LMFDB - Embedded newform 6864.2.a.bo.1.3

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:	$1$
Dimension:	$3$
Coefficient field:	3.3.892.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 8x + 10 $ x^3 - x^2 - 8*x + 10
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3432)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.91729$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +2.91729 q^{5} -4.51056 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +2.91729 q^{5} -4.51056 q^{7} +1.00000 q^{9} +1.00000 q^{11} +1.00000 q^{13} -2.91729 q^{15} -2.40673 q^{17} -2.00000 q^{19} +4.51056 q^{21} +5.32401 q^{23} +3.51056 q^{25} -1.00000 q^{27} -0.103829 q^{29} -2.40673 q^{31} -1.00000 q^{33} -13.1586 q^{35} +6.00000 q^{37} -1.00000 q^{39} -3.32401 q^{41} -7.73074 q^{43} +2.91729 q^{45} -0.813457 q^{47} +13.3451 q^{49} +2.40673 q^{51} +2.00000 q^{53} +2.91729 q^{55} +2.00000 q^{57} -8.13747 q^{59} -1.32401 q^{61} -4.51056 q^{63} +2.91729 q^{65} +12.5864 q^{67} -5.32401 q^{69} +10.6480 q^{71} -13.3662 q^{73} -3.51056 q^{75} -4.51056 q^{77} -3.59327 q^{79} +1.00000 q^{81} -5.02112 q^{83} -7.02112 q^{85} +0.103829 q^{87} -6.24130 q^{89} -4.51056 q^{91} +2.40673 q^{93} -5.83457 q^{95} +12.0422 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} - q^{5} - 5 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 - q^5 - 5 * q^7 + 3 * q^9	$ 3 q - 3 q^{3} - q^{5} - 5 q^{7} + 3 q^{9} + 3 q^{11} + 3 q^{13} + q^{15} - 6 q^{17} - 6 q^{19} + 5 q^{21} + 5 q^{23} + 2 q^{25} - 3 q^{27} + 7 q^{29} - 6 q^{31} - 3 q^{33} - 9 q^{35} + 18 q^{37} - 3 q^{39} + q^{41} - 11 q^{43} - q^{45} + 12 q^{49} + 6 q^{51} + 6 q^{53} - q^{55} + 6 q^{57} - 11 q^{59} + 7 q^{61} - 5 q^{63} - q^{65} - 11 q^{67} - 5 q^{69} + 10 q^{71} + 5 q^{73} - 2 q^{75} - 5 q^{77} - 12 q^{79} + 3 q^{81} + 2 q^{83} - 4 q^{85} - 7 q^{87} + 2 q^{89} - 5 q^{91} + 6 q^{93} + 2 q^{95} + 2 q^{97} + 3 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 - q^5 - 5 * q^7 + 3 * q^9 + 3 * q^11 + 3 * q^13 + q^15 - 6 * q^17 - 6 * q^19 + 5 * q^21 + 5 * q^23 + 2 * q^25 - 3 * q^27 + 7 * q^29 - 6 * q^31 - 3 * q^33 - 9 * q^35 + 18 * q^37 - 3 * q^39 + q^41 - 11 * q^43 - q^45 + 12 * q^49 + 6 * q^51 + 6 * q^53 - q^55 + 6 * q^57 - 11 * q^59 + 7 * q^61 - 5 * q^63 - q^65 - 11 * q^67 - 5 * q^69 + 10 * q^71 + 5 * q^73 - 2 * q^75 - 5 * q^77 - 12 * q^79 + 3 * q^81 + 2 * q^83 - 4 * q^85 - 7 * q^87 + 2 * q^89 - 5 * q^91 + 6 * q^93 + 2 * q^95 + 2 * q^97 + 3 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	2.91729	1.30465	0.652325	−	0.757939i	$-0.273794\pi$
$5$	2.91729	1.30465	0.652325	+	0.757939i	$0.273794\pi$
$6$	0	0
$7$	−4.51056	−1.70483	−0.852415	−	0.522865i	$-0.824863\pi$
$7$	−4.51056	−1.70483	−0.852415	+	0.522865i	$0.824863\pi$
$8$	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
$15$	−2.91729	−0.753240
$16$	0	0
$17$	−2.40673	−0.583717	−0.291859	−	0.956461i	$-0.594274\pi$
$17$	−2.40673	−0.583717	−0.291859	+	0.956461i	$0.594274\pi$
$18$	0	0
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	4.51056	0.984284
$22$	0	0
$23$	5.32401	1.11013	0.555067	−	0.831806i	$-0.312693\pi$
$23$	5.32401	1.11013	0.555067	+	0.831806i	$0.312693\pi$
$24$	0	0
$25$	3.51056	0.702112
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−0.103829	−0.0192806	−0.00964029	−	0.999954i	$-0.503069\pi$
$29$	−0.103829	−0.0192806	−0.00964029	+	0.999954i	$0.503069\pi$
$30$	0	0
$31$	−2.40673	−0.432261	−0.216131	−	0.976364i	$-0.569344\pi$
$31$	−2.40673	−0.432261	−0.216131	+	0.976364i	$0.569344\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	−13.1586	−2.22421
$36$	0	0
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−3.32401	−0.519124	−0.259562	−	0.965726i	$-0.583578\pi$
$41$	−3.32401	−0.519124	−0.259562	+	0.965726i	$0.583578\pi$
$42$	0	0
$43$	−7.73074	−1.17893	−0.589464	−	0.807795i	$-0.700661\pi$
$43$	−7.73074	−1.17893	−0.589464	+	0.807795i	$0.700661\pi$
$44$	0	0
$45$	2.91729	0.434883
$46$	0	0
$47$	−0.813457	−0.118655	−0.0593274	−	0.998239i	$-0.518896\pi$
$47$	−0.813457	−0.118655	−0.0593274	+	0.998239i	$0.518896\pi$
$48$	0	0
$49$	13.3451	1.90645
$50$	0	0
$51$	2.40673	0.337009
$52$	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
$55$	2.91729	0.393367
$56$	0	0
$57$	2.00000	0.264906
$58$	0	0
$59$	−8.13747	−1.05941	−0.529704	−	0.848182i	$-0.677697\pi$
$59$	−8.13747	−1.05941	−0.529704	+	0.848182i	$0.677697\pi$
$60$	0	0
$61$	−1.32401	−0.169523	−0.0847613	−	0.996401i	$-0.527013\pi$
$61$	−1.32401	−0.169523	−0.0847613	+	0.996401i	$0.527013\pi$
$62$	0	0
$63$	−4.51056	−0.568277
$64$	0	0
$65$	2.91729	0.361845
$66$	0	0
$67$	12.5864	1.53768	0.768839	−	0.639443i	$-0.220835\pi$
$67$	12.5864	1.53768	0.768839	+	0.639443i	$0.220835\pi$
$68$	0	0
