LMFDB - Embedded newform 3864.2.a.l.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3864,2,Mod(1,3864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3864, 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("3864.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3864 = 2^{3} \cdot 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3864.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:	$30.8541953410$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.86081$ of defining polynomial
Character	$\chi$	$=$	3864.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +0.462598 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +0.462598 q^{5} -1.00000 q^{7} +1.00000 q^{9} -0.398207 q^{11} -0.860806 q^{13} -0.462598 q^{15} -1.60179 q^{17} +6.11982 q^{19} +1.00000 q^{21} +1.00000 q^{23} -4.78600 q^{25} -1.00000 q^{27} -8.24860 q^{29} -1.32340 q^{31} +0.398207 q^{33} -0.462598 q^{35} +10.7666 q^{37} +0.860806 q^{39} +1.04502 q^{41} -2.86081 q^{43} +0.462598 q^{45} +2.27839 q^{47} +1.00000 q^{49} +1.60179 q^{51} -4.46260 q^{53} -0.184210 q^{55} -6.11982 q^{57} -1.90582 q^{59} -13.3580 q^{61} -1.00000 q^{63} -0.398207 q^{65} +2.33382 q^{67} -1.00000 q^{69} +14.8012 q^{71} -5.84143 q^{73} +4.78600 q^{75} +0.398207 q^{77} -1.75140 q^{79} +1.00000 q^{81} -8.76663 q^{83} -0.740987 q^{85} +8.24860 q^{87} +1.91478 q^{89} +0.860806 q^{91} +1.32340 q^{93} +2.83102 q^{95} -13.8414 q^{97} -0.398207 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} - q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 - q^5 - 3 * q^7 + 3 * q^9	$ 3 q - 3 q^{3} - q^{5} - 3 q^{7} + 3 q^{9} + 2 q^{11} + 3 q^{13} + q^{15} - 8 q^{17} + 4 q^{19} + 3 q^{21} + 3 q^{23} - 4 q^{25} - 3 q^{27} - 12 q^{29} + 4 q^{31} - 2 q^{33} + q^{35} + 2 q^{37} - 3 q^{39} - 16 q^{41} - 3 q^{43} - q^{45} + 18 q^{47} + 3 q^{49} + 8 q^{51} - 11 q^{53} + 13 q^{55} - 4 q^{57} + 19 q^{59} - 9 q^{61} - 3 q^{63} + 2 q^{65} + 3 q^{67} - 3 q^{69} - 9 q^{71} + 8 q^{73} + 4 q^{75} - 2 q^{77} - 18 q^{79} + 3 q^{81} + 4 q^{83} - 11 q^{85} + 12 q^{87} - 3 q^{89} - 3 q^{91} - 4 q^{93} - 21 q^{95} - 16 q^{97} + 2 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 - q^5 - 3 * q^7 + 3 * q^9 + 2 * q^11 + 3 * q^13 + q^15 - 8 * q^17 + 4 * q^19 + 3 * q^21 + 3 * q^23 - 4 * q^25 - 3 * q^27 - 12 * q^29 + 4 * q^31 - 2 * q^33 + q^35 + 2 * q^37 - 3 * q^39 - 16 * q^41 - 3 * q^43 - q^45 + 18 * q^47 + 3 * q^49 + 8 * q^51 - 11 * q^53 + 13 * q^55 - 4 * q^57 + 19 * q^59 - 9 * q^61 - 3 * q^63 + 2 * q^65 + 3 * q^67 - 3 * q^69 - 9 * q^71 + 8 * q^73 + 4 * q^75 - 2 * q^77 - 18 * q^79 + 3 * q^81 + 4 * q^83 - 11 * q^85 + 12 * q^87 - 3 * q^89 - 3 * q^91 - 4 * q^93 - 21 * q^95 - 16 * q^97 + 2 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	0.462598	0.206880	0.103440	−	0.994636i	$-0.467015\pi$
$5$	0.462598	0.206880	0.103440	+	0.994636i	$0.467015\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.398207	−0.120064	−0.0600320	−	0.998196i	$-0.519120\pi$
$11$	−0.398207	−0.120064	−0.0600320	+	0.998196i	$0.519120\pi$
$12$	0	0
$13$	−0.860806	−0.238745	−0.119372	−	0.992850i	$-0.538088\pi$
$13$	−0.860806	−0.238745	−0.119372	+	0.992850i	$0.538088\pi$
$14$	0	0
$15$	−0.462598	−0.119442
$16$	0	0
$17$	−1.60179	−0.388492	−0.194246	−	0.980953i	$-0.562226\pi$
$17$	−1.60179	−0.388492	−0.194246	+	0.980953i	$0.562226\pi$
$18$	0	0
$19$	6.11982	1.40398	0.701991	−	0.712185i	$-0.252294\pi$
$19$	6.11982	1.40398	0.701991	+	0.712185i	$0.252294\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−4.78600	−0.957201
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−8.24860	−1.53173	−0.765863	−	0.643003i	$-0.777688\pi$
$29$	−8.24860	−1.53173	−0.765863	+	0.643003i	$0.777688\pi$
$30$	0	0
$31$	−1.32340	−0.237690	−0.118845	−	0.992913i	$-0.537919\pi$
$31$	−1.32340	−0.237690	−0.118845	+	0.992913i	$0.537919\pi$
$32$	0	0
$33$	0.398207	0.0693190
$34$	0	0
$35$	−0.462598	−0.0781934
$36$	0	0
$37$	10.7666	1.77002	0.885011	−	0.465569i	$-0.154150\pi$
$37$	10.7666	1.77002	0.885011	+	0.465569i	$0.154150\pi$
$38$	0	0
$39$	0.860806	0.137839
$40$	0	0
$41$	1.04502	0.163204	0.0816020	−	0.996665i	$-0.473996\pi$
$41$	1.04502	0.163204	0.0816020	+	0.996665i	$0.473996\pi$
$42$	0	0
$43$	−2.86081	−0.436269	−0.218134	−	0.975919i	$-0.569997\pi$
$43$	−2.86081	−0.436269	−0.218134	+	0.975919i	$0.569997\pi$
$44$	0	0
$45$	0.462598	0.0689601
$46$	0	0
$47$	2.27839	0.332337	0.166169	−	0.986097i	$-0.446860\pi$
$47$	2.27839	0.332337	0.166169	+	0.986097i	$0.446860\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	1.60179	0.224296
$52$	0	0
$53$	−4.46260	−0.612985	−0.306493	−	0.951873i	$-0.599155\pi$
$53$	−4.46260	−0.612985	−0.306493	+	0.951873i	$0.599155\pi$
$54$	0	0
$55$	−0.184210	−0.0248389
$56$	0	0
$57$	−6.11982	−0.810590
$58$	0	0
$59$	−1.90582	−0.248117	−0.124058	−	0.992275i	$-0.539591\pi$
