LMFDB - Embedded newform 8001.2.a.v.1.12

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:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - 4 x^{18} - 22 x^{17} + 101 x^{16} + 178 x^{15} - 1035 x^{14} - 583 x^{13} + 5572 x^{12} + \cdots + 210 $ x^19 - 4x^18 - 22x^17 + 101x^16 + 178x^15 - 1035x^14 - 583x^13 + 5572x^12 + 21x^11 - 17032x^10 + 4985x^9 + 29792x^8 - 13249x^7 - 28600x^6 + 14000x^5 + 13725x^4 - 5723x^3 - 2913x^2 + 608x + 210
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 2667)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$-0.407951$ 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)

$f(q)$	$=$	$q+0.407951 q^{2} -1.83358 q^{4} +1.08992 q^{5} +1.00000 q^{7} -1.56391 q^{8} +O(q^{10})$	\(q+0.407951 q^{2} -1.83358 q^{4} +1.08992 q^{5} +1.00000 q^{7} -1.56391 q^{8} +0.444635 q^{10} -2.30086 q^{11} -1.28207 q^{13} +0.407951 q^{14} +3.02915 q^{16} -3.85945 q^{17} -5.48075 q^{19} -1.99845 q^{20} -0.938637 q^{22} -5.15478 q^{23} -3.81207 q^{25} -0.523023 q^{26} -1.83358 q^{28} +3.35002 q^{29} -0.261843 q^{31} +4.36357 q^{32} -1.57447 q^{34} +1.08992 q^{35} +4.76674 q^{37} -2.23588 q^{38} -1.70454 q^{40} -4.53932 q^{41} +3.17008 q^{43} +4.21880 q^{44} -2.10290 q^{46} -0.491884 q^{47} +1.00000 q^{49} -1.55514 q^{50} +2.35078 q^{52} +11.3199 q^{53} -2.50775 q^{55} -1.56391 q^{56} +1.36664 q^{58} -1.96040 q^{59} +0.0291708 q^{61} -0.106819 q^{62} -4.27818 q^{64} -1.39736 q^{65} +13.7705 q^{67} +7.07659 q^{68} +0.444635 q^{70} -1.95435 q^{71} +10.9732 q^{73} +1.94460 q^{74} +10.0494 q^{76} -2.30086 q^{77} +1.31077 q^{79} +3.30154 q^{80} -1.85182 q^{82} +9.68361 q^{83} -4.20649 q^{85} +1.29324 q^{86} +3.59834 q^{88} -13.3023 q^{89} -1.28207 q^{91} +9.45169 q^{92} -0.200664 q^{94} -5.97359 q^{95} -2.44284 q^{97} +0.407951 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q - 4 q^{2} + 22 q^{4} - 5 q^{5} + 19 q^{7} - 9 q^{8}+O(q^{10}) $ 19 * q - 4 * q^2 + 22 * q^4 - 5 * q^5 + 19 * q^7 - 9 * q^8	$ 19 q - 4 q^{2} + 22 q^{4} - 5 q^{5} + 19 q^{7} - 9 q^{8} + 9 q^{11} + 24 q^{13} - 4 q^{14} + 20 q^{16} - 17 q^{17} + 23 q^{19} - 5 q^{20} - 3 q^{22} + 17 q^{23} + 38 q^{25} - 28 q^{26} + 22 q^{28} - 2 q^{29} + 16 q^{31} - 17 q^{32} + 29 q^{34} - 5 q^{35} + 56 q^{37} - 2 q^{38} - 13 q^{40} + 7 q^{41} + 19 q^{43} + 29 q^{44} + 10 q^{46} - 25 q^{47} + 19 q^{49} + 9 q^{50} + 16 q^{52} - 18 q^{53} + 10 q^{55} - 9 q^{56} + 31 q^{58} - 11 q^{59} + 26 q^{61} - 26 q^{62} + 45 q^{64} - 27 q^{65} + 24 q^{67} - 14 q^{68} + 32 q^{71} + 51 q^{73} + 12 q^{76} + 9 q^{77} + 30 q^{79} + 30 q^{80} - 52 q^{82} - q^{83} + 44 q^{85} + 24 q^{86} - 30 q^{88} - 5 q^{89} + 24 q^{91} + 88 q^{92} + 7 q^{94} + 24 q^{95} + 5 q^{97} - 4 q^{98}+O(q^{100}) $ 19 * q - 4 * q^2 + 22 * q^4 - 5 * q^5 + 19 * q^7 - 9 * q^8 + 9 * q^11 + 24 * q^13 - 4 * q^14 + 20 * q^16 - 17 * q^17 + 23 * q^19 - 5 * q^20 - 3 * q^22 + 17 * q^23 + 38 * q^25 - 28 * q^26 + 22 * q^28 - 2 * q^29 + 16 * q^31 - 17 * q^32 + 29 * q^34 - 5 * q^35 + 56 * q^37 - 2 * q^38 - 13 * q^40 + 7 * q^41 + 19 * q^43 + 29 * q^44 + 10 * q^46 - 25 * q^47 + 19 * q^49 + 9 * q^50 + 16 * q^52 - 18 * q^53 + 10 * q^55 - 9 * q^56 + 31 * q^58 - 11 * q^59 + 26 * q^61 - 26 * q^62 + 45 * q^64 - 27 * q^65 + 24 * q^67 - 14 * q^68 + 32 * q^71 + 51 * q^73 + 12 * q^76 + 9 * q^77 + 30 * q^79 + 30 * q^80 - 52 * q^82 - q^83 + 44 * q^85 + 24 * q^86 - 30 * q^88 - 5 * q^89 + 24 * q^91 + 88 * q^92 + 7 * q^94 + 24 * q^95 + 5 * q^97 - 4 * 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 $

$2$	0.407951	0.288465	0.144232	−	0.989544i	$-0.453929\pi$
$2$	0.407951	0.288465	0.144232	+	0.989544i	$0.453929\pi$
$3$	0	0
$3$	0	0
$4$	−1.83358	−0.916788
$5$	1.08992	0.487428	0.243714	−	0.969847i	$-0.421634\pi$
$5$	1.08992	0.487428	0.243714	+	0.969847i	$0.421634\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.56391	−0.552926
$9$	0	0
$10$	0.444635	0.140606
$11$	−2.30086	−0.693734	−0.346867	−	0.937914i	$-0.612755\pi$
$11$	−2.30086	−0.693734	−0.346867	+	0.937914i	$0.612755\pi$
$12$	0	0
$13$	−1.28207	−0.355583	−0.177791	−	0.984068i	$-0.556895\pi$
$13$	−1.28207	−0.355583	−0.177791	+	0.984068i	$0.556895\pi$
$14$	0.407951	0.109030
$15$	0	0
$16$	3.02915	0.757288
$17$	−3.85945	−0.936053	−0.468027	−	0.883714i	$-0.655035\pi$
$17$	−3.85945	−0.936053	−0.468027	+	0.883714i	$0.655035\pi$
$18$	0	0
$19$	−5.48075	−1.25737	−0.628685	−	0.777660i	$-0.716406\pi$
$19$	−5.48075	−1.25737	−0.628685	+	0.777660i	$0.716406\pi$
$20$	−1.99845	−0.446868
$21$	0	0
$22$	−0.938637	−0.200118
$23$	−5.15478	−1.07485	−0.537423	−	0.843313i	$-0.680602\pi$
$23$	−5.15478	−1.07485	−0.537423	+	0.843313i	$0.680602\pi$
$24$	0	0
$25$	−3.81207	−0.762414
$26$	−0.523023	−0.102573
$27$	0	0
$28$	−1.83358	−0.346513
$29$	3.35002	0.622082	0.311041	−	0.950396i	$-0.399322\pi$
$29$	3.35002	0.622082	0.311041	+	0.950396i	$0.399322\pi$
$30$	0	0
$31$	−0.261843	−0.0470284	−0.0235142	−	0.999724i	$-0.507485\pi$
$31$	−0.261843	−0.0470284	−0.0235142	+	0.999724i	$0.507485\pi$
$32$	4.36357	0.771377
$33$	0	0
$34$	−1.57447	−0.270019
$35$	1.08992	0.184230
$36$	0	0
$37$	4.76674	0.783648	0.391824	−	0.920040i	$-0.371844\pi$
$37$	4.76674	0.783648	0.391824	+	0.920040i	$0.371844\pi$
$38$	−2.23588	−0.362707
$39$	0	0
$40$	−1.70454	−0.269512
$41$	−4.53932	−0.708923	−0.354461	−	0.935071i	$-0.615336\pi$
