LMFDB - Embedded newform 6021.2.a.r.1.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6021,2,Mod(1,6021)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6021, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6021.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6021 = 3^{3} \cdot 223 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6021.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:	$48.0779270570$
Analytic rank:	$0$
Dimension:	$35$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Character	$\chi$	$=$	6021.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.424438 q^{2} -1.81985 q^{4} -1.02884 q^{5} -4.02930 q^{7} -1.62129 q^{8} +O(q^{10})$	\(q+0.424438 q^{2} -1.81985 q^{4} -1.02884 q^{5} -4.02930 q^{7} -1.62129 q^{8} -0.436677 q^{10} +0.0504422 q^{11} -3.82631 q^{13} -1.71019 q^{14} +2.95157 q^{16} +0.468656 q^{17} +3.13071 q^{19} +1.87233 q^{20} +0.0214096 q^{22} -7.74707 q^{23} -3.94150 q^{25} -1.62403 q^{26} +7.33273 q^{28} -9.57779 q^{29} -10.6996 q^{31} +4.49534 q^{32} +0.198915 q^{34} +4.14548 q^{35} -5.80175 q^{37} +1.32879 q^{38} +1.66804 q^{40} -1.02595 q^{41} +11.0372 q^{43} -0.0917973 q^{44} -3.28815 q^{46} -8.77093 q^{47} +9.23524 q^{49} -1.67292 q^{50} +6.96331 q^{52} -2.50076 q^{53} -0.0518967 q^{55} +6.53266 q^{56} -4.06518 q^{58} -4.13983 q^{59} -2.10355 q^{61} -4.54131 q^{62} -3.99515 q^{64} +3.93664 q^{65} -11.3209 q^{67} -0.852885 q^{68} +1.75950 q^{70} +13.9288 q^{71} +1.63283 q^{73} -2.46248 q^{74} -5.69743 q^{76} -0.203246 q^{77} +10.3556 q^{79} -3.03668 q^{80} -0.435452 q^{82} -12.6229 q^{83} -0.482170 q^{85} +4.68461 q^{86} -0.0817813 q^{88} +1.89949 q^{89} +15.4173 q^{91} +14.0985 q^{92} -3.72271 q^{94} -3.22098 q^{95} -2.77745 q^{97} +3.91978 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 35 q + 4 q^{2} + 36 q^{4} + 10 q^{5} - 2 q^{7} + 15 q^{8}+O(q^{10}) $ 35 * q + 4 * q^2 + 36 * q^4 + 10 * q^5 - 2 * q^7 + 15 * q^8	$ 35 q + 4 q^{2} + 36 q^{4} + 10 q^{5} - 2 q^{7} + 15 q^{8} - 7 q^{10} + 34 q^{11} + 2 q^{13} + 18 q^{14} + 42 q^{16} + 20 q^{17} - 14 q^{19} + 27 q^{20} + 8 q^{22} + 8 q^{23} + 27 q^{25} + 28 q^{26} - 4 q^{28} + 23 q^{29} + 12 q^{31} + 29 q^{32} - 21 q^{34} + 41 q^{35} - 8 q^{37} + 18 q^{38} + 16 q^{40} + 50 q^{41} + 2 q^{43} + 83 q^{44} - 5 q^{46} + 21 q^{47} + 43 q^{49} + 39 q^{50} + 6 q^{52} + 37 q^{53} + 20 q^{55} + 33 q^{56} - 32 q^{58} + 81 q^{59} - 6 q^{61} + 26 q^{62} - q^{64} + 29 q^{65} + 12 q^{67} + 55 q^{68} + 50 q^{70} + 43 q^{71} - 20 q^{73} + 48 q^{74} - 15 q^{76} + 29 q^{77} + 28 q^{79} + 88 q^{80} - 6 q^{82} + 64 q^{83} - 67 q^{85} + 41 q^{86} + 10 q^{88} + 50 q^{89} + 2 q^{91} + 32 q^{92} + 15 q^{94} + 25 q^{95} + 9 q^{97} + 74 q^{98}+O(q^{100}) $ 35 * q + 4 * q^2 + 36 * q^4 + 10 * q^5 - 2 * q^7 + 15 * q^8 - 7 * q^10 + 34 * q^11 + 2 * q^13 + 18 * q^14 + 42 * q^16 + 20 * q^17 - 14 * q^19 + 27 * q^20 + 8 * q^22 + 8 * q^23 + 27 * q^25 + 28 * q^26 - 4 * q^28 + 23 * q^29 + 12 * q^31 + 29 * q^32 - 21 * q^34 + 41 * q^35 - 8 * q^37 + 18 * q^38 + 16 * q^40 + 50 * q^41 + 2 * q^43 + 83 * q^44 - 5 * q^46 + 21 * q^47 + 43 * q^49 + 39 * q^50 + 6 * q^52 + 37 * q^53 + 20 * q^55 + 33 * q^56 - 32 * q^58 + 81 * q^59 - 6 * q^61 + 26 * q^62 - q^64 + 29 * q^65 + 12 * q^67 + 55 * q^68 + 50 * q^70 + 43 * q^71 - 20 * q^73 + 48 * q^74 - 15 * q^76 + 29 * q^77 + 28 * q^79 + 88 * q^80 - 6 * q^82 + 64 * q^83 - 67 * q^85 + 41 * q^86 + 10 * q^88 + 50 * q^89 + 2 * q^91 + 32 * q^92 + 15 * q^94 + 25 * q^95 + 9 * q^97 + 74 * q^98

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 $

$2$	0.424438	0.300123	0.150061	−	0.988677i	$-0.452053\pi$
$2$	0.424438	0.300123	0.150061	+	0.988677i	$0.452053\pi$
$3$	0	0
$3$	0	0
$4$	−1.81985	−0.909926
$5$	−1.02884	−0.460109	−0.230055	−	0.973178i	$-0.573890\pi$
$5$	−1.02884	−0.460109	−0.230055	+	0.973178i	$0.573890\pi$
$6$	0	0
$7$	−4.02930	−1.52293	−0.761466	−	0.648205i	$-0.775520\pi$
$7$	−4.02930	−1.52293	−0.761466	+	0.648205i	$0.775520\pi$
$8$	−1.62129	−0.573213
$9$	0	0
$10$	−0.436677	−0.138089
$11$	0.0504422	0.0152089	0.00760444	−	0.999971i	$-0.497579\pi$
$11$	0.0504422	0.0152089	0.00760444	+	0.999971i	$0.497579\pi$
$12$	0	0
$13$	−3.82631	−1.06123	−0.530613	−	0.847614i	$-0.678038\pi$
$13$	−3.82631	−1.06123	−0.530613	+	0.847614i	$0.678038\pi$
$14$	−1.71019	−0.457067
$15$	0	0
$16$	2.95157	0.737892
$17$	0.468656	0.113666	0.0568329	−	0.998384i	$-0.481900\pi$
$17$	0.468656	0.113666	0.0568329	+	0.998384i	$0.481900\pi$
$18$	0	0
$19$	3.13071	0.718234	0.359117	−	0.933293i	$-0.383078\pi$
$19$	3.13071	0.718234	0.359117	+	0.933293i	$0.383078\pi$
$20$	1.87233	0.418665
$21$	0	0
$22$	0.0214096	0.00456453
$23$	−7.74707	−1.61538	−0.807688	−	0.589610i	$-0.799281\pi$
$23$	−7.74707	−1.61538	−0.807688	+	0.589610i	$0.799281\pi$
$24$	0	0
$25$	−3.94150	−0.788300
$26$	−1.62403	−0.318498
$27$	0	0
$28$	7.33273	1.38576
$29$	−9.57779	−1.77855	−0.889275	−	0.457372i	$-0.848791\pi$
$29$	−9.57779	−1.77855	−0.889275	+	0.457372i	$0.848791\pi$
$30$	0	0
$31$	−10.6996	−1.92170	−0.960852	−	0.277063i	$-0.910639\pi$
$31$	−10.6996	−1.92170	−0.960852	+	0.277063i	$0.910639\pi$
$32$	4.49534	0.794671
$33$	0	0
$34$	0.198915	0.0341137
$35$	4.14548	0.700715
$36$	0	0
$37$	−5.80175	−0.953802	−0.476901	−	0.878957i	$-0.658240\pi$
$37$	−5.80175	−0.953802	−0.476901	+	0.878957i	$0.658240\pi$
$38$	1.32879	0.215559
$39$	0	0