$69$	−5.32401	−0.640936
$70$	0	0
$71$	10.6480	1.26369	0.631844	−	0.775095i	$-0.282298\pi$
$71$	10.6480	1.26369	0.631844	+	0.775095i	$0.282298\pi$
$72$	0	0
$73$	−13.3662	−1.56440	−0.782200	−	0.623027i	$-0.785903\pi$
$73$	−13.3662	−1.56440	−0.782200	+	0.623027i	$0.785903\pi$
$74$	0	0
$75$	−3.51056	−0.405364
$76$	0	0
$77$	−4.51056	−0.514026
$78$	0	0
$79$	−3.59327	−0.404275	−0.202137	−	0.979357i	$-0.564789\pi$
$79$	−3.59327	−0.404275	−0.202137	+	0.979357i	$0.564789\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−5.02112	−0.551139	−0.275569	−	0.961281i	$-0.588866\pi$
$83$	−5.02112	−0.551139	−0.275569	+	0.961281i	$0.588866\pi$
$84$	0	0
$85$	−7.02112	−0.761547
$86$	0	0
$87$	0.103829	0.0111317
$88$	0	0
$89$	−6.24130	−0.661577	−0.330788	−	0.943705i	$-0.607315\pi$
$89$	−6.24130	−0.661577	−0.330788	+	0.943705i	$0.607315\pi$
$90$	0	0
$91$	−4.51056	−0.472835
$92$	0	0
$93$	2.40673	0.249566
$94$	0	0
$95$	−5.83457	−0.598614
$96$	0	0
$97$	12.0422	1.22270	0.611352	−	0.791359i	$-0.290626\pi$
$97$	12.0422	1.22270	0.611352	+	0.791359i	$0.290626\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	−10.6144	−1.05617	−0.528085	−	0.849191i	$-0.677090\pi$
$101$	−10.6144	−1.05617	−0.528085	+	0.849191i	$0.677090\pi$
$102$	0	0
$103$	0.137471	0.0135454	0.00677272	−	0.999977i	$-0.497844\pi$
$103$	0.137471	0.0135454	0.00677272	+	0.999977i	$0.497844\pi$
$104$	0	0
$105$	13.1586	1.28415
$106$	0	0
$107$	−9.15859	−0.885394	−0.442697	−	0.896671i	$-0.645978\pi$
$107$	−9.15859	−0.885394	−0.442697	+	0.896671i	$0.645978\pi$
$108$	0	0
$109$	−1.62691	−0.155830	−0.0779150	−	0.996960i	$-0.524826\pi$
$109$	−1.62691	−0.155830	−0.0779150	+	0.996960i	$0.524826\pi$
$110$	0	0
$111$	−6.00000	−0.569495
$112$	0	0
$113$	3.69710	0.347794	0.173897	−	0.984764i	$-0.444364\pi$
$113$	3.69710	0.347794	0.173897	+	0.984764i	$0.444364\pi$
$114$	0	0
$115$	15.5317	1.44834
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	10.8557	0.995139
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	3.32401	0.299716
$124$	0	0
$125$	−4.34513	−0.388640
$126$	0	0
$127$	12.8220	1.13777	0.568886	−	0.822416i	$-0.307374\pi$
$127$	12.8220	1.13777	0.568886	+	0.822416i	$0.307374\pi$
$128$	0	0
$129$	7.73074	0.680654
$130$	0	0
$131$	−7.32401	−0.639902	−0.319951	−	0.947434i	$-0.603666\pi$
$131$	−7.32401	−0.639902	−0.319951	+	0.947434i	$0.603666\pi$
$132$	0	0
$133$	9.02112	0.782230
$134$	0	0
$135$	−2.91729	−0.251080
$136$	0	0
$137$	−13.6355	−1.16496	−0.582480	−	0.812845i	$-0.697917\pi$
$137$	−13.6355	−1.16496	−0.582480	+	0.812845i	$0.697917\pi$
$138$	0	0
$139$	1.42784	0.121108	0.0605541	−	0.998165i	$-0.480713\pi$
$139$	1.42784	0.121108	0.0605541	+	0.998165i	$0.480713\pi$
$140$	0	0
$141$	0.813457	0.0685054
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	−0.302899	−0.0251544
$146$	0	0
$147$	−13.3451	−1.10069
$148$	0	0
$149$	−14.8557	−1.21703	−0.608513	−	0.793544i	$-0.708234\pi$
$149$	−14.8557	−1.21703	−0.608513	+	0.793544i	$0.708234\pi$
$150$	0	0
$151$	0.648029	0.0527358	0.0263679	−	0.999652i	$-0.491606\pi$
$151$	0.648029	0.0527358	0.0263679	+	0.999652i	$0.491606\pi$
$152$	0	0
$153$	−2.40673	−0.194572
$154$	0	0
$155$	−7.02112	−0.563950
$156$	0	0
$157$	−13.5037	−1.07771	−0.538857	−	0.842397i	$-0.681144\pi$
$157$	−13.5037	−1.07771	−0.538857	+	0.842397i	$0.681144\pi$
$158$	0	0
$159$	−2.00000	−0.158610
$160$	0	0
$161$	−24.0143	−1.89259
$162$	0	0
$163$	−5.73074	−0.448866	−0.224433	−	0.974489i	$-0.572053\pi$
$163$	−5.73074	−0.448866	−0.224433	+	0.974489i	$0.572053\pi$
$164$	0	0
$165$	−2.91729	−0.227110
$166$	0	0
$167$	−20.3451	−1.57435	−0.787177	−	0.616728i	$-0.788458\pi$
$167$	−20.3451	−1.57435	−0.787177	+	0.616728i	$0.788458\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−2.00000	−0.152944
$172$	0	0
$173$	4.10383	0.312008	0.156004	−	0.987756i	$-0.450139\pi$
$173$	4.10383	0.312008	0.156004	+	0.987756i	$0.450139\pi$
$174$	0	0
$175$	−15.8346	−1.19698
$176$	0	0
$177$	8.13747	0.611650
$178$	0	0
$179$	8.97888	0.671113	0.335557	−	0.942020i	$-0.391076\pi$
$179$	8.97888	0.671113	0.335557	+	0.942020i	$0.391076\pi$
$180$	0	0
$181$	−20.2077	−1.50202	−0.751012	−	0.660289i	$-0.770434\pi$
$181$	−20.2077	−1.50202	−0.751012	+	0.660289i	$0.770434\pi$
$182$	0	0
$183$	1.32401	0.0978740
$184$	0	0
$185$	17.5037	1.28690
$186$	0	0
$187$	−2.40673	−0.175997
$188$	0	0
$189$	4.51056	0.328095
$190$	0	0
$191$	−4.67599	−0.338342	−0.169171	−	0.985587i	$-0.554109\pi$
$191$	−4.67599	−0.338342	−0.169171	+	0.985587i	$0.554109\pi$
$192$	0	0
$193$	18.6480	1.34231	0.671157	−	0.741315i	$-0.265798\pi$
$193$	18.6480	1.34231	0.671157	+	0.741315i	$0.265798\pi$
$194$	0	0
$195$	−2.91729	−0.208911
$196$	0	0
$197$	−13.2288	−0.942511	−0.471256	−	0.881997i	$-0.656199\pi$
