LMFDB - Embedded newform 4212.2.a.m.1.3

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 = [6,0,0,0,1,0,0,0,0,0,-1] 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:	6.6.35342001.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 8x^{4} + 12x^{3} + 15x^{2} - 15x + 3 $ x^6 - 2x^5 - 8x^4 + 12x^3 + 15x^2 - 15*x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 468)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.07603$ 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)

$f(q)$	$=$	$q-0.697259 q^{5} +2.19680 q^{7} +4.09290 q^{11} -1.00000 q^{13} +1.99796 q^{17} +2.80943 q^{19} +2.93635 q^{23} -4.51383 q^{25} +0.760913 q^{29} +3.03128 q^{31} -1.53174 q^{35} +3.64391 q^{37} +7.46808 q^{41} -8.27547 q^{43} +5.55243 q^{47} -2.17407 q^{49} -8.15366 q^{53} -2.85381 q^{55} +4.33402 q^{59} -7.36764 q^{61} +0.697259 q^{65} -2.21914 q^{67} +3.60208 q^{71} +7.07867 q^{73} +8.99127 q^{77} +5.33597 q^{79} +17.8206 q^{83} -1.39310 q^{85} -1.14782 q^{89} -2.19680 q^{91} -1.95890 q^{95} -18.2277 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + q^{5} - q^{11} - 6 q^{13} + 6 q^{17} + 3 q^{19} + 7 q^{23} + 3 q^{25} + 10 q^{29} - 12 q^{31} + 13 q^{35} - 9 q^{37} + 12 q^{41} + 3 q^{43} + 17 q^{47} + 12 q^{49} + 24 q^{53} + 3 q^{55} + q^{59}+ \cdots - 12 q^{97}+O(q^{100}) $ 6 * q + q^5 - q^11 - 6 * q^13 + 6 * q^17 + 3 * q^19 + 7 * q^23 + 3 * q^25 + 10 * q^29 - 12 * q^31 + 13 * q^35 - 9 * q^37 + 12 * q^41 + 3 * q^43 + 17 * q^47 + 12 * q^49 + 24 * q^53 + 3 * q^55 + q^59 + 6 * q^61 - q^65 + 6 * q^67 + 17 * q^71 + 3 * q^73 + 41 * q^77 - 6 * q^79 - 6 * q^83 - 9 * q^85 + 21 * q^89 + 42 * q^95 - 12 * 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
$5$	−0.697259	−0.311824	−0.155912	−	0.987771i	$-0.549832\pi$
$5$	−0.697259	−0.311824	−0.155912	+	0.987771i	$0.549832\pi$
$6$	0	0
$7$	2.19680	0.830312	0.415156	−	0.909750i	$-0.363727\pi$
$7$	2.19680	0.830312	0.415156	+	0.909750i	$0.363727\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	4.09290	1.23405	0.617027	−	0.786942i	$-0.288337\pi$
$11$	4.09290	1.23405	0.617027	+	0.786942i	$0.288337\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.99796	0.484577	0.242289	−	0.970204i	$-0.422102\pi$
$17$	1.99796	0.484577	0.242289	+	0.970204i	$0.422102\pi$
$18$	0	0
$19$	2.80943	0.644527	0.322263	−	0.946650i	$-0.395556\pi$
$19$	2.80943	0.644527	0.322263	+	0.946650i	$0.395556\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.93635	0.612270	0.306135	−	0.951988i	$-0.400964\pi$
$23$	2.93635	0.612270	0.306135	+	0.951988i	$0.400964\pi$
$24$	0	0
$25$	−4.51383	−0.902766
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0.760913	0.141298	0.0706490	−	0.997501i	$-0.477493\pi$
$29$	0.760913	0.141298	0.0706490	+	0.997501i	$0.477493\pi$
$30$	0	0
$31$	3.03128	0.544434	0.272217	−	0.962236i	$-0.412243\pi$
$31$	3.03128	0.544434	0.272217	+	0.962236i	$0.412243\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.53174	−0.258911
$36$	0	0
$37$	3.64391	0.599054	0.299527	−	0.954088i	$-0.403171\pi$
$37$	3.64391	0.599054	0.299527	+	0.954088i	$0.403171\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	7.46808	1.16632	0.583159	−	0.812358i	$-0.301816\pi$
$41$	7.46808	1.16632	0.583159	+	0.812358i	$0.301816\pi$
$42$	0	0
$43$	−8.27547	−1.26200	−0.630999	−	0.775784i	$-0.717355\pi$
$43$	−8.27547	−1.26200	−0.630999	+	0.775784i	$0.717355\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.55243	0.809905	0.404953	−	0.914338i	$-0.367288\pi$
$47$	5.55243	0.809905	0.404953	+	0.914338i	$0.367288\pi$
$48$	0	0
$49$	−2.17407	−0.310582
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−8.15366	−1.11999	−0.559996	−	0.828496i	$-0.689197\pi$
$53$	−8.15366	−1.11999	−0.559996	+	0.828496i	$0.689197\pi$
$54$	0	0
$55$	−2.85381	−0.384808
$56$	0	0
$57$	0	0
$58$	0	0
$59$	4.33402	0.564241	0.282121	−	0.959379i	$-0.408962\pi$
$59$	4.33402	0.564241	0.282121	+	0.959379i	$0.408962\pi$
$60$	0	0
$61$	−7.36764	−0.943330	−0.471665	−	0.881778i	$-0.656347\pi$
$61$	−7.36764	−0.943330	−0.471665	+	0.881778i	$0.656347\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0.697259	0.0864844
$66$	0	0
$67$	−2.21914	−0.271111	−0.135556	−	0.990770i	$-0.543282\pi$
$67$	−2.21914	−0.271111	−0.135556	+	0.990770i	$0.543282\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	3.60208	0.427489	0.213744	−	0.976890i	$-0.431434\pi$
$71$	3.60208	0.427489	0.213744	+	0.976890i	$0.431434\pi$
$72$	0	0
$73$	7.07867	0.828496	0.414248	−	0.910164i	$-0.364045\pi$
