LMFDB - Embedded newform 6864.2.a.bs.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6864, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6864.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$54.8093159474$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
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			1.2
Root			$0.311108$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +1.31111 q^{5} -1.52543 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.31111 q^{5} -1.52543 q^{7} +1.00000 q^{9} +1.00000 q^{11} -1.00000 q^{13} -1.31111 q^{15} +2.21432 q^{17} +1.52543 q^{19} +1.52543 q^{21} -7.95407 q^{23} -3.28100 q^{25} -1.00000 q^{27} +7.39853 q^{29} -4.68889 q^{31} -1.00000 q^{33} -2.00000 q^{35} +8.85728 q^{37} +1.00000 q^{39} -3.52543 q^{41} +8.77631 q^{43} +1.31111 q^{45} -9.18421 q^{47} -4.67307 q^{49} -2.21432 q^{51} +3.67307 q^{53} +1.31111 q^{55} -1.52543 q^{57} -9.37778 q^{59} -11.4795 q^{61} -1.52543 q^{63} -1.31111 q^{65} -5.25088 q^{67} +7.95407 q^{69} +14.4701 q^{71} +3.13828 q^{73} +3.28100 q^{75} -1.52543 q^{77} +5.03011 q^{79} +1.00000 q^{81} -5.37778 q^{83} +2.90321 q^{85} -7.39853 q^{87} +0.688892 q^{89} +1.52543 q^{91} +4.68889 q^{93} +2.00000 q^{95} -12.8573 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + 4 q^{5} + 2 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 + 4 * q^5 + 2 * q^7 + 3 * q^9	$ 3 q - 3 q^{3} + 4 q^{5} + 2 q^{7} + 3 q^{9} + 3 q^{11} - 3 q^{13} - 4 q^{15} - 2 q^{19} - 2 q^{21} - 4 q^{23} - 3 q^{25} - 3 q^{27} + 2 q^{29} - 14 q^{31} - 3 q^{33} - 6 q^{35} + 3 q^{39} - 4 q^{41} + 6 q^{43} + 4 q^{45} - 14 q^{47} - q^{49} - 2 q^{53} + 4 q^{55} + 2 q^{57} - 28 q^{59} - 8 q^{61} + 2 q^{63} - 4 q^{65} - 2 q^{67} + 4 q^{69} - 10 q^{71} - 24 q^{73} + 3 q^{75} + 2 q^{77} + 22 q^{79} + 3 q^{81} - 16 q^{83} + 2 q^{85} - 2 q^{87} + 2 q^{89} - 2 q^{91} + 14 q^{93} + 6 q^{95} - 12 q^{97} + 3 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 + 4 * q^5 + 2 * q^7 + 3 * q^9 + 3 * q^11 - 3 * q^13 - 4 * q^15 - 2 * q^19 - 2 * q^21 - 4 * q^23 - 3 * q^25 - 3 * q^27 + 2 * q^29 - 14 * q^31 - 3 * q^33 - 6 * q^35 + 3 * q^39 - 4 * q^41 + 6 * q^43 + 4 * q^45 - 14 * q^47 - q^49 - 2 * q^53 + 4 * q^55 + 2 * q^57 - 28 * q^59 - 8 * q^61 + 2 * q^63 - 4 * q^65 - 2 * q^67 + 4 * q^69 - 10 * q^71 - 24 * q^73 + 3 * q^75 + 2 * q^77 + 22 * q^79 + 3 * q^81 - 16 * q^83 + 2 * q^85 - 2 * q^87 + 2 * q^89 - 2 * q^91 + 14 * q^93 + 6 * q^95 - 12 * 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
$5$	1.31111	0.586345	0.293173	−	0.956060i	$-0.405289\pi$
$5$	1.31111	0.586345	0.293173	+	0.956060i	$0.405289\pi$
$6$	0	0
$7$	−1.52543	−0.576557	−0.288279	−	0.957547i	$-0.593083\pi$
$7$	−1.52543	−0.576557	−0.288279	+	0.957547i	$0.593083\pi$
$8$	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
$15$	−1.31111	−0.338527
$16$	0	0
$17$	2.21432	0.537051	0.268526	−	0.963273i	$-0.413464\pi$
$17$	2.21432	0.537051	0.268526	+	0.963273i	$0.413464\pi$
$18$	0	0
$19$	1.52543	0.349957	0.174979	−	0.984572i	$-0.444014\pi$
$19$	1.52543	0.349957	0.174979	+	0.984572i	$0.444014\pi$
$20$	0	0
$21$	1.52543	0.332876
$22$	0	0
$23$	−7.95407	−1.65854	−0.829269	−	0.558850i	$-0.811243\pi$
$23$	−7.95407	−1.65854	−0.829269	+	0.558850i	$0.811243\pi$
$24$	0	0
$25$	−3.28100	−0.656199
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	7.39853	1.37387	0.686936	−	0.726718i	$-0.258955\pi$
$29$	7.39853	1.37387	0.686936	+	0.726718i	$0.258955\pi$
$30$	0	0
$31$	−4.68889	−0.842150	−0.421075	−	0.907026i	$-0.638347\pi$
$31$	−4.68889	−0.842150	−0.421075	+	0.907026i	$0.638347\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	−2.00000	−0.338062
$36$	0	0
$37$	8.85728	1.45613	0.728064	−	0.685509i	$-0.240420\pi$
$37$	8.85728	1.45613	0.728064	+	0.685509i	$0.240420\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	−3.52543	−0.550579	−0.275290	−	0.961361i	$-0.588774\pi$
$41$	−3.52543	−0.550579	−0.275290	+	0.961361i	$0.588774\pi$
$42$	0	0
$43$	8.77631	1.33838	0.669188	−	0.743094i	$-0.266642\pi$
$43$	8.77631	1.33838	0.669188	+	0.743094i	$0.266642\pi$
$44$	0	0
$45$	1.31111	0.195448
$46$	0	0
$47$	−9.18421	−1.33965	−0.669827	−	0.742517i	$-0.733632\pi$
$47$	−9.18421	−1.33965	−0.669827	+	0.742517i	$0.733632\pi$
$48$	0	0
$49$	−4.67307	−0.667582
$50$	0	0
$51$	−2.21432	−0.310067
$52$	0	0
$53$	3.67307	0.504535	0.252268	−	0.967658i	$-0.418824\pi$
$53$	3.67307	0.504535	0.252268	+	0.967658i	$0.418824\pi$
$54$	0	0
$55$	1.31111	0.176790
$56$	0	0
$57$	−1.52543	−0.202048
$58$	0	0
$59$	−9.37778	−1.22088	−0.610442	−	0.792061i	$-0.709008\pi$
$59$	−9.37778	−1.22088	−0.610442	+	0.792061i	$0.709008\pi$
$60$	0	0
$61$	−11.4795	−1.46980	−0.734899	−	0.678176i	$-0.762771\pi$
$61$	−11.4795	−1.46980	−0.734899	+	0.678176i	$0.762771\pi$
$62$	0	0
