LMFDB - Embedded newform 6003.2.a.w.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6003, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6003.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6003 = 3^{2} \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6003.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$47.9341963334$
Analytic rank:	$0$
Dimension:	$30$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Character	$\chi$	$=$	6003.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.480943 q^{2} -1.76869 q^{4} -1.51137 q^{5} +3.82960 q^{7} +1.81253 q^{8} +O(q^{10})$	\(q-0.480943 q^{2} -1.76869 q^{4} -1.51137 q^{5} +3.82960 q^{7} +1.81253 q^{8} +0.726880 q^{10} -4.55830 q^{11} -2.03350 q^{13} -1.84182 q^{14} +2.66567 q^{16} -3.09482 q^{17} -5.07355 q^{19} +2.67314 q^{20} +2.19228 q^{22} -1.00000 q^{23} -2.71578 q^{25} +0.977997 q^{26} -6.77339 q^{28} +1.00000 q^{29} +3.30853 q^{31} -4.90709 q^{32} +1.48843 q^{34} -5.78793 q^{35} +9.63233 q^{37} +2.44008 q^{38} -2.73939 q^{40} -7.83947 q^{41} +1.74208 q^{43} +8.06224 q^{44} +0.480943 q^{46} -6.78144 q^{47} +7.66585 q^{49} +1.30613 q^{50} +3.59664 q^{52} +8.59890 q^{53} +6.88926 q^{55} +6.94125 q^{56} -0.480943 q^{58} -2.86760 q^{59} -3.14004 q^{61} -1.59121 q^{62} -2.97131 q^{64} +3.07336 q^{65} -3.13434 q^{67} +5.47379 q^{68} +2.78366 q^{70} +5.22684 q^{71} +14.0074 q^{73} -4.63260 q^{74} +8.97355 q^{76} -17.4565 q^{77} -2.01553 q^{79} -4.02880 q^{80} +3.77034 q^{82} -0.169808 q^{83} +4.67740 q^{85} -0.837841 q^{86} -8.26204 q^{88} +4.44668 q^{89} -7.78750 q^{91} +1.76869 q^{92} +3.26148 q^{94} +7.66798 q^{95} -12.6221 q^{97} -3.68683 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q + q^{2} + 37 q^{4} + 10 q^{7} + 6 q^{8}+O(q^{10}) $ 30 * q + q^2 + 37 * q^4 + 10 * q^7 + 6 * q^8	$ 30 q + q^{2} + 37 q^{4} + 10 q^{7} + 6 q^{8} + 8 q^{10} + 36 q^{13} + 7 q^{14} + 47 q^{16} + 18 q^{17} + 16 q^{19} + 25 q^{22} - 30 q^{23} + 56 q^{25} + 11 q^{26} + 27 q^{28} + 30 q^{29} + 14 q^{31} - 7 q^{32} + 3 q^{34} - 22 q^{35} + 40 q^{37} + 6 q^{38} + 30 q^{40} + 14 q^{41} + 34 q^{43} + 5 q^{44} - q^{46} - 2 q^{47} + 74 q^{49} - 21 q^{50} + 71 q^{52} + 16 q^{53} + 22 q^{55} + 14 q^{56} + q^{58} - 32 q^{59} + 46 q^{61} + 20 q^{62} + 68 q^{64} + 12 q^{65} + 14 q^{67} + 27 q^{68} - 32 q^{71} + 50 q^{73} - 26 q^{74} + 56 q^{76} + 34 q^{77} + 16 q^{79} + 2 q^{80} + 38 q^{82} - 14 q^{83} + 38 q^{85} + 10 q^{86} + 40 q^{88} - 2 q^{89} + 32 q^{91} - 37 q^{92} + 29 q^{94} - 28 q^{95} + 56 q^{97} + 8 q^{98}+O(q^{100}) $ 30 * q + q^2 + 37 * q^4 + 10 * q^7 + 6 * q^8 + 8 * q^10 + 36 * q^13 + 7 * q^14 + 47 * q^16 + 18 * q^17 + 16 * q^19 + 25 * q^22 - 30 * q^23 + 56 * q^25 + 11 * q^26 + 27 * q^28 + 30 * q^29 + 14 * q^31 - 7 * q^32 + 3 * q^34 - 22 * q^35 + 40 * q^37 + 6 * q^38 + 30 * q^40 + 14 * q^41 + 34 * q^43 + 5 * q^44 - q^46 - 2 * q^47 + 74 * q^49 - 21 * q^50 + 71 * q^52 + 16 * q^53 + 22 * q^55 + 14 * q^56 + q^58 - 32 * q^59 + 46 * q^61 + 20 * q^62 + 68 * q^64 + 12 * q^65 + 14 * q^67 + 27 * q^68 - 32 * q^71 + 50 * q^73 - 26 * q^74 + 56 * q^76 + 34 * q^77 + 16 * q^79 + 2 * q^80 + 38 * q^82 - 14 * q^83 + 38 * q^85 + 10 * q^86 + 40 * q^88 - 2 * q^89 + 32 * q^91 - 37 * q^92 + 29 * q^94 - 28 * q^95 + 56 * q^97 + 8 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−0.480943	−0.340078	−0.170039	−	0.985437i	$-0.554389\pi$
$2$	−0.480943	−0.340078	−0.170039	+	0.985437i	$0.554389\pi$
$3$	0	0
$3$	0	0
$4$	−1.76869	−0.884347
$5$	−1.51137	−0.675903	−0.337952	−	0.941164i	$-0.609734\pi$
$5$	−1.51137	−0.675903	−0.337952	+	0.941164i	$0.609734\pi$
$6$	0	0
$7$	3.82960	1.44745	0.723727	−	0.690087i	$-0.242428\pi$
$7$	3.82960	1.44745	0.723727	+	0.690087i	$0.242428\pi$
$8$	1.81253	0.640825
$9$	0	0
$10$	0.726880	0.229860
$11$	−4.55830	−1.37438	−0.687190	−	0.726478i	$-0.741156\pi$
$11$	−4.55830	−1.37438	−0.687190	+	0.726478i	$0.741156\pi$
$12$	0	0
$13$	−2.03350	−0.563992	−0.281996	−	0.959416i	$-0.590996\pi$
$13$	−2.03350	−0.563992	−0.281996	+	0.959416i	$0.590996\pi$
$14$	−1.84182	−0.492247
$15$	0	0
$16$	2.66567	0.666417
$17$	−3.09482	−0.750604	−0.375302	−	0.926903i	$-0.622461\pi$
$17$	−3.09482	−0.750604	−0.375302	+	0.926903i	$0.622461\pi$
$18$	0	0
$19$	−5.07355	−1.16395	−0.581976	−	0.813206i	$-0.697720\pi$
$19$	−5.07355	−1.16395	−0.581976	+	0.813206i	$0.697720\pi$
$20$	2.67314	0.597733
$21$	0	0
$22$	2.19228	0.467396
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−2.71578	−0.543155
$26$	0.977997	0.191801
$27$	0	0
$28$	−6.77339	−1.28005
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	3.30853	0.594229	0.297115	−	0.954842i	$-0.403976\pi$
$31$	3.30853	0.594229	0.297115	+	0.954842i	$0.403976\pi$
$32$	−4.90709	−0.867458
$33$	0	0
$34$	1.48843	0.255264
$35$	−5.78793	−0.978338
$36$	0	0
$37$	9.63233	1.58355	0.791773	−	0.610816i	$-0.209158\pi$
$37$	9.63233	1.58355	0.791773	+	0.610816i	$0.209158\pi$
$38$	2.44008	0.395834
$39$	0	0
$40$	−2.73939	−0.433135
$41$	−7.83947	−1.22432	−0.612160	−	0.790734i	$-0.709699\pi$
$41$	−7.83947	−1.22432	−0.612160	+	0.790734i	$0.709699\pi$
$42$	0	0
$43$	1.74208	0.265665	0.132832	−	0.991139i	$-0.457593\pi$
$43$	1.74208	0.265665	0.132832	+	0.991139i	$0.457593\pi$
$44$	8.06224	1.21543
$45$	0	0
$46$	0.480943	0.0709111
