LMFDB - Embedded newform 8019.2.a.k.1.5

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			1.5
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)

$f(q)$	$=$	$q-2.47541 q^{2} +4.12764 q^{4} +0.628090 q^{5} +1.20891 q^{7} -5.26679 q^{8} +O(q^{10})$	\(q-2.47541 q^{2} +4.12764 q^{4} +0.628090 q^{5} +1.20891 q^{7} -5.26679 q^{8} -1.55478 q^{10} -1.00000 q^{11} -4.42101 q^{13} -2.99255 q^{14} +4.78216 q^{16} -3.47551 q^{17} +0.304740 q^{19} +2.59253 q^{20} +2.47541 q^{22} -5.79745 q^{23} -4.60550 q^{25} +10.9438 q^{26} +4.98996 q^{28} -2.63407 q^{29} -0.0660015 q^{31} -1.30422 q^{32} +8.60331 q^{34} +0.759306 q^{35} +1.28686 q^{37} -0.754356 q^{38} -3.30802 q^{40} -9.65643 q^{41} +10.1657 q^{43} -4.12764 q^{44} +14.3511 q^{46} -10.2630 q^{47} -5.53853 q^{49} +11.4005 q^{50} -18.2484 q^{52} -5.79582 q^{53} -0.628090 q^{55} -6.36708 q^{56} +6.52040 q^{58} -7.79869 q^{59} -0.694065 q^{61} +0.163381 q^{62} -6.33584 q^{64} -2.77680 q^{65} +5.77040 q^{67} -14.3457 q^{68} -1.87959 q^{70} +6.05033 q^{71} +16.2303 q^{73} -3.18550 q^{74} +1.25786 q^{76} -1.20891 q^{77} -4.12358 q^{79} +3.00363 q^{80} +23.9036 q^{82} -8.38519 q^{83} -2.18294 q^{85} -25.1643 q^{86} +5.26679 q^{88} +14.6749 q^{89} -5.34462 q^{91} -23.9298 q^{92} +25.4051 q^{94} +0.191404 q^{95} +0.549159 q^{97} +13.7101 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.47541	−1.75038	−0.875189	−	0.483781i	$-0.839263\pi$
$2$	−2.47541	−1.75038	−0.875189	+	0.483781i	$0.839263\pi$
$3$	0	0
$3$	0	0
$4$	4.12764	2.06382
$5$	0.628090	0.280891	0.140445	−	0.990088i	$-0.455147\pi$
$5$	0.628090	0.280891	0.140445	+	0.990088i	$0.455147\pi$
$6$	0	0
$7$	1.20891	0.456926	0.228463	−	0.973553i	$-0.426630\pi$
$7$	1.20891	0.456926	0.228463	+	0.973553i	$0.426630\pi$
$8$	−5.26679	−1.86209
$9$	0	0
$10$	−1.55478	−0.491665
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−4.42101	−1.22617	−0.613084	−	0.790018i	$-0.710071\pi$
$13$	−4.42101	−1.22617	−0.613084	+	0.790018i	$0.710071\pi$
$14$	−2.99255	−0.799793
$15$	0	0
$16$	4.78216	1.19554
$17$	−3.47551	−0.842935	−0.421468	−	0.906843i	$-0.638485\pi$
$17$	−3.47551	−0.842935	−0.421468	+	0.906843i	$0.638485\pi$
$18$	0	0
$19$	0.304740	0.0699122	0.0349561	−	0.999389i	$-0.488871\pi$
$19$	0.304740	0.0699122	0.0349561	+	0.999389i	$0.488871\pi$
$20$	2.59253	0.579708
$21$	0	0
$22$	2.47541	0.527759
$23$	−5.79745	−1.20885	−0.604426	−	0.796661i	$-0.706598\pi$
$23$	−5.79745	−1.20885	−0.604426	+	0.796661i	$0.706598\pi$
$24$	0	0
$25$	−4.60550	−0.921100
$26$	10.9438	2.14626
$27$	0	0
$28$	4.98996	0.943014
$29$	−2.63407	−0.489135	−0.244567	−	0.969632i	$-0.578646\pi$
$29$	−2.63407	−0.489135	−0.244567	+	0.969632i	$0.578646\pi$
$30$	0	0
$31$	−0.0660015	−0.0118542	−0.00592711	−	0.999982i	$-0.501887\pi$
$31$	−0.0660015	−0.0118542	−0.00592711	+	0.999982i	$0.501887\pi$
$32$	−1.30422	−0.230556
$33$	0	0
$34$	8.60331	1.47546
$35$	0.759306	0.128346
$36$	0	0
$37$	1.28686	0.211558	0.105779	−	0.994390i	$-0.466266\pi$
$37$	1.28686	0.211558	0.105779	+	0.994390i	$0.466266\pi$
$38$	−0.754356	−0.122373
$39$	0	0
$40$	−3.30802	−0.523044
$41$	−9.65643	−1.50808	−0.754040	−	0.656828i	$-0.771898\pi$
$41$	−9.65643	−1.50808	−0.754040	+	0.656828i	$0.771898\pi$
$42$	0	0
$43$	10.1657	1.55026	0.775128	−	0.631804i	$-0.217685\pi$
$43$	10.1657	1.55026	0.775128	+	0.631804i	$0.217685\pi$
$44$	−4.12764	−0.622266
$45$	0	0
$46$	14.3511	2.11595
$47$	−10.2630	−1.49701	−0.748507	−	0.663127i	$-0.769229\pi$
$47$	−10.2630	−1.49701	−0.748507	+	0.663127i	$0.769229\pi$
$48$	0	0
$49$	−5.53853	−0.791219
$50$	11.4005	1.61227
$51$	0	0
$52$	−18.2484	−2.53059
$53$	−5.79582	−0.796117	−0.398058	−	0.917360i	$-0.630316\pi$
$53$	−5.79582	−0.796117	−0.398058	+	0.917360i	$0.630316\pi$
$54$	0	0
$55$	−0.628090	−0.0846917
$56$	−6.36708	−0.850837
$57$	0	0
$58$	6.52040	0.856171
$59$	−7.79869	−1.01530	−0.507651	−	0.861563i	$-0.669486\pi$
$59$	−7.79869	−1.01530	−0.507651	+	0.861563i	$0.669486\pi$
$60$	0	0
$61$	−0.694065	−0.0888659	−0.0444330	−	0.999012i	$-0.514148\pi$
$61$	−0.694065	−0.0888659	−0.0444330	+	0.999012i	$0.514148\pi$
$62$	0.163381	0.0207493
$63$	0	0
$64$	−6.33584	−0.791980
$65$	−2.77680	−0.344419
$66$	0	0
$67$	5.77040	0.704967	0.352483	−	0.935818i	$-0.385337\pi$
$67$	5.77040	0.704967	0.352483	+	0.935818i	$0.385337\pi$
$68$	−14.3457	−1.73967
$69$	0	0
$70$	−1.87959	−0.224654
$71$	6.05033	0.718042	0.359021	−	0.933330i	$-0.383111\pi$
$71$	6.05033	0.718042	0.359021	+	0.933330i	$0.383111\pi$
$72$	0	0
$73$	16.2303	1.89961	0.949804	−	0.312846i	$-0.101282\pi$
$73$	16.2303	1.89961	0.949804	+	0.312846i	$0.101282\pi$
$74$	−3.18550	−0.370306
$75$	0	0
$76$	1.25786	0.144286
$77$	−1.20891	−0.137768
$78$	0	0
$79$	−4.12358	−0.463939	−0.231970	−	0.972723i	$-0.574517\pi$
$79$	−4.12358	−0.463939	−0.231970	+	0.972723i	$0.574517\pi$
$80$	3.00363	0.335816
$81$	0	0
$82$	23.9036	2.63971
