LMFDB - Embedded newform 9196.2.a.q.1.4

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			1.4
Root			$1.51360$ 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)

$f(q)$	$=$	$q-1.51360 q^{3} +1.33319 q^{5} +1.09646 q^{7} -0.709002 q^{9} -1.74738 q^{13} -2.01792 q^{15} -5.01936 q^{17} -1.00000 q^{19} -1.65961 q^{21} +5.31728 q^{23} -3.22261 q^{25} +5.61396 q^{27} -4.08844 q^{29} +6.84522 q^{31} +1.46179 q^{35} +11.7695 q^{37} +2.64484 q^{39} +3.70163 q^{41} -2.32772 q^{43} -0.945234 q^{45} -6.39330 q^{47} -5.79777 q^{49} +7.59732 q^{51} -2.97469 q^{53} +1.51360 q^{57} -11.9409 q^{59} +11.6153 q^{61} -0.777394 q^{63} -2.32959 q^{65} +8.95568 q^{67} -8.04826 q^{69} +4.16892 q^{71} +14.8230 q^{73} +4.87775 q^{75} -1.75830 q^{79} -6.37031 q^{81} -14.5404 q^{83} -6.69175 q^{85} +6.18828 q^{87} +1.68679 q^{89} -1.91593 q^{91} -10.3609 q^{93} -1.33319 q^{95} -17.3678 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.51360	−0.873880	−0.436940	−	0.899491i	$-0.643938\pi$
$3$	−1.51360	−0.873880	−0.436940	+	0.899491i	$0.643938\pi$
$4$	0	0
$5$	1.33319	0.596220	0.298110	−	0.954531i	$-0.403644\pi$
$5$	1.33319	0.596220	0.298110	+	0.954531i	$0.403644\pi$
$6$	0	0
$7$	1.09646	0.414424	0.207212	−	0.978296i	$-0.433561\pi$
$7$	1.09646	0.414424	0.207212	+	0.978296i	$0.433561\pi$
$8$	0	0
$9$	−0.709002	−0.236334
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−1.74738	−0.484635	−0.242318	−	0.970197i	$-0.577908\pi$
$13$	−1.74738	−0.484635	−0.242318	+	0.970197i	$0.577908\pi$
$14$	0	0
$15$	−2.01792	−0.521025
$16$	0	0
$17$	−5.01936	−1.21737	−0.608687	−	0.793411i	$-0.708303\pi$
$17$	−5.01936	−1.21737	−0.608687	+	0.793411i	$0.708303\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−1.65961	−0.362157
$22$	0	0
$23$	5.31728	1.10873	0.554365	−	0.832274i	$-0.312961\pi$
$23$	5.31728	1.10873	0.554365	+	0.832274i	$0.312961\pi$
$24$	0	0
$25$	−3.22261	−0.644521
$26$	0	0
$27$	5.61396	1.08041
$28$	0	0
$29$	−4.08844	−0.759205	−0.379602	−	0.925150i	$-0.623939\pi$
$29$	−4.08844	−0.759205	−0.379602	+	0.925150i	$0.623939\pi$
$30$	0	0
$31$	6.84522	1.22944	0.614719	−	0.788747i	$-0.289270\pi$
$31$	6.84522	1.22944	0.614719	+	0.788747i	$0.289270\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.46179	0.247088
$36$	0	0
$37$	11.7695	1.93489	0.967443	−	0.253088i	$-0.0814464\pi$
$37$	11.7695	1.93489	0.967443	+	0.253088i	$0.0814464\pi$
$38$	0	0
$39$	2.64484	0.423513
$40$	0	0
$41$	3.70163	0.578098	0.289049	−	0.957314i	$-0.406661\pi$
$41$	3.70163	0.578098	0.289049	+	0.957314i	$0.406661\pi$
$42$	0	0
$43$	−2.32772	−0.354974	−0.177487	−	0.984123i	$-0.556797\pi$
$43$	−2.32772	−0.354974	−0.177487	+	0.984123i	$0.556797\pi$
$44$	0	0
$45$	−0.945234	−0.140907
$46$	0	0
$47$	−6.39330	−0.932559	−0.466279	−	0.884637i	$-0.654406\pi$
$47$	−6.39330	−0.932559	−0.466279	+	0.884637i	$0.654406\pi$
$48$	0	0
$49$	−5.79777	−0.828253
$50$	0	0
$51$	7.59732	1.06384
$52$	0	0
$53$	−2.97469	−0.408606	−0.204303	−	0.978908i	$-0.565493\pi$
$53$	−2.97469	−0.408606	−0.204303	+	0.978908i	$0.565493\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.51360	0.200482
$58$	0	0
$59$	−11.9409	−1.55457	−0.777287	−	0.629147i	$-0.783404\pi$
$59$	−11.9409	−1.55457	−0.777287	+	0.629147i	$0.783404\pi$
$60$	0	0
$61$	11.6153	1.48719	0.743595	−	0.668630i	$-0.233119\pi$
$61$	11.6153	1.48719	0.743595	+	0.668630i	$0.233119\pi$
$62$	0	0
$63$	−0.777394	−0.0979424
$64$	0	0
$65$	−2.32959	−0.288950
$66$	0	0
$67$	8.95568	1.09411	0.547055	−	0.837097i	$-0.315749\pi$
$67$	8.95568	1.09411	0.547055	+	0.837097i	$0.315749\pi$
$68$	0	0
$69$	−8.04826	−0.968896
$70$	0	0
$71$	4.16892	0.494760	0.247380	−	0.968918i	$-0.420430\pi$
$71$	4.16892	0.494760	0.247380	+	0.968918i	$0.420430\pi$
$72$	0	0
$73$	14.8230	1.73490	0.867448	−	0.497527i	$-0.165758\pi$
$73$	14.8230	1.73490	0.867448	+	0.497527i	$0.165758\pi$
$74$	0	0
$75$	4.87775	0.563234
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−1.75830	−0.197824	−0.0989119	−	0.995096i	$-0.531536\pi$
$79$	−1.75830	−0.197824	−0.0989119	+	0.995096i	$0.531536\pi$
$80$	0	0
$81$	−6.37031	−0.707812
$82$	0	0
$83$	−14.5404	−1.59602	−0.798008	−	0.602646i	$-0.794113\pi$
$83$	−14.5404	−1.59602	−0.798008	+	0.602646i	$0.794113\pi$
$84$	0	0
$85$	−6.69175	−0.725823
$86$	0	0
$87$	6.18828	0.663454
$88$	0	0
$89$	1.68679	0.178800	0.0893998	−	0.995996i	$-0.471505\pi$
$89$	1.68679	0.178800	0.0893998	+	0.995996i	$0.471505\pi$
$90$	0	0
$91$	−1.91593	−0.200844