$63$	−1.52543	−0.192186
$64$	0	0
$65$	−1.31111	−0.162623
$66$	0	0
$67$	−5.25088	−0.641498	−0.320749	−	0.947164i	$-0.603935\pi$
$67$	−5.25088	−0.641498	−0.320749	+	0.947164i	$0.603935\pi$
$68$	0	0
$69$	7.95407	0.957557
$70$	0	0
$71$	14.4701	1.71729	0.858644	−	0.512572i	$-0.171307\pi$
$71$	14.4701	1.71729	0.858644	+	0.512572i	$0.171307\pi$
$72$	0	0
$73$	3.13828	0.367307	0.183654	−	0.982991i	$-0.441208\pi$
$73$	3.13828	0.367307	0.183654	+	0.982991i	$0.441208\pi$
$74$	0	0
$75$	3.28100	0.378857
$76$	0	0
$77$	−1.52543	−0.173839
$78$	0	0
$79$	5.03011	0.565932	0.282966	−	0.959130i	$-0.408682\pi$
$79$	5.03011	0.565932	0.282966	+	0.959130i	$0.408682\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−5.37778	−0.590289	−0.295144	−	0.955453i	$-0.595368\pi$
$83$	−5.37778	−0.590289	−0.295144	+	0.955453i	$0.595368\pi$
$84$	0	0
$85$	2.90321	0.314898
$86$	0	0
$87$	−7.39853	−0.793205
$88$	0	0
$89$	0.688892	0.0730224	0.0365112	−	0.999333i	$-0.488376\pi$
$89$	0.688892	0.0730224	0.0365112	+	0.999333i	$0.488376\pi$
$90$	0	0
$91$	1.52543	0.159908
$92$	0	0
$93$	4.68889	0.486215
$94$	0	0
$95$	2.00000	0.205196
$96$	0	0
$97$	−12.8573	−1.30546	−0.652729	−	0.757591i	$-0.726376\pi$
$97$	−12.8573	−1.30546	−0.652729	+	0.757591i	$0.726376\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	−16.0207	−1.59412	−0.797062	−	0.603898i	$-0.793614\pi$
$101$	−16.0207	−1.59412	−0.797062	+	0.603898i	$0.793614\pi$
$102$	0	0
$103$	7.61285	0.750116	0.375058	−	0.927001i	$-0.377623\pi$
$103$	7.61285	0.750116	0.375058	+	0.927001i	$0.377623\pi$
$104$	0	0
$105$	2.00000	0.195180
$106$	0	0
$107$	12.9906	1.25585	0.627926	−	0.778273i	$-0.283904\pi$
$107$	12.9906	1.25585	0.627926	+	0.778273i	$0.283904\pi$
$108$	0	0
$109$	3.82071	0.365958	0.182979	−	0.983117i	$-0.441426\pi$
$109$	3.82071	0.365958	0.182979	+	0.983117i	$0.441426\pi$
$110$	0	0
$111$	−8.85728	−0.840696
$112$	0	0
$113$	−18.3368	−1.72498	−0.862489	−	0.506075i	$-0.831096\pi$
$113$	−18.3368	−1.72498	−0.862489	+	0.506075i	$0.831096\pi$
$114$	0	0
$115$	−10.4286	−0.972476
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	−3.37778	−0.309641
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	3.52543	0.317877
$124$	0	0
$125$	−10.8573	−0.971105
$126$	0	0
$127$	20.5827	1.82642	0.913211	−	0.407486i	$-0.133595\pi$
$127$	20.5827	1.82642	0.913211	+	0.407486i	$0.133595\pi$
$128$	0	0
$129$	−8.77631	−0.772711
$130$	0	0
$131$	0.561993	0.0491015	0.0245508	−	0.999699i	$-0.492184\pi$
$131$	0.561993	0.0491015	0.0245508	+	0.999699i	$0.492184\pi$
$132$	0	0
$133$	−2.32693	−0.201770
$134$	0	0
$135$	−1.31111	−0.112842
$136$	0	0
$137$	5.07604	0.433676	0.216838	−	0.976208i	$-0.430426\pi$
$137$	5.07604	0.433676	0.216838	+	0.976208i	$0.430426\pi$
$138$	0	0
$139$	−18.5511	−1.57348	−0.786742	−	0.617282i	$-0.788234\pi$
$139$	−18.5511	−1.57348	−0.786742	+	0.617282i	$0.788234\pi$
$140$	0	0
$141$	9.18421	0.773450
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	9.70027	0.805563
$146$	0	0
$147$	4.67307	0.385428
$148$	0	0
$149$	−4.38271	−0.359045	−0.179523	−	0.983754i	$-0.557455\pi$
$149$	−4.38271	−0.359045	−0.179523	+	0.983754i	$0.557455\pi$
$150$	0	0
$151$	−1.23014	−0.100107	−0.0500537	−	0.998747i	$-0.515939\pi$
$151$	−1.23014	−0.100107	−0.0500537	+	0.998747i	$0.515939\pi$
$152$	0	0
$153$	2.21432	0.179017
$154$	0	0
$155$	−6.14764	−0.493791
$156$	0	0
$157$	7.62714	0.608712	0.304356	−	0.952558i	$-0.401559\pi$
$157$	7.62714	0.608712	0.304356	+	0.952558i	$0.401559\pi$
$158$	0	0
$159$	−3.67307	−0.291293
$160$	0	0
$161$	12.1334	0.956242
$162$	0	0
$163$	−7.25088	−0.567933	−0.283967	−	0.958834i	$-0.591650\pi$
$163$	−7.25088	−0.567933	−0.283967	+	0.958834i	$0.591650\pi$
$164$	0	0
$165$	−1.31111	−0.102070
$166$	0	0
$167$	−0.295286	−0.0228499	−0.0114250	−	0.999935i	$-0.503637\pi$
$167$	−0.295286	−0.0228499	−0.0114250	+	0.999935i	$0.503637\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	1.52543	0.116652
$172$	0	0
$173$	−17.5605	−1.33510	−0.667549	−	0.744566i	$-0.732656\pi$
$173$	−17.5605	−1.33510	−0.667549	+	0.744566i	$0.732656\pi$
$174$	0	0
$175$	5.00492	0.378337
$176$	0	0
$177$	9.37778	0.704877
$178$	0	0
$179$	−12.0874	−0.903456	−0.451728	−	0.892156i	$-0.649192\pi$
$179$	−12.0874	−0.903456	−0.451728	+	0.892156i	$0.649192\pi$
$180$	0	0
$181$	7.71900	0.573749	0.286875	−	0.957968i	$-0.407384\pi$
$181$	7.71900	0.573749	0.286875	+	0.957968i	$0.407384\pi$
$182$	0	0
$183$	11.4795	0.848589
$184$	0	0
$185$	11.6128	0.853794
$186$	0	0
$187$	2.21432	0.161927
$188$	0	0
$189$	1.52543	0.110959
$190$	0	0
$191$	16.6035	1.20139	0.600693	−	0.799480i	$-0.294891\pi$
$191$	16.6035	1.20139	0.600693	+	0.799480i	$0.294891\pi$
$192$	0	0
$193$	−7.52543	−0.541692	−0.270846	−	0.962623i	$-0.587303\pi$
$193$	−7.52543	−0.541692	−0.270846	+	0.962623i	$0.587303\pi$
