78$	0	0
$79$	−7.44274	−0.837374	−0.418687	−	0.908131i	$-0.637510\pi$
$79$	−7.44274	−0.837374	−0.418687	+	0.908131i	$0.637510\pi$
$80$	0	0
$81$	−10.0278	−1.11420
$82$	0	0
$83$	−7.36961	−0.808920	−0.404460	−	0.914556i	$-0.632541\pi$
$83$	−7.36961	−0.808920	−0.404460	+	0.914556i	$0.632541\pi$
$84$	0	0
$85$	−1.67000	−0.181137
$86$	0	0
$87$	−3.07681	−0.329869
$88$	0	0
$89$	10.6056	1.12419	0.562093	−	0.827074i	$-0.309996\pi$
$89$	10.6056	1.12419	0.562093	+	0.827074i	$0.309996\pi$
$90$	0	0
$91$	8.48528	0.889499
$92$	0	0
$93$	13.7125	1.42192
$94$	0	0
$95$	−10.8771	−1.11597
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	0	0
$99$	−1.84240	−0.185168
$100$	0	0
$101$	−1.67000	−0.166171	−0.0830856	−	0.996542i	$-0.526477\pi$
$101$	−1.67000	−0.166171	−0.0830856	+	0.996542i	$0.526477\pi$
$102$	0	0
$103$	15.6006	1.53717	0.768585	−	0.639748i	$-0.220961\pi$
$103$	15.6006	1.53717	0.768585	+	0.639748i	$0.220961\pi$
$104$	0	0
$105$	15.6333	1.52565
$106$	0	0
$107$	−3.81445	−0.368757	−0.184378	−	0.982855i	$-0.559027\pi$
$107$	−3.81445	−0.368757	−0.184378	+	0.982855i	$0.559027\pi$
$108$	0	0
$109$	−9.36127	−0.896647	−0.448323	−	0.893871i	$-0.647979\pi$
$109$	−9.36127	−0.896647	−0.448323	+	0.893871i	$0.647979\pi$
$110$	0	0
$111$	17.2472	1.63703
$112$	0	0
$113$	−3.21110	−0.302075	−0.151038	−	0.988528i	$-0.548261\pi$
$113$	−3.21110	−0.302075	−0.151038	+	0.988528i	$0.548261\pi$
$114$	0	0
$115$	−4.54118	−0.423468
$116$	0	0
$117$	−0.658729	−0.0608995
$118$	0	0
$119$	5.08101	0.465775
$120$	0	0
$121$	10.8167	0.983332
$122$	0	0
$123$	11.0544	0.996743
$124$	0	0
$125$	−12.0425	−1.07712
$126$	0	0
$127$	0.715076	0.0634527	0.0317264	−	0.999497i	$-0.489899\pi$
$127$	0.715076	0.0634527	0.0317264	+	0.999497i	$0.489899\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1.84240	0.160971	0.0804857	−	0.996756i	$-0.474353\pi$
$131$	1.84240	0.160971	0.0804857	+	0.996756i	$0.474353\pi$
$132$	0	0
$133$	33.0938	2.86960
$134$	0	0
$135$	8.01679	0.689975
$136$	0	0
$137$	19.8167	1.69305	0.846525	−	0.532348i	$-0.178690\pi$
$137$	19.8167	1.69305	0.846525	+	0.532348i	$0.178690\pi$
$138$	0	0
$139$	2.95807	0.250900	0.125450	−	0.992100i	$-0.459962\pi$
$139$	2.95807	0.250900	0.125450	+	0.992100i	$0.459962\pi$
$140$	0	0
$141$	−24.3913	−2.05411
$142$	0	0
$143$	7.80028	0.652292
$144$	0	0
$145$	2.78890	0.231605
$146$	0	0
$147$	−34.6679	−2.85936
$148$	0	0
$149$	−22.0625	−1.80743	−0.903716	−	0.428131i	$-0.859172\pi$
$149$	−22.0625	−1.80743	−0.903716	+	0.428131i	$0.859172\pi$
$150$	0	0
$151$	−5.43855	−0.442583	−0.221291	−	0.975208i	$-0.571027\pi$
$151$	−5.43855	−0.442583	−0.221291	+	0.975208i	$0.571027\pi$
$152$	0	0
$153$	−0.394449	−0.0318893
$154$	0	0
$155$	−12.4294	−0.998352
$156$	0	0
$157$	−11.0313	−0.880391	−0.440195	−	0.897902i	$-0.645091\pi$
$157$	−11.0313	−0.880391	−0.440195	+	0.897902i	$0.645091\pi$
$158$	0	0
$159$	14.1704	1.12379
$160$	0	0
$161$	13.8167	1.08890
$162$	0	0
$163$	12.0404	0.943080	0.471540	−	0.881845i	$-0.343698\pi$
$163$	12.0404	0.943080	0.471540	+	0.881845i	$0.343698\pi$
$164$	0	0
$165$	14.3713	1.11880
$166$	0	0
$167$	−4.36593	−0.337846	−0.168923	−	0.985629i	$-0.554029\pi$
$167$	−4.36593	−0.337846	−0.168923	+	0.985629i	$0.554029\pi$
$168$	0	0
$169$	−10.2111	−0.785469
$170$	0	0
$171$	−2.56914	−0.196467
$172$	0	0
$173$	12.7013	0.965659	0.482830	−	0.875714i	$-0.339609\pi$
$173$	12.7013	0.965659	0.482830	+	0.875714i	$0.339609\pi$
$174$	0	0
$175$	11.2346	0.849258
$176$	0	0
$177$	−10.4222	−0.783381
$178$	0	0
$179$	4.80048	0.358804	0.179402	−	0.983776i	$-0.442584\pi$
$179$	4.80048	0.358804	0.179402	+	0.983776i	$0.442584\pi$
$180$	0	0
$181$	−12.7013	−0.944078	−0.472039	−	0.881578i	$-0.656482\pi$
$181$	−12.7013	−0.944078	−0.472039	+	0.881578i	$0.656482\pi$
$182$	0	0
$183$	9.23043	0.682333
$184$	0	0
$185$	−15.6333	−1.14938
$186$	0	0
$187$	4.67083	0.341565
$188$	0	0
$189$	−24.3913	−1.77420
$190$	0	0
$191$	−13.2388	−0.957928	−0.478964	−	0.877835i	$-0.658988\pi$
$191$	−13.2388	−0.957928	−0.478964	+	0.877835i	$0.658988\pi$
