LMFDB - Embedded newform 6864.2.a.bh.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6864,2,Mod(1,6864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6864, 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("6864.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6864 = 2^{4} \cdot 3 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6864.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:	$54.8093159474$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 858)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	6864.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +0.561553 q^{5} +2.56155 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +0.561553 q^{5} +2.56155 q^{7} +1.00000 q^{9} -1.00000 q^{11} +1.00000 q^{13} +0.561553 q^{15} -3.12311 q^{17} -5.12311 q^{19} +2.56155 q^{21} +6.56155 q^{23} -4.68466 q^{25} +1.00000 q^{27} +1.68466 q^{29} +6.24621 q^{31} -1.00000 q^{33} +1.43845 q^{35} +8.24621 q^{37} +1.00000 q^{39} -3.43845 q^{41} -2.56155 q^{43} +0.561553 q^{45} +8.00000 q^{47} -0.438447 q^{49} -3.12311 q^{51} -4.24621 q^{53} -0.561553 q^{55} -5.12311 q^{57} +5.43845 q^{59} +4.56155 q^{61} +2.56155 q^{63} +0.561553 q^{65} +14.5616 q^{67} +6.56155 q^{69} +15.3693 q^{71} -7.43845 q^{73} -4.68466 q^{75} -2.56155 q^{77} -13.1231 q^{79} +1.00000 q^{81} +17.1231 q^{83} -1.75379 q^{85} +1.68466 q^{87} -17.3693 q^{89} +2.56155 q^{91} +6.24621 q^{93} -2.87689 q^{95} +14.4924 q^{97} -1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 3 q^{5} + q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 3 * q^5 + q^7 + 2 * q^9	$ 2 q + 2 q^{3} - 3 q^{5} + q^{7} + 2 q^{9} - 2 q^{11} + 2 q^{13} - 3 q^{15} + 2 q^{17} - 2 q^{19} + q^{21} + 9 q^{23} + 3 q^{25} + 2 q^{27} - 9 q^{29} - 4 q^{31} - 2 q^{33} + 7 q^{35} + 2 q^{39} - 11 q^{41} - q^{43} - 3 q^{45} + 16 q^{47} - 5 q^{49} + 2 q^{51} + 8 q^{53} + 3 q^{55} - 2 q^{57} + 15 q^{59} + 5 q^{61} + q^{63} - 3 q^{65} + 25 q^{67} + 9 q^{69} + 6 q^{71} - 19 q^{73} + 3 q^{75} - q^{77} - 18 q^{79} + 2 q^{81} + 26 q^{83} - 20 q^{85} - 9 q^{87} - 10 q^{89} + q^{91} - 4 q^{93} - 14 q^{95} - 4 q^{97} - 2 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 3 * q^5 + q^7 + 2 * q^9 - 2 * q^11 + 2 * q^13 - 3 * q^15 + 2 * q^17 - 2 * q^19 + q^21 + 9 * q^23 + 3 * q^25 + 2 * q^27 - 9 * q^29 - 4 * q^31 - 2 * q^33 + 7 * q^35 + 2 * q^39 - 11 * q^41 - q^43 - 3 * q^45 + 16 * q^47 - 5 * q^49 + 2 * q^51 + 8 * q^53 + 3 * q^55 - 2 * q^57 + 15 * q^59 + 5 * q^61 + q^63 - 3 * q^65 + 25 * q^67 + 9 * q^69 + 6 * q^71 - 19 * q^73 + 3 * q^75 - q^77 - 18 * q^79 + 2 * q^81 + 26 * q^83 - 20 * q^85 - 9 * q^87 - 10 * q^89 + q^91 - 4 * q^93 - 14 * q^95 - 4 * q^97 - 2 * 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
$5$	0.561553	0.251134	0.125567	−	0.992085i	$-0.459925\pi$
$5$	0.561553	0.251134	0.125567	+	0.992085i	$0.459925\pi$
$6$	0	0
$7$	2.56155	0.968176	0.484088	−	0.875019i	$-0.339151\pi$
$7$	2.56155	0.968176	0.484088	+	0.875019i	$0.339151\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0.561553	0.144992
$16$	0	0
$17$	−3.12311	−0.757464	−0.378732	−	0.925506i	$-0.623640\pi$
$17$	−3.12311	−0.757464	−0.378732	+	0.925506i	$0.623640\pi$
$18$	0	0
$19$	−5.12311	−1.17532	−0.587661	−	0.809108i	$-0.699951\pi$
$19$	−5.12311	−1.17532	−0.587661	+	0.809108i	$0.699951\pi$
$20$	0	0
$21$	2.56155	0.558977
$22$	0	0
$23$	6.56155	1.36818	0.684089	−	0.729398i	$-0.260200\pi$
$23$	6.56155	1.36818	0.684089	+	0.729398i	$0.260200\pi$
$24$	0	0
$25$	−4.68466	−0.936932
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	1.68466	0.312833	0.156417	−	0.987691i	$-0.450006\pi$
$29$	1.68466	0.312833	0.156417	+	0.987691i	$0.450006\pi$
$30$	0	0
$31$	6.24621	1.12185	0.560926	−	0.827866i	$-0.310445\pi$
$31$	6.24621	1.12185	0.560926	+	0.827866i	$0.310445\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	1.43845	0.243142
$36$	0	0
$37$	8.24621	1.35567	0.677834	−	0.735215i	$-0.262919\pi$
$37$	8.24621	1.35567	0.677834	+	0.735215i	$0.262919\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	−3.43845	−0.536995	−0.268498	−	0.963280i	$-0.586527\pi$
$41$	−3.43845	−0.536995	−0.268498	+	0.963280i	$0.586527\pi$
$42$	0	0
$43$	−2.56155	−0.390633	−0.195317	−	0.980740i	$-0.562573\pi$
$43$	−2.56155	−0.390633	−0.195317	+	0.980740i	$0.562573\pi$
$44$	0	0
$45$	0.561553	0.0837114
$46$	0	0
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	0	0
$49$	−0.438447	−0.0626353
$50$	0	0
$51$	−3.12311	−0.437322
$52$	0	0
$53$	−4.24621	−0.583262	−0.291631	−	0.956531i	$-0.594198\pi$
$53$	−4.24621	−0.583262	−0.291631	+	0.956531i	$0.594198\pi$
$54$	0	0
$55$	−0.561553	−0.0757198
$56$	0	0
$57$	−5.12311	−0.678572
$58$	0	0
$59$	5.43845	0.708026	0.354013	−	0.935241i	$-0.384817\pi$
$59$	5.43845	0.708026	0.354013	+	0.935241i	$0.384817\pi$
