LMFDB - Embedded newform 3800.2.d.j.3649.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3800,2,Mod(3649,3800)] 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("3800.3649"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3800, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3800 = 2^{3} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3800.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,0,0,0,0,-24,0,-10,0,0,0,0,0,0,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$30.3431527681$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.59105344.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 21x^{4} + 116x^{2} + 64 $ x^6 + 21x^4 + 116x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 152)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3649.1
Root			$3.08387i$ of defining polynomial
Character	$\chi$	$=$	3800.3649
Dual form			3800.2.d.j.3649.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.29707i q^{3} +1.78680i q^{7} -7.87067 q^{9} -5.08387 q^{11} +1.29707i q^{13} +0.213198i q^{17} +1.00000 q^{19} +5.89121 q^{21} +3.72347i q^{23} +16.0590i q^{27} -0.870674 q^{29} +16.7619i q^{33} -2.00000i q^{37} +4.27653 q^{39} +8.59414 q^{41} +3.67801i q^{43} -4.65748i q^{47} +3.80734 q^{49} +0.702929 q^{51} -11.0384i q^{53} -3.29707i q^{57} +4.70293 q^{59} +3.51027 q^{61} -14.0633i q^{63} +1.12933i q^{67} +12.2765 q^{69} +8.76189 q^{71} -6.80734i q^{73} -9.08387i q^{77} -14.5941 q^{79} +29.3355 q^{81} +9.74135i q^{83} +2.87067i q^{87} +6.76189 q^{89} -2.31761 q^{91} +4.16774i q^{97} +40.0135 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 24 q^{9} - 10 q^{11} + 6 q^{19} - 18 q^{21} + 18 q^{29} + 38 q^{39} + 16 q^{41} - 10 q^{49} + 22 q^{51} + 46 q^{59} + 6 q^{61} + 86 q^{69} - 24 q^{71} - 52 q^{79} + 94 q^{81} - 36 q^{89} + 34 q^{91}+ \cdots + 102 q^{99}+O(q^{100}) $ 6 * q - 24 * q^9 - 10 * q^11 + 6 * q^19 - 18 * q^21 + 18 * q^29 + 38 * q^39 + 16 * q^41 - 10 * q^49 + 22 * q^51 + 46 * q^59 + 6 * q^61 + 86 * q^69 - 24 * q^71 - 52 * q^79 + 94 * q^81 - 36 * q^89 + 34 * q^91 + 102 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3800\mathbb{Z}\right)^\times$.

$n$	$401$	$951$	$1901$	$1977$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	− 3.29707i	− 1.90356i	−0.306774	−	0.951782i	$-0.599250\pi$
$3$	− 3.29707i	− 1.90356i	0.306774	−	0.951782i	$-0.400750\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.78680i	0.675348i	0.941263	+	0.337674i	$0.109640\pi$
$7$	1.78680i	0.675348i	−0.941263	+	0.337674i	$0.890360\pi$
$8$	0	0
$9$	−7.87067	−2.62356
$10$	0	0
$11$	−5.08387	−1.53285	−0.766423	−	0.642337i	$-0.777965\pi$
$11$	−5.08387	−1.53285	−0.766423	+	0.642337i	$0.777965\pi$
$12$	0	0
$13$	1.29707i	0.359743i	0.983690	+	0.179871i	$0.0575681\pi$
$13$	1.29707i	0.359743i	−0.983690	+	0.179871i	$0.942432\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.213198i	0.0517082i	0.999666	+	0.0258541i	$0.00823053\pi$
$17$	0.213198i	0.0517082i	−0.999666	+	0.0258541i	$0.991769\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	5.89121	1.28557
$22$	0	0
$23$	3.72347i	0.776397i	0.921576	+	0.388198i	$0.126902\pi$
$23$	3.72347i	0.776397i	−0.921576	+	0.388198i	$0.873098\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	16.0590i	3.09055i
$28$	0	0
$29$	−0.870674	−0.161680	−0.0808401	−	0.996727i	$-0.525760\pi$
$29$	−0.870674	−0.161680	−0.0808401	+	0.996727i	$0.525760\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	16.7619i	2.91787i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 2.00000i	− 0.328798i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$	− 2.00000i	− 0.328798i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$	0	0
$39$	4.27653	0.684793
$40$	0	0
$41$	8.59414	1.34218	0.671090	−	0.741376i	$-0.265827\pi$
$41$	8.59414	1.34218	0.671090	+	0.741376i	$0.265827\pi$
$42$	0	0
$43$	3.67801i	0.560892i	0.959870	+	0.280446i	$0.0904823\pi$
$43$	3.67801i	0.560892i	−0.959870	+	0.280446i	$0.909518\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 4.65748i	− 0.679363i	−0.940541	−	0.339681i	$-0.889681\pi$
$47$	− 4.65748i	− 0.679363i	0.940541	−	0.339681i	$-0.110319\pi$
$48$	0	0
$49$	3.80734	0.543906
$50$	0	0
$51$	0.702929	0.0984298
$52$	0	0
$53$	− 11.0384i	− 1.51624i	−0.652113	−	0.758122i	$-0.726117\pi$
$53$	− 11.0384i	− 1.51624i	0.652113	−	0.758122i	$-0.273883\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 3.29707i	− 0.436708i
$58$	0	0
$59$	4.70293	0.612269	0.306135	−	0.951988i	$-0.400964\pi$
$59$	4.70293	0.612269	0.306135	+	0.951988i	$0.400964\pi$
$60$	0	0
$61$	3.51027	0.449444	0.224722	−	0.974423i	$-0.427853\pi$
$61$	3.51027	0.449444	0.224722	+	0.974423i	$0.427853\pi$
$62$	0	0
$63$	− 14.0633i	− 1.77181i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	1.12933i	0.137969i	0.997618	+	0.0689846i	$0.0219759\pi$
