LMFDB - Embedded newform 2300.4.a.c.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2300,4,Mod(1,2300)] 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("2300.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2300, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 2300 = 2^{2} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2300.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,0,3,0,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$135.704393013$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 80x^{3} + 121x^{2} + 1212x + 1044 $ x^5 - 2x^4 - 80x^3 + 121x^2 + 1212x + 1044
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 460)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$6.72467$ of defining polynomial
Character	$\chi$	$=$	2300.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-5.72467 q^{3} +12.6137 q^{7} +5.77185 q^{9} -21.9786 q^{11} -37.2893 q^{13} -62.5817 q^{17} -137.633 q^{19} -72.2093 q^{21} +23.0000 q^{23} +121.524 q^{27} -71.8975 q^{29} -219.731 q^{31} +125.820 q^{33} +296.223 q^{37} +213.469 q^{39} -305.282 q^{41} -140.234 q^{43} +185.509 q^{47} -183.895 q^{49} +358.260 q^{51} -312.176 q^{53} +787.903 q^{57} -161.771 q^{59} +835.780 q^{61} +72.8043 q^{63} +24.5018 q^{67} -131.667 q^{69} -973.973 q^{71} +343.447 q^{73} -277.231 q^{77} +1214.62 q^{79} -851.526 q^{81} -749.055 q^{83} +411.590 q^{87} +811.662 q^{89} -470.356 q^{91} +1257.89 q^{93} -1026.90 q^{97} -126.857 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 3 q^{3} - 8 q^{7} + 30 q^{9} + 7 q^{11} - 5 q^{13} + 24 q^{17} - 13 q^{19} + 115 q^{23} + 204 q^{27} - 253 q^{29} - 98 q^{31} + 473 q^{33} + 435 q^{37} - 410 q^{39} - 774 q^{41} + 498 q^{43} + 572 q^{47}+ \cdots + 111 q^{99}+O(q^{100}) $ 5 * q + 3 * q^3 - 8 * q^7 + 30 * q^9 + 7 * q^11 - 5 * q^13 + 24 * q^17 - 13 * q^19 + 115 * q^23 + 204 * q^27 - 253 * q^29 - 98 * q^31 + 473 * q^33 + 435 * q^37 - 410 * q^39 - 774 * q^41 + 498 * q^43 + 572 * q^47 - 683 * q^49 - 657 * q^51 + 665 * q^53 + 932 * q^57 - 763 * q^59 + 337 * q^61 + 1527 * q^63 + 305 * q^67 + 69 * q^69 - 1504 * q^71 + 1304 * q^73 - 182 * q^77 + 626 * q^79 - 959 * q^81 + 1703 * q^83 + 1354 * q^87 + 646 * q^89 - 767 * q^91 + 452 * q^93 + 233 * q^97 + 111 * 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
$3$	−5.72467	−1.10171	−0.550857	−	0.834600i	$-0.685699\pi$
$3$	−5.72467	−1.10171	−0.550857	+	0.834600i	$0.685699\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	12.6137	0.681076	0.340538	−	0.940231i	$-0.389391\pi$
$7$	12.6137	0.681076	0.340538	+	0.940231i	$0.389391\pi$
$8$	0	0
$9$	5.77185	0.213772
$10$	0	0
$11$	−21.9786	−0.602435	−0.301217	−	0.953555i	$-0.597393\pi$
$11$	−21.9786	−0.602435	−0.301217	+	0.953555i	$0.597393\pi$
$12$	0	0
$13$	−37.2893	−0.795554	−0.397777	−	0.917482i	$-0.630218\pi$
$13$	−37.2893	−0.795554	−0.397777	+	0.917482i	$0.630218\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−62.5817	−0.892841	−0.446420	−	0.894823i	$-0.647301\pi$
$17$	−62.5817	−0.892841	−0.446420	+	0.894823i	$0.647301\pi$
$18$	0	0
$19$	−137.633	−1.66185	−0.830925	−	0.556384i	$-0.812188\pi$
$19$	−137.633	−1.66185	−0.830925	+	0.556384i	$0.812188\pi$
$20$	0	0
$21$	−72.2093	−0.750350
$22$	0	0
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	121.524	0.866198
$28$	0	0
$29$	−71.8975	−0.460380	−0.230190	−	0.973146i	$-0.573935\pi$
$29$	−71.8975	−0.460380	−0.230190	+	0.973146i	$0.573935\pi$
$30$	0	0
$31$	−219.731	−1.27306	−0.636531	−	0.771251i	$-0.719631\pi$
$31$	−219.731	−1.27306	−0.636531	+	0.771251i	$0.719631\pi$
$32$	0	0
$33$	125.820	0.663710
$34$	0	0
$35$	0	0
$36$	0	0
$37$	296.223	1.31618	0.658091	−	0.752939i	$-0.271364\pi$
$37$	296.223	1.31618	0.658091	+	0.752939i	$0.271364\pi$
$38$	0	0
$39$	213.469	0.876473
$40$	0	0
$41$	−305.282	−1.16285	−0.581427	−	0.813599i	$-0.697505\pi$
$41$	−305.282	−1.16285	−0.581427	+	0.813599i	$0.697505\pi$
$42$	0	0
$43$	−140.234	−0.497338	−0.248669	−	0.968588i	$-0.579993\pi$
$43$	−140.234	−0.497338	−0.248669	+	0.968588i	$0.579993\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	185.509	0.575730	0.287865	−	0.957671i	$-0.407055\pi$
$47$	185.509	0.575730	0.287865	+	0.957671i	$0.407055\pi$
$48$	0	0
$49$	−183.895	−0.536136
$50$	0	0
$51$	358.260	0.983654
$52$	0	0
$53$	−312.176	−0.809069	−0.404534	−	0.914523i	$-0.632566\pi$
$53$	−312.176	−0.809069	−0.404534	+	0.914523i	$0.632566\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	787.903	1.83088
$58$	0	0
$59$	−161.771	−0.356964	−0.178482	−	0.983943i	$-0.557119\pi$
$59$	−161.771	−0.356964	−0.178482	+	0.983943i	$0.557119\pi$
$60$	0	0
$61$	835.780	1.75427	0.877137	−	0.480240i	$-0.159450\pi$
