LMFDB - Embedded newform 6003.2.a.s.1.3

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			1.3
Root			$2.45061$ 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)

$f(q)$	$=$	$q-2.45061 q^{2} +4.00547 q^{4} -1.21123 q^{5} +0.971463 q^{7} -4.91461 q^{8} +O(q^{10})$	\(q-2.45061 q^{2} +4.00547 q^{4} -1.21123 q^{5} +0.971463 q^{7} -4.91461 q^{8} +2.96824 q^{10} -3.60961 q^{11} -6.61576 q^{13} -2.38067 q^{14} +4.03285 q^{16} -4.82230 q^{17} -2.80468 q^{19} -4.85154 q^{20} +8.84573 q^{22} -1.00000 q^{23} -3.53293 q^{25} +16.2126 q^{26} +3.89117 q^{28} -1.00000 q^{29} -0.879818 q^{31} -0.0536852 q^{32} +11.8176 q^{34} -1.17666 q^{35} -3.58964 q^{37} +6.87317 q^{38} +5.95272 q^{40} -11.8028 q^{41} -11.1806 q^{43} -14.4582 q^{44} +2.45061 q^{46} -1.54804 q^{47} -6.05626 q^{49} +8.65781 q^{50} -26.4992 q^{52} -2.24340 q^{53} +4.37206 q^{55} -4.77437 q^{56} +2.45061 q^{58} -0.111475 q^{59} -7.12252 q^{61} +2.15609 q^{62} -7.93413 q^{64} +8.01320 q^{65} +6.15784 q^{67} -19.3156 q^{68} +2.88354 q^{70} -2.74676 q^{71} +4.91369 q^{73} +8.79680 q^{74} -11.2341 q^{76} -3.50660 q^{77} +5.76461 q^{79} -4.88470 q^{80} +28.9240 q^{82} +11.4558 q^{83} +5.84090 q^{85} +27.3993 q^{86} +17.7398 q^{88} -3.39528 q^{89} -6.42697 q^{91} -4.00547 q^{92} +3.79364 q^{94} +3.39711 q^{95} -7.46502 q^{97} +14.8415 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.45061	−1.73284	−0.866420	−	0.499316i	$-0.833585\pi$
$2$	−2.45061	−1.73284	−0.866420	+	0.499316i	$0.833585\pi$
$3$	0	0
$3$	0	0
$4$	4.00547	2.00273
$5$	−1.21123	−0.541678	−0.270839	−	0.962625i	$-0.587301\pi$
$5$	−1.21123	−0.541678	−0.270839	+	0.962625i	$0.587301\pi$
$6$	0	0
$7$	0.971463	0.367179	0.183589	−	0.983003i	$-0.441228\pi$
$7$	0.971463	0.367179	0.183589	+	0.983003i	$0.441228\pi$
$8$	−4.91461	−1.73758
$9$	0	0
$10$	2.96824	0.938641
$11$	−3.60961	−1.08834	−0.544169	−	0.838975i	$-0.683155\pi$
$11$	−3.60961	−1.08834	−0.544169	+	0.838975i	$0.683155\pi$
$12$	0	0
$13$	−6.61576	−1.83488	−0.917441	−	0.397872i	$-0.869749\pi$
$13$	−6.61576	−1.83488	−0.917441	+	0.397872i	$0.869749\pi$
$14$	−2.38067	−0.636262
$15$	0	0
$16$	4.03285	1.00821
$17$	−4.82230	−1.16958	−0.584790	−	0.811185i	$-0.698823\pi$
$17$	−4.82230	−1.16958	−0.584790	+	0.811185i	$0.698823\pi$
$18$	0	0
$19$	−2.80468	−0.643438	−0.321719	−	0.946835i	$-0.604261\pi$
$19$	−2.80468	−0.643438	−0.321719	+	0.946835i	$0.604261\pi$
$20$	−4.85154	−1.08484
$21$	0	0
$22$	8.84573	1.88592
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−3.53293	−0.706585
$26$	16.2126	3.17956
$27$	0	0
$28$	3.89117	0.735361
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−0.879818	−0.158020	−0.0790100	−	0.996874i	$-0.525176\pi$
$31$	−0.879818	−0.158020	−0.0790100	+	0.996874i	$0.525176\pi$
$32$	−0.0536852	−0.00949030
$33$	0	0
$34$	11.8176	2.02669
$35$	−1.17666	−0.198893
$36$	0	0
$37$	−3.58964	−0.590133	−0.295067	−	0.955477i	$-0.595342\pi$
$37$	−3.58964	−0.590133	−0.295067	+	0.955477i	$0.595342\pi$
$38$	6.87317	1.11498
$39$	0	0
$40$	5.95272	0.941208
$41$	−11.8028	−1.84329	−0.921643	−	0.388038i	$-0.873153\pi$
$41$	−11.8028	−1.84329	−0.921643	+	0.388038i	$0.873153\pi$
$42$	0	0
$43$	−11.1806	−1.70503	−0.852514	−	0.522704i	$-0.824923\pi$
$43$	−11.1806	−1.70503	−0.852514	+	0.522704i	$0.824923\pi$
$44$	−14.4582	−2.17965
$45$	0	0
$46$	2.45061	0.361322
$47$	−1.54804	−0.225805	−0.112903	−	0.993606i	$-0.536015\pi$
$47$	−1.54804	−0.225805	−0.112903	+	0.993606i	$0.536015\pi$
$48$	0	0
$49$	−6.05626	−0.865180
$50$	8.65781	1.22440
$51$	0	0
$52$	−26.4992	−3.67478
$53$	−2.24340	−0.308155	−0.154078	−	0.988059i	$-0.549241\pi$
$53$	−2.24340	−0.308155	−0.154078	+	0.988059i	$0.549241\pi$
$54$	0	0
$55$	4.37206	0.589529
$56$	−4.77437	−0.638002
$57$	0	0
$58$	2.45061	0.321780
$59$	−0.111475	−0.0145129	−0.00725643	−	0.999974i	$-0.502310\pi$
$59$	−0.111475	−0.0145129	−0.00725643	+	0.999974i	$0.502310\pi$
$60$	0	0
$61$	−7.12252	−0.911945	−0.455973	−	0.889994i	$-0.650709\pi$
$61$	−7.12252	−0.911945	−0.455973	+	0.889994i	$0.650709\pi$
$62$	2.15609	0.273823
$63$	0	0
$64$	−7.93413	−0.991766
$65$	8.01320	0.993915
$66$	0	0
$67$	6.15784	0.752300	0.376150	−	0.926559i	$-0.377248\pi$
$67$	6.15784	0.752300	0.376150	+	0.926559i	$0.377248\pi$
$68$	−19.3156	−2.34236
$69$	0	0
$70$	2.88354	0.344649
$71$	−2.74676	−0.325980	−0.162990	−	0.986628i	$-0.552114\pi$
$71$	−2.74676	−0.325980	−0.162990	+	0.986628i	$0.552114\pi$
$72$	0	0
$73$	4.91369	0.575104	0.287552	−	0.957765i	$-0.407159\pi$
$73$	4.91369	0.575104	0.287552	+	0.957765i	$0.407159\pi$
$74$	8.79680	1.02261
$75$	0	0
$76$	−11.2341	−1.28864
$77$	−3.50660	−0.399615
$78$	0	0
$79$	5.76461	0.648569	0.324285	−	0.945960i	$-0.394876\pi$
$79$	5.76461	0.648569	0.324285	+	0.945960i	$0.394876\pi$
$80$	−4.88470	−0.546126
$81$	0	0
$82$	28.9240	3.19412
$83$	11.4558	1.25743	0.628717	−	0.777634i	$-0.283580\pi$
