LMFDB - Embedded newform 9680.2.a.cy.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9680,2,Mod(1,9680)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9680, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 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("9680.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 9680 = 2^{4} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9680.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,-2,0,-6,0,6,0,10,0,0,0,6,0,2,0,-11] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$77.2951891566$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.45753625.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 13x^{4} + 11x^{3} + 41x^{2} - 30x - 20 $ x^6 - x^5 - 13x^4 + 11x^3 + 41x^2 - 30x - 20
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 440)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.36422$ of defining polynomial
Character	$\chi$	$=$	9680.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.843136 q^{3} -1.00000 q^{5} +4.75693 q^{7} -2.28912 q^{9} -3.78308 q^{13} +0.843136 q^{15} -4.98203 q^{17} +3.12115 q^{19} -4.01074 q^{21} -9.39001 q^{23} +1.00000 q^{25} +4.45945 q^{27} +2.60693 q^{29} +3.68869 q^{31} -4.75693 q^{35} -3.44757 q^{37} +3.18965 q^{39} -9.32837 q^{41} -11.7051 q^{43} +2.28912 q^{45} +1.66156 q^{47} +15.6284 q^{49} +4.20053 q^{51} -1.93994 q^{53} -2.63156 q^{57} -0.103915 q^{59} +10.1513 q^{61} -10.8892 q^{63} +3.78308 q^{65} +4.99277 q^{67} +7.91706 q^{69} +6.22929 q^{71} +7.30650 q^{73} -0.843136 q^{75} +10.5445 q^{79} +3.10744 q^{81} +5.69010 q^{83} +4.98203 q^{85} -2.19800 q^{87} +13.6371 q^{89} -17.9958 q^{91} -3.11007 q^{93} -3.12115 q^{95} +13.6967 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{3} - 6 q^{5} + 6 q^{7} + 10 q^{9} + 6 q^{13} + 2 q^{15} - 11 q^{17} - 11 q^{19} - 2 q^{21} - 18 q^{23} + 6 q^{25} + q^{27} + 6 q^{29} - q^{31} - 6 q^{35} + 4 q^{37} + 27 q^{39} + 4 q^{41} + 3 q^{43}+ \cdots + 20 q^{97}+O(q^{100}) $ 6 * q - 2 * q^3 - 6 * q^5 + 6 * q^7 + 10 * q^9 + 6 * q^13 + 2 * q^15 - 11 * q^17 - 11 * q^19 - 2 * q^21 - 18 * q^23 + 6 * q^25 + q^27 + 6 * q^29 - q^31 - 6 * q^35 + 4 * q^37 + 27 * q^39 + 4 * q^41 + 3 * q^43 - 10 * q^45 - 14 * q^47 + 8 * q^49 + 31 * q^51 + 14 * q^53 + 5 * q^57 - 2 * q^59 + 4 * q^61 - 16 * q^63 - 6 * q^65 - 11 * q^67 + 8 * q^69 - 7 * q^71 + 9 * q^73 - 2 * q^75 + 36 * q^79 + 30 * q^81 - 45 * q^83 + 11 * q^85 + 25 * q^87 + q^89 + 8 * q^91 + 55 * q^93 + 11 * q^95 + 20 * 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
$3$	−0.843136	−0.486785	−0.243392	−	0.969928i	$-0.578260\pi$
$3$	−0.843136	−0.486785	−0.243392	+	0.969928i	$0.578260\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	4.75693	1.79795	0.898975	−	0.438000i	$-0.144313\pi$
$7$	4.75693	1.79795	0.898975	+	0.438000i	$0.144313\pi$
$8$	0	0
$9$	−2.28912	−0.763040
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−3.78308	−1.04924	−0.524619	−	0.851337i	$-0.675792\pi$
$13$	−3.78308	−1.04924	−0.524619	+	0.851337i	$0.675792\pi$
$14$	0	0
$15$	0.843136	0.217697
$16$	0	0
$17$	−4.98203	−1.20832	−0.604160	−	0.796863i	$-0.706491\pi$
$17$	−4.98203	−1.20832	−0.604160	+	0.796863i	$0.706491\pi$
$18$	0	0
$19$	3.12115	0.716041	0.358021	−	0.933714i	$-0.383452\pi$
$19$	3.12115	0.716041	0.358021	+	0.933714i	$0.383452\pi$
$20$	0	0
$21$	−4.01074	−0.875215
$22$	0	0
$23$	−9.39001	−1.95795	−0.978976	−	0.203975i	$-0.934614\pi$
$23$	−9.39001	−1.95795	−0.978976	+	0.203975i	$0.934614\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	4.45945	0.858222
$28$	0	0
$29$	2.60693	0.484095	0.242047	−	0.970264i	$-0.422181\pi$
$29$	2.60693	0.484095	0.242047	+	0.970264i	$0.422181\pi$
$30$	0	0
$31$	3.68869	0.662508	0.331254	−	0.943542i	$-0.392528\pi$
$31$	3.68869	0.662508	0.331254	+	0.943542i	$0.392528\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−4.75693	−0.804068
$36$	0	0
$37$	−3.44757	−0.566776	−0.283388	−	0.959005i	$-0.591458\pi$
$37$	−3.44757	−0.566776	−0.283388	+	0.959005i	$0.591458\pi$
$38$	0	0
$39$	3.18965	0.510753
$40$	0	0
$41$	−9.32837	−1.45685	−0.728423	−	0.685127i	$-0.759747\pi$
$41$	−9.32837	−1.45685	−0.728423	+	0.685127i	$0.759747\pi$
$42$	0	0
$43$	−11.7051	−1.78500	−0.892502	−	0.451043i	$-0.851052\pi$
$43$	−11.7051	−1.78500	−0.892502	+	0.451043i	$0.851052\pi$
$44$	0	0
$45$	2.28912	0.341242
$46$	0	0
$47$	1.66156	0.242364	0.121182	−	0.992630i	$-0.461332\pi$
$47$	1.66156	0.242364	0.121182	+	0.992630i	$0.461332\pi$
$48$	0	0
$49$	15.6284	2.23262
$50$	0	0
$51$	4.20053	0.588192
$52$	0	0
$53$	−1.93994	−0.266472	−0.133236	−	0.991084i	$-0.542537\pi$
$53$	−1.93994	−0.266472	−0.133236	+	0.991084i	$0.542537\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−2.63156	−0.348558
