LMFDB - Embedded newform 5488.2.a.s.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5488,2,Mod(1,5488)] 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("5488.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5488, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5488 = 2^{4} \cdot 7^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5488.a (trivial)

Newform invariants

comment:select newform

sage:traces = [9,0,0,0,-13,0,0,0,13,0,-3] 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:	$43.8219006293$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 20x^{7} + 131x^{5} - 14x^{4} - 302x^{3} + 91x^{2} + 133x - 7 $ x^9 - 20x^7 + 131x^5 - 14x^4 - 302x^3 + 91x^2 + 133x - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2744)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.0511459$ of defining polynomial
Character	$\chi$	$=$	5488.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.0511459 q^{3} -0.342396 q^{5} -2.99738 q^{9} -2.07004 q^{11} +4.56820 q^{13} +0.0175121 q^{15} -3.03258 q^{17} +5.96658 q^{19} -0.364847 q^{23} -4.88277 q^{25} +0.306742 q^{27} -0.771564 q^{29} -1.14165 q^{31} +0.105874 q^{33} +8.56154 q^{37} -0.233645 q^{39} -7.12652 q^{41} +3.89506 q^{43} +1.02629 q^{45} +9.73147 q^{47} +0.155104 q^{51} +3.62458 q^{53} +0.708772 q^{55} -0.305166 q^{57} -13.4620 q^{59} -12.5942 q^{61} -1.56413 q^{65} +6.98529 q^{67} +0.0186604 q^{69} -13.4729 q^{71} -8.40566 q^{73} +0.249734 q^{75} +3.14740 q^{79} +8.97646 q^{81} +2.77912 q^{83} +1.03834 q^{85} +0.0394624 q^{87} -11.2101 q^{89} +0.0583907 q^{93} -2.04293 q^{95} -4.00431 q^{97} +6.20470 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 13 q^{5} + 13 q^{9} - 3 q^{11} - 20 q^{13} - 3 q^{15} - 10 q^{17} + 10 q^{19} + 4 q^{23} + 22 q^{25} + 3 q^{29} + 10 q^{31} - 7 q^{33} + 6 q^{37} + 20 q^{39} - 13 q^{41} + 8 q^{43} - 52 q^{45} - 10 q^{47}+ \cdots + 14 q^{99}+O(q^{100}) $ 9 * q - 13 * q^5 + 13 * q^9 - 3 * q^11 - 20 * q^13 - 3 * q^15 - 10 * q^17 + 10 * q^19 + 4 * q^23 + 22 * q^25 + 3 * q^29 + 10 * q^31 - 7 * q^33 + 6 * q^37 + 20 * q^39 - 13 * q^41 + 8 * q^43 - 52 * q^45 - 10 * q^47 + 23 * q^51 - 8 * q^53 + 25 * q^55 - q^57 - 7 * q^59 - 52 * q^61 + 28 * q^65 + 9 * q^67 - 42 * q^69 - 9 * q^71 - 23 * q^73 + 7 * q^75 - 10 * q^79 - 3 * q^81 - 18 * q^83 - 5 * q^85 + 7 * q^87 - 17 * q^89 - 53 * q^93 - 25 * q^95 - 24 * q^97 + 14 * 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
$3$	−0.0511459	−0.0295291	−0.0147646	−	0.999891i	$-0.504700\pi$
$3$	−0.0511459	−0.0295291	−0.0147646	+	0.999891i	$0.504700\pi$
$4$	0	0
$5$	−0.342396	−0.153124	−0.0765620	−	0.997065i	$-0.524394\pi$
$5$	−0.342396	−0.153124	−0.0765620	+	0.997065i	$0.524394\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−2.99738	−0.999128
$10$	0	0
$11$	−2.07004	−0.624140	−0.312070	−	0.950059i	$-0.601022\pi$
$11$	−2.07004	−0.624140	−0.312070	+	0.950059i	$0.601022\pi$
$12$	0	0
$13$	4.56820	1.26699	0.633495	−	0.773747i	$-0.281620\pi$
$13$	4.56820	1.26699	0.633495	+	0.773747i	$0.281620\pi$
$14$	0	0
$15$	0.0175121	0.00452162
$16$	0	0
$17$	−3.03258	−0.735509	−0.367755	−	0.929923i	$-0.619873\pi$
$17$	−3.03258	−0.735509	−0.367755	+	0.929923i	$0.619873\pi$
$18$	0	0
$19$	5.96658	1.36883	0.684413	−	0.729094i	$-0.260059\pi$
$19$	5.96658	1.36883	0.684413	+	0.729094i	$0.260059\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−0.364847	−0.0760759	−0.0380379	−	0.999276i	$-0.512111\pi$
$23$	−0.364847	−0.0760759	−0.0380379	+	0.999276i	$0.512111\pi$
$24$	0	0
$25$	−4.88277	−0.976553
$26$	0	0
$27$	0.306742	0.0590325
$28$	0	0
$29$	−0.771564	−0.143276	−0.0716379	−	0.997431i	$-0.522823\pi$
$29$	−0.771564	−0.143276	−0.0716379	+	0.997431i	$0.522823\pi$
$30$	0	0
$31$	−1.14165	−0.205046	−0.102523	−	0.994731i	$-0.532692\pi$
$31$	−1.14165	−0.205046	−0.102523	+	0.994731i	$0.532692\pi$
$32$	0	0
$33$	0.105874	0.0184303
$34$	0	0
$35$	0	0
$36$	0	0
$37$	8.56154	1.40751	0.703754	−	0.710444i	$-0.251506\pi$
$37$	8.56154	1.40751	0.703754	+	0.710444i	$0.251506\pi$
$38$	0	0
$39$	−0.233645	−0.0374131
$40$	0	0
$41$	−7.12652	−1.11298	−0.556488	−	0.830856i	$-0.687851\pi$
$41$	−7.12652	−1.11298	−0.556488	+	0.830856i	$0.687851\pi$
$42$	0	0
$43$	3.89506	0.593992	0.296996	−	0.954879i	$-0.404015\pi$
$43$	3.89506	0.593992	0.296996	+	0.954879i	$0.404015\pi$
$44$	0	0
$45$	1.02629	0.152990
$46$	0	0
$47$	9.73147	1.41948	0.709740	−	0.704464i	$-0.248812\pi$
$47$	9.73147	1.41948	0.709740	+	0.704464i	$0.248812\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0.155104	0.0217189
$52$	0	0
$53$	3.62458	0.497874	0.248937	−	0.968520i	$-0.419919\pi$
$53$	3.62458	0.497874	0.248937	+	0.968520i	$0.419919\pi$
$54$	0	0
$55$	0.708772	0.0955708
$56$	0	0
$57$	−0.305166	−0.0404202
$58$	0	0
