LMFDB - Embedded newform 6864.2.a.ce.1.5

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:	$5$
Coefficient field:	5.5.46437524.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 20x^{3} + 8x^{2} + 70x + 28 $ x^5 - x^4 - 20x^3 + 8x^2 + 70*x + 28
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 3432)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$4.26299$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +4.26299 q^{5} -0.932447 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +4.26299 q^{5} -0.932447 q^{7} +1.00000 q^{9} -1.00000 q^{11} +1.00000 q^{13} -4.26299 q^{15} +7.19544 q^{17} -6.90746 q^{19} +0.932447 q^{21} +2.93245 q^{23} +13.1731 q^{25} -1.00000 q^{27} -9.88514 q^{29} +8.23801 q^{31} +1.00000 q^{33} -3.97502 q^{35} -1.86489 q^{37} -1.00000 q^{39} +1.06755 q^{41} +5.30556 q^{43} +4.26299 q^{45} +11.1056 q^{47} -6.13054 q^{49} -7.19544 q^{51} +9.24066 q^{53} -4.26299 q^{55} +6.90746 q^{57} -9.26565 q^{59} -14.5010 q^{61} -0.932447 q^{63} +4.26299 q^{65} +2.26299 q^{67} -2.93245 q^{69} +7.23536 q^{71} -6.17311 q^{73} -13.1731 q^{75} +0.932447 q^{77} +2.62118 q^{79} +1.00000 q^{81} +8.52599 q^{83} +30.6741 q^{85} +9.88514 q^{87} +14.8991 q^{89} -0.932447 q^{91} -8.23801 q^{93} -29.4465 q^{95} +6.71468 q^{97} -1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{3} + q^{5} + 3 q^{7} + 5 q^{9}+O(q^{10}) $ 5 * q - 5 * q^3 + q^5 + 3 * q^7 + 5 * q^9	$ 5 q - 5 q^{3} + q^{5} + 3 q^{7} + 5 q^{9} - 5 q^{11} + 5 q^{13} - q^{15} + 8 q^{17} - 10 q^{19} - 3 q^{21} + 7 q^{23} + 16 q^{25} - 5 q^{27} - 3 q^{29} + 4 q^{31} + 5 q^{33} - 3 q^{35} + 6 q^{37} - 5 q^{39} + 13 q^{41} - 3 q^{43} + q^{45} - 2 q^{47} + 10 q^{49} - 8 q^{51} + 4 q^{53} - q^{55} + 10 q^{57} - 21 q^{59} - 15 q^{61} + 3 q^{63} + q^{65} - 9 q^{67} - 7 q^{69} + 4 q^{71} + 19 q^{73} - 16 q^{75} - 3 q^{77} - 8 q^{79} + 5 q^{81} + 2 q^{83} + 46 q^{85} + 3 q^{87} + 12 q^{89} + 3 q^{91} - 4 q^{93} + 32 q^{97} - 5 q^{99}+O(q^{100}) $ 5 * q - 5 * q^3 + q^5 + 3 * q^7 + 5 * q^9 - 5 * q^11 + 5 * q^13 - q^15 + 8 * q^17 - 10 * q^19 - 3 * q^21 + 7 * q^23 + 16 * q^25 - 5 * q^27 - 3 * q^29 + 4 * q^31 + 5 * q^33 - 3 * q^35 + 6 * q^37 - 5 * q^39 + 13 * q^41 - 3 * q^43 + q^45 - 2 * q^47 + 10 * q^49 - 8 * q^51 + 4 * q^53 - q^55 + 10 * q^57 - 21 * q^59 - 15 * q^61 + 3 * q^63 + q^65 - 9 * q^67 - 7 * q^69 + 4 * q^71 + 19 * q^73 - 16 * q^75 - 3 * q^77 - 8 * q^79 + 5 * q^81 + 2 * q^83 + 46 * q^85 + 3 * q^87 + 12 * q^89 + 3 * q^91 - 4 * q^93 + 32 * q^97 - 5 * 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
$5$	4.26299	1.90647	0.953234	−	0.302232i	$-0.0977319\pi$
$5$	4.26299	1.90647	0.953234	+	0.302232i	$0.0977319\pi$
$6$	0	0
$7$	−0.932447	−0.352432	−0.176216	−	0.984352i	$-0.556386\pi$
$7$	−0.932447	−0.352432	−0.176216	+	0.984352i	$0.556386\pi$
$8$	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
$15$	−4.26299	−1.10070
$16$	0	0
$17$	7.19544	1.74515	0.872575	−	0.488480i	$-0.162448\pi$
$17$	7.19544	1.74515	0.872575	+	0.488480i	$0.162448\pi$
$18$	0	0
$19$	−6.90746	−1.58468	−0.792341	−	0.610079i	$-0.791138\pi$
$19$	−6.90746	−1.58468	−0.792341	+	0.610079i	$0.791138\pi$
$20$	0	0
$21$	0.932447	0.203477
$22$	0	0
$23$	2.93245	0.611458	0.305729	−	0.952119i	$-0.401100\pi$
$23$	2.93245	0.611458	0.305729	+	0.952119i	$0.401100\pi$
$24$	0	0
$25$	13.1731	2.63462
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−9.88514	−1.83562	−0.917812	−	0.397016i	$-0.870046\pi$
$29$	−9.88514	−1.83562	−0.917812	+	0.397016i	$0.870046\pi$
$30$	0	0
$31$	8.23801	1.47959	0.739795	−	0.672832i	$-0.234922\pi$
$31$	8.23801	1.47959	0.739795	+	0.672832i	$0.234922\pi$
$32$	0	0
$33$	1.00000	0.174078
$34$	0	0
$35$	−3.97502	−0.671901
$36$	0	0
$37$	−1.86489	−0.306587	−0.153293	−	0.988181i	$-0.548988\pi$
$37$	−1.86489	−0.306587	−0.153293	+	0.988181i	$0.548988\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	1.06755	0.166724	0.0833619	−	0.996519i	$-0.473434\pi$
$41$	1.06755	0.166724	0.0833619	+	0.996519i	$0.473434\pi$
$42$	0	0
$43$	5.30556	0.809091	0.404545	−	0.914518i	$-0.367430\pi$
$43$	5.30556	0.809091	0.404545	+	0.914518i	$0.367430\pi$
$44$	0	0
$45$	4.26299	0.635490
$46$	0	0
$47$	11.1056	1.61991	0.809956	−	0.586490i	$-0.199491\pi$
$47$	11.1056	1.61991	0.809956	+	0.586490i	$0.199491\pi$
$48$	0	0
$49$	−6.13054	−0.875792
$50$	0	0
$51$	−7.19544	−1.00756
$52$	0	0
$53$	9.24066	1.26930	0.634651	−	0.772799i	$-0.281144\pi$
$53$	9.24066	1.26930	0.634651	+	0.772799i	$0.281144\pi$
$54$	0	0
$55$	−4.26299	−0.574822
$56$	0	0
$57$	6.90746	0.914916
$58$	0	0
$59$	−9.26565	−1.20628	−0.603142	−	0.797634i	$-0.706085\pi$
$59$	−9.26565	−1.20628	−0.603142	+	0.797634i	$0.706085\pi$
$60$	0	0
$61$	−14.5010	−1.85666	−0.928332	−	0.371753i	$-0.878757\pi$