$61$	835.780	1.75427	0.877137	+	0.480240i	$0.159450\pi$
$62$	0	0
$63$	72.8043	0.145595
$64$	0	0
$65$	0	0
$66$	0	0
$67$	24.5018	0.0446772	0.0223386	−	0.999750i	$-0.492889\pi$
$67$	24.5018	0.0446772	0.0223386	+	0.999750i	$0.492889\pi$
$68$	0	0
$69$	−131.667	−0.229723
$70$	0	0
$71$	−973.973	−1.62802	−0.814009	−	0.580852i	$-0.802719\pi$
$71$	−973.973	−1.62802	−0.814009	+	0.580852i	$0.802719\pi$
$72$	0	0
$73$	343.447	0.550650	0.275325	−	0.961351i	$-0.411214\pi$
$73$	343.447	0.550650	0.275325	+	0.961351i	$0.411214\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−277.231	−0.410304
$78$	0	0
$79$	1214.62	1.72981	0.864905	−	0.501935i	$-0.167378\pi$
$79$	1214.62	1.72981	0.864905	+	0.501935i	$0.167378\pi$
$80$	0	0
$81$	−851.526	−1.16807
$82$	0	0
$83$	−749.055	−0.990597	−0.495298	−	0.868723i	$-0.664941\pi$
$83$	−749.055	−0.990597	−0.495298	+	0.868723i	$0.664941\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	411.590	0.507207
$88$	0	0
$89$	811.662	0.966696	0.483348	−	0.875428i	$-0.339421\pi$
$89$	811.662	0.966696	0.483348	+	0.875428i	$0.339421\pi$
$90$	0	0
$91$	−470.356	−0.541833
$92$	0	0
$93$	1257.89	1.40255
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−1026.90	−1.07491	−0.537455	−	0.843293i	$-0.680614\pi$
$97$	−1026.90	−1.07491	−0.537455	+	0.843293i	$0.680614\pi$
$98$	0	0
$99$	−126.857	−0.128784
$100$	0	0
$101$	−788.330	−0.776651	−0.388326	−	0.921522i	$-0.626946\pi$
$101$	−788.330	−0.776651	−0.388326	+	0.921522i	$0.626946\pi$
$102$	0	0
$103$	−471.764	−0.451304	−0.225652	−	0.974208i	$-0.572451\pi$
$103$	−471.764	−0.451304	−0.225652	+	0.974208i	$0.572451\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1402.85	1.26746	0.633732	−	0.773553i	$-0.281522\pi$
$107$	1402.85	1.26746	0.633732	+	0.773553i	$0.281522\pi$
$108$	0	0
$109$	−2151.31	−1.89044	−0.945221	−	0.326432i	$-0.894154\pi$
$109$	−2151.31	−1.89044	−0.945221	+	0.326432i	$0.894154\pi$
$110$	0	0
$111$	−1695.78	−1.45005
$112$	0	0
$113$	−754.833	−0.628396	−0.314198	−	0.949358i	$-0.601736\pi$
$113$	−754.833	−0.628396	−0.314198	+	0.949358i	$0.601736\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−215.228	−0.170067
$118$	0	0
$119$	−789.387	−0.608092
$120$	0	0
$121$	−847.943	−0.637072
$122$	0	0
$123$	1747.64	1.28113
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−2161.28	−1.51010	−0.755050	−	0.655667i	$-0.772388\pi$
$127$	−2161.28	−1.51010	−0.755050	+	0.655667i	$0.772388\pi$
$128$	0	0
$129$	802.796	0.547924
$130$	0	0
$131$	397.736	0.265270	0.132635	−	0.991165i	$-0.457656\pi$
$131$	397.736	0.265270	0.132635	+	0.991165i	$0.457656\pi$
$132$	0	0
$133$	−1736.06	−1.13185
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1660.73	1.03566	0.517831	−	0.855483i	$-0.326740\pi$
$137$	1660.73	1.03566	0.517831	+	0.855483i	$0.326740\pi$
$138$	0	0
$139$	−997.853	−0.608897	−0.304449	−	0.952529i	$-0.598472\pi$
$139$	−997.853	−0.608897	−0.304449	+	0.952529i	$0.598472\pi$
$140$	0	0
$141$	−1061.98	−0.634289
$142$	0	0
$143$	819.566	0.479270
$144$	0	0
$145$	0	0
$146$	0	0
$147$	1052.74	0.590668
$148$	0	0
$149$	3014.50	1.65743	0.828715	−	0.559671i	$-0.189072\pi$
$149$	3014.50	1.65743	0.828715	+	0.559671i	$0.189072\pi$
$150$	0	0
$151$	−684.908	−0.369120	−0.184560	−	0.982821i	$-0.559086\pi$
$151$	−684.908	−0.369120	−0.184560	+	0.982821i	$0.559086\pi$
$152$	0	0
$153$	−361.212	−0.190864
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−727.598	−0.369864	−0.184932	−	0.982751i	$-0.559206\pi$
$157$	−727.598	−0.369864	−0.184932	+	0.982751i	$0.559206\pi$
$158$	0	0
$159$	1787.10	0.891362
$160$	0	0
$161$	290.115	0.142014
$162$	0	0
$163$	768.718	0.369390	0.184695	−	0.982796i	$-0.440870\pi$
$163$	768.718	0.369390	0.184695	+	0.982796i	$0.440870\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	243.397	0.112782	0.0563910	−	0.998409i	$-0.482041\pi$
$167$	243.397	0.112782	0.0563910	+	0.998409i	$0.482041\pi$
$168$	0	0
$169$	−806.505	−0.367094
$170$	0	0
$171$	−794.396	−0.355257
$172$	0	0
$173$	201.538	0.0885702	0.0442851	−	0.999019i	$-0.485899\pi$
$173$	201.538	0.0885702	0.0442851	+	0.999019i	$0.485899\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	926.088	0.393272
$178$	0	0
$179$	−722.564	−0.301715	−0.150857	−	0.988556i	$-0.548203\pi$
$179$	−722.564	−0.301715	−0.150857	+	0.988556i	$0.548203\pi$
$180$	0	0
$181$	2117.76	0.869678	0.434839	−	0.900508i	$-0.356805\pi$
$181$	2117.76	0.869678	0.434839	+	0.900508i	$0.356805\pi$
$182$	0	0
$183$	−4784.57	−1.93271
$184$	0	0
$185$	0	0
$186$	0	0
$187$	1375.46	0.537878
$188$	0	0
$189$	1532.87	0.589946
$190$	0	0
$191$	82.9624	0.0314290	0.0157145	−	0.999877i	$-0.494998\pi$