$83$	−8.38519	−0.920394	−0.460197	−	0.887817i	$-0.652221\pi$
$83$	−8.38519	−0.920394	−0.460197	+	0.887817i	$0.652221\pi$
$84$	0	0
$85$	−2.18294	−0.236773
$86$	−25.1643	−2.71353
$87$	0	0
$88$	5.26679	0.561442
$89$	14.6749	1.55554	0.777769	−	0.628550i	$-0.216351\pi$
$89$	14.6749	1.55554	0.777769	+	0.628550i	$0.216351\pi$
$90$	0	0
$91$	−5.34462	−0.560268
$92$	−23.9298	−2.49486
$93$	0	0
$94$	25.4051	2.62034
$95$	0.191404	0.0196377
$96$	0	0
$97$	0.549159	0.0557586	0.0278793	−	0.999611i	$-0.491125\pi$
$97$	0.549159	0.0557586	0.0278793	+	0.999611i	$0.491125\pi$
$98$	13.7101	1.38493
$99$	0	0
$100$	−19.0099	−1.90099
$101$	−4.43937	−0.441734	−0.220867	−	0.975304i	$-0.570889\pi$
$101$	−4.43937	−0.441734	−0.220867	+	0.975304i	$0.570889\pi$
$102$	0	0
$103$	−12.0784	−1.19012	−0.595062	−	0.803680i	$-0.702873\pi$
$103$	−12.0784	−1.19012	−0.595062	+	0.803680i	$0.702873\pi$
$104$	23.2845	2.28324
$105$	0	0
$106$	14.3470	1.39351
$107$	7.03977	0.680560	0.340280	−	0.940324i	$-0.389478\pi$
$107$	7.03977	0.680560	0.340280	+	0.940324i	$0.389478\pi$
$108$	0	0
$109$	7.29780	0.699003	0.349501	−	0.936936i	$-0.386351\pi$
$109$	7.29780	0.699003	0.349501	+	0.936936i	$0.386351\pi$
$110$	1.55478	0.148242
$111$	0	0
$112$	5.78121	0.546273
$113$	13.6220	1.28145	0.640723	−	0.767772i	$-0.278635\pi$
$113$	13.6220	1.28145	0.640723	+	0.767772i	$0.278635\pi$
$114$	0	0
$115$	−3.64133	−0.339555
$116$	−10.8725	−1.00949
$117$	0	0
$118$	19.3049	1.77716
$119$	−4.20159	−0.385159
$120$	0	0
$121$	1.00000	0.0909091
$122$	1.71809	0.155549
$123$	0	0
$124$	−0.272431	−0.0244650
$125$	−6.03312	−0.539619
$126$	0	0
$127$	13.3061	1.18072	0.590361	−	0.807139i	$-0.298985\pi$
$127$	13.3061	1.18072	0.590361	+	0.807139i	$0.298985\pi$
$128$	18.2922	1.61682
$129$	0	0
$130$	6.87370	0.602864
$131$	10.8208	0.945421	0.472710	−	0.881218i	$-0.343276\pi$
$131$	10.8208	0.945421	0.472710	+	0.881218i	$0.343276\pi$
$132$	0	0
$133$	0.368404	0.0319447
$134$	−14.2841	−1.23396
$135$	0	0
$136$	18.3048	1.56962
$137$	−7.89084	−0.674160	−0.337080	−	0.941476i	$-0.609439\pi$
$137$	−7.89084	−0.674160	−0.337080	+	0.941476i	$0.609439\pi$
$138$	0	0
$139$	20.6241	1.74932	0.874658	−	0.484741i	$-0.161086\pi$
$139$	20.6241	1.74932	0.874658	+	0.484741i	$0.161086\pi$
$140$	3.13415	0.264884
$141$	0	0
$142$	−14.9770	−1.25684
$143$	4.42101	0.369704
$144$	0	0
$145$	−1.65443	−0.137393
$146$	−40.1765	−3.32503
$147$	0	0
$148$	5.31169	0.436618
$149$	23.3358	1.91174	0.955870	−	0.293791i	$-0.0949170\pi$
$149$	23.3358	1.91174	0.955870	+	0.293791i	$0.0949170\pi$
$150$	0	0
$151$	17.3598	1.41272	0.706361	−	0.707852i	$-0.250335\pi$
$151$	17.3598	1.41272	0.706361	+	0.707852i	$0.250335\pi$
$152$	−1.60500	−0.130183
$153$	0	0
$154$	2.99255	0.241147
$155$	−0.0414549	−0.00332974
$156$	0	0
$157$	−6.60805	−0.527380	−0.263690	−	0.964607i	$-0.584940\pi$
$157$	−6.60805	−0.527380	−0.263690	+	0.964607i	$0.584940\pi$
$158$	10.2076	0.812069
$159$	0	0
$160$	−0.819170	−0.0647611
$161$	−7.00861	−0.552356
$162$	0	0
$163$	−20.7531	−1.62551	−0.812754	−	0.582607i	$-0.802033\pi$
$163$	−20.7531	−1.62551	−0.812754	+	0.582607i	$0.802033\pi$
$164$	−39.8583	−3.11241
$165$	0	0
$166$	20.7568	1.61104
$167$	20.2182	1.56453	0.782264	−	0.622947i	$-0.214065\pi$
$167$	20.2182	1.56453	0.782264	+	0.622947i	$0.214065\pi$
$168$	0	0
$169$	6.54536	0.503489
$170$	5.40366	0.414442
$171$	0	0
$172$	41.9604	3.19945
$173$	8.70763	0.662029	0.331014	−	0.943626i	$-0.392609\pi$
$173$	8.70763	0.662029	0.331014	+	0.943626i	$0.392609\pi$
$174$	0	0
$175$	−5.56765	−0.420875
$176$	−4.78216	−0.360469
$177$	0	0
$178$	−36.3264	−2.72278
$179$	2.50091	0.186927	0.0934634	−	0.995623i	$-0.470206\pi$
$179$	2.50091	0.186927	0.0934634	+	0.995623i	$0.470206\pi$
$180$	0	0
$181$	−2.46758	−0.183413	−0.0917067	−	0.995786i	$-0.529232\pi$
$181$	−2.46758	−0.183413	−0.0917067	+	0.995786i	$0.529232\pi$
$182$	13.2301	0.980681
$183$	0	0
$184$	30.5340	2.25099
$185$	0.808263	0.0594247
$186$	0	0
$187$	3.47551	0.254155
$188$	−42.3620	−3.08957
$189$	0	0
$190$	−0.473804	−0.0343734
$191$	22.3962	1.62053	0.810267	−	0.586061i	$-0.199322\pi$
$191$	22.3962	1.62053	0.810267	+	0.586061i	$0.199322\pi$
$192$	0	0
$193$	−23.7293	−1.70807	−0.854037	−	0.520213i	$-0.825853\pi$
$193$	−23.7293	−1.70807	−0.854037	+	0.520213i	$0.825853\pi$
$194$	−1.35939	−0.0975986
$195$	0	0
$196$	−22.8611	−1.63294
$197$	7.43239	0.529536	0.264768	−	0.964312i	$-0.414705\pi$
$197$	7.43239	0.529536	0.264768	+	0.964312i	$0.414705\pi$
$198$	0	0
$199$	26.2187	1.85860	0.929298	−	0.369332i	$-0.120413\pi$
$199$	26.2187	1.85860	0.929298	+	0.369332i	$0.120413\pi$
$200$	24.2562	1.71517
$201$	0	0
$202$	10.9892	0.773201
$203$	−3.18436	−0.223498
$204$	0	0
$205$	−6.06511	−0.423606
$206$	29.8991	2.08317
$207$	0	0
$208$	−21.1420	−1.46593
$209$	−0.304740	−0.0210793
$210$	0	0
$211$	10.8654	0.748005	0.374002	−	0.927428i	$-0.377985\pi$
$211$	10.8654	0.748005	0.374002	+	0.927428i	$0.377985\pi$
