LMFDB - Embedded newform 4368.2.a.s.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4368,2,Mod(1,4368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4368, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4368.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4368.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:	$34.8786556029$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 546)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	4368.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$

$q+1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} +1.00000 q^{13} -1.00000 q^{15} -1.00000 q^{17} -7.00000 q^{19} +1.00000 q^{21} -3.00000 q^{23} -4.00000 q^{25} +1.00000 q^{27} -3.00000 q^{29} -8.00000 q^{31} +1.00000 q^{33} -1.00000 q^{35} +7.00000 q^{37} +1.00000 q^{39} +8.00000 q^{41} -7.00000 q^{43} -1.00000 q^{45} -8.00000 q^{47} +1.00000 q^{49} -1.00000 q^{51} -10.0000 q^{53} -1.00000 q^{55} -7.00000 q^{57} -4.00000 q^{59} +7.00000 q^{61} +1.00000 q^{63} -1.00000 q^{65} -2.00000 q^{67} -3.00000 q^{69} -4.00000 q^{71} -1.00000 q^{73} -4.00000 q^{75} +1.00000 q^{77} -2.00000 q^{79} +1.00000 q^{81} +6.00000 q^{83} +1.00000 q^{85} -3.00000 q^{87} +14.0000 q^{89} +1.00000 q^{91} -8.00000 q^{93} +7.00000 q^{95} -14.0000 q^{97} +1.00000 q^{99} +O(q^{100})$

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
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.00000	0.301511	0.150756	−	0.988571i	$-0.451829\pi$
$11$	1.00000	0.301511	0.150756	+	0.988571i	$0.451829\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−1.00000	−0.242536	−0.121268	−	0.992620i	$-0.538696\pi$
$17$	−1.00000	−0.242536	−0.121268	+	0.992620i	$0.538696\pi$
$18$	0	0
$19$	−7.00000	−1.60591	−0.802955	−	0.596040i	$-0.796740\pi$
$19$	−7.00000	−1.60591	−0.802955	+	0.596040i	$0.796740\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	−3.00000	−0.625543	−0.312772	−	0.949828i	$-0.601257\pi$
$23$	−3.00000	−0.625543	−0.312772	+	0.949828i	$0.601257\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−3.00000	−0.557086	−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000	−0.557086	−0.278543	+	0.960424i	$0.589851\pi$
$30$	0	0
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	0	0
$33$	1.00000	0.174078
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	7.00000	1.15079	0.575396	−	0.817875i	$-0.304848\pi$
$37$	7.00000	1.15079	0.575396	+	0.817875i	$0.304848\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	8.00000	1.24939	0.624695	−	0.780869i	$-0.285223\pi$
$41$	8.00000	1.24939	0.624695	+	0.780869i	$0.285223\pi$
$42$	0	0
$43$	−7.00000	−1.06749	−0.533745	−	0.845645i	$-0.679216\pi$
$43$	−7.00000	−1.06749	−0.533745	+	0.845645i	$0.679216\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−1.00000	−0.140028
$52$	0	0
$53$	−10.0000	−1.37361	−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000	−1.37361	−0.686803	+	0.726844i	$0.740986\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	0	0
$57$	−7.00000	−0.927173
$58$	0	0
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	7.00000	0.896258	0.448129	−	0.893969i	$-0.352090\pi$
$61$	7.00000	0.896258	0.448129	+	0.893969i	$0.352090\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	−1.00000	−0.124035
$66$	0	0
$67$	−2.00000	−0.244339	−0.122169	−	0.992509i	$-0.538985\pi$
$67$	−2.00000	−0.244339	−0.122169	+	0.992509i	$0.538985\pi$
$68$	0	0
$69$	−3.00000	−0.361158
$70$	0	0
$71$	−4.00000	−0.474713	−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000	−0.474713	−0.237356	+	0.971423i	$0.576281\pi$
$72$	0	0
$73$	−1.00000	−0.117041	−0.0585206	−	0.998286i	$-0.518638\pi$
$73$	−1.00000	−0.117041	−0.0585206	+	0.998286i	$0.518638\pi$
$74$	0	0
$75$	−4.00000	−0.461880
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	−2.00000	−0.225018	−0.112509	−	0.993651i	$-0.535889\pi$
$79$	−2.00000	−0.225018	−0.112509	+	0.993651i	$0.535889\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	0	0
$85$	1.00000	0.108465
$86$	0	0
$87$	−3.00000	−0.321634
$88$	0	0
$89$	14.0000	1.48400	0.741999	−	0.670402i	$-0.233878\pi$
$89$	14.0000	1.48400	0.741999	+	0.670402i	$0.233878\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	−8.00000	−0.829561
