LMFDB - Embedded newform 6240.2.a.ce.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6240,2,Mod(1,6240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6240, 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("6240.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6240 = 2^{5} \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6240.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:	$49.8266508613$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.15317.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 5x + 2 $ x^4 - 2x^3 - 4x^2 + 5*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.329727$ of defining polynomial
Character	$\chi$	$=$	6240.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} -3.50407 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.00000 q^{5} -3.50407 q^{7} +1.00000 q^{9} -5.96772 q^{11} +1.00000 q^{13} -1.00000 q^{15} -1.50407 q^{17} -7.78256 q^{19} +3.50407 q^{21} -3.50407 q^{23} +1.00000 q^{25} -1.00000 q^{27} -5.78256 q^{29} +5.96772 q^{33} -3.50407 q^{35} -3.96772 q^{37} -1.00000 q^{39} +7.96772 q^{41} -1.68923 q^{43} +1.00000 q^{45} +5.27849 q^{49} +1.50407 q^{51} -5.65695 q^{53} -5.96772 q^{55} +7.78256 q^{57} +14.9436 q^{59} +3.04042 q^{61} -3.50407 q^{63} +1.00000 q^{65} +6.24621 q^{67} +3.50407 q^{69} -0.648810 q^{71} -3.47179 q^{73} -1.00000 q^{75} +20.9113 q^{77} -7.28663 q^{79} +1.00000 q^{81} +10.2462 q^{83} -1.50407 q^{85} +5.78256 q^{87} -5.28663 q^{89} -3.50407 q^{91} -7.78256 q^{95} -17.0692 q^{97} -5.96772 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} + 4 q^{5} - 3 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 + 4 * q^5 - 3 * q^7 + 4 * q^9	$ 4 q - 4 q^{3} + 4 q^{5} - 3 q^{7} + 4 q^{9} - 3 q^{11} + 4 q^{13} - 4 q^{15} + 5 q^{17} - 8 q^{19} + 3 q^{21} - 3 q^{23} + 4 q^{25} - 4 q^{27} + 3 q^{33} - 3 q^{35} + 5 q^{37} - 4 q^{39} + 11 q^{41} + 2 q^{43} + 4 q^{45} + 9 q^{49} - 5 q^{51} + 7 q^{53} - 3 q^{55} + 8 q^{57} - 4 q^{59} + 11 q^{61} - 3 q^{63} + 4 q^{65} - 8 q^{67} + 3 q^{69} + 5 q^{71} + 18 q^{73} - 4 q^{75} - q^{77} + 5 q^{79} + 4 q^{81} + 8 q^{83} + 5 q^{85} + 13 q^{89} - 3 q^{91} - 8 q^{95} - 11 q^{97} - 3 q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 + 4 * q^5 - 3 * q^7 + 4 * q^9 - 3 * q^11 + 4 * q^13 - 4 * q^15 + 5 * q^17 - 8 * q^19 + 3 * q^21 - 3 * q^23 + 4 * q^25 - 4 * q^27 + 3 * q^33 - 3 * q^35 + 5 * q^37 - 4 * q^39 + 11 * q^41 + 2 * q^43 + 4 * q^45 + 9 * q^49 - 5 * q^51 + 7 * q^53 - 3 * q^55 + 8 * q^57 - 4 * q^59 + 11 * q^61 - 3 * q^63 + 4 * q^65 - 8 * q^67 + 3 * q^69 + 5 * q^71 + 18 * q^73 - 4 * q^75 - q^77 + 5 * q^79 + 4 * q^81 + 8 * q^83 + 5 * q^85 + 13 * q^89 - 3 * q^91 - 8 * q^95 - 11 * q^97 - 3 * 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
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−3.50407	−1.32441	−0.662207	−	0.749321i	$-0.730380\pi$
$7$	−3.50407	−1.32441	−0.662207	+	0.749321i	$0.730380\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.96772	−1.79934	−0.899668	−	0.436576i	$-0.856191\pi$
$11$	−5.96772	−1.79934	−0.899668	+	0.436576i	$0.856191\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−1.50407	−0.364790	−0.182395	−	0.983225i	$-0.558385\pi$
$17$	−1.50407	−0.364790	−0.182395	+	0.983225i	$0.558385\pi$
$18$	0	0
$19$	−7.78256	−1.78544	−0.892721	−	0.450610i	$-0.851206\pi$
$19$	−7.78256	−1.78544	−0.892721	+	0.450610i	$0.851206\pi$
$20$	0	0
$21$	3.50407	0.764650
$22$	0	0
$23$	−3.50407	−0.730649	−0.365324	−	0.930880i	$-0.619042\pi$
$23$	−3.50407	−0.730649	−0.365324	+	0.930880i	$0.619042\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−5.78256	−1.07379	−0.536897	−	0.843648i	$-0.680404\pi$
$29$	−5.78256	−1.07379	−0.536897	+	0.843648i	$0.680404\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	5.96772	1.03885
$34$	0	0
$35$	−3.50407	−0.592296
$36$	0	0
$37$	−3.96772	−0.652289	−0.326145	−	0.945320i	$-0.605750\pi$
$37$	−3.96772	−0.652289	−0.326145	+	0.945320i	$0.605750\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	7.96772	1.24435	0.622174	−	0.782879i	$-0.286249\pi$
$41$	7.96772	1.24435	0.622174	+	0.782879i	$0.286249\pi$
$42$	0	0
$43$	−1.68923	−0.257605	−0.128802	−	0.991670i	$-0.541113\pi$
$43$	−1.68923	−0.257605	−0.128802	+	0.991670i	$0.541113\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	5.27849	0.754070
$50$	0	0
$51$	1.50407	0.210612
$52$	0	0
$53$	−5.65695	−0.777041	−0.388521	−	0.921440i	$-0.627014\pi$
$53$	−5.65695	−0.777041	−0.388521	+	0.921440i	$0.627014\pi$
$54$	0	0
$55$	−5.96772	−0.804687
$56$	0	0
$57$	7.78256	1.03083
$58$	0	0
$59$	14.9436	1.94549	0.972744	−	0.231882i	$-0.0744885\pi$
$59$	14.9436	1.94549	0.972744	+	0.231882i	$0.0744885\pi$
$60$	0	0
$61$	3.04042	0.389285	0.194643	−	0.980874i	$-0.437645\pi$