$212$	−23.9231	−1.64304
$213$	0	0
$214$	−17.4263	−1.19124
$215$	6.38499	0.435452
$216$	0	0
$217$	−0.0797899	−0.00541649
$218$	−18.0650	−1.22352
$219$	0	0
$220$	−2.59253	−0.174789
$221$	15.3653	1.03358
$222$	0	0
$223$	17.1630	1.14932	0.574661	−	0.818392i	$-0.305134\pi$
$223$	17.1630	1.14932	0.574661	+	0.818392i	$0.305134\pi$
$224$	−1.57669	−0.105347
$225$	0	0
$226$	−33.7199	−2.24302
$227$	−16.2949	−1.08153	−0.540764	−	0.841174i	$-0.681865\pi$
$227$	−16.2949	−1.08153	−0.540764	+	0.841174i	$0.681865\pi$
$228$	0	0
$229$	4.52590	0.299080	0.149540	−	0.988756i	$-0.452221\pi$
$229$	4.52590	0.299080	0.149540	+	0.988756i	$0.452221\pi$
$230$	9.01377	0.594350
$231$	0	0
$232$	13.8731	0.910813
$233$	10.8318	0.709616	0.354808	−	0.934939i	$-0.384546\pi$
$233$	10.8318	0.709616	0.354808	+	0.934939i	$0.384546\pi$
$234$	0	0
$235$	−6.44610	−0.420497
$236$	−32.1902	−2.09540
$237$	0	0
$238$	10.4006	0.674174
$239$	−7.07022	−0.457335	−0.228667	−	0.973505i	$-0.573437\pi$
$239$	−7.07022	−0.457335	−0.228667	+	0.973505i	$0.573437\pi$
$240$	0	0
$241$	−10.3542	−0.666972	−0.333486	−	0.942755i	$-0.608225\pi$
$241$	−10.3542	−0.666972	−0.333486	+	0.942755i	$0.608225\pi$
$242$	−2.47541	−0.159125
$243$	0	0
$244$	−2.86485	−0.183403
$245$	−3.47870	−0.222246
$246$	0	0
$247$	−1.34726	−0.0857241
$248$	0.347616	0.0220736
$249$	0	0
$250$	14.9344	0.944537
$251$	25.5377	1.61193	0.805963	−	0.591965i	$-0.201648\pi$
$251$	25.5377	1.61193	0.805963	+	0.591965i	$0.201648\pi$
$252$	0	0
$253$	5.79745	0.364483
$254$	−32.9379	−2.06671
$255$	0	0
$256$	−32.6091	−2.03807
$257$	8.24793	0.514492	0.257246	−	0.966346i	$-0.417185\pi$
$257$	8.24793	0.514492	0.257246	+	0.966346i	$0.417185\pi$
$258$	0	0
$259$	1.55570	0.0966663
$260$	−11.4616	−0.710820
$261$	0	0
$262$	−26.7860	−1.65484
$263$	−23.4264	−1.44454	−0.722268	−	0.691614i	$-0.756900\pi$
$263$	−23.4264	−1.44454	−0.722268	+	0.691614i	$0.756900\pi$
$264$	0	0
$265$	−3.64030	−0.223622
$266$	−0.911950	−0.0559153
$267$	0	0
$268$	23.8182	1.45493
$269$	1.84782	0.112663	0.0563317	−	0.998412i	$-0.482060\pi$
$269$	1.84782	0.112663	0.0563317	+	0.998412i	$0.482060\pi$
$270$	0	0
$271$	13.2049	0.802139	0.401069	−	0.916048i	$-0.368639\pi$
$271$	13.2049	0.802139	0.401069	+	0.916048i	$0.368639\pi$
$272$	−16.6205	−1.00776
$273$	0	0
$274$	19.5330	1.18003
$275$	4.60550	0.277722
$276$	0	0
$277$	9.95638	0.598221	0.299111	−	0.954218i	$-0.403310\pi$
$277$	9.95638	0.598221	0.299111	+	0.954218i	$0.403310\pi$
$278$	−51.0531	−3.06196
$279$	0	0
$280$	−3.99910	−0.238992
$281$	15.2595	0.910305	0.455152	−	0.890414i	$-0.349585\pi$
$281$	15.2595	0.910305	0.455152	+	0.890414i	$0.349585\pi$
$282$	0	0
$283$	7.89524	0.469323	0.234662	−	0.972077i	$-0.424602\pi$
$283$	7.89524	0.469323	0.234662	+	0.972077i	$0.424602\pi$
$284$	24.9736	1.48191
$285$	0	0
$286$	−10.9438	−0.647121
$287$	−11.6738	−0.689081
$288$	0	0
$289$	−4.92082	−0.289460
$290$	4.09540	0.240490
$291$	0	0
$292$	66.9927	3.92045
$293$	−29.8056	−1.74126	−0.870631	−	0.491937i	$-0.836289\pi$
$293$	−29.8056	−1.74126	−0.870631	+	0.491937i	$0.836289\pi$
$294$	0	0
$295$	−4.89828	−0.285189
$296$	−6.77760	−0.393940
$297$	0	0
$298$	−57.7655	−3.34627
$299$	25.6306	1.48226
$300$	0	0
$301$	12.2894	0.708352
$302$	−42.9726	−2.47280
$303$	0	0
$304$	1.45732	0.0835829
$305$	−0.435936	−0.0249616
$306$	0	0
$307$	−3.31989	−0.189476	−0.0947381	−	0.995502i	$-0.530201\pi$
$307$	−3.31989	−0.189476	−0.0947381	+	0.995502i	$0.530201\pi$
$308$	−4.98996	−0.284329
$309$	0	0
$310$	0.102618	0.00582830
$311$	9.02995	0.512041	0.256021	−	0.966671i	$-0.417588\pi$
$311$	9.02995	0.512041	0.256021	+	0.966671i	$0.417588\pi$
$312$	0	0
$313$	23.2854	1.31617	0.658084	−	0.752945i	$-0.271367\pi$
$313$	23.2854	1.31617	0.658084	+	0.752945i	$0.271367\pi$
$314$	16.3576	0.923114
$315$	0	0
$316$	−17.0207	−0.957488
$317$	−5.96772	−0.335181	−0.167590	−	0.985857i	$-0.553599\pi$
$317$	−5.96772	−0.335181	−0.167590	+	0.985857i	$0.553599\pi$
$318$	0	0
$319$	2.63407	0.147480
$320$	−3.97948	−0.222460
$321$	0	0
$322$	17.3492	0.966831
$323$	−1.05913	−0.0589315
$324$	0	0
$325$	20.3610	1.12942
$326$	51.3724	2.84525
$327$	0	0
$328$	50.8584	2.80818
$329$	−12.4071	−0.684024
$330$	0	0
$331$	10.9632	0.602591	0.301296	−	0.953531i	$-0.402581\pi$
$331$	10.9632	0.602591	0.301296	+	0.953531i	$0.402581\pi$
$332$	−34.6111	−1.89953
$333$	0	0
$334$	−50.0482	−2.73851
$335$	3.62433	0.198018
$336$	0	0
$337$	−1.62298	−0.0884093	−0.0442047	−	0.999022i	$-0.514075\pi$
$337$	−1.62298	−0.0884093	−0.0442047	+	0.999022i	$0.514075\pi$
$338$	−16.2024	−0.881296
$339$	0	0
$340$	−9.01038	−0.488657
$341$	0.0660015	0.00357418
$342$	0	0
$343$	−15.1580	−0.818454
$344$	−53.5407	−2.88672
$345$	0	0
$346$	−21.5549	−1.15880
$347$	−5.09566	−0.273549	−0.136775	−	0.990602i	$-0.543674\pi$
$347$	−5.09566	−0.273549	−0.136775	+	0.990602i	$0.543674\pi$
$348$	0	0
$349$	−14.4791	−0.775046	−0.387523	−	0.921860i	$-0.626669\pi$
