LMFDB - Embedded newform 6800.2.a.bz.1.4

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 = [5,0,-1,0,0,0,-1,0,6,0,-4,0,-3] 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:	$5$
Coefficient field:	5.5.1893456.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 10x^{3} + 10x^{2} + 23x - 25 $ x^5 - x^4 - 10x^3 + 10x^2 + 23*x - 25
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 425)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.96189$ 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)

$f(q)$	$=$	$q+1.96189 q^{3} +1.54475 q^{7} +0.849020 q^{9} -4.56006 q^{11} -1.09756 q^{13} +1.00000 q^{17} -4.67524 q^{19} +3.03063 q^{21} -0.529434 q^{23} -4.21999 q^{27} +8.06670 q^{29} +4.78005 q^{31} -8.94634 q^{33} -5.27917 q^{37} -2.15329 q^{39} -0.751460 q^{41} -9.49340 q^{43} -10.7419 q^{47} -4.61376 q^{49} +1.96189 q^{51} +0.0227951 q^{53} -9.17232 q^{57} +3.56962 q^{59} +3.92378 q^{61} +1.31152 q^{63} +9.75929 q^{67} -1.03869 q^{69} -1.21216 q^{71} +10.1135 q^{73} -7.04414 q^{77} -14.4151 q^{79} -10.8262 q^{81} +5.08949 q^{83} +15.8260 q^{87} -17.6123 q^{89} -1.69545 q^{91} +9.37794 q^{93} -6.78753 q^{97} -3.87158 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - q^{3} - q^{7} + 6 q^{9} - 4 q^{11} - 3 q^{13} + 5 q^{17} - 6 q^{19} - 5 q^{21} - 4 q^{23} + 5 q^{27} + 2 q^{29} - 21 q^{31} + 12 q^{33} - 2 q^{37} - 23 q^{39} - 8 q^{41} + 4 q^{43} + 2 q^{47} + 10 q^{49}+ \cdots + 14 q^{99}+O(q^{100}) $ 5 * q - q^3 - q^7 + 6 * q^9 - 4 * q^11 - 3 * q^13 + 5 * q^17 - 6 * q^19 - 5 * q^21 - 4 * q^23 + 5 * q^27 + 2 * q^29 - 21 * q^31 + 12 * q^33 - 2 * q^37 - 23 * q^39 - 8 * q^41 + 4 * q^43 + 2 * q^47 + 10 * q^49 - q^51 + 21 * q^53 - 20 * q^57 - 12 * q^59 - 2 * q^61 - 8 * q^63 + 12 * q^67 - 21 * q^71 + 22 * q^73 - 32 * q^77 - 41 * q^79 - 35 * q^81 + 8 * q^83 + 4 * q^87 - 10 * q^89 - 13 * q^91 + 9 * q^93 - 20 * q^97 + 14 * 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
$3$	1.96189	1.13270	0.566349	−	0.824165i	$-0.308355\pi$
$3$	1.96189	1.13270	0.566349	+	0.824165i	$0.308355\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.54475	0.583859	0.291930	−	0.956440i	$-0.405703\pi$
$7$	1.54475	0.583859	0.291930	+	0.956440i	$0.405703\pi$
$8$	0	0
$9$	0.849020	0.283007
$10$	0	0
$11$	−4.56006	−1.37491	−0.687455	−	0.726227i	$-0.741272\pi$
$11$	−4.56006	−1.37491	−0.687455	+	0.726227i	$0.741272\pi$
$12$	0	0
$13$	−1.09756	−0.304408	−0.152204	−	0.988349i	$-0.548637\pi$
$13$	−1.09756	−0.304408	−0.152204	+	0.988349i	$0.548637\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−4.67524	−1.07257	−0.536287	−	0.844036i	$-0.680174\pi$
$19$	−4.67524	−1.07257	−0.536287	+	0.844036i	$0.680174\pi$
$20$	0	0
$21$	3.03063	0.661337
$22$	0	0
$23$	−0.529434	−0.110395	−0.0551973	−	0.998475i	$-0.517579\pi$
$23$	−0.529434	−0.110395	−0.0551973	+	0.998475i	$0.517579\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−4.21999	−0.812138
$28$	0	0
$29$	8.06670	1.49795	0.748974	−	0.662599i	$-0.230547\pi$
$29$	8.06670	1.49795	0.748974	+	0.662599i	$0.230547\pi$
$30$	0	0
$31$	4.78005	0.858522	0.429261	−	0.903180i	$-0.358774\pi$
$31$	4.78005	0.858522	0.429261	+	0.903180i	$0.358774\pi$
$32$	0	0
$33$	−8.94634	−1.55736
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−5.27917	−0.867889	−0.433945	−	0.900939i	$-0.642879\pi$
$37$	−5.27917	−0.867889	−0.433945	+	0.900939i	$0.642879\pi$
$38$	0	0
$39$	−2.15329	−0.344803
$40$	0	0
$41$	−0.751460	−0.117358	−0.0586792	−	0.998277i	$-0.518689\pi$
$41$	−0.751460	−0.117358	−0.0586792	+	0.998277i	$0.518689\pi$
$42$	0	0
$43$	−9.49340	−1.44773	−0.723865	−	0.689941i	$-0.757636\pi$
$43$	−9.49340	−1.44773	−0.723865	+	0.689941i	$0.757636\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−10.7419	−1.56687	−0.783437	−	0.621472i	$-0.786535\pi$
$47$	−10.7419	−1.56687	−0.783437	+	0.621472i	$0.786535\pi$
$48$	0	0
$49$	−4.61376	−0.659108
$50$	0	0
$51$	1.96189	0.274720
$52$	0	0
$53$	0.0227951	0.00313115	0.00156557	−	0.999999i	$-0.499502\pi$
$53$	0.0227951	0.00313115	0.00156557	+	0.999999i	$0.499502\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−9.17232	−1.21490
$58$	0	0
$59$	3.56962	0.464725	0.232362	−	0.972629i	$-0.425354\pi$
$59$	3.56962	0.464725	0.232362	+	0.972629i	$0.425354\pi$
$60$	0	0
$61$	3.92378	0.502389	0.251195	−	0.967937i	$-0.419177\pi$
$61$	3.92378	0.502389	0.251195	+	0.967937i	$0.419177\pi$
$62$	0	0
$63$	1.31152	0.165236
$64$	0	0
$65$	0	0
$66$	0	0
$67$	9.75929	1.19229	0.596144	−	0.802878i	$-0.296699\pi$
$67$	9.75929	1.19229	0.596144	+	0.802878i	$0.296699\pi$
$68$	0	0
$69$	−1.03869	−0.125044
$70$	0	0
$71$	−1.21216	−0.143857	−0.0719285	−	0.997410i	$-0.522915\pi$
