LMFDB - Embedded newform 6003.2.a.o.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:	$1$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 4 x^{12} - 11 x^{11} + 58 x^{10} + 24 x^{9} - 298 x^{8} + 97 x^{7} + 641 x^{6} - 402 x^{5} + \cdots - 40 $ x^13 - 4x^12 - 11x^11 + 58x^10 + 24x^9 - 298x^8 + 97x^7 + 641x^6 - 402x^5 - 547x^4 + 352x^3 + 219x^2 - 88x - 40
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 667)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.30511$ 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)

$f(q)$	$=$	$q-1.30511 q^{2} -0.296685 q^{4} -0.572849 q^{5} +1.21746 q^{7} +2.99743 q^{8} +O(q^{10})$	\(q-1.30511 q^{2} -0.296685 q^{4} -0.572849 q^{5} +1.21746 q^{7} +2.99743 q^{8} +0.747631 q^{10} +3.82178 q^{11} +1.55748 q^{13} -1.58892 q^{14} -3.31861 q^{16} -3.24470 q^{17} +7.65905 q^{19} +0.169956 q^{20} -4.98785 q^{22} +1.00000 q^{23} -4.67184 q^{25} -2.03268 q^{26} -0.361201 q^{28} -1.00000 q^{29} -9.54445 q^{31} -1.66371 q^{32} +4.23469 q^{34} -0.697418 q^{35} -5.96920 q^{37} -9.99591 q^{38} -1.71707 q^{40} +2.82463 q^{41} -11.8023 q^{43} -1.13387 q^{44} -1.30511 q^{46} -0.392305 q^{47} -5.51780 q^{49} +6.09728 q^{50} -0.462081 q^{52} -11.2317 q^{53} -2.18930 q^{55} +3.64924 q^{56} +1.30511 q^{58} +13.0256 q^{59} +9.25323 q^{61} +12.4566 q^{62} +8.80854 q^{64} -0.892200 q^{65} -10.1462 q^{67} +0.962653 q^{68} +0.910208 q^{70} -10.3791 q^{71} -2.55044 q^{73} +7.79047 q^{74} -2.27233 q^{76} +4.65285 q^{77} +0.852395 q^{79} +1.90106 q^{80} -3.68645 q^{82} +0.606412 q^{83} +1.85872 q^{85} +15.4033 q^{86} +11.4555 q^{88} -3.82257 q^{89} +1.89616 q^{91} -0.296685 q^{92} +0.512002 q^{94} -4.38748 q^{95} +5.13466 q^{97} +7.20134 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 4 q^{2} + 12 q^{4} - 16 q^{5} + q^{7} - 6 q^{8}+O(q^{10}) $ 13 * q - 4 * q^2 + 12 * q^4 - 16 * q^5 + q^7 - 6 * q^8	$ 13 q - 4 q^{2} + 12 q^{4} - 16 q^{5} + q^{7} - 6 q^{8} + 10 q^{10} - 10 q^{11} + 7 q^{13} + 12 q^{14} + 2 q^{16} - 26 q^{17} - 25 q^{20} - 15 q^{22} + 13 q^{23} + 19 q^{25} + 15 q^{26} + 5 q^{28} - 13 q^{29} - 6 q^{31} - 16 q^{32} + 11 q^{34} - q^{35} + 15 q^{37} - 8 q^{38} + 14 q^{40} - 9 q^{41} + q^{43} - 29 q^{44} - 4 q^{46} - 15 q^{47} + 4 q^{49} - 31 q^{50} - 8 q^{52} - 43 q^{53} - 3 q^{55} + 5 q^{56} + 4 q^{58} + 9 q^{59} + 20 q^{61} - 11 q^{62} - 16 q^{64} + 25 q^{65} + q^{67} - 21 q^{68} - 2 q^{70} - 17 q^{71} + 26 q^{73} - 11 q^{74} + 8 q^{76} - 17 q^{77} + 5 q^{79} - 10 q^{80} - 25 q^{82} - 4 q^{83} + 20 q^{85} + 13 q^{86} - 32 q^{88} - 48 q^{89} - 9 q^{91} + 12 q^{92} - 65 q^{94} - 8 q^{95} + 30 q^{97} - 2 q^{98}+O(q^{100}) $ 13 * q - 4 * q^2 + 12 * q^4 - 16 * q^5 + q^7 - 6 * q^8 + 10 * q^10 - 10 * q^11 + 7 * q^13 + 12 * q^14 + 2 * q^16 - 26 * q^17 - 25 * q^20 - 15 * q^22 + 13 * q^23 + 19 * q^25 + 15 * q^26 + 5 * q^28 - 13 * q^29 - 6 * q^31 - 16 * q^32 + 11 * q^34 - q^35 + 15 * q^37 - 8 * q^38 + 14 * q^40 - 9 * q^41 + q^43 - 29 * q^44 - 4 * q^46 - 15 * q^47 + 4 * q^49 - 31 * q^50 - 8 * q^52 - 43 * q^53 - 3 * q^55 + 5 * q^56 + 4 * q^58 + 9 * q^59 + 20 * q^61 - 11 * q^62 - 16 * q^64 + 25 * q^65 + q^67 - 21 * q^68 - 2 * q^70 - 17 * q^71 + 26 * q^73 - 11 * q^74 + 8 * q^76 - 17 * q^77 + 5 * q^79 - 10 * q^80 - 25 * q^82 - 4 * q^83 + 20 * q^85 + 13 * q^86 - 32 * q^88 - 48 * q^89 - 9 * q^91 + 12 * q^92 - 65 * q^94 - 8 * q^95 + 30 * q^97 - 2 * 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 $

$2$	−1.30511	−0.922853	−0.461426	−	0.887178i	$-0.652662\pi$
$2$	−1.30511	−0.922853	−0.461426	+	0.887178i	$0.652662\pi$
$3$	0	0
$3$	0	0
$4$	−0.296685	−0.148343
$5$	−0.572849	−0.256186	−0.128093	−	0.991762i	$-0.540886\pi$
$5$	−0.572849	−0.256186	−0.128093	+	0.991762i	$0.540886\pi$
$6$	0	0
$7$	1.21746	0.460155	0.230078	−	0.973172i	$-0.426102\pi$
$7$	1.21746	0.460155	0.230078	+	0.973172i	$0.426102\pi$
$8$	2.99743	1.05975
$9$	0	0
$10$	0.747631	0.236422
$11$	3.82178	1.15231	0.576155	−	0.817340i	$-0.304552\pi$
$11$	3.82178	1.15231	0.576155	+	0.817340i	$0.304552\pi$
$12$	0	0
$13$	1.55748	0.431967	0.215984	−	0.976397i	$-0.430704\pi$
$13$	1.55748	0.431967	0.215984	+	0.976397i	$0.430704\pi$
$14$	−1.58892	−0.424656
$15$	0	0
$16$	−3.31861	−0.829652
$17$	−3.24470	−0.786954	−0.393477	−	0.919334i	$-0.628728\pi$
$17$	−3.24470	−0.786954	−0.393477	+	0.919334i	$0.628728\pi$
$18$	0	0
$19$	7.65905	1.75711	0.878553	−	0.477644i	$-0.158509\pi$
$19$	7.65905	1.75711	0.878553	+	0.477644i	$0.158509\pi$
$20$	0.169956	0.0380033
$21$	0	0
$22$	−4.98785	−1.06341
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−4.67184	−0.934369
$26$	−2.03268	−0.398642
$27$	0	0
$28$	−0.361201	−0.0682606
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−9.54445	−1.71423	−0.857117	−	0.515123i	$-0.827746\pi$
$31$	−9.54445	−1.71423	−0.857117	+	0.515123i	$0.827746\pi$
$32$	−1.66371	−0.294105
$33$	0	0
$34$	4.23469	0.726243
$35$	−0.697418	−0.117885
$36$	0	0
$37$	−5.96920	−0.981330	−0.490665	−	0.871348i	$-0.663246\pi$
$37$	−5.96920	−0.981330	−0.490665	+	0.871348i	$0.663246\pi$
$38$	−9.99591	−1.62155
$39$	0	0
$40$	−1.71707	−0.271493
$41$	2.82463	0.441133	0.220566	−	0.975372i	$-0.429209\pi$
$41$	2.82463	0.441133	0.220566	+	0.975372i	$0.429209\pi$
$42$	0	0
$43$	−11.8023	−1.79983	−0.899914	−	0.436067i	$-0.856371\pi$
