LMFDB - Embedded newform 6003.2.a.q.1.3

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:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 3 x^{15} - 22 x^{14} + 68 x^{13} + 187 x^{12} - 597 x^{11} - 795 x^{10} + 2592 x^{9} + \cdots + 64 $ x^16 - 3x^15 - 22x^14 + 68x^13 + 187x^12 - 597x^11 - 795x^10 + 2592x^9 + 1860x^8 - 5877x^7 - 2496x^6 + 6612x^5 + 1842x^4 - 3011x^3 - 505x^2 + 336*x + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 667)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.14844$ 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)

$f(q)$	$=$	$q-2.14844 q^{2} +2.61580 q^{4} +1.42439 q^{5} +3.61731 q^{7} -1.32302 q^{8} +O(q^{10})$	\(q-2.14844 q^{2} +2.61580 q^{4} +1.42439 q^{5} +3.61731 q^{7} -1.32302 q^{8} -3.06021 q^{10} -4.28818 q^{11} +0.939230 q^{13} -7.77158 q^{14} -2.38917 q^{16} -7.48700 q^{17} +4.97904 q^{19} +3.72592 q^{20} +9.21291 q^{22} -1.00000 q^{23} -2.97112 q^{25} -2.01788 q^{26} +9.46218 q^{28} +1.00000 q^{29} -2.22916 q^{31} +7.77905 q^{32} +16.0854 q^{34} +5.15245 q^{35} -0.191258 q^{37} -10.6972 q^{38} -1.88449 q^{40} +5.36788 q^{41} -6.26248 q^{43} -11.2170 q^{44} +2.14844 q^{46} -0.471038 q^{47} +6.08494 q^{49} +6.38329 q^{50} +2.45684 q^{52} +6.09535 q^{53} -6.10802 q^{55} -4.78578 q^{56} -2.14844 q^{58} -7.35600 q^{59} +9.04719 q^{61} +4.78922 q^{62} -11.9345 q^{64} +1.33783 q^{65} +8.32218 q^{67} -19.5845 q^{68} -11.0697 q^{70} -4.71689 q^{71} -10.9752 q^{73} +0.410907 q^{74} +13.0242 q^{76} -15.5117 q^{77} -7.80891 q^{79} -3.40311 q^{80} -11.5326 q^{82} +16.6667 q^{83} -10.6644 q^{85} +13.4546 q^{86} +5.67335 q^{88} -2.37020 q^{89} +3.39749 q^{91} -2.61580 q^{92} +1.01200 q^{94} +7.09207 q^{95} -11.3963 q^{97} -13.0731 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 3 q^{2} + 21 q^{4} - 16 q^{5} + q^{7} - 9 q^{8}+O(q^{10}) $ 16 * q - 3 * q^2 + 21 * q^4 - 16 * q^5 + q^7 - 9 * q^8	$ 16 q - 3 q^{2} + 21 q^{4} - 16 q^{5} + q^{7} - 9 q^{8} - 14 q^{10} - 4 q^{11} + 15 q^{13} - 8 q^{14} + 23 q^{16} - 20 q^{17} - 4 q^{19} - 25 q^{20} + 13 q^{22} - 16 q^{23} + 30 q^{25} - 25 q^{26} - 13 q^{28} + 16 q^{29} + 19 q^{32} - 23 q^{34} - 5 q^{35} + 5 q^{37} - 38 q^{38} - 20 q^{40} - 7 q^{41} - 17 q^{43} + 21 q^{44} + 3 q^{46} - 29 q^{47} + 31 q^{49} + 44 q^{50} + 20 q^{52} - 63 q^{53} + q^{55} + 19 q^{56} - 3 q^{58} - 11 q^{59} - 33 q^{62} + 29 q^{64} - 53 q^{65} - 13 q^{67} - 63 q^{68} - 46 q^{70} + 23 q^{71} - 38 q^{73} + 47 q^{74} - 56 q^{76} - 97 q^{77} - 27 q^{79} - 8 q^{80} + 9 q^{82} - 36 q^{83} + 6 q^{85} + 11 q^{86} - 24 q^{88} + 16 q^{89} - 47 q^{91} - 21 q^{92} + 37 q^{94} + 12 q^{95} - 30 q^{97} + 27 q^{98}+O(q^{100}) $ 16 * q - 3 * q^2 + 21 * q^4 - 16 * q^5 + q^7 - 9 * q^8 - 14 * q^10 - 4 * q^11 + 15 * q^13 - 8 * q^14 + 23 * q^16 - 20 * q^17 - 4 * q^19 - 25 * q^20 + 13 * q^22 - 16 * q^23 + 30 * q^25 - 25 * q^26 - 13 * q^28 + 16 * q^29 + 19 * q^32 - 23 * q^34 - 5 * q^35 + 5 * q^37 - 38 * q^38 - 20 * q^40 - 7 * q^41 - 17 * q^43 + 21 * q^44 + 3 * q^46 - 29 * q^47 + 31 * q^49 + 44 * q^50 + 20 * q^52 - 63 * q^53 + q^55 + 19 * q^56 - 3 * q^58 - 11 * q^59 - 33 * q^62 + 29 * q^64 - 53 * q^65 - 13 * q^67 - 63 * q^68 - 46 * q^70 + 23 * q^71 - 38 * q^73 + 47 * q^74 - 56 * q^76 - 97 * q^77 - 27 * q^79 - 8 * q^80 + 9 * q^82 - 36 * q^83 + 6 * q^85 + 11 * q^86 - 24 * q^88 + 16 * q^89 - 47 * q^91 - 21 * q^92 + 37 * q^94 + 12 * q^95 - 30 * q^97 + 27 * 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 $

$2$	−2.14844	−1.51918	−0.759589	−	0.650403i	$-0.774600\pi$
$2$	−2.14844	−1.51918	−0.759589	+	0.650403i	$0.774600\pi$
$3$	0	0
$3$	0	0
$4$	2.61580	1.30790
$5$	1.42439	0.637005	0.318502	−	0.947922i	$-0.396820\pi$
$5$	1.42439	0.637005	0.318502	+	0.947922i	$0.396820\pi$
$6$	0	0
$7$	3.61731	1.36721	0.683607	−	0.729850i	$-0.260410\pi$
$7$	3.61731	1.36721	0.683607	+	0.729850i	$0.260410\pi$
$8$	−1.32302	−0.467759
$9$	0	0
$10$	−3.06021	−0.967724
$11$	−4.28818	−1.29293	−0.646467	−	0.762942i	$-0.723754\pi$
$11$	−4.28818	−1.29293	−0.646467	+	0.762942i	$0.723754\pi$
$12$	0	0
$13$	0.939230	0.260496	0.130248	−	0.991481i	$-0.458423\pi$
$13$	0.939230	0.260496	0.130248	+	0.991481i	$0.458423\pi$
$14$	−7.77158	−2.07704
$15$	0	0
$16$	−2.38917	−0.597294
$17$	−7.48700	−1.81586	−0.907932	−	0.419119i	$-0.862339\pi$
$17$	−7.48700	−1.81586	−0.907932	+	0.419119i	$0.862339\pi$
$18$	0	0
$19$	4.97904	1.14227	0.571135	−	0.820856i	$-0.306503\pi$
$19$	4.97904	1.14227	0.571135	+	0.820856i	$0.306503\pi$
$20$	3.72592	0.833140
$21$	0	0
$22$	9.21291	1.96420
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−2.97112	−0.594225
$26$	−2.01788	−0.395739
$27$	0	0
$28$	9.46218	1.78818
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−2.22916	−0.400369	−0.200185	−	0.979758i	$-0.564154\pi$
$31$	−2.22916	−0.400369	−0.200185	+	0.979758i	$0.564154\pi$
$32$	7.77905	1.37515
$33$	0	0
$34$	16.0854	2.75862
$35$	5.15245	0.870923
$36$	0	0
$37$	−0.191258	−0.0314426	−0.0157213	−	0.999876i	$-0.505004\pi$
$37$	−0.191258	−0.0314426	−0.0157213	+	0.999876i	$0.505004\pi$
$38$	−10.6972	−1.73531
$39$	0	0
$40$	−1.88449	−0.297965
$41$	5.36788	0.838322	0.419161	−	0.907912i	$-0.362324\pi$
$41$	5.36788	0.838322	0.419161	+	0.907912i	$0.362324\pi$
$42$	0	0
$43$	−6.26248	−0.955019	−0.477510	−	0.878627i	$-0.658460\pi$