$83$	11.4558	1.25743	0.628717	+	0.777634i	$0.283580\pi$
$84$	0	0
$85$	5.84090	0.633535
$86$	27.3993	2.95454
$87$	0	0
$88$	17.7398	1.89107
$89$	−3.39528	−0.359899	−0.179949	−	0.983676i	$-0.557593\pi$
$89$	−3.39528	−0.359899	−0.179949	+	0.983676i	$0.557593\pi$
$90$	0	0
$91$	−6.42697	−0.673729
$92$	−4.00547	−0.417599
$93$	0	0
$94$	3.79364	0.391284
$95$	3.39711	0.348536
$96$	0	0
$97$	−7.46502	−0.757958	−0.378979	−	0.925405i	$-0.623725\pi$
$97$	−7.46502	−0.757958	−0.378979	+	0.925405i	$0.623725\pi$
$98$	14.8415	1.49922
$99$	0	0
$100$	−14.1510	−1.41510
$101$	−0.796755	−0.0792801	−0.0396400	−	0.999214i	$-0.512621\pi$
$101$	−0.796755	−0.0792801	−0.0396400	+	0.999214i	$0.512621\pi$
$102$	0	0
$103$	−0.447276	−0.0440714	−0.0220357	−	0.999757i	$-0.507015\pi$
$103$	−0.447276	−0.0440714	−0.0220357	+	0.999757i	$0.507015\pi$
$104$	32.5139	3.18825
$105$	0	0
$106$	5.49770	0.533984
$107$	12.7827	1.23575	0.617874	−	0.786277i	$-0.287994\pi$
$107$	12.7827	1.23575	0.617874	+	0.786277i	$0.287994\pi$
$108$	0	0
$109$	7.60998	0.728904	0.364452	−	0.931222i	$-0.381256\pi$
$109$	7.60998	0.728904	0.364452	+	0.931222i	$0.381256\pi$
$110$	−10.7142	−1.02156
$111$	0	0
$112$	3.91776	0.370194
$113$	7.01294	0.659722	0.329861	−	0.944030i	$-0.392998\pi$
$113$	7.01294	0.659722	0.329861	+	0.944030i	$0.392998\pi$
$114$	0	0
$115$	1.21123	0.112948
$116$	−4.00547	−0.371898
$117$	0	0
$118$	0.273182	0.0251485
$119$	−4.68469	−0.429444
$120$	0	0
$121$	2.02929	0.184481
$122$	17.4545	1.58026
$123$	0	0
$124$	−3.52408	−0.316472
$125$	10.3353	0.924419
$126$	0	0
$127$	16.0903	1.42778	0.713890	−	0.700258i	$-0.246932\pi$
$127$	16.0903	1.42778	0.713890	+	0.700258i	$0.246932\pi$
$128$	19.5508	1.72806
$129$	0	0
$130$	−19.6372	−1.72229
$131$	−4.15912	−0.363384	−0.181692	−	0.983356i	$-0.558157\pi$
$131$	−4.15912	−0.363384	−0.181692	+	0.983356i	$0.558157\pi$
$132$	0	0
$133$	−2.72465	−0.236257
$134$	−15.0904	−1.30362
$135$	0	0
$136$	23.6997	2.03224
$137$	−9.02194	−0.770797	−0.385398	−	0.922750i	$-0.625936\pi$
$137$	−9.02194	−0.770797	−0.385398	+	0.922750i	$0.625936\pi$
$138$	0	0
$139$	−2.93235	−0.248719	−0.124359	−	0.992237i	$-0.539688\pi$
$139$	−2.93235	−0.248719	−0.124359	+	0.992237i	$0.539688\pi$
$140$	−4.71309	−0.398329
$141$	0	0
$142$	6.73123	0.564872
$143$	23.8803	1.99697
$144$	0	0
$145$	1.21123	0.100587
$146$	−12.0415	−0.996562
$147$	0	0
$148$	−14.3782	−1.18188
$149$	−18.6407	−1.52711	−0.763555	−	0.645743i	$-0.776548\pi$
$149$	−18.6407	−1.52711	−0.763555	+	0.645743i	$0.776548\pi$
$150$	0	0
$151$	9.71968	0.790976	0.395488	−	0.918471i	$-0.370575\pi$
$151$	9.71968	0.790976	0.395488	+	0.918471i	$0.370575\pi$
$152$	13.7839	1.11802
$153$	0	0
$154$	8.59331	0.692468
$155$	1.06566	0.0855959
$156$	0	0
$157$	7.71197	0.615482	0.307741	−	0.951470i	$-0.400427\pi$
$157$	7.71197	0.615482	0.307741	+	0.951470i	$0.400427\pi$
$158$	−14.1268	−1.12387
$159$	0	0
$160$	0.0650251	0.00514068
$161$	−0.971463	−0.0765620
$162$	0	0
$163$	−11.9063	−0.932573	−0.466287	−	0.884634i	$-0.654408\pi$
$163$	−11.9063	−0.932573	−0.466287	+	0.884634i	$0.654408\pi$
$164$	−47.2757	−3.69161
$165$	0	0
$166$	−28.0736	−2.17893
$167$	−5.87801	−0.454854	−0.227427	−	0.973795i	$-0.573031\pi$
$167$	−5.87801	−0.454854	−0.227427	+	0.973795i	$0.573031\pi$
$168$	0	0
$169$	30.7683	2.36679
$170$	−14.3138	−1.09781
$171$	0	0
$172$	−44.7836	−3.41472
$173$	8.58330	0.652576	0.326288	−	0.945270i	$-0.394202\pi$
$173$	8.58330	0.652576	0.326288	+	0.945270i	$0.394202\pi$
$174$	0	0
$175$	−3.43211	−0.259443
$176$	−14.5570	−1.09728
$177$	0	0
$178$	8.32049	0.623647
$179$	18.7522	1.40161	0.700804	−	0.713354i	$-0.252825\pi$
$179$	18.7522	1.40161	0.700804	+	0.713354i	$0.252825\pi$
$180$	0	0
$181$	−17.2713	−1.28377	−0.641883	−	0.766802i	$-0.721847\pi$
$181$	−17.2713	−1.28377	−0.641883	+	0.766802i	$0.721847\pi$
$182$	15.7500	1.16747
$183$	0	0
$184$	4.91461	0.362310
$185$	4.34787	0.319662
$186$	0	0
$187$	17.4066	1.27290
$188$	−6.20064	−0.452228
$189$	0	0
$190$	−8.32498	−0.603957
$191$	−10.8221	−0.783063	−0.391532	−	0.920165i	$-0.628055\pi$
$191$	−10.8221	−0.783063	−0.391532	+	0.920165i	$0.628055\pi$
$192$	0	0
$193$	−10.0276	−0.721800	−0.360900	−	0.932605i	$-0.617530\pi$
$193$	−10.0276	−0.721800	−0.360900	+	0.932605i	$0.617530\pi$
$194$	18.2938	1.31342
$195$	0	0
$196$	−24.2582	−1.73273
$197$	−19.2703	−1.37295	−0.686476	−	0.727152i	$-0.740843\pi$
$197$	−19.2703	−1.37295	−0.686476	+	0.727152i	$0.740843\pi$
$198$	0	0
$199$	−26.3709	−1.86938	−0.934691	−	0.355461i	$-0.884324\pi$
$199$	−26.3709	−1.86938	−0.934691	+	0.355461i	$0.884324\pi$
$200$	17.3630	1.22775
$201$	0	0
$202$	1.95253	0.137380
$203$	−0.971463	−0.0681834
$204$	0	0
$205$	14.2959	0.998468
$206$	1.09610	0.0763687
$207$	0	0
$208$	−26.6803	−1.84995
$209$	10.1238	0.700278
$210$	0	0