$197$	−13.2288	−0.942511	−0.471256	+	0.881997i	$0.656199\pi$
$198$	0	0
$199$	−22.5106	−1.59573	−0.797866	−	0.602835i	$-0.794038\pi$
$199$	−22.5106	−1.59573	−0.797866	+	0.602835i	$0.794038\pi$
$200$	0	0
$201$	−12.5864	−0.887778
$202$	0	0
$203$	0.468327	0.0328701
$204$	0	0
$205$	−9.69710	−0.677275
$206$	0	0
$207$	5.32401	0.370045
$208$	0	0
$209$	−2.00000	−0.138343
$210$	0	0
$211$	1.01253	0.0697052	0.0348526	−	0.999392i	$-0.488904\pi$
$211$	1.01253	0.0697052	0.0348526	+	0.999392i	$0.488904\pi$
$212$	0	0
$213$	−10.6480	−0.729591
$214$	0	0
$215$	−22.5528	−1.53809
$216$	0	0
$217$	10.8557	0.736932
$218$	0	0
$219$	13.3662	0.903207
$220$	0	0
$221$	−2.40673	−0.161894
$222$	0	0
$223$	20.1181	1.34721	0.673604	−	0.739093i	$-0.264745\pi$
$223$	20.1181	1.34721	0.673604	+	0.739093i	$0.264745\pi$
$224$	0	0
$225$	3.51056	0.234037
$226$	0	0
$227$	15.5037	1.02902	0.514509	−	0.857485i	$-0.327974\pi$
$227$	15.5037	1.02902	0.514509	+	0.857485i	$0.327974\pi$
$228$	0	0
$229$	−20.1797	−1.33351	−0.666756	−	0.745276i	$-0.732318\pi$
$229$	−20.1797	−1.33351	−0.666756	+	0.745276i	$0.732318\pi$
$230$	0	0
$231$	4.51056	0.296773
$232$	0	0
$233$	17.4701	1.14450	0.572251	−	0.820078i	$-0.306070\pi$
$233$	17.4701	1.14450	0.572251	+	0.820078i	$0.306070\pi$
$234$	0	0
$235$	−2.37309	−0.154803
$236$	0	0
$237$	3.59327	0.233408
$238$	0	0
$239$	2.13747	0.138262	0.0691308	−	0.997608i	$-0.477977\pi$
$239$	2.13747	0.138262	0.0691308	+	0.997608i	$0.477977\pi$
$240$	0	0
$241$	15.7114	1.01206	0.506029	−	0.862516i	$-0.331113\pi$
$241$	15.7114	1.01206	0.506029	+	0.862516i	$0.331113\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	38.9316	2.48725
$246$	0	0
$247$	−2.00000	−0.127257
$248$	0	0
$249$	5.02112	0.318200
$250$	0	0
$251$	−11.0633	−0.698312	−0.349156	−	0.937065i	$-0.613532\pi$
$251$	−11.0633	−0.698312	−0.349156	+	0.937065i	$0.613532\pi$
$252$	0	0
$253$	5.32401	0.334718
$254$	0	0
$255$	7.02112	0.439679
$256$	0	0
$257$	−28.1797	−1.75780	−0.878901	−	0.477005i	$-0.841722\pi$
$257$	−28.1797	−1.75780	−0.878901	+	0.477005i	$0.841722\pi$
$258$	0	0
$259$	−27.0633	−1.68163
$260$	0	0
$261$	−0.103829	−0.00642686
$262$	0	0
$263$	9.22877	0.569071	0.284535	−	0.958666i	$-0.408161\pi$
$263$	9.22877	0.569071	0.284535	+	0.958666i	$0.408161\pi$
$264$	0	0
$265$	5.83457	0.358415
$266$	0	0
$267$	6.24130	0.381961
$268$	0	0
$269$	25.5459	1.55756	0.778782	−	0.627295i	$-0.215838\pi$
$269$	25.5459	1.55756	0.778782	+	0.627295i	$0.215838\pi$
$270$	0	0
$271$	−8.97888	−0.545428	−0.272714	−	0.962095i	$-0.587921\pi$
$271$	−8.97888	−0.545428	−0.272714	+	0.962095i	$0.587921\pi$
$272$	0	0
$273$	4.51056	0.272991
$274$	0	0
$275$	3.51056	0.211695
$276$	0	0
$277$	−7.69710	−0.462474	−0.231237	−	0.972897i	$-0.574277\pi$
$277$	−7.69710	−0.462474	−0.231237	+	0.972897i	$0.574277\pi$
$278$	0	0
$279$	−2.40673	−0.144087
$280$	0	0
$281$	5.69710	0.339861	0.169930	−	0.985456i	$-0.445646\pi$
$281$	5.69710	0.339861	0.169930	+	0.985456i	$0.445646\pi$
$282$	0	0
$283$	−29.0268	−1.72546	−0.862732	−	0.505661i	$-0.831249\pi$
$283$	−29.0268	−1.72546	−0.862732	+	0.505661i	$0.831249\pi$
$284$	0	0
$285$	5.83457	0.345610
$286$	0	0
$287$	14.9932	0.885018
$288$	0	0
$289$	−11.2077	−0.659274
$290$	0	0
$291$	−12.0422	−0.705928
$292$	0	0
$293$	−19.8768	−1.16122	−0.580608	−	0.814184i	$-0.697185\pi$
$293$	−19.8768	−1.16122	−0.580608	+	0.814184i	$0.697185\pi$
$294$	0	0
$295$	−23.7393	−1.38216
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	5.32401	0.307896
$300$	0	0
$301$	34.8700	2.00987
$302$	0	0
$303$	10.6144	0.609781
$304$	0	0
$305$	−3.86253	−0.221168
$306$	0	0
$307$	13.6691	0.780139	0.390070	−	0.920785i	$-0.372451\pi$
$307$	13.6691	0.780139	0.390070	+	0.920785i	$0.372451\pi$
$308$	0	0
$309$	−0.137471	−0.00782047
$310$	0	0
$311$	−24.6480	−1.39766	−0.698831	−	0.715287i	$-0.746296\pi$
$311$	−24.6480	−1.39766	−0.698831	+	0.715287i	$0.746296\pi$
$312$	0	0
$313$	−11.5317	−0.651809	−0.325904	−	0.945403i	$-0.605669\pi$
$313$	−11.5317	−0.651809	−0.325904	+	0.945403i	$0.605669\pi$
$314$	0	0
$315$	−13.1586	−0.741402
$316$	0	0
$317$	14.7941	0.830919	0.415459	−	0.909612i	$-0.363621\pi$
$317$	14.7941	0.830919	0.415459	+	0.909612i	$0.363621\pi$
$318$	0	0
$319$	−0.103829	−0.00581332
$320$	0	0
$321$	9.15859	0.511182
$322$	0	0
$323$	4.81346	0.267828
$324$	0	0
$325$	3.51056	0.194731
$326$	0	0
$327$	1.62691	0.0899685
$328$	0	0
$329$	3.66914	0.202286
$330$	0	0
$331$	9.33260	0.512966	0.256483	−	0.966549i	$-0.417436\pi$
$331$	9.33260	0.512966	0.256483	+	0.966549i	$0.417436\pi$
$332$	0	0
$333$	6.00000	0.328798
$334$	0	0
$335$	36.7182	2.00613
$336$	0	0
$337$	14.4826	0.788918	0.394459	−	0.918914i	$-0.370932\pi$
$337$	14.4826	0.788918	0.394459	+	0.918914i	$0.370932\pi$
$338$	0	0
$339$	−3.69710	−0.200799
