LMFDB - Embedded newform 8019.2.a.a.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8019,2,Mod(1,8019)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8019, 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("8019.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8019 = 3^{6} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8019.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:	$64.0320373809$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 297)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.347296$ of defining polynomial
Character	$\chi$	$=$	8019.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.347296 q^{2} -1.87939 q^{4} +2.41147 q^{5} -2.87939 q^{7} -1.34730 q^{8} +O(q^{10})$	\(q+0.347296 q^{2} -1.87939 q^{4} +2.41147 q^{5} -2.87939 q^{7} -1.34730 q^{8} +0.837496 q^{10} -1.00000 q^{11} -5.71688 q^{13} -1.00000 q^{14} +3.29086 q^{16} +3.65270 q^{17} +6.57398 q^{19} -4.53209 q^{20} -0.347296 q^{22} -3.75877 q^{23} +0.815207 q^{25} -1.98545 q^{26} +5.41147 q^{28} +7.14796 q^{29} -3.45336 q^{31} +3.83750 q^{32} +1.26857 q^{34} -6.94356 q^{35} +6.82295 q^{37} +2.28312 q^{38} -3.24897 q^{40} -4.98545 q^{41} +4.43376 q^{43} +1.87939 q^{44} -1.30541 q^{46} +7.57398 q^{47} +1.29086 q^{49} +0.283119 q^{50} +10.7442 q^{52} -10.7588 q^{53} -2.41147 q^{55} +3.87939 q^{56} +2.48246 q^{58} +6.06418 q^{59} -5.06418 q^{61} -1.19934 q^{62} -5.24897 q^{64} -13.7861 q^{65} +13.6878 q^{67} -6.86484 q^{68} -2.41147 q^{70} -3.92902 q^{71} +2.32770 q^{73} +2.36959 q^{74} -12.3550 q^{76} +2.87939 q^{77} +2.04963 q^{79} +7.93582 q^{80} -1.73143 q^{82} -14.5253 q^{83} +8.80840 q^{85} +1.53983 q^{86} +1.34730 q^{88} -8.83069 q^{89} +16.4611 q^{91} +7.06418 q^{92} +2.63041 q^{94} +15.8530 q^{95} +17.2121 q^{97} +0.448311 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{5} - 3 q^{7} - 3 q^{8}+O(q^{10}) $ 3 * q - 3 * q^5 - 3 * q^7 - 3 * q^8	$ 3 q - 3 q^{5} - 3 q^{7} - 3 q^{8} - 3 q^{11} - 9 q^{13} - 3 q^{14} - 6 q^{16} + 12 q^{17} + 12 q^{19} - 9 q^{20} + 6 q^{25} + 12 q^{26} + 6 q^{28} + 6 q^{29} + 3 q^{31} + 9 q^{32} - 6 q^{34} - 6 q^{35} + 15 q^{38} + 3 q^{40} + 3 q^{41} - 3 q^{43} - 6 q^{46} + 15 q^{47} - 12 q^{49} + 9 q^{50} + 3 q^{52} - 21 q^{53} + 3 q^{55} + 6 q^{56} + 30 q^{58} + 9 q^{59} - 6 q^{61} - 18 q^{62} - 3 q^{64} - 9 q^{65} - 3 q^{67} + 3 q^{68} + 3 q^{70} + 21 q^{71} + 3 q^{73} - 12 q^{76} + 3 q^{77} - 21 q^{79} + 33 q^{80} - 15 q^{82} + 3 q^{83} - 12 q^{85} - 24 q^{86} + 3 q^{88} + 18 q^{89} + 12 q^{91} + 12 q^{92} + 15 q^{94} - 3 q^{95} + 27 q^{97} + 3 q^{98}+O(q^{100}) $ 3 * q - 3 * q^5 - 3 * q^7 - 3 * q^8 - 3 * q^11 - 9 * q^13 - 3 * q^14 - 6 * q^16 + 12 * q^17 + 12 * q^19 - 9 * q^20 + 6 * q^25 + 12 * q^26 + 6 * q^28 + 6 * q^29 + 3 * q^31 + 9 * q^32 - 6 * q^34 - 6 * q^35 + 15 * q^38 + 3 * q^40 + 3 * q^41 - 3 * q^43 - 6 * q^46 + 15 * q^47 - 12 * q^49 + 9 * q^50 + 3 * q^52 - 21 * q^53 + 3 * q^55 + 6 * q^56 + 30 * q^58 + 9 * q^59 - 6 * q^61 - 18 * q^62 - 3 * q^64 - 9 * q^65 - 3 * q^67 + 3 * q^68 + 3 * q^70 + 21 * q^71 + 3 * q^73 - 12 * q^76 + 3 * q^77 - 21 * q^79 + 33 * q^80 - 15 * q^82 + 3 * q^83 - 12 * q^85 - 24 * q^86 + 3 * q^88 + 18 * q^89 + 12 * q^91 + 12 * q^92 + 15 * q^94 - 3 * q^95 + 27 * q^97 + 3 * 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 $

$2$	0.347296	0.245576	0.122788	−	0.992433i	$-0.460817\pi$
$2$	0.347296	0.245576	0.122788	+	0.992433i	$0.460817\pi$
$3$	0	0
$3$	0	0
$4$	−1.87939	−0.939693
$5$	2.41147	1.07844	0.539222	−	0.842164i	$-0.318718\pi$
$5$	2.41147	1.07844	0.539222	+	0.842164i	$0.318718\pi$
$6$	0	0
$7$	−2.87939	−1.08831	−0.544153	−	0.838986i	$-0.683149\pi$
$7$	−2.87939	−1.08831	−0.544153	+	0.838986i	$0.683149\pi$
$8$	−1.34730	−0.476341
$9$	0	0
$10$	0.837496	0.264840
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−5.71688	−1.58558	−0.792789	−	0.609496i	$-0.791372\pi$
$13$	−5.71688	−1.58558	−0.792789	+	0.609496i	$0.791372\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	3.29086	0.822715
$17$	3.65270	0.885911	0.442955	−	0.896544i	$-0.353930\pi$
$17$	3.65270	0.885911	0.442955	+	0.896544i	$0.353930\pi$
$18$	0	0
$19$	6.57398	1.50817	0.754087	−	0.656775i	$-0.228080\pi$
$19$	6.57398	1.50817	0.754087	+	0.656775i	$0.228080\pi$
$20$	−4.53209	−1.01341
$21$	0	0
$22$	−0.347296	−0.0740438
$23$	−3.75877	−0.783758	−0.391879	−	0.920017i	$-0.628175\pi$
$23$	−3.75877	−0.783758	−0.391879	+	0.920017i	$0.628175\pi$
$24$	0	0
$25$	0.815207	0.163041
$26$	−1.98545	−0.389379
$27$	0	0
$28$	5.41147	1.02267
$29$	7.14796	1.32734	0.663671	−	0.748025i	$-0.268997\pi$
$29$	7.14796	1.32734	0.663671	+	0.748025i	$0.268997\pi$
$30$	0	0
$31$	−3.45336	−0.620242	−0.310121	−	0.950697i	$-0.600370\pi$
$31$	−3.45336	−0.620242	−0.310121	+	0.950697i	$0.600370\pi$
$32$	3.83750	0.678380
$33$	0	0
$34$	1.26857	0.217558
$35$	−6.94356	−1.17368
$36$	0	0
$37$	6.82295	1.12169	0.560843	−	0.827922i	$-0.310477\pi$
$37$	6.82295	1.12169	0.560843	+	0.827922i	$0.310477\pi$
$38$	2.28312	0.370371
$39$	0	0
$40$	−3.24897	−0.513707
$41$	−4.98545	−0.778597	−0.389298	−	0.921112i	$-0.627283\pi$
$41$	−4.98545	−0.778597	−0.389298	+	0.921112i	$0.627283\pi$
$42$	0	0
$43$	4.43376	0.676142	0.338071	−	0.941121i	$-0.390226\pi$
$43$	4.43376	0.676142	0.338071	+	0.941121i	$0.390226\pi$
$44$	1.87939	0.283328
