LMFDB - Embedded newform 8019.2.a.k.1.12

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:	$0$
Dimension:	$51$
Twist minimal:	no (minimal twist has level 297)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
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)

$f(q)$	$=$	$q-1.75810 q^{2} +1.09093 q^{4} -0.0727216 q^{5} -2.24688 q^{7} +1.59824 q^{8} +O(q^{10})$	\(q-1.75810 q^{2} +1.09093 q^{4} -0.0727216 q^{5} -2.24688 q^{7} +1.59824 q^{8} +0.127852 q^{10} -1.00000 q^{11} +2.64801 q^{13} +3.95024 q^{14} -4.99173 q^{16} -7.69499 q^{17} -7.20375 q^{19} -0.0793340 q^{20} +1.75810 q^{22} +1.90545 q^{23} -4.99471 q^{25} -4.65548 q^{26} -2.45118 q^{28} -8.89496 q^{29} +0.137940 q^{31} +5.57950 q^{32} +13.5286 q^{34} +0.163396 q^{35} -10.8258 q^{37} +12.6649 q^{38} -0.116227 q^{40} -10.4427 q^{41} +5.64634 q^{43} -1.09093 q^{44} -3.34998 q^{46} -1.58582 q^{47} -1.95155 q^{49} +8.78122 q^{50} +2.88879 q^{52} -0.449984 q^{53} +0.0727216 q^{55} -3.59105 q^{56} +15.6383 q^{58} -4.16759 q^{59} +10.0797 q^{61} -0.242513 q^{62} +0.174134 q^{64} -0.192568 q^{65} -1.39463 q^{67} -8.39467 q^{68} -0.287268 q^{70} -5.39124 q^{71} +8.05143 q^{73} +19.0330 q^{74} -7.85877 q^{76} +2.24688 q^{77} +13.1394 q^{79} +0.363007 q^{80} +18.3593 q^{82} +10.3147 q^{83} +0.559592 q^{85} -9.92685 q^{86} -1.59824 q^{88} -13.5895 q^{89} -5.94975 q^{91} +2.07871 q^{92} +2.78803 q^{94} +0.523868 q^{95} -5.37188 q^{97} +3.43103 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 51 q + 60 q^{4} - 6 q^{5} + 12 q^{7}+O(q^{10}) $ 51 * q + 60 * q^4 - 6 * q^5 + 12 * q^7	$ 51 q + 60 q^{4} - 6 q^{5} + 12 q^{7} + 18 q^{10} - 51 q^{11} + 30 q^{13} - 12 q^{14} + 78 q^{16} + 30 q^{19} - 18 q^{20} - 3 q^{23} + 75 q^{25} - 9 q^{26} + 36 q^{28} + 42 q^{31} + 15 q^{32} + 42 q^{34} + 9 q^{35} + 48 q^{37} + 3 q^{38} + 54 q^{40} + 18 q^{43} - 60 q^{44} + 42 q^{46} - 30 q^{47} + 99 q^{49} + 30 q^{50} + 60 q^{52} - 18 q^{53} + 6 q^{55} - 21 q^{56} + 30 q^{58} - 24 q^{59} + 99 q^{61} + 114 q^{64} + 15 q^{65} + 39 q^{67} + 39 q^{68} + 48 q^{70} - 30 q^{71} + 69 q^{73} + 90 q^{76} - 12 q^{77} + 48 q^{79} - 42 q^{80} + 42 q^{82} + 21 q^{83} + 84 q^{85} - 24 q^{86} - 15 q^{89} + 69 q^{91} + 66 q^{92} + 66 q^{94} + 12 q^{95} + 72 q^{97} + 57 q^{98}+O(q^{100}) $ 51 * q + 60 * q^4 - 6 * q^5 + 12 * q^7 + 18 * q^10 - 51 * q^11 + 30 * q^13 - 12 * q^14 + 78 * q^16 + 30 * q^19 - 18 * q^20 - 3 * q^23 + 75 * q^25 - 9 * q^26 + 36 * q^28 + 42 * q^31 + 15 * q^32 + 42 * q^34 + 9 * q^35 + 48 * q^37 + 3 * q^38 + 54 * q^40 + 18 * q^43 - 60 * q^44 + 42 * q^46 - 30 * q^47 + 99 * q^49 + 30 * q^50 + 60 * q^52 - 18 * q^53 + 6 * q^55 - 21 * q^56 + 30 * q^58 - 24 * q^59 + 99 * q^61 + 114 * q^64 + 15 * q^65 + 39 * q^67 + 39 * q^68 + 48 * q^70 - 30 * q^71 + 69 * q^73 + 90 * q^76 - 12 * q^77 + 48 * q^79 - 42 * q^80 + 42 * q^82 + 21 * q^83 + 84 * q^85 - 24 * q^86 - 15 * q^89 + 69 * q^91 + 66 * q^92 + 66 * q^94 + 12 * q^95 + 72 * q^97 + 57 * 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 $

$2$	−1.75810	−1.24317	−0.621583	−	0.783348i	$-0.713510\pi$
$2$	−1.75810	−1.24317	−0.621583	+	0.783348i	$0.713510\pi$
$3$	0	0
$3$	0	0
$4$	1.09093	0.545464
$5$	−0.0727216	−0.0325221	−0.0162610	−	0.999868i	$-0.505176\pi$
$5$	−0.0727216	−0.0325221	−0.0162610	+	0.999868i	$0.505176\pi$
$6$	0	0
$7$	−2.24688	−0.849239	−0.424619	−	0.905372i	$-0.639592\pi$
$7$	−2.24688	−0.849239	−0.424619	+	0.905372i	$0.639592\pi$
$8$	1.59824	0.565064
$9$	0	0
$10$	0.127852	0.0404304
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	2.64801	0.734426	0.367213	−	0.930137i	$-0.380312\pi$
$13$	2.64801	0.734426	0.367213	+	0.930137i	$0.380312\pi$
$14$	3.95024	1.05575
$15$	0	0
$16$	−4.99173	−1.24793
$17$	−7.69499	−1.86631	−0.933154	−	0.359477i	$-0.882955\pi$
$17$	−7.69499	−1.86631	−0.933154	+	0.359477i	$0.882955\pi$
$18$	0	0
$19$	−7.20375	−1.65265	−0.826327	−	0.563191i	$-0.809574\pi$
$19$	−7.20375	−1.65265	−0.826327	+	0.563191i	$0.809574\pi$
$20$	−0.0793340	−0.0177396
$21$	0	0
$22$	1.75810	0.374829
$23$	1.90545	0.397314	0.198657	−	0.980069i	$-0.436342\pi$
$23$	1.90545	0.397314	0.198657	+	0.980069i	$0.436342\pi$
$24$	0	0
$25$	−4.99471	−0.998942
$26$	−4.65548	−0.913014
$27$	0	0
$28$	−2.45118	−0.463229
$29$	−8.89496	−1.65175	−0.825876	−	0.563851i	$-0.809319\pi$
$29$	−8.89496	−1.65175	−0.825876	+	0.563851i	$0.809319\pi$
$30$	0	0
$31$	0.137940	0.0247748	0.0123874	−	0.999923i	$-0.496057\pi$
$31$	0.137940	0.0247748	0.0123874	+	0.999923i	$0.496057\pi$
$32$	5.57950	0.986325
$33$	0	0
$34$	13.5286	2.32013
$35$	0.163396	0.0276190
$36$	0	0
$37$	−10.8258	−1.77976	−0.889879	−	0.456196i	$-0.849212\pi$
$37$	−10.8258	−1.77976	−0.889879	+	0.456196i	$0.849212\pi$
$38$	12.6649	2.05452
$39$	0	0
$40$	−0.116227	−0.0183771
$41$	−10.4427	−1.63087	−0.815435	−	0.578849i	$-0.803502\pi$
$41$	−10.4427	−1.63087	−0.815435	+	0.578849i	$0.803502\pi$
$42$	0	0
$43$	5.64634	0.861058	0.430529	−	0.902577i	$-0.358327\pi$
$43$	5.64634	0.861058	0.430529	+	0.902577i	$0.358327\pi$
$44$	−1.09093	−0.164464
$45$	0	0
$46$	−3.34998	−0.493928
$47$	−1.58582	−0.231316	−0.115658	−	0.993289i	$-0.536898\pi$
$47$	−1.58582	−0.231316	−0.115658	+	0.993289i	$0.536898\pi$
$48$	0	0
$49$	−1.95155	−0.278793
$50$	8.78122	1.24185
$51$	0	0
$52$	2.88879	0.400603
