LMFDB - Embedded newform 9984.2.a.bv.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9984,2,Mod(1,9984)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9984, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("9984.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9984 = 2^{8} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9984.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$79.7226413780$
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} - 2x^{7} - 14x^{6} + 24x^{5} + 65x^{4} - 82x^{3} - 126x^{2} + 84x + 92 $ x^8 - 2x^7 - 14x^6 + 24x^5 + 65x^4 - 82x^3 - 126x^2 + 84*x + 92
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 312)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.81444$ of defining polynomial
Character	$\chi$	$=$	9984.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -0.218531 q^{5} -4.47783 q^{7} +1.00000 q^{9} -3.58286 q^{11} +1.00000 q^{13} -0.218531 q^{15} -7.60367 q^{17} +4.65804 q^{19} -4.47783 q^{21} -5.21219 q^{23} -4.95224 q^{25} +1.00000 q^{27} -3.39240 q^{29} +1.71291 q^{31} -3.58286 q^{33} +0.978543 q^{35} -3.06703 q^{37} +1.00000 q^{39} -1.17387 q^{41} +6.53664 q^{43} -0.218531 q^{45} -6.69636 q^{47} +13.0509 q^{49} -7.60367 q^{51} +1.18362 q^{53} +0.782966 q^{55} +4.65804 q^{57} -9.33654 q^{59} +14.6809 q^{61} -4.47783 q^{63} -0.218531 q^{65} +2.10688 q^{67} -5.21219 q^{69} -11.9731 q^{71} +13.8518 q^{73} -4.95224 q^{75} +16.0434 q^{77} -2.12278 q^{79} +1.00000 q^{81} +11.9086 q^{83} +1.66164 q^{85} -3.39240 q^{87} +10.2015 q^{89} -4.47783 q^{91} +1.71291 q^{93} -1.01792 q^{95} +16.0227 q^{97} -3.58286 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{3} + 2 q^{5} + 2 q^{7} + 8 q^{9} + 8 q^{13} + 2 q^{15} + 8 q^{17} + 6 q^{19} + 2 q^{21} - 4 q^{23} + 16 q^{25} + 8 q^{27} + 4 q^{29} + 2 q^{31} - 4 q^{35} + 8 q^{37} + 8 q^{39} + 18 q^{41} + 16 q^{43}+ \cdots + 20 q^{97}+O(q^{100}) $ 8 * q + 8 * q^3 + 2 * q^5 + 2 * q^7 + 8 * q^9 + 8 * q^13 + 2 * q^15 + 8 * q^17 + 6 * q^19 + 2 * q^21 - 4 * q^23 + 16 * q^25 + 8 * q^27 + 4 * q^29 + 2 * q^31 - 4 * q^35 + 8 * q^37 + 8 * q^39 + 18 * q^41 + 16 * q^43 + 2 * q^45 - 12 * q^47 + 24 * q^49 + 8 * q^51 - 4 * q^53 + 12 * q^55 + 6 * q^57 - 4 * q^59 + 12 * q^61 + 2 * q^63 + 2 * q^65 + 22 * q^67 - 4 * q^69 + 16 * q^73 + 16 * q^75 - 20 * q^77 + 8 * q^81 + 16 * q^83 - 4 * q^85 + 4 * q^87 + 30 * q^89 + 2 * q^91 + 2 * q^93 + 12 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−0.218531	−0.0977299	−0.0488650	−	0.998805i	$-0.515560\pi$
$5$	−0.218531	−0.0977299	−0.0488650	+	0.998805i	$0.515560\pi$
$6$	0	0
$7$	−4.47783	−1.69246	−0.846230	−	0.532818i	$-0.821133\pi$
$7$	−4.47783	−1.69246	−0.846230	+	0.532818i	$0.821133\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.58286	−1.08027	−0.540137	−	0.841577i	$-0.681628\pi$
$11$	−3.58286	−1.08027	−0.540137	+	0.841577i	$0.681628\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−0.218531	−0.0564244
$16$	0	0
$17$	−7.60367	−1.84416	−0.922080	−	0.386998i	$-0.873512\pi$
$17$	−7.60367	−1.84416	−0.922080	+	0.386998i	$0.873512\pi$
$18$	0	0
$19$	4.65804	1.06863	0.534313	−	0.845286i	$-0.320570\pi$
$19$	4.65804	1.06863	0.534313	+	0.845286i	$0.320570\pi$
$20$	0	0
$21$	−4.47783	−0.977142
$22$	0	0
$23$	−5.21219	−1.08682	−0.543409	−	0.839468i	$-0.682867\pi$
$23$	−5.21219	−1.08682	−0.543409	+	0.839468i	$0.682867\pi$
$24$	0	0
$25$	−4.95224	−0.990449
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−3.39240	−0.629953	−0.314977	−	0.949099i	$-0.601997\pi$
$29$	−3.39240	−0.629953	−0.314977	+	0.949099i	$0.601997\pi$
$30$	0	0
$31$	1.71291	0.307647	0.153824	−	0.988098i	$-0.450841\pi$
$31$	1.71291	0.307647	0.153824	+	0.988098i	$0.450841\pi$
$32$	0	0
$33$	−3.58286	−0.623697
$34$	0	0
$35$	0.978543	0.165404
$36$	0	0
$37$	−3.06703	−0.504217	−0.252108	−	0.967699i	$-0.581124\pi$
$37$	−3.06703	−0.504217	−0.252108	+	0.967699i	$0.581124\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	−1.17387	−0.183328	−0.0916638	−	0.995790i	$-0.529219\pi$
$41$	−1.17387	−0.183328	−0.0916638	+	0.995790i	$0.529219\pi$
$42$	0	0
$43$	6.53664	0.996828	0.498414	−	0.866939i	$-0.333916\pi$
$43$	6.53664	0.996828	0.498414	+	0.866939i	$0.333916\pi$
$44$	0	0
$45$	−0.218531	−0.0325766
$46$	0	0
$47$	−6.69636	−0.976764	−0.488382	−	0.872630i	$-0.662413\pi$
$47$	−6.69636	−0.976764	−0.488382	+	0.872630i	$0.662413\pi$
$48$	0	0
$49$	13.0509	1.86442
$50$	0	0
$51$	−7.60367	−1.06473
$52$	0	0
$53$	1.18362	0.162583	0.0812914	−	0.996690i	$-0.474096\pi$
$53$	1.18362	0.162583	0.0812914	+	0.996690i	$0.474096\pi$
$54$	0	0
$55$	0.782966	0.105575
$56$	0	0
$57$	4.65804	0.616972
$58$	0	0
$59$	−9.33654	−1.21551	−0.607757	−	0.794123i	$-0.707930\pi$
$59$	−9.33654	−1.21551	−0.607757	+	0.794123i	$0.707930\pi$
$60$	0	0
$61$	14.6809	1.87970	0.939849	−	0.341590i	$-0.110965\pi$
