LMFDB - Embedded newform 8112.2.a.bm.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8112,2,Mod(1,8112)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8112, 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("8112.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8112 = 2^{4} \cdot 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8112.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:	$64.7746461197$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 39)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	8112.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -2.82843 q^{5} -2.82843 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -2.82843 q^{5} -2.82843 q^{7} +1.00000 q^{9} -2.00000 q^{11} +2.82843 q^{15} -3.65685 q^{17} +2.82843 q^{19} +2.82843 q^{21} +4.00000 q^{23} +3.00000 q^{25} -1.00000 q^{27} +2.00000 q^{29} -6.82843 q^{31} +2.00000 q^{33} +8.00000 q^{35} -3.65685 q^{37} -10.8284 q^{41} -9.65685 q^{43} -2.82843 q^{45} -0.343146 q^{47} +1.00000 q^{49} +3.65685 q^{51} -2.00000 q^{53} +5.65685 q^{55} -2.82843 q^{57} -3.65685 q^{59} -9.31371 q^{61} -2.82843 q^{63} +1.17157 q^{67} -4.00000 q^{69} +2.00000 q^{71} -11.6569 q^{73} -3.00000 q^{75} +5.65685 q^{77} -11.3137 q^{79} +1.00000 q^{81} -7.65685 q^{83} +10.3431 q^{85} -2.00000 q^{87} -9.17157 q^{89} +6.82843 q^{93} -8.00000 q^{95} +7.65685 q^{97} -2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + 2 * q^9	$ 2 q - 2 q^{3} + 2 q^{9} - 4 q^{11} + 4 q^{17} + 8 q^{23} + 6 q^{25} - 2 q^{27} + 4 q^{29} - 8 q^{31} + 4 q^{33} + 16 q^{35} + 4 q^{37} - 16 q^{41} - 8 q^{43} - 12 q^{47} + 2 q^{49} - 4 q^{51} - 4 q^{53} + 4 q^{59} + 4 q^{61} + 8 q^{67} - 8 q^{69} + 4 q^{71} - 12 q^{73} - 6 q^{75} + 2 q^{81} - 4 q^{83} + 32 q^{85} - 4 q^{87} - 24 q^{89} + 8 q^{93} - 16 q^{95} + 4 q^{97} - 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^9 - 4 * q^11 + 4 * q^17 + 8 * q^23 + 6 * q^25 - 2 * q^27 + 4 * q^29 - 8 * q^31 + 4 * q^33 + 16 * q^35 + 4 * q^37 - 16 * q^41 - 8 * q^43 - 12 * q^47 + 2 * q^49 - 4 * q^51 - 4 * q^53 + 4 * q^59 + 4 * q^61 + 8 * q^67 - 8 * q^69 + 4 * q^71 - 12 * q^73 - 6 * q^75 + 2 * q^81 - 4 * q^83 + 32 * q^85 - 4 * q^87 - 24 * q^89 + 8 * q^93 - 16 * q^95 + 4 * q^97 - 4 * q^99

Display 100 coefficients 10 coefficients

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
$5$	−2.82843	−1.26491	−0.632456	−	0.774597i	$-0.717953\pi$
$5$	−2.82843	−1.26491	−0.632456	+	0.774597i	$0.717953\pi$
$6$	0	0
$7$	−2.82843	−1.06904	−0.534522	−	0.845154i	$-0.679509\pi$
$7$	−2.82843	−1.06904	−0.534522	+	0.845154i	$0.679509\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	2.82843	0.730297
$16$	0	0
$17$	−3.65685	−0.886917	−0.443459	−	0.896295i	$-0.646249\pi$
$17$	−3.65685	−0.886917	−0.443459	+	0.896295i	$0.646249\pi$
$18$	0	0
$19$	2.82843	0.648886	0.324443	−	0.945905i	$-0.394823\pi$
$19$	2.82843	0.648886	0.324443	+	0.945905i	$0.394823\pi$
$20$	0	0
$21$	2.82843	0.617213
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	3.00000	0.600000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	−6.82843	−1.22642	−0.613211	−	0.789919i	$-0.710122\pi$
$31$	−6.82843	−1.22642	−0.613211	+	0.789919i	$0.710122\pi$
$32$	0	0
$33$	2.00000	0.348155
$34$	0	0
$35$	8.00000	1.35225
$36$	0	0
$37$	−3.65685	−0.601183	−0.300592	−	0.953753i	$-0.597184\pi$
$37$	−3.65685	−0.601183	−0.300592	+	0.953753i	$0.597184\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−10.8284	−1.69112	−0.845558	−	0.533883i	$-0.820732\pi$
$41$	−10.8284	−1.69112	−0.845558	+	0.533883i	$0.820732\pi$
$42$	0	0
$43$	−9.65685	−1.47266	−0.736328	−	0.676625i	$-0.763442\pi$
$43$	−9.65685	−1.47266	−0.736328	+	0.676625i	$0.763442\pi$
$44$	0	0
$45$	−2.82843	−0.421637
$46$	0	0
$47$	−0.343146	−0.0500530	−0.0250265	−	0.999687i	$-0.507967\pi$
$47$	−0.343146	−0.0500530	−0.0250265	+	0.999687i	$0.507967\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	3.65685	0.512062
$52$	0	0
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	0	0
$55$	5.65685	0.762770
$56$	0	0
$57$	−2.82843	−0.374634
$58$	0	0
$59$	−3.65685	−0.476082	−0.238041	−	0.971255i	$-0.576505\pi$
$59$	−3.65685	−0.476082	−0.238041	+	0.971255i	$0.576505\pi$
$60$	0	0
$61$	−9.31371	−1.19250	−0.596249	−	0.802799i	$-0.703343\pi$
$61$	−9.31371	−1.19250	−0.596249	+	0.802799i	$0.703343\pi$
$62$	0	0
$63$	−2.82843	−0.356348
$64$	0	0
$65$	0	0
$66$	0	0
$67$	1.17157	0.143130	0.0715652	−	0.997436i	$-0.477201\pi$
$67$	1.17157	0.143130	0.0715652	+	0.997436i	$0.477201\pi$
$68$	0	0
$69$	−4.00000	−0.481543
$70$	0	0
$71$	2.00000	0.237356	0.118678	−	0.992933i	$-0.462134\pi$
$71$	2.00000	0.237356	0.118678	+	0.992933i	$0.462134\pi$
$72$	0	0
$73$	−11.6569	−1.36433	−0.682166	−	0.731198i	$-0.738962\pi$
$73$	−11.6569	−1.36433	−0.682166	+	0.731198i	$0.738962\pi$
$74$	0	0
$75$	−3.00000	−0.346410
$76$	0	0
$77$	5.65685	0.644658
