LMFDB - Embedded newform 9196.2.a.q.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9196,2,Mod(1,9196)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9196, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9196.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 9196 = 2^{2} \cdot 11^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9196.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,0,-1,0,0,0,-2,0,9,0,0,0,5] 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:	$73.4304296988$
Analytic rank:	$1$
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} - x^{7} - 16x^{6} + 14x^{5} + 80x^{4} - 64x^{3} - 140x^{2} + 81x + 72 $ x^8 - x^7 - 16x^6 + 14x^5 + 80x^4 - 64x^3 - 140x^2 + 81x + 72
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-1.42458$ of defining polynomial
Character	$\chi$	$=$	9196.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.42458 q^{3} -2.11046 q^{5} +3.55881 q^{7} -0.970558 q^{9} +4.05605 q^{13} -3.00652 q^{15} +4.39668 q^{17} -1.00000 q^{19} +5.06982 q^{21} -8.50071 q^{23} -0.545973 q^{25} -5.65640 q^{27} -8.32874 q^{29} -0.583258 q^{31} -7.51070 q^{35} -2.92957 q^{37} +5.77818 q^{39} -0.892281 q^{41} -8.63194 q^{43} +2.04832 q^{45} -0.756169 q^{47} +5.66510 q^{49} +6.26344 q^{51} +1.53431 q^{53} -1.42458 q^{57} -13.9436 q^{59} +14.9005 q^{61} -3.45403 q^{63} -8.56011 q^{65} +1.09562 q^{67} -12.1100 q^{69} -4.46831 q^{71} +11.4022 q^{73} -0.777785 q^{75} -14.7613 q^{79} -5.14634 q^{81} -5.34261 q^{83} -9.27900 q^{85} -11.8650 q^{87} +1.99012 q^{89} +14.4347 q^{91} -0.830900 q^{93} +2.11046 q^{95} +2.89451 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - q^{3} - 2 q^{7} + 9 q^{9} + 5 q^{13} - 9 q^{15} - 3 q^{17} - 8 q^{19} - 5 q^{21} + q^{23} - q^{27} - q^{29} + 3 q^{31} - 8 q^{35} - 23 q^{37} - 9 q^{39} - 10 q^{41} - 2 q^{43} + 9 q^{45} - 8 q^{47}+ \cdots + 6 q^{97}+O(q^{100}) $ 8 * q - q^3 - 2 * q^7 + 9 * q^9 + 5 * q^13 - 9 * q^15 - 3 * q^17 - 8 * q^19 - 5 * q^21 + q^23 - q^27 - q^29 + 3 * q^31 - 8 * q^35 - 23 * q^37 - 9 * q^39 - 10 * q^41 - 2 * q^43 + 9 * q^45 - 8 * q^47 + 22 * q^49 - 9 * q^51 + 6 * q^53 + q^57 - 11 * q^59 - 9 * q^61 - 32 * q^63 - 4 * q^65 + 9 * q^67 - 27 * q^69 - 3 * q^71 + 6 * q^73 + 30 * q^75 - 10 * q^79 - 24 * q^81 - 34 * q^83 - 25 * q^85 - 24 * q^87 - 11 * q^89 - 13 * q^91 + 45 * q^93 + 6 * 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
$3$	1.42458	0.822484	0.411242	−	0.911526i	$-0.365095\pi$
$3$	1.42458	0.822484	0.411242	+	0.911526i	$0.365095\pi$
$4$	0	0
$5$	−2.11046	−0.943825	−0.471912	−	0.881645i	$-0.656436\pi$
$5$	−2.11046	−0.943825	−0.471912	+	0.881645i	$0.656436\pi$
$6$	0	0
$7$	3.55881	1.34510	0.672551	−	0.740051i	$-0.265199\pi$
$7$	3.55881	1.34510	0.672551	+	0.740051i	$0.265199\pi$
$8$	0	0
$9$	−0.970558	−0.323519
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	4.05605	1.12495	0.562473	−	0.826816i	$-0.309850\pi$
$13$	4.05605	1.12495	0.562473	+	0.826816i	$0.309850\pi$
$14$	0	0
$15$	−3.00652	−0.776281
$16$	0	0
$17$	4.39668	1.06635	0.533175	−	0.846005i	$-0.320999\pi$
$17$	4.39668	1.06635	0.533175	+	0.846005i	$0.320999\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	5.06982	1.10633
$22$	0	0
$23$	−8.50071	−1.77252	−0.886261	−	0.463187i	$-0.846706\pi$
$23$	−8.50071	−1.77252	−0.886261	+	0.463187i	$0.846706\pi$
$24$	0	0
$25$	−0.545973	−0.109195
$26$	0	0
$27$	−5.65640	−1.08857
$28$	0	0
$29$	−8.32874	−1.54661	−0.773304	−	0.634036i	$-0.781397\pi$
$29$	−8.32874	−1.54661	−0.773304	+	0.634036i	$0.781397\pi$
$30$	0	0
$31$	−0.583258	−0.104756	−0.0523781	−	0.998627i	$-0.516680\pi$
$31$	−0.583258	−0.104756	−0.0523781	+	0.998627i	$0.516680\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−7.51070	−1.26954
$36$	0	0
$37$	−2.92957	−0.481618	−0.240809	−	0.970573i	$-0.577413\pi$
$37$	−2.92957	−0.481618	−0.240809	+	0.970573i	$0.577413\pi$
$38$	0	0
$39$	5.77818	0.925250
$40$	0	0
$41$	−0.892281	−0.139351	−0.0696754	−	0.997570i	$-0.522196\pi$
$41$	−0.892281	−0.139351	−0.0696754	+	0.997570i	$0.522196\pi$
$42$	0	0
$43$	−8.63194	−1.31636	−0.658180	−	0.752861i	$-0.728673\pi$
$43$	−8.63194	−1.31636	−0.658180	+	0.752861i	$0.728673\pi$
$44$	0	0
$45$	2.04832	0.305346
$46$	0	0
$47$	−0.756169	−0.110299	−0.0551493	−	0.998478i	$-0.517563\pi$
$47$	−0.756169	−0.110299	−0.0551493	+	0.998478i	$0.517563\pi$
$48$	0	0
$49$	5.66510	0.809300
$50$	0	0
$51$	6.26344	0.877057
$52$	0	0
$53$	1.53431	0.210754	0.105377	−	0.994432i	$-0.466395\pi$
$53$	1.53431	0.210754	0.105377	+	0.994432i	$0.466395\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−1.42458	−0.188691
$58$	0	0
$59$	−13.9436	−1.81531	−0.907653	−	0.419722i	$-0.862127\pi$