$211$	8.00666	0.551201	0.275600	−	0.961272i	$-0.411123\pi$
$211$	8.00666	0.551201	0.275600	+	0.961272i	$0.411123\pi$
$212$	−8.98589	−0.617153
$213$	0	0
$214$	−31.3253	−2.14135
$215$	13.5423	0.923576
$216$	0	0
$217$	−0.854711	−0.0580216
$218$	−18.6491	−1.26307
$219$	0	0
$220$	17.5122	1.18067
$221$	31.9032	2.14604
$222$	0	0
$223$	−2.92914	−0.196150	−0.0980748	−	0.995179i	$-0.531268\pi$
$223$	−2.92914	−0.196150	−0.0980748	+	0.995179i	$0.531268\pi$
$224$	−0.0521532	−0.00348463
$225$	0	0
$226$	−17.1860	−1.14319
$227$	14.9020	0.989078	0.494539	−	0.869156i	$-0.335337\pi$
$227$	14.9020	0.989078	0.494539	+	0.869156i	$0.335337\pi$
$228$	0	0
$229$	11.0974	0.733339	0.366670	−	0.930351i	$-0.380498\pi$
$229$	11.0974	0.733339	0.366670	+	0.930351i	$0.380498\pi$
$230$	−2.96824	−0.195720
$231$	0	0
$232$	4.91461	0.322660
$233$	−11.2169	−0.734845	−0.367423	−	0.930054i	$-0.619760\pi$
$233$	−11.2169	−0.734845	−0.367423	+	0.930054i	$0.619760\pi$
$234$	0	0
$235$	1.87503	0.122314
$236$	−0.446511	−0.0290654
$237$	0	0
$238$	11.4803	0.744159
$239$	−18.4791	−1.19531	−0.597655	−	0.801753i	$-0.703901\pi$
$239$	−18.4791	−1.19531	−0.597655	+	0.801753i	$0.703901\pi$
$240$	0	0
$241$	−7.35820	−0.473984	−0.236992	−	0.971512i	$-0.576161\pi$
$241$	−7.35820	−0.473984	−0.236992	+	0.971512i	$0.576161\pi$
$242$	−4.97299	−0.319676
$243$	0	0
$244$	−28.5290	−1.82638
$245$	7.33551	0.468649
$246$	0	0
$247$	18.5551	1.18063
$248$	4.32397	0.274572
$249$	0	0
$250$	−25.3278	−1.60187
$251$	22.0099	1.38925	0.694627	−	0.719370i	$-0.255570\pi$
$251$	22.0099	1.38925	0.694627	+	0.719370i	$0.255570\pi$
$252$	0	0
$253$	3.60961	0.226934
$254$	−39.4309	−2.47411
$255$	0	0
$256$	−32.0430	−2.00269
$257$	13.2471	0.826328	0.413164	−	0.910657i	$-0.364424\pi$
$257$	13.2471	0.826328	0.413164	+	0.910657i	$0.364424\pi$
$258$	0	0
$259$	−3.48720	−0.216684
$260$	32.0966	1.99055
$261$	0	0
$262$	10.1924	0.629686
$263$	−11.0930	−0.684026	−0.342013	−	0.939695i	$-0.611109\pi$
$263$	−11.0930	−0.684026	−0.342013	+	0.939695i	$0.611109\pi$
$264$	0	0
$265$	2.71727	0.166921
$266$	6.67703	0.409395
$267$	0	0
$268$	24.6650	1.50666
$269$	15.7762	0.961894	0.480947	−	0.876750i	$-0.340293\pi$
$269$	15.7762	0.961894	0.480947	+	0.876750i	$0.340293\pi$
$270$	0	0
$271$	−4.31111	−0.261881	−0.130941	−	0.991390i	$-0.541800\pi$
$271$	−4.31111	−0.261881	−0.130941	+	0.991390i	$0.541800\pi$
$272$	−19.4476	−1.17918
$273$	0	0
$274$	22.1092	1.33567
$275$	12.7525	0.769004
$276$	0	0
$277$	0.634824	0.0381429	0.0190714	−	0.999818i	$-0.493929\pi$
$277$	0.634824	0.0381429	0.0190714	+	0.999818i	$0.493929\pi$
$278$	7.18604	0.430990
$279$	0	0
$280$	5.78285	0.345591
$281$	−27.9614	−1.66804	−0.834018	−	0.551738i	$-0.813965\pi$
$281$	−27.9614	−1.66804	−0.834018	+	0.551738i	$0.813965\pi$
$282$	0	0
$283$	−18.1799	−1.08068	−0.540340	−	0.841447i	$-0.681704\pi$
$283$	−18.1799	−1.08068	−0.540340	+	0.841447i	$0.681704\pi$
$284$	−11.0021	−0.652852
$285$	0	0
$286$	−58.5212	−3.46043
$287$	−11.4660	−0.676816
$288$	0	0
$289$	6.25456	0.367915
$290$	−2.96824	−0.174301
$291$	0	0
$292$	19.6816	1.15178
$293$	8.26939	0.483103	0.241551	−	0.970388i	$-0.422344\pi$
$293$	8.26939	0.483103	0.241551	+	0.970388i	$0.422344\pi$
$294$	0	0
$295$	0.135022	0.00786129
$296$	17.6417	1.02540
$297$	0	0
$298$	45.6811	2.64624
$299$	6.61576	0.382599
$300$	0	0
$301$	−10.8616	−0.626050
$302$	−23.8191	−1.37064
$303$	0	0
$304$	−11.3108	−0.648722
$305$	8.62700	0.493981
$306$	0	0
$307$	21.5875	1.23207	0.616033	−	0.787721i	$-0.288739\pi$
$307$	21.5875	1.23207	0.616033	+	0.787721i	$0.288739\pi$
$308$	−14.0456	−0.800322
$309$	0	0
$310$	−2.61151	−0.148324
$311$	−29.4192	−1.66821	−0.834104	−	0.551607i	$-0.814015\pi$
$311$	−29.4192	−1.66821	−0.834104	+	0.551607i	$0.814015\pi$
$312$	0	0
$313$	11.2268	0.634574	0.317287	−	0.948329i	$-0.397228\pi$
$313$	11.2268	0.634574	0.317287	+	0.948329i	$0.397228\pi$
$314$	−18.8990	−1.06653
$315$	0	0
$316$	23.0900	1.29891
$317$	−31.0877	−1.74606	−0.873030	−	0.487666i	$-0.837848\pi$
$317$	−31.0877	−1.74606	−0.873030	+	0.487666i	$0.837848\pi$
$318$	0	0
$319$	3.60961	0.202099
$320$	9.61004	0.537218
$321$	0	0
$322$	2.38067	0.132670
$323$	13.5250	0.752552
$324$	0	0
$325$	23.3730	1.29650
$326$	29.1776	1.61600
$327$	0	0
$328$	58.0062	3.20286
$329$	−1.50387	−0.0829108
$330$	0	0
$331$	19.1328	1.05164	0.525818	−	0.850597i	$-0.323759\pi$
$331$	19.1328	1.05164	0.525818	+	0.850597i	$0.323759\pi$
$332$	45.8858	2.51831
$333$	0	0
$334$	14.4047	0.788189
$335$	−7.45855	−0.407504
$336$	0	0
$337$	−19.3409	−1.05357	−0.526783	−	0.850000i	$-0.676602\pi$
$337$	−19.3409	−1.05357	−0.526783	+	0.850000i	$0.676602\pi$
$338$	−75.4009	−4.10127
$339$	0	0
$340$	23.3956	1.26880
$341$	3.17580	0.171979
$342$	0	0
$343$	−12.6837	−0.684854
$344$	54.9484	2.96262
$345$	0	0
$346$	−21.0343	−1.13081
$347$	26.3935	1.41688	0.708438	−	0.705773i	$-0.249400\pi$
