LMFDB - Embedded newform 6003.2.a.p.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6003,2,Mod(1,6003)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6003, 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("6003.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6003 = 3^{2} \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6003.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:	$47.9341963334$
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} - 2 x^{13} - 18 x^{12} + 34 x^{11} + 124 x^{10} - 216 x^{9} - 420 x^{8} + 647 x^{7} + 750 x^{6} + \cdots - 64 $ x^14 - 2x^13 - 18x^12 + 34x^11 + 124x^10 - 216x^9 - 420x^8 + 647x^7 + 750x^6 - 939x^5 - 717x^4 + 604x^3 + 352x^2 - 128*x - 64
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 2001)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.894954$ of defining polynomial
Character	$\chi$	$=$	6003.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.894954 q^{2} -1.19906 q^{4} -0.474904 q^{5} -1.99579 q^{7} +2.86301 q^{8} +O(q^{10})$	\(q-0.894954 q^{2} -1.19906 q^{4} -0.474904 q^{5} -1.99579 q^{7} +2.86301 q^{8} +0.425017 q^{10} -4.21327 q^{11} +5.77436 q^{13} +1.78614 q^{14} -0.164145 q^{16} +7.31225 q^{17} -1.82925 q^{19} +0.569438 q^{20} +3.77069 q^{22} +1.00000 q^{23} -4.77447 q^{25} -5.16779 q^{26} +2.39307 q^{28} +1.00000 q^{29} -0.108189 q^{31} -5.57912 q^{32} -6.54412 q^{34} +0.947812 q^{35} +10.9983 q^{37} +1.63710 q^{38} -1.35966 q^{40} +0.314460 q^{41} -9.14999 q^{43} +5.05196 q^{44} -0.894954 q^{46} -7.26676 q^{47} -3.01680 q^{49} +4.27293 q^{50} -6.92380 q^{52} +8.53731 q^{53} +2.00090 q^{55} -5.71398 q^{56} -0.894954 q^{58} +13.6287 q^{59} +0.283563 q^{61} +0.0968242 q^{62} +5.32134 q^{64} -2.74227 q^{65} -13.9767 q^{67} -8.76781 q^{68} -0.848248 q^{70} +0.728665 q^{71} -1.79211 q^{73} -9.84300 q^{74} +2.19338 q^{76} +8.40883 q^{77} -13.8676 q^{79} +0.0779533 q^{80} -0.281427 q^{82} +8.65060 q^{83} -3.47262 q^{85} +8.18882 q^{86} -12.0626 q^{88} +1.57598 q^{89} -11.5244 q^{91} -1.19906 q^{92} +6.50342 q^{94} +0.868720 q^{95} -3.65623 q^{97} +2.69990 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + 2 q^{2} + 12 q^{4} + 3 q^{5} - 3 q^{7} + 6 q^{8}+O(q^{10}) $ 14 * q + 2 * q^2 + 12 * q^4 + 3 * q^5 - 3 * q^7 + 6 * q^8	$ 14 q + 2 q^{2} + 12 q^{4} + 3 q^{5} - 3 q^{7} + 6 q^{8} - 5 q^{10} + 12 q^{11} + 13 q^{13} + 9 q^{14} + 14 q^{17} - 9 q^{19} + 2 q^{20} - 9 q^{22} + 14 q^{23} + 13 q^{25} + 16 q^{26} + 3 q^{28} + 14 q^{29} - 28 q^{31} + 4 q^{32} + 14 q^{34} + 9 q^{35} - 12 q^{37} - 2 q^{38} - 20 q^{40} + 25 q^{41} + 5 q^{43} + 37 q^{44} + 2 q^{46} + 17 q^{47} + 17 q^{49} + 44 q^{50} + 25 q^{52} + 17 q^{53} + q^{55} + 54 q^{56} + 2 q^{58} + 18 q^{59} - 13 q^{61} + 8 q^{62} + 20 q^{64} + 16 q^{65} + 2 q^{67} + 19 q^{68} + 14 q^{70} + 55 q^{71} + 19 q^{73} - 4 q^{74} - 32 q^{76} + 19 q^{77} - 68 q^{79} + 2 q^{80} - 12 q^{82} + 21 q^{83} + 16 q^{85} + 22 q^{86} - 25 q^{88} + 17 q^{89} - 30 q^{91} + 12 q^{92} + 16 q^{94} + 55 q^{95} + 25 q^{97} + 31 q^{98}+O(q^{100}) $ 14 * q + 2 * q^2 + 12 * q^4 + 3 * q^5 - 3 * q^7 + 6 * q^8 - 5 * q^10 + 12 * q^11 + 13 * q^13 + 9 * q^14 + 14 * q^17 - 9 * q^19 + 2 * q^20 - 9 * q^22 + 14 * q^23 + 13 * q^25 + 16 * q^26 + 3 * q^28 + 14 * q^29 - 28 * q^31 + 4 * q^32 + 14 * q^34 + 9 * q^35 - 12 * q^37 - 2 * q^38 - 20 * q^40 + 25 * q^41 + 5 * q^43 + 37 * q^44 + 2 * q^46 + 17 * q^47 + 17 * q^49 + 44 * q^50 + 25 * q^52 + 17 * q^53 + q^55 + 54 * q^56 + 2 * q^58 + 18 * q^59 - 13 * q^61 + 8 * q^62 + 20 * q^64 + 16 * q^65 + 2 * q^67 + 19 * q^68 + 14 * q^70 + 55 * q^71 + 19 * q^73 - 4 * q^74 - 32 * q^76 + 19 * q^77 - 68 * q^79 + 2 * q^80 - 12 * q^82 + 21 * q^83 + 16 * q^85 + 22 * q^86 - 25 * q^88 + 17 * q^89 - 30 * q^91 + 12 * q^92 + 16 * q^94 + 55 * q^95 + 25 * q^97 + 31 * 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 $

$2$	−0.894954	−0.632828	−0.316414	−	0.948621i	$-0.602479\pi$
$2$	−0.894954	−0.632828	−0.316414	+	0.948621i	$0.602479\pi$
$3$	0	0
$3$	0	0
$4$	−1.19906	−0.599529
$5$	−0.474904	−0.212384	−0.106192	−	0.994346i	$-0.533866\pi$
$5$	−0.474904	−0.212384	−0.106192	+	0.994346i	$0.533866\pi$
$6$	0	0
$7$	−1.99579	−0.754339	−0.377170	−	0.926144i	$-0.623103\pi$
$7$	−1.99579	−0.754339	−0.377170	+	0.926144i	$0.623103\pi$
$8$	2.86301	1.01223
$9$	0	0
$10$	0.425017	0.134402
$11$	−4.21327	−1.27035	−0.635175	−	0.772368i	$-0.719072\pi$
$11$	−4.21327	−1.27035	−0.635175	+	0.772368i	$0.719072\pi$
$12$	0	0
$13$	5.77436	1.60152	0.800760	−	0.598985i	$-0.204429\pi$
$13$	5.77436	1.60152	0.800760	+	0.598985i	$0.204429\pi$
$14$	1.78614	0.477367
$15$	0	0
$16$	−0.164145	−0.0410363
$17$	7.31225	1.77348	0.886740	−	0.462268i	$-0.152964\pi$
$17$	7.31225	1.77348	0.886740	+	0.462268i	$0.152964\pi$
$18$	0	0
$19$	−1.82925	−0.419659	−0.209830	−	0.977738i	$-0.567291\pi$
$19$	−1.82925	−0.419659	−0.209830	+	0.977738i	$0.567291\pi$
$20$	0.569438	0.127330
$21$	0	0
$22$	3.77069	0.803913
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−4.77447	−0.954893
$26$	−5.16779	−1.01349
$27$	0	0
$28$	2.39307	0.452248
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−0.108189	−0.0194313	−0.00971567	−	0.999953i	$-0.503093\pi$
$31$	−0.108189	−0.0194313	−0.00971567	+	0.999953i	$0.503093\pi$
$32$	−5.57912	−0.986258
$33$	0	0
$34$	−6.54412	−1.12231
$35$	0.947812	0.160209
$36$	0	0
