LMFDB - Embedded newform 6864.2.a.br.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:	$0$
Dimension:	$3$
Coefficient field:	3.3.564.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 3 $ x^3 - x^2 - 5*x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1716)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.571993$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +0.428007 q^{5} -4.67282 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +0.428007 q^{5} -4.67282 q^{7} +1.00000 q^{9} -1.00000 q^{11} -1.00000 q^{13} -0.428007 q^{15} +3.10083 q^{17} +0.672824 q^{19} +4.67282 q^{21} -6.96080 q^{23} -4.81681 q^{25} -1.00000 q^{27} -5.59046 q^{29} -2.42801 q^{31} +1.00000 q^{33} -2.00000 q^{35} -9.34565 q^{37} +1.00000 q^{39} +8.96080 q^{41} +4.24482 q^{43} +0.428007 q^{45} -1.14399 q^{47} +14.8353 q^{49} -3.10083 q^{51} +8.20166 q^{53} -0.428007 q^{55} -0.672824 q^{57} +3.14399 q^{59} -13.8353 q^{61} -4.67282 q^{63} -0.428007 q^{65} +8.62967 q^{67} +6.96080 q^{69} +5.05767 q^{71} -10.8745 q^{73} +4.81681 q^{75} +4.67282 q^{77} -7.30249 q^{79} +1.00000 q^{81} -8.48963 q^{83} +1.32718 q^{85} +5.59046 q^{87} -5.20561 q^{89} +4.67282 q^{91} +2.42801 q^{93} +0.287973 q^{95} +5.34565 q^{97} -1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + 2 q^{5} - 4 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 + 2 * q^5 - 4 * q^7 + 3 * q^9	$ 3 q - 3 q^{3} + 2 q^{5} - 4 q^{7} + 3 q^{9} - 3 q^{11} - 3 q^{13} - 2 q^{15} - 8 q^{19} + 4 q^{21} - 8 q^{23} - 3 q^{25} - 3 q^{27} + 14 q^{29} - 8 q^{31} + 3 q^{33} - 6 q^{35} - 8 q^{37} + 3 q^{39} + 14 q^{41} + 2 q^{43} + 2 q^{45} - 2 q^{47} + 3 q^{49} + 6 q^{53} - 2 q^{55} + 8 q^{57} + 8 q^{59} - 4 q^{63} - 2 q^{65} + 8 q^{67} + 8 q^{69} - 2 q^{71} - 4 q^{73} + 3 q^{75} + 4 q^{77} + 6 q^{79} + 3 q^{81} - 4 q^{83} + 14 q^{85} - 14 q^{87} + 8 q^{89} + 4 q^{91} + 8 q^{93} - 2 q^{95} - 4 q^{97} - 3 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 + 2 * q^5 - 4 * q^7 + 3 * q^9 - 3 * q^11 - 3 * q^13 - 2 * q^15 - 8 * q^19 + 4 * q^21 - 8 * q^23 - 3 * q^25 - 3 * q^27 + 14 * q^29 - 8 * q^31 + 3 * q^33 - 6 * q^35 - 8 * q^37 + 3 * q^39 + 14 * q^41 + 2 * q^43 + 2 * q^45 - 2 * q^47 + 3 * q^49 + 6 * q^53 - 2 * q^55 + 8 * q^57 + 8 * q^59 - 4 * q^63 - 2 * q^65 + 8 * q^67 + 8 * q^69 - 2 * q^71 - 4 * q^73 + 3 * q^75 + 4 * q^77 + 6 * q^79 + 3 * q^81 - 4 * q^83 + 14 * q^85 - 14 * q^87 + 8 * q^89 + 4 * q^91 + 8 * q^93 - 2 * q^95 - 4 * 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
$5$	0.428007	0.191410	0.0957052	−	0.995410i	$-0.469489\pi$
$5$	0.428007	0.191410	0.0957052	+	0.995410i	$0.469489\pi$
$6$	0	0
$7$	−4.67282	−1.76616	−0.883081	−	0.469221i	$-0.844535\pi$
$7$	−4.67282	−1.76616	−0.883081	+	0.469221i	$0.844535\pi$
$8$	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
$15$	−0.428007	−0.110511
$16$	0	0
$17$	3.10083	0.752062	0.376031	−	0.926607i	$-0.377289\pi$
$17$	3.10083	0.752062	0.376031	+	0.926607i	$0.377289\pi$
$18$	0	0
$19$	0.672824	0.154356	0.0771782	−	0.997017i	$-0.475409\pi$
$19$	0.672824	0.154356	0.0771782	+	0.997017i	$0.475409\pi$
$20$	0	0
$21$	4.67282	1.01969
$22$	0	0
$23$	−6.96080	−1.45143	−0.725713	−	0.687997i	$-0.758490\pi$
$23$	−6.96080	−1.45143	−0.725713	+	0.687997i	$0.758490\pi$
$24$	0	0
$25$	−4.81681	−0.963362
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−5.59046	−1.03812	−0.519062	−	0.854737i	$-0.673718\pi$
$29$	−5.59046	−1.03812	−0.519062	+	0.854737i	$0.673718\pi$
$30$	0	0
$31$	−2.42801	−0.436083	−0.218041	−	0.975940i	$-0.569967\pi$
$31$	−2.42801	−0.436083	−0.218041	+	0.975940i	$0.569967\pi$
$32$	0	0
$33$	1.00000	0.174078
$34$	0	0
$35$	−2.00000	−0.338062
$36$	0	0
$37$	−9.34565	−1.53641	−0.768207	−	0.640201i	$-0.778851\pi$
$37$	−9.34565	−1.53641	−0.768207	+	0.640201i	$0.778851\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	8.96080	1.39944	0.699721	−	0.714417i	$-0.253308\pi$
$41$	8.96080	1.39944	0.699721	+	0.714417i	$0.253308\pi$
$42$	0	0
$43$	4.24482	0.647329	0.323664	−	0.946172i	$-0.395085\pi$
$43$	4.24482	0.647329	0.323664	+	0.946172i	$0.395085\pi$
$44$	0	0
$45$	0.428007	0.0638035
$46$	0	0
$47$	−1.14399	−0.166868	−0.0834338	−	0.996513i	$-0.526589\pi$
$47$	−1.14399	−0.166868	−0.0834338	+	0.996513i	$0.526589\pi$
$48$	0	0
$49$	14.8353	2.11933
$50$	0	0
$51$	−3.10083	−0.434203
$52$	0	0
$53$	8.20166	1.12658	0.563292	−	0.826258i	$-0.309534\pi$
$53$	8.20166	1.12658	0.563292	+	0.826258i	$0.309534\pi$
$54$	0	0
$55$	−0.428007	−0.0577124
$56$	0	0
$57$	−0.672824	−0.0891177
$58$	0	0
$59$	3.14399	0.409312	0.204656	−	0.978834i	$-0.434392\pi$
$59$	3.14399	0.409312	0.204656	+	0.978834i	$0.434392\pi$
$60$	0	0
$61$	−13.8353	−1.77143	−0.885713	−	0.464233i	$-0.846330\pi$
$61$	−13.8353	−1.77143	−0.885713	+	0.464233i	$0.846330\pi$