$349$	−14.4791	−0.775046	−0.387523	+	0.921860i	$0.626669\pi$
$350$	13.7822	0.736689
$351$	0	0
$352$	1.30422	0.0695153
$353$	1.79365	0.0954666	0.0477333	−	0.998860i	$-0.484800\pi$
$353$	1.79365	0.0954666	0.0477333	+	0.998860i	$0.484800\pi$
$354$	0	0
$355$	3.80015	0.201691
$356$	60.5728	3.21035
$357$	0	0
$358$	−6.19077	−0.327192
$359$	15.8955	0.838930	0.419465	−	0.907772i	$-0.362218\pi$
$359$	15.8955	0.838930	0.419465	+	0.907772i	$0.362218\pi$
$360$	0	0
$361$	−18.9071	−0.995112
$362$	6.10826	0.321043
$363$	0	0
$364$	−22.0607	−1.15629
$365$	10.1941	0.533582
$366$	0	0
$367$	15.9421	0.832169	0.416085	−	0.909326i	$-0.363402\pi$
$367$	15.9421	0.832169	0.416085	+	0.909326i	$0.363402\pi$
$368$	−27.7244	−1.44523
$369$	0	0
$370$	−2.00078	−0.104016
$371$	−7.00663	−0.363766
$372$	0	0
$373$	−11.6462	−0.603018	−0.301509	−	0.953463i	$-0.597490\pi$
$373$	−11.6462	−0.603018	−0.301509	+	0.953463i	$0.597490\pi$
$374$	−8.60331	−0.444867
$375$	0	0
$376$	54.0531	2.78758
$377$	11.6453	0.599762
$378$	0	0
$379$	7.31201	0.375593	0.187796	−	0.982208i	$-0.439866\pi$
$379$	7.31201	0.375593	0.187796	+	0.982208i	$0.439866\pi$
$380$	0.790049	0.0405287
$381$	0	0
$382$	−55.4398	−2.83655
$383$	−23.8749	−1.21995	−0.609976	−	0.792420i	$-0.708821\pi$
$383$	−23.8749	−1.21995	−0.609976	+	0.792420i	$0.708821\pi$
$384$	0	0
$385$	−0.759306	−0.0386978
$386$	58.7397	2.98977
$387$	0	0
$388$	2.26673	0.115076
$389$	12.1272	0.614873	0.307436	−	0.951569i	$-0.400529\pi$
$389$	12.1272	0.614873	0.307436	+	0.951569i	$0.400529\pi$
$390$	0	0
$391$	20.1491	1.01898
$392$	29.1703	1.47332
$393$	0	0
$394$	−18.3982	−0.926888
$395$	−2.58998	−0.130316
$396$	0	0
$397$	−31.3721	−1.57452	−0.787260	−	0.616621i	$-0.788501\pi$
$397$	−31.3721	−1.57452	−0.787260	+	0.616621i	$0.788501\pi$
$398$	−64.9020	−3.25324
$399$	0	0
$400$	−22.0243	−1.10121
$401$	−19.3605	−0.966819	−0.483410	−	0.875394i	$-0.660602\pi$
$401$	−19.3605	−0.966819	−0.483410	+	0.875394i	$0.660602\pi$
$402$	0	0
$403$	0.291793	0.0145353
$404$	−18.3241	−0.911660
$405$	0	0
$406$	7.88259	0.391206
$407$	−1.28686	−0.0637871
$408$	0	0
$409$	−11.2793	−0.557723	−0.278862	−	0.960331i	$-0.589957\pi$
$409$	−11.2793	−0.557723	−0.278862	+	0.960331i	$0.589957\pi$
$410$	15.0136	0.741470
$411$	0	0
$412$	−49.8555	−2.45620
$413$	−9.42792	−0.463918
$414$	0	0
$415$	−5.26666	−0.258530
$416$	5.76599	0.282701
$417$	0	0
$418$	0.754356	0.0368968
$419$	8.76077	0.427992	0.213996	−	0.976835i	$-0.431352\pi$
$419$	8.76077	0.427992	0.213996	+	0.976835i	$0.431352\pi$
$420$	0	0
$421$	−40.1838	−1.95844	−0.979219	−	0.202808i	$-0.934993\pi$
$421$	−40.1838	−1.95844	−0.979219	+	0.202808i	$0.934993\pi$
$422$	−26.8963	−1.30929
$423$	0	0
$424$	30.5254	1.48244
$425$	16.0065	0.776428
$426$	0	0
$427$	−0.839063	−0.0406051
$428$	29.0577	1.40456
$429$	0	0
$430$	−15.8054	−0.762206
$431$	41.2625	1.98754	0.993772	−	0.111434i	$-0.0355443\pi$
$431$	41.2625	1.98754	0.993772	+	0.111434i	$0.0355443\pi$
$432$	0	0
$433$	−26.2223	−1.26016	−0.630081	−	0.776529i	$-0.716979\pi$
$433$	−26.2223	−1.26016	−0.630081	+	0.776529i	$0.716979\pi$
$434$	0.197513	0.00948091
$435$	0	0
$436$	30.1227	1.44262
$437$	−1.76672	−0.0845135
$438$	0	0
$439$	4.82234	0.230158	0.115079	−	0.993356i	$-0.463288\pi$
$439$	4.82234	0.230158	0.115079	+	0.993356i	$0.463288\pi$
$440$	3.30802	0.157704
$441$	0	0
$442$	−38.0353	−1.80916
$443$	−24.8594	−1.18111	−0.590553	−	0.806999i	$-0.701090\pi$
$443$	−24.8594	−1.18111	−0.590553	+	0.806999i	$0.701090\pi$
$444$	0	0
$445$	9.21718	0.436936
$446$	−42.4855	−2.01175
$447$	0	0
$448$	−7.65947	−0.361876
$449$	−5.11881	−0.241572	−0.120786	−	0.992679i	$-0.538541\pi$
$449$	−5.11881	−0.241572	−0.120786	+	0.992679i	$0.538541\pi$
$450$	0	0
$451$	9.65643	0.454703
$452$	56.2266	2.64468
$453$	0	0
$454$	40.3364	1.89308
$455$	−3.35690	−0.157374
$456$	0	0
$457$	−13.9089	−0.650631	−0.325316	−	0.945605i	$-0.605471\pi$
$457$	−13.9089	−0.650631	−0.325316	+	0.945605i	$0.605471\pi$
$458$	−11.2035	−0.523503
$459$	0	0
$460$	−15.0301	−0.700782
$461$	−24.6735	−1.14916	−0.574580	−	0.818449i	$-0.694834\pi$
$461$	−24.6735	−1.14916	−0.574580	+	0.818449i	$0.694834\pi$
$462$	0	0
$463$	26.8175	1.24631	0.623156	−	0.782097i	$-0.285850\pi$
$463$	26.8175	1.24631	0.623156	+	0.782097i	$0.285850\pi$
$464$	−12.5966	−0.584780
$465$	0	0
$466$	−26.8132	−1.24210
$467$	−13.9514	−0.645592	−0.322796	−	0.946469i	$-0.604623\pi$
$467$	−13.9514	−0.645592	−0.322796	+	0.946469i	$0.604623\pi$
$468$	0	0
$469$	6.97591	0.322117
$470$	15.9567	0.736029
$471$	0	0
$472$	41.0740	1.89059
$473$	−10.1657	−0.467420
$474$	0	0
$475$	−1.40348	−0.0643962
$476$	−17.3427	−0.794900
$477$	0	0
$478$	17.5017	0.800508
$479$	−1.79836	−0.0821691	−0.0410845	−	0.999156i	$-0.513081\pi$
$479$	−1.79836	−0.0821691	−0.0410845	+	0.999156i	$0.513081\pi$
$480$	0	0
$481$	−5.68921	−0.259406
$482$	25.6309	1.16745
$483$	0	0
