LMFDB - Embedded newform 6003.2.a.l.1.9

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:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 3x^{9} - 10x^{8} + 32x^{7} + 32x^{6} - 118x^{5} - 29x^{4} + 182x^{3} - 28x^{2} - 101x + 43 $ x^10 - 3x^9 - 10x^8 + 32x^7 + 32x^6 - 118x^5 - 29x^4 + 182x^3 - 28x^2 - 101*x + 43
Coefficient ring:	$\Z[a_1, a_2]$
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			1.9
Root			$2.37954$ 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)

$f(q)$	$=$	$q+2.37954 q^{2} +3.66223 q^{4} +2.05928 q^{5} +5.08987 q^{7} +3.95536 q^{8} +O(q^{10})$	\(q+2.37954 q^{2} +3.66223 q^{4} +2.05928 q^{5} +5.08987 q^{7} +3.95536 q^{8} +4.90014 q^{10} +4.57737 q^{11} -1.87097 q^{13} +12.1116 q^{14} +2.08748 q^{16} +0.272178 q^{17} +0.546501 q^{19} +7.54155 q^{20} +10.8920 q^{22} -1.00000 q^{23} -0.759381 q^{25} -4.45206 q^{26} +18.6403 q^{28} -1.00000 q^{29} -4.25284 q^{31} -2.94346 q^{32} +0.647660 q^{34} +10.4815 q^{35} +1.52870 q^{37} +1.30042 q^{38} +8.14517 q^{40} +8.02619 q^{41} -8.67751 q^{43} +16.7634 q^{44} -2.37954 q^{46} -8.58972 q^{47} +18.9068 q^{49} -1.80698 q^{50} -6.85194 q^{52} -2.34652 q^{53} +9.42606 q^{55} +20.1323 q^{56} -2.37954 q^{58} +1.14292 q^{59} -7.40397 q^{61} -10.1198 q^{62} -11.1791 q^{64} -3.85285 q^{65} -11.9170 q^{67} +0.996780 q^{68} +24.9411 q^{70} +6.20236 q^{71} -10.5699 q^{73} +3.63762 q^{74} +2.00141 q^{76} +23.2982 q^{77} -16.3652 q^{79} +4.29870 q^{80} +19.0987 q^{82} +15.9890 q^{83} +0.560490 q^{85} -20.6485 q^{86} +18.1051 q^{88} -9.38087 q^{89} -9.52302 q^{91} -3.66223 q^{92} -20.4396 q^{94} +1.12540 q^{95} -13.2320 q^{97} +44.9896 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 3 q^{2} + 9 q^{4} + 10 q^{5} + q^{7} + 9 q^{8}+O(q^{10}) $ 10 * q + 3 * q^2 + 9 * q^4 + 10 * q^5 + q^7 + 9 * q^8	$ 10 q + 3 q^{2} + 9 q^{4} + 10 q^{5} + q^{7} + 9 q^{8} - 6 q^{10} - 13 q^{13} + 12 q^{14} - 5 q^{16} + 22 q^{17} - 2 q^{19} - 3 q^{20} + 3 q^{22} - 10 q^{23} + 10 q^{25} + 25 q^{26} + 19 q^{28} - 10 q^{29} - 22 q^{31} + 31 q^{32} + 13 q^{34} + 15 q^{35} - 9 q^{37} + 10 q^{38} - 6 q^{40} + 25 q^{41} + 3 q^{43} + 27 q^{44} - 3 q^{46} + 17 q^{47} + 17 q^{49} - 2 q^{50} - 18 q^{52} + 43 q^{53} - 11 q^{55} + 7 q^{56} - 3 q^{58} + 7 q^{59} - 6 q^{61} - 3 q^{62} + 33 q^{64} - 11 q^{65} + 11 q^{67} + 51 q^{68} + 34 q^{70} + 17 q^{71} - 44 q^{73} - 9 q^{74} + 24 q^{76} + 71 q^{77} + 5 q^{79} - 38 q^{80} + 33 q^{82} + 32 q^{83} + 16 q^{85} + 9 q^{86} + 18 q^{88} + 10 q^{89} - 3 q^{91} - 9 q^{92} + 47 q^{94} + 8 q^{95} + 6 q^{97} + 73 q^{98}+O(q^{100}) $ 10 * q + 3 * q^2 + 9 * q^4 + 10 * q^5 + q^7 + 9 * q^8 - 6 * q^10 - 13 * q^13 + 12 * q^14 - 5 * q^16 + 22 * q^17 - 2 * q^19 - 3 * q^20 + 3 * q^22 - 10 * q^23 + 10 * q^25 + 25 * q^26 + 19 * q^28 - 10 * q^29 - 22 * q^31 + 31 * q^32 + 13 * q^34 + 15 * q^35 - 9 * q^37 + 10 * q^38 - 6 * q^40 + 25 * q^41 + 3 * q^43 + 27 * q^44 - 3 * q^46 + 17 * q^47 + 17 * q^49 - 2 * q^50 - 18 * q^52 + 43 * q^53 - 11 * q^55 + 7 * q^56 - 3 * q^58 + 7 * q^59 - 6 * q^61 - 3 * q^62 + 33 * q^64 - 11 * q^65 + 11 * q^67 + 51 * q^68 + 34 * q^70 + 17 * q^71 - 44 * q^73 - 9 * q^74 + 24 * q^76 + 71 * q^77 + 5 * q^79 - 38 * q^80 + 33 * q^82 + 32 * q^83 + 16 * q^85 + 9 * q^86 + 18 * q^88 + 10 * q^89 - 3 * q^91 - 9 * q^92 + 47 * q^94 + 8 * q^95 + 6 * q^97 + 73 * 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 $

$2$	2.37954	1.68259	0.841296	−	0.540575i	$-0.181793\pi$
$2$	2.37954	1.68259	0.841296	+	0.540575i	$0.181793\pi$
$3$	0	0
$3$	0	0
$4$	3.66223	1.83112
$5$	2.05928	0.920936	0.460468	−	0.887676i	$-0.347682\pi$
$5$	2.05928	0.920936	0.460468	+	0.887676i	$0.347682\pi$
$6$	0	0
$7$	5.08987	1.92379	0.961896	−	0.273416i	$-0.0881537\pi$
$7$	5.08987	1.92379	0.961896	+	0.273416i	$0.0881537\pi$
$8$	3.95536	1.39843
$9$	0	0
$10$	4.90014	1.54956
$11$	4.57737	1.38013	0.690064	−	0.723748i	$-0.257582\pi$
$11$	4.57737	1.38013	0.690064	+	0.723748i	$0.257582\pi$
$12$	0	0
$13$	−1.87097	−0.518915	−0.259457	−	0.965755i	$-0.583544\pi$
$13$	−1.87097	−0.518915	−0.259457	+	0.965755i	$0.583544\pi$
$14$	12.1116	3.23696
$15$	0	0
$16$	2.08748	0.521870
$17$	0.272178	0.0660129	0.0330065	−	0.999455i	$-0.489492\pi$
$17$	0.272178	0.0660129	0.0330065	+	0.999455i	$0.489492\pi$
$18$	0	0
$19$	0.546501	0.125376	0.0626879	−	0.998033i	$-0.480033\pi$
$19$	0.546501	0.125376	0.0626879	+	0.998033i	$0.480033\pi$
$20$	7.54155	1.68634
$21$	0	0
$22$	10.8920	2.32219
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−0.759381	−0.151876
$26$	−4.45206	−0.873122
$27$	0	0
$28$	18.6403	3.52269
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−4.25284	−0.763832	−0.381916	−	0.924197i	$-0.624736\pi$
$31$	−4.25284	−0.763832	−0.381916	+	0.924197i	$0.624736\pi$
$32$	−2.94346	−0.520335
$33$	0	0
$34$	0.647660	0.111073
$35$	10.4815	1.77169
$36$	0	0
$37$	1.52870	0.251318	0.125659	−	0.992074i	$-0.459896\pi$
$37$	1.52870	0.251318	0.125659	+	0.992074i	$0.459896\pi$
$38$	1.30042	0.210956
$39$	0	0
$40$	8.14517	1.28786
$41$	8.02619	1.25348	0.626740	−	0.779228i	$-0.284389\pi$
$41$	8.02619	1.25348	0.626740	+	0.779228i	$0.284389\pi$
$42$	0	0
$43$	−8.67751	−1.32331	−0.661654	−	0.749809i	$-0.730145\pi$
$43$	−8.67751	−1.32331	−0.661654	+	0.749809i	$0.730145\pi$