$191$	82.9624	0.0314290	0.0157145	+	0.999877i	$0.494998\pi$
$192$	0	0
$193$	−1350.43	−0.503657	−0.251829	−	0.967772i	$-0.581032\pi$
$193$	−1350.43	−0.503657	−0.251829	+	0.967772i	$0.581032\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−4317.69	−1.56154	−0.780769	−	0.624820i	$-0.785172\pi$
$197$	−4317.69	−1.56154	−0.780769	+	0.624820i	$0.785172\pi$
$198$	0	0
$199$	3026.27	1.07802	0.539011	−	0.842299i	$-0.318798\pi$
$199$	3026.27	1.07802	0.539011	+	0.842299i	$0.318798\pi$
$200$	0	0
$201$	−140.265	−0.0492215
$202$	0	0
$203$	−906.893	−0.313554
$204$	0	0
$205$	0	0
$206$	0	0
$207$	132.752	0.0445746
$208$	0	0
$209$	3024.97	1.00116
$210$	0	0
$211$	2292.01	0.747812	0.373906	−	0.927467i	$-0.378018\pi$
$211$	2292.01	0.747812	0.373906	+	0.927467i	$0.378018\pi$
$212$	0	0
$213$	5575.67	1.79361
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−2771.62	−0.867051
$218$	0	0
$219$	−1966.12	−0.606659
$220$	0	0
$221$	2333.63	0.710303
$222$	0	0
$223$	2997.29	0.900061	0.450030	−	0.893013i	$-0.351413\pi$
$223$	2997.29	0.900061	0.450030	+	0.893013i	$0.351413\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	5786.16	1.69181	0.845905	−	0.533333i	$-0.179061\pi$
$227$	5786.16	1.69181	0.845905	+	0.533333i	$0.179061\pi$
$228$	0	0
$229$	4516.35	1.30327	0.651635	−	0.758532i	$-0.274083\pi$
$229$	4516.35	1.30327	0.651635	+	0.758532i	$0.274083\pi$
$230$	0	0
$231$	1587.05	0.452037
$232$	0	0
$233$	−2990.75	−0.840903	−0.420452	−	0.907315i	$-0.638128\pi$
$233$	−2990.75	−0.840903	−0.420452	+	0.907315i	$0.638128\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−6953.28	−1.90575
$238$	0	0
$239$	4043.74	1.09443	0.547213	−	0.836994i	$-0.315689\pi$
$239$	4043.74	1.09443	0.547213	+	0.836994i	$0.315689\pi$
$240$	0	0
$241$	3653.64	0.976564	0.488282	−	0.872686i	$-0.337624\pi$
$241$	3653.64	0.976564	0.488282	+	0.872686i	$0.337624\pi$
$242$	0	0
$243$	1593.55	0.420684
$244$	0	0
$245$	0	0
$246$	0	0
$247$	5132.24	1.32209
$248$	0	0
$249$	4288.09	1.09135
$250$	0	0
$251$	−6799.45	−1.70987	−0.854935	−	0.518735i	$-0.826403\pi$
$251$	−6799.45	−1.70987	−0.854935	+	0.518735i	$0.826403\pi$
$252$	0	0
$253$	−505.507	−0.125616
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−179.925	−0.0436710	−0.0218355	−	0.999762i	$-0.506951\pi$
$257$	−179.925	−0.0436710	−0.0218355	+	0.999762i	$0.506951\pi$
$258$	0	0
$259$	3736.46	0.896419
$260$	0	0
$261$	−414.981	−0.0984165
$262$	0	0
$263$	1148.48	0.269271	0.134635	−	0.990895i	$-0.457014\pi$
$263$	1148.48	0.269271	0.134635	+	0.990895i	$0.457014\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−4646.50	−1.06502
$268$	0	0
$269$	2999.90	0.679953	0.339976	−	0.940434i	$-0.389581\pi$
$269$	2999.90	0.679953	0.339976	+	0.940434i	$0.389581\pi$
$270$	0	0
$271$	5587.83	1.25253	0.626266	−	0.779609i	$-0.284582\pi$
$271$	5587.83	1.25253	0.626266	+	0.779609i	$0.284582\pi$
$272$	0	0
$273$	2692.64	0.596944
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−6219.06	−1.34898	−0.674489	−	0.738285i	$-0.735636\pi$
$277$	−6219.06	−1.34898	−0.674489	+	0.738285i	$0.735636\pi$
$278$	0	0
$279$	−1268.26	−0.272145
$280$	0	0
$281$	−6770.91	−1.43743	−0.718717	−	0.695303i	$-0.755270\pi$
$281$	−6770.91	−1.43743	−0.718717	+	0.695303i	$0.755270\pi$
$282$	0	0
$283$	7811.42	1.64078	0.820390	−	0.571804i	$-0.193756\pi$
$283$	7811.42	1.64078	0.820390	+	0.571804i	$0.193756\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−3850.73	−0.791991
$288$	0	0
$289$	−996.531	−0.202836
$290$	0	0
$291$	5878.68	1.18424
$292$	0	0
$293$	8415.02	1.67785	0.838926	−	0.544246i	$-0.183184\pi$
$293$	8415.02	1.67785	0.838926	+	0.544246i	$0.183184\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−2670.93	−0.521828
$298$	0	0
$299$	−857.655	−0.165884
$300$	0	0
$301$	−1768.87	−0.338725
$302$	0	0
$303$	4512.93	0.855647
$304$	0	0
$305$	0	0
$306$	0	0
$307$	9633.23	1.79087	0.895435	−	0.445192i	$-0.146865\pi$
$307$	9633.23	1.79087	0.895435	+	0.445192i	$0.146865\pi$
$308$	0	0
$309$	2700.69	0.497207
$310$	0	0
$311$	−7000.64	−1.27643	−0.638216	−	0.769858i	$-0.720327\pi$
$311$	−7000.64	−1.27643	−0.638216	+	0.769858i	$0.720327\pi$
$312$	0	0
$313$	2954.79	0.533593	0.266796	−	0.963753i	$-0.414035\pi$
$313$	2954.79	0.533593	0.266796	+	0.963753i	$0.414035\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	4015.68	0.711492	0.355746	−	0.934583i	$-0.384227\pi$
$317$	4015.68	0.711492	0.355746	+	0.934583i	$0.384227\pi$
$318$	0	0
$319$	1580.20	0.277349
$320$	0	0
$321$	−8030.85	−1.39638
$322$	0	0
$323$	8613.30	1.48377
$324$	0	0
$325$	0	0
$326$	0	0
$327$	12315.5	2.08272
$328$	0	0
$329$	2339.96	0.392116
$330$	0	0
$331$	596.567	0.0990644	0.0495322	−	0.998773i	$-0.484227\pi$