$73$	7.07867	0.828496	0.414248	+	0.910164i	$0.364045\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	8.99127	1.02465
$78$	0	0
$79$	5.33597	0.600344	0.300172	−	0.953885i	$-0.402956\pi$
$79$	5.33597	0.600344	0.300172	+	0.953885i	$0.402956\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	17.8206	1.95606	0.978032	−	0.208457i	$-0.0668442\pi$
$83$	17.8206	1.95606	0.978032	+	0.208457i	$0.0668442\pi$
$84$	0	0
$85$	−1.39310	−0.151103
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−1.14782	−0.121669	−0.0608345	−	0.998148i	$-0.519376\pi$
$89$	−1.14782	−0.121669	−0.0608345	+	0.998148i	$0.519376\pi$
$90$	0	0
$91$	−2.19680	−0.230287
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.95890	−0.200979
$96$	0	0
$97$	−18.2277	−1.85074	−0.925370	−	0.379065i	$-0.876246\pi$
$97$	−18.2277	−1.85074	−0.925370	+	0.379065i	$0.876246\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−14.7038	−1.46308	−0.731542	−	0.681797i	$-0.761199\pi$
$101$	−14.7038	−1.46308	−0.731542	+	0.681797i	$0.761199\pi$
$102$	0	0
$103$	17.0305	1.67806	0.839032	−	0.544082i	$-0.183122\pi$
$103$	17.0305	1.67806	0.839032	+	0.544082i	$0.183122\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−5.28970	−0.511374	−0.255687	−	0.966760i	$-0.582302\pi$
$107$	−5.28970	−0.511374	−0.255687	+	0.966760i	$0.582302\pi$
$108$	0	0
$109$	−10.1767	−0.974749	−0.487374	−	0.873193i	$-0.662045\pi$
$109$	−10.1767	−0.974749	−0.487374	+	0.873193i	$0.662045\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	13.1912	1.24093	0.620463	−	0.784236i	$-0.286945\pi$
$113$	13.1912	1.24093	0.620463	+	0.784236i	$0.286945\pi$
$114$	0	0
$115$	−2.04739	−0.190920
$116$	0	0
$117$	0	0
$118$	0	0
$119$	4.38912	0.402350
$120$	0	0
$121$	5.75180	0.522891
$122$	0	0
$123$	0	0
$124$	0	0
$125$	6.63361	0.593328
$126$	0	0
$127$	−4.71403	−0.418302	−0.209151	−	0.977883i	$-0.567070\pi$
$127$	−4.71403	−0.418302	−0.209151	+	0.977883i	$0.567070\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−11.3230	−0.989296	−0.494648	−	0.869093i	$-0.664703\pi$
$131$	−11.3230	−0.989296	−0.494648	+	0.869093i	$0.664703\pi$
$132$	0	0
$133$	6.17175	0.535158
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2.57107	−0.219661	−0.109831	−	0.993950i	$-0.535031\pi$
$137$	−2.57107	−0.219661	−0.109831	+	0.993950i	$0.535031\pi$
$138$	0	0
$139$	1.92354	0.163153	0.0815764	−	0.996667i	$-0.474005\pi$
$139$	1.92354	0.163153	0.0815764	+	0.996667i	$0.474005\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−4.09290	−0.342265
$144$	0	0
$145$	−0.530554	−0.0440601
$146$	0	0
$147$	0	0
$148$	0	0
$149$	22.0303	1.80479	0.902395	−	0.430910i	$-0.141807\pi$
$149$	22.0303	1.80479	0.902395	+	0.430910i	$0.141807\pi$
$150$	0	0
$151$	14.7036	1.19656	0.598281	−	0.801286i	$-0.295850\pi$
$151$	14.7036	1.19656	0.598281	+	0.801286i	$0.295850\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−2.11359	−0.169767
$156$	0	0
$157$	4.83108	0.385562	0.192781	−	0.981242i	$-0.438249\pi$
$157$	4.83108	0.385562	0.192781	+	0.981242i	$0.438249\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	6.45056	0.508376
$162$	0	0
$163$	13.7385	1.07608	0.538042	−	0.842918i	$-0.319164\pi$
$163$	13.7385	1.07608	0.538042	+	0.842918i	$0.319164\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2.10366	0.162786	0.0813931	−	0.996682i	$-0.474063\pi$
$167$	2.10366	0.162786	0.0813931	+	0.996682i	$0.474063\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	21.5601	1.63918	0.819591	−	0.572950i	$-0.194201\pi$
$173$	21.5601	1.63918	0.819591	+	0.572950i	$0.194201\pi$
$174$	0	0
$175$	−9.91598	−0.749578
$176$	0	0
$177$	0	0
$178$	0	0
$179$	10.2922	0.769279	0.384639	−	0.923067i	$-0.374326\pi$
$179$	10.2922	0.769279	0.384639	+	0.923067i	$0.374326\pi$
$180$	0	0
$181$	21.0086	1.56156	0.780780	−	0.624806i	$-0.214822\pi$
$181$	21.0086	1.56156	0.780780	+	0.624806i	$0.214822\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−2.54075	−0.186799
$186$	0	0
$187$	8.17745	0.597995
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−7.48117	−0.541318	−0.270659	−	0.962675i	$-0.587242\pi$
$191$	−7.48117	−0.541318	−0.270659	+	0.962675i	$0.587242\pi$
$192$	0	0
$193$	−4.92072	−0.354201	−0.177100	−	0.984193i	$-0.556672\pi$
$193$	−4.92072	−0.354201	−0.177100	+	0.984193i	$0.556672\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0.0996564	0.00710023	0.00355011	−	0.999994i	$-0.498870\pi$
$197$	0.0996564	0.00710023	0.00355011	+	0.999994i	$0.498870\pi$
$198$	0	0
$199$	−22.9573	−1.62740	−0.813699	−	0.581287i	$-0.802550\pi$
