LMFDB - Embedded newform 4400.2.a.ca.1.1

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 = [3,0,3,0,0,0,3,0,6,0,3,0,3] 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:	$3$
Coefficient field:	3.3.837.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 1 $ x^3 - 6*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2200)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.52892$ 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)

$f(q)$	$=$	$q-1.52892 q^{3} -1.39543 q^{7} -0.662410 q^{9} +1.00000 q^{11} +6.05784 q^{13} +3.26193 q^{17} -1.00000 q^{19} +2.13349 q^{21} -0.528918 q^{23} +5.59952 q^{27} +1.60457 q^{29} -3.79085 q^{31} -1.52892 q^{33} -8.92434 q^{37} -9.26193 q^{39} +1.86651 q^{41} -0.866508 q^{43} -1.19133 q^{47} -5.05279 q^{49} -4.98723 q^{51} +10.7202 q^{53} +1.52892 q^{57} -13.1913 q^{59} -3.12844 q^{61} +0.924344 q^{63} +2.26698 q^{67} +0.808672 q^{69} -10.0400 q^{71} +13.3954 q^{73} -1.39543 q^{77} +2.58675 q^{79} -6.57398 q^{81} -0.128442 q^{83} -2.45326 q^{87} +4.12844 q^{89} -8.45326 q^{91} +5.79590 q^{93} -11.4354 q^{97} -0.662410 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} + 3 q^{7} + 6 q^{9} + 3 q^{11} + 3 q^{13} + 3 q^{17} - 3 q^{19} + 6 q^{21} + 6 q^{23} + 18 q^{27} + 12 q^{29} + 3 q^{31} + 3 q^{33} - 12 q^{37} - 21 q^{39} + 6 q^{41} - 3 q^{43} + 12 q^{47}+ \cdots + 6 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 + 3 * q^7 + 6 * q^9 + 3 * q^11 + 3 * q^13 + 3 * q^17 - 3 * q^19 + 6 * q^21 + 6 * q^23 + 18 * q^27 + 12 * q^29 + 3 * q^31 + 3 * q^33 - 12 * q^37 - 21 * q^39 + 6 * q^41 - 3 * q^43 + 12 * q^47 + 6 * q^49 + 9 * q^51 + 9 * q^53 - 3 * q^57 - 24 * q^59 - 3 * q^61 - 12 * q^63 + 6 * q^67 + 18 * q^69 + 15 * q^71 + 33 * q^73 + 3 * q^77 - 15 * q^79 + 27 * q^81 + 6 * q^83 + 15 * q^87 + 6 * q^89 - 3 * q^91 + 9 * q^93 + 18 * q^97 + 6 * 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
$3$	−1.52892	−0.882721	−0.441361	−	0.897330i	$-0.645504\pi$
$3$	−1.52892	−0.882721	−0.441361	+	0.897330i	$0.645504\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.39543	−0.527421	−0.263711	−	0.964602i	$-0.584946\pi$
$7$	−1.39543	−0.527421	−0.263711	+	0.964602i	$0.584946\pi$
$8$	0	0
$9$	−0.662410	−0.220803
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	6.05784	1.68014	0.840071	−	0.542477i	$-0.182513\pi$
$13$	6.05784	1.68014	0.840071	+	0.542477i	$0.182513\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.26193	0.791135	0.395568	−	0.918437i	$-0.370548\pi$
$17$	3.26193	0.791135	0.395568	+	0.918437i	$0.370548\pi$
$18$	0	0
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	0	0
$21$	2.13349	0.465566
$22$	0	0
$23$	−0.528918	−0.110287	−0.0551435	−	0.998478i	$-0.517562\pi$
$23$	−0.528918	−0.110287	−0.0551435	+	0.998478i	$0.517562\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.59952	1.07763
$28$	0	0
$29$	1.60457	0.297962	0.148981	−	0.988840i	$-0.452401\pi$
$29$	1.60457	0.297962	0.148981	+	0.988840i	$0.452401\pi$
$30$	0	0
$31$	−3.79085	−0.680857	−0.340429	−	0.940270i	$-0.610572\pi$
$31$	−3.79085	−0.680857	−0.340429	+	0.940270i	$0.610572\pi$
$32$	0	0
$33$	−1.52892	−0.266150
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−8.92434	−1.46715	−0.733577	−	0.679607i	$-0.762150\pi$
$37$	−8.92434	−1.46715	−0.733577	+	0.679607i	$0.762150\pi$
$38$	0	0
$39$	−9.26193	−1.48310
$40$	0	0
$41$	1.86651	0.291500	0.145750	−	0.989321i	$-0.453441\pi$
$41$	1.86651	0.291500	0.145750	+	0.989321i	$0.453441\pi$
$42$	0	0
$43$	−0.866508	−0.132141	−0.0660706	−	0.997815i	$-0.521046\pi$
$43$	−0.866508	−0.132141	−0.0660706	+	0.997815i	$0.521046\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−1.19133	−0.173773	−0.0868865	−	0.996218i	$-0.527692\pi$
$47$	−1.19133	−0.173773	−0.0868865	+	0.996218i	$0.527692\pi$
$48$	0	0
$49$	−5.05279	−0.721827
$50$	0	0
$51$	−4.98723	−0.698352
$52$	0	0
$53$	10.7202	1.47254	0.736270	−	0.676688i	$-0.236586\pi$
$53$	10.7202	1.47254	0.736270	+	0.676688i	$0.236586\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.52892	0.202510
$58$	0	0
$59$	−13.1913	−1.71736	−0.858682	−	0.512508i	$-0.828716\pi$
$59$	−13.1913	−1.71736	−0.858682	+	0.512508i	$0.828716\pi$
$60$	0	0
$61$	−3.12844	−0.400556	−0.200278	−	0.979739i	$-0.564185\pi$
$61$	−3.12844	−0.400556	−0.200278	+	0.979739i	$0.564185\pi$
$62$	0	0
$63$	0.924344	0.116456
$64$	0	0
$65$	0	0
$66$	0	0
$67$	2.26698	0.276956	0.138478	−	0.990365i	$-0.455779\pi$
$67$	2.26698	0.276956	0.138478	+	0.990365i	$0.455779\pi$
$68$	0	0
$69$	0.808672	0.0973527
$70$	0	0
$71$	−10.0400	−1.19153	−0.595765	−	0.803159i	$-0.703151\pi$
$71$	−10.0400	−1.19153	−0.595765	+	0.803159i	$0.703151\pi$
