LMFDB - Embedded newform 6864.2.a.bc.1.1

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:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 429)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -3.41421 q^{5} -0.828427 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -3.41421 q^{5} -0.828427 q^{7} +1.00000 q^{9} -1.00000 q^{11} -1.00000 q^{13} +3.41421 q^{15} -2.58579 q^{17} +6.00000 q^{19} +0.828427 q^{21} -4.82843 q^{23} +6.65685 q^{25} -1.00000 q^{27} -4.24264 q^{29} +4.24264 q^{31} +1.00000 q^{33} +2.82843 q^{35} -3.65685 q^{37} +1.00000 q^{39} -12.0000 q^{41} +9.07107 q^{43} -3.41421 q^{45} -8.48528 q^{47} -6.31371 q^{49} +2.58579 q^{51} -9.31371 q^{53} +3.41421 q^{55} -6.00000 q^{57} +5.17157 q^{59} -2.00000 q^{61} -0.828427 q^{63} +3.41421 q^{65} -11.0711 q^{67} +4.82843 q^{69} -5.65685 q^{71} -4.48528 q^{73} -6.65685 q^{75} +0.828427 q^{77} -5.07107 q^{79} +1.00000 q^{81} +11.3137 q^{83} +8.82843 q^{85} +4.24264 q^{87} -12.5858 q^{89} +0.828427 q^{91} -4.24264 q^{93} -20.4853 q^{95} +10.0000 q^{97} -1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 4 q^{5} + 4 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - 4 * q^5 + 4 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} - 4 q^{5} + 4 q^{7} + 2 q^{9} - 2 q^{11} - 2 q^{13} + 4 q^{15} - 8 q^{17} + 12 q^{19} - 4 q^{21} - 4 q^{23} + 2 q^{25} - 2 q^{27} + 2 q^{33} + 4 q^{37} + 2 q^{39} - 24 q^{41} + 4 q^{43} - 4 q^{45} + 10 q^{49} + 8 q^{51} + 4 q^{53} + 4 q^{55} - 12 q^{57} + 16 q^{59} - 4 q^{61} + 4 q^{63} + 4 q^{65} - 8 q^{67} + 4 q^{69} + 8 q^{73} - 2 q^{75} - 4 q^{77} + 4 q^{79} + 2 q^{81} + 12 q^{85} - 28 q^{89} - 4 q^{91} - 24 q^{95} + 20 q^{97} - 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 4 * q^5 + 4 * q^7 + 2 * q^9 - 2 * q^11 - 2 * q^13 + 4 * q^15 - 8 * q^17 + 12 * q^19 - 4 * q^21 - 4 * q^23 + 2 * q^25 - 2 * q^27 + 2 * q^33 + 4 * q^37 + 2 * q^39 - 24 * q^41 + 4 * q^43 - 4 * q^45 + 10 * q^49 + 8 * q^51 + 4 * q^53 + 4 * q^55 - 12 * q^57 + 16 * q^59 - 4 * q^61 + 4 * q^63 + 4 * q^65 - 8 * q^67 + 4 * q^69 + 8 * q^73 - 2 * q^75 - 4 * q^77 + 4 * q^79 + 2 * q^81 + 12 * q^85 - 28 * q^89 - 4 * q^91 - 24 * q^95 + 20 * 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
$5$	−3.41421	−1.52688	−0.763441	−	0.645877i	$-0.776492\pi$
$5$	−3.41421	−1.52688	−0.763441	+	0.645877i	$0.776492\pi$
$6$	0	0
$7$	−0.828427	−0.313116	−0.156558	−	0.987669i	$-0.550040\pi$
$7$	−0.828427	−0.313116	−0.156558	+	0.987669i	$0.550040\pi$
$8$	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
$15$	3.41421	0.881546
$16$	0	0
$17$	−2.58579	−0.627145	−0.313573	−	0.949564i	$-0.601526\pi$
$17$	−2.58579	−0.627145	−0.313573	+	0.949564i	$0.601526\pi$
$18$	0	0
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	0	0
$21$	0.828427	0.180778
$22$	0	0
$23$	−4.82843	−1.00680	−0.503398	−	0.864054i	$-0.667917\pi$
$23$	−4.82843	−1.00680	−0.503398	+	0.864054i	$0.667917\pi$
$24$	0	0
$25$	6.65685	1.33137
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−4.24264	−0.787839	−0.393919	−	0.919145i	$-0.628881\pi$
$29$	−4.24264	−0.787839	−0.393919	+	0.919145i	$0.628881\pi$
$30$	0	0
$31$	4.24264	0.762001	0.381000	−	0.924575i	$-0.375580\pi$
$31$	4.24264	0.762001	0.381000	+	0.924575i	$0.375580\pi$
$32$	0	0
$33$	1.00000	0.174078
$34$	0	0
$35$	2.82843	0.478091
$36$	0	0
$37$	−3.65685	−0.601183	−0.300592	−	0.953753i	$-0.597184\pi$
$37$	−3.65685	−0.601183	−0.300592	+	0.953753i	$0.597184\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	−12.0000	−1.87409	−0.937043	−	0.349215i	$-0.886448\pi$
$41$	−12.0000	−1.87409	−0.937043	+	0.349215i	$0.886448\pi$
$42$	0	0
$43$	9.07107	1.38332	0.691662	−	0.722221i	$-0.256879\pi$
$43$	9.07107	1.38332	0.691662	+	0.722221i	$0.256879\pi$
$44$	0	0
$45$	−3.41421	−0.508961
$46$	0	0
$47$	−8.48528	−1.23771	−0.618853	−	0.785507i	$-0.712402\pi$
$47$	−8.48528	−1.23771	−0.618853	+	0.785507i	$0.712402\pi$
$48$	0	0
$49$	−6.31371	−0.901958
$50$	0	0
$51$	2.58579	0.362083
$52$	0	0
$53$	−9.31371	−1.27934	−0.639668	−	0.768651i	$-0.720928\pi$
$53$	−9.31371	−1.27934	−0.639668	+	0.768651i	$0.720928\pi$
$54$	0	0
$55$	3.41421	0.460372
$56$	0	0
$57$	−6.00000	−0.794719
$58$	0	0
$59$	5.17157	0.673281	0.336641	−	0.941633i	$-0.390709\pi$
$59$	5.17157	0.673281	0.336641	+	0.941633i	$0.390709\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	0	0
$63$	−0.828427	−0.104372
$64$	0	0
$65$	3.41421	0.423481
$66$	0	0
$67$	−11.0711	−1.35255	−0.676273	−	0.736651i	$-0.736406\pi$
$67$	−11.0711	−1.35255	−0.676273	+	0.736651i	$0.736406\pi$
$68$	0	0
$69$	4.82843	0.581274
$70$	0	0
$71$	−5.65685	−0.671345	−0.335673	−	0.941979i	$-0.608964\pi$
$71$	−5.65685	−0.671345	−0.335673	+	0.941979i	$0.608964\pi$
$72$	0	0
