LMFDB - Embedded newform 6003.2.a.k.1.9

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:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - x^{9} - 17x^{8} + 23x^{7} + 69x^{6} - 88x^{5} - 106x^{4} + 101x^{3} + 60x^{2} - 23x - 11 $ x^10 - x^9 - 17x^8 + 23x^7 + 69x^6 - 88x^5 - 106x^4 + 101x^3 + 60x^2 - 23x - 11
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2001)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$-3.66000$ 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)

$f(q)$	$=$	$q+2.24431 q^{2} +3.03693 q^{4} +1.31370 q^{5} -2.96709 q^{7} +2.32720 q^{8} +O(q^{10})$	\(q+2.24431 q^{2} +3.03693 q^{4} +1.31370 q^{5} -2.96709 q^{7} +2.32720 q^{8} +2.94835 q^{10} -1.51682 q^{11} +0.902202 q^{13} -6.65908 q^{14} -0.850911 q^{16} -3.54263 q^{17} +0.438527 q^{19} +3.98961 q^{20} -3.40420 q^{22} +1.00000 q^{23} -3.27420 q^{25} +2.02482 q^{26} -9.01086 q^{28} -1.00000 q^{29} -1.36633 q^{31} -6.56410 q^{32} -7.95077 q^{34} -3.89786 q^{35} -10.9378 q^{37} +0.984190 q^{38} +3.05723 q^{40} -3.73744 q^{41} -6.23701 q^{43} -4.60646 q^{44} +2.24431 q^{46} -4.40935 q^{47} +1.80364 q^{49} -7.34832 q^{50} +2.73993 q^{52} +10.7318 q^{53} -1.99264 q^{55} -6.90501 q^{56} -2.24431 q^{58} -7.80539 q^{59} +8.66624 q^{61} -3.06647 q^{62} -13.0301 q^{64} +1.18522 q^{65} +1.02600 q^{67} -10.7587 q^{68} -8.74801 q^{70} +11.2518 q^{71} -2.88266 q^{73} -24.5479 q^{74} +1.33178 q^{76} +4.50053 q^{77} +16.1535 q^{79} -1.11784 q^{80} -8.38797 q^{82} -3.36143 q^{83} -4.65395 q^{85} -13.9978 q^{86} -3.52993 q^{88} +7.47996 q^{89} -2.67692 q^{91} +3.03693 q^{92} -9.89596 q^{94} +0.576092 q^{95} -15.6206 q^{97} +4.04792 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 3 q^{2} + 17 q^{4} - 6 q^{5} + 3 q^{7} + 6 q^{8}+O(q^{10}) $ 10 * q - 3 * q^2 + 17 * q^4 - 6 * q^5 + 3 * q^7 + 6 * q^8	$ 10 q - 3 q^{2} + 17 q^{4} - 6 q^{5} + 3 q^{7} + 6 q^{8} - 4 q^{10} - 9 q^{11} - 16 q^{13} - 16 q^{14} + 27 q^{16} + q^{19} - 21 q^{20} + 17 q^{22} + 10 q^{23} - 4 q^{25} - 28 q^{26} - 14 q^{28} - 10 q^{29} + 17 q^{31} - 21 q^{32} - 3 q^{34} - 29 q^{35} + q^{37} - 32 q^{38} + 13 q^{40} - 5 q^{43} - 33 q^{44} - 3 q^{46} - 15 q^{47} + 31 q^{49} + 22 q^{50} - 21 q^{52} - 35 q^{53} - 20 q^{55} - 18 q^{56} + 3 q^{58} - 49 q^{59} + 8 q^{61} - 15 q^{62} + 12 q^{64} + 3 q^{65} + 35 q^{67} + 18 q^{68} - 16 q^{70} - 30 q^{71} - 15 q^{73} - 23 q^{74} + 10 q^{76} - 23 q^{77} + 24 q^{79} - 23 q^{80} - 5 q^{82} - q^{83} + 10 q^{86} + 18 q^{88} - 15 q^{89} + 26 q^{91} + 17 q^{92} + 3 q^{94} - 7 q^{95} - 35 q^{97} - 3 q^{98}+O(q^{100}) $ 10 * q - 3 * q^2 + 17 * q^4 - 6 * q^5 + 3 * q^7 + 6 * q^8 - 4 * q^10 - 9 * q^11 - 16 * q^13 - 16 * q^14 + 27 * q^16 + q^19 - 21 * q^20 + 17 * q^22 + 10 * q^23 - 4 * q^25 - 28 * q^26 - 14 * q^28 - 10 * q^29 + 17 * q^31 - 21 * q^32 - 3 * q^34 - 29 * q^35 + q^37 - 32 * q^38 + 13 * q^40 - 5 * q^43 - 33 * q^44 - 3 * q^46 - 15 * q^47 + 31 * q^49 + 22 * q^50 - 21 * q^52 - 35 * q^53 - 20 * q^55 - 18 * q^56 + 3 * q^58 - 49 * q^59 + 8 * q^61 - 15 * q^62 + 12 * q^64 + 3 * q^65 + 35 * q^67 + 18 * q^68 - 16 * q^70 - 30 * q^71 - 15 * q^73 - 23 * q^74 + 10 * q^76 - 23 * q^77 + 24 * q^79 - 23 * q^80 - 5 * q^82 - q^83 + 10 * q^86 + 18 * q^88 - 15 * q^89 + 26 * q^91 + 17 * q^92 + 3 * q^94 - 7 * q^95 - 35 * q^97 - 3 * 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 $

$2$	2.24431	1.58697	0.793484	−	0.608591i	$-0.208265\pi$
$2$	2.24431	1.58697	0.793484	+	0.608591i	$0.208265\pi$
$3$	0	0
$3$	0	0
$4$	3.03693	1.51847
$5$	1.31370	0.587503	0.293752	−	0.955882i	$-0.405096\pi$
$5$	1.31370	0.587503	0.293752	+	0.955882i	$0.405096\pi$
$6$	0	0
$7$	−2.96709	−1.12146	−0.560728	−	0.828000i	$-0.689479\pi$
$7$	−2.96709	−1.12146	−0.560728	+	0.828000i	$0.689479\pi$
$8$	2.32720	0.822788
$9$	0	0
$10$	2.94835	0.932349
$11$	−1.51682	−0.457337	−0.228668	−	0.973504i	$-0.573437\pi$
$11$	−1.51682	−0.457337	−0.228668	+	0.973504i	$0.573437\pi$
$12$	0	0
$13$	0.902202	0.250226	0.125113	−	0.992143i	$-0.460071\pi$
$13$	0.902202	0.250226	0.125113	+	0.992143i	$0.460071\pi$
$14$	−6.65908	−1.77971
$15$	0	0
$16$	−0.850911	−0.212728
$17$	−3.54263	−0.859215	−0.429607	−	0.903016i	$-0.641348\pi$
$17$	−3.54263	−0.859215	−0.429607	+	0.903016i	$0.641348\pi$
$18$	0	0
$19$	0.438527	0.100605	0.0503025	−	0.998734i	$-0.483981\pi$
$19$	0.438527	0.100605	0.0503025	+	0.998734i	$0.483981\pi$
$20$	3.98961	0.892104
$21$	0	0
$22$	−3.40420	−0.725779
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−3.27420	−0.654840
$26$	2.02482	0.397100
$27$	0	0
$28$	−9.01086	−1.70289
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−1.36633	−0.245400	−0.122700	−	0.992444i	$-0.539155\pi$
$31$	−1.36633	−0.245400	−0.122700	+	0.992444i	$0.539155\pi$
$32$	−6.56410	−1.16038
$33$	0	0
$34$	−7.95077	−1.36355
$35$	−3.89786	−0.658859
$36$	0	0
$37$	−10.9378	−1.79817	−0.899084	−	0.437776i	$-0.855766\pi$
$37$	−10.9378	−1.79817	−0.899084	+	0.437776i	$0.855766\pi$
$38$	0.984190	0.159657
$39$	0	0
$40$	3.05723	0.483391
$41$	−3.73744	−0.583690	−0.291845	−	0.956466i	$-0.594269\pi$
$41$	−3.73744	−0.583690	−0.291845	+	0.956466i	$0.594269\pi$
$42$	0	0
$43$	−6.23701	−0.951135	−0.475568	−	0.879679i	$-0.657757\pi$
$43$	−6.23701	−0.951135	−0.475568	+	0.879679i	$0.657757\pi$