$47$	−6.78144	−0.989175	−0.494587	−	0.869128i	$-0.664681\pi$
$47$	−6.78144	−0.989175	−0.494587	+	0.869128i	$0.664681\pi$
$48$	0	0
$49$	7.66585	1.09512
$50$	1.30613	0.184715
$51$	0	0
$52$	3.59664	0.498764
$53$	8.59890	1.18115	0.590575	−	0.806983i	$-0.298901\pi$
$53$	8.59890	1.18115	0.590575	+	0.806983i	$0.298901\pi$
$54$	0	0
$55$	6.88926	0.928948
$56$	6.94125	0.927564
$57$	0	0
$58$	−0.480943	−0.0631509
$59$	−2.86760	−0.373330	−0.186665	−	0.982424i	$-0.559768\pi$
$59$	−2.86760	−0.373330	−0.186665	+	0.982424i	$0.559768\pi$
$60$	0	0
$61$	−3.14004	−0.402041	−0.201021	−	0.979587i	$-0.564426\pi$
$61$	−3.14004	−0.402041	−0.201021	+	0.979587i	$0.564426\pi$
$62$	−1.59121	−0.202084
$63$	0	0
$64$	−2.97131	−0.371413
$65$	3.07336	0.381204
$66$	0	0
$67$	−3.13434	−0.382921	−0.191460	−	0.981500i	$-0.561322\pi$
$67$	−3.13434	−0.382921	−0.191460	+	0.981500i	$0.561322\pi$
$68$	5.47379	0.663794
$69$	0	0
$70$	2.78366	0.332711
$71$	5.22684	0.620312	0.310156	−	0.950686i	$-0.399619\pi$
$71$	5.22684	0.620312	0.310156	+	0.950686i	$0.399619\pi$
$72$	0	0
$73$	14.0074	1.63944	0.819721	−	0.572763i	$-0.194128\pi$
$73$	14.0074	1.63944	0.819721	+	0.572763i	$0.194128\pi$
$74$	−4.63260	−0.538529
$75$	0	0
$76$	8.97355	1.02934
$77$	−17.4565	−1.98935
$78$	0	0
$79$	−2.01553	−0.226764	−0.113382	−	0.993551i	$-0.536168\pi$
$79$	−2.01553	−0.226764	−0.113382	+	0.993551i	$0.536168\pi$
$80$	−4.02880	−0.450433
$81$	0	0
$82$	3.77034	0.416364
$83$	−0.169808	−0.0186388	−0.00931941	−	0.999957i	$-0.502967\pi$
$83$	−0.169808	−0.0186388	−0.00931941	+	0.999957i	$0.502967\pi$
$84$	0	0
$85$	4.67740	0.507335
$86$	−0.837841	−0.0903467
$87$	0	0
$88$	−8.26204	−0.880737
$89$	4.44668	0.471347	0.235673	−	0.971832i	$-0.424270\pi$
$89$	4.44668	0.471347	0.235673	+	0.971832i	$0.424270\pi$
$90$	0	0
$91$	−7.78750	−0.816351
$92$	1.76869	0.184399
$93$	0	0
$94$	3.26148	0.336396
$95$	7.66798	0.786718
$96$	0	0
$97$	−12.6221	−1.28158	−0.640792	−	0.767715i	$-0.721394\pi$
$97$	−12.6221	−1.28158	−0.640792	+	0.767715i	$0.721394\pi$
$98$	−3.68683	−0.372426
$99$	0	0
$100$	4.80338	0.480338
$101$	−2.61569	−0.260271	−0.130135	−	0.991496i	$-0.541541\pi$
$101$	−2.61569	−0.260271	−0.130135	+	0.991496i	$0.541541\pi$
$102$	0	0
$103$	−15.4221	−1.51958	−0.759792	−	0.650166i	$-0.774699\pi$
$103$	−15.4221	−1.51958	−0.759792	+	0.650166i	$0.774699\pi$
$104$	−3.68577	−0.361420
$105$	0	0
$106$	−4.13558	−0.401683
$107$	−7.44129	−0.719376	−0.359688	−	0.933073i	$-0.617117\pi$
$107$	−7.44129	−0.719376	−0.359688	+	0.933073i	$0.617117\pi$
$108$	0	0
$109$	3.35061	0.320931	0.160465	−	0.987041i	$-0.448701\pi$
$109$	3.35061	0.320931	0.160465	+	0.987041i	$0.448701\pi$
$110$	−3.31334	−0.315915
$111$	0	0
$112$	10.2084	0.964607
$113$	3.11687	0.293211	0.146605	−	0.989195i	$-0.453165\pi$
$113$	3.11687	0.293211	0.146605	+	0.989195i	$0.453165\pi$
$114$	0	0
$115$	1.51137	0.140936
$116$	−1.76869	−0.164219
$117$	0	0
$118$	1.37915	0.126961
$119$	−11.8519	−1.08646
$120$	0	0
$121$	9.77813	0.888921
$122$	1.51018	0.136725
$123$	0	0
$124$	−5.85177	−0.525505
$125$	11.6614	1.04302
$126$	0	0
$127$	13.1857	1.17004	0.585022	−	0.811018i	$-0.301086\pi$
$127$	13.1857	1.17004	0.585022	+	0.811018i	$0.301086\pi$
$128$	11.2432	0.993768
$129$	0	0
$130$	−1.47811	−0.129639
$131$	−0.293403	−0.0256347	−0.0128174	−	0.999918i	$-0.504080\pi$
$131$	−0.293403	−0.0256347	−0.0128174	+	0.999918i	$0.504080\pi$
$132$	0	0
$133$	−19.4297	−1.68477
$134$	1.50744	0.130223
$135$	0	0
$136$	−5.60944	−0.481005
$137$	9.42909	0.805581	0.402791	−	0.915292i	$-0.368040\pi$
$137$	9.42909	0.805581	0.402791	+	0.915292i	$0.368040\pi$
$138$	0	0
$139$	4.18681	0.355120	0.177560	−	0.984110i	$-0.443180\pi$
$139$	4.18681	0.355120	0.177560	+	0.984110i	$0.443180\pi$
$140$	10.2371	0.865190
$141$	0	0
$142$	−2.51381	−0.210954
$143$	9.26931	0.775139
$144$	0	0
$145$	−1.51137	−0.125512
$146$	−6.73676	−0.557538
$147$	0	0
$148$	−17.0366	−1.40040
$149$	−3.07017	−0.251518	−0.125759	−	0.992061i	$-0.540137\pi$
$149$	−3.07017	−0.251518	−0.125759	+	0.992061i	$0.540137\pi$
$150$	0	0
$151$	−4.00783	−0.326153	−0.163076	−	0.986613i	$-0.552142\pi$
$151$	−4.00783	−0.326153	−0.163076	+	0.986613i	$0.552142\pi$
$152$	−9.19593	−0.745889
$153$	0	0
$154$	8.39557	0.676534
$155$	−5.00039	−0.401641
$156$	0	0
$157$	6.32743	0.504984	0.252492	−	0.967599i	$-0.418750\pi$
$157$	6.32743	0.504984	0.252492	+	0.967599i	$0.418750\pi$
$158$	0.969353	0.0771176
$159$	0	0
$160$	7.41640	0.586318
$161$	−3.82960	−0.301815
$162$	0	0
$163$	14.6695	1.14900	0.574501	−	0.818504i	$-0.305196\pi$
$163$	14.6695	1.14900	0.574501	+	0.818504i	$0.305196\pi$
$164$	13.8656	1.08272
$165$	0	0
$166$	0.0816678	0.00633865
$167$	9.34863	0.723419	0.361709	−	0.932291i	$-0.382193\pi$
$167$	9.34863	0.723419	0.361709	+	0.932291i	$0.382193\pi$
$168$	0	0
$169$	−8.86488	−0.681914
$170$	−2.24956	−0.172534
$171$	0	0
$172$	−3.08121	−0.234940
$173$	5.41582	0.411757	0.205879	−	0.978578i	$-0.433995\pi$
$173$	5.41582	0.411757	0.205879	+	0.978578i	$0.433995\pi$
$174$	0	0
$175$	−10.4003	−0.786192
$176$	−12.1509	−0.915910
$177$	0	0
$178$	−2.13860	−0.160295
$179$	7.39935	0.553054	0.276527	−	0.961006i	$-0.410817\pi$
$179$	7.39935	0.553054	0.276527	+	0.961006i	$0.410817\pi$
$180$	0	0
$181$	7.51076	0.558270	0.279135	−	0.960252i	$-0.409952\pi$
$181$	7.51076	0.558270	0.279135	+	0.960252i	$0.409952\pi$
