LMFDB - Embedded newform 9680.2.a.df.1.5

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)

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);

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")

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 = [8,0,-1,0,8,0,6,0,19,0,0,0,12,0,-1,0,2] 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:	$8$
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} - 21x^{6} + 15x^{5} + 140x^{4} - 60x^{3} - 295x^{2} + 50x + 100 $ x^8 - x^7 - 21x^6 + 15x^5 + 140x^4 - 60x^3 - 295x^2 + 50x + 100
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
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			1.5
Root			$-0.569801$ 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)

$f(q)$	$=$	$q+0.569801 q^{3} +1.00000 q^{5} +1.82111 q^{7} -2.67533 q^{9} -3.92663 q^{13} +0.569801 q^{15} -3.18309 q^{17} +8.21268 q^{19} +1.03767 q^{21} -0.734122 q^{23} +1.00000 q^{25} -3.23381 q^{27} -4.42950 q^{29} +0.396984 q^{31} +1.82111 q^{35} +2.99377 q^{37} -2.23740 q^{39} +12.0514 q^{41} -8.72957 q^{43} -2.67533 q^{45} +10.5107 q^{47} -3.68356 q^{49} -1.81373 q^{51} +9.84643 q^{53} +4.67959 q^{57} -10.0121 q^{59} +8.50878 q^{61} -4.87206 q^{63} -3.92663 q^{65} -1.11105 q^{67} -0.418304 q^{69} -5.87172 q^{71} +4.03588 q^{73} +0.569801 q^{75} -3.26768 q^{79} +6.18335 q^{81} +11.8570 q^{83} -3.18309 q^{85} -2.52394 q^{87} -6.56220 q^{89} -7.15083 q^{91} +0.226202 q^{93} +8.21268 q^{95} +14.5430 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - q^{3} + 8 q^{5} + 6 q^{7} + 19 q^{9} + 12 q^{13} - q^{15} + 2 q^{17} + 6 q^{19} + 6 q^{21} - 10 q^{23} + 8 q^{25} - 13 q^{27} + 8 q^{29} - 19 q^{31} + 6 q^{35} + 12 q^{37} + 21 q^{39} - 3 q^{41}+ \cdots + 21 q^{97}+O(q^{100}) $ 8 * q - q^3 + 8 * q^5 + 6 * q^7 + 19 * q^9 + 12 * q^13 - q^15 + 2 * q^17 + 6 * q^19 + 6 * q^21 - 10 * q^23 + 8 * q^25 - 13 * q^27 + 8 * q^29 - 19 * q^31 + 6 * q^35 + 12 * q^37 + 21 * q^39 - 3 * q^41 + 8 * q^43 + 19 * q^45 - 10 * q^47 + 22 * q^49 + 7 * q^51 + 28 * q^53 + 25 * q^57 - 25 * q^59 + 10 * q^61 + 64 * q^63 + 12 * q^65 + 2 * q^67 + 18 * q^69 - 25 * q^71 + 38 * q^73 - q^75 + 38 * q^79 + 32 * q^81 + 28 * q^83 + 2 * q^85 - 15 * q^87 + 12 * q^89 - 8 * q^91 - 15 * q^93 + 6 * q^95 + 21 * 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
$3$	0.569801	0.328975	0.164487	−	0.986379i	$-0.447403\pi$
$3$	0.569801	0.328975	0.164487	+	0.986379i	$0.447403\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.82111	0.688314	0.344157	−	0.938912i	$-0.388165\pi$
$7$	1.82111	0.688314	0.344157	+	0.938912i	$0.388165\pi$
$8$	0	0
$9$	−2.67533	−0.891776
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−3.92663	−1.08905	−0.544526	−	0.838744i	$-0.683291\pi$
$13$	−3.92663	−1.08905	−0.544526	+	0.838744i	$0.683291\pi$
$14$	0	0
$15$	0.569801	0.147122
$16$	0	0
$17$	−3.18309	−0.772014	−0.386007	−	0.922496i	$-0.626146\pi$
$17$	−3.18309	−0.772014	−0.386007	+	0.922496i	$0.626146\pi$
$18$	0	0
$19$	8.21268	1.88412	0.942059	−	0.335447i	$-0.108887\pi$
$19$	8.21268	1.88412	0.942059	+	0.335447i	$0.108887\pi$
$20$	0	0
$21$	1.03767	0.226438
$22$	0	0
$23$	−0.734122	−0.153075	−0.0765376	−	0.997067i	$-0.524387\pi$
$23$	−0.734122	−0.153075	−0.0765376	+	0.997067i	$0.524387\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−3.23381	−0.622346
$28$	0	0
$29$	−4.42950	−0.822538	−0.411269	−	0.911514i	$-0.634914\pi$
$29$	−4.42950	−0.822538	−0.411269	+	0.911514i	$0.634914\pi$
$30$	0	0
$31$	0.396984	0.0713004	0.0356502	−	0.999364i	$-0.488650\pi$
$31$	0.396984	0.0713004	0.0356502	+	0.999364i	$0.488650\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.82111	0.307824
$36$	0	0
$37$	2.99377	0.492172	0.246086	−	0.969248i	$-0.420855\pi$
$37$	2.99377	0.492172	0.246086	+	0.969248i	$0.420855\pi$
$38$	0	0
$39$	−2.23740	−0.358271
$40$	0	0
$41$	12.0514	1.88212	0.941060	−	0.338240i	$-0.109832\pi$
$41$	12.0514	1.88212	0.941060	+	0.338240i	$0.109832\pi$
$42$	0	0
$43$	−8.72957	−1.33125	−0.665624	−	0.746288i	$-0.731834\pi$
$43$	−8.72957	−1.33125	−0.665624	+	0.746288i	$0.731834\pi$
$44$	0	0
$45$	−2.67533	−0.398814
$46$	0	0
$47$	10.5107	1.53314	0.766570	−	0.642160i	$-0.221962\pi$
$47$	10.5107	1.53314	0.766570	+	0.642160i	$0.221962\pi$
$48$	0	0
$49$	−3.68356	−0.526223
$50$	0	0
$51$	−1.81373	−0.253973
$52$	0	0
$53$	9.84643	1.35251	0.676256	−	0.736667i	$-0.263602\pi$
$53$	9.84643	1.35251	0.676256	+	0.736667i	$0.263602\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	4.67959	0.619827
$58$	0	0
$59$	−10.0121	−1.30347	−0.651734	−	0.758448i	$-0.725958\pi$
$59$	−10.0121	−1.30347	−0.651734	+	0.758448i	$0.725958\pi$
$60$	0	0
$61$	8.50878	1.08944	0.544719	−	0.838619i	$-0.316636\pi$
