LMFDB - Embedded newform 8019.2.a.b.1.3

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			1.3
Root			$1.87939$ 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)

$f(q)$	$=$	$q+1.87939 q^{2} +1.53209 q^{4} +2.18479 q^{5} +0.532089 q^{7} -0.879385 q^{8} +O(q^{10})$	\(q+1.87939 q^{2} +1.53209 q^{4} +2.18479 q^{5} +0.532089 q^{7} -0.879385 q^{8} +4.10607 q^{10} +1.00000 q^{11} -5.57398 q^{13} +1.00000 q^{14} -4.71688 q^{16} -5.87939 q^{17} -1.29086 q^{19} +3.34730 q^{20} +1.87939 q^{22} -3.06418 q^{23} -0.226682 q^{25} -10.4757 q^{26} +0.815207 q^{28} +8.58172 q^{29} +7.82295 q^{31} -7.10607 q^{32} -11.0496 q^{34} +1.16250 q^{35} -2.36959 q^{37} -2.42602 q^{38} -1.92127 q^{40} -7.47565 q^{41} +4.14796 q^{43} +1.53209 q^{44} -5.75877 q^{46} +0.290859 q^{47} -6.71688 q^{49} -0.426022 q^{50} -8.53983 q^{52} +3.93582 q^{53} +2.18479 q^{55} -0.467911 q^{56} +16.1284 q^{58} -3.69459 q^{59} -2.69459 q^{61} +14.7023 q^{62} -3.92127 q^{64} -12.1780 q^{65} -11.3773 q^{67} -9.00774 q^{68} +2.18479 q^{70} -14.3131 q^{71} +11.0915 q^{73} -4.45336 q^{74} -1.97771 q^{76} +0.532089 q^{77} -12.7811 q^{79} -10.3054 q^{80} -14.0496 q^{82} -7.27126 q^{83} -12.8452 q^{85} +7.79561 q^{86} -0.879385 q^{88} -8.51249 q^{89} -2.96585 q^{91} -4.69459 q^{92} +0.546637 q^{94} -2.82026 q^{95} -0.887126 q^{97} -12.6236 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.87939	1.32893	0.664463	−	0.747321i	$-0.268660\pi$
$2$	1.87939	1.32893	0.664463	+	0.747321i	$0.268660\pi$
$3$	0	0
$3$	0	0
$4$	1.53209	0.766044
$5$	2.18479	0.977069	0.488534	−	0.872545i	$-0.337532\pi$
$5$	2.18479	0.977069	0.488534	+	0.872545i	$0.337532\pi$
$6$	0	0
$7$	0.532089	0.201111	0.100555	−	0.994931i	$-0.467938\pi$
$7$	0.532089	0.201111	0.100555	+	0.994931i	$0.467938\pi$
$8$	−0.879385	−0.310910
$9$	0	0
$10$	4.10607	1.29845
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−5.57398	−1.54594	−0.772972	−	0.634441i	$-0.781231\pi$
$13$	−5.57398	−1.54594	−0.772972	+	0.634441i	$0.781231\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	−4.71688	−1.17922
$17$	−5.87939	−1.42596	−0.712980	−	0.701184i	$-0.752655\pi$
$17$	−5.87939	−1.42596	−0.712980	+	0.701184i	$0.752655\pi$
$18$	0	0
$19$	−1.29086	−0.296143	−0.148072	−	0.988977i	$-0.547307\pi$
$19$	−1.29086	−0.296143	−0.148072	+	0.988977i	$0.547307\pi$
$20$	3.34730	0.748478
$21$	0	0
$22$	1.87939	0.400686
$23$	−3.06418	−0.638925	−0.319463	−	0.947599i	$-0.603502\pi$
$23$	−3.06418	−0.638925	−0.319463	+	0.947599i	$0.603502\pi$
$24$	0	0
$25$	−0.226682	−0.0453363
$26$	−10.4757	−2.05444
$27$	0	0
$28$	0.815207	0.154060
$29$	8.58172	1.59359	0.796793	−	0.604253i	$-0.206528\pi$
$29$	8.58172	1.59359	0.796793	+	0.604253i	$0.206528\pi$
$30$	0	0
$31$	7.82295	1.40504	0.702521	−	0.711663i	$-0.252057\pi$
$31$	7.82295	1.40504	0.702521	+	0.711663i	$0.252057\pi$
$32$	−7.10607	−1.25619
$33$	0	0
$34$	−11.0496	−1.89500
$35$	1.16250	0.196499
$36$	0	0
$37$	−2.36959	−0.389557	−0.194779	−	0.980847i	$-0.562399\pi$
$37$	−2.36959	−0.389557	−0.194779	+	0.980847i	$0.562399\pi$
$38$	−2.42602	−0.393553
$39$	0	0
$40$	−1.92127	−0.303780
$41$	−7.47565	−1.16750	−0.583750	−	0.811933i	$-0.698415\pi$
$41$	−7.47565	−1.16750	−0.583750	+	0.811933i	$0.698415\pi$
$42$	0	0
$43$	4.14796	0.632557	0.316279	−	0.948666i	$-0.397567\pi$
$43$	4.14796	0.632557	0.316279	+	0.948666i	$0.397567\pi$
$44$	1.53209	0.230971
$45$	0	0
$46$	−5.75877	−0.849084
$47$	0.290859	0.0424262	0.0212131	−	0.999775i	$-0.493247\pi$
$47$	0.290859	0.0424262	0.0212131	+	0.999775i	$0.493247\pi$
$48$	0	0
$49$	−6.71688	−0.959554
$50$	−0.426022	−0.0602486
$51$	0	0
$52$	−8.53983	−1.18426
$53$	3.93582	0.540627	0.270313	−	0.962772i	$-0.412873\pi$
$53$	3.93582	0.540627	0.270313	+	0.962772i	$0.412873\pi$
$54$	0	0
$55$	2.18479	0.294597
$56$	−0.467911	−0.0625273
$57$	0	0
$58$	16.1284	2.11776
$59$	−3.69459	−0.480995	−0.240498	−	0.970650i	$-0.577311\pi$
$59$	−3.69459	−0.480995	−0.240498	+	0.970650i	$0.577311\pi$
$60$	0	0
$61$	−2.69459	−0.345007	−0.172504	−	0.985009i	$-0.555186\pi$
$61$	−2.69459	−0.345007	−0.172504	+	0.985009i	$0.555186\pi$
$62$	14.7023	1.86720
$63$	0	0
$64$	−3.92127	−0.490159
$65$	−12.1780	−1.51049
$66$	0	0
$67$	−11.3773	−1.38996	−0.694981	−	0.719028i	$-0.744587\pi$
$67$	−11.3773	−1.38996	−0.694981	+	0.719028i	$0.744587\pi$
$68$	−9.00774	−1.09235
$69$	0	0
$70$	2.18479	0.261133
$71$	−14.3131	−1.69866	−0.849329	−	0.527864i	$-0.822993\pi$
$71$	−14.3131	−1.69866	−0.849329	+	0.527864i	$0.822993\pi$
$72$	0	0
$73$	11.0915	1.29816	0.649082	−	0.760718i	$-0.275153\pi$
$73$	11.0915	1.29816	0.649082	+	0.760718i	$0.275153\pi$
$74$	−4.45336	−0.517693
$75$	0	0
$76$	−1.97771	−0.226859
$77$	0.532089	0.0606372
$78$	0	0
$79$	−12.7811	−1.43798	−0.718991	−	0.695020i	$-0.755396\pi$
$79$	−12.7811	−1.43798	−0.718991	+	0.695020i	$0.755396\pi$
$80$	−10.3054	−1.15218
$81$	0	0
$82$	−14.0496	−1.55152
