LMFDB - Embedded newform 6003.2.a.s.1.7

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			1.7
Root			$1.20801$ 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)

$f(q)$	$=$	$q-1.20801 q^{2} -0.540713 q^{4} -2.72981 q^{5} -1.05258 q^{7} +3.06921 q^{8} +O(q^{10})$	\(q-1.20801 q^{2} -0.540713 q^{4} -2.72981 q^{5} -1.05258 q^{7} +3.06921 q^{8} +3.29764 q^{10} -0.0950928 q^{11} +1.29615 q^{13} +1.27153 q^{14} -2.62620 q^{16} -6.72257 q^{17} +1.49041 q^{19} +1.47604 q^{20} +0.114873 q^{22} -1.00000 q^{23} +2.45186 q^{25} -1.56577 q^{26} +0.569144 q^{28} -1.00000 q^{29} +3.03307 q^{31} -2.96593 q^{32} +8.12092 q^{34} +2.87334 q^{35} +5.28694 q^{37} -1.80043 q^{38} -8.37835 q^{40} +5.16061 q^{41} +0.852332 q^{43} +0.0514180 q^{44} +1.20801 q^{46} -9.07477 q^{47} -5.89208 q^{49} -2.96187 q^{50} -0.700848 q^{52} -8.68851 q^{53} +0.259585 q^{55} -3.23058 q^{56} +1.20801 q^{58} -7.14371 q^{59} -11.2692 q^{61} -3.66398 q^{62} +8.83528 q^{64} -3.53825 q^{65} +13.9103 q^{67} +3.63498 q^{68} -3.47102 q^{70} +10.8505 q^{71} -12.7611 q^{73} -6.38667 q^{74} -0.805885 q^{76} +0.100093 q^{77} -4.38026 q^{79} +7.16903 q^{80} -6.23406 q^{82} +2.83783 q^{83} +18.3513 q^{85} -1.02962 q^{86} -0.291859 q^{88} -2.67707 q^{89} -1.36430 q^{91} +0.540713 q^{92} +10.9624 q^{94} -4.06853 q^{95} -9.23470 q^{97} +7.11768 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.20801	−0.854192	−0.427096	−	0.904206i	$-0.640463\pi$
$2$	−1.20801	−0.854192	−0.427096	+	0.904206i	$0.640463\pi$
$3$	0	0
$3$	0	0
$4$	−0.540713	−0.270357
$5$	−2.72981	−1.22081	−0.610404	−	0.792090i	$-0.708993\pi$
$5$	−2.72981	−1.22081	−0.610404	+	0.792090i	$0.708993\pi$
$6$	0	0
$7$	−1.05258	−0.397838	−0.198919	−	0.980016i	$-0.563743\pi$
$7$	−1.05258	−0.397838	−0.198919	+	0.980016i	$0.563743\pi$
$8$	3.06921	1.08513
$9$	0	0
$10$	3.29764	1.04280
$11$	−0.0950928	−0.0286716	−0.0143358	−	0.999897i	$-0.504563\pi$
$11$	−0.0950928	−0.0286716	−0.0143358	+	0.999897i	$0.504563\pi$
$12$	0	0
$13$	1.29615	0.359488	0.179744	−	0.983713i	$-0.442473\pi$
$13$	1.29615	0.359488	0.179744	+	0.983713i	$0.442473\pi$
$14$	1.27153	0.339830
$15$	0	0
$16$	−2.62620	−0.656550
$17$	−6.72257	−1.63046	−0.815231	−	0.579136i	$-0.803390\pi$
$17$	−6.72257	−1.63046	−0.815231	+	0.579136i	$0.803390\pi$
$18$	0	0
$19$	1.49041	0.341923	0.170962	−	0.985278i	$-0.445313\pi$
$19$	1.49041	0.341923	0.170962	+	0.985278i	$0.445313\pi$
$20$	1.47604	0.330054
$21$	0	0
$22$	0.114873	0.0244910
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	2.45186	0.490372
$26$	−1.56577	−0.307072
$27$	0	0
$28$	0.569144	0.107558
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	3.03307	0.544756	0.272378	−	0.962190i	$-0.412190\pi$
$31$	3.03307	0.544756	0.272378	+	0.962190i	$0.412190\pi$
$32$	−2.96593	−0.524308
$33$	0	0
$34$	8.12092	1.39273
$35$	2.87334	0.485683
$36$	0	0
$37$	5.28694	0.869167	0.434584	−	0.900631i	$-0.356896\pi$
$37$	5.28694	0.869167	0.434584	+	0.900631i	$0.356896\pi$
$38$	−1.80043	−0.292068
$39$	0	0
$40$	−8.37835	−1.32473
$41$	5.16061	0.805951	0.402976	−	0.915211i	$-0.367976\pi$
$41$	5.16061	0.805951	0.402976	+	0.915211i	$0.367976\pi$
$42$	0	0
$43$	0.852332	0.129979	0.0649897	−	0.997886i	$-0.479299\pi$
$43$	0.852332	0.129979	0.0649897	+	0.997886i	$0.479299\pi$
$44$	0.0514180	0.00775155
$45$	0	0
$46$	1.20801	0.178111
$47$	−9.07477	−1.32369	−0.661845	−	0.749640i	$-0.730227\pi$
$47$	−9.07477	−1.32369	−0.661845	+	0.749640i	$0.730227\pi$
$48$	0	0
$49$	−5.89208	−0.841725
$50$	−2.96187	−0.418871
$51$	0	0
$52$	−0.700848	−0.0971901
$53$	−8.68851	−1.19346	−0.596729	−	0.802443i	$-0.703533\pi$
$53$	−8.68851	−1.19346	−0.596729	+	0.802443i	$0.703533\pi$
$54$	0	0
$55$	0.259585	0.0350025
$56$	−3.23058	−0.431705
$57$	0	0
$58$	1.20801	0.158619
$59$	−7.14371	−0.930031	−0.465016	−	0.885302i	$-0.653951\pi$
$59$	−7.14371	−0.930031	−0.465016	+	0.885302i	$0.653951\pi$
$60$	0	0
$61$	−11.2692	−1.44288	−0.721438	−	0.692479i	$-0.756519\pi$
$61$	−11.2692	−1.44288	−0.721438	+	0.692479i	$0.756519\pi$
$62$	−3.66398	−0.465326
$63$	0	0
$64$	8.83528	1.10441
$65$	−3.53825	−0.438866
$66$	0	0
$67$	13.9103	1.69941	0.849707	−	0.527256i	$-0.176779\pi$
$67$	13.9103	1.69941	0.849707	+	0.527256i	$0.176779\pi$
$68$	3.63498	0.440806
$69$	0	0
$70$	−3.47102	−0.414867
$71$	10.8505	1.28772	0.643859	−	0.765144i	$-0.277332\pi$
$71$	10.8505	1.28772	0.643859	+	0.765144i	$0.277332\pi$
$72$	0	0
$73$	−12.7611	−1.49357	−0.746786	−	0.665064i	$-0.768404\pi$
$73$	−12.7611	−1.49357	−0.746786	+	0.665064i	$0.768404\pi$
$74$	−6.38667	−0.742435
$75$	0	0
$76$	−0.805885	−0.0924413
$77$	0.100093	0.0114066
$78$	0	0
$79$	−4.38026	−0.492818	−0.246409	−	0.969166i	$-0.579251\pi$
$79$	−4.38026	−0.492818	−0.246409	+	0.969166i	$0.579251\pi$
$80$	7.16903	0.801522
$81$	0	0
$82$	−6.23406	−0.688437
$83$	2.83783	0.311492	0.155746	−	0.987797i	$-0.450222\pi$