$61$	3.04042	0.389285	0.194643	+	0.980874i	$0.437645\pi$
$62$	0	0
$63$	−3.50407	−0.441471
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	6.24621	0.763096	0.381548	−	0.924349i	$-0.375391\pi$
$67$	6.24621	0.763096	0.381548	+	0.924349i	$0.375391\pi$
$68$	0	0
$69$	3.50407	0.421840
$70$	0	0
$71$	−0.648810	−0.0769996	−0.0384998	−	0.999259i	$-0.512258\pi$
$71$	−0.648810	−0.0769996	−0.0384998	+	0.999259i	$0.512258\pi$
$72$	0	0
$73$	−3.47179	−0.406342	−0.203171	−	0.979143i	$-0.565125\pi$
$73$	−3.47179	−0.406342	−0.203171	+	0.979143i	$0.565125\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	20.9113	2.38306
$78$	0	0
$79$	−7.28663	−0.819810	−0.409905	−	0.912128i	$-0.634438\pi$
$79$	−7.28663	−0.819810	−0.409905	+	0.912128i	$0.634438\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	10.2462	1.12467	0.562334	−	0.826910i	$-0.309904\pi$
$83$	10.2462	1.12467	0.562334	+	0.826910i	$0.309904\pi$
$84$	0	0
$85$	−1.50407	−0.163139
$86$	0	0
$87$	5.78256	0.619956
$88$	0	0
$89$	−5.28663	−0.560381	−0.280191	−	0.959944i	$-0.590398\pi$
$89$	−5.28663	−0.560381	−0.280191	+	0.959944i	$0.590398\pi$
$90$	0	0
$91$	−3.50407	−0.367326
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−7.78256	−0.798474
$96$	0	0
$97$	−17.0692	−1.73311	−0.866557	−	0.499079i	$-0.833672\pi$
$97$	−17.0692	−1.73311	−0.866557	+	0.499079i	$0.833672\pi$
$98$	0	0
$99$	−5.96772	−0.599778
$100$	0	0
$101$	−0.855257	−0.0851013	−0.0425506	−	0.999094i	$-0.513548\pi$
$101$	−0.855257	−0.0851013	−0.0425506	+	0.999094i	$0.513548\pi$
$102$	0	0
$103$	13.6247	1.34248	0.671239	−	0.741241i	$-0.265762\pi$
$103$	13.6247	1.34248	0.671239	+	0.741241i	$0.265762\pi$
$104$	0	0
$105$	3.50407	0.341962
$106$	0	0
$107$	−9.59740	−0.927816	−0.463908	−	0.885883i	$-0.653553\pi$
$107$	−9.59740	−0.927816	−0.463908	+	0.885883i	$0.653553\pi$
$108$	0	0
$109$	−0.463651	−0.0444097	−0.0222049	−	0.999753i	$-0.507069\pi$
$109$	−0.463651	−0.0444097	−0.0222049	+	0.999753i	$0.507069\pi$
$110$	0	0
$111$	3.96772	0.376599
$112$	0	0
$113$	−14.1096	−1.32732	−0.663660	−	0.748034i	$-0.730998\pi$
$113$	−14.1096	−1.32732	−0.663660	+	0.748034i	$0.730998\pi$
$114$	0	0
$115$	−3.50407	−0.326756
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	5.27036	0.483133
$120$	0	0
$121$	24.6137	2.23761
$122$	0	0
$123$	−7.96772	−0.718425
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	16.2625	1.44306	0.721531	−	0.692382i	$-0.243439\pi$
$127$	16.2625	1.44306	0.721531	+	0.692382i	$0.243439\pi$
$128$	0	0
$129$	1.68923	0.148728
$130$	0	0
$131$	9.47179	0.827554	0.413777	−	0.910378i	$-0.364209\pi$
$131$	9.47179	0.827554	0.413777	+	0.910378i	$0.364209\pi$
$132$	0	0
$133$	27.2706	2.36466
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	7.31891	0.625297	0.312648	−	0.949869i	$-0.398784\pi$
$137$	7.31891	0.625297	0.312648	+	0.949869i	$0.398784\pi$
$138$	0	0
$139$	8.60554	0.729912	0.364956	−	0.931025i	$-0.381084\pi$
$139$	8.60554	0.729912	0.364956	+	0.931025i	$0.381084\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−5.96772	−0.499046
$144$	0	0
$145$	−5.78256	−0.480215
$146$	0	0
$147$	−5.27849	−0.435363
$148$	0	0
$149$	−7.34617	−0.601822	−0.300911	−	0.953652i	$-0.597291\pi$
$149$	−7.34617	−0.601822	−0.300911	+	0.953652i	$0.597291\pi$
$150$	0	0
$151$	14.0163	1.14063	0.570314	−	0.821427i	$-0.306822\pi$
$151$	14.0163	1.14063	0.570314	+	0.821427i	$0.306822\pi$
$152$	0	0
$153$	−1.50407	−0.121597
$154$	0	0
$155$	0	0
$156$	0	0
$157$	10.6974	0.853743	0.426871	−	0.904312i	$-0.359616\pi$
$157$	10.6974	0.853743	0.426871	+	0.904312i	$0.359616\pi$
$158$	0	0
$159$	5.65695	0.448625
$160$	0	0
$161$	12.2785	0.967681
$162$	0	0
$163$	−18.2948	−1.43296	−0.716478	−	0.697609i	$-0.754247\pi$
$163$	−18.2948	−1.43296	−0.716478	+	0.697609i	$0.754247\pi$
$164$	0	0
$165$	5.96772	0.464586
$166$	0	0
$167$	22.3220	1.72733	0.863665	−	0.504066i	$-0.168163\pi$
$167$	22.3220	1.72733	0.863665	+	0.504066i	$0.168163\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−7.78256	−0.595147
$172$	0	0
$173$	−10.6974	−0.813305	−0.406653	−	0.913583i	$-0.633304\pi$
$173$	−10.6974	−0.813305	−0.406653	+	0.913583i	$0.633304\pi$
$174$	0	0
$175$	−3.50407	−0.264883
$176$	0	0
$177$	−14.9436	−1.12323
$178$	0	0
$179$	−24.0450	−1.79721	−0.898605	−	0.438758i	$-0.855419\pi$
$179$	−24.0450	−1.79721	−0.898605	+	0.438758i	$0.855419\pi$
$180$	0	0
$181$	−21.9194	−1.62926	−0.814629	−	0.579982i	$-0.803060\pi$
$181$	−21.9194	−1.62926	−0.814629	+	0.579982i	$0.803060\pi$
$182$	0	0
$183$	−3.04042	−0.224754
$184$	0	0
$185$	−3.96772	−0.291713
$186$	0	0
$187$	8.97586	0.656380
$188$	0	0
$189$	3.50407	0.254883
$190$	0	0
$191$	2.08083	0.150564	0.0752819	−	0.997162i	$-0.476014\pi$
$191$	2.08083	0.150564	0.0752819	+	0.997162i	$0.476014\pi$
$192$	0	0
