LMFDB - Embedded newform 4900.2.a.be.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4900,2,Mod(1,4900)] 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("4900.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4900, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4900 = 2^{2} \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4900.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,0,0,-6,0,0,0,0,0,0,0,-2,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)

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

Self dual:	yes
Analytic conductor:	$39.1266969904$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{3}, \sqrt{19})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 11x^{2} + 16 $ x^4 - 11*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 140)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$3.04547$ of defining polynomial
Character	$\chi$	$=$	4900.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205 q^{3} -5.27492 q^{11} +2.62685 q^{13} +0.418627 q^{17} +3.27492 q^{19} -7.82300 q^{23} -5.19615 q^{27} +4.27492 q^{29} +3.27492 q^{31} -9.13642 q^{33} -9.97368 q^{37} +4.54983 q^{39} -3.72508 q^{41} -2.15068 q^{43} +6.50958 q^{47} +0.725083 q^{51} -5.67232 q^{53} +5.67232 q^{57} -3.27492 q^{59} -13.5498 q^{61} +3.52165 q^{67} -13.5498 q^{69} -4.54983 q^{71} +6.50958 q^{73} +7.27492 q^{79} -9.00000 q^{81} -7.40437 q^{83} +7.40437 q^{87} -7.00000 q^{89} +5.67232 q^{93} -6.92820 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{11} - 2 q^{19} + 2 q^{29} - 2 q^{31} - 12 q^{39} - 30 q^{41} + 18 q^{51} + 2 q^{59} - 24 q^{61} - 24 q^{69} + 12 q^{71} + 14 q^{79} - 36 q^{81} - 28 q^{89}+O(q^{100}) $ 4 * q - 6 * q^11 - 2 * q^19 + 2 * q^29 - 2 * q^31 - 12 * q^39 - 30 * q^41 + 18 * q^51 + 2 * q^59 - 24 * q^61 - 24 * q^69 + 12 * q^71 + 14 * q^79 - 36 * q^81 - 28 * q^89

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
$3$	1.73205	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$3$	1.73205	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−5.27492	−1.59045	−0.795224	−	0.606316i	$-0.792647\pi$
$11$	−5.27492	−1.59045	−0.795224	+	0.606316i	$0.792647\pi$
$12$	0	0
$13$	2.62685	0.728557	0.364278	−	0.931290i	$-0.381316\pi$
$13$	2.62685	0.728557	0.364278	+	0.931290i	$0.381316\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.418627	0.101532	0.0507659	−	0.998711i	$-0.483834\pi$
$17$	0.418627	0.101532	0.0507659	+	0.998711i	$0.483834\pi$
$18$	0	0
$19$	3.27492	0.751318	0.375659	−	0.926758i	$-0.377416\pi$
$19$	3.27492	0.751318	0.375659	+	0.926758i	$0.377416\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−7.82300	−1.63121	−0.815604	−	0.578610i	$-0.803595\pi$
$23$	−7.82300	−1.63121	−0.815604	+	0.578610i	$0.803595\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−5.19615	−1.00000
$28$	0	0
$29$	4.27492	0.793832	0.396916	−	0.917855i	$-0.370080\pi$
$29$	4.27492	0.793832	0.396916	+	0.917855i	$0.370080\pi$
$30$	0	0
$31$	3.27492	0.588192	0.294096	−	0.955776i	$-0.404981\pi$
$31$	3.27492	0.588192	0.294096	+	0.955776i	$0.404981\pi$
$32$	0	0
$33$	−9.13642	−1.59045
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−9.97368	−1.63966	−0.819831	−	0.572605i	$-0.805933\pi$
$37$	−9.97368	−1.63966	−0.819831	+	0.572605i	$0.805933\pi$
$38$	0	0
$39$	4.54983	0.728557
$40$	0	0
$41$	−3.72508	−0.581760	−0.290880	−	0.956760i	$-0.593948\pi$
$41$	−3.72508	−0.581760	−0.290880	+	0.956760i	$0.593948\pi$
$42$	0	0
$43$	−2.15068	−0.327975	−0.163988	−	0.986462i	$-0.552436\pi$
$43$	−2.15068	−0.327975	−0.163988	+	0.986462i	$0.552436\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.50958	0.949519	0.474760	−	0.880115i	$-0.342535\pi$
$47$	6.50958	0.949519	0.474760	+	0.880115i	$0.342535\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0.725083	0.101532
$52$	0	0
$53$	−5.67232	−0.779153	−0.389577	−	0.920994i	$-0.627379\pi$
$53$	−5.67232	−0.779153	−0.389577	+	0.920994i	$0.627379\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	5.67232	0.751318
$58$	0	0
$59$	−3.27492	−0.426358	−0.213179	−	0.977013i	$-0.568382\pi$
$59$	−3.27492	−0.426358	−0.213179	+	0.977013i	$0.568382\pi$
$60$	0	0
$61$	−13.5498	−1.73488	−0.867439	−	0.497543i	$-0.834236\pi$
$61$	−13.5498	−1.73488	−0.867439	+	0.497543i	$0.834236\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	3.52165	0.430237	0.215119	−	0.976588i	$-0.430986\pi$
$67$	3.52165	0.430237	0.215119	+	0.976588i	$0.430986\pi$
$68$	0	0
$69$	−13.5498	−1.63121
$70$	0	0
$71$	−4.54983	−0.539966	−0.269983	−	0.962865i	$-0.587018\pi$
