LMFDB - Embedded newform 4400.2.a.cb.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4400,2,Mod(1,4400)] 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("4400.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4400 = 2^{4} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4400.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,-1,0,0,0,-1,0,7,0,-4,0,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$35.1341768894$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.54764.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 9x^{2} + 3x + 2 $ x^4 - x^3 - 9x^2 + 3x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 440)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.339102$ of defining polynomial
Character	$\chi$	$=$	4400.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.339102 q^{3} +4.05237 q^{7} -2.88501 q^{9} -1.00000 q^{11} +4.00000 q^{13} +7.74350 q^{17} +7.06529 q^{19} +1.37417 q^{21} +2.72619 q^{23} -1.99562 q^{27} -4.73057 q^{29} -0.219729 q^{31} -0.339102 q^{33} +1.32618 q^{37} +1.35641 q^{39} -7.79587 q^{41} -11.1177 q^{43} +3.01292 q^{47} +9.42170 q^{49} +2.62583 q^{51} +5.03945 q^{53} +2.39585 q^{57} -10.9503 q^{59} +12.7306 q^{61} -11.6911 q^{63} -4.70034 q^{67} +0.924456 q^{69} +2.52860 q^{71} -4.10474 q^{73} -4.05237 q^{77} -13.9006 q^{79} +7.97831 q^{81} -3.63067 q^{83} -1.60415 q^{87} +9.88501 q^{89} +16.2095 q^{91} -0.0745107 q^{93} +12.4962 q^{97} +2.88501 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{3} - q^{7} + 7 q^{9} - 4 q^{11} + 16 q^{13} + 7 q^{17} + 9 q^{19} - 7 q^{21} - 6 q^{23} - 13 q^{27} + 3 q^{29} + 15 q^{31} + q^{33} + 5 q^{37} - 4 q^{39} + 10 q^{41} - 8 q^{43} + 10 q^{47} + 9 q^{49}+ \cdots - 7 q^{99}+O(q^{100}) $ 4 * q - q^3 - q^7 + 7 * q^9 - 4 * q^11 + 16 * q^13 + 7 * q^17 + 9 * q^19 - 7 * q^21 - 6 * q^23 - 13 * q^27 + 3 * q^29 + 15 * q^31 + q^33 + 5 * q^37 - 4 * q^39 + 10 * q^41 - 8 * q^43 + 10 * q^47 + 9 * q^49 + 23 * q^51 + 5 * q^53 - 15 * q^57 - 6 * q^59 + 29 * q^61 - 40 * q^63 - 6 * q^67 - 30 * q^69 + q^71 + 18 * q^73 + q^77 + 20 * q^79 + 44 * q^81 - 26 * q^83 - 31 * q^87 + 21 * q^89 - 4 * q^91 + 25 * q^93 - 4 * q^97 - 7 * q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0.339102	0.195781	0.0978903	−	0.995197i	$-0.468791\pi$
$3$	0.339102	0.195781	0.0978903	+	0.995197i	$0.468791\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.05237	1.53165	0.765826	−	0.643048i	$-0.222330\pi$
$7$	4.05237	1.53165	0.765826	+	0.643048i	$0.222330\pi$
$8$	0	0
$9$	−2.88501	−0.961670
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	7.74350	1.87807	0.939037	−	0.343816i	$-0.111720\pi$
$17$	7.74350	1.87807	0.939037	+	0.343816i	$0.111720\pi$
$18$	0	0
$19$	7.06529	1.62089	0.810445	−	0.585815i	$-0.199226\pi$
$19$	7.06529	1.62089	0.810445	+	0.585815i	$0.199226\pi$
$20$	0	0
$21$	1.37417	0.299868
$22$	0	0
$23$	2.72619	0.568450	0.284225	−	0.958758i	$-0.408264\pi$
$23$	2.72619	0.568450	0.284225	+	0.958758i	$0.408264\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.99562	−0.384057
$28$	0	0
$29$	−4.73057	−0.878445	−0.439223	−	0.898378i	$-0.644746\pi$
$29$	−4.73057	−0.878445	−0.439223	+	0.898378i	$0.644746\pi$
$30$	0	0
$31$	−0.219729	−0.0394646	−0.0197323	−	0.999805i	$-0.506281\pi$
$31$	−0.219729	−0.0394646	−0.0197323	+	0.999805i	$0.506281\pi$
$32$	0	0
$33$	−0.339102	−0.0590301
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.32618	0.218022	0.109011	−	0.994041i	$-0.465232\pi$
$37$	1.32618	0.218022	0.109011	+	0.994041i	$0.465232\pi$
$38$	0	0
$39$	1.35641	0.217199
$40$	0	0
$41$	−7.79587	−1.21751	−0.608755	−	0.793358i	$-0.708331\pi$
$41$	−7.79587	−1.21751	−0.608755	+	0.793358i	$0.708331\pi$
$42$	0	0
$43$	−11.1177	−1.69543	−0.847714	−	0.530454i	$-0.822022\pi$
$43$	−11.1177	−1.69543	−0.847714	+	0.530454i	$0.822022\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3.01292	0.439480	0.219740	−	0.975558i	$-0.429479\pi$
$47$	3.01292	0.439480	0.219740	+	0.975558i	$0.429479\pi$
$48$	0	0
$49$	9.42170	1.34596
$50$	0	0
$51$	2.62583	0.367690
$52$	0	0
$53$	5.03945	0.692221	0.346111	−	0.938194i	$-0.387502\pi$
$53$	5.03945	0.692221	0.346111	+	0.938194i	$0.387502\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.39585	0.317339
$58$	0	0
$59$	−10.9503	−1.42561	−0.712804	−	0.701363i	$-0.752575\pi$
$59$	−10.9503	−1.42561	−0.712804	+	0.701363i	$0.752575\pi$
$60$	0	0
$61$	12.7306	1.62998	0.814991	−	0.579473i	$-0.196742\pi$
$61$	12.7306	1.62998	0.814991	+	0.579473i	$0.196742\pi$
$62$	0	0
$63$	−11.6911	−1.47294
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−4.70034	−0.574238	−0.287119	−	0.957895i	$-0.592698\pi$
$67$	−4.70034	−0.574238	−0.287119	+	0.957895i	$0.592698\pi$
