LMFDB - Embedded newform 6864.2.a.bu.1.1

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:	3.3.564.1
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:	$\Z[a_1, \ldots, a_{7}]$
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			1.1
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)

$f(q)$	$=$	$q+1.00000 q^{3} -4.24482 q^{5} +2.67282 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -4.24482 q^{5} +2.67282 q^{7} +1.00000 q^{9} -1.00000 q^{11} -1.00000 q^{13} -4.24482 q^{15} -0.428007 q^{17} -6.67282 q^{19} +2.67282 q^{21} +7.81681 q^{23} +13.0185 q^{25} +1.00000 q^{27} -2.91764 q^{29} -1.75518 q^{31} -1.00000 q^{33} -11.3456 q^{35} +7.63362 q^{37} -1.00000 q^{39} -2.38485 q^{41} +11.7737 q^{43} -4.24482 q^{45} -6.48963 q^{47} +0.143987 q^{49} -0.428007 q^{51} -2.85601 q^{53} +4.24482 q^{55} -6.67282 q^{57} -12.4896 q^{59} +3.14399 q^{61} +2.67282 q^{63} +4.24482 q^{65} -13.3888 q^{67} +7.81681 q^{69} +7.34565 q^{71} +11.7305 q^{73} +13.0185 q^{75} -2.67282 q^{77} -6.42801 q^{79} +1.00000 q^{81} -1.79834 q^{83} +1.81681 q^{85} -2.91764 q^{87} +3.59046 q^{89} -2.67282 q^{91} -1.75518 q^{93} +28.3249 q^{95} -11.6336 q^{97} -1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} - 2 q^{5} - 2 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 - 2 * q^5 - 2 * q^7 + 3 * q^9	$ 3 q + 3 q^{3} - 2 q^{5} - 2 q^{7} + 3 q^{9} - 3 q^{11} - 3 q^{13} - 2 q^{15} - 2 q^{17} - 10 q^{19} - 2 q^{21} + 12 q^{23} + 9 q^{25} + 3 q^{27} + 12 q^{29} - 16 q^{31} - 3 q^{33} - 14 q^{35} - 3 q^{39} + 16 q^{43} - 2 q^{45} + 2 q^{47} - q^{49} - 2 q^{51} - 10 q^{53} + 2 q^{55} - 10 q^{57} - 16 q^{59} + 8 q^{61} - 2 q^{63} + 2 q^{65} - 28 q^{67} + 12 q^{69} + 2 q^{71} + 8 q^{73} + 9 q^{75} + 2 q^{77} - 20 q^{79} + 3 q^{81} - 24 q^{83} - 6 q^{85} + 12 q^{87} - 20 q^{89} + 2 q^{91} - 16 q^{93} + 22 q^{95} - 12 q^{97} - 3 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 - 2 * q^5 - 2 * q^7 + 3 * q^9 - 3 * q^11 - 3 * q^13 - 2 * q^15 - 2 * q^17 - 10 * q^19 - 2 * q^21 + 12 * q^23 + 9 * q^25 + 3 * q^27 + 12 * q^29 - 16 * q^31 - 3 * q^33 - 14 * q^35 - 3 * q^39 + 16 * q^43 - 2 * q^45 + 2 * q^47 - q^49 - 2 * q^51 - 10 * q^53 + 2 * q^55 - 10 * q^57 - 16 * q^59 + 8 * q^61 - 2 * q^63 + 2 * q^65 - 28 * q^67 + 12 * q^69 + 2 * q^71 + 8 * q^73 + 9 * q^75 + 2 * q^77 - 20 * q^79 + 3 * q^81 - 24 * q^83 - 6 * q^85 + 12 * q^87 - 20 * q^89 + 2 * q^91 - 16 * q^93 + 22 * 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
$5$	−4.24482	−1.89834	−0.949170	−	0.314764i	$-0.898075\pi$
$5$	−4.24482	−1.89834	−0.949170	+	0.314764i	$0.898075\pi$
$6$	0	0
$7$	2.67282	1.01023	0.505116	−	0.863051i	$-0.331450\pi$
$7$	2.67282	1.01023	0.505116	+	0.863051i	$0.331450\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−4.24482	−1.09601
$16$	0	0
$17$	−0.428007	−0.103807	−0.0519034	−	0.998652i	$-0.516529\pi$
$17$	−0.428007	−0.103807	−0.0519034	+	0.998652i	$0.516529\pi$
$18$	0	0
$19$	−6.67282	−1.53085	−0.765425	−	0.643525i	$-0.777471\pi$
$19$	−6.67282	−1.53085	−0.765425	+	0.643525i	$0.777471\pi$
$20$	0	0
$21$	2.67282	0.583258
$22$	0	0
$23$	7.81681	1.62992	0.814959	−	0.579519i	$-0.196760\pi$
$23$	7.81681	1.62992	0.814959	+	0.579519i	$0.196760\pi$
$24$	0	0
$25$	13.0185	2.60369
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−2.91764	−0.541792	−0.270896	−	0.962609i	$-0.587320\pi$
$29$	−2.91764	−0.541792	−0.270896	+	0.962609i	$0.587320\pi$
$30$	0	0
$31$	−1.75518	−0.315240	−0.157620	−	0.987500i	$-0.550382\pi$
$31$	−1.75518	−0.315240	−0.157620	+	0.987500i	$0.550382\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	−11.3456	−1.91776
$36$	0	0
$37$	7.63362	1.25496	0.627480	−	0.778633i	$-0.284087\pi$
$37$	7.63362	1.25496	0.627480	+	0.778633i	$0.284087\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−2.38485	−0.372451	−0.186226	−	0.982507i	$-0.559626\pi$
$41$	−2.38485	−0.372451	−0.186226	+	0.982507i	$0.559626\pi$
$42$	0	0
$43$	11.7737	1.79547	0.897733	−	0.440541i	$-0.145213\pi$
$43$	11.7737	1.79547	0.897733	+	0.440541i	$0.145213\pi$
$44$	0	0
$45$	−4.24482	−0.632780
$46$	0	0
$47$	−6.48963	−0.946610	−0.473305	−	0.880899i	$-0.656939\pi$
$47$	−6.48963	−0.946610	−0.473305	+	0.880899i	$0.656939\pi$
$48$	0	0
$49$	0.143987	0.0205695
$50$	0	0
$51$	−0.428007	−0.0599329
$52$	0	0
$53$	−2.85601	−0.392304	−0.196152	−	0.980574i	$-0.562845\pi$
$53$	−2.85601	−0.392304	−0.196152	+	0.980574i	$0.562845\pi$
$54$	0	0
$55$	4.24482	0.572371
$56$	0	0
$57$	−6.67282	−0.883837
$58$	0	0
$59$	−12.4896	−1.62601	−0.813006	−	0.582255i	$-0.802170\pi$
$59$	−12.4896	−1.62601	−0.813006	+	0.582255i	$0.802170\pi$
$60$	0	0
$61$	3.14399	0.402546	0.201273	−	0.979535i	$-0.435492\pi$
$61$	3.14399	0.402546	0.201273	+	0.979535i	$0.435492\pi$
$62$	0	0
$63$	2.67282	0.336744
$64$	0	0
$65$	4.24482	0.526505