$62$	0	0
$63$	−4.67282	−0.588720
$64$	0	0
$65$	−0.428007	−0.0530877
$66$	0	0
$67$	8.62967	1.05428	0.527141	−	0.849778i	$-0.323264\pi$
$67$	8.62967	1.05428	0.527141	+	0.849778i	$0.323264\pi$
$68$	0	0
$69$	6.96080	0.837981
$70$	0	0
$71$	5.05767	0.600236	0.300118	−	0.953902i	$-0.402974\pi$
$71$	5.05767	0.600236	0.300118	+	0.953902i	$0.402974\pi$
$72$	0	0
$73$	−10.8745	−1.27276	−0.636381	−	0.771375i	$-0.719569\pi$
$73$	−10.8745	−1.27276	−0.636381	+	0.771375i	$0.719569\pi$
$74$	0	0
$75$	4.81681	0.556197
$76$	0	0
$77$	4.67282	0.532518
$78$	0	0
$79$	−7.30249	−0.821594	−0.410797	−	0.911727i	$-0.634750\pi$
$79$	−7.30249	−0.821594	−0.410797	+	0.911727i	$0.634750\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−8.48963	−0.931858	−0.465929	−	0.884822i	$-0.654280\pi$
$83$	−8.48963	−0.931858	−0.465929	+	0.884822i	$0.654280\pi$
$84$	0	0
$85$	1.32718	0.143952
$86$	0	0
$87$	5.59046	0.599361
$88$	0	0
$89$	−5.20561	−0.551794	−0.275897	−	0.961187i	$-0.588975\pi$
$89$	−5.20561	−0.551794	−0.275897	+	0.961187i	$0.588975\pi$
$90$	0	0
$91$	4.67282	0.489845
$92$	0	0
$93$	2.42801	0.251773
$94$	0	0
$95$	0.287973	0.0295454
$96$	0	0
$97$	5.34565	0.542768	0.271384	−	0.962471i	$-0.412519\pi$
$97$	5.34565	0.542768	0.271384	+	0.962471i	$0.412519\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	0	0
$101$	−4.44648	−0.442441	−0.221221	−	0.975224i	$-0.571004\pi$
$101$	−4.44648	−0.442441	−0.221221	+	0.975224i	$0.571004\pi$
$102$	0	0
$103$	−12.9793	−1.27889	−0.639443	−	0.768839i	$-0.720835\pi$
$103$	−12.9793	−1.27889	−0.639443	+	0.768839i	$0.720835\pi$
$104$	0	0
$105$	2.00000	0.195180
$106$	0	0
$107$	−17.8353	−1.72420	−0.862101	−	0.506737i	$-0.830852\pi$
$107$	−17.8353	−1.72420	−0.862101	+	0.506737i	$0.830852\pi$
$108$	0	0
$109$	19.4504	1.86301	0.931507	−	0.363724i	$-0.118495\pi$
$109$	19.4504	1.86301	0.931507	+	0.363724i	$0.118495\pi$
$110$	0	0
$111$	9.34565	0.887050
$112$	0	0
$113$	11.8353	1.11337	0.556685	−	0.830724i	$-0.312073\pi$
$113$	11.8353	1.11337	0.556685	+	0.830724i	$0.312073\pi$
$114$	0	0
$115$	−2.97927	−0.277818
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	−14.4896	−1.32826
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−8.96080	−0.807968
$124$	0	0
$125$	−4.20166	−0.375808
$126$	0	0
$127$	9.38880	0.833122	0.416561	−	0.909108i	$-0.363235\pi$
$127$	9.38880	0.833122	0.416561	+	0.909108i	$0.363235\pi$
$128$	0	0
$129$	−4.24482	−0.373735
$130$	0	0
$131$	16.0369	1.40115	0.700577	−	0.713577i	$-0.252926\pi$
$131$	16.0369	1.40115	0.700577	+	0.713577i	$0.252926\pi$
$132$	0	0
$133$	−3.14399	−0.272618
$134$	0	0
$135$	−0.428007	−0.0368370
$136$	0	0
$137$	−5.20561	−0.444746	−0.222373	−	0.974962i	$-0.571380\pi$
$137$	−5.20561	−0.444746	−0.222373	+	0.974962i	$0.571380\pi$
$138$	0	0
$139$	−2.24482	−0.190403	−0.0952014	−	0.995458i	$-0.530350\pi$
$139$	−2.24482	−0.190403	−0.0952014	+	0.995458i	$0.530350\pi$
$140$	0	0
$141$	1.14399	0.0963410
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	−2.39276	−0.198708
$146$	0	0
$147$	−14.8353	−1.22359
$148$	0	0
$149$	−12.5944	−1.03177	−0.515887	−	0.856656i	$-0.672538\pi$
$149$	−12.5944	−1.03177	−0.515887	+	0.856656i	$0.672538\pi$
$150$	0	0
$151$	−10.0185	−0.815292	−0.407646	−	0.913140i	$-0.633650\pi$
$151$	−10.0185	−0.815292	−0.407646	+	0.913140i	$0.633650\pi$
$152$	0	0
$153$	3.10083	0.250687
$154$	0	0
$155$	−1.03920	−0.0834708
$156$	0	0
$157$	−8.47116	−0.676072	−0.338036	−	0.941133i	$-0.609763\pi$
$157$	−8.47116	−0.676072	−0.338036	+	0.941133i	$0.609763\pi$
$158$	0	0
$159$	−8.20166	−0.650434
$160$	0	0
$161$	32.5266	2.56345
$162$	0	0
$163$	−7.97531	−0.624675	−0.312337	−	0.949971i	$-0.601112\pi$
$163$	−7.97531	−0.624675	−0.312337	+	0.949971i	$0.601112\pi$
$164$	0	0
$165$	0.428007	0.0333203
$166$	0	0
$167$	21.7490	1.68299	0.841493	−	0.540268i	$-0.181677\pi$
$167$	21.7490	1.68299	0.841493	+	0.540268i	$0.181677\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0.672824	0.0514521
$172$	0	0
$173$	21.3025	1.61960	0.809799	−	0.586707i	$-0.199576\pi$
$173$	21.3025	1.61960	0.809799	+	0.586707i	$0.199576\pi$
$174$	0	0
$175$	22.5081	1.70145
$176$	0	0
$177$	−3.14399	−0.236316
$178$	0	0
$179$	15.8168	1.18220	0.591102	−	0.806597i	$-0.298693\pi$
$179$	15.8168	1.18220	0.591102	+	0.806597i	$0.298693\pi$
$180$	0	0
$181$	−15.9322	−1.18423	−0.592114	−	0.805854i	$-0.701706\pi$
$181$	−15.9322	−1.18423	−0.592114	+	0.805854i	$0.701706\pi$
$182$	0	0
$183$	13.8353	1.02273
$184$	0	0
$185$	−4.00000	−0.294086
$186$	0	0
$187$	−3.10083	−0.226755
$188$	0	0
$189$	4.67282	0.339898
$190$	0	0
$191$	20.8930	1.51176	0.755881	−	0.654709i	$-0.227209\pi$
$191$	20.8930	1.51176	0.755881	+	0.654709i	$0.227209\pi$
$192$	0	0
$193$	13.1625	0.947454	0.473727	−	0.880672i	$-0.342908\pi$