$94$	0	0
$95$	7.00000	0.718185
$96$	0	0
$97$	−14.0000	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	4.00000	0.398015	0.199007	−	0.979998i	$-0.436228\pi$
$101$	4.00000	0.398015	0.199007	+	0.979998i	$0.436228\pi$
$102$	0	0
$103$	5.00000	0.492665	0.246332	−	0.969185i	$-0.420775\pi$
$103$	5.00000	0.492665	0.246332	+	0.969185i	$0.420775\pi$
$104$	0	0
$105$	−1.00000	−0.0975900
$106$	0	0
$107$	18.0000	1.74013	0.870063	−	0.492941i	$-0.164078\pi$
$107$	18.0000	1.74013	0.870063	+	0.492941i	$0.164078\pi$
$108$	0	0
$109$	−11.0000	−1.05361	−0.526804	−	0.849987i	$-0.676610\pi$
$109$	−11.0000	−1.05361	−0.526804	+	0.849987i	$0.676610\pi$
$110$	0	0
$111$	7.00000	0.664411
$112$	0	0
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	3.00000	0.279751
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	−1.00000	−0.0916698
$120$	0	0
$121$	−10.0000	−0.909091
$122$	0	0
$123$	8.00000	0.721336
$124$	0	0
$125$	9.00000	0.804984
$126$	0	0
$127$	−18.0000	−1.59724	−0.798621	−	0.601834i	$-0.794437\pi$
$127$	−18.0000	−1.59724	−0.798621	+	0.601834i	$0.794437\pi$
$128$	0	0
$129$	−7.00000	−0.616316
$130$	0	0
$131$	5.00000	0.436852	0.218426	−	0.975854i	$-0.429908\pi$
$131$	5.00000	0.436852	0.218426	+	0.975854i	$0.429908\pi$
$132$	0	0
$133$	−7.00000	−0.606977
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−19.0000	−1.62328	−0.811640	−	0.584158i	$-0.801425\pi$
$137$	−19.0000	−1.62328	−0.811640	+	0.584158i	$0.801425\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	−8.00000	−0.673722
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	3.00000	0.249136
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	9.00000	0.732410	0.366205	−	0.930534i	$-0.380657\pi$
$151$	9.00000	0.732410	0.366205	+	0.930534i	$0.380657\pi$
$152$	0	0
$153$	−1.00000	−0.0808452
$154$	0	0
$155$	8.00000	0.642575
$156$	0	0
$157$	−13.0000	−1.03751	−0.518756	−	0.854922i	$-0.673605\pi$
$157$	−13.0000	−1.03751	−0.518756	+	0.854922i	$0.673605\pi$
$158$	0	0
$159$	−10.0000	−0.793052
$160$	0	0
$161$	−3.00000	−0.236433
$162$	0	0
$163$	−2.00000	−0.156652	−0.0783260	−	0.996928i	$-0.524958\pi$
$163$	−2.00000	−0.156652	−0.0783260	+	0.996928i	$0.524958\pi$
$164$	0	0
$165$	−1.00000	−0.0778499
$166$	0	0
$167$	−15.0000	−1.16073	−0.580367	−	0.814355i	$-0.697091\pi$
$167$	−15.0000	−1.16073	−0.580367	+	0.814355i	$0.697091\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−7.00000	−0.535303
$172$	0	0
$173$	14.0000	1.06440	0.532200	−	0.846619i	$-0.321365\pi$
$173$	14.0000	1.06440	0.532200	+	0.846619i	$0.321365\pi$
$174$	0	0
$175$	−4.00000	−0.302372
$176$	0	0
$177$	−4.00000	−0.300658
$178$	0	0
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	−22.0000	−1.63525	−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000	−1.63525	−0.817624	+	0.575753i	$0.804709\pi$
$182$	0	0
$183$	7.00000	0.517455
$184$	0	0
$185$	−7.00000	−0.514650
$186$	0	0
$187$	−1.00000	−0.0731272
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	−19.0000	−1.37479	−0.687396	−	0.726283i	$-0.741246\pi$
$191$	−19.0000	−1.37479	−0.687396	+	0.726283i	$0.741246\pi$
$192$	0	0
$193$	16.0000	1.15171	0.575853	−	0.817554i	$-0.304670\pi$
$193$	16.0000	1.15171	0.575853	+	0.817554i	$0.304670\pi$
$194$	0	0
$195$	−1.00000	−0.0716115
$196$	0	0
$197$	−12.0000	−0.854965	−0.427482	−	0.904024i	$-0.640599\pi$
$197$	−12.0000	−0.854965	−0.427482	+	0.904024i	$0.640599\pi$
$198$	0	0
$199$	−1.00000	−0.0708881	−0.0354441	−	0.999372i	$-0.511285\pi$
$199$	−1.00000	−0.0708881	−0.0354441	+	0.999372i	$0.511285\pi$
$200$	0	0
$201$	−2.00000	−0.141069
$202$	0	0
$203$	−3.00000	−0.210559
$204$	0	0
$205$	−8.00000	−0.558744
$206$	0	0
$207$	−3.00000	−0.208514
$208$	0	0
$209$	−7.00000	−0.484200
$210$	0	0
$211$	3.00000	0.206529	0.103264	−	0.994654i	$-0.467071\pi$
$211$	3.00000	0.206529	0.103264	+	0.994654i	$0.467071\pi$
$212$	0	0
$213$	−4.00000	−0.274075
$214$	0	0
$215$	7.00000	0.477396
$216$	0	0
$217$	−8.00000	−0.543075
$218$	0	0
$219$	−1.00000	−0.0675737
$220$	0	0
$221$	−1.00000	−0.0672673
$222$	0	0
$223$	−14.0000	−0.937509	−0.468755	−	0.883328i	$-0.655297\pi$
