LMFDB - Embedded newform 6003.2.a.j.1.4

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:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 5x^{5} + 18x^{4} + 4x^{3} - 26x^{2} + x + 8 $ x^7 - 3x^6 - 5x^5 + 18x^4 + 4x^3 - 26*x^2 + x + 8
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2001)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.743347$ 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)

$f(q)$	$=$	$q+0.743347 q^{2} -1.44743 q^{4} +3.36180 q^{5} -2.98340 q^{7} -2.56264 q^{8} +O(q^{10})$	\(q+0.743347 q^{2} -1.44743 q^{4} +3.36180 q^{5} -2.98340 q^{7} -2.56264 q^{8} +2.49898 q^{10} -1.73328 q^{11} +0.418547 q^{13} -2.21770 q^{14} +0.989937 q^{16} -3.04186 q^{17} +6.79263 q^{19} -4.86598 q^{20} -1.28843 q^{22} -1.00000 q^{23} +6.30168 q^{25} +0.311125 q^{26} +4.31828 q^{28} +1.00000 q^{29} -1.26899 q^{31} +5.86115 q^{32} -2.26116 q^{34} -10.0296 q^{35} -8.40619 q^{37} +5.04928 q^{38} -8.61508 q^{40} +5.67691 q^{41} +2.38122 q^{43} +2.50882 q^{44} -0.743347 q^{46} +1.01953 q^{47} +1.90067 q^{49} +4.68433 q^{50} -0.605819 q^{52} -1.08924 q^{53} -5.82695 q^{55} +7.64538 q^{56} +0.743347 q^{58} +0.434930 q^{59} -11.8388 q^{61} -0.943302 q^{62} +2.37699 q^{64} +1.40707 q^{65} -4.29594 q^{67} +4.40289 q^{68} -7.45546 q^{70} +7.01482 q^{71} -9.28137 q^{73} -6.24872 q^{74} -9.83189 q^{76} +5.17108 q^{77} +2.43604 q^{79} +3.32797 q^{80} +4.21991 q^{82} -15.3185 q^{83} -10.2261 q^{85} +1.77007 q^{86} +4.44179 q^{88} -11.8322 q^{89} -1.24869 q^{91} +1.44743 q^{92} +0.757862 q^{94} +22.8354 q^{95} +4.37241 q^{97} +1.41286 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 3 q^{2} + 5 q^{4} + 3 q^{5} - 5 q^{7} + 6 q^{8}+O(q^{10}) $ 7 * q + 3 * q^2 + 5 * q^4 + 3 * q^5 - 5 * q^7 + 6 * q^8	$ 7 q + 3 q^{2} + 5 q^{4} + 3 q^{5} - 5 q^{7} + 6 q^{8} + 3 q^{10} + 4 q^{11} - 18 q^{13} + 2 q^{14} - 7 q^{16} + 3 q^{17} - 4 q^{19} + 2 q^{20} - 26 q^{22} - 7 q^{23} - 8 q^{25} + 7 q^{26} - 6 q^{28} + 7 q^{29} - 22 q^{31} - 5 q^{32} + 9 q^{34} - 3 q^{35} - 25 q^{37} - 14 q^{38} - 10 q^{40} + 13 q^{41} - 2 q^{43} - 4 q^{44} - 3 q^{46} + 25 q^{47} - 8 q^{49} - 19 q^{50} - 12 q^{52} + 5 q^{53} - 15 q^{55} - 18 q^{56} + 3 q^{58} - 11 q^{59} - 33 q^{61} - 28 q^{62} - 14 q^{64} + 2 q^{65} + 8 q^{67} - 12 q^{68} - 22 q^{70} + 6 q^{71} + 15 q^{73} - 34 q^{74} - 28 q^{76} + q^{77} - 15 q^{79} + 12 q^{80} - 14 q^{82} - 21 q^{83} - 28 q^{85} + 12 q^{86} - 13 q^{88} - 8 q^{89} + 6 q^{91} - 5 q^{92} - 35 q^{94} + 25 q^{95} + 13 q^{97} - q^{98}+O(q^{100}) $ 7 * q + 3 * q^2 + 5 * q^4 + 3 * q^5 - 5 * q^7 + 6 * q^8 + 3 * q^10 + 4 * q^11 - 18 * q^13 + 2 * q^14 - 7 * q^16 + 3 * q^17 - 4 * q^19 + 2 * q^20 - 26 * q^22 - 7 * q^23 - 8 * q^25 + 7 * q^26 - 6 * q^28 + 7 * q^29 - 22 * q^31 - 5 * q^32 + 9 * q^34 - 3 * q^35 - 25 * q^37 - 14 * q^38 - 10 * q^40 + 13 * q^41 - 2 * q^43 - 4 * q^44 - 3 * q^46 + 25 * q^47 - 8 * q^49 - 19 * q^50 - 12 * q^52 + 5 * q^53 - 15 * q^55 - 18 * q^56 + 3 * q^58 - 11 * q^59 - 33 * q^61 - 28 * q^62 - 14 * q^64 + 2 * q^65 + 8 * q^67 - 12 * q^68 - 22 * q^70 + 6 * q^71 + 15 * q^73 - 34 * q^74 - 28 * q^76 + q^77 - 15 * q^79 + 12 * q^80 - 14 * q^82 - 21 * q^83 - 28 * q^85 + 12 * q^86 - 13 * q^88 - 8 * q^89 + 6 * q^91 - 5 * q^92 - 35 * q^94 + 25 * q^95 + 13 * q^97 - 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 $

$2$	0.743347	0.525626	0.262813	−	0.964847i	$-0.415350\pi$
$2$	0.743347	0.525626	0.262813	+	0.964847i	$0.415350\pi$
$3$	0	0
$3$	0	0
$4$	−1.44743	−0.723717
$5$	3.36180	1.50344	0.751721	−	0.659482i	$-0.229224\pi$
$5$	3.36180	1.50344	0.751721	+	0.659482i	$0.229224\pi$
$6$	0	0
$7$	−2.98340	−1.12762	−0.563809	−	0.825905i	$-0.690665\pi$
$7$	−2.98340	−1.12762	−0.563809	+	0.825905i	$0.690665\pi$
$8$	−2.56264	−0.906031
$9$	0	0
$10$	2.49898	0.790248
$11$	−1.73328	−0.522605	−0.261302	−	0.965257i	$-0.584152\pi$
$11$	−1.73328	−0.522605	−0.261302	+	0.965257i	$0.584152\pi$
$12$	0	0
$13$	0.418547	0.116084	0.0580420	−	0.998314i	$-0.481514\pi$
$13$	0.418547	0.116084	0.0580420	+	0.998314i	$0.481514\pi$
$14$	−2.21770	−0.592706
$15$	0	0
$16$	0.989937	0.247484
$17$	−3.04186	−0.737758	−0.368879	−	0.929477i	$-0.620258\pi$
$17$	−3.04186	−0.737758	−0.368879	+	0.929477i	$0.620258\pi$
$18$	0	0
$19$	6.79263	1.55834	0.779168	−	0.626815i	$-0.215642\pi$
$19$	6.79263	1.55834	0.779168	+	0.626815i	$0.215642\pi$
$20$	−4.86598	−1.08807
$21$	0	0
$22$	−1.28843	−0.274695
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	6.30168	1.26034
$26$	0.311125	0.0610167
$27$	0	0
$28$	4.31828	0.816077
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−1.26899	−0.227918	−0.113959	−	0.993485i	$-0.536353\pi$
$31$	−1.26899	−0.227918	−0.113959	+	0.993485i	$0.536353\pi$
$32$	5.86115	1.03611
$33$	0	0
$34$	−2.26116	−0.387785
$35$	−10.0296	−1.69531
$36$	0	0
$37$	−8.40619	−1.38197	−0.690985	−	0.722869i	$-0.742823\pi$
$37$	−8.40619	−1.38197	−0.690985	+	0.722869i	$0.742823\pi$
$38$	5.04928	0.819102
$39$	0	0
$40$	−8.61508	−1.36216
$41$	5.67691	0.886584	0.443292	−	0.896377i	$-0.353811\pi$
$41$	5.67691	0.886584	0.443292	+	0.896377i	$0.353811\pi$
$42$	0	0
$43$	2.38122	0.363133	0.181566	−	0.983379i	$-0.441883\pi$
$43$	2.38122	0.363133	0.181566	+	0.983379i	$0.441883\pi$
$44$	2.50882	0.378218
$45$	0	0
$46$	−0.743347	−0.109601
$47$	1.01953	0.148713	0.0743566	−	0.997232i	$-0.476310\pi$
$47$	1.01953	0.148713	0.0743566	+	0.997232i	$0.476310\pi$
$48$	0	0
$49$	1.90067	0.271525
$50$	4.68433	0.662465
$51$	0	0
$52$	−0.605819	−0.0840120
$53$	−1.08924	−0.149619	−0.0748095	−	0.997198i	$-0.523835\pi$
$53$	−1.08924	−0.149619	−0.0748095	+	0.997198i	$0.523835\pi$
$54$	0	0
$55$	−5.82695	−0.785706
