LMFDB - Embedded newform 6003.2.a.q.1.6

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			1.6
Root			$1.84222$ 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)

$f(q)$	$=$	$q-1.84222 q^{2} +1.39376 q^{4} -2.07719 q^{5} +4.15058 q^{7} +1.11683 q^{8} +O(q^{10})$	\(q-1.84222 q^{2} +1.39376 q^{4} -2.07719 q^{5} +4.15058 q^{7} +1.11683 q^{8} +3.82664 q^{10} -0.145774 q^{11} -6.13994 q^{13} -7.64627 q^{14} -4.84495 q^{16} +2.97373 q^{17} +3.57503 q^{19} -2.89511 q^{20} +0.268546 q^{22} -1.00000 q^{23} -0.685266 q^{25} +11.3111 q^{26} +5.78491 q^{28} +1.00000 q^{29} +5.64500 q^{31} +6.69180 q^{32} -5.47826 q^{34} -8.62156 q^{35} -8.51282 q^{37} -6.58598 q^{38} -2.31986 q^{40} +5.62371 q^{41} +7.38629 q^{43} -0.203173 q^{44} +1.84222 q^{46} -4.24940 q^{47} +10.2273 q^{49} +1.26241 q^{50} -8.55760 q^{52} -13.9662 q^{53} +0.302800 q^{55} +4.63548 q^{56} -1.84222 q^{58} -8.75550 q^{59} -7.03185 q^{61} -10.3993 q^{62} -2.63783 q^{64} +12.7539 q^{65} -4.17121 q^{67} +4.14467 q^{68} +15.8828 q^{70} +4.12290 q^{71} +8.24851 q^{73} +15.6825 q^{74} +4.98274 q^{76} -0.605045 q^{77} +10.2491 q^{79} +10.0639 q^{80} -10.3601 q^{82} -11.7911 q^{83} -6.17702 q^{85} -13.6071 q^{86} -0.162804 q^{88} +5.33589 q^{89} -25.4843 q^{91} -1.39376 q^{92} +7.82832 q^{94} -7.42604 q^{95} -7.69604 q^{97} -18.8410 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.84222	−1.30264	−0.651322	−	0.758802i	$-0.725785\pi$
$2$	−1.84222	−1.30264	−0.651322	+	0.758802i	$0.725785\pi$
$3$	0	0
$3$	0	0
$4$	1.39376	0.696880
$5$	−2.07719	−0.928949	−0.464475	−	0.885586i	$-0.653757\pi$
$5$	−2.07719	−0.928949	−0.464475	+	0.885586i	$0.653757\pi$
$6$	0	0
$7$	4.15058	1.56877	0.784386	−	0.620272i	$-0.212978\pi$
$7$	4.15058	1.56877	0.784386	+	0.620272i	$0.212978\pi$
$8$	1.11683	0.394858
$9$	0	0
$10$	3.82664	1.21009
$11$	−0.145774	−0.0439524	−0.0219762	−	0.999758i	$-0.506996\pi$
$11$	−0.145774	−0.0439524	−0.0219762	+	0.999758i	$0.506996\pi$
$12$	0	0
$13$	−6.13994	−1.70291	−0.851457	−	0.524425i	$-0.824280\pi$
$13$	−6.13994	−1.70291	−0.851457	+	0.524425i	$0.824280\pi$
$14$	−7.64627	−2.04355
$15$	0	0
$16$	−4.84495	−1.21124
$17$	2.97373	0.721236	0.360618	−	0.932714i	$-0.382566\pi$
$17$	2.97373	0.721236	0.360618	+	0.932714i	$0.382566\pi$
$18$	0	0
$19$	3.57503	0.820169	0.410084	−	0.912048i	$-0.365499\pi$
$19$	3.57503	0.820169	0.410084	+	0.912048i	$0.365499\pi$
$20$	−2.89511	−0.647366
$21$	0	0
$22$	0.268546	0.0572543
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−0.685266	−0.137053
$26$	11.3111	2.21829
$27$	0	0
$28$	5.78491	1.09325
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	5.64500	1.01387	0.506936	−	0.861984i	$-0.330778\pi$
$31$	5.64500	1.01387	0.506936	+	0.861984i	$0.330778\pi$
$32$	6.69180	1.18295
$33$	0	0
$34$	−5.47826	−0.939514
$35$	−8.62156	−1.45731
$36$	0	0
$37$	−8.51282	−1.39950	−0.699750	−	0.714388i	$-0.746705\pi$
$37$	−8.51282	−1.39950	−0.699750	+	0.714388i	$0.746705\pi$
$38$	−6.58598	−1.06839
$39$	0	0
$40$	−2.31986	−0.366803
$41$	5.62371	0.878275	0.439138	−	0.898420i	$-0.355284\pi$
$41$	5.62371	0.878275	0.439138	+	0.898420i	$0.355284\pi$
$42$	0	0
$43$	7.38629	1.12640	0.563200	−	0.826321i	$-0.309570\pi$
$43$	7.38629	1.12640	0.563200	+	0.826321i	$0.309570\pi$
$44$	−0.203173	−0.0306295
$45$	0	0
$46$	1.84222	0.271620
$47$	−4.24940	−0.619839	−0.309919	−	0.950763i	$-0.600302\pi$
$47$	−4.24940	−0.619839	−0.309919	+	0.950763i	$0.600302\pi$
$48$	0	0
$49$	10.2273	1.46105
$50$	1.26241	0.178531
$51$	0	0
$52$	−8.55760	−1.18673
$53$	−13.9662	−1.91841	−0.959204	−	0.282714i	$-0.908765\pi$
$53$	−13.9662	−1.91841	−0.959204	+	0.282714i	$0.908765\pi$
$54$	0	0
$55$	0.302800	0.0408295
$56$	4.63548	0.619442
$57$	0	0
$58$	−1.84222	−0.241895
$59$	−8.75550	−1.13987	−0.569935	−	0.821690i	$-0.693032\pi$
$59$	−8.75550	−1.13987	−0.569935	+	0.821690i	$0.693032\pi$
$60$	0	0
$61$	−7.03185	−0.900337	−0.450168	−	0.892944i	$-0.648636\pi$
$61$	−7.03185	−0.900337	−0.450168	+	0.892944i	$0.648636\pi$
$62$	−10.3993	−1.32071
$63$	0	0
$64$	−2.63783	−0.329729
$65$	12.7539	1.58192
$66$	0	0
$67$	−4.17121	−0.509595	−0.254798	−	0.966994i	$-0.582009\pi$
$67$	−4.17121	−0.509595	−0.254798	+	0.966994i	$0.582009\pi$
$68$	4.14467	0.502615
$69$	0	0
$70$	15.8828	1.89836
$71$	4.12290	0.489298	0.244649	−	0.969612i	$-0.421327\pi$
$71$	4.12290	0.489298	0.244649	+	0.969612i	$0.421327\pi$
$72$	0	0
$73$	8.24851	0.965415	0.482708	−	0.875782i	$-0.339653\pi$
$73$	8.24851	0.965415	0.482708	+	0.875782i	$0.339653\pi$
$74$	15.6825	1.82305
$75$	0	0
$76$	4.98274	0.571559
$77$	−0.605045	−0.0689513
$78$	0	0
$79$	10.2491	1.15311	0.576555	−	0.817058i	$-0.304397\pi$
$79$	10.2491	1.15311	0.576555	+	0.817058i	$0.304397\pi$
$80$	10.0639	1.12518
$81$	0	0
$82$	−10.3601	−1.14408
$83$	−11.7911	−1.29424	−0.647121	−	0.762387i	$-0.724027\pi$
$83$	−11.7911	−1.29424	−0.647121	+	0.762387i	$0.724027\pi$