$199$	−22.9573	−1.62740	−0.813699	+	0.581287i	$0.802550\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1.67157	0.117322
$204$	0	0
$205$	−5.20719	−0.363686
$206$	0	0
$207$	0	0
$208$	0	0
$209$	11.4987	0.795381
$210$	0	0
$211$	−12.7040	−0.874579	−0.437290	−	0.899321i	$-0.644062\pi$
$211$	−12.7040	−0.874579	−0.437290	+	0.899321i	$0.644062\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	5.77015	0.393521
$216$	0	0
$217$	6.65911	0.452050
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−1.99796	−0.134398
$222$	0	0
$223$	12.7102	0.851140	0.425570	−	0.904926i	$-0.360074\pi$
$223$	12.7102	0.851140	0.425570	+	0.904926i	$0.360074\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−15.9065	−1.05575	−0.527876	−	0.849321i	$-0.677012\pi$
$227$	−15.9065	−1.05575	−0.527876	+	0.849321i	$0.677012\pi$
$228$	0	0
$229$	−17.8495	−1.17953	−0.589765	−	0.807575i	$-0.700780\pi$
$229$	−17.8495	−1.17953	−0.589765	+	0.807575i	$0.700780\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	17.9134	1.17355	0.586773	−	0.809752i	$-0.300398\pi$
$233$	17.9134	1.17355	0.586773	+	0.809752i	$0.300398\pi$
$234$	0	0
$235$	−3.87148	−0.252548
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−21.9511	−1.41990	−0.709950	−	0.704252i	$-0.751282\pi$
$239$	−21.9511	−1.41990	−0.709950	+	0.704252i	$0.751282\pi$
$240$	0	0
$241$	12.1329	0.781549	0.390774	−	0.920486i	$-0.372207\pi$
$241$	12.1329	0.781549	0.390774	+	0.920486i	$0.372207\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.51589	0.0968467
$246$	0	0
$247$	−2.80943	−0.178759
$248$	0	0
$249$	0	0
$250$	0	0
$251$	28.4061	1.79298	0.896489	−	0.443067i	$-0.146110\pi$
$251$	28.4061	1.79298	0.896489	+	0.443067i	$0.146110\pi$
$252$	0	0
$253$	12.0182	0.755575
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−11.5592	−0.721044	−0.360522	−	0.932751i	$-0.617401\pi$
$257$	−11.5592	−0.721044	−0.360522	+	0.932751i	$0.617401\pi$
$258$	0	0
$259$	8.00493	0.497402
$260$	0	0
$261$	0	0
$262$	0	0
$263$	20.1537	1.24273	0.621364	−	0.783522i	$-0.286579\pi$
$263$	20.1537	1.24273	0.621364	+	0.783522i	$0.286579\pi$
$264$	0	0
$265$	5.68521	0.349240
$266$	0	0
$267$	0	0
$268$	0	0
$269$	16.2166	0.988745	0.494373	−	0.869250i	$-0.335398\pi$
$269$	16.2166	0.988745	0.494373	+	0.869250i	$0.335398\pi$
$270$	0	0
$271$	3.72713	0.226407	0.113204	−	0.993572i	$-0.463889\pi$
$271$	3.72713	0.226407	0.113204	+	0.993572i	$0.463889\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−18.4746	−1.11406
$276$	0	0
$277$	24.5407	1.47451	0.737254	−	0.675616i	$-0.236122\pi$
$277$	24.5407	1.47451	0.737254	+	0.675616i	$0.236122\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	11.9525	0.713025	0.356512	−	0.934291i	$-0.383966\pi$
$281$	11.9525	0.713025	0.356512	+	0.934291i	$0.383966\pi$
$282$	0	0
$283$	21.2645	1.26404	0.632021	−	0.774951i	$-0.282226\pi$
$283$	21.2645	1.26404	0.632021	+	0.774951i	$0.282226\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	16.4059	0.968409
$288$	0	0
$289$	−13.0081	−0.765185
$290$	0	0
$291$	0	0
$292$	0	0
$293$	7.18807	0.419932	0.209966	−	0.977709i	$-0.432665\pi$
$293$	7.18807	0.419932	0.209966	+	0.977709i	$0.432665\pi$
$294$	0	0
$295$	−3.02194	−0.175944
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−2.93635	−0.169813
$300$	0	0
$301$	−18.1796	−1.04785
$302$	0	0
$303$	0	0
$304$	0	0
$305$	5.13715	0.294153
$306$	0	0
$307$	−27.3920	−1.56335	−0.781673	−	0.623688i	$-0.785633\pi$
$307$	−27.3920	−1.56335	−0.781673	+	0.623688i	$0.785633\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	4.82049	0.273345	0.136673	−	0.990616i	$-0.456359\pi$
$311$	4.82049	0.273345	0.136673	+	0.990616i	$0.456359\pi$
$312$	0	0
$313$	−13.8976	−0.785536	−0.392768	−	0.919638i	$-0.628483\pi$
$313$	−13.8976	−0.785536	−0.392768	+	0.919638i	$0.628483\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−12.9004	−0.724559	−0.362280	−	0.932069i	$-0.618001\pi$
$317$	−12.9004	−0.724559	−0.362280	+	0.932069i	$0.618001\pi$
$318$	0	0
$319$	3.11434	0.174370
$320$	0	0
$321$	0	0
$322$	0	0
$323$	5.61313	0.312323
$324$	0	0
$325$	4.51383	0.250382
$326$	0	0
$327$	0	0
$328$	0	0
$329$	12.1976	0.672474
$330$	0	0
$331$	−20.5521	−1.12964	−0.564822	−	0.825213i	$-0.691055\pi$
$331$	−20.5521	−1.12964	−0.564822	+	0.825213i	$0.691055\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	1.54732	0.0845389
$336$	0	0
$337$	3.60690	0.196480	0.0982402	−	0.995163i	$-0.468679\pi$
$337$	3.60690	0.196480	0.0982402	+	0.995163i	$0.468679\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	12.4067	0.671861