$66$	0	0
$67$	−13.3888	−1.63570	−0.817851	−	0.575430i	$-0.804835\pi$
$67$	−13.3888	−1.63570	−0.817851	+	0.575430i	$0.804835\pi$
$68$	0	0
$69$	7.81681	0.941033
$70$	0	0
$71$	7.34565	0.871768	0.435884	−	0.900003i	$-0.356436\pi$
$71$	7.34565	0.871768	0.435884	+	0.900003i	$0.356436\pi$
$72$	0	0
$73$	11.7305	1.37295	0.686475	−	0.727153i	$-0.259157\pi$
$73$	11.7305	1.37295	0.686475	+	0.727153i	$0.259157\pi$
$74$	0	0
$75$	13.0185	1.50324
$76$	0	0
$77$	−2.67282	−0.304597
$78$	0	0
$79$	−6.42801	−0.723207	−0.361604	−	0.932332i	$-0.617771\pi$
$79$	−6.42801	−0.723207	−0.361604	+	0.932332i	$0.617771\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−1.79834	−0.197393	−0.0986967	−	0.995118i	$-0.531467\pi$
$83$	−1.79834	−0.197393	−0.0986967	+	0.995118i	$0.531467\pi$
$84$	0	0
$85$	1.81681	0.197061
$86$	0	0
$87$	−2.91764	−0.312804
$88$	0	0
$89$	3.59046	0.380588	0.190294	−	0.981727i	$-0.439056\pi$
$89$	3.59046	0.380588	0.190294	+	0.981727i	$0.439056\pi$
$90$	0	0
$91$	−2.67282	−0.280188
$92$	0	0
$93$	−1.75518	−0.182004
$94$	0	0
$95$	28.3249	2.90607
$96$	0	0
$97$	−11.6336	−1.18122	−0.590608	−	0.806959i	$-0.701112\pi$
$97$	−11.6336	−1.18122	−0.590608	+	0.806959i	$0.701112\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	0	0
$101$	7.48568	0.744853	0.372427	−	0.928062i	$-0.378526\pi$
$101$	7.48568	0.744853	0.372427	+	0.928062i	$0.378526\pi$
$102$	0	0
$103$	−1.71203	−0.168691	−0.0843455	−	0.996437i	$-0.526880\pi$
$103$	−1.71203	−0.168691	−0.0843455	+	0.996437i	$0.526880\pi$
$104$	0	0
$105$	−11.3456	−1.10722
$106$	0	0
$107$	−4.48963	−0.434029	−0.217015	−	0.976168i	$-0.569632\pi$
$107$	−4.48963	−0.434029	−0.217015	+	0.976168i	$0.569632\pi$
$108$	0	0
$109$	−10.0185	−0.959595	−0.479798	−	0.877379i	$-0.659290\pi$
$109$	−10.0185	−0.959595	−0.479798	+	0.877379i	$0.659290\pi$
$110$	0	0
$111$	7.63362	0.724551
$112$	0	0
$113$	−10.4896	−0.986782	−0.493391	−	0.869808i	$-0.664243\pi$
$113$	−10.4896	−0.986782	−0.493391	+	0.869808i	$0.664243\pi$
$114$	0	0
$115$	−33.1809	−3.09414
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	−1.14399	−0.104869
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−2.38485	−0.215035
$124$	0	0
$125$	−34.0369	−3.04436
$126$	0	0
$127$	−10.6297	−0.943230	−0.471615	−	0.881804i	$-0.656329\pi$
$127$	−10.6297	−0.943230	−0.471615	+	0.881804i	$0.656329\pi$
$128$	0	0
$129$	11.7737	1.03661
$130$	0	0
$131$	−19.6336	−1.71540	−0.857699	−	0.514153i	$-0.828106\pi$
$131$	−19.6336	−1.71540	−0.857699	+	0.514153i	$0.828106\pi$
$132$	0	0
$133$	−17.8353	−1.54652
$134$	0	0
$135$	−4.24482	−0.365336
$136$	0	0
$137$	−8.81286	−0.752933	−0.376467	−	0.926430i	$-0.622861\pi$
$137$	−8.81286	−0.752933	−0.376467	+	0.926430i	$0.622861\pi$
$138$	0	0
$139$	9.40727	0.797915	0.398957	−	0.916970i	$-0.369372\pi$
$139$	9.40727	0.797915	0.398957	+	0.916970i	$0.369372\pi$
$140$	0	0
$141$	−6.48963	−0.546526
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	12.3849	1.02851
$146$	0	0
$147$	0.143987	0.0118758
$148$	0	0
$149$	−4.67282	−0.382813	−0.191406	−	0.981511i	$-0.561305\pi$
$149$	−4.67282	−0.382813	−0.191406	+	0.981511i	$0.561305\pi$
$150$	0	0
$151$	5.73050	0.466341	0.233171	−	0.972436i	$-0.425090\pi$
$151$	5.73050	0.466341	0.233171	+	0.972436i	$0.425090\pi$
$152$	0	0
$153$	−0.428007	−0.0346023
$154$	0	0
$155$	7.45043	0.598433
$156$	0	0
$157$	8.38485	0.669184	0.334592	−	0.942363i	$-0.391402\pi$
$157$	8.38485	0.669184	0.334592	+	0.942363i	$0.391402\pi$
$158$	0	0
$159$	−2.85601	−0.226497
$160$	0	0
$161$	20.8930	1.64660
$162$	0	0
$163$	4.82076	0.377591	0.188796	−	0.982016i	$-0.439542\pi$
$163$	4.82076	0.377591	0.188796	+	0.982016i	$0.439542\pi$
$164$	0	0
$165$	4.24482	0.330459
$166$	0	0
$167$	−19.6336	−1.51930	−0.759648	−	0.650335i	$-0.774629\pi$
$167$	−19.6336	−1.51930	−0.759648	+	0.650335i	$0.774629\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−6.67282	−0.510284
$172$	0	0
$173$	4.42801	0.336655	0.168328	−	0.985731i	$-0.446163\pi$
$173$	4.42801	0.336655	0.168328	+	0.985731i	$0.446163\pi$
$174$	0	0
$175$	34.7961	2.63034
$176$	0	0
$177$	−12.4896	−0.938778
$178$	0	0
$179$	−13.2488	−0.990260	−0.495130	−	0.868819i	$-0.664879\pi$
$179$	−13.2488	−0.990260	−0.495130	+	0.868819i	$0.664879\pi$
$180$	0	0
$181$	11.4425	0.850516	0.425258	−	0.905072i	$-0.360183\pi$
$181$	11.4425	0.850516	0.425258	+	0.905072i	$0.360183\pi$
$182$	0	0
$183$	3.14399	0.232410
$184$	0	0
$185$	−32.4033	−2.38234
$186$	0	0
$187$	0.428007	0.0312990
$188$	0	0
$189$	2.67282	0.194419
$190$	0	0
$191$	−0.489634	−0.0354287	−0.0177143	−	0.999843i	$-0.505639\pi$
$191$	−0.489634	−0.0354287	−0.0177143	+	0.999843i	$0.505639\pi$
$192$	0	0
$193$	−1.03920	−0.0748035	−0.0374017	−	0.999300i	$-0.511908\pi$
$193$	−1.03920	−0.0748035	−0.0374017	+	0.999300i	$0.511908\pi$
$194$	0	0
