LMFDB - Embedded newform 8001.2.a.p.1.5

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:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 5 x^{13} - 9 x^{12} + 76 x^{11} - 12 x^{10} - 414 x^{9} + 331 x^{8} + 959 x^{7} - 1067 x^{6} + \cdots + 44 $ x^14 - 5x^13 - 9x^12 + 76x^11 - 12x^10 - 414x^9 + 331x^8 + 959x^7 - 1067x^6 - 875x^5 + 1134x^4 + 301x^3 - 418x^2 - 42*x + 44
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 5 $
Twist minimal:	no (minimal twist has level 2667)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.579209$ of defining polynomial
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)

$f(q)$	$=$	$q-0.579209 q^{2} -1.66452 q^{4} +1.12937 q^{5} -1.00000 q^{7} +2.12252 q^{8} +O(q^{10})$	\(q-0.579209 q^{2} -1.66452 q^{4} +1.12937 q^{5} -1.00000 q^{7} +2.12252 q^{8} -0.654144 q^{10} -5.13283 q^{11} -4.13855 q^{13} +0.579209 q^{14} +2.09965 q^{16} -6.27943 q^{17} -3.44855 q^{19} -1.87986 q^{20} +2.97298 q^{22} -2.79730 q^{23} -3.72452 q^{25} +2.39709 q^{26} +1.66452 q^{28} +0.885951 q^{29} +3.36249 q^{31} -5.46118 q^{32} +3.63710 q^{34} -1.12937 q^{35} -3.53011 q^{37} +1.99743 q^{38} +2.39712 q^{40} -9.71520 q^{41} -8.03235 q^{43} +8.54368 q^{44} +1.62022 q^{46} -7.44686 q^{47} +1.00000 q^{49} +2.15727 q^{50} +6.88869 q^{52} +12.8489 q^{53} -5.79688 q^{55} -2.12252 q^{56} -0.513151 q^{58} -6.04301 q^{59} +4.23118 q^{61} -1.94759 q^{62} -1.03613 q^{64} -4.67397 q^{65} -10.3911 q^{67} +10.4522 q^{68} +0.654144 q^{70} +11.1676 q^{71} +2.85739 q^{73} +2.04467 q^{74} +5.74016 q^{76} +5.13283 q^{77} -3.23967 q^{79} +2.37129 q^{80} +5.62713 q^{82} +15.3069 q^{83} -7.09182 q^{85} +4.65241 q^{86} -10.8945 q^{88} -7.28236 q^{89} +4.13855 q^{91} +4.65615 q^{92} +4.31329 q^{94} -3.89470 q^{95} +2.16988 q^{97} -0.579209 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + 5 q^{2} + 15 q^{4} + 4 q^{5} - 14 q^{7} + 12 q^{8}+O(q^{10}) $ 14 * q + 5 * q^2 + 15 * q^4 + 4 * q^5 - 14 * q^7 + 12 * q^8	$ 14 q + 5 q^{2} + 15 q^{4} + 4 q^{5} - 14 q^{7} + 12 q^{8} + 4 q^{10} + 3 q^{11} - 13 q^{13} - 5 q^{14} + 13 q^{16} + 5 q^{17} + 21 q^{19} + 3 q^{20} - 3 q^{22} + 10 q^{23} + 4 q^{25} + 6 q^{26} - 15 q^{28} + 15 q^{29} + 33 q^{31} + 29 q^{32} + 28 q^{34} - 4 q^{35} - 29 q^{37} + 15 q^{38} + 3 q^{40} + q^{41} - 25 q^{43} + 26 q^{44} - 4 q^{46} + 9 q^{47} + 14 q^{49} + 28 q^{50} - 13 q^{52} + 35 q^{53} + 14 q^{55} - 12 q^{56} - 23 q^{58} - 10 q^{59} + q^{61} + 43 q^{62} - 2 q^{64} + 24 q^{65} - 38 q^{67} + 2 q^{68} - 4 q^{70} + 10 q^{71} + 8 q^{73} + 25 q^{74} + 26 q^{76} - 3 q^{77} + 26 q^{79} + 48 q^{80} + 6 q^{82} + 30 q^{83} - 32 q^{85} + 50 q^{86} - 29 q^{88} - 4 q^{89} + 13 q^{91} + 32 q^{92} - 7 q^{94} + 32 q^{95} + 15 q^{97} + 5 q^{98}+O(q^{100}) $ 14 * q + 5 * q^2 + 15 * q^4 + 4 * q^5 - 14 * q^7 + 12 * q^8 + 4 * q^10 + 3 * q^11 - 13 * q^13 - 5 * q^14 + 13 * q^16 + 5 * q^17 + 21 * q^19 + 3 * q^20 - 3 * q^22 + 10 * q^23 + 4 * q^25 + 6 * q^26 - 15 * q^28 + 15 * q^29 + 33 * q^31 + 29 * q^32 + 28 * q^34 - 4 * q^35 - 29 * q^37 + 15 * q^38 + 3 * q^40 + q^41 - 25 * q^43 + 26 * q^44 - 4 * q^46 + 9 * q^47 + 14 * q^49 + 28 * q^50 - 13 * q^52 + 35 * q^53 + 14 * q^55 - 12 * q^56 - 23 * q^58 - 10 * q^59 + q^61 + 43 * q^62 - 2 * q^64 + 24 * q^65 - 38 * q^67 + 2 * q^68 - 4 * q^70 + 10 * q^71 + 8 * q^73 + 25 * q^74 + 26 * q^76 - 3 * q^77 + 26 * q^79 + 48 * q^80 + 6 * q^82 + 30 * q^83 - 32 * q^85 + 50 * q^86 - 29 * q^88 - 4 * q^89 + 13 * q^91 + 32 * q^92 - 7 * q^94 + 32 * q^95 + 15 * q^97 + 5 * 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 $

$2$	−0.579209	−0.409563	−0.204781	−	0.978808i	$-0.565648\pi$
$2$	−0.579209	−0.409563	−0.204781	+	0.978808i	$0.565648\pi$
$3$	0	0
$3$	0	0
$4$	−1.66452	−0.832258
$5$	1.12937	0.505071	0.252536	−	0.967588i	$-0.418736\pi$
$5$	1.12937	0.505071	0.252536	+	0.967588i	$0.418736\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	2.12252	0.750425
$9$	0	0
$10$	−0.654144	−0.206858
$11$	−5.13283	−1.54761	−0.773804	−	0.633426i	$-0.781648\pi$
$11$	−5.13283	−1.54761	−0.773804	+	0.633426i	$0.781648\pi$
$12$	0	0
$13$	−4.13855	−1.14783	−0.573914	−	0.818916i	$-0.694576\pi$
$13$	−4.13855	−1.14783	−0.573914	+	0.818916i	$0.694576\pi$
$14$	0.579209	0.154800
$15$	0	0
$16$	2.09965	0.524912
$17$	−6.27943	−1.52299	−0.761493	−	0.648173i	$-0.775533\pi$
$17$	−6.27943	−1.52299	−0.761493	+	0.648173i	$0.775533\pi$
$18$	0	0
$19$	−3.44855	−0.791151	−0.395575	−	0.918434i	$-0.629455\pi$
$19$	−3.44855	−0.791151	−0.395575	+	0.918434i	$0.629455\pi$
$20$	−1.87986	−0.420350
$21$	0	0
$22$	2.97298	0.633842
$23$	−2.79730	−0.583277	−0.291638	−	0.956529i	$-0.594200\pi$
$23$	−2.79730	−0.583277	−0.291638	+	0.956529i	$0.594200\pi$
$24$	0	0
$25$	−3.72452	−0.744903
$26$	2.39709	0.470107
$27$	0	0
$28$	1.66452	0.314564
$29$	0.885951	0.164517	0.0822585	−	0.996611i	$-0.473787\pi$
$29$	0.885951	0.164517	0.0822585	+	0.996611i	$0.473787\pi$
$30$	0	0
$31$	3.36249	0.603921	0.301961	−	0.953320i	$-0.402359\pi$
$31$	3.36249	0.603921	0.301961	+	0.953320i	$0.402359\pi$
$32$	−5.46118	−0.965409
$33$	0	0
$34$	3.63710	0.623758
$35$	−1.12937	−0.190899
$36$	0	0
$37$	−3.53011	−0.580346	−0.290173	−	0.956974i	$-0.593713\pi$
$37$	−3.53011	−0.580346	−0.290173	+	0.956974i	$0.593713\pi$
$38$	1.99743	0.324026
$39$	0	0
$40$	2.39712	0.379018