$192$	0	0
$193$	7.21110	0.519067	0.259533	−	0.965734i	$-0.416431\pi$
$193$	7.21110	0.519067	0.259533	+	0.965734i	$0.416431\pi$
$194$	0	0
$195$	5.13827	0.367959
$196$	0	0
$197$	−9.36127	−0.666963	−0.333481	−	0.942757i	$-0.608223\pi$
$197$	−9.36127	−0.666963	−0.333481	+	0.942757i	$0.608223\pi$
$198$	0	0
$199$	2.71927	0.192764	0.0963821	−	0.995344i	$-0.469273\pi$
$199$	2.71927	0.192764	0.0963821	+	0.995344i	$0.469273\pi$
$200$	0	0
$201$	15.6333	1.10269
$202$	0	0
$203$	−8.48528	−0.595550
$204$	0	0
$205$	−10.0200	−0.699827
$206$	0	0
$207$	−1.07261	−0.0745518
$208$	0	0
$209$	30.4222	2.10435
$210$	0	0
$211$	−24.4698	−1.68457	−0.842286	−	0.539031i	$-0.818791\pi$
$211$	−24.4698	−1.68457	−0.842286	+	0.539031i	$0.818791\pi$
$212$	0	0
$213$	−19.3813	−1.32798
$214$	0	0
$215$	0	0
$216$	0	0
$217$	37.8167	2.56716
$218$	0	0
$219$	13.2858	0.897768
$220$	0	0
$221$	1.67000	0.112336
$222$	0	0
$223$	−28.1243	−1.88334	−0.941672	−	0.336532i	$-0.890746\pi$
$223$	−28.1243	−1.88334	−0.941672	+	0.336532i	$0.890746\pi$
$224$	0	0
$225$	−0.872167	−0.0581444
$226$	0	0
$227$	17.6973	1.17461	0.587305	−	0.809365i	$-0.300189\pi$
$227$	17.6973	1.17461	0.587305	+	0.809365i	$0.300189\pi$
$228$	0	0
$229$	−27.0725	−1.78900	−0.894502	−	0.447065i	$-0.852469\pi$
$229$	−27.0725	−1.78900	−0.894502	+	0.447065i	$0.852469\pi$
$230$	0	0
$231$	−43.7249	−2.87689
$232$	0	0
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	22.1088	1.44222
$236$	0	0
$237$	13.7125	0.890725
$238$	0	0
$239$	17.9623	1.16188	0.580942	−	0.813945i	$-0.302684\pi$
$239$	17.9623	1.16188	0.580942	+	0.813945i	$0.302684\pi$
$240$	0	0
$241$	−4.78890	−0.308480	−0.154240	−	0.988033i	$-0.549293\pi$
$241$	−4.78890	−0.308480	−0.154240	+	0.988033i	$0.549293\pi$
$242$	0	0
$243$	4.07374	0.261331
$244$	0	0
$245$	31.4238	2.00759
$246$	0	0
$247$	10.8771	0.692093
$248$	0	0
$249$	13.5778	0.860458
$250$	0	0
$251$	19.7990	1.24970	0.624851	−	0.780744i	$-0.285160\pi$
$251$	19.7990	1.24970	0.624851	+	0.780744i	$0.285160\pi$
$252$	0	0
$253$	12.7013	0.798522
$254$	0	0
$255$	3.07681	0.192677
$256$	0	0
$257$	13.3944	0.835523	0.417761	−	0.908557i	$-0.362815\pi$
$257$	13.3944	0.835523	0.417761	+	0.908557i	$0.362815\pi$
$258$	0	0
$259$	47.5647	2.95553
$260$	0	0
$261$	0.658729	0.0407743
$262$	0	0
$263$	−10.1620	−0.626617	−0.313308	−	0.949651i	$-0.601437\pi$
$263$	−10.1620	−0.626617	−0.313308	+	0.949651i	$0.601437\pi$
$264$	0	0
$265$	−12.8444	−0.789026
$266$	0	0
$267$	−19.5397	−1.19581
$268$	0	0
$269$	11.6900	0.712752	0.356376	−	0.934343i	$-0.384012\pi$
$269$	11.6900	0.712752	0.356376	+	0.934343i	$0.384012\pi$
$270$	0	0
$271$	−17.9623	−1.09113	−0.545566	−	0.838068i	$-0.683685\pi$
$271$	−17.9623	−1.09113	−0.545566	+	0.838068i	$0.683685\pi$
$272$	0	0
$273$	−15.6333	−0.946171
$274$	0	0
$275$	10.3277	0.622783
$276$	0	0
$277$	9.36127	0.562464	0.281232	−	0.959640i	$-0.409257\pi$
$277$	9.36127	0.562464	0.281232	+	0.959640i	$0.409257\pi$
$278$	0	0
$279$	−2.93578	−0.175761
$280$	0	0
$281$	15.2111	0.907418	0.453709	−	0.891150i	$-0.350101\pi$
$281$	15.2111	0.907418	0.453709	+	0.891150i	$0.350101\pi$
$282$	0	0
$283$	−15.9845	−0.950182	−0.475091	−	0.879937i	$-0.657585\pi$
$283$	−15.9845	−0.950182	−0.475091	+	0.879937i	$0.657585\pi$
$284$	0	0
$285$	20.0400	1.18707
$286$	0	0
$287$	30.4861	1.79953
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−3.68481	−0.216007
$292$	0	0
$293$	6.68000	0.390250	0.195125	−	0.980778i	$-0.437489\pi$
$293$	6.68000	0.390250	0.195125	+	0.980778i	$0.437489\pi$
$294$	0	0
$295$	9.44694	0.550022
$296$	0	0
$297$	−22.4222	−1.30107
$298$	0	0
$299$	4.54118	0.262623
$300$	0	0
$301$	0	0
$302$	0	0
$303$	3.07681	0.176758
$304$	0	0
$305$	−8.36669	−0.479075
$306$	0	0
$307$	−18.9426	−1.08111	−0.540556	−	0.841308i	$-0.681786\pi$
$307$	−18.9426	−1.08111	−0.540556	+	0.841308i	$0.681786\pi$
$308$	0	0
$309$	−28.7425	−1.63511
$310$	0	0
$311$	11.2346	0.637058	0.318529	−	0.947913i	$-0.396811\pi$
$311$	11.2346	0.637058	0.318529	+	0.947913i	$0.396811\pi$
