LMFDB - Embedded newform 6864.2.a.bk.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:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
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			$-1.73205$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -2.73205 q^{5} +2.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -2.73205 q^{5} +2.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} -1.00000 q^{13} -2.73205 q^{15} +3.26795 q^{17} +7.46410 q^{19} +2.00000 q^{21} +2.00000 q^{23} +2.46410 q^{25} +1.00000 q^{27} +6.19615 q^{29} -2.19615 q^{31} +1.00000 q^{33} -5.46410 q^{35} -2.00000 q^{37} -1.00000 q^{39} +1.46410 q^{41} -4.19615 q^{43} -2.73205 q^{45} -5.46410 q^{47} -3.00000 q^{49} +3.26795 q^{51} +2.00000 q^{53} -2.73205 q^{55} +7.46410 q^{57} +6.53590 q^{59} -8.92820 q^{61} +2.00000 q^{63} +2.73205 q^{65} +1.80385 q^{67} +2.00000 q^{69} +6.92820 q^{71} -10.9282 q^{73} +2.46410 q^{75} +2.00000 q^{77} -13.6603 q^{79} +1.00000 q^{81} +13.8564 q^{83} -8.92820 q^{85} +6.19615 q^{87} +1.66025 q^{89} -2.00000 q^{91} -2.19615 q^{93} -20.3923 q^{95} -4.92820 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{5} + 4 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 2 * q^5 + 4 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} - 2 q^{5} + 4 q^{7} + 2 q^{9} + 2 q^{11} - 2 q^{13} - 2 q^{15} + 10 q^{17} + 8 q^{19} + 4 q^{21} + 4 q^{23} - 2 q^{25} + 2 q^{27} + 2 q^{29} + 6 q^{31} + 2 q^{33} - 4 q^{35} - 4 q^{37} - 2 q^{39} - 4 q^{41} + 2 q^{43} - 2 q^{45} - 4 q^{47} - 6 q^{49} + 10 q^{51} + 4 q^{53} - 2 q^{55} + 8 q^{57} + 20 q^{59} - 4 q^{61} + 4 q^{63} + 2 q^{65} + 14 q^{67} + 4 q^{69} - 8 q^{73} - 2 q^{75} + 4 q^{77} - 10 q^{79} + 2 q^{81} - 4 q^{85} + 2 q^{87} - 14 q^{89} - 4 q^{91} + 6 q^{93} - 20 q^{95} + 4 q^{97} + 2 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^5 + 4 * q^7 + 2 * q^9 + 2 * q^11 - 2 * q^13 - 2 * q^15 + 10 * q^17 + 8 * q^19 + 4 * q^21 + 4 * q^23 - 2 * q^25 + 2 * q^27 + 2 * q^29 + 6 * q^31 + 2 * q^33 - 4 * q^35 - 4 * q^37 - 2 * q^39 - 4 * q^41 + 2 * q^43 - 2 * q^45 - 4 * q^47 - 6 * q^49 + 10 * q^51 + 4 * q^53 - 2 * q^55 + 8 * q^57 + 20 * q^59 - 4 * q^61 + 4 * q^63 + 2 * q^65 + 14 * q^67 + 4 * q^69 - 8 * q^73 - 2 * q^75 + 4 * q^77 - 10 * q^79 + 2 * q^81 - 4 * q^85 + 2 * q^87 - 14 * q^89 - 4 * q^91 + 6 * q^93 - 20 * 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
$5$	−2.73205	−1.22181	−0.610905	−	0.791704i	$-0.709194\pi$
$5$	−2.73205	−1.22181	−0.610905	+	0.791704i	$0.709194\pi$
$6$	0	0
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	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
$15$	−2.73205	−0.705412
$16$	0	0
$17$	3.26795	0.792594	0.396297	−	0.918122i	$-0.370295\pi$
$17$	3.26795	0.792594	0.396297	+	0.918122i	$0.370295\pi$
$18$	0	0
$19$	7.46410	1.71238	0.856191	−	0.516659i	$-0.172825\pi$
$19$	7.46410	1.71238	0.856191	+	0.516659i	$0.172825\pi$
$20$	0	0
$21$	2.00000	0.436436
$22$	0	0
$23$	2.00000	0.417029	0.208514	−	0.978019i	$-0.433137\pi$
$23$	2.00000	0.417029	0.208514	+	0.978019i	$0.433137\pi$
$24$	0	0
$25$	2.46410	0.492820
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	6.19615	1.15060	0.575298	−	0.817944i	$-0.304886\pi$
$29$	6.19615	1.15060	0.575298	+	0.817944i	$0.304886\pi$
$30$	0	0
$31$	−2.19615	−0.394441	−0.197220	−	0.980359i	$-0.563191\pi$
$31$	−2.19615	−0.394441	−0.197220	+	0.980359i	$0.563191\pi$
$32$	0	0
$33$	1.00000	0.174078
$34$	0	0
$35$	−5.46410	−0.923602
$36$	0	0
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	1.46410	0.228654	0.114327	−	0.993443i	$-0.463529\pi$
$41$	1.46410	0.228654	0.114327	+	0.993443i	$0.463529\pi$
$42$	0	0
$43$	−4.19615	−0.639907	−0.319954	−	0.947433i	$-0.603667\pi$
$43$	−4.19615	−0.639907	−0.319954	+	0.947433i	$0.603667\pi$
$44$	0	0
$45$	−2.73205	−0.407270
$46$	0	0
$47$	−5.46410	−0.797021	−0.398511	−	0.917164i	$-0.630473\pi$
$47$	−5.46410	−0.797021	−0.398511	+	0.917164i	$0.630473\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	3.26795	0.457604
$52$	0	0
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	0	0
$55$	−2.73205	−0.368390
$56$	0	0
$57$	7.46410	0.988644
$58$	0	0
$59$	6.53590	0.850901	0.425451	−	0.904982i	$-0.360116\pi$
$59$	6.53590	0.850901	0.425451	+	0.904982i	$0.360116\pi$
$60$	0	0
$61$	−8.92820	−1.14314	−0.571570	−	0.820554i	$-0.693665\pi$
$61$	−8.92820	−1.14314	−0.571570	+	0.820554i	$0.693665\pi$
$62$	0	0
$63$	2.00000	0.251976
$64$	0	0
$65$	2.73205	0.338869
$66$	0	0
$67$	1.80385	0.220375	0.110188	−	0.993911i	$-0.464855\pi$
$67$	1.80385	0.220375	0.110188	+	0.993911i	$0.464855\pi$
$68$	0	0
$69$	2.00000	0.240772
$70$	0	0
$71$	6.92820	0.822226	0.411113	−	0.911584i	$-0.365140\pi$
$71$	6.92820	0.822226	0.411113	+	0.911584i	$0.365140\pi$