$194$	0	0
$195$	1.31111	0.0938904
$196$	0	0
$197$	12.2494	0.872730	0.436365	−	0.899770i	$-0.356266\pi$
$197$	12.2494	0.872730	0.436365	+	0.899770i	$0.356266\pi$
$198$	0	0
$199$	−11.6128	−0.823213	−0.411606	−	0.911362i	$-0.635032\pi$
$199$	−11.6128	−0.823213	−0.411606	+	0.911362i	$0.635032\pi$
$200$	0	0
$201$	5.25088	0.370369
$202$	0	0
$203$	−11.2859	−0.792116
$204$	0	0
$205$	−4.62222	−0.322830
$206$	0	0
$207$	−7.95407	−0.552846
$208$	0	0
$209$	1.52543	0.105516
$210$	0	0
$211$	−5.16346	−0.355468	−0.177734	−	0.984079i	$-0.556877\pi$
$211$	−5.16346	−0.355468	−0.177734	+	0.984079i	$0.556877\pi$
$212$	0	0
$213$	−14.4701	−0.991477
$214$	0	0
$215$	11.5067	0.784750
$216$	0	0
$217$	7.15257	0.485548
$218$	0	0
$219$	−3.13828	−0.212065
$220$	0	0
$221$	−2.21432	−0.148951
$222$	0	0
$223$	8.30174	0.555926	0.277963	−	0.960592i	$-0.410341\pi$
$223$	8.30174	0.555926	0.277963	+	0.960592i	$0.410341\pi$
$224$	0	0
$225$	−3.28100	−0.218733
$226$	0	0
$227$	5.27163	0.349890	0.174945	−	0.984578i	$-0.444025\pi$
$227$	5.27163	0.349890	0.174945	+	0.984578i	$0.444025\pi$
$228$	0	0
$229$	−25.5526	−1.68856	−0.844282	−	0.535898i	$-0.819973\pi$
$229$	−25.5526	−1.68856	−0.844282	+	0.535898i	$0.819973\pi$
$230$	0	0
$231$	1.52543	0.100366
$232$	0	0
$233$	2.40790	0.157747	0.0788733	−	0.996885i	$-0.474868\pi$
$233$	2.40790	0.157747	0.0788733	+	0.996885i	$0.474868\pi$
$234$	0	0
$235$	−12.0415	−0.785500
$236$	0	0
$237$	−5.03011	−0.326741
$238$	0	0
$239$	−17.8622	−1.15541	−0.577705	−	0.816246i	$-0.696052\pi$
$239$	−17.8622	−1.15541	−0.577705	+	0.816246i	$0.696052\pi$
$240$	0	0
$241$	−3.93978	−0.253783	−0.126892	−	0.991917i	$-0.540500\pi$
$241$	−3.93978	−0.253783	−0.126892	+	0.991917i	$0.540500\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−6.12690	−0.391433
$246$	0	0
$247$	−1.52543	−0.0970606
$248$	0	0
$249$	5.37778	0.340803
$250$	0	0
$251$	−23.3590	−1.47441	−0.737205	−	0.675669i	$-0.763855\pi$
$251$	−23.3590	−1.47441	−0.737205	+	0.675669i	$0.763855\pi$
$252$	0	0
$253$	−7.95407	−0.500068
$254$	0	0
$255$	−2.90321	−0.181806
$256$	0	0
$257$	1.70471	0.106337	0.0531686	−	0.998586i	$-0.483068\pi$
$257$	1.70471	0.106337	0.0531686	+	0.998586i	$0.483068\pi$
$258$	0	0
$259$	−13.5111	−0.839541
$260$	0	0
$261$	7.39853	0.457957
$262$	0	0
$263$	1.11108	0.0685120	0.0342560	−	0.999413i	$-0.489094\pi$
$263$	1.11108	0.0685120	0.0342560	+	0.999413i	$0.489094\pi$
$264$	0	0
$265$	4.81579	0.295832
$266$	0	0
$267$	−0.688892	−0.0421595
$268$	0	0
$269$	8.79706	0.536366	0.268183	−	0.963368i	$-0.413577\pi$
$269$	8.79706	0.536366	0.268183	+	0.963368i	$0.413577\pi$
$270$	0	0
$271$	−27.3733	−1.66281	−0.831406	−	0.555665i	$-0.812464\pi$
$271$	−27.3733	−1.66281	−0.831406	+	0.555665i	$0.812464\pi$
$272$	0	0
$273$	−1.52543	−0.0923231
$274$	0	0
$275$	−3.28100	−0.197852
$276$	0	0
$277$	−25.9081	−1.55667	−0.778334	−	0.627850i	$-0.783935\pi$
$277$	−25.9081	−1.55667	−0.778334	+	0.627850i	$0.783935\pi$
$278$	0	0
$279$	−4.68889	−0.280717
$280$	0	0
$281$	−5.76049	−0.343642	−0.171821	−	0.985128i	$-0.554965\pi$
$281$	−5.76049	−0.343642	−0.171821	+	0.985128i	$0.554965\pi$
$282$	0	0
$283$	3.03011	0.180121	0.0900607	−	0.995936i	$-0.471294\pi$
$283$	3.03011	0.180121	0.0900607	+	0.995936i	$0.471294\pi$
$284$	0	0
$285$	−2.00000	−0.118470
$286$	0	0
$287$	5.37778	0.317441
$288$	0	0
$289$	−12.0968	−0.711576
$290$	0	0
$291$	12.8573	0.753707
$292$	0	0
$293$	−9.07805	−0.530345	−0.265173	−	0.964201i	$-0.585429\pi$
$293$	−9.07805	−0.530345	−0.265173	+	0.964201i	$0.585429\pi$
$294$	0	0
$295$	−12.2953	−0.715859
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	7.95407	0.459996
$300$	0	0
$301$	−13.3876	−0.771650
$302$	0	0
$303$	16.0207	0.920368
$304$	0	0
$305$	−15.0509	−0.861809
$306$	0	0
$307$	5.43356	0.310110	0.155055	−	0.987906i	$-0.450445\pi$
$307$	5.43356	0.310110	0.155055	+	0.987906i	$0.450445\pi$
$308$	0	0
$309$	−7.61285	−0.433080
$310$	0	0
$311$	9.46520	0.536723	0.268361	−	0.963318i	$-0.413518\pi$
$311$	9.46520	0.536723	0.268361	+	0.963318i	$0.413518\pi$
$312$	0	0
$313$	−23.6400	−1.33621	−0.668107	−	0.744065i	$-0.732895\pi$
$313$	−23.6400	−1.33621	−0.668107	+	0.744065i	$0.732895\pi$
$314$	0	0
$315$	−2.00000	−0.112687
$316$	0	0
$317$	15.6479	0.878873	0.439436	−	0.898274i	$-0.355178\pi$
$317$	15.6479	0.878873	0.439436	+	0.898274i	$0.355178\pi$
$318$	0	0
$319$	7.39853	0.414238
$320$	0	0
$321$	−12.9906	−0.725066
$322$	0	0
$323$	3.37778	0.187945
$324$	0	0
$325$	3.28100	0.181997
$326$	0	0
$327$	−3.82071	−0.211286
$328$	0	0
$329$	14.0098	0.772388
$330$	0	0
$331$	19.6795	1.08168	0.540842	−	0.841124i	$-0.318106\pi$
$331$	19.6795	1.08168	0.540842	+	0.841124i	$0.318106\pi$
$332$	0	0
$333$	8.85728	0.485376
$334$	0	0
$335$	−6.88448	−0.376139
$336$	0	0
$337$	10.9906	0.598698	0.299349	−	0.954144i	$-0.403231\pi$
