LMFDB - Embedded newform 6800.2.a.bt.1.3

Error: no document with id 235638119 found in table mf_hecke_traces.

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6800,2,Mod(1,6800)] 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("6800.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6800, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6800 = 2^{4} \cdot 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6800.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,-4,0,0,0,-10,0,4,0,2,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$54.2982733745$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.6224.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} - 2x + 5 $ x^4 - 6x^2 - 2x + 5
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 85)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.796815$ of defining polynomial
Character	$\chi$	$=$	6800.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.203185 q^{3} -0.683735 q^{7} -2.95872 q^{9} -3.68135 q^{11} +4.43927 q^{13} -1.00000 q^{17} +1.03890 q^{19} +0.138925 q^{21} +4.52699 q^{23} +1.21072 q^{27} -3.69127 q^{29} +10.8921 q^{31} +0.747995 q^{33} +0.308729 q^{37} -0.901992 q^{39} -6.15198 q^{41} -7.88454 q^{43} -4.43927 q^{47} -6.53251 q^{49} +0.203185 q^{51} +11.4603 q^{53} -0.211089 q^{57} +2.00000 q^{59} +9.94089 q^{61} +2.02298 q^{63} -9.16944 q^{67} -0.919815 q^{69} +9.37262 q^{71} -2.26946 q^{73} +2.51707 q^{77} -7.42696 q^{79} +8.63015 q^{81} +8.92344 q^{83} +0.750010 q^{87} +11.5523 q^{89} -3.03528 q^{91} -2.21310 q^{93} -12.5500 q^{97} +10.8921 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} - 10 q^{7} + 4 q^{9} + 2 q^{11} + 6 q^{13} - 4 q^{17} - 4 q^{19} + 12 q^{21} - 4 q^{23} - 10 q^{27} - 4 q^{29} + 12 q^{31} + 2 q^{33} + 12 q^{37} + 22 q^{39} - 6 q^{41} - 18 q^{43} - 6 q^{47}+ \cdots + 12 q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 - 10 * q^7 + 4 * q^9 + 2 * q^11 + 6 * q^13 - 4 * q^17 - 4 * q^19 + 12 * q^21 - 4 * q^23 - 10 * q^27 - 4 * q^29 + 12 * q^31 + 2 * q^33 + 12 * q^37 + 22 * q^39 - 6 * q^41 - 18 * q^43 - 6 * q^47 + 8 * q^49 + 4 * q^51 + 8 * q^53 - 16 * q^57 + 8 * q^59 + 6 * q^61 - 16 * q^63 - 6 * q^67 + 4 * q^69 + 10 * q^71 + 2 * q^73 - 18 * q^77 + 12 * q^79 - 4 * q^81 + 14 * q^83 - 4 * q^87 + 24 * q^89 - 18 * q^91 - 18 * q^93 - 4 * q^97 + 12 * 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
$3$	−0.203185	−0.117309	−0.0586544	−	0.998278i	$-0.518681\pi$
$3$	−0.203185	−0.117309	−0.0586544	+	0.998278i	$0.518681\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−0.683735	−0.258427	−0.129214	−	0.991617i	$-0.541245\pi$
$7$	−0.683735	−0.258427	−0.129214	+	0.991617i	$0.541245\pi$
$8$	0	0
$9$	−2.95872	−0.986239
$10$	0	0
$11$	−3.68135	−1.10997	−0.554985	−	0.831861i	$-0.687276\pi$
$11$	−3.68135	−1.10997	−0.554985	+	0.831861i	$0.687276\pi$
$12$	0	0
$13$	4.43927	1.23123	0.615615	−	0.788047i	$-0.288908\pi$
$13$	4.43927	1.23123	0.615615	+	0.788047i	$0.288908\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	1.03890	0.238340	0.119170	−	0.992874i	$-0.461977\pi$
$19$	1.03890	0.238340	0.119170	+	0.992874i	$0.461977\pi$
$20$	0	0
$21$	0.138925	0.0303158
$22$	0	0
$23$	4.52699	0.943942	0.471971	−	0.881614i	$-0.343543\pi$
$23$	4.52699	0.943942	0.471971	+	0.881614i	$0.343543\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.21072	0.233003
$28$	0	0
$29$	−3.69127	−0.685452	−0.342726	−	0.939435i	$-0.611350\pi$
$29$	−3.69127	−0.685452	−0.342726	+	0.939435i	$0.611350\pi$
$30$	0	0
$31$	10.8921	1.95627	0.978137	−	0.207962i	$-0.0666830\pi$
$31$	10.8921	1.95627	0.978137	+	0.207962i	$0.0666830\pi$
$32$	0	0
$33$	0.747995	0.130209
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0.308729	0.0507548	0.0253774	−	0.999678i	$-0.491921\pi$
$37$	0.308729	0.0507548	0.0253774	+	0.999678i	$0.491921\pi$
$38$	0	0
$39$	−0.901992	−0.144434
$40$	0	0
$41$	−6.15198	−0.960778	−0.480389	−	0.877056i	$-0.659505\pi$
$41$	−6.15198	−0.960778	−0.480389	+	0.877056i	$0.659505\pi$
$42$	0	0
$43$	−7.88454	−1.20238	−0.601190	−	0.799106i	$-0.705307\pi$
$43$	−7.88454	−1.20238	−0.601190	+	0.799106i	$0.705307\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−4.43927	−0.647533	−0.323767	−	0.946137i	$-0.604949\pi$
$47$	−4.43927	−0.647533	−0.323767	+	0.946137i	$0.604949\pi$
$48$	0	0
$49$	−6.53251	−0.933215
$50$	0	0
$51$	0.203185	0.0284516
$52$	0	0
$53$	11.4603	1.57420	0.787100	−	0.616826i	$-0.211582\pi$
$53$	11.4603	1.57420	0.787100	+	0.616826i	$0.211582\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−0.211089	−0.0279594
$58$	0	0
$59$	2.00000	0.260378	0.130189	−	0.991489i	$-0.458442\pi$
$59$	2.00000	0.260378	0.130189	+	0.991489i	$0.458442\pi$
$60$	0	0
$61$	9.94089	1.27280	0.636400	−	0.771359i	$-0.280423\pi$