$44$	16.7634	2.52717
$45$	0	0
$46$	−2.37954	−0.350845
$47$	−8.58972	−1.25294	−0.626470	−	0.779446i	$-0.715501\pi$
$47$	−8.58972	−1.25294	−0.626470	+	0.779446i	$0.715501\pi$
$48$	0	0
$49$	18.9068	2.70097
$50$	−1.80698	−0.255546
$51$	0	0
$52$	−6.85194	−0.950193
$53$	−2.34652	−0.322319	−0.161160	−	0.986928i	$-0.551523\pi$
$53$	−2.34652	−0.322319	−0.161160	+	0.986928i	$0.551523\pi$
$54$	0	0
$55$	9.42606	1.27101
$56$	20.1323	2.69029
$57$	0	0
$58$	−2.37954	−0.312450
$59$	1.14292	0.148795	0.0743975	−	0.997229i	$-0.476297\pi$
$59$	1.14292	0.148795	0.0743975	+	0.997229i	$0.476297\pi$
$60$	0	0
$61$	−7.40397	−0.947981	−0.473990	−	0.880530i	$-0.657187\pi$
$61$	−7.40397	−0.947981	−0.473990	+	0.880530i	$0.657187\pi$
$62$	−10.1198	−1.28522
$63$	0	0
$64$	−11.1791	−1.39738
$65$	−3.85285	−0.477887
$66$	0	0
$67$	−11.9170	−1.45590	−0.727948	−	0.685632i	$-0.759526\pi$
$67$	−11.9170	−1.45590	−0.727948	+	0.685632i	$0.759526\pi$
$68$	0.996780	0.120877
$69$	0	0
$70$	24.9411	2.98103
$71$	6.20236	0.736085	0.368043	−	0.929809i	$-0.380028\pi$
$71$	6.20236	0.736085	0.368043	+	0.929809i	$0.380028\pi$
$72$	0	0
$73$	−10.5699	−1.23711	−0.618556	−	0.785741i	$-0.712282\pi$
$73$	−10.5699	−1.23711	−0.618556	+	0.785741i	$0.712282\pi$
$74$	3.63762	0.422865
$75$	0	0
$76$	2.00141	0.229578
$77$	23.2982	2.65508
$78$	0	0
$79$	−16.3652	−1.84123	−0.920614	−	0.390474i	$-0.872311\pi$
$79$	−16.3652	−1.84123	−0.920614	+	0.390474i	$0.872311\pi$
$80$	4.29870	0.480609
$81$	0	0
$82$	19.0987	2.10910
$83$	15.9890	1.75502	0.877509	−	0.479561i	$-0.159204\pi$
$83$	15.9890	1.75502	0.877509	+	0.479561i	$0.159204\pi$
$84$	0	0
$85$	0.560490	0.0607937
$86$	−20.6485	−2.22659
$87$	0	0
$88$	18.1051	1.93001
$89$	−9.38087	−0.994370	−0.497185	−	0.867645i	$-0.665633\pi$
$89$	−9.38087	−0.994370	−0.497185	+	0.867645i	$0.665633\pi$
$90$	0	0
$91$	−9.52302	−0.998284
$92$	−3.66223	−0.381814
$93$	0	0
$94$	−20.4396	−2.10819
$95$	1.12540	0.115463
$96$	0	0
$97$	−13.2320	−1.34350	−0.671751	−	0.740777i	$-0.734458\pi$
$97$	−13.2320	−1.34350	−0.671751	+	0.740777i	$0.734458\pi$
$98$	44.9896	4.54464
$99$	0	0
$100$	−2.78103	−0.278103
$101$	−6.02178	−0.599190	−0.299595	−	0.954067i	$-0.596852\pi$
$101$	−6.02178	−0.599190	−0.299595	+	0.954067i	$0.596852\pi$
$102$	0	0
$103$	15.3180	1.50933	0.754663	−	0.656113i	$-0.227800\pi$
$103$	15.3180	1.50933	0.754663	+	0.656113i	$0.227800\pi$
$104$	−7.40036	−0.725665
$105$	0	0
$106$	−5.58365	−0.542332
$107$	−0.735546	−0.0711079	−0.0355540	−	0.999368i	$-0.511320\pi$
$107$	−0.735546	−0.0711079	−0.0355540	+	0.999368i	$0.511320\pi$
$108$	0	0
$109$	8.53593	0.817594	0.408797	−	0.912625i	$-0.365948\pi$
$109$	8.53593	0.817594	0.408797	+	0.912625i	$0.365948\pi$
$110$	22.4297	2.13859
$111$	0	0
$112$	10.6250	1.00397
$113$	0.738224	0.0694463	0.0347231	−	0.999397i	$-0.488945\pi$
$113$	0.738224	0.0694463	0.0347231	+	0.999397i	$0.488945\pi$
$114$	0	0
$115$	−2.05928	−0.192029
$116$	−3.66223	−0.340030
$117$	0	0
$118$	2.71962	0.250361
$119$	1.38535	0.126995
$120$	0	0
$121$	9.95228	0.904753
$122$	−17.6181	−1.59507
$123$	0	0
$124$	−15.5749	−1.39867
$125$	−11.8602	−1.06080
$126$	0	0
$127$	−6.60130	−0.585771	−0.292885	−	0.956148i	$-0.594615\pi$
$127$	−6.60130	−0.585771	−0.292885	+	0.956148i	$0.594615\pi$
$128$	−20.7141	−1.83089
$129$	0	0
$130$	−9.16803	−0.804090
$131$	3.35350	0.292996	0.146498	−	0.989211i	$-0.453200\pi$
$131$	3.35350	0.292996	0.146498	+	0.989211i	$0.453200\pi$
$132$	0	0
$133$	2.78162	0.241197
$134$	−28.3571	−2.44968
$135$	0	0
$136$	1.07656	0.0923144
$137$	9.55553	0.816384	0.408192	−	0.912896i	$-0.366159\pi$
$137$	9.55553	0.816384	0.408192	+	0.912896i	$0.366159\pi$
$138$	0	0
$139$	−6.57246	−0.557469	−0.278734	−	0.960368i	$-0.589915\pi$
$139$	−6.57246	−0.557469	−0.278734	+	0.960368i	$0.589915\pi$
$140$	38.3855	3.24417
$141$	0	0
$142$	14.7588	1.23853
$143$	−8.56413	−0.716169
$144$	0	0
$145$	−2.05928	−0.171014
$146$	−25.1515	−2.08155
$147$	0	0
$148$	5.59847	0.460192
$149$	−16.8263	−1.37846	−0.689232	−	0.724540i	$-0.742052\pi$
$149$	−16.8263	−1.37846	−0.689232	+	0.724540i	$0.742052\pi$
$150$	0	0
$151$	19.5683	1.59245	0.796223	−	0.605003i	$-0.206828\pi$
$151$	19.5683	1.59245	0.796223	+	0.605003i	$0.206828\pi$
$152$	2.16160	0.175329
$153$	0	0
$154$	55.4391	4.46741
$155$	−8.75777	−0.703441
$156$	0	0
$157$	−12.3661	−0.986922	−0.493461	−	0.869768i	$-0.664268\pi$
$157$	−12.3661	−0.986922	−0.493461	+	0.869768i	$0.664268\pi$
$158$	−38.9417	−3.09804
$159$	0	0
$160$	−6.06140	−0.479195
$161$	−5.08987	−0.401138
$162$	0	0
$163$	10.0938	0.790611	0.395306	−	0.918550i	$-0.370639\pi$
$163$	10.0938	0.790611	0.395306	+	0.918550i	$0.370639\pi$
$164$	29.3938	2.29527
$165$	0	0
$166$	38.0465	2.95298
$167$	−1.30505	−0.100988	−0.0504940	−	0.998724i	$-0.516080\pi$
$167$	−1.30505	−0.100988	−0.0504940	+	0.998724i	$0.516080\pi$
$168$	0	0
$169$	−9.49946	−0.730728
$170$	1.33371	0.102291
$171$	0	0
$172$	−31.7791	−2.42313
$173$	−2.20103	−0.167342	−0.0836708	−	0.996493i	$-0.526664\pi$
$173$	−2.20103	−0.167342	−0.0836708	+	0.996493i	$0.526664\pi$
$174$	0	0
$175$	−3.86515	−0.292178
$176$	9.55516	0.720247
$177$	0	0
$178$	−22.3222	−1.67312
$179$	24.6440	1.84198	0.920991	−	0.389584i	$-0.127381\pi$
$179$	24.6440	1.84198	0.920991	+	0.389584i	$0.127381\pi$
$180$	0	0