$61$	14.6809	1.87970	0.939849	+	0.341590i	$0.110965\pi$
$62$	0	0
$63$	−4.47783	−0.564153
$64$	0	0
$65$	−0.218531	−0.0271054
$66$	0	0
$67$	2.10688	0.257397	0.128698	−	0.991684i	$-0.458920\pi$
$67$	2.10688	0.257397	0.128698	+	0.991684i	$0.458920\pi$
$68$	0	0
$69$	−5.21219	−0.627474
$70$	0	0
$71$	−11.9731	−1.42094	−0.710470	−	0.703728i	$-0.751518\pi$
$71$	−11.9731	−1.42094	−0.710470	+	0.703728i	$0.751518\pi$
$72$	0	0
$73$	13.8518	1.62123	0.810617	−	0.585576i	$-0.199132\pi$
$73$	13.8518	1.62123	0.810617	+	0.585576i	$0.199132\pi$
$74$	0	0
$75$	−4.95224	−0.571836
$76$	0	0
$77$	16.0434	1.82832
$78$	0	0
$79$	−2.12278	−0.238832	−0.119416	−	0.992844i	$-0.538102\pi$
$79$	−2.12278	−0.238832	−0.119416	+	0.992844i	$0.538102\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	11.9086	1.30713	0.653567	−	0.756869i	$-0.273272\pi$
$83$	11.9086	1.30713	0.653567	+	0.756869i	$0.273272\pi$
$84$	0	0
$85$	1.66164	0.180230
$86$	0	0
$87$	−3.39240	−0.363704
$88$	0	0
$89$	10.2015	1.08136	0.540679	−	0.841229i	$-0.318167\pi$
$89$	10.2015	1.08136	0.540679	+	0.841229i	$0.318167\pi$
$90$	0	0
$91$	−4.47783	−0.469404
$92$	0	0
$93$	1.71291	0.177620
$94$	0	0
$95$	−1.01792	−0.104437
$96$	0	0
$97$	16.0227	1.62686	0.813429	−	0.581665i	$-0.197598\pi$
$97$	16.0227	1.62686	0.813429	+	0.581665i	$0.197598\pi$
$98$	0	0
$99$	−3.58286	−0.360091
$100$	0	0
$101$	3.90916	0.388976	0.194488	−	0.980905i	$-0.437695\pi$
$101$	3.90916	0.388976	0.194488	+	0.980905i	$0.437695\pi$
$102$	0	0
$103$	−3.38273	−0.333310	−0.166655	−	0.986015i	$-0.553297\pi$
$103$	−3.38273	−0.333310	−0.166655	+	0.986015i	$0.553297\pi$
$104$	0	0
$105$	0.978543	0.0954960
$106$	0	0
$107$	2.01021	0.194334	0.0971672	−	0.995268i	$-0.469022\pi$
$107$	2.01021	0.194334	0.0971672	+	0.995268i	$0.469022\pi$
$108$	0	0
$109$	−16.0032	−1.53283	−0.766414	−	0.642347i	$-0.777961\pi$
$109$	−16.0032	−1.53283	−0.766414	+	0.642347i	$0.777961\pi$
$110$	0	0
$111$	−3.06703	−0.291110
$112$	0	0
$113$	5.90667	0.555653	0.277826	−	0.960631i	$-0.410386\pi$
$113$	5.90667	0.555653	0.277826	+	0.960631i	$0.410386\pi$
$114$	0	0
$115$	1.13902	0.106215
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	34.0479	3.12117
$120$	0	0
$121$	1.83691	0.166992
$122$	0	0
$123$	−1.17387	−0.105844
$124$	0	0
$125$	2.17487	0.194526
$126$	0	0
$127$	−6.89436	−0.611776	−0.305888	−	0.952068i	$-0.598953\pi$
$127$	−6.89436	−0.611776	−0.305888	+	0.952068i	$0.598953\pi$
$128$	0	0
$129$	6.53664	0.575519
$130$	0	0
$131$	−1.98979	−0.173849	−0.0869244	−	0.996215i	$-0.527704\pi$
$131$	−1.98979	−0.173849	−0.0869244	+	0.996215i	$0.527704\pi$
$132$	0	0
$133$	−20.8579	−1.80861
$134$	0	0
$135$	−0.218531	−0.0188081
$136$	0	0
$137$	0.961081	0.0821107	0.0410554	−	0.999157i	$-0.486928\pi$
$137$	0.961081	0.0821107	0.0410554	+	0.999157i	$0.486928\pi$
$138$	0	0
$139$	5.54489	0.470312	0.235156	−	0.971958i	$-0.424440\pi$
$139$	5.54489	0.470312	0.235156	+	0.971958i	$0.424440\pi$
$140$	0	0
$141$	−6.69636	−0.563935
$142$	0	0
$143$	−3.58286	−0.299614
$144$	0	0
$145$	0.741344	0.0615653
$146$	0	0
$147$	13.0509	1.07642
$148$	0	0
$149$	20.9305	1.71469	0.857347	−	0.514739i	$-0.172111\pi$
$149$	20.9305	1.71469	0.857347	+	0.514739i	$0.172111\pi$
$150$	0	0
$151$	−0.401184	−0.0326479	−0.0163239	−	0.999867i	$-0.505196\pi$
$151$	−0.401184	−0.0326479	−0.0163239	+	0.999867i	$0.505196\pi$
$152$	0	0
$153$	−7.60367	−0.614720
$154$	0	0
$155$	−0.374323	−0.0300663
$156$	0	0
$157$	−18.9079	−1.50901	−0.754507	−	0.656292i	$-0.772124\pi$
$157$	−18.9079	−1.50901	−0.754507	+	0.656292i	$0.772124\pi$
$158$	0	0
$159$	1.18362	0.0938672
$160$	0	0
$161$	23.3393	1.83939
$162$	0	0
$163$	18.5887	1.45598	0.727989	−	0.685589i	$-0.240455\pi$
$163$	18.5887	1.45598	0.727989	+	0.685589i	$0.240455\pi$
$164$	0	0
$165$	0.782966	0.0609538
$166$	0	0
$167$	−15.9204	−1.23196	−0.615979	−	0.787763i	$-0.711239\pi$
$167$	−15.9204	−1.23196	−0.615979	+	0.787763i	$0.711239\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	4.65804	0.356209
$172$	0	0
$173$	−3.39369	−0.258018	−0.129009	−	0.991643i	$-0.541180\pi$
$173$	−3.39369	−0.258018	−0.129009	+	0.991643i	$0.541180\pi$
$174$	0	0
$175$	22.1753	1.67629
$176$	0	0
$177$	−9.33654	−0.701777
$178$	0	0
$179$	−0.978543	−0.0731397	−0.0365699	−	0.999331i	$-0.511643\pi$
$179$	−0.978543	−0.0731397	−0.0365699	+	0.999331i	$0.511643\pi$
$180$	0	0
$181$	12.2039	0.907111	0.453555	−	0.891228i	$-0.350155\pi$
$181$	12.2039	0.907111	0.453555	+	0.891228i	$0.350155\pi$
$182$	0	0
$183$	14.6809	1.08524
$184$	0	0
$185$	0.670241	0.0492771
$186$	0	0
$187$	27.2429	1.99220
$188$	0	0
$189$	−4.47783	−0.325714
$190$	0	0
$191$	−9.11614	−0.659621	−0.329810	−	0.944047i	$-0.606985\pi$
$191$	−9.11614	−0.659621	−0.329810	+	0.944047i	$0.606985\pi$
$192$	0	0
$193$	7.98107	0.574490	0.287245	−	0.957857i	$-0.407261\pi$
