LMFDB - Embedded newform 8001.2.a.w.1.11

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:	$1$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 8 x^{19} + 152 x^{17} - 274 x^{16} - 1061 x^{15} + 3125 x^{14} + 2977 x^{13} - 15474 x^{12} + \cdots + 31 $ x^20 - 8x^19 + 152x^17 - 274x^16 - 1061x^15 + 3125x^14 + 2977x^13 - 15474x^12 - 56x^11 + 39579x^10 - 17664x^9 - 52271x^8 + 35701x^7 + 32493x^6 - 25504x^5 - 8607x^4 + 6812x^3 + 609x^2 - 425x + 31
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 889)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$0.170762$ 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)

$f(q)$	$=$	$q-0.170762 q^{2} -1.97084 q^{4} -2.09248 q^{5} +1.00000 q^{7} +0.678071 q^{8} +O(q^{10})$	\(q-0.170762 q^{2} -1.97084 q^{4} -2.09248 q^{5} +1.00000 q^{7} +0.678071 q^{8} +0.357317 q^{10} -0.105046 q^{11} +0.808011 q^{13} -0.170762 q^{14} +3.82589 q^{16} -2.11987 q^{17} +0.0869343 q^{19} +4.12394 q^{20} +0.0179379 q^{22} -5.49710 q^{23} -0.621542 q^{25} -0.137978 q^{26} -1.97084 q^{28} +6.62400 q^{29} +5.84253 q^{31} -2.00946 q^{32} +0.361995 q^{34} -2.09248 q^{35} +7.11832 q^{37} -0.0148451 q^{38} -1.41885 q^{40} -12.0211 q^{41} -7.06811 q^{43} +0.207029 q^{44} +0.938699 q^{46} -2.31361 q^{47} +1.00000 q^{49} +0.106136 q^{50} -1.59246 q^{52} -6.77897 q^{53} +0.219807 q^{55} +0.678071 q^{56} -1.13113 q^{58} -2.38189 q^{59} +6.80820 q^{61} -0.997685 q^{62} -7.30864 q^{64} -1.69074 q^{65} +10.7308 q^{67} +4.17793 q^{68} +0.357317 q^{70} +9.05001 q^{71} +11.3953 q^{73} -1.21554 q^{74} -0.171334 q^{76} -0.105046 q^{77} +10.2976 q^{79} -8.00559 q^{80} +2.05275 q^{82} +0.600098 q^{83} +4.43579 q^{85} +1.20697 q^{86} -0.0712287 q^{88} +10.3716 q^{89} +0.808011 q^{91} +10.8339 q^{92} +0.395079 q^{94} -0.181908 q^{95} -9.17979 q^{97} -0.170762 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 8 q^{2} + 24 q^{4} - 3 q^{5} + 20 q^{7} - 24 q^{8}+O(q^{10}) $ 20 * q - 8 * q^2 + 24 * q^4 - 3 * q^5 + 20 * q^7 - 24 * q^8	$ 20 q - 8 q^{2} + 24 q^{4} - 3 q^{5} + 20 q^{7} - 24 q^{8} - 8 q^{10} - 26 q^{11} - 4 q^{13} - 8 q^{14} + 24 q^{16} - 4 q^{17} + q^{19} + 2 q^{20} + q^{22} - 31 q^{23} + 27 q^{25} - 4 q^{26} + 24 q^{28} - 16 q^{29} + 6 q^{31} - 41 q^{32} - 10 q^{34} - 3 q^{35} + 2 q^{37} - 3 q^{38} - 38 q^{40} - 25 q^{41} + 13 q^{43} - 66 q^{44} + 20 q^{46} - 19 q^{47} + 20 q^{49} + 4 q^{50} + 20 q^{52} - 24 q^{53} - 3 q^{55} - 24 q^{56} + 12 q^{58} - 23 q^{59} - 27 q^{61} - 7 q^{62} + 2 q^{64} - 26 q^{65} + 9 q^{67} + 25 q^{68} - 8 q^{70} - 63 q^{71} - 21 q^{73} - 21 q^{74} - 10 q^{76} - 26 q^{77} + 18 q^{79} + 23 q^{80} - 42 q^{82} + q^{83} - 41 q^{85} + 12 q^{86} + 57 q^{88} + 16 q^{89} - 4 q^{91} - 17 q^{92} + 7 q^{94} - 75 q^{95} - 32 q^{97} - 8 q^{98}+O(q^{100}) $ 20 * q - 8 * q^2 + 24 * q^4 - 3 * q^5 + 20 * q^7 - 24 * q^8 - 8 * q^10 - 26 * q^11 - 4 * q^13 - 8 * q^14 + 24 * q^16 - 4 * q^17 + q^19 + 2 * q^20 + q^22 - 31 * q^23 + 27 * q^25 - 4 * q^26 + 24 * q^28 - 16 * q^29 + 6 * q^31 - 41 * q^32 - 10 * q^34 - 3 * q^35 + 2 * q^37 - 3 * q^38 - 38 * q^40 - 25 * q^41 + 13 * q^43 - 66 * q^44 + 20 * q^46 - 19 * q^47 + 20 * q^49 + 4 * q^50 + 20 * q^52 - 24 * q^53 - 3 * q^55 - 24 * q^56 + 12 * q^58 - 23 * q^59 - 27 * q^61 - 7 * q^62 + 2 * q^64 - 26 * q^65 + 9 * q^67 + 25 * q^68 - 8 * q^70 - 63 * q^71 - 21 * q^73 - 21 * q^74 - 10 * q^76 - 26 * q^77 + 18 * q^79 + 23 * q^80 - 42 * q^82 + q^83 - 41 * q^85 + 12 * q^86 + 57 * q^88 + 16 * q^89 - 4 * q^91 - 17 * q^92 + 7 * q^94 - 75 * q^95 - 32 * q^97 - 8 * 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 $

$2$	−0.170762	−0.120747	−0.0603737	−	0.998176i	$-0.519229\pi$
$2$	−0.170762	−0.120747	−0.0603737	+	0.998176i	$0.519229\pi$
$3$	0	0
$3$	0	0
$4$	−1.97084	−0.985420
$5$	−2.09248	−0.935784	−0.467892	−	0.883786i	$-0.654986\pi$
$5$	−2.09248	−0.935784	−0.467892	+	0.883786i	$0.654986\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0.678071	0.239734
$9$	0	0
$10$	0.357317	0.112993
$11$	−0.105046	−0.0316726	−0.0158363	−	0.999875i	$-0.505041\pi$
$11$	−0.105046	−0.0316726	−0.0158363	+	0.999875i	$0.505041\pi$
$12$	0	0
$13$	0.808011	0.224102	0.112051	−	0.993702i	$-0.464258\pi$
$13$	0.808011	0.224102	0.112051	+	0.993702i	$0.464258\pi$
$14$	−0.170762	−0.0456382
$15$	0	0
$16$	3.82589	0.956473
$17$	−2.11987	−0.514145	−0.257072	−	0.966392i	$-0.582758\pi$
$17$	−2.11987	−0.514145	−0.257072	+	0.966392i	$0.582758\pi$
$18$	0	0
$19$	0.0869343	0.0199441	0.00997205	−	0.999950i	$-0.496826\pi$
$19$	0.0869343	0.0199441	0.00997205	+	0.999950i	$0.496826\pi$
$20$	4.12394	0.922140
$21$	0	0
$22$	0.0179379	0.00382438
$23$	−5.49710	−1.14623	−0.573113	−	0.819477i	$-0.694264\pi$
$23$	−5.49710	−1.14623	−0.573113	+	0.819477i	$0.694264\pi$
$24$	0	0
$25$	−0.621542	−0.124308
$26$	−0.137978	−0.0270597
$27$	0	0
$28$	−1.97084	−0.372454
$29$	6.62400	1.23005	0.615023	−	0.788509i	$-0.289147\pi$
$29$	6.62400	1.23005	0.615023	+	0.788509i	$0.289147\pi$
$30$	0	0
$31$	5.84253	1.04935	0.524675	−	0.851303i	$-0.324187\pi$
$31$	5.84253	1.04935	0.524675	+	0.851303i	$0.324187\pi$
$32$	−2.00946	−0.355226
$33$	0	0
$34$	0.361995	0.0620816
$35$	−2.09248	−0.353693
$36$	0	0
$37$	7.11832	1.17024	0.585122	−	0.810945i	$-0.301046\pi$
$37$	7.11832	1.17024	0.585122	+	0.810945i	$0.301046\pi$
$38$	−0.0148451	−0.00240820
$39$	0	0
$40$	−1.41885	−0.224339