$182$	3.74534	0.277623
$183$	0	0
$184$	−1.81253	−0.133621
$185$	−14.5580	−1.07032
$186$	0	0
$187$	14.1071	1.03161
$188$	11.9943	0.874774
$189$	0	0
$190$	−3.68786	−0.267545
$191$	−22.8811	−1.65562	−0.827810	−	0.561008i	$-0.810414\pi$
$191$	−22.8811	−1.65562	−0.827810	+	0.561008i	$0.810414\pi$
$192$	0	0
$193$	−3.40826	−0.245332	−0.122666	−	0.992448i	$-0.539144\pi$
$193$	−3.40826	−0.245332	−0.122666	+	0.992448i	$0.539144\pi$
$194$	6.07052	0.435838
$195$	0	0
$196$	−13.5585	−0.968467
$197$	−19.0599	−1.35796	−0.678982	−	0.734155i	$-0.737578\pi$
$197$	−19.0599	−1.35796	−0.678982	+	0.734155i	$0.737578\pi$
$198$	0	0
$199$	13.1967	0.935492	0.467746	−	0.883863i	$-0.345066\pi$
$199$	13.1967	0.935492	0.467746	+	0.883863i	$0.345066\pi$
$200$	−4.92241	−0.348067
$201$	0	0
$202$	1.25800	0.0885123
$203$	3.82960	0.268785
$204$	0	0
$205$	11.8483	0.827521
$206$	7.41714	0.516777
$207$	0	0
$208$	−5.42064	−0.375853
$209$	23.1268	1.59971
$210$	0	0
$211$	11.2487	0.774393	0.387197	−	0.921997i	$-0.373443\pi$
$211$	11.2487	0.774393	0.387197	+	0.921997i	$0.373443\pi$
$212$	−15.2088	−1.04455
$213$	0	0
$214$	3.57883	0.244644
$215$	−2.63292	−0.179564
$216$	0	0
$217$	12.6703	0.860119
$218$	−1.61145	−0.109141
$219$	0	0
$220$	−12.1850	−0.821512
$221$	6.29332	0.423334
$222$	0	0
$223$	−5.42434	−0.363240	−0.181620	−	0.983369i	$-0.558134\pi$
$223$	−5.42434	−0.363240	−0.181620	+	0.983369i	$0.558134\pi$
$224$	−18.7922	−1.25561
$225$	0	0
$226$	−1.49904	−0.0997145
$227$	14.2301	0.944488	0.472244	−	0.881468i	$-0.343444\pi$
$227$	14.2301	0.944488	0.472244	+	0.881468i	$0.343444\pi$
$228$	0	0
$229$	17.6452	1.16603	0.583013	−	0.812463i	$-0.301874\pi$
$229$	17.6452	1.16603	0.583013	+	0.812463i	$0.301874\pi$
$230$	−0.726880	−0.0479291
$231$	0	0
$232$	1.81253	0.118998
$233$	−27.1596	−1.77929	−0.889643	−	0.456657i	$-0.849047\pi$
$233$	−27.1596	−1.77929	−0.889643	+	0.456657i	$0.849047\pi$
$234$	0	0
$235$	10.2492	0.668586
$236$	5.07192	0.330154
$237$	0	0
$238$	5.70010	0.369482
$239$	−23.7211	−1.53439	−0.767196	−	0.641412i	$-0.778349\pi$
$239$	−23.7211	−1.53439	−0.767196	+	0.641412i	$0.778349\pi$
$240$	0	0
$241$	21.4044	1.37878	0.689390	−	0.724390i	$-0.257878\pi$
$241$	21.4044	1.37878	0.689390	+	0.724390i	$0.257878\pi$
$242$	−4.70272	−0.302302
$243$	0	0
$244$	5.55378	0.355544
$245$	−11.5859	−0.740196
$246$	0	0
$247$	10.3171	0.656459
$248$	5.99679	0.380797
$249$	0	0
$250$	−5.60844	−0.354709
$251$	−1.11929	−0.0706487	−0.0353243	−	0.999376i	$-0.511246\pi$
$251$	−1.11929	−0.0706487	−0.0353243	+	0.999376i	$0.511246\pi$
$252$	0	0
$253$	4.55830	0.286578
$254$	−6.34158	−0.397906
$255$	0	0
$256$	0.535280	0.0334550
$257$	5.99320	0.373846	0.186923	−	0.982375i	$-0.440149\pi$
$257$	5.99320	0.373846	0.186923	+	0.982375i	$0.440149\pi$
$258$	0	0
$259$	36.8880	2.29211
$260$	−5.43584	−0.337116
$261$	0	0
$262$	0.141110	0.00871780
$263$	−22.6606	−1.39732	−0.698658	−	0.715456i	$-0.746219\pi$
$263$	−22.6606	−1.39732	−0.698658	+	0.715456i	$0.746219\pi$
$264$	0	0
$265$	−12.9961	−0.798342
$266$	9.34455	0.572951
$267$	0	0
$268$	5.54369	0.338635
$269$	15.1575	0.924169	0.462084	−	0.886836i	$-0.347102\pi$
$269$	15.1575	0.924169	0.462084	+	0.886836i	$0.347102\pi$
$270$	0	0
$271$	9.95810	0.604912	0.302456	−	0.953163i	$-0.402194\pi$
$271$	9.95810	0.604912	0.302456	+	0.953163i	$0.402194\pi$
$272$	−8.24976	−0.500215
$273$	0	0
$274$	−4.53485	−0.273960
$275$	12.3793	0.746502
$276$	0	0
$277$	−5.33062	−0.320286	−0.160143	−	0.987094i	$-0.551196\pi$
$277$	−5.33062	−0.320286	−0.160143	+	0.987094i	$0.551196\pi$
$278$	−2.01361	−0.120769
$279$	0	0
$280$	−10.4908	−0.626943
$281$	5.27221	0.314513	0.157257	−	0.987558i	$-0.449735\pi$
$281$	5.27221	0.314513	0.157257	+	0.987558i	$0.449735\pi$
$282$	0	0
$283$	25.8471	1.53645	0.768225	−	0.640180i	$-0.221140\pi$
$283$	25.8471	1.53645	0.768225	+	0.640180i	$0.221140\pi$
$284$	−9.24468	−0.548571
$285$	0	0
$286$	−4.45801	−0.263608
$287$	−30.0220	−1.77215
$288$	0	0
$289$	−7.42210	−0.436594
$290$	0.726880	0.0426839
$291$	0	0
$292$	−24.7748	−1.44984
$293$	3.67536	0.214717	0.107358	−	0.994220i	$-0.465761\pi$
$293$	3.67536	0.214717	0.107358	+	0.994220i	$0.465761\pi$
$294$	0	0
$295$	4.33400	0.252335
$296$	17.4589	1.01478
$297$	0	0
$298$	1.47658	0.0855358
$299$	2.03350	0.117600
$300$	0	0
$301$	6.67148	0.384538
$302$	1.92754	0.110917
$303$	0	0
$304$	−13.5244	−0.775677
$305$	4.74575	0.271741
$306$	0	0
$307$	32.5203	1.85603	0.928016	−	0.372540i	$-0.121513\pi$
$307$	32.5203	1.85603	0.928016	+	0.372540i	$0.121513\pi$
$308$	30.8752	1.75928
$309$	0	0
$310$	2.40490	0.136589
$311$	7.24369	0.410752	0.205376	−	0.978683i	$-0.434158\pi$
$311$	7.24369	0.410752	0.205376	+	0.978683i	$0.434158\pi$
$312$	0	0
$313$	−29.1361	−1.64687	−0.823434	−	0.567412i	$-0.807945\pi$
$313$	−29.1361	−1.64687	−0.823434	+	0.567412i	$0.807945\pi$
$314$	−3.04313	−0.171734
$315$	0	0
$316$	3.56485	0.200539
$317$	22.2779	1.25125	0.625626	−	0.780123i	$-0.284844\pi$
$317$	22.2779	1.25125	0.625626	+	0.780123i	$0.284844\pi$
$318$	0	0
$319$	−4.55830	−0.255216
$320$	4.49073	0.251039
$321$	0	0
$322$	1.84182	0.102641
$323$	15.7017	0.873666
$324$	0	0
$325$	5.52253	0.306335
$326$	−7.05518	−0.390750
$327$	0	0
$328$	−14.2092	−0.784574
$329$	−25.9702	−1.43178
$330$	0	0