$181$	1.45544	0.108182	0.0540908	−	0.998536i	$-0.482774\pi$
$181$	1.45544	0.108182	0.0540908	+	0.998536i	$0.482774\pi$
$182$	−22.6604	−1.67970
$183$	0	0
$184$	−3.95536	−0.291593
$185$	3.14803	0.231447
$186$	0	0
$187$	1.24586	0.0911063
$188$	−31.4575	−2.29428
$189$	0	0
$190$	2.67793	0.194277
$191$	0.592184	0.0428489	0.0214244	−	0.999770i	$-0.493180\pi$
$191$	0.592184	0.0428489	0.0214244	+	0.999770i	$0.493180\pi$
$192$	0	0
$193$	24.3728	1.75439	0.877197	−	0.480130i	$-0.159410\pi$
$193$	24.3728	1.75439	0.877197	+	0.480130i	$0.159410\pi$
$194$	−31.4861	−2.26057
$195$	0	0
$196$	69.2412	4.94580
$197$	18.2722	1.30184	0.650919	−	0.759147i	$-0.274384\pi$
$197$	18.2722	1.30184	0.650919	+	0.759147i	$0.274384\pi$
$198$	0	0
$199$	7.74033	0.548697	0.274349	−	0.961630i	$-0.411538\pi$
$199$	7.74033	0.548697	0.274349	+	0.961630i	$0.411538\pi$
$200$	−3.00362	−0.212388
$201$	0	0
$202$	−14.3291	−1.00819
$203$	−5.08987	−0.357239
$204$	0	0
$205$	16.5281	1.15438
$206$	36.4498	2.53958
$207$	0	0
$208$	−3.90562	−0.270806
$209$	2.50153	0.173035
$210$	0	0
$211$	13.6581	0.940262	0.470131	−	0.882597i	$-0.344207\pi$
$211$	13.6581	0.940262	0.470131	+	0.882597i	$0.344207\pi$
$212$	−8.59350	−0.590204
$213$	0	0
$214$	−1.75027	−0.119646
$215$	−17.8694	−1.21868
$216$	0	0
$217$	−21.6464	−1.46945
$218$	20.3116	1.37568
$219$	0	0
$220$	34.5204	2.32737
$221$	−0.509238	−0.0342551
$222$	0	0
$223$	0.497369	0.0333063	0.0166532	−	0.999861i	$-0.494699\pi$
$223$	0.497369	0.0333063	0.0166532	+	0.999861i	$0.494699\pi$
$224$	−14.9818	−1.00102
$225$	0	0
$226$	1.75664	0.116850
$227$	−12.6270	−0.838082	−0.419041	−	0.907967i	$-0.637634\pi$
$227$	−12.6270	−0.838082	−0.419041	+	0.907967i	$0.637634\pi$
$228$	0	0
$229$	5.22393	0.345207	0.172603	−	0.984991i	$-0.444782\pi$
$229$	5.22393	0.345207	0.172603	+	0.984991i	$0.444782\pi$
$230$	−4.90014	−0.323106
$231$	0	0
$232$	−3.95536	−0.259682
$233$	−14.1679	−0.928173	−0.464086	−	0.885790i	$-0.653617\pi$
$233$	−14.1679	−0.928173	−0.464086	+	0.885790i	$0.653617\pi$
$234$	0	0
$235$	−17.6886	−1.15388
$236$	4.18563	0.272461
$237$	0	0
$238$	3.29651	0.213681
$239$	−15.7660	−1.01981	−0.509907	−	0.860229i	$-0.670320\pi$
$239$	−15.7660	−1.01981	−0.509907	+	0.860229i	$0.670320\pi$
$240$	0	0
$241$	26.7382	1.72236	0.861179	−	0.508302i	$-0.169726\pi$
$241$	26.7382	1.72236	0.861179	+	0.508302i	$0.169726\pi$
$242$	23.6819	1.52233
$243$	0	0
$244$	−27.1150	−1.73586
$245$	38.9344	2.48743
$246$	0	0
$247$	−1.02249	−0.0650594
$248$	−16.8215	−1.06817
$249$	0	0
$250$	−28.2218	−1.78490
$251$	6.69588	0.422640	0.211320	−	0.977417i	$-0.432224\pi$
$251$	6.69588	0.422640	0.211320	+	0.977417i	$0.432224\pi$
$252$	0	0
$253$	−4.57737	−0.287777
$254$	−15.7081	−0.985613
$255$	0	0
$256$	−26.9321	−1.68326
$257$	10.6964	0.667224	0.333612	−	0.942711i	$-0.391733\pi$
$257$	10.6964	0.667224	0.333612	+	0.942711i	$0.391733\pi$
$258$	0	0
$259$	7.78092	0.483483
$260$	−14.1100	−0.875067
$261$	0	0
$262$	7.97979	0.492993
$263$	16.9191	1.04328	0.521638	−	0.853167i	$-0.325321\pi$
$263$	16.9191	1.04328	0.521638	+	0.853167i	$0.325321\pi$
$264$	0	0
$265$	−4.83213	−0.296836
$266$	6.61899	0.405836
$267$	0	0
$268$	−43.6429	−2.66591
$269$	6.78462	0.413666	0.206833	−	0.978376i	$-0.433684\pi$
$269$	6.78462	0.413666	0.206833	+	0.978376i	$0.433684\pi$
$270$	0	0
$271$	20.3901	1.23861	0.619305	−	0.785151i	$-0.287415\pi$
$271$	20.3901	1.23861	0.619305	+	0.785151i	$0.287415\pi$
$272$	0.568167	0.0344502
$273$	0	0
$274$	22.7378	1.37364
$275$	−3.47596	−0.209609
$276$	0	0
$277$	−6.68828	−0.401860	−0.200930	−	0.979606i	$-0.564396\pi$
$277$	−6.68828	−0.401860	−0.200930	+	0.979606i	$0.564396\pi$
$278$	−15.6395	−0.937993
$279$	0	0
$280$	41.4579	2.47758
$281$	−6.60817	−0.394211	−0.197105	−	0.980382i	$-0.563154\pi$
$281$	−6.60817	−0.394211	−0.197105	+	0.980382i	$0.563154\pi$
$282$	0	0
$283$	17.0146	1.01141	0.505706	−	0.862706i	$-0.331232\pi$
$283$	17.0146	1.01141	0.505706	+	0.862706i	$0.331232\pi$
$284$	22.7145	1.34786
$285$	0	0
$286$	−20.3787	−1.20502
$287$	40.8523	2.41144
$288$	0	0
$289$	−16.9259	−0.995642
$290$	−4.90014	−0.287746
$291$	0	0
$292$	−38.7094	−2.26530
$293$	28.2553	1.65069	0.825347	−	0.564625i	$-0.190979\pi$
$293$	28.2553	1.65069	0.825347	+	0.564625i	$0.190979\pi$
$294$	0	0
$295$	2.35358	0.137031
$296$	6.04657	0.351450
$297$	0	0
$298$	−40.0389	−2.31939
$299$	1.87097	0.108201
$300$	0	0
$301$	−44.1674	−2.54577
$302$	46.5637	2.67944
$303$	0	0
$304$	1.14081	0.0654299
$305$	−15.2468	−0.873030
$306$	0	0
$307$	−7.93383	−0.452808	−0.226404	−	0.974034i	$-0.572697\pi$
$307$	−7.93383	−0.452808	−0.226404	+	0.974034i	$0.572697\pi$
$308$	85.3235	4.86176
$309$	0	0
$310$	−20.8395	−1.18360
$311$	8.99087	0.509825	0.254913	−	0.966964i	$-0.417953\pi$
$311$	8.99087	0.509825	0.254913	+	0.966964i	$0.417953\pi$
$312$	0	0
$313$	18.2556	1.03187	0.515934	−	0.856628i	$-0.327445\pi$
$313$	18.2556	1.03187	0.515934	+	0.856628i	$0.327445\pi$
$314$	−29.4257	−1.66059
$315$	0	0
$316$	−59.9331	−3.37150
$317$	4.09860	0.230200	0.115100	−	0.993354i	$-0.463281\pi$
$317$	4.09860	0.230200	0.115100	+	0.993354i	$0.463281\pi$
$318$	0	0
$319$	−4.57737	−0.256283
$320$	−23.0208	−1.28690
$321$	0	0
$322$	−12.1116	−0.674952
$323$	0.148746	0.00827643
$324$	0	0
$325$	1.42078	0.0788108
$326$	24.0188	1.33028
$327$	0	0
