LMFDB - Embedded newform 6840.2.a.bn.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6840,2,Mod(1,6840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6840, 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("6840.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6840 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6840.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:	$54.6176749826$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1016.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x + 2 $ x^3 - x^2 - 6*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 2280)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.321637$ of defining polynomial
Character	$\chi$	$=$	6840.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} -4.21819 q^{7} +O(q^{10})$	$q+1.00000 q^{5} -4.21819 q^{7} -5.57491 q^{11} -2.21819 q^{13} -0.643274 q^{17} -1.00000 q^{19} -1.35673 q^{23} +1.00000 q^{25} -4.86146 q^{29} +2.64327 q^{31} -4.21819 q^{35} -2.21819 q^{37} -3.57491 q^{41} -0.218187 q^{43} +1.35673 q^{47} +10.7931 q^{49} -0.643274 q^{53} -5.57491 q^{55} +4.43637 q^{59} +13.0796 q^{61} -2.21819 q^{65} +1.28655 q^{67} -11.1498 q^{71} +10.0000 q^{73} +23.5160 q^{77} -1.28655 q^{79} -12.3662 q^{83} -0.643274 q^{85} -12.0113 q^{89} +9.35673 q^{91} -1.00000 q^{95} +5.78181 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{5}+O(q^{10}) $ 3 * q + 3 * q^5	$ 3 q + 3 q^{5} - 4 q^{11} + 6 q^{13} - 2 q^{17} - 3 q^{19} - 4 q^{23} + 3 q^{25} - 2 q^{29} + 8 q^{31} + 6 q^{37} + 2 q^{41} + 12 q^{43} + 4 q^{47} + 7 q^{49} - 2 q^{53} - 4 q^{55} - 12 q^{59} + 14 q^{61} + 6 q^{65} + 4 q^{67} - 8 q^{71} + 30 q^{73} + 20 q^{77} - 4 q^{79} - 12 q^{83} - 2 q^{85} + 2 q^{89} + 28 q^{91} - 3 q^{95} + 30 q^{97}+O(q^{100}) $ 3 * q + 3 * q^5 - 4 * q^11 + 6 * q^13 - 2 * q^17 - 3 * q^19 - 4 * q^23 + 3 * q^25 - 2 * q^29 + 8 * q^31 + 6 * q^37 + 2 * q^41 + 12 * q^43 + 4 * q^47 + 7 * q^49 - 2 * q^53 - 4 * q^55 - 12 * q^59 + 14 * q^61 + 6 * q^65 + 4 * q^67 - 8 * q^71 + 30 * q^73 + 20 * q^77 - 4 * q^79 - 12 * q^83 - 2 * q^85 + 2 * q^89 + 28 * q^91 - 3 * q^95 + 30 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−4.21819	−1.59432	−0.797162	−	0.603765i	$-0.793667\pi$
$7$	−4.21819	−1.59432	−0.797162	+	0.603765i	$0.793667\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−5.57491	−1.68090	−0.840450	−	0.541890i	$-0.817709\pi$
$11$	−5.57491	−1.68090	−0.840450	+	0.541890i	$0.817709\pi$
$12$	0	0
$13$	−2.21819	−0.615214	−0.307607	−	0.951513i	$-0.599528\pi$
$13$	−2.21819	−0.615214	−0.307607	+	0.951513i	$0.599528\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−0.643274	−0.156017	−0.0780085	−	0.996953i	$-0.524856\pi$
$17$	−0.643274	−0.156017	−0.0780085	+	0.996953i	$0.524856\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.35673	−0.282897	−0.141448	−	0.989946i	$-0.545176\pi$
$23$	−1.35673	−0.282897	−0.141448	+	0.989946i	$0.545176\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−4.86146	−0.902751	−0.451375	−	0.892334i	$-0.649066\pi$
$29$	−4.86146	−0.902751	−0.451375	+	0.892334i	$0.649066\pi$
$30$	0	0
$31$	2.64327	0.474746	0.237373	−	0.971419i	$-0.423714\pi$
$31$	2.64327	0.474746	0.237373	+	0.971419i	$0.423714\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−4.21819	−0.713004
$36$	0	0
$37$	−2.21819	−0.364668	−0.182334	−	0.983237i	$-0.558365\pi$
$37$	−2.21819	−0.364668	−0.182334	+	0.983237i	$0.558365\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−3.57491	−0.558308	−0.279154	−	0.960246i	$-0.590054\pi$
$41$	−3.57491	−0.558308	−0.279154	+	0.960246i	$0.590054\pi$
$42$	0	0
$43$	−0.218187	−0.0332732	−0.0166366	−	0.999862i	$-0.505296\pi$
$43$	−0.218187	−0.0332732	−0.0166366	+	0.999862i	$0.505296\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.35673	0.197899	0.0989494	−	0.995092i	$-0.468452\pi$
$47$	1.35673	0.197899	0.0989494	+	0.995092i	$0.468452\pi$
$48$	0	0
$49$	10.7931	1.54187
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−0.643274	−0.0883605	−0.0441803	−	0.999024i	$-0.514068\pi$
$53$	−0.643274	−0.0883605	−0.0441803	+	0.999024i	$0.514068\pi$
$54$	0	0
$55$	−5.57491	−0.751721
$56$	0	0
$57$	0	0
$58$	0	0
$59$	4.43637	0.577567	0.288783	−	0.957394i	$-0.406749\pi$
$59$	4.43637	0.577567	0.288783	+	0.957394i	$0.406749\pi$
$60$	0	0
$61$	13.0796	1.67468	0.837339	−	0.546685i	$-0.184110\pi$
$61$	13.0796	1.67468	0.837339	+	0.546685i	$0.184110\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−2.21819	−0.275132
$66$	0	0
$67$	1.28655	0.157177	0.0785885	−	0.996907i	$-0.474959\pi$