$41$	−12.0211	−1.87737	−0.938687	−	0.344770i	$-0.887957\pi$
$41$	−12.0211	−1.87737	−0.938687	+	0.344770i	$0.887957\pi$
$42$	0	0
$43$	−7.06811	−1.07788	−0.538939	−	0.842345i	$-0.681175\pi$
$43$	−7.06811	−1.07788	−0.538939	+	0.842345i	$0.681175\pi$
$44$	0.207029	0.0312108
$45$	0	0
$46$	0.938699	0.138404
$47$	−2.31361	−0.337475	−0.168738	−	0.985661i	$-0.553969\pi$
$47$	−2.31361	−0.337475	−0.168738	+	0.985661i	$0.553969\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0.106136	0.0150099
$51$	0	0
$52$	−1.59246	−0.220835
$53$	−6.77897	−0.931164	−0.465582	−	0.885005i	$-0.654155\pi$
$53$	−6.77897	−0.931164	−0.465582	+	0.885005i	$0.654155\pi$
$54$	0	0
$55$	0.219807	0.0296387
$56$	0.678071	0.0906110
$57$	0	0
$58$	−1.13113	−0.148525
$59$	−2.38189	−0.310096	−0.155048	−	0.987907i	$-0.549553\pi$
$59$	−2.38189	−0.310096	−0.155048	+	0.987907i	$0.549553\pi$
$60$	0	0
$61$	6.80820	0.871701	0.435851	−	0.900019i	$-0.356448\pi$
$61$	6.80820	0.871701	0.435851	+	0.900019i	$0.356448\pi$
$62$	−0.997685	−0.126706
$63$	0	0
$64$	−7.30864	−0.913580
$65$	−1.69074	−0.209711
$66$	0	0
$67$	10.7308	1.31097	0.655487	−	0.755206i	$-0.272463\pi$
$67$	10.7308	1.31097	0.655487	+	0.755206i	$0.272463\pi$
$68$	4.17793	0.506649
$69$	0	0
$70$	0.357317	0.0427075
$71$	9.05001	1.07404	0.537019	−	0.843570i	$-0.319550\pi$
$71$	9.05001	1.07404	0.537019	+	0.843570i	$0.319550\pi$
$72$	0	0
$73$	11.3953	1.33372	0.666862	−	0.745181i	$-0.267637\pi$
$73$	11.3953	1.33372	0.666862	+	0.745181i	$0.267637\pi$
$74$	−1.21554	−0.141304
$75$	0	0
$76$	−0.171334	−0.0196533
$77$	−0.105046	−0.0119711
$78$	0	0
$79$	10.2976	1.15858	0.579288	−	0.815123i	$-0.303331\pi$
$79$	10.2976	1.15858	0.579288	+	0.815123i	$0.303331\pi$
$80$	−8.00559	−0.895052
$81$	0	0
$82$	2.05275	0.226688
$83$	0.600098	0.0658693	0.0329347	−	0.999458i	$-0.489515\pi$
$83$	0.600098	0.0658693	0.0329347	+	0.999458i	$0.489515\pi$
$84$	0	0
$85$	4.43579	0.481129
$86$	1.20697	0.130151
$87$	0	0
$88$	−0.0712287	−0.00759301
$89$	10.3716	1.09939	0.549694	−	0.835366i	$-0.314744\pi$
$89$	10.3716	1.09939	0.549694	+	0.835366i	$0.314744\pi$
$90$	0	0
$91$	0.808011	0.0847026
$92$	10.8339	1.12951
$93$	0	0
$94$	0.395079	0.0407493
$95$	−0.181908	−0.0186634
$96$	0	0
$97$	−9.17979	−0.932066	−0.466033	−	0.884767i	$-0.654317\pi$
$97$	−9.17979	−0.932066	−0.466033	+	0.884767i	$0.654317\pi$
$98$	−0.170762	−0.0172496
$99$	0	0
$100$	1.22496	0.122496
$101$	10.1137	1.00635	0.503175	−	0.864184i	$-0.332165\pi$
$101$	10.1137	1.00635	0.503175	+	0.864184i	$0.332165\pi$
$102$	0	0
$103$	12.6686	1.24827	0.624137	−	0.781315i	$-0.285451\pi$
$103$	12.6686	1.24827	0.624137	+	0.781315i	$0.285451\pi$
$104$	0.547889	0.0537249
$105$	0	0
$106$	1.15759	0.112436
$107$	−14.6879	−1.41993	−0.709965	−	0.704237i	$-0.751289\pi$
$107$	−14.6879	−1.41993	−0.709965	+	0.704237i	$0.751289\pi$
$108$	0	0
$109$	−8.63491	−0.827074	−0.413537	−	0.910487i	$-0.635707\pi$
$109$	−8.63491	−0.827074	−0.413537	+	0.910487i	$0.635707\pi$
$110$	−0.0375347	−0.00357880
$111$	0	0
$112$	3.82589	0.361513
$113$	−5.56432	−0.523447	−0.261724	−	0.965143i	$-0.584291\pi$
$113$	−5.56432	−0.523447	−0.261724	+	0.965143i	$0.584291\pi$
$114$	0	0
$115$	11.5026	1.07262
$116$	−13.0548	−1.21211
$117$	0	0
$118$	0.406738	0.0374433
$119$	−2.11987	−0.194329
$120$	0	0
$121$	−10.9890	−0.998997
$122$	−1.16259	−0.105256
$123$	0	0
$124$	−11.5147	−1.03405
$125$	11.7629	1.05211
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	5.26696	0.465538
$129$	0	0
$130$	0.288716	0.0253220
$131$	17.9667	1.56976	0.784878	−	0.619650i	$-0.212726\pi$
$131$	17.9667	1.56976	0.784878	+	0.619650i	$0.212726\pi$
$132$	0	0
$133$	0.0869343	0.00753816
$134$	−1.83242	−0.158297
$135$	0	0
$136$	−1.43742	−0.123258
$137$	22.1414	1.89167	0.945833	−	0.324654i	$-0.105248\pi$
$137$	22.1414	1.89167	0.945833	+	0.324654i	$0.105248\pi$
$138$	0	0
$139$	−12.1606	−1.03145	−0.515724	−	0.856755i	$-0.672477\pi$
$139$	−12.1606	−1.03145	−0.515724	+	0.856755i	$0.672477\pi$
$140$	4.12394	0.348536
$141$	0	0
$142$	−1.54540	−0.129687
$143$	−0.0848785	−0.00709790
$144$	0	0
$145$	−13.8606	−1.15106
$146$	−1.94590	−0.161044
$147$	0	0
$148$	−14.0291	−1.15318
$149$	−10.4825	−0.858758	−0.429379	−	0.903124i	$-0.641268\pi$
$149$	−10.4825	−0.858758	−0.429379	+	0.903124i	$0.641268\pi$
$150$	0	0
$151$	3.13454	0.255085	0.127543	−	0.991833i	$-0.459291\pi$
$151$	3.13454	0.255085	0.127543	+	0.991833i	$0.459291\pi$
$152$	0.0589476	0.00478128
$153$	0	0
$154$	0.0179379	0.00144548
$155$	−12.2254	−0.981965
$156$	0	0
$157$	−17.7502	−1.41662	−0.708311	−	0.705900i	$-0.750543\pi$
$157$	−17.7502	−1.41662	−0.708311	+	0.705900i	$0.750543\pi$
$158$	−1.75845	−0.139895
$159$	0	0
$160$	4.20475	0.332414
$161$	−5.49710	−0.433232
$162$	0	0
$163$	5.49492	0.430395	0.215198	−	0.976571i	$-0.430960\pi$
$163$	5.49492	0.430395	0.215198	+	0.976571i	$0.430960\pi$
$164$	23.6916	1.85000
$165$	0	0
$166$	−0.102474	−0.00795354
$167$	−4.21229	−0.325957	−0.162978	−	0.986630i	$-0.552110\pi$
$167$	−4.21229	−0.325957	−0.162978	+	0.986630i	$0.552110\pi$
$168$	0	0
$169$	−12.3471	−0.949778
$170$	−0.757466	−0.0580950
$171$	0	0
$172$	13.9301	1.06216
$173$	7.19299	0.546873	0.273437	−	0.961890i	$-0.411840\pi$
$173$	7.19299	0.546873	0.273437	+	0.961890i	$0.411840\pi$
$174$	0	0
$175$	−0.621542	−0.0469841
$176$	−0.401895	−0.0302940
$177$	0	0
$178$	−1.77108	−0.132748
$179$	−22.8086	−1.70479	−0.852397	−	0.522895i	$-0.824852\pi$