$60$	0	0
$61$	4.56155	0.584047	0.292023	−	0.956411i	$-0.405671\pi$
$61$	4.56155	0.584047	0.292023	+	0.956411i	$0.405671\pi$
$62$	0	0
$63$	2.56155	0.322725
$64$	0	0
$65$	0.561553	0.0696521
$66$	0	0
$67$	14.5616	1.77898	0.889488	−	0.456958i	$-0.151061\pi$
$67$	14.5616	1.77898	0.889488	+	0.456958i	$0.151061\pi$
$68$	0	0
$69$	6.56155	0.789918
$70$	0	0
$71$	15.3693	1.82400	0.912001	−	0.410188i	$-0.134537\pi$
$71$	15.3693	1.82400	0.912001	+	0.410188i	$0.134537\pi$
$72$	0	0
$73$	−7.43845	−0.870604	−0.435302	−	0.900284i	$-0.643358\pi$
$73$	−7.43845	−0.870604	−0.435302	+	0.900284i	$0.643358\pi$
$74$	0	0
$75$	−4.68466	−0.540938
$76$	0	0
$77$	−2.56155	−0.291916
$78$	0	0
$79$	−13.1231	−1.47646	−0.738232	−	0.674546i	$-0.764339\pi$
$79$	−13.1231	−1.47646	−0.738232	+	0.674546i	$0.764339\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	17.1231	1.87951	0.939753	−	0.341856i	$-0.111055\pi$
$83$	17.1231	1.87951	0.939753	+	0.341856i	$0.111055\pi$
$84$	0	0
$85$	−1.75379	−0.190225
$86$	0	0
$87$	1.68466	0.180614
$88$	0	0
$89$	−17.3693	−1.84114	−0.920572	−	0.390573i	$-0.872277\pi$
$89$	−17.3693	−1.84114	−0.920572	+	0.390573i	$0.872277\pi$
$90$	0	0
$91$	2.56155	0.268524
$92$	0	0
$93$	6.24621	0.647702
$94$	0	0
$95$	−2.87689	−0.295163
$96$	0	0
$97$	14.4924	1.47148	0.735741	−	0.677263i	$-0.236834\pi$
$97$	14.4924	1.47148	0.735741	+	0.677263i	$0.236834\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	0	0
$101$	16.2462	1.61656	0.808279	−	0.588799i	$-0.200399\pi$
$101$	16.2462	1.61656	0.808279	+	0.588799i	$0.200399\pi$
$102$	0	0
$103$	9.43845	0.929998	0.464999	−	0.885311i	$-0.346055\pi$
$103$	9.43845	0.929998	0.464999	+	0.885311i	$0.346055\pi$
$104$	0	0
$105$	1.43845	0.140378
$106$	0	0
$107$	13.4384	1.29914	0.649572	−	0.760300i	$-0.274948\pi$
$107$	13.4384	1.29914	0.649572	+	0.760300i	$0.274948\pi$
$108$	0	0
$109$	−9.36932	−0.897418	−0.448709	−	0.893678i	$-0.648116\pi$
$109$	−9.36932	−0.897418	−0.448709	+	0.893678i	$0.648116\pi$
$110$	0	0
$111$	8.24621	0.782696
$112$	0	0
$113$	3.43845	0.323462	0.161731	−	0.986835i	$-0.448292\pi$
$113$	3.43845	0.323462	0.161731	+	0.986835i	$0.448292\pi$
$114$	0	0
$115$	3.68466	0.343596
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	−8.00000	−0.733359
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−3.43845	−0.310034
$124$	0	0
$125$	−5.43845	−0.486430
$126$	0	0
$127$	5.12311	0.454602	0.227301	−	0.973825i	$-0.427010\pi$
$127$	5.12311	0.454602	0.227301	+	0.973825i	$0.427010\pi$
$128$	0	0
$129$	−2.56155	−0.225532
$130$	0	0
$131$	−7.68466	−0.671412	−0.335706	−	0.941967i	$-0.608975\pi$
$131$	−7.68466	−0.671412	−0.335706	+	0.941967i	$0.608975\pi$
$132$	0	0
$133$	−13.1231	−1.13792
$134$	0	0
$135$	0.561553	0.0483308
$136$	0	0
$137$	18.4924	1.57991	0.789957	−	0.613162i	$-0.210103\pi$
$137$	18.4924	1.57991	0.789957	+	0.613162i	$0.210103\pi$
$138$	0	0
$139$	6.24621	0.529797	0.264898	−	0.964276i	$-0.414662\pi$
$139$	6.24621	0.529797	0.264898	+	0.964276i	$0.414662\pi$
$140$	0	0
$141$	8.00000	0.673722
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	0.946025	0.0785631
$146$	0	0
$147$	−0.438447	−0.0361625
$148$	0	0
$149$	7.12311	0.583548	0.291774	−	0.956487i	$-0.405755\pi$
$149$	7.12311	0.583548	0.291774	+	0.956487i	$0.405755\pi$
$150$	0	0
$151$	−6.24621	−0.508309	−0.254155	−	0.967164i	$-0.581797\pi$
$151$	−6.24621	−0.508309	−0.254155	+	0.967164i	$0.581797\pi$
$152$	0	0
$153$	−3.12311	−0.252488
$154$	0	0
$155$	3.50758	0.281735
$156$	0	0
$157$	8.24621	0.658119	0.329060	−	0.944309i	$-0.393268\pi$
$157$	8.24621	0.658119	0.329060	+	0.944309i	$0.393268\pi$
$158$	0	0
$159$	−4.24621	−0.336746
$160$	0	0
$161$	16.8078	1.32464
$162$	0	0
$163$	−8.80776	−0.689877	−0.344939	−	0.938625i	$-0.612100\pi$
$163$	−8.80776	−0.689877	−0.344939	+	0.938625i	$0.612100\pi$
$164$	0	0
$165$	−0.561553	−0.0437168
$166$	0	0
$167$	−13.9309	−1.07800	−0.539002	−	0.842305i	$-0.681198\pi$
$167$	−13.9309	−1.07800	−0.539002	+	0.842305i	$0.681198\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−5.12311	−0.391774
$172$	0	0
$173$	10.3153	0.784261	0.392130	−	0.919910i	$-0.371738\pi$
$173$	10.3153	0.784261	0.392130	+	0.919910i	$0.371738\pi$
$174$	0	0
$175$	−12.0000	−0.907115
$176$	0	0
$177$	5.43845	0.408779
$178$	0	0
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	0	0
$181$	−4.24621	−0.315618	−0.157809	−	0.987470i	$-0.550443\pi$
$181$	−4.24621	−0.315618	−0.157809	+	0.987470i	$0.550443\pi$
$182$	0	0
$183$	4.56155	0.337200
$184$	0	0
$185$	4.63068	0.340455
$186$	0	0
$187$	3.12311	0.228384
$188$	0	0
$189$	2.56155	0.186326
$190$	0	0
$191$	−14.5616	−1.05364	−0.526818	−	0.849978i	$-0.676615\pi$
$191$	−14.5616	−1.05364	−0.526818	+	0.849978i	$0.676615\pi$
$192$	0	0
$193$	−8.24621	−0.593575	−0.296788	−	0.954944i	$-0.595915\pi$