$83$	−7.27126	−0.798124	−0.399062	−	0.916924i	$-0.630664\pi$
$83$	−7.27126	−0.798124	−0.399062	+	0.916924i	$0.630664\pi$
$84$	0	0
$85$	−12.8452	−1.39326
$86$	7.79561	0.840622
$87$	0	0
$88$	−0.879385	−0.0937428
$89$	−8.51249	−0.902322	−0.451161	−	0.892443i	$-0.648990\pi$
$89$	−8.51249	−0.902322	−0.451161	+	0.892443i	$0.648990\pi$
$90$	0	0
$91$	−2.96585	−0.310906
$92$	−4.69459	−0.489445
$93$	0	0
$94$	0.546637	0.0563813
$95$	−2.82026	−0.289353
$96$	0	0
$97$	−0.887126	−0.0900740	−0.0450370	−	0.998985i	$-0.514341\pi$
$97$	−0.887126	−0.0900740	−0.0450370	+	0.998985i	$0.514341\pi$
$98$	−12.6236	−1.27518
$99$	0	0
$100$	−0.347296	−0.0347296
$101$	−14.9513	−1.48771	−0.743855	−	0.668341i	$-0.767005\pi$
$101$	−14.9513	−1.48771	−0.743855	+	0.668341i	$0.767005\pi$
$102$	0	0
$103$	−3.75103	−0.369600	−0.184800	−	0.982776i	$-0.559164\pi$
$103$	−3.75103	−0.369600	−0.184800	+	0.982776i	$0.559164\pi$
$104$	4.90167	0.480649
$105$	0	0
$106$	7.39693	0.718453
$107$	4.95811	0.479319	0.239659	−	0.970857i	$-0.422964\pi$
$107$	4.95811	0.479319	0.239659	+	0.970857i	$0.422964\pi$
$108$	0	0
$109$	4.21213	0.403449	0.201725	−	0.979442i	$-0.435345\pi$
$109$	4.21213	0.403449	0.201725	+	0.979442i	$0.435345\pi$
$110$	4.10607	0.391498
$111$	0	0
$112$	−2.50980	−0.237154
$113$	−5.43882	−0.511641	−0.255820	−	0.966724i	$-0.582346\pi$
$113$	−5.43882	−0.511641	−0.255820	+	0.966724i	$0.582346\pi$
$114$	0	0
$115$	−6.69459	−0.624274
$116$	13.1480	1.22076
$117$	0	0
$118$	−6.94356	−0.639207
$119$	−3.12836	−0.286776
$120$	0	0
$121$	1.00000	0.0909091
$122$	−5.06418	−0.458489
$123$	0	0
$124$	11.9855	1.07633
$125$	−11.4192	−1.02137
$126$	0	0
$127$	−5.61587	−0.498328	−0.249164	−	0.968461i	$-0.580156\pi$
$127$	−5.61587	−0.498328	−0.249164	+	0.968461i	$0.580156\pi$
$128$	6.84255	0.604802
$129$	0	0
$130$	−22.8871	−2.00733
$131$	4.50980	0.394023	0.197012	−	0.980401i	$-0.436876\pi$
$131$	4.50980	0.394023	0.197012	+	0.980401i	$0.436876\pi$
$132$	0	0
$133$	−0.686852	−0.0595576
$134$	−21.3824	−1.84716
$135$	0	0
$136$	5.17024	0.443345
$137$	15.6091	1.33357	0.666786	−	0.745249i	$-0.267670\pi$
$137$	15.6091	1.33357	0.666786	+	0.745249i	$0.267670\pi$
$138$	0	0
$139$	−17.2490	−1.46304	−0.731519	−	0.681821i	$-0.761188\pi$
$139$	−17.2490	−1.46304	−0.731519	+	0.681821i	$0.761188\pi$
$140$	1.78106	0.150527
$141$	0	0
$142$	−26.8999	−2.25739
$143$	−5.57398	−0.466119
$144$	0	0
$145$	18.7493	1.55704
$146$	20.8452	1.72516
$147$	0	0
$148$	−3.63041	−0.298418
$149$	24.3969	1.99867	0.999337	−	0.0364073i	$-0.0115914\pi$
$149$	24.3969	1.99867	0.999337	+	0.0364073i	$0.0115914\pi$
$150$	0	0
$151$	17.4611	1.42096	0.710482	−	0.703715i	$-0.248477\pi$
$151$	17.4611	1.42096	0.710482	+	0.703715i	$0.248477\pi$
$152$	1.13516	0.0920739
$153$	0	0
$154$	1.00000	0.0805823
$155$	17.0915	1.37282
$156$	0	0
$157$	19.1070	1.52490	0.762452	−	0.647044i	$-0.223995\pi$
$157$	19.1070	1.52490	0.762452	+	0.647044i	$0.223995\pi$
$158$	−24.0205	−1.91097
$159$	0	0
$160$	−15.5253	−1.22738
$161$	−1.63041	−0.128495
$162$	0	0
$163$	−15.9017	−1.24552	−0.622758	−	0.782415i	$-0.713988\pi$
$163$	−15.9017	−1.24552	−0.622758	+	0.782415i	$0.713988\pi$
$164$	−11.4534	−0.894357
$165$	0	0
$166$	−13.6655	−1.06065
$167$	11.1206	0.860539	0.430270	−	0.902700i	$-0.358419\pi$
$167$	11.1206	0.860539	0.430270	+	0.902700i	$0.358419\pi$
$168$	0	0
$169$	18.0692	1.38994
$170$	−24.1411	−1.85154
$171$	0	0
$172$	6.35504	0.484567
$173$	−1.09833	−0.0835042	−0.0417521	−	0.999128i	$-0.513294\pi$
$173$	−1.09833	−0.0835042	−0.0417521	+	0.999128i	$0.513294\pi$
$174$	0	0
$175$	−0.120615	−0.00911762
$176$	−4.71688	−0.355548
$177$	0	0
$178$	−15.9982	−1.19912
$179$	−10.0051	−0.747813	−0.373906	−	0.927466i	$-0.621982\pi$
$179$	−10.0051	−0.747813	−0.373906	+	0.927466i	$0.621982\pi$
$180$	0	0
$181$	6.68779	0.497099	0.248550	−	0.968619i	$-0.420046\pi$
$181$	6.68779	0.497099	0.248550	+	0.968619i	$0.420046\pi$
$182$	−5.57398	−0.413171
$183$	0	0
$184$	2.69459	0.198648
$185$	−5.17705	−0.380624
$186$	0	0
$187$	−5.87939	−0.429943
$188$	0.445622	0.0325004
$189$	0	0
$190$	−5.30035	−0.384528
$191$	−7.64321	−0.553043	−0.276522	−	0.961008i	$-0.589182\pi$
$191$	−7.64321	−0.553043	−0.276522	+	0.961008i	$0.589182\pi$
$192$	0	0
$193$	2.45336	0.176597	0.0882985	−	0.996094i	$-0.471857\pi$
$193$	2.45336	0.176597	0.0882985	+	0.996094i	$0.471857\pi$
$194$	−1.66725	−0.119702
$195$	0	0
$196$	−10.2909	−0.735061
$197$	−14.2686	−1.01659	−0.508297	−	0.861182i	$-0.669725\pi$
$197$	−14.2686	−1.01659	−0.508297	+	0.861182i	$0.669725\pi$
$198$	0	0
$199$	−18.5175	−1.31267	−0.656337	−	0.754468i	$-0.727895\pi$
$199$	−18.5175	−1.31267	−0.656337	+	0.754468i	$0.727895\pi$
$200$	0.199340	0.0140955
$201$	0	0
$202$	−28.0993	−1.97706
$203$	4.56624	0.320487
$204$	0	0
$205$	−16.3327	−1.14073
$206$	−7.04963	−0.491171
$207$	0	0
$208$	26.2918	1.82301
$209$	−1.29086	−0.0892906
$210$	0	0