$59$	−13.9436	−1.81531	−0.907653	+	0.419722i	$0.862127\pi$
$60$	0	0
$61$	14.9005	1.90781	0.953907	−	0.300101i	$-0.0970203\pi$
$61$	14.9005	1.90781	0.953907	+	0.300101i	$0.0970203\pi$
$62$	0	0
$63$	−3.45403	−0.435166
$64$	0	0
$65$	−8.56011	−1.06175
$66$	0	0
$67$	1.09562	0.133851	0.0669255	−	0.997758i	$-0.478681\pi$
$67$	1.09562	0.133851	0.0669255	+	0.997758i	$0.478681\pi$
$68$	0	0
$69$	−12.1100	−1.45787
$70$	0	0
$71$	−4.46831	−0.530291	−0.265146	−	0.964208i	$-0.585420\pi$
$71$	−4.46831	−0.530291	−0.265146	+	0.964208i	$0.585420\pi$
$72$	0	0
$73$	11.4022	1.33452	0.667261	−	0.744824i	$-0.267467\pi$
$73$	11.4022	1.33452	0.667261	+	0.744824i	$0.267467\pi$
$74$	0	0
$75$	−0.777785	−0.0898109
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−14.7613	−1.66077	−0.830387	−	0.557187i	$-0.811881\pi$
$79$	−14.7613	−1.66077	−0.830387	+	0.557187i	$0.811881\pi$
$80$	0	0
$81$	−5.14634	−0.571816
$82$	0	0
$83$	−5.34261	−0.586427	−0.293214	−	0.956047i	$-0.594725\pi$
$83$	−5.34261	−0.586427	−0.293214	+	0.956047i	$0.594725\pi$
$84$	0	0
$85$	−9.27900	−1.00645
$86$	0	0
$87$	−11.8650	−1.27206
$88$	0	0
$89$	1.99012	0.210953	0.105476	−	0.994422i	$-0.466363\pi$
$89$	1.99012	0.210953	0.105476	+	0.994422i	$0.466363\pi$
$90$	0	0
$91$	14.4347	1.51317
$92$	0	0
$93$	−0.830900	−0.0861603
$94$	0	0
$95$	2.11046	0.216528
$96$	0	0
$97$	2.89451	0.293893	0.146946	−	0.989144i	$-0.453056\pi$
$97$	2.89451	0.293893	0.146946	+	0.989144i	$0.453056\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−7.93765	−0.789826	−0.394913	−	0.918719i	$-0.629225\pi$
$101$	−7.93765	−0.789826	−0.394913	+	0.918719i	$0.629225\pi$
$102$	0	0
$103$	−10.6950	−1.05381	−0.526905	−	0.849924i	$-0.676648\pi$
$103$	−10.6950	−1.05381	−0.526905	+	0.849924i	$0.676648\pi$
$104$	0	0
$105$	−10.6996	−1.04418
$106$	0	0
$107$	−1.53788	−0.148672	−0.0743361	−	0.997233i	$-0.523684\pi$
$107$	−1.53788	−0.148672	−0.0743361	+	0.997233i	$0.523684\pi$
$108$	0	0
$109$	−5.73300	−0.549122	−0.274561	−	0.961570i	$-0.588533\pi$
$109$	−5.73300	−0.549122	−0.274561	+	0.961570i	$0.588533\pi$
$110$	0	0
$111$	−4.17341	−0.396123
$112$	0	0
$113$	−9.55906	−0.899241	−0.449620	−	0.893220i	$-0.648441\pi$
$113$	−9.55906	−0.899241	−0.449620	+	0.893220i	$0.648441\pi$
$114$	0	0
$115$	17.9404	1.67295
$116$	0	0
$117$	−3.93663	−0.363941
$118$	0	0
$119$	15.6469	1.43435
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−1.27113	−0.114614
$124$	0	0
$125$	11.7045	1.04689
$126$	0	0
$127$	3.07566	0.272921	0.136460	−	0.990646i	$-0.456427\pi$
$127$	3.07566	0.272921	0.136460	+	0.990646i	$0.456427\pi$
$128$	0	0
$129$	−12.2969	−1.08268
$130$	0	0
$131$	−17.0055	−1.48578	−0.742891	−	0.669413i	$-0.766546\pi$
$131$	−17.0055	−1.48578	−0.742891	+	0.669413i	$0.766546\pi$
$132$	0	0
$133$	−3.55881	−0.308588
$134$	0	0
$135$	11.9376	1.02742
$136$	0	0
$137$	13.0582	1.11564	0.557818	−	0.829964i	$-0.311639\pi$
$137$	13.0582	1.11564	0.557818	+	0.829964i	$0.311639\pi$
$138$	0	0
$139$	7.45893	0.632658	0.316329	−	0.948650i	$-0.397550\pi$
$139$	7.45893	0.632658	0.316329	+	0.948650i	$0.397550\pi$
$140$	0	0
$141$	−1.07723	−0.0907189
$142$	0	0
$143$	0	0
$144$	0	0
$145$	17.5774	1.45973
$146$	0	0
$147$	8.07041	0.665636
$148$	0	0
$149$	12.9688	1.06245	0.531224	−	0.847231i	$-0.321732\pi$
$149$	12.9688	1.06245	0.531224	+	0.847231i	$0.321732\pi$
$150$	0	0
$151$	−4.28916	−0.349047	−0.174524	−	0.984653i	$-0.555839\pi$
$151$	−4.28916	−0.349047	−0.174524	+	0.984653i	$0.555839\pi$
$152$	0	0
$153$	−4.26723	−0.344985
$154$	0	0
$155$	1.23094	0.0988715
$156$	0	0
$157$	19.5094	1.55702	0.778508	−	0.627634i	$-0.215977\pi$
$157$	19.5094	1.55702	0.778508	+	0.627634i	$0.215977\pi$
$158$	0	0
$159$	2.18576	0.173342
$160$	0	0
$161$	−30.2524	−2.38422
$162$	0	0
$163$	12.2011	0.955664	0.477832	−	0.878451i	$-0.341423\pi$
$163$	12.2011	0.955664	0.477832	+	0.878451i	$0.341423\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	14.8148	1.14641	0.573203	−	0.819413i	$-0.305701\pi$
$167$	14.8148	1.14641	0.573203	+	0.819413i	$0.305701\pi$
$168$	0	0
$169$	3.45152	0.265502
$170$	0	0
$171$	0.970558	0.0742204
$172$	0	0
$173$	−12.7016	−0.965683	−0.482841	−	0.875708i	$-0.660395\pi$
$173$	−12.7016	−0.965683	−0.482841	+	0.875708i	$0.660395\pi$
$174$	0	0
$175$	−1.94301	−0.146878
$176$	0	0
$177$	−19.8639	−1.49306
$178$	0	0
$179$	−0.870507	−0.0650647	−0.0325324	−	0.999471i	$-0.510357\pi$
$179$	−0.870507	−0.0650647	−0.0325324	+	0.999471i	$0.510357\pi$
$180$	0	0
$181$	23.0942	1.71658	0.858290	−	0.513165i	$-0.171527\pi$
$181$	23.0942	1.71658	0.858290	+	0.513165i	$0.171527\pi$
$182$	0	0
$183$	21.2270	1.56915
$184$	0	0
$185$	6.18272	0.454563
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−20.1300	−1.46424