$83$	2.83783	0.311492	0.155746	+	0.987797i	$0.450222\pi$
$84$	0	0
$85$	18.3513	1.99048
$86$	−1.02962	−0.111027
$87$	0	0
$88$	−0.291859	−0.0311123
$89$	−2.67707	−0.283769	−0.141884	−	0.989883i	$-0.545316\pi$
$89$	−2.67707	−0.283769	−0.141884	+	0.989883i	$0.545316\pi$
$90$	0	0
$91$	−1.36430	−0.143018
$92$	0.540713	0.0563733
$93$	0	0
$94$	10.9624	1.13069
$95$	−4.06853	−0.417423
$96$	0	0
$97$	−9.23470	−0.937642	−0.468821	−	0.883293i	$-0.655321\pi$
$97$	−9.23470	−0.937642	−0.468821	+	0.883293i	$0.655321\pi$
$98$	7.11768	0.718995
$99$	0	0
$100$	−1.32575	−0.132575
$101$	6.95395	0.691943	0.345972	−	0.938245i	$-0.387549\pi$
$101$	6.95395	0.691943	0.345972	+	0.938245i	$0.387549\pi$
$102$	0	0
$103$	17.3902	1.71351	0.856753	−	0.515727i	$-0.172478\pi$
$103$	17.3902	1.71351	0.856753	+	0.515727i	$0.172478\pi$
$104$	3.97816	0.390091
$105$	0	0
$106$	10.4958	1.01944
$107$	−15.0267	−1.45268	−0.726341	−	0.687335i	$-0.758781\pi$
$107$	−15.0267	−1.45268	−0.726341	+	0.687335i	$0.758781\pi$
$108$	0	0
$109$	−17.8192	−1.70677	−0.853383	−	0.521284i	$-0.825453\pi$
$109$	−17.8192	−1.70677	−0.853383	+	0.521284i	$0.825453\pi$
$110$	−0.313581	−0.0298988
$111$	0	0
$112$	2.76429	0.261200
$113$	6.63665	0.624323	0.312162	−	0.950029i	$-0.398947\pi$
$113$	6.63665	0.624323	0.312162	+	0.950029i	$0.398947\pi$
$114$	0	0
$115$	2.72981	0.254556
$116$	0.540713	0.0502040
$117$	0	0
$118$	8.62966	0.794425
$119$	7.07603	0.648659
$120$	0	0
$121$	−10.9910	−0.999178
$122$	13.6133	1.23249
$123$	0	0
$124$	−1.64002	−0.147279
$125$	6.95594	0.622158
$126$	0	0
$127$	−17.4909	−1.55206	−0.776032	−	0.630693i	$-0.782771\pi$
$127$	−17.4909	−1.55206	−0.776032	+	0.630693i	$0.782771\pi$
$128$	−4.74123	−0.419070
$129$	0	0
$130$	4.27424	0.374876
$131$	−20.0165	−1.74885	−0.874423	−	0.485164i	$-0.838760\pi$
$131$	−20.0165	−1.74885	−0.874423	+	0.485164i	$0.838760\pi$
$132$	0	0
$133$	−1.56877	−0.136030
$134$	−16.8038	−1.45162
$135$	0	0
$136$	−20.6329	−1.76926
$137$	10.8950	0.930823	0.465411	−	0.885094i	$-0.345906\pi$
$137$	10.8950	0.930823	0.465411	+	0.885094i	$0.345906\pi$
$138$	0	0
$139$	−8.06409	−0.683987	−0.341993	−	0.939702i	$-0.611102\pi$
$139$	−8.06409	−0.683987	−0.341993	+	0.939702i	$0.611102\pi$
$140$	−1.55365	−0.131308
$141$	0	0
$142$	−13.1075	−1.09996
$143$	−0.123255	−0.0103071
$144$	0	0
$145$	2.72981	0.226698
$146$	15.4155	1.27580
$147$	0	0
$148$	−2.85872	−0.234985
$149$	6.47063	0.530094	0.265047	−	0.964235i	$-0.414612\pi$
$149$	6.47063	0.530094	0.265047	+	0.964235i	$0.414612\pi$
$150$	0	0
$151$	−16.8594	−1.37200	−0.686001	−	0.727601i	$-0.740635\pi$
$151$	−16.8594	−1.37200	−0.686001	+	0.727601i	$0.740635\pi$
$152$	4.57437	0.371031
$153$	0	0
$154$	−0.120913	−0.00974344
$155$	−8.27971	−0.665043
$156$	0	0
$157$	15.2976	1.22088	0.610442	−	0.792061i	$-0.290992\pi$
$157$	15.2976	1.22088	0.610442	+	0.792061i	$0.290992\pi$
$158$	5.29140	0.420961
$159$	0	0
$160$	8.09644	0.640079
$161$	1.05258	0.0829549
$162$	0	0
$163$	13.4200	1.05114	0.525569	−	0.850751i	$-0.323852\pi$
$163$	13.4200	1.05114	0.525569	+	0.850751i	$0.323852\pi$
$164$	−2.79041	−0.217894
$165$	0	0
$166$	−3.42813	−0.266074
$167$	5.28341	0.408843	0.204421	−	0.978883i	$-0.434469\pi$
$167$	5.28341	0.408843	0.204421	+	0.978883i	$0.434469\pi$
$168$	0	0
$169$	−11.3200	−0.870768
$170$	−22.1686	−1.70025
$171$	0	0
$172$	−0.460867	−0.0351408
$173$	−19.7703	−1.50311	−0.751553	−	0.659672i	$-0.770695\pi$
$173$	−19.7703	−1.50311	−0.751553	+	0.659672i	$0.770695\pi$
$174$	0	0
$175$	−2.58078	−0.195088
$176$	0.249733	0.0188243
$177$	0	0
$178$	3.23392	0.242393
$179$	14.4299	1.07854	0.539270	−	0.842133i	$-0.318700\pi$
$179$	14.4299	1.07854	0.539270	+	0.842133i	$0.318700\pi$
$180$	0	0
$181$	−20.0238	−1.48836	−0.744179	−	0.667980i	$-0.767159\pi$
$181$	−20.0238	−1.48836	−0.744179	+	0.667980i	$0.767159\pi$
$182$	1.64809	0.122165
$183$	0	0
$184$	−3.06921	−0.226265
$185$	−14.4323	−1.06109
$186$	0	0
$187$	0.639268	0.0467479
$188$	4.90685	0.357869
$189$	0	0
$190$	4.91483	0.356559
$191$	−7.53689	−0.545350	−0.272675	−	0.962106i	$-0.587908\pi$
$191$	−7.53689	−0.545350	−0.272675	+	0.962106i	$0.587908\pi$
$192$	0	0
$193$	13.7002	0.986164	0.493082	−	0.869983i	$-0.335870\pi$
$193$	13.7002	0.986164	0.493082	+	0.869983i	$0.335870\pi$
$194$	11.1556	0.800926
$195$	0	0
$196$	3.18593	0.227566
$197$	14.2870	1.01791	0.508954	−	0.860794i	$-0.330032\pi$
$197$	14.2870	1.01791	0.508954	+	0.860794i	$0.330032\pi$
$198$	0	0
$199$	25.8998	1.83599	0.917993	−	0.396596i	$-0.129809\pi$
$199$	25.8998	1.83599	0.917993	+	0.396596i	$0.129809\pi$
$200$	7.52526	0.532116
$201$	0	0
$202$	−8.40043	−0.591052
$203$	1.05258	0.0738766
$204$	0	0
$205$	−14.0875	−0.983912
$206$	−21.0075	−1.46366
$207$	0	0
$208$	−3.40396	−0.236022
$209$	−0.141727	−0.00980348
$210$	0	0
$211$	1.77461	0.122169	0.0610846	−	0.998133i	$-0.480544\pi$