$342$	0	0
$343$	−20.1536	−1.08819
$344$	0	0
$345$	0	0
$346$	0	0
$347$	11.7383	0.630144	0.315072	−	0.949068i	$-0.397971\pi$
$347$	11.7383	0.630144	0.315072	+	0.949068i	$0.397971\pi$
$348$	0	0
$349$	32.9542	1.76400	0.881998	−	0.471253i	$-0.156198\pi$
$349$	32.9542	1.76400	0.881998	+	0.471253i	$0.156198\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	32.4321	1.72619	0.863095	−	0.505042i	$-0.168523\pi$
$353$	32.4321	1.72619	0.863095	+	0.505042i	$0.168523\pi$
$354$	0	0
$355$	−2.51159	−0.133301
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−11.1523	−0.588595	−0.294297	−	0.955714i	$-0.595086\pi$
$359$	−11.1523	−0.588595	−0.294297	+	0.955714i	$0.595086\pi$
$360$	0	0
$361$	−11.1071	−0.584586
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−4.93567	−0.258345
$366$	0	0
$367$	4.74909	0.247901	0.123950	−	0.992288i	$-0.460444\pi$
$367$	4.74909	0.247901	0.123950	+	0.992288i	$0.460444\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−17.9120	−0.929943
$372$	0	0
$373$	−18.8898	−0.978075	−0.489038	−	0.872263i	$-0.662652\pi$
$373$	−18.8898	−0.978075	−0.489038	+	0.872263i	$0.662652\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−0.760913	−0.0391890
$378$	0	0
$379$	−0.532771	−0.0273666	−0.0136833	−	0.999906i	$-0.504356\pi$
$379$	−0.532771	−0.0273666	−0.0136833	+	0.999906i	$0.504356\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	21.0635	1.07630	0.538148	−	0.842851i	$-0.319124\pi$
$383$	21.0635	1.07630	0.538148	+	0.842851i	$0.319124\pi$
$384$	0	0
$385$	−6.26925	−0.319510
$386$	0	0
$387$	0	0
$388$	0	0
$389$	0.528246	0.0267831	0.0133916	−	0.999910i	$-0.495737\pi$
$389$	0.528246	0.0267831	0.0133916	+	0.999910i	$0.495737\pi$
$390$	0	0
$391$	5.86671	0.296692
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−3.72056	−0.187201
$396$	0	0
$397$	5.91694	0.296963	0.148481	−	0.988915i	$-0.452562\pi$
$397$	5.91694	0.296963	0.148481	+	0.988915i	$0.452562\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	39.1394	1.95453	0.977263	−	0.212030i	$-0.0680075\pi$
$401$	39.1394	1.95453	0.977263	+	0.212030i	$0.0680075\pi$
$402$	0	0
$403$	−3.03128	−0.150999
$404$	0	0
$405$	0	0
$406$	0	0
$407$	14.9141	0.739266
$408$	0	0
$409$	−13.2087	−0.653131	−0.326565	−	0.945175i	$-0.605891\pi$
$409$	−13.2087	−0.653131	−0.326565	+	0.945175i	$0.605891\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	9.52097	0.468497
$414$	0	0
$415$	−12.4256	−0.609947
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−18.7373	−0.915379	−0.457689	−	0.889112i	$-0.651323\pi$
$419$	−18.7373	−0.915379	−0.457689	+	0.889112i	$0.651323\pi$
$420$	0	0
$421$	7.53862	0.367410	0.183705	−	0.982981i	$-0.441191\pi$
$421$	7.53862	0.367410	0.183705	+	0.982981i	$0.441191\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−9.01846	−0.437460
$426$	0	0
$427$	−16.1852	−0.783258
$428$	0	0
$429$	0	0
$430$	0	0
$431$	1.05377	0.0507584	0.0253792	−	0.999678i	$-0.491921\pi$
$431$	1.05377	0.0507584	0.0253792	+	0.999678i	$0.491921\pi$
$432$	0	0
$433$	−23.2509	−1.11737	−0.558685	−	0.829380i	$-0.688694\pi$
$433$	−23.2509	−1.11737	−0.558685	+	0.829380i	$0.688694\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	8.24945	0.394625
$438$	0	0
$439$	−25.6365	−1.22356	−0.611781	−	0.791027i	$-0.709546\pi$
$439$	−25.6365	−1.22356	−0.611781	+	0.791027i	$0.709546\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−37.4847	−1.78095	−0.890476	−	0.455031i	$-0.849628\pi$
$443$	−37.4847	−1.78095	−0.890476	+	0.455031i	$0.849628\pi$
$444$	0	0
$445$	0.800330	0.0379393
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−17.9727	−0.848182	−0.424091	−	0.905620i	$-0.639406\pi$
$449$	−17.9727	−0.848182	−0.424091	+	0.905620i	$0.639406\pi$
$450$	0	0
$451$	30.5661	1.43930
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1.53174	0.0718090
$456$	0	0
$457$	6.66781	0.311907	0.155953	−	0.987764i	$-0.450155\pi$
$457$	6.66781	0.311907	0.155953	+	0.987764i	$0.450155\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−36.5928	−1.70430	−0.852148	−	0.523300i	$-0.824701\pi$
$461$	−36.5928	−1.70430	−0.852148	+	0.523300i	$0.824701\pi$
$462$	0	0
$463$	23.6324	1.09829	0.549146	−	0.835726i	$-0.314953\pi$
$463$	23.6324	1.09829	0.549146	+	0.835726i	$0.314953\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−11.0351	−0.510643	−0.255322	−	0.966856i	$-0.582181\pi$
$467$	−11.0351	−0.510643	−0.255322	+	0.966856i	$0.582181\pi$
$468$	0	0
$469$	−4.87501	−0.225107
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−33.8707	−1.55737
$474$	0	0
