LMFDB - Embedded newform 4212.2.a.l.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4212,2,Mod(1,4212)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4212, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4212.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 4212 = 2^{2} \cdot 3^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4212.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,0,0,8,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$33.6329893314$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.232742592.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 17x^{4} + 37x^{2} - 12 $ x^6 - 17x^4 + 37x^2 - 12
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.627689$ of defining polynomial
Character	$\chi$	$=$	4212.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.627689 q^{5} -2.95044 q^{7} -5.46708 q^{11} +1.00000 q^{13} -0.356762 q^{17} -1.86325 q^{19} +2.47965 q^{23} -4.60601 q^{25} -0.747600 q^{29} +0.912810 q^{31} -1.85196 q^{35} +7.60601 q^{37} +4.09179 q^{41} +6.51882 q^{43} -1.85196 q^{47} +1.70512 q^{49} -3.58401 q^{53} -3.43163 q^{55} +9.91563 q^{59} +7.81370 q^{61} +0.627689 q^{65} +0.912810 q^{67} -7.70691 q^{71} +5.20769 q^{73} +16.1303 q^{77} +2.61793 q^{79} +3.22725 q^{83} -0.223935 q^{85} +15.2110 q^{89} -2.95044 q^{91} -1.16954 q^{95} +11.9009 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 8 q^{7} + 6 q^{13} + 2 q^{19} + 4 q^{25} + 18 q^{31} + 14 q^{37} + 20 q^{43} + 30 q^{49} - 14 q^{55} + 8 q^{61} + 18 q^{67} + 24 q^{73} + 48 q^{79} - 2 q^{85} + 8 q^{91} + 20 q^{97}+O(q^{100}) $ 6 * q + 8 * q^7 + 6 * q^13 + 2 * q^19 + 4 * q^25 + 18 * q^31 + 14 * q^37 + 20 * q^43 + 30 * q^49 - 14 * q^55 + 8 * q^61 + 18 * q^67 + 24 * q^73 + 48 * q^79 - 2 * q^85 + 8 * q^91 + 20 * q^97

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0.627689	0.280711	0.140356	−	0.990101i	$-0.455175\pi$
$5$	0.627689	0.280711	0.140356	+	0.990101i	$0.455175\pi$
$6$	0	0
$7$	−2.95044	−1.11516	−0.557581	−	0.830122i	$-0.688271\pi$
$7$	−2.95044	−1.11516	−0.557581	+	0.830122i	$0.688271\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−5.46708	−1.64839	−0.824193	−	0.566309i	$-0.808371\pi$
$11$	−5.46708	−1.64839	−0.824193	+	0.566309i	$0.808371\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−0.356762	−0.0865274	−0.0432637	−	0.999064i	$-0.513776\pi$
$17$	−0.356762	−0.0865274	−0.0432637	+	0.999064i	$0.513776\pi$
$18$	0	0
$19$	−1.86325	−0.427460	−0.213730	−	0.976893i	$-0.568561\pi$
$19$	−1.86325	−0.427460	−0.213730	+	0.976893i	$0.568561\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.47965	0.517043	0.258521	−	0.966006i	$-0.416765\pi$
$23$	2.47965	0.517043	0.258521	+	0.966006i	$0.416765\pi$
$24$	0	0
$25$	−4.60601	−0.921201
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−0.747600	−0.138826	−0.0694129	−	0.997588i	$-0.522113\pi$
$29$	−0.747600	−0.138826	−0.0694129	+	0.997588i	$0.522113\pi$
$30$	0	0
$31$	0.912810	0.163946	0.0819728	−	0.996635i	$-0.473878\pi$
$31$	0.912810	0.163946	0.0819728	+	0.996635i	$0.473878\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.85196	−0.313039
$36$	0	0
$37$	7.60601	1.25042	0.625210	−	0.780457i	$-0.285013\pi$
$37$	7.60601	1.25042	0.625210	+	0.780457i	$0.285013\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	4.09179	0.639030	0.319515	−	0.947581i	$-0.396480\pi$
$41$	4.09179	0.639030	0.319515	+	0.947581i	$0.396480\pi$
$42$	0	0
$43$	6.51882	0.994110	0.497055	−	0.867719i	$-0.334415\pi$
$43$	6.51882	0.994110	0.497055	+	0.867719i	$0.334415\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−1.85196	−0.270136	−0.135068	−	0.990836i	$-0.543125\pi$
$47$	−1.85196	−0.270136	−0.135068	+	0.990836i	$0.543125\pi$
$48$	0	0
$49$	1.70512	0.243588
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−3.58401	−0.492302	−0.246151	−	0.969231i	$-0.579166\pi$
$53$	−3.58401	−0.492302	−0.246151	+	0.969231i	$0.579166\pi$
$54$	0	0
$55$	−3.43163	−0.462721
$56$	0	0
$57$	0	0
$58$	0	0
$59$	9.91563	1.29091	0.645453	−	0.763800i	$-0.276669\pi$
$59$	9.91563	1.29091	0.645453	+	0.763800i	$0.276669\pi$
$60$	0	0
$61$	7.81370	1.00044	0.500221	−	0.865898i	$-0.333252\pi$
$61$	7.81370	1.00044	0.500221	+	0.865898i	$0.333252\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0.627689	0.0778553
$66$	0	0
$67$	0.912810	0.111518	0.0557588	−	0.998444i	$-0.482242\pi$
$67$	0.912810	0.111518	0.0557588	+	0.998444i	$0.482242\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−7.70691	−0.914642	−0.457321	−	0.889302i	$-0.651191\pi$