$68$	0	0
$69$	0.924456	0.111291
$70$	0	0
$71$	2.52860	0.300090	0.150045	−	0.988679i	$-0.452058\pi$
$71$	2.52860	0.300090	0.150045	+	0.988679i	$0.452058\pi$
$72$	0	0
$73$	−4.10474	−0.480423	−0.240212	−	0.970721i	$-0.577217\pi$
$73$	−4.10474	−0.480423	−0.240212	+	0.970721i	$0.577217\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−4.05237	−0.461810
$78$	0	0
$79$	−13.9006	−1.56394	−0.781970	−	0.623316i	$-0.785785\pi$
$79$	−13.9006	−1.56394	−0.781970	+	0.623316i	$0.785785\pi$
$80$	0	0
$81$	7.97831	0.886479
$82$	0	0
$83$	−3.63067	−0.398518	−0.199259	−	0.979947i	$-0.563853\pi$
$83$	−3.63067	−0.398518	−0.199259	+	0.979947i	$0.563853\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−1.60415	−0.171983
$88$	0	0
$89$	9.88501	1.04781	0.523904	−	0.851777i	$-0.324475\pi$
$89$	9.88501	1.04781	0.523904	+	0.851777i	$0.324475\pi$
$90$	0	0
$91$	16.2095	1.69922
$92$	0	0
$93$	−0.0745107	−0.00772640
$94$	0	0
$95$	0	0
$96$	0	0
$97$	12.4962	1.26880	0.634399	−	0.773006i	$-0.281248\pi$
$97$	12.4962	1.26880	0.634399	+	0.773006i	$0.281248\pi$
$98$	0	0
$99$	2.88501	0.289954
$100$	0	0
$101$	−0.643593	−0.0640399	−0.0320199	−	0.999487i	$-0.510194\pi$
$101$	−0.643593	−0.0640399	−0.0320199	+	0.999487i	$0.510194\pi$
$102$	0	0
$103$	−4.36933	−0.430523	−0.215261	−	0.976556i	$-0.569060\pi$
$103$	−4.36933	−0.430523	−0.215261	+	0.976556i	$0.569060\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−8.98708	−0.868814	−0.434407	−	0.900717i	$-0.643042\pi$
$107$	−8.98708	−0.868814	−0.434407	+	0.900717i	$0.643042\pi$
$108$	0	0
$109$	18.2095	1.74415	0.872076	−	0.489371i	$-0.162773\pi$
$109$	18.2095	1.74415	0.872076	+	0.489371i	$0.162773\pi$
$110$	0	0
$111$	0.449710	0.0426845
$112$	0	0
$113$	10.4173	0.979979	0.489989	−	0.871728i	$-0.337001\pi$
$113$	10.4173	0.979979	0.489989	+	0.871728i	$0.337001\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−11.5400	−1.06688
$118$	0	0
$119$	31.3795	2.87656
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−2.64359	−0.238365
$124$	0	0
$125$	0	0
$126$	0	0
$127$	9.77002	0.866949	0.433475	−	0.901166i	$-0.357287\pi$
$127$	9.77002	0.866949	0.433475	+	0.901166i	$0.357287\pi$
$128$	0	0
$129$	−3.77002	−0.331932
$130$	0	0
$131$	12.0870	1.05604	0.528022	−	0.849231i	$-0.322934\pi$
$131$	12.0870	1.05604	0.528022	+	0.849231i	$0.322934\pi$
$132$	0	0
$133$	28.6312	2.48264
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−4.49621	−0.384137	−0.192069	−	0.981381i	$-0.561520\pi$
$137$	−4.49621	−0.384137	−0.192069	+	0.981381i	$0.561520\pi$
$138$	0	0
$139$	8.33472	0.706942	0.353471	−	0.935446i	$-0.385001\pi$
$139$	8.33472	0.706942	0.353471	+	0.935446i	$0.385001\pi$
$140$	0	0
$141$	1.02169	0.0860416
$142$	0	0
$143$	−4.00000	−0.334497
$144$	0	0
$145$	0	0
$146$	0	0
$147$	3.19492	0.263512
$148$	0	0
$149$	−11.8136	−0.967810	−0.483905	−	0.875121i	$-0.660782\pi$
$149$	−11.8136	−0.967810	−0.483905	+	0.875121i	$0.660782\pi$
$150$	0	0
$151$	15.7700	1.28335	0.641673	−	0.766978i	$-0.278241\pi$
$151$	15.7700	1.28335	0.641673	+	0.766978i	$0.278241\pi$
$152$	0	0
$153$	−22.3401	−1.80609
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−0.0302288	−0.00241252	−0.00120626	−	0.999999i	$-0.500384\pi$
$157$	−0.0302288	−0.00241252	−0.00120626	+	0.999999i	$0.500384\pi$
$158$	0	0
$159$	1.70889	0.135523
$160$	0	0
$161$	11.0475	0.870668
$162$	0	0
$163$	14.0782	1.10269	0.551345	−	0.834277i	$-0.314115\pi$
$163$	14.0782	1.10269	0.551345	+	0.834277i	$0.314115\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−12.7306	−0.985121	−0.492561	−	0.870278i	$-0.663939\pi$
$167$	−12.7306	−0.985121	−0.492561	+	0.870278i	$0.663939\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	−20.3834	−1.55876
$172$	0	0
$173$	7.32180	0.556666	0.278333	−	0.960485i	$-0.410218\pi$
$173$	7.32180	0.556666	0.278333	+	0.960485i	$0.410218\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−3.71327	−0.279106
$178$	0	0
$179$	−6.56254	−0.490507	−0.245254	−	0.969459i	$-0.578871\pi$
$179$	−6.56254	−0.490507	−0.245254	+	0.969459i	$0.578871\pi$
$180$	0	0
$181$	−7.59390	−0.564450	−0.282225	−	0.959348i	$-0.591072\pi$
$181$	−7.59390	−0.564450	−0.282225	+	0.959348i	$0.591072\pi$
$182$	0	0
$183$	4.31696	0.319119
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−7.74350	−0.566261
$188$	0	0
$189$	−8.08698	−0.588241
$190$	0	0
$191$	12.6414	0.914702	0.457351	−	0.889286i	$-0.348798\pi$
$191$	12.6414	0.914702	0.457351	+	0.889286i	$0.348798\pi$
$192$	0	0
$193$	−4.42170	−0.318281	−0.159140	−	0.987256i	$-0.550872\pi$
$193$	−4.42170	−0.318281	−0.159140	+	0.987256i	$0.550872\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0.678204	0.0483200	0.0241600	−	0.999708i	$-0.492309\pi$
