LMFDB - Embedded newform 6003.2.a.s.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:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 2 x^{19} - 33 x^{18} + 64 x^{17} + 453 x^{16} - 846 x^{15} - 3353 x^{14} + 5985 x^{13} + \cdots - 256 $ x^20 - 2x^19 - 33x^18 + 64x^17 + 453x^16 - 846x^15 - 3353x^14 + 5985x^13 + 14484x^12 - 24566x^11 - 36791x^10 + 59410x^9 + 52109x^8 - 82362x^7 - 34967x^6 + 60661x^5 + 5201x^4 - 19624x^3 + 2768x^2 + 1408*x - 256
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	no (minimal twist has level 2001)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Root			$-1.19744$ 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)

$f(q)$	$=$	$q+1.19744 q^{2} -0.566132 q^{4} +2.06209 q^{5} -5.04192 q^{7} -3.07279 q^{8} +O(q^{10})$	\(q+1.19744 q^{2} -0.566132 q^{4} +2.06209 q^{5} -5.04192 q^{7} -3.07279 q^{8} +2.46923 q^{10} +3.25064 q^{11} -6.23948 q^{13} -6.03741 q^{14} -2.54723 q^{16} +1.68848 q^{17} -6.81012 q^{19} -1.16742 q^{20} +3.89245 q^{22} -1.00000 q^{23} -0.747782 q^{25} -7.47142 q^{26} +2.85439 q^{28} -1.00000 q^{29} +7.68803 q^{31} +3.09543 q^{32} +2.02185 q^{34} -10.3969 q^{35} +8.10347 q^{37} -8.15472 q^{38} -6.33638 q^{40} +3.55608 q^{41} -5.36003 q^{43} -1.84029 q^{44} -1.19744 q^{46} +2.48056 q^{47} +18.4210 q^{49} -0.895426 q^{50} +3.53237 q^{52} +12.9770 q^{53} +6.70311 q^{55} +15.4928 q^{56} -1.19744 q^{58} +6.25362 q^{59} -5.81696 q^{61} +9.20597 q^{62} +8.80106 q^{64} -12.8664 q^{65} +8.79413 q^{67} -0.955901 q^{68} -12.4497 q^{70} -3.86749 q^{71} +4.08466 q^{73} +9.70343 q^{74} +3.85542 q^{76} -16.3895 q^{77} +3.99104 q^{79} -5.25262 q^{80} +4.25820 q^{82} +10.7509 q^{83} +3.48179 q^{85} -6.41833 q^{86} -9.98854 q^{88} +6.20467 q^{89} +31.4590 q^{91} +0.566132 q^{92} +2.97033 q^{94} -14.0431 q^{95} -7.98764 q^{97} +22.0581 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 2 q^{2} + 30 q^{4} + q^{5} + 9 q^{7} - 6 q^{8}+O(q^{10}) $ 20 * q - 2 * q^2 + 30 * q^4 + q^5 + 9 * q^7 - 6 * q^8	$ 20 q - 2 q^{2} + 30 q^{4} + q^{5} + 9 q^{7} - 6 q^{8} + 7 q^{10} + 21 q^{13} + q^{14} + 58 q^{16} + 4 q^{17} + 7 q^{19} + 20 q^{20} + 7 q^{22} - 20 q^{23} + 47 q^{25} - 8 q^{26} + 11 q^{28} - 20 q^{29} + 28 q^{31} - 14 q^{32} + 16 q^{34} - 9 q^{35} + 14 q^{37} + 20 q^{38} + 34 q^{40} - 7 q^{41} + 3 q^{43} + q^{44} + 2 q^{46} - 3 q^{47} + 35 q^{49} + 24 q^{50} + 73 q^{52} + 19 q^{53} + 29 q^{55} + 30 q^{56} + 2 q^{58} - 20 q^{59} + 15 q^{61} - 12 q^{62} + 82 q^{64} + 28 q^{65} + 20 q^{67} + 23 q^{68} - 24 q^{70} - 63 q^{71} + 19 q^{73} - 16 q^{74} - 44 q^{76} + 7 q^{77} + 32 q^{79} + 56 q^{80} - 20 q^{82} + 21 q^{83} + 4 q^{85} + 6 q^{86} + 55 q^{88} + 13 q^{89} + 70 q^{91} - 30 q^{92} - 12 q^{94} - 9 q^{95} - 9 q^{97} - 31 q^{98}+O(q^{100}) $ 20 * q - 2 * q^2 + 30 * q^4 + q^5 + 9 * q^7 - 6 * q^8 + 7 * q^10 + 21 * q^13 + q^14 + 58 * q^16 + 4 * q^17 + 7 * q^19 + 20 * q^20 + 7 * q^22 - 20 * q^23 + 47 * q^25 - 8 * q^26 + 11 * q^28 - 20 * q^29 + 28 * q^31 - 14 * q^32 + 16 * q^34 - 9 * q^35 + 14 * q^37 + 20 * q^38 + 34 * q^40 - 7 * q^41 + 3 * q^43 + q^44 + 2 * q^46 - 3 * q^47 + 35 * q^49 + 24 * q^50 + 73 * q^52 + 19 * q^53 + 29 * q^55 + 30 * q^56 + 2 * q^58 - 20 * q^59 + 15 * q^61 - 12 * q^62 + 82 * q^64 + 28 * q^65 + 20 * q^67 + 23 * q^68 - 24 * q^70 - 63 * q^71 + 19 * q^73 - 16 * q^74 - 44 * q^76 + 7 * q^77 + 32 * q^79 + 56 * q^80 - 20 * q^82 + 21 * q^83 + 4 * q^85 + 6 * q^86 + 55 * q^88 + 13 * q^89 + 70 * q^91 - 30 * q^92 - 12 * q^94 - 9 * q^95 - 9 * q^97 - 31 * 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 $

$2$	1.19744	0.846720	0.423360	−	0.905962i	$-0.360851\pi$
$2$	1.19744	0.846720	0.423360	+	0.905962i	$0.360851\pi$
$3$	0	0
$3$	0	0
$4$	−0.566132	−0.283066
$5$	2.06209	0.922195	0.461098	−	0.887349i	$-0.347456\pi$
$5$	2.06209	0.922195	0.461098	+	0.887349i	$0.347456\pi$
$6$	0	0
$7$	−5.04192	−1.90567	−0.952834	−	0.303492i	$-0.901847\pi$
$7$	−5.04192	−1.90567	−0.952834	+	0.303492i	$0.901847\pi$
$8$	−3.07279	−1.08640
$9$	0	0
$10$	2.46923	0.780841
$11$	3.25064	0.980104	0.490052	−	0.871693i	$-0.336978\pi$
$11$	3.25064	0.980104	0.490052	+	0.871693i	$0.336978\pi$
$12$	0	0
$13$	−6.23948	−1.73052	−0.865261	−	0.501323i	$-0.832847\pi$
$13$	−6.23948	−1.73052	−0.865261	+	0.501323i	$0.832847\pi$
$14$	−6.03741	−1.61357
$15$	0	0
$16$	−2.54723	−0.636808
$17$	1.68848	0.409516	0.204758	−	0.978813i	$-0.434359\pi$
$17$	1.68848	0.409516	0.204758	+	0.978813i	$0.434359\pi$
$18$	0	0
$19$	−6.81012	−1.56235	−0.781174	−	0.624313i	$-0.785379\pi$
$19$	−6.81012	−1.56235	−0.781174	+	0.624313i	$0.785379\pi$
$20$	−1.16742	−0.261042
$21$	0	0
$22$	3.89245	0.829873
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−0.747782	−0.149556
$26$	−7.47142	−1.46527
$27$	0	0
$28$	2.85439	0.539430
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	7.68803	1.38081	0.690405	−	0.723423i	$-0.257432\pi$
$31$	7.68803	1.38081	0.690405	+	0.723423i	$0.257432\pi$
$32$	3.09543	0.547199
$33$	0	0
$34$	2.02185	0.346745
$35$	−10.3969	−1.75740
$36$	0	0
$37$	8.10347	1.33220	0.666101	−	0.745862i	$-0.267962\pi$
$37$	8.10347	1.33220	0.666101	+	0.745862i	$0.267962\pi$
$38$	−8.15472	−1.32287
$39$	0	0
$40$	−6.33638	−1.00187
$41$	3.55608	0.555366	0.277683	−	0.960673i	$-0.410434\pi$