$71$	−4.54983	−0.539966	−0.269983	+	0.962865i	$0.587018\pi$
$72$	0	0
$73$	6.50958	0.761888	0.380944	−	0.924598i	$-0.375599\pi$
$73$	6.50958	0.761888	0.380944	+	0.924598i	$0.375599\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	7.27492	0.818492	0.409246	−	0.912424i	$-0.365792\pi$
$79$	7.27492	0.818492	0.409246	+	0.912424i	$0.365792\pi$
$80$	0	0
$81$	−9.00000	−1.00000
$82$	0	0
$83$	−7.40437	−0.812736	−0.406368	−	0.913710i	$-0.633205\pi$
$83$	−7.40437	−0.812736	−0.406368	+	0.913710i	$0.633205\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	7.40437	0.793832
$88$	0	0
$89$	−7.00000	−0.741999	−0.370999	−	0.928633i	$-0.620985\pi$
$89$	−7.00000	−0.741999	−0.370999	+	0.928633i	$0.620985\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	5.67232	0.588192
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−6.92820	−0.703452	−0.351726	−	0.936103i	$-0.614405\pi$
$97$	−6.92820	−0.703452	−0.351726	+	0.936103i	$0.614405\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−13.5498	−1.34826	−0.674129	−	0.738613i	$-0.735481\pi$
$101$	−13.5498	−1.34826	−0.674129	+	0.738613i	$0.735481\pi$
$102$	0	0
$103$	11.2871	1.11215	0.556076	−	0.831132i	$-0.312306\pi$
$103$	11.2871	1.11215	0.556076	+	0.831132i	$0.312306\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	3.52165	0.340450	0.170225	−	0.985405i	$-0.445550\pi$
$107$	3.52165	0.340450	0.170225	+	0.985405i	$0.445550\pi$
$108$	0	0
$109$	−11.5498	−1.10627	−0.553137	−	0.833090i	$-0.686569\pi$
$109$	−11.5498	−1.10627	−0.553137	+	0.833090i	$0.686569\pi$
$110$	0	0
$111$	−17.2749	−1.63966
$112$	0	0
$113$	4.30136	0.404637	0.202319	−	0.979320i	$-0.435152\pi$
$113$	4.30136	0.404637	0.202319	+	0.979320i	$0.435152\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	16.8248	1.52952
$122$	0	0
$123$	−6.45203	−0.581760
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−15.6460	−1.38836	−0.694179	−	0.719802i	$-0.744232\pi$
$127$	−15.6460	−1.38836	−0.694179	+	0.719802i	$0.744232\pi$
$128$	0	0
$129$	−3.72508	−0.327975
$130$	0	0
$131$	−10.7251	−0.937055	−0.468527	−	0.883449i	$-0.655215\pi$
$131$	−10.7251	−0.937055	−0.468527	+	0.883449i	$0.655215\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−21.3183	−1.82135	−0.910674	−	0.413126i	$-0.864437\pi$
$137$	−21.3183	−1.82135	−0.910674	+	0.413126i	$0.864437\pi$
$138$	0	0
$139$	13.0997	1.11110	0.555550	−	0.831483i	$-0.312508\pi$
$139$	13.0997	1.11110	0.555550	+	0.831483i	$0.312508\pi$
$140$	0	0
$141$	11.2749	0.949519
$142$	0	0
$143$	−13.8564	−1.15873
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−7.54983	−0.618507	−0.309253	−	0.950980i	$-0.600079\pi$
$149$	−7.54983	−0.618507	−0.309253	+	0.950980i	$0.600079\pi$
$150$	0	0
$151$	−12.7251	−1.03555	−0.517776	−	0.855516i	$-0.673240\pi$
$151$	−12.7251	−1.03555	−0.517776	+	0.855516i	$0.673240\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−2.20822	−0.176235	−0.0881176	−	0.996110i	$-0.528085\pi$
$157$	−2.20822	−0.176235	−0.0881176	+	0.996110i	$0.528085\pi$
$158$	0	0
$159$	−9.82475	−0.779153
$160$	0	0
$161$	0	0
$162$	0	0
$163$	5.67232	0.444291	0.222145	−	0.975014i	$-0.428694\pi$
$163$	5.67232	0.444291	0.222145	+	0.975014i	$0.428694\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−0.476171	−0.0368472	−0.0184236	−	0.999830i	$-0.505865\pi$
$167$	−0.476171	−0.0368472	−0.0184236	+	0.999830i	$0.505865\pi$
$168$	0	0
$169$	−6.09967	−0.469205
$170$	0	0
$171$	0	0
$172$	0	0
$173$	20.4811	1.55715	0.778573	−	0.627553i	$-0.215944\pi$
$173$	20.4811	1.55715	0.778573	+	0.627553i	$0.215944\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−5.67232	−0.426358
$178$	0	0
$179$	−7.27492	−0.543753	−0.271876	−	0.962332i	$-0.587644\pi$
$179$	−7.27492	−0.543753	−0.271876	+	0.962332i	$0.587644\pi$
$180$	0	0
$181$	−24.2749	−1.80434	−0.902170	−	0.431380i	$-0.858027\pi$
$181$	−24.2749	−1.80434	−0.902170	+	0.431380i	$0.858027\pi$
$182$	0	0
$183$	−23.4690	−1.73488
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−2.20822	−0.161481
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0.175248	0.0126805	0.00634026	−	0.999980i	$-0.497982\pi$
$191$	0.175248	0.0126805	0.00634026	+	0.999980i	$0.497982\pi$
$192$	0	0
$193$	21.3183	1.53453	0.767263	−	0.641332i	$-0.221618\pi$
$193$	21.3183	1.53453	0.767263	+	0.641332i	$0.221618\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	8.60271	0.612918	0.306459	−	0.951884i	$-0.400856\pi$
$197$	8.60271	0.612918	0.306459	+	0.951884i	$0.400856\pi$
$198$	0	0
$199$	17.2749	1.22459	0.612293	−	0.790631i	$-0.290247\pi$
$199$	17.2749	1.22459	0.612293	+	0.790631i	$0.290247\pi$
