LMFDB - Embedded newform 8001.2.a.y.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8001,2,Mod(1,8001)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8001, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8001 = 3^{2} \cdot 7 \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8001.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:	$63.8883066572$
Analytic rank:	$0$
Dimension:	$28$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Character	$\chi$	$=$	8001.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.47474 q^{2} +4.12433 q^{4} -3.91787 q^{5} -1.00000 q^{7} -5.25716 q^{8} +O(q^{10})$	\(q-2.47474 q^{2} +4.12433 q^{4} -3.91787 q^{5} -1.00000 q^{7} -5.25716 q^{8} +9.69570 q^{10} -2.92151 q^{11} +1.07733 q^{13} +2.47474 q^{14} +4.76143 q^{16} -1.80341 q^{17} +1.65946 q^{19} -16.1586 q^{20} +7.22998 q^{22} -2.10778 q^{23} +10.3497 q^{25} -2.66610 q^{26} -4.12433 q^{28} +4.17547 q^{29} -1.66527 q^{31} -1.26898 q^{32} +4.46296 q^{34} +3.91787 q^{35} -1.01454 q^{37} -4.10672 q^{38} +20.5968 q^{40} +8.34018 q^{41} -2.95682 q^{43} -12.0493 q^{44} +5.21620 q^{46} -1.91717 q^{47} +1.00000 q^{49} -25.6128 q^{50} +4.44325 q^{52} -10.9277 q^{53} +11.4461 q^{55} +5.25716 q^{56} -10.3332 q^{58} -7.84084 q^{59} -10.6424 q^{61} +4.12110 q^{62} -6.38247 q^{64} -4.22082 q^{65} -8.08673 q^{67} -7.43785 q^{68} -9.69570 q^{70} +10.7204 q^{71} +0.791381 q^{73} +2.51073 q^{74} +6.84415 q^{76} +2.92151 q^{77} -0.860432 q^{79} -18.6546 q^{80} -20.6398 q^{82} +2.57850 q^{83} +7.06552 q^{85} +7.31734 q^{86} +15.3589 q^{88} +9.63697 q^{89} -1.07733 q^{91} -8.69317 q^{92} +4.74449 q^{94} -6.50154 q^{95} -5.99985 q^{97} -2.47474 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 30 q^{4} - 28 q^{7}+O(q^{10}) $ 28 * q + 30 * q^4 - 28 * q^7	$ 28 q + 30 q^{4} - 28 q^{7} + 4 q^{10} + 8 q^{13} + 42 q^{16} + 34 q^{19} - 10 q^{22} + 14 q^{25} - 30 q^{28} + 56 q^{31} - 6 q^{37} + 38 q^{40} + 18 q^{43} + 16 q^{46} + 28 q^{49} + 18 q^{52} + 48 q^{55} + 2 q^{58} + 36 q^{61} + 76 q^{64} - 4 q^{70} + 50 q^{73} + 132 q^{76} + 66 q^{79} - 36 q^{82} + 20 q^{85} + 6 q^{88} - 8 q^{91} + 54 q^{94} - 8 q^{97}+O(q^{100}) $ 28 * q + 30 * q^4 - 28 * q^7 + 4 * q^10 + 8 * q^13 + 42 * q^16 + 34 * q^19 - 10 * q^22 + 14 * q^25 - 30 * q^28 + 56 * q^31 - 6 * q^37 + 38 * q^40 + 18 * q^43 + 16 * q^46 + 28 * q^49 + 18 * q^52 + 48 * q^55 + 2 * q^58 + 36 * q^61 + 76 * q^64 - 4 * q^70 + 50 * q^73 + 132 * q^76 + 66 * q^79 - 36 * q^82 + 20 * q^85 + 6 * q^88 - 8 * q^91 + 54 * q^94 - 8 * q^97

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 $

$2$	−2.47474	−1.74990	−0.874952	−	0.484210i	$-0.839107\pi$
$2$	−2.47474	−1.74990	−0.874952	+	0.484210i	$0.839107\pi$
$3$	0	0
$3$	0	0
$4$	4.12433	2.06216
$5$	−3.91787	−1.75212	−0.876062	−	0.482199i	$-0.839838\pi$
$5$	−3.91787	−1.75212	−0.876062	+	0.482199i	$0.839838\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−5.25716	−1.85869
$9$	0	0
$10$	9.69570	3.06605
$11$	−2.92151	−0.880870	−0.440435	−	0.897785i	$-0.645176\pi$
$11$	−2.92151	−0.880870	−0.440435	+	0.897785i	$0.645176\pi$
$12$	0	0
$13$	1.07733	0.298797	0.149398	−	0.988777i	$-0.452266\pi$
$13$	1.07733	0.298797	0.149398	+	0.988777i	$0.452266\pi$
$14$	2.47474	0.661402
$15$	0	0
$16$	4.76143	1.19036
$17$	−1.80341	−0.437391	−0.218695	−	0.975793i	$-0.570180\pi$
$17$	−1.80341	−0.437391	−0.218695	+	0.975793i	$0.570180\pi$
$18$	0	0
$19$	1.65946	0.380706	0.190353	−	0.981716i	$-0.439037\pi$
$19$	1.65946	0.380706	0.190353	+	0.981716i	$0.439037\pi$
$20$	−16.1586	−3.61317
$21$	0	0
$22$	7.22998	1.54144
$23$	−2.10778	−0.439502	−0.219751	−	0.975556i	$-0.570525\pi$
$23$	−2.10778	−0.439502	−0.219751	+	0.975556i	$0.570525\pi$
$24$	0	0
$25$	10.3497	2.06994
$26$	−2.66610	−0.522865
$27$	0	0
$28$	−4.12433	−0.779425
$29$	4.17547	0.775365	0.387682	−	0.921793i	$-0.373276\pi$
$29$	4.17547	0.775365	0.387682	+	0.921793i	$0.373276\pi$
$30$	0	0
$31$	−1.66527	−0.299091	−0.149545	−	0.988755i	$-0.547781\pi$
$31$	−1.66527	−0.299091	−0.149545	+	0.988755i	$0.547781\pi$
$32$	−1.26898	−0.224325
$33$	0	0
$34$	4.46296	0.765392
$35$	3.91787	0.662240
$36$	0	0
$37$	−1.01454	−0.166790	−0.0833950	−	0.996517i	$-0.526576\pi$
$37$	−1.01454	−0.166790	−0.0833950	+	0.996517i	$0.526576\pi$
$38$	−4.10672	−0.666199
$39$	0	0
$40$	20.5968	3.25665
$41$	8.34018	1.30252	0.651259	−	0.758856i	$-0.274241\pi$
$41$	8.34018	1.30252	0.651259	+	0.758856i	$0.274241\pi$
$42$	0	0
$43$	−2.95682	−0.450910	−0.225455	−	0.974254i	$-0.572387\pi$
$43$	−2.95682	−0.450910	−0.225455	+	0.974254i	$0.572387\pi$
$44$	−12.0493	−1.81650
$45$	0	0
$46$	5.21620	0.769087
$47$	−1.91717	−0.279648	−0.139824	−	0.990176i	$-0.544654\pi$
$47$	−1.91717	−0.279648	−0.139824	+	0.990176i	$0.544654\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−25.6128	−3.62219
$51$	0	0
$52$	4.44325	0.616168
$53$	−10.9277	−1.50104	−0.750521	−	0.660847i	$-0.770197\pi$
$53$	−10.9277	−1.50104	−0.750521	+	0.660847i	$0.770197\pi$
$54$	0	0
$55$	11.4461	1.54339
$56$	5.25716	0.702517
$57$	0	0
$58$	−10.3332	−1.35681