$59$	−1.90582	−0.248117	−0.124058	+	0.992275i	$0.539591\pi$
$60$	0	0
$61$	−13.3580	−1.71032	−0.855159	−	0.518366i	$-0.826540\pi$
$61$	−13.3580	−1.71032	−0.855159	+	0.518366i	$0.826540\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	−0.398207	−0.0493916
$66$	0	0
$67$	2.33382	0.285121	0.142561	−	0.989786i	$-0.454466\pi$
$67$	2.33382	0.285121	0.142561	+	0.989786i	$0.454466\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	14.8012	1.75658	0.878292	−	0.478126i	$-0.158684\pi$
$71$	14.8012	1.75658	0.878292	+	0.478126i	$0.158684\pi$
$72$	0	0
$73$	−5.84143	−0.683688	−0.341844	−	0.939757i	$-0.611051\pi$
$73$	−5.84143	−0.683688	−0.341844	+	0.939757i	$0.611051\pi$
$74$	0	0
$75$	4.78600	0.552640
$76$	0	0
$77$	0.398207	0.0453799
$78$	0	0
$79$	−1.75140	−0.197048	−0.0985239	−	0.995135i	$-0.531412\pi$
$79$	−1.75140	−0.197048	−0.0985239	+	0.995135i	$0.531412\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−8.76663	−0.962262	−0.481131	−	0.876649i	$-0.659774\pi$
$83$	−8.76663	−0.962262	−0.481131	+	0.876649i	$0.659774\pi$
$84$	0	0
$85$	−0.740987	−0.0803713
$86$	0	0
$87$	8.24860	0.884343
$88$	0	0
$89$	1.91478	0.202967	0.101483	−	0.994837i	$-0.467641\pi$
$89$	1.91478	0.202967	0.101483	+	0.994837i	$0.467641\pi$
$90$	0	0
$91$	0.860806	0.0902370
$92$	0	0
$93$	1.32340	0.137231
$94$	0	0
$95$	2.83102	0.290456
$96$	0	0
$97$	−13.8414	−1.40538	−0.702692	−	0.711494i	$-0.748019\pi$
$97$	−13.8414	−1.40538	−0.702692	+	0.711494i	$0.748019\pi$
$98$	0	0
$99$	−0.398207	−0.0400214
$100$	0	0
$101$	−12.8310	−1.27673	−0.638367	−	0.769732i	$-0.720390\pi$
$101$	−12.8310	−1.27673	−0.638367	+	0.769732i	$0.720390\pi$
$102$	0	0
$103$	14.8864	1.46681	0.733403	−	0.679795i	$-0.237931\pi$
$103$	14.8864	1.46681	0.733403	+	0.679795i	$0.237931\pi$
$104$	0	0
$105$	0.462598	0.0451450
$106$	0	0
$107$	5.50761	0.532441	0.266221	−	0.963912i	$-0.414225\pi$
$107$	5.50761	0.532441	0.266221	+	0.963912i	$0.414225\pi$
$108$	0	0
$109$	−3.66618	−0.351157	−0.175578	−	0.984465i	$-0.556180\pi$
$109$	−3.66618	−0.351157	−0.175578	+	0.984465i	$0.556180\pi$
$110$	0	0
$111$	−10.7666	−1.02192
$112$	0	0
$113$	2.55263	0.240131	0.120066	−	0.992766i	$-0.461689\pi$
$113$	2.55263	0.240131	0.120066	+	0.992766i	$0.461689\pi$
$114$	0	0
$115$	0.462598	0.0431375
$116$	0	0
$117$	−0.860806	−0.0795815
$118$	0	0
$119$	1.60179	0.146836
$120$	0	0
$121$	−10.8414	−0.985585
$122$	0	0
$123$	−1.04502	−0.0942259
$124$	0	0
$125$	−4.52699	−0.404906
$126$	0	0
$127$	−12.1544	−1.07853	−0.539265	−	0.842136i	$-0.681298\pi$
$127$	−12.1544	−1.07853	−0.539265	+	0.842136i	$0.681298\pi$
$128$	0	0
$129$	2.86081	0.251880
$130$	0	0
$131$	−12.0990	−1.05709	−0.528547	−	0.848904i	$-0.677263\pi$
$131$	−12.0990	−1.05709	−0.528547	+	0.848904i	$0.677263\pi$
$132$	0	0
$133$	−6.11982	−0.530656
$134$	0	0
$135$	−0.462598	−0.0398141
$136$	0	0
$137$	−14.8954	−1.27260	−0.636300	−	0.771441i	$-0.719536\pi$
$137$	−14.8954	−1.27260	−0.636300	+	0.771441i	$0.719536\pi$
$138$	0	0
$139$	15.1994	1.28920	0.644600	−	0.764520i	$-0.277024\pi$
$139$	15.1994	1.28920	0.644600	+	0.764520i	$0.277024\pi$
$140$	0	0
$141$	−2.27839	−0.191875
$142$	0	0
$143$	0.342779	0.0286646
$144$	0	0
$145$	−3.81579	−0.316884
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	−16.5180	−1.35321	−0.676605	−	0.736346i	$-0.736549\pi$
$149$	−16.5180	−1.35321	−0.676605	+	0.736346i	$0.736549\pi$
$150$	0	0
$151$	−9.16484	−0.745824	−0.372912	−	0.927867i	$-0.621641\pi$
$151$	−9.16484	−0.745824	−0.372912	+	0.927867i	$0.621641\pi$
$152$	0	0
$153$	−1.60179	−0.129497
$154$	0	0
$155$	−0.612205	−0.0491735
$156$	0	0
$157$	23.1648	1.84876	0.924378	−	0.381479i	$-0.124585\pi$
$157$	23.1648	1.84876	0.924378	+	0.381479i	$0.124585\pi$
$158$	0	0
$159$	4.46260	0.353907
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	−19.0707	−1.49373	−0.746865	−	0.664976i	$-0.768442\pi$
$163$	−19.0707	−1.49373	−0.746865	+	0.664976i	$0.768442\pi$
$164$	0	0
$165$	0.184210	0.0143407
$166$	0	0
$167$	−18.7458	−1.45059	−0.725297	−	0.688436i	$-0.758298\pi$
$167$	−18.7458	−1.45059	−0.725297	+	0.688436i	$0.758298\pi$
$168$	0	0
$169$	−12.2590	−0.943001
$170$	0	0
$171$	6.11982	0.467994
$172$	0	0
$173$	19.3926	1.47439	0.737196	−	0.675678i	$-0.236149\pi$
$173$	19.3926	1.47439	0.737196	+	0.675678i	$0.236149\pi$
$174$	0	0
$175$	4.78600	0.361788
$176$	0	0
$177$	1.90582	0.143250
$178$	0	0
$179$	−14.8310	−1.10852	−0.554261	−	0.832343i	$-0.686999\pi$
$179$	−14.8310	−1.10852	−0.554261	+	0.832343i	$0.686999\pi$
$180$	0	0
$181$	1.60179	0.119060	0.0595302	−	0.998227i	$-0.481040\pi$
$181$	1.60179	0.119060	0.0595302	+	0.998227i	$0.481040\pi$
$182$	0	0
$183$	13.3580	0.987452
$184$	0	0
$185$	4.98062	0.366183
$186$	0	0
$187$	0.637846	0.0466439
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	−13.7216	−0.992861	−0.496430	−	0.868076i	$-0.665356\pi$