$84$	0	0
$85$	−6.17702	−0.669992
$86$	−13.6071	−1.46730
$87$	0	0
$88$	−0.162804	−0.0173549
$89$	5.33589	0.565603	0.282801	−	0.959178i	$-0.408736\pi$
$89$	5.33589	0.565603	0.282801	+	0.959178i	$0.408736\pi$
$90$	0	0
$91$	−25.4843	−2.67148
$92$	−1.39376	−0.145309
$93$	0	0
$94$	7.82832	0.807429
$95$	−7.42604	−0.761895
$96$	0	0
$97$	−7.69604	−0.781415	−0.390707	−	0.920515i	$-0.627770\pi$
$97$	−7.69604	−0.781415	−0.390707	+	0.920515i	$0.627770\pi$
$98$	−18.8410	−1.90322
$99$	0	0
$100$	−0.955096	−0.0955096
$101$	9.94345	0.989410	0.494705	−	0.869061i	$-0.335276\pi$
$101$	9.94345	0.989410	0.494705	+	0.869061i	$0.335276\pi$
$102$	0	0
$103$	−8.27531	−0.815390	−0.407695	−	0.913118i	$-0.633667\pi$
$103$	−8.27531	−0.815390	−0.407695	+	0.913118i	$0.633667\pi$
$104$	−6.85725	−0.672409
$105$	0	0
$106$	25.7288	2.49900
$107$	1.27773	0.123523	0.0617615	−	0.998091i	$-0.480328\pi$
$107$	1.27773	0.123523	0.0617615	+	0.998091i	$0.480328\pi$
$108$	0	0
$109$	−15.6336	−1.49742	−0.748712	−	0.662896i	$-0.769327\pi$
$109$	−15.6336	−1.49742	−0.748712	+	0.662896i	$0.769327\pi$
$110$	−0.557823	−0.0531863
$111$	0	0
$112$	−20.1094	−1.90016
$113$	−8.77553	−0.825532	−0.412766	−	0.910837i	$-0.635437\pi$
$113$	−8.77553	−0.825532	−0.412766	+	0.910837i	$0.635437\pi$
$114$	0	0
$115$	2.07719	0.193699
$116$	1.39376	0.129407
$117$	0	0
$118$	16.1295	1.48484
$119$	12.3427	1.13146
$120$	0	0
$121$	−10.9788	−0.998068
$122$	12.9542	1.17282
$123$	0	0
$124$	7.86777	0.706546
$125$	11.8094	1.05626
$126$	0	0
$127$	−8.11887	−0.720433	−0.360216	−	0.932869i	$-0.617297\pi$
$127$	−8.11887	−0.720433	−0.360216	+	0.932869i	$0.617297\pi$
$128$	−8.52414	−0.753435
$129$	0	0
$130$	−23.4953	−2.06068
$131$	−0.631685	−0.0551906	−0.0275953	−	0.999619i	$-0.508785\pi$
$131$	−0.631685	−0.0551906	−0.0275953	+	0.999619i	$0.508785\pi$
$132$	0	0
$133$	14.8385	1.28666
$134$	7.68428	0.663821
$135$	0	0
$136$	3.32114	0.284786
$137$	10.4588	0.893557	0.446779	−	0.894645i	$-0.352571\pi$
$137$	10.4588	0.893557	0.446779	+	0.894645i	$0.352571\pi$
$138$	0	0
$139$	−10.4118	−0.883121	−0.441561	−	0.897231i	$-0.645575\pi$
$139$	−10.4118	−0.883121	−0.441561	+	0.897231i	$0.645575\pi$
$140$	−12.0164	−1.01557
$141$	0	0
$142$	−7.59528	−0.637381
$143$	0.895042	0.0748471
$144$	0	0
$145$	−2.07719	−0.172502
$146$	−15.1955	−1.25759
$147$	0	0
$148$	−11.8648	−0.975283
$149$	−15.8403	−1.29768	−0.648842	−	0.760923i	$-0.724746\pi$
$149$	−15.8403	−1.29768	−0.648842	+	0.760923i	$0.724746\pi$
$150$	0	0
$151$	17.1020	1.39174	0.695871	−	0.718167i	$-0.255019\pi$
$151$	17.1020	1.39174	0.695871	+	0.718167i	$0.255019\pi$
$152$	3.99269	0.323850
$153$	0	0
$154$	1.11462	0.0898190
$155$	−11.7258	−0.941835
$156$	0	0
$157$	10.5280	0.840223	0.420111	−	0.907473i	$-0.361991\pi$
$157$	10.5280	0.840223	0.420111	+	0.907473i	$0.361991\pi$
$158$	−18.8810	−1.50209
$159$	0	0
$160$	−13.9002	−1.09890
$161$	−4.15058	−0.327112
$162$	0	0
$163$	21.7860	1.70641	0.853205	−	0.521575i	$-0.174655\pi$
$163$	21.7860	1.70641	0.853205	+	0.521575i	$0.174655\pi$
$164$	7.83809	0.612052
$165$	0	0
$166$	21.7218	1.68594
$167$	10.0955	0.781215	0.390608	−	0.920557i	$-0.372265\pi$
$167$	10.0955	0.781215	0.390608	+	0.920557i	$0.372265\pi$
$168$	0	0
$169$	24.6989	1.89992
$170$	11.3794	0.872761
$171$	0	0
$172$	10.2947	0.784965
$173$	14.4082	1.09543	0.547717	−	0.836663i	$-0.315497\pi$
$173$	14.4082	1.09543	0.547717	+	0.836663i	$0.315497\pi$
$174$	0	0
$175$	−2.84425	−0.215005
$176$	0.706266	0.0532368
$177$	0	0
$178$	−9.82985	−0.736779
$179$	−1.09484	−0.0818320	−0.0409160	−	0.999163i	$-0.513028\pi$
$179$	−1.09484	−0.0818320	−0.0409160	+	0.999163i	$0.513028\pi$
$180$	0	0
$181$	18.3262	1.36217	0.681087	−	0.732202i	$-0.261507\pi$
$181$	18.3262	1.36217	0.681087	+	0.732202i	$0.261507\pi$
$182$	46.9477	3.47999
$183$	0	0
$184$	−1.11683	−0.0823335
$185$	17.6828	1.30006
$186$	0	0
$187$	−0.433492	−0.0317001
$188$	−5.92264	−0.431953
$189$	0	0
$190$	13.6804	0.992478
$191$	9.02155	0.652777	0.326388	−	0.945236i	$-0.394168\pi$
$191$	9.02155	0.652777	0.326388	+	0.945236i	$0.394168\pi$
$192$	0	0
$193$	1.40684	0.101266	0.0506331	−	0.998717i	$-0.483876\pi$
$193$	1.40684	0.101266	0.0506331	+	0.998717i	$0.483876\pi$
$194$	14.1778	1.01790
$195$	0	0
$196$	14.2544	1.01817
$197$	−19.6544	−1.40032	−0.700160	−	0.713986i	$-0.746888\pi$
$197$	−19.6544	−1.40032	−0.700160	+	0.713986i	$0.746888\pi$
$198$	0	0
$199$	−2.23480	−0.158421	−0.0792103	−	0.996858i	$-0.525240\pi$
$199$	−2.23480	−0.158421	−0.0792103	+	0.996858i	$0.525240\pi$
$200$	−0.765323	−0.0541165
$201$	0	0
$202$	−18.3180	−1.28885
$203$	4.15058	0.291314
$204$	0	0
$205$	−11.6815	−0.815873
$206$	15.2449	1.06216
$207$	0	0
$208$	29.7477	2.06263
$209$	−0.521146	−0.0360484
$210$	0	0
$211$	2.91932	0.200974	0.100487	−	0.994938i	$-0.467960\pi$
$211$	2.91932	0.200974	0.100487	+	0.994938i	$0.467960\pi$
