LMFDB - Embedded newform 9196.2.a.l.1.5

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 = [6,0,0,0,-4,0,-4,0,2,0,0,0,-8] 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:	$6$
Coefficient field:	6.6.114134848.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 10x^{4} - 6x^{3} + 20x^{2} + 14x - 5 $ x^6 - 10x^4 - 6x^3 + 20x^2 + 14x - 5
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.50521$ 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)

$f(q)$	$=$	$q+1.50521 q^{3} -3.76432 q^{5} +4.72308 q^{7} -0.734348 q^{9} -0.374304 q^{13} -5.66608 q^{15} -2.41704 q^{17} +1.00000 q^{19} +7.10922 q^{21} -4.04039 q^{23} +9.17009 q^{25} -5.62097 q^{27} +8.79699 q^{29} -5.56277 q^{31} -17.7792 q^{35} -4.35412 q^{37} -0.563405 q^{39} +1.30956 q^{41} +7.35588 q^{43} +2.76432 q^{45} -1.25139 q^{47} +15.3075 q^{49} -3.63815 q^{51} -12.6976 q^{53} +1.50521 q^{57} -0.988733 q^{59} +14.7987 q^{61} -3.46838 q^{63} +1.40900 q^{65} -12.1018 q^{67} -6.08163 q^{69} -1.84949 q^{71} -8.51822 q^{73} +13.8029 q^{75} -2.81068 q^{79} -6.25769 q^{81} -16.5999 q^{83} +9.09851 q^{85} +13.2413 q^{87} -7.66431 q^{89} -1.76787 q^{91} -8.37313 q^{93} -3.76432 q^{95} +11.6055 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{5} - 4 q^{7} + 2 q^{9} - 8 q^{13} + 2 q^{15} - 6 q^{17} + 6 q^{19} + 20 q^{21} + 6 q^{23} + 2 q^{25} - 18 q^{27} + 2 q^{29} - 6 q^{31} - 14 q^{35} + 4 q^{37} - 6 q^{39} - 14 q^{41} + 10 q^{43}+ \cdots + 28 q^{97}+O(q^{100}) $ 6 * q - 4 * q^5 - 4 * q^7 + 2 * q^9 - 8 * q^13 + 2 * q^15 - 6 * q^17 + 6 * q^19 + 20 * q^21 + 6 * q^23 + 2 * q^25 - 18 * q^27 + 2 * q^29 - 6 * q^31 - 14 * q^35 + 4 * q^37 - 6 * q^39 - 14 * q^41 + 10 * q^43 - 2 * q^45 + 4 * q^47 + 20 * q^49 - 18 * q^51 + 20 * q^59 + 2 * q^61 - 12 * q^63 + 12 * q^65 + 10 * q^67 - 20 * q^69 - 32 * q^73 + 18 * q^75 - 6 * q^79 - 6 * q^81 - 36 * q^83 - 6 * q^85 + 28 * q^87 - 24 * q^89 - 16 * q^91 + 22 * q^93 - 4 * q^95 + 28 * 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
$3$	1.50521	0.869032	0.434516	−	0.900664i	$-0.356919\pi$
$3$	1.50521	0.869032	0.434516	+	0.900664i	$0.356919\pi$
$4$	0	0
$5$	−3.76432	−1.68345	−0.841727	−	0.539903i	$-0.818461\pi$
$5$	−3.76432	−1.68345	−0.841727	+	0.539903i	$0.818461\pi$
$6$	0	0
$7$	4.72308	1.78516	0.892578	−	0.450892i	$-0.148894\pi$
$7$	4.72308	1.78516	0.892578	+	0.450892i	$0.148894\pi$
$8$	0	0
$9$	−0.734348	−0.244783
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−0.374304	−0.103813	−0.0519066	−	0.998652i	$-0.516530\pi$
$13$	−0.374304	−0.103813	−0.0519066	+	0.998652i	$0.516530\pi$
$14$	0	0
$15$	−5.66608	−1.46298
$16$	0	0
$17$	−2.41704	−0.586218	−0.293109	−	0.956079i	$-0.594690\pi$
$17$	−2.41704	−0.586218	−0.293109	+	0.956079i	$0.594690\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	7.10922	1.55136
$22$	0	0
$23$	−4.04039	−0.842479	−0.421240	−	0.906949i	$-0.638405\pi$
$23$	−4.04039	−0.842479	−0.421240	+	0.906949i	$0.638405\pi$
$24$	0	0
$25$	9.17009	1.83402
$26$	0	0
$27$	−5.62097	−1.08176
$28$	0	0
$29$	8.79699	1.63356	0.816780	−	0.576949i	$-0.195757\pi$
$29$	8.79699	1.63356	0.816780	+	0.576949i	$0.195757\pi$
$30$	0	0
$31$	−5.56277	−0.999104	−0.499552	−	0.866284i	$-0.666502\pi$
$31$	−5.56277	−0.999104	−0.499552	+	0.866284i	$0.666502\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−17.7792	−3.00523
$36$	0	0
$37$	−4.35412	−0.715813	−0.357906	−	0.933757i	$-0.616509\pi$
$37$	−4.35412	−0.715813	−0.357906	+	0.933757i	$0.616509\pi$
$38$	0	0
$39$	−0.563405	−0.0902171
$40$	0	0
$41$	1.30956	0.204520	0.102260	−	0.994758i	$-0.467393\pi$
$41$	1.30956	0.204520	0.102260	+	0.994758i	$0.467393\pi$
$42$	0	0
$43$	7.35588	1.12176	0.560880	−	0.827897i	$-0.310463\pi$
$43$	7.35588	1.12176	0.560880	+	0.827897i	$0.310463\pi$
$44$	0	0
$45$	2.76432	0.412080
$46$	0	0
$47$	−1.25139	−0.182534	−0.0912672	−	0.995826i	$-0.529092\pi$
$47$	−1.25139	−0.182534	−0.0912672	+	0.995826i	$0.529092\pi$
$48$	0	0
$49$	15.3075	2.18678
$50$	0	0
$51$	−3.63815	−0.509443
$52$	0	0
$53$	−12.6976	−1.74414	−0.872072	−	0.489378i	$-0.837224\pi$
$53$	−12.6976	−1.74414	−0.872072	+	0.489378i	$0.837224\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.50521	0.199370
$58$	0	0
$59$	−0.988733	−0.128722	−0.0643610	−	0.997927i	$-0.520501\pi$
$59$	−0.988733	−0.128722	−0.0643610	+	0.997927i	$0.520501\pi$
$60$	0	0
$61$	14.7987	1.89478	0.947392	−	0.320076i	$-0.103709\pi$
$61$	14.7987	1.89478	0.947392	+	0.320076i	$0.103709\pi$
$62$	0	0
$63$	−3.46838	−0.436975
$64$	0	0
$65$	1.40900	0.174765
$66$	0	0
$67$	−12.1018	−1.47847	−0.739237	−	0.673446i	$-0.764814\pi$
$67$	−12.1018	−1.47847	−0.739237	+	0.673446i	$0.764814\pi$