$331$	2.57404	0.141482	0.0707409	−	0.997495i	$-0.477464\pi$
$331$	2.57404	0.141482	0.0707409	+	0.997495i	$0.477464\pi$
$332$	0.300338	0.0164832
$333$	0	0
$334$	−4.49616	−0.246019
$335$	4.73714	0.258817
$336$	0	0
$337$	29.9240	1.63007	0.815033	−	0.579415i	$-0.196719\pi$
$337$	29.9240	1.63007	0.815033	+	0.579415i	$0.196719\pi$
$338$	4.26350	0.231904
$339$	0	0
$340$	−8.27289	−0.448661
$341$	−15.0813	−0.816696
$342$	0	0
$343$	2.54993	0.137683
$344$	3.15757	0.170245
$345$	0	0
$346$	−2.60470	−0.140030
$347$	28.3110	1.51981	0.759906	−	0.650033i	$-0.225245\pi$
$347$	28.3110	1.51981	0.759906	+	0.650033i	$0.225245\pi$
$348$	0	0
$349$	32.2588	1.72678	0.863388	−	0.504540i	$-0.168338\pi$
$349$	32.2588	1.72678	0.863388	+	0.504540i	$0.168338\pi$
$350$	5.00197	0.267366
$351$	0	0
$352$	22.3680	1.19222
$353$	27.3821	1.45740	0.728701	−	0.684832i	$-0.240124\pi$
$353$	27.3821	1.45740	0.728701	+	0.684832i	$0.240124\pi$
$354$	0	0
$355$	−7.89966	−0.419271
$356$	−7.86481	−0.416834
$357$	0	0
$358$	−3.55866	−0.188081
$359$	22.3917	1.18179	0.590893	−	0.806750i	$-0.298775\pi$
$359$	22.3917	1.18179	0.590893	+	0.806750i	$0.298775\pi$
$360$	0	0
$361$	6.74087	0.354783
$362$	−3.61224	−0.189855
$363$	0	0
$364$	13.7737	0.721938
$365$	−21.1703	−1.10810
$366$	0	0
$367$	7.55651	0.394447	0.197223	−	0.980359i	$-0.436808\pi$
$367$	7.55651	0.394447	0.197223	+	0.980359i	$0.436808\pi$
$368$	−2.66567	−0.138958
$369$	0	0
$370$	7.00155	0.363993
$371$	32.9304	1.70966
$372$	0	0
$373$	−4.61281	−0.238842	−0.119421	−	0.992844i	$-0.538104\pi$
$373$	−4.61281	−0.238842	−0.119421	+	0.992844i	$0.538104\pi$
$374$	−6.78472	−0.350829
$375$	0	0
$376$	−12.2915	−0.633888
$377$	−2.03350	−0.104731
$378$	0	0
$379$	28.3480	1.45614	0.728069	−	0.685504i	$-0.240418\pi$
$379$	28.3480	1.45614	0.728069	+	0.685504i	$0.240418\pi$
$380$	−13.5623	−0.695732
$381$	0	0
$382$	11.0045	0.563040
$383$	−17.9810	−0.918786	−0.459393	−	0.888233i	$-0.651933\pi$
$383$	−17.9810	−0.918786	−0.459393	+	0.888233i	$0.651933\pi$
$384$	0	0
$385$	26.3831	1.34461
$386$	1.63918	0.0834321
$387$	0	0
$388$	22.3247	1.13336
$389$	17.4058	0.882507	0.441254	−	0.897382i	$-0.354534\pi$
$389$	17.4058	0.882507	0.441254	+	0.897382i	$0.354534\pi$
$390$	0	0
$391$	3.09482	0.156512
$392$	13.8945	0.701781
$393$	0	0
$394$	9.16673	0.461813
$395$	3.04620	0.153271
$396$	0	0
$397$	21.3372	1.07088	0.535441	−	0.844573i	$-0.320145\pi$
$397$	21.3372	1.07088	0.535441	+	0.844573i	$0.320145\pi$
$398$	−6.34688	−0.318140
$399$	0	0
$400$	−7.23935	−0.361968
$401$	14.1561	0.706920	0.353460	−	0.935450i	$-0.385005\pi$
$401$	14.1561	0.706920	0.353460	+	0.935450i	$0.385005\pi$
$402$	0	0
$403$	−6.72789	−0.335140
$404$	4.62635	0.230170
$405$	0	0
$406$	−1.84182	−0.0914079
$407$	−43.9071	−2.17639
$408$	0	0
$409$	−0.840710	−0.0415704	−0.0207852	−	0.999784i	$-0.506617\pi$
$409$	−0.840710	−0.0415704	−0.0207852	+	0.999784i	$0.506617\pi$
$410$	−5.69835	−0.281422
$411$	0	0
$412$	27.2770	1.34384
$413$	−10.9818	−0.540378
$414$	0	0
$415$	0.256641	0.0125980
$416$	9.97856	0.489239
$417$	0	0
$418$	−11.1226	−0.544026
$419$	−5.72839	−0.279850	−0.139925	−	0.990162i	$-0.544686\pi$
$419$	−5.72839	−0.279850	−0.139925	+	0.990162i	$0.544686\pi$
$420$	0	0
$421$	−11.4014	−0.555672	−0.277836	−	0.960629i	$-0.589617\pi$
$421$	−11.4014	−0.555672	−0.277836	+	0.960629i	$0.589617\pi$
$422$	−5.40999	−0.263354
$423$	0	0
$424$	15.5857	0.756910
$425$	8.40483	0.407694
$426$	0	0
$427$	−12.0251	−0.581936
$428$	13.1614	0.636178
$429$	0	0
$430$	1.26628	0.0610656
$431$	13.3430	0.642710	0.321355	−	0.946959i	$-0.395862\pi$
$431$	13.3430	0.642710	0.321355	+	0.946959i	$0.395862\pi$
$432$	0	0
$433$	3.77877	0.181596	0.0907980	−	0.995869i	$-0.471058\pi$
$433$	3.77877	0.181596	0.0907980	+	0.995869i	$0.471058\pi$
$434$	−6.09371	−0.292507
$435$	0	0
$436$	−5.92621	−0.283814
$437$	5.07355	0.242701
$438$	0	0
$439$	10.3726	0.495060	0.247530	−	0.968880i	$-0.420381\pi$
$439$	10.3726	0.495060	0.247530	+	0.968880i	$0.420381\pi$
$440$	12.4870	0.595293
$441$	0	0
$442$	−3.02672	−0.143967
$443$	10.7478	0.510645	0.255322	−	0.966856i	$-0.417818\pi$
$443$	10.7478	0.510645	0.255322	+	0.966856i	$0.417818\pi$
$444$	0	0
$445$	−6.72055	−0.318585
$446$	2.60879	0.123530
$447$	0	0
$448$	−11.3789	−0.537604
$449$	−16.4083	−0.774357	−0.387178	−	0.922005i	$-0.626550\pi$
$449$	−16.4083	−0.774357	−0.387178	+	0.922005i	$0.626550\pi$
$450$	0	0
$451$	35.7347	1.68268
$452$	−5.51279	−0.259300
$453$	0	0
$454$	−6.84389	−0.321199
$455$	11.7698	0.551774
$456$	0	0
$457$	−0.278452	−0.0130254	−0.00651272	−	0.999979i	$-0.502073\pi$
$457$	−0.278452	−0.0130254	−0.00651272	+	0.999979i	$0.502073\pi$
$458$	−8.48632	−0.396540
$459$	0	0
$460$	−2.67314	−0.124636
$461$	18.6860	0.870294	0.435147	−	0.900360i	$-0.356696\pi$
$461$	18.6860	0.870294	0.435147	+	0.900360i	$0.356696\pi$
$462$	0	0
$463$	−34.8292	−1.61865	−0.809325	−	0.587361i	$-0.800167\pi$
$463$	−34.8292	−1.61865	−0.809325	+	0.587361i	$0.800167\pi$
$464$	2.66567	0.123750
$465$	0	0
$466$	13.0622	0.605096
$467$	22.6213	1.04679	0.523395	−	0.852090i	$-0.324665\pi$
$467$	22.6213	1.04679	0.523395	+	0.852090i	$0.324665\pi$
$468$	0	0
$469$	−12.0033	−0.554260
$470$	−4.92929	−0.227371
$471$	0	0
$472$	−5.19761	−0.239239
$473$	−7.94093	−0.365125
$474$	0	0
$475$	13.7786	0.632206
$476$	20.9624	0.960811