$193$	13.1625	0.947454	0.473727	+	0.880672i	$0.342908\pi$
$194$	0	0
$195$	0.428007	0.0306502
$196$	0	0
$197$	16.8745	1.20226	0.601129	−	0.799152i	$-0.294718\pi$
$197$	16.8745	1.20226	0.601129	+	0.799152i	$0.294718\pi$
$198$	0	0
$199$	4.00000	0.283552	0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$200$	0	0
$201$	−8.62967	−0.608690
$202$	0	0
$203$	26.1233	1.83349
$204$	0	0
$205$	3.83528	0.267868
$206$	0	0
$207$	−6.96080	−0.483809
$208$	0	0
$209$	−0.672824	−0.0465402
$210$	0	0
$211$	19.0224	1.30956	0.654779	−	0.755821i	$-0.272762\pi$
$211$	19.0224	1.30956	0.654779	+	0.755821i	$0.272762\pi$
$212$	0	0
$213$	−5.05767	−0.346546
$214$	0	0
$215$	1.81681	0.123905
$216$	0	0
$217$	11.3456	0.770193
$218$	0	0
$219$	10.8745	0.734830
$220$	0	0
$221$	−3.10083	−0.208584
$222$	0	0
$223$	16.5434	1.10783	0.553913	−	0.832575i	$-0.313134\pi$
$223$	16.5434	1.10783	0.553913	+	0.832575i	$0.313134\pi$
$224$	0	0
$225$	−4.81681	−0.321121
$226$	0	0
$227$	−21.3641	−1.41799	−0.708993	−	0.705215i	$-0.750850\pi$
$227$	−21.3641	−1.41799	−0.708993	+	0.705215i	$0.750850\pi$
$228$	0	0
$229$	9.25934	0.611874	0.305937	−	0.952052i	$-0.401030\pi$
$229$	9.25934	0.611874	0.305937	+	0.952052i	$0.401030\pi$
$230$	0	0
$231$	−4.67282	−0.307449
$232$	0	0
$233$	−9.46721	−0.620218	−0.310109	−	0.950701i	$-0.600365\pi$
$233$	−9.46721	−0.620218	−0.310109	+	0.950701i	$0.600365\pi$
$234$	0	0
$235$	−0.489634	−0.0319402
$236$	0	0
$237$	7.30249	0.474348
$238$	0	0
$239$	−9.32718	−0.603325	−0.301662	−	0.953415i	$-0.597542\pi$
$239$	−9.32718	−0.603325	−0.301662	+	0.953415i	$0.597542\pi$
$240$	0	0
$241$	−5.54731	−0.357334	−0.178667	−	0.983910i	$-0.557178\pi$
$241$	−5.54731	−0.357334	−0.178667	+	0.983910i	$0.557178\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	6.34960	0.405661
$246$	0	0
$247$	−0.672824	−0.0428107
$248$	0	0
$249$	8.48963	0.538009
$250$	0	0
$251$	18.7776	1.18523	0.592616	−	0.805485i	$-0.298095\pi$
$251$	18.7776	1.18523	0.592616	+	0.805485i	$0.298095\pi$
$252$	0	0
$253$	6.96080	0.437622
$254$	0	0
$255$	−1.32718	−0.0831110
$256$	0	0
$257$	7.92159	0.494135	0.247068	−	0.968998i	$-0.420533\pi$
$257$	7.92159	0.494135	0.247068	+	0.968998i	$0.420533\pi$
$258$	0	0
$259$	43.6706	2.71356
$260$	0	0
$261$	−5.59046	−0.346041
$262$	0	0
$263$	−19.5473	−1.20534	−0.602669	−	0.797991i	$-0.705896\pi$
$263$	−19.5473	−1.20534	−0.602669	+	0.797991i	$0.705896\pi$
$264$	0	0
$265$	3.51037	0.215640
$266$	0	0
$267$	5.20561	0.318578
$268$	0	0
$269$	20.7776	1.26683	0.633417	−	0.773811i	$-0.281652\pi$
$269$	20.7776	1.26683	0.633417	+	0.773811i	$0.281652\pi$
$270$	0	0
$271$	16.1832	0.983059	0.491529	−	0.870861i	$-0.336438\pi$
$271$	16.1832	0.983059	0.491529	+	0.870861i	$0.336438\pi$
$272$	0	0
$273$	−4.67282	−0.282812
$274$	0	0
$275$	4.81681	0.290465
$276$	0	0
$277$	10.2096	0.613433	0.306717	−	0.951801i	$-0.400770\pi$
$277$	10.2096	0.613433	0.306717	+	0.951801i	$0.400770\pi$
$278$	0	0
$279$	−2.42801	−0.145361
$280$	0	0
$281$	16.5081	0.984791	0.492395	−	0.870372i	$-0.336121\pi$
$281$	16.5081	0.984791	0.492395	+	0.870372i	$0.336121\pi$
$282$	0	0
$283$	26.8577	1.59652	0.798262	−	0.602310i	$-0.205753\pi$
$283$	26.8577	1.59652	0.798262	+	0.602310i	$0.205753\pi$
$284$	0	0
$285$	−0.287973	−0.0170581
$286$	0	0
$287$	−41.8722	−2.47164
$288$	0	0
$289$	−7.38485	−0.434403
$290$	0	0
$291$	−5.34565	−0.313367
$292$	0	0
$293$	−25.4874	−1.48899	−0.744494	−	0.667629i	$-0.767309\pi$
$293$	−25.4874	−1.48899	−0.744494	+	0.667629i	$0.767309\pi$
$294$	0	0
$295$	1.34565	0.0783466
$296$	0	0
$297$	1.00000	0.0580259
$298$	0	0
$299$	6.96080	0.402553
$300$	0	0
$301$	−19.8353	−1.14329
$302$	0	0
$303$	4.44648	0.255443
$304$	0	0
$305$	−5.92159	−0.339069
$306$	0	0
$307$	0.635881	0.0362917	0.0181458	−	0.999835i	$-0.494224\pi$
$307$	0.635881	0.0362917	0.0181458	+	0.999835i	$0.494224\pi$
$308$	0	0
$309$	12.9793	0.738365
$310$	0	0
$311$	8.26950	0.468920	0.234460	−	0.972126i	$-0.424668\pi$
$311$	8.26950	0.468920	0.234460	+	0.972126i	$0.424668\pi$
$312$	0	0
$313$	−13.7305	−0.776094	−0.388047	−	0.921640i	$-0.626850\pi$
$313$	−13.7305	−0.776094	−0.388047	+	0.921640i	$0.626850\pi$
$314$	0	0
$315$	−2.00000	−0.112687
$316$	0	0
$317$	−34.5882	−1.94267	−0.971334	−	0.237721i	$-0.923600\pi$
$317$	−34.5882	−1.94267	−0.971334	+	0.237721i	$0.923600\pi$
$318$	0	0
$319$	5.59046	0.313006
$320$	0	0
$321$	17.8353	0.995468
$322$	0	0
$323$	2.08631	0.116086
$324$	0	0
$325$	4.81681	0.267189
$326$	0	0
$327$	−19.4504	−1.07561
$328$	0	0
$329$	5.34565	0.294715
$330$	0	0
$331$	7.08236	0.389282	0.194641	−	0.980875i	$-0.437646\pi$
$331$	7.08236	0.389282	0.194641	+	0.980875i	$0.437646\pi$
$332$	0	0
$333$	−9.34565	−0.512138
$334$	0	0
$335$	3.69356	0.201801
$336$	0	0
$337$	−34.1233	−1.85881	−0.929406	−	0.369059i	$-0.879680\pi$
