LMFDB - Embedded newform 7128.2.a.w.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7128,2,Mod(1,7128)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7128, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 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("7128.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 7128 = 2^{3} \cdot 3^{4} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7128.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,-4,0,5,0,0,0,-6] 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:	$56.9173665608$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.30796308.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 7x^{4} - x^{3} + 10x^{2} + 2x - 2 $ x^6 - 7x^4 - x^3 + 10x^2 + 2*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 792)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-2.16714$ of defining polynomial
Character	$\chi$	$=$	7128.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.69649 q^{5} +1.90201 q^{7} -1.00000 q^{11} -2.08396 q^{13} +0.310342 q^{17} -2.59850 q^{19} -0.226386 q^{23} -2.12192 q^{25} -4.42032 q^{29} -2.98597 q^{31} +3.22674 q^{35} +4.20720 q^{37} -5.62376 q^{41} -2.02393 q^{43} -1.48037 q^{47} -3.38235 q^{49} -3.56433 q^{53} -1.69649 q^{55} -9.87652 q^{59} -8.37392 q^{61} -3.53541 q^{65} +6.33735 q^{67} -0.355974 q^{71} +11.9857 q^{73} -1.90201 q^{77} -4.83688 q^{79} -2.90065 q^{83} +0.526492 q^{85} -3.43554 q^{89} -3.96371 q^{91} -4.40833 q^{95} -17.5014 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{5} + 5 q^{7} - 6 q^{11} + 3 q^{13} - 7 q^{17} + 5 q^{19} - 8 q^{23} - 2 q^{25} - 8 q^{29} + 4 q^{31} - 8 q^{35} + 3 q^{37} + 5 q^{43} - 17 q^{47} + 3 q^{49} - 14 q^{53} + 4 q^{55} - 4 q^{59}+ \cdots - 16 q^{97}+O(q^{100}) $ 6 * q - 4 * q^5 + 5 * q^7 - 6 * q^11 + 3 * q^13 - 7 * q^17 + 5 * q^19 - 8 * q^23 - 2 * q^25 - 8 * q^29 + 4 * q^31 - 8 * q^35 + 3 * q^37 + 5 * q^43 - 17 * q^47 + 3 * q^49 - 14 * q^53 + 4 * q^55 - 4 * q^59 + q^61 - 15 * q^65 - 4 * q^67 - 7 * q^71 - 12 * q^73 - 5 * q^77 + q^79 - 22 * q^83 - 3 * q^85 - 22 * q^89 + 11 * q^91 - 24 * q^95 - 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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.69649	0.758693	0.379347	−	0.925255i	$-0.376149\pi$
$5$	1.69649	0.758693	0.379347	+	0.925255i	$0.376149\pi$
$6$	0	0
$7$	1.90201	0.718893	0.359446	−	0.933166i	$-0.382966\pi$
$7$	1.90201	0.718893	0.359446	+	0.933166i	$0.382966\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−2.08396	−0.577985	−0.288993	−	0.957331i	$-0.593320\pi$
$13$	−2.08396	−0.577985	−0.288993	+	0.957331i	$0.593320\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.310342	0.0752691	0.0376345	−	0.999292i	$-0.488018\pi$
$17$	0.310342	0.0752691	0.0376345	+	0.999292i	$0.488018\pi$
$18$	0	0
$19$	−2.59850	−0.596137	−0.298069	−	0.954544i	$-0.596342\pi$
$19$	−2.59850	−0.596137	−0.298069	+	0.954544i	$0.596342\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−0.226386	−0.0472048	−0.0236024	−	0.999721i	$-0.507514\pi$
$23$	−0.226386	−0.0472048	−0.0236024	+	0.999721i	$0.507514\pi$
$24$	0	0
$25$	−2.12192	−0.424385
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−4.42032	−0.820833	−0.410416	−	0.911898i	$-0.634617\pi$
$29$	−4.42032	−0.820833	−0.410416	+	0.911898i	$0.634617\pi$
$30$	0	0
$31$	−2.98597	−0.536296	−0.268148	−	0.963378i	$-0.586412\pi$
$31$	−2.98597	−0.536296	−0.268148	+	0.963378i	$0.586412\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	3.22674	0.545419
$36$	0	0
$37$	4.20720	0.691659	0.345830	−	0.938297i	$-0.387598\pi$
$37$	4.20720	0.691659	0.345830	+	0.938297i	$0.387598\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−5.62376	−0.878283	−0.439142	−	0.898418i	$-0.644717\pi$
$41$	−5.62376	−0.878283	−0.439142	+	0.898418i	$0.644717\pi$
$42$	0	0
$43$	−2.02393	−0.308647	−0.154324	−	0.988020i	$-0.549320\pi$
$43$	−2.02393	−0.308647	−0.154324	+	0.988020i	$0.549320\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−1.48037	−0.215935	−0.107967	−	0.994154i	$-0.534434\pi$
$47$	−1.48037	−0.215935	−0.107967	+	0.994154i	$0.534434\pi$
$48$	0	0
$49$	−3.38235	−0.483193
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−3.56433	−0.489598	−0.244799	−	0.969574i	$-0.578722\pi$
$53$	−3.56433	−0.489598	−0.244799	+	0.969574i	$0.578722\pi$
$54$	0	0
$55$	−1.69649	−0.228755
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−9.87652	−1.28581	−0.642907	−	0.765945i	$-0.722272\pi$
$59$	−9.87652	−1.28581	−0.642907	+	0.765945i	$0.722272\pi$
$60$	0	0
$61$	−8.37392	−1.07217	−0.536085	−	0.844164i	$-0.680097\pi$