$61$	9.94089	1.27280	0.636400	+	0.771359i	$0.280423\pi$
$62$	0	0
$63$	2.02298	0.254871
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−9.16944	−1.12023	−0.560113	−	0.828417i	$-0.689242\pi$
$67$	−9.16944	−1.12023	−0.560113	+	0.828417i	$0.689242\pi$
$68$	0	0
$69$	−0.919815	−0.110733
$70$	0	0
$71$	9.37262	1.11233	0.556163	−	0.831073i	$-0.312273\pi$
$71$	9.37262	1.11233	0.556163	+	0.831073i	$0.312273\pi$
$72$	0	0
$73$	−2.26946	−0.265620	−0.132810	−	0.991141i	$-0.542400\pi$
$73$	−2.26946	−0.265620	−0.132810	+	0.991141i	$0.542400\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.51707	0.286846
$78$	0	0
$79$	−7.42696	−0.835599	−0.417799	−	0.908539i	$-0.637198\pi$
$79$	−7.42696	−0.835599	−0.417799	+	0.908539i	$0.637198\pi$
$80$	0	0
$81$	8.63015	0.958905
$82$	0	0
$83$	8.92344	0.979474	0.489737	−	0.871870i	$-0.337093\pi$
$83$	8.92344	0.979474	0.489737	+	0.871870i	$0.337093\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0.750010	0.0804096
$88$	0	0
$89$	11.5523	1.22455	0.612273	−	0.790646i	$-0.290255\pi$
$89$	11.5523	1.22455	0.612273	+	0.790646i	$0.290255\pi$
$90$	0	0
$91$	−3.03528	−0.318184
$92$	0	0
$93$	−2.21310	−0.229488
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−12.5500	−1.27426	−0.637128	−	0.770758i	$-0.719878\pi$
$97$	−12.5500	−1.27426	−0.637128	+	0.770758i	$0.719878\pi$
$98$	0	0
$99$	10.8921	1.09469
$100$	0	0
$101$	−6.78653	−0.675285	−0.337642	−	0.941274i	$-0.609629\pi$
$101$	−6.78653	−0.675285	−0.337642	+	0.941274i	$0.609629\pi$
$102$	0	0
$103$	−8.74799	−0.861966	−0.430983	−	0.902360i	$-0.641833\pi$
$103$	−8.74799	−0.861966	−0.430983	+	0.902360i	$0.641833\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−8.10078	−0.783132	−0.391566	−	0.920150i	$-0.628066\pi$
$107$	−8.10078	−0.783132	−0.391566	+	0.920150i	$0.628066\pi$
$108$	0	0
$109$	−3.11308	−0.298179	−0.149090	−	0.988824i	$-0.547634\pi$
$109$	−3.11308	−0.298179	−0.149090	+	0.988824i	$0.547634\pi$
$110$	0	0
$111$	−0.0627291	−0.00595399
$112$	0	0
$113$	−6.93287	−0.652190	−0.326095	−	0.945337i	$-0.605733\pi$
$113$	−6.93287	−0.652190	−0.326095	+	0.945337i	$0.605733\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−13.1345	−1.21429
$118$	0	0
$119$	0.683735	0.0626778
$120$	0	0
$121$	2.55235	0.232031
$122$	0	0
$123$	1.24999	0.112708
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−21.1496	−1.87672	−0.938362	−	0.345655i	$-0.887657\pi$
$127$	−21.1496	−1.87672	−0.938362	+	0.345655i	$0.887657\pi$
$128$	0	0
$129$	1.60202	0.141050
$130$	0	0
$131$	−8.71186	−0.761159	−0.380580	−	0.924748i	$-0.624275\pi$
$131$	−8.71186	−0.761159	−0.380580	+	0.924748i	$0.624275\pi$
$132$	0	0
$133$	−0.710332	−0.0615936
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0.243985	0.0208450	0.0104225	−	0.999946i	$-0.496682\pi$
$137$	0.243985	0.0208450	0.0104225	+	0.999946i	$0.496682\pi$
$138$	0	0
$139$	−12.6884	−1.07622	−0.538108	−	0.842876i	$-0.680861\pi$
$139$	−12.6884	−1.07622	−0.538108	+	0.842876i	$0.680861\pi$
$140$	0	0
$141$	0.901992	0.0759614
$142$	0	0
$143$	−16.3425	−1.36663
$144$	0	0
$145$	0	0
$146$	0	0
$147$	1.32731	0.109474
$148$	0	0
$149$	12.0103	0.983923	0.491961	−	0.870617i	$-0.336280\pi$
$149$	12.0103	0.983923	0.491961	+	0.870617i	$0.336280\pi$
$150$	0	0
$151$	−8.95633	−0.728856	−0.364428	−	0.931232i	$-0.618735\pi$
$151$	−8.95633	−0.728856	−0.364428	+	0.931232i	$0.618735\pi$
$152$	0	0
$153$	2.95872	0.239198
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−19.9127	−1.58920	−0.794602	−	0.607131i	$-0.792320\pi$
$157$	−19.9127	−1.58920	−0.794602	+	0.607131i	$0.792320\pi$
$158$	0	0
$159$	−2.32857	−0.184667
$160$	0	0
$161$	−3.09526	−0.243940
$162$	0	0
$163$	−22.4011	−1.75459	−0.877296	−	0.479951i	$-0.840655\pi$
$163$	−22.4011	−1.75459	−0.877296	+	0.479951i	$0.840655\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	4.58133	0.354514	0.177257	−	0.984165i	$-0.443278\pi$
$167$	4.58133	0.354514	0.177257	+	0.984165i	$0.443278\pi$
$168$	0	0
$169$	6.70708	0.515929
$170$	0	0
$171$	−3.07381	−0.235060
$172$	0	0
$173$	−8.09764	−0.615652	−0.307826	−	0.951443i	$-0.599601\pi$
$173$	−8.09764	−0.615652	−0.307826	+	0.951443i	$0.599601\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−0.406370	−0.0305446
$178$	0	0
$179$	−4.40637	−0.329348	−0.164674	−	0.986348i	$-0.552657\pi$
$179$	−4.40637	−0.329348	−0.164674	+	0.986348i	$0.552657\pi$
$180$	0	0
$181$	−5.93727	−0.441314	−0.220657	−	0.975351i	$-0.570820\pi$
$181$	−5.93727	−0.441314	−0.220657	+	0.975351i	$0.570820\pi$
$182$	0	0
$183$	−2.01984	−0.149311
$184$	0	0
$185$	0	0
$186$	0	0
$187$	3.68135	0.269207
$188$	0	0
$189$	−0.827812	−0.0602144
$190$	0	0
$191$	9.89759	0.716165	0.358082	−	0.933690i	$-0.383431\pi$
$191$	9.89759	0.716165	0.358082	+	0.933690i	$0.383431\pi$
$192$	0	0
$193$	−8.47578	−0.610100	−0.305050	−	0.952336i	$-0.598673\pi$