$45$	0	0
$46$	−1.30541	−0.192472
$47$	7.57398	1.10478	0.552389	−	0.833586i	$-0.313716\pi$
$47$	7.57398	1.10478	0.552389	+	0.833586i	$0.313716\pi$
$48$	0	0
$49$	1.29086	0.184408
$50$	0.283119	0.0400390
$51$	0	0
$52$	10.7442	1.48996
$53$	−10.7588	−1.47783	−0.738915	−	0.673798i	$-0.764662\pi$
$53$	−10.7588	−1.47783	−0.738915	+	0.673798i	$0.764662\pi$
$54$	0	0
$55$	−2.41147	−0.325163
$56$	3.87939	0.518405
$57$	0	0
$58$	2.48246	0.325963
$59$	6.06418	0.789489	0.394744	−	0.918791i	$-0.370833\pi$
$59$	6.06418	0.789489	0.394744	+	0.918791i	$0.370833\pi$
$60$	0	0
$61$	−5.06418	−0.648402	−0.324201	−	0.945988i	$-0.605095\pi$
$61$	−5.06418	−0.648402	−0.324201	+	0.945988i	$0.605095\pi$
$62$	−1.19934	−0.152316
$63$	0	0
$64$	−5.24897	−0.656121
$65$	−13.7861	−1.70996
$66$	0	0
$67$	13.6878	1.67223	0.836115	−	0.548555i	$-0.184822\pi$
$67$	13.6878	1.67223	0.836115	+	0.548555i	$0.184822\pi$
$68$	−6.86484	−0.832484
$69$	0	0
$70$	−2.41147	−0.288226
$71$	−3.92902	−0.466288	−0.233144	−	0.972442i	$-0.574901\pi$
$71$	−3.92902	−0.466288	−0.233144	+	0.972442i	$0.574901\pi$
$72$	0	0
$73$	2.32770	0.272436	0.136218	−	0.990679i	$-0.456505\pi$
$73$	2.32770	0.272436	0.136218	+	0.990679i	$0.456505\pi$
$74$	2.36959	0.275459
$75$	0	0
$76$	−12.3550	−1.41722
$77$	2.87939	0.328136
$78$	0	0
$79$	2.04963	0.230601	0.115301	−	0.993331i	$-0.463217\pi$
$79$	2.04963	0.230601	0.115301	+	0.993331i	$0.463217\pi$
$80$	7.93582	0.887252
$81$	0	0
$82$	−1.73143	−0.191204
$83$	−14.5253	−1.59436	−0.797178	−	0.603744i	$-0.793675\pi$
$83$	−14.5253	−1.59436	−0.797178	+	0.603744i	$0.793675\pi$
$84$	0	0
$85$	8.80840	0.955405
$86$	1.53983	0.166044
$87$	0	0
$88$	1.34730	0.143622
$89$	−8.83069	−0.936051	−0.468026	−	0.883715i	$-0.655034\pi$
$89$	−8.83069	−0.936051	−0.468026	+	0.883715i	$0.655034\pi$
$90$	0	0
$91$	16.4611	1.72559
$92$	7.06418	0.736491
$93$	0	0
$94$	2.63041	0.271307
$95$	15.8530	1.62648
$96$	0	0
$97$	17.2121	1.74763	0.873814	−	0.486261i	$-0.161640\pi$
$97$	17.2121	1.74763	0.873814	+	0.486261i	$0.161640\pi$
$98$	0.448311	0.0452862
$99$	0	0
$100$	−1.53209	−0.153209
$101$	−9.97090	−0.992142	−0.496071	−	0.868282i	$-0.665224\pi$
$101$	−9.97090	−0.992142	−0.496071	+	0.868282i	$0.665224\pi$
$102$	0	0
$103$	−15.1702	−1.49477	−0.747384	−	0.664392i	$-0.768691\pi$
$103$	−15.1702	−1.49477	−0.747384	+	0.664392i	$0.768691\pi$
$104$	7.70233	0.755276
$105$	0	0
$106$	−3.73648	−0.362919
$107$	−1.40373	−0.135704	−0.0678520	−	0.997695i	$-0.521615\pi$
$107$	−1.40373	−0.135704	−0.0678520	+	0.997695i	$0.521615\pi$
$108$	0	0
$109$	−2.32501	−0.222695	−0.111348	−	0.993782i	$-0.535517\pi$
$109$	−2.32501	−0.222695	−0.111348	+	0.993782i	$0.535517\pi$
$110$	−0.837496	−0.0798521
$111$	0	0
$112$	−9.47565	−0.895365
$113$	−2.14022	−0.201335	−0.100667	−	0.994920i	$-0.532098\pi$
$113$	−2.14022	−0.201335	−0.100667	+	0.994920i	$0.532098\pi$
$114$	0	0
$115$	−9.06418	−0.845239
$116$	−13.4338	−1.24729
$117$	0	0
$118$	2.10607	0.193879
$119$	−10.5175	−0.964141
$120$	0	0
$121$	1.00000	0.0909091
$122$	−1.75877	−0.159232
$123$	0	0
$124$	6.49020	0.582837
$125$	−10.0915	−0.902613
$126$	0	0
$127$	−9.31315	−0.826408	−0.413204	−	0.910638i	$-0.635590\pi$
$127$	−9.31315	−0.826408	−0.413204	+	0.910638i	$0.635590\pi$
$128$	−9.49794	−0.839507
$129$	0	0
$130$	−4.78787	−0.419924
$131$	−11.4757	−1.00263	−0.501316	−	0.865264i	$-0.667151\pi$
$131$	−11.4757	−1.00263	−0.501316	+	0.865264i	$0.667151\pi$
$132$	0	0
$133$	−18.9290	−1.64135
$134$	4.75372	0.410659
$135$	0	0
$136$	−4.92127	−0.421996
$137$	2.06149	0.176125	0.0880625	−	0.996115i	$-0.471932\pi$
$137$	2.06149	0.176125	0.0880625	+	0.996115i	$0.471932\pi$
$138$	0	0
$139$	−5.82976	−0.494473	−0.247237	−	0.968955i	$-0.579523\pi$
$139$	−5.82976	−0.494473	−0.247237	+	0.968955i	$0.579523\pi$
$140$	13.0496	1.10290
$141$	0	0
$142$	−1.36453	−0.114509
$143$	5.71688	0.478070
$144$	0	0
$145$	17.2371	1.43146
$146$	0.808400	0.0669037
$147$	0	0
$148$	−12.8229	−1.05404
$149$	−13.2635	−1.08659	−0.543295	−	0.839542i	$-0.682823\pi$
$149$	−13.2635	−1.08659	−0.543295	+	0.839542i	$0.682823\pi$
$150$	0	0
$151$	−0.495252	−0.0403031	−0.0201515	−	0.999797i	$-0.506415\pi$
$151$	−0.495252	−0.0403031	−0.0201515	+	0.999797i	$0.506415\pi$
$152$	−8.85710	−0.718405
$153$	0	0
$154$	1.00000	0.0805823
$155$	−8.32770	−0.668897
$156$	0	0
$157$	−21.4020	−1.70806	−0.854032	−	0.520221i	$-0.825850\pi$
$157$	−21.4020	−1.70806	−0.854032	+	0.520221i	$0.825850\pi$
$158$	0.711829	0.0566301
$159$	0	0
$160$	9.25402	0.731595
$161$	10.8229	0.852968
$162$	0	0
$163$	−3.29767	−0.258293	−0.129147	−	0.991626i	$-0.541224\pi$
$163$	−3.29767	−0.258293	−0.129147	+	0.991626i	$0.541224\pi$
$164$	9.36959	0.731642
$165$	0	0
$166$	−5.04458	−0.391535
$167$	−13.3473	−1.03284	−0.516422	−	0.856334i	$-0.672737\pi$
$167$	−13.3473	−1.03284	−0.516422	+	0.856334i	$0.672737\pi$
$168$	0	0
$169$	19.6827	1.51406
$170$	3.05913	0.234624
$171$	0	0
$172$	−8.33275	−0.635366
$173$	13.7023	1.04177	0.520885	−	0.853627i	$-0.325602\pi$
$173$	13.7023	1.04177	0.520885	+	0.853627i	$0.325602\pi$
$174$	0	0
$175$	−2.34730	−0.177439
$176$	−3.29086	−0.248058
$177$	0	0
$178$	−3.06687	−0.229871
$179$	18.4415	1.37838	0.689191	−	0.724579i	$-0.257966\pi$
$179$	18.4415	1.37838	0.689191	+	0.724579i	$0.257966\pi$
$180$	0	0
$181$	−12.3105	−0.915029	−0.457515	−	0.889202i	$-0.651260\pi$