$190$	0	0
$191$	−6.44949	−0.466669	−0.233334	−	0.972397i	$-0.574964\pi$
$191$	−6.44949	−0.466669	−0.233334	+	0.972397i	$0.574964\pi$
$192$	0	0
$193$	−21.7914	−1.56858	−0.784289	−	0.620396i	$-0.786972\pi$
$193$	−21.7914	−1.56858	−0.784289	+	0.620396i	$0.786972\pi$
$194$	0	0
$195$	−12.1946	−0.873274
$196$	0	0
$197$	19.7491	1.40706	0.703532	−	0.710663i	$-0.251605\pi$
$197$	19.7491	1.40706	0.703532	+	0.710663i	$0.251605\pi$
$198$	0	0
$199$	−11.6279	−0.824281	−0.412140	−	0.911120i	$-0.635219\pi$
$199$	−11.6279	−0.824281	−0.412140	+	0.911120i	$0.635219\pi$
$200$	0	0
$201$	1.56080	0.110090
$202$	0	0
$203$	−29.6404	−2.08034
$204$	0	0
$205$	1.88312	0.131523
$206$	0	0
$207$	8.25043	0.573445
$208$	0	0
$209$	0	0
$210$	0	0
$211$	23.6405	1.62748	0.813740	−	0.581230i	$-0.197428\pi$
$211$	23.6405	1.62748	0.813740	+	0.581230i	$0.197428\pi$
$212$	0	0
$213$	−6.36549	−0.436156
$214$	0	0
$215$	18.2173	1.24241
$216$	0	0
$217$	−2.07570	−0.140908
$218$	0	0
$219$	16.2433	1.09762
$220$	0	0
$221$	17.8331	1.19959
$222$	0	0
$223$	−19.1096	−1.27967	−0.639837	−	0.768511i	$-0.720998\pi$
$223$	−19.1096	−1.27967	−0.639837	+	0.768511i	$0.720998\pi$
$224$	0	0
$225$	0.529898	0.0353266
$226$	0	0
$227$	−20.0447	−1.33041	−0.665206	−	0.746660i	$-0.731656\pi$
$227$	−20.0447	−1.33041	−0.665206	+	0.746660i	$0.731656\pi$
$228$	0	0
$229$	−25.3615	−1.67593	−0.837967	−	0.545721i	$-0.816256\pi$
$229$	−25.3615	−1.67593	−0.837967	+	0.545721i	$0.816256\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	13.7132	0.898380	0.449190	−	0.893436i	$-0.351713\pi$
$233$	13.7132	0.898380	0.449190	+	0.893436i	$0.351713\pi$
$234$	0	0
$235$	1.59586	0.104103
$236$	0	0
$237$	−21.0287	−1.36596
$238$	0	0
$239$	−7.53415	−0.487344	−0.243672	−	0.969858i	$-0.578352\pi$
$239$	−7.53415	−0.487344	−0.243672	+	0.969858i	$0.578352\pi$
$240$	0	0
$241$	−21.0167	−1.35380	−0.676902	−	0.736073i	$-0.736678\pi$
$241$	−21.0167	−1.35380	−0.676902	+	0.736073i	$0.736678\pi$
$242$	0	0
$243$	9.63779	0.618264
$244$	0	0
$245$	−11.9559	−0.763837
$246$	0	0
$247$	−4.05605	−0.258080
$248$	0	0
$249$	−7.61100	−0.482327
$250$	0	0
$251$	19.6975	1.24329	0.621646	−	0.783298i	$-0.286464\pi$
$251$	19.6975	1.24329	0.621646	+	0.783298i	$0.286464\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−13.2187	−0.827788
$256$	0	0
$257$	−3.52523	−0.219898	−0.109949	−	0.993937i	$-0.535069\pi$
$257$	−3.52523	−0.219898	−0.109949	+	0.993937i	$0.535069\pi$
$258$	0	0
$259$	−10.4258	−0.647825
$260$	0	0
$261$	8.08352	0.500357
$262$	0	0
$263$	−5.91794	−0.364916	−0.182458	−	0.983214i	$-0.558405\pi$
$263$	−5.91794	−0.364916	−0.182458	+	0.983214i	$0.558405\pi$
$264$	0	0
$265$	−3.23810	−0.198915
$266$	0	0
$267$	2.83510	0.173505
$268$	0	0
$269$	−29.6651	−1.80871	−0.904356	−	0.426778i	$-0.859649\pi$
$269$	−29.6651	−1.80871	−0.904356	+	0.426778i	$0.859649\pi$
$270$	0	0
$271$	−0.413676	−0.0251290	−0.0125645	−	0.999921i	$-0.504000\pi$
$271$	−0.413676	−0.0251290	−0.0125645	+	0.999921i	$0.504000\pi$
$272$	0	0
$273$	20.5634	1.24456
$274$	0	0
$275$	0	0
$276$	0	0
$277$	0.205501	0.0123474	0.00617369	−	0.999981i	$-0.498035\pi$
$277$	0.205501	0.0123474	0.00617369	+	0.999981i	$0.498035\pi$
$278$	0	0
$279$	0.566085	0.0338906
$280$	0	0
$281$	−22.2026	−1.32449	−0.662247	−	0.749285i	$-0.730397\pi$
$281$	−22.2026	−1.32449	−0.662247	+	0.749285i	$0.730397\pi$
$282$	0	0
$283$	−1.33796	−0.0795335	−0.0397668	−	0.999209i	$-0.512661\pi$
$283$	−1.33796	−0.0795335	−0.0397668	+	0.999209i	$0.512661\pi$
$284$	0	0
$285$	3.00652	0.178091
$286$	0	0
$287$	−3.17545	−0.187441
$288$	0	0
$289$	2.33077	0.137104
$290$	0	0
$291$	4.12347	0.241722
$292$	0	0
$293$	−3.20807	−0.187417	−0.0937087	−	0.995600i	$-0.529872\pi$
$293$	−3.20807	−0.187417	−0.0937087	+	0.995600i	$0.529872\pi$
$294$	0	0
$295$	29.4274	1.71333
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−34.4793	−1.99399
$300$	0	0
$301$	−30.7194	−1.77064
$302$	0	0
$303$	−11.3079	−0.649619
$304$	0	0
$305$	−31.4469	−1.80064
$306$	0	0
$307$	−10.5284	−0.600887	−0.300443	−	0.953800i	$-0.597135\pi$
$307$	−10.5284	−0.600887	−0.300443	+	0.953800i	$0.597135\pi$
$308$	0	0
$309$	−15.2359	−0.866742
$310$	0	0
$311$	−32.8032	−1.86010	−0.930050	−	0.367433i	$-0.880237\pi$
$311$	−32.8032	−1.86010	−0.930050	+	0.367433i	$0.880237\pi$
$312$	0	0
$313$	−5.75883	−0.325508	−0.162754	−	0.986667i	$-0.552038\pi$
$313$	−5.75883	−0.325508	−0.162754	+	0.986667i	$0.552038\pi$
$314$	0	0
$315$	7.28957	0.410721
$316$	0	0
$317$	24.8593	1.39624	0.698118	−	0.715983i	$-0.254021\pi$
$317$	24.8593	1.39624	0.698118	+	0.715983i	$0.254021\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−2.19084	−0.122281
$322$	0	0
$323$	−4.39668	−0.244638
$324$	0	0