$41$	−4.53932	−0.708923	−0.354461	+	0.935071i	$0.615336\pi$
$42$	0	0
$43$	3.17008	0.483433	0.241716	−	0.970347i	$-0.422290\pi$
$43$	3.17008	0.483433	0.241716	+	0.970347i	$0.422290\pi$
$44$	4.21880	0.636007
$45$	0	0
$46$	−2.10290	−0.310056
$47$	−0.491884	−0.0717486	−0.0358743	−	0.999356i	$-0.511422\pi$
$47$	−0.491884	−0.0717486	−0.0358743	+	0.999356i	$0.511422\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−1.55514	−0.219930
$51$	0	0
$52$	2.35078	0.325994
$53$	11.3199	1.55490	0.777452	−	0.628943i	$-0.216512\pi$
$53$	11.3199	1.55490	0.777452	+	0.628943i	$0.216512\pi$
$54$	0	0
$55$	−2.50775	−0.338145
$56$	−1.56391	−0.208986
$57$	0	0
$58$	1.36664	0.179449
$59$	−1.96040	−0.255222	−0.127611	−	0.991824i	$-0.540731\pi$
$59$	−1.96040	−0.255222	−0.127611	+	0.991824i	$0.540731\pi$
$60$	0	0
$61$	0.0291708	0.00373494	0.00186747	−	0.999998i	$-0.499406\pi$
$61$	0.0291708	0.00373494	0.00186747	+	0.999998i	$0.499406\pi$
$62$	−0.106819	−0.0135660
$63$	0	0
$64$	−4.27818	−0.534773
$65$	−1.39736	−0.173321
$66$	0	0
$67$	13.7705	1.68234	0.841169	−	0.540772i	$-0.181868\pi$
$67$	13.7705	1.68234	0.841169	+	0.540772i	$0.181868\pi$
$68$	7.07659	0.858162
$69$	0	0
$70$	0.444635	0.0531440
$71$	−1.95435	−0.231938	−0.115969	−	0.993253i	$-0.536997\pi$
$71$	−1.95435	−0.231938	−0.115969	+	0.993253i	$0.536997\pi$
$72$	0	0
$73$	10.9732	1.28431	0.642155	−	0.766575i	$-0.278040\pi$
$73$	10.9732	1.28431	0.642155	+	0.766575i	$0.278040\pi$
$74$	1.94460	0.226055
$75$	0	0
$76$	10.0494	1.15274
$77$	−2.30086	−0.262207
$78$	0	0
$79$	1.31077	0.147473	0.0737364	−	0.997278i	$-0.476508\pi$
$79$	1.31077	0.147473	0.0737364	+	0.997278i	$0.476508\pi$
$80$	3.30154	0.369123
$81$	0	0
$82$	−1.85182	−0.204499
$83$	9.68361	1.06291	0.531457	−	0.847085i	$-0.321645\pi$
$83$	9.68361	1.06291	0.531457	+	0.847085i	$0.321645\pi$
$84$	0	0
$85$	−4.20649	−0.456258
$86$	1.29324	0.139453
$87$	0	0
$88$	3.59834	0.383584
$89$	−13.3023	−1.41004	−0.705022	−	0.709186i	$-0.749063\pi$
$89$	−13.3023	−1.41004	−0.705022	+	0.709186i	$0.749063\pi$
$90$	0	0
$91$	−1.28207	−0.134398
$92$	9.45169	0.985406
$93$	0	0
$94$	−0.200664	−0.0206970
$95$	−5.97359	−0.612877
$96$	0	0
$97$	−2.44284	−0.248033	−0.124016	−	0.992280i	$-0.539578\pi$
$97$	−2.44284	−0.248033	−0.124016	+	0.992280i	$0.539578\pi$
$98$	0.407951	0.0412093
$99$	0	0
$100$	6.98972	0.698972
$101$	−16.2404	−1.61598	−0.807991	−	0.589195i	$-0.799445\pi$
$101$	−16.2404	−1.61598	−0.807991	+	0.589195i	$0.799445\pi$
$102$	0	0
$103$	16.7605	1.65146	0.825730	−	0.564066i	$-0.190764\pi$
$103$	16.7605	1.65146	0.825730	+	0.564066i	$0.190764\pi$
$104$	2.00505	0.196611
$105$	0	0
$106$	4.61795	0.448535
$107$	7.03427	0.680028	0.340014	−	0.940420i	$-0.389568\pi$
$107$	7.03427	0.680028	0.340014	+	0.940420i	$0.389568\pi$
$108$	0	0
$109$	8.64818	0.828345	0.414173	−	0.910198i	$-0.364071\pi$
$109$	8.64818	0.828345	0.414173	+	0.910198i	$0.364071\pi$
$110$	−1.02304	−0.0975431
$111$	0	0
$112$	3.02915	0.286228
$113$	9.06671	0.852924	0.426462	−	0.904505i	$-0.359760\pi$
$113$	9.06671	0.852924	0.426462	+	0.904505i	$0.359760\pi$
$114$	0	0
$115$	−5.61831	−0.523910
$116$	−6.14251	−0.570318
$117$	0	0
$118$	−0.799745	−0.0736225
$119$	−3.85945	−0.353795
$120$	0	0
$121$	−5.70606	−0.518733
$122$	0.0119003	0.00107740
$123$	0	0
$124$	0.480109	0.0431150
$125$	−9.60447	−0.859050
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	−10.4724	−0.925640
$129$	0	0
$130$	−0.570054	−0.0499970
$131$	−13.6779	−1.19504	−0.597521	−	0.801853i	$-0.703847\pi$
$131$	−13.6779	−1.19504	−0.597521	+	0.801853i	$0.703847\pi$
$132$	0	0
$133$	−5.48075	−0.475241
$134$	5.61770	0.485296
$135$	0	0
$136$	6.03583	0.517568
$137$	21.0777	1.80079	0.900394	−	0.435076i	$-0.143278\pi$
$137$	21.0777	1.80079	0.900394	+	0.435076i	$0.143278\pi$
$138$	0	0
$139$	10.4381	0.885348	0.442674	−	0.896683i	$-0.354030\pi$
$139$	10.4381	0.885348	0.442674	+	0.896683i	$0.354030\pi$
$140$	−1.99845	−0.168900
$141$	0	0
$142$	−0.797278	−0.0669061
$143$	2.94986	0.246680
$144$	0	0
$145$	3.65125	0.303220
$146$	4.47651	0.370479
$147$	0	0
$148$	−8.74019	−0.718439
$149$	−0.369201	−0.0302461	−0.0151231	−	0.999886i	$-0.504814\pi$
$149$	−0.369201	−0.0302461	−0.0151231	+	0.999886i	$0.504814\pi$
$150$	0	0
$151$	10.3324	0.840841	0.420420	−	0.907329i	$-0.361883\pi$
$151$	10.3324	0.840841	0.420420	+	0.907329i	$0.361883\pi$
$152$	8.57140	0.695233
$153$	0	0
$154$	−0.938637	−0.0756375
$155$	−0.285388	−0.0229229
$156$	0	0
$157$	19.4698	1.55386	0.776929	−	0.629589i	$-0.216777\pi$
$157$	19.4698	1.55386	0.776929	+	0.629589i	$0.216777\pi$
$158$	0.534729	0.0425407
$159$	0	0
$160$	4.75595	0.375991
$161$	−5.15478	−0.406254
$162$	0	0
$163$	13.8845	1.08752	0.543760	−	0.839241i	$-0.317000\pi$
$163$	13.8845	1.08752	0.543760	+	0.839241i	$0.317000\pi$
$164$	8.32319	0.649932
$165$	0	0
$166$	3.95044	0.306614
$167$	8.27086	0.640018	0.320009	−	0.947414i	$-0.396314\pi$
$167$	8.27086	0.640018	0.320009	+	0.947414i	$0.396314\pi$
$168$	0	0
$169$	−11.3563	−0.873561
$170$	−1.71604	−0.131615
$171$	0	0
$172$	−5.81258	−0.443205
$173$	−17.6707	−1.34348	−0.671738	−	0.740789i	$-0.734452\pi$
$173$	−17.6707	−1.34348	−0.671738	+	0.740789i	$0.734452\pi$
$174$	0	0
$175$	−3.81207	−0.288165
$176$	−6.96965	−0.525357
$177$	0	0
$178$	−5.42670	−0.406748
$179$	−17.8016	−1.33056	−0.665279	−	0.746595i	$-0.731687\pi$