$67$	1.28655	0.157177	0.0785885	+	0.996907i	$0.474959\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−11.1498	−1.32324	−0.661620	−	0.749839i	$-0.730131\pi$
$71$	−11.1498	−1.32324	−0.661620	+	0.749839i	$0.730131\pi$
$72$	0	0
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	23.5160	2.67990
$78$	0	0
$79$	−1.28655	−0.144748	−0.0723740	−	0.997378i	$-0.523058\pi$
$79$	−1.28655	−0.144748	−0.0723740	+	0.997378i	$0.523058\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−12.3662	−1.35737	−0.678683	−	0.734431i	$-0.737449\pi$
$83$	−12.3662	−1.35737	−0.678683	+	0.734431i	$0.737449\pi$
$84$	0	0
$85$	−0.643274	−0.0697729
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−12.0113	−1.27319	−0.636597	−	0.771197i	$-0.719658\pi$
$89$	−12.0113	−1.27319	−0.636597	+	0.771197i	$0.719658\pi$
$90$	0	0
$91$	9.35673	0.980851
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.00000	−0.102598
$96$	0	0
$97$	5.78181	0.587054	0.293527	−	0.955951i	$-0.405171\pi$
$97$	5.78181	0.587054	0.293527	+	0.955951i	$0.405171\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	6.43637	0.640443	0.320222	−	0.947343i	$-0.396243\pi$
$101$	6.43637	0.640443	0.320222	+	0.947343i	$0.396243\pi$
$102$	0	0
$103$	−1.28655	−0.126767	−0.0633837	−	0.997989i	$-0.520189\pi$
$103$	−1.28655	−0.126767	−0.0633837	+	0.997989i	$0.520189\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.28655	0.124375	0.0621877	−	0.998064i	$-0.480192\pi$
$107$	1.28655	0.124375	0.0621877	+	0.998064i	$0.480192\pi$
$108$	0	0
$109$	18.4364	1.76588	0.882942	−	0.469482i	$-0.155559\pi$
$109$	18.4364	1.76588	0.882942	+	0.469482i	$0.155559\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−13.0796	−1.23043	−0.615215	−	0.788359i	$-0.710931\pi$
$113$	−13.0796	−1.23043	−0.615215	+	0.788359i	$0.710931\pi$
$114$	0	0
$115$	−1.35673	−0.126515
$116$	0	0
$117$	0	0
$118$	0	0
$119$	2.71345	0.248742
$120$	0	0
$121$	20.0796	1.82542
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−9.28655	−0.824048	−0.412024	−	0.911173i	$-0.635178\pi$
$127$	−9.28655	−0.824048	−0.412024	+	0.911173i	$0.635178\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−2.86146	−0.250007	−0.125004	−	0.992156i	$-0.539894\pi$
$131$	−2.86146	−0.250007	−0.125004	+	0.992156i	$0.539894\pi$
$132$	0	0
$133$	4.21819	0.365763
$134$	0	0
$135$	0	0
$136$	0	0
$137$	14.8727	1.27066	0.635332	−	0.772239i	$-0.280863\pi$
$137$	14.8727	1.27066	0.635332	+	0.772239i	$0.280863\pi$
$138$	0	0
$139$	0.850175	0.0721109	0.0360555	−	0.999350i	$-0.488521\pi$
$139$	0.850175	0.0721109	0.0360555	+	0.999350i	$0.488521\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	12.3662	1.03411
$144$	0	0
$145$	−4.86146	−0.403722
$146$	0	0
$147$	0	0
$148$	0	0
$149$	20.1593	1.65151	0.825757	−	0.564026i	$-0.190748\pi$
$149$	20.1593	1.65151	0.825757	+	0.564026i	$0.190748\pi$
$150$	0	0
$151$	11.5160	0.937161	0.468580	−	0.883421i	$-0.344766\pi$
$151$	11.5160	0.937161	0.468580	+	0.883421i	$0.344766\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2.64327	0.212313
$156$	0	0
$157$	9.14982	0.730236	0.365118	−	0.930961i	$-0.381029\pi$
$157$	9.14982	0.730236	0.365118	+	0.930961i	$0.381029\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	5.72292	0.451029
$162$	0	0
$163$	−5.50474	−0.431164	−0.215582	−	0.976486i	$-0.569165\pi$
$163$	−5.50474	−0.431164	−0.215582	+	0.976486i	$0.569165\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0.920352	0.0712190	0.0356095	−	0.999366i	$-0.488663\pi$
$167$	0.920352	0.0712190	0.0356095	+	0.999366i	$0.488663\pi$
$168$	0	0
$169$	−8.07965	−0.621511
$170$	0	0
$171$	0	0
$172$	0	0
$173$	16.6658	1.26708	0.633540	−	0.773710i	$-0.281601\pi$
$173$	16.6658	1.26708	0.633540	+	0.773710i	$0.281601\pi$
$174$	0	0
$175$	−4.21819	−0.318865
$176$	0	0
$177$	0	0
$178$	0	0
$179$	24.0226	1.79553	0.897766	−	0.440473i	$-0.145189\pi$
$179$	24.0226	1.79553	0.897766	+	0.440473i	$0.145189\pi$
$180$	0	0
$181$	−9.14982	−0.680101	−0.340051	−	0.940407i	$-0.610444\pi$
$181$	−9.14982	−0.680101	−0.340051	+	0.940407i	$0.610444\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−2.21819	−0.163084
$186$	0	0
$187$	3.58620	0.262249
$188$	0	0
$189$	0	0
$190$	0	0
$191$	16.2884	1.17858	0.589292	−	0.807920i	$-0.299407\pi$
$191$	16.2884	1.17858	0.589292	+	0.807920i	$0.299407\pi$
$192$	0	0
$193$	19.5047	1.40398	0.701991	−	0.712186i	$-0.252295\pi$
$193$	19.5047	1.40398	0.701991	+	0.712186i	$0.252295\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−9.51602	−0.677988	−0.338994	−	0.940788i	$-0.610087\pi$