$37$	10.9983	1.80811	0.904057	−	0.427411i	$-0.140574\pi$
$37$	10.9983	1.80811	0.904057	+	0.427411i	$0.140574\pi$
$38$	1.63710	0.265572
$39$	0	0
$40$	−1.35966	−0.214980
$41$	0.314460	0.0491104	0.0245552	−	0.999698i	$-0.492183\pi$
$41$	0.314460	0.0491104	0.0245552	+	0.999698i	$0.492183\pi$
$42$	0	0
$43$	−9.14999	−1.39536	−0.697681	−	0.716409i	$-0.745784\pi$
$43$	−9.14999	−1.39536	−0.697681	+	0.716409i	$0.745784\pi$
$44$	5.05196	0.761611
$45$	0	0
$46$	−0.894954	−0.131954
$47$	−7.26676	−1.05997	−0.529983	−	0.848008i	$-0.677802\pi$
$47$	−7.26676	−1.05997	−0.529983	+	0.848008i	$0.677802\pi$
$48$	0	0
$49$	−3.01680	−0.430972
$50$	4.27293	0.604283
$51$	0	0
$52$	−6.92380	−0.960158
$53$	8.53731	1.17269	0.586345	−	0.810062i	$-0.300566\pi$
$53$	8.53731	1.17269	0.586345	+	0.810062i	$0.300566\pi$
$54$	0	0
$55$	2.00090	0.269802
$56$	−5.71398	−0.763562
$57$	0	0
$58$	−0.894954	−0.117513
$59$	13.6287	1.77431	0.887155	−	0.461472i	$-0.152679\pi$
$59$	13.6287	1.77431	0.887155	+	0.461472i	$0.152679\pi$
$60$	0	0
$61$	0.283563	0.0363066	0.0181533	−	0.999835i	$-0.494221\pi$
$61$	0.283563	0.0363066	0.0181533	+	0.999835i	$0.494221\pi$
$62$	0.0968242	0.0122967
$63$	0	0
$64$	5.32134	0.665168
$65$	−2.74227	−0.340137
$66$	0	0
$67$	−13.9767	−1.70752	−0.853761	−	0.520665i	$-0.825684\pi$
$67$	−13.9767	−1.70752	−0.853761	+	0.520665i	$0.825684\pi$
$68$	−8.76781	−1.06325
$69$	0	0
$70$	−0.848248	−0.101385
$71$	0.728665	0.0864766	0.0432383	−	0.999065i	$-0.486233\pi$
$71$	0.728665	0.0864766	0.0432383	+	0.999065i	$0.486233\pi$
$72$	0	0
$73$	−1.79211	−0.209751	−0.104876	−	0.994485i	$-0.533444\pi$
$73$	−1.79211	−0.209751	−0.104876	+	0.994485i	$0.533444\pi$
$74$	−9.84300	−1.14423
$75$	0	0
$76$	2.19338	0.251598
$77$	8.40883	0.958275
$78$	0	0
$79$	−13.8676	−1.56022	−0.780111	−	0.625641i	$-0.784837\pi$
$79$	−13.8676	−1.56022	−0.780111	+	0.625641i	$0.784837\pi$
$80$	0.0779533	0.00871545
$81$	0	0
$82$	−0.281427	−0.0310784
$83$	8.65060	0.949527	0.474763	−	0.880114i	$-0.342534\pi$
$83$	8.65060	0.949527	0.474763	+	0.880114i	$0.342534\pi$
$84$	0	0
$85$	−3.47262	−0.376658
$86$	8.18882	0.883023
$87$	0	0
$88$	−12.0626	−1.28588
$89$	1.57598	0.167054	0.0835270	−	0.996506i	$-0.473382\pi$
$89$	1.57598	0.167054	0.0835270	+	0.996506i	$0.473382\pi$
$90$	0	0
$91$	−11.5244	−1.20809
$92$	−1.19906	−0.125010
$93$	0	0
$94$	6.50342	0.670776
$95$	0.868720	0.0891288
$96$	0	0
$97$	−3.65623	−0.371234	−0.185617	−	0.982622i	$-0.559428\pi$
$97$	−3.65623	−0.371234	−0.185617	+	0.982622i	$0.559428\pi$
$98$	2.69990	0.272731
$99$	0	0
$100$	5.72486	0.572486
$101$	3.45224	0.343511	0.171755	−	0.985140i	$-0.445056\pi$
$101$	3.45224	0.343511	0.171755	+	0.985140i	$0.445056\pi$
$102$	0	0
$103$	−6.03689	−0.594833	−0.297416	−	0.954748i	$-0.596125\pi$
$103$	−6.03689	−0.594833	−0.297416	+	0.954748i	$0.596125\pi$
$104$	16.5321	1.62110
$105$	0	0
$106$	−7.64050	−0.742111
$107$	−16.2660	−1.57249	−0.786246	−	0.617914i	$-0.787978\pi$
$107$	−16.2660	−1.57249	−0.786246	+	0.617914i	$0.787978\pi$
$108$	0	0
$109$	−17.6889	−1.69428	−0.847142	−	0.531366i	$-0.821679\pi$
$109$	−17.6889	−1.69428	−0.847142	+	0.531366i	$0.821679\pi$
$110$	−1.79072	−0.170738
$111$	0	0
$112$	0.327600	0.0309553
$113$	5.80082	0.545695	0.272848	−	0.962057i	$-0.412035\pi$
$113$	5.80082	0.545695	0.272848	+	0.962057i	$0.412035\pi$
$114$	0	0
$115$	−0.474904	−0.0442851
$116$	−1.19906	−0.111330
$117$	0	0
$118$	−12.1971	−1.12283
$119$	−14.5937	−1.33781
$120$	0	0
$121$	6.75168	0.613789
$122$	−0.253776	−0.0229758
$123$	0	0
$124$	0.129725	0.0116496
$125$	4.64194	0.415187
$126$	0	0
$127$	4.00908	0.355748	0.177874	−	0.984053i	$-0.443078\pi$
$127$	4.00908	0.355748	0.177874	+	0.984053i	$0.443078\pi$
$128$	6.39588	0.565321
$129$	0	0
$130$	2.45421	0.215248
$131$	16.7123	1.46016	0.730079	−	0.683363i	$-0.239483\pi$
$131$	16.7123	1.46016	0.730079	+	0.683363i	$0.239483\pi$
$132$	0	0
$133$	3.65081	0.316565
$134$	12.5085	1.08057
$135$	0	0
$136$	20.9350	1.79516
$137$	7.13547	0.609624	0.304812	−	0.952413i	$-0.401406\pi$
$137$	7.13547	0.609624	0.304812	+	0.952413i	$0.401406\pi$
$138$	0	0
$139$	−1.14377	−0.0970136	−0.0485068	−	0.998823i	$-0.515446\pi$
$139$	−1.14377	−0.0970136	−0.0485068	+	0.998823i	$0.515446\pi$
$140$	−1.13648	−0.0960501
$141$	0	0
$142$	−0.652121	−0.0547248
$143$	−24.3290	−2.03449
$144$	0	0
$145$	−0.474904	−0.0394387
$146$	1.60386	0.132736
$147$	0	0
$148$	−13.1876	−1.08402
$149$	−5.96369	−0.488565	−0.244282	−	0.969704i	$-0.578552\pi$
$149$	−5.96369	−0.488565	−0.244282	+	0.969704i	$0.578552\pi$
$150$	0	0
$151$	11.2886	0.918651	0.459326	−	0.888268i	$-0.348091\pi$
$151$	11.2886	0.918651	0.459326	+	0.888268i	$0.348091\pi$
$152$	−5.23717	−0.424790
$153$	0	0
$154$	−7.52551	−0.606423
$155$	0.0513795	0.00412690
$156$	0	0
$157$	10.8902	0.869130	0.434565	−	0.900641i	$-0.356902\pi$
$157$	10.8902	0.869130	0.434565	+	0.900641i	$0.356902\pi$
$158$	12.4108	0.987352
$159$	0	0
$160$	2.64955	0.209465
$161$	−1.99579	−0.157291
$162$	0	0
$163$	15.1857	1.18943	0.594717	−	0.803935i	$-0.297264\pi$
$163$	15.1857	1.18943	0.594717	+	0.803935i	$0.297264\pi$
$164$	−0.377055	−0.0294431
$165$	0	0
$166$	−7.74189	−0.600887
$167$	−7.84448	−0.607024	−0.303512	−	0.952828i	$-0.598159\pi$
$167$	−7.84448	−0.607024	−0.303512	+	0.952828i	$0.598159\pi$
$168$	0	0
$169$	20.3433	1.56487
$170$	3.10783	0.238360
$171$	0	0
$172$	10.9714	0.836559
$173$	−22.7107	−1.72666	−0.863331	−	0.504638i	$-0.831626\pi$