$337$	−34.1233	−1.85881	−0.929406	+	0.369059i	$0.879680\pi$
$338$	0	0
$339$	−11.8353	−0.642804
$340$	0	0
$341$	2.42801	0.131484
$342$	0	0
$343$	−36.6129	−1.97691
$344$	0	0
$345$	2.97927	0.160398
$346$	0	0
$347$	4.65435	0.249859	0.124929	−	0.992166i	$-0.460130\pi$
$347$	4.65435	0.249859	0.124929	+	0.992166i	$0.460130\pi$
$348$	0	0
$349$	8.00791	0.428653	0.214327	−	0.976762i	$-0.431244\pi$
$349$	8.00791	0.428653	0.214327	+	0.976762i	$0.431244\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	10.5927	0.563794	0.281897	−	0.959445i	$-0.409036\pi$
$353$	10.5927	0.563794	0.281897	+	0.959445i	$0.409036\pi$
$354$	0	0
$355$	2.16472	0.114891
$356$	0	0
$357$	14.4896	0.766873
$358$	0	0
$359$	7.03920	0.371515	0.185757	−	0.982596i	$-0.440526\pi$
$359$	7.03920	0.371515	0.185757	+	0.982596i	$0.440526\pi$
$360$	0	0
$361$	−18.5473	−0.976174
$362$	0	0
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	−4.65435	−0.243620
$366$	0	0
$367$	12.0863	0.630900	0.315450	−	0.948942i	$-0.397844\pi$
$367$	12.0863	0.630900	0.315450	+	0.948942i	$0.397844\pi$
$368$	0	0
$369$	8.96080	0.466480
$370$	0	0
$371$	−38.3249	−1.98973
$372$	0	0
$373$	15.2593	0.790098	0.395049	−	0.918660i	$-0.370728\pi$
$373$	15.2593	0.790098	0.395049	+	0.918660i	$0.370728\pi$
$374$	0	0
$375$	4.20166	0.216973
$376$	0	0
$377$	5.59046	0.287924
$378$	0	0
$379$	21.2840	1.09329	0.546643	−	0.837366i	$-0.315905\pi$
$379$	21.2840	1.09329	0.546643	+	0.837366i	$0.315905\pi$
$380$	0	0
$381$	−9.38880	−0.481003
$382$	0	0
$383$	3.91369	0.199980	0.0999901	−	0.994988i	$-0.468119\pi$
$383$	3.91369	0.199980	0.0999901	+	0.994988i	$0.468119\pi$
$384$	0	0
$385$	2.00000	0.101929
$386$	0	0
$387$	4.24482	0.215776
$388$	0	0
$389$	9.54731	0.484068	0.242034	−	0.970268i	$-0.422185\pi$
$389$	9.54731	0.484068	0.242034	+	0.970268i	$0.422185\pi$
$390$	0	0
$391$	−21.5843	−1.09156
$392$	0	0
$393$	−16.0369	−0.808957
$394$	0	0
$395$	−3.12552	−0.157262
$396$	0	0
$397$	21.7490	1.09155	0.545775	−	0.837932i	$-0.316235\pi$
$397$	21.7490	1.09155	0.545775	+	0.837932i	$0.316235\pi$
$398$	0	0
$399$	3.14399	0.157396
$400$	0	0
$401$	−18.3496	−0.916335	−0.458168	−	0.888866i	$-0.651494\pi$
$401$	−18.3496	−0.916335	−0.458168	+	0.888866i	$0.651494\pi$
$402$	0	0
$403$	2.42801	0.120948
$404$	0	0
$405$	0.428007	0.0212678
$406$	0	0
$407$	9.34565	0.463247
$408$	0	0
$409$	−2.87448	−0.142134	−0.0710671	−	0.997472i	$-0.522640\pi$
$409$	−2.87448	−0.142134	−0.0710671	+	0.997472i	$0.522640\pi$
$410$	0	0
$411$	5.20561	0.256774
$412$	0	0
$413$	−14.6913	−0.722911
$414$	0	0
$415$	−3.63362	−0.178367
$416$	0	0
$417$	2.24482	0.109929
$418$	0	0
$419$	2.66492	0.130190	0.0650949	−	0.997879i	$-0.479265\pi$
$419$	2.66492	0.130190	0.0650949	+	0.997879i	$0.479265\pi$
$420$	0	0
$421$	22.0369	1.07401	0.537007	−	0.843578i	$-0.319555\pi$
$421$	22.0369	1.07401	0.537007	+	0.843578i	$0.319555\pi$
$422$	0	0
$423$	−1.14399	−0.0556225
$424$	0	0
$425$	−14.9361	−0.724508
$426$	0	0
$427$	64.6498	3.12862
$428$	0	0
$429$	−1.00000	−0.0482805
$430$	0	0
$431$	−0.366380	−0.0176479	−0.00882394	−	0.999961i	$-0.502809\pi$
$431$	−0.366380	−0.0176479	−0.00882394	+	0.999961i	$0.502809\pi$
$432$	0	0
$433$	−21.8432	−1.04972	−0.524858	−	0.851190i	$-0.675882\pi$
$433$	−21.8432	−1.04972	−0.524858	+	0.851190i	$0.675882\pi$
$434$	0	0
$435$	2.39276	0.114724
$436$	0	0
$437$	−4.68339	−0.224037
$438$	0	0
$439$	−11.5984	−0.553560	−0.276780	−	0.960933i	$-0.589267\pi$
$439$	−11.5984	−0.553560	−0.276780	+	0.960933i	$0.589267\pi$
$440$	0	0
$441$	14.8353	0.706442
$442$	0	0
$443$	20.9793	0.996755	0.498378	−	0.866960i	$-0.333929\pi$
$443$	20.9793	0.996755	0.498378	+	0.866960i	$0.333929\pi$
$444$	0	0
$445$	−2.22804	−0.105619
$446$	0	0
$447$	12.5944	0.595695
$448$	0	0
$449$	6.34960	0.299656	0.149828	−	0.988712i	$-0.452128\pi$
$449$	6.34960	0.299656	0.149828	+	0.988712i	$0.452128\pi$
$450$	0	0
$451$	−8.96080	−0.421947
$452$	0	0
$453$	10.0185	0.470709
$454$	0	0
$455$	2.00000	0.0937614
$456$	0	0
$457$	6.20957	0.290471	0.145236	−	0.989397i	$-0.453606\pi$
$457$	6.20957	0.290471	0.145236	+	0.989397i	$0.453606\pi$
$458$	0	0
$459$	−3.10083	−0.144734
$460$	0	0
$461$	27.0761	1.26106	0.630531	−	0.776164i	$-0.282837\pi$
$461$	27.0761	1.26106	0.630531	+	0.776164i	$0.282837\pi$
$462$	0	0
$463$	14.3496	0.666882	0.333441	−	0.942771i	$-0.391790\pi$
$463$	14.3496	0.666882	0.333441	+	0.942771i	$0.391790\pi$
$464$	0	0
$465$	1.03920	0.0481919
$466$	0	0
$467$	−5.82472	−0.269536	−0.134768	−	0.990877i	$-0.543029\pi$
$467$	−5.82472	−0.269536	−0.134768	+	0.990877i	$0.543029\pi$
$468$	0	0
$469$	−40.3249	−1.86203
$470$	0	0
$471$	8.47116	0.390331
$472$	0	0
$473$	−4.24482	−0.195177
$474$	0	0
$475$	−3.24086	−0.148701
$476$	0	0
$477$	8.20166	0.375528
$478$	0	0
$479$	−7.65209	−0.349633	−0.174816	−	0.984601i	$-0.555933\pi$