$197$	−9.51602	−0.677988	−0.338994	+	0.940788i	$0.610087\pi$
$198$	0	0
$199$	−2.27708	−0.161418	−0.0807089	−	0.996738i	$-0.525718\pi$
$199$	−2.27708	−0.161418	−0.0807089	+	0.996738i	$0.525718\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	20.5066	1.43928
$204$	0	0
$205$	−3.57491	−0.249683
$206$	0	0
$207$	0	0
$208$	0	0
$209$	5.57491	0.385625
$210$	0	0
$211$	1.42690	0.0982320	0.0491160	−	0.998793i	$-0.484360\pi$
$211$	1.42690	0.0982320	0.0491160	+	0.998793i	$0.484360\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−0.218187	−0.0148802
$216$	0	0
$217$	−11.1498	−0.756899
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1.42690	0.0959839
$222$	0	0
$223$	−6.71345	−0.449566	−0.224783	−	0.974409i	$-0.572167\pi$
$223$	−6.71345	−0.449566	−0.224783	+	0.974409i	$0.572167\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	13.7931	0.915480	0.457740	−	0.889086i	$-0.348659\pi$
$227$	13.7931	0.915480	0.457740	+	0.889086i	$0.348659\pi$
$228$	0	0
$229$	−14.5066	−0.958620	−0.479310	−	0.877646i	$-0.659113\pi$
$229$	−14.5066	−0.958620	−0.479310	+	0.877646i	$0.659113\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−24.2996	−1.59192	−0.795961	−	0.605347i	$-0.793034\pi$
$233$	−24.2996	−1.59192	−0.795961	+	0.605347i	$0.793034\pi$
$234$	0	0
$235$	1.35673	0.0885030
$236$	0	0
$237$	0	0
$238$	0	0
$239$	5.57491	0.360611	0.180306	−	0.983611i	$-0.442291\pi$
$239$	5.57491	0.360611	0.180306	+	0.983611i	$0.442291\pi$
$240$	0	0
$241$	24.2996	1.56528	0.782639	−	0.622476i	$-0.213873\pi$
$241$	24.2996	1.56528	0.782639	+	0.622476i	$0.213873\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	10.7931	0.689546
$246$	0	0
$247$	2.21819	0.141140
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−14.0113	−0.884385	−0.442192	−	0.896920i	$-0.645799\pi$
$251$	−14.0113	−0.884385	−0.442192	+	0.896920i	$0.645799\pi$
$252$	0	0
$253$	7.56363	0.475521
$254$	0	0
$255$	0	0
$256$	0	0
$257$	10.9204	0.681193	0.340596	−	0.940210i	$-0.389371\pi$
$257$	10.9204	0.681193	0.340596	+	0.940210i	$0.389371\pi$
$258$	0	0
$259$	9.35673	0.581399
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−10.6658	−0.657684	−0.328842	−	0.944385i	$-0.606658\pi$
$263$	−10.6658	−0.657684	−0.328842	+	0.944385i	$0.606658\pi$
$264$	0	0
$265$	−0.643274	−0.0395160
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−16.0113	−0.976225	−0.488113	−	0.872781i	$-0.662314\pi$
$269$	−16.0113	−0.976225	−0.488113	+	0.872781i	$0.662314\pi$
$270$	0	0
$271$	−30.7360	−1.86708	−0.933540	−	0.358473i	$-0.883298\pi$
$271$	−30.7360	−1.86708	−0.933540	+	0.358473i	$0.883298\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−5.57491	−0.336180
$276$	0	0
$277$	20.2996	1.21969	0.609844	−	0.792522i	$-0.291232\pi$
$277$	20.2996	1.21969	0.609844	+	0.792522i	$0.291232\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	18.5844	1.10865	0.554326	−	0.832300i	$-0.312976\pi$
$281$	18.5844	1.10865	0.554326	+	0.832300i	$0.312976\pi$
$282$	0	0
$283$	−25.0909	−1.49150	−0.745751	−	0.666225i	$-0.767909\pi$
$283$	−25.0909	−1.49150	−0.745751	+	0.666225i	$0.767909\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	15.0796	0.890123
$288$	0	0
$289$	−16.5862	−0.975659
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−11.7931	−0.688960	−0.344480	−	0.938794i	$-0.611945\pi$
$293$	−11.7931	−0.688960	−0.344480	+	0.938794i	$0.611945\pi$
$294$	0	0
$295$	4.43637	0.258296
$296$	0	0
$297$	0	0
$298$	0	0
$299$	3.00947	0.174042
$300$	0	0
$301$	0.920352	0.0530482
$302$	0	0
$303$	0	0
$304$	0	0
$305$	13.0796	0.748938
$306$	0	0
$307$	−8.85018	−0.505106	−0.252553	−	0.967583i	$-0.581270\pi$
$307$	−8.85018	−0.505106	−0.252553	+	0.967583i	$0.581270\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	21.8709	1.24019	0.620093	−	0.784528i	$-0.287095\pi$
$311$	21.8709	1.24019	0.620093	+	0.784528i	$0.287095\pi$
$312$	0	0
$313$	−1.14982	−0.0649919	−0.0324960	−	0.999472i	$-0.510346\pi$
$313$	−1.14982	−0.0649919	−0.0324960	+	0.999472i	$0.510346\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−29.1022	−1.63454	−0.817272	−	0.576252i	$-0.804514\pi$
$317$	−29.1022	−1.63454	−0.817272	+	0.576252i	$0.804514\pi$
$318$	0	0
$319$	27.1022	1.51743
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0.643274	0.0357927
$324$	0	0
$325$	−2.21819	−0.123043
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−5.72292	−0.315515
$330$	0	0
$331$	−8.94292	−0.491548	−0.245774	−	0.969327i	$-0.579042\pi$
$331$	−8.94292	−0.491548	−0.245774	+	0.969327i	$0.579042\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	1.28655	0.0702917
$336$	0	0