$173$	−22.7107	−1.72666	−0.863331	+	0.504638i	$0.831626\pi$
$174$	0	0
$175$	9.52885	0.720314
$176$	0.691589	0.0521305
$177$	0	0
$178$	−1.41043	−0.105716
$179$	17.9552	1.34204	0.671019	−	0.741440i	$-0.265857\pi$
$179$	17.9552	1.34204	0.671019	+	0.741440i	$0.265857\pi$
$180$	0	0
$181$	20.7321	1.54101	0.770503	−	0.637437i	$-0.220005\pi$
$181$	20.7321	1.54101	0.770503	+	0.637437i	$0.220005\pi$
$182$	10.3138	0.764513
$183$	0	0
$184$	2.86301	0.211064
$185$	−5.22316	−0.384014
$186$	0	0
$187$	−30.8085	−2.25294
$188$	8.71327	0.635480
$189$	0	0
$190$	−0.777464	−0.0564032
$191$	1.02970	0.0745063	0.0372531	−	0.999306i	$-0.488139\pi$
$191$	1.02970	0.0745063	0.0372531	+	0.999306i	$0.488139\pi$
$192$	0	0
$193$	21.8114	1.57002	0.785008	−	0.619485i	$-0.212659\pi$
$193$	21.8114	1.57002	0.785008	+	0.619485i	$0.212659\pi$
$194$	3.27216	0.234927
$195$	0	0
$196$	3.61732	0.258380
$197$	−0.418534	−0.0298193	−0.0149096	−	0.999889i	$-0.504746\pi$
$197$	−0.418534	−0.0298193	−0.0149096	+	0.999889i	$0.504746\pi$
$198$	0	0
$199$	18.2933	1.29678	0.648388	−	0.761310i	$-0.275443\pi$
$199$	18.2933	1.29678	0.648388	+	0.761310i	$0.275443\pi$
$200$	−13.6693	−0.966568
$201$	0	0
$202$	−3.08959	−0.217383
$203$	−1.99579	−0.140077
$204$	0	0
$205$	−0.149338	−0.0104302
$206$	5.40274	0.376427
$207$	0	0
$208$	−0.947835	−0.0657205
$209$	7.70714	0.533114
$210$	0	0
$211$	−18.8816	−1.29986	−0.649930	−	0.759994i	$-0.725202\pi$
$211$	−18.8816	−1.29986	−0.649930	+	0.759994i	$0.725202\pi$
$212$	−10.2367	−0.703061
$213$	0	0
$214$	14.5573	0.995117
$215$	4.34537	0.296352
$216$	0	0
$217$	0.215923	0.0146578
$218$	15.8307	1.07219
$219$	0	0
$220$	−2.39920	−0.161754
$221$	42.2236	2.84027
$222$	0	0
$223$	9.85174	0.659722	0.329861	−	0.944030i	$-0.392998\pi$
$223$	9.85174	0.659722	0.329861	+	0.944030i	$0.392998\pi$
$224$	11.1348	0.743973
$225$	0	0
$226$	−5.19147	−0.345331
$227$	10.7086	0.710756	0.355378	−	0.934723i	$-0.384352\pi$
$227$	10.7086	0.710756	0.355378	+	0.934723i	$0.384352\pi$
$228$	0	0
$229$	−19.9167	−1.31613	−0.658067	−	0.752959i	$-0.728626\pi$
$229$	−19.9167	−1.31613	−0.658067	+	0.752959i	$0.728626\pi$
$230$	0.425017	0.0280248
$231$	0	0
$232$	2.86301	0.187966
$233$	18.7066	1.22551	0.612755	−	0.790273i	$-0.290061\pi$
$233$	18.7066	1.22551	0.612755	+	0.790273i	$0.290061\pi$
$234$	0	0
$235$	3.45102	0.225120
$236$	−16.3416	−1.06375
$237$	0	0
$238$	13.0607	0.846601
$239$	−10.1460	−0.656287	−0.328144	−	0.944628i	$-0.606423\pi$
$239$	−10.1460	−0.656287	−0.328144	+	0.944628i	$0.606423\pi$
$240$	0	0
$241$	21.3326	1.37415	0.687077	−	0.726585i	$-0.258894\pi$
$241$	21.3326	1.37415	0.687077	+	0.726585i	$0.258894\pi$
$242$	−6.04244	−0.388423
$243$	0	0
$244$	−0.340009	−0.0217668
$245$	1.43269	0.0915315
$246$	0	0
$247$	−10.5628	−0.672093
$248$	−0.309746	−0.0196689
$249$	0	0
$250$	−4.15432	−0.262742
$251$	−2.32556	−0.146788	−0.0733941	−	0.997303i	$-0.523383\pi$
$251$	−2.32556	−0.146788	−0.0733941	+	0.997303i	$0.523383\pi$
$252$	0	0
$253$	−4.21327	−0.264886
$254$	−3.58794	−0.225128
$255$	0	0
$256$	−16.3667	−1.02292
$257$	0.749025	0.0467229	0.0233614	−	0.999727i	$-0.492563\pi$
$257$	0.749025	0.0467229	0.0233614	+	0.999727i	$0.492563\pi$
$258$	0	0
$259$	−21.9504	−1.36393
$260$	3.28814	0.203922
$261$	0	0
$262$	−14.9567	−0.924029
$263$	21.6925	1.33761	0.668807	−	0.743436i	$-0.266805\pi$
$263$	21.6925	1.33761	0.668807	+	0.743436i	$0.266805\pi$
$264$	0	0
$265$	−4.05440	−0.249060
$266$	−3.26731	−0.200331
$267$	0	0
$268$	16.7588	1.02371
$269$	5.65243	0.344635	0.172317	−	0.985041i	$-0.444875\pi$
$269$	5.65243	0.344635	0.172317	+	0.985041i	$0.444875\pi$
$270$	0	0
$271$	−20.8246	−1.26500	−0.632502	−	0.774559i	$-0.717972\pi$
$271$	−20.8246	−1.26500	−0.632502	+	0.774559i	$0.717972\pi$
$272$	−1.20027	−0.0727772
$273$	0	0
$274$	−6.38591	−0.385787
$275$	20.1161	1.21305
$276$	0	0
$277$	−3.70261	−0.222468	−0.111234	−	0.993794i	$-0.535480\pi$
$277$	−3.70261	−0.222468	−0.111234	+	0.993794i	$0.535480\pi$
$278$	1.02362	0.0613929
$279$	0	0
$280$	2.71359	0.162168
$281$	14.8399	0.885272	0.442636	−	0.896701i	$-0.354043\pi$
$281$	14.8399	0.885272	0.442636	+	0.896701i	$0.354043\pi$
$282$	0	0
$283$	−18.7147	−1.11248	−0.556238	−	0.831023i	$-0.687756\pi$
$283$	−18.7147	−1.11248	−0.556238	+	0.831023i	$0.687756\pi$
$284$	−0.873711	−0.0518452
$285$	0	0
$286$	21.7733	1.28748
$287$	−0.627597	−0.0370459
$288$	0	0
$289$	36.4690	2.14523
$290$	0.425017	0.0249579
$291$	0	0
$292$	2.14885	0.125752
$293$	−32.2409	−1.88353	−0.941767	−	0.336265i	$-0.890836\pi$
$293$	−32.2409	−1.88353	−0.941767	+	0.336265i	$0.890836\pi$
$294$	0	0
$295$	−6.47234	−0.376834
$296$	31.4883	1.83022
$297$	0	0
$298$	5.33723	0.309177
$299$	5.77436	0.333940
$300$	0	0
$301$	18.2615	1.05258
$302$	−10.1028	−0.581348
$303$	0	0
$304$	0.300263	0.0172213
$305$	−0.134665	−0.00771092
$306$	0	0
$307$	−13.1519	−0.750618	−0.375309	−	0.926900i	$-0.622463\pi$
$307$	−13.1519	−0.750618	−0.375309	+	0.926900i	$0.622463\pi$
$308$	−10.0827	−0.574514
$309$	0	0
$310$	−0.0459823	−0.00261162
$311$	26.9063	1.52572	0.762859	−	0.646564i	$-0.223795\pi$
$311$	26.9063	1.52572	0.762859	+	0.646564i	$0.223795\pi$
$312$	0	0
$313$	−5.97393	−0.337667	−0.168833	−	0.985645i	$-0.554000\pi$
$313$	−5.97393	−0.337667	−0.168833	+	0.985645i	$0.554000\pi$
$314$	−9.74619	−0.550009
$315$	0	0
$316$	16.6280	0.935398