$61$	−8.37392	−1.07217	−0.536085	+	0.844164i	$0.680097\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−3.53541	−0.438514
$66$	0	0
$67$	6.33735	0.774230	0.387115	−	0.922031i	$-0.373472\pi$
$67$	6.33735	0.774230	0.387115	+	0.922031i	$0.373472\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−0.355974	−0.0422464	−0.0211232	−	0.999777i	$-0.506724\pi$
$71$	−0.355974	−0.0422464	−0.0211232	+	0.999777i	$0.506724\pi$
$72$	0	0
$73$	11.9857	1.40282	0.701412	−	0.712756i	$-0.252553\pi$
$73$	11.9857	1.40282	0.701412	+	0.712756i	$0.252553\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.90201	−0.216754
$78$	0	0
$79$	−4.83688	−0.544191	−0.272096	−	0.962270i	$-0.587717\pi$
$79$	−4.83688	−0.544191	−0.272096	+	0.962270i	$0.587717\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−2.90065	−0.318388	−0.159194	−	0.987247i	$-0.550890\pi$
$83$	−2.90065	−0.318388	−0.159194	+	0.987247i	$0.550890\pi$
$84$	0	0
$85$	0.526492	0.0571061
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−3.43554	−0.364167	−0.182083	−	0.983283i	$-0.558284\pi$
$89$	−3.43554	−0.364167	−0.182083	+	0.983283i	$0.558284\pi$
$90$	0	0
$91$	−3.96371	−0.415510
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−4.40833	−0.452285
$96$	0	0
$97$	−17.5014	−1.77699	−0.888497	−	0.458883i	$-0.848250\pi$
$97$	−17.5014	−1.77699	−0.888497	+	0.458883i	$0.848250\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−16.0290	−1.59495	−0.797474	−	0.603353i	$-0.793831\pi$
$101$	−16.0290	−1.59495	−0.797474	+	0.603353i	$0.793831\pi$
$102$	0	0
$103$	−2.12123	−0.209011	−0.104506	−	0.994524i	$-0.533326\pi$
$103$	−2.12123	−0.209011	−0.104506	+	0.994524i	$0.533326\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	20.4972	1.98154	0.990771	−	0.135548i	$-0.0432794\pi$
$107$	20.4972	1.98154	0.990771	+	0.135548i	$0.0432794\pi$
$108$	0	0
$109$	14.7125	1.40920	0.704599	−	0.709606i	$-0.251127\pi$
$109$	14.7125	1.40920	0.704599	+	0.709606i	$0.251127\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−7.39367	−0.695538	−0.347769	−	0.937580i	$-0.613061\pi$
$113$	−7.39367	−0.695538	−0.347769	+	0.937580i	$0.613061\pi$
$114$	0	0
$115$	−0.384062	−0.0358140
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0.590275	0.0541104
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−12.0823	−1.08067
$126$	0	0
$127$	−9.88973	−0.877572	−0.438786	−	0.898592i	$-0.644591\pi$
$127$	−9.88973	−0.877572	−0.438786	+	0.898592i	$0.644591\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−8.34239	−0.728878	−0.364439	−	0.931227i	$-0.618739\pi$
$131$	−8.34239	−0.728878	−0.364439	+	0.931227i	$0.618739\pi$
$132$	0	0
$133$	−4.94238	−0.428559
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−5.77681	−0.493546	−0.246773	−	0.969073i	$-0.579370\pi$
$137$	−5.77681	−0.493546	−0.246773	+	0.969073i	$0.579370\pi$
$138$	0	0
$139$	1.66196	0.140966	0.0704829	−	0.997513i	$-0.477546\pi$
$139$	1.66196	0.140966	0.0704829	+	0.997513i	$0.477546\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	2.08396	0.174269
$144$	0	0
$145$	−7.49902	−0.622760
$146$	0	0
$147$	0	0
$148$	0	0
$149$	5.19330	0.425451	0.212726	−	0.977112i	$-0.431766\pi$
$149$	5.19330	0.425451	0.212726	+	0.977112i	$0.431766\pi$
$150$	0	0
$151$	4.17735	0.339948	0.169974	−	0.985449i	$-0.445632\pi$
$151$	4.17735	0.339948	0.169974	+	0.985449i	$0.445632\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−5.06566	−0.406884
$156$	0	0
$157$	11.5337	0.920491	0.460246	−	0.887792i	$-0.347761\pi$
$157$	11.5337	0.920491	0.460246	+	0.887792i	$0.347761\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−0.430589	−0.0339352
$162$	0	0
$163$	20.5911	1.61282	0.806408	−	0.591359i	$-0.201408\pi$
$163$	20.5911	1.61282	0.806408	+	0.591359i	$0.201408\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−19.3048	−1.49385	−0.746926	−	0.664907i	$-0.768471\pi$
$167$	−19.3048	−1.49385	−0.746926	+	0.664907i	$0.768471\pi$
$168$	0	0
$169$	−8.65713	−0.665933
$170$	0	0
$171$	0	0
$172$	0	0
$173$	7.43329	0.565143	0.282571	−	0.959246i	$-0.408813\pi$
$173$	7.43329	0.565143	0.282571	+	0.959246i	$0.408813\pi$
$174$	0	0
$175$	−4.03592	−0.305087
$176$	0	0
$177$	0	0
$178$	0	0
$179$	3.12740	0.233752	0.116876	−	0.993146i	$-0.462712\pi$
$179$	3.12740	0.233752	0.116876	+	0.993146i	$0.462712\pi$
$180$	0	0
$181$	−15.8491	−1.17805	−0.589027	−	0.808114i	$-0.700489\pi$
$181$	−15.8491	−1.17805	−0.589027	+	0.808114i	$0.700489\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	7.13747	0.524757
$186$	0	0
$187$	−0.310342	−0.0226945
$188$	0	0
$189$	0	0
$190$	0	0
$191$	13.5481	0.980309	0.490155	−	0.871635i	$-0.336940\pi$
$191$	13.5481	0.980309	0.490155	+	0.871635i	$0.336940\pi$
$192$	0	0