$68$	0	0
$69$	−6.08163	−0.732142
$70$	0	0
$71$	−1.84949	−0.219494	−0.109747	−	0.993960i	$-0.535004\pi$
$71$	−1.84949	−0.219494	−0.109747	+	0.993960i	$0.535004\pi$
$72$	0	0
$73$	−8.51822	−0.996982	−0.498491	−	0.866895i	$-0.666112\pi$
$73$	−8.51822	−0.996982	−0.498491	+	0.866895i	$0.666112\pi$
$74$	0	0
$75$	13.8029	1.59382
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−2.81068	−0.316226	−0.158113	−	0.987421i	$-0.550541\pi$
$79$	−2.81068	−0.316226	−0.158113	+	0.987421i	$0.550541\pi$
$80$	0	0
$81$	−6.25769	−0.695299
$82$	0	0
$83$	−16.5999	−1.82208	−0.911041	−	0.412317i	$-0.864720\pi$
$83$	−16.5999	−1.82208	−0.911041	+	0.412317i	$0.864720\pi$
$84$	0	0
$85$	9.09851	0.986872
$86$	0	0
$87$	13.2413	1.41962
$88$	0	0
$89$	−7.66431	−0.812416	−0.406208	−	0.913781i	$-0.633149\pi$
$89$	−7.66431	−0.812416	−0.406208	+	0.913781i	$0.633149\pi$
$90$	0	0
$91$	−1.76787	−0.185323
$92$	0	0
$93$	−8.37313	−0.868254
$94$	0	0
$95$	−3.76432	−0.386211
$96$	0	0
$97$	11.6055	1.17836	0.589181	−	0.808001i	$-0.299451\pi$
$97$	11.6055	1.17836	0.589181	+	0.808001i	$0.299451\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−10.2253	−1.01746	−0.508729	−	0.860927i	$-0.669884\pi$
$101$	−10.2253	−1.01746	−0.508729	+	0.860927i	$0.669884\pi$
$102$	0	0
$103$	9.18375	0.904901	0.452451	−	0.891789i	$-0.350550\pi$
$103$	9.18375	0.904901	0.452451	+	0.891789i	$0.350550\pi$
$104$	0	0
$105$	−26.7614	−2.61164
$106$	0	0
$107$	−3.19620	−0.308988	−0.154494	−	0.987994i	$-0.549375\pi$
$107$	−3.19620	−0.308988	−0.154494	+	0.987994i	$0.549375\pi$
$108$	0	0
$109$	6.07063	0.581461	0.290730	−	0.956805i	$-0.406102\pi$
$109$	6.07063	0.581461	0.290730	+	0.956805i	$0.406102\pi$
$110$	0	0
$111$	−6.55386	−0.622065
$112$	0	0
$113$	14.1261	1.32887	0.664437	−	0.747344i	$-0.268671\pi$
$113$	14.1261	1.32887	0.664437	+	0.747344i	$0.268671\pi$
$114$	0	0
$115$	15.2093	1.41827
$116$	0	0
$117$	0.274869	0.0254117
$118$	0	0
$119$	−11.4159	−1.04649
$120$	0	0
$121$	0	0
$122$	0	0
$123$	1.97117	0.177734
$124$	0	0
$125$	−15.6976	−1.40403
$126$	0	0
$127$	−5.51357	−0.489250	−0.244625	−	0.969618i	$-0.578665\pi$
$127$	−5.51357	−0.489250	−0.244625	+	0.969618i	$0.578665\pi$
$128$	0	0
$129$	11.0721	0.974846
$130$	0	0
$131$	−8.00626	−0.699510	−0.349755	−	0.936841i	$-0.613735\pi$
$131$	−8.00626	−0.699510	−0.349755	+	0.936841i	$0.613735\pi$
$132$	0	0
$133$	4.72308	0.409543
$134$	0	0
$135$	21.1591	1.82109
$136$	0	0
$137$	−17.2250	−1.47163	−0.735816	−	0.677181i	$-0.763201\pi$
$137$	−17.2250	−1.47163	−0.735816	+	0.677181i	$0.763201\pi$
$138$	0	0
$139$	7.70353	0.653405	0.326702	−	0.945127i	$-0.394063\pi$
$139$	7.70353	0.653405	0.326702	+	0.945127i	$0.394063\pi$
$140$	0	0
$141$	−1.88361	−0.158628
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−33.1147	−2.75002
$146$	0	0
$147$	23.0410	1.90039
$148$	0	0
$149$	8.69755	0.712531	0.356266	−	0.934385i	$-0.384050\pi$
$149$	8.69755	0.712531	0.356266	+	0.934385i	$0.384050\pi$
$150$	0	0
$151$	−19.0525	−1.55047	−0.775236	−	0.631672i	$-0.782369\pi$
$151$	−19.0525	−1.55047	−0.775236	+	0.631672i	$0.782369\pi$
$152$	0	0
$153$	1.77495	0.143496
$154$	0	0
$155$	20.9401	1.68195
$156$	0	0
$157$	1.57730	0.125882	0.0629410	−	0.998017i	$-0.479952\pi$
$157$	1.57730	0.125882	0.0629410	+	0.998017i	$0.479952\pi$
$158$	0	0
$159$	−19.1125	−1.51572
$160$	0	0
$161$	−19.0831	−1.50396
$162$	0	0
$163$	−3.61351	−0.283032	−0.141516	−	0.989936i	$-0.545198\pi$
$163$	−3.61351	−0.283032	−0.141516	+	0.989936i	$0.545198\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−20.3660	−1.57597	−0.787984	−	0.615696i	$-0.788875\pi$
$167$	−20.3660	−1.57597	−0.787984	+	0.615696i	$0.788875\pi$
$168$	0	0
$169$	−12.8599	−0.989223
$170$	0	0
$171$	−0.734348	−0.0561570
$172$	0	0
$173$	21.8886	1.66416	0.832080	−	0.554655i	$-0.187150\pi$
$173$	21.8886	1.66416	0.832080	+	0.554655i	$0.187150\pi$
$174$	0	0
$175$	43.3111	3.27401
$176$	0	0
$177$	−1.48825	−0.111864
$178$	0	0
$179$	−16.0336	−1.19840	−0.599202	−	0.800598i	$-0.704515\pi$
$179$	−16.0336	−1.19840	−0.599202	+	0.800598i	$0.704515\pi$
$180$	0	0
$181$	−3.89824	−0.289754	−0.144877	−	0.989450i	$-0.546279\pi$
$181$	−3.89824	−0.289754	−0.144877	+	0.989450i	$0.546279\pi$
$182$	0	0
$183$	22.2752	1.64663
$184$	0	0
$185$	16.3903	1.20504
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−26.5483	−1.93110
$190$	0	0
$191$	4.25769	0.308076	0.154038	−	0.988065i	$-0.450772\pi$
$191$	4.25769	0.308076	0.154038	+	0.988065i	$0.450772\pi$
$192$	0	0
$193$	−2.87144	−0.206691	−0.103346	−	0.994646i	$-0.532955\pi$
$193$	−2.87144	−0.206691	−0.103346	+	0.994646i	$0.532955\pi$