$479$	−7.65209	−0.349633	−0.174816	+	0.984601i	$0.555933\pi$
$480$	0	0
$481$	9.34565	0.426125
$482$	0	0
$483$	−32.5266	−1.48001
$484$	0	0
$485$	2.28797	0.103892
$486$	0	0
$487$	42.8392	1.94123	0.970616	−	0.240636i	$-0.0773559\pi$
$487$	42.8392	1.94123	0.970616	+	0.240636i	$0.0773559\pi$
$488$	0	0
$489$	7.97531	0.360656
$490$	0	0
$491$	9.59668	0.433092	0.216546	−	0.976272i	$-0.430521\pi$
$491$	9.59668	0.433092	0.216546	+	0.976272i	$0.430521\pi$
$492$	0	0
$493$	−17.3351	−0.780733
$494$	0	0
$495$	−0.428007	−0.0192375
$496$	0	0
$497$	−23.6336	−1.06011
$498$	0	0
$499$	42.2218	1.89011	0.945054	−	0.326914i	$-0.106009\pi$
$499$	42.2218	1.89011	0.945054	+	0.326914i	$0.106009\pi$
$500$	0	0
$501$	−21.7490	−0.971672
$502$	0	0
$503$	−3.87675	−0.172856	−0.0864278	−	0.996258i	$-0.527545\pi$
$503$	−3.87675	−0.172856	−0.0864278	+	0.996258i	$0.527545\pi$
$504$	0	0
$505$	−1.90312	−0.0846878
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	18.3417	0.812981	0.406491	−	0.913655i	$-0.366752\pi$
$509$	18.3417	0.812981	0.406491	+	0.913655i	$0.366752\pi$
$510$	0	0
$511$	50.8145	2.24790
$512$	0	0
$513$	−0.672824	−0.0297059
$514$	0	0
$515$	−5.55521	−0.244792
$516$	0	0
$517$	1.14399	0.0503125
$518$	0	0
$519$	−21.3025	−0.935076
$520$	0	0
$521$	−38.4033	−1.68248	−0.841240	−	0.540662i	$-0.818174\pi$
$521$	−38.4033	−1.68248	−0.841240	+	0.540662i	$0.818174\pi$
$522$	0	0
$523$	−4.82076	−0.210797	−0.105399	−	0.994430i	$-0.533612\pi$
$523$	−4.82076	−0.210797	−0.105399	+	0.994430i	$0.533612\pi$
$524$	0	0
$525$	−22.5081	−0.982334
$526$	0	0
$527$	−7.52884	−0.327961
$528$	0	0
$529$	25.4527	1.10664
$530$	0	0
$531$	3.14399	0.136437
$532$	0	0
$533$	−8.96080	−0.388135
$534$	0	0
$535$	−7.63362	−0.330030
$536$	0	0
$537$	−15.8168	−0.682546
$538$	0	0
$539$	−14.8353	−0.639001
$540$	0	0
$541$	15.3087	0.658173	0.329086	−	0.944300i	$-0.393259\pi$
$541$	15.3087	0.658173	0.329086	+	0.944300i	$0.393259\pi$
$542$	0	0
$543$	15.9322	0.683714
$544$	0	0
$545$	8.32492	0.356600
$546$	0	0
$547$	19.2241	0.821963	0.410981	−	0.911644i	$-0.365186\pi$
$547$	19.2241	0.821963	0.410981	+	0.911644i	$0.365186\pi$
$548$	0	0
$549$	−13.8353	−0.590475
$550$	0	0
$551$	−3.76140	−0.160241
$552$	0	0
$553$	34.1233	1.45107
$554$	0	0
$555$	4.00000	0.169791
$556$	0	0
$557$	14.3434	0.607749	0.303874	−	0.952712i	$-0.401720\pi$
$557$	14.3434	0.607749	0.303874	+	0.952712i	$0.401720\pi$
$558$	0	0
$559$	−4.24482	−0.179537
$560$	0	0
$561$	3.10083	0.130917
$562$	0	0
$563$	−30.7282	−1.29504	−0.647520	−	0.762048i	$-0.724194\pi$
$563$	−30.7282	−1.29504	−0.647520	+	0.762048i	$0.724194\pi$
$564$	0	0
$565$	5.06558	0.213111
$566$	0	0
$567$	−4.67282	−0.196240
$568$	0	0
$569$	−0.985482	−0.0413135	−0.0206568	−	0.999787i	$-0.506576\pi$
$569$	−0.985482	−0.0413135	−0.0206568	+	0.999787i	$0.506576\pi$
$570$	0	0
$571$	9.10083	0.380858	0.190429	−	0.981701i	$-0.439012\pi$
$571$	9.10083	0.380858	0.190429	+	0.981701i	$0.439012\pi$
$572$	0	0
$573$	−20.8930	−0.872816
$574$	0	0
$575$	33.5288	1.39825
$576$	0	0
$577$	34.4403	1.43377	0.716883	−	0.697193i	$-0.245568\pi$
$577$	34.4403	1.43377	0.716883	+	0.697193i	$0.245568\pi$
$578$	0	0
$579$	−13.1625	−0.547013
$580$	0	0
$581$	39.6706	1.64581
$582$	0	0
$583$	−8.20166	−0.339678
$584$	0	0
$585$	−0.428007	−0.0176959
$586$	0	0
$587$	−7.54731	−0.311511	−0.155755	−	0.987796i	$-0.549781\pi$
$587$	−7.54731	−0.311511	−0.155755	+	0.987796i	$0.549781\pi$
$588$	0	0
$589$	−1.63362	−0.0673122
$590$	0	0
$591$	−16.8745	−0.694124
$592$	0	0
$593$	−39.6027	−1.62629	−0.813144	−	0.582062i	$-0.802246\pi$
$593$	−39.6027	−1.62629	−0.813144	+	0.582062i	$0.802246\pi$
$594$	0	0
$595$	−6.20166	−0.254243
$596$	0	0
$597$	−4.00000	−0.163709
$598$	0	0
$599$	40.7361	1.66443	0.832217	−	0.554450i	$-0.187071\pi$
$599$	40.7361	1.66443	0.832217	+	0.554450i	$0.187071\pi$
$600$	0	0
$601$	15.5552	0.634510	0.317255	−	0.948340i	$-0.397239\pi$
$601$	15.5552	0.634510	0.317255	+	0.948340i	$0.397239\pi$
$602$	0	0
$603$	8.62967	0.351427
$604$	0	0
$605$	0.428007	0.0174009
$606$	0	0
$607$	−41.3025	−1.67642	−0.838208	−	0.545350i	$-0.816397\pi$
$607$	−41.3025	−1.67642	−0.838208	+	0.545350i	$0.816397\pi$
$608$	0	0
$609$	−26.1233	−1.05857
$610$	0	0
$611$	1.14399	0.0462807
$612$	0	0
$613$	29.5288	1.19266	0.596329	−	0.802740i	$-0.296625\pi$
$613$	29.5288	1.19266	0.596329	+	0.802740i	$0.296625\pi$
$614$	0	0
$615$	−3.83528	−0.154653
$616$	0	0
$617$	11.0824	0.446159	0.223079	−	0.974800i	$-0.428389\pi$
$617$	11.0824	0.446159	0.223079	+	0.974800i	$0.428389\pi$
$618$	0	0
$619$	10.6297	0.427242	0.213621	−	0.976917i	$-0.431474\pi$
$619$	10.6297	0.427242	0.213621	+	0.976917i	$0.431474\pi$
$620$	0	0
$621$	6.96080	0.279327
$622$	0	0
$623$	24.3249	0.974557
$624$	0	0
$625$	22.2857	0.891428
$626$	0	0
$627$	0.672824	0.0268700