$200$	0	0
$201$	6.09967	0.430237
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−17.2749	−1.19493
$210$	0	0
$211$	25.6495	1.76578	0.882892	−	0.469576i	$-0.155593\pi$
$211$	25.6495	1.76578	0.882892	+	0.469576i	$0.155593\pi$
$212$	0	0
$213$	−7.88054	−0.539966
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	11.2749	0.761888
$220$	0	0
$221$	1.09967	0.0739717
$222$	0	0
$223$	−8.71780	−0.583787	−0.291893	−	0.956451i	$-0.594285\pi$
$223$	−8.71780	−0.583787	−0.291893	+	0.956451i	$0.594285\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−19.5287	−1.29617	−0.648084	−	0.761569i	$-0.724429\pi$
$227$	−19.5287	−1.29617	−0.648084	+	0.761569i	$0.724429\pi$
$228$	0	0
$229$	3.27492	0.216413	0.108206	−	0.994128i	$-0.465489\pi$
$229$	3.27492	0.216413	0.108206	+	0.994128i	$0.465489\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	14.2750	0.935189	0.467594	−	0.883943i	$-0.345121\pi$
$233$	14.2750	0.935189	0.467594	+	0.883943i	$0.345121\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	12.6005	0.818492
$238$	0	0
$239$	0.549834	0.0355658	0.0177829	−	0.999842i	$-0.494339\pi$
$239$	0.549834	0.0355658	0.0177829	+	0.999842i	$0.494339\pi$
$240$	0	0
$241$	9.82475	0.632868	0.316434	−	0.948615i	$-0.397514\pi$
$241$	9.82475	0.632868	0.316434	+	0.948615i	$0.397514\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	8.60271	0.547377
$248$	0	0
$249$	−12.8248	−0.812736
$250$	0	0
$251$	−20.5498	−1.29709	−0.648547	−	0.761175i	$-0.724623\pi$
$251$	−20.5498	−1.29709	−0.648547	+	0.761175i	$0.724623\pi$
$252$	0	0
$253$	41.2657	2.59435
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−11.6482	−0.726594	−0.363297	−	0.931673i	$-0.618349\pi$
$257$	−11.6482	−0.726594	−0.363297	+	0.931673i	$0.618349\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	0.779710	0.0480790	0.0240395	−	0.999711i	$-0.492347\pi$
$263$	0.779710	0.0480790	0.0240395	+	0.999711i	$0.492347\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−12.1244	−0.741999
$268$	0	0
$269$	−14.4502	−0.881042	−0.440521	−	0.897742i	$-0.645206\pi$
$269$	−14.4502	−0.881042	−0.440521	+	0.897742i	$0.645206\pi$
$270$	0	0
$271$	9.82475	0.596811	0.298406	−	0.954439i	$-0.403545\pi$
$271$	9.82475	0.596811	0.298406	+	0.954439i	$0.403545\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	14.2750	0.857704	0.428852	−	0.903375i	$-0.358918\pi$
$277$	14.2750	0.857704	0.428852	+	0.903375i	$0.358918\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	0	0
$283$	10.9260	0.649484	0.324742	−	0.945803i	$-0.394722\pi$
$283$	10.9260	0.649484	0.324742	+	0.945803i	$0.394722\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−16.8248	−0.989691
$290$	0	0
$291$	−12.0000	−0.703452
$292$	0	0
$293$	6.92820	0.404750	0.202375	−	0.979308i	$-0.435134\pi$
$293$	6.92820	0.404750	0.202375	+	0.979308i	$0.435134\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	27.4093	1.59045
$298$	0	0
$299$	−20.5498	−1.18843
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−23.4690	−1.34826
$304$	0	0
$305$	0	0
$306$	0	0
$307$	26.5145	1.51326	0.756631	−	0.653843i	$-0.226844\pi$
$307$	26.5145	1.51326	0.756631	+	0.653843i	$0.226844\pi$
$308$	0	0
$309$	19.5498	1.11215
$310$	0	0
$311$	−9.82475	−0.557111	−0.278555	−	0.960420i	$-0.589856\pi$
$311$	−9.82475	−0.557111	−0.278555	+	0.960420i	$0.589856\pi$
$312$	0	0
$313$	−33.5002	−1.89354	−0.946772	−	0.321904i	$-0.895677\pi$
$313$	−33.5002	−1.89354	−0.946772	+	0.321904i	$0.895677\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	25.6197	1.43894	0.719472	−	0.694521i	$-0.244384\pi$
$317$	25.6197	1.43894	0.719472	+	0.694521i	$0.244384\pi$
$318$	0	0
$319$	−22.5498	−1.26255
$320$	0	0
$321$	6.09967	0.340450
$322$	0	0
$323$	1.37097	0.0762827
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−20.0049	−1.10627
$328$	0	0
$329$	0	0
$330$	0	0
$331$	17.8248	0.979737	0.489868	−	0.871796i	$-0.337045\pi$
$331$	17.8248	0.979737	0.489868	+	0.871796i	$0.337045\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−4.30136	−0.234310	−0.117155	−	0.993114i	$-0.537377\pi$
$337$	−4.30136	−0.234310	−0.117155	+	0.993114i	$0.537377\pi$
$338$	0	0
$339$	7.45017	0.404637
$340$	0	0
$341$	−17.2749	−0.935489
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	12.1244	0.650870	0.325435	−	0.945564i	$-0.394489\pi$
$347$	12.1244	0.650870	0.325435	+	0.945564i	$0.394489\pi$
$348$	0	0
$349$	−3.72508	−0.199399	−0.0996996	−	0.995018i	$-0.531788\pi$
$349$	−3.72508	−0.199399	−0.0996996	+	0.995018i	$0.531788\pi$
$350$	0	0
$351$	−13.6495	−0.728557
$352$	0	0