$337$	10.9906	0.598698	0.299349	+	0.954144i	$0.403231\pi$
$338$	0	0
$339$	18.3368	0.995917
$340$	0	0
$341$	−4.68889	−0.253918
$342$	0	0
$343$	17.8064	0.961457
$344$	0	0
$345$	10.4286	0.561459
$346$	0	0
$347$	−17.3176	−0.929655	−0.464828	−	0.885401i	$-0.653884\pi$
$347$	−17.3176	−0.929655	−0.464828	+	0.885401i	$0.653884\pi$
$348$	0	0
$349$	−17.1842	−0.919850	−0.459925	−	0.887958i	$-0.652124\pi$
$349$	−17.1842	−0.919850	−0.459925	+	0.887958i	$0.652124\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	−14.4351	−0.768302	−0.384151	−	0.923270i	$-0.625506\pi$
$353$	−14.4351	−0.768302	−0.384151	+	0.923270i	$0.625506\pi$
$354$	0	0
$355$	18.9719	1.00692
$356$	0	0
$357$	3.37778	0.178771
$358$	0	0
$359$	16.1476	0.852240	0.426120	−	0.904667i	$-0.359880\pi$
$359$	16.1476	0.852240	0.426120	+	0.904667i	$0.359880\pi$
$360$	0	0
$361$	−16.6731	−0.877530
$362$	0	0
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	4.11462	0.215369
$366$	0	0
$367$	−26.7971	−1.39879	−0.699397	−	0.714733i	$-0.746548\pi$
$367$	−26.7971	−1.39879	−0.699397	+	0.714733i	$0.746548\pi$
$368$	0	0
$369$	−3.52543	−0.183526
$370$	0	0
$371$	−5.60300	−0.290893
$372$	0	0
$373$	13.3590	0.691705	0.345853	−	0.938289i	$-0.387590\pi$
$373$	13.3590	0.691705	0.345853	+	0.938289i	$0.387590\pi$
$374$	0	0
$375$	10.8573	0.560667
$376$	0	0
$377$	−7.39853	−0.381044
$378$	0	0
$379$	−6.85083	−0.351903	−0.175952	−	0.984399i	$-0.556300\pi$
$379$	−6.85083	−0.351903	−0.175952	+	0.984399i	$0.556300\pi$
$380$	0	0
$381$	−20.5827	−1.05449
$382$	0	0
$383$	−3.57136	−0.182488	−0.0912440	−	0.995829i	$-0.529084\pi$
$383$	−3.57136	−0.182488	−0.0912440	+	0.995829i	$0.529084\pi$
$384$	0	0
$385$	−2.00000	−0.101929
$386$	0	0
$387$	8.77631	0.446125
$388$	0	0
$389$	−8.35551	−0.423641	−0.211821	−	0.977309i	$-0.567939\pi$
$389$	−8.35551	−0.423641	−0.211821	+	0.977309i	$0.567939\pi$
$390$	0	0
$391$	−17.6128	−0.890720
$392$	0	0
$393$	−0.561993	−0.0283488
$394$	0	0
$395$	6.59502	0.331831
$396$	0	0
$397$	22.1847	1.11342	0.556709	−	0.830708i	$-0.312064\pi$
$397$	22.1847	1.11342	0.556709	+	0.830708i	$0.312064\pi$
$398$	0	0
$399$	2.32693	0.116492
$400$	0	0
$401$	20.6987	1.03365	0.516823	−	0.856092i	$-0.327115\pi$
$401$	20.6987	1.03365	0.516823	+	0.856092i	$0.327115\pi$
$402$	0	0
$403$	4.68889	0.233570
$404$	0	0
$405$	1.31111	0.0651495
$406$	0	0
$407$	8.85728	0.439039
$408$	0	0
$409$	20.8528	1.03111	0.515553	−	0.856858i	$-0.327586\pi$
$409$	20.8528	1.03111	0.515553	+	0.856858i	$0.327586\pi$
$410$	0	0
$411$	−5.07604	−0.250383
$412$	0	0
$413$	14.3051	0.703909
$414$	0	0
$415$	−7.05086	−0.346113
$416$	0	0
$417$	18.5511	0.908451
$418$	0	0
$419$	21.2400	1.03764	0.518821	−	0.854883i	$-0.326371\pi$
$419$	21.2400	1.03764	0.518821	+	0.854883i	$0.326371\pi$
$420$	0	0
$421$	−3.15257	−0.153647	−0.0768233	−	0.997045i	$-0.524478\pi$
$421$	−3.15257	−0.153647	−0.0768233	+	0.997045i	$0.524478\pi$
$422$	0	0
$423$	−9.18421	−0.446551
$424$	0	0
$425$	−7.26517	−0.352413
$426$	0	0
$427$	17.5111	0.847423
$428$	0	0
$429$	1.00000	0.0482805
$430$	0	0
$431$	−40.1116	−1.93211	−0.966053	−	0.258345i	$-0.916823\pi$
$431$	−40.1116	−1.93211	−0.966053	+	0.258345i	$0.916823\pi$
$432$	0	0
$433$	−36.1017	−1.73494	−0.867469	−	0.497492i	$-0.834254\pi$
$433$	−36.1017	−1.73494	−0.867469	+	0.497492i	$0.834254\pi$
$434$	0	0
$435$	−9.70027	−0.465092
$436$	0	0
$437$	−12.1334	−0.580417
$438$	0	0
$439$	−13.2050	−0.630238	−0.315119	−	0.949052i	$-0.602044\pi$
$439$	−13.2050	−0.630238	−0.315119	+	0.949052i	$0.602044\pi$
$440$	0	0
$441$	−4.67307	−0.222527
$442$	0	0
$443$	−13.9813	−0.664270	−0.332135	−	0.943232i	$-0.607769\pi$
$443$	−13.9813	−0.664270	−0.332135	+	0.943232i	$0.607769\pi$
$444$	0	0
$445$	0.903212	0.0428164
$446$	0	0
$447$	4.38271	0.207295
$448$	0	0
$449$	19.3210	0.911812	0.455906	−	0.890028i	$-0.349315\pi$
$449$	19.3210	0.911812	0.455906	+	0.890028i	$0.349315\pi$
$450$	0	0
$451$	−3.52543	−0.166006
$452$	0	0
$453$	1.23014	0.0577971
$454$	0	0
$455$	2.00000	0.0937614
$456$	0	0
$457$	−36.3970	−1.70258	−0.851290	−	0.524696i	$-0.824179\pi$
$457$	−36.3970	−1.70258	−0.851290	+	0.524696i	$0.824179\pi$
$458$	0	0
$459$	−2.21432	−0.103356
$460$	0	0
$461$	−23.5254	−1.09569	−0.547844	−	0.836580i	$-0.684551\pi$
$461$	−23.5254	−1.09569	−0.547844	+	0.836580i	$0.684551\pi$
$462$	0	0
$463$	36.3531	1.68947	0.844735	−	0.535184i	$-0.179758\pi$
$463$	36.3531	1.68947	0.844735	+	0.535184i	$0.179758\pi$
$464$	0	0
$465$	6.14764	0.285090
$466$	0	0
$467$	32.5161	1.50466	0.752332	−	0.658784i	$-0.228929\pi$
$467$	32.5161	1.50466	0.752332	+	0.658784i	$0.228929\pi$
$468$	0	0
$469$	8.00984	0.369860
$470$	0	0
$471$	−7.62714	−0.351440
$472$	0	0
$473$	8.77631	0.403535
$474$	0	0
$475$	−5.00492	−0.229642
$476$	0	0
$477$	3.67307	0.168178
$478$	0	0