$328$	31.7464	1.75290
$329$	−43.7206	−2.41039
$330$	0	0
$331$	−8.00388	−0.439933	−0.219967	−	0.975507i	$-0.570595\pi$
$331$	−8.00388	−0.439933	−0.219967	+	0.975507i	$0.570595\pi$
$332$	58.5553	3.21364
$333$	0	0
$334$	−3.10543	−0.169922
$335$	−24.5404	−1.34079
$336$	0	0
$337$	−3.94617	−0.214962	−0.107481	−	0.994207i	$-0.534278\pi$
$337$	−3.94617	−0.214962	−0.107481	+	0.994207i	$0.534278\pi$
$338$	−22.6044	−1.22952
$339$	0	0
$340$	2.05265	0.111320
$341$	−19.4668	−1.05419
$342$	0	0
$343$	60.6042	3.27232
$344$	−34.3226	−1.85055
$345$	0	0
$346$	−5.23746	−0.281568
$347$	24.9359	1.33863	0.669315	−	0.742979i	$-0.266588\pi$
$347$	24.9359	1.33863	0.669315	+	0.742979i	$0.266588\pi$
$348$	0	0
$349$	−8.80474	−0.471307	−0.235653	−	0.971837i	$-0.575723\pi$
$349$	−8.80474	−0.471307	−0.235653	+	0.971837i	$0.575723\pi$
$350$	−9.19730	−0.491616
$351$	0	0
$352$	−13.4733	−0.718129
$353$	8.66021	0.460937	0.230468	−	0.973080i	$-0.425974\pi$
$353$	8.66021	0.460937	0.230468	+	0.973080i	$0.425974\pi$
$354$	0	0
$355$	12.7724	0.677887
$356$	−34.3549	−1.82081
$357$	0	0
$358$	58.6416	3.09930
$359$	−22.8271	−1.20477	−0.602385	−	0.798205i	$-0.705783\pi$
$359$	−22.8271	−1.20477	−0.602385	+	0.798205i	$0.705783\pi$
$360$	0	0
$361$	−18.7013	−0.984281
$362$	3.46327	0.182026
$363$	0	0
$364$	−34.8755	−1.82797
$365$	−21.7663	−1.13930
$366$	0	0
$367$	31.3609	1.63702	0.818512	−	0.574489i	$-0.194799\pi$
$367$	31.3609	1.63702	0.818512	+	0.574489i	$0.194799\pi$
$368$	−2.08748	−0.108817
$369$	0	0
$370$	7.49087	0.389432
$371$	−11.9435	−0.620075
$372$	0	0
$373$	−23.9416	−1.23965	−0.619825	−	0.784740i	$-0.712796\pi$
$373$	−23.9416	−1.23965	−0.619825	+	0.784740i	$0.712796\pi$
$374$	2.96458	0.153295
$375$	0	0
$376$	−33.9754	−1.75215
$377$	1.87097	0.0963600
$378$	0	0
$379$	10.9308	0.561478	0.280739	−	0.959784i	$-0.409420\pi$
$379$	10.9308	0.561478	0.280739	+	0.959784i	$0.409420\pi$
$380$	4.12146	0.211426
$381$	0	0
$382$	1.40913	0.0720972
$383$	15.2443	0.778946	0.389473	−	0.921038i	$-0.372657\pi$
$383$	15.2443	0.778946	0.389473	+	0.921038i	$0.372657\pi$
$384$	0	0
$385$	47.9775	2.44516
$386$	57.9962	2.95193
$387$	0	0
$388$	−48.4585	−2.46011
$389$	28.9315	1.46688	0.733441	−	0.679753i	$-0.237913\pi$
$389$	28.9315	1.46688	0.733441	+	0.679753i	$0.237913\pi$
$390$	0	0
$391$	−0.272178	−0.0137647
$392$	74.7832	3.77712
$393$	0	0
$394$	43.4794	2.19046
$395$	−33.7005	−1.69565
$396$	0	0
$397$	19.3878	0.973048	0.486524	−	0.873667i	$-0.338265\pi$
$397$	19.3878	0.973048	0.486524	+	0.873667i	$0.338265\pi$
$398$	18.4185	0.923234
$399$	0	0
$400$	−1.58519	−0.0792596
$401$	−5.19885	−0.259618	−0.129809	−	0.991539i	$-0.541436\pi$
$401$	−5.19885	−0.259618	−0.129809	+	0.991539i	$0.541436\pi$
$402$	0	0
$403$	7.95695	0.396364
$404$	−22.0532	−1.09719
$405$	0	0
$406$	−12.1116	−0.601088
$407$	6.99744	0.346850
$408$	0	0
$409$	−25.6121	−1.26644	−0.633219	−	0.773972i	$-0.718267\pi$
$409$	−25.6121	−1.26644	−0.633219	+	0.773972i	$0.718267\pi$
$410$	39.3295	1.94234
$411$	0	0
$412$	56.0980	2.76375
$413$	5.81730	0.286251
$414$	0	0
$415$	32.9257	1.61626
$416$	5.50713	0.270009
$417$	0	0
$418$	5.95251	0.291147
$419$	−10.3556	−0.505907	−0.252953	−	0.967478i	$-0.581402\pi$
$419$	−10.3556	−0.505907	−0.252953	+	0.967478i	$0.581402\pi$
$420$	0	0
$421$	12.1491	0.592111	0.296055	−	0.955171i	$-0.404329\pi$
$421$	12.1491	0.592111	0.296055	+	0.955171i	$0.404329\pi$
$422$	32.5000	1.58208
$423$	0	0
$424$	−9.28132	−0.450741
$425$	−0.206687	−0.0100258
$426$	0	0
$427$	−37.6853	−1.82372
$428$	−2.69374	−0.130207
$429$	0	0
$430$	−42.5210	−2.05055
$431$	−28.2807	−1.36223	−0.681116	−	0.732175i	$-0.738505\pi$
$431$	−28.2807	−1.36223	−0.681116	+	0.732175i	$0.738505\pi$
$432$	0	0
$433$	−4.15742	−0.199793	−0.0998964	−	0.994998i	$-0.531851\pi$
$433$	−4.15742	−0.199793	−0.0998964	+	0.994998i	$0.531851\pi$
$434$	−51.5086	−2.47249
$435$	0	0
$436$	31.2605	1.49711
$437$	−0.546501	−0.0261427
$438$	0	0
$439$	18.6089	0.888153	0.444076	−	0.895989i	$-0.353532\pi$
$439$	18.6089	0.888153	0.444076	+	0.895989i	$0.353532\pi$
$440$	37.2834	1.77742
$441$	0	0
$442$	−1.21176	−0.0576373
$443$	7.39743	0.351462	0.175731	−	0.984438i	$-0.443771\pi$
$443$	7.39743	0.351462	0.175731	+	0.984438i	$0.443771\pi$
$444$	0	0
$445$	−19.3178	−0.915751
$446$	1.18351	0.0560409
$447$	0	0
$448$	−56.9000	−2.68827
$449$	−23.3610	−1.10247	−0.551236	−	0.834349i	$-0.685844\pi$
$449$	−23.3610	−1.10247	−0.551236	+	0.834349i	$0.685844\pi$
$450$	0	0
$451$	36.7388	1.72996
$452$	2.70355	0.127164
$453$	0	0
$454$	−30.0465	−1.41015
$455$	−19.6105	−0.919356
$456$	0	0
$457$	−6.43183	−0.300868	−0.150434	−	0.988620i	$-0.548067\pi$
$457$	−6.43183	−0.300868	−0.150434	+	0.988620i	$0.548067\pi$
$458$	12.4306	0.580842
$459$	0	0
$460$	−7.54155	−0.351626
$461$	28.5832	1.33125	0.665627	−	0.746285i	$-0.268164\pi$
$461$	28.5832	1.33125	0.665627	+	0.746285i	$0.268164\pi$
$462$	0	0
$463$	31.8120	1.47843	0.739214	−	0.673470i	$-0.235197\pi$
$463$	31.8120	1.47843	0.739214	+	0.673470i	$0.235197\pi$
$464$	−2.08748	−0.0969088
$465$	0	0
$466$	−33.7133	−1.56174
$467$	−34.9300	−1.61637	−0.808183	−	0.588931i	$-0.799549\pi$
$467$	−34.9300	−1.61637	−0.808183	+	0.588931i	$0.799549\pi$
$468$	0	0
$469$	−60.6561	−2.80084
$470$	−42.0908	−1.94151
$471$	0	0
$472$	4.52064	0.208079