$44$	−4.60646	−0.694450
$45$	0	0
$46$	2.24431	0.330906
$47$	−4.40935	−0.643170	−0.321585	−	0.946881i	$-0.604216\pi$
$47$	−4.40935	−0.643170	−0.321585	+	0.946881i	$0.604216\pi$
$48$	0	0
$49$	1.80364	0.257663
$50$	−7.34832	−1.03921
$51$	0	0
$52$	2.73993	0.379959
$53$	10.7318	1.47413	0.737065	−	0.675822i	$-0.236211\pi$
$53$	10.7318	1.47413	0.737065	+	0.675822i	$0.236211\pi$
$54$	0	0
$55$	−1.99264	−0.268687
$56$	−6.90501	−0.922720
$57$	0	0
$58$	−2.24431	−0.294692
$59$	−7.80539	−1.01618	−0.508088	−	0.861305i	$-0.669647\pi$
$59$	−7.80539	−1.01618	−0.508088	+	0.861305i	$0.669647\pi$
$60$	0	0
$61$	8.66624	1.10960	0.554799	−	0.831984i	$-0.312795\pi$
$61$	8.66624	1.10960	0.554799	+	0.831984i	$0.312795\pi$
$62$	−3.06647	−0.389442
$63$	0	0
$64$	−13.0301	−1.62876
$65$	1.18522	0.147009
$66$	0	0
$67$	1.02600	0.125346	0.0626729	−	0.998034i	$-0.480038\pi$
$67$	1.02600	0.125346	0.0626729	+	0.998034i	$0.480038\pi$
$68$	−10.7587	−1.30469
$69$	0	0
$70$	−8.74801	−1.04559
$71$	11.2518	1.33535	0.667674	−	0.744454i	$-0.267290\pi$
$71$	11.2518	1.33535	0.667674	+	0.744454i	$0.267290\pi$
$72$	0	0
$73$	−2.88266	−0.337390	−0.168695	−	0.985668i	$-0.553955\pi$
$73$	−2.88266	−0.337390	−0.168695	+	0.985668i	$0.553955\pi$
$74$	−24.5479	−2.85363
$75$	0	0
$76$	1.33178	0.152765
$77$	4.50053	0.512883
$78$	0	0
$79$	16.1535	1.81741	0.908703	−	0.417443i	$-0.137074\pi$
$79$	16.1535	1.81741	0.908703	+	0.417443i	$0.137074\pi$
$80$	−1.11784	−0.124978
$81$	0	0
$82$	−8.38797	−0.926296
$83$	−3.36143	−0.368965	−0.184483	−	0.982836i	$-0.559061\pi$
$83$	−3.36143	−0.368965	−0.184483	+	0.982836i	$0.559061\pi$
$84$	0	0
$85$	−4.65395	−0.504792
$86$	−13.9978	−1.50942
$87$	0	0
$88$	−3.52993	−0.376291
$89$	7.47996	0.792874	0.396437	−	0.918062i	$-0.370247\pi$
$89$	7.47996	0.792874	0.396437	+	0.918062i	$0.370247\pi$
$90$	0	0
$91$	−2.67692	−0.280617
$92$	3.03693	0.316622
$93$	0	0
$94$	−9.89596	−1.02069
$95$	0.576092	0.0591057
$96$	0	0
$97$	−15.6206	−1.58603	−0.793014	−	0.609203i	$-0.791489\pi$
$97$	−15.6206	−1.58603	−0.793014	+	0.609203i	$0.791489\pi$
$98$	4.04792	0.408902
$99$	0	0
$100$	−9.94352	−0.994352
$101$	3.63336	0.361533	0.180767	−	0.983526i	$-0.442142\pi$
$101$	3.63336	0.361533	0.180767	+	0.983526i	$0.442142\pi$
$102$	0	0
$103$	10.2810	1.01302	0.506510	−	0.862234i	$-0.330935\pi$
$103$	10.2810	1.01302	0.506510	+	0.862234i	$0.330935\pi$
$104$	2.09960	0.205883
$105$	0	0
$106$	24.0856	2.33940
$107$	−7.31175	−0.706854	−0.353427	−	0.935462i	$-0.614984\pi$
$107$	−7.31175	−0.706854	−0.353427	+	0.935462i	$0.614984\pi$
$108$	0	0
$109$	−3.37760	−0.323515	−0.161758	−	0.986831i	$-0.551716\pi$
$109$	−3.37760	−0.323515	−0.161758	+	0.986831i	$0.551716\pi$
$110$	−4.47210	−0.426398
$111$	0	0
$112$	2.52473	0.238565
$113$	2.02102	0.190122	0.0950610	−	0.995471i	$-0.469695\pi$
$113$	2.02102	0.190122	0.0950610	+	0.995471i	$0.469695\pi$
$114$	0	0
$115$	1.31370	0.122503
$116$	−3.03693	−0.281972
$117$	0	0
$118$	−17.5177	−1.61264
$119$	10.5113	0.963571
$120$	0	0
$121$	−8.69927	−0.790843
$122$	19.4497	1.76090
$123$	0	0
$124$	−4.14945	−0.372632
$125$	−10.8698	−0.972224
$126$	0	0
$127$	2.41766	0.214533	0.107266	−	0.994230i	$-0.465790\pi$
$127$	2.41766	0.214533	0.107266	+	0.994230i	$0.465790\pi$
$128$	−16.1153	−1.42441
$129$	0	0
$130$	2.66000	0.233298
$131$	−3.99026	−0.348630	−0.174315	−	0.984690i	$-0.555771\pi$
$131$	−3.99026	−0.348630	−0.174315	+	0.984690i	$0.555771\pi$
$132$	0	0
$133$	−1.30115	−0.112824
$134$	2.30266	0.198920
$135$	0	0
$136$	−8.24440	−0.706951
$137$	−6.48948	−0.554434	−0.277217	−	0.960807i	$-0.589412\pi$
$137$	−6.48948	−0.554434	−0.277217	+	0.960807i	$0.589412\pi$
$138$	0	0
$139$	−18.9540	−1.60766	−0.803830	−	0.594859i	$-0.797208\pi$
$139$	−18.9540	−1.60766	−0.803830	+	0.594859i	$0.797208\pi$
$140$	−11.8375	−1.00045
$141$	0	0
$142$	25.2526	2.11915
$143$	−1.36847	−0.114438
$144$	0	0
$145$	−1.31370	−0.109097
$146$	−6.46958	−0.535427
$147$	0	0
$148$	−33.2174	−2.73046
$149$	−22.8616	−1.87290	−0.936448	−	0.350807i	$-0.885907\pi$
$149$	−22.8616	−1.87290	−0.936448	+	0.350807i	$0.885907\pi$
$150$	0	0
$151$	16.8032	1.36743	0.683714	−	0.729750i	$-0.260364\pi$
$151$	16.8032	1.36743	0.683714	+	0.729750i	$0.260364\pi$
$152$	1.02054	0.0827765
$153$	0	0
$154$	10.1006	0.813929
$155$	−1.79495	−0.144174
$156$	0	0
$157$	4.66486	0.372296	0.186148	−	0.982522i	$-0.440400\pi$
$157$	4.66486	0.372296	0.186148	+	0.982522i	$0.440400\pi$
$158$	36.2534	2.88416
$159$	0	0
$160$	−8.62324	−0.681727
$161$	−2.96709	−0.233840
$162$	0	0
$163$	−6.11817	−0.479212	−0.239606	−	0.970870i	$-0.577018\pi$
$163$	−6.11817	−0.479212	−0.239606	+	0.970870i	$0.577018\pi$
$164$	−11.3503	−0.886312
$165$	0	0
$166$	−7.54410	−0.585536
$167$	−5.11459	−0.395779	−0.197889	−	0.980224i	$-0.563409\pi$
$167$	−5.11459	−0.395779	−0.197889	+	0.980224i	$0.563409\pi$
$168$	0	0
$169$	−12.1860	−0.937387
$170$	−10.4449	−0.801088
$171$	0	0
$172$	−18.9414	−1.44427
$173$	−10.5173	−0.799614	−0.399807	−	0.916599i	$-0.630923\pi$
$173$	−10.5173	−0.799614	−0.399807	+	0.916599i	$0.630923\pi$
$174$	0	0
$175$	9.71485	0.734374
$176$	1.29068	0.0972883
$177$	0	0
$178$	16.7873	1.25826
$179$	−10.1812	−0.760981	−0.380491	−	0.924785i	$-0.624245\pi$
$179$	−10.1812	−0.760981	−0.380491	+	0.924785i	$0.624245\pi$
$180$	0	0