$479$	−33.6499	−1.53750	−0.768751	−	0.639548i	$-0.779122\pi$
$479$	−33.6499	−1.53750	−0.768751	+	0.639548i	$0.779122\pi$
$480$	0	0
$481$	−8.85728	−0.403857
$482$	0	0
$483$	−12.1334	−0.552087
$484$	0	0
$485$	−16.8573	−0.765450
$486$	0	0
$487$	−17.0988	−0.774820	−0.387410	−	0.921907i	$-0.626630\pi$
$487$	−17.0988	−0.774820	−0.387410	+	0.921907i	$0.626630\pi$
$488$	0	0
$489$	7.25088	0.327896
$490$	0	0
$491$	−30.2034	−1.36306	−0.681531	−	0.731790i	$-0.738685\pi$
$491$	−30.2034	−1.36306	−0.681531	+	0.731790i	$0.738685\pi$
$492$	0	0
$493$	16.3827	0.737840
$494$	0	0
$495$	1.31111	0.0589299
$496$	0	0
$497$	−22.0731	−0.990115
$498$	0	0
$499$	−38.1367	−1.70724	−0.853618	−	0.520900i	$-0.825596\pi$
$499$	−38.1367	−1.70724	−0.853618	+	0.520900i	$0.825596\pi$
$500$	0	0
$501$	0.295286	0.0131924
$502$	0	0
$503$	23.8666	1.06416	0.532081	−	0.846694i	$-0.321410\pi$
$503$	23.8666	1.06416	0.532081	+	0.846694i	$0.321410\pi$
$504$	0	0
$505$	−21.0049	−0.934707
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	12.8222	0.568336	0.284168	−	0.958774i	$-0.408283\pi$
$509$	12.8222	0.568336	0.284168	+	0.958774i	$0.408283\pi$
$510$	0	0
$511$	−4.78721	−0.211774
$512$	0	0
$513$	−1.52543	−0.0673493
$514$	0	0
$515$	9.98126	0.439827
$516$	0	0
$517$	−9.18421	−0.403921
$518$	0	0
$519$	17.5605	0.770819
$520$	0	0
$521$	3.63158	0.159103	0.0795513	−	0.996831i	$-0.474651\pi$
$521$	3.63158	0.159103	0.0795513	+	0.996831i	$0.474651\pi$
$522$	0	0
$523$	9.40837	0.411399	0.205700	−	0.978615i	$-0.434053\pi$
$523$	9.40837	0.411399	0.205700	+	0.978615i	$0.434053\pi$
$524$	0	0
$525$	−5.00492	−0.218433
$526$	0	0
$527$	−10.3827	−0.452278
$528$	0	0
$529$	40.2672	1.75075
$530$	0	0
$531$	−9.37778	−0.406961
$532$	0	0
$533$	3.52543	0.152703
$534$	0	0
$535$	17.0321	0.736363
$536$	0	0
$537$	12.0874	0.521611
$538$	0	0
$539$	−4.67307	−0.201283
$540$	0	0
$541$	−30.0000	−1.28980	−0.644900	−	0.764267i	$-0.723101\pi$
$541$	−30.0000	−1.28980	−0.644900	+	0.764267i	$0.723101\pi$
$542$	0	0
$543$	−7.71900	−0.331254
$544$	0	0
$545$	5.00937	0.214578
$546$	0	0
$547$	−1.73530	−0.0741961	−0.0370981	−	0.999312i	$-0.511811\pi$
$547$	−1.73530	−0.0741961	−0.0370981	+	0.999312i	$0.511811\pi$
$548$	0	0
$549$	−11.4795	−0.489933
$550$	0	0
$551$	11.2859	0.480796
$552$	0	0
$553$	−7.67307	−0.326292
$554$	0	0
$555$	−11.6128	−0.492938
$556$	0	0
$557$	−21.2400	−0.899967	−0.449984	−	0.893037i	$-0.648570\pi$
$557$	−21.2400	−0.899967	−0.449984	+	0.893037i	$0.648570\pi$
$558$	0	0
$559$	−8.77631	−0.371198
$560$	0	0
$561$	−2.21432	−0.0934887
$562$	0	0
$563$	−22.7841	−0.960237	−0.480119	−	0.877204i	$-0.659406\pi$
$563$	−22.7841	−0.960237	−0.480119	+	0.877204i	$0.659406\pi$
$564$	0	0
$565$	−24.0415	−1.01143
$566$	0	0
$567$	−1.52543	−0.0640619
$568$	0	0
$569$	31.9704	1.34027	0.670134	−	0.742240i	$-0.266237\pi$
$569$	31.9704	1.34027	0.670134	+	0.742240i	$0.266237\pi$
$570$	0	0
$571$	−14.7032	−0.615309	−0.307655	−	0.951498i	$-0.599544\pi$
$571$	−14.7032	−0.615309	−0.307655	+	0.951498i	$0.599544\pi$
$572$	0	0
$573$	−16.6035	−0.693620
$574$	0	0
$575$	26.0973	1.08833
$576$	0	0
$577$	18.9491	0.788863	0.394432	−	0.918925i	$-0.370942\pi$
$577$	18.9491	0.788863	0.394432	+	0.918925i	$0.370942\pi$
$578$	0	0
$579$	7.52543	0.312746
$580$	0	0
$581$	8.20342	0.340335
$582$	0	0
$583$	3.67307	0.152123
$584$	0	0
$585$	−1.31111	−0.0542076
$586$	0	0
$587$	29.0736	1.20000	0.599998	−	0.800001i	$-0.295168\pi$
$587$	29.0736	1.20000	0.599998	+	0.800001i	$0.295168\pi$
$588$	0	0
$589$	−7.15257	−0.294716
$590$	0	0
$591$	−12.2494	−0.503871
$592$	0	0
$593$	0.312639	0.0128386	0.00641928	−	0.999979i	$-0.497957\pi$
$593$	0.312639	0.0128386	0.00641928	+	0.999979i	$0.497957\pi$
$594$	0	0
$595$	−4.42864	−0.181557
$596$	0	0
$597$	11.6128	0.475282
$598$	0	0
$599$	19.8666	0.811729	0.405865	−	0.913933i	$-0.366970\pi$
$599$	19.8666	0.811729	0.405865	+	0.913933i	$0.366970\pi$
$600$	0	0
$601$	−23.9813	−0.978216	−0.489108	−	0.872223i	$-0.662678\pi$
$601$	−23.9813	−0.978216	−0.489108	+	0.872223i	$0.662678\pi$
$602$	0	0
$603$	−5.25088	−0.213833
$604$	0	0
$605$	1.31111	0.0533041
$606$	0	0
$607$	−12.5412	−0.509034	−0.254517	−	0.967068i	$-0.581916\pi$
$607$	−12.5412	−0.509034	−0.254517	+	0.967068i	$0.581916\pi$
$608$	0	0
$609$	11.2859	0.457328
$610$	0	0
$611$	9.18421	0.371553
$612$	0	0
$613$	3.79213	0.153163	0.0765814	−	0.997063i	$-0.475599\pi$
$613$	3.79213	0.153163	0.0765814	+	0.997063i	$0.475599\pi$
$614$	0	0
$615$	4.62222	0.186386
$616$	0	0
$617$	0.278989	0.0112317	0.00561583	−	0.999984i	$-0.498212\pi$
$617$	0.278989	0.0112317	0.00561583	+	0.999984i	$0.498212\pi$
$618$	0	0
$619$	−9.69826	−0.389806	−0.194903	−	0.980823i	$-0.562439\pi$
$619$	−9.69826	−0.389806	−0.194903	+	0.980823i	$0.562439\pi$
$620$	0	0
$621$	7.95407	0.319186
$622$	0	0
$623$	−1.05086	−0.0421016
