LMFDB - Embedded newform 2816.2.c.t.1409.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2816,2,Mod(1409,2816)] 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("2816.1409"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2816, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2816 = 2^{8} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2816.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,-6,0,0,0,0,0,14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)

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

Self dual:	no
Analytic conductor:	$22.4858732092$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{17})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9x^{2} + 16 $ x^4 + 9*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 352)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1409.1
Root			$-2.56155i$ of defining polynomial
Character	$\chi$	$=$	2816.1409
Dual form			2816.2.c.t.1409.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.56155i q^{3} +0.561553i q^{5} -3.56155 q^{9} +1.00000i q^{11} +2.00000i q^{13} +1.43845 q^{15} +7.12311 q^{17} +1.12311i q^{19} -7.68466 q^{23} +4.68466 q^{25} +1.43845i q^{27} +7.12311i q^{29} +5.43845 q^{31} +2.56155 q^{33} +5.68466i q^{37} +5.12311 q^{39} +8.24621 q^{41} -1.12311i q^{43} -2.00000i q^{45} +4.00000 q^{47} -7.00000 q^{49} -18.2462i q^{51} -8.24621i q^{53} -0.561553 q^{55} +2.87689 q^{57} -0.315342i q^{59} +9.36932i q^{61} -1.12311 q^{65} -7.68466i q^{67} +19.6847i q^{69} +15.6847 q^{71} +6.00000 q^{73} -12.0000i q^{75} +13.1231 q^{79} -7.00000 q^{81} +11.3693i q^{83} +4.00000i q^{85} +18.2462 q^{87} -0.561553 q^{89} -13.9309i q^{93} -0.630683 q^{95} +5.68466 q^{97} -3.56155i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{9} + 14 q^{15} + 12 q^{17} - 6 q^{23} - 6 q^{25} + 30 q^{31} + 2 q^{33} + 4 q^{39} + 16 q^{47} - 28 q^{49} + 6 q^{55} + 28 q^{57} + 12 q^{65} + 38 q^{71} + 24 q^{73} + 36 q^{79} - 28 q^{81} + 40 q^{87}+ \cdots - 2 q^{97}+O(q^{100}) $ 4 * q - 6 * q^9 + 14 * q^15 + 12 * q^17 - 6 * q^23 - 6 * q^25 + 30 * q^31 + 2 * q^33 + 4 * q^39 + 16 * q^47 - 28 * q^49 + 6 * q^55 + 28 * q^57 + 12 * q^65 + 38 * q^71 + 24 * q^73 + 36 * q^79 - 28 * q^81 + 40 * q^87 + 6 * q^89 - 52 * q^95 - 2 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2816\mathbb{Z}\right)^\times$.

$n$	$1025$	$1541$	$2047$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	− 2.56155i	− 1.47891i	−0.673204	−	0.739457i	$-0.735083\pi$
$3$	− 2.56155i	− 1.47891i	0.673204	−	0.739457i	$-0.264917\pi$
$4$	0	0
$5$	0.561553i	0.251134i	0.992085	+	0.125567i	$0.0400750\pi$
$5$	0.561553i	0.251134i	−0.992085	+	0.125567i	$0.959925\pi$
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	0	0
$9$	−3.56155	−1.18718
$10$	0	0
$11$	1.00000i	0.301511i
$11$	1.00000i	0.301511i
$12$	0	0
$13$	2.00000i	0.554700i	0.960769	+	0.277350i	$0.0894562\pi$
$13$	2.00000i	0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$14$	0	0
$15$	1.43845	0.371405
$16$	0	0
$17$	7.12311	1.72761	0.863803	−	0.503829i	$-0.168076\pi$
$17$	7.12311	1.72761	0.863803	+	0.503829i	$0.168076\pi$
$18$	0	0
$19$	1.12311i	0.257658i	0.991667	+	0.128829i	$0.0411218\pi$
$19$	1.12311i	0.257658i	−0.991667	+	0.128829i	$0.958878\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−7.68466	−1.60236	−0.801181	−	0.598422i	$-0.795795\pi$
$23$	−7.68466	−1.60236	−0.801181	+	0.598422i	$0.795795\pi$
$24$	0	0
$25$	4.68466	0.936932
$26$	0	0
$27$	1.43845i	0.276829i
$28$	0	0
$29$	7.12311i	1.32273i	0.750065	+	0.661364i	$0.230022\pi$
$29$	7.12311i	1.32273i	−0.750065	+	0.661364i	$0.769978\pi$
$30$	0	0
$31$	5.43845	0.976774	0.488387	−	0.872627i	$-0.337585\pi$
$31$	5.43845	0.976774	0.488387	+	0.872627i	$0.337585\pi$
$32$	0	0
$33$	2.56155	0.445909
$34$	0	0
$35$	0	0
$36$	0	0
$37$	5.68466i	0.934552i	0.884111	+	0.467276i	$0.154765\pi$
$37$	5.68466i	0.934552i	−0.884111	+	0.467276i	$0.845235\pi$
$38$	0	0
$39$	5.12311	0.820353
$40$	0	0
$41$	8.24621	1.28784	0.643921	−	0.765092i	$-0.277307\pi$
$41$	8.24621	1.28784	0.643921	+	0.765092i	$0.277307\pi$
$42$	0	0
$43$	− 1.12311i	− 0.171272i	−0.996326	−	0.0856360i	$-0.972708\pi$
$43$	− 1.12311i	− 0.171272i	0.996326	−	0.0856360i	$-0.0272922\pi$
$44$	0	0
$45$	− 2.00000i	− 0.298142i
$46$	0	0
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	0	0
$51$	− 18.2462i	− 2.55498i
$52$	0	0
$53$	− 8.24621i	− 1.13270i	−0.824163	−	0.566352i	$-0.808354\pi$
$53$	− 8.24621i	− 1.13270i	0.824163	−	0.566352i	$-0.191646\pi$
$54$	0	0
$55$	−0.561553	−0.0757198
$56$	0	0
$57$	2.87689	0.381054
$58$	0	0
$59$	− 0.315342i	− 0.0410540i	−0.999789	−	0.0205270i	$-0.993466\pi$
$59$	− 0.315342i	− 0.0410540i	0.999789	−	0.0205270i	$-0.00653440\pi$
$60$	0	0
$61$	9.36932i	1.19962i	0.800143	+	0.599809i	$0.204757\pi$
$61$	9.36932i	1.19962i	−0.800143	+	0.599809i	$0.795243\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.12311	−0.139304