$317$	−26.3200	−1.47828	−0.739140	−	0.673552i	$-0.764768\pi$
$317$	−26.3200	−1.47828	−0.739140	+	0.673552i	$0.764768\pi$
$318$	0	0
$319$	−4.21327	−0.235898
$320$	−2.52713	−0.141271
$321$	0	0
$322$	1.78614	0.0995379
$323$	−13.3759	−0.744258
$324$	0	0
$325$	−27.5695	−1.52928
$326$	−13.5905	−0.752707
$327$	0	0
$328$	0.900301	0.0497108
$329$	14.5030	0.799574
$330$	0	0
$331$	7.51683	0.413162	0.206581	−	0.978429i	$-0.433766\pi$
$331$	7.51683	0.413162	0.206581	+	0.978429i	$0.433766\pi$
$332$	−10.3726	−0.569269
$333$	0	0
$334$	7.02044	0.384142
$335$	6.63758	0.362650
$336$	0	0
$337$	31.6701	1.72518	0.862590	−	0.505904i	$-0.168841\pi$
$337$	31.6701	1.72518	0.862590	+	0.505904i	$0.168841\pi$
$338$	−18.2063	−0.990292
$339$	0	0
$340$	4.16387	0.225818
$341$	0.455830	0.0246846
$342$	0	0
$343$	19.9915	1.07944
$344$	−26.1965	−1.41242
$345$	0	0
$346$	20.3250	1.09268
$347$	−12.4921	−0.670612	−0.335306	−	0.942109i	$-0.608840\pi$
$347$	−12.4921	−0.670612	−0.335306	+	0.942109i	$0.608840\pi$
$348$	0	0
$349$	35.3247	1.89089	0.945444	−	0.325784i	$-0.105628\pi$
$349$	35.3247	1.89089	0.945444	+	0.325784i	$0.105628\pi$
$350$	−8.52788	−0.455834
$351$	0	0
$352$	23.5063	1.25289
$353$	13.0353	0.693798	0.346899	−	0.937903i	$-0.387235\pi$
$353$	13.0353	0.693798	0.346899	+	0.937903i	$0.387235\pi$
$354$	0	0
$355$	−0.346046	−0.0183662
$356$	−1.88970	−0.100154
$357$	0	0
$358$	−16.0691	−0.849279
$359$	17.7976	0.939322	0.469661	−	0.882847i	$-0.344376\pi$
$359$	17.7976	0.939322	0.469661	+	0.882847i	$0.344376\pi$
$360$	0	0
$361$	−15.6538	−0.823886
$362$	−18.5543	−0.975191
$363$	0	0
$364$	13.8185	0.724285
$365$	0.851083	0.0445477
$366$	0	0
$367$	28.2055	1.47231	0.736157	−	0.676811i	$-0.236639\pi$
$367$	28.2055	1.47231	0.736157	+	0.676811i	$0.236639\pi$
$368$	−0.164145	−0.00855667
$369$	0	0
$370$	4.67448	0.243015
$371$	−17.0387	−0.884606
$372$	0	0
$373$	3.04203	0.157510	0.0787552	−	0.996894i	$-0.474905\pi$
$373$	3.04203	0.157510	0.0787552	+	0.996894i	$0.474905\pi$
$374$	27.5722	1.42572
$375$	0	0
$376$	−20.8048	−1.07293
$377$	5.77436	0.297395
$378$	0	0
$379$	−16.5577	−0.850511	−0.425255	−	0.905073i	$-0.639816\pi$
$379$	−16.5577	−0.850511	−0.425255	+	0.905073i	$0.639816\pi$
$380$	−1.04165	−0.0534353
$381$	0	0
$382$	−0.921531	−0.0471497
$383$	−10.9654	−0.560305	−0.280153	−	0.959955i	$-0.590385\pi$
$383$	−10.9654	−0.560305	−0.280153	+	0.959955i	$0.590385\pi$
$384$	0	0
$385$	−3.99339	−0.203522
$386$	−19.5202	−0.993550
$387$	0	0
$388$	4.38404	0.222566
$389$	8.65746	0.438951	0.219475	−	0.975618i	$-0.429565\pi$
$389$	8.65746	0.438951	0.219475	+	0.975618i	$0.429565\pi$
$390$	0	0
$391$	7.31225	0.369796
$392$	−8.63714	−0.436241
$393$	0	0
$394$	0.374568	0.0188705
$395$	6.58576	0.331366
$396$	0	0
$397$	−14.9378	−0.749705	−0.374853	−	0.927084i	$-0.622307\pi$
$397$	−14.9378	−0.749705	−0.374853	+	0.927084i	$0.622307\pi$
$398$	−16.3716	−0.820636
$399$	0	0
$400$	0.783706	0.0391853
$401$	4.02676	0.201087	0.100543	−	0.994933i	$-0.467942\pi$
$401$	4.02676	0.201087	0.100543	+	0.994933i	$0.467942\pi$
$402$	0	0
$403$	−0.624723	−0.0311197
$404$	−4.13943	−0.205945
$405$	0	0
$406$	1.78614	0.0886448
$407$	−46.3390	−2.29694
$408$	0	0
$409$	−0.983912	−0.0486513	−0.0243257	−	0.999704i	$-0.507744\pi$
$409$	−0.983912	−0.0486513	−0.0243257	+	0.999704i	$0.507744\pi$
$410$	0.133651	0.00660055
$411$	0	0
$412$	7.23858	0.356619
$413$	−27.2001	−1.33843
$414$	0	0
$415$	−4.10821	−0.201664
$416$	−32.2158	−1.57951
$417$	0	0
$418$	−6.89754	−0.337370
$419$	5.17889	0.253005	0.126503	−	0.991966i	$-0.459625\pi$
$419$	5.17889	0.253005	0.126503	+	0.991966i	$0.459625\pi$
$420$	0	0
$421$	−8.73075	−0.425511	−0.212755	−	0.977106i	$-0.568244\pi$
$421$	−8.73075	−0.425511	−0.212755	+	0.977106i	$0.568244\pi$
$422$	16.8981	0.822588
$423$	0	0
$424$	24.4424	1.18703
$425$	−34.9121	−1.69348
$426$	0	0
$427$	−0.565934	−0.0273875
$428$	19.5038	0.942754
$429$	0	0
$430$	−3.88891	−0.187540
$431$	10.7660	0.518580	0.259290	−	0.965800i	$-0.416512\pi$
$431$	10.7660	0.518580	0.259290	+	0.965800i	$0.416512\pi$
$432$	0	0
$433$	6.61490	0.317892	0.158946	−	0.987287i	$-0.449190\pi$
$433$	6.61490	0.317892	0.158946	+	0.987287i	$0.449190\pi$
$434$	−0.193241	−0.00927588
$435$	0	0
$436$	21.2100	1.01577
$437$	−1.82925	−0.0875050
$438$	0	0
$439$	−4.46205	−0.212962	−0.106481	−	0.994315i	$-0.533958\pi$
$439$	−4.46205	−0.212962	−0.106481	+	0.994315i	$0.533958\pi$
$440$	5.72860	0.273100
$441$	0	0
$442$	−37.7882	−1.79740
$443$	38.8894	1.84769	0.923845	−	0.382768i	$-0.125029\pi$
$443$	38.8894	1.84769	0.923845	+	0.382768i	$0.125029\pi$
$444$	0	0
$445$	−0.748442	−0.0354796
$446$	−8.81686	−0.417490
$447$	0	0
$448$	−10.6203	−0.501762
$449$	33.2095	1.56725	0.783627	−	0.621231i	$-0.213367\pi$
$449$	33.2095	1.56725	0.783627	+	0.621231i	$0.213367\pi$
$450$	0	0
$451$	−1.32491	−0.0623873
$452$	−6.95552	−0.327160
$453$	0	0
$454$	−9.58372	−0.449786
$455$	5.47301	0.256579
$456$	0	0
$457$	3.96363	0.185411	0.0927055	−	0.995694i	$-0.470449\pi$
$457$	3.96363	0.185411	0.0927055	+	0.995694i	$0.470449\pi$
$458$	17.8246	0.832887
$459$	0	0
$460$	0.569438	0.0265502
$461$	13.1510	0.612505	0.306253	−	0.951950i	$-0.400925\pi$
$461$	13.1510	0.612505	0.306253	+	0.951950i	$0.400925\pi$
$462$	0	0
$463$	40.8826	1.89997	0.949987	−	0.312289i	$-0.101096\pi$
$463$	40.8826	1.89997	0.949987	+	0.312289i	$0.101096\pi$
$464$	−0.164145	−0.00762026
$465$	0	0
$466$	−16.7415	−0.775537
