LMFDB - Embedded newform 6003.2.a.t.1.15

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6003,2,Mod(1,6003)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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

Newform invariants

comment:select newform

sage:traces = [22,-3,0,17,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$47.9341963334$
Analytic rank:	$1$
Dimension:	$22$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
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)

$f(q)$	$=$	\(q+1.05077 q^{2} -0.895876 q^{4} +3.10634 q^{5} +3.07400 q^{7} -3.04291 q^{8} +3.26406 q^{10} -3.20654 q^{11} -0.0429964 q^{13} +3.23008 q^{14} -1.40566 q^{16} -4.01221 q^{17} -7.31997 q^{19} -2.78289 q^{20} -3.36935 q^{22} +1.00000 q^{23} +4.64934 q^{25} -0.0451795 q^{26} -2.75393 q^{28} +1.00000 q^{29} +0.714345 q^{31} +4.60879 q^{32} -4.21592 q^{34} +9.54890 q^{35} -6.91091 q^{37} -7.69163 q^{38} -9.45231 q^{40} -5.11513 q^{41} -11.9603 q^{43} +2.87266 q^{44} +1.05077 q^{46} -5.42059 q^{47} +2.44951 q^{49} +4.88541 q^{50} +0.0385194 q^{52} +2.84097 q^{53} -9.96061 q^{55} -9.35392 q^{56} +1.05077 q^{58} -10.8989 q^{59} -8.48818 q^{61} +0.750615 q^{62} +7.65411 q^{64} -0.133561 q^{65} +3.06291 q^{67} +3.59444 q^{68} +10.0337 q^{70} -3.65634 q^{71} +9.89771 q^{73} -7.26180 q^{74} +6.55778 q^{76} -9.85692 q^{77} +3.31014 q^{79} -4.36644 q^{80} -5.37484 q^{82} +12.8843 q^{83} -12.4633 q^{85} -12.5676 q^{86} +9.75721 q^{88} +6.29940 q^{89} -0.132171 q^{91} -0.895876 q^{92} -5.69581 q^{94} -22.7383 q^{95} -0.736499 q^{97} +2.57387 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q - 3 q^{2} + 17 q^{4} - 6 q^{7} - 6 q^{8} - 12 q^{10} - 28 q^{13} - q^{14} + 3 q^{16} - 10 q^{17} - 8 q^{19} - 11 q^{22} + 22 q^{23} + 11 q^{26} - 21 q^{28} + 22 q^{29} - 18 q^{31} + 5 q^{32} - 33 q^{34}+ \cdots - 28 q^{98}+O(q^{100}) $ 22 * q - 3 * q^2 + 17 * q^4 - 6 * q^7 - 6 * q^8 - 12 * q^10 - 28 * q^13 - q^14 + 3 * q^16 - 10 * q^17 - 8 * q^19 - 11 * q^22 + 22 * q^23 + 11 * q^26 - 21 * q^28 + 22 * q^29 - 18 * q^31 + 5 * q^32 - 33 * q^34 + 2 * q^35 - 28 * q^37 + 14 * q^38 - 30 * q^40 - 10 * q^41 - 14 * q^43 + 37 * q^44 - 3 * q^46 - 18 * q^47 + 2 * q^49 + 7 * q^50 - 57 * q^52 + 20 * q^53 - 42 * q^55 - 2 * q^56 - 3 * q^58 - 20 * q^59 - 38 * q^61 + 4 * q^62 - 24 * q^64 + 12 * q^65 - 50 * q^67 + 11 * q^68 - 48 * q^70 + 12 * q^71 - 46 * q^73 - 6 * q^74 - 16 * q^76 - 14 * q^77 - 20 * q^79 - 58 * q^80 - 42 * q^82 + 22 * q^83 - 66 * q^85 + 22 * q^86 - 68 * q^88 - 14 * q^89 - 16 * q^91 + 17 * q^92 - 27 * q^94 - 20 * q^95 - 48 * q^97 - 28 * q^98

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 $

$2$	1.05077	0.743009	0.371504	−	0.928431i	$-0.378842\pi$
$2$	1.05077	0.743009	0.371504	+	0.928431i	$0.378842\pi$
$3$	0	0
$3$	0	0
$4$	−0.895876	−0.447938
$5$	3.10634	1.38920	0.694599	−	0.719398i	$-0.255582\pi$
$5$	3.10634	1.38920	0.694599	+	0.719398i	$0.255582\pi$
$6$	0	0
$7$	3.07400	1.16186	0.580932	−	0.813952i	$-0.302688\pi$
$7$	3.07400	1.16186	0.580932	+	0.813952i	$0.302688\pi$
$8$	−3.04291	−1.07583
$9$	0	0
$10$	3.26406	1.03219
$11$	−3.20654	−0.966809	−0.483404	−	0.875397i	$-0.660600\pi$
$11$	−3.20654	−0.966809	−0.483404	+	0.875397i	$0.660600\pi$
$12$	0	0
$13$	−0.0429964	−0.0119251	−0.00596253	−	0.999982i	$-0.501898\pi$
$13$	−0.0429964	−0.0119251	−0.00596253	+	0.999982i	$0.501898\pi$
$14$	3.23008	0.863276
$15$	0	0
$16$	−1.40566	−0.351414
$17$	−4.01221	−0.973104	−0.486552	−	0.873652i	$-0.661746\pi$
$17$	−4.01221	−0.973104	−0.486552	+	0.873652i	$0.661746\pi$
$18$	0	0
$19$	−7.31997	−1.67932	−0.839658	−	0.543116i	$-0.817244\pi$
$19$	−7.31997	−1.67932	−0.839658	+	0.543116i	$0.817244\pi$
$20$	−2.78289	−0.622274
$21$	0	0
$22$	−3.36935	−0.718347
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	4.64934	0.929869
$26$	−0.0451795	−0.00886042
$27$	0	0
$28$	−2.75393	−0.520443
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	0.714345	0.128300	0.0641501	−	0.997940i	$-0.479566\pi$
$31$	0.714345	0.128300	0.0641501	+	0.997940i	$0.479566\pi$
$32$	4.60879	0.814727
$33$	0	0
$34$	−4.21592	−0.723025
$35$	9.54890	1.61406
$36$	0	0
$37$	−6.91091	−1.13615	−0.568074	−	0.822978i	$-0.692311\pi$
$37$	−6.91091	−1.13615	−0.568074	+	0.822978i	$0.692311\pi$
$38$	−7.69163	−1.24775
$39$	0	0
$40$	−9.45231	−1.49454
$41$	−5.11513	−0.798848	−0.399424	−	0.916766i	$-0.630790\pi$
$41$	−5.11513	−0.798848	−0.399424	+	0.916766i	$0.630790\pi$
$42$	0	0
$43$	−11.9603	−1.82393	−0.911967	−	0.410263i	$-0.865437\pi$
$43$	−11.9603	−1.82393	−0.911967	+	0.410263i	$0.865437\pi$
$44$	2.87266	0.433070
$45$	0	0
$46$	1.05077	0.154928
$47$	−5.42059	−0.790675	−0.395337	−	0.918536i	$-0.629372\pi$
$47$	−5.42059	−0.790675	−0.395337	+	0.918536i	$0.629372\pi$
$48$	0	0
$49$	2.44951	0.349929
$50$	4.88541	0.690901
$51$	0	0
$52$	0.0385194	0.00534168
$53$	2.84097	0.390237	0.195119	−	0.980780i	$-0.437491\pi$
$53$	2.84097	0.390237	0.195119	+	0.980780i	$0.437491\pi$
$54$	0	0
$55$	−9.96061	−1.34309
$56$	−9.35392	−1.24997
$57$	0	0
$58$	1.05077	0.137973
$59$	−10.8989	−1.41892	−0.709458	−	0.704748i	$-0.751060\pi$