$66$	0	0
$67$	− 7.68466i	− 0.938830i	−0.882978	−	0.469415i	$-0.844465\pi$
$67$	− 7.68466i	− 0.938830i	0.882978	−	0.469415i	$-0.155535\pi$
$68$	0	0
$69$	19.6847i	2.36975i
$70$	0	0
$71$	15.6847	1.86143	0.930713	−	0.365750i	$-0.119187\pi$
$71$	15.6847	1.86143	0.930713	+	0.365750i	$0.119187\pi$
$72$	0	0
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	0	0
$75$	− 12.0000i	− 1.38564i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	13.1231	1.47646	0.738232	−	0.674546i	$-0.235661\pi$
$79$	13.1231	1.47646	0.738232	+	0.674546i	$0.235661\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	11.3693i	1.24794i	0.781446	+	0.623972i	$0.214482\pi$
$83$	11.3693i	1.24794i	−0.781446	+	0.623972i	$0.785518\pi$
$84$	0	0
$85$	4.00000i	0.433861i
$86$	0	0
$87$	18.2462	1.95620
$88$	0	0
$89$	−0.561553	−0.0595245	−0.0297622	−	0.999557i	$-0.509475\pi$
$89$	−0.561553	−0.0595245	−0.0297622	+	0.999557i	$0.509475\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	− 13.9309i	− 1.44456i
$94$	0	0
$95$	−0.630683	−0.0647067
$96$	0	0
$97$	5.68466	0.577190	0.288595	−	0.957451i	$-0.406812\pi$
$97$	5.68466	0.577190	0.288595	+	0.957451i	$0.406812\pi$
$98$	0	0
$99$	− 3.56155i	− 0.357950i
$100$	0	0
$101$	− 4.87689i	− 0.485269i	−0.970118	−	0.242635i	$-0.921988\pi$
$101$	− 4.87689i	− 0.485269i	0.970118	−	0.242635i	$-0.0780116\pi$
$102$	0	0
$103$	−14.2462	−1.40372	−0.701860	−	0.712314i	$-0.747647\pi$
$103$	−14.2462	−1.40372	−0.701860	+	0.712314i	$0.747647\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 12.0000i	− 1.16008i	−0.814587	−	0.580042i	$-0.803036\pi$
$107$	− 12.0000i	− 1.16008i	0.814587	−	0.580042i	$-0.196964\pi$
$108$	0	0
$109$	− 14.0000i	− 1.34096i	−0.741929	−	0.670478i	$-0.766089\pi$
$109$	− 14.0000i	− 1.34096i	0.741929	−	0.670478i	$-0.233911\pi$
$110$	0	0
$111$	14.5616	1.38212
$112$	0	0
$113$	11.4384	1.07604	0.538019	−	0.842933i	$-0.319173\pi$
$113$	11.4384	1.07604	0.538019	+	0.842933i	$0.319173\pi$
$114$	0	0
$115$	− 4.31534i	− 0.402408i
$116$	0	0
$117$	− 7.12311i	− 0.658531i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−1.00000	−0.0909091
$122$	0	0
$123$	− 21.1231i	− 1.90461i
$124$	0	0
$125$	5.43845i	0.486430i
$126$	0	0
$127$	−15.3693	−1.36381	−0.681903	−	0.731442i	$-0.738847\pi$
$127$	−15.3693	−1.36381	−0.681903	+	0.731442i	$0.738847\pi$
$128$	0	0
$129$	−2.87689	−0.253296
$130$	0	0
$131$	− 4.00000i	− 0.349482i	−0.984614	−	0.174741i	$-0.944091\pi$
$131$	− 4.00000i	− 0.349482i	0.984614	−	0.174741i	$-0.0559088\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−0.807764	−0.0695213
$136$	0	0
$137$	−13.6847	−1.16916	−0.584580	−	0.811336i	$-0.698741\pi$
$137$	−13.6847	−1.16916	−0.584580	+	0.811336i	$0.698741\pi$
$138$	0	0
$139$	12.0000i	1.01783i	0.860818	+	0.508913i	$0.169953\pi$
$139$	12.0000i	1.01783i	−0.860818	+	0.508913i	$0.830047\pi$
$140$	0	0
$141$	− 10.2462i	− 0.862887i
$142$	0	0
$143$	−2.00000	−0.167248
$144$	0	0
$145$	−4.00000	−0.332182
$146$	0	0
$147$	17.9309i	1.47891i
$148$	0	0
$149$	− 4.87689i	− 0.399531i	−0.979844	−	0.199765i	$-0.935982\pi$
$149$	− 4.87689i	− 0.399531i	0.979844	−	0.199765i	$-0.0640180\pi$
$150$	0	0
$151$	15.3693	1.25074	0.625369	−	0.780329i	$-0.284949\pi$
$151$	15.3693	1.25074	0.625369	+	0.780329i	$0.284949\pi$
$152$	0	0
$153$	−25.3693	−2.05099
$154$	0	0
$155$	3.05398i	0.245301i
$156$	0	0
$157$	1.68466i	0.134450i	0.997738	+	0.0672252i	$0.0214146\pi$
$157$	1.68466i	0.134450i	−0.997738	+	0.0672252i	$0.978585\pi$
$158$	0	0
$159$	−21.1231	−1.67517
$160$	0	0
$161$	0	0
$162$	0	0
$163$	12.0000i	0.939913i	0.882690	+	0.469956i	$0.155730\pi$
$163$	12.0000i	0.939913i	−0.882690	+	0.469956i	$0.844270\pi$
$164$	0	0
$165$	1.43845i	0.111983i
$166$	0	0
$167$	7.36932	0.570255	0.285127	−	0.958490i	$-0.407964\pi$
$167$	7.36932	0.570255	0.285127	+	0.958490i	$0.407964\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	− 4.00000i	− 0.305888i
$172$	0	0
$173$	18.0000i	1.36851i	0.729241	+	0.684257i	$0.239873\pi$
$173$	18.0000i	1.36851i	−0.729241	+	0.684257i	$0.760127\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−0.807764	−0.0607153
$178$	0	0
$179$	− 7.68466i	− 0.574378i	−0.957874	−	0.287189i	$-0.907279\pi$
$179$	− 7.68466i	− 0.574378i	0.957874	−	0.287189i	$-0.0927208\pi$
$180$	0	0
$181$	− 25.0540i	− 1.86225i	−0.364704	−	0.931124i	$-0.618830\pi$
$181$	− 25.0540i	− 1.86225i	0.364704	−	0.931124i	$-0.381170\pi$
$182$	0	0
$183$	24.0000	1.77413
$184$	0	0
$185$	−3.19224	−0.234698
$186$	0	0
$187$	7.12311i	0.520893i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−7.68466	−0.556042	−0.278021	−	0.960575i	$-0.589679\pi$
$191$	−7.68466	−0.556042	−0.278021	+	0.960575i	$0.589679\pi$
$192$	0	0
$193$	−13.3693	−0.962344	−0.481172	−	0.876626i	$-0.659789\pi$
$193$	−13.3693	−0.962344	−0.481172	+	0.876626i	$0.659789\pi$
$194$	0	0
$195$	2.87689i	0.206019i
$196$	0	0