$67$	1.12933i	0.137969i	−0.997618	+	0.0689846i	$0.978024\pi$
$68$	0	0
$69$	12.2765	1.47792
$70$	0	0
$71$	8.76189	1.03984	0.519922	−	0.854214i	$-0.325961\pi$
$71$	8.76189	1.03984	0.519922	+	0.854214i	$0.325961\pi$
$72$	0	0
$73$	− 6.80734i	− 0.796739i	−0.917225	−	0.398369i	$-0.869576\pi$
$73$	− 6.80734i	− 0.796739i	0.917225	−	0.398369i	$-0.130424\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 9.08387i	− 1.03520i
$78$	0	0
$79$	−14.5941	−1.64197	−0.820985	−	0.570950i	$-0.806575\pi$
$79$	−14.5941	−1.64197	−0.820985	+	0.570950i	$0.806575\pi$
$80$	0	0
$81$	29.3355	3.25950
$82$	0	0
$83$	9.74135i	1.06925i	0.845089	+	0.534626i	$0.179547\pi$
$83$	9.74135i	1.06925i	−0.845089	+	0.534626i	$0.820453\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	2.87067i	0.307769i
$88$	0	0
$89$	6.76189	0.716758	0.358379	−	0.933576i	$-0.383329\pi$
$89$	6.76189	0.716758	0.358379	+	0.933576i	$0.383329\pi$
$90$	0	0
$91$	−2.31761	−0.242951
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	4.16774i	0.423170i	0.977360	+	0.211585i	$0.0678626\pi$
$97$	4.16774i	0.423170i	−0.977360	+	0.211585i	$0.932137\pi$
$98$	0	0
$99$	40.0135	4.02151
$100$	0	0
$101$	19.1883	1.90931	0.954653	−	0.297722i	$-0.0962267\pi$
$101$	19.1883	1.90931	0.954653	+	0.297722i	$0.0962267\pi$
$102$	0	0
$103$	15.9091i	1.56757i	0.621033	+	0.783785i	$0.286713\pi$
$103$	15.9091i	1.56757i	−0.621033	+	0.783785i	$0.713287\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 3.29707i	− 0.318740i	−0.987219	−	0.159370i	$-0.949054\pi$
$107$	− 3.29707i	− 0.318740i	0.987219	−	0.159370i	$-0.0509463\pi$
$108$	0	0
$109$	−4.44428	−0.425685	−0.212842	−	0.977087i	$-0.568272\pi$
$109$	−4.44428	−0.425685	−0.212842	+	0.977087i	$0.568272\pi$
$110$	0	0
$111$	−6.59414	−0.625888
$112$	0	0
$113$	11.1883i	1.05251i	0.850328	+	0.526253i	$0.176403\pi$
$113$	11.1883i	1.05251i	−0.850328	+	0.526253i	$0.823597\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 10.2088i	− 0.943806i
$118$	0	0
$119$	−0.380943	−0.0349210
$120$	0	0
$121$	14.8458	1.34961
$122$	0	0
$123$	− 28.3355i	− 2.55493i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	16.7619i	1.48738i	0.668526	+	0.743688i	$0.266925\pi$
$127$	16.7619i	1.48738i	−0.668526	+	0.743688i	$0.733075\pi$
$128$	0	0
$129$	12.1267	1.06769
$130$	0	0
$131$	3.67801	0.321350	0.160675	−	0.987007i	$-0.448633\pi$
$131$	3.67801	0.321350	0.160675	+	0.987007i	$0.448633\pi$
$132$	0	0
$133$	1.78680i	0.154935i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	21.4015i	1.82845i	0.405205	+	0.914226i	$0.367200\pi$
$137$	21.4015i	1.82845i	−0.405205	+	0.914226i	$0.632800\pi$
$138$	0	0
$139$	10.8252	0.918183	0.459092	−	0.888389i	$-0.348175\pi$
$139$	10.8252	0.918183	0.459092	+	0.888389i	$0.348175\pi$
$140$	0	0
$141$	−15.3560	−1.29321
$142$	0	0
$143$	− 6.59414i	− 0.551430i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 12.5531i	− 1.03536i
$148$	0	0
$149$	21.2516	1.74100	0.870500	−	0.492168i	$-0.163795\pi$
$149$	21.2516	1.74100	0.870500	+	0.492168i	$0.163795\pi$
$150$	0	0
$151$	−15.9091	−1.29466	−0.647332	−	0.762208i	$-0.724115\pi$
$151$	−15.9091	−1.29466	−0.647332	+	0.762208i	$0.724115\pi$
$152$	0	0
$153$	− 1.67801i	− 0.135659i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	14.0000i	1.11732i	0.829396	+	0.558661i	$0.188685\pi$
$157$	14.0000i	1.11732i	−0.829396	+	0.558661i	$0.811315\pi$
$158$	0	0
$159$	−36.3944	−2.88627
$160$	0	0
$161$	−6.65310	−0.524338
$162$	0	0
$163$	− 9.18828i	− 0.719682i	−0.933014	−	0.359841i	$-0.882831\pi$
$163$	− 9.18828i	− 0.719682i	0.933014	−	0.359841i	$-0.117169\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 5.83226i	− 0.451313i	−0.974207	−	0.225657i	$-0.927547\pi$
$167$	− 5.83226i	− 0.451313i	0.974207	−	0.225657i	$-0.0724528\pi$
$168$	0	0
$169$	11.3176	0.870585
$170$	0	0
$171$	−7.87067	−0.601885
$172$	0	0
$173$	7.18828i	0.546515i	0.961941	+	0.273257i	$0.0881011\pi$
$173$	7.18828i	0.546515i	−0.961941	+	0.273257i	$0.911899\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 15.5059i	− 1.16549i
$178$	0	0
$179$	7.87333	0.588480	0.294240	−	0.955732i	$-0.404933\pi$
$179$	7.87333	0.588480	0.294240	+	0.955732i	$0.404933\pi$
$180$	0	0
$181$	−22.3355	−1.66018	−0.830092	−	0.557627i	$-0.811712\pi$
$181$	−22.3355	−1.66018	−0.830092	+	0.557627i	$0.811712\pi$
$182$	0	0
$183$	− 11.5736i	− 0.855545i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 1.08387i	− 0.0792606i
$188$	0	0
$189$	−28.6942	−2.08719
$190$	0	0
$191$	−23.7370	−1.71755	−0.858773	−	0.512356i	$-0.828773\pi$
$191$	−23.7370	−1.71755	−0.858773	+	0.512356i	$0.828773\pi$
$192$	0	0
$193$	7.31495i	0.526542i	0.964722	+	0.263271i	$0.0848013\pi$
$193$	7.31495i	0.526542i	−0.964722	+	0.263271i	$0.915199\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 8.59414i	− 0.612307i	−0.951982	−	0.306154i	$-0.900958\pi$