$72$	0	0
$73$	13.3954	1.56782	0.783908	−	0.620877i	$-0.213223\pi$
$73$	13.3954	1.56782	0.783908	+	0.620877i	$0.213223\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.39543	−0.159024
$78$	0	0
$79$	2.58675	0.291033	0.145516	−	0.989356i	$-0.453516\pi$
$79$	2.58675	0.291033	0.145516	+	0.989356i	$0.453516\pi$
$80$	0	0
$81$	−6.57398	−0.730443
$82$	0	0
$83$	−0.128442	−0.0140984	−0.00704918	−	0.999975i	$-0.502244\pi$
$83$	−0.128442	−0.0140984	−0.00704918	+	0.999975i	$0.502244\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−2.45326	−0.263017
$88$	0	0
$89$	4.12844	0.437614	0.218807	−	0.975768i	$-0.429783\pi$
$89$	4.12844	0.437614	0.218807	+	0.975768i	$0.429783\pi$
$90$	0	0
$91$	−8.45326	−0.886143
$92$	0	0
$93$	5.79590	0.601007
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−11.4354	−1.16109	−0.580547	−	0.814227i	$-0.697161\pi$
$97$	−11.4354	−1.16109	−0.580547	+	0.814227i	$0.697161\pi$
$98$	0	0
$99$	−0.662410	−0.0665747
$100$	0	0
$101$	14.2619	1.41912	0.709558	−	0.704647i	$-0.248895\pi$
$101$	14.2619	1.41912	0.709558	+	0.704647i	$0.248895\pi$
$102$	0	0
$103$	18.3598	1.80904	0.904522	−	0.426428i	$-0.140228\pi$
$103$	18.3598	1.80904	0.904522	+	0.426428i	$0.140228\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5.11567	0.494551	0.247275	−	0.968945i	$-0.420465\pi$
$107$	5.11567	0.494551	0.247275	+	0.968945i	$0.420465\pi$
$108$	0	0
$109$	8.24411	0.789643	0.394821	−	0.918758i	$-0.370806\pi$
$109$	8.24411	0.789643	0.394821	+	0.918758i	$0.370806\pi$
$110$	0	0
$111$	13.6446	1.29509
$112$	0	0
$113$	10.9243	1.02768	0.513838	−	0.857887i	$-0.328223\pi$
$113$	10.9243	1.02768	0.513838	+	0.857887i	$0.328223\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−4.01277	−0.370981
$118$	0	0
$119$	−4.55179	−0.417262
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−2.85374	−0.257313
$124$	0	0
$125$	0	0
$126$	0	0
$127$	8.47108	0.751687	0.375843	−	0.926683i	$-0.377353\pi$
$127$	8.47108	0.751687	0.375843	+	0.926683i	$0.377353\pi$
$128$	0	0
$129$	1.32482	0.116644
$130$	0	0
$131$	11.4354	0.999119	0.499560	−	0.866280i	$-0.333495\pi$
$131$	11.4354	0.999119	0.499560	+	0.866280i	$0.333495\pi$
$132$	0	0
$133$	1.39543	0.120999
$134$	0	0
$135$	0	0
$136$	0	0
$137$	6.66241	0.569208	0.284604	−	0.958645i	$-0.408138\pi$
$137$	6.66241	0.569208	0.284604	+	0.958645i	$0.408138\pi$
$138$	0	0
$139$	−12.8487	−1.08981	−0.544906	−	0.838497i	$-0.683435\pi$
$139$	−12.8487	−1.08981	−0.544906	+	0.838497i	$0.683435\pi$
$140$	0	0
$141$	1.82144	0.153393
$142$	0	0
$143$	6.05784	0.506582
$144$	0	0
$145$	0	0
$146$	0	0
$147$	7.72530	0.637172
$148$	0	0
$149$	−1.73302	−0.141974	−0.0709871	−	0.997477i	$-0.522615\pi$
$149$	−1.73302	−0.141974	−0.0709871	+	0.997477i	$0.522615\pi$
$150$	0	0
$151$	12.9243	1.05177	0.525884	−	0.850556i	$-0.323735\pi$
$151$	12.9243	1.05177	0.525884	+	0.850556i	$0.323735\pi$
$152$	0	0
$153$	−2.16074	−0.174685
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−12.6496	−1.00955	−0.504776	−	0.863251i	$-0.668425\pi$
$157$	−12.6496	−1.00955	−0.504776	+	0.863251i	$0.668425\pi$
$158$	0	0
$159$	−16.3904	−1.29984
$160$	0	0
$161$	0.738066	0.0581677
$162$	0	0
$163$	10.9771	0.859795	0.429898	−	0.902878i	$-0.358550\pi$
$163$	10.9771	0.859795	0.429898	+	0.902878i	$0.358550\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2.66746	0.206414	0.103207	−	0.994660i	$-0.467090\pi$
$167$	2.66746	0.206414	0.103207	+	0.994660i	$0.467090\pi$
$168$	0	0
$169$	23.6974	1.82288
$170$	0	0
$171$	0.662410	0.0506557
$172$	0	0
$173$	−15.1157	−1.14922	−0.574612	−	0.818426i	$-0.694847\pi$
$173$	−15.1157	−1.14922	−0.574612	+	0.818426i	$0.694847\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	20.1685	1.51595
$178$	0	0
$179$	−13.6624	−1.02118	−0.510588	−	0.859825i	$-0.670572\pi$
$179$	−13.6624	−1.02118	−0.510588	+	0.859825i	$0.670572\pi$
$180$	0	0
$181$	8.98723	0.668016	0.334008	−	0.942570i	$-0.391599\pi$
$181$	8.98723	0.668016	0.334008	+	0.942570i	$0.391599\pi$
$182$	0	0
$183$	4.78313	0.353579
$184$	0	0
$185$	0	0
$186$	0	0
$187$	3.26193	0.238536
$188$	0	0
$189$	−7.81372	−0.568365
$190$	0	0
$191$	4.24411	0.307093	0.153547	−	0.988141i	$-0.450930\pi$
$191$	4.24411	0.307093	0.153547	+	0.988141i	$0.450930\pi$
$192$	0	0
$193$	4.53397	0.326362	0.163181	−	0.986596i	$-0.447825\pi$
$193$	4.53397	0.326362	0.163181	+	0.986596i	$0.447825\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3.72797	0.265607	0.132803	−	0.991142i	$-0.457602\pi$
$197$	3.72797	0.265607	0.132803	+	0.991142i	$0.457602\pi$
$198$	0	0
$199$	12.5111	0.886888	0.443444	−	0.896302i	$-0.353756\pi$