$72$	0	0
$73$	−10.9282	−1.27905	−0.639525	−	0.768771i	$-0.720869\pi$
$73$	−10.9282	−1.27905	−0.639525	+	0.768771i	$0.720869\pi$
$74$	0	0
$75$	2.46410	0.284530
$76$	0	0
$77$	2.00000	0.227921
$78$	0	0
$79$	−13.6603	−1.53690	−0.768449	−	0.639911i	$-0.778971\pi$
$79$	−13.6603	−1.53690	−0.768449	+	0.639911i	$0.778971\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	13.8564	1.52094	0.760469	−	0.649374i	$-0.224969\pi$
$83$	13.8564	1.52094	0.760469	+	0.649374i	$0.224969\pi$
$84$	0	0
$85$	−8.92820	−0.968400
$86$	0	0
$87$	6.19615	0.664297
$88$	0	0
$89$	1.66025	0.175987	0.0879933	−	0.996121i	$-0.471955\pi$
$89$	1.66025	0.175987	0.0879933	+	0.996121i	$0.471955\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	0	0
$93$	−2.19615	−0.227730
$94$	0	0
$95$	−20.3923	−2.09221
$96$	0	0
$97$	−4.92820	−0.500383	−0.250192	−	0.968196i	$-0.580494\pi$
$97$	−4.92820	−0.500383	−0.250192	+	0.968196i	$0.580494\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	−2.19615	−0.218525	−0.109263	−	0.994013i	$-0.534849\pi$
$101$	−2.19615	−0.218525	−0.109263	+	0.994013i	$0.534849\pi$
$102$	0	0
$103$	15.3205	1.50957	0.754787	−	0.655970i	$-0.227740\pi$
$103$	15.3205	1.50957	0.754787	+	0.655970i	$0.227740\pi$
$104$	0	0
$105$	−5.46410	−0.533242
$106$	0	0
$107$	−12.3923	−1.19801	−0.599005	−	0.800746i	$-0.704437\pi$
$107$	−12.3923	−1.19801	−0.599005	+	0.800746i	$0.704437\pi$
$108$	0	0
$109$	−8.00000	−0.766261	−0.383131	−	0.923694i	$-0.625154\pi$
$109$	−8.00000	−0.766261	−0.383131	+	0.923694i	$0.625154\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	0	0
$113$	10.0000	0.940721	0.470360	−	0.882474i	$-0.344124\pi$
$113$	10.0000	0.940721	0.470360	+	0.882474i	$0.344124\pi$
$114$	0	0
$115$	−5.46410	−0.509530
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	6.53590	0.599145
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	1.46410	0.132014
$124$	0	0
$125$	6.92820	0.619677
$126$	0	0
$127$	11.1244	0.987127	0.493563	−	0.869710i	$-0.335694\pi$
$127$	11.1244	0.987127	0.493563	+	0.869710i	$0.335694\pi$
$128$	0	0
$129$	−4.19615	−0.369451
$130$	0	0
$131$	18.9282	1.65376	0.826882	−	0.562375i	$-0.190112\pi$
$131$	18.9282	1.65376	0.826882	+	0.562375i	$0.190112\pi$
$132$	0	0
$133$	14.9282	1.29444
$134$	0	0
$135$	−2.73205	−0.235137
$136$	0	0
$137$	16.5885	1.41725	0.708624	−	0.705587i	$-0.249316\pi$
$137$	16.5885	1.41725	0.708624	+	0.705587i	$0.249316\pi$
$138$	0	0
$139$	20.1962	1.71302	0.856508	−	0.516134i	$-0.172629\pi$
$139$	20.1962	1.71302	0.856508	+	0.516134i	$0.172629\pi$
$140$	0	0
$141$	−5.46410	−0.460160
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	−16.9282	−1.40581
$146$	0	0
$147$	−3.00000	−0.247436
$148$	0	0
$149$	8.00000	0.655386	0.327693	−	0.944784i	$-0.393729\pi$
$149$	8.00000	0.655386	0.327693	+	0.944784i	$0.393729\pi$
$150$	0	0
$151$	11.4641	0.932935	0.466468	−	0.884538i	$-0.345526\pi$
$151$	11.4641	0.932935	0.466468	+	0.884538i	$0.345526\pi$
$152$	0	0
$153$	3.26795	0.264198
$154$	0	0
$155$	6.00000	0.481932
$156$	0	0
$157$	19.3205	1.54194	0.770972	−	0.636869i	$-0.219771\pi$
$157$	19.3205	1.54194	0.770972	+	0.636869i	$0.219771\pi$
$158$	0	0
$159$	2.00000	0.158610
$160$	0	0
$161$	4.00000	0.315244
$162$	0	0
$163$	20.7321	1.62386	0.811930	−	0.583755i	$-0.198417\pi$
$163$	20.7321	1.62386	0.811930	+	0.583755i	$0.198417\pi$
$164$	0	0
$165$	−2.73205	−0.212690
$166$	0	0
$167$	−17.8564	−1.38177	−0.690885	−	0.722965i	$-0.742779\pi$
$167$	−17.8564	−1.38177	−0.690885	+	0.722965i	$0.742779\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	7.46410	0.570794
$172$	0	0
$173$	−5.12436	−0.389598	−0.194799	−	0.980843i	$-0.562405\pi$
$173$	−5.12436	−0.389598	−0.194799	+	0.980843i	$0.562405\pi$
$174$	0	0
$175$	4.92820	0.372537
$176$	0	0
$177$	6.53590	0.491268
$178$	0	0
$179$	19.8564	1.48414	0.742069	−	0.670324i	$-0.233845\pi$
$179$	19.8564	1.48414	0.742069	+	0.670324i	$0.233845\pi$
$180$	0	0
$181$	−12.3923	−0.921113	−0.460556	−	0.887630i	$-0.652350\pi$
$181$	−12.3923	−0.921113	−0.460556	+	0.887630i	$0.652350\pi$
$182$	0	0
$183$	−8.92820	−0.659992
$184$	0	0
$185$	5.46410	0.401729
$186$	0	0
$187$	3.26795	0.238976
$188$	0	0
$189$	2.00000	0.145479
$190$	0	0
$191$	5.07180	0.366982	0.183491	−	0.983021i	$-0.441260\pi$
$191$	5.07180	0.366982	0.183491	+	0.983021i	$0.441260\pi$
$192$	0	0
$193$	6.92820	0.498703	0.249351	−	0.968413i	$-0.419783\pi$
$193$	6.92820	0.498703	0.249351	+	0.968413i	$0.419783\pi$
$194$	0	0
$195$	2.73205	0.195646
$196$	0	0
$197$	8.00000	0.569976	0.284988	−	0.958531i	$-0.408010\pi$
$197$	8.00000	0.569976	0.284988	+	0.958531i	$0.408010\pi$
$198$	0	0
$199$	−17.4641	−1.23800	−0.618999	−	0.785392i	$-0.712461\pi$
$199$	−17.4641	−1.23800	−0.618999	+	0.785392i	$0.712461\pi$
$200$	0	0
$201$	1.80385	0.127234