$197$	8.24621i	0.587518i	0.955879	+	0.293759i	$0.0949064\pi$
$197$	8.24621i	0.587518i	−0.955879	+	0.293759i	$0.905094\pi$
$198$	0	0
$199$	−14.2462	−1.00989	−0.504944	−	0.863152i	$-0.668487\pi$
$199$	−14.2462	−1.00989	−0.504944	+	0.863152i	$0.668487\pi$
$200$	0	0
$201$	−19.6847	−1.38845
$202$	0	0
$203$	0	0
$204$	0	0
$205$	4.63068i	0.323421i
$206$	0	0
$207$	27.3693	1.90230
$208$	0	0
$209$	−1.12311	−0.0776868
$210$	0	0
$211$	− 16.4924i	− 1.13539i	−0.823241	−	0.567693i	$-0.807836\pi$
$211$	− 16.4924i	− 1.13539i	0.823241	−	0.567693i	$-0.192164\pi$
$212$	0	0
$213$	− 40.1771i	− 2.75289i
$214$	0	0
$215$	0.630683	0.0430122
$216$	0	0
$217$	0	0
$218$	0	0
$219$	− 15.3693i	− 1.03856i
$220$	0	0
$221$	14.2462i	0.958304i
$222$	0	0
$223$	7.68466	0.514603	0.257301	−	0.966331i	$-0.417167\pi$
$223$	7.68466	0.514603	0.257301	+	0.966331i	$0.417167\pi$
$224$	0	0
$225$	−16.6847	−1.11231
$226$	0	0
$227$	− 20.0000i	− 1.32745i	−0.747978	−	0.663723i	$-0.768975\pi$
$227$	− 20.0000i	− 1.32745i	0.747978	−	0.663723i	$-0.231025\pi$
$228$	0	0
$229$	21.6847i	1.43296i	0.697606	+	0.716481i	$0.254248\pi$
$229$	21.6847i	1.43296i	−0.697606	+	0.716481i	$0.745752\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−7.12311	−0.466650	−0.233325	−	0.972399i	$-0.574961\pi$
$233$	−7.12311	−0.466650	−0.233325	+	0.972399i	$0.574961\pi$
$234$	0	0
$235$	2.24621i	0.146527i
$236$	0	0
$237$	− 33.6155i	− 2.18356i
$238$	0	0
$239$	0.630683	0.0407955	0.0203977	−	0.999792i	$-0.493507\pi$
$239$	0.630683	0.0407955	0.0203977	+	0.999792i	$0.493507\pi$
$240$	0	0
$241$	2.63068	0.169457	0.0847286	−	0.996404i	$-0.472998\pi$
$241$	2.63068	0.169457	0.0847286	+	0.996404i	$0.472998\pi$
$242$	0	0
$243$	22.2462i	1.42710i
$244$	0	0
$245$	− 3.93087i	− 0.251134i
$246$	0	0
$247$	−2.24621	−0.142923
$248$	0	0
$249$	29.1231	1.84560
$250$	0	0
$251$	− 23.0540i	− 1.45515i	−0.686026	−	0.727577i	$-0.740646\pi$
$251$	− 23.0540i	− 1.45515i	0.686026	−	0.727577i	$-0.259354\pi$
$252$	0	0
$253$	− 7.68466i	− 0.483130i
$254$	0	0
$255$	10.2462	0.641643
$256$	0	0
$257$	−8.24621	−0.514385	−0.257192	−	0.966360i	$-0.582797\pi$
$257$	−8.24621	−0.514385	−0.257192	+	0.966360i	$0.582797\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	− 25.3693i	− 1.57032i
$262$	0	0
$263$	8.00000	0.493301	0.246651	−	0.969104i	$-0.420670\pi$
$263$	8.00000	0.493301	0.246651	+	0.969104i	$0.420670\pi$
$264$	0	0
$265$	4.63068	0.284461
$266$	0	0
$267$	1.43845i	0.0880315i
$268$	0	0
$269$	10.4924i	0.639734i	0.947462	+	0.319867i	$0.103638\pi$
$269$	10.4924i	0.639734i	−0.947462	+	0.319867i	$0.896362\pi$
$270$	0	0
$271$	13.1231	0.797172	0.398586	−	0.917131i	$-0.369501\pi$
$271$	13.1231	0.797172	0.398586	+	0.917131i	$0.369501\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4.68466i	0.282496i
$276$	0	0
$277$	− 2.00000i	− 0.120168i	−0.998193	−	0.0600842i	$-0.980863\pi$
$277$	− 2.00000i	− 0.120168i	0.998193	−	0.0600842i	$-0.0191369\pi$
$278$	0	0
$279$	−19.3693	−1.15961
$280$	0	0
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	0	0
$283$	9.75379i	0.579803i	0.957057	+	0.289901i	$0.0936225\pi$
$283$	9.75379i	0.579803i	−0.957057	+	0.289901i	$0.906378\pi$
$284$	0	0
$285$	1.61553i	0.0956956i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	33.7386	1.98463
$290$	0	0
$291$	− 14.5616i	− 0.853613i
$292$	0	0
$293$	10.4924i	0.612974i	0.951875	+	0.306487i	$0.0991536\pi$
$293$	10.4924i	0.612974i	−0.951875	+	0.306487i	$0.900846\pi$
$294$	0	0
$295$	0.177081	0.0103101
$296$	0	0
$297$	−1.43845	−0.0834672
$298$	0	0
$299$	− 15.3693i	− 0.888831i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−12.4924	−0.717671
$304$	0	0
$305$	−5.26137	−0.301265
$306$	0	0
$307$	− 27.3693i	− 1.56205i	−0.624500	−	0.781025i	$-0.714697\pi$
$307$	− 27.3693i	− 1.56205i	0.624500	−	0.781025i	$-0.285303\pi$
$308$	0	0
$309$	36.4924i	2.07598i
$310$	0	0
$311$	18.7386	1.06257	0.531285	−	0.847193i	$-0.321709\pi$
$311$	18.7386	1.06257	0.531285	+	0.847193i	$0.321709\pi$
$312$	0	0
$313$	−6.31534	−0.356964	−0.178482	−	0.983943i	$-0.557119\pi$
$313$	−6.31534	−0.356964	−0.178482	+	0.983943i	$0.557119\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 29.0540i	− 1.63183i	−0.578169	−	0.815917i	$-0.696233\pi$
$317$	− 29.0540i	− 1.63183i	0.578169	−	0.815917i	$-0.303767\pi$
$318$	0	0
$319$	−7.12311	−0.398817
$320$	0	0
$321$	−30.7386	−1.71566
$322$	0	0
$323$	8.00000i	0.445132i
$324$	0	0
$325$	9.36932i	0.519716i
$326$	0	0
$327$	−35.8617	−1.98316
$328$	0	0
$329$	0	0
$330$	0	0
$331$	16.3153i	0.896772i	0.893840	+	0.448386i	$0.148001\pi$
$331$	16.3153i	0.896772i	−0.893840	+	0.448386i	$0.851999\pi$
$332$	0	0
$333$	− 20.2462i	− 1.10949i
$334$	0	0
$335$	4.31534	0.235772
$336$	0	0
$337$	−6.63068	−0.361196	−0.180598	−	0.983557i	$-0.557803\pi$
$337$	−6.63068	−0.361196	−0.180598	+	0.983557i	$0.557803\pi$
$338$	0	0
$339$	− 29.3002i	− 1.59137i
$340$	0	0