$43$	−11.8023	−1.79983	−0.899914	+	0.436067i	$0.856371\pi$
$44$	−1.13387	−0.170937
$45$	0	0
$46$	−1.30511	−0.192428
$47$	−0.392305	−0.0572236	−0.0286118	−	0.999591i	$-0.509109\pi$
$47$	−0.392305	−0.0572236	−0.0286118	+	0.999591i	$0.509109\pi$
$48$	0	0
$49$	−5.51780	−0.788257
$50$	6.09728	0.862285
$51$	0	0
$52$	−0.462081	−0.0640792
$53$	−11.2317	−1.54279	−0.771394	−	0.636358i	$-0.780440\pi$
$53$	−11.2317	−1.54279	−0.771394	+	0.636358i	$0.780440\pi$
$54$	0	0
$55$	−2.18930	−0.295205
$56$	3.64924	0.487650
$57$	0	0
$58$	1.30511	0.171369
$59$	13.0256	1.69579	0.847896	−	0.530162i	$-0.177869\pi$
$59$	13.0256	1.69579	0.847896	+	0.530162i	$0.177869\pi$
$60$	0	0
$61$	9.25323	1.18475	0.592377	−	0.805661i	$-0.298190\pi$
$61$	9.25323	1.18475	0.592377	+	0.805661i	$0.298190\pi$
$62$	12.4566	1.58198
$63$	0	0
$64$	8.80854	1.10107
$65$	−0.892200	−0.110664
$66$	0	0
$67$	−10.1462	−1.23955	−0.619776	−	0.784779i	$-0.712777\pi$
$67$	−10.1462	−1.23955	−0.619776	+	0.784779i	$0.712777\pi$
$68$	0.962653	0.116739
$69$	0	0
$70$	0.910208	0.108791
$71$	−10.3791	−1.23178	−0.615889	−	0.787833i	$-0.711203\pi$
$71$	−10.3791	−1.23178	−0.615889	+	0.787833i	$0.711203\pi$
$72$	0	0
$73$	−2.55044	−0.298506	−0.149253	−	0.988799i	$-0.547687\pi$
$73$	−2.55044	−0.298506	−0.149253	+	0.988799i	$0.547687\pi$
$74$	7.79047	0.905624
$75$	0	0
$76$	−2.27233	−0.260654
$77$	4.65285	0.530241
$78$	0	0
$79$	0.852395	0.0959019	0.0479510	−	0.998850i	$-0.484731\pi$
$79$	0.852395	0.0959019	0.0479510	+	0.998850i	$0.484731\pi$
$80$	1.90106	0.212545
$81$	0	0
$82$	−3.68645	−0.407101
$83$	0.606412	0.0665624	0.0332812	−	0.999446i	$-0.489404\pi$
$83$	0.606412	0.0665624	0.0332812	+	0.999446i	$0.489404\pi$
$84$	0	0
$85$	1.85872	0.201606
$86$	15.4033	1.66098
$87$	0	0
$88$	11.4555	1.22116
$89$	−3.82257	−0.405192	−0.202596	−	0.979262i	$-0.564938\pi$
$89$	−3.82257	−0.405192	−0.202596	+	0.979262i	$0.564938\pi$
$90$	0	0
$91$	1.89616	0.198772
$92$	−0.296685	−0.0309316
$93$	0	0
$94$	0.512002	0.0528090
$95$	−4.38748	−0.450146
$96$	0	0
$97$	5.13466	0.521346	0.260673	−	0.965427i	$-0.416056\pi$
$97$	5.13466	0.521346	0.260673	+	0.965427i	$0.416056\pi$
$98$	7.20134	0.727445
$99$	0	0
$100$	1.38607	0.138607
$101$	11.9061	1.18470	0.592349	−	0.805682i	$-0.298201\pi$
$101$	11.9061	1.18470	0.592349	+	0.805682i	$0.298201\pi$
$102$	0	0
$103$	3.70761	0.365322	0.182661	−	0.983176i	$-0.441529\pi$
$103$	3.70761	0.365322	0.182661	+	0.983176i	$0.441529\pi$
$104$	4.66844	0.457778
$105$	0	0
$106$	14.6586	1.42377
$107$	−9.65894	−0.933765	−0.466882	−	0.884319i	$-0.654623\pi$
$107$	−9.65894	−0.933765	−0.466882	+	0.884319i	$0.654623\pi$
$108$	0	0
$109$	10.5404	1.00959	0.504794	−	0.863240i	$-0.331568\pi$
$109$	10.5404	1.00959	0.504794	+	0.863240i	$0.331568\pi$
$110$	2.85728	0.272431
$111$	0	0
$112$	−4.04026	−0.381769
$113$	−17.8383	−1.67809	−0.839045	−	0.544062i	$-0.816886\pi$
$113$	−17.8383	−1.67809	−0.839045	+	0.544062i	$0.816886\pi$
$114$	0	0
$115$	−0.572849	−0.0534184
$116$	0.296685	0.0275465
$117$	0	0
$118$	−16.9999	−1.56497
$119$	−3.95028	−0.362121
$120$	0	0
$121$	3.60600	0.327818
$122$	−12.0765	−1.09335
$123$	0	0
$124$	2.83170	0.254294
$125$	5.54050	0.495558
$126$	0	0
$127$	8.21224	0.728718	0.364359	−	0.931258i	$-0.381288\pi$
$127$	8.21224	0.728718	0.364359	+	0.931258i	$0.381288\pi$
$128$	−8.16870	−0.722018
$129$	0	0
$130$	1.16442	0.102126
$131$	−20.3674	−1.77951	−0.889754	−	0.456441i	$-0.849124\pi$
$131$	−20.3674	−1.77951	−0.889754	+	0.456441i	$0.849124\pi$
$132$	0	0
$133$	9.32456	0.808542
$134$	13.2419	1.14392
$135$	0	0
$136$	−9.72575	−0.833976
$137$	−3.66808	−0.313385	−0.156693	−	0.987647i	$-0.550083\pi$
$137$	−3.66808	−0.313385	−0.156693	+	0.987647i	$0.550083\pi$
$138$	0	0
$139$	−9.77997	−0.829526	−0.414763	−	0.909929i	$-0.636136\pi$
$139$	−9.77997	−0.829526	−0.414763	+	0.909929i	$0.636136\pi$
$140$	0.206914	0.0174874
$141$	0	0
$142$	13.5459	1.13675
$143$	5.95235	0.497760
$144$	0	0
$145$	0.572849	0.0475725
$146$	3.32860	0.275477
$147$	0	0
$148$	1.77097	0.145573
$149$	14.7501	1.20837	0.604187	−	0.796843i	$-0.293498\pi$
$149$	14.7501	1.20837	0.604187	+	0.796843i	$0.293498\pi$
$150$	0	0
$151$	2.30674	0.187720	0.0938601	−	0.995585i	$-0.470079\pi$
$151$	2.30674	0.187720	0.0938601	+	0.995585i	$0.470079\pi$
$152$	22.9575	1.86210
$153$	0	0
$154$	−6.07249	−0.489335
$155$	5.46752	0.439162
$156$	0	0
$157$	−8.57476	−0.684341	−0.342170	−	0.939638i	$-0.611162\pi$
$157$	−8.57476	−0.684341	−0.342170	+	0.939638i	$0.611162\pi$
$158$	−1.11247	−0.0885034
$159$	0	0
$160$	0.953053	0.0753454
$161$	1.21746	0.0959490
$162$	0	0
$163$	8.09710	0.634214	0.317107	−	0.948390i	$-0.397289\pi$
$163$	8.09710	0.634214	0.317107	+	0.948390i	$0.397289\pi$
$164$	−0.838025	−0.0654388
$165$	0	0
$166$	−0.791435	−0.0614273
$167$	−16.3123	−1.26228	−0.631141	−	0.775668i	$-0.717413\pi$
$167$	−16.3123	−1.26228	−0.631141	+	0.775668i	$0.717413\pi$
$168$	0	0
$169$	−10.5743	−0.813404
$170$	−2.42584	−0.186053
$171$	0	0
$172$	3.50156	0.266991
$173$	−8.46918	−0.643900	−0.321950	−	0.946757i	$-0.604338\pi$
$173$	−8.46918	−0.643900	−0.321950	+	0.946757i	$0.604338\pi$
$174$	0	0
$175$	−5.68777	−0.429955
$176$	−12.6830	−0.956016
$177$	0	0
$178$	4.98888	0.373932
$179$	−11.0527	−0.826120	−0.413060	−	0.910704i	$-0.635540\pi$
$179$	−11.0527	−0.826120	−0.413060	+	0.910704i	$0.635540\pi$
$180$	0	0