$331$	596.567	0.0990644	0.0495322	+	0.998773i	$0.484227\pi$
$332$	0	0
$333$	1709.75	0.281363
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−2162.05	−0.349479	−0.174739	−	0.984615i	$-0.555908\pi$
$337$	−2162.05	−0.349479	−0.174739	+	0.984615i	$0.555908\pi$
$338$	0	0
$339$	4321.17	0.692312
$340$	0	0
$341$	4829.38	0.766937
$342$	0	0
$343$	−6646.09	−1.04622
$344$	0	0
$345$	0	0
$346$	0	0
$347$	8014.61	1.23990	0.619952	−	0.784639i	$-0.287152\pi$
$347$	8014.61	1.23990	0.619952	+	0.784639i	$0.287152\pi$
$348$	0	0
$349$	3264.77	0.500743	0.250371	−	0.968150i	$-0.419447\pi$
$349$	3264.77	0.500743	0.250371	+	0.968150i	$0.419447\pi$
$350$	0	0
$351$	−4531.56	−0.689107
$352$	0	0
$353$	6659.27	1.00407	0.502036	−	0.864847i	$-0.332584\pi$
$353$	6659.27	1.00407	0.502036	+	0.864847i	$0.332584\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	4518.98	0.669943
$358$	0	0
$359$	670.397	0.0985577	0.0492789	−	0.998785i	$-0.484308\pi$
$359$	670.397	0.0985577	0.0492789	+	0.998785i	$0.484308\pi$
$360$	0	0
$361$	12083.8	1.76175
$362$	0	0
$363$	4854.19	0.701871
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−3528.04	−0.501804	−0.250902	−	0.968013i	$-0.580727\pi$
$367$	−3528.04	−0.501804	−0.250902	+	0.968013i	$0.580727\pi$
$368$	0	0
$369$	−1762.04	−0.248586
$370$	0	0
$371$	−3937.69	−0.551037
$372$	0	0
$373$	2718.66	0.377391	0.188695	−	0.982036i	$-0.439574\pi$
$373$	2718.66	0.377391	0.188695	+	0.982036i	$0.439574\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2681.01	0.366258
$378$	0	0
$379$	10773.8	1.46019	0.730096	−	0.683345i	$-0.239475\pi$
$379$	10773.8	1.46019	0.730096	+	0.683345i	$0.239475\pi$
$380$	0	0
$381$	12372.6	1.66370
$382$	0	0
$383$	6104.18	0.814384	0.407192	−	0.913343i	$-0.366508\pi$
$383$	6104.18	0.814384	0.407192	+	0.913343i	$0.366508\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−809.412	−0.106317
$388$	0	0
$389$	−5056.94	−0.659118	−0.329559	−	0.944135i	$-0.606900\pi$
$389$	−5056.94	−0.659118	−0.329559	+	0.944135i	$0.606900\pi$
$390$	0	0
$391$	−1439.38	−0.186170
$392$	0	0
$393$	−2276.91	−0.292251
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−8320.52	−1.05188	−0.525938	−	0.850523i	$-0.676286\pi$
$397$	−8320.52	−1.05188	−0.525938	+	0.850523i	$0.676286\pi$
$398$	0	0
$399$	9938.37	1.24697
$400$	0	0
$401$	1253.78	0.156137	0.0780685	−	0.996948i	$-0.475125\pi$
$401$	1253.78	0.156137	0.0780685	+	0.996948i	$0.475125\pi$
$402$	0	0
$403$	8193.64	1.01279
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−6510.55	−0.792914
$408$	0	0
$409$	−11283.0	−1.36408	−0.682041	−	0.731314i	$-0.738907\pi$
$409$	−11283.0	−1.36408	−0.682041	+	0.731314i	$0.738907\pi$
$410$	0	0
$411$	−9507.13	−1.14100
$412$	0	0
$413$	−2040.54	−0.243119
$414$	0	0
$415$	0	0
$416$	0	0
$417$	5712.38	0.670830
$418$	0	0
$419$	4178.10	0.487144	0.243572	−	0.969883i	$-0.421681\pi$
$419$	4178.10	0.487144	0.243572	+	0.969883i	$0.421681\pi$
$420$	0	0
$421$	−11693.0	−1.35364	−0.676821	−	0.736147i	$-0.736643\pi$
$421$	−11693.0	−1.35364	−0.676821	+	0.736147i	$0.736643\pi$
$422$	0	0
$423$	1070.73	0.123075
$424$	0	0
$425$	0	0
$426$	0	0
$427$	10542.3	1.19479
$428$	0	0
$429$	−4691.74	−0.528018
$430$	0	0
$431$	−11158.2	−1.24703	−0.623515	−	0.781812i	$-0.714296\pi$
$431$	−11158.2	−1.24703	−0.623515	+	0.781812i	$0.714296\pi$
$432$	0	0
$433$	13192.6	1.46419	0.732096	−	0.681201i	$-0.238542\pi$
$433$	13192.6	1.46419	0.732096	+	0.681201i	$0.238542\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−3165.56	−0.346520
$438$	0	0
$439$	−1659.40	−0.180407	−0.0902037	−	0.995923i	$-0.528752\pi$
$439$	−1659.40	−0.180407	−0.0902037	+	0.995923i	$0.528752\pi$
$440$	0	0
$441$	−1061.41	−0.114611
$442$	0	0
$443$	−5144.57	−0.551752	−0.275876	−	0.961193i	$-0.588968\pi$
$443$	−5144.57	−0.551752	−0.275876	+	0.961193i	$0.588968\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−17257.0	−1.82601
$448$	0	0
$449$	5469.59	0.574891	0.287446	−	0.957797i	$-0.407194\pi$
$449$	5469.59	0.574891	0.287446	+	0.957797i	$0.407194\pi$
$450$	0	0
$451$	6709.65	0.700544
$452$	0	0
$453$	3920.87	0.406664
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−6063.04	−0.620606	−0.310303	−	0.950638i	$-0.600431\pi$
$457$	−6063.04	−0.620606	−0.310303	+	0.950638i	$0.600431\pi$
$458$	0	0
$459$	−7605.19	−0.773377
$460$	0	0
$461$	4025.39	0.406684	0.203342	−	0.979108i	$-0.434820\pi$
$461$	4025.39	0.406684	0.203342	+	0.979108i	$0.434820\pi$
$462$	0	0
$463$	4617.49	0.463483	0.231742	−	0.972777i	$-0.425558\pi$
$463$	4617.49	0.463483	0.231742	+	0.972777i	$0.425558\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−4012.00	−0.397544	−0.198772	−	0.980046i	$-0.563695\pi$