$71$	−1.21216	−0.143857	−0.0719285	+	0.997410i	$0.522915\pi$
$72$	0	0
$73$	10.1135	1.18369	0.591845	−	0.806052i	$-0.298400\pi$
$73$	10.1135	1.18369	0.591845	+	0.806052i	$0.298400\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−7.04414	−0.802754
$78$	0	0
$79$	−14.4151	−1.62183	−0.810913	−	0.585166i	$-0.801029\pi$
$79$	−14.4151	−1.62183	−0.810913	+	0.585166i	$0.801029\pi$
$80$	0	0
$81$	−10.8262	−1.20291
$82$	0	0
$83$	5.08949	0.558644	0.279322	−	0.960197i	$-0.409890\pi$
$83$	5.08949	0.558644	0.279322	+	0.960197i	$0.409890\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	15.8260	1.69672
$88$	0	0
$89$	−17.6123	−1.86690	−0.933450	−	0.358707i	$-0.883218\pi$
$89$	−17.6123	−1.86690	−0.933450	+	0.358707i	$0.883218\pi$
$90$	0	0
$91$	−1.69545	−0.177732
$92$	0	0
$93$	9.37794	0.972447
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−6.78753	−0.689170	−0.344585	−	0.938755i	$-0.611980\pi$
$97$	−6.78753	−0.689170	−0.344585	+	0.938755i	$0.611980\pi$
$98$	0	0
$99$	−3.87158	−0.389108
$100$	0	0
$101$	−8.21768	−0.817690	−0.408845	−	0.912604i	$-0.634068\pi$
$101$	−8.21768	−0.817690	−0.408845	+	0.912604i	$0.634068\pi$
$102$	0	0
$103$	−17.2696	−1.70163	−0.850814	−	0.525466i	$-0.823891\pi$
$103$	−17.2696	−1.70163	−0.850814	+	0.525466i	$0.823891\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−15.3869	−1.48750	−0.743752	−	0.668455i	$-0.766956\pi$
$107$	−15.3869	−1.48750	−0.743752	+	0.668455i	$0.766956\pi$
$108$	0	0
$109$	17.2221	1.64957	0.824787	−	0.565443i	$-0.191295\pi$
$109$	17.2221	1.64957	0.824787	+	0.565443i	$0.191295\pi$
$110$	0	0
$111$	−10.3572	−0.983057
$112$	0	0
$113$	18.9562	1.78325	0.891624	−	0.452777i	$-0.149567\pi$
$113$	18.9562	1.78325	0.891624	+	0.452777i	$0.149567\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−0.931849	−0.0861495
$118$	0	0
$119$	1.54475	0.141607
$120$	0	0
$121$	9.79415	0.890377
$122$	0	0
$123$	−1.47428	−0.132932
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−15.6123	−1.38537	−0.692684	−	0.721241i	$-0.743572\pi$
$127$	−15.6123	−1.38537	−0.692684	+	0.721241i	$0.743572\pi$
$128$	0	0
$129$	−18.6250	−1.63984
$130$	0	0
$131$	4.79329	0.418791	0.209396	−	0.977831i	$-0.432850\pi$
$131$	4.79329	0.418791	0.209396	+	0.977831i	$0.432850\pi$
$132$	0	0
$133$	−7.22207	−0.626233
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−15.6750	−1.33921	−0.669603	−	0.742719i	$-0.733536\pi$
$137$	−15.6750	−1.33921	−0.669603	+	0.742719i	$0.733536\pi$
$138$	0	0
$139$	−8.33055	−0.706588	−0.353294	−	0.935512i	$-0.614938\pi$
$139$	−8.33055	−0.706588	−0.353294	+	0.935512i	$0.614938\pi$
$140$	0	0
$141$	−21.0745	−1.77480
$142$	0	0
$143$	5.00494	0.418534
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−9.05169	−0.746571
$148$	0	0
$149$	−3.10739	−0.254568	−0.127284	−	0.991866i	$-0.540626\pi$
$149$	−3.10739	−0.254568	−0.127284	+	0.991866i	$0.540626\pi$
$150$	0	0
$151$	−10.4088	−0.847056	−0.423528	−	0.905883i	$-0.639209\pi$
$151$	−10.4088	−0.847056	−0.423528	+	0.905883i	$0.639209\pi$
$152$	0	0
$153$	0.849020	0.0686392
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−0.661967	−0.0528307	−0.0264154	−	0.999651i	$-0.508409\pi$
$157$	−0.661967	−0.0528307	−0.0264154	+	0.999651i	$0.508409\pi$
$158$	0	0
$159$	0.0447215	0.00354664
$160$	0	0
$161$	−0.817841	−0.0644549
$162$	0	0
$163$	16.2543	1.27314	0.636569	−	0.771220i	$-0.280353\pi$
$163$	16.2543	1.27314	0.636569	+	0.771220i	$0.280353\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−19.9527	−1.54398	−0.771992	−	0.635632i	$-0.780740\pi$
$167$	−19.9527	−1.54398	−0.771992	+	0.635632i	$0.780740\pi$
$168$	0	0
$169$	−11.7954	−0.907336
$170$	0	0
$171$	−3.96937	−0.303546
$172$	0	0
$173$	3.91717	0.297817	0.148908	−	0.988851i	$-0.452424\pi$
$173$	3.91717	0.297817	0.148908	+	0.988851i	$0.452424\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	7.00321	0.526393
$178$	0	0
$179$	−3.14170	−0.234821	−0.117411	−	0.993083i	$-0.537459\pi$
$179$	−3.14170	−0.234821	−0.117411	+	0.993083i	$0.537459\pi$
$180$	0	0
$181$	0.782087	0.0581320	0.0290660	−	0.999577i	$-0.490747\pi$
$181$	0.782087	0.0581320	0.0290660	+	0.999577i	$0.490747\pi$
$182$	0	0
$183$	7.69804	0.569055
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−4.56006	−0.333465
$188$	0	0
$189$	−6.51882	−0.474174
$190$	0	0
$191$	−12.4154	−0.898348	−0.449174	−	0.893444i	$-0.648282\pi$
$191$	−12.4154	−0.898348	−0.449174	+	0.893444i	$0.648282\pi$
$192$	0	0
$193$	1.70465	0.122704	0.0613518	−	0.998116i	$-0.480459\pi$
$193$	1.70465	0.122704	0.0613518	+	0.998116i	$0.480459\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−16.2755	−1.15958	−0.579790	−	0.814766i	$-0.696866\pi$