$92$	0	0
$93$	−10.3609	−1.07438
$94$	0	0
$95$	−1.33319	−0.136782
$96$	0	0
$97$	−17.3678	−1.76344	−0.881718	−	0.471776i	$-0.843613\pi$
$97$	−17.3678	−1.76344	−0.881718	+	0.471776i	$0.843613\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1.58044	−0.157259	−0.0786297	−	0.996904i	$-0.525054\pi$
$101$	−1.58044	−0.157259	−0.0786297	+	0.996904i	$0.525054\pi$
$102$	0	0
$103$	7.95957	0.784280	0.392140	−	0.919906i	$-0.371735\pi$
$103$	7.95957	0.784280	0.392140	+	0.919906i	$0.371735\pi$
$104$	0	0
$105$	−2.21257	−0.215925
$106$	0	0
$107$	−1.48258	−0.143326	−0.0716632	−	0.997429i	$-0.522831\pi$
$107$	−1.48258	−0.143326	−0.0716632	+	0.997429i	$0.522831\pi$
$108$	0	0
$109$	3.54514	0.339563	0.169781	−	0.985482i	$-0.445694\pi$
$109$	3.54514	0.339563	0.169781	+	0.985482i	$0.445694\pi$
$110$	0	0
$111$	−17.8143	−1.69086
$112$	0	0
$113$	−8.09635	−0.761641	−0.380820	−	0.924649i	$-0.624358\pi$
$113$	−8.09635	−0.761641	−0.380820	+	0.924649i	$0.624358\pi$
$114$	0	0
$115$	7.08894	0.661047
$116$	0	0
$117$	1.23889	0.114536
$118$	0	0
$119$	−5.50354	−0.504508
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−5.60281	−0.505188
$124$	0	0
$125$	−10.9623	−0.980497
$126$	0	0
$127$	−11.9656	−1.06177	−0.530886	−	0.847444i	$-0.678141\pi$
$127$	−11.9656	−1.06177	−0.530886	+	0.847444i	$0.678141\pi$
$128$	0	0
$129$	3.52324	0.310204
$130$	0	0
$131$	11.5198	1.00649	0.503246	−	0.864143i	$-0.332139\pi$
$131$	11.5198	1.00649	0.503246	+	0.864143i	$0.332139\pi$
$132$	0	0
$133$	−1.09646	−0.0950753
$134$	0	0
$135$	7.48447	0.644161
$136$	0	0
$137$	19.3662	1.65456	0.827282	−	0.561787i	$-0.189886\pi$
$137$	19.3662	1.65456	0.827282	+	0.561787i	$0.189886\pi$
$138$	0	0
$139$	−9.96399	−0.845135	−0.422567	−	0.906332i	$-0.638871\pi$
$139$	−9.96399	−0.845135	−0.422567	+	0.906332i	$0.638871\pi$
$140$	0	0
$141$	9.67693	0.814944
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−5.45067	−0.452653
$146$	0	0
$147$	8.77553	0.723794
$148$	0	0
$149$	−15.3359	−1.25637	−0.628183	−	0.778066i	$-0.716201\pi$
$149$	−15.3359	−1.25637	−0.628183	+	0.778066i	$0.716201\pi$
$150$	0	0
$151$	−14.7954	−1.20403	−0.602014	−	0.798485i	$-0.705635\pi$
$151$	−14.7954	−1.20403	−0.602014	+	0.798485i	$0.705635\pi$
$152$	0	0
$153$	3.55873	0.287707
$154$	0	0
$155$	9.12597	0.733015
$156$	0	0
$157$	−18.9984	−1.51624	−0.758118	−	0.652117i	$-0.773881\pi$
$157$	−18.9984	−1.51624	−0.758118	+	0.652117i	$0.773881\pi$
$158$	0	0
$159$	4.50251	0.357072
$160$	0	0
$161$	5.83020	0.459484
$162$	0	0
$163$	3.78942	0.296810	0.148405	−	0.988927i	$-0.452586\pi$
$163$	3.78942	0.296810	0.148405	+	0.988927i	$0.452586\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−7.20831	−0.557796	−0.278898	−	0.960321i	$-0.589969\pi$
$167$	−7.20831	−0.557796	−0.278898	+	0.960321i	$0.589969\pi$
$168$	0	0
$169$	−9.94667	−0.765128
$170$	0	0
$171$	0.709002	0.0542187
$172$	0	0
$173$	−25.4398	−1.93415	−0.967077	−	0.254484i	$-0.918094\pi$
$173$	−25.4398	−1.93415	−0.967077	+	0.254484i	$0.918094\pi$
$174$	0	0
$175$	−3.53347	−0.267105
$176$	0	0
$177$	18.0738	1.35851
$178$	0	0
$179$	−11.6566	−0.871254	−0.435627	−	0.900127i	$-0.643473\pi$
$179$	−11.6566	−0.871254	−0.435627	+	0.900127i	$0.643473\pi$
$180$	0	0
$181$	4.04735	0.300837	0.150419	−	0.988622i	$-0.451938\pi$
$181$	4.04735	0.300837	0.150419	+	0.988622i	$0.451938\pi$
$182$	0	0
$183$	−17.5810	−1.29963
$184$	0	0
$185$	15.6909	1.15362
$186$	0	0
$187$	0	0
$188$	0	0
$189$	6.15550	0.447746
$190$	0	0
$191$	−14.5692	−1.05419	−0.527096	−	0.849806i	$-0.676719\pi$
$191$	−14.5692	−1.05419	−0.527096	+	0.849806i	$0.676719\pi$
$192$	0	0
$193$	−21.7467	−1.56536	−0.782681	−	0.622423i	$-0.786148\pi$
$193$	−21.7467	−1.56536	−0.782681	+	0.622423i	$0.786148\pi$
$194$	0	0
$195$	3.52607	0.252507
$196$	0	0
$197$	−9.96172	−0.709743	−0.354872	−	0.934915i	$-0.615475\pi$
$197$	−9.96172	−0.709743	−0.354872	+	0.934915i	$0.615475\pi$
$198$	0	0
$199$	15.3904	1.09100	0.545498	−	0.838112i	$-0.316341\pi$
$199$	15.3904	1.09100	0.545498	+	0.838112i	$0.316341\pi$
$200$	0	0
$201$	−13.5553	−0.956121
$202$	0	0
$203$	−4.48282	−0.314632
$204$	0	0
$205$	4.93498	0.344674
$206$	0	0
$207$	−3.76996	−0.262030
$208$	0	0
$209$	0	0
$210$	0	0
$211$	1.42952	0.0984124	0.0492062	−	0.998789i	$-0.484331\pi$
$211$	1.42952	0.0984124	0.0492062	+	0.998789i	$0.484331\pi$
$212$	0	0
$213$	−6.31010	−0.432361
$214$	0	0
$215$	−3.10329	−0.211642
$216$	0	0
$217$	7.50552	0.509508
$218$	0	0