$56$	7.64538	1.02166
$57$	0	0
$58$	0.743347	0.0976063
$59$	0.434930	0.0566231	0.0283115	−	0.999599i	$-0.490987\pi$
$59$	0.434930	0.0566231	0.0283115	+	0.999599i	$0.490987\pi$
$60$	0	0
$61$	−11.8388	−1.51580	−0.757901	−	0.652369i	$-0.773775\pi$
$61$	−11.8388	−1.51580	−0.757901	+	0.652369i	$0.773775\pi$
$62$	−0.943302	−0.119800
$63$	0	0
$64$	2.37699	0.297124
$65$	1.40707	0.174525
$66$	0	0
$67$	−4.29594	−0.524833	−0.262416	−	0.964955i	$-0.584519\pi$
$67$	−4.29594	−0.524833	−0.262416	+	0.964955i	$0.584519\pi$
$68$	4.40289	0.533929
$69$	0	0
$70$	−7.45546	−0.891098
$71$	7.01482	0.832506	0.416253	−	0.909249i	$-0.363343\pi$
$71$	7.01482	0.832506	0.416253	+	0.909249i	$0.363343\pi$
$72$	0	0
$73$	−9.28137	−1.08630	−0.543151	−	0.839635i	$-0.682769\pi$
$73$	−9.28137	−1.08630	−0.543151	+	0.839635i	$0.682769\pi$
$74$	−6.24872	−0.726399
$75$	0	0
$76$	−9.83189	−1.12780
$77$	5.17108	0.589299
$78$	0	0
$79$	2.43604	0.274076	0.137038	−	0.990566i	$-0.456242\pi$
$79$	2.43604	0.274076	0.137038	+	0.990566i	$0.456242\pi$
$80$	3.32797	0.372078
$81$	0	0
$82$	4.21991	0.466011
$83$	−15.3185	−1.68142	−0.840712	−	0.541482i	$-0.817864\pi$
$83$	−15.3185	−1.68142	−0.840712	+	0.541482i	$0.817864\pi$
$84$	0	0
$85$	−10.2261	−1.10918
$86$	1.77007	0.190872
$87$	0	0
$88$	4.44179	0.473496
$89$	−11.8322	−1.25421	−0.627105	−	0.778935i	$-0.715760\pi$
$89$	−11.8322	−1.25421	−0.627105	+	0.778935i	$0.715760\pi$
$90$	0	0
$91$	−1.24869	−0.130898
$92$	1.44743	0.150906
$93$	0	0
$94$	0.757862	0.0781675
$95$	22.8354	2.34287
$96$	0	0
$97$	4.37241	0.443951	0.221975	−	0.975052i	$-0.428750\pi$
$97$	4.37241	0.443951	0.221975	+	0.975052i	$0.428750\pi$
$98$	1.41286	0.142720
$99$	0	0
$100$	−9.12126	−0.912126
$101$	7.23477	0.719886	0.359943	−	0.932974i	$-0.382796\pi$
$101$	7.23477	0.719886	0.359943	+	0.932974i	$0.382796\pi$
$102$	0	0
$103$	−17.9998	−1.77358	−0.886788	−	0.462176i	$-0.847069\pi$
$103$	−17.9998	−1.77358	−0.886788	+	0.462176i	$0.847069\pi$
$104$	−1.07258	−0.105176
$105$	0	0
$106$	−0.809686	−0.0786437
$107$	5.18130	0.500895	0.250447	−	0.968130i	$-0.419422\pi$
$107$	5.18130	0.500895	0.250447	+	0.968130i	$0.419422\pi$
$108$	0	0
$109$	−11.3146	−1.08375	−0.541873	−	0.840460i	$-0.682285\pi$
$109$	−11.3146	−1.08375	−0.541873	+	0.840460i	$0.682285\pi$
$110$	−4.33145	−0.412987
$111$	0	0
$112$	−2.95338	−0.279068
$113$	−6.08524	−0.572451	−0.286226	−	0.958162i	$-0.592401\pi$
$113$	−6.08524	−0.572451	−0.286226	+	0.958162i	$0.592401\pi$
$114$	0	0
$115$	−3.36180	−0.313489
$116$	−1.44743	−0.134391
$117$	0	0
$118$	0.323304	0.0297625
$119$	9.07507	0.831910
$120$	0	0
$121$	−7.99572	−0.726884
$122$	−8.80034	−0.796745
$123$	0	0
$124$	1.83678	0.164948
$125$	4.37597	0.391399
$126$	0	0
$127$	−2.68174	−0.237966	−0.118983	−	0.992896i	$-0.537963\pi$
$127$	−2.68174	−0.237966	−0.118983	+	0.992896i	$0.537963\pi$