$181$	22.3254	1.65943	0.829716	−	0.558185i	$-0.188502\pi$
$181$	22.3254	1.65943	0.829716	+	0.558185i	$0.188502\pi$
$182$	−2.47470	−0.183437
$183$	0	0
$184$	2.99743	0.220973
$185$	3.41945	0.251403
$186$	0	0
$187$	−12.4005	−0.906815
$188$	0.116391	0.00848870
$189$	0	0
$190$	5.72614	0.415418
$191$	0.112731	0.00815694	0.00407847	−	0.999992i	$-0.498702\pi$
$191$	0.112731	0.00815694	0.00407847	+	0.999992i	$0.498702\pi$
$192$	0	0
$193$	−14.9894	−1.07896	−0.539480	−	0.841998i	$-0.681379\pi$
$193$	−14.9894	−1.07896	−0.539480	+	0.841998i	$0.681379\pi$
$194$	−6.70131	−0.481126
$195$	0	0
$196$	1.63705	0.116932
$197$	14.4378	1.02865	0.514326	−	0.857595i	$-0.328042\pi$
$197$	14.4378	1.02865	0.514326	+	0.857595i	$0.328042\pi$
$198$	0	0
$199$	−18.3497	−1.30078	−0.650389	−	0.759601i	$-0.725394\pi$
$199$	−18.3497	−1.30078	−0.650389	+	0.759601i	$0.725394\pi$
$200$	−14.0035	−0.990199
$201$	0	0
$202$	−15.5387	−1.09330
$203$	−1.21746	−0.0854487
$204$	0	0
$205$	−1.61808	−0.113012
$206$	−4.83884	−0.337138
$207$	0	0
$208$	−5.16866	−0.358382
$209$	29.2712	2.02473
$210$	0	0
$211$	8.48555	0.584170	0.292085	−	0.956392i	$-0.405651\pi$
$211$	8.48555	0.584170	0.292085	+	0.956392i	$0.405651\pi$
$212$	3.33227	0.228861
$213$	0	0
$214$	12.6060	0.861727
$215$	6.76091	0.461090
$216$	0	0
$217$	−11.6199	−0.788813
$218$	−13.7564	−0.931701
$219$	0	0
$220$	0.649533	0.0437915
$221$	−5.05355	−0.339938
$222$	0	0
$223$	7.93065	0.531076	0.265538	−	0.964100i	$-0.414450\pi$
$223$	7.93065	0.531076	0.265538	+	0.964100i	$0.414450\pi$
$224$	−2.02549	−0.135334
$225$	0	0
$226$	23.2810	1.54863
$227$	26.6251	1.76717	0.883584	−	0.468273i	$-0.155123\pi$
$227$	26.6251	1.76717	0.883584	+	0.468273i	$0.155123\pi$
$228$	0	0
$229$	22.1498	1.46370	0.731850	−	0.681465i	$-0.238657\pi$
$229$	22.1498	1.46370	0.731850	+	0.681465i	$0.238657\pi$
$230$	0.747631	0.0492973
$231$	0	0
$232$	−2.99743	−0.196791
$233$	−4.06717	−0.266449	−0.133225	−	0.991086i	$-0.542533\pi$
$233$	−4.06717	−0.266449	−0.133225	+	0.991086i	$0.542533\pi$
$234$	0	0
$235$	0.224732	0.0146599
$236$	−3.86451	−0.251558
$237$	0	0
$238$	5.15555	0.334185
$239$	−7.55588	−0.488749	−0.244375	−	0.969681i	$-0.578583\pi$
$239$	−7.55588	−0.488749	−0.244375	+	0.969681i	$0.578583\pi$
$240$	0	0
$241$	6.65618	0.428762	0.214381	−	0.976750i	$-0.431227\pi$
$241$	6.65618	0.428762	0.214381	+	0.976750i	$0.431227\pi$
$242$	−4.70623	−0.302528
$243$	0	0
$244$	−2.74530	−0.175750
$245$	3.16086	0.201940
$246$	0	0
$247$	11.9288	0.759012
$248$	−28.6088	−1.81666
$249$	0	0
$250$	−7.23097	−0.457327
$251$	3.90001	0.246166	0.123083	−	0.992396i	$-0.460722\pi$
$251$	3.90001	0.246166	0.123083	+	0.992396i	$0.460722\pi$
$252$	0	0
$253$	3.82178	0.240273
$254$	−10.7179	−0.672500
$255$	0	0
$256$	−6.95601	−0.434751
$257$	9.77946	0.610026	0.305013	−	0.952348i	$-0.401339\pi$
$257$	9.77946	0.610026	0.305013	+	0.952348i	$0.401339\pi$
$258$	0	0
$259$	−7.26724	−0.451564
$260$	0.264703	0.0164162
$261$	0	0
$262$	26.5817	1.64222
$263$	−14.2353	−0.877784	−0.438892	−	0.898540i	$-0.644629\pi$
$263$	−14.2353	−0.877784	−0.438892	+	0.898540i	$0.644629\pi$
$264$	0	0
$265$	6.43404	0.395240
$266$	−12.1696	−0.746165
$267$	0	0
$268$	3.01022	0.183878
$269$	19.0080	1.15894	0.579470	−	0.814993i	$-0.303259\pi$
$269$	19.0080	1.15894	0.579470	+	0.814993i	$0.303259\pi$
$270$	0	0
$271$	−26.6085	−1.61635	−0.808176	−	0.588941i	$-0.799545\pi$
$271$	−26.6085	−1.61635	−0.808176	+	0.588941i	$0.799545\pi$
$272$	10.7679	0.652898
$273$	0	0
$274$	4.78725	0.289208
$275$	−17.8548	−1.07668
$276$	0	0
$277$	−10.9717	−0.659224	−0.329612	−	0.944116i	$-0.606918\pi$
$277$	−10.9717	−0.659224	−0.329612	+	0.944116i	$0.606918\pi$
$278$	12.7639	0.765530
$279$	0	0
$280$	−2.09046	−0.124929
$281$	−28.0886	−1.67563	−0.837813	−	0.545957i	$-0.816166\pi$
$281$	−28.0886	−1.67563	−0.837813	+	0.545957i	$0.816166\pi$
$282$	0	0
$283$	−8.51705	−0.506286	−0.253143	−	0.967429i	$-0.581464\pi$
$283$	−8.51705	−0.506286	−0.253143	+	0.967429i	$0.581464\pi$
$284$	3.07934	0.182725
$285$	0	0
$286$	−7.76847	−0.459359
$287$	3.43886	0.202989
$288$	0	0
$289$	−6.47195	−0.380703
$290$	−0.747631	−0.0439024
$291$	0	0
$292$	0.756677	0.0442812
$293$	−25.2762	−1.47665	−0.738326	−	0.674444i	$-0.764383\pi$
$293$	−25.2762	−1.47665	−0.738326	+	0.674444i	$0.764383\pi$
$294$	0	0
$295$	−7.46171	−0.434438
$296$	−17.8923	−1.03997
$297$	0	0
$298$	−19.2505	−1.11515
$299$	1.55748	0.0900714
$300$	0	0
$301$	−14.3687	−0.828200
$302$	−3.01056	−0.173238
$303$	0	0
$304$	−25.4174	−1.45779
$305$	−5.30070	−0.303517
$306$	0	0
$307$	−10.8941	−0.621759	−0.310880	−	0.950449i	$-0.600624\pi$
$307$	−10.8941	−0.621759	−0.310880	+	0.950449i	$0.600624\pi$
$308$	−1.38043	−0.0786574
$309$	0	0
$310$	−7.13572	−0.405282
$311$	−15.0844	−0.855358	−0.427679	−	0.903931i	$-0.640669\pi$
$311$	−15.0844	−0.855358	−0.427679	+	0.903931i	$0.640669\pi$
$312$	0	0
$313$	3.08345	0.174287	0.0871434	−	0.996196i	$-0.472226\pi$
$313$	3.08345	0.174287	0.0871434	+	0.996196i	$0.472226\pi$
$314$	11.1910	0.631546
$315$	0	0
$316$	−0.252893	−0.0142263
$317$	−20.7123	−1.16332	−0.581659	−	0.813433i	$-0.697596\pi$
$317$	−20.7123	−1.16332	−0.581659	+	0.813433i	$0.697596\pi$
$318$	0	0
$319$	−3.82178	−0.213979
$320$	−5.04596	−0.282078
$321$	0	0
$322$	−1.58892	−0.0885468
$323$	−24.8513	−1.38276
$324$	0	0