$193$	7.98107	0.574490	0.287245	+	0.957857i	$0.407261\pi$
$194$	0	0
$195$	−0.218531	−0.0156493
$196$	0	0
$197$	12.6000	0.897713	0.448856	−	0.893604i	$-0.351831\pi$
$197$	12.6000	0.897713	0.448856	+	0.893604i	$0.351831\pi$
$198$	0	0
$199$	−19.1593	−1.35816	−0.679082	−	0.734062i	$-0.737622\pi$
$199$	−19.1593	−1.35816	−0.679082	+	0.734062i	$0.737622\pi$
$200$	0	0
$201$	2.10688	0.148608
$202$	0	0
$203$	15.1906	1.06617
$204$	0	0
$205$	0.256527	0.0179166
$206$	0	0
$207$	−5.21219	−0.362272
$208$	0	0
$209$	−16.6891	−1.15441
$210$	0	0
$211$	−17.4547	−1.20163	−0.600817	−	0.799386i	$-0.705158\pi$
$211$	−17.4547	−1.20163	−0.600817	+	0.799386i	$0.705158\pi$
$212$	0	0
$213$	−11.9731	−0.820380
$214$	0	0
$215$	−1.42846	−0.0974200
$216$	0	0
$217$	−7.67010	−0.520681
$218$	0	0
$219$	13.8518	0.936020
$220$	0	0
$221$	−7.60367	−0.511478
$222$	0	0
$223$	8.42115	0.563922	0.281961	−	0.959426i	$-0.409015\pi$
$223$	8.42115	0.563922	0.281961	+	0.959426i	$0.409015\pi$
$224$	0	0
$225$	−4.95224	−0.330150
$226$	0	0
$227$	−21.0875	−1.39963	−0.699813	−	0.714326i	$-0.746733\pi$
$227$	−21.0875	−1.39963	−0.699813	+	0.714326i	$0.746733\pi$
$228$	0	0
$229$	−12.0289	−0.794894	−0.397447	−	0.917625i	$-0.630104\pi$
$229$	−12.0289	−0.794894	−0.397447	+	0.917625i	$0.630104\pi$
$230$	0	0
$231$	16.0434	1.05558
$232$	0	0
$233$	14.3048	0.937140	0.468570	−	0.883426i	$-0.344769\pi$
$233$	14.3048	0.937140	0.468570	+	0.883426i	$0.344769\pi$
$234$	0	0
$235$	1.46336	0.0954591
$236$	0	0
$237$	−2.12278	−0.137890
$238$	0	0
$239$	5.22763	0.338147	0.169074	−	0.985603i	$-0.445922\pi$
$239$	5.22763	0.338147	0.169074	+	0.985603i	$0.445922\pi$
$240$	0	0
$241$	−2.04161	−0.131512	−0.0657559	−	0.997836i	$-0.520946\pi$
$241$	−2.04161	−0.131512	−0.0657559	+	0.997836i	$0.520946\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−2.85203	−0.182210
$246$	0	0
$247$	4.65804	0.296384
$248$	0	0
$249$	11.9086	0.754674
$250$	0	0
$251$	−22.1961	−1.40100	−0.700502	−	0.713650i	$-0.747041\pi$
$251$	−22.1961	−1.40100	−0.700502	+	0.713650i	$0.747041\pi$
$252$	0	0
$253$	18.6746	1.17406
$254$	0	0
$255$	1.66164	0.104056
$256$	0	0
$257$	−3.71742	−0.231886	−0.115943	−	0.993256i	$-0.536989\pi$
$257$	−3.71742	−0.231886	−0.115943	+	0.993256i	$0.536989\pi$
$258$	0	0
$259$	13.7336	0.853367
$260$	0	0
$261$	−3.39240	−0.209984
$262$	0	0
$263$	−2.09245	−0.129026	−0.0645130	−	0.997917i	$-0.520549\pi$
$263$	−2.09245	−0.129026	−0.0645130	+	0.997917i	$0.520549\pi$
$264$	0	0
$265$	−0.258657	−0.0158892
$266$	0	0
$267$	10.2015	0.624323
$268$	0	0
$269$	5.16604	0.314979	0.157490	−	0.987521i	$-0.449660\pi$
$269$	5.16604	0.314979	0.157490	+	0.987521i	$0.449660\pi$
$270$	0	0
$271$	12.0412	0.731453	0.365727	−	0.930722i	$-0.380821\pi$
$271$	12.0412	0.731453	0.365727	+	0.930722i	$0.380821\pi$
$272$	0	0
$273$	−4.47783	−0.271010
$274$	0	0
$275$	17.7432	1.06996
$276$	0	0
$277$	16.5584	0.994901	0.497450	−	0.867492i	$-0.334270\pi$
$277$	16.5584	0.994901	0.497450	+	0.867492i	$0.334270\pi$
$278$	0	0
$279$	1.71291	0.102549
$280$	0	0
$281$	−4.99219	−0.297809	−0.148905	−	0.988852i	$-0.547575\pi$
$281$	−4.99219	−0.297809	−0.148905	+	0.988852i	$0.547575\pi$
$282$	0	0
$283$	4.99175	0.296729	0.148364	−	0.988933i	$-0.452599\pi$
$283$	4.99175	0.296729	0.148364	+	0.988933i	$0.452599\pi$
$284$	0	0
$285$	−1.01792	−0.0602966
$286$	0	0
$287$	5.25639	0.310275
$288$	0	0
$289$	40.8158	2.40093
$290$	0	0
$291$	16.0227	0.939266
$292$	0	0
$293$	3.02283	0.176596	0.0882979	−	0.996094i	$-0.471857\pi$
$293$	3.02283	0.176596	0.0882979	+	0.996094i	$0.471857\pi$
$294$	0	0
$295$	2.04032	0.118792
$296$	0	0
$297$	−3.58286	−0.207899
$298$	0	0
$299$	−5.21219	−0.301429
$300$	0	0
$301$	−29.2699	−1.68709
$302$	0	0
$303$	3.90916	0.224575
$304$	0	0
$305$	−3.20823	−0.183703
$306$	0	0
$307$	−22.8741	−1.30549	−0.652747	−	0.757576i	$-0.726383\pi$
$307$	−22.8741	−1.30549	−0.652747	+	0.757576i	$0.726383\pi$
$308$	0	0
$309$	−3.38273	−0.192437
$310$	0	0
$311$	22.1117	1.25384	0.626920	−	0.779084i	$-0.284315\pi$
$311$	22.1117	1.25384	0.626920	+	0.779084i	$0.284315\pi$
$312$	0	0
$313$	15.6954	0.887155	0.443577	−	0.896236i	$-0.353709\pi$
$313$	15.6954	0.887155	0.443577	+	0.896236i	$0.353709\pi$
$314$	0	0
$315$	0.978543	0.0551347
$316$	0	0
$317$	24.6083	1.38214	0.691070	−	0.722788i	$-0.257140\pi$
$317$	24.6083	1.38214	0.691070	+	0.722788i	$0.257140\pi$
$318$	0	0
$319$	12.1545	0.680522
$320$	0	0
$321$	2.01021	0.112199
$322$	0	0
$323$	−35.4182	−1.97072
$324$	0	0
$325$	−4.95224	−0.274701
$326$	0	0
$327$	−16.0032	−0.884978
$328$	0	0
$329$	29.9851	1.65313
$330$	0	0
$331$	6.29568	0.346042	0.173021	−	0.984918i	$-0.444647\pi$
$331$	6.29568	0.346042	0.173021	+	0.984918i	$0.444647\pi$
$332$	0	0
$333$	−3.06703	−0.168072
$334$	0	0
$335$	−0.460419	−0.0251554
$336$	0	0
$337$	−16.8841	−0.919737	−0.459869	−	0.887987i	$-0.652104\pi$