$202$	0	0
$203$	12.3923	0.869769
$204$	0	0
$205$	−4.00000	−0.279372
$206$	0	0
$207$	2.00000	0.139010
$208$	0	0
$209$	7.46410	0.516303
$210$	0	0
$211$	−22.7321	−1.56494	−0.782469	−	0.622689i	$-0.786040\pi$
$211$	−22.7321	−1.56494	−0.782469	+	0.622689i	$0.786040\pi$
$212$	0	0
$213$	6.92820	0.474713
$214$	0	0
$215$	11.4641	0.781845
$216$	0	0
$217$	−4.39230	−0.298169
$218$	0	0
$219$	−10.9282	−0.738460
$220$	0	0
$221$	−3.26795	−0.219826
$222$	0	0
$223$	−21.5167	−1.44086	−0.720431	−	0.693527i	$-0.756056\pi$
$223$	−21.5167	−1.44086	−0.720431	+	0.693527i	$0.756056\pi$
$224$	0	0
$225$	2.46410	0.164273
$226$	0	0
$227$	−10.3923	−0.689761	−0.344881	−	0.938647i	$-0.612081\pi$
$227$	−10.3923	−0.689761	−0.344881	+	0.938647i	$0.612081\pi$
$228$	0	0
$229$	−8.53590	−0.564068	−0.282034	−	0.959404i	$-0.591009\pi$
$229$	−8.53590	−0.564068	−0.282034	+	0.959404i	$0.591009\pi$
$230$	0	0
$231$	2.00000	0.131590
$232$	0	0
$233$	−6.19615	−0.405923	−0.202962	−	0.979187i	$-0.565057\pi$
$233$	−6.19615	−0.405923	−0.202962	+	0.979187i	$0.565057\pi$
$234$	0	0
$235$	14.9282	0.973809
$236$	0	0
$237$	−13.6603	−0.887329
$238$	0	0
$239$	−1.60770	−0.103993	−0.0519966	−	0.998647i	$-0.516558\pi$
$239$	−1.60770	−0.103993	−0.0519966	+	0.998647i	$0.516558\pi$
$240$	0	0
$241$	22.7846	1.46769	0.733843	−	0.679319i	$-0.237725\pi$
$241$	22.7846	1.46769	0.733843	+	0.679319i	$0.237725\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	8.19615	0.523633
$246$	0	0
$247$	−7.46410	−0.474929
$248$	0	0
$249$	13.8564	0.878114
$250$	0	0
$251$	30.9282	1.95217	0.976085	−	0.217387i	$-0.0697534\pi$
$251$	30.9282	1.95217	0.976085	+	0.217387i	$0.0697534\pi$
$252$	0	0
$253$	2.00000	0.125739
$254$	0	0
$255$	−8.92820	−0.559106
$256$	0	0
$257$	−17.3205	−1.08042	−0.540212	−	0.841529i	$-0.681656\pi$
$257$	−17.3205	−1.08042	−0.540212	+	0.841529i	$0.681656\pi$
$258$	0	0
$259$	−4.00000	−0.248548
$260$	0	0
$261$	6.19615	0.383532
$262$	0	0
$263$	−2.53590	−0.156370	−0.0781851	−	0.996939i	$-0.524913\pi$
$263$	−2.53590	−0.156370	−0.0781851	+	0.996939i	$0.524913\pi$
$264$	0	0
$265$	−5.46410	−0.335657
$266$	0	0
$267$	1.66025	0.101606
$268$	0	0
$269$	17.3205	1.05605	0.528025	−	0.849229i	$-0.322933\pi$
$269$	17.3205	1.05605	0.528025	+	0.849229i	$0.322933\pi$
$270$	0	0
$271$	25.3205	1.53811	0.769056	−	0.639182i	$-0.220727\pi$
$271$	25.3205	1.53811	0.769056	+	0.639182i	$0.220727\pi$
$272$	0	0
$273$	−2.00000	−0.121046
$274$	0	0
$275$	2.46410	0.148591
$276$	0	0
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	0	0
$279$	−2.19615	−0.131480
$280$	0	0
$281$	−11.3205	−0.675325	−0.337662	−	0.941267i	$-0.609636\pi$
$281$	−11.3205	−0.675325	−0.337662	+	0.941267i	$0.609636\pi$
$282$	0	0
$283$	−20.1962	−1.20054	−0.600268	−	0.799799i	$-0.704940\pi$
$283$	−20.1962	−1.20054	−0.600268	+	0.799799i	$0.704940\pi$
$284$	0	0
$285$	−20.3923	−1.20794
$286$	0	0
$287$	2.92820	0.172846
$288$	0	0
$289$	−6.32051	−0.371795
$290$	0	0
$291$	−4.92820	−0.288896
$292$	0	0
$293$	26.9282	1.57316	0.786581	−	0.617487i	$-0.211849\pi$
$293$	26.9282	1.57316	0.786581	+	0.617487i	$0.211849\pi$
$294$	0	0
$295$	−17.8564	−1.03964
$296$	0	0
$297$	1.00000	0.0580259
$298$	0	0
$299$	−2.00000	−0.115663
$300$	0	0
$301$	−8.39230	−0.483724
$302$	0	0
$303$	−2.19615	−0.126166
$304$	0	0
$305$	24.3923	1.39670
$306$	0	0
$307$	27.4641	1.56746	0.783730	−	0.621102i	$-0.213315\pi$
$307$	27.4641	1.56746	0.783730	+	0.621102i	$0.213315\pi$
$308$	0	0
$309$	15.3205	0.871553
$310$	0	0
$311$	16.9282	0.959910	0.479955	−	0.877293i	$-0.340653\pi$
$311$	16.9282	0.959910	0.479955	+	0.877293i	$0.340653\pi$
$312$	0	0
$313$	−2.53590	−0.143337	−0.0716687	−	0.997428i	$-0.522832\pi$
$313$	−2.53590	−0.143337	−0.0716687	+	0.997428i	$0.522832\pi$
$314$	0	0
$315$	−5.46410	−0.307867
$316$	0	0
$317$	−7.12436	−0.400144	−0.200072	−	0.979781i	$-0.564118\pi$
$317$	−7.12436	−0.400144	−0.200072	+	0.979781i	$0.564118\pi$
$318$	0	0
$319$	6.19615	0.346918
$320$	0	0
$321$	−12.3923	−0.691671
$322$	0	0
$323$	24.3923	1.35722
$324$	0	0
$325$	−2.46410	−0.136684
$326$	0	0
$327$	−8.00000	−0.442401
$328$	0	0
$329$	−10.9282	−0.602491
$330$	0	0
$331$	21.5167	1.18266	0.591331	−	0.806429i	$-0.298603\pi$
$331$	21.5167	1.18266	0.591331	+	0.806429i	$0.298603\pi$
$332$	0	0
$333$	−2.00000	−0.109599
$334$	0	0
$335$	−4.92820	−0.269257
$336$	0	0
$337$	−25.3205	−1.37930	−0.689648	−	0.724145i	$-0.742235\pi$
$337$	−25.3205	−1.37930	−0.689648	+	0.724145i	$0.742235\pi$
$338$	0	0
$339$	10.0000	0.543125
$340$	0	0
$341$	−2.19615	−0.118928
$342$	0	0
$343$	−20.0000	−1.07990
$344$	0	0
$345$	−5.46410	−0.294177
$346$	0	0
$347$	−16.3923	−0.879985	−0.439993	−	0.898001i	$-0.645019\pi$
$347$	−16.3923	−0.879985	−0.439993	+	0.898001i	$0.645019\pi$
$348$	0	0