$197$	−16.2755	−1.15958	−0.579790	+	0.814766i	$0.696866\pi$
$198$	0	0
$199$	−17.1946	−1.21889	−0.609447	−	0.792827i	$-0.708608\pi$
$199$	−17.1946	−1.21889	−0.609447	+	0.792827i	$0.708608\pi$
$200$	0	0
$201$	19.1467	1.35050
$202$	0	0
$203$	12.4610	0.874591
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−0.449500	−0.0312424
$208$	0	0
$209$	21.3194	1.47469
$210$	0	0
$211$	−2.01038	−0.138400	−0.0692000	−	0.997603i	$-0.522045\pi$
$211$	−2.01038	−0.138400	−0.0692000	+	0.997603i	$0.522045\pi$
$212$	0	0
$213$	−2.37813	−0.162947
$214$	0	0
$215$	0	0
$216$	0	0
$217$	7.38397	0.501256
$218$	0	0
$219$	19.8415	1.34076
$220$	0	0
$221$	−1.09756	−0.0738298
$222$	0	0
$223$	14.1074	0.944701	0.472351	−	0.881411i	$-0.343406\pi$
$223$	14.1074	0.944701	0.472351	+	0.881411i	$0.343406\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	15.7117	1.04282	0.521410	−	0.853306i	$-0.325406\pi$
$227$	15.7117	1.04282	0.521410	+	0.853306i	$0.325406\pi$
$228$	0	0
$229$	16.5921	1.09644	0.548219	−	0.836335i	$-0.315306\pi$
$229$	16.5921	1.09644	0.548219	+	0.836335i	$0.315306\pi$
$230$	0	0
$231$	−13.8198	−0.909279
$232$	0	0
$233$	6.56378	0.430007	0.215004	−	0.976613i	$-0.431024\pi$
$233$	6.56378	0.430007	0.215004	+	0.976613i	$0.431024\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−28.2809	−1.83704
$238$	0	0
$239$	7.62182	0.493015	0.246507	−	0.969141i	$-0.420717\pi$
$239$	7.62182	0.493015	0.246507	+	0.969141i	$0.420717\pi$
$240$	0	0
$241$	6.89554	0.444181	0.222090	−	0.975026i	$-0.428712\pi$
$241$	6.89554	0.444181	0.222090	+	0.975026i	$0.428712\pi$
$242$	0	0
$243$	−8.57991	−0.550401
$244$	0	0
$245$	0	0
$246$	0	0
$247$	5.13136	0.326500
$248$	0	0
$249$	9.98504	0.632776
$250$	0	0
$251$	25.8326	1.63054	0.815270	−	0.579081i	$-0.196589\pi$
$251$	25.8326	1.63054	0.815270	+	0.579081i	$0.196589\pi$
$252$	0	0
$253$	2.41425	0.151783
$254$	0	0
$255$	0	0
$256$	0	0
$257$	17.8491	1.11340	0.556698	−	0.830715i	$-0.312068\pi$
$257$	17.8491	1.11340	0.556698	+	0.830715i	$0.312068\pi$
$258$	0	0
$259$	−8.15497	−0.506725
$260$	0	0
$261$	6.84879	0.423929
$262$	0	0
$263$	14.3446	0.884529	0.442264	−	0.896885i	$-0.354175\pi$
$263$	14.3446	0.884529	0.442264	+	0.896885i	$0.354175\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−34.5534	−2.11464
$268$	0	0
$269$	−3.91172	−0.238502	−0.119251	−	0.992864i	$-0.538049\pi$
$269$	−3.91172	−0.238502	−0.119251	+	0.992864i	$0.538049\pi$
$270$	0	0
$271$	−28.4490	−1.72816	−0.864078	−	0.503358i	$-0.832098\pi$
$271$	−28.4490	−1.72816	−0.864078	+	0.503358i	$0.832098\pi$
$272$	0	0
$273$	−3.32629	−0.201316
$274$	0	0
$275$	0	0
$276$	0	0
$277$	25.6839	1.54320	0.771598	−	0.636111i	$-0.219458\pi$
$277$	25.6839	1.54320	0.771598	+	0.636111i	$0.219458\pi$
$278$	0	0
$279$	4.05836	0.242967
$280$	0	0
$281$	−20.7667	−1.23884	−0.619420	−	0.785060i	$-0.712632\pi$
$281$	−20.7667	−1.23884	−0.619420	+	0.785060i	$0.712632\pi$
$282$	0	0
$283$	24.4689	1.45452	0.727262	−	0.686360i	$-0.240793\pi$
$283$	24.4689	1.45452	0.727262	+	0.686360i	$0.240793\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1.16082	−0.0685208
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−13.3164	−0.780621
$292$	0	0
$293$	−28.1952	−1.64718	−0.823591	−	0.567185i	$-0.808033\pi$
$293$	−28.1952	−1.64718	−0.823591	+	0.567185i	$0.808033\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	19.2434	1.11662
$298$	0	0
$299$	0.581085	0.0336050
$300$	0	0
$301$	−14.6649	−0.845271
$302$	0	0
$303$	−16.1222	−0.926196
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0.0120595	0.000688275 0	0.000344137	−	1.00000i	$-0.499890\pi$
$307$	0.0120595	0.000688275 0	0.000344137		1.00000i	$0.499890\pi$
$308$	0	0
$309$	−33.8812	−1.92743
$310$	0	0
$311$	28.5605	1.61951	0.809757	−	0.586765i	$-0.199599\pi$
$311$	28.5605	1.61951	0.809757	+	0.586765i	$0.199599\pi$
$312$	0	0
$313$	−4.11691	−0.232702	−0.116351	−	0.993208i	$-0.537120\pi$
$313$	−4.11691	−0.232702	−0.116351	+	0.993208i	$0.537120\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	20.6297	1.15868	0.579338	−	0.815087i	$-0.303311\pi$
$317$	20.6297	1.15868	0.579338	+	0.815087i	$0.303311\pi$
$318$	0	0
$319$	−36.7846	−2.05954
$320$	0	0
$321$	−30.1874	−1.68489
$322$	0	0
$323$	−4.67524	−0.260138
$324$	0	0
$325$	0	0
$326$	0	0
$327$	33.7878	1.86847
$328$	0	0
$329$	−16.5936	−0.914834
$330$	0	0
$331$	0.971759	0.0534127	0.0267063	−	0.999643i	$-0.491498\pi$
$331$	0.971759	0.0534127	0.0267063	+	0.999643i	$0.491498\pi$
$332$	0	0
$333$	−4.48212	−0.245618
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−6.01450	−0.327630	−0.163815	−	0.986491i	$-0.552380\pi$
$337$	−6.01450	−0.327630	−0.163815	+	0.986491i	$0.552380\pi$