$193$	15.6375	1.12561	0.562807	−	0.826588i	$-0.309721\pi$
$193$	15.6375	1.12561	0.562807	+	0.826588i	$0.309721\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1.36469	−0.0972301	−0.0486151	−	0.998818i	$-0.515481\pi$
$197$	−1.36469	−0.0972301	−0.0486151	+	0.998818i	$0.515481\pi$
$198$	0	0
$199$	−1.35834	−0.0962903	−0.0481451	−	0.998840i	$-0.515331\pi$
$199$	−1.35834	−0.0962903	−0.0481451	+	0.998840i	$0.515331\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−8.40750	−0.590091
$204$	0	0
$205$	−9.54064	−0.666347
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2.59850	0.179742
$210$	0	0
$211$	14.5575	1.00218	0.501089	−	0.865396i	$-0.332933\pi$
$211$	14.5575	1.00218	0.501089	+	0.865396i	$0.332933\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−3.43358	−0.234169
$216$	0	0
$217$	−5.67935	−0.385539
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−0.646740	−0.0435044
$222$	0	0
$223$	21.2005	1.41969	0.709846	−	0.704357i	$-0.248765\pi$
$223$	21.2005	1.41969	0.709846	+	0.704357i	$0.248765\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−11.9998	−0.796452	−0.398226	−	0.917287i	$-0.630374\pi$
$227$	−11.9998	−0.796452	−0.398226	+	0.917287i	$0.630374\pi$
$228$	0	0
$229$	−8.11625	−0.536337	−0.268168	−	0.963372i	$-0.586418\pi$
$229$	−8.11625	−0.536337	−0.268168	+	0.963372i	$0.586418\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	27.6506	1.81145	0.905726	−	0.423863i	$-0.139326\pi$
$233$	27.6506	1.81145	0.905726	+	0.423863i	$0.139326\pi$
$234$	0	0
$235$	−2.51144	−0.163828
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−7.22955	−0.467640	−0.233820	−	0.972280i	$-0.575123\pi$
$239$	−7.22955	−0.467640	−0.233820	+	0.972280i	$0.575123\pi$
$240$	0	0
$241$	−5.21847	−0.336151	−0.168076	−	0.985774i	$-0.553755\pi$
$241$	−5.21847	−0.336151	−0.168076	+	0.985774i	$0.553755\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−5.73812	−0.366595
$246$	0	0
$247$	5.41516	0.344559
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−12.0710	−0.761918	−0.380959	−	0.924592i	$-0.624406\pi$
$251$	−12.0710	−0.761918	−0.380959	+	0.924592i	$0.624406\pi$
$252$	0	0
$253$	0.226386	0.0142328
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−19.0542	−1.18857	−0.594283	−	0.804256i	$-0.702564\pi$
$257$	−19.0542	−1.18857	−0.594283	+	0.804256i	$0.702564\pi$
$258$	0	0
$259$	8.00214	0.497229
$260$	0	0
$261$	0	0
$262$	0	0
$263$	21.8171	1.34530	0.672649	−	0.739962i	$-0.265156\pi$
$263$	21.8171	1.34530	0.672649	+	0.739962i	$0.265156\pi$
$264$	0	0
$265$	−6.04685	−0.371455
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−11.0157	−0.671640	−0.335820	−	0.941926i	$-0.609013\pi$
$269$	−11.0157	−0.671640	−0.335820	+	0.941926i	$0.609013\pi$
$270$	0	0
$271$	14.7950	0.898733	0.449367	−	0.893347i	$-0.351650\pi$
$271$	14.7950	0.898733	0.449367	+	0.893347i	$0.351650\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2.12192	0.127957
$276$	0	0
$277$	9.78711	0.588050	0.294025	−	0.955798i	$-0.405005\pi$
$277$	9.78711	0.588050	0.294025	+	0.955798i	$0.405005\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−15.6494	−0.933566	−0.466783	−	0.884372i	$-0.654587\pi$
$281$	−15.6494	−0.933566	−0.466783	+	0.884372i	$0.654587\pi$
$282$	0	0
$283$	−6.74979	−0.401233	−0.200617	−	0.979670i	$-0.564295\pi$
$283$	−6.74979	−0.401233	−0.200617	+	0.979670i	$0.564295\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−10.6964	−0.631391
$288$	0	0
$289$	−16.9037	−0.994335
$290$	0	0
$291$	0	0
$292$	0	0
$293$	4.01370	0.234483	0.117241	−	0.993103i	$-0.462595\pi$
$293$	4.01370	0.234483	0.117241	+	0.993103i	$0.462595\pi$
$294$	0	0
$295$	−16.7554	−0.975538
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0.471779	0.0272837
$300$	0	0
$301$	−3.84955	−0.221884
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−14.2063	−0.813449
$306$	0	0
$307$	25.0796	1.43137	0.715683	−	0.698425i	$-0.246116\pi$
$307$	25.0796	1.43137	0.715683	+	0.698425i	$0.246116\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−24.1276	−1.36815	−0.684076	−	0.729411i	$-0.739794\pi$
$311$	−24.1276	−1.36815	−0.684076	+	0.729411i	$0.739794\pi$
$312$	0	0
$313$	−0.384342	−0.0217243	−0.0108622	−	0.999941i	$-0.503458\pi$
$313$	−0.384342	−0.0217243	−0.0108622	+	0.999941i	$0.503458\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−33.1855	−1.86388	−0.931942	−	0.362609i	$-0.881886\pi$
$317$	−33.1855	−1.86388	−0.931942	+	0.362609i	$0.881886\pi$
$318$	0	0
$319$	4.42032	0.247490
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−0.806425	−0.0448707
$324$	0	0
$325$	4.42199	0.245288
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−2.81569	−0.155234