$193$	10.4314	0.750866	0.375433	−	0.926849i	$-0.377494\pi$
$193$	10.4314	0.750866	0.375433	+	0.926849i	$0.377494\pi$
$194$	0	0
$195$	−1.00000	−0.0716115
$196$	0	0
$197$	−6.69736	−0.477167	−0.238584	−	0.971122i	$-0.576683\pi$
$197$	−6.69736	−0.477167	−0.238584	+	0.971122i	$0.576683\pi$
$198$	0	0
$199$	11.7701	0.834357	0.417179	−	0.908824i	$-0.363019\pi$
$199$	11.7701	0.834357	0.417179	+	0.908824i	$0.363019\pi$
$200$	0	0
$201$	−6.24621	−0.440574
$202$	0	0
$203$	20.2625	1.42215
$204$	0	0
$205$	7.96772	0.556490
$206$	0	0
$207$	−3.50407	−0.243550
$208$	0	0
$209$	46.4441	3.21261
$210$	0	0
$211$	−27.8709	−1.91871	−0.959355	−	0.282202i	$-0.908935\pi$
$211$	−27.8709	−1.91871	−0.959355	+	0.282202i	$0.908935\pi$
$212$	0	0
$213$	0.648810	0.0444558
$214$	0	0
$215$	−1.68923	−0.115204
$216$	0	0
$217$	0	0
$218$	0	0
$219$	3.47179	0.234602
$220$	0	0
$221$	−1.50407	−0.101175
$222$	0	0
$223$	−8.85024	−0.592656	−0.296328	−	0.955086i	$-0.595762\pi$
$223$	−8.85024	−0.592656	−0.296328	+	0.955086i	$0.595762\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	18.5733	1.23275	0.616375	−	0.787453i	$-0.288601\pi$
$227$	18.5733	1.23275	0.616375	+	0.787453i	$0.288601\pi$
$228$	0	0
$229$	17.3910	1.14923	0.574613	−	0.818425i	$-0.305152\pi$
$229$	17.3910	1.14923	0.574613	+	0.818425i	$0.305152\pi$
$230$	0	0
$231$	−20.9113	−1.37586
$232$	0	0
$233$	0.576765	0.0377852	0.0188926	−	0.999822i	$-0.493986\pi$
$233$	0.576765	0.0377852	0.0188926	+	0.999822i	$0.493986\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	7.28663	0.473317
$238$	0	0
$239$	21.5974	1.39702	0.698510	−	0.715600i	$-0.253847\pi$
$239$	21.5974	1.39702	0.698510	+	0.715600i	$0.253847\pi$
$240$	0	0
$241$	1.44302	0.0929528	0.0464764	−	0.998919i	$-0.485201\pi$
$241$	1.44302	0.0929528	0.0464764	+	0.998919i	$0.485201\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	5.27849	0.337230
$246$	0	0
$247$	−7.78256	−0.495192
$248$	0	0
$249$	−10.2462	−0.649327
$250$	0	0
$251$	4.54448	0.286845	0.143423	−	0.989662i	$-0.454189\pi$
$251$	4.54448	0.286845	0.143423	+	0.989662i	$0.454189\pi$
$252$	0	0
$253$	20.9113	1.31468
$254$	0	0
$255$	1.50407	0.0941884
$256$	0	0
$257$	−6.85024	−0.427306	−0.213653	−	0.976910i	$-0.568536\pi$
$257$	−6.85024	−0.427306	−0.213653	+	0.976910i	$0.568536\pi$
$258$	0	0
$259$	13.9032	0.863900
$260$	0	0
$261$	−5.78256	−0.357931
$262$	0	0
$263$	−11.4881	−0.708384	−0.354192	−	0.935173i	$-0.615244\pi$
$263$	−11.4881	−0.708384	−0.354192	+	0.935173i	$0.615244\pi$
$264$	0	0
$265$	−5.65695	−0.347503
$266$	0	0
$267$	5.28663	0.323536
$268$	0	0
$269$	19.1773	1.16926	0.584630	−	0.811300i	$-0.301240\pi$
$269$	19.1773	1.16926	0.584630	+	0.811300i	$0.301240\pi$
$270$	0	0
$271$	25.9517	1.57645	0.788227	−	0.615385i	$-0.210999\pi$
$271$	25.9517	1.57645	0.788227	+	0.615385i	$0.210999\pi$
$272$	0	0
$273$	3.50407	0.212076
$274$	0	0
$275$	−5.96772	−0.359867
$276$	0	0
$277$	−11.8759	−0.713553	−0.356777	−	0.934190i	$-0.616124\pi$
$277$	−11.8759	−0.713553	−0.356777	+	0.934190i	$0.616124\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	1.73752	0.103652	0.0518258	−	0.998656i	$-0.483496\pi$
$281$	1.73752	0.103652	0.0518258	+	0.998656i	$0.483496\pi$
$282$	0	0
$283$	−26.9003	−1.59906	−0.799528	−	0.600628i	$-0.794917\pi$
$283$	−26.9003	−1.59906	−0.799528	+	0.600628i	$0.794917\pi$
$284$	0	0
$285$	7.78256	0.460999
$286$	0	0
$287$	−27.9194	−1.64803
$288$	0	0
$289$	−14.7378	−0.866928
$290$	0	0
$291$	17.0692	1.00061
$292$	0	0
$293$	−11.8759	−0.693797	−0.346899	−	0.937903i	$-0.612765\pi$
$293$	−11.8759	−0.693797	−0.346899	+	0.937903i	$0.612765\pi$
$294$	0	0
$295$	14.9436	0.870049
$296$	0	0
$297$	5.96772	0.346282
$298$	0	0
$299$	−3.50407	−0.202645
$300$	0	0
$301$	5.91917	0.341175
$302$	0	0
$303$	0.855257	0.0491333
$304$	0	0
$305$	3.04042	0.174094
$306$	0	0
$307$	−17.2058	−0.981987	−0.490993	−	0.871163i	$-0.663366\pi$
$307$	−17.2058	−0.981987	−0.490993	+	0.871163i	$0.663366\pi$
$308$	0	0
$309$	−13.6247	−0.775080
$310$	0	0
$311$	−6.57326	−0.372735	−0.186368	−	0.982480i	$-0.559672\pi$
$311$	−6.57326	−0.372735	−0.186368	+	0.982480i	$0.559672\pi$
$312$	0	0
$313$	8.45115	0.477687	0.238844	−	0.971058i	$-0.423232\pi$
$313$	8.45115	0.477687	0.238844	+	0.971058i	$0.423232\pi$
$314$	0	0
$315$	−3.50407	−0.197432
$316$	0	0
$317$	1.37845	0.0774217	0.0387109	−	0.999250i	$-0.487675\pi$
$317$	1.37845	0.0774217	0.0387109	+	0.999250i	$0.487675\pi$
$318$	0	0
$319$	34.5087	1.93212
$320$	0	0
$321$	9.59740	0.535675
$322$	0	0
$323$	11.7055	0.651311
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	0.463651	0.0256400
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−1.76629	−0.0970840	−0.0485420	−	0.998821i	$-0.515457\pi$
$331$	−1.76629	−0.0970840	−0.0485420	+	0.998821i	$0.515457\pi$
$332$	0	0
$333$	−3.96772	−0.217430