$179$	−22.8086	−1.70479	−0.852397	+	0.522895i	$0.824852\pi$
$180$	0	0
$181$	−23.6385	−1.75704	−0.878518	−	0.477709i	$-0.841467\pi$
$181$	−23.6385	−1.75704	−0.878518	+	0.477709i	$0.841467\pi$
$182$	−0.137978	−0.0102276
$183$	0	0
$184$	−3.72742	−0.274789
$185$	−14.8949	−1.09510
$186$	0	0
$187$	0.222685	0.0162843
$188$	4.55977	0.332555
$189$	0	0
$190$	0.0310631	0.00225355
$191$	−13.0130	−0.941584	−0.470792	−	0.882244i	$-0.656032\pi$
$191$	−13.0130	−0.941584	−0.470792	+	0.882244i	$0.656032\pi$
$192$	0	0
$193$	8.54166	0.614842	0.307421	−	0.951574i	$-0.400534\pi$
$193$	8.54166	0.614842	0.307421	+	0.951574i	$0.400534\pi$
$194$	1.56756	0.112544
$195$	0	0
$196$	−1.97084	−0.140774
$197$	7.31749	0.521349	0.260675	−	0.965427i	$-0.416055\pi$
$197$	7.31749	0.521349	0.260675	+	0.965427i	$0.416055\pi$
$198$	0	0
$199$	−12.4704	−0.884006	−0.442003	−	0.897014i	$-0.645732\pi$
$199$	−12.4704	−0.884006	−0.442003	+	0.897014i	$0.645732\pi$
$200$	−0.421449	−0.0298010
$201$	0	0
$202$	−1.72704	−0.121514
$203$	6.62400	0.464914
$204$	0	0
$205$	25.1538	1.75682
$206$	−2.16332	−0.150726
$207$	0	0
$208$	3.09136	0.214347
$209$	−0.00913211	−0.000631682 0
$210$	0	0
$211$	10.2642	0.706618	0.353309	−	0.935507i	$-0.385056\pi$
$211$	10.2642	0.706618	0.353309	+	0.935507i	$0.385056\pi$
$212$	13.3603	0.917587
$213$	0	0
$214$	2.50814	0.171453
$215$	14.7899	1.00866
$216$	0	0
$217$	5.84253	0.396617
$218$	1.47452	0.0998670
$219$	0	0
$220$	−0.433204	−0.0292066
$221$	−1.71288	−0.115221
$222$	0	0
$223$	−10.7802	−0.721894	−0.360947	−	0.932586i	$-0.617546\pi$
$223$	−10.7802	−0.721894	−0.360947	+	0.932586i	$0.617546\pi$
$224$	−2.00946	−0.134263
$225$	0	0
$226$	0.950178	0.0632049
$227$	−16.1799	−1.07390	−0.536950	−	0.843614i	$-0.680423\pi$
$227$	−16.1799	−1.07390	−0.536950	+	0.843614i	$0.680423\pi$
$228$	0	0
$229$	−6.51569	−0.430569	−0.215284	−	0.976551i	$-0.569068\pi$
$229$	−6.51569	−0.430569	−0.215284	+	0.976551i	$0.569068\pi$
$230$	−1.96421	−0.129516
$231$	0	0
$232$	4.49154	0.294884
$233$	5.54516	0.363276	0.181638	−	0.983365i	$-0.441860\pi$
$233$	5.54516	0.363276	0.181638	+	0.983365i	$0.441860\pi$
$234$	0	0
$235$	4.84119	0.315804
$236$	4.69433	0.305575
$237$	0	0
$238$	0.361995	0.0234646
$239$	−20.4566	−1.32323	−0.661614	−	0.749844i	$-0.730128\pi$
$239$	−20.4566	−1.32323	−0.661614	+	0.749844i	$0.730128\pi$
$240$	0	0
$241$	29.9718	1.93065	0.965325	−	0.261051i	$-0.0840690\pi$
$241$	29.9718	1.93065	0.965325	+	0.261051i	$0.0840690\pi$
$242$	1.87650	0.120626
$243$	0	0
$244$	−13.4179	−0.858992
$245$	−2.09248	−0.133683
$246$	0	0
$247$	0.0702439	0.00446951
$248$	3.96165	0.251565
$249$	0	0
$250$	−2.00867	−0.127039
$251$	−31.1102	−1.96366	−0.981828	−	0.189772i	$-0.939225\pi$
$251$	−31.1102	−1.96366	−0.981828	+	0.189772i	$0.939225\pi$
$252$	0	0
$253$	0.577450	0.0363040
$254$	0.170762	0.0107146
$255$	0	0
$256$	13.7179	0.857368
$257$	−12.6641	−0.789964	−0.394982	−	0.918689i	$-0.629249\pi$
$257$	−12.6641	−0.789964	−0.394982	+	0.918689i	$0.629249\pi$
$258$	0	0
$259$	7.11832	0.442311
$260$	3.33219	0.206653
$261$	0	0
$262$	−3.06804	−0.189544
$263$	10.2863	0.634281	0.317141	−	0.948379i	$-0.397277\pi$
$263$	10.2863	0.634281	0.317141	+	0.948379i	$0.397277\pi$
$264$	0	0
$265$	14.1848	0.871368
$266$	−0.0148451	−0.000910212 0
$267$	0	0
$268$	−21.1487	−1.29186
$269$	20.6179	1.25709	0.628546	−	0.777772i	$-0.283650\pi$
$269$	20.6179	1.25709	0.628546	+	0.777772i	$0.283650\pi$
$270$	0	0
$271$	−5.78251	−0.351262	−0.175631	−	0.984456i	$-0.556197\pi$
$271$	−5.78251	−0.351262	−0.175631	+	0.984456i	$0.556197\pi$
$272$	−8.11041	−0.491766
$273$	0	0
$274$	−3.78092	−0.228414
$275$	0.0652906	0.00393717
$276$	0	0
$277$	0.736630	0.0442598	0.0221299	−	0.999755i	$-0.492955\pi$
$277$	0.736630	0.0442598	0.0221299	+	0.999755i	$0.492955\pi$
$278$	2.07657	0.124545
$279$	0	0
$280$	−1.41885	−0.0847923
$281$	−4.26864	−0.254646	−0.127323	−	0.991861i	$-0.540638\pi$
$281$	−4.26864	−0.254646	−0.127323	+	0.991861i	$0.540638\pi$
$282$	0	0
$283$	−10.8656	−0.645892	−0.322946	−	0.946417i	$-0.604673\pi$
$283$	−10.8656	−0.645892	−0.322946	+	0.946417i	$0.604673\pi$
$284$	−17.8361	−1.05838
$285$	0	0
$286$	0.0144941	0.000857052 0
$287$	−12.0211	−0.709581
$288$	0	0
$289$	−12.5061	−0.735655
$290$	2.36687	0.138987
$291$	0	0
$292$	−22.4584	−1.31428
$293$	−17.1990	−1.00478	−0.502388	−	0.864642i	$-0.667545\pi$
$293$	−17.1990	−1.00478	−0.502388	+	0.864642i	$0.667545\pi$
$294$	0	0
$295$	4.98406	0.290183
$296$	4.82672	0.280548
$297$	0	0
$298$	1.79001	0.103693
$299$	−4.44172	−0.256871
$300$	0	0
$301$	−7.06811	−0.407399
$302$	−0.535262	−0.0308008
$303$	0	0
$304$	0.332601	0.0190760
$305$	−14.2460	−0.815724
$306$	0	0
$307$	−12.2262	−0.697787	−0.348893	−	0.937162i	$-0.613442\pi$
$307$	−12.2262	−0.697787	−0.348893	+	0.937162i	$0.613442\pi$
$308$	0.207029	0.0117966
$309$	0	0
$310$	2.08763	0.118570
$311$	−6.40128	−0.362983	−0.181492	−	0.983392i	$-0.558093\pi$
$311$	−6.40128	−0.362983	−0.181492	+	0.983392i	$0.558093\pi$
$312$	0	0
$313$	0.142894	0.00807683	0.00403842	−	0.999992i	$-0.498715\pi$
$313$	0.142894	0.00807683	0.00403842	+	0.999992i	$0.498715\pi$
$314$	3.03107	0.171053
$315$	0	0
$316$	−20.2950	−1.14168
$317$	15.1694	0.851997	0.425999	−	0.904724i	$-0.359923\pi$
$317$	15.1694	0.851997	0.425999	+	0.904724i	$0.359923\pi$
$318$	0	0
$319$	−0.695826	−0.0389588
$320$	15.2932	0.854914
$321$	0	0
$322$	0.938699	0.0523117