$197$	0.678204	0.0483200	0.0241600	+	0.999708i	$0.492309\pi$
$198$	0	0
$199$	−4.42170	−0.313446	−0.156723	−	0.987643i	$-0.550093\pi$
$199$	−4.42170	−0.313446	−0.156723	+	0.987643i	$0.550093\pi$
$200$	0	0
$201$	−1.59390	−0.112425
$202$	0	0
$203$	−19.1700	−1.34547
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−7.86509	−0.546661
$208$	0	0
$209$	−7.06529	−0.488717
$210$	0	0
$211$	−11.1700	−0.768977	−0.384488	−	0.923130i	$-0.625622\pi$
$211$	−11.1700	−0.768977	−0.384488	+	0.923130i	$0.625622\pi$
$212$	0	0
$213$	0.857454	0.0587518
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−0.890425	−0.0604460
$218$	0	0
$219$	−1.39192	−0.0940576
$220$	0	0
$221$	30.9740	2.08354
$222$	0	0
$223$	19.4487	1.30238	0.651190	−	0.758915i	$-0.274270\pi$
$223$	19.4487	1.30238	0.651190	+	0.758915i	$0.274270\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−22.6047	−1.50032	−0.750162	−	0.661254i	$-0.770024\pi$
$227$	−22.6047	−1.50032	−0.750162	+	0.661254i	$0.770024\pi$
$228$	0	0
$229$	−17.0550	−1.12703	−0.563514	−	0.826106i	$-0.690551\pi$
$229$	−17.0550	−1.12703	−0.563514	+	0.826106i	$0.690551\pi$
$230$	0	0
$231$	−1.37417	−0.0904135
$232$	0	0
$233$	14.5006	0.949965	0.474983	−	0.879995i	$-0.342454\pi$
$233$	14.5006	0.949965	0.474983	+	0.879995i	$0.342454\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−4.71372	−0.306189
$238$	0	0
$239$	−13.9006	−0.899155	−0.449578	−	0.893241i	$-0.648426\pi$
$239$	−13.9006	−0.899155	−0.449578	+	0.893241i	$0.648426\pi$
$240$	0	0
$241$	15.0572	0.969920	0.484960	−	0.874536i	$-0.338834\pi$
$241$	15.0572	0.969920	0.484960	+	0.874536i	$0.338834\pi$
$242$	0	0
$243$	8.69231	0.557612
$244$	0	0
$245$	0	0
$246$	0	0
$247$	28.2612	1.79822
$248$	0	0
$249$	−1.23117	−0.0780220
$250$	0	0
$251$	−2.51084	−0.158483	−0.0792415	−	0.996855i	$-0.525250\pi$
$251$	−2.51084	−0.158483	−0.0792415	+	0.996855i	$0.525250\pi$
$252$	0	0
$253$	−2.72619	−0.171394
$254$	0	0
$255$	0	0
$256$	0	0
$257$	6.64359	0.414416	0.207208	−	0.978297i	$-0.433562\pi$
$257$	6.64359	0.414416	0.207208	+	0.978297i	$0.433562\pi$
$258$	0	0
$259$	5.37417	0.333934
$260$	0	0
$261$	13.6478	0.844775
$262$	0	0
$263$	−13.4088	−0.826821	−0.413410	−	0.910545i	$-0.635662\pi$
$263$	−13.4088	−0.826821	−0.413410	+	0.910545i	$0.635662\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	3.35203	0.205141
$268$	0	0
$269$	−19.4611	−1.18657	−0.593284	−	0.804994i	$-0.702169\pi$
$269$	−19.4611	−1.18657	−0.593284	+	0.804994i	$0.702169\pi$
$270$	0	0
$271$	−0.0530465	−0.00322235	−0.00161117	−	0.999999i	$-0.500513\pi$
$271$	−0.0530465	−0.00322235	−0.00161117	+	0.999999i	$0.500513\pi$
$272$	0	0
$273$	5.49666	0.332673
$274$	0	0
$275$	0	0
$276$	0	0
$277$	4.09597	0.246103	0.123052	−	0.992400i	$-0.460732\pi$
$277$	4.09597	0.246103	0.123052	+	0.992400i	$0.460732\pi$
$278$	0	0
$279$	0.633922	0.0379519
$280$	0	0
$281$	17.5400	1.04635	0.523176	−	0.852225i	$-0.324747\pi$
$281$	17.5400	1.04635	0.523176	+	0.852225i	$0.324747\pi$
$282$	0	0
$283$	−15.1264	−0.899173	−0.449586	−	0.893237i	$-0.648429\pi$
$283$	−15.1264	−0.899173	−0.449586	+	0.893237i	$0.648429\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−31.5917	−1.86480
$288$	0	0
$289$	42.9617	2.52716
$290$	0	0
$291$	4.23749	0.248406
$292$	0	0
$293$	−13.3564	−0.780290	−0.390145	−	0.920754i	$-0.627575\pi$
$293$	−13.3564	−0.780290	−0.390145	+	0.920754i	$0.627575\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1.99562	0.115797
$298$	0	0
$299$	10.9048	0.630639
$300$	0	0
$301$	−45.0529	−2.59680
$302$	0	0
$303$	−0.218243	−0.0125378
$304$	0	0
$305$	0	0
$306$	0	0
$307$	15.7959	0.901518	0.450759	−	0.892646i	$-0.351153\pi$
$307$	15.7959	0.901518	0.450759	+	0.892646i	$0.351153\pi$
$308$	0	0
$309$	−1.48165	−0.0842880
$310$	0	0
$311$	0.829968	0.0470631	0.0235316	−	0.999723i	$-0.492509\pi$
$311$	0.829968	0.0470631	0.0235316	+	0.999723i	$0.492509\pi$
$312$	0	0
$313$	−24.5050	−1.38510	−0.692552	−	0.721368i	$-0.743513\pi$
$313$	−24.5050	−1.38510	−0.692552	+	0.721368i	$0.743513\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−11.4655	−0.643968	−0.321984	−	0.946745i	$-0.604350\pi$
$317$	−11.4655	−0.643968	−0.321984	+	0.946745i	$0.604350\pi$
$318$	0	0
$319$	4.73057	0.264861
$320$	0	0
$321$	−3.04753	−0.170097
$322$	0	0
$323$	54.7101	3.04415
$324$	0	0
$325$	0	0
$326$	0	0
$327$	6.17487	0.341471
$328$	0	0
$329$	12.2095	0.673130
$330$	0	0
$331$	24.4114	1.34177	0.670887	−	0.741559i	$-0.265913\pi$
$331$	24.4114	1.34177	0.670887	+	0.741559i	$0.265913\pi$
$332$	0	0
$333$	−3.82604	−0.209666
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−14.3871	−0.783715	−0.391857	−	0.920026i	$-0.628167\pi$
$337$	−14.3871	−0.783715	−0.391857	+	0.920026i	$0.628167\pi$