$341$	5.43845i	0.294508i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−11.0540	−0.595126
$346$	0	0
$347$	27.3693i	1.46926i	0.678467	+	0.734631i	$0.262645\pi$
$347$	27.3693i	1.46926i	−0.678467	+	0.734631i	$0.737355\pi$
$348$	0	0
$349$	− 13.3693i	− 0.715643i	−0.933790	−	0.357822i	$-0.883520\pi$
$349$	− 13.3693i	− 0.715643i	0.933790	−	0.357822i	$-0.116480\pi$
$350$	0	0
$351$	−2.87689	−0.153557
$352$	0	0
$353$	24.5616	1.30728	0.653640	−	0.756806i	$-0.273241\pi$
$353$	24.5616	1.30728	0.653640	+	0.756806i	$0.273241\pi$
$354$	0	0
$355$	8.80776i	0.467468i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	15.3693	0.811162	0.405581	−	0.914059i	$-0.367069\pi$
$359$	15.3693	0.811162	0.405581	+	0.914059i	$0.367069\pi$
$360$	0	0
$361$	17.7386	0.933612
$362$	0	0
$363$	2.56155i	0.134447i
$364$	0	0
$365$	3.36932i	0.176358i
$366$	0	0
$367$	0.946025	0.0493821	0.0246910	−	0.999695i	$-0.492140\pi$
$367$	0.946025	0.0493821	0.0246910	+	0.999695i	$0.492140\pi$
$368$	0	0
$369$	−29.3693	−1.52891
$370$	0	0
$371$	0	0
$372$	0	0
$373$	21.3693i	1.10646i	0.833028	+	0.553231i	$0.186605\pi$
$373$	21.3693i	1.10646i	−0.833028	+	0.553231i	$0.813395\pi$
$374$	0	0
$375$	13.9309	0.719387
$376$	0	0
$377$	−14.2462	−0.733717
$378$	0	0
$379$	18.5616i	0.953443i	0.879054	+	0.476721i	$0.158175\pi$
$379$	18.5616i	0.953443i	−0.879054	+	0.476721i	$0.841825\pi$
$380$	0	0
$381$	39.3693i	2.01695i
$382$	0	0
$383$	−8.31534	−0.424894	−0.212447	−	0.977173i	$-0.568143\pi$
$383$	−8.31534	−0.424894	−0.212447	+	0.977173i	$0.568143\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	4.00000i	0.203331i
$388$	0	0
$389$	26.8078i	1.35921i	0.733579	+	0.679604i	$0.237848\pi$
$389$	26.8078i	1.35921i	−0.733579	+	0.679604i	$0.762152\pi$
$390$	0	0
$391$	−54.7386	−2.76825
$392$	0	0
$393$	−10.2462	−0.516853
$394$	0	0
$395$	7.36932i	0.370791i
$396$	0	0
$397$	14.0000i	0.702640i	0.936255	+	0.351320i	$0.114267\pi$
$397$	14.0000i	0.702640i	−0.936255	+	0.351320i	$0.885733\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	22.4924	1.12322	0.561609	−	0.827403i	$-0.310183\pi$
$401$	22.4924	1.12322	0.561609	+	0.827403i	$0.310183\pi$
$402$	0	0
$403$	10.8769i	0.541817i
$404$	0	0
$405$	− 3.93087i	− 0.195326i
$406$	0	0
$407$	−5.68466	−0.281778
$408$	0	0
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	0	0
$411$	35.0540i	1.72909i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−6.38447	−0.313401
$416$	0	0
$417$	30.7386	1.50528
$418$	0	0
$419$	18.7386i	0.915442i	0.889096	+	0.457721i	$0.151334\pi$
$419$	18.7386i	0.915442i	−0.889096	+	0.457721i	$0.848666\pi$
$420$	0	0
$421$	10.0000i	0.487370i	0.969854	+	0.243685i	$0.0783563\pi$
$421$	10.0000i	0.487370i	−0.969854	+	0.243685i	$0.921644\pi$
$422$	0	0
$423$	−14.2462	−0.692674
$424$	0	0
$425$	33.3693	1.61865
$426$	0	0
$427$	0	0
$428$	0	0
$429$	5.12311i	0.247346i
$430$	0	0
$431$	38.7386	1.86597	0.932987	−	0.359910i	$-0.117192\pi$
$431$	38.7386	1.86597	0.932987	+	0.359910i	$0.117192\pi$
$432$	0	0
$433$	5.68466	0.273187	0.136594	−	0.990627i	$-0.456385\pi$
$433$	5.68466	0.273187	0.136594	+	0.990627i	$0.456385\pi$
$434$	0	0
$435$	10.2462i	0.491268i
$436$	0	0
$437$	− 8.63068i	− 0.412862i
$438$	0	0
$439$	−26.2462	−1.25266	−0.626332	−	0.779557i	$-0.715444\pi$
$439$	−26.2462	−1.25266	−0.626332	+	0.779557i	$0.715444\pi$
$440$	0	0
$441$	24.9309	1.18718
$442$	0	0
$443$	− 8.31534i	− 0.395074i	−0.980295	−	0.197537i	$-0.936706\pi$
$443$	− 8.31534i	− 0.395074i	0.980295	−	0.197537i	$-0.0632942\pi$
$444$	0	0
$445$	− 0.315342i	− 0.0149486i
$446$	0	0
$447$	−12.4924	−0.590871
$448$	0	0
$449$	2.80776	0.132507	0.0662533	−	0.997803i	$-0.478895\pi$
$449$	2.80776	0.132507	0.0662533	+	0.997803i	$0.478895\pi$
$450$	0	0
$451$	8.24621i	0.388299i
$452$	0	0
$453$	− 39.3693i	− 1.84973i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−41.3693	−1.93518	−0.967588	−	0.252536i	$-0.918735\pi$
$457$	−41.3693	−1.93518	−0.967588	+	0.252536i	$0.918735\pi$
$458$	0	0
$459$	10.2462i	0.478252i
$460$	0	0
$461$	− 19.1231i	− 0.890652i	−0.895369	−	0.445326i	$-0.853088\pi$
$461$	− 19.1231i	− 0.890652i	0.895369	−	0.445326i	$-0.146912\pi$
$462$	0	0
$463$	−33.9309	−1.57690	−0.788451	−	0.615098i	$-0.789116\pi$
$463$	−33.9309	−1.57690	−0.788451	+	0.615098i	$0.789116\pi$
$464$	0	0
$465$	7.82292	0.362779
$466$	0	0
$467$	8.31534i	0.384788i	0.981318	+	0.192394i	$0.0616252\pi$
$467$	8.31534i	0.384788i	−0.981318	+	0.192394i	$0.938375\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	4.31534	0.198840
$472$	0	0
$473$	1.12311	0.0516405
$474$	0	0
$475$	5.26137i	0.241408i
$476$	0	0
$477$	29.3693i	1.34473i
$478$	0	0
$479$	−0.630683	−0.0288166	−0.0144083	−	0.999896i	$-0.504586\pi$
$479$	−0.630683	−0.0288166	−0.0144083	+	0.999896i	$0.504586\pi$
$480$	0	0
$481$	−11.3693	−0.518396
$482$	0	0
$483$	0	0
$484$	0	0
$485$	3.19224i	0.144952i
$486$	0	0
$487$	−31.6847	−1.43577	−0.717884	−	0.696162i	$-0.754889\pi$