$323$	−0.184290	−0.0102542
$324$	0	0
$325$	−0.502213	−0.0278577
$326$	−0.938326	−0.0519691
$327$	0	0
$328$	−8.15113	−0.450071
$329$	−2.31361	−0.127554
$330$	0	0
$331$	6.02629	0.331235	0.165617	−	0.986190i	$-0.447038\pi$
$331$	6.02629	0.331235	0.165617	+	0.986190i	$0.447038\pi$
$332$	−1.18270	−0.0649089
$333$	0	0
$334$	0.719301	0.0393584
$335$	−22.4539	−1.22679
$336$	0	0
$337$	−8.38437	−0.456726	−0.228363	−	0.973576i	$-0.573337\pi$
$337$	−8.38437	−0.456726	−0.228363	+	0.973576i	$0.573337\pi$
$338$	2.10842	0.114683
$339$	0	0
$340$	−8.74223	−0.474114
$341$	−0.613736	−0.0332356
$342$	0	0
$343$	1.00000	0.0539949
$344$	−4.79268	−0.258404
$345$	0	0
$346$	−1.22829	−0.0660335
$347$	−14.7304	−0.790772	−0.395386	−	0.918515i	$-0.629389\pi$
$347$	−14.7304	−0.790772	−0.395386	+	0.918515i	$0.629389\pi$
$348$	0	0
$349$	25.3999	1.35963	0.679813	−	0.733385i	$-0.262061\pi$
$349$	25.3999	1.35963	0.679813	+	0.733385i	$0.262061\pi$
$350$	0.106136	0.00567321
$351$	0	0
$352$	0.211086	0.0112509
$353$	7.28959	0.387986	0.193993	−	0.981003i	$-0.437856\pi$
$353$	7.28959	0.387986	0.193993	+	0.981003i	$0.437856\pi$
$354$	0	0
$355$	−18.9369	−1.00507
$356$	−20.4408	−1.08336
$357$	0	0
$358$	3.89485	0.205849
$359$	−1.73147	−0.0913834	−0.0456917	−	0.998956i	$-0.514549\pi$
$359$	−1.73147	−0.0913834	−0.0456917	+	0.998956i	$0.514549\pi$
$360$	0	0
$361$	−18.9924	−0.999602
$362$	4.03657	0.212157
$363$	0	0
$364$	−1.59246	−0.0834676
$365$	−23.8445	−1.24808
$366$	0	0
$367$	11.3441	0.592156	0.296078	−	0.955164i	$-0.404321\pi$
$367$	11.3441	0.592156	0.296078	+	0.955164i	$0.404321\pi$
$368$	−21.0313	−1.09633
$369$	0	0
$370$	2.54349	0.132230
$371$	−6.77897	−0.351947
$372$	0	0
$373$	−21.2246	−1.09897	−0.549485	−	0.835504i	$-0.685176\pi$
$373$	−21.2246	−1.09897	−0.549485	+	0.835504i	$0.685176\pi$
$374$	−0.0380262	−0.00196629
$375$	0	0
$376$	−1.56879	−0.0809044
$377$	5.35227	0.275656
$378$	0	0
$379$	−16.2256	−0.833455	−0.416728	−	0.909031i	$-0.636823\pi$
$379$	−16.2256	−0.833455	−0.416728	+	0.909031i	$0.636823\pi$
$380$	0.358511	0.0183913
$381$	0	0
$382$	2.22212	0.113694
$383$	35.8633	1.83253	0.916263	−	0.400576i	$-0.131190\pi$
$383$	35.8633	1.83253	0.916263	+	0.400576i	$0.131190\pi$
$384$	0	0
$385$	0.219807	0.0112024
$386$	−1.45860	−0.0742405
$387$	0	0
$388$	18.0919	0.918477
$389$	−33.5216	−1.69961	−0.849806	−	0.527096i	$-0.823281\pi$
$389$	−33.5216	−1.69961	−0.849806	+	0.527096i	$0.823281\pi$
$390$	0	0
$391$	11.6532	0.589326
$392$	0.678071	0.0342477
$393$	0	0
$394$	−1.24955	−0.0629515
$395$	−21.5476	−1.08418
$396$	0	0
$397$	31.2291	1.56734	0.783672	−	0.621174i	$-0.213344\pi$
$397$	31.2291	1.56734	0.783672	+	0.621174i	$0.213344\pi$
$398$	2.12948	0.106741
$399$	0	0
$400$	−2.37795	−0.118898
$401$	−15.4148	−0.769779	−0.384890	−	0.922963i	$-0.625760\pi$
$401$	−15.4148	−0.769779	−0.384890	+	0.922963i	$0.625760\pi$
$402$	0	0
$403$	4.72083	0.235161
$404$	−19.9325	−0.991678
$405$	0	0
$406$	−1.13113	−0.0561371
$407$	−0.747752	−0.0370647
$408$	0	0
$409$	−35.5778	−1.75921	−0.879605	−	0.475704i	$-0.842193\pi$
$409$	−35.5778	−1.75921	−0.879605	+	0.475704i	$0.842193\pi$
$410$	−4.29532	−0.212131
$411$	0	0
$412$	−24.9678	−1.23007
$413$	−2.38189	−0.117205
$414$	0	0
$415$	−1.25569	−0.0616394
$416$	−1.62367	−0.0796068
$417$	0	0
$418$	0.00155942	7.62738e−5 0
$419$	14.2947	0.698342	0.349171	−	0.937059i	$-0.386463\pi$
$419$	14.2947	0.698342	0.349171	+	0.937059i	$0.386463\pi$
$420$	0	0
$421$	−16.8025	−0.818901	−0.409451	−	0.912332i	$-0.634280\pi$
$421$	−16.8025	−0.818901	−0.409451	+	0.912332i	$0.634280\pi$
$422$	−1.75274	−0.0853223
$423$	0	0
$424$	−4.59662	−0.223232
$425$	1.31759	0.0639125
$426$	0	0
$427$	6.80820	0.329472
$428$	28.9474	1.39923
$429$	0	0
$430$	−2.52555	−0.121793
$431$	9.43818	0.454621	0.227311	−	0.973822i	$-0.427007\pi$
$431$	9.43818	0.454621	0.227311	+	0.973822i	$0.427007\pi$
$432$	0	0
$433$	−12.2811	−0.590192	−0.295096	−	0.955468i	$-0.595352\pi$
$433$	−12.2811	−0.590192	−0.295096	+	0.955468i	$0.595352\pi$
$434$	−0.997685	−0.0478904
$435$	0	0
$436$	17.0180	0.815016
$437$	−0.477887	−0.0228604
$438$	0	0
$439$	−17.6112	−0.840537	−0.420268	−	0.907400i	$-0.638064\pi$
$439$	−17.6112	−0.840537	−0.420268	+	0.907400i	$0.638064\pi$
$440$	0.149044	0.00710541
$441$	0	0
$442$	0.292496	0.0139126
$443$	−31.6272	−1.50265	−0.751326	−	0.659931i	$-0.770586\pi$
$443$	−31.6272	−1.50265	−0.751326	+	0.659931i	$0.770586\pi$
$444$	0	0
$445$	−21.7024	−1.02879
$446$	1.84085	0.0871667
$447$	0	0
$448$	−7.30864	−0.345301
$449$	−28.4968	−1.34485	−0.672424	−	0.740166i	$-0.734747\pi$
$449$	−28.4968	−1.34485	−0.672424	+	0.740166i	$0.734747\pi$
$450$	0	0
$451$	1.26277	0.0594614
$452$	10.9664	0.515816
$453$	0	0
$454$	2.76292	0.129670
$455$	−1.69074	−0.0792633
$456$	0	0
$457$	−3.02187	−0.141357	−0.0706786	−	0.997499i	$-0.522516\pi$
$457$	−3.02187	−0.141357	−0.0706786	+	0.997499i	$0.522516\pi$
$458$	1.11264	0.0519900
$459$	0	0
$460$	−22.6697	−1.05698
$461$	−9.77331	−0.455189	−0.227594	−	0.973756i	$-0.573086\pi$
$461$	−9.77331	−0.455189	−0.227594	+	0.973756i	$0.573086\pi$
$462$	0	0
$463$	13.4090	0.623170	0.311585	−	0.950218i	$-0.399140\pi$
$463$	13.4090	0.623170	0.311585	+	0.950218i	$0.399140\pi$
$464$	25.3427	1.17651
$465$	0	0
$466$	−0.946906	−0.0438646
$467$	−2.76155	−0.127789	−0.0638946	−	0.997957i	$-0.520352\pi$
$467$	−2.76155	−0.127789	−0.0638946	+	0.997957i	$0.520352\pi$