$58$	0	0
$59$	−0.103915	−0.0135285	−0.00676427	−	0.999977i	$-0.502153\pi$
$59$	−0.103915	−0.0135285	−0.00676427	+	0.999977i	$0.502153\pi$
$60$	0	0
$61$	10.1513	1.29974	0.649870	−	0.760045i	$-0.274823\pi$
$61$	10.1513	1.29974	0.649870	+	0.760045i	$0.274823\pi$
$62$	0	0
$63$	−10.8892	−1.37191
$64$	0	0
$65$	3.78308	0.469233
$66$	0	0
$67$	4.99277	0.609964	0.304982	−	0.952358i	$-0.401350\pi$
$67$	4.99277	0.609964	0.304982	+	0.952358i	$0.401350\pi$
$68$	0	0
$69$	7.91706	0.953102
$70$	0	0
$71$	6.22929	0.739281	0.369640	−	0.929175i	$-0.379481\pi$
$71$	6.22929	0.739281	0.369640	+	0.929175i	$0.379481\pi$
$72$	0	0
$73$	7.30650	0.855161	0.427580	−	0.903977i	$-0.359366\pi$
$73$	7.30650	0.855161	0.427580	+	0.903977i	$0.359366\pi$
$74$	0	0
$75$	−0.843136	−0.0973570
$76$	0	0
$77$	0	0
$78$	0	0
$79$	10.5445	1.18635	0.593175	−	0.805073i	$-0.297874\pi$
$79$	10.5445	1.18635	0.593175	+	0.805073i	$0.297874\pi$
$80$	0	0
$81$	3.10744	0.345271
$82$	0	0
$83$	5.69010	0.624569	0.312285	−	0.949989i	$-0.398906\pi$
$83$	5.69010	0.624569	0.312285	+	0.949989i	$0.398906\pi$
$84$	0	0
$85$	4.98203	0.540377
$86$	0	0
$87$	−2.19800	−0.235650
$88$	0	0
$89$	13.6371	1.44553	0.722767	−	0.691091i	$-0.242870\pi$
$89$	13.6371	1.44553	0.722767	+	0.691091i	$0.242870\pi$
$90$	0	0
$91$	−17.9958	−1.88648
$92$	0	0
$93$	−3.11007	−0.322499
$94$	0	0
$95$	−3.12115	−0.320223
$96$	0	0
$97$	13.6967	1.39069	0.695345	−	0.718676i	$-0.255252\pi$
$97$	13.6967	1.39069	0.695345	+	0.718676i	$0.255252\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−4.84287	−0.481884	−0.240942	−	0.970540i	$-0.577456\pi$
$101$	−4.84287	−0.481884	−0.240942	+	0.970540i	$0.577456\pi$
$102$	0	0
$103$	−0.904576	−0.0891305	−0.0445653	−	0.999006i	$-0.514190\pi$
$103$	−0.904576	−0.0891305	−0.0445653	+	0.999006i	$0.514190\pi$
$104$	0	0
$105$	4.01074	0.391408
$106$	0	0
$107$	−14.7915	−1.42995	−0.714974	−	0.699151i	$-0.753561\pi$
$107$	−14.7915	−1.42995	−0.714974	+	0.699151i	$0.753561\pi$
$108$	0	0
$109$	−3.39429	−0.325114	−0.162557	−	0.986699i	$-0.551974\pi$
$109$	−3.39429	−0.325114	−0.162557	+	0.986699i	$0.551974\pi$
$110$	0	0
$111$	2.90677	0.275898
$112$	0	0
$113$	−10.1755	−0.957231	−0.478616	−	0.878025i	$-0.658861\pi$
$113$	−10.1755	−0.957231	−0.478616	+	0.878025i	$0.658861\pi$
$114$	0	0
$115$	9.39001	0.875623
$116$	0	0
$117$	8.65993	0.800611
$118$	0	0
$119$	−23.6992	−2.17250
$120$	0	0
$121$	0	0
$122$	0	0
$123$	7.86509	0.709171
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−10.9587	−0.972430	−0.486215	−	0.873839i	$-0.661623\pi$
$127$	−10.9587	−0.972430	−0.486215	+	0.873839i	$0.661623\pi$
$128$	0	0
$129$	9.86896	0.868913
$130$	0	0
$131$	12.9021	1.12726	0.563631	−	0.826027i	$-0.309404\pi$
$131$	12.9021	1.12726	0.563631	+	0.826027i	$0.309404\pi$
$132$	0	0
$133$	14.8471	1.28741
$134$	0	0
$135$	−4.45945	−0.383808
$136$	0	0
$137$	18.8824	1.61324	0.806618	−	0.591073i	$-0.201295\pi$
$137$	18.8824	1.61324	0.806618	+	0.591073i	$0.201295\pi$
$138$	0	0
$139$	10.8892	0.923611	0.461806	−	0.886981i	$-0.347202\pi$
$139$	10.8892	0.923611	0.461806	+	0.886981i	$0.347202\pi$
$140$	0	0
$141$	−1.40092	−0.117979
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−2.60693	−0.216494
$146$	0	0
$147$	−13.1768	−1.08681
$148$	0	0
$149$	−1.69412	−0.138788	−0.0693941	−	0.997589i	$-0.522107\pi$
$149$	−1.69412	−0.138788	−0.0693941	+	0.997589i	$0.522107\pi$
$150$	0	0
$151$	−3.48984	−0.283999	−0.142000	−	0.989867i	$-0.545353\pi$
$151$	−3.48984	−0.283999	−0.142000	+	0.989867i	$0.545353\pi$
$152$	0	0
$153$	11.4045	0.921997
$154$	0	0
$155$	−3.68869	−0.296283
$156$	0	0
$157$	20.7003	1.65207	0.826033	−	0.563622i	$-0.190592\pi$
$157$	20.7003	1.65207	0.826033	+	0.563622i	$0.190592\pi$
$158$	0	0
$159$	1.63564	0.129714
$160$	0	0
$161$	−44.6676	−3.52030
$162$	0	0
$163$	−16.3831	−1.28323	−0.641613	−	0.767028i	$-0.721735\pi$
$163$	−16.3831	−1.28323	−0.641613	+	0.767028i	$0.721735\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	14.4012	1.11440	0.557199	−	0.830379i	$-0.311876\pi$
$167$	14.4012	1.11440	0.557199	+	0.830379i	$0.311876\pi$
$168$	0	0
$169$	1.31169	0.100900
$170$	0	0
$171$	−7.14470	−0.546369
$172$	0	0
$173$	8.12354	0.617621	0.308811	−	0.951124i	$-0.400069\pi$
$173$	8.12354	0.617621	0.308811	+	0.951124i	$0.400069\pi$
$174$	0	0
$175$	4.75693	0.359590
$176$	0	0
$177$	0.0876143	0.00658549
$178$	0	0
$179$	−2.99340	−0.223737	−0.111869	−	0.993723i	$-0.535684\pi$
$179$	−2.99340	−0.223737	−0.111869	+	0.993723i	$0.535684\pi$
$180$	0	0
$181$	−14.5264	−1.07974	−0.539870	−	0.841749i	$-0.681526\pi$
$181$	−14.5264	−1.07974	−0.539870	+	0.841749i	$0.681526\pi$
$182$	0	0
$183$	−8.55893	−0.632694
$184$	0	0
$185$	3.44757	0.253470
$186$	0	0
$187$	0	0
$188$	0	0
$189$	21.2133	1.54304
$190$	0	0