$193$	−8.47578	−0.610100	−0.305050	+	0.952336i	$0.598673\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	13.5381	0.964553	0.482276	−	0.876019i	$-0.339810\pi$
$197$	13.5381	0.964553	0.482276	+	0.876019i	$0.339810\pi$
$198$	0	0
$199$	9.15388	0.648901	0.324451	−	0.945903i	$-0.394821\pi$
$199$	9.15388	0.648901	0.324451	+	0.945903i	$0.394821\pi$
$200$	0	0
$201$	1.86309	0.131412
$202$	0	0
$203$	2.52385	0.177139
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−13.3941	−0.930952
$208$	0	0
$209$	−3.82456	−0.264550
$210$	0	0
$211$	−7.72465	−0.531787	−0.265893	−	0.964002i	$-0.585667\pi$
$211$	−7.72465	−0.531787	−0.265893	+	0.964002i	$0.585667\pi$
$212$	0	0
$213$	−1.90438	−0.130486
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−7.44729	−0.505555
$218$	0	0
$219$	0.461120	0.0311596
$220$	0	0
$221$	−4.43927	−0.298617
$222$	0	0
$223$	13.7194	0.918719	0.459359	−	0.888250i	$-0.348079\pi$
$223$	13.7194	0.918719	0.459359	+	0.888250i	$0.348079\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.80044	0.119499	0.0597496	−	0.998213i	$-0.480970\pi$
$227$	1.80044	0.119499	0.0597496	+	0.998213i	$0.480970\pi$
$228$	0	0
$229$	−3.24838	−0.214659	−0.107330	−	0.994223i	$-0.534230\pi$
$229$	−3.24838	−0.214659	−0.107330	+	0.994223i	$0.534230\pi$
$230$	0	0
$231$	−0.511430	−0.0336496
$232$	0	0
$233$	5.38254	0.352622	0.176311	−	0.984335i	$-0.443584\pi$
$233$	5.38254	0.352622	0.176311	+	0.984335i	$0.443584\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1.50905	0.0980231
$238$	0	0
$239$	3.62071	0.234204	0.117102	−	0.993120i	$-0.462639\pi$
$239$	3.62071	0.234204	0.117102	+	0.993120i	$0.462639\pi$
$240$	0	0
$241$	−7.66781	−0.493927	−0.246964	−	0.969025i	$-0.579433\pi$
$241$	−7.66781	−0.493927	−0.246964	+	0.969025i	$0.579433\pi$
$242$	0	0
$243$	−5.38568	−0.345491
$244$	0	0
$245$	0	0
$246$	0	0
$247$	4.61196	0.293452
$248$	0	0
$249$	−1.81311	−0.114901
$250$	0	0
$251$	6.04595	0.381617	0.190809	−	0.981627i	$-0.438889\pi$
$251$	6.04595	0.381617	0.190809	+	0.981627i	$0.438889\pi$
$252$	0	0
$253$	−16.6654	−1.04775
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−6.13054	−0.382412	−0.191206	−	0.981550i	$-0.561240\pi$
$257$	−6.13054	−0.382412	−0.191206	+	0.981550i	$0.561240\pi$
$258$	0	0
$259$	−0.211089	−0.0131164
$260$	0	0
$261$	10.9214	0.676019
$262$	0	0
$263$	12.3012	0.758525	0.379263	−	0.925289i	$-0.376178\pi$
$263$	12.3012	0.758525	0.379263	+	0.925289i	$0.376178\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−2.34726	−0.143650
$268$	0	0
$269$	9.92508	0.605143	0.302572	−	0.953127i	$-0.402155\pi$
$269$	9.92508	0.605143	0.302572	+	0.953127i	$0.402155\pi$
$270$	0	0
$271$	−20.3040	−1.23338	−0.616689	−	0.787207i	$-0.711526\pi$
$271$	−20.3040	−1.23338	−0.616689	+	0.787207i	$0.711526\pi$
$272$	0	0
$273$	0.616723	0.0373258
$274$	0	0
$275$	0	0
$276$	0	0
$277$	18.1671	1.09155	0.545776	−	0.837931i	$-0.316235\pi$
$277$	18.1671	1.09155	0.545776	+	0.837931i	$0.316235\pi$
$278$	0	0
$279$	−32.2265	−1.92935
$280$	0	0
$281$	10.6983	0.638208	0.319104	−	0.947720i	$-0.396618\pi$
$281$	10.6983	0.638208	0.319104	+	0.947720i	$0.396618\pi$
$282$	0	0
$283$	−10.0773	−0.599034	−0.299517	−	0.954091i	$-0.596826\pi$
$283$	−10.0773	−0.599034	−0.299517	+	0.954091i	$0.596826\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4.20632	0.248291
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	2.54996	0.149481
$292$	0	0
$293$	−20.5349	−1.19966	−0.599831	−	0.800127i	$-0.704765\pi$
$293$	−20.5349	−1.19966	−0.599831	+	0.800127i	$0.704765\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−4.45709	−0.258627
$298$	0	0
$299$	20.0965	1.16221
$300$	0	0
$301$	5.39093	0.310728
$302$	0	0
$303$	1.37892	0.0792169
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−13.1177	−0.748669	−0.374335	−	0.927294i	$-0.622129\pi$
$307$	−13.1177	−0.748669	−0.374335	+	0.927294i	$0.622129\pi$
$308$	0	0
$309$	1.77746	0.101116
$310$	0	0
$311$	12.3646	0.701132	0.350566	−	0.936538i	$-0.385989\pi$
$311$	12.3646	0.701132	0.350566	+	0.936538i	$0.385989\pi$
$312$	0	0
$313$	−5.16000	−0.291661	−0.145830	−	0.989310i	$-0.546585\pi$
$313$	−5.16000	−0.291661	−0.145830	+	0.989310i	$0.546585\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−23.7655	−1.33480	−0.667400	−	0.744699i	$-0.732593\pi$
$317$	−23.7655	−1.33480	−0.667400	+	0.744699i	$0.732593\pi$
$318$	0	0
$319$	13.5889	0.760830
$320$	0	0
$321$	1.64596	0.0918683
$322$	0	0
$323$	−1.03890	−0.0578060
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0.632531	0.0349790
$328$	0	0
$329$	3.03528	0.167340
$330$	0	0
$331$	−20.8341	−1.14514	−0.572572	−	0.819854i	$-0.694054\pi$
$331$	−20.8341	−1.14514	−0.572572	+	0.819854i	$0.694054\pi$
$332$	0	0
$333$	−0.913442	−0.0500563
$334$	0	0
$335$	0	0
$336$	0	0