$73$	−4.48528	−0.524962	−0.262481	−	0.964937i	$-0.584541\pi$
$73$	−4.48528	−0.524962	−0.262481	+	0.964937i	$0.584541\pi$
$74$	0	0
$75$	−6.65685	−0.768667
$76$	0	0
$77$	0.828427	0.0944080
$78$	0	0
$79$	−5.07107	−0.570540	−0.285270	−	0.958447i	$-0.592083\pi$
$79$	−5.07107	−0.570540	−0.285270	+	0.958447i	$0.592083\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	11.3137	1.24184	0.620920	−	0.783874i	$-0.286759\pi$
$83$	11.3137	1.24184	0.620920	+	0.783874i	$0.286759\pi$
$84$	0	0
$85$	8.82843	0.957577
$86$	0	0
$87$	4.24264	0.454859
$88$	0	0
$89$	−12.5858	−1.33409	−0.667045	−	0.745017i	$-0.732441\pi$
$89$	−12.5858	−1.33409	−0.667045	+	0.745017i	$0.732441\pi$
$90$	0	0
$91$	0.828427	0.0868428
$92$	0	0
$93$	−4.24264	−0.439941
$94$	0	0
$95$	−20.4853	−2.10175
$96$	0	0
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	0	0
$101$	9.89949	0.985037	0.492518	−	0.870302i	$-0.336076\pi$
$101$	9.89949	0.985037	0.492518	+	0.870302i	$0.336076\pi$
$102$	0	0
$103$	−8.48528	−0.836080	−0.418040	−	0.908429i	$-0.637283\pi$
$103$	−8.48528	−0.836080	−0.418040	+	0.908429i	$0.637283\pi$
$104$	0	0
$105$	−2.82843	−0.276026
$106$	0	0
$107$	8.48528	0.820303	0.410152	−	0.912017i	$-0.365476\pi$
$107$	8.48528	0.820303	0.410152	+	0.912017i	$0.365476\pi$
$108$	0	0
$109$	−1.17157	−0.112216	−0.0561082	−	0.998425i	$-0.517869\pi$
$109$	−1.17157	−0.112216	−0.0561082	+	0.998425i	$0.517869\pi$
$110$	0	0
$111$	3.65685	0.347093
$112$	0	0
$113$	−0.343146	−0.0322804	−0.0161402	−	0.999870i	$-0.505138\pi$
$113$	−0.343146	−0.0322804	−0.0161402	+	0.999870i	$0.505138\pi$
$114$	0	0
$115$	16.4853	1.53726
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	2.14214	0.196369
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	12.0000	1.08200
$124$	0	0
$125$	−5.65685	−0.505964
$126$	0	0
$127$	−5.75736	−0.510883	−0.255442	−	0.966825i	$-0.582221\pi$
$127$	−5.75736	−0.510883	−0.255442	+	0.966825i	$0.582221\pi$
$128$	0	0
$129$	−9.07107	−0.798663
$130$	0	0
$131$	−16.9706	−1.48272	−0.741362	−	0.671105i	$-0.765820\pi$
$131$	−16.9706	−1.48272	−0.741362	+	0.671105i	$0.765820\pi$
$132$	0	0
$133$	−4.97056	−0.431002
$134$	0	0
$135$	3.41421	0.293849
$136$	0	0
$137$	18.7279	1.60003	0.800017	−	0.599977i	$-0.204824\pi$
$137$	18.7279	1.60003	0.800017	+	0.599977i	$0.204824\pi$
$138$	0	0
$139$	8.58579	0.728237	0.364118	−	0.931353i	$-0.381370\pi$
$139$	8.58579	0.728237	0.364118	+	0.931353i	$0.381370\pi$
$140$	0	0
$141$	8.48528	0.714590
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	14.4853	1.20294
$146$	0	0
$147$	6.31371	0.520746
$148$	0	0
$149$	20.4853	1.67822	0.839110	−	0.543962i	$-0.183076\pi$
$149$	20.4853	1.67822	0.839110	+	0.543962i	$0.183076\pi$
$150$	0	0
$151$	−6.00000	−0.488273	−0.244137	−	0.969741i	$-0.578505\pi$
$151$	−6.00000	−0.488273	−0.244137	+	0.969741i	$0.578505\pi$
$152$	0	0
$153$	−2.58579	−0.209048
$154$	0	0
$155$	−14.4853	−1.16349
$156$	0	0
$157$	4.00000	0.319235	0.159617	−	0.987179i	$-0.448974\pi$
$157$	4.00000	0.319235	0.159617	+	0.987179i	$0.448974\pi$
$158$	0	0
$159$	9.31371	0.738625
$160$	0	0
$161$	4.00000	0.315244
$162$	0	0
$163$	−12.7279	−0.996928	−0.498464	−	0.866910i	$-0.666102\pi$
$163$	−12.7279	−0.996928	−0.498464	+	0.866910i	$0.666102\pi$
$164$	0	0
$165$	−3.41421	−0.265796
$166$	0	0
$167$	6.34315	0.490847	0.245424	−	0.969416i	$-0.421073\pi$
$167$	6.34315	0.490847	0.245424	+	0.969416i	$0.421073\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	6.00000	0.458831
$172$	0	0
$173$	8.92893	0.678854	0.339427	−	0.940632i	$-0.389767\pi$
$173$	8.92893	0.678854	0.339427	+	0.940632i	$0.389767\pi$
$174$	0	0
$175$	−5.51472	−0.416874
$176$	0	0
$177$	−5.17157	−0.388719
$178$	0	0
$179$	24.1421	1.80447	0.902234	−	0.431247i	$-0.141926\pi$
$179$	24.1421	1.80447	0.902234	+	0.431247i	$0.141926\pi$
$180$	0	0
$181$	−24.9706	−1.85605	−0.928024	−	0.372521i	$-0.878493\pi$
$181$	−24.9706	−1.85605	−0.928024	+	0.372521i	$0.878493\pi$
$182$	0	0
$183$	2.00000	0.147844
$184$	0	0
$185$	12.4853	0.917936
$186$	0	0
$187$	2.58579	0.189091
$188$	0	0
$189$	0.828427	0.0602592
$190$	0	0
$191$	19.3137	1.39749	0.698745	−	0.715370i	$-0.253742\pi$
$191$	19.3137	1.39749	0.698745	+	0.715370i	$0.253742\pi$
$192$	0	0
$193$	−18.8284	−1.35530	−0.677650	−	0.735385i	$-0.737002\pi$
$193$	−18.8284	−1.35530	−0.677650	+	0.735385i	$0.737002\pi$
$194$	0	0
$195$	−3.41421	−0.244497
$196$	0	0
$197$	−9.17157	−0.653448	−0.326724	−	0.945120i	$-0.605945\pi$
$197$	−9.17157	−0.653448	−0.326724	+	0.945120i	$0.605945\pi$
$198$	0	0
$199$	5.17157	0.366603	0.183302	−	0.983057i	$-0.441322\pi$
$199$	5.17157	0.366603	0.183302	+	0.983057i	$0.441322\pi$
$200$	0	0
$201$	11.0711	0.780893
$202$	0	0
$203$	3.51472	0.246685
$204$	0	0
$205$	40.9706	2.86151