$61$	−14.5010	−1.85666	−0.928332	+	0.371753i	$0.878757\pi$
$62$	0	0
$63$	−0.932447	−0.117477
$64$	0	0
$65$	4.26299	0.528759
$66$	0	0
$67$	2.26299	0.276469	0.138234	−	0.990400i	$-0.455857\pi$
$67$	2.26299	0.276469	0.138234	+	0.990400i	$0.455857\pi$
$68$	0	0
$69$	−2.93245	−0.353025
$70$	0	0
$71$	7.23536	0.858679	0.429339	−	0.903143i	$-0.358746\pi$
$71$	7.23536	0.858679	0.429339	+	0.903143i	$0.358746\pi$
$72$	0	0
$73$	−6.17311	−0.722508	−0.361254	−	0.932467i	$-0.617651\pi$
$73$	−6.17311	−0.722508	−0.361254	+	0.932467i	$0.617651\pi$
$74$	0	0
$75$	−13.1731	−1.52110
$76$	0	0
$77$	0.932447	0.106262
$78$	0	0
$79$	2.62118	0.294905	0.147453	−	0.989069i	$-0.452893\pi$
$79$	2.62118	0.294905	0.147453	+	0.989069i	$0.452893\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	8.52599	0.935849	0.467924	−	0.883769i	$-0.345002\pi$
$83$	8.52599	0.935849	0.467924	+	0.883769i	$0.345002\pi$
$84$	0	0
$85$	30.6741	3.32707
$86$	0	0
$87$	9.88514	1.05980
$88$	0	0
$89$	14.8991	1.57930	0.789651	−	0.613556i	$-0.210262\pi$
$89$	14.8991	1.57930	0.789651	+	0.613556i	$0.210262\pi$
$90$	0	0
$91$	−0.932447	−0.0977470
$92$	0	0
$93$	−8.23801	−0.854242
$94$	0	0
$95$	−29.4465	−3.02114
$96$	0	0
$97$	6.71468	0.681772	0.340886	−	0.940105i	$-0.389273\pi$
$97$	6.71468	0.681772	0.340886	+	0.940105i	$0.389273\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	0	0
$101$	12.5712	1.25088	0.625441	−	0.780271i	$-0.284919\pi$
$101$	12.5712	1.25088	0.625441	+	0.780271i	$0.284919\pi$
$102$	0	0
$103$	−3.26034	−0.321251	−0.160625	−	0.987015i	$-0.551351\pi$
$103$	−3.26034	−0.321251	−0.160625	+	0.987015i	$0.551351\pi$
$104$	0	0
$105$	3.97502	0.387922
$106$	0	0
$107$	12.3659	1.19546	0.597728	−	0.801699i	$-0.296070\pi$
$107$	12.3659	1.19546	0.597728	+	0.801699i	$0.296070\pi$
$108$	0	0
$109$	−12.1481	−1.16358	−0.581790	−	0.813339i	$-0.697647\pi$
$109$	−12.1481	−1.16358	−0.581790	+	0.813339i	$0.697647\pi$
$110$	0	0
$111$	1.86489	0.177008
$112$	0	0
$113$	0.734353	0.0690821	0.0345411	−	0.999403i	$-0.489003\pi$
$113$	0.734353	0.0690821	0.0345411	+	0.999403i	$0.489003\pi$
$114$	0	0
$115$	12.5010	1.16572
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	−6.70937	−0.615047
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−1.06755	−0.0962580
$124$	0	0
$125$	34.8419	3.11636
$126$	0	0
$127$	13.0603	1.15892	0.579459	−	0.815002i	$-0.303264\pi$
$127$	13.0603	1.15892	0.579459	+	0.815002i	$0.303264\pi$
$128$	0	0
$129$	−5.30556	−0.467129
$130$	0	0
$131$	11.0770	0.967798	0.483899	−	0.875124i	$-0.339220\pi$
$131$	11.0770	0.967798	0.483899	+	0.875124i	$0.339220\pi$
$132$	0	0
$133$	6.44085	0.558492
$134$	0	0
$135$	−4.26299	−0.366900
$136$	0	0
$137$	−6.10291	−0.521406	−0.260703	−	0.965419i	$-0.583954\pi$
$137$	−6.10291	−0.521406	−0.260703	+	0.965419i	$0.583954\pi$
$138$	0	0
$139$	−15.0157	−1.27361	−0.636807	−	0.771024i	$-0.719745\pi$
$139$	−15.0157	−1.27361	−0.636807	+	0.771024i	$0.719745\pi$
$140$	0	0
$141$	−11.1056	−0.935257
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	−42.1403	−3.49956
$146$	0	0
$147$	6.13054	0.505639
$148$	0	0
$149$	−12.9943	−1.06453	−0.532267	−	0.846577i	$-0.678660\pi$
$149$	−12.9943	−1.06453	−0.532267	+	0.846577i	$0.678660\pi$
$150$	0	0
$151$	0.192787	0.0156888	0.00784439	−	0.999969i	$-0.497503\pi$
$151$	0.192787	0.0156888	0.00784439	+	0.999969i	$0.497503\pi$
$152$	0	0
$153$	7.19544	0.581717
$154$	0	0
$155$	35.1186	2.82079
$156$	0	0
$157$	8.24637	0.658132	0.329066	−	0.944307i	$-0.393266\pi$
$157$	8.24637	0.658132	0.329066	+	0.944307i	$0.393266\pi$
$158$	0	0
$159$	−9.24066	−0.732832
$160$	0	0
$161$	−2.73435	−0.215497
$162$	0	0
$163$	−3.15743	−0.247309	−0.123655	−	0.992325i	$-0.539462\pi$
$163$	−3.15743	−0.247309	−0.123655	+	0.992325i	$0.539462\pi$
$164$	0	0
$165$	4.26299	0.331874
$166$	0	0
$167$	17.0753	1.32132	0.660662	−	0.750684i	$-0.270276\pi$
$167$	17.0753	1.32132	0.660662	+	0.750684i	$0.270276\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−6.90746	−0.528227
$172$	0	0
$173$	−15.3056	−1.16366	−0.581830	−	0.813310i	$-0.697663\pi$
$173$	−15.3056	−1.16366	−0.581830	+	0.813310i	$0.697663\pi$
$174$	0	0
$175$	−12.2832	−0.928525
$176$	0	0
$177$	9.26565	0.696449
$178$	0	0
$179$	2.24637	0.167902	0.0839509	−	0.996470i	$-0.473246\pi$
$179$	2.24637	0.167902	0.0839509	+	0.996470i	$0.473246\pi$
$180$	0	0
$181$	18.0613	1.34249	0.671243	−	0.741237i	$-0.265761\pi$
$181$	18.0613	1.34249	0.671243	+	0.741237i	$0.265761\pi$
$182$	0	0
$183$	14.5010	1.07195
$184$	0	0
$185$	−7.95003	−0.584498
$186$	0	0
$187$	−7.19544	−0.526183
$188$	0	0
$189$	0.932447	0.0678256
$190$	0	0
$191$	−4.41586	−0.319521	−0.159760	−	0.987156i	$-0.551072\pi$
$191$	−4.41586	−0.319521	−0.159760	+	0.987156i	$0.551072\pi$
$192$	0	0
$193$	8.38148	0.603312	0.301656	−	0.953417i	$-0.402461\pi$