$59$	−13.4620	−1.75260	−0.876299	−	0.481768i	$-0.839995\pi$
$59$	−13.4620	−1.75260	−0.876299	+	0.481768i	$0.839995\pi$
$60$	0	0
$61$	−12.5942	−1.61252	−0.806261	−	0.591560i	$-0.798512\pi$
$61$	−12.5942	−1.61252	−0.806261	+	0.591560i	$0.798512\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.56413	−0.194006
$66$	0	0
$67$	6.98529	0.853390	0.426695	−	0.904396i	$-0.359678\pi$
$67$	6.98529	0.853390	0.426695	+	0.904396i	$0.359678\pi$
$68$	0	0
$69$	0.0186604	0.00224645
$70$	0	0
$71$	−13.4729	−1.59894	−0.799472	−	0.600704i	$-0.794887\pi$
$71$	−13.4729	−1.59894	−0.799472	+	0.600704i	$0.794887\pi$
$72$	0	0
$73$	−8.40566	−0.983808	−0.491904	−	0.870650i	$-0.663699\pi$
$73$	−8.40566	−0.983808	−0.491904	+	0.870650i	$0.663699\pi$
$74$	0	0
$75$	0.249734	0.0288367
$76$	0	0
$77$	0	0
$78$	0	0
$79$	3.14740	0.354110	0.177055	−	0.984201i	$-0.443343\pi$
$79$	3.14740	0.354110	0.177055	+	0.984201i	$0.443343\pi$
$80$	0	0
$81$	8.97646	0.997385
$82$	0	0
$83$	2.77912	0.305048	0.152524	−	0.988300i	$-0.451260\pi$
$83$	2.77912	0.305048	0.152524	+	0.988300i	$0.451260\pi$
$84$	0	0
$85$	1.03834	0.112624
$86$	0	0
$87$	0.0394624	0.00423081
$88$	0	0
$89$	−11.2101	−1.18827	−0.594136	−	0.804365i	$-0.702506\pi$
$89$	−11.2101	−1.18827	−0.594136	+	0.804365i	$0.702506\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0.0583907	0.00605483
$94$	0	0
$95$	−2.04293	−0.209600
$96$	0	0
$97$	−4.00431	−0.406576	−0.203288	−	0.979119i	$-0.565163\pi$
$97$	−4.00431	−0.406576	−0.203288	+	0.979119i	$0.565163\pi$
$98$	0	0
$99$	6.20470	0.623596
$100$	0	0
$101$	−10.5814	−1.05289	−0.526446	−	0.850209i	$-0.676476\pi$
$101$	−10.5814	−1.05289	−0.526446	+	0.850209i	$0.676476\pi$
$102$	0	0
$103$	6.58810	0.649145	0.324573	−	0.945861i	$-0.394780\pi$
$103$	6.58810	0.649145	0.324573	+	0.945861i	$0.394780\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5.61435	0.542760	0.271380	−	0.962472i	$-0.412520\pi$
$107$	5.61435	0.542760	0.271380	+	0.962472i	$0.412520\pi$
$108$	0	0
$109$	1.88678	0.180720	0.0903602	−	0.995909i	$-0.471198\pi$
$109$	1.88678	0.180720	0.0903602	+	0.995909i	$0.471198\pi$
$110$	0	0
$111$	−0.437888	−0.0415625
$112$	0	0
$113$	−4.33640	−0.407934	−0.203967	−	0.978978i	$-0.565384\pi$
$113$	−4.33640	−0.407934	−0.203967	+	0.978978i	$0.565384\pi$
$114$	0	0
$115$	0.124922	0.0116490
$116$	0	0
$117$	−13.6926	−1.26588
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−6.71494	−0.610449
$122$	0	0
$123$	0.364492	0.0328652
$124$	0	0
$125$	3.38382	0.302658
$126$	0	0
$127$	12.6231	1.12012	0.560060	−	0.828452i	$-0.310778\pi$
$127$	12.6231	1.12012	0.560060	+	0.828452i	$0.310778\pi$
$128$	0	0
$129$	−0.199217	−0.0175401
$130$	0	0
$131$	11.8827	1.03820	0.519099	−	0.854714i	$-0.326268\pi$
$131$	11.8827	1.03820	0.519099	+	0.854714i	$0.326268\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−0.105027	−0.00903929
$136$	0	0
$137$	−12.1524	−1.03825	−0.519124	−	0.854699i	$-0.673742\pi$
$137$	−12.1524	−1.03825	−0.519124	+	0.854699i	$0.673742\pi$
$138$	0	0
$139$	−8.42138	−0.714292	−0.357146	−	0.934049i	$-0.616250\pi$
$139$	−8.42138	−0.714292	−0.357146	+	0.934049i	$0.616250\pi$
$140$	0	0
$141$	−0.497725	−0.0419160
$142$	0	0
$143$	−9.45634	−0.790779
$144$	0	0
$145$	0.264180	0.0219390
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−6.02530	−0.493612	−0.246806	−	0.969065i	$-0.579381\pi$
$149$	−6.02530	−0.493612	−0.246806	+	0.969065i	$0.579381\pi$
$150$	0	0
$151$	8.29219	0.674809	0.337405	−	0.941360i	$-0.390451\pi$
$151$	8.29219	0.674809	0.337405	+	0.941360i	$0.390451\pi$
$152$	0	0
$153$	9.08981	0.734868
$154$	0	0
$155$	0.390896	0.0313975
$156$	0	0
$157$	−21.3058	−1.70039	−0.850194	−	0.526469i	$-0.823516\pi$
$157$	−21.3058	−1.70039	−0.850194	+	0.526469i	$0.823516\pi$
$158$	0	0
$159$	−0.185382	−0.0147018
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−12.5494	−0.982945	−0.491472	−	0.870893i	$-0.663541\pi$
$163$	−12.5494	−0.982945	−0.491472	+	0.870893i	$0.663541\pi$
$164$	0	0
$165$	−0.0362508	−0.00282212
$166$	0	0
$167$	−8.31163	−0.643174	−0.321587	−	0.946880i	$-0.604216\pi$
$167$	−8.31163	−0.643174	−0.321587	+	0.946880i	$0.604216\pi$
$168$	0	0
$169$	7.86841	0.605262
$170$	0	0
$171$	−17.8841	−1.36763
$172$	0	0
$173$	−18.4798	−1.40499	−0.702496	−	0.711688i	$-0.747931\pi$
$173$	−18.4798	−1.40499	−0.702496	+	0.711688i	$0.747931\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0.688524	0.0517527
$178$	0	0
$179$	−14.9156	−1.11484	−0.557422	−	0.830229i	$-0.688210\pi$
$179$	−14.9156	−1.11484	−0.557422	+	0.830229i	$0.688210\pi$
$180$	0	0
$181$	3.85341	0.286422	0.143211	−	0.989692i	$-0.454257\pi$
$181$	3.85341	0.286422	0.143211	+	0.989692i	$0.454257\pi$
$182$	0	0
$183$	0.644142	0.0476164
$184$	0	0
$185$	−2.93143	−0.215523
$186$	0	0
$187$	6.27756	0.459061