$199$	12.5111	0.886888	0.443444	+	0.896302i	$0.353756\pi$
$200$	0	0
$201$	−3.46603	−0.244475
$202$	0	0
$203$	−2.23906	−0.157152
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0.350360	0.0243517
$208$	0	0
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	23.0400	1.58614	0.793070	−	0.609130i	$-0.208481\pi$
$211$	23.0400	1.58614	0.793070	+	0.609130i	$0.208481\pi$
$212$	0	0
$213$	15.3504	1.05179
$214$	0	0
$215$	0	0
$216$	0	0
$217$	5.28985	0.359099
$218$	0	0
$219$	−20.4805	−1.38394
$220$	0	0
$221$	19.7603	1.32922
$222$	0	0
$223$	−9.19133	−0.615497	−0.307748	−	0.951468i	$-0.599576\pi$
$223$	−9.19133	−0.615497	−0.307748	+	0.951468i	$0.599576\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	15.0350	0.997906	0.498953	−	0.866629i	$-0.333718\pi$
$227$	15.0350	0.997906	0.498953	+	0.866629i	$0.333718\pi$
$228$	0	0
$229$	−4.58675	−0.303101	−0.151551	−	0.988449i	$-0.548427\pi$
$229$	−4.58675	−0.303101	−0.151551	+	0.988449i	$0.548427\pi$
$230$	0	0
$231$	2.13349	0.140373
$232$	0	0
$233$	0.337590	0.0221163	0.0110581	−	0.999939i	$-0.496480\pi$
$233$	0.337590	0.0221163	0.0110581	+	0.999939i	$0.496480\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−3.95493	−0.256901
$238$	0	0
$239$	25.5111	1.65018	0.825088	−	0.565004i	$-0.191126\pi$
$239$	25.5111	1.65018	0.825088	+	0.565004i	$0.191126\pi$
$240$	0	0
$241$	−1.93711	−0.124781	−0.0623903	−	0.998052i	$-0.519872\pi$
$241$	−1.93711	−0.124781	−0.0623903	+	0.998052i	$0.519872\pi$
$242$	0	0
$243$	−6.74749	−0.432852
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−6.05784	−0.385451
$248$	0	0
$249$	0.196378	0.0124449
$250$	0	0
$251$	−17.3776	−1.09686	−0.548432	−	0.836195i	$-0.684775\pi$
$251$	−17.3776	−1.09686	−0.548432	+	0.836195i	$0.684775\pi$
$252$	0	0
$253$	−0.528918	−0.0332528
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−2.98218	−0.186023	−0.0930116	−	0.995665i	$-0.529649\pi$
$257$	−2.98218	−0.186023	−0.0930116	+	0.995665i	$0.529649\pi$
$258$	0	0
$259$	12.4533	0.773808
$260$	0	0
$261$	−1.06289	−0.0657910
$262$	0	0
$263$	22.7730	1.40425	0.702123	−	0.712056i	$-0.252236\pi$
$263$	22.7730	1.40425	0.702123	+	0.712056i	$0.252236\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−6.31205	−0.386291
$268$	0	0
$269$	−25.4933	−1.55435	−0.777176	−	0.629283i	$-0.783349\pi$
$269$	−25.4933	−1.55435	−0.777176	+	0.629283i	$0.783349\pi$
$270$	0	0
$271$	−9.18123	−0.557720	−0.278860	−	0.960332i	$-0.589957\pi$
$271$	−9.18123	−0.557720	−0.278860	+	0.960332i	$0.589957\pi$
$272$	0	0
$273$	12.9243	0.782217
$274$	0	0
$275$	0	0
$276$	0	0
$277$	24.0979	1.44790	0.723950	−	0.689853i	$-0.242325\pi$
$277$	24.0979	1.44790	0.723950	+	0.689853i	$0.242325\pi$
$278$	0	0
$279$	2.51110	0.150335
$280$	0	0
$281$	27.4126	1.63530	0.817648	−	0.575718i	$-0.195277\pi$
$281$	27.4126	1.63530	0.817648	+	0.575718i	$0.195277\pi$
$282$	0	0
$283$	17.6446	1.04886	0.524431	−	0.851453i	$-0.324278\pi$
$283$	17.6446	1.04886	0.524431	+	0.851453i	$0.324278\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2.60457	−0.153743
$288$	0	0
$289$	−6.35979	−0.374105
$290$	0	0
$291$	17.4839	1.02492
$292$	0	0
$293$	−25.1634	−1.47006	−0.735031	−	0.678033i	$-0.762832\pi$
$293$	−25.1634	−1.47006	−0.735031	+	0.678033i	$0.762832\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	5.59952	0.324917
$298$	0	0
$299$	−3.20410	−0.185298
$300$	0	0
$301$	1.20915	0.0696941
$302$	0	0
$303$	−21.8053	−1.25268
$304$	0	0
$305$	0	0
$306$	0	0
$307$	29.6089	1.68987	0.844936	−	0.534867i	$-0.179638\pi$
$307$	29.6089	1.68987	0.844936	+	0.534867i	$0.179638\pi$
$308$	0	0
$309$	−28.0706	−1.59688
$310$	0	0
$311$	−10.7330	−0.608614	−0.304307	−	0.952574i	$-0.598425\pi$
$311$	−10.7330	−0.608614	−0.304307	+	0.952574i	$0.598425\pi$
$312$	0	0
$313$	33.2892	1.88162	0.940808	−	0.338940i	$-0.110069\pi$
$313$	33.2892	1.88162	0.940808	+	0.338940i	$0.110069\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−18.5511	−1.04193	−0.520967	−	0.853577i	$-0.674429\pi$
$317$	−18.5511	−1.04193	−0.520967	+	0.853577i	$0.674429\pi$
$318$	0	0
$319$	1.60457	0.0898389
$320$	0	0
$321$	−7.82144	−0.436550
$322$	0	0
$323$	−3.26193	−0.181499
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−12.6046	−0.697035
$328$	0	0
$329$	1.66241	0.0916516
$330$	0	0
$331$	5.86651	0.322452	0.161226	−	0.986917i	$-0.448455\pi$
$331$	5.86651	0.322452	0.161226	+	0.986917i	$0.448455\pi$
$332$	0	0
$333$	5.91157	0.323952
$334$	0	0
$335$	0	0
$336$	0	0
$337$	7.58170	0.413002	0.206501	−	0.978446i	$-0.433792\pi$
$337$	7.58170	0.413002	0.206501	+	0.978446i	$0.433792\pi$
$338$	0	0
$339$	−16.7024	−0.907151
$340$	0	0
$341$	−3.79085	−0.205286