$219$	−22.4361	−1.51609
$220$	0	0
$221$	8.77071	0.589982
$222$	0	0
$223$	−0.795468	−0.0532685	−0.0266342	−	0.999645i	$-0.508479\pi$
$223$	−0.795468	−0.0532685	−0.0266342	+	0.999645i	$0.508479\pi$
$224$	0	0
$225$	2.28483	0.152322
$226$	0	0
$227$	23.1211	1.53460	0.767299	−	0.641289i	$-0.221600\pi$
$227$	23.1211	1.53460	0.767299	+	0.641289i	$0.221600\pi$
$228$	0	0
$229$	30.2298	1.99764	0.998821	−	0.0485484i	$-0.0154595\pi$
$229$	30.2298	1.99764	0.998821	+	0.0485484i	$0.0154595\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6.29900	0.412661	0.206331	−	0.978482i	$-0.433848\pi$
$233$	6.29900	0.412661	0.206331	+	0.978482i	$0.433848\pi$
$234$	0	0
$235$	−8.52348	−0.556011
$236$	0	0
$237$	2.66136	0.172874
$238$	0	0
$239$	−7.67899	−0.496713	−0.248356	−	0.968669i	$-0.579890\pi$
$239$	−7.67899	−0.496713	−0.248356	+	0.968669i	$0.579890\pi$
$240$	0	0
$241$	23.6984	1.52655	0.763274	−	0.646075i	$-0.223591\pi$
$241$	23.6984	1.52655	0.763274	+	0.646075i	$0.223591\pi$
$242$	0	0
$243$	−7.19975	−0.461864
$244$	0	0
$245$	−7.72953	−0.493821
$246$	0	0
$247$	1.74738	0.111183
$248$	0	0
$249$	22.0084	1.39473
$250$	0	0
$251$	17.2336	1.08777	0.543887	−	0.839158i	$-0.316952\pi$
$251$	17.2336	1.08777	0.543887	+	0.839158i	$0.316952\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	10.1287	0.634282
$256$	0	0
$257$	−6.95359	−0.433753	−0.216876	−	0.976199i	$-0.569587\pi$
$257$	−6.95359	−0.433753	−0.216876	+	0.976199i	$0.569587\pi$
$258$	0	0
$259$	12.9048	0.801863
$260$	0	0
$261$	2.89871	0.179426
$262$	0	0
$263$	6.02989	0.371819	0.185910	−	0.982567i	$-0.440477\pi$
$263$	6.02989	0.371819	0.185910	+	0.982567i	$0.440477\pi$
$264$	0	0
$265$	−3.96583	−0.243619
$266$	0	0
$267$	−2.55313	−0.156249
$268$	0	0
$269$	8.53764	0.520549	0.260274	−	0.965535i	$-0.416187\pi$
$269$	8.53764	0.520549	0.260274	+	0.965535i	$0.416187\pi$
$270$	0	0
$271$	−5.95424	−0.361694	−0.180847	−	0.983511i	$-0.557884\pi$
$271$	−5.95424	−0.361694	−0.180847	+	0.983511i	$0.557884\pi$
$272$	0	0
$273$	2.89997	0.175514
$274$	0	0
$275$	0	0
$276$	0	0
$277$	2.40742	0.144648	0.0723239	−	0.997381i	$-0.476958\pi$
$277$	2.40742	0.144648	0.0723239	+	0.997381i	$0.476958\pi$
$278$	0	0
$279$	−4.85327	−0.290558
$280$	0	0
$281$	31.2959	1.86696	0.933478	−	0.358634i	$-0.116757\pi$
$281$	31.2959	1.86696	0.933478	+	0.358634i	$0.116757\pi$
$282$	0	0
$283$	−26.2031	−1.55761	−0.778806	−	0.627265i	$-0.784174\pi$
$283$	−26.2031	−1.55761	−0.778806	+	0.627265i	$0.784174\pi$
$284$	0	0
$285$	2.01792	0.119531
$286$	0	0
$287$	4.05870	0.239577
$288$	0	0
$289$	8.19395	0.481997
$290$	0	0
$291$	26.2880	1.54103
$292$	0	0
$293$	6.04507	0.353157	0.176578	−	0.984287i	$-0.443497\pi$
$293$	6.04507	0.353157	0.176578	+	0.984287i	$0.443497\pi$
$294$	0	0
$295$	−15.9195	−0.926868
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−9.29130	−0.537330
$300$	0	0
$301$	−2.55225	−0.147109
$302$	0	0
$303$	2.39216	0.137426
$304$	0	0
$305$	15.4854	0.886693
$306$	0	0
$307$	−26.0572	−1.48716	−0.743581	−	0.668645i	$-0.766874\pi$
$307$	−26.0572	−1.48716	−0.743581	+	0.668645i	$0.766874\pi$
$308$	0	0
$309$	−12.0476	−0.685366
$310$	0	0
$311$	3.48226	0.197461	0.0987303	−	0.995114i	$-0.468522\pi$
$311$	3.48226	0.197461	0.0987303	+	0.995114i	$0.468522\pi$
$312$	0	0
$313$	19.6569	1.11107	0.555537	−	0.831492i	$-0.312513\pi$
$313$	19.6569	1.11107	0.555537	+	0.831492i	$0.312513\pi$
$314$	0	0
$315$	−1.03641	−0.0583953
$316$	0	0
$317$	24.7033	1.38747	0.693737	−	0.720228i	$-0.255963\pi$
$317$	24.7033	1.38747	0.693737	+	0.720228i	$0.255963\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	2.24404	0.125250
$322$	0	0
$323$	5.01936	0.279285
$324$	0	0
$325$	5.63111	0.312358
$326$	0	0
$327$	−5.36594	−0.296737
$328$	0	0
$329$	−7.01001	−0.386475
$330$	0	0
$331$	−14.3024	−0.786130	−0.393065	−	0.919511i	$-0.628585\pi$
$331$	−14.3024	−0.786130	−0.393065	+	0.919511i	$0.628585\pi$
$332$	0	0
$333$	−8.34457	−0.457279
$334$	0	0
$335$	11.9396	0.652331
$336$	0	0
$337$	−14.2921	−0.778539	−0.389270	−	0.921124i	$-0.627273\pi$
$337$	−14.2921	−0.778539	−0.389270	+	0.921124i	$0.627273\pi$
$338$	0	0
$339$	12.2547	0.665583
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−14.0323	−0.757671
$344$	0	0
$345$	−10.7299	−0.577676
$346$	0	0
$347$	−2.21533	−0.118925	−0.0594626	−	0.998231i	$-0.518939\pi$
$347$	−2.21533	−0.118925	−0.0594626	+	0.998231i	$0.518939\pi$
$348$	0	0
$349$	−15.0859	−0.807528	−0.403764	−	0.914863i	$-0.632298\pi$