$43$	−6.26248	−0.955019	−0.477510	+	0.878627i	$0.658460\pi$
$44$	−11.2170	−1.69103
$45$	0	0
$46$	2.14844	0.316771
$47$	−0.471038	−0.0687080	−0.0343540	−	0.999410i	$-0.510937\pi$
$47$	−0.471038	−0.0687080	−0.0343540	+	0.999410i	$0.510937\pi$
$48$	0	0
$49$	6.08494	0.869277
$50$	6.38329	0.902733
$51$	0	0
$52$	2.45684	0.340703
$53$	6.09535	0.837261	0.418631	−	0.908157i	$-0.362510\pi$
$53$	6.09535	0.837261	0.418631	+	0.908157i	$0.362510\pi$
$54$	0	0
$55$	−6.10802	−0.823606
$56$	−4.78578	−0.639527
$57$	0	0
$58$	−2.14844	−0.282104
$59$	−7.35600	−0.957670	−0.478835	−	0.877905i	$-0.658941\pi$
$59$	−7.35600	−0.957670	−0.478835	+	0.877905i	$0.658941\pi$
$60$	0	0
$61$	9.04719	1.15837	0.579187	−	0.815195i	$-0.303370\pi$
$61$	9.04719	1.15837	0.579187	+	0.815195i	$0.303370\pi$
$62$	4.78922	0.608232
$63$	0	0
$64$	−11.9345	−1.49181
$65$	1.33783	0.165937
$66$	0	0
$67$	8.32218	1.01672	0.508358	−	0.861146i	$-0.330253\pi$
$67$	8.32218	1.01672	0.508358	+	0.861146i	$0.330253\pi$
$68$	−19.5845	−2.37497
$69$	0	0
$70$	−11.0697	−1.32309
$71$	−4.71689	−0.559792	−0.279896	−	0.960030i	$-0.590300\pi$
$71$	−4.71689	−0.559792	−0.279896	+	0.960030i	$0.590300\pi$
$72$	0	0
$73$	−10.9752	−1.28455	−0.642274	−	0.766475i	$-0.722009\pi$
$73$	−10.9752	−1.28455	−0.642274	+	0.766475i	$0.722009\pi$
$74$	0.410907	0.0477670
$75$	0	0
$76$	13.0242	1.49398
$77$	−15.5117	−1.76772
$78$	0	0
$79$	−7.80891	−0.878571	−0.439286	−	0.898347i	$-0.644768\pi$
$79$	−7.80891	−0.878571	−0.439286	+	0.898347i	$0.644768\pi$
$80$	−3.40311	−0.380479
$81$	0	0
$82$	−11.5326	−1.27356
$83$	16.6667	1.82941	0.914707	−	0.404118i	$-0.132422\pi$
$83$	16.6667	1.82941	0.914707	+	0.404118i	$0.132422\pi$
$84$	0	0
$85$	−10.6644	−1.15671
$86$	13.4546	1.45084
$87$	0	0
$88$	5.67335	0.604781
$89$	−2.37020	−0.251241	−0.125621	−	0.992078i	$-0.540092\pi$
$89$	−2.37020	−0.251241	−0.125621	+	0.992078i	$0.540092\pi$
$90$	0	0
$91$	3.39749	0.356153
$92$	−2.61580	−0.272717
$93$	0	0
$94$	1.01200	0.104380
$95$	7.09207	0.727631
$96$	0	0
$97$	−11.3963	−1.15712	−0.578560	−	0.815640i	$-0.696385\pi$
$97$	−11.3963	−1.15712	−0.578560	+	0.815640i	$0.696385\pi$
$98$	−13.0731	−1.32059
$99$	0	0
$100$	−7.77188	−0.777188
$101$	−15.9434	−1.58643	−0.793216	−	0.608940i	$-0.791595\pi$
$101$	−15.9434	−1.58643	−0.793216	+	0.608940i	$0.791595\pi$
$102$	0	0
$103$	−8.88555	−0.875519	−0.437760	−	0.899092i	$-0.644228\pi$
$103$	−8.88555	−0.875519	−0.437760	+	0.899092i	$0.644228\pi$
$104$	−1.24262	−0.121849
$105$	0	0
$106$	−13.0955	−1.27195
$107$	3.21047	0.310368	0.155184	−	0.987886i	$-0.450403\pi$
$107$	3.21047	0.310368	0.155184	+	0.987886i	$0.450403\pi$
$108$	0	0
$109$	−0.712956	−0.0682888	−0.0341444	−	0.999417i	$-0.510871\pi$
$109$	−0.712956	−0.0682888	−0.0341444	+	0.999417i	$0.510871\pi$
$110$	13.1227	1.25120
$111$	0	0
$112$	−8.64239	−0.816629
$113$	−6.57373	−0.618405	−0.309202	−	0.950996i	$-0.600062\pi$
$113$	−6.57373	−0.618405	−0.309202	+	0.950996i	$0.600062\pi$
$114$	0	0
$115$	−1.42439	−0.132825
$116$	2.61580	0.242871
$117$	0	0
$118$	15.8039	1.45487
$119$	−27.0828	−2.48268
$120$	0	0
$121$	7.38848	0.671680
$122$	−19.4374	−1.75978
$123$	0	0
$124$	−5.83105	−0.523644
$125$	−11.3540	−1.01553
$126$	0	0
$127$	−20.9025	−1.85480	−0.927401	−	0.374070i	$-0.877962\pi$
$127$	−20.9025	−1.85480	−0.927401	+	0.374070i	$0.877962\pi$
$128$	10.0825	0.891172
$129$	0	0
$130$	−2.87424	−0.252088
$131$	5.49993	0.480531	0.240266	−	0.970707i	$-0.422765\pi$
$131$	5.49993	0.480531	0.240266	+	0.970707i	$0.422765\pi$
$132$	0	0
$133$	18.0107	1.56173
$134$	−17.8797	−1.54457
$135$	0	0
$136$	9.90545	0.849386
$137$	−2.30717	−0.197115	−0.0985573	−	0.995131i	$-0.531423\pi$
$137$	−2.30717	−0.197115	−0.0985573	+	0.995131i	$0.531423\pi$
$138$	0	0
$139$	−9.74965	−0.826954	−0.413477	−	0.910514i	$-0.635686\pi$
$139$	−9.74965	−0.826954	−0.413477	+	0.910514i	$0.635686\pi$
$140$	13.4778	1.13908
$141$	0	0
$142$	10.1340	0.850424
$143$	−4.02759	−0.336804
$144$	0	0
$145$	1.42439	0.118289
$146$	23.5795	1.95146
$147$	0	0
$148$	−0.500294	−0.0411239
$149$	20.0433	1.64201	0.821004	−	0.570923i	$-0.193415\pi$
$149$	20.0433	1.64201	0.821004	+	0.570923i	$0.193415\pi$
$150$	0	0
$151$	−22.1151	−1.79970	−0.899849	−	0.436202i	$-0.856323\pi$
$151$	−22.1151	−1.79970	−0.899849	+	0.436202i	$0.856323\pi$
$152$	−6.58737	−0.534306
$153$	0	0
$154$	33.3259	2.68548
$155$	−3.17519	−0.255037
$156$	0	0
$157$	17.2042	1.37304	0.686522	−	0.727109i	$-0.259137\pi$
$157$	17.2042	1.37304	0.686522	+	0.727109i	$0.259137\pi$
$158$	16.7770	1.33471
$159$	0	0
$160$	11.0804	0.875980
$161$	−3.61731	−0.285084
$162$	0	0
$163$	18.2181	1.42695	0.713475	−	0.700680i	$-0.247120\pi$
$163$	18.2181	1.42695	0.713475	+	0.700680i	$0.247120\pi$
$164$	14.0413	1.09644
$165$	0	0
$166$	−35.8076	−2.77920
$167$	−19.0888	−1.47714	−0.738569	−	0.674177i	$-0.764498\pi$
$167$	−19.0888	−1.47714	−0.738569	+	0.674177i	$0.764498\pi$
$168$	0	0
$169$	−12.1178	−0.932142
$170$	22.9118	1.75725
$171$	0	0
$172$	−16.3814	−1.24907
$173$	−4.81592	−0.366147	−0.183074	−	0.983099i	$-0.558605\pi$
$173$	−4.81592	−0.366147	−0.183074	+	0.983099i	$0.558605\pi$
$174$	0	0
$175$	−10.7475	−0.812433
$176$	10.2452	0.772262
$177$	0	0
$178$	5.09225	0.381680
$179$	7.48859	0.559723	0.279862	−	0.960040i	$-0.409711\pi$