$338$	0	0
$339$	37.1900	2.01988
$340$	0	0
$341$	−21.7973	−1.18039
$342$	0	0
$343$	−17.9403	−0.968686
$344$	0	0
$345$	0	0
$346$	0	0
$347$	5.33466	0.286380	0.143190	−	0.989695i	$-0.454264\pi$
$347$	5.33466	0.286380	0.143190	+	0.989695i	$0.454264\pi$
$348$	0	0
$349$	−20.7601	−1.11126	−0.555632	−	0.831429i	$-0.687524\pi$
$349$	−20.7601	−1.11126	−0.555632	+	0.831429i	$0.687524\pi$
$350$	0	0
$351$	4.63169	0.247221
$352$	0	0
$353$	15.0618	0.801659	0.400829	−	0.916153i	$-0.368722\pi$
$353$	15.0618	0.801659	0.400829	+	0.916153i	$0.368722\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	3.03063	0.160398
$358$	0	0
$359$	9.72949	0.513503	0.256751	−	0.966477i	$-0.417348\pi$
$359$	9.72949	0.513503	0.256751	+	0.966477i	$0.417348\pi$
$360$	0	0
$361$	2.85791	0.150416
$362$	0	0
$363$	19.2151	1.00853
$364$	0	0
$365$	0	0
$366$	0	0
$367$	8.62563	0.450254	0.225127	−	0.974329i	$-0.427720\pi$
$367$	8.62563	0.450254	0.225127	+	0.974329i	$0.427720\pi$
$368$	0	0
$369$	−0.638005	−0.0332132
$370$	0	0
$371$	0.0352126	0.00182815
$372$	0	0
$373$	15.0922	0.781444	0.390722	−	0.920509i	$-0.372225\pi$
$373$	15.0922	0.781444	0.390722	+	0.920509i	$0.372225\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−8.85368	−0.455988
$378$	0	0
$379$	10.7124	0.550261	0.275131	−	0.961407i	$-0.411279\pi$
$379$	10.7124	0.550261	0.275131	+	0.961407i	$0.411279\pi$
$380$	0	0
$381$	−30.6297	−1.56920
$382$	0	0
$383$	15.8670	0.810764	0.405382	−	0.914147i	$-0.367139\pi$
$383$	15.8670	0.810764	0.405382	+	0.914147i	$0.367139\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−8.06009	−0.409717
$388$	0	0
$389$	22.3496	1.13317	0.566585	−	0.824003i	$-0.308264\pi$
$389$	22.3496	1.13317	0.566585	+	0.824003i	$0.308264\pi$
$390$	0	0
$391$	−0.529434	−0.0267746
$392$	0	0
$393$	9.40391	0.474364
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−31.3003	−1.57092	−0.785458	−	0.618915i	$-0.787572\pi$
$397$	−31.3003	−1.57092	−0.785458	+	0.618915i	$0.787572\pi$
$398$	0	0
$399$	−14.1689	−0.709333
$400$	0	0
$401$	32.0373	1.59987	0.799934	−	0.600088i	$-0.204868\pi$
$401$	32.0373	1.59987	0.799934	+	0.600088i	$0.204868\pi$
$402$	0	0
$403$	−5.24639	−0.261341
$404$	0	0
$405$	0	0
$406$	0	0
$407$	24.0733	1.19327
$408$	0	0
$409$	−21.5037	−1.06329	−0.531645	−	0.846968i	$-0.678426\pi$
$409$	−21.5037	−1.06329	−0.531645	+	0.846968i	$0.678426\pi$
$410$	0	0
$411$	−30.7527	−1.51692
$412$	0	0
$413$	5.51416	0.271334
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−16.3436	−0.800352
$418$	0	0
$419$	16.7246	0.817048	0.408524	−	0.912748i	$-0.366044\pi$
$419$	16.7246	0.817048	0.408524	+	0.912748i	$0.366044\pi$
$420$	0	0
$421$	−16.9873	−0.827908	−0.413954	−	0.910298i	$-0.635853\pi$
$421$	−16.9873	−0.827908	−0.413954	+	0.910298i	$0.635853\pi$
$422$	0	0
$423$	−9.12012	−0.443435
$424$	0	0
$425$	0	0
$426$	0	0
$427$	6.06125	0.293325
$428$	0	0
$429$	9.81914	0.474073
$430$	0	0
$431$	17.0941	0.823392	0.411696	−	0.911321i	$-0.364936\pi$
$431$	17.0941	0.823392	0.411696	+	0.911321i	$0.364936\pi$
$432$	0	0
$433$	−9.95242	−0.478283	−0.239141	−	0.970985i	$-0.576866\pi$
$433$	−9.95242	−0.478283	−0.239141	+	0.970985i	$0.576866\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.47523	0.118406
$438$	0	0
$439$	−2.92877	−0.139782	−0.0698912	−	0.997555i	$-0.522265\pi$
$439$	−2.92877	−0.139782	−0.0698912	+	0.997555i	$0.522265\pi$
$440$	0	0
$441$	−3.91717	−0.186532
$442$	0	0
$443$	−0.778484	−0.0369869	−0.0184934	−	0.999829i	$-0.505887\pi$
$443$	−0.778484	−0.0369869	−0.0184934	+	0.999829i	$0.505887\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−6.09637	−0.288349
$448$	0	0
$449$	−31.2804	−1.47621	−0.738107	−	0.674684i	$-0.764280\pi$
$449$	−31.2804	−1.47621	−0.738107	+	0.674684i	$0.764280\pi$
$450$	0	0
$451$	3.42670	0.161357
$452$	0	0
$453$	−20.4209	−0.959460
$454$	0	0
$455$	0	0
$456$	0	0
$457$	3.59265	0.168057	0.0840285	−	0.996463i	$-0.473221\pi$
$457$	3.59265	0.168057	0.0840285	+	0.996463i	$0.473221\pi$
$458$	0	0
$459$	−4.21999	−0.196972
$460$	0	0
$461$	−6.60392	−0.307575	−0.153788	−	0.988104i	$-0.549147\pi$
$461$	−6.60392	−0.307575	−0.153788	+	0.988104i	$0.549147\pi$
$462$	0	0
$463$	10.9502	0.508898	0.254449	−	0.967086i	$-0.418106\pi$
$463$	10.9502	0.508898	0.254449	+	0.967086i	$0.418106\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	10.1288	0.468704	0.234352	−	0.972152i	$-0.424703\pi$
$467$	10.1288	0.468704	0.234352	+	0.972152i	$0.424703\pi$
$468$	0	0
$469$	15.0756	0.696128
$470$	0	0
$471$	−1.29871	−0.0598413
$472$	0	0
$473$	43.2905	1.99050
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0.0193535	0.000886135 0
$478$	0	0
$479$	7.73039	0.353211	0.176605	−	0.984282i	$-0.443488\pi$