$223$	−14.0000	−0.937509	−0.468755	+	0.883328i	$0.655297\pi$
$224$	0	0
$225$	−4.00000	−0.266667
$226$	0	0
$227$	22.0000	1.46019	0.730096	−	0.683345i	$-0.239475\pi$
$227$	22.0000	1.46019	0.730096	+	0.683345i	$0.239475\pi$
$228$	0	0
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	0	0
$231$	1.00000	0.0657952
$232$	0	0
$233$	12.0000	0.786146	0.393073	−	0.919507i	$-0.371412\pi$
$233$	12.0000	0.786146	0.393073	+	0.919507i	$0.371412\pi$
$234$	0	0
$235$	8.00000	0.521862
$236$	0	0
$237$	−2.00000	−0.129914
$238$	0	0
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	−7.00000	−0.445399
$248$	0	0
$249$	6.00000	0.380235
$250$	0	0
$251$	−7.00000	−0.441836	−0.220918	−	0.975292i	$-0.570905\pi$
$251$	−7.00000	−0.441836	−0.220918	+	0.975292i	$0.570905\pi$
$252$	0	0
$253$	−3.00000	−0.188608
$254$	0	0
$255$	1.00000	0.0626224
$256$	0	0
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	0	0
$259$	7.00000	0.434959
$260$	0	0
$261$	−3.00000	−0.185695
$262$	0	0
$263$	4.00000	0.246651	0.123325	−	0.992366i	$-0.460644\pi$
$263$	4.00000	0.246651	0.123325	+	0.992366i	$0.460644\pi$
$264$	0	0
$265$	10.0000	0.614295
$266$	0	0
$267$	14.0000	0.856786
$268$	0	0
$269$	−2.00000	−0.121942	−0.0609711	−	0.998140i	$-0.519420\pi$
$269$	−2.00000	−0.121942	−0.0609711	+	0.998140i	$0.519420\pi$
$270$	0	0
$271$	−2.00000	−0.121491	−0.0607457	−	0.998153i	$-0.519348\pi$
$271$	−2.00000	−0.121491	−0.0607457	+	0.998153i	$0.519348\pi$
$272$	0	0
$273$	1.00000	0.0605228
$274$	0	0
$275$	−4.00000	−0.241209
$276$	0	0
$277$	−14.0000	−0.841178	−0.420589	−	0.907251i	$-0.638177\pi$
$277$	−14.0000	−0.841178	−0.420589	+	0.907251i	$0.638177\pi$
$278$	0	0
$279$	−8.00000	−0.478947
$280$	0	0
$281$	10.0000	0.596550	0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000	0.596550	0.298275	+	0.954480i	$0.403589\pi$
$282$	0	0
$283$	−32.0000	−1.90220	−0.951101	−	0.308879i	$-0.900046\pi$
$283$	−32.0000	−1.90220	−0.951101	+	0.308879i	$0.900046\pi$
$284$	0	0
$285$	7.00000	0.414644
$286$	0	0
$287$	8.00000	0.472225
$288$	0	0
$289$	−16.0000	−0.941176
$290$	0	0
$291$	−14.0000	−0.820695
$292$	0	0
$293$	14.0000	0.817889	0.408944	−	0.912559i	$-0.365897\pi$
$293$	14.0000	0.817889	0.408944	+	0.912559i	$0.365897\pi$
$294$	0	0
$295$	4.00000	0.232889
$296$	0	0
$297$	1.00000	0.0580259
$298$	0	0
$299$	−3.00000	−0.173494
$300$	0	0
$301$	−7.00000	−0.403473
$302$	0	0
$303$	4.00000	0.229794
$304$	0	0
$305$	−7.00000	−0.400819
$306$	0	0
$307$	32.0000	1.82634	0.913168	−	0.407583i	$-0.133628\pi$
$307$	32.0000	1.82634	0.913168	+	0.407583i	$0.133628\pi$
$308$	0	0
$309$	5.00000	0.284440
$310$	0	0
$311$	−30.0000	−1.70114	−0.850572	−	0.525859i	$-0.823744\pi$
$311$	−30.0000	−1.70114	−0.850572	+	0.525859i	$0.823744\pi$
$312$	0	0
$313$	−24.0000	−1.35656	−0.678280	−	0.734803i	$-0.737274\pi$
$313$	−24.0000	−1.35656	−0.678280	+	0.734803i	$0.737274\pi$
$314$	0	0
$315$	−1.00000	−0.0563436
$316$	0	0
$317$	12.0000	0.673987	0.336994	−	0.941507i	$-0.390590\pi$
$317$	12.0000	0.673987	0.336994	+	0.941507i	$0.390590\pi$
$318$	0	0
$319$	−3.00000	−0.167968
$320$	0	0
$321$	18.0000	1.00466
$322$	0	0
$323$	7.00000	0.389490
$324$	0	0
$325$	−4.00000	−0.221880
$326$	0	0
$327$	−11.0000	−0.608301
$328$	0	0
$329$	−8.00000	−0.441054
$330$	0	0
$331$	10.0000	0.549650	0.274825	−	0.961494i	$-0.411380\pi$
$331$	10.0000	0.549650	0.274825	+	0.961494i	$0.411380\pi$
$332$	0	0
$333$	7.00000	0.383598
$334$	0	0
$335$	2.00000	0.109272
$336$	0	0
$337$	−21.0000	−1.14394	−0.571971	−	0.820274i	$-0.693821\pi$
$337$	−21.0000	−1.14394	−0.571971	+	0.820274i	$0.693821\pi$
$338$	0	0
$339$	6.00000	0.325875
$340$	0	0
$341$	−8.00000	−0.433224
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	3.00000	0.161515
$346$	0	0
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	0	0
$349$	−18.0000	−0.963518	−0.481759	−	0.876304i	$-0.660002\pi$
$349$	−18.0000	−0.963518	−0.481759	+	0.876304i	$0.660002\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	14.0000	0.745145	0.372572	−	0.928003i	$-0.378476\pi$
$353$	14.0000	0.745145	0.372572	+	0.928003i	$0.378476\pi$
$354$	0	0
$355$	4.00000	0.212298
$356$	0	0
$357$	−1.00000	−0.0529256
$358$	0	0
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	30.0000	1.57895