$194$	0	0
$195$	2.12084	0.151876
$196$	0	0
$197$	−27.2422	−1.94093	−0.970464	−	0.241246i	$-0.922444\pi$
$197$	−27.2422	−1.94093	−0.970464	+	0.241246i	$0.922444\pi$
$198$	0	0
$199$	−17.2390	−1.22204	−0.611020	−	0.791615i	$-0.709241\pi$
$199$	−17.2390	−1.22204	−0.611020	+	0.791615i	$0.709241\pi$
$200$	0	0
$201$	−18.2158	−1.28484
$202$	0	0
$203$	41.5489	2.91616
$204$	0	0
$205$	−4.92962	−0.344299
$206$	0	0
$207$	2.96705	0.206224
$208$	0	0
$209$	0	0
$210$	0	0
$211$	5.42981	0.373803	0.186902	−	0.982379i	$-0.440155\pi$
$211$	5.42981	0.373803	0.186902	+	0.982379i	$0.440155\pi$
$212$	0	0
$213$	−2.78387	−0.190748
$214$	0	0
$215$	−27.6899	−1.88843
$216$	0	0
$217$	−26.2734	−1.78356
$218$	0	0
$219$	−12.8217	−0.866410
$220$	0	0
$221$	0.904708	0.0608572
$222$	0	0
$223$	8.88179	0.594769	0.297384	−	0.954758i	$-0.403886\pi$
$223$	8.88179	0.594769	0.297384	+	0.954758i	$0.403886\pi$
$224$	0	0
$225$	−6.73404	−0.448936
$226$	0	0
$227$	26.5928	1.76503	0.882514	−	0.470287i	$-0.155850\pi$
$227$	26.5928	1.76503	0.882514	+	0.470287i	$0.155850\pi$
$228$	0	0
$229$	−11.2173	−0.741258	−0.370629	−	0.928781i	$-0.620858\pi$
$229$	−11.2173	−0.741258	−0.370629	+	0.928781i	$0.620858\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	19.7503	1.29389	0.646944	−	0.762537i	$-0.276047\pi$
$233$	19.7503	1.29389	0.646944	+	0.762537i	$0.276047\pi$
$234$	0	0
$235$	4.71064	0.307288
$236$	0	0
$237$	−4.23066	−0.274811
$238$	0	0
$239$	−12.1282	−0.784505	−0.392253	−	0.919858i	$-0.628304\pi$
$239$	−12.1282	−0.784505	−0.392253	+	0.919858i	$0.628304\pi$
$240$	0	0
$241$	−12.9849	−0.836429	−0.418215	−	0.908348i	$-0.637344\pi$
$241$	−12.9849	−0.836429	−0.418215	+	0.908348i	$0.637344\pi$
$242$	0	0
$243$	7.44379	0.477519
$244$	0	0
$245$	−57.6223	−3.68135
$246$	0	0
$247$	−0.374304	−0.0238164
$248$	0	0
$249$	−24.9864	−1.58345
$250$	0	0
$251$	0.0125045	0.000789279 0	0.000394639	−	1.00000i	$-0.499874\pi$
$251$	0.0125045	0.000789279 0	0.000394639		1.00000i	$0.499874\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	13.6952	0.857624
$256$	0	0
$257$	4.41016	0.275098	0.137549	−	0.990495i	$-0.456078\pi$
$257$	4.41016	0.275098	0.137549	+	0.990495i	$0.456078\pi$
$258$	0	0
$259$	−20.5649	−1.27784
$260$	0	0
$261$	−6.46005	−0.399867
$262$	0	0
$263$	20.4329	1.25994	0.629972	−	0.776618i	$-0.283066\pi$
$263$	20.4329	1.25994	0.629972	+	0.776618i	$0.283066\pi$
$264$	0	0
$265$	47.7976	2.93619
$266$	0	0
$267$	−11.5364	−0.706016
$268$	0	0
$269$	−0.354969	−0.0216429	−0.0108214	−	0.999941i	$-0.503445\pi$
$269$	−0.354969	−0.0216429	−0.0108214	+	0.999941i	$0.503445\pi$
$270$	0	0
$271$	−19.8613	−1.20649	−0.603245	−	0.797556i	$-0.706126\pi$
$271$	−19.8613	−1.20649	−0.603245	+	0.797556i	$0.706126\pi$
$272$	0	0
$273$	−2.66101	−0.161052
$274$	0	0
$275$	0	0
$276$	0	0
$277$	16.6660	1.00136	0.500682	−	0.865631i	$-0.333082\pi$
$277$	16.6660	1.00136	0.500682	+	0.865631i	$0.333082\pi$
$278$	0	0
$279$	4.08501	0.244563
$280$	0	0
$281$	−14.1650	−0.845013	−0.422506	−	0.906360i	$-0.638850\pi$
$281$	−14.1650	−0.845013	−0.422506	+	0.906360i	$0.638850\pi$
$282$	0	0
$283$	−12.5314	−0.744912	−0.372456	−	0.928050i	$-0.621484\pi$
$283$	−12.5314	−0.744912	−0.372456	+	0.928050i	$0.621484\pi$
$284$	0	0
$285$	−5.66608	−0.335630
$286$	0	0
$287$	6.18518	0.365100
$288$	0	0
$289$	−11.1579	−0.656348
$290$	0	0
$291$	17.4687	1.02403
$292$	0	0
$293$	−25.1270	−1.46794	−0.733969	−	0.679183i	$-0.762334\pi$
$293$	−25.1270	−1.46794	−0.733969	+	0.679183i	$0.762334\pi$
$294$	0	0
$295$	3.72191	0.216698
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1.51233	0.0874605
$300$	0	0
$301$	34.7424	2.00252
$302$	0	0
$303$	−15.3912	−0.884204
$304$	0	0
$305$	−55.7071	−3.18978
$306$	0	0
$307$	22.4129	1.27917	0.639587	−	0.768719i	$-0.279105\pi$
$307$	22.4129	1.27917	0.639587	+	0.768719i	$0.279105\pi$
$308$	0	0
$309$	13.8235	0.786389
$310$	0	0
$311$	16.5415	0.937980	0.468990	−	0.883204i	$-0.344618\pi$
$311$	16.5415	0.937980	0.468990	+	0.883204i	$0.344618\pi$
$312$	0	0
$313$	−30.8590	−1.74425	−0.872127	−	0.489280i	$-0.837260\pi$
$313$	−30.8590	−1.74425	−0.872127	+	0.489280i	$0.837260\pi$
$314$	0	0
$315$	13.0561	0.735628
$316$	0	0
$317$	−2.29769	−0.129051	−0.0645256	−	0.997916i	$-0.520553\pi$
$317$	−2.29769	−0.129051	−0.0645256	+	0.997916i	$0.520553\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−4.81095	−0.268521
$322$	0	0
$323$	−2.41704	−0.134488
$324$	0	0
$325$	−3.43240	−0.190395
$326$	0	0
$327$	9.13756	0.505308
$328$	0	0
$329$	−5.91043	−0.325852
$330$	0	0
$331$	12.0889	0.664464	0.332232	−	0.943198i	$-0.392198\pi$
$331$	12.0889	0.664464	0.332232	+	0.943198i	$0.392198\pi$
