LMFDB - Embedded newform 8001.2.a.ba.1.15

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:	$40$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
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)

$f(q)$	$=$	$q-1.07579 q^{2} -0.842680 q^{4} +1.06884 q^{5} +1.00000 q^{7} +3.05812 q^{8} +O(q^{10})$	\(q-1.07579 q^{2} -0.842680 q^{4} +1.06884 q^{5} +1.00000 q^{7} +3.05812 q^{8} -1.14985 q^{10} +1.63583 q^{11} +2.87629 q^{13} -1.07579 q^{14} -1.60453 q^{16} +0.537501 q^{17} +3.98482 q^{19} -0.900694 q^{20} -1.75980 q^{22} +7.38722 q^{23} -3.85757 q^{25} -3.09428 q^{26} -0.842680 q^{28} +4.43157 q^{29} +7.15367 q^{31} -4.39011 q^{32} -0.578237 q^{34} +1.06884 q^{35} +2.70089 q^{37} -4.28682 q^{38} +3.26866 q^{40} +6.22075 q^{41} -2.40870 q^{43} -1.37848 q^{44} -7.94709 q^{46} -10.9312 q^{47} +1.00000 q^{49} +4.14993 q^{50} -2.42379 q^{52} +12.9663 q^{53} +1.74845 q^{55} +3.05812 q^{56} -4.76743 q^{58} -5.89893 q^{59} +9.71661 q^{61} -7.69583 q^{62} +7.93189 q^{64} +3.07431 q^{65} +13.8723 q^{67} -0.452941 q^{68} -1.14985 q^{70} +13.0115 q^{71} -5.78761 q^{73} -2.90558 q^{74} -3.35793 q^{76} +1.63583 q^{77} -16.3142 q^{79} -1.71500 q^{80} -6.69221 q^{82} +3.30043 q^{83} +0.574505 q^{85} +2.59125 q^{86} +5.00256 q^{88} -14.8696 q^{89} +2.87629 q^{91} -6.22506 q^{92} +11.7597 q^{94} +4.25916 q^{95} +9.32468 q^{97} -1.07579 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q + 54 q^{4} + 40 q^{7}+O(q^{10}) $ 40 * q + 54 * q^4 + 40 * q^7	$ 40 q + 54 q^{4} + 40 q^{7} + 20 q^{10} + 10 q^{13} + 90 q^{16} + 38 q^{19} + 14 q^{22} + 84 q^{25} + 54 q^{28} + 66 q^{31} + 22 q^{34} + 40 q^{37} + 26 q^{40} + 38 q^{43} + 28 q^{46} + 40 q^{49} + 28 q^{52} + 60 q^{55} + 42 q^{58} + 54 q^{61} + 124 q^{64} + 48 q^{67} + 20 q^{70} + 16 q^{76} + 102 q^{79} + 48 q^{82} + 104 q^{85} + 48 q^{88} + 10 q^{91} - 10 q^{94} + 8 q^{97}+O(q^{100}) $ 40 * q + 54 * q^4 + 40 * q^7 + 20 * q^10 + 10 * q^13 + 90 * q^16 + 38 * q^19 + 14 * q^22 + 84 * q^25 + 54 * q^28 + 66 * q^31 + 22 * q^34 + 40 * q^37 + 26 * q^40 + 38 * q^43 + 28 * q^46 + 40 * q^49 + 28 * q^52 + 60 * q^55 + 42 * q^58 + 54 * q^61 + 124 * q^64 + 48 * q^67 + 20 * q^70 + 16 * q^76 + 102 * q^79 + 48 * q^82 + 104 * q^85 + 48 * q^88 + 10 * q^91 - 10 * 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 $

$2$	−1.07579	−0.760697	−0.380349	−	0.924843i	$-0.624196\pi$
$2$	−1.07579	−0.760697	−0.380349	+	0.924843i	$0.624196\pi$
$3$	0	0
$3$	0	0
$4$	−0.842680	−0.421340
$5$	1.06884	0.478002	0.239001	−	0.971019i	$-0.423180\pi$
$5$	1.06884	0.478002	0.239001	+	0.971019i	$0.423180\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	3.05812	1.08121
$9$	0	0
$10$	−1.14985	−0.363615
$11$	1.63583	0.493220	0.246610	−	0.969115i	$-0.420683\pi$
$11$	1.63583	0.493220	0.246610	+	0.969115i	$0.420683\pi$
$12$	0	0
$13$	2.87629	0.797739	0.398870	−	0.917008i	$-0.369403\pi$
$13$	2.87629	0.797739	0.398870	+	0.917008i	$0.369403\pi$
$14$	−1.07579	−0.287517
$15$	0	0
$16$	−1.60453	−0.401133
$17$	0.537501	0.130363	0.0651816	−	0.997873i	$-0.479237\pi$
$17$	0.537501	0.130363	0.0651816	+	0.997873i	$0.479237\pi$
$18$	0	0
$19$	3.98482	0.914181	0.457090	−	0.889420i	$-0.348892\pi$
$19$	3.98482	0.914181	0.457090	+	0.889420i	$0.348892\pi$
$20$	−0.900694	−0.201401
$21$	0	0
$22$	−1.75980	−0.375191
$23$	7.38722	1.54034	0.770171	−	0.637837i	$-0.220171\pi$
$23$	7.38722	1.54034	0.770171	+	0.637837i	$0.220171\pi$
$24$	0	0
$25$	−3.85757	−0.771514
$26$	−3.09428	−0.606838
$27$	0	0
$28$	−0.842680	−0.159251
$29$	4.43157	0.822922	0.411461	−	0.911427i	$-0.365019\pi$
$29$	4.43157	0.822922	0.411461	+	0.911427i	$0.365019\pi$
$30$	0	0
$31$	7.15367	1.28484	0.642418	−	0.766354i	$-0.277931\pi$
$31$	7.15367	1.28484	0.642418	+	0.766354i	$0.277931\pi$
$32$	−4.39011	−0.776069
$33$	0	0
$34$	−0.578237	−0.0991669
$35$	1.06884	0.180668
$36$	0	0
$37$	2.70089	0.444023	0.222012	−	0.975044i	$-0.428738\pi$
$37$	2.70089	0.444023	0.222012	+	0.975044i	$0.428738\pi$
$38$	−4.28682	−0.695415
$39$	0	0
$40$	3.26866	0.516820
$41$	6.22075	0.971518	0.485759	−	0.874093i	$-0.338543\pi$
$41$	6.22075	0.971518	0.485759	+	0.874093i	$0.338543\pi$
$42$	0	0
$43$	−2.40870	−0.367323	−0.183662	−	0.982990i	$-0.558795\pi$
$43$	−2.40870	−0.367323	−0.183662	+	0.982990i	$0.558795\pi$
$44$	−1.37848	−0.207813
$45$	0	0
$46$	−7.94709	−1.17173
$47$	−10.9312	−1.59449	−0.797243	−	0.603658i	$-0.793709\pi$
$47$	−10.9312	−1.59449	−0.797243	+	0.603658i	$0.793709\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	4.14993	0.586889
$51$	0	0
$52$	−2.42379	−0.336119
$53$	12.9663	1.78106	0.890531	−	0.454922i	$-0.150333\pi$
$53$	12.9663	1.78106	0.890531	+	0.454922i	$0.150333\pi$
$54$	0	0
$55$	1.74845	0.235760
$56$	3.05812	0.408659
$57$	0	0
$58$	−4.76743	−0.625995
$59$	−5.89893	−0.767976	−0.383988	−	0.923338i	$-0.625450\pi$
$59$	−5.89893	−0.767976	−0.383988	+	0.923338i	$0.625450\pi$
$60$	0	0
$61$	9.71661	1.24409	0.622043	−	0.782983i	$-0.286303\pi$
$61$	9.71661	1.24409	0.622043	+	0.782983i	$0.286303\pi$
$62$	−7.69583	−0.977371
$63$	0	0
$64$	7.93189	0.991486