$337$	20.3775	1.11003	0.555016	−	0.831840i	$-0.312712\pi$
$337$	20.3775	1.11003	0.555016	+	0.831840i	$0.312712\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−14.7360	−0.798000
$342$	0	0
$343$	−16.0000	−0.863919
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−7.07965	−0.380055	−0.190028	−	0.981779i	$-0.560858\pi$
$347$	−7.07965	−0.380055	−0.190028	+	0.981779i	$0.560858\pi$
$348$	0	0
$349$	36.1593	1.93556	0.967781	−	0.251792i	$-0.0810199\pi$
$349$	36.1593	1.93556	0.967781	+	0.251792i	$0.0810199\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−32.1593	−1.71167	−0.855833	−	0.517252i	$-0.826955\pi$
$353$	−32.1593	−1.71167	−0.855833	+	0.517252i	$0.826955\pi$
$354$	0	0
$355$	−11.1498	−0.591771
$356$	0	0
$357$	0	0
$358$	0	0
$359$	2.86146	0.151022	0.0755111	−	0.997145i	$-0.475941\pi$
$359$	2.86146	0.151022	0.0755111	+	0.997145i	$0.475941\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	10.0000	0.523424
$366$	0	0
$367$	−12.3585	−0.645111	−0.322555	−	0.946551i	$-0.604542\pi$
$367$	−12.3585	−0.645111	−0.322555	+	0.946551i	$0.604542\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	2.71345	0.140875
$372$	0	0
$373$	0.495265	0.0256438	0.0128219	−	0.999918i	$-0.495919\pi$
$373$	0.495265	0.0256438	0.0128219	+	0.999918i	$0.495919\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	10.7836	0.555385
$378$	0	0
$379$	2.78363	0.142985	0.0714927	−	0.997441i	$-0.477224\pi$
$379$	2.78363	0.142985	0.0714927	+	0.997441i	$0.477224\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	33.0131	1.68689	0.843445	−	0.537215i	$-0.180524\pi$
$383$	33.0131	1.68689	0.843445	+	0.537215i	$0.180524\pi$
$384$	0	0
$385$	23.5160	1.19849
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−30.4589	−1.54433	−0.772165	−	0.635422i	$-0.780826\pi$
$389$	−30.4589	−1.54433	−0.772165	+	0.635422i	$0.780826\pi$
$390$	0	0
$391$	0.872747	0.0441367
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−1.28655	−0.0647333
$396$	0	0
$397$	−15.2865	−0.767210	−0.383605	−	0.923497i	$-0.625318\pi$
$397$	−15.2865	−0.767210	−0.383605	+	0.923497i	$0.625318\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	36.0339	1.79944	0.899722	−	0.436462i	$-0.143769\pi$
$401$	36.0339	1.79944	0.899722	+	0.436462i	$0.143769\pi$
$402$	0	0
$403$	−5.86328	−0.292071
$404$	0	0
$405$	0	0
$406$	0	0
$407$	12.3662	0.612970
$408$	0	0
$409$	−34.0226	−1.68231	−0.841154	−	0.540796i	$-0.818123\pi$
$409$	−34.0226	−1.68231	−0.841154	+	0.540796i	$0.818123\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−18.7135	−0.920829
$414$	0	0
$415$	−12.3662	−0.607033
$416$	0	0
$417$	0	0
$418$	0	0
$419$	8.72474	0.426231	0.213116	−	0.977027i	$-0.431639\pi$
$419$	8.72474	0.426231	0.213116	+	0.977027i	$0.431639\pi$
$420$	0	0
$421$	35.0131	1.70643	0.853217	−	0.521556i	$-0.174648\pi$
$421$	35.0131	1.70643	0.853217	+	0.521556i	$0.174648\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−0.643274	−0.0312034
$426$	0	0
$427$	−55.1724	−2.66998
$428$	0	0
$429$	0	0
$430$	0	0
$431$	13.8633	0.667771	0.333885	−	0.942614i	$-0.391640\pi$
$431$	13.8633	0.667771	0.333885	+	0.942614i	$0.391640\pi$
$432$	0	0
$433$	−30.2408	−1.45328	−0.726639	−	0.687019i	$-0.758919\pi$
$433$	−30.2408	−1.45328	−0.726639	+	0.687019i	$0.758919\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.35673	0.0649010
$438$	0	0
$439$	31.5862	1.50753	0.753763	−	0.657146i	$-0.228236\pi$
$439$	31.5862	1.50753	0.753763	+	0.657146i	$0.228236\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−18.2295	−0.866108	−0.433054	−	0.901368i	$-0.642564\pi$
$443$	−18.2295	−0.866108	−0.433054	+	0.901368i	$0.642564\pi$
$444$	0	0
$445$	−12.0113	−0.569390
$446$	0	0
$447$	0	0
$448$	0	0
$449$	9.71164	0.458320	0.229160	−	0.973389i	$-0.426402\pi$
$449$	9.71164	0.458320	0.229160	+	0.973389i	$0.426402\pi$
$450$	0	0
$451$	19.9298	0.938459
$452$	0	0
$453$	0	0
$454$	0	0
$455$	9.35673	0.438650
$456$	0	0
$457$	−19.8633	−0.929165	−0.464582	−	0.885530i	$-0.653796\pi$
$457$	−19.8633	−0.929165	−0.464582	+	0.885530i	$0.653796\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	20.2996	0.945449	0.472724	−	0.881210i	$-0.343271\pi$
$461$	20.2996	0.945449	0.472724	+	0.881210i	$0.343271\pi$
$462$	0	0
$463$	32.2408	1.49836	0.749178	−	0.662369i	$-0.230449\pi$
$463$	32.2408	1.49836	0.749178	+	0.662369i	$0.230449\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	26.2295	1.21376	0.606878	−	0.794795i	$-0.292422\pi$
$467$	26.2295	1.21376	0.606878	+	0.794795i	$0.292422\pi$
$468$	0	0
$469$	−5.42690	−0.250591
$470$	0	0
$471$	0	0
$472$	0	0