$349$	−15.0859	−0.807528	−0.403764	+	0.914863i	$0.632298\pi$
$350$	0	0
$351$	−9.80971	−0.523604
$352$	0	0
$353$	−5.26372	−0.280159	−0.140080	−	0.990140i	$-0.544736\pi$
$353$	−5.26372	−0.280159	−0.140080	+	0.990140i	$0.544736\pi$
$354$	0	0
$355$	5.55797	0.294986
$356$	0	0
$357$	8.33018	0.440880
$358$	0	0
$359$	−24.4717	−1.29157	−0.645783	−	0.763521i	$-0.723469\pi$
$359$	−24.4717	−1.29157	−0.645783	+	0.763521i	$0.723469\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	19.7618	1.03438
$366$	0	0
$367$	−14.9723	−0.781545	−0.390773	−	0.920487i	$-0.627792\pi$
$367$	−14.9723	−0.781545	−0.390773	+	0.920487i	$0.627792\pi$
$368$	0	0
$369$	−2.62446	−0.136624
$370$	0	0
$371$	−3.26164	−0.169336
$372$	0	0
$373$	−8.06489	−0.417584	−0.208792	−	0.977960i	$-0.566953\pi$
$373$	−8.06489	−0.417584	−0.208792	+	0.977960i	$0.566953\pi$
$374$	0	0
$375$	16.5926	0.856837
$376$	0	0
$377$	7.14405	0.367937
$378$	0	0
$379$	−16.5659	−0.850931	−0.425465	−	0.904975i	$-0.639890\pi$
$379$	−16.5659	−0.850931	−0.425465	+	0.904975i	$0.639890\pi$
$380$	0	0
$381$	18.1111	0.927860
$382$	0	0
$383$	−24.6298	−1.25852	−0.629262	−	0.777193i	$-0.716643\pi$
$383$	−24.6298	−1.25852	−0.629262	+	0.777193i	$0.716643\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1.65036	0.0838923
$388$	0	0
$389$	21.7681	1.10369	0.551844	−	0.833948i	$-0.313925\pi$
$389$	21.7681	1.10369	0.551844	+	0.833948i	$0.313925\pi$
$390$	0	0
$391$	−26.6893	−1.34974
$392$	0	0
$393$	−17.4365	−0.879553
$394$	0	0
$395$	−2.34414	−0.117947
$396$	0	0
$397$	7.03872	0.353263	0.176632	−	0.984277i	$-0.443480\pi$
$397$	7.03872	0.353263	0.176632	+	0.984277i	$0.443480\pi$
$398$	0	0
$399$	1.65961	0.0830844
$400$	0	0
$401$	−11.6108	−0.579814	−0.289907	−	0.957055i	$-0.593624\pi$
$401$	−11.6108	−0.579814	−0.289907	+	0.957055i	$0.593624\pi$
$402$	0	0
$403$	−11.9612	−0.595829
$404$	0	0
$405$	−8.49283	−0.422012
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−2.80727	−0.138811	−0.0694053	−	0.997589i	$-0.522110\pi$
$409$	−2.80727	−0.138811	−0.0694053	+	0.997589i	$0.522110\pi$
$410$	0	0
$411$	−29.3127	−1.44589
$412$	0	0
$413$	−13.0927	−0.644252
$414$	0	0
$415$	−19.3851	−0.951578
$416$	0	0
$417$	15.0815	0.738546
$418$	0	0
$419$	−30.3695	−1.48365	−0.741823	−	0.670596i	$-0.766039\pi$
$419$	−30.3695	−1.48365	−0.741823	+	0.670596i	$0.766039\pi$
$420$	0	0
$421$	−9.40058	−0.458156	−0.229078	−	0.973408i	$-0.573571\pi$
$421$	−9.40058	−0.458156	−0.229078	+	0.973408i	$0.573571\pi$
$422$	0	0
$423$	4.53286	0.220395
$424$	0	0
$425$	16.1754	0.784623
$426$	0	0
$427$	12.7358	0.616327
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−38.8161	−1.86971	−0.934854	−	0.355033i	$-0.884470\pi$
$431$	−38.8161	−1.86971	−0.934854	+	0.355033i	$0.884470\pi$
$432$	0	0
$433$	2.19839	0.105648	0.0528239	−	0.998604i	$-0.483178\pi$
$433$	2.19839	0.105648	0.0528239	+	0.998604i	$0.483178\pi$
$434$	0	0
$435$	8.25015	0.395565
$436$	0	0
$437$	−5.31728	−0.254360
$438$	0	0
$439$	−23.1097	−1.10297	−0.551483	−	0.834186i	$-0.685938\pi$
$439$	−23.1097	−1.10297	−0.551483	+	0.834186i	$0.685938\pi$
$440$	0	0
$441$	4.11063	0.195744
$442$	0	0
$443$	24.2163	1.15055	0.575277	−	0.817959i	$-0.304894\pi$
$443$	24.2163	1.15055	0.575277	+	0.817959i	$0.304894\pi$
$444$	0	0
$445$	2.24881	0.106604
$446$	0	0
$447$	23.2125	1.09791
$448$	0	0
$449$	9.27826	0.437868	0.218934	−	0.975740i	$-0.429742\pi$
$449$	9.27826	0.437868	0.218934	+	0.975740i	$0.429742\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	22.3943	1.05218
$454$	0	0
$455$	−2.55430	−0.119748
$456$	0	0
$457$	0.155557	0.00727666	0.00363833	−	0.999993i	$-0.498842\pi$
$457$	0.155557	0.00727666	0.00363833	+	0.999993i	$0.498842\pi$
$458$	0	0
$459$	−28.1785	−1.31526
$460$	0	0
$461$	19.0078	0.885280	0.442640	−	0.896699i	$-0.354042\pi$
$461$	19.0078	0.885280	0.442640	+	0.896699i	$0.354042\pi$
$462$	0	0
$463$	−12.5796	−0.584623	−0.292311	−	0.956323i	$-0.594424\pi$
$463$	−12.5796	−0.584623	−0.292311	+	0.956323i	$0.594424\pi$
$464$	0	0
$465$	−13.8131	−0.640567
$466$	0	0
$467$	−28.1224	−1.30135	−0.650676	−	0.759356i	$-0.725514\pi$
$467$	−28.1224	−1.30135	−0.650676	+	0.759356i	$0.725514\pi$
$468$	0	0
$469$	9.81956	0.453425
$470$	0	0
$471$	28.7560	1.32501
$472$	0	0
$473$	0	0
$474$	0	0
$475$	3.22261	0.147863
$476$	0	0
$477$	2.10906	0.0965674
$478$	0	0
$479$	−40.9983	−1.87326	−0.936631	−	0.350317i	$-0.886074\pi$
$479$	−40.9983	−1.87326	−0.936631	+	0.350317i	$0.886074\pi$