$197$	− 8.59414i	− 0.612307i	0.951982	−	0.306154i	$-0.0990421\pi$
$198$	0	0
$199$	7.06599	0.500895	0.250447	−	0.968130i	$-0.419422\pi$
$199$	7.06599	0.500895	0.250447	+	0.968130i	$0.419422\pi$
$200$	0	0
$201$	3.72347	0.262633
$202$	0	0
$203$	− 1.55572i	− 0.109190i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 29.3062i	− 2.03692i
$208$	0	0
$209$	−5.08387	−0.351659
$210$	0	0
$211$	−5.55572	−0.382472	−0.191236	−	0.981544i	$-0.561250\pi$
$211$	−5.55572	−0.382472	−0.191236	+	0.981544i	$0.561250\pi$
$212$	0	0
$213$	− 28.8886i	− 1.97941i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−22.4443	−1.51664
$220$	0	0
$221$	−0.276533	−0.0186016
$222$	0	0
$223$	16.7619i	1.12246i	0.827660	+	0.561229i	$0.189671\pi$
$223$	16.7619i	1.12246i	−0.827660	+	0.561229i	$0.810329\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	8.27653i	0.549333i	0.961540	+	0.274666i	$0.0885674\pi$
$227$	8.27653i	0.549333i	−0.961540	+	0.274666i	$0.911433\pi$
$228$	0	0
$229$	12.6986	0.839144	0.419572	−	0.907722i	$-0.362180\pi$
$229$	12.6986	0.839144	0.419572	+	0.907722i	$0.362180\pi$
$230$	0	0
$231$	−29.9502	−1.97058
$232$	0	0
$233$	− 8.36306i	− 0.547882i	−0.961746	−	0.273941i	$-0.911673\pi$
$233$	− 8.36306i	− 0.547882i	0.961746	−	0.273941i	$-0.0883274\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	48.1179i	3.12559i
$238$	0	0
$239$	−6.97508	−0.451181	−0.225590	−	0.974222i	$-0.572431\pi$
$239$	−6.97508	−0.451181	−0.225590	+	0.974222i	$0.572431\pi$
$240$	0	0
$241$	−19.7413	−1.27165	−0.635826	−	0.771832i	$-0.719340\pi$
$241$	−19.7413	−1.27165	−0.635826	+	0.771832i	$0.719340\pi$
$242$	0	0
$243$	− 48.5443i	− 3.11412i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.29707i	0.0825306i
$248$	0	0
$249$	32.1179	2.03539
$250$	0	0
$251$	10.2722	0.648373	0.324186	−	0.945993i	$-0.394910\pi$
$251$	10.2722	0.648373	0.324186	+	0.945993i	$0.394910\pi$
$252$	0	0
$253$	− 18.9296i	− 1.19010i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	9.90909i	0.618112i	0.951044	+	0.309056i	$0.100013\pi$
$257$	9.90909i	0.618112i	−0.951044	+	0.309056i	$0.899987\pi$
$258$	0	0
$259$	3.57360	0.222053
$260$	0	0
$261$	6.85279	0.424177
$262$	0	0
$263$	− 23.6780i	− 1.46005i	−0.683421	−	0.730024i	$-0.739509\pi$
$263$	− 23.6780i	− 1.46005i	0.683421	−	0.730024i	$-0.260491\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 22.2944i	− 1.36440i
$268$	0	0
$269$	5.02054	0.306108	0.153054	−	0.988218i	$-0.451089\pi$
$269$	5.02054	0.306108	0.153054	+	0.988218i	$0.451089\pi$
$270$	0	0
$271$	4.27653	0.259781	0.129890	−	0.991528i	$-0.458537\pi$
$271$	4.27653	0.259781	0.129890	+	0.991528i	$0.458537\pi$
$272$	0	0
$273$	7.64132i	0.462474i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	7.93667i	0.476868i	0.971159	+	0.238434i	$0.0766341\pi$
$277$	7.93667i	0.476868i	−0.971159	+	0.238434i	$0.923366\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	22.8886	1.36542	0.682708	−	0.730691i	$-0.260802\pi$
$281$	22.8886	1.36542	0.682708	+	0.730691i	$0.260802\pi$
$282$	0	0
$283$	5.08387i	0.302205i	0.988518	+	0.151102i	$0.0482823\pi$
$283$	5.08387i	0.302205i	−0.988518	+	0.151102i	$0.951718\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	15.3560i	0.906438i
$288$	0	0
$289$	16.9545	0.997326
$290$	0	0
$291$	13.7413	0.805532
$292$	0	0
$293$	3.55572i	0.207728i	0.994592	+	0.103864i	$0.0331206\pi$
$293$	3.55572i	0.207728i	−0.994592	+	0.103864i	$0.966879\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 81.6417i	− 4.73733i
$298$	0	0
$299$	−4.82960	−0.279303
$300$	0	0
$301$	−6.57188	−0.378797
$302$	0	0
$303$	− 63.2651i	− 3.63449i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	16.3355i	0.932316i	0.884702	+	0.466158i	$0.154362\pi$
$307$	16.3355i	0.932316i	−0.884702	+	0.466158i	$0.845638\pi$
$308$	0	0
$309$	52.4534	2.98397
$310$	0	0
$311$	−16.3809	−0.928878	−0.464439	−	0.885605i	$-0.653744\pi$
$311$	−16.3809	−0.928878	−0.464439	+	0.885605i	$0.653744\pi$
$312$	0	0
$313$	30.0590i	1.69903i	0.527562	+	0.849516i	$0.323106\pi$
$313$	30.0590i	1.69903i	−0.527562	+	0.849516i	$0.676894\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 12.0179i	− 0.674991i	−0.941327	−	0.337496i	$-0.890420\pi$
$317$	− 12.0179i	− 0.674991i	0.941327	−	0.337496i	$-0.109580\pi$
$318$	0	0
$319$	4.42640	0.247831
$320$	0	0
$321$	−10.8707	−0.606742
$322$	0	0
$323$	0.213198i	0.0118627i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	14.6531i	0.810318i
$328$	0	0
$329$	8.32199	0.458806
$330$	0	0
$331$	−14.3176	−0.786967	−0.393483	−	0.919332i	$-0.628730\pi$
$331$	−14.3176	−0.786967	−0.393483	+	0.919332i	$0.628730\pi$
$332$	0	0
$333$	15.7413i	0.862621i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	11.6147i	0.632692i	0.948644	+	0.316346i	$0.102456\pi$