$211$	1.77461	0.122169	0.0610846	+	0.998133i	$0.480544\pi$
$212$	4.69799	0.322660
$213$	0	0
$214$	18.1523	1.24087
$215$	−2.32670	−0.158680
$216$	0	0
$217$	−3.19255	−0.216724
$218$	21.5257	1.45791
$219$	0	0
$220$	−0.140361	−0.00946315
$221$	−8.71348	−0.586132
$222$	0	0
$223$	11.2214	0.751443	0.375721	−	0.926733i	$-0.377395\pi$
$223$	11.2214	0.751443	0.375721	+	0.926733i	$0.377395\pi$
$224$	3.12188	0.208589
$225$	0	0
$226$	−8.01713	−0.533292
$227$	−0.0481254	−0.00319419	−0.00159710	−	0.999999i	$-0.500508\pi$
$227$	−0.0481254	−0.00319419	−0.00159710	+	0.999999i	$0.500508\pi$
$228$	0	0
$229$	−17.3588	−1.14710	−0.573552	−	0.819169i	$-0.694435\pi$
$229$	−17.3588	−1.14710	−0.573552	+	0.819169i	$0.694435\pi$
$230$	−3.29764	−0.217440
$231$	0	0
$232$	−3.06921	−0.201503
$233$	−5.44641	−0.356806	−0.178403	−	0.983957i	$-0.557093\pi$
$233$	−5.44641	−0.356806	−0.178403	+	0.983957i	$0.557093\pi$
$234$	0	0
$235$	24.7724	1.61597
$236$	3.86270	0.251440
$237$	0	0
$238$	−8.54792	−0.554079
$239$	0.363333	0.0235020	0.0117510	−	0.999931i	$-0.496259\pi$
$239$	0.363333	0.0235020	0.0117510	+	0.999931i	$0.496259\pi$
$240$	0	0
$241$	19.8759	1.28032	0.640159	−	0.768243i	$-0.278869\pi$
$241$	19.8759	1.28032	0.640159	+	0.768243i	$0.278869\pi$
$242$	13.2772	0.853489
$243$	0	0
$244$	6.09342	0.390091
$245$	16.0842	1.02758
$246$	0	0
$247$	1.93180	0.122917
$248$	9.30913	0.591130
$249$	0	0
$250$	−8.40284	−0.531442
$251$	24.4211	1.54144	0.770722	−	0.637172i	$-0.219896\pi$
$251$	24.4211	1.54144	0.770722	+	0.637172i	$0.219896\pi$
$252$	0	0
$253$	0.0950928	0.00597843
$254$	21.1291	1.32576
$255$	0	0
$256$	−11.9431	−0.746444
$257$	16.3921	1.02251	0.511255	−	0.859429i	$-0.329181\pi$
$257$	16.3921	1.02251	0.511255	+	0.859429i	$0.329181\pi$
$258$	0	0
$259$	−5.56492	−0.345787
$260$	1.91318	0.118650
$261$	0	0
$262$	24.1801	1.49385
$263$	−5.90552	−0.364150	−0.182075	−	0.983285i	$-0.558281\pi$
$263$	−5.90552	−0.364150	−0.182075	+	0.983285i	$0.558281\pi$
$264$	0	0
$265$	23.7180	1.45698
$266$	1.89509	0.116196
$267$	0	0
$268$	−7.52149	−0.459448
$269$	−20.8869	−1.27350	−0.636749	−	0.771071i	$-0.719721\pi$
$269$	−20.8869	−1.27350	−0.636749	+	0.771071i	$0.719721\pi$
$270$	0	0
$271$	19.0989	1.16018	0.580089	−	0.814553i	$-0.303018\pi$
$271$	19.0989	1.16018	0.580089	+	0.814553i	$0.303018\pi$
$272$	17.6548	1.07048
$273$	0	0
$274$	−13.1613	−0.795101
$275$	−0.233154	−0.0140597
$276$	0	0
$277$	−27.6649	−1.66222	−0.831110	−	0.556108i	$-0.812294\pi$
$277$	−27.6649	−1.66222	−0.831110	+	0.556108i	$0.812294\pi$
$278$	9.74149	0.584256
$279$	0	0
$280$	8.81887	0.527029
$281$	14.6598	0.874532	0.437266	−	0.899332i	$-0.355947\pi$
$281$	14.6598	0.874532	0.437266	+	0.899332i	$0.355947\pi$
$282$	0	0
$283$	10.6965	0.635842	0.317921	−	0.948117i	$-0.397015\pi$
$283$	10.6965	0.635842	0.317921	+	0.948117i	$0.397015\pi$
$284$	−5.86701	−0.348143
$285$	0	0
$286$	0.148893	0.00880423
$287$	−5.43195	−0.320638
$288$	0	0
$289$	28.1929	1.65841
$290$	−3.29764	−0.193644
$291$	0	0
$292$	6.90009	0.403797
$293$	30.1483	1.76128	0.880642	−	0.473782i	$-0.157112\pi$
$293$	30.1483	1.76128	0.880642	+	0.473782i	$0.157112\pi$
$294$	0	0
$295$	19.5010	1.13539
$296$	16.2267	0.943158
$297$	0	0
$298$	−7.81658	−0.452802
$299$	−1.29615	−0.0749585
$300$	0	0
$301$	−0.897147	−0.0517107
$302$	20.3664	1.17195
$303$	0	0
$304$	−3.91412	−0.224490
$305$	30.7628	1.76147
$306$	0	0
$307$	−9.53172	−0.544004	−0.272002	−	0.962297i	$-0.587686\pi$
$307$	−9.53172	−0.544004	−0.272002	+	0.962297i	$0.587686\pi$
$308$	−0.0541215	−0.00308386
$309$	0	0
$310$	10.0020	0.568074
$311$	31.4611	1.78400	0.891999	−	0.452038i	$-0.149303\pi$
$311$	31.4611	1.78400	0.891999	+	0.452038i	$0.149303\pi$
$312$	0	0
$313$	8.56299	0.484009	0.242004	−	0.970275i	$-0.422195\pi$
$313$	8.56299	0.484009	0.242004	+	0.970275i	$0.422195\pi$
$314$	−18.4797	−1.04287
$315$	0	0
$316$	2.36847	0.133237
$317$	−20.2256	−1.13598	−0.567990	−	0.823035i	$-0.692279\pi$
$317$	−20.2256	−1.13598	−0.567990	+	0.823035i	$0.692279\pi$
$318$	0	0
$319$	0.0950928	0.00532418
$320$	−24.1186	−1.34827
$321$	0	0
$322$	−1.27153	−0.0708594
$323$	−10.0194	−0.557493
$324$	0	0
$325$	3.17799	0.176283
$326$	−16.2115	−0.897874
$327$	0	0
$328$	15.8390	0.874560
$329$	9.55191	0.526614
$330$	0	0
$331$	−15.0603	−0.827788	−0.413894	−	0.910325i	$-0.635832\pi$
$331$	−15.0603	−0.827788	−0.413894	+	0.910325i	$0.635832\pi$
$332$	−1.53445	−0.0842141
$333$	0	0
$334$	−6.38241	−0.349230
$335$	−37.9725	−2.07466
$336$	0	0
$337$	−26.5968	−1.44882	−0.724409	−	0.689370i	$-0.757887\pi$
$337$	−26.5968	−1.44882	−0.724409	+	0.689370i	$0.757887\pi$
$338$	13.6746	0.743803
$339$	0	0
$340$	−9.92281	−0.538140
$341$	−0.288424	−0.0156190
$342$	0	0
$343$	13.5699	0.732708
$344$	2.61598	0.141044
$345$	0	0
$346$	23.8827	1.28394
$347$	9.67344	0.519298	0.259649	−	0.965703i	$-0.416393\pi$