$181$	21.6445	1.60882	0.804412	−	0.594072i	$-0.202481\pi$
$181$	21.6445	1.60882	0.804412	+	0.594072i	$0.202481\pi$
$182$	−6.00784	−0.445330
$183$	0	0
$184$	2.32720	0.171563
$185$	−14.3690	−1.05643
$186$	0	0
$187$	5.37352	0.392951
$188$	−13.3909	−0.976632
$189$	0	0
$190$	1.29293	0.0937989
$191$	−17.2119	−1.24541	−0.622705	−	0.782457i	$-0.713966\pi$
$191$	−17.2119	−1.24541	−0.622705	+	0.782457i	$0.713966\pi$
$192$	0	0
$193$	−7.97678	−0.574181	−0.287090	−	0.957903i	$-0.592688\pi$
$193$	−7.97678	−0.574181	−0.287090	+	0.957903i	$0.592688\pi$
$194$	−35.0574	−2.51698
$195$	0	0
$196$	5.47752	0.391252
$197$	−5.13962	−0.366183	−0.183092	−	0.983096i	$-0.558610\pi$
$197$	−5.13962	−0.366183	−0.183092	+	0.983096i	$0.558610\pi$
$198$	0	0
$199$	21.9501	1.55600	0.777999	−	0.628265i	$-0.216235\pi$
$199$	21.9501	1.55600	0.777999	+	0.628265i	$0.216235\pi$
$200$	−7.61970	−0.538794
$201$	0	0
$202$	8.15440	0.573741
$203$	2.96709	0.208249
$204$	0	0
$205$	−4.90986	−0.342920
$206$	23.0738	1.60763
$207$	0	0
$208$	−0.767694	−0.0532300
$209$	−0.665164	−0.0460104
$210$	0	0
$211$	7.00040	0.481927	0.240964	−	0.970534i	$-0.422537\pi$
$211$	7.00040	0.481927	0.240964	+	0.970534i	$0.422537\pi$
$212$	32.5918	2.23841
$213$	0	0
$214$	−16.4098	−1.12175
$215$	−8.19355	−0.558795
$216$	0	0
$217$	4.05403	0.275206
$218$	−7.58038	−0.513408
$219$	0	0
$220$	−6.05150	−0.407992
$221$	−3.19617	−0.214998
$222$	0	0
$223$	19.3205	1.29380	0.646899	−	0.762575i	$-0.276065\pi$
$223$	19.3205	1.29380	0.646899	+	0.762575i	$0.276065\pi$
$224$	19.4763	1.30131
$225$	0	0
$226$	4.53581	0.301717
$227$	4.35027	0.288738	0.144369	−	0.989524i	$-0.453885\pi$
$227$	4.35027	0.288738	0.144369	+	0.989524i	$0.453885\pi$
$228$	0	0
$229$	14.2691	0.942927	0.471464	−	0.881886i	$-0.343726\pi$
$229$	14.2691	0.942927	0.471464	+	0.881886i	$0.343726\pi$
$230$	2.94835	0.194408
$231$	0	0
$232$	−2.32720	−0.152788
$233$	−1.10104	−0.0721315	−0.0360657	−	0.999349i	$-0.511483\pi$
$233$	−1.10104	−0.0721315	−0.0360657	+	0.999349i	$0.511483\pi$
$234$	0	0
$235$	−5.79256	−0.377865
$236$	−23.7044	−1.54303
$237$	0	0
$238$	23.5907	1.52916
$239$	−3.57602	−0.231314	−0.115657	−	0.993289i	$-0.536897\pi$
$239$	−3.57602	−0.231314	−0.115657	+	0.993289i	$0.536897\pi$
$240$	0	0
$241$	23.8181	1.53426	0.767130	−	0.641492i	$-0.221684\pi$
$241$	23.8181	1.53426	0.767130	+	0.641492i	$0.221684\pi$
$242$	−19.5239	−1.25504
$243$	0	0
$244$	26.3188	1.68489
$245$	2.36943	0.151378
$246$	0	0
$247$	0.395640	0.0251740
$248$	−3.17972	−0.201912
$249$	0	0
$250$	−24.3952	−1.54289
$251$	11.0028	0.694489	0.347245	−	0.937775i	$-0.387117\pi$
$251$	11.0028	0.694489	0.347245	+	0.937775i	$0.387117\pi$
$252$	0	0
$253$	−1.51682	−0.0953614
$254$	5.42599	0.340457
$255$	0	0
$256$	−10.1076	−0.631727
$257$	−22.5544	−1.40691	−0.703453	−	0.710742i	$-0.748360\pi$
$257$	−22.5544	−1.40691	−0.703453	+	0.710742i	$0.748360\pi$
$258$	0	0
$259$	32.4536	2.01657
$260$	3.59943	0.223227
$261$	0	0
$262$	−8.95538	−0.553265
$263$	−10.0124	−0.617390	−0.308695	−	0.951161i	$-0.599892\pi$
$263$	−10.0124	−0.617390	−0.308695	+	0.951161i	$0.599892\pi$
$264$	0	0
$265$	14.0984	0.866056
$266$	−2.92018	−0.179048
$267$	0	0
$268$	3.11589	0.190333
$269$	12.9882	0.791902	0.395951	−	0.918272i	$-0.370415\pi$
$269$	12.9882	0.791902	0.395951	+	0.918272i	$0.370415\pi$
$270$	0	0
$271$	15.5385	0.943894	0.471947	−	0.881627i	$-0.343551\pi$
$271$	15.5385	0.943894	0.471947	+	0.881627i	$0.343551\pi$
$272$	3.01447	0.182779
$273$	0	0
$274$	−14.5644	−0.879868
$275$	4.96635	0.299482
$276$	0	0
$277$	17.5683	1.05557	0.527787	−	0.849377i	$-0.323022\pi$
$277$	17.5683	1.05557	0.527787	+	0.849377i	$0.323022\pi$
$278$	−42.5387	−2.55130
$279$	0	0
$280$	−9.07109	−0.542101
$281$	13.1371	0.783693	0.391846	−	0.920031i	$-0.371836\pi$
$281$	13.1371	0.783693	0.391846	+	0.920031i	$0.371836\pi$
$282$	0	0
$283$	−7.30957	−0.434509	−0.217254	−	0.976115i	$-0.569710\pi$
$283$	−7.30957	−0.434509	−0.217254	+	0.976115i	$0.569710\pi$
$284$	34.1710	2.02768
$285$	0	0
$286$	−3.07128	−0.181609
$287$	11.0893	0.654582
$288$	0	0
$289$	−4.44975	−0.261750
$290$	−2.94835	−0.173133
$291$	0	0
$292$	−8.75444	−0.512315
$293$	8.00780	0.467821	0.233910	−	0.972258i	$-0.424848\pi$
$293$	8.00780	0.467821	0.233910	+	0.972258i	$0.424848\pi$
$294$	0	0
$295$	−10.2539	−0.597007
$296$	−25.4545	−1.47951
$297$	0	0
$298$	−51.3085	−2.97222
$299$	0.902202	0.0521757
$300$	0	0
$301$	18.5058	1.06666
$302$	37.7117	2.17006
$303$	0	0
$304$	−0.373147	−0.0214015
$305$	11.3848	0.651893
$306$	0	0
$307$	−30.5296	−1.74242	−0.871208	−	0.490915i	$-0.836663\pi$
$307$	−30.5296	−1.74242	−0.871208	+	0.490915i	$0.836663\pi$
$308$	13.6678	0.778795
$309$	0	0
$310$	−4.02842	−0.228799
$311$	−26.8315	−1.52148	−0.760738	−	0.649060i	$-0.775163\pi$
$311$	−26.8315	−1.52148	−0.760738	+	0.649060i	$0.775163\pi$
$312$	0	0
$313$	7.67441	0.433783	0.216892	−	0.976196i	$-0.430408\pi$
$313$	7.67441	0.433783	0.216892	+	0.976196i	$0.430408\pi$
$314$	10.4694	0.590822
$315$	0	0
$316$	49.0569	2.75967
$317$	11.3137	0.635441	0.317721	−	0.948184i	$-0.397083\pi$
$317$	11.3137	0.635441	0.317721	+	0.948184i	$0.397083\pi$
$318$	0	0
$319$	1.51682	0.0849253
$320$	−17.1176	−0.956901
$321$	0	0
$322$	−6.65908	−0.371096
$323$	−1.55354	−0.0864412
$324$	0	0
$325$	−2.95399	−0.163858