$479$	7.73039	0.353211	0.176605	+	0.984282i	$0.443488\pi$
$480$	0	0
$481$	5.79420	0.264193
$482$	0	0
$483$	−1.60452	−0.0730080
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−13.5925	−0.615933	−0.307966	−	0.951397i	$-0.599648\pi$
$487$	−13.5925	−0.615933	−0.307966	+	0.951397i	$0.599648\pi$
$488$	0	0
$489$	31.8892	1.44208
$490$	0	0
$491$	−38.8592	−1.75369	−0.876846	−	0.480772i	$-0.840357\pi$
$491$	−38.8592	−1.75369	−0.876846	+	0.480772i	$0.840357\pi$
$492$	0	0
$493$	8.06670	0.363306
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1.87248	−0.0839922
$498$	0	0
$499$	26.6894	1.19478	0.597391	−	0.801950i	$-0.296204\pi$
$499$	26.6894	1.19478	0.597391	+	0.801950i	$0.296204\pi$
$500$	0	0
$501$	−39.1450	−1.74887
$502$	0	0
$503$	0.591853	0.0263894	0.0131947	−	0.999913i	$-0.495800\pi$
$503$	0.591853	0.0263894	0.0131947	+	0.999913i	$0.495800\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−23.1412	−1.02774
$508$	0	0
$509$	11.6397	0.515922	0.257961	−	0.966155i	$-0.416949\pi$
$509$	11.6397	0.515922	0.257961	+	0.966155i	$0.416949\pi$
$510$	0	0
$511$	15.6227	0.691109
$512$	0	0
$513$	19.7295	0.871078
$514$	0	0
$515$	0	0
$516$	0	0
$517$	48.9839	2.15431
$518$	0	0
$519$	7.68506	0.337337
$520$	0	0
$521$	−11.5584	−0.506382	−0.253191	−	0.967416i	$-0.581480\pi$
$521$	−11.5584	−0.506382	−0.253191	+	0.967416i	$0.581480\pi$
$522$	0	0
$523$	−14.5087	−0.634420	−0.317210	−	0.948355i	$-0.602746\pi$
$523$	−14.5087	−0.634420	−0.317210	+	0.948355i	$0.602746\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.78005	0.208222
$528$	0	0
$529$	−22.7197	−0.987813
$530$	0	0
$531$	3.03068	0.131520
$532$	0	0
$533$	0.824772	0.0357249
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−6.16367	−0.265982
$538$	0	0
$539$	21.0390	0.906214
$540$	0	0
$541$	−3.23241	−0.138972	−0.0694860	−	0.997583i	$-0.522136\pi$
$541$	−3.23241	−0.138972	−0.0694860	+	0.997583i	$0.522136\pi$
$542$	0	0
$543$	1.53437	0.0658461
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−38.7154	−1.65535	−0.827676	−	0.561206i	$-0.810338\pi$
$547$	−38.7154	−1.65535	−0.827676	+	0.561206i	$0.810338\pi$
$548$	0	0
$549$	3.33137	0.142179
$550$	0	0
$551$	−37.7138	−1.60666
$552$	0	0
$553$	−22.2677	−0.946919
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0.669168	0.0283536	0.0141768	−	0.999900i	$-0.495487\pi$
$557$	0.669168	0.0283536	0.0141768	+	0.999900i	$0.495487\pi$
$558$	0	0
$559$	10.4196	0.440701
$560$	0	0
$561$	−8.94634	−0.377715
$562$	0	0
$563$	42.4772	1.79020	0.895101	−	0.445864i	$-0.147104\pi$
$563$	42.4772	1.79020	0.895101	+	0.445864i	$0.147104\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−16.7238	−0.702333
$568$	0	0
$569$	36.3706	1.52474	0.762368	−	0.647143i	$-0.224036\pi$
$569$	36.3706	1.52474	0.762368	+	0.647143i	$0.224036\pi$
$570$	0	0
$571$	−26.5970	−1.11305	−0.556525	−	0.830831i	$-0.687866\pi$
$571$	−26.5970	−1.11305	−0.556525	+	0.830831i	$0.687866\pi$
$572$	0	0
$573$	−24.3577	−1.01756
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−2.30741	−0.0960586	−0.0480293	−	0.998846i	$-0.515294\pi$
$577$	−2.30741	−0.0960586	−0.0480293	+	0.998846i	$0.515294\pi$
$578$	0	0
$579$	3.34434	0.138986
$580$	0	0
$581$	7.86198	0.326170
$582$	0	0
$583$	−0.103947	−0.00430504
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−25.8915	−1.06866	−0.534329	−	0.845277i	$-0.679436\pi$
$587$	−25.8915	−1.06866	−0.534329	+	0.845277i	$0.679436\pi$
$588$	0	0
$589$	−22.3479	−0.920829
$590$	0	0
$591$	−31.9308	−1.31346
$592$	0	0
$593$	42.0620	1.72728	0.863640	−	0.504109i	$-0.168179\pi$
$593$	42.0620	1.72728	0.863640	+	0.504109i	$0.168179\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−33.7340	−1.38064
$598$	0	0
$599$	−23.6399	−0.965902	−0.482951	−	0.875647i	$-0.660435\pi$
$599$	−23.6399	−0.965902	−0.482951	+	0.875647i	$0.660435\pi$
$600$	0	0
$601$	−9.89731	−0.403720	−0.201860	−	0.979414i	$-0.564699\pi$
$601$	−9.89731	−0.403720	−0.201860	+	0.979414i	$0.564699\pi$
$602$	0	0
$603$	8.28583	0.337425
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−29.5988	−1.20138	−0.600688	−	0.799483i	$-0.705107\pi$
$607$	−29.5988	−1.20138	−0.600688	+	0.799483i	$0.705107\pi$
$608$	0	0
$609$	24.4471	0.990648
$610$	0	0
$611$	11.7899	0.476969
$612$	0	0
$613$	13.6657	0.551953	0.275977	−	0.961164i	$-0.410999\pi$
$613$	13.6657	0.551953	0.275977	+	0.961164i	$0.410999\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	41.8255	1.68383	0.841916	−	0.539609i	$-0.181428\pi$
$617$	41.8255	1.68383	0.841916	+	0.539609i	$0.181428\pi$
$618$	0	0
$619$	−17.8194	−0.716221	−0.358110	−	0.933679i	$-0.616579\pi$
$619$	−17.8194	−0.716221	−0.358110	+	0.933679i	$0.616579\pi$