$179$	−17.8016	−1.33056	−0.665279	+	0.746595i	$0.731687\pi$
$180$	0	0
$181$	24.5660	1.82598	0.912989	−	0.407984i	$-0.133768\pi$
$181$	24.5660	1.82598	0.912989	+	0.407984i	$0.133768\pi$
$182$	−0.523023	−0.0387690
$183$	0	0
$184$	8.06162	0.594311
$185$	5.19538	0.381972
$186$	0	0
$187$	8.88003	0.649372
$188$	0.901906	0.0657783
$189$	0	0
$190$	−2.43693	−0.176794
$191$	−23.1438	−1.67463	−0.837314	−	0.546723i	$-0.815875\pi$
$191$	−23.1438	−1.67463	−0.837314	+	0.546723i	$0.815875\pi$
$192$	0	0
$193$	4.89303	0.352208	0.176104	−	0.984372i	$-0.443651\pi$
$193$	4.89303	0.352208	0.176104	+	0.984372i	$0.443651\pi$
$194$	−0.996560	−0.0715488
$195$	0	0
$196$	−1.83358	−0.130970
$197$	−23.9211	−1.70431	−0.852155	−	0.523289i	$-0.824705\pi$
$197$	−23.9211	−1.70431	−0.852155	+	0.523289i	$0.824705\pi$
$198$	0	0
$199$	−12.8048	−0.907707	−0.453853	−	0.891076i	$-0.649951\pi$
$199$	−12.8048	−0.907707	−0.453853	+	0.891076i	$0.649951\pi$
$200$	5.96174	0.421559
$201$	0	0
$202$	−6.62530	−0.466154
$203$	3.35002	0.235125
$204$	0	0
$205$	−4.94751	−0.345549
$206$	6.83746	0.476388
$207$	0	0
$208$	−3.88359	−0.269279
$209$	12.6104	0.872281
$210$	0	0
$211$	−3.97912	−0.273934	−0.136967	−	0.990576i	$-0.543735\pi$
$211$	−3.97912	−0.273934	−0.136967	+	0.990576i	$0.543735\pi$
$212$	−20.7558	−1.42552
$213$	0	0
$214$	2.86964	0.196164
$215$	3.45514	0.235639
$216$	0	0
$217$	−0.261843	−0.0177751
$218$	3.52803	0.238949
$219$	0	0
$220$	4.59816	0.310008
$221$	4.94809	0.332844
$222$	0	0
$223$	1.15283	0.0771992	0.0385996	−	0.999255i	$-0.487710\pi$
$223$	1.15283	0.0771992	0.0385996	+	0.999255i	$0.487710\pi$
$224$	4.36357	0.291553
$225$	0	0
$226$	3.69877	0.246039
$227$	16.2210	1.07663	0.538314	−	0.842744i	$-0.319061\pi$
$227$	16.2210	1.07663	0.538314	+	0.842744i	$0.319061\pi$
$228$	0	0
$229$	−16.9029	−1.11698	−0.558488	−	0.829512i	$-0.688618\pi$
$229$	−16.9029	−1.11698	−0.558488	+	0.829512i	$0.688618\pi$
$230$	−2.29200	−0.151130
$231$	0	0
$232$	−5.23913	−0.343966
$233$	6.97836	0.457168	0.228584	−	0.973524i	$-0.426591\pi$
$233$	6.97836	0.457168	0.228584	+	0.973524i	$0.426591\pi$
$234$	0	0
$235$	−0.536115	−0.0349723
$236$	3.59453	0.233984
$237$	0	0
$238$	−1.57447	−0.102057
$239$	27.3643	1.77005	0.885025	−	0.465544i	$-0.154141\pi$
$239$	27.3643	1.77005	0.885025	+	0.465544i	$0.154141\pi$
$240$	0	0
$241$	−25.1019	−1.61695	−0.808477	−	0.588528i	$-0.799708\pi$
$241$	−25.1019	−1.61695	−0.808477	+	0.588528i	$0.799708\pi$
$242$	−2.32779	−0.149636
$243$	0	0
$244$	−0.0534869	−0.00342415
$245$	1.08992	0.0696325
$246$	0	0
$247$	7.02671	0.447099
$248$	0.409499	0.0260032
$249$	0	0
$250$	−3.91815	−0.247806
$251$	−7.05092	−0.445050	−0.222525	−	0.974927i	$-0.571430\pi$
$251$	−7.05092	−0.445050	−0.222525	+	0.974927i	$0.571430\pi$
$252$	0	0
$253$	11.8604	0.745658
$254$	0.407951	0.0255971
$255$	0	0
$256$	4.28413	0.267758
$257$	−0.0574366	−0.00358280	−0.00179140	−	0.999998i	$-0.500570\pi$
$257$	−0.0574366	−0.00358280	−0.00179140	+	0.999998i	$0.500570\pi$
$258$	0	0
$259$	4.76674	0.296191
$260$	2.56216	0.158899
$261$	0	0
$262$	−5.57990	−0.344728
$263$	14.6910	0.905885	0.452943	−	0.891540i	$-0.350374\pi$
$263$	14.6910	0.905885	0.452943	+	0.891540i	$0.350374\pi$
$264$	0	0
$265$	12.3378	0.757903
$266$	−2.23588	−0.137090
$267$	0	0
$268$	−25.2493	−1.54235
$269$	9.26332	0.564795	0.282397	−	0.959298i	$-0.408870\pi$
$269$	9.26332	0.564795	0.282397	+	0.959298i	$0.408870\pi$
$270$	0	0
$271$	13.1700	0.800024	0.400012	−	0.916510i	$-0.369006\pi$
$271$	13.1700	0.800024	0.400012	+	0.916510i	$0.369006\pi$
$272$	−11.6909	−0.708862
$273$	0	0
$274$	8.59866	0.519464
$275$	8.77103	0.528913
$276$	0	0
$277$	16.3591	0.982922	0.491461	−	0.870900i	$-0.336463\pi$
$277$	16.3591	0.982922	0.491461	+	0.870900i	$0.336463\pi$
$278$	4.25823	0.255392
$279$	0	0
$280$	−1.70454	−0.101866
$281$	7.16275	0.427294	0.213647	−	0.976911i	$-0.431466\pi$
$281$	7.16275	0.427294	0.213647	+	0.976911i	$0.431466\pi$
$282$	0	0
$283$	−2.99945	−0.178299	−0.0891494	−	0.996018i	$-0.528415\pi$
$283$	−2.99945	−0.178299	−0.0891494	+	0.996018i	$0.528415\pi$
$284$	3.58345	0.212638
$285$	0	0
$286$	1.20340	0.0711585
$287$	−4.53932	−0.267948
$288$	0	0
$289$	−2.10467	−0.123804
$290$	1.48953	0.0874684
$291$	0	0
$292$	−20.1201	−1.17744
$293$	7.96629	0.465396	0.232698	−	0.972549i	$-0.425245\pi$
$293$	7.96629	0.465396	0.232698	+	0.972549i	$0.425245\pi$
$294$	0	0
$295$	−2.13668	−0.124402
$296$	−7.45476	−0.433299
$297$	0	0
$298$	−0.150616	−0.00872495
$299$	6.60880	0.382197
$300$	0	0
$301$	3.17008	0.182720
$302$	4.21512	0.242553
$303$	0	0
$304$	−16.6020	−0.952191
$305$	0.0317939	0.00182051
$306$	0	0
$307$	9.84980	0.562158	0.281079	−	0.959685i	$-0.409308\pi$
$307$	9.84980	0.562158	0.281079	+	0.959685i	$0.409308\pi$
$308$	4.21880	0.240388
$309$	0	0
$310$	−0.116424	−0.00661246
$311$	31.7455	1.80012	0.900062	−	0.435762i	$-0.143521\pi$
$311$	31.7455	1.80012	0.900062	+	0.435762i	$0.143521\pi$
$312$	0	0
$313$	−12.9459	−0.731748	−0.365874	−	0.930664i	$-0.619230\pi$
$313$	−12.9459	−0.731748	−0.365874	+	0.930664i	$0.619230\pi$
$314$	7.94272	0.448233
$315$	0	0
$316$	−2.40339	−0.135201
$317$	4.65835	0.261639	0.130819	−	0.991406i	$-0.458239\pi$
$317$	4.65835	0.261639	0.130819	+	0.991406i	$0.458239\pi$
$318$	0	0
$319$	−7.70791	−0.431560
$320$	−4.66288	−0.260663
$321$	0	0