$211$	19.0223	1.30955	0.654774	−	0.755825i	$-0.272764\pi$
$211$	19.0223	1.30955	0.654774	+	0.755825i	$0.272764\pi$
$212$	6.03003	0.414144
$213$	0	0
$214$	9.31820	0.636979
$215$	9.06242	0.618052
$216$	0	0
$217$	4.16250	0.282569
$218$	7.91622	0.536154
$219$	0	0
$220$	3.34730	0.225675
$221$	32.7716	2.20445
$222$	0	0
$223$	13.2790	0.889228	0.444614	−	0.895722i	$-0.353341\pi$
$223$	13.2790	0.889228	0.444614	+	0.895722i	$0.353341\pi$
$224$	−3.78106	−0.252633
$225$	0	0
$226$	−10.2216	−0.679933
$227$	1.72193	0.114289	0.0571444	−	0.998366i	$-0.481800\pi$
$227$	1.72193	0.114289	0.0571444	+	0.998366i	$0.481800\pi$
$228$	0	0
$229$	−3.41828	−0.225886	−0.112943	−	0.993601i	$-0.536028\pi$
$229$	−3.41828	−0.225886	−0.112943	+	0.993601i	$0.536028\pi$
$230$	−12.5817	−0.829614
$231$	0	0
$232$	−7.54664	−0.495461
$233$	17.2763	1.13181	0.565904	−	0.824471i	$-0.308527\pi$
$233$	17.2763	1.13181	0.565904	+	0.824471i	$0.308527\pi$
$234$	0	0
$235$	0.635467	0.0414533
$236$	−5.66044	−0.368464
$237$	0	0
$238$	−5.87939	−0.381104
$239$	−17.1070	−1.10656	−0.553280	−	0.832995i	$-0.686624\pi$
$239$	−17.1070	−1.10656	−0.553280	+	0.832995i	$0.686624\pi$
$240$	0	0
$241$	21.8307	1.40624	0.703119	−	0.711072i	$-0.251790\pi$
$241$	21.8307	1.40624	0.703119	+	0.711072i	$0.251790\pi$
$242$	1.87939	0.120811
$243$	0	0
$244$	−4.12836	−0.264291
$245$	−14.6750	−0.937551
$246$	0	0
$247$	7.19522	0.457821
$248$	−6.87939	−0.436841
$249$	0	0
$250$	−21.4611	−1.35732
$251$	3.53209	0.222943	0.111472	−	0.993768i	$-0.464444\pi$
$251$	3.53209	0.222943	0.111472	+	0.993768i	$0.464444\pi$
$252$	0	0
$253$	−3.06418	−0.192643
$254$	−10.5544	−0.662241
$255$	0	0
$256$	20.7023	1.29390
$257$	5.22163	0.325716	0.162858	−	0.986649i	$-0.447929\pi$
$257$	5.22163	0.325716	0.162858	+	0.986649i	$0.447929\pi$
$258$	0	0
$259$	−1.26083	−0.0783442
$260$	−18.6578	−1.15710
$261$	0	0
$262$	8.47565	0.523628
$263$	−8.41653	−0.518985	−0.259493	−	0.965745i	$-0.583555\pi$
$263$	−8.41653	−0.518985	−0.259493	+	0.965745i	$0.583555\pi$
$264$	0	0
$265$	8.59896	0.528230
$266$	−1.29086	−0.0791477
$267$	0	0
$268$	−17.4311	−1.06477
$269$	−6.57903	−0.401131	−0.200565	−	0.979680i	$-0.564278\pi$
$269$	−6.57903	−0.401131	−0.200565	+	0.979680i	$0.564278\pi$
$270$	0	0
$271$	−7.10607	−0.431663	−0.215831	−	0.976431i	$-0.569246\pi$
$271$	−7.10607	−0.431663	−0.215831	+	0.976431i	$0.569246\pi$
$272$	27.7324	1.68152
$273$	0	0
$274$	29.3354	1.77222
$275$	−0.226682	−0.0136694
$276$	0	0
$277$	−4.35235	−0.261507	−0.130754	−	0.991415i	$-0.541740\pi$
$277$	−4.35235	−0.261507	−0.130754	+	0.991415i	$0.541740\pi$
$278$	−32.4175	−1.94427
$279$	0	0
$280$	−1.02229	−0.0610934
$281$	−20.4492	−1.21990	−0.609950	−	0.792440i	$-0.708810\pi$
$281$	−20.4492	−1.21990	−0.609950	+	0.792440i	$0.708810\pi$
$282$	0	0
$283$	10.2121	0.607048	0.303524	−	0.952824i	$-0.401837\pi$
$283$	10.2121	0.607048	0.303524	+	0.952824i	$0.401837\pi$
$284$	−21.9290	−1.30125
$285$	0	0
$286$	−10.4757	−0.619438
$287$	−3.97771	−0.234797
$288$	0	0
$289$	17.5672	1.03336
$290$	35.2371	2.06919
$291$	0	0
$292$	16.9932	0.994451
$293$	−12.5844	−0.735189	−0.367594	−	0.929986i	$-0.619819\pi$
$293$	−12.5844	−0.735189	−0.367594	+	0.929986i	$0.619819\pi$
$294$	0	0
$295$	−8.07192	−0.469965
$296$	2.08378	0.121117
$297$	0	0
$298$	45.8512	2.65609
$299$	17.0797	0.987742
$300$	0	0
$301$	2.20708	0.127214
$302$	32.8161	1.88836
$303$	0	0
$304$	6.08883	0.349218
$305$	−5.88713	−0.337096
$306$	0	0
$307$	−9.18304	−0.524104	−0.262052	−	0.965054i	$-0.584399\pi$
$307$	−9.18304	−0.524104	−0.262052	+	0.965054i	$0.584399\pi$
$308$	0.815207	0.0464508
$309$	0	0
$310$	32.1215	1.82438
$311$	28.1712	1.59744	0.798720	−	0.601702i	$-0.205511\pi$
$311$	28.1712	1.59744	0.798720	+	0.601702i	$0.205511\pi$
$312$	0	0
$313$	−7.03003	−0.397361	−0.198680	−	0.980064i	$-0.563666\pi$
$313$	−7.03003	−0.397361	−0.198680	+	0.980064i	$0.563666\pi$
$314$	35.9094	2.02649
$315$	0	0
$316$	−19.5817	−1.10156
$317$	19.7733	1.11058	0.555290	−	0.831657i	$-0.312607\pi$
$317$	19.7733	1.11058	0.555290	+	0.831657i	$0.312607\pi$
$318$	0	0
$319$	8.58172	0.480484
$320$	−8.56717	−0.478919
$321$	0	0
$322$	−3.06418	−0.170760
$323$	7.58946	0.422289
$324$	0	0
$325$	1.26352	0.0700874
$326$	−29.8854	−1.65520
$327$	0	0
$328$	6.57398	0.362987
$329$	0.154763	0.00853236
$330$	0	0
$331$	−11.8425	−0.650925	−0.325463	−	0.945555i	$-0.605520\pi$
$331$	−11.8425	−0.650925	−0.325463	+	0.945555i	$0.605520\pi$
$332$	−11.1402	−0.611399
$333$	0	0
$334$	20.8999	1.14359
$335$	−24.8571	−1.35809
$336$	0	0
$337$	11.1429	0.606993	0.303496	−	0.952833i	$-0.401846\pi$
$337$	11.1429	0.606993	0.303496	+	0.952833i	$0.401846\pi$
$338$	33.9590	1.84713
$339$	0	0
$340$	−19.6800	−1.06730
$341$	7.82295	0.423636
$342$	0	0
$343$	−7.29860	−0.394087
$344$	−3.64765	−0.196668
$345$	0	0
$346$	−2.06418	−0.110971
$347$	15.1310	0.812277	0.406138	−	0.913812i	$-0.366875\pi$