$53$	−0.449984	−0.0618100	−0.0309050	−	0.999522i	$-0.509839\pi$
$53$	−0.449984	−0.0618100	−0.0309050	+	0.999522i	$0.509839\pi$
$54$	0	0
$55$	0.0727216	0.00980578
$56$	−3.59105	−0.479875
$57$	0	0
$58$	15.6383	2.05340
$59$	−4.16759	−0.542573	−0.271287	−	0.962499i	$-0.587449\pi$
$59$	−4.16759	−0.542573	−0.271287	+	0.962499i	$0.587449\pi$
$60$	0	0
$61$	10.0797	1.29057	0.645287	−	0.763940i	$-0.276738\pi$
$61$	10.0797	1.29057	0.645287	+	0.763940i	$0.276738\pi$
$62$	−0.242513	−0.0307992
$63$	0	0
$64$	0.174134	0.0217668
$65$	−0.192568	−0.0238851
$66$	0	0
$67$	−1.39463	−0.170381	−0.0851907	−	0.996365i	$-0.527150\pi$
$67$	−1.39463	−0.170381	−0.0851907	+	0.996365i	$0.527150\pi$
$68$	−8.39467	−1.01800
$69$	0	0
$70$	−0.287268	−0.0343351
$71$	−5.39124	−0.639823	−0.319911	−	0.947447i	$-0.603653\pi$
$71$	−5.39124	−0.639823	−0.319911	+	0.947447i	$0.603653\pi$
$72$	0	0
$73$	8.05143	0.942348	0.471174	−	0.882040i	$-0.343830\pi$
$73$	8.05143	0.942348	0.471174	+	0.882040i	$0.343830\pi$
$74$	19.0330	2.21254
$75$	0	0
$76$	−7.85877	−0.901463
$77$	2.24688	0.256055
$78$	0	0
$79$	13.1394	1.47830	0.739151	−	0.673539i	$-0.235227\pi$
$79$	13.1394	1.47830	0.739151	+	0.673539i	$0.235227\pi$
$80$	0.363007	0.0405854
$81$	0	0
$82$	18.3593	2.02744
$83$	10.3147	1.13219	0.566095	−	0.824340i	$-0.308454\pi$
$83$	10.3147	1.13219	0.566095	+	0.824340i	$0.308454\pi$
$84$	0	0
$85$	0.559592	0.0606962
$86$	−9.92685	−1.07044
$87$	0	0
$88$	−1.59824	−0.170373
$89$	−13.5895	−1.44048	−0.720241	−	0.693724i	$-0.755969\pi$
$89$	−13.5895	−1.44048	−0.720241	+	0.693724i	$0.755969\pi$
$90$	0	0
$91$	−5.94975	−0.623703
$92$	2.07871	0.216721
$93$	0	0
$94$	2.78803	0.287564
$95$	0.523868	0.0537477
$96$	0	0
$97$	−5.37188	−0.545431	−0.272716	−	0.962095i	$-0.587922\pi$
$97$	−5.37188	−0.545431	−0.272716	+	0.962095i	$0.587922\pi$
$98$	3.43103	0.346587
$99$	0	0
$100$	−5.44887	−0.544887
$101$	−9.56837	−0.952089	−0.476044	−	0.879421i	$-0.657930\pi$
$101$	−9.56837	−0.952089	−0.476044	+	0.879421i	$0.657930\pi$
$102$	0	0
$103$	−6.75934	−0.666018	−0.333009	−	0.942924i	$-0.608064\pi$
$103$	−6.75934	−0.666018	−0.333009	+	0.942924i	$0.608064\pi$
$104$	4.23217	0.414998
$105$	0	0
$106$	0.791118	0.0768402
$107$	−7.46537	−0.721704	−0.360852	−	0.932623i	$-0.617514\pi$
$107$	−7.46537	−0.721704	−0.360852	+	0.932623i	$0.617514\pi$
$108$	0	0
$109$	10.7592	1.03055	0.515274	−	0.857026i	$-0.327690\pi$
$109$	10.7592	1.03055	0.515274	+	0.857026i	$0.327690\pi$
$110$	−0.127852	−0.0121902
$111$	0	0
$112$	11.2158	1.05979
$113$	6.81581	0.641177	0.320589	−	0.947219i	$-0.396119\pi$
$113$	6.81581	0.641177	0.320589	+	0.947219i	$0.396119\pi$
$114$	0	0
$115$	−0.138568	−0.0129215
$116$	−9.70376	−0.900971
$117$	0	0
$118$	7.32705	0.674509
$119$	17.2897	1.58494
$120$	0	0
$121$	1.00000	0.0909091
$122$	−17.7212	−1.60440
$123$	0	0
$124$	0.150483	0.0135138
$125$	0.726831	0.0650098
$126$	0	0
$127$	9.49526	0.842568	0.421284	−	0.906929i	$-0.361580\pi$
$127$	9.49526	0.842568	0.421284	+	0.906929i	$0.361580\pi$
$128$	−11.4651	−1.01338
$129$	0	0
$130$	0.338554	0.0296931
$131$	11.4358	0.999149	0.499575	−	0.866271i	$-0.333490\pi$
$131$	11.4358	0.999149	0.499575	+	0.866271i	$0.333490\pi$
$132$	0	0
$133$	16.1859	1.40350
$134$	2.45191	0.211813
$135$	0	0
$136$	−12.2985	−1.05458
$137$	5.06110	0.432399	0.216199	−	0.976349i	$-0.430634\pi$
$137$	5.06110	0.432399	0.216199	+	0.976349i	$0.430634\pi$
$138$	0	0
$139$	−11.0704	−0.938980	−0.469490	−	0.882938i	$-0.655562\pi$
$139$	−11.0704	−0.938980	−0.469490	+	0.882938i	$0.655562\pi$
$140$	0.178254	0.0150652
$141$	0	0
$142$	9.47836	0.795406
$143$	−2.64801	−0.221438
$144$	0	0
$145$	0.646856	0.0537184
$146$	−14.1552	−1.17150
$147$	0	0
$148$	−11.8102	−0.970794
$149$	−1.52602	−0.125016	−0.0625082	−	0.998044i	$-0.519910\pi$
$149$	−1.52602	−0.125016	−0.0625082	+	0.998044i	$0.519910\pi$
$150$	0	0
$151$	−0.273673	−0.0222712	−0.0111356	−	0.999938i	$-0.503545\pi$
$151$	−0.273673	−0.0222712	−0.0111356	+	0.999938i	$0.503545\pi$
$152$	−11.5133	−0.933856
$153$	0	0
$154$	−3.95024	−0.318319
$155$	−0.0100312	−0.000805729 0
$156$	0	0
$157$	3.67498	0.293295	0.146648	−	0.989189i	$-0.453152\pi$
$157$	3.67498	0.293295	0.146648	+	0.989189i	$0.453152\pi$
$158$	−23.1005	−1.83778
$159$	0	0
$160$	−0.405750	−0.0320773
$161$	−4.28131	−0.337415
$162$	0	0
$163$	9.90231	0.775609	0.387804	−	0.921742i	$-0.373234\pi$
$163$	9.90231	0.775609	0.387804	+	0.921742i	$0.373234\pi$
$164$	−11.3922	−0.889580
$165$	0	0
$166$	−18.1344	−1.40750
$167$	7.51218	0.581310	0.290655	−	0.956828i	$-0.406127\pi$
$167$	7.51218	0.581310	0.290655	+	0.956828i	$0.406127\pi$
$168$	0	0
$169$	−5.98804	−0.460618
$170$	−0.983820	−0.0754555
$171$	0	0
$172$	6.15975	0.469676
$173$	−2.37355	−0.180458	−0.0902289	−	0.995921i	$-0.528760\pi$
$173$	−2.37355	−0.180458	−0.0902289	+	0.995921i	$0.528760\pi$
$174$	0	0
$175$	11.2225	0.848341
$176$	4.99173	0.376266
$177$	0	0
$178$	23.8917	1.79076
$179$	−17.8713	−1.33576	−0.667882	−	0.744268i	$-0.732799\pi$
$179$	−17.8713	−1.33576	−0.667882	+	0.744268i	$0.732799\pi$
$180$	0	0
$181$	−7.72796	−0.574415	−0.287207	−	0.957868i	$-0.592727\pi$
$181$	−7.72796	−0.574415	−0.287207	+	0.957868i	$0.592727\pi$
$182$	10.4603	0.775367
$183$	0	0
$184$	3.04538	0.224508
$185$	0.787273	0.0578815
$186$	0	0
$187$	7.69499	0.562713
$188$	−1.73001	−0.126174
$189$	0	0