$337$	14.3080	0.779406	0.389703	−	0.920941i	$-0.372578\pi$
$337$	14.3080	0.779406	0.389703	+	0.920941i	$0.372578\pi$
$338$	0	0
$339$	1.40865	0.0765076
$340$	0	0
$341$	−40.0975	−2.17140
$342$	0	0
$343$	9.25264	0.499596
$344$	0	0
$345$	0	0
$346$	0	0
$347$	21.9988	1.18096	0.590478	−	0.807054i	$-0.298939\pi$
$347$	21.9988	1.18096	0.590478	+	0.807054i	$0.298939\pi$
$348$	0	0
$349$	−6.06199	−0.324491	−0.162246	−	0.986750i	$-0.551874\pi$
$349$	−6.06199	−0.324491	−0.162246	+	0.986750i	$0.551874\pi$
$350$	0	0
$351$	5.37471	0.286881
$352$	0	0
$353$	−20.8785	−1.11125	−0.555626	−	0.831432i	$-0.687521\pi$
$353$	−20.8785	−1.11125	−0.555626	+	0.831432i	$0.687521\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−0.138925	−0.00735267
$358$	0	0
$359$	−18.4842	−0.975557	−0.487779	−	0.872967i	$-0.662193\pi$
$359$	−18.4842	−0.975557	−0.487779	+	0.872967i	$0.662193\pi$
$360$	0	0
$361$	−17.9207	−0.943194
$362$	0	0
$363$	−0.518598	−0.0272193
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−23.6044	−1.23214	−0.616070	−	0.787691i	$-0.711276\pi$
$367$	−23.6044	−1.23214	−0.616070	+	0.787691i	$0.711276\pi$
$368$	0	0
$369$	18.2020	0.947556
$370$	0	0
$371$	−7.83583	−0.406816
$372$	0	0
$373$	18.0123	0.932643	0.466321	−	0.884615i	$-0.345579\pi$
$373$	18.0123	0.932643	0.466321	+	0.884615i	$0.345579\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−16.3865	−0.843949
$378$	0	0
$379$	5.74523	0.295112	0.147556	−	0.989054i	$-0.452859\pi$
$379$	5.74523	0.295112	0.147556	+	0.989054i	$0.452859\pi$
$380$	0	0
$381$	4.29728	0.220156
$382$	0	0
$383$	−35.3281	−1.80518	−0.902591	−	0.430499i	$-0.858337\pi$
$383$	−35.3281	−1.80518	−0.902591	+	0.430499i	$0.858337\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	23.3281	1.18583
$388$	0	0
$389$	−18.2627	−0.925955	−0.462977	−	0.886370i	$-0.653219\pi$
$389$	−18.2627	−0.925955	−0.462977	+	0.886370i	$0.653219\pi$
$390$	0	0
$391$	−4.52699	−0.228940
$392$	0	0
$393$	1.77012	0.0892907
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−4.64092	−0.232921	−0.116461	−	0.993195i	$-0.537155\pi$
$397$	−4.64092	−0.232921	−0.116461	+	0.993195i	$0.537155\pi$
$398$	0	0
$399$	0.144329	0.00722548
$400$	0	0
$401$	−27.4049	−1.36853	−0.684267	−	0.729232i	$-0.739878\pi$
$401$	−27.4049	−1.36853	−0.684267	+	0.729232i	$0.739878\pi$
$402$	0	0
$403$	48.3528	2.40862
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.13654	−0.0563363
$408$	0	0
$409$	1.38254	0.0683623	0.0341811	−	0.999416i	$-0.489118\pi$
$409$	1.38254	0.0683623	0.0341811	+	0.999416i	$0.489118\pi$
$410$	0	0
$411$	−0.0495740	−0.00244531
$412$	0	0
$413$	−1.36747	−0.0672888
$414$	0	0
$415$	0	0
$416$	0	0
$417$	2.57809	0.126250
$418$	0	0
$419$	33.2300	1.62339	0.811695	−	0.584081i	$-0.198545\pi$
$419$	33.2300	1.62339	0.811695	+	0.584081i	$0.198545\pi$
$420$	0	0
$421$	4.61109	0.224731	0.112365	−	0.993667i	$-0.464157\pi$
$421$	4.61109	0.224731	0.112365	+	0.993667i	$0.464157\pi$
$422$	0	0
$423$	13.1345	0.638622
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−6.79693	−0.328927
$428$	0	0
$429$	3.32055	0.160318
$430$	0	0
$431$	−7.33812	−0.353465	−0.176732	−	0.984259i	$-0.556553\pi$
$431$	−7.33812	−0.353465	−0.176732	+	0.984259i	$0.556553\pi$
$432$	0	0
$433$	−15.3487	−0.737610	−0.368805	−	0.929507i	$-0.620233\pi$
$433$	−15.3487	−0.737610	−0.368805	+	0.929507i	$0.620233\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4.70309	0.224979
$438$	0	0
$439$	5.60355	0.267443	0.133721	−	0.991019i	$-0.457307\pi$
$439$	5.60355	0.267443	0.133721	+	0.991019i	$0.457307\pi$
$440$	0	0
$441$	19.3278	0.920373
$442$	0	0
$443$	29.6338	1.40794	0.703971	−	0.710228i	$-0.251408\pi$
$443$	29.6338	1.40794	0.703971	+	0.710228i	$0.251408\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−2.44031	−0.115423
$448$	0	0
$449$	−34.4203	−1.62439	−0.812197	−	0.583383i	$-0.801729\pi$
$449$	−34.4203	−1.62439	−0.812197	+	0.583383i	$0.801729\pi$
$450$	0	0
$451$	22.6476	1.06643
$452$	0	0
$453$	1.81979	0.0855013
$454$	0	0
$455$	0	0
$456$	0	0
$457$	9.32504	0.436207	0.218103	−	0.975926i	$-0.430013\pi$
$457$	9.32504	0.436207	0.218103	+	0.975926i	$0.430013\pi$
$458$	0	0
$459$	−1.21072	−0.0565116
$460$	0	0
$461$	19.1715	0.892904	0.446452	−	0.894808i	$-0.352687\pi$
$461$	19.1715	0.892904	0.446452	+	0.894808i	$0.352687\pi$
$462$	0	0
$463$	−19.6385	−0.912680	−0.456340	−	0.889805i	$-0.650840\pi$
$463$	−19.6385	−0.912680	−0.456340	+	0.889805i	$0.650840\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	21.1027	0.976515	0.488258	−	0.872699i	$-0.337633\pi$
$467$	21.1027	0.976515	0.488258	+	0.872699i	$0.337633\pi$
$468$	0	0
$469$	6.26946	0.289497
$470$	0	0
$471$	4.04595	0.186428
$472$	0	0
$473$	29.0257	1.33461
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−33.9079	−1.55254
$478$	0	0