$65$	3.07431	0.381321
$66$	0	0
$67$	13.8723	1.69477	0.847387	−	0.530976i	$-0.178175\pi$
$67$	13.8723	1.69477	0.847387	+	0.530976i	$0.178175\pi$
$68$	−0.452941	−0.0549272
$69$	0	0
$70$	−1.14985	−0.137433
$71$	13.0115	1.54418	0.772089	−	0.635515i	$-0.219212\pi$
$71$	13.0115	1.54418	0.772089	+	0.635515i	$0.219212\pi$
$72$	0	0
$73$	−5.78761	−0.677389	−0.338694	−	0.940896i	$-0.609985\pi$
$73$	−5.78761	−0.677389	−0.338694	+	0.940896i	$0.609985\pi$
$74$	−2.90558	−0.337767
$75$	0	0
$76$	−3.35793	−0.385181
$77$	1.63583	0.186420
$78$	0	0
$79$	−16.3142	−1.83549	−0.917743	−	0.397174i	$-0.869991\pi$
$79$	−16.3142	−1.83549	−0.917743	+	0.397174i	$0.869991\pi$
$80$	−1.71500	−0.191742
$81$	0	0
$82$	−6.69221	−0.739031
$83$	3.30043	0.362270	0.181135	−	0.983458i	$-0.442023\pi$
$83$	3.30043	0.362270	0.181135	+	0.983458i	$0.442023\pi$
$84$	0	0
$85$	0.574505	0.0623138
$86$	2.59125	0.279422
$87$	0	0
$88$	5.00256	0.533274
$89$	−14.8696	−1.57617	−0.788085	−	0.615566i	$-0.788927\pi$
$89$	−14.8696	−1.57617	−0.788085	+	0.615566i	$0.788927\pi$
$90$	0	0
$91$	2.87629	0.301517
$92$	−6.22506	−0.649008
$93$	0	0
$94$	11.7597	1.21292
$95$	4.25916	0.436980
$96$	0	0
$97$	9.32468	0.946778	0.473389	−	0.880853i	$-0.343030\pi$
$97$	9.32468	0.946778	0.473389	+	0.880853i	$0.343030\pi$
$98$	−1.07579	−0.108671
$99$	0	0
$100$	3.25070	0.325070
$101$	−9.80905	−0.976037	−0.488019	−	0.872833i	$-0.662280\pi$
$101$	−9.80905	−0.976037	−0.488019	+	0.872833i	$0.662280\pi$
$102$	0	0
$103$	−8.74789	−0.861955	−0.430978	−	0.902363i	$-0.641831\pi$
$103$	−8.74789	−0.861955	−0.430978	+	0.902363i	$0.641831\pi$
$104$	8.79604	0.862523
$105$	0	0
$106$	−13.9490	−1.35485
$107$	−3.85750	−0.372918	−0.186459	−	0.982463i	$-0.559701\pi$
$107$	−3.85750	−0.372918	−0.186459	+	0.982463i	$0.559701\pi$
$108$	0	0
$109$	15.5504	1.48946	0.744728	−	0.667368i	$-0.232579\pi$
$109$	15.5504	1.48946	0.744728	+	0.667368i	$0.232579\pi$
$110$	−1.88096	−0.179342
$111$	0	0
$112$	−1.60453	−0.151614
$113$	−7.10816	−0.668680	−0.334340	−	0.942453i	$-0.608513\pi$
$113$	−7.10816	−0.668680	−0.334340	+	0.942453i	$0.608513\pi$
$114$	0	0
$115$	7.89580	0.736287
$116$	−3.73440	−0.346730
$117$	0	0
$118$	6.34600	0.584197
$119$	0.537501	0.0492726
$120$	0	0
$121$	−8.32407	−0.756734
$122$	−10.4530	−0.946372
$123$	0	0
$124$	−6.02825	−0.541353
$125$	−9.46737	−0.846787
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	0.247181	0.0218479
$129$	0	0
$130$	−3.30730	−0.290070
$131$	13.1545	1.14931	0.574657	−	0.818394i	$-0.305135\pi$
$131$	13.1545	1.14931	0.574657	+	0.818394i	$0.305135\pi$
$132$	0	0
$133$	3.98482	0.345528
$134$	−14.9237	−1.28921
$135$	0	0
$136$	1.64374	0.140950
$137$	−5.50274	−0.470131	−0.235065	−	0.971980i	$-0.575530\pi$
$137$	−5.50274	−0.470131	−0.235065	+	0.971980i	$0.575530\pi$
$138$	0	0
$139$	12.5381	1.06347	0.531735	−	0.846911i	$-0.321540\pi$
$139$	12.5381	1.06347	0.531735	+	0.846911i	$0.321540\pi$
$140$	−0.900694	−0.0761225
$141$	0	0
$142$	−13.9976	−1.17465
$143$	4.70511	0.393461
$144$	0	0
$145$	4.73666	0.393358
$146$	6.22625	0.515288
$147$	0	0
$148$	−2.27598	−0.187085
$149$	−16.4631	−1.34871	−0.674355	−	0.738407i	$-0.735578\pi$
$149$	−16.4631	−1.34871	−0.674355	+	0.738407i	$0.735578\pi$
$150$	0	0
$151$	8.91374	0.725390	0.362695	−	0.931908i	$-0.381857\pi$
$151$	8.91374	0.725390	0.362695	+	0.931908i	$0.381857\pi$
$152$	12.1861	0.988421
$153$	0	0
$154$	−1.75980	−0.141809
$155$	7.64616	0.614154
$156$	0	0
$157$	6.09118	0.486129	0.243065	−	0.970010i	$-0.421847\pi$
$157$	6.09118	0.486129	0.243065	+	0.970010i	$0.421847\pi$
$158$	17.5506	1.39625
$159$	0	0
$160$	−4.69234	−0.370962
$161$	7.38722	0.582195
$162$	0	0
$163$	1.47062	0.115187	0.0575937	−	0.998340i	$-0.481657\pi$
$163$	1.47062	0.115187	0.0575937	+	0.998340i	$0.481657\pi$
$164$	−5.24210	−0.409339
$165$	0	0
$166$	−3.55057	−0.275577
$167$	−16.6715	−1.29008	−0.645038	−	0.764150i	$-0.723159\pi$
$167$	−16.6715	−1.29008	−0.645038	+	0.764150i	$0.723159\pi$
$168$	0	0
$169$	−4.72696	−0.363612
$170$	−0.618046	−0.0474020
$171$	0	0
$172$	2.02976	0.154768
$173$	8.33016	0.633331	0.316665	−	0.948537i	$-0.397437\pi$
$173$	8.33016	0.633331	0.316665	+	0.948537i	$0.397437\pi$
$174$	0	0
$175$	−3.85757	−0.291605
$176$	−2.62474	−0.197847
$177$	0	0
$178$	15.9965	1.19899
$179$	−10.0622	−0.752081	−0.376040	−	0.926603i	$-0.622715\pi$
$179$	−10.0622	−0.752081	−0.376040	+	0.926603i	$0.622715\pi$
$180$	0	0
$181$	14.4216	1.07195	0.535976	−	0.844233i	$-0.319944\pi$
$181$	14.4216	1.07195	0.535976	+	0.844233i	$0.319944\pi$
$182$	−3.09428	−0.229363
$183$	0	0
$184$	22.5910	1.66543
$185$	2.88683	0.212244
$186$	0	0
$187$	0.879258	0.0642977
$188$	9.21154	0.671821
$189$	0	0
$190$	−4.58195	−0.332410
$191$	−13.0294	−0.942771	−0.471386	−	0.881927i	$-0.656246\pi$
$191$	−13.0294	−0.942771	−0.471386	+	0.881927i	$0.656246\pi$
$192$	0	0
$193$	−22.0872	−1.58987	−0.794937	−	0.606692i	$-0.792496\pi$
$193$	−22.0872	−1.58987	−0.794937	+	0.606692i	$0.792496\pi$
$194$	−10.0314	−0.720211
$195$	0	0
$196$	−0.842680	−0.0601914