$325$	−7.27630	−0.403617
$326$	−10.5676	−0.585286
$327$	0	0
$328$	8.46662	0.467491
$329$	−0.477615	−0.0263317
$330$	0	0
$331$	31.9172	1.75433	0.877165	−	0.480189i	$-0.159432\pi$
$331$	31.9172	1.75433	0.877165	+	0.480189i	$0.159432\pi$
$332$	−0.179913	−0.00987404
$333$	0	0
$334$	21.2893	1.16490
$335$	5.81222	0.317556
$336$	0	0
$337$	7.63778	0.416056	0.208028	−	0.978123i	$-0.433295\pi$
$337$	7.63778	0.416056	0.208028	+	0.978123i	$0.433295\pi$
$338$	13.8006	0.750652
$339$	0	0
$340$	−0.551455	−0.0299068
$341$	−36.4768	−1.97533
$342$	0	0
$343$	−15.2399	−0.822876
$344$	−35.3764	−1.90737
$345$	0	0
$346$	11.0532	0.594225
$347$	6.79169	0.364597	0.182298	−	0.983243i	$-0.441646\pi$
$347$	6.79169	0.364597	0.182298	+	0.983243i	$0.441646\pi$
$348$	0	0
$349$	13.0456	0.698316	0.349158	−	0.937064i	$-0.386468\pi$
$349$	13.0456	0.698316	0.349158	+	0.937064i	$0.386468\pi$
$350$	7.42317	0.396785
$351$	0	0
$352$	−6.35832	−0.338900
$353$	17.8334	0.949176	0.474588	−	0.880208i	$-0.342597\pi$
$353$	17.8334	0.949176	0.474588	+	0.880208i	$0.342597\pi$
$354$	0	0
$355$	5.94568	0.315564
$356$	1.13410	0.0601072
$357$	0	0
$358$	14.4250	0.762387
$359$	0.664618	0.0350772	0.0175386	−	0.999846i	$-0.494417\pi$
$359$	0.664618	0.0350772	0.0175386	+	0.999846i	$0.494417\pi$
$360$	0	0
$361$	39.6610	2.08742
$362$	−29.1371	−1.53141
$363$	0	0
$364$	−0.562564	−0.0294864
$365$	1.46101	0.0764730
$366$	0	0
$367$	−7.77513	−0.405859	−0.202929	−	0.979193i	$-0.565046\pi$
$367$	−7.77513	−0.405859	−0.202929	+	0.979193i	$0.565046\pi$
$368$	−3.31861	−0.172994
$369$	0	0
$370$	−4.46276	−0.232008
$371$	−13.6741	−0.709922
$372$	0	0
$373$	−6.25735	−0.323993	−0.161997	−	0.986791i	$-0.551793\pi$
$373$	−6.25735	−0.323993	−0.161997	+	0.986791i	$0.551793\pi$
$374$	16.1840	0.836857
$375$	0	0
$376$	−1.17591	−0.0606428
$377$	−1.55748	−0.0802143
$378$	0	0
$379$	−16.9416	−0.870231	−0.435116	−	0.900375i	$-0.643293\pi$
$379$	−16.9416	−0.870231	−0.435116	+	0.900375i	$0.643293\pi$
$380$	1.30170	0.0667758
$381$	0	0
$382$	−0.147127	−0.00752765
$383$	8.01241	0.409415	0.204707	−	0.978823i	$-0.434376\pi$
$383$	8.01241	0.409415	0.204707	+	0.978823i	$0.434376\pi$
$384$	0	0
$385$	−2.66538	−0.135840
$386$	19.5628	0.995722
$387$	0	0
$388$	−1.52338	−0.0773379
$389$	−11.0961	−0.562595	−0.281297	−	0.959621i	$-0.590765\pi$
$389$	−11.0961	−0.562595	−0.281297	+	0.959621i	$0.590765\pi$
$390$	0	0
$391$	−3.24470	−0.164091
$392$	−16.5392	−0.835357
$393$	0	0
$394$	−18.8429	−0.949294
$395$	−0.488293	−0.0245687
$396$	0	0
$397$	18.7848	0.942780	0.471390	−	0.881925i	$-0.343752\pi$
$397$	18.7848	0.942780	0.471390	+	0.881925i	$0.343752\pi$
$398$	23.9484	1.20043
$399$	0	0
$400$	15.5040	0.775201
$401$	8.32239	0.415600	0.207800	−	0.978171i	$-0.433370\pi$
$401$	8.32239	0.415600	0.207800	+	0.978171i	$0.433370\pi$
$402$	0	0
$403$	−14.8653	−0.740493
$404$	−3.53235	−0.175741
$405$	0	0
$406$	1.58892	0.0788566
$407$	−22.8130	−1.13080
$408$	0	0
$409$	−21.5434	−1.06525	−0.532627	−	0.846350i	$-0.678795\pi$
$409$	−21.5434	−1.06525	−0.532627	+	0.846350i	$0.678795\pi$
$410$	2.11178	0.104293
$411$	0	0
$412$	−1.09999	−0.0541928
$413$	15.8581	0.780328
$414$	0	0
$415$	−0.347382	−0.0170523
$416$	−2.59119	−0.127044
$417$	0	0
$418$	−38.2022	−1.86853
$419$	12.6654	0.618746	0.309373	−	0.950941i	$-0.399881\pi$
$419$	12.6654	0.618746	0.309373	+	0.950941i	$0.399881\pi$
$420$	0	0
$421$	0.796006	0.0387950	0.0193975	−	0.999812i	$-0.493825\pi$
$421$	0.796006	0.0387950	0.0193975	+	0.999812i	$0.493825\pi$
$422$	−11.0746	−0.539103
$423$	0	0
$424$	−33.6661	−1.63497
$425$	15.1587	0.735306
$426$	0	0
$427$	11.2654	0.545171
$428$	2.86566	0.138517
$429$	0	0
$430$	−8.82374	−0.425518
$431$	−32.6597	−1.57316	−0.786580	−	0.617488i	$-0.788151\pi$
$431$	−32.6597	−1.57316	−0.786580	+	0.617488i	$0.788151\pi$
$432$	0	0
$433$	10.5971	0.509266	0.254633	−	0.967038i	$-0.418045\pi$
$433$	10.5971	0.509266	0.254633	+	0.967038i	$0.418045\pi$
$434$	15.1653	0.727959
$435$	0	0
$436$	−3.12718	−0.149765
$437$	7.65905	0.366382
$438$	0	0
$439$	29.6100	1.41321	0.706604	−	0.707609i	$-0.250226\pi$
$439$	29.6100	1.41321	0.706604	+	0.707609i	$0.250226\pi$
$440$	−6.56228	−0.312844
$441$	0	0
$442$	6.59544	0.313713
$443$	34.0175	1.61622	0.808111	−	0.589031i	$-0.200490\pi$
$443$	34.0175	1.61622	0.808111	+	0.589031i	$0.200490\pi$
$444$	0	0
$445$	2.18975	0.103804
$446$	−10.3504	−0.490105
$447$	0	0
$448$	10.7240	0.506662
$449$	−38.7800	−1.83014	−0.915071	−	0.403292i	$-0.867866\pi$
$449$	−38.7800	−1.83014	−0.915071	+	0.403292i	$0.867866\pi$
$450$	0	0
$451$	10.7951	0.508322
$452$	5.29237	0.248932
$453$	0	0
$454$	−34.7487	−1.63084
$455$	−1.08621	−0.0509225
$456$	0	0
$457$	−20.0321	−0.937064	−0.468532	−	0.883447i	$-0.655217\pi$
$457$	−20.0321	−0.937064	−0.468532	+	0.883447i	$0.655217\pi$
$458$	−28.9080	−1.35078
$459$	0	0
$460$	0.169956	0.00792423
$461$	12.2868	0.572252	0.286126	−	0.958192i	$-0.407632\pi$
$461$	12.2868	0.572252	0.286126	+	0.958192i	$0.407632\pi$
$462$	0	0
$463$	−37.8812	−1.76049	−0.880245	−	0.474520i	$-0.842622\pi$
$463$	−37.8812	−1.76049	−0.880245	+	0.474520i	$0.842622\pi$
$464$	3.31861	0.154062
$465$	0	0
$466$	5.30811	0.245893
$467$	−0.782271	−0.0361992	−0.0180996	−	0.999836i	$-0.505762\pi$
$467$	−0.782271	−0.0361992	−0.0180996	+	0.999836i	$0.505762\pi$
$468$	0	0
$469$	−12.3525	−0.570386