$479$	28.6762	1.31025	0.655125	−	0.755521i	$-0.272616\pi$
$479$	28.6762	1.31025	0.655125	+	0.755521i	$0.272616\pi$
$480$	0	0
$481$	1.37053	0.0624909
$482$	0	0
$483$	0.628909	0.0286164
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−39.2139	−1.77695	−0.888475	−	0.458925i	$-0.848234\pi$
$487$	−39.2139	−1.77695	−0.888475	+	0.458925i	$0.848234\pi$
$488$	0	0
$489$	4.55157	0.205829
$490$	0	0
$491$	36.3499	1.64045	0.820224	−	0.572042i	$-0.193849\pi$
$491$	36.3499	1.64045	0.820224	+	0.572042i	$0.193849\pi$
$492$	0	0
$493$	3.69127	0.166246
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−6.40839	−0.287455
$498$	0	0
$499$	8.52061	0.381435	0.190718	−	0.981645i	$-0.438919\pi$
$499$	8.52061	0.381435	0.190718	+	0.981645i	$0.438919\pi$
$500$	0	0
$501$	−0.930856	−0.0415876
$502$	0	0
$503$	−18.6790	−0.832854	−0.416427	−	0.909169i	$-0.636718\pi$
$503$	−18.6790	−0.832854	−0.416427	+	0.909169i	$0.636718\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−1.36278	−0.0605231
$508$	0	0
$509$	3.53567	0.156716	0.0783579	−	0.996925i	$-0.475032\pi$
$509$	3.53567	0.156716	0.0783579	+	0.996925i	$0.475032\pi$
$510$	0	0
$511$	1.55171	0.0686436
$512$	0	0
$513$	1.25782	0.0555341
$514$	0	0
$515$	0	0
$516$	0	0
$517$	16.3425	0.718742
$518$	0	0
$519$	1.64532	0.0722215
$520$	0	0
$521$	−4.64206	−0.203373	−0.101686	−	0.994817i	$-0.532424\pi$
$521$	−4.64206	−0.203373	−0.101686	+	0.994817i	$0.532424\pi$
$522$	0	0
$523$	−18.3331	−0.801649	−0.400824	−	0.916155i	$-0.631276\pi$
$523$	−18.3331	−0.801649	−0.400824	+	0.916155i	$0.631276\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−10.8921	−0.474466
$528$	0	0
$529$	−2.50639	−0.108974
$530$	0	0
$531$	−5.91743	−0.256795
$532$	0	0
$533$	−27.3103	−1.18294
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0.895308	0.0386354
$538$	0	0
$539$	24.0485	1.03584
$540$	0	0
$541$	22.4428	0.964891	0.482445	−	0.875926i	$-0.339749\pi$
$541$	22.4428	0.964891	0.482445	+	0.875926i	$0.339749\pi$
$542$	0	0
$543$	1.20636	0.0517700
$544$	0	0
$545$	0	0
$546$	0	0
$547$	25.9778	1.11073	0.555365	−	0.831607i	$-0.312578\pi$
$547$	25.9778	1.11073	0.555365	+	0.831607i	$0.312578\pi$
$548$	0	0
$549$	−29.4123	−1.25529
$550$	0	0
$551$	−3.83486	−0.163371
$552$	0	0
$553$	5.07807	0.215942
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−4.86144	−0.205986	−0.102993	−	0.994682i	$-0.532842\pi$
$557$	−4.86144	−0.205986	−0.102993	+	0.994682i	$0.532842\pi$
$558$	0	0
$559$	−35.0015	−1.48041
$560$	0	0
$561$	−0.747995	−0.0315804
$562$	0	0
$563$	1.53365	0.0646357	0.0323179	−	0.999478i	$-0.489711\pi$
$563$	1.53365	0.0646357	0.0323179	+	0.999478i	$0.489711\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−5.90073	−0.247807
$568$	0	0
$569$	−16.5770	−0.694946	−0.347473	−	0.937690i	$-0.612960\pi$
$569$	−16.5770	−0.694946	−0.347473	+	0.937690i	$0.612960\pi$
$570$	0	0
$571$	14.2904	0.598036	0.299018	−	0.954248i	$-0.403341\pi$
$571$	14.2904	0.598036	0.299018	+	0.954248i	$0.403341\pi$
$572$	0	0
$573$	−2.01104	−0.0840125
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−28.5424	−1.18824	−0.594119	−	0.804377i	$-0.702499\pi$
$577$	−28.5424	−1.18824	−0.594119	+	0.804377i	$0.702499\pi$
$578$	0	0
$579$	1.72215	0.0715702
$580$	0	0
$581$	−6.10126	−0.253123
$582$	0	0
$583$	−42.1895	−1.74731
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−33.5281	−1.38385	−0.691927	−	0.721967i	$-0.743238\pi$
$587$	−33.5281	−1.38385	−0.691927	+	0.721967i	$0.743238\pi$
$588$	0	0
$589$	11.3158	0.466259
$590$	0	0
$591$	−2.75075	−0.113151
$592$	0	0
$593$	−42.4729	−1.74415	−0.872077	−	0.489368i	$-0.837227\pi$
$593$	−42.4729	−1.74415	−0.872077	+	0.489368i	$0.837227\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−1.85993	−0.0761219
$598$	0	0
$599$	7.00705	0.286300	0.143150	−	0.989701i	$-0.454277\pi$
$599$	7.00705	0.286300	0.143150	+	0.989701i	$0.454277\pi$
$600$	0	0
$601$	39.3146	1.60368	0.801839	−	0.597541i	$-0.203855\pi$
$601$	39.3146	1.60368	0.801839	+	0.597541i	$0.203855\pi$
$602$	0	0
$603$	27.1298	1.10481
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−18.7928	−0.762777	−0.381389	−	0.924415i	$-0.624554\pi$
$607$	−18.7928	−0.762777	−0.381389	+	0.924415i	$0.624554\pi$
$608$	0	0
$609$	−0.512808	−0.0207800
$610$	0	0
$611$	−19.7071	−0.797263
$612$	0	0
$613$	1.83560	0.0741392	0.0370696	−	0.999313i	$-0.488198\pi$
$613$	1.83560	0.0741392	0.0370696	+	0.999313i	$0.488198\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−29.0011	−1.16754	−0.583771	−	0.811918i	$-0.698423\pi$
$617$	−29.0011	−1.16754	−0.583771	+	0.811918i	$0.698423\pi$
$618$	0	0
$619$	−27.7941	−1.11714	−0.558569	−	0.829458i	$-0.688649\pi$
$619$	−27.7941	−1.11714	−0.558569	+	0.829458i	$0.688649\pi$
$620$	0	0
$621$	5.48092	0.219942
$622$	0	0
$623$	−7.89874	−0.316456