$41$	−9.71520	−1.51726	−0.758630	−	0.651522i	$-0.774131\pi$
$41$	−9.71520	−1.51726	−0.758630	+	0.651522i	$0.774131\pi$
$42$	0	0
$43$	−8.03235	−1.22492	−0.612461	−	0.790501i	$-0.709820\pi$
$43$	−8.03235	−1.22492	−0.612461	+	0.790501i	$0.709820\pi$
$44$	8.54368	1.28801
$45$	0	0
$46$	1.62022	0.238889
$47$	−7.44686	−1.08624	−0.543118	−	0.839657i	$-0.682756\pi$
$47$	−7.44686	−1.08624	−0.543118	+	0.839657i	$0.682756\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	2.15727	0.305085
$51$	0	0
$52$	6.88869	0.955289
$53$	12.8489	1.76493	0.882467	−	0.470374i	$-0.155881\pi$
$53$	12.8489	1.76493	0.882467	+	0.470374i	$0.155881\pi$
$54$	0	0
$55$	−5.79688	−0.781652
$56$	−2.12252	−0.283634
$57$	0	0
$58$	−0.513151	−0.0673800
$59$	−6.04301	−0.786733	−0.393367	−	0.919382i	$-0.628690\pi$
$59$	−6.04301	−0.786733	−0.393367	+	0.919382i	$0.628690\pi$
$60$	0	0
$61$	4.23118	0.541747	0.270873	−	0.962615i	$-0.412688\pi$
$61$	4.23118	0.541747	0.270873	+	0.962615i	$0.412688\pi$
$62$	−1.94759	−0.247344
$63$	0	0
$64$	−1.03613	−0.129517
$65$	−4.67397	−0.579734
$66$	0	0
$67$	−10.3911	−1.26947	−0.634735	−	0.772730i	$-0.718891\pi$
$67$	−10.3911	−1.26947	−0.634735	+	0.772730i	$0.718891\pi$
$68$	10.4522	1.26752
$69$	0	0
$70$	0.654144	0.0781851
$71$	11.1676	1.32535	0.662673	−	0.748909i	$-0.269422\pi$
$71$	11.1676	1.32535	0.662673	+	0.748909i	$0.269422\pi$
$72$	0	0
$73$	2.85739	0.334433	0.167216	−	0.985920i	$-0.446522\pi$
$73$	2.85739	0.334433	0.167216	+	0.985920i	$0.446522\pi$
$74$	2.04467	0.237688
$75$	0	0
$76$	5.74016	0.658442
$77$	5.13283	0.584940
$78$	0	0
$79$	−3.23967	−0.364492	−0.182246	−	0.983253i	$-0.558337\pi$
$79$	−3.23967	−0.364492	−0.182246	+	0.983253i	$0.558337\pi$
$80$	2.37129	0.265118
$81$	0	0
$82$	5.62713	0.621413
$83$	15.3069	1.68016	0.840078	−	0.542466i	$-0.182509\pi$
$83$	15.3069	1.68016	0.840078	+	0.542466i	$0.182509\pi$
$84$	0	0
$85$	−7.09182	−0.769216
$86$	4.65241	0.501682
$87$	0	0
$88$	−10.8945	−1.16136
$89$	−7.28236	−0.771929	−0.385964	−	0.922514i	$-0.626131\pi$
$89$	−7.28236	−0.771929	−0.385964	+	0.922514i	$0.626131\pi$
$90$	0	0
$91$	4.13855	0.433838
$92$	4.65615	0.485437
$93$	0	0
$94$	4.31329	0.444882
$95$	−3.89470	−0.399587
$96$	0	0
$97$	2.16988	0.220318	0.110159	−	0.993914i	$-0.464864\pi$
$97$	2.16988	0.220318	0.110159	+	0.993914i	$0.464864\pi$
$98$	−0.579209	−0.0585090
$99$	0	0
$100$	6.19952	0.619952
$101$	15.3334	1.52573	0.762864	−	0.646559i	$-0.223792\pi$
$101$	15.3334	1.52573	0.762864	+	0.646559i	$0.223792\pi$
$102$	0	0
$103$	−15.3380	−1.51129	−0.755647	−	0.654979i	$-0.772678\pi$
$103$	−15.3380	−1.51129	−0.755647	+	0.654979i	$0.772678\pi$
$104$	−8.78416	−0.861358
$105$	0	0
$106$	−7.44221	−0.722851
$107$	16.5612	1.60103	0.800514	−	0.599314i	$-0.204560\pi$
$107$	16.5612	1.60103	0.800514	+	0.599314i	$0.204560\pi$
$108$	0	0
$109$	−19.6618	−1.88326	−0.941628	−	0.336656i	$-0.890704\pi$
$109$	−19.6618	−1.88326	−0.941628	+	0.336656i	$0.890704\pi$
$110$	3.35761	0.320135
$111$	0	0
$112$	−2.09965	−0.198398
$113$	−10.6308	−1.00007	−0.500033	−	0.866006i	$-0.666679\pi$
$113$	−10.6308	−1.00007	−0.500033	+	0.866006i	$0.666679\pi$
$114$	0	0
$115$	−3.15919	−0.294596
$116$	−1.47468	−0.136921
$117$	0	0
$118$	3.50017	0.322217
$119$	6.27943	0.575634
$120$	0	0
$121$	15.3460	1.39509
$122$	−2.45074	−0.221879
$123$	0	0
$124$	−5.59692	−0.502619
$125$	−9.85324	−0.881300
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	11.5225	1.01845
$129$	0	0
$130$	2.70721	0.237438
$131$	17.4654	1.52596	0.762981	−	0.646421i	$-0.223735\pi$
$131$	17.4654	1.52596	0.762981	+	0.646421i	$0.223735\pi$
$132$	0	0
$133$	3.44855	0.299027
$134$	6.01860	0.519928
$135$	0	0
$136$	−13.3282	−1.14289
$137$	−8.39283	−0.717048	−0.358524	−	0.933520i	$-0.616720\pi$
$137$	−8.39283	−0.717048	−0.358524	+	0.933520i	$0.616720\pi$
$138$	0	0
$139$	12.1873	1.03371	0.516857	−	0.856071i	$-0.327102\pi$
$139$	12.1873	1.03371	0.516857	+	0.856071i	$0.327102\pi$
$140$	1.87986	0.158877
$141$	0	0
$142$	−6.46835	−0.542812
$143$	21.2425	1.77639
$144$	0	0
$145$	1.00057	0.0830928
$146$	−1.65503	−0.136971
$147$	0	0
$148$	5.87592	0.482998
$149$	14.9754	1.22684	0.613418	−	0.789759i	$-0.289794\pi$
$149$	14.9754	1.22684	0.613418	+	0.789759i	$0.289794\pi$
$150$	0	0
$151$	8.03102	0.653555	0.326778	−	0.945101i	$-0.394037\pi$
$151$	8.03102	0.653555	0.326778	+	0.945101i	$0.394037\pi$
$152$	−7.31961	−0.593699
$153$	0	0
$154$	−2.97298	−0.239570
$155$	3.79751	0.305023
$156$	0	0
$157$	−3.96554	−0.316485	−0.158242	−	0.987400i	$-0.550583\pi$
$157$	−3.96554	−0.316485	−0.158242	+	0.987400i	$0.550583\pi$
$158$	1.87645	0.149282
$159$	0	0
$160$	−6.16771	−0.487600
$161$	2.79730	0.220458
$162$	0	0
$163$	14.7955	1.15887	0.579437	−	0.815017i	$-0.303272\pi$
$163$	14.7955	1.15887	0.579437	+	0.815017i	$0.303272\pi$
$164$	16.1711	1.26275
$165$	0	0
$166$	−8.86593	−0.688129
$167$	−6.99524	−0.541308	−0.270654	−	0.962677i	$-0.587240\pi$
$167$	−6.99524	−0.541308	−0.270654	+	0.962677i	$0.587240\pi$
$168$	0	0
$169$	4.12760	0.317508
$170$	4.10765	0.315042
$171$	0	0
$172$	13.3700	1.01945
$173$	10.2879	0.782176	0.391088	−	0.920353i	$-0.372099\pi$
$173$	10.2879	0.782176	0.391088	+	0.920353i	$0.372099\pi$
$174$	0	0
$175$	3.72452	0.281547
$176$	−10.7771	−0.812358
$177$	0	0
$178$	4.21801	0.316153
$179$	1.98045	0.148026	0.0740129	−	0.997257i	$-0.476419\pi$