$193$	8.38148	0.603312	0.301656	+	0.953417i	$0.402461\pi$
$194$	0	0
$195$	−4.26299	−0.305279
$196$	0	0
$197$	−8.90384	−0.634373	−0.317186	−	0.948363i	$-0.602738\pi$
$197$	−8.90384	−0.634373	−0.317186	+	0.948363i	$0.602738\pi$
$198$	0	0
$199$	−9.79163	−0.694110	−0.347055	−	0.937845i	$-0.612818\pi$
$199$	−9.79163	−0.694110	−0.347055	+	0.937845i	$0.612818\pi$
$200$	0	0
$201$	−2.26299	−0.159619
$202$	0	0
$203$	9.21737	0.646932
$204$	0	0
$205$	4.55097	0.317853
$206$	0	0
$207$	2.93245	0.203819
$208$	0	0
$209$	6.90746	0.477799
$210$	0	0
$211$	−16.5659	−1.14044	−0.570222	−	0.821491i	$-0.693143\pi$
$211$	−16.5659	−1.14044	−0.570222	+	0.821491i	$0.693143\pi$
$212$	0	0
$213$	−7.23536	−0.495758
$214$	0	0
$215$	22.6176	1.54251
$216$	0	0
$217$	−7.68151	−0.521455
$218$	0	0
$219$	6.17311	0.417140
$220$	0	0
$221$	7.19544	0.484018
$222$	0	0
$223$	−26.4491	−1.77116	−0.885582	−	0.464482i	$-0.846240\pi$
$223$	−26.4491	−1.77116	−0.885582	+	0.464482i	$0.846240\pi$
$224$	0	0
$225$	13.1731	0.878207
$226$	0	0
$227$	1.89235	0.125600	0.0627999	−	0.998026i	$-0.479997\pi$
$227$	1.89235	0.125600	0.0627999	+	0.998026i	$0.479997\pi$
$228$	0	0
$229$	−15.5161	−1.02533	−0.512667	−	0.858588i	$-0.671342\pi$
$229$	−15.5161	−1.02533	−0.512667	+	0.858588i	$0.671342\pi$
$230$	0	0
$231$	−0.932447	−0.0613505
$232$	0	0
$233$	14.4308	0.945393	0.472696	−	0.881225i	$-0.343281\pi$
$233$	14.4308	0.945393	0.472696	+	0.881225i	$0.343281\pi$
$234$	0	0
$235$	47.3429	3.08831
$236$	0	0
$237$	−2.62118	−0.170264
$238$	0	0
$239$	−8.61227	−0.557082	−0.278541	−	0.960424i	$-0.589851\pi$
$239$	−8.61227	−0.557082	−0.278541	+	0.960424i	$0.589851\pi$
$240$	0	0
$241$	−1.24066	−0.0799182	−0.0399591	−	0.999201i	$-0.512723\pi$
$241$	−1.24066	−0.0799182	−0.0399591	+	0.999201i	$0.512723\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−26.1345	−1.66967
$246$	0	0
$247$	−6.90746	−0.439511
$248$	0	0
$249$	−8.52599	−0.540312
$250$	0	0
$251$	20.2558	1.27853	0.639267	−	0.768985i	$-0.279238\pi$
$251$	20.2558	1.27853	0.639267	+	0.768985i	$0.279238\pi$
$252$	0	0
$253$	−2.93245	−0.184361
$254$	0	0
$255$	−30.6741	−1.92089
$256$	0	0
$257$	9.92674	0.619213	0.309606	−	0.950865i	$-0.399803\pi$
$257$	9.92674	0.619213	0.309606	+	0.950865i	$0.399803\pi$
$258$	0	0
$259$	1.73892	0.108051
$260$	0	0
$261$	−9.88514	−0.611875
$262$	0	0
$263$	−17.6352	−1.08743	−0.543715	−	0.839270i	$-0.682983\pi$
$263$	−17.6352	−1.08743	−0.543715	+	0.839270i	$0.682983\pi$
$264$	0	0
$265$	39.3929	2.41989
$266$	0	0
$267$	−14.8991	−0.911810
$268$	0	0
$269$	17.0204	1.03775	0.518877	−	0.854849i	$-0.326350\pi$
$269$	17.0204	1.03775	0.518877	+	0.854849i	$0.326350\pi$
$270$	0	0
$271$	−16.6688	−1.01256	−0.506279	−	0.862370i	$-0.668979\pi$
$271$	−16.6688	−1.01256	−0.506279	+	0.862370i	$0.668979\pi$
$272$	0	0
$273$	0.932447	0.0564343
$274$	0	0
$275$	−13.1731	−0.794369
$276$	0	0
$277$	15.0753	0.905785	0.452893	−	0.891565i	$-0.350392\pi$
$277$	15.0753	0.905785	0.452893	+	0.891565i	$0.350392\pi$
$278$	0	0
$279$	8.23801	0.493197
$280$	0	0
$281$	−8.87717	−0.529568	−0.264784	−	0.964308i	$-0.585301\pi$
$281$	−8.87717	−0.529568	−0.264784	+	0.964308i	$0.585301\pi$
$282$	0	0
$283$	20.6813	1.22938	0.614689	−	0.788770i	$-0.289282\pi$
$283$	20.6813	1.22938	0.614689	+	0.788770i	$0.289282\pi$
$284$	0	0
$285$	29.4465	1.74426
$286$	0	0
$287$	−0.995437	−0.0587588
$288$	0	0
$289$	34.7744	2.04555
$290$	0	0
$291$	−6.71468	−0.393621
$292$	0	0
$293$	14.6688	0.856961	0.428480	−	0.903551i	$-0.359049\pi$
$293$	14.6688	0.856961	0.428480	+	0.903551i	$0.359049\pi$
$294$	0	0
$295$	−39.4994	−2.29974
$296$	0	0
$297$	1.00000	0.0580259
$298$	0	0
$299$	2.93245	0.169588
$300$	0	0
$301$	−4.94716	−0.285149
$302$	0	0
$303$	−12.5712	−0.722197
$304$	0	0
$305$	−61.8177	−3.53967
$306$	0	0
$307$	19.6892	1.12372	0.561862	−	0.827231i	$-0.310085\pi$
$307$	19.6892	1.12372	0.561862	+	0.827231i	$0.310085\pi$
$308$	0	0
$309$	3.26034	0.185474
$310$	0	0
$311$	−21.7761	−1.23481	−0.617403	−	0.786647i	$-0.711815\pi$
$311$	−21.7761	−1.23481	−0.617403	+	0.786647i	$0.711815\pi$
$312$	0	0
$313$	−18.7842	−1.06175	−0.530874	−	0.847451i	$-0.678136\pi$
$313$	−18.7842	−1.06175	−0.530874	+	0.847451i	$0.678136\pi$
$314$	0	0
$315$	−3.97502	−0.223967
$316$	0	0
$317$	−17.3739	−0.975813	−0.487907	−	0.872896i	$-0.662239\pi$
$317$	−17.3739	−0.975813	−0.487907	+	0.872896i	$0.662239\pi$
$318$	0	0
$319$	9.88514	0.553461
$320$	0	0
$321$	−12.3659	−0.690197
$322$	0	0
$323$	−49.7023	−2.76551
$324$	0	0
$325$	13.1731	0.730713
$326$	0	0
$327$	12.1481	0.671793
$328$	0	0
$329$	−10.3553	−0.570909
$330$	0	0
$331$	6.97767	0.383527	0.191764	−	0.981441i	$-0.438579\pi$
$331$	6.97767	0.383527	0.191764	+	0.981441i	$0.438579\pi$
$332$	0	0
$333$	−1.86489	−0.102196
$334$	0	0