$624$	0	0
$625$	2.16992	0.0867967
$626$	0	0
$627$	−1.52543	−0.0609197
$628$	0	0
$629$	19.6128	0.782015
$630$	0	0
$631$	−10.6889	−0.425518	−0.212759	−	0.977105i	$-0.568245\pi$
$631$	−10.6889	−0.425518	−0.212759	+	0.977105i	$0.568245\pi$
$632$	0	0
$633$	5.16346	0.205229
$634$	0	0
$635$	26.9862	1.07091
$636$	0	0
$637$	4.67307	0.185154
$638$	0	0
$639$	14.4701	0.572429
$640$	0	0
$641$	31.0321	1.22570	0.612848	−	0.790201i	$-0.290024\pi$
$641$	31.0321	1.22570	0.612848	+	0.790201i	$0.290024\pi$
$642$	0	0
$643$	−24.6099	−0.970521	−0.485261	−	0.874370i	$-0.661275\pi$
$643$	−24.6099	−0.970521	−0.485261	+	0.874370i	$0.661275\pi$
$644$	0	0
$645$	−11.5067	−0.453076
$646$	0	0
$647$	−24.3368	−0.956777	−0.478389	−	0.878148i	$-0.658779\pi$
$647$	−24.3368	−0.956777	−0.478389	+	0.878148i	$0.658779\pi$
$648$	0	0
$649$	−9.37778	−0.368110
$650$	0	0
$651$	−7.15257	−0.280331
$652$	0	0
$653$	−17.7877	−0.696086	−0.348043	−	0.937479i	$-0.613154\pi$
$653$	−17.7877	−0.696086	−0.348043	+	0.937479i	$0.613154\pi$
$654$	0	0
$655$	0.736833	0.0287904
$656$	0	0
$657$	3.13828	0.122436
$658$	0	0
$659$	−28.8256	−1.12289	−0.561444	−	0.827515i	$-0.689754\pi$
$659$	−28.8256	−1.12289	−0.561444	+	0.827515i	$0.689754\pi$
$660$	0	0
$661$	−39.1655	−1.52336	−0.761680	−	0.647953i	$-0.775625\pi$
$661$	−39.1655	−1.52336	−0.761680	+	0.647953i	$0.775625\pi$
$662$	0	0
$663$	2.21432	0.0859971
$664$	0	0
$665$	−3.05086	−0.118307
$666$	0	0
$667$	−58.8484	−2.27862
$668$	0	0
$669$	−8.30174	−0.320964
$670$	0	0
$671$	−11.4795	−0.443161
$672$	0	0
$673$	−46.8671	−1.80659	−0.903297	−	0.429015i	$-0.858861\pi$
$673$	−46.8671	−1.80659	−0.903297	+	0.429015i	$0.858861\pi$
$674$	0	0
$675$	3.28100	0.126286
$676$	0	0
$677$	22.9699	0.882805	0.441402	−	0.897309i	$-0.354481\pi$
$677$	22.9699	0.882805	0.441402	+	0.897309i	$0.354481\pi$
$678$	0	0
$679$	19.6128	0.752672
$680$	0	0
$681$	−5.27163	−0.202009
$682$	0	0
$683$	21.7333	0.831601	0.415801	−	0.909456i	$-0.363501\pi$
$683$	21.7333	0.831601	0.415801	+	0.909456i	$0.363501\pi$
$684$	0	0
$685$	6.65524	0.254284
$686$	0	0
$687$	25.5526	0.974893
$688$	0	0
$689$	−3.67307	−0.139933
$690$	0	0
$691$	−33.4445	−1.27229	−0.636144	−	0.771571i	$-0.719471\pi$
$691$	−33.4445	−1.27229	−0.636144	+	0.771571i	$0.719471\pi$
$692$	0	0
$693$	−1.52543	−0.0579462
$694$	0	0
$695$	−24.3225	−0.922604
$696$	0	0
$697$	−7.80642	−0.295689
$698$	0	0
$699$	−2.40790	−0.0910750
$700$	0	0
$701$	−31.7540	−1.19933	−0.599667	−	0.800250i	$-0.704700\pi$
$701$	−31.7540	−1.19933	−0.599667	+	0.800250i	$0.704700\pi$
$702$	0	0
$703$	13.5111	0.509582
$704$	0	0
$705$	12.0415	0.453509
$706$	0	0
$707$	24.4385	0.919104
$708$	0	0
$709$	25.3778	0.953083	0.476541	−	0.879152i	$-0.341890\pi$
$709$	25.3778	0.953083	0.476541	+	0.879152i	$0.341890\pi$
$710$	0	0
$711$	5.03011	0.188644
$712$	0	0
$713$	37.2958	1.39674
$714$	0	0
$715$	−1.31111	−0.0490327
$716$	0	0
$717$	17.8622	0.667076
$718$	0	0
$719$	11.6128	0.433086	0.216543	−	0.976273i	$-0.430522\pi$
$719$	11.6128	0.433086	0.216543	+	0.976273i	$0.430522\pi$
$720$	0	0
$721$	−11.6128	−0.432485
$722$	0	0
$723$	3.93978	0.146522
$724$	0	0
$725$	−24.2745	−0.901534
$726$	0	0
$727$	42.5303	1.57736	0.788682	−	0.614802i	$-0.210764\pi$
$727$	42.5303	1.57736	0.788682	+	0.614802i	$0.210764\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	19.4336	0.718776
$732$	0	0
$733$	16.2997	0.602044	0.301022	−	0.953617i	$-0.402672\pi$
$733$	16.2997	0.602044	0.301022	+	0.953617i	$0.402672\pi$
$734$	0	0
$735$	6.12690	0.225994
$736$	0	0
$737$	−5.25088	−0.193419
$738$	0	0
$739$	−27.5353	−1.01290	−0.506451	−	0.862269i	$-0.669043\pi$
$739$	−27.5353	−1.01290	−0.506451	+	0.862269i	$0.669043\pi$
$740$	0	0
$741$	1.52543	0.0560380
$742$	0	0
$743$	−34.1432	−1.25259	−0.626296	−	0.779585i	$-0.715430\pi$
$743$	−34.1432	−1.25259	−0.626296	+	0.779585i	$0.715430\pi$
$744$	0	0
$745$	−5.74620	−0.210525
$746$	0	0
$747$	−5.37778	−0.196763
$748$	0	0
$749$	−19.8163	−0.724071
$750$	0	0
$751$	−5.08250	−0.185463	−0.0927315	−	0.995691i	$-0.529560\pi$
$751$	−5.08250	−0.185463	−0.0927315	+	0.995691i	$0.529560\pi$
$752$	0	0
$753$	23.3590	0.851251
$754$	0	0
$755$	−1.61285	−0.0586975
$756$	0	0
$757$	−20.1062	−0.730771	−0.365385	−	0.930856i	$-0.619063\pi$
$757$	−20.1062	−0.730771	−0.365385	+	0.930856i	$0.619063\pi$
$758$	0	0
$759$	7.95407	0.288714
$760$	0	0
$761$	−7.47505	−0.270970	−0.135485	−	0.990779i	$-0.543259\pi$
$761$	−7.47505	−0.270970	−0.135485	+	0.990779i	$0.543259\pi$
$762$	0	0
$763$	−5.82822	−0.210996
$764$	0	0
$765$	2.90321	0.104966
$766$	0	0
$767$	9.37778	0.338612
$768$	0	0
$769$	−50.9403	−1.83695	−0.918476	−	0.395476i	$-0.870580\pi$
$769$	−50.9403	−1.83695	−0.918476	+	0.395476i	$0.870580\pi$
$770$	0	0
$771$	−1.70471	−0.0613938
$772$	0	0
$773$	47.7309	1.71676	0.858380	−	0.513015i	$-0.171471\pi$