$212$	−19.4656	−1.33690
$213$	0	0
$214$	−2.35386	−0.160906
$215$	−15.3428	−1.04637
$216$	0	0
$217$	23.4300	1.59053
$218$	28.8004	1.95061
$219$	0	0
$220$	0.422030	0.0284533
$221$	−18.2586	−1.22820
$222$	0	0
$223$	−14.3615	−0.961718	−0.480859	−	0.876798i	$-0.659675\pi$
$223$	−14.3615	−0.961718	−0.480859	+	0.876798i	$0.659675\pi$
$224$	27.7749	1.85579
$225$	0	0
$226$	16.1664	1.07537
$227$	2.25856	0.149906	0.0749529	−	0.997187i	$-0.476119\pi$
$227$	2.25856	0.149906	0.0749529	+	0.997187i	$0.476119\pi$
$228$	0	0
$229$	−4.91286	−0.324651	−0.162325	−	0.986737i	$-0.551899\pi$
$229$	−4.91286	−0.324651	−0.162325	+	0.986737i	$0.551899\pi$
$230$	−3.82664	−0.252321
$231$	0	0
$232$	1.11683	0.0733232
$233$	24.1358	1.58119	0.790595	−	0.612339i	$-0.209771\pi$
$233$	24.1358	1.58119	0.790595	+	0.612339i	$0.209771\pi$
$234$	0	0
$235$	8.82683	0.575799
$236$	−12.2031	−0.794352
$237$	0	0
$238$	−22.7380	−1.47388
$239$	−25.7125	−1.66321	−0.831603	−	0.555370i	$-0.812577\pi$
$239$	−25.7125	−1.66321	−0.831603	+	0.555370i	$0.812577\pi$
$240$	0	0
$241$	24.2176	1.56000	0.779998	−	0.625782i	$-0.215220\pi$
$241$	24.2176	1.56000	0.779998	+	0.625782i	$0.215220\pi$
$242$	20.2252	1.30013
$243$	0	0
$244$	−9.80071	−0.627426
$245$	−21.2442	−1.35724
$246$	0	0
$247$	−21.9505	−1.39668
$248$	6.30448	0.400335
$249$	0	0
$250$	−21.7555	−1.37594
$251$	−20.7820	−1.31175	−0.655875	−	0.754869i	$-0.727700\pi$
$251$	−20.7820	−1.31175	−0.655875	+	0.754869i	$0.727700\pi$
$252$	0	0
$253$	0.145774	0.00916471
$254$	14.9567	0.938467
$255$	0	0
$256$	20.9790	1.31119
$257$	−30.0878	−1.87683	−0.938413	−	0.345515i	$-0.887704\pi$
$257$	−30.0878	−1.87683	−0.938413	+	0.345515i	$0.887704\pi$
$258$	0	0
$259$	−35.3332	−2.19550
$260$	17.7758	1.10241
$261$	0	0
$262$	1.16370	0.0718936
$263$	−2.00117	−0.123397	−0.0616987	−	0.998095i	$-0.519652\pi$
$263$	−2.00117	−0.123397	−0.0616987	+	0.998095i	$0.519652\pi$
$264$	0	0
$265$	29.0106	1.78210
$266$	−27.3357	−1.67606
$267$	0	0
$268$	−5.81367	−0.355126
$269$	−12.8785	−0.785217	−0.392609	−	0.919706i	$-0.628427\pi$
$269$	−12.8785	−0.785217	−0.392609	+	0.919706i	$0.628427\pi$
$270$	0	0
$271$	−28.4267	−1.72680	−0.863399	−	0.504522i	$-0.831669\pi$
$271$	−28.4267	−1.72680	−0.863399	+	0.504522i	$0.831669\pi$
$272$	−14.4076	−0.873589
$273$	0	0
$274$	−19.2674	−1.16399
$275$	0.0998937	0.00602382
$276$	0	0
$277$	−14.6779	−0.881911	−0.440956	−	0.897529i	$-0.645360\pi$
$277$	−14.6779	−0.881911	−0.440956	+	0.897529i	$0.645360\pi$
$278$	19.1809	1.15039
$279$	0	0
$280$	−9.62879	−0.575430
$281$	13.6483	0.814188	0.407094	−	0.913386i	$-0.366542\pi$
$281$	13.6483	0.814188	0.407094	+	0.913386i	$0.366542\pi$
$282$	0	0
$283$	1.32653	0.0788543	0.0394272	−	0.999222i	$-0.487447\pi$
$283$	1.32653	0.0788543	0.0394272	+	0.999222i	$0.487447\pi$
$284$	5.74633	0.340982
$285$	0	0
$286$	−1.64886	−0.0974991
$287$	23.3417	1.37781
$288$	0	0
$289$	−8.15691	−0.479818
$290$	3.82664	0.224708
$291$	0	0
$292$	11.4964	0.672778
$293$	9.39817	0.549047	0.274524	−	0.961580i	$-0.411480\pi$
$293$	9.39817	0.549047	0.274524	+	0.961580i	$0.411480\pi$
$294$	0	0
$295$	18.1869	1.05888
$296$	−9.50734	−0.552603
$297$	0	0
$298$	29.1812	1.69042
$299$	6.13994	0.355082
$300$	0	0
$301$	30.6574	1.76706
$302$	−31.5056	−1.81294
$303$	0	0
$304$	−17.3209	−0.993420
$305$	14.6065	0.836367
$306$	0	0
$307$	19.1720	1.09420	0.547101	−	0.837067i	$-0.315731\pi$
$307$	19.1720	1.09420	0.547101	+	0.837067i	$0.315731\pi$
$308$	−0.843288	−0.0480508
$309$	0	0
$310$	21.6014	1.22688
$311$	11.1820	0.634076	0.317038	−	0.948413i	$-0.397312\pi$
$311$	11.1820	0.634076	0.317038	+	0.948413i	$0.397312\pi$
$312$	0	0
$313$	0.224634	0.0126971	0.00634854	−	0.999980i	$-0.497979\pi$
$313$	0.224634	0.0126971	0.00634854	+	0.999980i	$0.497979\pi$
$314$	−19.3948	−1.09451
$315$	0	0
$316$	14.2847	0.803579
$317$	5.11204	0.287121	0.143560	−	0.989642i	$-0.454145\pi$
$317$	5.11204	0.287121	0.143560	+	0.989642i	$0.454145\pi$
$318$	0	0
$319$	−0.145774	−0.00816175
$320$	5.47928	0.306301
$321$	0	0
$322$	7.64627	0.426110
$323$	10.6312	0.591536
$324$	0	0
$325$	4.20750	0.233390
$326$	−40.1345	−2.22284
$327$	0	0
$328$	6.28070	0.346794
$329$	−17.6375	−0.972386
$330$	0	0
$331$	−5.52003	−0.303408	−0.151704	−	0.988426i	$-0.548476\pi$
$331$	−5.52003	−0.303408	−0.151704	+	0.988426i	$0.548476\pi$
$332$	−16.4340	−0.901931
$333$	0	0
$334$	−18.5981	−1.01765
$335$	8.66442	0.473388
$336$	0	0
$337$	−6.12533	−0.333668	−0.166834	−	0.985985i	$-0.553354\pi$
$337$	−6.12533	−0.333668	−0.166834	+	0.985985i	$0.553354\pi$
$338$	−45.5007	−2.47491
$339$	0	0
$340$	−8.60928	−0.466904
$341$	−0.822892	−0.0445621
$342$	0	0
$343$	13.3953	0.723280
$344$	8.24921	0.444767
$345$	0	0
$346$	−26.5430	−1.42696
$347$	−26.8349	−1.44057	−0.720287	−	0.693676i	$-0.755990\pi$
$347$	−26.8349	−1.44057	−0.720287	+	0.693676i	$0.755990\pi$