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0.777092	0.0310341
$628$	0	0
$629$	−0.308729	−0.0123098
$630$	0	0
$631$	14.1785	0.564437	0.282219	−	0.959350i	$-0.408930\pi$
$631$	14.1785	0.564437	0.282219	+	0.959350i	$0.408930\pi$
$632$	0	0
$633$	1.56953	0.0623833
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−28.9995	−1.14900
$638$	0	0
$639$	−27.7309	−1.09702
$640$	0	0
$641$	−27.8734	−1.10093	−0.550466	−	0.834857i	$-0.685550\pi$
$641$	−27.8734	−1.10093	−0.550466	+	0.834857i	$0.685550\pi$
$642$	0	0
$643$	25.8277	1.01854	0.509272	−	0.860605i	$-0.329915\pi$
$643$	25.8277	1.01854	0.509272	+	0.860605i	$0.329915\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	4.40912	0.173340	0.0866702	−	0.996237i	$-0.472377\pi$
$647$	4.40912	0.173340	0.0866702	+	0.996237i	$0.472377\pi$
$648$	0	0
$649$	−7.36270	−0.289011
$650$	0	0
$651$	1.51318	0.0593060
$652$	0	0
$653$	30.9245	1.21017	0.605084	−	0.796161i	$-0.293139\pi$
$653$	30.9245	1.21017	0.605084	+	0.796161i	$0.293139\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	6.71469	0.261965
$658$	0	0
$659$	−39.7992	−1.55036	−0.775179	−	0.631742i	$-0.782340\pi$
$659$	−39.7992	−1.55036	−0.775179	+	0.631742i	$0.782340\pi$
$660$	0	0
$661$	−13.0969	−0.509409	−0.254704	−	0.967019i	$-0.581978\pi$
$661$	−13.0969	−0.509409	−0.254704	+	0.967019i	$0.581978\pi$
$662$	0	0
$663$	0.901992	0.0350305
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−16.7103	−0.647027
$668$	0	0
$669$	−2.78757	−0.107774
$670$	0	0
$671$	−36.5959	−1.41277
$672$	0	0
$673$	22.2830	0.858946	0.429473	−	0.903080i	$-0.358699\pi$
$673$	22.2830	0.858946	0.429473	+	0.903080i	$0.358699\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−10.5576	−0.405762	−0.202881	−	0.979203i	$-0.565030\pi$
$677$	−10.5576	−0.405762	−0.202881	+	0.979203i	$0.565030\pi$
$678$	0	0
$679$	8.58084	0.329303
$680$	0	0
$681$	−0.365822	−0.0140183
$682$	0	0
$683$	−1.80882	−0.0692128	−0.0346064	−	0.999401i	$-0.511018\pi$
$683$	−1.80882	−0.0692128	−0.0346064	+	0.999401i	$0.511018\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0.660022	0.0251814
$688$	0	0
$689$	50.8755	1.93820
$690$	0	0
$691$	−3.84649	−0.146327	−0.0731636	−	0.997320i	$-0.523310\pi$
$691$	−3.84649	−0.146327	−0.0731636	+	0.997320i	$0.523310\pi$
$692$	0	0
$693$	−7.44729	−0.282899
$694$	0	0
$695$	0	0
$696$	0	0
$697$	6.15198	0.233023
$698$	0	0
$699$	−1.09365	−0.0413657
$700$	0	0
$701$	43.9484	1.65991	0.829955	−	0.557830i	$-0.188366\pi$
$701$	43.9484	1.65991	0.829955	+	0.557830i	$0.188366\pi$
$702$	0	0
$703$	0.320739	0.0120969
$704$	0	0
$705$	0	0
$706$	0	0
$707$	4.64018	0.174512
$708$	0	0
$709$	45.9706	1.72646	0.863232	−	0.504808i	$-0.168437\pi$
$709$	45.9706	1.72646	0.863232	+	0.504808i	$0.168437\pi$
$710$	0	0
$711$	21.9743	0.824100
$712$	0	0
$713$	49.3083	1.84661
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−0.735674	−0.0274742
$718$	0	0
$719$	−6.24809	−0.233014	−0.116507	−	0.993190i	$-0.537170\pi$
$719$	−6.24809	−0.233014	−0.116507	+	0.993190i	$0.537170\pi$
$720$	0	0
$721$	5.98131	0.222755
$722$	0	0
$723$	1.55798	0.0579420
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−49.9606	−1.85294	−0.926469	−	0.376372i	$-0.877171\pi$
$727$	−49.9606	−1.85294	−0.926469	+	0.376372i	$0.877171\pi$
$728$	0	0
$729$	−24.7962	−0.918376
$730$	0	0
$731$	7.88454	0.291620
$732$	0	0
$733$	−37.4111	−1.38181	−0.690906	−	0.722945i	$-0.742788\pi$
$733$	−37.4111	−1.38181	−0.690906	+	0.722945i	$0.742788\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	33.7559	1.24342
$738$	0	0
$739$	−38.9348	−1.43224	−0.716120	−	0.697978i	$-0.754084\pi$
$739$	−38.9348	−1.43224	−0.716120	+	0.697978i	$0.754084\pi$
$740$	0	0
$741$	−0.937080	−0.0344245
$742$	0	0
$743$	−13.0253	−0.477850	−0.238925	−	0.971038i	$-0.576795\pi$
$743$	−13.0253	−0.477850	−0.238925	+	0.971038i	$0.576795\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−26.4019	−0.965996
$748$	0	0
$749$	5.53878	0.202383
$750$	0	0
$751$	42.3558	1.54559	0.772793	−	0.634659i	$-0.218859\pi$
$751$	42.3558	1.54559	0.772793	+	0.634659i	$0.218859\pi$
$752$	0	0
$753$	−1.22845	−0.0447671
$754$	0	0
$755$	0	0
$756$	0	0
$757$	4.20834	0.152955	0.0764773	−	0.997071i	$-0.475633\pi$
$757$	4.20834	0.152955	0.0764773	+	0.997071i	$0.475633\pi$
$758$	0	0
$759$	3.38616	0.122910
$760$	0	0
$761$	31.8563	1.15479	0.577395	−	0.816465i	$-0.304069\pi$
$761$	31.8563	1.15479	0.577395	+	0.816465i	$0.304069\pi$
$762$	0	0
$763$	2.12852	0.0770576
$764$	0	0
$765$	0	0
$766$	0	0
$767$	8.87853	0.320585
$768$	0	0
$769$	−7.43010	−0.267936	−0.133968	−	0.990986i	$-0.542772\pi$
$769$	−7.43010	−0.267936	−0.133968	+	0.990986i	$0.542772\pi$
$770$	0	0
$771$	1.24563	0.0448604
$772$	0	0
$773$	7.33384	0.263780	0.131890	−	0.991264i	$-0.457895\pi$
$773$	7.33384	0.263780	0.131890	+	0.991264i	$0.457895\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0.0428901	0.00153867