$337$	11.6147i	0.632692i	−0.948644	+	0.316346i	$0.897544\pi$
$338$	0	0
$339$	36.8886	2.00351
$340$	0	0
$341$	0	0
$342$	0	0
$343$	19.3106i	1.04267i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 2.91613i	− 0.156546i	−0.996932	−	0.0782730i	$-0.975059\pi$
$347$	− 2.91613i	− 0.156546i	0.996932	−	0.0782730i	$-0.0249406\pi$
$348$	0	0
$349$	−12.8252	−0.686518	−0.343259	−	0.939241i	$-0.611531\pi$
$349$	−12.8252	−0.686518	−0.343259	+	0.939241i	$0.611531\pi$
$350$	0	0
$351$	−20.8296	−1.11180
$352$	0	0
$353$	− 25.7592i	− 1.37103i	−0.728061	−	0.685513i	$-0.759578\pi$
$353$	− 25.7592i	− 1.37103i	0.728061	−	0.685513i	$-0.240422\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	1.25600i	0.0664743i
$358$	0	0
$359$	−22.4220	−1.18339	−0.591694	−	0.806162i	$-0.701541\pi$
$359$	−22.4220	−1.18339	−0.591694	+	0.806162i	$0.701541\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	− 48.9475i	− 2.56908i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	5.18828i	0.270826i	0.990789	+	0.135413i	$0.0432361\pi$
$367$	5.18828i	0.270826i	−0.990789	+	0.135413i	$0.956764\pi$
$368$	0	0
$369$	−67.6417	−3.52129
$370$	0	0
$371$	19.7235	1.02399
$372$	0	0
$373$	− 36.0091i	− 1.86448i	−0.361838	−	0.932241i	$-0.617851\pi$
$373$	− 36.0091i	− 1.86448i	0.361838	−	0.932241i	$-0.382149\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 1.12933i	− 0.0581632i
$378$	0	0
$379$	25.0384	1.28614	0.643069	−	0.765809i	$-0.277661\pi$
$379$	25.0384	1.28614	0.643069	+	0.765809i	$0.277661\pi$
$380$	0	0
$381$	55.2651	2.83132
$382$	0	0
$383$	21.7413i	1.11093i	0.831540	+	0.555466i	$0.187460\pi$
$383$	21.7413i	1.11093i	−0.831540	+	0.555466i	$0.812540\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 28.9484i	− 1.47153i
$388$	0	0
$389$	−11.5103	−0.583594	−0.291797	−	0.956480i	$-0.594253\pi$
$389$	−11.5103	−0.583594	−0.291797	+	0.956480i	$0.594253\pi$
$390$	0	0
$391$	−0.793836	−0.0401460
$392$	0	0
$393$	− 12.1267i	− 0.611710i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 7.21054i	− 0.361887i	−0.983494	−	0.180943i	$-0.942085\pi$
$397$	− 7.21054i	− 0.361887i	0.983494	−	0.180943i	$-0.0579151\pi$
$398$	0	0
$399$	5.89121	0.294929
$400$	0	0
$401$	1.23811	0.0618285	0.0309142	−	0.999522i	$-0.490158\pi$
$401$	1.23811	0.0618285	0.0309142	+	0.999522i	$0.490158\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	10.1677i	0.503996i
$408$	0	0
$409$	−32.8030	−1.62200	−0.811001	−	0.585045i	$-0.801077\pi$
$409$	−32.8030	−1.62200	−0.811001	+	0.585045i	$0.801077\pi$
$410$	0	0
$411$	70.5622	3.48058
$412$	0	0
$413$	8.40320i	0.413495i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 35.6915i	− 1.74782i
$418$	0	0
$419$	17.1883	0.839703	0.419851	−	0.907593i	$-0.362082\pi$
$419$	17.1883	0.839703	0.419851	+	0.907593i	$0.362082\pi$
$420$	0	0
$421$	19.8912	0.969438	0.484719	−	0.874670i	$-0.338922\pi$
$421$	19.8912	0.969438	0.484719	+	0.874670i	$0.338922\pi$
$422$	0	0
$423$	36.6575i	1.78235i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	6.27215i	0.303531i
$428$	0	0
$429$	−21.7413	−1.04968
$430$	0	0
$431$	8.55307	0.411987	0.205993	−	0.978553i	$-0.433957\pi$
$431$	8.55307	0.411987	0.205993	+	0.978553i	$0.433957\pi$
$432$	0	0
$433$	31.5238i	1.51494i	0.652872	+	0.757468i	$0.273564\pi$
$433$	31.5238i	1.51494i	−0.652872	+	0.757468i	$0.726436\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3.72347i	0.178118i
$438$	0	0
$439$	4.12667	0.196955	0.0984776	−	0.995139i	$-0.468603\pi$
$439$	4.12667	0.196955	0.0984776	+	0.995139i	$0.468603\pi$
$440$	0	0
$441$	−29.9663	−1.42697
$442$	0	0
$443$	− 37.7549i	− 1.79379i	−0.442246	−	0.896894i	$-0.645818\pi$
$443$	− 37.7549i	− 1.79379i	0.442246	−	0.896894i	$-0.354182\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 70.0681i	− 3.31411i
$448$	0	0
$449$	−33.3560	−1.57417	−0.787084	−	0.616846i	$-0.788410\pi$
$449$	−33.3560	−1.57417	−0.787084	+	0.616846i	$0.788410\pi$
$450$	0	0
$451$	−43.6915	−2.05735
$452$	0	0
$453$	52.4534i	2.46448i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	32.1223i	1.50262i	0.659951	+	0.751309i	$0.270577\pi$
$457$	32.1223i	1.50262i	−0.659951	+	0.751309i	$0.729423\pi$
$458$	0	0
$459$	−3.42374	−0.159807
$460$	0	0
$461$	−7.81000	−0.363748	−0.181874	−	0.983322i	$-0.558216\pi$
$461$	−7.81000	−0.363748	−0.181874	+	0.983322i	$0.558216\pi$
$462$	0	0
$463$	− 18.4897i	− 0.859291i	−0.902998	−	0.429645i	$-0.858639\pi$
$463$	− 18.4897i	− 0.859291i	0.902998	−	0.429645i	$-0.141361\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	21.0839i	0.975645i	0.872943	+	0.487823i	$0.162209\pi$
$467$	21.0839i	0.975645i	−0.872943	+	0.487823i	$0.837791\pi$
$468$	0	0
$469$	−2.01788	−0.0931771
$470$	0	0
$471$	46.1590	2.12689
$472$	0	0
$473$	− 18.6986i	− 0.859760i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	86.8798i	3.97795i
