LMFDB - Embedded newform 1184.2.a.p.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1184,2,Mod(1,1184)] 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("1184.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1184, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1184 = 2^{5} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1184.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,-2,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$9.45428759932$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 20x^{6} + 116x^{4} - 221x^{2} + 128 $ x^8 - 20x^6 + 116x^4 - 221*x^2 + 128
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$1.34313$ of defining polynomial
Character	$\chi$	$=$	1184.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.34313 q^{3} +3.42359 q^{5} +1.69567 q^{7} -1.19601 q^{9} -3.25519 q^{11} +2.67104 q^{13} +4.59832 q^{15} +2.00000 q^{17} +6.95547 q^{19} +2.27751 q^{21} -7.30801 q^{23} +6.72097 q^{25} -5.63578 q^{27} +2.67104 q^{29} +2.70969 q^{31} -4.37214 q^{33} +5.80529 q^{35} +1.00000 q^{37} +3.58755 q^{39} +8.94855 q^{41} +7.66056 q^{43} -4.09463 q^{45} -7.50097 q^{47} -4.12469 q^{49} +2.68626 q^{51} -8.66952 q^{53} -11.1444 q^{55} +9.34209 q^{57} -4.92743 q^{59} +0.576410 q^{61} -2.02804 q^{63} +9.14456 q^{65} -12.7842 q^{67} -9.81560 q^{69} -10.1872 q^{71} +3.80399 q^{73} +9.02712 q^{75} -5.51975 q^{77} -7.30801 q^{79} -3.98155 q^{81} -0.990583 q^{83} +6.84718 q^{85} +3.58755 q^{87} +18.7443 q^{89} +4.52922 q^{91} +3.63947 q^{93} +23.8127 q^{95} +2.00000 q^{97} +3.89323 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{5} + 16 q^{9} + 2 q^{13} + 16 q^{17} + 2 q^{21} + 22 q^{25} + 2 q^{29} + 30 q^{33} + 8 q^{37} + 36 q^{41} + 16 q^{45} + 42 q^{49} - 2 q^{53} + 36 q^{57} + 34 q^{61} + 12 q^{65} + 2 q^{69} + 56 q^{73}+ \cdots + 16 q^{97}+O(q^{100}) $ 8 * q - 2 * q^5 + 16 * q^9 + 2 * q^13 + 16 * q^17 + 2 * q^21 + 22 * q^25 + 2 * q^29 + 30 * q^33 + 8 * q^37 + 36 * q^41 + 16 * q^45 + 42 * q^49 - 2 * q^53 + 36 * q^57 + 34 * q^61 + 12 * q^65 + 2 * q^69 + 56 * q^73 - 14 * q^77 + 48 * q^81 - 4 * q^85 + 20 * q^89 - 12 * q^93 + 16 * q^97

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.34313	0.775456	0.387728	−	0.921774i	$-0.373260\pi$
$3$	1.34313	0.775456	0.387728	+	0.921774i	$0.373260\pi$
$4$	0	0
$5$	3.42359	1.53108	0.765538	−	0.643391i	$-0.222473\pi$
$5$	3.42359	1.53108	0.765538	+	0.643391i	$0.222473\pi$
$6$	0	0
$7$	1.69567	0.640905	0.320452	−	0.947265i	$-0.396165\pi$
$7$	1.69567	0.640905	0.320452	+	0.947265i	$0.396165\pi$
$8$	0	0
$9$	−1.19601	−0.398669
$10$	0	0
$11$	−3.25519	−0.981478	−0.490739	−	0.871307i	$-0.663273\pi$
$11$	−3.25519	−0.981478	−0.490739	+	0.871307i	$0.663273\pi$
$12$	0	0
$13$	2.67104	0.740814	0.370407	−	0.928870i	$-0.379218\pi$
$13$	2.67104	0.740814	0.370407	+	0.928870i	$0.379218\pi$
$14$	0	0
$15$	4.59832	1.18728
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	6.95547	1.59569	0.797847	−	0.602860i	$-0.205972\pi$
$19$	6.95547	1.59569	0.797847	+	0.602860i	$0.205972\pi$
$20$	0	0
$21$	2.27751	0.496993
$22$	0	0
$23$	−7.30801	−1.52383	−0.761913	−	0.647679i	$-0.775740\pi$
$23$	−7.30801	−1.52383	−0.761913	+	0.647679i	$0.775740\pi$
$24$	0	0
$25$	6.72097	1.34419
$26$	0	0
$27$	−5.63578	−1.08461
$28$	0	0
$29$	2.67104	0.496000	0.248000	−	0.968760i	$-0.420227\pi$
$29$	2.67104	0.496000	0.248000	+	0.968760i	$0.420227\pi$
$30$	0	0
$31$	2.70969	0.486675	0.243338	−	0.969942i	$-0.421758\pi$
$31$	2.70969	0.486675	0.243338	+	0.969942i	$0.421758\pi$
$32$	0	0
$33$	−4.37214	−0.761092
$34$	0	0
$35$	5.80529	0.981274
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	0	0
$39$	3.58755	0.574469
$40$	0	0
$41$	8.94855	1.39753	0.698765	−	0.715352i	$-0.253734\pi$
$41$	8.94855	1.39753	0.698765	+	0.715352i	$0.253734\pi$
$42$	0	0
$43$	7.66056	1.16822	0.584112	−	0.811673i	$-0.301443\pi$
$43$	7.66056	1.16822	0.584112	+	0.811673i	$0.301443\pi$
$44$	0	0
$45$	−4.09463	−0.610392
$46$	0	0
$47$	−7.50097	−1.09413	−0.547064	−	0.837091i	$-0.684255\pi$
$47$	−7.50097	−1.09413	−0.547064	+	0.837091i	$0.684255\pi$
$48$	0	0
$49$	−4.12469	−0.589241
$50$	0	0
$51$	2.68626	0.376151
$52$	0	0
$53$	−8.66952	−1.19085	−0.595425	−	0.803411i	$-0.703016\pi$
$53$	−8.66952	−1.19085	−0.595425	+	0.803411i	$0.703016\pi$
$54$	0	0
$55$	−11.1444	−1.50272
$56$	0	0
$57$	9.34209	1.23739
$58$	0	0
$59$	−4.92743	−0.641497	−0.320748	−	0.947164i	$-0.603934\pi$
$59$	−4.92743	−0.641497	−0.320748	+	0.947164i	$0.603934\pi$
$60$	0	0
$61$	0.576410	0.0738017	0.0369009	−	0.999319i	$-0.488251\pi$
$61$	0.576410	0.0738017	0.0369009	+	0.999319i	$0.488251\pi$
$62$	0	0
$63$	−2.02804	−0.255509
$64$	0	0
$65$	9.14456	1.13424