$337$	−16.8841	−0.919737	−0.459869	+	0.887987i	$0.652104\pi$
$338$	0	0
$339$	5.90667	0.320806
$340$	0	0
$341$	−6.13711	−0.332343
$342$	0	0
$343$	−27.0951	−1.46300
$344$	0	0
$345$	1.13902	0.0613230
$346$	0	0
$347$	29.1187	1.56317	0.781586	−	0.623798i	$-0.214411\pi$
$347$	29.1187	1.56317	0.781586	+	0.623798i	$0.214411\pi$
$348$	0	0
$349$	−7.81462	−0.418307	−0.209154	−	0.977883i	$-0.567071\pi$
$349$	−7.81462	−0.418307	−0.209154	+	0.977883i	$0.567071\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	11.3088	0.601908	0.300954	−	0.953639i	$-0.402695\pi$
$353$	11.3088	0.601908	0.300954	+	0.953639i	$0.402695\pi$
$354$	0	0
$355$	2.61648	0.138868
$356$	0	0
$357$	34.0479	1.80201
$358$	0	0
$359$	18.0361	0.951910	0.475955	−	0.879470i	$-0.342102\pi$
$359$	18.0361	0.951910	0.475955	+	0.879470i	$0.342102\pi$
$360$	0	0
$361$	2.69730	0.141963
$362$	0	0
$363$	1.83691	0.0964130
$364$	0	0
$365$	−3.02705	−0.158443
$366$	0	0
$367$	18.7918	0.980924	0.490462	−	0.871463i	$-0.336828\pi$
$367$	18.7918	0.980924	0.490462	+	0.871463i	$0.336828\pi$
$368$	0	0
$369$	−1.17387	−0.0611092
$370$	0	0
$371$	−5.30005	−0.275165
$372$	0	0
$373$	−4.04291	−0.209334	−0.104667	−	0.994507i	$-0.533378\pi$
$373$	−4.04291	−0.209334	−0.104667	+	0.994507i	$0.533378\pi$
$374$	0	0
$375$	2.17487	0.112310
$376$	0	0
$377$	−3.39240	−0.174718
$378$	0	0
$379$	18.3308	0.941590	0.470795	−	0.882243i	$-0.343967\pi$
$379$	18.3308	0.941590	0.470795	+	0.882243i	$0.343967\pi$
$380$	0	0
$381$	−6.89436	−0.353209
$382$	0	0
$383$	−1.24438	−0.0635851	−0.0317925	−	0.999494i	$-0.510122\pi$
$383$	−1.24438	−0.0635851	−0.0317925	+	0.999494i	$0.510122\pi$
$384$	0	0
$385$	−3.50599	−0.178682
$386$	0	0
$387$	6.53664	0.332276
$388$	0	0
$389$	18.8394	0.955196	0.477598	−	0.878579i	$-0.341508\pi$
$389$	18.8394	0.955196	0.477598	+	0.878579i	$0.341508\pi$
$390$	0	0
$391$	39.6318	2.00427
$392$	0	0
$393$	−1.98979	−0.100372
$394$	0	0
$395$	0.463893	0.0233410
$396$	0	0
$397$	−22.7274	−1.14065	−0.570327	−	0.821418i	$-0.693183\pi$
$397$	−22.7274	−1.14065	−0.570327	+	0.821418i	$0.693183\pi$
$398$	0	0
$399$	−20.8579	−1.04420
$400$	0	0
$401$	−0.720524	−0.0359813	−0.0179906	−	0.999838i	$-0.505727\pi$
$401$	−0.720524	−0.0359813	−0.0179906	+	0.999838i	$0.505727\pi$
$402$	0	0
$403$	1.71291	0.0853260
$404$	0	0
$405$	−0.218531	−0.0108589
$406$	0	0
$407$	10.9888	0.544692
$408$	0	0
$409$	13.4116	0.663163	0.331582	−	0.943427i	$-0.392418\pi$
$409$	13.4116	0.663163	0.331582	+	0.943427i	$0.392418\pi$
$410$	0	0
$411$	0.961081	0.0474067
$412$	0	0
$413$	41.8074	2.05721
$414$	0	0
$415$	−2.60238	−0.127746
$416$	0	0
$417$	5.54489	0.271535
$418$	0	0
$419$	−21.7292	−1.06154	−0.530770	−	0.847516i	$-0.678097\pi$
$419$	−21.7292	−1.06154	−0.530770	+	0.847516i	$0.678097\pi$
$420$	0	0
$421$	17.1215	0.834452	0.417226	−	0.908803i	$-0.363002\pi$
$421$	17.1215	0.834452	0.417226	+	0.908803i	$0.363002\pi$
$422$	0	0
$423$	−6.69636	−0.325588
$424$	0	0
$425$	37.6552	1.82655
$426$	0	0
$427$	−65.7386	−3.18131
$428$	0	0
$429$	−3.58286	−0.172982
$430$	0	0
$431$	35.9214	1.73027	0.865137	−	0.501536i	$-0.167232\pi$
$431$	35.9214	1.73027	0.865137	+	0.501536i	$0.167232\pi$
$432$	0	0
$433$	−20.7974	−0.999460	−0.499730	−	0.866181i	$-0.666567\pi$
$433$	−20.7974	−0.999460	−0.499730	+	0.866181i	$0.666567\pi$
$434$	0	0
$435$	0.741344	0.0355447
$436$	0	0
$437$	−24.2786	−1.16140
$438$	0	0
$439$	8.05946	0.384657	0.192328	−	0.981331i	$-0.438396\pi$
$439$	8.05946	0.384657	0.192328	+	0.981331i	$0.438396\pi$
$440$	0	0
$441$	13.0509	0.621473
$442$	0	0
$443$	7.84426	0.372692	0.186346	−	0.982484i	$-0.440335\pi$
$443$	7.84426	0.372692	0.186346	+	0.982484i	$0.440335\pi$
$444$	0	0
$445$	−2.22935	−0.105681
$446$	0	0
$447$	20.9305	0.989979
$448$	0	0
$449$	28.1265	1.32737	0.663685	−	0.748012i	$-0.268992\pi$
$449$	28.1265	1.32737	0.663685	+	0.748012i	$0.268992\pi$
$450$	0	0
$451$	4.20582	0.198044
$452$	0	0
$453$	−0.401184	−0.0188493
$454$	0	0
$455$	0.978543	0.0458748
$456$	0	0
$457$	−7.83861	−0.366675	−0.183337	−	0.983050i	$-0.558690\pi$
$457$	−7.83861	−0.366675	−0.183337	+	0.983050i	$0.558690\pi$
$458$	0	0
$459$	−7.60367	−0.354909
$460$	0	0
$461$	−11.2498	−0.523955	−0.261977	−	0.965074i	$-0.584375\pi$
$461$	−11.2498	−0.523955	−0.261977	+	0.965074i	$0.584375\pi$
$462$	0	0
$463$	−16.7795	−0.779811	−0.389905	−	0.920855i	$-0.627492\pi$
$463$	−16.7795	−0.779811	−0.389905	+	0.920855i	$0.627492\pi$
$464$	0	0
$465$	−0.374323	−0.0173588
$466$	0	0
$467$	−39.3481	−1.82081	−0.910407	−	0.413713i	$-0.864232\pi$
$467$	−39.3481	−1.82081	−0.910407	+	0.413713i	$0.864232\pi$
$468$	0	0
$469$	−9.43426	−0.435634
$470$	0	0
$471$	−18.9079	−0.871230
$472$	0	0
$473$	−23.4199	−1.07685
$474$	0	0
$475$	−23.0677	−1.05842
$476$	0	0
$477$	1.18362	0.0541942
$478$	0	0
$479$	12.7196	0.581173	0.290586	−	0.956849i	$-0.406150\pi$