$467$	−4012.00	−0.397544	−0.198772	+	0.980046i	$0.563695\pi$
$468$	0	0
$469$	309.058	0.0304285
$470$	0	0
$471$	4165.26	0.407484
$472$	0	0
$473$	3082.15	0.299614
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−1801.83	−0.172956
$478$	0	0
$479$	−7317.02	−0.697961	−0.348980	−	0.937130i	$-0.613472\pi$
$479$	−7317.02	−0.697961	−0.348980	+	0.937130i	$0.613472\pi$
$480$	0	0
$481$	−11046.0	−1.04709
$482$	0	0
$483$	−1660.81	−0.156459
$484$	0	0
$485$	0	0
$486$	0	0
$487$	3708.67	0.345084	0.172542	−	0.985002i	$-0.444802\pi$
$487$	3708.67	0.345084	0.172542	+	0.985002i	$0.444802\pi$
$488$	0	0
$489$	−4400.66	−0.406962
$490$	0	0
$491$	6689.69	0.614871	0.307435	−	0.951569i	$-0.400529\pi$
$491$	6689.69	0.614871	0.307435	+	0.951569i	$0.400529\pi$
$492$	0	0
$493$	4499.47	0.411046
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−12285.4	−1.10880
$498$	0	0
$499$	17381.9	1.55936	0.779678	−	0.626180i	$-0.215383\pi$
$499$	17381.9	1.55936	0.779678	+	0.626180i	$0.215383\pi$
$500$	0	0
$501$	−1393.37	−0.124253
$502$	0	0
$503$	−9771.20	−0.866156	−0.433078	−	0.901357i	$-0.642572\pi$
$503$	−9771.20	−0.866156	−0.433078	+	0.901357i	$0.642572\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	4616.97	0.404432
$508$	0	0
$509$	5820.46	0.506852	0.253426	−	0.967355i	$-0.418443\pi$
$509$	5820.46	0.506852	0.253426	+	0.967355i	$0.418443\pi$
$510$	0	0
$511$	4332.14	0.375035
$512$	0	0
$513$	−16725.7	−1.43949
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−4077.22	−0.346840
$518$	0	0
$519$	−1153.74	−0.0975789
$520$	0	0
$521$	−6357.64	−0.534613	−0.267307	−	0.963612i	$-0.586134\pi$
$521$	−6357.64	−0.534613	−0.267307	+	0.963612i	$0.586134\pi$
$522$	0	0
$523$	10556.2	0.882580	0.441290	−	0.897364i	$-0.354521\pi$
$523$	10556.2	0.882580	0.441290	+	0.897364i	$0.354521\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	13751.2	1.13664
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	−933.720	−0.0763089
$532$	0	0
$533$	11383.8	0.925113
$534$	0	0
$535$	0	0
$536$	0	0
$537$	4136.44	0.332403
$538$	0	0
$539$	4041.74	0.322987
$540$	0	0
$541$	−16883.4	−1.34172	−0.670862	−	0.741582i	$-0.734076\pi$
$541$	−16883.4	−1.34172	−0.670862	+	0.741582i	$0.734076\pi$
$542$	0	0
$543$	−12123.5	−0.958136
$544$	0	0
$545$	0	0
$546$	0	0
$547$	14318.4	1.11922	0.559608	−	0.828757i	$-0.310952\pi$
$547$	14318.4	1.11922	0.559608	+	0.828757i	$0.310952\pi$
$548$	0	0
$549$	4824.00	0.375015
$550$	0	0
$551$	9895.47	0.765084
$552$	0	0
$553$	15320.8	1.17813
$554$	0	0
$555$	0	0
$556$	0	0
$557$	11764.7	0.894946	0.447473	−	0.894297i	$-0.352324\pi$
$557$	11764.7	0.894946	0.447473	+	0.894297i	$0.352324\pi$
$558$	0	0
$559$	5229.25	0.395660
$560$	0	0
$561$	−7874.03	−0.592588
$562$	0	0
$563$	21201.3	1.58709	0.793543	−	0.608514i	$-0.208234\pi$
$563$	21201.3	1.58709	0.793543	+	0.608514i	$0.208234\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−10740.9	−0.795546
$568$	0	0
$569$	−17206.5	−1.26772	−0.633861	−	0.773447i	$-0.718531\pi$
$569$	−17206.5	−1.26772	−0.633861	+	0.773447i	$0.718531\pi$
$570$	0	0
$571$	12208.3	0.894746	0.447373	−	0.894348i	$-0.352360\pi$
$571$	12208.3	0.894746	0.447373	+	0.894348i	$0.352360\pi$
$572$	0	0
$573$	−474.932	−0.0346258
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−12432.0	−0.896967	−0.448484	−	0.893791i	$-0.648036\pi$
$577$	−12432.0	−0.896967	−0.448484	+	0.893791i	$0.648036\pi$
$578$	0	0
$579$	7730.75	0.554886
$580$	0	0
$581$	−9448.36	−0.674671
$582$	0	0
$583$	6861.17	0.487411
$584$	0	0
$585$	0	0
$586$	0	0
$587$	18732.7	1.31718	0.658588	−	0.752504i	$-0.271154\pi$
$587$	18732.7	1.31718	0.658588	+	0.752504i	$0.271154\pi$
$588$	0	0
$589$	30242.3	2.11564
$590$	0	0
$591$	24717.4	1.72037
$592$	0	0
$593$	−16320.1	−1.13016	−0.565081	−	0.825036i	$-0.691155\pi$
$593$	−16320.1	−1.13016	−0.565081	+	0.825036i	$0.691155\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−17324.4	−1.18767
$598$	0	0
$599$	−7015.52	−0.478542	−0.239271	−	0.970953i	$-0.576908\pi$
$599$	−7015.52	−0.478542	−0.239271	+	0.970953i	$0.576908\pi$
$600$	0	0
$601$	−4517.10	−0.306583	−0.153292	−	0.988181i	$-0.548987\pi$
$601$	−4517.10	−0.306583	−0.153292	+	0.988181i	$0.548987\pi$
$602$	0	0
$603$	141.421	0.00955074
$604$	0	0
$605$	0	0
$606$	0	0
$607$	9861.96	0.659448	0.329724	−	0.944077i	$-0.393044\pi$
$607$	9861.96	0.659448	0.329724	+	0.944077i	$0.393044\pi$
$608$	0	0
$609$	5191.67	0.345447
$610$	0	0
$611$	−6917.52	−0.458024
$612$	0	0
$613$	−20044.7	−1.32071	−0.660356	−	0.750953i	$-0.729595\pi$
$613$	−20044.7	−1.32071	−0.660356	+	0.750953i	$0.729595\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	3604.26	0.235174	0.117587	−	0.993063i	$-0.462484\pi$