$628$	0	0
$629$	−28.9793	−1.15548
$630$	0	0
$631$	−21.4936	−0.855646	−0.427823	−	0.903862i	$-0.640719\pi$
$631$	−21.4936	−0.855646	−0.427823	+	0.903862i	$0.640719\pi$
$632$	0	0
$633$	−19.0224	−0.756073
$634$	0	0
$635$	4.01847	0.159468
$636$	0	0
$637$	−14.8353	−0.587795
$638$	0	0
$639$	5.05767	0.200079
$640$	0	0
$641$	−45.1316	−1.78259	−0.891295	−	0.453424i	$-0.850202\pi$
$641$	−45.1316	−1.78259	−0.891295	+	0.453424i	$0.850202\pi$
$642$	0	0
$643$	−37.2795	−1.47016	−0.735080	−	0.677980i	$-0.762855\pi$
$643$	−37.2795	−1.47016	−0.735080	+	0.677980i	$0.762855\pi$
$644$	0	0
$645$	−1.81681	−0.0715368
$646$	0	0
$647$	41.4689	1.63031	0.815155	−	0.579242i	$-0.196652\pi$
$647$	41.4689	1.63031	0.815155	+	0.579242i	$0.196652\pi$
$648$	0	0
$649$	−3.14399	−0.123412
$650$	0	0
$651$	−11.3456	−0.444671
$652$	0	0
$653$	−10.3664	−0.405668	−0.202834	−	0.979213i	$-0.565015\pi$
$653$	−10.3664	−0.405668	−0.202834	+	0.979213i	$0.565015\pi$
$654$	0	0
$655$	6.86392	0.268196
$656$	0	0
$657$	−10.8745	−0.424254
$658$	0	0
$659$	−31.4320	−1.22442	−0.612208	−	0.790697i	$-0.709718\pi$
$659$	−31.4320	−1.22442	−0.612208	+	0.790697i	$0.709718\pi$
$660$	0	0
$661$	9.33774	0.363196	0.181598	−	0.983373i	$-0.441873\pi$
$661$	9.33774	0.363196	0.181598	+	0.983373i	$0.441873\pi$
$662$	0	0
$663$	3.10083	0.120426
$664$	0	0
$665$	−1.34565	−0.0521820
$666$	0	0
$667$	38.9141	1.50676
$668$	0	0
$669$	−16.5434	−0.639603
$670$	0	0
$671$	13.8353	0.534105
$672$	0	0
$673$	45.9216	1.77015	0.885074	−	0.465451i	$-0.154108\pi$
$673$	45.9216	1.77015	0.885074	+	0.465451i	$0.154108\pi$
$674$	0	0
$675$	4.81681	0.185399
$676$	0	0
$677$	−23.7921	−0.914406	−0.457203	−	0.889362i	$-0.651149\pi$
$677$	−23.7921	−0.914406	−0.457203	+	0.889362i	$0.651149\pi$
$678$	0	0
$679$	−24.9793	−0.958616
$680$	0	0
$681$	21.3641	0.818675
$682$	0	0
$683$	−37.8643	−1.44884	−0.724419	−	0.689360i	$-0.757892\pi$
$683$	−37.8643	−1.44884	−0.724419	+	0.689360i	$0.757892\pi$
$684$	0	0
$685$	−2.22804	−0.0851289
$686$	0	0
$687$	−9.25934	−0.353266
$688$	0	0
$689$	−8.20166	−0.312458
$690$	0	0
$691$	29.0409	1.10477	0.552384	−	0.833590i	$-0.313718\pi$
$691$	29.0409	1.10477	0.552384	+	0.833590i	$0.313718\pi$
$692$	0	0
$693$	4.67282	0.177506
$694$	0	0
$695$	−0.960797	−0.0364451
$696$	0	0
$697$	27.7859	1.05247
$698$	0	0
$699$	9.46721	0.358083
$700$	0	0
$701$	−10.9071	−0.411955	−0.205977	−	0.978557i	$-0.566037\pi$
$701$	−10.9071	−0.411955	−0.205977	+	0.978557i	$0.566037\pi$
$702$	0	0
$703$	−6.28797	−0.237155
$704$	0	0
$705$	0.489634	0.0184407
$706$	0	0
$707$	20.7776	0.781422
$708$	0	0
$709$	44.7361	1.68010	0.840051	−	0.542508i	$-0.182525\pi$
$709$	44.7361	1.68010	0.840051	+	0.542508i	$0.182525\pi$
$710$	0	0
$711$	−7.30249	−0.273865
$712$	0	0
$713$	16.9009	0.632942
$714$	0	0
$715$	0.428007	0.0160065
$716$	0	0
$717$	9.32718	0.348330
$718$	0	0
$719$	−36.6498	−1.36681	−0.683404	−	0.730040i	$-0.739501\pi$
$719$	−36.6498	−1.36681	−0.683404	+	0.730040i	$0.739501\pi$
$720$	0	0
$721$	60.6498	2.25872
$722$	0	0
$723$	5.54731	0.206307
$724$	0	0
$725$	26.9282	1.00009
$726$	0	0
$727$	9.46890	0.351182	0.175591	−	0.984463i	$-0.443816\pi$
$727$	9.46890	0.351182	0.175591	+	0.984463i	$0.443816\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	13.1625	0.486831
$732$	0	0
$733$	41.3720	1.52811	0.764055	−	0.645151i	$-0.223205\pi$
$733$	41.3720	1.52811	0.764055	+	0.645151i	$0.223205\pi$
$734$	0	0
$735$	−6.34960	−0.234209
$736$	0	0
$737$	−8.62967	−0.317878
$738$	0	0
$739$	5.82472	0.214266	0.107133	−	0.994245i	$-0.465833\pi$
$739$	5.82472	0.214266	0.107133	+	0.994245i	$0.465833\pi$
$740$	0	0
$741$	0.672824	0.0247168
$742$	0	0
$743$	−20.4527	−0.750336	−0.375168	−	0.926957i	$-0.622415\pi$
$743$	−20.4527	−0.750336	−0.375168	+	0.926957i	$0.622415\pi$
$744$	0	0
$745$	−5.39050	−0.197492
$746$	0	0
$747$	−8.48963	−0.310619
$748$	0	0
$749$	83.3411	3.04522
$750$	0	0
$751$	0.489634	0.0178670	0.00893350	−	0.999960i	$-0.497156\pi$
$751$	0.489634	0.0178670	0.00893350	+	0.999960i	$0.497156\pi$
$752$	0	0
$753$	−18.7776	−0.684294
$754$	0	0
$755$	−4.28797	−0.156055
$756$	0	0
$757$	−43.5658	−1.58343	−0.791713	−	0.610894i	$-0.790810\pi$
$757$	−43.5658	−1.58343	−0.791713	+	0.610894i	$0.790810\pi$
$758$	0	0
$759$	−6.96080	−0.252661
$760$	0	0
$761$	−24.8745	−0.901699	−0.450850	−	0.892600i	$-0.648879\pi$
$761$	−24.8745	−0.901699	−0.450850	+	0.892600i	$0.648879\pi$
$762$	0	0
$763$	−90.8884	−3.29038
$764$	0	0
$765$	1.32718	0.0479842
$766$	0	0
$767$	−3.14399	−0.113523
$768$	0	0
$769$	9.42405	0.339840	0.169920	−	0.985458i	$-0.445649\pi$
$769$	9.42405	0.339840	0.169920	+	0.985458i	$0.445649\pi$
$770$	0	0
$771$	−7.92159	−0.285289
$772$	0	0
$773$	19.0409	0.684853	0.342427	−	0.939545i	$-0.388751\pi$
$773$	19.0409	0.684853	0.342427	+	0.939545i	$0.388751\pi$