$59$	−10.8989	−1.41892	−0.709458	+	0.704748i	$0.751060\pi$
$60$	0	0
$61$	−8.48818	−1.08680	−0.543400	−	0.839474i	$-0.682863\pi$
$61$	−8.48818	−1.08680	−0.543400	+	0.839474i	$0.682863\pi$
$62$	0.750615	0.0953281
$63$	0	0
$64$	7.65411	0.956763
$65$	−0.133561	−0.0165662
$66$	0	0
$67$	3.06291	0.374194	0.187097	−	0.982341i	$-0.440092\pi$
$67$	3.06291	0.374194	0.187097	+	0.982341i	$0.440092\pi$
$68$	3.59444	0.435890
$69$	0	0
$70$	10.0337	1.19926
$71$	−3.65634	−0.433927	−0.216964	−	0.976180i	$-0.569615\pi$
$71$	−3.65634	−0.433927	−0.216964	+	0.976180i	$0.569615\pi$
$72$	0	0
$73$	9.89771	1.15844	0.579220	−	0.815172i	$-0.303357\pi$
$73$	9.89771	1.15844	0.579220	+	0.815172i	$0.303357\pi$
$74$	−7.26180	−0.844168
$75$	0	0
$76$	6.55778	0.752229
$77$	−9.85692	−1.12330
$78$	0	0
$79$	3.31014	0.372419	0.186210	−	0.982510i	$-0.440380\pi$
$79$	3.31014	0.372419	0.186210	+	0.982510i	$0.440380\pi$
$80$	−4.36644	−0.488183
$81$	0	0
$82$	−5.37484	−0.593551
$83$	12.8843	1.41423	0.707115	−	0.707098i	$-0.249996\pi$
$83$	12.8843	1.41423	0.707115	+	0.707098i	$0.249996\pi$
$84$	0	0
$85$	−12.4633	−1.35183
$86$	−12.5676	−1.35520
$87$	0	0
$88$	9.75721	1.04012
$89$	6.29940	0.667735	0.333867	−	0.942620i	$-0.391646\pi$
$89$	6.29940	0.667735	0.333867	+	0.942620i	$0.391646\pi$
$90$	0	0
$91$	−0.132171	−0.0138553
$92$	−0.895876	−0.0934015
$93$	0	0
$94$	−5.69581	−0.587478
$95$	−22.7383	−2.33290
$96$	0	0
$97$	−0.736499	−0.0747801	−0.0373901	−	0.999301i	$-0.511904\pi$
$97$	−0.736499	−0.0747801	−0.0373901	+	0.999301i	$0.511904\pi$
$98$	2.57387	0.260001
$99$	0	0
$100$	−4.16523	−0.416523
$101$	−7.08667	−0.705150	−0.352575	−	0.935784i	$-0.614694\pi$
$101$	−7.08667	−0.705150	−0.352575	+	0.935784i	$0.614694\pi$
$102$	0	0
$103$	10.0369	0.988963	0.494481	−	0.869188i	$-0.335358\pi$
$103$	10.0369	0.988963	0.494481	+	0.869188i	$0.335358\pi$
$104$	0.130834	0.0128293
$105$	0	0
$106$	2.98522	0.289950
$107$	19.2040	1.85652	0.928261	−	0.371929i	$-0.121304\pi$
$107$	19.2040	1.85652	0.928261	+	0.371929i	$0.121304\pi$
$108$	0	0
$109$	−13.5333	−1.29625	−0.648127	−	0.761533i	$-0.724447\pi$
$109$	−13.5333	−1.29625	−0.648127	+	0.761533i	$0.724447\pi$
$110$	−10.4663	−0.997926
$111$	0	0
$112$	−4.32099	−0.408295
$113$	−15.6808	−1.47512	−0.737562	−	0.675280i	$-0.764023\pi$
$113$	−15.6808	−1.47512	−0.737562	+	0.675280i	$0.764023\pi$
$114$	0	0
$115$	3.10634	0.289668
$116$	−0.895876	−0.0831800
$117$	0	0
$118$	−11.4523	−1.05427
$119$	−12.3336	−1.13062
$120$	0	0
$121$	−0.718091	−0.0652810
$122$	−8.91915	−0.807502
$123$	0	0
$124$	−0.639964	−0.0574705
$125$	−1.08926	−0.0974263
$126$	0	0
$127$	6.23438	0.553212	0.276606	−	0.960983i	$-0.410790\pi$
$127$	6.23438	0.553212	0.276606	+	0.960983i	$0.410790\pi$
$128$	−1.17486	−0.103843
$129$	0	0
$130$	−0.140343	−0.0123089
$131$	−14.7195	−1.28605	−0.643025	−	0.765845i	$-0.722321\pi$
$131$	−14.7195	−1.28605	−0.643025	+	0.765845i	$0.722321\pi$
$132$	0	0
$133$	−22.5016	−1.95114
$134$	3.21842	0.278029
$135$	0	0
$136$	12.2088	1.04690
$137$	14.3504	1.22604	0.613021	−	0.790067i	$-0.289954\pi$
$137$	14.3504	1.22604	0.613021	+	0.790067i	$0.289954\pi$
$138$	0	0
$139$	−6.86612	−0.582377	−0.291188	−	0.956666i	$-0.594051\pi$
$139$	−6.86612	−0.582377	−0.291188	+	0.956666i	$0.594051\pi$
$140$	−8.55463	−0.722998
$141$	0	0
$142$	−3.84198	−0.322412
$143$	0.137870	0.0115292
$144$	0	0
$145$	3.10634	0.257967
$146$	10.4002	0.860731
$147$	0	0
$148$	6.19132	0.508923
$149$	2.26630	0.185662	0.0928312	−	0.995682i	$-0.470408\pi$
$149$	2.26630	0.185662	0.0928312	+	0.995682i	$0.470408\pi$
$150$	0	0
$151$	17.1474	1.39543	0.697716	−	0.716375i	$-0.254200\pi$
$151$	17.1474	1.39543	0.697716	+	0.716375i	$0.254200\pi$
$152$	22.2740	1.80666
$153$	0	0
$154$	−10.3574	−0.834622
$155$	2.21900	0.178234
$156$	0	0
$157$	−18.3834	−1.46715	−0.733576	−	0.679608i	$-0.762150\pi$
$157$	−18.3834	−1.46715	−0.733576	+	0.679608i	$0.762150\pi$
$158$	3.47820	0.276711
$159$	0	0
$160$	14.3165	1.13182
$161$	3.07400	0.242266
$162$	0	0
$163$	12.5756	0.984995	0.492497	−	0.870314i	$-0.336084\pi$
$163$	12.5756	0.984995	0.492497	+	0.870314i	$0.336084\pi$
$164$	4.58252	0.357834
$165$	0	0
$166$	13.5384	1.05079
$167$	5.05267	0.390988	0.195494	−	0.980705i	$-0.437369\pi$
$167$	5.05267	0.390988	0.195494	+	0.980705i	$0.437369\pi$
$168$	0	0
$169$	−12.9982	−0.999858
$170$	−13.0961	−1.00442
$171$	0	0
$172$	10.7150	0.817009
$173$	−9.98395	−0.759066	−0.379533	−	0.925178i	$-0.623915\pi$
$173$	−9.98395	−0.759066	−0.379533	+	0.925178i	$0.623915\pi$
$174$	0	0
$175$	14.2921	1.08038
$176$	4.50729	0.339750
$177$	0	0
$178$	6.61924	0.496133
$179$	7.70479	0.575883	0.287942	−	0.957648i	$-0.407029\pi$
$179$	7.70479	0.575883	0.287942	+	0.957648i	$0.407029\pi$
$180$	0	0
$181$	−15.2801	−1.13576	−0.567881	−	0.823111i	$-0.692236\pi$
$181$	−15.2801	−1.13576	−0.567881	+	0.823111i	$0.692236\pi$
$182$	−0.138882	−0.0102946
$183$	0	0
$184$	−3.04291	−0.224326
$185$	−21.4676	−1.57833
$186$	0	0
$187$	12.8653	0.940806
$188$	4.85618	0.354173
$189$	0	0
$190$	−23.8928	−1.73337
$191$	8.19768	0.593163	0.296582	−	0.955007i	$-0.404153\pi$
$191$	8.19768	0.593163	0.296582	+	0.955007i	$0.404153\pi$