$338$	0	0
$339$	3.53253	0.191861
$340$	0	0
$341$	0.219729	0.0118990
$342$	0	0
$343$	9.81363	0.529886
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1.66528	0.0893969	0.0446985	−	0.999001i	$-0.485767\pi$
$347$	1.66528	0.0893969	0.0446985	+	0.999001i	$0.485767\pi$
$348$	0	0
$349$	6.73866	0.360712	0.180356	−	0.983601i	$-0.442275\pi$
$349$	6.73866	0.360712	0.180356	+	0.983601i	$0.442275\pi$
$350$	0	0
$351$	−7.98247	−0.426073
$352$	0	0
$353$	23.1486	1.23207	0.616037	−	0.787717i	$-0.288737\pi$
$353$	23.1486	1.23207	0.616037	+	0.787717i	$0.288737\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	10.6409	0.563174
$358$	0	0
$359$	−20.9481	−1.10560	−0.552800	−	0.833314i	$-0.686441\pi$
$359$	−20.9481	−1.10560	−0.552800	+	0.833314i	$0.686441\pi$
$360$	0	0
$361$	30.9184	1.62728
$362$	0	0
$363$	0.339102	0.0177982
$364$	0	0
$365$	0	0
$366$	0	0
$367$	13.4833	0.703822	0.351911	−	0.936033i	$-0.385532\pi$
$367$	13.4833	0.703822	0.351911	+	0.936033i	$0.385532\pi$
$368$	0	0
$369$	22.4912	1.17084
$370$	0	0
$371$	20.4217	1.06024
$372$	0	0
$373$	7.47823	0.387208	0.193604	−	0.981080i	$-0.437982\pi$
$373$	7.47823	0.387208	0.193604	+	0.981080i	$0.437982\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−18.9223	−0.974548
$378$	0	0
$379$	−7.33807	−0.376931	−0.188466	−	0.982080i	$-0.560351\pi$
$379$	−7.33807	−0.376931	−0.188466	+	0.982080i	$0.560351\pi$
$380$	0	0
$381$	3.31303	0.169732
$382$	0	0
$383$	20.2316	1.03379	0.516894	−	0.856050i	$-0.327088\pi$
$383$	20.2316	1.03379	0.516894	+	0.856050i	$0.327088\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	32.0746	1.63044
$388$	0	0
$389$	−0.406104	−0.0205903	−0.0102952	−	0.999947i	$-0.503277\pi$
$389$	−0.406104	−0.0205903	−0.0102952	+	0.999947i	$0.503277\pi$
$390$	0	0
$391$	21.1103	1.06759
$392$	0	0
$393$	4.09872	0.206753
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−6.74833	−0.338689	−0.169345	−	0.985557i	$-0.554165\pi$
$397$	−6.74833	−0.338689	−0.169345	+	0.985557i	$0.554165\pi$
$398$	0	0
$399$	9.70889	0.486052
$400$	0	0
$401$	−21.1700	−1.05718	−0.528590	−	0.848877i	$-0.677279\pi$
$401$	−21.1700	−1.05718	−0.528590	+	0.848877i	$0.677279\pi$
$402$	0	0
$403$	−0.878918	−0.0437820
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.32618	−0.0657362
$408$	0	0
$409$	3.25167	0.160785	0.0803923	−	0.996763i	$-0.474383\pi$
$409$	3.25167	0.160785	0.0803923	+	0.996763i	$0.474383\pi$
$410$	0	0
$411$	−1.52467	−0.0752066
$412$	0	0
$413$	−44.3747	−2.18354
$414$	0	0
$415$	0	0
$416$	0	0
$417$	2.82632	0.138405
$418$	0	0
$419$	−0.0516929	−0.00252536	−0.00126268	−	0.999999i	$-0.500402\pi$
$419$	−0.0516929	−0.00252536	−0.00126268	+	0.999999i	$0.500402\pi$
$420$	0	0
$421$	3.25167	0.158477	0.0792383	−	0.996856i	$-0.474751\pi$
$421$	3.25167	0.158477	0.0792383	+	0.996856i	$0.474751\pi$
$422$	0	0
$423$	−8.69231	−0.422635
$424$	0	0
$425$	0	0
$426$	0	0
$427$	51.5890	2.49657
$428$	0	0
$429$	−1.35641	−0.0654880
$430$	0	0
$431$	−8.28303	−0.398979	−0.199490	−	0.979900i	$-0.563928\pi$
$431$	−8.28303	−0.398979	−0.199490	+	0.979900i	$0.563928\pi$
$432$	0	0
$433$	−12.2263	−0.587557	−0.293779	−	0.955873i	$-0.594913\pi$
$433$	−12.2263	−0.587557	−0.293779	+	0.955873i	$0.594913\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	19.2613	0.921395
$438$	0	0
$439$	13.6706	0.652463	0.326232	−	0.945290i	$-0.394221\pi$
$439$	13.6706	0.652463	0.326232	+	0.945290i	$0.394221\pi$
$440$	0	0
$441$	−27.1817	−1.29437
$442$	0	0
$443$	5.27381	0.250566	0.125283	−	0.992121i	$-0.460016\pi$
$443$	5.27381	0.250566	0.125283	+	0.992121i	$0.460016\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−4.00602	−0.189478
$448$	0	0
$449$	10.9245	0.515557	0.257778	−	0.966204i	$-0.417010\pi$
$449$	10.9245	0.515557	0.257778	+	0.966204i	$0.417010\pi$
$450$	0	0
$451$	7.79587	0.367093
$452$	0	0
$453$	5.34764	0.251254
$454$	0	0
$455$	0	0
$456$	0	0
$457$	35.4843	1.65988	0.829942	−	0.557850i	$-0.188374\pi$
$457$	35.4843	1.65988	0.829942	+	0.557850i	$0.188374\pi$
$458$	0	0
$459$	−15.4531	−0.721287
$460$	0	0
$461$	−37.1657	−1.73098	−0.865490	−	0.500927i	$-0.832993\pi$
$461$	−37.1657	−1.73098	−0.865490	+	0.500927i	$0.832993\pi$
$462$	0	0
$463$	0.0309055	0.00143630	0.000718151	−	1.00000i	$-0.499771\pi$
$463$	0.0309055	0.00143630	0.000718151		1.00000i	$0.499771\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−35.3219	−1.63450	−0.817250	−	0.576283i	$-0.804503\pi$
$467$	−35.3219	−1.63450	−0.817250	+	0.576283i	$0.804503\pi$
$468$	0	0
$469$	−19.0475	−0.879533
$470$	0	0
$471$	−0.0102506	−0.000472324 0
$472$	0	0
$473$	11.1177	0.511191
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−14.5389	−0.665688
$478$	0	0