$487$	−31.6847	−1.43577	−0.717884	+	0.696162i	$0.754889\pi$
$488$	0	0
$489$	30.7386	1.39005
$490$	0	0
$491$	− 26.7386i	− 1.20670i	−0.797477	−	0.603349i	$-0.793833\pi$
$491$	− 26.7386i	− 1.20670i	0.797477	−	0.603349i	$-0.206167\pi$
$492$	0	0
$493$	50.7386i	2.28515i
$494$	0	0
$495$	2.00000	0.0898933
$496$	0	0
$497$	0	0
$498$	0	0
$499$	40.4924i	1.81269i	0.422539	+	0.906345i	$0.361139\pi$
$499$	40.4924i	1.81269i	−0.422539	+	0.906345i	$0.638861\pi$
$500$	0	0
$501$	− 18.8769i	− 0.843357i
$502$	0	0
$503$	15.3693	0.685284	0.342642	−	0.939466i	$-0.388678\pi$
$503$	15.3693	0.685284	0.342642	+	0.939466i	$0.388678\pi$
$504$	0	0
$505$	2.73863	0.121868
$506$	0	0
$507$	− 23.0540i	− 1.02386i
$508$	0	0
$509$	− 13.6847i	− 0.606562i	−0.952901	−	0.303281i	$-0.901918\pi$
$509$	− 13.6847i	− 0.606562i	0.952901	−	0.303281i	$-0.0980821\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−1.61553	−0.0713273
$514$	0	0
$515$	− 8.00000i	− 0.352522i
$516$	0	0
$517$	4.00000i	0.175920i
$518$	0	0
$519$	46.1080	2.02391
$520$	0	0
$521$	17.0540	0.747148	0.373574	−	0.927600i	$-0.378132\pi$
$521$	17.0540	0.747148	0.373574	+	0.927600i	$0.378132\pi$
$522$	0	0
$523$	34.1080i	1.49144i	0.666261	+	0.745718i	$0.267894\pi$
$523$	34.1080i	1.49144i	−0.666261	+	0.745718i	$0.732106\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	38.7386	1.68748
$528$	0	0
$529$	36.0540	1.56756
$530$	0	0
$531$	1.12311i	0.0487386i
$532$	0	0
$533$	16.4924i	0.714366i
$534$	0	0
$535$	6.73863	0.291337
$536$	0	0
$537$	−19.6847	−0.849456
$538$	0	0
$539$	− 7.00000i	− 0.301511i
$540$	0	0
$541$	18.0000i	0.773880i	0.922105	+	0.386940i	$0.126468\pi$
$541$	18.0000i	0.773880i	−0.922105	+	0.386940i	$0.873532\pi$
$542$	0	0
$543$	−64.1771	−2.75410
$544$	0	0
$545$	7.86174	0.336760
$546$	0	0
$547$	− 12.0000i	− 0.513083i	−0.966533	−	0.256541i	$-0.917417\pi$
$547$	− 12.0000i	− 0.513083i	0.966533	−	0.256541i	$-0.0825830\pi$
$548$	0	0
$549$	− 33.3693i	− 1.42417i
$550$	0	0
$551$	−8.00000	−0.340811
$552$	0	0
$553$	0	0
$554$	0	0
$555$	8.17708i	0.347098i
$556$	0	0
$557$	33.3693i	1.41390i	0.707262	+	0.706952i	$0.249930\pi$
$557$	33.3693i	1.41390i	−0.707262	+	0.706952i	$0.750070\pi$
$558$	0	0
$559$	2.24621	0.0950046
$560$	0	0
$561$	18.2462	0.770356
$562$	0	0
$563$	− 18.7386i	− 0.789739i	−0.918737	−	0.394870i	$-0.870790\pi$
$563$	− 18.7386i	− 0.789739i	0.918737	−	0.394870i	$-0.129210\pi$
$564$	0	0
$565$	6.42329i	0.270230i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	21.3693	0.895848	0.447924	−	0.894072i	$-0.352163\pi$
$569$	21.3693	0.895848	0.447924	+	0.894072i	$0.352163\pi$
$570$	0	0
$571$	12.0000i	0.502184i	0.967963	+	0.251092i	$0.0807897\pi$
$571$	12.0000i	0.502184i	−0.967963	+	0.251092i	$0.919210\pi$
$572$	0	0
$573$	19.6847i	0.822338i
$574$	0	0
$575$	−36.0000	−1.50130
$576$	0	0
$577$	5.68466	0.236655	0.118328	−	0.992975i	$-0.462247\pi$
$577$	5.68466	0.236655	0.118328	+	0.992975i	$0.462247\pi$
$578$	0	0
$579$	34.2462i	1.42322i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	8.24621	0.341523
$584$	0	0
$585$	4.00000	0.165380
$586$	0	0
$587$	28.0000i	1.15568i	0.816149	+	0.577842i	$0.196105\pi$
$587$	28.0000i	1.15568i	−0.816149	+	0.577842i	$0.803895\pi$
$588$	0	0
$589$	6.10795i	0.251674i
$590$	0	0
$591$	21.1231	0.868888
$592$	0	0
$593$	−25.8617	−1.06201	−0.531007	−	0.847367i	$-0.678186\pi$
$593$	−25.8617	−1.06201	−0.531007	+	0.847367i	$0.678186\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	36.4924i	1.49354i
$598$	0	0
$599$	−34.7386	−1.41938	−0.709691	−	0.704513i	$-0.751165\pi$
$599$	−34.7386	−1.41938	−0.709691	+	0.704513i	$0.751165\pi$
$600$	0	0
$601$	−32.1080	−1.30971	−0.654855	−	0.755754i	$-0.727270\pi$
$601$	−32.1080	−1.30971	−0.654855	+	0.755754i	$0.727270\pi$
$602$	0	0
$603$	27.3693i	1.11456i
$604$	0	0
$605$	− 0.561553i	− 0.0228304i
$606$	0	0
$607$	−37.1231	−1.50678	−0.753390	−	0.657574i	$-0.771583\pi$
$607$	−37.1231	−1.50678	−0.753390	+	0.657574i	$0.771583\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	8.00000i	0.323645i
$612$	0	0
$613$	− 17.3693i	− 0.701540i	−0.936462	−	0.350770i	$-0.885920\pi$
$613$	− 17.3693i	− 0.701540i	0.936462	−	0.350770i	$-0.114080\pi$
$614$	0	0
$615$	11.8617	0.478311
$616$	0	0
$617$	32.2462	1.29818	0.649092	−	0.760710i	$-0.275149\pi$
$617$	32.2462	1.29818	0.649092	+	0.760710i	$0.275149\pi$
$618$	0	0
$619$	− 7.68466i	− 0.308873i	−0.988003	−	0.154436i	$-0.950644\pi$
$619$	− 7.68466i	− 0.308873i	0.988003	−	0.154436i	$-0.0493561\pi$
$620$	0	0
$621$	− 11.0540i	− 0.443581i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	20.3693	0.814773
$626$	0	0
$627$	2.87689i	0.114892i
$628$	0	0
$629$	40.4924i	1.61454i
$630$	0	0
$631$	−38.4233	−1.52961	−0.764804	−	0.644264i	$-0.777164\pi$
$631$	−38.4233	−1.52961	−0.764804	+	0.644264i	$0.777164\pi$
$632$	0	0
$633$	−42.2462	−1.67914
$634$	0	0
$635$	− 8.63068i	− 0.342498i
$636$	0	0
$637$	− 14.0000i	− 0.554700i