$353$	8.18408	0.435595	0.217797	−	0.975994i	$-0.430113\pi$
$353$	8.18408	0.435595	0.217797	+	0.975994i	$0.430113\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	36.3746	1.91978	0.959889	−	0.280382i	$-0.0904610\pi$
$359$	36.3746	1.91978	0.959889	+	0.280382i	$0.0904610\pi$
$360$	0	0
$361$	−8.27492	−0.435522
$362$	0	0
$363$	29.1413	1.52952
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−6.03341	−0.314941	−0.157471	−	0.987524i	$-0.550334\pi$
$367$	−6.03341	−0.314941	−0.157471	+	0.987524i	$0.550334\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	9.97368	0.516417	0.258209	−	0.966089i	$-0.416868\pi$
$373$	9.97368	0.516417	0.258209	+	0.966089i	$0.416868\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	11.2296	0.578352
$378$	0	0
$379$	−21.6495	−1.11206	−0.556030	−	0.831162i	$-0.687676\pi$
$379$	−21.6495	−1.11206	−0.556030	+	0.831162i	$0.687676\pi$
$380$	0	0
$381$	−27.0997	−1.38836
$382$	0	0
$383$	−6.14849	−0.314173	−0.157087	−	0.987585i	$-0.550210\pi$
$383$	−6.14849	−0.314173	−0.157087	+	0.987585i	$0.550210\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	32.3746	1.64146	0.820728	−	0.571319i	$-0.193568\pi$
$389$	32.3746	1.64146	0.820728	+	0.571319i	$0.193568\pi$
$390$	0	0
$391$	−3.27492	−0.165620
$392$	0	0
$393$	−18.5764	−0.937055
$394$	0	0
$395$	0	0
$396$	0	0
$397$	10.8109	0.542585	0.271293	−	0.962497i	$-0.412549\pi$
$397$	10.8109	0.542585	0.271293	+	0.962497i	$0.412549\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	3.00000	0.149813	0.0749064	−	0.997191i	$-0.476134\pi$
$401$	3.00000	0.149813	0.0749064	+	0.997191i	$0.476134\pi$
$402$	0	0
$403$	8.60271	0.428532
$404$	0	0
$405$	0	0
$406$	0	0
$407$	52.6103	2.60780
$408$	0	0
$409$	−20.0997	−0.993865	−0.496932	−	0.867789i	$-0.665540\pi$
$409$	−20.0997	−0.993865	−0.496932	+	0.867789i	$0.665540\pi$
$410$	0	0
$411$	−36.9244	−1.82135
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	22.6893	1.11110
$418$	0	0
$419$	−13.0997	−0.639961	−0.319980	−	0.947424i	$-0.603676\pi$
$419$	−13.0997	−0.639961	−0.319980	+	0.947424i	$0.603676\pi$
$420$	0	0
$421$	−4.27492	−0.208347	−0.104173	−	0.994559i	$-0.533220\pi$
$421$	−4.27492	−0.208347	−0.104173	+	0.994559i	$0.533220\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−24.0000	−1.15873
$430$	0	0
$431$	−18.3746	−0.885073	−0.442536	−	0.896751i	$-0.645921\pi$
$431$	−18.3746	−0.885073	−0.442536	+	0.896751i	$0.645921\pi$
$432$	0	0
$433$	−18.1578	−0.872606	−0.436303	−	0.899800i	$-0.643712\pi$
$433$	−18.1578	−0.872606	−0.436303	+	0.899800i	$0.643712\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−25.6197	−1.22556
$438$	0	0
$439$	23.8248	1.13709	0.568547	−	0.822651i	$-0.307506\pi$
$439$	23.8248	1.13709	0.568547	+	0.822651i	$0.307506\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−12.1244	−0.576046	−0.288023	−	0.957624i	$-0.592998\pi$
$443$	−12.1244	−0.576046	−0.288023	+	0.957624i	$0.592998\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−13.0767	−0.618507
$448$	0	0
$449$	−3.17525	−0.149849	−0.0749246	−	0.997189i	$-0.523872\pi$
$449$	−3.17525	−0.149849	−0.0749246	+	0.997189i	$0.523872\pi$
$450$	0	0
$451$	19.6495	0.925259
$452$	0	0
$453$	−22.0405	−1.03555
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−1.37097	−0.0641312	−0.0320656	−	0.999486i	$-0.510209\pi$
$457$	−1.37097	−0.0641312	−0.0320656	+	0.999486i	$0.510209\pi$
$458$	0	0
$459$	−2.17525	−0.101532
$460$	0	0
$461$	−14.0000	−0.652045	−0.326023	−	0.945362i	$-0.605709\pi$
$461$	−14.0000	−0.652045	−0.326023	+	0.945362i	$0.605709\pi$
$462$	0	0
$463$	2.15068	0.0999505	0.0499752	−	0.998750i	$-0.484086\pi$
$463$	2.15068	0.0999505	0.0499752	+	0.998750i	$0.484086\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	15.7035	0.726673	0.363337	−	0.931658i	$-0.381637\pi$
$467$	15.7035	0.726673	0.363337	+	0.931658i	$0.381637\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−3.82475	−0.176235
$472$	0	0
$473$	11.3446	0.521627
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	9.82475	0.448904	0.224452	−	0.974485i	$-0.427941\pi$
$479$	9.82475	0.448904	0.224452	+	0.974485i	$0.427941\pi$
$480$	0	0
$481$	−26.1993	−1.19459
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	2.93039	0.132789	0.0663943	−	0.997793i	$-0.478850\pi$
$487$	2.93039	0.132789	0.0663943	+	0.997793i	$0.478850\pi$
$488$	0	0
$489$	9.82475	0.444291
$490$	0	0
$491$	−28.5498	−1.28844	−0.644218	−	0.764842i	$-0.722817\pi$
$491$	−28.5498	−1.28844	−0.644218	+	0.764842i	$0.722817\pi$
$492$	0	0
$493$	1.78959	0.0805993
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	1.62541	0.0727635	0.0363818	−	0.999338i	$-0.488417\pi$