$193$	−8.24621	−0.593575	−0.296788	+	0.954944i	$0.595915\pi$
$194$	0	0
$195$	0.561553	0.0402136
$196$	0	0
$197$	−15.6155	−1.11256	−0.556280	−	0.830995i	$-0.687772\pi$
$197$	−15.6155	−1.11256	−0.556280	+	0.830995i	$0.687772\pi$
$198$	0	0
$199$	−5.93087	−0.420428	−0.210214	−	0.977655i	$-0.567416\pi$
$199$	−5.93087	−0.420428	−0.210214	+	0.977655i	$0.567416\pi$
$200$	0	0
$201$	14.5616	1.02709
$202$	0	0
$203$	4.31534	0.302878
$204$	0	0
$205$	−1.93087	−0.134858
$206$	0	0
$207$	6.56155	0.456059
$208$	0	0
$209$	5.12311	0.354373
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	0	0
$213$	15.3693	1.05309
$214$	0	0
$215$	−1.43845	−0.0981013
$216$	0	0
$217$	16.0000	1.08615
$218$	0	0
$219$	−7.43845	−0.502644
$220$	0	0
$221$	−3.12311	−0.210083
$222$	0	0
$223$	−24.4924	−1.64013	−0.820067	−	0.572268i	$-0.806064\pi$
$223$	−24.4924	−1.64013	−0.820067	+	0.572268i	$0.806064\pi$
$224$	0	0
$225$	−4.68466	−0.312311
$226$	0	0
$227$	6.24621	0.414576	0.207288	−	0.978280i	$-0.433536\pi$
$227$	6.24621	0.414576	0.207288	+	0.978280i	$0.433536\pi$
$228$	0	0
$229$	−26.8078	−1.77151	−0.885753	−	0.464156i	$-0.846358\pi$
$229$	−26.8078	−1.77151	−0.885753	+	0.464156i	$0.846358\pi$
$230$	0	0
$231$	−2.56155	−0.168538
$232$	0	0
$233$	−0.246211	−0.0161298	−0.00806492	−	0.999967i	$-0.502567\pi$
$233$	−0.246211	−0.0161298	−0.00806492	+	0.999967i	$0.502567\pi$
$234$	0	0
$235$	4.49242	0.293053
$236$	0	0
$237$	−13.1231	−0.852437
$238$	0	0
$239$	29.9309	1.93607	0.968034	−	0.250821i	$-0.0807005\pi$
$239$	29.9309	1.93607	0.968034	+	0.250821i	$0.0807005\pi$
$240$	0	0
$241$	−16.2462	−1.04651	−0.523255	−	0.852176i	$-0.675283\pi$
$241$	−16.2462	−1.04651	−0.523255	+	0.852176i	$0.675283\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−0.246211	−0.0157299
$246$	0	0
$247$	−5.12311	−0.325975
$248$	0	0
$249$	17.1231	1.08513
$250$	0	0
$251$	−17.1231	−1.08080	−0.540400	−	0.841408i	$-0.681727\pi$
$251$	−17.1231	−1.08080	−0.540400	+	0.841408i	$0.681727\pi$
$252$	0	0
$253$	−6.56155	−0.412521
$254$	0	0
$255$	−1.75379	−0.109827
$256$	0	0
$257$	0.561553	0.0350287	0.0175144	−	0.999847i	$-0.494425\pi$
$257$	0.561553	0.0350287	0.0175144	+	0.999847i	$0.494425\pi$
$258$	0	0
$259$	21.1231	1.31253
$260$	0	0
$261$	1.68466	0.104278
$262$	0	0
$263$	−12.4924	−0.770316	−0.385158	−	0.922851i	$-0.625853\pi$
$263$	−12.4924	−0.770316	−0.385158	+	0.922851i	$0.625853\pi$
$264$	0	0
$265$	−2.38447	−0.146477
$266$	0	0
$267$	−17.3693	−1.06298
$268$	0	0
$269$	23.6155	1.43986	0.719932	−	0.694045i	$-0.244173\pi$
$269$	23.6155	1.43986	0.719932	+	0.694045i	$0.244173\pi$
$270$	0	0
$271$	−22.2462	−1.35136	−0.675681	−	0.737195i	$-0.736150\pi$
$271$	−22.2462	−1.35136	−0.675681	+	0.737195i	$0.736150\pi$
$272$	0	0
$273$	2.56155	0.155032
$274$	0	0
$275$	4.68466	0.282496
$276$	0	0
$277$	12.5616	0.754751	0.377375	−	0.926060i	$-0.376827\pi$
$277$	12.5616	0.754751	0.377375	+	0.926060i	$0.376827\pi$
$278$	0	0
$279$	6.24621	0.373951
$280$	0	0
$281$	−0.561553	−0.0334994	−0.0167497	−	0.999860i	$-0.505332\pi$
$281$	−0.561553	−0.0334994	−0.0167497	+	0.999860i	$0.505332\pi$
$282$	0	0
$283$	15.6847	0.932356	0.466178	−	0.884691i	$-0.345631\pi$
$283$	15.6847	0.932356	0.466178	+	0.884691i	$0.345631\pi$
$284$	0	0
$285$	−2.87689	−0.170413
$286$	0	0
$287$	−8.80776	−0.519906
$288$	0	0
$289$	−7.24621	−0.426248
$290$	0	0
$291$	14.4924	0.849561
$292$	0	0
$293$	−0.246211	−0.0143838	−0.00719191	−	0.999974i	$-0.502289\pi$
$293$	−0.246211	−0.0143838	−0.00719191	+	0.999974i	$0.502289\pi$
$294$	0	0
$295$	3.05398	0.177809
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	6.56155	0.379464
$300$	0	0
$301$	−6.56155	−0.378202
$302$	0	0
$303$	16.2462	0.933320
$304$	0	0
$305$	2.56155	0.146674
$306$	0	0
$307$	0.630683	0.0359950	0.0179975	−	0.999838i	$-0.494271\pi$
$307$	0.630683	0.0359950	0.0179975	+	0.999838i	$0.494271\pi$
$308$	0	0
$309$	9.43845	0.536935
$310$	0	0
$311$	16.0000	0.907277	0.453638	−	0.891186i	$-0.350126\pi$
$311$	16.0000	0.907277	0.453638	+	0.891186i	$0.350126\pi$
$312$	0	0
$313$	−3.93087	−0.222186	−0.111093	−	0.993810i	$-0.535435\pi$
$313$	−3.93087	−0.222186	−0.111093	+	0.993810i	$0.535435\pi$
$314$	0	0
$315$	1.43845	0.0810473
$316$	0	0
$317$	11.4384	0.642447	0.321224	−	0.947003i	$-0.395906\pi$
$317$	11.4384	0.642447	0.321224	+	0.947003i	$0.395906\pi$
$318$	0	0
$319$	−1.68466	−0.0943228
$320$	0	0
$321$	13.4384	0.750061
$322$	0	0
$323$	16.0000	0.890264
$324$	0	0
$325$	−4.68466	−0.259858
$326$	0	0
$327$	−9.36932	−0.518124
$328$	0	0
$329$	20.4924	1.12978
$330$	0	0
$331$	27.0540	1.48702	0.743510	−	0.668724i	$-0.233159\pi$
$331$	27.0540	1.48702	0.743510	+	0.668724i	$0.233159\pi$
$332$	0	0
$333$	8.24621	0.451890
$334$	0	0
$335$	8.17708	0.446762
$336$	0	0
$337$	−3.75379	−0.204482	−0.102241	−	0.994760i	$-0.532601\pi$