$477$	0	0
$478$	11.4085	0.521813
$479$	27.8673	1.27329	0.636644	−	0.771158i	$-0.280322\pi$
$479$	27.8673	1.27329	0.636644	+	0.771158i	$0.280322\pi$
$480$	0	0
$481$	−19.5874	−0.893106
$482$	−10.2943	−0.468893
$483$	0	0
$484$	−17.2945	−0.786114
$485$	19.0767	0.866226
$486$	0	0
$487$	−11.4508	−0.518885	−0.259443	−	0.965759i	$-0.583539\pi$
$487$	−11.4508	−0.518885	−0.259443	+	0.965759i	$0.583539\pi$
$488$	−5.69141	−0.257638
$489$	0	0
$490$	5.57215	0.251724
$491$	−22.6586	−1.02257	−0.511285	−	0.859411i	$-0.670830\pi$
$491$	−22.6586	−1.02257	−0.511285	+	0.859411i	$0.670830\pi$
$492$	0	0
$493$	−3.09482	−0.139384
$494$	−4.96191	−0.223247
$495$	0	0
$496$	8.81943	0.396004
$497$	20.0167	0.897872
$498$	0	0
$499$	11.8400	0.530032	0.265016	−	0.964244i	$-0.414623\pi$
$499$	11.8400	0.530032	0.265016	+	0.964244i	$0.414623\pi$
$500$	−20.6254	−0.922394
$501$	0	0
$502$	0.538312	0.0240261
$503$	33.5152	1.49437	0.747185	−	0.664616i	$-0.231405\pi$
$503$	33.5152	1.49437	0.747185	+	0.664616i	$0.231405\pi$
$504$	0	0
$505$	3.95326	0.175918
$506$	−2.19228	−0.0974588
$507$	0	0
$508$	−23.3215	−1.03472
$509$	−7.92984	−0.351484	−0.175742	−	0.984436i	$-0.556232\pi$
$509$	−7.92984	−0.351484	−0.175742	+	0.984436i	$0.556232\pi$
$510$	0	0
$511$	53.6428	2.37302
$512$	−22.7438	−1.00515
$513$	0	0
$514$	−2.88239	−0.127137
$515$	23.3084	1.02709
$516$	0	0
$517$	30.9119	1.35950
$518$	−17.7410	−0.779495
$519$	0	0
$520$	5.57055	0.244285
$521$	45.1228	1.97687	0.988433	−	0.151661i	$-0.0484621\pi$
$521$	45.1228	1.97687	0.988433	+	0.151661i	$0.0484621\pi$
$522$	0	0
$523$	20.7131	0.905720	0.452860	−	0.891582i	$-0.350404\pi$
$523$	20.7131	0.905720	0.452860	+	0.891582i	$0.350404\pi$
$524$	0.518940	0.0226700
$525$	0	0
$526$	10.8985	0.475196
$527$	−10.2393	−0.446031
$528$	0	0
$529$	1.00000	0.0434783
$530$	6.25037	0.271499
$531$	0	0
$532$	34.3651	1.48992
$533$	15.9416	0.690506
$534$	0	0
$535$	11.2465	0.486229
$536$	−5.68108	−0.245385
$537$	0	0
$538$	−7.28989	−0.314289
$539$	−34.9433	−1.50511
$540$	0	0
$541$	−16.2474	−0.698528	−0.349264	−	0.937024i	$-0.613568\pi$
$541$	−16.2474	−0.698528	−0.349264	+	0.937024i	$0.613568\pi$
$542$	−4.78928	−0.205717
$543$	0	0
$544$	15.1865	0.651118
$545$	−5.06400	−0.216918
$546$	0	0
$547$	19.7363	0.843864	0.421932	−	0.906627i	$-0.361352\pi$
$547$	19.7363	0.843864	0.421932	+	0.906627i	$0.361352\pi$
$548$	−16.6772	−0.712414
$549$	0	0
$550$	−5.95375	−0.253869
$551$	−5.07355	−0.216140
$552$	0	0
$553$	−7.71866	−0.328231
$554$	2.56372	0.108922
$555$	0	0
$556$	−7.40518	−0.314050
$557$	−35.9309	−1.52244	−0.761221	−	0.648492i	$-0.775400\pi$
$557$	−35.9309	−1.52244	−0.761221	+	0.648492i	$0.775400\pi$
$558$	0	0
$559$	−3.54252	−0.149833
$560$	−15.4287	−0.651981
$561$	0	0
$562$	−2.53563	−0.106959
$563$	−42.8625	−1.80644	−0.903220	−	0.429178i	$-0.858803\pi$
$563$	−42.8625	−1.80644	−0.903220	+	0.429178i	$0.858803\pi$
$564$	0	0
$565$	−4.71073	−0.198182
$566$	−12.4310	−0.522513
$567$	0	0
$568$	9.47378	0.397511
$569$	−2.91112	−0.122040	−0.0610202	−	0.998137i	$-0.519435\pi$
$569$	−2.91112	−0.122040	−0.0610202	+	0.998137i	$0.519435\pi$
$570$	0	0
$571$	−28.2792	−1.18345	−0.591724	−	0.806141i	$-0.701552\pi$
$571$	−28.2792	−1.18345	−0.591724	+	0.806141i	$0.701552\pi$
$572$	−16.3946	−0.685492
$573$	0	0
$574$	14.4389	0.602667
$575$	2.71578	0.113256
$576$	0	0
$577$	23.4164	0.974837	0.487419	−	0.873168i	$-0.337939\pi$
$577$	23.4164	0.974837	0.487419	+	0.873168i	$0.337939\pi$
$578$	3.56960	0.148476
$579$	0	0
$580$	2.67314	0.110996
$581$	−0.650296	−0.0269788
$582$	0	0
$583$	−39.1964	−1.62335
$584$	25.3888	1.05059
$585$	0	0
$586$	−1.76764	−0.0730204
$587$	−9.90376	−0.408772	−0.204386	−	0.978890i	$-0.565520\pi$
$587$	−9.90376	−0.408772	−0.204386	+	0.978890i	$0.565520\pi$
$588$	0	0
$589$	−16.7860	−0.691654
$590$	−2.08440	−0.0858136
$591$	0	0
$592$	25.6766	1.05530
$593$	−39.4781	−1.62117	−0.810585	−	0.585621i	$-0.800851\pi$
$593$	−39.4781	−1.62117	−0.810585	+	0.585621i	$0.800851\pi$
$594$	0	0
$595$	17.9126	0.734344
$596$	5.43019	0.222429
$597$	0	0
$598$	−0.977997	−0.0399933
$599$	28.8979	1.18074	0.590369	−	0.807134i	$-0.298982\pi$
$599$	28.8979	1.18074	0.590369	+	0.807134i	$0.298982\pi$
$600$	0	0
$601$	−8.30294	−0.338684	−0.169342	−	0.985557i	$-0.554164\pi$
$601$	−8.30294	−0.338684	−0.169342	+	0.985557i	$0.554164\pi$
$602$	−3.20860	−0.130773
$603$	0	0
$604$	7.08863	0.288432
$605$	−14.7783	−0.600824
$606$	0	0
$607$	−21.6030	−0.876840	−0.438420	−	0.898770i	$-0.644462\pi$
$607$	−21.6030	−0.876840	−0.438420	+	0.898770i	$0.644462\pi$
$608$	24.8963	1.00968
$609$	0	0
$610$	−2.28244	−0.0924131
$611$	13.7901	0.557886
$612$	0	0
$613$	−3.82158	−0.154352	−0.0771761	−	0.997017i	$-0.524590\pi$
$613$	−3.82158	−0.154352	−0.0771761	+	0.997017i	$0.524590\pi$
$614$	−15.6404	−0.631195
$615$	0	0
$616$	−31.6403	−1.27483
$617$	30.6641	1.23449	0.617246	−	0.786770i	$-0.288248\pi$
$617$	30.6641	1.23449	0.617246	+	0.786770i	$0.288248\pi$
$618$	0	0
$619$	8.05425	0.323728	0.161864	−	0.986813i	$-0.448249\pi$
$619$	8.05425	0.323728	0.161864	+	0.986813i	$0.448249\pi$
$620$	8.84416	0.355190
$621$	0	0
$622$	−3.48380	−0.139688
$623$	17.0290	0.682253
$624$	0	0
$625$	−4.04569	−0.161827
$626$	14.0128	0.560063
$627$	0	0
$628$	−11.1913	−0.446581
$629$	−29.8103	−1.18862
$630$	0	0