$467$	8.60638	0.398256	0.199128	−	0.979974i	$-0.436189\pi$
$467$	8.60638	0.398256	0.199128	+	0.979974i	$0.436189\pi$
$468$	0	0
$469$	27.8946	1.28805
$470$	−3.08850	−0.142462
$471$	0	0
$472$	39.0192	1.79600
$473$	38.5514	1.77260
$474$	0	0
$475$	8.73370	0.400730
$476$	17.4987	0.802053
$477$	0	0
$478$	9.08016	0.415317
$479$	31.9634	1.46045	0.730223	−	0.683208i	$-0.239416\pi$
$479$	31.9634	1.46045	0.730223	+	0.683208i	$0.239416\pi$
$480$	0	0
$481$	63.5084	2.89573
$482$	−19.0917	−0.869603
$483$	0	0
$484$	−8.09565	−0.367984
$485$	1.73636	0.0788441
$486$	0	0
$487$	−40.3554	−1.82868	−0.914339	−	0.404949i	$-0.867289\pi$
$487$	−40.3554	−1.82868	−0.914339	+	0.404949i	$0.867289\pi$
$488$	0.811844	0.0367505
$489$	0	0
$490$	−1.28219	−0.0579237
$491$	33.3566	1.50536	0.752682	−	0.658384i	$-0.228760\pi$
$491$	33.3566	1.50536	0.752682	+	0.658384i	$0.228760\pi$
$492$	0	0
$493$	7.31225	0.329327
$494$	9.45319	0.425319
$495$	0	0
$496$	0.0177587	0.000797391 0
$497$	−1.45426	−0.0652327
$498$	0	0
$499$	1.93558	0.0866486	0.0433243	−	0.999061i	$-0.486205\pi$
$499$	1.93558	0.0866486	0.0433243	+	0.999061i	$0.486205\pi$
$500$	−5.56595	−0.248917
$501$	0	0
$502$	2.08127	0.0928917
$503$	−40.5468	−1.80789	−0.903946	−	0.427647i	$-0.859343\pi$
$503$	−40.5468	−1.80789	−0.903946	+	0.427647i	$0.859343\pi$
$504$	0	0
$505$	−1.63948	−0.0729561
$506$	3.77069	0.167627
$507$	0	0
$508$	−4.80712	−0.213281
$509$	17.0589	0.756122	0.378061	−	0.925781i	$-0.376591\pi$
$509$	17.0589	0.756122	0.378061	+	0.925781i	$0.376591\pi$
$510$	0	0
$511$	3.57669	0.158223
$512$	1.85569	0.0820105
$513$	0	0
$514$	−0.670343	−0.0295675
$515$	2.86695	0.126333
$516$	0	0
$517$	30.6169	1.34653
$518$	19.6446	0.863134
$519$	0	0
$520$	−7.85115	−0.344296
$521$	−10.8295	−0.474448	−0.237224	−	0.971455i	$-0.576238\pi$
$521$	−10.8295	−0.474448	−0.237224	+	0.971455i	$0.576238\pi$
$522$	0	0
$523$	−13.6686	−0.597686	−0.298843	−	0.954302i	$-0.596601\pi$
$523$	−13.6686	−0.597686	−0.298843	+	0.954302i	$0.596601\pi$
$524$	−20.0390	−0.875407
$525$	0	0
$526$	−19.4138	−0.846480
$527$	−0.791105	−0.0344611
$528$	0	0
$529$	1.00000	0.0434783
$530$	3.62850	0.157612
$531$	0	0
$532$	−4.37753	−0.189790
$533$	1.81581	0.0786513
$534$	0	0
$535$	7.72478	0.333972
$536$	−40.0153	−1.72840
$537$	0	0
$538$	−5.05866	−0.218094
$539$	12.7106	0.547485
$540$	0	0
$541$	26.4643	1.13779	0.568895	−	0.822410i	$-0.307371\pi$
$541$	26.4643	1.13779	0.568895	+	0.822410i	$0.307371\pi$
$542$	18.6371	0.800530
$543$	0	0
$544$	−40.7959	−1.74911
$545$	8.40051	0.359839
$546$	0	0
$547$	8.91076	0.380996	0.190498	−	0.981688i	$-0.438990\pi$
$547$	8.91076	0.380996	0.190498	+	0.981688i	$0.438990\pi$
$548$	−8.55584	−0.365487
$549$	0	0
$550$	−18.0030	−0.767651
$551$	−1.82925	−0.0779288
$552$	0	0
$553$	27.6768	1.17694
$554$	3.31367	0.140784
$555$	0	0
$556$	1.37145	0.0581625
$557$	19.1693	0.812230	0.406115	−	0.913822i	$-0.366883\pi$
$557$	19.1693	0.812230	0.406115	+	0.913822i	$0.366883\pi$
$558$	0	0
$559$	−52.8354	−2.23470
$560$	−0.155579	−0.00657441
$561$	0	0
$562$	−13.2810	−0.560225
$563$	1.82432	0.0768861	0.0384430	−	0.999261i	$-0.487760\pi$
$563$	1.82432	0.0768861	0.0384430	+	0.999261i	$0.487760\pi$
$564$	0	0
$565$	−2.75483	−0.115897
$566$	16.7488	0.704006
$567$	0	0
$568$	2.08617	0.0875339
$569$	−11.6679	−0.489143	−0.244571	−	0.969631i	$-0.578647\pi$
$569$	−11.6679	−0.489143	−0.244571	+	0.969631i	$0.578647\pi$
$570$	0	0
$571$	31.7168	1.32731	0.663653	−	0.748041i	$-0.269005\pi$
$571$	31.7168	1.32731	0.663653	+	0.748041i	$0.269005\pi$
$572$	29.1719	1.21974
$573$	0	0
$574$	0.561670	0.0234437
$575$	−4.77447	−0.199109
$576$	0	0
$577$	13.0629	0.543817	0.271909	−	0.962323i	$-0.412345\pi$
$577$	13.0629	0.543817	0.271909	+	0.962323i	$0.412345\pi$
$578$	−32.6380	−1.35756
$579$	0	0
$580$	0.569438	0.0236446
$581$	−17.2648	−0.716265
$582$	0	0
$583$	−35.9700	−1.48973
$584$	−5.13084	−0.212316
$585$	0	0
$586$	28.8541	1.19195
$587$	42.1961	1.74162	0.870809	−	0.491621i	$-0.163595\pi$
$587$	42.1961	1.74162	0.870809	+	0.491621i	$0.163595\pi$
$588$	0	0
$589$	0.197905	0.00815454
$590$	5.79245	0.238471
$591$	0	0
$592$	−1.80533	−0.0741984
$593$	−10.2974	−0.422863	−0.211431	−	0.977393i	$-0.567812\pi$
$593$	−10.2974	−0.422863	−0.211431	+	0.977393i	$0.567812\pi$
$594$	0	0
$595$	6.93063	0.284128
$596$	7.15081	0.292909
$597$	0	0
$598$	−5.16779	−0.211327
$599$	−12.2322	−0.499795	−0.249898	−	0.968272i	$-0.580397\pi$
$599$	−12.2322	−0.499795	−0.249898	+	0.968272i	$0.580397\pi$
$600$	0	0
$601$	14.0743	0.574103	0.287051	−	0.957915i	$-0.407325\pi$
$601$	14.0743	0.574103	0.287051	+	0.957915i	$0.407325\pi$
$602$	−16.3432	−0.666099
$603$	0	0
$604$	−13.5357	−0.550758
$605$	−3.20640	−0.130359
$606$	0	0
$607$	−1.95760	−0.0794564	−0.0397282	−	0.999211i	$-0.512649\pi$
$607$	−1.95760	−0.0794564	−0.0397282	+	0.999211i	$0.512649\pi$
$608$	10.2056	0.413892
$609$	0	0
$610$	0.120519	0.00487969
$611$	−41.9609	−1.69756
$612$	0	0
$613$	−21.6541	−0.874599	−0.437300	−	0.899316i	$-0.644065\pi$
$613$	−21.6541	−0.874599	−0.437300	+	0.899316i	$0.644065\pi$
$614$	11.7703	0.475012
$615$	0	0
$616$	24.0746	0.969991
$617$	−19.0826	−0.768237	−0.384118	−	0.923284i	$-0.625495\pi$
$617$	−19.0826	−0.768237	−0.384118	+	0.923284i	$0.625495\pi$
$618$	0	0
$619$	10.6165	0.426713	0.213356	−	0.976974i	$-0.431560\pi$
$619$	10.6165	0.426713	0.213356	+	0.976974i	$0.431560\pi$