$195$	4.24482	0.303978
$196$	0	0
$197$	9.16246	0.652798	0.326399	−	0.945232i	$-0.394165\pi$
$197$	9.16246	0.652798	0.326399	+	0.945232i	$0.394165\pi$
$198$	0	0
$199$	8.40332	0.595696	0.297848	−	0.954613i	$-0.403731\pi$
$199$	8.40332	0.595696	0.297848	+	0.954613i	$0.403731\pi$
$200$	0	0
$201$	−13.3888	−0.944373
$202$	0	0
$203$	−7.79834	−0.547336
$204$	0	0
$205$	10.1233	0.707039
$206$	0	0
$207$	7.81681	0.543306
$208$	0	0
$209$	6.67282	0.461569
$210$	0	0
$211$	−14.4649	−0.995808	−0.497904	−	0.867232i	$-0.665897\pi$
$211$	−14.4649	−0.995808	−0.497904	+	0.867232i	$0.665897\pi$
$212$	0	0
$213$	7.34565	0.503315
$214$	0	0
$215$	−49.9770	−3.40840
$216$	0	0
$217$	−4.69129	−0.318466
$218$	0	0
$219$	11.7305	0.792674
$220$	0	0
$221$	0.428007	0.0287908
$222$	0	0
$223$	2.73445	0.183112	0.0915562	−	0.995800i	$-0.470816\pi$
$223$	2.73445	0.183112	0.0915562	+	0.995800i	$0.470816\pi$
$224$	0	0
$225$	13.0185	0.867898
$226$	0	0
$227$	−18.7961	−1.24754	−0.623770	−	0.781608i	$-0.714400\pi$
$227$	−18.7961	−1.24754	−0.623770	+	0.781608i	$0.714400\pi$
$228$	0	0
$229$	4.48963	0.296683	0.148342	−	0.988936i	$-0.452606\pi$
$229$	4.48963	0.296683	0.148342	+	0.988936i	$0.452606\pi$
$230$	0	0
$231$	−2.67282	−0.175859
$232$	0	0
$233$	−11.0040	−0.720893	−0.360446	−	0.932780i	$-0.617376\pi$
$233$	−11.0040	−0.720893	−0.360446	+	0.932780i	$0.617376\pi$
$234$	0	0
$235$	27.5473	1.79699
$236$	0	0
$237$	−6.42801	−0.417544
$238$	0	0
$239$	−25.2778	−1.63509	−0.817543	−	0.575868i	$-0.804664\pi$
$239$	−25.2778	−1.63509	−0.817543	+	0.575868i	$0.804664\pi$
$240$	0	0
$241$	15.2593	0.982940	0.491470	−	0.870894i	$-0.336460\pi$
$241$	15.2593	0.982940	0.491470	+	0.870894i	$0.336460\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−0.611196	−0.0390479
$246$	0	0
$247$	6.67282	0.424582
$248$	0	0
$249$	−1.79834	−0.113965
$250$	0	0
$251$	−7.91369	−0.499508	−0.249754	−	0.968309i	$-0.580350\pi$
$251$	−7.91369	−0.499508	−0.249754	+	0.968309i	$0.580350\pi$
$252$	0	0
$253$	−7.81681	−0.491439
$254$	0	0
$255$	1.81681	0.113773
$256$	0	0
$257$	−15.3456	−0.957235	−0.478618	−	0.878023i	$-0.658862\pi$
$257$	−15.3456	−0.957235	−0.478618	+	0.878023i	$0.658862\pi$
$258$	0	0
$259$	20.4033	1.26780
$260$	0	0
$261$	−2.91764	−0.180597
$262$	0	0
$263$	26.7776	1.65118	0.825589	−	0.564272i	$-0.190843\pi$
$263$	26.7776	1.65118	0.825589	+	0.564272i	$0.190843\pi$
$264$	0	0
$265$	12.1233	0.744726
$266$	0	0
$267$	3.59046	0.219733
$268$	0	0
$269$	6.85601	0.418019	0.209009	−	0.977914i	$-0.432976\pi$
$269$	6.85601	0.418019	0.209009	+	0.977914i	$0.432976\pi$
$270$	0	0
$271$	18.1832	1.10455	0.552275	−	0.833662i	$-0.313760\pi$
$271$	18.1832	1.10455	0.552275	+	0.833662i	$0.313760\pi$
$272$	0	0
$273$	−2.67282	−0.161767
$274$	0	0
$275$	−13.0185	−0.785043
$276$	0	0
$277$	−23.7490	−1.42694	−0.713469	−	0.700687i	$-0.752877\pi$
$277$	−23.7490	−1.42694	−0.713469	+	0.700687i	$0.752877\pi$
$278$	0	0
$279$	−1.75518	−0.105080
$280$	0	0
$281$	23.6442	1.41049	0.705247	−	0.708962i	$-0.250836\pi$
$281$	23.6442	1.41049	0.705247	+	0.708962i	$0.250836\pi$
$282$	0	0
$283$	−28.0616	−1.66809	−0.834045	−	0.551696i	$-0.813981\pi$
$283$	−28.0616	−1.66809	−0.834045	+	0.551696i	$0.813981\pi$
$284$	0	0
$285$	28.3249	1.67782
$286$	0	0
$287$	−6.37429	−0.376262
$288$	0	0
$289$	−16.8168	−0.989224
$290$	0	0
$291$	−11.6336	−0.681975
$292$	0	0
$293$	−27.4874	−1.60583	−0.802915	−	0.596094i	$-0.796719\pi$
$293$	−27.4874	−1.60583	−0.802915	+	0.596094i	$0.796719\pi$
$294$	0	0
$295$	53.0162	3.08672
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	−7.81681	−0.452058
$300$	0	0
$301$	31.4689	1.81384
$302$	0	0
$303$	7.48568	0.430041
$304$	0	0
$305$	−13.3456	−0.764170
$306$	0	0
$307$	−9.15455	−0.522478	−0.261239	−	0.965274i	$-0.584131\pi$
$307$	−9.15455	−0.522478	−0.261239	+	0.965274i	$0.584131\pi$
$308$	0	0
$309$	−1.71203	−0.0973938
$310$	0	0
$311$	−15.2409	−0.864230	−0.432115	−	0.901818i	$-0.642233\pi$
$311$	−15.2409	−0.864230	−0.432115	+	0.901818i	$0.642233\pi$
$312$	0	0
$313$	−8.87448	−0.501616	−0.250808	−	0.968037i	$-0.580696\pi$
$313$	−8.87448	−0.501616	−0.250808	+	0.968037i	$0.580696\pi$
$314$	0	0
$315$	−11.3456	−0.639255
$316$	0	0
$317$	7.01452	0.393975	0.196987	−	0.980406i	$-0.436884\pi$
$317$	7.01452	0.393975	0.196987	+	0.980406i	$0.436884\pi$
$318$	0	0
$319$	2.91764	0.163357
$320$	0	0
$321$	−4.48963	−0.250587
$322$	0	0
$323$	2.85601	0.158913
$324$	0	0
$325$	−13.0185	−0.722135
$326$	0	0
$327$	−10.0185	−0.554023
$328$	0	0
$329$	−17.3456	−0.956296
$330$	0	0
$331$	−24.3681	−1.33939	−0.669695	−	0.742636i	$-0.733575\pi$
$331$	−24.3681	−1.33939	−0.669695	+	0.742636i	$0.733575\pi$
$332$	0	0
$333$	7.63362	0.418320
$334$	0	0
$335$	56.8330	3.10512
$336$	0	0
$337$	−28.6050	−1.55821	−0.779106	−	0.626892i	$-0.784327\pi$