$59$	−7.84084	−1.02079	−0.510395	−	0.859940i	$-0.670501\pi$
$59$	−7.84084	−1.02079	−0.510395	+	0.859940i	$0.670501\pi$
$60$	0	0
$61$	−10.6424	−1.36262	−0.681311	−	0.731995i	$-0.738590\pi$
$61$	−10.6424	−1.36262	−0.681311	+	0.731995i	$0.738590\pi$
$62$	4.12110	0.523380
$63$	0	0
$64$	−6.38247	−0.797809
$65$	−4.22082	−0.523529
$66$	0	0
$67$	−8.08673	−0.987952	−0.493976	−	0.869476i	$-0.664457\pi$
$67$	−8.08673	−0.987952	−0.493976	+	0.869476i	$0.664457\pi$
$68$	−7.43785	−0.901972
$69$	0	0
$70$	−9.69570	−1.15886
$71$	10.7204	1.27227	0.636136	−	0.771577i	$-0.280532\pi$
$71$	10.7204	1.27227	0.636136	+	0.771577i	$0.280532\pi$
$72$	0	0
$73$	0.791381	0.0926241	0.0463121	−	0.998927i	$-0.485253\pi$
$73$	0.791381	0.0926241	0.0463121	+	0.998927i	$0.485253\pi$
$74$	2.51073	0.291866
$75$	0	0
$76$	6.84415	0.785078
$77$	2.92151	0.332937
$78$	0	0
$79$	−0.860432	−0.0968062	−0.0484031	−	0.998828i	$-0.515413\pi$
$79$	−0.860432	−0.0968062	−0.0484031	+	0.998828i	$0.515413\pi$
$80$	−18.6546	−2.08565
$81$	0	0
$82$	−20.6398	−2.27928
$83$	2.57850	0.283027	0.141513	−	0.989936i	$-0.454803\pi$
$83$	2.57850	0.283027	0.141513	+	0.989936i	$0.454803\pi$
$84$	0	0
$85$	7.06552	0.766363
$86$	7.31734	0.789049
$87$	0	0
$88$	15.3589	1.63726
$89$	9.63697	1.02152	0.510758	−	0.859724i	$-0.329365\pi$
$89$	9.63697	1.02152	0.510758	+	0.859724i	$0.329365\pi$
$90$	0	0
$91$	−1.07733	−0.112935
$92$	−8.69317	−0.906326
$93$	0	0
$94$	4.74449	0.489357
$95$	−6.50154	−0.667044
$96$	0	0
$97$	−5.99985	−0.609192	−0.304596	−	0.952482i	$-0.598521\pi$
$97$	−5.99985	−0.609192	−0.304596	+	0.952482i	$0.598521\pi$
$98$	−2.47474	−0.249986
$99$	0	0
$100$	42.6855	4.26855
$101$	−12.9224	−1.28582	−0.642912	−	0.765940i	$-0.722274\pi$
$101$	−12.9224	−1.28582	−0.642912	+	0.765940i	$0.722274\pi$
$102$	0	0
$103$	−4.66147	−0.459309	−0.229654	−	0.973272i	$-0.573760\pi$
$103$	−4.66147	−0.459309	−0.229654	+	0.973272i	$0.573760\pi$
$104$	−5.66367	−0.555369
$105$	0	0
$106$	27.0433	2.62668
$107$	−14.8610	−1.43667	−0.718333	−	0.695699i	$-0.755095\pi$
$107$	−14.8610	−1.43667	−0.718333	+	0.695699i	$0.755095\pi$
$108$	0	0
$109$	−4.33326	−0.415051	−0.207526	−	0.978230i	$-0.566541\pi$
$109$	−4.33326	−0.415051	−0.207526	+	0.978230i	$0.566541\pi$
$110$	−28.3261	−2.70079
$111$	0	0
$112$	−4.76143	−0.449913
$113$	9.36220	0.880722	0.440361	−	0.897821i	$-0.354850\pi$
$113$	9.36220	0.880722	0.440361	+	0.897821i	$0.354850\pi$
$114$	0	0
$115$	8.25799	0.770062
$116$	17.2210	1.59893
$117$	0	0
$118$	19.4040	1.78629
$119$	1.80341	0.165318
$120$	0	0
$121$	−2.46476	−0.224069
$122$	26.3372	2.38446
$123$	0	0
$124$	−6.86811	−0.616775
$125$	−20.9594	−1.87466
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	18.3329	1.62042
$129$	0	0
$130$	10.4454	0.916125
$131$	−20.7547	−1.81335	−0.906673	−	0.421834i	$-0.861386\pi$
$131$	−20.7547	−1.81335	−0.906673	+	0.421834i	$0.861386\pi$
$132$	0	0
$133$	−1.65946	−0.143893
$134$	20.0125	1.72882
$135$	0	0
$136$	9.48080	0.812972
$137$	−16.1255	−1.37770	−0.688849	−	0.724905i	$-0.741884\pi$
$137$	−16.1255	−1.37770	−0.688849	+	0.724905i	$0.741884\pi$
$138$	0	0
$139$	−0.0513424	−0.00435481	−0.00217740	−	0.999998i	$-0.500693\pi$
$139$	−0.0513424	−0.00435481	−0.00217740	+	0.999998i	$0.500693\pi$
$140$	16.1586	1.36565
$141$	0	0
$142$	−26.5301	−2.22635
$143$	−3.14742	−0.263201
$144$	0	0
$145$	−16.3589	−1.35853
$146$	−1.95846	−0.162083
$147$	0	0
$148$	−4.18431	−0.343948
$149$	−3.37272	−0.276304	−0.138152	−	0.990411i	$-0.544116\pi$
$149$	−3.37272	−0.276304	−0.138152	+	0.990411i	$0.544116\pi$
$150$	0	0
$151$	3.11866	0.253793	0.126897	−	0.991916i	$-0.459498\pi$
$151$	3.11866	0.253793	0.126897	+	0.991916i	$0.459498\pi$
$152$	−8.72403	−0.707612
$153$	0	0
$154$	−7.22998	−0.582609
$155$	6.52430	0.524044
$156$	0	0
$157$	−1.76450	−0.140822	−0.0704111	−	0.997518i	$-0.522431\pi$
$157$	−1.76450	−0.140822	−0.0704111	+	0.997518i	$0.522431\pi$
$158$	2.12934	0.169402
$159$	0	0
$160$	4.97168	0.393046
$161$	2.10778	0.166116
$162$	0	0
$163$	4.30450	0.337155	0.168577	−	0.985688i	$-0.446083\pi$
$163$	4.30450	0.337155	0.168577	+	0.985688i	$0.446083\pi$
$164$	34.3976	2.68601
$165$	0	0
$166$	−6.38111	−0.495270
$167$	14.4181	1.11571	0.557853	−	0.829940i	$-0.311625\pi$
$167$	14.4181	1.11571	0.557853	+	0.829940i	$0.311625\pi$
$168$	0	0
$169$	−11.8394	−0.910721
$170$	−17.4853	−1.34106
$171$	0	0
$172$	−12.1949	−0.929851
$173$	−6.77461	−0.515064	−0.257532	−	0.966270i	$-0.582909\pi$
$173$	−6.77461	−0.515064	−0.257532	+	0.966270i	$0.582909\pi$
$174$	0	0
$175$	−10.3497	−0.782363
$176$	−13.9106	−1.04855
$177$	0	0
$178$	−23.8490	−1.78756
$179$	4.64041	0.346841	0.173420	−	0.984848i	$-0.444518\pi$
$179$	4.64041	0.346841	0.173420	+	0.984848i	$0.444518\pi$
$180$	0	0
$181$	−10.2174	−0.759450	−0.379725	−	0.925099i	$-0.623981\pi$
$181$	−10.2174	−0.759450	−0.379725	+	0.925099i	$0.623981\pi$
$182$	2.66610	0.197625
$183$	0	0
$184$	11.0809	0.816896
$185$	3.97485	0.292237
$186$	0	0
$187$	5.26868	0.385284
$188$	−7.90703	−0.576680
$189$	0	0
$190$	16.0896	1.16726
$191$	−12.1642	−0.880169	−0.440085	−	0.897956i	$-0.645052\pi$