$330$	0	0
$331$	29.3043	1.61071	0.805356	−	0.592791i	$-0.201974\pi$
$331$	29.3043	1.61071	0.805356	+	0.592791i	$0.201974\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	10.7512	0.587403
$336$	0	0
$337$	2.58405	0.140762	0.0703810	−	0.997520i	$-0.477579\pi$
$337$	2.58405	0.140762	0.0703810	+	0.997520i	$0.477579\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	2.98597	0.161699
$342$	0	0
$343$	−19.7474	−1.06626
$344$	0	0
$345$	0	0
$346$	0	0
$347$	7.05366	0.378660	0.189330	−	0.981913i	$-0.439368\pi$
$347$	7.05366	0.378660	0.189330	+	0.981913i	$0.439368\pi$
$348$	0	0
$349$	6.56722	0.351535	0.175768	−	0.984432i	$-0.443759\pi$
$349$	6.56722	0.351535	0.175768	+	0.984432i	$0.443759\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	9.68561	0.515513	0.257757	−	0.966210i	$-0.417017\pi$
$353$	9.68561	0.515513	0.257757	+	0.966210i	$0.417017\pi$
$354$	0	0
$355$	−0.603907	−0.0320520
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−37.5152	−1.97998	−0.989990	−	0.141141i	$-0.954923\pi$
$359$	−37.5152	−1.97998	−0.989990	+	0.141141i	$0.954923\pi$
$360$	0	0
$361$	−12.2478	−0.644621
$362$	0	0
$363$	0	0
$364$	0	0
$365$	20.3337	1.06431
$366$	0	0
$367$	31.0993	1.62337	0.811684	−	0.584097i	$-0.198551\pi$
$367$	31.0993	1.62337	0.811684	+	0.584097i	$0.198551\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−6.77940	−0.351969
$372$	0	0
$373$	−20.7724	−1.07555	−0.537776	−	0.843088i	$-0.680735\pi$
$373$	−20.7724	−1.07555	−0.537776	+	0.843088i	$0.680735\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	9.21175	0.474429
$378$	0	0
$379$	18.8314	0.967303	0.483652	−	0.875261i	$-0.339310\pi$
$379$	18.8314	0.967303	0.483652	+	0.875261i	$0.339310\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	18.9966	0.970679	0.485339	−	0.874326i	$-0.338696\pi$
$383$	18.9966	0.970679	0.485339	+	0.874326i	$0.338696\pi$
$384$	0	0
$385$	−3.22674	−0.164450
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−3.22805	−0.163669	−0.0818343	−	0.996646i	$-0.526078\pi$
$389$	−3.22805	−0.163669	−0.0818343	+	0.996646i	$0.526078\pi$
$390$	0	0
$391$	−0.0702573	−0.00355306
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−8.20571	−0.412874
$396$	0	0
$397$	36.5699	1.83539	0.917694	−	0.397288i	$-0.130048\pi$
$397$	36.5699	1.83539	0.917694	+	0.397288i	$0.130048\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	13.3974	0.669032	0.334516	−	0.942390i	$-0.391427\pi$
$401$	13.3974	0.669032	0.334516	+	0.942390i	$0.391427\pi$
$402$	0	0
$403$	6.22263	0.309971
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−4.20720	−0.208543
$408$	0	0
$409$	−14.8940	−0.736460	−0.368230	−	0.929735i	$-0.620036\pi$
$409$	−14.8940	−0.736460	−0.368230	+	0.929735i	$0.620036\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−18.7853	−0.924362
$414$	0	0
$415$	−4.92093	−0.241559
$416$	0	0
$417$	0	0
$418$	0	0
$419$	37.1542	1.81510	0.907551	−	0.419941i	$-0.137949\pi$
$419$	37.1542	1.81510	0.907551	+	0.419941i	$0.137949\pi$
$420$	0	0
$421$	33.5625	1.63574	0.817868	−	0.575406i	$-0.195156\pi$
$421$	33.5625	1.63574	0.817868	+	0.575406i	$0.195156\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−0.658522	−0.0319430
$426$	0	0
$427$	−15.9273	−0.770776
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−11.9350	−0.574890	−0.287445	−	0.957797i	$-0.592806\pi$
$431$	−11.9350	−0.574890	−0.287445	+	0.957797i	$0.592806\pi$
$432$	0	0
$433$	−18.0875	−0.869230	−0.434615	−	0.900616i	$-0.643116\pi$
$433$	−18.0875	−0.869230	−0.434615	+	0.900616i	$0.643116\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0.588265	0.0281405
$438$	0	0
$439$	11.9900	0.572253	0.286126	−	0.958192i	$-0.407632\pi$
$439$	11.9900	0.572253	0.286126	+	0.958192i	$0.407632\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−18.6185	−0.884591	−0.442296	−	0.896869i	$-0.645836\pi$
$443$	−18.6185	−0.884591	−0.442296	+	0.896869i	$0.645836\pi$
$444$	0	0
$445$	−5.82836	−0.276291
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−33.1000	−1.56208	−0.781042	−	0.624478i	$-0.785312\pi$
$449$	−33.1000	−1.56208	−0.781042	+	0.624478i	$0.785312\pi$
$450$	0	0
$451$	5.62376	0.264812
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−6.72439	−0.315244
$456$	0	0
$457$	−34.0444	−1.59253	−0.796266	−	0.604947i	$-0.793194\pi$
$457$	−34.0444	−1.59253	−0.796266	+	0.604947i	$0.793194\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−18.1674	−0.846141	−0.423071	−	0.906097i	$-0.639048\pi$
$461$	−18.1674	−0.846141	−0.423071	+	0.906097i	$0.639048\pi$
$462$	0	0
$463$	17.3999	0.808644	0.404322	−	0.914617i	$-0.367508\pi$