$191$	−13.7216	−0.992861	−0.496430	+	0.868076i	$0.665356\pi$
$192$	0	0
$193$	22.6081	1.62736	0.813682	−	0.581310i	$-0.197460\pi$
$193$	22.6081	1.62736	0.813682	+	0.581310i	$0.197460\pi$
$194$	0	0
$195$	0.398207	0.0285162
$196$	0	0
$197$	7.07962	0.504402	0.252201	−	0.967675i	$-0.418846\pi$
$197$	7.07962	0.504402	0.252201	+	0.967675i	$0.418846\pi$
$198$	0	0
$199$	12.6814	0.898961	0.449481	−	0.893290i	$-0.351609\pi$
$199$	12.6814	0.898961	0.449481	+	0.893290i	$0.351609\pi$
$200$	0	0
$201$	−2.33382	−0.164615
$202$	0	0
$203$	8.24860	0.578938
$204$	0	0
$205$	0.483423	0.0337637
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	−2.43696	−0.168568
$210$	0	0
$211$	−3.47301	−0.239092	−0.119546	−	0.992829i	$-0.538144\pi$
$211$	−3.47301	−0.239092	−0.119546	+	0.992829i	$0.538144\pi$
$212$	0	0
$213$	−14.8012	−1.01416
$214$	0	0
$215$	−1.32340	−0.0902554
$216$	0	0
$217$	1.32340	0.0898385
$218$	0	0
$219$	5.84143	0.394727
$220$	0	0
$221$	1.37883	0.0927503
$222$	0	0
$223$	−7.38780	−0.494723	−0.247362	−	0.968923i	$-0.579564\pi$
$223$	−7.38780	−0.494723	−0.247362	+	0.968923i	$0.579564\pi$
$224$	0	0
$225$	−4.78600	−0.319067
$226$	0	0
$227$	18.6427	1.23736	0.618678	−	0.785644i	$-0.287668\pi$
$227$	18.6427	1.23736	0.618678	+	0.785644i	$0.287668\pi$
$228$	0	0
$229$	8.03460	0.530942	0.265471	−	0.964119i	$-0.414473\pi$
$229$	8.03460	0.530942	0.265471	+	0.964119i	$0.414473\pi$
$230$	0	0
$231$	−0.398207	−0.0262001
$232$	0	0
$233$	−28.2742	−1.85231	−0.926154	−	0.377147i	$-0.876905\pi$
$233$	−28.2742	−1.85231	−0.926154	+	0.377147i	$0.876905\pi$
$234$	0	0
$235$	1.05398	0.0687540
$236$	0	0
$237$	1.75140	0.113766
$238$	0	0
$239$	−16.8102	−1.08736	−0.543681	−	0.839292i	$-0.682970\pi$
$239$	−16.8102	−1.08736	−0.543681	+	0.839292i	$0.682970\pi$
$240$	0	0
$241$	1.84143	0.118617	0.0593085	−	0.998240i	$-0.481110\pi$
$241$	1.84143	0.118617	0.0593085	+	0.998240i	$0.481110\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0.462598	0.0295543
$246$	0	0
$247$	−5.26798	−0.335193
$248$	0	0
$249$	8.76663	0.555562
$250$	0	0
$251$	15.8116	0.998022	0.499011	−	0.866596i	$-0.333697\pi$
$251$	15.8116	0.998022	0.499011	+	0.866596i	$0.333697\pi$
$252$	0	0
$253$	−0.398207	−0.0250351
$254$	0	0
$255$	0.740987	0.0464024
$256$	0	0
$257$	−27.4045	−1.70944	−0.854722	−	0.519086i	$-0.826272\pi$
$257$	−27.4045	−1.70944	−0.854722	+	0.519086i	$0.826272\pi$
$258$	0	0
$259$	−10.7666	−0.669006
$260$	0	0
$261$	−8.24860	−0.510576
$262$	0	0
$263$	−5.56304	−0.343032	−0.171516	−	0.985181i	$-0.554867\pi$
$263$	−5.56304	−0.343032	−0.171516	+	0.985181i	$0.554867\pi$
$264$	0	0
$265$	−2.06439	−0.126815
$266$	0	0
$267$	−1.91478	−0.117183
$268$	0	0
$269$	−11.5076	−0.701632	−0.350816	−	0.936444i	$-0.614096\pi$
$269$	−11.5076	−0.701632	−0.350816	+	0.936444i	$0.614096\pi$
$270$	0	0
$271$	−27.2251	−1.65381	−0.826903	−	0.562345i	$-0.809899\pi$
$271$	−27.2251	−1.65381	−0.826903	+	0.562345i	$0.809899\pi$
$272$	0	0
$273$	−0.860806	−0.0520983
$274$	0	0
$275$	1.90582	0.114925
$276$	0	0
$277$	−3.07962	−0.185036	−0.0925182	−	0.995711i	$-0.529492\pi$
$277$	−3.07962	−0.185036	−0.0925182	+	0.995711i	$0.529492\pi$
$278$	0	0
$279$	−1.32340	−0.0792301
$280$	0	0
$281$	−23.1648	−1.38190	−0.690949	−	0.722903i	$-0.742807\pi$
$281$	−23.1648	−1.38190	−0.690949	+	0.722903i	$0.742807\pi$
$282$	0	0
$283$	−19.4391	−1.15553	−0.577767	−	0.816202i	$-0.696076\pi$
$283$	−19.4391	−1.15553	−0.577767	+	0.816202i	$0.696076\pi$
$284$	0	0
$285$	−2.83102	−0.167695
$286$	0	0
$287$	−1.04502	−0.0616853
$288$	0	0
$289$	−14.4343	−0.849074
$290$	0	0
$291$	13.8414	0.811399
$292$	0	0
$293$	−25.6829	−1.50041	−0.750204	−	0.661206i	$-0.770045\pi$
$293$	−25.6829	−1.50041	−0.750204	+	0.661206i	$0.770045\pi$
$294$	0	0
$295$	−0.881630	−0.0513305
$296$	0	0
$297$	0.398207	0.0231063
$298$	0	0
$299$	−0.860806	−0.0497817
$300$	0	0
$301$	2.86081	0.164894
$302$	0	0
$303$	12.8310	0.737123
$304$	0	0
$305$	−6.17939	−0.353831
$306$	0	0
$307$	28.2099	1.61002	0.805011	−	0.593260i	$-0.202160\pi$
$307$	28.2099	1.61002	0.805011	+	0.593260i	$0.202160\pi$
$308$	0	0
$309$	−14.8864	−0.846860
$310$	0	0
$311$	22.5437	1.27833	0.639167	−	0.769068i	$-0.279279\pi$
$311$	22.5437	1.27833	0.639167	+	0.769068i	$0.279279\pi$
$312$	0	0
$313$	1.20359	0.0680307	0.0340153	−	0.999421i	$-0.489170\pi$
$313$	1.20359	0.0680307	0.0340153	+	0.999421i	$0.489170\pi$
$314$	0	0
$315$	−0.462598	−0.0260645
$316$	0	0
$317$	0.831019	0.0466747	0.0233373	−	0.999728i	$-0.492571\pi$
$317$	0.831019	0.0466747	0.0233373	+	0.999728i	$0.492571\pi$
$318$	0	0
$319$	3.28465	0.183905
$320$	0	0
$321$	−5.50761	−0.307405
$322$	0	0
$323$	−9.80268	−0.545436
$324$	0	0
$325$	4.11982	0.228526
$326$	0	0
$327$	3.66618	0.202740
$328$	0	0
$329$	−2.27839	−0.125612
$330$	0	0
$331$	5.05398	0.277792	0.138896	−	0.990307i	$-0.455645\pi$
$331$	5.05398	0.277792	0.138896	+	0.990307i	$0.455645\pi$