$499$	1.62541	0.0727635	0.0363818	+	0.999338i	$0.488417\pi$
$500$	0	0
$501$	−0.824752	−0.0368472
$502$	0	0
$503$	31.7682	1.41647	0.708236	−	0.705975i	$-0.249491\pi$
$503$	31.7682	1.41647	0.708236	+	0.705975i	$0.249491\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−10.5649	−0.469205
$508$	0	0
$509$	−14.4502	−0.640492	−0.320246	−	0.947334i	$-0.603766\pi$
$509$	−14.4502	−0.640492	−0.320246	+	0.947334i	$0.603766\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−17.0170	−0.751318
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−34.3375	−1.51016
$518$	0	0
$519$	35.4743	1.55715
$520$	0	0
$521$	9.82475	0.430430	0.215215	−	0.976567i	$-0.430955\pi$
$521$	9.82475	0.430430	0.215215	+	0.976567i	$0.430955\pi$
$522$	0	0
$523$	7.34683	0.321254	0.160627	−	0.987015i	$-0.448648\pi$
$523$	7.34683	0.321254	0.160627	+	0.987015i	$0.448648\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.37097	0.0597203
$528$	0	0
$529$	38.1993	1.66084
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−9.78523	−0.423845
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−12.6005	−0.543753
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−17.5498	−0.754526	−0.377263	−	0.926106i	$-0.623135\pi$
$541$	−17.5498	−0.754526	−0.377263	+	0.926106i	$0.623135\pi$
$542$	0	0
$543$	−42.0454	−1.80434
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−20.5386	−0.878168	−0.439084	−	0.898446i	$-0.644697\pi$
$547$	−20.5386	−0.878168	−0.439084	+	0.898446i	$0.644697\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	14.0000	0.596420
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	9.97368	0.422598	0.211299	−	0.977421i	$-0.432231\pi$
$557$	9.97368	0.422598	0.211299	+	0.977421i	$0.432231\pi$
$558$	0	0
$559$	−5.64950	−0.238949
$560$	0	0
$561$	−3.82475	−0.161481
$562$	0	0
$563$	−22.6317	−0.953814	−0.476907	−	0.878954i	$-0.658242\pi$
$563$	−22.6317	−0.953814	−0.476907	+	0.878954i	$0.658242\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	8.37459	0.351081	0.175540	−	0.984472i	$-0.443833\pi$
$569$	8.37459	0.351081	0.175540	+	0.984472i	$0.443833\pi$
$570$	0	0
$571$	7.27492	0.304446	0.152223	−	0.988346i	$-0.451357\pi$
$571$	7.27492	0.304446	0.152223	+	0.988346i	$0.451357\pi$
$572$	0	0
$573$	0.303539	0.0126805
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−3.88273	−0.161640	−0.0808200	−	0.996729i	$-0.525754\pi$
$577$	−3.88273	−0.161640	−0.0808200	+	0.996729i	$0.525754\pi$
$578$	0	0
$579$	36.9244	1.53453
$580$	0	0
$581$	0	0
$582$	0	0
$583$	29.9210	1.23920
$584$	0	0
$585$	0	0
$586$	0	0
$587$	20.8997	0.862623	0.431311	−	0.902203i	$-0.358051\pi$
$587$	20.8997	0.862623	0.431311	+	0.902203i	$0.358051\pi$
$588$	0	0
$589$	10.7251	0.441919
$590$	0	0
$591$	14.9003	0.612918
$592$	0	0
$593$	33.3851	1.37096	0.685482	−	0.728090i	$-0.259592\pi$
$593$	33.3851	1.37096	0.685482	+	0.728090i	$0.259592\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	29.9210	1.22459
$598$	0	0
$599$	5.27492	0.215527	0.107764	−	0.994177i	$-0.465631\pi$
$599$	5.27492	0.215527	0.107764	+	0.994177i	$0.465631\pi$
$600$	0	0
$601$	14.0000	0.571072	0.285536	−	0.958368i	$-0.407828\pi$
$601$	14.0000	0.571072	0.285536	+	0.958368i	$0.407828\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	11.4022	0.462801	0.231400	−	0.972859i	$-0.425669\pi$
$607$	11.4022	0.462801	0.231400	+	0.972859i	$0.425669\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	17.0997	0.691779
$612$	0	0
$613$	−28.3616	−1.14551	−0.572757	−	0.819725i	$-0.694126\pi$
$613$	−28.3616	−1.14551	−0.572757	+	0.819725i	$0.694126\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−31.2920	−1.25977	−0.629884	−	0.776689i	$-0.716898\pi$
$617$	−31.2920	−1.25977	−0.629884	+	0.776689i	$0.716898\pi$
$618$	0	0
$619$	−8.92442	−0.358703	−0.179351	−	0.983785i	$-0.557400\pi$
$619$	−8.92442	−0.358703	−0.179351	+	0.983785i	$0.557400\pi$
$620$	0	0
$621$	40.6495	1.63121
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−29.9210	−1.19493
$628$	0	0
$629$	−4.17525	−0.166478
$630$	0	0
$631$	33.0997	1.31768	0.658839	−	0.752284i	$-0.271048\pi$
$631$	33.0997	1.31768	0.658839	+	0.752284i	$0.271048\pi$
$632$	0	0
$633$	44.4262	1.76578
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−2.09967	−0.0829319	−0.0414660	−	0.999140i	$-0.513203\pi$
$641$	−2.09967	−0.0829319	−0.0414660	+	0.999140i	$0.513203\pi$
$642$	0	0
$643$	−31.4071	−1.23857	−0.619287	−	0.785164i	$-0.712578\pi$
$643$	−31.4071	−1.23857	−0.619287	+	0.785164i	$0.712578\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	26.9331	1.05885	0.529425	−	0.848357i	$-0.322408\pi$