$66$	0	0
$67$	−12.7842	−1.56184	−0.780919	−	0.624632i	$-0.785249\pi$
$67$	−12.7842	−1.56184	−0.780919	+	0.624632i	$0.785249\pi$
$68$	0	0
$69$	−9.81560	−1.18166
$70$	0	0
$71$	−10.1872	−1.20900	−0.604501	−	0.796605i	$-0.706627\pi$
$71$	−10.1872	−1.20900	−0.604501	+	0.796605i	$0.706627\pi$
$72$	0	0
$73$	3.80399	0.445224	0.222612	−	0.974907i	$-0.428542\pi$
$73$	3.80399	0.445224	0.222612	+	0.974907i	$0.428542\pi$
$74$	0	0
$75$	9.02712	1.04236
$76$	0	0
$77$	−5.51975	−0.629034
$78$	0	0
$79$	−7.30801	−0.822216	−0.411108	−	0.911587i	$-0.634858\pi$
$79$	−7.30801	−0.822216	−0.411108	+	0.911587i	$0.634858\pi$
$80$	0	0
$81$	−3.98155	−0.442395
$82$	0	0
$83$	−0.990583	−0.108731	−0.0543653	−	0.998521i	$-0.517314\pi$
$83$	−0.990583	−0.108731	−0.0543653	+	0.998521i	$0.517314\pi$
$84$	0	0
$85$	6.84718	0.742681
$86$	0	0
$87$	3.58755	0.384626
$88$	0	0
$89$	18.7443	1.98689	0.993445	−	0.114310i	$-0.0364659\pi$
$89$	18.7443	1.98689	0.993445	+	0.114310i	$0.0364659\pi$
$90$	0	0
$91$	4.52922	0.474791
$92$	0	0
$93$	3.63947	0.377395
$94$	0	0
$95$	23.8127	2.44313
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	0	0
$99$	3.89323	0.391284
$100$	0	0
$101$	−1.43033	−0.142323	−0.0711615	−	0.997465i	$-0.522671\pi$
$101$	−1.43033	−0.142323	−0.0711615	+	0.997465i	$0.522671\pi$
$102$	0	0
$103$	15.7305	1.54997	0.774984	−	0.631980i	$-0.217758\pi$
$103$	15.7305	1.54997	0.774984	+	0.631980i	$0.217758\pi$
$104$	0	0
$105$	7.79726	0.760934
$106$	0	0
$107$	−9.11641	−0.881316	−0.440658	−	0.897675i	$-0.645255\pi$
$107$	−9.11641	−0.881316	−0.440658	+	0.897675i	$0.645255\pi$
$108$	0	0
$109$	−9.23919	−0.884954	−0.442477	−	0.896780i	$-0.645900\pi$
$109$	−9.23919	−0.884954	−0.442477	+	0.896780i	$0.645900\pi$
$110$	0	0
$111$	1.34313	0.127484
$112$	0	0
$113$	0.455168	0.0428186	0.0214093	−	0.999771i	$-0.493185\pi$
$113$	0.455168	0.0428186	0.0214093	+	0.999771i	$0.493185\pi$
$114$	0	0
$115$	−25.0196	−2.33309
$116$	0	0
$117$	−3.19458	−0.295339
$118$	0	0
$119$	3.39135	0.310884
$120$	0	0
$121$	−0.403720	−0.0367019
$122$	0	0
$123$	12.0191	1.08372
$124$	0	0
$125$	5.89189	0.526987
$126$	0	0
$127$	13.5786	1.20490	0.602452	−	0.798155i	$-0.294191\pi$
$127$	13.5786	1.20490	0.602452	+	0.798155i	$0.294191\pi$
$128$	0	0
$129$	10.2891	0.905906
$130$	0	0
$131$	−0.445082	−0.0388870	−0.0194435	−	0.999811i	$-0.506189\pi$
$131$	−0.445082	−0.0388870	−0.0194435	+	0.999811i	$0.506189\pi$
$132$	0	0
$133$	11.7942	1.02269
$134$	0	0
$135$	−19.2946	−1.66061
$136$	0	0
$137$	19.6129	1.67564	0.837820	−	0.545947i	$-0.183830\pi$
$137$	19.6129	1.67564	0.837820	+	0.545947i	$0.183830\pi$
$138$	0	0
$139$	−16.9497	−1.43766	−0.718829	−	0.695187i	$-0.755322\pi$
$139$	−16.9497	−1.43766	−0.718829	+	0.695187i	$0.755322\pi$
$140$	0	0
$141$	−10.0748	−0.848448
$142$	0	0
$143$	−8.69476	−0.727092
$144$	0	0
$145$	9.14456	0.759414
$146$	0	0
$147$	−5.53999	−0.456930
$148$	0	0
$149$	7.51670	0.615792	0.307896	−	0.951420i	$-0.400375\pi$
$149$	7.51670	0.615792	0.307896	+	0.951420i	$0.400375\pi$
$150$	0	0
$151$	−15.2742	−1.24300	−0.621500	−	0.783414i	$-0.713477\pi$
$151$	−15.2742	−1.24300	−0.621500	+	0.783414i	$0.713477\pi$
$152$	0	0
$153$	−2.39201	−0.193383
$154$	0	0
$155$	9.27688	0.745137
$156$	0	0
$157$	−4.11450	−0.328373	−0.164187	−	0.986429i	$-0.552500\pi$
$157$	−4.11450	−0.328373	−0.164187	+	0.986429i	$0.552500\pi$
$158$	0	0
$159$	−11.6443	−0.923451
$160$	0	0
$161$	−12.3920	−0.976627
$162$	0	0
$163$	18.8852	1.47921	0.739603	−	0.673043i	$-0.235013\pi$
$163$	18.8852	1.47921	0.739603	+	0.673043i	$0.235013\pi$
$164$	0	0
$165$	−14.9684	−1.16529
$166$	0	0
$167$	12.5646	0.972275	0.486137	−	0.873882i	$-0.338406\pi$
$167$	12.5646	0.972275	0.486137	+	0.873882i	$0.338406\pi$
$168$	0	0
$169$	−5.86553	−0.451194
$170$	0	0
$171$	−8.31878	−0.636153
$172$	0	0
$173$	−24.2640	−1.84476	−0.922380	−	0.386284i	$-0.873759\pi$
$173$	−24.2640	−1.84476	−0.922380	+	0.386284i	$0.873759\pi$
$174$	0	0
$175$	11.3966	0.861500
$176$	0	0
$177$	−6.61817	−0.497452
$178$	0	0
$179$	−2.28805	−0.171017	−0.0855083	−	0.996337i	$-0.527251\pi$
$179$	−2.28805	−0.171017	−0.0855083	+	0.996337i	$0.527251\pi$
$180$	0	0
$181$	−12.8588	−0.955786	−0.477893	−	0.878418i	$-0.658599\pi$
$181$	−12.8588	−0.955786	−0.477893	+	0.878418i	$0.658599\pi$
$182$	0	0
$183$	0.774193	0.0572300
$184$	0	0
$185$	3.42359	0.251707
$186$	0	0
$187$	−6.51039	−0.476087
$188$	0	0
$189$	−9.55644	−0.695129
$190$	0	0
$191$	−2.43738	−0.176363	−0.0881813	−	0.996104i	$-0.528105\pi$
$191$	−2.43738	−0.176363	−0.0881813	+	0.996104i	$0.528105\pi$
$192$	0	0
$193$	−18.1893	−1.30929	−0.654646	−	0.755936i	$-0.727182\pi$
$193$	−18.1893	−1.30929	−0.654646	+	0.755936i	$0.727182\pi$
