LMFDB - Embedded newform 6003.2.a.v.1.23

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:	$30$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.23
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)

$f(q)$	$=$	$q+1.59580 q^{2} +0.546582 q^{4} -1.23722 q^{5} +2.46504 q^{7} -2.31937 q^{8} +O(q^{10})$	\(q+1.59580 q^{2} +0.546582 q^{4} -1.23722 q^{5} +2.46504 q^{7} -2.31937 q^{8} -1.97436 q^{10} -4.04090 q^{11} +6.08460 q^{13} +3.93372 q^{14} -4.79441 q^{16} +5.58185 q^{17} -0.469978 q^{19} -0.676244 q^{20} -6.44848 q^{22} +1.00000 q^{23} -3.46928 q^{25} +9.70981 q^{26} +1.34735 q^{28} -1.00000 q^{29} -2.00112 q^{31} -3.01220 q^{32} +8.90752 q^{34} -3.04981 q^{35} +5.90581 q^{37} -0.749992 q^{38} +2.86957 q^{40} -8.01822 q^{41} +6.87624 q^{43} -2.20869 q^{44} +1.59580 q^{46} +8.37297 q^{47} -0.923569 q^{49} -5.53628 q^{50} +3.32573 q^{52} -2.29752 q^{53} +4.99950 q^{55} -5.71733 q^{56} -1.59580 q^{58} -6.71576 q^{59} +8.00054 q^{61} -3.19338 q^{62} +4.78196 q^{64} -7.52800 q^{65} +6.11908 q^{67} +3.05094 q^{68} -4.86688 q^{70} -0.713918 q^{71} +13.9116 q^{73} +9.42450 q^{74} -0.256882 q^{76} -9.96100 q^{77} +2.01641 q^{79} +5.93176 q^{80} -12.7955 q^{82} +7.66901 q^{83} -6.90599 q^{85} +10.9731 q^{86} +9.37233 q^{88} -0.733342 q^{89} +14.9988 q^{91} +0.546582 q^{92} +13.3616 q^{94} +0.581468 q^{95} -12.7405 q^{97} -1.47383 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q - q^{2} + 37 q^{4} + 10 q^{7} - 6 q^{8}+O(q^{10}) $ 30 * q - q^2 + 37 * q^4 + 10 * q^7 - 6 * q^8	$ 30 q - q^{2} + 37 q^{4} + 10 q^{7} - 6 q^{8} + 8 q^{10} + 36 q^{13} - 7 q^{14} + 47 q^{16} - 18 q^{17} + 16 q^{19} + 25 q^{22} + 30 q^{23} + 56 q^{25} - 11 q^{26} + 27 q^{28} - 30 q^{29} + 14 q^{31} + 7 q^{32} + 3 q^{34} + 22 q^{35} + 40 q^{37} - 6 q^{38} + 30 q^{40} - 14 q^{41} + 34 q^{43} - 5 q^{44} - q^{46} + 2 q^{47} + 74 q^{49} + 21 q^{50} + 71 q^{52} - 16 q^{53} + 22 q^{55} - 14 q^{56} + q^{58} + 32 q^{59} + 46 q^{61} - 20 q^{62} + 68 q^{64} - 12 q^{65} + 14 q^{67} - 27 q^{68} + 32 q^{71} + 50 q^{73} + 26 q^{74} + 56 q^{76} - 34 q^{77} + 16 q^{79} - 2 q^{80} + 38 q^{82} + 14 q^{83} + 38 q^{85} - 10 q^{86} + 40 q^{88} + 2 q^{89} + 32 q^{91} + 37 q^{92} + 29 q^{94} + 28 q^{95} + 56 q^{97} - 8 q^{98}+O(q^{100}) $ 30 * q - q^2 + 37 * q^4 + 10 * q^7 - 6 * q^8 + 8 * q^10 + 36 * q^13 - 7 * q^14 + 47 * q^16 - 18 * q^17 + 16 * q^19 + 25 * q^22 + 30 * q^23 + 56 * q^25 - 11 * q^26 + 27 * q^28 - 30 * q^29 + 14 * q^31 + 7 * q^32 + 3 * q^34 + 22 * q^35 + 40 * q^37 - 6 * q^38 + 30 * q^40 - 14 * q^41 + 34 * q^43 - 5 * q^44 - q^46 + 2 * q^47 + 74 * q^49 + 21 * q^50 + 71 * q^52 - 16 * q^53 + 22 * q^55 - 14 * q^56 + q^58 + 32 * q^59 + 46 * q^61 - 20 * q^62 + 68 * q^64 - 12 * q^65 + 14 * q^67 - 27 * q^68 + 32 * q^71 + 50 * q^73 + 26 * q^74 + 56 * q^76 - 34 * q^77 + 16 * q^79 - 2 * q^80 + 38 * q^82 + 14 * q^83 + 38 * q^85 - 10 * q^86 + 40 * q^88 + 2 * q^89 + 32 * q^91 + 37 * q^92 + 29 * q^94 + 28 * q^95 + 56 * q^97 - 8 * 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 $

$2$	1.59580	1.12840	0.564201	−	0.825637i	$-0.309184\pi$
$2$	1.59580	1.12840	0.564201	+	0.825637i	$0.309184\pi$
$3$	0	0
$3$	0	0
$4$	0.546582	0.273291
$5$	−1.23722	−0.553303	−0.276651	−	0.960970i	$-0.589225\pi$
$5$	−1.23722	−0.553303	−0.276651	+	0.960970i	$0.589225\pi$
$6$	0	0
$7$	2.46504	0.931698	0.465849	−	0.884864i	$-0.345749\pi$
$7$	2.46504	0.931698	0.465849	+	0.884864i	$0.345749\pi$
$8$	−2.31937	−0.820020
$9$	0	0
$10$	−1.97436	−0.624348
$11$	−4.04090	−1.21838	−0.609189	−	0.793025i	$-0.708505\pi$
$11$	−4.04090	−1.21838	−0.609189	+	0.793025i	$0.708505\pi$
$12$	0	0
$13$	6.08460	1.68756	0.843782	−	0.536686i	$-0.180324\pi$
$13$	6.08460	1.68756	0.843782	+	0.536686i	$0.180324\pi$
$14$	3.93372	1.05133
$15$	0	0
$16$	−4.79441	−1.19860
$17$	5.58185	1.35380	0.676899	−	0.736076i	$-0.263324\pi$
$17$	5.58185	1.35380	0.676899	+	0.736076i	$0.263324\pi$
$18$	0	0
$19$	−0.469978	−0.107820	−0.0539102	−	0.998546i	$-0.517168\pi$
$19$	−0.469978	−0.107820	−0.0539102	+	0.998546i	$0.517168\pi$
$20$	−0.676244	−0.151213
$21$	0	0
$22$	−6.44848	−1.37482
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−3.46928	−0.693856
$26$	9.70981	1.90425
$27$	0	0
$28$	1.34735	0.254625
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−2.00112	−0.359411	−0.179706	−	0.983720i	$-0.557514\pi$
$31$	−2.00112	−0.359411	−0.179706	+	0.983720i	$0.557514\pi$
$32$	−3.01220	−0.532486
$33$	0	0
$34$	8.90752	1.52763
$35$	−3.04981	−0.515511
$36$	0	0
$37$	5.90581	0.970909	0.485455	−	0.874262i	$-0.338654\pi$
$37$	5.90581	0.970909	0.485455	+	0.874262i	$0.338654\pi$
$38$	−0.749992	−0.121665
$39$	0	0
$40$	2.86957	0.453719
$41$	−8.01822	−1.25224	−0.626118	−	0.779728i	$-0.715357\pi$
$41$	−8.01822	−1.25224	−0.626118	+	0.779728i	$0.715357\pi$
$42$	0	0
$43$	6.87624	1.04862	0.524308	−	0.851529i	$-0.324324\pi$
$43$	6.87624	1.04862	0.524308	+	0.851529i	$0.324324\pi$
$44$	−2.20869	−0.332972
$45$	0	0
$46$	1.59580	0.235288
$47$	8.37297	1.22132	0.610662	−	0.791892i	$-0.290904\pi$
$47$	8.37297	1.22132	0.610662	+	0.791892i	$0.290904\pi$
$48$	0	0
$49$	−0.923569	−0.131938
$50$	−5.53628	−0.782948
$51$	0	0
$52$	3.32573	0.461196
$53$	−2.29752	−0.315589	−0.157795	−	0.987472i	$-0.550438\pi$