$192$	0	0
$193$	14.6875	1.05723	0.528615	−	0.848862i	$-0.322712\pi$
$193$	14.6875	1.05723	0.528615	+	0.848862i	$0.322712\pi$
$194$	−0.773893	−0.0555623
$195$	0	0
$196$	−2.19445	−0.156747
$197$	−6.90137	−0.491702	−0.245851	−	0.969308i	$-0.579067\pi$
$197$	−6.90137	−0.491702	−0.245851	+	0.969308i	$0.579067\pi$
$198$	0	0
$199$	14.1987	1.00652	0.503259	−	0.864136i	$-0.332134\pi$
$199$	14.1987	1.00652	0.503259	+	0.864136i	$0.332134\pi$
$200$	−14.1475	−1.00038
$201$	0	0
$202$	−7.44648	−0.523932
$203$	3.07400	0.215753
$204$	0	0
$205$	−15.8893	−1.10976
$206$	10.5465	0.734808
$207$	0	0
$208$	0.0604381	0.00419063
$209$	23.4718	1.62358
$210$	0	0
$211$	12.9150	0.889102	0.444551	−	0.895754i	$-0.353363\pi$
$211$	12.9150	0.889102	0.444551	+	0.895754i	$0.353363\pi$
$212$	−2.54516	−0.174802
$213$	0	0
$214$	20.1791	1.37941
$215$	−37.1529	−2.53380
$216$	0	0
$217$	2.19590	0.149067
$218$	−14.2204	−0.963128
$219$	0	0
$220$	8.92346	0.601620
$221$	0.172511	0.0116043
$222$	0	0
$223$	−6.99684	−0.468543	−0.234271	−	0.972171i	$-0.575270\pi$
$223$	−6.99684	−0.468543	−0.234271	+	0.972171i	$0.575270\pi$
$224$	14.1674	0.946603
$225$	0	0
$226$	−16.4769	−1.09603
$227$	20.8089	1.38114	0.690569	−	0.723267i	$-0.257360\pi$
$227$	20.8089	1.38114	0.690569	+	0.723267i	$0.257360\pi$
$228$	0	0
$229$	8.58224	0.567131	0.283565	−	0.958953i	$-0.408483\pi$
$229$	8.58224	0.567131	0.283565	+	0.958953i	$0.408483\pi$
$230$	3.26406	0.215226
$231$	0	0
$232$	−3.04291	−0.199777
$233$	20.1645	1.32102	0.660511	−	0.750817i	$-0.270340\pi$
$233$	20.1645	1.32102	0.660511	+	0.750817i	$0.270340\pi$
$234$	0	0
$235$	−16.8382	−1.09840
$236$	9.76406	0.635586
$237$	0	0
$238$	−12.9598	−0.840057
$239$	19.3093	1.24901	0.624507	−	0.781019i	$-0.285300\pi$
$239$	19.3093	1.24901	0.624507	+	0.781019i	$0.285300\pi$
$240$	0	0
$241$	−4.32842	−0.278818	−0.139409	−	0.990235i	$-0.544520\pi$
$241$	−4.32842	−0.278818	−0.139409	+	0.990235i	$0.544520\pi$
$242$	−0.754551	−0.0485044
$243$	0	0
$244$	7.60435	0.486819
$245$	7.60900	0.486121
$246$	0	0
$247$	0.314732	0.0200259
$248$	−2.17369	−0.138029
$249$	0	0
$250$	−1.14456	−0.0723886
$251$	7.41999	0.468345	0.234173	−	0.972195i	$-0.424762\pi$
$251$	7.41999	0.468345	0.234173	+	0.972195i	$0.424762\pi$
$252$	0	0
$253$	−3.20654	−0.201594
$254$	6.55092	0.411041
$255$	0	0
$256$	−16.5427	−1.03392
$257$	−24.9240	−1.55472	−0.777359	−	0.629057i	$-0.783441\pi$
$257$	−24.9240	−1.55472	−0.777359	+	0.629057i	$0.783441\pi$
$258$	0	0
$259$	−21.2442	−1.32005
$260$	0.119654	0.00742065
$261$	0	0
$262$	−15.4669	−0.955547
$263$	−8.09672	−0.499265	−0.249633	−	0.968341i	$-0.580310\pi$
$263$	−8.09672	−0.499265	−0.249633	+	0.968341i	$0.580310\pi$
$264$	0	0
$265$	8.82502	0.542117
$266$	−23.6441	−1.44971
$267$	0	0
$268$	−2.74398	−0.167615
$269$	10.3743	0.632533	0.316267	−	0.948670i	$-0.397571\pi$
$269$	10.3743	0.632533	0.316267	+	0.948670i	$0.397571\pi$
$270$	0	0
$271$	−31.9369	−1.94003	−0.970014	−	0.243050i	$-0.921852\pi$
$271$	−31.9369	−1.94003	−0.970014	+	0.243050i	$0.921852\pi$
$272$	5.63979	0.341962
$273$	0	0
$274$	15.0791	0.910960
$275$	−14.9083	−0.899005
$276$	0	0
$277$	26.8274	1.61190	0.805951	−	0.591982i	$-0.201655\pi$
$277$	26.8274	1.61190	0.805951	+	0.591982i	$0.201655\pi$
$278$	−7.21473	−0.432711
$279$	0	0
$280$	−29.0564	−1.73645
$281$	−1.09966	−0.0656000	−0.0328000	−	0.999462i	$-0.510442\pi$
$281$	−1.09966	−0.0656000	−0.0328000	+	0.999462i	$0.510442\pi$
$282$	0	0
$283$	−16.7079	−0.993182	−0.496591	−	0.867985i	$-0.665415\pi$
$283$	−16.7079	−0.993182	−0.496591	+	0.867985i	$0.665415\pi$
$284$	3.27562	0.194372
$285$	0	0
$286$	0.144870	0.00856633
$287$	−15.7239	−0.928154
$288$	0	0
$289$	−0.902157	−0.0530681
$290$	3.26406	0.191672
$291$	0	0
$292$	−8.86712	−0.518909
$293$	2.31072	0.134994	0.0674970	−	0.997719i	$-0.478499\pi$
$293$	2.31072	0.134994	0.0674970	+	0.997719i	$0.478499\pi$
$294$	0	0
$295$	−33.8557	−1.97115
$296$	21.0293	1.22230
$297$	0	0
$298$	2.38137	0.137949
$299$	−0.0429964	−0.00248655
$300$	0	0
$301$	−36.7661	−2.11916
$302$	18.0180	1.03682
$303$	0	0
$304$	10.2894	0.590135
$305$	−26.3672	−1.50978
$306$	0	0
$307$	−22.2607	−1.27049	−0.635243	−	0.772312i	$-0.719100\pi$
$307$	−22.2607	−1.27049	−0.635243	+	0.772312i	$0.719100\pi$
$308$	8.83058	0.503169
$309$	0	0
$310$	2.33166	0.132430
$311$	16.0897	0.912361	0.456181	−	0.889887i	$-0.349217\pi$
$311$	16.0897	0.912361	0.456181	+	0.889887i	$0.349217\pi$
$312$	0	0
$313$	9.19668	0.519827	0.259914	−	0.965632i	$-0.416306\pi$
$313$	9.19668	0.519827	0.259914	+	0.965632i	$0.416306\pi$
$314$	−19.3167	−1.09011
$315$	0	0
$316$	−2.96547	−0.166821
$317$	2.73116	0.153397	0.0766986	−	0.997054i	$-0.475562\pi$
$317$	2.73116	0.153397	0.0766986	+	0.997054i	$0.475562\pi$
$318$	0	0
$319$	−3.20654	−0.179532
$320$	23.7762	1.32913
$321$	0	0
$322$	3.23008	0.180005
$323$	29.3693	1.63415
$324$	0	0
$325$	−0.199905	−0.0110887
$326$	13.2141	0.731860
$327$	0	0
$328$	15.5649	0.859426
$329$	−16.6629	−0.918657
$330$	0	0
$331$	−24.6627	−1.35559	−0.677794	−	0.735252i	$-0.737064\pi$
$331$	−24.6627	−1.35559	−0.677794	+	0.735252i	$0.737064\pi$
$332$	−11.5427	−0.633487