$179$	7.48859	0.559723	0.279862	+	0.960040i	$0.409711\pi$
$180$	0	0
$181$	9.18783	0.682926	0.341463	−	0.939895i	$-0.389078\pi$
$181$	9.18783	0.682926	0.341463	+	0.939895i	$0.389078\pi$
$182$	−7.29930	−0.541060
$183$	0	0
$184$	1.32302	0.0975344
$185$	−0.272425	−0.0200291
$186$	0	0
$187$	32.1056	2.34779
$188$	−1.23214	−0.0898634
$189$	0	0
$190$	−15.2369	−1.10540
$191$	−11.7239	−0.848311	−0.424155	−	0.905589i	$-0.639429\pi$
$191$	−11.7239	−0.848311	−0.424155	+	0.905589i	$0.639429\pi$
$192$	0	0
$193$	−5.79568	−0.417182	−0.208591	−	0.978003i	$-0.566888\pi$
$193$	−5.79568	−0.417182	−0.208591	+	0.978003i	$0.566888\pi$
$194$	24.4843	1.75787
$195$	0	0
$196$	15.9170	1.13693
$197$	−19.8932	−1.41733	−0.708665	−	0.705545i	$-0.750702\pi$
$197$	−19.8932	−1.41733	−0.708665	+	0.705545i	$0.750702\pi$
$198$	0	0
$199$	6.73338	0.477317	0.238658	−	0.971104i	$-0.423292\pi$
$199$	6.73338	0.477317	0.238658	+	0.971104i	$0.423292\pi$
$200$	3.93086	0.277954
$201$	0	0
$202$	34.2536	2.41007
$203$	3.61731	0.253885
$204$	0	0
$205$	7.64594	0.534015
$206$	19.0901	1.33007
$207$	0	0
$208$	−2.24398	−0.155592
$209$	−21.3510	−1.47688
$210$	0	0
$211$	−12.5526	−0.864159	−0.432080	−	0.901835i	$-0.642220\pi$
$211$	−12.5526	−0.864159	−0.432080	+	0.901835i	$0.642220\pi$
$212$	15.9443	1.09506
$213$	0	0
$214$	−6.89751	−0.471504
$215$	−8.92019	−0.608352
$216$	0	0
$217$	−8.06357	−0.547391
$218$	1.53174	0.103743
$219$	0	0
$220$	−15.9774	−1.07720
$221$	−7.03201	−0.473024
$222$	0	0
$223$	25.5659	1.71202	0.856010	−	0.516959i	$-0.172936\pi$
$223$	25.5659	1.71202	0.856010	+	0.516959i	$0.172936\pi$
$224$	28.1392	1.88013
$225$	0	0
$226$	14.1233	0.939467
$227$	20.8928	1.38670	0.693351	−	0.720600i	$-0.256133\pi$
$227$	20.8928	1.38670	0.693351	+	0.720600i	$0.256133\pi$
$228$	0	0
$229$	18.1024	1.19624	0.598120	−	0.801406i	$-0.295915\pi$
$229$	18.1024	1.19624	0.598120	+	0.801406i	$0.295915\pi$
$230$	3.06021	0.201784
$231$	0	0
$232$	−1.32302	−0.0868606
$233$	−6.86606	−0.449810	−0.224905	−	0.974381i	$-0.572207\pi$
$233$	−6.86606	−0.449810	−0.224905	+	0.974381i	$0.572207\pi$
$234$	0	0
$235$	−0.670941	−0.0437673
$236$	−19.2419	−1.25254
$237$	0	0
$238$	58.1858	3.77163
$239$	−8.00068	−0.517521	−0.258760	−	0.965942i	$-0.583314\pi$
$239$	−8.00068	−0.517521	−0.258760	+	0.965942i	$0.583314\pi$
$240$	0	0
$241$	−10.7624	−0.693267	−0.346634	−	0.938001i	$-0.612675\pi$
$241$	−10.7624	−0.693267	−0.346634	+	0.938001i	$0.612675\pi$
$242$	−15.8737	−1.02040
$243$	0	0
$244$	23.6657	1.51504
$245$	8.66730	0.553734
$246$	0	0
$247$	4.67646	0.297556
$248$	2.94923	0.187276
$249$	0	0
$250$	24.3933	1.54277
$251$	−18.6479	−1.17705	−0.588523	−	0.808481i	$-0.700290\pi$
$251$	−18.6479	−1.17705	−0.588523	+	0.808481i	$0.700290\pi$
$252$	0	0
$253$	4.28818	0.269596
$254$	44.9079	2.81777
$255$	0	0
$256$	2.20739	0.137962
$257$	−6.50023	−0.405473	−0.202737	−	0.979233i	$-0.564983\pi$
$257$	−6.50023	−0.405473	−0.202737	+	0.979233i	$0.564983\pi$
$258$	0	0
$259$	−0.691840	−0.0429888
$260$	3.49949	0.217029
$261$	0	0
$262$	−11.8163	−0.730012
$263$	5.94459	0.366559	0.183280	−	0.983061i	$-0.441329\pi$
$263$	5.94459	0.366559	0.183280	+	0.983061i	$0.441329\pi$
$264$	0	0
$265$	8.68214	0.533340
$266$	−38.6950	−2.37254
$267$	0	0
$268$	21.7692	1.32977
$269$	17.3345	1.05691	0.528453	−	0.848963i	$-0.322772\pi$
$269$	17.3345	1.05691	0.528453	+	0.848963i	$0.322772\pi$
$270$	0	0
$271$	−24.0282	−1.45961	−0.729805	−	0.683656i	$-0.760389\pi$
$271$	−24.0282	−1.45961	−0.729805	+	0.683656i	$0.760389\pi$
$272$	17.8877	1.08460
$273$	0	0
$274$	4.95682	0.299452
$275$	12.7407	0.768294
$276$	0	0
$277$	−8.60153	−0.516816	−0.258408	−	0.966036i	$-0.583198\pi$
$277$	−8.60153	−0.516816	−0.258408	+	0.966036i	$0.583198\pi$
$278$	20.9466	1.25629
$279$	0	0
$280$	−6.81680	−0.407382
$281$	7.79943	0.465275	0.232638	−	0.972564i	$-0.425264\pi$
$281$	7.79943	0.465275	0.232638	+	0.972564i	$0.425264\pi$
$282$	0	0
$283$	16.4894	0.980191	0.490096	−	0.871669i	$-0.336962\pi$
$283$	16.4894	0.980191	0.490096	+	0.871669i	$0.336962\pi$
$284$	−12.3385	−0.732153
$285$	0	0
$286$	8.65304	0.511665
$287$	19.4173	1.14617
$288$	0	0
$289$	39.0551	2.29736
$290$	−3.06021	−0.179702
$291$	0	0
$292$	−28.7089	−1.68006
$293$	−24.2731	−1.41805	−0.709023	−	0.705185i	$-0.750864\pi$
$293$	−24.2731	−1.41805	−0.709023	+	0.705185i	$0.750864\pi$
$294$	0	0
$295$	−10.4778	−0.610041
$296$	0.253038	0.0147076
$297$	0	0
$298$	−43.0618	−2.49450
$299$	−0.939230	−0.0543171
$300$	0	0
$301$	−22.6533	−1.30572
$302$	47.5129	2.73406
$303$	0	0
$304$	−11.8958	−0.682270
$305$	12.8867	0.737890
$306$	0	0
$307$	−30.4031	−1.73520	−0.867598	−	0.497266i	$-0.834337\pi$
$307$	−30.4031	−1.73520	−0.867598	+	0.497266i	$0.834337\pi$
$308$	−40.5755	−2.31200
$309$	0	0
$310$	6.82170	0.387447
$311$	18.8544	1.06914	0.534568	−	0.845126i	$-0.320474\pi$
$311$	18.8544	1.06914	0.534568	+	0.845126i	$0.320474\pi$
$312$	0	0
$313$	−23.4073	−1.32306	−0.661531	−	0.749918i	$-0.730093\pi$
$313$	−23.4073	−1.32306	−0.661531	+	0.749918i	$0.730093\pi$
$314$	−36.9622	−2.08590
$315$	0	0
$316$	−20.4266	−1.14909
$317$	2.08214	0.116944	0.0584722	−	0.998289i	$-0.481377\pi$
$317$	2.08214	0.116944	0.0584722	+	0.998289i	$0.481377\pi$
$318$	0	0
$319$	−4.28818	−0.240092
$320$	−16.9993	−0.950291
$321$	0	0
$322$	7.77158	0.433093
$323$	−37.2780	−2.07421