$53$	−2.29752	−0.315589	−0.157795	+	0.987472i	$0.550438\pi$
$54$	0	0
$55$	4.99950	0.674132
$56$	−5.71733	−0.764011
$57$	0	0
$58$	−1.59580	−0.209539
$59$	−6.71576	−0.874318	−0.437159	−	0.899384i	$-0.644015\pi$
$59$	−6.71576	−0.874318	−0.437159	+	0.899384i	$0.644015\pi$
$60$	0	0
$61$	8.00054	1.02436	0.512182	−	0.858877i	$-0.328837\pi$
$61$	8.00054	1.02436	0.512182	+	0.858877i	$0.328837\pi$
$62$	−3.19338	−0.405560
$63$	0	0
$64$	4.78196	0.597744
$65$	−7.52800	−0.933734
$66$	0	0
$67$	6.11908	0.747565	0.373782	−	0.927516i	$-0.378061\pi$
$67$	6.11908	0.747565	0.373782	+	0.927516i	$0.378061\pi$
$68$	3.05094	0.369981
$69$	0	0
$70$	−4.86688	−0.581704
$71$	−0.713918	−0.0847265	−0.0423633	−	0.999102i	$-0.513489\pi$
$71$	−0.713918	−0.0847265	−0.0423633	+	0.999102i	$0.513489\pi$
$72$	0	0
$73$	13.9116	1.62822	0.814112	−	0.580708i	$-0.197224\pi$
$73$	13.9116	1.62822	0.814112	+	0.580708i	$0.197224\pi$
$74$	9.42450	1.09558
$75$	0	0
$76$	−0.256882	−0.0294664
$77$	−9.96100	−1.13516
$78$	0	0
$79$	2.01641	0.226864	0.113432	−	0.993546i	$-0.463816\pi$
$79$	2.01641	0.226864	0.113432	+	0.993546i	$0.463816\pi$
$80$	5.93176	0.663190
$81$	0	0
$82$	−12.7955	−1.41303
$83$	7.66901	0.841783	0.420891	−	0.907111i	$-0.361717\pi$
$83$	7.66901	0.841783	0.420891	+	0.907111i	$0.361717\pi$
$84$	0	0
$85$	−6.90599	−0.749060
$86$	10.9731	1.18326
$87$	0	0
$88$	9.37233	0.999094
$89$	−0.733342	−0.0777340	−0.0388670	−	0.999244i	$-0.512375\pi$
$89$	−0.733342	−0.0777340	−0.0388670	+	0.999244i	$0.512375\pi$
$90$	0	0
$91$	14.9988	1.57230
$92$	0.546582	0.0569851
$93$	0	0
$94$	13.3616	1.37814
$95$	0.581468	0.0596573
$96$	0	0
$97$	−12.7405	−1.29360	−0.646801	−	0.762659i	$-0.723894\pi$
$97$	−12.7405	−1.29360	−0.646801	+	0.762659i	$0.723894\pi$
$98$	−1.47383	−0.148880
$99$	0	0
$100$	−1.89625	−0.189625
$101$	5.49886	0.547157	0.273579	−	0.961850i	$-0.411793\pi$
$101$	5.49886	0.547157	0.273579	+	0.961850i	$0.411793\pi$
$102$	0	0
$103$	17.6277	1.73691	0.868456	−	0.495766i	$-0.165113\pi$
$103$	17.6277	1.73691	0.868456	+	0.495766i	$0.165113\pi$
$104$	−14.1124	−1.38384
$105$	0	0
$106$	−3.66639	−0.356111
$107$	1.85338	0.179173	0.0895864	−	0.995979i	$-0.471445\pi$
$107$	1.85338	0.179173	0.0895864	+	0.995979i	$0.471445\pi$
$108$	0	0
$109$	−1.48112	−0.141866	−0.0709330	−	0.997481i	$-0.522598\pi$
$109$	−1.48112	−0.141866	−0.0709330	+	0.997481i	$0.522598\pi$
$110$	7.97821	0.760692
$111$	0	0
$112$	−11.8184	−1.11674
$113$	10.0892	0.949111	0.474556	−	0.880225i	$-0.342609\pi$
$113$	10.0892	0.949111	0.474556	+	0.880225i	$0.342609\pi$
$114$	0	0
$115$	−1.23722	−0.115372
$116$	−0.546582	−0.0507489
$117$	0	0
$118$	−10.7170	−0.986582
$119$	13.7595	1.26133
$120$	0	0
$121$	5.32890	0.484445
$122$	12.7673	1.15589
$123$	0	0
$124$	−1.09377	−0.0982238
$125$	10.4784	0.937215
$126$	0	0
$127$	8.74706	0.776176	0.388088	−	0.921622i	$-0.373136\pi$
$127$	8.74706	0.776176	0.388088	+	0.921622i	$0.373136\pi$
$128$	13.6554	1.20698
$129$	0	0
$130$	−12.0132	−1.05363
$131$	1.66085	0.145109	0.0725546	−	0.997364i	$-0.476885\pi$
$131$	1.66085	0.145109	0.0725546	+	0.997364i	$0.476885\pi$
$132$	0	0
$133$	−1.15852	−0.100456
$134$	9.76484	0.843554
$135$	0	0
$136$	−12.9464	−1.11014
$137$	1.05944	0.0905145	0.0452572	−	0.998975i	$-0.485589\pi$
$137$	1.05944	0.0905145	0.0452572	+	0.998975i	$0.485589\pi$
$138$	0	0
$139$	−3.42956	−0.290892	−0.145446	−	0.989366i	$-0.546462\pi$
$139$	−3.42956	−0.290892	−0.145446	+	0.989366i	$0.546462\pi$
$140$	−1.66697	−0.140885
$141$	0	0
$142$	−1.13927	−0.0956056
$143$	−24.5873	−2.05609
$144$	0	0
$145$	1.23722	0.102746
$146$	22.2001	1.83729
$147$	0	0
$148$	3.22801	0.265341
$149$	16.0167	1.31214	0.656071	−	0.754699i	$-0.272217\pi$
$149$	16.0167	1.31214	0.656071	+	0.754699i	$0.272217\pi$
$150$	0	0
$151$	−8.43031	−0.686049	−0.343024	−	0.939326i	$-0.611451\pi$
$151$	−8.43031	−0.686049	−0.343024	+	0.939326i	$0.611451\pi$
$152$	1.09005	0.0884149
$153$	0	0
$154$	−15.8958	−1.28092
$155$	2.47583	0.198863
$156$	0	0
$157$	−3.27432	−0.261319	−0.130659	−	0.991427i	$-0.541709\pi$
$157$	−3.27432	−0.261319	−0.130659	+	0.991427i	$0.541709\pi$
$158$	3.21779	0.255994
$159$	0	0
$160$	3.72676	0.294626
$161$	2.46504	0.194273
$162$	0	0
$163$	−2.83003	−0.221665	−0.110833	−	0.993839i	$-0.535352\pi$
$163$	−2.83003	−0.221665	−0.110833	+	0.993839i	$0.535352\pi$
$164$	−4.38262	−0.342225
$165$	0	0
$166$	12.2382	0.949869
$167$	5.23171	0.404842	0.202421	−	0.979299i	$-0.435119\pi$
$167$	5.23171	0.404842	0.202421	+	0.979299i	$0.435119\pi$
$168$	0	0
$169$	24.0223	1.84787
$170$	−11.0206	−0.845241
$171$	0	0
$172$	3.75843	0.286577
$173$	−10.6333	−0.808434	−0.404217	−	0.914663i	$-0.632456\pi$
$173$	−10.6333	−0.808434	−0.404217	+	0.914663i	$0.632456\pi$
$174$	0	0
$175$	−8.55192	−0.646464
$176$	19.3738	1.46035
$177$	0	0
$178$	−1.17027	−0.0877153
$179$	25.2446	1.88687	0.943437	−	0.331553i	$-0.107573\pi$
$179$	25.2446	1.88687	0.943437	+	0.331553i	$0.107573\pi$
$180$	0	0
$181$	17.3550	1.28999	0.644993	−	0.764189i	$-0.276860\pi$
$181$	17.3550	1.28999	0.644993	+	0.764189i	$0.276860\pi$
$182$	23.9351	1.77419
$183$	0	0
$184$	−2.31937	−0.170986
$185$	−7.30680	−0.537207
$186$	0	0
$187$	−22.5557	−1.64944
$188$	4.57652	0.333777
$189$	0	0
$190$	0.927907	0.0673175