$620$	−0.0616070	−0.00247419
$621$	0	0
$622$	−24.0799	−0.965517
$623$	−3.14534	−0.126015
$624$	0	0
$625$	21.6679	0.866714
$626$	5.34639	0.213685
$627$	0	0
$628$	−13.0579	−0.521068
$629$	80.4225	3.20666
$630$	0	0
$631$	19.7674	0.786927	0.393463	−	0.919340i	$-0.371277\pi$
$631$	19.7674	0.786927	0.393463	+	0.919340i	$0.371277\pi$
$632$	−39.7029	−1.57930
$633$	0	0
$634$	23.5552	0.935497
$635$	−1.90393	−0.0755552
$636$	0	0
$637$	−17.4201	−0.690211
$638$	3.77069	0.149283
$639$	0	0
$640$	−3.03743	−0.120065
$641$	−30.5690	−1.20740	−0.603701	−	0.797210i	$-0.706308\pi$
$641$	−30.5690	−1.20740	−0.603701	+	0.797210i	$0.706308\pi$
$642$	0	0
$643$	8.22719	0.324449	0.162224	−	0.986754i	$-0.448133\pi$
$643$	8.22719	0.324449	0.162224	+	0.986754i	$0.448133\pi$
$644$	2.39307	0.0943003
$645$	0	0
$646$	11.9709	0.470987
$647$	−26.6031	−1.04587	−0.522937	−	0.852371i	$-0.675164\pi$
$647$	−26.6031	−1.04587	−0.522937	+	0.852371i	$0.675164\pi$
$648$	0	0
$649$	−57.4216	−2.25399
$650$	24.6734	0.967772
$651$	0	0
$652$	−18.2085	−0.713100
$653$	−21.9374	−0.858477	−0.429239	−	0.903191i	$-0.641218\pi$
$653$	−21.9374	−0.858477	−0.429239	+	0.903191i	$0.641218\pi$
$654$	0	0
$655$	−7.93673	−0.310114
$656$	−0.0516171	−0.00201531
$657$	0	0
$658$	−12.9795	−0.505993
$659$	30.1637	1.17501	0.587505	−	0.809220i	$-0.300110\pi$
$659$	30.1637	1.17501	0.587505	+	0.809220i	$0.300110\pi$
$660$	0	0
$661$	7.34972	0.285871	0.142936	−	0.989732i	$-0.454346\pi$
$661$	7.34972	0.285871	0.142936	+	0.989732i	$0.454346\pi$
$662$	−6.72722	−0.261461
$663$	0	0
$664$	24.7667	0.961136
$665$	−1.73379	−0.0672334
$666$	0	0
$667$	1.00000	0.0387202
$668$	9.40598	0.363928
$669$	0	0
$670$	−5.94033	−0.229495
$671$	−1.19473	−0.0461220
$672$	0	0
$673$	7.38112	0.284521	0.142261	−	0.989829i	$-0.454563\pi$
$673$	7.38112	0.284521	0.142261	+	0.989829i	$0.454563\pi$
$674$	−28.3433	−1.09174
$675$	0	0
$676$	−24.3928	−0.938184
$677$	30.9453	1.18933	0.594663	−	0.803975i	$-0.297286\pi$
$677$	30.9453	1.18933	0.594663	+	0.803975i	$0.297286\pi$
$678$	0	0
$679$	7.29709	0.280037
$680$	−9.94214	−0.381264
$681$	0	0
$682$	−0.407947	−0.0156211
$683$	19.0851	0.730273	0.365136	−	0.930954i	$-0.381022\pi$
$683$	19.0851	0.730273	0.365136	+	0.930954i	$0.381022\pi$
$684$	0	0
$685$	−3.38866	−0.129474
$686$	−17.8915	−0.683099
$687$	0	0
$688$	1.50193	0.0572605
$689$	49.2975	1.87809
$690$	0	0
$691$	−2.88191	−0.109633	−0.0548165	−	0.998496i	$-0.517457\pi$
$691$	−2.88191	−0.109633	−0.0548165	+	0.998496i	$0.517457\pi$
$692$	27.2314	1.03518
$693$	0	0
$694$	11.1799	0.424382
$695$	0.543183	0.0206041
$696$	0	0
$697$	2.29941	0.0870963
$698$	−31.6140	−1.19661
$699$	0	0
$700$	−11.4256	−0.431849
$701$	−24.0678	−0.909027	−0.454513	−	0.890740i	$-0.650187\pi$
$701$	−24.0678	−0.909027	−0.454513	+	0.890740i	$0.650187\pi$
$702$	0	0
$703$	−20.1187	−0.758792
$704$	−22.4203	−0.844996
$705$	0	0
$706$	−11.6660	−0.439055
$707$	−6.88996	−0.259124
$708$	0	0
$709$	−21.1696	−0.795041	−0.397521	−	0.917593i	$-0.630129\pi$
$709$	−21.1696	−0.795041	−0.397521	+	0.917593i	$0.630129\pi$
$710$	0.309695	0.0116227
$711$	0	0
$712$	4.51206	0.169097
$713$	−0.108189	−0.00405171
$714$	0	0
$715$	11.5539	0.432093
$716$	−21.5294	−0.804590
$717$	0	0
$718$	−15.9280	−0.594429
$719$	3.11667	0.116232	0.0581162	−	0.998310i	$-0.481491\pi$
$719$	3.11667	0.116232	0.0581162	+	0.998310i	$0.481491\pi$
$720$	0	0
$721$	12.0484	0.448706
$722$	14.0095	0.521378
$723$	0	0
$724$	−24.8590	−0.923877
$725$	−4.77447	−0.177319
$726$	0	0
$727$	−35.2277	−1.30652	−0.653262	−	0.757132i	$-0.726600\pi$
$727$	−35.2277	−1.30652	−0.653262	+	0.757132i	$0.726600\pi$
$728$	−32.9946	−1.22286
$729$	0	0
$730$	−0.761680	−0.0281910
$731$	−66.9070	−2.47465
$732$	0	0
$733$	−7.52531	−0.277954	−0.138977	−	0.990296i	$-0.544381\pi$
$733$	−7.52531	−0.277954	−0.138977	+	0.990296i	$0.544381\pi$
$734$	−25.2426	−0.931721
$735$	0	0
$736$	−5.57912	−0.205649
$737$	58.8875	2.16915
$738$	0	0
$739$	8.32114	0.306098	0.153049	−	0.988219i	$-0.451091\pi$
$739$	8.32114	0.306098	0.153049	+	0.988219i	$0.451091\pi$
$740$	6.26287	0.230228
$741$	0	0
$742$	15.2489	0.559803
$743$	50.1220	1.83880	0.919398	−	0.393328i	$-0.128676\pi$
$743$	50.1220	1.83880	0.919398	+	0.393328i	$0.128676\pi$
$744$	0	0
$745$	2.83218	0.103763
$746$	−2.72248	−0.0996769
$747$	0	0
$748$	36.9412	1.35070
$749$	32.4635	1.18619
$750$	0	0
$751$	6.96312	0.254088	0.127044	−	0.991897i	$-0.459451\pi$
$751$	6.96312	0.254088	0.127044	+	0.991897i	$0.459451\pi$
$752$	1.19281	0.0434971
$753$	0	0
$754$	−5.16779	−0.188200
$755$	−5.36100	−0.195107
$756$	0	0
$757$	−30.7302	−1.11691	−0.558455	−	0.829535i	$-0.688606\pi$
$757$	−30.7302	−1.11691	−0.558455	+	0.829535i	$0.688606\pi$
$758$	14.8184	0.538227
$759$	0	0
$760$	2.48715	0.0902185
$761$	−36.9406	−1.33910	−0.669548	−	0.742769i	$-0.733512\pi$
$761$	−36.9406	−1.33910	−0.669548	+	0.742769i	$0.733512\pi$
$762$	0	0
$763$	35.3033	1.27807
$764$	−1.23467	−0.0446687
$765$	0	0
$766$	9.81352	0.354577
$767$	78.6973	2.84159
$768$	0	0
$769$	−1.80160	−0.0649673	−0.0324837	−	0.999472i	$-0.510342\pi$
$769$	−1.80160	−0.0649673	−0.0324837	+	0.999472i	$0.510342\pi$
$770$	3.57390	0.128794
$771$	0	0
$772$	−26.1531	−0.941270
$773$	−18.7139	−0.673094	−0.336547	−	0.941667i	$-0.609259\pi$
$773$	−18.7139	−0.673094	−0.336547	+	0.941667i	$0.609259\pi$
$774$	0	0
$775$	0.516545	0.0185548
$776$	−10.4678	−0.375773
$777$	0	0