$71$	−7.70691	−0.914642	−0.457321	+	0.889302i	$0.651191\pi$
$72$	0	0
$73$	5.20769	0.609514	0.304757	−	0.952430i	$-0.401425\pi$
$73$	5.20769	0.609514	0.304757	+	0.952430i	$0.401425\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	16.1303	1.83822
$78$	0	0
$79$	2.61793	0.294540	0.147270	−	0.989096i	$-0.452951\pi$
$79$	2.61793	0.294540	0.147270	+	0.989096i	$0.452951\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	3.22725	0.354237	0.177118	−	0.984190i	$-0.443322\pi$
$83$	3.22725	0.354237	0.177118	+	0.984190i	$0.443322\pi$
$84$	0	0
$85$	−0.223935	−0.0242892
$86$	0	0
$87$	0	0
$88$	0	0
$89$	15.2110	1.61237	0.806184	−	0.591665i	$-0.201529\pi$
$89$	15.2110	1.61237	0.806184	+	0.591665i	$0.201529\pi$
$90$	0	0
$91$	−2.95044	−0.309291
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.16954	−0.119993
$96$	0	0
$97$	11.9009	1.20835	0.604176	−	0.796851i	$-0.293502\pi$
$97$	11.9009	1.20835	0.604176	+	0.796851i	$0.293502\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1.01853	0.101347	0.0506737	−	0.998715i	$-0.483863\pi$
$101$	1.01853	0.101347	0.0506737	+	0.998715i	$0.483863\pi$
$102$	0	0
$103$	−11.3325	−1.11663	−0.558313	−	0.829630i	$-0.688551\pi$
$103$	−11.3325	−1.11663	−0.558313	+	0.829630i	$0.688551\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−15.5678	−1.50500	−0.752498	−	0.658594i	$-0.771151\pi$
$107$	−15.5678	−1.50500	−0.752498	+	0.658594i	$0.771151\pi$
$108$	0	0
$109$	11.9009	1.13990	0.569949	−	0.821680i	$-0.306963\pi$
$109$	11.9009	1.13990	0.569949	+	0.821680i	$0.306963\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−4.12587	−0.388129	−0.194064	−	0.980989i	$-0.562167\pi$
$113$	−4.12587	−0.388129	−0.194064	+	0.980989i	$0.562167\pi$
$114$	0	0
$115$	1.55645	0.145140
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.05260	0.0964921
$120$	0	0
$121$	18.8890	1.71718
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−6.02959	−0.539303
$126$	0	0
$127$	−1.90089	−0.168676	−0.0843382	−	0.996437i	$-0.526878\pi$
$127$	−1.90089	−0.168676	−0.0843382	+	0.996437i	$0.526878\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	6.30051	0.550478	0.275239	−	0.961376i	$-0.411243\pi$
$131$	6.30051	0.550478	0.275239	+	0.961376i	$0.411243\pi$
$132$	0	0
$133$	5.49743	0.476687
$134$	0	0
$135$	0	0
$136$	0	0
$137$	12.5152	1.06925	0.534623	−	0.845091i	$-0.320454\pi$
$137$	12.5152	1.06925	0.534623	+	0.845091i	$0.320454\pi$
$138$	0	0
$139$	12.1248	1.02841	0.514207	−	0.857666i	$-0.328086\pi$
$139$	12.1248	1.02841	0.514207	+	0.857666i	$0.328086\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−5.46708	−0.457180
$144$	0	0
$145$	−0.469261	−0.0389700
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−5.26134	−0.431025	−0.215513	−	0.976501i	$-0.569142\pi$
$149$	−5.26134	−0.431025	−0.215513	+	0.976501i	$0.569142\pi$
$150$	0	0
$151$	10.1581	0.826657	0.413329	−	0.910582i	$-0.364366\pi$
$151$	10.1581	0.826657	0.413329	+	0.910582i	$0.364366\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0.572961	0.0460214
$156$	0	0
$157$	8.26157	0.659345	0.329672	−	0.944095i	$-0.393062\pi$
$157$	8.26157	0.659345	0.329672	+	0.944095i	$0.393062\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−7.31607	−0.576587
$162$	0	0
$163$	20.9385	1.64003	0.820016	−	0.572341i	$-0.193965\pi$
$163$	20.9385	1.64003	0.820016	+	0.572341i	$0.193965\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	13.6196	1.05391	0.526956	−	0.849892i	$-0.323333\pi$
$167$	13.6196	1.05391	0.526956	+	0.849892i	$0.323333\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−25.0956	−1.90798	−0.953990	−	0.299837i	$-0.903068\pi$
$173$	−25.0956	−1.90798	−0.953990	+	0.299837i	$0.903068\pi$
$174$	0	0
$175$	13.5898	1.02729
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−18.1333	−1.35535	−0.677673	−	0.735363i	$-0.737011\pi$
$179$	−18.1333	−1.35535	−0.677673	+	0.735363i	$0.737011\pi$
$180$	0	0
$181$	21.7641	1.61772	0.808858	−	0.588004i	$-0.200086\pi$
$181$	21.7641	1.61772	0.808858	+	0.588004i	$0.200086\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	4.77421	0.351007
$186$	0	0
$187$	1.95044	0.142631
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−10.4234	−0.754212	−0.377106	−	0.926170i	$-0.623081\pi$
$191$	−10.4234	−0.754212	−0.377106	+	0.926170i	$0.623081\pi$
$192$	0	0
$193$	12.5188	0.901124	0.450562	−	0.892745i	$-0.351224\pi$
$193$	12.5188	0.901124	0.450562	+	0.892745i	$0.351224\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−23.0068	−1.63916	−0.819582	−	0.572962i	$-0.805794\pi$