$337$	−3.75379	−0.204482	−0.102241	+	0.994760i	$0.532601\pi$
$338$	0	0
$339$	3.43845	0.186751
$340$	0	0
$341$	−6.24621	−0.338251
$342$	0	0
$343$	−19.0540	−1.02882
$344$	0	0
$345$	3.68466	0.198375
$346$	0	0
$347$	−8.49242	−0.455897	−0.227949	−	0.973673i	$-0.573202\pi$
$347$	−8.49242	−0.455897	−0.227949	+	0.973673i	$0.573202\pi$
$348$	0	0
$349$	33.8617	1.81258	0.906289	−	0.422659i	$-0.138903\pi$
$349$	33.8617	1.81258	0.906289	+	0.422659i	$0.138903\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	20.7386	1.10381	0.551903	−	0.833908i	$-0.313902\pi$
$353$	20.7386	1.10381	0.551903	+	0.833908i	$0.313902\pi$
$354$	0	0
$355$	8.63068	0.458069
$356$	0	0
$357$	−8.00000	−0.423405
$358$	0	0
$359$	−9.43845	−0.498142	−0.249071	−	0.968485i	$-0.580125\pi$
$359$	−9.43845	−0.498142	−0.249071	+	0.968485i	$0.580125\pi$
$360$	0	0
$361$	7.24621	0.381380
$362$	0	0
$363$	1.00000	0.0524864
$364$	0	0
$365$	−4.17708	−0.218638
$366$	0	0
$367$	16.0000	0.835193	0.417597	−	0.908633i	$-0.362873\pi$
$367$	16.0000	0.835193	0.417597	+	0.908633i	$0.362873\pi$
$368$	0	0
$369$	−3.43845	−0.178998
$370$	0	0
$371$	−10.8769	−0.564700
$372$	0	0
$373$	33.0540	1.71147	0.855735	−	0.517414i	$-0.173105\pi$
$373$	33.0540	1.71147	0.855735	+	0.517414i	$0.173105\pi$
$374$	0	0
$375$	−5.43845	−0.280840
$376$	0	0
$377$	1.68466	0.0867643
$378$	0	0
$379$	−12.4924	−0.641693	−0.320846	−	0.947131i	$-0.603967\pi$
$379$	−12.4924	−0.641693	−0.320846	+	0.947131i	$0.603967\pi$
$380$	0	0
$381$	5.12311	0.262465
$382$	0	0
$383$	−9.61553	−0.491331	−0.245665	−	0.969355i	$-0.579006\pi$
$383$	−9.61553	−0.491331	−0.245665	+	0.969355i	$0.579006\pi$
$384$	0	0
$385$	−1.43845	−0.0733101
$386$	0	0
$387$	−2.56155	−0.130211
$388$	0	0
$389$	−30.4924	−1.54603	−0.773014	−	0.634389i	$-0.781252\pi$
$389$	−30.4924	−1.54603	−0.773014	+	0.634389i	$0.781252\pi$
$390$	0	0
$391$	−20.4924	−1.03635
$392$	0	0
$393$	−7.68466	−0.387640
$394$	0	0
$395$	−7.36932	−0.370791
$396$	0	0
$397$	−32.5616	−1.63422	−0.817109	−	0.576484i	$-0.804424\pi$
$397$	−32.5616	−1.63422	−0.817109	+	0.576484i	$0.804424\pi$
$398$	0	0
$399$	−13.1231	−0.656977
$400$	0	0
$401$	−15.1231	−0.755212	−0.377606	−	0.925966i	$-0.623253\pi$
$401$	−15.1231	−0.755212	−0.377606	+	0.925966i	$0.623253\pi$
$402$	0	0
$403$	6.24621	0.311146
$404$	0	0
$405$	0.561553	0.0279038
$406$	0	0
$407$	−8.24621	−0.408750
$408$	0	0
$409$	−14.8078	−0.732197	−0.366098	−	0.930576i	$-0.619307\pi$
$409$	−14.8078	−0.732197	−0.366098	+	0.930576i	$0.619307\pi$
$410$	0	0
$411$	18.4924	0.912164
$412$	0	0
$413$	13.9309	0.685493
$414$	0	0
$415$	9.61553	0.472008
$416$	0	0
$417$	6.24621	0.305878
$418$	0	0
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	−0.561553	−0.0273684	−0.0136842	−	0.999906i	$-0.504356\pi$
$421$	−0.561553	−0.0273684	−0.0136842	+	0.999906i	$0.504356\pi$
$422$	0	0
$423$	8.00000	0.388973
$424$	0	0
$425$	14.6307	0.709692
$426$	0	0
$427$	11.6847	0.565460
$428$	0	0
$429$	−1.00000	−0.0482805
$430$	0	0
$431$	−8.00000	−0.385346	−0.192673	−	0.981263i	$-0.561716\pi$
$431$	−8.00000	−0.385346	−0.192673	+	0.981263i	$0.561716\pi$
$432$	0	0
$433$	−35.3002	−1.69642	−0.848209	−	0.529661i	$-0.822319\pi$
$433$	−35.3002	−1.69642	−0.848209	+	0.529661i	$0.822319\pi$
$434$	0	0
$435$	0.946025	0.0453584
$436$	0	0
$437$	−33.6155	−1.60805
$438$	0	0
$439$	22.7386	1.08526	0.542628	−	0.839973i	$-0.317429\pi$
$439$	22.7386	1.08526	0.542628	+	0.839973i	$0.317429\pi$
$440$	0	0
$441$	−0.438447	−0.0208784
$442$	0	0
$443$	−22.2462	−1.05695	−0.528475	−	0.848949i	$-0.677236\pi$
$443$	−22.2462	−1.05695	−0.528475	+	0.848949i	$0.677236\pi$
$444$	0	0
$445$	−9.75379	−0.462374
$446$	0	0
$447$	7.12311	0.336911
$448$	0	0
$449$	6.63068	0.312921	0.156461	−	0.987684i	$-0.449992\pi$
$449$	6.63068	0.312921	0.156461	+	0.987684i	$0.449992\pi$
$450$	0	0
$451$	3.43845	0.161910
$452$	0	0
$453$	−6.24621	−0.293473
$454$	0	0
$455$	1.43845	0.0674354
$456$	0	0
$457$	11.4384	0.535068	0.267534	−	0.963548i	$-0.413791\pi$
$457$	11.4384	0.535068	0.267534	+	0.963548i	$0.413791\pi$
$458$	0	0
$459$	−3.12311	−0.145774
$460$	0	0
$461$	−29.3693	−1.36787	−0.683933	−	0.729545i	$-0.739732\pi$
$461$	−29.3693	−1.36787	−0.683933	+	0.729545i	$0.739732\pi$
$462$	0	0
$463$	39.8617	1.85253	0.926266	−	0.376870i	$-0.123000\pi$
$463$	39.8617	1.85253	0.926266	+	0.376870i	$0.123000\pi$
$464$	0	0
$465$	3.50758	0.162660
$466$	0	0
$467$	25.7538	1.19174	0.595872	−	0.803080i	$-0.296807\pi$
$467$	25.7538	1.19174	0.595872	+	0.803080i	$0.296807\pi$
$468$	0	0
$469$	37.3002	1.72236
$470$	0	0
$471$	8.24621	0.379965
$472$	0	0
$473$	2.56155	0.117780
$474$	0	0
$475$	24.0000	1.10120
$476$	0	0
$477$	−4.24621	−0.194421
$478$	0	0
$479$	21.3002	0.973230	0.486615	−	0.873616i	$-0.338231\pi$
$479$	21.3002	0.973230	0.486615	+	0.873616i	$0.338231\pi$
$480$	0	0
$481$	8.24621	0.375995