$191$	7.96741	0.576502	0.288251	−	0.957555i	$-0.406926\pi$
$191$	7.96741	0.576502	0.288251	+	0.957555i	$0.406926\pi$
$192$	0	0
$193$	19.4962	1.40337	0.701684	−	0.712489i	$-0.252432\pi$
$193$	19.4962	1.40337	0.701684	+	0.712489i	$0.252432\pi$
$194$	0	0
$195$	−3.18965	−0.228416
$196$	0	0
$197$	−17.9751	−1.28067	−0.640336	−	0.768095i	$-0.721205\pi$
$197$	−17.9751	−1.28067	−0.640336	+	0.768095i	$0.721205\pi$
$198$	0	0
$199$	7.17717	0.508776	0.254388	−	0.967102i	$-0.418126\pi$
$199$	7.17717	0.508776	0.254388	+	0.967102i	$0.418126\pi$
$200$	0	0
$201$	−4.20959	−0.296921
$202$	0	0
$203$	12.4010	0.870378
$204$	0	0
$205$	9.32837	0.651522
$206$	0	0
$207$	21.4949	1.49400
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−4.43285	−0.305170	−0.152585	−	0.988290i	$-0.548760\pi$
$211$	−4.43285	−0.305170	−0.152585	+	0.988290i	$0.548760\pi$
$212$	0	0
$213$	−5.25214	−0.359871
$214$	0	0
$215$	11.7051	0.798278
$216$	0	0
$217$	17.5468	1.19116
$218$	0	0
$219$	−6.16037	−0.416280
$220$	0	0
$221$	18.8474	1.26781
$222$	0	0
$223$	15.2105	1.01857	0.509286	−	0.860597i	$-0.329909\pi$
$223$	15.2105	1.01857	0.509286	+	0.860597i	$0.329909\pi$
$224$	0	0
$225$	−2.28912	−0.152608
$226$	0	0
$227$	4.28398	0.284338	0.142169	−	0.989842i	$-0.454592\pi$
$227$	4.28398	0.284338	0.142169	+	0.989842i	$0.454592\pi$
$228$	0	0
$229$	−4.09655	−0.270708	−0.135354	−	0.990797i	$-0.543217\pi$
$229$	−4.09655	−0.270708	−0.135354	+	0.990797i	$0.543217\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	17.2531	1.13029	0.565143	−	0.824993i	$-0.308821\pi$
$233$	17.2531	1.13029	0.565143	+	0.824993i	$0.308821\pi$
$234$	0	0
$235$	−1.66156	−0.108388
$236$	0	0
$237$	−8.89046	−0.577498
$238$	0	0
$239$	−3.40626	−0.220332	−0.110166	−	0.993913i	$-0.535138\pi$
$239$	−3.40626	−0.220332	−0.110166	+	0.993913i	$0.535138\pi$
$240$	0	0
$241$	20.1839	1.30016	0.650080	−	0.759866i	$-0.274735\pi$
$241$	20.1839	1.30016	0.650080	+	0.759866i	$0.274735\pi$
$242$	0	0
$243$	−15.9983	−1.02629
$244$	0	0
$245$	−15.6284	−0.998460
$246$	0	0
$247$	−11.8076	−0.751298
$248$	0	0
$249$	−4.79753	−0.304031
$250$	0	0
$251$	−19.6971	−1.24327	−0.621635	−	0.783307i	$-0.713531\pi$
$251$	−19.6971	−1.24327	−0.621635	+	0.783307i	$0.713531\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−4.20053	−0.263047
$256$	0	0
$257$	−10.7134	−0.668285	−0.334143	−	0.942523i	$-0.608447\pi$
$257$	−10.7134	−0.668285	−0.334143	+	0.942523i	$0.608447\pi$
$258$	0	0
$259$	−16.3998	−1.01904
$260$	0	0
$261$	−5.96758	−0.369384
$262$	0	0
$263$	8.86317	0.546527	0.273263	−	0.961939i	$-0.411897\pi$
$263$	8.86317	0.546527	0.273263	+	0.961939i	$0.411897\pi$
$264$	0	0
$265$	1.93994	0.119170
$266$	0	0
$267$	−11.4980	−0.703665
$268$	0	0
$269$	−2.15481	−0.131381	−0.0656905	−	0.997840i	$-0.520925\pi$
$269$	−2.15481	−0.131381	−0.0656905	+	0.997840i	$0.520925\pi$
$270$	0	0
$271$	10.2552	0.622961	0.311481	−	0.950252i	$-0.399175\pi$
$271$	10.2552	0.622961	0.311481	+	0.950252i	$0.399175\pi$
$272$	0	0
$273$	15.1729	0.918309
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−14.8253	−0.890766	−0.445383	−	0.895340i	$-0.646933\pi$
$277$	−14.8253	−0.890766	−0.445383	+	0.895340i	$0.646933\pi$
$278$	0	0
$279$	−8.44386	−0.505521
$280$	0	0
$281$	16.8470	1.00501	0.502504	−	0.864575i	$-0.332412\pi$
$281$	16.8470	1.00501	0.502504	+	0.864575i	$0.332412\pi$
$282$	0	0
$283$	−4.89422	−0.290931	−0.145466	−	0.989363i	$-0.546468\pi$
$283$	−4.89422	−0.290931	−0.145466	+	0.989363i	$0.546468\pi$
$284$	0	0
$285$	2.63156	0.155880
$286$	0	0
$287$	−44.3744	−2.61934
$288$	0	0
$289$	7.82064	0.460037
$290$	0	0
$291$	−11.5482	−0.676967
$292$	0	0
$293$	−11.9575	−0.698565	−0.349283	−	0.937017i	$-0.613575\pi$
$293$	−11.9575	−0.698565	−0.349283	+	0.937017i	$0.613575\pi$
$294$	0	0
$295$	0.103915	0.00605015
$296$	0	0
$297$	0	0
$298$	0	0
$299$	35.5232	2.05436
$300$	0	0
$301$	−55.6801	−3.20935
$302$	0	0
$303$	4.08320	0.234574
$304$	0	0
$305$	−10.1513	−0.581262
$306$	0	0
$307$	16.8325	0.960680	0.480340	−	0.877082i	$-0.340513\pi$
$307$	16.8325	0.960680	0.480340	+	0.877082i	$0.340513\pi$
$308$	0	0
$309$	0.762681	0.0433874
$310$	0	0
$311$	−24.5949	−1.39465	−0.697324	−	0.716756i	$-0.745626\pi$
$311$	−24.5949	−1.39465	−0.697324	+	0.716756i	$0.745626\pi$
$312$	0	0
$313$	7.20995	0.407530	0.203765	−	0.979020i	$-0.434682\pi$
$313$	7.20995	0.407530	0.203765	+	0.979020i	$0.434682\pi$
$314$	0	0
$315$	10.8892	0.613536
$316$	0	0
$317$	−8.55901	−0.480722	−0.240361	−	0.970684i	$-0.577266\pi$
$317$	−8.55901	−0.480722	−0.240361	+	0.970684i	$0.577266\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	12.4712	0.696077
$322$	0	0
$323$	−15.5497	−0.865207
$324$	0	0
$325$	−3.78308	−0.209848
$326$	0	0
$327$	2.86185	0.158260
$328$	0	0
$329$	7.90393	0.435758