$194$	0	0
$195$	12.2823	0.879555
$196$	0	0
$197$	−3.33048	−0.237287	−0.118643	−	0.992937i	$-0.537855\pi$
$197$	−3.33048	−0.237287	−0.118643	+	0.992937i	$0.537855\pi$
$198$	0	0
$199$	7.61369	0.539720	0.269860	−	0.962900i	$-0.413023\pi$
$199$	7.61369	0.539720	0.269860	+	0.962900i	$0.413023\pi$
$200$	0	0
$201$	−17.1708	−1.21114
$202$	0	0
$203$	4.52922	0.317889
$204$	0	0
$205$	30.6362	2.13972
$206$	0	0
$207$	8.74043	0.607502
$208$	0	0
$209$	−22.6414	−1.56614
$210$	0	0
$211$	13.7748	0.948295	0.474147	−	0.880445i	$-0.342756\pi$
$211$	13.7748	0.948295	0.474147	+	0.880445i	$0.342756\pi$
$212$	0	0
$213$	−13.6828	−0.937527
$214$	0	0
$215$	26.2266	1.78864
$216$	0	0
$217$	4.59476	0.311912
$218$	0	0
$219$	5.10925	0.345251
$220$	0	0
$221$	5.34209	0.359348
$222$	0	0
$223$	−7.50097	−0.502302	−0.251151	−	0.967948i	$-0.580809\pi$
$223$	−7.50097	−0.502302	−0.251151	+	0.967948i	$0.580809\pi$
$224$	0	0
$225$	−8.03832	−0.535888
$226$	0	0
$227$	2.28805	0.151863	0.0759315	−	0.997113i	$-0.475807\pi$
$227$	2.28805	0.151863	0.0759315	+	0.997113i	$0.475807\pi$
$228$	0	0
$229$	−8.46678	−0.559500	−0.279750	−	0.960073i	$-0.590252\pi$
$229$	−8.46678	−0.559500	−0.279750	+	0.960073i	$0.590252\pi$
$230$	0	0
$231$	−7.41373	−0.487788
$232$	0	0
$233$	−18.0732	−1.18402	−0.592009	−	0.805932i	$-0.701665\pi$
$233$	−18.0732	−1.18402	−0.592009	+	0.805932i	$0.701665\pi$
$234$	0	0
$235$	−25.6802	−1.67519
$236$	0	0
$237$	−9.81560	−0.637592
$238$	0	0
$239$	0.409344	0.0264783	0.0132391	−	0.999912i	$-0.495786\pi$
$239$	0.409344	0.0264783	0.0132391	+	0.999912i	$0.495786\pi$
$240$	0	0
$241$	5.04992	0.325294	0.162647	−	0.986684i	$-0.447997\pi$
$241$	5.04992	0.325294	0.162647	+	0.986684i	$0.447997\pi$
$242$	0	0
$243$	11.5596	0.741548
$244$	0	0
$245$	−14.1212	−0.902173
$246$	0	0
$247$	18.5784	1.18211
$248$	0	0
$249$	−1.33048	−0.0843157
$250$	0	0
$251$	−24.9160	−1.57268	−0.786341	−	0.617793i	$-0.788027\pi$
$251$	−24.9160	−1.57268	−0.786341	+	0.617793i	$0.788027\pi$
$252$	0	0
$253$	23.7890	1.49560
$254$	0	0
$255$	9.19664	0.575916
$256$	0	0
$257$	−19.6944	−1.22850	−0.614250	−	0.789111i	$-0.710541\pi$
$257$	−19.6944	−1.22850	−0.614250	+	0.789111i	$0.710541\pi$
$258$	0	0
$259$	1.69567	0.105364
$260$	0	0
$261$	−3.19458	−0.197740
$262$	0	0
$263$	7.82015	0.482211	0.241106	−	0.970499i	$-0.422490\pi$
$263$	7.82015	0.482211	0.241106	+	0.970499i	$0.422490\pi$
$264$	0	0
$265$	−29.6809	−1.82328
$266$	0	0
$267$	25.1760	1.54075
$268$	0	0
$269$	−25.9835	−1.58424	−0.792120	−	0.610365i	$-0.791023\pi$
$269$	−25.9835	−1.58424	−0.792120	+	0.610365i	$0.791023\pi$
$270$	0	0
$271$	−2.40077	−0.145836	−0.0729181	−	0.997338i	$-0.523231\pi$
$271$	−2.40077	−0.145836	−0.0729181	+	0.997338i	$0.523231\pi$
$272$	0	0
$273$	6.08332	0.368180
$274$	0	0
$275$	−21.8780	−1.31930
$276$	0	0
$277$	−1.13143	−0.0679809	−0.0339904	−	0.999422i	$-0.510822\pi$
$277$	−1.13143	−0.0679809	−0.0339904	+	0.999422i	$0.510822\pi$
$278$	0	0
$279$	−3.24081	−0.194022
$280$	0	0
$281$	21.4285	1.27831	0.639157	−	0.769076i	$-0.279283\pi$
$281$	21.4285	1.27831	0.639157	+	0.769076i	$0.279283\pi$
$282$	0	0
$283$	1.19705	0.0711570	0.0355785	−	0.999367i	$-0.488673\pi$
$283$	1.19705	0.0711570	0.0355785	+	0.999367i	$0.488673\pi$
$284$	0	0
$285$	31.9835	1.89454
$286$	0	0
$287$	15.1738	0.895683
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	2.68626	0.157471
$292$	0	0
$293$	−28.1261	−1.64314	−0.821572	−	0.570104i	$-0.806903\pi$
$293$	−28.1261	−1.64314	−0.821572	+	0.570104i	$0.806903\pi$
$294$	0	0
$295$	−16.8695	−0.982181
$296$	0	0
$297$	18.3455	1.06452
$298$	0	0
$299$	−19.5200	−1.12887
$300$	0	0
$301$	12.9898	0.748720
$302$	0	0
$303$	−1.92112	−0.110365
$304$	0	0
$305$	1.97339	0.112996
$306$	0	0
$307$	26.4787	1.51122	0.755611	−	0.655021i	$-0.227340\pi$
$307$	26.4787	1.51122	0.755611	+	0.655021i	$0.227340\pi$
$308$	0	0
$309$	21.1280	1.20193
$310$	0	0
$311$	−17.8276	−1.01091	−0.505455	−	0.862853i	$-0.668675\pi$
$311$	−17.8276	−1.01091	−0.505455	+	0.862853i	$0.668675\pi$
$312$	0	0
$313$	−8.49186	−0.479988	−0.239994	−	0.970774i	$-0.577146\pi$
$313$	−8.49186	−0.479988	−0.239994	+	0.970774i	$0.577146\pi$
$314$	0	0
$315$	−6.94317	−0.391203
$316$	0	0
$317$	−8.49491	−0.477122	−0.238561	−	0.971128i	$-0.576676\pi$
$317$	−8.49491	−0.477122	−0.238561	+	0.971128i	$0.576676\pi$
$318$	0	0
$319$	−8.69476	−0.486813
$320$	0	0
$321$	−12.2445	−0.683422
$322$	0	0
$323$	13.9109	0.774025
$324$	0	0
$325$	17.9520	0.995798
$326$	0	0
$327$	−12.4094	−0.686243
$328$	0	0
$329$	−12.7192	−0.701232
$330$	0	0
$331$	7.38825	0.406095	0.203047	−	0.979169i	$-0.434915\pi$
$331$	7.38825	0.406095	0.203047	+	0.979169i	$0.434915\pi$
$332$	0	0