$128$	−9.95537	−0.879938
$129$	0	0
$130$	1.04594	0.0917350
$131$	−5.26968	−0.460414	−0.230207	−	0.973142i	$-0.573940\pi$
$131$	−5.26968	−0.460414	−0.230207	+	0.973142i	$0.573940\pi$
$132$	0	0
$133$	−20.2651	−1.75721
$134$	−3.19338	−0.275866
$135$	0	0
$136$	7.79518	0.668432
$137$	−11.0969	−0.948071	−0.474036	−	0.880506i	$-0.657203\pi$
$137$	−11.0969	−0.948071	−0.474036	+	0.880506i	$0.657203\pi$
$138$	0	0
$139$	−13.7752	−1.16840	−0.584200	−	0.811610i	$-0.698592\pi$
$139$	−13.7752	−1.16840	−0.584200	+	0.811610i	$0.698592\pi$
$140$	14.5172	1.22692
$141$	0	0
$142$	5.21445	0.437587
$143$	−0.725460	−0.0606660
$144$	0	0
$145$	3.36180	0.279182
$146$	−6.89928	−0.570989
$147$	0	0
$148$	12.1674	1.00016
$149$	−2.15134	−0.176244	−0.0881222	−	0.996110i	$-0.528087\pi$
$149$	−2.15134	−0.176244	−0.0881222	+	0.996110i	$0.528087\pi$
$150$	0	0
$151$	18.3565	1.49383	0.746915	−	0.664920i	$-0.231534\pi$
$151$	18.3565	1.49383	0.746915	+	0.664920i	$0.231534\pi$
$152$	−17.4071	−1.41190
$153$	0	0
$154$	3.84391	0.309751
$155$	−4.26610	−0.342661
$156$	0	0
$157$	−0.545808	−0.0435602	−0.0217801	−	0.999763i	$-0.506933\pi$
$157$	−0.545808	−0.0435602	−0.0217801	+	0.999763i	$0.506933\pi$
$158$	1.81082	0.144061
$159$	0	0
$160$	19.7040	1.55774
$161$	2.98340	0.235125
$162$	0	0
$163$	−0.544969	−0.0426853	−0.0213426	−	0.999772i	$-0.506794\pi$
$163$	−0.544969	−0.0426853	−0.0213426	+	0.999772i	$0.506794\pi$
$164$	−8.21695	−0.641636
$165$	0	0
$166$	−11.3870	−0.883800
$167$	−13.5418	−1.04790	−0.523950	−	0.851749i	$-0.675542\pi$
$167$	−13.5418	−1.04790	−0.523950	+	0.851749i	$0.675542\pi$
$168$	0	0
$169$	−12.8248	−0.986525
$170$	−7.60154	−0.583012
$171$	0	0
$172$	−3.44666	−0.262806
$173$	2.48390	0.188847	0.0944236	−	0.995532i	$-0.469899\pi$
$173$	2.48390	0.188847	0.0944236	+	0.995532i	$0.469899\pi$
$174$	0	0
$175$	−18.8004	−1.42118
$176$	−1.71584	−0.129337
$177$	0	0
$178$	−8.79543	−0.659245
$179$	8.19708	0.612679	0.306339	−	0.951922i	$-0.400896\pi$
$179$	8.19708	0.612679	0.306339	+	0.951922i	$0.400896\pi$
$180$	0	0
$181$	−8.21935	−0.610940	−0.305470	−	0.952202i	$-0.598814\pi$
$181$	−8.21935	−0.610940	−0.305470	+	0.952202i	$0.598814\pi$
$182$	−0.928211	−0.0688036
$183$	0	0
$184$	2.56264	0.188920
$185$	−28.2599	−2.07771
$186$	0	0
$187$	5.27240	0.385556
$188$	−1.47570	−0.107626
$189$	0	0
$190$	16.9747	1.23147
$191$	14.0910	1.01959	0.509794	−	0.860296i	$-0.329722\pi$
$191$	14.0910	1.01959	0.509794	+	0.860296i	$0.329722\pi$
$192$	0	0
$193$	0.272693	0.0196289	0.00981444	−	0.999952i	$-0.496876\pi$
$193$	0.272693	0.0196289	0.00981444	+	0.999952i	$0.496876\pi$
$194$	3.25022	0.233352
$195$	0	0
$196$	−2.75110	−0.196507
$197$	−23.5938	−1.68099	−0.840495	−	0.541819i	$-0.817736\pi$
$197$	−23.5938	−1.68099	−0.840495	+	0.541819i	$0.817736\pi$
$198$	0	0