$332$	0	0
$333$	10.7666	0.590008
$334$	0	0
$335$	1.07962	0.0589859
$336$	0	0
$337$	−15.1392	−0.824684	−0.412342	−	0.911029i	$-0.635289\pi$
$337$	−15.1392	−0.824684	−0.412342	+	0.911029i	$0.635289\pi$
$338$	0	0
$339$	−2.55263	−0.138640
$340$	0	0
$341$	0.526989	0.0285381
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−0.462598	−0.0249055
$346$	0	0
$347$	−29.2638	−1.57096	−0.785482	−	0.618884i	$-0.787585\pi$
$347$	−29.2638	−1.57096	−0.785482	+	0.618884i	$0.787585\pi$
$348$	0	0
$349$	−10.1842	−0.545148	−0.272574	−	0.962135i	$-0.587875\pi$
$349$	−10.1842	−0.545148	−0.272574	+	0.962135i	$0.587875\pi$
$350$	0	0
$351$	0.860806	0.0459464
$352$	0	0
$353$	−2.39821	−0.127644	−0.0638219	−	0.997961i	$-0.520329\pi$
$353$	−2.39821	−0.127644	−0.0638219	+	0.997961i	$0.520329\pi$
$354$	0	0
$355$	6.84703	0.363402
$356$	0	0
$357$	−1.60179	−0.0847759
$358$	0	0
$359$	−14.5914	−0.770104	−0.385052	−	0.922895i	$-0.625816\pi$
$359$	−14.5914	−0.770104	−0.385052	+	0.922895i	$0.625816\pi$
$360$	0	0
$361$	18.4522	0.971168
$362$	0	0
$363$	10.8414	0.569028
$364$	0	0
$365$	−2.70224	−0.141442
$366$	0	0
$367$	−0.453636	−0.0236796	−0.0118398	−	0.999930i	$-0.503769\pi$
$367$	−0.453636	−0.0236796	−0.0118398	+	0.999930i	$0.503769\pi$
$368$	0	0
$369$	1.04502	0.0544014
$370$	0	0
$371$	4.46260	0.231687
$372$	0	0
$373$	32.0602	1.66002	0.830008	−	0.557751i	$-0.188336\pi$
$373$	32.0602	1.66002	0.830008	+	0.557751i	$0.188336\pi$
$374$	0	0
$375$	4.52699	0.233773
$376$	0	0
$377$	7.10044	0.365691
$378$	0	0
$379$	−4.00000	−0.205466	−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000	−0.205466	−0.102733	+	0.994709i	$0.532759\pi$
$380$	0	0
$381$	12.1544	0.622690
$382$	0	0
$383$	5.62262	0.287302	0.143651	−	0.989628i	$-0.454116\pi$
$383$	5.62262	0.287302	0.143651	+	0.989628i	$0.454116\pi$
$384$	0	0
$385$	0.184210	0.00938822
$386$	0	0
$387$	−2.86081	−0.145423
$388$	0	0
$389$	−33.0963	−1.67805	−0.839024	−	0.544094i	$-0.816874\pi$
$389$	−33.0963	−1.67805	−0.839024	+	0.544094i	$0.816874\pi$
$390$	0	0
$391$	−1.60179	−0.0810061
$392$	0	0
$393$	12.0990	0.610314
$394$	0	0
$395$	−0.810194	−0.0407653
$396$	0	0
$397$	18.9252	0.949828	0.474914	−	0.880032i	$-0.342479\pi$
$397$	18.9252	0.949828	0.474914	+	0.880032i	$0.342479\pi$
$398$	0	0
$399$	6.11982	0.306374
$400$	0	0
$401$	−1.23337	−0.0615917	−0.0307958	−	0.999526i	$-0.509804\pi$
$401$	−1.23337	−0.0615917	−0.0307958	+	0.999526i	$0.509804\pi$
$402$	0	0
$403$	1.13919	0.0567473
$404$	0	0
$405$	0.462598	0.0229867
$406$	0	0
$407$	−4.28735	−0.212516
$408$	0	0
$409$	−5.47301	−0.270623	−0.135311	−	0.990803i	$-0.543204\pi$
$409$	−5.47301	−0.270623	−0.135311	+	0.990803i	$0.543204\pi$
$410$	0	0
$411$	14.8954	0.734736
$412$	0	0
$413$	1.90582	0.0937794
$414$	0	0
$415$	−4.05543	−0.199073
$416$	0	0
$417$	−15.1994	−0.744320
$418$	0	0
$419$	−12.0138	−0.586912	−0.293456	−	0.955973i	$-0.594805\pi$
$419$	−12.0138	−0.586912	−0.293456	+	0.955973i	$0.594805\pi$
$420$	0	0
$421$	33.4093	1.62827	0.814135	−	0.580676i	$-0.197212\pi$
$421$	33.4093	1.62827	0.814135	+	0.580676i	$0.197212\pi$
$422$	0	0
$423$	2.27839	0.110779
$424$	0	0
$425$	7.66618	0.371865
$426$	0	0
$427$	13.3580	0.646439
$428$	0	0
$429$	−0.342779	−0.0165495
$430$	0	0
$431$	−10.6212	−0.511604	−0.255802	−	0.966729i	$-0.582339\pi$
$431$	−10.6212	−0.511604	−0.255802	+	0.966729i	$0.582339\pi$
$432$	0	0
$433$	7.46405	0.358699	0.179350	−	0.983785i	$-0.442601\pi$
$433$	7.46405	0.358699	0.179350	+	0.983785i	$0.442601\pi$
$434$	0	0
$435$	3.81579	0.182953
$436$	0	0
$437$	6.11982	0.292751
$438$	0	0
$439$	21.8206	1.04144	0.520720	−	0.853727i	$-0.325663\pi$
$439$	21.8206	1.04144	0.520720	+	0.853727i	$0.325663\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	20.7368	0.985237	0.492619	−	0.870245i	$-0.336040\pi$
$443$	20.7368	0.985237	0.492619	+	0.870245i	$0.336040\pi$
$444$	0	0
$445$	0.885776	0.0419898
$446$	0	0
$447$	16.5180	0.781276
$448$	0	0
$449$	−8.46260	−0.399375	−0.199687	−	0.979860i	$-0.563993\pi$
$449$	−8.46260	−0.399375	−0.199687	+	0.979860i	$0.563993\pi$
$450$	0	0
$451$	−0.416133	−0.0195949
$452$	0	0
$453$	9.16484	0.430602
$454$	0	0
$455$	0.398207	0.0186683
$456$	0	0
$457$	−10.4834	−0.490394	−0.245197	−	0.969473i	$-0.578853\pi$
$457$	−10.4834	−0.490394	−0.245197	+	0.969473i	$0.578853\pi$
$458$	0	0
$459$	1.60179	0.0747653
$460$	0	0
$461$	−19.2382	−0.896012	−0.448006	−	0.894031i	$-0.647866\pi$
$461$	−19.2382	−0.896012	−0.448006	+	0.894031i	$0.647866\pi$
$462$	0	0
$463$	−9.63225	−0.447649	−0.223824	−	0.974630i	$-0.571854\pi$
$463$	−9.63225	−0.447649	−0.223824	+	0.974630i	$0.571854\pi$
$464$	0	0
$465$	0.612205	0.0283903
$466$	0	0
$467$	32.6475	1.51075	0.755373	−	0.655296i	$-0.227456\pi$
$467$	32.6475	1.51075	0.755373	+	0.655296i	$0.227456\pi$
$468$	0	0
$469$	−2.33382	−0.107766
$470$	0	0
$471$	−23.1648	−1.06738