$480$	0	0
$481$	−20.5657	−0.937714
$482$	0	0
$483$	−8.82461	−0.401534
$484$	0	0
$485$	−23.1546	−1.05140
$486$	0	0
$487$	20.9160	0.947793	0.473896	−	0.880581i	$-0.342847\pi$
$487$	20.9160	0.947793	0.473896	+	0.880581i	$0.342847\pi$
$488$	0	0
$489$	−5.73569	−0.259377
$490$	0	0
$491$	−28.6659	−1.29367	−0.646837	−	0.762628i	$-0.723909\pi$
$491$	−28.6659	−1.29367	−0.646837	+	0.762628i	$0.723909\pi$
$492$	0	0
$493$	20.5213	0.924235
$494$	0	0
$495$	0	0
$496$	0	0
$497$	4.57107	0.205040
$498$	0	0
$499$	−21.8290	−0.977202	−0.488601	−	0.872507i	$-0.662493\pi$
$499$	−21.8290	−0.977202	−0.488601	+	0.872507i	$0.662493\pi$
$500$	0	0
$501$	10.9105	0.487447
$502$	0	0
$503$	13.3778	0.596487	0.298244	−	0.954490i	$-0.403599\pi$
$503$	13.3778	0.596487	0.298244	+	0.954490i	$0.403599\pi$
$504$	0	0
$505$	−2.10702	−0.0937612
$506$	0	0
$507$	15.0553	0.668630
$508$	0	0
$509$	−36.5833	−1.62153	−0.810763	−	0.585374i	$-0.800948\pi$
$509$	−36.5833	−1.62153	−0.810763	+	0.585374i	$0.800948\pi$
$510$	0	0
$511$	16.2528	0.718982
$512$	0	0
$513$	−5.61396	−0.247862
$514$	0	0
$515$	10.6116	0.467604
$516$	0	0
$517$	0	0
$518$	0	0
$519$	38.5058	1.69022
$520$	0	0
$521$	−6.67114	−0.292268	−0.146134	−	0.989265i	$-0.546683\pi$
$521$	−6.67114	−0.292268	−0.146134	+	0.989265i	$0.546683\pi$
$522$	0	0
$523$	40.4462	1.76859	0.884295	−	0.466929i	$-0.154640\pi$
$523$	40.4462	1.76859	0.884295	+	0.466929i	$0.154640\pi$
$524$	0	0
$525$	5.34827	0.233418
$526$	0	0
$527$	−34.3586	−1.49668
$528$	0	0
$529$	5.27346	0.229281
$530$	0	0
$531$	8.46612	0.367398
$532$	0	0
$533$	−6.46815	−0.280167
$534$	0	0
$535$	−1.97656	−0.0854542
$536$	0	0
$537$	17.6435	0.761372
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−22.2525	−0.956711	−0.478355	−	0.878166i	$-0.658767\pi$
$541$	−22.2525	−0.956711	−0.478355	+	0.878166i	$0.658767\pi$
$542$	0	0
$543$	−6.12609	−0.262896
$544$	0	0
$545$	4.72634	0.202454
$546$	0	0
$547$	−32.3814	−1.38453	−0.692265	−	0.721643i	$-0.743387\pi$
$547$	−32.3814	−1.38453	−0.692265	+	0.721643i	$0.743387\pi$
$548$	0	0
$549$	−8.23529	−0.351474
$550$	0	0
$551$	4.08844	0.174173
$552$	0	0
$553$	−1.92790	−0.0819829
$554$	0	0
$555$	−23.7498	−1.00812
$556$	0	0
$557$	19.0577	0.807499	0.403750	−	0.914870i	$-0.367707\pi$
$557$	19.0577	0.807499	0.403750	+	0.914870i	$0.367707\pi$
$558$	0	0
$559$	4.06740	0.172033
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−34.5958	−1.45804	−0.729020	−	0.684492i	$-0.760024\pi$
$563$	−34.5958	−1.45804	−0.729020	+	0.684492i	$0.760024\pi$
$564$	0	0
$565$	−10.7940	−0.454106
$566$	0	0
$567$	−6.98480	−0.293334
$568$	0	0
$569$	−5.80870	−0.243514	−0.121757	−	0.992560i	$-0.538853\pi$
$569$	−5.80870	−0.243514	−0.121757	+	0.992560i	$0.538853\pi$
$570$	0	0
$571$	−10.7114	−0.448259	−0.224129	−	0.974559i	$-0.571954\pi$
$571$	−10.7114	−0.448259	−0.224129	+	0.974559i	$0.571954\pi$
$572$	0	0
$573$	22.0521	0.921238
$574$	0	0
$575$	−17.1355	−0.714600
$576$	0	0
$577$	10.6984	0.445381	0.222690	−	0.974889i	$-0.428516\pi$
$577$	10.6984	0.445381	0.222690	+	0.974889i	$0.428516\pi$
$578$	0	0
$579$	32.9159	1.36794
$580$	0	0
$581$	−15.9430	−0.661427
$582$	0	0
$583$	0	0
$584$	0	0
$585$	1.65168	0.0682886
$586$	0	0
$587$	−4.80938	−0.198504	−0.0992521	−	0.995062i	$-0.531645\pi$
$587$	−4.80938	−0.198504	−0.0992521	+	0.995062i	$0.531645\pi$
$588$	0	0
$589$	−6.84522	−0.282052
$590$	0	0
$591$	15.0781	0.620230
$592$	0	0
$593$	−3.60379	−0.147990	−0.0739949	−	0.997259i	$-0.523575\pi$
$593$	−3.60379	−0.147990	−0.0739949	+	0.997259i	$0.523575\pi$
$594$	0	0
$595$	−7.33726	−0.300798
$596$	0	0
$597$	−23.2950	−0.953399
$598$	0	0
$599$	−1.95864	−0.0800279	−0.0400139	−	0.999199i	$-0.512740\pi$
$599$	−1.95864	−0.0800279	−0.0400139	+	0.999199i	$0.512740\pi$
$600$	0	0
$601$	0.795699	0.0324573	0.0162286	−	0.999868i	$-0.494834\pi$
$601$	0.795699	0.0324573	0.0162286	+	0.999868i	$0.494834\pi$
$602$	0	0
$603$	−6.34959	−0.258575
$604$	0	0
$605$	0	0
$606$	0	0
$607$	6.28887	0.255257	0.127629	−	0.991822i	$-0.459263\pi$
$607$	6.28887	0.255257	0.127629	+	0.991822i	$0.459263\pi$
$608$	0	0
$609$	6.78522	0.274951
$610$	0	0
$611$	11.1715	0.451951
$612$	0	0
$613$	10.1880	0.411489	0.205745	−	0.978606i	$-0.434038\pi$
$613$	10.1880	0.411489	0.205745	+	0.978606i	$0.434038\pi$
$614$	0	0
$615$	−7.46960	−0.301203
$616$	0	0
$617$	−0.686744	−0.0276473	−0.0138236	−	0.999904i	$-0.504400\pi$
$617$	−0.686744	−0.0276473	−0.0138236	+	0.999904i	$0.504400\pi$
$618$	0	0