$631$	−23.6692	−0.942257	−0.471128	−	0.882065i	$-0.656153\pi$
$631$	−23.6692	−0.942257	−0.471128	+	0.882065i	$0.656153\pi$
$632$	−3.65319	−0.145316
$633$	0	0
$634$	−10.7144	−0.425523
$635$	−19.9284	−0.790836
$636$	0	0
$637$	−15.5885	−0.617639
$638$	2.19228	0.0867933
$639$	0	0
$640$	−16.9926	−0.671691
$641$	41.6923	1.64674	0.823372	−	0.567501i	$-0.192090\pi$
$641$	41.6923	1.64674	0.823372	+	0.567501i	$0.192090\pi$
$642$	0	0
$643$	−26.2442	−1.03497	−0.517486	−	0.855692i	$-0.673132\pi$
$643$	−26.2442	−1.03497	−0.517486	+	0.855692i	$0.673132\pi$
$644$	6.77339	0.266909
$645$	0	0
$646$	−7.55162	−0.297115
$647$	8.24798	0.324262	0.162131	−	0.986769i	$-0.448163\pi$
$647$	8.24798	0.324262	0.162131	+	0.986769i	$0.448163\pi$
$648$	0	0
$649$	13.0714	0.513098
$650$	−2.65602	−0.104178
$651$	0	0
$652$	−25.9458	−1.01612
$653$	−0.574006	−0.0224626	−0.0112313	−	0.999937i	$-0.503575\pi$
$653$	−0.574006	−0.0224626	−0.0112313	+	0.999937i	$0.503575\pi$
$654$	0	0
$655$	0.443439	0.0173266
$656$	−20.8974	−0.815907
$657$	0	0
$658$	12.4902	0.486918
$659$	50.4367	1.96474	0.982368	−	0.186957i	$-0.0598624\pi$
$659$	50.4367	1.96474	0.982368	+	0.186957i	$0.0598624\pi$
$660$	0	0
$661$	20.5906	0.800880	0.400440	−	0.916323i	$-0.368857\pi$
$661$	20.5906	0.800880	0.400440	+	0.916323i	$0.368857\pi$
$662$	−1.23796	−0.0481149
$663$	0	0
$664$	−0.307781	−0.0119442
$665$	29.3653	1.13874
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	−16.5349	−0.639753
$669$	0	0
$670$	−2.27829	−0.0880181
$671$	14.3133	0.552558
$672$	0	0
$673$	−45.0937	−1.73823	−0.869117	−	0.494607i	$-0.835312\pi$
$673$	−45.0937	−1.73823	−0.869117	+	0.494607i	$0.835312\pi$
$674$	−14.3917	−0.554349
$675$	0	0
$676$	15.6793	0.603048
$677$	−33.5470	−1.28932	−0.644658	−	0.764471i	$-0.723000\pi$
$677$	−33.5470	−1.28932	−0.644658	+	0.764471i	$0.723000\pi$
$678$	0	0
$679$	−48.3377	−1.85503
$680$	8.47791	0.325113
$681$	0	0
$682$	7.25323	0.277740
$683$	−25.6859	−0.982844	−0.491422	−	0.870922i	$-0.663523\pi$
$683$	−25.6859	−0.982844	−0.491422	+	0.870922i	$0.663523\pi$
$684$	0	0
$685$	−14.2508	−0.544495
$686$	−1.22637	−0.0468231
$687$	0	0
$688$	4.64381	0.177044
$689$	−17.4859	−0.666158
$690$	0	0
$691$	22.3465	0.850101	0.425050	−	0.905170i	$-0.360256\pi$
$691$	22.3465	0.850101	0.425050	+	0.905170i	$0.360256\pi$
$692$	−9.57893	−0.364136
$693$	0	0
$694$	−13.6160	−0.516855
$695$	−6.32779	−0.240027
$696$	0	0
$697$	24.2617	0.918979
$698$	−15.5147	−0.587238
$699$	0	0
$700$	18.3950	0.695266
$701$	−33.6787	−1.27203	−0.636013	−	0.771678i	$-0.719418\pi$
$701$	−33.6787	−1.27203	−0.636013	+	0.771678i	$0.719418\pi$
$702$	0	0
$703$	−48.8701	−1.84317
$704$	13.5441	0.510463
$705$	0	0
$706$	−13.1692	−0.495630
$707$	−10.0170	−0.376730
$708$	0	0
$709$	7.22302	0.271266	0.135633	−	0.990759i	$-0.456693\pi$
$709$	7.22302	0.271266	0.135633	+	0.990759i	$0.456693\pi$
$710$	3.79928	0.142585
$711$	0	0
$712$	8.05972	0.302051
$713$	−3.30853	−0.123905
$714$	0	0
$715$	−14.0093	−0.523919
$716$	−13.0872	−0.489091
$717$	0	0
$718$	−10.7691	−0.401900
$719$	27.3138	1.01863	0.509316	−	0.860579i	$-0.329898\pi$
$719$	27.3138	1.01863	0.509316	+	0.860579i	$0.329898\pi$
$720$	0	0
$721$	−59.0605	−2.19953
$722$	−3.24197	−0.120654
$723$	0	0
$724$	−13.2842	−0.493705
$725$	−2.71578	−0.100861
$726$	0	0
$727$	−41.5084	−1.53946	−0.769730	−	0.638370i	$-0.779609\pi$
$727$	−41.5084	−1.53946	−0.769730	+	0.638370i	$0.779609\pi$
$728$	−14.1150	−0.523138
$729$	0	0
$730$	10.1817	0.376842
$731$	−5.39143	−0.199409
$732$	0	0
$733$	−12.4918	−0.461394	−0.230697	−	0.973026i	$-0.574101\pi$
$733$	−12.4918	−0.461394	−0.230697	+	0.973026i	$0.574101\pi$
$734$	−3.63425	−0.134143
$735$	0	0
$736$	4.90709	0.180878
$737$	14.2873	0.526279
$738$	0	0
$739$	−30.0795	−1.10649	−0.553246	−	0.833018i	$-0.686611\pi$
$739$	−30.0795	−1.10649	−0.553246	+	0.833018i	$0.686611\pi$
$740$	25.7486	0.946537
$741$	0	0
$742$	−15.8376	−0.581417
$743$	−19.8015	−0.726445	−0.363223	−	0.931702i	$-0.618324\pi$
$743$	−19.8015	−0.726445	−0.363223	+	0.931702i	$0.618324\pi$
$744$	0	0
$745$	4.64015	0.170002
$746$	2.21850	0.0812249
$747$	0	0
$748$	−24.9512	−0.912306
$749$	−28.4972	−1.04126
$750$	0	0
$751$	−19.1667	−0.699405	−0.349702	−	0.936861i	$-0.613717\pi$
$751$	−19.1667	−0.699405	−0.349702	+	0.936861i	$0.613717\pi$
$752$	−18.0771	−0.659203
$753$	0	0
$754$	0.977997	0.0356166
$755$	6.05730	0.220448
$756$	0	0
$757$	−4.95766	−0.180189	−0.0900946	−	0.995933i	$-0.528717\pi$
$757$	−4.95766	−0.180189	−0.0900946	+	0.995933i	$0.528717\pi$
$758$	−13.6338	−0.495200
$759$	0	0
$760$	13.8984	0.504148
$761$	−21.5501	−0.781189	−0.390595	−	0.920563i	$-0.627731\pi$
$761$	−21.5501	−0.781189	−0.390595	+	0.920563i	$0.627731\pi$
$762$	0	0
$763$	12.8315	0.464532
$764$	40.4697	1.46414
$765$	0	0
$766$	8.64783	0.312459
$767$	5.83128	0.210555
$768$	0	0
$769$	38.5390	1.38975	0.694876	−	0.719130i	$-0.255459\pi$
$769$	38.5390	1.38975	0.694876	+	0.719130i	$0.255459\pi$
$770$	−12.6888	−0.457272
$771$	0	0
$772$	6.02818	0.216959
$773$	39.8782	1.43432	0.717160	−	0.696909i	$-0.245442\pi$
$773$	39.8782	1.43432	0.717160	+	0.696909i	$0.245442\pi$
$774$	0	0
$775$	−8.98522	−0.322759
$776$	−22.8779	−0.821270
$777$	0	0
$778$	−8.37118	−0.300121
$779$	39.7739	1.42505
$780$	0	0
$781$	−23.8255	−0.852544
$782$	−1.48843	−0.0532262