$472$	0	0
$473$	1.13919	0.0523802
$474$	0	0
$475$	−29.2895	−1.34389
$476$	0	0
$477$	−4.46260	−0.204328
$478$	0	0
$479$	7.90101	0.361006	0.180503	−	0.983574i	$-0.442227\pi$
$479$	7.90101	0.361006	0.180503	+	0.983574i	$0.442227\pi$
$480$	0	0
$481$	−9.26798	−0.422583
$482$	0	0
$483$	1.00000	0.0455016
$484$	0	0
$485$	−6.40302	−0.290746
$486$	0	0
$487$	20.0811	0.909960	0.454980	−	0.890502i	$-0.349646\pi$
$487$	20.0811	0.909960	0.454980	+	0.890502i	$0.349646\pi$
$488$	0	0
$489$	19.0707	0.862405
$490$	0	0
$491$	28.7112	1.29572	0.647859	−	0.761760i	$-0.275665\pi$
$491$	28.7112	1.29572	0.647859	+	0.761760i	$0.275665\pi$
$492$	0	0
$493$	13.2125	0.595063
$494$	0	0
$495$	−0.184210	−0.00827963
$496$	0	0
$497$	−14.8012	−0.663926
$498$	0	0
$499$	23.1994	1.03855	0.519275	−	0.854607i	$-0.326202\pi$
$499$	23.1994	1.03855	0.519275	+	0.854607i	$0.326202\pi$
$500$	0	0
$501$	18.7458	0.837501
$502$	0	0
$503$	−36.4238	−1.62406	−0.812030	−	0.583616i	$-0.801637\pi$
$503$	−36.4238	−1.62406	−0.812030	+	0.583616i	$0.801637\pi$
$504$	0	0
$505$	−5.93561	−0.264131
$506$	0	0
$507$	12.2590	0.544442
$508$	0	0
$509$	−20.2188	−0.896183	−0.448092	−	0.893988i	$-0.647896\pi$
$509$	−20.2188	−0.896183	−0.448092	+	0.893988i	$0.647896\pi$
$510$	0	0
$511$	5.84143	0.258410
$512$	0	0
$513$	−6.11982	−0.270197
$514$	0	0
$515$	6.88645	0.303453
$516$	0	0
$517$	−0.907271	−0.0399017
$518$	0	0
$519$	−19.3926	−0.851241
$520$	0	0
$521$	4.32967	0.189686	0.0948431	−	0.995492i	$-0.469765\pi$
$521$	4.32967	0.189686	0.0948431	+	0.995492i	$0.469765\pi$
$522$	0	0
$523$	−6.64681	−0.290645	−0.145322	−	0.989384i	$-0.546422\pi$
$523$	−6.64681	−0.290645	−0.145322	+	0.989384i	$0.546422\pi$
$524$	0	0
$525$	−4.78600	−0.208878
$526$	0	0
$527$	2.11982	0.0923408
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−1.90582	−0.0827056
$532$	0	0
$533$	−0.899556	−0.0389641
$534$	0	0
$535$	2.54781	0.110152
$536$	0	0
$537$	14.8310	0.640006
$538$	0	0
$539$	−0.398207	−0.0171520
$540$	0	0
$541$	−39.6620	−1.70520	−0.852602	−	0.522561i	$-0.824977\pi$
$541$	−39.6620	−1.70520	−0.852602	+	0.522561i	$0.824977\pi$
$542$	0	0
$543$	−1.60179	−0.0687395
$544$	0	0
$545$	−1.69597	−0.0726474
$546$	0	0
$547$	15.4570	0.660894	0.330447	−	0.943825i	$-0.392801\pi$
$547$	15.4570	0.660894	0.330447	+	0.943825i	$0.392801\pi$
$548$	0	0
$549$	−13.3580	−0.570106
$550$	0	0
$551$	−50.4799	−2.15052
$552$	0	0
$553$	1.75140	0.0744771
$554$	0	0
$555$	−4.98062	−0.211416
$556$	0	0
$557$	43.6025	1.84750	0.923748	−	0.383001i	$-0.125110\pi$
$557$	43.6025	1.84750	0.923748	+	0.383001i	$0.125110\pi$
$558$	0	0
$559$	2.46260	0.104157
$560$	0	0
$561$	−0.637846	−0.0269299
$562$	0	0
$563$	14.5616	0.613698	0.306849	−	0.951758i	$-0.400725\pi$
$563$	14.5616	0.613698	0.306849	+	0.951758i	$0.400725\pi$
$564$	0	0
$565$	1.18084	0.0496784
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−15.9100	−0.666981	−0.333490	−	0.942754i	$-0.608226\pi$
$569$	−15.9100	−0.666981	−0.333490	+	0.942754i	$0.608226\pi$
$570$	0	0
$571$	−30.4287	−1.27340	−0.636700	−	0.771112i	$-0.719701\pi$
$571$	−30.4287	−1.27340	−0.636700	+	0.771112i	$0.719701\pi$
$572$	0	0
$573$	13.7216	0.573229
$574$	0	0
$575$	−4.78600	−0.199590
$576$	0	0
$577$	9.74244	0.405583	0.202791	−	0.979222i	$-0.434999\pi$
$577$	9.74244	0.405583	0.202791	+	0.979222i	$0.434999\pi$
$578$	0	0
$579$	−22.6081	−0.939559
$580$	0	0
$581$	8.76663	0.363701
$582$	0	0
$583$	1.77704	0.0735975
$584$	0	0
$585$	−0.398207	−0.0164639
$586$	0	0
$587$	−20.7625	−0.856959	−0.428480	−	0.903551i	$-0.640951\pi$
$587$	−20.7625	−0.856959	−0.428480	+	0.903551i	$0.640951\pi$
$588$	0	0
$589$	−8.09899	−0.333713
$590$	0	0
$591$	−7.07962	−0.291217
$592$	0	0
$593$	11.1648	0.458485	0.229242	−	0.973369i	$-0.426375\pi$
$593$	11.1648	0.458485	0.229242	+	0.973369i	$0.426375\pi$
$594$	0	0
$595$	0.740987	0.0303775
$596$	0	0
$597$	−12.6814	−0.519016
$598$	0	0
$599$	14.2624	0.582745	0.291373	−	0.956610i	$-0.405888\pi$
$599$	14.2624	0.582745	0.291373	+	0.956610i	$0.405888\pi$
$600$	0	0
$601$	4.83102	0.197061	0.0985307	−	0.995134i	$-0.468586\pi$
$601$	4.83102	0.197061	0.0985307	+	0.995134i	$0.468586\pi$
$602$	0	0
$603$	2.33382	0.0950404
$604$	0	0
$605$	−5.01523	−0.203898
$606$	0	0
$607$	43.2984	1.75743	0.878715	−	0.477348i	$-0.158402\pi$
$607$	43.2984	1.75743	0.878715	+	0.477348i	$0.158402\pi$
$608$	0	0
$609$	−8.24860	−0.334250
$610$	0	0
$611$	−1.96125	−0.0793437
$612$	0	0
$613$	3.51176	0.141839	0.0709193	−	0.997482i	$-0.477407\pi$
$613$	3.51176	0.141839	0.0709193	+	0.997482i	$0.477407\pi$
$614$	0	0
$615$	−0.483423	−0.0194935
$616$	0	0
$617$	−41.5560	−1.67298	−0.836491	−	0.547981i	$-0.815397\pi$
$617$	−41.5560	−1.67298	−0.836491	+	0.547981i	$0.815397\pi$
$618$	0	0
$619$	38.8704	1.56233	0.781167	−	0.624322i	$-0.214625\pi$
$619$	38.8704	1.56233	0.781167	+	0.624322i	$0.214625\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	−1.91478	−0.0767142