$334$	0	0
$335$	6.24621	0.341267
$336$	0	0
$337$	−22.3058	−1.21507	−0.607536	−	0.794292i	$-0.707842\pi$
$337$	−22.3058	−1.21507	−0.607536	+	0.794292i	$0.707842\pi$
$338$	0	0
$339$	14.1096	0.766329
$340$	0	0
$341$	0	0
$342$	0	0
$343$	6.03228	0.325713
$344$	0	0
$345$	3.50407	0.188653
$346$	0	0
$347$	−13.2816	−0.712994	−0.356497	−	0.934296i	$-0.616029\pi$
$347$	−13.2816	−0.712994	−0.356497	+	0.934296i	$0.616029\pi$
$348$	0	0
$349$	−18.4154	−0.985752	−0.492876	−	0.870100i	$-0.664054\pi$
$349$	−18.4154	−0.985752	−0.492876	+	0.870100i	$0.664054\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	15.8759	0.844989	0.422494	−	0.906366i	$-0.361155\pi$
$353$	15.8759	0.844989	0.422494	+	0.906366i	$0.361155\pi$
$354$	0	0
$355$	−0.648810	−0.0344353
$356$	0	0
$357$	−5.27036	−0.278937
$358$	0	0
$359$	30.2462	1.59633	0.798167	−	0.602436i	$-0.205803\pi$
$359$	30.2462	1.59633	0.798167	+	0.602436i	$0.205803\pi$
$360$	0	0
$361$	41.5682	2.18780
$362$	0	0
$363$	−24.6137	−1.29188
$364$	0	0
$365$	−3.47179	−0.181722
$366$	0	0
$367$	−36.1979	−1.88952	−0.944758	−	0.327769i	$-0.893703\pi$
$367$	−36.1979	−1.88952	−0.944758	+	0.327769i	$0.893703\pi$
$368$	0	0
$369$	7.96772	0.414783
$370$	0	0
$371$	19.8223	1.02912
$372$	0	0
$373$	11.9405	0.618253	0.309127	−	0.951021i	$-0.399963\pi$
$373$	11.9405	0.618253	0.309127	+	0.951021i	$0.399963\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−5.78256	−0.297817
$378$	0	0
$379$	−5.67673	−0.291594	−0.145797	−	0.989315i	$-0.546575\pi$
$379$	−5.67673	−0.291594	−0.145797	+	0.989315i	$0.546575\pi$
$380$	0	0
$381$	−16.2625	−0.833152
$382$	0	0
$383$	−30.1384	−1.54000	−0.770000	−	0.638044i	$-0.779744\pi$
$383$	−30.1384	−1.54000	−0.770000	+	0.638044i	$0.779744\pi$
$384$	0	0
$385$	20.9113	1.06574
$386$	0	0
$387$	−1.68923	−0.0858682
$388$	0	0
$389$	13.1610	0.667290	0.333645	−	0.942699i	$-0.391721\pi$
$389$	13.1610	0.667290	0.333645	+	0.942699i	$0.391721\pi$
$390$	0	0
$391$	5.27036	0.266533
$392$	0	0
$393$	−9.47179	−0.477788
$394$	0	0
$395$	−7.28663	−0.366630
$396$	0	0
$397$	29.0171	1.45633	0.728164	−	0.685403i	$-0.240374\pi$
$397$	29.0171	1.45633	0.728164	+	0.685403i	$0.240374\pi$
$398$	0	0
$399$	−27.2706	−1.36524
$400$	0	0
$401$	38.9598	1.94556	0.972781	−	0.231727i	$-0.0744375\pi$
$401$	38.9598	1.94556	0.972781	+	0.231727i	$0.0744375\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	23.6782	1.17369
$408$	0	0
$409$	21.5006	1.06313	0.531567	−	0.847016i	$-0.321603\pi$
$409$	21.5006	1.06313	0.531567	+	0.847016i	$0.321603\pi$
$410$	0	0
$411$	−7.31891	−0.361015
$412$	0	0
$413$	−52.3633	−2.57663
$414$	0	0
$415$	10.2462	0.502967
$416$	0	0
$417$	−8.60554	−0.421415
$418$	0	0
$419$	16.1742	0.790160	0.395080	−	0.918647i	$-0.370717\pi$
$419$	16.1742	0.790160	0.395080	+	0.918647i	$0.370717\pi$
$420$	0	0
$421$	20.4637	0.997337	0.498669	−	0.866793i	$-0.333822\pi$
$421$	20.4637	0.997337	0.498669	+	0.866793i	$0.333822\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−1.50407	−0.0729580
$426$	0	0
$427$	−10.6538	−0.515575
$428$	0	0
$429$	5.96772	0.288124
$430$	0	0
$431$	35.2493	1.69790	0.848950	−	0.528473i	$-0.177235\pi$
$431$	35.2493	1.69790	0.848950	+	0.528473i	$0.177235\pi$
$432$	0	0
$433$	12.9436	0.622028	0.311014	−	0.950405i	$-0.399331\pi$
$433$	12.9436	0.622028	0.311014	+	0.950405i	$0.399331\pi$
$434$	0	0
$435$	5.78256	0.277253
$436$	0	0
$437$	27.2706	1.30453
$438$	0	0
$439$	−31.4845	−1.50268	−0.751338	−	0.659918i	$-0.770591\pi$
$439$	−31.4845	−1.50268	−0.751338	+	0.659918i	$0.770591\pi$
$440$	0	0
$441$	5.27849	0.251357
$442$	0	0
$443$	−23.9194	−1.13645	−0.568223	−	0.822874i	$-0.692369\pi$
$443$	−23.9194	−1.13645	−0.568223	+	0.822874i	$0.692369\pi$
$444$	0	0
$445$	−5.28663	−0.250610
$446$	0	0
$447$	7.34617	0.347462
$448$	0	0
$449$	−23.6982	−1.11839	−0.559194	−	0.829037i	$-0.688889\pi$
$449$	−23.6982	−1.11839	−0.559194	+	0.829037i	$0.688889\pi$
$450$	0	0
$451$	−47.5491	−2.23900
$452$	0	0
$453$	−14.0163	−0.658542
$454$	0	0
$455$	−3.50407	−0.164273
$456$	0	0
$457$	−11.0529	−0.517034	−0.258517	−	0.966007i	$-0.583234\pi$
$457$	−11.0529	−0.517034	−0.258517	+	0.966007i	$0.583234\pi$
$458$	0	0
$459$	1.50407	0.0702039
$460$	0	0
$461$	33.3874	1.55501	0.777504	−	0.628878i	$-0.216485\pi$
$461$	33.3874	1.55501	0.777504	+	0.628878i	$0.216485\pi$
$462$	0	0
$463$	23.4395	1.08933	0.544663	−	0.838655i	$-0.316658\pi$
$463$	23.4395	1.08933	0.544663	+	0.838655i	$0.316658\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	37.4037	1.73084	0.865419	−	0.501049i	$-0.167052\pi$
$467$	37.4037	1.73084	0.865419	+	0.501049i	$0.167052\pi$
$468$	0	0
$469$	−21.8871	−1.01065
$470$	0	0
$471$	−10.6974	−0.492908
$472$	0	0
$473$	10.0808	0.463517
$474$	0	0
$475$	−7.78256	−0.357088
$476$	0	0
$477$	−5.65695	−0.259014