$206$	0	0
$207$	−4.82843	−0.335599
$208$	0	0
$209$	−6.00000	−0.415029
$210$	0	0
$211$	−6.72792	−0.463169	−0.231585	−	0.972815i	$-0.574391\pi$
$211$	−6.72792	−0.463169	−0.231585	+	0.972815i	$0.574391\pi$
$212$	0	0
$213$	5.65685	0.387601
$214$	0	0
$215$	−30.9706	−2.11217
$216$	0	0
$217$	−3.51472	−0.238595
$218$	0	0
$219$	4.48528	0.303087
$220$	0	0
$221$	2.58579	0.173939
$222$	0	0
$223$	5.41421	0.362563	0.181281	−	0.983431i	$-0.441976\pi$
$223$	5.41421	0.362563	0.181281	+	0.983431i	$0.441976\pi$
$224$	0	0
$225$	6.65685	0.443790
$226$	0	0
$227$	−22.9706	−1.52461	−0.762305	−	0.647218i	$-0.775932\pi$
$227$	−22.9706	−1.52461	−0.762305	+	0.647218i	$0.775932\pi$
$228$	0	0
$229$	26.4853	1.75020	0.875098	−	0.483945i	$-0.160797\pi$
$229$	26.4853	1.75020	0.875098	+	0.483945i	$0.160797\pi$
$230$	0	0
$231$	−0.828427	−0.0545065
$232$	0	0
$233$	4.24264	0.277945	0.138972	−	0.990296i	$-0.455620\pi$
$233$	4.24264	0.277945	0.138972	+	0.990296i	$0.455620\pi$
$234$	0	0
$235$	28.9706	1.88983
$236$	0	0
$237$	5.07107	0.329401
$238$	0	0
$239$	15.6569	1.01276	0.506379	−	0.862311i	$-0.330984\pi$
$239$	15.6569	1.01276	0.506379	+	0.862311i	$0.330984\pi$
$240$	0	0
$241$	1.31371	0.0846234	0.0423117	−	0.999104i	$-0.486528\pi$
$241$	1.31371	0.0846234	0.0423117	+	0.999104i	$0.486528\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	21.5563	1.37718
$246$	0	0
$247$	−6.00000	−0.381771
$248$	0	0
$249$	−11.3137	−0.716977
$250$	0	0
$251$	7.31371	0.461637	0.230819	−	0.972997i	$-0.425860\pi$
$251$	7.31371	0.461637	0.230819	+	0.972997i	$0.425860\pi$
$252$	0	0
$253$	4.82843	0.303561
$254$	0	0
$255$	−8.82843	−0.552858
$256$	0	0
$257$	−20.8284	−1.29924	−0.649621	−	0.760258i	$-0.725072\pi$
$257$	−20.8284	−1.29924	−0.649621	+	0.760258i	$0.725072\pi$
$258$	0	0
$259$	3.02944	0.188240
$260$	0	0
$261$	−4.24264	−0.262613
$262$	0	0
$263$	−31.7990	−1.96081	−0.980405	−	0.196993i	$-0.936882\pi$
$263$	−31.7990	−1.96081	−0.980405	+	0.196993i	$0.936882\pi$
$264$	0	0
$265$	31.7990	1.95340
$266$	0	0
$267$	12.5858	0.770238
$268$	0	0
$269$	−12.1421	−0.740319	−0.370160	−	0.928968i	$-0.620697\pi$
$269$	−12.1421	−0.740319	−0.370160	+	0.928968i	$0.620697\pi$
$270$	0	0
$271$	4.34315	0.263827	0.131914	−	0.991261i	$-0.457888\pi$
$271$	4.34315	0.263827	0.131914	+	0.991261i	$0.457888\pi$
$272$	0	0
$273$	−0.828427	−0.0501387
$274$	0	0
$275$	−6.65685	−0.401423
$276$	0	0
$277$	8.34315	0.501291	0.250646	−	0.968079i	$-0.419357\pi$
$277$	8.34315	0.501291	0.250646	+	0.968079i	$0.419357\pi$
$278$	0	0
$279$	4.24264	0.254000
$280$	0	0
$281$	21.6569	1.29194	0.645970	−	0.763363i	$-0.276453\pi$
$281$	21.6569	1.29194	0.645970	+	0.763363i	$0.276453\pi$
$282$	0	0
$283$	26.7279	1.58881	0.794405	−	0.607388i	$-0.207783\pi$
$283$	26.7279	1.58881	0.794405	+	0.607388i	$0.207783\pi$
$284$	0	0
$285$	20.4853	1.21344
$286$	0	0
$287$	9.94113	0.586806
$288$	0	0
$289$	−10.3137	−0.606689
$290$	0	0
$291$	−10.0000	−0.586210
$292$	0	0
$293$	9.17157	0.535809	0.267905	−	0.963445i	$-0.413669\pi$
$293$	9.17157	0.535809	0.267905	+	0.963445i	$0.413669\pi$
$294$	0	0
$295$	−17.6569	−1.02802
$296$	0	0
$297$	1.00000	0.0580259
$298$	0	0
$299$	4.82843	0.279235
$300$	0	0
$301$	−7.51472	−0.433141
$302$	0	0
$303$	−9.89949	−0.568711
$304$	0	0
$305$	6.82843	0.390995
$306$	0	0
$307$	30.2843	1.72841	0.864207	−	0.503136i	$-0.167820\pi$
$307$	30.2843	1.72841	0.864207	+	0.503136i	$0.167820\pi$
$308$	0	0
$309$	8.48528	0.482711
$310$	0	0
$311$	−0.828427	−0.0469758	−0.0234879	−	0.999724i	$-0.507477\pi$
$311$	−0.828427	−0.0469758	−0.0234879	+	0.999724i	$0.507477\pi$
$312$	0	0
$313$	20.0000	1.13047	0.565233	−	0.824931i	$-0.308786\pi$
$313$	20.0000	1.13047	0.565233	+	0.824931i	$0.308786\pi$
$314$	0	0
$315$	2.82843	0.159364
$316$	0	0
$317$	−12.5858	−0.706888	−0.353444	−	0.935456i	$-0.614990\pi$
$317$	−12.5858	−0.706888	−0.353444	+	0.935456i	$0.614990\pi$
$318$	0	0
$319$	4.24264	0.237542
$320$	0	0
$321$	−8.48528	−0.473602
$322$	0	0
$323$	−15.5147	−0.863262
$324$	0	0
$325$	−6.65685	−0.369256
$326$	0	0
$327$	1.17157	0.0647881
$328$	0	0
$329$	7.02944	0.387545
$330$	0	0
$331$	20.2426	1.11264	0.556318	−	0.830969i	$-0.312214\pi$
$331$	20.2426	1.11264	0.556318	+	0.830969i	$0.312214\pi$
$332$	0	0
$333$	−3.65685	−0.200394
$334$	0	0
$335$	37.7990	2.06518
$336$	0	0
$337$	31.4558	1.71351	0.856755	−	0.515724i	$-0.172477\pi$
$337$	31.4558	1.71351	0.856755	+	0.515724i	$0.172477\pi$
$338$	0	0
$339$	0.343146	0.0186371
$340$	0	0
$341$	−4.24264	−0.229752
$342$	0	0
$343$	11.0294	0.595534
$344$	0	0
$345$	−16.4853	−0.887538
$346$	0	0
$347$	8.48528	0.455514	0.227757	−	0.973718i	$-0.426861\pi$
$347$	8.48528	0.455514	0.227757	+	0.973718i	$0.426861\pi$
$348$	0	0