$312$	0	0
$313$	26.8444	1.51734	0.758668	−	0.651478i	$-0.225851\pi$
$313$	26.8444	1.51734	0.758668	+	0.651478i	$0.225851\pi$
$314$	0	0
$315$	−3.34701	−0.188583
$316$	0	0
$317$	13.7125	0.770173	0.385086	−	0.922881i	$-0.374172\pi$
$317$	13.7125	0.770173	0.385086	+	0.922881i	$0.374172\pi$
$318$	0	0
$319$	−7.80028	−0.436732
$320$	0	0
$321$	7.02776	0.392251
$322$	0	0
$323$	6.51323	0.362406
$324$	0	0
$325$	3.69254	0.204825
$326$	0	0
$327$	17.2472	0.953774
$328$	0	0
$329$	−67.2666	−3.70853
$330$	0	0
$331$	26.0529	1.43200	0.715999	−	0.698101i	$-0.245971\pi$
$331$	26.0529	1.43200	0.715999	+	0.698101i	$0.245971\pi$
$332$	0	0
$333$	−3.69254	−0.202350
$334$	0	0
$335$	−14.1704	−0.774212
$336$	0	0
$337$	16.4222	0.894575	0.447287	−	0.894390i	$-0.352390\pi$
$337$	16.4222	0.894575	0.447287	+	0.894390i	$0.352390\pi$
$338$	0	0
$339$	5.91614	0.321321
$340$	0	0
$341$	34.7638	1.88257
$342$	0	0
$343$	−60.0405	−3.24188
$344$	0	0
$345$	8.36669	0.450448
$346$	0	0
$347$	−19.6693	−1.05591	−0.527953	−	0.849274i	$-0.677040\pi$
$347$	−19.6693	−1.05591	−0.527953	+	0.849274i	$0.677040\pi$
$348$	0	0
$349$	14.3713	0.769276	0.384638	−	0.923067i	$-0.374326\pi$
$349$	14.3713	0.769276	0.384638	+	0.923067i	$0.374326\pi$
$350$	0	0
$351$	−8.01679	−0.427905
$352$	0	0
$353$	15.2111	0.809605	0.404803	−	0.914404i	$-0.367340\pi$
$353$	15.2111	0.809605	0.404803	+	0.914404i	$0.367340\pi$
$354$	0	0
$355$	17.5677	0.932394
$356$	0	0
$357$	−9.36127	−0.495451
$358$	0	0
$359$	−28.1243	−1.48434	−0.742172	−	0.670209i	$-0.766204\pi$
$359$	−28.1243	−1.48434	−0.742172	+	0.670209i	$0.766204\pi$
$360$	0	0
$361$	23.4222	1.23275
$362$	0	0
$363$	−19.9286	−1.04598
$364$	0	0
$365$	−12.0425	−0.630335
$366$	0	0
$367$	−26.1201	−1.36346	−0.681730	−	0.731604i	$-0.738772\pi$
$367$	−26.1201	−1.36346	−0.681730	+	0.731604i	$0.738772\pi$
$368$	0	0
$369$	−2.36669	−0.123205
$370$	0	0
$371$	39.0794	2.02890
$372$	0	0
$373$	−11.6900	−0.605285	−0.302642	−	0.953104i	$-0.597869\pi$
$373$	−11.6900	−0.605285	−0.302642	+	0.953104i	$0.597869\pi$
$374$	0	0
$375$	22.1872	1.14574
$376$	0	0
$377$	−2.78890	−0.143636
$378$	0	0
$379$	1.58311	0.0813190	0.0406595	−	0.999173i	$-0.487054\pi$
$379$	1.58311	0.0813190	0.0406595	+	0.999173i	$0.487054\pi$
$380$	0	0
$381$	−1.31746	−0.0674954
$382$	0	0
$383$	−0.715076	−0.0365387	−0.0182693	−	0.999833i	$-0.505816\pi$
$383$	−0.715076	−0.0365387	−0.0182693	+	0.999833i	$0.505816\pi$
$384$	0	0
$385$	39.6333	2.01990
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−17.0525	−0.864598	−0.432299	−	0.901730i	$-0.642297\pi$
$389$	−17.0525	−0.864598	−0.432299	+	0.901730i	$0.642297\pi$
$390$	0	0
$391$	2.71927	0.137520
$392$	0	0
$393$	−3.39445	−0.171227
$394$	0	0
$395$	−12.4294	−0.625390
$396$	0	0
$397$	6.02127	0.302199	0.151099	−	0.988519i	$-0.451719\pi$
$397$	6.02127	0.302199	0.151099	+	0.988519i	$0.451719\pi$
$398$	0	0
$399$	−60.9721	−3.05242
$400$	0	0
$401$	21.6333	1.08032	0.540158	−	0.841564i	$-0.318365\pi$
$401$	21.6333	1.08032	0.540158	+	0.841564i	$0.318365\pi$
$402$	0	0
$403$	12.4294	0.619151
$404$	0	0
$405$	−16.7463	−0.832133
$406$	0	0
$407$	43.7249	2.16736
$408$	0	0
$409$	−20.4222	−1.00981	−0.504907	−	0.863174i	$-0.668473\pi$
$409$	−20.4222	−1.00981	−0.504907	+	0.863174i	$0.668473\pi$
$410$	0	0
$411$	−36.5103	−1.80092
$412$	0	0
$413$	−28.7425	−1.41433
$414$	0	0
$415$	−12.3072	−0.604139
$416$	0	0
$417$	−5.44996	−0.266886
$418$	0	0
$419$	17.1002	0.835400	0.417700	−	0.908585i	$-0.362836\pi$
$419$	17.1002	0.835400	0.417700	+	0.908585i	$0.362836\pi$
$420$	0	0
$421$	37.0925	1.80778	0.903890	−	0.427766i	$-0.140699\pi$
$421$	37.0925	1.80778	0.903890	+	0.427766i	$0.140699\pi$
$422$	0	0
$423$	5.22204	0.253904
$424$	0	0
$425$	2.21110	0.107254
$426$	0	0
$427$	25.4558	1.23189
$428$	0	0
$429$	−14.3713	−0.693851
$430$	0	0
$431$	6.72767	0.324060	0.162030	−	0.986786i	$-0.448196\pi$
$431$	6.72767	0.324060	0.162030	+	0.986786i	$0.448196\pi$
$432$	0	0
$433$	6.60555	0.317443	0.158721	−	0.987323i	$-0.449263\pi$
$433$	6.60555	0.317443	0.158721	+	0.987323i	$0.449263\pi$