$638$	0	0
$639$	−55.8617	−2.20986
$640$	0	0
$641$	−27.9309	−1.10320	−0.551602	−	0.834108i	$-0.685983\pi$
$641$	−27.9309	−1.10320	−0.551602	+	0.834108i	$0.685983\pi$
$642$	0	0
$643$	42.5616i	1.67846i	0.543774	+	0.839232i	$0.316995\pi$
$643$	42.5616i	1.67846i	−0.543774	+	0.839232i	$0.683005\pi$
$644$	0	0
$645$	− 1.61553i	− 0.0636114i
$646$	0	0
$647$	31.6847	1.24565	0.622826	−	0.782360i	$-0.285984\pi$
$647$	31.6847	1.24565	0.622826	+	0.782360i	$0.285984\pi$
$648$	0	0
$649$	0.315342	0.0123782
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 13.6847i	− 0.535522i	−0.963485	−	0.267761i	$-0.913716\pi$
$653$	− 13.6847i	− 0.535522i	0.963485	−	0.267761i	$-0.0862838\pi$
$654$	0	0
$655$	2.24621	0.0877667
$656$	0	0
$657$	−21.3693	−0.833696
$658$	0	0
$659$	12.6307i	0.492022i	0.969267	+	0.246011i	$0.0791199\pi$
$659$	12.6307i	0.492022i	−0.969267	+	0.246011i	$0.920880\pi$
$660$	0	0
$661$	21.0540i	0.818905i	0.912331	+	0.409452i	$0.134280\pi$
$661$	21.0540i	0.818905i	−0.912331	+	0.409452i	$0.865720\pi$
$662$	0	0
$663$	36.4924	1.41725
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 54.7386i	− 2.11949i
$668$	0	0
$669$	− 19.6847i	− 0.761053i
$670$	0	0
$671$	−9.36932	−0.361698
$672$	0	0
$673$	24.7386	0.953604	0.476802	−	0.879011i	$-0.341796\pi$
$673$	24.7386	0.953604	0.476802	+	0.879011i	$0.341796\pi$
$674$	0	0
$675$	6.73863i	0.259370i
$676$	0	0
$677$	− 37.8617i	− 1.45514i	−0.686031	−	0.727572i	$-0.740649\pi$
$677$	− 37.8617i	− 1.45514i	0.686031	−	0.727572i	$-0.259351\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−51.2311	−1.96318
$682$	0	0
$683$	− 12.0000i	− 0.459167i	−0.973289	−	0.229584i	$-0.926264\pi$
$683$	− 12.0000i	− 0.459167i	0.973289	−	0.229584i	$-0.0737364\pi$
$684$	0	0
$685$	− 7.68466i	− 0.293616i
$686$	0	0
$687$	55.5464	2.11923
$688$	0	0
$689$	16.4924	0.628311
$690$	0	0
$691$	25.3002i	0.962464i	0.876593	+	0.481232i	$0.159811\pi$
$691$	25.3002i	0.962464i	−0.876593	+	0.481232i	$0.840189\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−6.73863	−0.255611
$696$	0	0
$697$	58.7386	2.22488
$698$	0	0
$699$	18.2462i	0.690135i
$700$	0	0
$701$	− 47.6155i	− 1.79841i	−0.437524	−	0.899207i	$-0.644144\pi$
$701$	− 47.6155i	− 1.79841i	0.437524	−	0.899207i	$-0.355856\pi$
$702$	0	0
$703$	−6.38447	−0.240795
$704$	0	0
$705$	5.75379	0.216700
$706$	0	0
$707$	0	0
$708$	0	0
$709$	− 10.3153i	− 0.387401i	−0.981061	−	0.193700i	$-0.937951\pi$
$709$	− 10.3153i	− 0.387401i	0.981061	−	0.193700i	$-0.0620490\pi$
$710$	0	0
$711$	−46.7386	−1.75284
$712$	0	0
$713$	−41.7926	−1.56515
$714$	0	0
$715$	− 1.12311i	− 0.0420018i
$716$	0	0
$717$	− 1.61553i	− 0.0603330i
$718$	0	0
$719$	−16.3153	−0.608460	−0.304230	−	0.952599i	$-0.598399\pi$
$719$	−16.3153	−0.608460	−0.304230	+	0.952599i	$0.598399\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	− 6.73863i	− 0.250612i
$724$	0	0
$725$	33.3693i	1.23931i
$726$	0	0
$727$	16.3153	0.605103	0.302551	−	0.953133i	$-0.402162\pi$
$727$	16.3153	0.605103	0.302551	+	0.953133i	$0.402162\pi$
$728$	0	0
$729$	35.9848	1.33277
$730$	0	0
$731$	− 8.00000i	− 0.295891i
$732$	0	0
$733$	− 36.1080i	− 1.33368i	−0.745202	−	0.666839i	$-0.767647\pi$
$733$	− 36.1080i	− 1.33368i	0.745202	−	0.666839i	$-0.232353\pi$
$734$	0	0
$735$	−10.0691	−0.371405
$736$	0	0
$737$	7.68466	0.283068
$738$	0	0
$739$	− 29.6155i	− 1.08942i	−0.838623	−	0.544712i	$-0.816639\pi$
$739$	− 29.6155i	− 1.08942i	0.838623	−	0.544712i	$-0.183361\pi$
$740$	0	0
$741$	5.75379i	0.211371i
$742$	0	0
$743$	−30.7386	−1.12769	−0.563846	−	0.825880i	$-0.690679\pi$
$743$	−30.7386	−1.12769	−0.563846	+	0.825880i	$0.690679\pi$
$744$	0	0
$745$	2.73863	0.100336
$746$	0	0
$747$	− 40.4924i	− 1.48154i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−12.1771	−0.444348	−0.222174	−	0.975007i	$-0.571315\pi$
$751$	−12.1771	−0.444348	−0.222174	+	0.975007i	$0.571315\pi$
$752$	0	0
$753$	−59.0540	−2.15205
$754$	0	0
$755$	8.63068i	0.314103i
$756$	0	0
$757$	10.0000i	0.363456i	0.983349	+	0.181728i	$0.0581691\pi$
$757$	10.0000i	0.363456i	−0.983349	+	0.181728i	$0.941831\pi$
$758$	0	0
$759$	−19.6847	−0.714508
$760$	0	0
$761$	3.75379	0.136075	0.0680374	−	0.997683i	$-0.478326\pi$
$761$	3.75379	0.136075	0.0680374	+	0.997683i	$0.478326\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	− 14.2462i	− 0.515073i
$766$	0	0
$767$	0.630683	0.0227726
$768$	0	0
$769$	−38.0000	−1.37032	−0.685158	−	0.728395i	$-0.740267\pi$
$769$	−38.0000	−1.37032	−0.685158	+	0.728395i	$0.740267\pi$
$770$	0	0
$771$	21.1231i	0.760730i
$772$	0	0
$773$	− 8.24621i	− 0.296596i	−0.988943	−	0.148298i	$-0.952621\pi$
$773$	− 8.24621i	− 0.296596i	0.988943	−	0.148298i	$-0.0473794\pi$
$774$	0	0
$775$	25.4773	0.915170
$776$	0	0
$777$	0	0
$778$	0	0
$779$	9.26137i	0.331823i
$780$	0	0
$781$	15.6847i	0.561241i
$782$	0	0
$783$	−10.2462	−0.366170
$784$	0	0
$785$	−0.946025	−0.0337651
$786$	0	0