$324$	0	0
$325$	−2.79057	−0.154793
$326$	−39.1405	−2.16779
$327$	0	0
$328$	−7.10182	−0.392132
$329$	−1.70389	−0.0939386
$330$	0	0
$331$	−14.9198	−0.820067	−0.410034	−	0.912070i	$-0.634483\pi$
$331$	−14.9198	−0.820067	−0.410034	+	0.912070i	$0.634483\pi$
$332$	43.5970	2.39269
$333$	0	0
$334$	41.0113	2.24404
$335$	11.8540	0.647653
$336$	0	0
$337$	13.2112	0.719660	0.359830	−	0.933018i	$-0.382835\pi$
$337$	13.2112	0.719660	0.359830	+	0.933018i	$0.382835\pi$
$338$	26.0345	1.41609
$339$	0	0
$340$	−27.8959	−1.51287
$341$	9.55904	0.517651
$342$	0	0
$343$	−3.31007	−0.178727
$344$	8.28539	0.446718
$345$	0	0
$346$	10.3467	0.556243
$347$	25.0711	1.34589	0.672943	−	0.739695i	$-0.265030\pi$
$347$	25.0711	1.34589	0.672943	+	0.739695i	$0.265030\pi$
$348$	0	0
$349$	−15.3540	−0.821880	−0.410940	−	0.911662i	$-0.634799\pi$
$349$	−15.3540	−0.821880	−0.410940	+	0.911662i	$0.634799\pi$
$350$	23.0903	1.23423
$351$	0	0
$352$	−33.3579	−1.77798
$353$	−7.44772	−0.396402	−0.198201	−	0.980161i	$-0.563510\pi$
$353$	−7.44772	−0.396402	−0.198201	+	0.980161i	$0.563510\pi$
$354$	0	0
$355$	−6.71868	−0.356590
$356$	−6.19999	−0.328599
$357$	0	0
$358$	−16.0888	−0.850320
$359$	29.2314	1.54277	0.771386	−	0.636367i	$-0.219564\pi$
$359$	29.2314	1.54277	0.771386	+	0.636367i	$0.219564\pi$
$360$	0	0
$361$	5.79082	0.304780
$362$	−19.7395	−1.03749
$363$	0	0
$364$	8.88716	0.465814
$365$	−15.6329	−0.818263
$366$	0	0
$367$	−34.3083	−1.79088	−0.895438	−	0.445186i	$-0.853138\pi$
$367$	−34.3083	−1.79088	−0.895438	+	0.445186i	$0.853138\pi$
$368$	2.38917	0.124544
$369$	0	0
$370$	0.585290	0.0304278
$371$	22.0488	1.14472
$372$	0	0
$373$	21.0318	1.08899	0.544494	−	0.838765i	$-0.316722\pi$
$373$	21.0318	1.08899	0.544494	+	0.838765i	$0.316722\pi$
$374$	−68.9770	−3.56671
$375$	0	0
$376$	0.623194	0.0321388
$377$	0.939230	0.0483728
$378$	0	0
$379$	−27.7769	−1.42681	−0.713403	−	0.700754i	$-0.752847\pi$
$379$	−27.7769	−1.42681	−0.713403	+	0.700754i	$0.752847\pi$
$380$	18.5515	0.951671
$381$	0	0
$382$	25.1881	1.28874
$383$	11.7934	0.602617	0.301308	−	0.953527i	$-0.402577\pi$
$383$	11.7934	0.602617	0.301308	+	0.953527i	$0.402577\pi$
$384$	0	0
$385$	−22.0946	−1.12605
$386$	12.4517	0.633774
$387$	0	0
$388$	−29.8105	−1.51340
$389$	14.3127	0.725683	0.362842	−	0.931851i	$-0.381807\pi$
$389$	14.3127	0.725683	0.362842	+	0.931851i	$0.381807\pi$
$390$	0	0
$391$	7.48700	0.378634
$392$	−8.05050	−0.406612
$393$	0	0
$394$	42.7393	2.15318
$395$	−11.1229	−0.559654
$396$	0	0
$397$	−5.90410	−0.296318	−0.148159	−	0.988964i	$-0.547335\pi$
$397$	−5.90410	−0.296318	−0.148159	+	0.988964i	$0.547335\pi$
$398$	−14.4663	−0.725129
$399$	0	0
$400$	7.09853	0.354927
$401$	−20.6738	−1.03240	−0.516200	−	0.856468i	$-0.672654\pi$
$401$	−20.6738	−1.03240	−0.516200	+	0.856468i	$0.672654\pi$
$402$	0	0
$403$	−2.09369	−0.104294
$404$	−41.7049	−2.07490
$405$	0	0
$406$	−7.77158	−0.385697
$407$	0.820149	0.0406533
$408$	0	0
$409$	2.66079	0.131568	0.0657839	−	0.997834i	$-0.479045\pi$
$409$	2.66079	0.131568	0.0657839	+	0.997834i	$0.479045\pi$
$410$	−16.4269	−0.811265
$411$	0	0
$412$	−23.2429	−1.14509
$413$	−26.6090	−1.30934
$414$	0	0
$415$	23.7399	1.16535
$416$	7.30631	0.358221
$417$	0	0
$418$	45.8714	2.24364
$419$	−18.2654	−0.892325	−0.446163	−	0.894952i	$-0.647210\pi$
$419$	−18.2654	−0.892325	−0.446163	+	0.894952i	$0.647210\pi$
$420$	0	0
$421$	−8.51137	−0.414819	−0.207409	−	0.978254i	$-0.566503\pi$
$421$	−8.51137	−0.414819	−0.207409	+	0.978254i	$0.566503\pi$
$422$	26.9686	1.31281
$423$	0	0
$424$	−8.06428	−0.391636
$425$	22.2448	1.07903
$426$	0	0
$427$	32.7265	1.58375
$428$	8.39797	0.405931
$429$	0	0
$430$	19.1645	0.924195
$431$	−29.9998	−1.44504	−0.722520	−	0.691349i	$-0.757017\pi$
$431$	−29.9998	−1.44504	−0.722520	+	0.691349i	$0.757017\pi$
$432$	0	0
$433$	5.23836	0.251739	0.125870	−	0.992047i	$-0.459828\pi$
$433$	5.23836	0.251739	0.125870	+	0.992047i	$0.459828\pi$
$434$	17.3241	0.831584
$435$	0	0
$436$	−1.86495	−0.0893151
$437$	−4.97904	−0.238180
$438$	0	0
$439$	4.15170	0.198150	0.0990750	−	0.995080i	$-0.468412\pi$
$439$	4.15170	0.198150	0.0990750	+	0.995080i	$0.468412\pi$
$440$	8.08104	0.385249
$441$	0	0
$442$	15.1079	0.718608
$443$	0.571163	0.0271368	0.0135684	−	0.999908i	$-0.495681\pi$
$443$	0.571163	0.0271368	0.0135684	+	0.999908i	$0.495681\pi$
$444$	0	0
$445$	−3.37609	−0.160042
$446$	−54.9269	−2.60086
$447$	0	0
$448$	−43.1707	−2.03963
$449$	−1.55742	−0.0734992	−0.0367496	−	0.999325i	$-0.511700\pi$
$449$	−1.55742	−0.0734992	−0.0367496	+	0.999325i	$0.511700\pi$
$450$	0	0
$451$	−23.0184	−1.08390
$452$	−17.1956	−0.808813
$453$	0	0
$454$	−44.8869	−2.10665
$455$	4.83933	0.226871
$456$	0	0
$457$	17.4868	0.817999	0.408999	−	0.912535i	$-0.365878\pi$
$457$	17.4868	0.817999	0.408999	+	0.912535i	$0.365878\pi$
$458$	−38.8920	−1.81730
$459$	0	0
$460$	−3.72592	−0.173722
$461$	12.3367	0.574578	0.287289	−	0.957844i	$-0.407246\pi$
$461$	12.3367	0.574578	0.287289	+	0.957844i	$0.407246\pi$
$462$	0	0
$463$	36.6206	1.70191	0.850953	−	0.525242i	$-0.176025\pi$
$463$	36.6206	1.70191	0.850953	+	0.525242i	$0.176025\pi$
$464$	−2.38917	−0.110915
$465$	0	0
$466$	14.7513	0.683342
$467$	−32.9192	−1.52332	−0.761659	−	0.647978i	$-0.775615\pi$
$467$	−32.9192	−1.52332	−0.761659	+	0.647978i	$0.775615\pi$
$468$	0	0