$40$	1.66804	0.263740
$41$	−1.02595	−0.160227	−0.0801133	−	0.996786i	$-0.525528\pi$
$41$	−1.02595	−0.160227	−0.0801133	+	0.996786i	$0.525528\pi$
$42$	0	0
$43$	11.0372	1.68316	0.841579	−	0.540134i	$-0.181626\pi$
$43$	11.0372	1.68316	0.841579	+	0.540134i	$0.181626\pi$
$44$	−0.0917973	−0.0138390
$45$	0	0
$46$	−3.28815	−0.484811
$47$	−8.77093	−1.27937	−0.639686	−	0.768637i	$-0.720935\pi$
$47$	−8.77093	−1.27937	−0.639686	+	0.768637i	$0.720935\pi$
$48$	0	0
$49$	9.23524	1.31932
$50$	−1.67292	−0.236587
$51$	0	0
$52$	6.96331	0.965638
$53$	−2.50076	−0.343505	−0.171753	−	0.985140i	$-0.554943\pi$
$53$	−2.50076	−0.343505	−0.171753	+	0.985140i	$0.554943\pi$
$54$	0	0
$55$	−0.0518967	−0.00699774
$56$	6.53266	0.872963
$57$	0	0
$58$	−4.06518	−0.533784
$59$	−4.13983	−0.538959	−0.269480	−	0.963006i	$-0.586852\pi$
$59$	−4.13983	−0.538959	−0.269480	+	0.963006i	$0.586852\pi$
$60$	0	0
$61$	−2.10355	−0.269332	−0.134666	−	0.990891i	$-0.542996\pi$
$61$	−2.10355	−0.269332	−0.134666	+	0.990891i	$0.542996\pi$
$62$	−4.54131	−0.576747
$63$	0	0
$64$	−3.99515	−0.499393
$65$	3.93664	0.488280
$66$	0	0
$67$	−11.3209	−1.38306	−0.691532	−	0.722346i	$-0.743064\pi$
$67$	−11.3209	−1.38306	−0.691532	+	0.722346i	$0.743064\pi$
$68$	−0.852885	−0.103427
$69$	0	0
$70$	1.75950	0.210300
$71$	13.9288	1.65304	0.826520	−	0.562907i	$-0.190317\pi$
$71$	13.9288	1.65304	0.826520	+	0.562907i	$0.190317\pi$
$72$	0	0
$73$	1.63283	0.191108	0.0955542	−	0.995424i	$-0.469538\pi$
$73$	1.63283	0.191108	0.0955542	+	0.995424i	$0.469538\pi$
$74$	−2.46248	−0.286258
$75$	0	0
$76$	−5.69743	−0.653540
$77$	−0.203246	−0.0231621
$78$	0	0
$79$	10.3556	1.16510	0.582551	−	0.812794i	$-0.302055\pi$
$79$	10.3556	1.16510	0.582551	+	0.812794i	$0.302055\pi$
$80$	−3.03668	−0.339511
$81$	0	0
$82$	−0.435452	−0.0480876
$83$	−12.6229	−1.38554	−0.692771	−	0.721158i	$-0.743610\pi$
$83$	−12.6229	−1.38554	−0.692771	+	0.721158i	$0.743610\pi$
$84$	0	0
$85$	−0.482170	−0.0522987
$86$	4.68461	0.505154
$87$	0	0
$88$	−0.0817813	−0.00871792
$89$	1.89949	0.201346	0.100673	−	0.994920i	$-0.467900\pi$
$89$	1.89949	0.201346	0.100673	+	0.994920i	$0.467900\pi$
$90$	0	0
$91$	15.4173	1.61617
$92$	14.0985	1.46987
$93$	0	0
$94$	−3.72271	−0.383969
$95$	−3.22098	−0.330466
$96$	0	0
$97$	−2.77745	−0.282008	−0.141004	−	0.990009i	$-0.545033\pi$
$97$	−2.77745	−0.282008	−0.141004	+	0.990009i	$0.545033\pi$
$98$	3.91978	0.395958
$99$	0	0
$100$	7.17295	0.717295
$101$	−9.19234	−0.914672	−0.457336	−	0.889294i	$-0.651196\pi$
$101$	−9.19234	−0.914672	−0.457336	+	0.889294i	$0.651196\pi$
$102$	0	0
$103$	1.37765	0.135744	0.0678721	−	0.997694i	$-0.478379\pi$
$103$	1.37765	0.135744	0.0678721	+	0.997694i	$0.478379\pi$
$104$	6.20355	0.608308
$105$	0	0
$106$	−1.06142	−0.103094
$107$	−12.0873	−1.16852	−0.584262	−	0.811565i	$-0.698616\pi$
$107$	−12.0873	−1.16852	−0.584262	+	0.811565i	$0.698616\pi$
$108$	0	0
$109$	1.07642	0.103102	0.0515512	−	0.998670i	$-0.483583\pi$
$109$	1.07642	0.103102	0.0515512	+	0.998670i	$0.483583\pi$
$110$	−0.0220269	−0.00210018
$111$	0	0
$112$	−11.8927	−1.12376
$113$	14.7147	1.38424	0.692119	−	0.721783i	$-0.256677\pi$
$113$	14.7147	1.38424	0.692119	+	0.721783i	$0.256677\pi$
$114$	0	0
$115$	7.97046	0.743249
$116$	17.4302	1.61835
$117$	0	0
$118$	−1.75710	−0.161754
$119$	−1.88835	−0.173105
$120$	0	0
$121$	−10.9975	−0.999769
$122$	−0.892828	−0.0808328
$123$	0	0
$124$	19.4717	1.74861
$125$	9.19933	0.822813
$126$	0	0
$127$	−20.6670	−1.83390	−0.916951	−	0.399000i	$-0.869357\pi$
$127$	−20.6670	−1.83390	−0.916951	+	0.399000i	$0.869357\pi$
$128$	−10.6864	−0.944550
$129$	0	0
$130$	1.67086	0.146544
$131$	12.3585	1.07976	0.539882	−	0.841741i	$-0.318469\pi$
$131$	12.3585	1.07976	0.539882	+	0.841741i	$0.318469\pi$
$132$	0	0
$133$	−12.6146	−1.09382
$134$	−4.80501	−0.415089
$135$	0	0
$136$	−0.759827	−0.0651546
$137$	1.06655	0.0911213	0.0455606	−	0.998962i	$-0.485493\pi$
$137$	1.06655	0.0911213	0.0455606	+	0.998962i	$0.485493\pi$
$138$	0	0
$139$	−3.94767	−0.334837	−0.167418	−	0.985886i	$-0.553543\pi$
$139$	−3.94767	−0.334837	−0.167418	+	0.985886i	$0.553543\pi$
$140$	−7.54417	−0.637599
$141$	0	0
$142$	5.91189	0.496115
$143$	−0.193007	−0.0161401
$144$	0	0
$145$	9.85397	0.818327
$146$	0.693035	0.0573560
$147$	0	0
$148$	10.5583	0.867890
$149$	−7.30519	−0.598464	−0.299232	−	0.954180i	$-0.596730\pi$
$149$	−7.30519	−0.598464	−0.299232	+	0.954180i	$0.596730\pi$
$150$	0	0
$151$	−0.285565	−0.0232389	−0.0116195	−	0.999932i	$-0.503699\pi$
$151$	−0.285565	−0.0232389	−0.0116195	+	0.999932i	$0.503699\pi$
$152$	−5.07579	−0.411701
$153$	0	0
$154$	−0.0862655	−0.00695147
$155$	11.0081	0.884193
$156$	0	0
$157$	18.5910	1.48372	0.741860	−	0.670555i	$-0.233944\pi$
$157$	18.5910	1.48372	0.741860	+	0.670555i	$0.233944\pi$
$158$	4.39533	0.349674
$159$	0	0
$160$	−4.62496	−0.365635
$161$	31.2153	2.46011
$162$	0	0
$163$	−2.24060	−0.175497	−0.0877487	−	0.996143i	$-0.527967\pi$
$163$	−2.24060	−0.175497	−0.0877487	+	0.996143i	$0.527967\pi$
$164$	1.86708	0.145794
$165$	0	0
$166$	−5.35763	−0.415833
$167$	12.2225	0.945806	0.472903	−	0.881114i	$-0.343206\pi$
$167$	12.2225	0.945806	0.472903	+	0.881114i	$0.343206\pi$
$168$	0	0
$169$	1.64062	0.126201
$170$	−0.204651	−0.0156960
$171$	0	0
$172$	−20.0861	−1.53155
$173$	16.4486	1.25056	0.625281	−	0.780399i	$-0.284984\pi$
$173$	16.4486	1.25056	0.625281	+	0.780399i	$0.284984\pi$
$174$	0	0
$175$	15.8815	1.20053