$619$	25.4815	1.02419	0.512095	−	0.858929i	$-0.328870\pi$
$619$	25.4815	1.02419	0.512095	+	0.858929i	$0.328870\pi$
$620$	0	0
$621$	29.8510	1.19788
$622$	0	0
$623$	1.84950	0.0740988
$624$	0	0
$625$	1.49822	0.0599289
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−59.0751	−2.35548
$630$	0	0
$631$	−6.38938	−0.254357	−0.127179	−	0.991880i	$-0.540592\pi$
$631$	−6.38938	−0.254357	−0.127179	+	0.991880i	$0.540592\pi$
$632$	0	0
$633$	−2.16373	−0.0860007
$634$	0	0
$635$	−15.9523	−0.633050
$636$	0	0
$637$	10.1309	0.401401
$638$	0	0
$639$	−2.95578	−0.116929
$640$	0	0
$641$	−26.3450	−1.04056	−0.520282	−	0.853994i	$-0.674173\pi$
$641$	−26.3450	−1.04056	−0.520282	+	0.853994i	$0.674173\pi$
$642$	0	0
$643$	36.7133	1.44783	0.723915	−	0.689889i	$-0.242341\pi$
$643$	36.7133	1.44783	0.723915	+	0.689889i	$0.242341\pi$
$644$	0	0
$645$	4.69715	0.184950
$646$	0	0
$647$	−18.5628	−0.729780	−0.364890	−	0.931051i	$-0.618893\pi$
$647$	−18.5628	−0.729780	−0.364890	+	0.931051i	$0.618893\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−11.3604	−0.445249
$652$	0	0
$653$	0.551004	0.0215625	0.0107812	−	0.999942i	$-0.496568\pi$
$653$	0.551004	0.0215625	0.0107812	+	0.999942i	$0.496568\pi$
$654$	0	0
$655$	15.3581	0.600091
$656$	0	0
$657$	−10.5095	−0.410015
$658$	0	0
$659$	−1.55990	−0.0607651	−0.0303826	−	0.999538i	$-0.509673\pi$
$659$	−1.55990	−0.0607651	−0.0303826	+	0.999538i	$0.509673\pi$
$660$	0	0
$661$	−45.5441	−1.77146	−0.885730	−	0.464201i	$-0.846341\pi$
$661$	−45.5441	−1.77146	−0.885730	+	0.464201i	$0.846341\pi$
$662$	0	0
$663$	−13.2754	−0.515574
$664$	0	0
$665$	−1.46179	−0.0566858
$666$	0	0
$667$	−21.7394	−0.841752
$668$	0	0
$669$	1.20402	0.0465502
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−32.5268	−1.25382	−0.626909	−	0.779093i	$-0.715680\pi$
$673$	−32.5268	−1.25382	−0.626909	+	0.779093i	$0.715680\pi$
$674$	0	0
$675$	−18.0916	−0.696346
$676$	0	0
$677$	−46.9276	−1.80357	−0.901787	−	0.432180i	$-0.857745\pi$
$677$	−46.9276	−1.80357	−0.901787	+	0.432180i	$0.857745\pi$
$678$	0	0
$679$	−19.0432	−0.730810
$680$	0	0
$681$	−34.9961	−1.34105
$682$	0	0
$683$	−22.6833	−0.867952	−0.433976	−	0.900925i	$-0.642890\pi$
$683$	−22.6833	−0.867952	−0.433976	+	0.900925i	$0.642890\pi$
$684$	0	0
$685$	25.8188	0.986485
$686$	0	0
$687$	−45.7560	−1.74570
$688$	0	0
$689$	5.19791	0.198025
$690$	0	0
$691$	22.9500	0.873059	0.436530	−	0.899690i	$-0.356207\pi$
$691$	22.9500	0.873059	0.436530	+	0.899690i	$0.356207\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−13.2839	−0.503886
$696$	0	0
$697$	−18.5798	−0.703761
$698$	0	0
$699$	−9.53419	−0.360616
$700$	0	0
$701$	7.98081	0.301431	0.150715	−	0.988577i	$-0.451842\pi$
$701$	7.98081	0.301431	0.150715	+	0.988577i	$0.451842\pi$
$702$	0	0
$703$	−11.7695	−0.443893
$704$	0	0
$705$	12.9012	0.485886
$706$	0	0
$707$	−1.73289	−0.0651720
$708$	0	0
$709$	−1.22258	−0.0459149	−0.0229575	−	0.999736i	$-0.507308\pi$
$709$	−1.22258	−0.0459149	−0.0229575	+	0.999736i	$0.507308\pi$
$710$	0	0
$711$	1.24664	0.0467525
$712$	0	0
$713$	36.3979	1.36311
$714$	0	0
$715$	0	0
$716$	0	0
$717$	11.6230	0.434067
$718$	0	0
$719$	31.4841	1.17416	0.587079	−	0.809530i	$-0.300278\pi$
$719$	31.4841	1.17416	0.587079	+	0.809530i	$0.300278\pi$
$720$	0	0
$721$	8.72737	0.325024
$722$	0	0
$723$	−35.8700	−1.33402
$724$	0	0
$725$	13.1754	0.489323
$726$	0	0
$727$	−23.3533	−0.866127	−0.433064	−	0.901363i	$-0.642567\pi$
$727$	−23.3533	−0.866127	−0.433064	+	0.901363i	$0.642567\pi$
$728$	0	0
$729$	30.0085	1.11143
$730$	0	0
$731$	11.6836	0.432135
$732$	0	0
$733$	4.64897	0.171714	0.0858568	−	0.996307i	$-0.472637\pi$
$733$	4.64897	0.171714	0.0858568	+	0.996307i	$0.472637\pi$
$734$	0	0
$735$	11.6994	0.431540
$736$	0	0
$737$	0	0
$738$	0	0
$739$	10.3820	0.381908	0.190954	−	0.981599i	$-0.438842\pi$
$739$	10.3820	0.381908	0.190954	+	0.981599i	$0.438842\pi$
$740$	0	0
$741$	−2.64484	−0.0971606
$742$	0	0
$743$	20.8312	0.764221	0.382110	−	0.924117i	$-0.375197\pi$
$743$	20.8312	0.764221	0.382110	+	0.924117i	$0.375197\pi$
$744$	0	0
$745$	−20.4457	−0.749071
$746$	0	0
$747$	10.3092	0.377193
$748$	0	0
$749$	−1.62559	−0.0593979
$750$	0	0
$751$	−6.96440	−0.254135	−0.127067	−	0.991894i	$-0.540556\pi$
$751$	−6.96440	−0.254135	−0.127067	+	0.991894i	$0.540556\pi$
$752$	0	0
$753$	−26.0848	−0.950585
$754$	0	0
$755$	−19.7250	−0.717866
$756$	0	0
$757$	14.2001	0.516110	0.258055	−	0.966130i	$-0.416918\pi$