$41$	3.55608	0.555366	0.277683	+	0.960673i	$0.410434\pi$
$42$	0	0
$43$	−5.36003	−0.817397	−0.408699	−	0.912669i	$-0.634017\pi$
$43$	−5.36003	−0.817397	−0.408699	+	0.912669i	$0.634017\pi$
$44$	−1.84029	−0.277434
$45$	0	0
$46$	−1.19744	−0.176553
$47$	2.48056	0.361828	0.180914	−	0.983499i	$-0.442095\pi$
$47$	2.48056	0.361828	0.180914	+	0.983499i	$0.442095\pi$
$48$	0	0
$49$	18.4210	2.63157
$50$	−0.895426	−0.126632
$51$	0	0
$52$	3.53237	0.489852
$53$	12.9770	1.78252	0.891261	−	0.453491i	$-0.149822\pi$
$53$	12.9770	1.78252	0.891261	+	0.453491i	$0.149822\pi$
$54$	0	0
$55$	6.70311	0.903847
$56$	15.4928	2.07031
$57$	0	0
$58$	−1.19744	−0.157232
$59$	6.25362	0.814152	0.407076	−	0.913394i	$-0.366548\pi$
$59$	6.25362	0.814152	0.407076	+	0.913394i	$0.366548\pi$
$60$	0	0
$61$	−5.81696	−0.744786	−0.372393	−	0.928075i	$-0.621463\pi$
$61$	−5.81696	−0.744786	−0.372393	+	0.928075i	$0.621463\pi$
$62$	9.20597	1.16916
$63$	0	0
$64$	8.80106	1.10013
$65$	−12.8664	−1.59588
$66$	0	0
$67$	8.79413	1.07437	0.537187	−	0.843463i	$-0.319487\pi$
$67$	8.79413	1.07437	0.537187	+	0.843463i	$0.319487\pi$
$68$	−0.955901	−0.115920
$69$	0	0
$70$	−12.4497	−1.48802
$71$	−3.86749	−0.458986	−0.229493	−	0.973310i	$-0.573707\pi$
$71$	−3.86749	−0.458986	−0.229493	+	0.973310i	$0.573707\pi$
$72$	0	0
$73$	4.08466	0.478073	0.239037	−	0.971011i	$-0.423168\pi$
$73$	4.08466	0.478073	0.239037	+	0.971011i	$0.423168\pi$
$74$	9.70343	1.12800
$75$	0	0
$76$	3.85542	0.442248
$77$	−16.3895	−1.86775
$78$	0	0
$79$	3.99104	0.449027	0.224513	−	0.974471i	$-0.427921\pi$
$79$	3.99104	0.449027	0.224513	+	0.974471i	$0.427921\pi$
$80$	−5.25262	−0.587261
$81$	0	0
$82$	4.25820	0.470239
$83$	10.7509	1.18007	0.590033	−	0.807379i	$-0.299115\pi$
$83$	10.7509	1.18007	0.590033	+	0.807379i	$0.299115\pi$
$84$	0	0
$85$	3.48179	0.377653
$86$	−6.41833	−0.692106
$87$	0	0
$88$	−9.98854	−1.06478
$89$	6.20467	0.657694	0.328847	−	0.944383i	$-0.393340\pi$
$89$	6.20467	0.657694	0.328847	+	0.944383i	$0.393340\pi$
$90$	0	0
$91$	31.4590	3.29780
$92$	0.566132	0.0590233
$93$	0	0
$94$	2.97033	0.306366
$95$	−14.0431	−1.44079
$96$	0	0
$97$	−7.98764	−0.811022	−0.405511	−	0.914090i	$-0.632906\pi$
$97$	−7.98764	−0.811022	−0.405511	+	0.914090i	$0.632906\pi$
$98$	22.0581	2.22820
$99$	0	0
$100$	0.423343	0.0423343
$101$	−7.07558	−0.704047	−0.352023	−	0.935991i	$-0.614506\pi$
$101$	−7.07558	−0.704047	−0.352023	+	0.935991i	$0.614506\pi$
$102$	0	0
$103$	0.849617	0.0837153	0.0418576	−	0.999124i	$-0.486672\pi$
$103$	0.849617	0.0837153	0.0418576	+	0.999124i	$0.486672\pi$
$104$	19.1726	1.88003
$105$	0	0
$106$	15.5392	1.50930
$107$	−5.71817	−0.552797	−0.276398	−	0.961043i	$-0.589141\pi$
$107$	−5.71817	−0.552797	−0.276398	+	0.961043i	$0.589141\pi$
$108$	0	0
$109$	−10.9139	−1.04536	−0.522679	−	0.852529i	$-0.675068\pi$
$109$	−10.9139	−1.04536	−0.522679	+	0.852529i	$0.675068\pi$
$110$	8.02659	0.765305
$111$	0	0
$112$	12.8429	1.21354
$113$	−8.32904	−0.783530	−0.391765	−	0.920065i	$-0.628135\pi$
$113$	−8.32904	−0.783530	−0.391765	+	0.920065i	$0.628135\pi$
$114$	0	0
$115$	−2.06209	−0.192291
$116$	0.566132	0.0525640
$117$	0	0
$118$	7.48835	0.689359
$119$	−8.51317	−0.780401
$120$	0	0
$121$	−0.433356	−0.0393960
$122$	−6.96548	−0.630625
$123$	0	0
$124$	−4.35244	−0.390860
$125$	−11.8524	−1.06012
$126$	0	0
$127$	13.5960	1.20645	0.603226	−	0.797570i	$-0.293882\pi$
$127$	13.5960	1.20645	0.603226	+	0.797570i	$0.293882\pi$
$128$	4.34790	0.384304
$129$	0	0
$130$	−15.4067	−1.35126
$131$	−11.8536	−1.03566	−0.517829	−	0.855484i	$-0.673260\pi$
$131$	−11.8536	−1.03566	−0.517829	+	0.855484i	$0.673260\pi$
$132$	0	0
$133$	34.3361	2.97732
$134$	10.5305	0.909693
$135$	0	0
$136$	−5.18834	−0.444897
$137$	19.0233	1.62527	0.812633	−	0.582775i	$-0.198033\pi$
$137$	19.0233	1.62527	0.812633	+	0.582775i	$0.198033\pi$
$138$	0	0
$139$	−18.9710	−1.60910	−0.804550	−	0.593885i	$-0.797593\pi$
$139$	−18.9710	−1.60910	−0.804550	+	0.593885i	$0.797593\pi$
$140$	5.88602	0.497459
$141$	0	0
$142$	−4.63109	−0.388633
$143$	−20.2823	−1.69609
$144$	0	0
$145$	−2.06209	−0.171247
$146$	4.89115	0.404794
$147$	0	0
$148$	−4.58763	−0.377101
$149$	2.25885	0.185052	0.0925260	−	0.995710i	$-0.470506\pi$
$149$	2.25885	0.185052	0.0925260	+	0.995710i	$0.470506\pi$
$150$	0	0
$151$	8.59882	0.699762	0.349881	−	0.936794i	$-0.386222\pi$
$151$	8.59882	0.699762	0.349881	+	0.936794i	$0.386222\pi$
$152$	20.9261	1.69733
$153$	0	0
$154$	−19.6254	−1.58146
$155$	15.8534	1.27338
$156$	0	0
$157$	17.7686	1.41809	0.709044	−	0.705164i	$-0.249127\pi$
$157$	17.7686	1.41809	0.709044	+	0.705164i	$0.249127\pi$
$158$	4.77903	0.380200
$159$	0	0
$160$	6.38305	0.504625
$161$	5.04192	0.397359
$162$	0	0
$163$	−19.2559	−1.50824	−0.754120	−	0.656736i	$-0.771936\pi$
$163$	−19.2559	−1.50824	−0.754120	+	0.656736i	$0.771936\pi$
$164$	−2.01321	−0.157205
$165$	0	0
$166$	12.8736	0.999185
$167$	13.8503	1.07176	0.535882	−	0.844293i	$-0.319979\pi$
$167$	13.8503	1.07176	0.535882	+	0.844293i	$0.319979\pi$
$168$	0	0
$169$	25.9311	1.99470
$170$	4.16925	0.319767
$171$	0	0
$172$	3.03448	0.231377
$173$	7.38728	0.561645	0.280822	−	0.959760i	$-0.409393\pi$
$173$	7.38728	0.561645	0.280822	+	0.959760i	$0.409393\pi$
$174$	0	0
$175$	3.77026	0.285005
$176$	−8.28012	−0.624138
$177$	0	0
$178$	7.42974	0.556883