$181$	−12.3105	−0.915029	−0.457515	+	0.889202i	$0.651260\pi$
$182$	5.71688	0.423763
$183$	0	0
$184$	5.06418	0.373336
$185$	16.4534	1.20968
$186$	0	0
$187$	−3.65270	−0.267112
$188$	−14.2344	−1.03815
$189$	0	0
$190$	5.50568	0.399424
$191$	9.39961	0.680132	0.340066	−	0.940402i	$-0.389551\pi$
$191$	9.39961	0.680132	0.340066	+	0.940402i	$0.389551\pi$
$192$	0	0
$193$	0.369585	0.0266033	0.0133017	−	0.999912i	$-0.495766\pi$
$193$	0.369585	0.0266033	0.0133017	+	0.999912i	$0.495766\pi$
$194$	5.97771	0.429175
$195$	0	0
$196$	−2.42602	−0.173287
$197$	16.7811	1.19560	0.597800	−	0.801645i	$-0.296041\pi$
$197$	16.7811	1.19560	0.597800	+	0.801645i	$0.296041\pi$
$198$	0	0
$199$	−9.61081	−0.681293	−0.340646	−	0.940192i	$-0.610646\pi$
$199$	−9.61081	−0.681293	−0.340646	+	0.940192i	$0.610646\pi$
$200$	−1.09833	−0.0776634
$201$	0	0
$202$	−3.46286	−0.243646
$203$	−20.5817	−1.44455
$204$	0	0
$205$	−12.0223	−0.839673
$206$	−5.26857	−0.367079
$207$	0	0
$208$	−18.8135	−1.30448
$209$	−6.57398	−0.454732
$210$	0	0
$211$	8.64496	0.595144	0.297572	−	0.954699i	$-0.403823\pi$
$211$	8.64496	0.595144	0.297572	+	0.954699i	$0.403823\pi$
$212$	20.2199	1.38871
$213$	0	0
$214$	−0.487511	−0.0333256
$215$	10.6919	0.729182
$216$	0	0
$217$	9.94356	0.675013
$218$	−0.807467	−0.0546885
$219$	0	0
$220$	4.53209	0.305553
$221$	−20.8821	−1.40468
$222$	0	0
$223$	−24.3901	−1.63328	−0.816642	−	0.577145i	$-0.804167\pi$
$223$	−24.3901	−1.63328	−0.816642	+	0.577145i	$0.804167\pi$
$224$	−11.0496	−0.738284
$225$	0	0
$226$	−0.743289	−0.0494428
$227$	−2.15064	−0.142743	−0.0713716	−	0.997450i	$-0.522738\pi$
$227$	−2.15064	−0.142743	−0.0713716	+	0.997450i	$0.522738\pi$
$228$	0	0
$229$	−19.1480	−1.26533	−0.632666	−	0.774425i	$-0.718040\pi$
$229$	−19.1480	−1.26533	−0.632666	+	0.774425i	$0.718040\pi$
$230$	−3.14796	−0.207570
$231$	0	0
$232$	−9.63041	−0.632268
$233$	−3.91622	−0.256560	−0.128280	−	0.991738i	$-0.540946\pi$
$233$	−3.91622	−0.256560	−0.128280	+	0.991738i	$0.540946\pi$
$234$	0	0
$235$	18.2645	1.19144
$236$	−11.3969	−0.741877
$237$	0	0
$238$	−3.65270	−0.236770
$239$	−23.4020	−1.51375	−0.756874	−	0.653561i	$-0.773274\pi$
$239$	−23.4020	−1.51375	−0.756874	+	0.653561i	$0.773274\pi$
$240$	0	0
$241$	−5.31820	−0.342575	−0.171288	−	0.985221i	$-0.554793\pi$
$241$	−5.31820	−0.342575	−0.171288	+	0.985221i	$0.554793\pi$
$242$	0.347296	0.0223251
$243$	0	0
$244$	9.51754	0.609298
$245$	3.11287	0.198874
$246$	0	0
$247$	−37.5827	−2.39133
$248$	4.65270	0.295447
$249$	0	0
$250$	−3.50475	−0.221660
$251$	−0.120615	−0.00761314	−0.00380657	−	0.999993i	$-0.501212\pi$
$251$	−0.120615	−0.00761314	−0.00380657	+	0.999993i	$0.501212\pi$
$252$	0	0
$253$	3.75877	0.236312
$254$	−3.23442	−0.202946
$255$	0	0
$256$	7.19934	0.449959
$257$	4.25671	0.265526	0.132763	−	0.991148i	$-0.457615\pi$
$257$	4.25671	0.265526	0.132763	+	0.991148i	$0.457615\pi$
$258$	0	0
$259$	−19.6459	−1.22074
$260$	25.9094	1.60683
$261$	0	0
$262$	−3.98545	−0.246222
$263$	11.2148	0.691536	0.345768	−	0.938320i	$-0.387618\pi$
$263$	11.2148	0.691536	0.345768	+	0.938320i	$0.387618\pi$
$264$	0	0
$265$	−25.9445	−1.59376
$266$	−6.57398	−0.403076
$267$	0	0
$268$	−25.7246	−1.57138
$269$	15.1584	0.924223	0.462112	−	0.886822i	$-0.347092\pi$
$269$	15.1584	0.924223	0.462112	+	0.886822i	$0.347092\pi$
$270$	0	0
$271$	−3.83750	−0.233111	−0.116556	−	0.993184i	$-0.537185\pi$
$271$	−3.83750	−0.233111	−0.116556	+	0.993184i	$0.537185\pi$
$272$	12.0205	0.728852
$273$	0	0
$274$	0.715948	0.0432520
$275$	−0.815207	−0.0491589
$276$	0	0
$277$	−13.9736	−0.839592	−0.419796	−	0.907619i	$-0.637898\pi$
$277$	−13.9736	−0.839592	−0.419796	+	0.907619i	$0.637898\pi$
$278$	−2.02465	−0.121431
$279$	0	0
$280$	9.35504	0.559070
$281$	−27.3114	−1.62926	−0.814631	−	0.579980i	$-0.803060\pi$
$281$	−27.3114	−1.62926	−0.814631	+	0.579980i	$0.803060\pi$
$282$	0	0
$283$	3.67499	0.218456	0.109228	−	0.994017i	$-0.465162\pi$
$283$	3.67499	0.218456	0.109228	+	0.994017i	$0.465162\pi$
$284$	7.38413	0.438168
$285$	0	0
$286$	1.98545	0.117402
$287$	14.3550	0.847351
$288$	0	0
$289$	−3.65776	−0.215162
$290$	5.98639	0.351533
$291$	0	0
$292$	−4.37464	−0.256006
$293$	−27.4543	−1.60390	−0.801949	−	0.597393i	$-0.796203\pi$
$293$	−27.4543	−1.60390	−0.801949	+	0.597393i	$0.796203\pi$
$294$	0	0
$295$	14.6236	0.851419
$296$	−9.19253	−0.534305
$297$	0	0
$298$	−4.60637	−0.266840
$299$	21.4884	1.24271
$300$	0	0
$301$	−12.7665	−0.735849
$302$	−0.171999	−0.00989745
$303$	0	0
$304$	21.6340	1.24080
$305$	−12.2121	−0.699265
$306$	0	0
$307$	8.34461	0.476252	0.238126	−	0.971234i	$-0.423467\pi$
$307$	8.34461	0.476252	0.238126	+	0.971234i	$0.423467\pi$
$308$	−5.41147	−0.308347
$309$	0	0
$310$	−2.89218	−0.164265
$311$	19.1607	1.08651	0.543253	−	0.839569i	$-0.317192\pi$
$311$	19.1607	1.08651	0.543253	+	0.839569i	$0.317192\pi$
$312$	0	0
$313$	19.2199	1.08637	0.543186	−	0.839613i	$-0.317218\pi$
$313$	19.2199	1.08637	0.543186	+	0.839613i	$0.317218\pi$
$314$	−7.43283	−0.419459
$315$	0	0
$316$	−3.85204	−0.216694
$317$	−20.8152	−1.16910	−0.584549	−	0.811358i	$-0.698729\pi$
$317$	−20.8152	−1.16910	−0.584549	+	0.811358i	$0.698729\pi$
$318$	0	0
$319$	−7.14796	−0.400209
$320$	−12.6578	−0.707590
$321$	0	0
$322$	3.75877	0.209468
$323$	24.0128	1.33611
$324$	0	0
$325$	−4.66044	−0.258515
$326$	−1.14527	−0.0634305