$191$	−7.59838	−0.549800	−0.274900	−	0.961473i	$-0.588645\pi$
$191$	−7.59838	−0.549800	−0.274900	+	0.961473i	$0.588645\pi$
$192$	0	0
$193$	27.6252	1.98850	0.994251	−	0.107071i	$-0.0341471\pi$
$193$	27.6252	1.98850	0.994251	+	0.107071i	$0.0341471\pi$
$194$	−20.3313	−1.45970
$195$	0	0
$196$	−0.504806	−0.0360576
$197$	−9.20732	−0.655995	−0.327997	−	0.944679i	$-0.606374\pi$
$197$	−9.20732	−0.655995	−0.327997	+	0.944679i	$0.606374\pi$
$198$	0	0
$199$	−8.01668	−0.568287	−0.284144	−	0.958782i	$-0.591709\pi$
$199$	−8.01668	−0.568287	−0.284144	+	0.958782i	$0.591709\pi$
$200$	8.04653	0.568976
$201$	0	0
$202$	8.77509	0.617413
$203$	−2.46504	−0.173012
$204$	0	0
$205$	9.92033	0.692866
$206$	28.1304	1.95993
$207$	0	0
$208$	−29.1721	−2.02272
$209$	1.89914	0.131366
$210$	0	0
$211$	23.3826	1.60972	0.804862	−	0.593462i	$-0.202239\pi$
$211$	23.3826	1.60972	0.804862	+	0.593462i	$0.202239\pi$
$212$	−1.25579	−0.0862477
$213$	0	0
$214$	2.95762	0.202179
$215$	−8.50744	−0.580202
$216$	0	0
$217$	−4.93284	−0.334863
$218$	−2.36358	−0.160082
$219$	0	0
$220$	2.73264	0.184234
$221$	33.9633	2.28462
$222$	0	0
$223$	−4.51986	−0.302672	−0.151336	−	0.988482i	$-0.548358\pi$
$223$	−4.51986	−0.302672	−0.151336	+	0.988482i	$0.548358\pi$
$224$	−7.42519	−0.496117
$225$	0	0
$226$	16.1003	1.07098
$227$	−16.0461	−1.06501	−0.532507	−	0.846425i	$-0.678750\pi$
$227$	−16.0461	−1.06501	−0.532507	+	0.846425i	$0.678750\pi$
$228$	0	0
$229$	−12.3783	−0.817980	−0.408990	−	0.912539i	$-0.634119\pi$
$229$	−12.3783	−0.817980	−0.408990	+	0.912539i	$0.634119\pi$
$230$	−1.97436	−0.130186
$231$	0	0
$232$	2.31937	0.152274
$233$	12.8643	0.842767	0.421384	−	0.906882i	$-0.361545\pi$
$233$	12.8643	0.842767	0.421384	+	0.906882i	$0.361545\pi$
$234$	0	0
$235$	−10.3592	−0.675762
$236$	−3.67072	−0.238943
$237$	0	0
$238$	21.9574	1.42329
$239$	−14.5687	−0.942368	−0.471184	−	0.882035i	$-0.656173\pi$
$239$	−14.5687	−0.942368	−0.471184	+	0.882035i	$0.656173\pi$
$240$	0	0
$241$	28.8161	1.85621	0.928104	−	0.372322i	$-0.121438\pi$
$241$	28.8161	1.85621	0.928104	+	0.372322i	$0.121438\pi$
$242$	8.50386	0.546649
$243$	0	0
$244$	4.37295	0.279949
$245$	1.14266	0.0730019
$246$	0	0
$247$	−2.85963	−0.181954
$248$	4.64132	0.294724
$249$	0	0
$250$	16.7214	1.05756
$251$	0.648897	0.0409580	0.0204790	−	0.999790i	$-0.493481\pi$
$251$	0.648897	0.0409580	0.0204790	+	0.999790i	$0.493481\pi$
$252$	0	0
$253$	−4.04090	−0.254049
$254$	13.9586	0.875839
$255$	0	0
$256$	12.2275	0.764217
$257$	30.6508	1.91195	0.955973	−	0.293455i	$-0.0948050\pi$
$257$	30.6508	1.91195	0.955973	+	0.293455i	$0.0948050\pi$
$258$	0	0
$259$	14.5581	0.904595
$260$	−4.11467	−0.255181
$261$	0	0
$262$	2.65039	0.163741
$263$	−12.1422	−0.748721	−0.374361	−	0.927283i	$-0.622138\pi$
$263$	−12.1422	−0.748721	−0.374361	+	0.927283i	$0.622138\pi$
$264$	0	0
$265$	2.84255	0.174616
$266$	−1.84876	−0.113355
$267$	0	0
$268$	3.34458	0.204303
$269$	−19.5584	−1.19250	−0.596249	−	0.802800i	$-0.703343\pi$
$269$	−19.5584	−1.19250	−0.596249	+	0.802800i	$0.703343\pi$
$270$	0	0
$271$	10.2198	0.620809	0.310404	−	0.950605i	$-0.399536\pi$
$271$	10.2198	0.620809	0.310404	+	0.950605i	$0.399536\pi$
$272$	−26.7617	−1.62267
$273$	0	0
$274$	1.69066	0.102137
$275$	14.0190	0.845379
$276$	0	0
$277$	25.8559	1.55353	0.776765	−	0.629791i	$-0.216859\pi$
$277$	25.8559	1.55353	0.776765	+	0.629791i	$0.216859\pi$
$278$	−5.47290	−0.328243
$279$	0	0
$280$	7.07362	0.422729
$281$	−28.1037	−1.67653	−0.838263	−	0.545266i	$-0.816429\pi$
$281$	−28.1037	−1.67653	−0.838263	+	0.545266i	$0.816429\pi$
$282$	0	0
$283$	−32.1961	−1.91386	−0.956929	−	0.290321i	$-0.906238\pi$
$283$	−32.1961	−1.91386	−0.956929	+	0.290321i	$0.906238\pi$
$284$	−0.390215	−0.0231550
$285$	0	0
$286$	−39.2364	−2.32010
$287$	−19.7653	−1.16671
$288$	0	0
$289$	14.1570	0.832767
$290$	1.97436	0.115939
$291$	0	0
$292$	7.60381	0.444979
$293$	−15.0459	−0.878990	−0.439495	−	0.898245i	$-0.644842\pi$
$293$	−15.0459	−0.878990	−0.439495	+	0.898245i	$0.644842\pi$
$294$	0	0
$295$	8.30889	0.483762
$296$	−13.6977	−0.796165
$297$	0	0
$298$	25.5595	1.48062
$299$	6.08460	0.351881
$300$	0	0
$301$	16.9502	0.976994
$302$	−13.4531	−0.774139
$303$	0	0
$304$	2.25327	0.129234
$305$	−9.89844	−0.566783
$306$	0	0
$307$	−15.9566	−0.910688	−0.455344	−	0.890315i	$-0.650484\pi$
$307$	−15.9566	−0.910688	−0.455344	+	0.890315i	$0.650484\pi$
$308$	−5.44450	−0.310229
$309$	0	0
$310$	3.95093	0.224398
$311$	−6.18486	−0.350711	−0.175356	−	0.984505i	$-0.556108\pi$
$311$	−6.18486	−0.350711	−0.175356	+	0.984505i	$0.556108\pi$
$312$	0	0
$313$	−10.6715	−0.603187	−0.301594	−	0.953437i	$-0.597519\pi$
$313$	−10.6715	−0.603187	−0.301594	+	0.953437i	$0.597519\pi$
$314$	−5.22516	−0.294873
$315$	0	0
$316$	1.10214	0.0620000
$317$	19.7517	1.10937	0.554684	−	0.832061i	$-0.312839\pi$
$317$	19.7517	1.10937	0.554684	+	0.832061i	$0.312839\pi$
$318$	0	0
$319$	4.04090	0.226247
$320$	−5.91634	−0.330734
$321$	0	0
$322$	3.93372	0.219217
$323$	−2.62335	−0.145967
$324$	0	0
$325$	−21.1092	−1.17093
$326$	−4.51617	−0.250127
$327$	0	0
$328$	18.5972	1.02686
$329$	20.6397	1.13790
$330$	0	0
$331$	11.5464	0.634649	0.317325	−	0.948317i	$-0.397216\pi$
$331$	11.5464	0.634649	0.317325	+	0.948317i	$0.397216\pi$
$332$	4.19174	0.230052
$333$	0	0