$179$	−10.4050	−0.777705	−0.388853	−	0.921300i	$-0.627129\pi$
$179$	−10.4050	−0.777705	−0.388853	+	0.921300i	$0.627129\pi$
$180$	0	0
$181$	6.19248	0.460284	0.230142	−	0.973157i	$-0.426081\pi$
$181$	6.19248	0.460284	0.230142	+	0.973157i	$0.426081\pi$
$182$	37.6703	2.79231
$183$	0	0
$184$	3.07279	0.226529
$185$	16.7101	1.22855
$186$	0	0
$187$	5.48863	0.401368
$188$	−1.40433	−0.102421
$189$	0	0
$190$	−16.8158	−1.21994
$191$	−16.4242	−1.18842	−0.594208	−	0.804312i	$-0.702534\pi$
$191$	−16.4242	−1.18842	−0.594208	+	0.804312i	$0.702534\pi$
$192$	0	0
$193$	11.1785	0.804645	0.402323	−	0.915498i	$-0.368203\pi$
$193$	11.1785	0.804645	0.402323	+	0.915498i	$0.368203\pi$
$194$	−9.56474	−0.686708
$195$	0	0
$196$	−10.4287	−0.744908
$197$	17.0954	1.21799	0.608997	−	0.793173i	$-0.291572\pi$
$197$	17.0954	1.21799	0.608997	+	0.793173i	$0.291572\pi$
$198$	0	0
$199$	−1.17232	−0.0831035	−0.0415517	−	0.999136i	$-0.513230\pi$
$199$	−1.17232	−0.0831035	−0.0415517	+	0.999136i	$0.513230\pi$
$200$	2.29778	0.162478
$201$	0	0
$202$	−8.47260	−0.596130
$203$	5.04192	0.353874
$204$	0	0
$205$	7.33295	0.512156
$206$	1.01737	0.0708834
$207$	0	0
$208$	15.8934	1.10201
$209$	−22.1372	−1.53126
$210$	0	0
$211$	−11.7737	−0.810532	−0.405266	−	0.914199i	$-0.632821\pi$
$211$	−11.7737	−0.810532	−0.405266	+	0.914199i	$0.632821\pi$
$212$	−7.34667	−0.504571
$213$	0	0
$214$	−6.84718	−0.468064
$215$	−11.0529	−0.753800
$216$	0	0
$217$	−38.7625	−2.63137
$218$	−13.0687	−0.885126
$219$	0	0
$220$	−3.79484	−0.255848
$221$	−10.5352	−0.708676
$222$	0	0
$223$	4.01214	0.268673	0.134336	−	0.990936i	$-0.457110\pi$
$223$	4.01214	0.268673	0.134336	+	0.990936i	$0.457110\pi$
$224$	−15.6069	−1.04278
$225$	0	0
$226$	−9.97355	−0.663430
$227$	15.6398	1.03805	0.519024	−	0.854760i	$-0.326296\pi$
$227$	15.6398	1.03805	0.519024	+	0.854760i	$0.326296\pi$
$228$	0	0
$229$	10.1942	0.673652	0.336826	−	0.941567i	$-0.390647\pi$
$229$	10.1942	0.673652	0.336826	+	0.941567i	$0.390647\pi$
$230$	−2.46923	−0.162817
$231$	0	0
$232$	3.07279	0.201739
$233$	4.13073	0.270613	0.135307	−	0.990804i	$-0.456798\pi$
$233$	4.13073	0.270613	0.135307	+	0.990804i	$0.456798\pi$
$234$	0	0
$235$	5.11515	0.333676
$236$	−3.54037	−0.230459
$237$	0	0
$238$	−10.1940	−0.660781
$239$	4.45359	0.288079	0.144039	−	0.989572i	$-0.453991\pi$
$239$	4.45359	0.288079	0.144039	+	0.989572i	$0.453991\pi$
$240$	0	0
$241$	−11.9261	−0.768228	−0.384114	−	0.923286i	$-0.625493\pi$
$241$	−11.9261	−0.768228	−0.384114	+	0.923286i	$0.625493\pi$
$242$	−0.518919	−0.0333574
$243$	0	0
$244$	3.29317	0.210824
$245$	37.9858	2.42682
$246$	0	0
$247$	42.4916	2.70368
$248$	−23.6237	−1.50011
$249$	0	0
$250$	−14.1926	−0.897620
$251$	−2.06690	−0.130462	−0.0652308	−	0.997870i	$-0.520778\pi$
$251$	−2.06690	−0.130462	−0.0652308	+	0.997870i	$0.520778\pi$
$252$	0	0
$253$	−3.25064	−0.204366
$254$	16.2805	1.02153
$255$	0	0
$256$	−12.3958	−0.774734
$257$	−9.80955	−0.611903	−0.305951	−	0.952047i	$-0.598974\pi$
$257$	−9.80955	−0.611903	−0.305951	+	0.952047i	$0.598974\pi$
$258$	0	0
$259$	−40.8571	−2.53873
$260$	7.28407	0.451739
$261$	0	0
$262$	−14.1940	−0.876911
$263$	−24.1442	−1.48879	−0.744397	−	0.667738i	$-0.767263\pi$
$263$	−24.1442	−1.48879	−0.744397	+	0.667738i	$0.767263\pi$
$264$	0	0
$265$	26.7597	1.64383
$266$	41.1155	2.52095
$267$	0	0
$268$	−4.97863	−0.304119
$269$	14.0387	0.855952	0.427976	−	0.903790i	$-0.359227\pi$
$269$	14.0387	0.855952	0.427976	+	0.903790i	$0.359227\pi$
$270$	0	0
$271$	26.6094	1.61641	0.808204	−	0.588902i	$-0.200440\pi$
$271$	26.6094	1.61641	0.808204	+	0.588902i	$0.200440\pi$
$272$	−4.30094	−0.260783
$273$	0	0
$274$	22.7793	1.37615
$275$	−2.43077	−0.146581
$276$	0	0
$277$	−27.7848	−1.66943	−0.834715	−	0.550683i	$-0.814367\pi$
$277$	−27.7848	−1.66943	−0.834715	+	0.550683i	$0.814367\pi$
$278$	−22.7167	−1.36246
$279$	0	0
$280$	31.9476	1.90923
$281$	20.8344	1.24288	0.621438	−	0.783463i	$-0.286549\pi$
$281$	20.8344	1.24288	0.621438	+	0.783463i	$0.286549\pi$
$282$	0	0
$283$	2.22869	0.132482	0.0662410	−	0.997804i	$-0.478899\pi$
$283$	2.22869	0.132482	0.0662410	+	0.997804i	$0.478899\pi$
$284$	2.18951	0.129923
$285$	0	0
$286$	−24.2869	−1.43611
$287$	−17.9295	−1.05834
$288$	0	0
$289$	−14.1490	−0.832297
$290$	−2.46923	−0.144998
$291$	0	0
$292$	−2.31246	−0.135326
$293$	25.7063	1.50178	0.750889	−	0.660428i	$-0.229625\pi$
$293$	25.7063	1.50178	0.750889	+	0.660428i	$0.229625\pi$
$294$	0	0
$295$	12.8955	0.750807
$296$	−24.9003	−1.44730
$297$	0	0
$298$	2.70484	0.156687
$299$	6.23948	0.360839
$300$	0	0
$301$	27.0249	1.55769
$302$	10.2966	0.592502
$303$	0	0
$304$	17.3469	0.994915
$305$	−11.9951	−0.686838
$306$	0	0
$307$	32.2228	1.83905	0.919526	−	0.393029i	$-0.128573\pi$
$307$	32.2228	1.83905	0.919526	+	0.393029i	$0.128573\pi$
$308$	9.27860	0.528697
$309$	0	0
$310$	18.9835	1.07819
$311$	−13.1142	−0.743636	−0.371818	−	0.928306i	$-0.621266\pi$
$311$	−13.1142	−0.743636	−0.371818	+	0.928306i	$0.621266\pi$
$312$	0	0
$313$	−13.2453	−0.748668	−0.374334	−	0.927294i	$-0.622129\pi$
$313$	−13.2453	−0.748668	−0.374334	+	0.927294i	$0.622129\pi$
$314$	21.2769	1.20072
$315$	0	0
$316$	−2.25945	−0.127104
$317$	14.4154	0.809651	0.404826	−	0.914394i	$-0.367332\pi$
$317$	14.4154	0.809651	0.404826	+	0.914394i	$0.367332\pi$
$318$	0	0
$319$	−3.25064	−0.182001
$320$	18.1486	1.01454