$778$	0	0
$779$	−6.39130	−0.228992
$780$	0	0
$781$	−34.5039	−1.23465
$782$	0	0
$783$	−4.46910	−0.159713
$784$	0	0
$785$	0	0
$786$	0	0
$787$	35.7257	1.27348	0.636742	−	0.771077i	$-0.280282\pi$
$787$	35.7257	1.27348	0.636742	+	0.771077i	$0.280282\pi$
$788$	0	0
$789$	−2.49942	−0.0889817
$790$	0	0
$791$	4.74024	0.168544
$792$	0	0
$793$	44.1303	1.56711
$794$	0	0
$795$	0	0
$796$	0	0
$797$	28.2025	0.998985	0.499492	−	0.866318i	$-0.333520\pi$
$797$	28.2025	0.998985	0.499492	+	0.866318i	$0.333520\pi$
$798$	0	0
$799$	4.43927	0.157050
$800$	0	0
$801$	−34.1801	−1.20769
$802$	0	0
$803$	8.35468	0.294830
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−2.01663	−0.0709886
$808$	0	0
$809$	−41.9318	−1.47424	−0.737121	−	0.675761i	$-0.763815\pi$
$809$	−41.9318	−1.47424	−0.737121	+	0.675761i	$0.763815\pi$
$810$	0	0
$811$	−4.46604	−0.156824	−0.0784119	−	0.996921i	$-0.524985\pi$
$811$	−4.46604	−0.156824	−0.0784119	+	0.996921i	$0.524985\pi$
$812$	0	0
$813$	4.12546	0.144686
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−8.19125	−0.286576
$818$	0	0
$819$	8.98053	0.313805
$820$	0	0
$821$	19.6083	0.684334	0.342167	−	0.939639i	$-0.388839\pi$
$821$	19.6083	0.684334	0.342167	+	0.939639i	$0.388839\pi$
$822$	0	0
$823$	−46.9128	−1.63528	−0.817638	−	0.575732i	$-0.804717\pi$
$823$	−46.9128	−1.63528	−0.817638	+	0.575732i	$0.804717\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−14.0134	−0.487295	−0.243648	−	0.969864i	$-0.578344\pi$
$827$	−14.0134	−0.487295	−0.243648	+	0.969864i	$0.578344\pi$
$828$	0	0
$829$	−40.5206	−1.40734	−0.703669	−	0.710528i	$-0.748456\pi$
$829$	−40.5206	−1.40734	−0.703669	+	0.710528i	$0.748456\pi$
$830$	0	0
$831$	−3.69127	−0.128049
$832$	0	0
$833$	6.53251	0.226338
$834$	0	0
$835$	0	0
$836$	0	0
$837$	13.1873	0.455818
$838$	0	0
$839$	−8.68840	−0.299957	−0.149978	−	0.988689i	$-0.547920\pi$
$839$	−8.68840	−0.299957	−0.149978	+	0.988689i	$0.547920\pi$
$840$	0	0
$841$	−15.3745	−0.530156
$842$	0	0
$843$	−2.17374	−0.0748675
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−1.74513	−0.0599633
$848$	0	0
$849$	2.04756	0.0702720
$850$	0	0
$851$	1.39761	0.0479096
$852$	0	0
$853$	−23.1523	−0.792721	−0.396361	−	0.918095i	$-0.629727\pi$
$853$	−23.1523	−0.792721	−0.396361	+	0.918095i	$0.629727\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	43.6043	1.48950	0.744748	−	0.667346i	$-0.232570\pi$
$857$	43.6043	1.48950	0.744748	+	0.667346i	$0.232570\pi$
$858$	0	0
$859$	14.2532	0.486314	0.243157	−	0.969987i	$-0.421817\pi$
$859$	14.2532	0.486314	0.243157	+	0.969987i	$0.421817\pi$
$860$	0	0
$861$	−0.854661	−0.0291268
$862$	0	0
$863$	24.1162	0.820926	0.410463	−	0.911877i	$-0.365367\pi$
$863$	24.1162	0.820926	0.410463	+	0.911877i	$0.365367\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−0.203185	−0.00690052
$868$	0	0
$869$	27.3413	0.927489
$870$	0	0
$871$	−40.7056	−1.37926
$872$	0	0
$873$	37.1318	1.25672
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−1.56540	−0.0528599	−0.0264299	−	0.999651i	$-0.508414\pi$
$877$	−1.56540	−0.0528599	−0.0264299	+	0.999651i	$0.508414\pi$
$878$	0	0
$879$	4.17238	0.140731
$880$	0	0
$881$	−39.0973	−1.31722	−0.658610	−	0.752484i	$-0.728855\pi$
$881$	−39.0973	−1.31722	−0.658610	+	0.752484i	$0.728855\pi$
$882$	0	0
$883$	30.9249	1.04071	0.520354	−	0.853951i	$-0.325800\pi$
$883$	30.9249	1.04071	0.520354	+	0.853951i	$0.325800\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	45.1839	1.51713	0.758564	−	0.651599i	$-0.225901\pi$
$887$	45.1839	1.51713	0.758564	+	0.651599i	$0.225901\pi$
$888$	0	0
$889$	14.4607	0.484997
$890$	0	0
$891$	−31.7706	−1.06436
$892$	0	0
$893$	−4.61196	−0.154333
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−4.08330	−0.136338
$898$	0	0
$899$	−40.2056	−1.34093
$900$	0	0
$901$	−11.4603	−0.381799
$902$	0	0
$903$	−1.09536	−0.0364511
$904$	0	0
$905$	0	0
$906$	0	0
$907$	33.9862	1.12849	0.564246	−	0.825606i	$-0.309167\pi$
$907$	33.9862	1.12849	0.564246	+	0.825606i	$0.309167\pi$
$908$	0	0
$909$	20.0794	0.665992
$910$	0	0
$911$	−22.1997	−0.735509	−0.367754	−	0.929923i	$-0.619873\pi$
$911$	−22.1997	−0.735509	−0.367754	+	0.929923i	$0.619873\pi$
$912$	0	0
$913$	−32.8503	−1.08719
$914$	0	0
$915$	0	0
$916$	0	0
$917$	5.95660	0.196704
$918$	0	0
$919$	8.84668	0.291825	0.145913	−	0.989297i	$-0.453388\pi$
$919$	8.84668	0.291825	0.145913	+	0.989297i	$0.453388\pi$
$920$	0	0
$921$	2.66533	0.0878256
$922$	0	0
$923$	41.6076	1.36953
$924$	0	0
$925$	0	0
$926$	0	0
$927$	25.8828	0.850104
$928$	0	0
$929$	13.9941	0.459131	0.229566	−	0.973293i	$-0.426269\pi$
$929$	13.9941	0.459131	0.229566	+	0.973293i	$0.426269\pi$
$930$	0	0
$931$	−6.78663	−0.222423
$932$	0	0
$933$	−2.51230	−0.0822490
$934$	0	0
$935$	0	0
$936$	0	0