$191$	−12.1642	−0.880169	−0.440085	+	0.897956i	$0.645052\pi$
$192$	0	0
$193$	−16.7553	−1.20607	−0.603037	−	0.797714i	$-0.706043\pi$
$193$	−16.7553	−1.20607	−0.603037	+	0.797714i	$0.706043\pi$
$194$	14.8481	1.06603
$195$	0	0
$196$	4.12433	0.294595
$197$	−1.83706	−0.130885	−0.0654424	−	0.997856i	$-0.520846\pi$
$197$	−1.83706	−0.130885	−0.0654424	+	0.997856i	$0.520846\pi$
$198$	0	0
$199$	25.9546	1.83987	0.919936	−	0.392067i	$-0.128240\pi$
$199$	25.9546	1.83987	0.919936	+	0.392067i	$0.128240\pi$
$200$	−54.4099	−3.84736
$201$	0	0
$202$	31.9795	2.25007
$203$	−4.17547	−0.293060
$204$	0	0
$205$	−32.6757	−2.28217
$206$	11.5359	0.803746
$207$	0	0
$208$	5.12961	0.355675
$209$	−4.84813	−0.335352
$210$	0	0
$211$	4.84519	0.333557	0.166778	−	0.985994i	$-0.446664\pi$
$211$	4.84519	0.333557	0.166778	+	0.985994i	$0.446664\pi$
$212$	−45.0696	−3.09539
$213$	0	0
$214$	36.7771	2.51403
$215$	11.5844	0.790050
$216$	0	0
$217$	1.66527	0.113046
$218$	10.7237	0.726300
$219$	0	0
$220$	47.2075	3.18273
$221$	−1.94286	−0.130691
$222$	0	0
$223$	0.546699	0.0366097	0.0183048	−	0.999832i	$-0.494173\pi$
$223$	0.546699	0.0366097	0.0183048	+	0.999832i	$0.494173\pi$
$224$	1.26898	0.0847870
$225$	0	0
$226$	−23.1690	−1.54118
$227$	−17.6771	−1.17327	−0.586635	−	0.809851i	$-0.699548\pi$
$227$	−17.6771	−1.17327	−0.586635	+	0.809851i	$0.699548\pi$
$228$	0	0
$229$	11.5087	0.760515	0.380257	−	0.924881i	$-0.375835\pi$
$229$	11.5087	0.760515	0.380257	+	0.924881i	$0.375835\pi$
$230$	−20.4364	−1.34753
$231$	0	0
$232$	−21.9511	−1.44116
$233$	−3.70731	−0.242874	−0.121437	−	0.992599i	$-0.538750\pi$
$233$	−3.70731	−0.242874	−0.121437	+	0.992599i	$0.538750\pi$
$234$	0	0
$235$	7.51121	0.489977
$236$	−32.3382	−2.10504
$237$	0	0
$238$	−4.46296	−0.289291
$239$	27.4160	1.77339	0.886697	−	0.462351i	$-0.152994\pi$
$239$	27.4160	1.77339	0.886697	+	0.462351i	$0.152994\pi$
$240$	0	0
$241$	−3.10227	−0.199835	−0.0999175	−	0.994996i	$-0.531858\pi$
$241$	−3.10227	−0.199835	−0.0999175	+	0.994996i	$0.531858\pi$
$242$	6.09963	0.392099
$243$	0	0
$244$	−43.8928	−2.80995
$245$	−3.91787	−0.250303
$246$	0	0
$247$	1.78778	0.113754
$248$	8.75457	0.555916
$249$	0	0
$250$	51.8689	3.28048
$251$	10.9777	0.692907	0.346453	−	0.938067i	$-0.387386\pi$
$251$	10.9777	0.692907	0.346453	+	0.938067i	$0.387386\pi$
$252$	0	0
$253$	6.15790	0.387144
$254$	−2.47474	−0.155279
$255$	0	0
$256$	−32.6042	−2.03776
$257$	18.5226	1.15541	0.577704	−	0.816247i	$-0.303949\pi$
$257$	18.5226	1.15541	0.577704	+	0.816247i	$0.303949\pi$
$258$	0	0
$259$	1.01454	0.0630407
$260$	−17.4081	−1.07960
$261$	0	0
$262$	51.3624	3.17318
$263$	−5.23189	−0.322612	−0.161306	−	0.986904i	$-0.551571\pi$
$263$	−5.23189	−0.322612	−0.161306	+	0.986904i	$0.551571\pi$
$264$	0	0
$265$	42.8135	2.63001
$266$	4.10672	0.251799
$267$	0	0
$268$	−33.3523	−2.03732
$269$	−3.55339	−0.216654	−0.108327	−	0.994115i	$-0.534549\pi$
$269$	−3.55339	−0.216654	−0.108327	+	0.994115i	$0.534549\pi$
$270$	0	0
$271$	−3.04637	−0.185054	−0.0925269	−	0.995710i	$-0.529494\pi$
$271$	−3.04637	−0.185054	−0.0925269	+	0.995710i	$0.529494\pi$
$272$	−8.58680	−0.520651
$273$	0	0
$274$	39.9065	2.41084
$275$	−30.2367	−1.82334
$276$	0	0
$277$	−4.99969	−0.300402	−0.150201	−	0.988655i	$-0.547992\pi$
$277$	−4.99969	−0.300402	−0.150201	+	0.988655i	$0.547992\pi$
$278$	0.127059	0.00762050
$279$	0	0
$280$	−20.5968	−1.23090
$281$	−25.5299	−1.52299	−0.761493	−	0.648173i	$-0.775533\pi$
$281$	−25.5299	−1.52299	−0.761493	+	0.648173i	$0.775533\pi$
$282$	0	0
$283$	6.24103	0.370991	0.185495	−	0.982645i	$-0.440611\pi$
$283$	6.24103	0.370991	0.185495	+	0.982645i	$0.440611\pi$
$284$	44.2143	2.62363
$285$	0	0
$286$	7.78905	0.460576
$287$	−8.34018	−0.492305
$288$	0	0
$289$	−13.7477	−0.808689
$290$	40.4840	2.37731
$291$	0	0
$292$	3.26391	0.191006
$293$	23.7934	1.39003	0.695014	−	0.718996i	$-0.255398\pi$
$293$	23.7934	1.39003	0.695014	+	0.718996i	$0.255398\pi$
$294$	0	0
$295$	30.7194	1.78855
$296$	5.33361	0.310010
$297$	0	0
$298$	8.34659	0.483505
$299$	−2.27077	−0.131322
$300$	0	0
$301$	2.95682	0.170428
$302$	−7.71787	−0.444114
$303$	0	0
$304$	7.90139	0.453176
$305$	41.6956	2.38748
$306$	0	0
$307$	−11.1626	−0.637082	−0.318541	−	0.947909i	$-0.603193\pi$
$307$	−11.1626	−0.637082	−0.318541	+	0.947909i	$0.603193\pi$
$308$	12.0493	0.686572
$309$	0	0
$310$	−16.1459	−0.917027
$311$	−6.94373	−0.393743	−0.196871	−	0.980429i	$-0.563078\pi$
$311$	−6.94373	−0.393743	−0.196871	+	0.980429i	$0.563078\pi$
$312$	0	0
$313$	8.64852	0.488843	0.244422	−	0.969669i	$-0.421402\pi$
$313$	8.64852	0.488843	0.244422	+	0.969669i	$0.421402\pi$
$314$	4.36667	0.246425
$315$	0	0
$316$	−3.54871	−0.199630
$317$	15.5711	0.874558	0.437279	−	0.899326i	$-0.355942\pi$
$317$	15.5711	0.874558	0.437279	+	0.899326i	$0.355942\pi$
$318$	0	0
$319$	−12.1987	−0.682995
$320$	25.0057	1.39786
$321$	0	0
$322$	−5.21620	−0.290687
$323$	−2.99268	−0.166517
$324$	0	0
$325$	11.1500	0.618490
$326$	−10.6525	−0.589988
$327$	0	0
$328$	−43.8456	−2.42097
$329$	1.91717	0.105697
$330$	0	0
$331$	1.45717	0.0800934	0.0400467	−	0.999198i	$-0.487249\pi$