$332$	0	0
$333$	3.19744	0.175218
$334$	0	0
$335$	45.5551	2.48894
$336$	0	0
$337$	−30.0531	−1.63710	−0.818548	−	0.574437i	$-0.805221\pi$
$337$	−30.0531	−1.63710	−0.818548	+	0.574437i	$0.805221\pi$
$338$	0	0
$339$	21.2628	1.15484
$340$	0	0
$341$	0	0
$342$	0	0
$343$	39.2369	2.11860
$344$	0	0
$345$	22.8932	1.23253
$346$	0	0
$347$	−1.66553	−0.0894105	−0.0447052	−	0.999000i	$-0.514235\pi$
$347$	−1.66553	−0.0894105	−0.0447052	+	0.999000i	$0.514235\pi$
$348$	0	0
$349$	−24.6231	−1.31804	−0.659022	−	0.752124i	$-0.729030\pi$
$349$	−24.6231	−1.31804	−0.659022	+	0.752124i	$0.729030\pi$
$350$	0	0
$351$	2.10395	0.112301
$352$	0	0
$353$	1.61479	0.0859467	0.0429733	−	0.999076i	$-0.486317\pi$
$353$	1.61479	0.0859467	0.0429733	+	0.999076i	$0.486317\pi$
$354$	0	0
$355$	6.96208	0.369509
$356$	0	0
$357$	−17.1833	−0.909435
$358$	0	0
$359$	−7.07832	−0.373579	−0.186790	−	0.982400i	$-0.559808\pi$
$359$	−7.07832	−0.373579	−0.186790	+	0.982400i	$0.559808\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	32.0653	1.67837
$366$	0	0
$367$	32.4405	1.69338	0.846690	−	0.532086i	$-0.178592\pi$
$367$	32.4405	1.69338	0.846690	+	0.532086i	$0.178592\pi$
$368$	0	0
$369$	−0.961676	−0.0500628
$370$	0	0
$371$	−59.9716	−3.11357
$372$	0	0
$373$	23.6392	1.22399	0.611995	−	0.790861i	$-0.290367\pi$
$373$	23.6392	1.22399	0.611995	+	0.790861i	$0.290367\pi$
$374$	0	0
$375$	−23.6281	−1.22015
$376$	0	0
$377$	−3.29275	−0.169585
$378$	0	0
$379$	−35.4750	−1.82223	−0.911113	−	0.412156i	$-0.864776\pi$
$379$	−35.4750	−1.82223	−0.911113	+	0.412156i	$0.864776\pi$
$380$	0	0
$381$	−8.29907	−0.425174
$382$	0	0
$383$	−28.7917	−1.47119	−0.735593	−	0.677423i	$-0.763097\pi$
$383$	−28.7917	−1.47119	−0.735593	+	0.677423i	$0.763097\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−5.40177	−0.274587
$388$	0	0
$389$	3.54588	0.179783	0.0898916	−	0.995952i	$-0.471348\pi$
$389$	3.54588	0.179783	0.0898916	+	0.995952i	$0.471348\pi$
$390$	0	0
$391$	9.76578	0.493877
$392$	0	0
$393$	−12.0511	−0.607897
$394$	0	0
$395$	10.5803	0.532352
$396$	0	0
$397$	4.93712	0.247787	0.123893	−	0.992296i	$-0.460462\pi$
$397$	4.93712	0.247787	0.123893	+	0.992296i	$0.460462\pi$
$398$	0	0
$399$	7.10922	0.355906
$400$	0	0
$401$	13.9253	0.695395	0.347697	−	0.937607i	$-0.386964\pi$
$401$	13.9253	0.695395	0.347697	+	0.937607i	$0.386964\pi$
$402$	0	0
$403$	2.08217	0.103720
$404$	0	0
$405$	23.5559	1.17050
$406$	0	0
$407$	0	0
$408$	0	0
$409$	10.0953	0.499182	0.249591	−	0.968351i	$-0.419704\pi$
$409$	10.0953	0.499182	0.249591	+	0.968351i	$0.419704\pi$
$410$	0	0
$411$	−25.9272	−1.27890
$412$	0	0
$413$	−4.66987	−0.229789
$414$	0	0
$415$	62.4875	3.06739
$416$	0	0
$417$	11.5954	0.567830
$418$	0	0
$419$	−8.76469	−0.428183	−0.214092	−	0.976814i	$-0.568679\pi$
$419$	−8.76469	−0.428183	−0.214092	+	0.976814i	$0.568679\pi$
$420$	0	0
$421$	22.3390	1.08874	0.544369	−	0.838846i	$-0.316769\pi$
$421$	22.3390	1.08874	0.544369	+	0.838846i	$0.316769\pi$
$422$	0	0
$423$	0.918957	0.0446812
$424$	0	0
$425$	−22.1645	−1.07514
$426$	0	0
$427$	69.8956	3.38249
$428$	0	0
$429$	0	0
$430$	0	0
$431$	10.9849	0.529126	0.264563	−	0.964368i	$-0.414772\pi$
$431$	10.9849	0.529126	0.264563	+	0.964368i	$0.414772\pi$
$432$	0	0
$433$	−18.1660	−0.873004	−0.436502	−	0.899703i	$-0.643783\pi$
$433$	−18.1660	−0.873004	−0.436502	+	0.899703i	$0.643783\pi$
$434$	0	0
$435$	−49.8445	−2.38986
$436$	0	0
$437$	−4.04039	−0.193278
$438$	0	0
$439$	−10.1208	−0.483037	−0.241519	−	0.970396i	$-0.577645\pi$
$439$	−10.1208	−0.483037	−0.241519	+	0.970396i	$0.577645\pi$
$440$	0	0
$441$	−11.2410	−0.535287
$442$	0	0
$443$	16.1109	0.765453	0.382726	−	0.923862i	$-0.374985\pi$
$443$	16.1109	0.765453	0.382726	+	0.923862i	$0.374985\pi$
$444$	0	0
$445$	28.8509	1.36766
$446$	0	0
$447$	13.0916	0.619213
$448$	0	0
$449$	−0.396839	−0.0187280	−0.00936400	−	0.999956i	$-0.502981\pi$
$449$	−0.396839	−0.0187280	−0.00936400	+	0.999956i	$0.502981\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−28.6780	−1.34741
$454$	0	0
$455$	6.65482	0.311983
$456$	0	0
$457$	−0.319620	−0.0149512	−0.00747560	−	0.999972i	$-0.502380\pi$
$457$	−0.319620	−0.0149512	−0.00747560	+	0.999972i	$0.502380\pi$
$458$	0	0
$459$	13.5861	0.634146
$460$	0	0
$461$	−5.47580	−0.255033	−0.127517	−	0.991836i	$-0.540701\pi$
$461$	−5.47580	−0.255033	−0.127517	+	0.991836i	$0.540701\pi$
$462$	0	0
$463$	−14.3949	−0.668989	−0.334494	−	0.942398i	$-0.608565\pi$
$463$	−14.3949	−0.668989	−0.334494	+	0.942398i	$0.608565\pi$
$464$	0	0
$465$	31.5191	1.46167
$466$	0	0
$467$	9.78868	0.452966	0.226483	−	0.974015i	$-0.427277\pi$
$467$	9.78868	0.452966	0.226483	+	0.974015i	$0.427277\pi$