$199$	3.51044	0.248848	0.124424	−	0.992229i	$-0.460292\pi$
$199$	3.51044	0.248848	0.124424	+	0.992229i	$0.460292\pi$
$200$	−16.1489	−1.14190
$201$	0	0
$202$	5.37795	0.378391
$203$	−2.98340	−0.209394
$204$	0	0
$205$	19.0846	1.33293
$206$	−13.3801	−0.932238
$207$	0	0
$208$	0.414335	0.0287290
$209$	−11.7736	−0.814394
$210$	0	0
$211$	−0.138150	−0.00951064	−0.00475532	−	0.999989i	$-0.501514\pi$
$211$	−0.138150	−0.00951064	−0.00475532	+	0.999989i	$0.501514\pi$
$212$	1.57661	0.108282
$213$	0	0
$214$	3.85150	0.263283
$215$	8.00518	0.545949
$216$	0	0
$217$	3.78591	0.257004
$218$	−8.41070	−0.569645
$219$	0	0
$220$	8.43413	0.568629
$221$	−1.27316	−0.0856419
$222$	0	0
$223$	19.7576	1.32306	0.661532	−	0.749917i	$-0.269907\pi$
$223$	19.7576	1.32306	0.661532	+	0.749917i	$0.269907\pi$
$224$	−17.4862	−1.16834
$225$	0	0
$226$	−4.52345	−0.300895
$227$	21.2886	1.41297	0.706486	−	0.707727i	$-0.250279\pi$
$227$	21.2886	1.41297	0.706486	+	0.707727i	$0.250279\pi$
$228$	0	0
$229$	−9.98323	−0.659711	−0.329855	−	0.944031i	$-0.607000\pi$
$229$	−9.98323	−0.659711	−0.329855	+	0.944031i	$0.607000\pi$
$230$	−2.49898	−0.164778
$231$	0	0
$232$	−2.56264	−0.168246
$233$	11.7610	0.770490	0.385245	−	0.922814i	$-0.374117\pi$
$233$	11.7610	0.770490	0.385245	+	0.922814i	$0.374117\pi$
$234$	0	0
$235$	3.42744	0.223581
$236$	−0.629533	−0.0409791
$237$	0	0
$238$	6.74593	0.437274
$239$	14.7029	0.951053	0.475527	−	0.879701i	$-0.342258\pi$
$239$	14.7029	0.951053	0.475527	+	0.879701i	$0.342258\pi$
$240$	0	0
$241$	−6.48179	−0.417529	−0.208764	−	0.977966i	$-0.566944\pi$
$241$	−6.48179	−0.417529	−0.208764	+	0.977966i	$0.566944\pi$
$242$	−5.94360	−0.382069
$243$	0	0
$244$	17.1359	1.09701
$245$	6.38967	0.408221
$246$	0	0
$247$	2.84303	0.180898
$248$	3.25197	0.206501
$249$	0	0
$250$	3.25286	0.205729
$251$	−5.51951	−0.348389	−0.174194	−	0.984711i	$-0.555732\pi$
$251$	−5.51951	−0.348389	−0.174194	+	0.984711i	$0.555732\pi$
$252$	0	0
$253$	1.73328	0.108971
$254$	−1.99347	−0.125081
$255$	0	0
$256$	−12.1543	−0.759643
$257$	20.9651	1.30776	0.653882	−	0.756597i	$-0.273139\pi$
$257$	20.9651	1.30776	0.653882	+	0.756597i	$0.273139\pi$
$258$	0	0
$259$	25.0790	1.55833
$260$	−2.03664	−0.126307
$261$	0	0
$262$	−3.91720	−0.242005
$263$	−4.37099	−0.269527	−0.134763	−	0.990878i	$-0.543027\pi$
$263$	−4.37099	−0.269527	−0.134763	+	0.990878i	$0.543027\pi$
$264$	0	0
$265$	−3.66181	−0.224943
$266$	−15.0640	−0.923635
$267$	0	0
$268$	6.21809	0.379831
$269$	18.5392	1.13035	0.565177	−	0.824970i	$-0.308808\pi$
$269$	18.5392	1.13035	0.565177	+	0.824970i	$0.308808\pi$
$270$	0	0
$271$	−17.8222	−1.08262	−0.541311	−	0.840822i	$-0.682072\pi$
$271$	−17.8222	−1.08262	−0.541311	+	0.840822i	$0.682072\pi$
$272$	−3.01125	−0.182584
$273$	0	0
$274$	−8.24884	−0.498331
$275$	−10.9226	−0.658657
$276$	0	0