$434$	0	0
$435$	−5.13827	−0.246361
$436$	0	0
$437$	17.7113	0.847245
$438$	0	0
$439$	−26.8352	−1.28077	−0.640387	−	0.768052i	$-0.721226\pi$
$439$	−26.8352	−1.28077	−0.640387	+	0.768052i	$0.721226\pi$
$440$	0	0
$441$	7.42221	0.353438
$442$	0	0
$443$	28.8814	1.37219	0.686097	−	0.727510i	$-0.259322\pi$
$443$	28.8814	1.37219	0.686097	+	0.727510i	$0.259322\pi$
$444$	0	0
$445$	17.7113	0.839594
$446$	0	0
$447$	40.6481	1.92259
$448$	0	0
$449$	−18.0000	−0.849473	−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000	−0.849473	−0.424736	+	0.905317i	$0.639633\pi$
$450$	0	0
$451$	28.0250	1.31964
$452$	0	0
$453$	10.0200	0.470780
$454$	0	0
$455$	14.1704	0.664319
$456$	0	0
$457$	−3.81665	−0.178536	−0.0892678	−	0.996008i	$-0.528453\pi$
$457$	−3.81665	−0.178536	−0.0892678	+	0.996008i	$0.528453\pi$
$458$	0	0
$459$	−4.80048	−0.224067
$460$	0	0
$461$	−7.69127	−0.358218	−0.179109	−	0.983829i	$-0.557321\pi$
$461$	−7.69127	−0.358218	−0.179109	+	0.983829i	$0.557321\pi$
$462$	0	0
$463$	10.8771	0.505501	0.252751	−	0.967531i	$-0.418665\pi$
$463$	10.8771	0.505501	0.252751	+	0.967531i	$0.418665\pi$
$464$	0	0
$465$	22.8999	1.06196
$466$	0	0
$467$	25.1966	1.16596	0.582979	−	0.812487i	$-0.301887\pi$
$467$	25.1966	1.16596	0.582979	+	0.812487i	$0.301887\pi$
$468$	0	0
$469$	43.1138	1.99081
$470$	0	0
$471$	20.3240	0.936482
$472$	0	0
$473$	0	0
$474$	0	0
$475$	14.4014	0.660783
$476$	0	0
$477$	−3.03381	−0.138909
$478$	0	0
$479$	8.87290	0.405413	0.202706	−	0.979240i	$-0.435026\pi$
$479$	8.87290	0.405413	0.202706	+	0.979240i	$0.435026\pi$
$480$	0	0
$481$	15.6333	0.712817
$482$	0	0
$483$	−25.4558	−1.15828
$484$	0	0
$485$	3.34000	0.151662
$486$	0	0
$487$	−5.79609	−0.262646	−0.131323	−	0.991340i	$-0.541922\pi$
$487$	−5.79609	−0.262646	−0.131323	+	0.991340i	$0.541922\pi$
$488$	0	0
$489$	−22.1833	−1.00317
$490$	0	0
$491$	−17.8269	−0.804519	−0.402259	−	0.915526i	$-0.631775\pi$
$491$	−17.8269	−0.804519	−0.402259	+	0.915526i	$0.631775\pi$
$492$	0	0
$493$	−1.67000	−0.0752130
$494$	0	0
$495$	−3.07681	−0.138292
$496$	0	0
$497$	−53.4500	−2.39756
$498$	0	0
$499$	14.6096	0.654015	0.327007	−	0.945022i	$-0.393960\pi$
$499$	14.6096	0.654015	0.327007	+	0.945022i	$0.393960\pi$
$500$	0	0
$501$	8.04381	0.359371
$502$	0	0
$503$	−1.07261	−0.0478255	−0.0239127	−	0.999714i	$-0.507612\pi$
$503$	−1.07261	−0.0478255	−0.0239127	+	0.999714i	$0.507612\pi$
$504$	0	0
$505$	−2.78890	−0.124104
$506$	0	0
$507$	18.8130	0.835513
$508$	0	0
$509$	−15.3825	−0.681819	−0.340909	−	0.940096i	$-0.610735\pi$
$509$	−15.3825	−0.681819	−0.340909	+	0.940096i	$0.610735\pi$
$510$	0	0
$511$	36.6397	1.62084
$512$	0	0
$513$	−31.2666	−1.38045
$514$	0	0
$515$	26.0529	1.14803
$516$	0	0
$517$	−61.8363	−2.71956
$518$	0	0
$519$	−23.4008	−1.02718
$520$	0	0
$521$	−24.4222	−1.06996	−0.534978	−	0.844866i	$-0.679680\pi$
$521$	−24.4222	−1.06996	−0.534978	+	0.844866i	$0.679680\pi$
$522$	0	0
$523$	−8.48528	−0.371035	−0.185518	−	0.982641i	$-0.559396\pi$
$523$	−8.48528	−0.371035	−0.185518	+	0.982641i	$0.559396\pi$
$524$	0	0
$525$	−20.6987	−0.903366
$526$	0	0
$527$	7.44274	0.324211
$528$	0	0
$529$	−15.6056	−0.678502
$530$	0	0
$531$	2.23134	0.0968319
$532$	0	0
$533$	10.0200	0.434014
$534$	0	0
$535$	−6.37013	−0.275405
$536$	0	0
$537$	−8.84441	−0.381664
$538$	0	0
$539$	−87.8894	−3.78566
$540$	0	0
$541$	−28.0838	−1.20742	−0.603708	−	0.797205i	$-0.706311\pi$
$541$	−28.0838	−1.20742	−0.603708	+	0.797205i	$0.706311\pi$
$542$	0	0
$543$	23.4008	1.00423
$544$	0	0
$545$	−15.6333	−0.669657
$546$	0	0
$547$	−31.5801	−1.35027	−0.675135	−	0.737695i	$-0.735914\pi$
$547$	−31.5801	−1.35027	−0.675135	+	0.737695i	$0.735914\pi$
$548$	0	0
$549$	−1.97619	−0.0843416
$550$	0	0
$551$	−10.8771	−0.463380
$552$	0	0
$553$	37.8167	1.60813
$554$	0	0
$555$	28.8029	1.22261
$556$	0	0
$557$	23.7325	1.00558	0.502790	−	0.864409i	$-0.332307\pi$
$557$	23.7325	1.00558	0.502790	+	0.864409i	$0.332307\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−8.60555	−0.363327
$562$	0	0