$333$	0	0
$334$	5.30921	0.290507
$335$	9.51443	0.519829
$336$	0	0
$337$	−5.08897	−0.277214	−0.138607	−	0.990347i	$-0.544262\pi$
$337$	−5.08897	−0.277214	−0.138607	+	0.990347i	$0.544262\pi$
$338$	−13.6581	−0.742903
$339$	0	0
$340$	11.1656	0.605537
$341$	−2.29058	−0.124042
$342$	0	0
$343$	−13.9882	−0.755294
$344$	36.3942	1.96224
$345$	0	0
$346$	−10.4909	−0.563992
$347$	−26.6514	−1.43072	−0.715360	−	0.698756i	$-0.753737\pi$
$347$	−26.6514	−1.43072	−0.715360	+	0.698756i	$0.753737\pi$
$348$	0	0
$349$	−33.2848	−1.78170	−0.890848	−	0.454301i	$-0.849889\pi$
$349$	−33.2848	−1.78170	−0.890848	+	0.454301i	$0.849889\pi$
$350$	15.0178	0.802733
$351$	0	0
$352$	−14.7783	−0.787685
$353$	−28.5065	−1.51725	−0.758623	−	0.651530i	$-0.774127\pi$
$353$	−28.5065	−1.51725	−0.758623	+	0.651530i	$0.774127\pi$
$354$	0	0
$355$	−11.3578	−0.602811
$356$	−5.64348	−0.299104
$357$	0	0
$358$	8.09599	0.427886
$359$	12.3505	0.651832	0.325916	−	0.945399i	$-0.394327\pi$
$359$	12.3505	0.651832	0.325916	+	0.945399i	$0.394327\pi$
$360$	0	0
$361$	34.5819	1.82010
$362$	−16.0559	−0.843881
$363$	0	0
$364$	0.118409	0.00620631
$365$	30.7456	1.60930
$366$	0	0
$367$	−19.7884	−1.03294	−0.516472	−	0.856304i	$-0.672755\pi$
$367$	−19.7884	−1.03294	−0.516472	+	0.856304i	$0.672755\pi$
$368$	−1.40566	−0.0732748
$369$	0	0
$370$	−22.5576	−1.17272
$371$	8.73316	0.453403
$372$	0	0
$373$	0.00491154	0.000254310 0	0.000127155	−	1.00000i	$-0.499960\pi$
$373$	0.00491154	0.000254310 0	0.000127155		1.00000i	$0.499960\pi$
$374$	13.5185	0.699027
$375$	0	0
$376$	16.4944	0.850632
$377$	−0.0429964	−0.00221443
$378$	0	0
$379$	−29.3054	−1.50532	−0.752658	−	0.658411i	$-0.771229\pi$
$379$	−29.3054	−1.50532	−0.752658	+	0.658411i	$0.771229\pi$
$380$	20.3707	1.04499
$381$	0	0
$382$	8.61390	0.440726
$383$	2.27253	0.116121	0.0580604	−	0.998313i	$-0.481508\pi$
$383$	2.27253	0.116121	0.0580604	+	0.998313i	$0.481508\pi$
$384$	0	0
$385$	−30.6189	−1.56049
$386$	15.4332	0.785531
$387$	0	0
$388$	0.659811	0.0334968
$389$	29.8019	1.51101	0.755507	−	0.655140i	$-0.227391\pi$
$389$	29.8019	1.51101	0.755507	+	0.655140i	$0.227391\pi$
$390$	0	0
$391$	−4.01221	−0.202906
$392$	−7.45362	−0.376465
$393$	0	0
$394$	−7.25178	−0.365339
$395$	10.2824	0.517364
$396$	0	0
$397$	−13.7498	−0.690081	−0.345040	−	0.938588i	$-0.612135\pi$
$397$	−13.7498	−0.690081	−0.345040	+	0.938588i	$0.612135\pi$
$398$	14.9196	0.747852
$399$	0	0
$400$	−6.53537	−0.326769
$401$	11.4510	0.571836	0.285918	−	0.958254i	$-0.407701\pi$
$401$	11.4510	0.571836	0.285918	+	0.958254i	$0.407701\pi$
$402$	0	0
$403$	−0.0307143	−0.00152999
$404$	6.34877	0.315863
$405$	0	0
$406$	3.23008	0.160306
$407$	22.1601	1.09844
$408$	0	0
$409$	−6.99053	−0.345660	−0.172830	−	0.984952i	$-0.555291\pi$
$409$	−6.99053	−0.345660	−0.172830	+	0.984952i	$0.555291\pi$
$410$	−16.6961	−0.824560
$411$	0	0
$412$	−8.99179	−0.442994
$413$	−33.5033	−1.64859
$414$	0	0
$415$	40.0229	1.96464
$416$	−0.198161	−0.00971566
$417$	0	0
$418$	24.6635	1.20633
$419$	−18.8803	−0.922364	−0.461182	−	0.887305i	$-0.652575\pi$
$419$	−18.8803	−0.922364	−0.461182	+	0.887305i	$0.652575\pi$
$420$	0	0
$421$	5.75470	0.280467	0.140233	−	0.990118i	$-0.455215\pi$
$421$	5.75470	0.280467	0.140233	+	0.990118i	$0.455215\pi$
$422$	13.5707	0.660611
$423$	0	0
$424$	−8.64482	−0.419829
$425$	−18.6541	−0.904859
$426$	0	0
$427$	−26.0927	−1.26271
$428$	−17.2044	−0.831607
$429$	0	0
$430$	−39.0392	−1.88264
$431$	−19.6675	−0.947350	−0.473675	−	0.880700i	$-0.657073\pi$
$431$	−19.6675	−0.947350	−0.473675	+	0.880700i	$0.657073\pi$
$432$	0	0
$433$	−25.5240	−1.22660	−0.613302	−	0.789849i	$-0.710159\pi$
$433$	−25.5240	−1.22660	−0.613302	+	0.789849i	$0.710159\pi$
$434$	2.30739	0.110758
$435$	0	0
$436$	12.1241	0.580641
$437$	−7.31997	−0.350162
$438$	0	0
$439$	−8.99635	−0.429372	−0.214686	−	0.976683i	$-0.568873\pi$
$439$	−8.99635	−0.429372	−0.214686	+	0.976683i	$0.568873\pi$
$440$	30.3092	1.44493
$441$	0	0
$442$	0.181270	0.00862211
$443$	1.63232	0.0775541	0.0387770	−	0.999248i	$-0.487654\pi$
$443$	1.63232	0.0775541	0.0387770	+	0.999248i	$0.487654\pi$
$444$	0	0
$445$	19.5681	0.927615
$446$	−7.35209	−0.348131
$447$	0	0
$448$	23.5288	1.11163
$449$	36.0363	1.70066	0.850329	−	0.526251i	$-0.176403\pi$
$449$	36.0363	1.70066	0.850329	+	0.526251i	$0.176403\pi$
$450$	0	0
$451$	16.4019	0.772334
$452$	14.0480	0.660764
$453$	0	0
$454$	21.8655	1.02620
$455$	−0.410568	−0.0192477
$456$	0	0
$457$	−4.27483	−0.199968	−0.0999841	−	0.994989i	$-0.531879\pi$
$457$	−4.27483	−0.199968	−0.0999841	+	0.994989i	$0.531879\pi$
$458$	9.01799	0.421383
$459$	0	0
$460$	−2.78289	−0.129753
$461$	−23.0578	−1.07391	−0.536954	−	0.843612i	$-0.680425\pi$
$461$	−23.0578	−1.07391	−0.536954	+	0.843612i	$0.680425\pi$
$462$	0	0
$463$	34.6421	1.60996	0.804978	−	0.593305i	$-0.202177\pi$
$463$	34.6421	1.60996	0.804978	+	0.593305i	$0.202177\pi$
$464$	−1.40566	−0.0652559
$465$	0	0
$466$	21.1883	0.981531
$467$	6.60876	0.305817	0.152908	−	0.988240i	$-0.451136\pi$
$467$	6.60876	0.305817	0.152908	+	0.988240i	$0.451136\pi$
$468$	0	0
$469$	9.41539	0.434762
$470$	−17.6931	−0.816123
$471$	0	0