$348$	0	0
$349$	−12.5942	−0.674154	−0.337077	−	0.941477i	$-0.609438\pi$
$349$	−12.5942	−0.674154	−0.337077	+	0.941477i	$0.609438\pi$
$350$	5.23973	0.280075
$351$	0	0
$352$	−0.975488	−0.0519937
$353$	−10.7650	−0.572964	−0.286482	−	0.958086i	$-0.592486\pi$
$353$	−10.7650	−0.572964	−0.286482	+	0.958086i	$0.592486\pi$
$354$	0	0
$355$	−8.56407	−0.454533
$356$	7.43694	0.394157
$357$	0	0
$358$	2.01693	0.106598
$359$	−22.5070	−1.18787	−0.593937	−	0.804512i	$-0.702427\pi$
$359$	−22.5070	−1.18787	−0.593937	+	0.804512i	$0.702427\pi$
$360$	0	0
$361$	−6.21913	−0.327323
$362$	−33.7608	−1.77443
$363$	0	0
$364$	−35.5190	−1.86170
$365$	−17.1338	−0.896822
$366$	0	0
$367$	−7.86331	−0.410461	−0.205231	−	0.978714i	$-0.565794\pi$
$367$	−7.86331	−0.410461	−0.205231	+	0.978714i	$0.565794\pi$
$368$	4.84495	0.252561
$369$	0	0
$370$	−32.5755	−1.69352
$371$	−57.9680	−3.00955
$372$	0	0
$373$	−19.7398	−1.02209	−0.511044	−	0.859555i	$-0.670741\pi$
$373$	−19.7398	−1.02209	−0.511044	+	0.859555i	$0.670741\pi$
$374$	0.798586	0.0412939
$375$	0	0
$376$	−4.74584	−0.244748
$377$	−6.13994	−0.316223
$378$	0	0
$379$	−30.5619	−1.56986	−0.784931	−	0.619583i	$-0.787302\pi$
$379$	−30.5619	−1.56986	−0.784931	+	0.619583i	$0.787302\pi$
$380$	−10.3501	−0.530949
$381$	0	0
$382$	−16.6196	−0.850335
$383$	−24.8253	−1.26851	−0.634257	−	0.773123i	$-0.718694\pi$
$383$	−24.8253	−1.26851	−0.634257	+	0.773123i	$0.718694\pi$
$384$	0	0
$385$	1.25680	0.0640523
$386$	−2.59170	−0.131914
$387$	0	0
$388$	−10.7264	−0.544552
$389$	9.31041	0.472056	0.236028	−	0.971746i	$-0.424154\pi$
$389$	9.31041	0.472056	0.236028	+	0.971746i	$0.424154\pi$
$390$	0	0
$391$	−2.97373	−0.150388
$392$	11.4222	0.576906
$393$	0	0
$394$	36.2077	1.82412
$395$	−21.2893	−1.07118
$396$	0	0
$397$	0.692429	0.0347520	0.0173760	−	0.999849i	$-0.494469\pi$
$397$	0.692429	0.0347520	0.0173760	+	0.999849i	$0.494469\pi$
$398$	4.11698	0.206366
$399$	0	0
$400$	3.32008	0.166004
$401$	27.8834	1.39243	0.696215	−	0.717833i	$-0.254866\pi$
$401$	27.8834	1.39243	0.696215	+	0.717833i	$0.254866\pi$
$402$	0	0
$403$	−34.6600	−1.72654
$404$	13.8588	0.689500
$405$	0	0
$406$	−7.64627	−0.379478
$407$	1.24094	0.0615113
$408$	0	0
$409$	−2.99850	−0.148267	−0.0741333	−	0.997248i	$-0.523619\pi$
$409$	−2.99850	−0.148267	−0.0741333	+	0.997248i	$0.523619\pi$
$410$	21.5199	1.06279
$411$	0	0
$412$	−11.5338	−0.568229
$413$	−36.3404	−1.78820
$414$	0	0
$415$	24.4924	1.20228
$416$	−41.0873	−2.01447
$417$	0	0
$418$	0.960063	0.0469582
$419$	−19.7549	−0.965089	−0.482544	−	0.875872i	$-0.660287\pi$
$419$	−19.7549	−0.965089	−0.482544	+	0.875872i	$0.660287\pi$
$420$	0	0
$421$	12.6881	0.618381	0.309191	−	0.951000i	$-0.399942\pi$
$421$	12.6881	0.618381	0.309191	+	0.951000i	$0.399942\pi$
$422$	−5.37801	−0.261798
$423$	0	0
$424$	−15.5978	−0.757498
$425$	−2.03780	−0.0988478
$426$	0	0
$427$	−29.1863	−1.41242
$428$	1.78085	0.0860807
$429$	0	0
$430$	28.2647	1.36304
$431$	−26.4454	−1.27383	−0.636915	−	0.770934i	$-0.719790\pi$
$431$	−26.4454	−1.27383	−0.636915	+	0.770934i	$0.719790\pi$
$432$	0	0
$433$	9.73582	0.467873	0.233937	−	0.972252i	$-0.424839\pi$
$433$	9.73582	0.467873	0.233937	+	0.972252i	$0.424839\pi$
$434$	−43.1632	−2.07190
$435$	0	0
$436$	−21.7894	−1.04352
$437$	−3.57503	−0.171017
$438$	0	0
$439$	−4.42194	−0.211048	−0.105524	−	0.994417i	$-0.533652\pi$
$439$	−4.42194	−0.211048	−0.105524	+	0.994417i	$0.533652\pi$
$440$	0.338175	0.0161219
$441$	0	0
$442$	33.6362	1.59991
$443$	40.6046	1.92918	0.964592	−	0.263747i	$-0.0849584\pi$
$443$	40.6046	1.92918	0.964592	+	0.263747i	$0.0849584\pi$
$444$	0	0
$445$	−11.0837	−0.525416
$446$	26.4570	1.25278
$447$	0	0
$448$	−10.9485	−0.517269
$449$	15.9554	0.752981	0.376490	−	0.926421i	$-0.377131\pi$
$449$	15.9554	0.752981	0.376490	+	0.926421i	$0.377131\pi$
$450$	0	0
$451$	−0.819788	−0.0386023
$452$	−12.2310	−0.575297
$453$	0	0
$454$	−4.16075	−0.195274
$455$	52.9359	2.48167
$456$	0	0
$457$	−10.5150	−0.491871	−0.245935	−	0.969286i	$-0.579095\pi$
$457$	−10.5150	−0.491871	−0.245935	+	0.969286i	$0.579095\pi$
$458$	9.05055	0.422904
$459$	0	0
$460$	2.89511	0.134985
$461$	5.37096	0.250150	0.125075	−	0.992147i	$-0.460083\pi$
$461$	5.37096	0.250150	0.125075	+	0.992147i	$0.460083\pi$
$462$	0	0
$463$	35.2193	1.63678	0.818390	−	0.574663i	$-0.194867\pi$
$463$	35.2193	1.63678	0.818390	+	0.574663i	$0.194867\pi$
$464$	−4.84495	−0.224921
$465$	0	0
$466$	−44.4634	−2.05973
$467$	−22.8465	−1.05721	−0.528606	−	0.848868i	$-0.677285\pi$
$467$	−22.8465	−1.05721	−0.528606	+	0.848868i	$0.677285\pi$
$468$	0	0
$469$	−17.3130	−0.799439
$470$	−16.2609	−0.750061
$471$	0	0
$472$	−9.77837	−0.450086
$473$	−1.07673	−0.0495079
$474$	0	0
$475$	−2.44985	−0.112407
$476$	17.2028	0.788489
$477$	0	0
$478$	47.3681	2.16656
$479$	−34.1908	−1.56222	−0.781109	−	0.624395i	$-0.785346\pi$
$479$	−34.1908	−1.56222	−0.781109	+	0.624395i	$0.785346\pi$