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−18.9790	−1.37327	−0.686636	−	0.727001i	$-0.740914\pi$
$191$	−18.9790	−1.37327	−0.686636	+	0.727001i	$0.740914\pi$
$192$	0	0
$193$	7.74517	0.557510	0.278755	−	0.960362i	$-0.410078\pi$
$193$	7.74517	0.557510	0.278755	+	0.960362i	$0.410078\pi$
$194$	0	0
$195$	0.0799989	0.00572884
$196$	0	0
$197$	−8.12267	−0.578716	−0.289358	−	0.957221i	$-0.593442\pi$
$197$	−8.12267	−0.578716	−0.289358	+	0.957221i	$0.593442\pi$
$198$	0	0
$199$	−13.5620	−0.961382	−0.480691	−	0.876890i	$-0.659614\pi$
$199$	−13.5620	−0.961382	−0.480691	+	0.876890i	$0.659614\pi$
$200$	0	0
$201$	−0.357269	−0.0251998
$202$	0	0
$203$	0	0
$204$	0	0
$205$	2.44009	0.170423
$206$	0	0
$207$	1.09359	0.0760096
$208$	0	0
$209$	−12.3510	−0.854339
$210$	0	0
$211$	−25.4596	−1.75271	−0.876355	−	0.481666i	$-0.840032\pi$
$211$	−25.4596	−1.75271	−0.876355	+	0.481666i	$0.840032\pi$
$212$	0	0
$213$	0.689086	0.0472154
$214$	0	0
$215$	−1.33365	−0.0909544
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0.429915	0.0290510
$220$	0	0
$221$	−13.8534	−0.931882
$222$	0	0
$223$	11.5822	0.775604	0.387802	−	0.921743i	$-0.373234\pi$
$223$	11.5822	0.775604	0.387802	+	0.921743i	$0.373234\pi$
$224$	0	0
$225$	14.6355	0.975702
$226$	0	0
$227$	−1.29158	−0.0857251	−0.0428626	−	0.999081i	$-0.513648\pi$
$227$	−1.29158	−0.0857251	−0.0428626	+	0.999081i	$0.513648\pi$
$228$	0	0
$229$	−9.32706	−0.616349	−0.308175	−	0.951330i	$-0.599718\pi$
$229$	−9.32706	−0.616349	−0.308175	+	0.951330i	$0.599718\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	16.1827	1.06017	0.530084	−	0.847945i	$-0.322161\pi$
$233$	16.1827	1.06017	0.530084	+	0.847945i	$0.322161\pi$
$234$	0	0
$235$	−3.33201	−0.217357
$236$	0	0
$237$	−0.160977	−0.0104565
$238$	0	0
$239$	16.5483	1.07042	0.535209	−	0.844720i	$-0.320233\pi$
$239$	16.5483	1.07042	0.535209	+	0.844720i	$0.320233\pi$
$240$	0	0
$241$	−18.9203	−1.21876	−0.609381	−	0.792878i	$-0.708582\pi$
$241$	−18.9203	−1.21876	−0.609381	+	0.792878i	$0.708582\pi$
$242$	0	0
$243$	−1.37933	−0.0884844
$244$	0	0
$245$	0	0
$246$	0	0
$247$	27.2565	1.73429
$248$	0	0
$249$	−0.142141	−0.00900780
$250$	0	0
$251$	3.22125	0.203323	0.101662	−	0.994819i	$-0.467584\pi$
$251$	3.22125	0.203323	0.101662	+	0.994819i	$0.467584\pi$
$252$	0	0
$253$	0.755248	0.0474820
$254$	0	0
$255$	−0.0531070	−0.00332569
$256$	0	0
$257$	12.8497	0.801539	0.400770	−	0.916179i	$-0.368743\pi$
$257$	12.8497	0.801539	0.400770	+	0.916179i	$0.368743\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	2.31267	0.143151
$262$	0	0
$263$	−14.1853	−0.874703	−0.437352	−	0.899291i	$-0.644084\pi$
$263$	−14.1853	−0.874703	−0.437352	+	0.899291i	$0.644084\pi$
$264$	0	0
$265$	−1.24104	−0.0762364
$266$	0	0
$267$	0.573353	0.0350886
$268$	0	0
$269$	0.570801	0.0348023	0.0174012	−	0.999849i	$-0.494461\pi$
$269$	0.570801	0.0348023	0.0174012	+	0.999849i	$0.494461\pi$
$270$	0	0
$271$	18.4959	1.12355	0.561774	−	0.827291i	$-0.310119\pi$
$271$	18.4959	1.12355	0.561774	+	0.827291i	$0.310119\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	10.1075	0.609506
$276$	0	0
$277$	4.22545	0.253883	0.126941	−	0.991910i	$-0.459484\pi$
$277$	4.22545	0.253883	0.126941	+	0.991910i	$0.459484\pi$
$278$	0	0
$279$	3.42196	0.204867
$280$	0	0
$281$	−8.46608	−0.505044	−0.252522	−	0.967591i	$-0.581260\pi$
$281$	−8.46608	−0.505044	−0.252522	+	0.967591i	$0.581260\pi$
$282$	0	0
$283$	−10.8345	−0.644044	−0.322022	−	0.946732i	$-0.604363\pi$
$283$	−10.8345	−0.644044	−0.322022	+	0.946732i	$0.604363\pi$
$284$	0	0
$285$	0.104488	0.00618931
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−7.80345	−0.459026
$290$	0	0
$291$	0.204804	0.0120058
$292$	0	0
$293$	10.3369	0.603890	0.301945	−	0.953325i	$-0.402364\pi$
$293$	10.3369	0.603890	0.301945	+	0.953325i	$0.402364\pi$
$294$	0	0
$295$	4.60932	0.268365
$296$	0	0
$297$	−0.634967	−0.0368445
$298$	0	0
$299$	−1.66669	−0.0963874
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0.541197	0.0310910
$304$	0	0
$305$	4.31220	0.246916
$306$	0	0
$307$	19.0192	1.08548	0.542741	−	0.839900i	$-0.317387\pi$
$307$	19.0192	1.08548	0.542741	+	0.839900i	$0.317387\pi$
$308$	0	0
$309$	−0.336955	−0.0191687
$310$	0	0
$311$	−30.2980	−1.71804	−0.859021	−	0.511941i	$-0.828927\pi$
$311$	−30.2980	−1.71804	−0.859021	+	0.511941i	$0.828927\pi$
$312$	0	0
$313$	31.3723	1.77327	0.886635	−	0.462470i	$-0.153037\pi$
$313$	31.3723	1.77327	0.886635	+	0.462470i	$0.153037\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−3.74650	−0.210424	−0.105212	−	0.994450i	$-0.533552\pi$
$317$	−3.74650	−0.210424	−0.105212	+	0.994450i	$0.533552\pi$
$318$	0	0
$319$	1.59717	0.0894242
$320$	0	0
$321$	−0.287151	−0.0160272
$322$	0	0
$323$	−18.0941	−1.00678
$324$	0	0
$325$	−22.3054	−1.23728