$478$	0	0
$479$	25.5238	1.16621	0.583105	−	0.812396i	$-0.301837\pi$
$479$	25.5238	1.16621	0.583105	+	0.812396i	$0.301837\pi$
$480$	0	0
$481$	2.59414	0.118283
$482$	0	0
$483$	21.9357i	0.998110i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 10.4675i	− 0.474326i	−0.971470	−	0.237163i	$-0.923782\pi$
$487$	− 10.4675i	− 0.474326i	0.971470	−	0.237163i	$-0.0762176\pi$
$488$	0	0
$489$	−30.2944	−1.36996
$490$	0	0
$491$	12.6710	0.571833	0.285917	−	0.958254i	$-0.407702\pi$
$491$	12.6710	0.571833	0.285917	+	0.958254i	$0.407702\pi$
$492$	0	0
$493$	− 0.185626i	− 0.00836018i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	15.6558i	0.702257i
$498$	0	0
$499$	13.8458	0.619821	0.309911	−	0.950766i	$-0.399701\pi$
$499$	13.8458	0.619821	0.309911	+	0.950766i	$0.399701\pi$
$500$	0	0
$501$	−19.2294	−0.859104
$502$	0	0
$503$	9.76454i	0.435379i	0.976018	+	0.217690i	$0.0698521\pi$
$503$	9.76454i	0.435379i	−0.976018	+	0.217690i	$0.930148\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 37.3150i	− 1.65722i
$508$	0	0
$509$	28.1677	1.24851	0.624257	−	0.781219i	$-0.285402\pi$
$509$	28.1677	1.24851	0.624257	+	0.781219i	$0.285402\pi$
$510$	0	0
$511$	12.1634	0.538076
$512$	0	0
$513$	16.0590i	0.709020i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	23.6780i	1.04136i
$518$	0	0
$519$	23.7003	1.04033
$520$	0	0
$521$	−28.2035	−1.23562	−0.617809	−	0.786328i	$-0.711980\pi$
$521$	−28.2035	−1.23562	−0.617809	+	0.786328i	$0.711980\pi$
$522$	0	0
$523$	− 1.22023i	− 0.0533571i	−0.999644	−	0.0266785i	$-0.991507\pi$
$523$	− 1.22023i	− 0.0533571i	0.999644	−	0.0266785i	$-0.00849305\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	9.13579	0.397208
$530$	0	0
$531$	−37.0152	−1.60632
$532$	0	0
$533$	11.1472i	0.482839i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 25.9589i	− 1.12021i
$538$	0	0
$539$	−19.3560	−0.833723
$540$	0	0
$541$	12.1812	0.523713	0.261856	−	0.965107i	$-0.415665\pi$
$541$	12.1812	0.523713	0.261856	+	0.965107i	$0.415665\pi$
$542$	0	0
$543$	73.6417i	3.16027i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	12.0000i	0.513083i	0.966533	+	0.256541i	$0.0825830\pi$
$547$	12.0000i	0.513083i	−0.966533	+	0.256541i	$0.917417\pi$
$548$	0	0
$549$	−27.6282	−1.17914
$550$	0	0
$551$	−0.870674	−0.0370920
$552$	0	0
$553$	− 26.0768i	− 1.10890i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	8.60764i	0.364718i	0.983232	+	0.182359i	$0.0583732\pi$
$557$	8.60764i	0.364718i	−0.983232	+	0.182359i	$0.941627\pi$
$558$	0	0
$559$	−4.77064	−0.201777
$560$	0	0
$561$	−3.57360	−0.150878
$562$	0	0
$563$	19.3560i	0.815759i	0.913036	+	0.407880i	$0.133732\pi$
$563$	19.3560i	0.815759i	−0.913036	+	0.407880i	$0.866268\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	52.4167i	2.20129i
$568$	0	0
$569$	−9.23811	−0.387282	−0.193641	−	0.981072i	$-0.562030\pi$
$569$	−9.23811	−0.387282	−0.193641	+	0.981072i	$0.562030\pi$
$570$	0	0
$571$	14.3766	0.601640	0.300820	−	0.953681i	$-0.402740\pi$
$571$	14.3766	0.601640	0.300820	+	0.953681i	$0.402740\pi$
$572$	0	0
$573$	78.2625i	3.26946i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	5.19266i	0.216173i	0.994141	+	0.108087i	$0.0344724\pi$
$577$	5.19266i	0.216173i	−0.994141	+	0.108087i	$0.965528\pi$
$578$	0	0
$579$	24.1179	1.00231
$580$	0	0
$581$	−17.4059	−0.722117
$582$	0	0
$583$	56.1179i	2.32417i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	30.3079i	1.25094i	0.780248	+	0.625471i	$0.215093\pi$
$587$	30.3079i	1.25094i	−0.780248	+	0.625471i	$0.784907\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−28.3355	−1.16557
$592$	0	0
$593$	4.04107i	0.165947i	0.996552	+	0.0829735i	$0.0264417\pi$
$593$	4.04107i	0.165947i	−0.996552	+	0.0829735i	$0.973558\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 23.2971i	− 0.953486i
$598$	0	0
$599$	6.29441	0.257183	0.128591	−	0.991698i	$-0.458954\pi$
$599$	6.29441	0.257183	0.128591	+	0.991698i	$0.458954\pi$
$600$	0	0
$601$	−10.0358	−0.409367	−0.204684	−	0.978828i	$-0.565617\pi$
$601$	−10.0358	−0.409367	−0.204684	+	0.978828i	$0.565617\pi$
$602$	0	0
$603$	− 8.88856i	− 0.361970i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 45.3062i	− 1.83892i	−0.393182	−	0.919461i	$-0.628626\pi$
$607$	− 45.3062i	− 1.83892i	0.393182	−	0.919461i	$-0.371374\pi$
$608$	0	0
$609$	−5.12933	−0.207851
$610$	0	0
$611$	6.04107	0.244396
$612$	0	0
$613$	− 25.1607i	− 1.01623i	−0.861289	−	0.508116i	$-0.830342\pi$
$613$	− 25.1607i	− 1.01623i	0.861289	−	0.508116i	$-0.169658\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	6.01350i	0.242095i	0.992647	+	0.121047i	$0.0386253\pi$
$617$	6.01350i	0.242095i	−0.992647	+	0.121047i	$0.961375\pi$
$618$	0	0
$619$	−37.2240	−1.49616	−0.748080	−	0.663608i	$-0.769024\pi$
$619$	−37.2240	−1.49616	−0.748080	+	0.663608i	$0.769024\pi$