$478$	0	0
$479$	7.02414	0.320941	0.160471	−	0.987041i	$-0.448699\pi$
$479$	7.02414	0.320941	0.160471	+	0.987041i	$0.448699\pi$
$480$	0	0
$481$	−3.96772	−0.180912
$482$	0	0
$483$	−12.2785	−0.558691
$484$	0	0
$485$	−17.0692	−0.775072
$486$	0	0
$487$	−18.8180	−0.852723	−0.426362	−	0.904553i	$-0.640205\pi$
$487$	−18.8180	−0.852723	−0.426362	+	0.904553i	$0.640205\pi$
$488$	0	0
$489$	18.2948	0.827318
$490$	0	0
$491$	18.0288	0.813627	0.406814	−	0.913511i	$-0.366640\pi$
$491$	18.0288	0.813627	0.406814	+	0.913511i	$0.366640\pi$
$492$	0	0
$493$	8.69736	0.391710
$494$	0	0
$495$	−5.96772	−0.268229
$496$	0	0
$497$	2.27348	0.101979
$498$	0	0
$499$	−37.3640	−1.67264	−0.836320	−	0.548241i	$-0.815297\pi$
$499$	−37.3640	−1.67264	−0.836320	+	0.548241i	$0.815297\pi$
$500$	0	0
$501$	−22.3220	−0.997275
$502$	0	0
$503$	18.3991	0.820375	0.410187	−	0.912001i	$-0.365463\pi$
$503$	18.3991	0.820375	0.410187	+	0.912001i	$0.365463\pi$
$504$	0	0
$505$	−0.855257	−0.0380585
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	−16.8950	−0.748859	−0.374429	−	0.927255i	$-0.622161\pi$
$509$	−16.8950	−0.748859	−0.374429	+	0.927255i	$0.622161\pi$
$510$	0	0
$511$	12.1654	0.538165
$512$	0	0
$513$	7.78256	0.343608
$514$	0	0
$515$	13.6247	0.600374
$516$	0	0
$517$	0	0
$518$	0	0
$519$	10.6974	0.469562
$520$	0	0
$521$	41.5651	1.82100	0.910500	−	0.413508i	$-0.135697\pi$
$521$	41.5651	1.82100	0.910500	+	0.413508i	$0.135697\pi$
$522$	0	0
$523$	20.4279	0.893248	0.446624	−	0.894722i	$-0.352626\pi$
$523$	20.4279	0.893248	0.446624	+	0.894722i	$0.352626\pi$
$524$	0	0
$525$	3.50407	0.152930
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−10.7215	−0.466153
$530$	0	0
$531$	14.9436	0.648496
$532$	0	0
$533$	7.96772	0.345120
$534$	0	0
$535$	−9.59740	−0.414932
$536$	0	0
$537$	24.0450	1.03762
$538$	0	0
$539$	−31.5006	−1.35683
$540$	0	0
$541$	32.3991	1.39295	0.696473	−	0.717583i	$-0.254752\pi$
$541$	32.3991	1.39295	0.696473	+	0.717583i	$0.254752\pi$
$542$	0	0
$543$	21.9194	0.940653
$544$	0	0
$545$	−0.463651	−0.0198606
$546$	0	0
$547$	−20.6541	−0.883105	−0.441553	−	0.897235i	$-0.645572\pi$
$547$	−20.6541	−0.883105	−0.441553	+	0.897235i	$0.645572\pi$
$548$	0	0
$549$	3.04042	0.129762
$550$	0	0
$551$	45.0031	1.91720
$552$	0	0
$553$	25.5328	1.08577
$554$	0	0
$555$	3.96772	0.168420
$556$	0	0
$557$	−11.2976	−0.478696	−0.239348	−	0.970934i	$-0.576934\pi$
$557$	−11.2976	−0.478696	−0.239348	+	0.970934i	$0.576934\pi$
$558$	0	0
$559$	−1.68923	−0.0714467
$560$	0	0
$561$	−8.97586	−0.378961
$562$	0	0
$563$	7.49157	0.315732	0.157866	−	0.987461i	$-0.449539\pi$
$563$	7.49157	0.315732	0.157866	+	0.987461i	$0.449539\pi$
$564$	0	0
$565$	−14.1096	−0.593596
$566$	0	0
$567$	−3.50407	−0.147157
$568$	0	0
$569$	−24.2029	−1.01464	−0.507320	−	0.861758i	$-0.669364\pi$
$569$	−24.2029	−1.01464	−0.507320	+	0.861758i	$0.669364\pi$
$570$	0	0
$571$	17.1625	0.718229	0.359115	−	0.933293i	$-0.383079\pi$
$571$	17.1625	0.718229	0.359115	+	0.933293i	$0.383079\pi$
$572$	0	0
$573$	−2.08083	−0.0869281
$574$	0	0
$575$	−3.50407	−0.146130
$576$	0	0
$577$	−23.5203	−0.979165	−0.489582	−	0.871957i	$-0.662851\pi$
$577$	−23.5203	−0.979165	−0.489582	+	0.871957i	$0.662851\pi$
$578$	0	0
$579$	−10.4314	−0.433513
$580$	0	0
$581$	−35.9034	−1.48952
$582$	0	0
$583$	33.7591	1.39816
$584$	0	0
$585$	1.00000	0.0413449
$586$	0	0
$587$	−45.0820	−1.86073	−0.930366	−	0.366633i	$-0.880510\pi$
$587$	−45.0820	−1.86073	−0.930366	+	0.366633i	$0.880510\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	6.69736	0.275493
$592$	0	0
$593$	8.31077	0.341283	0.170641	−	0.985333i	$-0.445416\pi$
$593$	8.31077	0.341283	0.170641	+	0.985333i	$0.445416\pi$
$594$	0	0
$595$	5.27036	0.216064
$596$	0	0
$597$	−11.7701	−0.481717
$598$	0	0
$599$	39.8709	1.62908	0.814540	−	0.580107i	$-0.196989\pi$
$599$	39.8709	1.62908	0.814540	+	0.580107i	$0.196989\pi$
$600$	0	0
$601$	−3.57611	−0.145873	−0.0729363	−	0.997337i	$-0.523237\pi$
$601$	−3.57611	−0.145873	−0.0729363	+	0.997337i	$0.523237\pi$
$602$	0	0
$603$	6.24621	0.254365
$604$	0	0
$605$	24.6137	1.00069
$606$	0	0
$607$	−19.6730	−0.798501	−0.399250	−	0.916842i	$-0.630730\pi$
$607$	−19.6730	−0.798501	−0.399250	+	0.916842i	$0.630730\pi$
$608$	0	0
$609$	−20.2625	−0.821077
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−0.0322808	−0.00130381	−0.000651905	−	1.00000i	$-0.500208\pi$
$613$	−0.0322808	−0.00130381	−0.000651905		1.00000i	$0.500208\pi$
$614$	0	0
$615$	−7.96772	−0.321289
$616$	0	0
$617$	−19.3189	−0.777750	−0.388875	−	0.921291i	$-0.627136\pi$
$617$	−19.3189	−0.777750	−0.388875	+	0.921291i	$0.627136\pi$
$618$	0	0
$619$	−2.75815	−0.110860	−0.0554298	−	0.998463i	$-0.517653\pi$
$619$	−2.75815	−0.110860	−0.0554298	+	0.998463i	$0.517653\pi$
$620$	0	0
$621$	3.50407	0.140613