$337$	−28.6050	−1.55821	−0.779106	+	0.626892i	$0.784327\pi$
$338$	0	0
$339$	−10.4896	−0.569719
$340$	0	0
$341$	1.75518	0.0950485
$342$	0	0
$343$	−18.3249	−0.989452
$344$	0	0
$345$	−33.1809	−1.78640
$346$	0	0
$347$	1.42405	0.0764472	0.0382236	−	0.999269i	$-0.487830\pi$
$347$	1.42405	0.0764472	0.0382236	+	0.999269i	$0.487830\pi$
$348$	0	0
$349$	18.1233	0.970116	0.485058	−	0.874482i	$-0.338799\pi$
$349$	18.1233	0.970116	0.485058	+	0.874482i	$0.338799\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	−9.46721	−0.503889	−0.251944	−	0.967742i	$-0.581070\pi$
$353$	−9.46721	−0.503889	−0.251944	+	0.967742i	$0.581070\pi$
$354$	0	0
$355$	−31.1809	−1.65491
$356$	0	0
$357$	−1.14399	−0.0605462
$358$	0	0
$359$	5.04711	0.266376	0.133188	−	0.991091i	$-0.457479\pi$
$359$	5.04711	0.266376	0.133188	+	0.991091i	$0.457479\pi$
$360$	0	0
$361$	25.5266	1.34350
$362$	0	0
$363$	1.00000	0.0524864
$364$	0	0
$365$	−49.7938	−2.60633
$366$	0	0
$367$	−16.6992	−0.871691	−0.435846	−	0.900021i	$-0.643551\pi$
$367$	−16.6992	−0.871691	−0.435846	+	0.900021i	$0.643551\pi$
$368$	0	0
$369$	−2.38485	−0.124150
$370$	0	0
$371$	−7.63362	−0.396318
$372$	0	0
$373$	28.6050	1.48111	0.740555	−	0.671996i	$-0.234563\pi$
$373$	28.6050	1.48111	0.740555	+	0.671996i	$0.234563\pi$
$374$	0	0
$375$	−34.0369	−1.75766
$376$	0	0
$377$	2.91764	0.150266
$378$	0	0
$379$	31.8291	1.63495	0.817475	−	0.575965i	$-0.195373\pi$
$379$	31.8291	1.63495	0.817475	+	0.575965i	$0.195373\pi$
$380$	0	0
$381$	−10.6297	−0.544574
$382$	0	0
$383$	−1.83528	−0.0937785	−0.0468892	−	0.998900i	$-0.514931\pi$
$383$	−1.83528	−0.0937785	−0.0468892	+	0.998900i	$0.514931\pi$
$384$	0	0
$385$	11.3456	0.578228
$386$	0	0
$387$	11.7737	0.598488
$388$	0	0
$389$	19.4689	0.987113	0.493556	−	0.869714i	$-0.335697\pi$
$389$	19.4689	0.987113	0.493556	+	0.869714i	$0.335697\pi$
$390$	0	0
$391$	−3.34565	−0.169197
$392$	0	0
$393$	−19.6336	−0.990385
$394$	0	0
$395$	27.2857	1.37289
$396$	0	0
$397$	−21.0162	−1.05477	−0.527387	−	0.849625i	$-0.676828\pi$
$397$	−21.0162	−1.05477	−0.527387	+	0.849625i	$0.676828\pi$
$398$	0	0
$399$	−17.8353	−0.892881
$400$	0	0
$401$	−35.2162	−1.75861	−0.879306	−	0.476257i	$-0.841993\pi$
$401$	−35.2162	−1.75861	−0.879306	+	0.476257i	$0.841993\pi$
$402$	0	0
$403$	1.75518	0.0874319
$404$	0	0
$405$	−4.24482	−0.210927
$406$	0	0
$407$	−7.63362	−0.378385
$408$	0	0
$409$	−6.96080	−0.344189	−0.172095	−	0.985080i	$-0.555053\pi$
$409$	−6.96080	−0.344189	−0.172095	+	0.985080i	$0.555053\pi$
$410$	0	0
$411$	−8.81286	−0.434706
$412$	0	0
$413$	−33.3826	−1.64265
$414$	0	0
$415$	7.63362	0.374720
$416$	0	0
$417$	9.40727	0.460676
$418$	0	0
$419$	18.5944	0.908397	0.454198	−	0.890901i	$-0.349926\pi$
$419$	18.5944	0.908397	0.454198	+	0.890901i	$0.349926\pi$
$420$	0	0
$421$	−30.6129	−1.49198	−0.745990	−	0.665957i	$-0.768024\pi$
$421$	−30.6129	−1.49198	−0.745990	+	0.665957i	$0.768024\pi$
$422$	0	0
$423$	−6.48963	−0.315537
$424$	0	0
$425$	−5.57199	−0.270281
$426$	0	0
$427$	8.40332	0.406665
$428$	0	0
$429$	1.00000	0.0482805
$430$	0	0
$431$	8.20957	0.395441	0.197720	−	0.980258i	$-0.436646\pi$
$431$	8.20957	0.395441	0.197720	+	0.980258i	$0.436646\pi$
$432$	0	0
$433$	−28.1153	−1.35114	−0.675569	−	0.737297i	$-0.736102\pi$
$433$	−28.1153	−1.35114	−0.675569	+	0.737297i	$0.736102\pi$
$434$	0	0
$435$	12.3849	0.593808
$436$	0	0
$437$	−52.1602	−2.49516
$438$	0	0
$439$	16.7529	0.799573	0.399787	−	0.916608i	$-0.369084\pi$
$439$	16.7529	0.799573	0.399787	+	0.916608i	$0.369084\pi$
$440$	0	0
$441$	0.143987	0.00685650
$442$	0	0
$443$	−12.9793	−0.616664	−0.308332	−	0.951279i	$-0.599771\pi$
$443$	−12.9793	−0.616664	−0.308332	+	0.951279i	$0.599771\pi$
$444$	0	0
$445$	−15.2409	−0.722486
$446$	0	0
$447$	−4.67282	−0.221017
$448$	0	0
$449$	24.1954	1.14185	0.570927	−	0.821001i	$-0.306584\pi$
$449$	24.1954	1.14185	0.570927	+	0.821001i	$0.306584\pi$
$450$	0	0
$451$	2.38485	0.112298
$452$	0	0
$453$	5.73050	0.269242
$454$	0	0
$455$	11.3456	0.531892
$456$	0	0
$457$	−4.32492	−0.202311	−0.101156	−	0.994871i	$-0.532254\pi$
$457$	−4.32492	−0.202311	−0.101156	+	0.994871i	$0.532254\pi$
$458$	0	0
$459$	−0.428007	−0.0199776
$460$	0	0
$461$	−6.38485	−0.297372	−0.148686	−	0.988884i	$-0.547504\pi$
$461$	−6.38485	−0.297372	−0.148686	+	0.988884i	$0.547504\pi$
$462$	0	0
$463$	−30.6560	−1.42471	−0.712354	−	0.701821i	$-0.752371\pi$
$463$	−30.6560	−1.42471	−0.712354	+	0.701821i	$0.752371\pi$
$464$	0	0
$465$	7.45043	0.345505
$466$	0	0
$467$	−31.8538	−1.47402	−0.737008	−	0.675884i	$-0.763762\pi$
$467$	−31.8538	−1.47402	−0.737008	+	0.675884i	$0.763762\pi$
$468$	0	0
$469$	−35.7859	−1.65244
$470$	0	0
$471$	8.38485	0.386354
$472$	0	0
$473$	−11.7737	−0.541353
$474$	0	0
$475$	−86.8700	−3.98587
$476$	0	0
$477$	−2.85601	−0.130768
$478$	0	0
$479$	−29.8168	−1.36236	−0.681182	−	0.732114i	$-0.738534\pi$