$479$	35.0787	1.60279	0.801394	−	0.598137i	$-0.204092\pi$
$479$	35.0787	1.60279	0.801394	+	0.598137i	$0.204092\pi$
$480$	0	0
$481$	5.30471	0.241874
$482$	0	0
$483$	3.74624	0.170460
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−17.0663	−0.773346	−0.386673	−	0.922217i	$-0.626376\pi$
$487$	−17.0663	−0.773346	−0.386673	+	0.922217i	$0.626376\pi$
$488$	0	0
$489$	4.77395	0.215885
$490$	0	0
$491$	−15.0122	−0.677493	−0.338747	−	0.940878i	$-0.610003\pi$
$491$	−15.0122	−0.677493	−0.338747	+	0.940878i	$0.610003\pi$
$492$	0	0
$493$	−36.6312	−1.64979
$494$	0	0
$495$	0	0
$496$	0	0
$497$	10.2468	0.459633
$498$	0	0
$499$	25.4967	1.14139	0.570694	−	0.821163i	$-0.306674\pi$
$499$	25.4967	1.14139	0.570694	+	0.821163i	$0.306674\pi$
$500$	0	0
$501$	−4.31696	−0.192868
$502$	0	0
$503$	−36.5788	−1.63097	−0.815484	−	0.578779i	$-0.803529\pi$
$503$	−36.5788	−1.63097	−0.815484	+	0.578779i	$0.803529\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	1.01731	0.0451801
$508$	0	0
$509$	−14.1327	−0.626423	−0.313212	−	0.949683i	$-0.601405\pi$
$509$	−14.1327	−0.626423	−0.313212	+	0.949683i	$0.601405\pi$
$510$	0	0
$511$	−16.6339	−0.735841
$512$	0	0
$513$	−14.0996	−0.622514
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−3.01292	−0.132508
$518$	0	0
$519$	2.48283	0.108984
$520$	0	0
$521$	19.8551	0.869866	0.434933	−	0.900463i	$-0.356772\pi$
$521$	19.8551	0.869866	0.434933	+	0.900463i	$0.356772\pi$
$522$	0	0
$523$	−7.79587	−0.340889	−0.170445	−	0.985367i	$-0.554520\pi$
$523$	−7.79587	−0.340889	−0.170445	+	0.985367i	$0.554520\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.70147	−0.0741174
$528$	0	0
$529$	−15.5679	−0.676864
$530$	0	0
$531$	31.5917	1.37096
$532$	0	0
$533$	−31.1835	−1.35071
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−2.22537	−0.0960317
$538$	0	0
$539$	−9.42170	−0.405821
$540$	0	0
$541$	4.21757	0.181327	0.0906637	−	0.995882i	$-0.471101\pi$
$541$	4.21757	0.181327	0.0906637	+	0.995882i	$0.471101\pi$
$542$	0	0
$543$	−2.57510	−0.110508
$544$	0	0
$545$	0	0
$546$	0	0
$547$	20.4136	0.872823	0.436412	−	0.899747i	$-0.356249\pi$
$547$	20.4136	0.872823	0.436412	+	0.899747i	$0.356249\pi$
$548$	0	0
$549$	−36.7278	−1.56751
$550$	0	0
$551$	−33.4229	−1.42386
$552$	0	0
$553$	−56.3304	−2.39541
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−15.4782	−0.655834	−0.327917	−	0.944707i	$-0.606347\pi$
$557$	−15.4782	−0.655834	−0.327917	+	0.944707i	$0.606347\pi$
$558$	0	0
$559$	−44.4707	−1.88091
$560$	0	0
$561$	−2.62583	−0.110863
$562$	0	0
$563$	1.08305	0.0456452	0.0228226	−	0.999740i	$-0.492735\pi$
$563$	1.08305	0.0456452	0.0228226	+	0.999740i	$0.492735\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	32.3311	1.35778
$568$	0	0
$569$	19.0572	0.798920	0.399460	−	0.916751i	$-0.369198\pi$
$569$	19.0572	0.798920	0.399460	+	0.916751i	$0.369198\pi$
$570$	0	0
$571$	−23.9617	−1.00277	−0.501384	−	0.865225i	$-0.667176\pi$
$571$	−23.9617	−1.00277	−0.501384	+	0.865225i	$0.667176\pi$
$572$	0	0
$573$	4.28673	0.179081
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−1.57392	−0.0655230	−0.0327615	−	0.999463i	$-0.510430\pi$
$577$	−1.57392	−0.0655230	−0.0327615	+	0.999463i	$0.510430\pi$
$578$	0	0
$579$	−1.49941	−0.0623132
$580$	0	0
$581$	−14.7128	−0.610390
$582$	0	0
$583$	−5.03945	−0.208713
$584$	0	0
$585$	0	0
$586$	0	0
$587$	23.7488	0.980220	0.490110	−	0.871661i	$-0.336957\pi$
$587$	23.7488	0.980220	0.490110	+	0.871661i	$0.336957\pi$
$588$	0	0
$589$	−1.55245	−0.0639677
$590$	0	0
$591$	0.229980	0.00946012
$592$	0	0
$593$	14.1306	0.580274	0.290137	−	0.956985i	$-0.406299\pi$
$593$	14.1306	0.580274	0.290137	+	0.956985i	$0.406299\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−1.49941	−0.0613666
$598$	0	0
$599$	−23.8395	−0.974054	−0.487027	−	0.873387i	$-0.661919\pi$
$599$	−23.8395	−0.974054	−0.487027	+	0.873387i	$0.661919\pi$
$600$	0	0
$601$	28.2353	1.15174	0.575871	−	0.817540i	$-0.304663\pi$
$601$	28.2353	1.15174	0.575871	+	0.817540i	$0.304663\pi$
$602$	0	0
$603$	13.5605	0.552228
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−29.6699	−1.20427	−0.602133	−	0.798396i	$-0.705682\pi$
$607$	−29.6699	−1.20427	−0.602133	+	0.798396i	$0.705682\pi$
$608$	0	0
$609$	−6.50059	−0.263417
$610$	0	0
$611$	12.0517	0.487559
$612$	0	0
$613$	9.34023	0.377248	0.188624	−	0.982049i	$-0.439597\pi$
$613$	9.34023	0.377248	0.188624	+	0.982049i	$0.439597\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−17.7754	−0.715609	−0.357805	−	0.933796i	$-0.616475\pi$
$617$	−17.7754	−0.715609	−0.357805	+	0.933796i	$0.616475\pi$
$618$	0	0
$619$	8.64143	0.347328	0.173664	−	0.984805i	$-0.444439\pi$
$619$	8.64143	0.347328	0.173664	+	0.984805i	$0.444439\pi$
$620$	0	0
$621$	−5.44044	−0.218317