$347$	9.67344	0.519298	0.259649	+	0.965703i	$0.416393\pi$
$348$	0	0
$349$	20.8327	1.11515	0.557573	−	0.830128i	$-0.311733\pi$
$349$	20.8327	1.11515	0.557573	+	0.830128i	$0.311733\pi$
$350$	3.11760	0.166643
$351$	0	0
$352$	0.282039	0.0150327
$353$	34.5661	1.83977	0.919885	−	0.392189i	$-0.128282\pi$
$353$	34.5661	1.83977	0.919885	+	0.392189i	$0.128282\pi$
$354$	0	0
$355$	−29.6198	−1.57206
$356$	1.44753	0.0767188
$357$	0	0
$358$	−17.4314	−0.921280
$359$	−10.6404	−0.561579	−0.280790	−	0.959769i	$-0.590596\pi$
$359$	−10.6404	−0.561579	−0.280790	+	0.959769i	$0.590596\pi$
$360$	0	0
$361$	−16.7787	−0.883088
$362$	24.1890	1.27134
$363$	0	0
$364$	0.737698	0.0386659
$365$	34.8353	1.82336
$366$	0	0
$367$	2.37107	0.123769	0.0618844	−	0.998083i	$-0.480289\pi$
$367$	2.37107	0.123769	0.0618844	+	0.998083i	$0.480289\pi$
$368$	2.62620	0.136900
$369$	0	0
$370$	17.4344	0.906371
$371$	9.14535	0.474803
$372$	0	0
$373$	−10.9807	−0.568562	−0.284281	−	0.958741i	$-0.591755\pi$
$373$	−10.9807	−0.568562	−0.284281	+	0.958741i	$0.591755\pi$
$374$	−0.772242	−0.0399317
$375$	0	0
$376$	−27.8523	−1.43637
$377$	−1.29615	−0.0667553
$378$	0	0
$379$	−8.30310	−0.426502	−0.213251	−	0.976997i	$-0.568405\pi$
$379$	−8.30310	−0.426502	−0.213251	+	0.976997i	$0.568405\pi$
$380$	2.19991	0.112853
$381$	0	0
$382$	9.10464	0.465834
$383$	17.7406	0.906500	0.453250	−	0.891383i	$-0.350264\pi$
$383$	17.7406	0.906500	0.453250	+	0.891383i	$0.350264\pi$
$384$	0	0
$385$	−0.273234	−0.0139253
$386$	−16.5500	−0.842373
$387$	0	0
$388$	4.99333	0.253498
$389$	0.601566	0.0305006	0.0152503	−	0.999884i	$-0.495145\pi$
$389$	0.601566	0.0305006	0.0152503	+	0.999884i	$0.495145\pi$
$390$	0	0
$391$	6.72257	0.339975
$392$	−18.0840	−0.913380
$393$	0	0
$394$	−17.2588	−0.869488
$395$	11.9573	0.601636
$396$	0	0
$397$	32.5222	1.63224	0.816121	−	0.577881i	$-0.196120\pi$
$397$	32.5222	1.63224	0.816121	+	0.577881i	$0.196120\pi$
$398$	−31.2872	−1.56828
$399$	0	0
$400$	−6.43908	−0.321954
$401$	34.7143	1.73355	0.866774	−	0.498701i	$-0.166189\pi$
$401$	34.7143	1.73355	0.866774	+	0.498701i	$0.166189\pi$
$402$	0	0
$403$	3.93133	0.195834
$404$	−3.76009	−0.187072
$405$	0	0
$406$	−1.27153	−0.0631048
$407$	−0.502750	−0.0249204
$408$	0	0
$409$	37.2317	1.84099	0.920496	−	0.390753i	$-0.127785\pi$
$409$	37.2317	1.84099	0.920496	+	0.390753i	$0.127785\pi$
$410$	17.0178	0.840449
$411$	0	0
$412$	−9.40311	−0.463258
$413$	7.51932	0.370001
$414$	0	0
$415$	−7.74674	−0.380272
$416$	−3.84431	−0.188483
$417$	0	0
$418$	0.171208	0.00837405
$419$	11.2359	0.548911	0.274455	−	0.961600i	$-0.411502\pi$
$419$	11.2359	0.548911	0.274455	+	0.961600i	$0.411502\pi$
$420$	0	0
$421$	29.5588	1.44061	0.720304	−	0.693658i	$-0.244002\pi$
$421$	29.5588	1.44061	0.720304	+	0.693658i	$0.244002\pi$
$422$	−2.14374	−0.104356
$423$	0	0
$424$	−26.6668	−1.29506
$425$	−16.4828	−0.799533
$426$	0	0
$427$	11.8618	0.574030
$428$	8.12512	0.392742
$429$	0	0
$430$	2.81068	0.135543
$431$	10.8552	0.522876	0.261438	−	0.965220i	$-0.415803\pi$
$431$	10.8552	0.522876	0.261438	+	0.965220i	$0.415803\pi$
$432$	0	0
$433$	28.1930	1.35487	0.677434	−	0.735583i	$-0.263092\pi$
$433$	28.1930	1.35487	0.677434	+	0.735583i	$0.263092\pi$
$434$	3.85663	0.185124
$435$	0	0
$436$	9.63506	0.461436
$437$	−1.49041	−0.0712960
$438$	0	0
$439$	−23.1590	−1.10532	−0.552659	−	0.833408i	$-0.686387\pi$
$439$	−23.1590	−1.10532	−0.552659	+	0.833408i	$0.686387\pi$
$440$	0.796721	0.0379822
$441$	0	0
$442$	10.5260	0.500669
$443$	21.2269	1.00852	0.504261	−	0.863551i	$-0.331765\pi$
$443$	21.2269	1.00852	0.504261	+	0.863551i	$0.331765\pi$
$444$	0	0
$445$	7.30789	0.346427
$446$	−13.5556	−0.641876
$447$	0	0
$448$	−9.29983	−0.439376
$449$	9.30415	0.439090	0.219545	−	0.975602i	$-0.429543\pi$
$449$	9.30415	0.439090	0.219545	+	0.975602i	$0.429543\pi$
$450$	0	0
$451$	−0.490737	−0.0231079
$452$	−3.58853	−0.168790
$453$	0	0
$454$	0.0581359	0.00272845
$455$	3.72429	0.174597
$456$	0	0
$457$	−39.1723	−1.83240	−0.916202	−	0.400718i	$-0.868761\pi$
$457$	−39.1723	−1.83240	−0.916202	+	0.400718i	$0.868761\pi$
$458$	20.9696	0.979847
$459$	0	0
$460$	−1.47604	−0.0688209
$461$	−7.73012	−0.360027	−0.180014	−	0.983664i	$-0.557614\pi$
$461$	−7.73012	−0.360027	−0.180014	+	0.983664i	$0.557614\pi$
$462$	0	0
$463$	−0.502209	−0.0233396	−0.0116698	−	0.999932i	$-0.503715\pi$
$463$	−0.502209	−0.0233396	−0.0116698	+	0.999932i	$0.503715\pi$
$464$	2.62620	0.121918
$465$	0	0
$466$	6.57931	0.304781
$467$	17.9365	0.830001	0.415001	−	0.909821i	$-0.363781\pi$
$467$	17.9365	0.830001	0.415001	+	0.909821i	$0.363781\pi$
$468$	0	0
$469$	−14.6417	−0.676090
$470$	−29.9253	−1.38035
$471$	0	0
$472$	−21.9255	−1.00920
$473$	−0.0810507	−0.00372671
$474$	0	0
$475$	3.65427	0.167670
$476$	−3.82611	−0.175369
$477$	0	0
$478$	−0.438909	−0.0200752
$479$	17.7497	0.811004	0.405502	−	0.914094i	$-0.367097\pi$