$190$	−0.921015	−0.0668174
$191$	−14.4866	−1.04821	−0.524106	−	0.851653i	$-0.675600\pi$
$191$	−14.4866	−1.04821	−0.524106	+	0.851653i	$0.675600\pi$
$192$	0	0
$193$	13.3549	0.961310	0.480655	−	0.876910i	$-0.340399\pi$
$193$	13.3549	0.961310	0.480655	+	0.876910i	$0.340399\pi$
$194$	9.44431	0.678062
$195$	0	0
$196$	−2.12900	−0.152072
$197$	−25.1056	−1.78870	−0.894348	−	0.447371i	$-0.852360\pi$
$197$	−25.1056	−1.78870	−0.894348	+	0.447371i	$0.852360\pi$
$198$	0	0
$199$	−7.80698	−0.553422	−0.276711	−	0.960953i	$-0.589244\pi$
$199$	−7.80698	−0.553422	−0.276711	+	0.960953i	$0.589244\pi$
$200$	−7.98276	−0.564467
$201$	0	0
$202$	16.8222	1.18361
$203$	19.9859	1.40273
$204$	0	0
$205$	0.759407	0.0530393
$206$	11.8836	0.827971
$207$	0	0
$208$	−13.2182	−0.916515
$209$	7.20375	0.498294
$210$	0	0
$211$	19.4055	1.33593	0.667965	−	0.744192i	$-0.267165\pi$
$211$	19.4055	1.33593	0.667965	+	0.744192i	$0.267165\pi$
$212$	−0.490900	−0.0337151
$213$	0	0
$214$	13.1249	0.897199
$215$	−0.410611	−0.0280034
$216$	0	0
$217$	−0.309935	−0.0210397
$218$	−18.9158	−1.28114
$219$	0	0
$220$	0.0793340	0.00534870
$221$	−20.3764	−1.37067
$222$	0	0
$223$	−7.16175	−0.479586	−0.239793	−	0.970824i	$-0.577080\pi$
$223$	−7.16175	−0.479586	−0.239793	+	0.970824i	$0.577080\pi$
$224$	−12.5364	−0.837625
$225$	0	0
$226$	−11.9829	−0.797090
$227$	−5.46185	−0.362516	−0.181258	−	0.983436i	$-0.558017\pi$
$227$	−5.46185	−0.362516	−0.181258	+	0.983436i	$0.558017\pi$
$228$	0	0
$229$	5.23673	0.346053	0.173026	−	0.984917i	$-0.444645\pi$
$229$	5.23673	0.346053	0.173026	+	0.984917i	$0.444645\pi$
$230$	0.243616	0.0160636
$231$	0	0
$232$	−14.2163	−0.933346
$233$	−19.6756	−1.28899	−0.644495	−	0.764608i	$-0.722932\pi$
$233$	−19.6756	−1.28899	−0.644495	+	0.764608i	$0.722932\pi$
$234$	0	0
$235$	0.115323	0.00752286
$236$	−4.54653	−0.295954
$237$	0	0
$238$	−30.3970	−1.97035
$239$	23.1976	1.50053	0.750265	−	0.661137i	$-0.229926\pi$
$239$	23.1976	1.50053	0.750265	+	0.661137i	$0.229926\pi$
$240$	0	0
$241$	13.0484	0.840524	0.420262	−	0.907403i	$-0.361938\pi$
$241$	13.0484	0.840524	0.420262	+	0.907403i	$0.361938\pi$
$242$	−1.75810	−0.113015
$243$	0	0
$244$	10.9962	0.703962
$245$	0.141920	0.00906694
$246$	0	0
$247$	−19.0756	−1.21375
$248$	0.220462	0.0139994
$249$	0	0
$250$	−1.27784	−0.0808180
$251$	−17.8353	−1.12576	−0.562878	−	0.826540i	$-0.690306\pi$
$251$	−17.8353	−1.12576	−0.562878	+	0.826540i	$0.690306\pi$
$252$	0	0
$253$	−1.90545	−0.119795
$254$	−16.6936	−1.04745
$255$	0	0
$256$	19.8086	1.23804
$257$	10.0279	0.625522	0.312761	−	0.949832i	$-0.398746\pi$
$257$	10.0279	0.625522	0.312761	+	0.949832i	$0.398746\pi$
$258$	0	0
$259$	24.3243	1.51144
$260$	−0.210077	−0.0130284
$261$	0	0
$262$	−20.1053	−1.24211
$263$	−19.4329	−1.19829	−0.599143	−	0.800642i	$-0.704492\pi$
$263$	−19.4329	−1.19829	−0.599143	+	0.800642i	$0.704492\pi$
$264$	0	0
$265$	0.0327235	0.00201019
$266$	−28.4565	−1.74478
$267$	0	0
$268$	−1.52144	−0.0929369
$269$	17.2745	1.05325	0.526624	−	0.850098i	$-0.323458\pi$
$269$	17.2745	1.05325	0.526624	+	0.850098i	$0.323458\pi$
$270$	0	0
$271$	−6.07492	−0.369025	−0.184513	−	0.982830i	$-0.559071\pi$
$271$	−6.07492	−0.369025	−0.184513	+	0.982830i	$0.559071\pi$
$272$	38.4113	2.32903
$273$	0	0
$274$	−8.89793	−0.537544
$275$	4.99471	0.301192
$276$	0	0
$277$	−23.8124	−1.43075	−0.715374	−	0.698742i	$-0.753744\pi$
$277$	−23.8124	−1.43075	−0.715374	+	0.698742i	$0.753744\pi$
$278$	19.4629	1.16731
$279$	0	0
$280$	0.261147	0.0156065
$281$	−11.4555	−0.683380	−0.341690	−	0.939813i	$-0.610999\pi$
$281$	−11.4555	−0.683380	−0.341690	+	0.939813i	$0.610999\pi$
$282$	0	0
$283$	−10.6786	−0.634779	−0.317390	−	0.948295i	$-0.602806\pi$
$283$	−10.6786	−0.634779	−0.317390	+	0.948295i	$0.602806\pi$
$284$	−5.88145	−0.349000
$285$	0	0
$286$	4.65548	0.275284
$287$	23.4634	1.38500
$288$	0	0
$289$	42.2128	2.48311
$290$	−1.13724	−0.0667810
$291$	0	0
$292$	8.78352	0.514017
$293$	6.10475	0.356643	0.178322	−	0.983972i	$-0.442933\pi$
$293$	6.10475	0.356643	0.178322	+	0.983972i	$0.442933\pi$
$294$	0	0
$295$	0.303073	0.0176456
$296$	−17.3023	−1.00568
$297$	0	0
$298$	2.68290	0.155416
$299$	5.04566	0.291798
$300$	0	0
$301$	−12.6866	−0.731244
$302$	0.481145	0.0276868
$303$	0	0
$304$	35.9592	2.06240
$305$	−0.733013	−0.0419722
$306$	0	0
$307$	28.4688	1.62480	0.812401	−	0.583099i	$-0.198160\pi$
$307$	28.4688	1.62480	0.812401	+	0.583099i	$0.198160\pi$
$308$	2.45118	0.139669
$309$	0	0
$310$	0.0176360	0.00100166
$311$	−5.16206	−0.292714	−0.146357	−	0.989232i	$-0.546755\pi$
$311$	−5.16206	−0.292714	−0.146357	+	0.989232i	$0.546755\pi$
$312$	0	0
$313$	7.76192	0.438730	0.219365	−	0.975643i	$-0.429602\pi$
$313$	7.76192	0.438730	0.219365	+	0.975643i	$0.429602\pi$
$314$	−6.46099	−0.364615
$315$	0	0
$316$	14.3342	0.806361
$317$	6.33208	0.355645	0.177823	−	0.984063i	$-0.443095\pi$
$317$	6.33208	0.355645	0.177823	+	0.984063i	$0.443095\pi$
$318$	0	0
$319$	8.89496	0.498022
$320$	−0.0126633	−0.000707901 0
$321$	0	0
$322$	7.52699	0.419463
$323$	55.4328	3.08436
$324$	0	0
$325$	−13.2261	−0.733649
$326$	−17.4093	−0.964211
$327$	0	0
$328$	−16.6899	−0.921546
$329$	3.56314	0.196442
$330$	0	0
$331$	6.50897	0.357765	0.178883	−	0.983870i	$-0.442752\pi$
$331$	6.50897	0.357765	0.178883	+	0.983870i	$0.442752\pi$
$332$	11.2526	0.617568
$333$	0	0