$479$	−29.8168	−1.36236	−0.681182	+	0.732114i	$0.738534\pi$
$480$	0	0
$481$	−7.63362	−0.348063
$482$	0	0
$483$	20.8930	0.950662
$484$	0	0
$485$	49.3826	2.24235
$486$	0	0
$487$	19.4257	0.880265	0.440132	−	0.897933i	$-0.354932\pi$
$487$	19.4257	0.880265	0.440132	+	0.897933i	$0.354932\pi$
$488$	0	0
$489$	4.82076	0.218002
$490$	0	0
$491$	−16.7282	−0.754935	−0.377467	−	0.926023i	$-0.623205\pi$
$491$	−16.7282	−0.754935	−0.377467	+	0.926023i	$0.623205\pi$
$492$	0	0
$493$	1.24877	0.0562418
$494$	0	0
$495$	4.24482	0.190790
$496$	0	0
$497$	19.6336	0.880688
$498$	0	0
$499$	−18.0017	−0.805866	−0.402933	−	0.915229i	$-0.632009\pi$
$499$	−18.0017	−0.805866	−0.402933	+	0.915229i	$0.632009\pi$
$500$	0	0
$501$	−19.6336	−0.877165
$502$	0	0
$503$	−30.0448	−1.33963	−0.669817	−	0.742526i	$-0.733627\pi$
$503$	−30.0448	−1.33963	−0.669817	+	0.742526i	$0.733627\pi$
$504$	0	0
$505$	−31.7753	−1.41398
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	1.79213	0.0794346	0.0397173	−	0.999211i	$-0.487354\pi$
$509$	1.79213	0.0794346	0.0397173	+	0.999211i	$0.487354\pi$
$510$	0	0
$511$	31.3536	1.38700
$512$	0	0
$513$	−6.67282	−0.294612
$514$	0	0
$515$	7.26724	0.320233
$516$	0	0
$517$	6.48963	0.285414
$518$	0	0
$519$	4.42801	0.194368
$520$	0	0
$521$	−29.0946	−1.27466	−0.637329	−	0.770592i	$-0.719961\pi$
$521$	−29.0946	−1.27466	−0.637329	+	0.770592i	$0.719961\pi$
$522$	0	0
$523$	20.0695	0.877579	0.438790	−	0.898590i	$-0.355407\pi$
$523$	20.0695	0.877579	0.438790	+	0.898590i	$0.355407\pi$
$524$	0	0
$525$	34.7961	1.51863
$526$	0	0
$527$	0.751230	0.0327241
$528$	0	0
$529$	38.1025	1.65663
$530$	0	0
$531$	−12.4896	−0.542004
$532$	0	0
$533$	2.38485	0.103299
$534$	0	0
$535$	19.0577	0.823935
$536$	0	0
$537$	−13.2488	−0.571727
$538$	0	0
$539$	−0.143987	−0.00620194
$540$	0	0
$541$	24.1153	1.03680	0.518400	−	0.855138i	$-0.326528\pi$
$541$	24.1153	1.03680	0.518400	+	0.855138i	$0.326528\pi$
$542$	0	0
$543$	11.4425	0.491046
$544$	0	0
$545$	42.5266	1.82164
$546$	0	0
$547$	−32.5019	−1.38968	−0.694840	−	0.719164i	$-0.744525\pi$
$547$	−32.5019	−1.38968	−0.694840	+	0.719164i	$0.744525\pi$
$548$	0	0
$549$	3.14399	0.134182
$550$	0	0
$551$	19.4689	0.829403
$552$	0	0
$553$	−17.1809	−0.730607
$554$	0	0
$555$	−32.4033	−1.37544
$556$	0	0
$557$	41.6890	1.76642	0.883211	−	0.468977i	$-0.155377\pi$
$557$	41.6890	1.76642	0.883211	+	0.468977i	$0.155377\pi$
$558$	0	0
$559$	−11.7737	−0.497973
$560$	0	0
$561$	0.428007	0.0180705
$562$	0	0
$563$	5.74897	0.242290	0.121145	−	0.992635i	$-0.461343\pi$
$563$	5.74897	0.242290	0.121145	+	0.992635i	$0.461343\pi$
$564$	0	0
$565$	44.5266	1.87325
$566$	0	0
$567$	2.67282	0.112248
$568$	0	0
$569$	15.8521	0.664553	0.332276	−	0.943182i	$-0.392183\pi$
$569$	15.8521	0.664553	0.332276	+	0.943182i	$0.392183\pi$
$570$	0	0
$571$	5.57199	0.233181	0.116590	−	0.993180i	$-0.462804\pi$
$571$	5.57199	0.233181	0.116590	+	0.993180i	$0.462804\pi$
$572$	0	0
$573$	−0.489634	−0.0204548
$574$	0	0
$575$	101.763	4.24381
$576$	0	0
$577$	−2.20957	−0.0919855	−0.0459927	−	0.998942i	$-0.514645\pi$
$577$	−2.20957	−0.0919855	−0.0459927	+	0.998942i	$0.514645\pi$
$578$	0	0
$579$	−1.03920	−0.0431878
$580$	0	0
$581$	−4.80664	−0.199413
$582$	0	0
$583$	2.85601	0.118284
$584$	0	0
$585$	4.24482	0.175502
$586$	0	0
$587$	15.1809	0.626584	0.313292	−	0.949657i	$-0.398568\pi$
$587$	15.1809	0.626584	0.313292	+	0.949657i	$0.398568\pi$
$588$	0	0
$589$	11.7120	0.482586
$590$	0	0
$591$	9.16246	0.376893
$592$	0	0
$593$	−7.81681	−0.320998	−0.160499	−	0.987036i	$-0.551310\pi$
$593$	−7.81681	−0.320998	−0.160499	+	0.987036i	$0.551310\pi$
$594$	0	0
$595$	4.85601	0.199077
$596$	0	0
$597$	8.40332	0.343925
$598$	0	0
$599$	15.5104	0.633736	0.316868	−	0.948470i	$-0.397369\pi$
$599$	15.5104	0.633736	0.316868	+	0.948470i	$0.397369\pi$
$600$	0	0
$601$	−3.38259	−0.137979	−0.0689894	−	0.997617i	$-0.521977\pi$
$601$	−3.38259	−0.137979	−0.0689894	+	0.997617i	$0.521977\pi$
$602$	0	0
$603$	−13.3888	−0.545234
$604$	0	0
$605$	−4.24482	−0.172576
$606$	0	0
$607$	14.9097	0.605167	0.302584	−	0.953123i	$-0.402151\pi$
$607$	14.9097	0.605167	0.302584	+	0.953123i	$0.402151\pi$
$608$	0	0
$609$	−7.79834	−0.316005
$610$	0	0
$611$	6.48963	0.262542
$612$	0	0
$613$	−5.82472	−0.235258	−0.117629	−	0.993058i	$-0.537529\pi$
$613$	−5.82472	−0.235258	−0.117629	+	0.993058i	$0.537529\pi$
$614$	0	0
$615$	10.1233	0.408209
$616$	0	0
$617$	35.6353	1.43462	0.717312	−	0.696752i	$-0.245372\pi$
$617$	35.6353	1.43462	0.717312	+	0.696752i	$0.245372\pi$
$618$	0	0
$619$	−26.5697	−1.06793	−0.533964	−	0.845507i	$-0.679298\pi$
$619$	−26.5697	−1.06793	−0.533964	+	0.845507i	$0.679298\pi$
$620$	0	0
$621$	7.81681	0.313678
$622$	0	0
$623$	9.59668	0.384483
$624$	0	0
$625$	79.3882	3.17553
$626$	0	0
$627$	6.67282	0.266487
$628$	0	0