$468$	0	0
$469$	10.7308	0.495502
$470$	−0.826693	−0.0381325
$471$	0	0
$472$	−1.61509	−0.0743407
$473$	0.742478	0.0341392
$474$	0	0
$475$	−0.0540333	−0.00247922
$476$	4.17793	0.191495
$477$	0	0
$478$	3.49322	0.159776
$479$	−8.67556	−0.396396	−0.198198	−	0.980162i	$-0.563509\pi$
$479$	−8.67556	−0.396396	−0.198198	+	0.980162i	$0.563509\pi$
$480$	0	0
$481$	5.75168	0.262254
$482$	−5.11805	−0.233121
$483$	0	0
$484$	21.6575	0.984432
$485$	19.2085	0.872213
$486$	0	0
$487$	6.71387	0.304235	0.152117	−	0.988362i	$-0.451391\pi$
$487$	6.71387	0.304235	0.152117	+	0.988362i	$0.451391\pi$
$488$	4.61644	0.208977
$489$	0	0
$490$	0.357317	0.0161419
$491$	4.78940	0.216143	0.108071	−	0.994143i	$-0.465533\pi$
$491$	4.78940	0.216143	0.108071	+	0.994143i	$0.465533\pi$
$492$	0	0
$493$	−14.0420	−0.632422
$494$	−0.0119950	−0.000539681 0
$495$	0	0
$496$	22.3529	1.00367
$497$	9.05001	0.405948
$498$	0	0
$499$	25.3569	1.13513	0.567565	−	0.823329i	$-0.307886\pi$
$499$	25.3569	1.13513	0.567565	+	0.823329i	$0.307886\pi$
$500$	−23.1829	−1.03677
$501$	0	0
$502$	5.31245	0.237106
$503$	−4.99236	−0.222599	−0.111299	−	0.993787i	$-0.535501\pi$
$503$	−4.99236	−0.222599	−0.111299	+	0.993787i	$0.535501\pi$
$504$	0	0
$505$	−21.1627	−0.941727
$506$	−0.0986067	−0.00438360
$507$	0	0
$508$	1.97084	0.0874419
$509$	10.3690	0.459597	0.229798	−	0.973238i	$-0.426193\pi$
$509$	10.3690	0.459597	0.229798	+	0.973238i	$0.426193\pi$
$510$	0	0
$511$	11.3953	0.504100
$512$	−12.8764	−0.569063
$513$	0	0
$514$	2.16255	0.0953861
$515$	−26.5087	−1.16811
$516$	0	0
$517$	0.243036	0.0106887
$518$	−1.21554	−0.0534079
$519$	0	0
$520$	−1.14644	−0.0502749
$521$	2.16505	0.0948526	0.0474263	−	0.998875i	$-0.484898\pi$
$521$	2.16505	0.0948526	0.0474263	+	0.998875i	$0.484898\pi$
$522$	0	0
$523$	9.70712	0.424463	0.212231	−	0.977219i	$-0.431927\pi$
$523$	9.70712	0.424463	0.212231	+	0.977219i	$0.431927\pi$
$524$	−35.4095	−1.54687
$525$	0	0
$526$	−1.75652	−0.0765878
$527$	−12.3854	−0.539518
$528$	0	0
$529$	7.21815	0.313832
$530$	−2.42224	−0.105215
$531$	0	0
$532$	−0.171334	−0.00742825
$533$	−9.71315	−0.420723
$534$	0	0
$535$	30.7340	1.32875
$536$	7.27623	0.314285
$537$	0	0
$538$	−3.52076	−0.151791
$539$	−0.105046	−0.00452466
$540$	0	0
$541$	−42.9242	−1.84545	−0.922727	−	0.385453i	$-0.874045\pi$
$541$	−42.9242	−1.84545	−0.922727	+	0.385453i	$0.874045\pi$
$542$	0.987435	0.0424140
$543$	0	0
$544$	4.25980	0.182637
$545$	18.0683	0.773963
$546$	0	0
$547$	27.2813	1.16646	0.583232	−	0.812306i	$-0.301788\pi$
$547$	27.2813	1.16646	0.583232	+	0.812306i	$0.301788\pi$
$548$	−43.6371	−1.86409
$549$	0	0
$550$	−0.0111492	−0.000475403 0
$551$	0.575853	0.0245322
$552$	0	0
$553$	10.2976	0.437900
$554$	−0.125789	−0.00534425
$555$	0	0
$556$	23.9666	1.01641
$557$	26.2213	1.11103	0.555517	−	0.831505i	$-0.312520\pi$
$557$	26.2213	1.11103	0.555517	+	0.831505i	$0.312520\pi$
$558$	0	0
$559$	−5.71111	−0.241554
$560$	−8.00559	−0.338298
$561$	0	0
$562$	0.728924	0.0307478
$563$	−0.982962	−0.0414269	−0.0207134	−	0.999785i	$-0.506594\pi$
$563$	−0.982962	−0.0414269	−0.0207134	+	0.999785i	$0.506594\pi$
$564$	0	0
$565$	11.6432	0.489834
$566$	1.85543	0.0779897
$567$	0	0
$568$	6.13655	0.257484
$569$	−35.5211	−1.48912	−0.744560	−	0.667555i	$-0.767341\pi$
$569$	−35.5211	−1.48912	−0.744560	+	0.667555i	$0.767341\pi$
$570$	0	0
$571$	−33.2896	−1.39313	−0.696563	−	0.717495i	$-0.745288\pi$
$571$	−33.2896	−1.39313	−0.696563	+	0.717495i	$0.745288\pi$
$572$	0.167282	0.00699441
$573$	0	0
$574$	2.05275	0.0856800
$575$	3.41668	0.142485
$576$	0	0
$577$	−2.58118	−0.107456	−0.0537280	−	0.998556i	$-0.517110\pi$
$577$	−2.58118	−0.107456	−0.0537280	+	0.998556i	$0.517110\pi$
$578$	2.13558	0.0888284
$579$	0	0
$580$	27.3170	1.13428
$581$	0.600098	0.0248963
$582$	0	0
$583$	0.712105	0.0294924
$584$	7.72685	0.319739
$585$	0	0
$586$	2.93694	0.121324
$587$	−13.9735	−0.576748	−0.288374	−	0.957518i	$-0.593115\pi$
$587$	−13.9735	−0.576748	−0.288374	+	0.957518i	$0.593115\pi$
$588$	0	0
$589$	0.507916	0.0209283
$590$	−0.851090	−0.0350388
$591$	0	0
$592$	27.2339	1.11931
$593$	−22.4345	−0.921273	−0.460636	−	0.887589i	$-0.652379\pi$
$593$	−22.4345	−0.921273	−0.460636	+	0.887589i	$0.652379\pi$
$594$	0	0
$595$	4.43579	0.181850
$596$	20.6593	0.846237
$597$	0	0
$598$	0.758479	0.0310165
$599$	14.2568	0.582518	0.291259	−	0.956644i	$-0.405926\pi$
$599$	14.2568	0.582518	0.291259	+	0.956644i	$0.405926\pi$
$600$	0	0
$601$	23.0253	0.939222	0.469611	−	0.882873i	$-0.344394\pi$
$601$	23.0253	0.939222	0.469611	+	0.882873i	$0.344394\pi$
$602$	1.20697	0.0491924
$603$	0	0
$604$	−6.17767	−0.251366
$605$	22.9942	0.934845
$606$	0	0
$607$	−36.8434	−1.49543	−0.747715	−	0.664020i	$-0.768849\pi$
$607$	−36.8434	−1.49543	−0.747715	+	0.664020i	$0.768849\pi$
$608$	−0.174691	−0.00708465
$609$	0	0
$610$	2.43268	0.0984965
$611$	−1.86943	−0.0756289
$612$	0	0
$613$	−45.1182	−1.82231	−0.911154	−	0.412066i	$-0.864807\pi$
$613$	−45.1182	−1.82231	−0.911154	+	0.412066i	$0.864807\pi$
$614$	2.08778	0.0842559
$615$	0	0
$616$	−0.0712287	−0.00286989
$617$	−2.17969	−0.0877511	−0.0438755	−	0.999037i	$-0.513971\pi$
$617$	−2.17969	−0.0877511	−0.0438755	+	0.999037i	$0.513971\pi$
$618$	0	0
$619$	−4.55502	−0.183082	−0.0915408	−	0.995801i	$-0.529179\pi$
$619$	−4.55502	−0.183082	−0.0915408	+	0.995801i	$0.529179\pi$
$620$	24.0942	0.967648
$621$	0	0
$622$	1.09310	0.0438293
