LMFDB - Embedded newform 6003.2.a.o.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("6003.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$47.9341963334$
Analytic rank:	$1$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 4 x^{12} - 11 x^{11} + 58 x^{10} + 24 x^{9} - 298 x^{8} + 97 x^{7} + 641 x^{6} - 402 x^{5} + \cdots - 40 $ x^13 - 4x^12 - 11x^11 + 58x^10 + 24x^9 - 298x^8 + 97x^7 + 641x^6 - 402x^5 - 547x^4 + 352x^3 + 219x^2 - 88x - 40
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 667)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.60496$ of defining polynomial
Character	$\chi$	$=$	6003.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.60496 q^{2} +4.78583 q^{4} -1.68070 q^{5} +2.80821 q^{7} -7.25699 q^{8} +O(q^{10})$	\(q-2.60496 q^{2} +4.78583 q^{4} -1.68070 q^{5} +2.80821 q^{7} -7.25699 q^{8} +4.37815 q^{10} +1.84810 q^{11} -0.773187 q^{13} -7.31527 q^{14} +9.33253 q^{16} +0.882567 q^{17} -3.94266 q^{19} -8.04353 q^{20} -4.81425 q^{22} +1.00000 q^{23} -2.17526 q^{25} +2.01412 q^{26} +13.4396 q^{28} -1.00000 q^{29} +5.16975 q^{31} -9.79691 q^{32} -2.29905 q^{34} -4.71974 q^{35} +3.00050 q^{37} +10.2705 q^{38} +12.1968 q^{40} -1.65149 q^{41} -3.35090 q^{43} +8.84472 q^{44} -2.60496 q^{46} +1.98916 q^{47} +0.886016 q^{49} +5.66648 q^{50} -3.70034 q^{52} -1.44385 q^{53} -3.10610 q^{55} -20.3791 q^{56} +2.60496 q^{58} -2.83981 q^{59} +4.60276 q^{61} -13.4670 q^{62} +6.85553 q^{64} +1.29949 q^{65} -13.2444 q^{67} +4.22382 q^{68} +12.2947 q^{70} +14.4949 q^{71} -12.2449 q^{73} -7.81620 q^{74} -18.8689 q^{76} +5.18986 q^{77} -2.29854 q^{79} -15.6851 q^{80} +4.30208 q^{82} -6.46906 q^{83} -1.48333 q^{85} +8.72897 q^{86} -13.4117 q^{88} -16.1242 q^{89} -2.17127 q^{91} +4.78583 q^{92} -5.18168 q^{94} +6.62640 q^{95} +11.0455 q^{97} -2.30804 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 4 q^{2} + 12 q^{4} - 16 q^{5} + q^{7} - 6 q^{8}+O(q^{10}) $ 13 * q - 4 * q^2 + 12 * q^4 - 16 * q^5 + q^7 - 6 * q^8	$ 13 q - 4 q^{2} + 12 q^{4} - 16 q^{5} + q^{7} - 6 q^{8} + 10 q^{10} - 10 q^{11} + 7 q^{13} + 12 q^{14} + 2 q^{16} - 26 q^{17} - 25 q^{20} - 15 q^{22} + 13 q^{23} + 19 q^{25} + 15 q^{26} + 5 q^{28} - 13 q^{29} - 6 q^{31} - 16 q^{32} + 11 q^{34} - q^{35} + 15 q^{37} - 8 q^{38} + 14 q^{40} - 9 q^{41} + q^{43} - 29 q^{44} - 4 q^{46} - 15 q^{47} + 4 q^{49} - 31 q^{50} - 8 q^{52} - 43 q^{53} - 3 q^{55} + 5 q^{56} + 4 q^{58} + 9 q^{59} + 20 q^{61} - 11 q^{62} - 16 q^{64} + 25 q^{65} + q^{67} - 21 q^{68} - 2 q^{70} - 17 q^{71} + 26 q^{73} - 11 q^{74} + 8 q^{76} - 17 q^{77} + 5 q^{79} - 10 q^{80} - 25 q^{82} - 4 q^{83} + 20 q^{85} + 13 q^{86} - 32 q^{88} - 48 q^{89} - 9 q^{91} + 12 q^{92} - 65 q^{94} - 8 q^{95} + 30 q^{97} - 2 q^{98}+O(q^{100}) $ 13 * q - 4 * q^2 + 12 * q^4 - 16 * q^5 + q^7 - 6 * q^8 + 10 * q^10 - 10 * q^11 + 7 * q^13 + 12 * q^14 + 2 * q^16 - 26 * q^17 - 25 * q^20 - 15 * q^22 + 13 * q^23 + 19 * q^25 + 15 * q^26 + 5 * q^28 - 13 * q^29 - 6 * q^31 - 16 * q^32 + 11 * q^34 - q^35 + 15 * q^37 - 8 * q^38 + 14 * q^40 - 9 * q^41 + q^43 - 29 * q^44 - 4 * q^46 - 15 * q^47 + 4 * q^49 - 31 * q^50 - 8 * q^52 - 43 * q^53 - 3 * q^55 + 5 * q^56 + 4 * q^58 + 9 * q^59 + 20 * q^61 - 11 * q^62 - 16 * q^64 + 25 * q^65 + q^67 - 21 * q^68 - 2 * q^70 - 17 * q^71 + 26 * q^73 - 11 * q^74 + 8 * q^76 - 17 * q^77 + 5 * q^79 - 10 * q^80 - 25 * q^82 - 4 * q^83 + 20 * q^85 + 13 * q^86 - 32 * q^88 - 48 * q^89 - 9 * q^91 + 12 * q^92 - 65 * q^94 - 8 * q^95 + 30 * q^97 - 2 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.60496	−1.84199	−0.920994	−	0.389578i	$-0.872621\pi$
$2$	−2.60496	−1.84199	−0.920994	+	0.389578i	$0.872621\pi$
$3$	0	0
$3$	0	0
$4$	4.78583	2.39292
$5$	−1.68070	−0.751630	−0.375815	−	0.926695i	$-0.622637\pi$
$5$	−1.68070	−0.751630	−0.375815	+	0.926695i	$0.622637\pi$
$6$	0	0
$7$	2.80821	1.06140	0.530701	−	0.847559i	$-0.321929\pi$
$7$	2.80821	1.06140	0.530701	+	0.847559i	$0.321929\pi$
$8$	−7.25699	−2.56573
$9$	0	0
$10$	4.37815	1.38449
$11$	1.84810	0.557225	0.278612	−	0.960404i	$-0.410126\pi$
$11$	1.84810	0.557225	0.278612	+	0.960404i	$0.410126\pi$
$12$	0	0
$13$	−0.773187	−0.214443	−0.107222	−	0.994235i	$-0.534195\pi$
$13$	−0.773187	−0.214443	−0.107222	+	0.994235i	$0.534195\pi$
$14$	−7.31527	−1.95509
$15$	0	0
$16$	9.33253	2.33313
$17$	0.882567	0.214054	0.107027	−	0.994256i	$-0.465867\pi$
$17$	0.882567	0.214054	0.107027	+	0.994256i	$0.465867\pi$
$18$	0	0
$19$	−3.94266	−0.904507	−0.452254	−	0.891889i	$-0.649380\pi$
$19$	−3.94266	−0.904507	−0.452254	+	0.891889i	$0.649380\pi$
$20$	−8.04353	−1.79859
$21$	0	0
$22$	−4.81425	−1.02640
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−2.17526	−0.435052
$26$	2.01412	0.395002
$27$	0	0
$28$	13.4396	2.53985
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	5.16975	0.928514	0.464257	−	0.885700i	$-0.346321\pi$
$31$	5.16975	0.928514	0.464257	+	0.885700i	$0.346321\pi$
$32$	−9.79691	−1.73187
$33$	0	0
$34$	−2.29905	−0.394284
$35$	−4.71974	−0.797781
$36$	0	0
$37$	3.00050	0.493280	0.246640	−	0.969107i	$-0.420674\pi$
$37$	3.00050	0.493280	0.246640	+	0.969107i	$0.420674\pi$
$38$	10.2705	1.66609
$39$	0	0
$40$	12.1968	1.92848
$41$	−1.65149	−0.257920	−0.128960	−	0.991650i	$-0.541164\pi$
$41$	−1.65149	−0.257920	−0.128960	+	0.991650i	$0.541164\pi$