$197$	−23.0068	−1.63916	−0.819582	+	0.572962i	$0.805794\pi$
$198$	0	0
$199$	−15.8299	−1.12215	−0.561077	−	0.827763i	$-0.689613\pi$
$199$	−15.8299	−1.12215	−0.561077	+	0.827763i	$0.689613\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	2.20575	0.154813
$204$	0	0
$205$	2.56837	0.179383
$206$	0	0
$207$	0	0
$208$	0	0
$209$	10.1866	0.704619
$210$	0	0
$211$	−11.1625	−0.768455	−0.384228	−	0.923238i	$-0.625532\pi$
$211$	−11.1625	−0.768455	−0.384228	+	0.923238i	$0.625532\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	4.09179	0.279058
$216$	0	0
$217$	−2.69320	−0.182826
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−0.356762	−0.0239984
$222$	0	0
$223$	−14.5565	−0.974772	−0.487386	−	0.873187i	$-0.662050\pi$
$223$	−14.5565	−0.974772	−0.487386	+	0.873187i	$0.662050\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−21.0349	−1.39613	−0.698067	−	0.716032i	$-0.745956\pi$
$227$	−21.0349	−1.39613	−0.698067	+	0.716032i	$0.745956\pi$
$228$	0	0
$229$	17.9548	1.18648	0.593242	−	0.805024i	$-0.297848\pi$
$229$	17.9548	1.18648	0.593242	+	0.805024i	$0.297848\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	8.57442	0.561729	0.280864	−	0.959747i	$-0.409379\pi$
$233$	8.57442	0.561729	0.280864	+	0.959747i	$0.409379\pi$
$234$	0	0
$235$	−1.16246	−0.0758303
$236$	0	0
$237$	0	0
$238$	0	0
$239$	8.27239	0.535096	0.267548	−	0.963545i	$-0.413787\pi$
$239$	8.27239	0.535096	0.267548	+	0.963545i	$0.413787\pi$
$240$	0	0
$241$	−0.888965	−0.0572633	−0.0286316	−	0.999590i	$-0.509115\pi$
$241$	−0.888965	−0.0572633	−0.0286316	+	0.999590i	$0.509115\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.07028	0.0683780
$246$	0	0
$247$	−1.86325	−0.118556
$248$	0	0
$249$	0	0
$250$	0	0
$251$	19.2170	1.21297	0.606483	−	0.795096i	$-0.292580\pi$
$251$	19.2170	1.21297	0.606483	+	0.795096i	$0.292580\pi$
$252$	0	0
$253$	−13.5565	−0.852287
$254$	0	0
$255$	0	0
$256$	0	0
$257$	12.9371	0.806997	0.403498	−	0.914980i	$-0.367794\pi$
$257$	12.9371	0.806997	0.403498	+	0.914980i	$0.367794\pi$
$258$	0	0
$259$	−22.4411	−1.39442
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−24.8217	−1.53057	−0.765285	−	0.643692i	$-0.777402\pi$
$263$	−24.8217	−1.53057	−0.765285	+	0.643692i	$0.777402\pi$
$264$	0	0
$265$	−2.24965	−0.138195
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−18.8468	−1.14911	−0.574555	−	0.818466i	$-0.694825\pi$
$269$	−18.8468	−1.14911	−0.574555	+	0.818466i	$0.694825\pi$
$270$	0	0
$271$	30.8770	1.87565	0.937823	−	0.347113i	$-0.112838\pi$
$271$	30.8770	1.87565	0.937823	+	0.347113i	$0.112838\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	25.1814	1.51850
$276$	0	0
$277$	21.2215	1.27508	0.637538	−	0.770419i	$-0.279953\pi$
$277$	21.2215	1.27508	0.637538	+	0.770419i	$0.279953\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	0.442595	0.0264030	0.0132015	−	0.999913i	$-0.495798\pi$
$281$	0.442595	0.0264030	0.0132015	+	0.999913i	$0.495798\pi$
$282$	0	0
$283$	−13.1129	−0.779481	−0.389741	−	0.920925i	$-0.627435\pi$
$283$	−13.1129	−0.779481	−0.389741	+	0.920925i	$0.627435\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−12.0726	−0.712623
$288$	0	0
$289$	−16.8727	−0.992513
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−3.22428	−0.188364	−0.0941822	−	0.995555i	$-0.530024\pi$
$293$	−3.22428	−0.188364	−0.0941822	+	0.995555i	$0.530024\pi$
$294$	0	0
$295$	6.22394	0.362372
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2.47965	0.143402
$300$	0	0
$301$	−19.2334	−1.10859
$302$	0	0
$303$	0	0
$304$	0	0
$305$	4.90457	0.280835
$306$	0	0
$307$	1.13675	0.0648775	0.0324388	−	0.999474i	$-0.489673\pi$
$307$	1.13675	0.0648775	0.0324388	+	0.999474i	$0.489673\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	15.7395	0.892504	0.446252	−	0.894907i	$-0.352759\pi$
$311$	15.7395	0.892504	0.446252	+	0.894907i	$0.352759\pi$
$312$	0	0
$313$	4.92473	0.278362	0.139181	−	0.990267i	$-0.455553\pi$
$313$	4.92473	0.278362	0.139181	+	0.990267i	$0.455553\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	15.4374	0.867053	0.433527	−	0.901141i	$-0.357269\pi$
$317$	15.4374	0.867053	0.433527	+	0.901141i	$0.357269\pi$
$318$	0	0
$319$	4.08719	0.228839
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0.664737	0.0369870
$324$	0	0
$325$	−4.60601	−0.255495
$326$	0	0
$327$	0	0
$328$	0	0
$329$	5.46411	0.301246
$330$	0	0
$331$	11.7308	0.644785	0.322392	−	0.946606i	$-0.395513\pi$
$331$	11.7308	0.644785	0.322392	+	0.946606i	$0.395513\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0.572961	0.0313042
$336$	0	0