$623$	10.3716	0.415530
$624$	0	0
$625$	−21.5060	−0.860239
$626$	−0.0244009	−0.000975256 0
$627$	0	0
$628$	34.9829	1.39597
$629$	−15.0899	−0.601675
$630$	0	0
$631$	4.01294	0.159753	0.0798763	−	0.996805i	$-0.474547\pi$
$631$	4.01294	0.159753	0.0798763	+	0.996805i	$0.474547\pi$
$632$	6.98253	0.277750
$633$	0	0
$634$	−2.59036	−0.102876
$635$	2.09248	0.0830374
$636$	0	0
$637$	0.808011	0.0320146
$638$	0.118821	0.00470417
$639$	0	0
$640$	−11.0210	−0.435643
$641$	−15.3614	−0.606739	−0.303369	−	0.952873i	$-0.598112\pi$
$641$	−15.3614	−0.606739	−0.303369	+	0.952873i	$0.598112\pi$
$642$	0	0
$643$	−16.6240	−0.655585	−0.327792	−	0.944750i	$-0.606305\pi$
$643$	−16.6240	−0.655585	−0.327792	+	0.944750i	$0.606305\pi$
$644$	10.8339	0.426916
$645$	0	0
$646$	0.0314698	0.00123816
$647$	−3.33572	−0.131141	−0.0655704	−	0.997848i	$-0.520887\pi$
$647$	−3.33572	−0.131141	−0.0655704	+	0.997848i	$0.520887\pi$
$648$	0	0
$649$	0.250209	0.00982156
$650$	0.0857591	0.00336375
$651$	0	0
$652$	−10.8296	−0.424120
$653$	2.02411	0.0792094	0.0396047	−	0.999215i	$-0.487390\pi$
$653$	2.02411	0.0792094	0.0396047	+	0.999215i	$0.487390\pi$
$654$	0	0
$655$	−37.5949	−1.46895
$656$	−45.9913	−1.79566
$657$	0	0
$658$	0.395079	0.0154018
$659$	7.73013	0.301123	0.150562	−	0.988601i	$-0.451892\pi$
$659$	7.73013	0.301123	0.150562	+	0.988601i	$0.451892\pi$
$660$	0	0
$661$	4.41299	0.171646	0.0858228	−	0.996310i	$-0.472648\pi$
$661$	4.41299	0.171646	0.0858228	+	0.996310i	$0.472648\pi$
$662$	−1.02906	−0.0399957
$663$	0	0
$664$	0.406909	0.0157911
$665$	−0.181908	−0.00705409
$666$	0	0
$667$	−36.4128	−1.40991
$668$	8.30175	0.321205
$669$	0	0
$670$	3.83429	0.148132
$671$	−0.715176	−0.0276091
$672$	0	0
$673$	−45.8987	−1.76927	−0.884633	−	0.466289i	$-0.845591\pi$
$673$	−45.8987	−1.76927	−0.884633	+	0.466289i	$0.845591\pi$
$674$	1.43174	0.0551484
$675$	0	0
$676$	24.3342	0.935931
$677$	−14.7478	−0.566804	−0.283402	−	0.959001i	$-0.591463\pi$
$677$	−14.7478	−0.566804	−0.283402	+	0.959001i	$0.591463\pi$
$678$	0	0
$679$	−9.17979	−0.352288
$680$	3.00778	0.115343
$681$	0	0
$682$	0.104803	0.00401312
$683$	0.702351	0.0268747	0.0134373	−	0.999910i	$-0.495723\pi$
$683$	0.702351	0.0268747	0.0134373	+	0.999910i	$0.495723\pi$
$684$	0	0
$685$	−46.3303	−1.77019
$686$	−0.170762	−0.00651974
$687$	0	0
$688$	−27.0418	−1.03096
$689$	−5.47749	−0.208676
$690$	0	0
$691$	9.88041	0.375868	0.187934	−	0.982182i	$-0.439821\pi$
$691$	9.88041	0.375868	0.187934	+	0.982182i	$0.439821\pi$
$692$	−14.1762	−0.538900
$693$	0	0
$694$	2.51541	0.0954835
$695$	25.4457	0.965212
$696$	0	0
$697$	25.4831	0.965243
$698$	−4.33735	−0.164171
$699$	0	0
$700$	1.22496	0.0462991
$701$	8.29961	0.313472	0.156736	−	0.987641i	$-0.449903\pi$
$701$	8.29961	0.313472	0.156736	+	0.987641i	$0.449903\pi$
$702$	0	0
$703$	0.618826	0.0233395
$704$	0.767745	0.0289355
$705$	0	0
$706$	−1.24479	−0.0468482
$707$	10.1137	0.380365
$708$	0	0
$709$	40.1970	1.50963	0.754815	−	0.655938i	$-0.227727\pi$
$709$	40.1970	1.50963	0.754815	+	0.655938i	$0.227727\pi$
$710$	3.23372	0.121359
$711$	0	0
$712$	7.03268	0.263561
$713$	−32.1170	−1.20279
$714$	0	0
$715$	0.177606	0.00664210
$716$	44.9521	1.67994
$717$	0	0
$718$	0.295670	0.0110343
$719$	−15.5900	−0.581408	−0.290704	−	0.956813i	$-0.593889\pi$
$719$	−15.5900	−0.581408	−0.290704	+	0.956813i	$0.593889\pi$
$720$	0	0
$721$	12.6686	0.471803
$722$	3.24320	0.120699
$723$	0	0
$724$	46.5877	1.73142
$725$	−4.11709	−0.152905
$726$	0	0
$727$	−31.1166	−1.15405	−0.577025	−	0.816727i	$-0.695786\pi$
$727$	−31.1166	−1.15405	−0.577025	+	0.816727i	$0.695786\pi$
$728$	0.547889	0.0203061
$729$	0	0
$730$	4.07174	0.150702
$731$	14.9835	0.554185
$732$	0	0
$733$	1.46591	0.0541446	0.0270723	−	0.999633i	$-0.491382\pi$
$733$	1.46591	0.0541446	0.0270723	+	0.999633i	$0.491382\pi$
$734$	−1.93714	−0.0715012
$735$	0	0
$736$	11.0462	0.407169
$737$	−1.12723	−0.0415220
$738$	0	0
$739$	−31.4380	−1.15647	−0.578234	−	0.815871i	$-0.696258\pi$
$739$	−31.4380	−1.15647	−0.578234	+	0.815871i	$0.696258\pi$
$740$	29.3555	1.07913
$741$	0	0
$742$	1.15759	0.0424966
$743$	22.3714	0.820728	0.410364	−	0.911922i	$-0.365402\pi$
$743$	22.3714	0.820728	0.410364	+	0.911922i	$0.365402\pi$
$744$	0	0
$745$	21.9343	0.803612
$746$	3.62437	0.132698
$747$	0	0
$748$	−0.438876	−0.0160469
$749$	−14.6879	−0.536683
$750$	0	0
$751$	41.1115	1.50018	0.750090	−	0.661335i	$-0.230010\pi$
$751$	41.1115	1.50018	0.750090	+	0.661335i	$0.230010\pi$
$752$	−8.85164	−0.322786
$753$	0	0
$754$	−0.913966	−0.0332847
$755$	−6.55895	−0.238705
$756$	0	0
$757$	−9.51476	−0.345820	−0.172910	−	0.984938i	$-0.555317\pi$
$757$	−9.51476	−0.345820	−0.172910	+	0.984938i	$0.555317\pi$
$758$	2.77073	0.100638
$759$	0	0
$760$	−0.123346	−0.00447425
$761$	−35.3516	−1.28150	−0.640748	−	0.767752i	$-0.721375\pi$
$761$	−35.3516	−1.28150	−0.640748	+	0.767752i	$0.721375\pi$
$762$	0	0
$763$	−8.63491	−0.312605
$764$	25.6465	0.927856
$765$	0	0
$766$	−6.12410	−0.221273
$767$	−1.92460	−0.0694932
$768$	0	0
$769$	7.40477	0.267023	0.133511	−	0.991047i	$-0.457375\pi$
$769$	7.40477	0.267023	0.133511	+	0.991047i	$0.457375\pi$
$770$	−0.0375347	−0.00135266
$771$	0	0
$772$	−16.8342	−0.605878
$773$	38.4610	1.38335	0.691673	−	0.722210i	$-0.256874\pi$
$773$	38.4610	1.38335	0.691673	+	0.722210i	$0.256874\pi$
$774$	0	0
$775$	−3.63138	−0.130443
$776$	−6.22454	−0.223448
$777$	0	0
$778$	5.72423	0.205224