$347$	15.1310	0.812277	0.406138	+	0.913812i	$0.366875\pi$
$348$	0	0
$349$	2.03415	0.108885	0.0544427	−	0.998517i	$-0.482662\pi$
$349$	2.03415	0.108885	0.0544427	+	0.998517i	$0.482662\pi$
$350$	−0.226682	−0.0121166
$351$	0	0
$352$	−7.10607	−0.378755
$353$	−34.9249	−1.85886	−0.929432	−	0.368993i	$-0.879703\pi$
$353$	−34.9249	−1.85886	−0.929432	+	0.368993i	$0.879703\pi$
$354$	0	0
$355$	−31.2713	−1.65971
$356$	−13.0419	−0.691219
$357$	0	0
$358$	−18.8033	−0.993788
$359$	21.0506	1.11101	0.555503	−	0.831514i	$-0.312526\pi$
$359$	21.0506	1.11101	0.555503	+	0.831514i	$0.312526\pi$
$360$	0	0
$361$	−17.3337	−0.912299
$362$	12.5689	0.660608
$363$	0	0
$364$	−4.54395	−0.238168
$365$	24.2327	1.26840
$366$	0	0
$367$	−36.1043	−1.88463	−0.942315	−	0.334728i	$-0.891356\pi$
$367$	−36.1043	−1.88463	−0.942315	+	0.334728i	$0.891356\pi$
$368$	14.4534	0.753434
$369$	0	0
$370$	−9.72967	−0.505822
$371$	2.09421	0.108726
$372$	0	0
$373$	−9.55262	−0.494616	−0.247308	−	0.968937i	$-0.579546\pi$
$373$	−9.55262	−0.494616	−0.247308	+	0.968937i	$0.579546\pi$
$374$	−11.0496	−0.571363
$375$	0	0
$376$	−0.255777	−0.0131907
$377$	−47.8343	−2.46359
$378$	0	0
$379$	−4.12061	−0.211662	−0.105831	−	0.994384i	$-0.533750\pi$
$379$	−4.12061	−0.211662	−0.105831	+	0.994384i	$0.533750\pi$
$380$	−4.32089	−0.221657
$381$	0	0
$382$	−14.3645	−0.734953
$383$	−10.2558	−0.524046	−0.262023	−	0.965062i	$-0.584390\pi$
$383$	−10.2558	−0.524046	−0.262023	+	0.965062i	$0.584390\pi$
$384$	0	0
$385$	1.16250	0.0592467
$386$	4.61081	0.234684
$387$	0	0
$388$	−1.35916	−0.0690007
$389$	−3.79292	−0.192309	−0.0961543	−	0.995366i	$-0.530654\pi$
$389$	−3.79292	−0.192309	−0.0961543	+	0.995366i	$0.530654\pi$
$390$	0	0
$391$	18.0155	0.911082
$392$	5.90673	0.298335
$393$	0	0
$394$	−26.8161	−1.35098
$395$	−27.9240	−1.40501
$396$	0	0
$397$	34.5303	1.73303	0.866514	−	0.499153i	$-0.166355\pi$
$397$	34.5303	1.73303	0.866514	+	0.499153i	$0.166355\pi$
$398$	−34.8016	−1.74445
$399$	0	0
$400$	1.06923	0.0534615
$401$	35.7401	1.78478	0.892388	−	0.451269i	$-0.149029\pi$
$401$	35.7401	1.78478	0.892388	+	0.451269i	$0.149029\pi$
$402$	0	0
$403$	−43.6049	−2.17212
$404$	−22.9067	−1.13965
$405$	0	0
$406$	8.58172	0.425904
$407$	−2.36959	−0.117456
$408$	0	0
$409$	−9.74329	−0.481775	−0.240887	−	0.970553i	$-0.577438\pi$
$409$	−9.74329	−0.481775	−0.240887	+	0.970553i	$0.577438\pi$
$410$	−30.6955	−1.51594
$411$	0	0
$412$	−5.74691	−0.283130
$413$	−1.96585	−0.0967332
$414$	0	0
$415$	−15.8862	−0.779823
$416$	39.6091	1.94199
$417$	0	0
$418$	−2.42602	−0.118661
$419$	−22.9418	−1.12078	−0.560390	−	0.828229i	$-0.689349\pi$
$419$	−22.9418	−1.12078	−0.560390	+	0.828229i	$0.689349\pi$
$420$	0	0
$421$	−10.7811	−0.525437	−0.262718	−	0.964873i	$-0.584619\pi$
$421$	−10.7811	−0.525437	−0.262718	+	0.964873i	$0.584619\pi$
$422$	35.7502	1.74029
$423$	0	0
$424$	−3.46110	−0.168086
$425$	1.33275	0.0646478
$426$	0	0
$427$	−1.43376	−0.0693846
$428$	7.59627	0.367179
$429$	0	0
$430$	17.0318	0.821346
$431$	19.8348	0.955409	0.477705	−	0.878521i	$-0.341469\pi$
$431$	19.8348	0.955409	0.477705	+	0.878521i	$0.341469\pi$
$432$	0	0
$433$	−24.6851	−1.18629	−0.593145	−	0.805096i	$-0.702114\pi$
$433$	−24.6851	−1.18629	−0.593145	+	0.805096i	$0.702114\pi$
$434$	7.82295	0.375514
$435$	0	0
$436$	6.45336	0.309060
$437$	3.95542	0.189214
$438$	0	0
$439$	−11.5808	−0.552721	−0.276360	−	0.961054i	$-0.589128\pi$
$439$	−11.5808	−0.552721	−0.276360	+	0.961054i	$0.589128\pi$
$440$	−1.92127	−0.0915932
$441$	0	0
$442$	61.5904	2.92956
$443$	15.9632	0.758433	0.379216	−	0.925308i	$-0.376194\pi$
$443$	15.9632	0.758433	0.379216	+	0.925308i	$0.376194\pi$
$444$	0	0
$445$	−18.5980	−0.881631
$446$	24.9564	1.18172
$447$	0	0
$448$	−2.08647	−0.0985763
$449$	−29.4834	−1.39141	−0.695704	−	0.718329i	$-0.744907\pi$
$449$	−29.4834	−1.39141	−0.695704	+	0.718329i	$0.744907\pi$
$450$	0	0
$451$	−7.47565	−0.352015
$452$	−8.33275	−0.391940
$453$	0	0
$454$	3.23618	0.151881
$455$	−6.47977	−0.303776
$456$	0	0
$457$	−21.7861	−1.01911	−0.509556	−	0.860438i	$-0.670190\pi$
$457$	−21.7861	−1.01911	−0.509556	+	0.860438i	$0.670190\pi$
$458$	−6.42427	−0.300186
$459$	0	0
$460$	−10.2567	−0.478222
$461$	9.78013	0.455506	0.227753	−	0.973719i	$-0.426862\pi$
$461$	9.78013	0.455506	0.227753	+	0.973719i	$0.426862\pi$
$462$	0	0
$463$	41.4662	1.92710	0.963548	−	0.267536i	$-0.0862094\pi$
$463$	41.4662	1.92710	0.963548	+	0.267536i	$0.0862094\pi$
$464$	−40.4789	−1.87919
$465$	0	0
$466$	32.4688	1.50409
$467$	10.1411	0.469276	0.234638	−	0.972083i	$-0.424609\pi$
$467$	10.1411	0.469276	0.234638	+	0.972083i	$0.424609\pi$
$468$	0	0
$469$	−6.05375	−0.279536
$470$	1.19429	0.0550884
$471$	0	0
$472$	3.24897	0.149546
$473$	4.14796	0.190723
$474$	0	0
$475$	0.292614	0.0134261
$476$	−4.79292	−0.219683
$477$	0	0
$478$	−32.1506	−1.47054
$479$	23.7597	1.08561	0.542804	−	0.839859i	$-0.317363\pi$
$479$	23.7597	1.08561	0.542804	+	0.839859i	$0.317363\pi$