$470$	−0.293300	−0.0135289
$471$	0	0
$472$	39.0434	1.79712
$473$	−45.1057	−2.07396
$474$	0	0
$475$	−35.7819	−1.64179
$476$	1.17199	0.0537180
$477$	0	0
$478$	9.86126	0.451044
$479$	−7.13285	−0.325908	−0.162954	−	0.986634i	$-0.552102\pi$
$479$	−7.13285	−0.325908	−0.162954	+	0.986634i	$0.552102\pi$
$480$	0	0
$481$	−9.29691	−0.423903
$482$	−8.68705	−0.395684
$483$	0	0
$484$	−1.06985	−0.0486294
$485$	−2.94139	−0.133561
$486$	0	0
$487$	−18.0659	−0.818644	−0.409322	−	0.912390i	$-0.634235\pi$
$487$	−18.0659	−0.818644	−0.409322	+	0.912390i	$0.634235\pi$
$488$	27.7359	1.25555
$489$	0	0
$490$	−4.12528	−0.186361
$491$	1.19496	0.0539278	0.0269639	−	0.999636i	$-0.491416\pi$
$491$	1.19496	0.0539278	0.0269639	+	0.999636i	$0.491416\pi$
$492$	0	0
$493$	3.24470	0.146134
$494$	−15.5684	−0.700457
$495$	0	0
$496$	31.6743	1.42222
$497$	−12.6362	−0.566809
$498$	0	0
$499$	22.2895	0.997815	0.498907	−	0.866655i	$-0.333735\pi$
$499$	22.2895	0.997815	0.498907	+	0.866655i	$0.333735\pi$
$500$	−1.64379	−0.0735123
$501$	0	0
$502$	−5.08994	−0.227175
$503$	13.3652	0.595924	0.297962	−	0.954578i	$-0.403693\pi$
$503$	13.3652	0.595924	0.297962	+	0.954578i	$0.403693\pi$
$504$	0	0
$505$	−6.82037	−0.303503
$506$	−4.98785	−0.221737
$507$	0	0
$508$	−2.43645	−0.108100
$509$	−1.42116	−0.0629917	−0.0314958	−	0.999504i	$-0.510027\pi$
$509$	−1.42116	−0.0629917	−0.0314958	+	0.999504i	$0.510027\pi$
$510$	0	0
$511$	−3.10505	−0.137359
$512$	25.4158	1.12323
$513$	0	0
$514$	−12.7633	−0.562964
$515$	−2.12390	−0.0935901
$516$	0	0
$517$	−1.49930	−0.0659393
$518$	9.48456	0.416727
$519$	0	0
$520$	−2.67431	−0.117276
$521$	0.459592	0.0201351	0.0100675	−	0.999949i	$-0.496795\pi$
$521$	0.459592	0.0201351	0.0100675	+	0.999949i	$0.496795\pi$
$522$	0	0
$523$	−28.7937	−1.25906	−0.629530	−	0.776976i	$-0.716753\pi$
$523$	−28.7937	−1.25906	−0.629530	+	0.776976i	$0.716753\pi$
$524$	6.04271	0.263977
$525$	0	0
$526$	18.5786	0.810066
$527$	30.9688	1.34902
$528$	0	0
$529$	1.00000	0.0434783
$530$	−8.39714	−0.364748
$531$	0	0
$532$	−2.76646	−0.119941
$533$	4.39930	0.190555
$534$	0	0
$535$	5.53311	0.239217
$536$	−30.4124	−1.31362
$537$	0	0
$538$	−24.8076	−1.06953
$539$	−21.0878	−0.908317
$540$	0	0
$541$	−10.0629	−0.432639	−0.216320	−	0.976323i	$-0.569405\pi$
$541$	−10.0629	−0.432639	−0.216320	+	0.976323i	$0.569405\pi$
$542$	34.7271	1.49165
$543$	0	0
$544$	5.39822	0.231447
$545$	−6.03806	−0.258642
$546$	0	0
$547$	37.1617	1.58892	0.794460	−	0.607317i	$-0.207754\pi$
$547$	37.1617	1.58892	0.794460	+	0.607317i	$0.207754\pi$
$548$	1.08827	0.0464884
$549$	0	0
$550$	23.3024	0.993620
$551$	−7.65905	−0.326287
$552$	0	0
$553$	1.03775	0.0441298
$554$	14.3193	0.608367
$555$	0	0
$556$	2.90157	0.123054
$557$	−23.4235	−0.992486	−0.496243	−	0.868184i	$-0.665287\pi$
$557$	−23.4235	−0.992486	−0.496243	+	0.868184i	$0.665287\pi$
$558$	0	0
$559$	−18.3818	−0.777467
$560$	2.31446	0.0978037
$561$	0	0
$562$	36.6588	1.54636
$563$	−21.1477	−0.891267	−0.445634	−	0.895215i	$-0.647022\pi$
$563$	−21.1477	−0.891267	−0.445634	+	0.895215i	$0.647022\pi$
$564$	0	0
$565$	10.2187	0.429903
$566$	11.1157	0.467228
$567$	0	0
$568$	−31.1107	−1.30538
$569$	7.90059	0.331210	0.165605	−	0.986192i	$-0.447042\pi$
$569$	7.90059	0.331210	0.165605	+	0.986192i	$0.447042\pi$
$570$	0	0
$571$	20.5706	0.860851	0.430425	−	0.902626i	$-0.358364\pi$
$571$	20.5706	0.860851	0.430425	+	0.902626i	$0.358364\pi$
$572$	−1.76597	−0.0738391
$573$	0	0
$574$	−4.48809	−0.187329
$575$	−4.67184	−0.194829
$576$	0	0
$577$	34.1557	1.42192	0.710959	−	0.703233i	$-0.248261\pi$
$577$	34.1557	1.42192	0.710959	+	0.703233i	$0.248261\pi$
$578$	8.44661	0.351333
$579$	0	0
$580$	−0.169956	−0.00705703
$581$	0.738280	0.0306290
$582$	0	0
$583$	−42.9249	−1.77777
$584$	−7.64476	−0.316342
$585$	0	0
$586$	32.9883	1.36273
$587$	26.8588	1.10858	0.554290	−	0.832323i	$-0.312990\pi$
$587$	26.8588	1.10858	0.554290	+	0.832323i	$0.312990\pi$
$588$	0	0
$589$	−73.1014	−3.01209
$590$	9.73837	0.400922
$591$	0	0
$592$	19.8094	0.814163
$593$	31.3103	1.28576	0.642881	−	0.765966i	$-0.277739\pi$
$593$	31.3103	1.28576	0.642881	+	0.765966i	$0.277739\pi$
$594$	0	0
$595$	2.26291	0.0927703
$596$	−4.37613	−0.179253
$597$	0	0
$598$	−2.03268	−0.0831226
$599$	−36.5278	−1.49248	−0.746242	−	0.665675i	$-0.768144\pi$
$599$	−36.5278	−1.49248	−0.746242	+	0.665675i	$0.768144\pi$
$600$	0	0
$601$	19.7339	0.804964	0.402482	−	0.915428i	$-0.368148\pi$
$601$	19.7339	0.804964	0.402482	+	0.915428i	$0.368148\pi$
$602$	18.7528	0.764307
$603$	0	0
$604$	−0.684377	−0.0278469
$605$	−2.06569	−0.0839823
$606$	0	0
$607$	−22.6610	−0.919782	−0.459891	−	0.887975i	$-0.652112\pi$
$607$	−22.6610	−0.919782	−0.459891	+	0.887975i	$0.652112\pi$
$608$	−12.7424	−0.516773
$609$	0	0
$610$	6.91800	0.280102
$611$	−0.611008	−0.0247187
$612$	0	0
$613$	−10.0972	−0.407821	−0.203911	−	0.978989i	$-0.565365\pi$
$613$	−10.0972	−0.407821	−0.203911	+	0.978989i	$0.565365\pi$
$614$	14.2180	0.573792
$615$	0	0
$616$	13.9466	0.561924
$617$	10.9420	0.440509	0.220254	−	0.975442i	$-0.429311\pi$
$617$	10.9420	0.440509	0.220254	+	0.975442i	$0.429311\pi$
$618$	0	0
$619$	−36.8923	−1.48283	−0.741413	−	0.671049i	$-0.765844\pi$
$619$	−36.8923	−1.48283	−0.741413	+	0.671049i	$0.765844\pi$
$620$	−1.62213	−0.0651464
$621$	0	0
$622$	19.6868	0.789369
$623$	−4.65381	−0.186451