$326$	0	0
$327$	−0.0965009	−0.00533651
$328$	0	0
$329$	0	0
$330$	0	0
$331$	4.60640	0.253191	0.126595	−	0.991954i	$-0.459595\pi$
$331$	4.60640	0.253191	0.126595	+	0.991954i	$0.459595\pi$
$332$	0	0
$333$	−25.6622	−1.40628
$334$	0	0
$335$	−2.39174	−0.130674
$336$	0	0
$337$	−22.6282	−1.23264	−0.616319	−	0.787497i	$-0.711377\pi$
$337$	−22.6282	−1.23264	−0.616319	+	0.787497i	$0.711377\pi$
$338$	0	0
$339$	0.221789	0.0120459
$340$	0	0
$341$	2.36326	0.127977
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−0.00638926	−0.000343986 0
$346$	0	0
$347$	31.2638	1.67833	0.839164	−	0.543879i	$-0.183045\pi$
$347$	31.2638	1.67833	0.839164	+	0.543879i	$0.183045\pi$
$348$	0	0
$349$	−9.45243	−0.505977	−0.252989	−	0.967469i	$-0.581414\pi$
$349$	−9.45243	−0.505977	−0.252989	+	0.967469i	$0.581414\pi$
$350$	0	0
$351$	1.40126	0.0747935
$352$	0	0
$353$	−0.358715	−0.0190925	−0.00954623	−	0.999954i	$-0.503039\pi$
$353$	−0.358715	−0.0190925	−0.00954623	+	0.999954i	$0.503039\pi$
$354$	0	0
$355$	4.61307	0.244837
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−19.0511	−1.00548	−0.502739	−	0.864438i	$-0.667674\pi$
$359$	−19.0511	−1.00548	−0.502739	+	0.864438i	$0.667674\pi$
$360$	0	0
$361$	16.6000	0.873685
$362$	0	0
$363$	0.343442	0.0180260
$364$	0	0
$365$	2.87806	0.150645
$366$	0	0
$367$	22.2022	1.15894	0.579472	−	0.814992i	$-0.303259\pi$
$367$	22.2022	1.15894	0.579472	+	0.814992i	$0.303259\pi$
$368$	0	0
$369$	21.3609	1.11200
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−31.6567	−1.63912	−0.819560	−	0.572994i	$-0.805782\pi$
$373$	−31.6567	−1.63912	−0.819560	+	0.572994i	$0.805782\pi$
$374$	0	0
$375$	−0.173068	−0.00893721
$376$	0	0
$377$	−3.52465	−0.181529
$378$	0	0
$379$	16.0332	0.823572	0.411786	−	0.911281i	$-0.364905\pi$
$379$	16.0332	0.823572	0.411786	+	0.911281i	$0.364905\pi$
$380$	0	0
$381$	−0.645620	−0.0330761
$382$	0	0
$383$	−19.6681	−1.00499	−0.502496	−	0.864580i	$-0.667585\pi$
$383$	−19.6681	−1.00499	−0.502496	+	0.864580i	$0.667585\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−11.6750	−0.593474
$388$	0	0
$389$	30.3100	1.53678	0.768390	−	0.639982i	$-0.221058\pi$
$389$	30.3100	1.53678	0.768390	+	0.639982i	$0.221058\pi$
$390$	0	0
$391$	1.10643	0.0559545
$392$	0	0
$393$	−0.607753	−0.0306571
$394$	0	0
$395$	−1.07766	−0.0542227
$396$	0	0
$397$	−16.2003	−0.813069	−0.406535	−	0.913635i	$-0.633263\pi$
$397$	−16.2003	−0.813069	−0.406535	+	0.913635i	$0.633263\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	8.16007	0.407495	0.203747	−	0.979024i	$-0.434688\pi$
$401$	8.16007	0.407495	0.203747	+	0.979024i	$0.434688\pi$
$402$	0	0
$403$	−5.21527	−0.259791
$404$	0	0
$405$	−3.07350	−0.152724
$406$	0	0
$407$	−17.7227	−0.878482
$408$	0	0
$409$	−26.3946	−1.30513	−0.652565	−	0.757733i	$-0.726307\pi$
$409$	−26.3946	−1.30513	−0.652565	+	0.757733i	$0.726307\pi$
$410$	0	0
$411$	0.621544	0.0306585
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−0.951559	−0.0467102
$416$	0	0
$417$	0.430719	0.0210924
$418$	0	0
$419$	−27.1902	−1.32833	−0.664164	−	0.747587i	$-0.731212\pi$
$419$	−27.1902	−1.32833	−0.664164	+	0.747587i	$0.731212\pi$
$420$	0	0
$421$	23.7404	1.15703	0.578517	−	0.815670i	$-0.303632\pi$
$421$	23.7404	1.15703	0.578517	+	0.815670i	$0.303632\pi$
$422$	0	0
$423$	−29.1689	−1.41824
$424$	0	0
$425$	14.8074	0.718264
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0.483653	0.0233510
$430$	0	0
$431$	26.4493	1.27402	0.637009	−	0.770856i	$-0.280171\pi$
$431$	26.4493	1.27402	0.637009	+	0.770856i	$0.280171\pi$
$432$	0	0
$433$	38.0388	1.82803	0.914014	−	0.405684i	$-0.132966\pi$
$433$	38.0388	1.82803	0.914014	+	0.405684i	$0.132966\pi$
$434$	0	0
$435$	−0.0135117	−0.000647838 0
$436$	0	0
$437$	−2.17689	−0.104135
$438$	0	0
$439$	−24.5018	−1.16941	−0.584705	−	0.811246i	$-0.698790\pi$
$439$	−24.5018	−1.16941	−0.584705	+	0.811246i	$0.698790\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	6.67050	0.316925	0.158463	−	0.987365i	$-0.449346\pi$
$443$	6.67050	0.316925	0.158463	+	0.987365i	$0.449346\pi$
$444$	0	0
$445$	3.83830	0.181953
$446$	0	0
$447$	0.308170	0.0145759
$448$	0	0
$449$	22.6007	1.06659	0.533296	−	0.845928i	$-0.320953\pi$
$449$	22.6007	1.06659	0.533296	+	0.845928i	$0.320953\pi$
$450$	0	0
$451$	14.7522	0.694652
$452$	0	0
$453$	−0.424112	−0.0199265
$454$	0	0
$455$	0	0
$456$	0	0
$457$	41.2215	1.92826	0.964130	−	0.265431i	$-0.0855141\pi$
$457$	41.2215	1.92826	0.964130	+	0.265431i	$0.0855141\pi$
$458$	0	0
$459$	−0.930219	−0.0434189
$460$	0	0
$461$	−21.8689	−1.01854	−0.509269	−	0.860608i	$-0.670084\pi$
$461$	−21.8689	−1.01854	−0.509269	+	0.860608i	$0.670084\pi$
$462$	0	0
$463$	−18.6568	−0.867055	−0.433527	−	0.901140i	$-0.642731\pi$
$463$	−18.6568	−0.867055	−0.433527	+	0.901140i	$0.642731\pi$