$773$	47.7309	1.71676	0.858380	+	0.513015i	$0.171471\pi$
$774$	0	0
$775$	15.3842	0.552618
$776$	0	0
$777$	13.5111	0.484709
$778$	0	0
$779$	−5.37778	−0.192679
$780$	0	0
$781$	14.4701	0.517782
$782$	0	0
$783$	−7.39853	−0.264402
$784$	0	0
$785$	10.0000	0.356915
$786$	0	0
$787$	17.6084	0.627672	0.313836	−	0.949477i	$-0.398386\pi$
$787$	17.6084	0.627672	0.313836	+	0.949477i	$0.398386\pi$
$788$	0	0
$789$	−1.11108	−0.0395554
$790$	0	0
$791$	27.9714	0.994549
$792$	0	0
$793$	11.4795	0.407649
$794$	0	0
$795$	−4.81579	−0.170799
$796$	0	0
$797$	7.32741	0.259550	0.129775	−	0.991543i	$-0.458574\pi$
$797$	7.32741	0.259550	0.129775	+	0.991543i	$0.458574\pi$
$798$	0	0
$799$	−20.3368	−0.719463
$800$	0	0
$801$	0.688892	0.0243408
$802$	0	0
$803$	3.13828	0.110747
$804$	0	0
$805$	15.9081	0.560688
$806$	0	0
$807$	−8.79706	−0.309671
$808$	0	0
$809$	19.2968	0.678440	0.339220	−	0.940707i	$-0.389837\pi$
$809$	19.2968	0.678440	0.339220	+	0.940707i	$0.389837\pi$
$810$	0	0
$811$	−31.6271	−1.11058	−0.555290	−	0.831657i	$-0.687393\pi$
$811$	−31.6271	−1.11058	−0.555290	+	0.831657i	$0.687393\pi$
$812$	0	0
$813$	27.3733	0.960025
$814$	0	0
$815$	−9.50669	−0.333005
$816$	0	0
$817$	13.3876	0.468374
$818$	0	0
$819$	1.52543	0.0533028
$820$	0	0
$821$	37.4652	1.30754	0.653772	−	0.756691i	$-0.273185\pi$
$821$	37.4652	1.30754	0.653772	+	0.756691i	$0.273185\pi$
$822$	0	0
$823$	−28.8988	−1.00735	−0.503674	−	0.863894i	$-0.668019\pi$
$823$	−28.8988	−1.00735	−0.503674	+	0.863894i	$0.668019\pi$
$824$	0	0
$825$	3.28100	0.114230
$826$	0	0
$827$	−19.7319	−0.686146	−0.343073	−	0.939309i	$-0.611468\pi$
$827$	−19.7319	−0.686146	−0.343073	+	0.939309i	$0.611468\pi$
$828$	0	0
$829$	−3.44785	−0.119749	−0.0598744	−	0.998206i	$-0.519070\pi$
$829$	−3.44785	−0.119749	−0.0598744	+	0.998206i	$0.519070\pi$
$830$	0	0
$831$	25.9081	0.898743
$832$	0	0
$833$	−10.3477	−0.358526
$834$	0	0
$835$	−0.387152	−0.0133980
$836$	0	0
$837$	4.68889	0.162072
$838$	0	0
$839$	52.2578	1.80414	0.902070	−	0.431590i	$-0.142047\pi$
$839$	52.2578	1.80414	0.902070	+	0.431590i	$0.142047\pi$
$840$	0	0
$841$	25.7382	0.887525
$842$	0	0
$843$	5.76049	0.198402
$844$	0	0
$845$	1.31111	0.0451035
$846$	0	0
$847$	−1.52543	−0.0524143
$848$	0	0
$849$	−3.03011	−0.103993
$850$	0	0
$851$	−70.4514	−2.41504
$852$	0	0
$853$	31.2988	1.07165	0.535826	−	0.844329i	$-0.320000\pi$
$853$	31.2988	1.07165	0.535826	+	0.844329i	$0.320000\pi$
$854$	0	0
$855$	2.00000	0.0683986
$856$	0	0
$857$	37.1131	1.26776	0.633879	−	0.773432i	$-0.281462\pi$
$857$	37.1131	1.26776	0.633879	+	0.773432i	$0.281462\pi$
$858$	0	0
$859$	16.6953	0.569638	0.284819	−	0.958581i	$-0.408067\pi$
$859$	16.6953	0.569638	0.284819	+	0.958581i	$0.408067\pi$
$860$	0	0
$861$	−5.37778	−0.183274
$862$	0	0
$863$	41.8765	1.42549	0.712746	−	0.701422i	$-0.247451\pi$
$863$	41.8765	1.42549	0.712746	+	0.701422i	$0.247451\pi$
$864$	0	0
$865$	−23.0237	−0.782828
$866$	0	0
$867$	12.0968	0.410828
$868$	0	0
$869$	5.03011	0.170635
$870$	0	0
$871$	5.25088	0.177919
$872$	0	0
$873$	−12.8573	−0.435153
$874$	0	0
$875$	16.5620	0.559898
$876$	0	0
$877$	−6.54909	−0.221147	−0.110573	−	0.993868i	$-0.535269\pi$
$877$	−6.54909	−0.221147	−0.110573	+	0.993868i	$0.535269\pi$
$878$	0	0
$879$	9.07805	0.306195
$880$	0	0
$881$	−15.6316	−0.526641	−0.263321	−	0.964708i	$-0.584818\pi$
$881$	−15.6316	−0.526641	−0.263321	+	0.964708i	$0.584818\pi$
$882$	0	0
$883$	−19.2257	−0.646996	−0.323498	−	0.946229i	$-0.604859\pi$
$883$	−19.2257	−0.646996	−0.323498	+	0.946229i	$0.604859\pi$
$884$	0	0
$885$	12.2953	0.413302
$886$	0	0
$887$	−8.88892	−0.298461	−0.149230	−	0.988802i	$-0.547680\pi$
$887$	−8.88892	−0.298461	−0.149230	+	0.988802i	$0.547680\pi$
$888$	0	0
$889$	−31.3975	−1.05304
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	−14.0098	−0.468822
$894$	0	0
$895$	−15.8479	−0.529737
$896$	0	0
$897$	−7.95407	−0.265579
$898$	0	0
$899$	−34.6909	−1.15701
$900$	0	0
$901$	8.13335	0.270961
$902$	0	0
$903$	13.3876	0.445512
$904$	0	0
$905$	10.1204	0.336415
$906$	0	0
$907$	15.5714	0.517039	0.258519	−	0.966006i	$-0.416765\pi$
$907$	15.5714	0.517039	0.258519	+	0.966006i	$0.416765\pi$
$908$	0	0
$909$	−16.0207	−0.531375
$910$	0	0
$911$	42.5215	1.40880	0.704399	−	0.709804i	$-0.251216\pi$
$911$	42.5215	1.40880	0.704399	+	0.709804i	$0.251216\pi$
$912$	0	0
$913$	−5.37778	−0.177979
$914$	0	0
$915$	15.0509	0.497566
$916$	0	0
$917$	−0.857279	−0.0283098
$918$	0	0
$919$	14.5640	0.480422	0.240211	−	0.970721i	$-0.422783\pi$
$919$	14.5640	0.480422	0.240211	+	0.970721i	$0.422783\pi$
$920$	0	0
$921$	−5.43356	−0.179042
$922$	0	0
$923$	−14.4701	−0.476290
$924$	0	0
$925$	−29.0607	−0.955510
$926$	0	0
$927$	7.61285	0.250039
$928$	0	0
$929$	−39.5689	−1.29821	−0.649107	−	0.760697i	$-0.724857\pi$