$473$	1.21637	0.0559288
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−33.1611	−1.51517	−0.757585	−	0.652737i	$-0.773621\pi$
$479$	−33.1611	−1.51517	−0.757585	+	0.652737i	$0.773621\pi$
$480$	0	0
$481$	4.92035	0.224349
$482$	0	0
$483$	0	0
$484$	0	0
$485$	5.78181	0.262539
$486$	0	0
$487$	−19.5636	−0.886513	−0.443256	−	0.896395i	$-0.646177\pi$
$487$	−19.5636	−0.886513	−0.443256	+	0.896395i	$0.646177\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	26.8615	1.21224	0.606120	−	0.795373i	$-0.292725\pi$
$491$	26.8615	1.21224	0.606120	+	0.795373i	$0.292725\pi$
$492$	0	0
$493$	3.12725	0.140844
$494$	0	0
$495$	0	0
$496$	0	0
$497$	47.0320	2.10968
$498$	0	0
$499$	40.0226	1.79166	0.895828	−	0.444401i	$-0.146583\pi$
$499$	40.0226	1.79166	0.895828	+	0.444401i	$0.146583\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	17.6527	0.787097	0.393549	−	0.919304i	$-0.371247\pi$
$503$	17.6527	0.787097	0.393549	+	0.919304i	$0.371247\pi$
$504$	0	0
$505$	6.43637	0.286415
$506$	0	0
$507$	0	0
$508$	0	0
$509$	16.8615	0.747371	0.373686	−	0.927555i	$-0.378094\pi$
$509$	16.8615	0.747371	0.373686	+	0.927555i	$0.378094\pi$
$510$	0	0
$511$	−42.1819	−1.86602
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−1.28655	−0.0566921
$516$	0	0
$517$	−7.56363	−0.332648
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−3.57491	−0.156620	−0.0783099	−	0.996929i	$-0.524952\pi$
$521$	−3.57491	−0.156620	−0.0783099	+	0.996929i	$0.524952\pi$
$522$	0	0
$523$	−28.4364	−1.24344	−0.621718	−	0.783241i	$-0.713565\pi$
$523$	−28.4364	−1.24344	−0.621718	+	0.783241i	$0.713565\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.70035	−0.0740684
$528$	0	0
$529$	−21.1593	−0.919969
$530$	0	0
$531$	0	0
$532$	0	0
$533$	7.92982	0.343479
$534$	0	0
$535$	1.28655	0.0556224
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−60.1706	−2.59173
$540$	0	0
$541$	0.573097	0.0246394	0.0123197	−	0.999924i	$-0.496078\pi$
$541$	0.573097	0.0246394	0.0123197	+	0.999924i	$0.496078\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	18.4364	0.789727
$546$	0	0
$547$	−8.85018	−0.378406	−0.189203	−	0.981938i	$-0.560590\pi$
$547$	−8.85018	−0.378406	−0.189203	+	0.981938i	$0.560590\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	4.86146	0.207105
$552$	0	0
$553$	5.42690	0.230775
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−8.29965	−0.351667	−0.175834	−	0.984420i	$-0.556262\pi$
$557$	−8.29965	−0.351667	−0.175834	+	0.984420i	$0.556262\pi$
$558$	0	0
$559$	0.483979	0.0204701
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−17.3793	−0.732450	−0.366225	−	0.930526i	$-0.619350\pi$
$563$	−17.3793	−0.732450	−0.366225	+	0.930526i	$0.619350\pi$
$564$	0	0
$565$	−13.0796	−0.550265
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−19.4346	−0.814739	−0.407370	−	0.913263i	$-0.633554\pi$
$569$	−19.4346	−0.814739	−0.407370	+	0.913263i	$0.633554\pi$
$570$	0	0
$571$	32.1629	1.34598	0.672988	−	0.739653i	$-0.265010\pi$
$571$	32.1629	1.34598	0.672988	+	0.739653i	$0.265010\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1.35673	−0.0565794
$576$	0	0
$577$	−31.1688	−1.29757	−0.648786	−	0.760971i	$-0.724723\pi$
$577$	−31.1688	−1.29757	−0.648786	+	0.760971i	$0.724723\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	52.1629	2.16408
$582$	0	0
$583$	3.58620	0.148525
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−23.5160	−0.970610	−0.485305	−	0.874345i	$-0.661291\pi$
$587$	−23.5160	−0.970610	−0.485305	+	0.874345i	$0.661291\pi$
$588$	0	0
$589$	−2.64327	−0.108914
$590$	0	0
$591$	0	0
$592$	0	0
$593$	31.0131	1.27356	0.636778	−	0.771047i	$-0.280267\pi$
$593$	31.0131	1.27356	0.636778	+	0.771047i	$0.280267\pi$
$594$	0	0
$595$	2.71345	0.111241
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−39.1724	−1.60054	−0.800270	−	0.599639i	$-0.795311\pi$
$599$	−39.1724	−1.60054	−0.800270	+	0.599639i	$0.795311\pi$
$600$	0	0
$601$	−3.86328	−0.157586	−0.0787932	−	0.996891i	$-0.525107\pi$
$601$	−3.86328	−0.157586	−0.0787932	+	0.996891i	$0.525107\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	20.0796	0.816354
$606$	0	0
$607$	8.02257	0.325626	0.162813	−	0.986657i	$-0.447943\pi$
$607$	8.02257	0.325626	0.162813	+	0.986657i	$0.447943\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−3.00947	−0.121750
$612$	0	0
$613$	19.8633	0.802270	0.401135	−	0.916019i	$-0.368616\pi$
$613$	19.8633	0.802270	0.401135	+	0.916019i	$0.368616\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−18.3888	−0.740304	−0.370152	−	0.928971i	$-0.620694\pi$