$464$	0	0
$465$	−0.0199927	−0.000927140 0
$466$	0	0
$467$	−38.6977	−1.79072	−0.895359	−	0.445345i	$-0.853081\pi$
$467$	−38.6977	−1.79072	−0.895359	+	0.445345i	$0.853081\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	1.08971	0.0502110
$472$	0	0
$473$	−8.06293	−0.370734
$474$	0	0
$475$	−29.1334	−1.33673
$476$	0	0
$477$	−10.8642	−0.497440
$478$	0	0
$479$	−9.04999	−0.413505	−0.206752	−	0.978393i	$-0.566289\pi$
$479$	−9.04999	−0.413505	−0.206752	+	0.978393i	$0.566289\pi$
$480$	0	0
$481$	39.1108	1.78330
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1.37106	0.0622565
$486$	0	0
$487$	−29.4815	−1.33593	−0.667967	−	0.744191i	$-0.732835\pi$
$487$	−29.4815	−1.33593	−0.667967	+	0.744191i	$0.732835\pi$
$488$	0	0
$489$	0.641851	0.0290255
$490$	0	0
$491$	43.9409	1.98302	0.991512	−	0.130012i	$-0.0415016\pi$
$491$	43.9409	1.98302	0.991512	+	0.130012i	$0.0415016\pi$
$492$	0	0
$493$	2.33983	0.105381
$494$	0	0
$495$	−2.12446	−0.0954875
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−22.3222	−0.999280	−0.499640	−	0.866233i	$-0.666534\pi$
$499$	−22.3222	−0.999280	−0.499640	+	0.866233i	$0.666534\pi$
$500$	0	0
$501$	0.425106	0.0189923
$502$	0	0
$503$	−36.1482	−1.61177	−0.805885	−	0.592073i	$-0.798310\pi$
$503$	−36.1482	−1.61177	−0.805885	+	0.592073i	$0.798310\pi$
$504$	0	0
$505$	3.62304	0.161223
$506$	0	0
$507$	−0.402437	−0.0178729
$508$	0	0
$509$	−16.3922	−0.726570	−0.363285	−	0.931678i	$-0.618345\pi$
$509$	−16.3922	−0.726570	−0.363285	+	0.931678i	$0.618345\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	1.83020	0.0808052
$514$	0	0
$515$	−2.25574	−0.0993997
$516$	0	0
$517$	−20.1445	−0.885954
$518$	0	0
$519$	0.945165	0.0414881
$520$	0	0
$521$	11.8391	0.518679	0.259339	−	0.965786i	$-0.416495\pi$
$521$	11.8391	0.518679	0.259339	+	0.965786i	$0.416495\pi$
$522$	0	0
$523$	−27.0376	−1.18227	−0.591135	−	0.806573i	$-0.701320\pi$
$523$	−27.0376	−1.18227	−0.591135	+	0.806573i	$0.701320\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3.46214	0.150813
$528$	0	0
$529$	−22.8669	−0.994212
$530$	0	0
$531$	40.3507	1.75107
$532$	0	0
$533$	−32.5553	−1.41013
$534$	0	0
$535$	−1.92233	−0.0831095
$536$	0	0
$537$	0.762872	0.0329204
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−11.9847	−0.515262	−0.257631	−	0.966243i	$-0.582942\pi$
$541$	−11.9847	−0.515262	−0.257631	+	0.966243i	$0.582942\pi$
$542$	0	0
$543$	−0.197086	−0.00845778
$544$	0	0
$545$	−0.646024	−0.0276726
$546$	0	0
$547$	16.0040	0.684284	0.342142	−	0.939648i	$-0.388848\pi$
$547$	16.0040	0.684284	0.342142	+	0.939648i	$0.388848\pi$
$548$	0	0
$549$	37.7497	1.61112
$550$	0	0
$551$	−4.60359	−0.196120
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0.149931	0.00636421
$556$	0	0
$557$	16.4355	0.696394	0.348197	−	0.937421i	$-0.386794\pi$
$557$	16.4355	0.696394	0.348197	+	0.937421i	$0.386794\pi$
$558$	0	0
$559$	17.7934	0.752581
$560$	0	0
$561$	−0.321072	−0.0135557
$562$	0	0
$563$	−13.0302	−0.549159	−0.274580	−	0.961564i	$-0.588539\pi$
$563$	−13.0302	−0.549159	−0.274580	+	0.961564i	$0.588539\pi$
$564$	0	0
$565$	1.48476	0.0624645
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−40.2493	−1.68734	−0.843668	−	0.536865i	$-0.819608\pi$
$569$	−40.2493	−1.68734	−0.843668	+	0.536865i	$0.819608\pi$
$570$	0	0
$571$	−35.3843	−1.48079	−0.740394	−	0.672173i	$-0.765361\pi$
$571$	−35.3843	−1.48079	−0.740394	+	0.672173i	$0.765361\pi$
$572$	0	0
$573$	0.970699	0.0405515
$574$	0	0
$575$	1.78146	0.0742921
$576$	0	0
$577$	−15.3580	−0.639364	−0.319682	−	0.947525i	$-0.603576\pi$
$577$	−15.3580	−0.639364	−0.319682	+	0.947525i	$0.603576\pi$
$578$	0	0
$579$	−0.396134	−0.0164628
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−7.50301	−0.310743
$584$	0	0
$585$	4.68830	0.193837
$586$	0	0
$587$	1.66914	0.0688928	0.0344464	−	0.999407i	$-0.489033\pi$
$587$	1.66914	0.0688928	0.0344464	+	0.999407i	$0.489033\pi$
$588$	0	0
$589$	−6.81173	−0.280672
$590$	0	0
$591$	0.415441	0.0170890
$592$	0	0
$593$	42.6294	1.75058	0.875289	−	0.483600i	$-0.160671\pi$
$593$	42.6294	1.75058	0.875289	+	0.483600i	$0.160671\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0.693639	0.0283888
$598$	0	0
$599$	36.5904	1.49504	0.747522	−	0.664237i	$-0.231243\pi$
$599$	36.5904	1.49504	0.747522	+	0.664237i	$0.231243\pi$
$600$	0	0
$601$	14.7372	0.601143	0.300571	−	0.953759i	$-0.402823\pi$
$601$	14.7372	0.601143	0.300571	+	0.953759i	$0.402823\pi$
$602$	0	0
$603$	−20.9376	−0.852645
$604$	0	0
$605$	2.29917	0.0934744
$606$	0	0
$607$	−36.8630	−1.49622	−0.748111	−	0.663573i	$-0.769039\pi$
$607$	−36.8630	−1.49622	−0.748111	+	0.663573i	$0.769039\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	44.4552	1.79847
$612$	0	0
$613$	−37.0948	−1.49824	−0.749122	−	0.662432i	$-0.769524\pi$