$629$	−3.26724	−0.130273
$630$	0	0
$631$	−14.4834	−0.576576	−0.288288	−	0.957544i	$-0.593086\pi$
$631$	−14.4834	−0.576576	−0.288288	+	0.957544i	$0.593086\pi$
$632$	0	0
$633$	−14.4649	−0.574930
$634$	0	0
$635$	45.1210	1.79057
$636$	0	0
$637$	−0.143987	−0.00570495
$638$	0	0
$639$	7.34565	0.290589
$640$	0	0
$641$	19.0162	0.751095	0.375548	−	0.926803i	$-0.377455\pi$
$641$	19.0162	0.751095	0.375548	+	0.926803i	$0.377455\pi$
$642$	0	0
$643$	−11.5905	−0.457083	−0.228542	−	0.973534i	$-0.573396\pi$
$643$	−11.5905	−0.457083	−0.228542	+	0.973534i	$0.573396\pi$
$644$	0	0
$645$	−49.9770	−1.96784
$646$	0	0
$647$	−9.22239	−0.362570	−0.181285	−	0.983431i	$-0.558026\pi$
$647$	−9.22239	−0.362570	−0.181285	+	0.983431i	$0.558026\pi$
$648$	0	0
$649$	12.4896	0.490261
$650$	0	0
$651$	−4.69129	−0.183866
$652$	0	0
$653$	35.0162	1.37029	0.685145	−	0.728407i	$-0.259739\pi$
$653$	35.0162	1.37029	0.685145	+	0.728407i	$0.259739\pi$
$654$	0	0
$655$	83.3411	3.25641
$656$	0	0
$657$	11.7305	0.457650
$658$	0	0
$659$	4.81455	0.187548	0.0937741	−	0.995593i	$-0.470107\pi$
$659$	4.81455	0.187548	0.0937741	+	0.995593i	$0.470107\pi$
$660$	0	0
$661$	7.43196	0.289070	0.144535	−	0.989500i	$-0.453831\pi$
$661$	7.43196	0.289070	0.144535	+	0.989500i	$0.453831\pi$
$662$	0	0
$663$	0.428007	0.0166224
$664$	0	0
$665$	75.7075	2.93581
$666$	0	0
$667$	−22.8066	−0.883077
$668$	0	0
$669$	2.73445	0.105720
$670$	0	0
$671$	−3.14399	−0.121372
$672$	0	0
$673$	−9.13608	−0.352170	−0.176085	−	0.984375i	$-0.556343\pi$
$673$	−9.13608	−0.352170	−0.176085	+	0.984375i	$0.556343\pi$
$674$	0	0
$675$	13.0185	0.501081
$676$	0	0
$677$	−8.50641	−0.326928	−0.163464	−	0.986549i	$-0.552267\pi$
$677$	−8.50641	−0.326928	−0.163464	+	0.986549i	$0.552267\pi$
$678$	0	0
$679$	−31.0946	−1.19330
$680$	0	0
$681$	−18.7961	−0.720267
$682$	0	0
$683$	25.7075	0.983670	0.491835	−	0.870688i	$-0.336326\pi$
$683$	25.7075	0.983670	0.491835	+	0.870688i	$0.336326\pi$
$684$	0	0
$685$	37.4090	1.42932
$686$	0	0
$687$	4.48963	0.171290
$688$	0	0
$689$	2.85601	0.108805
$690$	0	0
$691$	14.7424	0.560826	0.280413	−	0.959879i	$-0.409529\pi$
$691$	14.7424	0.560826	0.280413	+	0.959879i	$0.409529\pi$
$692$	0	0
$693$	−2.67282	−0.101532
$694$	0	0
$695$	−39.9322	−1.51471
$696$	0	0
$697$	1.02073	0.0386630
$698$	0	0
$699$	−11.0040	−0.416208
$700$	0	0
$701$	−33.3210	−1.25852	−0.629258	−	0.777197i	$-0.716641\pi$
$701$	−33.3210	−1.25852	−0.629258	+	0.777197i	$0.716641\pi$
$702$	0	0
$703$	−50.9378	−1.92116
$704$	0	0
$705$	27.5473	1.03749
$706$	0	0
$707$	20.0079	0.752475
$708$	0	0
$709$	1.25934	0.0472953	0.0236477	−	0.999720i	$-0.492472\pi$
$709$	1.25934	0.0472953	0.0236477	+	0.999720i	$0.492472\pi$
$710$	0	0
$711$	−6.42801	−0.241069
$712$	0	0
$713$	−13.7199	−0.513816
$714$	0	0
$715$	−4.24482	−0.158747
$716$	0	0
$717$	−25.2778	−0.944017
$718$	0	0
$719$	15.2672	0.569372	0.284686	−	0.958621i	$-0.408111\pi$
$719$	15.2672	0.569372	0.284686	+	0.958621i	$0.408111\pi$
$720$	0	0
$721$	−4.57595	−0.170417
$722$	0	0
$723$	15.2593	0.567501
$724$	0	0
$725$	−37.9832	−1.41066
$726$	0	0
$727$	24.1233	0.894682	0.447341	−	0.894363i	$-0.352371\pi$
$727$	24.1233	0.894682	0.447341	+	0.894363i	$0.352371\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−5.03920	−0.186382
$732$	0	0
$733$	47.5737	1.75717	0.878587	−	0.477582i	$-0.158487\pi$
$733$	47.5737	1.75717	0.878587	+	0.477582i	$0.158487\pi$
$734$	0	0
$735$	−0.611196	−0.0225443
$736$	0	0
$737$	13.3888	0.493183
$738$	0	0
$739$	17.7305	0.652227	0.326113	−	0.945331i	$-0.394261\pi$
$739$	17.7305	0.652227	0.326113	+	0.945331i	$0.394261\pi$
$740$	0	0
$741$	6.67282	0.245132
$742$	0	0
$743$	−5.99209	−0.219829	−0.109914	−	0.993941i	$-0.535058\pi$
$743$	−5.99209	−0.219829	−0.109914	+	0.993941i	$0.535058\pi$
$744$	0	0
$745$	19.8353	0.726708
$746$	0	0
$747$	−1.79834	−0.0657978
$748$	0	0
$749$	−12.0000	−0.438470
$750$	0	0
$751$	−45.5058	−1.66053	−0.830266	−	0.557367i	$-0.811811\pi$
$751$	−45.5058	−1.66053	−0.830266	+	0.557367i	$0.811811\pi$
$752$	0	0
$753$	−7.91369	−0.288391
$754$	0	0
$755$	−24.3249	−0.885274
$756$	0	0
$757$	46.1708	1.67810	0.839052	−	0.544051i	$-0.183110\pi$
$757$	46.1708	1.67810	0.839052	+	0.544051i	$0.183110\pi$
$758$	0	0
$759$	−7.81681	−0.283732
$760$	0	0
$761$	−4.01057	−0.145383	−0.0726914	−	0.997354i	$-0.523159\pi$
$761$	−4.01057	−0.145383	−0.0726914	+	0.997354i	$0.523159\pi$
$762$	0	0
$763$	−26.7776	−0.969414
$764$	0	0
$765$	1.81681	0.0656869
$766$	0	0
$767$	12.4896	0.450975
$768$	0	0
$769$	41.8432	1.50890	0.754452	−	0.656355i	$-0.227903\pi$
$769$	41.8432	1.50890	0.754452	+	0.656355i	$0.227903\pi$
$770$	0	0
$771$	−15.3456	−0.552660
$772$	0	0
$773$	−26.9440	−0.969109	−0.484554	−	0.874761i	$-0.661018\pi$
$773$	−26.9440	−0.969109	−0.484554	+	0.874761i	$0.661018\pi$
$774$	0	0
$775$	−22.8498	−0.820789