$334$	8.34877	0.456824
$335$	−7.57067	−0.413630
$336$	0	0
$337$	−11.3595	−0.618791	−0.309395	−	0.950934i	$-0.600127\pi$
$337$	−11.3595	−0.618791	−0.309395	+	0.950934i	$0.600127\pi$
$338$	38.3349	2.08514
$339$	0	0
$340$	−3.77469	−0.204711
$341$	8.08632	0.437899
$342$	0	0
$343$	−19.5319	−1.05463
$344$	−15.9485	−0.859886
$345$	0	0
$346$	−16.9686	−0.912239
$347$	−0.330557	−0.0177452	−0.00887262	−	0.999961i	$-0.502824\pi$
$347$	−0.330557	−0.0177452	−0.00887262	+	0.999961i	$0.502824\pi$
$348$	0	0
$349$	−13.3266	−0.713358	−0.356679	−	0.934227i	$-0.616091\pi$
$349$	−13.3266	−0.713358	−0.356679	+	0.934227i	$0.616091\pi$
$350$	−13.6472	−0.729472
$351$	0	0
$352$	12.1720	0.648770
$353$	14.4688	0.770094	0.385047	−	0.922897i	$-0.374185\pi$
$353$	14.4688	0.770094	0.385047	+	0.922897i	$0.374185\pi$
$354$	0	0
$355$	0.883276	0.0468794
$356$	−0.400831	−0.0212440
$357$	0	0
$358$	40.2854	2.12915
$359$	10.3405	0.545752	0.272876	−	0.962049i	$-0.412025\pi$
$359$	10.3405	0.545752	0.272876	+	0.962049i	$0.412025\pi$
$360$	0	0
$361$	−18.7791	−0.988375
$362$	27.6951	1.45562
$363$	0	0
$364$	8.19807	0.429696
$365$	−17.2117	−0.900901
$366$	0	0
$367$	−4.70969	−0.245844	−0.122922	−	0.992416i	$-0.539226\pi$
$367$	−4.70969	−0.245844	−0.122922	+	0.992416i	$0.539226\pi$
$368$	−4.79441	−0.249926
$369$	0	0
$370$	−11.6602	−0.606185
$371$	−5.66349	−0.294034
$372$	0	0
$373$	−5.70376	−0.295329	−0.147665	−	0.989037i	$-0.547176\pi$
$373$	−5.70376	−0.295329	−0.147665	+	0.989037i	$0.547176\pi$
$374$	−35.9944	−1.86123
$375$	0	0
$376$	−19.4200	−1.00151
$377$	−6.08460	−0.313373
$378$	0	0
$379$	−6.98605	−0.358849	−0.179425	−	0.983772i	$-0.557424\pi$
$379$	−6.98605	−0.358849	−0.179425	+	0.983772i	$0.557424\pi$
$380$	0.317820	0.0163038
$381$	0	0
$382$	−12.1255	−0.620395
$383$	−27.4430	−1.40227	−0.701136	−	0.713028i	$-0.747323\pi$
$383$	−27.4430	−1.40227	−0.701136	+	0.713028i	$0.747323\pi$
$384$	0	0
$385$	12.3240	0.628088
$386$	44.0843	2.24383
$387$	0	0
$388$	−6.96373	−0.353530
$389$	30.4907	1.54594	0.772969	−	0.634444i	$-0.218771\pi$
$389$	30.4907	1.54594	0.772969	+	0.634444i	$0.218771\pi$
$390$	0	0
$391$	5.58185	0.282286
$392$	2.14209	0.108192
$393$	0	0
$394$	−14.6931	−0.740226
$395$	−2.49475	−0.125525
$396$	0	0
$397$	−23.9935	−1.20420	−0.602099	−	0.798422i	$-0.705669\pi$
$397$	−23.9935	−1.20420	−0.602099	+	0.798422i	$0.705669\pi$
$398$	−12.7930	−0.641257
$399$	0	0
$400$	16.6332	0.831658
$401$	21.0309	1.05023	0.525116	−	0.851031i	$-0.324022\pi$
$401$	21.0309	1.05023	0.525116	+	0.851031i	$0.324022\pi$
$402$	0	0
$403$	−12.1760	−0.606529
$404$	3.00558	0.149533
$405$	0	0
$406$	−3.93372	−0.195227
$407$	−23.8648	−1.18293
$408$	0	0
$409$	23.0923	1.14184	0.570920	−	0.821006i	$-0.306587\pi$
$409$	23.0923	1.14184	0.570920	+	0.821006i	$0.306587\pi$
$410$	15.8309	0.781831
$411$	0	0
$412$	9.63500	0.474682
$413$	−16.5546	−0.814600
$414$	0	0
$415$	−9.48827	−0.465761
$416$	−18.3280	−0.898605
$417$	0	0
$418$	3.03064	0.148234
$419$	−40.4641	−1.97680	−0.988401	−	0.151867i	$-0.951471\pi$
$419$	−40.4641	−1.97680	−0.988401	+	0.151867i	$0.951471\pi$
$420$	0	0
$421$	−15.9778	−0.778709	−0.389354	−	0.921088i	$-0.627302\pi$
$421$	−15.9778	−0.778709	−0.389354	+	0.921088i	$0.627302\pi$
$422$	37.3140	1.81642
$423$	0	0
$424$	5.32880	0.258789
$425$	−19.3650	−0.939340
$426$	0	0
$427$	19.7217	0.954398
$428$	1.01302	0.0489663
$429$	0	0
$430$	−13.5762	−0.654702
$431$	8.02554	0.386576	0.193288	−	0.981142i	$-0.438085\pi$
$431$	8.02554	0.386576	0.193288	+	0.981142i	$0.438085\pi$
$432$	0	0
$433$	−15.3653	−0.738408	−0.369204	−	0.929348i	$-0.620370\pi$
$433$	−15.3653	−0.738408	−0.369204	+	0.929348i	$0.620370\pi$
$434$	−7.87183	−0.377860
$435$	0	0
$436$	−0.809556	−0.0387707
$437$	−0.469978	−0.0224821
$438$	0	0
$439$	2.13940	0.102108	0.0510541	−	0.998696i	$-0.483742\pi$
$439$	2.13940	0.102108	0.0510541	+	0.998696i	$0.483742\pi$
$440$	−11.5957	−0.552802
$441$	0	0
$442$	54.1987	2.57797
$443$	−23.7135	−1.12666	−0.563331	−	0.826232i	$-0.690480\pi$
$443$	−23.7135	−1.12666	−0.563331	+	0.826232i	$0.690480\pi$
$444$	0	0
$445$	0.907307	0.0430105
$446$	−7.21280	−0.341536
$447$	0	0
$448$	11.7877	0.556917
$449$	−32.8609	−1.55080	−0.775400	−	0.631470i	$-0.782452\pi$
$449$	−32.8609	−1.55080	−0.775400	+	0.631470i	$0.782452\pi$
$450$	0	0
$451$	32.4009	1.52570
$452$	5.51457	0.259384
$453$	0	0
$454$	−25.6063	−1.20176
$455$	−18.5568	−0.869958
$456$	0	0
$457$	−17.6100	−0.823760	−0.411880	−	0.911238i	$-0.635128\pi$
$457$	−17.6100	−0.823760	−0.411880	+	0.911238i	$0.635128\pi$
$458$	−19.7533	−0.923011
$459$	0	0
$460$	−0.676244	−0.0315300
$461$	−15.0249	−0.699778	−0.349889	−	0.936791i	$-0.613781\pi$
$461$	−15.0249	−0.699778	−0.349889	+	0.936791i	$0.613781\pi$
$462$	0	0
$463$	−1.01242	−0.0470510	−0.0235255	−	0.999723i	$-0.507489\pi$
$463$	−1.01242	−0.0470510	−0.0235255	+	0.999723i	$0.507489\pi$
$464$	4.79441	0.222575
$465$	0	0
$466$	20.5288	0.950980
$467$	31.8850	1.47546	0.737731	−	0.675095i	$-0.235897\pi$
$467$	31.8850	1.47546	0.737731	+	0.675095i	$0.235897\pi$
$468$	0	0
$469$	15.0838	0.696505
$470$	−16.5313	−0.762531
$471$	0	0
$472$	15.5763	0.716958
$473$	−27.7862	−1.27761
$474$	0	0
$475$	1.63049	0.0748118
$476$	7.52069	0.344710
$477$	0	0
$478$	−23.2487	−1.06337