$330$	0	0
$331$	25.4094	1.39662	0.698312	−	0.715793i	$-0.253935\pi$
$331$	25.4094	1.39662	0.698312	+	0.715793i	$0.253935\pi$
$332$	0	0
$333$	7.89189	0.432473
$334$	0	0
$335$	−4.99277	−0.272784
$336$	0	0
$337$	18.9403	1.03175	0.515873	−	0.856665i	$-0.327468\pi$
$337$	18.9403	1.03175	0.515873	+	0.856665i	$0.327468\pi$
$338$	0	0
$339$	8.57934	0.465966
$340$	0	0
$341$	0	0
$342$	0	0
$343$	41.0446	2.21620
$344$	0	0
$345$	−7.91706	−0.426240
$346$	0	0
$347$	−7.89209	−0.423670	−0.211835	−	0.977305i	$-0.567944\pi$
$347$	−7.89209	−0.423670	−0.211835	+	0.977305i	$0.567944\pi$
$348$	0	0
$349$	−7.45858	−0.399249	−0.199624	−	0.979873i	$-0.563972\pi$
$349$	−7.45858	−0.399249	−0.199624	+	0.979873i	$0.563972\pi$
$350$	0	0
$351$	−16.8705	−0.900478
$352$	0	0
$353$	−2.03405	−0.108262	−0.0541308	−	0.998534i	$-0.517239\pi$
$353$	−2.03405	−0.108262	−0.0541308	+	0.998534i	$0.517239\pi$
$354$	0	0
$355$	−6.22929	−0.330616
$356$	0	0
$357$	19.9816	1.05754
$358$	0	0
$359$	16.3025	0.860413	0.430206	−	0.902731i	$-0.358441\pi$
$359$	16.3025	0.860413	0.430206	+	0.902731i	$0.358441\pi$
$360$	0	0
$361$	−9.25841	−0.487285
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−7.30650	−0.382440
$366$	0	0
$367$	0.770896	0.0402404	0.0201202	−	0.999798i	$-0.493595\pi$
$367$	0.770896	0.0402404	0.0201202	+	0.999798i	$0.493595\pi$
$368$	0	0
$369$	21.3538	1.11163
$370$	0	0
$371$	−9.22817	−0.479103
$372$	0	0
$373$	5.01958	0.259904	0.129952	−	0.991520i	$-0.458518\pi$
$373$	5.01958	0.259904	0.129952	+	0.991520i	$0.458518\pi$
$374$	0	0
$375$	0.843136	0.0435394
$376$	0	0
$377$	−9.86222	−0.507930
$378$	0	0
$379$	−3.65481	−0.187735	−0.0938676	−	0.995585i	$-0.529923\pi$
$379$	−3.65481	−0.187735	−0.0938676	+	0.995585i	$0.529923\pi$
$380$	0	0
$381$	9.23970	0.473364
$382$	0	0
$383$	−5.48146	−0.280090	−0.140045	−	0.990145i	$-0.544725\pi$
$383$	−5.48146	−0.280090	−0.140045	+	0.990145i	$0.544725\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	26.7943	1.36203
$388$	0	0
$389$	0.840557	0.0426179	0.0213090	−	0.999773i	$-0.493217\pi$
$389$	0.840557	0.0426179	0.0213090	+	0.999773i	$0.493217\pi$
$390$	0	0
$391$	46.7813	2.36583
$392$	0	0
$393$	−10.8782	−0.548734
$394$	0	0
$395$	−10.5445	−0.530552
$396$	0	0
$397$	3.61935	0.181650	0.0908249	−	0.995867i	$-0.471050\pi$
$397$	3.61935	0.181650	0.0908249	+	0.995867i	$0.471050\pi$
$398$	0	0
$399$	−12.5181	−0.626690
$400$	0	0
$401$	10.0879	0.503768	0.251884	−	0.967757i	$-0.418950\pi$
$401$	10.0879	0.503768	0.251884	+	0.967757i	$0.418950\pi$
$402$	0	0
$403$	−13.9546	−0.695129
$404$	0	0
$405$	−3.10744	−0.154410
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−12.3840	−0.612352	−0.306176	−	0.951975i	$-0.599050\pi$
$409$	−12.3840	−0.612352	−0.306176	+	0.951975i	$0.599050\pi$
$410$	0	0
$411$	−15.9205	−0.785299
$412$	0	0
$413$	−0.494315	−0.0243237
$414$	0	0
$415$	−5.69010	−0.279316
$416$	0	0
$417$	−9.18109	−0.449600
$418$	0	0
$419$	0.612942	0.0299442	0.0149721	−	0.999888i	$-0.495234\pi$
$419$	0.612942	0.0299442	0.0149721	+	0.999888i	$0.495234\pi$
$420$	0	0
$421$	−33.5449	−1.63488	−0.817440	−	0.576013i	$-0.804608\pi$
$421$	−33.5449	−1.63488	−0.817440	+	0.576013i	$0.804608\pi$
$422$	0	0
$423$	−3.80352	−0.184933
$424$	0	0
$425$	−4.98203	−0.241664
$426$	0	0
$427$	48.2890	2.33687
$428$	0	0
$429$	0	0
$430$	0	0
$431$	8.72012	0.420034	0.210017	−	0.977698i	$-0.432648\pi$
$431$	8.72012	0.420034	0.210017	+	0.977698i	$0.432648\pi$
$432$	0	0
$433$	−11.3703	−0.546420	−0.273210	−	0.961954i	$-0.588085\pi$
$433$	−11.3703	−0.546420	−0.273210	+	0.961954i	$0.588085\pi$
$434$	0	0
$435$	2.19800	0.105386
$436$	0	0
$437$	−29.3076	−1.40197
$438$	0	0
$439$	24.3176	1.16061	0.580307	−	0.814398i	$-0.302933\pi$
$439$	24.3176	1.16061	0.580307	+	0.814398i	$0.302933\pi$
$440$	0	0
$441$	−35.7752	−1.70358
$442$	0	0
$443$	0.227915	0.0108285	0.00541427	−	0.999985i	$-0.498277\pi$
$443$	0.227915	0.0108285	0.00541427	+	0.999985i	$0.498277\pi$
$444$	0	0
$445$	−13.6371	−0.646463
$446$	0	0
$447$	1.42838	0.0675600
$448$	0	0
$449$	40.4036	1.90676	0.953382	−	0.301767i	$-0.0975764\pi$
$449$	40.4036	1.90676	0.953382	+	0.301767i	$0.0975764\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	2.94241	0.138247
$454$	0	0
$455$	17.9958	0.843658
$456$	0	0
$457$	−4.54698	−0.212699	−0.106349	−	0.994329i	$-0.533916\pi$
$457$	−4.54698	−0.212699	−0.106349	+	0.994329i	$0.533916\pi$
$458$	0	0
$459$	−22.2171	−1.03701
$460$	0	0
$461$	36.5792	1.70366	0.851832	−	0.523815i	$-0.175492\pi$
$461$	36.5792	1.70366	0.851832	+	0.523815i	$0.175492\pi$
$462$	0	0
$463$	5.18502	0.240968	0.120484	−	0.992715i	$-0.461555\pi$
$463$	5.18502	0.240968	0.120484	+	0.992715i	$0.461555\pi$
$464$	0	0
$465$	3.11007	0.144226
$466$	0	0