$61$	8.50878	1.08944	0.544719	+	0.838619i	$0.316636\pi$
$62$	0	0
$63$	−4.87206	−0.613822
$64$	0	0
$65$	−3.92663	−0.487039
$66$	0	0
$67$	−1.11105	−0.135737	−0.0678684	−	0.997694i	$-0.521620\pi$
$67$	−1.11105	−0.135737	−0.0678684	+	0.997694i	$0.521620\pi$
$68$	0	0
$69$	−0.418304	−0.0503579
$70$	0	0
$71$	−5.87172	−0.696845	−0.348422	−	0.937338i	$-0.613283\pi$
$71$	−5.87172	−0.696845	−0.348422	+	0.937338i	$0.613283\pi$
$72$	0	0
$73$	4.03588	0.472364	0.236182	−	0.971709i	$-0.424104\pi$
$73$	4.03588	0.472364	0.236182	+	0.971709i	$0.424104\pi$
$74$	0	0
$75$	0.569801	0.0657950
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−3.26768	−0.367643	−0.183821	−	0.982960i	$-0.558847\pi$
$79$	−3.26768	−0.367643	−0.183821	+	0.982960i	$0.558847\pi$
$80$	0	0
$81$	6.18335	0.687039
$82$	0	0
$83$	11.8570	1.30148	0.650739	−	0.759302i	$-0.274459\pi$
$83$	11.8570	1.30148	0.650739	+	0.759302i	$0.274459\pi$
$84$	0	0
$85$	−3.18309	−0.345255
$86$	0	0
$87$	−2.52394	−0.270594
$88$	0	0
$89$	−6.56220	−0.695592	−0.347796	−	0.937570i	$-0.613070\pi$
$89$	−6.56220	−0.695592	−0.347796	+	0.937570i	$0.613070\pi$
$90$	0	0
$91$	−7.15083	−0.749610
$92$	0	0
$93$	0.226202	0.0234560
$94$	0	0
$95$	8.21268	0.842603
$96$	0	0
$97$	14.5430	1.47662	0.738310	−	0.674462i	$-0.235624\pi$
$97$	14.5430	1.47662	0.738310	+	0.674462i	$0.235624\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−9.89649	−0.984737	−0.492369	−	0.870387i	$-0.663869\pi$
$101$	−9.89649	−0.984737	−0.492369	+	0.870387i	$0.663869\pi$
$102$	0	0
$103$	−7.21119	−0.710539	−0.355270	−	0.934764i	$-0.615611\pi$
$103$	−7.21119	−0.710539	−0.355270	+	0.934764i	$0.615611\pi$
$104$	0	0
$105$	1.03767	0.101266
$106$	0	0
$107$	9.77736	0.945213	0.472606	−	0.881274i	$-0.343313\pi$
$107$	9.77736	0.945213	0.472606	+	0.881274i	$0.343313\pi$
$108$	0	0
$109$	18.7329	1.79429	0.897143	−	0.441741i	$-0.145639\pi$
$109$	18.7329	1.79429	0.897143	+	0.441741i	$0.145639\pi$
$110$	0	0
$111$	1.70585	0.161912
$112$	0	0
$113$	12.8614	1.20990	0.604948	−	0.796265i	$-0.293194\pi$
$113$	12.8614	1.20990	0.604948	+	0.796265i	$0.293194\pi$
$114$	0	0
$115$	−0.734122	−0.0684573
$116$	0	0
$117$	10.5050	0.971190
$118$	0	0
$119$	−5.79676	−0.531388
$120$	0	0
$121$	0	0
$122$	0	0
$123$	6.86693	0.619170
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	2.84780	0.252701	0.126350	−	0.991986i	$-0.459674\pi$
$127$	2.84780	0.252701	0.126350	+	0.991986i	$0.459674\pi$
$128$	0	0
$129$	−4.97412	−0.437947
$130$	0	0
$131$	9.49411	0.829504	0.414752	−	0.909934i	$-0.363868\pi$
$131$	9.49411	0.829504	0.414752	+	0.909934i	$0.363868\pi$
$132$	0	0
$133$	14.9562	1.29687
$134$	0	0
$135$	−3.23381	−0.278322
$136$	0	0
$137$	−7.03836	−0.601328	−0.300664	−	0.953730i	$-0.597208\pi$
$137$	−7.03836	−0.601328	−0.300664	+	0.953730i	$0.597208\pi$
$138$	0	0
$139$	−3.73184	−0.316531	−0.158265	−	0.987397i	$-0.550590\pi$
$139$	−3.73184	−0.316531	−0.158265	+	0.987397i	$0.550590\pi$
$140$	0	0
$141$	5.98900	0.504365
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−4.42950	−0.367850
$146$	0	0
$147$	−2.09890	−0.173114
$148$	0	0
$149$	12.2751	1.00561	0.502806	−	0.864399i	$-0.332301\pi$
$149$	12.2751	1.00561	0.502806	+	0.864399i	$0.332301\pi$
$150$	0	0
$151$	5.80255	0.472205	0.236102	−	0.971728i	$-0.424130\pi$
$151$	5.80255	0.472205	0.236102	+	0.971728i	$0.424130\pi$
$152$	0	0
$153$	8.51581	0.688463
$154$	0	0
$155$	0.396984	0.0318865
$156$	0	0
$157$	−20.7356	−1.65488	−0.827440	−	0.561553i	$-0.810204\pi$
$157$	−20.7356	−1.65488	−0.827440	+	0.561553i	$0.810204\pi$
$158$	0	0
$159$	5.61051	0.444942
$160$	0	0
$161$	−1.33692	−0.105364
$162$	0	0
$163$	11.8323	0.926778	0.463389	−	0.886155i	$-0.346633\pi$
$163$	11.8323	0.926778	0.463389	+	0.886155i	$0.346633\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	18.1848	1.40718	0.703592	−	0.710604i	$-0.251578\pi$
$167$	18.1848	1.40718	0.703592	+	0.710604i	$0.251578\pi$
$168$	0	0
$169$	2.41846	0.186035
$170$	0	0
$171$	−21.9716	−1.68021
$172$	0	0
$173$	−10.2152	−0.776649	−0.388324	−	0.921523i	$-0.626946\pi$
$173$	−10.2152	−0.776649	−0.388324	+	0.921523i	$0.626946\pi$
$174$	0	0
$175$	1.82111	0.137663
$176$	0	0
$177$	−5.70492	−0.428808
$178$	0	0
$179$	−1.72353	−0.128823	−0.0644114	−	0.997923i	$-0.520517\pi$
$179$	−1.72353	−0.128823	−0.0644114	+	0.997923i	$0.520517\pi$
$180$	0	0
$181$	13.9521	1.03705	0.518525	−	0.855062i	$-0.326481\pi$
$181$	13.9521	1.03705	0.518525	+	0.855062i	$0.326481\pi$
$182$	0	0
$183$	4.84831	0.358398
$184$	0	0
$185$	2.99377	0.220106
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−5.88911	−0.428370
$190$	0	0
$191$	−24.6929	−1.78672	−0.893359	−	0.449344i	$-0.851658\pi$
$191$	−24.6929	−1.78672	−0.893359	+	0.449344i	$0.851658\pi$
$192$	0	0