$620$	0	0
$621$	−59.7950	−2.39949
$622$	0	0
$623$	12.0821i	0.484061i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	16.7619i	0.669405i
$628$	0	0
$629$	0.426396	0.0170015
$630$	0	0
$631$	−12.0135	−0.478250	−0.239125	−	0.970989i	$-0.576861\pi$
$631$	−12.0135	−0.478250	−0.239125	+	0.970989i	$0.576861\pi$
$632$	0	0
$633$	18.3176i	0.728060i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	4.93839i	0.195666i
$638$	0	0
$639$	−68.9619	−2.72809
$640$	0	0
$641$	6.97946	0.275672	0.137836	−	0.990455i	$-0.455985\pi$
$641$	6.97946	0.275672	0.137836	+	0.990455i	$0.455985\pi$
$642$	0	0
$643$	− 40.7754i	− 1.60802i	−0.594613	−	0.804012i	$-0.702695\pi$
$643$	− 40.7754i	− 1.60802i	0.594613	−	0.804012i	$-0.297305\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	24.9340i	0.980257i	0.871650	+	0.490129i	$0.163050\pi$
$647$	24.9340i	0.980257i	−0.871650	+	0.490129i	$0.836950\pi$
$648$	0	0
$649$	−23.9091	−0.938514
$650$	0	0
$651$	0	0
$652$	0	0
$653$	36.3988i	1.42440i	0.701979	+	0.712198i	$0.252300\pi$
$653$	36.3988i	1.42440i	−0.701979	+	0.712198i	$0.747700\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	53.5783i	2.09029i
$658$	0	0
$659$	−11.8501	−0.461616	−0.230808	−	0.972999i	$-0.574137\pi$
$659$	−11.8501	−0.461616	−0.230808	+	0.972999i	$0.574137\pi$
$660$	0	0
$661$	34.4854	1.34132	0.670662	−	0.741763i	$-0.266010\pi$
$661$	34.4854	1.34132	0.670662	+	0.741763i	$0.266010\pi$
$662$	0	0
$663$	0.911749i	0.0354094i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 3.24193i	− 0.125528i
$668$	0	0
$669$	55.2651	2.13667
$670$	0	0
$671$	−17.8458	−0.688928
$672$	0	0
$673$	− 17.7003i	− 0.682295i	−0.940010	−	0.341148i	$-0.889184\pi$
$673$	− 17.7003i	− 0.682295i	0.940010	−	0.341148i	$-0.110816\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 26.0590i	− 1.00153i	−0.865584	−	0.500764i	$-0.833053\pi$
$677$	− 26.0590i	− 1.00153i	0.865584	−	0.500764i	$-0.166947\pi$
$678$	0	0
$679$	−7.44693	−0.285787
$680$	0	0
$681$	27.2883	1.04569
$682$	0	0
$683$	12.0000i	0.459167i	0.973289	+	0.229584i	$0.0737364\pi$
$683$	12.0000i	0.459167i	−0.973289	+	0.229584i	$0.926264\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 41.8680i	− 1.59736i
$688$	0	0
$689$	14.3176	0.545457
$690$	0	0
$691$	13.9367	0.530176	0.265088	−	0.964224i	$-0.414599\pi$
$691$	13.9367	0.530176	0.265088	+	0.964224i	$0.414599\pi$
$692$	0	0
$693$	71.4962i	2.71592i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	1.83226i	0.0694016i
$698$	0	0
$699$	−27.5736	−1.04293
$700$	0	0
$701$	6.55307	0.247506	0.123753	−	0.992313i	$-0.460507\pi$
$701$	6.55307	0.247506	0.123753	+	0.992313i	$0.460507\pi$
$702$	0	0
$703$	− 2.00000i	− 0.0754314i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	34.2857i	1.28944i
$708$	0	0
$709$	−23.5238	−0.883454	−0.441727	−	0.897150i	$-0.645634\pi$
$709$	−23.5238	−0.883454	−0.441727	+	0.897150i	$0.645634\pi$
$710$	0	0
$711$	114.866	4.30780
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	22.9973i	0.858852i
$718$	0	0
$719$	14.8842	0.555086	0.277543	−	0.960713i	$-0.410480\pi$
$719$	14.8842	0.555086	0.277543	+	0.960713i	$0.410480\pi$
$720$	0	0
$721$	−28.4264	−1.05865
$722$	0	0
$723$	65.0886i	2.42067i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	1.32464i	0.0491283i	0.999698	+	0.0245641i	$0.00781979\pi$
$727$	1.32464i	0.0491283i	−0.999698	+	0.0245641i	$0.992180\pi$
$728$	0	0
$729$	−72.0475	−2.66843
$730$	0	0
$731$	−0.784146	−0.0290027
$732$	0	0
$733$	13.7824i	0.509065i	0.967064	+	0.254533i	$0.0819217\pi$
$733$	13.7824i	0.509065i	−0.967064	+	0.254533i	$0.918078\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 5.74135i	− 0.211485i
$738$	0	0
$739$	−23.5513	−0.866350	−0.433175	−	0.901310i	$-0.642607\pi$
$739$	−23.5513	−0.866350	−0.433175	+	0.901310i	$0.642607\pi$
$740$	0	0
$741$	4.27653	0.157102
$742$	0	0
$743$	14.0411i	0.515117i	0.966263	+	0.257559i	$0.0829180\pi$
$743$	14.0411i	0.515117i	−0.966263	+	0.257559i	$0.917082\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 76.6710i	− 2.80524i
$748$	0	0
$749$	5.89121	0.215260
$750$	0	0
$751$	−17.7056	−0.646086	−0.323043	−	0.946384i	$-0.604706\pi$
$751$	−17.7056	−0.646086	−0.323043	+	0.946384i	$0.604706\pi$
$752$	0	0
$753$	− 33.8680i	− 1.23422i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	30.3220i	1.10207i	0.834482	+	0.551036i	$0.185767\pi$
$757$	30.3220i	1.10207i	−0.834482	+	0.551036i	$0.814233\pi$
$758$	0	0
$759$	−62.4123	−2.26542
$760$	0	0
$761$	12.5487	0.454890	0.227445	−	0.973791i	$-0.426963\pi$
$761$	12.5487	0.454890	0.227445	+	0.973791i	$0.426963\pi$
$762$	0	0
$763$	− 7.94104i	− 0.287485i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	6.10003i	0.220259i
$768$	0	0
$769$	50.0725	1.80566	0.902830	−	0.429999i	$-0.141486\pi$
$769$	50.0725	1.80566	0.902830	+	0.429999i	$0.141486\pi$