$480$	0	0
$481$	52.2682	2.38323
$482$	−44.6141	−2.03212
$483$	0	0
$484$	−15.3017	−0.695533
$485$	15.9862	0.725895
$486$	0	0
$487$	−12.5516	−0.568769	−0.284384	−	0.958710i	$-0.591789\pi$
$487$	−12.5516	−0.568769	−0.284384	+	0.958710i	$0.591789\pi$
$488$	−7.85336	−0.355505
$489$	0	0
$490$	39.1363	1.76800
$491$	−7.98049	−0.360155	−0.180077	−	0.983652i	$-0.557635\pi$
$491$	−7.98049	−0.360155	−0.180077	+	0.983652i	$0.557635\pi$
$492$	0	0
$493$	2.97373	0.133930
$494$	40.4376	1.81937
$495$	0	0
$496$	−27.3498	−1.22804
$497$	17.1124	0.767598
$498$	0	0
$499$	21.0779	0.943575	0.471788	−	0.881712i	$-0.343609\pi$
$499$	21.0779	0.943575	0.471788	+	0.881712i	$0.343609\pi$
$500$	16.4595	0.736089
$501$	0	0
$502$	38.2850	1.70874
$503$	2.26233	0.100873	0.0504363	−	0.998727i	$-0.483939\pi$
$503$	2.26233	0.100873	0.0504363	+	0.998727i	$0.483939\pi$
$504$	0	0
$505$	−20.6545	−0.919112
$506$	−0.268546	−0.0119383
$507$	0	0
$508$	−11.3157	−0.502055
$509$	−23.6889	−1.04999	−0.524996	−	0.851104i	$-0.675933\pi$
$509$	−23.6889	−1.04999	−0.524996	+	0.851104i	$0.675933\pi$
$510$	0	0
$511$	34.2361	1.51452
$512$	−21.5995	−0.954572
$513$	0	0
$514$	55.4283	2.44484
$515$	17.1894	0.757456
$516$	0	0
$517$	0.619451	0.0272434
$518$	65.0913	2.85995
$519$	0	0
$520$	14.2438	0.624633
$521$	27.7808	1.21710	0.608550	−	0.793516i	$-0.291752\pi$
$521$	27.7808	1.21710	0.608550	+	0.793516i	$0.291752\pi$
$522$	0	0
$523$	−37.7042	−1.64869	−0.824345	−	0.566088i	$-0.808456\pi$
$523$	−37.7042	−1.64869	−0.824345	+	0.566088i	$0.808456\pi$
$524$	−0.880417	−0.0384612
$525$	0	0
$526$	3.68659	0.160743
$527$	16.7867	0.731241
$528$	0	0
$529$	1.00000	0.0434783
$530$	−53.4437	−2.32145
$531$	0	0
$532$	20.6813	0.896646
$533$	−34.5292	−1.49563
$534$	0	0
$535$	−2.65410	−0.114747
$536$	−4.65852	−0.201218
$537$	0	0
$538$	23.7250	1.02286
$539$	−1.49088	−0.0642166
$540$	0	0
$541$	−36.7177	−1.57862	−0.789308	−	0.613997i	$-0.789561\pi$
$541$	−36.7177	−1.57862	−0.789308	+	0.613997i	$0.789561\pi$
$542$	52.3681	2.24940
$543$	0	0
$544$	19.8996	0.853189
$545$	32.4739	1.39103
$546$	0	0
$547$	−14.3544	−0.613749	−0.306874	−	0.951750i	$-0.599283\pi$
$547$	−14.3544	−0.613749	−0.306874	+	0.951750i	$0.599283\pi$
$548$	14.5771	0.622702
$549$	0	0
$550$	−0.184026	−0.00784689
$551$	3.57503	0.152302
$552$	0	0
$553$	42.5396	1.80897
$554$	27.0399	1.14882
$555$	0	0
$556$	−14.5116	−0.615429
$557$	−21.5152	−0.911630	−0.455815	−	0.890074i	$-0.650652\pi$
$557$	−21.5152	−0.911630	−0.455815	+	0.890074i	$0.650652\pi$
$558$	0	0
$559$	−45.3514	−1.91816
$560$	41.7711	1.76515
$561$	0	0
$562$	−25.1431	−1.06060
$563$	28.3505	1.19483	0.597415	−	0.801933i	$-0.296195\pi$
$563$	28.3505	1.19483	0.597415	+	0.801933i	$0.296195\pi$
$564$	0	0
$565$	18.2285	0.766878
$566$	−2.44376	−0.102719
$567$	0	0
$568$	4.60456	0.193203
$569$	32.2071	1.35019	0.675096	−	0.737730i	$-0.264102\pi$
$569$	32.2071	1.35019	0.675096	+	0.737730i	$0.264102\pi$
$570$	0	0
$571$	−41.6110	−1.74137	−0.870683	−	0.491845i	$-0.836323\pi$
$571$	−41.6110	−1.74137	−0.870683	+	0.491845i	$0.836323\pi$
$572$	1.24747	0.0521595
$573$	0	0
$574$	−43.0004	−1.79480
$575$	0.685266	0.0285776
$576$	0	0
$577$	−43.6017	−1.81516	−0.907581	−	0.419878i	$-0.862073\pi$
$577$	−43.6017	−1.81516	−0.907581	+	0.419878i	$0.862073\pi$
$578$	15.0268	0.625032
$579$	0	0
$580$	−2.89511	−0.120213
$581$	−48.9400	−2.03037
$582$	0	0
$583$	2.03591	0.0843186
$584$	9.21215	0.381202
$585$	0	0
$586$	−17.3135	−0.715212
$587$	−10.2996	−0.425109	−0.212555	−	0.977149i	$-0.568178\pi$
$587$	−10.2996	−0.425109	−0.212555	+	0.977149i	$0.568178\pi$
$588$	0	0
$589$	20.1811	0.831546
$590$	−33.5042	−1.37934
$591$	0	0
$592$	41.2442	1.69513
$593$	−43.2869	−1.77758	−0.888789	−	0.458317i	$-0.848453\pi$
$593$	−43.2869	−1.77758	−0.888789	+	0.458317i	$0.848453\pi$
$594$	0	0
$595$	−25.6382	−1.05107
$596$	−22.0775	−0.904330
$597$	0	0
$598$	−11.3111	−0.462545
$599$	6.46895	0.264314	0.132157	−	0.991229i	$-0.457810\pi$
$599$	6.46895	0.264314	0.132157	+	0.991229i	$0.457810\pi$
$600$	0	0
$601$	0.823955	0.0336098	0.0168049	−	0.999859i	$-0.494651\pi$
$601$	0.823955	0.0336098	0.0168049	+	0.999859i	$0.494651\pi$
$602$	−56.4776	−2.30185
$603$	0	0
$604$	23.8361	0.969876
$605$	22.8050	0.927155
$606$	0	0
$607$	−29.0739	−1.18007	−0.590036	−	0.807377i	$-0.700887\pi$
$607$	−29.0739	−1.18007	−0.590036	+	0.807377i	$0.700887\pi$
$608$	23.9234	0.970222
$609$	0	0
$610$	−26.9084	−1.08949
$611$	26.0911	1.05553
$612$	0	0
$613$	42.2174	1.70514	0.852572	−	0.522611i	$-0.175042\pi$
$613$	42.2174	1.70514	0.852572	+	0.522611i	$0.175042\pi$
$614$	−35.3189	−1.42535
$615$	0	0
$616$	−0.675731	−0.0272260
$617$	−30.0536	−1.20991	−0.604957	−	0.796258i	$-0.706810\pi$
$617$	−30.0536	−1.20991	−0.604957	+	0.796258i	$0.706810\pi$
$618$	0	0
$619$	15.7840	0.634413	0.317207	−	0.948356i	$-0.397255\pi$