$472$	33.1644	1.52651
$473$	38.3513	1.76340
$474$	0	0
$475$	−34.0330	−1.56154
$476$	11.0493	0.506445
$477$	0	0
$478$	20.2897	0.928029
$479$	14.1027	0.644368	0.322184	−	0.946677i	$-0.395583\pi$
$479$	14.1027	0.644368	0.322184	+	0.946677i	$0.395583\pi$
$480$	0	0
$481$	0.297144	0.0135486
$482$	−4.54819	−0.207164
$483$	0	0
$484$	0.643321	0.0292418
$485$	−2.28781	−0.103884
$486$	0	0
$487$	−16.1971	−0.733962	−0.366981	−	0.930228i	$-0.619609\pi$
$487$	−16.1971	−0.733962	−0.366981	+	0.930228i	$0.619609\pi$
$488$	25.8287	1.16921
$489$	0	0
$490$	7.99533	0.361192
$491$	−35.1704	−1.58722	−0.793609	−	0.608428i	$-0.791800\pi$
$491$	−35.1704	−1.58722	−0.793609	+	0.608428i	$0.791800\pi$
$492$	0	0
$493$	−4.01221	−0.180701
$494$	0.330712	0.0148794
$495$	0	0
$496$	−1.00412	−0.0450864
$497$	−11.2396	−0.504165
$498$	0	0
$499$	−29.0912	−1.30230	−0.651150	−	0.758949i	$-0.725713\pi$
$499$	−29.0912	−1.30230	−0.651150	+	0.758949i	$0.725713\pi$
$500$	0.975841	0.0436409
$501$	0	0
$502$	7.79672	0.347985
$503$	42.0281	1.87394	0.936970	−	0.349411i	$-0.113618\pi$
$503$	42.0281	1.87394	0.936970	+	0.349411i	$0.113618\pi$
$504$	0	0
$505$	−22.0136	−0.979592
$506$	−3.36935	−0.149786
$507$	0	0
$508$	−5.58523	−0.247804
$509$	−1.26068	−0.0558787	−0.0279394	−	0.999610i	$-0.508895\pi$
$509$	−1.26068	−0.0558787	−0.0279394	+	0.999610i	$0.508895\pi$
$510$	0	0
$511$	30.4256	1.34595
$512$	−15.0329	−0.664368
$513$	0	0
$514$	−26.1895	−1.15517
$515$	31.1779	1.37386
$516$	0	0
$517$	17.3814	0.764431
$518$	−22.3228	−0.980808
$519$	0	0
$520$	0.406415	0.0178225
$521$	−27.7996	−1.21792	−0.608961	−	0.793200i	$-0.708413\pi$
$521$	−27.7996	−1.21792	−0.608961	+	0.793200i	$0.708413\pi$
$522$	0	0
$523$	34.2875	1.49929	0.749644	−	0.661842i	$-0.230225\pi$
$523$	34.2875	1.49929	0.749644	+	0.661842i	$0.230225\pi$
$524$	13.1869	0.576070
$525$	0	0
$526$	−8.50782	−0.370958
$527$	−2.86610	−0.124849
$528$	0	0
$529$	1.00000	0.0434783
$530$	9.27310	0.402798
$531$	0	0
$532$	20.1586	0.873988
$533$	0.219932	0.00952631
$534$	0	0
$535$	59.6542	2.57908
$536$	−9.32015	−0.402569
$537$	0	0
$538$	10.9011	0.469978
$539$	−7.85444	−0.338315
$540$	0	0
$541$	39.2929	1.68933	0.844666	−	0.535293i	$-0.179799\pi$
$541$	39.2929	1.68933	0.844666	+	0.535293i	$0.179799\pi$
$542$	−33.5584	−1.44146
$543$	0	0
$544$	−18.4914	−0.792814
$545$	−42.0390	−1.80075
$546$	0	0
$547$	−25.1700	−1.07619	−0.538095	−	0.842884i	$-0.680856\pi$
$547$	−25.1700	−1.07619	−0.538095	+	0.842884i	$0.680856\pi$
$548$	−12.8562	−0.549190
$549$	0	0
$550$	−15.6653	−0.667969
$551$	−7.31997	−0.311841
$552$	0	0
$553$	10.1754	0.432701
$554$	28.1895	1.19766
$555$	0	0
$556$	6.15119	0.260869
$557$	−18.8437	−0.798433	−0.399217	−	0.916857i	$-0.630718\pi$
$557$	−18.8437	−0.798433	−0.399217	+	0.916857i	$0.630718\pi$
$558$	0	0
$559$	0.514251	0.0217505
$560$	−13.4225	−0.567203
$561$	0	0
$562$	−1.15549	−0.0487414
$563$	18.4647	0.778196	0.389098	−	0.921196i	$-0.372787\pi$
$563$	18.4647	0.778196	0.389098	+	0.921196i	$0.372787\pi$
$564$	0	0
$565$	−48.7098	−2.04924
$566$	−17.5562	−0.737943
$567$	0	0
$568$	11.1259	0.466832
$569$	−24.5751	−1.03024	−0.515121	−	0.857117i	$-0.672253\pi$
$569$	−24.5751	−1.03024	−0.515121	+	0.857117i	$0.672253\pi$
$570$	0	0
$571$	32.1662	1.34612	0.673058	−	0.739590i	$-0.264981\pi$
$571$	32.1662	1.34612	0.673058	+	0.739590i	$0.264981\pi$
$572$	−0.123514	−0.00516438
$573$	0	0
$574$	−16.5223	−0.689626
$575$	4.64934	0.193891
$576$	0	0
$577$	3.94261	0.164133	0.0820664	−	0.996627i	$-0.473848\pi$
$577$	3.94261	0.164133	0.0820664	+	0.996627i	$0.473848\pi$
$578$	−0.947962	−0.0394300
$579$	0	0
$580$	−2.78289	−0.115553
$581$	39.6063	1.64314
$582$	0	0
$583$	−9.10969	−0.377285
$584$	−30.1178	−1.24628
$585$	0	0
$586$	2.42805	0.100302
$587$	−31.9270	−1.31777	−0.658885	−	0.752244i	$-0.728971\pi$
$587$	−31.9270	−1.31777	−0.658885	+	0.752244i	$0.728971\pi$
$588$	0	0
$589$	−5.22898	−0.215456
$590$	−35.5746	−1.46458
$591$	0	0
$592$	9.71436	0.399258
$593$	9.21942	0.378596	0.189298	−	0.981920i	$-0.439379\pi$
$593$	9.21942	0.378596	0.189298	+	0.981920i	$0.439379\pi$
$594$	0	0
$595$	−38.3122	−1.57065
$596$	−2.03032	−0.0831652
$597$	0	0
$598$	−0.0451795	−0.00184753
$599$	26.7668	1.09366	0.546831	−	0.837243i	$-0.315834\pi$
$599$	26.7668	1.09366	0.546831	+	0.837243i	$0.315834\pi$
$600$	0	0
$601$	−22.6788	−0.925088	−0.462544	−	0.886596i	$-0.653063\pi$
$601$	−22.6788	−0.925088	−0.462544	+	0.886596i	$0.653063\pi$
$602$	−38.6329	−1.57456
$603$	0	0
$604$	−15.3619	−0.625067
$605$	−2.23064	−0.0906882
$606$	0	0
$607$	−29.7375	−1.20701	−0.603503	−	0.797360i	$-0.706229\pi$
$607$	−29.7375	−1.20701	−0.603503	+	0.797360i	$0.706229\pi$
$608$	−33.7362	−1.36818
$609$	0	0
$610$	−27.7059	−1.12178
$611$	0.233066	0.00942884
$612$	0	0
$613$	−10.4654	−0.422695	−0.211347	−	0.977411i	$-0.567785\pi$
$613$	−10.4654	−0.422695	−0.211347	+	0.977411i	$0.567785\pi$
$614$	−23.3910	−0.943982
$615$	0	0
$616$	29.9937	1.20848
$617$	−1.84782	−0.0743905	−0.0371953	−	0.999308i	$-0.511842\pi$
$617$	−1.84782	−0.0743905	−0.0371953	+	0.999308i	$0.511842\pi$
$618$	0	0
$619$	4.67510	0.187908	0.0939540	−	0.995577i	$-0.470049\pi$