$78$	0	0
$79$	−11.3137	−1.27289	−0.636446	−	0.771321i	$-0.719596\pi$
$79$	−11.3137	−1.27289	−0.636446	+	0.771321i	$0.719596\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−7.65685	−0.840449	−0.420224	−	0.907420i	$-0.638049\pi$
$83$	−7.65685	−0.840449	−0.420224	+	0.907420i	$0.638049\pi$
$84$	0	0
$85$	10.3431	1.12187
$86$	0	0
$87$	−2.00000	−0.214423
$88$	0	0
$89$	−9.17157	−0.972185	−0.486092	−	0.873907i	$-0.661578\pi$
$89$	−9.17157	−0.972185	−0.486092	+	0.873907i	$0.661578\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	6.82843	0.708075
$94$	0	0
$95$	−8.00000	−0.820783
$96$	0	0
$97$	7.65685	0.777436	0.388718	−	0.921357i	$-0.372918\pi$
$97$	7.65685	0.777436	0.388718	+	0.921357i	$0.372918\pi$
$98$	0	0
$99$	−2.00000	−0.201008
$100$	0	0
$101$	−3.65685	−0.363871	−0.181935	−	0.983311i	$-0.558236\pi$
$101$	−3.65685	−0.363871	−0.181935	+	0.983311i	$0.558236\pi$
$102$	0	0
$103$	−13.6569	−1.34565	−0.672825	−	0.739802i	$-0.734919\pi$
$103$	−13.6569	−1.34565	−0.672825	+	0.739802i	$0.734919\pi$
$104$	0	0
$105$	−8.00000	−0.780720
$106$	0	0
$107$	−11.3137	−1.09374	−0.546869	−	0.837218i	$-0.684180\pi$
$107$	−11.3137	−1.09374	−0.546869	+	0.837218i	$0.684180\pi$
$108$	0	0
$109$	17.3137	1.65835	0.829176	−	0.558987i	$-0.188810\pi$
$109$	17.3137	1.65835	0.829176	+	0.558987i	$0.188810\pi$
$110$	0	0
$111$	3.65685	0.347093
$112$	0	0
$113$	17.3137	1.62874	0.814368	−	0.580348i	$-0.197084\pi$
$113$	17.3137	1.62874	0.814368	+	0.580348i	$0.197084\pi$
$114$	0	0
$115$	−11.3137	−1.05501
$116$	0	0
$117$	0	0
$118$	0	0
$119$	10.3431	0.948155
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	10.8284	0.976366
$124$	0	0
$125$	5.65685	0.505964
$126$	0	0
$127$	5.65685	0.501965	0.250982	−	0.967992i	$-0.419246\pi$
$127$	5.65685	0.501965	0.250982	+	0.967992i	$0.419246\pi$
$128$	0	0
$129$	9.65685	0.850239
$130$	0	0
$131$	8.00000	0.698963	0.349482	−	0.936943i	$-0.386358\pi$
$131$	8.00000	0.698963	0.349482	+	0.936943i	$0.386358\pi$
$132$	0	0
$133$	−8.00000	−0.693688
$134$	0	0
$135$	2.82843	0.243432
$136$	0	0
$137$	5.17157	0.441837	0.220919	−	0.975292i	$-0.429094\pi$
$137$	5.17157	0.441837	0.220919	+	0.975292i	$0.429094\pi$
$138$	0	0
$139$	−15.3137	−1.29889	−0.649446	−	0.760408i	$-0.724999\pi$
$139$	−15.3137	−1.29889	−0.649446	+	0.760408i	$0.724999\pi$
$140$	0	0
$141$	0.343146	0.0288981
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−5.65685	−0.469776
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	14.8284	1.21479	0.607396	−	0.794399i	$-0.292214\pi$
$149$	14.8284	1.21479	0.607396	+	0.794399i	$0.292214\pi$
$150$	0	0
$151$	−20.4853	−1.66707	−0.833534	−	0.552468i	$-0.813686\pi$
$151$	−20.4853	−1.66707	−0.833534	+	0.552468i	$0.813686\pi$
$152$	0	0
$153$	−3.65685	−0.295639
$154$	0	0
$155$	19.3137	1.55131
$156$	0	0
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	0	0
$159$	2.00000	0.158610
$160$	0	0
$161$	−11.3137	−0.891645
$162$	0	0
$163$	13.1716	1.03168	0.515839	−	0.856686i	$-0.327480\pi$
$163$	13.1716	1.03168	0.515839	+	0.856686i	$0.327480\pi$
$164$	0	0
$165$	−5.65685	−0.440386
$166$	0	0
$167$	7.65685	0.592505	0.296253	−	0.955110i	$-0.404263\pi$
$167$	7.65685	0.592505	0.296253	+	0.955110i	$0.404263\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	2.82843	0.216295
$172$	0	0
$173$	−0.343146	−0.0260889	−0.0130444	−	0.999915i	$-0.504152\pi$
$173$	−0.343146	−0.0260889	−0.0130444	+	0.999915i	$0.504152\pi$
$174$	0	0
$175$	−8.48528	−0.641427
$176$	0	0
$177$	3.65685	0.274866
$178$	0	0
$179$	0.686292	0.0512958	0.0256479	−	0.999671i	$-0.491835\pi$
$179$	0.686292	0.0512958	0.0256479	+	0.999671i	$0.491835\pi$
$180$	0	0
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	0	0
$183$	9.31371	0.688489
$184$	0	0
$185$	10.3431	0.760443
$186$	0	0
$187$	7.31371	0.534831
$188$	0	0
$189$	2.82843	0.205738
$190$	0	0
$191$	19.3137	1.39749	0.698745	−	0.715370i	$-0.253742\pi$
$191$	19.3137	1.39749	0.698745	+	0.715370i	$0.253742\pi$
$192$	0	0
$193$	17.3137	1.24627	0.623134	−	0.782115i	$-0.285859\pi$
$193$	17.3137	1.24627	0.623134	+	0.782115i	$0.285859\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	16.4853	1.17453	0.587264	−	0.809396i	$-0.300205\pi$
$197$	16.4853	1.17453	0.587264	+	0.809396i	$0.300205\pi$
$198$	0	0
$199$	−10.3431	−0.733206	−0.366603	−	0.930377i	$-0.619479\pi$
$199$	−10.3431	−0.733206	−0.366603	+	0.930377i	$0.619479\pi$
$200$	0	0
$201$	−1.17157	−0.0826364
$202$	0	0
$203$	−5.65685	−0.397033
$204$	0	0
$205$	30.6274	2.13911
$206$	0	0
$207$	4.00000	0.278019
$208$	0	0
$209$	−5.65685	−0.391293
$210$	0	0
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	0	0
$213$	−2.00000	−0.137038
$214$	0	0
$215$	27.3137	1.86278
$216$	0	0
$217$	19.3137	1.31110
$218$	0	0
$219$	11.6569	0.787697
$220$	0	0
$221$	0	0
$222$	0	0
$223$	4.48528	0.300357	0.150178	−	0.988659i	$-0.452015\pi$