$757$	14.2001	0.516110	0.258055	+	0.966130i	$0.416918\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1.12777	0.0408817	0.0204409	−	0.999791i	$-0.493493\pi$
$761$	1.12777	0.0408817	0.0204409	+	0.999791i	$0.493493\pi$
$762$	0	0
$763$	3.88711	0.140723
$764$	0	0
$765$	4.74447	0.171537
$766$	0	0
$767$	20.8653	0.753401
$768$	0	0
$769$	−42.8668	−1.54582	−0.772908	−	0.634518i	$-0.781199\pi$
$769$	−42.8668	−1.54582	−0.772908	+	0.634518i	$0.781199\pi$
$770$	0	0
$771$	10.5250	0.379048
$772$	0	0
$773$	12.4787	0.448827	0.224413	−	0.974494i	$-0.427953\pi$
$773$	12.4787	0.448827	0.224413	+	0.974494i	$0.427953\pi$
$774$	0	0
$775$	−22.0594	−0.792398
$776$	0	0
$777$	−19.5327	−0.700732
$778$	0	0
$779$	−3.70163	−0.132625
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−22.9524	−0.820250
$784$	0	0
$785$	−25.3285	−0.904011
$786$	0	0
$787$	31.7045	1.13014	0.565072	−	0.825042i	$-0.308848\pi$
$787$	31.7045	1.13014	0.565072	+	0.825042i	$0.308848\pi$
$788$	0	0
$789$	−9.12687	−0.324925
$790$	0	0
$791$	−8.87734	−0.315642
$792$	0	0
$793$	−20.2964	−0.720745
$794$	0	0
$795$	6.00270	0.212894
$796$	0	0
$797$	24.7478	0.876610	0.438305	−	0.898826i	$-0.355579\pi$
$797$	24.7478	0.876610	0.438305	+	0.898826i	$0.355579\pi$
$798$	0	0
$799$	32.0903	1.13527
$800$	0	0
$801$	−1.19594	−0.0422564
$802$	0	0
$803$	0	0
$804$	0	0
$805$	7.77275	0.273954
$806$	0	0
$807$	−12.9226	−0.454897
$808$	0	0
$809$	8.16399	0.287031	0.143515	−	0.989648i	$-0.454159\pi$
$809$	8.16399	0.287031	0.143515	+	0.989648i	$0.454159\pi$
$810$	0	0
$811$	47.9887	1.68511	0.842555	−	0.538610i	$-0.181050\pi$
$811$	47.9887	1.68511	0.842555	+	0.538610i	$0.181050\pi$
$812$	0	0
$813$	9.01237	0.316078
$814$	0	0
$815$	5.05202	0.176964
$816$	0	0
$817$	2.32772	0.0814365
$818$	0	0
$819$	1.35840	0.0474664
$820$	0	0
$821$	−35.5472	−1.24061	−0.620304	−	0.784362i	$-0.712991\pi$
$821$	−35.5472	−1.24061	−0.620304	+	0.784362i	$0.712991\pi$
$822$	0	0
$823$	0.615346	0.0214496	0.0107248	−	0.999942i	$-0.496586\pi$
$823$	0.615346	0.0214496	0.0107248	+	0.999942i	$0.496586\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	8.89005	0.309137	0.154569	−	0.987982i	$-0.450601\pi$
$827$	8.89005	0.309137	0.154569	+	0.987982i	$0.450601\pi$
$828$	0	0
$829$	6.39629	0.222152	0.111076	−	0.993812i	$-0.464570\pi$
$829$	6.39629	0.222152	0.111076	+	0.993812i	$0.464570\pi$
$830$	0	0
$831$	−3.64388	−0.126405
$832$	0	0
$833$	29.1011	1.00829
$834$	0	0
$835$	−9.61005	−0.332569
$836$	0	0
$837$	38.4288	1.32829
$838$	0	0
$839$	−23.3227	−0.805189	−0.402594	−	0.915378i	$-0.631891\pi$
$839$	−23.3227	−0.805189	−0.402594	+	0.915378i	$0.631891\pi$
$840$	0	0
$841$	−12.2846	−0.423609
$842$	0	0
$843$	−47.3696	−1.63150
$844$	0	0
$845$	−13.2608	−0.456185
$846$	0	0
$847$	0	0
$848$	0	0
$849$	39.6611	1.36117
$850$	0	0
$851$	62.5815	2.14527
$852$	0	0
$853$	36.9318	1.26452	0.632260	−	0.774756i	$-0.282127\pi$
$853$	36.9318	1.26452	0.632260	+	0.774756i	$0.282127\pi$
$854$	0	0
$855$	0.945234	0.0323263
$856$	0	0
$857$	−24.3510	−0.831814	−0.415907	−	0.909407i	$-0.636536\pi$
$857$	−24.3510	−0.831814	−0.415907	+	0.909407i	$0.636536\pi$
$858$	0	0
$859$	16.7328	0.570917	0.285459	−	0.958391i	$-0.407854\pi$
$859$	16.7328	0.570917	0.285459	+	0.958391i	$0.407854\pi$
$860$	0	0
$861$	−6.14326	−0.209362
$862$	0	0
$863$	−26.9920	−0.918818	−0.459409	−	0.888225i	$-0.651939\pi$
$863$	−26.9920	−0.918818	−0.459409	+	0.888225i	$0.651939\pi$
$864$	0	0
$865$	−33.9161	−1.15318
$866$	0	0
$867$	−12.4024	−0.421208
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−15.6489	−0.530244
$872$	0	0
$873$	12.3138	0.416760
$874$	0	0
$875$	−12.0197	−0.406341
$876$	0	0
$877$	−8.75009	−0.295470	−0.147735	−	0.989027i	$-0.547198\pi$
$877$	−8.75009	−0.295470	−0.147735	+	0.989027i	$0.547198\pi$
$878$	0	0
$879$	−9.14984	−0.308616
$880$	0	0
$881$	49.2795	1.66027	0.830134	−	0.557564i	$-0.188264\pi$
$881$	49.2795	1.66027	0.830134	+	0.557564i	$0.188264\pi$
$882$	0	0
$883$	43.1865	1.45334	0.726670	−	0.686987i	$-0.241067\pi$
$883$	43.1865	1.45334	0.726670	+	0.686987i	$0.241067\pi$
$884$	0	0
$885$	24.0958	0.809972
$886$	0	0
$887$	−42.4943	−1.42682	−0.713409	−	0.700748i	$-0.752850\pi$
$887$	−42.4943	−1.42682	−0.713409	+	0.700748i	$0.752850\pi$
$888$	0	0
$889$	−13.1198	−0.440023
$890$	0	0
$891$	0	0
$892$	0	0
$893$	6.39330	0.213944
$894$	0	0
$895$	−15.5404	−0.519460
$896$	0	0
$897$	14.0633	0.469561
$898$	0	0
$899$	−27.9863	−0.933394
$900$	0	0
$901$	14.9310	0.497425
$902$	0	0