$331$	1.45717	0.0800934	0.0400467	+	0.999198i	$0.487249\pi$
$332$	10.6346	0.583648
$333$	0	0
$334$	−35.6810	−1.95238
$335$	31.6827	1.73101
$336$	0	0
$337$	−11.8360	−0.644746	−0.322373	−	0.946613i	$-0.604481\pi$
$337$	−11.8360	−0.644746	−0.322373	+	0.946613i	$0.604481\pi$
$338$	29.2993	1.59367
$339$	0	0
$340$	29.1405	1.58037
$341$	4.86510	0.263460
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	15.5444	0.838100
$345$	0	0
$346$	16.7654	0.901313
$347$	−16.8491	−0.904505	−0.452252	−	0.891890i	$-0.649379\pi$
$347$	−16.8491	−0.904505	−0.452252	+	0.891890i	$0.649379\pi$
$348$	0	0
$349$	15.6507	0.837764	0.418882	−	0.908041i	$-0.362422\pi$
$349$	15.6507	0.837764	0.418882	+	0.908041i	$0.362422\pi$
$350$	25.6128	1.36906
$351$	0	0
$352$	3.70733	0.197601
$353$	−32.9837	−1.75554	−0.877772	−	0.479078i	$-0.840971\pi$
$353$	−32.9837	−1.75554	−0.877772	+	0.479078i	$0.840971\pi$
$354$	0	0
$355$	−42.0009	−2.22918
$356$	39.7460	2.10654
$357$	0	0
$358$	−11.4838	−0.606938
$359$	18.1175	0.956205	0.478103	−	0.878304i	$-0.341325\pi$
$359$	18.1175	0.956205	0.478103	+	0.878304i	$0.341325\pi$
$360$	0	0
$361$	−16.2462	−0.855063
$362$	25.2853	1.32896
$363$	0	0
$364$	−4.44325	−0.232889
$365$	−3.10052	−0.162289
$366$	0	0
$367$	15.4460	0.806276	0.403138	−	0.915139i	$-0.367920\pi$
$367$	15.4460	0.806276	0.403138	+	0.915139i	$0.367920\pi$
$368$	−10.0360	−0.523165
$369$	0	0
$370$	−9.83671	−0.511386
$371$	10.9277	0.567340
$372$	0	0
$373$	4.96287	0.256968	0.128484	−	0.991712i	$-0.458989\pi$
$373$	4.96287	0.256968	0.128484	+	0.991712i	$0.458989\pi$
$374$	−13.0386	−0.674211
$375$	0	0
$376$	10.0789	0.519777
$377$	4.49834	0.231676
$378$	0	0
$379$	−14.3351	−0.736347	−0.368173	−	0.929757i	$-0.620017\pi$
$379$	−14.3351	−0.736347	−0.368173	+	0.929757i	$0.620017\pi$
$380$	−26.8145	−1.37555
$381$	0	0
$382$	30.1032	1.54021
$383$	5.38582	0.275202	0.137601	−	0.990488i	$-0.456061\pi$
$383$	5.38582	0.275202	0.137601	+	0.990488i	$0.456061\pi$
$384$	0	0
$385$	−11.4461	−0.583347
$386$	41.4650	2.11051
$387$	0	0
$388$	−24.7453	−1.25625
$389$	1.12686	0.0571338	0.0285669	−	0.999592i	$-0.490906\pi$
$389$	1.12686	0.0571338	0.0285669	+	0.999592i	$0.490906\pi$
$390$	0	0
$391$	3.80119	0.192234
$392$	−5.25716	−0.265527
$393$	0	0
$394$	4.54623	0.229036
$395$	3.37106	0.169616
$396$	0	0
$397$	1.61381	0.0809946	0.0404973	−	0.999180i	$-0.487106\pi$
$397$	1.61381	0.0809946	0.0404973	+	0.999180i	$0.487106\pi$
$398$	−64.2308	−3.21960
$399$	0	0
$400$	49.2793	2.46396
$401$	18.3637	0.917041	0.458521	−	0.888684i	$-0.348379\pi$
$401$	18.3637	0.917041	0.458521	+	0.888684i	$0.348379\pi$
$402$	0	0
$403$	−1.79404	−0.0893673
$404$	−53.2961	−2.65158
$405$	0	0
$406$	10.3332	0.512827
$407$	2.96400	0.146920
$408$	0	0
$409$	25.8853	1.27995	0.639974	−	0.768397i	$-0.278945\pi$
$409$	25.8853	1.27995	0.639974	+	0.768397i	$0.278945\pi$
$410$	80.8639	3.99358
$411$	0	0
$412$	−19.2255	−0.947170
$413$	7.84084	0.385823
$414$	0	0
$415$	−10.1022	−0.495898
$416$	−1.36710	−0.0670277
$417$	0	0
$418$	11.9979	0.586834
$419$	−14.6688	−0.716617	−0.358308	−	0.933603i	$-0.616646\pi$
$419$	−14.6688	−0.716617	−0.358308	+	0.933603i	$0.616646\pi$
$420$	0	0
$421$	−23.3016	−1.13565	−0.567826	−	0.823149i	$-0.692215\pi$
$421$	−23.3016	−1.13565	−0.567826	+	0.823149i	$0.692215\pi$
$422$	−11.9906	−0.583692
$423$	0	0
$424$	57.4489	2.78996
$425$	−18.6647	−0.905372
$426$	0	0
$427$	10.6424	0.515022
$428$	−61.2916	−2.96264
$429$	0	0
$430$	−28.6684	−1.38251
$431$	20.0873	0.967573	0.483786	−	0.875186i	$-0.339261\pi$
$431$	20.0873	0.967573	0.483786	+	0.875186i	$0.339261\pi$
$432$	0	0
$433$	−5.43036	−0.260966	−0.130483	−	0.991451i	$-0.541653\pi$
$433$	−5.43036	−0.260966	−0.130483	+	0.991451i	$0.541653\pi$
$434$	−4.12110	−0.197819
$435$	0	0
$436$	−17.8718	−0.855904
$437$	−3.49777	−0.167321
$438$	0	0
$439$	−24.4762	−1.16819	−0.584093	−	0.811687i	$-0.698550\pi$
$439$	−24.4762	−1.16819	−0.584093	+	0.811687i	$0.698550\pi$
$440$	−60.1740	−2.86868
$441$	0	0
$442$	4.80807	0.228697
$443$	20.0224	0.951291	0.475646	−	0.879637i	$-0.342214\pi$
$443$	20.0224	0.951291	0.475646	+	0.879637i	$0.342214\pi$
$444$	0	0
$445$	−37.7564	−1.78982
$446$	−1.35294	−0.0640634
$447$	0	0
$448$	6.38247	0.301544
$449$	8.34826	0.393979	0.196989	−	0.980406i	$-0.436884\pi$
$449$	8.34826	0.393979	0.196989	+	0.980406i	$0.436884\pi$
$450$	0	0
$451$	−24.3660	−1.14735
$452$	38.6128	1.81619
$453$	0	0
$454$	43.7462	2.05311
$455$	4.22082	0.197875
$456$	0	0
$457$	−28.0929	−1.31413	−0.657066	−	0.753833i	$-0.728203\pi$
$457$	−28.0929	−1.31413	−0.657066	+	0.753833i	$0.728203\pi$
$458$	−28.4810	−1.33083
$459$	0	0
$460$	34.0587	1.58799
$461$	22.0587	1.02738	0.513688	−	0.857977i	$-0.328279\pi$
$461$	22.0587	1.02738	0.513688	+	0.857977i	$0.328279\pi$
$462$	0	0
$463$	−16.9156	−0.786136	−0.393068	−	0.919509i	$-0.628586\pi$
$463$	−16.9156	−0.786136	−0.393068	+	0.919509i	$0.628586\pi$
$464$	19.8812	0.922961
$465$	0	0
$466$	9.17461	0.425006
$467$	8.73499	0.404207	0.202104	−	0.979364i	$-0.435222\pi$
$467$	8.73499	0.404207	0.202104	+	0.979364i	$0.435222\pi$