$469$	30.1039	1.39007
$470$	1.44148	0.0664904
$471$	0	0
$472$	9.73215	0.447958
$473$	26.8546	1.23478
$474$	0	0
$475$	−14.7933	−0.678765
$476$	−70.8433	−3.24710
$477$	0	0
$478$	17.1890	0.786206
$479$	32.8624	1.50152	0.750761	−	0.660573i	$-0.229687\pi$
$479$	32.8624	1.50152	0.750761	+	0.660573i	$0.229687\pi$
$480$	0	0
$481$	−0.179635	−0.00819066
$482$	23.1224	1.05320
$483$	0	0
$484$	19.3268	0.878492
$485$	−16.2328	−0.737092
$486$	0	0
$487$	−14.1334	−0.640445	−0.320223	−	0.947342i	$-0.603758\pi$
$487$	−14.1334	−0.640445	−0.320223	+	0.947342i	$0.603758\pi$
$488$	−11.9696	−0.541839
$489$	0	0
$490$	−18.6212	−0.841220
$491$	−34.7675	−1.56904	−0.784519	−	0.620105i	$-0.787090\pi$
$491$	−34.7675	−1.56904	−0.784519	+	0.620105i	$0.787090\pi$
$492$	0	0
$493$	−7.48700	−0.337197
$494$	−10.0471	−0.452041
$495$	0	0
$496$	5.32585	0.239138
$497$	−17.0625	−0.765356
$498$	0	0
$499$	−5.83942	−0.261408	−0.130704	−	0.991421i	$-0.541724\pi$
$499$	−5.83942	−0.261408	−0.130704	+	0.991421i	$0.541724\pi$
$500$	−29.6997	−1.32821
$501$	0	0
$502$	40.0639	1.78814
$503$	7.00546	0.312358	0.156179	−	0.987729i	$-0.450082\pi$
$503$	7.00546	0.312358	0.156179	+	0.987729i	$0.450082\pi$
$504$	0	0
$505$	−22.7096	−1.01056
$506$	−9.21291	−0.409564
$507$	0	0
$508$	−54.6770	−2.42590
$509$	−19.2678	−0.854032	−0.427016	−	0.904244i	$-0.640435\pi$
$509$	−19.2678	−0.854032	−0.427016	+	0.904244i	$0.640435\pi$
$510$	0	0
$511$	−39.7006	−1.75625
$512$	−24.9074	−1.10076
$513$	0	0
$514$	13.9654	0.615986
$515$	−12.6565	−0.557710
$516$	0	0
$517$	2.01990	0.0888350
$518$	1.48638	0.0653077
$519$	0	0
$520$	−1.76997	−0.0776184
$521$	24.3594	1.06720	0.533602	−	0.845736i	$-0.320838\pi$
$521$	24.3594	1.06720	0.533602	+	0.845736i	$0.320838\pi$
$522$	0	0
$523$	16.2475	0.710454	0.355227	−	0.934780i	$-0.384404\pi$
$523$	16.2475	0.710454	0.355227	+	0.934780i	$0.384404\pi$
$524$	14.3867	0.628488
$525$	0	0
$526$	−12.7716	−0.556869
$527$	16.6897	0.727015
$528$	0	0
$529$	1.00000	0.0434783
$530$	−18.6531	−0.810238
$531$	0	0
$532$	47.1125	2.04259
$533$	5.04167	0.218379
$534$	0	0
$535$	4.57295	0.197706
$536$	−11.0104	−0.475578
$537$	0	0
$538$	−37.2423	−1.60563
$539$	−26.0933	−1.12392
$540$	0	0
$541$	−9.28757	−0.399304	−0.199652	−	0.979867i	$-0.563981\pi$
$541$	−9.28757	−0.399304	−0.199652	+	0.979867i	$0.563981\pi$
$542$	51.6232	2.21741
$543$	0	0
$544$	−58.2417	−2.49709
$545$	−1.01552	−0.0435003
$546$	0	0
$547$	−13.4050	−0.573157	−0.286579	−	0.958057i	$-0.592518\pi$
$547$	−13.4050	−0.573157	−0.286579	+	0.958057i	$0.592518\pi$
$548$	−6.03510	−0.257807
$549$	0	0
$550$	−27.3727	−1.16718
$551$	4.97904	0.212114
$552$	0	0
$553$	−28.2473	−1.20120
$554$	18.4799	0.785135
$555$	0	0
$556$	−25.5032	−1.08158
$557$	−26.1238	−1.10690	−0.553450	−	0.832883i	$-0.686689\pi$
$557$	−26.1238	−1.10690	−0.553450	+	0.832883i	$0.686689\pi$
$558$	0	0
$559$	−5.88191	−0.248778
$560$	−12.3101	−0.520197
$561$	0	0
$562$	−16.7566	−0.706836
$563$	−29.3871	−1.23852	−0.619260	−	0.785186i	$-0.712567\pi$
$563$	−29.3871	−1.23852	−0.619260	+	0.785186i	$0.712567\pi$
$564$	0	0
$565$	−9.36353	−0.393927
$566$	−35.4265	−1.48909
$567$	0	0
$568$	6.24055	0.261848
$569$	−16.9654	−0.711228	−0.355614	−	0.934633i	$-0.615728\pi$
$569$	−16.9654	−0.711228	−0.355614	+	0.934633i	$0.615728\pi$
$570$	0	0
$571$	−16.3163	−0.682818	−0.341409	−	0.939915i	$-0.610904\pi$
$571$	−16.3163	−0.682818	−0.341409	+	0.939915i	$0.610904\pi$
$572$	−10.5354	−0.440506
$573$	0	0
$574$	−41.7169	−1.74123
$575$	2.97112	0.123904
$576$	0	0
$577$	−11.6604	−0.485427	−0.242714	−	0.970098i	$-0.578038\pi$
$577$	−11.6604	−0.485427	−0.242714	+	0.970098i	$0.578038\pi$
$578$	−83.9076	−3.49010
$579$	0	0
$580$	3.72592	0.154710
$581$	60.2888	2.50120
$582$	0	0
$583$	−26.1380	−1.08252
$584$	14.5204	0.600858
$585$	0	0
$586$	52.1493	2.15427
$587$	4.57966	0.189023	0.0945114	−	0.995524i	$-0.469871\pi$
$587$	4.57966	0.189023	0.0945114	+	0.995524i	$0.469871\pi$
$588$	0	0
$589$	−11.0991	−0.457329
$590$	22.5109	0.926760
$591$	0	0
$592$	0.456949	0.0187805
$593$	27.8865	1.14516	0.572581	−	0.819848i	$-0.305942\pi$
$593$	27.8865	1.14516	0.572581	+	0.819848i	$0.305942\pi$
$594$	0	0
$595$	−38.5764	−1.58148
$596$	52.4292	2.14759
$597$	0	0
$598$	2.01788	0.0825173
$599$	10.3105	0.421277	0.210639	−	0.977564i	$-0.432446\pi$
$599$	10.3105	0.421277	0.210639	+	0.977564i	$0.432446\pi$
$600$	0	0
$601$	4.82829	0.196950	0.0984751	−	0.995140i	$-0.468604\pi$
$601$	4.82829	0.196950	0.0984751	+	0.995140i	$0.468604\pi$
$602$	48.6694	1.98362
$603$	0	0
$604$	−57.8487	−2.35383
$605$	10.5240	0.427863
$606$	0	0
$607$	18.8619	0.765581	0.382790	−	0.923835i	$-0.374963\pi$
$607$	18.8619	0.765581	0.382790	+	0.923835i	$0.374963\pi$
$608$	38.7322	1.57080
$609$	0	0
$610$	−27.6863	−1.12099
$611$	−0.442413	−0.0178981
$612$	0	0
$613$	4.42684	0.178798	0.0893991	−	0.995996i	$-0.471505\pi$
$613$	4.42684	0.178798	0.0893991	+	0.995996i	$0.471505\pi$
$614$	65.3193	2.63607
$615$	0	0
$616$	20.5223	0.826866
$617$	−17.2207	−0.693280	−0.346640	−	0.937998i	$-0.612677\pi$
$617$	−17.2207	−0.693280	−0.346640	+	0.937998i	$0.612677\pi$
$618$	0	0
$619$	−39.0725	−1.57046	−0.785228	−	0.619207i	$-0.787454\pi$
$619$	−39.0725	−1.57046	−0.785228	+	0.619207i	$0.787454\pi$
$620$	−8.30567	−0.333564
$621$	0	0
$622$	−40.5076	−1.62421