$349$	−14.9706	−0.801356	−0.400678	−	0.916219i	$-0.631225\pi$
$349$	−14.9706	−0.801356	−0.400678	+	0.916219i	$0.631225\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	−33.5563	−1.78602	−0.893012	−	0.450033i	$-0.851412\pi$
$353$	−33.5563	−1.78602	−0.893012	+	0.450033i	$0.851412\pi$
$354$	0	0
$355$	19.3137	1.02507
$356$	0	0
$357$	−2.14214	−0.113374
$358$	0	0
$359$	19.6569	1.03745	0.518725	−	0.854941i	$-0.326407\pi$
$359$	19.6569	1.03745	0.518725	+	0.854941i	$0.326407\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	0	0
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	15.3137	0.801556
$366$	0	0
$367$	−28.9706	−1.51225	−0.756126	−	0.654427i	$-0.772910\pi$
$367$	−28.9706	−1.51225	−0.756126	+	0.654427i	$0.772910\pi$
$368$	0	0
$369$	−12.0000	−0.624695
$370$	0	0
$371$	7.71573	0.400581
$372$	0	0
$373$	−2.48528	−0.128683	−0.0643415	−	0.997928i	$-0.520495\pi$
$373$	−2.48528	−0.128683	−0.0643415	+	0.997928i	$0.520495\pi$
$374$	0	0
$375$	5.65685	0.292119
$376$	0	0
$377$	4.24264	0.218507
$378$	0	0
$379$	−8.92893	−0.458648	−0.229324	−	0.973350i	$-0.573652\pi$
$379$	−8.92893	−0.458648	−0.229324	+	0.973350i	$0.573652\pi$
$380$	0	0
$381$	5.75736	0.294958
$382$	0	0
$383$	−2.82843	−0.144526	−0.0722629	−	0.997386i	$-0.523022\pi$
$383$	−2.82843	−0.144526	−0.0722629	+	0.997386i	$0.523022\pi$
$384$	0	0
$385$	−2.82843	−0.144150
$386$	0	0
$387$	9.07107	0.461108
$388$	0	0
$389$	−2.48528	−0.126009	−0.0630044	−	0.998013i	$-0.520068\pi$
$389$	−2.48528	−0.126009	−0.0630044	+	0.998013i	$0.520068\pi$
$390$	0	0
$391$	12.4853	0.631408
$392$	0	0
$393$	16.9706	0.856052
$394$	0	0
$395$	17.3137	0.871147
$396$	0	0
$397$	−20.6274	−1.03526	−0.517630	−	0.855604i	$-0.673186\pi$
$397$	−20.6274	−1.03526	−0.517630	+	0.855604i	$0.673186\pi$
$398$	0	0
$399$	4.97056	0.248839
$400$	0	0
$401$	1.07107	0.0534866	0.0267433	−	0.999642i	$-0.491486\pi$
$401$	1.07107	0.0534866	0.0267433	+	0.999642i	$0.491486\pi$
$402$	0	0
$403$	−4.24264	−0.211341
$404$	0	0
$405$	−3.41421	−0.169654
$406$	0	0
$407$	3.65685	0.181264
$408$	0	0
$409$	28.4853	1.40851	0.704253	−	0.709949i	$-0.251282\pi$
$409$	28.4853	1.40851	0.704253	+	0.709949i	$0.251282\pi$
$410$	0	0
$411$	−18.7279	−0.923780
$412$	0	0
$413$	−4.28427	−0.210815
$414$	0	0
$415$	−38.6274	−1.89615
$416$	0	0
$417$	−8.58579	−0.420448
$418$	0	0
$419$	32.1421	1.57025	0.785123	−	0.619340i	$-0.212600\pi$
$419$	32.1421	1.57025	0.785123	+	0.619340i	$0.212600\pi$
$420$	0	0
$421$	36.6274	1.78511	0.892556	−	0.450937i	$-0.148910\pi$
$421$	36.6274	1.78511	0.892556	+	0.450937i	$0.148910\pi$
$422$	0	0
$423$	−8.48528	−0.412568
$424$	0	0
$425$	−17.2132	−0.834963
$426$	0	0
$427$	1.65685	0.0801808
$428$	0	0
$429$	−1.00000	−0.0482805
$430$	0	0
$431$	−23.3137	−1.12298	−0.561491	−	0.827483i	$-0.689772\pi$
$431$	−23.3137	−1.12298	−0.561491	+	0.827483i	$0.689772\pi$
$432$	0	0
$433$	14.6863	0.705778	0.352889	−	0.935665i	$-0.385199\pi$
$433$	14.6863	0.705778	0.352889	+	0.935665i	$0.385199\pi$
$434$	0	0
$435$	−14.4853	−0.694516
$436$	0	0
$437$	−28.9706	−1.38585
$438$	0	0
$439$	21.0711	1.00567	0.502834	−	0.864383i	$-0.332291\pi$
$439$	21.0711	1.00567	0.502834	+	0.864383i	$0.332291\pi$
$440$	0	0
$441$	−6.31371	−0.300653
$442$	0	0
$443$	25.6569	1.21899	0.609497	−	0.792788i	$-0.291371\pi$
$443$	25.6569	1.21899	0.609497	+	0.792788i	$0.291371\pi$
$444$	0	0
$445$	42.9706	2.03700
$446$	0	0
$447$	−20.4853	−0.968921
$448$	0	0
$449$	4.58579	0.216417	0.108208	−	0.994128i	$-0.465489\pi$
$449$	4.58579	0.216417	0.108208	+	0.994128i	$0.465489\pi$
$450$	0	0
$451$	12.0000	0.565058
$452$	0	0
$453$	6.00000	0.281905
$454$	0	0
$455$	−2.82843	−0.132599
$456$	0	0
$457$	−14.0000	−0.654892	−0.327446	−	0.944870i	$-0.606188\pi$
$457$	−14.0000	−0.654892	−0.327446	+	0.944870i	$0.606188\pi$
$458$	0	0
$459$	2.58579	0.120694
$460$	0	0
$461$	−19.3137	−0.899529	−0.449765	−	0.893147i	$-0.648492\pi$
$461$	−19.3137	−0.899529	−0.449765	+	0.893147i	$0.648492\pi$
$462$	0	0
$463$	−15.7574	−0.732307	−0.366153	−	0.930555i	$-0.619325\pi$
$463$	−15.7574	−0.732307	−0.366153	+	0.930555i	$0.619325\pi$
$464$	0	0
$465$	14.4853	0.671739
$466$	0	0
$467$	−25.7990	−1.19383	−0.596917	−	0.802303i	$-0.703608\pi$
$467$	−25.7990	−1.19383	−0.596917	+	0.802303i	$0.703608\pi$
$468$	0	0
$469$	9.17157	0.423504
$470$	0	0
$471$	−4.00000	−0.184310
$472$	0	0
$473$	−9.07107	−0.417088
$474$	0	0
$475$	39.9411	1.83262
$476$	0	0
$477$	−9.31371	−0.426445
$478$	0	0
$479$	39.9411	1.82496	0.912478	−	0.409125i	$-0.134166\pi$
$479$	39.9411	1.82496	0.912478	+	0.409125i	$0.134166\pi$
$480$	0	0
$481$	3.65685	0.166738
$482$	0	0
$483$	−4.00000	−0.182006
$484$	0	0
$485$	−34.1421	−1.55031
$486$	0	0
$487$	20.7279	0.939272	0.469636	−	0.882860i	$-0.344385\pi$