$783$	0	0
$784$	20.4346	0.729807
$785$	−9.56306	−0.341320
$786$	0	0
$787$	−54.2889	−1.93519	−0.967595	−	0.252506i	$-0.918745\pi$
$787$	−54.2889	−1.93519	−0.967595	+	0.252506i	$0.918745\pi$
$788$	33.7112	1.20091
$789$	0	0
$790$	−1.46505	−0.0521240
$791$	11.9364	0.424409
$792$	0	0
$793$	6.38528	0.226748
$794$	−10.2620	−0.364183
$795$	0	0
$796$	−23.3410	−0.827300
$797$	6.62914	0.234816	0.117408	−	0.993084i	$-0.462541\pi$
$797$	6.62914	0.234816	0.117408	+	0.993084i	$0.462541\pi$
$798$	0	0
$799$	20.9873	0.742478
$800$	13.3265	0.471164
$801$	0	0
$802$	−6.80825	−0.240408
$803$	−63.8500	−2.25322
$804$	0	0
$805$	5.78793	0.203998
$806$	3.23573	0.113974
$807$	0	0
$808$	−4.74100	−0.166788
$809$	−18.7279	−0.658439	−0.329219	−	0.944253i	$-0.606786\pi$
$809$	−18.7279	−0.658439	−0.329219	+	0.944253i	$0.606786\pi$
$810$	0	0
$811$	26.1555	0.918443	0.459222	−	0.888322i	$-0.348128\pi$
$811$	26.1555	0.918443	0.459222	+	0.888322i	$0.348128\pi$
$812$	−6.77339	−0.237700
$813$	0	0
$814$	21.1168	0.740143
$815$	−22.1710	−0.776615
$816$	0	0
$817$	−8.83853	−0.309221
$818$	0.404333	0.0141372
$819$	0	0
$820$	−20.9560	−0.731816
$821$	−37.4288	−1.30628	−0.653138	−	0.757239i	$-0.726548\pi$
$821$	−37.4288	−1.30628	−0.653138	+	0.757239i	$0.726548\pi$
$822$	0	0
$823$	31.8575	1.11048	0.555241	−	0.831690i	$-0.312626\pi$
$823$	31.8575	1.11048	0.555241	+	0.831690i	$0.312626\pi$
$824$	−27.9529	−0.973787
$825$	0	0
$826$	5.28161	0.183771
$827$	39.9645	1.38970	0.694851	−	0.719154i	$-0.255470\pi$
$827$	39.9645	1.38970	0.694851	+	0.719154i	$0.255470\pi$
$828$	0	0
$829$	−17.3349	−0.602066	−0.301033	−	0.953614i	$-0.597331\pi$
$829$	−17.3349	−0.602066	−0.301033	+	0.953614i	$0.597331\pi$
$830$	−0.123430	−0.00428431
$831$	0	0
$832$	6.04215	0.209474
$833$	−23.7244	−0.822002
$834$	0	0
$835$	−14.1292	−0.488961
$836$	−40.9042	−1.41470
$837$	0	0
$838$	2.75503	0.0951709
$839$	17.4594	0.602765	0.301382	−	0.953503i	$-0.402552\pi$
$839$	17.4594	0.602765	0.301382	+	0.953503i	$0.402552\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	5.48344	0.188972
$843$	0	0
$844$	−19.8955	−0.684833
$845$	13.3981	0.460907
$846$	0	0
$847$	37.4463	1.28667
$848$	22.9218	0.787138
$849$	0	0
$850$	−4.04224	−0.138648
$851$	−9.63233	−0.330192
$852$	0	0
$853$	32.0237	1.09647	0.548235	−	0.836324i	$-0.315300\pi$
$853$	32.0237	1.09647	0.548235	+	0.836324i	$0.315300\pi$
$854$	5.78339	0.197904
$855$	0	0
$856$	−13.4875	−0.460994
$857$	6.53401	0.223197	0.111599	−	0.993753i	$-0.464403\pi$
$857$	6.53401	0.223197	0.111599	+	0.993753i	$0.464403\pi$
$858$	0	0
$859$	−2.81937	−0.0961958	−0.0480979	−	0.998843i	$-0.515316\pi$
$859$	−2.81937	−0.0961958	−0.0480979	+	0.998843i	$0.515316\pi$
$860$	4.65683	0.158797
$861$	0	0
$862$	−6.41722	−0.218571
$863$	1.36845	0.0465825	0.0232913	−	0.999729i	$-0.492585\pi$
$863$	1.36845	0.0465825	0.0232913	+	0.999729i	$0.492585\pi$
$864$	0	0
$865$	−8.18528	−0.278308
$866$	−1.81737	−0.0617567
$867$	0	0
$868$	−22.4100	−0.760644
$869$	9.18738	0.311661
$870$	0	0
$871$	6.37369	0.215964
$872$	6.07307	0.205660
$873$	0	0
$874$	−2.44008	−0.0825371
$875$	44.6583	1.50973
$876$	0	0
$877$	−30.4787	−1.02919	−0.514597	−	0.857432i	$-0.672058\pi$
$877$	−30.4787	−1.02919	−0.514597	+	0.857432i	$0.672058\pi$
$878$	−4.98865	−0.168359
$879$	0	0
$880$	18.3645	0.619066
$881$	−26.0641	−0.878122	−0.439061	−	0.898457i	$-0.644689\pi$
$881$	−26.0641	−0.878122	−0.439061	+	0.898457i	$0.644689\pi$
$882$	0	0
$883$	−36.7473	−1.23664	−0.618322	−	0.785925i	$-0.712187\pi$
$883$	−36.7473	−1.23664	−0.618322	+	0.785925i	$0.712187\pi$
$884$	−11.1310	−0.374374
$885$	0	0
$886$	−5.16909	−0.173659
$887$	24.6091	0.826293	0.413147	−	0.910665i	$-0.364430\pi$
$887$	24.6091	0.826293	0.413147	+	0.910665i	$0.364430\pi$
$888$	0	0
$889$	50.4961	1.69358
$890$	3.23220	0.108344
$891$	0	0
$892$	9.59399	0.321231
$893$	34.4059	1.15135
$894$	0	0
$895$	−11.1831	−0.373811
$896$	43.0570	1.43843
$897$	0	0
$898$	7.89147	0.263342
$899$	3.30853	0.110346
$900$	0	0
$901$	−26.6120	−0.886575
$902$	−17.1863	−0.572242
$903$	0	0
$904$	5.64941	0.187897
$905$	−11.3515	−0.377336
$906$	0	0
$907$	53.1771	1.76572	0.882859	−	0.469638i	$-0.155616\pi$
$907$	53.1771	1.76572	0.882859	+	0.469638i	$0.155616\pi$
$908$	−25.1688	−0.835255
$909$	0	0
$910$	−5.66058	−0.187646
$911$	−32.6481	−1.08168	−0.540840	−	0.841126i	$-0.681893\pi$
$911$	−32.6481	−1.08168	−0.540840	+	0.841126i	$0.681893\pi$
$912$	0	0
$913$	0.774035	0.0256168
$914$	0.133920	0.00442966
$915$	0	0
$916$	−31.2089	−1.03117
$917$	−1.12362	−0.0371051
$918$	0	0
$919$	50.2070	1.65618	0.828089	−	0.560597i	$-0.189428\pi$
$919$	50.2070	1.65618	0.828089	+	0.560597i	$0.189428\pi$
$920$	2.73939	0.0903150
$921$	0	0
$922$	−8.98689	−0.295968
$923$	−10.6288	−0.349851
$924$	0	0
$925$	−26.1593	−0.860111
$926$	16.7509	0.550467
$927$	0	0
$928$	−4.90709	−0.161083
$929$	4.68086	0.153574	0.0767870	−	0.997048i	$-0.475534\pi$
$929$	4.68086	0.153574	0.0767870	+	0.997048i	$0.475534\pi$
$930$	0	0
$931$	−38.8930	−1.27467
$932$	48.0371	1.57351
$933$	0	0
$934$	−10.8796	−0.355990
$935$	−21.3210	−0.697272
$936$	0	0
$937$	−56.6795	−1.85164	−0.925819	−	0.377966i	$-0.876624\pi$
$937$	−56.6795	−1.85164	−0.925819	+	0.377966i	$0.876624\pi$
$938$	5.77289	0.188492
$939$	0	0
$940$	−18.1278	−0.591262