$623$	−8.57377	−0.343501
$624$	0	0
$625$	−1.31681	−0.0526723
$626$	50.2893	2.00997
$627$	0	0
$628$	45.0028	1.79581
$629$	1.43195	0.0570955
$630$	0	0
$631$	31.4897	1.25359	0.626793	−	0.779186i	$-0.284367\pi$
$631$	31.4897	1.25359	0.626793	+	0.779186i	$0.284367\pi$
$632$	10.3314	0.410959
$633$	0	0
$634$	−4.47335	−0.177659
$635$	−29.7733	−1.18152
$636$	0	0
$637$	5.71516	0.226443
$638$	9.21291	0.364742
$639$	0	0
$640$	14.3613	0.567681
$641$	22.9920	0.908130	0.454065	−	0.890969i	$-0.349973\pi$
$641$	22.9920	0.908130	0.454065	+	0.890969i	$0.349973\pi$
$642$	0	0
$643$	10.1036	0.398446	0.199223	−	0.979954i	$-0.436158\pi$
$643$	10.1036	0.398446	0.199223	+	0.979954i	$0.436158\pi$
$644$	−9.46218	−0.372862
$645$	0	0
$646$	80.0897	3.15109
$647$	−27.7945	−1.09271	−0.546357	−	0.837553i	$-0.683986\pi$
$647$	−27.7945	−1.09271	−0.546357	+	0.837553i	$0.683986\pi$
$648$	0	0
$649$	31.5439	1.23820
$650$	5.99537	0.235158
$651$	0	0
$652$	47.6550	1.86631
$653$	−24.8044	−0.970670	−0.485335	−	0.874328i	$-0.661302\pi$
$653$	−24.8044	−0.970670	−0.485335	+	0.874328i	$0.661302\pi$
$654$	0	0
$655$	7.83402	0.306101
$656$	−12.8248	−0.500725
$657$	0	0
$658$	3.66071	0.142710
$659$	14.3327	0.558322	0.279161	−	0.960244i	$-0.409944\pi$
$659$	14.3327	0.558322	0.279161	+	0.960244i	$0.409944\pi$
$660$	0	0
$661$	31.8919	1.24045	0.620226	−	0.784423i	$-0.287041\pi$
$661$	31.8919	1.24045	0.620226	+	0.784423i	$0.287041\pi$
$662$	32.0544	1.24583
$663$	0	0
$664$	−22.0505	−0.855724
$665$	25.6542	0.994828
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	−49.9327	−1.93195
$669$	0	0
$670$	−25.4676	−0.983901
$671$	−38.7960	−1.49770
$672$	0	0
$673$	−16.2702	−0.627172	−0.313586	−	0.949560i	$-0.601530\pi$
$673$	−16.2702	−0.627172	−0.313586	+	0.949560i	$0.601530\pi$
$674$	−28.3835	−1.09329
$675$	0	0
$676$	−31.6979	−1.21915
$677$	42.9053	1.64898	0.824492	−	0.565874i	$-0.191461\pi$
$677$	42.9053	1.64898	0.824492	+	0.565874i	$0.191461\pi$
$678$	0	0
$679$	−41.2240	−1.58203
$680$	14.1092	0.541063
$681$	0	0
$682$	−20.5370	−0.786404
$683$	−25.5812	−0.978838	−0.489419	−	0.872049i	$-0.662791\pi$
$683$	−25.5812	−0.978838	−0.489419	+	0.872049i	$0.662791\pi$
$684$	0	0
$685$	−3.28630	−0.125563
$686$	7.11148	0.271518
$687$	0	0
$688$	14.9622	0.570427
$689$	5.72494	0.218103
$690$	0	0
$691$	37.5806	1.42963	0.714816	−	0.699312i	$-0.246510\pi$
$691$	37.5806	1.42963	0.714816	+	0.699312i	$0.246510\pi$
$692$	−12.5975	−0.478885
$693$	0	0
$694$	−53.8637	−2.04464
$695$	−13.8873	−0.526774
$696$	0	0
$697$	−40.1893	−1.52228
$698$	32.9872	1.24858
$699$	0	0
$700$	−28.1133	−1.06258
$701$	5.02646	0.189847	0.0949234	−	0.995485i	$-0.469739\pi$
$701$	5.02646	0.189847	0.0949234	+	0.995485i	$0.469739\pi$
$702$	0	0
$703$	−0.952281	−0.0359160
$704$	51.1772	1.92881
$705$	0	0
$706$	16.0010	0.602205
$707$	−57.6724	−2.16899
$708$	0	0
$709$	5.13423	0.192820	0.0964101	−	0.995342i	$-0.469264\pi$
$709$	5.13423	0.192820	0.0964101	+	0.995342i	$0.469264\pi$
$710$	14.4347	0.541724
$711$	0	0
$712$	3.13583	0.117520
$713$	2.22916	0.0834827
$714$	0	0
$715$	−5.73684	−0.214546
$716$	19.5887	0.732064
$717$	0	0
$718$	−62.8019	−2.34375
$719$	−3.09716	−0.115505	−0.0577523	−	0.998331i	$-0.518393\pi$
$719$	−3.09716	−0.115505	−0.0577523	+	0.998331i	$0.518393\pi$
$720$	0	0
$721$	−32.1418	−1.19702
$722$	−12.4412	−0.463015
$723$	0	0
$724$	24.0336	0.893201
$725$	−2.97112	−0.110345
$726$	0	0
$727$	51.8440	1.92279	0.961393	−	0.275177i	$-0.0887366\pi$
$727$	51.8440	1.92279	0.961393	+	0.275177i	$0.0887366\pi$
$728$	−4.49495	−0.166594
$729$	0	0
$730$	33.5864	1.24309
$731$	46.8872	1.73418
$732$	0	0
$733$	−47.7628	−1.76416	−0.882080	−	0.471099i	$-0.843858\pi$
$733$	−47.7628	−1.76416	−0.882080	+	0.471099i	$0.843858\pi$
$734$	73.7093	2.72066
$735$	0	0
$736$	−7.77905	−0.286739
$737$	−35.6870	−1.31455
$738$	0	0
$739$	10.8623	0.399576	0.199788	−	0.979839i	$-0.435975\pi$
$739$	10.8623	0.399576	0.199788	+	0.979839i	$0.435975\pi$
$740$	−0.712612	−0.0261961
$741$	0	0
$742$	−47.3706	−1.73903
$743$	26.6848	0.978970	0.489485	−	0.872012i	$-0.337185\pi$
$743$	26.6848	0.978970	0.489485	+	0.872012i	$0.337185\pi$
$744$	0	0
$745$	28.5493	1.04597
$746$	−45.1857	−1.65437
$747$	0	0
$748$	83.9819	3.07068
$749$	11.6133	0.424340
$750$	0	0
$751$	−23.0568	−0.841354	−0.420677	−	0.907210i	$-0.638207\pi$
$751$	−23.0568	−0.841354	−0.420677	+	0.907210i	$0.638207\pi$
$752$	1.12539	0.0410389
$753$	0	0
$754$	−2.01788	−0.0734869
$755$	−31.5004	−1.14642
$756$	0	0
$757$	−7.36545	−0.267702	−0.133851	−	0.991001i	$-0.542734\pi$
$757$	−7.36545	−0.267702	−0.133851	+	0.991001i	$0.542734\pi$
$758$	59.6771	2.16757
$759$	0	0
$760$	−9.38296	−0.340356
$761$	−29.2835	−1.06153	−0.530764	−	0.847520i	$-0.678095\pi$
$761$	−29.2835	−1.06153	−0.530764	+	0.847520i	$0.678095\pi$
$762$	0	0
$763$	−2.57898	−0.0933655
$764$	−30.6674	−1.10951
$765$	0	0
$766$	−25.3375	−0.915482
$767$	−6.90898	−0.249469
$768$	0	0
$769$	−33.3173	−1.20145	−0.600727	−	0.799454i	$-0.705122\pi$
$769$	−33.3173	−1.20145	−0.600727	+	0.799454i	$0.705122\pi$
$770$	47.4690	1.71066
$771$	0	0
$772$	−15.1604	−0.545634
$773$	−8.75575	−0.314922	−0.157461	−	0.987525i	$-0.550331\pi$
$773$	−8.75575	−0.314922	−0.157461	+	0.987525i	$0.550331\pi$
$774$	0	0
$775$	6.62311	0.237909
$776$	15.0776	0.541253
$777$	0	0