$333$	−1.19601	−0.0655407
$334$	0	0
$335$	−43.7678	−2.39129
$336$	0	0
$337$	17.4984	0.953196	0.476598	−	0.879121i	$-0.341870\pi$
$337$	17.4984	0.953196	0.476598	+	0.879121i	$0.341870\pi$
$338$	0	0
$339$	0.611349	0.0332039
$340$	0	0
$341$	−8.82057	−0.477661
$342$	0	0
$343$	−18.8638	−1.01855
$344$	0	0
$345$	−33.6046	−1.80921
$346$	0	0
$347$	32.3165	1.73484	0.867421	−	0.497575i	$-0.165776\pi$
$347$	32.3165	1.73484	0.867421	+	0.497575i	$0.165776\pi$
$348$	0	0
$349$	35.1363	1.88080	0.940401	−	0.340066i	$-0.110450\pi$
$349$	35.1363	1.88080	0.940401	+	0.340066i	$0.110450\pi$
$350$	0	0
$351$	−15.0534	−0.803491
$352$	0	0
$353$	−1.64773	−0.0876997	−0.0438498	−	0.999038i	$-0.513962\pi$
$353$	−1.64773	−0.0876997	−0.0438498	+	0.999038i	$0.513962\pi$
$354$	0	0
$355$	−34.8769	−1.85107
$356$	0	0
$357$	4.55502	0.241077
$358$	0	0
$359$	19.0448	1.00515	0.502574	−	0.864534i	$-0.332386\pi$
$359$	19.0448	1.00515	0.502574	+	0.864534i	$0.332386\pi$
$360$	0	0
$361$	29.3785	1.54624
$362$	0	0
$363$	−0.542248	−0.0284607
$364$	0	0
$365$	13.0233	0.681672
$366$	0	0
$367$	3.77726	0.197171	0.0985856	−	0.995129i	$-0.468568\pi$
$367$	3.77726	0.197171	0.0985856	+	0.995129i	$0.468568\pi$
$368$	0	0
$369$	−10.7025	−0.557151
$370$	0	0
$371$	−14.7007	−0.763221
$372$	0	0
$373$	−34.7162	−1.79753	−0.898767	−	0.438426i	$-0.855536\pi$
$373$	−34.7162	−1.79753	−0.898767	+	0.438426i	$0.855536\pi$
$374$	0	0
$375$	7.91356	0.408655
$376$	0	0
$377$	7.13447	0.367444
$378$	0	0
$379$	12.5678	0.645565	0.322782	−	0.946473i	$-0.395382\pi$
$379$	12.5678	0.645565	0.322782	+	0.946473i	$0.395382\pi$
$380$	0	0
$381$	18.2378	0.934349
$382$	0	0
$383$	−24.6905	−1.26163	−0.630814	−	0.775934i	$-0.717279\pi$
$383$	−24.6905	−1.26163	−0.630814	+	0.775934i	$0.717279\pi$
$384$	0	0
$385$	−18.8973	−0.963098
$386$	0	0
$387$	−9.16207	−0.465734
$388$	0	0
$389$	−11.0233	−0.558904	−0.279452	−	0.960160i	$-0.590153\pi$
$389$	−11.0233	−0.558904	−0.279452	+	0.960160i	$0.590153\pi$
$390$	0	0
$391$	−14.6160	−0.739164
$392$	0	0
$393$	−0.597803	−0.0301552
$394$	0	0
$395$	−25.0196	−1.25887
$396$	0	0
$397$	22.0116	1.10473	0.552365	−	0.833602i	$-0.313725\pi$
$397$	22.0116	1.10473	0.552365	+	0.833602i	$0.313725\pi$
$398$	0	0
$399$	15.8411	0.793049
$400$	0	0
$401$	17.4022	0.869024	0.434512	−	0.900666i	$-0.356921\pi$
$401$	17.4022	0.869024	0.434512	+	0.900666i	$0.356921\pi$
$402$	0	0
$403$	7.23771	0.360536
$404$	0	0
$405$	−13.6312	−0.677340
$406$	0	0
$407$	−3.25519	−0.161354
$408$	0	0
$409$	6.35227	0.314100	0.157050	−	0.987591i	$-0.449802\pi$
$409$	6.35227	0.314100	0.157050	+	0.987591i	$0.449802\pi$
$410$	0	0
$411$	26.3426	1.29938
$412$	0	0
$413$	−8.35532	−0.411138
$414$	0	0
$415$	−3.39135	−0.166475
$416$	0	0
$417$	−22.7657	−1.11484
$418$	0	0
$419$	18.6454	0.910888	0.455444	−	0.890264i	$-0.349481\pi$
$419$	18.6454	0.910888	0.455444	+	0.890264i	$0.349481\pi$
$420$	0	0
$421$	23.0233	1.12209	0.561044	−	0.827786i	$-0.310400\pi$
$421$	23.0233	1.12209	0.561044	+	0.827786i	$0.310400\pi$
$422$	0	0
$423$	8.97120	0.436195
$424$	0	0
$425$	13.4419	0.652030
$426$	0	0
$427$	0.977404	0.0472999
$428$	0	0
$429$	−11.6782	−0.563828
$430$	0	0
$431$	37.7113	1.81649	0.908245	−	0.418439i	$-0.137423\pi$
$431$	37.7113	1.81649	0.908245	+	0.418439i	$0.137423\pi$
$432$	0	0
$433$	24.7458	1.18921	0.594604	−	0.804019i	$-0.297309\pi$
$433$	24.7458	1.18921	0.594604	+	0.804019i	$0.297309\pi$
$434$	0	0
$435$	12.2823	0.588892
$436$	0	0
$437$	−50.8307	−2.43156
$438$	0	0
$439$	−6.41785	−0.306307	−0.153154	−	0.988202i	$-0.548943\pi$
$439$	−6.41785	−0.306307	−0.153154	+	0.988202i	$0.548943\pi$
$440$	0	0
$441$	4.93315	0.234912
$442$	0	0
$443$	−22.0145	−1.04594	−0.522971	−	0.852350i	$-0.675176\pi$
$443$	−22.0145	−1.04594	−0.522971	+	0.852350i	$0.675176\pi$
$444$	0	0
$445$	64.1727	3.04208
$446$	0	0
$447$	10.0959	0.477519
$448$	0	0
$449$	37.0334	1.74771	0.873857	−	0.486183i	$-0.161611\pi$
$449$	37.0334	1.74771	0.873857	+	0.486183i	$0.161611\pi$
$450$	0	0
$451$	−29.1293	−1.37164
$452$	0	0
$453$	−20.5153	−0.963892
$454$	0	0
$455$	15.5062	0.726942
$456$	0	0
$457$	−19.5314	−0.913638	−0.456819	−	0.889560i	$-0.651011\pi$
$457$	−19.5314	−0.913638	−0.456819	+	0.889560i	$0.651011\pi$
$458$	0	0
$459$	−11.2716	−0.526111
$460$	0	0
$461$	−22.1495	−1.03161	−0.515803	−	0.856707i	$-0.672506\pi$
$461$	−22.1495	−1.03161	−0.515803	+	0.856707i	$0.672506\pi$
$462$	0	0
$463$	−23.7201	−1.10237	−0.551184	−	0.834384i	$-0.685824\pi$
$463$	−23.7201	−1.10237	−0.551184	+	0.834384i	$0.685824\pi$
$464$	0	0
$465$	12.4600	0.577820
$466$	0	0
$467$	12.7608	0.590498	0.295249	−	0.955420i	$-0.404597\pi$
$467$	12.7608	0.590498	0.295249	+	0.955420i	$0.404597\pi$
$468$	0	0