$770$	0	0
$771$	32.6710	1.17662
$772$	0	0
$773$	− 11.8003i	− 0.424427i	−0.977223	−	0.212214i	$-0.931933\pi$
$773$	− 11.8003i	− 0.424427i	0.977223	−	0.212214i	$-0.0680673\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 11.7824i	− 0.422692i
$778$	0	0
$779$	8.59414	0.307917
$780$	0	0
$781$	−44.5443	−1.59392
$782$	0	0
$783$	− 13.9821i	− 0.499680i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	32.9475i	1.17445i	0.809423	+	0.587226i	$0.199780\pi$
$787$	32.9475i	1.17445i	−0.809423	+	0.587226i	$0.800220\pi$
$788$	0	0
$789$	−78.0681	−2.77930
$790$	0	0
$791$	−19.9912	−0.710807
$792$	0	0
$793$	4.55307i	0.161684i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 18.8208i	− 0.666668i	−0.942809	−	0.333334i	$-0.891826\pi$
$797$	− 18.8208i	− 0.666668i	0.942809	−	0.333334i	$-0.108174\pi$
$798$	0	0
$799$	0.992965	0.0351286
$800$	0	0
$801$	−53.2206	−1.88046
$802$	0	0
$803$	34.6076i	1.22128i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 16.5531i	− 0.582696i
$808$	0	0
$809$	39.2607	1.38033	0.690167	−	0.723650i	$-0.257537\pi$
$809$	39.2607	1.38033	0.690167	+	0.723650i	$0.257537\pi$
$810$	0	0
$811$	33.9182	1.19103	0.595515	−	0.803344i	$-0.296948\pi$
$811$	33.9182	1.19103	0.595515	+	0.803344i	$0.296948\pi$
$812$	0	0
$813$	− 14.1000i	− 0.494510i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	3.67801i	0.128677i
$818$	0	0
$819$	18.2411	0.637397
$820$	0	0
$821$	38.6487	1.34885	0.674425	−	0.738344i	$-0.264392\pi$
$821$	38.6487	1.34885	0.674425	+	0.738344i	$0.264392\pi$
$822$	0	0
$823$	8.47185i	0.295310i	0.989039	+	0.147655i	$0.0471725\pi$
$823$	8.47185i	0.295310i	−0.989039	+	0.147655i	$0.952827\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 44.7299i	− 1.55541i	−0.628628	−	0.777706i	$-0.716383\pi$
$827$	− 44.7299i	− 1.55541i	0.628628	−	0.777706i	$-0.283617\pi$
$828$	0	0
$829$	38.8566	1.34955	0.674773	−	0.738025i	$-0.264242\pi$
$829$	38.8566	1.34955	0.674773	+	0.738025i	$0.264242\pi$
$830$	0	0
$831$	26.1677	0.907749
$832$	0	0
$833$	0.811718i	0.0281244i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	7.53784	0.260235	0.130118	−	0.991499i	$-0.458465\pi$
$839$	7.53784	0.260235	0.130118	+	0.991499i	$0.458465\pi$
$840$	0	0
$841$	−28.2419	−0.973860
$842$	0	0
$843$	− 75.4652i	− 2.59916i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	26.5264i	0.911459i
$848$	0	0
$849$	16.7619	0.575266
$850$	0	0
$851$	7.44693	0.255278
$852$	0	0
$853$	43.5238i	1.49023i	0.666938	+	0.745113i	$0.267604\pi$
$853$	43.5238i	1.49023i	−0.666938	+	0.745113i	$0.732396\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 14.5531i	− 0.497123i	−0.968616	−	0.248562i	$-0.920042\pi$
$857$	− 14.5531i	− 0.497123i	0.968616	−	0.248562i	$-0.0799579\pi$
$858$	0	0
$859$	24.3132	0.829557	0.414778	−	0.909922i	$-0.363859\pi$
$859$	24.3132	0.829557	0.414778	+	0.909922i	$0.363859\pi$
$860$	0	0
$861$	50.6299	1.72546
$862$	0	0
$863$	12.4264i	0.422999i	0.977378	+	0.211500i	$0.0678347\pi$
$863$	12.4264i	0.422999i	−0.977378	+	0.211500i	$0.932165\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 55.9003i	− 1.89847i
$868$	0	0
$869$	74.1948	2.51688
$870$	0	0
$871$	−1.46482	−0.0496334
$872$	0	0
$873$	− 32.8030i	− 1.11021i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	54.6120i	1.84412i	0.387051	+	0.922058i	$0.373494\pi$
$877$	54.6120i	1.84412i	−0.387051	+	0.922058i	$0.626506\pi$
$878$	0	0
$879$	11.7235	0.395423
$880$	0	0
$881$	35.4551	1.19451	0.597257	−	0.802050i	$-0.296257\pi$
$881$	35.4551	1.19451	0.597257	+	0.802050i	$0.296257\pi$
$882$	0	0
$883$	− 21.5460i	− 0.725082i	−0.931968	−	0.362541i	$-0.881909\pi$
$883$	− 21.5460i	− 0.725082i	0.931968	−	0.362541i	$-0.118091\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	49.6147i	1.66590i	0.553350	+	0.832949i	$0.313349\pi$
$887$	49.6147i	1.66590i	−0.553350	+	0.832949i	$0.686651\pi$
$888$	0	0
$889$	−29.9502	−1.00450
$890$	0	0
$891$	−149.138	−4.99631
$892$	0	0
$893$	− 4.65748i	− 0.155856i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	15.9235i	0.531671i
$898$	0	0
$899$	0	0
$900$	0	0
$901$	2.35337	0.0784022
$902$	0	0
$903$	21.6680i	0.721064i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 11.0883i	− 0.368179i	−0.982909	−	0.184090i	$-0.941066\pi$
$907$	− 11.0883i	− 0.368179i	0.982909	−	0.184090i	$-0.0589337\pi$
$908$	0	0
$909$	−151.025	−5.00917
$910$	0	0
$911$	12.5443	0.415612	0.207806	−	0.978170i	$-0.433368\pi$
$911$	12.5443	0.415612	0.207806	+	0.978170i	$0.433368\pi$
$912$	0	0
$913$	− 49.5238i	− 1.63900i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	6.57188i	0.217023i
$918$	0	0
$919$	−42.8707	−1.41417	−0.707087	−	0.707127i	$-0.749991\pi$
$919$	−42.8707	−1.41417	−0.707087	+	0.707127i	$0.749991\pi$
$920$	0	0
$921$	53.8593	1.77472
$922$	0	0