$327$	0	0
$328$	6.71688	0.370878
$329$	−21.8084	−1.20234
$330$	0	0
$331$	−14.4979	−0.796879	−0.398439	−	0.917195i	$-0.630448\pi$
$331$	−14.4979	−0.796879	−0.398439	+	0.917195i	$0.630448\pi$
$332$	27.2986	1.49821
$333$	0	0
$334$	−4.63547	−0.253641
$335$	33.0077	1.80341
$336$	0	0
$337$	2.99226	0.162999	0.0814994	−	0.996673i	$-0.474029\pi$
$337$	2.99226	0.162999	0.0814994	+	0.996673i	$0.474029\pi$
$338$	6.83574	0.371815
$339$	0	0
$340$	−16.5544	−0.897787
$341$	3.45336	0.187010
$342$	0	0
$343$	16.4388	0.887613
$344$	−5.97359	−0.322075
$345$	0	0
$346$	4.75877	0.255833
$347$	22.8239	1.22525	0.612625	−	0.790374i	$-0.290114\pi$
$347$	22.8239	1.22525	0.612625	+	0.790374i	$0.290114\pi$
$348$	0	0
$349$	21.4611	1.14879	0.574393	−	0.818579i	$-0.305238\pi$
$349$	21.4611	1.14879	0.574393	+	0.818579i	$0.305238\pi$
$350$	−0.815207	−0.0435747
$351$	0	0
$352$	−3.83750	−0.204539
$353$	−25.2968	−1.34642	−0.673208	−	0.739454i	$-0.735084\pi$
$353$	−25.2968	−1.34642	−0.673208	+	0.739454i	$0.735084\pi$
$354$	0	0
$355$	−9.47472	−0.502866
$356$	16.5963	0.879600
$357$	0	0
$358$	6.40467	0.338497
$359$	28.5080	1.50460	0.752299	−	0.658822i	$-0.228945\pi$
$359$	28.5080	1.50460	0.752299	+	0.658822i	$0.228945\pi$
$360$	0	0
$361$	24.2172	1.27459
$362$	−4.27538	−0.224709
$363$	0	0
$364$	−30.9368	−1.62153
$365$	5.61318	0.293807
$366$	0	0
$367$	−19.9044	−1.03900	−0.519500	−	0.854471i	$-0.673882\pi$
$367$	−19.9044	−1.03900	−0.519500	+	0.854471i	$0.673882\pi$
$368$	−12.3696	−0.644809
$369$	0	0
$370$	5.71419	0.297067
$371$	30.9786	1.60833
$372$	0	0
$373$	17.1676	0.888902	0.444451	−	0.895803i	$-0.353399\pi$
$373$	17.1676	0.888902	0.444451	+	0.895803i	$0.353399\pi$
$374$	−1.26857	−0.0655962
$375$	0	0
$376$	−10.2044	−0.526251
$377$	−40.8640	−2.10460
$378$	0	0
$379$	−6.34730	−0.326039	−0.163019	−	0.986623i	$-0.552123\pi$
$379$	−6.34730	−0.326039	−0.163019	+	0.986623i	$0.552123\pi$
$380$	−29.7939	−1.52839
$381$	0	0
$382$	3.26445	0.167024
$383$	20.2044	1.03240	0.516198	−	0.856469i	$-0.327347\pi$
$383$	20.2044	1.03240	0.516198	+	0.856469i	$0.327347\pi$
$384$	0	0
$385$	6.94356	0.353877
$386$	0.128356	0.00653313
$387$	0	0
$388$	−32.3482	−1.64223
$389$	18.7665	0.951500	0.475750	−	0.879581i	$-0.342177\pi$
$389$	18.7665	0.951500	0.475750	+	0.879581i	$0.342177\pi$
$390$	0	0
$391$	−13.7297	−0.694339
$392$	−1.73917	−0.0878414
$393$	0	0
$394$	5.82800	0.293610
$395$	4.94263	0.248691
$396$	0	0
$397$	18.1875	0.912803	0.456402	−	0.889774i	$-0.349138\pi$
$397$	18.1875	0.912803	0.456402	+	0.889774i	$0.349138\pi$
$398$	−3.33780	−0.167309
$399$	0	0
$400$	2.68273	0.134137
$401$	19.8854	0.993028	0.496514	−	0.868029i	$-0.334613\pi$
$401$	19.8854	0.993028	0.496514	+	0.868029i	$0.334613\pi$
$402$	0	0
$403$	19.7425	0.983442
$404$	18.7392	0.932309
$405$	0	0
$406$	−7.14796	−0.354747
$407$	−6.82295	−0.338201
$408$	0	0
$409$	−37.0351	−1.83127	−0.915633	−	0.402014i	$-0.868310\pi$
$409$	−37.0351	−1.83127	−0.915633	+	0.402014i	$0.868310\pi$
$410$	−4.17530	−0.206203
$411$	0	0
$412$	28.5107	1.40462
$413$	−17.4611	−0.859205
$414$	0	0
$415$	−35.0273	−1.71942
$416$	−21.9385	−1.07562
$417$	0	0
$418$	−2.28312	−0.111671
$419$	0.960799	0.0469381	0.0234691	−	0.999725i	$-0.492529\pi$
$419$	0.960799	0.0469381	0.0234691	+	0.999725i	$0.492529\pi$
$420$	0	0
$421$	4.04963	0.197367	0.0986834	−	0.995119i	$-0.468537\pi$
$421$	4.04963	0.197367	0.0986834	+	0.995119i	$0.468537\pi$
$422$	3.00236	0.146153
$423$	0	0
$424$	14.4953	0.703952
$425$	2.97771	0.144440
$426$	0	0
$427$	14.5817	0.705659
$428$	2.63816	0.127520
$429$	0	0
$430$	3.71326	0.179069
$431$	−38.3628	−1.84787	−0.923935	−	0.382550i	$-0.875046\pi$
$431$	−38.3628	−1.84787	−0.923935	+	0.382550i	$0.875046\pi$
$432$	0	0
$433$	−29.9959	−1.44151	−0.720755	−	0.693190i	$-0.756205\pi$
$433$	−29.9959	−1.44151	−0.720755	+	0.693190i	$0.756205\pi$
$434$	3.45336	0.165767
$435$	0	0
$436$	4.36959	0.209265
$437$	−24.7101	−1.18204
$438$	0	0
$439$	−33.0915	−1.57937	−0.789686	−	0.613511i	$-0.789757\pi$
$439$	−33.0915	−1.57937	−0.789686	+	0.613511i	$0.789757\pi$
$440$	3.24897	0.154889
$441$	0	0
$442$	−7.25227	−0.344955
$443$	−20.8452	−0.990387	−0.495194	−	0.868783i	$-0.664903\pi$
$443$	−20.8452	−0.990387	−0.495194	+	0.868783i	$0.664903\pi$
$444$	0	0
$445$	−21.2950	−1.00948
$446$	−8.47060	−0.401095
$447$	0	0
$448$	15.1138	0.714060
$449$	1.14971	0.0542582	0.0271291	−	0.999632i	$-0.491363\pi$
$449$	1.14971	0.0542582	0.0271291	+	0.999632i	$0.491363\pi$
$450$	0	0
$451$	4.98545	0.234756
$452$	4.02229	0.189193
$453$	0	0
$454$	−0.746911	−0.0350543
$455$	39.6955	1.86095
$456$	0	0
$457$	−15.3919	−0.720001	−0.360001	−	0.932952i	$-0.617224\pi$
$457$	−15.3919	−0.720001	−0.360001	+	0.932952i	$0.617224\pi$
$458$	−6.65002	−0.310735
$459$	0	0
$460$	17.0351	0.794265
$461$	−32.1898	−1.49923	−0.749615	−	0.661874i	$-0.769761\pi$
$461$	−32.1898	−1.49923	−0.749615	+	0.661874i	$0.769761\pi$
$462$	0	0
$463$	31.9463	1.48467	0.742334	−	0.670030i	$-0.233719\pi$
$463$	31.9463	1.48467	0.742334	+	0.670030i	$0.233719\pi$
$464$	23.5229	1.09202
$465$	0	0
$466$	−1.36009	−0.0630049
$467$	10.9409	0.506283	0.253142	−	0.967429i	$-0.418536\pi$
$467$	10.9409	0.506283	0.253142	+	0.967429i	$0.418536\pi$
$468$	0	0
$469$	−39.4124	−1.81990
$470$	6.34318	0.292589
$471$	0	0
$472$	−8.17024	−0.376066