$480$	0	0
$481$	13.2080	0.602234
$482$	41.0283	1.86879
$483$	0	0
$484$	1.53209	0.0696404
$485$	−1.93819	−0.0880085
$486$	0	0
$487$	−16.0865	−0.728947	−0.364474	−	0.931214i	$-0.618751\pi$
$487$	−16.0865	−0.728947	−0.364474	+	0.931214i	$0.618751\pi$
$488$	2.36959	0.107266
$489$	0	0
$490$	−27.5800	−1.24594
$491$	−1.98276	−0.0894809	−0.0447404	−	0.998999i	$-0.514246\pi$
$491$	−1.98276	−0.0894809	−0.0447404	+	0.998999i	$0.514246\pi$
$492$	0	0
$493$	−50.4552	−2.27239
$494$	13.5226	0.608410
$495$	0	0
$496$	−36.8999	−1.65686
$497$	−7.61587	−0.341618
$498$	0	0
$499$	17.0669	0.764018	0.382009	−	0.924159i	$-0.375232\pi$
$499$	17.0669	0.764018	0.382009	+	0.924159i	$0.375232\pi$
$500$	−17.4953	−0.782411
$501$	0	0
$502$	6.63816	0.296275
$503$	−11.2121	−0.499924	−0.249962	−	0.968256i	$-0.580418\pi$
$503$	−11.2121	−0.499924	−0.249962	+	0.968256i	$0.580418\pi$
$504$	0	0
$505$	−32.6655	−1.45360
$506$	−5.75877	−0.256009
$507$	0	0
$508$	−8.60401	−0.381741
$509$	33.3874	1.47987	0.739936	−	0.672677i	$-0.234856\pi$
$509$	33.3874	1.47987	0.739936	+	0.672677i	$0.234856\pi$
$510$	0	0
$511$	5.90167	0.261075
$512$	25.2226	1.11469
$513$	0	0
$514$	9.81345	0.432853
$515$	−8.19522	−0.361125
$516$	0	0
$517$	0.290859	0.0127920
$518$	−2.36959	−0.104114
$519$	0	0
$520$	10.7091	0.469627
$521$	−27.5868	−1.20860	−0.604299	−	0.796757i	$-0.706547\pi$
$521$	−27.5868	−1.20860	−0.604299	+	0.796757i	$0.706547\pi$
$522$	0	0
$523$	42.7128	1.86770	0.933849	−	0.357667i	$-0.116428\pi$
$523$	42.7128	1.86770	0.933849	+	0.357667i	$0.116428\pi$
$524$	6.90941	0.301839
$525$	0	0
$526$	−15.8179	−0.689693
$527$	−45.9941	−2.00354
$528$	0	0
$529$	−13.6108	−0.591775
$530$	16.1607	0.701978
$531$	0	0
$532$	−1.05232	−0.0456238
$533$	41.6691	1.80489
$534$	0	0
$535$	10.8324	0.468327
$536$	10.0051	0.432152
$537$	0	0
$538$	−12.3645	−0.533073
$539$	−6.71688	−0.289317
$540$	0	0
$541$	−27.4593	−1.18057	−0.590285	−	0.807195i	$-0.700985\pi$
$541$	−27.4593	−1.18057	−0.590285	+	0.807195i	$0.700985\pi$
$542$	−13.3550	−0.573648
$543$	0	0
$544$	41.7793	1.79127
$545$	9.20264	0.394198
$546$	0	0
$547$	29.3378	1.25439	0.627197	−	0.778861i	$-0.284202\pi$
$547$	29.3378	1.25439	0.627197	+	0.778861i	$0.284202\pi$
$548$	23.9145	1.02158
$549$	0	0
$550$	−0.426022	−0.0181656
$551$	−11.0778	−0.471930
$552$	0	0
$553$	−6.80066	−0.289193
$554$	−8.17974	−0.347524
$555$	0	0
$556$	−26.4270	−1.12075
$557$	43.9154	1.86076	0.930378	−	0.366603i	$-0.119479\pi$
$557$	43.9154	1.86076	0.930378	+	0.366603i	$0.119479\pi$
$558$	0	0
$559$	−23.1206	−0.977898
$560$	−5.48339	−0.231716
$561$	0	0
$562$	−38.4320	−1.62116
$563$	27.8135	1.17220	0.586099	−	0.810240i	$-0.300663\pi$
$563$	27.8135	1.17220	0.586099	+	0.810240i	$0.300663\pi$
$564$	0	0
$565$	−11.8827	−0.499908
$566$	19.1925	0.806722
$567$	0	0
$568$	12.5868	0.528129
$569$	−26.1762	−1.09736	−0.548682	−	0.836031i	$-0.684870\pi$
$569$	−26.1762	−1.09736	−0.548682	+	0.836031i	$0.684870\pi$
$570$	0	0
$571$	−29.1043	−1.21798	−0.608989	−	0.793179i	$-0.708425\pi$
$571$	−29.1043	−1.21798	−0.608989	+	0.793179i	$0.708425\pi$
$572$	−8.53983	−0.357068
$573$	0	0
$574$	−7.47565	−0.312028
$575$	0.694593	0.0289665
$576$	0	0
$577$	−4.27807	−0.178098	−0.0890491	−	0.996027i	$-0.528383\pi$
$577$	−4.27807	−0.178098	−0.0890491	+	0.996027i	$0.528383\pi$
$578$	33.0155	1.37326
$579$	0	0
$580$	28.7256	1.19276
$581$	−3.86896	−0.160511
$582$	0	0
$583$	3.93582	0.163005
$584$	−9.75372	−0.403612
$585$	0	0
$586$	−23.6509	−0.977012
$587$	−33.9837	−1.40266	−0.701329	−	0.712838i	$-0.747409\pi$
$587$	−33.9837	−1.40266	−0.701329	+	0.712838i	$0.747409\pi$
$588$	0	0
$589$	−10.0983	−0.416094
$590$	−15.1702	−0.624549
$591$	0	0
$592$	11.1771	0.459374
$593$	9.34730	0.383847	0.191924	−	0.981410i	$-0.438527\pi$
$593$	9.34730	0.383847	0.191924	+	0.981410i	$0.438527\pi$
$594$	0	0
$595$	−6.83481	−0.280200
$596$	37.3783	1.53107
$597$	0	0
$598$	32.0993	1.31264
$599$	34.2249	1.39839	0.699196	−	0.714930i	$-0.253541\pi$
$599$	34.2249	1.39839	0.699196	+	0.714930i	$0.253541\pi$
$600$	0	0
$601$	−39.5449	−1.61307	−0.806535	−	0.591187i	$-0.798660\pi$
$601$	−39.5449	−1.61307	−0.806535	+	0.591187i	$0.798660\pi$
$602$	4.14796	0.169058
$603$	0	0
$604$	26.7520	1.08852
$605$	2.18479	0.0888244
$606$	0	0
$607$	21.9881	0.892471	0.446236	−	0.894916i	$-0.352764\pi$
$607$	21.9881	0.892471	0.446236	+	0.894916i	$0.352764\pi$
$608$	9.17293	0.372012
$609$	0	0
$610$	−11.0642	−0.447975
$611$	−1.62124	−0.0655885
$612$	0	0
$613$	−26.3277	−1.06337	−0.531683	−	0.846944i	$-0.678440\pi$
$613$	−26.3277	−1.06337	−0.531683	+	0.846944i	$0.678440\pi$
$614$	−17.2585	−0.696495
$615$	0	0
$616$	−0.467911	−0.0188527
$617$	−0.667252	−0.0268625	−0.0134313	−	0.999910i	$-0.504275\pi$
$617$	−0.667252	−0.0268625	−0.0134313	+	0.999910i	$0.504275\pi$
$618$	0	0
$619$	0.310460	0.0124784	0.00623922	−	0.999981i	$-0.498014\pi$