$179$	1.98045	0.148026	0.0740129	+	0.997257i	$0.476419\pi$
$180$	0	0
$181$	−8.89832	−0.661407	−0.330704	−	0.943735i	$-0.607286\pi$
$181$	−8.89832	−0.661407	−0.330704	+	0.943735i	$0.607286\pi$
$182$	−2.39709	−0.177684
$183$	0	0
$184$	−5.93733	−0.437705
$185$	−3.98681	−0.293116
$186$	0	0
$187$	32.2313	2.35698
$188$	12.3954	0.904029
$189$	0	0
$190$	2.25584	0.163656
$191$	−10.3347	−0.747796	−0.373898	−	0.927470i	$-0.621979\pi$
$191$	−10.3347	−0.747796	−0.373898	+	0.927470i	$0.621979\pi$
$192$	0	0
$193$	−26.5768	−1.91304	−0.956519	−	0.291671i	$-0.905789\pi$
$193$	−26.5768	−1.91304	−0.956519	+	0.291671i	$0.905789\pi$
$194$	−1.25681	−0.0902340
$195$	0	0
$196$	−1.66452	−0.118894
$197$	−14.1163	−1.00574	−0.502872	−	0.864361i	$-0.667723\pi$
$197$	−14.1163	−1.00574	−0.502872	+	0.864361i	$0.667723\pi$
$198$	0	0
$199$	−10.1775	−0.721462	−0.360731	−	0.932670i	$-0.617473\pi$
$199$	−10.1775	−0.721462	−0.360731	+	0.932670i	$0.617473\pi$
$200$	−7.90537	−0.558994
$201$	0	0
$202$	−8.88124	−0.624882
$203$	−0.885951	−0.0621816
$204$	0	0
$205$	−10.9721	−0.766324
$206$	8.88389	0.618970
$207$	0	0
$208$	−8.68950	−0.602509
$209$	17.7008	1.22439
$210$	0	0
$211$	−1.88806	−0.129979	−0.0649896	−	0.997886i	$-0.520701\pi$
$211$	−1.88806	−0.129979	−0.0649896	+	0.997886i	$0.520701\pi$
$212$	−21.3872	−1.46888
$213$	0	0
$214$	−9.59238	−0.655721
$215$	−9.07152	−0.618672
$216$	0	0
$217$	−3.36249	−0.228261
$218$	11.3883	0.771311
$219$	0	0
$220$	9.64901	0.650536
$221$	25.9877	1.74812
$222$	0	0
$223$	−18.6537	−1.24914	−0.624571	−	0.780968i	$-0.714726\pi$
$223$	−18.6537	−1.24914	−0.624571	+	0.780968i	$0.714726\pi$
$224$	5.46118	0.364890
$225$	0	0
$226$	6.15748	0.409590
$227$	−0.968602	−0.0642884	−0.0321442	−	0.999483i	$-0.510234\pi$
$227$	−0.968602	−0.0642884	−0.0321442	+	0.999483i	$0.510234\pi$
$228$	0	0
$229$	21.5925	1.42687	0.713435	−	0.700722i	$-0.247139\pi$
$229$	21.5925	1.42687	0.713435	+	0.700722i	$0.247139\pi$
$230$	1.82983	0.120656
$231$	0	0
$232$	1.88045	0.123458
$233$	−12.3536	−0.809313	−0.404656	−	0.914469i	$-0.632609\pi$
$233$	−12.3536	−0.809313	−0.404656	+	0.914469i	$0.632609\pi$
$234$	0	0
$235$	−8.41028	−0.548626
$236$	10.0587	0.654765
$237$	0	0
$238$	−3.63710	−0.235758
$239$	−12.2491	−0.792329	−0.396165	−	0.918180i	$-0.629659\pi$
$239$	−12.2491	−0.792329	−0.396165	+	0.918180i	$0.629659\pi$
$240$	0	0
$241$	−23.6646	−1.52437	−0.762185	−	0.647360i	$-0.775873\pi$
$241$	−23.6646	−1.52437	−0.762185	+	0.647360i	$0.775873\pi$
$242$	−8.88852	−0.571376
$243$	0	0
$244$	−7.04287	−0.450873
$245$	1.12937	0.0721530
$246$	0	0
$247$	14.2720	0.908104
$248$	7.13696	0.453198
$249$	0	0
$250$	5.70709	0.360948
$251$	−19.5387	−1.23327	−0.616637	−	0.787248i	$-0.711505\pi$
$251$	−19.5387	−1.23327	−0.616637	+	0.787248i	$0.711505\pi$
$252$	0	0
$253$	14.3581	0.902683
$254$	0.579209	0.0363428
$255$	0	0
$256$	−4.60167	−0.287605
$257$	13.5965	0.848125	0.424063	−	0.905633i	$-0.360604\pi$
$257$	13.5965	0.848125	0.424063	+	0.905633i	$0.360604\pi$
$258$	0	0
$259$	3.53011	0.219350
$260$	7.77990	0.482489
$261$	0	0
$262$	−10.1161	−0.624977
$263$	−11.9585	−0.737392	−0.368696	−	0.929550i	$-0.620196\pi$
$263$	−11.9585	−0.737392	−0.368696	+	0.929550i	$0.620196\pi$
$264$	0	0
$265$	14.5112	0.891417
$266$	−1.99743	−0.122470
$267$	0	0
$268$	17.2961	1.05653
$269$	−9.19379	−0.560555	−0.280278	−	0.959919i	$-0.590427\pi$
$269$	−9.19379	−0.560555	−0.280278	+	0.959919i	$0.590427\pi$
$270$	0	0
$271$	25.5228	1.55040	0.775199	−	0.631717i	$-0.217650\pi$
$271$	25.5228	1.55040	0.775199	+	0.631717i	$0.217650\pi$
$272$	−13.1846	−0.799434
$273$	0	0
$274$	4.86121	0.293676
$275$	19.1173	1.15282
$276$	0	0
$277$	−26.8759	−1.61482	−0.807408	−	0.589994i	$-0.799130\pi$
$277$	−26.8759	−1.61482	−0.807408	+	0.589994i	$0.799130\pi$
$278$	−7.05901	−0.423371
$279$	0	0
$280$	−2.39712	−0.143255
$281$	5.80878	0.346523	0.173261	−	0.984876i	$-0.444569\pi$
$281$	5.80878	0.346523	0.173261	+	0.984876i	$0.444569\pi$
$282$	0	0
$283$	10.3065	0.612656	0.306328	−	0.951926i	$-0.400900\pi$
$283$	10.3065	0.612656	0.306328	+	0.951926i	$0.400900\pi$
$284$	−18.5886	−1.10303
$285$	0	0
$286$	−12.3038	−0.727541
$287$	9.71520	0.573470
$288$	0	0
$289$	22.4313	1.31949
$290$	−0.579539	−0.0340317
$291$	0	0
$292$	−4.75618	−0.278334
$293$	−27.7789	−1.62286	−0.811431	−	0.584448i	$-0.801311\pi$
$293$	−27.7789	−1.62286	−0.811431	+	0.584448i	$0.801311\pi$
$294$	0	0
$295$	−6.82482	−0.397356
$296$	−7.49273	−0.435506
$297$	0	0
$298$	−8.67391	−0.502466
$299$	11.5768	0.669501
$300$	0	0
$301$	8.03235	0.462977
$302$	−4.65164	−0.267672
$303$	0	0
$304$	−7.24073	−0.415285
$305$	4.77858	0.273621
$306$	0	0
$307$	−4.68696	−0.267499	−0.133749	−	0.991015i	$-0.542702\pi$
$307$	−4.68696	−0.267499	−0.133749	+	0.991015i	$0.542702\pi$
$308$	−8.54368	−0.486822
$309$	0	0
$310$	−2.19955	−0.124926
$311$	20.1141	1.14057	0.570284	−	0.821448i	$-0.306833\pi$
$311$	20.1141	1.14057	0.570284	+	0.821448i	$0.306833\pi$
$312$	0	0
$313$	6.99687	0.395487	0.197743	−	0.980254i	$-0.436639\pi$
$313$	6.99687	0.395487	0.197743	+	0.980254i	$0.436639\pi$
$314$	2.29688	0.129620
$315$	0	0
$316$	5.39249	0.303351
$317$	25.4086	1.42709	0.713545	−	0.700609i	$-0.247088\pi$
$317$	25.4086	1.42709	0.713545	+	0.700609i	$0.247088\pi$
$318$	0	0
$319$	−4.54744	−0.254608
$320$	−1.17018	−0.0654150
$321$	0	0