$484$	4.12764	0.187620
$485$	0.344921	0.0156621
$486$	0	0
$487$	−14.3269	−0.649214	−0.324607	−	0.945849i	$-0.605232\pi$
$487$	−14.3269	−0.649214	−0.324607	+	0.945849i	$0.605232\pi$
$488$	3.65549	0.165476
$489$	0	0
$490$	8.61120	0.389014
$491$	−4.32441	−0.195158	−0.0975789	−	0.995228i	$-0.531110\pi$
$491$	−4.32441	−0.195158	−0.0975789	+	0.995228i	$0.531110\pi$
$492$	0	0
$493$	9.15475	0.412309
$494$	3.33502	0.150050
$495$	0	0
$496$	−0.315630	−0.0141722
$497$	7.31431	0.328092
$498$	0	0
$499$	−41.6956	−1.86655	−0.933275	−	0.359163i	$-0.883062\pi$
$499$	−41.6956	−1.86655	−0.933275	+	0.359163i	$0.883062\pi$
$500$	−24.9026	−1.11368
$501$	0	0
$502$	−63.2163	−2.82148
$503$	−2.26703	−0.101082	−0.0505410	−	0.998722i	$-0.516095\pi$
$503$	−2.26703	−0.101082	−0.0505410	+	0.998722i	$0.516095\pi$
$504$	0	0
$505$	−2.78832	−0.124079
$506$	−14.3511	−0.637983
$507$	0	0
$508$	54.9227	2.43680
$509$	−1.37753	−0.0610581	−0.0305290	−	0.999534i	$-0.509719\pi$
$509$	−1.37753	−0.0610581	−0.0305290	+	0.999534i	$0.509719\pi$
$510$	0	0
$511$	19.6209	0.867980
$512$	44.1363	1.95057
$513$	0	0
$514$	−20.4170	−0.900555
$515$	−7.58635	−0.334295
$516$	0	0
$517$	10.2630	0.451367
$518$	−3.85098	−0.169203
$519$	0	0
$520$	14.6248	0.641340
$521$	34.4184	1.50790	0.753949	−	0.656933i	$-0.228147\pi$
$521$	34.4184	1.50790	0.753949	+	0.656933i	$0.228147\pi$
$522$	0	0
$523$	−0.108698	−0.00475305	−0.00237653	−	0.999997i	$-0.500756\pi$
$523$	−0.108698	−0.00475305	−0.00237653	+	0.999997i	$0.500756\pi$
$524$	44.6646	1.95118
$525$	0	0
$526$	57.9900	2.52848
$527$	0.229389	0.00999234
$528$	0	0
$529$	10.6105	0.461325
$530$	9.01122	0.391423
$531$	0	0
$532$	1.52064	0.0659281
$533$	42.6912	1.84916
$534$	0	0
$535$	4.42161	0.191163
$536$	−30.3915	−1.31271
$537$	0	0
$538$	−4.57410	−0.197203
$539$	5.53853	0.238561
$540$	0	0
$541$	−23.1842	−0.996768	−0.498384	−	0.866956i	$-0.666073\pi$
$541$	−23.1842	−0.996768	−0.498384	+	0.866956i	$0.666073\pi$
$542$	−32.6874	−1.40405
$543$	0	0
$544$	4.53284	0.194344
$545$	4.58368	0.196343
$546$	0	0
$547$	−41.8604	−1.78982	−0.894912	−	0.446243i	$-0.852762\pi$
$547$	−41.8604	−1.78982	−0.894912	+	0.446243i	$0.852762\pi$
$548$	−32.5706	−1.39135
$549$	0	0
$550$	−11.4005	−0.486119
$551$	−0.802707	−0.0341965
$552$	0	0
$553$	−4.98505	−0.211986
$554$	−24.6461	−1.04711
$555$	0	0
$556$	85.1291	3.61028
$557$	7.31999	0.310158	0.155079	−	0.987902i	$-0.450437\pi$
$557$	7.31999	0.310158	0.155079	+	0.987902i	$0.450437\pi$
$558$	0	0
$559$	−44.9427	−1.90088
$560$	3.63112	0.153443
$561$	0	0
$562$	−37.7735	−1.59338
$563$	10.0271	0.422593	0.211297	−	0.977422i	$-0.432231\pi$
$563$	10.0271	0.422593	0.211297	+	0.977422i	$0.432231\pi$
$564$	0	0
$565$	8.55583	0.359946
$566$	−19.5439	−0.821493
$567$	0	0
$568$	−31.8658	−1.33706
$569$	−3.79700	−0.159179	−0.0795893	−	0.996828i	$-0.525361\pi$
$569$	−3.79700	−0.159179	−0.0795893	+	0.996828i	$0.525361\pi$
$570$	0	0
$571$	−8.56664	−0.358503	−0.179251	−	0.983803i	$-0.557368\pi$
$571$	−8.56664	−0.358503	−0.179251	+	0.983803i	$0.557368\pi$
$572$	18.2484	0.763003
$573$	0	0
$574$	28.8973	1.20615
$575$	26.7002	1.11347
$576$	0	0
$577$	15.8168	0.658462	0.329231	−	0.944249i	$-0.393210\pi$
$577$	15.8168	0.658462	0.329231	+	0.944249i	$0.393210\pi$
$578$	12.1810	0.506664
$579$	0	0
$580$	−6.82892	−0.283555
$581$	−10.1370	−0.420552
$582$	0	0
$583$	5.79582	0.240038
$584$	−85.4813	−3.53724
$585$	0	0
$586$	73.7810	3.04787
$587$	32.2426	1.33080	0.665398	−	0.746489i	$-0.268262\pi$
$587$	32.2426	1.33080	0.665398	+	0.746489i	$0.268262\pi$
$588$	0	0
$589$	−0.0201133	−0.000828754 0
$590$	12.1252	0.499188
$591$	0	0
$592$	6.15396	0.252926
$593$	37.6652	1.54672	0.773362	−	0.633965i	$-0.218573\pi$
$593$	37.6652	1.54672	0.773362	+	0.633965i	$0.218573\pi$
$594$	0	0
$595$	−2.63898	−0.108188
$596$	96.3217	3.94549
$597$	0	0
$598$	−63.4462	−2.59451
$599$	−1.63016	−0.0666066	−0.0333033	−	0.999445i	$-0.510603\pi$
$599$	−1.63016	−0.0666066	−0.0333033	+	0.999445i	$0.510603\pi$
$600$	0	0
$601$	8.51115	0.347177	0.173589	−	0.984818i	$-0.444464\pi$
$601$	8.51115	0.347177	0.173589	+	0.984818i	$0.444464\pi$
$602$	−30.4214	−1.23988
$603$	0	0
$604$	71.6552	2.91561
$605$	0.628090	0.0255355
$606$	0	0
$607$	21.9766	0.892003	0.446002	−	0.895032i	$-0.352848\pi$
$607$	21.9766	0.892003	0.446002	+	0.895032i	$0.352848\pi$
$608$	−0.397449	−0.0161187
$609$	0	0
$610$	1.07912	0.0436922
$611$	45.3729	1.83559
$612$	0	0
$613$	29.6027	1.19564	0.597822	−	0.801629i	$-0.296033\pi$
$613$	29.6027	1.19564	0.597822	+	0.801629i	$0.296033\pi$
$614$	8.21808	0.331655
$615$	0	0
$616$	6.36708	0.256537
$617$	−30.5568	−1.23017	−0.615085	−	0.788460i	$-0.710878\pi$
$617$	−30.5568	−1.23017	−0.615085	+	0.788460i	$0.710878\pi$
$618$	0	0
$619$	0.357262	0.0143596	0.00717979	−	0.999974i	$-0.497715\pi$
$619$	0.357262	0.0143596	0.00717979	+	0.999974i	$0.497715\pi$
$620$	−0.171111	−0.00687198
$621$	0	0