$42$	0	0
$43$	−3.35090	−0.511007	−0.255504	−	0.966808i	$-0.582241\pi$
$43$	−3.35090	−0.511007	−0.255504	+	0.966808i	$0.582241\pi$
$44$	8.84472	1.33339
$45$	0	0
$46$	−2.60496	−0.384081
$47$	1.98916	0.290148	0.145074	−	0.989421i	$-0.453658\pi$
$47$	1.98916	0.290148	0.145074	+	0.989421i	$0.453658\pi$
$48$	0	0
$49$	0.886016	0.126574
$50$	5.66648	0.801361
$51$	0	0
$52$	−3.70034	−0.513145
$53$	−1.44385	−0.198328	−0.0991642	−	0.995071i	$-0.531617\pi$
$53$	−1.44385	−0.198328	−0.0991642	+	0.995071i	$0.531617\pi$
$54$	0	0
$55$	−3.10610	−0.418827
$56$	−20.3791	−2.72327
$57$	0	0
$58$	2.60496	0.342048
$59$	−2.83981	−0.369711	−0.184856	−	0.982766i	$-0.559182\pi$
$59$	−2.83981	−0.369711	−0.184856	+	0.982766i	$0.559182\pi$
$60$	0	0
$61$	4.60276	0.589323	0.294661	−	0.955602i	$-0.404793\pi$
$61$	4.60276	0.589323	0.294661	+	0.955602i	$0.404793\pi$
$62$	−13.4670	−1.71031
$63$	0	0
$64$	6.85553	0.856942
$65$	1.29949	0.161182
$66$	0	0
$67$	−13.2444	−1.61806	−0.809032	−	0.587764i	$-0.800008\pi$
$67$	−13.2444	−1.61806	−0.809032	+	0.587764i	$0.800008\pi$
$68$	4.22382	0.512213
$69$	0	0
$70$	12.2947	1.46950
$71$	14.4949	1.72023	0.860115	−	0.510099i	$-0.170391\pi$
$71$	14.4949	1.72023	0.860115	+	0.510099i	$0.170391\pi$
$72$	0	0
$73$	−12.2449	−1.43316	−0.716579	−	0.697506i	$-0.754293\pi$
$73$	−12.2449	−1.43316	−0.716579	+	0.697506i	$0.754293\pi$
$74$	−7.81620	−0.908615
$75$	0	0
$76$	−18.8689	−2.16441
$77$	5.18986	0.591439
$78$	0	0
$79$	−2.29854	−0.258606	−0.129303	−	0.991605i	$-0.541274\pi$
$79$	−2.29854	−0.258606	−0.129303	+	0.991605i	$0.541274\pi$
$80$	−15.6851	−1.75365
$81$	0	0
$82$	4.30208	0.475085
$83$	−6.46906	−0.710072	−0.355036	−	0.934853i	$-0.615531\pi$
$83$	−6.46906	−0.710072	−0.355036	+	0.934853i	$0.615531\pi$
$84$	0	0
$85$	−1.48333	−0.160889
$86$	8.72897	0.941269
$87$	0	0
$88$	−13.4117	−1.42969
$89$	−16.1242	−1.70916	−0.854582	−	0.519316i	$-0.826187\pi$
$89$	−16.1242	−1.70916	−0.854582	+	0.519316i	$0.826187\pi$
$90$	0	0
$91$	−2.17127	−0.227611
$92$	4.78583	0.498958
$93$	0	0
$94$	−5.18168	−0.534449
$95$	6.62640	0.679855
$96$	0	0
$97$	11.0455	1.12150	0.560750	−	0.827985i	$-0.310513\pi$
$97$	11.0455	1.12150	0.560750	+	0.827985i	$0.310513\pi$
$98$	−2.30804	−0.233147
$99$	0	0
$100$	−10.4104	−1.04104
$101$	4.27667	0.425544	0.212772	−	0.977102i	$-0.431751\pi$
$101$	4.27667	0.425544	0.212772	+	0.977102i	$0.431751\pi$
$102$	0	0
$103$	−10.9818	−1.08207	−0.541036	−	0.840999i	$-0.681968\pi$
$103$	−10.9818	−1.08207	−0.541036	+	0.840999i	$0.681968\pi$
$104$	5.61101	0.550205
$105$	0	0
$106$	3.76118	0.365318
$107$	3.61453	0.349430	0.174715	−	0.984619i	$-0.444100\pi$
$107$	3.61453	0.349430	0.174715	+	0.984619i	$0.444100\pi$
$108$	0	0
$109$	11.3108	1.08338	0.541689	−	0.840579i	$-0.317785\pi$
$109$	11.3108	1.08338	0.541689	+	0.840579i	$0.317785\pi$
$110$	8.09128	0.771473
$111$	0	0
$112$	26.2077	2.47639
$113$	−8.28187	−0.779093	−0.389546	−	0.921007i	$-0.627368\pi$
$113$	−8.28187	−0.779093	−0.389546	+	0.921007i	$0.627368\pi$
$114$	0	0
$115$	−1.68070	−0.156726
$116$	−4.78583	−0.444353
$117$	0	0
$118$	7.39759	0.681003
$119$	2.47843	0.227197
$120$	0	0
$121$	−7.58451	−0.689501
$122$	−11.9900	−1.08553
$123$	0	0
$124$	24.7416	2.22186
$125$	12.0594	1.07863
$126$	0	0
$127$	14.2573	1.26514	0.632568	−	0.774505i	$-0.282001\pi$
$127$	14.2573	1.26514	0.632568	+	0.774505i	$0.282001\pi$
$128$	1.73541	0.153390
$129$	0	0
$130$	−3.38513	−0.296895
$131$	−6.25663	−0.546645	−0.273322	−	0.961923i	$-0.588123\pi$
$131$	−6.25663	−0.546645	−0.273322	+	0.961923i	$0.588123\pi$
$132$	0	0
$133$	−11.0718	−0.960046
$134$	34.5013	2.98045
$135$	0	0
$136$	−6.40478	−0.549205
$137$	−9.16909	−0.783369	−0.391684	−	0.920100i	$-0.628107\pi$
$137$	−9.16909	−0.783369	−0.391684	+	0.920100i	$0.628107\pi$
$138$	0	0
$139$	4.57593	0.388126	0.194063	−	0.980989i	$-0.437833\pi$
$139$	4.57593	0.388126	0.194063	+	0.980989i	$0.437833\pi$
$140$	−22.5879	−1.90902
$141$	0	0
$142$	−37.7587	−3.16864
$143$	−1.42893	−0.119493
$144$	0	0
$145$	1.68070	0.139574
$146$	31.8975	2.63986
$147$	0	0
$148$	14.3599	1.18038
$149$	−11.8201	−0.968344	−0.484172	−	0.874973i	$-0.660879\pi$
$149$	−11.8201	−0.968344	−0.484172	+	0.874973i	$0.660879\pi$
$150$	0	0
$151$	−19.5875	−1.59401	−0.797003	−	0.603975i	$-0.793583\pi$
$151$	−19.5875	−1.59401	−0.797003	+	0.603975i	$0.793583\pi$
$152$	28.6118	2.32072
$153$	0	0
$154$	−13.5194	−1.08942
$155$	−8.68878	−0.697899
$156$	0	0
$157$	13.1463	1.04919	0.524595	−	0.851352i	$-0.324217\pi$
$157$	13.1463	1.04919	0.524595	+	0.851352i	$0.324217\pi$
$158$	5.98761	0.476348
$159$	0	0
$160$	16.4656	1.30172
$161$	2.80821	0.221318
$162$	0	0
$163$	−17.5673	−1.37598	−0.687988	−	0.725722i	$-0.741506\pi$
$163$	−17.5673	−1.37598	−0.687988	+	0.725722i	$0.741506\pi$
$164$	−7.90377	−0.617181
$165$	0	0
$166$	16.8517	1.30794
$167$	2.69139	0.208266	0.104133	−	0.994563i	$-0.466793\pi$
$167$	2.69139	0.208266	0.104133	+	0.994563i	$0.466793\pi$
$168$	0	0
$169$	−12.4022	−0.954014
$170$	3.86401	0.296356
$171$	0	0
$172$	−16.0368	−1.22280
$173$	−4.04664	−0.307661	−0.153830	−	0.988097i	$-0.549161\pi$
$173$	−4.04664	−0.307661	−0.153830	+	0.988097i	$0.549161\pi$
$174$	0	0
$175$	−6.10858	−0.461765
$176$	17.2475	1.30008
$177$	0	0
$178$	42.0030	3.14826
$179$	−18.9925	−1.41956	−0.709782	−	0.704422i	$-0.751206\pi$