$774$	0	0
$775$	11.6952	0.420106
$776$	0	0
$777$	−43.6706	−1.56667
$778$	0	0
$779$	6.02904	0.216013
$780$	0	0
$781$	−5.05767	−0.180978
$782$	0	0
$783$	5.59046	0.199787
$784$	0	0
$785$	−3.62571	−0.129407
$786$	0	0
$787$	16.0969	0.573792	0.286896	−	0.957962i	$-0.407377\pi$
$787$	16.0969	0.573792	0.286896	+	0.957962i	$0.407377\pi$
$788$	0	0
$789$	19.5473	0.695903
$790$	0	0
$791$	−55.3042	−1.96639
$792$	0	0
$793$	13.8353	0.491305
$794$	0	0
$795$	−3.51037	−0.124500
$796$	0	0
$797$	−7.95854	−0.281906	−0.140953	−	0.990016i	$-0.545017\pi$
$797$	−7.95854	−0.281906	−0.140953	+	0.990016i	$0.545017\pi$
$798$	0	0
$799$	−3.54731	−0.125495
$800$	0	0
$801$	−5.20561	−0.183931
$802$	0	0
$803$	10.8745	0.383752
$804$	0	0
$805$	13.9216	0.490672
$806$	0	0
$807$	−20.7776	−0.731406
$808$	0	0
$809$	−39.5042	−1.38889	−0.694446	−	0.719545i	$-0.744350\pi$
$809$	−39.5042	−1.38889	−0.694446	+	0.719545i	$0.744350\pi$
$810$	0	0
$811$	−16.2695	−0.571299	−0.285650	−	0.958334i	$-0.592209\pi$
$811$	−16.2695	−0.571299	−0.285650	+	0.958334i	$0.592209\pi$
$812$	0	0
$813$	−16.1832	−0.567569
$814$	0	0
$815$	−3.41349	−0.119569
$816$	0	0
$817$	2.85601	0.0999193
$818$	0	0
$819$	4.67282	0.163282
$820$	0	0
$821$	48.3355	1.68692	0.843460	−	0.537192i	$-0.180515\pi$
$821$	48.3355	1.68692	0.843460	+	0.537192i	$0.180515\pi$
$822$	0	0
$823$	−38.0448	−1.32616	−0.663080	−	0.748549i	$-0.730751\pi$
$823$	−38.0448	−1.32616	−0.663080	+	0.748549i	$0.730751\pi$
$824$	0	0
$825$	−4.81681	−0.167700
$826$	0	0
$827$	−27.0761	−0.941530	−0.470765	−	0.882259i	$-0.656022\pi$
$827$	−27.0761	−0.941530	−0.470765	+	0.882259i	$0.656022\pi$
$828$	0	0
$829$	−13.4399	−0.466786	−0.233393	−	0.972383i	$-0.574983\pi$
$829$	−13.4399	−0.466786	−0.233393	+	0.972383i	$0.574983\pi$
$830$	0	0
$831$	−10.2096	−0.354166
$832$	0	0
$833$	46.0017	1.59386
$834$	0	0
$835$	9.30871	0.322141
$836$	0	0
$837$	2.42801	0.0839242
$838$	0	0
$839$	−12.1938	−0.420975	−0.210488	−	0.977597i	$-0.567505\pi$
$839$	−12.1938	−0.420975	−0.210488	+	0.977597i	$0.567505\pi$
$840$	0	0
$841$	2.25329	0.0776997
$842$	0	0
$843$	−16.5081	−0.568569
$844$	0	0
$845$	0.428007	0.0147239
$846$	0	0
$847$	−4.67282	−0.160560
$848$	0	0
$849$	−26.8577	−0.921754
$850$	0	0
$851$	65.0532	2.22999
$852$	0	0
$853$	−39.4195	−1.34970	−0.674850	−	0.737955i	$-0.735791\pi$
$853$	−39.4195	−1.34970	−0.674850	+	0.737955i	$0.735791\pi$
$854$	0	0
$855$	0.287973	0.00984847
$856$	0	0
$857$	−4.03525	−0.137842	−0.0689208	−	0.997622i	$-0.521956\pi$
$857$	−4.03525	−0.137842	−0.0689208	+	0.997622i	$0.521956\pi$
$858$	0	0
$859$	−2.20166	−0.0751197	−0.0375598	−	0.999294i	$-0.511958\pi$
$859$	−2.20166	−0.0751197	−0.0375598	+	0.999294i	$0.511958\pi$
$860$	0	0
$861$	41.8722	1.42700
$862$	0	0
$863$	−39.9137	−1.35868	−0.679339	−	0.733825i	$-0.737733\pi$
$863$	−39.9137	−1.35868	−0.679339	+	0.733825i	$0.737733\pi$
$864$	0	0
$865$	9.11761	0.310008
$866$	0	0
$867$	7.38485	0.250803
$868$	0	0
$869$	7.30249	0.247720
$870$	0	0
$871$	−8.62967	−0.292405
$872$	0	0
$873$	5.34565	0.180923
$874$	0	0
$875$	19.6336	0.663738
$876$	0	0
$877$	−23.1025	−0.780117	−0.390058	−	0.920790i	$-0.627545\pi$
$877$	−23.1025	−0.780117	−0.390058	+	0.920790i	$0.627545\pi$
$878$	0	0
$879$	25.4874	0.859668
$880$	0	0
$881$	−26.2465	−0.884267	−0.442134	−	0.896949i	$-0.645778\pi$
$881$	−26.2465	−0.884267	−0.442134	+	0.896949i	$0.645778\pi$
$882$	0	0
$883$	−10.6913	−0.359791	−0.179895	−	0.983686i	$-0.557576\pi$
$883$	−10.6913	−0.359791	−0.179895	+	0.983686i	$0.557576\pi$
$884$	0	0
$885$	−1.34565	−0.0452334
$886$	0	0
$887$	−34.9299	−1.17283	−0.586416	−	0.810010i	$-0.699461\pi$
$887$	−34.9299	−1.17283	−0.586416	+	0.810010i	$0.699461\pi$
$888$	0	0
$889$	−43.8722	−1.47143
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	−0.769701	−0.0257571
$894$	0	0
$895$	6.76970	0.226286
$896$	0	0
$897$	−6.96080	−0.232414
$898$	0	0
$899$	13.5737	0.452708
$900$	0	0
$901$	25.4320	0.847261
$902$	0	0
$903$	19.8353	0.660077
$904$	0	0
$905$	−6.81907	−0.226674
$906$	0	0
$907$	45.2963	1.50404	0.752019	−	0.659141i	$-0.229080\pi$
$907$	45.2963	1.50404	0.752019	+	0.659141i	$0.229080\pi$
$908$	0	0
$909$	−4.44648	−0.147480
$910$	0	0
$911$	−21.2224	−0.703129	−0.351565	−	0.936164i	$-0.614350\pi$
$911$	−21.2224	−0.703129	−0.351565	+	0.936164i	$0.614350\pi$
$912$	0	0
$913$	8.48963	0.280966
$914$	0	0
$915$	5.92159	0.195762
$916$	0	0
$917$	−74.9378	−2.47466
$918$	0	0
$919$	−38.6112	−1.27367	−0.636833	−	0.771002i	$-0.719756\pi$
$919$	−38.6112	−1.27367	−0.636833	+	0.771002i	$0.719756\pi$
$920$	0	0
$921$	−0.635881	−0.0209530
$922$	0	0
$923$	−5.05767	−0.166475
$924$	0	0
$925$	45.0162	1.48012
$926$	0	0
$927$	−12.9793	−0.426295
$928$	0	0
$929$	49.1193	1.61155	0.805776	−	0.592220i	$-0.201749\pi$