$349$	23.8564	1.27700	0.638502	−	0.769620i	$-0.279554\pi$
$349$	23.8564	1.27700	0.638502	+	0.769620i	$0.279554\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	−28.9808	−1.54249	−0.771245	−	0.636538i	$-0.780366\pi$
$353$	−28.9808	−1.54249	−0.771245	+	0.636538i	$0.780366\pi$
$354$	0	0
$355$	−18.9282	−1.00460
$356$	0	0
$357$	6.53590	0.345916
$358$	0	0
$359$	23.4641	1.23839	0.619194	−	0.785238i	$-0.287459\pi$
$359$	23.4641	1.23839	0.619194	+	0.785238i	$0.287459\pi$
$360$	0	0
$361$	36.7128	1.93225
$362$	0	0
$363$	1.00000	0.0524864
$364$	0	0
$365$	29.8564	1.56276
$366$	0	0
$367$	−17.0718	−0.891141	−0.445570	−	0.895247i	$-0.646999\pi$
$367$	−17.0718	−0.891141	−0.445570	+	0.895247i	$0.646999\pi$
$368$	0	0
$369$	1.46410	0.0762181
$370$	0	0
$371$	4.00000	0.207670
$372$	0	0
$373$	32.2487	1.66977	0.834887	−	0.550421i	$-0.185533\pi$
$373$	32.2487	1.66977	0.834887	+	0.550421i	$0.185533\pi$
$374$	0	0
$375$	6.92820	0.357771
$376$	0	0
$377$	−6.19615	−0.319118
$378$	0	0
$379$	−1.80385	−0.0926574	−0.0463287	−	0.998926i	$-0.514752\pi$
$379$	−1.80385	−0.0926574	−0.0463287	+	0.998926i	$0.514752\pi$
$380$	0	0
$381$	11.1244	0.569918
$382$	0	0
$383$	−3.60770	−0.184345	−0.0921723	−	0.995743i	$-0.529381\pi$
$383$	−3.60770	−0.184345	−0.0921723	+	0.995743i	$0.529381\pi$
$384$	0	0
$385$	−5.46410	−0.278476
$386$	0	0
$387$	−4.19615	−0.213302
$388$	0	0
$389$	15.4641	0.784061	0.392031	−	0.919952i	$-0.371773\pi$
$389$	15.4641	0.784061	0.392031	+	0.919952i	$0.371773\pi$
$390$	0	0
$391$	6.53590	0.330535
$392$	0	0
$393$	18.9282	0.954802
$394$	0	0
$395$	37.3205	1.87780
$396$	0	0
$397$	11.0718	0.555678	0.277839	−	0.960628i	$-0.410382\pi$
$397$	11.0718	0.555678	0.277839	+	0.960628i	$0.410382\pi$
$398$	0	0
$399$	14.9282	0.747345
$400$	0	0
$401$	15.8038	0.789206	0.394603	−	0.918852i	$-0.370882\pi$
$401$	15.8038	0.789206	0.394603	+	0.918852i	$0.370882\pi$
$402$	0	0
$403$	2.19615	0.109398
$404$	0	0
$405$	−2.73205	−0.135757
$406$	0	0
$407$	−2.00000	−0.0991363
$408$	0	0
$409$	27.7128	1.37031	0.685155	−	0.728397i	$-0.259734\pi$
$409$	27.7128	1.37031	0.685155	+	0.728397i	$0.259734\pi$
$410$	0	0
$411$	16.5885	0.818248
$412$	0	0
$413$	13.0718	0.643221
$414$	0	0
$415$	−37.8564	−1.85830
$416$	0	0
$417$	20.1962	0.989010
$418$	0	0
$419$	−26.0000	−1.27018	−0.635092	−	0.772437i	$-0.719038\pi$
$419$	−26.0000	−1.27018	−0.635092	+	0.772437i	$0.719038\pi$
$420$	0	0
$421$	−4.92820	−0.240186	−0.120093	−	0.992763i	$-0.538319\pi$
$421$	−4.92820	−0.240186	−0.120093	+	0.992763i	$0.538319\pi$
$422$	0	0
$423$	−5.46410	−0.265674
$424$	0	0
$425$	8.05256	0.390606
$426$	0	0
$427$	−17.8564	−0.864132
$428$	0	0
$429$	−1.00000	−0.0482805
$430$	0	0
$431$	−28.7846	−1.38651	−0.693253	−	0.720694i	$-0.743823\pi$
$431$	−28.7846	−1.38651	−0.693253	+	0.720694i	$0.743823\pi$
$432$	0	0
$433$	−6.00000	−0.288342	−0.144171	−	0.989553i	$-0.546051\pi$
$433$	−6.00000	−0.288342	−0.144171	+	0.989553i	$0.546051\pi$
$434$	0	0
$435$	−16.9282	−0.811645
$436$	0	0
$437$	14.9282	0.714113
$438$	0	0
$439$	−35.9090	−1.71384	−0.856921	−	0.515448i	$-0.827625\pi$
$439$	−35.9090	−1.71384	−0.856921	+	0.515448i	$0.827625\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	0	0
$443$	−36.7846	−1.74769	−0.873845	−	0.486205i	$-0.838381\pi$
$443$	−36.7846	−1.74769	−0.873845	+	0.486205i	$0.838381\pi$
$444$	0	0
$445$	−4.53590	−0.215022
$446$	0	0
$447$	8.00000	0.378387
$448$	0	0
$449$	8.19615	0.386800	0.193400	−	0.981120i	$-0.438048\pi$
$449$	8.19615	0.386800	0.193400	+	0.981120i	$0.438048\pi$
$450$	0	0
$451$	1.46410	0.0689419
$452$	0	0
$453$	11.4641	0.538630
$454$	0	0
$455$	5.46410	0.256161
$456$	0	0
$457$	10.7846	0.504483	0.252241	−	0.967664i	$-0.418832\pi$
$457$	10.7846	0.504483	0.252241	+	0.967664i	$0.418832\pi$
$458$	0	0
$459$	3.26795	0.152535
$460$	0	0
$461$	−10.5359	−0.490706	−0.245353	−	0.969434i	$-0.578904\pi$
$461$	−10.5359	−0.490706	−0.245353	+	0.969434i	$0.578904\pi$
$462$	0	0
$463$	25.5167	1.18586	0.592930	−	0.805254i	$-0.297971\pi$
$463$	25.5167	1.18586	0.592930	+	0.805254i	$0.297971\pi$
$464$	0	0
$465$	6.00000	0.278243
$466$	0	0
$467$	19.0718	0.882538	0.441269	−	0.897375i	$-0.354529\pi$
$467$	19.0718	0.882538	0.441269	+	0.897375i	$0.354529\pi$
$468$	0	0
$469$	3.60770	0.166588
$470$	0	0
$471$	19.3205	0.890242
$472$	0	0
$473$	−4.19615	−0.192939
$474$	0	0
$475$	18.3923	0.843897
$476$	0	0
$477$	2.00000	0.0915737
$478$	0	0
$479$	−27.4641	−1.25487	−0.627433	−	0.778670i	$-0.715895\pi$
$479$	−27.4641	−1.25487	−0.627433	+	0.778670i	$0.715895\pi$
$480$	0	0
$481$	2.00000	0.0911922
$482$	0	0
$483$	4.00000	0.182006
$484$	0	0
$485$	13.4641	0.611373
$486$	0	0
$487$	8.33975	0.377910	0.188955	−	0.981986i	$-0.439490\pi$
$487$	8.33975	0.377910	0.188955	+	0.981986i	$0.439490\pi$
$488$	0	0