$778$	−7.74803	−0.277780
$779$	−0.575226	−0.0206096
$780$	0	0
$781$	−3.07006	−0.109856
$782$	−6.54412	−0.234017
$783$	0	0
$784$	0.495194	0.0176855
$785$	−5.17179	−0.184589
$786$	0	0
$787$	11.5837	0.412916	0.206458	−	0.978455i	$-0.433806\pi$
$787$	11.5837	0.412916	0.206458	+	0.978455i	$0.433806\pi$
$788$	0.501846	0.0178775
$789$	0	0
$790$	−5.89395	−0.209697
$791$	−11.5772	−0.411639
$792$	0	0
$793$	1.63740	0.0581457
$794$	13.3686	0.474434
$795$	0	0
$796$	−21.9347	−0.777455
$797$	4.87547	0.172698	0.0863491	−	0.996265i	$-0.472480\pi$
$797$	4.87547	0.172698	0.0863491	+	0.996265i	$0.472480\pi$
$798$	0	0
$799$	−53.1364	−1.87983
$800$	26.6373	0.941771
$801$	0	0
$802$	−3.60377	−0.127253
$803$	7.55067	0.266457
$804$	0	0
$805$	0.947812	0.0334060
$806$	0.559098	0.0196934
$807$	0	0
$808$	9.88379	0.347711
$809$	−18.2484	−0.641580	−0.320790	−	0.947150i	$-0.603948\pi$
$809$	−18.2484	−0.641580	−0.320790	+	0.947150i	$0.603948\pi$
$810$	0	0
$811$	29.9782	1.05268	0.526338	−	0.850275i	$-0.323565\pi$
$811$	29.9782	1.05268	0.526338	+	0.850275i	$0.323565\pi$
$812$	2.39307	0.0839804
$813$	0	0
$814$	41.4713	1.45357
$815$	−7.21175	−0.252616
$816$	0	0
$817$	16.7376	0.585576
$818$	0.880556	0.0307879
$819$	0	0
$820$	0.179065	0.00625323
$821$	−4.24158	−0.148032	−0.0740161	−	0.997257i	$-0.523582\pi$
$821$	−4.24158	−0.148032	−0.0740161	+	0.997257i	$0.523582\pi$
$822$	0	0
$823$	27.7238	0.966392	0.483196	−	0.875512i	$-0.339476\pi$
$823$	27.7238	0.966392	0.483196	+	0.875512i	$0.339476\pi$
$824$	−17.2837	−0.602106
$825$	0	0
$826$	24.3429	0.846997
$827$	−19.7909	−0.688197	−0.344098	−	0.938934i	$-0.611815\pi$
$827$	−19.7909	−0.688197	−0.344098	+	0.938934i	$0.611815\pi$
$828$	0	0
$829$	48.9443	1.69991	0.849953	−	0.526859i	$-0.176630\pi$
$829$	48.9443	1.69991	0.849953	+	0.526859i	$0.176630\pi$
$830$	3.67666	0.127619
$831$	0	0
$832$	30.7274	1.06528
$833$	−22.0596	−0.764321
$834$	0	0
$835$	3.72538	0.128922
$836$	−9.24131	−0.319617
$837$	0	0
$838$	−4.63487	−0.160109
$839$	30.7814	1.06269	0.531347	−	0.847154i	$-0.321686\pi$
$839$	30.7814	1.06269	0.531347	+	0.847154i	$0.321686\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	7.81362	0.269275
$843$	0	0
$844$	22.6401	0.779304
$845$	−9.66112	−0.332353
$846$	0	0
$847$	−13.4750	−0.463005
$848$	−1.40136	−0.0481229
$849$	0	0
$850$	31.2447	1.07168
$851$	10.9983	0.377018
$852$	0	0
$853$	−40.1642	−1.37520	−0.687598	−	0.726092i	$-0.741335\pi$
$853$	−40.1642	−1.37520	−0.687598	+	0.726092i	$0.741335\pi$
$854$	0.506485	0.0173316
$855$	0	0
$856$	−46.5696	−1.59172
$857$	−25.4718	−0.870102	−0.435051	−	0.900406i	$-0.643270\pi$
$857$	−25.4718	−0.870102	−0.435051	+	0.900406i	$0.643270\pi$
$858$	0	0
$859$	−2.41538	−0.0824117	−0.0412059	−	0.999151i	$-0.513120\pi$
$859$	−2.41538	−0.0824117	−0.0412059	+	0.999151i	$0.513120\pi$
$860$	−5.21035	−0.177672
$861$	0	0
$862$	−9.63507	−0.328172
$863$	−20.6291	−0.702224	−0.351112	−	0.936333i	$-0.614196\pi$
$863$	−20.6291	−0.702224	−0.351112	+	0.936333i	$0.614196\pi$
$864$	0	0
$865$	10.7854	0.366715
$866$	−5.92003	−0.201171
$867$	0	0
$868$	−0.258904	−0.00878779
$869$	58.4278	1.98203
$870$	0	0
$871$	−80.7064	−2.73463
$872$	−50.6433	−1.71500
$873$	0	0
$874$	1.63710	0.0553756
$875$	−9.26435	−0.313192
$876$	0	0
$877$	−1.94556	−0.0656969	−0.0328485	−	0.999460i	$-0.510458\pi$
$877$	−1.94556	−0.0656969	−0.0328485	+	0.999460i	$0.510458\pi$
$878$	3.99333	0.134768
$879$	0	0
$880$	−0.328439	−0.0110717
$881$	27.7483	0.934863	0.467431	−	0.884029i	$-0.345180\pi$
$881$	27.7483	0.934863	0.467431	+	0.884029i	$0.345180\pi$
$882$	0	0
$883$	23.2234	0.781531	0.390765	−	0.920490i	$-0.372210\pi$
$883$	23.2234	0.781531	0.390765	+	0.920490i	$0.372210\pi$
$884$	−50.6285	−1.70282
$885$	0	0
$886$	−34.8042	−1.16927
$887$	−42.6578	−1.43231	−0.716154	−	0.697943i	$-0.754099\pi$
$887$	−42.6578	−1.43231	−0.716154	+	0.697943i	$0.754099\pi$
$888$	0	0
$889$	−8.00130	−0.268355
$890$	0.669821	0.0224525
$891$	0	0
$892$	−11.8128	−0.395522
$893$	13.2927	0.444825
$894$	0	0
$895$	−8.52702	−0.285027
$896$	−12.7649	−0.426444
$897$	0	0
$898$	−29.7210	−0.991803
$899$	−0.108189	−0.00360831
$900$	0	0
$901$	62.4269	2.07974
$902$	1.18573	0.0394805
$903$	0	0
$904$	16.6078	0.552367
$905$	−9.84577	−0.327284
$906$	0	0
$907$	7.09985	0.235747	0.117873	−	0.993029i	$-0.462392\pi$
$907$	7.09985	0.235747	0.117873	+	0.993029i	$0.462392\pi$
$908$	−12.8403	−0.426119
$909$	0	0
$910$	−4.89809	−0.162370
$911$	−7.31959	−0.242509	−0.121254	−	0.992621i	$-0.538692\pi$
$911$	−7.31959	−0.242509	−0.121254	+	0.992621i	$0.538692\pi$
$912$	0	0
$913$	−36.4473	−1.20623
$914$	−3.54727	−0.117333
$915$	0	0
$916$	23.8813	0.789061
$917$	−33.3543	−1.10145
$918$	0	0
$919$	30.3798	1.00214	0.501068	−	0.865408i	$-0.332940\pi$
$919$	30.3798	1.00214	0.501068	+	0.865408i	$0.332940\pi$
$920$	−1.35966	−0.0448265
$921$	0	0
$922$	−11.7696	−0.387610
$923$	4.20758	0.138494
$924$	0	0
$925$	−52.5112	−1.72656
$926$	−36.5880	−1.20236
$927$	0	0
$928$	−5.57912	−0.183143
$929$	53.5343	1.75640	0.878202	−	0.478289i	$-0.158743\pi$
$929$	53.5343	1.75640	0.878202	+	0.478289i	$0.158743\pi$
$930$	0	0
$931$	5.51850	0.180861
$932$	−22.4303	−0.734729
$933$	0	0
$934$	−7.70231	−0.252027
$935$	14.6311	0.478488
$936$	0	0
$937$	−42.5867	−1.39125	−0.695624	−	0.718406i	$-0.744872\pi$
$937$	−42.5867	−1.39125	−0.695624	+	0.718406i	$0.744872\pi$
$938$	−24.9643	−0.815115