$469$	−21.6778	−1.00099
$470$	0	0
$471$	−5.52631	−0.254639
$472$	0	0
$473$	−24.9366	−1.14659
$474$	0	0
$475$	46.7475	2.14492
$476$	0	0
$477$	10.3688	0.474755
$478$	0	0
$479$	−9.99427	−0.456650	−0.228325	−	0.973585i	$-0.573325\pi$
$479$	−9.99427	−0.456650	−0.228325	+	0.973585i	$0.573325\pi$
$480$	0	0
$481$	2.67104	0.121789
$482$	0	0
$483$	−16.6441	−0.757331
$484$	0	0
$485$	6.84718	0.310914
$486$	0	0
$487$	6.52269	0.295571	0.147786	−	0.989019i	$-0.452785\pi$
$487$	6.52269	0.295571	0.147786	+	0.989019i	$0.452785\pi$
$488$	0	0
$489$	25.3653	1.14706
$490$	0	0
$491$	−30.8607	−1.39272	−0.696361	−	0.717691i	$-0.745199\pi$
$491$	−30.8607	−1.39272	−0.696361	+	0.717691i	$0.745199\pi$
$492$	0	0
$493$	5.34209	0.240596
$494$	0	0
$495$	13.3288	0.599086
$496$	0	0
$497$	−17.2742	−0.774855
$498$	0	0
$499$	18.6129	0.833229	0.416615	−	0.909083i	$-0.363216\pi$
$499$	18.6129	0.833229	0.416615	+	0.909083i	$0.363216\pi$
$500$	0	0
$501$	16.8758	0.753956
$502$	0	0
$503$	31.8949	1.42212	0.711061	−	0.703130i	$-0.248215\pi$
$503$	31.8949	1.42212	0.711061	+	0.703130i	$0.248215\pi$
$504$	0	0
$505$	−4.89686	−0.217907
$506$	0	0
$507$	−7.87816	−0.349881
$508$	0	0
$509$	5.26732	0.233470	0.116735	−	0.993163i	$-0.462757\pi$
$509$	5.26732	0.233470	0.116735	+	0.993163i	$0.462757\pi$
$510$	0	0
$511$	6.45034	0.285346
$512$	0	0
$513$	−39.1995	−1.73070
$514$	0	0
$515$	53.8547	2.37312
$516$	0	0
$517$	24.4171	1.07386
$518$	0	0
$519$	−32.5897	−1.43053
$520$	0	0
$521$	−17.2245	−0.754621	−0.377310	−	0.926087i	$-0.623151\pi$
$521$	−17.2245	−0.754621	−0.377310	+	0.926087i	$0.623151\pi$
$522$	0	0
$523$	−32.7493	−1.43203	−0.716014	−	0.698086i	$-0.754035\pi$
$523$	−32.7493	−1.43203	−0.716014	+	0.698086i	$0.754035\pi$
$524$	0	0
$525$	15.3071	0.668055
$526$	0	0
$527$	5.41939	0.236072
$528$	0	0
$529$	30.4071	1.32205
$530$	0	0
$531$	5.89324	0.255745
$532$	0	0
$533$	23.9020	1.03531
$534$	0	0
$535$	−31.2108	−1.34936
$536$	0	0
$537$	−3.07314	−0.132616
$538$	0	0
$539$	13.4267	0.578327
$540$	0	0
$541$	32.5201	1.39815	0.699075	−	0.715048i	$-0.253595\pi$
$541$	32.5201	1.39815	0.699075	+	0.715048i	$0.253595\pi$
$542$	0	0
$543$	−17.2710	−0.741170
$544$	0	0
$545$	−31.6312	−1.35493
$546$	0	0
$547$	−36.7989	−1.57341	−0.786703	−	0.617332i	$-0.788214\pi$
$547$	−36.7989	−1.57341	−0.786703	+	0.617332i	$0.788214\pi$
$548$	0	0
$549$	−0.689390	−0.0294224
$550$	0	0
$551$	18.5784	0.791465
$552$	0	0
$553$	−12.3920	−0.526962
$554$	0	0
$555$	4.59832	0.195188
$556$	0	0
$557$	−23.8021	−1.00853	−0.504264	−	0.863549i	$-0.668236\pi$
$557$	−23.8021	−1.00853	−0.504264	+	0.863549i	$0.668236\pi$
$558$	0	0
$559$	20.4617	0.865437
$560$	0	0
$561$	−8.74428	−0.369184
$562$	0	0
$563$	0.717395	0.0302346	0.0151173	−	0.999886i	$-0.495188\pi$
$563$	0.717395	0.0302346	0.0151173	+	0.999886i	$0.495188\pi$
$564$	0	0
$565$	1.55831	0.0655586
$566$	0	0
$567$	−6.75142	−0.283533
$568$	0	0
$569$	−1.81073	−0.0759099	−0.0379549	−	0.999279i	$-0.512084\pi$
$569$	−1.81073	−0.0759099	−0.0379549	+	0.999279i	$0.512084\pi$
$570$	0	0
$571$	26.1999	1.09643	0.548216	−	0.836337i	$-0.315307\pi$
$571$	26.1999	1.09643	0.548216	+	0.836337i	$0.315307\pi$
$572$	0	0
$573$	−3.27372	−0.136761
$574$	0	0
$575$	−49.1169	−2.04832
$576$	0	0
$577$	23.0966	0.961522	0.480761	−	0.876852i	$-0.340360\pi$
$577$	23.0966	0.961522	0.480761	+	0.876852i	$0.340360\pi$
$578$	0	0
$579$	−24.4305	−1.01530
$580$	0	0
$581$	−1.67971	−0.0696859
$582$	0	0
$583$	28.2210	1.16879
$584$	0	0
$585$	−10.9369	−0.452187
$586$	0	0
$587$	12.8480	0.530294	0.265147	−	0.964208i	$-0.414580\pi$
$587$	12.8480	0.530294	0.265147	+	0.964208i	$0.414580\pi$
$588$	0	0
$589$	18.8472	0.776585
$590$	0	0
$591$	−4.47326	−0.184005
$592$	0	0
$593$	−1.89037	−0.0776280	−0.0388140	−	0.999246i	$-0.512358\pi$
$593$	−1.89037	−0.0776280	−0.0388140	+	0.999246i	$0.512358\pi$
$594$	0	0
$595$	11.6106	0.475988
$596$	0	0
$597$	10.2262	0.418529
$598$	0	0
$599$	−21.8916	−0.894465	−0.447232	−	0.894418i	$-0.647590\pi$
$599$	−21.8916	−0.894465	−0.447232	+	0.894418i	$0.647590\pi$
$600$	0	0
$601$	15.0285	0.613027	0.306513	−	0.951866i	$-0.400838\pi$
$601$	15.0285	0.613027	0.306513	+	0.951866i	$0.400838\pi$
$602$	0	0
$603$	15.2900	0.622656
$604$	0	0
$605$	−1.38217	−0.0561933
$606$	0	0
$607$	39.8377	1.61696	0.808481	−	0.588523i	$-0.200290\pi$
$607$	39.8377	1.61696	0.808481	+	0.588523i	$0.200290\pi$
$608$	0	0
$609$	6.08332	0.246509
$610$	0	0
$611$	−20.0354	−0.810546
$612$	0	0
$613$	7.91176	0.319553	0.159776	−	0.987153i	$-0.448923\pi$
$613$	7.91176	0.319553	0.159776	+	0.987153i	$0.448923\pi$
$614$	0	0
$615$	41.1483	1.65926
$616$	0	0
$617$	15.4037	0.620130	0.310065	−	0.950715i	$-0.399649\pi$