$778$	−30.7500	−1.10244
$779$	26.7269	0.957590
$780$	0	0
$781$	20.2269	0.723774
$782$	−16.0854	−0.575212
$783$	0	0
$784$	−14.5380	−0.519214
$785$	24.5054	0.874635
$786$	0	0
$787$	8.42865	0.300449	0.150224	−	0.988652i	$-0.452000\pi$
$787$	8.42865	0.300449	0.150224	+	0.988652i	$0.452000\pi$
$788$	−52.0366	−1.85373
$789$	0	0
$790$	23.8969	0.850215
$791$	−23.7792	−0.845492
$792$	0	0
$793$	8.49739	0.301751
$794$	12.6846	0.450160
$795$	0	0
$796$	17.6132	0.624284
$797$	−49.8884	−1.76714	−0.883568	−	0.468303i	$-0.844866\pi$
$797$	−49.8884	−1.76714	−0.883568	+	0.468303i	$0.844866\pi$
$798$	0	0
$799$	3.52666	0.124764
$800$	−23.1125	−0.817151
$801$	0	0
$802$	44.4164	1.56840
$803$	47.0635	1.66084
$804$	0	0
$805$	−5.15245	−0.181600
$806$	4.49818	0.158442
$807$	0	0
$808$	21.0935	0.742067
$809$	14.5866	0.512839	0.256420	−	0.966566i	$-0.417457\pi$
$809$	14.5866	0.512839	0.256420	+	0.966566i	$0.417457\pi$
$810$	0	0
$811$	−24.9937	−0.877646	−0.438823	−	0.898574i	$-0.644604\pi$
$811$	−24.9937	−0.877646	−0.438823	+	0.898574i	$0.644604\pi$
$812$	9.46218	0.332057
$813$	0	0
$814$	−1.76204	−0.0617596
$815$	25.9496	0.908975
$816$	0	0
$817$	−31.1811	−1.09089
$818$	−5.71656	−0.199875
$819$	0	0
$820$	20.0003	0.698440
$821$	1.14274	0.0398818	0.0199409	−	0.999801i	$-0.493652\pi$
$821$	1.14274	0.0398818	0.0199409	+	0.999801i	$0.493652\pi$
$822$	0	0
$823$	45.5226	1.58682	0.793409	−	0.608688i	$-0.208304\pi$
$823$	45.5226	1.58682	0.793409	+	0.608688i	$0.208304\pi$
$824$	11.7558	0.409532
$825$	0	0
$826$	57.1678	1.98912
$827$	−53.4312	−1.85799	−0.928993	−	0.370097i	$-0.879324\pi$
$827$	−53.4312	−1.85799	−0.928993	+	0.370097i	$0.879324\pi$
$828$	0	0
$829$	−17.8514	−0.620004	−0.310002	−	0.950736i	$-0.600330\pi$
$829$	−17.8514	−0.620004	−0.310002	+	0.950736i	$0.600330\pi$
$830$	−51.0038	−1.77037
$831$	0	0
$832$	−11.2092	−0.388610
$833$	−45.5579	−1.57849
$834$	0	0
$835$	−27.1899	−0.940945
$836$	−55.8501	−1.93161
$837$	0	0
$838$	39.2423	1.35560
$839$	18.4282	0.636210	0.318105	−	0.948055i	$-0.396953\pi$
$839$	18.4282	0.636210	0.318105	+	0.948055i	$0.396953\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	18.2862	0.630184
$843$	0	0
$844$	−32.8353	−1.13024
$845$	−17.2605	−0.593779
$846$	0	0
$847$	26.7264	0.918331
$848$	−14.5629	−0.500091
$849$	0	0
$850$	−47.7916	−1.63924
$851$	0.191258	0.00655624
$852$	0	0
$853$	−30.0943	−1.03041	−0.515205	−	0.857067i	$-0.672284\pi$
$853$	−30.0943	−1.03041	−0.515205	+	0.857067i	$0.672284\pi$
$854$	−70.3110	−2.40599
$855$	0	0
$856$	−4.24752	−0.145177
$857$	24.9732	0.853070	0.426535	−	0.904471i	$-0.359734\pi$
$857$	24.9732	0.853070	0.426535	+	0.904471i	$0.359734\pi$
$858$	0	0
$859$	22.8073	0.778175	0.389087	−	0.921201i	$-0.372790\pi$
$859$	22.8073	0.778175	0.389087	+	0.921201i	$0.372790\pi$
$860$	−23.3335	−0.795665
$861$	0	0
$862$	64.4529	2.19527
$863$	−26.3898	−0.898319	−0.449159	−	0.893452i	$-0.648276\pi$
$863$	−26.3898	−0.898319	−0.449159	+	0.893452i	$0.648276\pi$
$864$	0	0
$865$	−6.85973	−0.233238
$866$	−11.2543	−0.382437
$867$	0	0
$868$	−21.0927	−0.715933
$869$	33.4860	1.13594
$870$	0	0
$871$	7.81644	0.264850
$872$	0.943256	0.0319427
$873$	0	0
$874$	10.6972	0.361837
$875$	−41.0708	−1.38845
$876$	0	0
$877$	17.9650	0.606636	0.303318	−	0.952889i	$-0.401906\pi$
$877$	17.9650	0.606636	0.303318	+	0.952889i	$0.401906\pi$
$878$	−8.91970	−0.301025
$879$	0	0
$880$	14.5931	0.491935
$881$	−34.8937	−1.17560	−0.587799	−	0.809007i	$-0.700005\pi$
$881$	−34.8937	−1.17560	−0.587799	+	0.809007i	$0.700005\pi$
$882$	0	0
$883$	24.3100	0.818095	0.409048	−	0.912513i	$-0.365861\pi$
$883$	24.3100	0.818095	0.409048	+	0.912513i	$0.365861\pi$
$884$	−18.3944	−0.618669
$885$	0	0
$886$	−1.22711	−0.0412256
$887$	15.3916	0.516799	0.258400	−	0.966038i	$-0.416805\pi$
$887$	15.3916	0.516799	0.258400	+	0.966038i	$0.416805\pi$
$888$	0	0
$889$	−75.6110	−2.53591
$890$	7.25333	0.243132
$891$	0	0
$892$	66.8755	2.23916
$893$	−2.34532	−0.0784831
$894$	0	0
$895$	10.6666	0.356547
$896$	36.4714	1.21842
$897$	0	0
$898$	3.34603	0.111658
$899$	−2.22916	−0.0743467
$900$	0	0
$901$	−45.6359	−1.52035
$902$	49.4538	1.64663
$903$	0	0
$904$	8.69718	0.289264
$905$	13.0870	0.435027
$906$	0	0
$907$	−48.0004	−1.59383	−0.796913	−	0.604094i	$-0.793535\pi$
$907$	−48.0004	−1.59383	−0.796913	+	0.604094i	$0.793535\pi$
$908$	54.6514	1.81367
$909$	0	0
$910$	−10.3970	−0.344658
$911$	45.0598	1.49290	0.746448	−	0.665444i	$-0.231758\pi$
$911$	45.0598	1.49290	0.746448	+	0.665444i	$0.231758\pi$
$912$	0	0
$913$	−71.4700	−2.36531
$914$	−37.5694	−1.24269
$915$	0	0
$916$	47.3523	1.56457
$917$	19.8950	0.656989
$918$	0	0
$919$	−27.3705	−0.902870	−0.451435	−	0.892304i	$-0.649088\pi$
$919$	−27.3705	−0.902870	−0.451435	+	0.892304i	$0.649088\pi$
$920$	1.88449	0.0621299
$921$	0	0
$922$	−26.5047	−0.872886
$923$	−4.43025	−0.145823
$924$	0	0
$925$	0.568251	0.0186840
$926$	−78.6774	−2.58550
$927$	0	0
$928$	7.77905	0.255360
$929$	−23.4108	−0.768084	−0.384042	−	0.923316i	$-0.625468\pi$
$929$	−23.4108	−0.768084	−0.384042	+	0.923316i	$0.625468\pi$
$930$	0	0
$931$	30.2971	0.992948
$932$	−17.9603	−0.588308
$933$	0	0
$934$	70.7249	2.31419
$935$	45.7307	1.49556
$936$	0	0
$937$	−16.8103	−0.549168	−0.274584	−	0.961563i	$-0.588540\pi$
$937$	−16.8103	−0.549168	−0.274584	+	0.961563i	$0.588540\pi$