$489$	20.7321	0.937536
$490$	0	0
$491$	−2.53590	−0.114443	−0.0572217	−	0.998361i	$-0.518224\pi$
$491$	−2.53590	−0.114443	−0.0572217	+	0.998361i	$0.518224\pi$
$492$	0	0
$493$	20.2487	0.911956
$494$	0	0
$495$	−2.73205	−0.122797
$496$	0	0
$497$	13.8564	0.621545
$498$	0	0
$499$	17.8038	0.797010	0.398505	−	0.917166i	$-0.369529\pi$
$499$	17.8038	0.797010	0.398505	+	0.917166i	$0.369529\pi$
$500$	0	0
$501$	−17.8564	−0.797765
$502$	0	0
$503$	21.4641	0.957037	0.478518	−	0.878077i	$-0.341174\pi$
$503$	21.4641	0.957037	0.478518	+	0.878077i	$0.341174\pi$
$504$	0	0
$505$	6.00000	0.266996
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	−41.6603	−1.84656	−0.923279	−	0.384130i	$-0.874502\pi$
$509$	−41.6603	−1.84656	−0.923279	+	0.384130i	$0.874502\pi$
$510$	0	0
$511$	−21.8564	−0.966870
$512$	0	0
$513$	7.46410	0.329548
$514$	0	0
$515$	−41.8564	−1.84441
$516$	0	0
$517$	−5.46410	−0.240311
$518$	0	0
$519$	−5.12436	−0.224934
$520$	0	0
$521$	−40.9282	−1.79310	−0.896549	−	0.442945i	$-0.853934\pi$
$521$	−40.9282	−1.79310	−0.896549	+	0.442945i	$0.853934\pi$
$522$	0	0
$523$	21.2679	0.929982	0.464991	−	0.885315i	$-0.346057\pi$
$523$	21.2679	0.929982	0.464991	+	0.885315i	$0.346057\pi$
$524$	0	0
$525$	4.92820	0.215084
$526$	0	0
$527$	−7.17691	−0.312631
$528$	0	0
$529$	−19.0000	−0.826087
$530$	0	0
$531$	6.53590	0.283634
$532$	0	0
$533$	−1.46410	−0.0634173
$534$	0	0
$535$	33.8564	1.46374
$536$	0	0
$537$	19.8564	0.856867
$538$	0	0
$539$	−3.00000	−0.129219
$540$	0	0
$541$	−14.7846	−0.635640	−0.317820	−	0.948151i	$-0.602951\pi$
$541$	−14.7846	−0.635640	−0.317820	+	0.948151i	$0.602951\pi$
$542$	0	0
$543$	−12.3923	−0.531805
$544$	0	0
$545$	21.8564	0.936226
$546$	0	0
$547$	−12.9808	−0.555017	−0.277509	−	0.960723i	$-0.589509\pi$
$547$	−12.9808	−0.555017	−0.277509	+	0.960723i	$0.589509\pi$
$548$	0	0
$549$	−8.92820	−0.381046
$550$	0	0
$551$	46.2487	1.97026
$552$	0	0
$553$	−27.3205	−1.16179
$554$	0	0
$555$	5.46410	0.231938
$556$	0	0
$557$	−24.0000	−1.01691	−0.508456	−	0.861088i	$-0.669784\pi$
$557$	−24.0000	−1.01691	−0.508456	+	0.861088i	$0.669784\pi$
$558$	0	0
$559$	4.19615	0.177478
$560$	0	0
$561$	3.26795	0.137973
$562$	0	0
$563$	39.7128	1.67370	0.836848	−	0.547436i	$-0.184396\pi$
$563$	39.7128	1.67370	0.836848	+	0.547436i	$0.184396\pi$
$564$	0	0
$565$	−27.3205	−1.14938
$566$	0	0
$567$	2.00000	0.0839921
$568$	0	0
$569$	15.6603	0.656512	0.328256	−	0.944589i	$-0.393539\pi$
$569$	15.6603	0.656512	0.328256	+	0.944589i	$0.393539\pi$
$570$	0	0
$571$	−16.5885	−0.694205	−0.347103	−	0.937827i	$-0.612834\pi$
$571$	−16.5885	−0.694205	−0.347103	+	0.937827i	$0.612834\pi$
$572$	0	0
$573$	5.07180	0.211877
$574$	0	0
$575$	4.92820	0.205520
$576$	0	0
$577$	−8.92820	−0.371686	−0.185843	−	0.982579i	$-0.559502\pi$
$577$	−8.92820	−0.371686	−0.185843	+	0.982579i	$0.559502\pi$
$578$	0	0
$579$	6.92820	0.287926
$580$	0	0
$581$	27.7128	1.14972
$582$	0	0
$583$	2.00000	0.0828315
$584$	0	0
$585$	2.73205	0.112956
$586$	0	0
$587$	−2.53590	−0.104668	−0.0523339	−	0.998630i	$-0.516666\pi$
$587$	−2.53590	−0.104668	−0.0523339	+	0.998630i	$0.516666\pi$
$588$	0	0
$589$	−16.3923	−0.675433
$590$	0	0
$591$	8.00000	0.329076
$592$	0	0
$593$	36.0000	1.47834	0.739171	−	0.673517i	$-0.235217\pi$
$593$	36.0000	1.47834	0.739171	+	0.673517i	$0.235217\pi$
$594$	0	0
$595$	−17.8564	−0.732041
$596$	0	0
$597$	−17.4641	−0.714758
$598$	0	0
$599$	27.7128	1.13231	0.566157	−	0.824297i	$-0.308429\pi$
$599$	27.7128	1.13231	0.566157	+	0.824297i	$0.308429\pi$
$600$	0	0
$601$	0.143594	0.00585730	0.00292865	−	0.999996i	$-0.499068\pi$
$601$	0.143594	0.00585730	0.00292865	+	0.999996i	$0.499068\pi$
$602$	0	0
$603$	1.80385	0.0734584
$604$	0	0
$605$	−2.73205	−0.111074
$606$	0	0
$607$	−23.8038	−0.966168	−0.483084	−	0.875574i	$-0.660483\pi$
$607$	−23.8038	−0.966168	−0.483084	+	0.875574i	$0.660483\pi$
$608$	0	0
$609$	12.3923	0.502162
$610$	0	0
$611$	5.46410	0.221054
$612$	0	0
$613$	−9.85641	−0.398097	−0.199048	−	0.979990i	$-0.563785\pi$
$613$	−9.85641	−0.398097	−0.199048	+	0.979990i	$0.563785\pi$
$614$	0	0
$615$	−4.00000	−0.161296
$616$	0	0
$617$	6.33975	0.255229	0.127614	−	0.991824i	$-0.459268\pi$
$617$	6.33975	0.255229	0.127614	+	0.991824i	$0.459268\pi$
$618$	0	0
$619$	−2.58846	−0.104039	−0.0520194	−	0.998646i	$-0.516566\pi$
$619$	−2.58846	−0.104039	−0.0520194	+	0.998646i	$0.516566\pi$
$620$	0	0
$621$	2.00000	0.0802572
$622$	0	0
$623$	3.32051	0.133033
$624$	0	0
$625$	−31.2487	−1.24995
$626$	0	0
$627$	7.46410	0.298088
$628$	0	0
$629$	−6.53590	−0.260603
$630$	0	0
$631$	23.6603	0.941900	0.470950	−	0.882160i	$-0.343911\pi$
$631$	23.6603	0.941900	0.470950	+	0.882160i	$0.343911\pi$
$632$	0	0
$633$	−22.7321	−0.903518
$634$	0	0
$635$	−30.3923	−1.20608
$636$	0	0
$637$	3.00000	0.118864
$638$	0	0