$197$	−12.9842	−0.925085	−0.462542	−	0.886597i	$-0.653063\pi$
$197$	−12.9842	−0.925085	−0.462542	+	0.886597i	$0.653063\pi$
$198$	0	0
$199$	−19.0596	−1.35110	−0.675549	−	0.737315i	$-0.736093\pi$
$199$	−19.0596	−1.35110	−0.675549	+	0.737315i	$0.736093\pi$
$200$	−11.7969	−0.834168
$201$	0	0
$202$	10.5525	0.742469
$203$	4.43157	0.311035
$204$	0	0
$205$	6.64901	0.464387
$206$	9.41088	0.655687
$207$	0	0
$208$	−4.61510	−0.319999
$209$	6.51848	0.450893
$210$	0	0
$211$	16.7384	1.15232	0.576160	−	0.817337i	$-0.304550\pi$
$211$	16.7384	1.15232	0.576160	+	0.817337i	$0.304550\pi$
$212$	−10.9265	−0.750433
$213$	0	0
$214$	4.14985	0.283678
$215$	−2.57452	−0.175581
$216$	0	0
$217$	7.15367	0.485623
$218$	−16.7289	−1.13303
$219$	0	0
$220$	−1.47338	−0.0993352
$221$	1.54601	0.103996
$222$	0	0
$223$	−5.69250	−0.381198	−0.190599	−	0.981668i	$-0.561043\pi$
$223$	−5.69250	−0.381198	−0.190599	+	0.981668i	$0.561043\pi$
$224$	−4.39011	−0.293326
$225$	0	0
$226$	7.64688	0.508663
$227$	−3.35522	−0.222694	−0.111347	−	0.993782i	$-0.535516\pi$
$227$	−3.35522	−0.222694	−0.111347	+	0.993782i	$0.535516\pi$
$228$	0	0
$229$	0.00440077	0.000290811 0	0.000145405	−	1.00000i	$-0.499954\pi$
$229$	0.00440077	0.000290811 0	0.000145405		1.00000i	$0.499954\pi$
$230$	−8.49420	−0.560091
$231$	0	0
$232$	13.5523	0.889751
$233$	−15.9301	−1.04361	−0.521807	−	0.853064i	$-0.674742\pi$
$233$	−15.9301	−1.04361	−0.521807	+	0.853064i	$0.674742\pi$
$234$	0	0
$235$	−11.6838	−0.762168
$236$	4.97091	0.323579
$237$	0	0
$238$	−0.578237	−0.0374815
$239$	23.7320	1.53510	0.767549	−	0.640991i	$-0.221476\pi$
$239$	23.7320	1.53510	0.767549	+	0.640991i	$0.221476\pi$
$240$	0	0
$241$	20.8434	1.34264	0.671321	−	0.741167i	$-0.265727\pi$
$241$	20.8434	1.34264	0.671321	+	0.741167i	$0.265727\pi$
$242$	8.95494	0.575645
$243$	0	0
$244$	−8.18799	−0.524183
$245$	1.06884	0.0682860
$246$	0	0
$247$	11.4615	0.729278
$248$	21.8768	1.38918
$249$	0	0
$250$	10.1849	0.644149
$251$	−15.0751	−0.951531	−0.475766	−	0.879572i	$-0.657829\pi$
$251$	−15.0751	−0.951531	−0.475766	+	0.879572i	$0.657829\pi$
$252$	0	0
$253$	12.0842	0.759728
$254$	1.07579	0.0675010
$255$	0	0
$256$	−16.1297	−1.00811
$257$	−3.52841	−0.220096	−0.110048	−	0.993926i	$-0.535100\pi$
$257$	−3.52841	−0.220096	−0.110048	+	0.993926i	$0.535100\pi$
$258$	0	0
$259$	2.70089	0.167825
$260$	−2.59066	−0.160666
$261$	0	0
$262$	−14.1515	−0.874280
$263$	−10.1738	−0.627343	−0.313671	−	0.949532i	$-0.601559\pi$
$263$	−10.1738	−0.627343	−0.313671	+	0.949532i	$0.601559\pi$
$264$	0	0
$265$	13.8590	0.851351
$266$	−4.28682	−0.262842
$267$	0	0
$268$	−11.6899	−0.714075
$269$	−8.88704	−0.541852	−0.270926	−	0.962600i	$-0.587330\pi$
$269$	−8.88704	−0.541852	−0.270926	+	0.962600i	$0.587330\pi$
$270$	0	0
$271$	−0.478981	−0.0290960	−0.0145480	−	0.999894i	$-0.504631\pi$
$271$	−0.478981	−0.0290960	−0.0145480	+	0.999894i	$0.504631\pi$
$272$	−0.862437	−0.0522929
$273$	0	0
$274$	5.91978	0.357627
$275$	−6.31032	−0.380526
$276$	0	0
$277$	−3.54457	−0.212973	−0.106486	−	0.994314i	$-0.533960\pi$
$277$	−3.54457	−0.212973	−0.106486	+	0.994314i	$0.533960\pi$
$278$	−13.4884	−0.808979
$279$	0	0
$280$	3.26866	0.195340
$281$	2.35435	0.140449	0.0702244	−	0.997531i	$-0.477628\pi$
$281$	2.35435	0.140449	0.0702244	+	0.997531i	$0.477628\pi$
$282$	0	0
$283$	3.40843	0.202610	0.101305	−	0.994855i	$-0.467698\pi$
$283$	3.40843	0.202610	0.101305	+	0.994855i	$0.467698\pi$
$284$	−10.9645	−0.650623
$285$	0	0
$286$	−5.06170	−0.299305
$287$	6.22075	0.367199
$288$	0	0
$289$	−16.7111	−0.983005
$290$	−5.09565	−0.299227
$291$	0	0
$292$	4.87710	0.285411
$293$	−31.0615	−1.81463	−0.907315	−	0.420451i	$-0.861872\pi$
$293$	−31.0615	−1.81463	−0.907315	+	0.420451i	$0.861872\pi$
$294$	0	0
$295$	−6.30504	−0.367094
$296$	8.25964	0.480082
$297$	0	0
$298$	17.7108	1.02596
$299$	21.2478	1.22879
$300$	0	0
$301$	−2.40870	−0.138835
$302$	−9.58930	−0.551802
$303$	0	0
$304$	−6.39377	−0.366708
$305$	10.3856	0.594675
$306$	0	0
$307$	−17.8537	−1.01896	−0.509482	−	0.860481i	$-0.670163\pi$
$307$	−17.8537	−1.01896	−0.509482	+	0.860481i	$0.670163\pi$
$308$	−1.37848	−0.0785461
$309$	0	0
$310$	−8.22565	−0.467185
$311$	−30.6824	−1.73984	−0.869921	−	0.493191i	$-0.835830\pi$
$311$	−30.6824	−1.73984	−0.869921	+	0.493191i	$0.835830\pi$
$312$	0	0
$313$	22.3385	1.26265	0.631323	−	0.775520i	$-0.282512\pi$
$313$	22.3385	1.26265	0.631323	+	0.775520i	$0.282512\pi$
$314$	−6.55282	−0.369797
$315$	0	0
$316$	13.7476	0.773364
$317$	−11.3176	−0.635662	−0.317831	−	0.948147i	$-0.602954\pi$
$317$	−11.3176	−0.635662	−0.317831	+	0.948147i	$0.602954\pi$
$318$	0	0
$319$	7.24929	0.405882
$320$	8.47796	0.473932
$321$	0	0
$322$	−7.94709	−0.442874
$323$	2.14185	0.119175
$324$	0	0
$325$	−11.0955	−0.615467
$326$	−1.58207	−0.0876228
$327$	0	0
$328$	19.0238	1.05041
$329$	−10.9312	−0.602659
$330$	0	0
$331$	17.6807	0.971819	0.485909	−	0.874009i	$-0.338489\pi$
$331$	17.6807	0.971819	0.485909	+	0.874009i	$0.338489\pi$
$332$	−2.78121	−0.152639
$333$	0	0
$334$	17.9350	0.981358
$335$	14.8274	0.810105
$336$	0	0
$337$	27.1610	1.47955	0.739776	−	0.672853i	$-0.234931\pi$