$325$	−2.21449	−0.122838
$326$	0	0
$327$	−8.16715	−0.451644
$328$	0	0
$329$	−2.69106	−0.148363
$330$	0	0
$331$	31.9606	1.75671	0.878357	−	0.478006i	$-0.158640\pi$
$331$	31.9606	1.75671	0.878357	+	0.478006i	$0.158640\pi$
$332$	0	0
$333$	2.84331	0.155813
$334$	0	0
$335$	−2.31225	−0.126332
$336$	0	0
$337$	−1.49742	−0.0815696	−0.0407848	−	0.999168i	$-0.512986\pi$
$337$	−1.49742	−0.0815696	−0.0407848	+	0.999168i	$0.512986\pi$
$338$	0	0
$339$	−13.6177	−0.739611
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−4.75066	−0.256512
$344$	0	0
$345$	25.5576	1.37598
$346$	0	0
$347$	−13.9215	−0.747346	−0.373673	−	0.927561i	$-0.621902\pi$
$347$	−13.9215	−0.747346	−0.373673	+	0.927561i	$0.621902\pi$
$348$	0	0
$349$	−0.0849581	−0.00454770	−0.00227385	−	0.999997i	$-0.500724\pi$
$349$	−0.0849581	−0.00454770	−0.00227385	+	0.999997i	$0.500724\pi$
$350$	0	0
$351$	−22.9426	−1.22459
$352$	0	0
$353$	−4.96951	−0.264500	−0.132250	−	0.991216i	$-0.542220\pi$
$353$	−4.96951	−0.264500	−0.132250	+	0.991216i	$0.542220\pi$
$354$	0	0
$355$	9.43018	0.500502
$356$	0	0
$357$	22.2904	1.17973
$358$	0	0
$359$	30.7695	1.62395	0.811975	−	0.583692i	$-0.198393\pi$
$359$	30.7695	1.62395	0.811975	+	0.583692i	$0.198393\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−24.0638	−1.25956
$366$	0	0
$367$	−27.4654	−1.43368	−0.716841	−	0.697236i	$-0.754413\pi$
$367$	−27.4654	−1.43368	−0.716841	+	0.697236i	$0.754413\pi$
$368$	0	0
$369$	0.866010	0.0450827
$370$	0	0
$371$	5.46031	0.283485
$372$	0	0
$373$	−8.92750	−0.462248	−0.231124	−	0.972924i	$-0.574240\pi$
$373$	−8.92750	−0.462248	−0.231124	+	0.972924i	$0.574240\pi$
$374$	0	0
$375$	16.6741	0.861047
$376$	0	0
$377$	−33.7818	−1.73985
$378$	0	0
$379$	11.5337	0.592448	0.296224	−	0.955118i	$-0.404273\pi$
$379$	11.5337	0.592448	0.296224	+	0.955118i	$0.404273\pi$
$380$	0	0
$381$	4.38154	0.224473
$382$	0	0
$383$	15.7010	0.802283	0.401142	−	0.916016i	$-0.368614\pi$
$383$	15.7010	0.802283	0.401142	+	0.916016i	$0.368614\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	8.37780	0.425868
$388$	0	0
$389$	−0.994188	−0.0504073	−0.0252037	−	0.999682i	$-0.508023\pi$
$389$	−0.994188	−0.0504073	−0.0252037	+	0.999682i	$0.508023\pi$
$390$	0	0
$391$	−37.3749	−1.89013
$392$	0	0
$393$	−24.2258	−1.22203
$394$	0	0
$395$	31.1530	1.56748
$396$	0	0
$397$	24.4782	1.22852	0.614262	−	0.789102i	$-0.289454\pi$
$397$	24.4782	1.22852	0.614262	+	0.789102i	$0.289454\pi$
$398$	0	0
$399$	−5.06982	−0.253808
$400$	0	0
$401$	17.0301	0.850441	0.425221	−	0.905090i	$-0.360196\pi$
$401$	17.0301	0.850441	0.425221	+	0.905090i	$0.360196\pi$
$402$	0	0
$403$	−2.36572	−0.117845
$404$	0	0
$405$	10.8611	0.539694
$406$	0	0
$407$	0	0
$408$	0	0
$409$	12.1977	0.603138	0.301569	−	0.953444i	$-0.402490\pi$
$409$	12.1977	0.603138	0.301569	+	0.953444i	$0.402490\pi$
$410$	0	0
$411$	18.6025	0.917593
$412$	0	0
$413$	−49.6226	−2.44177
$414$	0	0
$415$	11.2753	0.553485
$416$	0	0
$417$	10.6259	0.520351
$418$	0	0
$419$	−6.93005	−0.338555	−0.169278	−	0.985568i	$-0.554143\pi$
$419$	−6.93005	−0.338555	−0.169278	+	0.985568i	$0.554143\pi$
$420$	0	0
$421$	−38.3418	−1.86866	−0.934332	−	0.356403i	$-0.884003\pi$
$421$	−38.3418	−1.86866	−0.934332	+	0.356403i	$0.884003\pi$
$422$	0	0
$423$	0.733906	0.0356837
$424$	0	0
$425$	−2.40047	−0.116440
$426$	0	0
$427$	53.0280	2.56621
$428$	0	0
$429$	0	0
$430$	0	0
$431$	24.5731	1.18364	0.591822	−	0.806068i	$-0.298409\pi$
$431$	24.5731	1.18364	0.591822	+	0.806068i	$0.298409\pi$
$432$	0	0
$433$	−34.6709	−1.66618	−0.833088	−	0.553140i	$-0.813430\pi$
$433$	−34.6709	−1.66618	−0.833088	+	0.553140i	$0.813430\pi$
$434$	0	0
$435$	25.0406	1.20060
$436$	0	0
$437$	8.50071	0.406644
$438$	0	0
$439$	22.5238	1.07500	0.537502	−	0.843263i	$-0.319368\pi$
$439$	22.5238	1.07500	0.537502	+	0.843263i	$0.319368\pi$
$440$	0	0
$441$	−5.49830	−0.261824
$442$	0	0
$443$	22.7214	1.07953	0.539763	−	0.841817i	$-0.318514\pi$
$443$	22.7214	1.07953	0.539763	+	0.841817i	$0.318514\pi$
$444$	0	0
$445$	−4.20007	−0.199102
$446$	0	0
$447$	18.4752	0.873847
$448$	0	0
$449$	−12.1007	−0.571068	−0.285534	−	0.958369i	$-0.592171\pi$
$449$	−12.1007	−0.571068	−0.285534	+	0.958369i	$0.592171\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−6.11027	−0.287086
$454$	0	0
$455$	−30.4638	−1.42816
$456$	0	0
$457$	9.84939	0.460735	0.230367	−	0.973104i	$-0.426007\pi$
$457$	9.84939	0.460735	0.230367	+	0.973104i	$0.426007\pi$
$458$	0	0
$459$	−24.8694	−1.16080
$460$	0	0
$461$	−15.9844	−0.744468	−0.372234	−	0.928139i	$-0.621408\pi$
$461$	−15.9844	−0.744468	−0.372234	+	0.928139i	$0.621408\pi$
$462$	0	0
$463$	−28.9974	−1.34763	−0.673813	−	0.738902i	$-0.735345\pi$
$463$	−28.9974	−1.34763	−0.673813	+	0.738902i	$0.735345\pi$