$639$	6.92820	0.274075
$640$	0	0
$641$	−7.85641	−0.310309	−0.155155	−	0.987890i	$-0.549588\pi$
$641$	−7.85641	−0.310309	−0.155155	+	0.987890i	$0.549588\pi$
$642$	0	0
$643$	8.05256	0.317562	0.158781	−	0.987314i	$-0.449244\pi$
$643$	8.05256	0.317562	0.158781	+	0.987314i	$0.449244\pi$
$644$	0	0
$645$	11.4641	0.451399
$646$	0	0
$647$	−18.9282	−0.744144	−0.372072	−	0.928204i	$-0.621353\pi$
$647$	−18.9282	−0.744144	−0.372072	+	0.928204i	$0.621353\pi$
$648$	0	0
$649$	6.53590	0.256556
$650$	0	0
$651$	−4.39230	−0.172148
$652$	0	0
$653$	2.67949	0.104857	0.0524283	−	0.998625i	$-0.483304\pi$
$653$	2.67949	0.104857	0.0524283	+	0.998625i	$0.483304\pi$
$654$	0	0
$655$	−51.7128	−2.02059
$656$	0	0
$657$	−10.9282	−0.426350
$658$	0	0
$659$	38.9282	1.51643	0.758214	−	0.652006i	$-0.226072\pi$
$659$	38.9282	1.51643	0.758214	+	0.652006i	$0.226072\pi$
$660$	0	0
$661$	−11.4641	−0.445902	−0.222951	−	0.974830i	$-0.571569\pi$
$661$	−11.4641	−0.445902	−0.222951	+	0.974830i	$0.571569\pi$
$662$	0	0
$663$	−3.26795	−0.126917
$664$	0	0
$665$	−40.7846	−1.58156
$666$	0	0
$667$	12.3923	0.479832
$668$	0	0
$669$	−21.5167	−0.831882
$670$	0	0
$671$	−8.92820	−0.344669
$672$	0	0
$673$	−16.5359	−0.637412	−0.318706	−	0.947854i	$-0.603248\pi$
$673$	−16.5359	−0.637412	−0.318706	+	0.947854i	$0.603248\pi$
$674$	0	0
$675$	2.46410	0.0948433
$676$	0	0
$677$	46.3013	1.77950	0.889751	−	0.456446i	$-0.150878\pi$
$677$	46.3013	1.77950	0.889751	+	0.456446i	$0.150878\pi$
$678$	0	0
$679$	−9.85641	−0.378254
$680$	0	0
$681$	−10.3923	−0.398234
$682$	0	0
$683$	−16.7846	−0.642245	−0.321123	−	0.947038i	$-0.604060\pi$
$683$	−16.7846	−0.642245	−0.321123	+	0.947038i	$0.604060\pi$
$684$	0	0
$685$	−45.3205	−1.73161
$686$	0	0
$687$	−8.53590	−0.325665
$688$	0	0
$689$	−2.00000	−0.0761939
$690$	0	0
$691$	32.4449	1.23426	0.617130	−	0.786861i	$-0.288295\pi$
$691$	32.4449	1.23426	0.617130	+	0.786861i	$0.288295\pi$
$692$	0	0
$693$	2.00000	0.0759737
$694$	0	0
$695$	−55.1769	−2.09298
$696$	0	0
$697$	4.78461	0.181230
$698$	0	0
$699$	−6.19615	−0.234360
$700$	0	0
$701$	−44.0526	−1.66384	−0.831921	−	0.554894i	$-0.812759\pi$
$701$	−44.0526	−1.66384	−0.831921	+	0.554894i	$0.812759\pi$
$702$	0	0
$703$	−14.9282	−0.563028
$704$	0	0
$705$	14.9282	0.562229
$706$	0	0
$707$	−4.39230	−0.165190
$708$	0	0
$709$	−11.4641	−0.430543	−0.215272	−	0.976554i	$-0.569064\pi$
$709$	−11.4641	−0.430543	−0.215272	+	0.976554i	$0.569064\pi$
$710$	0	0
$711$	−13.6603	−0.512300
$712$	0	0
$713$	−4.39230	−0.164493
$714$	0	0
$715$	2.73205	0.102173
$716$	0	0
$717$	−1.60770	−0.0600405
$718$	0	0
$719$	8.00000	0.298350	0.149175	−	0.988811i	$-0.452338\pi$
$719$	8.00000	0.298350	0.149175	+	0.988811i	$0.452338\pi$
$720$	0	0
$721$	30.6410	1.14113
$722$	0	0
$723$	22.7846	0.847369
$724$	0	0
$725$	15.2679	0.567037
$726$	0	0
$727$	−4.67949	−0.173553	−0.0867764	−	0.996228i	$-0.527657\pi$
$727$	−4.67949	−0.173553	−0.0867764	+	0.996228i	$0.527657\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−13.7128	−0.507187
$732$	0	0
$733$	−9.07180	−0.335074	−0.167537	−	0.985866i	$-0.553581\pi$
$733$	−9.07180	−0.335074	−0.167537	+	0.985866i	$0.553581\pi$
$734$	0	0
$735$	8.19615	0.302320
$736$	0	0
$737$	1.80385	0.0664456
$738$	0	0
$739$	2.00000	0.0735712	0.0367856	−	0.999323i	$-0.488288\pi$
$739$	2.00000	0.0735712	0.0367856	+	0.999323i	$0.488288\pi$
$740$	0	0
$741$	−7.46410	−0.274201
$742$	0	0
$743$	−10.1436	−0.372132	−0.186066	−	0.982537i	$-0.559574\pi$
$743$	−10.1436	−0.372132	−0.186066	+	0.982537i	$0.559574\pi$
$744$	0	0
$745$	−21.8564	−0.800757
$746$	0	0
$747$	13.8564	0.506979
$748$	0	0
$749$	−24.7846	−0.905610
$750$	0	0
$751$	44.7846	1.63421	0.817107	−	0.576486i	$-0.195577\pi$
$751$	44.7846	1.63421	0.817107	+	0.576486i	$0.195577\pi$
$752$	0	0
$753$	30.9282	1.12709
$754$	0	0
$755$	−31.3205	−1.13987
$756$	0	0
$757$	−2.53590	−0.0921688	−0.0460844	−	0.998938i	$-0.514674\pi$
$757$	−2.53590	−0.0921688	−0.0460844	+	0.998938i	$0.514674\pi$
$758$	0	0
$759$	2.00000	0.0725954
$760$	0	0
$761$	−14.6410	−0.530736	−0.265368	−	0.964147i	$-0.585494\pi$
$761$	−14.6410	−0.530736	−0.265368	+	0.964147i	$0.585494\pi$
$762$	0	0
$763$	−16.0000	−0.579239
$764$	0	0
$765$	−8.92820	−0.322800
$766$	0	0
$767$	−6.53590	−0.235998
$768$	0	0
$769$	43.8564	1.58150	0.790751	−	0.612138i	$-0.209690\pi$
$769$	43.8564	1.58150	0.790751	+	0.612138i	$0.209690\pi$
$770$	0	0
$771$	−17.3205	−0.623783
$772$	0	0
$773$	−4.48334	−0.161255	−0.0806273	−	0.996744i	$-0.525692\pi$
$773$	−4.48334	−0.161255	−0.0806273	+	0.996744i	$0.525692\pi$
$774$	0	0
$775$	−5.41154	−0.194388
$776$	0	0
$777$	−4.00000	−0.143499
$778$	0	0
$779$	10.9282	0.391544
$780$	0	0
$781$	6.92820	0.247911
$782$	0	0
$783$	6.19615	0.221432
$784$	0	0
$785$	−52.7846	−1.88396
$786$	0	0