$475$	−12.6813	−0.581857
$476$	0	0
$477$	0	0
$478$	0	0
$479$	4.93162	0.225332	0.112666	−	0.993633i	$-0.464061\pi$
$479$	4.93162	0.225332	0.112666	+	0.993633i	$0.464061\pi$
$480$	0	0
$481$	−3.64391	−0.166148
$482$	0	0
$483$	0	0
$484$	0	0
$485$	12.7094	0.577105
$486$	0	0
$487$	15.3352	0.694904	0.347452	−	0.937698i	$-0.387047\pi$
$487$	15.3352	0.694904	0.347452	+	0.937698i	$0.387047\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−34.7657	−1.56895	−0.784477	−	0.620158i	$-0.787068\pi$
$491$	−34.7657	−1.56895	−0.784477	+	0.620158i	$0.787068\pi$
$492$	0	0
$493$	1.52028	0.0684698
$494$	0	0
$495$	0	0
$496$	0	0
$497$	7.91305	0.354949
$498$	0	0
$499$	30.4715	1.36409	0.682046	−	0.731309i	$-0.261090\pi$
$499$	30.4715	1.36409	0.682046	+	0.731309i	$0.261090\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−26.2790	−1.17172	−0.585862	−	0.810411i	$-0.699244\pi$
$503$	−26.2790	−1.17172	−0.585862	+	0.810411i	$0.699244\pi$
$504$	0	0
$505$	10.2524	0.456224
$506$	0	0
$507$	0	0
$508$	0	0
$509$	0.631599	0.0279952	0.0139976	−	0.999902i	$-0.495544\pi$
$509$	0.631599	0.0279952	0.0139976	+	0.999902i	$0.495544\pi$
$510$	0	0
$511$	15.5504	0.687910
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−11.8747	−0.523260
$516$	0	0
$517$	22.7255	0.999467
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−19.0092	−0.832809	−0.416405	−	0.909179i	$-0.636710\pi$
$521$	−19.0092	−0.832809	−0.416405	+	0.909179i	$0.636710\pi$
$522$	0	0
$523$	−22.3923	−0.979146	−0.489573	−	0.871962i	$-0.662847\pi$
$523$	−22.3923	−0.979146	−0.489573	+	0.871962i	$0.662847\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	6.05638	0.263820
$528$	0	0
$529$	−14.3779	−0.625125
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−7.46808	−0.323479
$534$	0	0
$535$	3.68829	0.159459
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−8.89825	−0.383275
$540$	0	0
$541$	36.4172	1.56570	0.782850	−	0.622211i	$-0.213765\pi$
$541$	36.4172	1.56570	0.782850	+	0.622211i	$0.213765\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	7.09578	0.303950
$546$	0	0
$547$	18.1753	0.777121	0.388561	−	0.921423i	$-0.372972\pi$
$547$	18.1753	0.777121	0.388561	+	0.921423i	$0.372972\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	2.13773	0.0910703
$552$	0	0
$553$	11.7221	0.498473
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−14.6307	−0.619922	−0.309961	−	0.950749i	$-0.600316\pi$
$557$	−14.6307	−0.619922	−0.309961	+	0.950749i	$0.600316\pi$
$558$	0	0
$559$	8.27547	0.350015
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−24.4851	−1.03192	−0.515962	−	0.856611i	$-0.672565\pi$
$563$	−24.4851	−1.03192	−0.515962	+	0.856611i	$0.672565\pi$
$564$	0	0
$565$	−9.19771	−0.386950
$566$	0	0
$567$	0	0
$568$	0	0
$569$	21.7225	0.910655	0.455328	−	0.890324i	$-0.349522\pi$
$569$	21.7225	0.910655	0.455328	+	0.890324i	$0.349522\pi$
$570$	0	0
$571$	−28.4620	−1.19110	−0.595549	−	0.803319i	$-0.703065\pi$
$571$	−28.4620	−1.19110	−0.595549	+	0.803319i	$0.703065\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−13.2542	−0.552737
$576$	0	0
$577$	−33.7463	−1.40488	−0.702439	−	0.711744i	$-0.747906\pi$
$577$	−33.7463	−1.40488	−0.702439	+	0.711744i	$0.747906\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	39.1482	1.62414
$582$	0	0
$583$	−33.3721	−1.38213
$584$	0	0
$585$	0	0
$586$	0	0
$587$	6.95846	0.287206	0.143603	−	0.989635i	$-0.454131\pi$
$587$	6.95846	0.287206	0.143603	+	0.989635i	$0.454131\pi$
$588$	0	0
$589$	8.51615	0.350902
$590$	0	0
$591$	0	0
$592$	0	0
$593$	0.939651	0.0385869	0.0192934	−	0.999814i	$-0.493858\pi$
$593$	0.939651	0.0385869	0.0192934	+	0.999814i	$0.493858\pi$
$594$	0	0
$595$	−3.06036	−0.125462
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−21.8754	−0.893804	−0.446902	−	0.894583i	$-0.647473\pi$
$599$	−21.8754	−0.893804	−0.446902	+	0.894583i	$0.647473\pi$
$600$	0	0
$601$	−15.2309	−0.621280	−0.310640	−	0.950528i	$-0.600543\pi$
$601$	−15.2309	−0.621280	−0.310640	+	0.950528i	$0.600543\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−4.01049	−0.163050
$606$	0	0
$607$	−17.5919	−0.714033	−0.357016	−	0.934098i	$-0.616206\pi$
$607$	−17.5919	−0.714033	−0.357016	+	0.934098i	$0.616206\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−5.55243	−0.224627
$612$	0	0
$613$	−0.225209	−0.00909610	−0.00454805	−	0.999990i	$-0.501448\pi$
$613$	−0.225209	−0.00909610	−0.00454805	+	0.999990i	$0.501448\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	13.2993	0.535410	0.267705	−	0.963501i	$-0.413735\pi$