$334$	−13.2072	−0.722665
$335$	0.101420	0.00554116
$336$	0	0
$337$	7.53316	0.410358	0.205179	−	0.978725i	$-0.434222\pi$
$337$	7.53316	0.410358	0.205179	+	0.978725i	$0.434222\pi$
$338$	10.5276	0.572625
$339$	0	0
$340$	0.610474	0.0331076
$341$	−0.137940	−0.00746989
$342$	0	0
$343$	20.1130	1.08600
$344$	9.02422	0.486553
$345$	0	0
$346$	4.17295	0.224339
$347$	9.84087	0.528286	0.264143	−	0.964484i	$-0.414911\pi$
$347$	9.84087	0.528286	0.264143	+	0.964484i	$0.414911\pi$
$348$	0	0
$349$	29.9507	1.60322	0.801611	−	0.597846i	$-0.203976\pi$
$349$	29.9507	1.60322	0.801611	+	0.597846i	$0.203976\pi$
$350$	−19.7303	−1.05463
$351$	0	0
$352$	−5.57950	−0.297388
$353$	7.52270	0.400393	0.200196	−	0.979756i	$-0.435842\pi$
$353$	7.52270	0.400393	0.200196	+	0.979756i	$0.435842\pi$
$354$	0	0
$355$	0.392060	0.0208084
$356$	−14.8251	−0.785731
$357$	0	0
$358$	31.4196	1.66058
$359$	−17.5749	−0.927565	−0.463783	−	0.885949i	$-0.653508\pi$
$359$	−17.5749	−0.927565	−0.463783	+	0.885949i	$0.653508\pi$
$360$	0	0
$361$	32.8940	1.73126
$362$	13.5866	0.714094
$363$	0	0
$364$	−6.49075	−0.340208
$365$	−0.585512	−0.0306471
$366$	0	0
$367$	−22.7420	−1.18712	−0.593561	−	0.804789i	$-0.702278\pi$
$367$	−22.7420	−1.18712	−0.593561	+	0.804789i	$0.702278\pi$
$368$	−9.51151	−0.495822
$369$	0	0
$370$	−1.38411	−0.0719563
$371$	1.01106	0.0524915
$372$	0	0
$373$	−28.0093	−1.45027	−0.725134	−	0.688607i	$-0.758222\pi$
$373$	−28.0093	−1.45027	−0.725134	+	0.688607i	$0.758222\pi$
$374$	−13.5286	−0.699546
$375$	0	0
$376$	−2.53452	−0.130708
$377$	−23.5540	−1.21309
$378$	0	0
$379$	−3.57543	−0.183657	−0.0918286	−	0.995775i	$-0.529271\pi$
$379$	−3.57543	−0.183657	−0.0918286	+	0.995775i	$0.529271\pi$
$380$	0.571502	0.0293175
$381$	0	0
$382$	25.4689	1.30310
$383$	0.678864	0.0346883	0.0173442	−	0.999850i	$-0.494479\pi$
$383$	0.678864	0.0346883	0.0173442	+	0.999850i	$0.494479\pi$
$384$	0	0
$385$	−0.163396	−0.00832745
$386$	−23.4794	−1.19507
$387$	0	0
$388$	−5.86033	−0.297513
$389$	−15.4473	−0.783208	−0.391604	−	0.920134i	$-0.628080\pi$
$389$	−15.4473	−0.783208	−0.391604	+	0.920134i	$0.628080\pi$
$390$	0	0
$391$	−14.6624	−0.741511
$392$	−3.11906	−0.157536
$393$	0	0
$394$	44.1382	2.22365
$395$	−0.955521	−0.0480775
$396$	0	0
$397$	−8.67719	−0.435496	−0.217748	−	0.976005i	$-0.569871\pi$
$397$	−8.67719	−0.435496	−0.217748	+	0.976005i	$0.569871\pi$
$398$	13.7255	0.687996
$399$	0	0
$400$	24.9323	1.24661
$401$	29.1358	1.45497	0.727487	−	0.686121i	$-0.240688\pi$
$401$	29.1358	1.45497	0.727487	+	0.686121i	$0.240688\pi$
$402$	0	0
$403$	0.365268	0.0181953
$404$	−10.4384	−0.519330
$405$	0	0
$406$	−35.1372	−1.74383
$407$	10.8258	0.536617
$408$	0	0
$409$	32.2586	1.59509	0.797543	−	0.603262i	$-0.206133\pi$
$409$	32.2586	1.59509	0.797543	+	0.603262i	$0.206133\pi$
$410$	−1.33512	−0.0659367
$411$	0	0
$412$	−7.37395	−0.363289
$413$	9.36404	0.460774
$414$	0	0
$415$	−0.750104	−0.0368212
$416$	14.7746	0.724383
$417$	0	0
$418$	−12.6649	−0.619462
$419$	28.3657	1.38575	0.692877	−	0.721056i	$-0.256343\pi$
$419$	28.3657	1.38575	0.692877	+	0.721056i	$0.256343\pi$
$420$	0	0
$421$	−7.75854	−0.378128	−0.189064	−	0.981965i	$-0.560545\pi$
$421$	−7.75854	−0.378128	−0.189064	+	0.981965i	$0.560545\pi$
$422$	−34.1169	−1.66078
$423$	0	0
$424$	−0.719184	−0.0349266
$425$	38.4342	1.86433
$426$	0	0
$427$	−22.6479	−1.09601
$428$	−8.14418	−0.393664
$429$	0	0
$430$	0.721896	0.0348129
$431$	−26.4230	−1.27275	−0.636376	−	0.771379i	$-0.719567\pi$
$431$	−26.4230	−1.27275	−0.636376	+	0.771379i	$0.719567\pi$
$432$	0	0
$433$	13.1284	0.630912	0.315456	−	0.948940i	$-0.397843\pi$
$433$	13.1284	0.630912	0.315456	+	0.948940i	$0.397843\pi$
$434$	0.544897	0.0261559
$435$	0	0
$436$	11.7375	0.562126
$437$	−13.7264	−0.656623
$438$	0	0
$439$	−16.9206	−0.807574	−0.403787	−	0.914853i	$-0.632306\pi$
$439$	−16.9206	−0.807574	−0.403787	+	0.914853i	$0.632306\pi$
$440$	0.116227	0.00554089
$441$	0	0
$442$	35.8238	1.70397
$443$	−13.3231	−0.632998	−0.316499	−	0.948593i	$-0.602507\pi$
$443$	−13.3231	−0.632998	−0.316499	+	0.948593i	$0.602507\pi$
$444$	0	0
$445$	0.988249	0.0468475
$446$	12.5911	0.596206
$447$	0	0
$448$	−0.391258	−0.0184852
$449$	−3.88453	−0.183322	−0.0916612	−	0.995790i	$-0.529218\pi$
$449$	−3.88453	−0.183322	−0.0916612	+	0.995790i	$0.529218\pi$
$450$	0	0
$451$	10.4427	0.491726
$452$	7.43555	0.349739
$453$	0	0
$454$	9.60250	0.450667
$455$	0.432675	0.0202841
$456$	0	0
$457$	15.0243	0.702807	0.351403	−	0.936224i	$-0.385705\pi$
$457$	15.0243	0.702807	0.351403	+	0.936224i	$0.385705\pi$
$458$	−9.20671	−0.430201
$459$	0	0
$460$	−0.151167	−0.00704821
$461$	−9.51304	−0.443066	−0.221533	−	0.975153i	$-0.571106\pi$
$461$	−9.51304	−0.443066	−0.221533	+	0.975153i	$0.571106\pi$
$462$	0	0
$463$	−16.5202	−0.767759	−0.383880	−	0.923383i	$-0.625412\pi$
$463$	−16.5202	−0.767759	−0.383880	+	0.923383i	$0.625412\pi$
$464$	44.4013	2.06128
$465$	0	0
$466$	34.5917	1.60243
$467$	2.31369	0.107065	0.0535325	−	0.998566i	$-0.482952\pi$
$467$	2.31369	0.107065	0.0535325	+	0.998566i	$0.482952\pi$
$468$	0	0
$469$	3.13356	0.144695
$470$	−0.202750	−0.00935217
$471$	0	0
$472$	−6.66081	−0.306589
$473$	−5.64634	−0.259619
$474$	0	0
$475$	35.9807	1.65091
$476$	18.8618	0.864528
$477$	0	0
$478$	−40.7839	−1.86541
$479$	−21.9695	−1.00381	−0.501906	−	0.864922i	$-0.667368\pi$