$362$	0	0
$363$	−10.0000	−0.524864
$364$	0	0
$365$	1.00000	0.0523424
$366$	0	0
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	0	0
$369$	8.00000	0.416463
$370$	0	0
$371$	−10.0000	−0.519174
$372$	0	0
$373$	−12.0000	−0.621336	−0.310668	−	0.950518i	$-0.600553\pi$
$373$	−12.0000	−0.621336	−0.310668	+	0.950518i	$0.600553\pi$
$374$	0	0
$375$	9.00000	0.464758
$376$	0	0
$377$	−3.00000	−0.154508
$378$	0	0
$379$	−12.0000	−0.616399	−0.308199	−	0.951322i	$-0.599726\pi$
$379$	−12.0000	−0.616399	−0.308199	+	0.951322i	$0.599726\pi$
$380$	0	0
$381$	−18.0000	−0.922168
$382$	0	0
$383$	9.00000	0.459879	0.229939	−	0.973205i	$-0.426147\pi$
$383$	9.00000	0.459879	0.229939	+	0.973205i	$0.426147\pi$
$384$	0	0
$385$	−1.00000	−0.0509647
$386$	0	0
$387$	−7.00000	−0.355830
$388$	0	0
$389$	−2.00000	−0.101404	−0.0507020	−	0.998714i	$-0.516146\pi$
$389$	−2.00000	−0.101404	−0.0507020	+	0.998714i	$0.516146\pi$
$390$	0	0
$391$	3.00000	0.151717
$392$	0	0
$393$	5.00000	0.252217
$394$	0	0
$395$	2.00000	0.100631
$396$	0	0
$397$	18.0000	0.903394	0.451697	−	0.892171i	$-0.350819\pi$
$397$	18.0000	0.903394	0.451697	+	0.892171i	$0.350819\pi$
$398$	0	0
$399$	−7.00000	−0.350438
$400$	0	0
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	0	0
$403$	−8.00000	−0.398508
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	7.00000	0.346977
$408$	0	0
$409$	5.00000	0.247234	0.123617	−	0.992330i	$-0.460551\pi$
$409$	5.00000	0.247234	0.123617	+	0.992330i	$0.460551\pi$
$410$	0	0
$411$	−19.0000	−0.937201
$412$	0	0
$413$	−4.00000	−0.196827
$414$	0	0
$415$	−6.00000	−0.294528
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−1.00000	−0.0488532	−0.0244266	−	0.999702i	$-0.507776\pi$
$419$	−1.00000	−0.0488532	−0.0244266	+	0.999702i	$0.507776\pi$
$420$	0	0
$421$	34.0000	1.65706	0.828529	−	0.559946i	$-0.189178\pi$
$421$	34.0000	1.65706	0.828529	+	0.559946i	$0.189178\pi$
$422$	0	0
$423$	−8.00000	−0.388973
$424$	0	0
$425$	4.00000	0.194029
$426$	0	0
$427$	7.00000	0.338754
$428$	0	0
$429$	1.00000	0.0482805
$430$	0	0
$431$	38.0000	1.83040	0.915198	−	0.403005i	$-0.132034\pi$
$431$	38.0000	1.83040	0.915198	+	0.403005i	$0.132034\pi$
$432$	0	0
$433$	12.0000	0.576683	0.288342	−	0.957528i	$-0.406896\pi$
$433$	12.0000	0.576683	0.288342	+	0.957528i	$0.406896\pi$
$434$	0	0
$435$	3.00000	0.143839
$436$	0	0
$437$	21.0000	1.00457
$438$	0	0
$439$	17.0000	0.811366	0.405683	−	0.914014i	$-0.367034\pi$
$439$	17.0000	0.811366	0.405683	+	0.914014i	$0.367034\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	10.0000	0.475114	0.237557	−	0.971374i	$-0.423653\pi$
$443$	10.0000	0.475114	0.237557	+	0.971374i	$0.423653\pi$
$444$	0	0
$445$	−14.0000	−0.663664
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−39.0000	−1.84052	−0.920262	−	0.391303i	$-0.872024\pi$
$449$	−39.0000	−1.84052	−0.920262	+	0.391303i	$0.872024\pi$
$450$	0	0
$451$	8.00000	0.376705
$452$	0	0
$453$	9.00000	0.422857
$454$	0	0
$455$	−1.00000	−0.0468807
$456$	0	0
$457$	−12.0000	−0.561336	−0.280668	−	0.959805i	$-0.590556\pi$
$457$	−12.0000	−0.561336	−0.280668	+	0.959805i	$0.590556\pi$
$458$	0	0
$459$	−1.00000	−0.0466760
$460$	0	0
$461$	21.0000	0.978068	0.489034	−	0.872265i	$-0.337349\pi$
$461$	21.0000	0.978068	0.489034	+	0.872265i	$0.337349\pi$
$462$	0	0
$463$	17.0000	0.790057	0.395029	−	0.918669i	$-0.370735\pi$
$463$	17.0000	0.790057	0.395029	+	0.918669i	$0.370735\pi$
$464$	0	0
$465$	8.00000	0.370991
$466$	0	0
$467$	25.0000	1.15686	0.578431	−	0.815731i	$-0.303665\pi$
$467$	25.0000	1.15686	0.578431	+	0.815731i	$0.303665\pi$
$468$	0	0
$469$	−2.00000	−0.0923514
$470$	0	0
$471$	−13.0000	−0.599008
$472$	0	0
$473$	−7.00000	−0.321860
$474$	0	0
$475$	28.0000	1.28473
$476$	0	0
$477$	−10.0000	−0.457869
$478$	0	0
$479$	−5.00000	−0.228456	−0.114228	−	0.993455i	$-0.536439\pi$
$479$	−5.00000	−0.228456	−0.114228	+	0.993455i	$0.536439\pi$
$480$	0	0
$481$	7.00000	0.319173
$482$	0	0
$483$	−3.00000	−0.136505
$484$	0	0
$485$	14.0000	0.635707
$486$	0	0
$487$	−16.0000	−0.725029	−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000	−0.725029	−0.362515	+	0.931978i	$0.618082\pi$
$488$	0	0
$489$	−2.00000	−0.0904431