$563$	18.6833	0.787408	0.393704	−	0.919237i	$-0.371193\pi$
$563$	18.6833	0.787408	0.393704	+	0.919237i	$0.371193\pi$
$564$	0	0
$565$	−5.36254	−0.225604
$566$	0	0
$567$	50.9511	2.13975
$568$	0	0
$569$	−9.63331	−0.403849	−0.201925	−	0.979401i	$-0.564720\pi$
$569$	−9.63331	−0.403849	−0.201925	+	0.979401i	$0.564720\pi$
$570$	0	0
$571$	−31.5801	−1.32159	−0.660794	−	0.750567i	$-0.729780\pi$
$571$	−31.5801	−1.32159	−0.660794	+	0.750567i	$0.729780\pi$
$572$	0	0
$573$	24.3913	1.01896
$574$	0	0
$575$	6.01259	0.250742
$576$	0	0
$577$	14.6056	0.608037	0.304019	−	0.952666i	$-0.401672\pi$
$577$	14.6056	0.608037	0.304019	+	0.952666i	$0.401672\pi$
$578$	0	0
$579$	−13.2858	−0.552137
$580$	0	0
$581$	37.4451	1.55348
$582$	0	0
$583$	35.9246	1.48784
$584$	0	0
$585$	−1.10008	−0.0454826
$586$	0	0
$587$	−30.8534	−1.27346	−0.636728	−	0.771088i	$-0.719713\pi$
$587$	−30.8534	−1.27346	−0.636728	+	0.771088i	$0.719713\pi$
$588$	0	0
$589$	48.4763	1.99743
$590$	0	0
$591$	17.2472	0.709456
$592$	0	0
$593$	−21.6333	−0.888373	−0.444187	−	0.895934i	$-0.646507\pi$
$593$	−21.6333	−0.888373	−0.444187	+	0.895934i	$0.646507\pi$
$594$	0	0
$595$	8.48528	0.347863
$596$	0	0
$597$	−5.01000	−0.205046
$598$	0	0
$599$	−31.2011	−1.27484	−0.637422	−	0.770515i	$-0.719999\pi$
$599$	−31.2011	−1.27484	−0.637422	+	0.770515i	$0.719999\pi$
$600$	0	0
$601$	4.78890	0.195343	0.0976716	−	0.995219i	$-0.468861\pi$
$601$	4.78890	0.195343	0.0976716	+	0.995219i	$0.468861\pi$
$602$	0	0
$603$	−3.34701	−0.136301
$604$	0	0
$605$	18.0638	0.734398
$606$	0	0
$607$	6.72767	0.273068	0.136534	−	0.990635i	$-0.456404\pi$
$607$	6.72767	0.273068	0.136534	+	0.990635i	$0.456404\pi$
$608$	0	0
$609$	15.6333	0.633494
$610$	0	0
$611$	−22.1088	−0.894428
$612$	0	0
$613$	−22.0625	−0.891097	−0.445549	−	0.895258i	$-0.646991\pi$
$613$	−22.0625	−0.891097	−0.445549	+	0.895258i	$0.646991\pi$
$614$	0	0
$615$	18.4609	0.744414
$616$	0	0
$617$	6.00000	0.241551	0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000	0.241551	0.120775	+	0.992680i	$0.461462\pi$
$618$	0	0
$619$	8.09635	0.325420	0.162710	−	0.986674i	$-0.447977\pi$
$619$	8.09635	0.325420	0.162710	+	0.986674i	$0.447977\pi$
$620$	0	0
$621$	−13.0538	−0.523831
$622$	0	0
$623$	−53.8869	−2.15893
$624$	0	0
$625$	−9.05551	−0.362221
$626$	0	0
$627$	−56.0500	−2.23842
$628$	0	0
$629$	9.36127	0.373258
$630$	0	0
$631$	7.80028	0.310524	0.155262	−	0.987873i	$-0.450378\pi$
$631$	7.80028	0.310524	0.155262	+	0.987873i	$0.450378\pi$
$632$	0	0
$633$	45.0833	1.79190
$634$	0	0
$635$	1.19418	0.0473894
$636$	0	0
$637$	−31.4238	−1.24506
$638$	0	0
$639$	4.14943	0.164149
$640$	0	0
$641$	−24.4222	−0.964619	−0.482310	−	0.876001i	$-0.660202\pi$
$641$	−24.4222	−0.964619	−0.482310	+	0.876001i	$0.660202\pi$
$642$	0	0
$643$	−11.4434	−0.451282	−0.225641	−	0.974211i	$-0.572448\pi$
$643$	−11.4434	−0.451282	−0.225641	+	0.974211i	$0.572448\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	34.9930	1.37572	0.687859	−	0.725845i	$-0.258551\pi$
$647$	34.9930	1.37572	0.687859	+	0.725845i	$0.258551\pi$
$648$	0	0
$649$	−26.4222	−1.03716
$650$	0	0
$651$	−69.6735	−2.73072
$652$	0	0
$653$	−23.7325	−0.928726	−0.464363	−	0.885645i	$-0.653717\pi$
$653$	−23.7325	−0.928726	−0.464363	+	0.885645i	$0.653717\pi$
$654$	0	0
$655$	3.07681	0.120221
$656$	0	0
$657$	−2.84441	−0.110971
$658$	0	0
$659$	9.34166	0.363899	0.181950	−	0.983308i	$-0.441759\pi$
$659$	9.34166	0.363899	0.181950	+	0.983308i	$0.441759\pi$
$660$	0	0
$661$	−25.4025	−0.988044	−0.494022	−	0.869449i	$-0.664474\pi$
$661$	−25.4025	−0.988044	−0.494022	+	0.869449i	$0.664474\pi$
$662$	0	0
$663$	−3.07681	−0.119493
$664$	0	0
$665$	55.2666	2.14315
$666$	0	0
$667$	−4.54118	−0.175835
$668$	0	0
$669$	51.8163	2.00334
$670$	0	0
$671$	23.4008	0.903380
$672$	0	0
$673$	19.5778	0.754669	0.377334	−	0.926077i	$-0.376841\pi$
$673$	19.5778	0.754669	0.377334	+	0.926077i	$0.376841\pi$
$674$	0	0
$675$	−10.6143	−0.408546
$676$	0	0
$677$	2.68127	0.103050	0.0515248	−	0.998672i	$-0.483592\pi$