$467$	10.0802	0.466454	0.233227	−	0.972422i	$-0.425071\pi$
$467$	10.0802	0.466454	0.233227	+	0.972422i	$0.425071\pi$
$468$	0	0
$469$	23.7503	1.09668
$470$	0	0
$471$	−17.4532	−0.804201
$472$	0	0
$473$	0	0
$474$	0	0
$475$	3.12115	0.143208
$476$	0	0
$477$	4.44077	0.203329
$478$	0	0
$479$	2.56337	0.117123	0.0585617	−	0.998284i	$-0.481349\pi$
$479$	2.56337	0.117123	0.0585617	+	0.998284i	$0.481349\pi$
$480$	0	0
$481$	13.0424	0.594683
$482$	0	0
$483$	37.6609	1.71363
$484$	0	0
$485$	−13.6967	−0.621935
$486$	0	0
$487$	−2.41271	−0.109330	−0.0546651	−	0.998505i	$-0.517409\pi$
$487$	−2.41271	−0.109330	−0.0546651	+	0.998505i	$0.517409\pi$
$488$	0	0
$489$	13.8132	0.624656
$490$	0	0
$491$	15.0326	0.678412	0.339206	−	0.940712i	$-0.389842\pi$
$491$	15.0326	0.678412	0.339206	+	0.940712i	$0.389842\pi$
$492$	0	0
$493$	−12.9878	−0.584941
$494$	0	0
$495$	0	0
$496$	0	0
$497$	29.6323	1.32919
$498$	0	0
$499$	−7.43047	−0.332634	−0.166317	−	0.986072i	$-0.553187\pi$
$499$	−7.43047	−0.332634	−0.166317	+	0.986072i	$0.553187\pi$
$500$	0	0
$501$	−12.1422	−0.542472
$502$	0	0
$503$	15.1978	0.677637	0.338818	−	0.940852i	$-0.389973\pi$
$503$	15.1978	0.677637	0.338818	+	0.940852i	$0.389973\pi$
$504$	0	0
$505$	4.84287	0.215505
$506$	0	0
$507$	−1.10594	−0.0491164
$508$	0	0
$509$	−23.0293	−1.02076	−0.510379	−	0.859950i	$-0.670495\pi$
$509$	−23.0293	−1.02076	−0.510379	+	0.859950i	$0.670495\pi$
$510$	0	0
$511$	34.7565	1.53754
$512$	0	0
$513$	13.9186	0.614522
$514$	0	0
$515$	0.904576	0.0398604
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−6.84925	−0.300649
$520$	0	0
$521$	1.10634	0.0484698	0.0242349	−	0.999706i	$-0.492285\pi$
$521$	1.10634	0.0484698	0.0242349	+	0.999706i	$0.492285\pi$
$522$	0	0
$523$	−1.16929	−0.0511294	−0.0255647	−	0.999673i	$-0.508138\pi$
$523$	−1.16929	−0.0511294	−0.0255647	+	0.999673i	$0.508138\pi$
$524$	0	0
$525$	−4.01074	−0.175043
$526$	0	0
$527$	−18.3772	−0.800522
$528$	0	0
$529$	65.1723	2.83358
$530$	0	0
$531$	0.237873	0.0103228
$532$	0	0
$533$	35.2900	1.52858
$534$	0	0
$535$	14.7915	0.639492
$536$	0	0
$537$	2.52384	0.108912
$538$	0	0
$539$	0	0
$540$	0	0
$541$	3.43127	0.147522	0.0737609	−	0.997276i	$-0.476500\pi$
$541$	3.43127	0.147522	0.0737609	+	0.997276i	$0.476500\pi$
$542$	0	0
$543$	12.2477	0.525601
$544$	0	0
$545$	3.39429	0.145395
$546$	0	0
$547$	−41.9440	−1.79340	−0.896699	−	0.442642i	$-0.854041\pi$
$547$	−41.9440	−1.79340	−0.896699	+	0.442642i	$0.854041\pi$
$548$	0	0
$549$	−23.2376	−0.991755
$550$	0	0
$551$	8.13662	0.346632
$552$	0	0
$553$	50.1595	2.13300
$554$	0	0
$555$	−2.90677	−0.123385
$556$	0	0
$557$	32.5066	1.37735	0.688675	−	0.725070i	$-0.258193\pi$
$557$	32.5066	1.37735	0.688675	+	0.725070i	$0.258193\pi$
$558$	0	0
$559$	44.2812	1.87289
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−37.2583	−1.57025	−0.785124	−	0.619338i	$-0.787401\pi$
$563$	−37.2583	−1.57025	−0.785124	+	0.619338i	$0.787401\pi$
$564$	0	0
$565$	10.1755	0.428087
$566$	0	0
$567$	14.7819	0.620780
$568$	0	0
$569$	37.5360	1.57359	0.786796	−	0.617213i	$-0.211738\pi$
$569$	37.5360	1.57359	0.786796	+	0.617213i	$0.211738\pi$
$570$	0	0
$571$	−21.6905	−0.907718	−0.453859	−	0.891073i	$-0.649953\pi$
$571$	−21.6905	−0.907718	−0.453859	+	0.891073i	$0.649953\pi$
$572$	0	0
$573$	−6.71762	−0.280632
$574$	0	0
$575$	−9.39001	−0.391590
$576$	0	0
$577$	−26.9306	−1.12114	−0.560568	−	0.828108i	$-0.689417\pi$
$577$	−26.9306	−1.12114	−0.560568	+	0.828108i	$0.689417\pi$
$578$	0	0
$579$	−16.4380	−0.683138
$580$	0	0
$581$	27.0674	1.12294
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−8.65993	−0.358044
$586$	0	0
$587$	15.4757	0.638751	0.319376	−	0.947628i	$-0.396527\pi$
$587$	15.4757	0.638751	0.319376	+	0.947628i	$0.396527\pi$
$588$	0	0
$589$	11.5130	0.474383
$590$	0	0
$591$	15.1554	0.623412
$592$	0	0
$593$	−24.7646	−1.01696	−0.508481	−	0.861073i	$-0.669793\pi$
$593$	−24.7646	−1.01696	−0.508481	+	0.861073i	$0.669793\pi$
$594$	0	0
$595$	23.6992	0.971571
$596$	0	0
$597$	−6.05133	−0.247665
$598$	0	0
$599$	−12.9401	−0.528716	−0.264358	−	0.964425i	$-0.585160\pi$
$599$	−12.9401	−0.528716	−0.264358	+	0.964425i	$0.585160\pi$
$600$	0	0
$601$	18.7917	0.766531	0.383266	−	0.923638i	$-0.374799\pi$
$601$	18.7917	0.766531	0.383266	+	0.923638i	$0.374799\pi$
$602$	0	0
$603$	−11.4291	−0.465427
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−14.3465	−0.582308	−0.291154	−	0.956676i	$-0.594039\pi$
$607$	−14.3465	−0.582308	−0.291154	+	0.956676i	$0.594039\pi$
$608$	0	0
$609$	−10.4557	−0.423687
$610$	0	0
$611$	−6.28582	−0.254297
$612$	0	0
$613$	39.4276	1.59247	0.796233	−	0.604990i	$-0.206823\pi$
$613$	39.4276	1.59247	0.796233	+	0.604990i	$0.206823\pi$
$614$	0	0
$615$	−7.86509	−0.317151
$616$	0	0
$617$	32.9316	1.32578	0.662888	−	0.748718i	$-0.269330\pi$