$490$	0	0
$491$	30.0000	1.35388	0.676941	−	0.736038i	$-0.263305\pi$
$491$	30.0000	1.35388	0.676941	+	0.736038i	$0.263305\pi$
$492$	0	0
$493$	3.00000	0.135113
$494$	0	0
$495$	−1.00000	−0.0449467
$496$	0	0
$497$	−4.00000	−0.179425
$498$	0	0
$499$	−12.0000	−0.537194	−0.268597	−	0.963253i	$-0.586560\pi$
$499$	−12.0000	−0.537194	−0.268597	+	0.963253i	$0.586560\pi$
$500$	0	0
$501$	−15.0000	−0.670151
$502$	0	0
$503$	−34.0000	−1.51599	−0.757993	−	0.652263i	$-0.773820\pi$
$503$	−34.0000	−1.51599	−0.757993	+	0.652263i	$0.773820\pi$
$504$	0	0
$505$	−4.00000	−0.177998
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	−3.00000	−0.132973	−0.0664863	−	0.997787i	$-0.521179\pi$
$509$	−3.00000	−0.132973	−0.0664863	+	0.997787i	$0.521179\pi$
$510$	0	0
$511$	−1.00000	−0.0442374
$512$	0	0
$513$	−7.00000	−0.309058
$514$	0	0
$515$	−5.00000	−0.220326
$516$	0	0
$517$	−8.00000	−0.351840
$518$	0	0
$519$	14.0000	0.614532
$520$	0	0
$521$	−35.0000	−1.53338	−0.766689	−	0.642019i	$-0.778097\pi$
$521$	−35.0000	−1.53338	−0.766689	+	0.642019i	$0.778097\pi$
$522$	0	0
$523$	20.0000	0.874539	0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000	0.874539	0.437269	+	0.899331i	$0.355946\pi$
$524$	0	0
$525$	−4.00000	−0.174574
$526$	0	0
$527$	8.00000	0.348485
$528$	0	0
$529$	−14.0000	−0.608696
$530$	0	0
$531$	−4.00000	−0.173585
$532$	0	0
$533$	8.00000	0.346518
$534$	0	0
$535$	−18.0000	−0.778208
$536$	0	0
$537$	−12.0000	−0.517838
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	−43.0000	−1.84871	−0.924357	−	0.381528i	$-0.875398\pi$
$541$	−43.0000	−1.84871	−0.924357	+	0.381528i	$0.875398\pi$
$542$	0	0
$543$	−22.0000	−0.944110
$544$	0	0
$545$	11.0000	0.471188
$546$	0	0
$547$	4.00000	0.171028	0.0855138	−	0.996337i	$-0.472747\pi$
$547$	4.00000	0.171028	0.0855138	+	0.996337i	$0.472747\pi$
$548$	0	0
$549$	7.00000	0.298753
$550$	0	0
$551$	21.0000	0.894630
$552$	0	0
$553$	−2.00000	−0.0850487
$554$	0	0
$555$	−7.00000	−0.297133
$556$	0	0
$557$	−24.0000	−1.01691	−0.508456	−	0.861088i	$-0.669784\pi$
$557$	−24.0000	−1.01691	−0.508456	+	0.861088i	$0.669784\pi$
$558$	0	0
$559$	−7.00000	−0.296068
$560$	0	0
$561$	−1.00000	−0.0422200
$562$	0	0
$563$	−19.0000	−0.800755	−0.400377	−	0.916350i	$-0.631121\pi$
$563$	−19.0000	−0.800755	−0.400377	+	0.916350i	$0.631121\pi$
$564$	0	0
$565$	−6.00000	−0.252422
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	0	0
$571$	−16.0000	−0.669579	−0.334790	−	0.942293i	$-0.608665\pi$
$571$	−16.0000	−0.669579	−0.334790	+	0.942293i	$0.608665\pi$
$572$	0	0
$573$	−19.0000	−0.793736
$574$	0	0
$575$	12.0000	0.500435
$576$	0	0
$577$	42.0000	1.74848	0.874241	−	0.485491i	$-0.161359\pi$
$577$	42.0000	1.74848	0.874241	+	0.485491i	$0.161359\pi$
$578$	0	0
$579$	16.0000	0.664937
$580$	0	0
$581$	6.00000	0.248922
$582$	0	0
$583$	−10.0000	−0.414158
$584$	0	0
$585$	−1.00000	−0.0413449
$586$	0	0
$587$	38.0000	1.56843	0.784214	−	0.620491i	$-0.213066\pi$
$587$	38.0000	1.56843	0.784214	+	0.620491i	$0.213066\pi$
$588$	0	0
$589$	56.0000	2.30744
$590$	0	0
$591$	−12.0000	−0.493614
$592$	0	0
$593$	30.0000	1.23195	0.615976	−	0.787765i	$-0.288762\pi$
$593$	30.0000	1.23195	0.615976	+	0.787765i	$0.288762\pi$
$594$	0	0
$595$	1.00000	0.0409960
$596$	0	0
$597$	−1.00000	−0.0409273
$598$	0	0
$599$	3.00000	0.122577	0.0612883	−	0.998120i	$-0.480479\pi$
$599$	3.00000	0.122577	0.0612883	+	0.998120i	$0.480479\pi$
$600$	0	0
$601$	−28.0000	−1.14214	−0.571072	−	0.820900i	$-0.693472\pi$
$601$	−28.0000	−1.14214	−0.571072	+	0.820900i	$0.693472\pi$
$602$	0	0
$603$	−2.00000	−0.0814463
$604$	0	0
$605$	10.0000	0.406558
$606$	0	0
$607$	1.00000	0.0405887	0.0202944	−	0.999794i	$-0.493540\pi$
$607$	1.00000	0.0405887	0.0202944	+	0.999794i	$0.493540\pi$
$608$	0	0
$609$	−3.00000	−0.121566
$610$	0	0
$611$	−8.00000	−0.323645
$612$	0	0
$613$	37.0000	1.49442	0.747208	−	0.664590i	$-0.231394\pi$
$613$	37.0000	1.49442	0.747208	+	0.664590i	$0.231394\pi$
$614$	0	0
$615$	−8.00000	−0.322591
$616$	0	0
$617$	−33.0000	−1.32853	−0.664265	−	0.747497i	$-0.731255\pi$
$617$	−33.0000	−1.32853	−0.664265	+	0.747497i	$0.731255\pi$
$618$	0	0
$619$	1.00000	0.0401934	0.0200967	−	0.999798i	$-0.493603\pi$