$479$	−35.2220	−1.60934	−0.804668	−	0.593726i	$-0.797657\pi$
$479$	−35.2220	−1.60934	−0.804668	+	0.593726i	$0.797657\pi$
$480$	0	0
$481$	35.9345	1.63847
$482$	45.9847	2.09455
$483$	0	0
$484$	2.91268	0.132395
$485$	15.7628	0.715754
$486$	0	0
$487$	−7.49271	−0.339527	−0.169763	−	0.985485i	$-0.554300\pi$
$487$	−7.49271	−0.339527	−0.169763	+	0.985485i	$0.554300\pi$
$488$	−18.5562	−0.839998
$489$	0	0
$490$	1.82346	0.0823755
$491$	16.3875	0.739557	0.369778	−	0.929120i	$-0.379434\pi$
$491$	16.3875	0.739557	0.369778	+	0.929120i	$0.379434\pi$
$492$	0	0
$493$	−5.58185	−0.251394
$494$	−4.56340	−0.205317
$495$	0	0
$496$	9.59418	0.430791
$497$	−1.75984	−0.0789396
$498$	0	0
$499$	−28.5285	−1.27711	−0.638555	−	0.769576i	$-0.720468\pi$
$499$	−28.5285	−1.27711	−0.638555	+	0.769576i	$0.720468\pi$
$500$	5.72730	0.256133
$501$	0	0
$502$	1.03551	0.0462171
$503$	1.07113	0.0477591	0.0238796	−	0.999715i	$-0.492398\pi$
$503$	1.07113	0.0477591	0.0238796	+	0.999715i	$0.492398\pi$
$504$	0	0
$505$	−6.80332	−0.302744
$506$	−6.44848	−0.286670
$507$	0	0
$508$	4.78099	0.212122
$509$	19.7132	0.873772	0.436886	−	0.899517i	$-0.356081\pi$
$509$	19.7132	0.873772	0.436886	+	0.899517i	$0.356081\pi$
$510$	0	0
$511$	34.2926	1.51701
$512$	−7.79828	−0.344639
$513$	0	0
$514$	48.9126	2.15744
$515$	−21.8094	−0.961038
$516$	0	0
$517$	−33.8344	−1.48803
$518$	23.2318	1.02075
$519$	0	0
$520$	17.4602	0.765680
$521$	11.5600	0.506453	0.253226	−	0.967407i	$-0.418508\pi$
$521$	11.5600	0.506453	0.253226	+	0.967407i	$0.418508\pi$
$522$	0	0
$523$	30.9631	1.35392	0.676961	−	0.736019i	$-0.263297\pi$
$523$	30.9631	1.35392	0.676961	+	0.736019i	$0.263297\pi$
$524$	0.907791	0.0396570
$525$	0	0
$526$	−19.3766	−0.844859
$527$	−11.1699	−0.486570
$528$	0	0
$529$	1.00000	0.0434783
$530$	4.53614	0.197037
$531$	0	0
$532$	−0.633224	−0.0274538
$533$	−48.7877	−2.11323
$534$	0	0
$535$	−2.29304	−0.0991368
$536$	−14.1924	−0.613018
$537$	0	0
$538$	−31.2113	−1.34562
$539$	3.73205	0.160751
$540$	0	0
$541$	−20.0458	−0.861837	−0.430918	−	0.902391i	$-0.641810\pi$
$541$	−20.0458	−0.861837	−0.430918	+	0.902391i	$0.641810\pi$
$542$	16.3088	0.700522
$543$	0	0
$544$	−16.8136	−0.720878
$545$	1.83248	0.0784948
$546$	0	0
$547$	4.01462	0.171653	0.0858264	−	0.996310i	$-0.472647\pi$
$547$	4.01462	0.171653	0.0858264	+	0.996310i	$0.472647\pi$
$548$	0.579073	0.0247368
$549$	0	0
$550$	22.3716	0.953927
$551$	0.469978	0.0200217
$552$	0	0
$553$	4.97054	0.211369
$554$	41.2608	1.75301
$555$	0	0
$556$	−1.87454	−0.0794981
$557$	−0.128470	−0.00544345	−0.00272172	−	0.999996i	$-0.500866\pi$
$557$	−0.128470	−0.00544345	−0.00272172	+	0.999996i	$0.500866\pi$
$558$	0	0
$559$	41.8391	1.76961
$560$	14.6220	0.617893
$561$	0	0
$562$	−44.8479	−1.89180
$563$	31.2296	1.31617	0.658086	−	0.752943i	$-0.271366\pi$
$563$	31.2296	1.31617	0.658086	+	0.752943i	$0.271366\pi$
$564$	0	0
$565$	−12.4826	−0.525146
$566$	−51.3786	−2.15960
$567$	0	0
$568$	1.65584	0.0694774
$569$	−18.6330	−0.781138	−0.390569	−	0.920574i	$-0.627722\pi$
$569$	−18.6330	−0.781138	−0.390569	+	0.920574i	$0.627722\pi$
$570$	0	0
$571$	−4.08799	−0.171077	−0.0855386	−	0.996335i	$-0.527261\pi$
$571$	−4.08799	−0.171077	−0.0855386	+	0.996335i	$0.527261\pi$
$572$	−13.4390	−0.561911
$573$	0	0
$574$	−31.5414	−1.31651
$575$	−3.46928	−0.144679
$576$	0	0
$577$	40.8197	1.69935	0.849674	−	0.527308i	$-0.176799\pi$
$577$	40.8197	1.69935	0.849674	+	0.527308i	$0.176799\pi$
$578$	22.5918	0.939696
$579$	0	0
$580$	0.676244	0.0280795
$581$	18.9044	0.784287
$582$	0	0
$583$	9.28407	0.384507
$584$	−32.2660	−1.33518
$585$	0	0
$586$	−24.0102	−0.991854
$587$	−0.932895	−0.0385047	−0.0192524	−	0.999815i	$-0.506129\pi$
$587$	−0.932895	−0.0385047	−0.0192524	+	0.999815i	$0.506129\pi$
$588$	0	0
$589$	0.940481	0.0387519
$590$	13.2593	0.545879
$591$	0	0
$592$	−28.3149	−1.16374
$593$	6.06982	0.249257	0.124629	−	0.992203i	$-0.460226\pi$
$593$	6.06982	0.249257	0.124629	+	0.992203i	$0.460226\pi$
$594$	0	0
$595$	−17.0236	−0.697898
$596$	8.75446	0.358597
$597$	0	0
$598$	9.70981	0.397064
$599$	−43.7176	−1.78625	−0.893126	−	0.449807i	$-0.851493\pi$
$599$	−43.7176	−1.78625	−0.893126	+	0.449807i	$0.851493\pi$
$600$	0	0
$601$	12.9621	0.528735	0.264367	−	0.964422i	$-0.414837\pi$
$601$	12.9621	0.528735	0.264367	+	0.964422i	$0.414837\pi$
$602$	27.0492	1.10244
$603$	0	0
$604$	−4.60786	−0.187491
$605$	−6.59304	−0.268045
$606$	0	0
$607$	6.54188	0.265527	0.132763	−	0.991148i	$-0.457615\pi$
$607$	6.54188	0.265527	0.132763	+	0.991148i	$0.457615\pi$
$608$	1.41567	0.0574129
$609$	0	0
$610$	−15.7960	−0.639559
$611$	50.9462	2.06106
$612$	0	0
$613$	−2.99617	−0.121014	−0.0605072	−	0.998168i	$-0.519272\pi$
$613$	−2.99617	−0.121014	−0.0605072	+	0.998168i	$0.519272\pi$
$614$	−25.4635	−1.02762
$615$	0	0
$616$	23.1032	0.930854
$617$	−11.7247	−0.472017	−0.236008	−	0.971751i	$-0.575839\pi$
$617$	−11.7247	−0.472017	−0.236008	+	0.971751i	$0.575839\pi$
$618$	0	0
$619$	41.9741	1.68708	0.843540	−	0.537066i	$-0.180467\pi$
$619$	41.9741	1.68708	0.843540	+	0.537066i	$0.180467\pi$
$620$	1.35324	0.0543475
$621$	0	0
$622$	−9.86981	−0.395743
$623$	−1.80772	−0.0724247
$624$	0	0
$625$	4.38230	0.175292
$626$	−17.0296	−0.680638
$627$	0	0
$628$	−1.78968	−0.0714161
$629$	32.9653	1.31441
$630$	0	0