$617$	32.9316	1.32578	0.662888	+	0.748718i	$0.269330\pi$
$618$	0	0
$619$	41.2089	1.65632	0.828162	−	0.560489i	$-0.189387\pi$
$619$	41.2089	1.65632	0.828162	+	0.560489i	$0.189387\pi$
$620$	0	0
$621$	−41.8743	−1.68036
$622$	0	0
$623$	64.8709	2.59900
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	17.1759	0.684847
$630$	0	0
$631$	−7.79380	−0.310266	−0.155133	−	0.987894i	$-0.549581\pi$
$631$	−7.79380	−0.310266	−0.155133	+	0.987894i	$0.549581\pi$
$632$	0	0
$633$	3.73749	0.148552
$634$	0	0
$635$	10.9587	0.434884
$636$	0	0
$637$	−59.1234	−2.34255
$638$	0	0
$639$	−14.2596	−0.564101
$640$	0	0
$641$	−16.9699	−0.670269	−0.335135	−	0.942170i	$-0.608782\pi$
$641$	−16.9699	−0.670269	−0.335135	+	0.942170i	$0.608782\pi$
$642$	0	0
$643$	−0.346553	−0.0136667	−0.00683336	−	0.999977i	$-0.502175\pi$
$643$	−0.346553	−0.0136667	−0.00683336	+	0.999977i	$0.502175\pi$
$644$	0	0
$645$	−9.86896	−0.388590
$646$	0	0
$647$	−8.23214	−0.323639	−0.161819	−	0.986820i	$-0.551736\pi$
$647$	−8.23214	−0.323639	−0.161819	+	0.986820i	$0.551736\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−14.7944	−0.579837
$652$	0	0
$653$	32.9455	1.28926	0.644629	−	0.764496i	$-0.277012\pi$
$653$	32.9455	1.28926	0.644629	+	0.764496i	$0.277012\pi$
$654$	0	0
$655$	−12.9021	−0.504127
$656$	0	0
$657$	−16.7255	−0.652522
$658$	0	0
$659$	18.7687	0.731127	0.365563	−	0.930786i	$-0.380876\pi$
$659$	18.7687	0.731127	0.365563	+	0.930786i	$0.380876\pi$
$660$	0	0
$661$	8.66066	0.336860	0.168430	−	0.985714i	$-0.446130\pi$
$661$	8.66066	0.336860	0.168430	+	0.985714i	$0.446130\pi$
$662$	0	0
$663$	−15.8909	−0.617153
$664$	0	0
$665$	−14.8471	−0.575746
$666$	0	0
$667$	−24.4791	−0.947834
$668$	0	0
$669$	−12.8246	−0.495826
$670$	0	0
$671$	0	0
$672$	0	0
$673$	11.7088	0.451341	0.225670	−	0.974204i	$-0.427543\pi$
$673$	11.7088	0.451341	0.225670	+	0.974204i	$0.427543\pi$
$674$	0	0
$675$	4.45945	0.171644
$676$	0	0
$677$	31.0721	1.19420	0.597099	−	0.802167i	$-0.296320\pi$
$677$	31.0721	1.19420	0.597099	+	0.802167i	$0.296320\pi$
$678$	0	0
$679$	65.1542	2.50039
$680$	0	0
$681$	−3.61198	−0.138411
$682$	0	0
$683$	31.0210	1.18698	0.593492	−	0.804840i	$-0.297749\pi$
$683$	31.0210	1.18698	0.593492	+	0.804840i	$0.297749\pi$
$684$	0	0
$685$	−18.8824	−0.721461
$686$	0	0
$687$	3.45395	0.131776
$688$	0	0
$689$	7.33896	0.279592
$690$	0	0
$691$	−41.9523	−1.59594	−0.797969	−	0.602698i	$-0.794092\pi$
$691$	−41.9523	−1.59594	−0.797969	+	0.602698i	$0.794092\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−10.8892	−0.413051
$696$	0	0
$697$	46.4742	1.76034
$698$	0	0
$699$	−14.5467	−0.550206
$700$	0	0
$701$	9.77487	0.369192	0.184596	−	0.982815i	$-0.440902\pi$
$701$	9.77487	0.369192	0.184596	+	0.982815i	$0.440902\pi$
$702$	0	0
$703$	−10.7604	−0.405835
$704$	0	0
$705$	1.40092	0.0527618
$706$	0	0
$707$	−23.0372	−0.866404
$708$	0	0
$709$	17.6328	0.662214	0.331107	−	0.943593i	$-0.392578\pi$
$709$	17.6328	0.662214	0.331107	+	0.943593i	$0.392578\pi$
$710$	0	0
$711$	−24.1377	−0.905233
$712$	0	0
$713$	−34.6368	−1.29716
$714$	0	0
$715$	0	0
$716$	0	0
$717$	2.87194	0.107255
$718$	0	0
$719$	−51.4088	−1.91723	−0.958613	−	0.284713i	$-0.908102\pi$
$719$	−51.4088	−1.91723	−0.958613	+	0.284713i	$0.908102\pi$
$720$	0	0
$721$	−4.30300	−0.160252
$722$	0	0
$723$	−17.0178	−0.632898
$724$	0	0
$725$	2.60693	0.0968189
$726$	0	0
$727$	17.3527	0.643577	0.321788	−	0.946812i	$-0.395716\pi$
$727$	17.3527	0.643577	0.321788	+	0.946812i	$0.395716\pi$
$728$	0	0
$729$	4.16647	0.154314
$730$	0	0
$731$	58.3150	2.15686
$732$	0	0
$733$	−42.8536	−1.58284	−0.791418	−	0.611276i	$-0.790657\pi$
$733$	−42.8536	−1.58284	−0.791418	+	0.611276i	$0.790657\pi$
$734$	0	0
$735$	13.1768	0.486035
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−24.5873	−0.904458	−0.452229	−	0.891902i	$-0.649371\pi$
$739$	−24.5873	−0.904458	−0.452229	+	0.891902i	$0.649371\pi$
$740$	0	0
$741$	9.95539	0.365720
$742$	0	0
$743$	−4.69027	−0.172069	−0.0860347	−	0.996292i	$-0.527420\pi$
$743$	−4.69027	−0.172069	−0.0860347	+	0.996292i	$0.527420\pi$
$744$	0	0
$745$	1.69412	0.0620679
$746$	0	0
$747$	−13.0253	−0.476571
$748$	0	0
$749$	−70.3621	−2.57097
$750$	0	0
$751$	48.2922	1.76221	0.881103	−	0.472924i	$-0.156801\pi$
$751$	48.2922	1.76221	0.881103	+	0.472924i	$0.156801\pi$
$752$	0	0
$753$	16.6073	0.605205
$754$	0	0
$755$	3.48984	0.127008
$756$	0	0
$757$	38.8242	1.41109	0.705544	−	0.708666i	$-0.250703\pi$
$757$	38.8242	1.41109	0.705544	+	0.708666i	$0.250703\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	27.8876	1.01093	0.505463	−	0.862848i	$-0.331322\pi$
$761$	27.8876	1.01093	0.505463	+	0.862848i	$0.331322\pi$
$762$	0	0
$763$	−16.1464	−0.584538
$764$	0	0
$765$	−11.4045	−0.412330
$766$	0	0
$767$	0.393118	0.0141947