$179$	−18.9925	−1.41956	−0.709782	+	0.704422i	$0.751206\pi$
$180$	0	0
$181$	−1.08134	−0.0803754	−0.0401877	−	0.999192i	$-0.512796\pi$
$181$	−1.08134	−0.0803754	−0.0401877	+	0.999192i	$0.512796\pi$
$182$	5.65607	0.419256
$183$	0	0
$184$	−7.25699	−0.534992
$185$	−5.04293	−0.370764
$186$	0	0
$187$	1.63108	0.119276
$188$	9.51977	0.694301
$189$	0	0
$190$	−17.2615	−1.25228
$191$	8.42595	0.609680	0.304840	−	0.952404i	$-0.401397\pi$
$191$	8.42595	0.609680	0.304840	+	0.952404i	$0.401397\pi$
$192$	0	0
$193$	−0.304423	−0.0219129	−0.0109564	−	0.999940i	$-0.503488\pi$
$193$	−0.304423	−0.0219129	−0.0109564	+	0.999940i	$0.503488\pi$
$194$	−28.7731	−2.06579
$195$	0	0
$196$	4.24033	0.302880
$197$	12.1145	0.863119	0.431560	−	0.902084i	$-0.357963\pi$
$197$	12.1145	0.863119	0.431560	+	0.902084i	$0.357963\pi$
$198$	0	0
$199$	13.0890	0.927857	0.463928	−	0.885873i	$-0.346440\pi$
$199$	13.0890	0.927857	0.463928	+	0.885873i	$0.346440\pi$
$200$	15.7859	1.11623
$201$	0	0
$202$	−11.1406	−0.783847
$203$	−2.80821	−0.197097
$204$	0	0
$205$	2.77566	0.193860
$206$	28.6073	1.99316
$207$	0	0
$208$	−7.21579	−0.500325
$209$	−7.28644	−0.504014
$210$	0	0
$211$	20.5324	1.41351	0.706755	−	0.707458i	$-0.250158\pi$
$211$	20.5324	1.41351	0.706755	+	0.707458i	$0.250158\pi$
$212$	−6.91004	−0.474583
$213$	0	0
$214$	−9.41571	−0.643645
$215$	5.63184	0.384088
$216$	0	0
$217$	14.5177	0.985527
$218$	−29.4642	−1.99557
$219$	0	0
$220$	−14.8653	−1.00222
$221$	−0.682389	−0.0459025
$222$	0	0
$223$	4.71330	0.315626	0.157813	−	0.987469i	$-0.449556\pi$
$223$	4.71330	0.315626	0.157813	+	0.987469i	$0.449556\pi$
$224$	−27.5117	−1.83821
$225$	0	0
$226$	21.5740	1.43508
$227$	−15.8873	−1.05448	−0.527240	−	0.849716i	$-0.676773\pi$
$227$	−15.8873	−1.05448	−0.527240	+	0.849716i	$0.676773\pi$
$228$	0	0
$229$	24.2731	1.60401	0.802006	−	0.597316i	$-0.203766\pi$
$229$	24.2731	1.60401	0.802006	+	0.597316i	$0.203766\pi$
$230$	4.37815	0.288687
$231$	0	0
$232$	7.25699	0.476445
$233$	−15.0023	−0.982833	−0.491416	−	0.870925i	$-0.663521\pi$
$233$	−15.0023	−0.982833	−0.491416	+	0.870925i	$0.663521\pi$
$234$	0	0
$235$	−3.34317	−0.218084
$236$	−13.5908	−0.884688
$237$	0	0
$238$	−6.45621	−0.418494
$239$	−3.22723	−0.208752	−0.104376	−	0.994538i	$-0.533285\pi$
$239$	−3.22723	−0.208752	−0.104376	+	0.994538i	$0.533285\pi$
$240$	0	0
$241$	18.2846	1.17782	0.588909	−	0.808200i	$-0.299558\pi$
$241$	18.2846	1.17782	0.588909	+	0.808200i	$0.299558\pi$
$242$	19.7574	1.27005
$243$	0	0
$244$	22.0280	1.41020
$245$	−1.48912	−0.0951366
$246$	0	0
$247$	3.04841	0.193966
$248$	−37.5168	−2.38232
$249$	0	0
$250$	−31.4144	−1.98682
$251$	−8.58591	−0.541938	−0.270969	−	0.962588i	$-0.587344\pi$
$251$	−8.58591	−0.541938	−0.270969	+	0.962588i	$0.587344\pi$
$252$	0	0
$253$	1.84810	0.116189
$254$	−37.1399	−2.33036
$255$	0	0
$256$	−18.2317	−1.13948
$257$	3.17251	0.197896	0.0989478	−	0.995093i	$-0.468452\pi$
$257$	3.17251	0.197896	0.0989478	+	0.995093i	$0.468452\pi$
$258$	0	0
$259$	8.42603	0.523568
$260$	6.21915	0.385695
$261$	0	0
$262$	16.2983	1.00691
$263$	−0.895148	−0.0551972	−0.0275986	−	0.999619i	$-0.508786\pi$
$263$	−0.895148	−0.0551972	−0.0275986	+	0.999619i	$0.508786\pi$
$264$	0	0
$265$	2.42668	0.149070
$266$	28.8416	1.76839
$267$	0	0
$268$	−63.3856	−3.87189
$269$	−16.1610	−0.985355	−0.492677	−	0.870212i	$-0.663982\pi$
$269$	−16.1610	−0.985355	−0.492677	+	0.870212i	$0.663982\pi$
$270$	0	0
$271$	−7.97606	−0.484511	−0.242255	−	0.970212i	$-0.577887\pi$
$271$	−7.97606	−0.484511	−0.242255	+	0.970212i	$0.577887\pi$
$272$	8.23658	0.499416
$273$	0	0
$274$	23.8852	1.44295
$275$	−4.02011	−0.242422
$276$	0	0
$277$	3.58237	0.215244	0.107622	−	0.994192i	$-0.465676\pi$
$277$	3.58237	0.215244	0.107622	+	0.994192i	$0.465676\pi$
$278$	−11.9201	−0.714922
$279$	0	0
$280$	34.2511	2.04689
$281$	−4.75298	−0.283539	−0.141770	−	0.989900i	$-0.545279\pi$
$281$	−4.75298	−0.283539	−0.141770	+	0.989900i	$0.545279\pi$
$282$	0	0
$283$	−11.3753	−0.676193	−0.338096	−	0.941111i	$-0.609783\pi$
$283$	−11.3753	−0.676193	−0.338096	+	0.941111i	$0.609783\pi$
$284$	69.3703	4.11637
$285$	0	0
$286$	3.72231	0.220105
$287$	−4.63773	−0.273757
$288$	0	0
$289$	−16.2211	−0.954181
$290$	−4.37815	−0.257094
$291$	0	0
$292$	−58.6021	−3.42943
$293$	13.1847	0.770257	0.385128	−	0.922863i	$-0.374157\pi$
$293$	13.1847	0.770257	0.385128	+	0.922863i	$0.374157\pi$
$294$	0	0
$295$	4.77285	0.277886
$296$	−21.7746	−1.26562
$297$	0	0
$298$	30.7910	1.78368
$299$	−0.773187	−0.0447146
$300$	0	0
$301$	−9.41001	−0.542384
$302$	51.0247	2.93614
$303$	0	0
$304$	−36.7949	−2.11033
$305$	−7.73584	−0.442953
$306$	0	0
$307$	−22.1761	−1.26566	−0.632829	−	0.774292i	$-0.718106\pi$
$307$	−22.1761	−1.26566	−0.632829	+	0.774292i	$0.718106\pi$
$308$	24.8378	1.41526
$309$	0	0
$310$	22.6339	1.28552
$311$	31.3611	1.77832	0.889161	−	0.457594i	$-0.151289\pi$
$311$	31.3611	1.77832	0.889161	+	0.457594i	$0.151289\pi$
$312$	0	0
$313$	21.5552	1.21837	0.609185	−	0.793028i	$-0.291497\pi$
$313$	21.5552	1.21837	0.609185	+	0.793028i	$0.291497\pi$
$314$	−34.2457	−1.93260
$315$	0	0
$316$	−11.0004	−0.618822
$317$	−25.8705	−1.45303	−0.726515	−	0.687151i	$-0.758861\pi$
$317$	−25.8705	−1.45303	−0.726515	+	0.687151i	$0.758861\pi$
$318$	0	0
$319$	−1.84810	−0.103474
$320$	−11.5221	−0.644103