$321$	0	0
$322$	6.03741	0.336452
$323$	−11.4987	−0.639806
$324$	0	0
$325$	4.66577	0.258810
$326$	−23.0579	−1.27706
$327$	0	0
$328$	−10.9271	−0.603348
$329$	−12.5068	−0.689523
$330$	0	0
$331$	24.5366	1.34865	0.674326	−	0.738434i	$-0.264434\pi$
$331$	24.5366	1.34865	0.674326	+	0.738434i	$0.264434\pi$
$332$	−6.08643	−0.334036
$333$	0	0
$334$	16.5849	0.907484
$335$	18.1343	0.990782
$336$	0	0
$337$	29.2378	1.59268	0.796342	−	0.604846i	$-0.206765\pi$
$337$	29.2378	1.59268	0.796342	+	0.604846i	$0.206765\pi$
$338$	31.0510	1.68895
$339$	0	0
$340$	−1.97115	−0.106901
$341$	24.9910	1.35334
$342$	0	0
$343$	−57.5838	−3.10923
$344$	16.4703	0.888018
$345$	0	0
$346$	8.84585	0.475556
$347$	28.3632	1.52262	0.761309	−	0.648389i	$-0.224557\pi$
$347$	28.3632	1.52262	0.761309	+	0.648389i	$0.224557\pi$
$348$	0	0
$349$	−1.53730	−0.0822900	−0.0411450	−	0.999153i	$-0.513101\pi$
$349$	−1.53730	−0.0822900	−0.0411450	+	0.999153i	$0.513101\pi$
$350$	4.51467	0.241319
$351$	0	0
$352$	10.0621	0.536312
$353$	27.5147	1.46446	0.732230	−	0.681057i	$-0.238479\pi$
$353$	27.5147	1.46446	0.732230	+	0.681057i	$0.238479\pi$
$354$	0	0
$355$	−7.97511	−0.423275
$356$	−3.51266	−0.186171
$357$	0	0
$358$	−12.4594	−0.658498
$359$	−9.89302	−0.522134	−0.261067	−	0.965321i	$-0.584074\pi$
$359$	−9.89302	−0.522134	−0.261067	+	0.965321i	$0.584074\pi$
$360$	0	0
$361$	27.3777	1.44093
$362$	7.41514	0.389731
$363$	0	0
$364$	−17.8099	−0.933494
$365$	8.42294	0.440877
$366$	0	0
$367$	−33.5227	−1.74987	−0.874936	−	0.484239i	$-0.839096\pi$
$367$	−33.5227	−1.74987	−0.874936	+	0.484239i	$0.839096\pi$
$368$	2.54723	0.132784
$369$	0	0
$370$	20.0094	1.04024
$371$	−65.4288	−3.39690
$372$	0	0
$373$	3.10888	0.160972	0.0804858	−	0.996756i	$-0.474353\pi$
$373$	3.10888	0.160972	0.0804858	+	0.996756i	$0.474353\pi$
$374$	6.57231	0.339846
$375$	0	0
$376$	−7.62227	−0.393088
$377$	6.23948	0.321350
$378$	0	0
$379$	4.02318	0.206657	0.103328	−	0.994647i	$-0.467051\pi$
$379$	4.02318	0.206657	0.103328	+	0.994647i	$0.467051\pi$
$380$	7.95023	0.407838
$381$	0	0
$382$	−19.6671	−1.00625
$383$	−24.5468	−1.25428	−0.627142	−	0.778905i	$-0.715775\pi$
$383$	−24.5468	−1.25428	−0.627142	+	0.778905i	$0.715775\pi$
$384$	0	0
$385$	−33.7966	−1.72243
$386$	13.3856	0.681309
$387$	0	0
$388$	4.52206	0.229573
$389$	4.99909	0.253464	0.126732	−	0.991937i	$-0.459551\pi$
$389$	4.99909	0.253464	0.126732	+	0.991937i	$0.459551\pi$
$390$	0	0
$391$	−1.68848	−0.0853899
$392$	−56.6039	−2.85893
$393$	0	0
$394$	20.4707	1.03130
$395$	8.22988	0.414090
$396$	0	0
$397$	22.3687	1.12265	0.561326	−	0.827595i	$-0.310292\pi$
$397$	22.3687	1.12265	0.561326	+	0.827595i	$0.310292\pi$
$398$	−1.40378	−0.0703653
$399$	0	0
$400$	1.90477	0.0952387
$401$	−23.8839	−1.19270	−0.596352	−	0.802723i	$-0.703384\pi$
$401$	−23.8839	−1.19270	−0.596352	+	0.802723i	$0.703384\pi$
$402$	0	0
$403$	−47.9693	−2.38952
$404$	4.00571	0.199292
$405$	0	0
$406$	6.03741	0.299632
$407$	26.3414	1.30570
$408$	0	0
$409$	26.6761	1.31905	0.659523	−	0.751684i	$-0.270758\pi$
$409$	26.6761	1.31905	0.659523	+	0.751684i	$0.270758\pi$
$410$	8.78079	0.433652
$411$	0	0
$412$	−0.480995	−0.0236969
$413$	−31.5303	−1.55150
$414$	0	0
$415$	22.1694	1.08825
$416$	−19.3139	−0.946940
$417$	0	0
$418$	−26.5080	−1.29655
$419$	28.5412	1.39433	0.697165	−	0.716911i	$-0.254445\pi$
$419$	28.5412	1.39433	0.697165	+	0.716911i	$0.254445\pi$
$420$	0	0
$421$	8.17571	0.398460	0.199230	−	0.979953i	$-0.436156\pi$
$421$	8.17571	0.398460	0.199230	+	0.979953i	$0.436156\pi$
$422$	−14.0983	−0.686293
$423$	0	0
$424$	−39.8755	−1.93653
$425$	−1.26261	−0.0612457
$426$	0	0
$427$	29.3287	1.41931
$428$	3.23724	0.156478
$429$	0	0
$430$	−13.2352	−0.638257
$431$	−23.2858	−1.12164	−0.560819	−	0.827939i	$-0.689513\pi$
$431$	−23.2858	−1.12164	−0.560819	+	0.827939i	$0.689513\pi$
$432$	0	0
$433$	−32.2493	−1.54980	−0.774901	−	0.632083i	$-0.782200\pi$
$433$	−32.2493	−1.54980	−0.774901	+	0.632083i	$0.782200\pi$
$434$	−46.4158	−2.22803
$435$	0	0
$436$	6.17869	0.295905
$437$	6.81012	0.325772
$438$	0	0
$439$	−7.65572	−0.365387	−0.182694	−	0.983170i	$-0.558482\pi$
$439$	−7.65572	−0.365387	−0.182694	+	0.983170i	$0.558482\pi$
$440$	−20.5973	−0.981937
$441$	0	0
$442$	−12.6153	−0.600050
$443$	−13.5058	−0.641682	−0.320841	−	0.947133i	$-0.603965\pi$
$443$	−13.5058	−0.641682	−0.320841	+	0.947133i	$0.603965\pi$
$444$	0	0
$445$	12.7946	0.606522
$446$	4.80430	0.227490
$447$	0	0
$448$	−44.3743	−2.09649
$449$	13.4126	0.632978	0.316489	−	0.948596i	$-0.397496\pi$
$449$	13.4126	0.632978	0.316489	+	0.948596i	$0.397496\pi$
$450$	0	0
$451$	11.5595	0.544316
$452$	4.71534	0.221791
$453$	0	0
$454$	18.7277	0.878936
$455$	64.8713	3.04121
$456$	0	0
$457$	−20.5991	−0.963587	−0.481793	−	0.876285i	$-0.660014\pi$
$457$	−20.5991	−0.963587	−0.481793	+	0.876285i	$0.660014\pi$
$458$	12.2070	0.570394
$459$	0	0
$460$	1.16742	0.0544310
$461$	19.4004	0.903567	0.451783	−	0.892128i	$-0.350788\pi$
$461$	19.4004	0.903567	0.451783	+	0.892128i	$0.350788\pi$
$462$	0	0
$463$	−16.7434	−0.778132	−0.389066	−	0.921210i	$-0.627202\pi$
$463$	−16.7434	−0.778132	−0.389066	+	0.921210i	$0.627202\pi$
$464$	2.54723	0.118252
$465$	0	0
$466$	4.94631	0.229133
$467$	−14.4444	−0.668409	−0.334205	−	0.942501i	$-0.608468\pi$
$467$	−14.4444	−0.668409	−0.334205	+	0.942501i	$0.608468\pi$