$622$	0	0
$623$	18.5247	0.742177
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−46.4441	−1.85480
$628$	0	0
$629$	5.96772	0.237949
$630$	0	0
$631$	−5.71051	−0.227332	−0.113666	−	0.993519i	$-0.536259\pi$
$631$	−5.71051	−0.227332	−0.113666	+	0.993519i	$0.536259\pi$
$632$	0	0
$633$	27.8709	1.10777
$634$	0	0
$635$	16.2625	0.645357
$636$	0	0
$637$	5.27849	0.209141
$638$	0	0
$639$	−0.648810	−0.0256665
$640$	0	0
$641$	2.74064	0.108249	0.0541243	−	0.998534i	$-0.482763\pi$
$641$	2.74064	0.108249	0.0541243	+	0.998534i	$0.482763\pi$
$642$	0	0
$643$	−12.5842	−0.496274	−0.248137	−	0.968725i	$-0.579818\pi$
$643$	−12.5842	−0.496274	−0.248137	+	0.968725i	$0.579818\pi$
$644$	0	0
$645$	1.68923	0.0665133
$646$	0	0
$647$	21.3587	0.839696	0.419848	−	0.907594i	$-0.362083\pi$
$647$	21.3587	0.839696	0.419848	+	0.907594i	$0.362083\pi$
$648$	0	0
$649$	−89.1791	−3.50058
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−23.6247	−0.924505	−0.462252	−	0.886748i	$-0.652959\pi$
$653$	−23.6247	−0.924505	−0.462252	+	0.886748i	$0.652959\pi$
$654$	0	0
$655$	9.47179	0.370093
$656$	0	0
$657$	−3.47179	−0.135447
$658$	0	0
$659$	−1.16603	−0.0454221	−0.0227110	−	0.999742i	$-0.507230\pi$
$659$	−1.16603	−0.0454221	−0.0227110	+	0.999742i	$0.507230\pi$
$660$	0	0
$661$	28.5608	1.11088	0.555442	−	0.831555i	$-0.312549\pi$
$661$	28.5608	1.11088	0.555442	+	0.831555i	$0.312549\pi$
$662$	0	0
$663$	1.50407	0.0584132
$664$	0	0
$665$	27.2706	1.05751
$666$	0	0
$667$	20.2625	0.784566
$668$	0	0
$669$	8.85024	0.342170
$670$	0	0
$671$	−18.1444	−0.700455
$672$	0	0
$673$	32.4441	1.25063	0.625315	−	0.780373i	$-0.284971\pi$
$673$	32.4441	1.25063	0.625315	+	0.780373i	$0.284971\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	24.6651	0.947956	0.473978	−	0.880537i	$-0.342818\pi$
$677$	24.6651	0.947956	0.473978	+	0.880537i	$0.342818\pi$
$678$	0	0
$679$	59.8116	2.29536
$680$	0	0
$681$	−18.5733	−0.711729
$682$	0	0
$683$	−7.05141	−0.269815	−0.134907	−	0.990858i	$-0.543074\pi$
$683$	−7.05141	−0.269815	−0.134907	+	0.990858i	$0.543074\pi$
$684$	0	0
$685$	7.31891	0.279641
$686$	0	0
$687$	−17.3910	−0.663506
$688$	0	0
$689$	−5.65695	−0.215512
$690$	0	0
$691$	10.7261	0.408041	0.204021	−	0.978967i	$-0.434599\pi$
$691$	10.7261	0.408041	0.204021	+	0.978967i	$0.434599\pi$
$692$	0	0
$693$	20.9113	0.794354
$694$	0	0
$695$	8.60554	0.326427
$696$	0	0
$697$	−11.9840	−0.453926
$698$	0	0
$699$	−0.576765	−0.0218153
$700$	0	0
$701$	29.8151	1.12610	0.563050	−	0.826423i	$-0.309628\pi$
$701$	29.8151	1.12610	0.563050	+	0.826423i	$0.309628\pi$
$702$	0	0
$703$	30.8790	1.16462
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2.99688	0.112709
$708$	0	0
$709$	17.3910	0.653131	0.326565	−	0.945175i	$-0.394109\pi$
$709$	17.3910	0.653131	0.326565	+	0.945175i	$0.394109\pi$
$710$	0	0
$711$	−7.28663	−0.273270
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−5.96772	−0.223180
$716$	0	0
$717$	−21.5974	−0.806570
$718$	0	0
$719$	33.0890	1.23401	0.617005	−	0.786959i	$-0.288346\pi$
$719$	33.0890	1.23401	0.617005	+	0.786959i	$0.288346\pi$
$720$	0	0
$721$	−47.7418	−1.77800
$722$	0	0
$723$	−1.44302	−0.0536663
$724$	0	0
$725$	−5.78256	−0.214759
$726$	0	0
$727$	−14.6165	−0.542097	−0.271049	−	0.962566i	$-0.587370\pi$
$727$	−14.6165	−0.542097	−0.271049	+	0.962566i	$0.587370\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	2.54071	0.0939716
$732$	0	0
$733$	−46.3473	−1.71188	−0.855938	−	0.517078i	$-0.827020\pi$
$733$	−46.3473	−1.71188	−0.855938	+	0.517078i	$0.827020\pi$
$734$	0	0
$735$	−5.27849	−0.194700
$736$	0	0
$737$	−37.2756	−1.37307
$738$	0	0
$739$	−1.74129	−0.0640544	−0.0320272	−	0.999487i	$-0.510196\pi$
$739$	−1.74129	−0.0640544	−0.0320272	+	0.999487i	$0.510196\pi$
$740$	0	0
$741$	7.78256	0.285899
$742$	0	0
$743$	19.6297	0.720143	0.360072	−	0.932925i	$-0.382752\pi$
$743$	19.6297	0.720143	0.360072	+	0.932925i	$0.382752\pi$
$744$	0	0
$745$	−7.34617	−0.269143
$746$	0	0
$747$	10.2462	0.374889
$748$	0	0
$749$	33.6299	1.22881
$750$	0	0
$751$	−34.9921	−1.27688	−0.638440	−	0.769671i	$-0.720420\pi$
$751$	−34.9921	−1.27688	−0.638440	+	0.769671i	$0.720420\pi$
$752$	0	0
$753$	−4.54448	−0.165610
$754$	0	0
$755$	14.0163	0.510104
$756$	0	0
$757$	−31.0494	−1.12851	−0.564255	−	0.825601i	$-0.690836\pi$
$757$	−31.0494	−1.12851	−0.564255	+	0.825601i	$0.690836\pi$
$758$	0	0
$759$	−20.9113	−0.759032
$760$	0	0
$761$	54.9598	1.99229	0.996146	−	0.0877057i	$-0.0279535\pi$
$761$	54.9598	1.99229	0.996146	+	0.0877057i	$0.0279535\pi$
$762$	0	0
$763$	1.62467	0.0588168
$764$	0	0
$765$	−1.50407	−0.0543797
$766$	0	0
$767$	14.9436	0.539581
$768$	0	0
$769$	−3.49131	−0.125900	−0.0629499	−	0.998017i	$-0.520051\pi$
$769$	−3.49131	−0.125900	−0.0629499	+	0.998017i	$0.520051\pi$
$770$	0	0
$771$	6.85024	0.246705