$321$	0	0
$322$	−7.31527	−0.407664
$323$	−3.47966	−0.193613
$324$	0	0
$325$	1.68188	0.0932942
$326$	45.7621	2.53453
$327$	0	0
$328$	11.9849	0.661754
$329$	5.58596	0.307964
$330$	0	0
$331$	31.0454	1.70641	0.853205	−	0.521576i	$-0.174656\pi$
$331$	31.0454	1.70641	0.853205	+	0.521576i	$0.174656\pi$
$332$	−30.9599	−1.69914
$333$	0	0
$334$	−7.01098	−0.383624
$335$	22.2599	1.21619
$336$	0	0
$337$	21.2178	1.15581	0.577903	−	0.816105i	$-0.303871\pi$
$337$	21.2178	1.15581	0.577903	+	0.816105i	$0.303871\pi$
$338$	32.3072	1.75728
$339$	0	0
$340$	−7.09895	−0.384995
$341$	9.55424	0.517391
$342$	0	0
$343$	−17.1693	−0.927056
$344$	24.3174	1.31111
$345$	0	0
$346$	10.5414	0.566707
$347$	25.1938	1.35248	0.676238	−	0.736684i	$-0.263609\pi$
$347$	25.1938	1.35248	0.676238	+	0.736684i	$0.263609\pi$
$348$	0	0
$349$	27.4503	1.46938	0.734691	−	0.678402i	$-0.237327\pi$
$349$	27.4503	1.46938	0.734691	+	0.678402i	$0.237327\pi$
$350$	15.9126	0.850566
$351$	0	0
$352$	−18.1057	−0.965038
$353$	−15.5298	−0.826567	−0.413283	−	0.910602i	$-0.635618\pi$
$353$	−15.5298	−0.826567	−0.413283	+	0.910602i	$0.635618\pi$
$354$	0	0
$355$	−24.3616	−1.29298
$356$	−77.1679	−4.08989
$357$	0	0
$358$	49.4747	2.61482
$359$	−8.30533	−0.438339	−0.219169	−	0.975687i	$-0.570335\pi$
$359$	−8.30533	−0.438339	−0.219169	+	0.975687i	$0.570335\pi$
$360$	0	0
$361$	−3.45547	−0.181867
$362$	2.81685	0.148050
$363$	0	0
$364$	−10.3913	−0.544653
$365$	20.5800	1.07720
$366$	0	0
$367$	−13.6556	−0.712815	−0.356407	−	0.934331i	$-0.615998\pi$
$367$	−13.6556	−0.712815	−0.356407	+	0.934331i	$0.615998\pi$
$368$	9.33253	0.486492
$369$	0	0
$370$	13.1367	0.682942
$371$	−4.05463	−0.210506
$372$	0	0
$373$	−33.1536	−1.71663	−0.858315	−	0.513124i	$-0.828488\pi$
$373$	−33.1536	−1.71663	−0.858315	+	0.513124i	$0.828488\pi$
$374$	−4.24889	−0.219705
$375$	0	0
$376$	−14.4353	−0.744443
$377$	0.773187	0.0398212
$378$	0	0
$379$	−19.4221	−0.997649	−0.498824	−	0.866703i	$-0.666235\pi$
$379$	−19.4221	−0.997649	−0.498824	+	0.866703i	$0.666235\pi$
$380$	31.7129	1.62684
$381$	0	0
$382$	−21.9493	−1.12302
$383$	−10.5047	−0.536764	−0.268382	−	0.963313i	$-0.586489\pi$
$383$	−10.5047	−0.536764	−0.268382	+	0.963313i	$0.586489\pi$
$384$	0	0
$385$	−8.72257	−0.444543
$386$	0.793011	0.0403632
$387$	0	0
$388$	52.8619	2.68365
$389$	17.3721	0.880802	0.440401	−	0.897801i	$-0.354836\pi$
$389$	17.3721	0.880802	0.440401	+	0.897801i	$0.354836\pi$
$390$	0	0
$391$	0.882567	0.0446333
$392$	−6.42981	−0.324755
$393$	0	0
$394$	−31.5577	−1.58985
$395$	3.86314	0.194376
$396$	0	0
$397$	−12.6473	−0.634750	−0.317375	−	0.948300i	$-0.602801\pi$
$397$	−12.6473	−0.634750	−0.317375	+	0.948300i	$0.602801\pi$
$398$	−34.0964	−1.70910
$399$	0	0
$400$	−20.3007	−1.01503
$401$	26.3988	1.31829	0.659146	−	0.752015i	$-0.270918\pi$
$401$	26.3988	1.31829	0.659146	+	0.752015i	$0.270918\pi$
$402$	0	0
$403$	−3.99718	−0.199114
$404$	20.4674	1.01829
$405$	0	0
$406$	7.31527	0.363051
$407$	5.54525	0.274868
$408$	0	0
$409$	9.29368	0.459543	0.229771	−	0.973245i	$-0.426202\pi$
$409$	9.29368	0.459543	0.229771	+	0.973245i	$0.426202\pi$
$410$	−7.23049	−0.357088
$411$	0	0
$412$	−52.5572	−2.58931
$413$	−7.97476	−0.392412
$414$	0	0
$415$	10.8725	0.533711
$416$	7.57484	0.371387
$417$	0	0
$418$	18.9809	0.928387
$419$	−25.3396	−1.23792	−0.618961	−	0.785421i	$-0.712446\pi$
$419$	−25.3396	−1.23792	−0.618961	+	0.785421i	$0.712446\pi$
$420$	0	0
$421$	−4.02340	−0.196089	−0.0980443	−	0.995182i	$-0.531259\pi$
$421$	−4.02340	−0.196089	−0.0980443	+	0.995182i	$0.531259\pi$
$422$	−53.4862	−2.60367
$423$	0	0
$424$	10.4780	0.508858
$425$	−1.91981	−0.0931247
$426$	0	0
$427$	12.9255	0.625508
$428$	17.2985	0.836156
$429$	0	0
$430$	−14.6707	−0.707486
$431$	35.5666	1.71318	0.856591	−	0.515996i	$-0.172578\pi$
$431$	35.5666	1.71318	0.856591	+	0.515996i	$0.172578\pi$
$432$	0	0
$433$	30.5767	1.46942	0.734712	−	0.678380i	$-0.237317\pi$
$433$	30.5767	1.46942	0.734712	+	0.678380i	$0.237317\pi$
$434$	−37.8181	−1.81533
$435$	0	0
$436$	54.1316	2.59243
$437$	−3.94266	−0.188603
$438$	0	0
$439$	−31.7051	−1.51320	−0.756600	−	0.653878i	$-0.773141\pi$
$439$	−31.7051	−1.51320	−0.756600	+	0.653878i	$0.773141\pi$
$440$	22.5410	1.07460
$441$	0	0
$442$	1.77760	0.0845517
$443$	−22.0093	−1.04569	−0.522846	−	0.852427i	$-0.675130\pi$
$443$	−22.0093	−1.04569	−0.522846	+	0.852427i	$0.675130\pi$
$444$	0	0
$445$	27.0999	1.28466
$446$	−12.2780	−0.581379
$447$	0	0
$448$	19.2517	0.909559
$449$	−2.97968	−0.140620	−0.0703098	−	0.997525i	$-0.522399\pi$
$449$	−2.97968	−0.140620	−0.0703098	+	0.997525i	$0.522399\pi$
$450$	0	0
$451$	−3.05213	−0.143719
$452$	−39.6356	−1.86430
$453$	0	0
$454$	41.3859	1.94234
$455$	3.64924	0.171079
$456$	0	0
$457$	8.73038	0.408390	0.204195	−	0.978930i	$-0.434542\pi$
$457$	8.73038	0.408390	0.204195	+	0.978930i	$0.434542\pi$
$458$	−63.2305	−2.95457
$459$	0	0
$460$	−8.04353	−0.375031
$461$	12.7674	0.594639	0.297319	−	0.954778i	$-0.403907\pi$
$461$	12.7674	0.594639	0.297319	+	0.954778i	$0.403907\pi$
$462$	0	0
$463$	31.1939	1.44970	0.724851	−	0.688906i	$-0.241909\pi$
$463$	31.1939	1.44970	0.724851	+	0.688906i	$0.241909\pi$
$464$	−9.33253	−0.433252
$465$	0	0
$466$	39.0804	1.81037
$467$	−22.5391	−1.04299	−0.521493	−	0.853256i	$-0.674625\pi$
$467$	−22.5391	−1.04299	−0.521493	+	0.853256i	$0.674625\pi$