$322$	−2.10290	−0.117190
$323$	21.1527	1.17697
$324$	0	0
$325$	4.88735	0.271101
$326$	5.66421	0.313712
$327$	0	0
$328$	7.09910	0.391982
$329$	−0.491884	−0.0271184
$330$	0	0
$331$	14.0354	0.771458	0.385729	−	0.922612i	$-0.373950\pi$
$331$	14.0354	0.771458	0.385729	+	0.922612i	$0.373950\pi$
$332$	−17.7556	−0.974467
$333$	0	0
$334$	3.37410	0.184623
$335$	15.0088	0.820018
$336$	0	0
$337$	−19.7961	−1.07836	−0.539182	−	0.842190i	$-0.681266\pi$
$337$	−19.7961	−1.07836	−0.539182	+	0.842190i	$0.681266\pi$
$338$	−4.63281	−0.251992
$339$	0	0
$340$	7.71293	0.418292
$341$	0.602463	0.0326252
$342$	0	0
$343$	1.00000	0.0539949
$344$	−4.95772	−0.267303
$345$	0	0
$346$	−7.20877	−0.387546
$347$	21.2045	1.13832	0.569158	−	0.822228i	$-0.307269\pi$
$347$	21.2045	1.13832	0.569158	+	0.822228i	$0.307269\pi$
$348$	0	0
$349$	−3.55610	−0.190354	−0.0951769	−	0.995460i	$-0.530342\pi$
$349$	−3.55610	−0.190354	−0.0951769	+	0.995460i	$0.530342\pi$
$350$	−1.55514	−0.0831256
$351$	0	0
$352$	−10.0399	−0.535131
$353$	15.4845	0.824156	0.412078	−	0.911149i	$-0.364803\pi$
$353$	15.4845	0.824156	0.412078	+	0.911149i	$0.364803\pi$
$354$	0	0
$355$	−2.13009	−0.113053
$356$	24.3908	1.29271
$357$	0	0
$358$	−7.26220	−0.383819
$359$	−14.4272	−0.761436	−0.380718	−	0.924691i	$-0.624323\pi$
$359$	−14.4272	−0.761436	−0.380718	+	0.924691i	$0.624323\pi$
$360$	0	0
$361$	11.0386	0.580979
$362$	10.0217	0.526731
$363$	0	0
$364$	2.35078	0.123214
$365$	11.9599	0.626009
$366$	0	0
$367$	37.0573	1.93438	0.967188	−	0.254063i	$-0.0817671\pi$
$367$	37.0573	1.93438	0.967188	+	0.254063i	$0.0817671\pi$
$368$	−15.6146	−0.813969
$369$	0	0
$370$	2.11946	0.110185
$371$	11.3199	0.587698
$372$	0	0
$373$	29.5070	1.52782	0.763908	−	0.645325i	$-0.223278\pi$
$373$	29.5070	1.52782	0.763908	+	0.645325i	$0.223278\pi$
$374$	3.62262	0.187321
$375$	0	0
$376$	0.769262	0.0396717
$377$	−4.29496	−0.221202
$378$	0	0
$379$	20.2256	1.03892	0.519460	−	0.854495i	$-0.326133\pi$
$379$	20.2256	1.03892	0.519460	+	0.854495i	$0.326133\pi$
$380$	10.9530	0.561878
$381$	0	0
$382$	−9.44154	−0.483071
$383$	1.33594	0.0682635	0.0341317	−	0.999417i	$-0.489133\pi$
$383$	1.33594	0.0682635	0.0341317	+	0.999417i	$0.489133\pi$
$384$	0	0
$385$	−2.50775	−0.127807
$386$	1.99612	0.101600
$387$	0	0
$388$	4.47914	0.227394
$389$	31.0104	1.57229	0.786144	−	0.618044i	$-0.212075\pi$
$389$	31.0104	1.57229	0.786144	+	0.618044i	$0.212075\pi$
$390$	0	0
$391$	19.8946	1.00611
$392$	−1.56391	−0.0789894
$393$	0	0
$394$	−9.75865	−0.491634
$395$	1.42863	0.0718823
$396$	0	0
$397$	−1.70559	−0.0856012	−0.0428006	−	0.999084i	$-0.513628\pi$
$397$	−1.70559	−0.0856012	−0.0428006	+	0.999084i	$0.513628\pi$
$398$	−5.22372	−0.261842
$399$	0	0
$400$	−11.5473	−0.577367
$401$	−31.1528	−1.55570	−0.777849	−	0.628451i	$-0.783689\pi$
$401$	−31.1528	−1.55570	−0.777849	+	0.628451i	$0.783689\pi$
$402$	0	0
$403$	0.335701	0.0167225
$404$	29.7780	1.48151
$405$	0	0
$406$	1.36664	0.0678253
$407$	−10.9676	−0.543643
$408$	0	0
$409$	28.7531	1.42175	0.710875	−	0.703318i	$-0.248299\pi$
$409$	28.7531	1.42175	0.710875	+	0.703318i	$0.248299\pi$
$410$	−2.01834	−0.0996787
$411$	0	0
$412$	−30.7316	−1.51404
$413$	−1.96040	−0.0964647
$414$	0	0
$415$	10.5544	0.518094
$416$	−5.59441	−0.274288
$417$	0	0
$418$	5.14443	0.251622
$419$	10.0450	0.490730	0.245365	−	0.969431i	$-0.421092\pi$
$419$	10.0450	0.490730	0.245365	+	0.969431i	$0.421092\pi$
$420$	0	0
$421$	−24.2888	−1.18376	−0.591882	−	0.806025i	$-0.701615\pi$
$421$	−24.2888	−1.18376	−0.591882	+	0.806025i	$0.701615\pi$
$422$	−1.62329	−0.0790203
$423$	0	0
$424$	−17.7033	−0.859747
$425$	14.7125	0.713660
$426$	0	0
$427$	0.0291708	0.00141167
$428$	−12.8979	−0.623442
$429$	0	0
$430$	1.40953	0.0679735
$431$	−8.94786	−0.431003	−0.215502	−	0.976503i	$-0.569139\pi$
$431$	−8.94786	−0.431003	−0.215502	+	0.976503i	$0.569139\pi$
$432$	0	0
$433$	−12.7200	−0.611282	−0.305641	−	0.952147i	$-0.598871\pi$
$433$	−12.7200	−0.611282	−0.305641	+	0.952147i	$0.598871\pi$
$434$	−0.106819	−0.00512748
$435$	0	0
$436$	−15.8571	−0.759417
$437$	28.2521	1.35148
$438$	0	0
$439$	20.2200	0.965048	0.482524	−	0.875883i	$-0.339720\pi$
$439$	20.2200	0.965048	0.482524	+	0.875883i	$0.339720\pi$
$440$	3.92190	0.186969
$441$	0	0
$442$	2.01858	0.0960140
$443$	8.70900	0.413777	0.206889	−	0.978364i	$-0.433666\pi$
$443$	8.70900	0.413777	0.206889	+	0.978364i	$0.433666\pi$
$444$	0	0
$445$	−14.4985	−0.687295
$446$	0.470298	0.0222693
$447$	0	0
$448$	−4.27818	−0.202125
$449$	2.31692	0.109342	0.0546711	−	0.998504i	$-0.482589\pi$
$449$	2.31692	0.109342	0.0546711	+	0.998504i	$0.482589\pi$
$450$	0	0
$451$	10.4443	0.491804
$452$	−16.6245	−0.781951
$453$	0	0
$454$	6.61739	0.310570
$455$	−1.39736	−0.0655092
$456$	0	0
$457$	−2.94767	−0.137886	−0.0689430	−	0.997621i	$-0.521963\pi$
$457$	−2.94767	−0.137886	−0.0689430	+	0.997621i	$0.521963\pi$
$458$	−6.89557	−0.322209
$459$	0	0
$460$	10.3016	0.480314
$461$	−14.1185	−0.657564	−0.328782	−	0.944406i	$-0.606638\pi$
$461$	−14.1185	−0.657564	−0.328782	+	0.944406i	$0.606638\pi$
$462$	0	0
$463$	−37.5227	−1.74383	−0.871913	−	0.489661i	$-0.837121\pi$
$463$	−37.5227	−1.74383	−0.871913	+	0.489661i	$0.837121\pi$
$464$	10.1477	0.471096
$465$	0	0
$466$	2.84683	0.131877
$467$	−30.4306	−1.40816	−0.704081	−	0.710120i	$-0.748641\pi$
$467$	−30.4306	−1.40816	−0.704081	+	0.710120i	$0.748641\pi$