$335$	9.64712	0.527079
$336$	0	0
$337$	12.0053	0.653971	0.326985	−	0.945029i	$-0.393967\pi$
$337$	12.0053	0.653971	0.326985	+	0.945029i	$0.393967\pi$
$338$	0	0
$339$	−0.734353	−0.0398846
$340$	0	0
$341$	−8.23801	−0.446113
$342$	0	0
$343$	12.2435	0.661089
$344$	0	0
$345$	−12.5010	−0.673031
$346$	0	0
$347$	−9.67982	−0.519640	−0.259820	−	0.965657i	$-0.583663\pi$
$347$	−9.67982	−0.519640	−0.259820	+	0.965657i	$0.583663\pi$
$348$	0	0
$349$	6.14041	0.328689	0.164344	−	0.986403i	$-0.447449\pi$
$349$	6.14041	0.328689	0.164344	+	0.986403i	$0.447449\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	17.7489	0.944678	0.472339	−	0.881417i	$-0.343410\pi$
$353$	17.7489	0.944678	0.472339	+	0.881417i	$0.343410\pi$
$354$	0	0
$355$	30.8443	1.63704
$356$	0	0
$357$	6.70937	0.355098
$358$	0	0
$359$	11.8305	0.624390	0.312195	−	0.950018i	$-0.398936\pi$
$359$	11.8305	0.624390	0.312195	+	0.950018i	$0.398936\pi$
$360$	0	0
$361$	28.7131	1.51121
$362$	0	0
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	−26.3159	−1.37744
$366$	0	0
$367$	−11.9833	−0.625522	−0.312761	−	0.949832i	$-0.601254\pi$
$367$	−11.9833	−0.625522	−0.312761	+	0.949832i	$0.601254\pi$
$368$	0	0
$369$	1.06755	0.0555746
$370$	0	0
$371$	−8.61643	−0.447343
$372$	0	0
$373$	24.3176	1.25912	0.629559	−	0.776952i	$-0.283235\pi$
$373$	24.3176	1.25912	0.629559	+	0.776952i	$0.283235\pi$
$374$	0	0
$375$	−34.8419	−1.79923
$376$	0	0
$377$	−9.88514	−0.509110
$378$	0	0
$379$	−19.8327	−1.01874	−0.509369	−	0.860548i	$-0.670121\pi$
$379$	−19.8327	−1.01874	−0.509369	+	0.860548i	$0.670121\pi$
$380$	0	0
$381$	−13.0603	−0.669101
$382$	0	0
$383$	31.3929	1.60410	0.802051	−	0.597256i	$-0.203742\pi$
$383$	31.3929	1.60410	0.802051	+	0.597256i	$0.203742\pi$
$384$	0	0
$385$	3.97502	0.202586
$386$	0	0
$387$	5.30556	0.269697
$388$	0	0
$389$	−32.3778	−1.64162	−0.820809	−	0.571202i	$-0.806477\pi$
$389$	−32.3778	−1.64162	−0.820809	+	0.571202i	$0.806477\pi$
$390$	0	0
$391$	21.1003	1.06709
$392$	0	0
$393$	−11.0770	−0.558759
$394$	0	0
$395$	11.1741	0.562228
$396$	0	0
$397$	28.9787	1.45440	0.727200	−	0.686426i	$-0.240821\pi$
$397$	28.9787	1.45440	0.727200	+	0.686426i	$0.240821\pi$
$398$	0	0
$399$	−6.44085	−0.322446
$400$	0	0
$401$	16.8622	0.842060	0.421030	−	0.907047i	$-0.361669\pi$
$401$	16.8622	0.842060	0.421030	+	0.907047i	$0.361669\pi$
$402$	0	0
$403$	8.23801	0.410365
$404$	0	0
$405$	4.26299	0.211830
$406$	0	0
$407$	1.86489	0.0924394
$408$	0	0
$409$	−10.1231	−0.500557	−0.250279	−	0.968174i	$-0.580522\pi$
$409$	−10.1231	−0.500557	−0.250279	+	0.968174i	$0.580522\pi$
$410$	0	0
$411$	6.10291	0.301034
$412$	0	0
$413$	8.63973	0.425133
$414$	0	0
$415$	36.3462	1.78417
$416$	0	0
$417$	15.0157	0.735321
$418$	0	0
$419$	−7.18298	−0.350912	−0.175456	−	0.984487i	$-0.556140\pi$
$419$	−7.18298	−0.350912	−0.175456	+	0.984487i	$0.556140\pi$
$420$	0	0
$421$	21.0270	1.02479	0.512396	−	0.858749i	$-0.328758\pi$
$421$	21.0270	1.02479	0.512396	+	0.858749i	$0.328758\pi$
$422$	0	0
$423$	11.1056	0.539971
$424$	0	0
$425$	94.7863	4.59781
$426$	0	0
$427$	13.5214	0.654348
$428$	0	0
$429$	1.00000	0.0482805
$430$	0	0
$431$	11.2443	0.541618	0.270809	−	0.962633i	$-0.412709\pi$
$431$	11.2443	0.541618	0.270809	+	0.962633i	$0.412709\pi$
$432$	0	0
$433$	−16.7199	−0.803508	−0.401754	−	0.915748i	$-0.631599\pi$
$433$	−16.7199	−0.803508	−0.401754	+	0.915748i	$0.631599\pi$
$434$	0	0
$435$	42.1403	2.02047
$436$	0	0
$437$	−20.2558	−0.968965
$438$	0	0
$439$	21.5548	1.02875	0.514376	−	0.857564i	$-0.328023\pi$
$439$	21.5548	1.02875	0.514376	+	0.857564i	$0.328023\pi$
$440$	0	0
$441$	−6.13054	−0.291931
$442$	0	0
$443$	14.5706	0.692272	0.346136	−	0.938184i	$-0.387494\pi$
$443$	14.5706	0.692272	0.346136	+	0.938184i	$0.387494\pi$
$444$	0	0
$445$	63.5148	3.01089
$446$	0	0
$447$	12.9943	0.614609
$448$	0	0
$449$	35.5494	1.67768	0.838839	−	0.544379i	$-0.183235\pi$
$449$	35.5494	1.67768	0.838839	+	0.544379i	$0.183235\pi$
$450$	0	0
$451$	−1.06755	−0.0502691
$452$	0	0
$453$	−0.192787	−0.00905792
$454$	0	0
$455$	−3.97502	−0.186352
$456$	0	0
$457$	−17.0770	−0.798826	−0.399413	−	0.916771i	$-0.630786\pi$
$457$	−17.0770	−0.798826	−0.399413	+	0.916771i	$0.630786\pi$
$458$	0	0
$459$	−7.19544	−0.335854
$460$	0	0
$461$	−19.2947	−0.898645	−0.449322	−	0.893370i	$-0.648335\pi$
$461$	−19.2947	−0.898645	−0.449322	+	0.893370i	$0.648335\pi$
$462$	0	0
$463$	−10.2416	−0.475969	−0.237984	−	0.971269i	$-0.576487\pi$
$463$	−10.2416	−0.475969	−0.237984	+	0.971269i	$0.576487\pi$
$464$	0	0
$465$	−35.1186	−1.62859
$466$	0	0
$467$	−19.3888	−0.897206	−0.448603	−	0.893731i	$-0.648078\pi$
$467$	−19.3888	−0.897206	−0.448603	+	0.893731i	$0.648078\pi$
$468$	0	0
$469$	−2.11012	−0.0974364
$470$	0	0
$471$	−8.24637	−0.379973
$472$	0	0
$473$	−5.30556	−0.243950
$474$	0	0