$468$	0	0
$469$	−37.1931	−1.71742
$470$	8.70883	0.401708
$471$	0	0
$472$	20.6085	0.948581
$473$	−6.19281	−0.284746
$474$	0	0
$475$	8.57631	0.393508
$476$	11.8613	0.543664
$477$	0	0
$478$	8.40682	0.384519
$479$	24.5397	1.12125	0.560623	−	0.828071i	$-0.310562\pi$
$479$	24.5397	1.12125	0.560623	+	0.828071i	$0.310562\pi$
$480$	0	0
$481$	−2.31995	−0.105781
$482$	−47.6308	−2.16952
$483$	0	0
$484$	−36.2982	−1.64992
$485$	−18.5641	−0.842953
$486$	0	0
$487$	−23.2452	−1.05334	−0.526671	−	0.850069i	$-0.676560\pi$
$487$	−23.2452	−1.05334	−0.526671	+	0.850069i	$0.676560\pi$
$488$	−33.4022	−1.51205
$489$	0	0
$490$	3.87911	0.175240
$491$	5.76174	0.260024	0.130012	−	0.991512i	$-0.458498\pi$
$491$	5.76174	0.260024	0.130012	+	0.991512i	$0.458498\pi$
$492$	0	0
$493$	−0.882567	−0.0397488
$494$	−7.94100	−0.357282
$495$	0	0
$496$	48.2468	2.16635
$497$	40.7047	1.82586
$498$	0	0
$499$	−30.8637	−1.38165	−0.690825	−	0.723022i	$-0.742752\pi$
$499$	−30.8637	−1.38165	−0.690825	+	0.723022i	$0.742752\pi$
$500$	57.7144	2.58107
$501$	0	0
$502$	22.3660	0.998243
$503$	−35.6442	−1.58930	−0.794648	−	0.607071i	$-0.792344\pi$
$503$	−35.6442	−1.58930	−0.794648	+	0.607071i	$0.792344\pi$
$504$	0	0
$505$	−7.18778	−0.319852
$506$	−4.81425	−0.214019
$507$	0	0
$508$	68.2333	3.02736
$509$	3.06341	0.135783	0.0678917	−	0.997693i	$-0.478373\pi$
$509$	3.06341	0.135783	0.0678917	+	0.997693i	$0.478373\pi$
$510$	0	0
$511$	−34.3862	−1.52116
$512$	44.0222	1.94553
$513$	0	0
$514$	−8.26427	−0.364521
$515$	18.4571	0.813318
$516$	0	0
$517$	3.67617	0.161678
$518$	−21.9495	−0.964406
$519$	0	0
$520$	−9.43040	−0.413550
$521$	−30.5468	−1.33828	−0.669140	−	0.743136i	$-0.733337\pi$
$521$	−30.5468	−1.33828	−0.669140	+	0.743136i	$0.733337\pi$
$522$	0	0
$523$	−25.4624	−1.11339	−0.556696	−	0.830716i	$-0.687931\pi$
$523$	−25.4624	−1.11339	−0.556696	+	0.830716i	$0.687931\pi$
$524$	−29.9432	−1.30807
$525$	0	0
$526$	2.33183	0.101672
$527$	4.56265	0.198752
$528$	0	0
$529$	1.00000	0.0434783
$530$	−6.32140	−0.274584
$531$	0	0
$532$	−52.9877	−2.29731
$533$	1.27691	0.0553093
$534$	0	0
$535$	−6.07492	−0.262642
$536$	96.1147	4.15152
$537$	0	0
$538$	42.0989	1.81501
$539$	1.63745	0.0705300
$540$	0	0
$541$	9.43243	0.405532	0.202766	−	0.979227i	$-0.435007\pi$
$541$	9.43243	0.405532	0.202766	+	0.979227i	$0.435007\pi$
$542$	20.7773	0.892463
$543$	0	0
$544$	−8.64643	−0.370712
$545$	−19.0100	−0.814299
$546$	0	0
$547$	41.5690	1.77736	0.888681	−	0.458527i	$-0.151623\pi$
$547$	41.5690	1.77736	0.888681	+	0.458527i	$0.151623\pi$
$548$	−43.8817	−1.87454
$549$	0	0
$550$	10.4722	0.446538
$551$	3.94266	0.167963
$552$	0	0
$553$	−6.45477	−0.274485
$554$	−9.33195	−0.396477
$555$	0	0
$556$	21.8997	0.928752
$557$	−29.1407	−1.23473	−0.617366	−	0.786676i	$-0.711800\pi$
$557$	−29.1407	−1.23473	−0.617366	+	0.786676i	$0.711800\pi$
$558$	0	0
$559$	2.59087	0.109582
$560$	−44.0471	−1.86133
$561$	0	0
$562$	12.3813	0.522276
$563$	−28.9802	−1.22137	−0.610685	−	0.791873i	$-0.709106\pi$
$563$	−28.9802	−1.22137	−0.610685	+	0.791873i	$0.709106\pi$
$564$	0	0
$565$	13.9193	0.585590
$566$	29.6323	1.24554
$567$	0	0
$568$	−105.190	−4.41365
$569$	34.2223	1.43467	0.717337	−	0.696726i	$-0.245361\pi$
$569$	34.2223	1.43467	0.717337	+	0.696726i	$0.245361\pi$
$570$	0	0
$571$	−26.0169	−1.08877	−0.544386	−	0.838835i	$-0.683237\pi$
$571$	−26.0169	−1.08877	−0.544386	+	0.838835i	$0.683237\pi$
$572$	−6.83862	−0.285937
$573$	0	0
$574$	12.0811	0.504256
$575$	−2.17526	−0.0907147
$576$	0	0
$577$	−28.2981	−1.17806	−0.589032	−	0.808110i	$-0.700491\pi$
$577$	−28.2981	−1.17806	−0.589032	+	0.808110i	$0.700491\pi$
$578$	42.2553	1.75759
$579$	0	0
$580$	8.04353	0.333989
$581$	−18.1665	−0.753672
$582$	0	0
$583$	−2.66839	−0.110513
$584$	88.8612	3.67710
$585$	0	0
$586$	−34.3456	−1.41880
$587$	−17.1615	−0.708331	−0.354165	−	0.935183i	$-0.615235\pi$
$587$	−17.1615	−0.708331	−0.354165	+	0.935183i	$0.615235\pi$
$588$	0	0
$589$	−20.3825	−0.839848
$590$	−12.4331	−0.511863
$591$	0	0
$592$	28.0023	1.15089
$593$	17.0152	0.698731	0.349365	−	0.936987i	$-0.386397\pi$
$593$	17.0152	0.698731	0.349365	+	0.936987i	$0.386397\pi$
$594$	0	0
$595$	−4.16548	−0.170768
$596$	−56.5692	−2.31717
$597$	0	0
$598$	2.01412	0.0823636
$599$	6.72206	0.274656	0.137328	−	0.990526i	$-0.456149\pi$
$599$	6.72206	0.274656	0.137328	+	0.990526i	$0.456149\pi$
$600$	0	0
$601$	−45.1047	−1.83986	−0.919930	−	0.392082i	$-0.871755\pi$
$601$	−45.1047	−1.83986	−0.919930	+	0.392082i	$0.871755\pi$
$602$	24.5127	0.999064
$603$	0	0
$604$	−93.7424	−3.81432
$605$	12.7472	0.518249
$606$	0	0
$607$	−22.7337	−0.922732	−0.461366	−	0.887210i	$-0.652640\pi$
$607$	−22.7337	−0.922732	−0.461366	+	0.887210i	$0.652640\pi$
$608$	38.6258	1.56648
$609$	0	0
$610$	20.1516	0.815913
$611$	−1.53799	−0.0622204
$612$	0	0
$613$	−25.6809	−1.03724	−0.518620	−	0.855005i	$-0.673554\pi$
$613$	−25.6809	−1.03724	−0.518620	+	0.855005i	$0.673554\pi$
$614$	57.7679	2.33132
$615$	0	0
$616$	−37.6628	−1.51748
$617$	−22.8029	−0.918009	−0.459004	−	0.888434i	$-0.651794\pi$
$617$	−22.8029	−0.918009	−0.459004	+	0.888434i	$0.651794\pi$
$618$	0	0
$619$	17.3140	0.695909	0.347955	−	0.937511i	$-0.386876\pi$
$619$	17.3140	0.695909	0.347955	+	0.937511i	$0.386876\pi$
$620$	−41.5830	−1.67001
$621$	0	0
$622$	−81.6944	−3.27565