$613$	−37.0948	−1.49824	−0.749122	+	0.662432i	$0.769524\pi$
$614$	0	0
$615$	−0.124801	−0.00503245
$616$	0	0
$617$	14.9852	0.603279	0.301640	−	0.953422i	$-0.402466\pi$
$617$	14.9852	0.603279	0.301640	+	0.953422i	$0.402466\pi$
$618$	0	0
$619$	21.9345	0.881621	0.440811	−	0.897600i	$-0.354691\pi$
$619$	21.9345	0.881621	0.440811	+	0.897600i	$0.354691\pi$
$620$	0	0
$621$	−0.111914	−0.00449095
$622$	0	0
$623$	0	0
$624$	0	0
$625$	23.2552	0.930209
$626$	0	0
$627$	0.631705	0.0252279
$628$	0	0
$629$	−25.9636	−1.03524
$630$	0	0
$631$	−5.77408	−0.229862	−0.114931	−	0.993373i	$-0.536665\pi$
$631$	−5.77408	−0.229862	−0.114931	+	0.993373i	$0.536665\pi$
$632$	0	0
$633$	1.30215	0.0517560
$634$	0	0
$635$	−4.32210	−0.171517
$636$	0	0
$637$	0	0
$638$	0	0
$639$	40.3836	1.59755
$640$	0	0
$641$	−19.9157	−0.786621	−0.393311	−	0.919406i	$-0.628670\pi$
$641$	−19.9157	−0.786621	−0.393311	+	0.919406i	$0.628670\pi$
$642$	0	0
$643$	10.0697	0.397112	0.198556	−	0.980090i	$-0.436375\pi$
$643$	10.0697	0.397112	0.198556	+	0.980090i	$0.436375\pi$
$644$	0	0
$645$	0.0682109	0.00268580
$646$	0	0
$647$	−3.16255	−0.124333	−0.0621663	−	0.998066i	$-0.519801\pi$
$647$	−3.16255	−0.124333	−0.0621663	+	0.998066i	$0.519801\pi$
$648$	0	0
$649$	27.8668	1.09387
$650$	0	0
$651$	0	0
$652$	0	0
$653$	22.4828	0.879821	0.439910	−	0.898042i	$-0.355010\pi$
$653$	22.4828	0.879821	0.439910	+	0.898042i	$0.355010\pi$
$654$	0	0
$655$	−4.06859	−0.158973
$656$	0	0
$657$	25.1950	0.982950
$658$	0	0
$659$	−18.4940	−0.720426	−0.360213	−	0.932870i	$-0.617296\pi$
$659$	−18.4940	−0.720426	−0.360213	+	0.932870i	$0.617296\pi$
$660$	0	0
$661$	−25.1019	−0.976350	−0.488175	−	0.872746i	$-0.662337\pi$
$661$	−25.1019	−0.976350	−0.488175	+	0.872746i	$0.662337\pi$
$662$	0	0
$663$	0.708546	0.0275177
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0.281503	0.0108998
$668$	0	0
$669$	−0.592384	−0.0229029
$670$	0	0
$671$	26.0705	1.00644
$672$	0	0
$673$	−23.6027	−0.909816	−0.454908	−	0.890538i	$-0.650328\pi$
$673$	−23.6027	−0.909816	−0.454908	+	0.890538i	$0.650328\pi$
$674$	0	0
$675$	−1.49775	−0.0576483
$676$	0	0
$677$	−14.0872	−0.541413	−0.270707	−	0.962662i	$-0.587257\pi$
$677$	−14.0872	−0.541413	−0.270707	+	0.962662i	$0.587257\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0.0660590	0.00253139
$682$	0	0
$683$	−9.53028	−0.364666	−0.182333	−	0.983237i	$-0.558365\pi$
$683$	−9.53028	−0.364666	−0.182333	+	0.983237i	$0.558365\pi$
$684$	0	0
$685$	4.16092	0.158981
$686$	0	0
$687$	0.477041	0.0182002
$688$	0	0
$689$	16.5578	0.630801
$690$	0	0
$691$	−33.4538	−1.27264	−0.636321	−	0.771425i	$-0.719545\pi$
$691$	−33.4538	−1.27264	−0.636321	+	0.771425i	$0.719545\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	2.88344	0.109375
$696$	0	0
$697$	21.6118	0.818603
$698$	0	0
$699$	−0.827682	−0.0313058
$700$	0	0
$701$	31.8333	1.20233	0.601164	−	0.799125i	$-0.294704\pi$
$701$	31.8333	1.20233	0.601164	+	0.799125i	$0.294704\pi$
$702$	0	0
$703$	51.0831	1.92663
$704$	0	0
$705$	0.170419	0.00641835
$706$	0	0
$707$	0	0
$708$	0	0
$709$	24.9975	0.938802	0.469401	−	0.882985i	$-0.344470\pi$
$709$	24.9975	0.938802	0.469401	+	0.882985i	$0.344470\pi$
$710$	0	0
$711$	−9.43396	−0.353801
$712$	0	0
$713$	0.416527	0.0155991
$714$	0	0
$715$	3.23781	0.121087
$716$	0	0
$717$	−0.846376	−0.0316085
$718$	0	0
$719$	−18.7817	−0.700438	−0.350219	−	0.936668i	$-0.613893\pi$
$719$	−18.7817	−0.700438	−0.350219	+	0.936668i	$0.613893\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0.967695	0.0359889
$724$	0	0
$725$	3.76737	0.139916
$726$	0	0
$727$	16.3132	0.605021	0.302511	−	0.953146i	$-0.402175\pi$
$727$	16.3132	0.605021	0.302511	+	0.953146i	$0.402175\pi$
$728$	0	0
$729$	−26.8588	−0.994772
$730$	0	0
$731$	−11.8121	−0.436886
$732$	0	0
$733$	−24.4567	−0.903328	−0.451664	−	0.892188i	$-0.649169\pi$
$733$	−24.4567	−0.903328	−0.451664	+	0.892188i	$0.649169\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−14.4598	−0.532635
$738$	0	0
$739$	36.6708	1.34896	0.674479	−	0.738294i	$-0.264368\pi$
$739$	36.6708	1.34896	0.674479	+	0.738294i	$0.264368\pi$
$740$	0	0
$741$	−1.39406	−0.0512120
$742$	0	0
$743$	7.45052	0.273333	0.136667	−	0.990617i	$-0.456361\pi$
$743$	7.45052	0.273333	0.136667	+	0.990617i	$0.456361\pi$
$744$	0	0
$745$	2.06304	0.0755839
$746$	0	0
$747$	−8.33009	−0.304782
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−33.2124	−1.21194	−0.605969	−	0.795488i	$-0.707214\pi$
$751$	−33.2124	−1.21194	−0.605969	+	0.795488i	$0.707214\pi$
$752$	0	0
$753$	−0.164754	−0.00600395
$754$	0	0
$755$	−2.83921	−0.103329
$756$	0	0
$757$	36.5598	1.32879	0.664395	−	0.747382i	$-0.268689\pi$
$757$	36.5598	1.32879	0.664395	+	0.747382i	$0.268689\pi$
$758$	0	0
$759$	−0.0386278	−0.00140210
$760$	0	0