$342$	0	0
$343$	16.8188	0.908128
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−25.3954	−1.36330	−0.681649	−	0.731679i	$-0.738737\pi$
$347$	−25.3954	−1.36330	−0.681649	+	0.731679i	$0.738737\pi$
$348$	0	0
$349$	−21.2670	−1.13840	−0.569198	−	0.822201i	$-0.692746\pi$
$349$	−21.2670	−1.13840	−0.569198	+	0.822201i	$0.692746\pi$
$350$	0	0
$351$	33.9210	1.81057
$352$	0	0
$353$	−8.88433	−0.472865	−0.236433	−	0.971648i	$-0.575978\pi$
$353$	−8.88433	−0.472865	−0.236433	+	0.971648i	$0.575978\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	6.95931	0.368326
$358$	0	0
$359$	−11.1990	−0.591063	−0.295532	−	0.955333i	$-0.595497\pi$
$359$	−11.1990	−0.591063	−0.295532	+	0.955333i	$0.595497\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	0	0
$363$	−1.52892	−0.0802474
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−1.58675	−0.0828279	−0.0414139	−	0.999142i	$-0.513186\pi$
$367$	−1.58675	−0.0828279	−0.0414139	+	0.999142i	$0.513186\pi$
$368$	0	0
$369$	−1.23639	−0.0643641
$370$	0	0
$371$	−14.9593	−0.776649
$372$	0	0
$373$	−33.2714	−1.72273	−0.861363	−	0.507990i	$-0.830389\pi$
$373$	−33.2714	−1.72273	−0.861363	+	0.507990i	$0.830389\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	9.72025	0.500618
$378$	0	0
$379$	24.0979	1.23782	0.618912	−	0.785461i	$-0.287574\pi$
$379$	24.0979	1.23782	0.618912	+	0.785461i	$0.287574\pi$
$380$	0	0
$381$	−12.9516	−0.663530
$382$	0	0
$383$	15.3648	0.785106	0.392553	−	0.919729i	$-0.371592\pi$
$383$	15.3648	0.785106	0.392553	+	0.919729i	$0.371592\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0.573984	0.0291772
$388$	0	0
$389$	−1.88433	−0.0955392	−0.0477696	−	0.998858i	$-0.515211\pi$
$389$	−1.88433	−0.0955392	−0.0477696	+	0.998858i	$0.515211\pi$
$390$	0	0
$391$	−1.72530	−0.0872519
$392$	0	0
$393$	−17.4839	−0.881944
$394$	0	0
$395$	0	0
$396$	0	0
$397$	7.27203	0.364973	0.182486	−	0.983208i	$-0.441585\pi$
$397$	7.27203	0.364973	0.182486	+	0.983208i	$0.441585\pi$
$398$	0	0
$399$	−2.13349	−0.106808
$400$	0	0
$401$	12.8665	0.642523	0.321261	−	0.946991i	$-0.395893\pi$
$401$	12.8665	0.642523	0.321261	+	0.946991i	$0.395893\pi$
$402$	0	0
$403$	−22.9644	−1.14394
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−8.92434	−0.442363
$408$	0	0
$409$	5.15569	0.254932	0.127466	−	0.991843i	$-0.459316\pi$
$409$	5.15569	0.254932	0.127466	+	0.991843i	$0.459316\pi$
$410$	0	0
$411$	−10.1863	−0.502452
$412$	0	0
$413$	18.4075	0.905775
$414$	0	0
$415$	0	0
$416$	0	0
$417$	19.6446	0.962000
$418$	0	0
$419$	39.0145	1.90598	0.952991	−	0.302999i	$-0.0979878\pi$
$419$	39.0145	1.90598	0.952991	+	0.302999i	$0.0979878\pi$
$420$	0	0
$421$	28.9415	1.41052	0.705261	−	0.708948i	$-0.250830\pi$
$421$	28.9415	1.41052	0.705261	+	0.708948i	$0.250830\pi$
$422$	0	0
$423$	0.789147	0.0383697
$424$	0	0
$425$	0	0
$426$	0	0
$427$	4.36551	0.211262
$428$	0	0
$429$	−9.26193	−0.447170
$430$	0	0
$431$	17.2969	0.833162	0.416581	−	0.909099i	$-0.363228\pi$
$431$	17.2969	0.833162	0.416581	+	0.909099i	$0.363228\pi$
$432$	0	0
$433$	17.8010	0.855459	0.427730	−	0.903907i	$-0.359314\pi$
$433$	17.8010	0.855459	0.427730	+	0.903907i	$0.359314\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0.528918	0.0253016
$438$	0	0
$439$	28.9694	1.38263	0.691317	−	0.722551i	$-0.257031\pi$
$439$	28.9694	1.38263	0.691317	+	0.722551i	$0.257031\pi$
$440$	0	0
$441$	3.34702	0.159382
$442$	0	0
$443$	−5.70748	−0.271170	−0.135585	−	0.990766i	$-0.543291\pi$
$443$	−5.70748	−0.271170	−0.135585	+	0.990766i	$0.543291\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	2.64964	0.125324
$448$	0	0
$449$	21.8436	1.03087	0.515433	−	0.856930i	$-0.327631\pi$
$449$	21.8436	1.03087	0.515433	+	0.856930i	$0.327631\pi$
$450$	0	0
$451$	1.86651	0.0878904
$452$	0	0
$453$	−19.7603	−0.928418
$454$	0	0
$455$	0	0
$456$	0	0
$457$	19.2441	0.900202	0.450101	−	0.892978i	$-0.351388\pi$
$457$	19.2441	0.900202	0.450101	+	0.892978i	$0.351388\pi$
$458$	0	0
$459$	18.2653	0.852550
$460$	0	0
$461$	10.6574	0.496363	0.248181	−	0.968714i	$-0.420167\pi$
$461$	10.6574	0.496363	0.248181	+	0.968714i	$0.420167\pi$
$462$	0	0
$463$	−6.98990	−0.324848	−0.162424	−	0.986721i	$-0.551931\pi$
$463$	−6.98990	−0.324848	−0.162424	+	0.986721i	$0.551931\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−10.3904	−0.480809	−0.240405	−	0.970673i	$-0.577280\pi$
$467$	−10.3904	−0.480809	−0.240405	+	0.970673i	$0.577280\pi$
$468$	0	0
$469$	−3.16341	−0.146073
$470$	0	0
$471$	19.3403	0.891152
$472$	0	0
$473$	−0.866508	−0.0398421
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−7.10120	−0.325141
$478$	0	0
$479$	37.0044	1.69077	0.845387	−	0.534155i	$-0.179370\pi$