$622$	−22.3528	−0.896266
$623$	17.7407	0.710765
$624$	0	0
$625$	19.2382	0.769527
$626$	−57.6408	−2.30379
$627$	0	0
$628$	−27.2757	−1.08842
$629$	−4.47249	−0.178330
$630$	0	0
$631$	−33.8695	−1.34832	−0.674161	−	0.738584i	$-0.735495\pi$
$631$	−33.8695	−1.34832	−0.674161	+	0.738584i	$0.735495\pi$
$632$	21.7180	0.863897
$633$	0	0
$634$	14.7725	0.586693
$635$	8.35741	0.331654
$636$	0	0
$637$	24.4859	0.970168
$638$	−6.52040	−0.258145
$639$	0	0
$640$	11.4892	0.454150
$641$	−29.3183	−1.15800	−0.579001	−	0.815327i	$-0.696557\pi$
$641$	−29.3183	−1.15800	−0.579001	+	0.815327i	$0.696557\pi$
$642$	0	0
$643$	−49.6960	−1.95982	−0.979909	−	0.199444i	$-0.936087\pi$
$643$	−49.6960	−1.95982	−0.979909	+	0.199444i	$0.936087\pi$
$644$	−28.9291	−1.13996
$645$	0	0
$646$	2.62177	0.103152
$647$	−11.7824	−0.463213	−0.231607	−	0.972810i	$-0.574398\pi$
$647$	−11.7824	−0.463213	−0.231607	+	0.972810i	$0.574398\pi$
$648$	0	0
$649$	7.79869	0.306125
$650$	−50.4017	−1.97692
$651$	0	0
$652$	−85.6614	−3.35476
$653$	−9.84470	−0.385253	−0.192626	−	0.981272i	$-0.561701\pi$
$653$	−9.84470	−0.385253	−0.192626	+	0.981272i	$0.561701\pi$
$654$	0	0
$655$	6.79646	0.265560
$656$	−46.1786	−1.80297
$657$	0	0
$658$	30.7126	1.19730
$659$	−30.2458	−1.17821	−0.589105	−	0.808057i	$-0.700519\pi$
$659$	−30.2458	−1.17821	−0.589105	+	0.808057i	$0.700519\pi$
$660$	0	0
$661$	44.9184	1.74712	0.873561	−	0.486714i	$-0.161805\pi$
$661$	44.9184	1.74712	0.873561	+	0.486714i	$0.161805\pi$
$662$	−27.1384	−1.05476
$663$	0	0
$664$	44.1630	1.71386
$665$	0.231391	0.00897296
$666$	0	0
$667$	15.2709	0.591292
$668$	83.4533	3.22891
$669$	0	0
$670$	−8.97170	−0.346607
$671$	0.694065	0.0267941
$672$	0	0
$673$	20.6959	0.797768	0.398884	−	0.917001i	$-0.369398\pi$
$673$	20.6959	0.797768	0.398884	+	0.917001i	$0.369398\pi$
$674$	4.01754	0.154750
$675$	0	0
$676$	27.0169	1.03911
$677$	22.7501	0.874358	0.437179	−	0.899375i	$-0.355978\pi$
$677$	22.7501	0.874358	0.437179	+	0.899375i	$0.355978\pi$
$678$	0	0
$679$	0.663884	0.0254775
$680$	11.4971	0.440892
$681$	0	0
$682$	−0.163381	−0.00625616
$683$	−42.5450	−1.62794	−0.813970	−	0.580907i	$-0.802698\pi$
$683$	−42.5450	−1.62794	−0.813970	+	0.580907i	$0.802698\pi$
$684$	0	0
$685$	−4.95616	−0.189365
$686$	37.5222	1.43260
$687$	0	0
$688$	48.6141	1.85339
$689$	25.6234	0.976174
$690$	0	0
$691$	35.7601	1.36038	0.680189	−	0.733037i	$-0.261898\pi$
$691$	35.7601	1.36038	0.680189	+	0.733037i	$0.261898\pi$
$692$	35.9420	1.36631
$693$	0	0
$694$	12.6138	0.478815
$695$	12.9538	0.491366
$696$	0	0
$697$	33.5610	1.27121
$698$	35.8416	1.35662
$699$	0	0
$700$	−22.9813	−0.868610
$701$	15.8328	0.597998	0.298999	−	0.954253i	$-0.403347\pi$
$701$	15.8328	0.597998	0.298999	+	0.954253i	$0.403347\pi$
$702$	0	0
$703$	0.392157	0.0147905
$704$	6.33584	0.238791
$705$	0	0
$706$	−4.44003	−0.167103
$707$	−5.36680	−0.201839
$708$	0	0
$709$	30.9075	1.16076	0.580378	−	0.814347i	$-0.302905\pi$
$709$	30.9075	1.16076	0.580378	+	0.814347i	$0.302905\pi$
$710$	−9.40693	−0.353036
$711$	0	0
$712$	−77.2897	−2.89655
$713$	0.382640	0.0143300
$714$	0	0
$715$	2.77680	0.103846
$716$	10.3229	0.385784
$717$	0	0
$718$	−39.3477	−1.46844
$719$	10.1193	0.377387	0.188694	−	0.982036i	$-0.439575\pi$
$719$	10.1193	0.377387	0.188694	+	0.982036i	$0.439575\pi$
$720$	0	0
$721$	−14.6018	−0.543798
$722$	46.8029	1.74182
$723$	0	0
$724$	−10.1853	−0.378533
$725$	12.1312	0.450542
$726$	0	0
$727$	5.61639	0.208300	0.104150	−	0.994562i	$-0.466788\pi$
$727$	5.61639	0.208300	0.104150	+	0.994562i	$0.466788\pi$
$728$	28.1490	1.04327
$729$	0	0
$730$	−25.2345	−0.933970
$731$	−35.3310	−1.30677
$732$	0	0
$733$	18.6829	0.690068	0.345034	−	0.938590i	$-0.387867\pi$
$733$	18.6829	0.690068	0.345034	+	0.938590i	$0.387867\pi$
$734$	−39.4631	−1.45661
$735$	0	0
$736$	7.56117	0.278709
$737$	−5.77040	−0.212555
$738$	0	0
$739$	−23.0833	−0.849131	−0.424566	−	0.905397i	$-0.639573\pi$
$739$	−23.0833	−0.849131	−0.424566	+	0.905397i	$0.639573\pi$
$740$	3.33622	0.122642
$741$	0	0
$742$	17.3443	0.636729
$743$	−38.5196	−1.41315	−0.706573	−	0.707640i	$-0.749760\pi$
$743$	−38.5196	−1.41315	−0.706573	+	0.707640i	$0.749760\pi$
$744$	0	0
$745$	14.6570	0.536990
$746$	28.8291	1.05551
$747$	0	0
$748$	14.3457	0.524530
$749$	8.51046	0.310966
$750$	0	0
$751$	3.17354	0.115804	0.0579020	−	0.998322i	$-0.481559\pi$
$751$	3.17354	0.115804	0.0579020	+	0.998322i	$0.481559\pi$
$752$	−49.0794	−1.78974
$753$	0	0
$754$	−28.8268	−1.04981
$755$	10.9035	0.396820
$756$	0	0
$757$	28.7216	1.04390	0.521952	−	0.852975i	$-0.325204\pi$
$757$	28.7216	1.04390	0.521952	+	0.852975i	$0.325204\pi$
$758$	−18.1002	−0.657429
$759$	0	0
$760$	−1.00809	−0.0365671
$761$	14.3319	0.519530	0.259765	−	0.965672i	$-0.416355\pi$
$761$	14.3319	0.519530	0.259765	+	0.965672i	$0.416355\pi$
$762$	0	0
$763$	8.82240	0.319392
$764$	92.4437	3.34450
$765$	0	0
$766$	59.1002	2.13538
$767$	34.4781	1.24493
$768$	0	0