$768$	0	0
$769$	38.2240	1.37839	0.689197	−	0.724574i	$-0.257964\pi$
$769$	38.2240	1.37839	0.689197	+	0.724574i	$0.257964\pi$
$770$	0	0
$771$	9.03288	0.325311
$772$	0	0
$773$	8.39582	0.301977	0.150988	−	0.988536i	$-0.451754\pi$
$773$	8.39582	0.301977	0.150988	+	0.988536i	$0.451754\pi$
$774$	0	0
$775$	3.68869	0.132502
$776$	0	0
$777$	13.8273	0.496051
$778$	0	0
$779$	−29.1153	−1.04316
$780$	0	0
$781$	0	0
$782$	0	0
$783$	11.6255	0.415460
$784$	0	0
$785$	−20.7003	−0.738826
$786$	0	0
$787$	35.4248	1.26276	0.631379	−	0.775474i	$-0.282489\pi$
$787$	35.4248	1.26276	0.631379	+	0.775474i	$0.282489\pi$
$788$	0	0
$789$	−7.47286	−0.266041
$790$	0	0
$791$	−48.4042	−1.72105
$792$	0	0
$793$	−38.4032	−1.36374
$794$	0	0
$795$	−1.63564	−0.0580101
$796$	0	0
$797$	53.3896	1.89116	0.945579	−	0.325392i	$-0.105496\pi$
$797$	53.3896	1.89116	0.945579	+	0.325392i	$0.105496\pi$
$798$	0	0
$799$	−8.27796	−0.292853
$800$	0	0
$801$	−31.2171	−1.10300
$802$	0	0
$803$	0	0
$804$	0	0
$805$	44.6676	1.57433
$806$	0	0
$807$	1.81680	0.0639543
$808$	0	0
$809$	3.84086	0.135038	0.0675188	−	0.997718i	$-0.478492\pi$
$809$	3.84086	0.135038	0.0675188	+	0.997718i	$0.478492\pi$
$810$	0	0
$811$	23.7250	0.833097	0.416548	−	0.909114i	$-0.363240\pi$
$811$	23.7250	0.833097	0.416548	+	0.909114i	$0.363240\pi$
$812$	0	0
$813$	−8.64657	−0.303248
$814$	0	0
$815$	16.3831	0.573877
$816$	0	0
$817$	−36.5333	−1.27814
$818$	0	0
$819$	41.1947	1.43946
$820$	0	0
$821$	−0.152300	−0.00531531	−0.00265765	−	0.999996i	$-0.500846\pi$
$821$	−0.152300	−0.00531531	−0.00265765	+	0.999996i	$0.500846\pi$
$822$	0	0
$823$	50.1736	1.74894	0.874472	−	0.485077i	$-0.161208\pi$
$823$	50.1736	1.74894	0.874472	+	0.485077i	$0.161208\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	22.2010	0.772006	0.386003	−	0.922498i	$-0.373856\pi$
$827$	22.2010	0.772006	0.386003	+	0.922498i	$0.373856\pi$
$828$	0	0
$829$	8.18588	0.284307	0.142154	−	0.989845i	$-0.454597\pi$
$829$	8.18588	0.284307	0.142154	+	0.989845i	$0.454597\pi$
$830$	0	0
$831$	12.4998	0.433612
$832$	0	0
$833$	−77.8610	−2.69773
$834$	0	0
$835$	−14.4012	−0.498374
$836$	0	0
$837$	16.4495	0.568579
$838$	0	0
$839$	−7.95392	−0.274600	−0.137300	−	0.990530i	$-0.543842\pi$
$839$	−7.95392	−0.274600	−0.137300	+	0.990530i	$0.543842\pi$
$840$	0	0
$841$	−22.2039	−0.765653
$842$	0	0
$843$	−14.2043	−0.489223
$844$	0	0
$845$	−1.31169	−0.0451237
$846$	0	0
$847$	0	0
$848$	0	0
$849$	4.12650	0.141621
$850$	0	0
$851$	32.3727	1.10972
$852$	0	0
$853$	5.87193	0.201051	0.100525	−	0.994934i	$-0.467948\pi$
$853$	5.87193	0.201051	0.100525	+	0.994934i	$0.467948\pi$
$854$	0	0
$855$	7.14470	0.244343
$856$	0	0
$857$	−30.9089	−1.05583	−0.527914	−	0.849298i	$-0.677026\pi$
$857$	−30.9089	−1.05583	−0.527914	+	0.849298i	$0.677026\pi$
$858$	0	0
$859$	10.9133	0.372358	0.186179	−	0.982516i	$-0.440390\pi$
$859$	10.9133	0.372358	0.186179	+	0.982516i	$0.440390\pi$
$860$	0	0
$861$	37.4137	1.27505
$862$	0	0
$863$	12.3339	0.419850	0.209925	−	0.977718i	$-0.432678\pi$
$863$	12.3339	0.419850	0.209925	+	0.977718i	$0.432678\pi$
$864$	0	0
$865$	−8.12354	−0.276209
$866$	0	0
$867$	−6.59386	−0.223939
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−18.8881	−0.639997
$872$	0	0
$873$	−31.3534	−1.06115
$874$	0	0
$875$	−4.75693	−0.160814
$876$	0	0
$877$	45.3557	1.53155	0.765777	−	0.643106i	$-0.222355\pi$
$877$	45.3557	1.53155	0.765777	+	0.643106i	$0.222355\pi$
$878$	0	0
$879$	10.0818	0.340051
$880$	0	0
$881$	−24.4098	−0.822388	−0.411194	−	0.911548i	$-0.634888\pi$
$881$	−24.4098	−0.822388	−0.411194	+	0.911548i	$0.634888\pi$
$882$	0	0
$883$	−34.9932	−1.17761	−0.588807	−	0.808274i	$-0.700402\pi$
$883$	−34.9932	−1.17761	−0.588807	+	0.808274i	$0.700402\pi$
$884$	0	0
$885$	−0.0876143	−0.00294512
$886$	0	0
$887$	−44.8014	−1.50428	−0.752142	−	0.659001i	$-0.770979\pi$
$887$	−44.8014	−1.50428	−0.752142	+	0.659001i	$0.770979\pi$
$888$	0	0
$889$	−52.1299	−1.74838
$890$	0	0
$891$	0	0
$892$	0	0
$893$	5.18599	0.173542
$894$	0	0
$895$	2.99340	0.100058
$896$	0	0
$897$	−29.9509	−1.00003
$898$	0	0
$899$	9.61615	0.320717
$900$	0	0
$901$	9.66486	0.321983
$902$	0	0
$903$	46.9459	1.56226
$904$	0	0
$905$	14.5264	0.482874
$906$	0	0
$907$	−6.93726	−0.230348	−0.115174	−	0.993345i	$-0.536743\pi$
$907$	−6.93726	−0.230348	−0.115174	+	0.993345i	$0.536743\pi$
$908$	0	0
$909$	11.0859	0.367697
$910$	0	0
$911$	26.7451	0.886106	0.443053	−	0.896496i	$-0.353895\pi$
$911$	26.7451	0.886106	0.443053	+	0.896496i	$0.353895\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	8.55893	0.282950
$916$	0	0
$917$	61.3744	2.02676
$918$	0	0
$919$	55.1263	1.81845	0.909225	−	0.416305i	$-0.136675\pi$
$919$	55.1263	1.81845	0.909225	+	0.416305i	$0.136675\pi$
$920$	0	0
$921$	−14.1921	−0.467645