$624$	0	0
$625$	21.8358	0.873433
$626$	0	0
$627$	2.43696	0.0973227
$628$	0	0
$629$	−17.2459	−0.687639
$630$	0	0
$631$	29.5035	1.17451	0.587257	−	0.809400i	$-0.300208\pi$
$631$	29.5035	1.17451	0.587257	+	0.809400i	$0.300208\pi$
$632$	0	0
$633$	3.47301	0.138040
$634$	0	0
$635$	−5.62262	−0.223127
$636$	0	0
$637$	−0.860806	−0.0341064
$638$	0	0
$639$	14.8012	0.585528
$640$	0	0
$641$	33.4868	1.32265	0.661324	−	0.750100i	$-0.269995\pi$
$641$	33.4868	1.32265	0.661324	+	0.750100i	$0.269995\pi$
$642$	0	0
$643$	−5.56719	−0.219548	−0.109774	−	0.993957i	$-0.535013\pi$
$643$	−5.56719	−0.219548	−0.109774	+	0.993957i	$0.535013\pi$
$644$	0	0
$645$	1.32340	0.0521090
$646$	0	0
$647$	24.4418	0.960905	0.480453	−	0.877021i	$-0.340472\pi$
$647$	24.4418	0.960905	0.480453	+	0.877021i	$0.340472\pi$
$648$	0	0
$649$	0.758912	0.0297899
$650$	0	0
$651$	−1.32340	−0.0518683
$652$	0	0
$653$	22.6122	0.884884	0.442442	−	0.896797i	$-0.354112\pi$
$653$	22.6122	0.884884	0.442442	+	0.896797i	$0.354112\pi$
$654$	0	0
$655$	−5.59698	−0.218692
$656$	0	0
$657$	−5.84143	−0.227896
$658$	0	0
$659$	34.1890	1.33182	0.665908	−	0.746034i	$-0.268044\pi$
$659$	34.1890	1.33182	0.665908	+	0.746034i	$0.268044\pi$
$660$	0	0
$661$	16.0090	0.622676	0.311338	−	0.950299i	$-0.399223\pi$
$661$	16.0090	0.622676	0.311338	+	0.950299i	$0.399223\pi$
$662$	0	0
$663$	−1.37883	−0.0535494
$664$	0	0
$665$	−2.83102	−0.109782
$666$	0	0
$667$	−8.24860	−0.319387
$668$	0	0
$669$	7.38780	0.285629
$670$	0	0
$671$	5.31926	0.205348
$672$	0	0
$673$	18.5783	0.716140	0.358070	−	0.933695i	$-0.383435\pi$
$673$	18.5783	0.716140	0.358070	+	0.933695i	$0.383435\pi$
$674$	0	0
$675$	4.78600	0.184213
$676$	0	0
$677$	43.5768	1.67479	0.837397	−	0.546596i	$-0.184077\pi$
$677$	43.5768	1.67479	0.837397	+	0.546596i	$0.184077\pi$
$678$	0	0
$679$	13.8414	0.531185
$680$	0	0
$681$	−18.6427	−0.714388
$682$	0	0
$683$	24.9646	0.955245	0.477622	−	0.878565i	$-0.341499\pi$
$683$	24.9646	0.955245	0.477622	+	0.878565i	$0.341499\pi$
$684$	0	0
$685$	−6.89059	−0.263276
$686$	0	0
$687$	−8.03460	−0.306539
$688$	0	0
$689$	3.84143	0.146347
$690$	0	0
$691$	−8.15442	−0.310209	−0.155104	−	0.987898i	$-0.549571\pi$
$691$	−8.15442	−0.310209	−0.155104	+	0.987898i	$0.549571\pi$
$692$	0	0
$693$	0.398207	0.0151266
$694$	0	0
$695$	7.03124	0.266710
$696$	0	0
$697$	−1.67390	−0.0634034
$698$	0	0
$699$	28.2742	1.06943
$700$	0	0
$701$	−16.5318	−0.624398	−0.312199	−	0.950017i	$-0.601066\pi$
$701$	−16.5318	−0.624398	−0.312199	+	0.950017i	$0.601066\pi$
$702$	0	0
$703$	65.8898	2.48508
$704$	0	0
$705$	−1.05398	−0.0396951
$706$	0	0
$707$	12.8310	0.482560
$708$	0	0
$709$	−4.13360	−0.155241	−0.0776203	−	0.996983i	$-0.524732\pi$
$709$	−4.13360	−0.155241	−0.0776203	+	0.996983i	$0.524732\pi$
$710$	0	0
$711$	−1.75140	−0.0656826
$712$	0	0
$713$	−1.32340	−0.0495619
$714$	0	0
$715$	0.158569	0.00593015
$716$	0	0
$717$	16.8102	0.627788
$718$	0	0
$719$	16.0119	0.597142	0.298571	−	0.954387i	$-0.403490\pi$
$719$	16.0119	0.597142	0.298571	+	0.954387i	$0.403490\pi$
$720$	0	0
$721$	−14.8864	−0.554400
$722$	0	0
$723$	−1.84143	−0.0684835
$724$	0	0
$725$	39.4778	1.46617
$726$	0	0
$727$	−0.269425	−0.00999244	−0.00499622	−	0.999988i	$-0.501590\pi$
$727$	−0.269425	−0.00999244	−0.00499622	+	0.999988i	$0.501590\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	4.58242	0.169487
$732$	0	0
$733$	48.1592	1.77880	0.889401	−	0.457128i	$-0.151122\pi$
$733$	48.1592	1.77880	0.889401	+	0.457128i	$0.151122\pi$
$734$	0	0
$735$	−0.462598	−0.0170632
$736$	0	0
$737$	−0.929343	−0.0342328
$738$	0	0
$739$	−43.6114	−1.60427	−0.802136	−	0.597141i	$-0.796303\pi$
$739$	−43.6114	−1.60427	−0.802136	+	0.597141i	$0.796303\pi$
$740$	0	0
$741$	5.26798	0.193524
$742$	0	0
$743$	−41.1786	−1.51070	−0.755348	−	0.655323i	$-0.772532\pi$
$743$	−41.1786	−1.51070	−0.755348	+	0.655323i	$0.772532\pi$
$744$	0	0
$745$	−7.64121	−0.279952
$746$	0	0
$747$	−8.76663	−0.320754
$748$	0	0
$749$	−5.50761	−0.201244
$750$	0	0
$751$	−45.1607	−1.64794	−0.823968	−	0.566636i	$-0.808245\pi$
$751$	−45.1607	−1.64794	−0.823968	+	0.566636i	$0.808245\pi$
$752$	0	0
$753$	−15.8116	−0.576208
$754$	0	0
$755$	−4.23964	−0.154296
$756$	0	0
$757$	32.3774	1.17678	0.588388	−	0.808579i	$-0.299763\pi$
$757$	32.3774	1.17678	0.588388	+	0.808579i	$0.299763\pi$
$758$	0	0
$759$	0.398207	0.0144540
$760$	0	0
$761$	35.5333	1.28808	0.644040	−	0.764992i	$-0.277257\pi$
$761$	35.5333	1.28808	0.644040	+	0.764992i	$0.277257\pi$
$762$	0	0
$763$	3.66618	0.132725
$764$	0	0
$765$	−0.740987	−0.0267904
$766$	0	0
$767$	1.64054	0.0592366
$768$	0	0
$769$	−12.4585	−0.449263	−0.224632	−	0.974444i	$-0.572118\pi$
$769$	−12.4585	−0.449263	−0.224632	+	0.974444i	$0.572118\pi$
$770$	0	0
$771$	27.4045	0.986948
$772$	0	0
$773$	−0.359457	−0.0129288	−0.00646439	−	0.999979i	$-0.502058\pi$