$463$	17.3999	0.808644	0.404322	+	0.914617i	$0.367508\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−23.2774	−1.07715	−0.538575	−	0.842578i	$-0.681037\pi$
$467$	−23.2774	−1.07715	−0.538575	+	0.842578i	$0.681037\pi$
$468$	0	0
$469$	12.0537	0.556589
$470$	0	0
$471$	0	0
$472$	0	0
$473$	2.02393	0.0930606
$474$	0	0
$475$	5.51382	0.252991
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−31.9473	−1.45971	−0.729855	−	0.683602i	$-0.760412\pi$
$479$	−31.9473	−1.45971	−0.729855	+	0.683602i	$0.760412\pi$
$480$	0	0
$481$	−8.76762	−0.399769
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−29.6909	−1.34819
$486$	0	0
$487$	−29.7548	−1.34832	−0.674159	−	0.738586i	$-0.735494\pi$
$487$	−29.7548	−1.34832	−0.674159	+	0.738586i	$0.735494\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	1.76954	0.0798580	0.0399290	−	0.999203i	$-0.487287\pi$
$491$	1.76954	0.0798580	0.0399290	+	0.999203i	$0.487287\pi$
$492$	0	0
$493$	−1.37181	−0.0617833
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−0.677067	−0.0303706
$498$	0	0
$499$	−15.8916	−0.711404	−0.355702	−	0.934599i	$-0.615758\pi$
$499$	−15.8916	−0.711404	−0.355702	+	0.934599i	$0.615758\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−14.5272	−0.647735	−0.323867	−	0.946102i	$-0.604983\pi$
$503$	−14.5272	−0.647735	−0.323867	+	0.946102i	$0.604983\pi$
$504$	0	0
$505$	−27.1931	−1.21008
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−29.9661	−1.32822	−0.664112	−	0.747633i	$-0.731190\pi$
$509$	−29.9661	−1.32822	−0.664112	+	0.747633i	$0.731190\pi$
$510$	0	0
$511$	22.7970	1.00848
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−3.59865	−0.158575
$516$	0	0
$517$	1.48037	0.0651068
$518$	0	0
$519$	0	0
$520$	0	0
$521$	11.3725	0.498237	0.249118	−	0.968473i	$-0.419859\pi$
$521$	11.3725	0.498237	0.249118	+	0.968473i	$0.419859\pi$
$522$	0	0
$523$	13.1069	0.573124	0.286562	−	0.958062i	$-0.407488\pi$
$523$	13.1069	0.573124	0.286562	+	0.958062i	$0.407488\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−0.926672	−0.0403665
$528$	0	0
$529$	−22.9487	−0.997772
$530$	0	0
$531$	0	0
$532$	0	0
$533$	11.7197	0.507635
$534$	0	0
$535$	34.7733	1.50338
$536$	0	0
$537$	0	0
$538$	0	0
$539$	3.38235	0.145688
$540$	0	0
$541$	−25.4013	−1.09209	−0.546043	−	0.837757i	$-0.683866\pi$
$541$	−25.4013	−1.09209	−0.546043	+	0.837757i	$0.683866\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	24.9595	1.06915
$546$	0	0
$547$	−4.41718	−0.188865	−0.0944325	−	0.995531i	$-0.530104\pi$
$547$	−4.41718	−0.188865	−0.0944325	+	0.995531i	$0.530104\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	11.4862	0.489329
$552$	0	0
$553$	−9.19979	−0.391215
$554$	0	0
$555$	0	0
$556$	0	0
$557$	4.65681	0.197315	0.0986577	−	0.995121i	$-0.468545\pi$
$557$	4.65681	0.197315	0.0986577	+	0.995121i	$0.468545\pi$
$558$	0	0
$559$	4.21779	0.178394
$560$	0	0
$561$	0	0
$562$	0	0
$563$	5.85475	0.246748	0.123374	−	0.992360i	$-0.460629\pi$
$563$	5.85475	0.246748	0.123374	+	0.992360i	$0.460629\pi$
$564$	0	0
$565$	−12.5433	−0.527700
$566$	0	0
$567$	0	0
$568$	0	0
$569$	31.3201	1.31301	0.656504	−	0.754323i	$-0.272035\pi$
$569$	31.3201	1.31301	0.656504	+	0.754323i	$0.272035\pi$
$570$	0	0
$571$	5.95396	0.249165	0.124583	−	0.992209i	$-0.460241\pi$
$571$	5.95396	0.249165	0.124583	+	0.992209i	$0.460241\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0.480374	0.0200330
$576$	0	0
$577$	−15.4452	−0.642993	−0.321497	−	0.946911i	$-0.604186\pi$
$577$	−15.4452	−0.642993	−0.321497	+	0.946911i	$0.604186\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−5.51707	−0.228887
$582$	0	0
$583$	3.56433	0.147619
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−40.0682	−1.65379	−0.826896	−	0.562355i	$-0.809895\pi$
$587$	−40.0682	−1.65379	−0.826896	+	0.562355i	$0.809895\pi$
$588$	0	0
$589$	7.75904	0.319706
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−20.1031	−0.825537	−0.412769	−	0.910836i	$-0.635438\pi$
$593$	−20.1031	−0.825537	−0.412769	+	0.910836i	$0.635438\pi$
$594$	0	0
$595$	1.00139	0.0410532
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−24.2378	−0.990332	−0.495166	−	0.868799i	$-0.664893\pi$
$599$	−24.2378	−0.990332	−0.495166	+	0.868799i	$0.664893\pi$
$600$	0	0
$601$	28.1516	1.14833	0.574164	−	0.818740i	$-0.305327\pi$
$601$	28.1516	1.14833	0.574164	+	0.818740i	$0.305327\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	1.69649	0.0689721
$606$	0	0
$607$	27.5473	1.11811	0.559056	−	0.829130i	$-0.311164\pi$
$607$	27.5473	1.11811	0.559056	+	0.829130i	$0.311164\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	3.08504	0.124807
$612$	0	0
$613$	−34.2328	−1.38265	−0.691325	−	0.722544i	$-0.742973\pi$