$617$	13.2993	0.535410	0.267705	+	0.963501i	$0.413735\pi$
$618$	0	0
$619$	6.13123	0.246435	0.123217	−	0.992380i	$-0.460679\pi$
$619$	6.13123	0.246435	0.123217	+	0.992380i	$0.460679\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−2.52154	−0.101023
$624$	0	0
$625$	17.9438	0.717752
$626$	0	0
$627$	0	0
$628$	0	0
$629$	7.28039	0.290288
$630$	0	0
$631$	23.6772	0.942573	0.471287	−	0.881980i	$-0.343790\pi$
$631$	23.6772	0.942573	0.471287	+	0.881980i	$0.343790\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	3.28690	0.130437
$636$	0	0
$637$	2.17407	0.0861398
$638$	0	0
$639$	0	0
$640$	0	0
$641$	44.6477	1.76348	0.881738	−	0.471739i	$-0.156374\pi$
$641$	44.6477	1.76348	0.881738	+	0.471739i	$0.156374\pi$
$642$	0	0
$643$	17.6700	0.696835	0.348418	−	0.937339i	$-0.386719\pi$
$643$	17.6700	0.696835	0.348418	+	0.937339i	$0.386719\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−30.0945	−1.18314	−0.591568	−	0.806255i	$-0.701491\pi$
$647$	−30.0945	−1.18314	−0.591568	+	0.806255i	$0.701491\pi$
$648$	0	0
$649$	17.7387	0.696305
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−6.25555	−0.244799	−0.122399	−	0.992481i	$-0.539059\pi$
$653$	−6.25555	−0.244799	−0.122399	+	0.992481i	$0.539059\pi$
$654$	0	0
$655$	7.89507	0.308486
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−39.7387	−1.54800	−0.774000	−	0.633185i	$-0.781747\pi$
$659$	−39.7387	−1.54800	−0.774000	+	0.633185i	$0.781747\pi$
$660$	0	0
$661$	−1.36977	−0.0532779	−0.0266389	−	0.999645i	$-0.508480\pi$
$661$	−1.36977	−0.0532779	−0.0266389	+	0.999645i	$0.508480\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−4.30331	−0.166875
$666$	0	0
$667$	2.23430	0.0865126
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−30.1550	−1.16412
$672$	0	0
$673$	34.1483	1.31632	0.658161	−	0.752878i	$-0.271335\pi$
$673$	34.1483	1.31632	0.658161	+	0.752878i	$0.271335\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	14.5143	0.557829	0.278915	−	0.960316i	$-0.410025\pi$
$677$	14.5143	0.557829	0.278915	+	0.960316i	$0.410025\pi$
$678$	0	0
$679$	−40.0425	−1.53669
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−41.5040	−1.58810	−0.794052	−	0.607849i	$-0.792032\pi$
$683$	−41.5040	−1.58810	−0.794052	+	0.607849i	$0.792032\pi$
$684$	0	0
$685$	1.79270	0.0684956
$686$	0	0
$687$	0	0
$688$	0	0
$689$	8.15366	0.310630
$690$	0	0
$691$	−0.959263	−0.0364921	−0.0182460	−	0.999834i	$-0.505808\pi$
$691$	−0.959263	−0.0364921	−0.0182460	+	0.999834i	$0.505808\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1.34121	−0.0508749
$696$	0	0
$697$	14.9210	0.565171
$698$	0	0
$699$	0	0
$700$	0	0
$701$	35.3399	1.33477	0.667385	−	0.744713i	$-0.267414\pi$
$701$	35.3399	1.33477	0.667385	+	0.744713i	$0.267414\pi$
$702$	0	0
$703$	10.2373	0.386106
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−32.3013	−1.21482
$708$	0	0
$709$	−49.5501	−1.86089	−0.930446	−	0.366428i	$-0.880581\pi$
$709$	−49.5501	−1.86089	−0.930446	+	0.366428i	$0.880581\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	8.90088	0.333341
$714$	0	0
$715$	2.85381	0.106726
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−3.76966	−0.140585	−0.0702924	−	0.997526i	$-0.522393\pi$
$719$	−3.76966	−0.140585	−0.0702924	+	0.997526i	$0.522393\pi$
$720$	0	0
$721$	37.4126	1.39332
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−3.43463	−0.127559
$726$	0	0
$727$	−8.28166	−0.307150	−0.153575	−	0.988137i	$-0.549079\pi$
$727$	−8.28166	−0.307150	−0.153575	+	0.988137i	$0.549079\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−16.5341	−0.611535
$732$	0	0
$733$	−26.3425	−0.972983	−0.486491	−	0.873685i	$-0.661724\pi$
$733$	−26.3425	−0.972983	−0.486491	+	0.873685i	$0.661724\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−9.08271	−0.334566
$738$	0	0
$739$	33.3366	1.22631	0.613154	−	0.789964i	$-0.289901\pi$
$739$	33.3366	1.22631	0.613154	+	0.789964i	$0.289901\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	42.1726	1.54716	0.773581	−	0.633698i	$-0.218464\pi$
$743$	42.1726	1.54716	0.773581	+	0.633698i	$0.218464\pi$
$744$	0	0
$745$	−15.3608	−0.562776
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−11.6204	−0.424600
$750$	0	0
$751$	5.97976	0.218205	0.109102	−	0.994031i	$-0.465202\pi$
$751$	5.97976	0.218205	0.109102	+	0.994031i	$0.465202\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−10.2522	−0.373117
$756$	0	0
$757$	26.0167	0.945593	0.472796	−	0.881172i	$-0.343245\pi$
$757$	26.0167	0.945593	0.472796	+	0.881172i	$0.343245\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−21.4536	−0.777693	−0.388847	−	0.921302i	$-0.627126\pi$