$622$	0	0
$623$	40.0577	1.60488
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−2.39585	−0.0956812
$628$	0	0
$629$	10.2693	0.409462
$630$	0	0
$631$	−20.0414	−0.797837	−0.398919	−	0.916986i	$-0.630614\pi$
$631$	−20.0414	−0.797837	−0.398919	+	0.916986i	$0.630614\pi$
$632$	0	0
$633$	−3.78778	−0.150551
$634$	0	0
$635$	0	0
$636$	0	0
$637$	37.6868	1.49321
$638$	0	0
$639$	−7.29504	−0.288587
$640$	0	0
$641$	48.0113	1.89633	0.948166	−	0.317777i	$-0.102936\pi$
$641$	48.0113	1.89633	0.948166	+	0.317777i	$0.102936\pi$
$642$	0	0
$643$	−35.9396	−1.41732	−0.708660	−	0.705550i	$-0.750700\pi$
$643$	−35.9396	−1.41732	−0.708660	+	0.705550i	$0.750700\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−32.9172	−1.29411	−0.647055	−	0.762443i	$-0.724000\pi$
$647$	−32.9172	−1.29411	−0.647055	+	0.762443i	$0.724000\pi$
$648$	0	0
$649$	10.9503	0.429837
$650$	0	0
$651$	−0.301945	−0.0118341
$652$	0	0
$653$	−3.34461	−0.130885	−0.0654424	−	0.997856i	$-0.520846\pi$
$653$	−3.34461	−0.130885	−0.0654424	+	0.997856i	$0.520846\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	11.8422	0.462009
$658$	0	0
$659$	8.65168	0.337022	0.168511	−	0.985700i	$-0.446104\pi$
$659$	8.65168	0.337022	0.168511	+	0.985700i	$0.446104\pi$
$660$	0	0
$661$	−29.0550	−1.13011	−0.565055	−	0.825053i	$-0.691145\pi$
$661$	−29.0550	−1.13011	−0.565055	+	0.825053i	$0.691145\pi$
$662$	0	0
$663$	10.5033	0.407916
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−12.8964	−0.499352
$668$	0	0
$669$	6.59508	0.254981
$670$	0	0
$671$	−12.7306	−0.491458
$672$	0	0
$673$	5.71765	0.220399	0.110200	−	0.993909i	$-0.464851\pi$
$673$	5.71765	0.220399	0.110200	+	0.993909i	$0.464851\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−10.0701	−0.387026	−0.193513	−	0.981098i	$-0.561988\pi$
$677$	−10.0701	−0.387026	−0.193513	+	0.981098i	$0.561988\pi$
$678$	0	0
$679$	50.6393	1.94336
$680$	0	0
$681$	−7.66528	−0.293734
$682$	0	0
$683$	−3.79120	−0.145066	−0.0725331	−	0.997366i	$-0.523108\pi$
$683$	−3.79120	−0.145066	−0.0725331	+	0.997366i	$0.523108\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−5.78340	−0.220650
$688$	0	0
$689$	20.1578	0.767950
$690$	0	0
$691$	−27.5637	−1.04857	−0.524287	−	0.851542i	$-0.675668\pi$
$691$	−27.5637	−1.04857	−0.524287	+	0.851542i	$0.675668\pi$
$692$	0	0
$693$	11.6911	0.444109
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−60.3673	−2.28657
$698$	0	0
$699$	4.91718	0.185985
$700$	0	0
$701$	13.6830	0.516801	0.258401	−	0.966038i	$-0.416805\pi$
$701$	13.6830	0.516801	0.258401	+	0.966038i	$0.416805\pi$
$702$	0	0
$703$	9.36984	0.353390
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−2.60808	−0.0980868
$708$	0	0
$709$	14.3584	0.539241	0.269621	−	0.962967i	$-0.413102\pi$
$709$	14.3584	0.539241	0.269621	+	0.962967i	$0.413102\pi$
$710$	0	0
$711$	40.1034	1.50399
$712$	0	0
$713$	−0.599025	−0.0224336
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−4.71372	−0.176037
$718$	0	0
$719$	−48.8332	−1.82117	−0.910585	−	0.413323i	$-0.864368\pi$
$719$	−48.8332	−1.82117	−0.910585	+	0.413323i	$0.864368\pi$
$720$	0	0
$721$	−17.7061	−0.659411
$722$	0	0
$723$	5.10593	0.189891
$724$	0	0
$725$	0	0
$726$	0	0
$727$	32.2566	1.19633	0.598165	−	0.801373i	$-0.295897\pi$
$727$	32.2566	1.19633	0.598165	+	0.801373i	$0.295897\pi$
$728$	0	0
$729$	−20.9874	−0.777310
$730$	0	0
$731$	−86.0896	−3.18414
$732$	0	0
$733$	−18.8176	−0.695042	−0.347521	−	0.937672i	$-0.612976\pi$
$733$	−18.8176	−0.695042	−0.347521	+	0.937672i	$0.612976\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4.70034	0.173139
$738$	0	0
$739$	14.3089	0.526360	0.263180	−	0.964747i	$-0.415229\pi$
$739$	14.3089	0.526360	0.263180	+	0.964747i	$0.415229\pi$
$740$	0	0
$741$	9.58342	0.352056
$742$	0	0
$743$	−43.8448	−1.60851	−0.804255	−	0.594284i	$-0.797435\pi$
$743$	−43.8448	−1.60851	−0.804255	+	0.594284i	$0.797435\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	10.4745	0.383243
$748$	0	0
$749$	−36.4190	−1.33072
$750$	0	0
$751$	−23.0903	−0.842578	−0.421289	−	0.906926i	$-0.638422\pi$
$751$	−23.0903	−0.842578	−0.421289	+	0.906926i	$0.638422\pi$
$752$	0	0
$753$	−0.851432	−0.0310279
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−19.0270	−0.691549	−0.345775	−	0.938318i	$-0.612384\pi$
$757$	−19.0270	−0.691549	−0.345775	+	0.938318i	$0.612384\pi$
$758$	0	0
$759$	−0.924456	−0.0335556
$760$	0	0
$761$	−35.9537	−1.30332	−0.651659	−	0.758512i	$-0.725927\pi$
$761$	−35.9537	−1.30332	−0.651659	+	0.758512i	$0.725927\pi$
$762$	0	0
$763$	73.7915	2.67143
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−43.8012	−1.58157
$768$	0	0
$769$	−17.4665	−0.629858	−0.314929	−	0.949115i	$-0.601981\pi$