$337$	14.7685	0.804489	0.402245	−	0.915532i	$-0.368230\pi$
$337$	14.7685	0.804489	0.402245	+	0.915532i	$0.368230\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−4.99041	−0.270246
$342$	0	0
$343$	15.6222	0.843522
$344$	0	0
$345$	0	0
$346$	0	0
$347$	15.6536	0.840331	0.420166	−	0.907447i	$-0.361972\pi$
$347$	15.6536	0.840331	0.420166	+	0.907447i	$0.361972\pi$
$348$	0	0
$349$	−0.398315	−0.0213213	−0.0106606	−	0.999943i	$-0.503393\pi$
$349$	−0.398315	−0.0213213	−0.0106606	+	0.999943i	$0.503393\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−3.51883	−0.187288	−0.0936442	−	0.995606i	$-0.529852\pi$
$353$	−3.51883	−0.187288	−0.0936442	+	0.995606i	$0.529852\pi$
$354$	0	0
$355$	−4.83754	−0.256750
$356$	0	0
$357$	0	0
$358$	0	0
$359$	35.1105	1.85306	0.926530	−	0.376222i	$-0.122777\pi$
$359$	35.1105	1.85306	0.926530	+	0.376222i	$0.122777\pi$
$360$	0	0
$361$	−15.5283	−0.817278
$362$	0	0
$363$	0	0
$364$	0	0
$365$	3.26881	0.171097
$366$	0	0
$367$	6.93420	0.361962	0.180981	−	0.983487i	$-0.442073\pi$
$367$	6.93420	0.361962	0.180981	+	0.983487i	$0.442073\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	10.5744	0.548997
$372$	0	0
$373$	−22.3368	−1.15656	−0.578279	−	0.815839i	$-0.696275\pi$
$373$	−22.3368	−1.15656	−0.578279	+	0.815839i	$0.696275\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−0.747600	−0.0385034
$378$	0	0
$379$	−28.9009	−1.48454	−0.742269	−	0.670102i	$-0.766251\pi$
$379$	−28.9009	−1.48454	−0.742269	+	0.670102i	$0.766251\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	5.19615	0.265511	0.132755	−	0.991149i	$-0.457617\pi$
$383$	5.19615	0.265511	0.132755	+	0.991149i	$0.457617\pi$
$384$	0	0
$385$	10.1248	0.516009
$386$	0	0
$387$	0	0
$388$	0	0
$389$	8.69433	0.440820	0.220410	−	0.975407i	$-0.429260\pi$
$389$	8.69433	0.440820	0.220410	+	0.975407i	$0.429260\pi$
$390$	0	0
$391$	−0.884644	−0.0447384
$392$	0	0
$393$	0	0
$394$	0	0
$395$	1.64325	0.0826807
$396$	0	0
$397$	30.5941	1.53547	0.767737	−	0.640766i	$-0.221383\pi$
$397$	30.5941	1.53547	0.767737	+	0.640766i	$0.221383\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−18.0475	−0.901247	−0.450624	−	0.892714i	$-0.648798\pi$
$401$	−18.0475	−0.901247	−0.450624	+	0.892714i	$0.648798\pi$
$402$	0	0
$403$	0.912810	0.0454703
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−41.5826	−2.06118
$408$	0	0
$409$	−1.73083	−0.0855840	−0.0427920	−	0.999084i	$-0.513625\pi$
$409$	−1.73083	−0.0855840	−0.0427920	+	0.999084i	$0.513625\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−29.2555	−1.43957
$414$	0	0
$415$	2.02571	0.0994382
$416$	0	0
$417$	0	0
$418$	0	0
$419$	10.8719	0.531129	0.265565	−	0.964093i	$-0.414442\pi$
$419$	10.8719	0.531129	0.265565	+	0.964093i	$0.414442\pi$
$420$	0	0
$421$	19.1367	0.932668	0.466334	−	0.884609i	$-0.345575\pi$
$421$	19.1367	0.932668	0.466334	+	0.884609i	$0.345575\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1.64325	0.0797091
$426$	0	0
$427$	−23.0539	−1.11566
$428$	0	0
$429$	0	0
$430$	0	0
$431$	17.4404	0.840075	0.420038	−	0.907507i	$-0.362017\pi$
$431$	17.4404	0.840075	0.420038	+	0.907507i	$0.362017\pi$
$432$	0	0
$433$	8.03763	0.386264	0.193132	−	0.981173i	$-0.438135\pi$
$433$	8.03763	0.386264	0.193132	+	0.981173i	$0.438135\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−4.62022	−0.221015
$438$	0	0
$439$	−25.3582	−1.21028	−0.605141	−	0.796118i	$-0.706883\pi$
$439$	−25.3582	−1.21028	−0.605141	+	0.796118i	$0.706883\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	30.9505	1.47050	0.735252	−	0.677794i	$-0.237064\pi$
$443$	30.9505	1.47050	0.735252	+	0.677794i	$0.237064\pi$
$444$	0	0
$445$	9.54781	0.452610
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−6.02959	−0.284554	−0.142277	−	0.989827i	$-0.545442\pi$
$449$	−6.02959	−0.284554	−0.142277	+	0.989827i	$0.545442\pi$
$450$	0	0
$451$	−22.3701	−1.05337
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−1.85196	−0.0868213
$456$	0	0
$457$	−7.90089	−0.369588	−0.184794	−	0.982777i	$-0.559162\pi$
$457$	−7.90089	−0.369588	−0.184794	+	0.982777i	$0.559162\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−2.86752	−0.133554	−0.0667768	−	0.997768i	$-0.521272\pi$
$461$	−2.86752	−0.133554	−0.0667768	+	0.997768i	$0.521272\pi$
$462$	0	0
$463$	23.3959	1.08730	0.543649	−	0.839313i	$-0.317042\pi$
$463$	23.3959	1.08730	0.543649	+	0.839313i	$0.317042\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	11.7099	0.541869	0.270935	−	0.962598i	$-0.412667\pi$
$467$	11.7099	0.541869	0.270935	+	0.962598i	$0.412667\pi$