$340$	0	0
$341$	−2.40673	−0.130332
$342$	0	0
$343$	−28.6201	−1.54534
$344$	0	0
$345$	−15.5317	−0.836197
$346$	0	0
$347$	−2.44037	−0.131006	−0.0655030	−	0.997852i	$-0.520865\pi$
$347$	−2.44037	−0.131006	−0.0655030	+	0.997852i	$0.520865\pi$
$348$	0	0
$349$	26.3594	1.41099	0.705493	−	0.708717i	$-0.250726\pi$
$349$	26.3594	1.41099	0.705493	+	0.708717i	$0.250726\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	−10.8471	−0.577333	−0.288666	−	0.957430i	$-0.593212\pi$
$353$	−10.8471	−0.577333	−0.288666	+	0.957430i	$0.593212\pi$
$354$	0	0
$355$	31.0633	1.64867
$356$	0	0
$357$	−10.8557	−0.574544
$358$	0	0
$359$	−8.51056	−0.449170	−0.224585	−	0.974454i	$-0.572103\pi$
$359$	−8.51056	−0.449170	−0.224585	+	0.974454i	$0.572103\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	−38.9932	−2.04100
$366$	0	0
$367$	−28.4153	−1.48327	−0.741634	−	0.670805i	$-0.765949\pi$
$367$	−28.4153	−1.48327	−0.741634	+	0.670805i	$0.765949\pi$
$368$	0	0
$369$	−3.32401	−0.173041
$370$	0	0
$371$	−9.02112	−0.468353
$372$	0	0
$373$	26.8277	1.38909	0.694544	−	0.719451i	$-0.255606\pi$
$373$	26.8277	1.38909	0.694544	+	0.719451i	$0.255606\pi$
$374$	0	0
$375$	4.34513	0.224382
$376$	0	0
$377$	−0.103829	−0.00534747
$378$	0	0
$379$	−0.847099	−0.0435126	−0.0217563	−	0.999763i	$-0.506926\pi$
$379$	−0.847099	−0.0435126	−0.0217563	+	0.999763i	$0.506926\pi$
$380$	0	0
$381$	−12.8220	−0.656893
$382$	0	0
$383$	5.83457	0.298133	0.149066	−	0.988827i	$-0.452373\pi$
$383$	5.83457	0.298133	0.149066	+	0.988827i	$0.452373\pi$
$384$	0	0
$385$	−13.1586	−0.670624
$386$	0	0
$387$	−7.73074	−0.392976
$388$	0	0
$389$	−36.1095	−1.83083	−0.915413	−	0.402517i	$-0.868135\pi$
$389$	−36.1095	−1.83083	−0.915413	+	0.402517i	$0.868135\pi$
$390$	0	0
$391$	−12.8135	−0.648004
$392$	0	0
$393$	7.32401	0.369448
$394$	0	0
$395$	−10.4826	−0.527437
$396$	0	0
$397$	−33.4085	−1.67672	−0.838362	−	0.545114i	$-0.816486\pi$
$397$	−33.4085	−1.67672	−0.838362	+	0.545114i	$0.816486\pi$
$398$	0	0
$399$	−9.02112	−0.451621
$400$	0	0
$401$	−13.4951	−0.673915	−0.336957	−	0.941520i	$-0.609398\pi$
$401$	−13.4951	−0.673915	−0.336957	+	0.941520i	$0.609398\pi$
$402$	0	0
$403$	−2.40673	−0.119888
$404$	0	0
$405$	2.91729	0.144961
$406$	0	0
$407$	6.00000	0.297409
$408$	0	0
$409$	−39.6834	−1.96222	−0.981109	−	0.193454i	$-0.938031\pi$
$409$	−39.6834	−1.96222	−0.981109	+	0.193454i	$0.938031\pi$
$410$	0	0
$411$	13.6355	0.672590
$412$	0	0
$413$	36.7045	1.80611
$414$	0	0
$415$	−14.6480	−0.719043
$416$	0	0
$417$	−1.42784	−0.0699218
$418$	0	0
$419$	14.2749	0.697377	0.348688	−	0.937239i	$-0.386627\pi$
$419$	14.2749	0.697377	0.348688	+	0.937239i	$0.386627\pi$
$420$	0	0
$421$	10.3451	0.504191	0.252095	−	0.967702i	$-0.418880\pi$
$421$	10.3451	0.504191	0.252095	+	0.967702i	$0.418880\pi$
$422$	0	0
$423$	−0.813457	−0.0395516
$424$	0	0
$425$	−8.44896	−0.409835
$426$	0	0
$427$	5.97204	0.289007
$428$	0	0
$429$	−1.00000	−0.0482805
$430$	0	0
$431$	−23.6691	−1.14010	−0.570051	−	0.821609i	$-0.693077\pi$
$431$	−23.6691	−1.14010	−0.570051	+	0.821609i	$0.693077\pi$
$432$	0	0
$433$	0.993158	0.0477281	0.0238641	−	0.999715i	$-0.492403\pi$
$433$	0.993158	0.0477281	0.0238641	+	0.999715i	$0.492403\pi$
$434$	0	0
$435$	0.302899	0.0145229
$436$	0	0
$437$	−10.6480	−0.509364
$438$	0	0
$439$	−7.80093	−0.372318	−0.186159	−	0.982520i	$-0.559604\pi$
$439$	−7.80093	−0.372318	−0.186159	+	0.982520i	$0.559604\pi$
$440$	0	0
$441$	13.3451	0.635482
$442$	0	0
$443$	−11.3383	−0.538698	−0.269349	−	0.963043i	$-0.586809\pi$
$443$	−11.3383	−0.538698	−0.269349	+	0.963043i	$0.586809\pi$
$444$	0	0
$445$	−18.2077	−0.863126
$446$	0	0
$447$	14.8557	0.702650
$448$	0	0
$449$	−7.38561	−0.348549	−0.174274	−	0.984697i	$-0.555758\pi$
$449$	−7.38561	−0.348549	−0.174274	+	0.984697i	$0.555758\pi$
$450$	0	0
$451$	−3.32401	−0.156522
$452$	0	0
$453$	−0.648029	−0.0304471
$454$	0	0
$455$	−13.1586	−0.616884
$456$	0	0
$457$	5.92981	0.277385	0.138692	−	0.990335i	$-0.455710\pi$
$457$	5.92981	0.277385	0.138692	+	0.990335i	$0.455710\pi$
$458$	0	0
$459$	2.40673	0.112336
$460$	0	0
$461$	35.3383	1.64587	0.822934	−	0.568137i	$-0.192336\pi$
$461$	35.3383	1.64587	0.822934	+	0.568137i	$0.192336\pi$
$462$	0	0
$463$	4.24130	0.197110	0.0985550	−	0.995132i	$-0.468578\pi$
$463$	4.24130	0.197110	0.0985550	+	0.995132i	$0.468578\pi$
$464$	0	0
$465$	7.02112	0.325596
$466$	0	0
$467$	−0.232712	−0.0107686	−0.00538432	−	0.999986i	$-0.501714\pi$
$467$	−0.232712	−0.0107686	−0.00538432	+	0.999986i	$0.501714\pi$
$468$	0	0
$469$	−56.7718	−2.62148
$470$	0	0
$471$	13.5037	0.622218
$472$	0	0
$473$	−7.73074	−0.355460
$474$	0	0
$475$	−7.02112	−0.322151
$476$	0	0
$477$	2.00000	0.0915737
$478$	0	0
$479$	−21.8066	−0.996370	−0.498185	−	0.867071i	$-0.666000\pi$