$776$	0	0
$777$	20.4033	0.731965
$778$	0	0
$779$	15.9137	0.570167
$780$	0	0
$781$	−7.34565	−0.262848
$782$	0	0
$783$	−2.91764	−0.104268
$784$	0	0
$785$	−35.5922	−1.27034
$786$	0	0
$787$	−5.53674	−0.197364	−0.0986818	−	0.995119i	$-0.531463\pi$
$787$	−5.53674	−0.197364	−0.0986818	+	0.995119i	$0.531463\pi$
$788$	0	0
$789$	26.7776	0.953308
$790$	0	0
$791$	−28.0369	−0.996879
$792$	0	0
$793$	−3.14399	−0.111646
$794$	0	0
$795$	12.1233	0.429968
$796$	0	0
$797$	14.2465	0.504637	0.252319	−	0.967644i	$-0.418807\pi$
$797$	14.2465	0.504637	0.252319	+	0.967644i	$0.418807\pi$
$798$	0	0
$799$	2.77761	0.0982647
$800$	0	0
$801$	3.59046	0.126863
$802$	0	0
$803$	−11.7305	−0.413960
$804$	0	0
$805$	−88.6868	−3.12580
$806$	0	0
$807$	6.85601	0.241343
$808$	0	0
$809$	40.6745	1.43004	0.715020	−	0.699104i	$-0.246418\pi$
$809$	40.6745	1.43004	0.715020	+	0.699104i	$0.246418\pi$
$810$	0	0
$811$	−21.9031	−0.769123	−0.384561	−	0.923099i	$-0.625647\pi$
$811$	−21.9031	−0.769123	−0.384561	+	0.923099i	$0.625647\pi$
$812$	0	0
$813$	18.1832	0.637712
$814$	0	0
$815$	−20.4633	−0.716797
$816$	0	0
$817$	−78.5635	−2.74859
$818$	0	0
$819$	−2.67282	−0.0933960
$820$	0	0
$821$	16.2201	0.566087	0.283043	−	0.959107i	$-0.408656\pi$
$821$	16.2201	0.566087	0.283043	+	0.959107i	$0.408656\pi$
$822$	0	0
$823$	47.5473	1.65739	0.828697	−	0.559697i	$-0.189082\pi$
$823$	47.5473	1.65739	0.828697	+	0.559697i	$0.189082\pi$
$824$	0	0
$825$	−13.0185	−0.453245
$826$	0	0
$827$	27.4090	0.953103	0.476552	−	0.879147i	$-0.341887\pi$
$827$	27.4090	0.953103	0.476552	+	0.879147i	$0.341887\pi$
$828$	0	0
$829$	6.97927	0.242400	0.121200	−	0.992628i	$-0.461326\pi$
$829$	6.97927	0.242400	0.121200	+	0.992628i	$0.461326\pi$
$830$	0	0
$831$	−23.7490	−0.823843
$832$	0	0
$833$	−0.0616272	−0.00213526
$834$	0	0
$835$	83.3411	2.88414
$836$	0	0
$837$	−1.75518	−0.0606680
$838$	0	0
$839$	−33.1888	−1.14581	−0.572903	−	0.819623i	$-0.694183\pi$
$839$	−33.1888	−1.14581	−0.572903	+	0.819623i	$0.694183\pi$
$840$	0	0
$841$	−20.4874	−0.706461
$842$	0	0
$843$	23.6442	0.814349
$844$	0	0
$845$	−4.24482	−0.146026
$846$	0	0
$847$	2.67282	0.0918393
$848$	0	0
$849$	−28.0616	−0.963073
$850$	0	0
$851$	59.6706	2.04548
$852$	0	0
$853$	6.61289	0.226421	0.113210	−	0.993571i	$-0.463887\pi$
$853$	6.61289	0.226421	0.113210	+	0.993571i	$0.463887\pi$
$854$	0	0
$855$	28.3249	0.968692
$856$	0	0
$857$	11.0409	0.377150	0.188575	−	0.982059i	$-0.439613\pi$
$857$	11.0409	0.377150	0.188575	+	0.982059i	$0.439613\pi$
$858$	0	0
$859$	8.52658	0.290923	0.145462	−	0.989364i	$-0.453533\pi$
$859$	8.52658	0.290923	0.145462	+	0.989364i	$0.453533\pi$
$860$	0	0
$861$	−6.37429	−0.217235
$862$	0	0
$863$	17.2593	0.587515	0.293757	−	0.955880i	$-0.405094\pi$
$863$	17.2593	0.587515	0.293757	+	0.955880i	$0.405094\pi$
$864$	0	0
$865$	−18.7961	−0.639086
$866$	0	0
$867$	−16.8168	−0.571129
$868$	0	0
$869$	6.42801	0.218055
$870$	0	0
$871$	13.3888	0.453662
$872$	0	0
$873$	−11.6336	−0.393738
$874$	0	0
$875$	−90.9747	−3.07551
$876$	0	0
$877$	12.1647	0.410773	0.205387	−	0.978681i	$-0.434155\pi$
$877$	12.1647	0.410773	0.205387	+	0.978681i	$0.434155\pi$
$878$	0	0
$879$	−27.4874	−0.927126
$880$	0	0
$881$	1.09462	0.0368786	0.0184393	−	0.999830i	$-0.494130\pi$
$881$	1.09462	0.0368786	0.0184393	+	0.999830i	$0.494130\pi$
$882$	0	0
$883$	40.9793	1.37906	0.689531	−	0.724256i	$-0.257817\pi$
$883$	40.9793	1.37906	0.689531	+	0.724256i	$0.257817\pi$
$884$	0	0
$885$	53.0162	1.78212
$886$	0	0
$887$	−32.0818	−1.07720	−0.538601	−	0.842561i	$-0.681047\pi$
$887$	−32.0818	−1.07720	−0.538601	+	0.842561i	$0.681047\pi$
$888$	0	0
$889$	−28.4112	−0.952882
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	43.3042	1.44912
$894$	0	0
$895$	56.2386	1.87985
$896$	0	0
$897$	−7.81681	−0.260996
$898$	0	0
$899$	5.12099	0.170795
$900$	0	0
$901$	1.22239	0.0407238
$902$	0	0
$903$	31.4689	1.04722
$904$	0	0
$905$	−48.5714	−1.61457
$906$	0	0
$907$	2.81455	0.0934556	0.0467278	−	0.998908i	$-0.485121\pi$
$907$	2.81455	0.0934556	0.0467278	+	0.998908i	$0.485121\pi$
$908$	0	0
$909$	7.48568	0.248284
$910$	0	0
$911$	−7.18093	−0.237915	−0.118957	−	0.992899i	$-0.537955\pi$
$911$	−7.18093	−0.237915	−0.118957	+	0.992899i	$0.537955\pi$
$912$	0	0
$913$	1.79834	0.0595163
$914$	0	0
$915$	−13.3456	−0.441193
$916$	0	0
$917$	−52.4772	−1.73295
$918$	0	0
$919$	−24.0123	−0.792091	−0.396046	−	0.918231i	$-0.629618\pi$
$919$	−24.0123	−0.792091	−0.396046	+	0.918231i	$0.629618\pi$
$920$	0	0
$921$	−9.15455	−0.301653
$922$	0	0
$923$	−7.34565	−0.241785
$924$	0	0
$925$	99.3781	3.26753
$926$	0	0
$927$	−1.71203	−0.0562303
$928$	0	0
$929$	−22.0017	−0.721852	−0.360926	−	0.932594i	$-0.617539\pi$
$929$	−22.0017	−0.721852	−0.360926	+	0.932594i	$0.617539\pi$
$930$	0	0
$931$	−0.960797	−0.0314888
$932$	0	0
$933$	−15.2409	−0.498963