$623$	−45.2801	−1.81411
$624$	0	0
$625$	−9.39192	−0.375677
$626$	−56.1504	−2.24422
$627$	0	0
$628$	62.9161	2.51063
$629$	2.64814	0.105588
$630$	0	0
$631$	−4.67782	−0.186221	−0.0931105	−	0.995656i	$-0.529681\pi$
$631$	−4.67782	−0.186221	−0.0931105	+	0.995656i	$0.529681\pi$
$632$	16.6805	0.663513
$633$	0	0
$634$	67.3916	2.67646
$635$	−23.9623	−0.950913
$636$	0	0
$637$	−0.685056	−0.0271429
$638$	4.81425	0.190598
$639$	0	0
$640$	−2.91670	−0.115293
$641$	27.9708	1.10478	0.552390	−	0.833586i	$-0.313716\pi$
$641$	27.9708	1.10478	0.552390	+	0.833586i	$0.313716\pi$
$642$	0	0
$643$	−29.9724	−1.18199	−0.590997	−	0.806673i	$-0.701266\pi$
$643$	−29.9724	−1.18199	−0.590997	+	0.806673i	$0.701266\pi$
$644$	13.4396	0.529594
$645$	0	0
$646$	9.06438	0.356633
$647$	2.35510	0.0925886	0.0462943	−	0.998928i	$-0.485259\pi$
$647$	2.35510	0.0925886	0.0462943	+	0.998928i	$0.485259\pi$
$648$	0	0
$649$	−5.24826	−0.206012
$650$	−4.38125	−0.171847
$651$	0	0
$652$	−84.0741	−3.29260
$653$	−5.15341	−0.201668	−0.100834	−	0.994903i	$-0.532151\pi$
$653$	−5.15341	−0.201668	−0.100834	+	0.994903i	$0.532151\pi$
$654$	0	0
$655$	10.5155	0.410875
$656$	−15.4126	−0.601761
$657$	0	0
$658$	−14.5512	−0.567266
$659$	−20.5225	−0.799444	−0.399722	−	0.916636i	$-0.630893\pi$
$659$	−20.5225	−0.799444	−0.399722	+	0.916636i	$0.630893\pi$
$660$	0	0
$661$	−12.3073	−0.478697	−0.239348	−	0.970934i	$-0.576934\pi$
$661$	−12.3073	−0.478697	−0.239348	+	0.970934i	$0.576934\pi$
$662$	−80.8721	−3.14318
$663$	0	0
$664$	46.9459	1.82186
$665$	18.6083	0.721599
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	12.8806	0.498364
$669$	0	0
$670$	−57.9861	−2.24020
$671$	8.50638	0.328385
$672$	0	0
$673$	6.69894	0.258225	0.129113	−	0.991630i	$-0.458787\pi$
$673$	6.69894	0.258225	0.129113	+	0.991630i	$0.458787\pi$
$674$	−55.2716	−2.12898
$675$	0	0
$676$	−59.3548	−2.28288
$677$	−19.1249	−0.735030	−0.367515	−	0.930018i	$-0.619791\pi$
$677$	−19.1249	−0.735030	−0.367515	+	0.930018i	$0.619791\pi$
$678$	0	0
$679$	31.0180	1.19036
$680$	10.7645	0.412799
$681$	0	0
$682$	−24.8884	−0.953028
$683$	−7.36478	−0.281805	−0.140903	−	0.990023i	$-0.545000\pi$
$683$	−7.36478	−0.281805	−0.140903	+	0.990023i	$0.545000\pi$
$684$	0	0
$685$	15.4105	0.588803
$686$	44.7254	1.70763
$687$	0	0
$688$	−31.2724	−1.19225
$689$	1.11637	0.0425302
$690$	0	0
$691$	−49.1019	−1.86792	−0.933961	−	0.357374i	$-0.883672\pi$
$691$	−49.1019	−1.86792	−0.933961	+	0.357374i	$0.883672\pi$
$692$	−19.3666	−0.736206
$693$	0	0
$694$	−65.6290	−2.49124
$695$	−7.69075	−0.291727
$696$	0	0
$697$	−1.45755	−0.0552088
$698$	−71.5071	−2.70658
$699$	0	0
$700$	−29.2347	−1.10497
$701$	−36.2294	−1.36837	−0.684183	−	0.729310i	$-0.739841\pi$
$701$	−36.2294	−1.36837	−0.684183	+	0.729310i	$0.739841\pi$
$702$	0	0
$703$	−11.8300	−0.446175
$704$	12.6697	0.477509
$705$	0	0
$706$	40.4545	1.52253
$707$	12.0098	0.451674
$708$	0	0
$709$	−45.7281	−1.71735	−0.858677	−	0.512518i	$-0.828713\pi$
$709$	−45.7281	−1.71735	−0.858677	+	0.512518i	$0.828713\pi$
$710$	63.4609	2.38165
$711$	0	0
$712$	117.013	4.38526
$713$	5.16975	0.193609
$714$	0	0
$715$	2.40160	0.0898147
$716$	−90.8947	−3.39690
$717$	0	0
$718$	21.6351	0.807414
$719$	−33.7657	−1.25925	−0.629624	−	0.776900i	$-0.716791\pi$
$719$	−33.7657	−1.25925	−0.629624	+	0.776900i	$0.716791\pi$
$720$	0	0
$721$	−30.8392	−1.14851
$722$	9.00137	0.334996
$723$	0	0
$724$	−5.17512	−0.192332
$725$	2.17526	0.0807872
$726$	0	0
$727$	−27.4791	−1.01914	−0.509571	−	0.860429i	$-0.670196\pi$
$727$	−27.4791	−1.01914	−0.509571	+	0.860429i	$0.670196\pi$
$728$	15.7569	0.583988
$729$	0	0
$730$	−53.6100	−1.98420
$731$	−2.95739	−0.109383
$732$	0	0
$733$	−15.6367	−0.577554	−0.288777	−	0.957396i	$-0.593249\pi$
$733$	−15.6367	−0.577554	−0.288777	+	0.957396i	$0.593249\pi$
$734$	35.5723	1.31300
$735$	0	0
$736$	−9.79691	−0.361119
$737$	−24.4771	−0.901625
$738$	0	0
$739$	4.37819	0.161054	0.0805271	−	0.996752i	$-0.474340\pi$
$739$	4.37819	0.161054	0.0805271	+	0.996752i	$0.474340\pi$
$740$	−24.1346	−0.887207
$741$	0	0
$742$	10.5622	0.387750
$743$	7.80342	0.286280	0.143140	−	0.989702i	$-0.454280\pi$
$743$	7.80342	0.286280	0.143140	+	0.989702i	$0.454280\pi$
$744$	0	0
$745$	19.8661	0.727836
$746$	86.3640	3.16201
$747$	0	0
$748$	7.80606	0.285418
$749$	10.1503	0.370885
$750$	0	0
$751$	36.6586	1.33769	0.668845	−	0.743402i	$-0.266789\pi$
$751$	36.6586	1.33769	0.668845	+	0.743402i	$0.266789\pi$
$752$	18.5639	0.676954
$753$	0	0
$754$	−2.01412	−0.0733501
$755$	32.9206	1.19810
$756$	0	0
$757$	−15.5724	−0.565989	−0.282994	−	0.959122i	$-0.591328\pi$
$757$	−15.5724	−0.565989	−0.282994	+	0.959122i	$0.591328\pi$
$758$	50.5940	1.83766
$759$	0	0
$760$	−48.0878	−1.74433
$761$	−27.3332	−0.990827	−0.495414	−	0.868657i	$-0.664983\pi$
$761$	−27.3332	−0.990827	−0.495414	+	0.868657i	$0.664983\pi$
$762$	0	0
$763$	31.7630	1.14990
$764$	40.3252	1.45891
$765$	0	0
$766$	27.3643	0.988713
$767$	2.19570	0.0792822
$768$	0	0
$769$	23.2168	0.837219	0.418610	−	0.908166i	$-0.362518\pi$
$769$	23.2168	0.837219	0.418610	+	0.908166i	$0.362518\pi$
$770$	22.7220	0.818843
$771$	0	0
$772$	−1.45692	−0.0524356
$773$	7.03205	0.252925	0.126463	−	0.991971i	$-0.459638\pi$
$773$	7.03205	0.252925	0.126463	+	0.991971i	$0.459638\pi$
$774$	0	0
$775$	−11.2456	−0.403952
$776$	−80.1570	−2.87747
$777$	0	0