$464$	0	0
$465$	1.75358	0.0813203
$466$	0	0
$467$	−17.9635	−0.831253	−0.415627	−	0.909535i	$-0.636438\pi$
$467$	−17.9635	−0.831253	−0.415627	+	0.909535i	$0.636438\pi$
$468$	0	0
$469$	3.89909	0.180043
$470$	0	0
$471$	27.7927	1.28062
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0.545973	0.0250510
$476$	0	0
$477$	−1.48914	−0.0681829
$478$	0	0
$479$	−1.72715	−0.0789157	−0.0394578	−	0.999221i	$-0.512563\pi$
$479$	−1.72715	−0.0789157	−0.0394578	+	0.999221i	$0.512563\pi$
$480$	0	0
$481$	−11.8825	−0.541793
$482$	0	0
$483$	−43.0971	−1.96099
$484$	0	0
$485$	−6.10873	−0.277383
$486$	0	0
$487$	−25.2254	−1.14307	−0.571536	−	0.820577i	$-0.693652\pi$
$487$	−25.2254	−1.14307	−0.571536	+	0.820577i	$0.693652\pi$
$488$	0	0
$489$	17.3815	0.786019
$490$	0	0
$491$	−37.2934	−1.68303	−0.841515	−	0.540234i	$-0.818336\pi$
$491$	−37.2934	−1.68303	−0.841515	+	0.540234i	$0.818336\pi$
$492$	0	0
$493$	−36.6188	−1.64923
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−15.9019	−0.713296
$498$	0	0
$499$	34.4064	1.54024	0.770121	−	0.637898i	$-0.220196\pi$
$499$	34.4064	1.54024	0.770121	+	0.637898i	$0.220196\pi$
$500$	0	0
$501$	21.1050	0.942901
$502$	0	0
$503$	30.8729	1.37655	0.688277	−	0.725448i	$-0.258367\pi$
$503$	30.8729	1.37655	0.688277	+	0.725448i	$0.258367\pi$
$504$	0	0
$505$	16.7521	0.745457
$506$	0	0
$507$	4.91699	0.218371
$508$	0	0
$509$	−1.07504	−0.0476505	−0.0238253	−	0.999716i	$-0.507585\pi$
$509$	−1.07504	−0.0476505	−0.0238253	+	0.999716i	$0.507585\pi$
$510$	0	0
$511$	40.5781	1.79507
$512$	0	0
$513$	5.65640	0.249736
$514$	0	0
$515$	22.5713	0.994612
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−18.0945	−0.794259
$520$	0	0
$521$	−7.68815	−0.336824	−0.168412	−	0.985717i	$-0.553864\pi$
$521$	−7.68815	−0.336824	−0.168412	+	0.985717i	$0.553864\pi$
$522$	0	0
$523$	3.26038	0.142567	0.0712833	−	0.997456i	$-0.477291\pi$
$523$	3.26038	0.142567	0.0712833	+	0.997456i	$0.477291\pi$
$524$	0	0
$525$	−2.76799	−0.120805
$526$	0	0
$527$	−2.56440	−0.111707
$528$	0	0
$529$	49.2621	2.14183
$530$	0	0
$531$	13.5331	0.587286
$532$	0	0
$533$	−3.61913	−0.156762
$534$	0	0
$535$	3.24562	0.140321
$536$	0	0
$537$	−1.24011	−0.0535147
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−22.6124	−0.972183	−0.486092	−	0.873908i	$-0.661578\pi$
$541$	−22.6124	−0.972183	−0.486092	+	0.873908i	$0.661578\pi$
$542$	0	0
$543$	32.8997	1.41186
$544$	0	0
$545$	12.0993	0.518275
$546$	0	0
$547$	3.18772	0.136297	0.0681485	−	0.997675i	$-0.478291\pi$
$547$	3.18772	0.136297	0.0681485	+	0.997675i	$0.478291\pi$
$548$	0	0
$549$	−14.4618	−0.617215
$550$	0	0
$551$	8.32874	0.354816
$552$	0	0
$553$	−52.5325	−2.23391
$554$	0	0
$555$	8.80781	0.373871
$556$	0	0
$557$	18.5279	0.785054	0.392527	−	0.919741i	$-0.371601\pi$
$557$	18.5279	0.785054	0.392527	+	0.919741i	$0.371601\pi$
$558$	0	0
$559$	−35.0116	−1.48083
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−6.91255	−0.291329	−0.145665	−	0.989334i	$-0.546532\pi$
$563$	−6.91255	−0.291329	−0.145665	+	0.989334i	$0.546532\pi$
$564$	0	0
$565$	20.1740	0.848726
$566$	0	0
$567$	−18.3148	−0.769151
$568$	0	0
$569$	−18.8169	−0.788844	−0.394422	−	0.918929i	$-0.629055\pi$
$569$	−18.8169	−0.788844	−0.394422	+	0.918929i	$0.629055\pi$
$570$	0	0
$571$	7.93105	0.331904	0.165952	−	0.986134i	$-0.446930\pi$
$571$	7.93105	0.331904	0.165952	+	0.986134i	$0.446930\pi$
$572$	0	0
$573$	−9.18785	−0.383828
$574$	0	0
$575$	4.64116	0.193550
$576$	0	0
$577$	1.48485	0.0618152	0.0309076	−	0.999522i	$-0.490160\pi$
$577$	1.48485	0.0618152	0.0309076	+	0.999522i	$0.490160\pi$
$578$	0	0
$579$	−31.0437	−1.29013
$580$	0	0
$581$	−19.0133	−0.788805
$582$	0	0
$583$	0	0
$584$	0	0
$585$	8.30808	0.343497
$586$	0	0
$587$	−16.5713	−0.683971	−0.341986	−	0.939705i	$-0.611099\pi$
$587$	−16.5713	−0.683971	−0.341986	+	0.939705i	$0.611099\pi$
$588$	0	0
$589$	0.583258	0.0240327
$590$	0	0
$591$	28.1343	1.15729
$592$	0	0
$593$	12.1402	0.498540	0.249270	−	0.968434i	$-0.419809\pi$
$593$	12.1402	0.498540	0.249270	+	0.968434i	$0.419809\pi$
$594$	0	0
$595$	−33.0221	−1.35378
$596$	0	0
$597$	−16.5649	−0.677958
$598$	0	0
$599$	29.0057	1.18514	0.592570	−	0.805519i	$-0.298113\pi$
$599$	29.0057	1.18514	0.592570	+	0.805519i	$0.298113\pi$
$600$	0	0
$601$	−4.43059	−0.180728	−0.0903638	−	0.995909i	$-0.528803\pi$
$601$	−4.43059	−0.180728	−0.0903638	+	0.995909i	$0.528803\pi$
$602$	0	0
$603$	−1.06336	−0.0433034
$604$	0	0
$605$	0	0
$606$	0	0
$607$	5.86365	0.237998	0.118999	−	0.992894i	$-0.462031\pi$
$607$	5.86365	0.237998	0.118999	+	0.992894i	$0.462031\pi$
$608$	0	0
$609$	−42.2252	−1.71105
$610$	0	0
$611$	−3.06706	−0.124080
$612$	0	0
$613$	−16.6576	−0.672794	−0.336397	−	0.941720i	$-0.609208\pi$