$193$	2.66652	0.191940	0.0959702	−	0.995384i	$-0.469405\pi$
$193$	2.66652	0.191940	0.0959702	+	0.995384i	$0.469405\pi$
$194$	0	0
$195$	−2.23740	−0.160224
$196$	0	0
$197$	21.4277	1.52666	0.763330	−	0.646009i	$-0.223563\pi$
$197$	21.4277	1.52666	0.763330	+	0.646009i	$0.223563\pi$
$198$	0	0
$199$	5.78067	0.409781	0.204891	−	0.978785i	$-0.434316\pi$
$199$	5.78067	0.409781	0.204891	+	0.978785i	$0.434316\pi$
$200$	0	0
$201$	−0.633079	−0.0446540
$202$	0	0
$203$	−8.06661	−0.566165
$204$	0	0
$205$	12.0514	0.841709
$206$	0	0
$207$	1.96402	0.136509
$208$	0	0
$209$	0	0
$210$	0	0
$211$	11.5772	0.797006	0.398503	−	0.917167i	$-0.369530\pi$
$211$	11.5772	0.797006	0.398503	+	0.917167i	$0.369530\pi$
$212$	0	0
$213$	−3.34571	−0.229244
$214$	0	0
$215$	−8.72957	−0.595352
$216$	0	0
$217$	0.722951	0.0490771
$218$	0	0
$219$	2.29965	0.155396
$220$	0	0
$221$	12.4988	0.840763
$222$	0	0
$223$	11.8308	0.792252	0.396126	−	0.918196i	$-0.370354\pi$
$223$	11.8308	0.792252	0.396126	+	0.918196i	$0.370354\pi$
$224$	0	0
$225$	−2.67533	−0.178355
$226$	0	0
$227$	6.18986	0.410835	0.205418	−	0.978674i	$-0.434145\pi$
$227$	6.18986	0.410835	0.205418	+	0.978674i	$0.434145\pi$
$228$	0	0
$229$	17.7699	1.17427	0.587133	−	0.809491i	$-0.300257\pi$
$229$	17.7699	1.17427	0.587133	+	0.809491i	$0.300257\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−17.0269	−1.11547	−0.557733	−	0.830020i	$-0.688329\pi$
$233$	−17.0269	−1.11547	−0.557733	+	0.830020i	$0.688329\pi$
$234$	0	0
$235$	10.5107	0.685641
$236$	0	0
$237$	−1.86193	−0.120945
$238$	0	0
$239$	−1.18624	−0.0767313	−0.0383656	−	0.999264i	$-0.512215\pi$
$239$	−1.18624	−0.0767313	−0.0383656	+	0.999264i	$0.512215\pi$
$240$	0	0
$241$	10.3606	0.667388	0.333694	−	0.942682i	$-0.391705\pi$
$241$	10.3606	0.667388	0.333694	+	0.942682i	$0.391705\pi$
$242$	0	0
$243$	13.2247	0.848365
$244$	0	0
$245$	−3.68356	−0.235334
$246$	0	0
$247$	−32.2482	−2.05190
$248$	0	0
$249$	6.75614	0.428153
$250$	0	0
$251$	−19.6814	−1.24228	−0.621139	−	0.783701i	$-0.713330\pi$
$251$	−19.6814	−1.24228	−0.621139	+	0.783701i	$0.713330\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−1.81373	−0.113580
$256$	0	0
$257$	13.4421	0.838496	0.419248	−	0.907872i	$-0.362294\pi$
$257$	13.4421	0.838496	0.419248	+	0.907872i	$0.362294\pi$
$258$	0	0
$259$	5.45197	0.338769
$260$	0	0
$261$	11.8504	0.733519
$262$	0	0
$263$	1.60780	0.0991410	0.0495705	−	0.998771i	$-0.484215\pi$
$263$	1.60780	0.0991410	0.0495705	+	0.998771i	$0.484215\pi$
$264$	0	0
$265$	9.84643	0.604861
$266$	0	0
$267$	−3.73915	−0.228832
$268$	0	0
$269$	9.01686	0.549767	0.274884	−	0.961477i	$-0.411361\pi$
$269$	9.01686	0.549767	0.274884	+	0.961477i	$0.411361\pi$
$270$	0	0
$271$	19.3857	1.17759	0.588797	−	0.808281i	$-0.299602\pi$
$271$	19.3857	1.17759	0.588797	+	0.808281i	$0.299602\pi$
$272$	0	0
$273$	−4.07455	−0.246603
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1.92697	0.115781	0.0578903	−	0.998323i	$-0.481563\pi$
$277$	1.92697	0.115781	0.0578903	+	0.998323i	$0.481563\pi$
$278$	0	0
$279$	−1.06206	−0.0635840
$280$	0	0
$281$	−2.19286	−0.130815	−0.0654076	−	0.997859i	$-0.520835\pi$
$281$	−2.19286	−0.130815	−0.0654076	+	0.997859i	$0.520835\pi$
$282$	0	0
$283$	28.9913	1.72335	0.861677	−	0.507458i	$-0.169415\pi$
$283$	28.9913	1.72335	0.861677	+	0.507458i	$0.169415\pi$
$284$	0	0
$285$	4.67959	0.277195
$286$	0	0
$287$	21.9470	1.29549
$288$	0	0
$289$	−6.86792	−0.403995
$290$	0	0
$291$	8.28662	0.485770
$292$	0	0
$293$	6.09890	0.356301	0.178151	−	0.984003i	$-0.442989\pi$
$293$	6.09890	0.356301	0.178151	+	0.984003i	$0.442989\pi$
$294$	0	0
$295$	−10.0121	−0.582929
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2.88263	0.166707
$300$	0	0
$301$	−15.8975	−0.916317
$302$	0	0
$303$	−5.63903	−0.323954
$304$	0	0
$305$	8.50878	0.487212
$306$	0	0
$307$	−15.2560	−0.870706	−0.435353	−	0.900260i	$-0.643376\pi$
$307$	−15.2560	−0.870706	−0.435353	+	0.900260i	$0.643376\pi$
$308$	0	0
$309$	−4.10894	−0.233750
$310$	0	0
$311$	3.46576	0.196525	0.0982627	−	0.995161i	$-0.468671\pi$
$311$	3.46576	0.196525	0.0982627	+	0.995161i	$0.468671\pi$
$312$	0	0
$313$	13.0464	0.737428	0.368714	−	0.929543i	$-0.379798\pi$
$313$	13.0464	0.737428	0.368714	+	0.929543i	$0.379798\pi$
$314$	0	0
$315$	−4.87206	−0.274510
$316$	0	0
$317$	−30.2860	−1.70103	−0.850517	−	0.525948i	$-0.823711\pi$
$317$	−30.2860	−1.70103	−0.850517	+	0.525948i	$0.823711\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	5.57115	0.310951
$322$	0	0
$323$	−26.1417	−1.45456
$324$	0	0
$325$	−3.92663	−0.217810
$326$	0	0
$327$	10.6740	0.590275
$328$	0	0
$329$	19.1411	1.05528
$330$	0	0
$331$	−4.19183	−0.230404	−0.115202	−	0.993342i	$-0.536752\pi$
$331$	−4.19183	−0.230404	−0.115202	+	0.993342i	$0.536752\pi$