$223$	4.48528	0.300357	0.150178	+	0.988659i	$0.452015\pi$
$224$	0	0
$225$	3.00000	0.200000
$226$	0	0
$227$	5.31371	0.352683	0.176342	−	0.984329i	$-0.443574\pi$
$227$	5.31371	0.352683	0.176342	+	0.984329i	$0.443574\pi$
$228$	0	0
$229$	−21.3137	−1.40845	−0.704225	−	0.709977i	$-0.748705\pi$
$229$	−21.3137	−1.40845	−0.704225	+	0.709977i	$0.748705\pi$
$230$	0	0
$231$	−5.65685	−0.372194
$232$	0	0
$233$	−26.9706	−1.76690	−0.883450	−	0.468525i	$-0.844786\pi$
$233$	−26.9706	−1.76690	−0.883450	+	0.468525i	$0.844786\pi$
$234$	0	0
$235$	0.970563	0.0633125
$236$	0	0
$237$	11.3137	0.734904
$238$	0	0
$239$	2.00000	0.129369	0.0646846	−	0.997906i	$-0.479396\pi$
$239$	2.00000	0.129369	0.0646846	+	0.997906i	$0.479396\pi$
$240$	0	0
$241$	−11.6569	−0.750884	−0.375442	−	0.926846i	$-0.622509\pi$
$241$	−11.6569	−0.750884	−0.375442	+	0.926846i	$0.622509\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−2.82843	−0.180702
$246$	0	0
$247$	0	0
$248$	0	0
$249$	7.65685	0.485233
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	−8.00000	−0.502956
$254$	0	0
$255$	−10.3431	−0.647713
$256$	0	0
$257$	−15.6569	−0.976648	−0.488324	−	0.872662i	$-0.662392\pi$
$257$	−15.6569	−0.976648	−0.488324	+	0.872662i	$0.662392\pi$
$258$	0	0
$259$	10.3431	0.642692
$260$	0	0
$261$	2.00000	0.123797
$262$	0	0
$263$	−12.0000	−0.739952	−0.369976	−	0.929041i	$-0.620634\pi$
$263$	−12.0000	−0.739952	−0.369976	+	0.929041i	$0.620634\pi$
$264$	0	0
$265$	5.65685	0.347498
$266$	0	0
$267$	9.17157	0.561291
$268$	0	0
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	0	0
$271$	11.7990	0.716738	0.358369	−	0.933580i	$-0.383333\pi$
$271$	11.7990	0.716738	0.358369	+	0.933580i	$0.383333\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−6.00000	−0.361814
$276$	0	0
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	0	0
$279$	−6.82843	−0.408807
$280$	0	0
$281$	−26.8284	−1.60045	−0.800225	−	0.599700i	$-0.795287\pi$
$281$	−26.8284	−1.60045	−0.800225	+	0.599700i	$0.795287\pi$
$282$	0	0
$283$	4.97056	0.295469	0.147735	−	0.989027i	$-0.452802\pi$
$283$	4.97056	0.295469	0.147735	+	0.989027i	$0.452802\pi$
$284$	0	0
$285$	8.00000	0.473879
$286$	0	0
$287$	30.6274	1.80788
$288$	0	0
$289$	−3.62742	−0.213377
$290$	0	0
$291$	−7.65685	−0.448853
$292$	0	0
$293$	26.1421	1.52724	0.763620	−	0.645666i	$-0.223420\pi$
$293$	26.1421	1.52724	0.763620	+	0.645666i	$0.223420\pi$
$294$	0	0
$295$	10.3431	0.602201
$296$	0	0
$297$	2.00000	0.116052
$298$	0	0
$299$	0	0
$300$	0	0
$301$	27.3137	1.57434
$302$	0	0
$303$	3.65685	0.210081
$304$	0	0
$305$	26.3431	1.50840
$306$	0	0
$307$	−17.1716	−0.980033	−0.490017	−	0.871713i	$-0.663009\pi$
$307$	−17.1716	−0.980033	−0.490017	+	0.871713i	$0.663009\pi$
$308$	0	0
$309$	13.6569	0.776911
$310$	0	0
$311$	−34.6274	−1.96354	−0.981770	−	0.190071i	$-0.939128\pi$
$311$	−34.6274	−1.96354	−0.981770	+	0.190071i	$0.939128\pi$
$312$	0	0
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	0	0
$315$	8.00000	0.450749
$316$	0	0
$317$	8.48528	0.476581	0.238290	−	0.971194i	$-0.423413\pi$
$317$	8.48528	0.476581	0.238290	+	0.971194i	$0.423413\pi$
$318$	0	0
$319$	−4.00000	−0.223957
$320$	0	0
$321$	11.3137	0.631470
$322$	0	0
$323$	−10.3431	−0.575508
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−17.3137	−0.957450
$328$	0	0
$329$	0.970563	0.0535089
$330$	0	0
$331$	−2.14214	−0.117742	−0.0588712	−	0.998266i	$-0.518750\pi$
$331$	−2.14214	−0.117742	−0.0588712	+	0.998266i	$0.518750\pi$
$332$	0	0
$333$	−3.65685	−0.200394
$334$	0	0
$335$	−3.31371	−0.181047
$336$	0	0
$337$	−13.3137	−0.725244	−0.362622	−	0.931936i	$-0.618118\pi$
$337$	−13.3137	−0.725244	−0.362622	+	0.931936i	$0.618118\pi$
$338$	0	0
$339$	−17.3137	−0.940352
$340$	0	0
$341$	13.6569	0.739560
$342$	0	0
$343$	16.9706	0.916324
$344$	0	0
$345$	11.3137	0.609110
$346$	0	0
$347$	31.3137	1.68101	0.840504	−	0.541805i	$-0.182259\pi$
$347$	31.3137	1.68101	0.840504	+	0.541805i	$0.182259\pi$
$348$	0	0
$349$	7.65685	0.409862	0.204931	−	0.978776i	$-0.434303\pi$
$349$	7.65685	0.409862	0.204931	+	0.978776i	$0.434303\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−17.4558	−0.929081	−0.464540	−	0.885552i	$-0.653780\pi$
$353$	−17.4558	−0.929081	−0.464540	+	0.885552i	$0.653780\pi$
$354$	0	0
$355$	−5.65685	−0.300235
$356$	0	0
$357$	−10.3431	−0.547417
$358$	0	0
$359$	1.02944	0.0543316	0.0271658	−	0.999631i	$-0.491352\pi$
$359$	1.02944	0.0543316	0.0271658	+	0.999631i	$0.491352\pi$
$360$	0	0
$361$	−11.0000	−0.578947
$362$	0	0
$363$	7.00000	0.367405
$364$	0	0
$365$	32.9706	1.72576
$366$	0	0
$367$	24.0000	1.25279	0.626395	−	0.779506i	$-0.284530\pi$
$367$	24.0000	1.25279	0.626395	+	0.779506i	$0.284530\pi$
$368$	0	0
$369$	−10.8284	−0.563705
$370$	0	0
$371$	5.65685	0.293689
$372$	0	0
$373$	10.0000	0.517780	0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000	0.517780	0.258890	+	0.965907i	$0.416643\pi$
$374$	0	0