$473$	−39.7201	−1.82633
$474$	0	0
$475$	−0.415002	−0.0190416
$476$	5.07349	0.232543
$477$	0	0
$478$	−37.5158	−1.71593
$479$	4.20878	0.192304	0.0961521	−	0.995367i	$-0.469346\pi$
$479$	4.20878	0.192304	0.0961521	+	0.995367i	$0.469346\pi$
$480$	0	0
$481$	−2.86017	−0.130412
$482$	63.6247	2.89803
$483$	0	0
$484$	36.4476	1.65671
$485$	−27.2483	−1.23728
$486$	0	0
$487$	−11.1540	−0.505437	−0.252719	−	0.967540i	$-0.581325\pi$
$487$	−11.1540	−0.505437	−0.252719	+	0.967540i	$0.581325\pi$
$488$	−29.2853	−1.32568
$489$	0	0
$490$	92.6461	4.18532
$491$	−0.298683	−0.0134794	−0.00673969	−	0.999977i	$-0.502145\pi$
$491$	−0.298683	−0.0134794	−0.00673969	+	0.999977i	$0.502145\pi$
$492$	0	0
$493$	−0.272178	−0.0122583
$494$	−2.43306	−0.109468
$495$	0	0
$496$	−8.87771	−0.398621
$497$	31.5692	1.41607
$498$	0	0
$499$	−19.6615	−0.880168	−0.440084	−	0.897957i	$-0.645051\pi$
$499$	−19.6615	−0.880168	−0.440084	+	0.897957i	$0.645051\pi$
$500$	−43.4346	−1.94246
$501$	0	0
$502$	15.9331	0.711131
$503$	0.660795	0.0294634	0.0147317	−	0.999891i	$-0.495311\pi$
$503$	0.660795	0.0294634	0.0147317	+	0.999891i	$0.495311\pi$
$504$	0	0
$505$	−12.4005	−0.551816
$506$	−10.8920	−0.484211
$507$	0	0
$508$	−24.1755	−1.07261
$509$	−34.5373	−1.53084	−0.765420	−	0.643531i	$-0.777469\pi$
$509$	−34.5373	−1.53084	−0.765420	+	0.643531i	$0.777469\pi$
$510$	0	0
$511$	−53.7994	−2.37995
$512$	−22.6579	−1.00135
$513$	0	0
$514$	25.4526	1.12267
$515$	31.5440	1.38999
$516$	0	0
$517$	−39.3183	−1.72922
$518$	18.5150	0.813504
$519$	0	0
$520$	−15.2394	−0.668292
$521$	−2.47755	−0.108543	−0.0542717	−	0.998526i	$-0.517284\pi$
$521$	−2.47755	−0.108543	−0.0542717	+	0.998526i	$0.517284\pi$
$522$	0	0
$523$	21.1107	0.923108	0.461554	−	0.887112i	$-0.347292\pi$
$523$	21.1107	0.923108	0.461554	+	0.887112i	$0.347292\pi$
$524$	12.2813	0.536510
$525$	0	0
$526$	40.2597	1.75541
$527$	−1.15753	−0.0504228
$528$	0	0
$529$	1.00000	0.0434783
$530$	−11.4983	−0.499453
$531$	0	0
$532$	10.1869	0.441660
$533$	−15.0168	−0.650449
$534$	0	0
$535$	−1.51469	−0.0654859
$536$	−47.1361	−2.03597
$537$	0	0
$538$	16.1443	0.696030
$539$	86.5434	3.72769
$540$	0	0
$541$	27.0220	1.16177	0.580884	−	0.813986i	$-0.302707\pi$
$541$	27.0220	1.16177	0.580884	+	0.813986i	$0.302707\pi$
$542$	48.5191	2.08407
$543$	0	0
$544$	−0.801146	−0.0343488
$545$	17.5778	0.752952
$546$	0	0
$547$	−40.1096	−1.71496	−0.857481	−	0.514516i	$-0.827972\pi$
$547$	−40.1096	−1.71496	−0.857481	+	0.514516i	$0.827972\pi$
$548$	34.9946	1.49489
$549$	0	0
$550$	−8.27121	−0.352686
$551$	−0.546501	−0.0232817
$552$	0	0
$553$	−83.2968	−3.54214
$554$	−15.9151	−0.676166
$555$	0	0
$556$	−24.0699	−1.02079
$557$	−12.7652	−0.540880	−0.270440	−	0.962737i	$-0.587169\pi$
$557$	−12.7652	−0.540880	−0.270440	+	0.962737i	$0.587169\pi$
$558$	0	0
$559$	16.2354	0.686684
$560$	21.8798	0.924592
$561$	0	0
$562$	−15.7244	−0.663296
$563$	−1.70334	−0.0717873	−0.0358937	−	0.999356i	$-0.511428\pi$
$563$	−1.70334	−0.0717873	−0.0358937	+	0.999356i	$0.511428\pi$
$564$	0	0
$565$	1.52021	0.0639556
$566$	40.4870	1.70179
$567$	0	0
$568$	24.5325	1.02936
$569$	−11.6091	−0.486678	−0.243339	−	0.969941i	$-0.578243\pi$
$569$	−11.6091	−0.486678	−0.243339	+	0.969941i	$0.578243\pi$
$570$	0	0
$571$	33.4149	1.39837	0.699185	−	0.714941i	$-0.253546\pi$
$571$	33.4149	1.39837	0.699185	+	0.714941i	$0.253546\pi$
$572$	−31.3638	−1.31139
$573$	0	0
$574$	97.2099	4.05746
$575$	0.759381	0.0316684
$576$	0	0
$577$	40.6873	1.69384	0.846918	−	0.531723i	$-0.178455\pi$
$577$	40.6873	1.69384	0.846918	+	0.531723i	$0.178455\pi$
$578$	−40.2760	−1.67526
$579$	0	0
$580$	−7.54155	−0.313146
$581$	81.3818	3.37629
$582$	0	0
$583$	−10.7409	−0.444842
$584$	−41.8077	−1.73001
$585$	0	0
$586$	67.2349	2.77745
$587$	13.3233	0.549910	0.274955	−	0.961457i	$-0.411337\pi$
$587$	13.3233	0.549910	0.274955	+	0.961457i	$0.411337\pi$
$588$	0	0
$589$	−2.32418	−0.0957661
$590$	5.60045	0.230567
$591$	0	0
$592$	3.19114	0.131155
$593$	−4.89062	−0.200834	−0.100417	−	0.994945i	$-0.532018\pi$
$593$	−4.89062	−0.200834	−0.100417	+	0.994945i	$0.532018\pi$
$594$	0	0
$595$	2.85283	0.116954
$596$	−61.6218	−2.52413
$597$	0	0
$598$	4.45206	0.182058
$599$	−40.1284	−1.63960	−0.819800	−	0.572650i	$-0.805915\pi$
$599$	−40.1284	−1.63960	−0.819800	+	0.572650i	$0.805915\pi$
$600$	0	0
$601$	−37.1812	−1.51665	−0.758326	−	0.651876i	$-0.773982\pi$
$601$	−37.1812	−1.51665	−0.758326	+	0.651876i	$0.773982\pi$
$602$	−105.098	−4.28349
$603$	0	0
$604$	71.6637	2.91595
$605$	20.4945	0.833220
$606$	0	0
$607$	−11.2868	−0.458117	−0.229058	−	0.973413i	$-0.573565\pi$
$607$	−11.2868	−0.458117	−0.229058	+	0.973413i	$0.573565\pi$
$608$	−1.60860	−0.0652374
$609$	0	0
$610$	−36.2805	−1.46895
$611$	16.0711	0.650169
$612$	0	0
$613$	44.5301	1.79855	0.899277	−	0.437380i	$-0.144093\pi$
$613$	44.5301	1.79855	0.899277	+	0.437380i	$0.144093\pi$
$614$	−18.8789	−0.761891
$615$	0	0
$616$	92.1527	3.71294
$617$	−22.2432	−0.895477	−0.447739	−	0.894164i	$-0.647771\pi$
$617$	−22.2432	−0.895477	−0.447739	+	0.894164i	$0.647771\pi$
$618$	0	0
$619$	−19.3967	−0.779621	−0.389811	−	0.920895i	$-0.627460\pi$
$619$	−19.3967	−0.779621	−0.389811	+	0.920895i	$0.627460\pi$
$620$	−32.0730	−1.28808
$621$	0	0
$622$	21.3942	0.857828
$623$	−47.7474	−1.91296
$624$	0	0