$778$	−45.2538	−1.62243
$779$	6.51127	0.233290
$780$	0	0
$781$	26.7881	0.958555
$782$	−2.29905	−0.0822140
$783$	0	0
$784$	8.26877	0.295313
$785$	−22.0950	−0.788603
$786$	0	0
$787$	−38.4072	−1.36907	−0.684535	−	0.728980i	$-0.739995\pi$
$787$	−38.4072	−1.36907	−0.684535	+	0.728980i	$0.739995\pi$
$788$	57.9778	2.06537
$789$	0	0
$790$	−10.0633	−0.358038
$791$	−23.2572	−0.826931
$792$	0	0
$793$	−3.55879	−0.126376
$794$	32.9457	1.16920
$795$	0	0
$796$	62.6419	2.22028
$797$	−29.2736	−1.03693	−0.518463	−	0.855100i	$-0.673496\pi$
$797$	−29.2736	−1.03693	−0.518463	+	0.855100i	$0.673496\pi$
$798$	0	0
$799$	1.75556	0.0621074
$800$	21.3108	0.753452
$801$	0	0
$802$	−68.7678	−2.42828
$803$	−22.6299	−0.798591
$804$	0	0
$805$	−4.71974	−0.166349
$806$	10.4125	0.366765
$807$	0	0
$808$	−31.0357	−1.09183
$809$	53.7604	1.89011	0.945057	−	0.326905i	$-0.106006\pi$
$809$	53.7604	1.89011	0.945057	+	0.326905i	$0.106006\pi$
$810$	0	0
$811$	−15.2539	−0.535636	−0.267818	−	0.963470i	$-0.586303\pi$
$811$	−15.2539	−0.535636	−0.267818	+	0.963470i	$0.586303\pi$
$812$	−13.4396	−0.471637
$813$	0	0
$814$	−14.4452	−0.506303
$815$	29.5253	1.03423
$816$	0	0
$817$	13.2114	0.462210
$818$	−24.2097	−0.846472
$819$	0	0
$820$	13.2838	0.463892
$821$	−32.7377	−1.14256	−0.571278	−	0.820757i	$-0.693552\pi$
$821$	−32.7377	−1.14256	−0.571278	+	0.820757i	$0.693552\pi$
$822$	0	0
$823$	−34.5678	−1.20496	−0.602479	−	0.798135i	$-0.705820\pi$
$823$	−34.5678	−1.20496	−0.602479	+	0.798135i	$0.705820\pi$
$824$	79.6950	2.77631
$825$	0	0
$826$	20.7740	0.722818
$827$	−45.1215	−1.56903	−0.784513	−	0.620112i	$-0.787087\pi$
$827$	−45.1215	−1.56903	−0.784513	+	0.620112i	$0.787087\pi$
$828$	0	0
$829$	−48.9447	−1.69992	−0.849960	−	0.526847i	$-0.823374\pi$
$829$	−48.9447	−1.69992	−0.849960	+	0.526847i	$0.823374\pi$
$830$	−28.3225	−0.983089
$831$	0	0
$832$	−5.30061	−0.183766
$833$	0.781968	0.0270936
$834$	0	0
$835$	−4.52341	−0.156539
$836$	−34.8717	−1.20606
$837$	0	0
$838$	66.0088	2.28024
$839$	54.6452	1.88656	0.943281	−	0.331995i	$-0.107722\pi$
$839$	54.6452	1.88656	0.943281	+	0.331995i	$0.107722\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	10.4808	0.361193
$843$	0	0
$844$	98.2648	3.38241
$845$	20.8443	0.717065
$846$	0	0
$847$	−21.2989	−0.731837
$848$	−13.4748	−0.462726
$849$	0	0
$850$	5.00104	0.171534
$851$	3.00050	0.102856
$852$	0	0
$853$	−18.9241	−0.647948	−0.323974	−	0.946066i	$-0.605019\pi$
$853$	−18.9241	−0.647948	−0.323974	+	0.946066i	$0.605019\pi$
$854$	−33.6704	−1.15218
$855$	0	0
$856$	−26.2306	−0.896543
$857$	−14.1409	−0.483046	−0.241523	−	0.970395i	$-0.577647\pi$
$857$	−14.1409	−0.483046	−0.241523	+	0.970395i	$0.577647\pi$
$858$	0	0
$859$	55.3730	1.88930	0.944652	−	0.328075i	$-0.106400\pi$
$859$	55.3730	1.88930	0.944652	+	0.328075i	$0.106400\pi$
$860$	26.9530	0.919091
$861$	0	0
$862$	−92.6496	−3.15566
$863$	9.62971	0.327799	0.163900	−	0.986477i	$-0.447593\pi$
$863$	9.62971	0.327799	0.163900	+	0.986477i	$0.447593\pi$
$864$	0	0
$865$	6.80117	0.231247
$866$	−79.6512	−2.70666
$867$	0	0
$868$	69.4794	2.35828
$869$	−4.24794	−0.144101
$870$	0	0
$871$	10.2404	0.346983
$872$	−82.0824	−2.77966
$873$	0	0
$874$	10.2705	0.347404
$875$	33.8654	1.14486
$876$	0	0
$877$	6.67453	0.225383	0.112691	−	0.993630i	$-0.464053\pi$
$877$	6.67453	0.225383	0.112691	+	0.993630i	$0.464053\pi$
$878$	82.5905	2.78730
$879$	0	0
$880$	−28.9878	−0.977178
$881$	56.9064	1.91722	0.958612	−	0.284715i	$-0.0918991\pi$
$881$	56.9064	1.91722	0.958612	+	0.284715i	$0.0918991\pi$
$882$	0	0
$883$	8.65659	0.291318	0.145659	−	0.989335i	$-0.453470\pi$
$883$	8.65659	0.291318	0.145659	+	0.989335i	$0.453470\pi$
$884$	−3.26580	−0.109841
$885$	0	0
$886$	57.3334	1.92615
$887$	−21.9095	−0.735648	−0.367824	−	0.929895i	$-0.619897\pi$
$887$	−21.9095	−0.735648	−0.367824	+	0.929895i	$0.619897\pi$
$888$	0	0
$889$	40.0376	1.34282
$890$	−70.5943	−2.36633
$891$	0	0
$892$	22.5571	0.755267
$893$	−7.84256	−0.262441
$894$	0	0
$895$	31.9205	1.06699
$896$	4.87339	0.162809
$897$	0	0
$898$	7.76194	0.259019
$899$	−5.16975	−0.172421
$900$	0	0
$901$	−1.27430	−0.0424530
$902$	7.95069	0.264729
$903$	0	0
$904$	60.1015	1.99895
$905$	1.81740	0.0604126
$906$	0	0
$907$	−50.4844	−1.67631	−0.838153	−	0.545435i	$-0.816365\pi$
$907$	−50.4844	−1.67631	−0.838153	+	0.545435i	$0.816365\pi$
$908$	−76.0342	−2.52328
$909$	0	0
$910$	−9.50614	−0.315125
$911$	−17.2002	−0.569867	−0.284933	−	0.958547i	$-0.591971\pi$
$911$	−17.2002	−0.569867	−0.284933	+	0.958547i	$0.591971\pi$
$912$	0	0
$913$	−11.9555	−0.395670
$914$	−22.7423	−0.752249
$915$	0	0
$916$	116.167	3.83827
$917$	−17.5699	−0.580210
$918$	0	0
$919$	−12.9679	−0.427771	−0.213886	−	0.976859i	$-0.568612\pi$
$919$	−12.9679	−0.427771	−0.213886	+	0.976859i	$0.568612\pi$
$920$	12.1968	0.402116
$921$	0	0
$922$	−33.2587	−1.09532
$923$	−11.2073	−0.368892
$924$	0	0
$925$	−6.52688	−0.214603
$926$	−81.2588	−2.67033
$927$	0	0
$928$	9.79691	0.321599
$929$	47.6705	1.56402	0.782010	−	0.623266i	$-0.214195\pi$
$929$	47.6705	1.56402	0.782010	+	0.623266i	$0.214195\pi$
$930$	0	0
$931$	−3.49326	−0.114487
$932$	−71.7985	−2.35184
$933$	0	0
$934$	58.7136	1.92117
$935$	−2.74134	−0.0896515
$936$	0	0
$937$	−12.0522	−0.393727	−0.196863	−	0.980431i	$-0.563076\pi$
$937$	−12.0522	−0.393727	−0.196863	+	0.980431i	$0.563076\pi$