$937$	5.19129	0.169592	0.0847960	−	0.996398i	$-0.472976\pi$
$937$	5.19129	0.169592	0.0847960	+	0.996398i	$0.472976\pi$
$938$	0	0
$939$	1.04843	0.0342144
$940$	0	0
$941$	−26.2746	−0.856527	−0.428264	−	0.903654i	$-0.640875\pi$
$941$	−26.2746	−0.856527	−0.428264	+	0.903654i	$0.640875\pi$
$942$	0	0
$943$	−27.8499	−0.906919
$944$	0	0
$945$	0	0
$946$	0	0
$947$	13.9416	0.453040	0.226520	−	0.974007i	$-0.427265\pi$
$947$	13.9416	0.453040	0.226520	+	0.974007i	$0.427265\pi$
$948$	0	0
$949$	−10.0747	−0.327040
$950$	0	0
$951$	4.82878	0.156584
$952$	0	0
$953$	−19.3298	−0.626154	−0.313077	−	0.949728i	$-0.601360\pi$
$953$	−19.3298	−0.626154	−0.313077	+	0.949728i	$0.601360\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−2.76105	−0.0892521
$958$	0	0
$959$	−0.166821	−0.00538692
$960$	0	0
$961$	87.6372	2.82701
$962$	0	0
$963$	23.9679	0.772355
$964$	0	0
$965$	0	0
$966$	0	0
$967$	27.0663	0.870393	0.435197	−	0.900335i	$-0.356679\pi$
$967$	27.0663	0.870393	0.435197	+	0.900335i	$0.356679\pi$
$968$	0	0
$969$	0.211089	0.00678115
$970$	0	0
$971$	−9.36344	−0.300487	−0.150244	−	0.988649i	$-0.548006\pi$
$971$	−9.36344	−0.300487	−0.150244	+	0.988649i	$0.548006\pi$
$972$	0	0
$973$	8.67550	0.278124
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−14.1127	−0.451506	−0.225753	−	0.974185i	$-0.572484\pi$
$977$	−14.1127	−0.451506	−0.225753	+	0.974185i	$0.572484\pi$
$978$	0	0
$979$	−42.5282	−1.35921
$980$	0	0
$981$	9.21072	0.294076
$982$	0	0
$983$	−48.9970	−1.56276	−0.781381	−	0.624054i	$-0.785485\pi$
$983$	−48.9970	−1.56276	−0.781381	+	0.624054i	$0.785485\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−0.616723	−0.0196305
$988$	0	0
$989$	−35.6932	−1.13498
$990$	0	0
$991$	20.7542	0.659279	0.329639	−	0.944107i	$-0.393073\pi$
$991$	20.7542	0.659279	0.329639	+	0.944107i	$0.393073\pi$
$992$	0	0
$993$	4.23317	0.134336
$994$	0	0
$995$	0	0
$996$	0	0
$997$	9.54268	0.302220	0.151110	−	0.988517i	$-0.451715\pi$
$997$	9.54268	0.302220	0.151110	+	0.988517i	$0.451715\pi$
$998$	0	0
$999$	0.373785	0.0118260

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6800.2.a.bt.1.3		4
4.3	odd	2		425.2.a.h.1.4		4
5.2	odd	4		1360.2.e.d.1089.5		8
5.3	odd	4		1360.2.e.d.1089.4		8
5.4	even	2		6800.2.a.bw.1.2		4
12.11	even	2		3825.2.a.bh.1.1		4
20.3	even	4		85.2.b.a.69.1	✓	8
20.7	even	4		85.2.b.a.69.8	yes	8
20.19	odd	2		425.2.a.g.1.1		4
60.23	odd	4		765.2.b.c.154.8		8
60.47	odd	4		765.2.b.c.154.1		8
60.59	even	2		3825.2.a.bj.1.4		4
68.67	odd	2		7225.2.a.w.1.4		4
340.67	even	4		1445.2.b.e.579.8		8
340.203	even	4		1445.2.b.e.579.1		8
340.339	odd	2		7225.2.a.v.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.2.b.a.69.1	✓	8	20.3	even	4
85.2.b.a.69.8	yes	8	20.7	even	4
425.2.a.g.1.1		4	20.19	odd	2
425.2.a.h.1.4		4	4.3	odd	2
765.2.b.c.154.1		8	60.47	odd	4
765.2.b.c.154.8		8	60.23	odd	4
1360.2.e.d.1089.4		8	5.3	odd	4
1360.2.e.d.1089.5		8	5.2	odd	4
1445.2.b.e.579.1		8	340.203	even	4
1445.2.b.e.579.8		8	340.67	even	4
3825.2.a.bh.1.1		4	12.11	even	2
3825.2.a.bj.1.4		4	60.59	even	2
6800.2.a.bt.1.3		4	1.1	even	1	trivial
6800.2.a.bw.1.2		4	5.4	even	2
7225.2.a.v.1.1		4	340.339	odd	2
7225.2.a.w.1.4		4	68.67	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:	\( N \)	\(=\)	\( 6800 = 2^{4} \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6800.a (trivial)

\(f(q)\)	\(=\)	\(q-0.203185 q^{3} -0.683735 q^{7} -2.95872 q^{9} -3.68135 q^{11} +4.43927 q^{13} -1.00000 q^{17} +1.03890 q^{19} +0.138925 q^{21} +4.52699 q^{23} +1.21072 q^{27} -3.69127 q^{29} +10.8921 q^{31} +0.747995 q^{33} +0.308729 q^{37} -0.901992 q^{39} -6.15198 q^{41} -7.88454 q^{43} -4.43927 q^{47} -6.53251 q^{49} +0.203185 q^{51} +11.4603 q^{53} -0.211089 q^{57} +2.00000 q^{59} +9.94089 q^{61} +2.02298 q^{63} -9.16944 q^{67} -0.919815 q^{69} +9.37262 q^{71} -2.26946 q^{73} +2.51707 q^{77} -7.42696 q^{79} +8.63015 q^{81} +8.92344 q^{83} +0.750010 q^{87} +11.5523 q^{89} -3.03528 q^{91} -2.21310 q^{93} -12.5500 q^{97} +10.8921 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} - 10 q^{7} + 4 q^{9} + 2 q^{11} + 6 q^{13} - 4 q^{17} - 4 q^{19} + 12 q^{21} - 4 q^{23} - 10 q^{27} - 4 q^{29} + 12 q^{31} + 2 q^{33} + 12 q^{37} + 22 q^{39} - 6 q^{41} - 18 q^{43} - 6 q^{47}+ \cdots + 12 q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 - 10 * q^7 + 4 * q^9 + 2 * q^11 + 6 * q^13 - 4 * q^17 - 4 * q^19 + 12 * q^21 - 4 * q^23 - 10 * q^27 - 4 * q^29 + 12 * q^31 + 2 * q^33 + 12 * q^37 + 22 * q^39 - 6 * q^41 - 18 * q^43 - 6 * q^47 + 8 * q^49 + 4 * q^51 + 8 * q^53 - 16 * q^57 + 8 * q^59 + 6 * q^61 - 16 * q^63 - 6 * q^67 + 4 * q^69 + 10 * q^71 + 2 * q^73 - 18 * q^77 + 12 * q^79 - 4 * q^81 + 14 * q^83 - 4 * q^87 + 24 * q^89 - 18 * q^91 - 18 * q^93 - 4 * q^97 + 12 * q^99

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