$619$	4.67510	0.187908	0.0939540	+	0.995577i	$0.470049\pi$
$620$	−1.98795	−0.0798378
$621$	0	0
$622$	16.9066	0.677892
$623$	19.3644	0.775817
$624$	0	0
$625$	−26.6303	−1.06521
$626$	9.66363	0.386236
$627$	0	0
$628$	16.4692	0.657193
$629$	27.7281	1.10559
$630$	0	0
$631$	−42.2191	−1.68072	−0.840358	−	0.542031i	$-0.817655\pi$
$631$	−42.2191	−1.68072	−0.840358	+	0.542031i	$0.817655\pi$
$632$	−10.0724	−0.400660
$633$	0	0
$634$	2.86983	0.113975
$635$	19.3661	0.768520
$636$	0	0
$637$	−0.105320	−0.00417293
$638$	−3.36935	−0.133394
$639$	0	0
$640$	−3.64950	−0.144259
$641$	−29.1294	−1.15054	−0.575271	−	0.817963i	$-0.695103\pi$
$641$	−29.1294	−1.15054	−0.575271	+	0.817963i	$0.695103\pi$
$642$	0	0
$643$	13.8177	0.544916	0.272458	−	0.962168i	$-0.412163\pi$
$643$	13.8177	0.544916	0.272458	+	0.962168i	$0.412163\pi$
$644$	−2.75393	−0.108520
$645$	0	0
$646$	30.8604	1.21419
$647$	42.2780	1.66212	0.831059	−	0.556184i	$-0.187735\pi$
$647$	42.2780	1.66212	0.831059	+	0.556184i	$0.187735\pi$
$648$	0	0
$649$	34.9478	1.37182
$650$	−0.210055	−0.00823903
$651$	0	0
$652$	−11.2661	−0.441216
$653$	29.2866	1.14607	0.573037	−	0.819529i	$-0.305765\pi$
$653$	29.2866	1.14607	0.573037	+	0.819529i	$0.305765\pi$
$654$	0	0
$655$	−45.7238	−1.78658
$656$	7.19010	0.280726
$657$	0	0
$658$	−17.5090	−0.682570
$659$	−6.15767	−0.239869	−0.119934	−	0.992782i	$-0.538268\pi$
$659$	−6.15767	−0.239869	−0.119934	+	0.992782i	$0.538268\pi$
$660$	0	0
$661$	−20.9869	−0.816295	−0.408148	−	0.912916i	$-0.633825\pi$
$661$	−20.9869	−0.816295	−0.408148	+	0.912916i	$0.633825\pi$
$662$	−25.9150	−1.00721
$663$	0	0
$664$	−39.2056	−1.52147
$665$	−69.8976	−2.71051
$666$	0	0
$667$	1.00000	0.0387202
$668$	−4.52657	−0.175138
$669$	0	0
$670$	9.99751	0.386237
$671$	27.2177	1.05073
$672$	0	0
$673$	18.9550	0.730660	0.365330	−	0.930878i	$-0.380956\pi$
$673$	18.9550	0.730660	0.365330	+	0.930878i	$0.380956\pi$
$674$	−5.34736	−0.205972
$675$	0	0
$676$	11.6447	0.447874
$677$	−2.63505	−0.101273	−0.0506366	−	0.998717i	$-0.516125\pi$
$677$	−2.63505	−0.101273	−0.0506366	+	0.998717i	$0.516125\pi$
$678$	0	0
$679$	−2.26400	−0.0868844
$680$	37.9247	1.45434
$681$	0	0
$682$	−2.40688	−0.0921641
$683$	18.5862	0.711180	0.355590	−	0.934642i	$-0.384280\pi$
$683$	18.5862	0.711180	0.355590	+	0.934642i	$0.384280\pi$
$684$	0	0
$685$	44.5774	1.70321
$686$	−14.6985	−0.561190
$687$	0	0
$688$	16.8121	0.640956
$689$	−0.122152	−0.00465360
$690$	0	0
$691$	43.6567	1.66078	0.830389	−	0.557184i	$-0.188118\pi$
$691$	43.6567	1.66078	0.830389	+	0.557184i	$0.188118\pi$
$692$	8.94438	0.340014
$693$	0	0
$694$	−28.0045	−1.06304
$695$	−21.3285	−0.809036
$696$	0	0
$697$	20.5230	0.777363
$698$	−34.9748	−1.32382
$699$	0	0
$700$	−12.8039	−0.483944
$701$	−22.0312	−0.832107	−0.416054	−	0.909340i	$-0.636587\pi$
$701$	−22.0312	−0.832107	−0.416054	+	0.909340i	$0.636587\pi$
$702$	0	0
$703$	50.5877	1.90795
$704$	−24.5432	−0.925007
$705$	0	0
$706$	−29.9538	−1.12733
$707$	−21.7844	−0.819288
$708$	0	0
$709$	−33.9056	−1.27335	−0.636676	−	0.771131i	$-0.719691\pi$
$709$	−33.9056	−1.27335	−0.636676	+	0.771131i	$0.719691\pi$
$710$	−11.9345	−0.447894
$711$	0	0
$712$	−19.1685	−0.718369
$713$	0.714345	0.0267524
$714$	0	0
$715$	0.428270	0.0160164
$716$	−6.90253	−0.257960
$717$	0	0
$718$	12.9775	0.484317
$719$	−10.7651	−0.401472	−0.200736	−	0.979645i	$-0.564333\pi$
$719$	−10.7651	−0.401472	−0.200736	+	0.979645i	$0.564333\pi$
$720$	0	0
$721$	30.8534	1.14904
$722$	36.3378	1.35235
$723$	0	0
$724$	13.6891	0.508750
$725$	4.64934	0.172672
$726$	0	0
$727$	48.9953	1.81714	0.908568	−	0.417737i	$-0.137177\pi$
$727$	48.9953	1.81714	0.908568	+	0.417737i	$0.137177\pi$
$728$	0.402185	0.0149060
$729$	0	0
$730$	32.3067	1.19572
$731$	47.9874	1.77488
$732$	0	0
$733$	30.4364	1.12420	0.562098	−	0.827071i	$-0.309995\pi$
$733$	30.4364	1.12420	0.562098	+	0.827071i	$0.309995\pi$
$734$	−20.7931	−0.767487
$735$	0	0
$736$	4.60879	0.169882
$737$	−9.82134	−0.361774
$738$	0	0
$739$	2.83114	0.104145	0.0520727	−	0.998643i	$-0.483417\pi$
$739$	2.83114	0.104145	0.0520727	+	0.998643i	$0.483417\pi$
$740$	19.2323	0.706995
$741$	0	0
$742$	9.17657	0.336883
$743$	18.5983	0.682307	0.341153	−	0.940008i	$-0.389182\pi$
$743$	18.5983	0.682307	0.341153	+	0.940008i	$0.389182\pi$
$744$	0	0
$745$	7.03989	0.257922
$746$	0.00516091	0.000188954 0
$747$	0	0
$748$	−11.5257	−0.421422
$749$	59.0332	2.15703
$750$	0	0
$751$	−24.7942	−0.904752	−0.452376	−	0.891827i	$-0.649424\pi$
$751$	−24.7942	−0.904752	−0.452376	+	0.891827i	$0.649424\pi$
$752$	7.61948	0.277854
$753$	0	0
$754$	−0.0451795	−0.00164534
$755$	53.2655	1.93853
$756$	0	0
$757$	−2.63450	−0.0957527	−0.0478763	−	0.998853i	$-0.515245\pi$
$757$	−2.63450	−0.0957527	−0.0478763	+	0.998853i	$0.515245\pi$
$758$	−30.7933	−1.11846
$759$	0	0
$760$	69.1906	2.50981
$761$	22.7357	0.824168	0.412084	−	0.911146i	$-0.364801\pi$
$761$	22.7357	0.824168	0.412084	+	0.911146i	$0.364801\pi$
$762$	0	0
$763$	−41.6014	−1.50607
$764$	−7.34410	−0.265700
$765$	0	0
$766$	2.38791	0.0862788
$767$	0.468613	0.0169206
$768$	0	0
$769$	0.477039	0.0172025	0.00860123	−	0.999963i	$-0.497262\pi$