$619$	15.7840	0.634413	0.317207	+	0.948356i	$0.397255\pi$
$620$	−16.3429	−0.656346
$621$	0	0
$622$	−20.5997	−0.825974
$623$	22.1470	0.887302
$624$	0	0
$625$	−21.1041	−0.844163
$626$	−0.413825	−0.0165398
$627$	0	0
$628$	14.6734	0.585534
$629$	−25.3149	−1.00937
$630$	0	0
$631$	−10.9521	−0.435995	−0.217998	−	0.975949i	$-0.569952\pi$
$631$	−10.9521	−0.435995	−0.217998	+	0.975949i	$0.569952\pi$
$632$	11.4464	0.455314
$633$	0	0
$634$	−9.41749	−0.374016
$635$	16.8645	0.669246
$636$	0	0
$637$	−62.7953	−2.48804
$638$	0.268546	0.0106319
$639$	0	0
$640$	17.7063	0.699903
$641$	11.7178	0.462824	0.231412	−	0.972856i	$-0.425666\pi$
$641$	11.7178	0.462824	0.231412	+	0.972856i	$0.425666\pi$
$642$	0	0
$643$	−14.3263	−0.564975	−0.282488	−	0.959271i	$-0.591160\pi$
$643$	−14.3263	−0.564975	−0.282488	+	0.959271i	$0.591160\pi$
$644$	−5.78491	−0.227958
$645$	0	0
$646$	−19.5850	−0.770560
$647$	11.0142	0.433014	0.216507	−	0.976281i	$-0.430534\pi$
$647$	11.0142	0.433014	0.216507	+	0.976281i	$0.430534\pi$
$648$	0	0
$649$	1.27632	0.0501000
$650$	−7.75111	−0.304024
$651$	0	0
$652$	30.3644	1.18916
$653$	−37.6448	−1.47316	−0.736578	−	0.676353i	$-0.763560\pi$
$653$	−37.6448	−1.47316	−0.736578	+	0.676353i	$0.763560\pi$
$654$	0	0
$655$	1.31213	0.0512692
$656$	−27.2466	−1.06380
$657$	0	0
$658$	32.4921	1.26667
$659$	28.4006	1.10633	0.553166	−	0.833071i	$-0.313419\pi$
$659$	28.4006	1.10633	0.553166	+	0.833071i	$0.313419\pi$
$660$	0	0
$661$	18.0622	0.702538	0.351269	−	0.936275i	$-0.385750\pi$
$661$	18.0622	0.702538	0.351269	+	0.936275i	$0.385750\pi$
$662$	10.1691	0.395233
$663$	0	0
$664$	−13.1686	−0.511041
$665$	−30.8224	−1.19524
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	14.0707	0.544413
$669$	0	0
$670$	−15.9617	−0.616656
$671$	1.02506	0.0395720
$672$	0	0
$673$	−41.4922	−1.59941	−0.799703	−	0.600396i	$-0.795010\pi$
$673$	−41.4922	−1.59941	−0.799703	+	0.600396i	$0.795010\pi$
$674$	11.2842	0.434650
$675$	0	0
$676$	34.4243	1.32401
$677$	0.747122	0.0287142	0.0143571	−	0.999897i	$-0.495430\pi$
$677$	0.747122	0.0287142	0.0143571	+	0.999897i	$0.495430\pi$
$678$	0	0
$679$	−31.9431	−1.22586
$680$	−6.89866	−0.264551
$681$	0	0
$682$	1.51594	0.0580485
$683$	27.1608	1.03928	0.519639	−	0.854386i	$-0.326067\pi$
$683$	27.1608	1.03928	0.519639	+	0.854386i	$0.326067\pi$
$684$	0	0
$685$	−21.7250	−0.830069
$686$	−24.6771	−0.942176
$687$	0	0
$688$	−35.7863	−1.36434
$689$	85.7518	3.26688
$690$	0	0
$691$	−14.5087	−0.551939	−0.275969	−	0.961166i	$-0.588999\pi$
$691$	−14.5087	−0.551939	−0.275969	+	0.961166i	$0.588999\pi$
$692$	20.0816	0.763386
$693$	0	0
$694$	49.4357	1.87655
$695$	21.6274	0.820375
$696$	0	0
$697$	16.7234	0.633444
$698$	23.2013	0.878182
$699$	0	0
$700$	−3.96421	−0.149833
$701$	22.3077	0.842551	0.421276	−	0.906933i	$-0.361582\pi$
$701$	22.3077	0.842551	0.421276	+	0.906933i	$0.361582\pi$
$702$	0	0
$703$	−30.4336	−1.14783
$704$	0.384526	0.0144924
$705$	0	0
$706$	19.8315	0.746367
$707$	41.2711	1.55216
$708$	0	0
$709$	−37.1956	−1.39691	−0.698455	−	0.715654i	$-0.746129\pi$
$709$	−37.1956	−1.39691	−0.698455	+	0.715654i	$0.746129\pi$
$710$	15.7769	0.592095
$711$	0	0
$712$	5.95926	0.223333
$713$	−5.64500	−0.211407
$714$	0	0
$715$	−1.85917	−0.0695292
$716$	−1.52594	−0.0570271
$717$	0	0
$718$	41.4627	1.54738
$719$	−29.9152	−1.11565	−0.557824	−	0.829959i	$-0.688364\pi$
$719$	−29.9152	−1.11565	−0.557824	+	0.829959i	$0.688364\pi$
$720$	0	0
$721$	−34.3473	−1.27916
$722$	11.4570	0.426385
$723$	0	0
$724$	25.5423	0.949272
$725$	−0.685266	−0.0254501
$726$	0	0
$727$	−9.84613	−0.365173	−0.182586	−	0.983190i	$-0.558447\pi$
$727$	−9.84613	−0.365173	−0.182586	+	0.983190i	$0.558447\pi$
$728$	−28.4616	−1.05486
$729$	0	0
$730$	31.5641	1.16824
$731$	21.9649	0.812400
$732$	0	0
$733$	24.9293	0.920783	0.460392	−	0.887716i	$-0.347709\pi$
$733$	24.9293	0.920783	0.460392	+	0.887716i	$0.347709\pi$
$734$	14.4859	0.534685
$735$	0	0
$736$	−6.69180	−0.246663
$737$	0.608053	0.0223979
$738$	0	0
$739$	11.8686	0.436595	0.218297	−	0.975882i	$-0.429950\pi$
$739$	11.8686	0.436595	0.218297	+	0.975882i	$0.429950\pi$
$740$	24.6455	0.905988
$741$	0	0
$742$	106.790	3.92037
$743$	12.3354	0.452540	0.226270	−	0.974065i	$-0.427347\pi$
$743$	12.3354	0.452540	0.226270	+	0.974065i	$0.427347\pi$
$744$	0	0
$745$	32.9033	1.20548
$746$	36.3650	1.33142
$747$	0	0
$748$	−0.604183	−0.0220911
$749$	5.30333	0.193780
$750$	0	0
$751$	−44.0869	−1.60875	−0.804376	−	0.594120i	$-0.797500\pi$
$751$	−44.0869	−1.60875	−0.804376	+	0.594120i	$0.797500\pi$
$752$	20.5882	0.750773
$753$	0	0
$754$	11.3111	0.411926
$755$	−35.5242	−1.29286
$756$	0	0
$757$	−41.7631	−1.51791	−0.758953	−	0.651145i	$-0.774289\pi$
$757$	−41.7631	−1.51791	−0.758953	+	0.651145i	$0.774289\pi$
$758$	56.3017	2.04497
$759$	0	0
$760$	−8.29359	−0.300840
$761$	−7.73853	−0.280522	−0.140261	−	0.990115i	$-0.544794\pi$