$479$	−21.9695	−1.00381	−0.501906	+	0.864922i	$0.667368\pi$
$480$	0	0
$481$	−28.6670	−1.30710
$482$	−22.9405	−1.04491
$483$	0	0
$484$	1.09093	0.0495876
$485$	0.390651	0.0177386
$486$	0	0
$487$	−40.1822	−1.82083	−0.910416	−	0.413695i	$-0.864238\pi$
$487$	−40.1822	−1.82083	−0.910416	+	0.413695i	$0.864238\pi$
$488$	16.1098	0.729258
$489$	0	0
$490$	−0.249510	−0.0112717
$491$	−4.74433	−0.214109	−0.107054	−	0.994253i	$-0.534142\pi$
$491$	−4.74433	−0.214109	−0.107054	+	0.994253i	$0.534142\pi$
$492$	0	0
$493$	68.4466	3.08268
$494$	33.5369	1.50890
$495$	0	0
$496$	−0.688561	−0.0309173
$497$	12.1134	0.543362
$498$	0	0
$499$	−34.4530	−1.54233	−0.771164	−	0.636637i	$-0.780325\pi$
$499$	−34.4530	−1.54233	−0.771164	+	0.636637i	$0.780325\pi$
$500$	0.792920	0.0354605
$501$	0	0
$502$	31.3564	1.39950
$503$	29.7479	1.32639	0.663196	−	0.748445i	$-0.269199\pi$
$503$	29.7479	1.32639	0.663196	+	0.748445i	$0.269199\pi$
$504$	0	0
$505$	0.695827	0.0309639
$506$	3.34998	0.148925
$507$	0	0
$508$	10.3586	0.459590
$509$	−37.0698	−1.64309	−0.821545	−	0.570143i	$-0.806888\pi$
$509$	−37.0698	−1.64309	−0.821545	+	0.570143i	$0.806888\pi$
$510$	0	0
$511$	−18.0905	−0.800279
$512$	−11.8953	−0.525705
$513$	0	0
$514$	−17.6300	−0.777628
$515$	0.491550	0.0216603
$516$	0	0
$517$	1.58582	0.0697443
$518$	−42.7647	−1.87897
$519$	0	0
$520$	−0.307770	−0.0134966
$521$	34.2942	1.50246	0.751228	−	0.660043i	$-0.229462\pi$
$521$	34.2942	1.50246	0.751228	+	0.660043i	$0.229462\pi$
$522$	0	0
$523$	−27.9637	−1.22277	−0.611383	−	0.791334i	$-0.709387\pi$
$523$	−27.9637	−1.22277	−0.611383	+	0.791334i	$0.709387\pi$
$524$	12.4756	0.545000
$525$	0	0
$526$	34.1651	1.48967
$527$	−1.06145	−0.0462374
$528$	0	0
$529$	−19.3692	−0.842141
$530$	−0.0575314	−0.00249900
$531$	0	0
$532$	17.6577	0.765557
$533$	−27.6523	−1.19775
$534$	0	0
$535$	0.542893	0.0234713
$536$	−2.22896	−0.0962764
$537$	0	0
$538$	−30.3704	−1.30936
$539$	1.95155	0.0840593
$540$	0	0
$541$	1.57342	0.0676467	0.0338234	−	0.999428i	$-0.489232\pi$
$541$	1.57342	0.0676467	0.0338234	+	0.999428i	$0.489232\pi$
$542$	10.6803	0.458760
$543$	0	0
$544$	−42.9341	−1.84079
$545$	−0.782428	−0.0335155
$546$	0	0
$547$	1.01405	0.0433577	0.0216788	−	0.999765i	$-0.493099\pi$
$547$	1.01405	0.0433577	0.0216788	+	0.999765i	$0.493099\pi$
$548$	5.52129	0.235858
$549$	0	0
$550$	−8.78122	−0.374432
$551$	64.0771	2.72978
$552$	0	0
$553$	−29.5227	−1.25543
$554$	41.8646	1.77866
$555$	0	0
$556$	−12.0770	−0.512180
$557$	29.1458	1.23495	0.617473	−	0.786592i	$-0.288157\pi$
$557$	29.1458	1.23495	0.617473	+	0.786592i	$0.288157\pi$
$558$	0	0
$559$	14.9516	0.632384
$560$	−0.815631	−0.0344667
$561$	0	0
$562$	20.1400	0.849555
$563$	−11.9335	−0.502936	−0.251468	−	0.967866i	$-0.580913\pi$
$563$	−11.9335	−0.502936	−0.251468	+	0.967866i	$0.580913\pi$
$564$	0	0
$565$	−0.495656	−0.0208524
$566$	18.7742	0.789136
$567$	0	0
$568$	−8.61651	−0.361541
$569$	9.22575	0.386763	0.193382	−	0.981124i	$-0.438054\pi$
$569$	9.22575	0.386763	0.193382	+	0.981124i	$0.438054\pi$
$570$	0	0
$571$	−6.39851	−0.267769	−0.133885	−	0.990997i	$-0.542745\pi$
$571$	−6.39851	−0.267769	−0.133885	+	0.990997i	$0.542745\pi$
$572$	−2.88879	−0.120786
$573$	0	0
$574$	−41.2510	−1.72178
$575$	−9.51719	−0.396894
$576$	0	0
$577$	−8.93463	−0.371954	−0.185977	−	0.982554i	$-0.559545\pi$
$577$	−8.93463	−0.371954	−0.185977	+	0.982554i	$0.559545\pi$
$578$	−74.2145	−3.08691
$579$	0	0
$580$	0.705673	0.0293015
$581$	−23.1759	−0.961499
$582$	0	0
$583$	0.449984	0.0186364
$584$	12.8681	0.532487
$585$	0	0
$586$	−10.7328	−0.443367
$587$	−4.91864	−0.203014	−0.101507	−	0.994835i	$-0.532366\pi$
$587$	−4.91864	−0.203014	−0.101507	+	0.994835i	$0.532366\pi$
$588$	0	0
$589$	−0.993688	−0.0409442
$590$	−0.532834	−0.0219364
$591$	0	0
$592$	54.0397	2.22102
$593$	−2.61107	−0.107224	−0.0536120	−	0.998562i	$-0.517073\pi$
$593$	−2.61107	−0.107224	−0.0536120	+	0.998562i	$0.517073\pi$
$594$	0	0
$595$	−1.25733	−0.0515456
$596$	−1.66478	−0.0681919
$597$	0	0
$598$	−8.87079	−0.362754
$599$	1.40119	0.0572509	0.0286254	−	0.999590i	$-0.490887\pi$
$599$	1.40119	0.0572509	0.0286254	+	0.999590i	$0.490887\pi$
$600$	0	0
$601$	−15.2117	−0.620497	−0.310249	−	0.950655i	$-0.600412\pi$
$601$	−15.2117	−0.620497	−0.310249	+	0.950655i	$0.600412\pi$
$602$	22.3044	0.909059
$603$	0	0
$604$	−0.298557	−0.0121481
$605$	−0.0727216	−0.00295655
$606$	0	0
$607$	−45.2123	−1.83511	−0.917555	−	0.397608i	$-0.869840\pi$
$607$	−45.2123	−1.83511	−0.917555	+	0.397608i	$0.869840\pi$
$608$	−40.1933	−1.63005
$609$	0	0
$610$	1.28871	0.0521784
$611$	−4.19927	−0.169884
$612$	0	0
$613$	14.6218	0.590568	0.295284	−	0.955410i	$-0.404586\pi$
$613$	14.6218	0.590568	0.295284	+	0.955410i	$0.404586\pi$
$614$	−50.0511	−2.01990
$615$	0	0
$616$	3.59105	0.144688
$617$	1.70137	0.0684944	0.0342472	−	0.999413i	$-0.489097\pi$
$617$	1.70137	0.0684944	0.0342472	+	0.999413i	$0.489097\pi$
$618$	0	0
$619$	−37.0392	−1.48873	−0.744365	−	0.667773i	$-0.767248\pi$
$619$	−37.0392	−1.48873	−0.744365	+	0.667773i	$0.767248\pi$
$620$	−0.0109434	−0.000439496 0
$621$	0	0
$622$	9.07544	0.363892
$623$	30.5339	1.22331
$624$	0	0
$625$	24.9207	0.996828
$626$	−13.6463	−0.545414
$627$	0	0
$628$	4.00914	0.159982
$629$	83.3047	3.32158
$630$	0	0
$631$	6.53656	0.260217	0.130108	−	0.991500i	$-0.458468\pi$