$787$	23.0718	0.822421	0.411210	−	0.911540i	$-0.365106\pi$
$787$	23.0718	0.822421	0.411210	+	0.911540i	$0.365106\pi$
$788$	0	0
$789$	−2.53590	−0.0902804
$790$	0	0
$791$	20.0000	0.711118
$792$	0	0
$793$	8.92820	0.317050
$794$	0	0
$795$	−5.46410	−0.193792
$796$	0	0
$797$	−29.6077	−1.04876	−0.524379	−	0.851485i	$-0.675703\pi$
$797$	−29.6077	−1.04876	−0.524379	+	0.851485i	$0.675703\pi$
$798$	0	0
$799$	−17.8564	−0.631714
$800$	0	0
$801$	1.66025	0.0586622
$802$	0	0
$803$	−10.9282	−0.385648
$804$	0	0
$805$	−10.9282	−0.385169
$806$	0	0
$807$	17.3205	0.609711
$808$	0	0
$809$	−36.8372	−1.29513	−0.647563	−	0.762012i	$-0.724212\pi$
$809$	−36.8372	−1.29513	−0.647563	+	0.762012i	$0.724212\pi$
$810$	0	0
$811$	−1.60770	−0.0564538	−0.0282269	−	0.999602i	$-0.508986\pi$
$811$	−1.60770	−0.0564538	−0.0282269	+	0.999602i	$0.508986\pi$
$812$	0	0
$813$	25.3205	0.888029
$814$	0	0
$815$	−56.6410	−1.98405
$816$	0	0
$817$	−31.3205	−1.09577
$818$	0	0
$819$	−2.00000	−0.0698857
$820$	0	0
$821$	−47.3205	−1.65150	−0.825749	−	0.564038i	$-0.809247\pi$
$821$	−47.3205	−1.65150	−0.825749	+	0.564038i	$0.809247\pi$
$822$	0	0
$823$	−32.3923	−1.12912	−0.564562	−	0.825390i	$-0.690955\pi$
$823$	−32.3923	−1.12912	−0.564562	+	0.825390i	$0.690955\pi$
$824$	0	0
$825$	2.46410	0.0857890
$826$	0	0
$827$	28.2487	0.982304	0.491152	−	0.871074i	$-0.336576\pi$
$827$	28.2487	0.982304	0.491152	+	0.871074i	$0.336576\pi$
$828$	0	0
$829$	−20.1436	−0.699616	−0.349808	−	0.936821i	$-0.613753\pi$
$829$	−20.1436	−0.699616	−0.349808	+	0.936821i	$0.613753\pi$
$830$	0	0
$831$	−10.0000	−0.346896
$832$	0	0
$833$	−9.80385	−0.339683
$834$	0	0
$835$	48.7846	1.68826
$836$	0	0
$837$	−2.19615	−0.0759101
$838$	0	0
$839$	31.7128	1.09485	0.547424	−	0.836855i	$-0.315609\pi$
$839$	31.7128	1.09485	0.547424	+	0.836855i	$0.315609\pi$
$840$	0	0
$841$	9.39230	0.323873
$842$	0	0
$843$	−11.3205	−0.389899
$844$	0	0
$845$	−2.73205	−0.0939854
$846$	0	0
$847$	2.00000	0.0687208
$848$	0	0
$849$	−20.1962	−0.693130
$850$	0	0
$851$	−4.00000	−0.137118
$852$	0	0
$853$	−39.8564	−1.36466	−0.682329	−	0.731046i	$-0.739033\pi$
$853$	−39.8564	−1.36466	−0.682329	+	0.731046i	$0.739033\pi$
$854$	0	0
$855$	−20.3923	−0.697402
$856$	0	0
$857$	42.1962	1.44139	0.720697	−	0.693251i	$-0.243822\pi$
$857$	42.1962	1.44139	0.720697	+	0.693251i	$0.243822\pi$
$858$	0	0
$859$	53.9615	1.84114	0.920572	−	0.390574i	$-0.127723\pi$
$859$	53.9615	1.84114	0.920572	+	0.390574i	$0.127723\pi$
$860$	0	0
$861$	2.92820	0.0997929
$862$	0	0
$863$	23.3205	0.793839	0.396920	−	0.917853i	$-0.370079\pi$
$863$	23.3205	0.793839	0.396920	+	0.917853i	$0.370079\pi$
$864$	0	0
$865$	14.0000	0.476014
$866$	0	0
$867$	−6.32051	−0.214656
$868$	0	0
$869$	−13.6603	−0.463392
$870$	0	0
$871$	−1.80385	−0.0611210
$872$	0	0
$873$	−4.92820	−0.166794
$874$	0	0
$875$	13.8564	0.468432
$876$	0	0
$877$	−11.8564	−0.400362	−0.200181	−	0.979759i	$-0.564153\pi$
$877$	−11.8564	−0.400362	−0.200181	+	0.979759i	$0.564153\pi$
$878$	0	0
$879$	26.9282	0.908266
$880$	0	0
$881$	40.6410	1.36923	0.684615	−	0.728905i	$-0.259970\pi$
$881$	40.6410	1.36923	0.684615	+	0.728905i	$0.259970\pi$
$882$	0	0
$883$	−46.2487	−1.55639	−0.778197	−	0.628021i	$-0.783865\pi$
$883$	−46.2487	−1.55639	−0.778197	+	0.628021i	$0.783865\pi$
$884$	0	0
$885$	−17.8564	−0.600237
$886$	0	0
$887$	−10.6410	−0.357290	−0.178645	−	0.983914i	$-0.557171\pi$
$887$	−10.6410	−0.357290	−0.178645	+	0.983914i	$0.557171\pi$
$888$	0	0
$889$	22.2487	0.746198
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	−40.7846	−1.36480
$894$	0	0
$895$	−54.2487	−1.81333
$896$	0	0
$897$	−2.00000	−0.0667781
$898$	0	0
$899$	−13.6077	−0.453842
$900$	0	0
$901$	6.53590	0.217742
$902$	0	0
$903$	−8.39230	−0.279278
$904$	0	0
$905$	33.8564	1.12543
$906$	0	0
$907$	−10.9282	−0.362865	−0.181433	−	0.983403i	$-0.558073\pi$
$907$	−10.9282	−0.362865	−0.181433	+	0.983403i	$0.558073\pi$
$908$	0	0
$909$	−2.19615	−0.0728418
$910$	0	0
$911$	−2.14359	−0.0710204	−0.0355102	−	0.999369i	$-0.511306\pi$
$911$	−2.14359	−0.0710204	−0.0355102	+	0.999369i	$0.511306\pi$
$912$	0	0
$913$	13.8564	0.458580
$914$	0	0
$915$	24.3923	0.806385
$916$	0	0
$917$	37.8564	1.25013
$918$	0	0
$919$	−0.875644	−0.0288848	−0.0144424	−	0.999896i	$-0.504597\pi$
$919$	−0.875644	−0.0288848	−0.0144424	+	0.999896i	$0.504597\pi$
$920$	0	0
$921$	27.4641	0.904973
$922$	0	0
$923$	−6.92820	−0.228045
$924$	0	0
$925$	−4.92820	−0.162038
$926$	0	0
$927$	15.3205	0.503192
$928$	0	0
$929$	−51.1244	−1.67734	−0.838668	−	0.544643i	$-0.816665\pi$
$929$	−51.1244	−1.67734	−0.838668	+	0.544643i	$0.816665\pi$
$930$	0	0
$931$	−22.3923	−0.733878
$932$	0	0
$933$	16.9282	0.554204
$934$	0	0
$935$	−8.92820	−0.291983
$936$	0	0
$937$	33.0333	1.07915	0.539576	−	0.841937i	$-0.318585\pi$