$761$	−7.73853	−0.280522	−0.140261	+	0.990115i	$0.544794\pi$
$762$	0	0
$763$	−64.8884	−2.34912
$764$	12.5739	0.454907
$765$	0	0
$766$	45.7336	1.65242
$767$	53.7583	1.94110
$768$	0	0
$769$	−4.76866	−0.171962	−0.0859811	−	0.996297i	$-0.527402\pi$
$769$	−4.76866	−0.171962	−0.0859811	+	0.996297i	$0.527402\pi$
$770$	−2.31529	−0.0834373
$771$	0	0
$772$	1.96079	0.0705704
$773$	−21.7146	−0.781021	−0.390511	−	0.920598i	$-0.627701\pi$
$773$	−21.7146	−0.781021	−0.390511	+	0.920598i	$0.627701\pi$
$774$	0	0
$775$	−3.86833	−0.138954
$776$	−8.59514	−0.308548
$777$	0	0
$778$	−17.1518	−0.614921
$779$	20.1049	0.720334
$780$	0	0
$781$	−0.601010	−0.0215058
$782$	5.47826	0.195902
$783$	0	0
$784$	−49.5510	−1.76968
$785$	−21.8686	−0.780525
$786$	0	0
$787$	20.6919	0.737587	0.368794	−	0.929511i	$-0.379771\pi$
$787$	20.6919	0.737587	0.368794	+	0.929511i	$0.379771\pi$
$788$	−27.3935	−0.975854
$789$	0	0
$790$	39.2195	1.39537
$791$	−36.4236	−1.29507
$792$	0	0
$793$	43.1752	1.53320
$794$	−1.27560	−0.0452695
$795$	0	0
$796$	−3.11477	−0.110400
$797$	28.7124	1.01705	0.508523	−	0.861048i	$-0.330192\pi$
$797$	28.7124	1.01705	0.508523	+	0.861048i	$0.330192\pi$
$798$	0	0
$799$	−12.6366	−0.447050
$800$	−4.58566	−0.162128
$801$	0	0
$802$	−51.3672	−1.81384
$803$	−1.20242	−0.0424323
$804$	0	0
$805$	8.62156	0.303870
$806$	63.8511	2.24906
$807$	0	0
$808$	11.1051	0.390676
$809$	2.01995	0.0710178	0.0355089	−	0.999369i	$-0.488695\pi$
$809$	2.01995	0.0710178	0.0355089	+	0.999369i	$0.488695\pi$
$810$	0	0
$811$	10.0804	0.353970	0.176985	−	0.984214i	$-0.443366\pi$
$811$	10.0804	0.353970	0.176985	+	0.984214i	$0.443366\pi$
$812$	5.78491	0.203011
$813$	0	0
$814$	−2.28609	−0.0801273
$815$	−45.2537	−1.58517
$816$	0	0
$817$	26.4063	0.923838
$818$	5.52389	0.193138
$819$	0	0
$820$	−16.2812	−0.568565
$821$	−15.6006	−0.544464	−0.272232	−	0.962232i	$-0.587762\pi$
$821$	−15.6006	−0.544464	−0.272232	+	0.962232i	$0.587762\pi$
$822$	0	0
$823$	−34.5435	−1.20411	−0.602056	−	0.798454i	$-0.705652\pi$
$823$	−34.5435	−1.20411	−0.602056	+	0.798454i	$0.705652\pi$
$824$	−9.24208	−0.321963
$825$	0	0
$826$	66.9469	2.32938
$827$	−6.81132	−0.236853	−0.118426	−	0.992963i	$-0.537785\pi$
$827$	−6.81132	−0.236853	−0.118426	+	0.992963i	$0.537785\pi$
$828$	0	0
$829$	25.9918	0.902733	0.451366	−	0.892339i	$-0.350937\pi$
$829$	25.9918	0.902733	0.451366	+	0.892339i	$0.350937\pi$
$830$	−45.1203	−1.56615
$831$	0	0
$832$	16.1961	0.561500
$833$	30.4134	1.05376
$834$	0	0
$835$	−20.9704	−0.725710
$836$	−0.726351	−0.0251214
$837$	0	0
$838$	36.3927	1.25717
$839$	−28.5857	−0.986887	−0.493444	−	0.869778i	$-0.664262\pi$
$839$	−28.5857	−0.986887	−0.493444	+	0.869778i	$0.664262\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−23.3743	−0.805530
$843$	0	0
$844$	4.06883	0.140055
$845$	−51.3044	−1.76493
$846$	0	0
$847$	−45.5682	−1.56574
$848$	67.6657	2.32365
$849$	0	0
$850$	3.75407	0.128763
$851$	8.51282	0.291816
$852$	0	0
$853$	−50.1708	−1.71782	−0.858908	−	0.512130i	$-0.828857\pi$
$853$	−50.1708	−1.71782	−0.858908	+	0.512130i	$0.828857\pi$
$854$	53.7675	1.83988
$855$	0	0
$856$	1.42700	0.0487740
$857$	13.5371	0.462420	0.231210	−	0.972904i	$-0.425732\pi$
$857$	13.5371	0.462420	0.231210	+	0.972904i	$0.425732\pi$
$858$	0	0
$859$	55.2656	1.88564	0.942819	−	0.333304i	$-0.108164\pi$
$859$	55.2656	1.88564	0.942819	+	0.333304i	$0.108164\pi$
$860$	−21.3841	−0.729192
$861$	0	0
$862$	48.7182	1.65935
$863$	−55.3839	−1.88529	−0.942644	−	0.333799i	$-0.891669\pi$
$863$	−55.3839	−1.88529	−0.942644	+	0.333799i	$0.891669\pi$
$864$	0	0
$865$	−29.9286	−1.01760
$866$	−17.9355	−0.609472
$867$	0	0
$868$	32.6558	1.10841
$869$	−1.49404	−0.0506820
$870$	0	0
$871$	25.6110	0.867796
$872$	−17.4600	−0.591269
$873$	0	0
$874$	6.58598	0.222774
$875$	49.0159	1.65704
$876$	0	0
$877$	53.5484	1.80820	0.904101	−	0.427318i	$-0.140542\pi$
$877$	53.5484	1.80820	0.904101	+	0.427318i	$0.140542\pi$
$878$	8.14617	0.274920
$879$	0	0
$880$	−1.46705	−0.0494543
$881$	−16.9053	−0.569555	−0.284778	−	0.958594i	$-0.591920\pi$
$881$	−16.9053	−0.569555	−0.284778	+	0.958594i	$0.591920\pi$
$882$	0	0
$883$	−44.7662	−1.50650	−0.753252	−	0.657732i	$-0.771516\pi$
$883$	−44.7662	−1.50650	−0.753252	+	0.657732i	$0.771516\pi$
$884$	−25.4480	−0.855910
$885$	0	0
$886$	−74.8025	−2.51304
$887$	−7.46392	−0.250614	−0.125307	−	0.992118i	$-0.539992\pi$
$887$	−7.46392	−0.250614	−0.125307	+	0.992118i	$0.539992\pi$
$888$	0	0
$889$	−33.6980	−1.13020
$890$	20.4185	0.684430
$891$	0	0
$892$	−20.0165	−0.670202
$893$	−15.1918	−0.508373
$894$	0	0
$895$	2.27419	0.0760178
$896$	−35.3802	−1.18197
$897$	0	0
$898$	−29.3933	−0.980866
$899$	5.64500	0.188271
$900$	0	0
$901$	−41.5318	−1.38363
$902$	1.51023	0.0502850
$903$	0	0
$904$	−9.80074	−0.325968
$905$	−38.0670	−1.26539
$906$	0	0
$907$	27.3474	0.908054	0.454027	−	0.890988i	$-0.349987\pi$