$922$	0	0
$923$	−23.5659	−0.775681
$924$	0	0
$925$	−3.44757	−0.113355
$926$	0	0
$927$	2.07068	0.0680102
$928$	0	0
$929$	13.6870	0.449054	0.224527	−	0.974468i	$-0.427916\pi$
$929$	13.6870	0.449054	0.224527	+	0.974468i	$0.427916\pi$
$930$	0	0
$931$	48.7785	1.59865
$932$	0	0
$933$	20.7368	0.678894
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−18.9104	−0.617774	−0.308887	−	0.951099i	$-0.599957\pi$
$937$	−18.9104	−0.617774	−0.308887	+	0.951099i	$0.599957\pi$
$938$	0	0
$939$	−6.07897	−0.198380
$940$	0	0
$941$	10.9993	0.358569	0.179284	−	0.983797i	$-0.442622\pi$
$941$	10.9993	0.358569	0.179284	+	0.983797i	$0.442622\pi$
$942$	0	0
$943$	87.5935	2.85244
$944$	0	0
$945$	−21.2133	−0.690068
$946$	0	0
$947$	11.9776	0.389219	0.194610	−	0.980881i	$-0.437656\pi$
$947$	11.9776	0.389219	0.194610	+	0.980881i	$0.437656\pi$
$948$	0	0
$949$	−27.6411	−0.897267
$950$	0	0
$951$	7.21642	0.234008
$952$	0	0
$953$	26.2762	0.851170	0.425585	−	0.904918i	$-0.360068\pi$
$953$	26.2762	0.851170	0.425585	+	0.904918i	$0.360068\pi$
$954$	0	0
$955$	−7.96741	−0.257819
$956$	0	0
$957$	0	0
$958$	0	0
$959$	89.8224	2.90052
$960$	0	0
$961$	−17.3936	−0.561083
$962$	0	0
$963$	33.8595	1.09111
$964$	0	0
$965$	−19.4962	−0.627605
$966$	0	0
$967$	−11.2656	−0.362277	−0.181138	−	0.983458i	$-0.557978\pi$
$967$	−11.2656	−0.362277	−0.181138	+	0.983458i	$0.557978\pi$
$968$	0	0
$969$	13.1105	0.421170
$970$	0	0
$971$	57.0037	1.82934	0.914669	−	0.404204i	$-0.132452\pi$
$971$	57.0037	1.82934	0.914669	+	0.404204i	$0.132452\pi$
$972$	0	0
$973$	51.7992	1.66061
$974$	0	0
$975$	3.18965	0.102151
$976$	0	0
$977$	−34.9823	−1.11918	−0.559591	−	0.828769i	$-0.689042\pi$
$977$	−34.9823	−1.11918	−0.559591	+	0.828769i	$0.689042\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	7.76993	0.248075
$982$	0	0
$983$	−17.9629	−0.572927	−0.286464	−	0.958091i	$-0.592480\pi$
$983$	−17.9629	−0.572927	−0.286464	+	0.958091i	$0.592480\pi$
$984$	0	0
$985$	17.9751	0.572734
$986$	0	0
$987$	−6.66409	−0.212120
$988$	0	0
$989$	109.911	3.49495
$990$	0	0
$991$	−47.2159	−1.49986	−0.749931	−	0.661516i	$-0.769913\pi$
$991$	−47.2159	−1.49986	−0.749931	+	0.661516i	$0.769913\pi$
$992$	0	0
$993$	−21.4235	−0.679856
$994$	0	0
$995$	−7.17717	−0.227532
$996$	0	0
$997$	20.1669	0.638693	0.319347	−	0.947638i	$-0.396537\pi$
$997$	20.1669	0.638693	0.319347	+	0.947638i	$0.396537\pi$
$998$	0	0
$999$	−15.3742	−0.486420

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9680.2.a.cy.1.3		6
4.3	odd	2		4840.2.a.be.1.4		6
11.2	odd	10		880.2.bo.j.81.2		12
11.6	odd	10		880.2.bo.j.641.2		12
11.10	odd	2		9680.2.a.cx.1.3		6
44.35	even	10		440.2.y.b.81.2	✓	12
44.39	even	10		440.2.y.b.201.2	yes	12
44.43	even	2		4840.2.a.bf.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
440.2.y.b.81.2	✓	12	44.35	even	10
440.2.y.b.201.2	yes	12	44.39	even	10
880.2.bo.j.81.2		12	11.2	odd	10
880.2.bo.j.641.2		12	11.6	odd	10
4840.2.a.be.1.4		6	4.3	odd	2
4840.2.a.bf.1.4		6	44.43	even	2
9680.2.a.cx.1.3		6	11.10	odd	2
9680.2.a.cy.1.3		6	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 9680 = 2^{4} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9680.a (trivial)

\(f(q)\)	\(=\)	\(q-0.843136 q^{3} -1.00000 q^{5} +4.75693 q^{7} -2.28912 q^{9} -3.78308 q^{13} +0.843136 q^{15} -4.98203 q^{17} +3.12115 q^{19} -4.01074 q^{21} -9.39001 q^{23} +1.00000 q^{25} +4.45945 q^{27} +2.60693 q^{29} +3.68869 q^{31} -4.75693 q^{35} -3.44757 q^{37} +3.18965 q^{39} -9.32837 q^{41} -11.7051 q^{43} +2.28912 q^{45} +1.66156 q^{47} +15.6284 q^{49} +4.20053 q^{51} -1.93994 q^{53} -2.63156 q^{57} -0.103915 q^{59} +10.1513 q^{61} -10.8892 q^{63} +3.78308 q^{65} +4.99277 q^{67} +7.91706 q^{69} +6.22929 q^{71} +7.30650 q^{73} -0.843136 q^{75} +10.5445 q^{79} +3.10744 q^{81} +5.69010 q^{83} +4.98203 q^{85} -2.19800 q^{87} +13.6371 q^{89} -17.9958 q^{91} -3.11007 q^{93} -3.12115 q^{95} +13.6967 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{3} - 6 q^{5} + 6 q^{7} + 10 q^{9} + 6 q^{13} + 2 q^{15} - 11 q^{17} - 11 q^{19} - 2 q^{21} - 18 q^{23} + 6 q^{25} + q^{27} + 6 q^{29} - q^{31} - 6 q^{35} + 4 q^{37} + 27 q^{39} + 4 q^{41} + 3 q^{43}+ \cdots + 20 q^{97}+O(q^{100}) \) 6 * q - 2 * q^3 - 6 * q^5 + 6 * q^7 + 10 * q^9 + 6 * q^13 + 2 * q^15 - 11 * q^17 - 11 * q^19 - 2 * q^21 - 18 * q^23 + 6 * q^25 + q^27 + 6 * q^29 - q^31 - 6 * q^35 + 4 * q^37 + 27 * q^39 + 4 * q^41 + 3 * q^43 - 10 * q^45 - 14 * q^47 + 8 * q^49 + 31 * q^51 + 14 * q^53 + 5 * q^57 - 2 * q^59 + 4 * q^61 - 16 * q^63 - 6 * q^65 - 11 * q^67 + 8 * q^69 - 7 * q^71 + 9 * q^73 - 2 * q^75 + 36 * q^79 + 30 * q^81 - 45 * q^83 + 11 * q^85 + 25 * q^87 + q^89 + 8 * q^91 + 55 * q^93 + 11 * q^95 + 20 * q^97

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