$479$	37.0044	1.69077	0.845387	+	0.534155i	$0.179370\pi$
$480$	0	0
$481$	−54.0622	−2.46502
$482$	0	0
$483$	−1.12844	−0.0513459
$484$	0	0
$485$	0	0
$486$	0	0
$487$	24.0578	1.09016	0.545082	−	0.838383i	$-0.316498\pi$
$487$	24.0578	1.09016	0.545082	+	0.838383i	$0.316498\pi$
$488$	0	0
$489$	−16.7831	−0.758959
$490$	0	0
$491$	0.314720	0.0142031	0.00710156	−	0.999975i	$-0.497739\pi$
$491$	0.314720	0.0142031	0.00710156	+	0.999975i	$0.497739\pi$
$492$	0	0
$493$	5.23401	0.235728
$494$	0	0
$495$	0	0
$496$	0	0
$497$	14.0101	0.628439
$498$	0	0
$499$	−39.6268	−1.77394	−0.886969	−	0.461829i	$-0.847193\pi$
$499$	−39.6268	−1.77394	−0.886969	+	0.461829i	$0.847193\pi$
$500$	0	0
$501$	−4.07833	−0.182206
$502$	0	0
$503$	21.2892	0.949238	0.474619	−	0.880191i	$-0.342586\pi$
$503$	21.2892	0.949238	0.474619	+	0.880191i	$0.342586\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−36.2313	−1.60909
$508$	0	0
$509$	12.3019	0.545274	0.272637	−	0.962117i	$-0.412104\pi$
$509$	12.3019	0.545274	0.272637	+	0.962117i	$0.412104\pi$
$510$	0	0
$511$	−18.6923	−0.826900
$512$	0	0
$513$	−5.59952	−0.247225
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−1.19133	−0.0523945
$518$	0	0
$519$	23.1106	1.01444
$520$	0	0
$521$	36.4227	1.59571	0.797853	−	0.602852i	$-0.205969\pi$
$521$	36.4227	1.59571	0.797853	+	0.602852i	$0.205969\pi$
$522$	0	0
$523$	−23.6675	−1.03491	−0.517453	−	0.855712i	$-0.673120\pi$
$523$	−23.6675	−1.03491	−0.517453	+	0.855712i	$0.673120\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−12.3655	−0.538650
$528$	0	0
$529$	−22.7202	−0.987837
$530$	0	0
$531$	8.73807	0.379200
$532$	0	0
$533$	11.3070	0.489761
$534$	0	0
$535$	0	0
$536$	0	0
$537$	20.8887	0.901414
$538$	0	0
$539$	−5.05279	−0.217639
$540$	0	0
$541$	12.9015	0.554678	0.277339	−	0.960772i	$-0.410548\pi$
$541$	12.9015	0.554678	0.277339	+	0.960772i	$0.410548\pi$
$542$	0	0
$543$	−13.7407	−0.589671
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−35.2841	−1.50864	−0.754320	−	0.656507i	$-0.772033\pi$
$547$	−35.2841	−1.50864	−0.754320	+	0.656507i	$0.772033\pi$
$548$	0	0
$549$	2.07231	0.0884441
$550$	0	0
$551$	−1.60457	−0.0683571
$552$	0	0
$553$	−3.60962	−0.153497
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−24.9745	−1.05820	−0.529101	−	0.848559i	$-0.677471\pi$
$557$	−24.9745	−1.05820	−0.529101	+	0.848559i	$0.677471\pi$
$558$	0	0
$559$	−5.24916	−0.222016
$560$	0	0
$561$	−4.98723	−0.210561
$562$	0	0
$563$	40.8258	1.72060	0.860302	−	0.509785i	$-0.170275\pi$
$563$	40.8258	1.72060	0.860302	+	0.509785i	$0.170275\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	9.17351	0.385251
$568$	0	0
$569$	−18.6046	−0.779944	−0.389972	−	0.920827i	$-0.627515\pi$
$569$	−18.6046	−0.779944	−0.389972	+	0.920827i	$0.627515\pi$
$570$	0	0
$571$	9.44554	0.395284	0.197642	−	0.980274i	$-0.436672\pi$
$571$	9.44554	0.395284	0.197642	+	0.980274i	$0.436672\pi$
$572$	0	0
$573$	−6.48890	−0.271078
$574$	0	0
$575$	0	0
$576$	0	0
$577$	13.5646	0.564700	0.282350	−	0.959311i	$-0.408886\pi$
$577$	13.5646	0.564700	0.282350	+	0.959311i	$0.408886\pi$
$578$	0	0
$579$	−6.93206	−0.288087
$580$	0	0
$581$	0.179232	0.00743578
$582$	0	0
$583$	10.7202	0.443987
$584$	0	0
$585$	0	0
$586$	0	0
$587$	35.2263	1.45394	0.726972	−	0.686667i	$-0.240927\pi$
$587$	35.2263	1.45394	0.726972	+	0.686667i	$0.240927\pi$
$588$	0	0
$589$	3.79085	0.156199
$590$	0	0
$591$	−5.69975	−0.234457
$592$	0	0
$593$	−39.7952	−1.63419	−0.817097	−	0.576500i	$-0.804418\pi$
$593$	−39.7952	−1.63419	−0.817097	+	0.576500i	$0.804418\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−19.1284	−0.782875
$598$	0	0
$599$	−19.1207	−0.781252	−0.390626	−	0.920550i	$-0.627741\pi$
$599$	−19.1207	−0.781252	−0.390626	+	0.920550i	$0.627741\pi$
$600$	0	0
$601$	−22.1964	−0.905409	−0.452705	−	0.891661i	$-0.649541\pi$
$601$	−22.1964	−0.905409	−0.452705	+	0.891661i	$0.649541\pi$
$602$	0	0
$603$	−1.50167	−0.0611528
$604$	0	0
$605$	0	0
$606$	0	0
$607$	12.9321	0.524896	0.262448	−	0.964946i	$-0.415470\pi$
$607$	12.9321	0.524896	0.262448	+	0.964946i	$0.415470\pi$
$608$	0	0
$609$	3.42335	0.138721
$610$	0	0
$611$	−7.21687	−0.291963
$612$	0	0
$613$	−18.4711	−0.746040	−0.373020	−	0.927823i	$-0.621678\pi$
$613$	−18.4711	−0.746040	−0.373020	+	0.927823i	$0.621678\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	22.6096	0.910229	0.455115	−	0.890433i	$-0.349598\pi$
$617$	22.6096	0.910229	0.455115	+	0.890433i	$0.349598\pi$
$618$	0	0
$619$	37.1557	1.49341	0.746707	−	0.665154i	$-0.231634\pi$
$619$	37.1557	1.49341	0.746707	+	0.665154i	$0.231634\pi$
$620$	0	0
$621$	−2.96169	−0.118848