$468$	0	0
$469$	−57.1579	−2.63931
$470$	0	0
$471$	2.37416	0.109396
$472$	0	0
$473$	0	0
$474$	0	0
$475$	9.17009	0.420753
$476$	0	0
$477$	9.32442	0.426936
$478$	0	0
$479$	−17.1474	−0.783487	−0.391743	−	0.920075i	$-0.628128\pi$
$479$	−17.1474	−0.783487	−0.391743	+	0.920075i	$0.628128\pi$
$480$	0	0
$481$	1.62976	0.0743108
$482$	0	0
$483$	−28.7240	−1.30699
$484$	0	0
$485$	−43.6868	−1.98372
$486$	0	0
$487$	−38.1522	−1.72884	−0.864421	−	0.502769i	$-0.832314\pi$
$487$	−38.1522	−1.72884	−0.864421	+	0.502769i	$0.832314\pi$
$488$	0	0
$489$	−5.43909	−0.245964
$490$	0	0
$491$	5.88366	0.265526	0.132763	−	0.991148i	$-0.457615\pi$
$491$	5.88366	0.265526	0.132763	+	0.991148i	$0.457615\pi$
$492$	0	0
$493$	−21.2627	−0.957623
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−8.73530	−0.391832
$498$	0	0
$499$	−26.9072	−1.20453	−0.602265	−	0.798296i	$-0.705735\pi$
$499$	−26.9072	−1.20453	−0.602265	+	0.798296i	$0.705735\pi$
$500$	0	0
$501$	−30.6551	−1.36957
$502$	0	0
$503$	−29.2264	−1.30314	−0.651570	−	0.758588i	$-0.725889\pi$
$503$	−29.2264	−1.30314	−0.651570	+	0.758588i	$0.725889\pi$
$504$	0	0
$505$	38.4914	1.71284
$506$	0	0
$507$	−19.3568	−0.859667
$508$	0	0
$509$	19.7242	0.874258	0.437129	−	0.899399i	$-0.355995\pi$
$509$	19.7242	0.874258	0.437129	+	0.899399i	$0.355995\pi$
$510$	0	0
$511$	−40.2322	−1.77977
$512$	0	0
$513$	−5.62097	−0.248172
$514$	0	0
$515$	−34.5705	−1.52336
$516$	0	0
$517$	0	0
$518$	0	0
$519$	32.9469	1.44621
$520$	0	0
$521$	−18.5672	−0.813445	−0.406722	−	0.913552i	$-0.633328\pi$
$521$	−18.5672	−0.813445	−0.406722	+	0.913552i	$0.633328\pi$
$522$	0	0
$523$	16.0658	0.702510	0.351255	−	0.936280i	$-0.385755\pi$
$523$	16.0658	0.702510	0.351255	+	0.936280i	$0.385755\pi$
$524$	0	0
$525$	65.1922	2.84522
$526$	0	0
$527$	13.4455	0.585693
$528$	0	0
$529$	−6.67527	−0.290229
$530$	0	0
$531$	0.726074	0.0315089
$532$	0	0
$533$	−0.490175	−0.0212318
$534$	0	0
$535$	12.0315	0.520168
$536$	0	0
$537$	−24.1338	−1.04145
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−1.54553	−0.0664474	−0.0332237	−	0.999448i	$-0.510577\pi$
$541$	−1.54553	−0.0664474	−0.0332237	+	0.999448i	$0.510577\pi$
$542$	0	0
$543$	−5.86766	−0.251806
$544$	0	0
$545$	−22.8518	−0.978862
$546$	0	0
$547$	−32.6301	−1.39516	−0.697582	−	0.716505i	$-0.745741\pi$
$547$	−32.6301	−1.39516	−0.697582	+	0.716505i	$0.745741\pi$
$548$	0	0
$549$	−10.8674	−0.463810
$550$	0	0
$551$	8.79699	0.374764
$552$	0	0
$553$	−13.2751	−0.564513
$554$	0	0
$555$	24.6708	1.04722
$556$	0	0
$557$	−3.75374	−0.159051	−0.0795255	−	0.996833i	$-0.525341\pi$
$557$	−3.75374	−0.159051	−0.0795255	+	0.996833i	$0.525341\pi$
$558$	0	0
$559$	−2.75333	−0.116454
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−12.1901	−0.513753	−0.256877	−	0.966444i	$-0.582693\pi$
$563$	−12.1901	−0.513753	−0.256877	+	0.966444i	$0.582693\pi$
$564$	0	0
$565$	−53.1753	−2.23710
$566$	0	0
$567$	−29.5556	−1.24122
$568$	0	0
$569$	8.10125	0.339622	0.169811	−	0.985477i	$-0.445684\pi$
$569$	8.10125	0.339622	0.169811	+	0.985477i	$0.445684\pi$
$570$	0	0
$571$	−28.3681	−1.18717	−0.593585	−	0.804771i	$-0.702288\pi$
$571$	−28.3681	−1.18717	−0.593585	+	0.804771i	$0.702288\pi$
$572$	0	0
$573$	6.40871	0.267728
$574$	0	0
$575$	−37.0507	−1.54512
$576$	0	0
$577$	4.88199	0.203240	0.101620	−	0.994823i	$-0.467597\pi$
$577$	4.88199	0.203240	0.101620	+	0.994823i	$0.467597\pi$
$578$	0	0
$579$	−4.32212	−0.179621
$580$	0	0
$581$	−78.4029	−3.25270
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−1.03470	−0.0427794
$586$	0	0
$587$	−25.6285	−1.05780	−0.528902	−	0.848683i	$-0.677396\pi$
$587$	−25.6285	−1.05780	−0.528902	+	0.848683i	$0.677396\pi$
$588$	0	0
$589$	−5.56277	−0.229210
$590$	0	0
$591$	−41.0052	−1.68673
$592$	0	0
$593$	2.42516	0.0995893	0.0497946	−	0.998759i	$-0.484143\pi$
$593$	2.42516	0.0995893	0.0497946	+	0.998759i	$0.484143\pi$
$594$	0	0
$595$	42.9730	1.76172
$596$	0	0
$597$	−25.9483	−1.06199
$598$	0	0
$599$	1.93817	0.0791914	0.0395957	−	0.999216i	$-0.487393\pi$
$599$	1.93817	0.0791914	0.0395957	+	0.999216i	$0.487393\pi$
$600$	0	0
$601$	−23.0321	−0.939499	−0.469750	−	0.882800i	$-0.655656\pi$
$601$	−23.0321	−0.939499	−0.469750	+	0.882800i	$0.655656\pi$
$602$	0	0
$603$	8.88695	0.361905
$604$	0	0
$605$	0	0
$606$	0	0
$607$	17.0707	0.692878	0.346439	−	0.938073i	$-0.387391\pi$
$607$	17.0707	0.692878	0.346439	+	0.938073i	$0.387391\pi$
$608$	0	0
$609$	62.5397	2.53424
$610$	0	0
$611$	0.468401	0.0189495
$612$	0	0
$613$	−24.0509	−0.971408	−0.485704	−	0.874123i	$-0.661437\pi$
$613$	−24.0509	−0.971408	−0.485704	+	0.874123i	$0.661437\pi$
$614$	0	0