$613$	−34.2328	−1.38265	−0.691325	+	0.722544i	$0.742973\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0.213745	0.00860507	0.00430253	−	0.999991i	$-0.498630\pi$
$617$	0.213745	0.00860507	0.00430253	+	0.999991i	$0.498630\pi$
$618$	0	0
$619$	30.4544	1.22406	0.612032	−	0.790833i	$-0.290352\pi$
$619$	30.4544	1.22406	0.612032	+	0.790833i	$0.290352\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−6.53444	−0.261797
$624$	0	0
$625$	−9.88783	−0.395513
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1.30567	0.0520605
$630$	0	0
$631$	−27.6272	−1.09982	−0.549910	−	0.835224i	$-0.685338\pi$
$631$	−27.6272	−1.09982	−0.549910	+	0.835224i	$0.685338\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−16.7778	−0.665808
$636$	0	0
$637$	7.04867	0.279279
$638$	0	0
$639$	0	0
$640$	0	0
$641$	23.2740	0.919268	0.459634	−	0.888108i	$-0.347981\pi$
$641$	23.2740	0.919268	0.459634	+	0.888108i	$0.347981\pi$
$642$	0	0
$643$	−14.4221	−0.568754	−0.284377	−	0.958713i	$-0.591787\pi$
$643$	−14.4221	−0.568754	−0.284377	+	0.958713i	$0.591787\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−0.518760	−0.0203946	−0.0101973	−	0.999948i	$-0.503246\pi$
$647$	−0.518760	−0.0203946	−0.0101973	+	0.999948i	$0.503246\pi$
$648$	0	0
$649$	9.87652	0.387687
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−28.7953	−1.12685	−0.563424	−	0.826168i	$-0.690516\pi$
$653$	−28.7953	−1.12685	−0.563424	+	0.826168i	$0.690516\pi$
$654$	0	0
$655$	−14.1528	−0.552995
$656$	0	0
$657$	0	0
$658$	0	0
$659$	30.1118	1.17299	0.586494	−	0.809953i	$-0.300508\pi$
$659$	30.1118	1.17299	0.586494	+	0.809953i	$0.300508\pi$
$660$	0	0
$661$	−48.2495	−1.87669	−0.938344	−	0.345703i	$-0.887640\pi$
$661$	−48.2495	−1.87669	−0.938344	+	0.345703i	$0.887640\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−8.38470	−0.325145
$666$	0	0
$667$	1.00070	0.0387472
$668$	0	0
$669$	0	0
$670$	0	0
$671$	8.37392	0.323272
$672$	0	0
$673$	2.59141	0.0998916	0.0499458	−	0.998752i	$-0.484095\pi$
$673$	2.59141	0.0998916	0.0499458	+	0.998752i	$0.484095\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−12.0027	−0.461301	−0.230651	−	0.973037i	$-0.574085\pi$
$677$	−12.0027	−0.461301	−0.230651	+	0.973037i	$0.574085\pi$
$678$	0	0
$679$	−33.2878	−1.27747
$680$	0	0
$681$	0	0
$682$	0	0
$683$	10.4104	0.398341	0.199171	−	0.979965i	$-0.436175\pi$
$683$	10.4104	0.398341	0.199171	+	0.979965i	$0.436175\pi$
$684$	0	0
$685$	−9.80030	−0.374450
$686$	0	0
$687$	0	0
$688$	0	0
$689$	7.42791	0.282981
$690$	0	0
$691$	−46.0665	−1.75245	−0.876226	−	0.481900i	$-0.839947\pi$
$691$	−46.0665	−1.75245	−0.876226	+	0.481900i	$0.839947\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	2.81950	0.106950
$696$	0	0
$697$	−1.74529	−0.0661075
$698$	0	0
$699$	0	0
$700$	0	0
$701$	18.3855	0.694410	0.347205	−	0.937789i	$-0.387131\pi$
$701$	18.3855	0.694410	0.347205	+	0.937789i	$0.387131\pi$
$702$	0	0
$703$	−10.9324	−0.412324
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−30.4874	−1.14660
$708$	0	0
$709$	22.7877	0.855809	0.427904	−	0.903824i	$-0.359252\pi$
$709$	22.7877	0.855809	0.427904	+	0.903824i	$0.359252\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0.675982	0.0253157
$714$	0	0
$715$	3.53541	0.132217
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−8.21384	−0.306325	−0.153162	−	0.988201i	$-0.548946\pi$
$719$	−8.21384	−0.306325	−0.153162	+	0.988201i	$0.548946\pi$
$720$	0	0
$721$	−4.03461	−0.150257
$722$	0	0
$723$	0	0
$724$	0	0
$725$	9.37958	0.348349
$726$	0	0
$727$	−25.8441	−0.958504	−0.479252	−	0.877677i	$-0.659092\pi$
$727$	−25.8441	−0.958504	−0.479252	+	0.877677i	$0.659092\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−0.628113	−0.0232316
$732$	0	0
$733$	34.2198	1.26394	0.631968	−	0.774994i	$-0.282247\pi$
$733$	34.2198	1.26394	0.631968	+	0.774994i	$0.282247\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−6.33735	−0.233439
$738$	0	0
$739$	38.8028	1.42738	0.713692	−	0.700460i	$-0.247022\pi$
$739$	38.8028	1.42738	0.713692	+	0.700460i	$0.247022\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−1.35099	−0.0495631	−0.0247816	−	0.999693i	$-0.507889\pi$
$743$	−1.35099	−0.0495631	−0.0247816	+	0.999693i	$0.507889\pi$
$744$	0	0
$745$	8.81037	0.322787
$746$	0	0
$747$	0	0
$748$	0	0
$749$	38.9860	1.42452
$750$	0	0
$751$	18.5343	0.676326	0.338163	−	0.941088i	$-0.390194\pi$
$751$	18.5343	0.676326	0.338163	+	0.941088i	$0.390194\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	7.08683	0.257916
$756$	0	0
$757$	−17.5676	−0.638504	−0.319252	−	0.947670i	$-0.603432\pi$
$757$	−17.5676	−0.638504	−0.319252	+	0.947670i	$0.603432\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−11.8030	−0.427857	−0.213928	−	0.976849i	$-0.568626\pi$