$337$	27.1610	1.47955	0.739776	+	0.672853i	$0.234931\pi$
$338$	5.08521	0.276599
$339$	0	0
$340$	−0.484124	−0.0262553
$341$	11.7022	0.633707
$342$	0	0
$343$	1.00000	0.0539949
$344$	−7.36609	−0.397153
$345$	0	0
$346$	−8.96149	−0.481773
$347$	20.0182	1.07464	0.537318	−	0.843380i	$-0.319438\pi$
$347$	20.0182	1.07464	0.537318	+	0.843380i	$0.319438\pi$
$348$	0	0
$349$	17.2446	0.923080	0.461540	−	0.887119i	$-0.347297\pi$
$349$	17.2446	0.923080	0.461540	+	0.887119i	$0.347297\pi$
$350$	4.14993	0.221823
$351$	0	0
$352$	−7.18145	−0.382773
$353$	−16.3658	−0.871066	−0.435533	−	0.900173i	$-0.643440\pi$
$353$	−16.3658	−0.871066	−0.435533	+	0.900173i	$0.643440\pi$
$354$	0	0
$355$	13.9072	0.738120
$356$	12.5303	0.664103
$357$	0	0
$358$	10.8247	0.572106
$359$	−15.6327	−0.825063	−0.412532	−	0.910943i	$-0.635355\pi$
$359$	−15.6327	−0.825063	−0.412532	+	0.910943i	$0.635355\pi$
$360$	0	0
$361$	−3.12120	−0.164274
$362$	−15.5146	−0.815431
$363$	0	0
$364$	−2.42379	−0.127041
$365$	−6.18606	−0.323793
$366$	0	0
$367$	28.2137	1.47274	0.736372	−	0.676577i	$-0.236537\pi$
$367$	28.2137	1.47274	0.736372	+	0.676577i	$0.236537\pi$
$368$	−11.8530	−0.617882
$369$	0	0
$370$	−3.10562	−0.161453
$371$	12.9663	0.673178
$372$	0	0
$373$	32.8603	1.70144	0.850721	−	0.525617i	$-0.176166\pi$
$373$	32.8603	1.70144	0.850721	+	0.525617i	$0.176166\pi$
$374$	−0.945896	−0.0489111
$375$	0	0
$376$	−33.4291	−1.72397
$377$	12.7465	0.656477
$378$	0	0
$379$	32.3821	1.66336	0.831679	−	0.555257i	$-0.187380\pi$
$379$	32.3821	1.66336	0.831679	+	0.555257i	$0.187380\pi$
$380$	−3.58910	−0.184117
$381$	0	0
$382$	14.0168	0.717163
$383$	9.34998	0.477762	0.238881	−	0.971049i	$-0.423219\pi$
$383$	9.34998	0.477762	0.238881	+	0.971049i	$0.423219\pi$
$384$	0	0
$385$	1.74845	0.0891090
$386$	23.7612	1.20941
$387$	0	0
$388$	−7.85772	−0.398915
$389$	−6.99602	−0.354712	−0.177356	−	0.984147i	$-0.556754\pi$
$389$	−6.99602	−0.354712	−0.177356	+	0.984147i	$0.556754\pi$
$390$	0	0
$391$	3.97064	0.200804
$392$	3.05812	0.154458
$393$	0	0
$394$	13.9682	0.703709
$395$	−17.4373	−0.877366
$396$	0	0
$397$	−13.8064	−0.692923	−0.346461	−	0.938064i	$-0.612617\pi$
$397$	−13.8064	−0.692923	−0.346461	+	0.938064i	$0.612617\pi$
$398$	20.5041	1.02778
$399$	0	0
$400$	6.18959	0.309480
$401$	22.7928	1.13822	0.569110	−	0.822261i	$-0.307288\pi$
$401$	22.7928	1.13822	0.569110	+	0.822261i	$0.307288\pi$
$402$	0	0
$403$	20.5760	1.02496
$404$	8.26589	0.411243
$405$	0	0
$406$	−4.76743	−0.236604
$407$	4.41818	0.219001
$408$	0	0
$409$	−30.5164	−1.50894	−0.754469	−	0.656336i	$-0.772105\pi$
$409$	−30.5164	−1.50894	−0.754469	+	0.656336i	$0.772105\pi$
$410$	−7.15293	−0.353258
$411$	0	0
$412$	7.37167	0.363176
$413$	−5.89893	−0.290268
$414$	0	0
$415$	3.52765	0.173166
$416$	−12.6272	−0.619100
$417$	0	0
$418$	−7.01250	−0.342993
$419$	−19.2494	−0.940392	−0.470196	−	0.882562i	$-0.655817\pi$
$419$	−19.2494	−0.940392	−0.470196	+	0.882562i	$0.655817\pi$
$420$	0	0
$421$	32.5460	1.58620	0.793098	−	0.609094i	$-0.208467\pi$
$421$	32.5460	1.58620	0.793098	+	0.609094i	$0.208467\pi$
$422$	−18.0070	−0.876567
$423$	0	0
$424$	39.6526	1.92570
$425$	−2.07345	−0.100577
$426$	0	0
$427$	9.71661	0.470220
$428$	3.25063	0.157125
$429$	0	0
$430$	2.76964	0.133564
$431$	−13.6010	−0.655135	−0.327567	−	0.944828i	$-0.606229\pi$
$431$	−13.6010	−0.655135	−0.327567	+	0.944828i	$0.606229\pi$
$432$	0	0
$433$	2.94371	0.141466	0.0707328	−	0.997495i	$-0.477466\pi$
$433$	2.94371	0.141466	0.0707328	+	0.997495i	$0.477466\pi$
$434$	−7.69583	−0.369412
$435$	0	0
$436$	−13.1040	−0.627567
$437$	29.4368	1.40815
$438$	0	0
$439$	−31.6977	−1.51285	−0.756423	−	0.654082i	$-0.773055\pi$
$439$	−31.6977	−1.51285	−0.756423	+	0.654082i	$0.773055\pi$
$440$	5.34696	0.254906
$441$	0	0
$442$	−1.66318	−0.0791093
$443$	−3.16977	−0.150600	−0.0753001	−	0.997161i	$-0.523991\pi$
$443$	−3.16977	−0.150600	−0.0753001	+	0.997161i	$0.523991\pi$
$444$	0	0
$445$	−15.8933	−0.753413
$446$	6.12393	0.289976
$447$	0	0
$448$	7.93189	0.374746
$449$	26.5058	1.25089	0.625444	−	0.780269i	$-0.284918\pi$
$449$	26.5058	1.25089	0.625444	+	0.780269i	$0.284918\pi$
$450$	0	0
$451$	10.1761	0.479172
$452$	5.98990	0.281741
$453$	0	0
$454$	3.60951	0.169403
$455$	3.07431	0.144126
$456$	0	0
$457$	−19.1466	−0.895642	−0.447821	−	0.894123i	$-0.647800\pi$
$457$	−19.1466	−0.895642	−0.447821	+	0.894123i	$0.647800\pi$
$458$	−0.00473429	−0.000221219 0
$459$	0	0
$460$	−6.65363	−0.310227
$461$	18.6597	0.869071	0.434536	−	0.900655i	$-0.356913\pi$
$461$	18.6597	0.869071	0.434536	+	0.900655i	$0.356913\pi$
$462$	0	0
$463$	−7.76783	−0.361002	−0.180501	−	0.983575i	$-0.557772\pi$
$463$	−7.76783	−0.361002	−0.180501	+	0.983575i	$0.557772\pi$
$464$	−7.11060	−0.330101
$465$	0	0
$466$	17.1374	0.793874
$467$	34.6788	1.60475	0.802373	−	0.596823i	$-0.203571\pi$
$467$	34.6788	1.60475	0.802373	+	0.596823i	$0.203571\pi$
$468$	0	0
$469$	13.8723	0.640564
$470$	12.5693	0.579779
$471$	0	0
$472$	−18.0397	−0.830342
$473$	−3.94021	−0.181171
$474$	0	0
$475$	−15.3717	−0.705303
$476$	−0.452941	−0.0207605