$631$	6.53656	0.260217	0.130108	+	0.991500i	$0.458468\pi$
$632$	21.0000	0.835336
$633$	0	0
$634$	−11.1325	−0.442126
$635$	−0.690510	−0.0274021
$636$	0	0
$637$	−5.16773	−0.204753
$638$	−15.6383	−0.619125
$639$	0	0
$640$	0.833763	0.0329574
$641$	8.17162	0.322760	0.161380	−	0.986892i	$-0.448406\pi$
$641$	8.17162	0.322760	0.161380	+	0.986892i	$0.448406\pi$
$642$	0	0
$643$	14.4304	0.569080	0.284540	−	0.958664i	$-0.408159\pi$
$643$	14.4304	0.569080	0.284540	+	0.958664i	$0.408159\pi$
$644$	−4.67061	−0.184048
$645$	0	0
$646$	−97.4565	−3.83438
$647$	−2.28290	−0.0897499	−0.0448749	−	0.998993i	$-0.514289\pi$
$647$	−2.28290	−0.0897499	−0.0448749	+	0.998993i	$0.514289\pi$
$648$	0	0
$649$	4.16759	0.163592
$650$	23.2528	0.912049
$651$	0	0
$652$	10.8027	0.423067
$653$	24.3168	0.951591	0.475796	−	0.879556i	$-0.342160\pi$
$653$	24.3168	0.951591	0.475796	+	0.879556i	$0.342160\pi$
$654$	0	0
$655$	−0.831629	−0.0324944
$656$	52.1270	2.03522
$657$	0	0
$658$	−6.26436	−0.244210
$659$	20.9723	0.816966	0.408483	−	0.912766i	$-0.366058\pi$
$659$	20.9723	0.816966	0.408483	+	0.912766i	$0.366058\pi$
$660$	0	0
$661$	39.0557	1.51909	0.759545	−	0.650455i	$-0.225422\pi$
$661$	39.0557	1.51909	0.759545	+	0.650455i	$0.225422\pi$
$662$	−11.4434	−0.444762
$663$	0	0
$664$	16.4855	0.639760
$665$	−1.17707	−0.0456447
$666$	0	0
$667$	−16.9489	−0.656265
$668$	8.19525	0.317084
$669$	0	0
$670$	−0.178307	−0.00688858
$671$	−10.0797	−0.389123
$672$	0	0
$673$	1.52689	0.0588572	0.0294286	−	0.999567i	$-0.490631\pi$
$673$	1.52689	0.0588572	0.0294286	+	0.999567i	$0.490631\pi$
$674$	−13.2441	−0.510143
$675$	0	0
$676$	−6.53252	−0.251251
$677$	−15.4637	−0.594318	−0.297159	−	0.954828i	$-0.596039\pi$
$677$	−15.4637	−0.594318	−0.297159	+	0.954828i	$0.596039\pi$
$678$	0	0
$679$	12.0699	0.463202
$680$	0.894363	0.0342973
$681$	0	0
$682$	0.242513	0.00928632
$683$	−6.60633	−0.252784	−0.126392	−	0.991980i	$-0.540340\pi$
$683$	−6.60633	−0.252784	−0.126392	+	0.991980i	$0.540340\pi$
$684$	0	0
$685$	−0.368051	−0.0140625
$686$	−35.3608	−1.35008
$687$	0	0
$688$	−28.1850	−1.07454
$689$	−1.19156	−0.0453949
$690$	0	0
$691$	30.4832	1.15963	0.579817	−	0.814747i	$-0.303124\pi$
$691$	30.4832	1.15963	0.579817	+	0.814747i	$0.303124\pi$
$692$	−2.58937	−0.0984332
$693$	0	0
$694$	−17.3013	−0.656747
$695$	0.805058	0.0305376
$696$	0	0
$697$	80.3561	3.04370
$698$	−52.6564	−1.99307
$699$	0	0
$700$	12.2429	0.462739
$701$	6.98480	0.263812	0.131906	−	0.991262i	$-0.457890\pi$
$701$	6.98480	0.263812	0.131906	+	0.991262i	$0.457890\pi$
$702$	0	0
$703$	77.9867	2.94133
$704$	−0.174134	−0.00656293
$705$	0	0
$706$	−13.2257	−0.497755
$707$	21.4989	0.808551
$708$	0	0
$709$	12.8550	0.482780	0.241390	−	0.970428i	$-0.422397\pi$
$709$	12.8550	0.482780	0.241390	+	0.970428i	$0.422397\pi$
$710$	−0.689281	−0.0258683
$711$	0	0
$712$	−21.7193	−0.813965
$713$	0.262839	0.00984339
$714$	0	0
$715$	0.192568	0.00720162
$716$	−19.4963	−0.728610
$717$	0	0
$718$	30.8984	1.15312
$719$	45.4903	1.69650	0.848251	−	0.529594i	$-0.177656\pi$
$719$	45.4903	1.69650	0.848251	+	0.529594i	$0.177656\pi$
$720$	0	0
$721$	15.1874	0.565608
$722$	−57.8311	−2.15225
$723$	0	0
$724$	−8.43065	−0.313323
$725$	44.4278	1.65001
$726$	0	0
$727$	−15.9488	−0.591508	−0.295754	−	0.955264i	$-0.595571\pi$
$727$	−15.9488	−0.591508	−0.295754	+	0.955264i	$0.595571\pi$
$728$	−9.50915	−0.352432
$729$	0	0
$730$	1.02939	0.0380995
$731$	−43.4485	−1.60700
$732$	0	0
$733$	0.736171	0.0271911	0.0135956	−	0.999908i	$-0.495672\pi$
$733$	0.736171	0.0271911	0.0135956	+	0.999908i	$0.495672\pi$
$734$	39.9828	1.47579
$735$	0	0
$736$	10.6315	0.391881
$737$	1.39463	0.0513719
$738$	0	0
$739$	−43.0176	−1.58243	−0.791214	−	0.611540i	$-0.790550\pi$
$739$	−43.0176	−1.58243	−0.791214	+	0.611540i	$0.790550\pi$
$740$	0.858858	0.0315722
$741$	0	0
$742$	−1.77754	−0.0652557
$743$	52.0700	1.91026	0.955132	−	0.296181i	$-0.0957131\pi$
$743$	52.0700	1.91026	0.955132	+	0.296181i	$0.0957131\pi$
$744$	0	0
$745$	0.110975	0.00406579
$746$	49.2433	1.80293
$747$	0	0
$748$	8.39467	0.306940
$749$	16.7737	0.612899
$750$	0	0
$751$	−45.5107	−1.66071	−0.830355	−	0.557235i	$-0.811862\pi$
$751$	−45.5107	−1.66071	−0.830355	+	0.557235i	$0.811862\pi$
$752$	7.91598	0.288666
$753$	0	0
$754$	41.4103	1.50807
$755$	0.0199019	0.000724305 0
$756$	0	0
$757$	7.81233	0.283944	0.141972	−	0.989871i	$-0.454656\pi$
$757$	7.81233	0.283944	0.141972	+	0.989871i	$0.454656\pi$
$758$	6.28597	0.228317
$759$	0	0
$760$	0.837269	0.0303709
$761$	18.5666	0.673040	0.336520	−	0.941676i	$-0.390750\pi$
$761$	18.5666	0.673040	0.336520	+	0.941676i	$0.390750\pi$
$762$	0	0
$763$	−24.1746	−0.875181
$764$	−15.8038	−0.571762
$765$	0	0
$766$	−1.19351	−0.0431234
$767$	−11.0358	−0.398480
$768$	0	0
$769$	−10.1752	−0.366927	−0.183464	−	0.983026i	$-0.558731\pi$
$769$	−10.1752	−0.366927	−0.183464	+	0.983026i	$0.558731\pi$
$770$	0.287268	0.0103524
$771$	0	0
$772$	14.5693	0.524360
$773$	−11.9571	−0.430066	−0.215033	−	0.976607i	$-0.568986\pi$
$773$	−11.9571	−0.430066	−0.215033	+	0.976607i	$0.568986\pi$
$774$	0	0
$775$	−0.688972	−0.0247486
$776$	−8.58556	−0.308204
$777$	0	0
$778$	27.1579	0.973658
$779$	75.2263	2.69526
$780$	0	0
$781$	5.39124	0.192914
$782$	25.7781	0.921822
$783$	0	0
$784$	9.74163	0.347915
$785$	−0.267250	−0.00953857
$786$	0	0