$482$	0	0
$483$	16.8078	0.764780
$484$	0	0
$485$	8.13826	0.369539
$486$	0	0
$487$	11.3693	0.515193	0.257596	−	0.966253i	$-0.417069\pi$
$487$	11.3693	0.515193	0.257596	+	0.966253i	$0.417069\pi$
$488$	0	0
$489$	−8.80776	−0.398301
$490$	0	0
$491$	−28.1771	−1.27161	−0.635807	−	0.771848i	$-0.719333\pi$
$491$	−28.1771	−1.27161	−0.635807	+	0.771848i	$0.719333\pi$
$492$	0	0
$493$	−5.26137	−0.236960
$494$	0	0
$495$	−0.561553	−0.0252399
$496$	0	0
$497$	39.3693	1.76596
$498$	0	0
$499$	−29.3002	−1.31166	−0.655828	−	0.754910i	$-0.727680\pi$
$499$	−29.3002	−1.31166	−0.655828	+	0.754910i	$0.727680\pi$
$500$	0	0
$501$	−13.9309	−0.622385
$502$	0	0
$503$	28.4924	1.27041	0.635207	−	0.772342i	$-0.280915\pi$
$503$	28.4924	1.27041	0.635207	+	0.772342i	$0.280915\pi$
$504$	0	0
$505$	9.12311	0.405973
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	2.00000	0.0886484	0.0443242	−	0.999017i	$-0.485887\pi$
$509$	2.00000	0.0886484	0.0443242	+	0.999017i	$0.485887\pi$
$510$	0	0
$511$	−19.0540	−0.842898
$512$	0	0
$513$	−5.12311	−0.226191
$514$	0	0
$515$	5.30019	0.233554
$516$	0	0
$517$	−8.00000	−0.351840
$518$	0	0
$519$	10.3153	0.452793
$520$	0	0
$521$	34.8078	1.52496	0.762478	−	0.647014i	$-0.223983\pi$
$521$	34.8078	1.52496	0.762478	+	0.647014i	$0.223983\pi$
$522$	0	0
$523$	−4.00000	−0.174908	−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000	−0.174908	−0.0874539	+	0.996169i	$0.527873\pi$
$524$	0	0
$525$	−12.0000	−0.523723
$526$	0	0
$527$	−19.5076	−0.849763
$528$	0	0
$529$	20.0540	0.871912
$530$	0	0
$531$	5.43845	0.236009
$532$	0	0
$533$	−3.43845	−0.148936
$534$	0	0
$535$	7.54640	0.326259
$536$	0	0
$537$	−4.00000	−0.172613
$538$	0	0
$539$	0.438447	0.0188853
$540$	0	0
$541$	44.7386	1.92346	0.961732	−	0.273992i	$-0.0883441\pi$
$541$	44.7386	1.92346	0.961732	+	0.273992i	$0.0883441\pi$
$542$	0	0
$543$	−4.24621	−0.182222
$544$	0	0
$545$	−5.26137	−0.225372
$546$	0	0
$547$	1.93087	0.0825580	0.0412790	−	0.999148i	$-0.486857\pi$
$547$	1.93087	0.0825580	0.0412790	+	0.999148i	$0.486857\pi$
$548$	0	0
$549$	4.56155	0.194682
$550$	0	0
$551$	−8.63068	−0.367679
$552$	0	0
$553$	−33.6155	−1.42948
$554$	0	0
$555$	4.63068	0.196562
$556$	0	0
$557$	−8.24621	−0.349403	−0.174702	−	0.984621i	$-0.555896\pi$
$557$	−8.24621	−0.349403	−0.174702	+	0.984621i	$0.555896\pi$
$558$	0	0
$559$	−2.56155	−0.108342
$560$	0	0
$561$	3.12311	0.131858
$562$	0	0
$563$	10.7386	0.452579	0.226290	−	0.974060i	$-0.427340\pi$
$563$	10.7386	0.452579	0.226290	+	0.974060i	$0.427340\pi$
$564$	0	0
$565$	1.93087	0.0812323
$566$	0	0
$567$	2.56155	0.107575
$568$	0	0
$569$	−26.4924	−1.11062	−0.555310	−	0.831643i	$-0.687400\pi$
$569$	−26.4924	−1.11062	−0.555310	+	0.831643i	$0.687400\pi$
$570$	0	0
$571$	−26.5616	−1.11157	−0.555783	−	0.831327i	$-0.687582\pi$
$571$	−26.5616	−1.11157	−0.555783	+	0.831327i	$0.687582\pi$
$572$	0	0
$573$	−14.5616	−0.608318
$574$	0	0
$575$	−30.7386	−1.28189
$576$	0	0
$577$	4.87689	0.203028	0.101514	−	0.994834i	$-0.467631\pi$
$577$	4.87689	0.203028	0.101514	+	0.994834i	$0.467631\pi$
$578$	0	0
$579$	−8.24621	−0.342701
$580$	0	0
$581$	43.8617	1.81969
$582$	0	0
$583$	4.24621	0.175860
$584$	0	0
$585$	0.561553	0.0232174
$586$	0	0
$587$	18.5616	0.766117	0.383059	−	0.923724i	$-0.374871\pi$
$587$	18.5616	0.766117	0.383059	+	0.923724i	$0.374871\pi$
$588$	0	0
$589$	−32.0000	−1.31854
$590$	0	0
$591$	−15.6155	−0.642337
$592$	0	0
$593$	10.4924	0.430872	0.215436	−	0.976518i	$-0.430883\pi$
$593$	10.4924	0.430872	0.215436	+	0.976518i	$0.430883\pi$
$594$	0	0
$595$	−4.49242	−0.184171
$596$	0	0
$597$	−5.93087	−0.242734
$598$	0	0
$599$	8.80776	0.359875	0.179938	−	0.983678i	$-0.442410\pi$
$599$	8.80776	0.359875	0.179938	+	0.983678i	$0.442410\pi$
$600$	0	0
$601$	−33.8617	−1.38125	−0.690625	−	0.723213i	$-0.742664\pi$
$601$	−33.8617	−1.38125	−0.690625	+	0.723213i	$0.742664\pi$
$602$	0	0
$603$	14.5616	0.592992
$604$	0	0
$605$	0.561553	0.0228304
$606$	0	0
$607$	−5.12311	−0.207940	−0.103970	−	0.994580i	$-0.533155\pi$
$607$	−5.12311	−0.207940	−0.103970	+	0.994580i	$0.533155\pi$
$608$	0	0
$609$	4.31534	0.174866
$610$	0	0
$611$	8.00000	0.323645
$612$	0	0
$613$	−29.8617	−1.20610	−0.603052	−	0.797702i	$-0.706049\pi$
$613$	−29.8617	−1.20610	−0.603052	+	0.797702i	$0.706049\pi$
$614$	0	0
$615$	−1.93087	−0.0778602
$616$	0	0
$617$	−12.2462	−0.493014	−0.246507	−	0.969141i	$-0.579283\pi$
$617$	−12.2462	−0.493014	−0.246507	+	0.969141i	$0.579283\pi$
$618$	0	0
$619$	10.0691	0.404713	0.202356	−	0.979312i	$-0.435140\pi$
$619$	10.0691	0.404713	0.202356	+	0.979312i	$0.435140\pi$
$620$	0	0
$621$	6.56155	0.263306
$622$	0	0
$623$	−44.4924	−1.78255
$624$	0	0
$625$	20.3693	0.814773
$626$	0	0
$627$	5.12311	0.204597
$628$	0	0
$629$	−25.7538	−1.02687
$630$	0	0
$631$	−17.1231	−0.681660	−0.340830	−	0.940125i	$-0.610708\pi$