$176$	0.148883	0.0112225
$177$	0	0
$178$	0.806216	0.0604284
$179$	18.2072	1.36087	0.680434	−	0.732809i	$-0.261791\pi$
$179$	18.2072	1.36087	0.680434	+	0.732809i	$0.261791\pi$
$180$	0	0
$181$	18.3353	1.36285	0.681425	−	0.731887i	$-0.261360\pi$
$181$	18.3353	1.36285	0.681425	+	0.731887i	$0.261360\pi$
$182$	6.54370	0.485051
$183$	0	0
$184$	12.5603	0.925954
$185$	5.96905	0.438853
$186$	0	0
$187$	0.0236400	0.00172873
$188$	15.9618	1.16413
$189$	0	0
$190$	−1.36711	−0.0991804
$191$	6.35566	0.459879	0.229940	−	0.973205i	$-0.426147\pi$
$191$	6.35566	0.459879	0.229940	+	0.973205i	$0.426147\pi$
$192$	0	0
$193$	−1.60064	−0.115216	−0.0576082	−	0.998339i	$-0.518347\pi$
$193$	−1.60064	−0.115216	−0.0576082	+	0.998339i	$0.518347\pi$
$194$	−1.17886	−0.0846369
$195$	0	0
$196$	−16.8068	−1.20048
$197$	23.3882	1.66634	0.833170	−	0.553017i	$-0.186523\pi$
$197$	23.3882	1.66634	0.833170	+	0.553017i	$0.186523\pi$
$198$	0	0
$199$	−13.1598	−0.932871	−0.466436	−	0.884555i	$-0.654462\pi$
$199$	−13.1598	−0.932871	−0.466436	+	0.884555i	$0.654462\pi$
$200$	6.39031	0.451863
$201$	0	0
$202$	−3.90158	−0.274514
$203$	38.5918	2.70861
$204$	0	0
$205$	1.05553	0.0737217
$206$	0.584728	0.0407400
$207$	0	0
$208$	−11.2936	−0.783071
$209$	0.157920	0.0109235
$210$	0	0
$211$	−10.1048	−0.695645	−0.347823	−	0.937560i	$-0.613079\pi$
$211$	−10.1048	−0.695645	−0.347823	+	0.937560i	$0.613079\pi$
$212$	4.55101	0.312564
$213$	0	0
$214$	−5.13031	−0.350701
$215$	−11.3555	−0.774437
$216$	0	0
$217$	43.1118	2.92662
$218$	0.456873	0.0309434
$219$	0	0
$220$	0.0944443	0.00636743
$221$	−1.79322	−0.120625
$222$	0	0
$223$	1.00000	0.0669650
$223$	1.00000	0.0669650
$224$	−18.1131	−1.21023
$225$	0	0
$226$	6.24545	0.415441
$227$	7.82851	0.519597	0.259798	−	0.965663i	$-0.416344\pi$
$227$	7.82851	0.519597	0.259798	+	0.965663i	$0.416344\pi$
$228$	0	0
$229$	13.5094	0.892729	0.446364	−	0.894851i	$-0.352719\pi$
$229$	13.5094	0.892729	0.446364	+	0.894851i	$0.352719\pi$
$230$	3.38297	0.223066
$231$	0	0
$232$	15.5284	1.01949
$233$	−26.7417	−1.75191	−0.875953	−	0.482397i	$-0.839766\pi$
$233$	−26.7417	−1.75191	−0.875953	+	0.482397i	$0.839766\pi$
$234$	0	0
$235$	9.02384	0.588650
$236$	7.53387	0.490413
$237$	0	0
$238$	−0.801489	−0.0519528
$239$	9.99996	0.646844	0.323422	−	0.946255i	$-0.395167\pi$
$239$	9.99996	0.646844	0.323422	+	0.946255i	$0.395167\pi$
$240$	0	0
$241$	1.09994	0.0708535	0.0354268	−	0.999372i	$-0.488721\pi$
$241$	1.09994	0.0708535	0.0354268	+	0.999372i	$0.488721\pi$
$242$	−4.66774	−0.300053
$243$	0	0
$244$	3.82816	0.245073
$245$	−9.50154	−0.607031
$246$	0	0
$247$	−11.9791	−0.762209
$248$	17.3471	1.10154
$249$	0	0
$250$	3.90454	0.246945
$251$	−2.52838	−0.159590	−0.0797950	−	0.996811i	$-0.525427\pi$
$251$	−2.52838	−0.159590	−0.0797950	+	0.996811i	$0.525427\pi$
$252$	0	0
$253$	−0.390779	−0.0245681
$254$	−8.77187	−0.550396
$255$	0	0
$256$	3.45459	0.215912
$257$	−0.948122	−0.0591423	−0.0295711	−	0.999563i	$-0.509414\pi$
$257$	−0.948122	−0.0591423	−0.0295711	+	0.999563i	$0.509414\pi$
$258$	0	0
$259$	23.3770	1.45258
$260$	−7.16410	−0.444299
$261$	0	0
$262$	5.24540	0.324062
$263$	16.3050	1.00541	0.502706	−	0.864458i	$-0.332338\pi$
$263$	16.3050	1.00541	0.502706	+	0.864458i	$0.332338\pi$
$264$	0	0
$265$	2.57287	0.158050
$266$	−5.35410	−0.328281
$267$	0	0
$268$	20.6023	1.25849
$269$	−5.90391	−0.359967	−0.179984	−	0.983670i	$-0.557605\pi$
$269$	−5.90391	−0.359967	−0.179984	+	0.983670i	$0.557605\pi$
$270$	0	0
$271$	−2.42626	−0.147385	−0.0736923	−	0.997281i	$-0.523478\pi$
$271$	−2.42626	−0.147385	−0.0736923	+	0.997281i	$0.523478\pi$
$272$	1.38327	0.0838731
$273$	0	0
$274$	0.452683	0.0273476
$275$	−0.198818	−0.0119892
$276$	0	0
$277$	9.63587	0.578964	0.289482	−	0.957184i	$-0.406517\pi$
$277$	9.63587	0.578964	0.289482	+	0.957184i	$0.406517\pi$
$278$	−1.67554	−0.100492
$279$	0	0
$280$	−6.72103	−0.401658
$281$	0.972089	0.0579900	0.0289950	−	0.999580i	$-0.490769\pi$
$281$	0.972089	0.0579900	0.0289950	+	0.999580i	$0.490769\pi$
$282$	0	0
$283$	−19.7242	−1.17248	−0.586240	−	0.810137i	$-0.699393\pi$
$283$	−19.7242	−1.17248	−0.586240	+	0.810137i	$0.699393\pi$
$284$	−25.3483	−1.50414
$285$	0	0
$286$	−0.0819195	−0.00484400
$287$	4.13386	0.244014
$288$	0	0
$289$	−16.7804	−0.987080
$290$	4.18240	0.245599
$291$	0	0
$292$	−2.97151	−0.173895
$293$	−28.7455	−1.67933	−0.839664	−	0.543106i	$-0.817248\pi$
$293$	−28.7455	−1.67933	−0.839664	+	0.543106i	$0.817248\pi$
$294$	0	0
$295$	4.25920	0.247980
$296$	9.40632	0.546731
$297$	0	0
$298$	−3.10060	−0.179613
$299$	29.6427	1.71428
$300$	0	0
$301$	−44.4722	−2.56333
$302$	−0.121204	−0.00697453
$303$	0	0
$304$	9.24051	0.529979
$305$	2.16421	0.123922
$306$	0	0
$307$	21.8530	1.24722	0.623610	−	0.781736i	$-0.285666\pi$
$307$	21.8530	1.24722	0.623610	+	0.781736i	$0.285666\pi$
$308$	0.369879	0.0210758
$309$	0	0
$310$	4.67226	0.265367
$311$	−22.6392	−1.28375	−0.641876	−	0.766809i	$-0.721844\pi$
$311$	−22.6392	−1.28375	−0.641876	+	0.766809i	$0.721844\pi$
$312$	0	0
$313$	−7.95695	−0.449753	−0.224877	−	0.974387i	$-0.572198\pi$
$313$	−7.95695	−0.449753	−0.224877	+	0.974387i	$0.572198\pi$
$314$	7.89070	0.445298
$315$	0	0
$316$	−18.8458	−1.06016
$317$	−25.8793	−1.45353	−0.726763	−	0.686889i	$-0.758976\pi$
$317$	−25.8793	−1.45353	−0.726763	+	0.686889i	$0.758976\pi$
$318$	0	0
$319$	−0.483124	−0.0270498
$320$	4.11035	0.229775
$321$	0	0