$769$	−9.56588	−0.344954	−0.172477	−	0.985014i	$-0.555177\pi$
$769$	−9.56588	−0.344954	−0.172477	+	0.985014i	$0.555177\pi$
$770$	1.87959	0.0677358
$771$	0	0
$772$	−97.9462	−3.52516
$773$	31.6015	1.13663	0.568314	−	0.822812i	$-0.307596\pi$
$773$	31.6015	1.13663	0.568314	+	0.822812i	$0.307596\pi$
$774$	0	0
$775$	0.303970	0.0109189
$776$	−2.89230	−0.103828
$777$	0	0
$778$	−30.0197	−1.07626
$779$	−2.94270	−0.105433
$780$	0	0
$781$	−6.05033	−0.216498
$782$	−49.8773	−1.78361
$783$	0	0
$784$	−26.4862	−0.945934
$785$	−4.15045	−0.148136
$786$	0	0
$787$	21.5122	0.766829	0.383414	−	0.923576i	$-0.374748\pi$
$787$	21.5122	0.766829	0.383414	+	0.923576i	$0.374748\pi$
$788$	30.6783	1.09287
$789$	0	0
$790$	6.41127	0.228103
$791$	16.4678	0.585526
$792$	0	0
$793$	3.06847	0.108965
$794$	77.6587	2.75600
$795$	0	0
$796$	108.222	3.83581
$797$	3.64971	0.129279	0.0646396	−	0.997909i	$-0.479410\pi$
$797$	3.64971	0.129279	0.0646396	+	0.997909i	$0.479410\pi$
$798$	0	0
$799$	35.6692	1.26189
$800$	6.00660	0.212366
$801$	0	0
$802$	47.9252	1.69230
$803$	−16.2303	−0.572753
$804$	0	0
$805$	−4.40204	−0.155152
$806$	−0.722308	−0.0254422
$807$	0	0
$808$	23.3812	0.822548
$809$	16.2484	0.571264	0.285632	−	0.958339i	$-0.407797\pi$
$809$	16.2484	0.571264	0.285632	+	0.958339i	$0.407797\pi$
$810$	0	0
$811$	−17.3955	−0.610839	−0.305419	−	0.952218i	$-0.598797\pi$
$811$	−17.3955	−0.610839	−0.305419	+	0.952218i	$0.598797\pi$
$812$	−13.1439	−0.461261
$813$	0	0
$814$	3.18550	0.111652
$815$	−13.0348	−0.456590
$816$	0	0
$817$	3.09790	0.108382
$818$	27.9208	0.976227
$819$	0	0
$820$	−25.0346	−0.874247
$821$	42.4682	1.48215	0.741075	−	0.671422i	$-0.234316\pi$
$821$	42.4682	1.48215	0.741075	+	0.671422i	$0.234316\pi$
$822$	0	0
$823$	−11.8705	−0.413780	−0.206890	−	0.978364i	$-0.566334\pi$
$823$	−11.8705	−0.413780	−0.206890	+	0.978364i	$0.566334\pi$
$824$	63.6146	2.21612
$825$	0	0
$826$	23.3380	0.812031
$827$	−8.70675	−0.302763	−0.151382	−	0.988475i	$-0.548372\pi$
$827$	−8.70675	−0.302763	−0.151382	+	0.988475i	$0.548372\pi$
$828$	0	0
$829$	28.0094	0.972807	0.486403	−	0.873734i	$-0.338309\pi$
$829$	28.0094	0.972807	0.486403	+	0.873734i	$0.338309\pi$
$830$	13.0371	0.452525
$831$	0	0
$832$	28.0108	0.971101
$833$	19.2492	0.666946
$834$	0	0
$835$	12.6988	0.439461
$836$	−1.25786	−0.0435040
$837$	0	0
$838$	−21.6865	−0.749147
$839$	−36.0034	−1.24298	−0.621488	−	0.783424i	$-0.713472\pi$
$839$	−36.0034	−1.24298	−0.621488	+	0.783424i	$0.713472\pi$
$840$	0	0
$841$	−22.0617	−0.760747
$842$	99.4712	3.42800
$843$	0	0
$844$	44.8485	1.54375
$845$	4.11108	0.141425
$846$	0	0
$847$	1.20891	0.0415387
$848$	−27.7165	−0.951790
$849$	0	0
$850$	−39.6226	−1.35904
$851$	−7.46049	−0.255742
$852$	0	0
$853$	−7.57045	−0.259207	−0.129604	−	0.991566i	$-0.541370\pi$
$853$	−7.57045	−0.259207	−0.129604	+	0.991566i	$0.541370\pi$
$854$	2.07702	0.0710743
$855$	0	0
$856$	−37.0770	−1.26727
$857$	−34.0865	−1.16437	−0.582187	−	0.813055i	$-0.697803\pi$
$857$	−34.0865	−1.16437	−0.582187	+	0.813055i	$0.697803\pi$
$858$	0	0
$859$	−6.57958	−0.224492	−0.112246	−	0.993680i	$-0.535804\pi$
$859$	−6.57958	−0.224492	−0.112246	+	0.993680i	$0.535804\pi$
$860$	26.3550	0.898696
$861$	0	0
$862$	−102.141	−3.47895
$863$	7.19930	0.245067	0.122533	−	0.992464i	$-0.460898\pi$
$863$	7.19930	0.245067	0.122533	+	0.992464i	$0.460898\pi$
$864$	0	0
$865$	5.46918	0.185958
$866$	64.9109	2.20576
$867$	0	0
$868$	−0.329345	−0.0111787
$869$	4.12358	0.139883
$870$	0	0
$871$	−25.5110	−0.864408
$872$	−38.4360	−1.30161
$873$	0	0
$874$	4.37335	0.147931
$875$	−7.29352	−0.246566
$876$	0	0
$877$	−30.7462	−1.03823	−0.519113	−	0.854706i	$-0.673738\pi$
$877$	−30.7462	−1.03823	−0.519113	+	0.854706i	$0.673738\pi$
$878$	−11.9373	−0.402863
$879$	0	0
$880$	−3.00363	−0.101252
$881$	−1.81471	−0.0611391	−0.0305696	−	0.999533i	$-0.509732\pi$
$881$	−1.81471	−0.0611391	−0.0305696	+	0.999533i	$0.509732\pi$
$882$	0	0
$883$	7.16879	0.241249	0.120624	−	0.992698i	$-0.461510\pi$
$883$	7.16879	0.241249	0.120624	+	0.992698i	$0.461510\pi$
$884$	63.4224	2.13313
$885$	0	0
$886$	61.5372	2.06738
$887$	51.3599	1.72450	0.862249	−	0.506485i	$-0.169055\pi$
$887$	51.3599	1.72450	0.862249	+	0.506485i	$0.169055\pi$
$888$	0	0
$889$	16.0859	0.539502
$890$	−22.8163	−0.764803
$891$	0	0
$892$	70.8429	2.37200
$893$	−3.12755	−0.104659
$894$	0	0
$895$	1.57080	0.0525060
$896$	22.1137	0.738767
$897$	0	0
$898$	12.6712	0.422842
$899$	0.173853	0.00579831
$900$	0	0
$901$	20.1434	0.671075
$902$	−23.9036	−0.795903
$903$	0	0
$904$	−71.7440	−2.38617
$905$	−1.54986	−0.0515191
$906$	0	0
$907$	−42.8167	−1.42171	−0.710853	−	0.703341i	$-0.751691\pi$
$907$	−42.8167	−1.42171	−0.710853	+	0.703341i	$0.751691\pi$
$908$	−67.2594	−2.23208
$909$	0	0
$910$	8.30970	0.275464
$911$	−4.34991	−0.144119	−0.0720595	−	0.997400i	$-0.522957\pi$
$911$	−4.34991	−0.144119	−0.0720595	+	0.997400i	$0.522957\pi$
$912$	0	0