$677$	2.68127	0.103050	0.0515248	+	0.998672i	$0.483592\pi$
$678$	0	0
$679$	−10.1620	−0.389982
$680$	0	0
$681$	−32.6056	−1.24945
$682$	0	0
$683$	−30.1267	−1.15277	−0.576383	−	0.817180i	$-0.695536\pi$
$683$	−30.1267	−1.15277	−0.576383	+	0.817180i	$0.695536\pi$
$684$	0	0
$685$	33.0938	1.26445
$686$	0	0
$687$	49.8785	1.90298
$688$	0	0
$689$	12.8444	0.489333
$690$	0	0
$691$	−45.9816	−1.74922	−0.874611	−	0.484826i	$-0.838883\pi$
$691$	−45.9816	−1.74922	−0.874611	+	0.484826i	$0.838883\pi$
$692$	0	0
$693$	9.36127	0.355605
$694$	0	0
$695$	4.93998	0.187384
$696$	0	0
$697$	6.00000	0.227266
$698$	0	0
$699$	−11.0544	−0.418116
$700$	0	0
$701$	29.0951	1.09891	0.549453	−	0.835525i	$-0.314836\pi$
$701$	29.0951	1.09891	0.549453	+	0.835525i	$0.314836\pi$
$702$	0	0
$703$	60.9721	2.29961
$704$	0	0
$705$	−40.7334	−1.53411
$706$	0	0
$707$	8.48528	0.319122
$708$	0	0
$709$	−32.4351	−1.21812	−0.609062	−	0.793122i	$-0.708454\pi$
$709$	−32.4351	−1.21812	−0.609062	+	0.793122i	$0.708454\pi$
$710$	0	0
$711$	−2.93578	−0.110100
$712$	0	0
$713$	20.2389	0.757951
$714$	0	0
$715$	13.0265	0.487162
$716$	0	0
$717$	−33.0938	−1.23591
$718$	0	0
$719$	−15.9581	−0.595137	−0.297568	−	0.954700i	$-0.596176\pi$
$719$	−15.9581	−0.595137	−0.297568	+	0.954700i	$0.596176\pi$
$720$	0	0
$721$	−79.2666	−2.95204
$722$	0	0
$723$	8.82308	0.328134
$724$	0	0
$725$	−3.69254	−0.137137
$726$	0	0
$727$	−11.8087	−0.437960	−0.218980	−	0.975729i	$-0.570273\pi$
$727$	−11.8087	−0.437960	−0.218980	+	0.975729i	$0.570273\pi$
$728$	0	0
$729$	22.5778	0.836215
$730$	0	0
$731$	0	0
$732$	0	0
$733$	38.7625	1.43173	0.715863	−	0.698241i	$-0.246033\pi$
$733$	38.7625	1.43173	0.715863	+	0.698241i	$0.246033\pi$
$734$	0	0
$735$	−57.8953	−2.13550
$736$	0	0
$737$	39.6333	1.45991
$738$	0	0
$739$	36.5103	1.34305	0.671526	−	0.740981i	$-0.265639\pi$
$739$	36.5103	1.34305	0.671526	+	0.740981i	$0.265639\pi$
$740$	0	0
$741$	−20.0400	−0.736187
$742$	0	0
$743$	−40.0740	−1.47017	−0.735087	−	0.677973i	$-0.762859\pi$
$743$	−40.0740	−1.47017	−0.735087	+	0.677973i	$0.762859\pi$
$744$	0	0
$745$	−36.8444	−1.34987
$746$	0	0
$747$	−2.90693	−0.106359
$748$	0	0
$749$	19.3813	0.708176
$750$	0	0
$751$	−6.51116	−0.237596	−0.118798	−	0.992918i	$-0.537904\pi$
$751$	−6.51116	−0.237596	−0.118798	+	0.992918i	$0.537904\pi$
$752$	0	0
$753$	−36.4777	−1.32932
$754$	0	0
$755$	−9.08237	−0.330541
$756$	0	0
$757$	−32.4351	−1.17887	−0.589436	−	0.807815i	$-0.700650\pi$
$757$	−32.4351	−1.17887	−0.589436	+	0.807815i	$0.700650\pi$
$758$	0	0
$759$	−23.4008	−0.849397
$760$	0	0
$761$	−4.18335	−0.151646	−0.0758231	−	0.997121i	$-0.524158\pi$
$761$	−4.18335	−0.151646	−0.0758231	+	0.997121i	$0.524158\pi$
$762$	0	0
$763$	47.5647	1.72196
$764$	0	0
$765$	−0.658729	−0.0238164
$766$	0	0
$767$	−9.44694	−0.341109
$768$	0	0
$769$	−19.4500	−0.701384	−0.350692	−	0.936491i	$-0.614054\pi$
$769$	−19.4500	−0.701384	−0.350692	+	0.936491i	$0.614054\pi$
$770$	0	0
$771$	−24.6780	−0.888755
$772$	0	0
$773$	−7.03254	−0.252943	−0.126471	−	0.991970i	$-0.540365\pi$
$773$	−7.03254	−0.252943	−0.126471	+	0.991970i	$0.540365\pi$
$774$	0	0
$775$	16.4567	0.591141
$776$	0	0
$777$	−87.6333	−3.14383
$778$	0	0
$779$	39.0794	1.40016
$780$	0	0
$781$	−49.1351	−1.75819
$782$	0	0
$783$	8.01679	0.286497
$784$	0	0
$785$	−18.4222	−0.657517
$786$	0	0
$787$	−10.0684	−0.358899	−0.179450	−	0.983767i	$-0.557432\pi$
$787$	−10.0684	−0.358899	−0.179450	+	0.983767i	$0.557432\pi$
$788$	0	0
$789$	18.7225	0.666540
$790$	0	0
$791$	16.3156	0.580117
$792$	0	0
$793$	8.36669	0.297110
$794$	0	0
$795$	23.6646	0.839296
$796$	0	0
$797$	−38.7625	−1.37304	−0.686520	−	0.727111i	$-0.740862\pi$
$797$	−38.7625	−1.37304	−0.686520	+	0.727111i	$0.740862\pi$
$798$	0	0
$799$	−13.2388	−0.468356
$800$	0	0
$801$	4.18335	0.147811
$802$	0	0
$803$	33.6818	1.18861
$804$	0	0
$805$	23.0738	0.813245
$806$	0	0
$807$	−21.5377	−0.758162
$808$	0	0
$809$	−32.7889	−1.15280	−0.576398	−	0.817169i	$-0.695542\pi$
$809$	−32.7889	−1.15280	−0.576398	+	0.817169i	$0.695542\pi$
$810$	0	0