$624$	0	0
$625$	20.1854	0.807414
$626$	−4.02424	−0.160841
$627$	0	0
$628$	2.54401	0.101517
$629$	19.3682	0.772262
$630$	0	0
$631$	−6.52074	−0.259587	−0.129793	−	0.991541i	$-0.541431\pi$
$631$	−6.52074	−0.259587	−0.129793	+	0.991541i	$0.541431\pi$
$632$	2.55499	0.101632
$633$	0	0
$634$	27.0318	1.07357
$635$	−4.70437	−0.186687
$636$	0	0
$637$	−8.59386	−0.340501
$638$	4.98785	0.197471
$639$	0	0
$640$	4.67943	0.184971
$641$	27.5826	1.08945	0.544723	−	0.838616i	$-0.316635\pi$
$641$	27.5826	1.08945	0.544723	+	0.838616i	$0.316635\pi$
$642$	0	0
$643$	13.6254	0.537333	0.268666	−	0.963233i	$-0.413417\pi$
$643$	13.6254	0.537333	0.268666	+	0.963233i	$0.413417\pi$
$644$	−0.361201	−0.0142333
$645$	0	0
$646$	32.4337	1.27609
$647$	21.2032	0.833583	0.416792	−	0.909002i	$-0.363154\pi$
$647$	21.2032	0.833583	0.416792	+	0.909002i	$0.363154\pi$
$648$	0	0
$649$	49.7811	1.95408
$650$	9.49638	0.372479
$651$	0	0
$652$	−2.40229	−0.0940810
$653$	−17.3572	−0.679239	−0.339620	−	0.940563i	$-0.610298\pi$
$653$	−17.3572	−0.679239	−0.339620	+	0.940563i	$0.610298\pi$
$654$	0	0
$655$	11.6674	0.455884
$656$	−9.37383	−0.365987
$657$	0	0
$658$	0.623340	0.0243003
$659$	19.9119	0.775658	0.387829	−	0.921731i	$-0.373225\pi$
$659$	19.9119	0.775658	0.387829	+	0.921731i	$0.373225\pi$
$660$	0	0
$661$	6.21683	0.241807	0.120903	−	0.992664i	$-0.461421\pi$
$661$	6.21683	0.241807	0.120903	+	0.992664i	$0.461421\pi$
$662$	−41.6555	−1.61899
$663$	0	0
$664$	1.81768	0.0705395
$665$	−5.34156	−0.207137
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	4.83962	0.187250
$669$	0	0
$670$	−7.58559	−0.293057
$671$	35.3638	1.36520
$672$	0	0
$673$	−30.8251	−1.18822	−0.594109	−	0.804384i	$-0.702495\pi$
$673$	−30.8251	−1.18822	−0.594109	+	0.804384i	$0.702495\pi$
$674$	−9.96815	−0.383959
$675$	0	0
$676$	3.13723	0.120663
$677$	−2.78347	−0.106977	−0.0534887	−	0.998568i	$-0.517034\pi$
$677$	−2.78347	−0.106977	−0.0534887	+	0.998568i	$0.517034\pi$
$678$	0	0
$679$	6.25123	0.239900
$680$	5.57138	0.213653
$681$	0	0
$682$	47.6062	1.82294
$683$	32.3027	1.23603	0.618013	−	0.786168i	$-0.287938\pi$
$683$	32.3027	1.23603	0.618013	+	0.786168i	$0.287938\pi$
$684$	0	0
$685$	2.10125	0.0802848
$686$	19.8897	0.759393
$687$	0	0
$688$	39.1671	1.49323
$689$	−17.4931	−0.666434
$690$	0	0
$691$	−28.1794	−1.07200	−0.535998	−	0.844219i	$-0.680065\pi$
$691$	−28.1794	−1.07200	−0.535998	+	0.844219i	$0.680065\pi$
$692$	2.51268	0.0955179
$693$	0	0
$694$	−8.86391	−0.336469
$695$	5.60244	0.212513
$696$	0	0
$697$	−9.16506	−0.347151
$698$	−17.0260	−0.644443
$699$	0	0
$700$	1.68748	0.0637806
$701$	−18.3177	−0.691850	−0.345925	−	0.938262i	$-0.612435\pi$
$701$	−18.3177	−0.691850	−0.345925	+	0.938262i	$0.612435\pi$
$702$	0	0
$703$	−45.7184	−1.72430
$704$	33.6643	1.26877
$705$	0	0
$706$	−23.2746	−0.875950
$707$	14.4951	0.545145
$708$	0	0
$709$	−27.7519	−1.04224	−0.521122	−	0.853482i	$-0.674486\pi$
$709$	−27.7519	−1.04224	−0.521122	+	0.853482i	$0.674486\pi$
$710$	−7.75977	−0.291219
$711$	0	0
$712$	−11.4579	−0.429402
$713$	−9.54445	−0.357442
$714$	0	0
$715$	−3.40979	−0.127519
$716$	3.27918	0.122549
$717$	0	0
$718$	−0.867400	−0.0323711
$719$	−35.7583	−1.33356	−0.666780	−	0.745255i	$-0.732328\pi$
$719$	−35.7583	−1.33356	−0.666780	+	0.745255i	$0.732328\pi$
$720$	0	0
$721$	4.51385	0.168105
$722$	−51.7621	−1.92638
$723$	0	0
$724$	−6.62361	−0.246165
$725$	4.67184	0.173508
$726$	0	0
$727$	−17.5411	−0.650564	−0.325282	−	0.945617i	$-0.605459\pi$
$727$	−17.5411	−0.650564	−0.325282	+	0.945617i	$0.605459\pi$
$728$	5.68362	0.210649
$729$	0	0
$730$	−1.90679	−0.0705733
$731$	38.2948	1.41638
$732$	0	0
$733$	−2.71604	−0.100319	−0.0501595	−	0.998741i	$-0.515973\pi$
$733$	−2.71604	−0.100319	−0.0501595	+	0.998741i	$0.515973\pi$
$734$	10.1474	0.374548
$735$	0	0
$736$	−1.66371	−0.0613251
$737$	−38.7764	−1.42835
$738$	0	0
$739$	−50.4189	−1.85469	−0.927344	−	0.374210i	$-0.877914\pi$
$739$	−50.4189	−1.85469	−0.927344	+	0.374210i	$0.877914\pi$
$740$	−1.01450	−0.0372938
$741$	0	0
$742$	17.8462	0.655153
$743$	22.6523	0.831033	0.415517	−	0.909586i	$-0.363601\pi$
$743$	22.6523	0.831033	0.415517	+	0.909586i	$0.363601\pi$
$744$	0	0
$745$	−8.44956	−0.309568
$746$	8.16654	0.298998
$747$	0	0
$748$	3.67905	0.134519
$749$	−11.7593	−0.429677
$750$	0	0
$751$	−12.3305	−0.449945	−0.224972	−	0.974365i	$-0.572229\pi$
$751$	−12.3305	−0.449945	−0.224972	+	0.974365i	$0.572229\pi$
$752$	1.30191	0.0474757
$753$	0	0
$754$	2.03268	0.0740260
$755$	−1.32141	−0.0480912
$756$	0	0
$757$	−5.85052	−0.212641	−0.106320	−	0.994332i	$-0.533907\pi$
$757$	−5.85052	−0.212641	−0.106320	+	0.994332i	$0.533907\pi$
$758$	22.1107	0.803095
$759$	0	0
$760$	−13.1511	−0.477042
$761$	15.9306	0.577484	0.288742	−	0.957407i	$-0.406763\pi$
$761$	15.9306	0.577484	0.288742	+	0.957407i	$0.406763\pi$
$762$	0	0
$763$	12.8325	0.464567
$764$	−0.0334457	−0.00121002
$765$	0	0
$766$	−10.4571	−0.377830
$767$	20.2872	0.732527
$768$	0	0
$769$	−13.5190	−0.487506	−0.243753	−	0.969837i	$-0.578379\pi$
$769$	−13.5190	−0.487506	−0.243753	+	0.969837i	$0.578379\pi$
$770$	3.47861	0.125361
$771$	0	0
$772$	4.44714	0.160056
$773$	−48.7672	−1.75403	−0.877017	−	0.480460i	$-0.840470\pi$
$773$	−48.7672	−1.75403	−0.877017	+	0.480460i	$0.840470\pi$
$774$	0	0
$775$	44.5902	1.60173
$776$	15.3908	0.552497
$777$	0	0