$647$	26.9331	1.05885	0.529425	+	0.848357i	$0.322408\pi$
$648$	0	0
$649$	17.2749	0.678100
$650$	0	0
$651$	0	0
$652$	0	0
$653$	28.3616	1.10988	0.554938	−	0.831892i	$-0.312742\pi$
$653$	28.3616	1.10988	0.554938	+	0.831892i	$0.312742\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−40.5498	−1.57960	−0.789799	−	0.613366i	$-0.789815\pi$
$659$	−40.5498	−1.57960	−0.789799	+	0.613366i	$0.789815\pi$
$660$	0	0
$661$	−0.450166	−0.0175094	−0.00875471	−	0.999962i	$-0.502787\pi$
$661$	−0.450166	−0.0175094	−0.00875471	+	0.999962i	$0.502787\pi$
$662$	0	0
$663$	1.90468	0.0739717
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−33.4427	−1.29491
$668$	0	0
$669$	−15.0997	−0.583787
$670$	0	0
$671$	71.4743	2.75923
$672$	0	0
$673$	−31.2920	−1.20622	−0.603109	−	0.797659i	$-0.706072\pi$
$673$	−31.2920	−1.20622	−0.603109	+	0.797659i	$0.706072\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	46.4043	1.78346	0.891731	−	0.452566i	$-0.149491\pi$
$677$	46.4043	1.78346	0.891731	+	0.452566i	$0.149491\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−33.8248	−1.29617
$682$	0	0
$683$	19.1676	0.733430	0.366715	−	0.930333i	$-0.380482\pi$
$683$	19.1676	0.733430	0.366715	+	0.930333i	$0.380482\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	5.67232	0.216413
$688$	0	0
$689$	−14.9003	−0.567657
$690$	0	0
$691$	30.3746	1.15550	0.577752	−	0.816212i	$-0.303930\pi$
$691$	30.3746	1.15550	0.577752	+	0.816212i	$0.303930\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−1.55942	−0.0590672
$698$	0	0
$699$	24.7251	0.935189
$700$	0	0
$701$	8.82475	0.333306	0.166653	−	0.986016i	$-0.446704\pi$
$701$	8.82475	0.333306	0.166653	+	0.986016i	$0.446704\pi$
$702$	0	0
$703$	−32.6630	−1.23191
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−10.4502	−0.392464	−0.196232	−	0.980557i	$-0.562871\pi$
$709$	−10.4502	−0.392464	−0.196232	+	0.980557i	$0.562871\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−25.6197	−0.959465
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0.952341	0.0355658
$718$	0	0
$719$	−30.3746	−1.13278	−0.566390	−	0.824137i	$-0.691661\pi$
$719$	−30.3746	−1.13278	−0.566390	+	0.824137i	$0.691661\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	17.0170	0.632868
$724$	0	0
$725$	0	0
$726$	0	0
$727$	3.10302	0.115085	0.0575423	−	0.998343i	$-0.481674\pi$
$727$	3.10302	0.115085	0.0575423	+	0.998343i	$0.481674\pi$
$728$	0	0
$729$	27.0000	1.00000
$730$	0	0
$731$	−0.900331	−0.0332999
$732$	0	0
$733$	37.6865	1.39198	0.695991	−	0.718050i	$-0.254965\pi$
$733$	37.6865	1.39198	0.695991	+	0.718050i	$0.254965\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−18.5764	−0.684270
$738$	0	0
$739$	−20.9244	−0.769717	−0.384859	−	0.922976i	$-0.625750\pi$
$739$	−20.9244	−0.769717	−0.384859	+	0.922976i	$0.625750\pi$
$740$	0	0
$741$	14.9003	0.547377
$742$	0	0
$743$	−6.45203	−0.236702	−0.118351	−	0.992972i	$-0.537761\pi$
$743$	−6.45203	−0.236702	−0.118351	+	0.992972i	$0.537761\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−14.7251	−0.537326	−0.268663	−	0.963234i	$-0.586582\pi$
$751$	−14.7251	−0.537326	−0.268663	+	0.963234i	$0.586582\pi$
$752$	0	0
$753$	−35.5934	−1.29709
$754$	0	0
$755$	0	0
$756$	0	0
$757$	35.5934	1.29366	0.646831	−	0.762633i	$-0.276094\pi$
$757$	35.5934	1.29366	0.646831	+	0.762633i	$0.276094\pi$
$758$	0	0
$759$	71.4743	2.59435
$760$	0	0
$761$	22.9244	0.831010	0.415505	−	0.909591i	$-0.363605\pi$
$761$	22.9244	0.831010	0.415505	+	0.909591i	$0.363605\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−8.60271	−0.310626
$768$	0	0
$769$	−14.0000	−0.504853	−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000	−0.504853	−0.252426	+	0.967616i	$0.581229\pi$
$770$	0	0
$771$	−20.1752	−0.726594
$772$	0	0
$773$	−40.3133	−1.44997	−0.724985	−	0.688765i	$-0.758153\pi$
$773$	−40.3133	−1.44997	−0.724985	+	0.688765i	$0.758153\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−12.1993	−0.437087
$780$	0	0
$781$	24.0000	0.858788
$782$	0	0
$783$	−22.2131	−0.793832
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−1.73205	−0.0617409	−0.0308705	−	0.999523i	$-0.509828\pi$
$787$	−1.73205	−0.0617409	−0.0308705	+	0.999523i	$0.509828\pi$
$788$	0	0
$789$	1.35050	0.0480790
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−35.5934	−1.26396
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−46.8229	−1.65855	−0.829276	−	0.558839i	$-0.811247\pi$
$797$	−46.8229	−1.65855	−0.829276	+	0.558839i	$0.811247\pi$
$798$	0	0
$799$	2.72508	0.0964065