$907$	27.3474	0.908054	0.454027	+	0.890988i	$0.349987\pi$
$908$	3.14789	0.104466
$909$	0	0
$910$	−97.5194	−3.23274
$911$	−9.45160	−0.313146	−0.156573	−	0.987666i	$-0.550045\pi$
$911$	−9.45160	−0.313146	−0.156573	+	0.987666i	$0.550045\pi$
$912$	0	0
$913$	1.71883	0.0568850
$914$	19.3709	0.640732
$915$	0	0
$916$	−6.84734	−0.226243
$917$	−2.62186	−0.0865815
$918$	0	0
$919$	−14.3653	−0.473867	−0.236934	−	0.971526i	$-0.576142\pi$
$919$	−14.3653	−0.473867	−0.236934	+	0.971526i	$0.576142\pi$
$920$	2.31986	0.0764837
$921$	0	0
$922$	−9.89446	−0.325857
$923$	−25.3144	−0.833233
$924$	0	0
$925$	5.83355	0.191806
$926$	−64.8816	−2.13214
$927$	0	0
$928$	6.69180	0.219669
$929$	−16.5830	−0.544069	−0.272035	−	0.962287i	$-0.587697\pi$
$929$	−16.5830	−0.544069	−0.272035	+	0.962287i	$0.587697\pi$
$930$	0	0
$931$	36.5631	1.19831
$932$	33.6395	1.10190
$933$	0	0
$934$	42.0882	1.37717
$935$	0.900447	0.0294478
$936$	0	0
$937$	41.9256	1.36965	0.684825	−	0.728707i	$-0.259879\pi$
$937$	41.9256	1.36965	0.684825	+	0.728707i	$0.259879\pi$
$938$	31.8942	1.04138
$939$	0	0
$940$	12.3025	0.401263
$941$	−16.8417	−0.549023	−0.274511	−	0.961584i	$-0.588516\pi$
$941$	−16.8417	−0.549023	−0.274511	+	0.961584i	$0.588516\pi$
$942$	0	0
$943$	−5.62371	−0.183133
$944$	42.4200	1.38065
$945$	0	0
$946$	1.98356	0.0644912
$947$	−16.6787	−0.541984	−0.270992	−	0.962582i	$-0.587352\pi$
$947$	−16.6787	−0.541984	−0.270992	+	0.962582i	$0.587352\pi$
$948$	0	0
$949$	−50.6454	−1.64402
$950$	4.51315	0.146426
$951$	0	0
$952$	13.7847	0.446764
$953$	11.0915	0.359289	0.179644	−	0.983732i	$-0.442505\pi$
$953$	11.0915	0.359289	0.179644	+	0.983732i	$0.442505\pi$
$954$	0	0
$955$	−18.7395	−0.606396
$956$	−35.8371	−1.15905
$957$	0	0
$958$	62.9869	2.03501
$959$	43.4102	1.40179
$960$	0	0
$961$	0.866004	0.0279356
$962$	−96.2894	−3.10449
$963$	0	0
$964$	33.7536	1.08713
$965$	−2.92227	−0.0940712
$966$	0	0
$967$	−18.9893	−0.610654	−0.305327	−	0.952248i	$-0.598766\pi$
$967$	−18.9893	−0.610654	−0.305327	+	0.952248i	$0.598766\pi$
$968$	−12.2614	−0.394095
$969$	0	0
$970$	−29.4500	−0.945582
$971$	44.4160	1.42538	0.712688	−	0.701481i	$-0.247477\pi$
$971$	44.4160	1.42538	0.712688	+	0.701481i	$0.247477\pi$
$972$	0	0
$973$	−43.2152	−1.38542
$974$	23.1228	0.740903
$975$	0	0
$976$	34.0690	1.09052
$977$	37.2234	1.19088	0.595441	−	0.803399i	$-0.296977\pi$
$977$	37.2234	1.19088	0.595441	+	0.803399i	$0.296977\pi$
$978$	0	0
$979$	−0.777831	−0.0248596
$980$	−29.6092	−0.945833
$981$	0	0
$982$	14.7018	0.469153
$983$	−36.7550	−1.17230	−0.586152	−	0.810201i	$-0.699358\pi$
$983$	−36.7550	−1.17230	−0.586152	+	0.810201i	$0.699358\pi$
$984$	0	0
$985$	40.8261	1.30083
$986$	−5.47826	−0.174463
$987$	0	0
$988$	−30.5937	−0.973316
$989$	−7.38629	−0.234870
$990$	0	0
$991$	15.0774	0.478948	0.239474	−	0.970903i	$-0.423025\pi$
$991$	15.0774	0.478948	0.239474	+	0.970903i	$0.423025\pi$
$992$	37.7752	1.19936
$993$	0	0
$994$	−31.5248	−0.999907
$995$	4.64211	0.147165
$996$	0	0
$997$	6.64401	0.210418	0.105209	−	0.994450i	$-0.466449\pi$
$997$	6.64401	0.210418	0.105209	+	0.994450i	$0.466449\pi$
$998$	−38.8300	−1.22914
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.q.1.6		16
3.2	odd	2		667.2.a.d.1.11	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
667.2.a.d.1.11	✓	16	3.2	odd	2
6003.2.a.q.1.6		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)

\(f(q)\)	\(=\)	\(q-1.84222 q^{2} +1.39376 q^{4} -2.07719 q^{5} +4.15058 q^{7} +1.11683 q^{8} +O(q^{10})\)	\(q-1.84222 q^{2} +1.39376 q^{4} -2.07719 q^{5} +4.15058 q^{7} +1.11683 q^{8} +3.82664 q^{10} -0.145774 q^{11} -6.13994 q^{13} -7.64627 q^{14} -4.84495 q^{16} +2.97373 q^{17} +3.57503 q^{19} -2.89511 q^{20} +0.268546 q^{22} -1.00000 q^{23} -0.685266 q^{25} +11.3111 q^{26} +5.78491 q^{28} +1.00000 q^{29} +5.64500 q^{31} +6.69180 q^{32} -5.47826 q^{34} -8.62156 q^{35} -8.51282 q^{37} -6.58598 q^{38} -2.31986 q^{40} +5.62371 q^{41} +7.38629 q^{43} -0.203173 q^{44} +1.84222 q^{46} -4.24940 q^{47} +10.2273 q^{49} +1.26241 q^{50} -8.55760 q^{52} -13.9662 q^{53} +0.302800 q^{55} +4.63548 q^{56} -1.84222 q^{58} -8.75550 q^{59} -7.03185 q^{61} -10.3993 q^{62} -2.63783 q^{64} +12.7539 q^{65} -4.17121 q^{67} +4.14467 q^{68} +15.8828 q^{70} +4.12290 q^{71} +8.24851 q^{73} +15.6825 q^{74} +4.98274 q^{76} -0.605045 q^{77} +10.2491 q^{79} +10.0639 q^{80} -10.3601 q^{82} -11.7911 q^{83} -6.17702 q^{85} -13.6071 q^{86} -0.162804 q^{88} +5.33589 q^{89} -25.4843 q^{91} -1.39376 q^{92} +7.82832 q^{94} -7.42604 q^{95} -7.69604 q^{97} -18.8410 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 3 q^{2} + 21 q^{4} - 16 q^{5} + q^{7} - 9 q^{8}+O(q^{10}) \) 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

Modular forms

Varieties

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Database

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