$479$	−21.8066	−0.996370	−0.498185	+	0.867071i	$0.666000\pi$
$480$	0	0
$481$	6.00000	0.273576
$482$	0	0
$483$	24.0143	1.09269
$484$	0	0
$485$	35.1306	1.59520
$486$	0	0
$487$	−4.57216	−0.207184	−0.103592	−	0.994620i	$-0.533034\pi$
$487$	−4.57216	−0.207184	−0.103592	+	0.994620i	$0.533034\pi$
$488$	0	0
$489$	5.73074	0.259153
$490$	0	0
$491$	35.2008	1.58859	0.794295	−	0.607532i	$-0.207840\pi$
$491$	35.2008	1.58859	0.794295	+	0.607532i	$0.207840\pi$
$492$	0	0
$493$	0.249889	0.0112544
$494$	0	0
$495$	2.91729	0.131122
$496$	0	0
$497$	−48.0285	−2.15437
$498$	0	0
$499$	−27.4421	−1.22848	−0.614239	−	0.789120i	$-0.710537\pi$
$499$	−27.4421	−1.22848	−0.614239	+	0.789120i	$0.710537\pi$
$500$	0	0
$501$	20.3451	0.908953
$502$	0	0
$503$	38.5248	1.71774	0.858869	−	0.512196i	$-0.171168\pi$
$503$	38.5248	1.71774	0.858869	+	0.512196i	$0.171168\pi$
$504$	0	0
$505$	−30.9652	−1.37793
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	−16.4912	−0.730959	−0.365480	−	0.930819i	$-0.619095\pi$
$509$	−16.4912	−0.730959	−0.365480	+	0.930819i	$0.619095\pi$
$510$	0	0
$511$	60.2892	2.66704
$512$	0	0
$513$	2.00000	0.0883022
$514$	0	0
$515$	0.401043	0.0176721
$516$	0	0
$517$	−0.813457	−0.0357758
$518$	0	0
$519$	−4.10383	−0.180138
$520$	0	0
$521$	−43.3662	−1.89991	−0.949955	−	0.312387i	$-0.898871\pi$
$521$	−43.3662	−1.89991	−0.949955	+	0.312387i	$0.898871\pi$
$522$	0	0
$523$	−25.9104	−1.13298	−0.566492	−	0.824067i	$-0.691700\pi$
$523$	−25.9104	−1.13298	−0.566492	+	0.824067i	$0.691700\pi$
$524$	0	0
$525$	15.8346	0.691077
$526$	0	0
$527$	5.79234	0.252318
$528$	0	0
$529$	5.34513	0.232397
$530$	0	0
$531$	−8.13747	−0.353136
$532$	0	0
$533$	−3.32401	−0.143979
$534$	0	0
$535$	−26.7182	−1.15513
$536$	0	0
$537$	−8.97888	−0.387467
$538$	0	0
$539$	13.3451	0.574815
$540$	0	0
$541$	7.62691	0.327907	0.163953	−	0.986468i	$-0.447575\pi$
$541$	7.62691	0.327907	0.163953	+	0.986468i	$0.447575\pi$
$542$	0	0
$543$	20.2077	0.867194
$544$	0	0
$545$	−4.74617	−0.203304
$546$	0	0
$547$	−34.1883	−1.46179	−0.730893	−	0.682492i	$-0.760896\pi$
$547$	−34.1883	−1.46179	−0.730893	+	0.682492i	$0.760896\pi$
$548$	0	0
$549$	−1.32401	−0.0565076
$550$	0	0
$551$	0.207658	0.00884654
$552$	0	0
$553$	16.2077	0.689220
$554$	0	0
$555$	−17.5037	−0.742991
$556$	0	0
$557$	17.8346	0.755675	0.377838	−	0.925872i	$-0.376668\pi$
$557$	17.8346	0.755675	0.377838	+	0.925872i	$0.376668\pi$
$558$	0	0
$559$	−7.73074	−0.326976
$560$	0	0
$561$	2.40673	0.101612
$562$	0	0
$563$	34.5921	1.45788	0.728942	−	0.684576i	$-0.240012\pi$
$563$	34.5921	1.45788	0.728942	+	0.684576i	$0.240012\pi$
$564$	0	0
$565$	10.7855	0.453749
$566$	0	0
$567$	−4.51056	−0.189426
$568$	0	0
$569$	−11.0297	−0.462389	−0.231195	−	0.972908i	$-0.574263\pi$
$569$	−11.0297	−0.462389	−0.231195	+	0.972908i	$0.574263\pi$
$570$	0	0
$571$	43.7593	1.83127	0.915635	−	0.402011i	$-0.131689\pi$
$571$	43.7593	1.83127	0.915635	+	0.402011i	$0.131689\pi$
$572$	0	0
$573$	4.67599	0.195342
$574$	0	0
$575$	18.6903	0.779438
$576$	0	0
$577$	−13.7251	−0.571382	−0.285691	−	0.958322i	$-0.592223\pi$
$577$	−13.7251	−0.571382	−0.285691	+	0.958322i	$0.592223\pi$
$578$	0	0
$579$	−18.6480	−0.774986
$580$	0	0
$581$	22.6480	0.939599
$582$	0	0
$583$	2.00000	0.0828315
$584$	0	0
$585$	2.91729	0.120615
$586$	0	0
$587$	22.1797	0.915454	0.457727	−	0.889093i	$-0.348664\pi$
$587$	22.1797	0.915454	0.457727	+	0.889093i	$0.348664\pi$
$588$	0	0
$589$	4.81346	0.198335
$590$	0	0
$591$	13.2288	0.544159
$592$	0	0
$593$	7.46149	0.306406	0.153203	−	0.988195i	$-0.451041\pi$
$593$	7.46149	0.306406	0.153203	+	0.988195i	$0.451041\pi$
$594$	0	0
$595$	31.6691	1.29831
$596$	0	0
$597$	22.5106	0.921296
$598$	0	0
$599$	9.97204	0.407447	0.203723	−	0.979029i	$-0.434696\pi$
$599$	9.97204	0.407447	0.203723	+	0.979029i	$0.434696\pi$
$600$	0	0
$601$	24.7325	1.00886	0.504430	−	0.863453i	$-0.331703\pi$
$601$	24.7325	1.00886	0.504430	+	0.863453i	$0.331703\pi$
$602$	0	0
$603$	12.5864	0.512559
$604$	0	0
$605$	2.91729	0.118605
$606$	0	0
$607$	−40.2276	−1.63279	−0.816394	−	0.577495i	$-0.804030\pi$
$607$	−40.2276	−1.63279	−0.816394	+	0.577495i	$0.804030\pi$
$608$	0	0
$609$	−0.468327	−0.0189776
$610$	0	0
$611$	−0.813457	−0.0329089
$612$	0	0
$613$	−17.2961	−0.698581	−0.349291	−	0.937014i	$-0.613577\pi$
$613$	−17.2961	−0.698581	−0.349291	+	0.937014i	$0.613577\pi$
$614$	0	0
$615$	9.69710	0.391025
$616$	0	0
$617$	−13.7028	−0.551653	−0.275827	−	0.961207i	$-0.588952\pi$
$617$	−13.7028	−0.551653	−0.275827	+	0.961207i	$0.588952\pi$
$618$	0	0
$619$	4.51915	0.181640	0.0908199	−	0.995867i	$-0.471051\pi$
$619$	4.51915	0.181640	0.0908199	+	0.995867i	$0.471051\pi$
$620$	0	0
$621$	−5.32401	−0.213645
$622$	0	0
$623$	28.1517	1.12788
$624$	0	0
$625$	−30.2288	−1.20915
$626$	0	0