$617$	−18.3888	−0.740304	−0.370152	+	0.928971i	$0.620694\pi$
$618$	0	0
$619$	12.4364	0.499860	0.249930	−	0.968264i	$-0.419592\pi$
$619$	12.4364	0.499860	0.249930	+	0.968264i	$0.419592\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	50.6658	2.02988
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1.42690	0.0568943
$630$	0	0
$631$	42.6182	1.69661	0.848303	−	0.529512i	$-0.177625\pi$
$631$	42.6182	1.69661	0.848303	+	0.529512i	$0.177625\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−9.28655	−0.368525
$636$	0	0
$637$	−23.9411	−0.948581
$638$	0	0
$639$	0	0
$640$	0	0
$641$	3.16111	0.124856	0.0624282	−	0.998049i	$-0.480116\pi$
$641$	3.16111	0.124856	0.0624282	+	0.998049i	$0.480116\pi$
$642$	0	0
$643$	36.5368	1.44087	0.720435	−	0.693523i	$-0.243942\pi$
$643$	36.5368	1.44087	0.720435	+	0.693523i	$0.243942\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	17.6527	0.694001	0.347001	−	0.937865i	$-0.387200\pi$
$647$	17.6527	0.694001	0.347001	+	0.937865i	$0.387200\pi$
$648$	0	0
$649$	−24.7324	−0.970831
$650$	0	0
$651$	0	0
$652$	0	0
$653$	5.51602	0.215859	0.107929	−	0.994159i	$-0.465578\pi$
$653$	5.51602	0.215859	0.107929	+	0.994159i	$0.465578\pi$
$654$	0	0
$655$	−2.86146	−0.111807
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−18.2996	−0.712853	−0.356427	−	0.934323i	$-0.616005\pi$
$659$	−18.2996	−0.712853	−0.356427	+	0.934323i	$0.616005\pi$
$660$	0	0
$661$	7.72292	0.300387	0.150193	−	0.988657i	$-0.452010\pi$
$661$	7.72292	0.300387	0.150193	+	0.988657i	$0.452010\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	4.21819	0.163574
$666$	0	0
$667$	6.59567	0.255385
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−72.9179	−2.81496
$672$	0	0
$673$	39.9637	1.54049	0.770243	−	0.637750i	$-0.220135\pi$
$673$	39.9637	1.54049	0.770243	+	0.637750i	$0.220135\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	48.2295	1.85361	0.926805	−	0.375544i	$-0.122544\pi$
$677$	48.2295	1.85361	0.926805	+	0.375544i	$0.122544\pi$
$678$	0	0
$679$	−24.3888	−0.935955
$680$	0	0
$681$	0	0
$682$	0	0
$683$	15.5862	0.596389	0.298195	−	0.954505i	$-0.403616\pi$
$683$	15.5862	0.596389	0.298195	+	0.954505i	$0.403616\pi$
$684$	0	0
$685$	14.8727	0.568258
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1.42690	0.0543607
$690$	0	0
$691$	33.8633	1.28822	0.644110	−	0.764933i	$-0.277228\pi$
$691$	33.8633	1.28822	0.644110	+	0.764933i	$0.277228\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0.850175	0.0322490
$696$	0	0
$697$	2.29965	0.0871054
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−31.3317	−1.18338	−0.591691	−	0.806165i	$-0.701539\pi$
$701$	−31.3317	−1.18338	−0.591691	+	0.806165i	$0.701539\pi$
$702$	0	0
$703$	2.21819	0.0836605
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−27.1498	−1.02107
$708$	0	0
$709$	−0.939294	−0.0352759	−0.0176380	−	0.999844i	$-0.505615\pi$
$709$	−0.939294	−0.0352759	−0.0176380	+	0.999844i	$0.505615\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−3.58620	−0.134304
$714$	0	0
$715$	12.3662	0.462470
$716$	0	0
$717$	0	0
$718$	0	0
$719$	33.1611	1.23670	0.618350	−	0.785903i	$-0.287801\pi$
$719$	33.1611	1.23670	0.618350	+	0.785903i	$0.287801\pi$
$720$	0	0
$721$	5.42690	0.202108
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−4.86146	−0.180550
$726$	0	0
$727$	−32.2408	−1.19574	−0.597872	−	0.801592i	$-0.703987\pi$
$727$	−32.2408	−1.19574	−0.597872	+	0.801592i	$0.703987\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0.140354	0.00519118
$732$	0	0
$733$	17.5862	0.649561	0.324781	−	0.945789i	$-0.394710\pi$
$733$	17.5862	0.649561	0.324781	+	0.945789i	$0.394710\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−7.17240	−0.264199
$738$	0	0
$739$	14.5731	0.536080	0.268040	−	0.963408i	$-0.413624\pi$
$739$	14.5731	0.536080	0.268040	+	0.963408i	$0.413624\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	27.5862	1.01204	0.506020	−	0.862522i	$-0.331116\pi$
$743$	27.5862	1.01204	0.506020	+	0.862522i	$0.331116\pi$
$744$	0	0
$745$	20.1593	0.738579
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−5.42690	−0.198295
$750$	0	0
$751$	−11.5160	−0.420226	−0.210113	−	0.977677i	$-0.567383\pi$
$751$	−11.5160	−0.420226	−0.210113	+	0.977677i	$0.567383\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	11.5160	0.419111
$756$	0	0
$757$	−13.2902	−0.483040	−0.241520	−	0.970396i	$-0.577646\pi$
$757$	−13.2902	−0.483040	−0.241520	+	0.970396i	$0.577646\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	7.84070	0.284225	0.142113	−	0.989850i	$-0.454610\pi$
$761$	7.84070	0.284225	0.142113	+	0.989850i	$0.454610\pi$