$322$	13.2489	0.738334
$323$	1.46723	0.0816386
$324$	0	0
$325$	15.0814	0.836564
$326$	−0.950996	−0.0526708
$327$	0	0
$328$	1.66336	0.0918439
$329$	35.3407	1.94839
$330$	0	0
$331$	−18.7103	−1.02841	−0.514205	−	0.857667i	$-0.671913\pi$
$331$	−18.7103	−1.02841	−0.514205	+	0.857667i	$0.671913\pi$
$332$	22.9718	1.26074
$333$	0	0
$334$	5.18770	0.283858
$335$	11.6473	0.636360
$336$	0	0
$337$	−28.9305	−1.57594	−0.787972	−	0.615711i	$-0.788869\pi$
$337$	−28.9305	−1.57594	−0.787972	+	0.615711i	$0.788869\pi$
$338$	0.696341	0.0378759
$339$	0	0
$340$	0.877478	0.0475879
$341$	−0.539710	−0.0292270
$342$	0	0
$343$	−9.00644	−0.486302
$344$	−17.8945	−0.964808
$345$	0	0
$346$	6.98140	0.375323
$347$	15.1785	0.814823	0.407411	−	0.913245i	$-0.366432\pi$
$347$	15.1785	0.814823	0.407411	+	0.913245i	$0.366432\pi$
$348$	0	0
$349$	−19.1677	−1.02602	−0.513011	−	0.858382i	$-0.671470\pi$
$349$	−19.1677	−1.02602	−0.513011	+	0.858382i	$0.671470\pi$
$350$	6.74070	0.360305
$351$	0	0
$352$	0.226754	0.0120861
$353$	15.7507	0.838325	0.419163	−	0.907911i	$-0.362324\pi$
$353$	15.7507	0.838325	0.419163	+	0.907911i	$0.362324\pi$
$354$	0	0
$355$	−14.3304	−0.760579
$356$	−3.45679	−0.183210
$357$	0	0
$358$	7.72781	0.408428
$359$	5.65945	0.298694	0.149347	−	0.988785i	$-0.452283\pi$
$359$	5.65945	0.298694	0.149347	+	0.988785i	$0.452283\pi$
$360$	0	0
$361$	−9.19865	−0.484140
$362$	7.78219	0.409023
$363$	0	0
$364$	−28.0573	−1.47060
$365$	−1.67991	−0.0879307
$366$	0	0
$367$	−8.06425	−0.420950	−0.210475	−	0.977599i	$-0.567501\pi$
$367$	−8.06425	−0.420950	−0.210475	+	0.977599i	$0.567501\pi$
$368$	−22.8660	−1.19197
$369$	0	0
$370$	2.53349	0.131710
$371$	10.0763	0.523135
$372$	0	0
$373$	−4.26093	−0.220623	−0.110311	−	0.993897i	$-0.535185\pi$
$373$	−4.26093	−0.220623	−0.110311	+	0.993897i	$0.535185\pi$
$374$	0.0100337	0.000518831 0
$375$	0	0
$376$	14.2202	0.733352
$377$	36.6475	1.88744
$378$	0	0
$379$	−4.93854	−0.253676	−0.126838	−	0.991923i	$-0.540483\pi$
$379$	−4.93854	−0.253676	−0.126838	+	0.991923i	$0.540483\pi$
$380$	5.86172	0.300700
$381$	0	0
$382$	2.69758	0.138020
$383$	−1.87823	−0.0959733	−0.0479866	−	0.998848i	$-0.515280\pi$
$383$	−1.87823	−0.0959733	−0.0479866	+	0.998848i	$0.515280\pi$
$384$	0	0
$385$	0.209107	0.0106571
$386$	−0.679371	−0.0345791
$387$	0	0
$388$	5.05455	0.256606
$389$	8.39934	0.425863	0.212932	−	0.977067i	$-0.431699\pi$
$389$	8.39934	0.425863	0.212932	+	0.977067i	$0.431699\pi$
$390$	0	0
$391$	−3.63071	−0.183613
$392$	−14.9730	−0.756251
$393$	0	0
$394$	9.92684	0.500107
$395$	−10.6543	−0.536074
$396$	0	0
$397$	−21.4422	−1.07615	−0.538075	−	0.842897i	$-0.680848\pi$
$397$	−21.4422	−1.07615	−0.538075	+	0.842897i	$0.680848\pi$
$398$	−5.58550	−0.279976
$399$	0	0
$400$	−11.6336	−0.581680
$401$	32.2112	1.60855	0.804276	−	0.594256i	$-0.202553\pi$
$401$	32.2112	1.60855	0.804276	+	0.594256i	$0.202553\pi$
$402$	0	0
$403$	40.9399	2.03936
$404$	16.7287	0.832284
$405$	0	0
$406$	16.3798	0.812916
$407$	−0.292653	−0.0145063
$408$	0	0
$409$	−39.4866	−1.95249	−0.976244	−	0.216673i	$-0.930480\pi$
$409$	−39.4866	−1.95249	−0.976244	+	0.216673i	$0.930480\pi$
$410$	0.448009	0.0221256
$411$	0	0
$412$	−2.50713	−0.123517
$413$	16.6806	0.820798
$414$	0	0
$415$	12.9869	0.637501
$416$	−17.2005	−0.843326
$417$	0	0
$418$	0.0670271	0.00327840
$419$	3.25102	0.158823	0.0794114	−	0.996842i	$-0.474696\pi$
$419$	3.25102	0.158823	0.0794114	+	0.996842i	$0.474696\pi$
$420$	0	0
$421$	−15.7350	−0.766875	−0.383438	−	0.923567i	$-0.625260\pi$
$421$	−15.7350	−0.766875	−0.383438	+	0.923567i	$0.625260\pi$
$422$	−4.28887	−0.208779
$423$	0	0
$424$	4.05445	0.196901
$425$	−1.84721	−0.0896027
$426$	0	0
$427$	8.47584	0.410175
$428$	21.9971	1.06327
$429$	0	0
$430$	−4.81969	−0.232426
$431$	−18.3505	−0.883912	−0.441956	−	0.897037i	$-0.645715\pi$
$431$	−18.3505	−0.883912	−0.441956	+	0.897037i	$0.645715\pi$
$432$	0	0
$433$	24.7470	1.18927	0.594633	−	0.803997i	$-0.297297\pi$
$433$	24.7470	1.18927	0.594633	+	0.803997i	$0.297297\pi$
$434$	18.2983	0.878346
$435$	0	0
$436$	−1.95893	−0.0938155
$437$	−24.2538	−1.16022
$438$	0	0
$439$	−19.7417	−0.942218	−0.471109	−	0.882075i	$-0.656146\pi$
$439$	−19.7417	−0.942218	−0.471109	+	0.882075i	$0.656146\pi$
$440$	0.0841395	0.00401119
$441$	0	0
$442$	−0.761111	−0.0362024
$443$	−6.33294	−0.300887	−0.150444	−	0.988619i	$-0.548070\pi$
$443$	−6.33294	−0.300887	−0.150444	+	0.988619i	$0.548070\pi$
$444$	0	0
$445$	−1.95426	−0.0926410
$446$	0.424438	0.0200977
$447$	0	0
$448$	16.0976	0.760542
$449$	30.5912	1.44369	0.721845	−	0.692055i	$-0.243294\pi$
$449$	30.5912	1.44369	0.721845	+	0.692055i	$0.243294\pi$
$450$	0	0
$451$	−0.0517511	−0.00243687
$452$	−26.7785	−1.25955
$453$	0	0
$454$	3.32272	0.155943
$455$	−15.8619	−0.743617
$456$	0	0
$457$	−6.17602	−0.288902	−0.144451	−	0.989512i	$-0.546142\pi$
$457$	−6.17602	−0.288902	−0.144451	+	0.989512i	$0.546142\pi$
$458$	5.73392	0.267928
$459$	0	0
$460$	−14.5051	−0.676302
$461$	10.2787	0.478729	0.239364	−	0.970930i	$-0.423061\pi$
$461$	10.2787	0.478729	0.239364	+	0.970930i	$0.423061\pi$
$462$	0	0
$463$	−23.1767	−1.07711	−0.538557	−	0.842589i	$-0.681030\pi$
$463$	−23.1767	−1.07711	−0.538557	+	0.842589i	$0.681030\pi$
$464$	−28.2695	−1.31238
$465$	0	0
$466$	−11.3502	−0.525787
$467$	−7.53738	−0.348788	−0.174394	−	0.984676i	$-0.555797\pi$
$467$	−7.53738	−0.348788	−0.174394	+	0.984676i	$0.555797\pi$
$468$	0	0