$929$	49.1193	1.61155	0.805776	+	0.592220i	$0.201749\pi$
$930$	0	0
$931$	9.98153	0.327131
$932$	0	0
$933$	−8.26950	−0.270731
$934$	0	0
$935$	−1.32718	−0.0434033
$936$	0	0
$937$	28.2465	0.922773	0.461387	−	0.887199i	$-0.347352\pi$
$937$	28.2465	0.922773	0.461387	+	0.887199i	$0.347352\pi$
$938$	0	0
$939$	13.7305	0.448078
$940$	0	0
$941$	49.2073	1.60411	0.802056	−	0.597249i	$-0.203739\pi$
$941$	49.2073	1.60411	0.802056	+	0.597249i	$0.203739\pi$
$942$	0	0
$943$	−62.3743	−2.03119
$944$	0	0
$945$	2.00000	0.0650600
$946$	0	0
$947$	−2.81907	−0.0916075	−0.0458038	−	0.998950i	$-0.514585\pi$
$947$	−2.81907	−0.0916075	−0.0458038	+	0.998950i	$0.514585\pi$
$948$	0	0
$949$	10.8745	0.353001
$950$	0	0
$951$	34.5882	1.12160
$952$	0	0
$953$	−52.6481	−1.70544	−0.852720	−	0.522368i	$-0.825049\pi$
$953$	−52.6481	−1.70544	−0.852720	+	0.522368i	$0.825049\pi$
$954$	0	0
$955$	8.94233	0.289367
$956$	0	0
$957$	−5.59046	−0.180714
$958$	0	0
$959$	24.3249	0.785492
$960$	0	0
$961$	−25.1048	−0.809832
$962$	0	0
$963$	−17.8353	−0.574734
$964$	0	0
$965$	5.63362	0.181353
$966$	0	0
$967$	−22.4712	−0.722624	−0.361312	−	0.932445i	$-0.617671\pi$
$967$	−22.4712	−0.722624	−0.361312	+	0.932445i	$0.617671\pi$
$968$	0	0
$969$	−2.08631	−0.0670220
$970$	0	0
$971$	−13.8617	−0.444842	−0.222421	−	0.974951i	$-0.571396\pi$
$971$	−13.8617	−0.444842	−0.222421	+	0.974951i	$0.571396\pi$
$972$	0	0
$973$	10.4896	0.336282
$974$	0	0
$975$	−4.81681	−0.154261
$976$	0	0
$977$	10.3417	0.330860	0.165430	−	0.986222i	$-0.447099\pi$
$977$	10.3417	0.330860	0.165430	+	0.986222i	$0.447099\pi$
$978$	0	0
$979$	5.20561	0.166372
$980$	0	0
$981$	19.4504	0.621004
$982$	0	0
$983$	2.90086	0.0925231	0.0462616	−	0.998929i	$-0.485269\pi$
$983$	2.90086	0.0925231	0.0462616	+	0.998929i	$0.485269\pi$
$984$	0	0
$985$	7.22239	0.230125
$986$	0	0
$987$	−5.34565	−0.170154
$988$	0	0
$989$	−29.5473	−0.939550
$990$	0	0
$991$	27.9506	0.887881	0.443941	−	0.896056i	$-0.353580\pi$
$991$	27.9506	0.887881	0.443941	+	0.896056i	$0.353580\pi$
$992$	0	0
$993$	−7.08236	−0.224752
$994$	0	0
$995$	1.71203	0.0542749
$996$	0	0
$997$	−0.568040	−0.0179900	−0.00899501	−	0.999960i	$-0.502863\pi$
$997$	−0.568040	−0.0179900	−0.00899501	+	0.999960i	$0.502863\pi$
$998$	0	0
$999$	9.34565	0.295683

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6864.2.a.br.1.2		3
4.3	odd	2		1716.2.a.g.1.2	✓	3
12.11	even	2		5148.2.a.m.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1716.2.a.g.1.2	✓	3	4.3	odd	2
5148.2.a.m.1.2		3	12.11	even	2
6864.2.a.br.1.2		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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +0.428007 q^{5} -4.67282 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +0.428007 q^{5} -4.67282 q^{7} +1.00000 q^{9} -1.00000 q^{11} -1.00000 q^{13} -0.428007 q^{15} +3.10083 q^{17} +0.672824 q^{19} +4.67282 q^{21} -6.96080 q^{23} -4.81681 q^{25} -1.00000 q^{27} -5.59046 q^{29} -2.42801 q^{31} +1.00000 q^{33} -2.00000 q^{35} -9.34565 q^{37} +1.00000 q^{39} +8.96080 q^{41} +4.24482 q^{43} +0.428007 q^{45} -1.14399 q^{47} +14.8353 q^{49} -3.10083 q^{51} +8.20166 q^{53} -0.428007 q^{55} -0.672824 q^{57} +3.14399 q^{59} -13.8353 q^{61} -4.67282 q^{63} -0.428007 q^{65} +8.62967 q^{67} +6.96080 q^{69} +5.05767 q^{71} -10.8745 q^{73} +4.81681 q^{75} +4.67282 q^{77} -7.30249 q^{79} +1.00000 q^{81} -8.48963 q^{83} +1.32718 q^{85} +5.59046 q^{87} -5.20561 q^{89} +4.67282 q^{91} +2.42801 q^{93} +0.287973 q^{95} +5.34565 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 2 q^{5} - 4 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 + 2 * q^5 - 4 * q^7 + 3 * q^9	\( 3 q - 3 q^{3} + 2 q^{5} - 4 q^{7} + 3 q^{9} - 3 q^{11} - 3 q^{13} - 2 q^{15} - 8 q^{19} + 4 q^{21} - 8 q^{23} - 3 q^{25} - 3 q^{27} + 14 q^{29} - 8 q^{31} + 3 q^{33} - 6 q^{35} - 8 q^{37} + 3 q^{39} + 14 q^{41} + 2 q^{43} + 2 q^{45} - 2 q^{47} + 3 q^{49} + 6 q^{53} - 2 q^{55} + 8 q^{57} + 8 q^{59} - 4 q^{63} - 2 q^{65} + 8 q^{67} + 8 q^{69} - 2 q^{71} - 4 q^{73} + 3 q^{75} + 4 q^{77} + 6 q^{79} + 3 q^{81} - 4 q^{83} + 14 q^{85} - 14 q^{87} + 8 q^{89} + 4 q^{91} + 8 q^{93} - 2 q^{95} - 4 q^{97} - 3 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 + 2 * q^5 - 4 * q^7 + 3 * q^9 - 3 * q^11 - 3 * q^13 - 2 * q^15 - 8 * q^19 + 4 * q^21 - 8 * q^23 - 3 * q^25 - 3 * q^27 + 14 * q^29 - 8 * q^31 + 3 * q^33 - 6 * q^35 - 8 * q^37 + 3 * q^39 + 14 * q^41 + 2 * q^43 + 2 * q^45 - 2 * q^47 + 3 * q^49 + 6 * q^53 - 2 * q^55 + 8 * q^57 + 8 * q^59 - 4 * q^63 - 2 * q^65 + 8 * q^67 + 8 * q^69 - 2 * q^71 - 4 * q^73 + 3 * q^75 + 4 * q^77 + 6 * q^79 + 3 * q^81 - 4 * q^83 + 14 * q^85 - 14 * q^87 + 8 * q^89 + 4 * q^91 + 8 * q^93 - 2 * q^95 - 4 * q^97 - 3 * q^99

Introduction

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