$773$	−0.359457	−0.0129288	−0.00646439	+	0.999979i	$0.502058\pi$
$774$	0	0
$775$	6.33382	0.227517
$776$	0	0
$777$	10.7666	0.386251
$778$	0	0
$779$	6.39531	0.229136
$780$	0	0
$781$	−5.89396	−0.210902
$782$	0	0
$783$	8.24860	0.294781
$784$	0	0
$785$	10.7160	0.382471
$786$	0	0
$787$	−40.6731	−1.44984	−0.724920	−	0.688833i	$-0.758123\pi$
$787$	−40.6731	−1.44984	−0.724920	+	0.688833i	$0.758123\pi$
$788$	0	0
$789$	5.56304	0.198050
$790$	0	0
$791$	−2.55263	−0.0907611
$792$	0	0
$793$	11.4987	0.408329
$794$	0	0
$795$	2.06439	0.0732164
$796$	0	0
$797$	−42.1109	−1.49164	−0.745822	−	0.666146i	$-0.767943\pi$
$797$	−42.1109	−1.49164	−0.745822	+	0.666146i	$0.767943\pi$
$798$	0	0
$799$	−3.64951	−0.129110
$800$	0	0
$801$	1.91478	0.0676556
$802$	0	0
$803$	2.32610	0.0820863
$804$	0	0
$805$	−0.462598	−0.0163045
$806$	0	0
$807$	11.5076	0.405087
$808$	0	0
$809$	−14.5408	−0.511226	−0.255613	−	0.966779i	$-0.582277\pi$
$809$	−14.5408	−0.511226	−0.255613	+	0.966779i	$0.582277\pi$
$810$	0	0
$811$	−52.0990	−1.82944	−0.914722	−	0.404085i	$-0.867590\pi$
$811$	−52.0990	−1.82944	−0.914722	+	0.404085i	$0.867590\pi$
$812$	0	0
$813$	27.2251	0.954825
$814$	0	0
$815$	−8.82206	−0.309023
$816$	0	0
$817$	−17.5076	−0.612514
$818$	0	0
$819$	0.860806	0.0300790
$820$	0	0
$821$	24.9557	0.870958	0.435479	−	0.900199i	$-0.356579\pi$
$821$	24.9557	0.870958	0.435479	+	0.900199i	$0.356579\pi$
$822$	0	0
$823$	17.4958	0.609864	0.304932	−	0.952374i	$-0.401366\pi$
$823$	17.4958	0.609864	0.304932	+	0.952374i	$0.401366\pi$
$824$	0	0
$825$	−1.90582	−0.0663522
$826$	0	0
$827$	0.570556	0.0198402	0.00992009	−	0.999951i	$-0.496842\pi$
$827$	0.570556	0.0198402	0.00992009	+	0.999951i	$0.496842\pi$
$828$	0	0
$829$	52.0186	1.80668	0.903340	−	0.428925i	$-0.141107\pi$
$829$	52.0186	1.80668	0.903340	+	0.428925i	$0.141107\pi$
$830$	0	0
$831$	3.07962	0.106831
$832$	0	0
$833$	−1.60179	−0.0554988
$834$	0	0
$835$	−8.67178	−0.300099
$836$	0	0
$837$	1.32340	0.0457435
$838$	0	0
$839$	−5.75555	−0.198703	−0.0993517	−	0.995052i	$-0.531677\pi$
$839$	−5.75555	−0.198703	−0.0993517	+	0.995052i	$0.531677\pi$
$840$	0	0
$841$	39.0394	1.34619
$842$	0	0
$843$	23.1648	0.797839
$844$	0	0
$845$	−5.67100	−0.195088
$846$	0	0
$847$	10.8414	0.372516
$848$	0	0
$849$	19.4391	0.667147
$850$	0	0
$851$	10.7666	0.369075
$852$	0	0
$853$	50.2313	1.71989	0.859944	−	0.510388i	$-0.170498\pi$
$853$	50.2313	1.71989	0.859944	+	0.510388i	$0.170498\pi$
$854$	0	0
$855$	2.83102	0.0968188
$856$	0	0
$857$	−43.3449	−1.48063	−0.740317	−	0.672258i	$-0.765324\pi$
$857$	−43.3449	−1.48063	−0.740317	+	0.672258i	$0.765324\pi$
$858$	0	0
$859$	10.4667	0.357121	0.178560	−	0.983929i	$-0.442856\pi$
$859$	10.4667	0.357121	0.178560	+	0.983929i	$0.442856\pi$
$860$	0	0
$861$	1.04502	0.0356140
$862$	0	0
$863$	−18.6981	−0.636490	−0.318245	−	0.948008i	$-0.603094\pi$
$863$	−18.6981	−0.636490	−0.318245	+	0.948008i	$0.603094\pi$
$864$	0	0
$865$	8.97099	0.305023
$866$	0	0
$867$	14.4343	0.490213
$868$	0	0
$869$	0.697420	0.0236584
$870$	0	0
$871$	−2.00896	−0.0680711
$872$	0	0
$873$	−13.8414	−0.468461
$874$	0	0
$875$	4.52699	0.153040
$876$	0	0
$877$	−11.3115	−0.381964	−0.190982	−	0.981594i	$-0.561167\pi$
$877$	−11.3115	−0.381964	−0.190982	+	0.981594i	$0.561167\pi$
$878$	0	0
$879$	25.6829	0.866261
$880$	0	0
$881$	−32.7577	−1.10363	−0.551817	−	0.833965i	$-0.686065\pi$
$881$	−32.7577	−1.10363	−0.551817	+	0.833965i	$0.686065\pi$
$882$	0	0
$883$	29.7293	1.00047	0.500236	−	0.865889i	$-0.333247\pi$
$883$	29.7293	1.00047	0.500236	+	0.865889i	$0.333247\pi$
$884$	0	0
$885$	0.881630	0.0296357
$886$	0	0
$887$	35.5797	1.19465	0.597325	−	0.801999i	$-0.296230\pi$
$887$	35.5797	1.19465	0.597325	+	0.801999i	$0.296230\pi$
$888$	0	0
$889$	12.1544	0.407646
$890$	0	0
$891$	−0.398207	−0.0133405
$892$	0	0
$893$	13.9433	0.466596
$894$	0	0
$895$	−6.86081	−0.229331
$896$	0	0
$897$	0.860806	0.0287415
$898$	0	0
$899$	10.9162	0.364077
$900$	0	0
$901$	7.14816	0.238140
$902$	0	0
$903$	−2.86081	−0.0952017
$904$	0	0
$905$	0.740987	0.0246312
$906$	0	0
$907$	13.3490	0.443248	0.221624	−	0.975132i	$-0.428864\pi$
$907$	13.3490	0.443248	0.221624	+	0.975132i	$0.428864\pi$
$908$	0	0
$909$	−12.8310	−0.425578
$910$	0	0
$911$	2.79352	0.0925533	0.0462767	−	0.998929i	$-0.485264\pi$
$911$	2.79352	0.0925533	0.0462767	+	0.998929i	$0.485264\pi$
$912$	0	0
$913$	3.49094	0.115533
$914$	0	0
$915$	6.17939	0.204284
$916$	0	0
$917$	12.0990	0.399544
$918$	0	0
$919$	1.32340	0.0436551	0.0218275	−	0.999762i	$-0.493052\pi$
$919$	1.32340	0.0436551	0.0218275	+	0.999762i	$0.493052\pi$
$920$	0	0
$921$	−28.2099	−0.929546
$922$	0	0
$923$	−12.7410	−0.419375
$924$	0	0
$925$	−51.5291	−1.69427
$926$	0	0
$927$	14.8864	0.488935
$928$	0	0
$929$	45.3699	1.48854	0.744269	−	0.667880i	$-0.232798\pi$
$929$	45.3699	1.48854	0.744269	+	0.667880i	$0.232798\pi$