$787$	12.0000i	0.427754i	0.976861	+	0.213877i	$0.0686091\pi$
$787$	12.0000i	0.427754i	−0.976861	+	0.213877i	$0.931391\pi$
$788$	0	0
$789$	− 20.4924i	− 0.729550i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−18.7386	−0.665428
$794$	0	0
$795$	− 11.8617i	− 0.420693i
$796$	0	0
$797$	− 24.5616i	− 0.870015i	−0.900427	−	0.435007i	$-0.856746\pi$
$797$	− 24.5616i	− 0.870015i	0.900427	−	0.435007i	$-0.143254\pi$
$798$	0	0
$799$	28.4924	1.00799
$800$	0	0
$801$	2.00000	0.0706665
$802$	0	0
$803$	6.00000i	0.211735i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	26.8769	0.946111
$808$	0	0
$809$	8.24621	0.289921	0.144961	−	0.989437i	$-0.453694\pi$
$809$	8.24621	0.289921	0.144961	+	0.989437i	$0.453694\pi$
$810$	0	0
$811$	51.3693i	1.80382i	0.431923	+	0.901910i	$0.357835\pi$
$811$	51.3693i	1.80382i	−0.431923	+	0.901910i	$0.642165\pi$
$812$	0	0
$813$	− 33.6155i	− 1.17895i
$814$	0	0
$815$	−6.73863	−0.236044
$816$	0	0
$817$	1.26137	0.0441296
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 0.738634i	− 0.0257785i	−0.999917	−	0.0128892i	$-0.995897\pi$
$821$	− 0.738634i	− 0.0257785i	0.999917	−	0.0128892i	$-0.00410289\pi$
$822$	0	0
$823$	18.5616	0.647015	0.323508	−	0.946226i	$-0.395138\pi$
$823$	18.5616	0.647015	0.323508	+	0.946226i	$0.395138\pi$
$824$	0	0
$825$	12.0000	0.417786
$826$	0	0
$827$	3.36932i	0.117163i	0.998283	+	0.0585813i	$0.0186577\pi$
$827$	3.36932i	0.117163i	−0.998283	+	0.0585813i	$0.981342\pi$
$828$	0	0
$829$	17.6847i	0.614214i	0.951675	+	0.307107i	$0.0993609\pi$
$829$	17.6847i	0.614214i	−0.951675	+	0.307107i	$0.900639\pi$
$830$	0	0
$831$	−5.12311	−0.177719
$832$	0	0
$833$	−49.8617	−1.72761
$834$	0	0
$835$	4.13826i	0.143210i
$836$	0	0
$837$	7.82292i	0.270400i
$838$	0	0
$839$	−30.4233	−1.05033	−0.525164	−	0.851001i	$-0.675996\pi$
$839$	−30.4233	−1.05033	−0.525164	+	0.851001i	$0.675996\pi$
$840$	0	0
$841$	−21.7386	−0.749608
$842$	0	0
$843$	− 15.3693i	− 0.529347i
$844$	0	0
$845$	5.05398i	0.173862i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	24.9848	0.857478
$850$	0	0
$851$	− 43.6847i	− 1.49749i
$852$	0	0
$853$	− 40.7386i	− 1.39486i	−0.716651	−	0.697432i	$-0.754326\pi$
$853$	− 40.7386i	− 1.39486i	0.716651	−	0.697432i	$-0.245674\pi$
$854$	0	0
$855$	2.24621	0.0768188
$856$	0	0
$857$	−4.87689	−0.166592	−0.0832958	−	0.996525i	$-0.526545\pi$
$857$	−4.87689	−0.166592	−0.0832958	+	0.996525i	$0.526545\pi$
$858$	0	0
$859$	57.9309i	1.97658i	0.152601	+	0.988288i	$0.451235\pi$
$859$	57.9309i	1.97658i	−0.152601	+	0.988288i	$0.548765\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−18.7386	−0.637871	−0.318935	−	0.947777i	$-0.603325\pi$
$863$	−18.7386	−0.637871	−0.318935	+	0.947777i	$0.603325\pi$
$864$	0	0
$865$	−10.1080	−0.343681
$866$	0	0
$867$	− 86.4233i	− 2.93509i
$868$	0	0
$869$	13.1231i	0.445171i
$870$	0	0
$871$	15.3693	0.520769
$872$	0	0
$873$	−20.2462	−0.685230
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 7.26137i	− 0.245199i	−0.992456	−	0.122599i	$-0.960877\pi$
$877$	− 7.26137i	− 0.245199i	0.992456	−	0.122599i	$-0.0391230\pi$
$878$	0	0
$879$	26.8769	0.906535
$880$	0	0
$881$	−19.3002	−0.650240	−0.325120	−	0.945673i	$-0.605405\pi$
$881$	−19.3002	−0.650240	−0.325120	+	0.945673i	$0.605405\pi$
$882$	0	0
$883$	36.0000i	1.21150i	0.795656	+	0.605748i	$0.207126\pi$
$883$	36.0000i	1.21150i	−0.795656	+	0.605748i	$0.792874\pi$
$884$	0	0
$885$	− 0.453602i	− 0.0152477i
$886$	0	0
$887$	22.1080	0.742312	0.371156	−	0.928570i	$-0.378961\pi$
$887$	22.1080	0.742312	0.371156	+	0.928570i	$0.378961\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	− 7.00000i	− 0.234509i
$892$	0	0
$893$	4.49242i	0.150333i
$894$	0	0
$895$	4.31534	0.144246
$896$	0	0
$897$	−39.3693	−1.31450
$898$	0	0
$899$	38.7386i	1.29201i
$900$	0	0
$901$	− 58.7386i	− 1.95687i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	14.0691	0.467674
$906$	0	0
$907$	− 16.4924i	− 0.547622i	−0.961784	−	0.273811i	$-0.911716\pi$
$907$	− 16.4924i	− 0.547622i	0.961784	−	0.273811i	$-0.0882843\pi$
$908$	0	0
$909$	17.3693i	0.576104i
$910$	0	0
$911$	−26.7386	−0.885890	−0.442945	−	0.896549i	$-0.646066\pi$
$911$	−26.7386	−0.885890	−0.442945	+	0.896549i	$0.646066\pi$
$912$	0	0
$913$	−11.3693	−0.376269
$914$	0	0
$915$	13.4773i	0.445545i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	17.6155	0.581083	0.290541	−	0.956862i	$-0.406165\pi$
$919$	17.6155	0.581083	0.290541	+	0.956862i	$0.406165\pi$
$920$	0	0
$921$	−70.1080	−2.31014
$922$	0	0
$923$	31.3693i	1.03253i
$924$	0	0
$925$	26.6307i	0.875611i
$926$	0	0
$927$	50.7386	1.66648
$928$	0	0
$929$	22.4924	0.737952	0.368976	−	0.929439i	$-0.379708\pi$
$929$	22.4924	0.737952	0.368976	+	0.929439i	$0.379708\pi$
$930$	0	0
$931$	− 7.86174i	− 0.257658i
$932$	0	0
$933$	− 48.0000i	− 1.57145i
$934$	0	0
$935$	−4.00000	−0.130814
$936$	0	0
$937$	20.7386	0.677502	0.338751	−	0.940876i	$-0.389996\pi$
$937$	20.7386	0.677502	0.338751	+	0.940876i	$0.389996\pi$