$779$	−1.04504	−0.0374425
$780$	0	0
$781$	−0.950669	−0.0340176
$782$	−1.98992	−0.0711595
$783$	0	0
$784$	3.82589	0.136639
$785$	37.1419	1.32565
$786$	0	0
$787$	−0.555186	−0.0197902	−0.00989512	−	0.999951i	$-0.503150\pi$
$787$	−0.555186	−0.0197902	−0.00989512	+	0.999951i	$0.503150\pi$
$788$	−14.4216	−0.513748
$789$	0	0
$790$	3.67952	0.130911
$791$	−5.56432	−0.197845
$792$	0	0
$793$	5.50110	0.195350
$794$	−5.33276	−0.189253
$795$	0	0
$796$	24.5772	0.871117
$797$	42.6739	1.51159	0.755794	−	0.654809i	$-0.227251\pi$
$797$	42.6739	1.51159	0.755794	+	0.654809i	$0.227251\pi$
$798$	0	0
$799$	4.90457	0.173511
$800$	1.24896	0.0441575
$801$	0	0
$802$	2.63227	0.0929488
$803$	−1.19704	−0.0422425
$804$	0	0
$805$	11.5026	0.405412
$806$	−0.806141	−0.0283951
$807$	0	0
$808$	6.85780	0.241257
$809$	−37.6590	−1.32402	−0.662011	−	0.749494i	$-0.730297\pi$
$809$	−37.6590	−1.32402	−0.662011	+	0.749494i	$0.730297\pi$
$810$	0	0
$811$	−53.5990	−1.88211	−0.941057	−	0.338248i	$-0.890166\pi$
$811$	−53.5990	−1.88211	−0.941057	+	0.338248i	$0.890166\pi$
$812$	−13.0548	−0.458135
$813$	0	0
$814$	0.127688	0.00447546
$815$	−11.4980	−0.402757
$816$	0	0
$817$	−0.614461	−0.0214973
$818$	6.07536	0.212420
$819$	0	0
$820$	−49.5741	−1.73120
$821$	−12.7495	−0.444962	−0.222481	−	0.974937i	$-0.571415\pi$
$821$	−12.7495	−0.444962	−0.222481	+	0.974937i	$0.571415\pi$
$822$	0	0
$823$	−54.0883	−1.88540	−0.942700	−	0.333642i	$-0.891722\pi$
$823$	−54.0883	−1.88540	−0.942700	+	0.333642i	$0.891722\pi$
$824$	8.59020	0.299254
$825$	0	0
$826$	0.406738	0.0141522
$827$	42.6925	1.48456	0.742281	−	0.670089i	$-0.233744\pi$
$827$	42.6925	1.48456	0.742281	+	0.670089i	$0.233744\pi$
$828$	0	0
$829$	−10.9337	−0.379744	−0.189872	−	0.981809i	$-0.560807\pi$
$829$	−10.9337	−0.379744	−0.189872	+	0.981809i	$0.560807\pi$
$830$	0.214425	0.00744280
$831$	0	0
$832$	−5.90546	−0.204735
$833$	−2.11987	−0.0734493
$834$	0	0
$835$	8.81412	0.305025
$836$	0.0179979	0.000622472 0
$837$	0	0
$838$	−2.44100	−0.0843230
$839$	34.6305	1.19558	0.597788	−	0.801654i	$-0.296046\pi$
$839$	34.6305	1.19558	0.597788	+	0.801654i	$0.296046\pi$
$840$	0	0
$841$	14.8774	0.513014
$842$	2.86923	0.0988802
$843$	0	0
$844$	−20.2291	−0.696316
$845$	25.8361	0.888787
$846$	0	0
$847$	−10.9890	−0.377585
$848$	−25.9356	−0.890633
$849$	0	0
$850$	−0.224995	−0.00771726
$851$	−39.1301	−1.34136
$852$	0	0
$853$	−31.8641	−1.09101	−0.545503	−	0.838109i	$-0.683661\pi$
$853$	−31.8641	−1.09101	−0.545503	+	0.838109i	$0.683661\pi$
$854$	−1.16259	−0.0397829
$855$	0	0
$856$	−9.95941	−0.340406
$857$	15.1325	0.516918	0.258459	−	0.966022i	$-0.416785\pi$
$857$	15.1325	0.516918	0.258459	+	0.966022i	$0.416785\pi$
$858$	0	0
$859$	−5.42054	−0.184946	−0.0924731	−	0.995715i	$-0.529477\pi$
$859$	−5.42054	−0.184946	−0.0924731	+	0.995715i	$0.529477\pi$
$860$	−29.1485	−0.993954
$861$	0	0
$862$	−1.61169	−0.0548943
$863$	−34.1292	−1.16177	−0.580886	−	0.813985i	$-0.697294\pi$
$863$	−34.1292	−1.16177	−0.580886	+	0.813985i	$0.697294\pi$
$864$	0	0
$865$	−15.0512	−0.511755
$866$	2.09715	0.0712642
$867$	0	0
$868$	−11.5147	−0.390834
$869$	−1.08173	−0.0366951
$870$	0	0
$871$	8.67060	0.293792
$872$	−5.85508	−0.198278
$873$	0	0
$874$	0.0816051	0.00276033
$875$	11.7629	0.397660
$876$	0	0
$877$	35.6664	1.20437	0.602184	−	0.798357i	$-0.294297\pi$
$877$	35.6664	1.20437	0.602184	+	0.798357i	$0.294297\pi$
$878$	3.00733	0.101493
$879$	0	0
$880$	0.840956	0.0283486
$881$	−46.1962	−1.55639	−0.778195	−	0.628023i	$-0.783864\pi$
$881$	−46.1962	−1.55639	−0.778195	+	0.628023i	$0.783864\pi$
$882$	0	0
$883$	5.83935	0.196510	0.0982549	−	0.995161i	$-0.468674\pi$
$883$	5.83935	0.196510	0.0982549	+	0.995161i	$0.468674\pi$
$884$	3.37582	0.113541
$885$	0	0
$886$	5.40074	0.181441
$887$	30.1007	1.01068	0.505341	−	0.862920i	$-0.331367\pi$
$887$	30.1007	1.01068	0.505341	+	0.862920i	$0.331367\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	3.70595	0.124224
$891$	0	0
$892$	21.2460	0.711369
$893$	−0.201132	−0.00673064
$894$	0	0
$895$	47.7264	1.59532
$896$	5.26696	0.175957
$897$	0	0
$898$	4.86619	0.162387
$899$	38.7009	1.29075
$900$	0	0
$901$	14.3706	0.478753
$902$	−0.215633	−0.00717980
$903$	0	0
$904$	−3.77300	−0.125488
$905$	49.4630	1.64421
$906$	0	0
$907$	25.6126	0.850454	0.425227	−	0.905087i	$-0.360194\pi$
$907$	25.6126	0.850454	0.425227	+	0.905087i	$0.360194\pi$
$908$	31.8880	1.05824
$909$	0	0
$910$	0.288716	0.00957083
$911$	−25.1999	−0.834908	−0.417454	−	0.908698i	$-0.637078\pi$
$911$	−25.1999	−0.834908	−0.417454	+	0.908698i	$0.637078\pi$
$912$	0	0
$913$	−0.0630380	−0.00208625
$914$	0.516023	0.0170685
$915$	0	0
$916$	12.8414	0.424291
$917$	17.9667	0.593312
$918$	0	0
$919$	36.1759	1.19333	0.596667	−	0.802489i	$-0.296492\pi$
$919$	36.1759	1.19333	0.596667	+	0.802489i	$0.296492\pi$
$920$	7.79955	0.257143
$921$	0	0
$922$	1.66892	0.0549628
$923$	7.31251	0.240694
$924$	0	0
$925$	−4.42433	−0.145471
$926$	−2.28976	−0.0752461
$927$	0	0
$928$	−13.3107	−0.436944
$929$	29.7844	0.977195	0.488597	−	0.872509i	$-0.337509\pi$
$929$	29.7844	0.977195	0.488597	+	0.872509i	$0.337509\pi$
$930$	0	0
$931$	0.0869343	0.00284916
$932$	−10.9286	−0.357979
$933$	0	0
$934$	0.471569	0.0154302
$935$	−0.465962	−0.0152386
$936$	0	0
$937$	19.6196	0.640945	0.320472	−	0.947258i	$-0.396158\pi$
$937$	19.6196	0.640945	0.320472	+	0.947258i	$0.396158\pi$
$938$	−1.83242	−0.0598305
$939$	0	0
$940$	−9.54120	−0.311200