$619$	0.310460	0.0124784	0.00623922	+	0.999981i	$0.498014\pi$
$620$	26.1857	1.05164
$621$	0	0
$622$	52.9445	2.12288
$623$	−4.52940	−0.181467
$624$	0	0
$625$	−23.8152	−0.952608
$626$	−13.2121	−0.528063
$627$	0	0
$628$	29.2736	1.16814
$629$	13.9317	0.555493
$630$	0	0
$631$	−31.3131	−1.24656	−0.623278	−	0.782000i	$-0.714200\pi$
$631$	−31.3131	−1.24656	−0.623278	+	0.782000i	$0.714200\pi$
$632$	11.2395	0.447082
$633$	0	0
$634$	37.1617	1.47588
$635$	−12.2695	−0.486900
$636$	0	0
$637$	37.4397	1.48342
$638$	16.1284	0.638528
$639$	0	0
$640$	14.9495	0.590933
$641$	−34.9513	−1.38049	−0.690247	−	0.723574i	$-0.742498\pi$
$641$	−34.9513	−1.38049	−0.690247	+	0.723574i	$0.742498\pi$
$642$	0	0
$643$	−13.1165	−0.517264	−0.258632	−	0.965976i	$-0.583272\pi$
$643$	−13.1165	−0.517264	−0.258632	+	0.965976i	$0.583272\pi$
$644$	−2.49794	−0.0984326
$645$	0	0
$646$	14.2635	0.561191
$647$	20.6108	0.810295	0.405147	−	0.914251i	$-0.367220\pi$
$647$	20.6108	0.810295	0.405147	+	0.914251i	$0.367220\pi$
$648$	0	0
$649$	−3.69459	−0.145025
$650$	2.37464	0.0931409
$651$	0	0
$652$	−24.3628	−0.954120
$653$	6.20945	0.242994	0.121497	−	0.992592i	$-0.461230\pi$
$653$	6.20945	0.242994	0.121497	+	0.992592i	$0.461230\pi$
$654$	0	0
$655$	9.85298	0.384988
$656$	35.2618	1.37674
$657$	0	0
$658$	0.290859	0.0113389
$659$	16.4807	0.641997	0.320999	−	0.947080i	$-0.395982\pi$
$659$	16.4807	0.641997	0.320999	+	0.947080i	$0.395982\pi$
$660$	0	0
$661$	−46.7306	−1.81761	−0.908805	−	0.417221i	$-0.863004\pi$
$661$	−46.7306	−1.81761	−0.908805	+	0.417221i	$0.863004\pi$
$662$	−22.2567	−0.865032
$663$	0	0
$664$	6.39424	0.248145
$665$	−1.50063	−0.0581919
$666$	0	0
$667$	−26.2959	−1.01818
$668$	17.0378	0.659211
$669$	0	0
$670$	−46.7161	−1.80480
$671$	−2.69459	−0.104024
$672$	0	0
$673$	−4.09327	−0.157784	−0.0788921	−	0.996883i	$-0.525138\pi$
$673$	−4.09327	−0.157784	−0.0788921	+	0.996883i	$0.525138\pi$
$674$	20.9418	0.806648
$675$	0	0
$676$	27.6837	1.06476
$677$	35.8999	1.37975	0.689873	−	0.723930i	$-0.257666\pi$
$677$	35.8999	1.37975	0.689873	+	0.723930i	$0.257666\pi$
$678$	0	0
$679$	−0.472030	−0.0181148
$680$	11.2959	0.433178
$681$	0	0
$682$	14.7023	0.562981
$683$	8.87021	0.339409	0.169705	−	0.985495i	$-0.445719\pi$
$683$	8.87021	0.339409	0.169705	+	0.985495i	$0.445719\pi$
$684$	0	0
$685$	34.1026	1.30299
$686$	−13.7169	−0.523713
$687$	0	0
$688$	−19.5654	−0.745924
$689$	−21.9382	−0.835778
$690$	0	0
$691$	−32.3277	−1.22980	−0.614902	−	0.788604i	$-0.710804\pi$
$691$	−32.3277	−1.22980	−0.614902	+	0.788604i	$0.710804\pi$
$692$	−1.68273	−0.0639679
$693$	0	0
$694$	28.4371	1.07946
$695$	−37.6854	−1.42949
$696$	0	0
$697$	43.9522	1.66481
$698$	3.82295	0.144701
$699$	0	0
$700$	−0.184793	−0.00698450
$701$	15.9905	0.603953	0.301977	−	0.953315i	$-0.402354\pi$
$701$	15.9905	0.603953	0.301977	+	0.953315i	$0.402354\pi$
$702$	0	0
$703$	3.05880	0.115365
$704$	−3.92127	−0.147789
$705$	0	0
$706$	−65.6373	−2.47029
$707$	−7.95542	−0.299194
$708$	0	0
$709$	49.5577	1.86118	0.930589	−	0.366066i	$-0.119296\pi$
$709$	49.5577	1.86118	0.930589	+	0.366066i	$0.119296\pi$
$710$	−58.7707	−2.20563
$711$	0	0
$712$	7.48576	0.280541
$713$	−23.9709	−0.897717
$714$	0	0
$715$	−12.1780	−0.455431
$716$	−15.3286	−0.572858
$717$	0	0
$718$	39.5621	1.47645
$719$	30.4483	1.13553	0.567765	−	0.823191i	$-0.307808\pi$
$719$	30.4483	1.13553	0.567765	+	0.823191i	$0.307808\pi$
$720$	0	0
$721$	−1.99588	−0.0743305
$722$	−32.5767	−1.21238
$723$	0	0
$724$	10.2463	0.380800
$725$	−1.94532	−0.0722473
$726$	0	0
$727$	−15.3942	−0.570941	−0.285470	−	0.958388i	$-0.592150\pi$
$727$	−15.3942	−0.570941	−0.285470	+	0.958388i	$0.592150\pi$
$728$	2.60813	0.0966636
$729$	0	0
$730$	45.5425	1.68560
$731$	−24.3874	−0.902002
$732$	0	0
$733$	34.0283	1.25686	0.628432	−	0.777865i	$-0.283697\pi$
$733$	34.0283	1.25686	0.628432	+	0.777865i	$0.283697\pi$
$734$	−67.8539	−2.50453
$735$	0	0
$736$	21.7743	0.802610
$737$	−11.3773	−0.419089
$738$	0	0
$739$	41.1857	1.51504	0.757521	−	0.652811i	$-0.226410\pi$
$739$	41.1857	1.51504	0.757521	+	0.652811i	$0.226410\pi$
$740$	−7.93170	−0.291575
$741$	0	0
$742$	3.93582	0.144489
$743$	38.2576	1.40354	0.701768	−	0.712405i	$-0.252394\pi$
$743$	38.2576	1.40354	0.701768	+	0.712405i	$0.252394\pi$
$744$	0	0
$745$	53.3022	1.95284
$746$	−17.9531	−0.657308
$747$	0	0
$748$	−9.00774	−0.329356
$749$	2.63816	0.0963961
$750$	0	0
$751$	−16.6254	−0.606668	−0.303334	−	0.952884i	$-0.598100\pi$
$751$	−16.6254	−0.606668	−0.303334	+	0.952884i	$0.598100\pi$
$752$	−1.37195	−0.0500298
$753$	0	0
$754$	−89.8991	−3.27393
$755$	38.1489	1.38838
$756$	0	0
$757$	18.8220	0.684098	0.342049	−	0.939682i	$-0.388879\pi$
$757$	18.8220	0.684098	0.342049	+	0.939682i	$0.388879\pi$
$758$	−7.74422	−0.281283
$759$	0	0
$760$	2.48009	0.0899625
$761$	11.3396	0.411059	0.205529	−	0.978651i	$-0.434108\pi$
$761$	11.3396	0.411059	0.205529	+	0.978651i	$0.434108\pi$
$762$	0	0