$620$	0	0
$621$	2.23421	0.0896556
$622$	0	0
$623$	−27.2066	−1.09001
$624$	0	0
$625$	0	0
$626$	0	0
$627$	41.8263	1.67038
$628$	0	0
$629$	−5.27917	−0.210494
$630$	0	0
$631$	−32.6557	−1.30000	−0.650001	−	0.759934i	$-0.725231\pi$
$631$	−32.6557	−1.30000	−0.650001	+	0.759934i	$0.725231\pi$
$632$	0	0
$633$	−3.94414	−0.156766
$634$	0	0
$635$	0	0
$636$	0	0
$637$	5.06387	0.200638
$638$	0	0
$639$	−1.02915	−0.0407124
$640$	0	0
$641$	25.0356	0.988845	0.494422	−	0.869222i	$-0.335380\pi$
$641$	25.0356	0.988845	0.494422	+	0.869222i	$0.335380\pi$
$642$	0	0
$643$	12.4048	0.489196	0.244598	−	0.969625i	$-0.421344\pi$
$643$	12.4048	0.489196	0.244598	+	0.969625i	$0.421344\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	37.9570	1.49224	0.746121	−	0.665811i	$-0.231914\pi$
$647$	37.9570	1.49224	0.746121	+	0.665811i	$0.231914\pi$
$648$	0	0
$649$	−16.2777	−0.638955
$650$	0	0
$651$	14.4865	0.567773
$652$	0	0
$653$	−36.9847	−1.44732	−0.723662	−	0.690155i	$-0.757543\pi$
$653$	−36.9847	−1.44732	−0.723662	+	0.690155i	$0.757543\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	8.58652	0.334992
$658$	0	0
$659$	−29.0956	−1.13341	−0.566703	−	0.823922i	$-0.691781\pi$
$659$	−29.0956	−1.13341	−0.566703	+	0.823922i	$0.691781\pi$
$660$	0	0
$661$	33.6207	1.30769	0.653847	−	0.756627i	$-0.273154\pi$
$661$	33.6207	1.30769	0.653847	+	0.756627i	$0.273154\pi$
$662$	0	0
$663$	−2.15329	−0.0836269
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−4.27078	−0.165365
$668$	0	0
$669$	27.6772	1.07006
$670$	0	0
$671$	−17.8927	−0.690740
$672$	0	0
$673$	13.1865	0.508304	0.254152	−	0.967164i	$-0.418204\pi$
$673$	13.1865	0.508304	0.254152	+	0.967164i	$0.418204\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−1.37696	−0.0529208	−0.0264604	−	0.999650i	$-0.508424\pi$
$677$	−1.37696	−0.0529208	−0.0264604	+	0.999650i	$0.508424\pi$
$678$	0	0
$679$	−10.4850	−0.402378
$680$	0	0
$681$	30.8246	1.18120
$682$	0	0
$683$	10.4181	0.398637	0.199319	−	0.979935i	$-0.436127\pi$
$683$	10.4181	0.398637	0.199319	+	0.979935i	$0.436127\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	32.5520	1.24193
$688$	0	0
$689$	−0.0250190	−0.000953146 0
$690$	0	0
$691$	−4.39455	−0.167177	−0.0835883	−	0.996500i	$-0.526638\pi$
$691$	−4.39455	−0.167177	−0.0835883	+	0.996500i	$0.526638\pi$
$692$	0	0
$693$	−5.98061	−0.227185
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−0.751460	−0.0284636
$698$	0	0
$699$	12.8774	0.487069
$700$	0	0
$701$	−1.18383	−0.0447127	−0.0223563	−	0.999750i	$-0.507117\pi$
$701$	−1.18383	−0.0447127	−0.0223563	+	0.999750i	$0.507117\pi$
$702$	0	0
$703$	24.6814	0.930876
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−12.6942	−0.477416
$708$	0	0
$709$	−13.3650	−0.501932	−0.250966	−	0.967996i	$-0.580748\pi$
$709$	−13.3650	−0.501932	−0.250966	+	0.967996i	$0.580748\pi$
$710$	0	0
$711$	−12.2387	−0.458987
$712$	0	0
$713$	−2.53072	−0.0947762
$714$	0	0
$715$	0	0
$716$	0	0
$717$	14.9532	0.558437
$718$	0	0
$719$	−14.1925	−0.529292	−0.264646	−	0.964346i	$-0.585255\pi$
$719$	−14.1925	−0.529292	−0.264646	+	0.964346i	$0.585255\pi$
$720$	0	0
$721$	−26.6772	−0.993512
$722$	0	0
$723$	13.5283	0.503123
$724$	0	0
$725$	0	0
$726$	0	0
$727$	33.0852	1.22706	0.613530	−	0.789671i	$-0.289749\pi$
$727$	33.0852	1.22706	0.613530	+	0.789671i	$0.289749\pi$
$728$	0	0
$729$	15.6458	0.579475
$730$	0	0
$731$	−9.49340	−0.351126
$732$	0	0
$733$	−14.0451	−0.518767	−0.259383	−	0.965774i	$-0.583519\pi$
$733$	−14.0451	−0.518767	−0.259383	+	0.965774i	$0.583519\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−44.5030	−1.63929
$738$	0	0
$739$	5.14831	0.189384	0.0946918	−	0.995507i	$-0.469813\pi$
$739$	5.14831	0.189384	0.0946918	+	0.995507i	$0.469813\pi$
$740$	0	0
$741$	10.0672	0.369827
$742$	0	0
$743$	−26.8978	−0.986784	−0.493392	−	0.869807i	$-0.664243\pi$
$743$	−26.8978	−0.986784	−0.493392	+	0.869807i	$0.664243\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	4.32108	0.158100
$748$	0	0
$749$	−23.7688	−0.868494
$750$	0	0
$751$	−13.5733	−0.495295	−0.247648	−	0.968850i	$-0.579657\pi$
$751$	−13.5733	−0.495295	−0.247648	+	0.968850i	$0.579657\pi$
$752$	0	0
$753$	50.6808	1.84691
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−9.44665	−0.343344	−0.171672	−	0.985154i	$-0.554917\pi$
$757$	−9.44665	−0.343344	−0.171672	+	0.985154i	$0.554917\pi$
$758$	0	0
$759$	4.73650	0.171924
$760$	0	0
$761$	−26.6325	−0.965427	−0.482713	−	0.875778i	$-0.660349\pi$
$761$	−26.6325	−0.965427	−0.482713	+	0.875778i	$0.660349\pi$
$762$	0	0
$763$	26.6037	0.963120
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−3.91787	−0.141466
$768$	0	0