$473$	−4.43376	−0.203865
$474$	0	0
$475$	5.35916	0.245895
$476$	19.7665	0.905997
$477$	0	0
$478$	−8.12742	−0.371740
$479$	17.9341	0.819428	0.409714	−	0.912214i	$-0.365628\pi$
$479$	17.9341	0.819428	0.409714	+	0.912214i	$0.365628\pi$
$480$	0	0
$481$	−39.0060	−1.77852
$482$	−1.84699	−0.0841282
$483$	0	0
$484$	−1.87939	−0.0854266
$485$	41.5066	1.88472
$486$	0	0
$487$	1.11381	0.0504714	0.0252357	−	0.999682i	$-0.491966\pi$
$487$	1.11381	0.0504714	0.0252357	+	0.999682i	$0.491966\pi$
$488$	6.82295	0.308860
$489$	0	0
$490$	1.08109	0.0488387
$491$	20.7965	0.938535	0.469267	−	0.883056i	$-0.344518\pi$
$491$	20.7965	0.938535	0.469267	+	0.883056i	$0.344518\pi$
$492$	0	0
$493$	26.1094	1.17591
$494$	−13.0523	−0.587252
$495$	0	0
$496$	−11.3645	−0.510283
$497$	11.3131	0.507464
$498$	0	0
$499$	−14.0651	−0.629641	−0.314820	−	0.949151i	$-0.601944\pi$
$499$	−14.0651	−0.629641	−0.314820	+	0.949151i	$0.601944\pi$
$500$	18.9659	0.848179
$501$	0	0
$502$	−0.0418891	−0.00186960
$503$	4.67499	0.208448	0.104224	−	0.994554i	$-0.466764\pi$
$503$	4.67499	0.208448	0.104224	+	0.994554i	$0.466764\pi$
$504$	0	0
$505$	−24.0446	−1.06997
$506$	1.30541	0.0580324
$507$	0	0
$508$	17.5030	0.776570
$509$	−25.1952	−1.11676	−0.558379	−	0.829586i	$-0.688577\pi$
$509$	−25.1952	−1.11676	−0.558379	+	0.829586i	$0.688577\pi$
$510$	0	0
$511$	−6.70233	−0.296494
$512$	21.4962	0.950006
$513$	0	0
$514$	1.47834	0.0652068
$515$	−36.5827	−1.61202
$516$	0	0
$517$	−7.57398	−0.333103
$518$	−6.82295	−0.299783
$519$	0	0
$520$	18.5740	0.814523
$521$	20.2935	0.889076	0.444538	−	0.895760i	$-0.353368\pi$
$521$	20.2935	0.889076	0.444538	+	0.895760i	$0.353368\pi$
$522$	0	0
$523$	−10.9718	−0.479765	−0.239882	−	0.970802i	$-0.577109\pi$
$523$	−10.9718	−0.479765	−0.239882	+	0.970802i	$0.577109\pi$
$524$	21.5672	0.942166
$525$	0	0
$526$	3.89487	0.169824
$527$	−12.6141	−0.549479
$528$	0	0
$529$	−8.87164	−0.385724
$530$	−9.01043	−0.391388
$531$	0	0
$532$	35.5749	1.54237
$533$	28.5012	1.23453
$534$	0	0
$535$	−3.38507	−0.146349
$536$	−18.4415	−0.796552
$537$	0	0
$538$	5.26445	0.226967
$539$	−1.29086	−0.0556012
$540$	0	0
$541$	3.42839	0.147398	0.0736989	−	0.997281i	$-0.476520\pi$
$541$	3.42839	0.147398	0.0736989	+	0.997281i	$0.476520\pi$
$542$	−1.33275	−0.0572464
$543$	0	0
$544$	14.0172	0.600984
$545$	−5.60670	−0.240164
$546$	0	0
$547$	33.4638	1.43081	0.715404	−	0.698711i	$-0.246243\pi$
$547$	33.4638	1.43081	0.715404	+	0.698711i	$0.246243\pi$
$548$	−3.87433	−0.165503
$549$	0	0
$550$	−0.283119	−0.0120722
$551$	46.9905	2.00186
$552$	0	0
$553$	−5.90167	−0.250965
$554$	−4.85298	−0.206183
$555$	0	0
$556$	10.9564	0.464653
$557$	13.3651	0.566299	0.283150	−	0.959076i	$-0.408621\pi$
$557$	13.3651	0.566299	0.283150	+	0.959076i	$0.408621\pi$
$558$	0	0
$559$	−25.3473	−1.07208
$560$	−22.8503	−0.965601
$561$	0	0
$562$	−9.48515	−0.400107
$563$	−19.4783	−0.820914	−0.410457	−	0.911880i	$-0.634631\pi$
$563$	−19.4783	−0.820914	−0.410457	+	0.911880i	$0.634631\pi$
$564$	0	0
$565$	−5.16107	−0.217128
$566$	1.27631	0.0536474
$567$	0	0
$568$	5.29355	0.222112
$569$	−12.7192	−0.533219	−0.266609	−	0.963805i	$-0.585903\pi$
$569$	−12.7192	−0.533219	−0.266609	+	0.963805i	$0.585903\pi$
$570$	0	0
$571$	−12.9044	−0.540031	−0.270015	−	0.962856i	$-0.587029\pi$
$571$	−12.9044	−0.540031	−0.270015	+	0.962856i	$0.587029\pi$
$572$	−10.7442	−0.449239
$573$	0	0
$574$	4.98545	0.208089
$575$	−3.06418	−0.127785
$576$	0	0
$577$	−3.84936	−0.160251	−0.0801254	−	0.996785i	$-0.525532\pi$
$577$	−3.84936	−0.160251	−0.0801254	+	0.996785i	$0.525532\pi$
$578$	−1.27033	−0.0528386
$579$	0	0
$580$	−32.3952	−1.34514
$581$	41.8239	1.73515
$582$	0	0
$583$	10.7588	0.445583
$584$	−3.13610	−0.129773
$585$	0	0
$586$	−9.53478	−0.393878
$587$	15.5571	0.642109	0.321054	−	0.947061i	$-0.395963\pi$
$587$	15.5571	0.642109	0.321054	+	0.947061i	$0.395963\pi$
$588$	0	0
$589$	−22.7023	−0.935433
$590$	5.07873	0.209088
$591$	0	0
$592$	22.4534	0.922828
$593$	−10.5321	−0.432501	−0.216251	−	0.976338i	$-0.569383\pi$
$593$	−10.5321	−0.432501	−0.216251	+	0.976338i	$0.569383\pi$
$594$	0	0
$595$	−25.3628	−1.03977
$596$	24.9273	1.02106
$597$	0	0
$598$	7.46286	0.305179
$599$	−20.2517	−0.827460	−0.413730	−	0.910400i	$-0.635774\pi$
$599$	−20.2517	−0.827460	−0.413730	+	0.910400i	$0.635774\pi$
$600$	0	0
$601$	−28.6973	−1.17059	−0.585293	−	0.810822i	$-0.699021\pi$
$601$	−28.6973	−1.17059	−0.585293	+	0.810822i	$0.699021\pi$
$602$	−4.43376	−0.180707
$603$	0	0
$604$	0.930770	0.0378725
$605$	2.41147	0.0980404
$606$	0	0
$607$	−7.81614	−0.317247	−0.158624	−	0.987339i	$-0.550706\pi$
$607$	−7.81614	−0.317247	−0.158624	+	0.987339i	$0.550706\pi$
$608$	25.2276	1.02311
$609$	0	0
$610$	−4.24123	−0.171722
$611$	−43.2995	−1.75171
$612$	0	0
$613$	−13.5808	−0.548523	−0.274261	−	0.961655i	$-0.588433\pi$
$613$	−13.5808	−0.548523	−0.274261	+	0.961655i	$0.588433\pi$
$614$	2.89805	0.116956
$615$	0	0
$616$	−3.87939	−0.156305
$617$	4.97771	0.200395	0.100198	−	0.994968i	$-0.468053\pi$
$617$	4.97771	0.200395	0.100198	+	0.994968i	$0.468053\pi$
$618$	0	0
$619$	6.37733	0.256326	0.128163	−	0.991753i	$-0.459092\pi$
$619$	6.37733	0.256326	0.128163	+	0.991753i	$0.459092\pi$
$620$	15.6509	0.628557
$621$	0	0
$622$	6.65446	0.266819
$623$	25.4270	1.01871
$624$	0	0
$625$	−28.4115	−1.13646
$626$	6.67499	0.266786