$477$	0	0
$478$	−25.5306	−1.16774
$479$	−30.8095	−1.40772	−0.703862	−	0.710337i	$-0.748543\pi$
$479$	−30.8095	−1.40772	−0.703862	+	0.710337i	$0.748543\pi$
$480$	0	0
$481$	7.76853	0.354215
$482$	−22.4231	−1.02134
$483$	0	0
$484$	7.01453	0.318842
$485$	9.96664	0.452562
$486$	0	0
$487$	−31.6190	−1.43279	−0.716397	−	0.697693i	$-0.754210\pi$
$487$	−31.6190	−1.43279	−0.716397	+	0.697693i	$0.754210\pi$
$488$	29.7146	1.34512
$489$	0	0
$490$	−1.14985	−0.0519450
$491$	16.3953	0.739910	0.369955	−	0.929050i	$-0.379373\pi$
$491$	16.3953	0.739910	0.369955	+	0.929050i	$0.379373\pi$
$492$	0	0
$493$	2.38197	0.107279
$494$	−12.3301	−0.554759
$495$	0	0
$496$	−11.4783	−0.515390
$497$	13.0115	0.583644
$498$	0	0
$499$	−37.1236	−1.66188	−0.830940	−	0.556363i	$-0.812197\pi$
$499$	−37.1236	−1.66188	−0.830940	+	0.556363i	$0.812197\pi$
$500$	7.97796	0.356785
$501$	0	0
$502$	16.2176	0.723827
$503$	−27.5974	−1.23051	−0.615254	−	0.788329i	$-0.710946\pi$
$503$	−27.5974	−1.23051	−0.615254	+	0.788329i	$0.710946\pi$
$504$	0	0
$505$	−10.4844	−0.466548
$506$	−13.0001	−0.577923
$507$	0	0
$508$	0.842680	0.0373879
$509$	30.3692	1.34609	0.673045	−	0.739602i	$-0.264986\pi$
$509$	30.3692	1.34609	0.673045	+	0.739602i	$0.264986\pi$
$510$	0	0
$511$	−5.78761	−0.256029
$512$	16.8578	0.745015
$513$	0	0
$514$	3.79582	0.167426
$515$	−9.35014	−0.412016
$516$	0	0
$517$	−17.8816	−0.786433
$518$	−2.90558	−0.127664
$519$	0	0
$520$	9.40160	0.412288
$521$	5.33540	0.233748	0.116874	−	0.993147i	$-0.462713\pi$
$521$	5.33540	0.233748	0.116874	+	0.993147i	$0.462713\pi$
$522$	0	0
$523$	−3.88610	−0.169927	−0.0849637	−	0.996384i	$-0.527077\pi$
$523$	−3.88610	−0.169927	−0.0849637	+	0.996384i	$0.527077\pi$
$524$	−11.0850	−0.484252
$525$	0	0
$526$	10.9448	0.477218
$527$	3.84510	0.167495
$528$	0	0
$529$	31.5711	1.37266
$530$	−14.9093	−0.647621
$531$	0	0
$532$	−3.35793	−0.145585
$533$	17.8927	0.775018
$534$	0	0
$535$	−4.12306	−0.178256
$536$	42.4232	1.83240
$537$	0	0
$538$	9.56057	0.412186
$539$	1.63583	0.0704600
$540$	0	0
$541$	−15.2236	−0.654514	−0.327257	−	0.944935i	$-0.606124\pi$
$541$	−15.2236	−0.654514	−0.327257	+	0.944935i	$0.606124\pi$
$542$	0.515282	0.0221333
$543$	0	0
$544$	−2.35969	−0.101171
$545$	16.6209	0.711963
$546$	0	0
$547$	−41.8131	−1.78780	−0.893899	−	0.448269i	$-0.852041\pi$
$547$	−41.8131	−1.78780	−0.893899	+	0.448269i	$0.852041\pi$
$548$	4.63705	0.198085
$549$	0	0
$550$	6.78857	0.289465
$551$	17.6590	0.752300
$552$	0	0
$553$	−16.3142	−0.693749
$554$	3.81321	0.162008
$555$	0	0
$556$	−10.5656	−0.448082
$557$	11.0989	0.470275	0.235137	−	0.971962i	$-0.424446\pi$
$557$	11.0989	0.470275	0.235137	+	0.971962i	$0.424446\pi$
$558$	0	0
$559$	−6.92811	−0.293028
$560$	−1.71500	−0.0724718
$561$	0	0
$562$	−2.53278	−0.106839
$563$	32.8791	1.38569	0.692844	−	0.721088i	$-0.256358\pi$
$563$	32.8791	1.38569	0.692844	+	0.721088i	$0.256358\pi$
$564$	0	0
$565$	−7.59752	−0.319630
$566$	−3.66674	−0.154125
$567$	0	0
$568$	39.7907	1.66958
$569$	−18.3934	−0.771090	−0.385545	−	0.922689i	$-0.625987\pi$
$569$	−18.3934	−0.771090	−0.385545	+	0.922689i	$0.625987\pi$
$570$	0	0
$571$	14.5142	0.607401	0.303701	−	0.952768i	$-0.401778\pi$
$571$	14.5142	0.607401	0.303701	+	0.952768i	$0.401778\pi$
$572$	−3.96490	−0.165781
$573$	0	0
$574$	−6.69221	−0.279327
$575$	−28.4967	−1.18840
$576$	0	0
$577$	18.7104	0.778922	0.389461	−	0.921043i	$-0.372661\pi$
$577$	18.7104	0.778922	0.389461	+	0.921043i	$0.372661\pi$
$578$	17.9776	0.747769
$579$	0	0
$580$	−3.99149	−0.165738
$581$	3.30043	0.136925
$582$	0	0
$583$	21.2107	0.878456
$584$	−17.6992	−0.732399
$585$	0	0
$586$	33.4156	1.38038
$587$	6.47637	0.267308	0.133654	−	0.991028i	$-0.457329\pi$
$587$	6.47637	0.267308	0.133654	+	0.991028i	$0.457329\pi$
$588$	0	0
$589$	28.5061	1.17457
$590$	6.78289	0.279247
$591$	0	0
$592$	−4.33366	−0.178112
$593$	31.3723	1.28831	0.644153	−	0.764897i	$-0.277210\pi$
$593$	31.3723	1.28831	0.644153	+	0.764897i	$0.277210\pi$
$594$	0	0
$595$	0.574505	0.0235524
$596$	13.8731	0.568266
$597$	0	0
$598$	−22.8581	−0.934738
$599$	7.74267	0.316357	0.158178	−	0.987411i	$-0.449438\pi$
$599$	7.74267	0.316357	0.158178	+	0.987411i	$0.449438\pi$
$600$	0	0
$601$	33.7405	1.37630	0.688152	−	0.725566i	$-0.258422\pi$
$601$	33.7405	1.37630	0.688152	+	0.725566i	$0.258422\pi$
$602$	2.59125	0.105611
$603$	0	0
$604$	−7.51143	−0.305636
$605$	−8.89714	−0.361720
$606$	0	0
$607$	42.7590	1.73553	0.867767	−	0.496972i	$-0.165555\pi$
$607$	42.7590	1.73553	0.867767	+	0.496972i	$0.165555\pi$
$608$	−17.4938	−0.709467
$609$	0	0
$610$	−11.1727	−0.452368
$611$	−31.4414	−1.27198
$612$	0	0
$613$	1.24787	0.0504012	0.0252006	−	0.999682i	$-0.491978\pi$
$613$	1.24787	0.0504012	0.0252006	+	0.999682i	$0.491978\pi$
$614$	19.2068	0.775123
$615$	0	0
$616$	5.00256	0.201559
$617$	31.4523	1.26622	0.633111	−	0.774061i	$-0.281778\pi$
$617$	31.4523	1.26622	0.633111	+	0.774061i	$0.281778\pi$
$618$	0	0
$619$	30.0735	1.20876	0.604378	−	0.796697i	$-0.293422\pi$
$619$	30.0735	1.20876	0.604378	+	0.796697i	$0.293422\pi$
$620$	−6.44326	−0.258768