$929$	−39.5689	−1.29821	−0.649107	+	0.760697i	$0.724857\pi$
$930$	0	0
$931$	−7.12843	−0.233625
$932$	0	0
$933$	−9.46520	−0.309877
$934$	0	0
$935$	2.90321	0.0949452
$936$	0	0
$937$	31.9625	1.04417	0.522085	−	0.852893i	$-0.325154\pi$
$937$	31.9625	1.04417	0.522085	+	0.852893i	$0.325154\pi$
$938$	0	0
$939$	23.6400	0.771464
$940$	0	0
$941$	50.1891	1.63612	0.818059	−	0.575134i	$-0.195050\pi$
$941$	50.1891	1.63612	0.818059	+	0.575134i	$0.195050\pi$
$942$	0	0
$943$	28.0415	0.913156
$944$	0	0
$945$	2.00000	0.0650600
$946$	0	0
$947$	0.742662	0.0241333	0.0120666	−	0.999927i	$-0.496159\pi$
$947$	0.742662	0.0241333	0.0120666	+	0.999927i	$0.496159\pi$
$948$	0	0
$949$	−3.13828	−0.101873
$950$	0	0
$951$	−15.6479	−0.507417
$952$	0	0
$953$	42.3861	1.37302	0.686510	−	0.727120i	$-0.259142\pi$
$953$	42.3861	1.37302	0.686510	+	0.727120i	$0.259142\pi$
$954$	0	0
$955$	21.7690	0.704427
$956$	0	0
$957$	−7.39853	−0.239160
$958$	0	0
$959$	−7.74314	−0.250039
$960$	0	0
$961$	−9.01429	−0.290784
$962$	0	0
$963$	12.9906	0.418617
$964$	0	0
$965$	−9.86665	−0.317619
$966$	0	0
$967$	33.2083	1.06791	0.533954	−	0.845513i	$-0.320705\pi$
$967$	33.2083	1.06791	0.533954	+	0.845513i	$0.320705\pi$
$968$	0	0
$969$	−3.37778	−0.108510
$970$	0	0
$971$	−29.2904	−0.939973	−0.469986	−	0.882674i	$-0.655741\pi$
$971$	−29.2904	−0.939973	−0.469986	+	0.882674i	$0.655741\pi$
$972$	0	0
$973$	28.2983	0.907203
$974$	0	0
$975$	−3.28100	−0.105076
$976$	0	0
$977$	−28.3847	−0.908107	−0.454054	−	0.890974i	$-0.650023\pi$
$977$	−28.3847	−0.908107	−0.454054	+	0.890974i	$0.650023\pi$
$978$	0	0
$979$	0.688892	0.0220171
$980$	0	0
$981$	3.82071	0.121986
$982$	0	0
$983$	−44.2578	−1.41161	−0.705803	−	0.708409i	$-0.749413\pi$
$983$	−44.2578	−1.41161	−0.705803	+	0.708409i	$0.749413\pi$
$984$	0	0
$985$	16.0602	0.511721
$986$	0	0
$987$	−14.0098	−0.445938
$988$	0	0
$989$	−69.8074	−2.21975
$990$	0	0
$991$	−32.4197	−1.02985	−0.514924	−	0.857236i	$-0.672180\pi$
$991$	−32.4197	−1.02985	−0.514924	+	0.857236i	$0.672180\pi$
$992$	0	0
$993$	−19.6795	−0.624511
$994$	0	0
$995$	−15.2257	−0.482687
$996$	0	0
$997$	42.5205	1.34664	0.673319	−	0.739352i	$-0.264868\pi$
$997$	42.5205	1.34664	0.673319	+	0.739352i	$0.264868\pi$
$998$	0	0
$999$	−8.85728	−0.280232

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6864.2.a.bs.1.2		3
4.3	odd	2		429.2.a.g.1.1	✓	3
12.11	even	2		1287.2.a.h.1.3		3
44.43	even	2		4719.2.a.q.1.3		3
52.51	odd	2		5577.2.a.j.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.a.g.1.1	✓	3	4.3	odd	2
1287.2.a.h.1.3		3	12.11	even	2
4719.2.a.q.1.3		3	44.43	even	2
5577.2.a.j.1.3		3	52.51	odd	2
6864.2.a.bs.1.2		3	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.31111 q^{5} -1.52543 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.31111 q^{5} -1.52543 q^{7} +1.00000 q^{9} +1.00000 q^{11} -1.00000 q^{13} -1.31111 q^{15} +2.21432 q^{17} +1.52543 q^{19} +1.52543 q^{21} -7.95407 q^{23} -3.28100 q^{25} -1.00000 q^{27} +7.39853 q^{29} -4.68889 q^{31} -1.00000 q^{33} -2.00000 q^{35} +8.85728 q^{37} +1.00000 q^{39} -3.52543 q^{41} +8.77631 q^{43} +1.31111 q^{45} -9.18421 q^{47} -4.67307 q^{49} -2.21432 q^{51} +3.67307 q^{53} +1.31111 q^{55} -1.52543 q^{57} -9.37778 q^{59} -11.4795 q^{61} -1.52543 q^{63} -1.31111 q^{65} -5.25088 q^{67} +7.95407 q^{69} +14.4701 q^{71} +3.13828 q^{73} +3.28100 q^{75} -1.52543 q^{77} +5.03011 q^{79} +1.00000 q^{81} -5.37778 q^{83} +2.90321 q^{85} -7.39853 q^{87} +0.688892 q^{89} +1.52543 q^{91} +4.68889 q^{93} +2.00000 q^{95} -12.8573 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 4 q^{5} + 2 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 + 4 * q^5 + 2 * q^7 + 3 * q^9	\( 3 q - 3 q^{3} + 4 q^{5} + 2 q^{7} + 3 q^{9} + 3 q^{11} - 3 q^{13} - 4 q^{15} - 2 q^{19} - 2 q^{21} - 4 q^{23} - 3 q^{25} - 3 q^{27} + 2 q^{29} - 14 q^{31} - 3 q^{33} - 6 q^{35} + 3 q^{39} - 4 q^{41} + 6 q^{43} + 4 q^{45} - 14 q^{47} - q^{49} - 2 q^{53} + 4 q^{55} + 2 q^{57} - 28 q^{59} - 8 q^{61} + 2 q^{63} - 4 q^{65} - 2 q^{67} + 4 q^{69} - 10 q^{71} - 24 q^{73} + 3 q^{75} + 2 q^{77} + 22 q^{79} + 3 q^{81} - 16 q^{83} + 2 q^{85} - 2 q^{87} + 2 q^{89} - 2 q^{91} + 14 q^{93} + 6 q^{95} - 12 q^{97} + 3 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 + 4 * q^5 + 2 * q^7 + 3 * q^9 + 3 * q^11 - 3 * q^13 - 4 * q^15 - 2 * q^19 - 2 * q^21 - 4 * q^23 - 3 * q^25 - 3 * q^27 + 2 * q^29 - 14 * q^31 - 3 * q^33 - 6 * q^35 + 3 * q^39 - 4 * q^41 + 6 * q^43 + 4 * q^45 - 14 * q^47 - q^49 - 2 * q^53 + 4 * q^55 + 2 * q^57 - 28 * q^59 - 8 * q^61 + 2 * q^63 - 4 * q^65 - 2 * q^67 + 4 * q^69 - 10 * q^71 - 24 * q^73 + 3 * q^75 + 2 * q^77 + 22 * q^79 + 3 * q^81 - 16 * q^83 + 2 * q^85 - 2 * q^87 + 2 * q^89 - 2 * q^91 + 14 * q^93 + 6 * q^95 - 12 * q^97 + 3 * q^99

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