$631$	−17.1231	−0.681660	−0.340830	+	0.940125i	$0.610708\pi$
$632$	0	0
$633$	−4.00000	−0.158986
$634$	0	0
$635$	2.87689	0.114166
$636$	0	0
$637$	−0.438447	−0.0173719
$638$	0	0
$639$	15.3693	0.608001
$640$	0	0
$641$	−1.68466	−0.0665400	−0.0332700	−	0.999446i	$-0.510592\pi$
$641$	−1.68466	−0.0665400	−0.0332700	+	0.999446i	$0.510592\pi$
$642$	0	0
$643$	34.2462	1.35054	0.675269	−	0.737571i	$-0.264027\pi$
$643$	34.2462	1.35054	0.675269	+	0.737571i	$0.264027\pi$
$644$	0	0
$645$	−1.43845	−0.0566388
$646$	0	0
$647$	16.9848	0.667743	0.333872	−	0.942619i	$-0.391645\pi$
$647$	16.9848	0.667743	0.333872	+	0.942619i	$0.391645\pi$
$648$	0	0
$649$	−5.43845	−0.213478
$650$	0	0
$651$	16.0000	0.627089
$652$	0	0
$653$	26.4924	1.03673	0.518364	−	0.855160i	$-0.326541\pi$
$653$	26.4924	1.03673	0.518364	+	0.855160i	$0.326541\pi$
$654$	0	0
$655$	−4.31534	−0.168614
$656$	0	0
$657$	−7.43845	−0.290201
$658$	0	0
$659$	−38.2462	−1.48986	−0.744930	−	0.667142i	$-0.767517\pi$
$659$	−38.2462	−1.48986	−0.744930	+	0.667142i	$0.767517\pi$
$660$	0	0
$661$	8.24621	0.320740	0.160370	−	0.987057i	$-0.448731\pi$
$661$	8.24621	0.320740	0.160370	+	0.987057i	$0.448731\pi$
$662$	0	0
$663$	−3.12311	−0.121291
$664$	0	0
$665$	−7.36932	−0.285770
$666$	0	0
$667$	11.0540	0.428012
$668$	0	0
$669$	−24.4924	−0.946932
$670$	0	0
$671$	−4.56155	−0.176097
$672$	0	0
$673$	−6.63068	−0.255594	−0.127797	−	0.991800i	$-0.540791\pi$
$673$	−6.63068	−0.255594	−0.127797	+	0.991800i	$0.540791\pi$
$674$	0	0
$675$	−4.68466	−0.180313
$676$	0	0
$677$	14.0000	0.538064	0.269032	−	0.963131i	$-0.413296\pi$
$677$	14.0000	0.538064	0.269032	+	0.963131i	$0.413296\pi$
$678$	0	0
$679$	37.1231	1.42465
$680$	0	0
$681$	6.24621	0.239355
$682$	0	0
$683$	−14.0691	−0.538340	−0.269170	−	0.963093i	$-0.586749\pi$
$683$	−14.0691	−0.538340	−0.269170	+	0.963093i	$0.586749\pi$
$684$	0	0
$685$	10.3845	0.396770
$686$	0	0
$687$	−26.8078	−1.02278
$688$	0	0
$689$	−4.24621	−0.161768
$690$	0	0
$691$	−40.0000	−1.52167	−0.760836	−	0.648944i	$-0.775211\pi$
$691$	−40.0000	−1.52167	−0.760836	+	0.648944i	$0.775211\pi$
$692$	0	0
$693$	−2.56155	−0.0973053
$694$	0	0
$695$	3.50758	0.133050
$696$	0	0
$697$	10.7386	0.406755
$698$	0	0
$699$	−0.246211	−0.00931256
$700$	0	0
$701$	3.93087	0.148467	0.0742334	−	0.997241i	$-0.476349\pi$
$701$	3.93087	0.148467	0.0742334	+	0.997241i	$0.476349\pi$
$702$	0	0
$703$	−42.2462	−1.59335
$704$	0	0
$705$	4.49242	0.169194
$706$	0	0
$707$	41.6155	1.56511
$708$	0	0
$709$	−10.8078	−0.405894	−0.202947	−	0.979190i	$-0.565052\pi$
$709$	−10.8078	−0.405894	−0.202947	+	0.979190i	$0.565052\pi$
$710$	0	0
$711$	−13.1231	−0.492155
$712$	0	0
$713$	40.9848	1.53489
$714$	0	0
$715$	−0.561553	−0.0210009
$716$	0	0
$717$	29.9309	1.11779
$718$	0	0
$719$	−19.0540	−0.710593	−0.355297	−	0.934754i	$-0.615620\pi$
$719$	−19.0540	−0.710593	−0.355297	+	0.934754i	$0.615620\pi$
$720$	0	0
$721$	24.1771	0.900402
$722$	0	0
$723$	−16.2462	−0.604203
$724$	0	0
$725$	−7.89205	−0.293103
$726$	0	0
$727$	50.2462	1.86353	0.931764	−	0.363063i	$-0.118269\pi$
$727$	50.2462	1.86353	0.931764	+	0.363063i	$0.118269\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	8.00000	0.295891
$732$	0	0
$733$	−2.00000	−0.0738717	−0.0369358	−	0.999318i	$-0.511760\pi$
$733$	−2.00000	−0.0738717	−0.0369358	+	0.999318i	$0.511760\pi$
$734$	0	0
$735$	−0.246211	−0.00908164
$736$	0	0
$737$	−14.5616	−0.536382
$738$	0	0
$739$	22.7386	0.836454	0.418227	−	0.908343i	$-0.362652\pi$
$739$	22.7386	0.836454	0.418227	+	0.908343i	$0.362652\pi$
$740$	0	0
$741$	−5.12311	−0.188202
$742$	0	0
$743$	30.5616	1.12119	0.560597	−	0.828089i	$-0.310572\pi$
$743$	30.5616	1.12119	0.560597	+	0.828089i	$0.310572\pi$
$744$	0	0
$745$	4.00000	0.146549
$746$	0	0
$747$	17.1231	0.626502
$748$	0	0
$749$	34.4233	1.25780
$750$	0	0
$751$	−53.9309	−1.96797	−0.983983	−	0.178264i	$-0.942952\pi$
$751$	−53.9309	−1.96797	−0.983983	+	0.178264i	$0.942952\pi$
$752$	0	0
$753$	−17.1231	−0.624001
$754$	0	0
$755$	−3.50758	−0.127654
$756$	0	0
$757$	22.0000	0.799604	0.399802	−	0.916602i	$-0.369079\pi$
$757$	22.0000	0.799604	0.399802	+	0.916602i	$0.369079\pi$
$758$	0	0
$759$	−6.56155	−0.238169
$760$	0	0
$761$	3.93087	0.142494	0.0712470	−	0.997459i	$-0.477302\pi$
$761$	3.93087	0.142494	0.0712470	+	0.997459i	$0.477302\pi$
$762$	0	0
$763$	−24.0000	−0.868858
$764$	0	0
$765$	−1.75379	−0.0634084
$766$	0	0
$767$	5.43845	0.196371
$768$	0	0
$769$	−7.43845	−0.268237	−0.134119	−	0.990965i	$-0.542820\pi$
$769$	−7.43845	−0.268237	−0.134119	+	0.990965i	$0.542820\pi$
$770$	0	0
$771$	0.561553	0.0202238
$772$	0	0
$773$	3.26137	0.117303	0.0586516	−	0.998279i	$-0.481320\pi$
$773$	3.26137	0.117303	0.0586516	+	0.998279i	$0.481320\pi$
$774$	0	0
$775$	−29.2614	−1.05110
$776$	0	0
$777$	21.1231	0.757787
$778$	0	0
$779$	17.6155	0.631142
$780$	0	0