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−34.3375	−1.21174
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−25.0284	−0.881042
$808$	0	0
$809$	−17.1993	−0.604697	−0.302348	−	0.953198i	$-0.597771\pi$
$809$	−17.1993	−0.604697	−0.302348	+	0.953198i	$0.597771\pi$
$810$	0	0
$811$	7.45017	0.261611	0.130805	−	0.991408i	$-0.458244\pi$
$811$	7.45017	0.261611	0.130805	+	0.991408i	$0.458244\pi$
$812$	0	0
$813$	17.0170	0.596811
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−7.04329	−0.246414
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−20.3746	−0.711078	−0.355539	−	0.934661i	$-0.615703\pi$
$821$	−20.3746	−0.711078	−0.355539	+	0.934661i	$0.615703\pi$
$822$	0	0
$823$	46.1583	1.60898	0.804488	−	0.593968i	$-0.202440\pi$
$823$	46.1583	1.60898	0.804488	+	0.593968i	$0.202440\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−15.0547	−0.523505	−0.261752	−	0.965135i	$-0.584300\pi$
$827$	−15.0547	−0.523505	−0.261752	+	0.965135i	$0.584300\pi$
$828$	0	0
$829$	−50.9244	−1.76868	−0.884339	−	0.466845i	$-0.845391\pi$
$829$	−50.9244	−1.76868	−0.884339	+	0.466845i	$0.845391\pi$
$830$	0	0
$831$	24.7251	0.857704
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−17.0170	−0.588192
$838$	0	0
$839$	41.0997	1.41892	0.709459	−	0.704747i	$-0.248939\pi$
$839$	41.0997	1.41892	0.709459	+	0.704747i	$0.248939\pi$
$840$	0	0
$841$	−10.7251	−0.369830
$842$	0	0
$843$	10.3923	0.357930
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	18.9244	0.649484
$850$	0	0
$851$	78.0241	2.67463
$852$	0	0
$853$	13.1342	0.449708	0.224854	−	0.974392i	$-0.427810\pi$
$853$	13.1342	0.449708	0.224854	+	0.974392i	$0.427810\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	37.6865	1.28735	0.643673	−	0.765301i	$-0.277410\pi$
$857$	37.6865	1.28735	0.643673	+	0.765301i	$0.277410\pi$
$858$	0	0
$859$	−2.37459	−0.0810198	−0.0405099	−	0.999179i	$-0.512898\pi$
$859$	−2.37459	−0.0810198	−0.0405099	+	0.999179i	$0.512898\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−16.4257	−0.559138	−0.279569	−	0.960126i	$-0.590192\pi$
$863$	−16.4257	−0.559138	−0.279569	+	0.960126i	$0.590192\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−29.1413	−0.989691
$868$	0	0
$869$	−38.3746	−1.30177
$870$	0	0
$871$	9.25083	0.313452
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−22.8777	−0.772527	−0.386263	−	0.922389i	$-0.626234\pi$
$877$	−22.8777	−0.772527	−0.386263	+	0.922389i	$0.626234\pi$
$878$	0	0
$879$	12.0000	0.404750
$880$	0	0
$881$	−43.0241	−1.44952	−0.724759	−	0.689002i	$-0.758049\pi$
$881$	−43.0241	−1.44952	−0.724759	+	0.689002i	$0.758049\pi$
$882$	0	0
$883$	55.5407	1.86909	0.934547	−	0.355840i	$-0.115805\pi$
$883$	55.5407	1.86909	0.934547	+	0.355840i	$0.115805\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−39.2301	−1.31722	−0.658609	−	0.752486i	$-0.728855\pi$
$887$	−39.2301	−1.31722	−0.658609	+	0.752486i	$0.728855\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	47.4743	1.59045
$892$	0	0
$893$	21.3183	0.713391
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−35.5934	−1.18843
$898$	0	0
$899$	14.0000	0.466926
$900$	0	0
$901$	−2.37459	−0.0791089
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−41.8569	−1.38984	−0.694918	−	0.719089i	$-0.744560\pi$
$907$	−41.8569	−1.38984	−0.694918	+	0.719089i	$0.744560\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−25.0997	−0.831589	−0.415795	−	0.909459i	$-0.636496\pi$
$911$	−25.0997	−0.831589	−0.415795	+	0.909459i	$0.636496\pi$
$912$	0	0
$913$	39.0575	1.29261
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−46.9244	−1.54789	−0.773947	−	0.633250i	$-0.781720\pi$
$919$	−46.9244	−1.54789	−0.773947	+	0.633250i	$0.781720\pi$
$920$	0	0
$921$	45.9244	1.51326
$922$	0	0
$923$	−11.9517	−0.393396
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−48.0997	−1.57810	−0.789049	−	0.614330i	$-0.789427\pi$
$929$	−48.0997	−1.57810	−0.789049	+	0.614330i	$0.789427\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−17.0170	−0.557111
$934$	0	0
$935$	0	0
$936$	0	0
$937$	24.3638	0.795931	0.397965	−	0.917400i	$-0.369716\pi$
$937$	24.3638	0.795931	0.397965	+	0.917400i	$0.369716\pi$
$938$	0	0
$939$	−58.0241	−1.89354
$940$	0	0
$941$	−3.27492	−0.106759	−0.0533796	−	0.998574i	$-0.516999\pi$
$941$	−3.27492	−0.106759	−0.0533796	+	0.998574i	$0.516999\pi$
$942$	0	0
$943$	29.1413	0.948972
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−10.5649	−0.343314	−0.171657	−	0.985157i	$-0.554912\pi$
$947$	−10.5649	−0.343314	−0.171657	+	0.985157i	$0.554912\pi$
$948$	0	0