$468$	0	0
$469$	−44.3393	−2.04740
$470$	6.12510	0.282530
$471$	0	0
$472$	−19.2161	−0.884492
$473$	−17.4235	−0.801134
$474$	0	0
$475$	5.09248	0.233659
$476$	4.81958	0.220905
$477$	0	0
$478$	5.33292	0.243922
$479$	42.0985	1.92353	0.961765	−	0.273877i	$-0.0883063\pi$
$479$	42.0985	1.92353	0.961765	+	0.273877i	$0.0883063\pi$
$480$	0	0
$481$	−50.5614	−2.30540
$482$	−14.2808	−0.650474
$483$	0	0
$484$	0.245337	0.0111517
$485$	−16.4712	−0.747920
$486$	0	0
$487$	−33.3917	−1.51312	−0.756561	−	0.653923i	$-0.773122\pi$
$487$	−33.3917	−1.51312	−0.756561	+	0.653923i	$0.773122\pi$
$488$	17.8743	0.809133
$489$	0	0
$490$	45.4858	2.05484
$491$	18.5636	0.837765	0.418883	−	0.908040i	$-0.362422\pi$
$491$	18.5636	0.837765	0.418883	+	0.908040i	$0.362422\pi$
$492$	0	0
$493$	−1.68848	−0.0760452
$494$	50.8812	2.28926
$495$	0	0
$496$	−19.5832	−0.879311
$497$	19.4996	0.874675
$498$	0	0
$499$	−40.8090	−1.82686	−0.913431	−	0.406993i	$-0.866577\pi$
$499$	−40.8090	−1.82686	−0.913431	+	0.406993i	$0.866577\pi$
$500$	6.71005	0.300082
$501$	0	0
$502$	−2.47499	−0.110464
$503$	19.9372	0.888955	0.444478	−	0.895790i	$-0.353389\pi$
$503$	19.9372	0.888955	0.444478	+	0.895790i	$0.353389\pi$
$504$	0	0
$505$	−14.5905	−0.649268
$506$	−3.89245	−0.173041
$507$	0	0
$508$	−7.69714	−0.341506
$509$	5.56812	0.246803	0.123401	−	0.992357i	$-0.460620\pi$
$509$	5.56812	0.246803	0.123401	+	0.992357i	$0.460620\pi$
$510$	0	0
$511$	−20.5946	−0.911049
$512$	−23.5390	−1.04029
$513$	0	0
$514$	−11.7464	−0.518110
$515$	1.75199	0.0772018
$516$	0	0
$517$	8.06342	0.354629
$518$	−48.9240	−2.14960
$519$	0	0
$520$	39.5357	1.73376
$521$	14.0001	0.613353	0.306677	−	0.951814i	$-0.400783\pi$
$521$	14.0001	0.613353	0.306677	+	0.951814i	$0.400783\pi$
$522$	0	0
$523$	−24.8212	−1.08536	−0.542678	−	0.839941i	$-0.682590\pi$
$523$	−24.8212	−1.08536	−0.542678	+	0.839941i	$0.682590\pi$
$524$	6.71072	0.293159
$525$	0	0
$526$	−28.9112	−1.26059
$527$	12.9811	0.565464
$528$	0	0
$529$	1.00000	0.0434783
$530$	32.0431	1.39187
$531$	0	0
$532$	−19.4388	−0.842777
$533$	−22.1881	−0.961072
$534$	0	0
$535$	−11.7914	−0.509786
$536$	−27.0225	−1.16720
$537$	0	0
$538$	16.8105	0.724752
$539$	59.8800	2.57921
$540$	0	0
$541$	6.70658	0.288338	0.144169	−	0.989553i	$-0.453949\pi$
$541$	6.70658	0.288338	0.144169	+	0.989553i	$0.453949\pi$
$542$	31.8633	1.36865
$543$	0	0
$544$	5.22656	0.224087
$545$	−22.5054	−0.964025
$546$	0	0
$547$	30.7909	1.31653	0.658263	−	0.752788i	$-0.271292\pi$
$547$	30.7909	1.31653	0.658263	+	0.752788i	$0.271292\pi$
$548$	−10.7697	−0.460058
$549$	0	0
$550$	−2.91070	−0.124113
$551$	6.81012	0.290121
$552$	0	0
$553$	−20.1225	−0.855695
$554$	−33.2707	−1.41354
$555$	0	0
$556$	10.7401	0.455481
$557$	−16.1106	−0.682629	−0.341315	−	0.939949i	$-0.610872\pi$
$557$	−16.1106	−0.682629	−0.341315	+	0.939949i	$0.610872\pi$
$558$	0	0
$559$	33.4438	1.41452
$560$	26.4833	1.11912
$561$	0	0
$562$	24.9480	1.05237
$563$	12.0500	0.507846	0.253923	−	0.967224i	$-0.418279\pi$
$563$	12.0500	0.507846	0.253923	+	0.967224i	$0.418279\pi$
$564$	0	0
$565$	−17.1752	−0.722568
$566$	2.66873	0.112175
$567$	0	0
$568$	11.8840	0.498641
$569$	5.73467	0.240410	0.120205	−	0.992749i	$-0.461645\pi$
$569$	5.73467	0.240410	0.120205	+	0.992749i	$0.461645\pi$
$570$	0	0
$571$	15.9360	0.666900	0.333450	−	0.942768i	$-0.391787\pi$
$571$	15.9360	0.666900	0.333450	+	0.942768i	$0.391787\pi$
$572$	11.4825	0.480105
$573$	0	0
$574$	−21.4695	−0.896120
$575$	0.747782	0.0311847
$576$	0	0
$577$	25.1482	1.04693	0.523467	−	0.852046i	$-0.324638\pi$
$577$	25.1482	1.04693	0.523467	+	0.852046i	$0.324638\pi$
$578$	−16.9427	−0.704722
$579$	0	0
$580$	1.16742	0.0484743
$581$	−54.2053	−2.24881
$582$	0	0
$583$	42.1834	1.74706
$584$	−12.5513	−0.519378
$585$	0	0
$586$	30.7818	1.27159
$587$	20.5545	0.848376	0.424188	−	0.905574i	$-0.360560\pi$
$587$	20.5545	0.848376	0.424188	+	0.905574i	$0.360560\pi$
$588$	0	0
$589$	−52.3564	−2.15731
$590$	15.4417	0.635723
$591$	0	0
$592$	−20.6414	−0.848356
$593$	4.65816	0.191288	0.0956439	−	0.995416i	$-0.469509\pi$
$593$	4.65816	0.191288	0.0956439	+	0.995416i	$0.469509\pi$
$594$	0	0
$595$	−17.5549	−0.719682
$596$	−1.27880	−0.0523819
$597$	0	0
$598$	7.47142	0.305529
$599$	−16.0992	−0.657797	−0.328899	−	0.944365i	$-0.606677\pi$
$599$	−16.0992	−0.657797	−0.328899	+	0.944365i	$0.606677\pi$
$600$	0	0
$601$	−25.2142	−1.02851	−0.514254	−	0.857638i	$-0.671931\pi$
$601$	−25.2142	−1.02851	−0.514254	+	0.857638i	$0.671931\pi$
$602$	32.3607	1.31892
$603$	0	0
$604$	−4.86807	−0.198079
$605$	−0.893620	−0.0363308
$606$	0	0
$607$	28.0469	1.13839	0.569195	−	0.822203i	$-0.307255\pi$
$607$	28.0469	1.13839	0.569195	+	0.822203i	$0.307255\pi$
$608$	−21.0802	−0.854916
$609$	0	0
$610$	−14.3635	−0.581559
$611$	−15.4774	−0.626150
$612$	0	0
$613$	31.0124	1.25258	0.626290	−	0.779590i	$-0.284572\pi$
$613$	31.0124	1.25258	0.626290	+	0.779590i	$0.284572\pi$
$614$	38.5849	1.55716
$615$	0	0
$616$	50.3615	2.02912
$617$	−30.9966	−1.24787	−0.623937	−	0.781475i	$-0.714468\pi$
$617$	−30.9966	−1.24787	−0.623937	+	0.781475i	$0.714468\pi$
$618$	0	0
$619$	−0.176704	−0.00710234	−0.00355117	−	0.999994i	$-0.501130\pi$
$619$	−0.176704	−0.00710234	−0.00355117	+	0.999994i	$0.501130\pi$
$620$	−8.97512	−0.360450
$621$	0	0
$622$	−15.7035	−0.629652