$468$	0	0
$469$	8.08673	0.373411
$470$	−18.5883	−0.857414
$471$	0	0
$472$	41.2205	1.89733
$473$	8.63838	0.397193
$474$	0	0
$475$	17.1749	0.788037
$476$	7.43785	0.340913
$477$	0	0
$478$	−67.8474	−3.10327
$479$	−20.8341	−0.951936	−0.475968	−	0.879463i	$-0.657902\pi$
$479$	−20.8341	−0.951936	−0.475968	+	0.879463i	$0.657902\pi$
$480$	0	0
$481$	−1.09299	−0.0498363
$482$	7.67732	0.349692
$483$	0	0
$484$	−10.1655	−0.462067
$485$	23.5066	1.06738
$486$	0	0
$487$	29.7910	1.34996	0.674979	−	0.737837i	$-0.264153\pi$
$487$	29.7910	1.34996	0.674979	+	0.737837i	$0.264153\pi$
$488$	55.9488	2.53268
$489$	0	0
$490$	9.69570	0.438007
$491$	−28.1990	−1.27260	−0.636302	−	0.771440i	$-0.719537\pi$
$491$	−28.1990	−1.27260	−0.636302	+	0.771440i	$0.719537\pi$
$492$	0	0
$493$	−7.53007	−0.339137
$494$	−4.42428	−0.199058
$495$	0	0
$496$	−7.92905	−0.356025
$497$	−10.7204	−0.480874
$498$	0	0
$499$	24.4485	1.09446	0.547232	−	0.836981i	$-0.315681\pi$
$499$	24.4485	1.09446	0.547232	+	0.836981i	$0.315681\pi$
$500$	−86.4433	−3.86586
$501$	0	0
$502$	−27.1670	−1.21252
$503$	−41.0214	−1.82905	−0.914526	−	0.404526i	$-0.867436\pi$
$503$	−41.0214	−1.82905	−0.914526	+	0.404526i	$0.867436\pi$
$504$	0	0
$505$	50.6281	2.25292
$506$	−15.2392	−0.677465
$507$	0	0
$508$	4.12433	0.182987
$509$	17.3479	0.768932	0.384466	−	0.923139i	$-0.374386\pi$
$509$	17.3479	0.768932	0.384466	+	0.923139i	$0.374386\pi$
$510$	0	0
$511$	−0.791381	−0.0350086
$512$	44.0210	1.94547
$513$	0	0
$514$	−45.8386	−2.02185
$515$	18.2630	0.804766
$516$	0	0
$517$	5.60103	0.246333
$518$	−2.51073	−0.110315
$519$	0	0
$520$	22.1895	0.973075
$521$	16.6929	0.731331	0.365666	−	0.930746i	$-0.380841\pi$
$521$	16.6929	0.731331	0.365666	+	0.930746i	$0.380841\pi$
$522$	0	0
$523$	−7.90355	−0.345598	−0.172799	−	0.984957i	$-0.555281\pi$
$523$	−7.90355	−0.345598	−0.172799	+	0.984957i	$0.555281\pi$
$524$	−85.5992	−3.73942
$525$	0	0
$526$	12.9475	0.564540
$527$	3.00316	0.130820
$528$	0	0
$529$	−18.5573	−0.806838
$530$	−105.952	−4.60227
$531$	0	0
$532$	−6.84415	−0.296732
$533$	8.98510	0.389188
$534$	0	0
$535$	58.2234	2.51722
$536$	42.5132	1.83629
$537$	0	0
$538$	8.79370	0.379123
$539$	−2.92151	−0.125839
$540$	0	0
$541$	5.39321	0.231872	0.115936	−	0.993257i	$-0.463013\pi$
$541$	5.39321	0.231872	0.115936	+	0.993257i	$0.463013\pi$
$542$	7.53897	0.323826
$543$	0	0
$544$	2.28848	0.0981179
$545$	16.9772	0.727221
$546$	0	0
$547$	−1.17116	−0.0500753	−0.0250377	−	0.999687i	$-0.507971\pi$
$547$	−1.17116	−0.0500753	−0.0250377	+	0.999687i	$0.507971\pi$
$548$	−66.5070	−2.84104
$549$	0	0
$550$	74.8280	3.19068
$551$	6.92901	0.295186
$552$	0	0
$553$	0.860432	0.0365893
$554$	12.3729	0.525675
$555$	0	0
$556$	−0.211753	−0.00898033
$557$	22.2198	0.941482	0.470741	−	0.882271i	$-0.343987\pi$
$557$	22.2198	0.941482	0.470741	+	0.882271i	$0.343987\pi$
$558$	0	0
$559$	−3.18546	−0.134730
$560$	18.6546	0.788303
$561$	0	0
$562$	63.1798	2.66508
$563$	−18.9663	−0.799334	−0.399667	−	0.916661i	$-0.630874\pi$
$563$	−18.9663	−0.799334	−0.399667	+	0.916661i	$0.630874\pi$
$564$	0	0
$565$	−36.6799	−1.54313
$566$	−15.4449	−0.649198
$567$	0	0
$568$	−56.3586	−2.36475
$569$	5.83703	0.244701	0.122351	−	0.992487i	$-0.460957\pi$
$569$	5.83703	0.244701	0.122351	+	0.992487i	$0.460957\pi$
$570$	0	0
$571$	15.6464	0.654781	0.327390	−	0.944889i	$-0.393831\pi$
$571$	15.6464	0.654781	0.327390	+	0.944889i	$0.393831\pi$
$572$	−12.9810	−0.542763
$573$	0	0
$574$	20.6398	0.861487
$575$	−21.8148	−0.909742
$576$	0	0
$577$	−6.24872	−0.260138	−0.130069	−	0.991505i	$-0.541520\pi$
$577$	−6.24872	−0.260138	−0.130069	+	0.991505i	$0.541520\pi$
$578$	34.0220	1.41513
$579$	0	0
$580$	−67.4696	−2.80152
$581$	−2.57850	−0.106974
$582$	0	0
$583$	31.9256	1.32222
$584$	−4.16041	−0.172159
$585$	0	0
$586$	−58.8825	−2.43242
$587$	25.7862	1.06431	0.532155	−	0.846647i	$-0.321382\pi$
$587$	25.7862	1.06431	0.532155	+	0.846647i	$0.321382\pi$
$588$	0	0
$589$	−2.76344	−0.113866
$590$	−76.0224	−3.12979
$591$	0	0
$592$	−4.83068	−0.198540
$593$	−32.9415	−1.35274	−0.676372	−	0.736560i	$-0.736449\pi$
$593$	−32.9415	−1.35274	−0.676372	+	0.736560i	$0.736449\pi$
$594$	0	0
$595$	−7.06552	−0.289658
$596$	−13.9102	−0.569784
$597$	0	0
$598$	5.61955	0.229800
$599$	33.7434	1.37872	0.689358	−	0.724421i	$-0.257893\pi$
$599$	33.7434	1.37872	0.689358	+	0.724421i	$0.257893\pi$
$600$	0	0
$601$	9.65133	0.393686	0.196843	−	0.980435i	$-0.436931\pi$
$601$	9.65133	0.393686	0.196843	+	0.980435i	$0.436931\pi$
$602$	−7.31734	−0.298233
$603$	0	0
$604$	12.8624	0.523363
$605$	9.65659	0.392596
$606$	0	0
$607$	16.4010	0.665696	0.332848	−	0.942980i	$-0.391990\pi$
$607$	16.4010	0.665696	0.332848	+	0.942980i	$0.391990\pi$
$608$	−2.10581	−0.0854020
$609$	0	0
$610$	−103.186	−4.17786
$611$	−2.06542	−0.0835578
$612$	0	0
$613$	5.86335	0.236819	0.118409	−	0.992965i	$-0.462221\pi$
$613$	5.86335	0.236819	0.118409	+	0.992965i	$0.462221\pi$
$614$	27.6244	1.11483
$615$	0	0
$616$	−15.3589	−0.618826
$617$	−19.4634	−0.783567	−0.391784	−	0.920057i	$-0.628142\pi$
$617$	−19.4634	−0.783567	−0.391784	+	0.920057i	$0.628142\pi$