$332$	0	0
$333$	−8.00930	−0.438907
$334$	0	0
$335$	−1.11105	−0.0607033
$336$	0	0
$337$	26.5352	1.44546	0.722731	−	0.691129i	$-0.242887\pi$
$337$	26.5352	1.44546	0.722731	+	0.691129i	$0.242887\pi$
$338$	0	0
$339$	7.32842	0.398025
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−19.4559	−1.05052
$344$	0	0
$345$	−0.418304	−0.0225207
$346$	0	0
$347$	−1.13932	−0.0611619	−0.0305809	−	0.999532i	$-0.509736\pi$
$347$	−1.13932	−0.0611619	−0.0305809	+	0.999532i	$0.509736\pi$
$348$	0	0
$349$	16.2421	0.869420	0.434710	−	0.900571i	$-0.356851\pi$
$349$	16.2421	0.869420	0.434710	+	0.900571i	$0.356851\pi$
$350$	0	0
$351$	12.6980	0.677768
$352$	0	0
$353$	−0.432655	−0.0230279	−0.0115140	−	0.999934i	$-0.503665\pi$
$353$	−0.432655	−0.0230279	−0.0115140	+	0.999934i	$0.503665\pi$
$354$	0	0
$355$	−5.87172	−0.311638
$356$	0	0
$357$	−3.30300	−0.174813
$358$	0	0
$359$	−0.0977164	−0.00515727	−0.00257864	−	0.999997i	$-0.500821\pi$
$359$	−0.0977164	−0.00515727	−0.00257864	+	0.999997i	$0.500821\pi$
$360$	0	0
$361$	48.4481	2.54990
$362$	0	0
$363$	0	0
$364$	0	0
$365$	4.03588	0.211248
$366$	0	0
$367$	−14.3906	−0.751185	−0.375592	−	0.926785i	$-0.622561\pi$
$367$	−14.3906	−0.751185	−0.375592	+	0.926785i	$0.622561\pi$
$368$	0	0
$369$	−32.2416	−1.67843
$370$	0	0
$371$	17.9314	0.930953
$372$	0	0
$373$	−13.8834	−0.718855	−0.359428	−	0.933173i	$-0.617028\pi$
$373$	−13.8834	−0.718855	−0.359428	+	0.933173i	$0.617028\pi$
$374$	0	0
$375$	0.569801	0.0294244
$376$	0	0
$377$	17.3930	0.895787
$378$	0	0
$379$	−17.8304	−0.915885	−0.457943	−	0.888982i	$-0.651414\pi$
$379$	−17.8304	−0.915885	−0.457943	+	0.888982i	$0.651414\pi$
$380$	0	0
$381$	1.62268	0.0831322
$382$	0	0
$383$	−8.39655	−0.429044	−0.214522	−	0.976719i	$-0.568819\pi$
$383$	−8.39655	−0.429044	−0.214522	+	0.976719i	$0.568819\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	23.3545	1.18717
$388$	0	0
$389$	−7.89641	−0.400364	−0.200182	−	0.979759i	$-0.564153\pi$
$389$	−7.89641	−0.400364	−0.200182	+	0.979759i	$0.564153\pi$
$390$	0	0
$391$	2.33678	0.118176
$392$	0	0
$393$	5.40975	0.272886
$394$	0	0
$395$	−3.26768	−0.164415
$396$	0	0
$397$	−36.2170	−1.81768	−0.908840	−	0.417146i	$-0.863030\pi$
$397$	−36.2170	−1.81768	−0.908840	+	0.417146i	$0.863030\pi$
$398$	0	0
$399$	8.52205	0.426636
$400$	0	0
$401$	−12.4872	−0.623582	−0.311791	−	0.950151i	$-0.600929\pi$
$401$	−12.4872	−0.623582	−0.311791	+	0.950151i	$0.600929\pi$
$402$	0	0
$403$	−1.55881	−0.0776499
$404$	0	0
$405$	6.18335	0.307253
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−9.24803	−0.457286	−0.228643	−	0.973510i	$-0.573429\pi$
$409$	−9.24803	−0.457286	−0.228643	+	0.973510i	$0.573429\pi$
$410$	0	0
$411$	−4.01047	−0.197822
$412$	0	0
$413$	−18.2332	−0.897196
$414$	0	0
$415$	11.8570	0.582038
$416$	0	0
$417$	−2.12641	−0.104131
$418$	0	0
$419$	−28.9794	−1.41574	−0.707869	−	0.706344i	$-0.750343\pi$
$419$	−28.9794	−1.41574	−0.707869	+	0.706344i	$0.750343\pi$
$420$	0	0
$421$	22.8185	1.11210	0.556052	−	0.831148i	$-0.312316\pi$
$421$	22.8185	1.11210	0.556052	+	0.831148i	$0.312316\pi$
$422$	0	0
$423$	−28.1195	−1.36722
$424$	0	0
$425$	−3.18309	−0.154403
$426$	0	0
$427$	15.4954	0.749876
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−25.6107	−1.23362	−0.616811	−	0.787111i	$-0.711576\pi$
$431$	−25.6107	−1.23362	−0.616811	+	0.787111i	$0.711576\pi$
$432$	0	0
$433$	2.63416	0.126590	0.0632948	−	0.997995i	$-0.479839\pi$
$433$	2.63416	0.126590	0.0632948	+	0.997995i	$0.479839\pi$
$434$	0	0
$435$	−2.52394	−0.121013
$436$	0	0
$437$	−6.02911	−0.288412
$438$	0	0
$439$	7.16551	0.341991	0.170996	−	0.985272i	$-0.445302\pi$
$439$	7.16551	0.341991	0.170996	+	0.985272i	$0.445302\pi$
$440$	0	0
$441$	9.85473	0.469273
$442$	0	0
$443$	−15.3877	−0.731092	−0.365546	−	0.930793i	$-0.619118\pi$
$443$	−15.3877	−0.731092	−0.365546	+	0.930793i	$0.619118\pi$
$444$	0	0
$445$	−6.56220	−0.311078
$446$	0	0
$447$	6.99435	0.330821
$448$	0	0
$449$	−4.63933	−0.218943	−0.109472	−	0.993990i	$-0.534916\pi$
$449$	−4.63933	−0.218943	−0.109472	+	0.993990i	$0.534916\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	3.30630	0.155344
$454$	0	0
$455$	−7.15083	−0.335236
$456$	0	0
$457$	12.8616	0.601639	0.300820	−	0.953681i	$-0.402740\pi$
$457$	12.8616	0.601639	0.300820	+	0.953681i	$0.402740\pi$
$458$	0	0
$459$	10.2935	0.480460
$460$	0	0
$461$	−19.6157	−0.913594	−0.456797	−	0.889571i	$-0.651003\pi$
$461$	−19.6157	−0.913594	−0.456797	+	0.889571i	$0.651003\pi$
$462$	0	0
$463$	3.15807	0.146768	0.0733839	−	0.997304i	$-0.476620\pi$
$463$	3.15807	0.146768	0.0733839	+	0.997304i	$0.476620\pi$
$464$	0	0
$465$	0.226202	0.0104899
$466$	0	0
$467$	28.9593	1.34008	0.670038	−	0.742327i	$-0.266278\pi$