$475$	−90.9928	−4.17504
$476$	0	0
$477$	9.24066	0.423101
$478$	0	0
$479$	6.57481	0.300411	0.150205	−	0.988655i	$-0.452007\pi$
$479$	6.57481	0.300411	0.150205	+	0.988655i	$0.452007\pi$
$480$	0	0
$481$	−1.86489	−0.0850319
$482$	0	0
$483$	2.73435	0.124417
$484$	0	0
$485$	28.6246	1.29978
$486$	0	0
$487$	−9.17293	−0.415665	−0.207833	−	0.978164i	$-0.566641\pi$
$487$	−9.17293	−0.415665	−0.207833	+	0.978164i	$0.566641\pi$
$488$	0	0
$489$	3.15743	0.142784
$490$	0	0
$491$	35.0359	1.58115	0.790574	−	0.612366i	$-0.209782\pi$
$491$	35.0359	1.58115	0.790574	+	0.612366i	$0.209782\pi$
$492$	0	0
$493$	−71.1279	−3.20344
$494$	0	0
$495$	−4.26299	−0.191607
$496$	0	0
$497$	−6.74659	−0.302626
$498$	0	0
$499$	−3.77218	−0.168866	−0.0844330	−	0.996429i	$-0.526908\pi$
$499$	−3.77218	−0.168866	−0.0844330	+	0.996429i	$0.526908\pi$
$500$	0	0
$501$	−17.0753	−0.762867
$502$	0	0
$503$	−29.5280	−1.31659	−0.658294	−	0.752761i	$-0.728722\pi$
$503$	−29.5280	−1.31659	−0.658294	+	0.752761i	$0.728722\pi$
$504$	0	0
$505$	53.5910	2.38477
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	−41.0530	−1.81964	−0.909821	−	0.415000i	$-0.863782\pi$
$509$	−41.0530	−1.81964	−0.909821	+	0.415000i	$0.863782\pi$
$510$	0	0
$511$	5.75610	0.254635
$512$	0	0
$513$	6.90746	0.304972
$514$	0	0
$515$	−13.8988	−0.612455
$516$	0	0
$517$	−11.1056	−0.488422
$518$	0	0
$519$	15.3056	0.671840
$520$	0	0
$521$	−20.9508	−0.917870	−0.458935	−	0.888470i	$-0.651769\pi$
$521$	−20.9508	−0.917870	−0.458935	+	0.888470i	$0.651769\pi$
$522$	0	0
$523$	11.3342	0.495608	0.247804	−	0.968810i	$-0.420291\pi$
$523$	11.3342	0.495608	0.247804	+	0.968810i	$0.420291\pi$
$524$	0	0
$525$	12.2832	0.536084
$526$	0	0
$527$	59.2761	2.58211
$528$	0	0
$529$	−14.4008	−0.626120
$530$	0	0
$531$	−9.26565	−0.402095
$532$	0	0
$533$	1.06755	0.0462408
$534$	0	0
$535$	52.7157	2.27910
$536$	0	0
$537$	−2.24637	−0.0969381
$538$	0	0
$539$	6.13054	0.264061
$540$	0	0
$541$	10.2702	0.441551	0.220775	−	0.975325i	$-0.429141\pi$
$541$	10.2702	0.441551	0.220775	+	0.975325i	$0.429141\pi$
$542$	0	0
$543$	−18.0613	−0.775085
$544$	0	0
$545$	−51.7874	−2.21833
$546$	0	0
$547$	27.6482	1.18215	0.591075	−	0.806617i	$-0.298704\pi$
$547$	27.6482	1.18215	0.591075	+	0.806617i	$0.298704\pi$
$548$	0	0
$549$	−14.5010	−0.618888
$550$	0	0
$551$	68.2812	2.90888
$552$	0	0
$553$	−2.44411	−0.103934
$554$	0	0
$555$	7.95003	0.337460
$556$	0	0
$557$	4.37786	0.185496	0.0927479	−	0.995690i	$-0.470435\pi$
$557$	4.37786	0.185496	0.0927479	+	0.995690i	$0.470435\pi$
$558$	0	0
$559$	5.30556	0.224401
$560$	0	0
$561$	7.19544	0.303792
$562$	0	0
$563$	−22.2873	−0.939299	−0.469649	−	0.882853i	$-0.655620\pi$
$563$	−22.2873	−0.939299	−0.469649	+	0.882853i	$0.655620\pi$
$564$	0	0
$565$	3.13054	0.131703
$566$	0	0
$567$	−0.932447	−0.0391591
$568$	0	0
$569$	28.0177	1.17456	0.587281	−	0.809383i	$-0.300198\pi$
$569$	28.0177	1.17456	0.587281	+	0.809383i	$0.300198\pi$
$570$	0	0
$571$	−20.3296	−0.850767	−0.425384	−	0.905013i	$-0.639861\pi$
$571$	−20.3296	−0.850767	−0.425384	+	0.905013i	$0.639861\pi$
$572$	0	0
$573$	4.41586	0.184475
$574$	0	0
$575$	38.6295	1.61096
$576$	0	0
$577$	12.1351	0.505191	0.252596	−	0.967572i	$-0.418716\pi$
$577$	12.1351	0.505191	0.252596	+	0.967572i	$0.418716\pi$
$578$	0	0
$579$	−8.38148	−0.348322
$580$	0	0
$581$	−7.95003	−0.329823
$582$	0	0
$583$	−9.24066	−0.382709
$584$	0	0
$585$	4.26299	0.176253
$586$	0	0
$587$	4.89550	0.202059	0.101030	−	0.994883i	$-0.467786\pi$
$587$	4.89550	0.202059	0.101030	+	0.994883i	$0.467786\pi$
$588$	0	0
$589$	−56.9038	−2.34468
$590$	0	0
$591$	8.90384	0.366255
$592$	0	0
$593$	−40.7646	−1.67400	−0.837000	−	0.547203i	$-0.815693\pi$
$593$	−40.7646	−1.67400	−0.837000	+	0.547203i	$0.815693\pi$
$594$	0	0
$595$	−28.6020	−1.17257
$596$	0	0
$597$	9.79163	0.400745
$598$	0	0
$599$	24.1457	0.986565	0.493282	−	0.869869i	$-0.335797\pi$
$599$	24.1457	0.986565	0.493282	+	0.869869i	$0.335797\pi$
$600$	0	0
$601$	17.6798	0.721175	0.360588	−	0.932725i	$-0.382576\pi$
$601$	17.6798	0.721175	0.360588	+	0.932725i	$0.382576\pi$
$602$	0	0
$603$	2.26299	0.0921562
$604$	0	0
$605$	4.26299	0.173315
$606$	0	0
$607$	12.9604	0.526047	0.263023	−	0.964789i	$-0.415280\pi$
$607$	12.9604	0.526047	0.263023	+	0.964789i	$0.415280\pi$
$608$	0	0
$609$	−9.21737	−0.373507
$610$	0	0
$611$	11.1056	0.449283
$612$	0	0
$613$	41.2001	1.66406	0.832028	−	0.554733i	$-0.187180\pi$
$613$	41.2001	1.66406	0.832028	+	0.554733i	$0.187180\pi$
$614$	0	0
$615$	−4.55097	−0.183513
$616$	0	0
$617$	12.2469	0.493043	0.246522	−	0.969137i	$-0.420712\pi$
$617$	12.2469	0.493043	0.246522	+	0.969137i	$0.420712\pi$
$618$	0	0
$619$	−31.7095	−1.27451	−0.637256	−	0.770653i	$-0.719930\pi$
$619$	−31.7095	−1.27451	−0.637256	+	0.770653i	$0.719930\pi$
$620$	0	0