$769$	0.477039	0.0172025	0.00860123	+	0.999963i	$0.497262\pi$
$770$	−32.1736	−1.15946
$771$	0	0
$772$	−13.1582	−0.473573
$773$	4.06587	0.146239	0.0731197	−	0.997323i	$-0.476704\pi$
$773$	4.06587	0.146239	0.0731197	+	0.997323i	$0.476704\pi$
$774$	0	0
$775$	3.32123	0.119302
$776$	2.24110	0.0804507
$777$	0	0
$778$	31.3150	1.12270
$779$	37.4426	1.34152
$780$	0	0
$781$	11.7242	0.419525
$782$	−4.21592	−0.150761
$783$	0	0
$784$	−3.44316	−0.122970
$785$	−57.1049	−2.03816
$786$	0	0
$787$	−22.2517	−0.793188	−0.396594	−	0.917994i	$-0.629808\pi$
$787$	−22.2517	−0.793188	−0.396594	+	0.917994i	$0.629808\pi$
$788$	6.18277	0.220252
$789$	0	0
$790$	10.8045	0.384406
$791$	−48.2028	−1.71389
$792$	0	0
$793$	0.364961	0.0129601
$794$	−14.4479	−0.512736
$795$	0	0
$796$	−12.7203	−0.450858
$797$	55.3496	1.96058	0.980291	−	0.197559i	$-0.0633015\pi$
$797$	55.3496	1.96058	0.980291	+	0.197559i	$0.0633015\pi$
$798$	0	0
$799$	21.7486	0.769409
$800$	21.4279	0.757589
$801$	0	0
$802$	12.0324	0.424879
$803$	−31.7374	−1.11999
$804$	0	0
$805$	9.54890	0.336555
$806$	−0.0322737	−0.00113679
$807$	0	0
$808$	21.5641	0.758621
$809$	10.4303	0.366711	0.183356	−	0.983047i	$-0.441304\pi$
$809$	10.4303	0.366711	0.183356	+	0.983047i	$0.441304\pi$
$810$	0	0
$811$	−28.7760	−1.01046	−0.505232	−	0.862984i	$-0.668593\pi$
$811$	−28.7760	−1.01046	−0.505232	+	0.862984i	$0.668593\pi$
$812$	−2.75393	−0.0966439
$813$	0	0
$814$	23.2853	0.816148
$815$	39.0640	1.36835
$816$	0	0
$817$	87.5493	3.06296
$818$	−7.34546	−0.256828
$819$	0	0
$820$	14.2349	0.497103
$821$	19.8123	0.691453	0.345727	−	0.938335i	$-0.387632\pi$
$821$	19.8123	0.691453	0.345727	+	0.938335i	$0.387632\pi$
$822$	0	0
$823$	−18.5275	−0.645828	−0.322914	−	0.946428i	$-0.604662\pi$
$823$	−18.5275	−0.645828	−0.322914	+	0.946428i	$0.604662\pi$
$824$	−30.5413	−1.06396
$825$	0	0
$826$	−35.2043	−1.22492
$827$	6.68508	0.232463	0.116232	−	0.993222i	$-0.462919\pi$
$827$	6.68508	0.232463	0.116232	+	0.993222i	$0.462919\pi$
$828$	0	0
$829$	32.6286	1.13324	0.566620	−	0.823980i	$-0.308251\pi$
$829$	32.6286	1.13324	0.566620	+	0.823980i	$0.308251\pi$
$830$	42.0549	1.45975
$831$	0	0
$832$	−0.329099	−0.0114095
$833$	−9.82794	−0.340518
$834$	0	0
$835$	15.6953	0.543159
$836$	−21.0278	−0.727262
$837$	0	0
$838$	−19.8389	−0.685325
$839$	−14.7983	−0.510894	−0.255447	−	0.966823i	$-0.582223\pi$
$839$	−14.7983	−0.510894	−0.255447	+	0.966823i	$0.582223\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	6.04688	0.208389
$843$	0	0
$844$	−11.5702	−0.398262
$845$	−40.3767	−1.38900
$846$	0	0
$847$	−2.20742	−0.0758477
$848$	−3.99343	−0.137135
$849$	0	0
$850$	−19.6013	−0.672318
$851$	−6.91091	−0.236903
$852$	0	0
$853$	16.3532	0.559922	0.279961	−	0.960011i	$-0.409679\pi$
$853$	16.3532	0.559922	0.279961	+	0.960011i	$0.409679\pi$
$854$	−27.4175	−0.938208
$855$	0	0
$856$	−58.4361	−1.99730
$857$	47.5474	1.62419	0.812094	−	0.583527i	$-0.198328\pi$
$857$	47.5474	1.62419	0.812094	+	0.583527i	$0.198328\pi$
$858$	0	0
$859$	5.87247	0.200366	0.100183	−	0.994969i	$-0.468057\pi$
$859$	5.87247	0.200366	0.100183	+	0.994969i	$0.468057\pi$
$860$	33.2843	1.13499
$861$	0	0
$862$	−20.6661	−0.703889
$863$	30.9619	1.05396	0.526978	−	0.849879i	$-0.323325\pi$
$863$	30.9619	1.05396	0.526978	+	0.849879i	$0.323325\pi$
$864$	0	0
$865$	−31.0135	−1.05449
$866$	−26.8199	−0.911377
$867$	0	0
$868$	−1.96725	−0.0667729
$869$	−10.6141	−0.360058
$870$	0	0
$871$	−0.131694	−0.00446228
$872$	41.1805	1.39455
$873$	0	0
$874$	−7.69163	−0.260173
$875$	−3.34839	−0.113196
$876$	0	0
$877$	−17.6154	−0.594830	−0.297415	−	0.954748i	$-0.596124\pi$
$877$	−17.6154	−0.594830	−0.297415	+	0.954748i	$0.596124\pi$
$878$	−9.45312	−0.319027
$879$	0	0
$880$	14.0012	0.471980
$881$	−4.21972	−0.142166	−0.0710831	−	0.997470i	$-0.522646\pi$
$881$	−4.21972	−0.142166	−0.0710831	+	0.997470i	$0.522646\pi$
$882$	0	0
$883$	6.73658	0.226704	0.113352	−	0.993555i	$-0.463841\pi$
$883$	6.73658	0.226704	0.113352	+	0.993555i	$0.463841\pi$
$884$	−0.154548	−0.00519801
$885$	0	0
$886$	1.71520	0.0576234
$887$	56.2580	1.88896	0.944479	−	0.328571i	$-0.106567\pi$
$887$	56.2580	1.88896	0.944479	+	0.328571i	$0.106567\pi$
$888$	0	0
$889$	19.1645	0.642757
$890$	20.5616	0.689226
$891$	0	0
$892$	6.26829	0.209878
$893$	39.6786	1.32779
$894$	0	0
$895$	23.9337	0.800015
$896$	−3.61151	−0.120652
$897$	0	0
$898$	37.8660	1.26360
$899$	0.714345	0.0238247
$900$	0	0
$901$	−11.3986	−0.379742
$902$	17.2346	0.573851
$903$	0	0
$904$	47.7152	1.58698
$905$	−47.4652	−1.57780
$906$	0	0
$907$	32.6879	1.08538	0.542691	−	0.839932i	$-0.317405\pi$
$907$	32.6879	1.08538	0.542691	+	0.839932i	$0.317405\pi$
$908$	−18.6422	−0.618664
$909$	0	0
$910$	−0.431414	−0.0143012
$911$	−27.6041	−0.914565	−0.457283	−	0.889321i	$-0.651177\pi$
$911$	−27.6041	−0.914565	−0.457283	+	0.889321i	$0.651177\pi$
$912$	0	0
$913$	−41.3139	−1.36729
$914$	−4.49188	−0.148578
$915$	0	0
$916$	−7.68862	−0.254039
$917$	−45.2479	−1.49422
$918$	0	0
$919$	13.4419	0.443409	0.221704	−	0.975114i	$-0.428838\pi$
$919$	13.4419	0.443409	0.221704	+	0.975114i	$0.428838\pi$
$920$	−9.45231	−0.311633
$921$	0	0