$618$	0	0
$619$	2.68085	0.107752	0.0538762	−	0.998548i	$-0.482842\pi$
$619$	2.68085	0.107752	0.0538762	+	0.998548i	$0.482842\pi$
$620$	26.9083	1.08067
$621$	0	0
$622$	17.1839	0.689012
$623$	−9.63697	−0.386097
$624$	0	0
$625$	30.3676	1.21470
$626$	−21.4028	−0.855429
$627$	0	0
$628$	−7.27736	−0.290398
$629$	1.82964	0.0729524
$630$	0	0
$631$	12.2135	0.486210	0.243105	−	0.970000i	$-0.421834\pi$
$631$	12.2135	0.486210	0.243105	+	0.970000i	$0.421834\pi$
$632$	4.52343	0.179932
$633$	0	0
$634$	−38.5343	−1.53039
$635$	−3.91787	−0.155476
$636$	0	0
$637$	1.07733	0.0426852
$638$	30.1885	1.19518
$639$	0	0
$640$	−71.8259	−2.83917
$641$	43.6238	1.72304	0.861518	−	0.507727i	$-0.169514\pi$
$641$	43.6238	1.72304	0.861518	+	0.507727i	$0.169514\pi$
$642$	0	0
$643$	11.5418	0.455163	0.227582	−	0.973759i	$-0.426918\pi$
$643$	11.5418	0.455163	0.227582	+	0.973759i	$0.426918\pi$
$644$	8.69317	0.342559
$645$	0	0
$646$	7.40610	0.291389
$647$	29.9157	1.17611	0.588053	−	0.808822i	$-0.299895\pi$
$647$	29.9157	1.17611	0.588053	+	0.808822i	$0.299895\pi$
$648$	0	0
$649$	22.9071	0.899183
$650$	−27.5933	−1.08230
$651$	0	0
$652$	17.7532	0.695268
$653$	−44.6270	−1.74639	−0.873194	−	0.487372i	$-0.837955\pi$
$653$	−44.6270	−1.74639	−0.873194	+	0.487372i	$0.837955\pi$
$654$	0	0
$655$	81.3141	3.17721
$656$	39.7112	1.55046
$657$	0	0
$658$	−4.74449	−0.184959
$659$	24.1468	0.940626	0.470313	−	0.882500i	$-0.344141\pi$
$659$	24.1468	0.940626	0.470313	+	0.882500i	$0.344141\pi$
$660$	0	0
$661$	32.6973	1.27178	0.635889	−	0.771780i	$-0.280634\pi$
$661$	32.6973	1.27178	0.635889	+	0.771780i	$0.280634\pi$
$662$	−3.60612	−0.140156
$663$	0	0
$664$	−13.5556	−0.526058
$665$	6.50154	0.252119
$666$	0	0
$667$	−8.80096	−0.340774
$668$	59.4649	2.30077
$669$	0	0
$670$	−78.4065	−3.02911
$671$	31.0919	1.20029
$672$	0	0
$673$	−16.2136	−0.624987	−0.312494	−	0.949920i	$-0.601164\pi$
$673$	−16.2136	−0.624987	−0.312494	+	0.949920i	$0.601164\pi$
$674$	29.2909	1.12824
$675$	0	0
$676$	−48.8294	−1.87806
$677$	−13.0452	−0.501367	−0.250683	−	0.968069i	$-0.580655\pi$
$677$	−13.0452	−0.501367	−0.250683	+	0.968069i	$0.580655\pi$
$678$	0	0
$679$	5.99985	0.230253
$680$	−37.1445	−1.42443
$681$	0	0
$682$	−12.0399	−0.461030
$683$	15.7653	0.603241	0.301620	−	0.953428i	$-0.402472\pi$
$683$	15.7653	0.603241	0.301620	+	0.953428i	$0.402472\pi$
$684$	0	0
$685$	63.1777	2.41390
$686$	2.47474	0.0944859
$687$	0	0
$688$	−14.0787	−0.536744
$689$	−11.7728	−0.448506
$690$	0	0
$691$	−2.70125	−0.102761	−0.0513803	−	0.998679i	$-0.516362\pi$
$691$	−2.70125	−0.102761	−0.0513803	+	0.998679i	$0.516362\pi$
$692$	−27.9407	−1.06215
$693$	0	0
$694$	41.6970	1.58280
$695$	0.201153	0.00763016
$696$	0	0
$697$	−15.0408	−0.569709
$698$	−38.7314	−1.46601
$699$	0	0
$700$	−42.6855	−1.61336
$701$	−40.0401	−1.51229	−0.756147	−	0.654401i	$-0.772921\pi$
$701$	−40.0401	−1.51229	−0.756147	+	0.654401i	$0.772921\pi$
$702$	0	0
$703$	−1.68359	−0.0634979
$704$	18.6465	0.702766
$705$	0	0
$706$	81.6260	3.07203
$707$	12.9224	0.485996
$708$	0	0
$709$	−19.6420	−0.737672	−0.368836	−	0.929494i	$-0.620244\pi$
$709$	−19.6420	−0.737672	−0.368836	+	0.929494i	$0.620244\pi$
$710$	103.941	3.90085
$711$	0	0
$712$	−50.6631	−1.89868
$713$	3.51001	0.131451
$714$	0	0
$715$	12.3312	0.461160
$716$	19.1386	0.715242
$717$	0	0
$718$	−44.8361	−1.67327
$719$	−28.0246	−1.04514	−0.522570	−	0.852596i	$-0.675027\pi$
$719$	−28.0246	−1.04514	−0.522570	+	0.852596i	$0.675027\pi$
$720$	0	0
$721$	4.66147	0.173602
$722$	40.2051	1.49628
$723$	0	0
$724$	−42.1397	−1.56611
$725$	43.2148	1.60496
$726$	0	0
$727$	−26.8831	−0.997040	−0.498520	−	0.866878i	$-0.666123\pi$
$727$	−26.8831	−0.997040	−0.498520	+	0.866878i	$0.666123\pi$
$728$	5.66367	0.209910
$729$	0	0
$730$	7.67299	0.283990
$731$	5.33235	0.197224
$732$	0	0
$733$	−22.0712	−0.815219	−0.407610	−	0.913156i	$-0.633638\pi$
$733$	−22.0712	−0.815219	−0.407610	+	0.913156i	$0.633638\pi$
$734$	−38.2249	−1.41091
$735$	0	0
$736$	2.67472	0.0985915
$737$	23.6255	0.870257
$738$	0	0
$739$	−6.58344	−0.242176	−0.121088	−	0.992642i	$-0.538638\pi$
$739$	−6.58344	−0.242176	−0.121088	+	0.992642i	$0.538638\pi$
$740$	16.3936	0.602640
$741$	0	0
$742$	−27.0433	−0.992791
$743$	−45.5817	−1.67223	−0.836115	−	0.548554i	$-0.815178\pi$
$743$	−45.5817	−1.67223	−0.836115	+	0.548554i	$0.815178\pi$
$744$	0	0
$745$	13.2139	0.484118
$746$	−12.2818	−0.449669
$747$	0	0
$748$	21.7298	0.794520
$749$	14.8610	0.543009
$750$	0	0
$751$	41.4489	1.51249	0.756246	−	0.654288i	$-0.227031\pi$
$751$	41.4489	1.51249	0.756246	+	0.654288i	$0.227031\pi$
$752$	−9.12846	−0.332881
$753$	0	0
$754$	−11.1322	−0.405411
$755$	−12.2185	−0.444677
$756$	0	0
$757$	−37.7510	−1.37208	−0.686042	−	0.727562i	$-0.740654\pi$
$757$	−37.7510	−1.37208	−0.686042	+	0.727562i	$0.740654\pi$
$758$	35.4757	1.28854
$759$	0	0
$760$	34.1796	1.23982
$761$	−6.62486	−0.240151	−0.120076	−	0.992765i	$-0.538314\pi$
$761$	−6.62486	−0.240151	−0.120076	+	0.992765i	$0.538314\pi$
$762$	0	0
$763$	4.33326	0.156875
$764$	−50.1691	−1.81505
$765$	0	0
$766$	−13.3285	−0.481578