$468$	0	0
$469$	13.7705	0.635864
$470$	−0.218709	−0.0100883
$471$	0	0
$472$	3.06588	0.141119
$473$	−7.29390	−0.335374
$474$	0	0
$475$	20.8930	0.958637
$476$	7.07659	0.324355
$477$	0	0
$478$	11.1633	0.510597
$479$	−25.2836	−1.15524	−0.577619	−	0.816306i	$-0.696018\pi$
$479$	−25.2836	−1.15524	−0.577619	+	0.816306i	$0.696018\pi$
$480$	0	0
$481$	−6.11131	−0.278652
$482$	−10.2403	−0.466434
$483$	0	0
$484$	10.4625	0.475568
$485$	−2.66251	−0.120898
$486$	0	0
$487$	−23.6931	−1.07364	−0.536819	−	0.843697i	$-0.680374\pi$
$487$	−23.6931	−1.07364	−0.536819	+	0.843697i	$0.680374\pi$
$488$	−0.0456206	−0.00206515
$489$	0	0
$490$	0.444635	0.0200865
$491$	−29.3480	−1.32445	−0.662227	−	0.749303i	$-0.730389\pi$
$491$	−29.3480	−1.32445	−0.662227	+	0.749303i	$0.730389\pi$
$492$	0	0
$493$	−12.9292	−0.582302
$494$	2.86656	0.128972
$495$	0	0
$496$	−0.793162	−0.0356140
$497$	−1.95435	−0.0876645
$498$	0	0
$499$	23.3986	1.04746	0.523732	−	0.851883i	$-0.324539\pi$
$499$	23.3986	1.04746	0.523732	+	0.851883i	$0.324539\pi$
$500$	17.6105	0.787566
$501$	0	0
$502$	−2.87643	−0.128381
$503$	15.4011	0.686701	0.343351	−	0.939207i	$-0.388438\pi$
$503$	15.4011	0.686701	0.343351	+	0.939207i	$0.388438\pi$
$504$	0	0
$505$	−17.7008	−0.787675
$506$	4.83847	0.215096
$507$	0	0
$508$	−1.83358	−0.0813518
$509$	9.83148	0.435773	0.217886	−	0.975974i	$-0.430084\pi$
$509$	9.83148	0.435773	0.217886	+	0.975974i	$0.430084\pi$
$510$	0	0
$511$	10.9732	0.485424
$512$	22.6926	1.00288
$513$	0	0
$514$	−0.0234313	−0.00103351
$515$	18.2676	0.804967
$516$	0	0
$517$	1.13175	0.0497745
$518$	1.94460	0.0854407
$519$	0	0
$520$	2.18534	0.0958337
$521$	0.581283	0.0254665	0.0127332	−	0.999919i	$-0.495947\pi$
$521$	0.581283	0.0254665	0.0127332	+	0.999919i	$0.495947\pi$
$522$	0	0
$523$	9.39466	0.410800	0.205400	−	0.978678i	$-0.434150\pi$
$523$	9.39466	0.410800	0.205400	+	0.978678i	$0.434150\pi$
$524$	25.0794	1.09560
$525$	0	0
$526$	5.99320	0.261316
$527$	1.01057	0.0440211
$528$	0	0
$529$	3.57179	0.155295
$530$	5.03321	0.218628
$531$	0	0
$532$	10.0494	0.435695
$533$	5.81974	0.252081
$534$	0	0
$535$	7.66680	0.331465
$536$	−21.5359	−0.930209
$537$	0	0
$538$	3.77898	0.162923
$539$	−2.30086	−0.0991049
$540$	0	0
$541$	16.0235	0.688905	0.344452	−	0.938804i	$-0.388065\pi$
$541$	16.0235	0.688905	0.344452	+	0.938804i	$0.388065\pi$
$542$	5.37274	0.230779
$543$	0	0
$544$	−16.8410	−0.722050
$545$	9.42584	0.403758
$546$	0	0
$547$	−21.8696	−0.935077	−0.467538	−	0.883973i	$-0.654859\pi$
$547$	−21.8696	−0.935077	−0.467538	+	0.883973i	$0.654859\pi$
$548$	−38.6475	−1.65094
$549$	0	0
$550$	3.57815	0.152573
$551$	−18.3606	−0.782188
$552$	0	0
$553$	1.31077	0.0557395
$554$	6.67370	0.283539
$555$	0	0
$556$	−19.1390	−0.811676
$557$	−9.18821	−0.389317	−0.194658	−	0.980871i	$-0.562360\pi$
$557$	−9.18821	−0.389317	−0.194658	+	0.980871i	$0.562360\pi$
$558$	0	0
$559$	−4.06427	−0.171900
$560$	3.30154	0.139515
$561$	0	0
$562$	2.92205	0.123259
$563$	26.1377	1.10157	0.550786	−	0.834647i	$-0.314328\pi$
$563$	26.1377	1.10157	0.550786	+	0.834647i	$0.314328\pi$
$564$	0	0
$565$	9.88200	0.415739
$566$	−1.22363	−0.0514330
$567$	0	0
$568$	3.05643	0.128245
$569$	17.4717	0.732451	0.366225	−	0.930526i	$-0.380650\pi$
$569$	17.4717	0.732451	0.366225	+	0.930526i	$0.380650\pi$
$570$	0	0
$571$	23.0247	0.963553	0.481776	−	0.876294i	$-0.339992\pi$
$571$	23.0247	0.963553	0.481776	+	0.876294i	$0.339992\pi$
$572$	−5.40880	−0.226153
$573$	0	0
$574$	−1.85182	−0.0772935
$575$	19.6504	0.819478
$576$	0	0
$577$	4.55690	0.189706	0.0948531	−	0.995491i	$-0.469762\pi$
$577$	4.55690	0.189706	0.0948531	+	0.995491i	$0.469762\pi$
$578$	−0.858603	−0.0357132
$579$	0	0
$580$	−6.69485	−0.277989
$581$	9.68361	0.401744
$582$	0	0
$583$	−26.0454	−1.07869
$584$	−17.1610	−0.710129
$585$	0	0
$586$	3.24986	0.134250
$587$	3.20341	0.132219	0.0661094	−	0.997812i	$-0.478941\pi$
$587$	3.20341	0.132219	0.0661094	+	0.997812i	$0.478941\pi$
$588$	0	0
$589$	1.43509	0.0591321
$590$	−0.871660	−0.0358857
$591$	0	0
$592$	14.4392	0.593447
$593$	36.9825	1.51869	0.759344	−	0.650689i	$-0.225520\pi$
$593$	36.9825	1.51869	0.759344	+	0.650689i	$0.225520\pi$
$594$	0	0
$595$	−4.20649	−0.172449
$596$	0.676958	0.0277293
$597$	0	0
$598$	2.69607	0.110250
$599$	38.3476	1.56684	0.783421	−	0.621491i	$-0.213473\pi$
$599$	38.3476	1.56684	0.783421	+	0.621491i	$0.213473\pi$
$600$	0	0
$601$	−33.1185	−1.35093	−0.675467	−	0.737391i	$-0.736058\pi$
$601$	−33.1185	−1.35093	−0.675467	+	0.737391i	$0.736058\pi$
$602$	1.29324	0.0527084
$603$	0	0
$604$	−18.9453	−0.770873
$605$	−6.21916	−0.252845
$606$	0	0
$607$	−1.37747	−0.0559096	−0.0279548	−	0.999609i	$-0.508899\pi$
$607$	−1.37747	−0.0559096	−0.0279548	+	0.999609i	$0.508899\pi$
$608$	−23.9156	−0.969907
$609$	0	0
$610$	0.0129704	0.000525154 0
$611$	0.630630	0.0255126
$612$	0	0
$613$	6.71060	0.271039	0.135519	−	0.990775i	$-0.456730\pi$
$613$	6.71060	0.271039	0.135519	+	0.990775i	$0.456730\pi$
$614$	4.01824	0.162163
$615$	0	0
$616$	3.59834	0.144981
$617$	20.8740	0.840356	0.420178	−	0.907442i	$-0.361968\pi$
$617$	20.8740	0.840356	0.420178	+	0.907442i	$0.361968\pi$
$618$	0	0
$619$	10.0657	0.404576	0.202288	−	0.979326i	$-0.435162\pi$
$619$	10.0657	0.404576	0.202288	+	0.979326i	$0.435162\pi$
$620$	0.523281	0.0210155
$621$	0	0
$622$	12.9506	0.519273