$277$	0.652922	0.0392303	0.0196152	−	0.999808i	$-0.493756\pi$
$277$	0.652922	0.0392303	0.0196152	+	0.999808i	$0.493756\pi$
$278$	−10.2398	−0.614142
$279$	0	0
$280$	25.7022	1.53600
$281$	24.5072	1.46198	0.730988	−	0.682390i	$-0.239060\pi$
$281$	24.5072	1.46198	0.730988	+	0.682390i	$0.239060\pi$
$282$	0	0
$283$	−29.4332	−1.74962	−0.874810	−	0.484466i	$-0.839014\pi$
$283$	−29.4332	−1.74962	−0.874810	+	0.484466i	$0.839014\pi$
$284$	−10.1535	−0.602499
$285$	0	0
$286$	−0.539269	−0.0318876
$287$	−16.9365	−0.999729
$288$	0	0
$289$	−7.74711	−0.455713
$290$	2.49898	0.146745
$291$	0	0
$292$	13.4342	0.786176
$293$	−10.4542	−0.610739	−0.305369	−	0.952234i	$-0.598780\pi$
$293$	−10.4542	−0.610739	−0.305369	+	0.952234i	$0.598780\pi$
$294$	0	0
$295$	1.46215	0.0851294
$296$	21.5421	1.25211
$297$	0	0
$298$	−1.59919	−0.0926387
$299$	−0.418547	−0.0242052
$300$	0	0
$301$	−7.10413	−0.409476
$302$	13.6452	0.785195
$303$	0	0
$304$	6.72428	0.385664
$305$	−39.7996	−2.27892
$306$	0	0
$307$	−29.3141	−1.67304	−0.836522	−	0.547933i	$-0.815415\pi$
$307$	−29.3141	−1.67304	−0.836522	+	0.547933i	$0.815415\pi$
$308$	−7.48480	−0.426486
$309$	0	0
$310$	−3.17119	−0.180112
$311$	14.2023	0.805338	0.402669	−	0.915346i	$-0.368083\pi$
$311$	14.2023	0.805338	0.402669	+	0.915346i	$0.368083\pi$
$312$	0	0
$313$	−18.3008	−1.03443	−0.517213	−	0.855857i	$-0.673030\pi$
$313$	−18.3008	−1.03443	−0.517213	+	0.855857i	$0.673030\pi$
$314$	−0.405725	−0.0228964
$315$	0	0
$316$	−3.52600	−0.198353
$317$	−1.64584	−0.0924397	−0.0462198	−	0.998931i	$-0.514717\pi$
$317$	−1.64584	−0.0924397	−0.0462198	+	0.998931i	$0.514717\pi$
$318$	0	0
$319$	−1.73328	−0.0970453
$320$	7.99097	0.446709
$321$	0	0
$322$	2.21770	0.123588
$323$	−20.6622	−1.14968
$324$	0	0
$325$	2.63754	0.146305
$326$	−0.405101	−0.0224365
$327$	0	0
$328$	−14.5479	−0.803272
$329$	−3.04165	−0.167692
$330$	0	0
$331$	−14.4258	−0.792914	−0.396457	−	0.918053i	$-0.629760\pi$
$331$	−14.4258	−0.792914	−0.396457	+	0.918053i	$0.629760\pi$
$332$	22.1725	1.21688
$333$	0	0
$334$	−10.0663	−0.550803
$335$	−14.4421	−0.789055
$336$	0	0
$337$	−7.79363	−0.424546	−0.212273	−	0.977210i	$-0.568087\pi$
$337$	−7.79363	−0.424546	−0.212273	+	0.977210i	$0.568087\pi$
$338$	−9.53329	−0.518543
$339$	0	0
$340$	14.8016	0.802730
$341$	2.19953	0.119111
$342$	0	0
$343$	15.2133	0.821443
$344$	−6.10222	−0.329009
$345$	0	0
$346$	1.84640	0.0992630
$347$	27.5605	1.47953	0.739764	−	0.672867i	$-0.234937\pi$
$347$	27.5605	1.47953	0.739764	+	0.672867i	$0.234937\pi$
$348$	0	0
$349$	−29.9855	−1.60509	−0.802543	−	0.596594i	$-0.796520\pi$
$349$	−29.9855	−1.60509	−0.802543	+	0.596594i	$0.796520\pi$
$350$	−13.9752	−0.747008
$351$	0	0
$352$	−10.1590	−0.541479
$353$	29.6920	1.58034	0.790172	−	0.612885i	$-0.209991\pi$
$353$	29.6920	1.58034	0.790172	+	0.612885i	$0.209991\pi$
$354$	0	0