$625$	−20.6264	−0.825057
$626$	43.4401	1.73621
$627$	0	0
$628$	−45.2875	−1.80717
$629$	0.416080	0.0165902
$630$	0	0
$631$	−34.9451	−1.39114	−0.695572	−	0.718456i	$-0.744849\pi$
$631$	−34.9451	−1.39114	−0.695572	+	0.718456i	$0.744849\pi$
$632$	−64.7302	−2.57483
$633$	0	0
$634$	9.75280	0.387333
$635$	−13.5939	−0.539458
$636$	0	0
$637$	−35.3742	−1.40158
$638$	−10.8920	−0.431220
$639$	0	0
$640$	−42.6561	−1.68613
$641$	26.1304	1.03209	0.516044	−	0.856562i	$-0.327404\pi$
$641$	26.1304	1.03209	0.516044	+	0.856562i	$0.327404\pi$
$642$	0	0
$643$	33.8053	1.33315	0.666576	−	0.745437i	$-0.267759\pi$
$643$	33.8053	1.33315	0.666576	+	0.745437i	$0.267759\pi$
$644$	−18.6403	−0.734531
$645$	0	0
$646$	0.353947	0.0139259
$647$	41.8834	1.64660	0.823302	−	0.567603i	$-0.192129\pi$
$647$	41.8834	1.64660	0.823302	+	0.567603i	$0.192129\pi$
$648$	0	0
$649$	5.23155	0.205356
$650$	3.38081	0.132606
$651$	0	0
$652$	36.9660	1.44770
$653$	17.5154	0.685432	0.342716	−	0.939439i	$-0.388653\pi$
$653$	17.5154	0.685432	0.342716	+	0.939439i	$0.388653\pi$
$654$	0	0
$655$	6.90578	0.269831
$656$	16.7545	0.654154
$657$	0	0
$658$	−104.035	−4.05571
$659$	−14.8699	−0.579247	−0.289624	−	0.957141i	$-0.593530\pi$
$659$	−14.8699	−0.579247	−0.289624	+	0.957141i	$0.593530\pi$
$660$	0	0
$661$	−27.7449	−1.07915	−0.539577	−	0.841936i	$-0.681416\pi$
$661$	−27.7449	−1.07915	−0.539577	+	0.841936i	$0.681416\pi$
$662$	−19.0456	−0.740228
$663$	0	0
$664$	63.2421	2.45427
$665$	5.72812	0.222127
$666$	0	0
$667$	1.00000	0.0387202
$668$	−4.77941	−0.184921
$669$	0	0
$670$	−58.3951	−2.25600
$671$	−33.8907	−1.30833
$672$	0	0
$673$	16.8217	0.648427	0.324214	−	0.945984i	$-0.394900\pi$
$673$	16.8217	0.648427	0.324214	+	0.945984i	$0.394900\pi$
$674$	−9.39010	−0.361693
$675$	0	0
$676$	−34.7892	−1.33805
$677$	28.3544	1.08975	0.544874	−	0.838518i	$-0.316578\pi$
$677$	28.3544	1.08975	0.544874	+	0.838518i	$0.316578\pi$
$678$	0	0
$679$	−67.3491	−2.58462
$680$	2.21694	0.0850157
$681$	0	0
$682$	−46.3221	−1.77377
$683$	−18.3962	−0.703912	−0.351956	−	0.936017i	$-0.614483\pi$
$683$	−18.3962	−0.703912	−0.351956	+	0.936017i	$0.614483\pi$
$684$	0	0
$685$	19.6775	0.751837
$686$	144.210	5.50598
$687$	0	0
$688$	−18.1141	−0.690595
$689$	4.39028	0.167256
$690$	0	0
$691$	−48.7956	−1.85627	−0.928136	−	0.372240i	$-0.878590\pi$
$691$	−48.7956	−1.85627	−0.928136	+	0.372240i	$0.878590\pi$
$692$	−8.06070	−0.306422
$693$	0	0
$694$	59.3361	2.25237
$695$	−13.5345	−0.513393
$696$	0	0
$697$	2.18456	0.0827460
$698$	−20.9513	−0.793017
$699$	0	0
$700$	−14.1551	−0.535012
$701$	11.8246	0.446610	0.223305	−	0.974749i	$-0.428315\pi$
$701$	11.8246	0.446610	0.223305	+	0.974749i	$0.428315\pi$
$702$	0	0
$703$	0.835438	0.0315091
$704$	−51.1706	−1.92857
$705$	0	0
$706$	20.6074	0.775568
$707$	−30.6501	−1.15272
$708$	0	0
$709$	18.6243	0.699448	0.349724	−	0.936853i	$-0.386275\pi$
$709$	18.6243	0.699448	0.349724	+	0.936853i	$0.386275\pi$
$710$	30.3924	1.14061
$711$	0	0
$712$	−37.1047	−1.39056
$713$	4.25284	0.159270
$714$	0	0
$715$	−17.6359	−0.659546
$716$	90.2522	3.37288
$717$	0	0
$718$	−54.3182	−2.02714
$719$	46.0282	1.71656	0.858282	−	0.513178i	$-0.171532\pi$
$719$	46.0282	1.71656	0.858282	+	0.513178i	$0.171532\pi$
$720$	0	0
$721$	77.9666	2.90363
$722$	−44.5007	−1.65614
$723$	0	0
$724$	5.33014	0.198093
$725$	0.759381	0.0282027
$726$	0	0
$727$	−9.43522	−0.349933	−0.174966	−	0.984574i	$-0.555982\pi$
$727$	−9.43522	−0.349933	−0.174966	+	0.984574i	$0.555982\pi$
$728$	−37.6669	−1.39603
$729$	0	0
$730$	−51.7939	−1.91698
$731$	−2.36183	−0.0873555
$732$	0	0
$733$	−35.1930	−1.29988	−0.649941	−	0.759985i	$-0.725206\pi$
$733$	−35.1930	−1.29988	−0.649941	+	0.759985i	$0.725206\pi$
$734$	74.6246	2.75444
$735$	0	0
$736$	2.94346	0.108497
$737$	−54.5486	−2.00932
$738$	0	0
$739$	−20.3793	−0.749666	−0.374833	−	0.927092i	$-0.622300\pi$
$739$	−20.3793	−0.749666	−0.374833	+	0.927092i	$0.622300\pi$
$740$	11.5288	0.423807
$741$	0	0
$742$	−28.4201	−1.04333
$743$	44.2871	1.62474	0.812368	−	0.583146i	$-0.198178\pi$
$743$	44.2871	1.62474	0.812368	+	0.583146i	$0.198178\pi$
$744$	0	0
$745$	−34.6500	−1.26948
$746$	−56.9701	−2.08583
$747$	0	0
$748$	4.56263	0.166826
$749$	−3.74384	−0.136797
$750$	0	0
$751$	−31.1048	−1.13503	−0.567514	−	0.823363i	$-0.692095\pi$
$751$	−31.1048	−1.13503	−0.567514	+	0.823363i	$0.692095\pi$
$752$	−17.9309	−0.653871
$753$	0	0
$754$	4.45206	0.162135
$755$	40.2966	1.46654
$756$	0	0
$757$	−40.6495	−1.47743	−0.738715	−	0.674018i	$-0.764567\pi$
$757$	−40.6495	−1.47743	−0.738715	+	0.674018i	$0.764567\pi$
$758$	26.0104	0.944739
$759$	0	0
$760$	4.45134	0.161467
$761$	−18.6533	−0.676180	−0.338090	−	0.941114i	$-0.609781\pi$
$761$	−18.6533	−0.676180	−0.338090	+	0.941114i	$0.609781\pi$
$762$	0	0
$763$	43.4468	1.57288
$764$	2.16871	0.0784613
$765$	0	0
$766$	36.2744	1.31065
$767$	−2.13837	−0.0772119
$768$	0	0
$769$	17.4326	0.628637	0.314318	−	0.949318i	$-0.398224\pi$
$769$	17.4326	0.628637	0.314318	+	0.949318i	$0.398224\pi$
$770$	114.165	4.11420
$771$	0	0
$772$	89.2590	3.21250
$773$	−46.9728	−1.68949	−0.844747	−	0.535166i	$-0.820249\pi$
$773$	−46.9728	−1.68949	−0.844747	+	0.535166i	$0.820249\pi$
$774$	0	0
$775$	3.22952	0.116008
$776$	−52.3371	−1.87879
$777$	0	0
$778$	68.8437	2.46817
$779$	4.38632	0.157156
$780$	0	0