$468$	0	0
$469$	−2.69320	−0.124360
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−35.6389	−1.63868
$474$	0	0
$475$	8.58216	0.393777
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−2.08881	−0.0954403	−0.0477201	−	0.998861i	$-0.515196\pi$
$479$	−2.08881	−0.0954403	−0.0477201	+	0.998861i	$0.515196\pi$
$480$	0	0
$481$	7.60601	0.346804
$482$	0	0
$483$	0	0
$484$	0	0
$485$	7.47006	0.339198
$486$	0	0
$487$	18.0138	0.816283	0.408141	−	0.912919i	$-0.366177\pi$
$487$	18.0138	0.816283	0.408141	+	0.912919i	$0.366177\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−8.33757	−0.376269	−0.188135	−	0.982143i	$-0.560244\pi$
$491$	−8.33757	−0.376269	−0.188135	+	0.982143i	$0.560244\pi$
$492$	0	0
$493$	0.266715	0.0120122
$494$	0	0
$495$	0	0
$496$	0	0
$497$	22.7388	1.01997
$498$	0	0
$499$	24.1300	1.08021	0.540103	−	0.841599i	$-0.318385\pi$
$499$	24.1300	1.08021	0.540103	+	0.841599i	$0.318385\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	21.6521	0.965420	0.482710	−	0.875780i	$-0.339652\pi$
$503$	21.6521	0.965420	0.482710	+	0.875780i	$0.339652\pi$
$504$	0	0
$505$	0.639319	0.0284493
$506$	0	0
$507$	0	0
$508$	0	0
$509$	11.3768	0.504266	0.252133	−	0.967693i	$-0.418868\pi$
$509$	11.3768	0.504266	0.252133	+	0.967693i	$0.418868\pi$
$510$	0	0
$511$	−15.3650	−0.679708
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−7.11330	−0.313449
$516$	0	0
$517$	10.1248	0.445289
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−29.2185	−1.28008	−0.640042	−	0.768340i	$-0.721083\pi$
$521$	−29.2185	−1.28008	−0.640042	+	0.768340i	$0.721083\pi$
$522$	0	0
$523$	9.01379	0.394145	0.197073	−	0.980389i	$-0.436856\pi$
$523$	9.01379	0.394145	0.197073	+	0.980389i	$0.436856\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−0.325656	−0.0141858
$528$	0	0
$529$	−16.8513	−0.732667
$530$	0	0
$531$	0	0
$532$	0	0
$533$	4.09179	0.177235
$534$	0	0
$535$	−9.77174	−0.422469
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−9.32202	−0.401528
$540$	0	0
$541$	19.7027	0.847084	0.423542	−	0.905877i	$-0.360787\pi$
$541$	19.7027	0.847084	0.423542	+	0.905877i	$0.360787\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	7.47006	0.319982
$546$	0	0
$547$	−26.3488	−1.12659	−0.563296	−	0.826255i	$-0.690467\pi$
$547$	−26.3488	−1.12659	−0.563296	+	0.826255i	$0.690467\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1.39297	0.0593425
$552$	0	0
$553$	−7.72405	−0.328460
$554$	0	0
$555$	0	0
$556$	0	0
$557$	9.40785	0.398624	0.199312	−	0.979936i	$-0.436129\pi$
$557$	9.40785	0.398624	0.199312	+	0.979936i	$0.436129\pi$
$558$	0	0
$559$	6.51882	0.275717
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−19.1859	−0.808589	−0.404295	−	0.914629i	$-0.632483\pi$
$563$	−19.1859	−0.808589	−0.404295	+	0.914629i	$0.632483\pi$
$564$	0	0
$565$	−2.58976	−0.108952
$566$	0	0
$567$	0	0
$568$	0	0
$569$	46.8676	1.96479	0.982396	−	0.186810i	$-0.0598149\pi$
$569$	46.8676	1.96479	0.982396	+	0.186810i	$0.0598149\pi$
$570$	0	0
$571$	−30.4197	−1.27303	−0.636513	−	0.771266i	$-0.719624\pi$
$571$	−30.4197	−1.27303	−0.636513	+	0.771266i	$0.719624\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−11.4213	−0.476301
$576$	0	0
$577$	20.7685	0.864602	0.432301	−	0.901729i	$-0.357702\pi$
$577$	20.7685	0.864602	0.432301	+	0.901729i	$0.357702\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−9.52182	−0.395032
$582$	0	0
$583$	19.5941	0.811504
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−32.4012	−1.33734	−0.668670	−	0.743559i	$-0.733136\pi$
$587$	−32.4012	−1.33734	−0.668670	+	0.743559i	$0.733136\pi$
$588$	0	0
$589$	−1.70080	−0.0700802
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−3.67282	−0.150825	−0.0754123	−	0.997152i	$-0.524027\pi$
$593$	−3.67282	−0.150825	−0.0754123	+	0.997152i	$0.524027\pi$
$594$	0	0
$595$	0.660709	0.0270864
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−29.6329	−1.21077	−0.605384	−	0.795933i	$-0.706981\pi$
$599$	−29.6329	−1.21077	−0.605384	+	0.795933i	$0.706981\pi$
$600$	0	0
$601$	−34.2077	−1.39536	−0.697680	−	0.716409i	$-0.745784\pi$
$601$	−34.2077	−1.39536	−0.697680	+	0.716409i	$0.745784\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	11.8564	0.482031
$606$	0	0
$607$	31.7565	1.28896	0.644479	−	0.764622i	$-0.277074\pi$
$607$	31.7565	1.28896	0.644479	+	0.764622i	$0.277074\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−1.85196	−0.0749224
$612$	0	0
$613$	8.22394	0.332162	0.166081	−	0.986112i	$-0.446889\pi$