$469$	45.6152	2.10631
$470$	3.83006	0.176667
$471$	0	0
$472$	6.71186	0.308938
$473$	0.556741	0.0255990
$474$	0	0
$475$	−12.3397	−0.566184
$476$	3.43653	0.157513
$477$	0	0
$478$	4.24436	0.194133
$479$	25.0339	1.14383	0.571915	−	0.820313i	$-0.306201\pi$
$479$	25.0339	1.14383	0.571915	+	0.820313i	$0.306201\pi$
$480$	0	0
$481$	22.1993	1.01220
$482$	0.466857	0.0212648
$483$	0	0
$484$	20.0137	0.909716
$485$	2.85754	0.129754
$486$	0	0
$487$	11.4487	0.518790	0.259395	−	0.965771i	$-0.416477\pi$
$487$	11.4487	0.518790	0.259395	+	0.965771i	$0.416477\pi$
$488$	3.41047	0.154385
$489$	0	0
$490$	−4.03281	−0.182184
$491$	2.08913	0.0942810	0.0471405	−	0.998888i	$-0.484989\pi$
$491$	2.08913	0.0942810	0.0471405	+	0.998888i	$0.484989\pi$
$492$	0	0
$493$	−4.48869	−0.202160
$494$	−5.08436	−0.228756
$495$	0	0
$496$	−31.5806	−1.41801
$497$	−56.1231	−2.51747
$498$	0	0
$499$	21.8065	0.976195	0.488098	−	0.872789i	$-0.337691\pi$
$499$	21.8065	0.976195	0.488098	+	0.872789i	$0.337691\pi$
$500$	−16.7414	−0.748699
$501$	0	0
$502$	−1.07314	−0.0478966
$503$	8.07297	0.359956	0.179978	−	0.983671i	$-0.442397\pi$
$503$	8.07297	0.359956	0.179978	+	0.983671i	$0.442397\pi$
$504$	0	0
$505$	9.45740	0.420849
$506$	−0.165861	−0.00737344
$507$	0	0
$508$	37.6109	1.66872
$509$	−6.05436	−0.268355	−0.134177	−	0.990957i	$-0.542839\pi$
$509$	−6.05436	−0.268355	−0.134177	+	0.990957i	$0.542839\pi$
$510$	0	0
$511$	−6.57916	−0.291045
$512$	22.8390	1.00935
$513$	0	0
$514$	−0.402419	−0.0177499
$515$	−1.41738	−0.0624572
$516$	0	0
$517$	−0.442424	−0.0194578
$518$	9.92208	0.435951
$519$	0	0
$520$	−6.38243	−0.279888
$521$	17.5440	0.768617	0.384309	−	0.923205i	$-0.374440\pi$
$521$	17.5440	0.768617	0.384309	+	0.923205i	$0.374440\pi$
$522$	0	0
$523$	−27.9833	−1.22362	−0.611811	−	0.791004i	$-0.709559\pi$
$523$	−27.9833	−1.22362	−0.611811	+	0.791004i	$0.709559\pi$
$524$	−22.4906	−0.982505
$525$	0	0
$526$	6.92047	0.301747
$527$	−5.01443	−0.218432
$528$	0	0
$529$	37.0171	1.60944
$530$	1.09202	0.0474344
$531$	0	0
$532$	22.9566	0.995297
$533$	3.92560	0.170037
$534$	0	0
$535$	12.4359	0.537649
$536$	18.3544	0.792790
$537$	0	0
$538$	−2.50584	−0.108034
$539$	0.465845	0.0200654
$540$	0	0
$541$	32.4550	1.39535	0.697674	−	0.716415i	$-0.254218\pi$
$541$	32.4550	1.39535	0.697674	+	0.716415i	$0.254218\pi$
$542$	−1.02980	−0.0442335
$543$	0	0
$544$	2.10677	0.0903269
$545$	−1.10746	−0.0474383
$546$	0	0
$547$	4.78304	0.204508	0.102254	−	0.994758i	$-0.467395\pi$
$547$	4.78304	0.204508	0.102254	+	0.994758i	$0.467395\pi$
$548$	−1.94096	−0.0829137
$549$	0	0
$550$	−0.0843857	−0.00359822
$551$	−29.9853	−1.27742
$552$	0	0
$553$	−41.7260	−1.77437
$554$	4.08983	0.173760
$555$	0	0
$556$	7.18418	0.304677
$557$	−46.1871	−1.95701	−0.978504	−	0.206226i	$-0.933882\pi$
$557$	−46.1871	−1.95701	−0.978504	+	0.206226i	$0.933882\pi$
$558$	0	0
$559$	−42.2317	−1.78621
$560$	12.2357	0.517052
$561$	0	0
$562$	0.412591	0.0174041
$563$	−9.51906	−0.401180	−0.200590	−	0.979675i	$-0.564286\pi$
$563$	−9.51906	−0.401180	−0.200590	+	0.979675i	$0.564286\pi$
$564$	0	0
$565$	−15.1390	−0.636900
$566$	−8.37169	−0.351888
$567$	0	0
$568$	−22.5826	−0.947543
$569$	−6.92821	−0.290445	−0.145223	−	0.989399i	$-0.546390\pi$
$569$	−6.92821	−0.290445	−0.145223	+	0.989399i	$0.546390\pi$
$570$	0	0
$571$	2.08192	0.0871255	0.0435628	−	0.999051i	$-0.486129\pi$
$571$	2.08192	0.0871255	0.0435628	+	0.999051i	$0.486129\pi$
$572$	0.351244	0.0146863
$573$	0	0
$574$	1.75457	0.0732342
$575$	30.5351	1.27340
$576$	0	0
$577$	17.6255	0.733758	0.366879	−	0.930269i	$-0.380426\pi$
$577$	17.6255	0.733758	0.366879	+	0.930269i	$0.380426\pi$
$578$	−7.12222	−0.296245
$579$	0	0
$580$	−17.9328	−0.744617
$581$	50.8614	2.11009
$582$	0	0
$583$	−0.126143	−0.00522433
$584$	−2.64729	−0.109546
$585$	0	0
$586$	−12.2007	−0.504005
$587$	11.7990	0.486999	0.243499	−	0.969901i	$-0.421705\pi$
$587$	11.7990	0.486999	0.243499	+	0.969901i	$0.421705\pi$
$588$	0	0
$589$	−33.4973	−1.38023
$590$	1.80776	0.0744245
$591$	0	0
$592$	−17.1243	−0.703803
$593$	12.4446	0.511037	0.255518	−	0.966804i	$-0.417754\pi$
$593$	12.4446	0.511037	0.255518	+	0.966804i	$0.417754\pi$
$594$	0	0
$595$	1.94281	0.0796473
$596$	13.2944	0.544558
$597$	0	0
$598$	12.5815	0.514495
$599$	20.8230	0.850805	0.425402	−	0.905004i	$-0.360133\pi$
$599$	20.8230	0.850805	0.425402	+	0.905004i	$0.360133\pi$
$600$	0	0
$601$	1.49372	0.0609300	0.0304650	−	0.999536i	$-0.490301\pi$
$601$	1.49372	0.0609300	0.0304650	+	0.999536i	$0.490301\pi$
$602$	−18.8757	−0.769315
$603$	0	0
$604$	0.519685	0.0211457
$605$	11.3146	0.460003
$606$	0	0
$607$	1.24316	0.0504582	0.0252291	−	0.999682i	$-0.491968\pi$
$607$	1.24316	0.0504582	0.0252291	+	0.999682i	$0.491968\pi$
$608$	14.0736	0.570760
$609$	0	0
$610$	0.918572	0.0371919
$611$	33.5602	1.35770
$612$	0	0
$613$	−15.8243	−0.639139	−0.319570	−	0.947563i	$-0.603538\pi$
$613$	−15.8243	−0.639139	−0.319570	+	0.947563i	$0.603538\pi$
$614$	9.27526	0.374319
$615$	0	0
$616$	0.329521	0.0132768
$617$	−31.6249	−1.27317	−0.636585	−	0.771207i	$-0.719654\pi$
$617$	−31.6249	−1.27317	−0.636585	+	0.771207i	$0.719654\pi$
$618$	0	0
$619$	1.81320	0.0728786	0.0364393	−	0.999336i	$-0.488398\pi$
$619$	1.81320	0.0728786	0.0364393	+	0.999336i	$0.488398\pi$
$620$	−20.0332	−0.804551
$621$	0	0
$622$	−9.60894	−0.385283
$623$	−7.65361	−0.306636
$624$	0	0
$625$	10.2429	0.409716
$626$	−3.37723	−0.134981