$617$	15.4037	0.620130	0.310065	+	0.950715i	$0.399649\pi$
$618$	0	0
$619$	42.9600	1.72671	0.863353	−	0.504600i	$-0.168360\pi$
$619$	42.9600	1.72671	0.863353	+	0.504600i	$0.168360\pi$
$620$	0	0
$621$	41.1863	1.65275
$622$	0	0
$623$	31.7842	1.27341
$624$	0	0
$625$	−13.4334	−0.537337
$626$	0	0
$627$	−30.4103	−1.21447
$628$	0	0
$629$	2.00000	0.0797452
$630$	0	0
$631$	40.0919	1.59603	0.798016	−	0.602636i	$-0.205883\pi$
$631$	40.0919	1.59603	0.798016	+	0.602636i	$0.205883\pi$
$632$	0	0
$633$	18.5013	0.735361
$634$	0	0
$635$	46.4875	1.84480
$636$	0	0
$637$	−11.0172	−0.436518
$638$	0	0
$639$	12.1840	0.481991
$640$	0	0
$641$	−45.9084	−1.81327	−0.906636	−	0.421914i	$-0.861358\pi$
$641$	−45.9084	−1.81327	−0.906636	+	0.421914i	$0.861358\pi$
$642$	0	0
$643$	−46.9548	−1.85172	−0.925859	−	0.377870i	$-0.876657\pi$
$643$	−46.9548	−1.85172	−0.925859	+	0.377870i	$0.876657\pi$
$644$	0	0
$645$	35.2257	1.38701
$646$	0	0
$647$	−15.6860	−0.616679	−0.308340	−	0.951276i	$-0.599773\pi$
$647$	−15.6860	−0.616679	−0.308340	+	0.951276i	$0.599773\pi$
$648$	0	0
$649$	16.0397	0.629615
$650$	0	0
$651$	6.17135	0.241874
$652$	0	0
$653$	−0.418620	−0.0163819	−0.00819093	−	0.999966i	$-0.502607\pi$
$653$	−0.418620	−0.0163819	−0.00819093	+	0.999966i	$0.502607\pi$
$654$	0	0
$655$	−1.52378	−0.0595390
$656$	0	0
$657$	−4.54960	−0.177497
$658$	0	0
$659$	19.5759	0.762571	0.381285	−	0.924457i	$-0.375482\pi$
$659$	19.5759	0.762571	0.381285	+	0.924457i	$0.375482\pi$
$660$	0	0
$661$	14.3188	0.556936	0.278468	−	0.960446i	$-0.410173\pi$
$661$	14.3188	0.556936	0.278468	+	0.960446i	$0.410173\pi$
$662$	0	0
$663$	7.17511	0.278658
$664$	0	0
$665$	40.3785	1.56581
$666$	0	0
$667$	−19.5200	−0.755818
$668$	0	0
$669$	−10.0748	−0.389513
$670$	0	0
$671$	−1.87633	−0.0724348
$672$	0	0
$673$	39.0214	1.50417	0.752083	−	0.659068i	$-0.229049\pi$
$673$	39.0214	1.50417	0.752083	+	0.659068i	$0.229049\pi$
$674$	0	0
$675$	−37.8779	−1.45792
$676$	0	0
$677$	24.6560	0.947609	0.473804	−	0.880630i	$-0.342880\pi$
$677$	24.6560	0.947609	0.473804	+	0.880630i	$0.342880\pi$
$678$	0	0
$679$	3.39135	0.130148
$680$	0	0
$681$	3.07314	0.117763
$682$	0	0
$683$	−18.3587	−0.702477	−0.351238	−	0.936286i	$-0.614239\pi$
$683$	−18.3587	−0.702477	−0.351238	+	0.936286i	$0.614239\pi$
$684$	0	0
$685$	67.1464	2.56553
$686$	0	0
$687$	−11.3720	−0.433868
$688$	0	0
$689$	−23.1567	−0.882199
$690$	0	0
$691$	35.4701	1.34935	0.674674	−	0.738116i	$-0.264284\pi$
$691$	35.4701	1.34935	0.674674	+	0.738116i	$0.264284\pi$
$692$	0	0
$693$	6.60165	0.250776
$694$	0	0
$695$	−58.0290	−2.20116
$696$	0	0
$697$	17.8971	0.677901
$698$	0	0
$699$	−24.2747	−0.918153
$700$	0	0
$701$	23.2973	0.879926	0.439963	−	0.898016i	$-0.354992\pi$
$701$	23.2973	0.879926	0.439963	+	0.898016i	$0.354992\pi$
$702$	0	0
$703$	6.95547	0.262330
$704$	0	0
$705$	−34.4919	−1.29904
$706$	0	0
$707$	−2.42537	−0.0912155
$708$	0	0
$709$	−7.50307	−0.281784	−0.140892	−	0.990025i	$-0.544997\pi$
$709$	−7.50307	−0.281784	−0.140892	+	0.990025i	$0.544997\pi$
$710$	0	0
$711$	8.74043	0.327792
$712$	0	0
$713$	−19.8025	−0.741608
$714$	0	0
$715$	−29.7673	−1.11323
$716$	0	0
$717$	0.549802	0.0205327
$718$	0	0
$719$	−22.8690	−0.852868	−0.426434	−	0.904519i	$-0.640230\pi$
$719$	−22.8690	−0.852868	−0.426434	+	0.904519i	$0.640230\pi$
$720$	0	0
$721$	26.6737	0.993382
$722$	0	0
$723$	6.78270	0.252251
$724$	0	0
$725$	17.9520	0.666720
$726$	0	0
$727$	36.9482	1.37033	0.685167	−	0.728386i	$-0.259729\pi$
$727$	36.9482	1.37033	0.685167	+	0.728386i	$0.259729\pi$
$728$	0	0
$729$	27.4707	1.01743
$730$	0	0
$731$	15.3211	0.566672
$732$	0	0
$733$	−45.6031	−1.68439	−0.842194	−	0.539175i	$-0.818736\pi$
$733$	−45.6031	−1.68439	−0.842194	+	0.539175i	$0.818736\pi$
$734$	0	0
$735$	−18.9666	−0.699595
$736$	0	0
$737$	41.6150	1.53291
$738$	0	0
$739$	39.4485	1.45114	0.725569	−	0.688150i	$-0.241577\pi$
$739$	39.4485	1.45114	0.725569	+	0.688150i	$0.241577\pi$
$740$	0	0
$741$	24.9531	0.916676
$742$	0	0
$743$	−51.7596	−1.89887	−0.949437	−	0.313957i	$-0.898345\pi$
$743$	−51.7596	−1.89887	−0.949437	+	0.313957i	$0.898345\pi$
$744$	0	0
$745$	25.7341	0.942824
$746$	0	0
$747$	1.18474	0.0433475
$748$	0	0
$749$	−15.4585	−0.564840
$750$	0	0
$751$	−1.74255	−0.0635864	−0.0317932	−	0.999494i	$-0.510122\pi$
$751$	−1.74255	−0.0635864	−0.0317932	+	0.999494i	$0.510122\pi$
$752$	0	0
$753$	−33.4654	−1.21955
$754$	0	0
$755$	−52.2928	−1.90313
$756$	0	0
$757$	−32.8054	−1.19233	−0.596167	−	0.802861i	$-0.703310\pi$
$757$	−32.8054	−1.19233	−0.596167	+	0.802861i	$0.703310\pi$
$758$	0	0
$759$	31.9517	1.15977
$760$	0	0
$761$	9.15130	0.331734	0.165867	−	0.986148i	$-0.446958\pi$
$761$	9.15130	0.331734	0.165867	+	0.986148i	$0.446958\pi$