$613$	8.22394	0.332162	0.166081	+	0.986112i	$0.446889\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−41.0172	−1.65129	−0.825645	−	0.564191i	$-0.809188\pi$
$617$	−41.0172	−1.65129	−0.825645	+	0.564191i	$0.809188\pi$
$618$	0	0
$619$	4.38207	0.176130	0.0880652	−	0.996115i	$-0.471932\pi$
$619$	4.38207	0.176130	0.0880652	+	0.996115i	$0.471932\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−44.8793	−1.79805
$624$	0	0
$625$	19.2453	0.769813
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−2.71353	−0.108196
$630$	0	0
$631$	−41.2754	−1.64315	−0.821573	−	0.570103i	$-0.806903\pi$
$631$	−41.2754	−1.64315	−0.821573	+	0.570103i	$0.806903\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−1.19317	−0.0473494
$636$	0	0
$637$	1.70512	0.0675593
$638$	0	0
$639$	0	0
$640$	0	0
$641$	14.8513	0.586592	0.293296	−	0.956022i	$-0.405248\pi$
$641$	14.8513	0.586592	0.293296	+	0.956022i	$0.405248\pi$
$642$	0	0
$643$	−16.3564	−0.645032	−0.322516	−	0.946564i	$-0.604529\pi$
$643$	−16.3564	−0.645032	−0.322516	+	0.946564i	$0.604529\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	23.8608	0.938067	0.469033	−	0.883180i	$-0.344602\pi$
$647$	23.8608	0.938067	0.469033	+	0.883180i	$0.344602\pi$
$648$	0	0
$649$	−54.2096	−2.12791
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−23.4494	−0.917644	−0.458822	−	0.888528i	$-0.651728\pi$
$653$	−23.4494	−0.917644	−0.458822	+	0.888528i	$0.651728\pi$
$654$	0	0
$655$	3.95476	0.154525
$656$	0	0
$657$	0	0
$658$	0	0
$659$	9.11330	0.355004	0.177502	−	0.984120i	$-0.443198\pi$
$659$	9.11330	0.355004	0.177502	+	0.984120i	$0.443198\pi$
$660$	0	0
$661$	31.2548	1.21567	0.607835	−	0.794063i	$-0.292038\pi$
$661$	31.2548	1.21567	0.607835	+	0.794063i	$0.292038\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	3.45068	0.133811
$666$	0	0
$667$	−1.85379	−0.0717789
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−42.7181	−1.64911
$672$	0	0
$673$	16.1367	0.622026	0.311013	−	0.950406i	$-0.399332\pi$
$673$	16.1367	0.622026	0.311013	+	0.950406i	$0.399332\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	41.6655	1.60134	0.800668	−	0.599109i	$-0.204478\pi$
$677$	41.6655	1.60134	0.800668	+	0.599109i	$0.204478\pi$
$678$	0	0
$679$	−35.1129	−1.34751
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−11.1369	−0.426143	−0.213071	−	0.977037i	$-0.568347\pi$
$683$	−11.1369	−0.426143	−0.213071	+	0.977037i	$0.568347\pi$
$684$	0	0
$685$	7.85565	0.300149
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−3.58401	−0.136540
$690$	0	0
$691$	−22.9223	−0.872005	−0.436002	−	0.899946i	$-0.643606\pi$
$691$	−22.9223	−0.872005	−0.436002	+	0.899946i	$0.643606\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	7.61062	0.288687
$696$	0	0
$697$	−1.45979	−0.0552936
$698$	0	0
$699$	0	0
$700$	0	0
$701$	13.1635	0.497180	0.248590	−	0.968609i	$-0.420033\pi$
$701$	13.1635	0.497180	0.248590	+	0.968609i	$0.420033\pi$
$702$	0	0
$703$	−14.1719	−0.534504
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−3.00511	−0.113019
$708$	0	0
$709$	−6.39831	−0.240294	−0.120147	−	0.992756i	$-0.538337\pi$
$709$	−6.39831	−0.240294	−0.120147	+	0.992756i	$0.538337\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	2.26345	0.0847669
$714$	0	0
$715$	−3.43163	−0.128336
$716$	0	0
$717$	0	0
$718$	0	0
$719$	21.5116	0.802246	0.401123	−	0.916024i	$-0.368620\pi$
$719$	21.5116	0.802246	0.401123	+	0.916024i	$0.368620\pi$
$720$	0	0
$721$	33.4359	1.24522
$722$	0	0
$723$	0	0
$724$	0	0
$725$	3.44345	0.127887
$726$	0	0
$727$	4.02571	0.149305	0.0746527	−	0.997210i	$-0.476215\pi$
$727$	4.02571	0.149305	0.0746527	+	0.997210i	$0.476215\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−2.32566	−0.0860178
$732$	0	0
$733$	−27.9480	−1.03228	−0.516141	−	0.856503i	$-0.672632\pi$
$733$	−27.9480	−1.03228	−0.516141	+	0.856503i	$0.672632\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−4.99041	−0.183824
$738$	0	0
$739$	−30.8299	−1.13410	−0.567049	−	0.823684i	$-0.691915\pi$
$739$	−30.8299	−1.13410	−0.567049	+	0.823684i	$0.691915\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	13.6196	0.499653	0.249827	−	0.968291i	$-0.419626\pi$
$743$	13.6196	0.499653	0.249827	+	0.968291i	$0.419626\pi$
$744$	0	0
$745$	−3.30248	−0.120994
$746$	0	0
$747$	0	0
$748$	0	0
$749$	45.9319	1.67832
$750$	0	0
$751$	−6.94098	−0.253280	−0.126640	−	0.991949i	$-0.540419\pi$
$751$	−6.94098	−0.253280	−0.126640	+	0.991949i	$0.540419\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	6.37615	0.232052
$756$	0	0
$757$	−17.3701	−0.631329	−0.315664	−	0.948871i	$-0.602227\pi$