$627$	0	0
$628$	40.2226	1.60505
$629$	24.9222	0.993714
$630$	0	0
$631$	−13.0710	−0.520348	−0.260174	−	0.965562i	$-0.583780\pi$
$631$	−13.0710	−0.520348	−0.260174	+	0.965562i	$0.583780\pi$
$632$	−2.76146	−0.109845
$633$	0	0
$634$	−7.22905	−0.287102
$635$	−22.4584	−0.891235
$636$	0	0
$637$	−7.37969	−0.292394
$638$	−2.48246	−0.0982815
$639$	0	0
$640$	−22.9040	−0.905362
$641$	10.0291	0.396125	0.198063	−	0.980189i	$-0.436535\pi$
$641$	10.0291	0.396125	0.198063	+	0.980189i	$0.436535\pi$
$642$	0	0
$643$	30.3337	1.19624	0.598122	−	0.801405i	$-0.295914\pi$
$643$	30.3337	1.19624	0.598122	+	0.801405i	$0.295914\pi$
$644$	−20.3405	−0.801528
$645$	0	0
$646$	8.33956	0.328115
$647$	−15.8716	−0.623979	−0.311989	−	0.950086i	$-0.600995\pi$
$647$	−15.8716	−0.623979	−0.311989	+	0.950086i	$0.600995\pi$
$648$	0	0
$649$	−6.06418	−0.238040
$650$	−1.61856	−0.0634850
$651$	0	0
$652$	6.19759	0.242716
$653$	−23.9813	−0.938462	−0.469231	−	0.883075i	$-0.655469\pi$
$653$	−23.9813	−0.938462	−0.469231	+	0.883075i	$0.655469\pi$
$654$	0	0
$655$	−27.6732	−1.08128
$656$	−16.4064	−0.640563
$657$	0	0
$658$	−7.57398	−0.295264
$659$	−12.4561	−0.485219	−0.242609	−	0.970124i	$-0.578003\pi$
$659$	−12.4561	−0.485219	−0.242609	+	0.970124i	$0.578003\pi$
$660$	0	0
$661$	5.95367	0.231571	0.115785	−	0.993274i	$-0.463062\pi$
$661$	5.95367	0.231571	0.115785	+	0.993274i	$0.463062\pi$
$662$	−5.03508	−0.195694
$663$	0	0
$664$	19.5699	0.759458
$665$	−45.6468	−1.77011
$666$	0	0
$667$	−26.8675	−1.04031
$668$	25.0847	0.970557
$669$	0	0
$670$	11.4635	0.442872
$671$	5.06418	0.195500
$672$	0	0
$673$	−8.26083	−0.318432	−0.159216	−	0.987244i	$-0.550897\pi$
$673$	−8.26083	−0.318432	−0.159216	+	0.987244i	$0.550897\pi$
$674$	1.03920	0.0400285
$675$	0	0
$676$	−36.9914	−1.42275
$677$	−10.3645	−0.398341	−0.199171	−	0.979965i	$-0.563825\pi$
$677$	−10.3645	−0.398341	−0.199171	+	0.979965i	$0.563825\pi$
$678$	0	0
$679$	−49.5604	−1.90195
$680$	−11.8675	−0.455099
$681$	0	0
$682$	1.19934	0.0459251
$683$	47.4698	1.81638	0.908190	−	0.418558i	$-0.137464\pi$
$683$	47.4698	1.81638	0.908190	+	0.418558i	$0.137464\pi$
$684$	0	0
$685$	4.97123	0.189941
$686$	5.70914	0.217976
$687$	0	0
$688$	14.5909	0.556272
$689$	61.5066	2.34322
$690$	0	0
$691$	−19.5808	−0.744888	−0.372444	−	0.928055i	$-0.621480\pi$
$691$	−19.5808	−0.744888	−0.372444	+	0.928055i	$0.621480\pi$
$692$	−25.7520	−0.978943
$693$	0	0
$694$	7.92665	0.300892
$695$	−14.0583	−0.533262
$696$	0	0
$697$	−18.2104	−0.689767
$698$	7.45336	0.282114
$699$	0	0
$700$	4.41147	0.166738
$701$	−18.9317	−0.715041	−0.357520	−	0.933905i	$-0.616378\pi$
$701$	−18.9317	−0.715041	−0.357520	+	0.933905i	$0.616378\pi$
$702$	0	0
$703$	44.8539	1.69170
$704$	5.24897	0.197828
$705$	0	0
$706$	−8.78550	−0.330647
$707$	28.7101	1.07975
$708$	0	0
$709$	31.2739	1.17452	0.587259	−	0.809399i	$-0.300207\pi$
$709$	31.2739	1.17452	0.587259	+	0.809399i	$0.300207\pi$
$710$	−3.29054	−0.123492
$711$	0	0
$712$	11.8976	0.445880
$713$	12.9804	0.486120
$714$	0	0
$715$	13.7861	0.515571
$716$	−34.6587	−1.29526
$717$	0	0
$718$	9.90074	0.369492
$719$	−19.9281	−0.743192	−0.371596	−	0.928395i	$-0.621189\pi$
$719$	−19.9281	−0.743192	−0.371596	+	0.928395i	$0.621189\pi$
$720$	0	0
$721$	43.6810	1.62676
$722$	8.41054	0.313008
$723$	0	0
$724$	23.1361	0.859846
$725$	5.82707	0.216412
$726$	0	0
$727$	−28.5699	−1.05960	−0.529799	−	0.848123i	$-0.677733\pi$
$727$	−28.5699	−1.05960	−0.529799	+	0.848123i	$0.677733\pi$
$728$	−22.1780	−0.821971
$729$	0	0
$730$	1.94944	0.0721519
$731$	16.1952	0.599002
$732$	0	0
$733$	−5.15301	−0.190331	−0.0951654	−	0.995461i	$-0.530338\pi$
$733$	−5.15301	−0.190331	−0.0951654	+	0.995461i	$0.530338\pi$
$734$	−6.91271	−0.255153
$735$	0	0
$736$	−14.4243	−0.531686
$737$	−13.6878	−0.504196
$738$	0	0
$739$	−0.650949	−0.0239455	−0.0119728	−	0.999928i	$-0.503811\pi$
$739$	−0.650949	−0.0239455	−0.0119728	+	0.999928i	$0.503811\pi$
$740$	−30.9222	−1.13672
$741$	0	0
$742$	10.7588	0.394967
$743$	26.2746	0.963920	0.481960	−	0.876193i	$-0.339925\pi$
$743$	26.2746	0.963920	0.481960	+	0.876193i	$0.339925\pi$
$744$	0	0
$745$	−31.9846	−1.17183
$746$	5.96223	0.218293
$747$	0	0
$748$	6.86484	0.251003
$749$	4.04189	0.147687
$750$	0	0
$751$	−17.3814	−0.634258	−0.317129	−	0.948382i	$-0.602719\pi$
$751$	−17.3814	−0.634258	−0.317129	+	0.948382i	$0.602719\pi$
$752$	24.9249	0.908917
$753$	0	0
$754$	−14.1919	−0.516839
$755$	−1.19429	−0.0434646
$756$	0	0
$757$	44.7861	1.62778	0.813889	−	0.581020i	$-0.197346\pi$
$757$	44.7861	1.62778	0.813889	+	0.581020i	$0.197346\pi$
$758$	−2.20439	−0.0800672
$759$	0	0
$760$	−21.3587	−0.774760
$761$	−28.3969	−1.02939	−0.514694	−	0.857374i	$-0.672094\pi$
$761$	−28.3969	−1.02939	−0.514694	+	0.857374i	$0.672094\pi$
$762$	0	0
$763$	6.69459	0.242361
$764$	−17.6655	−0.639115
$765$	0	0
$766$	7.01691	0.253531
$767$	−34.6682	−1.25180
$768$	0	0
$769$	−49.8462	−1.79750	−0.898749	−	0.438463i	$-0.855523\pi$
$769$	−49.8462	−1.79750	−0.898749	+	0.438463i	$0.855523\pi$
$770$	2.41147	0.0869035
$771$	0	0
$772$	−0.694593	−0.0249989
$773$	−32.2959	−1.16160	−0.580802	−	0.814045i	$-0.697261\pi$
$773$	−32.2959	−1.16160	−0.580802	+	0.814045i	$0.697261\pi$
$774$	0	0
$775$	−2.81521	−0.101125
$776$	−23.1898	−0.832467
$777$	0	0
$778$	6.51754	0.233665
$779$	−32.7743	−1.17426
$780$	0	0
$781$	3.92902	0.140591