$778$	14.4816	0.519192
$779$	21.6340	0.775117
$780$	0	0
$781$	−39.6668	−1.41939
$782$	4.23469	0.151432
$783$	0	0
$784$	18.3114	0.653979
$785$	4.91204	0.175318
$786$	0	0
$787$	6.25934	0.223122	0.111561	−	0.993758i	$-0.464415\pi$
$787$	6.25934	0.223122	0.111561	+	0.993758i	$0.464415\pi$
$788$	−4.28349	−0.152593
$789$	0	0
$790$	0.637277	0.0226733
$791$	−21.7174	−0.772182
$792$	0	0
$793$	14.4117	0.511775
$794$	−24.5162	−0.870047
$795$	0	0
$796$	5.44410	0.192961
$797$	−27.7945	−0.984530	−0.492265	−	0.870445i	$-0.663831\pi$
$797$	−27.7945	−0.984530	−0.492265	+	0.870445i	$0.663831\pi$
$798$	0	0
$799$	1.27291	0.0450324
$800$	7.77258	0.274802
$801$	0	0
$802$	−10.8616	−0.383538
$803$	−9.74721	−0.343972
$804$	0	0
$805$	−0.697418	−0.0245808
$806$	19.4008	0.683366
$807$	0	0
$808$	35.6876	1.25549
$809$	−7.21091	−0.253522	−0.126761	−	0.991933i	$-0.540458\pi$
$809$	−7.21091	−0.253522	−0.126761	+	0.991933i	$0.540458\pi$
$810$	0	0
$811$	5.68402	0.199593	0.0997964	−	0.995008i	$-0.468181\pi$
$811$	5.68402	0.199593	0.0997964	+	0.995008i	$0.468181\pi$
$812$	0.361201	0.0126757
$813$	0	0
$814$	29.7735	1.04356
$815$	−4.63841	−0.162477
$816$	0	0
$817$	−90.3941	−3.16249
$818$	28.1166	0.983073
$819$	0	0
$820$	0.480062	0.0167645
$821$	−17.6765	−0.616915	−0.308458	−	0.951238i	$-0.599813\pi$
$821$	−17.6765	−0.616915	−0.308458	+	0.951238i	$0.599813\pi$
$822$	0	0
$823$	−48.5960	−1.69395	−0.846975	−	0.531633i	$-0.821578\pi$
$823$	−48.5960	−1.69395	−0.846975	+	0.531633i	$0.821578\pi$
$824$	11.1133	0.387150
$825$	0	0
$826$	−20.6966	−0.720128
$827$	48.9103	1.70078	0.850389	−	0.526154i	$-0.176367\pi$
$827$	48.9103	1.70078	0.850389	+	0.526154i	$0.176367\pi$
$828$	0	0
$829$	38.9522	1.35287	0.676434	−	0.736504i	$-0.263525\pi$
$829$	38.9522	1.35287	0.676434	+	0.736504i	$0.263525\pi$
$830$	0.453372	0.0157368
$831$	0	0
$832$	13.7191	0.475625
$833$	17.9036	0.620322
$834$	0	0
$835$	9.34447	0.323379
$836$	−8.68433	−0.300354
$837$	0	0
$838$	−16.5298	−0.571012
$839$	−27.9458	−0.964797	−0.482399	−	0.875952i	$-0.660234\pi$
$839$	−27.9458	−0.964797	−0.482399	+	0.875952i	$0.660234\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−1.03888	−0.0358020
$843$	0	0
$844$	−2.51754	−0.0866573
$845$	6.05745	0.208383
$846$	0	0
$847$	4.39015	0.150847
$848$	37.2735	1.27998
$849$	0	0
$850$	−19.7838	−0.678579
$851$	−5.96920	−0.204622
$852$	0	0
$853$	19.3309	0.661878	0.330939	−	0.943652i	$-0.392635\pi$
$853$	19.3309	0.661878	0.330939	+	0.943652i	$0.392635\pi$
$854$	−14.7026	−0.503113
$855$	0	0
$856$	−28.9520	−0.989558
$857$	42.6767	1.45781	0.728904	−	0.684616i	$-0.240030\pi$
$857$	42.6767	1.45781	0.728904	+	0.684616i	$0.240030\pi$
$858$	0	0
$859$	−10.8383	−0.369800	−0.184900	−	0.982757i	$-0.559196\pi$
$859$	−10.8383	−0.369800	−0.184900	+	0.982757i	$0.559196\pi$
$860$	−2.00586	−0.0683993
$861$	0	0
$862$	42.6245	1.45180
$863$	−35.3269	−1.20254	−0.601271	−	0.799045i	$-0.705339\pi$
$863$	−35.3269	−1.20254	−0.601271	+	0.799045i	$0.705339\pi$
$864$	0	0
$865$	4.85156	0.164958
$866$	−13.8304	−0.469977
$867$	0	0
$868$	3.44747	0.117015
$869$	3.25767	0.110509
$870$	0	0
$871$	−15.8025	−0.535446
$872$	31.5941	1.06991
$873$	0	0
$874$	−9.99591	−0.338117
$875$	6.74532	0.228033
$876$	0	0
$877$	16.6037	0.560667	0.280333	−	0.959903i	$-0.409555\pi$
$877$	16.6037	0.560667	0.280333	+	0.959903i	$0.409555\pi$
$878$	−38.6443	−1.30418
$879$	0	0
$880$	7.26543	0.244918
$881$	−23.2009	−0.781659	−0.390830	−	0.920463i	$-0.627812\pi$
$881$	−23.2009	−0.781659	−0.390830	+	0.920463i	$0.627812\pi$
$882$	0	0
$883$	25.1560	0.846568	0.423284	−	0.905997i	$-0.360877\pi$
$883$	25.1560	0.846568	0.423284	+	0.905997i	$0.360877\pi$
$884$	1.49931	0.0504274
$885$	0	0
$886$	−44.3966	−1.49153
$887$	17.2591	0.579502	0.289751	−	0.957102i	$-0.406427\pi$
$887$	17.2591	0.579502	0.289751	+	0.957102i	$0.406427\pi$
$888$	0	0
$889$	9.99804	0.335324
$890$	−2.85787	−0.0957961
$891$	0	0
$892$	−2.35291	−0.0787812
$893$	−3.00469	−0.100548
$894$	0	0
$895$	6.33154	0.211640
$896$	−9.94504	−0.332240
$897$	0	0
$898$	50.6122	1.68895
$899$	9.54445	0.318325
$900$	0	0
$901$	36.4433	1.21410
$902$	−14.0888	−0.469106
$903$	0	0
$904$	−53.4692	−1.77836
$905$	−12.7891	−0.425123
$906$	0	0
$907$	−28.5679	−0.948581	−0.474290	−	0.880368i	$-0.657295\pi$
$907$	−28.5679	−0.948581	−0.474290	+	0.880368i	$0.657295\pi$
$908$	−7.89927	−0.262146
$909$	0	0
$910$	1.41763	0.0469940
$911$	−13.1206	−0.434705	−0.217353	−	0.976093i	$-0.569742\pi$
$911$	−13.1206	−0.434705	−0.217353	+	0.976093i	$0.569742\pi$
$912$	0	0
$913$	2.31757	0.0767005
$914$	26.1442	0.864772
$915$	0	0
$916$	−6.57152	−0.217129
$917$	−24.7964	−0.818850
$918$	0	0
$919$	−2.49407	−0.0822717	−0.0411358	−	0.999154i	$-0.513098\pi$
$919$	−2.49407	−0.0822717	−0.0411358	+	0.999154i	$0.513098\pi$
$920$	−1.71707	−0.0566102
$921$	0	0
$922$	−16.0356	−0.528104
$923$	−16.1653	−0.532088
$924$	0	0
$925$	27.8872	0.916925
$926$	49.4392	1.62467
$927$	0	0
$928$	1.66371	0.0546139
$929$	14.5108	0.476083	0.238042	−	0.971255i	$-0.423495\pi$
$929$	14.5108	0.476083	0.238042	+	0.971255i	$0.423495\pi$
$930$	0	0
$931$	−42.2611	−1.38505
$932$	1.20667	0.0395258
$933$	0	0
$934$	1.02095	0.0334065
$935$	7.10362	0.232313
$936$	0	0
$937$	−36.4431	−1.19054	−0.595271	−	0.803525i	$-0.702955\pi$
$937$	−36.4431	−1.19054	−0.595271	+	0.803525i	$0.702955\pi$