$487$	20.7279	0.939272	0.469636	+	0.882860i	$0.344385\pi$
$488$	0	0
$489$	12.7279	0.575577
$490$	0	0
$491$	5.17157	0.233390	0.116695	−	0.993168i	$-0.462770\pi$
$491$	5.17157	0.233390	0.116695	+	0.993168i	$0.462770\pi$
$492$	0	0
$493$	10.9706	0.494089
$494$	0	0
$495$	3.41421	0.153457
$496$	0	0
$497$	4.68629	0.210209
$498$	0	0
$499$	−22.3848	−1.00208	−0.501040	−	0.865424i	$-0.667049\pi$
$499$	−22.3848	−1.00208	−0.501040	+	0.865424i	$0.667049\pi$
$500$	0	0
$501$	−6.34315	−0.283391
$502$	0	0
$503$	−20.4853	−0.913394	−0.456697	−	0.889622i	$-0.650968\pi$
$503$	−20.4853	−0.913394	−0.456697	+	0.889622i	$0.650968\pi$
$504$	0	0
$505$	−33.7990	−1.50404
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	15.8995	0.704733	0.352366	−	0.935862i	$-0.385377\pi$
$509$	15.8995	0.704733	0.352366	+	0.935862i	$0.385377\pi$
$510$	0	0
$511$	3.71573	0.164374
$512$	0	0
$513$	−6.00000	−0.264906
$514$	0	0
$515$	28.9706	1.27660
$516$	0	0
$517$	8.48528	0.373182
$518$	0	0
$519$	−8.92893	−0.391937
$520$	0	0
$521$	34.9706	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$521$	34.9706	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$522$	0	0
$523$	6.24264	0.272972	0.136486	−	0.990642i	$-0.456419\pi$
$523$	6.24264	0.272972	0.136486	+	0.990642i	$0.456419\pi$
$524$	0	0
$525$	5.51472	0.240682
$526$	0	0
$527$	−10.9706	−0.477885
$528$	0	0
$529$	0.313708	0.0136395
$530$	0	0
$531$	5.17157	0.224427
$532$	0	0
$533$	12.0000	0.519778
$534$	0	0
$535$	−28.9706	−1.25251
$536$	0	0
$537$	−24.1421	−1.04181
$538$	0	0
$539$	6.31371	0.271951
$540$	0	0
$541$	34.9706	1.50350	0.751751	−	0.659447i	$-0.229210\pi$
$541$	34.9706	1.50350	0.751751	+	0.659447i	$0.229210\pi$
$542$	0	0
$543$	24.9706	1.07159
$544$	0	0
$545$	4.00000	0.171341
$546$	0	0
$547$	−14.2426	−0.608971	−0.304486	−	0.952517i	$-0.598485\pi$
$547$	−14.2426	−0.608971	−0.304486	+	0.952517i	$0.598485\pi$
$548$	0	0
$549$	−2.00000	−0.0853579
$550$	0	0
$551$	−25.4558	−1.08446
$552$	0	0
$553$	4.20101	0.178645
$554$	0	0
$555$	−12.4853	−0.529971
$556$	0	0
$557$	17.1716	0.727583	0.363791	−	0.931480i	$-0.381482\pi$
$557$	17.1716	0.727583	0.363791	+	0.931480i	$0.381482\pi$
$558$	0	0
$559$	−9.07107	−0.383665
$560$	0	0
$561$	−2.58579	−0.109172
$562$	0	0
$563$	6.34315	0.267332	0.133666	−	0.991026i	$-0.457325\pi$
$563$	6.34315	0.267332	0.133666	+	0.991026i	$0.457325\pi$
$564$	0	0
$565$	1.17157	0.0492884
$566$	0	0
$567$	−0.828427	−0.0347907
$568$	0	0
$569$	−24.7279	−1.03665	−0.518324	−	0.855184i	$-0.673444\pi$
$569$	−24.7279	−1.03665	−0.518324	+	0.855184i	$0.673444\pi$
$570$	0	0
$571$	−17.7574	−0.743122	−0.371561	−	0.928408i	$-0.621177\pi$
$571$	−17.7574	−0.743122	−0.371561	+	0.928408i	$0.621177\pi$
$572$	0	0
$573$	−19.3137	−0.806842
$574$	0	0
$575$	−32.1421	−1.34042
$576$	0	0
$577$	−3.65685	−0.152237	−0.0761184	−	0.997099i	$-0.524253\pi$
$577$	−3.65685	−0.152237	−0.0761184	+	0.997099i	$0.524253\pi$
$578$	0	0
$579$	18.8284	0.782483
$580$	0	0
$581$	−9.37258	−0.388840
$582$	0	0
$583$	9.31371	0.385734
$584$	0	0
$585$	3.41421	0.141160
$586$	0	0
$587$	13.8579	0.571975	0.285988	−	0.958233i	$-0.407678\pi$
$587$	13.8579	0.571975	0.285988	+	0.958233i	$0.407678\pi$
$588$	0	0
$589$	25.4558	1.04889
$590$	0	0
$591$	9.17157	0.377268
$592$	0	0
$593$	28.7696	1.18142	0.590712	−	0.806883i	$-0.298847\pi$
$593$	28.7696	1.18142	0.590712	+	0.806883i	$0.298847\pi$
$594$	0	0
$595$	−7.31371	−0.299833
$596$	0	0
$597$	−5.17157	−0.211658
$598$	0	0
$599$	18.3431	0.749481	0.374740	−	0.927130i	$-0.377732\pi$
$599$	18.3431	0.749481	0.374740	+	0.927130i	$0.377732\pi$
$600$	0	0
$601$	5.02944	0.205155	0.102578	−	0.994725i	$-0.467291\pi$
$601$	5.02944	0.205155	0.102578	+	0.994725i	$0.467291\pi$
$602$	0	0
$603$	−11.0711	−0.450849
$604$	0	0
$605$	−3.41421	−0.138808
$606$	0	0
$607$	20.5858	0.835551	0.417776	−	0.908550i	$-0.362810\pi$
$607$	20.5858	0.835551	0.417776	+	0.908550i	$0.362810\pi$
$608$	0	0
$609$	−3.51472	−0.142424
$610$	0	0
$611$	8.48528	0.343278
$612$	0	0
$613$	35.7990	1.44591	0.722954	−	0.690896i	$-0.242784\pi$
$613$	35.7990	1.44591	0.722954	+	0.690896i	$0.242784\pi$
$614$	0	0
$615$	−40.9706	−1.65209
$616$	0	0
$617$	−47.0122	−1.89264	−0.946320	−	0.323232i	$-0.895231\pi$
$617$	−47.0122	−1.89264	−0.946320	+	0.323232i	$0.895231\pi$
$618$	0	0
$619$	−21.2132	−0.852631	−0.426315	−	0.904575i	$-0.640189\pi$
$619$	−21.2132	−0.852631	−0.426315	+	0.904575i	$0.640189\pi$
$620$	0	0
$621$	4.82843	0.193758
$622$	0	0
$623$	10.4264	0.417725
$624$	0	0
$625$	−13.9706	−0.558823
$626$	0	0
$627$	6.00000	0.239617
$628$	0	0
$629$	9.45584	0.377029
$630$	0	0
$631$	16.5269	0.657926	0.328963	−	0.944343i	$-0.393301\pi$
$631$	16.5269	0.657926	0.328963	+	0.944343i	$0.393301\pi$
$632$	0	0
$633$	6.72792	0.267411