$623$	−13.3023	−0.532946
$624$	0	0
$625$	8.59224	0.343689
$626$	−5.28131	−0.211084
$627$	0	0
$628$	−35.6993	−1.42456
$629$	−18.3970	−0.733536
$630$	0	0
$631$	−45.6084	−1.81564	−0.907821	−	0.419359i	$-0.862255\pi$
$631$	−45.6084	−1.81564	−0.907821	+	0.419359i	$0.862255\pi$
$632$	−2.04992	−0.0815416
$633$	0	0
$634$	1.90038	0.0754736
$635$	1.08992	0.0432522
$636$	0	0
$637$	−1.28207	−0.0507975
$638$	−3.14445	−0.124490
$639$	0	0
$640$	−11.4141	−0.451183
$641$	36.7175	1.45025	0.725126	−	0.688616i	$-0.241781\pi$
$641$	36.7175	1.45025	0.725126	+	0.688616i	$0.241781\pi$
$642$	0	0
$643$	−45.1998	−1.78250	−0.891252	−	0.453508i	$-0.850172\pi$
$643$	−45.1998	−1.78250	−0.891252	+	0.453508i	$0.850172\pi$
$644$	9.45169	0.372449
$645$	0	0
$646$	8.62925	0.339513
$647$	−4.64930	−0.182783	−0.0913915	−	0.995815i	$-0.529131\pi$
$647$	−4.64930	−0.182783	−0.0913915	+	0.995815i	$0.529131\pi$
$648$	0	0
$649$	4.51059	0.177056
$650$	1.99380	0.0782032
$651$	0	0
$652$	−25.4584	−0.997026
$653$	−38.9603	−1.52463	−0.762317	−	0.647204i	$-0.775938\pi$
$653$	−38.9603	−1.52463	−0.762317	+	0.647204i	$0.775938\pi$
$654$	0	0
$655$	−14.9078	−0.582496
$656$	−13.7503	−0.536859
$657$	0	0
$658$	−0.200664	−0.00782271
$659$	19.9316	0.776425	0.388212	−	0.921570i	$-0.373093\pi$
$659$	19.9316	0.776425	0.388212	+	0.921570i	$0.373093\pi$
$660$	0	0
$661$	−36.2288	−1.40914	−0.704569	−	0.709635i	$-0.748860\pi$
$661$	−36.2288	−1.40914	−0.704569	+	0.709635i	$0.748860\pi$
$662$	5.72578	0.222539
$663$	0	0
$664$	−15.1443	−0.587713
$665$	−5.97359	−0.231646
$666$	0	0
$667$	−17.2686	−0.668643
$668$	−15.1652	−0.586761
$669$	0	0
$670$	6.12286	0.236547
$671$	−0.0671179	−0.00259106
$672$	0	0
$673$	10.0220	0.386320	0.193160	−	0.981167i	$-0.438126\pi$
$673$	10.0220	0.386320	0.193160	+	0.981167i	$0.438126\pi$
$674$	−8.07585	−0.311070
$675$	0	0
$676$	20.8226	0.800870
$677$	−2.27813	−0.0875555	−0.0437778	−	0.999041i	$-0.513939\pi$
$677$	−2.27813	−0.0875555	−0.0437778	+	0.999041i	$0.513939\pi$
$678$	0	0
$679$	−2.44284	−0.0937477
$680$	6.57858	0.252277
$681$	0	0
$682$	0.245775	0.00941123
$683$	34.7013	1.32781	0.663904	−	0.747817i	$-0.268898\pi$
$683$	34.7013	1.32781	0.663904	+	0.747817i	$0.268898\pi$
$684$	0	0
$685$	22.9730	0.877754
$686$	0.407951	0.0155756
$687$	0	0
$688$	9.60266	0.366098
$689$	−14.5129	−0.552897
$690$	0	0
$691$	25.5923	0.973576	0.486788	−	0.873520i	$-0.338168\pi$
$691$	25.5923	0.973576	0.486788	+	0.873520i	$0.338168\pi$
$692$	32.4005	1.23168
$693$	0	0
$694$	8.65038	0.328364
$695$	11.3767	0.431543
$696$	0	0
$697$	17.5193	0.663590
$698$	−1.45072	−0.0549104
$699$	0	0
$700$	6.98972	0.264187
$701$	34.4738	1.30206	0.651028	−	0.759054i	$-0.274338\pi$
$701$	34.4738	1.30206	0.651028	+	0.759054i	$0.274338\pi$
$702$	0	0
$703$	−26.1253	−0.985335
$704$	9.84349	0.370990
$705$	0	0
$706$	6.31691	0.237740
$707$	−16.2404	−0.610784
$708$	0	0
$709$	−5.12099	−0.192323	−0.0961614	−	0.995366i	$-0.530656\pi$
$709$	−5.12099	−0.192323	−0.0961614	+	0.995366i	$0.530656\pi$
$710$	−0.868971	−0.0326119
$711$	0	0
$712$	20.8037	0.779650
$713$	1.34974	0.0505483
$714$	0	0
$715$	3.21512	0.120239
$716$	32.6407	1.21984
$717$	0	0
$718$	−5.88557	−0.219648
$719$	9.38376	0.349955	0.174978	−	0.984572i	$-0.444015\pi$
$719$	9.38376	0.349955	0.174978	+	0.984572i	$0.444015\pi$
$720$	0	0
$721$	16.7605	0.624193
$722$	4.50321	0.167592
$723$	0	0
$724$	−45.0437	−1.67404
$725$	−12.7705	−0.474284
$726$	0	0
$727$	−17.8615	−0.662445	−0.331222	−	0.943553i	$-0.607461\pi$
$727$	−17.8615	−0.662445	−0.331222	+	0.943553i	$0.607461\pi$
$728$	2.00505	0.0743120
$729$	0	0
$730$	4.87904	0.180582
$731$	−12.2348	−0.452519
$732$	0	0
$733$	20.4202	0.754238	0.377119	−	0.926165i	$-0.376915\pi$
$733$	20.4202	0.754238	0.377119	+	0.926165i	$0.376915\pi$
$734$	15.1176	0.557999
$735$	0	0
$736$	−22.4932	−0.829112
$737$	−31.6840	−1.16710
$738$	0	0
$739$	51.1576	1.88186	0.940931	−	0.338597i	$-0.109952\pi$
$739$	51.1576	1.88186	0.940931	+	0.338597i	$0.109952\pi$
$740$	−9.52612	−0.350187
$741$	0	0
$742$	4.61795	0.169530
$743$	−32.3657	−1.18738	−0.593691	−	0.804693i	$-0.702330\pi$
$743$	−32.3657	−1.18738	−0.593691	+	0.804693i	$0.702330\pi$
$744$	0	0
$745$	−0.402400	−0.0147428
$746$	12.0374	0.440722
$747$	0	0
$748$	−16.2822	−0.595337
$749$	7.03427	0.257027
$750$	0	0
$751$	−50.9969	−1.86090	−0.930452	−	0.366414i	$-0.880585\pi$
$751$	−50.9969	−1.86090	−0.930452	+	0.366414i	$0.880585\pi$
$752$	−1.48999	−0.0543344
$753$	0	0
$754$	−1.75213	−0.0638090
$755$	11.2615	0.409849
$756$	0	0
$757$	37.7384	1.37162	0.685812	−	0.727779i	$-0.259447\pi$
$757$	37.7384	1.37162	0.685812	+	0.727779i	$0.259447\pi$
$758$	8.25106	0.299692
$759$	0	0
$760$	9.34216	0.338876
$761$	−43.9468	−1.59307	−0.796535	−	0.604593i	$-0.793336\pi$
$761$	−43.9468	−1.59307	−0.796535	+	0.604593i	$0.793336\pi$
$762$	0	0
$763$	8.64818	0.313085
$764$	42.4359	1.53528
$765$	0	0
$766$	0.544999	0.0196916
$767$	2.51337	0.0907525
$768$	0	0
$769$	42.1961	1.52163	0.760816	−	0.648968i	$-0.224799\pi$
$769$	42.1961	1.52163	0.760816	+	0.648968i	$0.224799\pi$
$770$	−1.02304	−0.0368678
$771$	0	0
$772$	−8.97174	−0.322900
$773$	40.4493	1.45486	0.727430	−	0.686182i	$-0.240715\pi$
$773$	40.4493	1.45486	0.727430	+	0.686182i	$0.240715\pi$
$774$	0	0
$775$	0.998164	0.0358551
$776$	3.82039	0.137144
$777$	0	0
$778$	12.6507	0.453550