$941$	7.55777	0.246376	0.123188	−	0.992383i	$-0.460688\pi$
$941$	7.55777	0.246376	0.123188	+	0.992383i	$0.460688\pi$
$942$	0	0
$943$	66.0810	2.15189
$944$	−9.11287	−0.296599
$945$	0	0
$946$	−0.126787	−0.00412222
$947$	45.6186	1.48241	0.741203	−	0.671281i	$-0.234256\pi$
$947$	45.6186	1.48241	0.741203	+	0.671281i	$0.234256\pi$
$948$	0	0
$949$	9.20756	0.298890
$950$	0.00922686	0.000299359 0
$951$	0	0
$952$	−1.43742	−0.0465872
$953$	0.279229	0.00904511	0.00452255	−	0.999990i	$-0.498560\pi$
$953$	0.279229	0.00904511	0.00452255	+	0.999990i	$0.498560\pi$
$954$	0	0
$955$	27.2293	0.881120
$956$	40.3167	1.30394
$957$	0	0
$958$	1.48146	0.0478638
$959$	22.1414	0.714982
$960$	0	0
$961$	3.13518	0.101135
$962$	−0.982172	−0.0316665
$963$	0	0
$964$	−59.0695	−1.90250
$965$	−17.8732	−0.575359
$966$	0	0
$967$	54.5935	1.75561	0.877804	−	0.479021i	$-0.159008\pi$
$967$	54.5935	1.75561	0.877804	+	0.479021i	$0.159008\pi$
$968$	−7.45129	−0.239494
$969$	0	0
$970$	−3.28009	−0.105317
$971$	−13.5864	−0.436007	−0.218004	−	0.975948i	$-0.569954\pi$
$971$	−13.5864	−0.436007	−0.218004	+	0.975948i	$0.569954\pi$
$972$	0	0
$973$	−12.1606	−0.389850
$974$	−1.14648	−0.0367355
$975$	0	0
$976$	26.0474	0.833758
$977$	19.3476	0.618984	0.309492	−	0.950902i	$-0.399841\pi$
$977$	19.3476	0.618984	0.309492	+	0.950902i	$0.399841\pi$
$978$	0	0
$979$	−1.08950	−0.0348205
$980$	4.12394	0.131734
$981$	0	0
$982$	−0.817850	−0.0260986
$983$	52.9492	1.68882	0.844409	−	0.535699i	$-0.179952\pi$
$983$	52.9492	1.68882	0.844409	+	0.535699i	$0.179952\pi$
$984$	0	0
$985$	−15.3117	−0.487870
$986$	2.39785	0.0763633
$987$	0	0
$988$	−0.138439	−0.00440435
$989$	38.8542	1.23549
$990$	0	0
$991$	39.0942	1.24187	0.620934	−	0.783863i	$-0.286754\pi$
$991$	39.0942	1.24187	0.620934	+	0.783863i	$0.286754\pi$
$992$	−11.7403	−0.372756
$993$	0	0
$994$	−1.54540	−0.0490172
$995$	26.0941	0.827239
$996$	0	0
$997$	25.4016	0.804476	0.402238	−	0.915535i	$-0.368232\pi$
$997$	25.4016	0.804476	0.402238	+	0.915535i	$0.368232\pi$
$998$	−4.33000	−0.137064
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.w.1.11		20
3.2	odd	2		889.2.a.d.1.10	✓	20
21.20	even	2		6223.2.a.l.1.10		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
889.2.a.d.1.10	✓	20	3.2	odd	2
6223.2.a.l.1.10		20	21.20	even	2
8001.2.a.w.1.11		20	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)

\(f(q)\)	\(=\)	\(q-0.170762 q^{2} -1.97084 q^{4} -2.09248 q^{5} +1.00000 q^{7} +0.678071 q^{8} +O(q^{10})\)	\(q-0.170762 q^{2} -1.97084 q^{4} -2.09248 q^{5} +1.00000 q^{7} +0.678071 q^{8} +0.357317 q^{10} -0.105046 q^{11} +0.808011 q^{13} -0.170762 q^{14} +3.82589 q^{16} -2.11987 q^{17} +0.0869343 q^{19} +4.12394 q^{20} +0.0179379 q^{22} -5.49710 q^{23} -0.621542 q^{25} -0.137978 q^{26} -1.97084 q^{28} +6.62400 q^{29} +5.84253 q^{31} -2.00946 q^{32} +0.361995 q^{34} -2.09248 q^{35} +7.11832 q^{37} -0.0148451 q^{38} -1.41885 q^{40} -12.0211 q^{41} -7.06811 q^{43} +0.207029 q^{44} +0.938699 q^{46} -2.31361 q^{47} +1.00000 q^{49} +0.106136 q^{50} -1.59246 q^{52} -6.77897 q^{53} +0.219807 q^{55} +0.678071 q^{56} -1.13113 q^{58} -2.38189 q^{59} +6.80820 q^{61} -0.997685 q^{62} -7.30864 q^{64} -1.69074 q^{65} +10.7308 q^{67} +4.17793 q^{68} +0.357317 q^{70} +9.05001 q^{71} +11.3953 q^{73} -1.21554 q^{74} -0.171334 q^{76} -0.105046 q^{77} +10.2976 q^{79} -8.00559 q^{80} +2.05275 q^{82} +0.600098 q^{83} +4.43579 q^{85} +1.20697 q^{86} -0.0712287 q^{88} +10.3716 q^{89} +0.808011 q^{91} +10.8339 q^{92} +0.395079 q^{94} -0.181908 q^{95} -9.17979 q^{97} -0.170762 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 8 q^{2} + 24 q^{4} - 3 q^{5} + 20 q^{7} - 24 q^{8}+O(q^{10}) \) 20 * q - 8 * q^2 + 24 * q^4 - 3 * q^5 + 20 * q^7 - 24 * q^8	\( 20 q - 8 q^{2} + 24 q^{4} - 3 q^{5} + 20 q^{7} - 24 q^{8} - 8 q^{10} - 26 q^{11} - 4 q^{13} - 8 q^{14} + 24 q^{16} - 4 q^{17} + q^{19} + 2 q^{20} + q^{22} - 31 q^{23} + 27 q^{25} - 4 q^{26} + 24 q^{28} - 16 q^{29} + 6 q^{31} - 41 q^{32} - 10 q^{34} - 3 q^{35} + 2 q^{37} - 3 q^{38} - 38 q^{40} - 25 q^{41} + 13 q^{43} - 66 q^{44} + 20 q^{46} - 19 q^{47} + 20 q^{49} + 4 q^{50} + 20 q^{52} - 24 q^{53} - 3 q^{55} - 24 q^{56} + 12 q^{58} - 23 q^{59} - 27 q^{61} - 7 q^{62} + 2 q^{64} - 26 q^{65} + 9 q^{67} + 25 q^{68} - 8 q^{70} - 63 q^{71} - 21 q^{73} - 21 q^{74} - 10 q^{76} - 26 q^{77} + 18 q^{79} + 23 q^{80} - 42 q^{82} + q^{83} - 41 q^{85} + 12 q^{86} + 57 q^{88} + 16 q^{89} - 4 q^{91} - 17 q^{92} + 7 q^{94} - 75 q^{95} - 32 q^{97} - 8 q^{98}+O(q^{100}) \) 20 * q - 8 * q^2 + 24 * q^4 - 3 * q^5 + 20 * q^7 - 24 * q^8 - 8 * q^10 - 26 * q^11 - 4 * q^13 - 8 * q^14 + 24 * q^16 - 4 * q^17 + q^19 + 2 * q^20 + q^22 - 31 * q^23 + 27 * q^25 - 4 * q^26 + 24 * q^28 - 16 * q^29 + 6 * q^31 - 41 * q^32 - 10 * q^34 - 3 * q^35 + 2 * q^37 - 3 * q^38 - 38 * q^40 - 25 * q^41 + 13 * q^43 - 66 * q^44 + 20 * q^46 - 19 * q^47 + 20 * q^49 + 4 * q^50 + 20 * q^52 - 24 * q^53 - 3 * q^55 - 24 * q^56 + 12 * q^58 - 23 * q^59 - 27 * q^61 - 7 * q^62 + 2 * q^64 - 26 * q^65 + 9 * q^67 + 25 * q^68 - 8 * q^70 - 63 * q^71 - 21 * q^73 - 21 * q^74 - 10 * q^76 - 26 * q^77 + 18 * q^79 + 23 * q^80 - 42 * q^82 + q^83 - 41 * q^85 + 12 * q^86 + 57 * q^88 + 16 * q^89 - 4 * q^91 - 17 * q^92 + 7 * q^94 - 75 * q^95 - 32 * q^97 - 8 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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