$772$	0	0
$773$	17.7305	0.637722	0.318861	−	0.947802i	$-0.396700\pi$
$773$	17.7305	0.637722	0.318861	+	0.947802i	$0.396700\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−13.9032	−0.498773
$778$	0	0
$779$	−62.0093	−2.22171
$780$	0	0
$781$	3.87192	0.138548
$782$	0	0
$783$	5.78256	0.206652
$784$	0	0
$785$	10.6974	0.381805
$786$	0	0
$787$	−11.3026	−0.402895	−0.201448	−	0.979499i	$-0.564565\pi$
$787$	−11.3026	−0.402895	−0.201448	+	0.979499i	$0.564565\pi$
$788$	0	0
$789$	11.4881	0.408986
$790$	0	0
$791$	49.4410	1.75792
$792$	0	0
$793$	3.04042	0.107968
$794$	0	0
$795$	5.65695	0.200631
$796$	0	0
$797$	−31.7378	−1.12421	−0.562105	−	0.827066i	$-0.690008\pi$
$797$	−31.7378	−1.12421	−0.562105	+	0.827066i	$0.690008\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−5.28663	−0.186794
$802$	0	0
$803$	20.7187	0.731145
$804$	0	0
$805$	12.2785	0.432760
$806$	0	0
$807$	−19.1773	−0.675072
$808$	0	0
$809$	18.9848	0.667472	0.333736	−	0.942667i	$-0.391691\pi$
$809$	18.9848	0.667472	0.333736	+	0.942667i	$0.391691\pi$
$810$	0	0
$811$	−6.38783	−0.224307	−0.112154	−	0.993691i	$-0.535775\pi$
$811$	−6.38783	−0.224307	−0.112154	+	0.993691i	$0.535775\pi$
$812$	0	0
$813$	−25.9517	−0.910166
$814$	0	0
$815$	−18.2948	−0.640838
$816$	0	0
$817$	13.1465	0.459938
$818$	0	0
$819$	−3.50407	−0.122442
$820$	0	0
$821$	12.0003	0.418812	0.209406	−	0.977829i	$-0.432847\pi$
$821$	12.0003	0.418812	0.209406	+	0.977829i	$0.432847\pi$
$822$	0	0
$823$	−14.7386	−0.513756	−0.256878	−	0.966444i	$-0.582694\pi$
$823$	−14.7386	−0.513756	−0.256878	+	0.966444i	$0.582694\pi$
$824$	0	0
$825$	5.96772	0.207769
$826$	0	0
$827$	−36.7549	−1.27809	−0.639047	−	0.769168i	$-0.720671\pi$
$827$	−36.7549	−1.27809	−0.639047	+	0.769168i	$0.720671\pi$
$828$	0	0
$829$	−6.62155	−0.229976	−0.114988	−	0.993367i	$-0.536683\pi$
$829$	−6.62155	−0.229976	−0.114988	+	0.993367i	$0.536683\pi$
$830$	0	0
$831$	11.8759	0.411970
$832$	0	0
$833$	−7.93921	−0.275077
$834$	0	0
$835$	22.3220	0.772486
$836$	0	0
$837$	0	0
$838$	0	0
$839$	29.8782	1.03151	0.515754	−	0.856737i	$-0.327512\pi$
$839$	29.8782	1.03151	0.515754	+	0.856737i	$0.327512\pi$
$840$	0	0
$841$	4.43800	0.153034
$842$	0	0
$843$	−1.73752	−0.0598433
$844$	0	0
$845$	1.00000	0.0344010
$846$	0	0
$847$	−86.2480	−2.96352
$848$	0	0
$849$	26.9003	0.923216
$850$	0	0
$851$	13.9032	0.476594
$852$	0	0
$853$	39.3787	1.34830	0.674151	−	0.738594i	$-0.264510\pi$
$853$	39.3787	1.34830	0.674151	+	0.738594i	$0.264510\pi$
$854$	0	0
$855$	−7.78256	−0.266158
$856$	0	0
$857$	1.99649	0.0681988	0.0340994	−	0.999418i	$-0.489144\pi$
$857$	1.99649	0.0681988	0.0340994	+	0.999418i	$0.489144\pi$
$858$	0	0
$859$	−35.4520	−1.20961	−0.604803	−	0.796375i	$-0.706748\pi$
$859$	−35.4520	−1.20961	−0.604803	+	0.796375i	$0.706748\pi$
$860$	0	0
$861$	27.9194	0.951492
$862$	0	0
$863$	16.5320	0.562755	0.281378	−	0.959597i	$-0.409209\pi$
$863$	16.5320	0.562755	0.281378	+	0.959597i	$0.409209\pi$
$864$	0	0
$865$	−10.6974	−0.363721
$866$	0	0
$867$	14.7378	0.500521
$868$	0	0
$869$	43.4845	1.47511
$870$	0	0
$871$	6.24621	0.211645
$872$	0	0
$873$	−17.0692	−0.577704
$874$	0	0
$875$	−3.50407	−0.118459
$876$	0	0
$877$	−10.7406	−0.362686	−0.181343	−	0.983420i	$-0.558044\pi$
$877$	−10.7406	−0.362686	−0.181343	+	0.983420i	$0.558044\pi$
$878$	0	0
$879$	11.8759	0.400564
$880$	0	0
$881$	49.6622	1.67316	0.836581	−	0.547843i	$-0.184551\pi$
$881$	49.6622	1.67316	0.836581	+	0.547843i	$0.184551\pi$
$882$	0	0
$883$	28.6541	0.964287	0.482143	−	0.876092i	$-0.339858\pi$
$883$	28.6541	0.964287	0.482143	+	0.876092i	$0.339858\pi$
$884$	0	0
$885$	−14.9436	−0.502323
$886$	0	0
$887$	−4.06105	−0.136357	−0.0681784	−	0.997673i	$-0.521719\pi$
$887$	−4.06105	−0.136357	−0.0681784	+	0.997673i	$0.521719\pi$
$888$	0	0
$889$	−56.9848	−1.91121
$890$	0	0
$891$	−5.96772	−0.199926
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−24.0450	−0.803737
$896$	0	0
$897$	3.50407	0.116997
$898$	0	0
$899$	0	0
$900$	0	0
$901$	8.50843	0.283457
$902$	0	0
$903$	−5.91917	−0.196978
$904$	0	0
$905$	−21.9194	−0.728626
$906$	0	0
$907$	42.0626	1.39666	0.698332	−	0.715774i	$-0.253926\pi$
$907$	42.0626	1.39666	0.698332	+	0.715774i	$0.253926\pi$
$908$	0	0
$909$	−0.855257	−0.0283671
$910$	0	0
$911$	−27.5006	−0.911134	−0.455567	−	0.890201i	$-0.650563\pi$
$911$	−27.5006	−0.911134	−0.455567	+	0.890201i	$0.650563\pi$
$912$	0	0
$913$	−61.1465	−2.02365
$914$	0	0
$915$	−3.04042	−0.100513
$916$	0	0
$917$	−33.1898	−1.09602
$918$	0	0
$919$	−57.8961	−1.90982	−0.954909	−	0.296900i	$-0.904047\pi$
$919$	−57.8961	−1.90982	−0.954909	+	0.296900i	$0.904047\pi$
$920$	0	0
$921$	17.2058	0.566950
$922$	0	0
$923$	−0.648810	−0.0213559
$924$	0	0
$925$	−3.96772	−0.130458
$926$	0	0
$927$	13.6247	0.447493
$928$	0	0