$757$	−17.3701	−0.631329	−0.315664	+	0.948871i	$0.602227\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−46.5596	−1.68779	−0.843893	−	0.536512i	$-0.819742\pi$
$761$	−46.5596	−1.68779	−0.843893	+	0.536512i	$0.819742\pi$
$762$	0	0
$763$	−35.1129	−1.27117
$764$	0	0
$765$	0	0
$766$	0	0
$767$	9.91563	0.358033
$768$	0	0
$769$	−45.2873	−1.63310	−0.816551	−	0.577274i	$-0.804116\pi$
$769$	−45.2873	−1.63310	−0.816551	+	0.577274i	$0.804116\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	32.6233	1.17338	0.586690	−	0.809812i	$-0.300431\pi$
$773$	32.6233	1.17338	0.586690	+	0.809812i	$0.300431\pi$
$774$	0	0
$775$	−4.20441	−0.151027
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−7.62405	−0.273160
$780$	0	0
$781$	42.1343	1.50768
$782$	0	0
$783$	0	0
$784$	0	0
$785$	5.18570	0.185085
$786$	0	0
$787$	7.19577	0.256501	0.128251	−	0.991742i	$-0.459064\pi$
$787$	7.19577	0.256501	0.128251	+	0.991742i	$0.459064\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	12.1731	0.432827
$792$	0	0
$793$	7.81370	0.277473
$794$	0	0
$795$	0	0
$796$	0	0
$797$	38.6945	1.37063	0.685314	−	0.728247i	$-0.259665\pi$
$797$	38.6945	1.37063	0.685314	+	0.728247i	$0.259665\pi$
$798$	0	0
$799$	0.660709	0.0233742
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−28.4709	−1.00472
$804$	0	0
$805$	−4.59222	−0.161854
$806$	0	0
$807$	0	0
$808$	0	0
$809$	23.7855	0.836252	0.418126	−	0.908389i	$-0.362687\pi$
$809$	23.7855	0.836252	0.418126	+	0.908389i	$0.362687\pi$
$810$	0	0
$811$	−36.9266	−1.29667	−0.648334	−	0.761356i	$-0.724534\pi$
$811$	−36.9266	−1.29667	−0.648334	+	0.761356i	$0.724534\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	13.1429	0.460375
$816$	0	0
$817$	−12.1462	−0.424942
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−51.0216	−1.78067	−0.890333	−	0.455310i	$-0.849529\pi$
$821$	−51.0216	−1.78067	−0.890333	+	0.455310i	$0.849529\pi$
$822$	0	0
$823$	−14.5616	−0.507585	−0.253793	−	0.967259i	$-0.581678\pi$
$823$	−14.5616	−0.507585	−0.253793	+	0.967259i	$0.581678\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	8.78016	0.305316	0.152658	−	0.988279i	$-0.451217\pi$
$827$	8.78016	0.305316	0.152658	+	0.988279i	$0.451217\pi$
$828$	0	0
$829$	9.98808	0.346900	0.173450	−	0.984843i	$-0.444508\pi$
$829$	9.98808	0.346900	0.173450	+	0.984843i	$0.444508\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−0.608321	−0.0210771
$834$	0	0
$835$	8.54885	0.295845
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−5.44643	−0.188032	−0.0940158	−	0.995571i	$-0.529970\pi$
$839$	−5.44643	−0.188032	−0.0940158	+	0.995571i	$0.529970\pi$
$840$	0	0
$841$	−28.4411	−0.980727
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0.627689	0.0215932
$846$	0	0
$847$	−55.7308	−1.91493
$848$	0	0
$849$	0	0
$850$	0	0
$851$	18.8602	0.646521
$852$	0	0
$853$	0.667485	0.0228543	0.0114271	−	0.999935i	$-0.496363\pi$
$853$	0.667485	0.0228543	0.0114271	+	0.999935i	$0.496363\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−51.5975	−1.76254	−0.881269	−	0.472615i	$-0.843310\pi$
$857$	−51.5975	−1.76254	−0.881269	+	0.472615i	$0.843310\pi$
$858$	0	0
$859$	−23.7479	−0.810268	−0.405134	−	0.914257i	$-0.632775\pi$
$859$	−23.7479	−0.810268	−0.405134	+	0.914257i	$0.632775\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−1.41982	−0.0483313	−0.0241656	−	0.999708i	$-0.507693\pi$
$863$	−1.41982	−0.0483313	−0.0241656	+	0.999708i	$0.507693\pi$
$864$	0	0
$865$	−15.7522	−0.535592
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−14.3124	−0.485516
$870$	0	0
$871$	0.912810	0.0309294
$872$	0	0
$873$	0	0
$874$	0	0
$875$	17.7900	0.601410
$876$	0	0
$877$	−5.23772	−0.176865	−0.0884327	−	0.996082i	$-0.528186\pi$
$877$	−5.23772	−0.176865	−0.0884327	+	0.996082i	$0.528186\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	28.9816	0.976416	0.488208	−	0.872727i	$-0.337651\pi$
$881$	28.9816	0.976416	0.488208	+	0.872727i	$0.337651\pi$
$882$	0	0
$883$	−47.9095	−1.61228	−0.806142	−	0.591722i	$-0.798448\pi$
$883$	−47.9095	−1.61228	−0.806142	+	0.591722i	$0.798448\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	23.1371	0.776869	0.388434	−	0.921476i	$-0.373016\pi$
$887$	23.1371	0.776869	0.388434	+	0.921476i	$0.373016\pi$
$888$	0	0
$889$	5.60846	0.188102
$890$	0	0
$891$	0	0
$892$	0	0
$893$	3.45068	0.115472
$894$	0	0
$895$	−11.3821	−0.380461
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−0.682417	−0.0227599
$900$	0	0
$901$	1.27864	0.0425976
$902$	0	0
$903$	0	0
$904$	0	0
$905$	13.6611	0.454111
$906$	0	0
$907$	23.1343	0.768161	0.384081	−	0.923300i	$-0.374518\pi$