$761$	46.1954	1.67458	0.837291	−	0.546757i	$-0.184138\pi$
$761$	46.1954	1.67458	0.837291	+	0.546757i	$0.184138\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−3.11231	−0.112526
$766$	0	0
$767$	−61.4969	−2.22052
$768$	0	0
$769$	−19.5242	−0.704060	−0.352030	−	0.935989i	$-0.614509\pi$
$769$	−19.5242	−0.704060	−0.352030	+	0.935989i	$0.614509\pi$
$770$	0	0
$771$	−0.657207	−0.0236687
$772$	0	0
$773$	5.64922	0.203188	0.101594	−	0.994826i	$-0.467606\pi$
$773$	5.64922	0.203188	0.101594	+	0.994826i	$0.467606\pi$
$774$	0	0
$775$	5.57440	0.200238
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−42.5209	−1.52347
$780$	0	0
$781$	27.8895	0.997964
$782$	0	0
$783$	−0.236671	−0.00845793
$784$	0	0
$785$	7.29502	0.260370
$786$	0	0
$787$	30.8459	1.09954	0.549768	−	0.835317i	$-0.314716\pi$
$787$	30.8459	1.09954	0.549768	+	0.835317i	$0.314716\pi$
$788$	0	0
$789$	0.725521	0.0258292
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−57.5328	−2.04305
$794$	0	0
$795$	0.0634741	0.00225119
$796$	0	0
$797$	4.59861	0.162891	0.0814456	−	0.996678i	$-0.474046\pi$
$797$	4.59861	0.162891	0.0814456	+	0.996678i	$0.474046\pi$
$798$	0	0
$799$	−29.5115	−1.04404
$800$	0	0
$801$	33.6011	1.18724
$802$	0	0
$803$	17.4000	0.614034
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−0.0291941	−0.00102768
$808$	0	0
$809$	−29.8037	−1.04784	−0.523921	−	0.851767i	$-0.675531\pi$
$809$	−29.8037	−1.04784	−0.523921	+	0.851767i	$0.675531\pi$
$810$	0	0
$811$	7.25941	0.254912	0.127456	−	0.991844i	$-0.459319\pi$
$811$	7.25941	0.254912	0.127456	+	0.991844i	$0.459319\pi$
$812$	0	0
$813$	−0.945992	−0.0331774
$814$	0	0
$815$	4.29686	0.150512
$816$	0	0
$817$	23.2402	0.813072
$818$	0	0
$819$	0	0
$820$	0	0
$821$	42.3594	1.47835	0.739177	−	0.673512i	$-0.235215\pi$
$821$	42.3594	1.47835	0.739177	+	0.673512i	$0.235215\pi$
$822$	0	0
$823$	35.1716	1.22601	0.613003	−	0.790081i	$-0.289962\pi$
$823$	35.1716	1.22601	0.613003	+	0.790081i	$0.289962\pi$
$824$	0	0
$825$	−0.516958	−0.0179982
$826$	0	0
$827$	−19.5363	−0.679343	−0.339671	−	0.940544i	$-0.610316\pi$
$827$	−19.5363	−0.679343	−0.339671	+	0.940544i	$0.610316\pi$
$828$	0	0
$829$	11.2043	0.389141	0.194570	−	0.980889i	$-0.437669\pi$
$829$	11.2043	0.389141	0.194570	+	0.980889i	$0.437669\pi$
$830$	0	0
$831$	−0.216115	−0.00749693
$832$	0	0
$833$	0	0
$834$	0	0
$835$	2.84587	0.0984853
$836$	0	0
$837$	−0.350191	−0.0121044
$838$	0	0
$839$	−47.0494	−1.62433	−0.812164	−	0.583430i	$-0.801710\pi$
$839$	−47.0494	−1.62433	−0.812164	+	0.583430i	$0.801710\pi$
$840$	0	0
$841$	−28.4047	−0.979472
$842$	0	0
$843$	0.433006	0.0149135
$844$	0	0
$845$	−2.69411	−0.0926801
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0.554141	0.0190181
$850$	0	0
$851$	−3.12365	−0.107077
$852$	0	0
$853$	37.9168	1.29825	0.649124	−	0.760683i	$-0.275136\pi$
$853$	37.9168	1.29825	0.649124	+	0.760683i	$0.275136\pi$
$854$	0	0
$855$	6.12345	0.209417
$856$	0	0
$857$	48.3340	1.65106	0.825529	−	0.564360i	$-0.190877\pi$
$857$	48.3340	1.65106	0.825529	+	0.564360i	$0.190877\pi$
$858$	0	0
$859$	23.7594	0.810661	0.405331	−	0.914170i	$-0.367156\pi$
$859$	23.7594	0.810661	0.405331	+	0.914170i	$0.367156\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−5.09654	−0.173488	−0.0867440	−	0.996231i	$-0.527646\pi$
$863$	−5.09654	−0.173488	−0.0867440	+	0.996231i	$0.527646\pi$
$864$	0	0
$865$	6.32740	0.215138
$866$	0	0
$867$	0.399115	0.0135546
$868$	0	0
$869$	−6.51523	−0.221014
$870$	0	0
$871$	31.9102	1.08124
$872$	0	0
$873$	12.0024	0.406221
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−7.79930	−0.263364	−0.131682	−	0.991292i	$-0.542038\pi$
$877$	−7.79930	−0.263364	−0.131682	+	0.991292i	$0.542038\pi$
$878$	0	0
$879$	−0.528692	−0.0178323
$880$	0	0
$881$	−27.5192	−0.927144	−0.463572	−	0.886059i	$-0.653432\pi$
$881$	−27.5192	−0.927144	−0.463572	+	0.886059i	$0.653432\pi$
$882$	0	0
$883$	42.0423	1.41484	0.707419	−	0.706795i	$-0.249859\pi$
$883$	42.0423	1.41484	0.707419	+	0.706795i	$0.249859\pi$
$884$	0	0
$885$	−0.235748	−0.00792458
$886$	0	0
$887$	15.5806	0.523144	0.261572	−	0.965184i	$-0.415759\pi$
$887$	15.5806	0.523144	0.261572	+	0.965184i	$0.415759\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−18.5816	−0.622508
$892$	0	0
$893$	58.0635	1.94302
$894$	0	0
$895$	5.10704	0.170710
$896$	0	0
$897$	0.0852446	0.00284623
$898$	0	0
$899$	0.880855	0.0293781
$900$	0	0
$901$	−10.9918	−0.366191
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1.31939	−0.0438580
$906$	0	0
$907$	32.2447	1.07067	0.535334	−	0.844640i	$-0.320186\pi$
$907$	32.2447	1.07067	0.535334	+	0.844640i	$0.320186\pi$
$908$	0	0
$909$	31.7166	1.05197
$910$	0	0
$911$	41.5953	1.37811	0.689056	−	0.724708i	$-0.258025\pi$
$911$	41.5953	1.37811	0.689056	+	0.724708i	$0.258025\pi$
$912$	0	0
$913$	−5.75289	−0.190393