$326$	−13.7311	−0.760493
$327$	0	0
$328$	−8.69775	−0.480253
$329$	13.0830	0.721287
$330$	0	0
$331$	34.5401	1.89849	0.949247	−	0.314531i	$-0.101847\pi$
$331$	34.5401	1.89849	0.949247	+	0.314531i	$0.101847\pi$
$332$	−10.2084	−0.560261
$333$	0	0
$334$	−11.4787	−0.628088
$335$	1.34785	0.0736411
$336$	0	0
$337$	22.5432	1.22801	0.614004	−	0.789303i	$-0.289558\pi$
$337$	22.5432	1.22801	0.614004	+	0.789303i	$0.289558\pi$
$338$	−27.3492	−1.48760
$339$	0	0
$340$	−14.1337	−0.766509
$341$	2.07247	0.112231
$342$	0	0
$343$	15.4181	0.832498
$344$	−14.5147	−0.782583
$345$	0	0
$346$	−23.6041	−1.26896
$347$	−15.7661	−0.846367	−0.423184	−	0.906044i	$-0.639087\pi$
$347$	−15.7661	−0.846367	−0.423184	+	0.906044i	$0.639087\pi$
$348$	0	0
$349$	−18.6754	−0.999673	−0.499836	−	0.866120i	$-0.666607\pi$
$349$	−18.6754	−0.999673	−0.499836	+	0.866120i	$0.666607\pi$
$350$	21.8031	1.16543
$351$	0	0
$352$	9.95653	0.530685
$353$	22.1152	1.17708	0.588538	−	0.808470i	$-0.299704\pi$
$353$	22.1152	1.17708	0.588538	+	0.808470i	$0.299704\pi$
$354$	0	0
$355$	14.7815	0.784521
$356$	22.7161	1.20395
$357$	0	0
$358$	−22.8499	−1.20765
$359$	31.7410	1.67523	0.837613	−	0.546264i	$-0.183951\pi$
$359$	31.7410	1.67523	0.837613	+	0.546264i	$0.183951\pi$
$360$	0	0
$361$	−18.8077	−0.989879
$362$	48.5770	2.55315
$363$	0	0
$364$	−8.12961	−0.426108
$365$	−3.78694	−0.198218
$366$	0	0
$367$	7.88341	0.411511	0.205755	−	0.978603i	$-0.434035\pi$
$367$	7.88341	0.411511	0.205755	+	0.978603i	$0.434035\pi$
$368$	−0.850911	−0.0443568
$369$	0	0
$370$	−32.2485	−1.67652
$371$	−31.8423	−1.65317
$372$	0	0
$373$	−19.3349	−1.00112	−0.500561	−	0.865701i	$-0.666873\pi$
$373$	−19.3349	−1.00112	−0.500561	+	0.865701i	$0.666873\pi$
$374$	12.0598	0.623600
$375$	0	0
$376$	−10.2614	−0.529193
$377$	−0.902202	−0.0464658
$378$	0	0
$379$	−13.7522	−0.706405	−0.353202	−	0.935547i	$-0.614907\pi$
$379$	−13.7522	−0.706405	−0.353202	+	0.935547i	$0.614907\pi$
$380$	1.74955	0.0897500
$381$	0	0
$382$	−38.6289	−1.97642
$383$	−9.78378	−0.499928	−0.249964	−	0.968255i	$-0.580419\pi$
$383$	−9.78378	−0.499928	−0.249964	+	0.968255i	$0.580419\pi$
$384$	0	0
$385$	5.91234	0.301321
$386$	−17.9024	−0.911206
$387$	0	0
$388$	−47.4386	−2.40833
$389$	−20.3246	−1.03050	−0.515250	−	0.857040i	$-0.672301\pi$
$389$	−20.3246	−1.03050	−0.515250	+	0.857040i	$0.672301\pi$
$390$	0	0
$391$	−3.54263	−0.179159
$392$	4.19742	0.212002
$393$	0	0
$394$	−11.5349	−0.581120
$395$	21.2208	1.06773
$396$	0	0
$397$	−15.0836	−0.757024	−0.378512	−	0.925596i	$-0.623564\pi$
$397$	−15.0836	−0.757024	−0.378512	+	0.925596i	$0.623564\pi$
$398$	49.2628	2.46932
$399$	0	0
$400$	2.78605	0.139303
$401$	−31.7667	−1.58635	−0.793176	−	0.608993i	$-0.791574\pi$
$401$	−31.7667	−1.58635	−0.793176	+	0.608993i	$0.791574\pi$
$402$	0	0
$403$	−1.23271	−0.0614055
$404$	11.0343	0.548976
$405$	0	0
$406$	6.65908	0.330484
$407$	16.5907	0.822369
$408$	0	0
$409$	−2.67330	−0.132186	−0.0660931	−	0.997813i	$-0.521053\pi$
$409$	−2.67330	−0.132186	−0.0660931	+	0.997813i	$0.521053\pi$
$410$	−11.0193	−0.544202
$411$	0	0
$412$	31.2228	1.53824
$413$	23.1593	1.13960
$414$	0	0
$415$	−4.41591	−0.216768
$416$	−5.92215	−0.290357
$417$	0	0
$418$	−1.49283	−0.0730169
$419$	−28.1776	−1.37657	−0.688284	−	0.725442i	$-0.741636\pi$
$419$	−28.1776	−1.37657	−0.688284	+	0.725442i	$0.741636\pi$
$420$	0	0
$421$	27.5851	1.34441	0.672207	−	0.740363i	$-0.265346\pi$
$421$	27.5851	1.34441	0.672207	+	0.740363i	$0.265346\pi$
$422$	15.7111	0.764803
$423$	0	0
$424$	24.9751	1.21290
$425$	11.5993	0.562648
$426$	0	0
$427$	−25.7135	−1.24437
$428$	−22.2053	−1.07333
$429$	0	0
$430$	−18.3889	−0.886790
$431$	−5.24056	−0.252429	−0.126215	−	0.992003i	$-0.540283\pi$
$431$	−5.24056	−0.252429	−0.126215	+	0.992003i	$0.540283\pi$
$432$	0	0
$433$	−21.8854	−1.05174	−0.525872	−	0.850564i	$-0.676261\pi$
$433$	−21.8854	−1.05174	−0.525872	+	0.850564i	$0.676261\pi$
$434$	9.09851	0.436742
$435$	0	0
$436$	−10.2575	−0.491247
$437$	0.438527	0.0209776
$438$	0	0
$439$	−13.9271	−0.664702	−0.332351	−	0.943156i	$-0.607842\pi$
$439$	−13.9271	−0.664702	−0.332351	+	0.943156i	$0.607842\pi$
$440$	−4.63726	−0.221072
$441$	0	0
$442$	−7.17320	−0.341194
$443$	27.5227	1.30764	0.653821	−	0.756649i	$-0.273165\pi$
$443$	27.5227	1.30764	0.653821	+	0.756649i	$0.273165\pi$
$444$	0	0
$445$	9.82640	0.465816
$446$	43.3613	2.05322
$447$	0	0
$448$	38.6614	1.82658
$449$	−35.1874	−1.66060	−0.830298	−	0.557320i	$-0.811830\pi$
$449$	−35.1874	−1.66060	−0.830298	+	0.557320i	$0.811830\pi$
$450$	0	0
$451$	5.66900	0.266943
$452$	6.13771	0.288694
$453$	0	0
$454$	9.76336	0.458217
$455$	−3.51666	−0.164864
$456$	0	0
$457$	−18.3507	−0.858409	−0.429205	−	0.903207i	$-0.641206\pi$
$457$	−18.3507	−0.858409	−0.429205	+	0.903207i	$0.641206\pi$
$458$	32.0242	1.49639
$459$	0	0
$460$	3.98961	0.186016
$461$	24.7369	1.15211	0.576056	−	0.817410i	$-0.304591\pi$
$461$	24.7369	1.15211	0.576056	+	0.817410i	$0.304591\pi$
$462$	0	0
$463$	12.6364	0.587264	0.293632	−	0.955919i	$-0.405136\pi$
$463$	12.6364	0.587264	0.293632	+	0.955919i	$0.405136\pi$
$464$	0.850911	0.0395026
$465$	0	0
$466$	−2.47107	−0.114470
$467$	20.5427	0.950603	0.475301	−	0.879823i	$-0.342339\pi$
$467$	20.5427	0.950603	0.475301	+	0.879823i	$0.342339\pi$
$468$	0	0
$469$	−3.04424	−0.140570