$619$	1.00000	0.0401934	0.0200967	+	0.999798i	$0.493603\pi$
$620$	0	0
$621$	−3.00000	−0.120386
$622$	0	0
$623$	14.0000	0.560898
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	−7.00000	−0.279553
$628$	0	0
$629$	−7.00000	−0.279108
$630$	0	0
$631$	−27.0000	−1.07485	−0.537427	−	0.843311i	$-0.680603\pi$
$631$	−27.0000	−1.07485	−0.537427	+	0.843311i	$0.680603\pi$
$632$	0	0
$633$	3.00000	0.119239
$634$	0	0
$635$	18.0000	0.714308
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	−4.00000	−0.158238
$640$	0	0
$641$	50.0000	1.97488	0.987441	−	0.157991i	$-0.0505015\pi$
$641$	50.0000	1.97488	0.987441	+	0.157991i	$0.0505015\pi$
$642$	0	0
$643$	−17.0000	−0.670415	−0.335207	−	0.942144i	$-0.608806\pi$
$643$	−17.0000	−0.670415	−0.335207	+	0.942144i	$0.608806\pi$
$644$	0	0
$645$	7.00000	0.275625
$646$	0	0
$647$	−12.0000	−0.471769	−0.235884	−	0.971781i	$-0.575799\pi$
$647$	−12.0000	−0.471769	−0.235884	+	0.971781i	$0.575799\pi$
$648$	0	0
$649$	−4.00000	−0.157014
$650$	0	0
$651$	−8.00000	−0.313545
$652$	0	0
$653$	35.0000	1.36966	0.684828	−	0.728705i	$-0.259877\pi$
$653$	35.0000	1.36966	0.684828	+	0.728705i	$0.259877\pi$
$654$	0	0
$655$	−5.00000	−0.195366
$656$	0	0
$657$	−1.00000	−0.0390137
$658$	0	0
$659$	−6.00000	−0.233727	−0.116863	−	0.993148i	$-0.537284\pi$
$659$	−6.00000	−0.233727	−0.116863	+	0.993148i	$0.537284\pi$
$660$	0	0
$661$	22.0000	0.855701	0.427850	−	0.903850i	$-0.359271\pi$
$661$	22.0000	0.855701	0.427850	+	0.903850i	$0.359271\pi$
$662$	0	0
$663$	−1.00000	−0.0388368
$664$	0	0
$665$	7.00000	0.271448
$666$	0	0
$667$	9.00000	0.348481
$668$	0	0
$669$	−14.0000	−0.541271
$670$	0	0
$671$	7.00000	0.270232
$672$	0	0
$673$	7.00000	0.269830	0.134915	−	0.990857i	$-0.456924\pi$
$673$	7.00000	0.269830	0.134915	+	0.990857i	$0.456924\pi$
$674$	0	0
$675$	−4.00000	−0.153960
$676$	0	0
$677$	−22.0000	−0.845529	−0.422764	−	0.906240i	$-0.638940\pi$
$677$	−22.0000	−0.845529	−0.422764	+	0.906240i	$0.638940\pi$
$678$	0	0
$679$	−14.0000	−0.537271
$680$	0	0
$681$	22.0000	0.843042
$682$	0	0
$683$	19.0000	0.727015	0.363507	−	0.931591i	$-0.381579\pi$
$683$	19.0000	0.727015	0.363507	+	0.931591i	$0.381579\pi$
$684$	0	0
$685$	19.0000	0.725953
$686$	0	0
$687$	10.0000	0.381524
$688$	0	0
$689$	−10.0000	−0.380970
$690$	0	0
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	0	0
$693$	1.00000	0.0379869
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−8.00000	−0.303022
$698$	0	0
$699$	12.0000	0.453882
$700$	0	0
$701$	10.0000	0.377695	0.188847	−	0.982006i	$-0.439525\pi$
$701$	10.0000	0.377695	0.188847	+	0.982006i	$0.439525\pi$
$702$	0	0
$703$	−49.0000	−1.84807
$704$	0	0
$705$	8.00000	0.301297
$706$	0	0
$707$	4.00000	0.150435
$708$	0	0
$709$	−6.00000	−0.225335	−0.112667	−	0.993633i	$-0.535939\pi$
$709$	−6.00000	−0.225335	−0.112667	+	0.993633i	$0.535939\pi$
$710$	0	0
$711$	−2.00000	−0.0750059
$712$	0	0
$713$	24.0000	0.898807
$714$	0	0
$715$	−1.00000	−0.0373979
$716$	0	0
$717$	−24.0000	−0.896296
$718$	0	0
$719$	42.0000	1.56634	0.783168	−	0.621810i	$-0.213603\pi$
$719$	42.0000	1.56634	0.783168	+	0.621810i	$0.213603\pi$
$720$	0	0
$721$	5.00000	0.186210
$722$	0	0
$723$	10.0000	0.371904
$724$	0	0
$725$	12.0000	0.445669
$726$	0	0
$727$	−23.0000	−0.853023	−0.426511	−	0.904482i	$-0.640258\pi$
$727$	−23.0000	−0.853023	−0.426511	+	0.904482i	$0.640258\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	7.00000	0.258904
$732$	0	0
$733$	−32.0000	−1.18195	−0.590973	−	0.806691i	$-0.701256\pi$
$733$	−32.0000	−1.18195	−0.590973	+	0.806691i	$0.701256\pi$
$734$	0	0
$735$	−1.00000	−0.0368856
$736$	0	0
$737$	−2.00000	−0.0736709
$738$	0	0
$739$	−22.0000	−0.809283	−0.404642	−	0.914475i	$-0.632604\pi$
$739$	−22.0000	−0.809283	−0.404642	+	0.914475i	$0.632604\pi$
$740$	0	0
$741$	−7.00000	−0.257151
$742$	0	0
$743$	−6.00000	−0.220119	−0.110059	−	0.993925i	$-0.535104\pi$
$743$	−6.00000	−0.220119	−0.110059	+	0.993925i	$0.535104\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	6.00000	0.219529
$748$	0	0
$749$	18.0000	0.657706
$750$	0	0
$751$	28.0000	1.02173	0.510867	−	0.859660i	$-0.329324\pi$
$751$	28.0000	1.02173	0.510867	+	0.859660i	$0.329324\pi$
$752$	0	0
$753$	−7.00000	−0.255094