$621$	0	0
$622$	33.0078	1.32349
$623$	−14.8696	−0.595736
$624$	0	0
$625$	9.16871	0.366748
$626$	−24.0315	−0.960491
$627$	0	0
$628$	−5.13292	−0.204826
$629$	1.45173	0.0578842
$630$	0	0
$631$	11.0743	0.440863	0.220431	−	0.975403i	$-0.429254\pi$
$631$	11.0743	0.440863	0.220431	+	0.975403i	$0.429254\pi$
$632$	−49.8907	−1.98454
$633$	0	0
$634$	12.1754	0.483546
$635$	−1.06884	−0.0424158
$636$	0	0
$637$	2.87629	0.113963
$638$	−7.79870	−0.308753
$639$	0	0
$640$	0.264198	0.0104433
$641$	−16.2581	−0.642157	−0.321078	−	0.947053i	$-0.604045\pi$
$641$	−16.2581	−0.642157	−0.321078	+	0.947053i	$0.604045\pi$
$642$	0	0
$643$	24.8807	0.981200	0.490600	−	0.871385i	$-0.336778\pi$
$643$	24.8807	0.981200	0.490600	+	0.871385i	$0.336778\pi$
$644$	−6.22506	−0.245302
$645$	0	0
$646$	−2.30417	−0.0906564
$647$	−12.1535	−0.477805	−0.238902	−	0.971044i	$-0.576788\pi$
$647$	−12.1535	−0.477805	−0.238902	+	0.971044i	$0.576788\pi$
$648$	0	0
$649$	−9.64963	−0.378781
$650$	11.9364	0.468184
$651$	0	0
$652$	−1.23926	−0.0485331
$653$	44.4765	1.74050	0.870249	−	0.492611i	$-0.163958\pi$
$653$	44.4765	1.74050	0.870249	+	0.492611i	$0.163958\pi$
$654$	0	0
$655$	14.0601	0.549374
$656$	−9.98139	−0.389708
$657$	0	0
$658$	11.7597	0.458441
$659$	0.975019	0.0379814	0.0189907	−	0.999820i	$-0.493955\pi$
$659$	0.975019	0.0379814	0.0189907	+	0.999820i	$0.493955\pi$
$660$	0	0
$661$	−18.2270	−0.708947	−0.354474	−	0.935066i	$-0.615340\pi$
$661$	−18.2270	−0.708947	−0.354474	+	0.935066i	$0.615340\pi$
$662$	−19.0207	−0.739260
$663$	0	0
$664$	10.0931	0.391689
$665$	4.25916	0.165163
$666$	0	0
$667$	32.7370	1.26758
$668$	14.0487	0.543561
$669$	0	0
$670$	−15.9511	−0.616245
$671$	15.8947	0.613608
$672$	0	0
$673$	7.89734	0.304420	0.152210	−	0.988348i	$-0.451361\pi$
$673$	7.89734	0.304420	0.152210	+	0.988348i	$0.451361\pi$
$674$	−29.2195	−1.12549
$675$	0	0
$676$	3.98331	0.153204
$677$	34.8206	1.33827	0.669133	−	0.743143i	$-0.266666\pi$
$677$	34.8206	1.33827	0.669133	+	0.743143i	$0.266666\pi$
$678$	0	0
$679$	9.32468	0.357848
$680$	1.75691	0.0673743
$681$	0	0
$682$	−12.5890	−0.482059
$683$	−32.8277	−1.25612	−0.628059	−	0.778166i	$-0.716150\pi$
$683$	−32.8277	−1.25612	−0.628059	+	0.778166i	$0.716150\pi$
$684$	0	0
$685$	−5.88158	−0.224724
$686$	−1.07579	−0.0410738
$687$	0	0
$688$	3.86483	0.147345
$689$	37.2949	1.42082
$690$	0	0
$691$	17.1140	0.651046	0.325523	−	0.945534i	$-0.394460\pi$
$691$	17.1140	0.651046	0.325523	+	0.945534i	$0.394460\pi$
$692$	−7.01966	−0.266847
$693$	0	0
$694$	−21.5354	−0.817472
$695$	13.4013	0.508341
$696$	0	0
$697$	3.34366	0.126650
$698$	−18.5515	−0.702184
$699$	0	0
$700$	3.25070	0.122865
$701$	−3.01925	−0.114035	−0.0570177	−	0.998373i	$-0.518159\pi$
$701$	−3.01925	−0.114035	−0.0570177	+	0.998373i	$0.518159\pi$
$702$	0	0
$703$	10.7626	0.405917
$704$	12.9752	0.489021
$705$	0	0
$706$	17.6062	0.662617
$707$	−9.80905	−0.368907
$708$	0	0
$709$	−11.0304	−0.414257	−0.207128	−	0.978314i	$-0.566412\pi$
$709$	−11.0304	−0.414257	−0.207128	+	0.978314i	$0.566412\pi$
$710$	−14.9612	−0.561486
$711$	0	0
$712$	−45.4729	−1.70417
$713$	52.8457	1.97909
$714$	0	0
$715$	5.02903	0.188075
$716$	8.47917	0.316882
$717$	0	0
$718$	16.8175	0.627623
$719$	−13.2398	−0.493760	−0.246880	−	0.969046i	$-0.579405\pi$
$719$	−13.2398	−0.493760	−0.246880	+	0.969046i	$0.579405\pi$
$720$	0	0
$721$	−8.74789	−0.325788
$722$	3.35775	0.124962
$723$	0	0
$724$	−12.1528	−0.451656
$725$	−17.0951	−0.634896
$726$	0	0
$727$	33.0309	1.22505	0.612524	−	0.790452i	$-0.290154\pi$
$727$	33.0309	1.22505	0.612524	+	0.790452i	$0.290154\pi$
$728$	8.79604	0.326003
$729$	0	0
$730$	6.65489	0.246309
$731$	−1.29468	−0.0478854
$732$	0	0
$733$	−45.5976	−1.68419	−0.842094	−	0.539331i	$-0.818677\pi$
$733$	−45.5976	−1.68419	−0.842094	+	0.539331i	$0.818677\pi$
$734$	−30.3520	−1.12031
$735$	0	0
$736$	−32.4307	−1.19541
$737$	22.6927	0.835897
$738$	0	0
$739$	27.4813	1.01092	0.505458	−	0.862851i	$-0.331324\pi$
$739$	27.4813	1.01092	0.505458	+	0.862851i	$0.331324\pi$
$740$	−2.43267	−0.0894268
$741$	0	0
$742$	−13.9490	−0.512085
$743$	24.0064	0.880711	0.440355	−	0.897824i	$-0.354852\pi$
$743$	24.0064	0.880711	0.440355	+	0.897824i	$0.354852\pi$
$744$	0	0
$745$	−17.5965	−0.644686
$746$	−35.3507	−1.29428
$747$	0	0
$748$	−0.740933	−0.0270912
$749$	−3.85750	−0.140950
$750$	0	0
$751$	2.29809	0.0838584	0.0419292	−	0.999121i	$-0.486650\pi$
$751$	2.29809	0.0838584	0.0419292	+	0.999121i	$0.486650\pi$
$752$	17.5395	0.639601
$753$	0	0
$754$	−13.7125	−0.499380
$755$	9.52741	0.346738
$756$	0	0
$757$	−49.6049	−1.80292	−0.901460	−	0.432863i	$-0.857503\pi$
$757$	−49.6049	−1.80292	−0.901460	+	0.432863i	$0.857503\pi$
$758$	−34.8363	−1.26531
$759$	0	0
$760$	13.0250	0.472467
$761$	29.5034	1.06950	0.534748	−	0.845011i	$-0.320406\pi$
$761$	29.5034	1.06950	0.534748	+	0.845011i	$0.320406\pi$
$762$	0	0
$763$	15.5504	0.562962
$764$	10.9796	0.397227
$765$	0	0
$766$	−10.0586	−0.363432
$767$	−16.9670	−0.612644
$768$	0	0
$769$	−39.2438	−1.41517	−0.707584	−	0.706629i	$-0.750215\pi$
$769$	−39.2438	−1.41517	−0.707584	+	0.706629i	$0.750215\pi$