$470$	−13.0003	−0.599659
$471$	0	0
$472$	−18.1647	−0.836097
$473$	9.46039	0.434989
$474$	0	0
$475$	−1.43582	−0.0658801
$476$	31.9222	1.46315
$477$	0	0
$478$	−8.02570	−0.367087
$479$	14.3319	0.654842	0.327421	−	0.944879i	$-0.393821\pi$
$479$	14.3319	0.654842	0.327421	+	0.944879i	$0.393821\pi$
$480$	0	0
$481$	−9.86814	−0.449948
$482$	53.4553	2.43482
$483$	0	0
$484$	−26.4191	−1.20087
$485$	−20.5207	−0.931797
$486$	0	0
$487$	−6.62111	−0.300031	−0.150016	−	0.988684i	$-0.547932\pi$
$487$	−6.62111	−0.300031	−0.150016	+	0.988684i	$0.547932\pi$
$488$	20.1680	0.912964
$489$	0	0
$490$	5.31775	0.240231
$491$	16.2298	0.732440	0.366220	−	0.930528i	$-0.380652\pi$
$491$	16.2298	0.732440	0.366220	+	0.930528i	$0.380652\pi$
$492$	0	0
$493$	3.54263	0.159552
$494$	0.887939	0.0399503
$495$	0	0
$496$	1.16263	0.0522035
$497$	−33.3852	−1.49753
$498$	0	0
$499$	−11.1031	−0.497043	−0.248521	−	0.968626i	$-0.579945\pi$
$499$	−11.1031	−0.497043	−0.248521	+	0.968626i	$0.579945\pi$
$500$	−33.0108	−1.47629
$501$	0	0
$502$	24.6937	1.10213
$503$	−7.73894	−0.345062	−0.172531	−	0.985004i	$-0.555195\pi$
$503$	−7.73894	−0.345062	−0.172531	+	0.985004i	$0.555195\pi$
$504$	0	0
$505$	4.77314	0.212402
$506$	−3.40420	−0.151335
$507$	0	0
$508$	7.34227	0.325761
$509$	−12.1628	−0.539105	−0.269553	−	0.962986i	$-0.586876\pi$
$509$	−12.1628	−0.539105	−0.269553	+	0.962986i	$0.586876\pi$
$510$	0	0
$511$	8.55312	0.378368
$512$	9.54594	0.421875
$513$	0	0
$514$	−50.6191	−2.23271
$515$	13.5062	0.595152
$516$	0	0
$517$	6.68817	0.294146
$518$	72.8359	3.20022
$519$	0	0
$520$	2.75824	0.120957
$521$	19.3594	0.848152	0.424076	−	0.905627i	$-0.360599\pi$
$521$	19.3594	0.848152	0.424076	+	0.905627i	$0.360599\pi$
$522$	0	0
$523$	4.25739	0.186163	0.0930814	−	0.995659i	$-0.470328\pi$
$523$	4.25739	0.186163	0.0930814	+	0.995659i	$0.470328\pi$
$524$	−12.1181	−0.529383
$525$	0	0
$526$	−22.4709	−0.979777
$527$	4.84041	0.210852
$528$	0	0
$529$	1.00000	0.0434783
$530$	31.6411	1.37440
$531$	0	0
$532$	−3.95150	−0.171319
$533$	−3.37192	−0.146054
$534$	0	0
$535$	−9.60543	−0.415279
$536$	2.38770	0.103133
$537$	0	0
$538$	29.1495	1.25672
$539$	−2.73579	−0.117839
$540$	0	0
$541$	16.3992	0.705057	0.352528	−	0.935801i	$-0.385322\pi$
$541$	16.3992	0.705057	0.352528	+	0.935801i	$0.385322\pi$
$542$	34.8731	1.49793
$543$	0	0
$544$	23.2542	0.997016
$545$	−4.43714	−0.190066
$546$	0	0
$547$	−16.7707	−0.717065	−0.358533	−	0.933517i	$-0.616723\pi$
$547$	−16.7707	−0.717065	−0.358533	+	0.933517i	$0.616723\pi$
$548$	−19.7081	−0.841889
$549$	0	0
$550$	11.1460	0.475269
$551$	−0.438527	−0.0186819
$552$	0	0
$553$	−47.9288	−2.03814
$554$	39.4286	1.67516
$555$	0	0
$556$	−57.5621	−2.44118
$557$	−36.1688	−1.53252	−0.766261	−	0.642530i	$-0.777885\pi$
$557$	−36.1688	−1.53252	−0.766261	+	0.642530i	$0.777885\pi$
$558$	0	0
$559$	−5.62705	−0.237999
$560$	3.31674	0.140158
$561$	0	0
$562$	29.4837	1.24370
$563$	−15.4439	−0.650883	−0.325441	−	0.945562i	$-0.605513\pi$
$563$	−15.4439	−0.650883	−0.325441	+	0.945562i	$0.605513\pi$
$564$	0	0
$565$	2.65501	0.111697
$566$	−16.4049	−0.689551
$567$	0	0
$568$	26.1852	1.09871
$569$	18.1051	0.759007	0.379503	−	0.925190i	$-0.376095\pi$
$569$	18.1051	0.759007	0.379503	+	0.925190i	$0.376095\pi$
$570$	0	0
$571$	40.4246	1.69171	0.845857	−	0.533409i	$-0.179089\pi$
$571$	40.4246	1.69171	0.845857	+	0.533409i	$0.179089\pi$
$572$	−4.15596	−0.173769
$573$	0	0
$574$	24.8879	1.03880
$575$	−3.27420	−0.136544
$576$	0	0
$577$	−5.89094	−0.245243	−0.122621	−	0.992454i	$-0.539130\pi$
$577$	−5.89094	−0.245243	−0.122621	+	0.992454i	$0.539130\pi$
$578$	−9.98663	−0.415389
$579$	0	0
$580$	−3.98961	−0.165660
$581$	9.97368	0.413778
$582$	0	0
$583$	−16.2782	−0.674174
$584$	−6.70851	−0.277600
$585$	0	0
$586$	17.9720	0.742416
$587$	13.3990	0.553036	0.276518	−	0.961009i	$-0.410819\pi$
$587$	13.3990	0.553036	0.276518	+	0.961009i	$0.410819\pi$
$588$	0	0
$589$	−0.599173	−0.0246885
$590$	−23.0130	−0.947430
$591$	0	0
$592$	9.30713	0.382521
$593$	−7.11166	−0.292041	−0.146020	−	0.989282i	$-0.546647\pi$
$593$	−7.11166	−0.292041	−0.146020	+	0.989282i	$0.546647\pi$
$594$	0	0
$595$	13.8087	0.566101
$596$	−69.4291	−2.84393
$597$	0	0
$598$	2.02482	0.0828011
$599$	−0.867674	−0.0354522	−0.0177261	−	0.999843i	$-0.505643\pi$
$599$	−0.867674	−0.0354522	−0.0177261	+	0.999843i	$0.505643\pi$
$600$	0	0
$601$	2.04317	0.0833424	0.0416712	−	0.999131i	$-0.486732\pi$
$601$	2.04317	0.0833424	0.0416712	+	0.999131i	$0.486732\pi$
$602$	41.5327	1.69275
$603$	0	0
$604$	51.0302	2.07639
$605$	−11.4282	−0.464623
$606$	0	0
$607$	45.7094	1.85529	0.927644	−	0.373465i	$-0.121830\pi$
$607$	45.7094	1.85529	0.927644	+	0.373465i	$0.121830\pi$
$608$	−2.87853	−0.116740
$609$	0	0
$610$	25.5511	1.03453
$611$	−3.97813	−0.160938
$612$	0	0
$613$	21.0161	0.848831	0.424416	−	0.905467i	$-0.360480\pi$
$613$	21.0161	0.848831	0.424416	+	0.905467i	$0.360480\pi$
$614$	−68.5179	−2.76516
$615$	0	0
$616$	10.4736	0.421994
$617$	16.5056	0.664489	0.332244	−	0.943193i	$-0.392194\pi$
$617$	16.5056	0.664489	0.332244	+	0.943193i	$0.392194\pi$
$618$	0	0
$619$	30.3608	1.22030	0.610152	−	0.792284i	$-0.291108\pi$
$619$	30.3608	1.22030	0.610152	+	0.792284i	$0.291108\pi$
$620$	−5.45113	−0.218923
$621$	0	0
$622$	−60.2182	−2.41453
$623$	−22.1937	−0.889173