$467$	28.9593	1.34008	0.670038	+	0.742327i	$0.266278\pi$
$468$	0	0
$469$	−2.02335	−0.0934296
$470$	0	0
$471$	−11.8152	−0.544414
$472$	0	0
$473$	0	0
$474$	0	0
$475$	8.21268	0.376824
$476$	0	0
$477$	−26.3424	−1.20614
$478$	0	0
$479$	21.2299	0.970021	0.485011	−	0.874508i	$-0.338816\pi$
$479$	21.2299	0.970021	0.485011	+	0.874508i	$0.338816\pi$
$480$	0	0
$481$	−11.7554	−0.536001
$482$	0	0
$483$	−0.761777	−0.0346620
$484$	0	0
$485$	14.5430	0.660364
$486$	0	0
$487$	13.9771	0.633365	0.316682	−	0.948532i	$-0.397431\pi$
$487$	13.9771	0.633365	0.316682	+	0.948532i	$0.397431\pi$
$488$	0	0
$489$	6.74206	0.304887
$490$	0	0
$491$	17.0953	0.771501	0.385751	−	0.922603i	$-0.373943\pi$
$491$	17.0953	0.771501	0.385751	+	0.922603i	$0.373943\pi$
$492$	0	0
$493$	14.0995	0.635010
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−10.6930	−0.479648
$498$	0	0
$499$	−40.9501	−1.83318	−0.916590	−	0.399828i	$-0.869070\pi$
$499$	−40.9501	−1.83318	−0.916590	+	0.399828i	$0.869070\pi$
$500$	0	0
$501$	10.3617	0.462928
$502$	0	0
$503$	−13.4389	−0.599212	−0.299606	−	0.954063i	$-0.596855\pi$
$503$	−13.4389	−0.599212	−0.299606	+	0.954063i	$0.596855\pi$
$504$	0	0
$505$	−9.89649	−0.440388
$506$	0	0
$507$	1.37804	0.0612009
$508$	0	0
$509$	18.3442	0.813094	0.406547	−	0.913630i	$-0.366733\pi$
$509$	18.3442	0.813094	0.406547	+	0.913630i	$0.366733\pi$
$510$	0	0
$511$	7.34978	0.325135
$512$	0	0
$513$	−26.5582	−1.17257
$514$	0	0
$515$	−7.21119	−0.317763
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−5.82064	−0.255498
$520$	0	0
$521$	2.87412	0.125918	0.0629588	−	0.998016i	$-0.479946\pi$
$521$	2.87412	0.125918	0.0629588	+	0.998016i	$0.479946\pi$
$522$	0	0
$523$	−37.6986	−1.64845	−0.824223	−	0.566266i	$-0.808387\pi$
$523$	−37.6986	−1.64845	−0.824223	+	0.566266i	$0.808387\pi$
$524$	0	0
$525$	1.03767	0.0452876
$526$	0	0
$527$	−1.26364	−0.0550449
$528$	0	0
$529$	−22.4611	−0.976568
$530$	0	0
$531$	26.7857	1.16240
$532$	0	0
$533$	−47.3216	−2.04973
$534$	0	0
$535$	9.77736	0.422712
$536$	0	0
$537$	−0.982071	−0.0423795
$538$	0	0
$539$	0	0
$540$	0	0
$541$	12.5509	0.539605	0.269802	−	0.962916i	$-0.413042\pi$
$541$	12.5509	0.539605	0.269802	+	0.962916i	$0.413042\pi$
$542$	0	0
$543$	7.94992	0.341164
$544$	0	0
$545$	18.7329	0.802429
$546$	0	0
$547$	−35.5307	−1.51918	−0.759592	−	0.650400i	$-0.774601\pi$
$547$	−35.5307	−1.51918	−0.759592	+	0.650400i	$0.774601\pi$
$548$	0	0
$549$	−22.7638	−0.971534
$550$	0	0
$551$	−36.3781	−1.54976
$552$	0	0
$553$	−5.95080	−0.253054
$554$	0	0
$555$	1.70585	0.0724093
$556$	0	0
$557$	39.8419	1.68815	0.844077	−	0.536222i	$-0.180149\pi$
$557$	39.8419	1.68815	0.844077	+	0.536222i	$0.180149\pi$
$558$	0	0
$559$	34.2778	1.44980
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−32.2632	−1.35973	−0.679866	−	0.733336i	$-0.737962\pi$
$563$	−32.2632	−1.35973	−0.679866	+	0.733336i	$0.737962\pi$
$564$	0	0
$565$	12.8614	0.541082
$566$	0	0
$567$	11.2606	0.472899
$568$	0	0
$569$	17.8955	0.750216	0.375108	−	0.926981i	$-0.377606\pi$
$569$	17.8955	0.750216	0.375108	+	0.926981i	$0.377606\pi$
$570$	0	0
$571$	11.9435	0.499819	0.249910	−	0.968269i	$-0.419599\pi$
$571$	11.9435	0.499819	0.249910	+	0.968269i	$0.419599\pi$
$572$	0	0
$573$	−14.0701	−0.587785
$574$	0	0
$575$	−0.734122	−0.0306150
$576$	0	0
$577$	11.9451	0.497282	0.248641	−	0.968596i	$-0.420016\pi$
$577$	11.9451	0.497282	0.248641	+	0.968596i	$0.420016\pi$
$578$	0	0
$579$	1.51939	0.0631435
$580$	0	0
$581$	21.5929	0.895825
$582$	0	0
$583$	0	0
$584$	0	0
$585$	10.5050	0.434330
$586$	0	0
$587$	−15.8502	−0.654208	−0.327104	−	0.944988i	$-0.606073\pi$
$587$	−15.8502	−0.654208	−0.327104	+	0.944988i	$0.606073\pi$
$588$	0	0
$589$	3.26030	0.134338
$590$	0	0
$591$	12.2095	0.502233
$592$	0	0
$593$	40.0924	1.64640	0.823198	−	0.567754i	$-0.192188\pi$
$593$	40.0924	1.64640	0.823198	+	0.567754i	$0.192188\pi$
$594$	0	0
$595$	−5.79676	−0.237644
$596$	0	0
$597$	3.29383	0.134808
$598$	0	0
$599$	−32.5610	−1.33041	−0.665204	−	0.746662i	$-0.731655\pi$
$599$	−32.5610	−1.33041	−0.665204	+	0.746662i	$0.731655\pi$
$600$	0	0
$601$	5.80180	0.236660	0.118330	−	0.992974i	$-0.462246\pi$
$601$	5.80180	0.236660	0.118330	+	0.992974i	$0.462246\pi$
$602$	0	0
$603$	2.97243	0.121047
$604$	0	0
$605$	0	0
$606$	0	0
$607$	14.0588	0.570630	0.285315	−	0.958434i	$-0.407902\pi$
$607$	14.0588	0.570630	0.285315	+	0.958434i	$0.407902\pi$
$608$	0	0
$609$	−4.59636	−0.186254
$610$	0	0
$611$	−41.2716	−1.66967
$612$	0	0
$613$	30.8706	1.24685	0.623426	−	0.781882i	$-0.285740\pi$
$613$	30.8706	1.24685	0.623426	+	0.781882i	$0.285740\pi$
$614$	0	0
$615$	6.86693	0.276901
$616$	0	0
$617$	35.9658	1.44793	0.723963	−	0.689838i	$-0.242318\pi$
$617$	35.9658	1.44793	0.723963	+	0.689838i	$0.242318\pi$
$618$	0	0