$627$	2.00000	0.0798723
$628$	0	0
$629$	−14.4404	−0.575775
$630$	0	0
$631$	−40.7239	−1.62119	−0.810596	−	0.585605i	$-0.800857\pi$
$631$	−40.7239	−1.62119	−0.810596	+	0.585605i	$0.800857\pi$
$632$	0	0
$633$	−1.01253	−0.0402443
$634$	0	0
$635$	37.4056	1.48439
$636$	0	0
$637$	13.3451	0.528753
$638$	0	0
$639$	10.6480	0.421230
$640$	0	0
$641$	22.7604	0.898984	0.449492	−	0.893284i	$-0.351605\pi$
$641$	22.7604	0.898984	0.449492	+	0.893284i	$0.351605\pi$
$642$	0	0
$643$	−7.30465	−0.288067	−0.144034	−	0.989573i	$-0.546007\pi$
$643$	−7.30465	−0.288067	−0.144034	+	0.989573i	$0.546007\pi$
$644$	0	0
$645$	22.5528	0.888015
$646$	0	0
$647$	44.0845	1.73314	0.866569	−	0.499056i	$-0.166320\pi$
$647$	44.0845	1.73314	0.866569	+	0.499056i	$0.166320\pi$
$648$	0	0
$649$	−8.13747	−0.319424
$650$	0	0
$651$	−10.8557	−0.425468
$652$	0	0
$653$	−34.5671	−1.35271	−0.676357	−	0.736574i	$-0.736442\pi$
$653$	−34.5671	−1.35271	−0.676357	+	0.736574i	$0.736442\pi$
$654$	0	0
$655$	−21.3662	−0.834848
$656$	0	0
$657$	−13.3662	−0.521467
$658$	0	0
$659$	−2.78840	−0.108621	−0.0543104	−	0.998524i	$-0.517296\pi$
$659$	−2.78840	−0.108621	−0.0543104	+	0.998524i	$0.517296\pi$
$660$	0	0
$661$	−3.22877	−0.125585	−0.0627924	−	0.998027i	$-0.520001\pi$
$661$	−3.22877	−0.125585	−0.0627924	+	0.998027i	$0.520001\pi$
$662$	0	0
$663$	2.40673	0.0934696
$664$	0	0
$665$	26.3172	1.02054
$666$	0	0
$667$	−0.552788	−0.0214040
$668$	0	0
$669$	−20.1181	−0.777811
$670$	0	0
$671$	−1.32401	−0.0511130
$672$	0	0
$673$	−7.83457	−0.302001	−0.151000	−	0.988534i	$-0.548249\pi$
$673$	−7.83457	−0.302001	−0.151000	+	0.988534i	$0.548249\pi$
$674$	0	0
$675$	−3.51056	−0.135121
$676$	0	0
$677$	−11.4951	−0.441794	−0.220897	−	0.975297i	$-0.570898\pi$
$677$	−11.4951	−0.441794	−0.220897	+	0.975297i	$0.570898\pi$
$678$	0	0
$679$	−54.3172	−2.08450
$680$	0	0
$681$	−15.5037	−0.594104
$682$	0	0
$683$	10.7182	0.410121	0.205061	−	0.978749i	$-0.434261\pi$
$683$	10.7182	0.410121	0.205061	+	0.978749i	$0.434261\pi$
$684$	0	0
$685$	−39.7787	−1.51986
$686$	0	0
$687$	20.1797	0.769904
$688$	0	0
$689$	2.00000	0.0761939
$690$	0	0
$691$	11.3046	0.430049	0.215024	−	0.976609i	$-0.431017\pi$
$691$	11.3046	0.430049	0.215024	+	0.976609i	$0.431017\pi$
$692$	0	0
$693$	−4.51056	−0.171342
$694$	0	0
$695$	4.16543	0.158004
$696$	0	0
$697$	8.00000	0.303022
$698$	0	0
$699$	−17.4701	−0.660779
$700$	0	0
$701$	−1.24814	−0.0471417	−0.0235708	−	0.999722i	$-0.507504\pi$
$701$	−1.24814	−0.0471417	−0.0235708	+	0.999722i	$0.507504\pi$
$702$	0	0
$703$	−12.0000	−0.452589
$704$	0	0
$705$	2.37309	0.0893756
$706$	0	0
$707$	47.8768	1.80059
$708$	0	0
$709$	37.6161	1.41270	0.706352	−	0.707861i	$-0.250340\pi$
$709$	37.6161	1.41270	0.706352	+	0.707861i	$0.250340\pi$
$710$	0	0
$711$	−3.59327	−0.134758
$712$	0	0
$713$	−12.8135	−0.479868
$714$	0	0
$715$	2.91729	0.109100
$716$	0	0
$717$	−2.13747	−0.0798253
$718$	0	0
$719$	−44.9795	−1.67745	−0.838726	−	0.544554i	$-0.816699\pi$
$719$	−44.9795	−1.67745	−0.838726	+	0.544554i	$0.816699\pi$
$720$	0	0
$721$	−0.620072	−0.0230927
$722$	0	0
$723$	−15.7114	−0.584312
$724$	0	0
$725$	−0.364498	−0.0135371
$726$	0	0
$727$	−15.1865	−0.563238	−0.281619	−	0.959526i	$-0.590871\pi$
$727$	−15.1865	−0.563238	−0.281619	+	0.959526i	$0.590871\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	18.6058	0.688160
$732$	0	0
$733$	4.69026	0.173239	0.0866193	−	0.996241i	$-0.472394\pi$
$733$	4.69026	0.173239	0.0866193	+	0.996241i	$0.472394\pi$
$734$	0	0
$735$	−38.9316	−1.43601
$736$	0	0
$737$	12.5864	0.463627
$738$	0	0
$739$	−24.7998	−0.912274	−0.456137	−	0.889909i	$-0.650767\pi$
$739$	−24.7998	−0.912274	−0.456137	+	0.889909i	$0.650767\pi$
$740$	0	0
$741$	2.00000	0.0734718
$742$	0	0
$743$	34.6623	1.27164	0.635818	−	0.771839i	$-0.280663\pi$
$743$	34.6623	1.27164	0.635818	+	0.771839i	$0.280663\pi$
$744$	0	0
$745$	−43.3383	−1.58779
$746$	0	0
$747$	−5.02112	−0.183713
$748$	0	0
$749$	41.3103	1.50945
$750$	0	0
$751$	−43.8238	−1.59915	−0.799576	−	0.600564i	$-0.794943\pi$
$751$	−43.8238	−1.59915	−0.799576	+	0.600564i	$0.794943\pi$
$752$	0	0
$753$	11.0633	0.403171
$754$	0	0
$755$	1.89049	0.0688018
$756$	0	0
$757$	−4.95383	−0.180050	−0.0900250	−	0.995940i	$-0.528695\pi$
$757$	−4.95383	−0.180050	−0.0900250	+	0.995940i	$0.528695\pi$
$758$	0	0
$759$	−5.32401	−0.193249
$760$	0	0
$761$	−39.6161	−1.43608	−0.718042	−	0.696000i	$-0.754961\pi$
$761$	−39.6161	−1.43608	−0.718042	+	0.696000i	$0.754961\pi$
$762$	0	0
$763$	7.33829	0.265664
$764$	0	0
$765$	−7.02112	−0.253849
$766$	0	0
$767$	−8.13747	−0.293827
$768$	0	0
$769$	23.3662	0.842608	0.421304	−	0.906919i	$-0.361572\pi$
$769$	23.3662	0.842608	0.421304	+	0.906919i	$0.361572\pi$
$770$	0	0
$771$	28.1797	1.01487
$772$	0	0
$773$	34.4490	1.23904	0.619521	−	0.784980i	$-0.287327\pi$