$479$	17.7497	0.811004	0.405502	+	0.914094i	$0.367097\pi$
$480$	0	0
$481$	6.85268	0.312456
$482$	−24.0102	−1.09364
$483$	0	0
$484$	5.94296	0.270134
$485$	25.2090	1.14468
$486$	0	0
$487$	0.542296	0.0245738	0.0122869	−	0.999925i	$-0.496089\pi$
$487$	0.542296	0.0245738	0.0122869	+	0.999925i	$0.496089\pi$
$488$	−34.5876	−1.56571
$489$	0	0
$490$	−19.4299	−0.877754
$491$	−11.3071	−0.510282	−0.255141	−	0.966904i	$-0.582122\pi$
$491$	−11.3071	−0.510282	−0.255141	+	0.966904i	$0.582122\pi$
$492$	0	0
$493$	6.72257	0.302769
$494$	−2.33363	−0.104995
$495$	0	0
$496$	−7.96547	−0.357660
$497$	−11.4210	−0.512303
$498$	0	0
$499$	23.1334	1.03560	0.517798	−	0.855503i	$-0.326752\pi$
$499$	23.1334	1.03560	0.517798	+	0.855503i	$0.326752\pi$
$500$	−3.76117	−0.168205
$501$	0	0
$502$	−29.5009	−1.31669
$503$	23.1463	1.03204	0.516022	−	0.856575i	$-0.327412\pi$
$503$	23.1463	1.03204	0.516022	+	0.856575i	$0.327412\pi$
$504$	0	0
$505$	−18.9829	−0.844730
$506$	−0.114873	−0.00510673
$507$	0	0
$508$	9.45755	0.419611
$509$	−2.91911	−0.129387	−0.0646937	−	0.997905i	$-0.520607\pi$
$509$	−2.91911	−0.129387	−0.0646937	+	0.997905i	$0.520607\pi$
$510$	0	0
$511$	13.4321	0.594199
$512$	23.9099	1.05668
$513$	0	0
$514$	−19.8018	−0.873419
$515$	−47.4719	−2.09186
$516$	0	0
$517$	0.862945	0.0379523
$518$	6.72248	0.295369
$519$	0	0
$520$	−10.8596	−0.476226
$521$	−21.6716	−0.949450	−0.474725	−	0.880134i	$-0.657452\pi$
$521$	−21.6716	−0.949450	−0.474725	+	0.880134i	$0.657452\pi$
$522$	0	0
$523$	14.3768	0.628654	0.314327	−	0.949315i	$-0.398221\pi$
$523$	14.3768	0.628654	0.314327	+	0.949315i	$0.398221\pi$
$524$	10.8232	0.472812
$525$	0	0
$526$	7.13392	0.311054
$527$	−20.3900	−0.888204
$528$	0	0
$529$	1.00000	0.0434783
$530$	−28.6515	−1.24454
$531$	0	0
$532$	0.848257	0.0367766
$533$	6.68894	0.289730
$534$	0	0
$535$	41.0199	1.77345
$536$	42.6936	1.84408
$537$	0	0
$538$	25.2316	1.08781
$539$	0.560294	0.0241336
$540$	0	0
$541$	11.6330	0.500142	0.250071	−	0.968227i	$-0.419546\pi$
$541$	11.6330	0.500142	0.250071	+	0.968227i	$0.419546\pi$
$542$	−23.0717	−0.991015
$543$	0	0
$544$	19.9387	0.854864
$545$	48.6429	2.08363
$546$	0	0
$547$	−9.07153	−0.387871	−0.193935	−	0.981014i	$-0.562125\pi$
$547$	−9.07153	−0.387871	−0.193935	+	0.981014i	$0.562125\pi$
$548$	−5.89107	−0.251654
$549$	0	0
$550$	0.281652	0.0120097
$551$	−1.49041	−0.0634936
$552$	0	0
$553$	4.61057	0.196061
$554$	33.4194	1.41985
$555$	0	0
$556$	4.36036	0.184920
$557$	−19.3611	−0.820355	−0.410178	−	0.912006i	$-0.634533\pi$
$557$	−19.3611	−0.820355	−0.410178	+	0.912006i	$0.634533\pi$
$558$	0	0
$559$	1.10475	0.0467261
$560$	−7.54597	−0.318876
$561$	0	0
$562$	−17.7092	−0.747017
$563$	−17.8870	−0.753845	−0.376923	−	0.926245i	$-0.623018\pi$
$563$	−17.8870	−0.753845	−0.376923	+	0.926245i	$0.623018\pi$
$564$	0	0
$565$	−18.1168	−0.762179
$566$	−12.9215	−0.543130
$567$	0	0
$568$	33.3024	1.39734
$569$	−10.4808	−0.439378	−0.219689	−	0.975570i	$-0.570504\pi$
$569$	−10.4808	−0.439378	−0.219689	+	0.975570i	$0.570504\pi$
$570$	0	0
$571$	19.1917	0.803149	0.401575	−	0.915826i	$-0.368463\pi$
$571$	19.1917	0.803149	0.401575	+	0.915826i	$0.368463\pi$
$572$	0.0666456	0.00278659
$573$	0	0
$574$	6.56184	0.273886
$575$	−2.45186	−0.102250
$576$	0	0
$577$	−18.9756	−0.789966	−0.394983	−	0.918689i	$-0.629249\pi$
$577$	−18.9756	−0.789966	−0.394983	+	0.918689i	$0.629249\pi$
$578$	−34.0573	−1.41660
$579$	0	0
$580$	−1.47604	−0.0612894
$581$	−2.98704	−0.123923
$582$	0	0
$583$	0.826215	0.0342183
$584$	−39.1664	−1.62072
$585$	0	0
$586$	−36.4195	−1.50447
$587$	−43.8136	−1.80838	−0.904190	−	0.427130i	$-0.859525\pi$
$587$	−43.8136	−1.80838	−0.904190	+	0.427130i	$0.859525\pi$
$588$	0	0
$589$	4.52052	0.186265
$590$	−23.5573	−0.969840
$591$	0	0
$592$	−13.8846	−0.570652
$593$	−13.1914	−0.541707	−0.270853	−	0.962621i	$-0.587306\pi$
$593$	−13.1914	−0.541707	−0.270853	+	0.962621i	$0.587306\pi$
$594$	0	0
$595$	−19.3162	−0.791888
$596$	−3.49875	−0.143315
$597$	0	0
$598$	1.56577	0.0640289
$599$	−26.1663	−1.06913	−0.534564	−	0.845128i	$-0.679524\pi$
$599$	−26.1663	−1.06913	−0.534564	+	0.845128i	$0.679524\pi$
$600$	0	0
$601$	−3.58376	−0.146185	−0.0730923	−	0.997325i	$-0.523287\pi$
$601$	−3.58376	−0.146185	−0.0730923	+	0.997325i	$0.523287\pi$
$602$	1.08376	0.0441708
$603$	0	0
$604$	9.11612	0.370930
$605$	30.0032	1.21980
$606$	0	0
$607$	37.6954	1.53001	0.765005	−	0.644024i	$-0.222736\pi$
$607$	37.6954	1.53001	0.765005	+	0.644024i	$0.222736\pi$
$608$	−4.42046	−0.179273
$609$	0	0
$610$	−37.1618	−1.50464
$611$	−11.7623	−0.475851
$612$	0	0
$613$	−27.3706	−1.10549	−0.552745	−	0.833351i	$-0.686419\pi$
$613$	−27.3706	−1.10549	−0.552745	+	0.833351i	$0.686419\pi$
$614$	11.5144	0.464684
$615$	0	0
$616$	0.307205	0.0123776
$617$	−37.2694	−1.50041	−0.750205	−	0.661205i	$-0.770045\pi$