$619$	−8.85192	−0.355789	−0.177894	−	0.984050i	$-0.556929\pi$
$619$	−8.85192	−0.355789	−0.177894	+	0.984050i	$0.556929\pi$
$620$	0	0
$621$	2.37401	0.0952658
$622$	0	0
$623$	−11.9505	−0.478786
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−9.52943	−0.379963
$630$	0	0
$631$	45.5152	1.81193	0.905967	−	0.423349i	$-0.139146\pi$
$631$	45.5152	1.81193	0.905967	+	0.423349i	$0.139146\pi$
$632$	0	0
$633$	6.59669	0.262195
$634$	0	0
$635$	2.84780	0.113011
$636$	0	0
$637$	14.4640	0.573085
$638$	0	0
$639$	15.7088	0.621429
$640$	0	0
$641$	23.7684	0.938796	0.469398	−	0.882987i	$-0.344471\pi$
$641$	23.7684	0.938796	0.469398	+	0.882987i	$0.344471\pi$
$642$	0	0
$643$	26.6580	1.05129	0.525644	−	0.850705i	$-0.323824\pi$
$643$	26.6580	1.05129	0.525644	+	0.850705i	$0.323824\pi$
$644$	0	0
$645$	−4.97412	−0.195856
$646$	0	0
$647$	38.0457	1.49573	0.747865	−	0.663851i	$-0.231079\pi$
$647$	38.0457	1.49573	0.747865	+	0.663851i	$0.231079\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0.411938	0.0161451
$652$	0	0
$653$	−7.85962	−0.307571	−0.153785	−	0.988104i	$-0.549146\pi$
$653$	−7.85962	−0.307571	−0.153785	+	0.988104i	$0.549146\pi$
$654$	0	0
$655$	9.49411	0.370966
$656$	0	0
$657$	−10.7973	−0.421243
$658$	0	0
$659$	−27.5518	−1.07327	−0.536633	−	0.843816i	$-0.680304\pi$
$659$	−27.5518	−1.07327	−0.536633	+	0.843816i	$0.680304\pi$
$660$	0	0
$661$	39.6062	1.54050	0.770251	−	0.637741i	$-0.220131\pi$
$661$	39.6062	1.54050	0.770251	+	0.637741i	$0.220131\pi$
$662$	0	0
$663$	7.12185	0.276590
$664$	0	0
$665$	14.9562	0.579976
$666$	0	0
$667$	3.25180	0.125910
$668$	0	0
$669$	6.74123	0.260631
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−43.1226	−1.66225	−0.831127	−	0.556082i	$-0.812304\pi$
$673$	−43.1226	−1.66225	−0.831127	+	0.556082i	$0.812304\pi$
$674$	0	0
$675$	−3.23381	−0.124469
$676$	0	0
$677$	−21.6325	−0.831406	−0.415703	−	0.909500i	$-0.636464\pi$
$677$	−21.6325	−0.831406	−0.415703	+	0.909500i	$0.636464\pi$
$678$	0	0
$679$	26.4844	1.01638
$680$	0	0
$681$	3.52699	0.135154
$682$	0	0
$683$	1.01316	0.0387675	0.0193838	−	0.999812i	$-0.493830\pi$
$683$	1.01316	0.0387675	0.0193838	+	0.999812i	$0.493830\pi$
$684$	0	0
$685$	−7.03836	−0.268922
$686$	0	0
$687$	10.1253	0.386304
$688$	0	0
$689$	−38.6633	−1.47296
$690$	0	0
$691$	17.1815	0.653616	0.326808	−	0.945091i	$-0.394027\pi$
$691$	17.1815	0.653616	0.326808	+	0.945091i	$0.394027\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−3.73184	−0.141557
$696$	0	0
$697$	−38.3609	−1.45302
$698$	0	0
$699$	−9.70192	−0.366960
$700$	0	0
$701$	17.4446	0.658875	0.329437	−	0.944177i	$-0.393141\pi$
$701$	17.4446	0.658875	0.329437	+	0.944177i	$0.393141\pi$
$702$	0	0
$703$	24.5868	0.927310
$704$	0	0
$705$	5.98900	0.225559
$706$	0	0
$707$	−18.0226	−0.677809
$708$	0	0
$709$	−17.0730	−0.641191	−0.320595	−	0.947216i	$-0.603883\pi$
$709$	−17.0730	−0.641191	−0.320595	+	0.947216i	$0.603883\pi$
$710$	0	0
$711$	8.74212	0.327855
$712$	0	0
$713$	−0.291435	−0.0109143
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−0.675919	−0.0252427
$718$	0	0
$719$	25.2508	0.941695	0.470848	−	0.882215i	$-0.343948\pi$
$719$	25.2508	0.941695	0.470848	+	0.882215i	$0.343948\pi$
$720$	0	0
$721$	−13.1324	−0.489075
$722$	0	0
$723$	5.90351	0.219554
$724$	0	0
$725$	−4.42950	−0.164508
$726$	0	0
$727$	49.9360	1.85202	0.926012	−	0.377493i	$-0.123214\pi$
$727$	49.9360	1.85202	0.926012	+	0.377493i	$0.123214\pi$
$728$	0	0
$729$	−11.0146	−0.407949
$730$	0	0
$731$	27.7870	1.02774
$732$	0	0
$733$	14.9515	0.552246	0.276123	−	0.961122i	$-0.410950\pi$
$733$	14.9515	0.552246	0.276123	+	0.961122i	$0.410950\pi$
$734$	0	0
$735$	−2.09890	−0.0774190
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−8.68720	−0.319564	−0.159782	−	0.987152i	$-0.551079\pi$
$739$	−8.68720	−0.319564	−0.159782	+	0.987152i	$0.551079\pi$
$740$	0	0
$741$	−18.3751	−0.675025
$742$	0	0
$743$	11.9221	0.437380	0.218690	−	0.975794i	$-0.429822\pi$
$743$	11.9221	0.437380	0.218690	+	0.975794i	$0.429822\pi$
$744$	0	0
$745$	12.2751	0.449724
$746$	0	0
$747$	−31.7214	−1.16063
$748$	0	0
$749$	17.8056	0.650604
$750$	0	0
$751$	−52.3281	−1.90948	−0.954739	−	0.297444i	$-0.903866\pi$
$751$	−52.3281	−1.90948	−0.954739	+	0.297444i	$0.903866\pi$
$752$	0	0
$753$	−11.2145	−0.408678
$754$	0	0
$755$	5.80255	0.211176
$756$	0	0
$757$	−45.0769	−1.63835	−0.819174	−	0.573545i	$-0.805568\pi$
$757$	−45.0769	−1.63835	−0.819174	+	0.573545i	$0.805568\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	22.7150	0.823417	0.411708	−	0.911316i	$-0.364932\pi$
$761$	22.7150	0.823417	0.411708	+	0.911316i	$0.364932\pi$
$762$	0	0
$763$	34.1146	1.23503
$764$	0	0
$765$	8.51581	0.307890
$766$	0	0
$767$	39.3140	1.41954
$768$	0	0