$627$	0	0
$628$	−33.8328	−1.35008
$629$	−2.71903	−0.108415
$630$	0	0
$631$	−8.91919	−0.355067	−0.177534	−	0.984115i	$-0.556812\pi$
$631$	−8.91919	−0.355067	−0.177534	+	0.984115i	$0.556812\pi$
$632$	−16.7895	−0.667851
$633$	0	0
$634$	−10.9841	−0.436236
$635$	21.2630	0.843795
$636$	0	0
$637$	−35.3369	−1.40010
$638$	−0.205056	−0.00811825
$639$	0	0
$640$	10.9945	0.434596
$641$	−18.7684	−0.741307	−0.370653	−	0.928771i	$-0.620866\pi$
$641$	−18.7684	−0.741307	−0.370653	+	0.928771i	$0.620866\pi$
$642$	0	0
$643$	−15.5211	−0.612094	−0.306047	−	0.952016i	$-0.599007\pi$
$643$	−15.5211	−0.612094	−0.306047	+	0.952016i	$0.599007\pi$
$644$	−56.8072	−2.23852
$645$	0	0
$646$	0.622746	0.0245016
$647$	−4.06440	−0.159788	−0.0798940	−	0.996803i	$-0.525458\pi$
$647$	−4.06440	−0.159788	−0.0798940	+	0.996803i	$0.525458\pi$
$648$	0	0
$649$	−0.208822	−0.00819697
$650$	6.40111	0.251072
$651$	0	0
$652$	4.07756	0.159690
$653$	−44.3605	−1.73596	−0.867981	−	0.496598i	$-0.834582\pi$
$653$	−44.3605	−1.73596	−0.867981	+	0.496598i	$0.834582\pi$
$654$	0	0
$655$	−12.7148	−0.496809
$656$	−3.02816	−0.118230
$657$	0	0
$658$	14.9999	0.584758
$659$	48.0083	1.87014	0.935070	−	0.354464i	$-0.115337\pi$
$659$	48.0083	1.87014	0.935070	+	0.354464i	$0.115337\pi$
$660$	0	0
$661$	0.361230	0.0140502	0.00702510	−	0.999975i	$-0.497764\pi$
$661$	0.361230	0.0140502	0.00702510	+	0.999975i	$0.497764\pi$
$662$	−7.94135	−0.308649
$663$	0	0
$664$	20.4654	0.794210
$665$	12.9783	0.503277
$666$	0	0
$667$	74.1998	2.87303
$668$	−22.2432	−0.860614
$669$	0	0
$670$	4.94356	0.190986
$671$	−0.106108	−0.00409624
$672$	0	0
$673$	−22.8568	−0.881066	−0.440533	−	0.897736i	$-0.645211\pi$
$673$	−22.8568	−0.881066	−0.440533	+	0.897736i	$0.645211\pi$
$674$	−12.2792	−0.472977
$675$	0	0
$676$	−2.98568	−0.114834
$677$	−43.0577	−1.65484	−0.827420	−	0.561583i	$-0.810192\pi$
$677$	−43.0577	−1.65484	−0.827420	+	0.561583i	$0.810192\pi$
$678$	0	0
$679$	11.1912	0.429478
$680$	0.781737	0.0299782
$681$	0	0
$682$	−0.229074	−0.00877168
$683$	17.5965	0.673311	0.336656	−	0.941628i	$-0.390704\pi$
$683$	17.5965	0.673311	0.336656	+	0.941628i	$0.390704\pi$
$684$	0	0
$685$	−1.09730	−0.0419257
$686$	−3.82267	−0.145950
$687$	0	0
$688$	32.5771	1.24199
$689$	9.56866	0.364537
$690$	0	0
$691$	−35.6563	−1.35643	−0.678216	−	0.734863i	$-0.737246\pi$
$691$	−35.6563	−1.35643	−0.678216	+	0.734863i	$0.737246\pi$
$692$	−29.9340	−1.13792
$693$	0	0
$694$	6.44231	0.244547
$695$	4.06150	0.154062
$696$	0	0
$697$	−0.480818	−0.0182123
$698$	−8.13548	−0.307933
$699$	0	0
$700$	−28.9019	−1.09239
$701$	−50.1979	−1.89595	−0.947974	−	0.318348i	$-0.896872\pi$
$701$	−50.1979	−1.89595	−0.947974	+	0.318348i	$0.896872\pi$
$702$	0	0
$703$	−18.1636	−0.685053
$704$	−0.201524	−0.00759521
$705$	0	0
$706$	6.68519	0.251601
$707$	37.0387	1.39298
$708$	0	0
$709$	20.5597	0.772136	0.386068	−	0.922470i	$-0.373833\pi$
$709$	20.5597	0.772136	0.386068	+	0.922470i	$0.373833\pi$
$710$	−6.08236	−0.228267
$711$	0	0
$712$	−3.07963	−0.115414
$713$	82.8905	3.10427
$714$	0	0
$715$	0.198573	0.00742619
$716$	−33.1344	−1.23829
$717$	0	0
$718$	2.40208	0.0896450
$719$	19.3922	0.723208	0.361604	−	0.932332i	$-0.382229\pi$
$719$	19.3922	0.723208	0.361604	+	0.932332i	$0.382229\pi$
$720$	0	0
$721$	−5.55098	−0.206729
$722$	−3.90426	−0.145301
$723$	0	0
$724$	−33.3675	−1.24009
$725$	37.7508	1.40203
$726$	0	0
$727$	−35.9729	−1.33416	−0.667081	−	0.744985i	$-0.732457\pi$
$727$	−35.9729	−1.33416	−0.667081	+	0.744985i	$0.732457\pi$
$728$	−24.9960	−0.926412
$729$	0	0
$730$	−0.713019	−0.0263900
$731$	5.17265	0.191318
$732$	0	0
$733$	−22.8330	−0.843358	−0.421679	−	0.906745i	$-0.638559\pi$
$733$	−22.8330	−0.843358	−0.421679	+	0.906745i	$0.638559\pi$
$734$	−3.42277	−0.126337
$735$	0	0
$736$	−34.8257	−1.28369
$737$	−0.571049	−0.0210349
$738$	0	0
$739$	26.4242	0.972028	0.486014	−	0.873951i	$-0.338450\pi$
$739$	26.4242	0.972028	0.486014	+	0.873951i	$0.338450\pi$
$740$	−10.8628	−0.399324
$741$	0	0
$742$	4.27676	0.157005
$743$	−9.65034	−0.354037	−0.177018	−	0.984208i	$-0.556645\pi$
$743$	−9.65034	−0.354037	−0.177018	+	0.984208i	$0.556645\pi$
$744$	0	0
$745$	7.51583	0.275359
$746$	−1.80850	−0.0662139
$747$	0	0
$748$	−0.0430213	−0.00157302
$749$	48.7034	1.77958
$750$	0	0
$751$	−34.0702	−1.24324	−0.621620	−	0.783319i	$-0.713525\pi$
$751$	−34.0702	−1.24324	−0.621620	+	0.783319i	$0.713525\pi$
$752$	−25.8880	−0.944038
$753$	0	0
$754$	15.5546	0.566465
$755$	0.293799	0.0106924
$756$	0	0
$757$	−40.1662	−1.45987	−0.729933	−	0.683519i	$-0.760449\pi$
$757$	−40.1662	−1.45987	−0.729933	+	0.683519i	$0.760449\pi$
$758$	−2.09610	−0.0761339
$759$	0	0
$760$	5.22215	0.189427
$761$	38.2103	1.38512	0.692562	−	0.721358i	$-0.256482\pi$
$761$	38.2103	1.38512	0.692562	+	0.721358i	$0.256482\pi$
$762$	0	0
$763$	−4.33722	−0.157018
$764$	−11.5664	−0.418456
$765$	0	0
$766$	−0.797193	−0.0288038
$767$	15.8402	0.571958
$768$	0	0
$769$	14.7111	0.530496	0.265248	−	0.964180i	$-0.414546\pi$
$769$	14.7111	0.530496	0.265248	+	0.964180i	$0.414546\pi$
$770$	0.0887530	0.00319843
$771$	0	0
$772$	2.91292	0.104838
$773$	−21.6013	−0.776945	−0.388472	−	0.921460i	$-0.626997\pi$
$773$	−21.6013	−0.776945	−0.388472	+	0.921460i	$0.626997\pi$
$774$	0	0
$775$	42.1724	1.51488
$776$	4.50305	0.161650
$777$	0	0
$778$	3.56500	0.127811
$779$	−3.21195	−0.115080
$780$	0	0
$781$	0.702597	0.0251409
$782$	−1.54101	−0.0551065