$773$	34.4490	1.23904	0.619521	+	0.784980i	$0.287327\pi$
$774$	0	0
$775$	−8.44896	−0.303496
$776$	0	0
$777$	27.0633	0.970892
$778$	0	0
$779$	6.64803	0.238190
$780$	0	0
$781$	10.6480	0.381016
$782$	0	0
$783$	0.103829	0.00371055
$784$	0	0
$785$	−39.3942	−1.40604
$786$	0	0
$787$	−7.41926	−0.264468	−0.132234	−	0.991219i	$-0.542215\pi$
$787$	−7.41926	−0.264468	−0.132234	+	0.991219i	$0.542215\pi$
$788$	0	0
$789$	−9.22877	−0.328553
$790$	0	0
$791$	−16.6760	−0.592930
$792$	0	0
$793$	−1.32401	−0.0470171
$794$	0	0
$795$	−5.83457	−0.206931
$796$	0	0
$797$	−14.5385	−0.514981	−0.257490	−	0.966281i	$-0.582896\pi$
$797$	−14.5385	−0.514981	−0.257490	+	0.966281i	$0.582896\pi$
$798$	0	0
$799$	1.95777	0.0692609
$800$	0	0
$801$	−6.24130	−0.220526
$802$	0	0
$803$	−13.3662	−0.471685
$804$	0	0
$805$	−70.0565	−2.46917
$806$	0	0
$807$	−25.5459	−0.899260
$808$	0	0
$809$	−4.65662	−0.163718	−0.0818590	−	0.996644i	$-0.526086\pi$
$809$	−4.65662	−0.163718	−0.0818590	+	0.996644i	$0.526086\pi$
$810$	0	0
$811$	10.6903	0.375386	0.187693	−	0.982228i	$-0.439899\pi$
$811$	10.6903	0.375386	0.187693	+	0.982228i	$0.439899\pi$
$812$	0	0
$813$	8.97888	0.314903
$814$	0	0
$815$	−16.7182	−0.585614
$816$	0	0
$817$	15.4615	0.540929
$818$	0	0
$819$	−4.51056	−0.157612
$820$	0	0
$821$	15.7251	0.548808	0.274404	−	0.961614i	$-0.411519\pi$
$821$	15.7251	0.548808	0.274404	+	0.961614i	$0.411519\pi$
$822$	0	0
$823$	43.2008	1.50589	0.752943	−	0.658086i	$-0.228634\pi$
$823$	43.2008	1.50589	0.752943	+	0.658086i	$0.228634\pi$
$824$	0	0
$825$	−3.51056	−0.122222
$826$	0	0
$827$	6.53851	0.227366	0.113683	−	0.993517i	$-0.463735\pi$
$827$	6.53851	0.227366	0.113683	+	0.993517i	$0.463735\pi$
$828$	0	0
$829$	−31.2961	−1.08696	−0.543479	−	0.839423i	$-0.682893\pi$
$829$	−31.2961	−1.08696	−0.543479	+	0.839423i	$0.682893\pi$
$830$	0	0
$831$	7.69710	0.267009
$832$	0	0
$833$	−32.1181	−1.11283
$834$	0	0
$835$	−59.3526	−2.05398
$836$	0	0
$837$	2.40673	0.0831887
$838$	0	0
$839$	−30.4016	−1.04958	−0.524790	−	0.851231i	$-0.675856\pi$
$839$	−30.4016	−1.04958	−0.524790	+	0.851231i	$0.675856\pi$
$840$	0	0
$841$	−28.9892	−0.999628
$842$	0	0
$843$	−5.69710	−0.196219
$844$	0	0
$845$	2.91729	0.100358
$846$	0	0
$847$	−4.51056	−0.154985
$848$	0	0
$849$	29.0268	0.996197
$850$	0	0
$851$	31.9441	1.09503
$852$	0	0
$853$	−39.6555	−1.35778	−0.678889	−	0.734241i	$-0.737538\pi$
$853$	−39.6555	−1.35778	−0.678889	+	0.734241i	$0.737538\pi$
$854$	0	0
$855$	−5.83457	−0.199538
$856$	0	0
$857$	7.30465	0.249522	0.124761	−	0.992187i	$-0.460184\pi$
$857$	7.30465	0.249522	0.124761	+	0.992187i	$0.460184\pi$
$858$	0	0
$859$	36.5671	1.24765	0.623826	−	0.781563i	$-0.285577\pi$
$859$	36.5671	1.24765	0.623826	+	0.781563i	$0.285577\pi$
$860$	0	0
$861$	−14.9932	−0.510965
$862$	0	0
$863$	19.5174	0.664380	0.332190	−	0.943212i	$-0.392212\pi$
$863$	19.5174	0.664380	0.332190	+	0.943212i	$0.392212\pi$
$864$	0	0
$865$	11.9720	0.407062
$866$	0	0
$867$	11.2077	0.380632
$868$	0	0
$869$	−3.59327	−0.121893
$870$	0	0
$871$	12.5864	0.426475
$872$	0	0
$873$	12.0422	0.407568
$874$	0	0
$875$	19.5990	0.662566
$876$	0	0
$877$	−10.2749	−0.346960	−0.173480	−	0.984837i	$-0.555501\pi$
$877$	−10.2749	−0.346960	−0.173480	+	0.984837i	$0.555501\pi$
$878$	0	0
$879$	19.8768	0.670428
$880$	0	0
$881$	20.2470	0.682138	0.341069	−	0.940038i	$-0.389211\pi$
$881$	20.2470	0.682138	0.341069	+	0.940038i	$0.389211\pi$
$882$	0	0
$883$	25.5037	0.858268	0.429134	−	0.903241i	$-0.358819\pi$
$883$	25.5037	0.858268	0.429134	+	0.903241i	$0.358819\pi$
$884$	0	0
$885$	23.7393	0.797989
$886$	0	0
$887$	−2.70394	−0.0907895	−0.0453947	−	0.998969i	$-0.514455\pi$
$887$	−2.70394	−0.0907895	−0.0453947	+	0.998969i	$0.514455\pi$
$888$	0	0
$889$	−57.8346	−1.93971
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	1.62691	0.0544426
$894$	0	0
$895$	26.1940	0.875568
$896$	0	0
$897$	−5.32401	−0.177764
$898$	0	0
$899$	0.249889	0.00833425
$900$	0	0
$901$	−4.81346	−0.160359
$902$	0	0
$903$	−34.8700	−1.16040
$904$	0	0
$905$	−58.9515	−1.95961
$906$	0	0
$907$	45.2961	1.50403	0.752015	−	0.659145i	$-0.229082\pi$
$907$	45.2961	1.50403	0.752015	+	0.659145i	$0.229082\pi$
$908$	0	0
$909$	−10.6144	−0.352057
$910$	0	0
$911$	−6.50765	−0.215608	−0.107804	−	0.994172i	$-0.534382\pi$
$911$	−6.50765	−0.215608	−0.107804	+	0.994172i	$0.534382\pi$
$912$	0	0
$913$	−5.02112	−0.166175
$914$	0	0
$915$	3.86253	0.127691
$916$	0	0
$917$	33.0354	1.09092
$918$	0	0
$919$	−8.94524	−0.295076	−0.147538	−	0.989056i	$-0.547135\pi$
$919$	−8.94524	−0.295076	−0.147538	+	0.989056i	$0.547135\pi$
$920$	0	0
$921$	−13.6691	−0.450414
$922$	0	0
$923$	10.6480	0.350484
$924$	0	0
$925$	21.0633	0.692559
$926$	0	0
$927$	0.137471	0.00451515
$928$	0	0