$631$	19.6939	0.784000	0.392000	−	0.919965i	$-0.371783\pi$
$631$	19.6939	0.784000	0.392000	+	0.919965i	$0.371783\pi$
$632$	−4.67680	−0.186033
$633$	0	0
$634$	31.5198	1.25181
$635$	−10.8221	−0.429461
$636$	0	0
$637$	−5.61955	−0.222655
$638$	6.44848	0.255298
$639$	0	0
$640$	−16.8948	−0.667827
$641$	2.43221	0.0960667	0.0480333	−	0.998846i	$-0.484705\pi$
$641$	2.43221	0.0960667	0.0480333	+	0.998846i	$0.484705\pi$
$642$	0	0
$643$	1.93996	0.0765047	0.0382524	−	0.999268i	$-0.487821\pi$
$643$	1.93996	0.0765047	0.0382524	+	0.999268i	$0.487821\pi$
$644$	1.34735	0.0530929
$645$	0	0
$646$	−4.18634	−0.164709
$647$	26.6233	1.04667	0.523335	−	0.852127i	$-0.324688\pi$
$647$	26.6233	1.04667	0.523335	+	0.852127i	$0.324688\pi$
$648$	0	0
$649$	27.1377	1.06525
$650$	−33.6861	−1.32128
$651$	0	0
$652$	−1.54684	−0.0605791
$653$	33.9855	1.32995	0.664977	−	0.746864i	$-0.268441\pi$
$653$	33.9855	1.32995	0.664977	+	0.746864i	$0.268441\pi$
$654$	0	0
$655$	−2.05484	−0.0802893
$656$	38.4427	1.50093
$657$	0	0
$658$	32.9369	1.28401
$659$	0.545221	0.0212388	0.0106194	−	0.999944i	$-0.496620\pi$
$659$	0.545221	0.0212388	0.0106194	+	0.999944i	$0.496620\pi$
$660$	0	0
$661$	33.0032	1.28368	0.641838	−	0.766841i	$-0.278172\pi$
$661$	33.0032	1.28368	0.641838	+	0.766841i	$0.278172\pi$
$662$	18.4258	0.716139
$663$	0	0
$664$	−17.7872	−0.690279
$665$	1.43334	0.0555826
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	2.85956	0.110640
$669$	0	0
$670$	−12.0813	−0.466741
$671$	−32.3294	−1.24806
$672$	0	0
$673$	50.7564	1.95652	0.978259	−	0.207389i	$-0.0664964\pi$
$673$	50.7564	1.95652	0.978259	+	0.207389i	$0.0664964\pi$
$674$	−18.1275	−0.698245
$675$	0	0
$676$	13.1302	0.505007
$677$	−33.9999	−1.30672	−0.653362	−	0.757046i	$-0.726642\pi$
$677$	−33.9999	−1.30672	−0.653362	+	0.757046i	$0.726642\pi$
$678$	0	0
$679$	−31.4059	−1.20525
$680$	16.0175	0.614244
$681$	0	0
$682$	12.9042	0.494126
$683$	10.9943	0.420686	0.210343	−	0.977628i	$-0.432542\pi$
$683$	10.9943	0.420686	0.210343	+	0.977628i	$0.432542\pi$
$684$	0	0
$685$	−1.31077	−0.0500819
$686$	−31.1691	−1.19004
$687$	0	0
$688$	−32.9675	−1.25687
$689$	−13.9795	−0.532577
$690$	0	0
$691$	−43.5308	−1.65599	−0.827994	−	0.560736i	$-0.810518\pi$
$691$	−43.5308	−1.65599	−0.827994	+	0.560736i	$0.810518\pi$
$692$	−5.81197	−0.220938
$693$	0	0
$694$	−0.527504	−0.0200238
$695$	4.24313	0.160951
$696$	0	0
$697$	−44.7565	−1.69527
$698$	−21.2666	−0.804954
$699$	0	0
$700$	−4.67433	−0.176673
$701$	49.4520	1.86778	0.933889	−	0.357564i	$-0.116393\pi$
$701$	49.4520	1.86778	0.933889	+	0.357564i	$0.116393\pi$
$702$	0	0
$703$	−2.77560	−0.104684
$704$	−19.3234	−0.728279
$705$	0	0
$706$	23.0893	0.868976
$707$	13.5549	0.509785
$708$	0	0
$709$	−7.65262	−0.287400	−0.143700	−	0.989621i	$-0.545900\pi$
$709$	−7.65262	−0.287400	−0.143700	+	0.989621i	$0.545900\pi$
$710$	1.40953	0.0528988
$711$	0	0
$712$	1.70089	0.0637435
$713$	−2.00112	−0.0749424
$714$	0	0
$715$	30.4199	1.13764
$716$	13.7983	0.515666
$717$	0	0
$718$	16.5014	0.615828
$719$	41.8360	1.56022	0.780110	−	0.625642i	$-0.215163\pi$
$719$	41.8360	1.56022	0.780110	+	0.625642i	$0.215163\pi$
$720$	0	0
$721$	43.4531	1.61828
$722$	−29.9677	−1.11528
$723$	0	0
$724$	9.48592	0.352542
$725$	3.46928	0.128846
$726$	0	0
$727$	−49.6803	−1.84254	−0.921270	−	0.388924i	$-0.872847\pi$
$727$	−49.6803	−1.84254	−0.921270	+	0.388924i	$0.872847\pi$
$728$	−34.7877	−1.28932
$729$	0	0
$730$	−27.4664	−1.01658
$731$	38.3821	1.41961
$732$	0	0
$733$	3.24207	0.119749	0.0598744	−	0.998206i	$-0.480930\pi$
$733$	3.24207	0.119749	0.0598744	+	0.998206i	$0.480930\pi$
$734$	−7.51573	−0.277411
$735$	0	0
$736$	−3.01220	−0.111031
$737$	−24.7266	−0.910817
$738$	0	0
$739$	−6.53356	−0.240341	−0.120170	−	0.992753i	$-0.538344\pi$
$739$	−6.53356	−0.240341	−0.120170	+	0.992753i	$0.538344\pi$
$740$	−3.99377	−0.146814
$741$	0	0
$742$	−9.03781	−0.331788
$743$	−13.1158	−0.481173	−0.240587	−	0.970628i	$-0.577340\pi$
$743$	−13.1158	−0.481173	−0.240587	+	0.970628i	$0.577340\pi$
$744$	0	0
$745$	−19.8163	−0.726012
$746$	−9.10207	−0.333250
$747$	0	0
$748$	−12.3285	−0.450776
$749$	4.56865	0.166935
$750$	0	0
$751$	−3.48871	−0.127305	−0.0636525	−	0.997972i	$-0.520275\pi$
$751$	−3.48871	−0.127305	−0.0636525	+	0.997972i	$0.520275\pi$
$752$	−40.1435	−1.46388
$753$	0	0
$754$	−9.70981	−0.353610
$755$	10.4302	0.379593
$756$	0	0
$757$	−18.2975	−0.665032	−0.332516	−	0.943098i	$-0.607898\pi$
$757$	−18.2975	−0.665032	−0.332516	+	0.943098i	$0.607898\pi$
$758$	−11.1484	−0.404926
$759$	0	0
$760$	−1.34864	−0.0489202
$761$	−29.1438	−1.05646	−0.528231	−	0.849101i	$-0.677145\pi$
$761$	−29.1438	−1.05646	−0.528231	+	0.849101i	$0.677145\pi$
$762$	0	0
$763$	−3.65103	−0.132176
$764$	−4.15314	−0.150255
$765$	0	0
$766$	−43.7936	−1.58233
$767$	−40.8627	−1.47547
$768$	0	0
$769$	−15.4216	−0.556117	−0.278059	−	0.960564i	$-0.589691\pi$
$769$	−15.4216	−0.556117	−0.278059	+	0.960564i	$0.589691\pi$
$770$	19.6666	0.708735
$771$	0	0
$772$	15.0994	0.543440
$773$	−14.1320	−0.508291	−0.254146	−	0.967166i	$-0.581794\pi$
$773$	−14.1320	−0.508291	−0.254146	+	0.967166i	$0.581794\pi$
$774$	0	0
$775$	6.94243	0.249380
$776$	29.5499	1.06078
$777$	0	0
$778$	48.6570	1.74444
$779$	3.76839	0.135017
$780$	0	0
$781$	2.88488	0.103229