$941$	−5.57428	−0.181716	−0.0908582	−	0.995864i	$-0.528961\pi$
$941$	−5.57428	−0.181716	−0.0908582	+	0.995864i	$0.528961\pi$
$942$	0	0
$943$	7.83947	0.255288
$944$	−7.64408	−0.248794
$945$	0	0
$946$	3.81913	0.124171
$947$	−25.1058	−0.815828	−0.407914	−	0.913020i	$-0.633744\pi$
$947$	−25.1058	−0.815828	−0.407914	+	0.913020i	$0.633744\pi$
$948$	0	0
$949$	−28.4841	−0.924631
$950$	−6.62672	−0.214999
$951$	0	0
$952$	−21.4819	−0.696233
$953$	18.7777	0.608269	0.304134	−	0.952629i	$-0.401633\pi$
$953$	18.7777	0.608269	0.304134	+	0.952629i	$0.401633\pi$
$954$	0	0
$955$	34.5817	1.11904
$956$	41.9554	1.35694
$957$	0	0
$958$	−13.4026	−0.433017
$959$	36.1097	1.16604
$960$	0	0
$961$	−20.0536	−0.646892
$962$	9.42039	0.303726
$963$	0	0
$964$	−37.8579	−1.21932
$965$	5.15113	0.165821
$966$	0	0
$967$	−4.19383	−0.134864	−0.0674322	−	0.997724i	$-0.521481\pi$
$967$	−4.19383	−0.134864	−0.0674322	+	0.997724i	$0.521481\pi$
$968$	17.7231	0.569642
$969$	0	0
$970$	−9.17478	−0.294584
$971$	−5.18080	−0.166260	−0.0831300	−	0.996539i	$-0.526492\pi$
$971$	−5.18080	−0.166260	−0.0831300	+	0.996539i	$0.526492\pi$
$972$	0	0
$973$	16.0338	0.514020
$974$	5.50718	0.176461
$975$	0	0
$976$	−8.37031	−0.267927
$977$	−22.3619	−0.715421	−0.357711	−	0.933832i	$-0.616443\pi$
$977$	−22.3619	−0.715421	−0.357711	+	0.933832i	$0.616443\pi$
$978$	0	0
$979$	−20.2693	−0.647810
$980$	20.4919	0.654590
$981$	0	0
$982$	10.8975	0.347753
$983$	−29.6255	−0.944907	−0.472454	−	0.881355i	$-0.656632\pi$
$983$	−29.6255	−0.944907	−0.472454	+	0.881355i	$0.656632\pi$
$984$	0	0
$985$	28.8065	0.917852
$986$	1.48843	0.0474013
$987$	0	0
$988$	−18.2477	−0.580537
$989$	−1.74208	−0.0553950
$990$	0	0
$991$	−32.5311	−1.03339	−0.516693	−	0.856171i	$-0.672837\pi$
$991$	−32.5311	−1.03339	−0.516693	+	0.856171i	$0.672837\pi$
$992$	−16.2352	−0.515469
$993$	0	0
$994$	−9.62689	−0.305346
$995$	−19.9451	−0.632302
$996$	0	0
$997$	0.00782078	0.000247687 0	0.000123843	−	1.00000i	$-0.499961\pi$
$997$	0.00782078	0.000247687 0	0.000123843		1.00000i	$0.499961\pi$
$998$	−5.69437	−0.180252
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.w.1.14	yes	30
3.2	odd	2		6003.2.a.v.1.17	✓	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6003.2.a.v.1.17	✓	30	3.2	odd	2
6003.2.a.w.1.14	yes	30	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6003 = 3^{2} \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6003.a (trivial)

\(f(q)\)	\(=\)	\(q-0.480943 q^{2} -1.76869 q^{4} -1.51137 q^{5} +3.82960 q^{7} +1.81253 q^{8} +O(q^{10})\)	\(q-0.480943 q^{2} -1.76869 q^{4} -1.51137 q^{5} +3.82960 q^{7} +1.81253 q^{8} +0.726880 q^{10} -4.55830 q^{11} -2.03350 q^{13} -1.84182 q^{14} +2.66567 q^{16} -3.09482 q^{17} -5.07355 q^{19} +2.67314 q^{20} +2.19228 q^{22} -1.00000 q^{23} -2.71578 q^{25} +0.977997 q^{26} -6.77339 q^{28} +1.00000 q^{29} +3.30853 q^{31} -4.90709 q^{32} +1.48843 q^{34} -5.78793 q^{35} +9.63233 q^{37} +2.44008 q^{38} -2.73939 q^{40} -7.83947 q^{41} +1.74208 q^{43} +8.06224 q^{44} +0.480943 q^{46} -6.78144 q^{47} +7.66585 q^{49} +1.30613 q^{50} +3.59664 q^{52} +8.59890 q^{53} +6.88926 q^{55} +6.94125 q^{56} -0.480943 q^{58} -2.86760 q^{59} -3.14004 q^{61} -1.59121 q^{62} -2.97131 q^{64} +3.07336 q^{65} -3.13434 q^{67} +5.47379 q^{68} +2.78366 q^{70} +5.22684 q^{71} +14.0074 q^{73} -4.63260 q^{74} +8.97355 q^{76} -17.4565 q^{77} -2.01553 q^{79} -4.02880 q^{80} +3.77034 q^{82} -0.169808 q^{83} +4.67740 q^{85} -0.837841 q^{86} -8.26204 q^{88} +4.44668 q^{89} -7.78750 q^{91} +1.76869 q^{92} +3.26148 q^{94} +7.66798 q^{95} -12.6221 q^{97} -3.68683 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + q^{2} + 37 q^{4} + 10 q^{7} + 6 q^{8}+O(q^{10}) \) 30 * q + q^2 + 37 * q^4 + 10 * q^7 + 6 * q^8	\( 30 q + q^{2} + 37 q^{4} + 10 q^{7} + 6 q^{8} + 8 q^{10} + 36 q^{13} + 7 q^{14} + 47 q^{16} + 18 q^{17} + 16 q^{19} + 25 q^{22} - 30 q^{23} + 56 q^{25} + 11 q^{26} + 27 q^{28} + 30 q^{29} + 14 q^{31} - 7 q^{32} + 3 q^{34} - 22 q^{35} + 40 q^{37} + 6 q^{38} + 30 q^{40} + 14 q^{41} + 34 q^{43} + 5 q^{44} - q^{46} - 2 q^{47} + 74 q^{49} - 21 q^{50} + 71 q^{52} + 16 q^{53} + 22 q^{55} + 14 q^{56} + q^{58} - 32 q^{59} + 46 q^{61} + 20 q^{62} + 68 q^{64} + 12 q^{65} + 14 q^{67} + 27 q^{68} - 32 q^{71} + 50 q^{73} - 26 q^{74} + 56 q^{76} + 34 q^{77} + 16 q^{79} + 2 q^{80} + 38 q^{82} - 14 q^{83} + 38 q^{85} + 10 q^{86} + 40 q^{88} - 2 q^{89} + 32 q^{91} - 37 q^{92} + 29 q^{94} - 28 q^{95} + 56 q^{97} + 8 q^{98}+O(q^{100}) \) 30 * q + q^2 + 37 * q^4 + 10 * q^7 + 6 * q^8 + 8 * q^10 + 36 * q^13 + 7 * q^14 + 47 * q^16 + 18 * q^17 + 16 * q^19 + 25 * q^22 - 30 * q^23 + 56 * q^25 + 11 * q^26 + 27 * q^28 + 30 * q^29 + 14 * q^31 - 7 * q^32 + 3 * q^34 - 22 * q^35 + 40 * q^37 + 6 * q^38 + 30 * q^40 + 14 * q^41 + 34 * q^43 + 5 * q^44 - q^46 - 2 * q^47 + 74 * q^49 - 21 * q^50 + 71 * q^52 + 16 * q^53 + 22 * q^55 + 14 * q^56 + q^58 - 32 * q^59 + 46 * q^61 + 20 * q^62 + 68 * q^64 + 12 * q^65 + 14 * q^67 + 27 * q^68 - 32 * q^71 + 50 * q^73 - 26 * q^74 + 56 * q^76 + 34 * q^77 + 16 * q^79 + 2 * q^80 + 38 * q^82 - 14 * q^83 + 38 * q^85 + 10 * q^86 + 40 * q^88 - 2 * q^89 + 32 * q^91 - 37 * q^92 + 29 * q^94 - 28 * q^95 + 56 * q^97 + 8 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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