$347$	26.3935	1.41688	0.708438	+	0.705773i	$0.249400\pi$
$348$	0	0
$349$	−13.3300	−0.713541	−0.356770	−	0.934192i	$-0.616122\pi$
$349$	−13.3300	−0.713541	−0.356770	+	0.934192i	$0.616122\pi$
$350$	8.41074	0.449573
$351$	0	0
$352$	0.193783	0.0103287
$353$	11.0751	0.589468	0.294734	−	0.955579i	$-0.404769\pi$
$353$	11.0751	0.589468	0.294734	+	0.955579i	$0.404769\pi$
$354$	0	0
$355$	3.32695	0.176576
$356$	−13.5997	−0.720781
$357$	0	0
$358$	−45.9543	−2.42876
$359$	−4.23971	−0.223763	−0.111882	−	0.993722i	$-0.535688\pi$
$359$	−4.23971	−0.223763	−0.111882	+	0.993722i	$0.535688\pi$
$360$	0	0
$361$	−11.1338	−0.585987
$362$	42.3252	2.22456
$363$	0	0
$364$	−25.7430	−1.34930
$365$	−5.95160	−0.311521
$366$	0	0
$367$	13.6881	0.714516	0.357258	−	0.934006i	$-0.383712\pi$
$367$	13.6881	0.714516	0.357258	+	0.934006i	$0.383712\pi$
$368$	−4.03285	−0.210227
$369$	0	0
$370$	−10.6549	−0.553923
$371$	−2.17939	−0.113148
$372$	0	0
$373$	−30.5174	−1.58013	−0.790064	−	0.613024i	$-0.789953\pi$
$373$	−30.5174	−1.58013	−0.790064	+	0.613024i	$0.789953\pi$
$374$	−42.6568	−2.20573
$375$	0	0
$376$	7.60803	0.392354
$377$	6.61576	0.340729
$378$	0	0
$379$	−18.7133	−0.961237	−0.480618	−	0.876930i	$-0.659588\pi$
$379$	−18.7133	−0.961237	−0.480618	+	0.876930i	$0.659588\pi$
$380$	13.6070	0.698025
$381$	0	0
$382$	26.5208	1.35692
$383$	−26.8095	−1.36990	−0.684951	−	0.728589i	$-0.740176\pi$
$383$	−26.8095	−1.36990	−0.684951	+	0.728589i	$0.740176\pi$
$384$	0	0
$385$	4.24730	0.216462
$386$	24.5736	1.25076
$387$	0	0
$388$	−29.9009	−1.51799
$389$	−26.1690	−1.32682	−0.663410	−	0.748256i	$-0.730892\pi$
$389$	−26.1690	−1.32682	−0.663410	+	0.748256i	$0.730892\pi$
$390$	0	0
$391$	4.82230	0.243874
$392$	29.7642	1.50332
$393$	0	0
$394$	47.2239	2.37911
$395$	−6.98226	−0.351316
$396$	0	0
$397$	19.8435	0.995914	0.497957	−	0.867202i	$-0.334084\pi$
$397$	19.8435	0.995914	0.497957	+	0.867202i	$0.334084\pi$
$398$	64.6246	3.23934
$399$	0	0
$400$	−14.2477	−0.712387
$401$	0.873341	0.0436125	0.0218063	−	0.999762i	$-0.493058\pi$
$401$	0.873341	0.0436125	0.0218063	+	0.999762i	$0.493058\pi$
$402$	0	0
$403$	5.82066	0.289948
$404$	−3.19138	−0.158777
$405$	0	0
$406$	2.38067	0.118151
$407$	12.9572	0.642265
$408$	0	0
$409$	−35.4908	−1.75491	−0.877455	−	0.479660i	$-0.840760\pi$
$409$	−35.4908	−1.75491	−0.877455	+	0.479660i	$0.840760\pi$
$410$	−35.0336	−1.73018
$411$	0	0
$412$	−1.79155	−0.0882633
$413$	−0.108294	−0.00532881
$414$	0	0
$415$	−13.8756	−0.681125
$416$	0.355169	0.0174136
$417$	0	0
$418$	−24.8095	−1.21347
$419$	25.3204	1.23698	0.618492	−	0.785791i	$-0.287744\pi$
$419$	25.3204	1.23698	0.618492	+	0.785791i	$0.287744\pi$
$420$	0	0
$421$	−2.35782	−0.114913	−0.0574566	−	0.998348i	$-0.518299\pi$
$421$	−2.35782	−0.114913	−0.0574566	+	0.998348i	$0.518299\pi$
$422$	−19.6212	−0.955143
$423$	0	0
$424$	11.0255	0.535444
$425$	17.0368	0.826407
$426$	0	0
$427$	−6.91927	−0.334847
$428$	51.2006	2.47488
$429$	0	0
$430$	−33.1868	−1.60041
$431$	−26.6339	−1.28291	−0.641455	−	0.767161i	$-0.721669\pi$
$431$	−26.6339	−1.28291	−0.641455	+	0.767161i	$0.721669\pi$
$432$	0	0
$433$	30.9755	1.48859	0.744294	−	0.667852i	$-0.232786\pi$
$433$	30.9755	1.48859	0.744294	+	0.667852i	$0.232786\pi$
$434$	2.09456	0.100542
$435$	0	0
$436$	30.4815	1.45980
$437$	2.80468	0.134166
$438$	0	0
$439$	14.0374	0.669970	0.334985	−	0.942224i	$-0.391269\pi$
$439$	14.0374	0.669970	0.334985	+	0.942224i	$0.391269\pi$
$440$	−21.4870	−1.02435
$441$	0	0
$442$	−78.1821	−3.71874
$443$	−33.4135	−1.58752	−0.793762	−	0.608229i	$-0.791880\pi$
$443$	−33.4135	−1.58752	−0.793762	+	0.608229i	$0.791880\pi$
$444$	0	0
$445$	4.11246	0.194949
$446$	7.17817	0.339896
$447$	0	0
$448$	−7.70772	−0.364155
$449$	−7.37954	−0.348262	−0.174131	−	0.984722i	$-0.555712\pi$
$449$	−7.37954	−0.348262	−0.174131	+	0.984722i	$0.555712\pi$
$450$	0	0
$451$	42.6035	2.00612
$452$	28.0901	1.32125
$453$	0	0
$454$	−36.5188	−1.71391
$455$	7.78453	0.364944
$456$	0	0
$457$	−31.2019	−1.45956	−0.729781	−	0.683681i	$-0.760378\pi$
$457$	−31.2019	−1.45956	−0.729781	+	0.683681i	$0.760378\pi$
$458$	−27.1954	−1.27076
$459$	0	0
$460$	4.85154	0.226204
$461$	−40.8683	−1.90343	−0.951714	−	0.306986i	$-0.900680\pi$
$461$	−40.8683	−1.90343	−0.951714	+	0.306986i	$0.900680\pi$
$462$	0	0
$463$	−9.18712	−0.426962	−0.213481	−	0.976947i	$-0.568480\pi$
$463$	−9.18712	−0.426962	−0.213481	+	0.976947i	$0.568480\pi$
$464$	−4.03285	−0.187220
$465$	0	0
$466$	27.4883	1.27337
$467$	15.8601	0.733917	0.366958	−	0.930237i	$-0.380399\pi$
$467$	15.8601	0.733917	0.366958	+	0.930237i	$0.380399\pi$
$468$	0	0
$469$	5.98212	0.276229
$470$	−4.59497	−0.211950
$471$	0	0
$472$	0.547859	0.0252172
$473$	40.3577	1.85565
$474$	0	0
$475$	9.90873	0.454644
$476$	−18.7644	−0.860063
$477$	0	0
$478$	45.2849	2.07128