$613$	−16.6576	−0.672794	−0.336397	+	0.941720i	$0.609208\pi$
$614$	0	0
$615$	2.68266	0.108175
$616$	0	0
$617$	7.81920	0.314789	0.157395	−	0.987536i	$-0.449691\pi$
$617$	7.81920	0.314789	0.157395	+	0.987536i	$0.449691\pi$
$618$	0	0
$619$	−27.8786	−1.12054	−0.560268	−	0.828311i	$-0.689302\pi$
$619$	−27.8786	−1.12054	−0.560268	+	0.828311i	$0.689302\pi$
$620$	0	0
$621$	48.0834	1.92952
$622$	0	0
$623$	7.08247	0.283753
$624$	0	0
$625$	−21.9720	−0.878882
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−12.8804	−0.513573
$630$	0	0
$631$	31.0238	1.23504	0.617518	−	0.786557i	$-0.288138\pi$
$631$	31.0238	1.23504	0.617518	+	0.786557i	$0.288138\pi$
$632$	0	0
$633$	33.6779	1.33858
$634$	0	0
$635$	−6.49105	−0.257589
$636$	0	0
$637$	22.9779	0.910418
$638$	0	0
$639$	4.33676	0.171559
$640$	0	0
$641$	−24.5560	−0.969905	−0.484952	−	0.874541i	$-0.661163\pi$
$641$	−24.5560	−0.969905	−0.484952	+	0.874541i	$0.661163\pi$
$642$	0	0
$643$	−23.7262	−0.935669	−0.467834	−	0.883816i	$-0.654966\pi$
$643$	−23.7262	−0.935669	−0.467834	+	0.883816i	$0.654966\pi$
$644$	0	0
$645$	25.9522	1.02187
$646$	0	0
$647$	−43.1207	−1.69525	−0.847625	−	0.530596i	$-0.821968\pi$
$647$	−43.1207	−1.69525	−0.847625	+	0.530596i	$0.821968\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−2.95701	−0.115894
$652$	0	0
$653$	−14.3592	−0.561917	−0.280959	−	0.959720i	$-0.590652\pi$
$653$	−14.3592	−0.561917	−0.280959	+	0.959720i	$0.590652\pi$
$654$	0	0
$655$	35.8895	1.40232
$656$	0	0
$657$	−11.0665	−0.431744
$658$	0	0
$659$	−34.4316	−1.34126	−0.670632	−	0.741790i	$-0.733977\pi$
$659$	−34.4316	−1.34126	−0.670632	+	0.741790i	$0.733977\pi$
$660$	0	0
$661$	39.5074	1.53666	0.768330	−	0.640054i	$-0.221088\pi$
$661$	39.5074	1.53666	0.768330	+	0.640054i	$0.221088\pi$
$662$	0	0
$663$	25.4048	0.986641
$664$	0	0
$665$	7.51070	0.291253
$666$	0	0
$667$	70.8002	2.74139
$668$	0	0
$669$	−27.2233	−1.05251
$670$	0	0
$671$	0	0
$672$	0	0
$673$	28.4691	1.09740	0.548701	−	0.836019i	$-0.315123\pi$
$673$	28.4691	1.09740	0.548701	+	0.836019i	$0.315123\pi$
$674$	0	0
$675$	3.08824	0.118866
$676$	0	0
$677$	−18.5874	−0.714370	−0.357185	−	0.934034i	$-0.616263\pi$
$677$	−18.5874	−0.714370	−0.357185	+	0.934034i	$0.616263\pi$
$678$	0	0
$679$	10.3010	0.395316
$680$	0	0
$681$	−28.5553	−1.09424
$682$	0	0
$683$	30.1245	1.15268	0.576340	−	0.817210i	$-0.304480\pi$
$683$	30.1245	1.15268	0.576340	+	0.817210i	$0.304480\pi$
$684$	0	0
$685$	−27.5587	−1.05296
$686$	0	0
$687$	−36.1296	−1.37843
$688$	0	0
$689$	6.22324	0.237086
$690$	0	0
$691$	−0.959889	−0.0365159	−0.0182579	−	0.999833i	$-0.505812\pi$
$691$	−0.959889	−0.0365159	−0.0182579	+	0.999833i	$0.505812\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−15.7417	−0.597118
$696$	0	0
$697$	−3.92307	−0.148597
$698$	0	0
$699$	19.5356	0.738904
$700$	0	0
$701$	18.0325	0.681079	0.340539	−	0.940230i	$-0.389390\pi$
$701$	18.0325	0.681079	0.340539	+	0.940230i	$0.389390\pi$
$702$	0	0
$703$	2.92957	0.110491
$704$	0	0
$705$	2.27344	0.0856228
$706$	0	0
$707$	−28.2486	−1.06240
$708$	0	0
$709$	−2.59735	−0.0975455	−0.0487727	−	0.998810i	$-0.515531\pi$
$709$	−2.59735	−0.0975455	−0.0487727	+	0.998810i	$0.515531\pi$
$710$	0	0
$711$	14.3267	0.537292
$712$	0	0
$713$	4.95811	0.185683
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−10.7330	−0.400833
$718$	0	0
$719$	8.62999	0.321844	0.160922	−	0.986967i	$-0.448553\pi$
$719$	8.62999	0.321844	0.160922	+	0.986967i	$0.448553\pi$
$720$	0	0
$721$	−38.0614	−1.41748
$722$	0	0
$723$	−29.9401	−1.11348
$724$	0	0
$725$	4.54727	0.168881
$726$	0	0
$727$	−37.2249	−1.38059	−0.690297	−	0.723526i	$-0.742520\pi$
$727$	−37.2249	−1.38059	−0.690297	+	0.723526i	$0.742520\pi$
$728$	0	0
$729$	29.1689	1.08033
$730$	0	0
$731$	−37.9519	−1.40370
$732$	0	0
$733$	29.5022	1.08969	0.544845	−	0.838537i	$-0.316589\pi$
$733$	29.5022	1.08969	0.544845	+	0.838537i	$0.316589\pi$
$734$	0	0
$735$	−17.0323	−0.628244
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−40.7293	−1.49825	−0.749125	−	0.662429i	$-0.769526\pi$
$739$	−40.7293	−1.49825	−0.749125	+	0.662429i	$0.769526\pi$
$740$	0	0
$741$	−5.77818	−0.212267
$742$	0	0
$743$	4.26188	0.156353	0.0781766	−	0.996940i	$-0.475090\pi$
$743$	4.26188	0.156353	0.0781766	+	0.996940i	$0.475090\pi$
$744$	0	0
$745$	−27.3702	−1.00277
$746$	0	0
$747$	5.18531	0.189721
$748$	0	0
$749$	−5.47301	−0.199979
$750$	0	0
$751$	12.2072	0.445446	0.222723	−	0.974882i	$-0.428505\pi$
$751$	12.2072	0.445446	0.222723	+	0.974882i	$0.428505\pi$
$752$	0	0
$753$	28.0607	1.02259
$754$	0	0
$755$	9.05209	0.329439
$756$	0	0
$757$	30.2463	1.09932	0.549659	−	0.835389i	$-0.314758\pi$