$769$	−10.5711	−0.381202	−0.190601	−	0.981668i	$-0.561044\pi$
$769$	−10.5711	−0.381202	−0.190601	+	0.981668i	$0.561044\pi$
$770$	0	0
$771$	7.65933	0.275844
$772$	0	0
$773$	−8.71357	−0.313405	−0.156703	−	0.987646i	$-0.550086\pi$
$773$	−8.71357	−0.313405	−0.156703	+	0.987646i	$0.550086\pi$
$774$	0	0
$775$	0.396984	0.0142601
$776$	0	0
$777$	3.10654	0.111446
$778$	0	0
$779$	98.9747	3.54614
$780$	0	0
$781$	0	0
$782$	0	0
$783$	14.3242	0.511904
$784$	0	0
$785$	−20.7356	−0.740085
$786$	0	0
$787$	52.0753	1.85629	0.928143	−	0.372225i	$-0.121405\pi$
$787$	52.0753	1.85629	0.928143	+	0.372225i	$0.121405\pi$
$788$	0	0
$789$	0.916124	0.0326149
$790$	0	0
$791$	23.4220	0.832789
$792$	0	0
$793$	−33.4109	−1.18646
$794$	0	0
$795$	5.61051	0.198984
$796$	0	0
$797$	48.7161	1.72561	0.862806	−	0.505535i	$-0.168705\pi$
$797$	48.7161	1.72561	0.862806	+	0.505535i	$0.168705\pi$
$798$	0	0
$799$	−33.4565	−1.18361
$800$	0	0
$801$	17.5560	0.620312
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−1.33692	−0.0471201
$806$	0	0
$807$	5.13781	0.180860
$808$	0	0
$809$	−15.0005	−0.527389	−0.263695	−	0.964606i	$-0.584941\pi$
$809$	−15.0005	−0.527389	−0.263695	+	0.964606i	$0.584941\pi$
$810$	0	0
$811$	−37.2387	−1.30763	−0.653814	−	0.756656i	$-0.726832\pi$
$811$	−37.2387	−1.30763	−0.653814	+	0.756656i	$0.726832\pi$
$812$	0	0
$813$	11.0460	0.387399
$814$	0	0
$815$	11.8323	0.414468
$816$	0	0
$817$	−71.6932	−2.50823
$818$	0	0
$819$	19.1308	0.668484
$820$	0	0
$821$	−17.9737	−0.627285	−0.313642	−	0.949541i	$-0.601549\pi$
$821$	−17.9737	−0.627285	−0.313642	+	0.949541i	$0.601549\pi$
$822$	0	0
$823$	18.3732	0.640449	0.320224	−	0.947342i	$-0.396242\pi$
$823$	18.3732	0.640449	0.320224	+	0.947342i	$0.396242\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−38.6104	−1.34262	−0.671308	−	0.741178i	$-0.734267\pi$
$827$	−38.6104	−1.34262	−0.671308	+	0.741178i	$0.734267\pi$
$828$	0	0
$829$	52.7170	1.83094	0.915469	−	0.402388i	$-0.131820\pi$
$829$	52.7170	1.83094	0.915469	+	0.402388i	$0.131820\pi$
$830$	0	0
$831$	1.09799	0.0380889
$832$	0	0
$833$	11.7251	0.406252
$834$	0	0
$835$	18.1848	0.629312
$836$	0	0
$837$	−1.28377	−0.0443736
$838$	0	0
$839$	25.8498	0.892435	0.446217	−	0.894925i	$-0.352771\pi$
$839$	25.8498	0.892435	0.446217	+	0.894925i	$0.352771\pi$
$840$	0	0
$841$	−9.37951	−0.323431
$842$	0	0
$843$	−1.24949	−0.0430349
$844$	0	0
$845$	2.41846	0.0831975
$846$	0	0
$847$	0	0
$848$	0	0
$849$	16.5193	0.566940
$850$	0	0
$851$	−2.19779	−0.0753393
$852$	0	0
$853$	−25.3912	−0.869380	−0.434690	−	0.900580i	$-0.643142\pi$
$853$	−25.3912	−0.869380	−0.434690	+	0.900580i	$0.643142\pi$
$854$	0	0
$855$	−21.9716	−0.751413
$856$	0	0
$857$	4.42328	0.151096	0.0755482	−	0.997142i	$-0.475929\pi$
$857$	4.42328	0.151096	0.0755482	+	0.997142i	$0.475929\pi$
$858$	0	0
$859$	13.1491	0.448643	0.224321	−	0.974515i	$-0.427983\pi$
$859$	13.1491	0.448643	0.224321	+	0.974515i	$0.427983\pi$
$860$	0	0
$861$	12.5054	0.426184
$862$	0	0
$863$	−15.4387	−0.525539	−0.262770	−	0.964859i	$-0.584636\pi$
$863$	−15.4387	−0.525539	−0.262770	+	0.964859i	$0.584636\pi$
$864$	0	0
$865$	−10.2152	−0.347328
$866$	0	0
$867$	−3.91335	−0.132904
$868$	0	0
$869$	0	0
$870$	0	0
$871$	4.36270	0.147824
$872$	0	0
$873$	−38.9073	−1.31681
$874$	0	0
$875$	1.82111	0.0615647
$876$	0	0
$877$	7.02185	0.237111	0.118556	−	0.992947i	$-0.462174\pi$
$877$	7.02185	0.237111	0.118556	+	0.992947i	$0.462174\pi$
$878$	0	0
$879$	3.47516	0.117214
$880$	0	0
$881$	22.0522	0.742958	0.371479	−	0.928441i	$-0.378851\pi$
$881$	22.0522	0.742958	0.371479	+	0.928441i	$0.378851\pi$
$882$	0	0
$883$	−12.0558	−0.405709	−0.202855	−	0.979209i	$-0.565022\pi$
$883$	−12.0558	−0.405709	−0.202855	+	0.979209i	$0.565022\pi$
$884$	0	0
$885$	−5.70492	−0.191769
$886$	0	0
$887$	33.1393	1.11271	0.556354	−	0.830946i	$-0.312200\pi$
$887$	33.1393	1.11271	0.556354	+	0.830946i	$0.312200\pi$
$888$	0	0
$889$	5.18614	0.173938
$890$	0	0
$891$	0	0
$892$	0	0
$893$	86.3209	2.88862
$894$	0	0
$895$	−1.72353	−0.0576113
$896$	0	0
$897$	1.64253	0.0548423
$898$	0	0
$899$	−1.75844	−0.0586473
$900$	0	0
$901$	−31.3421	−1.04416
$902$	0	0
$903$	−9.05841	−0.301445
$904$	0	0
$905$	13.9521	0.463783
$906$	0	0
$907$	21.2434	0.705376	0.352688	−	0.935741i	$-0.385268\pi$
$907$	21.2434	0.705376	0.352688	+	0.935741i	$0.385268\pi$
$908$	0	0
$909$	26.4763	0.878165
$910$	0	0
$911$	−47.9763	−1.58952	−0.794762	−	0.606921i	$-0.792404\pi$
$911$	−47.9763	−1.58952	−0.794762	+	0.606921i	$0.792404\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	4.84831	0.160280
$916$	0	0
$917$	17.2898	0.570960
$918$	0	0
$919$	6.85646	0.226174	0.113087	−	0.993585i	$-0.463926\pi$
$919$	6.85646	0.226174	0.113087	+	0.993585i	$0.463926\pi$
$920$	0	0
$921$	−8.69289	−0.286440