$634$	0	0
$635$	19.6569	0.780058
$636$	0	0
$637$	6.31371	0.250158
$638$	0	0
$639$	−5.65685	−0.223782
$640$	0	0
$641$	16.3431	0.645515	0.322758	−	0.946482i	$-0.395390\pi$
$641$	16.3431	0.645515	0.322758	+	0.946482i	$0.395390\pi$
$642$	0	0
$643$	20.7279	0.817429	0.408715	−	0.912662i	$-0.365977\pi$
$643$	20.7279	0.817429	0.408715	+	0.912662i	$0.365977\pi$
$644$	0	0
$645$	30.9706	1.21946
$646$	0	0
$647$	−38.6274	−1.51860	−0.759300	−	0.650740i	$-0.774459\pi$
$647$	−38.6274	−1.51860	−0.759300	+	0.650740i	$0.774459\pi$
$648$	0	0
$649$	−5.17157	−0.203002
$650$	0	0
$651$	3.51472	0.137753
$652$	0	0
$653$	−32.8284	−1.28468	−0.642338	−	0.766422i	$-0.722035\pi$
$653$	−32.8284	−1.28468	−0.642338	+	0.766422i	$0.722035\pi$
$654$	0	0
$655$	57.9411	2.26395
$656$	0	0
$657$	−4.48528	−0.174987
$658$	0	0
$659$	−28.2843	−1.10180	−0.550899	−	0.834572i	$-0.685715\pi$
$659$	−28.2843	−1.10180	−0.550899	+	0.834572i	$0.685715\pi$
$660$	0	0
$661$	39.4558	1.53465	0.767327	−	0.641256i	$-0.221586\pi$
$661$	39.4558	1.53465	0.767327	+	0.641256i	$0.221586\pi$
$662$	0	0
$663$	−2.58579	−0.100424
$664$	0	0
$665$	16.9706	0.658090
$666$	0	0
$667$	20.4853	0.793193
$668$	0	0
$669$	−5.41421	−0.209326
$670$	0	0
$671$	2.00000	0.0772091
$672$	0	0
$673$	1.51472	0.0583881	0.0291941	−	0.999574i	$-0.490706\pi$
$673$	1.51472	0.0583881	0.0291941	+	0.999574i	$0.490706\pi$
$674$	0	0
$675$	−6.65685	−0.256222
$676$	0	0
$677$	5.61522	0.215811	0.107905	−	0.994161i	$-0.465586\pi$
$677$	5.61522	0.215811	0.107905	+	0.994161i	$0.465586\pi$
$678$	0	0
$679$	−8.28427	−0.317921
$680$	0	0
$681$	22.9706	0.880234
$682$	0	0
$683$	−2.62742	−0.100535	−0.0502677	−	0.998736i	$-0.516007\pi$
$683$	−2.62742	−0.100535	−0.0502677	+	0.998736i	$0.516007\pi$
$684$	0	0
$685$	−63.9411	−2.44306
$686$	0	0
$687$	−26.4853	−1.01048
$688$	0	0
$689$	9.31371	0.354824
$690$	0	0
$691$	41.8995	1.59393	0.796966	−	0.604024i	$-0.206437\pi$
$691$	41.8995	1.59393	0.796966	+	0.604024i	$0.206437\pi$
$692$	0	0
$693$	0.828427	0.0314693
$694$	0	0
$695$	−29.3137	−1.11193
$696$	0	0
$697$	31.0294	1.17532
$698$	0	0
$699$	−4.24264	−0.160471
$700$	0	0
$701$	−43.0711	−1.62677	−0.813386	−	0.581725i	$-0.802378\pi$
$701$	−43.0711	−1.62677	−0.813386	+	0.581725i	$0.802378\pi$
$702$	0	0
$703$	−21.9411	−0.827525
$704$	0	0
$705$	−28.9706	−1.09109
$706$	0	0
$707$	−8.20101	−0.308431
$708$	0	0
$709$	−14.4853	−0.544006	−0.272003	−	0.962296i	$-0.587686\pi$
$709$	−14.4853	−0.544006	−0.272003	+	0.962296i	$0.587686\pi$
$710$	0	0
$711$	−5.07107	−0.190180
$712$	0	0
$713$	−20.4853	−0.767180
$714$	0	0
$715$	−3.41421	−0.127684
$716$	0	0
$717$	−15.6569	−0.584716
$718$	0	0
$719$	−45.2548	−1.68772	−0.843860	−	0.536563i	$-0.819722\pi$
$719$	−45.2548	−1.68772	−0.843860	+	0.536563i	$0.819722\pi$
$720$	0	0
$721$	7.02944	0.261790
$722$	0	0
$723$	−1.31371	−0.0488573
$724$	0	0
$725$	−28.2426	−1.04891
$726$	0	0
$727$	−1.17157	−0.0434512	−0.0217256	−	0.999764i	$-0.506916\pi$
$727$	−1.17157	−0.0434512	−0.0217256	+	0.999764i	$0.506916\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−23.4558	−0.867546
$732$	0	0
$733$	−32.4853	−1.19987	−0.599936	−	0.800048i	$-0.704807\pi$
$733$	−32.4853	−1.19987	−0.599936	+	0.800048i	$0.704807\pi$
$734$	0	0
$735$	−21.5563	−0.795118
$736$	0	0
$737$	11.0711	0.407808
$738$	0	0
$739$	−41.7990	−1.53760	−0.768800	−	0.639489i	$-0.779146\pi$
$739$	−41.7990	−1.53760	−0.768800	+	0.639489i	$0.779146\pi$
$740$	0	0
$741$	6.00000	0.220416
$742$	0	0
$743$	44.2843	1.62463	0.812316	−	0.583217i	$-0.198206\pi$
$743$	44.2843	1.62463	0.812316	+	0.583217i	$0.198206\pi$
$744$	0	0
$745$	−69.9411	−2.56244
$746$	0	0
$747$	11.3137	0.413947
$748$	0	0
$749$	−7.02944	−0.256850
$750$	0	0
$751$	−11.0294	−0.402470	−0.201235	−	0.979543i	$-0.564495\pi$
$751$	−11.0294	−0.402470	−0.201235	+	0.979543i	$0.564495\pi$
$752$	0	0
$753$	−7.31371	−0.266526
$754$	0	0
$755$	20.4853	0.745536
$756$	0	0
$757$	−50.6274	−1.84008	−0.920042	−	0.391819i	$-0.871846\pi$
$757$	−50.6274	−1.84008	−0.920042	+	0.391819i	$0.871846\pi$
$758$	0	0
$759$	−4.82843	−0.175261
$760$	0	0
$761$	37.4558	1.35777	0.678887	−	0.734243i	$-0.262463\pi$
$761$	37.4558	1.35777	0.678887	+	0.734243i	$0.262463\pi$
$762$	0	0
$763$	0.970563	0.0351367
$764$	0	0
$765$	8.82843	0.319192
$766$	0	0
$767$	−5.17157	−0.186735
$768$	0	0
$769$	−35.9411	−1.29607	−0.648035	−	0.761610i	$-0.724409\pi$
$769$	−35.9411	−1.29607	−0.648035	+	0.761610i	$0.724409\pi$
$770$	0	0
$771$	20.8284	0.750117
$772$	0	0
$773$	6.04163	0.217302	0.108651	−	0.994080i	$-0.465347\pi$
$773$	6.04163	0.217302	0.108651	+	0.994080i	$0.465347\pi$
$774$	0	0
$775$	28.2426	1.01451
$776$	0	0
$777$	−3.02944	−0.108680
$778$	0	0
$779$	−72.0000	−2.57967
$780$	0	0
$781$	5.65685	0.202418