$937$	33.0333	1.07915	0.539576	+	0.841937i	$0.318585\pi$
$938$	0	0
$939$	−2.53590	−0.0827559
$940$	0	0
$941$	−16.3923	−0.534374	−0.267187	−	0.963645i	$-0.586094\pi$
$941$	−16.3923	−0.534374	−0.267187	+	0.963645i	$0.586094\pi$
$942$	0	0
$943$	2.92820	0.0953554
$944$	0	0
$945$	−5.46410	−0.177747
$946$	0	0
$947$	−0.679492	−0.0220805	−0.0110403	−	0.999939i	$-0.503514\pi$
$947$	−0.679492	−0.0220805	−0.0110403	+	0.999939i	$0.503514\pi$
$948$	0	0
$949$	10.9282	0.354744
$950$	0	0
$951$	−7.12436	−0.231023
$952$	0	0
$953$	7.26795	0.235432	0.117716	−	0.993047i	$-0.462443\pi$
$953$	7.26795	0.235432	0.117716	+	0.993047i	$0.462443\pi$
$954$	0	0
$955$	−13.8564	−0.448383
$956$	0	0
$957$	6.19615	0.200293
$958$	0	0
$959$	33.1769	1.07134
$960$	0	0
$961$	−26.1769	−0.844417
$962$	0	0
$963$	−12.3923	−0.399336
$964$	0	0
$965$	−18.9282	−0.609320
$966$	0	0
$967$	−27.0718	−0.870570	−0.435285	−	0.900293i	$-0.643352\pi$
$967$	−27.0718	−0.870570	−0.435285	+	0.900293i	$0.643352\pi$
$968$	0	0
$969$	24.3923	0.783594
$970$	0	0
$971$	−3.85641	−0.123758	−0.0618790	−	0.998084i	$-0.519709\pi$
$971$	−3.85641	−0.123758	−0.0618790	+	0.998084i	$0.519709\pi$
$972$	0	0
$973$	40.3923	1.29492
$974$	0	0
$975$	−2.46410	−0.0789144
$976$	0	0
$977$	−17.6603	−0.565002	−0.282501	−	0.959267i	$-0.591164\pi$
$977$	−17.6603	−0.565002	−0.282501	+	0.959267i	$0.591164\pi$
$978$	0	0
$979$	1.66025	0.0530619
$980$	0	0
$981$	−8.00000	−0.255420
$982$	0	0
$983$	33.0718	1.05483	0.527413	−	0.849609i	$-0.323162\pi$
$983$	33.0718	1.05483	0.527413	+	0.849609i	$0.323162\pi$
$984$	0	0
$985$	−21.8564	−0.696403
$986$	0	0
$987$	−10.9282	−0.347849
$988$	0	0
$989$	−8.39230	−0.266860
$990$	0	0
$991$	5.17691	0.164450	0.0822251	−	0.996614i	$-0.473797\pi$
$991$	5.17691	0.164450	0.0822251	+	0.996614i	$0.473797\pi$
$992$	0	0
$993$	21.5167	0.682811
$994$	0	0
$995$	47.7128	1.51260
$996$	0	0
$997$	41.3205	1.30863	0.654317	−	0.756221i	$-0.272956\pi$
$997$	41.3205	1.30863	0.654317	+	0.756221i	$0.272956\pi$
$998$	0	0
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6864.2.a.bk.1.1		2
4.3	odd	2		429.2.a.d.1.2	✓	2
12.11	even	2		1287.2.a.f.1.1		2
44.43	even	2		4719.2.a.n.1.1		2
52.51	odd	2		5577.2.a.h.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.a.d.1.2	✓	2	4.3	odd	2
1287.2.a.f.1.1		2	12.11	even	2
4719.2.a.n.1.1		2	44.43	even	2
5577.2.a.h.1.1		2	52.51	odd	2
6864.2.a.bk.1.1		2	1.1	even	1	trivial

Overview	Random
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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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -2.73205 q^{5} +2.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -2.73205 q^{5} +2.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} -1.00000 q^{13} -2.73205 q^{15} +3.26795 q^{17} +7.46410 q^{19} +2.00000 q^{21} +2.00000 q^{23} +2.46410 q^{25} +1.00000 q^{27} +6.19615 q^{29} -2.19615 q^{31} +1.00000 q^{33} -5.46410 q^{35} -2.00000 q^{37} -1.00000 q^{39} +1.46410 q^{41} -4.19615 q^{43} -2.73205 q^{45} -5.46410 q^{47} -3.00000 q^{49} +3.26795 q^{51} +2.00000 q^{53} -2.73205 q^{55} +7.46410 q^{57} +6.53590 q^{59} -8.92820 q^{61} +2.00000 q^{63} +2.73205 q^{65} +1.80385 q^{67} +2.00000 q^{69} +6.92820 q^{71} -10.9282 q^{73} +2.46410 q^{75} +2.00000 q^{77} -13.6603 q^{79} +1.00000 q^{81} +13.8564 q^{83} -8.92820 q^{85} +6.19615 q^{87} +1.66025 q^{89} -2.00000 q^{91} -2.19615 q^{93} -20.3923 q^{95} -4.92820 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{5} + 4 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 2 * q^5 + 4 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} - 2 q^{5} + 4 q^{7} + 2 q^{9} + 2 q^{11} - 2 q^{13} - 2 q^{15} + 10 q^{17} + 8 q^{19} + 4 q^{21} + 4 q^{23} - 2 q^{25} + 2 q^{27} + 2 q^{29} + 6 q^{31} + 2 q^{33} - 4 q^{35} - 4 q^{37} - 2 q^{39} - 4 q^{41} + 2 q^{43} - 2 q^{45} - 4 q^{47} - 6 q^{49} + 10 q^{51} + 4 q^{53} - 2 q^{55} + 8 q^{57} + 20 q^{59} - 4 q^{61} + 4 q^{63} + 2 q^{65} + 14 q^{67} + 4 q^{69} - 8 q^{73} - 2 q^{75} + 4 q^{77} - 10 q^{79} + 2 q^{81} - 4 q^{85} + 2 q^{87} - 14 q^{89} - 4 q^{91} + 6 q^{93} - 20 q^{95} + 4 q^{97} + 2 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^5 + 4 * q^7 + 2 * q^9 + 2 * q^11 - 2 * q^13 - 2 * q^15 + 10 * q^17 + 8 * q^19 + 4 * q^21 + 4 * q^23 - 2 * q^25 + 2 * q^27 + 2 * q^29 + 6 * q^31 + 2 * q^33 - 4 * q^35 - 4 * q^37 - 2 * q^39 - 4 * q^41 + 2 * q^43 - 2 * q^45 - 4 * q^47 - 6 * q^49 + 10 * q^51 + 4 * q^53 - 2 * q^55 + 8 * q^57 + 20 * q^59 - 4 * q^61 + 4 * q^63 + 2 * q^65 + 14 * q^67 + 4 * q^69 - 8 * q^73 - 2 * q^75 + 4 * q^77 - 10 * q^79 + 2 * q^81 - 4 * q^85 + 2 * q^87 - 14 * q^89 - 4 * q^91 + 6 * q^93 - 20 * q^95 + 4 * q^97 + 2 * q^99

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