$769$	−17.9587	−0.647609	−0.323805	−	0.946124i	$-0.604962\pi$
$769$	−17.9587	−0.647609	−0.323805	+	0.946124i	$0.604962\pi$
$770$	0	0
$771$	35.0180	1.26114
$772$	0	0
$773$	−17.0705	−0.613984	−0.306992	−	0.951712i	$-0.599323\pi$
$773$	−17.0705	−0.613984	−0.306992	+	0.951712i	$0.599323\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−15.9992	−0.573967
$778$	0	0
$779$	3.51326	0.125876
$780$	0	0
$781$	5.52752	0.197790
$782$	0	0
$783$	−34.0414	−1.21654
$784$	0	0
$785$	0	0
$786$	0	0
$787$	18.3219	0.653104	0.326552	−	0.945179i	$-0.394113\pi$
$787$	18.3219	0.653104	0.326552	+	0.945179i	$0.394113\pi$
$788$	0	0
$789$	28.1426	1.00190
$790$	0	0
$791$	29.2825	1.04117
$792$	0	0
$793$	−4.30658	−0.152931
$794$	0	0
$795$	0	0
$796$	0	0
$797$	16.4573	0.582949	0.291474	−	0.956579i	$-0.405854\pi$
$797$	16.4573	0.582949	0.291474	+	0.956579i	$0.405854\pi$
$798$	0	0
$799$	−10.7419	−0.380023
$800$	0	0
$801$	−14.9532	−0.528345
$802$	0	0
$803$	−46.1180	−1.62747
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−7.67438	−0.270151
$808$	0	0
$809$	19.3869	0.681607	0.340804	−	0.940134i	$-0.389301\pi$
$809$	19.3869	0.681607	0.340804	+	0.940134i	$0.389301\pi$
$810$	0	0
$811$	27.4015	0.962196	0.481098	−	0.876667i	$-0.340238\pi$
$811$	27.4015	0.962196	0.481098	+	0.876667i	$0.340238\pi$
$812$	0	0
$813$	−55.8139	−1.95748
$814$	0	0
$815$	0	0
$816$	0	0
$817$	44.3840	1.55280
$818$	0	0
$819$	−1.43947	−0.0502992
$820$	0	0
$821$	20.6646	0.721198	0.360599	−	0.932721i	$-0.382572\pi$
$821$	20.6646	0.721198	0.360599	+	0.932721i	$0.382572\pi$
$822$	0	0
$823$	−0.983957	−0.0342986	−0.0171493	−	0.999853i	$-0.505459\pi$
$823$	−0.983957	−0.0342986	−0.0171493	+	0.999853i	$0.505459\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−15.5237	−0.539811	−0.269906	−	0.962887i	$-0.586992\pi$
$827$	−15.5237	−0.539811	−0.269906	+	0.962887i	$0.586992\pi$
$828$	0	0
$829$	−41.6910	−1.44799	−0.723994	−	0.689806i	$-0.757696\pi$
$829$	−41.6910	−1.44799	−0.723994	+	0.689806i	$0.757696\pi$
$830$	0	0
$831$	50.3890	1.74798
$832$	0	0
$833$	−4.61376	−0.159857
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−20.1718	−0.697238
$838$	0	0
$839$	−11.0633	−0.381949	−0.190974	−	0.981595i	$-0.561165\pi$
$839$	−11.0633	−0.381949	−0.190974	+	0.981595i	$0.561165\pi$
$840$	0	0
$841$	36.0716	1.24385
$842$	0	0
$843$	−40.7421	−1.40323
$844$	0	0
$845$	0	0
$846$	0	0
$847$	15.1295	0.519855
$848$	0	0
$849$	48.0053	1.64754
$850$	0	0
$851$	2.79497	0.0958103
$852$	0	0
$853$	16.9575	0.580615	0.290307	−	0.956933i	$-0.406242\pi$
$853$	16.9575	0.580615	0.290307	+	0.956933i	$0.406242\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−21.7394	−0.742605	−0.371302	−	0.928512i	$-0.621089\pi$
$857$	−21.7394	−0.742605	−0.371302	+	0.928512i	$0.621089\pi$
$858$	0	0
$859$	−31.5036	−1.07489	−0.537444	−	0.843300i	$-0.680610\pi$
$859$	−31.5036	−1.07489	−0.537444	+	0.843300i	$0.680610\pi$
$860$	0	0
$861$	−2.27740	−0.0776134
$862$	0	0
$863$	12.4798	0.424817	0.212408	−	0.977181i	$-0.431869\pi$
$863$	12.4798	0.424817	0.212408	+	0.977181i	$0.431869\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	1.96189	0.0666293
$868$	0	0
$869$	65.7338	2.22987
$870$	0	0
$871$	−10.7114	−0.362942
$872$	0	0
$873$	−5.76275	−0.195039
$874$	0	0
$875$	0	0
$876$	0	0
$877$	14.5835	0.492450	0.246225	−	0.969213i	$-0.420810\pi$
$877$	14.5835	0.492450	0.246225	+	0.969213i	$0.420810\pi$
$878$	0	0
$879$	−55.3159	−1.86576
$880$	0	0
$881$	−12.4663	−0.420001	−0.210001	−	0.977701i	$-0.567347\pi$
$881$	−12.4663	−0.420001	−0.210001	+	0.977701i	$0.567347\pi$
$882$	0	0
$883$	52.0737	1.75242	0.876209	−	0.481931i	$-0.160064\pi$
$883$	52.0737	1.75242	0.876209	+	0.481931i	$0.160064\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	46.1318	1.54895	0.774477	−	0.632602i	$-0.218013\pi$
$887$	46.1318	1.54895	0.774477	+	0.632602i	$0.218013\pi$
$888$	0	0
$889$	−24.1171	−0.808860
$890$	0	0
$891$	49.3682	1.65390
$892$	0	0
$893$	50.2212	1.68059
$894$	0	0
$895$	0	0
$896$	0	0
$897$	1.14003	0.0380643
$898$	0	0
$899$	38.5592	1.28602
$900$	0	0
$901$	0.0227951	0.000759414 0
$902$	0	0
$903$	−28.7710	−0.957437
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−2.91050	−0.0966415	−0.0483207	−	0.998832i	$-0.515387\pi$
$907$	−2.91050	−0.0966415	−0.0483207	+	0.998832i	$0.515387\pi$
$908$	0	0
$909$	−6.97697	−0.231412
$910$	0	0
$911$	41.7029	1.38168	0.690839	−	0.723009i	$-0.257241\pi$
$911$	41.7029	1.38168	0.690839	+	0.723009i	$0.257241\pi$
$912$	0	0
$913$	−23.2084	−0.768086
$914$	0	0
$915$	0	0
$916$	0	0
$917$	7.40441	0.244515
$918$	0	0
$919$	2.60146	0.0858144	0.0429072	−	0.999079i	$-0.486338\pi$
$919$	2.60146	0.0858144	0.0429072	+	0.999079i	$0.486338\pi$