$770$	−1.88096	−0.0677850
$771$	0	0
$772$	18.6125	0.669877
$773$	35.6778	1.28324	0.641621	−	0.767022i	$-0.278262\pi$
$773$	35.6778	1.28324	0.641621	+	0.767022i	$0.278262\pi$
$774$	0	0
$775$	−27.5958	−0.991270
$776$	28.5160	1.02367
$777$	0	0
$778$	7.52624	0.269829
$779$	24.7886	0.888143
$780$	0	0
$781$	21.2845	0.761620
$782$	−4.27157	−0.152751
$783$	0	0
$784$	−1.60453	−0.0573047
$785$	6.51053	0.232371
$786$	0	0
$787$	16.4232	0.585423	0.292712	−	0.956201i	$-0.405442\pi$
$787$	16.4232	0.585423	0.292712	+	0.956201i	$0.405442\pi$
$788$	10.9415	0.389775
$789$	0	0
$790$	18.7588	0.667410
$791$	−7.10816	−0.252737
$792$	0	0
$793$	27.9478	0.992455
$794$	14.8528	0.527104
$795$	0	0
$796$	16.0611	0.569271
$797$	1.89933	0.0672778	0.0336389	−	0.999434i	$-0.489290\pi$
$797$	1.89933	0.0672778	0.0336389	+	0.999434i	$0.489290\pi$
$798$	0	0
$799$	−5.87556	−0.207862
$800$	16.9351	0.598748
$801$	0	0
$802$	−24.5203	−0.865841
$803$	−9.46753	−0.334102
$804$	0	0
$805$	7.89580	0.278290
$806$	−22.1354	−0.779687
$807$	0	0
$808$	−29.9973	−1.05530
$809$	−54.1503	−1.90382	−0.951912	−	0.306371i	$-0.900885\pi$
$809$	−54.1503	−1.90382	−0.951912	+	0.306371i	$0.900885\pi$
$810$	0	0
$811$	20.0329	0.703450	0.351725	−	0.936103i	$-0.385595\pi$
$811$	20.0329	0.703450	0.351725	+	0.936103i	$0.385595\pi$
$812$	−3.73440	−0.131052
$813$	0	0
$814$	−4.75303	−0.166594
$815$	1.57186	0.0550598
$816$	0	0
$817$	−9.59823	−0.335800
$818$	32.8292	1.14784
$819$	0	0
$820$	−5.60299	−0.195665
$821$	−16.0267	−0.559334	−0.279667	−	0.960097i	$-0.590224\pi$
$821$	−16.0267	−0.559334	−0.279667	+	0.960097i	$0.590224\pi$
$822$	0	0
$823$	−19.5675	−0.682079	−0.341040	−	0.940049i	$-0.610779\pi$
$823$	−19.5675	−0.682079	−0.341040	+	0.940049i	$0.610779\pi$
$824$	−26.7521	−0.931954
$825$	0	0
$826$	6.34600	0.220806
$827$	29.3096	1.01919	0.509597	−	0.860413i	$-0.329794\pi$
$827$	29.3096	1.01919	0.509597	+	0.860413i	$0.329794\pi$
$828$	0	0
$829$	−30.7346	−1.06746	−0.533728	−	0.845656i	$-0.679210\pi$
$829$	−30.7346	−1.06746	−0.533728	+	0.845656i	$0.679210\pi$
$830$	−3.79500	−0.131727
$831$	0	0
$832$	22.8144	0.790947
$833$	0.537501	0.0186233
$834$	0	0
$835$	−17.8192	−0.616659
$836$	−5.49299	−0.189979
$837$	0	0
$838$	20.7082	0.715354
$839$	24.8218	0.856944	0.428472	−	0.903555i	$-0.359052\pi$
$839$	24.8218	0.856944	0.428472	+	0.903555i	$0.359052\pi$
$840$	0	0
$841$	−9.36116	−0.322799
$842$	−35.0126	−1.20662
$843$	0	0
$844$	−14.1051	−0.485519
$845$	−5.05239	−0.173807
$846$	0	0
$847$	−8.32407	−0.286018
$848$	−20.8049	−0.714443
$849$	0	0
$850$	2.23059	0.0765086
$851$	19.9521	0.683948
$852$	0	0
$853$	6.68047	0.228735	0.114367	−	0.993439i	$-0.463516\pi$
$853$	6.68047	0.228735	0.114367	+	0.993439i	$0.463516\pi$
$854$	−10.4530	−0.357695
$855$	0	0
$856$	−11.7967	−0.403203
$857$	−31.3738	−1.07171	−0.535854	−	0.844311i	$-0.680010\pi$
$857$	−31.3738	−1.07171	−0.535854	+	0.844311i	$0.680010\pi$
$858$	0	0
$859$	−5.50268	−0.187749	−0.0938745	−	0.995584i	$-0.529925\pi$
$859$	−5.50268	−0.187749	−0.0938745	+	0.995584i	$0.529925\pi$
$860$	2.16950	0.0739793
$861$	0	0
$862$	14.6317	0.498359
$863$	−2.38157	−0.0810696	−0.0405348	−	0.999178i	$-0.512906\pi$
$863$	−2.38157	−0.0810696	−0.0405348	+	0.999178i	$0.512906\pi$
$864$	0	0
$865$	8.90365	0.302733
$866$	−3.16681	−0.107612
$867$	0	0
$868$	−6.02825	−0.204612
$869$	−26.6871	−0.905299
$870$	0	0
$871$	39.9008	1.35199
$872$	47.5550	1.61041
$873$	0	0
$874$	−31.6677	−1.07118
$875$	−9.46737	−0.320055
$876$	0	0
$877$	53.8721	1.81913	0.909566	−	0.415559i	$-0.136414\pi$
$877$	53.8721	1.81913	0.909566	+	0.415559i	$0.136414\pi$
$878$	34.1000	1.15082
$879$	0	0
$880$	−2.80544	−0.0945712
$881$	−20.3102	−0.684269	−0.342135	−	0.939651i	$-0.611150\pi$
$881$	−20.3102	−0.684269	−0.342135	+	0.939651i	$0.611150\pi$
$882$	0	0
$883$	−50.7024	−1.70627	−0.853135	−	0.521690i	$-0.825302\pi$
$883$	−50.7024	−1.70627	−0.853135	+	0.521690i	$0.825302\pi$
$884$	−1.30279	−0.0438176
$885$	0	0
$886$	3.41000	0.114561
$887$	27.3602	0.918664	0.459332	−	0.888265i	$-0.348089\pi$
$887$	27.3602	0.918664	0.459332	+	0.888265i	$0.348089\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	17.0978	0.573119
$891$	0	0
$892$	4.79696	0.160614
$893$	−43.5591	−1.45765
$894$	0	0
$895$	−10.7549	−0.359496
$896$	0.247181	0.00825773
$897$	0	0
$898$	−28.5147	−0.951546
$899$	31.7020	1.05732
$900$	0	0
$901$	6.96942	0.232185
$902$	−10.9473	−0.364505
$903$	0	0
$904$	−21.7376	−0.722983
$905$	15.4145	0.512395
$906$	0	0
$907$	52.3236	1.73738	0.868689	−	0.495358i	$-0.164963\pi$
$907$	52.3236	1.73738	0.868689	+	0.495358i	$0.164963\pi$
$908$	2.82738	0.0938298
$909$	0	0
$910$	−3.30730	−0.109636
$911$	−8.75537	−0.290078	−0.145039	−	0.989426i	$-0.546331\pi$
$911$	−8.75537	−0.290078	−0.145039	+	0.989426i	$0.546331\pi$
$912$	0	0
$913$	5.39894	0.178679
$914$	20.5977	0.681312
$915$	0	0
$916$	−0.00370844	−0.000122530 0
$917$	13.1545	0.434400
$918$	0	0
$919$	−13.9562	−0.460371	−0.230186	−	0.973147i	$-0.573933\pi$
$919$	−13.9562	−0.460371	−0.230186	+	0.973147i	$0.573933\pi$