$624$	0	0
$625$	2.09137	0.0836548
$626$	17.2238	0.688400
$627$	0	0
$628$	14.1669	0.565319
$629$	38.7487	1.54501
$630$	0	0
$631$	0.182379	0.00726040	0.00363020	−	0.999993i	$-0.498844\pi$
$631$	0.182379	0.00726040	0.00363020	+	0.999993i	$0.498844\pi$
$632$	37.5923	1.49534
$633$	0	0
$634$	25.3915	1.00842
$635$	3.17608	0.126039
$636$	0	0
$637$	1.62725	0.0644738
$638$	3.40420	0.134774
$639$	0	0
$640$	−21.1706	−0.836843
$641$	7.86588	0.310684	0.155342	−	0.987861i	$-0.450352\pi$
$641$	7.86588	0.310684	0.155342	+	0.987861i	$0.450352\pi$
$642$	0	0
$643$	11.3455	0.447423	0.223711	−	0.974655i	$-0.428183\pi$
$643$	11.3455	0.447423	0.223711	+	0.974655i	$0.428183\pi$
$644$	−9.01086	−0.355077
$645$	0	0
$646$	−3.48663	−0.137179
$647$	−22.3414	−0.878330	−0.439165	−	0.898406i	$-0.644726\pi$
$647$	−22.3414	−0.878330	−0.439165	+	0.898406i	$0.644726\pi$
$648$	0	0
$649$	11.8393	0.464735
$650$	−6.62967	−0.260037
$651$	0	0
$652$	−18.5804	−0.727667
$653$	−39.1961	−1.53386	−0.766931	−	0.641729i	$-0.778217\pi$
$653$	−39.1961	−1.53386	−0.766931	+	0.641729i	$0.778217\pi$
$654$	0	0
$655$	−5.24199	−0.204821
$656$	3.18023	0.124167
$657$	0	0
$658$	29.3622	1.14466
$659$	15.0318	0.585557	0.292778	−	0.956180i	$-0.405420\pi$
$659$	15.0318	0.585557	0.292778	+	0.956180i	$0.405420\pi$
$660$	0	0
$661$	−39.2171	−1.52537	−0.762685	−	0.646770i	$-0.776119\pi$
$661$	−39.2171	−1.52537	−0.762685	+	0.646770i	$0.776119\pi$
$662$	77.5187	3.01285
$663$	0	0
$664$	−7.82271	−0.303580
$665$	−1.70932	−0.0662845
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	−15.5327	−0.600976
$669$	0	0
$670$	3.02500	0.116866
$671$	−13.1451	−0.507460
$672$	0	0
$673$	11.9758	0.461635	0.230818	−	0.972997i	$-0.425860\pi$
$673$	11.9758	0.461635	0.230818	+	0.972997i	$0.425860\pi$
$674$	50.5941	1.94881
$675$	0	0
$676$	−37.0081	−1.42339
$677$	−22.1732	−0.852185	−0.426093	−	0.904680i	$-0.640110\pi$
$677$	−22.1732	−0.852185	−0.426093	+	0.904680i	$0.640110\pi$
$678$	0	0
$679$	46.3477	1.77866
$680$	−10.8306	−0.415336
$681$	0	0
$682$	4.65127	0.178106
$683$	3.66223	0.140131	0.0700657	−	0.997542i	$-0.477679\pi$
$683$	3.66223	0.140131	0.0700657	+	0.997542i	$0.477679\pi$
$684$	0	0
$685$	−8.52522	−0.325732
$686$	34.6030	1.32115
$687$	0	0
$688$	5.30715	0.202333
$689$	9.68228	0.368865
$690$	0	0
$691$	14.7825	0.562354	0.281177	−	0.959656i	$-0.409275\pi$
$691$	14.7825	0.562354	0.281177	+	0.959656i	$0.409275\pi$
$692$	−31.9403	−1.21419
$693$	0	0
$694$	−35.3840	−1.34316
$695$	−24.8999	−0.944505
$696$	0	0
$697$	13.2404	0.501515
$698$	−41.9135	−1.58645
$699$	0	0
$700$	29.5033	1.11512
$701$	−19.8395	−0.749328	−0.374664	−	0.927161i	$-0.622242\pi$
$701$	−19.8395	−0.749328	−0.374664	+	0.927161i	$0.622242\pi$
$702$	0	0
$703$	−4.79653	−0.180905
$704$	19.7642	0.744891
$705$	0	0
$706$	49.6335	1.86798
$707$	−10.7805	−0.405443
$708$	0	0
$709$	17.7266	0.665735	0.332868	−	0.942974i	$-0.391984\pi$
$709$	17.7266	0.665735	0.332868	+	0.942974i	$0.391984\pi$
$710$	33.1743	1.24501
$711$	0	0
$712$	17.4073	0.652367
$713$	−1.36633	−0.0511695
$714$	0	0
$715$	−1.79776	−0.0672325
$716$	−30.9197	−1.15552
$717$	0	0
$718$	71.2367	2.65853
$719$	−18.0624	−0.673612	−0.336806	−	0.941574i	$-0.609347\pi$
$719$	−18.0624	−0.673612	−0.336806	+	0.941574i	$0.609347\pi$
$720$	0	0
$721$	−30.5048	−1.13606
$722$	−42.2103	−1.57091
$723$	0	0
$724$	65.7329	2.44294
$725$	3.27420	0.121601
$726$	0	0
$727$	−20.3044	−0.753048	−0.376524	−	0.926407i	$-0.622881\pi$
$727$	−20.3044	−0.753048	−0.376524	+	0.926407i	$0.622881\pi$
$728$	−6.22971	−0.230888
$729$	0	0
$730$	−8.49908	−0.314565
$731$	22.0954	0.817229
$732$	0	0
$733$	−6.63182	−0.244952	−0.122476	−	0.992471i	$-0.539083\pi$
$733$	−6.63182	−0.244952	−0.122476	+	0.992471i	$0.539083\pi$
$734$	17.6928	0.653054
$735$	0	0
$736$	−6.56410	−0.241956
$737$	−1.55625	−0.0573253
$738$	0	0
$739$	26.4937	0.974588	0.487294	−	0.873238i	$-0.337984\pi$
$739$	26.4937	0.974588	0.487294	+	0.873238i	$0.337984\pi$
$740$	−43.6377	−1.60415
$741$	0	0
$742$	−71.4641	−2.62353
$743$	41.8653	1.53589	0.767945	−	0.640516i	$-0.221279\pi$
$743$	41.8653	1.53589	0.767945	+	0.640516i	$0.221279\pi$
$744$	0	0
$745$	−30.0332	−1.10033
$746$	−43.3935	−1.58875
$747$	0	0
$748$	16.3190	0.596682
$749$	21.6946	0.792705
$750$	0	0
$751$	−13.9711	−0.509811	−0.254905	−	0.966966i	$-0.582044\pi$
$751$	−13.9711	−0.509811	−0.254905	+	0.966966i	$0.582044\pi$
$752$	3.75197	0.136820
$753$	0	0
$754$	−2.02482	−0.0737397
$755$	22.0744	0.803368
$756$	0	0
$757$	1.79857	0.0653703	0.0326852	−	0.999466i	$-0.489594\pi$
$757$	1.79857	0.0653703	0.0326852	+	0.999466i	$0.489594\pi$
$758$	−30.8643	−1.12104
$759$	0	0
$760$	1.34068	0.0486315
$761$	22.2173	0.805378	0.402689	−	0.915337i	$-0.368076\pi$
$761$	22.2173	0.805378	0.402689	+	0.915337i	$0.368076\pi$
$762$	0	0
$763$	10.0216	0.362808
$764$	−52.2714	−1.89111
$765$	0	0
$766$	−21.9578	−0.793369
$767$	−7.04204	−0.254273
$768$	0	0
$769$	−41.3008	−1.48934	−0.744672	−	0.667431i	$-0.767394\pi$
$769$	−41.3008	−1.48934	−0.744672	+	0.667431i	$0.767394\pi$
$770$	13.2691	0.478186
$771$	0	0
$772$	−24.2249	−0.871874
$773$	19.5525	0.703254	0.351627	−	0.936140i	$-0.385629\pi$
$773$	19.5525	0.703254	0.351627	+	0.936140i	$0.385629\pi$
$774$	0	0
$775$	4.47364	0.160698
$776$	−36.3521	−1.30497