$761$	−21.4536	−0.777693	−0.388847	+	0.921302i	$0.627126\pi$
$762$	0	0
$763$	−22.3561	−0.809346
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−4.33402	−0.156492
$768$	0	0
$769$	28.9538	1.04410	0.522051	−	0.852915i	$-0.325167\pi$
$769$	28.9538	1.04410	0.522051	+	0.852915i	$0.325167\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−31.4888	−1.13257	−0.566287	−	0.824208i	$-0.691621\pi$
$773$	−31.4888	−1.13257	−0.566287	+	0.824208i	$0.691621\pi$
$774$	0	0
$775$	−13.6827	−0.491496
$776$	0	0
$777$	0	0
$778$	0	0
$779$	20.9810	0.751723
$780$	0	0
$781$	14.7429	0.527544
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−3.36852	−0.120227
$786$	0	0
$787$	−40.6417	−1.44872	−0.724361	−	0.689421i	$-0.757865\pi$
$787$	−40.6417	−1.44872	−0.724361	+	0.689421i	$0.757865\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	28.9785	1.03036
$792$	0	0
$793$	7.36764	0.261633
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−47.3529	−1.67733	−0.838663	−	0.544651i	$-0.816662\pi$
$797$	−47.3529	−1.67733	−0.838663	+	0.544651i	$0.816662\pi$
$798$	0	0
$799$	11.0935	0.392462
$800$	0	0
$801$	0	0
$802$	0	0
$803$	28.9723	1.02241
$804$	0	0
$805$	−4.49771	−0.158524
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−12.5477	−0.441155	−0.220577	−	0.975369i	$-0.570794\pi$
$809$	−12.5477	−0.441155	−0.220577	+	0.975369i	$0.570794\pi$
$810$	0	0
$811$	−16.5528	−0.581246	−0.290623	−	0.956838i	$-0.593863\pi$
$811$	−16.5528	−0.581246	−0.290623	+	0.956838i	$0.593863\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−9.57930	−0.335548
$816$	0	0
$817$	−23.2493	−0.813391
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−44.7485	−1.56173	−0.780867	−	0.624697i	$-0.785223\pi$
$821$	−44.7485	−1.56173	−0.780867	+	0.624697i	$0.785223\pi$
$822$	0	0
$823$	52.2195	1.82026	0.910129	−	0.414326i	$-0.135983\pi$
$823$	52.2195	1.82026	0.910129	+	0.414326i	$0.135983\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−52.5104	−1.82597	−0.912983	−	0.407997i	$-0.866227\pi$
$827$	−52.5104	−1.82597	−0.912983	+	0.407997i	$0.866227\pi$
$828$	0	0
$829$	−41.3146	−1.43492	−0.717458	−	0.696601i	$-0.754695\pi$
$829$	−41.3146	−1.43492	−0.717458	+	0.696601i	$0.754695\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−4.34371	−0.150501
$834$	0	0
$835$	−1.46680	−0.0507606
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−25.3840	−0.876352	−0.438176	−	0.898889i	$-0.644375\pi$
$839$	−25.3840	−0.876352	−0.438176	+	0.898889i	$0.644375\pi$
$840$	0	0
$841$	−28.4210	−0.980035
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−0.697259	−0.0239864
$846$	0	0
$847$	12.6355	0.434163
$848$	0	0
$849$	0	0
$850$	0	0
$851$	10.6998	0.366783
$852$	0	0
$853$	−26.8535	−0.919448	−0.459724	−	0.888062i	$-0.652052\pi$
$853$	−26.8535	−0.919448	−0.459724	+	0.888062i	$0.652052\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	23.4735	0.801841	0.400921	−	0.916113i	$-0.368690\pi$
$857$	23.4735	0.801841	0.400921	+	0.916113i	$0.368690\pi$
$858$	0	0
$859$	−23.9387	−0.816778	−0.408389	−	0.912808i	$-0.633909\pi$
$859$	−23.9387	−0.816778	−0.408389	+	0.912808i	$0.633909\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−20.8970	−0.711343	−0.355671	−	0.934611i	$-0.615748\pi$
$863$	−20.8970	−0.711343	−0.355671	+	0.934611i	$0.615748\pi$
$864$	0	0
$865$	−15.0330	−0.511136
$866$	0	0
$867$	0	0
$868$	0	0
$869$	21.8396	0.740857
$870$	0	0
$871$	2.21914	0.0751927
$872$	0	0
$873$	0	0
$874$	0	0
$875$	14.5727	0.492647
$876$	0	0
$877$	−14.6208	−0.493709	−0.246855	−	0.969053i	$-0.579397\pi$
$877$	−14.6208	−0.493709	−0.246855	+	0.969053i	$0.579397\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	12.4625	0.419874	0.209937	−	0.977715i	$-0.432674\pi$
$881$	12.4625	0.419874	0.209937	+	0.977715i	$0.432674\pi$
$882$	0	0
$883$	−8.07304	−0.271679	−0.135840	−	0.990731i	$-0.543373\pi$
$883$	−8.07304	−0.271679	−0.135840	+	0.990731i	$0.543373\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	52.8526	1.77462	0.887309	−	0.461176i	$-0.152572\pi$
$887$	52.8526	1.77462	0.887309	+	0.461176i	$0.152572\pi$
$888$	0	0
$889$	−10.3558	−0.347322
$890$	0	0
$891$	0	0
$892$	0	0
$893$	15.5991	0.522005
$894$	0	0
$895$	−7.17636	−0.239879
$896$	0	0
$897$	0	0
$898$	0	0
$899$	2.30654	0.0769274
$900$	0	0
$901$	−16.2907	−0.542722
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−14.6485	−0.486932
$906$	0	0
$907$	23.4654	0.779154	0.389577	−	0.920994i	$-0.372621\pi$