$623$	−31.2835	−1.25335
$624$	0	0
$625$	−20.7019	−0.828077
$626$	−15.8605	−0.633911
$627$	0	0
$628$	−10.0594	−0.401412
$629$	13.6825	0.545558
$630$	0	0
$631$	−9.35919	−0.372583	−0.186292	−	0.982494i	$-0.559647\pi$
$631$	−9.35919	−0.372583	−0.186292	+	0.982494i	$0.559647\pi$
$632$	−12.2636	−0.487821
$633$	0	0
$634$	17.2617	0.685548
$635$	28.0362	1.11258
$636$	0	0
$637$	−114.937	−4.55399
$638$	−3.89245	−0.154104
$639$	0	0
$640$	8.96577	0.354403
$641$	−24.6606	−0.974037	−0.487019	−	0.873392i	$-0.661916\pi$
$641$	−24.6606	−0.974037	−0.487019	+	0.873392i	$0.661916\pi$
$642$	0	0
$643$	7.81464	0.308179	0.154090	−	0.988057i	$-0.450756\pi$
$643$	7.81464	0.308179	0.154090	+	0.988057i	$0.450756\pi$
$644$	−2.85439	−0.112479
$645$	0	0
$646$	−13.7691	−0.541736
$647$	13.5259	0.531757	0.265878	−	0.964007i	$-0.414338\pi$
$647$	13.5259	0.531757	0.265878	+	0.964007i	$0.414338\pi$
$648$	0	0
$649$	20.3283	0.797954
$650$	5.58699	0.219140
$651$	0	0
$652$	10.9014	0.426932
$653$	22.2051	0.868951	0.434476	−	0.900684i	$-0.356934\pi$
$653$	22.2051	0.868951	0.434476	+	0.900684i	$0.356934\pi$
$654$	0	0
$655$	−24.4433	−0.955078
$656$	−9.05815	−0.353661
$657$	0	0
$658$	−14.9762	−0.583833
$659$	−2.41065	−0.0939057	−0.0469529	−	0.998897i	$-0.514951\pi$
$659$	−2.41065	−0.0939057	−0.0469529	+	0.998897i	$0.514951\pi$
$660$	0	0
$661$	−30.1307	−1.17195	−0.585975	−	0.810329i	$-0.699288\pi$
$661$	−30.1307	−1.17195	−0.585975	+	0.810329i	$0.699288\pi$
$662$	29.3811	1.14193
$663$	0	0
$664$	−33.0353	−1.28202
$665$	70.8041	2.74567
$666$	0	0
$667$	1.00000	0.0387202
$668$	−7.84107	−0.303380
$669$	0	0
$670$	21.7148	0.838914
$671$	−18.9088	−0.729968
$672$	0	0
$673$	48.4182	1.86638	0.933192	−	0.359377i	$-0.117011\pi$
$673$	48.4182	1.86638	0.933192	+	0.359377i	$0.117011\pi$
$674$	35.0106	1.34856
$675$	0	0
$676$	−14.6804	−0.564632
$677$	−16.8459	−0.647442	−0.323721	−	0.946153i	$-0.604934\pi$
$677$	−16.8459	−0.647442	−0.323721	+	0.946153i	$0.604934\pi$
$678$	0	0
$679$	40.2731	1.54554
$680$	−10.6988	−0.410282
$681$	0	0
$682$	29.9253	1.14590
$683$	−28.0812	−1.07450	−0.537248	−	0.843424i	$-0.680536\pi$
$683$	−28.0812	−1.07450	−0.537248	+	0.843424i	$0.680536\pi$
$684$	0	0
$685$	39.2277	1.49881
$686$	−68.9533	−2.63265
$687$	0	0
$688$	13.6532	0.520525
$689$	−80.9695	−3.08469
$690$	0	0
$691$	−18.2999	−0.696161	−0.348081	−	0.937465i	$-0.613166\pi$
$691$	−18.2999	−0.696161	−0.348081	+	0.937465i	$0.613166\pi$
$692$	−4.18218	−0.158982
$693$	0	0
$694$	33.9633	1.28923
$695$	−39.1199	−1.48390
$696$	0	0
$697$	6.00435	0.227431
$698$	−1.84083	−0.0696766
$699$	0	0
$700$	−2.13446	−0.0806751
$701$	−34.8853	−1.31760	−0.658800	−	0.752318i	$-0.728936\pi$
$701$	−34.8853	−1.31760	−0.658800	+	0.752318i	$0.728936\pi$
$702$	0	0
$703$	−55.1856	−2.08136
$704$	28.6090	1.07824
$705$	0	0
$706$	32.9473	1.23999
$707$	35.6745	1.34168
$708$	0	0
$709$	41.4053	1.55501	0.777503	−	0.628879i	$-0.216486\pi$
$709$	41.4053	1.55501	0.777503	+	0.628879i	$0.216486\pi$
$710$	−9.54973	−0.358395
$711$	0	0
$712$	−19.0657	−0.714517
$713$	−7.68803	−0.287919
$714$	0	0
$715$	−41.8239	−1.56413
$716$	5.89060	0.220142
$717$	0	0
$718$	−11.8463	−0.442101
$719$	−31.3446	−1.16895	−0.584477	−	0.811410i	$-0.698700\pi$
$719$	−31.3446	−1.16895	−0.584477	+	0.811410i	$0.698700\pi$
$720$	0	0
$721$	−4.28371	−0.159534
$722$	32.7832	1.22007
$723$	0	0
$724$	−3.50576	−0.130291
$725$	0.747782	0.0277719
$726$	0	0
$727$	23.6364	0.876627	0.438314	−	0.898822i	$-0.355576\pi$
$727$	23.6364	0.876627	0.438314	+	0.898822i	$0.355576\pi$
$728$	−96.6670	−3.58272
$729$	0	0
$730$	10.0860	0.373299
$731$	−9.05029	−0.334737
$732$	0	0
$733$	22.6643	0.837126	0.418563	−	0.908188i	$-0.362534\pi$
$733$	22.6643	0.837126	0.418563	+	0.908188i	$0.362534\pi$
$734$	−40.1415	−1.48165
$735$	0	0
$736$	−3.09543	−0.114099
$737$	28.5865	1.05300
$738$	0	0
$739$	−28.5965	−1.05194	−0.525970	−	0.850503i	$-0.676298\pi$
$739$	−28.5965	−1.05194	−0.525970	+	0.850503i	$0.676298\pi$
$740$	−9.46011	−0.347761
$741$	0	0
$742$	−78.3472	−2.87622
$743$	51.2185	1.87903	0.939513	−	0.342514i	$-0.111278\pi$
$743$	51.2185	1.87903	0.939513	+	0.342514i	$0.111278\pi$
$744$	0	0
$745$	4.65795	0.170654
$746$	3.72270	0.136298
$747$	0	0
$748$	−3.10729	−0.113614
$749$	28.8306	1.05345
$750$	0	0
$751$	31.2908	1.14182	0.570908	−	0.821014i	$-0.306591\pi$
$751$	31.2908	1.14182	0.570908	+	0.821014i	$0.306591\pi$
$752$	−6.31857	−0.230415
$753$	0	0
$754$	7.47142	0.272093
$755$	17.7315	0.645317
$756$	0	0
$757$	−4.16334	−0.151319	−0.0756596	−	0.997134i	$-0.524106\pi$
$757$	−4.16334	−0.151319	−0.0756596	+	0.997134i	$0.524106\pi$
$758$	4.81753	0.174980
$759$	0	0
$760$	43.1515	1.56527
$761$	30.3090	1.09870	0.549351	−	0.835592i	$-0.314875\pi$
$761$	30.3090	1.09870	0.549351	+	0.835592i	$0.314875\pi$
$762$	0	0
$763$	55.0269	1.99211
$764$	9.29828	0.336400
$765$	0	0
$766$	−29.3934	−1.06203
$767$	−39.0194	−1.40891
$768$	0	0
$769$	0.202559	0.00730447	0.00365224	−	0.999993i	$-0.498837\pi$
$769$	0.202559	0.00730447	0.00365224	+	0.999993i	$0.498837\pi$
$770$	−40.4694	−1.45842
$771$	0	0
$772$	−6.32850	−0.227768
$773$	50.4003	1.81277	0.906386	−	0.422450i	$-0.138830\pi$
$773$	50.4003	1.81277	0.906386	+	0.422450i	$0.138830\pi$
$774$	0	0
$775$	−5.74897	−0.206509
$776$	24.5444	0.881092
$777$	0	0