$355$	23.5824	1.25162
$356$	17.1263	0.907694
$357$	0	0
$358$	6.09328	0.322040
$359$	−2.09906	−0.110784	−0.0553921	−	0.998465i	$-0.517641\pi$
$359$	−2.09906	−0.110784	−0.0553921	+	0.998465i	$0.517641\pi$
$360$	0	0
$361$	27.1398	1.42841
$362$	−6.10983	−0.321126
$363$	0	0
$364$	1.80740	0.0947335
$365$	−31.2021	−1.63319
$366$	0	0
$367$	19.4918	1.01746	0.508732	−	0.860925i	$-0.330114\pi$
$367$	19.4918	1.01746	0.508732	+	0.860925i	$0.330114\pi$
$368$	−0.989937	−0.0516041
$369$	0	0
$370$	−21.0069	−1.09210
$371$	3.24965	0.168713
$372$	0	0
$373$	−33.4309	−1.73099	−0.865494	−	0.500919i	$-0.832995\pi$
$373$	−33.4309	−1.73099	−0.865494	+	0.500919i	$0.832995\pi$
$374$	3.91923	0.202658
$375$	0	0
$376$	−2.61268	−0.134739
$377$	0.418547	0.0215562
$378$	0	0
$379$	23.4822	1.20620	0.603100	−	0.797665i	$-0.293932\pi$
$379$	23.4822	1.20620	0.603100	+	0.797665i	$0.293932\pi$
$380$	−33.0528	−1.69557
$381$	0	0
$382$	10.4745	0.535922
$383$	−22.0092	−1.12462	−0.562310	−	0.826927i	$-0.690087\pi$
$383$	−22.0092	−1.12462	−0.562310	+	0.826927i	$0.690087\pi$
$384$	0	0
$385$	17.3841	0.885977
$386$	0.202706	0.0103174
$387$	0	0
$388$	−6.32877	−0.321295
$389$	−14.7724	−0.748991	−0.374496	−	0.927229i	$-0.622184\pi$
$389$	−14.7724	−0.748991	−0.374496	+	0.927229i	$0.622184\pi$
$390$	0	0
$391$	3.04186	0.153833
$392$	−4.87074	−0.246010
$393$	0	0
$394$	−17.5384	−0.883572
$395$	8.18946	0.412056
$396$	0	0
$397$	21.7744	1.09283	0.546413	−	0.837516i	$-0.315993\pi$
$397$	21.7744	1.09283	0.546413	+	0.837516i	$0.315993\pi$
$398$	2.60947	0.130801
$399$	0	0
$400$	6.23826	0.311913
$401$	2.37651	0.118677	0.0593387	−	0.998238i	$-0.481101\pi$
$401$	2.37651	0.118677	0.0593387	+	0.998238i	$0.481101\pi$
$402$	0	0
$403$	−0.531133	−0.0264576
$404$	−10.4719	−0.520994
$405$	0	0
$406$	−2.21770	−0.110063
$407$	14.5703	0.722224
$408$	0	0
$409$	0.143213	0.00708141	0.00354071	−	0.999994i	$-0.498873\pi$
$409$	0.143213	0.00708141	0.00354071	+	0.999994i	$0.498873\pi$
$410$	14.1865	0.700621
$411$	0	0
$412$	26.0536	1.28357
$413$	−1.29757	−0.0638492
$414$	0	0
$415$	−51.4977	−2.52792
$416$	2.45316	0.120276
$417$	0	0
$418$	−8.75185	−0.428067
$419$	1.85725	0.0907324	0.0453662	−	0.998970i	$-0.485555\pi$
$419$	1.85725	0.0907324	0.0453662	+	0.998970i	$0.485555\pi$
$420$	0	0
$421$	16.7959	0.818584	0.409292	−	0.912403i	$-0.365776\pi$
$421$	16.7959	0.818584	0.409292	+	0.912403i	$0.365776\pi$
$422$	−0.102693	−0.00499904
$423$	0	0
$424$	2.79134	0.135559
$425$	−19.1688	−0.929823
$426$	0	0
$427$	35.3199	1.70925
$428$	−7.49959	−0.362506
$429$	0	0
$430$	5.95063	0.286965
$431$	−15.4833	−0.745803	−0.372901	−	0.927871i	$-0.621637\pi$
$431$	−15.4833	−0.745803	−0.372901	+	0.927871i	$0.621637\pi$
$432$	0	0
$433$	−10.8387	−0.520873	−0.260436	−	0.965491i	$-0.583866\pi$
$433$	−10.8387	−0.520873	−0.260436	+	0.965491i	$0.583866\pi$