$617$	−37.2694	−1.50041	−0.750205	+	0.661205i	$0.770045\pi$
$618$	0	0
$619$	14.1754	0.569759	0.284879	−	0.958563i	$-0.408046\pi$
$619$	14.1754	0.569759	0.284879	+	0.958563i	$0.408046\pi$
$620$	4.47695	0.179799
$621$	0	0
$622$	−38.0053	−1.52388
$623$	2.81783	0.112894
$624$	0	0
$625$	−31.2477	−1.24991
$626$	−10.3442	−0.413436
$627$	0	0
$628$	−8.27163	−0.330074
$629$	−35.5418	−1.41714
$630$	0	0
$631$	−29.1365	−1.15991	−0.579953	−	0.814650i	$-0.696929\pi$
$631$	−29.1365	−1.15991	−0.579953	+	0.814650i	$0.696929\pi$
$632$	−13.4439	−0.534770
$633$	0	0
$634$	24.4327	0.970345
$635$	47.7468	1.89477
$636$	0	0
$637$	−7.63704	−0.302590
$638$	−0.114873	−0.00454787
$639$	0	0
$640$	12.9427	0.511604
$641$	13.6557	0.539367	0.269684	−	0.962949i	$-0.413081\pi$
$641$	13.6557	0.539367	0.269684	+	0.962949i	$0.413081\pi$
$642$	0	0
$643$	16.8084	0.662859	0.331429	−	0.943480i	$-0.392469\pi$
$643$	16.8084	0.662859	0.331429	+	0.943480i	$0.392469\pi$
$644$	−0.569144	−0.0224274
$645$	0	0
$646$	12.1035	0.476206
$647$	15.6785	0.616386	0.308193	−	0.951324i	$-0.400276\pi$
$647$	15.6785	0.616386	0.308193	+	0.951324i	$0.400276\pi$
$648$	0	0
$649$	0.679315	0.0266655
$650$	−3.83904	−0.150579
$651$	0	0
$652$	−7.25640	−0.284182
$653$	−31.6408	−1.23820	−0.619099	−	0.785313i	$-0.712502\pi$
$653$	−31.6408	−1.23820	−0.619099	+	0.785313i	$0.712502\pi$
$654$	0	0
$655$	54.6411	2.13501
$656$	−13.5528	−0.529148
$657$	0	0
$658$	−11.5388	−0.449829
$659$	16.4824	0.642062	0.321031	−	0.947069i	$-0.395971\pi$
$659$	16.4824	0.642062	0.321031	+	0.947069i	$0.395971\pi$
$660$	0	0
$661$	2.37429	0.0923490	0.0461745	−	0.998933i	$-0.485297\pi$
$661$	2.37429	0.0923490	0.0461745	+	0.998933i	$0.485297\pi$
$662$	18.1930	0.707089
$663$	0	0
$664$	8.70989	0.338009
$665$	4.28245	0.166066
$666$	0	0
$667$	1.00000	0.0387202
$668$	−2.85681	−0.110533
$669$	0	0
$670$	45.8711	1.77215
$671$	1.07162	0.0413695
$672$	0	0
$673$	−11.7196	−0.451757	−0.225879	−	0.974155i	$-0.572525\pi$
$673$	−11.7196	−0.451757	−0.225879	+	0.974155i	$0.572525\pi$
$674$	32.1291	1.23757
$675$	0	0
$676$	6.12087	0.235418
$677$	46.0678	1.77053	0.885265	−	0.465086i	$-0.153977\pi$
$677$	46.0678	1.77053	0.885265	+	0.465086i	$0.153977\pi$
$678$	0	0
$679$	9.72026	0.373029
$680$	56.3240	2.15993
$681$	0	0
$682$	0.348418	0.0133416
$683$	15.3863	0.588740	0.294370	−	0.955692i	$-0.404890\pi$
$683$	15.3863	0.588740	0.294370	+	0.955692i	$0.404890\pi$
$684$	0	0
$685$	−29.7413	−1.13636
$686$	−16.3926	−0.625873
$687$	0	0
$688$	−2.23840	−0.0853380
$689$	−11.2616	−0.429034
$690$	0	0
$691$	21.9023	0.833204	0.416602	−	0.909089i	$-0.363221\pi$
$691$	21.9023	0.833204	0.416602	+	0.909089i	$0.363221\pi$
$692$	10.6901	0.406375
$693$	0	0
$694$	−11.6856	−0.443580
$695$	22.0134	0.835017
$696$	0	0
$697$	−34.6925	−1.31407
$698$	−25.1660	−0.952549
$699$	0	0
$700$	1.39546	0.0527434
$701$	−32.8062	−1.23907	−0.619536	−	0.784968i	$-0.712679\pi$
$701$	−32.8062	−1.23907	−0.619536	+	0.784968i	$0.712679\pi$
$702$	0	0
$703$	7.87970	0.297189
$704$	−0.840172	−0.0316652
$705$	0	0
$706$	−41.7562	−1.57152
$707$	−7.31958	−0.275281
$708$	0	0
$709$	12.0699	0.453295	0.226647	−	0.973977i	$-0.427224\pi$
$709$	12.0699	0.453295	0.226647	+	0.973977i	$0.427224\pi$
$710$	35.7810	1.34284
$711$	0	0
$712$	−8.21647	−0.307925
$713$	−3.03307	−0.113590
$714$	0	0
$715$	0.336462	0.0125830
$716$	−7.80244	−0.291591
$717$	0	0
$718$	12.8537	0.479696
$719$	−25.7940	−0.961955	−0.480978	−	0.876733i	$-0.659718\pi$
$719$	−25.7940	−0.961955	−0.480978	+	0.876733i	$0.659718\pi$
$720$	0	0
$721$	−18.3045	−0.681697
$722$	20.2688	0.754327
$723$	0	0
$724$	10.8272	0.402388
$725$	−2.45186	−0.0910598
$726$	0	0
$727$	−13.2122	−0.490014	−0.245007	−	0.969521i	$-0.578790\pi$
$727$	−13.2122	−0.490014	−0.245007	+	0.969521i	$0.578790\pi$
$728$	−4.18733	−0.155193
$729$	0	0
$730$	−42.0814	−1.55750
$731$	−5.72986	−0.211926
$732$	0	0
$733$	24.8191	0.916714	0.458357	−	0.888768i	$-0.348438\pi$
$733$	24.8191	0.916714	0.458357	+	0.888768i	$0.348438\pi$
$734$	−2.86427	−0.105722
$735$	0	0
$736$	2.96593	0.109326
$737$	−1.32277	−0.0487248
$738$	0	0
$739$	−20.0451	−0.737370	−0.368685	−	0.929554i	$-0.620192\pi$
$739$	−20.0451	−0.737370	−0.368685	+	0.929554i	$0.620192\pi$
$740$	7.80376	0.286872
$741$	0	0
$742$	−11.0477	−0.405572
$743$	19.4968	0.715269	0.357635	−	0.933862i	$-0.383583\pi$
$743$	19.4968	0.715269	0.357635	+	0.933862i	$0.383583\pi$
$744$	0	0
$745$	−17.6636	−0.647143
$746$	13.2648	0.485660
$747$	0	0
$748$	−0.345661	−0.0126386
$749$	15.8167	0.577931
$750$	0	0
$751$	−5.18641	−0.189255	−0.0946274	−	0.995513i	$-0.530166\pi$
$751$	−5.18641	−0.189255	−0.0946274	+	0.995513i	$0.530166\pi$
$752$	23.8322	0.869070
$753$	0	0
$754$	1.56577	0.0570218
$755$	46.0230	1.67495
$756$	0	0
$757$	33.3241	1.21119	0.605593	−	0.795774i	$-0.292936\pi$
$757$	33.3241	1.21119	0.605593	+	0.795774i	$0.292936\pi$