$907$	23.1343	0.768161	0.384081	+	0.923300i	$0.374518\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−6.27689	−0.207963	−0.103981	−	0.994579i	$-0.533158\pi$
$911$	−6.27689	−0.207963	−0.103981	+	0.994579i	$0.533158\pi$
$912$	0	0
$913$	−17.6436	−0.583919
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−18.5893	−0.613873
$918$	0	0
$919$	−2.14621	−0.0707970	−0.0353985	−	0.999373i	$-0.511270\pi$
$919$	−2.14621	−0.0707970	−0.0353985	+	0.999373i	$0.511270\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−7.70691	−0.253676
$924$	0	0
$925$	−35.0333	−1.15189
$926$	0	0
$927$	0	0
$928$	0	0
$929$	57.7100	1.89340	0.946702	−	0.322112i	$-0.104393\pi$
$929$	57.7100	1.89340	0.946702	+	0.322112i	$0.104393\pi$
$930$	0	0
$931$	−3.17707	−0.104124
$932$	0	0
$933$	0	0
$934$	0	0
$935$	1.22427	0.0400380
$936$	0	0
$937$	54.8822	1.79292	0.896461	−	0.443122i	$-0.146129\pi$
$937$	54.8822	1.79292	0.896461	+	0.443122i	$0.146129\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−58.9032	−1.92019	−0.960094	−	0.279678i	$-0.909772\pi$
$941$	−58.9032	−1.92019	−0.960094	+	0.279678i	$0.909772\pi$
$942$	0	0
$943$	10.1462	0.330406
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−45.0913	−1.46527	−0.732635	−	0.680622i	$-0.761710\pi$
$947$	−45.0913	−1.46527	−0.732635	+	0.680622i	$0.761710\pi$
$948$	0	0
$949$	5.20769	0.169049
$950$	0	0
$951$	0	0
$952$	0	0
$953$	52.2636	1.69298	0.846491	−	0.532402i	$-0.178711\pi$
$953$	52.2636	1.69298	0.846491	+	0.532402i	$0.178711\pi$
$954$	0	0
$955$	−6.54266	−0.211716
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−36.9254	−1.19238
$960$	0	0
$961$	−30.1668	−0.973122
$962$	0	0
$963$	0	0
$964$	0	0
$965$	7.85793	0.252956
$966$	0	0
$967$	16.4573	0.529232	0.264616	−	0.964354i	$-0.414755\pi$
$967$	16.4573	0.529232	0.264616	+	0.964354i	$0.414755\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	42.8380	1.37474	0.687369	−	0.726309i	$-0.258766\pi$
$971$	42.8380	1.37474	0.687369	+	0.726309i	$0.258766\pi$
$972$	0	0
$973$	−35.7736	−1.14685
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−24.3791	−0.779956	−0.389978	−	0.920824i	$-0.627517\pi$
$977$	−24.3791	−0.779956	−0.389978	+	0.920824i	$0.627517\pi$
$978$	0	0
$979$	−83.1600	−2.65781
$980$	0	0
$981$	0	0
$982$	0	0
$983$	18.5863	0.592812	0.296406	−	0.955062i	$-0.404212\pi$
$983$	18.5863	0.592812	0.296406	+	0.955062i	$0.404212\pi$
$984$	0	0
$985$	−14.4411	−0.460132
$986$	0	0
$987$	0	0
$988$	0	0
$989$	16.1644	0.513998
$990$	0	0
$991$	47.2634	1.50137	0.750686	−	0.660659i	$-0.229723\pi$
$991$	47.2634	1.50137	0.750686	+	0.660659i	$0.229723\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−9.93628	−0.315001
$996$	0	0
$997$	44.0395	1.39474	0.697372	−	0.716709i	$-0.254352\pi$
$997$	44.0395	1.39474	0.697372	+	0.716709i	$0.254352\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4212.2.a.l.1.4	yes	6
3.2	odd	2	inner	4212.2.a.l.1.3	✓	6
9.2	odd	6		4212.2.i.y.1405.4		12
9.4	even	3		4212.2.i.y.2809.3		12
9.5	odd	6		4212.2.i.y.2809.4		12
9.7	even	3		4212.2.i.y.1405.3		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4212.2.a.l.1.3	✓	6	3.2	odd	2	inner
4212.2.a.l.1.4	yes	6	1.1	even	1	trivial
4212.2.i.y.1405.3		12	9.7	even	3
4212.2.i.y.1405.4		12	9.2	odd	6
4212.2.i.y.2809.3		12	9.4	even	3
4212.2.i.y.2809.4		12	9.5	odd	6

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:	\( N \)	\(=\)	\( 4212 = 2^{2} \cdot 3^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4212.a (trivial)

\(f(q)\)	\(=\)	\(q+0.627689 q^{5} -2.95044 q^{7} -5.46708 q^{11} +1.00000 q^{13} -0.356762 q^{17} -1.86325 q^{19} +2.47965 q^{23} -4.60601 q^{25} -0.747600 q^{29} +0.912810 q^{31} -1.85196 q^{35} +7.60601 q^{37} +4.09179 q^{41} +6.51882 q^{43} -1.85196 q^{47} +1.70512 q^{49} -3.58401 q^{53} -3.43163 q^{55} +9.91563 q^{59} +7.81370 q^{61} +0.627689 q^{65} +0.912810 q^{67} -7.70691 q^{71} +5.20769 q^{73} +16.1303 q^{77} +2.61793 q^{79} +3.22725 q^{83} -0.223935 q^{85} +15.2110 q^{89} -2.95044 q^{91} -1.16954 q^{95} +11.9009 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 8 q^{7} + 6 q^{13} + 2 q^{19} + 4 q^{25} + 18 q^{31} + 14 q^{37} + 20 q^{43} + 30 q^{49} - 14 q^{55} + 8 q^{61} + 18 q^{67} + 24 q^{73} + 48 q^{79} - 2 q^{85} + 8 q^{91} + 20 q^{97}+O(q^{100}) \) 6 * q + 8 * q^7 + 6 * q^13 + 2 * q^19 + 4 * q^25 + 18 * q^31 + 14 * q^37 + 20 * q^43 + 30 * q^49 - 14 * q^55 + 8 * q^61 + 18 * q^67 + 24 * q^73 + 48 * q^79 - 2 * q^85 + 8 * q^91 + 20 * q^97

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