$479$	16.4171	0.750115	0.375057	−	0.927002i	$-0.377623\pi$
$479$	16.4171	0.750115	0.375057	+	0.927002i	$0.377623\pi$
$480$	0	0
$481$	23.7482	1.08282
$482$	18.0321	0.821338
$483$	0	0
$484$	8.12825	0.369466
$485$	9.04184	0.410569
$486$	0	0
$487$	35.6972	1.61759	0.808797	−	0.588088i	$-0.200119\pi$
$487$	35.6972	1.61759	0.808797	+	0.588088i	$0.200119\pi$
$488$	35.0044	1.58458
$489$	0	0
$490$	−17.9764	−0.812093
$491$	−34.2249	−1.54455	−0.772273	−	0.635290i	$-0.780880\pi$
$491$	−34.2249	−1.54455	−0.772273	+	0.635290i	$0.780880\pi$
$492$	0	0
$493$	4.82230	0.217185
$494$	−45.4712	−2.04585
$495$	0	0
$496$	−3.54817	−0.159318
$497$	−2.66838	−0.119693
$498$	0	0
$499$	30.1196	1.34834	0.674169	−	0.738577i	$-0.264502\pi$
$499$	30.1196	1.34834	0.674169	+	0.738577i	$0.264502\pi$
$500$	41.3978	1.85137
$501$	0	0
$502$	−53.9376	−2.40735
$503$	−8.55512	−0.381454	−0.190727	−	0.981643i	$-0.561085\pi$
$503$	−8.55512	−0.381454	−0.190727	+	0.981643i	$0.561085\pi$
$504$	0	0
$505$	0.965052	0.0429442
$506$	−8.84573	−0.393241
$507$	0	0
$508$	64.4491	2.85946
$509$	6.87162	0.304579	0.152290	−	0.988336i	$-0.451335\pi$
$509$	6.87162	0.304579	0.152290	+	0.988336i	$0.451335\pi$
$510$	0	0
$511$	4.77347	0.211166
$512$	39.4233	1.74228
$513$	0	0
$514$	−32.4633	−1.43189
$515$	0.541753	0.0238725
$516$	0	0
$517$	5.58783	0.245752
$518$	8.54576	0.375479
$519$	0	0
$520$	−39.3818	−1.72700
$521$	19.3761	0.848881	0.424440	−	0.905456i	$-0.360471\pi$
$521$	19.3761	0.848881	0.424440	+	0.905456i	$0.360471\pi$
$522$	0	0
$523$	0.309818	0.0135474	0.00677370	−	0.999977i	$-0.497844\pi$
$523$	0.309818	0.0135474	0.00677370	+	0.999977i	$0.497844\pi$
$524$	−16.6592	−0.727761
$525$	0	0
$526$	27.1847	1.18531
$527$	4.24274	0.184817
$528$	0	0
$529$	1.00000	0.0434783
$530$	−6.65897	−0.289247
$531$	0	0
$532$	−10.9135	−0.473160
$533$	78.0845	3.38221
$534$	0	0
$535$	−15.4827	−0.669378
$536$	−30.2634	−1.30718
$537$	0	0
$538$	−38.6613	−1.66681
$539$	21.8607	0.941609
$540$	0	0
$541$	10.5011	0.451478	0.225739	−	0.974188i	$-0.427520\pi$
$541$	10.5011	0.451478	0.225739	+	0.974188i	$0.427520\pi$
$542$	10.5648	0.453799
$543$	0	0
$544$	0.258886	0.0110997
$545$	−9.21742	−0.394831
$546$	0	0
$547$	−35.2453	−1.50698	−0.753489	−	0.657460i	$-0.771631\pi$
$547$	−35.2453	−1.50698	−0.753489	+	0.657460i	$0.771631\pi$
$548$	−36.1371	−1.54370
$549$	0	0
$550$	−31.2513	−1.33256
$551$	2.80468	0.119483
$552$	0	0
$553$	5.60011	0.238141
$554$	−1.55570	−0.0660955
$555$	0	0
$556$	−11.7455	−0.498118
$557$	36.6005	1.55081	0.775407	−	0.631462i	$-0.217545\pi$
$557$	36.6005	1.55081	0.775407	+	0.631462i	$0.217545\pi$
$558$	0	0
$559$	73.9683	3.12852
$560$	−4.74530	−0.200526
$561$	0	0
$562$	68.5223	2.89044
$563$	−44.7796	−1.88724	−0.943618	−	0.331037i	$-0.892602\pi$
$563$	−44.7796	−1.88724	−0.943618	+	0.331037i	$0.892602\pi$
$564$	0	0
$565$	−8.49427	−0.357357
$566$	44.5517	1.87265
$567$	0	0
$568$	13.4993	0.566417
$569$	−7.11731	−0.298373	−0.149187	−	0.988809i	$-0.547665\pi$
$569$	−7.11731	−0.298373	−0.149187	+	0.988809i	$0.547665\pi$
$570$	0	0
$571$	0.486328	0.0203522	0.0101761	−	0.999948i	$-0.496761\pi$
$571$	0.486328	0.0203522	0.0101761	+	0.999948i	$0.496761\pi$
$572$	95.6519	3.99941
$573$	0	0
$574$	28.0986	1.17281
$575$	3.53293	0.147333
$576$	0	0
$577$	17.2741	0.719129	0.359565	−	0.933120i	$-0.382925\pi$
$577$	17.2741	0.719129	0.359565	+	0.933120i	$0.382925\pi$
$578$	−15.3275	−0.637538
$579$	0	0
$580$	4.85154	0.201449
$581$	11.1289	0.461703
$582$	0	0
$583$	8.09782	0.335377
$584$	−24.1489	−0.999288
$585$	0	0
$586$	−20.2650	−0.837140
$587$	32.8644	1.35646	0.678230	−	0.734850i	$-0.262747\pi$
$587$	32.8644	1.35646	0.678230	+	0.734850i	$0.262747\pi$
$588$	0	0
$589$	2.46761	0.101676
$590$	−0.330886	−0.0136224
$591$	0	0
$592$	−14.4765	−0.594979
$593$	27.4183	1.12594	0.562968	−	0.826479i	$-0.309659\pi$
$593$	27.4183	1.12594	0.562968	+	0.826479i	$0.309659\pi$
$594$	0	0
$595$	5.67422	0.232621
$596$	−74.6649	−3.05839
$597$	0	0
$598$	−16.2126	−0.662983
$599$	−19.4591	−0.795076	−0.397538	−	0.917586i	$-0.630135\pi$
$599$	−19.4591	−0.795076	−0.397538	+	0.917586i	$0.630135\pi$
$600$	0	0
$601$	−36.0765	−1.47159	−0.735796	−	0.677203i	$-0.763192\pi$
$601$	−36.0765	−1.47159	−0.735796	+	0.677203i	$0.763192\pi$
$602$	26.6174	1.08484
$603$	0	0
$604$	38.9319	1.58412
$605$	−2.45793	−0.0999291
$606$	0	0
$607$	9.87912	0.400981	0.200491	−	0.979696i	$-0.435746\pi$
$607$	9.87912	0.400981	0.200491	+	0.979696i	$0.435746\pi$
$608$	0.150570	0.00610642
$609$	0	0
$610$	−21.1414	−0.855989
$611$	10.2415	0.414326
$612$	0	0
$613$	−29.1959	−1.17921	−0.589605	−	0.807692i	$-0.700717\pi$
$613$	−29.1959	−1.17921	−0.589605	+	0.807692i	$0.700717\pi$
$614$	−52.9025	−2.13497
$615$	0	0
$616$	17.2336	0.694362
$617$	−11.1507	−0.448911	−0.224456	−	0.974484i	$-0.572060\pi$
$617$	−11.1507	−0.448911	−0.224456	+	0.974484i	$0.572060\pi$