$782$	0	0
$783$	4.24264	0.151620
$784$	0	0
$785$	−13.6569	−0.487434
$786$	0	0
$787$	45.7990	1.63256	0.816279	−	0.577658i	$-0.196033\pi$
$787$	45.7990	1.63256	0.816279	+	0.577658i	$0.196033\pi$
$788$	0	0
$789$	31.7990	1.13207
$790$	0	0
$791$	0.284271	0.0101075
$792$	0	0
$793$	2.00000	0.0710221
$794$	0	0
$795$	−31.7990	−1.12779
$796$	0	0
$797$	34.7696	1.23160	0.615800	−	0.787902i	$-0.288833\pi$
$797$	34.7696	1.23160	0.615800	+	0.787902i	$0.288833\pi$
$798$	0	0
$799$	21.9411	0.776221
$800$	0	0
$801$	−12.5858	−0.444697
$802$	0	0
$803$	4.48528	0.158282
$804$	0	0
$805$	−13.6569	−0.481341
$806$	0	0
$807$	12.1421	0.427423
$808$	0	0
$809$	4.92893	0.173292	0.0866460	−	0.996239i	$-0.472385\pi$
$809$	4.92893	0.173292	0.0866460	+	0.996239i	$0.472385\pi$
$810$	0	0
$811$	32.6274	1.14570	0.572852	−	0.819659i	$-0.305837\pi$
$811$	32.6274	1.14570	0.572852	+	0.819659i	$0.305837\pi$
$812$	0	0
$813$	−4.34315	−0.152321
$814$	0	0
$815$	43.4558	1.52219
$816$	0	0
$817$	54.4264	1.90414
$818$	0	0
$819$	0.828427	0.0289476
$820$	0	0
$821$	−28.9706	−1.01108	−0.505540	−	0.862803i	$-0.668707\pi$
$821$	−28.9706	−1.01108	−0.505540	+	0.862803i	$0.668707\pi$
$822$	0	0
$823$	−36.4853	−1.27180	−0.635898	−	0.771773i	$-0.719370\pi$
$823$	−36.4853	−1.27180	−0.635898	+	0.771773i	$0.719370\pi$
$824$	0	0
$825$	6.65685	0.231762
$826$	0	0
$827$	5.31371	0.184776	0.0923879	−	0.995723i	$-0.470550\pi$
$827$	5.31371	0.184776	0.0923879	+	0.995723i	$0.470550\pi$
$828$	0	0
$829$	−37.3137	−1.29596	−0.647979	−	0.761658i	$-0.724386\pi$
$829$	−37.3137	−1.29596	−0.647979	+	0.761658i	$0.724386\pi$
$830$	0	0
$831$	−8.34315	−0.289421
$832$	0	0
$833$	16.3259	0.565659
$834$	0	0
$835$	−21.6569	−0.749466
$836$	0	0
$837$	−4.24264	−0.146647
$838$	0	0
$839$	18.6274	0.643090	0.321545	−	0.946894i	$-0.395798\pi$
$839$	18.6274	0.643090	0.321545	+	0.946894i	$0.395798\pi$
$840$	0	0
$841$	−11.0000	−0.379310
$842$	0	0
$843$	−21.6569	−0.745902
$844$	0	0
$845$	−3.41421	−0.117453
$846$	0	0
$847$	−0.828427	−0.0284651
$848$	0	0
$849$	−26.7279	−0.917300
$850$	0	0
$851$	17.6569	0.605269
$852$	0	0
$853$	13.0294	0.446119	0.223060	−	0.974805i	$-0.428396\pi$
$853$	13.0294	0.446119	0.223060	+	0.974805i	$0.428396\pi$
$854$	0	0
$855$	−20.4853	−0.700582
$856$	0	0
$857$	−16.2426	−0.554838	−0.277419	−	0.960749i	$-0.589479\pi$
$857$	−16.2426	−0.554838	−0.277419	+	0.960749i	$0.589479\pi$
$858$	0	0
$859$	0.485281	0.0165576	0.00827879	−	0.999966i	$-0.497365\pi$
$859$	0.485281	0.0165576	0.00827879	+	0.999966i	$0.497365\pi$
$860$	0	0
$861$	−9.94113	−0.338793
$862$	0	0
$863$	52.7696	1.79630	0.898148	−	0.439693i	$-0.144913\pi$
$863$	52.7696	1.79630	0.898148	+	0.439693i	$0.144913\pi$
$864$	0	0
$865$	−30.4853	−1.03653
$866$	0	0
$867$	10.3137	0.350272
$868$	0	0
$869$	5.07107	0.172024
$870$	0	0
$871$	11.0711	0.375129
$872$	0	0
$873$	10.0000	0.338449
$874$	0	0
$875$	4.68629	0.158426
$876$	0	0
$877$	−13.0294	−0.439973	−0.219986	−	0.975503i	$-0.570601\pi$
$877$	−13.0294	−0.439973	−0.219986	+	0.975503i	$0.570601\pi$
$878$	0	0
$879$	−9.17157	−0.309349
$880$	0	0
$881$	46.2843	1.55936	0.779678	−	0.626180i	$-0.215383\pi$
$881$	46.2843	1.55936	0.779678	+	0.626180i	$0.215383\pi$
$882$	0	0
$883$	−28.4853	−0.958606	−0.479303	−	0.877649i	$-0.659111\pi$
$883$	−28.4853	−0.958606	−0.479303	+	0.877649i	$0.659111\pi$
$884$	0	0
$885$	17.6569	0.593529
$886$	0	0
$887$	47.5980	1.59818	0.799092	−	0.601209i	$-0.205314\pi$
$887$	47.5980	1.59818	0.799092	+	0.601209i	$0.205314\pi$
$888$	0	0
$889$	4.76955	0.159966
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	−50.9117	−1.70369
$894$	0	0
$895$	−82.4264	−2.75521
$896$	0	0
$897$	−4.82843	−0.161216
$898$	0	0
$899$	−18.0000	−0.600334
$900$	0	0
$901$	24.0833	0.802330
$902$	0	0
$903$	7.51472	0.250074
$904$	0	0
$905$	85.2548	2.83397
$906$	0	0
$907$	−13.6569	−0.453468	−0.226734	−	0.973957i	$-0.572805\pi$
$907$	−13.6569	−0.453468	−0.226734	+	0.973957i	$0.572805\pi$
$908$	0	0
$909$	9.89949	0.328346
$910$	0	0
$911$	−5.65685	−0.187420	−0.0937100	−	0.995600i	$-0.529873\pi$
$911$	−5.65685	−0.187420	−0.0937100	+	0.995600i	$0.529873\pi$
$912$	0	0
$913$	−11.3137	−0.374429
$914$	0	0
$915$	−6.82843	−0.225741
$916$	0	0
$917$	14.0589	0.464265
$918$	0	0
$919$	21.2721	0.701701	0.350851	−	0.936431i	$-0.385893\pi$
$919$	21.2721	0.701701	0.350851	+	0.936431i	$0.385893\pi$
$920$	0	0
$921$	−30.2843	−0.997901
$922$	0	0
$923$	5.65685	0.186198
$924$	0	0
$925$	−24.3431	−0.800398
$926$	0	0
$927$	−8.48528	−0.278693
$928$	0	0
$929$	−3.89949	−0.127938	−0.0639691	−	0.997952i	$-0.520376\pi$
$929$	−3.89949	−0.127938	−0.0639691	+	0.997952i	$0.520376\pi$
$930$	0	0
$931$	−37.8823	−1.24154
$932$	0	0
$933$	0.828427	0.0271215
$934$	0	0
$935$	−8.82843	−0.288720