$782$	−4.76827	−0.170513
$783$	0	0
$784$	4.24804	0.151716
$785$	−51.6103	−1.84205
$786$	0	0
$787$	35.9317	1.28083	0.640413	−	0.768030i	$-0.278763\pi$
$787$	35.9317	1.28083	0.640413	+	0.768030i	$0.278763\pi$
$788$	−31.5381	−1.12350
$789$	0	0
$790$	1.71656	0.0610724
$791$	6.16250	0.219113
$792$	0	0
$793$	28.9513	1.02809
$794$	6.31645	0.224162
$795$	0	0
$796$	18.0624	0.640206
$797$	−27.1516	−0.961758	−0.480879	−	0.876787i	$-0.659682\pi$
$797$	−27.1516	−0.961758	−0.480879	+	0.876787i	$0.659682\pi$
$798$	0	0
$799$	27.6655	0.978735
$800$	3.12836	0.110604
$801$	0	0
$802$	6.90612	0.243863
$803$	−2.32770	−0.0821426
$804$	0	0
$805$	26.0993	0.919878
$806$	6.85649	0.241509
$807$	0	0
$808$	13.4338	0.472598
$809$	−47.9728	−1.68663	−0.843316	−	0.537417i	$-0.819400\pi$
$809$	−47.9728	−1.68663	−0.843316	+	0.537417i	$0.819400\pi$
$810$	0	0
$811$	10.7229	0.376531	0.188265	−	0.982118i	$-0.439713\pi$
$811$	10.7229	0.376531	0.188265	+	0.982118i	$0.439713\pi$
$812$	38.6810	1.35744
$813$	0	0
$814$	−2.36959	−0.0830539
$815$	−7.95224	−0.278555
$816$	0	0
$817$	29.1475	1.01974
$818$	−12.8621	−0.449714
$819$	0	0
$820$	22.5945	0.789035
$821$	7.53807	0.263081	0.131540	−	0.991311i	$-0.458008\pi$
$821$	7.53807	0.263081	0.131540	+	0.991311i	$0.458008\pi$
$822$	0	0
$823$	31.5111	1.09841	0.549203	−	0.835689i	$-0.314931\pi$
$823$	31.5111	1.09841	0.549203	+	0.835689i	$0.314931\pi$
$824$	20.4388	0.712020
$825$	0	0
$826$	−6.06418	−0.211000
$827$	3.61411	0.125675	0.0628375	−	0.998024i	$-0.479985\pi$
$827$	3.61411	0.125675	0.0628375	+	0.998024i	$0.479985\pi$
$828$	0	0
$829$	−24.9581	−0.866831	−0.433416	−	0.901194i	$-0.642692\pi$
$829$	−24.9581	−0.866831	−0.433416	+	0.901194i	$0.642692\pi$
$830$	−12.1649	−0.422249
$831$	0	0
$832$	30.0077	1.04033
$833$	4.71513	0.163369
$834$	0	0
$835$	−32.1867	−1.11387
$836$	12.3550	0.427308
$837$	0	0
$838$	0.333682	0.0115269
$839$	−31.9198	−1.10200	−0.550998	−	0.834507i	$-0.685753\pi$
$839$	−31.9198	−1.10200	−0.550998	+	0.834507i	$0.685753\pi$
$840$	0	0
$841$	22.0933	0.761837
$842$	1.40642	0.0484685
$843$	0	0
$844$	−16.2472	−0.559252
$845$	47.4644	1.63283
$846$	0	0
$847$	−2.87939	−0.0989368
$848$	−35.4056	−1.21583
$849$	0	0
$850$	1.03415	0.0354710
$851$	−25.6459	−0.879130
$852$	0	0
$853$	−33.0145	−1.13040	−0.565198	−	0.824955i	$-0.691200\pi$
$853$	−33.0145	−1.13040	−0.565198	+	0.824955i	$0.691200\pi$
$854$	5.06418	0.173293
$855$	0	0
$856$	1.89124	0.0646414
$857$	−51.1884	−1.74856	−0.874281	−	0.485419i	$-0.838667\pi$
$857$	−51.1884	−1.74856	−0.874281	+	0.485419i	$0.838667\pi$
$858$	0	0
$859$	−2.68416	−0.0915825	−0.0457912	−	0.998951i	$-0.514581\pi$
$859$	−2.68416	−0.0915825	−0.0457912	+	0.998951i	$0.514581\pi$
$860$	−20.0942	−0.685207
$861$	0	0
$862$	−13.3233	−0.453792
$863$	−50.3432	−1.71370	−0.856851	−	0.515564i	$-0.827582\pi$
$863$	−50.3432	−1.71370	−0.856851	+	0.515564i	$0.827582\pi$
$864$	0	0
$865$	33.0428	1.12349
$866$	−10.4175	−0.354000
$867$	0	0
$868$	−18.6878	−0.634305
$869$	−2.04963	−0.0695289
$870$	0	0
$871$	−78.2514	−2.65145
$872$	3.13247	0.106079
$873$	0	0
$874$	−8.58172	−0.290281
$875$	29.0574	0.982318
$876$	0	0
$877$	0.325008	0.0109747	0.00548736	−	0.999985i	$-0.498253\pi$
$877$	0.325008	0.0109747	0.00548736	+	0.999985i	$0.498253\pi$
$878$	−11.4926	−0.387855
$879$	0	0
$880$	−7.93582	−0.267517
$881$	32.1320	1.08255	0.541277	−	0.840844i	$-0.317941\pi$
$881$	32.1320	1.08255	0.541277	+	0.840844i	$0.317941\pi$
$882$	0	0
$883$	48.9359	1.64683	0.823413	−	0.567443i	$-0.192067\pi$
$883$	48.9359	1.64683	0.823413	+	0.567443i	$0.192067\pi$
$884$	39.2455	1.31997
$885$	0	0
$886$	−7.23947	−0.243215
$887$	7.62361	0.255976	0.127988	−	0.991776i	$-0.459148\pi$
$887$	7.62361	0.255976	0.127988	+	0.991776i	$0.459148\pi$
$888$	0	0
$889$	26.8161	0.899385
$890$	−7.39567	−0.247903
$891$	0	0
$892$	45.8384	1.53478
$893$	49.7912	1.66620
$894$	0	0
$895$	44.4712	1.48651
$896$	27.3482	0.913640
$897$	0	0
$898$	0.399290	0.0133245
$899$	−24.6845	−0.823274
$900$	0	0
$901$	−39.2986	−1.30923
$902$	1.73143	0.0576503
$903$	0	0
$904$	2.88350	0.0959039
$905$	−29.6864	−0.986808
$906$	0	0
$907$	−22.1539	−0.735610	−0.367805	−	0.929903i	$-0.619891\pi$
$907$	−22.1539	−0.735610	−0.367805	+	0.929903i	$0.619891\pi$
$908$	4.04189	0.134135
$909$	0	0
$910$	13.7861	0.457005
$911$	−11.6313	−0.385364	−0.192682	−	0.981261i	$-0.561719\pi$
$911$	−11.6313	−0.385364	−0.192682	+	0.981261i	$0.561719\pi$
$912$	0	0
$913$	14.5253	0.480717
$914$	−5.34554	−0.176815
$915$	0	0
$916$	35.9864	1.18902
$917$	33.0428	1.09117
$918$	0	0
$919$	−47.1029	−1.55378	−0.776890	−	0.629636i	$-0.783204\pi$
$919$	−47.1029	−1.55378	−0.776890	+	0.629636i	$0.783204\pi$
$920$	12.2121	0.402622
$921$	0	0
$922$	−11.1794	−0.368174
$923$	22.4617	0.739336
$924$	0	0
$925$	5.56212	0.182881
$926$	11.0948	0.364598
$927$	0	0
$928$	27.4303	0.900442
$929$	−40.5895	−1.33170	−0.665848	−	0.746087i	$-0.731930\pi$
$929$	−40.5895	−1.33170	−0.665848	+	0.746087i	$0.731930\pi$
$930$	0	0
$931$	8.48608	0.278120
$932$	7.36009	0.241088
$933$	0	0
$934$	3.79973	0.124331
$935$	−8.80840	−0.288065
$936$	0	0
$937$	−51.3296	−1.67686	−0.838432	−	0.545006i	$-0.816527\pi$
$937$	−51.3296	−1.67686	−0.838432	+	0.545006i	$0.816527\pi$
$938$	−13.6878	−0.446922
$939$	0	0
$940$	−34.3259	−1.11959