$934$	0	0
$935$	−1.81681	−0.0594160
$936$	0	0
$937$	−18.4975	−0.604288	−0.302144	−	0.953262i	$-0.597702\pi$
$937$	−18.4975	−0.604288	−0.302144	+	0.953262i	$0.597702\pi$
$938$	0	0
$939$	−8.87448	−0.289608
$940$	0	0
$941$	34.4218	1.12212	0.561059	−	0.827776i	$-0.310394\pi$
$941$	34.4218	1.12212	0.561059	+	0.827776i	$0.310394\pi$
$942$	0	0
$943$	−18.6419	−0.607065
$944$	0	0
$945$	−11.3456	−0.369074
$946$	0	0
$947$	41.5473	1.35011	0.675053	−	0.737769i	$-0.264121\pi$
$947$	41.5473	1.35011	0.675053	+	0.737769i	$0.264121\pi$
$948$	0	0
$949$	−11.7305	−0.380788
$950$	0	0
$951$	7.01452	0.227461
$952$	0	0
$953$	−3.53053	−0.114365	−0.0571825	−	0.998364i	$-0.518212\pi$
$953$	−3.53053	−0.114365	−0.0571825	+	0.998364i	$0.518212\pi$
$954$	0	0
$955$	2.07841	0.0672557
$956$	0	0
$957$	2.91764	0.0943139
$958$	0	0
$959$	−23.5552	−0.760638
$960$	0	0
$961$	−27.9193	−0.900624
$962$	0	0
$963$	−4.48963	−0.144676
$964$	0	0
$965$	4.41123	0.142002
$966$	0	0
$967$	−44.5450	−1.43247	−0.716236	−	0.697858i	$-0.754137\pi$
$967$	−44.5450	−1.43247	−0.716236	+	0.697858i	$0.754137\pi$
$968$	0	0
$969$	2.85601	0.0917484
$970$	0	0
$971$	14.5450	0.466773	0.233386	−	0.972384i	$-0.425019\pi$
$971$	14.5450	0.466773	0.233386	+	0.972384i	$0.425019\pi$
$972$	0	0
$973$	25.1440	0.806079
$974$	0	0
$975$	−13.0185	−0.416925
$976$	0	0
$977$	38.3681	1.22750	0.613752	−	0.789499i	$-0.289660\pi$
$977$	38.3681	1.22750	0.613752	+	0.789499i	$0.289660\pi$
$978$	0	0
$979$	−3.59046	−0.114752
$980$	0	0
$981$	−10.0185	−0.319865
$982$	0	0
$983$	34.6544	1.10530	0.552651	−	0.833413i	$-0.313616\pi$
$983$	34.6544	1.10530	0.552651	+	0.833413i	$0.313616\pi$
$984$	0	0
$985$	−38.8930	−1.23923
$986$	0	0
$987$	−17.3456	−0.552118
$988$	0	0
$989$	92.0324	2.92646
$990$	0	0
$991$	1.06558	0.0338493	0.0169246	−	0.999857i	$-0.494612\pi$
$991$	1.06558	0.0338493	0.0169246	+	0.999857i	$0.494612\pi$
$992$	0	0
$993$	−24.3681	−0.773297
$994$	0	0
$995$	−35.6706	−1.13083
$996$	0	0
$997$	−8.53110	−0.270183	−0.135091	−	0.990833i	$-0.543133\pi$
$997$	−8.53110	−0.270183	−0.135091	+	0.990833i	$0.543133\pi$
$998$	0	0
$999$	7.63362	0.241517

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6864.2.a.bu.1.1		3
4.3	odd	2		429.2.a.e.1.2	✓	3
12.11	even	2		1287.2.a.j.1.2		3
44.43	even	2		4719.2.a.u.1.2		3
52.51	odd	2		5577.2.a.l.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.a.e.1.2	✓	3	4.3	odd	2
1287.2.a.j.1.2		3	12.11	even	2
4719.2.a.u.1.2		3	44.43	even	2
5577.2.a.l.1.2		3	52.51	odd	2
6864.2.a.bu.1.1		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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -4.24482 q^{5} +2.67282 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -4.24482 q^{5} +2.67282 q^{7} +1.00000 q^{9} -1.00000 q^{11} -1.00000 q^{13} -4.24482 q^{15} -0.428007 q^{17} -6.67282 q^{19} +2.67282 q^{21} +7.81681 q^{23} +13.0185 q^{25} +1.00000 q^{27} -2.91764 q^{29} -1.75518 q^{31} -1.00000 q^{33} -11.3456 q^{35} +7.63362 q^{37} -1.00000 q^{39} -2.38485 q^{41} +11.7737 q^{43} -4.24482 q^{45} -6.48963 q^{47} +0.143987 q^{49} -0.428007 q^{51} -2.85601 q^{53} +4.24482 q^{55} -6.67282 q^{57} -12.4896 q^{59} +3.14399 q^{61} +2.67282 q^{63} +4.24482 q^{65} -13.3888 q^{67} +7.81681 q^{69} +7.34565 q^{71} +11.7305 q^{73} +13.0185 q^{75} -2.67282 q^{77} -6.42801 q^{79} +1.00000 q^{81} -1.79834 q^{83} +1.81681 q^{85} -2.91764 q^{87} +3.59046 q^{89} -2.67282 q^{91} -1.75518 q^{93} +28.3249 q^{95} -11.6336 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} - 2 q^{5} - 2 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 - 2 * q^5 - 2 * q^7 + 3 * q^9	\( 3 q + 3 q^{3} - 2 q^{5} - 2 q^{7} + 3 q^{9} - 3 q^{11} - 3 q^{13} - 2 q^{15} - 2 q^{17} - 10 q^{19} - 2 q^{21} + 12 q^{23} + 9 q^{25} + 3 q^{27} + 12 q^{29} - 16 q^{31} - 3 q^{33} - 14 q^{35} - 3 q^{39} + 16 q^{43} - 2 q^{45} + 2 q^{47} - q^{49} - 2 q^{51} - 10 q^{53} + 2 q^{55} - 10 q^{57} - 16 q^{59} + 8 q^{61} - 2 q^{63} + 2 q^{65} - 28 q^{67} + 12 q^{69} + 2 q^{71} + 8 q^{73} + 9 q^{75} + 2 q^{77} - 20 q^{79} + 3 q^{81} - 24 q^{83} - 6 q^{85} + 12 q^{87} - 20 q^{89} + 2 q^{91} - 16 q^{93} + 22 q^{95} - 12 q^{97} - 3 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 - 2 * q^5 - 2 * q^7 + 3 * q^9 - 3 * q^11 - 3 * q^13 - 2 * q^15 - 2 * q^17 - 10 * q^19 - 2 * q^21 + 12 * q^23 + 9 * q^25 + 3 * q^27 + 12 * q^29 - 16 * q^31 - 3 * q^33 - 14 * q^35 - 3 * q^39 + 16 * q^43 - 2 * q^45 + 2 * q^47 - q^49 - 2 * q^51 - 10 * q^53 + 2 * q^55 - 10 * q^57 - 16 * q^59 + 8 * q^61 - 2 * q^63 + 2 * q^65 - 28 * q^67 + 12 * q^69 + 2 * q^71 + 8 * q^73 + 9 * q^75 + 2 * q^77 - 20 * q^79 + 3 * q^81 - 24 * q^83 - 6 * q^85 + 12 * q^87 - 20 * q^89 + 2 * q^91 - 16 * q^93 + 22 * q^95 - 12 * q^97 - 3 * q^99

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