$781$	28.3905	1.01589
$782$	−0.647660	−0.0231603
$783$	0	0
$784$	39.4676	1.40956
$785$	−25.4652	−0.908893
$786$	0	0
$787$	−48.1005	−1.71460	−0.857300	−	0.514818i	$-0.827860\pi$
$787$	−48.1005	−1.71460	−0.857300	+	0.514818i	$0.827860\pi$
$788$	66.9169	2.38382
$789$	0	0
$790$	−80.1917	−2.85309
$791$	3.75747	0.133600
$792$	0	0
$793$	13.8526	0.491921
$794$	46.1342	1.63724
$795$	0	0
$796$	28.3469	1.00473
$797$	6.84219	0.242363	0.121181	−	0.992630i	$-0.461332\pi$
$797$	6.84219	0.242363	0.121181	+	0.992630i	$0.461332\pi$
$798$	0	0
$799$	−2.33794	−0.0827102
$800$	2.23521	0.0790265
$801$	0	0
$802$	−12.3709	−0.436832
$803$	−48.3823	−1.70737
$804$	0	0
$805$	−10.4815	−0.369423
$806$	18.9339	0.666919
$807$	0	0
$808$	−23.8183	−0.837924
$809$	−2.15661	−0.0758223	−0.0379112	−	0.999281i	$-0.512070\pi$
$809$	−2.15661	−0.0758223	−0.0379112	+	0.999281i	$0.512070\pi$
$810$	0	0
$811$	30.0010	1.05348	0.526739	−	0.850027i	$-0.323414\pi$
$811$	30.0010	1.05348	0.526739	+	0.850027i	$0.323414\pi$
$812$	−18.6403	−0.654146
$813$	0	0
$814$	16.6507	0.583608
$815$	20.7860	0.728102
$816$	0	0
$817$	−4.74226	−0.165911
$818$	−60.9452	−2.13090
$819$	0	0
$820$	60.5299	2.11380
$821$	35.2369	1.22977	0.614887	−	0.788615i	$-0.289201\pi$
$821$	35.2369	1.22977	0.614887	+	0.788615i	$0.289201\pi$
$822$	0	0
$823$	−36.9231	−1.28706	−0.643530	−	0.765421i	$-0.722531\pi$
$823$	−36.9231	−1.28706	−0.643530	+	0.765421i	$0.722531\pi$
$824$	60.5881	2.11069
$825$	0	0
$826$	13.8425	0.481643
$827$	−45.8002	−1.59263	−0.796314	−	0.604884i	$-0.793220\pi$
$827$	−45.8002	−1.59263	−0.796314	+	0.604884i	$0.793220\pi$
$828$	0	0
$829$	13.3071	0.462176	0.231088	−	0.972933i	$-0.425771\pi$
$829$	13.3071	0.462176	0.231088	+	0.972933i	$0.425771\pi$
$830$	78.3482	2.71951
$831$	0	0
$832$	20.9157	0.725122
$833$	5.14603	0.178299
$834$	0	0
$835$	−2.68747	−0.0930036
$836$	9.16120	0.316847
$837$	0	0
$838$	−24.6417	−0.851234
$839$	−5.43546	−0.187653	−0.0938264	−	0.995589i	$-0.529910\pi$
$839$	−5.43546	−0.187653	−0.0938264	+	0.995589i	$0.529910\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	28.9093	0.996281
$843$	0	0
$844$	50.0191	1.72173
$845$	−19.5620	−0.672954
$846$	0	0
$847$	50.6559	1.74056
$848$	−4.89831	−0.168209
$849$	0	0
$850$	−0.491821	−0.0168693
$851$	−1.52870	−0.0524033
$852$	0	0
$853$	4.95255	0.169572	0.0847861	−	0.996399i	$-0.472979\pi$
$853$	4.95255	0.169572	0.0847861	+	0.996399i	$0.472979\pi$
$854$	−89.6738	−3.06857
$855$	0	0
$856$	−2.90935	−0.0994394
$857$	−13.5239	−0.461967	−0.230983	−	0.972958i	$-0.574194\pi$
$857$	−13.5239	−0.461967	−0.230983	+	0.972958i	$0.574194\pi$
$858$	0	0
$859$	−34.5149	−1.17763	−0.588817	−	0.808267i	$-0.700406\pi$
$859$	−34.5149	−1.17763	−0.588817	+	0.808267i	$0.700406\pi$
$860$	−65.4419	−2.23155
$861$	0	0
$862$	−67.2951	−2.29208
$863$	−2.25993	−0.0769289	−0.0384645	−	0.999260i	$-0.512247\pi$
$863$	−2.25993	−0.0769289	−0.0384645	+	0.999260i	$0.512247\pi$
$864$	0	0
$865$	−4.53254	−0.154111
$866$	−9.89276	−0.336170
$867$	0	0
$868$	−79.2742	−2.69074
$869$	−74.9095	−2.54113
$870$	0	0
$871$	22.2964	0.755486
$872$	33.7626	1.14335
$873$	0	0
$874$	−1.30042	−0.0439874
$875$	−60.3667	−2.04077
$876$	0	0
$877$	−37.1470	−1.25436	−0.627182	−	0.778873i	$-0.715792\pi$
$877$	−37.1470	−1.25436	−0.627182	+	0.778873i	$0.715792\pi$
$878$	44.2806	1.49440
$879$	0	0
$880$	19.6767	0.663302
$881$	−3.76794	−0.126945	−0.0634726	−	0.997984i	$-0.520218\pi$
$881$	−3.76794	−0.126945	−0.0634726	+	0.997984i	$0.520218\pi$
$882$	0	0
$883$	31.1564	1.04850	0.524248	−	0.851566i	$-0.324347\pi$
$883$	31.1564	1.04850	0.524248	+	0.851566i	$0.324347\pi$
$884$	−1.86495	−0.0627250
$885$	0	0
$886$	17.6025	0.591368
$887$	49.0264	1.64614	0.823072	−	0.567937i	$-0.192258\pi$
$887$	49.0264	1.64614	0.823072	+	0.567937i	$0.192258\pi$
$888$	0	0
$889$	−33.5998	−1.12690
$890$	−45.9676	−1.54084
$891$	0	0
$892$	1.82148	0.0609877
$893$	−4.69429	−0.157088
$894$	0	0
$895$	50.7489	1.69635
$896$	−105.432	−3.52225
$897$	0	0
$898$	−55.5885	−1.85501
$899$	4.25284	0.141840
$900$	0	0
$901$	−0.638672	−0.0212773
$902$	87.4217	2.91082
$903$	0	0
$904$	2.91994	0.0971157
$905$	2.99714	0.0996284
$906$	0	0
$907$	9.47164	0.314501	0.157250	−	0.987559i	$-0.449737\pi$
$907$	9.47164	0.314501	0.157250	+	0.987559i	$0.449737\pi$
$908$	−46.2429	−1.53463
$909$	0	0
$910$	−46.6641	−1.54690
$911$	−19.2798	−0.638769	−0.319385	−	0.947625i	$-0.603476\pi$
$911$	−19.2798	−0.638769	−0.319385	+	0.947625i	$0.603476\pi$
$912$	0	0
$913$	73.1874	2.42215
$914$	−15.3048	−0.506239
$915$	0	0
$916$	19.1312	0.632114
$917$	17.0689	0.563664
$918$	0	0
$919$	−17.8590	−0.589115	−0.294558	−	0.955634i	$-0.595172\pi$
$919$	−17.8590	−0.589115	−0.294558	+	0.955634i	$0.595172\pi$
$920$	−8.14517	−0.268538
$921$	0	0
$922$	68.0150	2.23996
$923$	−11.6045	−0.381965
$924$	0	0
$925$	−1.16087	−0.0381691
$926$	75.6980	2.48759
$927$	0	0
$928$	2.94346	0.0966238
$929$	39.6211	1.29993	0.649963	−	0.759966i	$-0.274784\pi$
$929$	39.6211	1.29993	0.649963	+	0.759966i	$0.274784\pi$
$930$	0	0
$931$	10.3326	0.338637
$932$	−51.8863	−1.69959
$933$	0	0
$934$	−83.1174	−2.71968
$935$	2.56557	0.0839031
$936$	0	0
$937$	−37.4465	−1.22332	−0.611662	−	0.791119i	$-0.709499\pi$
$937$	−37.4465	−1.22332	−0.611662	+	0.791119i	$0.709499\pi$
$938$	−144.334	−4.71267