$754$	0	0
$755$	−9.00000	−0.327544
$756$	0	0
$757$	18.0000	0.654221	0.327111	−	0.944986i	$-0.393925\pi$
$757$	18.0000	0.654221	0.327111	+	0.944986i	$0.393925\pi$
$758$	0	0
$759$	−3.00000	−0.108893
$760$	0	0
$761$	16.0000	0.580000	0.290000	−	0.957027i	$-0.406345\pi$
$761$	16.0000	0.580000	0.290000	+	0.957027i	$0.406345\pi$
$762$	0	0
$763$	−11.0000	−0.398227
$764$	0	0
$765$	1.00000	0.0361551
$766$	0	0
$767$	−4.00000	−0.144432
$768$	0	0
$769$	5.00000	0.180305	0.0901523	−	0.995928i	$-0.471265\pi$
$769$	5.00000	0.180305	0.0901523	+	0.995928i	$0.471265\pi$
$770$	0	0
$771$	18.0000	0.648254
$772$	0	0
$773$	−27.0000	−0.971123	−0.485561	−	0.874203i	$-0.661385\pi$
$773$	−27.0000	−0.971123	−0.485561	+	0.874203i	$0.661385\pi$
$774$	0	0
$775$	32.0000	1.14947
$776$	0	0
$777$	7.00000	0.251124
$778$	0	0
$779$	−56.0000	−2.00641
$780$	0	0
$781$	−4.00000	−0.143131
$782$	0	0
$783$	−3.00000	−0.107211
$784$	0	0
$785$	13.0000	0.463990
$786$	0	0
$787$	−55.0000	−1.96054	−0.980269	−	0.197667i	$-0.936663\pi$
$787$	−55.0000	−1.96054	−0.980269	+	0.197667i	$0.936663\pi$
$788$	0	0
$789$	4.00000	0.142404
$790$	0	0
$791$	6.00000	0.213335
$792$	0	0
$793$	7.00000	0.248577
$794$	0	0
$795$	10.0000	0.354663
$796$	0	0
$797$	16.0000	0.566749	0.283375	−	0.959009i	$-0.408546\pi$
$797$	16.0000	0.566749	0.283375	+	0.959009i	$0.408546\pi$
$798$	0	0
$799$	8.00000	0.283020
$800$	0	0
$801$	14.0000	0.494666
$802$	0	0
$803$	−1.00000	−0.0352892
$804$	0	0
$805$	3.00000	0.105736
$806$	0	0
$807$	−2.00000	−0.0704033
$808$	0	0
$809$	−10.0000	−0.351581	−0.175791	−	0.984428i	$-0.556248\pi$
$809$	−10.0000	−0.351581	−0.175791	+	0.984428i	$0.556248\pi$
$810$	0	0
$811$	45.0000	1.58016	0.790082	−	0.613001i	$-0.210038\pi$
$811$	45.0000	1.58016	0.790082	+	0.613001i	$0.210038\pi$
$812$	0	0
$813$	−2.00000	−0.0701431
$814$	0	0
$815$	2.00000	0.0700569
$816$	0	0
$817$	49.0000	1.71429
$818$	0	0
$819$	1.00000	0.0349428
$820$	0	0
$821$	−24.0000	−0.837606	−0.418803	−	0.908077i	$-0.637550\pi$
$821$	−24.0000	−0.837606	−0.418803	+	0.908077i	$0.637550\pi$
$822$	0	0
$823$	−20.0000	−0.697156	−0.348578	−	0.937280i	$-0.613335\pi$
$823$	−20.0000	−0.697156	−0.348578	+	0.937280i	$0.613335\pi$
$824$	0	0
$825$	−4.00000	−0.139262
$826$	0	0
$827$	41.0000	1.42571	0.712855	−	0.701312i	$-0.247402\pi$
$827$	41.0000	1.42571	0.712855	+	0.701312i	$0.247402\pi$
$828$	0	0
$829$	7.00000	0.243120	0.121560	−	0.992584i	$-0.461210\pi$
$829$	7.00000	0.243120	0.121560	+	0.992584i	$0.461210\pi$
$830$	0	0
$831$	−14.0000	−0.485655
$832$	0	0
$833$	−1.00000	−0.0346479
$834$	0	0
$835$	15.0000	0.519096
$836$	0	0
$837$	−8.00000	−0.276520
$838$	0	0
$839$	4.00000	0.138095	0.0690477	−	0.997613i	$-0.478004\pi$
$839$	4.00000	0.138095	0.0690477	+	0.997613i	$0.478004\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	0	0
$843$	10.0000	0.344418
$844$	0	0
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	−10.0000	−0.343604
$848$	0	0
$849$	−32.0000	−1.09824
$850$	0	0
$851$	−21.0000	−0.719871
$852$	0	0
$853$	−44.0000	−1.50653	−0.753266	−	0.657716i	$-0.771523\pi$
$853$	−44.0000	−1.50653	−0.753266	+	0.657716i	$0.771523\pi$
$854$	0	0
$855$	7.00000	0.239395
$856$	0	0
$857$	−18.0000	−0.614868	−0.307434	−	0.951569i	$-0.599470\pi$
$857$	−18.0000	−0.614868	−0.307434	+	0.951569i	$0.599470\pi$
$858$	0	0
$859$	44.0000	1.50126	0.750630	−	0.660722i	$-0.229750\pi$
$859$	44.0000	1.50126	0.750630	+	0.660722i	$0.229750\pi$
$860$	0	0
$861$	8.00000	0.272639
$862$	0	0
$863$	−46.0000	−1.56586	−0.782929	−	0.622111i	$-0.786275\pi$
$863$	−46.0000	−1.56586	−0.782929	+	0.622111i	$0.786275\pi$
$864$	0	0
$865$	−14.0000	−0.476014
$866$	0	0
$867$	−16.0000	−0.543388
$868$	0	0
$869$	−2.00000	−0.0678454
$870$	0	0
$871$	−2.00000	−0.0677674
$872$	0	0
$873$	−14.0000	−0.473828
$874$	0	0
$875$	9.00000	0.304256
$876$	0	0
$877$	34.0000	1.14810	0.574049	−	0.818821i	$-0.305372\pi$
$877$	34.0000	1.14810	0.574049	+	0.818821i	$0.305372\pi$
$878$	0	0
$879$	14.0000	0.472208
$880$	0	0
$881$	−49.0000	−1.65085	−0.825426	−	0.564510i	$-0.809065\pi$
$881$	−49.0000	−1.65085	−0.825426	+	0.564510i	$0.809065\pi$
$882$	0	0
$883$	−31.0000	−1.04323	−0.521617	−	0.853180i	$-0.674671\pi$