$922$	−24.2285	−0.797923
$923$	0.157209	0.00517461
$924$	0	0
$925$	−32.1312	−1.05647
$926$	36.4010	1.19621
$927$	0	0
$928$	4.60879	0.151291
$929$	29.7370	0.975640	0.487820	−	0.872944i	$-0.337792\pi$
$929$	29.7370	0.975640	0.487820	+	0.872944i	$0.337792\pi$
$930$	0	0
$931$	−17.9303	−0.587642
$932$	−18.0649	−0.591735
$933$	0	0
$934$	6.94431	0.227225
$935$	39.9641	1.30696
$936$	0	0
$937$	−1.43663	−0.0469325	−0.0234663	−	0.999725i	$-0.507470\pi$
$937$	−1.43663	−0.0469325	−0.0234663	+	0.999725i	$0.507470\pi$
$938$	9.89344	0.323032
$939$	0	0
$940$	15.0849	0.492016
$941$	6.73354	0.219507	0.109754	−	0.993959i	$-0.464994\pi$
$941$	6.73354	0.219507	0.109754	+	0.993959i	$0.464994\pi$
$942$	0	0
$943$	−5.11513	−0.166571
$944$	15.3201	0.498627
$945$	0	0
$946$	40.2985	1.31022
$947$	−14.0154	−0.455440	−0.227720	−	0.973727i	$-0.573127\pi$
$947$	−14.0154	−0.455440	−0.227720	+	0.973727i	$0.573127\pi$
$948$	0	0
$949$	−0.425566	−0.0138144
$950$	−35.7610	−1.16024
$951$	0	0
$952$	37.5299	1.21635
$953$	−12.8306	−0.415624	−0.207812	−	0.978169i	$-0.566634\pi$
$953$	−12.8306	−0.415624	−0.207812	+	0.978169i	$0.566634\pi$
$954$	0	0
$955$	25.4648	0.824021
$956$	−17.2987	−0.559481
$957$	0	0
$958$	14.8187	0.478771
$959$	44.1134	1.42449
$960$	0	0
$961$	−30.4897	−0.983539
$962$	0.312231	0.0100667
$963$	0	0
$964$	3.87773	0.124893
$965$	45.6244	1.46870
$966$	0	0
$967$	30.6661	0.986156	0.493078	−	0.869985i	$-0.335872\pi$
$967$	30.6661	0.986156	0.493078	+	0.869985i	$0.335872\pi$
$968$	2.18509	0.0702313
$969$	0	0
$970$	−2.40397	−0.0771870
$971$	−12.6876	−0.407166	−0.203583	−	0.979058i	$-0.565259\pi$
$971$	−12.6876	−0.407166	−0.203583	+	0.979058i	$0.565259\pi$
$972$	0	0
$973$	−21.1065	−0.676643
$974$	−17.0195	−0.545340
$975$	0	0
$976$	11.9315	0.381916
$977$	−34.7063	−1.11035	−0.555177	−	0.831732i	$-0.687350\pi$
$977$	−34.7063	−1.11035	−0.555177	+	0.831732i	$0.687350\pi$
$978$	0	0
$979$	−20.1993	−0.645572
$980$	−6.81671	−0.217752
$981$	0	0
$982$	−36.9561	−1.17932
$983$	59.0415	1.88313	0.941566	−	0.336828i	$-0.109354\pi$
$983$	59.0415	1.88313	0.941566	+	0.336828i	$0.109354\pi$
$984$	0	0
$985$	−21.4380	−0.683072
$986$	−4.21592	−0.134262
$987$	0	0
$988$	−0.281961	−0.00897037
$989$	−11.9603	−0.380317
$990$	0	0
$991$	−48.1877	−1.53073	−0.765367	−	0.643594i	$-0.777442\pi$
$991$	−48.1877	−1.53073	−0.765367	+	0.643594i	$0.777442\pi$
$992$	3.29227	0.104530
$993$	0	0
$994$	−11.8103	−0.374599
$995$	44.1059	1.39825
$996$	0	0
$997$	38.0886	1.20628	0.603139	−	0.797636i	$-0.293916\pi$
$997$	38.0886	1.20628	0.603139	+	0.797636i	$0.293916\pi$
$998$	−30.5682	−0.967620
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.t.1.15	✓	22
3.2	odd	2		6003.2.a.u.1.8	yes	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6003.2.a.t.1.15	✓	22	1.1	even	1	trivial
6003.2.a.u.1.8	yes	22	3.2	odd	2

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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+1.05077 q^{2} -0.895876 q^{4} +3.10634 q^{5} +3.07400 q^{7} -3.04291 q^{8} +3.26406 q^{10} -3.20654 q^{11} -0.0429964 q^{13} +3.23008 q^{14} -1.40566 q^{16} -4.01221 q^{17} -7.31997 q^{19} -2.78289 q^{20} -3.36935 q^{22} +1.00000 q^{23} +4.64934 q^{25} -0.0451795 q^{26} -2.75393 q^{28} +1.00000 q^{29} +0.714345 q^{31} +4.60879 q^{32} -4.21592 q^{34} +9.54890 q^{35} -6.91091 q^{37} -7.69163 q^{38} -9.45231 q^{40} -5.11513 q^{41} -11.9603 q^{43} +2.87266 q^{44} +1.05077 q^{46} -5.42059 q^{47} +2.44951 q^{49} +4.88541 q^{50} +0.0385194 q^{52} +2.84097 q^{53} -9.96061 q^{55} -9.35392 q^{56} +1.05077 q^{58} -10.8989 q^{59} -8.48818 q^{61} +0.750615 q^{62} +7.65411 q^{64} -0.133561 q^{65} +3.06291 q^{67} +3.59444 q^{68} +10.0337 q^{70} -3.65634 q^{71} +9.89771 q^{73} -7.26180 q^{74} +6.55778 q^{76} -9.85692 q^{77} +3.31014 q^{79} -4.36644 q^{80} -5.37484 q^{82} +12.8843 q^{83} -12.4633 q^{85} -12.5676 q^{86} +9.75721 q^{88} +6.29940 q^{89} -0.132171 q^{91} -0.895876 q^{92} -5.69581 q^{94} -22.7383 q^{95} -0.736499 q^{97} +2.57387 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q - 3 q^{2} + 17 q^{4} - 6 q^{7} - 6 q^{8} - 12 q^{10} - 28 q^{13} - q^{14} + 3 q^{16} - 10 q^{17} - 8 q^{19} - 11 q^{22} + 22 q^{23} + 11 q^{26} - 21 q^{28} + 22 q^{29} - 18 q^{31} + 5 q^{32} - 33 q^{34}+ \cdots - 28 q^{98}+O(q^{100}) \) 22 * q - 3 * q^2 + 17 * q^4 - 6 * q^7 - 6 * q^8 - 12 * q^10 - 28 * q^13 - q^14 + 3 * q^16 - 10 * q^17 - 8 * q^19 - 11 * q^22 + 22 * q^23 + 11 * q^26 - 21 * q^28 + 22 * q^29 - 18 * q^31 + 5 * q^32 - 33 * q^34 + 2 * q^35 - 28 * q^37 + 14 * q^38 - 30 * q^40 - 10 * q^41 - 14 * q^43 + 37 * q^44 - 3 * q^46 - 18 * q^47 + 2 * q^49 + 7 * q^50 - 57 * q^52 + 20 * q^53 - 42 * q^55 - 2 * q^56 - 3 * q^58 - 20 * q^59 - 38 * q^61 + 4 * q^62 - 24 * q^64 + 12 * q^65 - 50 * q^67 + 11 * q^68 - 48 * q^70 + 12 * q^71 - 46 * q^73 - 6 * q^74 - 16 * q^76 - 14 * q^77 - 20 * q^79 - 58 * q^80 - 42 * q^82 + 22 * q^83 - 66 * q^85 + 22 * q^86 - 68 * q^88 - 14 * q^89 - 16 * q^91 + 17 * q^92 - 27 * q^94 - 20 * q^95 - 48 * q^97 - 28 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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