$923$	11.3648i	0.374076i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 125.215i	− 4.11261i
$928$	0	0
$929$	27.9182	0.915967	0.457984	−	0.888961i	$-0.348572\pi$
$929$	27.9182	0.915967	0.457984	+	0.888961i	$0.348572\pi$
$930$	0	0
$931$	3.80734	0.124781
$932$	0	0
$933$	54.0091i	1.76818i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 10.8982i	− 0.356030i	−0.984028	−	0.178015i	$-0.943032\pi$
$937$	− 10.8982i	− 0.356030i	0.984028	−	0.178015i	$-0.0569676\pi$
$938$	0	0
$939$	99.1065	3.23422
$940$	0	0
$941$	−30.9475	−1.00886	−0.504430	−	0.863453i	$-0.668297\pi$
$941$	−30.9475	−1.00886	−0.504430	+	0.863453i	$0.668297\pi$
$942$	0	0
$943$	32.0000i	1.04206i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 18.2944i	− 0.594489i	−0.954801	−	0.297244i	$-0.903932\pi$
$947$	− 18.2944i	− 0.594489i	0.954801	−	0.297244i	$-0.0960676\pi$
$948$	0	0
$949$	8.82960	0.286621
$950$	0	0
$951$	−39.6238	−1.28489
$952$	0	0
$953$	23.0616i	0.747039i	0.927622	+	0.373519i	$0.121849\pi$
$953$	23.0616i	0.747039i	−0.927622	+	0.373519i	$0.878151\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 14.5941i	− 0.471762i
$958$	0	0
$959$	−38.2402	−1.23484
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	25.9502i	0.836232i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	26.3766i	0.848213i	0.905612	+	0.424107i	$0.139412\pi$
$967$	26.3766i	0.848213i	−0.905612	+	0.424107i	$0.860588\pi$
$968$	0	0
$969$	0.702929	0.0225813
$970$	0	0
$971$	20.0000	0.641831	0.320915	−	0.947108i	$-0.396010\pi$
$971$	20.0000	0.641831	0.320915	+	0.947108i	$0.396010\pi$
$972$	0	0
$973$	19.3425i	0.620093i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 26.4264i	− 0.845455i	−0.906257	−	0.422728i	$-0.861073\pi$
$977$	− 26.4264i	− 0.845455i	0.906257	−	0.422728i	$-0.138927\pi$
$978$	0	0
$979$	−34.3766	−1.09868
$980$	0	0
$981$	34.9795	1.11681
$982$	0	0
$983$	− 25.4059i	− 0.810321i	−0.914246	−	0.405161i	$-0.867216\pi$
$983$	− 25.4059i	− 0.810321i	0.914246	−	0.405161i	$-0.132784\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 27.4382i	− 0.873367i
$988$	0	0
$989$	−13.6950	−0.435474
$990$	0	0
$991$	29.6504	0.941877	0.470939	−	0.882166i	$-0.343915\pi$
$991$	29.6504	0.941877	0.470939	+	0.882166i	$0.343915\pi$
$992$	0	0
$993$	47.2062i	1.49804i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 38.7754i	− 1.22803i	−0.789295	−	0.614014i	$-0.789554\pi$
$997$	− 38.7754i	− 1.22803i	0.789295	−	0.614014i	$-0.210446\pi$
$998$	0	0
$999$	32.1179	1.01617

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3800.2.d.j.3649.1		6
5.2	odd	4		3800.2.a.r.1.1		3
5.3	odd	4		152.2.a.c.1.3	✓	3
5.4	even	2	inner	3800.2.d.j.3649.6		6
15.8	even	4		1368.2.a.n.1.3		3
20.3	even	4		304.2.a.g.1.1		3
20.7	even	4		7600.2.a.bv.1.3		3
35.13	even	4		7448.2.a.bf.1.1		3
40.3	even	4		1216.2.a.v.1.3		3
40.13	odd	4		1216.2.a.u.1.1		3
60.23	odd	4		2736.2.a.bd.1.3		3
95.18	even	4		2888.2.a.o.1.1		3
380.303	odd	4		5776.2.a.bp.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.2.a.c.1.3	✓	3	5.3	odd	4
304.2.a.g.1.1		3	20.3	even	4
1216.2.a.u.1.1		3	40.13	odd	4
1216.2.a.v.1.3		3	40.3	even	4
1368.2.a.n.1.3		3	15.8	even	4
2736.2.a.bd.1.3		3	60.23	odd	4
2888.2.a.o.1.1		3	95.18	even	4
3800.2.a.r.1.1		3	5.2	odd	4
3800.2.d.j.3649.1		6	1.1	even	1	trivial
3800.2.d.j.3649.6		6	5.4	even	2	inner
5776.2.a.bp.1.3		3	380.303	odd	4
7448.2.a.bf.1.1		3	35.13	even	4
7600.2.a.bv.1.3		3	20.7	even	4

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:	\( N \)	\(=\)	\( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3800.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.29707i q^{3} +1.78680i q^{7} -7.87067 q^{9} -5.08387 q^{11} +1.29707i q^{13} +0.213198i q^{17} +1.00000 q^{19} +5.89121 q^{21} +3.72347i q^{23} +16.0590i q^{27} -0.870674 q^{29} +16.7619i q^{33} -2.00000i q^{37} +4.27653 q^{39} +8.59414 q^{41} +3.67801i q^{43} -4.65748i q^{47} +3.80734 q^{49} +0.702929 q^{51} -11.0384i q^{53} -3.29707i q^{57} +4.70293 q^{59} +3.51027 q^{61} -14.0633i q^{63} +1.12933i q^{67} +12.2765 q^{69} +8.76189 q^{71} -6.80734i q^{73} -9.08387i q^{77} -14.5941 q^{79} +29.3355 q^{81} +9.74135i q^{83} +2.87067i q^{87} +6.76189 q^{89} -2.31761 q^{91} +4.16774i q^{97} +40.0135 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 24 q^{9} - 10 q^{11} + 6 q^{19} - 18 q^{21} + 18 q^{29} + 38 q^{39} + 16 q^{41} - 10 q^{49} + 22 q^{51} + 46 q^{59} + 6 q^{61} + 86 q^{69} - 24 q^{71} - 52 q^{79} + 94 q^{81} - 36 q^{89} + 34 q^{91}+ \cdots + 102 q^{99}+O(q^{100}) \) 6 * q - 24 * q^9 - 10 * q^11 + 6 * q^19 - 18 * q^21 + 18 * q^29 + 38 * q^39 + 16 * q^41 - 10 * q^49 + 22 * q^51 + 46 * q^59 + 6 * q^61 + 86 * q^69 - 24 * q^71 - 52 * q^79 + 94 * q^81 - 36 * q^89 + 34 * q^91 + 102 * q^99

Introduction

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