$920$	0	0
$921$	0.0236595	0.000779608 0
$922$	0	0
$923$	1.33042	0.0437912
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−14.6623	−0.481572
$928$	0	0
$929$	5.89744	0.193489	0.0967443	−	0.995309i	$-0.469157\pi$
$929$	5.89744	0.193489	0.0967443	+	0.995309i	$0.469157\pi$
$930$	0	0
$931$	21.5704	0.706943
$932$	0	0
$933$	56.0325	1.83442
$934$	0	0
$935$	0	0
$936$	0	0
$937$	58.2446	1.90277	0.951383	−	0.308010i	$-0.0996629\pi$
$937$	58.2446	1.90277	0.951383	+	0.308010i	$0.0996629\pi$
$938$	0	0
$939$	−8.07694	−0.263581
$940$	0	0
$941$	28.2631	0.921350	0.460675	−	0.887569i	$-0.347607\pi$
$941$	28.2631	0.921350	0.460675	+	0.887569i	$0.347607\pi$
$942$	0	0
$943$	0.397848	0.0129557
$944$	0	0
$945$	0	0
$946$	0	0
$947$	31.3759	1.01958	0.509789	−	0.860299i	$-0.329723\pi$
$947$	31.3759	1.01958	0.509789	+	0.860299i	$0.329723\pi$
$948$	0	0
$949$	−11.1001	−0.360325
$950$	0	0
$951$	40.4731	1.31243
$952$	0	0
$953$	27.2735	0.883476	0.441738	−	0.897144i	$-0.354362\pi$
$953$	27.2735	0.883476	0.441738	+	0.897144i	$0.354362\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−72.1675	−2.33284
$958$	0	0
$959$	−24.2139	−0.781908
$960$	0	0
$961$	−8.15111	−0.262939
$962$	0	0
$963$	−13.0638	−0.420974
$964$	0	0
$965$	0	0
$966$	0	0
$967$	1.30807	0.0420648	0.0210324	−	0.999779i	$-0.493305\pi$
$967$	1.30807	0.0420648	0.0210324	+	0.999779i	$0.493305\pi$
$968$	0	0
$969$	−9.17232	−0.294657
$970$	0	0
$971$	41.0902	1.31865	0.659324	−	0.751859i	$-0.270843\pi$
$971$	41.0902	1.31865	0.659324	+	0.751859i	$0.270843\pi$
$972$	0	0
$973$	−12.8686	−0.412548
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−60.7556	−1.94375	−0.971873	−	0.235507i	$-0.924325\pi$
$977$	−60.7556	−1.94375	−0.971873	+	0.235507i	$0.924325\pi$
$978$	0	0
$979$	80.3132	2.56682
$980$	0	0
$981$	14.6219	0.466840
$982$	0	0
$983$	−9.13830	−0.291466	−0.145733	−	0.989324i	$-0.546554\pi$
$983$	−9.13830	−0.291466	−0.145733	+	0.989324i	$0.546554\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−32.5548	−1.03623
$988$	0	0
$989$	5.02613	0.159822
$990$	0	0
$991$	−33.1886	−1.05427	−0.527135	−	0.849782i	$-0.676734\pi$
$991$	−33.1886	−1.05427	−0.527135	+	0.849782i	$0.676734\pi$
$992$	0	0
$993$	1.90649	0.0605005
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−30.0746	−0.952471	−0.476235	−	0.879318i	$-0.657999\pi$
$997$	−30.0746	−0.952471	−0.476235	+	0.879318i	$0.657999\pi$
$998$	0	0
$999$	22.2780	0.704846

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6800.2.a.bz.1.4		5
4.3	odd	2		425.2.a.i.1.3	✓	5
5.4	even	2		6800.2.a.cd.1.2		5
12.11	even	2		3825.2.a.bq.1.3		5
20.3	even	4		425.2.b.f.324.5		10
20.7	even	4		425.2.b.f.324.6		10
20.19	odd	2		425.2.a.j.1.3	yes	5
60.59	even	2		3825.2.a.bl.1.3		5
68.67	odd	2		7225.2.a.x.1.3		5
340.339	odd	2		7225.2.a.y.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
425.2.a.i.1.3	✓	5	4.3	odd	2
425.2.a.j.1.3	yes	5	20.19	odd	2
425.2.b.f.324.5		10	20.3	even	4
425.2.b.f.324.6		10	20.7	even	4
3825.2.a.bl.1.3		5	60.59	even	2
3825.2.a.bq.1.3		5	12.11	even	2
6800.2.a.bz.1.4		5	1.1	even	1	trivial
6800.2.a.cd.1.2		5	5.4	even	2
7225.2.a.x.1.3		5	68.67	odd	2
7225.2.a.y.1.3		5	340.339	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)

\(f(q)\)	\(=\)	\(q+1.96189 q^{3} +1.54475 q^{7} +0.849020 q^{9} -4.56006 q^{11} -1.09756 q^{13} +1.00000 q^{17} -4.67524 q^{19} +3.03063 q^{21} -0.529434 q^{23} -4.21999 q^{27} +8.06670 q^{29} +4.78005 q^{31} -8.94634 q^{33} -5.27917 q^{37} -2.15329 q^{39} -0.751460 q^{41} -9.49340 q^{43} -10.7419 q^{47} -4.61376 q^{49} +1.96189 q^{51} +0.0227951 q^{53} -9.17232 q^{57} +3.56962 q^{59} +3.92378 q^{61} +1.31152 q^{63} +9.75929 q^{67} -1.03869 q^{69} -1.21216 q^{71} +10.1135 q^{73} -7.04414 q^{77} -14.4151 q^{79} -10.8262 q^{81} +5.08949 q^{83} +15.8260 q^{87} -17.6123 q^{89} -1.69545 q^{91} +9.37794 q^{93} -6.78753 q^{97} -3.87158 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - q^{3} - q^{7} + 6 q^{9} - 4 q^{11} - 3 q^{13} + 5 q^{17} - 6 q^{19} - 5 q^{21} - 4 q^{23} + 5 q^{27} + 2 q^{29} - 21 q^{31} + 12 q^{33} - 2 q^{37} - 23 q^{39} - 8 q^{41} + 4 q^{43} + 2 q^{47} + 10 q^{49}+ \cdots + 14 q^{99}+O(q^{100}) \) 5 * q - q^3 - q^7 + 6 * q^9 - 4 * q^11 - 3 * q^13 + 5 * q^17 - 6 * q^19 - 5 * q^21 - 4 * q^23 + 5 * q^27 + 2 * q^29 - 21 * q^31 + 12 * q^33 - 2 * q^37 - 23 * q^39 - 8 * q^41 + 4 * q^43 + 2 * q^47 + 10 * q^49 - q^51 + 21 * q^53 - 20 * q^57 - 12 * q^59 - 2 * q^61 - 8 * q^63 + 12 * q^67 - 21 * q^71 + 22 * q^73 - 32 * q^77 - 41 * q^79 - 35 * q^81 + 8 * q^83 + 4 * q^87 - 10 * q^89 - 13 * q^91 + 9 * q^93 - 20 * q^97 + 14 * q^99

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