$617$	3604.26	0.235174	0.117587	+	0.993063i	$0.462484\pi$
$618$	0	0
$619$	14864.7	0.965204	0.482602	−	0.875840i	$-0.339692\pi$
$619$	14864.7	0.965204	0.482602	+	0.875840i	$0.339692\pi$
$620$	0	0
$621$	2795.06	0.180615
$622$	0	0
$623$	10238.1	0.658393
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−17317.0	−1.10299
$628$	0	0
$629$	−18538.1	−1.17514
$630$	0	0
$631$	12677.7	0.799828	0.399914	−	0.916553i	$-0.369040\pi$
$631$	12677.7	0.799828	0.399914	+	0.916553i	$0.369040\pi$
$632$	0	0
$633$	−13121.0	−0.823875
$634$	0	0
$635$	0	0
$636$	0	0
$637$	6857.31	0.426525
$638$	0	0
$639$	−5621.62	−0.348025
$640$	0	0
$641$	−19168.5	−1.18114	−0.590571	−	0.806986i	$-0.701097\pi$
$641$	−19168.5	−1.18114	−0.590571	+	0.806986i	$0.701097\pi$
$642$	0	0
$643$	−31298.6	−1.91959	−0.959793	−	0.280707i	$-0.909431\pi$
$643$	−31298.6	−1.91959	−0.959793	+	0.280707i	$0.909431\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	26340.3	1.60053	0.800267	−	0.599643i	$-0.204691\pi$
$647$	26340.3	1.60053	0.800267	+	0.599643i	$0.204691\pi$
$648$	0	0
$649$	3555.50	0.215047
$650$	0	0
$651$	15866.6	0.955242
$652$	0	0
$653$	−27113.8	−1.62488	−0.812439	−	0.583046i	$-0.801861\pi$
$653$	−27113.8	−1.62488	−0.812439	+	0.583046i	$0.801861\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	1982.33	0.117714
$658$	0	0
$659$	−24125.5	−1.42609	−0.713046	−	0.701117i	$-0.752685\pi$
$659$	−24125.5	−1.42609	−0.713046	+	0.701117i	$0.752685\pi$
$660$	0	0
$661$	1776.69	0.104546	0.0522732	−	0.998633i	$-0.483353\pi$
$661$	1776.69	0.104546	0.0522732	+	0.998633i	$0.483353\pi$
$662$	0	0
$663$	−13359.3	−0.782550
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1653.64	−0.0959960
$668$	0	0
$669$	−17158.5	−0.991609
$670$	0	0
$671$	−18369.2	−1.05684
$672$	0	0
$673$	14137.6	0.809756	0.404878	−	0.914371i	$-0.367314\pi$
$673$	14137.6	0.809756	0.404878	+	0.914371i	$0.367314\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	9057.23	0.514177	0.257088	−	0.966388i	$-0.417237\pi$
$677$	9057.23	0.514177	0.257088	+	0.966388i	$0.417237\pi$
$678$	0	0
$679$	−12953.0	−0.732094
$680$	0	0
$681$	−33123.9	−1.86389
$682$	0	0
$683$	−23206.6	−1.30011	−0.650054	−	0.759888i	$-0.725254\pi$
$683$	−23206.6	−1.30011	−0.650054	+	0.759888i	$0.725254\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−25854.6	−1.43583
$688$	0	0
$689$	11640.8	0.643658
$690$	0	0
$691$	2060.79	0.113453	0.0567265	−	0.998390i	$-0.481934\pi$
$691$	2060.79	0.113453	0.0567265	+	0.998390i	$0.481934\pi$
$692$	0	0
$693$	−1600.13	−0.0877115
$694$	0	0
$695$	0	0
$696$	0	0
$697$	19105.0	1.03824
$698$	0	0
$699$	17121.1	0.926434
$700$	0	0
$701$	4057.66	0.218624	0.109312	−	0.994007i	$-0.465135\pi$
$701$	4057.66	0.218624	0.109312	+	0.994007i	$0.465135\pi$
$702$	0	0
$703$	−40770.0	−2.18730
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−9943.76	−0.528958
$708$	0	0
$709$	18777.0	0.994622	0.497311	−	0.867572i	$-0.334321\pi$
$709$	18777.0	0.994622	0.497311	+	0.867572i	$0.334321\pi$
$710$	0	0
$711$	7010.58	0.369785
$712$	0	0
$713$	−5053.82	−0.265452
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−23149.1	−1.20574
$718$	0	0
$719$	−2217.63	−0.115026	−0.0575129	−	0.998345i	$-0.518317\pi$
$719$	−2217.63	−0.115026	−0.0575129	+	0.998345i	$0.518317\pi$
$720$	0	0
$721$	−5950.69	−0.307372
$722$	0	0
$723$	−20915.9	−1.07589
$724$	0	0
$725$	0	0
$726$	0	0
$727$	23162.5	1.18164	0.590818	−	0.806805i	$-0.298805\pi$
$727$	23162.5	1.18164	0.590818	+	0.806805i	$0.298805\pi$
$728$	0	0
$729$	13868.6	0.704600
$730$	0	0
$731$	8776.11	0.444044
$732$	0	0
$733$	−11686.4	−0.588875	−0.294438	−	0.955671i	$-0.595132\pi$
$733$	−11686.4	−0.588875	−0.294438	+	0.955671i	$0.595132\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−538.514	−0.0269151
$738$	0	0
$739$	−7408.28	−0.368766	−0.184383	−	0.982854i	$-0.559029\pi$
$739$	−7408.28	−0.368766	−0.184383	+	0.982854i	$0.559029\pi$
$740$	0	0
$741$	−29380.4	−1.45657
$742$	0	0
$743$	7008.68	0.346061	0.173031	−	0.984916i	$-0.444644\pi$
$743$	7008.68	0.346061	0.173031	+	0.984916i	$0.444644\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−4323.43	−0.211762
$748$	0	0
$749$	17695.1	0.863238
$750$	0	0
$751$	10916.9	0.530443	0.265222	−	0.964187i	$-0.414555\pi$
$751$	10916.9	0.530443	0.265222	+	0.964187i	$0.414555\pi$
$752$	0	0
$753$	38924.6	1.88379
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−14124.7	−0.678165	−0.339083	−	0.940757i	$-0.610117\pi$
$757$	−14124.7	−0.678165	−0.339083	+	0.940757i	$0.610117\pi$
$758$	0	0
$759$	2893.86	0.138393
$760$	0	0
$761$	6947.27	0.330931	0.165465	−	0.986216i	$-0.447087\pi$
$761$	6947.27	0.330931	0.165465	+	0.986216i	$0.447087\pi$
$762$	0	0