$761$	−11.8030	−0.427857	−0.213928	+	0.976849i	$0.568626\pi$
$762$	0	0
$763$	27.9833	1.01306
$764$	0	0
$765$	0	0
$766$	0	0
$767$	20.5822	0.743181
$768$	0	0
$769$	−9.74474	−0.351404	−0.175702	−	0.984443i	$-0.556220\pi$
$769$	−9.74474	−0.351404	−0.175702	+	0.984443i	$0.556220\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	31.8568	1.14581	0.572905	−	0.819622i	$-0.305816\pi$
$773$	31.8568	1.14581	0.572905	+	0.819622i	$0.305816\pi$
$774$	0	0
$775$	6.33599	0.227596
$776$	0	0
$777$	0	0
$778$	0	0
$779$	14.6133	0.523577
$780$	0	0
$781$	0.355974	0.0127378
$782$	0	0
$783$	0	0
$784$	0	0
$785$	19.5668	0.698370
$786$	0	0
$787$	−6.04780	−0.215581	−0.107790	−	0.994174i	$-0.534378\pi$
$787$	−6.04780	−0.215581	−0.107790	+	0.994174i	$0.534378\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−14.0628	−0.500017
$792$	0	0
$793$	17.4509	0.619699
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−21.7843	−0.771638	−0.385819	−	0.922574i	$-0.626081\pi$
$797$	−21.7843	−0.771638	−0.385819	+	0.922574i	$0.626081\pi$
$798$	0	0
$799$	−0.459423	−0.0162532
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−11.9857	−0.422967
$804$	0	0
$805$	−0.730491	−0.0257464
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1.07452	0.0377779	0.0188890	−	0.999822i	$-0.493987\pi$
$809$	1.07452	0.0377779	0.0188890	+	0.999822i	$0.493987\pi$
$810$	0	0
$811$	6.38538	0.224221	0.112110	−	0.993696i	$-0.464239\pi$
$811$	6.38538	0.224221	0.112110	+	0.993696i	$0.464239\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	34.9325	1.22363
$816$	0	0
$817$	5.25920	0.183996
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−0.582948	−0.0203450	−0.0101725	−	0.999948i	$-0.503238\pi$
$821$	−0.582948	−0.0203450	−0.0101725	+	0.999948i	$0.503238\pi$
$822$	0	0
$823$	24.9978	0.871369	0.435685	−	0.900099i	$-0.356506\pi$
$823$	24.9978	0.871369	0.435685	+	0.900099i	$0.356506\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−9.64913	−0.335533	−0.167767	−	0.985827i	$-0.553655\pi$
$827$	−9.64913	−0.335533	−0.167767	+	0.985827i	$0.553655\pi$
$828$	0	0
$829$	−8.07397	−0.280421	−0.140210	−	0.990122i	$-0.544778\pi$
$829$	−8.07397	−0.280421	−0.140210	+	0.990122i	$0.544778\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−1.04969	−0.0363695
$834$	0	0
$835$	−32.7504	−1.13338
$836$	0	0
$837$	0	0
$838$	0	0
$839$	40.8914	1.41173	0.705864	−	0.708347i	$-0.250559\pi$
$839$	40.8914	1.41173	0.705864	+	0.708347i	$0.250559\pi$
$840$	0	0
$841$	−9.46079	−0.326234
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−14.6867	−0.505239
$846$	0	0
$847$	1.90201	0.0653539
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−0.952452	−0.0326496
$852$	0	0
$853$	−23.1338	−0.792086	−0.396043	−	0.918232i	$-0.629617\pi$
$853$	−23.1338	−0.792086	−0.396043	+	0.918232i	$0.629617\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−36.1845	−1.23604	−0.618019	−	0.786163i	$-0.712064\pi$
$857$	−36.1845	−1.23604	−0.618019	+	0.786163i	$0.712064\pi$
$858$	0	0
$859$	−4.73165	−0.161442	−0.0807208	−	0.996737i	$-0.525722\pi$
$859$	−4.73165	−0.161442	−0.0807208	+	0.996737i	$0.525722\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	51.6542	1.75833	0.879164	−	0.476519i	$-0.158102\pi$
$863$	51.6542	1.75833	0.879164	+	0.476519i	$0.158102\pi$
$864$	0	0
$865$	12.6105	0.428770
$866$	0	0
$867$	0	0
$868$	0	0
$869$	4.83688	0.164080
$870$	0	0
$871$	−13.2068	−0.447494
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−22.9806	−0.776887
$876$	0	0
$877$	15.2455	0.514805	0.257402	−	0.966304i	$-0.417133\pi$
$877$	15.2455	0.514805	0.257402	+	0.966304i	$0.417133\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	29.7320	1.00170	0.500849	−	0.865535i	$-0.333021\pi$
$881$	29.7320	1.00170	0.500849	+	0.865535i	$0.333021\pi$
$882$	0	0
$883$	−41.5017	−1.39664	−0.698322	−	0.715783i	$-0.746070\pi$
$883$	−41.5017	−1.39664	−0.698322	+	0.715783i	$0.746070\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	16.5422	0.555434	0.277717	−	0.960663i	$-0.410422\pi$
$887$	16.5422	0.555434	0.277717	+	0.960663i	$0.410422\pi$
$888$	0	0
$889$	−18.8104	−0.630880
$890$	0	0
$891$	0	0
$892$	0	0
$893$	3.84675	0.128727
$894$	0	0
$895$	5.30559	0.177346
$896$	0	0
$897$	0	0
$898$	0	0
$899$	13.1989	0.440209
$900$	0	0
$901$	−1.10616	−0.0368516
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−26.8878	−0.893781
$906$	0	0
$907$	30.7025	1.01946	0.509729	−	0.860335i	$-0.329746\pi$
$907$	30.7025	1.01946	0.509729	+	0.860335i	$0.329746\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−2.07472	−0.0687387	−0.0343693	−	0.999409i	$-0.510942\pi$