$767$	−8.44715	−0.305009
$768$	0	0
$769$	31.7004	1.14315	0.571573	−	0.820551i	$-0.306333\pi$
$769$	31.7004	1.14315	0.571573	+	0.820551i	$0.306333\pi$
$770$	28.3261	1.02080
$771$	0	0
$772$	−69.1044	−2.48712
$773$	16.5751	0.596167	0.298083	−	0.954540i	$-0.403653\pi$
$773$	16.5751	0.596167	0.298083	+	0.954540i	$0.403653\pi$
$774$	0	0
$775$	−17.2350	−0.619099
$776$	31.5421	1.13230
$777$	0	0
$778$	−2.78867	−0.0999787
$779$	13.8402	0.495876
$780$	0	0
$781$	−31.3197	−1.12071
$782$	−9.40694	−0.336391
$783$	0	0
$784$	4.76143	0.170051
$785$	6.91306	0.246738
$786$	0	0
$787$	1.37024	0.0488439	0.0244220	−	0.999702i	$-0.492225\pi$
$787$	1.37024	0.0488439	0.0244220	+	0.999702i	$0.492225\pi$
$788$	−7.57662	−0.269906
$789$	0	0
$790$	−8.34249	−0.296812
$791$	−9.36220	−0.332882
$792$	0	0
$793$	−11.4654	−0.407147
$794$	−3.99375	−0.141733
$795$	0	0
$796$	107.045	3.79412
$797$	−53.0731	−1.87995	−0.939974	−	0.341247i	$-0.889151\pi$
$797$	−53.0731	−1.87995	−0.939974	+	0.341247i	$0.889151\pi$
$798$	0	0
$799$	3.45744	0.122315
$800$	−13.1335	−0.464340
$801$	0	0
$802$	−45.4454	−1.60473
$803$	−2.31203	−0.0815898
$804$	0	0
$805$	−8.25799	−0.291056
$806$	4.43977	0.156384
$807$	0	0
$808$	67.9349	2.38994
$809$	21.6799	0.762224	0.381112	−	0.924529i	$-0.375541\pi$
$809$	21.6799	0.762224	0.381112	+	0.924529i	$0.375541\pi$
$810$	0	0
$811$	17.4530	0.612858	0.306429	−	0.951893i	$-0.400866\pi$
$811$	17.4530	0.612858	0.306429	+	0.951893i	$0.400866\pi$
$812$	−17.2210	−0.604338
$813$	0	0
$814$	−7.33513	−0.257096
$815$	−16.8645	−0.590737
$816$	0	0
$817$	−4.90671	−0.171664
$818$	−64.0594	−2.23979
$819$	0	0
$820$	−134.765	−4.70621
$821$	8.63391	0.301325	0.150663	−	0.988585i	$-0.451859\pi$
$821$	8.63391	0.301325	0.150663	+	0.988585i	$0.451859\pi$
$822$	0	0
$823$	−8.78037	−0.306064	−0.153032	−	0.988221i	$-0.548904\pi$
$823$	−8.78037	−0.306064	−0.153032	+	0.988221i	$0.548904\pi$
$824$	24.5061	0.853711
$825$	0	0
$826$	−19.4040	−0.675153
$827$	16.6464	0.578854	0.289427	−	0.957200i	$-0.406535\pi$
$827$	16.6464	0.578854	0.289427	+	0.957200i	$0.406535\pi$
$828$	0	0
$829$	9.16649	0.318365	0.159183	−	0.987249i	$-0.449114\pi$
$829$	9.16649	0.318365	0.159183	+	0.987249i	$0.449114\pi$
$830$	25.0003	0.867774
$831$	0	0
$832$	−6.87601	−0.238383
$833$	−1.80341	−0.0624844
$834$	0	0
$835$	−56.4882	−1.95485
$836$	−19.9953	−0.691551
$837$	0	0
$838$	36.3014	1.25401
$839$	8.36702	0.288862	0.144431	−	0.989515i	$-0.453865\pi$
$839$	8.36702	0.288862	0.144431	+	0.989515i	$0.453865\pi$
$840$	0	0
$841$	−11.5655	−0.398810
$842$	57.6654	1.98728
$843$	0	0
$844$	19.9832	0.687849
$845$	46.3851	1.59570
$846$	0	0
$847$	2.46476	0.0846900
$848$	−52.0317	−1.78678
$849$	0	0
$850$	46.1903	1.58431
$851$	2.13843	0.0733045
$852$	0	0
$853$	37.1322	1.27138	0.635691	−	0.771943i	$-0.280715\pi$
$853$	37.1322	1.27138	0.635691	+	0.771943i	$0.280715\pi$
$854$	−26.3372	−0.901240
$855$	0	0
$856$	78.1266	2.67031
$857$	1.07976	0.0368841	0.0184420	−	0.999830i	$-0.494129\pi$
$857$	1.07976	0.0368841	0.0184420	+	0.999830i	$0.494129\pi$
$858$	0	0
$859$	34.3372	1.17157	0.585785	−	0.810466i	$-0.300786\pi$
$859$	34.3372	1.17157	0.585785	+	0.810466i	$0.300786\pi$
$860$	47.7779	1.62921
$861$	0	0
$862$	−49.7109	−1.69316
$863$	50.6643	1.72463	0.862317	−	0.506369i	$-0.169013\pi$
$863$	50.6643	1.72463	0.862317	+	0.506369i	$0.169013\pi$
$864$	0	0
$865$	26.5420	0.902456
$866$	13.4387	0.456666
$867$	0	0
$868$	6.86811	0.233119
$869$	2.51376	0.0852736
$870$	0	0
$871$	−8.71205	−0.295197
$872$	22.7807	0.771450
$873$	0	0
$874$	8.65606	0.292796
$875$	20.9594	0.708556
$876$	0	0
$877$	48.4477	1.63596	0.817981	−	0.575245i	$-0.195093\pi$
$877$	48.4477	1.63596	0.817981	+	0.575245i	$0.195093\pi$
$878$	60.5722	2.04421
$879$	0	0
$880$	54.4998	1.83719
$881$	49.7366	1.67567	0.837835	−	0.545924i	$-0.183821\pi$
$881$	49.7366	1.67567	0.837835	+	0.545924i	$0.183821\pi$
$882$	0	0
$883$	45.6694	1.53690	0.768449	−	0.639911i	$-0.221029\pi$
$883$	45.6694	1.53690	0.768449	+	0.639911i	$0.221029\pi$
$884$	−8.01299	−0.269506
$885$	0	0
$886$	−49.5501	−1.66467
$887$	24.5906	0.825673	0.412836	−	0.910805i	$-0.364538\pi$
$887$	24.5906	0.825673	0.412836	+	0.910805i	$0.364538\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	93.4371	3.13202
$891$	0	0
$892$	2.25477	0.0754952
$893$	−3.18146	−0.106464
$894$	0	0
$895$	−18.1805	−0.607708
$896$	−18.3329	−0.612459
$897$	0	0
$898$	−20.6598	−0.689425
$899$	−6.95327	−0.231904
$900$	0	0
$901$	19.7072	0.656542
$902$	60.2994	2.00775
$903$	0	0
$904$	−49.2186	−1.63699
$905$	40.0302	1.33065
$906$	0	0
$907$	−16.8704	−0.560172	−0.280086	−	0.959975i	$-0.590363\pi$
$907$	−16.8704	−0.560172	−0.280086	+	0.959975i	$0.590363\pi$
$908$	−72.9062	−2.41948
$909$	0	0
$910$	−10.4454	−0.346263
$911$	−8.76847	−0.290512	−0.145256	−	0.989394i	$-0.546401\pi$
$911$	−8.76847	−0.290512	−0.145256	+	0.989394i	$0.546401\pi$
$912$	0	0
$913$	−7.53312	−0.249310
$914$	69.5227	2.29960
$915$	0	0
$916$	47.4656	1.56831
$917$	20.7547	0.685380
$918$	0	0
$919$	17.8373	0.588397	0.294199	−	0.955744i	$-0.404947\pi$