$763$	−27136.0	−1.28753
$764$	0	0
$765$	0	0
$766$	0	0
$767$	6032.35	0.283984
$768$	0	0
$769$	19409.1	0.910158	0.455079	−	0.890451i	$-0.349611\pi$
$769$	19409.1	0.910158	0.455079	+	0.890451i	$0.349611\pi$
$770$	0	0
$771$	1030.01	0.0481129
$772$	0	0
$773$	32794.2	1.52591	0.762954	−	0.646453i	$-0.223748\pi$
$773$	32794.2	1.52591	0.762954	+	0.646453i	$0.223748\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−21390.0	−0.987597
$778$	0	0
$779$	42016.8	1.93249
$780$	0	0
$781$	21406.5	0.980775
$782$	0	0
$783$	−8737.29	−0.398781
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−1234.92	−0.0559341	−0.0279670	−	0.999609i	$-0.508903\pi$
$787$	−1234.92	−0.0559341	−0.0279670	+	0.999609i	$0.508903\pi$
$788$	0	0
$789$	−6574.66	−0.296659
$790$	0	0
$791$	−9521.24	−0.427985
$792$	0	0
$793$	−31165.7	−1.39562
$794$	0	0
$795$	0	0
$796$	0	0
$797$	19464.3	0.865070	0.432535	−	0.901617i	$-0.357619\pi$
$797$	19464.3	0.865070	0.432535	+	0.901617i	$0.357619\pi$
$798$	0	0
$799$	−11609.5	−0.514035
$800$	0	0
$801$	4684.79	0.206653
$802$	0	0
$803$	−7548.48	−0.331731
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−17173.4	−0.749113
$808$	0	0
$809$	−33587.0	−1.45965	−0.729824	−	0.683635i	$-0.760398\pi$
$809$	−33587.0	−1.45965	−0.729824	+	0.683635i	$0.760398\pi$
$810$	0	0
$811$	−2612.03	−0.113096	−0.0565479	−	0.998400i	$-0.518009\pi$
$811$	−2612.03	−0.113096	−0.0565479	+	0.998400i	$0.518009\pi$
$812$	0	0
$813$	−31988.5	−1.37993
$814$	0	0
$815$	0	0
$816$	0	0
$817$	19300.9	0.826502
$818$	0	0
$819$	−2714.83	−0.115829
$820$	0	0
$821$	−17811.2	−0.757146	−0.378573	−	0.925572i	$-0.623585\pi$
$821$	−17811.2	−0.757146	−0.378573	+	0.925572i	$0.623585\pi$
$822$	0	0
$823$	−30535.0	−1.29330	−0.646649	−	0.762788i	$-0.723830\pi$
$823$	−30535.0	−1.29330	−0.646649	+	0.762788i	$0.723830\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	3344.05	0.140609	0.0703047	−	0.997526i	$-0.477603\pi$
$827$	3344.05	0.140609	0.0703047	+	0.997526i	$0.477603\pi$
$828$	0	0
$829$	−24905.6	−1.04344	−0.521718	−	0.853118i	$-0.674709\pi$
$829$	−24905.6	−1.04344	−0.521718	+	0.853118i	$0.674709\pi$
$830$	0	0
$831$	35602.1	1.48619
$832$	0	0
$833$	11508.4	0.478684
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−26702.7	−1.10272
$838$	0	0
$839$	−38233.9	−1.57328	−0.786639	−	0.617413i	$-0.788181\pi$
$839$	−38233.9	−1.57328	−0.786639	+	0.617413i	$0.788181\pi$
$840$	0	0
$841$	−19219.7	−0.788050
$842$	0	0
$843$	38761.2	1.58364
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−10695.7	−0.433894
$848$	0	0
$849$	−44717.8	−1.80767
$850$	0	0
$851$	6813.12	0.274443
$852$	0	0
$853$	−21297.4	−0.854875	−0.427438	−	0.904045i	$-0.640584\pi$
$853$	−21297.4	−0.854875	−0.427438	+	0.904045i	$0.640584\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	2100.08	0.0837074	0.0418537	−	0.999124i	$-0.486674\pi$
$857$	2100.08	0.0837074	0.0418537	+	0.999124i	$0.486674\pi$
$858$	0	0
$859$	−16555.4	−0.657582	−0.328791	−	0.944403i	$-0.606641\pi$
$859$	−16555.4	−0.657582	−0.328791	+	0.944403i	$0.606641\pi$
$860$	0	0
$861$	22044.2	0.872547
$862$	0	0
$863$	40225.0	1.58664	0.793322	−	0.608802i	$-0.208350\pi$
$863$	40225.0	1.58664	0.793322	+	0.608802i	$0.208350\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	5704.81	0.223467
$868$	0	0
$869$	−26695.5	−1.04210
$870$	0	0
$871$	−913.656	−0.0355431
$872$	0	0
$873$	−5927.13	−0.229786
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−35335.8	−1.36055	−0.680277	−	0.732955i	$-0.738141\pi$
$877$	−35335.8	−1.36055	−0.680277	+	0.732955i	$0.738141\pi$
$878$	0	0
$879$	−48173.2	−1.84851
$880$	0	0
$881$	28460.9	1.08839	0.544196	−	0.838958i	$-0.316835\pi$
$881$	28460.9	1.08839	0.544196	+	0.838958i	$0.316835\pi$
$882$	0	0
$883$	35713.3	1.36110	0.680548	−	0.732703i	$-0.261742\pi$
$883$	35713.3	1.36110	0.680548	+	0.732703i	$0.261742\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	354.759	0.0134292	0.00671458	−	0.999977i	$-0.497863\pi$
$887$	354.759	0.0134292	0.00671458	+	0.999977i	$0.497863\pi$
$888$	0	0
$889$	−27261.8	−1.02849
$890$	0	0
$891$	18715.3	0.703688
$892$	0	0
$893$	−25532.2	−0.956777
$894$	0	0
$895$	0	0
$896$	0	0
$897$	4909.79	0.182757
$898$	0	0
$899$	15798.1	0.586093
$900$	0	0
$901$	19536.5	0.722370
$902$	0	0
$903$	10126.2	0.373178
$904$	0	0
$905$	0	0
$906$	0	0
$907$	41414.8	1.51616	0.758080	−	0.652162i	$-0.226138\pi$
$907$	41414.8	1.51616	0.758080	+	0.652162i	$0.226138\pi$
$908$	0	0
$909$	−4550.12	−0.166026
$910$	0	0
$911$	4521.49	0.164439	0.0822194	−	0.996614i	$-0.473799\pi$
$911$	4521.49	0.164439	0.0822194	+	0.996614i	$0.473799\pi$
$912$	0	0
$913$	16463.2	0.596770
$914$	0	0
$915$	0	0
$916$	0	0
$917$	5016.92	0.180669