$929$	−18.2413	−0.598478	−0.299239	−	0.954178i	$-0.596733\pi$
$929$	−18.2413	−0.598478	−0.299239	+	0.954178i	$0.596733\pi$
$930$	0	0
$931$	−26.6903	−0.874738
$932$	0	0
$933$	24.6480	0.806940
$934$	0	0
$935$	−7.02112	−0.229615
$936$	0	0
$937$	−16.4963	−0.538910	−0.269455	−	0.963013i	$-0.586844\pi$
$937$	−16.4963	−0.538910	−0.269455	+	0.963013i	$0.586844\pi$
$938$	0	0
$939$	11.5317	0.376322
$940$	0	0
$941$	−0.330856	−0.0107856	−0.00539280	−	0.999985i	$-0.501717\pi$
$941$	−0.330856	−0.0107856	−0.00539280	+	0.999985i	$0.501717\pi$
$942$	0	0
$943$	−17.6971	−0.576297
$944$	0	0
$945$	13.1586	0.428049
$946$	0	0
$947$	−12.0673	−0.392134	−0.196067	−	0.980590i	$-0.562817\pi$
$947$	−12.0673	−0.392134	−0.196067	+	0.980590i	$0.562817\pi$
$948$	0	0
$949$	−13.3662	−0.433887
$950$	0	0
$951$	−14.7941	−0.479731
$952$	0	0
$953$	30.8893	1.00060	0.500302	−	0.865851i	$-0.333222\pi$
$953$	30.8893	1.00060	0.500302	+	0.865851i	$0.333222\pi$
$954$	0	0
$955$	−13.6412	−0.441418
$956$	0	0
$957$	0.103829	0.00335632
$958$	0	0
$959$	61.5037	1.98606
$960$	0	0
$961$	−25.2077	−0.813150
$962$	0	0
$963$	−9.15859	−0.295131
$964$	0	0
$965$	54.4016	1.75125
$966$	0	0
$967$	−32.8700	−1.05703	−0.528513	−	0.848925i	$-0.677250\pi$
$967$	−32.8700	−1.05703	−0.528513	+	0.848925i	$0.677250\pi$
$968$	0	0
$969$	−4.81346	−0.154631
$970$	0	0
$971$	−31.3805	−1.00705	−0.503524	−	0.863981i	$-0.667964\pi$
$971$	−31.3805	−1.00705	−0.503524	+	0.863981i	$0.667964\pi$
$972$	0	0
$973$	−6.44037	−0.206469
$974$	0	0
$975$	−3.51056	−0.112428
$976$	0	0
$977$	25.9104	0.828949	0.414474	−	0.910061i	$-0.363965\pi$
$977$	25.9104	0.828949	0.414474	+	0.910061i	$0.363965\pi$
$978$	0	0
$979$	−6.24130	−0.199473
$980$	0	0
$981$	−1.62691	−0.0519434
$982$	0	0
$983$	57.2401	1.82568	0.912839	−	0.408321i	$-0.133885\pi$
$983$	57.2401	1.82568	0.912839	+	0.408321i	$0.133885\pi$
$984$	0	0
$985$	−38.5921	−1.22965
$986$	0	0
$987$	−3.66914	−0.116790
$988$	0	0
$989$	−41.1586	−1.30877
$990$	0	0
$991$	−26.8700	−0.853552	−0.426776	−	0.904357i	$-0.640351\pi$
$991$	−26.8700	−0.853552	−0.426776	+	0.904357i	$0.640351\pi$
$992$	0	0
$993$	−9.33260	−0.296161
$994$	0	0
$995$	−65.6697	−2.08187
$996$	0	0
$997$	−46.9122	−1.48572	−0.742862	−	0.669445i	$-0.766532\pi$
$997$	−46.9122	−1.48572	−0.742862	+	0.669445i	$0.766532\pi$
$998$	0	0
$999$	−6.00000	−0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6864.2.a.bo.1.3		3
4.3	odd	2		3432.2.a.o.1.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3432.2.a.o.1.3	✓	3	4.3	odd	2
6864.2.a.bo.1.3		3	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +2.91729 q^{5} -4.51056 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +2.91729 q^{5} -4.51056 q^{7} +1.00000 q^{9} +1.00000 q^{11} +1.00000 q^{13} -2.91729 q^{15} -2.40673 q^{17} -2.00000 q^{19} +4.51056 q^{21} +5.32401 q^{23} +3.51056 q^{25} -1.00000 q^{27} -0.103829 q^{29} -2.40673 q^{31} -1.00000 q^{33} -13.1586 q^{35} +6.00000 q^{37} -1.00000 q^{39} -3.32401 q^{41} -7.73074 q^{43} +2.91729 q^{45} -0.813457 q^{47} +13.3451 q^{49} +2.40673 q^{51} +2.00000 q^{53} +2.91729 q^{55} +2.00000 q^{57} -8.13747 q^{59} -1.32401 q^{61} -4.51056 q^{63} +2.91729 q^{65} +12.5864 q^{67} -5.32401 q^{69} +10.6480 q^{71} -13.3662 q^{73} -3.51056 q^{75} -4.51056 q^{77} -3.59327 q^{79} +1.00000 q^{81} -5.02112 q^{83} -7.02112 q^{85} +0.103829 q^{87} -6.24130 q^{89} -4.51056 q^{91} +2.40673 q^{93} -5.83457 q^{95} +12.0422 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} - q^{5} - 5 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 - q^5 - 5 * q^7 + 3 * q^9	\( 3 q - 3 q^{3} - q^{5} - 5 q^{7} + 3 q^{9} + 3 q^{11} + 3 q^{13} + q^{15} - 6 q^{17} - 6 q^{19} + 5 q^{21} + 5 q^{23} + 2 q^{25} - 3 q^{27} + 7 q^{29} - 6 q^{31} - 3 q^{33} - 9 q^{35} + 18 q^{37} - 3 q^{39} + q^{41} - 11 q^{43} - q^{45} + 12 q^{49} + 6 q^{51} + 6 q^{53} - q^{55} + 6 q^{57} - 11 q^{59} + 7 q^{61} - 5 q^{63} - q^{65} - 11 q^{67} - 5 q^{69} + 10 q^{71} + 5 q^{73} - 2 q^{75} - 5 q^{77} - 12 q^{79} + 3 q^{81} + 2 q^{83} - 4 q^{85} - 7 q^{87} + 2 q^{89} - 5 q^{91} + 6 q^{93} + 2 q^{95} + 2 q^{97} + 3 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 - q^5 - 5 * q^7 + 3 * q^9 + 3 * q^11 + 3 * q^13 + q^15 - 6 * q^17 - 6 * q^19 + 5 * q^21 + 5 * q^23 + 2 * q^25 - 3 * q^27 + 7 * q^29 - 6 * q^31 - 3 * q^33 - 9 * q^35 + 18 * q^37 - 3 * q^39 + q^41 - 11 * q^43 - q^45 + 12 * q^49 + 6 * q^51 + 6 * q^53 - q^55 + 6 * q^57 - 11 * q^59 + 7 * q^61 - 5 * q^63 - q^65 - 11 * q^67 - 5 * q^69 + 10 * q^71 + 5 * q^73 - 2 * q^75 - 5 * q^77 - 12 * q^79 + 3 * q^81 + 2 * q^83 - 4 * q^85 - 7 * q^87 + 2 * q^89 - 5 * q^91 + 6 * q^93 + 2 * q^95 + 2 * q^97 + 3 * q^99

Introduction

L-functions

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