$621$	−2.93245	−0.117675
$622$	0	0
$623$	−13.8926	−0.556596
$624$	0	0
$625$	82.6653	3.30661
$626$	0	0
$627$	−6.90746	−0.275858
$628$	0	0
$629$	−13.4187	−0.535040
$630$	0	0
$631$	−40.7230	−1.62116	−0.810578	−	0.585631i	$-0.800847\pi$
$631$	−40.7230	−1.62116	−0.810578	+	0.585631i	$0.800847\pi$
$632$	0	0
$633$	16.5659	0.658436
$634$	0	0
$635$	55.6761	2.20944
$636$	0	0
$637$	−6.13054	−0.242901
$638$	0	0
$639$	7.23536	0.286226
$640$	0	0
$641$	−12.7176	−0.502316	−0.251158	−	0.967946i	$-0.580811\pi$
$641$	−12.7176	−0.502316	−0.251158	+	0.967946i	$0.580811\pi$
$642$	0	0
$643$	−15.8810	−0.626284	−0.313142	−	0.949706i	$-0.601382\pi$
$643$	−15.8810	−0.626284	−0.313142	+	0.949706i	$0.601382\pi$
$644$	0	0
$645$	−22.6176	−0.890566
$646$	0	0
$647$	−19.9071	−0.782627	−0.391314	−	0.920257i	$-0.627979\pi$
$647$	−19.9071	−0.782627	−0.391314	+	0.920257i	$0.627979\pi$
$648$	0	0
$649$	9.26565	0.363708
$650$	0	0
$651$	7.68151	0.301062
$652$	0	0
$653$	−40.0171	−1.56599	−0.782995	−	0.622027i	$-0.786309\pi$
$653$	−40.0171	−1.56599	−0.782995	+	0.622027i	$0.786309\pi$
$654$	0	0
$655$	47.2210	1.84508
$656$	0	0
$657$	−6.17311	−0.240836
$658$	0	0
$659$	20.3409	0.792370	0.396185	−	0.918171i	$-0.370334\pi$
$659$	20.3409	0.792370	0.396185	+	0.918171i	$0.370334\pi$
$660$	0	0
$661$	−6.39981	−0.248924	−0.124462	−	0.992224i	$-0.539720\pi$
$661$	−6.39981	−0.248924	−0.124462	+	0.992224i	$0.539720\pi$
$662$	0	0
$663$	−7.19544	−0.279448
$664$	0	0
$665$	27.4573	1.06475
$666$	0	0
$667$	−28.9876	−1.12241
$668$	0	0
$669$	26.4491	1.02258
$670$	0	0
$671$	14.5010	0.559805
$672$	0	0
$673$	−5.99107	−0.230939	−0.115469	−	0.993311i	$-0.536837\pi$
$673$	−5.99107	−0.230939	−0.115469	+	0.993311i	$0.536837\pi$
$674$	0	0
$675$	−13.1731	−0.507033
$676$	0	0
$677$	−6.36407	−0.244591	−0.122296	−	0.992494i	$-0.539026\pi$
$677$	−6.36407	−0.244591	−0.122296	+	0.992494i	$0.539026\pi$
$678$	0	0
$679$	−6.26108	−0.240278
$680$	0	0
$681$	−1.89235	−0.0725151
$682$	0	0
$683$	19.6548	0.752072	0.376036	−	0.926605i	$-0.377287\pi$
$683$	19.6548	0.752072	0.376036	+	0.926605i	$0.377287\pi$
$684$	0	0
$685$	−26.0166	−0.994045
$686$	0	0
$687$	15.5161	0.591977
$688$	0	0
$689$	9.24066	0.352041
$690$	0	0
$691$	−30.5044	−1.16044	−0.580221	−	0.814459i	$-0.697034\pi$
$691$	−30.5044	−1.16044	−0.580221	+	0.814459i	$0.697034\pi$
$692$	0	0
$693$	0.932447	0.0354207
$694$	0	0
$695$	−64.0117	−2.42810
$696$	0	0
$697$	7.68151	0.290958
$698$	0	0
$699$	−14.4308	−0.545823
$700$	0	0
$701$	32.6705	1.23395	0.616974	−	0.786983i	$-0.288358\pi$
$701$	32.6705	1.23395	0.616974	+	0.786983i	$0.288358\pi$
$702$	0	0
$703$	12.8817	0.485842
$704$	0	0
$705$	−47.3429	−1.78304
$706$	0	0
$707$	−11.7220	−0.440851
$708$	0	0
$709$	12.7623	0.479298	0.239649	−	0.970860i	$-0.422968\pi$
$709$	12.7623	0.479298	0.239649	+	0.970860i	$0.422968\pi$
$710$	0	0
$711$	2.62118	0.0983018
$712$	0	0
$713$	24.1575	0.904707
$714$	0	0
$715$	−4.26299	−0.159427
$716$	0	0
$717$	8.61227	0.321631
$718$	0	0
$719$	40.8419	1.52315	0.761573	−	0.648079i	$-0.224427\pi$
$719$	40.8419	1.52315	0.761573	+	0.648079i	$0.224427\pi$
$720$	0	0
$721$	3.04010	0.113219
$722$	0	0
$723$	1.24066	0.0461408
$724$	0	0
$725$	−130.218	−4.83617
$726$	0	0
$727$	36.9721	1.37122	0.685610	−	0.727969i	$-0.259535\pi$
$727$	36.9721	1.37122	0.685610	+	0.727969i	$0.259535\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	38.1759	1.41199
$732$	0	0
$733$	45.4203	1.67764	0.838819	−	0.544410i	$-0.183246\pi$
$733$	45.4203	1.67764	0.838819	+	0.544410i	$0.183246\pi$
$734$	0	0
$735$	26.1345	0.963984
$736$	0	0
$737$	−2.26299	−0.0833584
$738$	0	0
$739$	−16.3762	−0.602407	−0.301204	−	0.953560i	$-0.597388\pi$
$739$	−16.3762	−0.602407	−0.301204	+	0.953560i	$0.597388\pi$
$740$	0	0
$741$	6.90746	0.253752
$742$	0	0
$743$	30.8031	1.13006	0.565029	−	0.825071i	$-0.308865\pi$
$743$	30.8031	1.13006	0.565029	+	0.825071i	$0.308865\pi$
$744$	0	0
$745$	−55.3946	−2.02950
$746$	0	0
$747$	8.52599	0.311950
$748$	0	0
$749$	−11.5305	−0.421317
$750$	0	0
$751$	15.3402	0.559771	0.279885	−	0.960033i	$-0.409704\pi$
$751$	15.3402	0.559771	0.279885	+	0.960033i	$0.409704\pi$
$752$	0	0
$753$	−20.2558	−0.738162
$754$	0	0
$755$	0.821849	0.0299102
$756$	0	0
$757$	43.4988	1.58099	0.790496	−	0.612467i	$-0.209823\pi$
$757$	43.4988	1.58099	0.790496	+	0.612467i	$0.209823\pi$
$758$	0	0
$759$	2.93245	0.106441
$760$	0	0
$761$	36.9532	1.33955	0.669776	−	0.742563i	$-0.266390\pi$
$761$	36.9532	1.33955	0.669776	+	0.742563i	$0.266390\pi$
$762$	0	0
$763$	11.3275	0.410083
$764$	0	0
$765$	30.6741	1.10902
$766$	0	0
$767$	−9.26565	−0.334563
$768$	0	0
$769$	−41.6934	−1.50350	−0.751751	−	0.659447i	$-0.770790\pi$
$769$	−41.6934	−1.50350	−0.751751	+	0.659447i	$0.770790\pi$
$770$	0	0
$771$	−9.92674	−0.357503
$772$	0	0