$782$	8.90752	0.318532
$783$	0	0
$784$	4.42797	0.158142
$785$	4.05106	0.144588
$786$	0	0
$787$	−27.3022	−0.973219	−0.486609	−	0.873620i	$-0.661767\pi$
$787$	−27.3022	−0.973219	−0.486609	+	0.873620i	$0.661767\pi$
$788$	−5.03256	−0.179277
$789$	0	0
$790$	−3.98113	−0.141642
$791$	24.8703	0.884285
$792$	0	0
$793$	48.6800	1.72868
$794$	−38.2888	−1.35882
$795$	0	0
$796$	−4.38177	−0.155308
$797$	−31.0268	−1.09902	−0.549512	−	0.835486i	$-0.685186\pi$
$797$	−31.0268	−1.09902	−0.549512	+	0.835486i	$0.685186\pi$
$798$	0	0
$799$	46.7367	1.65342
$800$	10.4502	0.369469
$801$	0	0
$802$	33.5611	1.18508
$803$	−56.2152	−1.98379
$804$	0	0
$805$	−3.04981	−0.107492
$806$	−19.4305	−0.684409
$807$	0	0
$808$	−12.7539	−0.448680
$809$	6.99136	0.245803	0.122902	−	0.992419i	$-0.460780\pi$
$809$	6.99136	0.245803	0.122902	+	0.992419i	$0.460780\pi$
$810$	0	0
$811$	−14.7374	−0.517499	−0.258750	−	0.965944i	$-0.583310\pi$
$811$	−14.7374	−0.517499	−0.258750	+	0.965944i	$0.583310\pi$
$812$	−1.34735	−0.0472826
$813$	0	0
$814$	−38.0835	−1.33483
$815$	3.50138	0.122648
$816$	0	0
$817$	−3.23168	−0.113062
$818$	36.8507	1.28845
$819$	0	0
$820$	5.42227	0.189354
$821$	−32.5264	−1.13518	−0.567590	−	0.823312i	$-0.692124\pi$
$821$	−32.5264	−1.13518	−0.567590	+	0.823312i	$0.692124\pi$
$822$	0	0
$823$	−48.1514	−1.67845	−0.839226	−	0.543783i	$-0.816991\pi$
$823$	−48.1514	−1.67845	−0.839226	+	0.543783i	$0.816991\pi$
$824$	−40.8852	−1.42430
$825$	0	0
$826$	−26.4179	−0.919197
$827$	−39.9441	−1.38899	−0.694497	−	0.719496i	$-0.744373\pi$
$827$	−39.9441	−1.38899	−0.694497	+	0.719496i	$0.744373\pi$
$828$	0	0
$829$	−32.4169	−1.12589	−0.562944	−	0.826495i	$-0.690331\pi$
$829$	−32.4169	−1.12589	−0.562944	+	0.826495i	$0.690331\pi$
$830$	−15.1414	−0.525565
$831$	0	0
$832$	29.0963	1.00873
$833$	−5.15522	−0.178618
$834$	0	0
$835$	−6.47279	−0.224000
$836$	1.03803	0.0359012
$837$	0	0
$838$	−64.5727	−2.23063
$839$	−26.8967	−0.928577	−0.464289	−	0.885684i	$-0.653690\pi$
$839$	−26.8967	−0.928577	−0.464289	+	0.885684i	$0.653690\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−25.4973	−0.878696
$843$	0	0
$844$	12.7805	0.439923
$845$	−29.7210	−1.02243
$846$	0	0
$847$	13.1360	0.451357
$848$	11.0153	0.378266
$849$	0	0
$850$	−30.9027	−1.05995
$851$	5.90581	0.202449
$852$	0	0
$853$	42.2014	1.44495	0.722474	−	0.691398i	$-0.243005\pi$
$853$	42.2014	1.44495	0.722474	+	0.691398i	$0.243005\pi$
$854$	31.4718	1.07694
$855$	0	0
$856$	−4.29866	−0.146925
$857$	−28.4265	−0.971030	−0.485515	−	0.874228i	$-0.661368\pi$
$857$	−28.4265	−0.971030	−0.485515	+	0.874228i	$0.661368\pi$
$858$	0	0
$859$	−19.8157	−0.676103	−0.338051	−	0.941128i	$-0.609768\pi$
$859$	−19.8157	−0.676103	−0.338051	+	0.941128i	$0.609768\pi$
$860$	−4.65001	−0.158564
$861$	0	0
$862$	12.8072	0.436214
$863$	14.3006	0.486797	0.243399	−	0.969926i	$-0.421738\pi$
$863$	14.3006	0.486797	0.243399	+	0.969926i	$0.421738\pi$
$864$	0	0
$865$	13.1557	0.447309
$866$	−24.5199	−0.833221
$867$	0	0
$868$	−2.69620	−0.0915150
$869$	−8.14813	−0.276406
$870$	0	0
$871$	37.2322	1.26156
$872$	3.43527	0.116333
$873$	0	0
$874$	−0.749992	−0.0253689
$875$	25.8297	0.873202
$876$	0	0
$877$	40.1559	1.35597	0.677984	−	0.735077i	$-0.262854\pi$
$877$	40.1559	1.35597	0.677984	+	0.735077i	$0.262854\pi$
$878$	3.41407	0.115219
$879$	0	0
$880$	−23.9697	−0.808017
$881$	−34.8681	−1.17474	−0.587368	−	0.809320i	$-0.699836\pi$
$881$	−34.8681	−1.17474	−0.587368	+	0.809320i	$0.699836\pi$
$882$	0	0
$883$	51.2214	1.72374	0.861868	−	0.507132i	$-0.169295\pi$
$883$	51.2214	1.72374	0.861868	+	0.507132i	$0.169295\pi$
$884$	18.5637	0.624366
$885$	0	0
$886$	−37.8420	−1.27133
$887$	2.84962	0.0956809	0.0478405	−	0.998855i	$-0.484766\pi$
$887$	2.84962	0.0956809	0.0478405	+	0.998855i	$0.484766\pi$
$888$	0	0
$889$	21.5619	0.723162
$890$	1.44788	0.0485331
$891$	0	0
$892$	−2.47047	−0.0827176
$893$	−3.93511	−0.131684
$894$	0	0
$895$	−31.2332	−1.04401
$896$	33.6612	1.12454
$897$	0	0
$898$	−52.4394	−1.74993
$899$	2.00112	0.0667410
$900$	0	0
$901$	−12.8244	−0.427244
$902$	51.7053	1.72160
$903$	0	0
$904$	−23.4005	−0.778290
$905$	−21.4720	−0.713753
$906$	0	0
$907$	34.5438	1.14701	0.573504	−	0.819203i	$-0.305584\pi$
$907$	34.5438	1.14701	0.573504	+	0.819203i	$0.305584\pi$
$908$	−8.77049	−0.291059
$909$	0	0
$910$	−29.6130	−0.981663
$911$	57.6795	1.91101	0.955503	−	0.294980i	$-0.0953130\pi$
$911$	57.6795	1.91101	0.955503	+	0.294980i	$0.0953130\pi$
$912$	0	0
$913$	−30.9897	−1.02561
$914$	−28.1020	−0.929533
$915$	0	0
$916$	−6.76575	−0.223547
$917$	4.09407	0.135198
$918$	0	0
$919$	−28.5597	−0.942096	−0.471048	−	0.882108i	$-0.656124\pi$
$919$	−28.5597	−0.942096	−0.471048	+	0.882108i	$0.656124\pi$
$920$	2.86957	0.0946070
$921$	0	0
$922$	−23.9767	−0.789631
$923$	−4.34391	−0.142981
$924$	0	0
$925$	−20.4889	−0.673671
$926$	−1.61562	−0.0530925
$927$	0	0
$928$	3.01220	0.0988802
$929$	−2.84560	−0.0933609	−0.0466805	−	0.998910i	$-0.514864\pi$
$929$	−2.84560	−0.0933609	−0.0466805	+	0.998910i	$0.514864\pi$
$930$	0	0
$931$	0.434057	0.0142257
$932$	7.03139	0.230321
$933$	0	0
$934$	50.8821	1.66491
$935$	27.9064	0.912638
$936$	0	0
$937$	28.5068	0.931276	0.465638	−	0.884975i	$-0.345825\pi$
$937$	28.5068	0.931276	0.465638	+	0.884975i	$0.345825\pi$
$938$	24.0707	0.785938