$757$	30.2463	1.09932	0.549659	+	0.835389i	$0.314758\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	19.8327	0.718934	0.359467	−	0.933158i	$-0.382959\pi$
$761$	19.8327	0.718934	0.359467	+	0.933158i	$0.382959\pi$
$762$	0	0
$763$	−20.4026	−0.738625
$764$	0	0
$765$	9.00580	0.325606
$766$	0	0
$767$	−56.5560	−2.04212
$768$	0	0
$769$	33.2319	1.19837	0.599186	−	0.800610i	$-0.295491\pi$
$769$	33.2319	1.19837	0.599186	+	0.800610i	$0.295491\pi$
$770$	0	0
$771$	−5.02199	−0.180862
$772$	0	0
$773$	20.1676	0.725378	0.362689	−	0.931910i	$-0.381859\pi$
$773$	20.1676	0.725378	0.362689	+	0.931910i	$0.381859\pi$
$774$	0	0
$775$	0.318443	0.0114388
$776$	0	0
$777$	−14.8524	−0.532826
$778$	0	0
$779$	0.892281	0.0319693
$780$	0	0
$781$	0	0
$782$	0	0
$783$	47.1106	1.68360
$784$	0	0
$785$	−41.1737	−1.46955
$786$	0	0
$787$	13.3980	0.477587	0.238794	−	0.971070i	$-0.423248\pi$
$787$	13.3980	0.477587	0.238794	+	0.971070i	$0.423248\pi$
$788$	0	0
$789$	−8.43061	−0.300138
$790$	0	0
$791$	−34.0188	−1.20957
$792$	0	0
$793$	60.4372	2.14619
$794$	0	0
$795$	−4.61294	−0.163604
$796$	0	0
$797$	−46.1047	−1.63311	−0.816556	−	0.577266i	$-0.804119\pi$
$797$	−46.1047	−1.63311	−0.816556	+	0.577266i	$0.804119\pi$
$798$	0	0
$799$	−3.32463	−0.117617
$800$	0	0
$801$	−1.93153	−0.0682473
$802$	0	0
$803$	0	0
$804$	0	0
$805$	63.8463	2.25029
$806$	0	0
$807$	−42.2605	−1.48764
$808$	0	0
$809$	−55.2692	−1.94316	−0.971581	−	0.236709i	$-0.923931\pi$
$809$	−55.2692	−1.94316	−0.971581	+	0.236709i	$0.923931\pi$
$810$	0	0
$811$	−28.4936	−1.00055	−0.500273	−	0.865868i	$-0.666767\pi$
$811$	−28.4936	−1.00055	−0.500273	+	0.865868i	$0.666767\pi$
$812$	0	0
$813$	−0.589317	−0.0206682
$814$	0	0
$815$	−25.7499	−0.901979
$816$	0	0
$817$	8.63194	0.301994
$818$	0	0
$819$	−14.0097	−0.489538
$820$	0	0
$821$	33.6192	1.17332	0.586659	−	0.809834i	$-0.300443\pi$
$821$	33.6192	1.17332	0.586659	+	0.809834i	$0.300443\pi$
$822$	0	0
$823$	6.71808	0.234178	0.117089	−	0.993121i	$-0.462644\pi$
$823$	6.71808	0.234178	0.117089	+	0.993121i	$0.462644\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−12.2762	−0.426885	−0.213443	−	0.976956i	$-0.568468\pi$
$827$	−12.2762	−0.426885	−0.213443	+	0.976956i	$0.568468\pi$
$828$	0	0
$829$	47.6507	1.65498	0.827488	−	0.561483i	$-0.189769\pi$
$829$	47.6507	1.65498	0.827488	+	0.561483i	$0.189769\pi$
$830$	0	0
$831$	0.292754	0.0101555
$832$	0	0
$833$	24.9076	0.862997
$834$	0	0
$835$	−31.2661	−1.08201
$836$	0	0
$837$	3.29914	0.114035
$838$	0	0
$839$	−5.93886	−0.205032	−0.102516	−	0.994731i	$-0.532689\pi$
$839$	−5.93886	−0.205032	−0.102516	+	0.994731i	$0.532689\pi$
$840$	0	0
$841$	40.3678	1.39199
$842$	0	0
$843$	−31.6294	−1.08938
$844$	0	0
$845$	−7.28429	−0.250587
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−1.90604	−0.0654151
$850$	0	0
$851$	24.9034	0.853677
$852$	0	0
$853$	37.6031	1.28751	0.643754	−	0.765233i	$-0.277376\pi$
$853$	37.6031	1.28751	0.643754	+	0.765233i	$0.277376\pi$
$854$	0	0
$855$	−2.04832	−0.0700511
$856$	0	0
$857$	3.66558	0.125214	0.0626069	−	0.998038i	$-0.480059\pi$
$857$	3.66558	0.125214	0.0626069	+	0.998038i	$0.480059\pi$
$858$	0	0
$859$	27.5818	0.941077	0.470539	−	0.882379i	$-0.344060\pi$
$859$	27.5818	0.941077	0.470539	+	0.882379i	$0.344060\pi$
$860$	0	0
$861$	−4.52370	−0.154167
$862$	0	0
$863$	50.2521	1.71060	0.855301	−	0.518131i	$-0.173372\pi$
$863$	50.2521	1.71060	0.855301	+	0.518131i	$0.173372\pi$
$864$	0	0
$865$	26.8061	0.911436
$866$	0	0
$867$	3.32038	0.112766
$868$	0	0
$869$	0	0
$870$	0	0
$871$	4.44388	0.150575
$872$	0	0
$873$	−2.80929	−0.0950799
$874$	0	0
$875$	41.6542	1.40817
$876$	0	0
$877$	28.5790	0.965044	0.482522	−	0.875884i	$-0.339721\pi$
$877$	28.5790	0.965044	0.482522	+	0.875884i	$0.339721\pi$
$878$	0	0
$879$	−4.57017	−0.154148
$880$	0	0
$881$	16.9395	0.570706	0.285353	−	0.958422i	$-0.407889\pi$
$881$	16.9395	0.570706	0.285353	+	0.958422i	$0.407889\pi$
$882$	0	0
$883$	−55.0730	−1.85335	−0.926677	−	0.375860i	$-0.877347\pi$
$883$	−55.0730	−1.85335	−0.926677	+	0.375860i	$0.877347\pi$
$884$	0	0
$885$	41.9218	1.40919
$886$	0	0
$887$	−17.2072	−0.577761	−0.288881	−	0.957365i	$-0.593283\pi$
$887$	−17.2072	−0.577761	−0.288881	+	0.957365i	$0.593283\pi$
$888$	0	0
$889$	10.9457	0.367106
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0.756169	0.0253042
$894$	0	0
$895$	1.83717	0.0614097
$896$	0	0
$897$	−49.1187	−1.64003
$898$	0	0
$899$	4.85780	0.162017
$900$	0	0
$901$	6.74587	0.224737
$902$	0	0
$903$	−43.7624	−1.45632
$904$	0	0
$905$	−48.7393	−1.62015
$906$	0	0
$907$	−29.3310	−0.973918	−0.486959	−	0.873425i	$-0.661894\pi$
$907$	−29.3310	−0.973918	−0.486959	+	0.873425i	$0.661894\pi$
$908$	0	0
$909$	7.70395	0.255524
$910$	0	0