$939$	0	0
$940$	−64.7798	−2.11288
$941$	15.4626	0.504068	0.252034	−	0.967718i	$-0.418901\pi$
$941$	15.4626	0.504068	0.252034	+	0.967718i	$0.418901\pi$
$942$	0	0
$943$	−8.02619	−0.261369
$944$	2.38582	0.0776517
$945$	0	0
$946$	−94.5159	−3.07298
$947$	32.7987	1.06581	0.532907	−	0.846174i	$-0.321100\pi$
$947$	32.7987	1.06581	0.532907	+	0.846174i	$0.321100\pi$
$948$	0	0
$949$	19.7760	0.641955
$950$	−0.987516	−0.0320392
$951$	0	0
$952$	5.47957	0.177594
$953$	16.0845	0.521027	0.260513	−	0.965470i	$-0.416108\pi$
$953$	16.0845	0.521027	0.260513	+	0.965470i	$0.416108\pi$
$954$	0	0
$955$	1.21947	0.0394611
$956$	−57.7386	−1.86740
$957$	0	0
$958$	10.0150	0.323570
$959$	48.6364	1.57055
$960$	0	0
$961$	−12.9134	−0.416560
$962$	−6.80589	−0.219431
$963$	0	0
$964$	97.9215	3.15384
$965$	50.1904	1.61569
$966$	0	0
$967$	24.9701	0.802986	0.401493	−	0.915862i	$-0.368491\pi$
$967$	24.9701	0.802986	0.401493	+	0.915862i	$0.368491\pi$
$968$	39.3648	1.26523
$969$	0	0
$970$	−64.8385	−2.08184
$971$	−1.43669	−0.0461056	−0.0230528	−	0.999734i	$-0.507339\pi$
$971$	−1.43669	−0.0461056	−0.0230528	+	0.999734i	$0.507339\pi$
$972$	0	0
$973$	−33.4530	−1.07245
$974$	−26.5415	−0.850445
$975$	0	0
$976$	−15.4556	−0.494723
$977$	−45.3505	−1.45089	−0.725446	−	0.688279i	$-0.758366\pi$
$977$	−45.3505	−1.45089	−0.725446	+	0.688279i	$0.758366\pi$
$978$	0	0
$979$	−42.9397	−1.37236
$980$	142.587	4.55476
$981$	0	0
$982$	−0.710729	−0.0226803
$983$	60.0472	1.91521	0.957604	−	0.288086i	$-0.0930190\pi$
$983$	60.0472	1.91521	0.957604	+	0.288086i	$0.0930190\pi$
$984$	0	0
$985$	37.6274	1.19891
$986$	−0.647660	−0.0206257
$987$	0	0
$988$	−3.74459	−0.119131
$989$	8.67751	0.275929
$990$	0	0
$991$	11.5981	0.368425	0.184212	−	0.982886i	$-0.441027\pi$
$991$	11.5981	0.368425	0.184212	+	0.982886i	$0.441027\pi$
$992$	12.5181	0.397449
$993$	0	0
$994$	75.1204	2.38268
$995$	15.9395	0.505315
$996$	0	0
$997$	22.6532	0.717433	0.358717	−	0.933446i	$-0.383214\pi$
$997$	22.6532	0.717433	0.358717	+	0.933446i	$0.383214\pi$
$998$	−46.7853	−1.48096
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.l.1.9		10
3.2	odd	2		667.2.a.a.1.2	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
667.2.a.a.1.2	✓	10	3.2	odd	2
6003.2.a.l.1.9		10	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)

\(f(q)\)	\(=\)	\(q+2.37954 q^{2} +3.66223 q^{4} +2.05928 q^{5} +5.08987 q^{7} +3.95536 q^{8} +O(q^{10})\)	\(q+2.37954 q^{2} +3.66223 q^{4} +2.05928 q^{5} +5.08987 q^{7} +3.95536 q^{8} +4.90014 q^{10} +4.57737 q^{11} -1.87097 q^{13} +12.1116 q^{14} +2.08748 q^{16} +0.272178 q^{17} +0.546501 q^{19} +7.54155 q^{20} +10.8920 q^{22} -1.00000 q^{23} -0.759381 q^{25} -4.45206 q^{26} +18.6403 q^{28} -1.00000 q^{29} -4.25284 q^{31} -2.94346 q^{32} +0.647660 q^{34} +10.4815 q^{35} +1.52870 q^{37} +1.30042 q^{38} +8.14517 q^{40} +8.02619 q^{41} -8.67751 q^{43} +16.7634 q^{44} -2.37954 q^{46} -8.58972 q^{47} +18.9068 q^{49} -1.80698 q^{50} -6.85194 q^{52} -2.34652 q^{53} +9.42606 q^{55} +20.1323 q^{56} -2.37954 q^{58} +1.14292 q^{59} -7.40397 q^{61} -10.1198 q^{62} -11.1791 q^{64} -3.85285 q^{65} -11.9170 q^{67} +0.996780 q^{68} +24.9411 q^{70} +6.20236 q^{71} -10.5699 q^{73} +3.63762 q^{74} +2.00141 q^{76} +23.2982 q^{77} -16.3652 q^{79} +4.29870 q^{80} +19.0987 q^{82} +15.9890 q^{83} +0.560490 q^{85} -20.6485 q^{86} +18.1051 q^{88} -9.38087 q^{89} -9.52302 q^{91} -3.66223 q^{92} -20.4396 q^{94} +1.12540 q^{95} -13.2320 q^{97} +44.9896 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 3 q^{2} + 9 q^{4} + 10 q^{5} + q^{7} + 9 q^{8}+O(q^{10}) \) 10 * q + 3 * q^2 + 9 * q^4 + 10 * q^5 + q^7 + 9 * q^8	\( 10 q + 3 q^{2} + 9 q^{4} + 10 q^{5} + q^{7} + 9 q^{8} - 6 q^{10} - 13 q^{13} + 12 q^{14} - 5 q^{16} + 22 q^{17} - 2 q^{19} - 3 q^{20} + 3 q^{22} - 10 q^{23} + 10 q^{25} + 25 q^{26} + 19 q^{28} - 10 q^{29} - 22 q^{31} + 31 q^{32} + 13 q^{34} + 15 q^{35} - 9 q^{37} + 10 q^{38} - 6 q^{40} + 25 q^{41} + 3 q^{43} + 27 q^{44} - 3 q^{46} + 17 q^{47} + 17 q^{49} - 2 q^{50} - 18 q^{52} + 43 q^{53} - 11 q^{55} + 7 q^{56} - 3 q^{58} + 7 q^{59} - 6 q^{61} - 3 q^{62} + 33 q^{64} - 11 q^{65} + 11 q^{67} + 51 q^{68} + 34 q^{70} + 17 q^{71} - 44 q^{73} - 9 q^{74} + 24 q^{76} + 71 q^{77} + 5 q^{79} - 38 q^{80} + 33 q^{82} + 32 q^{83} + 16 q^{85} + 9 q^{86} + 18 q^{88} + 10 q^{89} - 3 q^{91} - 9 q^{92} + 47 q^{94} + 8 q^{95} + 6 q^{97} + 73 q^{98}+O(q^{100}) \) 10 * q + 3 * q^2 + 9 * q^4 + 10 * q^5 + q^7 + 9 * q^8 - 6 * q^10 - 13 * q^13 + 12 * q^14 - 5 * q^16 + 22 * q^17 - 2 * q^19 - 3 * q^20 + 3 * q^22 - 10 * q^23 + 10 * q^25 + 25 * q^26 + 19 * q^28 - 10 * q^29 - 22 * q^31 + 31 * q^32 + 13 * q^34 + 15 * q^35 - 9 * q^37 + 10 * q^38 - 6 * q^40 + 25 * q^41 + 3 * q^43 + 27 * q^44 - 3 * q^46 + 17 * q^47 + 17 * q^49 - 2 * q^50 - 18 * q^52 + 43 * q^53 - 11 * q^55 + 7 * q^56 - 3 * q^58 + 7 * q^59 - 6 * q^61 - 3 * q^62 + 33 * q^64 - 11 * q^65 + 11 * q^67 + 51 * q^68 + 34 * q^70 + 17 * q^71 - 44 * q^73 - 9 * q^74 + 24 * q^76 + 71 * q^77 + 5 * q^79 - 38 * q^80 + 33 * q^82 + 32 * q^83 + 16 * q^85 + 9 * q^86 + 18 * q^88 + 10 * q^89 - 3 * q^91 - 9 * q^92 + 47 * q^94 + 8 * q^95 + 6 * q^97 + 73 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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