$779$	24.8789	0.891378
$780$	0	0
$781$	4.49667	0.160904
$782$	8.11603	0.290229
$783$	0	0
$784$	3.02915	0.108184
$785$	21.2205	0.757393
$786$	0	0
$787$	17.1152	0.610090	0.305045	−	0.952338i	$-0.401329\pi$
$787$	17.1152	0.610090	0.305045	+	0.952338i	$0.401329\pi$
$788$	43.8612	1.56249
$789$	0	0
$790$	0.582812	0.0207355
$791$	9.06671	0.322375
$792$	0	0
$793$	−0.0373991	−0.00132808
$794$	−0.695798	−0.0246929
$795$	0	0
$796$	23.4785	0.832175
$797$	−6.60740	−0.234046	−0.117023	−	0.993129i	$-0.537335\pi$
$797$	−6.60740	−0.234046	−0.117023	+	0.993129i	$0.537335\pi$
$798$	0	0
$799$	1.89840	0.0671605
$800$	−16.6342	−0.588109
$801$	0	0
$802$	−12.7088	−0.448764
$803$	−25.2477	−0.890970
$804$	0	0
$805$	−5.61831	−0.198019
$806$	0.136950	0.00482385
$807$	0	0
$808$	25.3986	0.893519
$809$	8.40122	0.295371	0.147686	−	0.989034i	$-0.452818\pi$
$809$	8.40122	0.295371	0.147686	+	0.989034i	$0.452818\pi$
$810$	0	0
$811$	40.8644	1.43494	0.717472	−	0.696588i	$-0.245299\pi$
$811$	40.8644	1.43494	0.717472	+	0.696588i	$0.245299\pi$
$812$	−6.14251	−0.215560
$813$	0	0
$814$	−4.47424	−0.156822
$815$	15.1331	0.530088
$816$	0	0
$817$	−17.3744	−0.607854
$818$	11.7299	0.410125
$819$	0	0
$820$	9.07163	0.316795
$821$	−5.24790	−0.183153	−0.0915764	−	0.995798i	$-0.529191\pi$
$821$	−5.24790	−0.183153	−0.0915764	+	0.995798i	$0.529191\pi$
$822$	0	0
$823$	−13.5425	−0.472060	−0.236030	−	0.971746i	$-0.575846\pi$
$823$	−13.5425	−0.472060	−0.236030	+	0.971746i	$0.575846\pi$
$824$	−26.2119	−0.913135
$825$	0	0
$826$	−0.799745	−0.0278267
$827$	21.6919	0.754301	0.377150	−	0.926152i	$-0.376904\pi$
$827$	21.6919	0.754301	0.377150	+	0.926152i	$0.376904\pi$
$828$	0	0
$829$	51.7300	1.79666	0.898328	−	0.439325i	$-0.144782\pi$
$829$	51.7300	1.79666	0.898328	+	0.439325i	$0.144782\pi$
$830$	4.30567	0.149452
$831$	0	0
$832$	5.48494	0.190156
$833$	−3.85945	−0.133722
$834$	0	0
$835$	9.01459	0.311963
$836$	−23.1222	−0.799697
$837$	0	0
$838$	4.09787	0.141558
$839$	21.3671	0.737673	0.368836	−	0.929494i	$-0.379756\pi$
$839$	21.3671	0.737673	0.368836	+	0.929494i	$0.379756\pi$
$840$	0	0
$841$	−17.7774	−0.613014
$842$	−9.90865	−0.341474
$843$	0	0
$844$	7.29601	0.251139
$845$	−12.3775	−0.425798
$846$	0	0
$847$	−5.70606	−0.196062
$848$	34.2896	1.17751
$849$	0	0
$850$	6.00197	0.205866
$851$	−24.5715	−0.842301
$852$	0	0
$853$	−33.7067	−1.15410	−0.577048	−	0.816710i	$-0.695796\pi$
$853$	−33.7067	−1.15410	−0.577048	+	0.816710i	$0.695796\pi$
$854$	0.0119003	0.000407219 0
$855$	0	0
$856$	−11.0010	−0.376005
$857$	−18.6766	−0.637982	−0.318991	−	0.947758i	$-0.603344\pi$
$857$	−18.6766	−0.637982	−0.318991	+	0.947758i	$0.603344\pi$
$858$	0	0
$859$	−37.7733	−1.28881	−0.644405	−	0.764684i	$-0.722895\pi$
$859$	−37.7733	−1.28881	−0.644405	+	0.764684i	$0.722895\pi$
$860$	−6.33526	−0.216031
$861$	0	0
$862$	−3.65029	−0.124329
$863$	19.9811	0.680166	0.340083	−	0.940395i	$-0.389545\pi$
$863$	19.9811	0.680166	0.340083	+	0.940395i	$0.389545\pi$
$864$	0	0
$865$	−19.2597	−0.654848
$866$	−5.18912	−0.176334
$867$	0	0
$868$	0.480109	0.0162960
$869$	−3.01589	−0.102307
$870$	0	0
$871$	−17.6548	−0.598211
$872$	−13.5250	−0.458014
$873$	0	0
$874$	11.5255	0.389855
$875$	−9.60447	−0.324690
$876$	0	0
$877$	−20.6716	−0.698030	−0.349015	−	0.937117i	$-0.613484\pi$
$877$	−20.6716	−0.698030	−0.349015	+	0.937117i	$0.613484\pi$
$878$	8.24877	0.278382
$879$	0	0
$880$	−7.59637	−0.256074
$881$	5.74429	0.193530	0.0967651	−	0.995307i	$-0.469150\pi$
$881$	5.74429	0.193530	0.0967651	+	0.995307i	$0.469150\pi$
$882$	0	0
$883$	0.269530	0.00907042	0.00453521	−	0.999990i	$-0.498556\pi$
$883$	0.269530	0.00907042	0.00453521	+	0.999990i	$0.498556\pi$
$884$	−9.07270	−0.305148
$885$	0	0
$886$	3.55285	0.119360
$887$	−7.13439	−0.239550	−0.119775	−	0.992801i	$-0.538217\pi$
$887$	−7.13439	−0.239550	−0.119775	+	0.992801i	$0.538217\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	−5.91468	−0.198260
$891$	0	0
$892$	−2.11380	−0.0707753
$893$	2.69589	0.0902145
$894$	0	0
$895$	−19.4024	−0.648551
$896$	−10.4724	−0.349859
$897$	0	0
$898$	0.945190	0.0315414
$899$	−0.877178	−0.0292555
$900$	0	0
$901$	−43.6884	−1.45547
$902$	4.26077	0.141868
$903$	0	0
$904$	−14.1795	−0.471604
$905$	26.7750	0.890033
$906$	0	0
$907$	18.4478	0.612549	0.306275	−	0.951943i	$-0.400917\pi$
$907$	18.4478	0.612549	0.306275	+	0.951943i	$0.400917\pi$
$908$	−29.7425	−0.987040
$909$	0	0
$910$	−0.570054	−0.0188971
$911$	−48.1253	−1.59446	−0.797231	−	0.603675i	$-0.793703\pi$
$911$	−48.1253	−1.59446	−0.797231	+	0.603675i	$0.793703\pi$
$912$	0	0
$913$	−22.2806	−0.737380
$914$	−1.20250	−0.0397753
$915$	0	0
$916$	30.9928	1.02403
$917$	−13.6779	−0.451683
$918$	0	0
$919$	−44.9642	−1.48323	−0.741615	−	0.670825i	$-0.765940\pi$
$919$	−44.9642	−1.48323	−0.741615	+	0.670825i	$0.765940\pi$
$920$	8.78654	0.289684
$921$	0	0
$922$	−5.75966	−0.189684
$923$	2.50561	0.0824733
$924$	0	0
$925$	−18.1712	−0.597464
$926$	−15.3074	−0.503033
$927$	0	0
$928$	14.6180	0.479860
$929$	−12.6354	−0.414553	−0.207276	−	0.978282i	$-0.566460\pi$
$929$	−12.6354	−0.414553	−0.207276	+	0.978282i	$0.566460\pi$
$930$	0	0
$931$	−5.48075	−0.179624
$932$	−12.7954	−0.419126
$933$	0	0
$934$	−12.4142	−0.406205
$935$	9.67854	0.316522
$936$	0	0
$937$	46.3648	1.51467	0.757335	−	0.653026i	$-0.226501\pi$
$937$	46.3648	1.51467	0.757335	+	0.653026i	$0.226501\pi$