$773$	29.4376	1.05880	0.529399	−	0.848373i	$-0.322417\pi$
$773$	29.4376	1.05880	0.529399	+	0.848373i	$0.322417\pi$
$774$	0	0
$775$	108.520	3.89816
$776$	0	0
$777$	−1.73892	−0.0623833
$778$	0	0
$779$	−7.37408	−0.264204
$780$	0	0
$781$	−7.23536	−0.258901
$782$	0	0
$783$	9.88514	0.353266
$784$	0	0
$785$	35.1542	1.25471
$786$	0	0
$787$	9.04257	0.322333	0.161166	−	0.986927i	$-0.448474\pi$
$787$	9.04257	0.322333	0.161166	+	0.986927i	$0.448474\pi$
$788$	0	0
$789$	17.6352	0.627829
$790$	0	0
$791$	−0.684746	−0.0243467
$792$	0	0
$793$	−14.5010	−0.514946
$794$	0	0
$795$	−39.3929	−1.39712
$796$	0	0
$797$	0.0588934	0.00208611	0.00104306	−	0.999999i	$-0.499668\pi$
$797$	0.0588934	0.00208611	0.00104306	+	0.999999i	$0.499668\pi$
$798$	0	0
$799$	79.9094	2.82699
$800$	0	0
$801$	14.8991	0.526434
$802$	0	0
$803$	6.17311	0.217844
$804$	0	0
$805$	−11.6565	−0.410839
$806$	0	0
$807$	−17.0204	−0.599147
$808$	0	0
$809$	−2.09350	−0.0736036	−0.0368018	−	0.999323i	$-0.511717\pi$
$809$	−2.09350	−0.0736036	−0.0368018	+	0.999323i	$0.511717\pi$
$810$	0	0
$811$	18.9574	0.665685	0.332843	−	0.942982i	$-0.391992\pi$
$811$	18.9574	0.665685	0.332843	+	0.942982i	$0.391992\pi$
$812$	0	0
$813$	16.6688	0.584601
$814$	0	0
$815$	−13.4601	−0.471487
$816$	0	0
$817$	−36.6480	−1.28215
$818$	0	0
$819$	−0.932447	−0.0325823
$820$	0	0
$821$	−27.8280	−0.971202	−0.485601	−	0.874181i	$-0.661399\pi$
$821$	−27.8280	−0.971202	−0.485601	+	0.874181i	$0.661399\pi$
$822$	0	0
$823$	34.9734	1.21910	0.609548	−	0.792749i	$-0.291351\pi$
$823$	34.9734	1.21910	0.609548	+	0.792749i	$0.291351\pi$
$824$	0	0
$825$	13.1731	0.458629
$826$	0	0
$827$	−16.6889	−0.580330	−0.290165	−	0.956977i	$-0.593710\pi$
$827$	−16.6889	−0.580330	−0.290165	+	0.956977i	$0.593710\pi$
$828$	0	0
$829$	5.64047	0.195902	0.0979509	−	0.995191i	$-0.468771\pi$
$829$	5.64047	0.195902	0.0979509	+	0.995191i	$0.468771\pi$
$830$	0	0
$831$	−15.0753	−0.522955
$832$	0	0
$833$	−44.1119	−1.52839
$834$	0	0
$835$	72.7918	2.51906
$836$	0	0
$837$	−8.23801	−0.284747
$838$	0	0
$839$	41.2279	1.42335	0.711673	−	0.702511i	$-0.247938\pi$
$839$	41.2279	1.42335	0.711673	+	0.702511i	$0.247938\pi$
$840$	0	0
$841$	68.7159	2.36951
$842$	0	0
$843$	8.87717	0.305746
$844$	0	0
$845$	4.26299	0.146651
$846$	0	0
$847$	−0.932447	−0.0320393
$848$	0	0
$849$	−20.6813	−0.709781
$850$	0	0
$851$	−5.46871	−0.187465
$852$	0	0
$853$	−25.4679	−0.872004	−0.436002	−	0.899946i	$-0.643606\pi$
$853$	−25.4679	−0.872004	−0.436002	+	0.899946i	$0.643606\pi$
$854$	0	0
$855$	−29.4465	−1.00705
$856$	0	0
$857$	−40.9881	−1.40013	−0.700064	−	0.714080i	$-0.746845\pi$
$857$	−40.9881	−1.40013	−0.700064	+	0.714080i	$0.746845\pi$
$858$	0	0
$859$	18.3909	0.627489	0.313744	−	0.949507i	$-0.398416\pi$
$859$	18.3909	0.627489	0.313744	+	0.949507i	$0.398416\pi$
$860$	0	0
$861$	0.995437	0.0339244
$862$	0	0
$863$	−25.1424	−0.855858	−0.427929	−	0.903812i	$-0.640757\pi$
$863$	−25.1424	−0.855858	−0.427929	+	0.903812i	$0.640757\pi$
$864$	0	0
$865$	−65.2475	−2.21848
$866$	0	0
$867$	−34.7744	−1.18100
$868$	0	0
$869$	−2.62118	−0.0889173
$870$	0	0
$871$	2.26299	0.0766786
$872$	0	0
$873$	6.71468	0.227257
$874$	0	0
$875$	−32.4883	−1.09830
$876$	0	0
$877$	27.0852	0.914603	0.457301	−	0.889312i	$-0.348816\pi$
$877$	27.0852	0.914603	0.457301	+	0.889312i	$0.348816\pi$
$878$	0	0
$879$	−14.6688	−0.494767
$880$	0	0
$881$	−29.5305	−0.994910	−0.497455	−	0.867490i	$-0.665732\pi$
$881$	−29.5305	−0.994910	−0.497455	+	0.867490i	$0.665732\pi$
$882$	0	0
$883$	51.0501	1.71797	0.858986	−	0.511999i	$-0.171095\pi$
$883$	51.0501	1.71797	0.858986	+	0.511999i	$0.171095\pi$
$884$	0	0
$885$	39.4994	1.32776
$886$	0	0
$887$	−35.9422	−1.20682	−0.603411	−	0.797430i	$-0.706192\pi$
$887$	−35.9422	−1.20682	−0.603411	+	0.797430i	$0.706192\pi$
$888$	0	0
$889$	−12.1781	−0.408440
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	−76.7113	−2.56704
$894$	0	0
$895$	9.57627	0.320099
$896$	0	0
$897$	−2.93245	−0.0979116
$898$	0	0
$899$	−81.4338	−2.71597
$900$	0	0
$901$	66.4906	2.21512
$902$	0	0
$903$	4.94716	0.164631
$904$	0	0
$905$	76.9952	2.55941
$906$	0	0
$907$	−36.3318	−1.20638	−0.603189	−	0.797599i	$-0.706103\pi$
$907$	−36.3318	−1.20638	−0.603189	+	0.797599i	$0.706103\pi$
$908$	0	0
$909$	12.5712	0.416961
$910$	0	0
$911$	−1.32580	−0.0439258	−0.0219629	−	0.999759i	$-0.506992\pi$
$911$	−1.32580	−0.0439258	−0.0219629	+	0.999759i	$0.506992\pi$
$912$	0	0
$913$	−8.52599	−0.282169
$914$	0	0
$915$	61.8177	2.04363
$916$	0	0
$917$	−10.3287	−0.341083
$918$	0	0
$919$	−45.7826	−1.51023	−0.755115	−	0.655592i	$-0.772419\pi$
$919$	−45.7826	−1.51023	−0.755115	+	0.655592i	$0.772419\pi$
$920$	0	0
$921$	−19.6892	−0.648782
$922$	0	0
$923$	7.23536	0.238155
$924$	0	0
$925$	−24.5665	−0.807740
$926$	0	0
$927$	−3.26034	−0.107084