$783$	0	0
$784$	27.2584	0.973516
$785$	−19.1270	−0.682673
$786$	0	0
$787$	−52.3048	−1.86446	−0.932232	−	0.361860i	$-0.882142\pi$
$787$	−52.3048	−1.86446	−0.932232	+	0.361860i	$0.882142\pi$
$788$	−42.5631	−1.51625
$789$	0	0
$790$	−4.52207	−0.160888
$791$	−59.2897	−2.10810
$792$	0	0
$793$	8.04884	0.285823
$794$	−9.10086	−0.322977
$795$	0	0
$796$	23.9488	0.848844
$797$	−35.3226	−1.25119	−0.625596	−	0.780147i	$-0.715144\pi$
$797$	−35.3226	−1.25119	−0.625596	+	0.780147i	$0.715144\pi$
$798$	0	0
$799$	−4.11055	−0.145421
$800$	−17.7184	−0.626439
$801$	0	0
$802$	13.6717	0.482763
$803$	0.0823635	0.00290654
$804$	0	0
$805$	−32.1154	−1.13192
$806$	17.3765	0.612059
$807$	0	0
$808$	14.9035	0.524302
$809$	7.01231	0.246540	0.123270	−	0.992373i	$-0.460662\pi$
$809$	7.01231	0.246540	0.123270	+	0.992373i	$0.460662\pi$
$810$	0	0
$811$	−7.78299	−0.273298	−0.136649	−	0.990620i	$-0.543633\pi$
$811$	−7.78299	−0.273298	−0.136649	+	0.990620i	$0.543633\pi$
$812$	−70.2313	−2.46464
$813$	0	0
$814$	−0.124213	−0.00435366
$815$	2.30521	0.0807479
$816$	0	0
$817$	34.5543	1.20890
$818$	−16.7596	−0.585986
$819$	0	0
$820$	−1.92092	−0.0670813
$821$	−34.5506	−1.20582	−0.602912	−	0.797808i	$-0.705993\pi$
$821$	−34.5506	−1.20582	−0.602912	+	0.797808i	$0.705993\pi$
$822$	0	0
$823$	−11.2118	−0.390817	−0.195409	−	0.980722i	$-0.562603\pi$
$823$	−11.2118	−0.390817	−0.195409	+	0.980722i	$0.562603\pi$
$824$	−2.23358	−0.0778103
$825$	0	0
$826$	7.07987	0.246340
$827$	3.51072	0.122080	0.0610398	−	0.998135i	$-0.480558\pi$
$827$	3.51072	0.122080	0.0610398	+	0.998135i	$0.480558\pi$
$828$	0	0
$829$	−32.7661	−1.13801	−0.569007	−	0.822333i	$-0.692672\pi$
$829$	−32.7661	−1.13801	−0.569007	+	0.822333i	$0.692672\pi$
$830$	5.51212	0.191329
$831$	0	0
$832$	15.2866	0.529969
$833$	4.32815	0.149961
$834$	0	0
$835$	−12.5749	−0.435174
$836$	−0.287391	−0.00993961
$837$	0	0
$838$	1.37986	0.0476664
$839$	49.8123	1.71971	0.859856	−	0.510537i	$-0.170553\pi$
$839$	49.8123	1.71971	0.859856	+	0.510537i	$0.170553\pi$
$840$	0	0
$841$	62.7340	2.16324
$842$	−6.67851	−0.230157
$843$	0	0
$844$	18.3893	0.632986
$845$	−1.68793	−0.0580664
$846$	0	0
$847$	44.3120	1.52258
$848$	−7.38115	−0.253470
$849$	0	0
$850$	−0.784024	−0.0268918
$851$	44.9466	1.54075
$852$	0	0
$853$	38.6148	1.32215	0.661073	−	0.750321i	$-0.270101\pi$
$853$	38.6148	1.32215	0.661073	+	0.750321i	$0.270101\pi$
$854$	3.59747	0.123103
$855$	0	0
$856$	19.5970	0.669813
$857$	−31.3590	−1.07120	−0.535602	−	0.844471i	$-0.679915\pi$
$857$	−31.3590	−1.07120	−0.535602	+	0.844471i	$0.679915\pi$
$858$	0	0
$859$	−11.6214	−0.396517	−0.198258	−	0.980150i	$-0.563529\pi$
$859$	−11.6214	−0.396517	−0.198258	+	0.980150i	$0.563529\pi$
$860$	20.6653	0.704680
$861$	0	0
$862$	−7.78864	−0.265282
$863$	51.5415	1.75449	0.877246	−	0.480041i	$-0.159378\pi$
$863$	51.5415	1.75449	0.877246	+	0.480041i	$0.159378\pi$
$864$	0	0
$865$	−16.9229	−0.575395
$866$	10.5036	0.356926
$867$	0	0
$868$	−78.4572	−2.66301
$869$	0.522361	0.0177199
$870$	0	0
$871$	43.3171	1.46774
$872$	−1.74519	−0.0590996
$873$	0	0
$874$	−10.2942	−0.348208
$875$	−37.0668	−1.25309
$876$	0	0
$877$	−12.2654	−0.414175	−0.207087	−	0.978322i	$-0.566398\pi$
$877$	−12.2654	−0.414175	−0.207087	+	0.978322i	$0.566398\pi$
$878$	−8.37911	−0.282781
$879$	0	0
$880$	−0.153177	−0.00516358
$881$	−4.00361	−0.134885	−0.0674425	−	0.997723i	$-0.521484\pi$
$881$	−4.00361	−0.134885	−0.0674425	+	0.997723i	$0.521484\pi$
$882$	0	0
$883$	29.1976	0.982577	0.491289	−	0.870997i	$-0.336526\pi$
$883$	29.1976	0.982577	0.491289	+	0.870997i	$0.336526\pi$
$884$	3.26340	0.109760
$885$	0	0
$886$	−2.68794	−0.0903031
$887$	10.0374	0.337025	0.168512	−	0.985700i	$-0.446104\pi$
$887$	10.0374	0.337025	0.168512	+	0.985700i	$0.446104\pi$
$888$	0	0
$889$	83.2736	2.79291
$890$	−0.829463	−0.0278037
$891$	0	0
$892$	−1.81985	−0.0609332
$893$	−27.4592	−0.918888
$894$	0	0
$895$	−18.7322	−0.626148
$896$	43.0585	1.43848
$897$	0	0
$898$	12.9841	0.433284
$899$	102.478	3.41785
$900$	0	0
$901$	−1.17199	−0.0390448
$902$	−0.0219651	−0.000731359 0
$903$	0	0
$904$	−23.8567	−0.793462
$905$	−18.8640	−0.627060
$906$	0	0
$907$	44.0352	1.46217	0.731083	−	0.682289i	$-0.239015\pi$
$907$	44.0352	1.46217	0.731083	+	0.682289i	$0.239015\pi$
$908$	−14.2467	−0.472795
$909$	0	0
$910$	−6.73239	−0.223176
$911$	1.74341	0.0577618	0.0288809	−	0.999583i	$-0.490806\pi$
$911$	1.74341	0.0577618	0.0288809	+	0.999583i	$0.490806\pi$
$912$	0	0
$913$	−0.636726	−0.0210725
$914$	−2.62134	−0.0867062
$915$	0	0
$916$	−24.5852	−0.812317
$917$	−49.7959	−1.64441
$918$	0	0
$919$	15.2029	0.501498	0.250749	−	0.968052i	$-0.419323\pi$
$919$	15.2029	0.501498	0.250749	+	0.968052i	$0.419323\pi$
$920$	−12.9224	−0.426040
$921$	0	0
$922$	4.36269	0.143677
$923$	−53.2957	−1.75425
$924$	0	0
$925$	22.8676	0.751882
$926$	−9.83708	−0.323266
$927$	0	0
$928$	−43.0554	−1.41336
$929$	27.5649	0.904374	0.452187	−	0.891923i	$-0.350644\pi$
$929$	27.5649	0.904374	0.452187	+	0.891923i	$0.350644\pi$
$930$	0	0
$931$	28.9129	0.947581
$932$	48.6659	1.59410
$933$	0	0
$934$	−3.19915	−0.104679
$935$	−0.0243217	−0.000795404 0
$936$	0	0
$937$	−36.2422	−1.18398	−0.591990	−	0.805945i	$-0.701658\pi$
$937$	−36.2422	−1.18398	−0.591990	+	0.805945i	$0.701658\pi$
$938$	19.3608	0.632152
$939$	0	0
$940$	−16.4221	−0.535628
$941$	15.8111	0.515429	0.257714	−	0.966221i	$-0.417031\pi$