$622$	0	0
$623$	−5.76094	−0.230807
$624$	0	0
$625$	0	0
$626$	0	0
$627$	1.52892	0.0610591
$628$	0	0
$629$	−29.1106	−1.16072
$630$	0	0
$631$	24.1506	0.961422	0.480711	−	0.876879i	$-0.340379\pi$
$631$	24.1506	0.961422	0.480711	+	0.876879i	$0.340379\pi$
$632$	0	0
$633$	−35.2263	−1.40012
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−30.6089	−1.21277
$638$	0	0
$639$	6.65061	0.263094
$640$	0	0
$641$	−34.3948	−1.35851	−0.679256	−	0.733902i	$-0.737697\pi$
$641$	−34.3948	−1.35851	−0.679256	+	0.733902i	$0.737697\pi$
$642$	0	0
$643$	44.4983	1.75484	0.877421	−	0.479720i	$-0.159262\pi$
$643$	44.4983	1.75484	0.877421	+	0.479720i	$0.159262\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−40.2135	−1.58096	−0.790478	−	0.612490i	$-0.790168\pi$
$647$	−40.2135	−1.58096	−0.790478	+	0.612490i	$0.790168\pi$
$648$	0	0
$649$	−13.1913	−0.517805
$650$	0	0
$651$	−8.08775	−0.316984
$652$	0	0
$653$	−31.3420	−1.22651	−0.613253	−	0.789887i	$-0.710139\pi$
$653$	−31.3420	−1.22651	−0.613253	+	0.789887i	$0.710139\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−8.87326	−0.346179
$658$	0	0
$659$	−32.3598	−1.26056	−0.630279	−	0.776369i	$-0.717060\pi$
$659$	−32.3598	−1.26056	−0.630279	+	0.776369i	$0.717060\pi$
$660$	0	0
$661$	−22.8087	−0.887155	−0.443577	−	0.896236i	$-0.646291\pi$
$661$	−22.8087	−0.887155	−0.443577	+	0.896236i	$0.646291\pi$
$662$	0	0
$663$	−30.2118	−1.17333
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−0.848688	−0.0328613
$668$	0	0
$669$	14.0528	0.543312
$670$	0	0
$671$	−3.12844	−0.120772
$672$	0	0
$673$	27.7152	1.06834	0.534171	−	0.845376i	$-0.320624\pi$
$673$	27.7152	1.06834	0.534171	+	0.845376i	$0.320624\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−16.0400	−0.616468	−0.308234	−	0.951311i	$-0.599738\pi$
$677$	−16.0400	−0.616468	−0.308234	+	0.951311i	$0.599738\pi$
$678$	0	0
$679$	15.9573	0.612385
$680$	0	0
$681$	−22.9872	−0.880873
$682$	0	0
$683$	35.8208	1.37064	0.685322	−	0.728240i	$-0.259662\pi$
$683$	35.8208	1.37064	0.685322	+	0.728240i	$0.259662\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	7.01277	0.267554
$688$	0	0
$689$	64.9415	2.47407
$690$	0	0
$691$	−38.5333	−1.46588	−0.732938	−	0.680296i	$-0.761851\pi$
$691$	−38.5333	−1.46588	−0.732938	+	0.680296i	$0.761851\pi$
$692$	0	0
$693$	0.924344	0.0351129
$694$	0	0
$695$	0	0
$696$	0	0
$697$	6.08843	0.230616
$698$	0	0
$699$	−0.516148	−0.0195225
$700$	0	0
$701$	−52.7596	−1.99270	−0.996351	−	0.0853497i	$-0.972799\pi$
$701$	−52.7596	−1.99270	−0.996351	+	0.0853497i	$0.972799\pi$
$702$	0	0
$703$	8.92434	0.336588
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−19.9015	−0.748472
$708$	0	0
$709$	5.04774	0.189572	0.0947859	−	0.995498i	$-0.469783\pi$
$709$	5.04774	0.189572	0.0947859	+	0.995498i	$0.469783\pi$
$710$	0	0
$711$	−1.71349	−0.0642609
$712$	0	0
$713$	2.00505	0.0750897
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−39.0044	−1.45665
$718$	0	0
$719$	−16.2848	−0.607321	−0.303660	−	0.952780i	$-0.598209\pi$
$719$	−16.2848	−0.607321	−0.303660	+	0.952780i	$0.598209\pi$
$720$	0	0
$721$	−25.6197	−0.954128
$722$	0	0
$723$	2.96169	0.110146
$724$	0	0
$725$	0	0
$726$	0	0
$727$	9.68461	0.359182	0.179591	−	0.983741i	$-0.442523\pi$
$727$	9.68461	0.359182	0.179591	+	0.983741i	$0.442523\pi$
$728$	0	0
$729$	30.0383	1.11253
$730$	0	0
$731$	−2.82649	−0.104542
$732$	0	0
$733$	13.8964	0.513276	0.256638	−	0.966508i	$-0.417385\pi$
$733$	13.8964	0.513276	0.256638	+	0.966508i	$0.417385\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	2.26698	0.0835054
$738$	0	0
$739$	17.9092	0.658800	0.329400	−	0.944190i	$-0.393153\pi$
$739$	17.9092	0.658800	0.329400	+	0.944190i	$0.393153\pi$
$740$	0	0
$741$	9.26193	0.340246
$742$	0	0
$743$	−9.77808	−0.358723	−0.179362	−	0.983783i	$-0.557403\pi$
$743$	−9.77808	−0.358723	−0.179362	+	0.983783i	$0.557403\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0.0850814	0.00311296
$748$	0	0
$749$	−7.13854	−0.260837
$750$	0	0
$751$	−24.5212	−0.894791	−0.447396	−	0.894336i	$-0.647648\pi$
$751$	−24.5212	−0.894791	−0.447396	+	0.894336i	$0.647648\pi$
$752$	0	0
$753$	26.5689	0.968226
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−43.9822	−1.59856	−0.799280	−	0.600959i	$-0.794785\pi$
$757$	−43.9822	−1.59856	−0.799280	+	0.600959i	$0.794785\pi$
$758$	0	0
$759$	0.808672	0.0293529
$760$	0	0
$761$	24.2313	0.878386	0.439193	−	0.898393i	$-0.355265\pi$
$761$	24.2313	0.878386	0.439193	+	0.898393i	$0.355265\pi$
$762$	0	0
$763$	−11.5041	−0.416475
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−79.9109	−2.88542
$768$	0	0
$769$	−23.5282	−0.848450	−0.424225	−	0.905557i	$-0.639453\pi$