$762$	0	0
$763$	−15.6667	−0.567171
$764$	0	0
$765$	−8.18927	−0.296084
$766$	0	0
$767$	−13.1614	−0.475230
$768$	0	0
$769$	−45.6177	−1.64502	−0.822509	−	0.568753i	$-0.807426\pi$
$769$	−45.6177	−1.64502	−0.822509	+	0.568753i	$0.807426\pi$
$770$	0	0
$771$	−26.4521	−0.952648
$772$	0	0
$773$	25.6428	0.922308	0.461154	−	0.887320i	$-0.347436\pi$
$773$	25.6428	0.922308	0.461154	+	0.887320i	$0.347436\pi$
$774$	0	0
$775$	18.2118	0.654186
$776$	0	0
$777$	2.27751	0.0817052
$778$	0	0
$779$	62.2414	2.23003
$780$	0	0
$781$	33.1614	1.18661
$782$	0	0
$783$	−15.0534	−0.537965
$784$	0	0
$785$	−14.0864	−0.502764
$786$	0	0
$787$	−0.789747	−0.0281515	−0.0140757	−	0.999901i	$-0.504481\pi$
$787$	−0.789747	−0.0281515	−0.0140757	+	0.999901i	$0.504481\pi$
$788$	0	0
$789$	10.5035	0.373933
$790$	0	0
$791$	0.771817	0.0274427
$792$	0	0
$793$	1.53962	0.0546734
$794$	0	0
$795$	−39.8652	−1.41387
$796$	0	0
$797$	46.6410	1.65211	0.826055	−	0.563589i	$-0.190580\pi$
$797$	46.6410	1.65211	0.826055	+	0.563589i	$0.190580\pi$
$798$	0	0
$799$	−15.0019	−0.530730
$800$	0	0
$801$	−22.4183	−0.792111
$802$	0	0
$803$	−12.3827	−0.436977
$804$	0	0
$805$	−42.4252	−1.49529
$806$	0	0
$807$	−34.8991	−1.22851
$808$	0	0
$809$	1.75752	0.0617910	0.0308955	−	0.999523i	$-0.490164\pi$
$809$	1.75752	0.0617910	0.0308955	+	0.999523i	$0.490164\pi$
$810$	0	0
$811$	−18.5763	−0.652303	−0.326151	−	0.945318i	$-0.605752\pi$
$811$	−18.5763	−0.652303	−0.326151	+	0.945318i	$0.605752\pi$
$812$	0	0
$813$	−3.22454	−0.113089
$814$	0	0
$815$	64.6553	2.26478
$816$	0	0
$817$	53.2828	1.86413
$818$	0	0
$819$	−5.41697	−0.189284
$820$	0	0
$821$	36.6927	1.28059	0.640293	−	0.768131i	$-0.278813\pi$
$821$	36.6927	1.28059	0.640293	+	0.768131i	$0.278813\pi$
$822$	0	0
$823$	28.3419	0.987936	0.493968	−	0.869480i	$-0.335546\pi$
$823$	28.3419	0.987936	0.493968	+	0.869480i	$0.335546\pi$
$824$	0	0
$825$	−29.3850	−1.02306
$826$	0	0
$827$	−25.9666	−0.902947	−0.451474	−	0.892284i	$-0.649102\pi$
$827$	−25.9666	−0.902947	−0.451474	+	0.892284i	$0.649102\pi$
$828$	0	0
$829$	−32.1889	−1.11797	−0.558984	−	0.829179i	$-0.688809\pi$
$829$	−32.1889	−1.11797	−0.558984	+	0.829179i	$0.688809\pi$
$830$	0	0
$831$	−1.51965	−0.0527161
$832$	0	0
$833$	−8.24938	−0.285824
$834$	0	0
$835$	43.0159	1.48863
$836$	0	0
$837$	−15.2712	−0.527851
$838$	0	0
$839$	31.4855	1.08700	0.543501	−	0.839409i	$-0.317099\pi$
$839$	31.4855	1.08700	0.543501	+	0.839409i	$0.317099\pi$
$840$	0	0
$841$	−21.8655	−0.753984
$842$	0	0
$843$	28.7812	0.991276
$844$	0	0
$845$	−20.0812	−0.690813
$846$	0	0
$847$	−0.684578	−0.0235224
$848$	0	0
$849$	1.60779	0.0551791
$850$	0	0
$851$	−7.30801	−0.250515
$852$	0	0
$853$	−3.79239	−0.129849	−0.0649244	−	0.997890i	$-0.520681\pi$
$853$	−3.79239	−0.129849	−0.0649244	+	0.997890i	$0.520681\pi$
$854$	0	0
$855$	−28.4801	−0.973999
$856$	0	0
$857$	−10.5782	−0.361346	−0.180673	−	0.983543i	$-0.557827\pi$
$857$	−10.5782	−0.361346	−0.180673	+	0.983543i	$0.557827\pi$
$858$	0	0
$859$	2.69856	0.0920737	0.0460368	−	0.998940i	$-0.485341\pi$
$859$	2.69856	0.0920737	0.0460368	+	0.998940i	$0.485341\pi$
$860$	0	0
$861$	20.3804	0.694562
$862$	0	0
$863$	48.5155	1.65149	0.825743	−	0.564046i	$-0.190756\pi$
$863$	48.5155	1.65149	0.825743	+	0.564046i	$0.190756\pi$
$864$	0	0
$865$	−83.0701	−2.82447
$866$	0	0
$867$	−17.4607	−0.592995
$868$	0	0
$869$	23.7890	0.806986
$870$	0	0
$871$	−34.1471	−1.15703
$872$	0	0
$873$	−2.39201	−0.0809573
$874$	0	0
$875$	9.99073	0.337748
$876$	0	0
$877$	−27.1626	−0.917214	−0.458607	−	0.888639i	$-0.651651\pi$
$877$	−27.1626	−0.917214	−0.458607	+	0.888639i	$0.651651\pi$
$878$	0	0
$879$	−37.7770	−1.27419
$880$	0	0
$881$	17.8757	0.602248	0.301124	−	0.953585i	$-0.402638\pi$
$881$	17.8757	0.602248	0.301124	+	0.953585i	$0.402638\pi$
$882$	0	0
$883$	8.27841	0.278591	0.139295	−	0.990251i	$-0.455516\pi$
$883$	8.27841	0.278591	0.139295	+	0.990251i	$0.455516\pi$
$884$	0	0
$885$	−22.6579	−0.761637
$886$	0	0
$887$	27.5833	0.926155	0.463077	−	0.886318i	$-0.346745\pi$
$887$	27.5833	0.926155	0.463077	+	0.886318i	$0.346745\pi$
$888$	0	0
$889$	23.0248	0.772228
$890$	0	0
$891$	12.9607	0.434200
$892$	0	0
$893$	−52.1727	−1.74589
$894$	0	0
$895$	−7.83333	−0.261839
$896$	0	0
$897$	−26.2179	−0.875390
$898$	0	0
$899$	7.23771	0.241391
$900$	0	0
$901$	−17.3390	−0.577647
$902$	0	0
$903$	17.4470	0.580599
$904$	0	0
$905$	−44.0232	−1.46338
$906$	0	0
$907$	32.7962	1.08898	0.544490	−	0.838768i	$-0.316723\pi$
$907$	32.7962	1.08898	0.544490	+	0.838768i	$0.316723\pi$
$908$	0	0
$909$	1.71068	0.0567397
$910$	0	0
$911$	4.63527	0.153573	0.0767866	−	0.997048i	$-0.475534\pi$
$911$	4.63527	0.153573	0.0767866	+	0.997048i	$0.475534\pi$
$912$	0	0