$762$	0	0
$763$	−77.7681	−2.81539
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−9.84070	−0.355327
$768$	0	0
$769$	−52.2996	−1.88597	−0.942987	−	0.332830i	$-0.891996\pi$
$769$	−52.2996	−1.88597	−0.942987	+	0.332830i	$0.891996\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	25.2426	0.907912	0.453956	−	0.891024i	$-0.350012\pi$
$773$	25.2426	0.907912	0.453956	+	0.891024i	$0.350012\pi$
$774$	0	0
$775$	2.64327	0.0949492
$776$	0	0
$777$	0	0
$778$	0	0
$779$	3.57491	0.128085
$780$	0	0
$781$	62.1593	2.22423
$782$	0	0
$783$	0	0
$784$	0	0
$785$	9.14982	0.326571
$786$	0	0
$787$	−17.8633	−0.636757	−0.318379	−	0.947964i	$-0.603138\pi$
$787$	−17.8633	−0.636757	−0.318379	+	0.947964i	$0.603138\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	55.1724	1.96170
$792$	0	0
$793$	−29.0131	−1.03029
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−6.50655	−0.230474	−0.115237	−	0.993338i	$-0.536763\pi$
$797$	−6.50655	−0.230474	−0.115237	+	0.993338i	$0.536763\pi$
$798$	0	0
$799$	−0.872747	−0.0308756
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−55.7491	−1.96734
$804$	0	0
$805$	5.72292	0.201707
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−3.84070	−0.135032	−0.0675160	−	0.997718i	$-0.521507\pi$
$809$	−3.84070	−0.135032	−0.0675160	+	0.997718i	$0.521507\pi$
$810$	0	0
$811$	−7.44584	−0.261459	−0.130729	−	0.991418i	$-0.541732\pi$
$811$	−7.44584	−0.261459	−0.130729	+	0.991418i	$0.541732\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−5.50474	−0.192822
$816$	0	0
$817$	0.218187	0.00763339
$818$	0	0
$819$	0	0
$820$	0	0
$821$	30.2960	1.05734	0.528669	−	0.848828i	$-0.322691\pi$
$821$	30.2960	1.05734	0.528669	+	0.848828i	$0.322691\pi$
$822$	0	0
$823$	32.3811	1.12873	0.564367	−	0.825524i	$-0.309120\pi$
$823$	32.3811	1.12873	0.564367	+	0.825524i	$0.309120\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	11.9524	0.415625	0.207813	−	0.978169i	$-0.433366\pi$
$827$	11.9524	0.415625	0.207813	+	0.978169i	$0.433366\pi$
$828$	0	0
$829$	−22.5767	−0.784122	−0.392061	−	0.919939i	$-0.628238\pi$
$829$	−22.5767	−0.784122	−0.392061	+	0.919939i	$0.628238\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−6.94292	−0.240558
$834$	0	0
$835$	0.920352	0.0318501
$836$	0	0
$837$	0	0
$838$	0	0
$839$	3.14982	0.108744	0.0543720	−	0.998521i	$-0.482684\pi$
$839$	3.14982	0.108744	0.0543720	+	0.998521i	$0.482684\pi$
$840$	0	0
$841$	−5.36620	−0.185041
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−8.07965	−0.277948
$846$	0	0
$847$	−84.6997	−2.91032
$848$	0	0
$849$	0	0
$850$	0	0
$851$	3.00947	0.103163
$852$	0	0
$853$	11.5826	0.396580	0.198290	−	0.980143i	$-0.436461\pi$
$853$	11.5826	0.396580	0.198290	+	0.980143i	$0.436461\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−3.91088	−0.133593	−0.0667966	−	0.997767i	$-0.521278\pi$
$857$	−3.91088	−0.133593	−0.0667966	+	0.997767i	$0.521278\pi$
$858$	0	0
$859$	−5.01310	−0.171045	−0.0855224	−	0.996336i	$-0.527256\pi$
$859$	−5.01310	−0.171045	−0.0855224	+	0.996336i	$0.527256\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−40.7324	−1.38655	−0.693273	−	0.720675i	$-0.743832\pi$
$863$	−40.7324	−1.38655	−0.693273	+	0.720675i	$0.743832\pi$
$864$	0	0
$865$	16.6658	0.566656
$866$	0	0
$867$	0	0
$868$	0	0
$869$	7.17240	0.243307
$870$	0	0
$871$	−2.85381	−0.0966975
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−4.21819	−0.142601
$876$	0	0
$877$	36.2371	1.22364	0.611820	−	0.790997i	$-0.290438\pi$
$877$	36.2371	1.22364	0.611820	+	0.790997i	$0.290438\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	18.0226	0.607196	0.303598	−	0.952800i	$-0.401812\pi$
$881$	18.0226	0.607196	0.303598	+	0.952800i	$0.401812\pi$
$882$	0	0
$883$	−52.6771	−1.77273	−0.886363	−	0.462990i	$-0.846776\pi$
$883$	−52.6771	−1.77273	−0.886363	+	0.462990i	$0.846776\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−32.8727	−1.10376	−0.551879	−	0.833924i	$-0.686089\pi$
$887$	−32.8727	−1.10376	−0.551879	+	0.833924i	$0.686089\pi$
$888$	0	0
$889$	39.1724	1.31380
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−1.35673	−0.0454011
$894$	0	0
$895$	24.0226	0.802986
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−12.8502	−0.428577
$900$	0	0
$901$	0.413802	0.0137857
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−9.14982	−0.304150
$906$	0	0
$907$	−26.5957	−0.883095	−0.441547	−	0.897238i	$-0.645570\pi$
$907$	−26.5957	−0.883095	−0.441547	+	0.897238i	$0.645570\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−6.15930	−0.204067	−0.102033	−	0.994781i	$-0.532535\pi$