$322$	−1.62022	−0.0902914
$323$	21.6549	1.20491
$324$	0	0
$325$	15.4141	0.855020
$326$	−8.56970	−0.474632
$327$	0	0
$328$	−20.6207	−1.13859
$329$	7.44686	0.410558
$330$	0	0
$331$	16.5654	0.910518	0.455259	−	0.890359i	$-0.349547\pi$
$331$	16.5654	0.910518	0.455259	+	0.890359i	$0.349547\pi$
$332$	−25.4787	−1.39832
$333$	0	0
$334$	4.05171	0.221700
$335$	−11.7354	−0.641173
$336$	0	0
$337$	−32.0698	−1.74696	−0.873478	−	0.486864i	$-0.838141\pi$
$337$	−32.0698	−1.74696	−0.873478	+	0.486864i	$0.838141\pi$
$338$	−2.39074	−0.130039
$339$	0	0
$340$	11.8045	0.640186
$341$	−17.2591	−0.934633
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−17.0488	−0.919211
$345$	0	0
$346$	−5.95885	−0.320350
$347$	21.1905	1.13756	0.568782	−	0.822489i	$-0.307415\pi$
$347$	21.1905	1.13756	0.568782	+	0.822489i	$0.307415\pi$
$348$	0	0
$349$	6.70223	0.358762	0.179381	−	0.983780i	$-0.442591\pi$
$349$	6.70223	0.358762	0.179381	+	0.983780i	$0.442591\pi$
$350$	−2.15727	−0.115311
$351$	0	0
$352$	28.0313	1.49407
$353$	24.7878	1.31932	0.659662	−	0.751563i	$-0.270700\pi$
$353$	24.7878	1.31932	0.659662	+	0.751563i	$0.270700\pi$
$354$	0	0
$355$	12.6123	0.669394
$356$	12.1216	0.642444
$357$	0	0
$358$	−1.14709	−0.0606258
$359$	−5.17658	−0.273209	−0.136605	−	0.990626i	$-0.543619\pi$
$359$	−5.17658	−0.273209	−0.136605	+	0.990626i	$0.543619\pi$
$360$	0	0
$361$	−7.10754	−0.374081
$362$	5.15399	0.270888
$363$	0	0
$364$	−6.88869	−0.361065
$365$	3.22706	0.168912
$366$	0	0
$367$	7.83880	0.409182	0.204591	−	0.978848i	$-0.434414\pi$
$367$	7.83880	0.409182	0.204591	+	0.978848i	$0.434414\pi$
$368$	−5.87334	−0.306169
$369$	0	0
$370$	2.30920	0.120049
$371$	−12.8489	−0.667082
$372$	0	0
$373$	−14.8126	−0.766967	−0.383483	−	0.923548i	$-0.625276\pi$
$373$	−14.8126	−0.766967	−0.383483	+	0.923548i	$0.625276\pi$
$374$	−18.6686	−0.965333
$375$	0	0
$376$	−15.8061	−0.815138
$377$	−3.66655	−0.188837
$378$	0	0
$379$	−2.85017	−0.146403	−0.0732017	−	0.997317i	$-0.523322\pi$
$379$	−2.85017	−0.146403	−0.0732017	+	0.997317i	$0.523322\pi$
$380$	6.48279	0.332560
$381$	0	0
$382$	5.98598	0.306269
$383$	−10.7694	−0.550293	−0.275147	−	0.961402i	$-0.588726\pi$
$383$	−10.7694	−0.550293	−0.275147	+	0.961402i	$0.588726\pi$
$384$	0	0
$385$	5.79688	0.295437
$386$	15.3935	0.783509
$387$	0	0
$388$	−3.61180	−0.183361
$389$	23.4878	1.19088	0.595440	−	0.803400i	$-0.296978\pi$
$389$	23.4878	1.19088	0.595440	+	0.803400i	$0.296978\pi$
$390$	0	0
$391$	17.5654	0.888322
$392$	2.12252	0.107204
$393$	0	0
$394$	8.17629	0.411916
$395$	−3.65880	−0.184094
$396$	0	0
$397$	−8.94150	−0.448761	−0.224381	−	0.974502i	$-0.572036\pi$
$397$	−8.94150	−0.448761	−0.224381	+	0.974502i	$0.572036\pi$
$398$	5.89489	0.295484
$399$	0	0
$400$	−7.82018	−0.391009
$401$	−2.98205	−0.148917	−0.0744583	−	0.997224i	$-0.523723\pi$
$401$	−2.98205	−0.148917	−0.0744583	+	0.997224i	$0.523723\pi$
$402$	0	0
$403$	−13.9158	−0.693197
$404$	−25.5227	−1.26980
$405$	0	0
$406$	0.513151	0.0254673
$407$	18.1194	0.898147
$408$	0	0
$409$	−5.61097	−0.277445	−0.138722	−	0.990331i	$-0.544300\pi$
$409$	−5.61097	−0.277445	−0.138722	+	0.990331i	$0.544300\pi$
$410$	6.35514	0.313858
$411$	0	0
$412$	25.5303	1.25779
$413$	6.04301	0.297357
$414$	0	0
$415$	17.2873	0.848598
$416$	22.6014	1.10812
$417$	0	0
$418$	−10.2525	−0.501465
$419$	−30.7222	−1.50088	−0.750438	−	0.660941i	$-0.770158\pi$
$419$	−30.7222	−1.50088	−0.750438	+	0.660941i	$0.770158\pi$
$420$	0	0
$421$	−5.94793	−0.289884	−0.144942	−	0.989440i	$-0.546300\pi$
$421$	−5.94793	−0.289884	−0.144942	+	0.989440i	$0.546300\pi$
$422$	1.09358	0.0532346
$423$	0	0
$424$	27.2721	1.32445
$425$	23.3878	1.13448
$426$	0	0
$427$	−4.23118	−0.204761
$428$	−27.5663	−1.33247
$429$	0	0
$430$	5.25431	0.253385
$431$	10.9229	0.526139	0.263070	−	0.964777i	$-0.415265\pi$
$431$	10.9229	0.526139	0.263070	+	0.964777i	$0.415265\pi$
$432$	0	0
$433$	5.90994	0.284014	0.142007	−	0.989866i	$-0.454645\pi$
$433$	5.90994	0.284014	0.142007	+	0.989866i	$0.454645\pi$
$434$	1.94759	0.0934871
$435$	0	0
$436$	32.7273	1.56736
$437$	9.64661	0.461460
$438$	0	0
$439$	−34.1527	−1.63002	−0.815011	−	0.579446i	$-0.803269\pi$
$439$	−34.1527	−1.63002	−0.815011	+	0.579446i	$0.803269\pi$
$440$	−12.3040	−0.586571
$441$	0	0
$442$	−15.0523	−0.715967
$443$	−19.9377	−0.947269	−0.473635	−	0.880721i	$-0.657058\pi$
$443$	−19.9377	−0.947269	−0.473635	+	0.880721i	$0.657058\pi$
$444$	0	0
$445$	−8.22450	−0.389879
$446$	10.8044	0.511602
$447$	0	0
$448$	1.03613	0.0489526
$449$	30.1418	1.42248	0.711239	−	0.702950i	$-0.248134\pi$
$449$	30.1418	1.42248	0.711239	+	0.702950i	$0.248134\pi$
$450$	0	0
$451$	49.8665	2.34812
$452$	17.6952	0.832313
$453$	0	0
$454$	0.561023	0.0263301
$455$	4.67397	0.219119
$456$	0	0
$457$	34.8120	1.62844	0.814219	−	0.580558i	$-0.197166\pi$
$457$	34.8120	1.62844	0.814219	+	0.580558i	$0.197166\pi$
$458$	−12.5065	−0.584393
$459$	0	0
$460$	5.25853	0.245180
$461$	−1.41335	−0.0658261	−0.0329131	−	0.999458i	$-0.510478\pi$
$461$	−1.41335	−0.0658261	−0.0329131	+	0.999458i	$0.510478\pi$
$462$	0	0
$463$	−29.7433	−1.38229	−0.691144	−	0.722717i	$-0.742893\pi$
$463$	−29.7433	−1.38229	−0.691144	+	0.722717i	$0.742893\pi$
$464$	1.86019	0.0863570
$465$	0	0
$466$	7.15533	0.331464
$467$	−17.2795	−0.799598	−0.399799	−	0.916603i	$-0.630920\pi$
$467$	−17.2795	−0.799598	−0.399799	+	0.916603i	$0.630920\pi$