$375$	−5.65685	−0.292119
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−16.4853	−0.846792	−0.423396	−	0.905945i	$-0.639162\pi$
$379$	−16.4853	−0.846792	−0.423396	+	0.905945i	$0.639162\pi$
$380$	0	0
$381$	−5.65685	−0.289809
$382$	0	0
$383$	−2.97056	−0.151789	−0.0758943	−	0.997116i	$-0.524181\pi$
$383$	−2.97056	−0.151789	−0.0758943	+	0.997116i	$0.524181\pi$
$384$	0	0
$385$	−16.0000	−0.815436
$386$	0	0
$387$	−9.65685	−0.490885
$388$	0	0
$389$	6.97056	0.353422	0.176711	−	0.984263i	$-0.443454\pi$
$389$	6.97056	0.353422	0.176711	+	0.984263i	$0.443454\pi$
$390$	0	0
$391$	−14.6274	−0.739740
$392$	0	0
$393$	−8.00000	−0.403547
$394$	0	0
$395$	32.0000	1.61009
$396$	0	0
$397$	2.97056	0.149088	0.0745441	−	0.997218i	$-0.476250\pi$
$397$	2.97056	0.149088	0.0745441	+	0.997218i	$0.476250\pi$
$398$	0	0
$399$	8.00000	0.400501
$400$	0	0
$401$	2.14214	0.106973	0.0534866	−	0.998569i	$-0.482967\pi$
$401$	2.14214	0.106973	0.0534866	+	0.998569i	$0.482967\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−2.82843	−0.140546
$406$	0	0
$407$	7.31371	0.362527
$408$	0	0
$409$	1.02944	0.0509024	0.0254512	−	0.999676i	$-0.491898\pi$
$409$	1.02944	0.0509024	0.0254512	+	0.999676i	$0.491898\pi$
$410$	0	0
$411$	−5.17157	−0.255095
$412$	0	0
$413$	10.3431	0.508953
$414$	0	0
$415$	21.6569	1.06309
$416$	0	0
$417$	15.3137	0.749916
$418$	0	0
$419$	30.6274	1.49625	0.748124	−	0.663559i	$-0.230955\pi$
$419$	30.6274	1.49625	0.748124	+	0.663559i	$0.230955\pi$
$420$	0	0
$421$	−14.6863	−0.715766	−0.357883	−	0.933766i	$-0.616501\pi$
$421$	−14.6863	−0.715766	−0.357883	+	0.933766i	$0.616501\pi$
$422$	0	0
$423$	−0.343146	−0.0166843
$424$	0	0
$425$	−10.9706	−0.532150
$426$	0	0
$427$	26.3431	1.27483
$428$	0	0
$429$	0	0
$430$	0	0
$431$	19.6569	0.946837	0.473419	−	0.880838i	$-0.343020\pi$
$431$	19.6569	0.946837	0.473419	+	0.880838i	$0.343020\pi$
$432$	0	0
$433$	1.31371	0.0631328	0.0315664	−	0.999502i	$-0.489950\pi$
$433$	1.31371	0.0631328	0.0315664	+	0.999502i	$0.489950\pi$
$434$	0	0
$435$	5.65685	0.271225
$436$	0	0
$437$	11.3137	0.541208
$438$	0	0
$439$	16.9706	0.809961	0.404980	−	0.914325i	$-0.367278\pi$
$439$	16.9706	0.809961	0.404980	+	0.914325i	$0.367278\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−41.9411	−1.99268	−0.996342	−	0.0854611i	$-0.972764\pi$
$443$	−41.9411	−1.99268	−0.996342	+	0.0854611i	$0.972764\pi$
$444$	0	0
$445$	25.9411	1.22973
$446$	0	0
$447$	−14.8284	−0.701361
$448$	0	0
$449$	−7.79899	−0.368057	−0.184029	−	0.982921i	$-0.558914\pi$
$449$	−7.79899	−0.368057	−0.184029	+	0.982921i	$0.558914\pi$
$450$	0	0
$451$	21.6569	1.01978
$452$	0	0
$453$	20.4853	0.962482
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−3.65685	−0.171060	−0.0855302	−	0.996336i	$-0.527258\pi$
$457$	−3.65685	−0.171060	−0.0855302	+	0.996336i	$0.527258\pi$
$458$	0	0
$459$	3.65685	0.170687
$460$	0	0
$461$	−10.8284	−0.504330	−0.252165	−	0.967684i	$-0.581143\pi$
$461$	−10.8284	−0.504330	−0.252165	+	0.967684i	$0.581143\pi$
$462$	0	0
$463$	−7.51472	−0.349239	−0.174619	−	0.984636i	$-0.555869\pi$
$463$	−7.51472	−0.349239	−0.174619	+	0.984636i	$0.555869\pi$
$464$	0	0
$465$	−19.3137	−0.895652
$466$	0	0
$467$	8.00000	0.370196	0.185098	−	0.982720i	$-0.440740\pi$
$467$	8.00000	0.370196	0.185098	+	0.982720i	$0.440740\pi$
$468$	0	0
$469$	−3.31371	−0.153013
$470$	0	0
$471$	10.0000	0.460776
$472$	0	0
$473$	19.3137	0.888045
$474$	0	0
$475$	8.48528	0.389331
$476$	0	0
$477$	−2.00000	−0.0915737
$478$	0	0
$479$	2.68629	0.122740	0.0613699	−	0.998115i	$-0.480453\pi$
$479$	2.68629	0.122740	0.0613699	+	0.998115i	$0.480453\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	11.3137	0.514792
$484$	0	0
$485$	−21.6569	−0.983387
$486$	0	0
$487$	31.7990	1.44095	0.720475	−	0.693481i	$-0.243924\pi$
$487$	31.7990	1.44095	0.720475	+	0.693481i	$0.243924\pi$
$488$	0	0
$489$	−13.1716	−0.595639
$490$	0	0
$491$	14.6274	0.660126	0.330063	−	0.943959i	$-0.392930\pi$
$491$	14.6274	0.660126	0.330063	+	0.943959i	$0.392930\pi$
$492$	0	0
$493$	−7.31371	−0.329393
$494$	0	0
$495$	5.65685	0.254257
$496$	0	0
$497$	−5.65685	−0.253745
$498$	0	0
$499$	−2.14214	−0.0958952	−0.0479476	−	0.998850i	$-0.515268\pi$
$499$	−2.14214	−0.0958952	−0.0479476	+	0.998850i	$0.515268\pi$
$500$	0	0
$501$	−7.65685	−0.342083
$502$	0	0
$503$	−15.3137	−0.682805	−0.341402	−	0.939917i	$-0.610902\pi$
$503$	−15.3137	−0.682805	−0.341402	+	0.939917i	$0.610902\pi$
$504$	0	0
$505$	10.3431	0.460264
$506$	0	0
$507$	0	0
$508$	0	0
$509$	27.7990	1.23217	0.616084	−	0.787680i	$-0.288718\pi$
$509$	27.7990	1.23217	0.616084	+	0.787680i	$0.288718\pi$
$510$	0	0
$511$	32.9706	1.45853
$512$	0	0
$513$	−2.82843	−0.124878
$514$	0	0
$515$	38.6274	1.70213
$516$	0	0
$517$	0.686292	0.0301831
$518$	0	0
$519$	0.343146	0.0150624
$520$	0	0
$521$	2.68629	0.117689	0.0588443	−	0.998267i	$-0.481258\pi$
$521$	2.68629	0.117689	0.0588443	+	0.998267i	$0.481258\pi$