$918$	0	0
$919$	−31097.0	−1.11621	−0.558104	−	0.829771i	$-0.688471\pi$
$919$	−31097.0	−1.11621	−0.558104	+	0.829771i	$0.688471\pi$
$920$	0	0
$921$	−55147.0	−1.97303
$922$	0	0
$923$	36318.8	1.29518
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−2722.95	−0.0964761
$928$	0	0
$929$	11243.2	0.397070	0.198535	−	0.980094i	$-0.436382\pi$
$929$	11243.2	0.397070	0.198535	+	0.980094i	$0.436382\pi$
$930$	0	0
$931$	25310.0	0.890978
$932$	0	0
$933$	40076.4	1.40626
$934$	0	0
$935$	0	0
$936$	0	0
$937$	31275.3	1.09042	0.545209	−	0.838300i	$-0.316450\pi$
$937$	31275.3	1.09042	0.545209	+	0.838300i	$0.316450\pi$
$938$	0	0
$939$	−16915.2	−0.587866
$940$	0	0
$941$	−19726.6	−0.683387	−0.341694	−	0.939811i	$-0.611001\pi$
$941$	−19726.6	−0.683387	−0.341694	+	0.939811i	$0.611001\pi$
$942$	0	0
$943$	−7021.48	−0.242472
$944$	0	0
$945$	0	0
$946$	0	0
$947$	9073.85	0.311363	0.155681	−	0.987807i	$-0.450243\pi$
$947$	9073.85	0.311363	0.155681	+	0.987807i	$0.450243\pi$
$948$	0	0
$949$	−12806.9	−0.438072
$950$	0	0
$951$	−22988.4	−0.783860
$952$	0	0
$953$	−20332.8	−0.691125	−0.345563	−	0.938396i	$-0.612312\pi$
$953$	−20332.8	−0.691125	−0.345563	+	0.938396i	$0.612312\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−9046.14	−0.305559
$958$	0	0
$959$	20947.9	0.705364
$960$	0	0
$961$	18490.8	0.620686
$962$	0	0
$963$	8097.03	0.270948
$964$	0	0
$965$	0	0
$966$	0	0
$967$	10303.5	0.342646	0.171323	−	0.985215i	$-0.445196\pi$
$967$	10303.5	0.342646	0.171323	+	0.985215i	$0.445196\pi$
$968$	0	0
$969$	−49308.3	−1.63469
$970$	0	0
$971$	−937.151	−0.0309728	−0.0154864	−	0.999880i	$-0.504930\pi$
$971$	−937.151	−0.0309728	−0.0154864	+	0.999880i	$0.504930\pi$
$972$	0	0
$973$	−12586.6	−0.414705
$974$	0	0
$975$	0	0
$976$	0	0
$977$	23339.0	0.764258	0.382129	−	0.924109i	$-0.375191\pi$
$977$	23339.0	0.764258	0.382129	+	0.924109i	$0.375191\pi$
$978$	0	0
$979$	−17839.2	−0.582372
$980$	0	0
$981$	−12417.0	−0.404124
$982$	0	0
$983$	−26638.6	−0.864334	−0.432167	−	0.901794i	$-0.642251\pi$
$983$	−26638.6	−0.864334	−0.432167	+	0.901794i	$0.642251\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−13395.5	−0.431999
$988$	0	0
$989$	−3225.39	−0.103702
$990$	0	0
$991$	−24631.4	−0.789548	−0.394774	−	0.918778i	$-0.629177\pi$
$991$	−24631.4	−0.789548	−0.394774	+	0.918778i	$0.629177\pi$
$992$	0	0
$993$	−3415.15	−0.109141
$994$	0	0
$995$	0	0
$996$	0	0
$997$	31438.3	0.998656	0.499328	−	0.866413i	$-0.333580\pi$
$997$	31438.3	0.998656	0.499328	+	0.866413i	$0.333580\pi$
$998$	0	0
$999$	35998.2	1.14007

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2300.4.a.c.1.2		5
5.2	odd	4		2300.4.c.d.1749.8		10
5.3	odd	4		2300.4.c.d.1749.3		10
5.4	even	2		460.4.a.b.1.4	✓	5
20.19	odd	2		1840.4.a.o.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
460.4.a.b.1.4	✓	5	5.4	even	2
1840.4.a.o.1.2		5	20.19	odd	2
2300.4.a.c.1.2		5	1.1	even	1	trivial
2300.4.c.d.1749.3		10	5.3	odd	4
2300.4.c.d.1749.8		10	5.2	odd	4

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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:	\( N \)	\(=\)	\( 2300 = 2^{2} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2300.a (trivial)

\(f(q)\)	\(=\)	\(q-5.72467 q^{3} +12.6137 q^{7} +5.77185 q^{9} -21.9786 q^{11} -37.2893 q^{13} -62.5817 q^{17} -137.633 q^{19} -72.2093 q^{21} +23.0000 q^{23} +121.524 q^{27} -71.8975 q^{29} -219.731 q^{31} +125.820 q^{33} +296.223 q^{37} +213.469 q^{39} -305.282 q^{41} -140.234 q^{43} +185.509 q^{47} -183.895 q^{49} +358.260 q^{51} -312.176 q^{53} +787.903 q^{57} -161.771 q^{59} +835.780 q^{61} +72.8043 q^{63} +24.5018 q^{67} -131.667 q^{69} -973.973 q^{71} +343.447 q^{73} -277.231 q^{77} +1214.62 q^{79} -851.526 q^{81} -749.055 q^{83} +411.590 q^{87} +811.662 q^{89} -470.356 q^{91} +1257.89 q^{93} -1026.90 q^{97} -126.857 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 3 q^{3} - 8 q^{7} + 30 q^{9} + 7 q^{11} - 5 q^{13} + 24 q^{17} - 13 q^{19} + 115 q^{23} + 204 q^{27} - 253 q^{29} - 98 q^{31} + 473 q^{33} + 435 q^{37} - 410 q^{39} - 774 q^{41} + 498 q^{43} + 572 q^{47}+ \cdots + 111 q^{99}+O(q^{100}) \) 5 * q + 3 * q^3 - 8 * q^7 + 30 * q^9 + 7 * q^11 - 5 * q^13 + 24 * q^17 - 13 * q^19 + 115 * q^23 + 204 * q^27 - 253 * q^29 - 98 * q^31 + 473 * q^33 + 435 * q^37 - 410 * q^39 - 774 * q^41 + 498 * q^43 + 572 * q^47 - 683 * q^49 - 657 * q^51 + 665 * q^53 + 932 * q^57 - 763 * q^59 + 337 * q^61 + 1527 * q^63 + 305 * q^67 + 69 * q^69 - 1504 * q^71 + 1304 * q^73 - 182 * q^77 + 626 * q^79 - 959 * q^81 + 1703 * q^83 + 1354 * q^87 + 646 * q^89 - 767 * q^91 + 452 * q^93 + 233 * q^97 + 111 * q^99

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