$618$	0	0
$619$	18.2017	0.731589	0.365795	−	0.930696i	$-0.380797\pi$
$619$	18.2017	0.731589	0.365795	+	0.930696i	$0.380797\pi$
$620$	4.26847	0.171426
$621$	0	0
$622$	72.0948	2.89074
$623$	−3.29839	−0.132147
$624$	0	0
$625$	5.14619	0.205848
$626$	−27.5124	−1.09962
$627$	0	0
$628$	30.8901	1.23265
$629$	17.3103	0.690208
$630$	0	0
$631$	28.7794	1.14569	0.572844	−	0.819664i	$-0.305840\pi$
$631$	28.7794	1.14569	0.572844	+	0.819664i	$0.305840\pi$
$632$	−28.3308	−1.12694
$633$	0	0
$634$	76.1838	3.02564
$635$	−19.4890	−0.773397
$636$	0	0
$637$	40.0668	1.58750
$638$	−8.84573	−0.350206
$639$	0	0
$640$	−23.6805	−0.936053
$641$	−46.7463	−1.84637	−0.923184	−	0.384358i	$-0.874423\pi$
$641$	−46.7463	−1.84637	−0.923184	+	0.384358i	$0.874423\pi$
$642$	0	0
$643$	−16.3867	−0.646228	−0.323114	−	0.946360i	$-0.604730\pi$
$643$	−16.3867	−0.646228	−0.323114	+	0.946360i	$0.604730\pi$
$644$	−3.89117	−0.153333
$645$	0	0
$646$	−33.1445	−1.30405
$647$	−3.45161	−0.135697	−0.0678485	−	0.997696i	$-0.521613\pi$
$647$	−3.45161	−0.135697	−0.0678485	+	0.997696i	$0.521613\pi$
$648$	0	0
$649$	0.402383	0.0157949
$650$	−57.2780	−2.24663
$651$	0	0
$652$	−47.6903	−1.86770
$653$	−22.1264	−0.865874	−0.432937	−	0.901424i	$-0.642523\pi$
$653$	−22.1264	−0.865874	−0.432937	+	0.901424i	$0.642523\pi$
$654$	0	0
$655$	5.03764	0.196837
$656$	−47.5988	−1.85842
$657$	0	0
$658$	3.68538	0.143671
$659$	8.83575	0.344192	0.172096	−	0.985080i	$-0.444946\pi$
$659$	8.83575	0.344192	0.172096	+	0.985080i	$0.444946\pi$
$660$	0	0
$661$	6.69279	0.260319	0.130160	−	0.991493i	$-0.458451\pi$
$661$	6.69279	0.260319	0.130160	+	0.991493i	$0.458451\pi$
$662$	−46.8870	−1.82232
$663$	0	0
$664$	−56.3007	−2.18489
$665$	3.30017	0.127975
$666$	0	0
$667$	1.00000	0.0387202
$668$	−23.5442	−0.910952
$669$	0	0
$670$	18.2780	0.706140
$671$	25.7095	0.992505
$672$	0	0
$673$	20.8594	0.804071	0.402035	−	0.915624i	$-0.368303\pi$
$673$	20.8594	0.804071	0.402035	+	0.915624i	$0.368303\pi$
$674$	47.3969	1.82566
$675$	0	0
$676$	123.241	4.74005
$677$	20.5616	0.790248	0.395124	−	0.918628i	$-0.370702\pi$
$677$	20.5616	0.790248	0.395124	+	0.918628i	$0.370702\pi$
$678$	0	0
$679$	−7.25199	−0.278306
$680$	−28.7058	−1.10082
$681$	0	0
$682$	−7.78263	−0.298012
$683$	−33.8369	−1.29473	−0.647366	−	0.762180i	$-0.724129\pi$
$683$	−33.8369	−1.29473	−0.647366	+	0.762180i	$0.724129\pi$
$684$	0	0
$685$	10.9276	0.417523
$686$	31.0827	1.18674
$687$	0	0
$688$	−45.0897	−1.71903
$689$	14.8418	0.565428
$690$	0	0
$691$	22.3353	0.849674	0.424837	−	0.905270i	$-0.360331\pi$
$691$	22.3353	0.849674	0.424837	+	0.905270i	$0.360331\pi$
$692$	34.3801	1.30694
$693$	0	0
$694$	−64.6800	−2.45522
$695$	3.55175	0.134726
$696$	0	0
$697$	56.9166	2.15587
$698$	32.6667	1.23645
$699$	0	0
$700$	−13.7472	−0.519595
$701$	31.9852	1.20806	0.604031	−	0.796961i	$-0.293560\pi$
$701$	31.9852	1.20806	0.604031	+	0.796961i	$0.293560\pi$
$702$	0	0
$703$	10.0678	0.379714
$704$	28.6391	1.07938
$705$	0	0
$706$	−27.1407	−1.02145
$707$	−0.774018	−0.0291099
$708$	0	0
$709$	42.6363	1.60124	0.800619	−	0.599174i	$-0.204504\pi$
$709$	42.6363	1.60124	0.800619	+	0.599174i	$0.204504\pi$
$710$	−8.15305	−0.305979
$711$	0	0
$712$	16.6865	0.625352
$713$	0.879818	0.0329494
$714$	0	0
$715$	−28.9245	−1.08172
$716$	75.1115	2.80705
$717$	0	0
$718$	10.3899	0.387746
$719$	−5.03168	−0.187650	−0.0938250	−	0.995589i	$-0.529909\pi$
$719$	−5.03168	−0.187650	−0.0938250	+	0.995589i	$0.529909\pi$
$720$	0	0
$721$	−0.434512	−0.0161821
$722$	27.2845	1.01542
$723$	0	0
$724$	−69.1797	−2.57104
$725$	3.53293	0.131210
$726$	0	0
$727$	−0.687407	−0.0254945	−0.0127473	−	0.999919i	$-0.504058\pi$
$727$	−0.687407	−0.0254945	−0.0127473	+	0.999919i	$0.504058\pi$
$728$	31.5861	1.17066
$729$	0	0
$730$	14.5850	0.539816
$731$	53.9163	1.99416
$732$	0	0
$733$	29.8891	1.10398	0.551989	−	0.833851i	$-0.313869\pi$
$733$	29.8891	1.10398	0.551989	+	0.833851i	$0.313869\pi$
$734$	−33.5443	−1.23814
$735$	0	0
$736$	0.0536852	0.00197886
$737$	−22.2274	−0.818757
$738$	0	0
$739$	8.44047	0.310488	0.155244	−	0.987876i	$-0.450384\pi$
$739$	8.44047	0.310488	0.155244	+	0.987876i	$0.450384\pi$
$740$	17.4153	0.640198
$741$	0	0
$742$	5.34081	0.196067
$743$	41.2959	1.51500	0.757500	−	0.652835i	$-0.226421\pi$
$743$	41.2959	1.51500	0.757500	+	0.652835i	$0.226421\pi$
$744$	0	0
$745$	22.5782	0.827201
$746$	74.7860	2.73811
$747$	0	0
$748$	69.7217	2.54928
$749$	12.4179	0.453741
$750$	0	0
$751$	23.1167	0.843540	0.421770	−	0.906703i	$-0.361409\pi$
$751$	23.1167	0.843540	0.421770	+	0.906703i	$0.361409\pi$
$752$	−6.24301	−0.227659
$753$	0	0
$754$	−16.2126	−0.590429
$755$	−11.7728	−0.428454
$756$	0	0
$757$	−21.5153	−0.781988	−0.390994	−	0.920393i	$-0.627869\pi$
$757$	−21.5153	−0.781988	−0.390994	+	0.920393i	$0.627869\pi$
$758$	45.8589	1.66567
$759$	0	0