$522$	0	0
$523$	−7.31371	−0.319806	−0.159903	−	0.987133i	$-0.551118\pi$
$523$	−7.31371	−0.319806	−0.159903	+	0.987133i	$0.551118\pi$
$524$	0	0
$525$	8.48528	0.370328
$526$	0	0
$527$	24.9706	1.08773
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	−3.65685	−0.158694
$532$	0	0
$533$	0	0
$534$	0	0
$535$	32.0000	1.38348
$536$	0	0
$537$	−0.686292	−0.0296157
$538$	0	0
$539$	−2.00000	−0.0861461
$540$	0	0
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	0	0
$543$	−14.0000	−0.600798
$544$	0	0
$545$	−48.9706	−2.09767
$546$	0	0
$547$	−0.686292	−0.0293437	−0.0146719	−	0.999892i	$-0.504670\pi$
$547$	−0.686292	−0.0293437	−0.0146719	+	0.999892i	$0.504670\pi$
$548$	0	0
$549$	−9.31371	−0.397499
$550$	0	0
$551$	5.65685	0.240990
$552$	0	0
$553$	32.0000	1.36078
$554$	0	0
$555$	−10.3431	−0.439042
$556$	0	0
$557$	−31.7990	−1.34737	−0.673683	−	0.739020i	$-0.735289\pi$
$557$	−31.7990	−1.34737	−0.673683	+	0.739020i	$0.735289\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−7.31371	−0.308785
$562$	0	0
$563$	−4.00000	−0.168580	−0.0842900	−	0.996441i	$-0.526862\pi$
$563$	−4.00000	−0.168580	−0.0842900	+	0.996441i	$0.526862\pi$
$564$	0	0
$565$	−48.9706	−2.06021
$566$	0	0
$567$	−2.82843	−0.118783
$568$	0	0
$569$	−9.02944	−0.378534	−0.189267	−	0.981926i	$-0.560611\pi$
$569$	−9.02944	−0.378534	−0.189267	+	0.981926i	$0.560611\pi$
$570$	0	0
$571$	−20.9706	−0.877591	−0.438795	−	0.898587i	$-0.644595\pi$
$571$	−20.9706	−0.877591	−0.438795	+	0.898587i	$0.644595\pi$
$572$	0	0
$573$	−19.3137	−0.806842
$574$	0	0
$575$	12.0000	0.500435
$576$	0	0
$577$	−35.9411	−1.49625	−0.748124	−	0.663559i	$-0.769045\pi$
$577$	−35.9411	−1.49625	−0.748124	+	0.663559i	$0.769045\pi$
$578$	0	0
$579$	−17.3137	−0.719533
$580$	0	0
$581$	21.6569	0.898478
$582$	0	0
$583$	4.00000	0.165663
$584$	0	0
$585$	0	0
$586$	0	0
$587$	22.9706	0.948097	0.474048	−	0.880499i	$-0.342792\pi$
$587$	22.9706	0.948097	0.474048	+	0.880499i	$0.342792\pi$
$588$	0	0
$589$	−19.3137	−0.795807
$590$	0	0
$591$	−16.4853	−0.678114
$592$	0	0
$593$	3.51472	0.144332	0.0721661	−	0.997393i	$-0.477009\pi$
$593$	3.51472	0.144332	0.0721661	+	0.997393i	$0.477009\pi$
$594$	0	0
$595$	−29.2548	−1.19933
$596$	0	0
$597$	10.3431	0.423317
$598$	0	0
$599$	0.686292	0.0280411	0.0140206	−	0.999902i	$-0.495537\pi$
$599$	0.686292	0.0280411	0.0140206	+	0.999902i	$0.495537\pi$
$600$	0	0
$601$	44.6274	1.82039	0.910195	−	0.414180i	$-0.135931\pi$
$601$	44.6274	1.82039	0.910195	+	0.414180i	$0.135931\pi$
$602$	0	0
$603$	1.17157	0.0477101
$604$	0	0
$605$	19.7990	0.804943
$606$	0	0
$607$	25.9411	1.05292	0.526459	−	0.850201i	$-0.323519\pi$
$607$	25.9411	1.05292	0.526459	+	0.850201i	$0.323519\pi$
$608$	0	0
$609$	5.65685	0.229227
$610$	0	0
$611$	0	0
$612$	0	0
$613$	36.3431	1.46789	0.733943	−	0.679211i	$-0.237678\pi$
$613$	36.3431	1.46789	0.733943	+	0.679211i	$0.237678\pi$
$614$	0	0
$615$	−30.6274	−1.23502
$616$	0	0
$617$	29.1716	1.17440	0.587202	−	0.809441i	$-0.300230\pi$
$617$	29.1716	1.17440	0.587202	+	0.809441i	$0.300230\pi$
$618$	0	0
$619$	−15.7990	−0.635015	−0.317508	−	0.948256i	$-0.602846\pi$
$619$	−15.7990	−0.635015	−0.317508	+	0.948256i	$0.602846\pi$
$620$	0	0
$621$	−4.00000	−0.160514
$622$	0	0
$623$	25.9411	1.03931
$624$	0	0
$625$	−31.0000	−1.24000
$626$	0	0
$627$	5.65685	0.225913
$628$	0	0
$629$	13.3726	0.533200
$630$	0	0
$631$	−19.1127	−0.760865	−0.380432	−	0.924809i	$-0.624225\pi$
$631$	−19.1127	−0.760865	−0.380432	+	0.924809i	$0.624225\pi$
$632$	0	0
$633$	−12.0000	−0.476957
$634$	0	0
$635$	−16.0000	−0.634941
$636$	0	0
$637$	0	0
$638$	0	0
$639$	2.00000	0.0791188
$640$	0	0
$641$	−26.2843	−1.03817	−0.519083	−	0.854724i	$-0.673727\pi$
$641$	−26.2843	−1.03817	−0.519083	+	0.854724i	$0.673727\pi$
$642$	0	0
$643$	17.1716	0.677181	0.338590	−	0.940934i	$-0.390050\pi$
$643$	17.1716	0.677181	0.338590	+	0.940934i	$0.390050\pi$
$644$	0	0
$645$	−27.3137	−1.07548
$646$	0	0
$647$	11.3137	0.444788	0.222394	−	0.974957i	$-0.428613\pi$
$647$	11.3137	0.444788	0.222394	+	0.974957i	$0.428613\pi$
$648$	0	0
$649$	7.31371	0.287088
$650$	0	0
$651$	−19.3137	−0.756964
$652$	0	0
$653$	−2.68629	−0.105123	−0.0525614	−	0.998618i	$-0.516739\pi$
$653$	−2.68629	−0.105123	−0.0525614	+	0.998618i	$0.516739\pi$
$654$	0	0
$655$	−22.6274	−0.884126
$656$	0	0
$657$	−11.6569	−0.454777
$658$	0	0
$659$	24.6863	0.961641	0.480821	−	0.876819i	$-0.340339\pi$
$659$	24.6863	0.961641	0.480821	+	0.876819i	$0.340339\pi$
$660$	0	0
$661$	1.02944	0.0400405	0.0200202	−	0.999800i	$-0.493627\pi$
$661$	1.02944	0.0400405	0.0200202	+	0.999800i	$0.493627\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	22.6274	0.877454
$666$	0	0
$667$	8.00000	0.309761
$668$	0	0
$669$	−4.48528	−0.173411
$670$	0	0
$671$	18.6274	0.719103
$672$	0	0
$673$	−28.6274	−1.10351	−0.551753	−	0.834008i	$-0.686041\pi$
$673$	−28.6274	−1.10351	−0.551753	+	0.834008i	$0.686041\pi$