$781$	−15.3693	−0.549957
$782$	0	0
$783$	1.68466	0.0602048
$784$	0	0
$785$	4.63068	0.165276
$786$	0	0
$787$	−24.0000	−0.855508	−0.427754	−	0.903895i	$-0.640695\pi$
$787$	−24.0000	−0.855508	−0.427754	+	0.903895i	$0.640695\pi$
$788$	0	0
$789$	−12.4924	−0.444742
$790$	0	0
$791$	8.80776	0.313168
$792$	0	0
$793$	4.56155	0.161985
$794$	0	0
$795$	−2.38447	−0.0845685
$796$	0	0
$797$	9.86174	0.349321	0.174660	−	0.984629i	$-0.444117\pi$
$797$	9.86174	0.349321	0.174660	+	0.984629i	$0.444117\pi$
$798$	0	0
$799$	−24.9848	−0.883900
$800$	0	0
$801$	−17.3693	−0.613715
$802$	0	0
$803$	7.43845	0.262497
$804$	0	0
$805$	9.43845	0.332662
$806$	0	0
$807$	23.6155	0.831306
$808$	0	0
$809$	−20.7386	−0.729132	−0.364566	−	0.931178i	$-0.618783\pi$
$809$	−20.7386	−0.729132	−0.364566	+	0.931178i	$0.618783\pi$
$810$	0	0
$811$	−46.7386	−1.64122	−0.820608	−	0.571492i	$-0.806365\pi$
$811$	−46.7386	−1.64122	−0.820608	+	0.571492i	$0.806365\pi$
$812$	0	0
$813$	−22.2462	−0.780209
$814$	0	0
$815$	−4.94602	−0.173252
$816$	0	0
$817$	13.1231	0.459119
$818$	0	0
$819$	2.56155	0.0895079
$820$	0	0
$821$	52.2462	1.82341	0.911703	−	0.410851i	$-0.134768\pi$
$821$	52.2462	1.82341	0.911703	+	0.410851i	$0.134768\pi$
$822$	0	0
$823$	−24.1771	−0.842760	−0.421380	−	0.906884i	$-0.638454\pi$
$823$	−24.1771	−0.842760	−0.421380	+	0.906884i	$0.638454\pi$
$824$	0	0
$825$	4.68466	0.163099
$826$	0	0
$827$	−10.7386	−0.373419	−0.186709	−	0.982415i	$-0.559782\pi$
$827$	−10.7386	−0.373419	−0.186709	+	0.982415i	$0.559782\pi$
$828$	0	0
$829$	−42.9848	−1.49293	−0.746463	−	0.665427i	$-0.768249\pi$
$829$	−42.9848	−1.49293	−0.746463	+	0.665427i	$0.768249\pi$
$830$	0	0
$831$	12.5616	0.435755
$832$	0	0
$833$	1.36932	0.0474440
$834$	0	0
$835$	−7.82292	−0.270723
$836$	0	0
$837$	6.24621	0.215901
$838$	0	0
$839$	−21.1231	−0.729251	−0.364625	−	0.931154i	$-0.618803\pi$
$839$	−21.1231	−0.729251	−0.364625	+	0.931154i	$0.618803\pi$
$840$	0	0
$841$	−26.1619	−0.902135
$842$	0	0
$843$	−0.561553	−0.0193409
$844$	0	0
$845$	0.561553	0.0193180
$846$	0	0
$847$	2.56155	0.0880160
$848$	0	0
$849$	15.6847	0.538296
$850$	0	0
$851$	54.1080	1.85480
$852$	0	0
$853$	−3.61553	−0.123793	−0.0618967	−	0.998083i	$-0.519715\pi$
$853$	−3.61553	−0.123793	−0.0618967	+	0.998083i	$0.519715\pi$
$854$	0	0
$855$	−2.87689	−0.0983877
$856$	0	0
$857$	8.38447	0.286408	0.143204	−	0.989693i	$-0.454259\pi$
$857$	8.38447	0.286408	0.143204	+	0.989693i	$0.454259\pi$
$858$	0	0
$859$	48.4924	1.65454	0.827270	−	0.561804i	$-0.189893\pi$
$859$	48.4924	1.65454	0.827270	+	0.561804i	$0.189893\pi$
$860$	0	0
$861$	−8.80776	−0.300168
$862$	0	0
$863$	−2.24621	−0.0764619	−0.0382310	−	0.999269i	$-0.512172\pi$
$863$	−2.24621	−0.0764619	−0.0382310	+	0.999269i	$0.512172\pi$
$864$	0	0
$865$	5.79261	0.196955
$866$	0	0
$867$	−7.24621	−0.246094
$868$	0	0
$869$	13.1231	0.445171
$870$	0	0
$871$	14.5616	0.493399
$872$	0	0
$873$	14.4924	0.490494
$874$	0	0
$875$	−13.9309	−0.470949
$876$	0	0
$877$	14.6307	0.494043	0.247022	−	0.969010i	$-0.420548\pi$
$877$	14.6307	0.494043	0.247022	+	0.969010i	$0.420548\pi$
$878$	0	0
$879$	−0.246211	−0.00830450
$880$	0	0
$881$	−2.31534	−0.0780058	−0.0390029	−	0.999239i	$-0.512418\pi$
$881$	−2.31534	−0.0780058	−0.0390029	+	0.999239i	$0.512418\pi$
$882$	0	0
$883$	−27.3693	−0.921051	−0.460525	−	0.887647i	$-0.652339\pi$
$883$	−27.3693	−0.921051	−0.460525	+	0.887647i	$0.652339\pi$
$884$	0	0
$885$	3.05398	0.102658
$886$	0	0
$887$	−16.0000	−0.537227	−0.268614	−	0.963248i	$-0.586566\pi$
$887$	−16.0000	−0.537227	−0.268614	+	0.963248i	$0.586566\pi$
$888$	0	0
$889$	13.1231	0.440135
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	−40.9848	−1.37151
$894$	0	0
$895$	−2.24621	−0.0750826
$896$	0	0
$897$	6.56155	0.219084
$898$	0	0
$899$	10.5227	0.350953
$900$	0	0
$901$	13.2614	0.441800
$902$	0	0
$903$	−6.56155	−0.218355
$904$	0	0
$905$	−2.38447	−0.0792625
$906$	0	0
$907$	−1.75379	−0.0582336	−0.0291168	−	0.999576i	$-0.509269\pi$
$907$	−1.75379	−0.0582336	−0.0291168	+	0.999576i	$0.509269\pi$
$908$	0	0
$909$	16.2462	0.538853
$910$	0	0
$911$	−8.00000	−0.265052	−0.132526	−	0.991180i	$-0.542309\pi$
$911$	−8.00000	−0.265052	−0.132526	+	0.991180i	$0.542309\pi$
$912$	0	0
$913$	−17.1231	−0.566692
$914$	0	0
$915$	2.56155	0.0846823
$916$	0	0
$917$	−19.6847	−0.650045
$918$	0	0
$919$	−18.2462	−0.601887	−0.300943	−	0.953642i	$-0.597302\pi$
$919$	−18.2462	−0.601887	−0.300943	+	0.953642i	$0.597302\pi$
$920$	0	0
$921$	0.630683	0.0207817
$922$	0	0
$923$	15.3693	0.505887
$924$	0	0
$925$	−38.6307	−1.27017
$926$	0	0
$927$	9.43845	0.309999
$928$	0	0
$929$	−23.1231	−0.758644	−0.379322	−	0.925265i	$-0.623843\pi$
$929$	−23.1231	−0.758644	−0.379322	+	0.925265i	$0.623843\pi$
$930$	0	0
$931$	2.24621	0.0736166
$932$	0	0
$933$	16.0000	0.523816
$934$	0	0
$935$	1.75379	0.0573550