$769$	−23.5282	−0.848450	−0.424225	+	0.905557i	$0.639453\pi$
$770$	0	0
$771$	4.55951	0.164207
$772$	0	0
$773$	−2.74579	−0.0987591	−0.0493795	−	0.998780i	$-0.515724\pi$
$773$	−2.74579	−0.0987591	−0.0493795	+	0.998780i	$0.515724\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−19.0400	−0.683057
$778$	0	0
$779$	−1.86651	−0.0668746
$780$	0	0
$781$	−10.0400	−0.359260
$782$	0	0
$783$	8.98485	0.321092
$784$	0	0
$785$	0	0
$786$	0	0
$787$	31.4482	1.12101	0.560504	−	0.828152i	$-0.310608\pi$
$787$	31.4482	1.12101	0.560504	+	0.828152i	$0.310608\pi$
$788$	0	0
$789$	−34.8181	−1.23956
$790$	0	0
$791$	−15.2441	−0.542018
$792$	0	0
$793$	−18.9516	−0.672991
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−12.1685	−0.431029	−0.215514	−	0.976501i	$-0.569143\pi$
$797$	−12.1685	−0.431029	−0.215514	+	0.976501i	$0.569143\pi$
$798$	0	0
$799$	−3.88603	−0.137478
$800$	0	0
$801$	−2.73472	−0.0966266
$802$	0	0
$803$	13.3954	0.472714
$804$	0	0
$805$	0	0
$806$	0	0
$807$	38.9771	1.37206
$808$	0	0
$809$	49.1480	1.72795	0.863975	−	0.503534i	$-0.167967\pi$
$809$	49.1480	1.72795	0.863975	+	0.503534i	$0.167967\pi$
$810$	0	0
$811$	−29.7253	−1.04380	−0.521898	−	0.853008i	$-0.674776\pi$
$811$	−29.7253	−1.04380	−0.521898	+	0.853008i	$0.674776\pi$
$812$	0	0
$813$	14.0373	0.492311
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0.866508	0.0303153
$818$	0	0
$819$	5.59952	0.195663
$820$	0	0
$821$	49.3648	1.72284	0.861422	−	0.507889i	$-0.169574\pi$
$821$	49.3648	1.72284	0.861422	+	0.507889i	$0.169574\pi$
$822$	0	0
$823$	−37.5817	−1.31002	−0.655008	−	0.755622i	$-0.727335\pi$
$823$	−37.5817	−1.31002	−0.655008	+	0.755622i	$0.727335\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−35.3749	−1.23011	−0.615054	−	0.788485i	$-0.710866\pi$
$827$	−35.3749	−1.23011	−0.615054	+	0.788485i	$0.710866\pi$
$828$	0	0
$829$	44.1786	1.53438	0.767192	−	0.641417i	$-0.221653\pi$
$829$	44.1786	1.53438	0.767192	+	0.641417i	$0.221653\pi$
$830$	0	0
$831$	−36.8436	−1.27809
$832$	0	0
$833$	−16.4819	−0.571062
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−21.2270	−0.733711
$838$	0	0
$839$	−47.7424	−1.64825	−0.824126	−	0.566406i	$-0.808333\pi$
$839$	−47.7424	−1.64825	−0.824126	+	0.566406i	$0.808333\pi$
$840$	0	0
$841$	−26.4253	−0.911219
$842$	0	0
$843$	−41.9116	−1.44351
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−1.39543	−0.0479474
$848$	0	0
$849$	−26.9771	−0.925853
$850$	0	0
$851$	4.72025	0.161808
$852$	0	0
$853$	−39.7502	−1.36102	−0.680510	−	0.732739i	$-0.738242\pi$
$853$	−39.7502	−1.36102	−0.680510	+	0.732739i	$0.738242\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	12.1080	0.413600	0.206800	−	0.978383i	$-0.433695\pi$
$857$	12.1080	0.413600	0.206800	+	0.978383i	$0.433695\pi$
$858$	0	0
$859$	55.4049	1.89039	0.945195	−	0.326508i	$-0.105872\pi$
$859$	55.4049	1.89039	0.945195	+	0.326508i	$0.105872\pi$
$860$	0	0
$861$	3.98218	0.135712
$862$	0	0
$863$	−10.4660	−0.356268	−0.178134	−	0.984006i	$-0.557006\pi$
$863$	−10.4660	−0.356268	−0.178134	+	0.984006i	$0.557006\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	9.72359	0.330230
$868$	0	0
$869$	2.58675	0.0877496
$870$	0	0
$871$	13.7330	0.465326
$872$	0	0
$873$	7.57495	0.256373
$874$	0	0
$875$	0	0
$876$	0	0
$877$	21.4959	0.725867	0.362933	−	0.931815i	$-0.381775\pi$
$877$	21.4959	0.725867	0.362933	+	0.931815i	$0.381775\pi$
$878$	0	0
$879$	38.4728	1.29765
$880$	0	0
$881$	−38.3521	−1.29211	−0.646057	−	0.763289i	$-0.723583\pi$
$881$	−38.3521	−1.29211	−0.646057	+	0.763289i	$0.723583\pi$
$882$	0	0
$883$	−25.8665	−0.870477	−0.435239	−	0.900315i	$-0.643336\pi$
$883$	−25.8665	−0.870477	−0.435239	+	0.900315i	$0.643336\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	7.19133	0.241461	0.120731	−	0.992685i	$-0.461476\pi$
$887$	7.19133	0.241461	0.120731	+	0.992685i	$0.461476\pi$
$888$	0	0
$889$	−11.8208	−0.396456
$890$	0	0
$891$	−6.57398	−0.220237
$892$	0	0
$893$	1.19133	0.0398663
$894$	0	0
$895$	0	0
$896$	0	0
$897$	4.89880	0.163566
$898$	0	0
$899$	−6.08270	−0.202869
$900$	0	0
$901$	34.9687	1.16498
$902$	0	0
$903$	−1.84869	−0.0615205
$904$	0	0
$905$	0	0
$906$	0	0
$907$	14.0356	0.466046	0.233023	−	0.972471i	$-0.425138\pi$
$907$	14.0356	0.466046	0.233023	+	0.972471i	$0.425138\pi$
$908$	0	0
$909$	−9.44725	−0.313345
$910$	0	0
$911$	−18.9166	−0.626736	−0.313368	−	0.949632i	$-0.601457\pi$
$911$	−18.9166	−0.626736	−0.313368	+	0.949632i	$0.601457\pi$
$912$	0	0
$913$	−0.128442	−0.00425082
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−15.9573	−0.526957
$918$	0	0
$919$	−32.7451	−1.08016	−0.540081	−	0.841613i	$-0.681606\pi$