$769$	−17.4665	−0.629858	−0.314929	+	0.949115i	$0.601981\pi$
$770$	0	0
$771$	2.25285	0.0811346
$772$	0	0
$773$	10.5361	0.378958	0.189479	−	0.981885i	$-0.439320\pi$
$773$	10.5361	0.378958	0.189479	+	0.981885i	$0.439320\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	1.82239	0.0653779
$778$	0	0
$779$	−55.0801	−1.97345
$780$	0	0
$781$	−2.52860	−0.0904805
$782$	0	0
$783$	9.44042	0.337373
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−37.8748	−1.35009	−0.675045	−	0.737777i	$-0.735876\pi$
$787$	−37.8748	−1.35009	−0.675045	+	0.737777i	$0.735876\pi$
$788$	0	0
$789$	−4.54694	−0.161875
$790$	0	0
$791$	42.2148	1.50099
$792$	0	0
$793$	50.9223	1.80830
$794$	0	0
$795$	0	0
$796$	0	0
$797$	29.9574	1.06114	0.530572	−	0.847640i	$-0.321977\pi$
$797$	29.9574	1.06114	0.530572	+	0.847640i	$0.321977\pi$
$798$	0	0
$799$	23.3306	0.825376
$800$	0	0
$801$	−28.5184	−1.00765
$802$	0	0
$803$	4.10474	0.144853
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−6.59931	−0.232307
$808$	0	0
$809$	−2.59589	−0.0912667	−0.0456333	−	0.998958i	$-0.514531\pi$
$809$	−2.59589	−0.0912667	−0.0456333	+	0.998958i	$0.514531\pi$
$810$	0	0
$811$	−39.2012	−1.37654	−0.688271	−	0.725454i	$-0.741630\pi$
$811$	−39.2012	−1.37654	−0.688271	+	0.725454i	$0.741630\pi$
$812$	0	0
$813$	−0.0179882	−0.000630873 0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−78.5495	−2.74810
$818$	0	0
$819$	−46.7645	−1.63408
$820$	0	0
$821$	−35.7578	−1.24796	−0.623979	−	0.781441i	$-0.714485\pi$
$821$	−35.7578	−1.24796	−0.623979	+	0.781441i	$0.714485\pi$
$822$	0	0
$823$	17.3093	0.603365	0.301683	−	0.953408i	$-0.402452\pi$
$823$	17.3093	0.603365	0.301683	+	0.953408i	$0.402452\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	25.9957	0.903958	0.451979	−	0.892029i	$-0.350718\pi$
$827$	25.9957	0.903958	0.451979	+	0.892029i	$0.350718\pi$
$828$	0	0
$829$	−46.6902	−1.62162	−0.810808	−	0.585312i	$-0.800972\pi$
$829$	−46.6902	−1.62162	−0.810808	+	0.585312i	$0.800972\pi$
$830$	0	0
$831$	1.38895	0.0481822
$832$	0	0
$833$	72.9569	2.52781
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0.438496	0.0151566
$838$	0	0
$839$	9.45364	0.326376	0.163188	−	0.986595i	$-0.447822\pi$
$839$	9.45364	0.326376	0.163188	+	0.986595i	$0.447822\pi$
$840$	0	0
$841$	−6.62168	−0.228334
$842$	0	0
$843$	5.94786	0.204855
$844$	0	0
$845$	0	0
$846$	0	0
$847$	4.05237	0.139241
$848$	0	0
$849$	−5.12940	−0.176041
$850$	0	0
$851$	3.61542	0.123935
$852$	0	0
$853$	21.5733	0.738656	0.369328	−	0.929299i	$-0.379588\pi$
$853$	21.5733	0.738656	0.369328	+	0.929299i	$0.379588\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	19.5783	0.668782	0.334391	−	0.942434i	$-0.391469\pi$
$857$	19.5783	0.668782	0.334391	+	0.942434i	$0.391469\pi$
$858$	0	0
$859$	28.3598	0.967622	0.483811	−	0.875172i	$-0.339252\pi$
$859$	28.3598	0.967622	0.483811	+	0.875172i	$0.339252\pi$
$860$	0	0
$861$	−10.7128	−0.365092
$862$	0	0
$863$	9.05630	0.308280	0.154140	−	0.988049i	$-0.450739\pi$
$863$	9.05630	0.308280	0.154140	+	0.988049i	$0.450739\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	14.5684	0.494769
$868$	0	0
$869$	13.9006	0.471546
$870$	0	0
$871$	−18.8014	−0.637060
$872$	0	0
$873$	−36.0517	−1.22016
$874$	0	0
$875$	0	0
$876$	0	0
$877$	53.2624	1.79854	0.899271	−	0.437392i	$-0.144098\pi$
$877$	53.2624	1.79854	0.899271	+	0.437392i	$0.144098\pi$
$878$	0	0
$879$	−4.52918	−0.152766
$880$	0	0
$881$	33.7557	1.13726	0.568629	−	0.822594i	$-0.307474\pi$
$881$	33.7557	1.13726	0.568629	+	0.822594i	$0.307474\pi$
$882$	0	0
$883$	0.157109	0.00528714	0.00264357	−	0.999997i	$-0.499159\pi$
$883$	0.157109	0.00528714	0.00264357	+	0.999997i	$0.499159\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	8.63050	0.289784	0.144892	−	0.989447i	$-0.453717\pi$
$887$	8.63050	0.289784	0.144892	+	0.989447i	$0.453717\pi$
$888$	0	0
$889$	39.5917	1.32786
$890$	0	0
$891$	−7.97831	−0.267284
$892$	0	0
$893$	21.2872	0.712348
$894$	0	0
$895$	0	0
$896$	0	0
$897$	3.69783	0.123467
$898$	0	0
$899$	1.03945	0.0346675
$900$	0	0
$901$	39.0229	1.30004
$902$	0	0
$903$	−15.2775	−0.508404
$904$	0	0
$905$	0	0
$906$	0	0
$907$	6.69596	0.222336	0.111168	−	0.993802i	$-0.464541\pi$
$907$	6.69596	0.222336	0.111168	+	0.993802i	$0.464541\pi$
$908$	0	0
$909$	1.85677	0.0615852
$910$	0	0
$911$	29.9345	0.991776	0.495888	−	0.868387i	$-0.334843\pi$
$911$	29.9345	0.991776	0.495888	+	0.868387i	$0.334843\pi$
$912$	0	0
$913$	3.63067	0.120158
$914$	0	0
$915$	0	0
$916$	0	0
$917$	48.9809	1.61749
$918$	0	0
$919$	−35.3618	−1.16648	−0.583238	−	0.812301i	$-0.698215\pi$
$919$	−35.3618	−1.16648	−0.583238	+	0.812301i	$0.698215\pi$
$920$	0	0
$921$	5.35641	0.176500
$922$	0	0
$923$	10.1144	0.332920
$924$	0	0
$925$	0	0
$926$	0	0