$758$	10.0302	0.364314
$759$	0	0
$760$	−12.4872	−0.452957
$761$	−8.82878	−0.320043	−0.160021	−	0.987114i	$-0.551156\pi$
$761$	−8.82878	−0.320043	−0.160021	+	0.987114i	$0.551156\pi$
$762$	0	0
$763$	18.7561	0.679016
$764$	4.07530	0.147439
$765$	0	0
$766$	−21.4308	−0.774325
$767$	−9.25934	−0.334335
$768$	0	0
$769$	28.8580	1.04065	0.520323	−	0.853970i	$-0.325812\pi$
$769$	28.8580	1.04065	0.520323	+	0.853970i	$0.325812\pi$
$770$	0.330069	0.0118949
$771$	0	0
$772$	−7.40790	−0.266616
$773$	5.63551	0.202695	0.101348	−	0.994851i	$-0.467685\pi$
$773$	5.63551	0.202695	0.101348	+	0.994851i	$0.467685\pi$
$774$	0	0
$775$	7.43667	0.267133
$776$	−28.3432	−1.01746
$777$	0	0
$778$	−0.726698	−0.0260534
$779$	7.69142	0.275574
$780$	0	0
$781$	−1.03181	−0.0369209
$782$	−8.12092	−0.290404
$783$	0	0
$784$	15.4738	0.552635
$785$	−41.7596	−1.49046
$786$	0	0
$787$	−28.0128	−0.998548	−0.499274	−	0.866444i	$-0.666400\pi$
$787$	−28.0128	−0.998548	−0.499274	+	0.866444i	$0.666400\pi$
$788$	−7.72518	−0.275198
$789$	0	0
$790$	−14.4445	−0.513912
$791$	−6.98560	−0.248379
$792$	0	0
$793$	−14.6066	−0.518697
$794$	−39.2871	−1.39425
$795$	0	0
$796$	−14.0044	−0.496371
$797$	55.2666	1.95764	0.978822	−	0.204715i	$-0.0656269\pi$
$797$	55.2666	1.95764	0.978822	+	0.204715i	$0.0656269\pi$
$798$	0	0
$799$	61.0057	2.15823
$800$	−7.27205	−0.257106
$801$	0	0
$802$	−41.9352	−1.48078
$803$	1.21349	0.0428231
$804$	0	0
$805$	−2.87334	−0.101272
$806$	−4.74908	−0.167279
$807$	0	0
$808$	21.3431	0.750847
$809$	−5.09682	−0.179195	−0.0895974	−	0.995978i	$-0.528558\pi$
$809$	−5.09682	−0.179195	−0.0895974	+	0.995978i	$0.528558\pi$
$810$	0	0
$811$	−44.0762	−1.54773	−0.773863	−	0.633353i	$-0.781678\pi$
$811$	−44.0762	−1.54773	−0.773863	+	0.633353i	$0.781678\pi$
$812$	−0.569144	−0.0199730
$813$	0	0
$814$	0.607327	0.0212868
$815$	−36.6341	−1.28324
$816$	0	0
$817$	1.27032	0.0444430
$818$	−44.9763	−1.57256
$819$	0	0
$820$	7.61728	0.266007
$821$	0.900882	0.0314410	0.0157205	−	0.999876i	$-0.494996\pi$
$821$	0.900882	0.0314410	0.0157205	+	0.999876i	$0.494996\pi$
$822$	0	0
$823$	16.8019	0.585678	0.292839	−	0.956162i	$-0.405400\pi$
$823$	16.8019	0.585678	0.292839	+	0.956162i	$0.405400\pi$
$824$	53.3741	1.85937
$825$	0	0
$826$	−9.08340	−0.316052
$827$	45.8691	1.59502	0.797512	−	0.603303i	$-0.206149\pi$
$827$	45.8691	1.59502	0.797512	+	0.603303i	$0.206149\pi$
$828$	0	0
$829$	5.68550	0.197466	0.0987328	−	0.995114i	$-0.468521\pi$
$829$	5.68550	0.197466	0.0987328	+	0.995114i	$0.468521\pi$
$830$	9.35813	0.324826
$831$	0	0
$832$	11.4519	0.397023
$833$	39.6099	1.37240
$834$	0	0
$835$	−14.4227	−0.499119
$836$	0.0766338	0.00265044
$837$	0	0
$838$	−13.5731	−0.468875
$839$	36.1884	1.24936	0.624681	−	0.780880i	$-0.285229\pi$
$839$	36.1884	1.24936	0.624681	+	0.780880i	$0.285229\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−35.7073	−1.23056
$843$	0	0
$844$	−0.959555	−0.0330293
$845$	30.9014	1.06304
$846$	0	0
$847$	11.5689	0.397511
$848$	22.8178	0.783566
$849$	0	0
$850$	19.9114	0.682954
$851$	−5.28694	−0.181234
$852$	0	0
$853$	53.4179	1.82899	0.914497	−	0.404593i	$-0.132587\pi$
$853$	53.4179	1.82899	0.914497	+	0.404593i	$0.132587\pi$
$854$	−14.3291	−0.490332
$855$	0	0
$856$	−46.1199	−1.57635
$857$	−48.9105	−1.67075	−0.835376	−	0.549679i	$-0.814750\pi$
$857$	−48.9105	−1.67075	−0.835376	+	0.549679i	$0.814750\pi$
$858$	0	0
$859$	−34.0344	−1.16124	−0.580619	−	0.814175i	$-0.697190\pi$
$859$	−34.0344	−1.16124	−0.580619	+	0.814175i	$0.697190\pi$
$860$	1.25808	0.0429002
$861$	0	0
$862$	−13.1132	−0.446636
$863$	45.5204	1.54953	0.774766	−	0.632248i	$-0.217868\pi$
$863$	45.5204	1.54953	0.774766	+	0.632248i	$0.217868\pi$
$864$	0	0
$865$	53.9691	1.83500
$866$	−34.0574	−1.15732
$867$	0	0
$868$	1.72626	0.0585929
$869$	0.416531	0.0141299
$870$	0	0
$871$	18.0299	0.610919
$872$	−54.6907	−1.85206
$873$	0	0
$874$	1.80043	0.0609004
$875$	−7.32168	−0.247518
$876$	0	0
$877$	23.1936	0.783192	0.391596	−	0.920137i	$-0.371923\pi$
$877$	23.1936	0.783192	0.391596	+	0.920137i	$0.371923\pi$
$878$	27.9763	0.944153
$879$	0	0
$880$	−0.681723	−0.0229809
$881$	22.6964	0.764661	0.382331	−	0.924026i	$-0.375122\pi$
$881$	22.6964	0.764661	0.382331	+	0.924026i	$0.375122\pi$
$882$	0	0
$883$	43.9771	1.47995	0.739974	−	0.672636i	$-0.234838\pi$
$883$	43.9771	1.47995	0.739974	+	0.672636i	$0.234838\pi$
$884$	4.71150	0.158465
$885$	0	0
$886$	−25.6423	−0.861470
$887$	−15.2109	−0.510733	−0.255367	−	0.966844i	$-0.582196\pi$
$887$	−15.2109	−0.510733	−0.255367	+	0.966844i	$0.582196\pi$
$888$	0	0
$889$	18.4105	0.617470
$890$	−8.82799	−0.295915
$891$	0	0
$892$	−6.06758	−0.203158
$893$	−13.5251	−0.452601
$894$	0	0
$895$	−39.3909	−1.31669
$896$	4.99052	0.166722
$897$	0	0
$898$	−11.2395	−0.375067
$899$	−3.03307	−0.101159
$900$	0	0
$901$	58.4091	1.94589
$902$	0.592814	0.0197386