$949$	17.0997	0.555079
$950$	0	0
$951$	44.3746	1.43894
$952$	0	0
$953$	22.6893	0.734978	0.367489	−	0.930028i	$-0.380218\pi$
$953$	22.6893	0.734978	0.367489	+	0.930028i	$0.380218\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−39.0575	−1.26255
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−20.2749	−0.654030
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−2.15068	−0.0691611	−0.0345806	−	0.999402i	$-0.511010\pi$
$967$	−2.15068	−0.0691611	−0.0345806	+	0.999402i	$0.511010\pi$
$968$	0	0
$969$	2.37459	0.0762827
$970$	0	0
$971$	−36.9244	−1.18496	−0.592481	−	0.805585i	$-0.701851\pi$
$971$	−36.9244	−1.18496	−0.592481	+	0.805585i	$0.701851\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−29.9210	−0.957259	−0.478629	−	0.878017i	$-0.658866\pi$
$977$	−29.9210	−0.957259	−0.478629	+	0.878017i	$0.658866\pi$
$978$	0	0
$979$	36.9244	1.18011
$980$	0	0
$981$	0	0
$982$	0	0
$983$	45.9281	1.46488	0.732440	−	0.680832i	$-0.238382\pi$
$983$	45.9281	1.46488	0.732440	+	0.680832i	$0.238382\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	16.8248	0.534996
$990$	0	0
$991$	33.4743	1.06334	0.531672	−	0.846950i	$-0.321564\pi$
$991$	33.4743	1.06334	0.531672	+	0.846950i	$0.321564\pi$
$992$	0	0
$993$	30.8734	0.979737
$994$	0	0
$995$	0	0
$996$	0	0
$997$	0.418627	0.0132580	0.00662902	−	0.999978i	$-0.497890\pi$
$997$	0.418627	0.0132580	0.00662902	+	0.999978i	$0.497890\pi$
$998$	0	0
$999$	51.8248	1.63966

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4900.2.a.be.1.3		4
5.2	odd	4		980.2.e.f.589.1		4
5.3	odd	4		980.2.e.f.589.3		4
5.4	even	2	inner	4900.2.a.be.1.1		4
7.2	even	3		700.2.i.f.501.2		8
7.4	even	3		700.2.i.f.401.2		8
7.6	odd	2		4900.2.a.bf.1.1		4
35.2	odd	12		140.2.q.b.109.2	yes	4
35.3	even	12		980.2.q.b.569.2		4
35.4	even	6		700.2.i.f.401.3		8
35.9	even	6		700.2.i.f.501.3		8
35.12	even	12		980.2.q.b.949.1		4
35.13	even	4		980.2.e.c.589.2		4
35.17	even	12		980.2.q.g.569.1		4
35.18	odd	12		140.2.q.b.9.1	yes	4
35.23	odd	12		140.2.q.a.109.2	yes	4
35.27	even	4		980.2.e.c.589.4		4
35.32	odd	12		140.2.q.a.9.2	✓	4
35.33	even	12		980.2.q.g.949.1		4
35.34	odd	2		4900.2.a.bf.1.3		4
105.2	even	12		1260.2.bm.b.109.1		4
105.23	even	12		1260.2.bm.a.109.1		4
105.32	even	12		1260.2.bm.a.289.1		4
105.53	even	12		1260.2.bm.b.289.2		4
140.23	even	12		560.2.bw.e.529.2		4
140.67	even	12		560.2.bw.e.289.2		4
140.107	even	12		560.2.bw.a.529.2		4
140.123	even	12		560.2.bw.a.289.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.2.q.a.9.2	✓	4	35.32	odd	12
140.2.q.a.109.2	yes	4	35.23	odd	12
140.2.q.b.9.1	yes	4	35.18	odd	12
140.2.q.b.109.2	yes	4	35.2	odd	12
560.2.bw.a.289.1		4	140.123	even	12
560.2.bw.a.529.2		4	140.107	even	12
560.2.bw.e.289.2		4	140.67	even	12
560.2.bw.e.529.2		4	140.23	even	12
700.2.i.f.401.2		8	7.4	even	3
700.2.i.f.401.3		8	35.4	even	6
700.2.i.f.501.2		8	7.2	even	3
700.2.i.f.501.3		8	35.9	even	6
980.2.e.c.589.2		4	35.13	even	4
980.2.e.c.589.4		4	35.27	even	4
980.2.e.f.589.1		4	5.2	odd	4
980.2.e.f.589.3		4	5.3	odd	4
980.2.q.b.569.2		4	35.3	even	12
980.2.q.b.949.1		4	35.12	even	12
980.2.q.g.569.1		4	35.17	even	12
980.2.q.g.949.1		4	35.33	even	12
1260.2.bm.a.109.1		4	105.23	even	12
1260.2.bm.a.289.1		4	105.32	even	12
1260.2.bm.b.109.1		4	105.2	even	12
1260.2.bm.b.289.2		4	105.53	even	12
4900.2.a.be.1.1		4	5.4	even	2	inner
4900.2.a.be.1.3		4	1.1	even	1	trivial
4900.2.a.bf.1.1		4	7.6	odd	2
4900.2.a.bf.1.3		4	35.34	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:	\( N \)	\(=\)	\( 4900 = 2^{2} \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4900.a (trivial)

\(f(q)\)	\(=\)	\(q+1.73205 q^{3} -5.27492 q^{11} +2.62685 q^{13} +0.418627 q^{17} +3.27492 q^{19} -7.82300 q^{23} -5.19615 q^{27} +4.27492 q^{29} +3.27492 q^{31} -9.13642 q^{33} -9.97368 q^{37} +4.54983 q^{39} -3.72508 q^{41} -2.15068 q^{43} +6.50958 q^{47} +0.725083 q^{51} -5.67232 q^{53} +5.67232 q^{57} -3.27492 q^{59} -13.5498 q^{61} +3.52165 q^{67} -13.5498 q^{69} -4.54983 q^{71} +6.50958 q^{73} +7.27492 q^{79} -9.00000 q^{81} -7.40437 q^{83} +7.40437 q^{87} -7.00000 q^{89} +5.67232 q^{93} -6.92820 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{11} - 2 q^{19} + 2 q^{29} - 2 q^{31} - 12 q^{39} - 30 q^{41} + 18 q^{51} + 2 q^{59} - 24 q^{61} - 24 q^{69} + 12 q^{71} + 14 q^{79} - 36 q^{81} - 28 q^{89}+O(q^{100}) \) 4 * q - 6 * q^11 - 2 * q^19 + 2 * q^29 - 2 * q^31 - 12 * q^39 - 30 * q^41 + 18 * q^51 + 2 * q^59 - 24 * q^61 - 24 * q^69 + 12 * q^71 + 14 * q^79 - 36 * q^81 - 28 * q^89

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