$479$	12.7196	0.581173	0.290586	+	0.956849i	$0.406150\pi$
$480$	0	0
$481$	−3.06703	−0.139845
$482$	0	0
$483$	23.3393	1.06197
$484$	0	0
$485$	−3.50145	−0.158993
$486$	0	0
$487$	36.3811	1.64859	0.824293	−	0.566164i	$-0.191573\pi$
$487$	36.3811	1.64859	0.824293	+	0.566164i	$0.191573\pi$
$488$	0	0
$489$	18.5887	0.840609
$490$	0	0
$491$	−21.3927	−0.965440	−0.482720	−	0.875775i	$-0.660351\pi$
$491$	−21.3927	−0.965440	−0.482720	+	0.875775i	$0.660351\pi$
$492$	0	0
$493$	25.7947	1.16173
$494$	0	0
$495$	0.782966	0.0351917
$496$	0	0
$497$	53.6133	2.40488
$498$	0	0
$499$	8.39795	0.375944	0.187972	−	0.982174i	$-0.439809\pi$
$499$	8.39795	0.375944	0.187972	+	0.982174i	$0.439809\pi$
$500$	0	0
$501$	−15.9204	−0.711271
$502$	0	0
$503$	−37.1422	−1.65609	−0.828044	−	0.560663i	$-0.810547\pi$
$503$	−37.1422	−1.65609	−0.828044	+	0.560663i	$0.810547\pi$
$504$	0	0
$505$	−0.854272	−0.0380146
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	−6.89677	−0.305694	−0.152847	−	0.988250i	$-0.548844\pi$
$509$	−6.89677	−0.305694	−0.152847	+	0.988250i	$0.548844\pi$
$510$	0	0
$511$	−62.0261	−2.74387
$512$	0	0
$513$	4.65804	0.205657
$514$	0	0
$515$	0.739231	0.0325744
$516$	0	0
$517$	23.9921	1.05517
$518$	0	0
$519$	−3.39369	−0.148967
$520$	0	0
$521$	7.62254	0.333949	0.166975	−	0.985961i	$-0.446600\pi$
$521$	7.62254	0.333949	0.166975	+	0.985961i	$0.446600\pi$
$522$	0	0
$523$	−14.7801	−0.646287	−0.323144	−	0.946350i	$-0.604740\pi$
$523$	−14.7801	−0.646287	−0.323144	+	0.946350i	$0.604740\pi$
$524$	0	0
$525$	22.1753	0.967809
$526$	0	0
$527$	−13.0244	−0.567351
$528$	0	0
$529$	4.16695	0.181172
$530$	0	0
$531$	−9.33654	−0.405171
$532$	0	0
$533$	−1.17387	−0.0508460
$534$	0	0
$535$	−0.439293	−0.0189923
$536$	0	0
$537$	−0.978543	−0.0422273
$538$	0	0
$539$	−46.7597	−2.01408
$540$	0	0
$541$	37.4916	1.61189	0.805945	−	0.591990i	$-0.201658\pi$
$541$	37.4916	1.61189	0.805945	+	0.591990i	$0.201658\pi$
$542$	0	0
$543$	12.2039	0.523721
$544$	0	0
$545$	3.49719	0.149803
$546$	0	0
$547$	−15.7598	−0.673840	−0.336920	−	0.941533i	$-0.609385\pi$
$547$	−15.7598	−0.673840	−0.336920	+	0.941533i	$0.609385\pi$
$548$	0	0
$549$	14.6809	0.626566
$550$	0	0
$551$	−15.8019	−0.673185
$552$	0	0
$553$	9.50545	0.404213
$554$	0	0
$555$	0.670241	0.0284501
$556$	0	0
$557$	−24.0153	−1.01756	−0.508781	−	0.860896i	$-0.669904\pi$
$557$	−24.0153	−1.01756	−0.508781	+	0.860896i	$0.669904\pi$
$558$	0	0
$559$	6.53664	0.276470
$560$	0	0
$561$	27.2429	1.15020
$562$	0	0
$563$	26.9917	1.13757	0.568783	−	0.822488i	$-0.307414\pi$
$563$	26.9917	1.13757	0.568783	+	0.822488i	$0.307414\pi$
$564$	0	0
$565$	−1.29079	−0.0543039
$566$	0	0
$567$	−4.47783	−0.188051
$568$	0	0
$569$	39.0770	1.63819	0.819096	−	0.573657i	$-0.194476\pi$
$569$	39.0770	1.63819	0.819096	+	0.573657i	$0.194476\pi$
$570$	0	0
$571$	−38.5630	−1.61381	−0.806907	−	0.590679i	$-0.798860\pi$
$571$	−38.5630	−1.61381	−0.806907	+	0.590679i	$0.798860\pi$
$572$	0	0
$573$	−9.11614	−0.380832
$574$	0	0
$575$	25.8120	1.07644
$576$	0	0
$577$	29.4706	1.22688	0.613438	−	0.789743i	$-0.289786\pi$
$577$	29.4706	1.22688	0.613438	+	0.789743i	$0.289786\pi$
$578$	0	0
$579$	7.98107	0.331682
$580$	0	0
$581$	−53.3244	−2.21227
$582$	0	0
$583$	−4.24075	−0.175634
$584$	0	0
$585$	−0.218531	−0.00903514
$586$	0	0
$587$	−7.19553	−0.296991	−0.148496	−	0.988913i	$-0.547443\pi$
$587$	−7.19553	−0.296991	−0.148496	+	0.988913i	$0.547443\pi$
$588$	0	0
$589$	7.97879	0.328760
$590$	0	0
$591$	12.6000	0.518295
$592$	0	0
$593$	7.68075	0.315411	0.157705	−	0.987486i	$-0.449590\pi$
$593$	7.68075	0.315411	0.157705	+	0.987486i	$0.449590\pi$
$594$	0	0
$595$	−7.44052	−0.305032
$596$	0	0
$597$	−19.1593	−0.784137
$598$	0	0
$599$	24.3642	0.995493	0.497747	−	0.867322i	$-0.334161\pi$
$599$	24.3642	0.995493	0.497747	+	0.867322i	$0.334161\pi$
$600$	0	0
$601$	−8.34674	−0.340471	−0.170235	−	0.985403i	$-0.554453\pi$
$601$	−8.34674	−0.340471	−0.170235	+	0.985403i	$0.554453\pi$
$602$	0	0
$603$	2.10688	0.0857989
$604$	0	0
$605$	−0.401422	−0.0163201
$606$	0	0
$607$	−25.7099	−1.04353	−0.521767	−	0.853088i	$-0.674727\pi$
$607$	−25.7099	−1.04353	−0.521767	+	0.853088i	$0.674727\pi$
$608$	0	0
$609$	15.1906	0.615554
$610$	0	0
$611$	−6.69636	−0.270906
$612$	0	0
$613$	12.4181	0.501564	0.250782	−	0.968044i	$-0.419312\pi$
$613$	12.4181	0.501564	0.250782	+	0.968044i	$0.419312\pi$
$614$	0	0
$615$	0.256527	0.0103442
$616$	0	0
$617$	−25.8886	−1.04224	−0.521118	−	0.853485i	$-0.674485\pi$
$617$	−25.8886	−1.04224	−0.521118	+	0.853485i	$0.674485\pi$
$618$	0	0
$619$	35.4019	1.42292	0.711461	−	0.702726i	$-0.248034\pi$
$619$	35.4019	1.42292	0.711461	+	0.702726i	$0.248034\pi$
$620$	0	0
$621$	−5.21219	−0.209158
$622$	0	0
$623$	−45.6806	−1.83016
$624$	0	0
$625$	24.2859	0.971438
$626$	0	0
$627$	−16.6891	−0.666499
$628$	0	0
$629$	23.3207	0.929857
$630$	0	0