$468$	0	0
$469$	10.3911	0.479815
$470$	4.87131	0.224697
$471$	0	0
$472$	−12.8264	−0.590384
$473$	41.2287	1.89570
$474$	0	0
$475$	12.8442	0.589331
$476$	−10.4522	−0.479077
$477$	0	0
$478$	7.09480	0.324509
$479$	27.5827	1.26029	0.630144	−	0.776479i	$-0.282996\pi$
$479$	27.5827	1.26029	0.630144	+	0.776479i	$0.282996\pi$
$480$	0	0
$481$	14.6095	0.666137
$482$	13.7067	0.624325
$483$	0	0
$484$	−25.5436	−1.16107
$485$	2.45060	0.111276
$486$	0	0
$487$	10.9254	0.495077	0.247538	−	0.968878i	$-0.420378\pi$
$487$	10.9254	0.495077	0.247538	+	0.968878i	$0.420378\pi$
$488$	8.98077	0.406540
$489$	0	0
$490$	−0.654144	−0.0295512
$491$	10.4127	0.469919	0.234959	−	0.972005i	$-0.424504\pi$
$491$	10.4127	0.469919	0.234959	+	0.972005i	$0.424504\pi$
$492$	0	0
$493$	−5.56327	−0.250557
$494$	−8.26646	−0.371926
$495$	0	0
$496$	7.06005	0.317006
$497$	−11.1676	−0.500934
$498$	0	0
$499$	24.9461	1.11674	0.558371	−	0.829591i	$-0.311427\pi$
$499$	24.9461	1.11674	0.558371	+	0.829591i	$0.311427\pi$
$500$	16.4009	0.733469
$501$	0	0
$502$	11.3170	0.505103
$503$	13.7735	0.614129	0.307065	−	0.951689i	$-0.400653\pi$
$503$	13.7735	0.614129	0.307065	+	0.951689i	$0.400653\pi$
$504$	0	0
$505$	17.3171	0.770601
$506$	−8.31632	−0.369706
$507$	0	0
$508$	1.66452	0.0738510
$509$	−10.9329	−0.484591	−0.242296	−	0.970202i	$-0.577900\pi$
$509$	−10.9329	−0.484591	−0.242296	+	0.970202i	$0.577900\pi$
$510$	0	0
$511$	−2.85739	−0.126404
$512$	−20.3797	−0.900662
$513$	0	0
$514$	−7.87521	−0.347361
$515$	−17.3223	−0.763311
$516$	0	0
$517$	38.2235	1.68107
$518$	−2.04467	−0.0898376
$519$	0	0
$520$	−9.92060	−0.435047
$521$	20.8068	0.911562	0.455781	−	0.890092i	$-0.349360\pi$
$521$	20.8068	0.911562	0.455781	+	0.890092i	$0.349360\pi$
$522$	0	0
$523$	−11.2355	−0.491295	−0.245648	−	0.969359i	$-0.579001\pi$
$523$	−11.2355	−0.491295	−0.245648	+	0.969359i	$0.579001\pi$
$524$	−29.0715	−1.27000
$525$	0	0
$526$	6.92646	0.302008
$527$	−21.1145	−0.919763
$528$	0	0
$529$	−15.1751	−0.659788
$530$	−8.40503	−0.365091
$531$	0	0
$532$	−5.74016	−0.248868
$533$	40.2068	1.74155
$534$	0	0
$535$	18.7037	0.808633
$536$	−22.0553	−0.952642
$537$	0	0
$538$	5.32513	0.229583
$539$	−5.13283	−0.221087
$540$	0	0
$541$	30.0869	1.29354	0.646768	−	0.762687i	$-0.276120\pi$
$541$	30.0869	1.29354	0.646768	+	0.762687i	$0.276120\pi$
$542$	−14.7830	−0.634985
$543$	0	0
$544$	34.2931	1.47030
$545$	−22.2055	−0.951178
$546$	0	0
$547$	15.6586	0.669515	0.334758	−	0.942304i	$-0.391346\pi$
$547$	15.6586	0.669515	0.334758	+	0.942304i	$0.391346\pi$
$548$	13.9700	0.596769
$549$	0	0
$550$	−11.0729	−0.472151
$551$	−3.05524	−0.130158
$552$	0	0
$553$	3.23967	0.137765
$554$	15.5668	0.661368
$555$	0	0
$556$	−20.2860	−0.860318
$557$	−15.1041	−0.639980	−0.319990	−	0.947421i	$-0.603680\pi$
$557$	−15.1041	−0.639980	−0.319990	+	0.947421i	$0.603680\pi$
$558$	0	0
$559$	33.2423	1.40600
$560$	−2.37129	−0.100205
$561$	0	0
$562$	−3.36450	−0.141923
$563$	−14.1971	−0.598334	−0.299167	−	0.954201i	$-0.596709\pi$
$563$	−14.1971	−0.598334	−0.299167	+	0.954201i	$0.596709\pi$
$564$	0	0
$565$	−12.0062	−0.505104
$566$	−5.96960	−0.250921
$567$	0	0
$568$	23.7034	0.994572
$569$	−6.62014	−0.277531	−0.138765	−	0.990325i	$-0.544313\pi$
$569$	−6.62014	−0.277531	−0.138765	+	0.990325i	$0.544313\pi$
$570$	0	0
$571$	19.0461	0.797055	0.398527	−	0.917156i	$-0.369521\pi$
$571$	19.0461	0.797055	0.398527	+	0.917156i	$0.369521\pi$
$572$	−35.3585	−1.47841
$573$	0	0
$574$	−5.62713	−0.234872
$575$	10.4186	0.434485
$576$	0	0
$577$	45.2822	1.88512	0.942562	−	0.334032i	$-0.108409\pi$
$577$	45.2822	1.88512	0.942562	+	0.334032i	$0.108409\pi$
$578$	−12.9924	−0.540412
$579$	0	0
$580$	−1.66546	−0.0691547
$581$	−15.3069	−0.635039
$582$	0	0
$583$	−65.9513	−2.73142
$584$	6.06488	0.250967
$585$	0	0
$586$	16.0898	0.664664
$587$	40.9606	1.69062	0.845312	−	0.534273i	$-0.179415\pi$
$587$	40.9606	1.69062	0.845312	+	0.534273i	$0.179415\pi$
$588$	0	0
$589$	−11.5957	−0.477793
$590$	3.95300	0.162742
$591$	0	0
$592$	−7.41198	−0.304631
$593$	26.4038	1.08427	0.542137	−	0.840290i	$-0.317615\pi$
$593$	26.4038	1.08427	0.542137	+	0.840290i	$0.317615\pi$
$594$	0	0
$595$	7.09182	0.290736
$596$	−24.9269	−1.02104
$597$	0	0
$598$	−6.70536	−0.274203
$599$	−26.6019	−1.08692	−0.543462	−	0.839433i	$-0.682887\pi$
$599$	−26.6019	−1.08692	−0.543462	+	0.839433i	$0.682887\pi$
$600$	0	0
$601$	−7.81198	−0.318657	−0.159329	−	0.987226i	$-0.550933\pi$
$601$	−7.81198	−0.318657	−0.159329	+	0.987226i	$0.550933\pi$
$602$	−4.65241	−0.189618
$603$	0	0
$604$	−13.3678	−0.543927
$605$	17.3313	0.704619
$606$	0	0
$607$	−33.3766	−1.35471	−0.677357	−	0.735655i	$-0.736875\pi$
$607$	−33.3766	−1.35471	−0.677357	+	0.735655i	$0.736875\pi$
$608$	18.8331	0.763784
$609$	0	0
$610$	−2.76780	−0.112065
$611$	30.8192	1.24681
$612$	0	0
$613$	−18.6564	−0.753524	−0.376762	−	0.926310i	$-0.622962\pi$
$613$	−18.6564	−0.753524	−0.376762	+	0.926310i	$0.622962\pi$
$614$	2.71473	0.109558
$615$	0	0
$616$	10.8945	0.438954
$617$	−11.4262	−0.460003	−0.230002	−	0.973190i	$-0.573873\pi$
$617$	−11.4262	−0.460003	−0.230002	+	0.973190i	$0.573873\pi$
$618$	0	0
$619$	−36.3702	−1.46184	−0.730921	−	0.682462i	$-0.760909\pi$
$619$	−36.3702	−1.46184	−0.730921	+	0.682462i	$0.760909\pi$
$620$	−6.32102	−0.253858
$621$	0	0
$622$	−11.6503	−0.467134