$778$	5.98612	0.214613
$779$	−24.2173	−0.867675
$780$	0	0
$781$	−12.5718	−0.449854
$782$	−2.02185	−0.0723013
$783$	0	0
$784$	−46.9225	−1.67580
$785$	36.6405	1.30775
$786$	0	0
$787$	−10.7921	−0.384697	−0.192349	−	0.981327i	$-0.561610\pi$
$787$	−10.7921	−0.384697	−0.192349	+	0.981327i	$0.561610\pi$
$788$	−9.67822	−0.344772
$789$	0	0
$790$	9.85480	0.350618
$791$	41.9944	1.49315
$792$	0	0
$793$	36.2948	1.28887
$794$	26.7852	0.950571
$795$	0	0
$796$	0.663687	0.0235238
$797$	52.6370	1.86450	0.932249	−	0.361816i	$-0.117843\pi$
$797$	52.6370	1.86450	0.932249	+	0.361816i	$0.117843\pi$
$798$	0	0
$799$	4.18838	0.148174
$800$	−2.31470	−0.0818372
$801$	0	0
$802$	−28.5996	−1.00989
$803$	13.2778	0.468562
$804$	0	0
$805$	10.3969	0.366443
$806$	−57.4405	−2.02326
$807$	0	0
$808$	21.7418	0.764874
$809$	−41.3605	−1.45416	−0.727080	−	0.686553i	$-0.759123\pi$
$809$	−41.3605	−1.45416	−0.727080	+	0.686553i	$0.759123\pi$
$810$	0	0
$811$	11.8526	0.416203	0.208101	−	0.978107i	$-0.433272\pi$
$811$	11.8526	0.416203	0.208101	+	0.978107i	$0.433272\pi$
$812$	−2.85439	−0.100170
$813$	0	0
$814$	31.5423	1.10556
$815$	−39.7075	−1.39089
$816$	0	0
$817$	36.5024	1.27706
$818$	31.9431	1.11686
$819$	0	0
$820$	−4.15142	−0.144974
$821$	−23.3747	−0.815781	−0.407891	−	0.913031i	$-0.633736\pi$
$821$	−23.3747	−0.815781	−0.407891	+	0.913031i	$0.633736\pi$
$822$	0	0
$823$	22.8021	0.794832	0.397416	−	0.917638i	$-0.369907\pi$
$823$	22.8021	0.794832	0.397416	+	0.917638i	$0.369907\pi$
$824$	−2.61070	−0.0909480
$825$	0	0
$826$	−37.7557	−1.31369
$827$	33.2724	1.15699	0.578497	−	0.815684i	$-0.303639\pi$
$827$	33.2724	1.15699	0.578497	+	0.815684i	$0.303639\pi$
$828$	0	0
$829$	39.2694	1.36388	0.681942	−	0.731407i	$-0.261136\pi$
$829$	39.2694	1.36388	0.681942	+	0.731407i	$0.261136\pi$
$830$	26.5465	0.921443
$831$	0	0
$832$	−54.9140	−1.90380
$833$	31.1034	1.07767
$834$	0	0
$835$	28.5605	0.988376
$836$	12.5326	0.433449
$837$	0	0
$838$	34.1765	1.18061
$839$	−34.4182	−1.18825	−0.594124	−	0.804374i	$-0.702501\pi$
$839$	−34.4182	−1.18825	−0.594124	+	0.804374i	$0.702501\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	9.78994	0.337384
$843$	0	0
$844$	6.66544	0.229434
$845$	53.4724	1.83951
$846$	0	0
$847$	2.18495	0.0750758
$848$	−33.0553	−1.13512
$849$	0	0
$850$	−1.51191	−0.0518579
$851$	−8.10347	−0.277783
$852$	0	0
$853$	29.4613	1.00873	0.504367	−	0.863489i	$-0.331726\pi$
$853$	29.4613	1.00873	0.504367	+	0.863489i	$0.331726\pi$
$854$	35.1194	1.20176
$855$	0	0
$856$	17.5708	0.600557
$857$	−18.1437	−0.619776	−0.309888	−	0.950773i	$-0.600292\pi$
$857$	−18.1437	−0.619776	−0.309888	+	0.950773i	$0.600292\pi$
$858$	0	0
$859$	22.3321	0.761960	0.380980	−	0.924583i	$-0.375587\pi$
$859$	22.3321	0.761960	0.380980	+	0.924583i	$0.375587\pi$
$860$	6.25738	0.213375
$861$	0	0
$862$	−27.8834	−0.949712
$863$	1.86768	0.0635766	0.0317883	−	0.999495i	$-0.489880\pi$
$863$	1.86768	0.0635766	0.0317883	+	0.999495i	$0.489880\pi$
$864$	0	0
$865$	15.2332	0.517946
$866$	−38.6167	−1.31225
$867$	0	0
$868$	21.9447	0.744850
$869$	12.9734	0.440093
$870$	0	0
$871$	−54.8708	−1.85923
$872$	33.5361	1.13567
$873$	0	0
$874$	8.15472	0.275838
$875$	59.7591	2.02023
$876$	0	0
$877$	−13.3782	−0.451749	−0.225875	−	0.974156i	$-0.572524\pi$
$877$	−13.3782	−0.451749	−0.225875	+	0.974156i	$0.572524\pi$
$878$	−9.16728	−0.309381
$879$	0	0
$880$	−17.0744	−0.575577
$881$	9.97548	0.336083	0.168041	−	0.985780i	$-0.446256\pi$
$881$	9.97548	0.336083	0.168041	+	0.985780i	$0.446256\pi$
$882$	0	0
$883$	−6.25218	−0.210403	−0.105201	−	0.994451i	$-0.533549\pi$
$883$	−6.25218	−0.210403	−0.105201	+	0.994451i	$0.533549\pi$
$884$	5.96432	0.200602
$885$	0	0
$886$	−16.1725	−0.543325
$887$	−41.2664	−1.38559	−0.692795	−	0.721134i	$-0.743621\pi$
$887$	−41.2664	−1.38559	−0.692795	+	0.721134i	$0.743621\pi$
$888$	0	0
$889$	−68.5501	−2.29910
$890$	15.3208	0.513554
$891$	0	0
$892$	−2.27140	−0.0760521
$893$	−16.8929	−0.565301
$894$	0	0
$895$	−21.4560	−0.717196
$896$	−21.9218	−0.732356
$897$	0	0
$898$	16.0608	0.535955
$899$	−7.68803	−0.256410
$900$	0	0
$901$	21.9113	0.729971
$902$	13.8419	0.460883
$903$	0	0
$904$	25.5934	0.851225
$905$	12.7695	0.424471
$906$	0	0
$907$	23.6124	0.784038	0.392019	−	0.919957i	$-0.371777\pi$
$907$	23.6124	0.784038	0.392019	+	0.919957i	$0.371777\pi$
$908$	−8.85417	−0.293836
$909$	0	0
$910$	77.6796	2.57505
$911$	6.19372	0.205207	0.102603	−	0.994722i	$-0.467283\pi$
$911$	6.19372	0.205207	0.102603	+	0.994722i	$0.467283\pi$
$912$	0	0
$913$	34.9473	1.15659
$914$	−24.6663	−0.815888
$915$	0	0
$916$	−5.77126	−0.190688
$917$	59.7651	1.97362
$918$	0	0
$919$	41.9811	1.38483	0.692414	−	0.721500i	$-0.256547\pi$
$919$	41.9811	1.38483	0.692414	+	0.721500i	$0.256547\pi$
$920$	6.33638	0.208904
$921$	0	0
$922$	23.2309	0.765068
$923$	24.1311	0.794285
$924$	0	0
$925$	−6.05962	−0.199239
$926$	−20.0492	−0.658859
$927$	0	0
$928$	−3.09543	−0.101612
$929$	45.9424	1.50732	0.753661	−	0.657264i	$-0.228286\pi$
$929$	45.9424	1.50732	0.753661	+	0.657264i	$0.228286\pi$
$930$	0	0
$931$	−125.449	−4.11143
$932$	−2.33854	−0.0766014
$933$	0	0
$934$	−17.2964	−0.565955
$935$	11.3180	0.370140
$936$	0	0
$937$	−26.1173	−0.853215	−0.426608	−	0.904437i	$-0.640291\pi$
$937$	−26.1173	−0.853215	−0.426608	+	0.904437i	$0.640291\pi$
$938$	−53.0938	−1.73357