$913$	8.38519	0.277509
$914$	34.4302	1.13885
$915$	0	0
$916$	18.6813	0.617248
$917$	13.0814	0.431987
$918$	0	0
$919$	49.2751	1.62543	0.812717	−	0.582658i	$-0.197987\pi$
$919$	49.2751	1.62543	0.812717	+	0.582658i	$0.197987\pi$
$920$	19.1781	0.632283
$921$	0	0
$922$	61.0770	2.01146
$923$	−26.7486	−0.880440
$924$	0	0
$925$	−5.92662	−0.194866
$926$	−66.3841	−2.18152
$927$	0	0
$928$	3.43542	0.112773
$929$	49.8125	1.63430	0.817148	−	0.576428i	$-0.195554\pi$
$929$	49.8125	1.63430	0.817148	+	0.576428i	$0.195554\pi$
$930$	0	0
$931$	−1.68781	−0.0553158
$932$	44.7099	1.46452
$933$	0	0
$934$	34.5353	1.13003
$935$	2.18294	0.0713896
$936$	0	0
$937$	−25.8544	−0.844628	−0.422314	−	0.906450i	$-0.638782\pi$
$937$	−25.8544	−0.844628	−0.422314	+	0.906450i	$0.638782\pi$
$938$	−17.2682	−0.563827
$939$	0	0
$940$	−26.6072	−0.867831
$941$	−28.1941	−0.919100	−0.459550	−	0.888152i	$-0.651989\pi$
$941$	−28.1941	−0.919100	−0.459550	+	0.888152i	$0.651989\pi$
$942$	0	0
$943$	55.9827	1.82305
$944$	−37.2946	−1.21384
$945$	0	0
$946$	25.1643	0.818161
$947$	21.5946	0.701729	0.350864	−	0.936426i	$-0.385888\pi$
$947$	21.5946	0.701729	0.350864	+	0.936426i	$0.385888\pi$
$948$	0	0
$949$	−71.7542	−2.32924
$950$	3.47419	0.112718
$951$	0	0
$952$	22.1289	0.717201
$953$	−15.6533	−0.507060	−0.253530	−	0.967327i	$-0.581592\pi$
$953$	−15.6533	−0.507060	−0.253530	+	0.967327i	$0.581592\pi$
$954$	0	0
$955$	14.0669	0.455193
$956$	−29.1834	−0.943857
$957$	0	0
$958$	4.45167	0.143827
$959$	−9.53933	−0.308041
$960$	0	0
$961$	−30.9956	−0.999859
$962$	14.0831	0.454058
$963$	0	0
$964$	−42.7384	−1.37651
$965$	−14.9042	−0.479782
$966$	0	0
$967$	−14.7775	−0.475212	−0.237606	−	0.971362i	$-0.576363\pi$
$967$	−14.7775	−0.475212	−0.237606	+	0.971362i	$0.576363\pi$
$968$	−5.26679	−0.169281
$969$	0	0
$970$	−0.853821	−0.0274145
$971$	−12.4319	−0.398959	−0.199480	−	0.979902i	$-0.563925\pi$
$971$	−12.4319	−0.398959	−0.199480	+	0.979902i	$0.563925\pi$
$972$	0	0
$973$	24.9327	0.799307
$974$	35.4649	1.13637
$975$	0	0
$976$	−3.31913	−0.106243
$977$	38.2596	1.22403	0.612016	−	0.790845i	$-0.290359\pi$
$977$	38.2596	1.22403	0.612016	+	0.790845i	$0.290359\pi$
$978$	0	0
$979$	−14.6749	−0.469012
$980$	−14.3588	−0.458676
$981$	0	0
$982$	10.7047	0.341600
$983$	6.94901	0.221639	0.110820	−	0.993841i	$-0.464652\pi$
$983$	6.94901	0.221639	0.110820	+	0.993841i	$0.464652\pi$
$984$	0	0
$985$	4.66821	0.148742
$986$	−22.6617	−0.721697
$987$	0	0
$988$	−5.56101	−0.176919
$989$	−58.9352	−1.87403
$990$	0	0
$991$	−39.2350	−1.24634	−0.623170	−	0.782086i	$-0.714156\pi$
$991$	−39.2350	−1.24634	−0.623170	+	0.782086i	$0.714156\pi$
$992$	0.0860807	0.00273306
$993$	0	0
$994$	−18.1059	−0.574285
$995$	16.4677	0.522062
$996$	0	0
$997$	−18.4773	−0.585183	−0.292592	−	0.956237i	$-0.594518\pi$
$997$	−18.4773	−0.585183	−0.292592	+	0.956237i	$0.594518\pi$
$998$	103.214	3.26717
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8019.2.a.k.1.5		51
3.2	odd	2		8019.2.a.l.1.47		51
27.4	even	9		891.2.j.c.694.2		102
27.7	even	9		891.2.j.c.199.2		102
27.20	odd	18		297.2.j.c.265.16	yes	102
27.23	odd	18		297.2.j.c.232.16	✓	102

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
297.2.j.c.232.16	✓	102	27.23	odd	18
297.2.j.c.265.16	yes	102	27.20	odd	18
891.2.j.c.199.2		102	27.7	even	9
891.2.j.c.694.2		102	27.4	even	9
8019.2.a.k.1.5		51	1.1	even	1	trivial
8019.2.a.l.1.47		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)

\(f(q)\)	\(=\)	\(q-2.47541 q^{2} +4.12764 q^{4} +0.628090 q^{5} +1.20891 q^{7} -5.26679 q^{8} +O(q^{10})\)	\(q-2.47541 q^{2} +4.12764 q^{4} +0.628090 q^{5} +1.20891 q^{7} -5.26679 q^{8} -1.55478 q^{10} -1.00000 q^{11} -4.42101 q^{13} -2.99255 q^{14} +4.78216 q^{16} -3.47551 q^{17} +0.304740 q^{19} +2.59253 q^{20} +2.47541 q^{22} -5.79745 q^{23} -4.60550 q^{25} +10.9438 q^{26} +4.98996 q^{28} -2.63407 q^{29} -0.0660015 q^{31} -1.30422 q^{32} +8.60331 q^{34} +0.759306 q^{35} +1.28686 q^{37} -0.754356 q^{38} -3.30802 q^{40} -9.65643 q^{41} +10.1657 q^{43} -4.12764 q^{44} +14.3511 q^{46} -10.2630 q^{47} -5.53853 q^{49} +11.4005 q^{50} -18.2484 q^{52} -5.79582 q^{53} -0.628090 q^{55} -6.36708 q^{56} +6.52040 q^{58} -7.79869 q^{59} -0.694065 q^{61} +0.163381 q^{62} -6.33584 q^{64} -2.77680 q^{65} +5.77040 q^{67} -14.3457 q^{68} -1.87959 q^{70} +6.05033 q^{71} +16.2303 q^{73} -3.18550 q^{74} +1.25786 q^{76} -1.20891 q^{77} -4.12358 q^{79} +3.00363 q^{80} +23.9036 q^{82} -8.38519 q^{83} -2.18294 q^{85} -25.1643 q^{86} +5.26679 q^{88} +14.6749 q^{89} -5.34462 q^{91} -23.9298 q^{92} +25.4051 q^{94} +0.191404 q^{95} +0.549159 q^{97} +13.7101 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 51 q + 60 q^{4} - 6 q^{5} + 12 q^{7}+O(q^{10}) \) 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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Database

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