$760$	−16.6955	−0.605609
$761$	−8.35122	−0.302732	−0.151366	−	0.988478i	$-0.548367\pi$
$761$	−8.35122	−0.302732	−0.151366	+	0.988478i	$0.548367\pi$
$762$	0	0
$763$	7.39281	0.267638
$764$	−43.3478	−1.56827
$765$	0	0
$766$	65.6995	2.37382
$767$	0.737494	0.0266294
$768$	0	0
$769$	−48.6804	−1.75546	−0.877731	−	0.479154i	$-0.840944\pi$
$769$	−48.6804	−1.75546	−0.877731	+	0.479154i	$0.840944\pi$
$770$	−10.4085	−0.375095
$771$	0	0
$772$	−40.1651	−1.44557
$773$	−35.2809	−1.26896	−0.634482	−	0.772937i	$-0.718787\pi$
$773$	−35.2809	−1.26896	−0.634482	+	0.772937i	$0.718787\pi$
$774$	0	0
$775$	3.10833	0.111655
$776$	36.6877	1.31701
$777$	0	0
$778$	64.1299	2.29917
$779$	33.1031	1.18604
$780$	0	0
$781$	9.91474	0.354777
$782$	−11.8176	−0.422595
$783$	0	0
$784$	−24.4240	−0.872284
$785$	−9.34096	−0.333393
$786$	0	0
$787$	18.7092	0.666909	0.333455	−	0.942766i	$-0.391786\pi$
$787$	18.7092	0.666909	0.333455	+	0.942766i	$0.391786\pi$
$788$	−77.1866	−2.74966
$789$	0	0
$790$	17.1108	0.608774
$791$	6.81282	0.242236
$792$	0	0
$793$	47.1209	1.67331
$794$	−48.6285	−1.72576
$795$	0	0
$796$	−105.628	−3.74388
$797$	11.0424	0.391141	0.195571	−	0.980690i	$-0.437344\pi$
$797$	11.0424	0.391141	0.195571	+	0.980690i	$0.437344\pi$
$798$	0	0
$799$	7.46512	0.264097
$800$	0.189666	0.00670570
$801$	0	0
$802$	−2.14021	−0.0755736
$803$	−17.7365	−0.625907
$804$	0	0
$805$	1.17666	0.0414720
$806$	−14.2642	−0.502433
$807$	0	0
$808$	3.91574	0.137755
$809$	13.3665	0.469941	0.234970	−	0.972003i	$-0.424501\pi$
$809$	13.3665	0.469941	0.234970	+	0.972003i	$0.424501\pi$
$810$	0	0
$811$	20.6937	0.726654	0.363327	−	0.931662i	$-0.381641\pi$
$811$	20.6937	0.726654	0.363327	+	0.931662i	$0.381641\pi$
$812$	−3.89117	−0.136553
$813$	0	0
$814$	−31.7530	−1.11294
$815$	14.4212	0.505154
$816$	0	0
$817$	31.3581	1.09708
$818$	86.9741	3.04098
$819$	0	0
$820$	57.2617	1.99967
$821$	−37.8007	−1.31925	−0.659626	−	0.751594i	$-0.729285\pi$
$821$	−37.8007	−1.31925	−0.659626	+	0.751594i	$0.729285\pi$
$822$	0	0
$823$	−35.3953	−1.23380	−0.616902	−	0.787040i	$-0.711612\pi$
$823$	−35.3953	−1.23380	−0.616902	+	0.787040i	$0.711612\pi$
$824$	2.19819	0.0765775
$825$	0	0
$826$	0.265387	0.00923398
$827$	−50.9326	−1.77110	−0.885549	−	0.464545i	$-0.846218\pi$
$827$	−50.9326	−1.77110	−0.885549	+	0.464545i	$0.846218\pi$
$828$	0	0
$829$	15.8607	0.550865	0.275432	−	0.961320i	$-0.411179\pi$
$829$	15.8607	0.550865	0.275432	+	0.961320i	$0.411179\pi$
$830$	34.0035	1.18028
$831$	0	0
$832$	52.4903	1.81977
$833$	29.2051	1.01190
$834$	0	0
$835$	7.11961	0.246384
$836$	40.5506	1.40247
$837$	0	0
$838$	−62.0504	−2.14350
$839$	−44.4147	−1.53336	−0.766682	−	0.642027i	$-0.778094\pi$
$839$	−44.4147	−1.53336	−0.766682	+	0.642027i	$0.778094\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	5.77809	0.199126
$843$	0	0
$844$	32.0704	1.10391
$845$	−37.2674	−1.28204
$846$	0	0
$847$	1.97138	0.0677374
$848$	−9.04730	−0.310686
$849$	0	0
$850$	−41.7505	−1.43203
$851$	3.58964	0.123051
$852$	0	0
$853$	−57.4686	−1.96769	−0.983844	−	0.179027i	$-0.942705\pi$
$853$	−57.4686	−1.96769	−0.983844	+	0.179027i	$0.942705\pi$
$854$	16.9564	0.580236
$855$	0	0
$856$	−62.8220	−2.14721
$857$	−30.6152	−1.04580	−0.522898	−	0.852396i	$-0.675149\pi$
$857$	−30.6152	−1.04580	−0.522898	+	0.852396i	$0.675149\pi$
$858$	0	0
$859$	−1.13987	−0.0388920	−0.0194460	−	0.999811i	$-0.506190\pi$
$859$	−1.13987	−0.0388920	−0.0194460	+	0.999811i	$0.506190\pi$
$860$	54.2432	1.84968
$861$	0	0
$862$	65.2692	2.22308
$863$	44.0565	1.49970	0.749851	−	0.661607i	$-0.230125\pi$
$863$	44.0565	1.49970	0.749851	+	0.661607i	$0.230125\pi$
$864$	0	0
$865$	−10.3963	−0.353486
$866$	−75.9088	−2.57949
$867$	0	0
$868$	−3.42352	−0.116202
$869$	−20.8080	−0.705863
$870$	0	0
$871$	−40.7388	−1.38038
$872$	−37.4001	−1.26653
$873$	0	0
$874$	−6.87317	−0.232488
$875$	10.0404	0.339427
$876$	0	0
$877$	−17.3140	−0.584652	−0.292326	−	0.956319i	$-0.594429\pi$
$877$	−17.3140	−0.584652	−0.292326	+	0.956319i	$0.594429\pi$
$878$	−34.4002	−1.16095
$879$	0	0
$880$	17.6319	0.594370
$881$	38.0049	1.28042	0.640209	−	0.768201i	$-0.278848\pi$
$881$	38.0049	1.28042	0.640209	+	0.768201i	$0.278848\pi$
$882$	0	0
$883$	7.68227	0.258529	0.129264	−	0.991610i	$-0.458738\pi$
$883$	7.68227	0.258529	0.129264	+	0.991610i	$0.458738\pi$
$884$	127.787	4.29795
$885$	0	0
$886$	81.8833	2.75092
$887$	−45.7949	−1.53764	−0.768821	−	0.639465i	$-0.779156\pi$
$887$	−45.7949	−1.53764	−0.768821	+	0.639465i	$0.779156\pi$
$888$	0	0
$889$	15.6311	0.524250
$890$	−10.0780	−0.337816
$891$	0	0
$892$	−11.7326	−0.392836
$893$	4.34177	0.145292
$894$	0	0
$895$	−22.7132	−0.759220
$896$	18.9929	0.634508
$897$	0	0
$898$	18.0843	0.603482
$899$	0.879818	0.0293436
$900$	0	0
$901$	10.8184	0.360412
$902$	−104.404	−3.47629
$903$	0	0
$904$	−34.4659	−1.14632
$905$	20.9195	0.695388
$906$	0	0