$615$	−7.42010	−0.299207
$616$	0	0
$617$	−7.68237	−0.309281	−0.154640	−	0.987971i	$-0.549422\pi$
$617$	−7.68237	−0.309281	−0.154640	+	0.987971i	$0.549422\pi$
$618$	0	0
$619$	38.9380	1.56505	0.782525	−	0.622619i	$-0.213931\pi$
$619$	38.9380	1.56505	0.782525	+	0.622619i	$0.213931\pi$
$620$	0	0
$621$	22.7109	0.911357
$622$	0	0
$623$	−36.1992	−1.45029
$624$	0	0
$625$	13.2401	0.529605
$626$	0	0
$627$	0	0
$628$	0	0
$629$	10.5241	0.419623
$630$	0	0
$631$	−22.9866	−0.915082	−0.457541	−	0.889188i	$-0.651270\pi$
$631$	−22.9866	−0.915082	−0.457541	+	0.889188i	$0.651270\pi$
$632$	0	0
$633$	8.17299	0.324847
$634$	0	0
$635$	20.7548	0.823630
$636$	0	0
$637$	−5.72965	−0.227017
$638$	0	0
$639$	1.35817	0.0537284
$640$	0	0
$641$	6.15460	0.243092	0.121546	−	0.992586i	$-0.461215\pi$
$641$	6.15460	0.243092	0.121546	+	0.992586i	$0.461215\pi$
$642$	0	0
$643$	−27.9222	−1.10114	−0.550572	−	0.834788i	$-0.685590\pi$
$643$	−27.9222	−1.10114	−0.550572	+	0.834788i	$0.685590\pi$
$644$	0	0
$645$	−41.6790	−1.64111
$646$	0	0
$647$	41.1422	1.61747	0.808733	−	0.588176i	$-0.200154\pi$
$647$	41.1422	1.61747	0.808733	+	0.588176i	$0.200154\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−39.5470	−1.54997
$652$	0	0
$653$	−9.38105	−0.367109	−0.183554	−	0.983010i	$-0.558760\pi$
$653$	−9.38105	−0.367109	−0.183554	+	0.983010i	$0.558760\pi$
$654$	0	0
$655$	30.1381	1.17759
$656$	0	0
$657$	6.25533	0.244044
$658$	0	0
$659$	−17.8237	−0.694313	−0.347157	−	0.937807i	$-0.612853\pi$
$659$	−17.8237	−0.694313	−0.347157	+	0.937807i	$0.612853\pi$
$660$	0	0
$661$	27.7143	1.07796	0.538980	−	0.842319i	$-0.318810\pi$
$661$	27.7143	1.07796	0.538980	+	0.842319i	$0.318810\pi$
$662$	0	0
$663$	1.36177	0.0528869
$664$	0	0
$665$	−17.7792	−0.689447
$666$	0	0
$667$	−35.5432	−1.37624
$668$	0	0
$669$	13.3689	0.516873
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−10.5455	−0.406500	−0.203250	−	0.979127i	$-0.565150\pi$
$673$	−10.5455	−0.406500	−0.203250	+	0.979127i	$0.565150\pi$
$674$	0	0
$675$	−51.5448	−1.98396
$676$	0	0
$677$	43.8910	1.68687	0.843433	−	0.537234i	$-0.180531\pi$
$677$	43.8910	1.68687	0.843433	+	0.537234i	$0.180531\pi$
$678$	0	0
$679$	54.8138	2.10356
$680$	0	0
$681$	40.0277	1.53387
$682$	0	0
$683$	−37.4188	−1.43179	−0.715895	−	0.698208i	$-0.753981\pi$
$683$	−37.4188	−1.43179	−0.715895	+	0.698208i	$0.753981\pi$
$684$	0	0
$685$	64.8404	2.47743
$686$	0	0
$687$	−16.8843	−0.644177
$688$	0	0
$689$	4.75274	0.181065
$690$	0	0
$691$	−22.2844	−0.847737	−0.423869	−	0.905724i	$-0.639328\pi$
$691$	−22.2844	−0.847737	−0.423869	+	0.905724i	$0.639328\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−28.9985	−1.09998
$696$	0	0
$697$	−3.16527	−0.119893
$698$	0	0
$699$	29.7284	1.12443
$700$	0	0
$701$	7.51432	0.283812	0.141906	−	0.989880i	$-0.454677\pi$
$701$	7.51432	0.283812	0.141906	+	0.989880i	$0.454677\pi$
$702$	0	0
$703$	−4.35412	−0.164219
$704$	0	0
$705$	7.09049	0.267043
$706$	0	0
$707$	−48.2950	−1.81632
$708$	0	0
$709$	−35.3022	−1.32580	−0.662900	−	0.748708i	$-0.730675\pi$
$709$	−35.3022	−1.32580	−0.662900	+	0.748708i	$0.730675\pi$
$710$	0	0
$711$	2.06402	0.0774066
$712$	0	0
$713$	22.4758	0.841724
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−18.2554	−0.681761
$718$	0	0
$719$	2.73664	0.102060	0.0510298	−	0.998697i	$-0.483750\pi$
$719$	2.73664	0.102060	0.0510298	+	0.998697i	$0.483750\pi$
$720$	0	0
$721$	43.3756	1.61539
$722$	0	0
$723$	−19.5449	−0.726884
$724$	0	0
$725$	80.6692	2.99598
$726$	0	0
$727$	−23.4681	−0.870382	−0.435191	−	0.900338i	$-0.643319\pi$
$727$	−23.4681	−0.870382	−0.435191	+	0.900338i	$0.643319\pi$
$728$	0	0
$729$	29.9775	1.11028
$730$	0	0
$731$	−17.7795	−0.657597
$732$	0	0
$733$	−0.490093	−0.0181020	−0.00905100	−	0.999959i	$-0.502881\pi$
$733$	−0.490093	−0.0181020	−0.00905100	+	0.999959i	$0.502881\pi$
$734$	0	0
$735$	−86.7335	−3.19921
$736$	0	0
$737$	0	0
$738$	0	0
$739$	52.1774	1.91938	0.959689	−	0.281064i	$-0.0906873\pi$
$739$	52.1774	1.91938	0.959689	+	0.281064i	$0.0906873\pi$
$740$	0	0
$741$	−0.563405	−0.0206972
$742$	0	0
$743$	−11.5533	−0.423850	−0.211925	−	0.977286i	$-0.567973\pi$
$743$	−11.5533	−0.423850	−0.211925	+	0.977286i	$0.567973\pi$
$744$	0	0
$745$	−32.7404	−1.19951
$746$	0	0
$747$	12.1901	0.446014
$748$	0	0
$749$	−15.0959	−0.551593
$750$	0	0
$751$	−4.40464	−0.160727	−0.0803637	−	0.996766i	$-0.525608\pi$
$751$	−4.40464	−0.160727	−0.0803637	+	0.996766i	$0.525608\pi$
$752$	0	0
$753$	0.0188219	0.000685909 0
$754$	0	0
$755$	71.7198	2.61015
$756$	0	0
$757$	44.2986	1.61006	0.805030	−	0.593234i	$-0.202149\pi$