$911$	−6.15930	−0.204067	−0.102033	+	0.994781i	$0.532535\pi$
$912$	0	0
$913$	68.9405	2.28160
$914$	0	0
$915$	0	0
$916$	0	0
$917$	12.0702	0.398592
$918$	0	0
$919$	−33.8858	−1.11779	−0.558895	−	0.829238i	$-0.688775\pi$
$919$	−33.8858	−1.11779	−0.558895	+	0.829238i	$0.688775\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	24.7324	0.814077
$924$	0	0
$925$	−2.21819	−0.0729335
$926$	0	0
$927$	0	0
$928$	0	0
$929$	34.8953	1.14488	0.572439	−	0.819947i	$-0.305997\pi$
$929$	34.8953	1.14488	0.572439	+	0.819947i	$0.305997\pi$
$930$	0	0
$931$	−10.7931	−0.353730
$932$	0	0
$933$	0	0
$934$	0	0
$935$	3.58620	0.117281
$936$	0	0
$937$	−6.29602	−0.205682	−0.102841	−	0.994698i	$-0.532793\pi$
$937$	−6.29602	−0.205682	−0.102841	+	0.994698i	$0.532793\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−10.8651	−0.354192	−0.177096	−	0.984194i	$-0.556670\pi$
$941$	−10.8651	−0.354192	−0.177096	+	0.984194i	$0.556670\pi$
$942$	0	0
$943$	4.85018	0.157943
$944$	0	0
$945$	0	0
$946$	0	0
$947$	9.79310	0.318233	0.159116	−	0.987260i	$-0.449135\pi$
$947$	9.79310	0.318233	0.159116	+	0.987260i	$0.449135\pi$
$948$	0	0
$949$	−22.1819	−0.720054
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−1.35310	−0.0438311	−0.0219155	−	0.999760i	$-0.506976\pi$
$953$	−1.35310	−0.0438311	−0.0219155	+	0.999760i	$0.506976\pi$
$954$	0	0
$955$	16.2884	0.527079
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−62.7360	−2.02585
$960$	0	0
$961$	−24.0131	−0.774616
$962$	0	0
$963$	0	0
$964$	0	0
$965$	19.5047	0.627880
$966$	0	0
$967$	21.9637	0.706304	0.353152	−	0.935566i	$-0.385110\pi$
$967$	21.9637	0.706304	0.353152	+	0.935566i	$0.385110\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	13.7455	0.441114	0.220557	−	0.975374i	$-0.429213\pi$
$971$	13.7455	0.441114	0.220557	+	0.975374i	$0.429213\pi$
$972$	0	0
$973$	−3.58620	−0.114968
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−48.3698	−1.54749	−0.773744	−	0.633498i	$-0.781618\pi$
$977$	−48.3698	−1.54749	−0.773744	+	0.633498i	$0.781618\pi$
$978$	0	0
$979$	66.9619	2.14011
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−14.0665	−0.448653	−0.224327	−	0.974514i	$-0.572018\pi$
$983$	−14.0665	−0.448653	−0.224327	+	0.974514i	$0.572018\pi$
$984$	0	0
$985$	−9.51602	−0.303206
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0.296019	0.00941287
$990$	0	0
$991$	−33.2865	−1.05738	−0.528691	−	0.848814i	$-0.677317\pi$
$991$	−33.2865	−1.05738	−0.528691	+	0.848814i	$0.677317\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−2.27708	−0.0721882
$996$	0	0
$997$	−48.5957	−1.53904	−0.769520	−	0.638623i	$-0.779505\pi$
$997$	−48.5957	−1.53904	−0.769520	+	0.638623i	$0.779505\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6840.2.a.bn.1.1		3
3.2	odd	2		2280.2.a.t.1.1	✓	3
12.11	even	2		4560.2.a.bq.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2280.2.a.t.1.1	✓	3	3.2	odd	2
4560.2.a.bq.1.3		3	12.11	even	2
6840.2.a.bn.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:	\( N \)	\(=\)	\( 6840 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6840.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} -4.21819 q^{7} +O(q^{10})\)	\(q+1.00000 q^{5} -4.21819 q^{7} -5.57491 q^{11} -2.21819 q^{13} -0.643274 q^{17} -1.00000 q^{19} -1.35673 q^{23} +1.00000 q^{25} -4.86146 q^{29} +2.64327 q^{31} -4.21819 q^{35} -2.21819 q^{37} -3.57491 q^{41} -0.218187 q^{43} +1.35673 q^{47} +10.7931 q^{49} -0.643274 q^{53} -5.57491 q^{55} +4.43637 q^{59} +13.0796 q^{61} -2.21819 q^{65} +1.28655 q^{67} -11.1498 q^{71} +10.0000 q^{73} +23.5160 q^{77} -1.28655 q^{79} -12.3662 q^{83} -0.643274 q^{85} -12.0113 q^{89} +9.35673 q^{91} -1.00000 q^{95} +5.78181 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{5}+O(q^{10}) \) 3 * q + 3 * q^5	\( 3 q + 3 q^{5} - 4 q^{11} + 6 q^{13} - 2 q^{17} - 3 q^{19} - 4 q^{23} + 3 q^{25} - 2 q^{29} + 8 q^{31} + 6 q^{37} + 2 q^{41} + 12 q^{43} + 4 q^{47} + 7 q^{49} - 2 q^{53} - 4 q^{55} - 12 q^{59} + 14 q^{61} + 6 q^{65} + 4 q^{67} - 8 q^{71} + 30 q^{73} + 20 q^{77} - 4 q^{79} - 12 q^{83} - 2 q^{85} + 2 q^{89} + 28 q^{91} - 3 q^{95} + 30 q^{97}+O(q^{100}) \) 3 * q + 3 * q^5 - 4 * q^11 + 6 * q^13 - 2 * q^17 - 3 * q^19 - 4 * q^23 + 3 * q^25 - 2 * q^29 + 8 * q^31 + 6 * q^37 + 2 * q^41 + 12 * q^43 + 4 * q^47 + 7 * q^49 - 2 * q^53 - 4 * q^55 - 12 * q^59 + 14 * q^61 + 6 * q^65 + 4 * q^67 - 8 * q^71 + 30 * q^73 + 20 * q^77 - 4 * q^79 - 12 * q^83 - 2 * q^85 + 2 * q^89 + 28 * q^91 - 3 * q^95 + 30 * q^97

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