$903$	0	0
$904$	20.3692	0.677471
$905$	54.6612	1.81700
$906$	0	0
$907$	−7.95460	−0.264128	−0.132064	−	0.991241i	$-0.542160\pi$
$907$	−7.95460	−0.264128	−0.132064	+	0.991241i	$0.542160\pi$
$908$	0.0260220	0.000863572 0
$909$	0	0
$910$	−4.49898	−0.149140
$911$	−0.300857	−0.00996784	−0.00498392	−	0.999988i	$-0.501586\pi$
$911$	−0.300857	−0.00996784	−0.00498392	+	0.999988i	$0.501586\pi$
$912$	0	0
$913$	−0.269857	−0.00893098
$914$	47.3205	1.56522
$915$	0	0
$916$	9.38616	0.310128
$917$	21.0689	0.695757
$918$	0	0
$919$	−4.08143	−0.134634	−0.0673170	−	0.997732i	$-0.521444\pi$
$919$	−4.08143	−0.134634	−0.0673170	+	0.997732i	$0.521444\pi$
$920$	8.37835	0.276226
$921$	0	0
$922$	9.33805	0.307532
$923$	14.0639	0.462920
$924$	0	0
$925$	12.9628	0.426215
$926$	0.606673	0.0199365
$927$	0	0
$928$	2.96593	0.0973616
$929$	−22.6934	−0.744547	−0.372274	−	0.928123i	$-0.621422\pi$
$929$	−22.6934	−0.744547	−0.372274	+	0.928123i	$0.621422\pi$
$930$	0	0
$931$	−8.78161	−0.287806
$932$	2.94495	0.0964649
$933$	0	0
$934$	−21.6674	−0.708980
$935$	−1.74508	−0.0570702
$936$	0	0
$937$	36.1651	1.18146	0.590731	−	0.806869i	$-0.298839\pi$
$937$	36.1651	1.18146	0.590731	+	0.806869i	$0.298839\pi$
$938$	17.6873	0.577511
$939$	0	0
$940$	−13.3948	−0.436889
$941$	−34.0213	−1.10906	−0.554532	−	0.832163i	$-0.687103\pi$
$941$	−34.0213	−1.10906	−0.554532	+	0.832163i	$0.687103\pi$
$942$	0	0
$943$	−5.16061	−0.168052
$944$	18.7608	0.610612
$945$	0	0
$946$	0.0979099	0.00318333
$947$	−53.2033	−1.72888	−0.864438	−	0.502740i	$-0.832325\pi$
$947$	−53.2033	−1.72888	−0.864438	+	0.502740i	$0.832325\pi$
$948$	0	0
$949$	−16.5403	−0.536922
$950$	−4.41440	−0.143222
$951$	0	0
$952$	21.7178	0.703878
$953$	−31.5026	−1.02047	−0.510234	−	0.860036i	$-0.670441\pi$
$953$	−31.5026	−1.02047	−0.510234	+	0.860036i	$0.670441\pi$
$954$	0	0
$955$	20.5743	0.665768
$956$	−0.196459	−0.00635394
$957$	0	0
$958$	−21.4418	−0.692753
$959$	−11.4679	−0.370316
$960$	0	0
$961$	−21.8005	−0.703241
$962$	−8.27811	−0.266897
$963$	0	0
$964$	−10.7472	−0.346142
$965$	−37.3990	−1.20392
$966$	0	0
$967$	−1.81425	−0.0583423	−0.0291711	−	0.999574i	$-0.509287\pi$
$967$	−1.81425	−0.0583423	−0.0291711	+	0.999574i	$0.509287\pi$
$968$	−33.7335	−1.08424
$969$	0	0
$970$	−30.4527	−0.977777
$971$	51.7766	1.66159	0.830795	−	0.556579i	$-0.187886\pi$
$971$	51.7766	1.66159	0.830795	+	0.556579i	$0.187886\pi$
$972$	0	0
$973$	8.48809	0.272116
$974$	−0.655099	−0.0209907
$975$	0	0
$976$	29.5953	0.947321
$977$	27.9076	0.892843	0.446421	−	0.894823i	$-0.352698\pi$
$977$	27.9076	0.892843	0.446421	+	0.894823i	$0.352698\pi$
$978$	0	0
$979$	0.254570	0.00813609
$980$	−8.69697	−0.277814
$981$	0	0
$982$	13.6591	0.435879
$983$	7.30632	0.233036	0.116518	−	0.993189i	$-0.462827\pi$
$983$	7.30632	0.233036	0.116518	+	0.993189i	$0.462827\pi$
$984$	0	0
$985$	−39.0008	−1.24267
$986$	−8.12092	−0.258623
$987$	0	0
$988$	−1.04455	−0.0332316
$989$	−0.852332	−0.0271026
$990$	0	0
$991$	11.8211	0.375508	0.187754	−	0.982216i	$-0.439879\pi$
$991$	11.8211	0.375508	0.187754	+	0.982216i	$0.439879\pi$
$992$	−8.99590	−0.285620
$993$	0	0
$994$	13.7967	0.437605
$995$	−70.7015	−2.24139
$996$	0	0
$997$	−19.3020	−0.611300	−0.305650	−	0.952144i	$-0.598874\pi$
$997$	−19.3020	−0.611300	−0.305650	+	0.952144i	$0.598874\pi$
$998$	−27.9454	−0.884597
$999$	0	0

Twists

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

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

\(f(q)\)	\(=\)	\(q-1.20801 q^{2} -0.540713 q^{4} -2.72981 q^{5} -1.05258 q^{7} +3.06921 q^{8} +O(q^{10})\)	\(q-1.20801 q^{2} -0.540713 q^{4} -2.72981 q^{5} -1.05258 q^{7} +3.06921 q^{8} +3.29764 q^{10} -0.0950928 q^{11} +1.29615 q^{13} +1.27153 q^{14} -2.62620 q^{16} -6.72257 q^{17} +1.49041 q^{19} +1.47604 q^{20} +0.114873 q^{22} -1.00000 q^{23} +2.45186 q^{25} -1.56577 q^{26} +0.569144 q^{28} -1.00000 q^{29} +3.03307 q^{31} -2.96593 q^{32} +8.12092 q^{34} +2.87334 q^{35} +5.28694 q^{37} -1.80043 q^{38} -8.37835 q^{40} +5.16061 q^{41} +0.852332 q^{43} +0.0514180 q^{44} +1.20801 q^{46} -9.07477 q^{47} -5.89208 q^{49} -2.96187 q^{50} -0.700848 q^{52} -8.68851 q^{53} +0.259585 q^{55} -3.23058 q^{56} +1.20801 q^{58} -7.14371 q^{59} -11.2692 q^{61} -3.66398 q^{62} +8.83528 q^{64} -3.53825 q^{65} +13.9103 q^{67} +3.63498 q^{68} -3.47102 q^{70} +10.8505 q^{71} -12.7611 q^{73} -6.38667 q^{74} -0.805885 q^{76} +0.100093 q^{77} -4.38026 q^{79} +7.16903 q^{80} -6.23406 q^{82} +2.83783 q^{83} +18.3513 q^{85} -1.02962 q^{86} -0.291859 q^{88} -2.67707 q^{89} -1.36430 q^{91} +0.540713 q^{92} +10.9624 q^{94} -4.06853 q^{95} -9.23470 q^{97} +7.11768 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 2 q^{2} + 30 q^{4} + q^{5} + 9 q^{7} - 6 q^{8}+O(q^{10}) \) 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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