$763$	2.24123	0.0811380
$764$	−11.7101	−0.423656
$765$	0	0
$766$	−19.2746	−0.696418
$767$	20.5936	0.743591
$768$	0	0
$769$	−16.2017	−0.584248	−0.292124	−	0.956380i	$-0.594362\pi$
$769$	−16.2017	−0.584248	−0.292124	+	0.956380i	$0.594362\pi$
$770$	2.18479	0.0787345
$771$	0	0
$772$	3.75877	0.135281
$773$	0.836563	0.0300891	0.0150445	−	0.999887i	$-0.495211\pi$
$773$	0.836563	0.0300891	0.0150445	+	0.999887i	$0.495211\pi$
$774$	0	0
$775$	−1.77332	−0.0636995
$776$	0.780126	0.0280049
$777$	0	0
$778$	−7.12836	−0.255564
$779$	9.65002	0.345748
$780$	0	0
$781$	−14.3131	−0.512165
$782$	33.8580	1.21076
$783$	0	0
$784$	31.6827	1.13153
$785$	41.7448	1.48994
$786$	0	0
$787$	32.9905	1.17598	0.587992	−	0.808867i	$-0.299919\pi$
$787$	32.9905	1.17598	0.587992	+	0.808867i	$0.299919\pi$
$788$	−21.8607	−0.778756
$789$	0	0
$790$	−52.4799	−1.86715
$791$	−2.89393	−0.102896
$792$	0	0
$793$	15.0196	0.533362
$794$	64.8958	2.30307
$795$	0	0
$796$	−28.3705	−1.00557
$797$	−2.03952	−0.0722437	−0.0361218	−	0.999347i	$-0.511500\pi$
$797$	−2.03952	−0.0722437	−0.0361218	+	0.999347i	$0.511500\pi$
$798$	0	0
$799$	−1.71007	−0.0604981
$800$	1.61081	0.0569509
$801$	0	0
$802$	67.1694	2.37183
$803$	11.0915	0.391411
$804$	0	0
$805$	−3.56212	−0.125548
$806$	−81.9505	−2.88658
$807$	0	0
$808$	13.1480	0.462543
$809$	−34.4742	−1.21205	−0.606025	−	0.795446i	$-0.707237\pi$
$809$	−34.4742	−1.21205	−0.606025	+	0.795446i	$0.707237\pi$
$810$	0	0
$811$	−41.6340	−1.46197	−0.730984	−	0.682394i	$-0.760939\pi$
$811$	−41.6340	−1.46197	−0.730984	+	0.682394i	$0.760939\pi$
$812$	6.99588	0.245507
$813$	0	0
$814$	−4.45336	−0.156090
$815$	−34.7419	−1.21695
$816$	0	0
$817$	−5.35443	−0.187328
$818$	−18.3114	−0.640243
$819$	0	0
$820$	−25.0232	−0.873849
$821$	45.8607	1.60055	0.800275	−	0.599633i	$-0.204687\pi$
$821$	45.8607	1.60055	0.800275	+	0.599633i	$0.204687\pi$
$822$	0	0
$823$	51.3070	1.78845	0.894224	−	0.447620i	$-0.147728\pi$
$823$	51.3070	1.78845	0.894224	+	0.447620i	$0.147728\pi$
$824$	3.29860	0.114912
$825$	0	0
$826$	−3.69459	−0.128551
$827$	54.9941	1.91233	0.956167	−	0.292823i	$-0.0945948\pi$
$827$	54.9941	1.91233	0.956167	+	0.292823i	$0.0945948\pi$
$828$	0	0
$829$	−31.6382	−1.09884	−0.549419	−	0.835547i	$-0.685151\pi$
$829$	−31.6382	−1.09884	−0.549419	+	0.835547i	$0.685151\pi$
$830$	−29.8563	−1.03633
$831$	0	0
$832$	21.8571	0.757758
$833$	39.4911	1.36829
$834$	0	0
$835$	24.2962	0.840806
$836$	−1.97771	−0.0684006
$837$	0	0
$838$	−43.1165	−1.48943
$839$	31.8185	1.09850	0.549248	−	0.835659i	$-0.314914\pi$
$839$	31.8185	1.09850	0.549248	+	0.835659i	$0.314914\pi$
$840$	0	0
$841$	44.6459	1.53951
$842$	−20.2618	−0.698266
$843$	0	0
$844$	29.1438	1.00317
$845$	39.4775	1.35807
$846$	0	0
$847$	0.532089	0.0182828
$848$	−18.5648	−0.637518
$849$	0	0
$850$	2.50475	0.0859121
$851$	7.26083	0.248898
$852$	0	0
$853$	−45.4757	−1.55706	−0.778528	−	0.627609i	$-0.784033\pi$
$853$	−45.4757	−1.55706	−0.778528	+	0.627609i	$0.784033\pi$
$854$	−2.69459	−0.0922071
$855$	0	0
$856$	−4.36009	−0.149025
$857$	38.7689	1.32432	0.662160	−	0.749363i	$-0.269640\pi$
$857$	38.7689	1.32432	0.662160	+	0.749363i	$0.269640\pi$
$858$	0	0
$859$	8.91952	0.304330	0.152165	−	0.988355i	$-0.451375\pi$
$859$	8.91952	0.304330	0.152165	+	0.988355i	$0.451375\pi$
$860$	13.8844	0.473455
$861$	0	0
$862$	37.2772	1.26967
$863$	42.8057	1.45712	0.728562	−	0.684980i	$-0.240189\pi$
$863$	42.8057	1.45712	0.728562	+	0.684980i	$0.240189\pi$
$864$	0	0
$865$	−2.39961	−0.0815893
$866$	−46.3928	−1.57649
$867$	0	0
$868$	6.37733	0.216461
$869$	−12.7811	−0.433568
$870$	0	0
$871$	63.4170	2.14880
$872$	−3.70409	−0.125436
$873$	0	0
$874$	7.43376	0.251451
$875$	−6.07604	−0.205408
$876$	0	0
$877$	−6.21213	−0.209769	−0.104884	−	0.994484i	$-0.533447\pi$
$877$	−6.21213	−0.209769	−0.104884	+	0.994484i	$0.533447\pi$
$878$	−21.7648	−0.734525
$879$	0	0
$880$	−10.3054	−0.347395
$881$	−13.9314	−0.469360	−0.234680	−	0.972073i	$-0.575404\pi$
$881$	−13.9314	−0.469360	−0.234680	+	0.972073i	$0.575404\pi$
$882$	0	0
$883$	−50.2826	−1.69214	−0.846072	−	0.533068i	$-0.821039\pi$
$883$	−50.2826	−1.69214	−0.846072	+	0.533068i	$0.821039\pi$
$884$	50.2089	1.68871
$885$	0	0
$886$	30.0009	1.00790
$887$	15.0719	0.506065	0.253033	−	0.967458i	$-0.418572\pi$
$887$	15.0719	0.506065	0.253033	+	0.967458i	$0.418572\pi$
$888$	0	0
$889$	−2.98814	−0.100219
$890$	−34.9528	−1.17162
$891$	0	0
$892$	20.3446	0.681188
$893$	−0.375459	−0.0125642
$894$	0	0
$895$	−21.8590	−0.730665
$896$	3.64084	0.121632
$897$	0	0
$898$	−55.4107	−1.84908
$899$	67.1343	2.23906
$900$	0	0
$901$	−23.1402	−0.770912
$902$	−14.0496	−0.467802
$903$	0	0
$904$	4.78281	0.159074
$905$	14.6114	0.485700
$906$	0	0
$907$	45.7897	1.52042	0.760212	−	0.649676i	$-0.225095\pi$
$907$	45.7897	1.52042	0.760212	+	0.649676i	$0.225095\pi$
$908$	2.63816	0.0875503
$909$	0	0
$910$	−12.1780	−0.403696