$936$	0	0
$937$	−33.8617	−1.10621	−0.553107	−	0.833110i	$-0.686558\pi$
$937$	−33.8617	−1.10621	−0.553107	+	0.833110i	$0.686558\pi$
$938$	0	0
$939$	−3.93087	−0.128279
$940$	0	0
$941$	−25.8617	−0.843069	−0.421534	−	0.906812i	$-0.638508\pi$
$941$	−25.8617	−0.843069	−0.421534	+	0.906812i	$0.638508\pi$
$942$	0	0
$943$	−22.5616	−0.734705
$944$	0	0
$945$	1.43845	0.0467927
$946$	0	0
$947$	−52.9848	−1.72178	−0.860888	−	0.508794i	$-0.830091\pi$
$947$	−52.9848	−1.72178	−0.860888	+	0.508794i	$0.830091\pi$
$948$	0	0
$949$	−7.43845	−0.241462
$950$	0	0
$951$	11.4384	0.370917
$952$	0	0
$953$	−4.38447	−0.142027	−0.0710135	−	0.997475i	$-0.522623\pi$
$953$	−4.38447	−0.142027	−0.0710135	+	0.997475i	$0.522623\pi$
$954$	0	0
$955$	−8.17708	−0.264604
$956$	0	0
$957$	−1.68466	−0.0544573
$958$	0	0
$959$	47.3693	1.52964
$960$	0	0
$961$	8.01515	0.258553
$962$	0	0
$963$	13.4384	0.433048
$964$	0	0
$965$	−4.63068	−0.149067
$966$	0	0
$967$	−22.0691	−0.709695	−0.354848	−	0.934924i	$-0.615467\pi$
$967$	−22.0691	−0.709695	−0.354848	+	0.934924i	$0.615467\pi$
$968$	0	0
$969$	16.0000	0.513994
$970$	0	0
$971$	45.6155	1.46387	0.731936	−	0.681373i	$-0.238617\pi$
$971$	45.6155	1.46387	0.731936	+	0.681373i	$0.238617\pi$
$972$	0	0
$973$	16.0000	0.512936
$974$	0	0
$975$	−4.68466	−0.150029
$976$	0	0
$977$	22.6307	0.724020	0.362010	−	0.932174i	$-0.382091\pi$
$977$	22.6307	0.724020	0.362010	+	0.932174i	$0.382091\pi$
$978$	0	0
$979$	17.3693	0.555126
$980$	0	0
$981$	−9.36932	−0.299139
$982$	0	0
$983$	−15.3693	−0.490205	−0.245103	−	0.969497i	$-0.578822\pi$
$983$	−15.3693	−0.490205	−0.245103	+	0.969497i	$0.578822\pi$
$984$	0	0
$985$	−8.76894	−0.279402
$986$	0	0
$987$	20.4924	0.652281
$988$	0	0
$989$	−16.8078	−0.534456
$990$	0	0
$991$	−37.3002	−1.18488	−0.592440	−	0.805615i	$-0.701835\pi$
$991$	−37.3002	−1.18488	−0.592440	+	0.805615i	$0.701835\pi$
$992$	0	0
$993$	27.0540	0.858532
$994$	0	0
$995$	−3.33050	−0.105584
$996$	0	0
$997$	14.1771	0.448993	0.224496	−	0.974475i	$-0.427926\pi$
$997$	14.1771	0.448993	0.224496	+	0.974475i	$0.427926\pi$
$998$	0	0
$999$	8.24621	0.260899

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6864.2.a.bh.1.2		2
4.3	odd	2		858.2.a.n.1.2	✓	2
12.11	even	2		2574.2.a.bg.1.1		2
44.43	even	2		9438.2.a.bv.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
858.2.a.n.1.2	✓	2	4.3	odd	2
2574.2.a.bg.1.1		2	12.11	even	2
6864.2.a.bh.1.2		2	1.1	even	1	trivial
9438.2.a.bv.1.2		2	44.43	even	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:	\( N \)	\(=\)	\( 6864 = 2^{4} \cdot 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6864.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +0.561553 q^{5} +2.56155 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +0.561553 q^{5} +2.56155 q^{7} +1.00000 q^{9} -1.00000 q^{11} +1.00000 q^{13} +0.561553 q^{15} -3.12311 q^{17} -5.12311 q^{19} +2.56155 q^{21} +6.56155 q^{23} -4.68466 q^{25} +1.00000 q^{27} +1.68466 q^{29} +6.24621 q^{31} -1.00000 q^{33} +1.43845 q^{35} +8.24621 q^{37} +1.00000 q^{39} -3.43845 q^{41} -2.56155 q^{43} +0.561553 q^{45} +8.00000 q^{47} -0.438447 q^{49} -3.12311 q^{51} -4.24621 q^{53} -0.561553 q^{55} -5.12311 q^{57} +5.43845 q^{59} +4.56155 q^{61} +2.56155 q^{63} +0.561553 q^{65} +14.5616 q^{67} +6.56155 q^{69} +15.3693 q^{71} -7.43845 q^{73} -4.68466 q^{75} -2.56155 q^{77} -13.1231 q^{79} +1.00000 q^{81} +17.1231 q^{83} -1.75379 q^{85} +1.68466 q^{87} -17.3693 q^{89} +2.56155 q^{91} +6.24621 q^{93} -2.87689 q^{95} +14.4924 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 3 q^{5} + q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 3 * q^5 + q^7 + 2 * q^9	\( 2 q + 2 q^{3} - 3 q^{5} + q^{7} + 2 q^{9} - 2 q^{11} + 2 q^{13} - 3 q^{15} + 2 q^{17} - 2 q^{19} + q^{21} + 9 q^{23} + 3 q^{25} + 2 q^{27} - 9 q^{29} - 4 q^{31} - 2 q^{33} + 7 q^{35} + 2 q^{39} - 11 q^{41} - q^{43} - 3 q^{45} + 16 q^{47} - 5 q^{49} + 2 q^{51} + 8 q^{53} + 3 q^{55} - 2 q^{57} + 15 q^{59} + 5 q^{61} + q^{63} - 3 q^{65} + 25 q^{67} + 9 q^{69} + 6 q^{71} - 19 q^{73} + 3 q^{75} - q^{77} - 18 q^{79} + 2 q^{81} + 26 q^{83} - 20 q^{85} - 9 q^{87} - 10 q^{89} + q^{91} - 4 q^{93} - 14 q^{95} - 4 q^{97} - 2 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 3 * q^5 + q^7 + 2 * q^9 - 2 * q^11 + 2 * q^13 - 3 * q^15 + 2 * q^17 - 2 * q^19 + q^21 + 9 * q^23 + 3 * q^25 + 2 * q^27 - 9 * q^29 - 4 * q^31 - 2 * q^33 + 7 * q^35 + 2 * q^39 - 11 * q^41 - q^43 - 3 * q^45 + 16 * q^47 - 5 * q^49 + 2 * q^51 + 8 * q^53 + 3 * q^55 - 2 * q^57 + 15 * q^59 + 5 * q^61 + q^63 - 3 * q^65 + 25 * q^67 + 9 * q^69 + 6 * q^71 - 19 * q^73 + 3 * q^75 - q^77 - 18 * q^79 + 2 * q^81 + 26 * q^83 - 20 * q^85 - 9 * q^87 - 10 * q^89 + q^91 - 4 * q^93 - 14 * q^95 - 4 * q^97 - 2 * q^99

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