$787$	−9.13901	−0.325771	−0.162885	−	0.986645i	$-0.552080\pi$
$787$	−9.13901	−0.325771	−0.162885	+	0.986645i	$0.552080\pi$
$788$	−27.3883	−0.975670
$789$	0	0
$790$	1.67990	0.0597683
$791$	−15.3143	−0.544513
$792$	0	0
$793$	26.6912	0.947832
$794$	15.2554	0.541394
$795$	0	0
$796$	−8.51685	−0.301872
$797$	21.6322	0.766251	0.383126	−	0.923696i	$-0.374848\pi$
$797$	21.6322	0.766251	0.383126	+	0.923696i	$0.374848\pi$
$798$	0	0
$799$	12.2029	0.431706
$800$	−27.8680	−0.985282
$801$	0	0
$802$	−51.2238	−1.80878
$803$	−8.05143	−0.284129
$804$	0	0
$805$	0.311344	0.0109734
$806$	−0.642178	−0.0226198
$807$	0	0
$808$	−15.2926	−0.537991
$809$	12.9913	0.456751	0.228375	−	0.973573i	$-0.426659\pi$
$809$	12.9913	0.456751	0.228375	+	0.973573i	$0.426659\pi$
$810$	0	0
$811$	4.57477	0.160642	0.0803209	−	0.996769i	$-0.474405\pi$
$811$	4.57477	0.160642	0.0803209	+	0.996769i	$0.474405\pi$
$812$	21.8031	0.765140
$813$	0	0
$814$	−19.0330	−0.667105
$815$	−0.720112	−0.0252244
$816$	0	0
$817$	−40.6748	−1.42303
$818$	−56.7140	−1.98296
$819$	0	0
$820$	0.828458	0.0289310
$821$	18.5878	0.648720	0.324360	−	0.945934i	$-0.394851\pi$
$821$	18.5878	0.648720	0.324360	+	0.945934i	$0.394851\pi$
$822$	0	0
$823$	5.99976	0.209139	0.104569	−	0.994518i	$-0.466654\pi$
$823$	5.99976	0.209139	0.104569	+	0.994518i	$0.466654\pi$
$824$	−10.8031	−0.376343
$825$	0	0
$826$	−16.4630	−0.572820
$827$	0.218685	0.00760441	0.00380221	−	0.999993i	$-0.498790\pi$
$827$	0.218685	0.00760441	0.00380221	+	0.999993i	$0.498790\pi$
$828$	0	0
$829$	1.86763	0.0648656	0.0324328	−	0.999474i	$-0.489675\pi$
$829$	1.86763	0.0648656	0.0324328	+	0.999474i	$0.489675\pi$
$830$	1.31876	0.0457748
$831$	0	0
$832$	0.461109	0.0159861
$833$	15.0172	0.520314
$834$	0	0
$835$	−0.546298	−0.0189054
$836$	7.85877	0.271801
$837$	0	0
$838$	−49.8698	−1.72272
$839$	−12.7543	−0.440328	−0.220164	−	0.975463i	$-0.570659\pi$
$839$	−12.7543	−0.440328	−0.220164	+	0.975463i	$0.570659\pi$
$840$	0	0
$841$	50.1203	1.72829
$842$	13.6403	0.470077
$843$	0	0
$844$	21.1700	0.728702
$845$	0.435460	0.0149803
$846$	0	0
$847$	−2.24688	−0.0772035
$848$	2.24620	0.0771348
$849$	0	0
$850$	−67.5714	−2.31768
$851$	−20.6281	−0.707124
$852$	0	0
$853$	25.3504	0.867982	0.433991	−	0.900917i	$-0.357105\pi$
$853$	25.3504	0.867982	0.433991	+	0.900917i	$0.357105\pi$
$854$	39.8173	1.36252
$855$	0	0
$856$	−11.9315	−0.407809
$857$	−3.02875	−0.103460	−0.0517300	−	0.998661i	$-0.516474\pi$
$857$	−3.02875	−0.103460	−0.0517300	+	0.998661i	$0.516474\pi$
$858$	0	0
$859$	−17.1015	−0.583497	−0.291748	−	0.956495i	$-0.594237\pi$
$859$	−17.1015	−0.583497	−0.291748	+	0.956495i	$0.594237\pi$
$860$	−0.447947	−0.0152749
$861$	0	0
$862$	46.4544	1.58224
$863$	3.68728	0.125517	0.0627583	−	0.998029i	$-0.480010\pi$
$863$	3.68728	0.125517	0.0627583	+	0.998029i	$0.480010\pi$
$864$	0	0
$865$	0.172608	0.00586886
$866$	−23.0811	−0.784328
$867$	0	0
$868$	−0.338116	−0.0114764
$869$	−13.1394	−0.445725
$870$	0	0
$871$	−3.69300	−0.125133
$872$	17.1959	0.582325
$873$	0	0
$874$	24.1324	0.816292
$875$	−1.63310	−0.0552088
$876$	0	0
$877$	40.0581	1.35267	0.676334	−	0.736596i	$-0.263568\pi$
$877$	40.0581	1.35267	0.676334	+	0.736596i	$0.263568\pi$
$878$	29.7481	1.00395
$879$	0	0
$880$	−0.363007	−0.0122370
$881$	−28.1713	−0.949114	−0.474557	−	0.880225i	$-0.657392\pi$
$881$	−28.1713	−0.949114	−0.474557	+	0.880225i	$0.657392\pi$
$882$	0	0
$883$	4.35711	0.146629	0.0733143	−	0.997309i	$-0.476642\pi$
$883$	4.35711	0.146629	0.0733143	+	0.997309i	$0.476642\pi$
$884$	−22.2292	−0.747648
$885$	0	0
$886$	23.4233	0.786922
$887$	−22.7515	−0.763920	−0.381960	−	0.924179i	$-0.624751\pi$
$887$	−22.7515	−0.763920	−0.381960	+	0.924179i	$0.624751\pi$
$888$	0	0
$889$	−21.3347	−0.715542
$890$	−1.73744	−0.0582393
$891$	0	0
$892$	−7.81295	−0.261597
$893$	11.4238	0.382284
$894$	0	0
$895$	1.29963	0.0434418
$896$	25.7607	0.860606
$897$	0	0
$898$	6.82941	0.227900
$899$	−1.22697	−0.0409219
$900$	0	0
$901$	3.46262	0.115357
$902$	−18.3593	−0.611297
$903$	0	0
$904$	10.8933	0.362306
$905$	0.561990	0.0186812
$906$	0	0
$907$	−45.5728	−1.51322	−0.756611	−	0.653866i	$-0.773146\pi$
$907$	−45.5728	−1.51322	−0.756611	+	0.653866i	$0.773146\pi$
$908$	−5.95848	−0.197739
$909$	0	0
$910$	−0.760688	−0.0252166
$911$	−7.24336	−0.239983	−0.119992	−	0.992775i	$-0.538287\pi$
$911$	−7.24336	−0.239983	−0.119992	+	0.992775i	$0.538287\pi$
$912$	0	0
$913$	−10.3147	−0.341368
$914$	−26.4143	−0.873706
$915$	0	0
$916$	5.71289	0.188759
$917$	−25.6948	−0.848516
$918$	0	0
$919$	−37.4389	−1.23499	−0.617497	−	0.786573i	$-0.711853\pi$
$919$	−37.4389	−1.23499	−0.617497	+	0.786573i	$0.711853\pi$
$920$	−0.221465	−0.00730147
$921$	0	0
$922$	16.7249	0.550805
$923$	−14.2761	−0.469902
$924$	0	0
$925$	54.0720	1.77788
$926$	29.0442	0.954453
$927$	0	0
$928$	−49.6294	−1.62916
$929$	24.2736	0.796391	0.398196	−	0.917301i	$-0.369637\pi$
$929$	24.2736	0.796391	0.398196	+	0.917301i	$0.369637\pi$
$930$	0	0
$931$	14.0585	0.460749
$932$	−21.4646	−0.703098
$933$	0	0
$934$	−4.06771	−0.133100
$935$	−0.559592	−0.0183006
$936$	0	0
$937$	−39.9239	−1.30426	−0.652129	−	0.758108i	$-0.726124\pi$
$937$	−39.9239	−1.30426	−0.652129	+	0.758108i	$0.726124\pi$
$938$	−5.50913	−0.179879
$939$	0	0
$940$	0.125809	0.00410345
$941$	−28.3493	−0.924159	−0.462080	−	0.886838i	$-0.652897\pi$