$922$	0	0
$923$	23.0561	0.758901
$924$	0	0
$925$	2.99377	0.0984344
$926$	0	0
$927$	19.2923	0.633642
$928$	0	0
$929$	−40.2466	−1.32045	−0.660224	−	0.751069i	$-0.729538\pi$
$929$	−40.2466	−1.32045	−0.660224	+	0.751069i	$0.729538\pi$
$930$	0	0
$931$	−30.2519	−0.991467
$932$	0	0
$933$	1.97480	0.0646519
$934$	0	0
$935$	0	0
$936$	0	0
$937$	47.4498	1.55012	0.775058	−	0.631890i	$-0.217721\pi$
$937$	47.4498	1.55012	0.775058	+	0.631890i	$0.217721\pi$
$938$	0	0
$939$	7.43387	0.242595
$940$	0	0
$941$	18.5398	0.604382	0.302191	−	0.953247i	$-0.402282\pi$
$941$	18.5398	0.604382	0.302191	+	0.953247i	$0.402282\pi$
$942$	0	0
$943$	−8.84724	−0.288106
$944$	0	0
$945$	−5.88911	−0.191573
$946$	0	0
$947$	57.2726	1.86111	0.930555	−	0.366153i	$-0.119325\pi$
$947$	57.2726	1.86111	0.930555	+	0.366153i	$0.119325\pi$
$948$	0	0
$949$	−15.8474	−0.514430
$950$	0	0
$951$	−17.2570	−0.559597
$952$	0	0
$953$	1.12981	0.0365983	0.0182991	−	0.999833i	$-0.494175\pi$
$953$	1.12981	0.0365983	0.0182991	+	0.999833i	$0.494175\pi$
$954$	0	0
$955$	−24.6929	−0.799044
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−12.8176	−0.413903
$960$	0	0
$961$	−30.8424	−0.994916
$962$	0	0
$963$	−26.1576	−0.842918
$964$	0	0
$965$	2.66652	0.0858383
$966$	0	0
$967$	54.1275	1.74062	0.870311	−	0.492502i	$-0.163917\pi$
$967$	54.1275	1.74062	0.870311	+	0.492502i	$0.163917\pi$
$968$	0	0
$969$	−14.8956	−0.478515
$970$	0	0
$971$	−20.1605	−0.646980	−0.323490	−	0.946232i	$-0.604856\pi$
$971$	−20.1605	−0.646980	−0.323490	+	0.946232i	$0.604856\pi$
$972$	0	0
$973$	−6.79609	−0.217873
$974$	0	0
$975$	−2.23740	−0.0716542
$976$	0	0
$977$	−42.3633	−1.35532	−0.677662	−	0.735374i	$-0.737007\pi$
$977$	−42.3633	−1.35532	−0.677662	+	0.735374i	$0.737007\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−50.1166	−1.60010
$982$	0	0
$983$	−52.7171	−1.68141	−0.840706	−	0.541491i	$-0.817860\pi$
$983$	−52.7171	−1.68141	−0.840706	+	0.541491i	$0.817860\pi$
$984$	0	0
$985$	21.4277	0.682743
$986$	0	0
$987$	10.9066	0.347162
$988$	0	0
$989$	6.40858	0.203781
$990$	0	0
$991$	−25.8605	−0.821487	−0.410744	−	0.911751i	$-0.634731\pi$
$991$	−25.8605	−0.821487	−0.410744	+	0.911751i	$0.634731\pi$
$992$	0	0
$993$	−2.38851	−0.0757971
$994$	0	0
$995$	5.78067	0.183260
$996$	0	0
$997$	0.222068	0.00703296	0.00351648	−	0.999994i	$-0.498881\pi$
$997$	0.222068	0.00703296	0.00351648	+	0.999994i	$0.498881\pi$
$998$	0	0
$999$	−9.68126	−0.306301

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9680.2.a.df.1.5		8
4.3	odd	2		4840.2.a.bg.1.4		8
11.3	even	5		880.2.bo.k.801.2		16
11.4	even	5		880.2.bo.k.401.2		16
11.10	odd	2		9680.2.a.de.1.5		8
44.3	odd	10		440.2.y.d.361.3	✓	16
44.15	odd	10		440.2.y.d.401.3	yes	16
44.43	even	2		4840.2.a.bh.1.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
440.2.y.d.361.3	✓	16	44.3	odd	10
440.2.y.d.401.3	yes	16	44.15	odd	10
880.2.bo.k.401.2		16	11.4	even	5
880.2.bo.k.801.2		16	11.3	even	5
4840.2.a.bg.1.4		8	4.3	odd	2
4840.2.a.bh.1.4		8	44.43	even	2
9680.2.a.de.1.5		8	11.10	odd	2
9680.2.a.df.1.5		8	1.1	even	1	trivial

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

\(f(q)\)	\(=\)	\(q+0.569801 q^{3} +1.00000 q^{5} +1.82111 q^{7} -2.67533 q^{9} -3.92663 q^{13} +0.569801 q^{15} -3.18309 q^{17} +8.21268 q^{19} +1.03767 q^{21} -0.734122 q^{23} +1.00000 q^{25} -3.23381 q^{27} -4.42950 q^{29} +0.396984 q^{31} +1.82111 q^{35} +2.99377 q^{37} -2.23740 q^{39} +12.0514 q^{41} -8.72957 q^{43} -2.67533 q^{45} +10.5107 q^{47} -3.68356 q^{49} -1.81373 q^{51} +9.84643 q^{53} +4.67959 q^{57} -10.0121 q^{59} +8.50878 q^{61} -4.87206 q^{63} -3.92663 q^{65} -1.11105 q^{67} -0.418304 q^{69} -5.87172 q^{71} +4.03588 q^{73} +0.569801 q^{75} -3.26768 q^{79} +6.18335 q^{81} +11.8570 q^{83} -3.18309 q^{85} -2.52394 q^{87} -6.56220 q^{89} -7.15083 q^{91} +0.226202 q^{93} +8.21268 q^{95} +14.5430 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - q^{3} + 8 q^{5} + 6 q^{7} + 19 q^{9} + 12 q^{13} - q^{15} + 2 q^{17} + 6 q^{19} + 6 q^{21} - 10 q^{23} + 8 q^{25} - 13 q^{27} + 8 q^{29} - 19 q^{31} + 6 q^{35} + 12 q^{37} + 21 q^{39} - 3 q^{41}+ \cdots + 21 q^{97}+O(q^{100}) \) 8 * q - q^3 + 8 * q^5 + 6 * q^7 + 19 * q^9 + 12 * q^13 - q^15 + 2 * q^17 + 6 * q^19 + 6 * q^21 - 10 * q^23 + 8 * q^25 - 13 * q^27 + 8 * q^29 - 19 * q^31 + 6 * q^35 + 12 * q^37 + 21 * q^39 - 3 * q^41 + 8 * q^43 + 19 * q^45 - 10 * q^47 + 22 * q^49 + 7 * q^51 + 28 * q^53 + 25 * q^57 - 25 * q^59 + 10 * q^61 + 64 * q^63 + 12 * q^65 + 2 * q^67 + 18 * q^69 - 25 * q^71 + 38 * q^73 - q^75 + 38 * q^79 + 32 * q^81 + 28 * q^83 + 2 * q^85 - 15 * q^87 + 12 * q^89 - 8 * q^91 - 15 * q^93 + 6 * q^95 + 21 * q^97

Introduction

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