$920$	24.1463	0.796080
$921$	0	0
$922$	−20.0739	−0.661100
$923$	37.4248	1.23185
$924$	0	0
$925$	−10.4189	−0.342570
$926$	8.35654	0.274613
$927$	0	0
$928$	−19.4551	−0.638644
$929$	−46.0364	−1.51041	−0.755203	−	0.655491i	$-0.772462\pi$
$929$	−46.0364	−1.51041	−0.755203	+	0.655491i	$0.772462\pi$
$930$	0	0
$931$	3.98482	0.130597
$932$	13.4239	0.439716
$933$	0	0
$934$	−37.3071	−1.22073
$935$	0.939791	0.0307344
$936$	0	0
$937$	60.0525	1.96183	0.980915	−	0.194437i	$-0.0622881\pi$
$937$	60.0525	1.96183	0.980915	+	0.194437i	$0.0622881\pi$
$938$	−14.9237	−0.487275
$939$	0	0
$940$	9.84571	0.321132
$941$	−10.5532	−0.344024	−0.172012	−	0.985095i	$-0.555027\pi$
$941$	−10.5532	−0.344024	−0.172012	+	0.985095i	$0.555027\pi$
$942$	0	0
$943$	45.9541	1.49647
$944$	9.46503	0.308060
$945$	0	0
$946$	4.23884	0.137816
$947$	−32.9749	−1.07154	−0.535770	−	0.844364i	$-0.679979\pi$
$947$	−32.9749	−1.07154	−0.535770	+	0.844364i	$0.679979\pi$
$948$	0	0
$949$	−16.6468	−0.540380
$950$	16.5367	0.536522
$951$	0	0
$952$	1.64374	0.0532740
$953$	−27.5200	−0.891461	−0.445731	−	0.895167i	$-0.647056\pi$
$953$	−27.5200	−0.891461	−0.445731	+	0.895167i	$0.647056\pi$
$954$	0	0
$955$	−13.9264	−0.450646
$956$	−19.9985	−0.646798
$957$	0	0
$958$	33.1445	1.07085
$959$	−5.50274	−0.177693
$960$	0	0
$961$	20.1750	0.650805
$962$	−8.35729	−0.269450
$963$	0	0
$964$	−17.5643	−0.565709
$965$	−23.6078	−0.759963
$966$	0	0
$967$	13.6019	0.437407	0.218704	−	0.975791i	$-0.429817\pi$
$967$	13.6019	0.437407	0.218704	+	0.975791i	$0.429817\pi$
$968$	−25.4560	−0.818187
$969$	0	0
$970$	−10.7220	−0.344262
$971$	13.3938	0.429828	0.214914	−	0.976633i	$-0.431053\pi$
$971$	13.3938	0.429828	0.214914	+	0.976633i	$0.431053\pi$
$972$	0	0
$973$	12.5381	0.401954
$974$	34.0154	1.08992
$975$	0	0
$976$	−15.5906	−0.499043
$977$	−10.4967	−0.335820	−0.167910	−	0.985802i	$-0.553702\pi$
$977$	−10.4967	−0.335820	−0.167910	+	0.985802i	$0.553702\pi$
$978$	0	0
$979$	−24.3240	−0.777399
$980$	−0.900694	−0.0287716
$981$	0	0
$982$	−17.6379	−0.562848
$983$	−45.3976	−1.44796	−0.723980	−	0.689821i	$-0.757689\pi$
$983$	−45.3976	−1.44796	−0.723980	+	0.689821i	$0.757689\pi$
$984$	0	0
$985$	−13.8781	−0.442192
$986$	−2.56250	−0.0816066
$987$	0	0
$988$	−9.65837	−0.307274
$989$	−17.7936	−0.565803
$990$	0	0
$991$	38.7511	1.23097	0.615484	−	0.788149i	$-0.288961\pi$
$991$	38.7511	1.23097	0.615484	+	0.788149i	$0.288961\pi$
$992$	−31.4054	−0.997121
$993$	0	0
$994$	−13.9976	−0.443977
$995$	−20.3717	−0.645827
$996$	0	0
$997$	−27.9293	−0.884531	−0.442266	−	0.896884i	$-0.645825\pi$
$997$	−27.9293	−0.884531	−0.442266	+	0.896884i	$0.645825\pi$
$998$	39.9371	1.26419
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.ba.1.15	✓	40
3.2	odd	2	inner	8001.2.a.ba.1.26	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8001.2.a.ba.1.15	✓	40	1.1	even	1	trivial
8001.2.a.ba.1.26	yes	40	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)

\(f(q)\)	\(=\)	\(q-1.07579 q^{2} -0.842680 q^{4} +1.06884 q^{5} +1.00000 q^{7} +3.05812 q^{8} +O(q^{10})\)	\(q-1.07579 q^{2} -0.842680 q^{4} +1.06884 q^{5} +1.00000 q^{7} +3.05812 q^{8} -1.14985 q^{10} +1.63583 q^{11} +2.87629 q^{13} -1.07579 q^{14} -1.60453 q^{16} +0.537501 q^{17} +3.98482 q^{19} -0.900694 q^{20} -1.75980 q^{22} +7.38722 q^{23} -3.85757 q^{25} -3.09428 q^{26} -0.842680 q^{28} +4.43157 q^{29} +7.15367 q^{31} -4.39011 q^{32} -0.578237 q^{34} +1.06884 q^{35} +2.70089 q^{37} -4.28682 q^{38} +3.26866 q^{40} +6.22075 q^{41} -2.40870 q^{43} -1.37848 q^{44} -7.94709 q^{46} -10.9312 q^{47} +1.00000 q^{49} +4.14993 q^{50} -2.42379 q^{52} +12.9663 q^{53} +1.74845 q^{55} +3.05812 q^{56} -4.76743 q^{58} -5.89893 q^{59} +9.71661 q^{61} -7.69583 q^{62} +7.93189 q^{64} +3.07431 q^{65} +13.8723 q^{67} -0.452941 q^{68} -1.14985 q^{70} +13.0115 q^{71} -5.78761 q^{73} -2.90558 q^{74} -3.35793 q^{76} +1.63583 q^{77} -16.3142 q^{79} -1.71500 q^{80} -6.69221 q^{82} +3.30043 q^{83} +0.574505 q^{85} +2.59125 q^{86} +5.00256 q^{88} -14.8696 q^{89} +2.87629 q^{91} -6.22506 q^{92} +11.7597 q^{94} +4.25916 q^{95} +9.32468 q^{97} -1.07579 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 54 q^{4} + 40 q^{7}+O(q^{10}) \) 40 * q + 54 * q^4 + 40 * q^7	\( 40 q + 54 q^{4} + 40 q^{7} + 20 q^{10} + 10 q^{13} + 90 q^{16} + 38 q^{19} + 14 q^{22} + 84 q^{25} + 54 q^{28} + 66 q^{31} + 22 q^{34} + 40 q^{37} + 26 q^{40} + 38 q^{43} + 28 q^{46} + 40 q^{49} + 28 q^{52} + 60 q^{55} + 42 q^{58} + 54 q^{61} + 124 q^{64} + 48 q^{67} + 20 q^{70} + 16 q^{76} + 102 q^{79} + 48 q^{82} + 104 q^{85} + 48 q^{88} + 10 q^{91} - 10 q^{94} + 8 q^{97}+O(q^{100}) \) 40 * q + 54 * q^4 + 40 * q^7 + 20 * q^10 + 10 * q^13 + 90 * q^16 + 38 * q^19 + 14 * q^22 + 84 * q^25 + 54 * q^28 + 66 * q^31 + 22 * q^34 + 40 * q^37 + 26 * q^40 + 38 * q^43 + 28 * q^46 + 40 * q^49 + 28 * q^52 + 60 * q^55 + 42 * q^58 + 54 * q^61 + 124 * q^64 + 48 * q^67 + 20 * q^70 + 16 * q^76 + 102 * q^79 + 48 * q^82 + 104 * q^85 + 48 * q^88 + 10 * q^91 - 10 * q^94 + 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more