$941$	34.1780	1.11417	0.557085	−	0.830455i	$-0.311920\pi$
$941$	34.1780	1.11417	0.557085	+	0.830455i	$0.311920\pi$
$942$	0	0
$943$	18.7392	0.610231
$944$	19.9564	0.649524
$945$	0	0
$946$	−1.53983	−0.0500642
$947$	33.0300	1.07333	0.536666	−	0.843795i	$-0.319684\pi$
$947$	33.0300	1.07333	0.536666	+	0.843795i	$0.319684\pi$
$948$	0	0
$949$	−13.3072	−0.431969
$950$	1.86122	0.0603858
$951$	0	0
$952$	14.1702	0.459260
$953$	24.3301	0.788128	0.394064	−	0.919083i	$-0.371069\pi$
$953$	24.3301	0.788128	0.394064	+	0.919083i	$0.371069\pi$
$954$	0	0
$955$	22.6669	0.733485
$956$	43.9813	1.42246
$957$	0	0
$958$	6.22844	0.201232
$959$	−5.93582	−0.191678
$960$	0	0
$961$	−19.0743	−0.615299
$962$	−13.5466	−0.436761
$963$	0	0
$964$	9.99495	0.321916
$965$	0.891245	0.0286902
$966$	0	0
$967$	17.3601	0.558263	0.279131	−	0.960253i	$-0.409953\pi$
$967$	17.3601	0.558263	0.279131	+	0.960253i	$0.409953\pi$
$968$	−1.34730	−0.0433037
$969$	0	0
$970$	14.4151	0.462841
$971$	35.7796	1.14822	0.574111	−	0.818777i	$-0.305348\pi$
$971$	35.7796	1.14822	0.574111	+	0.818777i	$0.305348\pi$
$972$	0	0
$973$	16.7861	0.538138
$974$	0.386821	0.0123946
$975$	0	0
$976$	−16.6655	−0.533450
$977$	−7.94087	−0.254051	−0.127026	−	0.991899i	$-0.540543\pi$
$977$	−7.94087	−0.254051	−0.127026	+	0.991899i	$0.540543\pi$
$978$	0	0
$979$	8.83069	0.282230
$980$	−5.85029	−0.186881
$981$	0	0
$982$	7.22256	0.230481
$983$	52.3773	1.67058	0.835289	−	0.549812i	$-0.185301\pi$
$983$	52.3773	1.67058	0.835289	+	0.549812i	$0.185301\pi$
$984$	0	0
$985$	40.4671	1.28939
$986$	9.06769	0.288774
$987$	0	0
$988$	70.6323	2.24711
$989$	−16.6655	−0.529932
$990$	0	0
$991$	40.4671	1.28548	0.642740	−	0.766085i	$-0.277798\pi$
$991$	40.4671	1.28548	0.642740	+	0.766085i	$0.277798\pi$
$992$	−13.2523	−0.420760
$993$	0	0
$994$	3.92902	0.124621
$995$	−23.1762	−0.734736
$996$	0	0
$997$	10.4929	0.332313	0.166157	−	0.986099i	$-0.446864\pi$
$997$	10.4929	0.332313	0.166157	+	0.986099i	$0.446864\pi$
$998$	−4.88476	−0.154624
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8019.2.a.a.1.2		3
3.2	odd	2		8019.2.a.b.1.2		3
27.4	even	9		297.2.j.a.232.1	✓	6
27.7	even	9		297.2.j.a.265.1	yes	6
27.20	odd	18		891.2.j.a.199.1		6
27.23	odd	18		891.2.j.a.694.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
297.2.j.a.232.1	✓	6	27.4	even	9
297.2.j.a.265.1	yes	6	27.7	even	9
891.2.j.a.199.1		6	27.20	odd	18
891.2.j.a.694.1		6	27.23	odd	18
8019.2.a.a.1.2		3	1.1	even	1	trivial
8019.2.a.b.1.2		3	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8019 = 3^{6} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8019.a (trivial)

\(f(q)\)	\(=\)	\(q+0.347296 q^{2} -1.87939 q^{4} +2.41147 q^{5} -2.87939 q^{7} -1.34730 q^{8} +O(q^{10})\)	\(q+0.347296 q^{2} -1.87939 q^{4} +2.41147 q^{5} -2.87939 q^{7} -1.34730 q^{8} +0.837496 q^{10} -1.00000 q^{11} -5.71688 q^{13} -1.00000 q^{14} +3.29086 q^{16} +3.65270 q^{17} +6.57398 q^{19} -4.53209 q^{20} -0.347296 q^{22} -3.75877 q^{23} +0.815207 q^{25} -1.98545 q^{26} +5.41147 q^{28} +7.14796 q^{29} -3.45336 q^{31} +3.83750 q^{32} +1.26857 q^{34} -6.94356 q^{35} +6.82295 q^{37} +2.28312 q^{38} -3.24897 q^{40} -4.98545 q^{41} +4.43376 q^{43} +1.87939 q^{44} -1.30541 q^{46} +7.57398 q^{47} +1.29086 q^{49} +0.283119 q^{50} +10.7442 q^{52} -10.7588 q^{53} -2.41147 q^{55} +3.87939 q^{56} +2.48246 q^{58} +6.06418 q^{59} -5.06418 q^{61} -1.19934 q^{62} -5.24897 q^{64} -13.7861 q^{65} +13.6878 q^{67} -6.86484 q^{68} -2.41147 q^{70} -3.92902 q^{71} +2.32770 q^{73} +2.36959 q^{74} -12.3550 q^{76} +2.87939 q^{77} +2.04963 q^{79} +7.93582 q^{80} -1.73143 q^{82} -14.5253 q^{83} +8.80840 q^{85} +1.53983 q^{86} +1.34730 q^{88} -8.83069 q^{89} +16.4611 q^{91} +7.06418 q^{92} +2.63041 q^{94} +15.8530 q^{95} +17.2121 q^{97} +0.448311 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{5} - 3 q^{7} - 3 q^{8}+O(q^{10}) \) 3 * q - 3 * q^5 - 3 * q^7 - 3 * q^8	\( 3 q - 3 q^{5} - 3 q^{7} - 3 q^{8} - 3 q^{11} - 9 q^{13} - 3 q^{14} - 6 q^{16} + 12 q^{17} + 12 q^{19} - 9 q^{20} + 6 q^{25} + 12 q^{26} + 6 q^{28} + 6 q^{29} + 3 q^{31} + 9 q^{32} - 6 q^{34} - 6 q^{35} + 15 q^{38} + 3 q^{40} + 3 q^{41} - 3 q^{43} - 6 q^{46} + 15 q^{47} - 12 q^{49} + 9 q^{50} + 3 q^{52} - 21 q^{53} + 3 q^{55} + 6 q^{56} + 30 q^{58} + 9 q^{59} - 6 q^{61} - 18 q^{62} - 3 q^{64} - 9 q^{65} - 3 q^{67} + 3 q^{68} + 3 q^{70} + 21 q^{71} + 3 q^{73} - 12 q^{76} + 3 q^{77} - 21 q^{79} + 33 q^{80} - 15 q^{82} + 3 q^{83} - 12 q^{85} - 24 q^{86} + 3 q^{88} + 18 q^{89} + 12 q^{91} + 12 q^{92} + 15 q^{94} - 3 q^{95} + 27 q^{97} + 3 q^{98}+O(q^{100}) \) 3 * q - 3 * q^5 - 3 * q^7 - 3 * q^8 - 3 * q^11 - 9 * q^13 - 3 * q^14 - 6 * q^16 + 12 * q^17 + 12 * q^19 - 9 * q^20 + 6 * q^25 + 12 * q^26 + 6 * q^28 + 6 * q^29 + 3 * q^31 + 9 * q^32 - 6 * q^34 - 6 * q^35 + 15 * q^38 + 3 * q^40 + 3 * q^41 - 3 * q^43 - 6 * q^46 + 15 * q^47 - 12 * q^49 + 9 * q^50 + 3 * q^52 - 21 * q^53 + 3 * q^55 + 6 * q^56 + 30 * q^58 + 9 * q^59 - 6 * q^61 - 18 * q^62 - 3 * q^64 - 9 * q^65 - 3 * q^67 + 3 * q^68 + 3 * q^70 + 21 * q^71 + 3 * q^73 - 12 * q^76 + 3 * q^77 - 21 * q^79 + 33 * q^80 - 15 * q^82 + 3 * q^83 - 12 * q^85 - 24 * q^86 + 3 * q^88 + 18 * q^89 + 12 * q^91 + 12 * q^92 + 15 * q^94 - 3 * q^95 + 27 * q^97 + 3 * q^98

Introduction

L-functions

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