$777$	0	0
$778$	−45.6148	−1.63537
$779$	−1.63897	−0.0587221
$780$	0	0
$781$	−17.0670	−0.610704
$782$	−7.95077	−0.284319
$783$	0	0
$784$	−1.53474	−0.0548120
$785$	6.12821	0.218725
$786$	0	0
$787$	13.6255	0.485695	0.242848	−	0.970064i	$-0.421919\pi$
$787$	13.6255	0.485695	0.242848	+	0.970064i	$0.421919\pi$
$788$	−15.6087	−0.556036
$789$	0	0
$790$	47.6260	1.69446
$791$	−5.99656	−0.213213
$792$	0	0
$793$	7.81870	0.277650
$794$	−33.8523	−1.20137
$795$	0	0
$796$	66.6608	2.36273
$797$	−14.3368	−0.507836	−0.253918	−	0.967226i	$-0.581719\pi$
$797$	−14.3368	−0.507836	−0.253918	+	0.967226i	$0.581719\pi$
$798$	0	0
$799$	15.6207	0.552621
$800$	21.4922	0.759863
$801$	0	0
$802$	−71.2943	−2.51749
$803$	4.37246	0.154301
$804$	0	0
$805$	−3.89786	−0.137382
$806$	−2.76658	−0.0974486
$807$	0	0
$808$	8.45555	0.297465
$809$	45.8588	1.61231	0.806155	−	0.591705i	$-0.201545\pi$
$809$	45.8588	1.61231	0.806155	+	0.591705i	$0.201545\pi$
$810$	0	0
$811$	13.4624	0.472728	0.236364	−	0.971665i	$-0.424044\pi$
$811$	13.4624	0.472728	0.236364	+	0.971665i	$0.424044\pi$
$812$	9.01086	0.316219
$813$	0	0
$814$	37.2346	1.30507
$815$	−8.03742	−0.281539
$816$	0	0
$817$	−2.73510	−0.0956889
$818$	−5.99972	−0.209775
$819$	0	0
$820$	−14.9109	−0.520712
$821$	−48.9229	−1.70742	−0.853711	−	0.520748i	$-0.825653\pi$
$821$	−48.9229	−1.70742	−0.853711	+	0.520748i	$0.825653\pi$
$822$	0	0
$823$	6.18455	0.215580	0.107790	−	0.994174i	$-0.465623\pi$
$823$	6.18455	0.215580	0.107790	+	0.994174i	$0.465623\pi$
$824$	23.9260	0.833500
$825$	0	0
$826$	51.9767	1.80850
$827$	21.6665	0.753417	0.376709	−	0.926332i	$-0.377056\pi$
$827$	21.6665	0.753417	0.376709	+	0.926332i	$0.377056\pi$
$828$	0	0
$829$	−51.9818	−1.80540	−0.902702	−	0.430267i	$-0.858420\pi$
$829$	−51.9818	−1.80540	−0.902702	+	0.430267i	$0.858420\pi$
$830$	−9.91066	−0.344004
$831$	0	0
$832$	−11.7558	−0.407557
$833$	−6.38963	−0.221387
$834$	0	0
$835$	−6.71902	−0.232521
$836$	−2.02006	−0.0698652
$837$	0	0
$838$	−63.2394	−2.18457
$839$	29.1361	1.00589	0.502946	−	0.864318i	$-0.332250\pi$
$839$	29.1361	1.00589	0.502946	+	0.864318i	$0.332250\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	61.9095	2.13354
$843$	0	0
$844$	21.2597	0.731790
$845$	−16.0088	−0.550718
$846$	0	0
$847$	25.8115	0.886895
$848$	−9.13183	−0.313588
$849$	0	0
$850$	26.0324	0.892904
$851$	−10.9378	−0.374944
$852$	0	0
$853$	−6.26515	−0.214515	−0.107257	−	0.994231i	$-0.534207\pi$
$853$	−6.26515	−0.214515	−0.107257	+	0.994231i	$0.534207\pi$
$854$	−57.7092	−1.97477
$855$	0	0
$856$	−17.0159	−0.581591
$857$	−32.0046	−1.09326	−0.546628	−	0.837376i	$-0.684089\pi$
$857$	−32.0046	−1.09326	−0.546628	+	0.837376i	$0.684089\pi$
$858$	0	0
$859$	−2.27931	−0.0777690	−0.0388845	−	0.999244i	$-0.512380\pi$
$859$	−2.27931	−0.0777690	−0.0388845	+	0.999244i	$0.512380\pi$
$860$	−24.8832	−0.848511
$861$	0	0
$862$	−11.7615	−0.400597
$863$	−23.3854	−0.796048	−0.398024	−	0.917375i	$-0.630304\pi$
$863$	−23.3854	−0.796048	−0.398024	+	0.917375i	$0.630304\pi$
$864$	0	0
$865$	−13.8165	−0.469776
$866$	−49.1176	−1.66908
$867$	0	0
$868$	12.3118	0.417890
$869$	−24.5018	−0.831167
$870$	0	0
$871$	0.925659	0.0313648
$872$	−7.86033	−0.266185
$873$	0	0
$874$	0.984190	0.0332907
$875$	32.2517	1.09031
$876$	0	0
$877$	−37.0607	−1.25145	−0.625725	−	0.780044i	$-0.715197\pi$
$877$	−37.0607	−1.25145	−0.625725	+	0.780044i	$0.715197\pi$
$878$	−31.2566	−1.05486
$879$	0	0
$880$	1.69556	0.0571572
$881$	−20.0136	−0.674274	−0.337137	−	0.941456i	$-0.609459\pi$
$881$	−20.0136	−0.674274	−0.337137	+	0.941456i	$0.609459\pi$
$882$	0	0
$883$	32.4866	1.09326	0.546631	−	0.837374i	$-0.315910\pi$
$883$	32.4866	1.09326	0.546631	+	0.837374i	$0.315910\pi$
$884$	−9.70655	−0.326467
$885$	0	0
$886$	61.7695	2.07519
$887$	−19.3042	−0.648170	−0.324085	−	0.946028i	$-0.605056\pi$
$887$	−19.3042	−0.648170	−0.324085	+	0.946028i	$0.605056\pi$
$888$	0	0
$889$	−7.17343	−0.240589
$890$	22.0535	0.739235
$891$	0	0
$892$	58.6751	1.96459
$893$	−1.93362	−0.0647061
$894$	0	0
$895$	−13.3751	−0.447079
$896$	47.8156	1.59741
$897$	0	0
$898$	−78.9715	−2.63531
$899$	1.36633	0.0455697
$900$	0	0
$901$	−38.0189	−1.26659
$902$	12.7230	0.423630
$903$	0	0
$904$	4.70332	0.156430
$905$	28.4343	0.945189
$906$	0	0
$907$	−10.6991	−0.355259	−0.177629	−	0.984097i	$-0.556843\pi$
$907$	−10.6991	−0.355259	−0.177629	+	0.984097i	$0.556843\pi$
$908$	13.2115	0.438438
$909$	0	0
$910$	−7.89248	−0.261633
$911$	−30.1038	−0.997384	−0.498692	−	0.866779i	$-0.666186\pi$
$911$	−30.1038	−0.997384	−0.498692	+	0.866779i	$0.666186\pi$
$912$	0	0
$913$	5.09867	0.168741
$914$	−41.1847	−1.36227
$915$	0	0
$916$	43.3342	1.43180
$917$	11.8395	0.390973
$918$	0	0
$919$	7.97055	0.262924	0.131462	−	0.991321i	$-0.458033\pi$
$919$	7.97055	0.262924	0.131462	+	0.991321i	$0.458033\pi$
$920$	3.05723	0.100794
$921$	0	0
$922$	55.5173	1.82837
$923$	10.1514	0.334138
$924$	0	0
$925$	35.8126	1.17751
$926$	28.3600	0.931969
$927$	0	0
$928$	6.56410	0.215477
$929$	−45.5813	−1.49548	−0.747738	−	0.663994i	$-0.768860\pi$
$929$	−45.5813	−1.49548	−0.747738	+	0.663994i	$0.768860\pi$
$930$	0	0
$931$	0.790944	0.0259221
$932$	−3.34378	−0.109529
$933$	0	0
$934$	46.1042	1.50858
$935$	7.05918	0.230860
$936$	0	0
$937$	42.9955	1.40460	0.702301	−	0.711880i	$-0.252156\pi$