$939$	0	0
$940$	−4.13797	−0.134966
$941$	−8.63427	−0.281469	−0.140735	−	0.990047i	$-0.544946\pi$
$941$	−8.63427	−0.281469	−0.140735	+	0.990047i	$0.544946\pi$
$942$	0	0
$943$	0.314460	0.0102402
$944$	−2.23709	−0.0728112
$945$	0	0
$946$	−34.5018	−1.12175
$947$	−6.11497	−0.198710	−0.0993549	−	0.995052i	$-0.531678\pi$
$947$	−6.11497	−0.198710	−0.0993549	+	0.995052i	$0.531678\pi$
$948$	0	0
$949$	−10.3483	−0.335921
$950$	−7.81626	−0.253593
$951$	0	0
$952$	−41.7820	−1.35416
$953$	39.6506	1.28441	0.642204	−	0.766533i	$-0.278020\pi$
$953$	39.6506	1.28441	0.642204	+	0.766533i	$0.278020\pi$
$954$	0	0
$955$	−0.489008	−0.0158239
$956$	12.1656	0.393463
$957$	0	0
$958$	−28.6058	−0.924212
$959$	−14.2409	−0.459863
$960$	0	0
$961$	−30.9883	−0.999622
$962$	−56.8371	−1.83250
$963$	0	0
$964$	−25.5790	−0.823845
$965$	−10.3583	−0.333446
$966$	0	0
$967$	38.2387	1.22967	0.614837	−	0.788655i	$-0.289222\pi$
$967$	38.2387	1.22967	0.614837	+	0.788655i	$0.289222\pi$
$968$	19.3301	0.621294
$969$	0	0
$970$	−1.55396	−0.0498948
$971$	−1.26038	−0.0404475	−0.0202237	−	0.999795i	$-0.506438\pi$
$971$	−1.26038	−0.0404475	−0.0202237	+	0.999795i	$0.506438\pi$
$972$	0	0
$973$	2.28274	0.0731812
$974$	36.1162	1.15724
$975$	0	0
$976$	−0.0465456	−0.00148989
$977$	−22.8546	−0.731185	−0.365592	−	0.930775i	$-0.619134\pi$
$977$	−22.8546	−0.731185	−0.365592	+	0.930775i	$0.619134\pi$
$978$	0	0
$979$	−6.64006	−0.212217
$980$	−1.71788	−0.0548757
$981$	0	0
$982$	−29.8526	−0.952636
$983$	49.7031	1.58528	0.792642	−	0.609688i	$-0.208705\pi$
$983$	49.7031	1.58528	0.792642	+	0.609688i	$0.208705\pi$
$984$	0	0
$985$	0.198763	0.00633313
$986$	−6.54412	−0.208407
$987$	0	0
$988$	12.6654	0.402939
$989$	−9.14999	−0.290953
$990$	0	0
$991$	−12.5663	−0.399181	−0.199590	−	0.979879i	$-0.563961\pi$
$991$	−12.5663	−0.399181	−0.199590	+	0.979879i	$0.563961\pi$
$992$	0.603599	0.0191643
$993$	0	0
$994$	1.30150	0.0412811
$995$	−8.68756	−0.275414
$996$	0	0
$997$	24.4153	0.773242	0.386621	−	0.922239i	$-0.373642\pi$
$997$	24.4153	0.773242	0.386621	+	0.922239i	$0.373642\pi$
$998$	−1.73226	−0.0548336
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.p.1.5		14
3.2	odd	2		2001.2.a.m.1.10	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.2.a.m.1.10	✓	14	3.2	odd	2
6003.2.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:	\( N \)	\(=\)	\( 6003 = 3^{2} \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6003.a (trivial)

\(f(q)\)	\(=\)	\(q-0.894954 q^{2} -1.19906 q^{4} -0.474904 q^{5} -1.99579 q^{7} +2.86301 q^{8} +O(q^{10})\)	\(q-0.894954 q^{2} -1.19906 q^{4} -0.474904 q^{5} -1.99579 q^{7} +2.86301 q^{8} +0.425017 q^{10} -4.21327 q^{11} +5.77436 q^{13} +1.78614 q^{14} -0.164145 q^{16} +7.31225 q^{17} -1.82925 q^{19} +0.569438 q^{20} +3.77069 q^{22} +1.00000 q^{23} -4.77447 q^{25} -5.16779 q^{26} +2.39307 q^{28} +1.00000 q^{29} -0.108189 q^{31} -5.57912 q^{32} -6.54412 q^{34} +0.947812 q^{35} +10.9983 q^{37} +1.63710 q^{38} -1.35966 q^{40} +0.314460 q^{41} -9.14999 q^{43} +5.05196 q^{44} -0.894954 q^{46} -7.26676 q^{47} -3.01680 q^{49} +4.27293 q^{50} -6.92380 q^{52} +8.53731 q^{53} +2.00090 q^{55} -5.71398 q^{56} -0.894954 q^{58} +13.6287 q^{59} +0.283563 q^{61} +0.0968242 q^{62} +5.32134 q^{64} -2.74227 q^{65} -13.9767 q^{67} -8.76781 q^{68} -0.848248 q^{70} +0.728665 q^{71} -1.79211 q^{73} -9.84300 q^{74} +2.19338 q^{76} +8.40883 q^{77} -13.8676 q^{79} +0.0779533 q^{80} -0.281427 q^{82} +8.65060 q^{83} -3.47262 q^{85} +8.18882 q^{86} -12.0626 q^{88} +1.57598 q^{89} -11.5244 q^{91} -1.19906 q^{92} +6.50342 q^{94} +0.868720 q^{95} -3.65623 q^{97} +2.69990 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 2 q^{2} + 12 q^{4} + 3 q^{5} - 3 q^{7} + 6 q^{8}+O(q^{10}) \) 14 * q + 2 * q^2 + 12 * q^4 + 3 * q^5 - 3 * q^7 + 6 * q^8	\( 14 q + 2 q^{2} + 12 q^{4} + 3 q^{5} - 3 q^{7} + 6 q^{8} - 5 q^{10} + 12 q^{11} + 13 q^{13} + 9 q^{14} + 14 q^{17} - 9 q^{19} + 2 q^{20} - 9 q^{22} + 14 q^{23} + 13 q^{25} + 16 q^{26} + 3 q^{28} + 14 q^{29} - 28 q^{31} + 4 q^{32} + 14 q^{34} + 9 q^{35} - 12 q^{37} - 2 q^{38} - 20 q^{40} + 25 q^{41} + 5 q^{43} + 37 q^{44} + 2 q^{46} + 17 q^{47} + 17 q^{49} + 44 q^{50} + 25 q^{52} + 17 q^{53} + q^{55} + 54 q^{56} + 2 q^{58} + 18 q^{59} - 13 q^{61} + 8 q^{62} + 20 q^{64} + 16 q^{65} + 2 q^{67} + 19 q^{68} + 14 q^{70} + 55 q^{71} + 19 q^{73} - 4 q^{74} - 32 q^{76} + 19 q^{77} - 68 q^{79} + 2 q^{80} - 12 q^{82} + 21 q^{83} + 16 q^{85} + 22 q^{86} - 25 q^{88} + 17 q^{89} - 30 q^{91} + 12 q^{92} + 16 q^{94} + 55 q^{95} + 25 q^{97} + 31 q^{98}+O(q^{100}) \) 14 * q + 2 * q^2 + 12 * q^4 + 3 * q^5 - 3 * q^7 + 6 * q^8 - 5 * q^10 + 12 * q^11 + 13 * q^13 + 9 * q^14 + 14 * q^17 - 9 * q^19 + 2 * q^20 - 9 * q^22 + 14 * q^23 + 13 * q^25 + 16 * q^26 + 3 * q^28 + 14 * q^29 - 28 * q^31 + 4 * q^32 + 14 * q^34 + 9 * q^35 - 12 * q^37 - 2 * q^38 - 20 * q^40 + 25 * q^41 + 5 * q^43 + 37 * q^44 + 2 * q^46 + 17 * q^47 + 17 * q^49 + 44 * q^50 + 25 * q^52 + 17 * q^53 + q^55 + 54 * q^56 + 2 * q^58 + 18 * q^59 - 13 * q^61 + 8 * q^62 + 20 * q^64 + 16 * q^65 + 2 * q^67 + 19 * q^68 + 14 * q^70 + 55 * q^71 + 19 * q^73 - 4 * q^74 - 32 * q^76 + 19 * q^77 - 68 * q^79 + 2 * q^80 - 12 * q^82 + 21 * q^83 + 16 * q^85 + 22 * q^86 - 25 * q^88 + 17 * q^89 - 30 * q^91 + 12 * q^92 + 16 * q^94 + 55 * q^95 + 25 * q^97 + 31 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more