$938$	96.8866	3.16346
$939$	0	0
$940$	−15.9998	−0.521857
$941$	−15.0969	−0.492146	−0.246073	−	0.969251i	$-0.579140\pi$
$941$	−15.0969	−0.492146	−0.246073	+	0.969251i	$0.579140\pi$
$942$	0	0
$943$	−1.65149	−0.0537800
$944$	−26.5026	−0.862585
$945$	0	0
$946$	16.1320	0.524498
$947$	−15.8760	−0.515900	−0.257950	−	0.966158i	$-0.583047\pi$
$947$	−15.8760	−0.515900	−0.257950	+	0.966158i	$0.583047\pi$
$948$	0	0
$949$	9.46760	0.307331
$950$	−22.3410	−0.724837
$951$	0	0
$952$	−17.9859	−0.582927
$953$	−24.7437	−0.801528	−0.400764	−	0.916181i	$-0.631255\pi$
$953$	−24.7437	−0.801528	−0.400764	+	0.916181i	$0.631255\pi$
$954$	0	0
$955$	−14.1615	−0.458254
$956$	−15.4450	−0.499527
$957$	0	0
$958$	−63.9250	−2.06532
$959$	−25.7487	−0.831469
$960$	0	0
$961$	−4.27369	−0.137861
$962$	6.04339	0.194847
$963$	0	0
$964$	87.5072	2.81842
$965$	0.511643	0.0164704
$966$	0	0
$967$	4.50132	0.144753	0.0723763	−	0.997377i	$-0.476942\pi$
$967$	4.50132	0.144753	0.0723763	+	0.997377i	$0.476942\pi$
$968$	55.0407	1.76908
$969$	0	0
$970$	48.3588	1.55271
$971$	9.14396	0.293444	0.146722	−	0.989178i	$-0.453128\pi$
$971$	9.14396	0.293444	0.146722	+	0.989178i	$0.453128\pi$
$972$	0	0
$973$	12.8502	0.411957
$974$	60.5530	1.94024
$975$	0	0
$976$	42.9554	1.37497
$977$	−26.5953	−0.850858	−0.425429	−	0.904992i	$-0.639877\pi$
$977$	−26.5953	−0.850858	−0.425429	+	0.904992i	$0.639877\pi$
$978$	0	0
$979$	−29.7993	−0.952389
$980$	−7.12670	−0.227654
$981$	0	0
$982$	−15.0091	−0.478961
$983$	14.8855	0.474773	0.237387	−	0.971415i	$-0.423709\pi$
$983$	14.8855	0.474773	0.237387	+	0.971415i	$0.423709\pi$
$984$	0	0
$985$	−20.3607	−0.648746
$986$	2.29905	0.0732168
$987$	0	0
$988$	14.5892	0.464144
$989$	−3.35090	−0.106552
$990$	0	0
$991$	−14.8656	−0.472222	−0.236111	−	0.971726i	$-0.575873\pi$
$991$	−14.8656	−0.472222	−0.236111	+	0.971726i	$0.575873\pi$
$992$	−50.6476	−1.60806
$993$	0	0
$994$	−106.034	−3.36320
$995$	−21.9987	−0.697405
$996$	0	0
$997$	−3.64966	−0.115586	−0.0577930	−	0.998329i	$-0.518406\pi$
$997$	−3.64966	−0.115586	−0.0577930	+	0.998329i	$0.518406\pi$
$998$	80.3989	2.54498
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.o.1.1		13
3.2	odd	2		667.2.a.c.1.13	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
667.2.a.c.1.13	✓	13	3.2	odd	2
6003.2.a.o.1.1		13	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-2.60496 q^{2} +4.78583 q^{4} -1.68070 q^{5} +2.80821 q^{7} -7.25699 q^{8} +O(q^{10})\)	\(q-2.60496 q^{2} +4.78583 q^{4} -1.68070 q^{5} +2.80821 q^{7} -7.25699 q^{8} +4.37815 q^{10} +1.84810 q^{11} -0.773187 q^{13} -7.31527 q^{14} +9.33253 q^{16} +0.882567 q^{17} -3.94266 q^{19} -8.04353 q^{20} -4.81425 q^{22} +1.00000 q^{23} -2.17526 q^{25} +2.01412 q^{26} +13.4396 q^{28} -1.00000 q^{29} +5.16975 q^{31} -9.79691 q^{32} -2.29905 q^{34} -4.71974 q^{35} +3.00050 q^{37} +10.2705 q^{38} +12.1968 q^{40} -1.65149 q^{41} -3.35090 q^{43} +8.84472 q^{44} -2.60496 q^{46} +1.98916 q^{47} +0.886016 q^{49} +5.66648 q^{50} -3.70034 q^{52} -1.44385 q^{53} -3.10610 q^{55} -20.3791 q^{56} +2.60496 q^{58} -2.83981 q^{59} +4.60276 q^{61} -13.4670 q^{62} +6.85553 q^{64} +1.29949 q^{65} -13.2444 q^{67} +4.22382 q^{68} +12.2947 q^{70} +14.4949 q^{71} -12.2449 q^{73} -7.81620 q^{74} -18.8689 q^{76} +5.18986 q^{77} -2.29854 q^{79} -15.6851 q^{80} +4.30208 q^{82} -6.46906 q^{83} -1.48333 q^{85} +8.72897 q^{86} -13.4117 q^{88} -16.1242 q^{89} -2.17127 q^{91} +4.78583 q^{92} -5.18168 q^{94} +6.62640 q^{95} +11.0455 q^{97} -2.30804 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 4 q^{2} + 12 q^{4} - 16 q^{5} + q^{7} - 6 q^{8}+O(q^{10}) \) 13 * q - 4 * q^2 + 12 * q^4 - 16 * q^5 + q^7 - 6 * q^8	\( 13 q - 4 q^{2} + 12 q^{4} - 16 q^{5} + q^{7} - 6 q^{8} + 10 q^{10} - 10 q^{11} + 7 q^{13} + 12 q^{14} + 2 q^{16} - 26 q^{17} - 25 q^{20} - 15 q^{22} + 13 q^{23} + 19 q^{25} + 15 q^{26} + 5 q^{28} - 13 q^{29} - 6 q^{31} - 16 q^{32} + 11 q^{34} - q^{35} + 15 q^{37} - 8 q^{38} + 14 q^{40} - 9 q^{41} + q^{43} - 29 q^{44} - 4 q^{46} - 15 q^{47} + 4 q^{49} - 31 q^{50} - 8 q^{52} - 43 q^{53} - 3 q^{55} + 5 q^{56} + 4 q^{58} + 9 q^{59} + 20 q^{61} - 11 q^{62} - 16 q^{64} + 25 q^{65} + q^{67} - 21 q^{68} - 2 q^{70} - 17 q^{71} + 26 q^{73} - 11 q^{74} + 8 q^{76} - 17 q^{77} + 5 q^{79} - 10 q^{80} - 25 q^{82} - 4 q^{83} + 20 q^{85} + 13 q^{86} - 32 q^{88} - 48 q^{89} - 9 q^{91} + 12 q^{92} - 65 q^{94} - 8 q^{95} + 30 q^{97} - 2 q^{98}+O(q^{100}) \) 13 * q - 4 * q^2 + 12 * q^4 - 16 * q^5 + q^7 - 6 * q^8 + 10 * q^10 - 10 * q^11 + 7 * q^13 + 12 * q^14 + 2 * q^16 - 26 * q^17 - 25 * q^20 - 15 * q^22 + 13 * q^23 + 19 * q^25 + 15 * q^26 + 5 * q^28 - 13 * q^29 - 6 * q^31 - 16 * q^32 + 11 * q^34 - q^35 + 15 * q^37 - 8 * q^38 + 14 * q^40 - 9 * q^41 + q^43 - 29 * q^44 - 4 * q^46 - 15 * q^47 + 4 * q^49 - 31 * q^50 - 8 * q^52 - 43 * q^53 - 3 * q^55 + 5 * q^56 + 4 * q^58 + 9 * q^59 + 20 * q^61 - 11 * q^62 - 16 * q^64 + 25 * q^65 + q^67 - 21 * q^68 - 2 * q^70 - 17 * q^71 + 26 * q^73 - 11 * q^74 + 8 * q^76 - 17 * q^77 + 5 * q^79 - 10 * q^80 - 25 * q^82 - 4 * q^83 + 20 * q^85 + 13 * q^86 - 32 * q^88 - 48 * q^89 - 9 * q^91 + 12 * q^92 - 65 * q^94 - 8 * q^95 + 30 * q^97 - 2 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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