$919$	17.8373	0.588397	0.294199	+	0.955744i	$0.404947\pi$
$920$	−43.4136	−1.43130
$921$	0	0
$922$	−54.5895	−1.79781
$923$	11.5493	0.380151
$924$	0	0
$925$	−10.5002	−0.345245
$926$	41.8618	1.37566
$927$	0	0
$928$	−5.29857	−0.173934
$929$	−50.1283	−1.64466	−0.822328	−	0.569014i	$-0.807325\pi$
$929$	−50.1283	−1.64466	−0.822328	+	0.569014i	$0.807325\pi$
$930$	0	0
$931$	1.65946	0.0543865
$932$	−15.2902	−0.500846
$933$	0	0
$934$	−21.6168	−0.707324
$935$	−20.6420	−0.675066
$936$	0	0
$937$	42.9789	1.40406	0.702030	−	0.712147i	$-0.252277\pi$
$937$	42.9789	1.40406	0.702030	+	0.712147i	$0.252277\pi$
$938$	−20.0125	−0.653433
$939$	0	0
$940$	30.9787	1.01041
$941$	58.3405	1.90185	0.950923	−	0.309428i	$-0.100137\pi$
$941$	58.3405	1.90185	0.950923	+	0.309428i	$0.100137\pi$
$942$	0	0
$943$	−17.5793	−0.572459
$944$	−37.3336	−1.21511
$945$	0	0
$946$	−21.3777	−0.695050
$947$	2.95540	0.0960375	0.0480187	−	0.998846i	$-0.484709\pi$
$947$	2.95540	0.0960375	0.0480187	+	0.998846i	$0.484709\pi$
$948$	0	0
$949$	0.852575	0.0276758
$950$	−42.5033	−1.37899
$951$	0	0
$952$	−9.48080	−0.307275
$953$	49.3795	1.59956	0.799779	−	0.600294i	$-0.204950\pi$
$953$	49.3795	1.59956	0.799779	+	0.600294i	$0.204950\pi$
$954$	0	0
$955$	47.6577	1.54217
$956$	113.073	3.65703
$957$	0	0
$958$	51.5590	1.66580
$959$	16.1255	0.520721
$960$	0	0
$961$	−28.2269	−0.910545
$962$	2.70488	0.0872087
$963$	0	0
$964$	−12.7948	−0.412093
$965$	65.6451	2.11319
$966$	0	0
$967$	−11.3137	−0.363825	−0.181912	−	0.983315i	$-0.558229\pi$
$967$	−11.3137	−0.363825	−0.181912	+	0.983315i	$0.558229\pi$
$968$	12.9576	0.416473
$969$	0	0
$970$	−58.1727	−1.86781
$971$	16.2686	0.522085	0.261043	−	0.965327i	$-0.415934\pi$
$971$	16.2686	0.522085	0.261043	+	0.965327i	$0.415934\pi$
$972$	0	0
$973$	0.0513424	0.00164596
$974$	−73.7249	−2.36230
$975$	0	0
$976$	−50.6731	−1.62201
$977$	26.0290	0.832740	0.416370	−	0.909195i	$-0.363302\pi$
$977$	26.0290	0.832740	0.416370	+	0.909195i	$0.363302\pi$
$978$	0	0
$979$	−28.1545	−0.899823
$980$	−16.1586	−0.516167
$981$	0	0
$982$	69.7852	2.22693
$983$	15.7174	0.501306	0.250653	−	0.968077i	$-0.419355\pi$
$983$	15.7174	0.501306	0.250653	+	0.968077i	$0.419355\pi$
$984$	0	0
$985$	7.19734	0.229326
$986$	18.6350	0.593458
$987$	0	0
$988$	7.37339	0.234579
$989$	6.23231	0.198176
$990$	0	0
$991$	3.48742	0.110782	0.0553908	−	0.998465i	$-0.482360\pi$
$991$	3.48742	0.110782	0.0553908	+	0.998465i	$0.482360\pi$
$992$	2.11319	0.0670937
$993$	0	0
$994$	26.5301	0.841483
$995$	−101.687	−3.22368
$996$	0	0
$997$	36.3874	1.15240	0.576200	−	0.817309i	$-0.304535\pi$
$997$	36.3874	1.15240	0.576200	+	0.817309i	$0.304535\pi$
$998$	−60.5036	−1.91521
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.y.1.3	✓	28
3.2	odd	2	inner	8001.2.a.y.1.26	yes	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8001.2.a.y.1.3	✓	28	1.1	even	1	trivial
8001.2.a.y.1.26	yes	28	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8001.a (trivial)

\(f(q)\)	\(=\)	\(q-2.47474 q^{2} +4.12433 q^{4} -3.91787 q^{5} -1.00000 q^{7} -5.25716 q^{8} +O(q^{10})\)	\(q-2.47474 q^{2} +4.12433 q^{4} -3.91787 q^{5} -1.00000 q^{7} -5.25716 q^{8} +9.69570 q^{10} -2.92151 q^{11} +1.07733 q^{13} +2.47474 q^{14} +4.76143 q^{16} -1.80341 q^{17} +1.65946 q^{19} -16.1586 q^{20} +7.22998 q^{22} -2.10778 q^{23} +10.3497 q^{25} -2.66610 q^{26} -4.12433 q^{28} +4.17547 q^{29} -1.66527 q^{31} -1.26898 q^{32} +4.46296 q^{34} +3.91787 q^{35} -1.01454 q^{37} -4.10672 q^{38} +20.5968 q^{40} +8.34018 q^{41} -2.95682 q^{43} -12.0493 q^{44} +5.21620 q^{46} -1.91717 q^{47} +1.00000 q^{49} -25.6128 q^{50} +4.44325 q^{52} -10.9277 q^{53} +11.4461 q^{55} +5.25716 q^{56} -10.3332 q^{58} -7.84084 q^{59} -10.6424 q^{61} +4.12110 q^{62} -6.38247 q^{64} -4.22082 q^{65} -8.08673 q^{67} -7.43785 q^{68} -9.69570 q^{70} +10.7204 q^{71} +0.791381 q^{73} +2.51073 q^{74} +6.84415 q^{76} +2.92151 q^{77} -0.860432 q^{79} -18.6546 q^{80} -20.6398 q^{82} +2.57850 q^{83} +7.06552 q^{85} +7.31734 q^{86} +15.3589 q^{88} +9.63697 q^{89} -1.07733 q^{91} -8.69317 q^{92} +4.74449 q^{94} -6.50154 q^{95} -5.99985 q^{97} -2.47474 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 30 q^{4} - 28 q^{7}+O(q^{10}) \) 28 * q + 30 * q^4 - 28 * q^7	\( 28 q + 30 q^{4} - 28 q^{7} + 4 q^{10} + 8 q^{13} + 42 q^{16} + 34 q^{19} - 10 q^{22} + 14 q^{25} - 30 q^{28} + 56 q^{31} - 6 q^{37} + 38 q^{40} + 18 q^{43} + 16 q^{46} + 28 q^{49} + 18 q^{52} + 48 q^{55} + 2 q^{58} + 36 q^{61} + 76 q^{64} - 4 q^{70} + 50 q^{73} + 132 q^{76} + 66 q^{79} - 36 q^{82} + 20 q^{85} + 6 q^{88} - 8 q^{91} + 54 q^{94} - 8 q^{97}+O(q^{100}) \) 28 * q + 30 * q^4 - 28 * q^7 + 4 * q^10 + 8 * q^13 + 42 * q^16 + 34 * q^19 - 10 * q^22 + 14 * q^25 - 30 * q^28 + 56 * q^31 - 6 * q^37 + 38 * q^40 + 18 * q^43 + 16 * q^46 + 28 * q^49 + 18 * q^52 + 48 * q^55 + 2 * q^58 + 36 * q^61 + 76 * q^64 - 4 * q^70 + 50 * q^73 + 132 * q^76 + 66 * q^79 - 36 * q^82 + 20 * q^85 + 6 * q^88 - 8 * q^91 + 54 * q^94 - 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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