$674$	0	0
$675$	−3.00000	−0.115470
$676$	0	0
$677$	49.3137	1.89528	0.947640	−	0.319341i	$-0.103462\pi$
$677$	49.3137	1.89528	0.947640	+	0.319341i	$0.103462\pi$
$678$	0	0
$679$	−21.6569	−0.831114
$680$	0	0
$681$	−5.31371	−0.203622
$682$	0	0
$683$	−19.9411	−0.763026	−0.381513	−	0.924363i	$-0.624597\pi$
$683$	−19.9411	−0.763026	−0.381513	+	0.924363i	$0.624597\pi$
$684$	0	0
$685$	−14.6274	−0.558885
$686$	0	0
$687$	21.3137	0.813169
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−34.1421	−1.29883	−0.649414	−	0.760435i	$-0.724986\pi$
$691$	−34.1421	−1.29883	−0.649414	+	0.760435i	$0.724986\pi$
$692$	0	0
$693$	5.65685	0.214886
$694$	0	0
$695$	43.3137	1.64298
$696$	0	0
$697$	39.5980	1.49988
$698$	0	0
$699$	26.9706	1.02012
$700$	0	0
$701$	38.9706	1.47190	0.735949	−	0.677037i	$-0.236736\pi$
$701$	38.9706	1.47190	0.735949	+	0.677037i	$0.236736\pi$
$702$	0	0
$703$	−10.3431	−0.390099
$704$	0	0
$705$	−0.970563	−0.0365535
$706$	0	0
$707$	10.3431	0.388994
$708$	0	0
$709$	−40.6274	−1.52579	−0.762897	−	0.646520i	$-0.776224\pi$
$709$	−40.6274	−1.52579	−0.762897	+	0.646520i	$0.776224\pi$
$710$	0	0
$711$	−11.3137	−0.424297
$712$	0	0
$713$	−27.3137	−1.02291
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−2.00000	−0.0746914
$718$	0	0
$719$	−37.9411	−1.41497	−0.707483	−	0.706731i	$-0.750169\pi$
$719$	−37.9411	−1.41497	−0.707483	+	0.706731i	$0.750169\pi$
$720$	0	0
$721$	38.6274	1.43856
$722$	0	0
$723$	11.6569	0.433523
$724$	0	0
$725$	6.00000	0.222834
$726$	0	0
$727$	21.6569	0.803208	0.401604	−	0.915813i	$-0.368453\pi$
$727$	21.6569	0.803208	0.401604	+	0.915813i	$0.368453\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	35.3137	1.30612
$732$	0	0
$733$	−8.62742	−0.318661	−0.159330	−	0.987225i	$-0.550934\pi$
$733$	−8.62742	−0.318661	−0.159330	+	0.987225i	$0.550934\pi$
$734$	0	0
$735$	2.82843	0.104328
$736$	0	0
$737$	−2.34315	−0.0863109
$738$	0	0
$739$	10.1421	0.373084	0.186542	−	0.982447i	$-0.440272\pi$
$739$	10.1421	0.373084	0.186542	+	0.982447i	$0.440272\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	2.00000	0.0733729	0.0366864	−	0.999327i	$-0.488320\pi$
$743$	2.00000	0.0733729	0.0366864	+	0.999327i	$0.488320\pi$
$744$	0	0
$745$	−41.9411	−1.53660
$746$	0	0
$747$	−7.65685	−0.280150
$748$	0	0
$749$	32.0000	1.16925
$750$	0	0
$751$	−32.9706	−1.20311	−0.601556	−	0.798830i	$-0.705453\pi$
$751$	−32.9706	−1.20311	−0.601556	+	0.798830i	$0.705453\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	57.9411	2.10869
$756$	0	0
$757$	−15.9411	−0.579390	−0.289695	−	0.957119i	$-0.593554\pi$
$757$	−15.9411	−0.579390	−0.289695	+	0.957119i	$0.593554\pi$
$758$	0	0
$759$	8.00000	0.290382
$760$	0	0
$761$	−15.5147	−0.562408	−0.281204	−	0.959648i	$-0.590734\pi$
$761$	−15.5147	−0.562408	−0.281204	+	0.959648i	$0.590734\pi$
$762$	0	0
$763$	−48.9706	−1.77285
$764$	0	0
$765$	10.3431	0.373957
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−42.0000	−1.51456	−0.757279	−	0.653091i	$-0.773472\pi$
$769$	−42.0000	−1.51456	−0.757279	+	0.653091i	$0.773472\pi$
$770$	0	0
$771$	15.6569	0.563868
$772$	0	0
$773$	−5.85786	−0.210693	−0.105346	−	0.994436i	$-0.533595\pi$
$773$	−5.85786	−0.210693	−0.105346	+	0.994436i	$0.533595\pi$
$774$	0	0
$775$	−20.4853	−0.735853
$776$	0	0
$777$	−10.3431	−0.371058
$778$	0	0
$779$	−30.6274	−1.09734
$780$	0	0
$781$	−4.00000	−0.143131
$782$	0	0
$783$	−2.00000	−0.0714742
$784$	0	0
$785$	28.2843	1.00951
$786$	0	0
$787$	32.7696	1.16811	0.584054	−	0.811715i	$-0.301466\pi$
$787$	32.7696	1.16811	0.584054	+	0.811715i	$0.301466\pi$
$788$	0	0
$789$	12.0000	0.427211
$790$	0	0
$791$	−48.9706	−1.74119
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−5.65685	−0.200628
$796$	0	0
$797$	35.6569	1.26303	0.631515	−	0.775363i	$-0.282433\pi$
$797$	35.6569	1.26303	0.631515	+	0.775363i	$0.282433\pi$
$798$	0	0
$799$	1.25483	0.0443928
$800$	0	0
$801$	−9.17157	−0.324062
$802$	0	0
$803$	23.3137	0.822723
$804$	0	0
$805$	32.0000	1.12785
$806$	0	0
$807$	−18.0000	−0.633630
$808$	0	0
$809$	41.3137	1.45251	0.726256	−	0.687424i	$-0.241259\pi$
$809$	41.3137	1.45251	0.726256	+	0.687424i	$0.241259\pi$
$810$	0	0
$811$	−1.85786	−0.0652384	−0.0326192	−	0.999468i	$-0.510385\pi$
$811$	−1.85786	−0.0652384	−0.0326192	+	0.999468i	$0.510385\pi$
$812$	0	0
$813$	−11.7990	−0.413809
$814$	0	0
$815$	−37.2548	−1.30498
$816$	0	0
$817$	−27.3137	−0.955586
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−15.7990	−0.551389	−0.275694	−	0.961245i	$-0.588908\pi$
$821$	−15.7990	−0.551389	−0.275694	+	0.961245i	$0.588908\pi$
$822$	0	0
$823$	−48.9706	−1.70701	−0.853503	−	0.521088i	$-0.825527\pi$
$823$	−48.9706	−1.70701	−0.853503	+	0.521088i	$0.825527\pi$
$824$	0	0
$825$	6.00000	0.208893
$826$	0	0
$827$	−26.0000	−0.904109	−0.452054	−	0.891990i	$-0.649309\pi$
$827$	−26.0000	−0.904109	−0.452054	+	0.891990i	$0.649309\pi$