$937$	42.9955	1.40460	0.702301	+	0.711880i	$0.252156\pi$
$938$	−6.83221	−0.223080
$939$	0	0
$940$	−17.5916	−0.573775
$941$	22.3893	0.729870	0.364935	−	0.931033i	$-0.381091\pi$
$941$	22.3893	0.729870	0.364935	+	0.931033i	$0.381091\pi$
$942$	0	0
$943$	−3.73744	−0.121708
$944$	6.64170	0.216169
$945$	0	0
$946$	21.2321	0.690314
$947$	14.3161	0.465211	0.232605	−	0.972571i	$-0.425275\pi$
$947$	14.3161	0.465211	0.232605	+	0.972571i	$0.425275\pi$
$948$	0	0
$949$	−2.60074	−0.0844237
$950$	−3.22243	−0.104550
$951$	0	0
$952$	24.4619	0.792815
$953$	−46.4935	−1.50607	−0.753036	−	0.657979i	$-0.771411\pi$
$953$	−46.4935	−1.50607	−0.753036	+	0.657979i	$0.771411\pi$
$954$	0	0
$955$	−22.6112	−0.731682
$956$	−10.8601	−0.351242
$957$	0	0
$958$	32.1653	1.03921
$959$	19.2549	0.621773
$960$	0	0
$961$	−29.1331	−0.939779
$962$	−22.1472	−0.714053
$963$	0	0
$964$	72.3340	2.32972
$965$	−10.4791	−0.337333
$966$	0	0
$967$	−6.58785	−0.211851	−0.105926	−	0.994374i	$-0.533781\pi$
$967$	−6.58785	−0.211851	−0.105926	+	0.994374i	$0.533781\pi$
$968$	−20.2449	−0.650696
$969$	0	0
$970$	−46.0548	−1.47873
$971$	−7.43166	−0.238493	−0.119247	−	0.992865i	$-0.538048\pi$
$971$	−7.43166	−0.238493	−0.119247	+	0.992865i	$0.538048\pi$
$972$	0	0
$973$	56.2383	1.80292
$974$	−14.8598	−0.476140
$975$	0	0
$976$	−7.37420	−0.236042
$977$	−36.1933	−1.15793	−0.578963	−	0.815354i	$-0.696542\pi$
$977$	−36.1933	−1.15793	−0.578963	+	0.815354i	$0.696542\pi$
$978$	0	0
$979$	−11.3457	−0.362610
$980$	7.19581	0.229862
$981$	0	0
$982$	36.4247	1.16236
$983$	−25.3261	−0.807779	−0.403889	−	0.914808i	$-0.632342\pi$
$983$	−25.3261	−0.807779	−0.403889	+	0.914808i	$0.632342\pi$
$984$	0	0
$985$	−6.75191	−0.215134
$986$	7.95077	0.253204
$987$	0	0
$988$	1.20153	0.0382258
$989$	−6.23701	−0.198325
$990$	0	0
$991$	−23.1884	−0.736603	−0.368302	−	0.929706i	$-0.620061\pi$
$991$	−23.1884	−0.736603	−0.368302	+	0.929706i	$0.620061\pi$
$992$	8.96874	0.284758
$993$	0	0
$994$	−74.9268	−2.37654
$995$	28.8357	0.914154
$996$	0	0
$997$	7.28351	0.230671	0.115336	−	0.993327i	$-0.463206\pi$
$997$	7.28351	0.230671	0.115336	+	0.993327i	$0.463206\pi$
$998$	−24.9188	−0.788791
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.k.1.9		10
3.2	odd	2		2001.2.a.k.1.2	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.2.a.k.1.2	✓	10	3.2	odd	2
6003.2.a.k.1.9		10	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)

\(f(q)\)	\(=\)	\(q+2.24431 q^{2} +3.03693 q^{4} +1.31370 q^{5} -2.96709 q^{7} +2.32720 q^{8} +O(q^{10})\)	\(q+2.24431 q^{2} +3.03693 q^{4} +1.31370 q^{5} -2.96709 q^{7} +2.32720 q^{8} +2.94835 q^{10} -1.51682 q^{11} +0.902202 q^{13} -6.65908 q^{14} -0.850911 q^{16} -3.54263 q^{17} +0.438527 q^{19} +3.98961 q^{20} -3.40420 q^{22} +1.00000 q^{23} -3.27420 q^{25} +2.02482 q^{26} -9.01086 q^{28} -1.00000 q^{29} -1.36633 q^{31} -6.56410 q^{32} -7.95077 q^{34} -3.89786 q^{35} -10.9378 q^{37} +0.984190 q^{38} +3.05723 q^{40} -3.73744 q^{41} -6.23701 q^{43} -4.60646 q^{44} +2.24431 q^{46} -4.40935 q^{47} +1.80364 q^{49} -7.34832 q^{50} +2.73993 q^{52} +10.7318 q^{53} -1.99264 q^{55} -6.90501 q^{56} -2.24431 q^{58} -7.80539 q^{59} +8.66624 q^{61} -3.06647 q^{62} -13.0301 q^{64} +1.18522 q^{65} +1.02600 q^{67} -10.7587 q^{68} -8.74801 q^{70} +11.2518 q^{71} -2.88266 q^{73} -24.5479 q^{74} +1.33178 q^{76} +4.50053 q^{77} +16.1535 q^{79} -1.11784 q^{80} -8.38797 q^{82} -3.36143 q^{83} -4.65395 q^{85} -13.9978 q^{86} -3.52993 q^{88} +7.47996 q^{89} -2.67692 q^{91} +3.03693 q^{92} -9.89596 q^{94} +0.576092 q^{95} -15.6206 q^{97} +4.04792 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 3 q^{2} + 17 q^{4} - 6 q^{5} + 3 q^{7} + 6 q^{8}+O(q^{10}) \) 10 * q - 3 * q^2 + 17 * q^4 - 6 * q^5 + 3 * q^7 + 6 * q^8	\( 10 q - 3 q^{2} + 17 q^{4} - 6 q^{5} + 3 q^{7} + 6 q^{8} - 4 q^{10} - 9 q^{11} - 16 q^{13} - 16 q^{14} + 27 q^{16} + q^{19} - 21 q^{20} + 17 q^{22} + 10 q^{23} - 4 q^{25} - 28 q^{26} - 14 q^{28} - 10 q^{29} + 17 q^{31} - 21 q^{32} - 3 q^{34} - 29 q^{35} + q^{37} - 32 q^{38} + 13 q^{40} - 5 q^{43} - 33 q^{44} - 3 q^{46} - 15 q^{47} + 31 q^{49} + 22 q^{50} - 21 q^{52} - 35 q^{53} - 20 q^{55} - 18 q^{56} + 3 q^{58} - 49 q^{59} + 8 q^{61} - 15 q^{62} + 12 q^{64} + 3 q^{65} + 35 q^{67} + 18 q^{68} - 16 q^{70} - 30 q^{71} - 15 q^{73} - 23 q^{74} + 10 q^{76} - 23 q^{77} + 24 q^{79} - 23 q^{80} - 5 q^{82} - q^{83} + 10 q^{86} + 18 q^{88} - 15 q^{89} + 26 q^{91} + 17 q^{92} + 3 q^{94} - 7 q^{95} - 35 q^{97} - 3 q^{98}+O(q^{100}) \) 10 * q - 3 * q^2 + 17 * q^4 - 6 * q^5 + 3 * q^7 + 6 * q^8 - 4 * q^10 - 9 * q^11 - 16 * q^13 - 16 * q^14 + 27 * q^16 + q^19 - 21 * q^20 + 17 * q^22 + 10 * q^23 - 4 * q^25 - 28 * q^26 - 14 * q^28 - 10 * q^29 + 17 * q^31 - 21 * q^32 - 3 * q^34 - 29 * q^35 + q^37 - 32 * q^38 + 13 * q^40 - 5 * q^43 - 33 * q^44 - 3 * q^46 - 15 * q^47 + 31 * q^49 + 22 * q^50 - 21 * q^52 - 35 * q^53 - 20 * q^55 - 18 * q^56 + 3 * q^58 - 49 * q^59 + 8 * q^61 - 15 * q^62 + 12 * q^64 + 3 * q^65 + 35 * q^67 + 18 * q^68 - 16 * q^70 - 30 * q^71 - 15 * q^73 - 23 * q^74 + 10 * q^76 - 23 * q^77 + 24 * q^79 - 23 * q^80 - 5 * q^82 - q^83 + 10 * q^86 + 18 * q^88 - 15 * q^89 + 26 * q^91 + 17 * q^92 + 3 * q^94 - 7 * q^95 - 35 * q^97 - 3 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more