$914$	0	0
$915$	−0.220552	−0.00729121
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−31.4125	−1.03620	−0.518101	−	0.855320i	$-0.673361\pi$
$919$	−31.4125	−1.03620	−0.518101	+	0.855320i	$0.673361\pi$
$920$	0	0
$921$	−0.972753	−0.0320533
$922$	0	0
$923$	−61.5470	−2.02584
$924$	0	0
$925$	−41.8040	−1.37451
$926$	0	0
$927$	−19.7471	−0.648579
$928$	0	0
$929$	44.6965	1.46645	0.733223	−	0.679989i	$-0.238015\pi$
$929$	44.6965	1.46645	0.733223	+	0.679989i	$0.238015\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	1.54962	0.0507322
$934$	0	0
$935$	−2.14941	−0.0702932
$936$	0	0
$937$	−2.85775	−0.0933586	−0.0466793	−	0.998910i	$-0.514864\pi$
$937$	−2.85775	−0.0933586	−0.0466793	+	0.998910i	$0.514864\pi$
$938$	0	0
$939$	−1.60457	−0.0523631
$940$	0	0
$941$	−12.1864	−0.397265	−0.198632	−	0.980074i	$-0.563650\pi$
$941$	−12.1864	−0.397265	−0.198632	+	0.980074i	$0.563650\pi$
$942$	0	0
$943$	2.60009	0.0846706
$944$	0	0
$945$	0	0
$946$	0	0
$947$	8.81341	0.286397	0.143199	−	0.989694i	$-0.454261\pi$
$947$	8.81341	0.286397	0.143199	+	0.989694i	$0.454261\pi$
$948$	0	0
$949$	−38.3987	−1.24647
$950$	0	0
$951$	0.191618	0.00621365
$952$	0	0
$953$	55.2630	1.79014	0.895071	−	0.445923i	$-0.147125\pi$
$953$	55.2630	1.79014	0.895071	+	0.445923i	$0.147125\pi$
$954$	0	0
$955$	6.49833	0.210281
$956$	0	0
$957$	−0.0816886	−0.00264062
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−29.6966	−0.957956
$962$	0	0
$963$	−16.8284	−0.542286
$964$	0	0
$965$	−2.65191	−0.0853681
$966$	0	0
$967$	−3.33608	−0.107281	−0.0536406	−	0.998560i	$-0.517083\pi$
$967$	−3.33608	−0.107281	−0.0536406	+	0.998560i	$0.517083\pi$
$968$	0	0
$969$	0.925441	0.0297294
$970$	0	0
$971$	26.7478	0.858377	0.429189	−	0.903215i	$-0.358800\pi$
$971$	26.7478	0.858377	0.429189	+	0.903215i	$0.358800\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	1.14083	0.0365358
$976$	0	0
$977$	12.6696	0.405337	0.202668	−	0.979247i	$-0.435039\pi$
$977$	12.6696	0.405337	0.202668	+	0.979247i	$0.435039\pi$
$978$	0	0
$979$	23.2054	0.741648
$980$	0	0
$981$	−5.65539	−0.180563
$982$	0	0
$983$	33.9244	1.08202	0.541010	−	0.841016i	$-0.318042\pi$
$983$	33.9244	1.08202	0.541010	+	0.841016i	$0.318042\pi$
$984$	0	0
$985$	2.78117	0.0886153
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1.42110	−0.0451885
$990$	0	0
$991$	51.9178	1.64922	0.824611	−	0.565700i	$-0.191394\pi$
$991$	51.9178	1.64922	0.824611	+	0.565700i	$0.191394\pi$
$992$	0	0
$993$	−0.235599	−0.00747650
$994$	0	0
$995$	4.64356	0.147211
$996$	0	0
$997$	21.6197	0.684703	0.342351	−	0.939572i	$-0.388777\pi$
$997$	21.6197	0.684703	0.342351	+	0.939572i	$0.388777\pi$
$998$	0	0
$999$	2.62618	0.0830887

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5488.2.a.s.1.5		9
4.3	odd	2		2744.2.a.d.1.5	✓	9
7.6	odd	2		5488.2.a.t.1.5		9
28.27	even	2		2744.2.a.e.1.5	yes	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2744.2.a.d.1.5	✓	9	4.3	odd	2
2744.2.a.e.1.5	yes	9	28.27	even	2
5488.2.a.s.1.5		9	1.1	even	1	trivial
5488.2.a.t.1.5		9	7.6	odd	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:	\( N \)	\(=\)	\( 5488 = 2^{4} \cdot 7^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5488.a (trivial)

\(f(q)\)	\(=\)	\(q-0.0511459 q^{3} -0.342396 q^{5} -2.99738 q^{9} -2.07004 q^{11} +4.56820 q^{13} +0.0175121 q^{15} -3.03258 q^{17} +5.96658 q^{19} -0.364847 q^{23} -4.88277 q^{25} +0.306742 q^{27} -0.771564 q^{29} -1.14165 q^{31} +0.105874 q^{33} +8.56154 q^{37} -0.233645 q^{39} -7.12652 q^{41} +3.89506 q^{43} +1.02629 q^{45} +9.73147 q^{47} +0.155104 q^{51} +3.62458 q^{53} +0.708772 q^{55} -0.305166 q^{57} -13.4620 q^{59} -12.5942 q^{61} -1.56413 q^{65} +6.98529 q^{67} +0.0186604 q^{69} -13.4729 q^{71} -8.40566 q^{73} +0.249734 q^{75} +3.14740 q^{79} +8.97646 q^{81} +2.77912 q^{83} +1.03834 q^{85} +0.0394624 q^{87} -11.2101 q^{89} +0.0583907 q^{93} -2.04293 q^{95} -4.00431 q^{97} +6.20470 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 13 q^{5} + 13 q^{9} - 3 q^{11} - 20 q^{13} - 3 q^{15} - 10 q^{17} + 10 q^{19} + 4 q^{23} + 22 q^{25} + 3 q^{29} + 10 q^{31} - 7 q^{33} + 6 q^{37} + 20 q^{39} - 13 q^{41} + 8 q^{43} - 52 q^{45} - 10 q^{47}+ \cdots + 14 q^{99}+O(q^{100}) \) 9 * q - 13 * q^5 + 13 * q^9 - 3 * q^11 - 20 * q^13 - 3 * q^15 - 10 * q^17 + 10 * q^19 + 4 * q^23 + 22 * q^25 + 3 * q^29 + 10 * q^31 - 7 * q^33 + 6 * q^37 + 20 * q^39 - 13 * q^41 + 8 * q^43 - 52 * q^45 - 10 * q^47 + 23 * q^51 - 8 * q^53 + 25 * q^55 - q^57 - 7 * q^59 - 52 * q^61 + 28 * q^65 + 9 * q^67 - 42 * q^69 - 9 * q^71 - 23 * q^73 + 7 * q^75 - 10 * q^79 - 3 * q^81 - 18 * q^83 - 5 * q^85 + 7 * q^87 - 17 * q^89 - 53 * q^93 - 25 * q^95 - 24 * q^97 + 14 * q^99

Introduction

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