$928$	0	0
$929$	5.19134	0.170322	0.0851612	−	0.996367i	$-0.472859\pi$
$929$	5.19134	0.170322	0.0851612	+	0.996367i	$0.472859\pi$
$930$	0	0
$931$	42.3465	1.38785
$932$	0	0
$933$	21.7761	0.712916
$934$	0	0
$935$	−30.6741	−1.00315
$936$	0	0
$937$	−11.9797	−0.391358	−0.195679	−	0.980668i	$-0.562691\pi$
$937$	−11.9797	−0.391358	−0.195679	+	0.980668i	$0.562691\pi$
$938$	0	0
$939$	18.7842	0.613000
$940$	0	0
$941$	−13.9725	−0.455492	−0.227746	−	0.973721i	$-0.573136\pi$
$941$	−13.9725	−0.455492	−0.227746	+	0.973721i	$0.573136\pi$
$942$	0	0
$943$	3.13054	0.101944
$944$	0	0
$945$	3.97502	0.129307
$946$	0	0
$947$	−46.2427	−1.50268	−0.751342	−	0.659913i	$-0.770593\pi$
$947$	−46.2427	−1.50268	−0.751342	+	0.659913i	$0.770593\pi$
$948$	0	0
$949$	−6.17311	−0.200388
$950$	0	0
$951$	17.3739	0.563386
$952$	0	0
$953$	−3.73984	−0.121145	−0.0605726	−	0.998164i	$-0.519293\pi$
$953$	−3.73984	−0.121145	−0.0605726	+	0.998164i	$0.519293\pi$
$954$	0	0
$955$	−18.8248	−0.609156
$956$	0	0
$957$	−9.88514	−0.319541
$958$	0	0
$959$	5.69064	0.183760
$960$	0	0
$961$	36.8648	1.18919
$962$	0	0
$963$	12.3659	0.398486
$964$	0	0
$965$	35.7302	1.15020
$966$	0	0
$967$	−23.7172	−0.762695	−0.381347	−	0.924432i	$-0.624540\pi$
$967$	−23.7172	−0.762695	−0.381347	+	0.924432i	$0.624540\pi$
$968$	0	0
$969$	49.7023	1.59667
$970$	0	0
$971$	11.4388	0.367087	0.183544	−	0.983012i	$-0.441243\pi$
$971$	11.4388	0.367087	0.183544	+	0.983012i	$0.441243\pi$
$972$	0	0
$973$	14.0013	0.448862
$974$	0	0
$975$	−13.1731	−0.421877
$976$	0	0
$977$	−8.59867	−0.275096	−0.137548	−	0.990495i	$-0.543922\pi$
$977$	−8.59867	−0.275096	−0.137548	+	0.990495i	$0.543922\pi$
$978$	0	0
$979$	−14.8991	−0.476177
$980$	0	0
$981$	−12.1481	−0.387860
$982$	0	0
$983$	−59.6986	−1.90409	−0.952045	−	0.305957i	$-0.901024\pi$
$983$	−59.6986	−1.90409	−0.952045	+	0.305957i	$0.901024\pi$
$984$	0	0
$985$	−37.9570	−1.20941
$986$	0	0
$987$	10.3553	0.329614
$988$	0	0
$989$	15.5583	0.494725
$990$	0	0
$991$	−15.4661	−0.491298	−0.245649	−	0.969359i	$-0.579001\pi$
$991$	−15.4661	−0.491298	−0.245649	+	0.969359i	$0.579001\pi$
$992$	0	0
$993$	−6.97767	−0.221430
$994$	0	0
$995$	−41.7417	−1.32330
$996$	0	0
$997$	−0.587830	−0.0186168	−0.00930839	−	0.999957i	$-0.502963\pi$
$997$	−0.587830	−0.0186168	−0.00930839	+	0.999957i	$0.502963\pi$
$998$	0	0
$999$	1.86489	0.0590027

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6864.2.a.ce.1.5		5
4.3	odd	2		3432.2.a.x.1.5	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3432.2.a.x.1.5	✓	5	4.3	odd	2
6864.2.a.ce.1.5		5	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +4.26299 q^{5} -0.932447 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +4.26299 q^{5} -0.932447 q^{7} +1.00000 q^{9} -1.00000 q^{11} +1.00000 q^{13} -4.26299 q^{15} +7.19544 q^{17} -6.90746 q^{19} +0.932447 q^{21} +2.93245 q^{23} +13.1731 q^{25} -1.00000 q^{27} -9.88514 q^{29} +8.23801 q^{31} +1.00000 q^{33} -3.97502 q^{35} -1.86489 q^{37} -1.00000 q^{39} +1.06755 q^{41} +5.30556 q^{43} +4.26299 q^{45} +11.1056 q^{47} -6.13054 q^{49} -7.19544 q^{51} +9.24066 q^{53} -4.26299 q^{55} +6.90746 q^{57} -9.26565 q^{59} -14.5010 q^{61} -0.932447 q^{63} +4.26299 q^{65} +2.26299 q^{67} -2.93245 q^{69} +7.23536 q^{71} -6.17311 q^{73} -13.1731 q^{75} +0.932447 q^{77} +2.62118 q^{79} +1.00000 q^{81} +8.52599 q^{83} +30.6741 q^{85} +9.88514 q^{87} +14.8991 q^{89} -0.932447 q^{91} -8.23801 q^{93} -29.4465 q^{95} +6.71468 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{3} + q^{5} + 3 q^{7} + 5 q^{9}+O(q^{10}) \) 5 * q - 5 * q^3 + q^5 + 3 * q^7 + 5 * q^9	\( 5 q - 5 q^{3} + q^{5} + 3 q^{7} + 5 q^{9} - 5 q^{11} + 5 q^{13} - q^{15} + 8 q^{17} - 10 q^{19} - 3 q^{21} + 7 q^{23} + 16 q^{25} - 5 q^{27} - 3 q^{29} + 4 q^{31} + 5 q^{33} - 3 q^{35} + 6 q^{37} - 5 q^{39} + 13 q^{41} - 3 q^{43} + q^{45} - 2 q^{47} + 10 q^{49} - 8 q^{51} + 4 q^{53} - q^{55} + 10 q^{57} - 21 q^{59} - 15 q^{61} + 3 q^{63} + q^{65} - 9 q^{67} - 7 q^{69} + 4 q^{71} + 19 q^{73} - 16 q^{75} - 3 q^{77} - 8 q^{79} + 5 q^{81} + 2 q^{83} + 46 q^{85} + 3 q^{87} + 12 q^{89} + 3 q^{91} - 4 q^{93} + 32 q^{97} - 5 q^{99}+O(q^{100}) \) 5 * q - 5 * q^3 + q^5 + 3 * q^7 + 5 * q^9 - 5 * q^11 + 5 * q^13 - q^15 + 8 * q^17 - 10 * q^19 - 3 * q^21 + 7 * q^23 + 16 * q^25 - 5 * q^27 - 3 * q^29 + 4 * q^31 + 5 * q^33 - 3 * q^35 + 6 * q^37 - 5 * q^39 + 13 * q^41 - 3 * q^43 + q^45 - 2 * q^47 + 10 * q^49 - 8 * q^51 + 4 * q^53 - q^55 + 10 * q^57 - 21 * q^59 - 15 * q^61 + 3 * q^63 + q^65 - 9 * q^67 - 7 * q^69 + 4 * q^71 + 19 * q^73 - 16 * q^75 - 3 * q^77 - 8 * q^79 + 5 * q^81 + 2 * q^83 + 46 * q^85 + 3 * q^87 + 12 * q^89 + 3 * q^91 - 4 * q^93 + 32 * q^97 - 5 * q^99

Introduction

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