$911$	47.1223	1.56123	0.780616	−	0.625011i	$-0.214906\pi$
$911$	47.1223	1.56123	0.780616	+	0.625011i	$0.214906\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−44.7988	−1.48100
$916$	0	0
$917$	−60.5194	−1.99853
$918$	0	0
$919$	17.1897	0.567036	0.283518	−	0.958967i	$-0.408498\pi$
$919$	17.1897	0.567036	0.283518	+	0.958967i	$0.408498\pi$
$920$	0	0
$921$	−14.9986	−0.494220
$922$	0	0
$923$	−18.1237	−0.596549
$924$	0	0
$925$	1.59946	0.0525900
$926$	0	0
$927$	10.3801	0.340928
$928$	0	0
$929$	−23.4132	−0.768161	−0.384081	−	0.923300i	$-0.625482\pi$
$929$	−23.4132	−0.768161	−0.384081	+	0.923300i	$0.625482\pi$
$930$	0	0
$931$	−5.66510	−0.185666
$932$	0	0
$933$	−46.7310	−1.52990
$934$	0	0
$935$	0	0
$936$	0	0
$937$	1.96478	0.0641864	0.0320932	−	0.999485i	$-0.489783\pi$
$937$	1.96478	0.0641864	0.0320932	+	0.999485i	$0.489783\pi$
$938$	0	0
$939$	−8.20394	−0.267726
$940$	0	0
$941$	39.3081	1.28141	0.640704	−	0.767788i	$-0.278642\pi$
$941$	39.3081	1.28141	0.640704	+	0.767788i	$0.278642\pi$
$942$	0	0
$943$	7.58502	0.247002
$944$	0	0
$945$	42.4835	1.38199
$946$	0	0
$947$	−9.10650	−0.295922	−0.147961	−	0.988993i	$-0.547271\pi$
$947$	−9.10650	−0.295922	−0.147961	+	0.988993i	$0.547271\pi$
$948$	0	0
$949$	46.2477	1.50126
$950$	0	0
$951$	35.4141	1.14838
$952$	0	0
$953$	13.9088	0.450550	0.225275	−	0.974295i	$-0.427672\pi$
$953$	13.9088	0.450550	0.225275	+	0.974295i	$0.427672\pi$
$954$	0	0
$955$	13.6114	0.440454
$956$	0	0
$957$	0	0
$958$	0	0
$959$	46.4715	1.50064
$960$	0	0
$961$	−30.6598	−0.989026
$962$	0	0
$963$	1.49260	0.0480984
$964$	0	0
$965$	45.9898	1.48046
$966$	0	0
$967$	15.3620	0.494009	0.247004	−	0.969014i	$-0.420554\pi$
$967$	15.3620	0.494009	0.247004	+	0.969014i	$0.420554\pi$
$968$	0	0
$969$	−6.26344	−0.201211
$970$	0	0
$971$	−8.30207	−0.266426	−0.133213	−	0.991087i	$-0.542529\pi$
$971$	−8.30207	−0.266426	−0.133213	+	0.991087i	$0.542529\pi$
$972$	0	0
$973$	26.5449	0.850990
$974$	0	0
$975$	−3.15473	−0.101032
$976$	0	0
$977$	−15.1961	−0.486165	−0.243082	−	0.970006i	$-0.578159\pi$
$977$	−15.1961	−0.486165	−0.243082	+	0.970006i	$0.578159\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	5.56421	0.177652
$982$	0	0
$983$	−37.2279	−1.18739	−0.593693	−	0.804692i	$-0.702331\pi$
$983$	−37.2279	−1.18739	−0.593693	+	0.804692i	$0.702331\pi$
$984$	0	0
$985$	−41.6796	−1.32802
$986$	0	0
$987$	−3.83364	−0.122026
$988$	0	0
$989$	73.3777	2.33327
$990$	0	0
$991$	0.913096	0.0290055	0.0145027	−	0.999895i	$-0.495383\pi$
$991$	0.913096	0.0290055	0.0145027	+	0.999895i	$0.495383\pi$
$992$	0	0
$993$	45.5306	1.44487
$994$	0	0
$995$	24.5402	0.777976
$996$	0	0
$997$	−26.2337	−0.830829	−0.415414	−	0.909632i	$-0.636363\pi$
$997$	−26.2337	−0.830829	−0.415414	+	0.909632i	$0.636363\pi$
$998$	0	0
$999$	16.5708	0.524276

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9196.2.a.q.1.6	✓	8
11.10	odd	2		9196.2.a.r.1.6	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9196.2.a.q.1.6	✓	8	1.1	even	1	trivial
9196.2.a.r.1.6	yes	8	11.10	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:	\( N \)	\(=\)	\( 9196 = 2^{2} \cdot 11^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9196.a (trivial)

\(f(q)\)	\(=\)	\(q+1.42458 q^{3} -2.11046 q^{5} +3.55881 q^{7} -0.970558 q^{9} +4.05605 q^{13} -3.00652 q^{15} +4.39668 q^{17} -1.00000 q^{19} +5.06982 q^{21} -8.50071 q^{23} -0.545973 q^{25} -5.65640 q^{27} -8.32874 q^{29} -0.583258 q^{31} -7.51070 q^{35} -2.92957 q^{37} +5.77818 q^{39} -0.892281 q^{41} -8.63194 q^{43} +2.04832 q^{45} -0.756169 q^{47} +5.66510 q^{49} +6.26344 q^{51} +1.53431 q^{53} -1.42458 q^{57} -13.9436 q^{59} +14.9005 q^{61} -3.45403 q^{63} -8.56011 q^{65} +1.09562 q^{67} -12.1100 q^{69} -4.46831 q^{71} +11.4022 q^{73} -0.777785 q^{75} -14.7613 q^{79} -5.14634 q^{81} -5.34261 q^{83} -9.27900 q^{85} -11.8650 q^{87} +1.99012 q^{89} +14.4347 q^{91} -0.830900 q^{93} +2.11046 q^{95} +2.89451 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - q^{3} - 2 q^{7} + 9 q^{9} + 5 q^{13} - 9 q^{15} - 3 q^{17} - 8 q^{19} - 5 q^{21} + q^{23} - q^{27} - q^{29} + 3 q^{31} - 8 q^{35} - 23 q^{37} - 9 q^{39} - 10 q^{41} - 2 q^{43} + 9 q^{45} - 8 q^{47}+ \cdots + 6 q^{97}+O(q^{100}) \) 8 * q - q^3 - 2 * q^7 + 9 * q^9 + 5 * q^13 - 9 * q^15 - 3 * q^17 - 8 * q^19 - 5 * q^21 + q^23 - q^27 - q^29 + 3 * q^31 - 8 * q^35 - 23 * q^37 - 9 * q^39 - 10 * q^41 - 2 * q^43 + 9 * q^45 - 8 * q^47 + 22 * q^49 - 9 * q^51 + 6 * q^53 + q^57 - 11 * q^59 - 9 * q^61 - 32 * q^63 - 4 * q^65 + 9 * q^67 - 27 * q^69 - 3 * q^71 + 6 * q^73 + 30 * q^75 - 10 * q^79 - 24 * q^81 - 34 * q^83 - 25 * q^85 - 24 * q^87 - 11 * q^89 - 13 * q^91 + 45 * q^93 + 6 * q^97

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