$939$	0	0
$940$	−5.66217	−0.184680
$941$	−5.49695	−0.179195	−0.0895977	−	0.995978i	$-0.528558\pi$
$941$	−5.49695	−0.179195	−0.0895977	+	0.995978i	$0.528558\pi$
$942$	0	0
$943$	−8.01822	−0.261109
$944$	32.1981	1.04796
$945$	0	0
$946$	−44.3413	−1.44166
$947$	−54.9545	−1.78578	−0.892891	−	0.450273i	$-0.851327\pi$
$947$	−54.9545	−1.78578	−0.892891	+	0.450273i	$0.851327\pi$
$948$	0	0
$949$	84.6462	2.74773
$950$	2.60193	0.0844178
$951$	0	0
$952$	−31.9133	−1.03432
$953$	−10.0498	−0.325546	−0.162773	−	0.986664i	$-0.552044\pi$
$953$	−10.0498	−0.325546	−0.162773	+	0.986664i	$0.552044\pi$
$954$	0	0
$955$	9.40089	0.304206
$956$	−7.96296	−0.257541
$957$	0	0
$958$	−56.2074	−1.81598
$959$	2.61158	0.0843322
$960$	0	0
$961$	−26.9955	−0.870824
$962$	57.3443	1.84886
$963$	0	0
$964$	15.7504	0.507285
$965$	−34.1785	−1.10024
$966$	0	0
$967$	−41.9191	−1.34803	−0.674014	−	0.738718i	$-0.735431\pi$
$967$	−41.9191	−1.34803	−0.674014	+	0.738718i	$0.735431\pi$
$968$	−12.3597	−0.397255
$969$	0	0
$970$	25.1544	0.807658
$971$	−25.6453	−0.822998	−0.411499	−	0.911410i	$-0.634995\pi$
$971$	−25.6453	−0.822998	−0.411499	+	0.911410i	$0.634995\pi$
$972$	0	0
$973$	−8.45402	−0.271023
$974$	−11.9569	−0.383123
$975$	0	0
$976$	−38.3579	−1.22781
$977$	−18.9681	−0.606843	−0.303422	−	0.952856i	$-0.598129\pi$
$977$	−18.9681	−0.606843	−0.303422	+	0.952856i	$0.598129\pi$
$978$	0	0
$979$	2.96336	0.0947095
$980$	0.624558	0.0199508
$981$	0	0
$982$	26.1512	0.834517
$983$	−1.42191	−0.0453519	−0.0226760	−	0.999743i	$-0.507219\pi$
$983$	−1.42191	−0.0453519	−0.0226760	+	0.999743i	$0.507219\pi$
$984$	0	0
$985$	11.3915	0.362964
$986$	−8.90752	−0.283673
$987$	0	0
$988$	−1.56302	−0.0497264
$989$	6.87624	0.218652
$990$	0	0
$991$	−43.7528	−1.38985	−0.694927	−	0.719080i	$-0.744563\pi$
$991$	−43.7528	−1.38985	−0.694927	+	0.719080i	$0.744563\pi$
$992$	6.02776	0.191381
$993$	0	0
$994$	−2.80835	−0.0890756
$995$	9.91842	0.314435
$996$	0	0
$997$	0.221132	0.00700333	0.00350167	−	0.999994i	$-0.498885\pi$
$997$	0.221132	0.00700333	0.00350167	+	0.999994i	$0.498885\pi$
$998$	−45.5258	−1.44109
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.v.1.23	✓	30
3.2	odd	2		6003.2.a.w.1.8	yes	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6003.2.a.v.1.23	✓	30	1.1	even	1	trivial
6003.2.a.w.1.8	yes	30	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+1.59580 q^{2} +0.546582 q^{4} -1.23722 q^{5} +2.46504 q^{7} -2.31937 q^{8} +O(q^{10})\)	\(q+1.59580 q^{2} +0.546582 q^{4} -1.23722 q^{5} +2.46504 q^{7} -2.31937 q^{8} -1.97436 q^{10} -4.04090 q^{11} +6.08460 q^{13} +3.93372 q^{14} -4.79441 q^{16} +5.58185 q^{17} -0.469978 q^{19} -0.676244 q^{20} -6.44848 q^{22} +1.00000 q^{23} -3.46928 q^{25} +9.70981 q^{26} +1.34735 q^{28} -1.00000 q^{29} -2.00112 q^{31} -3.01220 q^{32} +8.90752 q^{34} -3.04981 q^{35} +5.90581 q^{37} -0.749992 q^{38} +2.86957 q^{40} -8.01822 q^{41} +6.87624 q^{43} -2.20869 q^{44} +1.59580 q^{46} +8.37297 q^{47} -0.923569 q^{49} -5.53628 q^{50} +3.32573 q^{52} -2.29752 q^{53} +4.99950 q^{55} -5.71733 q^{56} -1.59580 q^{58} -6.71576 q^{59} +8.00054 q^{61} -3.19338 q^{62} +4.78196 q^{64} -7.52800 q^{65} +6.11908 q^{67} +3.05094 q^{68} -4.86688 q^{70} -0.713918 q^{71} +13.9116 q^{73} +9.42450 q^{74} -0.256882 q^{76} -9.96100 q^{77} +2.01641 q^{79} +5.93176 q^{80} -12.7955 q^{82} +7.66901 q^{83} -6.90599 q^{85} +10.9731 q^{86} +9.37233 q^{88} -0.733342 q^{89} +14.9988 q^{91} +0.546582 q^{92} +13.3616 q^{94} +0.581468 q^{95} -12.7405 q^{97} -1.47383 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - q^{2} + 37 q^{4} + 10 q^{7} - 6 q^{8}+O(q^{10}) \) 30 * q - q^2 + 37 * q^4 + 10 * q^7 - 6 * q^8	\( 30 q - q^{2} + 37 q^{4} + 10 q^{7} - 6 q^{8} + 8 q^{10} + 36 q^{13} - 7 q^{14} + 47 q^{16} - 18 q^{17} + 16 q^{19} + 25 q^{22} + 30 q^{23} + 56 q^{25} - 11 q^{26} + 27 q^{28} - 30 q^{29} + 14 q^{31} + 7 q^{32} + 3 q^{34} + 22 q^{35} + 40 q^{37} - 6 q^{38} + 30 q^{40} - 14 q^{41} + 34 q^{43} - 5 q^{44} - q^{46} + 2 q^{47} + 74 q^{49} + 21 q^{50} + 71 q^{52} - 16 q^{53} + 22 q^{55} - 14 q^{56} + q^{58} + 32 q^{59} + 46 q^{61} - 20 q^{62} + 68 q^{64} - 12 q^{65} + 14 q^{67} - 27 q^{68} + 32 q^{71} + 50 q^{73} + 26 q^{74} + 56 q^{76} - 34 q^{77} + 16 q^{79} - 2 q^{80} + 38 q^{82} + 14 q^{83} + 38 q^{85} - 10 q^{86} + 40 q^{88} + 2 q^{89} + 32 q^{91} + 37 q^{92} + 29 q^{94} + 28 q^{95} + 56 q^{97} - 8 q^{98}+O(q^{100}) \) 30 * q - q^2 + 37 * q^4 + 10 * q^7 - 6 * q^8 + 8 * q^10 + 36 * q^13 - 7 * q^14 + 47 * q^16 - 18 * q^17 + 16 * q^19 + 25 * q^22 + 30 * q^23 + 56 * q^25 - 11 * q^26 + 27 * q^28 - 30 * q^29 + 14 * q^31 + 7 * q^32 + 3 * q^34 + 22 * q^35 + 40 * q^37 - 6 * q^38 + 30 * q^40 - 14 * q^41 + 34 * q^43 - 5 * q^44 - q^46 + 2 * q^47 + 74 * q^49 + 21 * q^50 + 71 * q^52 - 16 * q^53 + 22 * q^55 - 14 * q^56 + q^58 + 32 * q^59 + 46 * q^61 - 20 * q^62 + 68 * q^64 - 12 * q^65 + 14 * q^67 - 27 * q^68 + 32 * q^71 + 50 * q^73 + 26 * q^74 + 56 * q^76 - 34 * q^77 + 16 * q^79 - 2 * q^80 + 38 * q^82 + 14 * q^83 + 38 * q^85 - 10 * q^86 + 40 * q^88 + 2 * q^89 + 32 * q^91 + 37 * q^92 + 29 * q^94 + 28 * q^95 + 56 * q^97 - 8 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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