$938$	16.1214	0.526383
$939$	0	0
$940$	−0.0666745	−0.00217468
$941$	6.09402	0.198659	0.0993296	−	0.995055i	$-0.468330\pi$
$941$	6.09402	0.198659	0.0993296	+	0.995055i	$0.468330\pi$
$942$	0	0
$943$	2.82463	0.0919825
$944$	−43.2270	−1.40692
$945$	0	0
$946$	58.8679	1.91396
$947$	11.8081	0.383711	0.191855	−	0.981423i	$-0.438550\pi$
$947$	11.8081	0.383711	0.191855	+	0.981423i	$0.438550\pi$
$948$	0	0
$949$	−3.97226	−0.128945
$950$	46.6993	1.51513
$951$	0	0
$952$	−11.8407	−0.383758
$953$	−28.4697	−0.922225	−0.461112	−	0.887342i	$-0.652549\pi$
$953$	−28.4697	−0.922225	−0.461112	+	0.887342i	$0.652549\pi$
$954$	0	0
$955$	−0.0645779	−0.00208969
$956$	2.24172	0.0725023
$957$	0	0
$958$	9.30916	0.300765
$959$	−4.46573	−0.144206
$960$	0	0
$961$	60.0964	1.93860
$962$	12.1335	0.391200
$963$	0	0
$964$	−1.97479	−0.0636037
$965$	8.58666	0.276414
$966$	0	0
$967$	13.8752	0.446196	0.223098	−	0.974796i	$-0.428383\pi$
$967$	13.8752	0.446196	0.223098	+	0.974796i	$0.428383\pi$
$968$	10.8087	0.347406
$969$	0	0
$970$	3.83883	0.123258
$971$	13.7112	0.440013	0.220007	−	0.975498i	$-0.429392\pi$
$971$	13.7112	0.440013	0.220007	+	0.975498i	$0.429392\pi$
$972$	0	0
$973$	−11.9067	−0.381711
$974$	23.5780	0.755488
$975$	0	0
$976$	−30.7078	−0.982934
$977$	−35.8383	−1.14657	−0.573284	−	0.819356i	$-0.694331\pi$
$977$	−35.8383	−1.14657	−0.573284	+	0.819356i	$0.694331\pi$
$978$	0	0
$979$	−14.6090	−0.466906
$980$	−0.937782	−0.0299563
$981$	0	0
$982$	−1.55956	−0.0497675
$983$	−9.92432	−0.316537	−0.158268	−	0.987396i	$-0.550591\pi$
$983$	−9.92432	−0.316537	−0.158268	+	0.987396i	$0.550591\pi$
$984$	0	0
$985$	−8.27068	−0.263526
$986$	−4.23469	−0.134860
$987$	0	0
$988$	−3.53910	−0.112594
$989$	−11.8023	−0.375290
$990$	0	0
$991$	52.4826	1.66716	0.833582	−	0.552396i	$-0.186286\pi$
$991$	52.4826	1.66716	0.833582	+	0.552396i	$0.186286\pi$
$992$	15.8792	0.504164
$993$	0	0
$994$	16.4916	0.523081
$995$	10.5116	0.333241
$996$	0	0
$997$	3.72613	0.118008	0.0590038	−	0.998258i	$-0.481208\pi$
$997$	3.72613	0.118008	0.0590038	+	0.998258i	$0.481208\pi$
$998$	−29.0903	−0.920836
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.o.1.5		13
3.2	odd	2		667.2.a.c.1.9	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
667.2.a.c.1.9	✓	13	3.2	odd	2
6003.2.a.o.1.5		13	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)

\(f(q)\)	\(=\)	\(q-1.30511 q^{2} -0.296685 q^{4} -0.572849 q^{5} +1.21746 q^{7} +2.99743 q^{8} +O(q^{10})\)	\(q-1.30511 q^{2} -0.296685 q^{4} -0.572849 q^{5} +1.21746 q^{7} +2.99743 q^{8} +0.747631 q^{10} +3.82178 q^{11} +1.55748 q^{13} -1.58892 q^{14} -3.31861 q^{16} -3.24470 q^{17} +7.65905 q^{19} +0.169956 q^{20} -4.98785 q^{22} +1.00000 q^{23} -4.67184 q^{25} -2.03268 q^{26} -0.361201 q^{28} -1.00000 q^{29} -9.54445 q^{31} -1.66371 q^{32} +4.23469 q^{34} -0.697418 q^{35} -5.96920 q^{37} -9.99591 q^{38} -1.71707 q^{40} +2.82463 q^{41} -11.8023 q^{43} -1.13387 q^{44} -1.30511 q^{46} -0.392305 q^{47} -5.51780 q^{49} +6.09728 q^{50} -0.462081 q^{52} -11.2317 q^{53} -2.18930 q^{55} +3.64924 q^{56} +1.30511 q^{58} +13.0256 q^{59} +9.25323 q^{61} +12.4566 q^{62} +8.80854 q^{64} -0.892200 q^{65} -10.1462 q^{67} +0.962653 q^{68} +0.910208 q^{70} -10.3791 q^{71} -2.55044 q^{73} +7.79047 q^{74} -2.27233 q^{76} +4.65285 q^{77} +0.852395 q^{79} +1.90106 q^{80} -3.68645 q^{82} +0.606412 q^{83} +1.85872 q^{85} +15.4033 q^{86} +11.4555 q^{88} -3.82257 q^{89} +1.89616 q^{91} -0.296685 q^{92} +0.512002 q^{94} -4.38748 q^{95} +5.13466 q^{97} +7.20134 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 4 q^{2} + 12 q^{4} - 16 q^{5} + q^{7} - 6 q^{8}+O(q^{10}) \) 13 * q - 4 * q^2 + 12 * q^4 - 16 * q^5 + q^7 - 6 * q^8	\( 13 q - 4 q^{2} + 12 q^{4} - 16 q^{5} + q^{7} - 6 q^{8} + 10 q^{10} - 10 q^{11} + 7 q^{13} + 12 q^{14} + 2 q^{16} - 26 q^{17} - 25 q^{20} - 15 q^{22} + 13 q^{23} + 19 q^{25} + 15 q^{26} + 5 q^{28} - 13 q^{29} - 6 q^{31} - 16 q^{32} + 11 q^{34} - q^{35} + 15 q^{37} - 8 q^{38} + 14 q^{40} - 9 q^{41} + q^{43} - 29 q^{44} - 4 q^{46} - 15 q^{47} + 4 q^{49} - 31 q^{50} - 8 q^{52} - 43 q^{53} - 3 q^{55} + 5 q^{56} + 4 q^{58} + 9 q^{59} + 20 q^{61} - 11 q^{62} - 16 q^{64} + 25 q^{65} + q^{67} - 21 q^{68} - 2 q^{70} - 17 q^{71} + 26 q^{73} - 11 q^{74} + 8 q^{76} - 17 q^{77} + 5 q^{79} - 10 q^{80} - 25 q^{82} - 4 q^{83} + 20 q^{85} + 13 q^{86} - 32 q^{88} - 48 q^{89} - 9 q^{91} + 12 q^{92} - 65 q^{94} - 8 q^{95} + 30 q^{97} - 2 q^{98}+O(q^{100}) \) 13 * q - 4 * q^2 + 12 * q^4 - 16 * q^5 + q^7 - 6 * q^8 + 10 * q^10 - 10 * q^11 + 7 * q^13 + 12 * q^14 + 2 * q^16 - 26 * q^17 - 25 * q^20 - 15 * q^22 + 13 * q^23 + 19 * q^25 + 15 * q^26 + 5 * q^28 - 13 * q^29 - 6 * q^31 - 16 * q^32 + 11 * q^34 - q^35 + 15 * q^37 - 8 * q^38 + 14 * q^40 - 9 * q^41 + q^43 - 29 * q^44 - 4 * q^46 - 15 * q^47 + 4 * q^49 - 31 * q^50 - 8 * q^52 - 43 * q^53 - 3 * q^55 + 5 * q^56 + 4 * q^58 + 9 * q^59 + 20 * q^61 - 11 * q^62 - 16 * q^64 + 25 * q^65 + q^67 - 21 * q^68 - 2 * q^70 - 17 * q^71 + 26 * q^73 - 11 * q^74 + 8 * q^76 - 17 * q^77 + 5 * q^79 - 10 * q^80 - 25 * q^82 - 4 * q^83 + 20 * q^85 + 13 * q^86 - 32 * q^88 - 48 * q^89 - 9 * q^91 + 12 * q^92 - 65 * q^94 - 8 * q^95 + 30 * q^97 - 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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