$828$	0	0
$829$	5.31371	0.184553	0.0922764	−	0.995733i	$-0.470586\pi$
$829$	5.31371	0.184553	0.0922764	+	0.995733i	$0.470586\pi$
$830$	0	0
$831$	2.00000	0.0693792
$832$	0	0
$833$	−3.65685	−0.126702
$834$	0	0
$835$	−21.6569	−0.749466
$836$	0	0
$837$	6.82843	0.236025
$838$	0	0
$839$	−47.2548	−1.63142	−0.815709	−	0.578462i	$-0.803653\pi$
$839$	−47.2548	−1.63142	−0.815709	+	0.578462i	$0.803653\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	26.8284	0.924020
$844$	0	0
$845$	0	0
$846$	0	0
$847$	19.7990	0.680301
$848$	0	0
$849$	−4.97056	−0.170589
$850$	0	0
$851$	−14.6274	−0.501421
$852$	0	0
$853$	7.65685	0.262166	0.131083	−	0.991371i	$-0.458155\pi$
$853$	7.65685	0.262166	0.131083	+	0.991371i	$0.458155\pi$
$854$	0	0
$855$	−8.00000	−0.273594
$856$	0	0
$857$	29.5980	1.01105	0.505524	−	0.862813i	$-0.331299\pi$
$857$	29.5980	1.01105	0.505524	+	0.862813i	$0.331299\pi$
$858$	0	0
$859$	23.3137	0.795453	0.397727	−	0.917504i	$-0.369799\pi$
$859$	23.3137	0.795453	0.397727	+	0.917504i	$0.369799\pi$
$860$	0	0
$861$	−30.6274	−1.04378
$862$	0	0
$863$	−39.6569	−1.34994	−0.674968	−	0.737847i	$-0.735842\pi$
$863$	−39.6569	−1.34994	−0.674968	+	0.737847i	$0.735842\pi$
$864$	0	0
$865$	0.970563	0.0330001
$866$	0	0
$867$	3.62742	0.123194
$868$	0	0
$869$	22.6274	0.767583
$870$	0	0
$871$	0	0
$872$	0	0
$873$	7.65685	0.259145
$874$	0	0
$875$	−16.0000	−0.540899
$876$	0	0
$877$	14.2843	0.482346	0.241173	−	0.970482i	$-0.422468\pi$
$877$	14.2843	0.482346	0.241173	+	0.970482i	$0.422468\pi$
$878$	0	0
$879$	−26.1421	−0.881752
$880$	0	0
$881$	53.5980	1.80576	0.902881	−	0.429891i	$-0.141448\pi$
$881$	53.5980	1.80576	0.902881	+	0.429891i	$0.141448\pi$
$882$	0	0
$883$	51.5980	1.73641	0.868205	−	0.496205i	$-0.165274\pi$
$883$	51.5980	1.73641	0.868205	+	0.496205i	$0.165274\pi$
$884$	0	0
$885$	−10.3431	−0.347681
$886$	0	0
$887$	8.00000	0.268614	0.134307	−	0.990940i	$-0.457119\pi$
$887$	8.00000	0.268614	0.134307	+	0.990940i	$0.457119\pi$
$888$	0	0
$889$	−16.0000	−0.536623
$890$	0	0
$891$	−2.00000	−0.0670025
$892$	0	0
$893$	−0.970563	−0.0324786
$894$	0	0
$895$	−1.94113	−0.0648847
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−13.6569	−0.455482
$900$	0	0
$901$	7.31371	0.243655
$902$	0	0
$903$	−27.3137	−0.908943
$904$	0	0
$905$	−39.5980	−1.31628
$906$	0	0
$907$	−20.9706	−0.696316	−0.348158	−	0.937436i	$-0.613193\pi$
$907$	−20.9706	−0.696316	−0.348158	+	0.937436i	$0.613193\pi$
$908$	0	0
$909$	−3.65685	−0.121290
$910$	0	0
$911$	40.0000	1.32526	0.662630	−	0.748947i	$-0.269440\pi$
$911$	40.0000	1.32526	0.662630	+	0.748947i	$0.269440\pi$
$912$	0	0
$913$	15.3137	0.506810
$914$	0	0
$915$	−26.3431	−0.870878
$916$	0	0
$917$	−22.6274	−0.747223
$918$	0	0
$919$	19.3137	0.637100	0.318550	−	0.947906i	$-0.396804\pi$
$919$	19.3137	0.637100	0.318550	+	0.947906i	$0.396804\pi$
$920$	0	0
$921$	17.1716	0.565823
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−10.9706	−0.360710
$926$	0	0
$927$	−13.6569	−0.448550
$928$	0	0
$929$	27.7990	0.912055	0.456028	−	0.889966i	$-0.349272\pi$
$929$	27.7990	0.912055	0.456028	+	0.889966i	$0.349272\pi$
$930$	0	0
$931$	2.82843	0.0926980
$932$	0	0
$933$	34.6274	1.13365
$934$	0	0
$935$	−20.6863	−0.676514
$936$	0	0
$937$	1.31371	0.0429170	0.0214585	−	0.999770i	$-0.493169\pi$
$937$	1.31371	0.0429170	0.0214585	+	0.999770i	$0.493169\pi$
$938$	0	0
$939$	−6.00000	−0.195803
$940$	0	0
$941$	−5.85786	−0.190961	−0.0954805	−	0.995431i	$-0.530439\pi$
$941$	−5.85786	−0.190961	−0.0954805	+	0.995431i	$0.530439\pi$
$942$	0	0
$943$	−43.3137	−1.41049
$944$	0	0
$945$	−8.00000	−0.260240
$946$	0	0
$947$	−54.9706	−1.78630	−0.893152	−	0.449756i	$-0.851511\pi$
$947$	−54.9706	−1.78630	−0.893152	+	0.449756i	$0.851511\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−8.48528	−0.275154
$952$	0	0
$953$	51.6569	1.67333	0.836665	−	0.547715i	$-0.184502\pi$
$953$	51.6569	1.67333	0.836665	+	0.547715i	$0.184502\pi$
$954$	0	0
$955$	−54.6274	−1.76770
$956$	0	0
$957$	4.00000	0.129302
$958$	0	0
$959$	−14.6274	−0.472344
$960$	0	0
$961$	15.6274	0.504110
$962$	0	0
$963$	−11.3137	−0.364579
$964$	0	0
$965$	−48.9706	−1.57642
$966$	0	0
$967$	−10.1421	−0.326149	−0.163075	−	0.986614i	$-0.552141\pi$
$967$	−10.1421	−0.326149	−0.163075	+	0.986614i	$0.552141\pi$
$968$	0	0
$969$	10.3431	0.332270
$970$	0	0
$971$	−7.31371	−0.234708	−0.117354	−	0.993090i	$-0.537441\pi$
$971$	−7.31371	−0.234708	−0.117354	+	0.993090i	$0.537441\pi$
$972$	0	0
$973$	43.3137	1.38857
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−13.8579	−0.443352	−0.221676	−	0.975120i	$-0.571153\pi$
$977$	−13.8579	−0.443352	−0.221676	+	0.975120i	$0.571153\pi$
$978$	0	0
$979$	18.3431	0.586249
$980$	0	0
$981$	17.3137	0.552784
$982$	0	0
$983$	2.68629	0.0856794	0.0428397	−	0.999082i	$-0.486360\pi$
$983$	2.68629	0.0856794	0.0428397	+	0.999082i	$0.486360\pi$
$984$	0	0