$941$	15.8111	0.515429	0.257714	+	0.966221i	$0.417031\pi$
$942$	0	0
$943$	7.94811	0.258826
$944$	−12.2190	−0.397694
$945$	0	0
$946$	0.236302	0.00768283
$947$	15.7512	0.511844	0.255922	−	0.966697i	$-0.417621\pi$
$947$	15.7512	0.511844	0.255922	+	0.966697i	$0.417621\pi$
$948$	0	0
$949$	−6.24771	−0.202809
$950$	−5.23743	−0.169925
$951$	0	0
$952$	3.06157	0.0992260
$953$	−39.2425	−1.27119	−0.635595	−	0.772022i	$-0.719245\pi$
$953$	−39.2425	−1.27119	−0.635595	+	0.772022i	$0.719245\pi$
$954$	0	0
$955$	−6.53893	−0.211595
$956$	−18.1985	−0.588580
$957$	0	0
$958$	10.6254	0.343290
$959$	−4.29744	−0.138771
$960$	0	0
$961$	83.4813	2.69295
$962$	9.42222	0.303784
$963$	0	0
$964$	−2.00173	−0.0644715
$965$	1.64679	0.0530121
$966$	0	0
$967$	41.1325	1.32273	0.661366	−	0.750063i	$-0.269977\pi$
$967$	41.1325	1.32273	0.661366	+	0.750063i	$0.269977\pi$
$968$	17.8301	0.573080
$969$	0	0
$970$	1.21285	0.0389422
$971$	−48.4565	−1.55504	−0.777521	−	0.628857i	$-0.783523\pi$
$971$	−48.4565	−1.55504	−0.777521	+	0.628857i	$0.783523\pi$
$972$	0	0
$973$	15.9063	0.509934
$974$	4.85926	0.155701
$975$	0	0
$976$	−6.20878	−0.198738
$977$	−32.3388	−1.03461	−0.517305	−	0.855801i	$-0.673065\pi$
$977$	−32.3388	−1.03461	−0.517305	+	0.855801i	$0.673065\pi$
$978$	0	0
$979$	0.0958144	0.00306224
$980$	17.2914	0.552353
$981$	0	0
$982$	0.886704	0.0282959
$983$	8.75781	0.279331	0.139665	−	0.990199i	$-0.455397\pi$
$983$	8.75781	0.279331	0.139665	+	0.990199i	$0.455397\pi$
$984$	0	0
$985$	−24.0626	−0.766698
$986$	−1.90517	−0.0606729
$987$	0	0
$988$	21.8001	0.693554
$989$	−85.5061	−2.71893
$990$	0	0
$991$	−54.5038	−1.73137	−0.865685	−	0.500589i	$-0.833117\pi$
$991$	−54.5038	−1.73137	−0.865685	+	0.500589i	$0.833117\pi$
$992$	−48.0983	−1.52712
$993$	0	0
$994$	−23.8208	−0.755549
$995$	13.5392	0.429223
$996$	0	0
$997$	−14.9955	−0.474912	−0.237456	−	0.971398i	$-0.576314\pi$
$997$	−14.9955	−0.474912	−0.237456	+	0.971398i	$0.576314\pi$
$998$	9.25552	0.292978
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6021.2.a.r.1.20	yes	35
3.2	odd	2		6021.2.a.q.1.16	✓	35

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6021.2.a.q.1.16	✓	35	3.2	odd	2
6021.2.a.r.1.20	yes	35	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:	\( N \)	\(=\)	\( 6021 = 3^{3} \cdot 223 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6021.a (trivial)

\(f(q)\)	\(=\)	\(q+0.424438 q^{2} -1.81985 q^{4} -1.02884 q^{5} -4.02930 q^{7} -1.62129 q^{8} +O(q^{10})\)	\(q+0.424438 q^{2} -1.81985 q^{4} -1.02884 q^{5} -4.02930 q^{7} -1.62129 q^{8} -0.436677 q^{10} +0.0504422 q^{11} -3.82631 q^{13} -1.71019 q^{14} +2.95157 q^{16} +0.468656 q^{17} +3.13071 q^{19} +1.87233 q^{20} +0.0214096 q^{22} -7.74707 q^{23} -3.94150 q^{25} -1.62403 q^{26} +7.33273 q^{28} -9.57779 q^{29} -10.6996 q^{31} +4.49534 q^{32} +0.198915 q^{34} +4.14548 q^{35} -5.80175 q^{37} +1.32879 q^{38} +1.66804 q^{40} -1.02595 q^{41} +11.0372 q^{43} -0.0917973 q^{44} -3.28815 q^{46} -8.77093 q^{47} +9.23524 q^{49} -1.67292 q^{50} +6.96331 q^{52} -2.50076 q^{53} -0.0518967 q^{55} +6.53266 q^{56} -4.06518 q^{58} -4.13983 q^{59} -2.10355 q^{61} -4.54131 q^{62} -3.99515 q^{64} +3.93664 q^{65} -11.3209 q^{67} -0.852885 q^{68} +1.75950 q^{70} +13.9288 q^{71} +1.63283 q^{73} -2.46248 q^{74} -5.69743 q^{76} -0.203246 q^{77} +10.3556 q^{79} -3.03668 q^{80} -0.435452 q^{82} -12.6229 q^{83} -0.482170 q^{85} +4.68461 q^{86} -0.0817813 q^{88} +1.89949 q^{89} +15.4173 q^{91} +14.0985 q^{92} -3.72271 q^{94} -3.22098 q^{95} -2.77745 q^{97} +3.91978 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 35 q + 4 q^{2} + 36 q^{4} + 10 q^{5} - 2 q^{7} + 15 q^{8}+O(q^{10}) \) 35 * q + 4 * q^2 + 36 * q^4 + 10 * q^5 - 2 * q^7 + 15 * q^8	\( 35 q + 4 q^{2} + 36 q^{4} + 10 q^{5} - 2 q^{7} + 15 q^{8} - 7 q^{10} + 34 q^{11} + 2 q^{13} + 18 q^{14} + 42 q^{16} + 20 q^{17} - 14 q^{19} + 27 q^{20} + 8 q^{22} + 8 q^{23} + 27 q^{25} + 28 q^{26} - 4 q^{28} + 23 q^{29} + 12 q^{31} + 29 q^{32} - 21 q^{34} + 41 q^{35} - 8 q^{37} + 18 q^{38} + 16 q^{40} + 50 q^{41} + 2 q^{43} + 83 q^{44} - 5 q^{46} + 21 q^{47} + 43 q^{49} + 39 q^{50} + 6 q^{52} + 37 q^{53} + 20 q^{55} + 33 q^{56} - 32 q^{58} + 81 q^{59} - 6 q^{61} + 26 q^{62} - q^{64} + 29 q^{65} + 12 q^{67} + 55 q^{68} + 50 q^{70} + 43 q^{71} - 20 q^{73} + 48 q^{74} - 15 q^{76} + 29 q^{77} + 28 q^{79} + 88 q^{80} - 6 q^{82} + 64 q^{83} - 67 q^{85} + 41 q^{86} + 10 q^{88} + 50 q^{89} + 2 q^{91} + 32 q^{92} + 15 q^{94} + 25 q^{95} + 9 q^{97} + 74 q^{98}+O(q^{100}) \) 35 * q + 4 * q^2 + 36 * q^4 + 10 * q^5 - 2 * q^7 + 15 * q^8 - 7 * q^10 + 34 * q^11 + 2 * q^13 + 18 * q^14 + 42 * q^16 + 20 * q^17 - 14 * q^19 + 27 * q^20 + 8 * q^22 + 8 * q^23 + 27 * q^25 + 28 * q^26 - 4 * q^28 + 23 * q^29 + 12 * q^31 + 29 * q^32 - 21 * q^34 + 41 * q^35 - 8 * q^37 + 18 * q^38 + 16 * q^40 + 50 * q^41 + 2 * q^43 + 83 * q^44 - 5 * q^46 + 21 * q^47 + 43 * q^49 + 39 * q^50 + 6 * q^52 + 37 * q^53 + 20 * q^55 + 33 * q^56 - 32 * q^58 + 81 * q^59 - 6 * q^61 + 26 * q^62 - q^64 + 29 * q^65 + 12 * q^67 + 55 * q^68 + 50 * q^70 + 43 * q^71 - 20 * q^73 + 48 * q^74 - 15 * q^76 + 29 * q^77 + 28 * q^79 + 88 * q^80 - 6 * q^82 + 64 * q^83 - 67 * q^85 + 41 * q^86 + 10 * q^88 + 50 * q^89 + 2 * q^91 + 32 * q^92 + 15 * q^94 + 25 * q^95 + 9 * q^97 + 74 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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