$907$	23.4654	0.779154	0.389577	+	0.920994i	$0.372621\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−42.7566	−1.41659	−0.708294	−	0.705917i	$-0.750535\pi$
$911$	−42.7566	−1.41659	−0.708294	+	0.705917i	$0.750535\pi$
$912$	0	0
$913$	72.9378	2.41389
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−24.8744	−0.821425
$918$	0	0
$919$	53.1680	1.75385	0.876925	−	0.480627i	$-0.159591\pi$
$919$	53.1680	1.75385	0.876925	+	0.480627i	$0.159591\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−3.60208	−0.118564
$924$	0	0
$925$	−16.4480	−0.540806
$926$	0	0
$927$	0	0
$928$	0	0
$929$	12.1903	0.399951	0.199975	−	0.979801i	$-0.435914\pi$
$929$	12.1903	0.399951	0.199975	+	0.979801i	$0.435914\pi$
$930$	0	0
$931$	−6.10789	−0.200178
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−5.70180	−0.186469
$936$	0	0
$937$	−55.1572	−1.80191	−0.900953	−	0.433917i	$-0.857131\pi$
$937$	−55.1572	−1.80191	−0.900953	+	0.433917i	$0.857131\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−53.2529	−1.73599	−0.867997	−	0.496569i	$-0.834593\pi$
$941$	−53.2529	−1.73599	−0.867997	+	0.496569i	$0.834593\pi$
$942$	0	0
$943$	21.9289	0.714103
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−49.9061	−1.62173	−0.810865	−	0.585234i	$-0.801003\pi$
$947$	−49.9061	−1.62173	−0.810865	+	0.585234i	$0.801003\pi$
$948$	0	0
$949$	−7.07867	−0.229783
$950$	0	0
$951$	0	0
$952$	0	0
$953$	22.8217	0.739267	0.369633	−	0.929178i	$-0.379483\pi$
$953$	22.8217	0.739267	0.369633	+	0.929178i	$0.379483\pi$
$954$	0	0
$955$	5.21631	0.168796
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−5.64812	−0.182387
$960$	0	0
$961$	−21.8113	−0.703592
$962$	0	0
$963$	0	0
$964$	0	0
$965$	3.43102	0.110448
$966$	0	0
$967$	42.5788	1.36924	0.684622	−	0.728899i	$-0.259967\pi$
$967$	42.5788	1.36924	0.684622	+	0.728899i	$0.259967\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−32.7844	−1.05210	−0.526051	−	0.850453i	$-0.676328\pi$
$971$	−32.7844	−1.05210	−0.526051	+	0.850453i	$0.676328\pi$
$972$	0	0
$973$	4.22564	0.135468
$974$	0	0
$975$	0	0
$976$	0	0
$977$	39.8627	1.27532	0.637660	−	0.770318i	$-0.279902\pi$
$977$	39.8627	1.27532	0.637660	+	0.770318i	$0.279902\pi$
$978$	0	0
$979$	−4.69792	−0.150146
$980$	0	0
$981$	0	0
$982$	0	0
$983$	0.770487	0.0245747	0.0122874	−	0.999925i	$-0.496089\pi$
$983$	0.770487	0.0245747	0.0122874	+	0.999925i	$0.496089\pi$
$984$	0	0
$985$	−0.0694864	−0.00221402
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−24.2997	−0.772684
$990$	0	0
$991$	5.38396	0.171027	0.0855135	−	0.996337i	$-0.472747\pi$
$991$	5.38396	0.171027	0.0855135	+	0.996337i	$0.472747\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	16.0072	0.507461
$996$	0	0
$997$	38.7889	1.22846	0.614229	−	0.789128i	$-0.289467\pi$
$997$	38.7889	1.22846	0.614229	+	0.789128i	$0.289467\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4212.2.a.m.1.3		6
3.2	odd	2		4212.2.a.k.1.4		6
9.2	odd	6		1404.2.i.c.469.3		12
9.4	even	3		468.2.i.c.313.6	yes	12
9.5	odd	6		1404.2.i.c.937.3		12
9.7	even	3		468.2.i.c.157.6	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
468.2.i.c.157.6	✓	12	9.7	even	3
468.2.i.c.313.6	yes	12	9.4	even	3
1404.2.i.c.469.3		12	9.2	odd	6
1404.2.i.c.937.3		12	9.5	odd	6
4212.2.a.k.1.4		6	3.2	odd	2
4212.2.a.m.1.3		6	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q-0.697259 q^{5} +2.19680 q^{7} +4.09290 q^{11} -1.00000 q^{13} +1.99796 q^{17} +2.80943 q^{19} +2.93635 q^{23} -4.51383 q^{25} +0.760913 q^{29} +3.03128 q^{31} -1.53174 q^{35} +3.64391 q^{37} +7.46808 q^{41} -8.27547 q^{43} +5.55243 q^{47} -2.17407 q^{49} -8.15366 q^{53} -2.85381 q^{55} +4.33402 q^{59} -7.36764 q^{61} +0.697259 q^{65} -2.21914 q^{67} +3.60208 q^{71} +7.07867 q^{73} +8.99127 q^{77} +5.33597 q^{79} +17.8206 q^{83} -1.39310 q^{85} -1.14782 q^{89} -2.19680 q^{91} -1.95890 q^{95} -18.2277 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + q^{5} - q^{11} - 6 q^{13} + 6 q^{17} + 3 q^{19} + 7 q^{23} + 3 q^{25} + 10 q^{29} - 12 q^{31} + 13 q^{35} - 9 q^{37} + 12 q^{41} + 3 q^{43} + 17 q^{47} + 12 q^{49} + 24 q^{53} + 3 q^{55} + q^{59}+ \cdots - 12 q^{97}+O(q^{100}) \) 6 * q + q^5 - q^11 - 6 * q^13 + 6 * q^17 + 3 * q^19 + 7 * q^23 + 3 * q^25 + 10 * q^29 - 12 * q^31 + 13 * q^35 - 9 * q^37 + 12 * q^41 + 3 * q^43 + 17 * q^47 + 12 * q^49 + 24 * q^53 + 3 * q^55 + q^59 + 6 * q^61 - q^65 + 6 * q^67 + 17 * q^71 + 3 * q^73 + 41 * q^77 - 6 * q^79 - 6 * q^83 - 9 * q^85 + 21 * q^89 + 42 * q^95 - 12 * q^97

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