$623$	7.28236	0.291762
$624$	0	0
$625$	7.49460	0.299784
$626$	−4.05265	−0.161977
$627$	0	0
$628$	6.60071	0.263397
$629$	22.1671	0.883858
$630$	0	0
$631$	35.1539	1.39945	0.699727	−	0.714410i	$-0.253305\pi$
$631$	35.1539	1.39945	0.699727	+	0.714410i	$0.253305\pi$
$632$	−6.87628	−0.273524
$633$	0	0
$634$	−14.7169	−0.584483
$635$	−1.12937	−0.0448178
$636$	0	0
$637$	−4.13855	−0.163975
$638$	2.63392	0.104278
$639$	0	0
$640$	13.0132	0.514392
$641$	32.4678	1.28240	0.641201	−	0.767373i	$-0.278437\pi$
$641$	32.4678	1.28240	0.641201	+	0.767373i	$0.278437\pi$
$642$	0	0
$643$	22.7793	0.898328	0.449164	−	0.893449i	$-0.351722\pi$
$643$	22.7793	0.898328	0.449164	+	0.893449i	$0.351722\pi$
$644$	−4.65615	−0.183478
$645$	0	0
$646$	−12.5427	−0.493487
$647$	47.2544	1.85776	0.928881	−	0.370379i	$-0.120772\pi$
$647$	47.2544	1.85776	0.928881	+	0.370379i	$0.120772\pi$
$648$	0	0
$649$	31.0178	1.21755
$650$	−8.92799	−0.350184
$651$	0	0
$652$	−24.6274	−0.964483
$653$	2.80371	0.109718	0.0548588	−	0.998494i	$-0.482529\pi$
$653$	2.80371	0.109718	0.0548588	+	0.998494i	$0.482529\pi$
$654$	0	0
$655$	19.7250	0.770720
$656$	−20.3985	−0.796428
$657$	0	0
$658$	−4.31329	−0.168149
$659$	35.1669	1.36991	0.684955	−	0.728586i	$-0.259822\pi$
$659$	35.1669	1.36991	0.684955	+	0.728586i	$0.259822\pi$
$660$	0	0
$661$	−30.7793	−1.19718	−0.598589	−	0.801057i	$-0.704272\pi$
$661$	−30.7793	−1.19718	−0.598589	+	0.801057i	$0.704272\pi$
$662$	−9.59484	−0.372914
$663$	0	0
$664$	32.4893	1.26083
$665$	3.89470	0.151030
$666$	0	0
$667$	−2.47827	−0.0959590
$668$	11.6437	0.450508
$669$	0	0
$670$	6.79725	0.262601
$671$	−21.7179	−0.838411
$672$	0	0
$673$	9.16788	0.353396	0.176698	−	0.984265i	$-0.443458\pi$
$673$	9.16788	0.353396	0.176698	+	0.984265i	$0.443458\pi$
$674$	18.5752	0.715488
$675$	0	0
$676$	−6.87046	−0.264248
$677$	28.5471	1.09715	0.548577	−	0.836100i	$-0.315170\pi$
$677$	28.5471	1.09715	0.548577	+	0.836100i	$0.315170\pi$
$678$	0	0
$679$	−2.16988	−0.0832723
$680$	−15.0525	−0.577239
$681$	0	0
$682$	9.99663	0.382791
$683$	−24.3416	−0.931405	−0.465703	−	0.884941i	$-0.654198\pi$
$683$	−24.3416	−0.931405	−0.465703	+	0.884941i	$0.654198\pi$
$684$	0	0
$685$	−9.47864	−0.362160
$686$	0.579209	0.0221143
$687$	0	0
$688$	−16.8651	−0.642976
$689$	−53.1759	−2.02584
$690$	0	0
$691$	−3.77547	−0.143626	−0.0718128	−	0.997418i	$-0.522878\pi$
$691$	−3.77547	−0.143626	−0.0718128	+	0.997418i	$0.522878\pi$
$692$	−17.1244	−0.650972
$693$	0	0
$694$	−12.2737	−0.465903
$695$	13.7640	0.522100
$696$	0	0
$697$	61.0059	2.31076
$698$	−3.88199	−0.146936
$699$	0	0
$700$	−6.19952	−0.234320
$701$	−10.9687	−0.414284	−0.207142	−	0.978311i	$-0.566416\pi$
$701$	−10.9687	−0.414284	−0.207142	+	0.978311i	$0.566416\pi$
$702$	0	0
$703$	12.1737	0.459141
$704$	5.31829	0.200441
$705$	0	0
$706$	−14.3573	−0.540346
$707$	−15.3334	−0.576671
$708$	0	0
$709$	−30.3975	−1.14160	−0.570802	−	0.821088i	$-0.693367\pi$
$709$	−30.3975	−1.14160	−0.570802	+	0.821088i	$0.693367\pi$
$710$	−7.30519	−0.274159
$711$	0	0
$712$	−15.4570	−0.579274
$713$	−9.40589	−0.352253
$714$	0	0
$715$	23.9907	0.897201
$716$	−3.29649	−0.123196
$717$	0	0
$718$	2.99832	0.111896
$719$	−7.47705	−0.278847	−0.139423	−	0.990233i	$-0.544525\pi$
$719$	−7.47705	−0.278847	−0.139423	+	0.990233i	$0.544525\pi$
$720$	0	0
$721$	15.3380	0.571216
$722$	4.11675	0.153210
$723$	0	0
$724$	14.8114	0.550462
$725$	−3.29974	−0.122549
$726$	0	0
$727$	−30.1763	−1.11918	−0.559588	−	0.828771i	$-0.689041\pi$
$727$	−30.1763	−1.11918	−0.559588	+	0.828771i	$0.689041\pi$
$728$	8.78416	0.325563
$729$	0	0
$730$	−1.86915	−0.0691802
$731$	50.4386	1.86554
$732$	0	0
$733$	19.3325	0.714061	0.357030	−	0.934093i	$-0.383789\pi$
$733$	19.3325	0.714061	0.357030	+	0.934093i	$0.383789\pi$
$734$	−4.54030	−0.167586
$735$	0	0
$736$	15.2765	0.563101
$737$	53.3356	1.96464
$738$	0	0
$739$	35.3818	1.30154	0.650770	−	0.759275i	$-0.274446\pi$
$739$	35.3818	1.30154	0.650770	+	0.759275i	$0.274446\pi$
$740$	6.63611	0.243948
$741$	0	0
$742$	7.44221	0.273212
$743$	13.6795	0.501852	0.250926	−	0.968006i	$-0.419265\pi$
$743$	13.6795	0.501852	0.250926	+	0.968006i	$0.419265\pi$
$744$	0	0
$745$	16.9129	0.619639
$746$	8.57959	0.314121
$747$	0	0
$748$	−53.6495	−1.96162
$749$	−16.5612	−0.605132
$750$	0	0
$751$	−38.4578	−1.40334	−0.701672	−	0.712500i	$-0.747563\pi$
$751$	−38.4578	−1.40334	−0.701672	+	0.712500i	$0.747563\pi$
$752$	−15.6358	−0.570178
$753$	0	0
$754$	2.12370	0.0773406
$755$	9.07002	0.330092
$756$	0	0
$757$	−39.6721	−1.44191	−0.720954	−	0.692983i	$-0.756296\pi$
$757$	−39.6721	−1.44191	−0.720954	+	0.692983i	$0.756296\pi$
$758$	1.65085	0.0599614
$759$	0	0
$760$	−8.26658	−0.299860
$761$	1.20599	0.0437172	0.0218586	−	0.999761i	$-0.493042\pi$
$761$	1.20599	0.0437172	0.0218586	+	0.999761i	$0.493042\pi$
$762$	0	0
$763$	19.6618	0.711804
$764$	17.2024	0.622359
$765$	0	0
$766$	6.23777	0.225380
$767$	25.0093	0.903034
$768$	0	0
$769$	4.10301	0.147958	0.0739791	−	0.997260i	$-0.476430\pi$
$769$	4.10301	0.147958	0.0739791	+	0.997260i	$0.476430\pi$
$770$	−3.35761	−0.121000
$771$	0	0
$772$	44.2375	1.59214
$773$	−14.9068	−0.536159	−0.268080	−	0.963397i	$-0.586389\pi$
$773$	−14.9068	−0.536159	−0.268080	+	0.963397i	$0.586389\pi$
$774$	0	0
$775$	−12.5237	−0.449863
$776$	4.60562	0.165332
$777$	0	0
$778$	−13.6044	−0.487740
$779$	33.5033	1.20038