$938$	−64.6765	−2.11176
$939$	0	0
$940$	−1.75505	−0.0572434
$941$	−42.4050	−1.38236	−0.691181	−	0.722682i	$-0.742909\pi$
$941$	−42.4050	−1.38236	−0.691181	+	0.722682i	$0.742909\pi$
$942$	0	0
$943$	−5.36788	−0.174802
$944$	17.5748	0.572010
$945$	0	0
$946$	−57.6956	−1.87585
$947$	−37.7128	−1.22550	−0.612751	−	0.790276i	$-0.709937\pi$
$947$	−37.7128	−1.22550	−0.612751	+	0.790276i	$0.709937\pi$
$948$	0	0
$949$	−10.3082	−0.334619
$950$	31.7826	1.03116
$951$	0	0
$952$	35.8311	1.16129
$953$	−47.6248	−1.54272	−0.771360	−	0.636399i	$-0.780423\pi$
$953$	−47.6248	−1.54272	−0.771360	+	0.636399i	$0.780423\pi$
$954$	0	0
$955$	−16.6993	−0.540378
$956$	−20.9282	−0.676866
$957$	0	0
$958$	−70.6030	−2.28108
$959$	−8.34574	−0.269498
$960$	0	0
$961$	−26.0308	−0.839705
$962$	0.385936	0.0124431
$963$	0	0
$964$	−28.1523	−0.906726
$965$	−8.25529	−0.265747
$966$	0	0
$967$	4.85286	0.156057	0.0780287	−	0.996951i	$-0.475137\pi$
$967$	4.85286	0.156057	0.0780287	+	0.996951i	$0.475137\pi$
$968$	−9.77511	−0.314184
$969$	0	0
$970$	34.8751	1.11977
$971$	31.9786	1.02624	0.513122	−	0.858316i	$-0.328489\pi$
$971$	31.9786	1.02624	0.513122	+	0.858316i	$0.328489\pi$
$972$	0	0
$973$	−35.2675	−1.13062
$974$	30.3648	0.972950
$975$	0	0
$976$	−21.6153	−0.691889
$977$	−24.3468	−0.778925	−0.389462	−	0.921042i	$-0.627339\pi$
$977$	−24.3468	−0.778925	−0.389462	+	0.921042i	$0.627339\pi$
$978$	0	0
$979$	10.1639	0.324838
$980$	22.6720	0.724230
$981$	0	0
$982$	74.6961	2.38365
$983$	−1.54152	−0.0491667	−0.0245834	−	0.999698i	$-0.507826\pi$
$983$	−1.54152	−0.0491667	−0.0245834	+	0.999698i	$0.507826\pi$
$984$	0	0
$985$	−28.3356	−0.902846
$986$	16.0854	0.512263
$987$	0	0
$988$	12.2327	0.389174
$989$	6.26248	0.199135
$990$	0	0
$991$	8.91405	0.283164	0.141582	−	0.989927i	$-0.454781\pi$
$991$	8.91405	0.283164	0.141582	+	0.989927i	$0.454781\pi$
$992$	−17.3407	−0.550569
$993$	0	0
$994$	36.6577	1.16271
$995$	9.59094	0.304053
$996$	0	0
$997$	43.7974	1.38708	0.693538	−	0.720420i	$-0.256051\pi$
$997$	43.7974	1.38708	0.693538	+	0.720420i	$0.256051\pi$
$998$	12.5456	0.397126
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.q.1.3		16
3.2	odd	2		667.2.a.d.1.14	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
667.2.a.d.1.14	✓	16	3.2	odd	2
6003.2.a.q.1.3		16	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)

\(f(q)\)	\(=\)	\(q-2.14844 q^{2} +2.61580 q^{4} +1.42439 q^{5} +3.61731 q^{7} -1.32302 q^{8} +O(q^{10})\)	\(q-2.14844 q^{2} +2.61580 q^{4} +1.42439 q^{5} +3.61731 q^{7} -1.32302 q^{8} -3.06021 q^{10} -4.28818 q^{11} +0.939230 q^{13} -7.77158 q^{14} -2.38917 q^{16} -7.48700 q^{17} +4.97904 q^{19} +3.72592 q^{20} +9.21291 q^{22} -1.00000 q^{23} -2.97112 q^{25} -2.01788 q^{26} +9.46218 q^{28} +1.00000 q^{29} -2.22916 q^{31} +7.77905 q^{32} +16.0854 q^{34} +5.15245 q^{35} -0.191258 q^{37} -10.6972 q^{38} -1.88449 q^{40} +5.36788 q^{41} -6.26248 q^{43} -11.2170 q^{44} +2.14844 q^{46} -0.471038 q^{47} +6.08494 q^{49} +6.38329 q^{50} +2.45684 q^{52} +6.09535 q^{53} -6.10802 q^{55} -4.78578 q^{56} -2.14844 q^{58} -7.35600 q^{59} +9.04719 q^{61} +4.78922 q^{62} -11.9345 q^{64} +1.33783 q^{65} +8.32218 q^{67} -19.5845 q^{68} -11.0697 q^{70} -4.71689 q^{71} -10.9752 q^{73} +0.410907 q^{74} +13.0242 q^{76} -15.5117 q^{77} -7.80891 q^{79} -3.40311 q^{80} -11.5326 q^{82} +16.6667 q^{83} -10.6644 q^{85} +13.4546 q^{86} +5.67335 q^{88} -2.37020 q^{89} +3.39749 q^{91} -2.61580 q^{92} +1.01200 q^{94} +7.09207 q^{95} -11.3963 q^{97} -13.0731 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 3 q^{2} + 21 q^{4} - 16 q^{5} + q^{7} - 9 q^{8}+O(q^{10}) \) 16 * q - 3 * q^2 + 21 * q^4 - 16 * q^5 + q^7 - 9 * q^8	\( 16 q - 3 q^{2} + 21 q^{4} - 16 q^{5} + q^{7} - 9 q^{8} - 14 q^{10} - 4 q^{11} + 15 q^{13} - 8 q^{14} + 23 q^{16} - 20 q^{17} - 4 q^{19} - 25 q^{20} + 13 q^{22} - 16 q^{23} + 30 q^{25} - 25 q^{26} - 13 q^{28} + 16 q^{29} + 19 q^{32} - 23 q^{34} - 5 q^{35} + 5 q^{37} - 38 q^{38} - 20 q^{40} - 7 q^{41} - 17 q^{43} + 21 q^{44} + 3 q^{46} - 29 q^{47} + 31 q^{49} + 44 q^{50} + 20 q^{52} - 63 q^{53} + q^{55} + 19 q^{56} - 3 q^{58} - 11 q^{59} - 33 q^{62} + 29 q^{64} - 53 q^{65} - 13 q^{67} - 63 q^{68} - 46 q^{70} + 23 q^{71} - 38 q^{73} + 47 q^{74} - 56 q^{76} - 97 q^{77} - 27 q^{79} - 8 q^{80} + 9 q^{82} - 36 q^{83} + 6 q^{85} + 11 q^{86} - 24 q^{88} + 16 q^{89} - 47 q^{91} - 21 q^{92} + 37 q^{94} + 12 q^{95} - 30 q^{97} + 27 q^{98}+O(q^{100}) \) 16 * q - 3 * q^2 + 21 * q^4 - 16 * q^5 + q^7 - 9 * q^8 - 14 * q^10 - 4 * q^11 + 15 * q^13 - 8 * q^14 + 23 * q^16 - 20 * q^17 - 4 * q^19 - 25 * q^20 + 13 * q^22 - 16 * q^23 + 30 * q^25 - 25 * q^26 - 13 * q^28 + 16 * q^29 + 19 * q^32 - 23 * q^34 - 5 * q^35 + 5 * q^37 - 38 * q^38 - 20 * q^40 - 7 * q^41 - 17 * q^43 + 21 * q^44 + 3 * q^46 - 29 * q^47 + 31 * q^49 + 44 * q^50 + 20 * q^52 - 63 * q^53 + q^55 + 19 * q^56 - 3 * q^58 - 11 * q^59 - 33 * q^62 + 29 * q^64 - 53 * q^65 - 13 * q^67 - 63 * q^68 - 46 * q^70 + 23 * q^71 - 38 * q^73 + 47 * q^74 - 56 * q^76 - 97 * q^77 - 27 * q^79 - 8 * q^80 + 9 * q^82 - 36 * q^83 + 6 * q^85 + 11 * q^86 - 24 * q^88 + 16 * q^89 - 47 * q^91 - 21 * q^92 + 37 * q^94 + 12 * q^95 - 30 * q^97 + 27 * q^98

Introduction

L-functions

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