$938$	0	0
$939$	16.1771i	0.527919i
$940$	0	0
$941$	− 8.24621i	− 0.268819i	−0.990926	−	0.134409i	$-0.957086\pi$
$941$	− 8.24621i	− 0.268819i	0.990926	−	0.134409i	$-0.0429137\pi$
$942$	0	0
$943$	−63.3693	−2.06359
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 31.6847i	− 1.02961i	−0.857306	−	0.514807i	$-0.827864\pi$
$947$	− 31.6847i	− 1.02961i	0.857306	−	0.514807i	$-0.172136\pi$
$948$	0	0
$949$	12.0000i	0.389536i
$950$	0	0
$951$	−74.4233	−2.41334
$952$	0	0
$953$	47.6155	1.54242	0.771209	−	0.636582i	$-0.219652\pi$
$953$	47.6155	1.54242	0.771209	+	0.636582i	$0.219652\pi$
$954$	0	0
$955$	− 4.31534i	− 0.139641i
$956$	0	0
$957$	18.2462i	0.589816i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−1.42329	−0.0459127
$962$	0	0
$963$	42.7386i	1.37723i
$964$	0	0
$965$	− 7.50758i	− 0.241677i
$966$	0	0
$967$	−30.7386	−0.988488	−0.494244	−	0.869323i	$-0.664555\pi$
$967$	−30.7386	−0.988488	−0.494244	+	0.869323i	$0.664555\pi$
$968$	0	0
$969$	20.4924	0.658311
$970$	0	0
$971$	16.3153i	0.523584i	0.965124	+	0.261792i	$0.0843135\pi$
$971$	16.3153i	0.523584i	−0.965124	+	0.261792i	$0.915687\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	24.0000	0.768615
$976$	0	0
$977$	−50.0388	−1.60088	−0.800442	−	0.599410i	$-0.795402\pi$
$977$	−50.0388	−1.60088	−0.800442	+	0.599410i	$0.795402\pi$
$978$	0	0
$979$	− 0.561553i	− 0.0179473i
$980$	0	0
$981$	49.8617i	1.59196i
$982$	0	0
$983$	16.3153	0.520379	0.260189	−	0.965558i	$-0.416215\pi$
$983$	16.3153	0.520379	0.260189	+	0.965558i	$0.416215\pi$
$984$	0	0
$985$	−4.63068	−0.147546
$986$	0	0
$987$	0	0
$988$	0	0
$989$	8.63068i	0.274440i
$990$	0	0
$991$	−5.26137	−0.167133	−0.0835664	−	0.996502i	$-0.526631\pi$
$991$	−5.26137	−0.167133	−0.0835664	+	0.996502i	$0.526631\pi$
$992$	0	0
$993$	41.7926	1.32625
$994$	0	0
$995$	− 8.00000i	− 0.253617i
$996$	0	0
$997$	− 25.3693i	− 0.803454i	−0.915759	−	0.401727i	$-0.868410\pi$
$997$	− 25.3693i	− 0.803454i	0.915759	−	0.401727i	$-0.131590\pi$
$998$	0	0
$999$	−8.17708	−0.258711

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2816.2.c.t.1409.1		4
4.3	odd	2		2816.2.c.s.1409.4		4
8.3	odd	2		2816.2.c.s.1409.1		4
8.5	even	2	inner	2816.2.c.t.1409.4		4
16.3	odd	4		704.2.a.n.1.1		2
16.5	even	4		352.2.a.g.1.1	✓	2
16.11	odd	4		352.2.a.h.1.2	yes	2
16.13	even	4		704.2.a.o.1.2		2
48.5	odd	4		3168.2.a.bd.1.2		2
48.11	even	4		3168.2.a.bc.1.2		2
48.29	odd	4		6336.2.a.cv.1.1		2
48.35	even	4		6336.2.a.cw.1.1		2
80.59	odd	4		8800.2.a.bd.1.1		2
80.69	even	4		8800.2.a.be.1.2		2
176.21	odd	4		3872.2.a.p.1.1		2
176.43	even	4		3872.2.a.ba.1.2		2
176.109	odd	4		7744.2.a.cm.1.2		2
176.131	even	4		7744.2.a.bw.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
352.2.a.g.1.1	✓	2	16.5	even	4
352.2.a.h.1.2	yes	2	16.11	odd	4
704.2.a.n.1.1		2	16.3	odd	4
704.2.a.o.1.2		2	16.13	even	4
2816.2.c.s.1409.1		4	8.3	odd	2
2816.2.c.s.1409.4		4	4.3	odd	2
2816.2.c.t.1409.1		4	1.1	even	1	trivial
2816.2.c.t.1409.4		4	8.5	even	2	inner
3168.2.a.bc.1.2		2	48.11	even	4
3168.2.a.bd.1.2		2	48.5	odd	4
3872.2.a.p.1.1		2	176.21	odd	4
3872.2.a.ba.1.2		2	176.43	even	4
6336.2.a.cv.1.1		2	48.29	odd	4
6336.2.a.cw.1.1		2	48.35	even	4
7744.2.a.bw.1.1		2	176.131	even	4
7744.2.a.cm.1.2		2	176.109	odd	4
8800.2.a.bd.1.1		2	80.59	odd	4
8800.2.a.be.1.2		2	80.69	even	4

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:	\( N \)	\(=\)	\( 2816 = 2^{8} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2816.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.56155i q^{3} +0.561553i q^{5} -3.56155 q^{9} +1.00000i q^{11} +2.00000i q^{13} +1.43845 q^{15} +7.12311 q^{17} +1.12311i q^{19} -7.68466 q^{23} +4.68466 q^{25} +1.43845i q^{27} +7.12311i q^{29} +5.43845 q^{31} +2.56155 q^{33} +5.68466i q^{37} +5.12311 q^{39} +8.24621 q^{41} -1.12311i q^{43} -2.00000i q^{45} +4.00000 q^{47} -7.00000 q^{49} -18.2462i q^{51} -8.24621i q^{53} -0.561553 q^{55} +2.87689 q^{57} -0.315342i q^{59} +9.36932i q^{61} -1.12311 q^{65} -7.68466i q^{67} +19.6847i q^{69} +15.6847 q^{71} +6.00000 q^{73} -12.0000i q^{75} +13.1231 q^{79} -7.00000 q^{81} +11.3693i q^{83} +4.00000i q^{85} +18.2462 q^{87} -0.561553 q^{89} -13.9309i q^{93} -0.630683 q^{95} +5.68466 q^{97} -3.56155i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{9} + 14 q^{15} + 12 q^{17} - 6 q^{23} - 6 q^{25} + 30 q^{31} + 2 q^{33} + 4 q^{39} + 16 q^{47} - 28 q^{49} + 6 q^{55} + 28 q^{57} + 12 q^{65} + 38 q^{71} + 24 q^{73} + 36 q^{79} - 28 q^{81} + 40 q^{87}+ \cdots - 2 q^{97}+O(q^{100}) \) 4 * q - 6 * q^9 + 14 * q^15 + 12 * q^17 - 6 * q^23 - 6 * q^25 + 30 * q^31 + 2 * q^33 + 4 * q^39 + 16 * q^47 - 28 * q^49 + 6 * q^55 + 28 * q^57 + 12 * q^65 + 38 * q^71 + 24 * q^73 + 36 * q^79 - 28 * q^81 + 40 * q^87 + 6 * q^89 - 52 * q^95 - 2 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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