$780$	0	0
$781$	−57.3212	−2.05111
$782$	−10.1741	−0.363824
$783$	0	0
$784$	2.09965	0.0749875
$785$	−4.47858	−0.159847
$786$	0	0
$787$	−27.7332	−0.988582	−0.494291	−	0.869296i	$-0.664572\pi$
$787$	−27.7332	−0.988582	−0.494291	+	0.869296i	$0.664572\pi$
$788$	23.4968	0.837040
$789$	0	0
$790$	2.11921	0.0753982
$791$	10.6308	0.377989
$792$	0	0
$793$	−17.5109	−0.621832
$794$	5.17900	0.183796
$795$	0	0
$796$	16.9406	0.600443
$797$	10.3220	0.365624	0.182812	−	0.983148i	$-0.441480\pi$
$797$	10.3220	0.365624	0.182812	+	0.983148i	$0.441480\pi$
$798$	0	0
$799$	46.7620	1.65432
$800$	20.3403	0.719136
$801$	0	0
$802$	1.72723	0.0609907
$803$	−14.6665	−0.517570
$804$	0	0
$805$	3.15919	0.111347
$806$	8.06018	0.283908
$807$	0	0
$808$	32.5454	1.14494
$809$	42.8113	1.50517	0.752583	−	0.658497i	$-0.228808\pi$
$809$	42.8113	1.50517	0.752583	+	0.658497i	$0.228808\pi$
$810$	0	0
$811$	22.2947	0.782872	0.391436	−	0.920205i	$-0.371978\pi$
$811$	22.2947	0.782872	0.391436	+	0.920205i	$0.371978\pi$
$812$	1.47468	0.0517511
$813$	0	0
$814$	−10.4949	−0.367848
$815$	16.7097	0.585314
$816$	0	0
$817$	27.6999	0.969097
$818$	3.24993	0.113631
$819$	0	0
$820$	18.2632	0.637779
$821$	−32.3517	−1.12908	−0.564541	−	0.825405i	$-0.690947\pi$
$821$	−32.3517	−1.12908	−0.564541	+	0.825405i	$0.690947\pi$
$822$	0	0
$823$	32.6652	1.13864	0.569319	−	0.822117i	$-0.307207\pi$
$823$	32.6652	1.13864	0.569319	+	0.822117i	$0.307207\pi$
$824$	−32.5552	−1.13411
$825$	0	0
$826$	−3.50017	−0.121786
$827$	37.2899	1.29670	0.648348	−	0.761344i	$-0.275460\pi$
$827$	37.2899	1.29670	0.648348	+	0.761344i	$0.275460\pi$
$828$	0	0
$829$	−5.55728	−0.193012	−0.0965062	−	0.995332i	$-0.530767\pi$
$829$	−5.55728	−0.193012	−0.0965062	+	0.995332i	$0.530767\pi$
$830$	−10.0129	−0.347554
$831$	0	0
$832$	4.28808	0.148663
$833$	−6.27943	−0.217569
$834$	0	0
$835$	−7.90024	−0.273399
$836$	−29.4633	−1.01901
$837$	0	0
$838$	17.7946	0.614703
$839$	−36.4344	−1.25786	−0.628928	−	0.777464i	$-0.716506\pi$
$839$	−36.4344	−1.25786	−0.628928	+	0.777464i	$0.716506\pi$
$840$	0	0
$841$	−28.2151	−0.972934
$842$	3.44510	0.118726
$843$	0	0
$844$	3.14270	0.108176
$845$	4.66160	0.160364
$846$	0	0
$847$	−15.3460	−0.527294
$848$	26.9782	0.926435
$849$	0	0
$850$	−13.5465	−0.464639
$851$	9.87476	0.338502
$852$	0	0
$853$	10.7354	0.367574	0.183787	−	0.982966i	$-0.441164\pi$
$853$	10.7354	0.367574	0.183787	+	0.982966i	$0.441164\pi$
$854$	2.45074	0.0838625
$855$	0	0
$856$	35.1514	1.20145
$857$	−24.5647	−0.839113	−0.419556	−	0.907729i	$-0.637814\pi$
$857$	−24.5647	−0.839113	−0.419556	+	0.907729i	$0.637814\pi$
$858$	0	0
$859$	−8.19427	−0.279585	−0.139792	−	0.990181i	$-0.544644\pi$
$859$	−8.19427	−0.279585	−0.139792	+	0.990181i	$0.544644\pi$
$860$	15.0997	0.514895
$861$	0	0
$862$	−6.32667	−0.215487
$863$	44.0732	1.50027	0.750134	−	0.661286i	$-0.229989\pi$
$863$	44.0732	1.50027	0.750134	+	0.661286i	$0.229989\pi$
$864$	0	0
$865$	11.6189	0.395054
$866$	−3.42309	−0.116321
$867$	0	0
$868$	5.59692	0.189972
$869$	16.6287	0.564090
$870$	0	0
$871$	43.0039	1.45713
$872$	−41.7325	−1.41324
$873$	0	0
$874$	−5.58740	−0.188997
$875$	9.85324	0.333100
$876$	0	0
$877$	−46.6339	−1.57472	−0.787358	−	0.616497i	$-0.788551\pi$
$877$	−46.6339	−1.57472	−0.787358	+	0.616497i	$0.788551\pi$
$878$	19.7816	0.667596
$879$	0	0
$880$	−12.1714	−0.410299
$881$	−36.7215	−1.23718	−0.618589	−	0.785715i	$-0.712295\pi$
$881$	−36.7215	−1.23718	−0.618589	+	0.785715i	$0.712295\pi$
$882$	0	0
$883$	19.6988	0.662917	0.331458	−	0.943470i	$-0.392459\pi$
$883$	19.6988	0.662917	0.331458	+	0.943470i	$0.392459\pi$
$884$	−43.2570	−1.45489
$885$	0	0
$886$	11.5481	0.387966
$887$	−43.5744	−1.46309	−0.731543	−	0.681795i	$-0.761199\pi$
$887$	−43.5744	−1.46309	−0.731543	+	0.681795i	$0.761199\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	4.76371	0.159680
$891$	0	0
$892$	31.0493	1.03961
$893$	25.6808	0.859376
$894$	0	0
$895$	2.23667	0.0747635
$896$	−11.5225	−0.384940
$897$	0	0
$898$	−17.4584	−0.582594
$899$	2.97900	0.0993553
$900$	0	0
$901$	−80.6839	−2.68797
$902$	−28.8831	−0.961703
$903$	0	0
$904$	−22.5642	−0.750474
$905$	−10.0495	−0.334058
$906$	0	0
$907$	21.6583	0.719153	0.359576	−	0.933116i	$-0.382921\pi$
$907$	21.6583	0.719153	0.359576	+	0.933116i	$0.382921\pi$
$908$	1.61225	0.0535045
$909$	0	0
$910$	−2.70721	−0.0897430
$911$	−43.1069	−1.42820	−0.714098	−	0.700046i	$-0.753163\pi$
$911$	−43.1069	−1.42820	−0.714098	+	0.700046i	$0.753163\pi$
$912$	0	0
$913$	−78.5680	−2.60022
$914$	−20.1634	−0.666947
$915$	0	0
$916$	−35.9410	−1.18752
$917$	−17.4654	−0.576760
$918$	0	0
$919$	6.73107	0.222037	0.111019	−	0.993818i	$-0.464589\pi$
$919$	6.73107	0.222037	0.111019	+	0.993818i	$0.464589\pi$
$920$	−6.70546	−0.221072
$921$	0	0
$922$	0.818624	0.0269599
$923$	−46.2175	−1.52127
$924$	0	0
$925$	13.1479	0.432301
$926$	17.2276	0.566133
$927$	0	0
$928$	−4.83834	−0.158826
$929$	45.7005	1.49939	0.749693	−	0.661786i	$-0.230201\pi$
$929$	45.7005	1.49939	0.749693	+	0.661786i	$0.230201\pi$
$930$	0	0
$931$	−3.44855	−0.113022
$932$	20.5628	0.673557
$933$	0	0
$934$	10.0084	0.327486
$935$	36.4011	1.19044
$936$	0	0
$937$	−39.9095	−1.30379	−0.651893	−	0.758311i	$-0.726025\pi$
$937$	−39.9095	−1.30379	−0.651893	+	0.758311i	$0.726025\pi$
$938$	−6.01860	−0.196514
$939$	0	0
$940$	13.9991	0.456599