$883$	−31.0000	−1.04323	−0.521617	+	0.853180i	$0.674671\pi$
$884$	0	0
$885$	4.00000	0.134459
$886$	0	0
$887$	36.0000	1.20876	0.604381	−	0.796696i	$-0.293421\pi$
$887$	36.0000	1.20876	0.604381	+	0.796696i	$0.293421\pi$
$888$	0	0
$889$	−18.0000	−0.603701
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	56.0000	1.87397
$894$	0	0
$895$	12.0000	0.401116
$896$	0	0
$897$	−3.00000	−0.100167
$898$	0	0
$899$	24.0000	0.800445
$900$	0	0
$901$	10.0000	0.333148
$902$	0	0
$903$	−7.00000	−0.232945
$904$	0	0
$905$	22.0000	0.731305
$906$	0	0
$907$	−4.00000	−0.132818	−0.0664089	−	0.997792i	$-0.521154\pi$
$907$	−4.00000	−0.132818	−0.0664089	+	0.997792i	$0.521154\pi$
$908$	0	0
$909$	4.00000	0.132672
$910$	0	0
$911$	−13.0000	−0.430709	−0.215355	−	0.976536i	$-0.569091\pi$
$911$	−13.0000	−0.430709	−0.215355	+	0.976536i	$0.569091\pi$
$912$	0	0
$913$	6.00000	0.198571
$914$	0	0
$915$	−7.00000	−0.231413
$916$	0	0
$917$	5.00000	0.165115
$918$	0	0
$919$	−38.0000	−1.25350	−0.626752	−	0.779219i	$-0.715616\pi$
$919$	−38.0000	−1.25350	−0.626752	+	0.779219i	$0.715616\pi$
$920$	0	0
$921$	32.0000	1.05444
$922$	0	0
$923$	−4.00000	−0.131662
$924$	0	0
$925$	−28.0000	−0.920634
$926$	0	0
$927$	5.00000	0.164222
$928$	0	0
$929$	28.0000	0.918650	0.459325	−	0.888268i	$-0.348091\pi$
$929$	28.0000	0.918650	0.459325	+	0.888268i	$0.348091\pi$
$930$	0	0
$931$	−7.00000	−0.229416
$932$	0	0
$933$	−30.0000	−0.982156
$934$	0	0
$935$	1.00000	0.0327035
$936$	0	0
$937$	2.00000	0.0653372	0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000	0.0653372	0.0326686	+	0.999466i	$0.489599\pi$
$938$	0	0
$939$	−24.0000	−0.783210
$940$	0	0
$941$	−46.0000	−1.49956	−0.749779	−	0.661689i	$-0.769840\pi$
$941$	−46.0000	−1.49956	−0.749779	+	0.661689i	$0.769840\pi$
$942$	0	0
$943$	−24.0000	−0.781548
$944$	0	0
$945$	−1.00000	−0.0325300
$946$	0	0
$947$	13.0000	0.422443	0.211222	−	0.977438i	$-0.432256\pi$
$947$	13.0000	0.422443	0.211222	+	0.977438i	$0.432256\pi$
$948$	0	0
$949$	−1.00000	−0.0324614
$950$	0	0
$951$	12.0000	0.389127
$952$	0	0
$953$	36.0000	1.16615	0.583077	−	0.812417i	$-0.301849\pi$
$953$	36.0000	1.16615	0.583077	+	0.812417i	$0.301849\pi$
$954$	0	0
$955$	19.0000	0.614826
$956$	0	0
$957$	−3.00000	−0.0969762
$958$	0	0
$959$	−19.0000	−0.613542
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	18.0000	0.580042
$964$	0	0
$965$	−16.0000	−0.515058
$966$	0	0
$967$	−13.0000	−0.418052	−0.209026	−	0.977910i	$-0.567029\pi$
$967$	−13.0000	−0.418052	−0.209026	+	0.977910i	$0.567029\pi$
$968$	0	0
$969$	7.00000	0.224872
$970$	0	0
$971$	40.0000	1.28366	0.641831	−	0.766846i	$-0.278175\pi$
$971$	40.0000	1.28366	0.641831	+	0.766846i	$0.278175\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−4.00000	−0.128103
$976$	0	0
$977$	−45.0000	−1.43968	−0.719839	−	0.694141i	$-0.755784\pi$
$977$	−45.0000	−1.43968	−0.719839	+	0.694141i	$0.755784\pi$
$978$	0	0
$979$	14.0000	0.447442
$980$	0	0
$981$	−11.0000	−0.351203
$982$	0	0
$983$	3.00000	0.0956851	0.0478426	−	0.998855i	$-0.484765\pi$
$983$	3.00000	0.0956851	0.0478426	+	0.998855i	$0.484765\pi$
$984$	0	0
$985$	12.0000	0.382352
$986$	0	0
$987$	−8.00000	−0.254643
$988$	0	0
$989$	21.0000	0.667761
$990$	0	0
$991$	−42.0000	−1.33417	−0.667087	−	0.744980i	$-0.732459\pi$
$991$	−42.0000	−1.33417	−0.667087	+	0.744980i	$0.732459\pi$
$992$	0	0
$993$	10.0000	0.317340
$994$	0	0
$995$	1.00000	0.0317021
$996$	0	0
$997$	−14.0000	−0.443384	−0.221692	−	0.975117i	$-0.571158\pi$
$997$	−14.0000	−0.443384	−0.221692	+	0.975117i	$0.571158\pi$
$998$	0	0
$999$	7.00000	0.221470

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4368.2.a.s.1.1		1
4.3	odd	2		546.2.a.a.1.1	✓	1
12.11	even	2		1638.2.a.r.1.1		1
28.27	even	2		3822.2.a.n.1.1		1
52.51	odd	2		7098.2.a.t.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.2.a.a.1.1	✓	1	4.3	odd	2
1638.2.a.r.1.1		1	12.11	even	2
3822.2.a.n.1.1		1	28.27	even	2
4368.2.a.s.1.1		1	1.1	even	1	trivial
7098.2.a.t.1.1		1	52.51	odd	2

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:	\( N \)	\(=\)	\( 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4368.a (trivial)

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