$631$	15.1930	0.604824	0.302412	−	0.953177i	$-0.402208\pi$
$631$	15.1930	0.604824	0.302412	+	0.953177i	$0.402208\pi$
$632$	0	0
$633$	−17.4547	−0.693764
$634$	0	0
$635$	1.50663	0.0597888
$636$	0	0
$637$	13.0509	0.517097
$638$	0	0
$639$	−11.9731	−0.473647
$640$	0	0
$641$	33.9555	1.34116	0.670582	−	0.741836i	$-0.266045\pi$
$641$	33.9555	1.34116	0.670582	+	0.741836i	$0.266045\pi$
$642$	0	0
$643$	11.8070	0.465623	0.232812	−	0.972522i	$-0.425207\pi$
$643$	11.8070	0.465623	0.232812	+	0.972522i	$0.425207\pi$
$644$	0	0
$645$	−1.42846	−0.0562454
$646$	0	0
$647$	−19.7307	−0.775695	−0.387848	−	0.921724i	$-0.626781\pi$
$647$	−19.7307	−0.775695	−0.387848	+	0.921724i	$0.626781\pi$
$648$	0	0
$649$	33.4515	1.31309
$650$	0	0
$651$	−7.67010	−0.300615
$652$	0	0
$653$	−14.9225	−0.583963	−0.291982	−	0.956424i	$-0.594315\pi$
$653$	−14.9225	−0.583963	−0.291982	+	0.956424i	$0.594315\pi$
$654$	0	0
$655$	0.434830	0.0169902
$656$	0	0
$657$	13.8518	0.540411
$658$	0	0
$659$	34.7477	1.35358	0.676789	−	0.736177i	$-0.263371\pi$
$659$	34.7477	1.35358	0.676789	+	0.736177i	$0.263371\pi$
$660$	0	0
$661$	−42.5576	−1.65530	−0.827650	−	0.561245i	$-0.810323\pi$
$661$	−42.5576	−1.65530	−0.827650	+	0.561245i	$0.810323\pi$
$662$	0	0
$663$	−7.60367	−0.295302
$664$	0	0
$665$	4.55809	0.176755
$666$	0	0
$667$	17.6818	0.684644
$668$	0	0
$669$	8.42115	0.325581
$670$	0	0
$671$	−52.5997	−2.03059
$672$	0	0
$673$	−30.8129	−1.18775	−0.593875	−	0.804557i	$-0.702403\pi$
$673$	−30.8129	−1.18775	−0.593875	+	0.804557i	$0.702403\pi$
$674$	0	0
$675$	−4.95224	−0.190612
$676$	0	0
$677$	−28.1235	−1.08087	−0.540436	−	0.841385i	$-0.681741\pi$
$677$	−28.1235	−1.08087	−0.540436	+	0.841385i	$0.681741\pi$
$678$	0	0
$679$	−71.7468	−2.75339
$680$	0	0
$681$	−21.0875	−0.808075
$682$	0	0
$683$	10.1192	0.387199	0.193600	−	0.981081i	$-0.437984\pi$
$683$	10.1192	0.387199	0.193600	+	0.981081i	$0.437984\pi$
$684$	0	0
$685$	−0.210026	−0.00802468
$686$	0	0
$687$	−12.0289	−0.458932
$688$	0	0
$689$	1.18362	0.0450923
$690$	0	0
$691$	25.1776	0.957803	0.478901	−	0.877869i	$-0.341035\pi$
$691$	25.1776	0.957803	0.478901	+	0.877869i	$0.341035\pi$
$692$	0	0
$693$	16.0434	0.609440
$694$	0	0
$695$	−1.21173	−0.0459635
$696$	0	0
$697$	8.92572	0.338086
$698$	0	0
$699$	14.3048	0.541058
$700$	0	0
$701$	7.53851	0.284725	0.142363	−	0.989815i	$-0.454530\pi$
$701$	7.53851	0.284725	0.142363	+	0.989815i	$0.454530\pi$
$702$	0	0
$703$	−14.2863	−0.538820
$704$	0	0
$705$	1.46336	0.0551133
$706$	0	0
$707$	−17.5045	−0.658326
$708$	0	0
$709$	13.5851	0.510201	0.255100	−	0.966915i	$-0.417892\pi$
$709$	13.5851	0.510201	0.255100	+	0.966915i	$0.417892\pi$
$710$	0	0
$711$	−2.12278	−0.0796105
$712$	0	0
$713$	−8.92800	−0.334356
$714$	0	0
$715$	0.782966	0.0292813
$716$	0	0
$717$	5.22763	0.195229
$718$	0	0
$719$	20.6991	0.771947	0.385973	−	0.922510i	$-0.373866\pi$
$719$	20.6991	0.771947	0.385973	+	0.922510i	$0.373866\pi$
$720$	0	0
$721$	15.1473	0.564114
$722$	0	0
$723$	−2.04161	−0.0759284
$724$	0	0
$725$	16.8000	0.623936
$726$	0	0
$727$	15.2057	0.563947	0.281974	−	0.959422i	$-0.409011\pi$
$727$	15.2057	0.563947	0.281974	+	0.959422i	$0.409011\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−49.7025	−1.83831
$732$	0	0
$733$	−43.4506	−1.60488	−0.802442	−	0.596730i	$-0.796466\pi$
$733$	−43.4506	−1.60488	−0.802442	+	0.596730i	$0.796466\pi$
$734$	0	0
$735$	−2.85203	−0.105199
$736$	0	0
$737$	−7.54868	−0.278059
$738$	0	0
$739$	4.52267	0.166369	0.0831845	−	0.996534i	$-0.473491\pi$
$739$	4.52267	0.166369	0.0831845	+	0.996534i	$0.473491\pi$
$740$	0	0
$741$	4.65804	0.171117
$742$	0	0
$743$	−36.1002	−1.32439	−0.662194	−	0.749332i	$-0.730375\pi$
$743$	−36.1002	−1.32439	−0.662194	+	0.749332i	$0.730375\pi$
$744$	0	0
$745$	−4.57396	−0.167577
$746$	0	0
$747$	11.9086	0.435711
$748$	0	0
$749$	−9.00137	−0.328903
$750$	0	0
$751$	−17.6770	−0.645043	−0.322522	−	0.946562i	$-0.604531\pi$
$751$	−17.6770	−0.645043	−0.322522	+	0.946562i	$0.604531\pi$
$752$	0	0
$753$	−22.1961	−0.808871
$754$	0	0
$755$	0.0876710	0.00319067
$756$	0	0
$757$	−30.7650	−1.11817	−0.559086	−	0.829110i	$-0.688848\pi$
$757$	−30.7650	−1.11817	−0.559086	+	0.829110i	$0.688848\pi$
$758$	0	0
$759$	18.6746	0.677844
$760$	0	0
$761$	36.6808	1.32968	0.664838	−	0.746987i	$-0.268500\pi$
$761$	36.6808	1.32968	0.664838	+	0.746987i	$0.268500\pi$
$762$	0	0
$763$	71.6595	2.59425
$764$	0	0
$765$	1.66164	0.0600766
$766$	0	0
$767$	−9.33654	−0.337123
$768$	0	0
$769$	−43.9664	−1.58547	−0.792735	−	0.609566i	$-0.791344\pi$
$769$	−43.9664	−1.58547	−0.792735	+	0.609566i	$0.791344\pi$
$770$	0	0
$771$	−3.71742	−0.133880
$772$	0	0
$773$	−18.1079	−0.651296	−0.325648	−	0.945491i	$-0.605582\pi$
$773$	−18.1079	−0.651296	−0.325648	+	0.945491i	$0.605582\pi$
$774$	0	0
$775$	−8.48274	−0.304709
$776$	0	0
$777$	13.7336	0.492691
$778$	0	0