$911$	−19.2148	−0.636615	−0.318308	−	0.947987i	$-0.603115\pi$
$911$	−19.2148	−0.636615	−0.318308	+	0.947987i	$0.603115\pi$
$912$	0	0
$913$	−7.27126	−0.240644
$914$	−40.9445	−1.35432
$915$	0	0
$916$	−5.23711	−0.173039
$917$	2.39961	0.0792423
$918$	0	0
$919$	−3.98008	−0.131291	−0.0656453	−	0.997843i	$-0.520911\pi$
$919$	−3.98008	−0.131291	−0.0656453	+	0.997843i	$0.520911\pi$
$920$	5.88713	0.194093
$921$	0	0
$922$	18.3806	0.605334
$923$	79.7812	2.62603
$924$	0	0
$925$	0.537141	0.0176611
$926$	77.9309	2.56097
$927$	0	0
$928$	−60.9823	−2.00184
$929$	16.4233	0.538832	0.269416	−	0.963024i	$-0.413169\pi$
$929$	16.4233	0.538832	0.269416	+	0.963024i	$0.413169\pi$
$930$	0	0
$931$	8.67055	0.284166
$932$	26.4688	0.867016
$933$	0	0
$934$	19.0591	0.623634
$935$	−12.8452	−0.420084
$936$	0	0
$937$	−2.56860	−0.0839125	−0.0419563	−	0.999119i	$-0.513359\pi$
$937$	−2.56860	−0.0839125	−0.0419563	+	0.999119i	$0.513359\pi$
$938$	−11.3773	−0.371483
$939$	0	0
$940$	0.973593	0.0317551
$941$	−14.6081	−0.476211	−0.238106	−	0.971239i	$-0.576526\pi$
$941$	−14.6081	−0.476211	−0.238106	+	0.971239i	$0.576526\pi$
$942$	0	0
$943$	22.9067	0.745946
$944$	17.4270	0.567199
$945$	0	0
$946$	7.79561	0.253457
$947$	−29.1898	−0.948543	−0.474271	−	0.880379i	$-0.657288\pi$
$947$	−29.1898	−0.948543	−0.474271	+	0.880379i	$0.657288\pi$
$948$	0	0
$949$	−61.8239	−2.00689
$950$	0.549935	0.0178422
$951$	0	0
$952$	2.75103	0.0891614
$953$	5.65869	0.183303	0.0916515	−	0.995791i	$-0.470785\pi$
$953$	5.65869	0.183303	0.0916515	+	0.995791i	$0.470785\pi$
$954$	0	0
$955$	−16.6988	−0.540361
$956$	−26.2094	−0.847674
$957$	0	0
$958$	44.6536	1.44269
$959$	8.30541	0.268196
$960$	0	0
$961$	30.1985	0.974146
$962$	24.8229	0.800324
$963$	0	0
$964$	33.4466	1.07724
$965$	5.36009	0.172547
$966$	0	0
$967$	−16.4688	−0.529602	−0.264801	−	0.964303i	$-0.585306\pi$
$967$	−16.4688	−0.529602	−0.264801	+	0.964303i	$0.585306\pi$
$968$	−0.879385	−0.0282645
$969$	0	0
$970$	−3.64260	−0.116957
$971$	−43.2573	−1.38819	−0.694097	−	0.719882i	$-0.744196\pi$
$971$	−43.2573	−1.38819	−0.694097	+	0.719882i	$0.744196\pi$
$972$	0	0
$973$	−9.17799	−0.294233
$974$	−30.2327	−0.968717
$975$	0	0
$976$	12.7101	0.406840
$977$	−13.1411	−0.420423	−0.210211	−	0.977656i	$-0.567415\pi$
$977$	−13.1411	−0.420423	−0.210211	+	0.977656i	$0.567415\pi$
$978$	0	0
$979$	−8.51249	−0.272060
$980$	−22.4834	−0.718206
$981$	0	0
$982$	−3.72638	−0.118913
$983$	−46.3105	−1.47707	−0.738537	−	0.674213i	$-0.764483\pi$
$983$	−46.3105	−1.47707	−0.738537	+	0.674213i	$0.764483\pi$
$984$	0	0
$985$	−31.1739	−0.993282
$986$	−94.8248	−3.01984
$987$	0	0
$988$	11.0237	0.350711
$989$	−12.7101	−0.404157
$990$	0	0
$991$	−31.1739	−0.990271	−0.495135	−	0.868816i	$-0.664881\pi$
$991$	−31.1739	−0.990271	−0.495135	+	0.868816i	$0.664881\pi$
$992$	−55.5904	−1.76500
$993$	0	0
$994$	−14.3131	−0.453985
$995$	−40.4570	−1.28257
$996$	0	0
$997$	31.2891	0.990936	0.495468	−	0.868626i	$-0.334997\pi$
$997$	31.2891	0.990936	0.495468	+	0.868626i	$0.334997\pi$
$998$	32.0752	1.01532
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8019.2.a.b.1.3		3
3.2	odd	2		8019.2.a.a.1.1		3
27.5	odd	18		297.2.j.a.133.1	yes	6
27.11	odd	18		297.2.j.a.67.1	✓	6
27.16	even	9		891.2.j.a.496.1		6
27.22	even	9		891.2.j.a.397.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
297.2.j.a.67.1	✓	6	27.11	odd	18
297.2.j.a.133.1	yes	6	27.5	odd	18
891.2.j.a.397.1		6	27.22	even	9
891.2.j.a.496.1		6	27.16	even	9
8019.2.a.a.1.1		3	3.2	odd	2
8019.2.a.b.1.3		3	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.87939 q^{2} +1.53209 q^{4} +2.18479 q^{5} +0.532089 q^{7} -0.879385 q^{8} +O(q^{10})\)	\(q+1.87939 q^{2} +1.53209 q^{4} +2.18479 q^{5} +0.532089 q^{7} -0.879385 q^{8} +4.10607 q^{10} +1.00000 q^{11} -5.57398 q^{13} +1.00000 q^{14} -4.71688 q^{16} -5.87939 q^{17} -1.29086 q^{19} +3.34730 q^{20} +1.87939 q^{22} -3.06418 q^{23} -0.226682 q^{25} -10.4757 q^{26} +0.815207 q^{28} +8.58172 q^{29} +7.82295 q^{31} -7.10607 q^{32} -11.0496 q^{34} +1.16250 q^{35} -2.36959 q^{37} -2.42602 q^{38} -1.92127 q^{40} -7.47565 q^{41} +4.14796 q^{43} +1.53209 q^{44} -5.75877 q^{46} +0.290859 q^{47} -6.71688 q^{49} -0.426022 q^{50} -8.53983 q^{52} +3.93582 q^{53} +2.18479 q^{55} -0.467911 q^{56} +16.1284 q^{58} -3.69459 q^{59} -2.69459 q^{61} +14.7023 q^{62} -3.92127 q^{64} -12.1780 q^{65} -11.3773 q^{67} -9.00774 q^{68} +2.18479 q^{70} -14.3131 q^{71} +11.0915 q^{73} -4.45336 q^{74} -1.97771 q^{76} +0.532089 q^{77} -12.7811 q^{79} -10.3054 q^{80} -14.0496 q^{82} -7.27126 q^{83} -12.8452 q^{85} +7.79561 q^{86} -0.879385 q^{88} -8.51249 q^{89} -2.96585 q^{91} -4.69459 q^{92} +0.546637 q^{94} -2.82026 q^{95} -0.887126 q^{97} -12.6236 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{5} - 3 q^{7} + 3 q^{8}+O(q^{10}) \) 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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