$757$	44.2986	1.61006	0.805030	+	0.593234i	$0.202149\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	38.6170	1.39987	0.699933	−	0.714209i	$-0.253213\pi$
$761$	38.6170	1.39987	0.699933	+	0.714209i	$0.253213\pi$
$762$	0	0
$763$	28.6721	1.03800
$764$	0	0
$765$	−6.68147	−0.241569
$766$	0	0
$767$	0.370087	0.0133631
$768$	0	0
$769$	33.7369	1.21659	0.608293	−	0.793713i	$-0.291855\pi$
$769$	33.7369	1.21659	0.608293	+	0.793713i	$0.291855\pi$
$770$	0	0
$771$	6.63821	0.239069
$772$	0	0
$773$	−46.5746	−1.67517	−0.837586	−	0.546305i	$-0.816034\pi$
$773$	−46.5746	−1.67517	−0.837586	+	0.546305i	$0.816034\pi$
$774$	0	0
$775$	−51.0112	−1.83237
$776$	0	0
$777$	−30.9544	−1.11048
$778$	0	0
$779$	1.30956	0.0469200
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−49.4476	−1.76711
$784$	0	0
$785$	−5.93745	−0.211917
$786$	0	0
$787$	−43.7448	−1.55934	−0.779668	−	0.626194i	$-0.784612\pi$
$787$	−43.7448	−1.55934	−0.779668	+	0.626194i	$0.784612\pi$
$788$	0	0
$789$	30.7557	1.09493
$790$	0	0
$791$	66.7189	2.37225
$792$	0	0
$793$	−5.53922	−0.196704
$794$	0	0
$795$	71.9454	2.55164
$796$	0	0
$797$	40.6996	1.44165	0.720827	−	0.693115i	$-0.243762\pi$
$797$	40.6996	1.44165	0.720827	+	0.693115i	$0.243762\pi$
$798$	0	0
$799$	3.02467	0.107005
$800$	0	0
$801$	5.62827	0.198865
$802$	0	0
$803$	0	0
$804$	0	0
$805$	71.8348	2.53184
$806$	0	0
$807$	−0.534303	−0.0188084
$808$	0	0
$809$	−5.83239	−0.205056	−0.102528	−	0.994730i	$-0.532693\pi$
$809$	−5.83239	−0.205056	−0.102528	+	0.994730i	$0.532693\pi$
$810$	0	0
$811$	42.9571	1.50843	0.754214	−	0.656629i	$-0.228018\pi$
$811$	42.9571	1.50843	0.754214	+	0.656629i	$0.228018\pi$
$812$	0	0
$813$	−29.8955	−1.04848
$814$	0	0
$815$	13.6024	0.476472
$816$	0	0
$817$	7.35588	0.257350
$818$	0	0
$819$	1.29823	0.0453638
$820$	0	0
$821$	24.2449	0.846153	0.423076	−	0.906094i	$-0.360950\pi$
$821$	24.2449	0.846153	0.423076	+	0.906094i	$0.360950\pi$
$822$	0	0
$823$	−0.501066	−0.0174661	−0.00873303	−	0.999962i	$-0.502780\pi$
$823$	−0.501066	−0.0174661	−0.00873303	+	0.999962i	$0.502780\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	38.8655	1.35148	0.675742	−	0.737138i	$-0.263823\pi$
$827$	38.8655	1.35148	0.675742	+	0.737138i	$0.263823\pi$
$828$	0	0
$829$	−2.45369	−0.0852203	−0.0426102	−	0.999092i	$-0.513567\pi$
$829$	−2.45369	−0.0852203	−0.0426102	+	0.999092i	$0.513567\pi$
$830$	0	0
$831$	25.0859	0.870219
$832$	0	0
$833$	−36.9988	−1.28193
$834$	0	0
$835$	76.6641	2.65307
$836$	0	0
$837$	31.2682	1.08079
$838$	0	0
$839$	31.9540	1.10317	0.551586	−	0.834118i	$-0.314023\pi$
$839$	31.9540	1.10317	0.551586	+	0.834118i	$0.314023\pi$
$840$	0	0
$841$	48.3870	1.66852
$842$	0	0
$843$	−21.3213	−0.734344
$844$	0	0
$845$	48.4087	1.66531
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−18.8623	−0.647353
$850$	0	0
$851$	17.5923	0.603057
$852$	0	0
$853$	43.4431	1.48746	0.743731	−	0.668479i	$-0.233054\pi$
$853$	43.4431	1.48746	0.743731	+	0.668479i	$0.233054\pi$
$854$	0	0
$855$	2.76432	0.0945377
$856$	0	0
$857$	−29.4476	−1.00591	−0.502955	−	0.864312i	$-0.667754\pi$
$857$	−29.4476	−1.00591	−0.502955	+	0.864312i	$0.667754\pi$
$858$	0	0
$859$	−47.7033	−1.62762	−0.813808	−	0.581134i	$-0.802609\pi$
$859$	−47.7033	−1.62762	−0.813808	+	0.581134i	$0.802609\pi$
$860$	0	0
$861$	9.30998	0.317283
$862$	0	0
$863$	15.8972	0.541146	0.270573	−	0.962699i	$-0.412787\pi$
$863$	15.8972	0.541146	0.270573	+	0.962699i	$0.412787\pi$
$864$	0	0
$865$	−82.3957	−2.80154
$866$	0	0
$867$	−16.7950	−0.570388
$868$	0	0
$869$	0	0
$870$	0	0
$871$	4.52976	0.153485
$872$	0	0
$873$	−8.52248	−0.288442
$874$	0	0
$875$	−74.1408	−2.50642
$876$	0	0
$877$	42.0102	1.41858	0.709291	−	0.704916i	$-0.249015\pi$
$877$	42.0102	1.41858	0.709291	+	0.704916i	$0.249015\pi$
$878$	0	0
$879$	−37.8214	−1.27569
$880$	0	0
$881$	−49.6270	−1.67198	−0.835989	−	0.548746i	$-0.815105\pi$
$881$	−49.6270	−1.67198	−0.835989	+	0.548746i	$0.815105\pi$
$882$	0	0
$883$	58.2922	1.96169	0.980844	−	0.194797i	$-0.0624048\pi$
$883$	58.2922	1.96169	0.980844	+	0.194797i	$0.0624048\pi$
$884$	0	0
$885$	5.60224	0.188317
$886$	0	0
$887$	46.9783	1.57738	0.788689	−	0.614792i	$-0.210760\pi$
$887$	46.9783	1.57738	0.788689	+	0.614792i	$0.210760\pi$
$888$	0	0
$889$	−26.0410	−0.873388
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−1.25139	−0.0418762
$894$	0	0
$895$	60.3554	2.01746
$896$	0	0
$897$	2.27638	0.0760060
$898$	0	0
$899$	−48.9357	−1.63210
$900$	0	0
$901$	30.6905	1.02245
$902$	0	0
$903$	52.2945	1.74025
$904$	0	0
$905$	14.6742	0.487788
$906$	0	0
$907$	−46.4709	−1.54304	−0.771521	−	0.636204i	$-0.780504\pi$
$907$	−46.4709	−1.54304	−0.771521	+	0.636204i	$0.780504\pi$
$908$	0	0
$909$	7.50894	0.249056