$913$	3.22454	0.106717
$914$	0	0
$915$	2.65052	0.0876234
$916$	0	0
$917$	−0.754715	−0.0249229
$918$	0	0
$919$	4.76696	0.157248	0.0786238	−	0.996904i	$-0.474947\pi$
$919$	4.76696	0.157248	0.0786238	+	0.996904i	$0.474947\pi$
$920$	0	0
$921$	35.5644	1.17189
$922$	0	0
$923$	−27.2105	−0.895645
$924$	0	0
$925$	6.72097	0.220984
$926$	0	0
$927$	−18.8137	−0.617924
$928$	0	0
$929$	35.4499	1.16307	0.581536	−	0.813521i	$-0.302452\pi$
$929$	35.4499	1.16307	0.581536	+	0.813521i	$0.302452\pi$
$930$	0	0
$931$	−28.6891	−0.940249
$932$	0	0
$933$	−23.9448	−0.783916
$934$	0	0
$935$	−22.2889	−0.728925
$936$	0	0
$937$	4.39354	0.143531	0.0717653	−	0.997422i	$-0.477137\pi$
$937$	4.39354	0.143531	0.0717653	+	0.997422i	$0.477137\pi$
$938$	0	0
$939$	−11.4057	−0.372210
$940$	0	0
$941$	−21.2422	−0.692477	−0.346239	−	0.938146i	$-0.612541\pi$
$941$	−21.2422	−0.692477	−0.346239	+	0.938146i	$0.612541\pi$
$942$	0	0
$943$	−65.3961	−2.12959
$944$	0	0
$945$	−32.7173	−1.06429
$946$	0	0
$947$	23.8004	0.773408	0.386704	−	0.922204i	$-0.373614\pi$
$947$	23.8004	0.773408	0.386704	+	0.922204i	$0.373614\pi$
$948$	0	0
$949$	10.1606	0.329828
$950$	0	0
$951$	−11.4098	−0.369987
$952$	0	0
$953$	43.7477	1.41713	0.708564	−	0.705647i	$-0.249343\pi$
$953$	43.7477	1.41713	0.708564	+	0.705647i	$0.249343\pi$
$954$	0	0
$955$	−8.34459	−0.270025
$956$	0	0
$957$	−11.6782	−0.377502
$958$	0	0
$959$	33.2570	1.07393
$960$	0	0
$961$	−23.6576	−0.763147
$962$	0	0
$963$	10.9033	0.351353
$964$	0	0
$965$	−62.2726	−2.00463
$966$	0	0
$967$	1.72579	0.0554975	0.0277488	−	0.999615i	$-0.491166\pi$
$967$	1.72579	0.0554975	0.0277488	+	0.999615i	$0.491166\pi$
$968$	0	0
$969$	18.6842	0.600222
$970$	0	0
$971$	−18.2036	−0.584181	−0.292090	−	0.956391i	$-0.594351\pi$
$971$	−18.2036	−0.584181	−0.292090	+	0.956391i	$0.594351\pi$
$972$	0	0
$973$	−28.7412	−0.921402
$974$	0	0
$975$	24.1118	0.772197
$976$	0	0
$977$	39.6381	1.26814	0.634068	−	0.773278i	$-0.281384\pi$
$977$	39.6381	1.26814	0.634068	+	0.773278i	$0.281384\pi$
$978$	0	0
$979$	−61.0163	−1.95009
$980$	0	0
$981$	11.0501	0.352803
$982$	0	0
$983$	−45.4500	−1.44963	−0.724815	−	0.688943i	$-0.758075\pi$
$983$	−45.4500	−1.44963	−0.724815	+	0.688943i	$0.758075\pi$
$984$	0	0
$985$	−11.4022	−0.363304
$986$	0	0
$987$	−17.0835	−0.543774
$988$	0	0
$989$	−55.9835	−1.78017
$990$	0	0
$991$	−29.4118	−0.934298	−0.467149	−	0.884179i	$-0.654719\pi$
$991$	−29.4118	−0.934298	−0.467149	+	0.884179i	$0.654719\pi$
$992$	0	0
$993$	9.92337	0.314909
$994$	0	0
$995$	26.0661	0.826352
$996$	0	0
$997$	−14.2290	−0.450637	−0.225319	−	0.974285i	$-0.572342\pi$
$997$	−14.2290	−0.450637	−0.225319	+	0.974285i	$0.572342\pi$
$998$	0	0
$999$	−5.63578	−0.178308

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1184.2.a.p.1.6	yes	8
4.3	odd	2	inner	1184.2.a.p.1.3	✓	8
8.3	odd	2		2368.2.a.bj.1.6		8
8.5	even	2		2368.2.a.bj.1.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1184.2.a.p.1.3	✓	8	4.3	odd	2	inner
1184.2.a.p.1.6	yes	8	1.1	even	1	trivial
2368.2.a.bj.1.3		8	8.5	even	2
2368.2.a.bj.1.6		8	8.3	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:	\( N \)	\(=\)	\( 1184 = 2^{5} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1184.a (trivial)

\(f(q)\)	\(=\)	\(q+1.34313 q^{3} +3.42359 q^{5} +1.69567 q^{7} -1.19601 q^{9} -3.25519 q^{11} +2.67104 q^{13} +4.59832 q^{15} +2.00000 q^{17} +6.95547 q^{19} +2.27751 q^{21} -7.30801 q^{23} +6.72097 q^{25} -5.63578 q^{27} +2.67104 q^{29} +2.70969 q^{31} -4.37214 q^{33} +5.80529 q^{35} +1.00000 q^{37} +3.58755 q^{39} +8.94855 q^{41} +7.66056 q^{43} -4.09463 q^{45} -7.50097 q^{47} -4.12469 q^{49} +2.68626 q^{51} -8.66952 q^{53} -11.1444 q^{55} +9.34209 q^{57} -4.92743 q^{59} +0.576410 q^{61} -2.02804 q^{63} +9.14456 q^{65} -12.7842 q^{67} -9.81560 q^{69} -10.1872 q^{71} +3.80399 q^{73} +9.02712 q^{75} -5.51975 q^{77} -7.30801 q^{79} -3.98155 q^{81} -0.990583 q^{83} +6.84718 q^{85} +3.58755 q^{87} +18.7443 q^{89} +4.52922 q^{91} +3.63947 q^{93} +23.8127 q^{95} +2.00000 q^{97} +3.89323 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{5} + 16 q^{9} + 2 q^{13} + 16 q^{17} + 2 q^{21} + 22 q^{25} + 2 q^{29} + 30 q^{33} + 8 q^{37} + 36 q^{41} + 16 q^{45} + 42 q^{49} - 2 q^{53} + 36 q^{57} + 34 q^{61} + 12 q^{65} + 2 q^{69} + 56 q^{73}+ \cdots + 16 q^{97}+O(q^{100}) \) 8 * q - 2 * q^5 + 16 * q^9 + 2 * q^13 + 16 * q^17 + 2 * q^21 + 22 * q^25 + 2 * q^29 + 30 * q^33 + 8 * q^37 + 36 * q^41 + 16 * q^45 + 42 * q^49 - 2 * q^53 + 36 * q^57 + 34 * q^61 + 12 * q^65 + 2 * q^69 + 56 * q^73 - 14 * q^77 + 48 * q^81 - 4 * q^85 + 20 * q^89 - 12 * q^93 + 16 * q^97

Introduction

L-functions

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