$811$	−42.0375	−1.47614	−0.738068	−	0.674727i	$-0.764261\pi$
$811$	−42.0375	−1.47614	−0.738068	+	0.674727i	$0.764261\pi$
$812$	0	0
$813$	33.0938	1.16065
$814$	0	0
$815$	20.1075	0.704336
$816$	0	0
$817$	0	0
$818$	0	0
$819$	3.34701	0.116954
$820$	0	0
$821$	−28.0838	−0.980131	−0.490066	−	0.871686i	$-0.663027\pi$
$821$	−28.0838	−0.980131	−0.490066	+	0.871686i	$0.663027\pi$
$822$	0	0
$823$	−33.9204	−1.18239	−0.591195	−	0.806528i	$-0.701344\pi$
$823$	−33.9204	−1.18239	−0.591195	+	0.806528i	$0.701344\pi$
$824$	0	0
$825$	−19.0278	−0.662461
$826$	0	0
$827$	28.1546	0.979032	0.489516	−	0.871994i	$-0.337173\pi$
$827$	28.1546	0.979032	0.489516	+	0.871994i	$0.337173\pi$
$828$	0	0
$829$	−53.1338	−1.84541	−0.922706	−	0.385504i	$-0.874028\pi$
$829$	−53.1338	−1.84541	−0.922706	+	0.385504i	$0.874028\pi$
$830$	0	0
$831$	−17.2472	−0.598300
$832$	0	0
$833$	−18.8167	−0.651958
$834$	0	0
$835$	−7.29111	−0.252319
$836$	0	0
$837$	−35.7287	−1.23496
$838$	0	0
$839$	−36.2821	−1.25260	−0.626299	−	0.779583i	$-0.715431\pi$
$839$	−36.2821	−1.25260	−0.626299	+	0.779583i	$0.715431\pi$
$840$	0	0
$841$	−26.2111	−0.903831
$842$	0	0
$843$	−28.0250	−0.965232
$844$	0	0
$845$	−17.0525	−0.586625
$846$	0	0
$847$	−54.9595	−1.88843
$848$	0	0
$849$	29.4500	1.01072
$850$	0	0
$851$	25.4558	0.872615
$852$	0	0
$853$	54.4976	1.86596	0.932981	−	0.359925i	$-0.117198\pi$
$853$	54.4976	1.86596	0.932981	+	0.359925i	$0.117198\pi$
$854$	0	0
$855$	−4.29046	−0.146730
$856$	0	0
$857$	−36.4222	−1.24416	−0.622079	−	0.782954i	$-0.713712\pi$
$857$	−36.4222	−1.24416	−0.622079	+	0.782954i	$0.713712\pi$
$858$	0	0
$859$	3.94410	0.134571	0.0672854	−	0.997734i	$-0.478566\pi$
$859$	3.94410	0.134571	0.0672854	+	0.997734i	$0.478566\pi$
$860$	0	0
$861$	−56.1676	−1.91419
$862$	0	0
$863$	−6.15362	−0.209472	−0.104736	−	0.994500i	$-0.533400\pi$
$863$	−6.15362	−0.209472	−0.104736	+	0.994500i	$0.533400\pi$
$864$	0	0
$865$	21.2111	0.721199
$866$	0	0
$867$	−1.84240	−0.0625713
$868$	0	0
$869$	34.7638	1.17928
$870$	0	0
$871$	14.1704	0.480146
$872$	0	0
$873$	0.788897	0.0267001
$874$	0	0
$875$	61.1882	2.06854
$876$	0	0
$877$	6.02127	0.203324	0.101662	−	0.994819i	$-0.467584\pi$
$877$	6.02127	0.203324	0.101662	+	0.994819i	$0.467584\pi$
$878$	0	0
$879$	−12.3072	−0.415113
$880$	0	0
$881$	6.00000	0.202145	0.101073	−	0.994879i	$-0.467773\pi$
$881$	6.00000	0.202145	0.101073	+	0.994879i	$0.467773\pi$
$882$	0	0
$883$	4.54118	0.152823	0.0764115	−	0.997076i	$-0.475654\pi$
$883$	4.54118	0.152823	0.0764115	+	0.997076i	$0.475654\pi$
$884$	0	0
$885$	−17.4051	−0.585065
$886$	0	0
$887$	37.2137	1.24951	0.624757	−	0.780819i	$-0.285198\pi$
$887$	37.2137	1.24951	0.624757	+	0.780819i	$0.285198\pi$
$888$	0	0
$889$	−3.63331	−0.121857
$890$	0	0
$891$	46.8379	1.56913
$892$	0	0
$893$	−86.2276	−2.88550
$894$	0	0
$895$	8.01679	0.267972
$896$	0	0
$897$	−8.36669	−0.279356
$898$	0	0
$899$	−12.4294	−0.414543
$900$	0	0
$901$	7.69127	0.256233
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−21.2111	−0.705081
$906$	0	0
$907$	−12.0404	−0.399796	−0.199898	−	0.979817i	$-0.564061\pi$
$907$	−12.0404	−0.399796	−0.199898	+	0.979817i	$0.564061\pi$
$908$	0	0
$909$	−0.658729	−0.0218487
$910$	0	0
$911$	−39.5755	−1.31119	−0.655597	−	0.755111i	$-0.727583\pi$
$911$	−39.5755	−1.31119	−0.655597	+	0.755111i	$0.727583\pi$
$912$	0	0
$913$	34.4222	1.13921
$914$	0	0
$915$	15.4148	0.509598
$916$	0	0
$917$	−9.36127	−0.309136
$918$	0	0
$919$	34.9930	1.15431	0.577157	−	0.816634i	$-0.304162\pi$
$919$	34.9930	1.15431	0.577157	+	0.816634i	$0.304162\pi$
$920$	0	0
$921$	34.8999	1.14999
$922$	0	0
$923$	−17.5677	−0.578246
$924$	0	0
$925$	20.6987	0.680570
$926$	0	0
$927$	6.15362	0.202112
$928$	0	0
$929$	−44.7889	−1.46948	−0.734738	−	0.678351i	$-0.762695\pi$
$929$	−44.7889	−1.46948	−0.734738	+	0.678351i	$0.762695\pi$
$930$	0	0
$931$	−122.557	−4.01665
$932$	0	0
$933$	−20.6987	−0.677646
$934$	0	0
$935$	7.80028	0.255097
$936$	0	0
$937$	46.8444	1.53034	0.765170	−	0.643828i	$-0.222655\pi$
$937$	46.8444	1.53034	0.765170	+	0.643828i	$0.222655\pi$