$938$	5.61770	0.183424
$939$	0	0
$940$	0.983007	0.0320622
$941$	−2.85670	−0.0931259	−0.0465629	−	0.998915i	$-0.514827\pi$
$941$	−2.85670	−0.0931259	−0.0465629	+	0.998915i	$0.514827\pi$
$942$	0	0
$943$	23.3992	0.761983
$944$	−5.93834	−0.193276
$945$	0	0
$946$	−2.97555	−0.0967436
$947$	−10.6467	−0.345971	−0.172985	−	0.984924i	$-0.555341\pi$
$947$	−10.6467	−0.345971	−0.172985	+	0.984924i	$0.555341\pi$
$948$	0	0
$949$	−14.0684	−0.456679
$950$	8.52332	0.276533
$951$	0	0
$952$	6.03583	0.195622
$953$	26.1036	0.845577	0.422789	−	0.906228i	$-0.361051\pi$
$953$	26.1036	0.845577	0.422789	+	0.906228i	$0.361051\pi$
$954$	0	0
$955$	−25.2249	−0.816260
$956$	−50.1745	−1.62276
$957$	0	0
$958$	−10.3145	−0.333246
$959$	21.0777	0.680634
$960$	0	0
$961$	−30.9314	−0.997788
$962$	−2.49311	−0.0803812
$963$	0	0
$964$	46.0262	1.48240
$965$	5.33302	0.171676
$966$	0	0
$967$	−13.5836	−0.436818	−0.218409	−	0.975857i	$-0.570087\pi$
$967$	−13.5836	−0.436818	−0.218409	+	0.975857i	$0.570087\pi$
$968$	8.92377	0.286821
$969$	0	0
$970$	−1.08617	−0.0348749
$971$	39.0472	1.25308	0.626542	−	0.779388i	$-0.284470\pi$
$971$	39.0472	1.25308	0.626542	+	0.779388i	$0.284470\pi$
$972$	0	0
$973$	10.4381	0.334630
$974$	−9.66564	−0.309707
$975$	0	0
$976$	0.0883629	0.00282843
$977$	−13.9595	−0.446604	−0.223302	−	0.974749i	$-0.571684\pi$
$977$	−13.9595	−0.446604	−0.223302	+	0.974749i	$0.571684\pi$
$978$	0	0
$979$	30.6067	0.978196
$980$	−1.99845	−0.0638383
$981$	0	0
$982$	−11.9725	−0.382059
$983$	−11.1853	−0.356755	−0.178378	−	0.983962i	$-0.557085\pi$
$983$	−11.1853	−0.356755	−0.178378	+	0.983962i	$0.557085\pi$
$984$	0	0
$985$	−26.0722	−0.830728
$986$	−5.27448	−0.167974
$987$	0	0
$988$	−12.8840	−0.409895
$989$	−16.3411	−0.519616
$990$	0	0
$991$	36.4779	1.15876	0.579379	−	0.815059i	$-0.303295\pi$
$991$	36.4779	1.15876	0.579379	+	0.815059i	$0.303295\pi$
$992$	−1.14257	−0.0362766
$993$	0	0
$994$	−0.797278	−0.0252881
$995$	−13.9562	−0.442442
$996$	0	0
$997$	−34.7863	−1.10169	−0.550846	−	0.834607i	$-0.685695\pi$
$997$	−34.7863	−1.10169	−0.550846	+	0.834607i	$0.685695\pi$
$998$	9.54548	0.302157
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.v.1.12		19
3.2	odd	2		2667.2.a.q.1.8	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.q.1.8	✓	19	3.2	odd	2
8001.2.a.v.1.12		19	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)

\(f(q)\)	\(=\)	\(q+0.407951 q^{2} -1.83358 q^{4} +1.08992 q^{5} +1.00000 q^{7} -1.56391 q^{8} +O(q^{10})\)	\(q+0.407951 q^{2} -1.83358 q^{4} +1.08992 q^{5} +1.00000 q^{7} -1.56391 q^{8} +0.444635 q^{10} -2.30086 q^{11} -1.28207 q^{13} +0.407951 q^{14} +3.02915 q^{16} -3.85945 q^{17} -5.48075 q^{19} -1.99845 q^{20} -0.938637 q^{22} -5.15478 q^{23} -3.81207 q^{25} -0.523023 q^{26} -1.83358 q^{28} +3.35002 q^{29} -0.261843 q^{31} +4.36357 q^{32} -1.57447 q^{34} +1.08992 q^{35} +4.76674 q^{37} -2.23588 q^{38} -1.70454 q^{40} -4.53932 q^{41} +3.17008 q^{43} +4.21880 q^{44} -2.10290 q^{46} -0.491884 q^{47} +1.00000 q^{49} -1.55514 q^{50} +2.35078 q^{52} +11.3199 q^{53} -2.50775 q^{55} -1.56391 q^{56} +1.36664 q^{58} -1.96040 q^{59} +0.0291708 q^{61} -0.106819 q^{62} -4.27818 q^{64} -1.39736 q^{65} +13.7705 q^{67} +7.07659 q^{68} +0.444635 q^{70} -1.95435 q^{71} +10.9732 q^{73} +1.94460 q^{74} +10.0494 q^{76} -2.30086 q^{77} +1.31077 q^{79} +3.30154 q^{80} -1.85182 q^{82} +9.68361 q^{83} -4.20649 q^{85} +1.29324 q^{86} +3.59834 q^{88} -13.3023 q^{89} -1.28207 q^{91} +9.45169 q^{92} -0.200664 q^{94} -5.97359 q^{95} -2.44284 q^{97} +0.407951 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 4 q^{2} + 22 q^{4} - 5 q^{5} + 19 q^{7} - 9 q^{8}+O(q^{10}) \) 19 * q - 4 * q^2 + 22 * q^4 - 5 * q^5 + 19 * q^7 - 9 * q^8	\( 19 q - 4 q^{2} + 22 q^{4} - 5 q^{5} + 19 q^{7} - 9 q^{8} + 9 q^{11} + 24 q^{13} - 4 q^{14} + 20 q^{16} - 17 q^{17} + 23 q^{19} - 5 q^{20} - 3 q^{22} + 17 q^{23} + 38 q^{25} - 28 q^{26} + 22 q^{28} - 2 q^{29} + 16 q^{31} - 17 q^{32} + 29 q^{34} - 5 q^{35} + 56 q^{37} - 2 q^{38} - 13 q^{40} + 7 q^{41} + 19 q^{43} + 29 q^{44} + 10 q^{46} - 25 q^{47} + 19 q^{49} + 9 q^{50} + 16 q^{52} - 18 q^{53} + 10 q^{55} - 9 q^{56} + 31 q^{58} - 11 q^{59} + 26 q^{61} - 26 q^{62} + 45 q^{64} - 27 q^{65} + 24 q^{67} - 14 q^{68} + 32 q^{71} + 51 q^{73} + 12 q^{76} + 9 q^{77} + 30 q^{79} + 30 q^{80} - 52 q^{82} - q^{83} + 44 q^{85} + 24 q^{86} - 30 q^{88} - 5 q^{89} + 24 q^{91} + 88 q^{92} + 7 q^{94} + 24 q^{95} + 5 q^{97} - 4 q^{98}+O(q^{100}) \) 19 * q - 4 * q^2 + 22 * q^4 - 5 * q^5 + 19 * q^7 - 9 * q^8 + 9 * q^11 + 24 * q^13 - 4 * q^14 + 20 * q^16 - 17 * q^17 + 23 * q^19 - 5 * q^20 - 3 * q^22 + 17 * q^23 + 38 * q^25 - 28 * q^26 + 22 * q^28 - 2 * q^29 + 16 * q^31 - 17 * q^32 + 29 * q^34 - 5 * q^35 + 56 * q^37 - 2 * q^38 - 13 * q^40 + 7 * q^41 + 19 * q^43 + 29 * q^44 + 10 * q^46 - 25 * q^47 + 19 * q^49 + 9 * q^50 + 16 * q^52 - 18 * q^53 + 10 * q^55 - 9 * q^56 + 31 * q^58 - 11 * q^59 + 26 * q^61 - 26 * q^62 + 45 * q^64 - 27 * q^65 + 24 * q^67 - 14 * q^68 + 32 * q^71 + 51 * q^73 + 12 * q^76 + 9 * q^77 + 30 * q^79 + 30 * q^80 - 52 * q^82 - q^83 + 44 * q^85 + 24 * q^86 - 30 * q^88 - 5 * q^89 + 24 * q^91 + 88 * q^92 + 7 * q^94 + 24 * q^95 + 5 * q^97 - 4 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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