$939$	0	0
$940$	−2.89585	−0.0944522
$941$	−32.5296	−1.06044	−0.530218	−	0.847861i	$-0.677890\pi$
$941$	−32.5296	−1.06044	−0.530218	+	0.847861i	$0.677890\pi$
$942$	0	0
$943$	−3.55608	−0.115802
$944$	−15.9294	−0.518458
$945$	0	0
$946$	−20.8637	−0.678336
$947$	−3.46449	−0.112581	−0.0562905	−	0.998414i	$-0.517927\pi$
$947$	−3.46449	−0.112581	−0.0562905	+	0.998414i	$0.517927\pi$
$948$	0	0
$949$	−25.4862	−0.827316
$950$	6.09795	0.197844
$951$	0	0
$952$	26.1592	0.847826
$953$	−0.809283	−0.0262153	−0.0131076	−	0.999914i	$-0.504172\pi$
$953$	−0.809283	−0.0262153	−0.0131076	+	0.999914i	$0.504172\pi$
$954$	0	0
$955$	−33.8683	−1.09595
$956$	−2.52132	−0.0815453
$957$	0	0
$958$	50.4105	1.62869
$959$	−95.9138	−3.09722
$960$	0	0
$961$	28.1058	0.906638
$962$	−60.5444	−1.95203
$963$	0	0
$964$	6.75175	0.217459
$965$	23.0511	0.742040
$966$	0	0
$967$	23.6129	0.759341	0.379671	−	0.925122i	$-0.376037\pi$
$967$	23.6129	0.759341	0.379671	+	0.925122i	$0.376037\pi$
$968$	1.33162	0.0427997
$969$	0	0
$970$	−19.7234	−0.633279
$971$	−23.4909	−0.753859	−0.376930	−	0.926242i	$-0.623020\pi$
$971$	−23.4909	−0.753859	−0.376930	+	0.926242i	$0.623020\pi$
$972$	0	0
$973$	95.6504	3.06641
$974$	−39.9846	−1.28119
$975$	0	0
$976$	14.8172	0.474286
$977$	49.8145	1.59371	0.796854	−	0.604172i	$-0.206496\pi$
$977$	49.8145	1.59371	0.796854	+	0.604172i	$0.206496\pi$
$978$	0	0
$979$	20.1691	0.644609
$980$	−21.5049	−0.686950
$981$	0	0
$982$	22.2289	0.709352
$983$	38.4490	1.22633	0.613167	−	0.789954i	$-0.289895\pi$
$983$	38.4490	1.22633	0.613167	+	0.789954i	$0.289895\pi$
$984$	0	0
$985$	35.2522	1.12323
$986$	−2.02185	−0.0643889
$987$	0	0
$988$	−24.0559	−0.765319
$989$	5.36003	0.170439
$990$	0	0
$991$	−7.36344	−0.233907	−0.116954	−	0.993137i	$-0.537313\pi$
$991$	−7.36344	−0.233907	−0.116954	+	0.993137i	$0.537313\pi$
$992$	23.7977	0.755579
$993$	0	0
$994$	23.3496	0.740605
$995$	−2.41743	−0.0766376
$996$	0	0
$997$	13.9655	0.442290	0.221145	−	0.975241i	$-0.429021\pi$
$997$	13.9655	0.442290	0.221145	+	0.975241i	$0.429021\pi$
$998$	−48.8664	−1.54684
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.s.1.14		20
3.2	odd	2		2001.2.a.o.1.7	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.2.a.o.1.7	✓	20	3.2	odd	2
6003.2.a.s.1.14		20	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)

\(f(q)\)	\(=\)	\(q+1.19744 q^{2} -0.566132 q^{4} +2.06209 q^{5} -5.04192 q^{7} -3.07279 q^{8} +O(q^{10})\)	\(q+1.19744 q^{2} -0.566132 q^{4} +2.06209 q^{5} -5.04192 q^{7} -3.07279 q^{8} +2.46923 q^{10} +3.25064 q^{11} -6.23948 q^{13} -6.03741 q^{14} -2.54723 q^{16} +1.68848 q^{17} -6.81012 q^{19} -1.16742 q^{20} +3.89245 q^{22} -1.00000 q^{23} -0.747782 q^{25} -7.47142 q^{26} +2.85439 q^{28} -1.00000 q^{29} +7.68803 q^{31} +3.09543 q^{32} +2.02185 q^{34} -10.3969 q^{35} +8.10347 q^{37} -8.15472 q^{38} -6.33638 q^{40} +3.55608 q^{41} -5.36003 q^{43} -1.84029 q^{44} -1.19744 q^{46} +2.48056 q^{47} +18.4210 q^{49} -0.895426 q^{50} +3.53237 q^{52} +12.9770 q^{53} +6.70311 q^{55} +15.4928 q^{56} -1.19744 q^{58} +6.25362 q^{59} -5.81696 q^{61} +9.20597 q^{62} +8.80106 q^{64} -12.8664 q^{65} +8.79413 q^{67} -0.955901 q^{68} -12.4497 q^{70} -3.86749 q^{71} +4.08466 q^{73} +9.70343 q^{74} +3.85542 q^{76} -16.3895 q^{77} +3.99104 q^{79} -5.25262 q^{80} +4.25820 q^{82} +10.7509 q^{83} +3.48179 q^{85} -6.41833 q^{86} -9.98854 q^{88} +6.20467 q^{89} +31.4590 q^{91} +0.566132 q^{92} +2.97033 q^{94} -14.0431 q^{95} -7.98764 q^{97} +22.0581 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 2 q^{2} + 30 q^{4} + q^{5} + 9 q^{7} - 6 q^{8}+O(q^{10}) \) 20 * q - 2 * q^2 + 30 * q^4 + q^5 + 9 * q^7 - 6 * q^8	\( 20 q - 2 q^{2} + 30 q^{4} + q^{5} + 9 q^{7} - 6 q^{8} + 7 q^{10} + 21 q^{13} + q^{14} + 58 q^{16} + 4 q^{17} + 7 q^{19} + 20 q^{20} + 7 q^{22} - 20 q^{23} + 47 q^{25} - 8 q^{26} + 11 q^{28} - 20 q^{29} + 28 q^{31} - 14 q^{32} + 16 q^{34} - 9 q^{35} + 14 q^{37} + 20 q^{38} + 34 q^{40} - 7 q^{41} + 3 q^{43} + q^{44} + 2 q^{46} - 3 q^{47} + 35 q^{49} + 24 q^{50} + 73 q^{52} + 19 q^{53} + 29 q^{55} + 30 q^{56} + 2 q^{58} - 20 q^{59} + 15 q^{61} - 12 q^{62} + 82 q^{64} + 28 q^{65} + 20 q^{67} + 23 q^{68} - 24 q^{70} - 63 q^{71} + 19 q^{73} - 16 q^{74} - 44 q^{76} + 7 q^{77} + 32 q^{79} + 56 q^{80} - 20 q^{82} + 21 q^{83} + 4 q^{85} + 6 q^{86} + 55 q^{88} + 13 q^{89} + 70 q^{91} - 30 q^{92} - 12 q^{94} - 9 q^{95} - 9 q^{97} - 31 q^{98}+O(q^{100}) \) 20 * q - 2 * q^2 + 30 * q^4 + q^5 + 9 * q^7 - 6 * q^8 + 7 * q^10 + 21 * q^13 + q^14 + 58 * q^16 + 4 * q^17 + 7 * q^19 + 20 * q^20 + 7 * q^22 - 20 * q^23 + 47 * q^25 - 8 * q^26 + 11 * q^28 - 20 * q^29 + 28 * q^31 - 14 * q^32 + 16 * q^34 - 9 * q^35 + 14 * q^37 + 20 * q^38 + 34 * q^40 - 7 * q^41 + 3 * q^43 + q^44 + 2 * q^46 - 3 * q^47 + 35 * q^49 + 24 * q^50 + 73 * q^52 + 19 * q^53 + 29 * q^55 + 30 * q^56 + 2 * q^58 - 20 * q^59 + 15 * q^61 - 12 * q^62 + 82 * q^64 + 28 * q^65 + 20 * q^67 + 23 * q^68 - 24 * q^70 - 63 * q^71 + 19 * q^73 - 16 * q^74 - 44 * q^76 + 7 * q^77 + 32 * q^79 + 56 * q^80 - 20 * q^82 + 21 * q^83 + 4 * q^85 + 6 * q^86 + 55 * q^88 + 13 * q^89 + 70 * q^91 - 30 * q^92 - 12 * q^94 - 9 * q^95 - 9 * q^97 - 31 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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