$938$	0	0
$939$	−49.4582	−1.61401
$940$	0	0
$941$	0.658729	0.0214740	0.0107370	−	0.999942i	$-0.496582\pi$
$941$	0.658729	0.0214740	0.0107370	+	0.999942i	$0.496582\pi$
$942$	0	0
$943$	16.3156	0.531310
$944$	0	0
$945$	−40.7334	−1.32506
$946$	0	0
$947$	1.24531	0.0404673	0.0202336	−	0.999795i	$-0.493559\pi$
$947$	1.24531	0.0404673	0.0202336	+	0.999795i	$0.493559\pi$
$948$	0	0
$949$	12.0425	0.390917
$950$	0	0
$951$	−25.2640	−0.819242
$952$	0	0
$953$	31.8167	1.03064	0.515321	−	0.856997i	$-0.327673\pi$
$953$	31.8167	1.03064	0.515321	+	0.856997i	$0.327673\pi$
$954$	0	0
$955$	−22.1088	−0.715425
$956$	0	0
$957$	14.3713	0.464557
$958$	0	0
$959$	−100.689	−3.25140
$960$	0	0
$961$	24.3944	0.786918
$962$	0	0
$963$	−1.50461	−0.0484852
$964$	0	0
$965$	12.0425	0.387663
$966$	0	0
$967$	−12.5238	−0.402737	−0.201368	−	0.979516i	$-0.564539\pi$
$967$	−12.5238	−0.402737	−0.201368	+	0.979516i	$0.564539\pi$
$968$	0	0
$969$	−12.0000	−0.385496
$970$	0	0
$971$	54.3372	1.74376	0.871882	−	0.489716i	$-0.162900\pi$
$971$	54.3372	1.74376	0.871882	+	0.489716i	$0.162900\pi$
$972$	0	0
$973$	−15.0300	−0.481839
$974$	0	0
$975$	−6.80315	−0.217875
$976$	0	0
$977$	−15.2111	−0.486646	−0.243323	−	0.969945i	$-0.578238\pi$
$977$	−15.2111	−0.486646	−0.243323	+	0.969945i	$0.578238\pi$
$978$	0	0
$979$	−49.5367	−1.58320
$980$	0	0
$981$	−3.69254	−0.117894
$982$	0	0
$983$	−5.08101	−0.162059	−0.0810295	−	0.996712i	$-0.525821\pi$
$983$	−5.08101	−0.162059	−0.0810295	+	0.996712i	$0.525821\pi$
$984$	0	0
$985$	−15.6333	−0.498119
$986$	0	0
$987$	123.932	3.94481
$988$	0	0
$989$	0	0
$990$	0	0
$991$	2.93578	0.0932582	0.0466291	−	0.998912i	$-0.485152\pi$
$991$	2.93578	0.0932582	0.0466291	+	0.998912i	$0.485152\pi$
$992$	0	0
$993$	−48.0000	−1.52323
$994$	0	0
$995$	4.54118	0.143965
$996$	0	0
$997$	28.0838	0.889423	0.444711	−	0.895674i	$-0.353306\pi$
$997$	28.0838	0.889423	0.444711	+	0.895674i	$0.353306\pi$
$998$	0	0
$999$	−44.9385	−1.42179

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4352.2.a.bd.1.4		8
4.3	odd	2	inner	4352.2.a.bd.1.6		8
8.3	odd	2	inner	4352.2.a.bd.1.3		8
8.5	even	2	inner	4352.2.a.bd.1.5		8
16.3	odd	4		2176.2.c.h.1089.5	yes	8
16.5	even	4		2176.2.c.h.1089.6	yes	8
16.11	odd	4		2176.2.c.h.1089.4	yes	8
16.13	even	4		2176.2.c.h.1089.3	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2176.2.c.h.1089.3	✓	8	16.13	even	4
2176.2.c.h.1089.4	yes	8	16.11	odd	4
2176.2.c.h.1089.5	yes	8	16.3	odd	4
2176.2.c.h.1089.6	yes	8	16.5	even	4
4352.2.a.bd.1.3		8	8.3	odd	2	inner
4352.2.a.bd.1.4		8	1.1	even	1	trivial
4352.2.a.bd.1.5		8	8.5	even	2	inner
4352.2.a.bd.1.6		8	4.3	odd	2	inner

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:	\( N \)	\(=\)	\( 4352 = 2^{8} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4352.a (trivial)

\(f(q)\)	\(=\)	\(q-1.84240 q^{3} +1.67000 q^{5} -5.08101 q^{7} +0.394449 q^{9} -4.67083 q^{11} -1.67000 q^{13} -3.07681 q^{15} -1.00000 q^{17} -6.51323 q^{19} +9.36127 q^{21} -2.71927 q^{23} -2.21110 q^{25} +4.80048 q^{27} +1.67000 q^{29} -7.44274 q^{31} +8.60555 q^{33} -8.48528 q^{35} -9.36127 q^{37} +3.07681 q^{39} -6.00000 q^{41} +0.658729 q^{45} +13.2388 q^{47} +18.8167 q^{49} +1.84240 q^{51} -7.69127 q^{53} -7.80028 q^{55} +12.0000 q^{57} +5.65685 q^{59} -5.01000 q^{61} -2.00420 q^{63} -2.78890 q^{65} -8.48528 q^{67} +5.01000 q^{69} +10.5196 q^{71} -7.21110 q^{73} +4.07374 q^{75} +23.7325 q^{77} -7.44274 q^{79} -10.0278 q^{81} -7.36961 q^{83} -1.67000 q^{85} -3.07681 q^{87} +10.6056 q^{89} +8.48528 q^{91} +13.7125 q^{93} -10.8771 q^{95} +2.00000 q^{97} -1.84240 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 32 q^{9} - 8 q^{17} + 40 q^{25} + 40 q^{33} - 48 q^{41} + 64 q^{49} + 96 q^{57} - 80 q^{65} + 64 q^{81} + 56 q^{89} + 16 q^{97}+O(q^{100}) \) 8 * q + 32 * q^9 - 8 * q^17 + 40 * q^25 + 40 * q^33 - 48 * q^41 + 64 * q^49 + 96 * q^57 - 80 * q^65 + 64 * q^81 + 56 * q^89 + 16 * q^97

Introduction

L-functions

Modular forms

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