$779$	−5.46793	−0.195909
$780$	0	0
$781$	42.8978	1.53500
$782$	0	0
$783$	−3.39240	−0.121235
$784$	0	0
$785$	4.13196	0.147476
$786$	0	0
$787$	34.2703	1.22160	0.610801	−	0.791784i	$-0.290847\pi$
$787$	34.2703	1.22160	0.610801	+	0.791784i	$0.290847\pi$
$788$	0	0
$789$	−2.09245	−0.0744931
$790$	0	0
$791$	−26.4491	−0.940420
$792$	0	0
$793$	14.6809	0.521334
$794$	0	0
$795$	−0.258657	−0.00917363
$796$	0	0
$797$	−23.3407	−0.826770	−0.413385	−	0.910556i	$-0.635654\pi$
$797$	−23.3407	−0.826770	−0.413385	+	0.910556i	$0.635654\pi$
$798$	0	0
$799$	50.9169	1.80131
$800$	0	0
$801$	10.2015	0.360453
$802$	0	0
$803$	−49.6292	−1.75138
$804$	0	0
$805$	−5.10035	−0.179764
$806$	0	0
$807$	5.16604	0.181853
$808$	0	0
$809$	−48.3125	−1.69858	−0.849289	−	0.527928i	$-0.822969\pi$
$809$	−48.3125	−1.69858	−0.849289	+	0.527928i	$0.822969\pi$
$810$	0	0
$811$	25.5793	0.898210	0.449105	−	0.893479i	$-0.351743\pi$
$811$	25.5793	0.898210	0.449105	+	0.893479i	$0.351743\pi$
$812$	0	0
$813$	12.0412	0.422305
$814$	0	0
$815$	−4.06220	−0.142293
$816$	0	0
$817$	30.4479	1.06524
$818$	0	0
$819$	−4.47783	−0.156468
$820$	0	0
$821$	−49.7894	−1.73766	−0.868832	−	0.495108i	$-0.835129\pi$
$821$	−49.7894	−1.73766	−0.868832	+	0.495108i	$0.835129\pi$
$822$	0	0
$823$	20.4403	0.712504	0.356252	−	0.934390i	$-0.384054\pi$
$823$	20.4403	0.712504	0.356252	+	0.934390i	$0.384054\pi$
$824$	0	0
$825$	17.7432	0.617740
$826$	0	0
$827$	−36.4382	−1.26708	−0.633540	−	0.773710i	$-0.718399\pi$
$827$	−36.4382	−1.26708	−0.633540	+	0.773710i	$0.718399\pi$
$828$	0	0
$829$	45.7206	1.58794	0.793971	−	0.607956i	$-0.208010\pi$
$829$	45.7206	1.58794	0.793971	+	0.607956i	$0.208010\pi$
$830$	0	0
$831$	16.5584	0.574406
$832$	0	0
$833$	−99.2351	−3.43829
$834$	0	0
$835$	3.47910	0.120399
$836$	0	0
$837$	1.71291	0.0592067
$838$	0	0
$839$	−17.2215	−0.594554	−0.297277	−	0.954791i	$-0.596078\pi$
$839$	−17.2215	−0.594554	−0.297277	+	0.954791i	$0.596078\pi$
$840$	0	0
$841$	−17.4916	−0.603159
$842$	0	0
$843$	−4.99219	−0.171940
$844$	0	0
$845$	−0.218531	−0.00751769
$846$	0	0
$847$	−8.22539	−0.282628
$848$	0	0
$849$	4.99175	0.171316
$850$	0	0
$851$	15.9860	0.547991
$852$	0	0
$853$	39.1565	1.34069	0.670347	−	0.742048i	$-0.266145\pi$
$853$	39.1565	1.34069	0.670347	+	0.742048i	$0.266145\pi$
$854$	0	0
$855$	−1.01792	−0.0348123
$856$	0	0
$857$	−41.8926	−1.43102	−0.715511	−	0.698601i	$-0.753806\pi$
$857$	−41.8926	−1.43102	−0.715511	+	0.698601i	$0.753806\pi$
$858$	0	0
$859$	−10.2063	−0.348236	−0.174118	−	0.984725i	$-0.555707\pi$
$859$	−10.2063	−0.348236	−0.174118	+	0.984725i	$0.555707\pi$
$860$	0	0
$861$	5.25639	0.179137
$862$	0	0
$863$	18.6974	0.636466	0.318233	−	0.948013i	$-0.396911\pi$
$863$	18.6974	0.636466	0.318233	+	0.948013i	$0.396911\pi$
$864$	0	0
$865$	0.741626	0.0252160
$866$	0	0
$867$	40.8158	1.38618
$868$	0	0
$869$	7.60564	0.258004
$870$	0	0
$871$	2.10688	0.0713890
$872$	0	0
$873$	16.0227	0.542286
$874$	0	0
$875$	−9.73870	−0.329228
$876$	0	0
$877$	0.583295	0.0196965	0.00984824	−	0.999952i	$-0.496865\pi$
$877$	0.583295	0.0196965	0.00984824	+	0.999952i	$0.496865\pi$
$878$	0	0
$879$	3.02283	0.101958
$880$	0	0
$881$	30.7506	1.03601	0.518007	−	0.855377i	$-0.326674\pi$
$881$	30.7506	1.03601	0.518007	+	0.855377i	$0.326674\pi$
$882$	0	0
$883$	2.72627	0.0917464	0.0458732	−	0.998947i	$-0.485393\pi$
$883$	2.72627	0.0917464	0.0458732	+	0.998947i	$0.485393\pi$
$884$	0	0
$885$	2.04032	0.0685846
$886$	0	0
$887$	5.52841	0.185626	0.0928130	−	0.995684i	$-0.470414\pi$
$887$	5.52841	0.185626	0.0928130	+	0.995684i	$0.470414\pi$
$888$	0	0
$889$	30.8718	1.03541
$890$	0	0
$891$	−3.58286	−0.120030
$892$	0	0
$893$	−31.1919	−1.04380
$894$	0	0
$895$	0.213842	0.00714794
$896$	0	0
$897$	−5.21219	−0.174030
$898$	0	0
$899$	−5.81087	−0.193803
$900$	0	0
$901$	−8.99986	−0.299829
$902$	0	0
$903$	−29.2699	−0.974043
$904$	0	0
$905$	−2.66693	−0.0886519
$906$	0	0
$907$	−41.3739	−1.37380	−0.686900	−	0.726752i	$-0.741029\pi$
$907$	−41.3739	−1.37380	−0.686900	+	0.726752i	$0.741029\pi$
$908$	0	0
$909$	3.90916	0.129659
$910$	0	0
$911$	29.6950	0.983839	0.491920	−	0.870641i	$-0.336295\pi$
$911$	29.6950	0.983839	0.491920	+	0.870641i	$0.336295\pi$
$912$	0	0
$913$	−42.6667	−1.41206
$914$	0	0
$915$	−3.20823	−0.106061
$916$	0	0
$917$	8.90994	0.294232
$918$	0	0
$919$	52.9480	1.74659	0.873297	−	0.487188i	$-0.161977\pi$
$919$	52.9480	1.74659	0.873297	+	0.487188i	$0.161977\pi$
$920$	0	0
$921$	−22.8741	−0.753727
$922$	0	0
$923$	−11.9731	−0.394098
$924$	0	0
$925$	15.1887	0.499401
$926$	0	0
$927$	−3.38273	−0.111103
$928$	0	0
$929$	35.6986	1.17123	0.585617	−	0.810588i	$-0.300852\pi$
$929$	35.6986	1.17123	0.585617	+	0.810588i	$0.300852\pi$
$930$	0	0
$931$	60.7918	1.99237
$932$	0	0
$933$	22.1117	0.723905
$934$	0	0
$935$	−5.95342	−0.194697
$936$	0	0
$937$	59.7049	1.95047	0.975237	−	0.221163i	$-0.0709853\pi$