$927$	12.6056	0.414021
$928$	0	0
$929$	−27.2134	−0.892843	−0.446421	−	0.894823i	$-0.647302\pi$
$929$	−27.2134	−0.892843	−0.446421	+	0.894823i	$0.647302\pi$
$930$	0	0
$931$	66.5671	2.18165
$932$	0	0
$933$	0.281444	0.00921405
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−16.3659	−0.534651	−0.267326	−	0.963606i	$-0.586140\pi$
$937$	−16.3659	−0.534651	−0.267326	+	0.963606i	$0.586140\pi$
$938$	0	0
$939$	−8.30968	−0.271176
$940$	0	0
$941$	−18.0134	−0.587221	−0.293611	−	0.955925i	$-0.594857\pi$
$941$	−18.0134	−0.587221	−0.293611	+	0.955925i	$0.594857\pi$
$942$	0	0
$943$	−21.2530	−0.692094
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−47.9751	−1.55898	−0.779491	−	0.626414i	$-0.784522\pi$
$947$	−47.9751	−1.55898	−0.779491	+	0.626414i	$0.784522\pi$
$948$	0	0
$949$	−16.4190	−0.532982
$950$	0	0
$951$	−3.88798	−0.126076
$952$	0	0
$953$	40.7788	1.32096	0.660478	−	0.750845i	$-0.270354\pi$
$953$	40.7788	1.32096	0.660478	+	0.750845i	$0.270354\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	1.60415	0.0518547
$958$	0	0
$959$	−18.2203	−0.588364
$960$	0	0
$961$	−30.9517	−0.998443
$962$	0	0
$963$	25.9278	0.835512
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−0.835313	−0.0268618	−0.0134309	−	0.999910i	$-0.504275\pi$
$967$	−0.835313	−0.0268618	−0.0134309	+	0.999910i	$0.504275\pi$
$968$	0	0
$969$	18.5523	0.595985
$970$	0	0
$971$	−27.3542	−0.877839	−0.438920	−	0.898526i	$-0.644639\pi$
$971$	−27.3542	−0.877839	−0.438920	+	0.898526i	$0.644639\pi$
$972$	0	0
$973$	33.7754	1.08279
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−4.40765	−0.141013	−0.0705066	−	0.997511i	$-0.522462\pi$
$977$	−4.40765	−0.141013	−0.0705066	+	0.997511i	$0.522462\pi$
$978$	0	0
$979$	−9.88501	−0.315926
$980$	0	0
$981$	−52.5345	−1.67730
$982$	0	0
$983$	−28.3968	−0.905718	−0.452859	−	0.891582i	$-0.649596\pi$
$983$	−28.3968	−0.905718	−0.452859	+	0.891582i	$0.649596\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	4.14026	0.131786
$988$	0	0
$989$	−30.3089	−0.963766
$990$	0	0
$991$	6.76451	0.214882	0.107441	−	0.994211i	$-0.465734\pi$
$991$	6.76451	0.214882	0.107441	+	0.994211i	$0.465734\pi$
$992$	0	0
$993$	8.27797	0.262693
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−33.5747	−1.06332	−0.531660	−	0.846958i	$-0.678432\pi$
$997$	−33.5747	−1.06332	−0.531660	+	0.846958i	$0.678432\pi$
$998$	0	0
$999$	−2.64655	−0.0837330

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4400.2.a.cb.1.3		4
4.3	odd	2		2200.2.a.y.1.2		4
5.2	odd	4		880.2.b.j.529.4		8
5.3	odd	4		880.2.b.j.529.5		8
5.4	even	2		4400.2.a.ce.1.2		4
20.3	even	4		440.2.b.d.89.4	✓	8
20.7	even	4		440.2.b.d.89.5	yes	8
20.19	odd	2		2200.2.a.x.1.3		4
60.23	odd	4		3960.2.d.f.3169.4		8
60.47	odd	4		3960.2.d.f.3169.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
440.2.b.d.89.4	✓	8	20.3	even	4
440.2.b.d.89.5	yes	8	20.7	even	4
880.2.b.j.529.4		8	5.2	odd	4
880.2.b.j.529.5		8	5.3	odd	4
2200.2.a.x.1.3		4	20.19	odd	2
2200.2.a.y.1.2		4	4.3	odd	2
3960.2.d.f.3169.3		8	60.47	odd	4
3960.2.d.f.3169.4		8	60.23	odd	4
4400.2.a.cb.1.3		4	1.1	even	1	trivial
4400.2.a.ce.1.2		4	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4400 = 2^{4} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4400.a (trivial)

\(f(q)\)	\(=\)	\(q+0.339102 q^{3} +4.05237 q^{7} -2.88501 q^{9} -1.00000 q^{11} +4.00000 q^{13} +7.74350 q^{17} +7.06529 q^{19} +1.37417 q^{21} +2.72619 q^{23} -1.99562 q^{27} -4.73057 q^{29} -0.219729 q^{31} -0.339102 q^{33} +1.32618 q^{37} +1.35641 q^{39} -7.79587 q^{41} -11.1177 q^{43} +3.01292 q^{47} +9.42170 q^{49} +2.62583 q^{51} +5.03945 q^{53} +2.39585 q^{57} -10.9503 q^{59} +12.7306 q^{61} -11.6911 q^{63} -4.70034 q^{67} +0.924456 q^{69} +2.52860 q^{71} -4.10474 q^{73} -4.05237 q^{77} -13.9006 q^{79} +7.97831 q^{81} -3.63067 q^{83} -1.60415 q^{87} +9.88501 q^{89} +16.2095 q^{91} -0.0745107 q^{93} +12.4962 q^{97} +2.88501 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{3} - q^{7} + 7 q^{9} - 4 q^{11} + 16 q^{13} + 7 q^{17} + 9 q^{19} - 7 q^{21} - 6 q^{23} - 13 q^{27} + 3 q^{29} + 15 q^{31} + q^{33} + 5 q^{37} - 4 q^{39} + 10 q^{41} - 8 q^{43} + 10 q^{47} + 9 q^{49}+ \cdots - 7 q^{99}+O(q^{100}) \) 4 * q - q^3 - q^7 + 7 * q^9 - 4 * q^11 + 16 * q^13 + 7 * q^17 + 9 * q^19 - 7 * q^21 - 6 * q^23 - 13 * q^27 + 3 * q^29 + 15 * q^31 + q^33 + 5 * q^37 - 4 * q^39 + 10 * q^41 - 8 * q^43 + 10 * q^47 + 9 * q^49 + 23 * q^51 + 5 * q^53 - 15 * q^57 - 6 * q^59 + 29 * q^61 - 40 * q^63 - 6 * q^67 - 30 * q^69 + q^71 + 18 * q^73 + q^77 + 20 * q^79 + 44 * q^81 - 26 * q^83 - 31 * q^87 + 21 * q^89 - 4 * q^91 + 25 * q^93 - 4 * q^97 - 7 * q^99

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