$910$	0	0
$911$	−44.3741	−1.47018	−0.735090	−	0.677969i	$-0.762860\pi$
$911$	−44.3741	−1.47018	−0.735090	+	0.677969i	$0.762860\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−83.8509	−2.77202
$916$	0	0
$917$	−37.8142	−1.24874
$918$	0	0
$919$	32.0627	1.05765	0.528826	−	0.848730i	$-0.322632\pi$
$919$	32.0627	1.05765	0.528826	+	0.848730i	$0.322632\pi$
$920$	0	0
$921$	33.7361	1.11164
$922$	0	0
$923$	0.692272	0.0227864
$924$	0	0
$925$	−39.9277	−1.31281
$926$	0	0
$927$	−6.74406	−0.221504
$928$	0	0
$929$	35.7205	1.17195	0.585976	−	0.810328i	$-0.300711\pi$
$929$	35.7205	1.17195	0.585976	+	0.810328i	$0.300711\pi$
$930$	0	0
$931$	15.3075	0.501683
$932$	0	0
$933$	24.8983	0.815135
$934$	0	0
$935$	0	0
$936$	0	0
$937$	43.2264	1.41214	0.706072	−	0.708140i	$-0.250466\pi$
$937$	43.2264	1.41214	0.706072	+	0.708140i	$0.250466\pi$
$938$	0	0
$939$	−46.4492	−1.51581
$940$	0	0
$941$	−12.1950	−0.397547	−0.198773	−	0.980045i	$-0.563696\pi$
$941$	−12.1950	−0.397547	−0.198773	+	0.980045i	$0.563696\pi$
$942$	0	0
$943$	−5.29115	−0.172303
$944$	0	0
$945$	99.9363	3.25093
$946$	0	0
$947$	46.7136	1.51799	0.758994	−	0.651098i	$-0.225691\pi$
$947$	46.7136	1.51799	0.758994	+	0.651098i	$0.225691\pi$
$948$	0	0
$949$	3.18840	0.103500
$950$	0	0
$951$	−3.45851	−0.112150
$952$	0	0
$953$	24.1844	0.783408	0.391704	−	0.920091i	$-0.371886\pi$
$953$	24.1844	0.783408	0.391704	+	0.920091i	$0.371886\pi$
$954$	0	0
$955$	−16.0273	−0.518631
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−81.3551	−2.62709
$960$	0	0
$961$	−0.0555403	−0.00179162
$962$	0	0
$963$	2.34712	0.0756350
$964$	0	0
$965$	10.8090	0.347955
$966$	0	0
$967$	24.7122	0.794692	0.397346	−	0.917669i	$-0.369931\pi$
$967$	24.7122	0.794692	0.397346	+	0.917669i	$0.369931\pi$
$968$	0	0
$969$	−3.63815	−0.116874
$970$	0	0
$971$	−23.2303	−0.745496	−0.372748	−	0.927933i	$-0.621584\pi$
$971$	−23.2303	−0.745496	−0.372748	+	0.927933i	$0.621584\pi$
$972$	0	0
$973$	36.3844	1.16643
$974$	0	0
$975$	−5.16648	−0.165460
$976$	0	0
$977$	−40.8058	−1.30549	−0.652747	−	0.757576i	$-0.726383\pi$
$977$	−40.8058	−1.30549	−0.652747	+	0.757576i	$0.726383\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−4.45795	−0.142331
$982$	0	0
$983$	20.5743	0.656218	0.328109	−	0.944640i	$-0.393589\pi$
$983$	20.5743	0.656218	0.328109	+	0.944640i	$0.393589\pi$
$984$	0	0
$985$	102.548	3.26746
$986$	0	0
$987$	−8.89642	−0.283176
$988$	0	0
$989$	−29.7206	−0.945060
$990$	0	0
$991$	19.9994	0.635303	0.317651	−	0.948208i	$-0.397106\pi$
$991$	19.9994	0.635303	0.317651	+	0.948208i	$0.397106\pi$
$992$	0	0
$993$	18.1963	0.577441
$994$	0	0
$995$	64.8931	2.05725
$996$	0	0
$997$	47.6591	1.50938	0.754689	−	0.656082i	$-0.227788\pi$
$997$	47.6591	1.50938	0.754689	+	0.656082i	$0.227788\pi$
$998$	0	0
$999$	24.4744	0.774335

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9196.2.a.l.1.5	✓	6
11.10	odd	2		9196.2.a.m.1.5	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9196.2.a.l.1.5	✓	6	1.1	even	1	trivial
9196.2.a.m.1.5	yes	6	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)

\(f(q)\)	\(=\)	\(q+1.50521 q^{3} -3.76432 q^{5} +4.72308 q^{7} -0.734348 q^{9} -0.374304 q^{13} -5.66608 q^{15} -2.41704 q^{17} +1.00000 q^{19} +7.10922 q^{21} -4.04039 q^{23} +9.17009 q^{25} -5.62097 q^{27} +8.79699 q^{29} -5.56277 q^{31} -17.7792 q^{35} -4.35412 q^{37} -0.563405 q^{39} +1.30956 q^{41} +7.35588 q^{43} +2.76432 q^{45} -1.25139 q^{47} +15.3075 q^{49} -3.63815 q^{51} -12.6976 q^{53} +1.50521 q^{57} -0.988733 q^{59} +14.7987 q^{61} -3.46838 q^{63} +1.40900 q^{65} -12.1018 q^{67} -6.08163 q^{69} -1.84949 q^{71} -8.51822 q^{73} +13.8029 q^{75} -2.81068 q^{79} -6.25769 q^{81} -16.5999 q^{83} +9.09851 q^{85} +13.2413 q^{87} -7.66431 q^{89} -1.76787 q^{91} -8.37313 q^{93} -3.76432 q^{95} +11.6055 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{5} - 4 q^{7} + 2 q^{9} - 8 q^{13} + 2 q^{15} - 6 q^{17} + 6 q^{19} + 20 q^{21} + 6 q^{23} + 2 q^{25} - 18 q^{27} + 2 q^{29} - 6 q^{31} - 14 q^{35} + 4 q^{37} - 6 q^{39} - 14 q^{41} + 10 q^{43}+ \cdots + 28 q^{97}+O(q^{100}) \) 6 * q - 4 * q^5 - 4 * q^7 + 2 * q^9 - 8 * q^13 + 2 * q^15 - 6 * q^17 + 6 * q^19 + 20 * q^21 + 6 * q^23 + 2 * q^25 - 18 * q^27 + 2 * q^29 - 6 * q^31 - 14 * q^35 + 4 * q^37 - 6 * q^39 - 14 * q^41 + 10 * q^43 - 2 * q^45 + 4 * q^47 + 20 * q^49 - 18 * q^51 + 20 * q^59 + 2 * q^61 - 12 * q^63 + 12 * q^65 + 10 * q^67 - 20 * q^69 - 32 * q^73 + 18 * q^75 - 6 * q^79 - 6 * q^81 - 36 * q^83 - 6 * q^85 + 28 * q^87 - 24 * q^89 - 16 * q^91 + 22 * q^93 - 4 * q^95 + 28 * q^97

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