$903$	3.86310	0.128556
$904$	0	0
$905$	5.39588	0.179365
$906$	0	0
$907$	−21.8256	−0.724706	−0.362353	−	0.932041i	$-0.618027\pi$
$907$	−21.8256	−0.724706	−0.362353	+	0.932041i	$0.618027\pi$
$908$	0	0
$909$	1.12053	0.0371657
$910$	0	0
$911$	10.1765	0.337161	0.168581	−	0.985688i	$-0.446082\pi$
$911$	10.1765	0.337161	0.168581	+	0.985688i	$0.446082\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−23.4388	−0.774863
$916$	0	0
$917$	12.6311	0.417114
$918$	0	0
$919$	−36.3424	−1.19882	−0.599412	−	0.800441i	$-0.704599\pi$
$919$	−36.3424	−1.19882	−0.599412	+	0.800441i	$0.704599\pi$
$920$	0	0
$921$	39.4403	1.29960
$922$	0	0
$923$	−7.28469	−0.239778
$924$	0	0
$925$	−37.9283	−1.24708
$926$	0	0
$927$	−5.64335	−0.185352
$928$	0	0
$929$	27.3907	0.898658	0.449329	−	0.893366i	$-0.351663\pi$
$929$	27.3907	0.898658	0.449329	+	0.893366i	$0.351663\pi$
$930$	0	0
$931$	5.79777	0.190014
$932$	0	0
$933$	−5.27076	−0.172557
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−9.34995	−0.305450	−0.152725	−	0.988269i	$-0.548805\pi$
$937$	−9.34995	−0.305450	−0.152725	+	0.988269i	$0.548805\pi$
$938$	0	0
$939$	−29.7528	−0.970945
$940$	0	0
$941$	−39.4676	−1.28661	−0.643303	−	0.765611i	$-0.722437\pi$
$941$	−39.4676	−1.28661	−0.643303	+	0.765611i	$0.722437\pi$
$942$	0	0
$943$	19.6826	0.640954
$944$	0	0
$945$	8.20644	0.266956
$946$	0	0
$947$	33.7506	1.09675	0.548374	−	0.836233i	$-0.315247\pi$
$947$	33.7506	1.09675	0.548374	+	0.836233i	$0.315247\pi$
$948$	0	0
$949$	−25.9013	−0.840793
$950$	0	0
$951$	−37.3910	−1.21249
$952$	0	0
$953$	36.6455	1.18706	0.593531	−	0.804811i	$-0.297733\pi$
$953$	36.6455	1.18706	0.593531	+	0.804811i	$0.297733\pi$
$954$	0	0
$955$	−19.4235	−0.628531
$956$	0	0
$957$	0	0
$958$	0	0
$959$	21.2343	0.685691
$960$	0	0
$961$	15.8570	0.511516
$962$	0	0
$963$	1.05115	0.0338729
$964$	0	0
$965$	−28.9925	−0.933301
$966$	0	0
$967$	−46.7639	−1.50383	−0.751913	−	0.659262i	$-0.770869\pi$
$967$	−46.7639	−1.50383	−0.751913	+	0.659262i	$0.770869\pi$
$968$	0	0
$969$	−7.59732	−0.244061
$970$	0	0
$971$	−20.7415	−0.665626	−0.332813	−	0.942993i	$-0.607998\pi$
$971$	−20.7415	−0.665626	−0.332813	+	0.942993i	$0.607998\pi$
$972$	0	0
$973$	−10.9251	−0.350244
$974$	0	0
$975$	−8.52327	−0.272963
$976$	0	0
$977$	−25.7757	−0.824637	−0.412319	−	0.911040i	$-0.635281\pi$
$977$	−25.7757	−0.824637	−0.412319	+	0.911040i	$0.635281\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−2.51351	−0.0802502
$982$	0	0
$983$	35.9982	1.14816	0.574082	−	0.818798i	$-0.305359\pi$
$983$	35.9982	1.14816	0.574082	+	0.818798i	$0.305359\pi$
$984$	0	0
$985$	−13.2809	−0.423163
$986$	0	0
$987$	10.6104	0.337732
$988$	0	0
$989$	−12.3771	−0.393570
$990$	0	0
$991$	−3.65666	−0.116158	−0.0580788	−	0.998312i	$-0.518497\pi$
$991$	−3.65666	−0.116158	−0.0580788	+	0.998312i	$0.518497\pi$
$992$	0	0
$993$	21.6482	0.686984
$994$	0	0
$995$	20.5183	0.650474
$996$	0	0
$997$	51.1900	1.62121	0.810603	−	0.585597i	$-0.199140\pi$
$997$	51.1900	1.62121	0.810603	+	0.585597i	$0.199140\pi$
$998$	0	0
$999$	66.0733	2.09047

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9196.2.a.q.1.4	✓	8
11.10	odd	2		9196.2.a.r.1.4	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9196.2.a.q.1.4	✓	8	1.1	even	1	trivial
9196.2.a.r.1.4	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)

\(f(q)\)	\(=\)	\(q-1.51360 q^{3} +1.33319 q^{5} +1.09646 q^{7} -0.709002 q^{9} -1.74738 q^{13} -2.01792 q^{15} -5.01936 q^{17} -1.00000 q^{19} -1.65961 q^{21} +5.31728 q^{23} -3.22261 q^{25} +5.61396 q^{27} -4.08844 q^{29} +6.84522 q^{31} +1.46179 q^{35} +11.7695 q^{37} +2.64484 q^{39} +3.70163 q^{41} -2.32772 q^{43} -0.945234 q^{45} -6.39330 q^{47} -5.79777 q^{49} +7.59732 q^{51} -2.97469 q^{53} +1.51360 q^{57} -11.9409 q^{59} +11.6153 q^{61} -0.777394 q^{63} -2.32959 q^{65} +8.95568 q^{67} -8.04826 q^{69} +4.16892 q^{71} +14.8230 q^{73} +4.87775 q^{75} -1.75830 q^{79} -6.37031 q^{81} -14.5404 q^{83} -6.69175 q^{85} +6.18828 q^{87} +1.68679 q^{89} -1.91593 q^{91} -10.3609 q^{93} -1.33319 q^{95} -17.3678 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - q^{3} - 2 q^{7} + 9 q^{9} + 5 q^{13} - 9 q^{15} - 3 q^{17} - 8 q^{19} - 5 q^{21} + q^{23} - q^{27} - q^{29} + 3 q^{31} - 8 q^{35} - 23 q^{37} - 9 q^{39} - 10 q^{41} - 2 q^{43} + 9 q^{45} - 8 q^{47}+ \cdots + 6 q^{97}+O(q^{100}) \) 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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