$941$	−20.8011	−0.678095	−0.339048	−	0.940769i	$-0.610105\pi$
$941$	−20.8011	−0.678095	−0.339048	+	0.940769i	$0.610105\pi$
$942$	0	0
$943$	27.1763	0.884982
$944$	−12.6882	−0.412966
$945$	0	0
$946$	−23.8800	−0.776407
$947$	−9.56592	−0.310851	−0.155425	−	0.987848i	$-0.549675\pi$
$947$	−9.56592	−0.310851	−0.155425	+	0.987848i	$0.549675\pi$
$948$	0	0
$949$	−11.8255	−0.383871
$950$	−7.43946	−0.241368
$951$	0	0
$952$	13.3282	0.431970
$953$	−0.693898	−0.0224776	−0.0112388	−	0.999937i	$-0.503577\pi$
$953$	−0.693898	−0.0224776	−0.0112388	+	0.999937i	$0.503577\pi$
$954$	0	0
$955$	−11.6718	−0.377690
$956$	20.3889	0.659423
$957$	0	0
$958$	−15.9762	−0.516167
$959$	8.39283	0.271019
$960$	0	0
$961$	−19.6937	−0.635279
$962$	−8.46197	−0.272825
$963$	0	0
$964$	39.3901	1.26867
$965$	−30.0151	−0.966220
$966$	0	0
$967$	41.8670	1.34635	0.673176	−	0.739482i	$-0.264930\pi$
$967$	41.8670	1.34635	0.673176	+	0.739482i	$0.264930\pi$
$968$	32.5721	1.04691
$969$	0	0
$970$	−1.41941	−0.0455746
$971$	−35.5827	−1.14190	−0.570951	−	0.820984i	$-0.693426\pi$
$971$	−35.5827	−1.14190	−0.570951	+	0.820984i	$0.693426\pi$
$972$	0	0
$973$	−12.1873	−0.390708
$974$	−6.32809	−0.202765
$975$	0	0
$976$	8.88399	0.284370
$977$	−39.3467	−1.25881	−0.629406	−	0.777076i	$-0.716702\pi$
$977$	−39.3467	−1.25881	−0.629406	+	0.777076i	$0.716702\pi$
$978$	0	0
$979$	37.3791	1.19464
$980$	−1.87986	−0.0600500
$981$	0	0
$982$	−6.03114	−0.192461
$983$	−22.2577	−0.709909	−0.354954	−	0.934884i	$-0.615504\pi$
$983$	−22.2577	−0.709909	−0.354954	+	0.934884i	$0.615504\pi$
$984$	0	0
$985$	−15.9426	−0.507973
$986$	3.22230	0.102619
$987$	0	0
$988$	−23.7559	−0.755777
$989$	22.4689	0.714468
$990$	0	0
$991$	22.4055	0.711734	0.355867	−	0.934537i	$-0.384186\pi$
$991$	22.4055	0.711734	0.355867	+	0.934537i	$0.384186\pi$
$992$	−18.3632	−0.583031
$993$	0	0
$994$	6.46835	0.205164
$995$	−11.4942	−0.364390
$996$	0	0
$997$	16.5738	0.524899	0.262449	−	0.964946i	$-0.415470\pi$
$997$	16.5738	0.524899	0.262449	+	0.964946i	$0.415470\pi$
$998$	−14.4490	−0.457376
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.p.1.5		14
3.2	odd	2		2667.2.a.m.1.10	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.m.1.10	✓	14	3.2	odd	2
8001.2.a.p.1.5		14	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-0.579209 q^{2} -1.66452 q^{4} +1.12937 q^{5} -1.00000 q^{7} +2.12252 q^{8} +O(q^{10})\)	\(q-0.579209 q^{2} -1.66452 q^{4} +1.12937 q^{5} -1.00000 q^{7} +2.12252 q^{8} -0.654144 q^{10} -5.13283 q^{11} -4.13855 q^{13} +0.579209 q^{14} +2.09965 q^{16} -6.27943 q^{17} -3.44855 q^{19} -1.87986 q^{20} +2.97298 q^{22} -2.79730 q^{23} -3.72452 q^{25} +2.39709 q^{26} +1.66452 q^{28} +0.885951 q^{29} +3.36249 q^{31} -5.46118 q^{32} +3.63710 q^{34} -1.12937 q^{35} -3.53011 q^{37} +1.99743 q^{38} +2.39712 q^{40} -9.71520 q^{41} -8.03235 q^{43} +8.54368 q^{44} +1.62022 q^{46} -7.44686 q^{47} +1.00000 q^{49} +2.15727 q^{50} +6.88869 q^{52} +12.8489 q^{53} -5.79688 q^{55} -2.12252 q^{56} -0.513151 q^{58} -6.04301 q^{59} +4.23118 q^{61} -1.94759 q^{62} -1.03613 q^{64} -4.67397 q^{65} -10.3911 q^{67} +10.4522 q^{68} +0.654144 q^{70} +11.1676 q^{71} +2.85739 q^{73} +2.04467 q^{74} +5.74016 q^{76} +5.13283 q^{77} -3.23967 q^{79} +2.37129 q^{80} +5.62713 q^{82} +15.3069 q^{83} -7.09182 q^{85} +4.65241 q^{86} -10.8945 q^{88} -7.28236 q^{89} +4.13855 q^{91} +4.65615 q^{92} +4.31329 q^{94} -3.89470 q^{95} +2.16988 q^{97} -0.579209 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 5 q^{2} + 15 q^{4} + 4 q^{5} - 14 q^{7} + 12 q^{8}+O(q^{10}) \) 14 * q + 5 * q^2 + 15 * q^4 + 4 * q^5 - 14 * q^7 + 12 * q^8	\( 14 q + 5 q^{2} + 15 q^{4} + 4 q^{5} - 14 q^{7} + 12 q^{8} + 4 q^{10} + 3 q^{11} - 13 q^{13} - 5 q^{14} + 13 q^{16} + 5 q^{17} + 21 q^{19} + 3 q^{20} - 3 q^{22} + 10 q^{23} + 4 q^{25} + 6 q^{26} - 15 q^{28} + 15 q^{29} + 33 q^{31} + 29 q^{32} + 28 q^{34} - 4 q^{35} - 29 q^{37} + 15 q^{38} + 3 q^{40} + q^{41} - 25 q^{43} + 26 q^{44} - 4 q^{46} + 9 q^{47} + 14 q^{49} + 28 q^{50} - 13 q^{52} + 35 q^{53} + 14 q^{55} - 12 q^{56} - 23 q^{58} - 10 q^{59} + q^{61} + 43 q^{62} - 2 q^{64} + 24 q^{65} - 38 q^{67} + 2 q^{68} - 4 q^{70} + 10 q^{71} + 8 q^{73} + 25 q^{74} + 26 q^{76} - 3 q^{77} + 26 q^{79} + 48 q^{80} + 6 q^{82} + 30 q^{83} - 32 q^{85} + 50 q^{86} - 29 q^{88} - 4 q^{89} + 13 q^{91} + 32 q^{92} - 7 q^{94} + 32 q^{95} + 15 q^{97} + 5 q^{98}+O(q^{100}) \) 14 * q + 5 * q^2 + 15 * q^4 + 4 * q^5 - 14 * q^7 + 12 * q^8 + 4 * q^10 + 3 * q^11 - 13 * q^13 - 5 * q^14 + 13 * q^16 + 5 * q^17 + 21 * q^19 + 3 * q^20 - 3 * q^22 + 10 * q^23 + 4 * q^25 + 6 * q^26 - 15 * q^28 + 15 * q^29 + 33 * q^31 + 29 * q^32 + 28 * q^34 - 4 * q^35 - 29 * q^37 + 15 * q^38 + 3 * q^40 + q^41 - 25 * q^43 + 26 * q^44 - 4 * q^46 + 9 * q^47 + 14 * q^49 + 28 * q^50 - 13 * q^52 + 35 * q^53 + 14 * q^55 - 12 * q^56 - 23 * q^58 - 10 * q^59 + q^61 + 43 * q^62 - 2 * q^64 + 24 * q^65 - 38 * q^67 + 2 * q^68 - 4 * q^70 + 10 * q^71 + 8 * q^73 + 25 * q^74 + 26 * q^76 - 3 * q^77 + 26 * q^79 + 48 * q^80 + 6 * q^82 + 30 * q^83 - 32 * q^85 + 50 * q^86 - 29 * q^88 - 4 * q^89 + 13 * q^91 + 32 * q^92 - 7 * q^94 + 32 * q^95 + 15 * q^97 + 5 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

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Database

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