$941$	−28.3493	−0.924159	−0.462080	+	0.886838i	$0.652897\pi$
$942$	0	0
$943$	−19.8980	−0.647968
$944$	20.8035	0.677095
$945$	0	0
$946$	9.92685	0.322750
$947$	−43.5574	−1.41542	−0.707712	−	0.706501i	$-0.750273\pi$
$947$	−43.5574	−1.41542	−0.707712	+	0.706501i	$0.750273\pi$
$948$	0	0
$949$	21.3203	0.692085
$950$	−63.2577	−2.05235
$951$	0	0
$952$	27.6331	0.895594
$953$	37.3580	1.21015	0.605073	−	0.796170i	$-0.293144\pi$
$953$	37.3580	1.21015	0.605073	+	0.796170i	$0.293144\pi$
$954$	0	0
$955$	1.05349	0.0340900
$956$	25.3070	0.818485
$957$	0	0
$958$	38.6246	1.24791
$959$	−11.3716	−0.367210
$960$	0	0
$961$	−30.9810	−0.999386
$962$	50.3995	1.62494
$963$	0	0
$964$	14.2349	0.458475
$965$	−0.971193	−0.0312638
$966$	0	0
$967$	57.8232	1.85947	0.929734	−	0.368233i	$-0.120037\pi$
$967$	57.8232	1.85947	0.929734	+	0.368233i	$0.120037\pi$
$968$	1.59824	0.0513695
$969$	0	0
$970$	−0.686806	−0.0220520
$971$	−13.1517	−0.422060	−0.211030	−	0.977480i	$-0.567682\pi$
$971$	−13.1517	−0.422060	−0.211030	+	0.977480i	$0.567682\pi$
$972$	0	0
$973$	24.8738	0.797419
$974$	70.6445	2.26360
$975$	0	0
$976$	−50.3152	−1.61055
$977$	55.2072	1.76623	0.883117	−	0.469153i	$-0.155441\pi$
$977$	55.2072	1.76623	0.883117	+	0.469153i	$0.155441\pi$
$978$	0	0
$979$	13.5895	0.434322
$980$	0.154824	0.00494569
$981$	0	0
$982$	8.34102	0.266173
$983$	−50.3290	−1.60525	−0.802623	−	0.596487i	$-0.796563\pi$
$983$	−50.3290	−1.60525	−0.802623	+	0.596487i	$0.796563\pi$
$984$	0	0
$985$	1.82572	0.0581722
$986$	−120.336	−3.83228
$987$	0	0
$988$	−20.8101	−0.662058
$989$	10.7588	0.342111
$990$	0	0
$991$	34.5053	1.09610	0.548048	−	0.836447i	$-0.315371\pi$
$991$	34.5053	1.09610	0.548048	+	0.836447i	$0.315371\pi$
$992$	0.769637	0.0244360
$993$	0	0
$994$	−21.2967	−0.675490
$995$	0.567736	0.0179984
$996$	0	0
$997$	41.0143	1.29894	0.649468	−	0.760389i	$-0.274991\pi$
$997$	41.0143	1.29894	0.649468	+	0.760389i	$0.274991\pi$
$998$	60.5719	1.91737
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8019.2.a.k.1.12		51
3.2	odd	2		8019.2.a.l.1.40		51
27.5	odd	18		297.2.j.c.133.13	yes	102
27.11	odd	18		297.2.j.c.67.13	✓	102
27.16	even	9		891.2.j.c.496.5		102
27.22	even	9		891.2.j.c.397.5		102

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
297.2.j.c.67.13	✓	102	27.11	odd	18
297.2.j.c.133.13	yes	102	27.5	odd	18
891.2.j.c.397.5		102	27.22	even	9
891.2.j.c.496.5		102	27.16	even	9
8019.2.a.k.1.12		51	1.1	even	1	trivial
8019.2.a.l.1.40		51	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)

\(f(q)\)	\(=\)	\(q-1.75810 q^{2} +1.09093 q^{4} -0.0727216 q^{5} -2.24688 q^{7} +1.59824 q^{8} +O(q^{10})\)	\(q-1.75810 q^{2} +1.09093 q^{4} -0.0727216 q^{5} -2.24688 q^{7} +1.59824 q^{8} +0.127852 q^{10} -1.00000 q^{11} +2.64801 q^{13} +3.95024 q^{14} -4.99173 q^{16} -7.69499 q^{17} -7.20375 q^{19} -0.0793340 q^{20} +1.75810 q^{22} +1.90545 q^{23} -4.99471 q^{25} -4.65548 q^{26} -2.45118 q^{28} -8.89496 q^{29} +0.137940 q^{31} +5.57950 q^{32} +13.5286 q^{34} +0.163396 q^{35} -10.8258 q^{37} +12.6649 q^{38} -0.116227 q^{40} -10.4427 q^{41} +5.64634 q^{43} -1.09093 q^{44} -3.34998 q^{46} -1.58582 q^{47} -1.95155 q^{49} +8.78122 q^{50} +2.88879 q^{52} -0.449984 q^{53} +0.0727216 q^{55} -3.59105 q^{56} +15.6383 q^{58} -4.16759 q^{59} +10.0797 q^{61} -0.242513 q^{62} +0.174134 q^{64} -0.192568 q^{65} -1.39463 q^{67} -8.39467 q^{68} -0.287268 q^{70} -5.39124 q^{71} +8.05143 q^{73} +19.0330 q^{74} -7.85877 q^{76} +2.24688 q^{77} +13.1394 q^{79} +0.363007 q^{80} +18.3593 q^{82} +10.3147 q^{83} +0.559592 q^{85} -9.92685 q^{86} -1.59824 q^{88} -13.5895 q^{89} -5.94975 q^{91} +2.07871 q^{92} +2.78803 q^{94} +0.523868 q^{95} -5.37188 q^{97} +3.43103 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 51 q + 60 q^{4} - 6 q^{5} + 12 q^{7}+O(q^{10}) \) 51 * q + 60 * q^4 - 6 * q^5 + 12 * q^7	\( 51 q + 60 q^{4} - 6 q^{5} + 12 q^{7} + 18 q^{10} - 51 q^{11} + 30 q^{13} - 12 q^{14} + 78 q^{16} + 30 q^{19} - 18 q^{20} - 3 q^{23} + 75 q^{25} - 9 q^{26} + 36 q^{28} + 42 q^{31} + 15 q^{32} + 42 q^{34} + 9 q^{35} + 48 q^{37} + 3 q^{38} + 54 q^{40} + 18 q^{43} - 60 q^{44} + 42 q^{46} - 30 q^{47} + 99 q^{49} + 30 q^{50} + 60 q^{52} - 18 q^{53} + 6 q^{55} - 21 q^{56} + 30 q^{58} - 24 q^{59} + 99 q^{61} + 114 q^{64} + 15 q^{65} + 39 q^{67} + 39 q^{68} + 48 q^{70} - 30 q^{71} + 69 q^{73} + 90 q^{76} - 12 q^{77} + 48 q^{79} - 42 q^{80} + 42 q^{82} + 21 q^{83} + 84 q^{85} - 24 q^{86} - 15 q^{89} + 69 q^{91} + 66 q^{92} + 66 q^{94} + 12 q^{95} + 72 q^{97} + 57 q^{98}+O(q^{100}) \) 51 * q + 60 * q^4 - 6 * q^5 + 12 * q^7 + 18 * q^10 - 51 * q^11 + 30 * q^13 - 12 * q^14 + 78 * q^16 + 30 * q^19 - 18 * q^20 - 3 * q^23 + 75 * q^25 - 9 * q^26 + 36 * q^28 + 42 * q^31 + 15 * q^32 + 42 * q^34 + 9 * q^35 + 48 * q^37 + 3 * q^38 + 54 * q^40 + 18 * q^43 - 60 * q^44 + 42 * q^46 - 30 * q^47 + 99 * q^49 + 30 * q^50 + 60 * q^52 - 18 * q^53 + 6 * q^55 - 21 * q^56 + 30 * q^58 - 24 * q^59 + 99 * q^61 + 114 * q^64 + 15 * q^65 + 39 * q^67 + 39 * q^68 + 48 * q^70 - 30 * q^71 + 69 * q^73 + 90 * q^76 - 12 * q^77 + 48 * q^79 - 42 * q^80 + 42 * q^82 + 21 * q^83 + 84 * q^85 - 24 * q^86 - 15 * q^89 + 69 * q^91 + 66 * q^92 + 66 * q^94 + 12 * q^95 + 72 * q^97 + 57 * q^98

Introduction

L-functions

Modular forms

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