$936$	0	0
$937$	−11.1716	−0.364959	−0.182480	−	0.983210i	$-0.558412\pi$
$937$	−11.1716	−0.364959	−0.182480	+	0.983210i	$0.558412\pi$
$938$	0	0
$939$	−20.0000	−0.652675
$940$	0	0
$941$	−6.62742	−0.216048	−0.108024	−	0.994148i	$-0.534452\pi$
$941$	−6.62742	−0.216048	−0.108024	+	0.994148i	$0.534452\pi$
$942$	0	0
$943$	57.9411	1.88682
$944$	0	0
$945$	−2.82843	−0.0920087
$946$	0	0
$947$	−11.1127	−0.361114	−0.180557	−	0.983565i	$-0.557790\pi$
$947$	−11.1127	−0.361114	−0.180557	+	0.983565i	$0.557790\pi$
$948$	0	0
$949$	4.48528	0.145598
$950$	0	0
$951$	12.5858	0.408122
$952$	0	0
$953$	−45.8995	−1.48683	−0.743415	−	0.668830i	$-0.766795\pi$
$953$	−45.8995	−1.48683	−0.743415	+	0.668830i	$0.766795\pi$
$954$	0	0
$955$	−65.9411	−2.13380
$956$	0	0
$957$	−4.24264	−0.137145
$958$	0	0
$959$	−15.5147	−0.500996
$960$	0	0
$961$	−13.0000	−0.419355
$962$	0	0
$963$	8.48528	0.273434
$964$	0	0
$965$	64.2843	2.06938
$966$	0	0
$967$	−36.1421	−1.16225	−0.581126	−	0.813813i	$-0.697388\pi$
$967$	−36.1421	−1.16225	−0.581126	+	0.813813i	$0.697388\pi$
$968$	0	0
$969$	15.5147	0.498405
$970$	0	0
$971$	8.82843	0.283318	0.141659	−	0.989916i	$-0.454756\pi$
$971$	8.82843	0.283318	0.141659	+	0.989916i	$0.454756\pi$
$972$	0	0
$973$	−7.11270	−0.228023
$974$	0	0
$975$	6.65685	0.213190
$976$	0	0
$977$	0.301515	0.00964633	0.00482316	−	0.999988i	$-0.498465\pi$
$977$	0.301515	0.00964633	0.00482316	+	0.999988i	$0.498465\pi$
$978$	0	0
$979$	12.5858	0.402243
$980$	0	0
$981$	−1.17157	−0.0374054
$982$	0	0
$983$	28.9706	0.924017	0.462009	−	0.886875i	$-0.347129\pi$
$983$	28.9706	0.924017	0.462009	+	0.886875i	$0.347129\pi$
$984$	0	0
$985$	31.3137	0.997738
$986$	0	0
$987$	−7.02944	−0.223749
$988$	0	0
$989$	−43.7990	−1.39273
$990$	0	0
$991$	−1.45584	−0.0462464	−0.0231232	−	0.999733i	$-0.507361\pi$
$991$	−1.45584	−0.0462464	−0.0231232	+	0.999733i	$0.507361\pi$
$992$	0	0
$993$	−20.2426	−0.642381
$994$	0	0
$995$	−17.6569	−0.559760
$996$	0	0
$997$	38.7696	1.22784	0.613922	−	0.789367i	$-0.289591\pi$
$997$	38.7696	1.22784	0.613922	+	0.789367i	$0.289591\pi$
$998$	0	0
$999$	3.65685	0.115698

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6864.2.a.bc.1.1		2
4.3	odd	2		429.2.a.c.1.1	✓	2
12.11	even	2		1287.2.a.g.1.2		2
44.43	even	2		4719.2.a.o.1.2		2
52.51	odd	2		5577.2.a.i.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.a.c.1.1	✓	2	4.3	odd	2
1287.2.a.g.1.2		2	12.11	even	2
4719.2.a.o.1.2		2	44.43	even	2
5577.2.a.i.1.2		2	52.51	odd	2
6864.2.a.bc.1.1		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6864 = 2^{4} \cdot 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6864.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -3.41421 q^{5} -0.828427 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -3.41421 q^{5} -0.828427 q^{7} +1.00000 q^{9} -1.00000 q^{11} -1.00000 q^{13} +3.41421 q^{15} -2.58579 q^{17} +6.00000 q^{19} +0.828427 q^{21} -4.82843 q^{23} +6.65685 q^{25} -1.00000 q^{27} -4.24264 q^{29} +4.24264 q^{31} +1.00000 q^{33} +2.82843 q^{35} -3.65685 q^{37} +1.00000 q^{39} -12.0000 q^{41} +9.07107 q^{43} -3.41421 q^{45} -8.48528 q^{47} -6.31371 q^{49} +2.58579 q^{51} -9.31371 q^{53} +3.41421 q^{55} -6.00000 q^{57} +5.17157 q^{59} -2.00000 q^{61} -0.828427 q^{63} +3.41421 q^{65} -11.0711 q^{67} +4.82843 q^{69} -5.65685 q^{71} -4.48528 q^{73} -6.65685 q^{75} +0.828427 q^{77} -5.07107 q^{79} +1.00000 q^{81} +11.3137 q^{83} +8.82843 q^{85} +4.24264 q^{87} -12.5858 q^{89} +0.828427 q^{91} -4.24264 q^{93} -20.4853 q^{95} +10.0000 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 4 q^{5} + 4 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 4 * q^5 + 4 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} - 4 q^{5} + 4 q^{7} + 2 q^{9} - 2 q^{11} - 2 q^{13} + 4 q^{15} - 8 q^{17} + 12 q^{19} - 4 q^{21} - 4 q^{23} + 2 q^{25} - 2 q^{27} + 2 q^{33} + 4 q^{37} + 2 q^{39} - 24 q^{41} + 4 q^{43} - 4 q^{45} + 10 q^{49} + 8 q^{51} + 4 q^{53} + 4 q^{55} - 12 q^{57} + 16 q^{59} - 4 q^{61} + 4 q^{63} + 4 q^{65} - 8 q^{67} + 4 q^{69} + 8 q^{73} - 2 q^{75} - 4 q^{77} + 4 q^{79} + 2 q^{81} + 12 q^{85} - 28 q^{89} - 4 q^{91} - 24 q^{95} + 20 q^{97} - 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 4 * q^5 + 4 * q^7 + 2 * q^9 - 2 * q^11 - 2 * q^13 + 4 * q^15 - 8 * q^17 + 12 * q^19 - 4 * q^21 - 4 * q^23 + 2 * q^25 - 2 * q^27 + 2 * q^33 + 4 * q^37 + 2 * q^39 - 24 * q^41 + 4 * q^43 - 4 * q^45 + 10 * q^49 + 8 * q^51 + 4 * q^53 + 4 * q^55 - 12 * q^57 + 16 * q^59 - 4 * q^61 + 4 * q^63 + 4 * q^65 - 8 * q^67 + 4 * q^69 + 8 * q^73 - 2 * q^75 - 4 * q^77 + 4 * q^79 + 2 * q^81 + 12 * q^85 - 28 * q^89 - 4 * q^91 - 24 * q^95 + 20 * q^97 - 2 * q^99

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