$985$	−46.6274	−1.48567
$986$	0	0
$987$	−0.970563	−0.0308934
$988$	0	0
$989$	−38.6274	−1.22828
$990$	0	0
$991$	−27.3137	−0.867649	−0.433824	−	0.900998i	$-0.642836\pi$
$991$	−27.3137	−0.867649	−0.433824	+	0.900998i	$0.642836\pi$
$992$	0	0
$993$	2.14214	0.0679786
$994$	0	0
$995$	29.2548	0.927441
$996$	0	0
$997$	51.2548	1.62326	0.811628	−	0.584174i	$-0.198581\pi$
$997$	51.2548	1.62326	0.811628	+	0.584174i	$0.198581\pi$
$998$	0	0
$999$	3.65685	0.115698

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8112.2.a.bm.1.1		2
4.3	odd	2		507.2.a.h.1.2		2
12.11	even	2		1521.2.a.f.1.1		2
13.12	even	2		624.2.a.k.1.2		2
39.38	odd	2		1872.2.a.w.1.1		2
52.3	odd	6		507.2.e.d.22.1		4
52.7	even	12		507.2.j.f.361.4		8
52.11	even	12		507.2.j.f.316.1		8
52.15	even	12		507.2.j.f.316.4		8
52.19	even	12		507.2.j.f.361.1		8
52.23	odd	6		507.2.e.h.22.2		4
52.31	even	4		507.2.b.e.337.1		4
52.35	odd	6		507.2.e.d.484.1		4
52.43	odd	6		507.2.e.h.484.2		4
52.47	even	4		507.2.b.e.337.4		4
52.51	odd	2		39.2.a.b.1.1	✓	2
104.51	odd	2		2496.2.a.bf.1.1		2
104.77	even	2		2496.2.a.bi.1.1		2
156.47	odd	4		1521.2.b.j.1351.1		4
156.83	odd	4		1521.2.b.j.1351.4		4
156.155	even	2		117.2.a.c.1.2		2
260.103	even	4		975.2.c.h.274.4		4
260.207	even	4		975.2.c.h.274.1		4
260.259	odd	2		975.2.a.l.1.2		2
312.77	odd	2		7488.2.a.co.1.2		2
312.155	even	2		7488.2.a.cl.1.2		2
364.363	even	2		1911.2.a.h.1.1		2
468.103	odd	6		1053.2.e.m.703.2		4
468.155	even	6		1053.2.e.e.352.1		4
468.259	odd	6		1053.2.e.m.352.2		4
468.311	even	6		1053.2.e.e.703.1		4
572.571	even	2		4719.2.a.p.1.2		2
780.467	odd	4		2925.2.c.u.2224.4		4
780.623	odd	4		2925.2.c.u.2224.1		4
780.779	even	2		2925.2.a.v.1.1		2
1092.1091	odd	2		5733.2.a.u.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.2.a.b.1.1	✓	2	52.51	odd	2
117.2.a.c.1.2		2	156.155	even	2
507.2.a.h.1.2		2	4.3	odd	2
507.2.b.e.337.1		4	52.31	even	4
507.2.b.e.337.4		4	52.47	even	4
507.2.e.d.22.1		4	52.3	odd	6
507.2.e.d.484.1		4	52.35	odd	6
507.2.e.h.22.2		4	52.23	odd	6
507.2.e.h.484.2		4	52.43	odd	6
507.2.j.f.316.1		8	52.11	even	12
507.2.j.f.316.4		8	52.15	even	12
507.2.j.f.361.1		8	52.19	even	12
507.2.j.f.361.4		8	52.7	even	12
624.2.a.k.1.2		2	13.12	even	2
975.2.a.l.1.2		2	260.259	odd	2
975.2.c.h.274.1		4	260.207	even	4
975.2.c.h.274.4		4	260.103	even	4
1053.2.e.e.352.1		4	468.155	even	6
1053.2.e.e.703.1		4	468.311	even	6
1053.2.e.m.352.2		4	468.259	odd	6
1053.2.e.m.703.2		4	468.103	odd	6
1521.2.a.f.1.1		2	12.11	even	2
1521.2.b.j.1351.1		4	156.47	odd	4
1521.2.b.j.1351.4		4	156.83	odd	4
1872.2.a.w.1.1		2	39.38	odd	2
1911.2.a.h.1.1		2	364.363	even	2
2496.2.a.bf.1.1		2	104.51	odd	2
2496.2.a.bi.1.1		2	104.77	even	2
2925.2.a.v.1.1		2	780.779	even	2
2925.2.c.u.2224.1		4	780.623	odd	4
2925.2.c.u.2224.4		4	780.467	odd	4
4719.2.a.p.1.2		2	572.571	even	2
5733.2.a.u.1.2		2	1092.1091	odd	2
7488.2.a.cl.1.2		2	312.155	even	2
7488.2.a.co.1.2		2	312.77	odd	2
8112.2.a.bm.1.1		2	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:	\( N \)	\(=\)	\( 8112 = 2^{4} \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8112.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -2.82843 q^{5} -2.82843 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -2.82843 q^{5} -2.82843 q^{7} +1.00000 q^{9} -2.00000 q^{11} +2.82843 q^{15} -3.65685 q^{17} +2.82843 q^{19} +2.82843 q^{21} +4.00000 q^{23} +3.00000 q^{25} -1.00000 q^{27} +2.00000 q^{29} -6.82843 q^{31} +2.00000 q^{33} +8.00000 q^{35} -3.65685 q^{37} -10.8284 q^{41} -9.65685 q^{43} -2.82843 q^{45} -0.343146 q^{47} +1.00000 q^{49} +3.65685 q^{51} -2.00000 q^{53} +5.65685 q^{55} -2.82843 q^{57} -3.65685 q^{59} -9.31371 q^{61} -2.82843 q^{63} +1.17157 q^{67} -4.00000 q^{69} +2.00000 q^{71} -11.6569 q^{73} -3.00000 q^{75} +5.65685 q^{77} -11.3137 q^{79} +1.00000 q^{81} -7.65685 q^{83} +10.3431 q^{85} -2.00000 q^{87} -9.17157 q^{89} +6.82843 q^{93} -8.00000 q^{95} +7.65685 q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 2 * q^9	\( 2 q - 2 q^{3} + 2 q^{9} - 4 q^{11} + 4 q^{17} + 8 q^{23} + 6 q^{25} - 2 q^{27} + 4 q^{29} - 8 q^{31} + 4 q^{33} + 16 q^{35} + 4 q^{37} - 16 q^{41} - 8 q^{43} - 12 q^{47} + 2 q^{49} - 4 q^{51} - 4 q^{53} + 4 q^{59} + 4 q^{61} + 8 q^{67} - 8 q^{69} + 4 q^{71} - 12 q^{73} - 6 q^{75} + 2 q^{81} - 4 q^{83} + 32 q^{85} - 4 q^{87} - 24 q^{89} + 8 q^{93} - 16 q^{95} + 4 q^{97} - 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^9 - 4 * q^11 + 4 * q^17 + 8 * q^23 + 6 * q^25 - 2 * q^27 + 4 * q^29 - 8 * q^31 + 4 * q^33 + 16 * q^35 + 4 * q^37 - 16 * q^41 - 8 * q^43 - 12 * q^47 + 2 * q^49 - 4 * q^51 - 4 * q^53 + 4 * q^59 + 4 * q^61 + 8 * q^67 - 8 * q^69 + 4 * q^71 - 12 * q^73 - 6 * q^75 + 2 * q^81 - 4 * q^83 + 32 * q^85 - 4 * q^87 - 24 * q^89 + 8 * q^93 - 16 * q^95 + 4 * q^97 - 4 * q^99

Introduction

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