$930$	0	0
$931$	6.11982	0.200569
$932$	0	0
$933$	−22.5437	−0.738047
$934$	0	0
$935$	0.295066	0.00964970
$936$	0	0
$937$	55.4439	1.81127	0.905637	−	0.424055i	$-0.139394\pi$
$937$	55.4439	1.81127	0.905637	+	0.424055i	$0.139394\pi$
$938$	0	0
$939$	−1.20359	−0.0392775
$940$	0	0
$941$	−7.83247	−0.255331	−0.127666	−	0.991817i	$-0.540748\pi$
$941$	−7.83247	−0.255331	−0.127666	+	0.991817i	$0.540748\pi$
$942$	0	0
$943$	1.04502	0.0340304
$944$	0	0
$945$	0.462598	0.0150483
$946$	0	0
$947$	−10.3476	−0.336252	−0.168126	−	0.985766i	$-0.553771\pi$
$947$	−10.3476	−0.336252	−0.168126	+	0.985766i	$0.553771\pi$
$948$	0	0
$949$	5.02834	0.163227
$950$	0	0
$951$	−0.831019	−0.0269476
$952$	0	0
$953$	9.28880	0.300894	0.150447	−	0.988618i	$-0.451929\pi$
$953$	9.28880	0.300894	0.150447	+	0.988618i	$0.451929\pi$
$954$	0	0
$955$	−6.34760	−0.205403
$956$	0	0
$957$	−3.28465	−0.106178
$958$	0	0
$959$	14.8954	0.480998
$960$	0	0
$961$	−29.2486	−0.943503
$962$	0	0
$963$	5.50761	0.177480
$964$	0	0
$965$	10.4585	0.336669
$966$	0	0
$967$	−14.0900	−0.453105	−0.226552	−	0.973999i	$-0.572745\pi$
$967$	−14.0900	−0.453105	−0.226552	+	0.973999i	$0.572745\pi$
$968$	0	0
$969$	9.80268	0.314907
$970$	0	0
$971$	−3.97436	−0.127543	−0.0637716	−	0.997965i	$-0.520313\pi$
$971$	−3.97436	−0.127543	−0.0637716	+	0.997965i	$0.520313\pi$
$972$	0	0
$973$	−15.1994	−0.487272
$974$	0	0
$975$	−4.11982	−0.131940
$976$	0	0
$977$	−9.83728	−0.314723	−0.157361	−	0.987541i	$-0.550299\pi$
$977$	−9.83728	−0.314723	−0.157361	+	0.987541i	$0.550299\pi$
$978$	0	0
$979$	−0.762481	−0.0243690
$980$	0	0
$981$	−3.66618	−0.117052
$982$	0	0
$983$	−0.269425	−0.00859334	−0.00429667	−	0.999991i	$-0.501368\pi$
$983$	−0.269425	−0.00859334	−0.00429667	+	0.999991i	$0.501368\pi$
$984$	0	0
$985$	3.27502	0.104351
$986$	0	0
$987$	2.27839	0.0725219
$988$	0	0
$989$	−2.86081	−0.0909683
$990$	0	0
$991$	−9.99518	−0.317507	−0.158754	−	0.987318i	$-0.550748\pi$
$991$	−9.99518	−0.317507	−0.158754	+	0.987318i	$0.550748\pi$
$992$	0	0
$993$	−5.05398	−0.160383
$994$	0	0
$995$	5.86640	0.185977
$996$	0	0
$997$	48.9162	1.54919	0.774596	−	0.632456i	$-0.217953\pi$
$997$	48.9162	1.54919	0.774596	+	0.632456i	$0.217953\pi$
$998$	0	0
$999$	−10.7666	−0.340641

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3864.2.a.l.1.2	✓	3
4.3	odd	2		7728.2.a.bw.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3864.2.a.l.1.2	✓	3	1.1	even	1	trivial
7728.2.a.bw.1.2		3	4.3	odd	2

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:	\( N \)	\(=\)	\( 3864 = 2^{3} \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3864.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +0.462598 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +0.462598 q^{5} -1.00000 q^{7} +1.00000 q^{9} -0.398207 q^{11} -0.860806 q^{13} -0.462598 q^{15} -1.60179 q^{17} +6.11982 q^{19} +1.00000 q^{21} +1.00000 q^{23} -4.78600 q^{25} -1.00000 q^{27} -8.24860 q^{29} -1.32340 q^{31} +0.398207 q^{33} -0.462598 q^{35} +10.7666 q^{37} +0.860806 q^{39} +1.04502 q^{41} -2.86081 q^{43} +0.462598 q^{45} +2.27839 q^{47} +1.00000 q^{49} +1.60179 q^{51} -4.46260 q^{53} -0.184210 q^{55} -6.11982 q^{57} -1.90582 q^{59} -13.3580 q^{61} -1.00000 q^{63} -0.398207 q^{65} +2.33382 q^{67} -1.00000 q^{69} +14.8012 q^{71} -5.84143 q^{73} +4.78600 q^{75} +0.398207 q^{77} -1.75140 q^{79} +1.00000 q^{81} -8.76663 q^{83} -0.740987 q^{85} +8.24860 q^{87} +1.91478 q^{89} +0.860806 q^{91} +1.32340 q^{93} +2.83102 q^{95} -13.8414 q^{97} -0.398207 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} - q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 - q^5 - 3 * q^7 + 3 * q^9	\( 3 q - 3 q^{3} - q^{5} - 3 q^{7} + 3 q^{9} + 2 q^{11} + 3 q^{13} + q^{15} - 8 q^{17} + 4 q^{19} + 3 q^{21} + 3 q^{23} - 4 q^{25} - 3 q^{27} - 12 q^{29} + 4 q^{31} - 2 q^{33} + q^{35} + 2 q^{37} - 3 q^{39} - 16 q^{41} - 3 q^{43} - q^{45} + 18 q^{47} + 3 q^{49} + 8 q^{51} - 11 q^{53} + 13 q^{55} - 4 q^{57} + 19 q^{59} - 9 q^{61} - 3 q^{63} + 2 q^{65} + 3 q^{67} - 3 q^{69} - 9 q^{71} + 8 q^{73} + 4 q^{75} - 2 q^{77} - 18 q^{79} + 3 q^{81} + 4 q^{83} - 11 q^{85} + 12 q^{87} - 3 q^{89} - 3 q^{91} - 4 q^{93} - 21 q^{95} - 16 q^{97} + 2 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 - q^5 - 3 * q^7 + 3 * q^9 + 2 * q^11 + 3 * q^13 + q^15 - 8 * q^17 + 4 * q^19 + 3 * q^21 + 3 * q^23 - 4 * q^25 - 3 * q^27 - 12 * q^29 + 4 * q^31 - 2 * q^33 + q^35 + 2 * q^37 - 3 * q^39 - 16 * q^41 - 3 * q^43 - q^45 + 18 * q^47 + 3 * q^49 + 8 * q^51 - 11 * q^53 + 13 * q^55 - 4 * q^57 + 19 * q^59 - 9 * q^61 - 3 * q^63 + 2 * q^65 + 3 * q^67 - 3 * q^69 - 9 * q^71 + 8 * q^73 + 4 * q^75 - 2 * q^77 - 18 * q^79 + 3 * q^81 + 4 * q^83 - 11 * q^85 + 12 * q^87 - 3 * q^89 - 3 * q^91 - 4 * q^93 - 21 * q^95 - 16 * q^97 + 2 * q^99

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