$919$	−32.7451	−1.08016	−0.540081	+	0.841613i	$0.681606\pi$
$920$	0	0
$921$	−45.2697	−1.49169
$922$	0	0
$923$	−60.8208	−2.00194
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−12.1617	−0.399443
$928$	0	0
$929$	−9.37256	−0.307504	−0.153752	−	0.988110i	$-0.549136\pi$
$929$	−9.37256	−0.307504	−0.153752	+	0.988110i	$0.549136\pi$
$930$	0	0
$931$	5.05279	0.165598
$932$	0	0
$933$	16.4099	0.537236
$934$	0	0
$935$	0	0
$936$	0	0
$937$	49.6617	1.62238	0.811189	−	0.584784i	$-0.198821\pi$
$937$	49.6617	1.62238	0.811189	+	0.584784i	$0.198821\pi$
$938$	0	0
$939$	−50.8964	−1.66094
$940$	0	0
$941$	18.0178	0.587364	0.293682	−	0.955903i	$-0.405119\pi$
$941$	18.0178	0.587364	0.293682	+	0.955903i	$0.405119\pi$
$942$	0	0
$943$	−0.987230	−0.0321486
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−29.6096	−0.962183	−0.481092	−	0.876670i	$-0.659760\pi$
$947$	−29.6096	−0.962183	−0.481092	+	0.876670i	$0.659760\pi$
$948$	0	0
$949$	81.1473	2.63415
$950$	0	0
$951$	28.3631	0.919738
$952$	0	0
$953$	0.649640	0.0210439	0.0105219	−	0.999945i	$-0.496651\pi$
$953$	0.649640	0.0210439	0.0105219	+	0.999945i	$0.496651\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−2.45326	−0.0793027
$958$	0	0
$959$	−9.29690	−0.300213
$960$	0	0
$961$	−16.6294	−0.536434
$962$	0	0
$963$	−3.38867	−0.109198
$964$	0	0
$965$	0	0
$966$	0	0
$967$	30.7381	0.988470	0.494235	−	0.869328i	$-0.335448\pi$
$967$	30.7381	0.988470	0.494235	+	0.869328i	$0.335448\pi$
$968$	0	0
$969$	4.98723	0.160213
$970$	0	0
$971$	−24.4634	−0.785067	−0.392533	−	0.919738i	$-0.628401\pi$
$971$	−24.4634	−0.785067	−0.392533	+	0.919738i	$0.628401\pi$
$972$	0	0
$973$	17.9294	0.574790
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−2.14359	−0.0685796	−0.0342898	−	0.999412i	$-0.510917\pi$
$977$	−2.14359	−0.0685796	−0.0342898	+	0.999412i	$0.510917\pi$
$978$	0	0
$979$	4.12844	0.131946
$980$	0	0
$981$	−5.46098	−0.174356
$982$	0	0
$983$	−6.56388	−0.209355	−0.104678	−	0.994506i	$-0.533381\pi$
$983$	−6.56388	−0.209355	−0.104678	+	0.994506i	$0.533381\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−2.54169	−0.0809028
$988$	0	0
$989$	0.458312	0.0145735
$990$	0	0
$991$	−27.6345	−0.877839	−0.438919	−	0.898527i	$-0.644639\pi$
$991$	−27.6345	−0.877839	−0.438919	+	0.898527i	$0.644639\pi$
$992$	0	0
$993$	−8.96941	−0.284636
$994$	0	0
$995$	0	0
$996$	0	0
$997$	19.2008	0.608094	0.304047	−	0.952657i	$-0.401662\pi$
$997$	19.2008	0.608094	0.304047	+	0.952657i	$0.401662\pi$
$998$	0	0
$999$	−49.9721	−1.58105

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4400.2.a.ca.1.1		3
4.3	odd	2		2200.2.a.t.1.3	✓	3
5.2	odd	4		4400.2.b.bc.4049.5		6
5.3	odd	4		4400.2.b.bc.4049.2		6
5.4	even	2		4400.2.a.bx.1.3		3
20.3	even	4		2200.2.b.l.1849.5		6
20.7	even	4		2200.2.b.l.1849.2		6
20.19	odd	2		2200.2.a.w.1.1	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2200.2.a.t.1.3	✓	3	4.3	odd	2
2200.2.a.w.1.1	yes	3	20.19	odd	2
2200.2.b.l.1849.2		6	20.7	even	4
2200.2.b.l.1849.5		6	20.3	even	4
4400.2.a.bx.1.3		3	5.4	even	2
4400.2.a.ca.1.1		3	1.1	even	1	trivial
4400.2.b.bc.4049.2		6	5.3	odd	4
4400.2.b.bc.4049.5		6	5.2	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.52892 q^{3} -1.39543 q^{7} -0.662410 q^{9} +1.00000 q^{11} +6.05784 q^{13} +3.26193 q^{17} -1.00000 q^{19} +2.13349 q^{21} -0.528918 q^{23} +5.59952 q^{27} +1.60457 q^{29} -3.79085 q^{31} -1.52892 q^{33} -8.92434 q^{37} -9.26193 q^{39} +1.86651 q^{41} -0.866508 q^{43} -1.19133 q^{47} -5.05279 q^{49} -4.98723 q^{51} +10.7202 q^{53} +1.52892 q^{57} -13.1913 q^{59} -3.12844 q^{61} +0.924344 q^{63} +2.26698 q^{67} +0.808672 q^{69} -10.0400 q^{71} +13.3954 q^{73} -1.39543 q^{77} +2.58675 q^{79} -6.57398 q^{81} -0.128442 q^{83} -2.45326 q^{87} +4.12844 q^{89} -8.45326 q^{91} +5.79590 q^{93} -11.4354 q^{97} -0.662410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} + 3 q^{7} + 6 q^{9} + 3 q^{11} + 3 q^{13} + 3 q^{17} - 3 q^{19} + 6 q^{21} + 6 q^{23} + 18 q^{27} + 12 q^{29} + 3 q^{31} + 3 q^{33} - 12 q^{37} - 21 q^{39} + 6 q^{41} - 3 q^{43} + 12 q^{47}+ \cdots + 6 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 + 3 * q^7 + 6 * q^9 + 3 * q^11 + 3 * q^13 + 3 * q^17 - 3 * q^19 + 6 * q^21 + 6 * q^23 + 18 * q^27 + 12 * q^29 + 3 * q^31 + 3 * q^33 - 12 * q^37 - 21 * q^39 + 6 * q^41 - 3 * q^43 + 12 * q^47 + 6 * q^49 + 9 * q^51 + 9 * q^53 - 3 * q^57 - 24 * q^59 - 3 * q^61 - 12 * q^63 + 6 * q^67 + 18 * q^69 + 15 * q^71 + 33 * q^73 + 3 * q^77 - 15 * q^79 + 27 * q^81 + 6 * q^83 + 15 * q^87 + 6 * q^89 - 3 * q^91 + 9 * q^93 + 18 * q^97 + 6 * q^99

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