$911$	−2.07472	−0.0687387	−0.0343693	+	0.999409i	$0.510942\pi$
$912$	0	0
$913$	2.90065	0.0959976
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−15.8673	−0.523985
$918$	0	0
$919$	51.3954	1.69538	0.847688	−	0.530495i	$-0.177994\pi$
$919$	51.3954	1.69538	0.847688	+	0.530495i	$0.177994\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0.741835	0.0244178
$924$	0	0
$925$	−8.92735	−0.293529
$926$	0	0
$927$	0	0
$928$	0	0
$929$	25.9209	0.850437	0.425218	−	0.905091i	$-0.360197\pi$
$929$	25.9209	0.850437	0.425218	+	0.905091i	$0.360197\pi$
$930$	0	0
$931$	8.78904	0.288049
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−0.526492	−0.0172181
$936$	0	0
$937$	−26.1718	−0.854994	−0.427497	−	0.904017i	$-0.640605\pi$
$937$	−26.1718	−0.854994	−0.427497	+	0.904017i	$0.640605\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	53.3116	1.73791	0.868954	−	0.494893i	$-0.164793\pi$
$941$	53.3116	1.73791	0.868954	+	0.494893i	$0.164793\pi$
$942$	0	0
$943$	1.27314	0.0414592
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−19.5995	−0.636897	−0.318448	−	0.947940i	$-0.603162\pi$
$947$	−19.5995	−0.636897	−0.318448	+	0.947940i	$0.603162\pi$
$948$	0	0
$949$	−24.9777	−0.810811
$950$	0	0
$951$	0	0
$952$	0	0
$953$	22.3925	0.725366	0.362683	−	0.931913i	$-0.381861\pi$
$953$	22.3925	0.725366	0.362683	+	0.931913i	$0.381861\pi$
$954$	0	0
$955$	22.9843	0.743754
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−10.9876	−0.354807
$960$	0	0
$961$	−22.0840	−0.712387
$962$	0	0
$963$	0	0
$964$	0	0
$965$	26.5289	0.853996
$966$	0	0
$967$	28.8139	0.926593	0.463297	−	0.886203i	$-0.346666\pi$
$967$	28.8139	0.926593	0.463297	+	0.886203i	$0.346666\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	8.45080	0.271199	0.135599	−	0.990764i	$-0.456704\pi$
$971$	8.45080	0.271199	0.135599	+	0.990764i	$0.456704\pi$
$972$	0	0
$973$	3.16107	0.101339
$974$	0	0
$975$	0	0
$976$	0	0
$977$	36.2572	1.15997	0.579985	−	0.814627i	$-0.303058\pi$
$977$	36.2572	1.15997	0.579985	+	0.814627i	$0.303058\pi$
$978$	0	0
$979$	3.43554	0.109800
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−31.5204	−1.00534	−0.502672	−	0.864477i	$-0.667650\pi$
$983$	−31.5204	−1.00534	−0.502672	+	0.864477i	$0.667650\pi$
$984$	0	0
$985$	−2.31518	−0.0737678
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0.458191	0.0145696
$990$	0	0
$991$	−35.9833	−1.14305	−0.571524	−	0.820586i	$-0.693647\pi$
$991$	−35.9833	−1.14305	−0.571524	+	0.820586i	$0.693647\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−2.30441	−0.0730548
$996$	0	0
$997$	4.84166	0.153337	0.0766684	−	0.997057i	$-0.475572\pi$
$997$	4.84166	0.153337	0.0766684	+	0.997057i	$0.475572\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7128.2.a.w.1.5		6
3.2	odd	2		7128.2.a.ba.1.2		6
9.2	odd	6		792.2.q.f.265.6	✓	12
9.4	even	3		2376.2.q.f.1585.2		12
9.5	odd	6		792.2.q.f.529.6	yes	12
9.7	even	3		2376.2.q.f.793.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
792.2.q.f.265.6	✓	12	9.2	odd	6
792.2.q.f.529.6	yes	12	9.5	odd	6
2376.2.q.f.793.2		12	9.7	even	3
2376.2.q.f.1585.2		12	9.4	even	3
7128.2.a.w.1.5		6	1.1	even	1	trivial
7128.2.a.ba.1.2		6	3.2	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:	\( N \)	\(=\)	\( 7128 = 2^{3} \cdot 3^{4} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7128.a (trivial)

\(f(q)\)	\(=\)	\(q+1.69649 q^{5} +1.90201 q^{7} -1.00000 q^{11} -2.08396 q^{13} +0.310342 q^{17} -2.59850 q^{19} -0.226386 q^{23} -2.12192 q^{25} -4.42032 q^{29} -2.98597 q^{31} +3.22674 q^{35} +4.20720 q^{37} -5.62376 q^{41} -2.02393 q^{43} -1.48037 q^{47} -3.38235 q^{49} -3.56433 q^{53} -1.69649 q^{55} -9.87652 q^{59} -8.37392 q^{61} -3.53541 q^{65} +6.33735 q^{67} -0.355974 q^{71} +11.9857 q^{73} -1.90201 q^{77} -4.83688 q^{79} -2.90065 q^{83} +0.526492 q^{85} -3.43554 q^{89} -3.96371 q^{91} -4.40833 q^{95} -17.5014 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{5} + 5 q^{7} - 6 q^{11} + 3 q^{13} - 7 q^{17} + 5 q^{19} - 8 q^{23} - 2 q^{25} - 8 q^{29} + 4 q^{31} - 8 q^{35} + 3 q^{37} + 5 q^{43} - 17 q^{47} + 3 q^{49} - 14 q^{53} + 4 q^{55} - 4 q^{59}+ \cdots - 16 q^{97}+O(q^{100}) \) 6 * q - 4 * q^5 + 5 * q^7 - 6 * q^11 + 3 * q^13 - 7 * q^17 + 5 * q^19 - 8 * q^23 - 2 * q^25 - 8 * q^29 + 4 * q^31 - 8 * q^35 + 3 * q^37 + 5 * q^43 - 17 * q^47 + 3 * q^49 - 14 * q^53 + 4 * q^55 - 4 * q^59 + q^61 - 15 * q^65 - 4 * q^67 - 7 * q^71 - 12 * q^73 - 5 * q^77 + q^79 - 22 * q^83 - 3 * q^85 - 22 * q^89 + 11 * q^91 - 24 * q^95 - 16 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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