$907$	−7.34775	−0.243978	−0.121989	−	0.992531i	$-0.538927\pi$
$907$	−7.34775	−0.243978	−0.121989	+	0.992531i	$0.538927\pi$
$908$	59.6893	1.98086
$909$	0	0
$910$	−19.0768	−0.632390
$911$	−8.12967	−0.269348	−0.134674	−	0.990890i	$-0.542999\pi$
$911$	−8.12967	−0.269348	−0.134674	+	0.990890i	$0.542999\pi$
$912$	0	0
$913$	−41.3509	−1.36851
$914$	76.4635	2.52919
$915$	0	0
$916$	44.4504	1.46868
$917$	−4.04043	−0.133427
$918$	0	0
$919$	16.4766	0.543514	0.271757	−	0.962366i	$-0.412395\pi$
$919$	16.4766	0.543514	0.271757	+	0.962366i	$0.412395\pi$
$920$	−5.95272	−0.196255
$921$	0	0
$922$	100.152	3.29834
$923$	18.1719	0.598136
$924$	0	0
$925$	12.6819	0.416979
$926$	22.5140	0.739857
$927$	0	0
$928$	0.0536852	0.00176230
$929$	−54.4912	−1.78780	−0.893898	−	0.448270i	$-0.852040\pi$
$929$	−54.4912	−1.78780	−0.893898	+	0.448270i	$0.852040\pi$
$930$	0	0
$931$	16.9859	0.556690
$932$	−44.9291	−1.47170
$933$	0	0
$934$	−38.8668	−1.27176
$935$	−21.0834	−0.689501
$936$	0	0
$937$	48.2496	1.57624	0.788122	−	0.615519i	$-0.211054\pi$
$937$	48.2496	1.57624	0.788122	+	0.615519i	$0.211054\pi$
$938$	−14.6598	−0.478660
$939$	0	0
$940$	7.51039	0.244962
$941$	−5.22563	−0.170351	−0.0851753	−	0.996366i	$-0.527145\pi$
$941$	−5.22563	−0.170351	−0.0851753	+	0.996366i	$0.527145\pi$
$942$	0	0
$943$	11.8028	0.384352
$944$	−0.449563	−0.0146320
$945$	0	0
$946$	−98.9007	−3.21554
$947$	−30.3485	−0.986192	−0.493096	−	0.869975i	$-0.664135\pi$
$947$	−30.3485	−0.986192	−0.493096	+	0.869975i	$0.664135\pi$
$948$	0	0
$949$	−32.5078	−1.05525
$950$	−24.2824	−0.787825
$951$	0	0
$952$	23.0234	0.746194
$953$	46.2230	1.49731	0.748654	−	0.662960i	$-0.230700\pi$
$953$	46.2230	1.49731	0.748654	+	0.662960i	$0.230700\pi$
$954$	0	0
$955$	13.1081	0.424168
$956$	−74.0173	−2.39389
$957$	0	0
$958$	−40.2317	−1.29983
$959$	−8.76449	−0.283020
$960$	0	0
$961$	−30.2259	−0.975030
$962$	−58.1975	−1.87636
$963$	0	0
$964$	−29.4731	−0.949263
$965$	12.1457	0.390983
$966$	0	0
$967$	8.43018	0.271096	0.135548	−	0.990771i	$-0.456720\pi$
$967$	8.43018	0.271096	0.135548	+	0.990771i	$0.456720\pi$
$968$	−9.97317	−0.320550
$969$	0	0
$970$	−22.1580	−0.711450
$971$	18.9726	0.608861	0.304430	−	0.952535i	$-0.401534\pi$
$971$	18.9726	0.608861	0.304430	+	0.952535i	$0.401534\pi$
$972$	0	0
$973$	−2.84867	−0.0913243
$974$	−87.4797	−2.80303
$975$	0	0
$976$	−28.7240	−0.919434
$977$	28.2590	0.904085	0.452042	−	0.891996i	$-0.350696\pi$
$977$	28.2590	0.904085	0.452042	+	0.891996i	$0.350696\pi$
$978$	0	0
$979$	12.2556	0.391692
$980$	29.3822	0.938579
$981$	0	0
$982$	83.8717	2.67645
$983$	20.4747	0.653043	0.326522	−	0.945190i	$-0.394123\pi$
$983$	20.4747	0.653043	0.326522	+	0.945190i	$0.394123\pi$
$984$	0	0
$985$	23.3407	0.743698
$986$	−11.8176	−0.376348
$987$	0	0
$988$	74.3219	2.36449
$989$	11.1806	0.355523
$990$	0	0
$991$	−28.5762	−0.907753	−0.453877	−	0.891065i	$-0.649959\pi$
$991$	−28.5762	−0.907753	−0.453877	+	0.891065i	$0.649959\pi$
$992$	0.0472332	0.00149966
$993$	0	0
$994$	6.53914	0.207409
$995$	31.9412	1.01260
$996$	0	0
$997$	−16.8259	−0.532881	−0.266441	−	0.963851i	$-0.585848\pi$
$997$	−16.8259	−0.532881	−0.266441	+	0.963851i	$0.585848\pi$
$998$	−73.8113	−2.33646
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.s.1.3		20
3.2	odd	2		2001.2.a.o.1.18	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.2.a.o.1.18	✓	20	3.2	odd	2
6003.2.a.s.1.3		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)

\(f(q)\)	\(=\)	\(q-2.45061 q^{2} +4.00547 q^{4} -1.21123 q^{5} +0.971463 q^{7} -4.91461 q^{8} +O(q^{10})\)	\(q-2.45061 q^{2} +4.00547 q^{4} -1.21123 q^{5} +0.971463 q^{7} -4.91461 q^{8} +2.96824 q^{10} -3.60961 q^{11} -6.61576 q^{13} -2.38067 q^{14} +4.03285 q^{16} -4.82230 q^{17} -2.80468 q^{19} -4.85154 q^{20} +8.84573 q^{22} -1.00000 q^{23} -3.53293 q^{25} +16.2126 q^{26} +3.89117 q^{28} -1.00000 q^{29} -0.879818 q^{31} -0.0536852 q^{32} +11.8176 q^{34} -1.17666 q^{35} -3.58964 q^{37} +6.87317 q^{38} +5.95272 q^{40} -11.8028 q^{41} -11.1806 q^{43} -14.4582 q^{44} +2.45061 q^{46} -1.54804 q^{47} -6.05626 q^{49} +8.65781 q^{50} -26.4992 q^{52} -2.24340 q^{53} +4.37206 q^{55} -4.77437 q^{56} +2.45061 q^{58} -0.111475 q^{59} -7.12252 q^{61} +2.15609 q^{62} -7.93413 q^{64} +8.01320 q^{65} +6.15784 q^{67} -19.3156 q^{68} +2.88354 q^{70} -2.74676 q^{71} +4.91369 q^{73} +8.79680 q^{74} -11.2341 q^{76} -3.50660 q^{77} +5.76461 q^{79} -4.88470 q^{80} +28.9240 q^{82} +11.4558 q^{83} +5.84090 q^{85} +27.3993 q^{86} +17.7398 q^{88} -3.39528 q^{89} -6.42697 q^{91} -4.00547 q^{92} +3.79364 q^{94} +3.39711 q^{95} -7.46502 q^{97} +14.8415 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 2 q^{2} + 30 q^{4} + q^{5} + 9 q^{7} - 6 q^{8}+O(q^{10}) \) 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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