$929$	−22.0273	−0.722691	−0.361346	−	0.932432i	$-0.617683\pi$
$929$	−22.0273	−0.722691	−0.361346	+	0.932432i	$0.617683\pi$
$930$	0	0
$931$	−41.0802	−1.34635
$932$	0	0
$933$	6.57326	0.215199
$934$	0	0
$935$	8.97586	0.293542
$936$	0	0
$937$	−8.82147	−0.288185	−0.144092	−	0.989564i	$-0.546026\pi$
$937$	−8.82147	−0.288185	−0.144092	+	0.989564i	$0.546026\pi$
$938$	0	0
$939$	−8.45115	−0.275793
$940$	0	0
$941$	3.24907	0.105917	0.0529583	−	0.998597i	$-0.483135\pi$
$941$	3.24907	0.105917	0.0529583	+	0.998597i	$0.483135\pi$
$942$	0	0
$943$	−27.9194	−0.909182
$944$	0	0
$945$	3.50407	0.113987
$946$	0	0
$947$	−29.4955	−0.958476	−0.479238	−	0.877685i	$-0.659087\pi$
$947$	−29.4955	−0.958476	−0.479238	+	0.877685i	$0.659087\pi$
$948$	0	0
$949$	−3.47179	−0.112699
$950$	0	0
$951$	−1.37845	−0.0446994
$952$	0	0
$953$	−44.0128	−1.42571	−0.712857	−	0.701310i	$-0.752599\pi$
$953$	−44.0128	−1.42571	−0.712857	+	0.701310i	$0.752599\pi$
$954$	0	0
$955$	2.08083	0.0673342
$956$	0	0
$957$	−34.5087	−1.11551
$958$	0	0
$959$	−25.6460	−0.828151
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	−9.59740	−0.309272
$964$	0	0
$965$	10.4314	0.335798
$966$	0	0
$967$	−15.3014	−0.492060	−0.246030	−	0.969262i	$-0.579126\pi$
$967$	−15.3014	−0.492060	−0.246030	+	0.969262i	$0.579126\pi$
$968$	0	0
$969$	−11.7055	−0.376035
$970$	0	0
$971$	−60.0776	−1.92798	−0.963991	−	0.265936i	$-0.914319\pi$
$971$	−60.0776	−1.92798	−0.963991	+	0.265936i	$0.914319\pi$
$972$	0	0
$973$	−30.1544	−0.966705
$974$	0	0
$975$	−1.00000	−0.0320256
$976$	0	0
$977$	−21.6084	−0.691314	−0.345657	−	0.938361i	$-0.612344\pi$
$977$	−21.6084	−0.691314	−0.345657	+	0.938361i	$0.612344\pi$
$978$	0	0
$979$	31.5491	1.00831
$980$	0	0
$981$	−0.463651	−0.0148032
$982$	0	0
$983$	51.5551	1.64435	0.822176	−	0.569233i	$-0.192760\pi$
$983$	51.5551	1.64435	0.822176	+	0.569233i	$0.192760\pi$
$984$	0	0
$985$	−6.69736	−0.213396
$986$	0	0
$987$	0	0
$988$	0	0
$989$	5.91917	0.188219
$990$	0	0
$991$	−25.3675	−0.805824	−0.402912	−	0.915239i	$-0.632002\pi$
$991$	−25.3675	−0.805824	−0.402912	+	0.915239i	$0.632002\pi$
$992$	0	0
$993$	1.76629	0.0560515
$994$	0	0
$995$	11.7701	0.373136
$996$	0	0
$997$	15.4160	0.488230	0.244115	−	0.969746i	$-0.421503\pi$
$997$	15.4160	0.488230	0.244115	+	0.969746i	$0.421503\pi$
$998$	0	0
$999$	3.96772	0.125533

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6240.2.a.ce.1.1	✓	4
4.3	odd	2		6240.2.a.cf.1.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6240.2.a.ce.1.1	✓	4	1.1	even	1	trivial
6240.2.a.cf.1.4	yes	4	4.3	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:	\( N \)	\(=\)	\( 6240 = 2^{5} \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6240.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} -3.50407 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.00000 q^{5} -3.50407 q^{7} +1.00000 q^{9} -5.96772 q^{11} +1.00000 q^{13} -1.00000 q^{15} -1.50407 q^{17} -7.78256 q^{19} +3.50407 q^{21} -3.50407 q^{23} +1.00000 q^{25} -1.00000 q^{27} -5.78256 q^{29} +5.96772 q^{33} -3.50407 q^{35} -3.96772 q^{37} -1.00000 q^{39} +7.96772 q^{41} -1.68923 q^{43} +1.00000 q^{45} +5.27849 q^{49} +1.50407 q^{51} -5.65695 q^{53} -5.96772 q^{55} +7.78256 q^{57} +14.9436 q^{59} +3.04042 q^{61} -3.50407 q^{63} +1.00000 q^{65} +6.24621 q^{67} +3.50407 q^{69} -0.648810 q^{71} -3.47179 q^{73} -1.00000 q^{75} +20.9113 q^{77} -7.28663 q^{79} +1.00000 q^{81} +10.2462 q^{83} -1.50407 q^{85} +5.78256 q^{87} -5.28663 q^{89} -3.50407 q^{91} -7.78256 q^{95} -17.0692 q^{97} -5.96772 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + 4 q^{5} - 3 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 + 4 * q^5 - 3 * q^7 + 4 * q^9	\( 4 q - 4 q^{3} + 4 q^{5} - 3 q^{7} + 4 q^{9} - 3 q^{11} + 4 q^{13} - 4 q^{15} + 5 q^{17} - 8 q^{19} + 3 q^{21} - 3 q^{23} + 4 q^{25} - 4 q^{27} + 3 q^{33} - 3 q^{35} + 5 q^{37} - 4 q^{39} + 11 q^{41} + 2 q^{43} + 4 q^{45} + 9 q^{49} - 5 q^{51} + 7 q^{53} - 3 q^{55} + 8 q^{57} - 4 q^{59} + 11 q^{61} - 3 q^{63} + 4 q^{65} - 8 q^{67} + 3 q^{69} + 5 q^{71} + 18 q^{73} - 4 q^{75} - q^{77} + 5 q^{79} + 4 q^{81} + 8 q^{83} + 5 q^{85} + 13 q^{89} - 3 q^{91} - 8 q^{95} - 11 q^{97} - 3 q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 + 4 * q^5 - 3 * q^7 + 4 * q^9 - 3 * q^11 + 4 * q^13 - 4 * q^15 + 5 * q^17 - 8 * q^19 + 3 * q^21 - 3 * q^23 + 4 * q^25 - 4 * q^27 + 3 * q^33 - 3 * q^35 + 5 * q^37 - 4 * q^39 + 11 * q^41 + 2 * q^43 + 4 * q^45 + 9 * q^49 - 5 * q^51 + 7 * q^53 - 3 * q^55 + 8 * q^57 - 4 * q^59 + 11 * q^61 - 3 * q^63 + 4 * q^65 - 8 * q^67 + 3 * q^69 + 5 * q^71 + 18 * q^73 - 4 * q^75 - q^77 + 5 * q^79 + 4 * q^81 + 8 * q^83 + 5 * q^85 + 13 * q^89 - 3 * q^91 - 8 * q^95 - 11 * q^97 - 3 * q^99

Introduction

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