$937$	59.7049	1.95047	0.975237	+	0.221163i	$0.0709853\pi$
$938$	0	0
$939$	15.6954	0.512199
$940$	0	0
$941$	22.1999	0.723697	0.361849	−	0.932237i	$-0.382146\pi$
$941$	22.1999	0.723697	0.361849	+	0.932237i	$0.382146\pi$
$942$	0	0
$943$	6.11844	0.199244
$944$	0	0
$945$	0.978543	0.0318320
$946$	0	0
$947$	−8.89010	−0.288889	−0.144445	−	0.989513i	$-0.546140\pi$
$947$	−8.89010	−0.288889	−0.144445	+	0.989513i	$0.546140\pi$
$948$	0	0
$949$	13.8518	0.449649
$950$	0	0
$951$	24.6083	0.797979
$952$	0	0
$953$	8.55786	0.277216	0.138608	−	0.990347i	$-0.455737\pi$
$953$	8.55786	0.277216	0.138608	+	0.990347i	$0.455737\pi$
$954$	0	0
$955$	1.99216	0.0644647
$956$	0	0
$957$	12.1545	0.392900
$958$	0	0
$959$	−4.30356	−0.138969
$960$	0	0
$961$	−28.0659	−0.905353
$962$	0	0
$963$	2.01021	0.0647781
$964$	0	0
$965$	−1.74411	−0.0561449
$966$	0	0
$967$	−47.2162	−1.51837	−0.759186	−	0.650874i	$-0.774403\pi$
$967$	−47.2162	−1.51837	−0.759186	+	0.650874i	$0.774403\pi$
$968$	0	0
$969$	−35.4182	−1.13780
$970$	0	0
$971$	−28.5324	−0.915648	−0.457824	−	0.889043i	$-0.651371\pi$
$971$	−28.5324	−0.915648	−0.457824	+	0.889043i	$0.651371\pi$
$972$	0	0
$973$	−24.8291	−0.795984
$974$	0	0
$975$	−4.95224	−0.158599
$976$	0	0
$977$	−42.2487	−1.35165	−0.675827	−	0.737060i	$-0.736213\pi$
$977$	−42.2487	−1.35165	−0.675827	+	0.737060i	$0.736213\pi$
$978$	0	0
$979$	−36.5507	−1.16816
$980$	0	0
$981$	−16.0032	−0.510942
$982$	0	0
$983$	28.6115	0.912565	0.456283	−	0.889835i	$-0.349181\pi$
$983$	28.6115	0.912565	0.456283	+	0.889835i	$0.349181\pi$
$984$	0	0
$985$	−2.75349	−0.0877334
$986$	0	0
$987$	29.9851	0.954438
$988$	0	0
$989$	−34.0702	−1.08337
$990$	0	0
$991$	−51.5651	−1.63802	−0.819009	−	0.573780i	$-0.805476\pi$
$991$	−51.5651	−1.63802	−0.819009	+	0.573780i	$0.805476\pi$
$992$	0	0
$993$	6.29568	0.199788
$994$	0	0
$995$	4.18689	0.132733
$996$	0	0
$997$	37.0106	1.17214	0.586069	−	0.810261i	$-0.300675\pi$
$997$	37.0106	1.17214	0.586069	+	0.810261i	$0.300675\pi$
$998$	0	0
$999$	−3.06703	−0.0970366

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9984.2.a.bv.1.4		8
4.3	odd	2		9984.2.a.bt.1.4		8
8.3	odd	2		9984.2.a.bu.1.5		8
8.5	even	2		9984.2.a.bs.1.5		8
16.3	odd	4		312.2.g.b.157.8	yes	16
16.5	even	4		1248.2.g.b.625.4		16
16.11	odd	4		312.2.g.b.157.7	✓	16
16.13	even	4		1248.2.g.b.625.13		16
48.5	odd	4		3744.2.g.e.1873.9		16
48.11	even	4		936.2.g.e.469.10		16
48.29	odd	4		3744.2.g.e.1873.8		16
48.35	even	4		936.2.g.e.469.9		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.2.g.b.157.7	✓	16	16.11	odd	4
312.2.g.b.157.8	yes	16	16.3	odd	4
936.2.g.e.469.9		16	48.35	even	4
936.2.g.e.469.10		16	48.11	even	4
1248.2.g.b.625.4		16	16.5	even	4
1248.2.g.b.625.13		16	16.13	even	4
3744.2.g.e.1873.8		16	48.29	odd	4
3744.2.g.e.1873.9		16	48.5	odd	4
9984.2.a.bs.1.5		8	8.5	even	2
9984.2.a.bt.1.4		8	4.3	odd	2
9984.2.a.bu.1.5		8	8.3	odd	2
9984.2.a.bv.1.4		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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 9984 = 2^{8} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9984.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -0.218531 q^{5} -4.47783 q^{7} +1.00000 q^{9} -3.58286 q^{11} +1.00000 q^{13} -0.218531 q^{15} -7.60367 q^{17} +4.65804 q^{19} -4.47783 q^{21} -5.21219 q^{23} -4.95224 q^{25} +1.00000 q^{27} -3.39240 q^{29} +1.71291 q^{31} -3.58286 q^{33} +0.978543 q^{35} -3.06703 q^{37} +1.00000 q^{39} -1.17387 q^{41} +6.53664 q^{43} -0.218531 q^{45} -6.69636 q^{47} +13.0509 q^{49} -7.60367 q^{51} +1.18362 q^{53} +0.782966 q^{55} +4.65804 q^{57} -9.33654 q^{59} +14.6809 q^{61} -4.47783 q^{63} -0.218531 q^{65} +2.10688 q^{67} -5.21219 q^{69} -11.9731 q^{71} +13.8518 q^{73} -4.95224 q^{75} +16.0434 q^{77} -2.12278 q^{79} +1.00000 q^{81} +11.9086 q^{83} +1.66164 q^{85} -3.39240 q^{87} +10.2015 q^{89} -4.47783 q^{91} +1.71291 q^{93} -1.01792 q^{95} +16.0227 q^{97} -3.58286 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{3} + 2 q^{5} + 2 q^{7} + 8 q^{9} + 8 q^{13} + 2 q^{15} + 8 q^{17} + 6 q^{19} + 2 q^{21} - 4 q^{23} + 16 q^{25} + 8 q^{27} + 4 q^{29} + 2 q^{31} - 4 q^{35} + 8 q^{37} + 8 q^{39} + 18 q^{41} + 16 q^{43}+ \cdots + 20 q^{97}+O(q^{100}) \) 8 * q + 8 * q^3 + 2 * q^5 + 2 * q^7 + 8 * q^9 + 8 * q^13 + 2 * q^15 + 8 * q^17 + 6 * q^19 + 2 * q^21 - 4 * q^23 + 16 * q^25 + 8 * q^27 + 4 * q^29 + 2 * q^31 - 4 * q^35 + 8 * q^37 + 8 * q^39 + 18 * q^41 + 16 * q^43 + 2 * q^45 - 12 * q^47 + 24 * q^49 + 8 * q^51 - 4 * q^53 + 12 * q^55 + 6 * q^57 - 4 * q^59 + 12 * q^61 + 2 * q^63 + 2 * q^65 + 22 * q^67 - 4 * q^69 + 16 * q^73 + 16 * q^75 - 20 * q^77 + 8 * q^81 + 16 * q^83 - 4 * q^85 + 4 * q^87 + 30 * q^89 + 2 * q^91 + 2 * q^93 + 12 * q^95 + 20 * q^97

Introduction

L-functions

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