$434$	2.81425	0.135088
$435$	0	0
$436$	16.3772	0.784326
$437$	−6.79263	−0.324936
$438$	0	0
$439$	13.0587	0.623260	0.311630	−	0.950204i	$-0.399125\pi$
$439$	13.0587	0.623260	0.311630	+	0.950204i	$0.399125\pi$
$440$	14.9324	0.711873
$441$	0	0
$442$	−0.946399	−0.0450156
$443$	13.9516	0.662861	0.331431	−	0.943480i	$-0.392469\pi$
$443$	13.9516	0.662861	0.331431	+	0.943480i	$0.392469\pi$
$444$	0	0
$445$	−39.7774	−1.88563
$446$	14.6867	0.695437
$447$	0	0
$448$	−7.09153	−0.335043
$449$	−0.108496	−0.00512024	−0.00256012	−	0.999997i	$-0.500815\pi$
$449$	−0.108496	−0.00512024	−0.00256012	+	0.999997i	$0.500815\pi$
$450$	0	0
$451$	−9.83969	−0.463333
$452$	8.80799	0.414293
$453$	0	0
$454$	15.8248	0.742695
$455$	−4.19785	−0.196798
$456$	0	0
$457$	28.6297	1.33924	0.669621	−	0.742703i	$-0.266457\pi$
$457$	28.6297	1.33924	0.669621	+	0.742703i	$0.266457\pi$
$458$	−7.42101	−0.346761
$459$	0	0
$460$	4.86598	0.226878
$461$	−27.3672	−1.27462	−0.637310	−	0.770608i	$-0.719953\pi$
$461$	−27.3672	−1.27462	−0.637310	+	0.770608i	$0.719953\pi$
$462$	0	0
$463$	−9.24333	−0.429574	−0.214787	−	0.976661i	$-0.568906\pi$
$463$	−9.24333	−0.429574	−0.214787	+	0.976661i	$0.568906\pi$
$464$	0.989937	0.0459567
$465$	0	0
$466$	8.74253	0.404990
$467$	−27.1433	−1.25604	−0.628022	−	0.778196i	$-0.716135\pi$
$467$	−27.1433	−1.25604	−0.628022	+	0.778196i	$0.716135\pi$
$468$	0	0
$469$	12.8165	0.591811
$470$	2.54778	0.117520
$471$	0	0
$472$	−1.11457	−0.0513022
$473$	−4.12733	−0.189775
$474$	0	0
$475$	42.8050	1.96403
$476$	−13.1356	−0.602068
$477$	0	0
$478$	10.9294	0.499898
$479$	15.3490	0.701315	0.350657	−	0.936504i	$-0.385958\pi$
$479$	15.3490	0.701315	0.350657	+	0.936504i	$0.385958\pi$
$480$	0	0
$481$	−3.51838	−0.160424
$482$	−4.81822	−0.219464
$483$	0	0
$484$	11.5733	0.526059
$485$	14.6991	0.667454
$486$	0	0
$487$	−23.2996	−1.05581	−0.527904	−	0.849304i	$-0.677022\pi$
$487$	−23.2996	−1.05581	−0.527904	+	0.849304i	$0.677022\pi$
$488$	30.3386	1.37336
$489$	0	0
$490$	4.74974	0.214572
$491$	31.9633	1.44248	0.721242	−	0.692683i	$-0.243571\pi$
$491$	31.9633	1.44248	0.721242	+	0.692683i	$0.243571\pi$
$492$	0	0
$493$	−3.04186	−0.136998
$494$	2.11336	0.0950846
$495$	0	0
$496$	−1.25622	−0.0564061
$497$	−20.9280	−0.938750
$498$	0	0
$499$	20.4662	0.916194	0.458097	−	0.888902i	$-0.348531\pi$
$499$	20.4662	0.916194	0.458097	+	0.888902i	$0.348531\pi$
$500$	−6.33393	−0.283262
$501$	0	0
$502$	−4.10292	−0.183122
$503$	0.0476041	0.00212256	0.00106128	−	0.999999i	$-0.499662\pi$
$503$	0.0476041	0.00212256	0.00106128	+	0.999999i	$0.499662\pi$
$504$	0	0
$505$	24.3218	1.08231
$506$	1.28843	0.0572778
$507$	0	0
$508$	3.88165	0.172220
$509$	−11.6626	−0.516935	−0.258467	−	0.966020i	$-0.583217\pi$
$509$	−11.6626	−0.516935	−0.258467	+	0.966020i	$0.583217\pi$
$510$	0	0
$511$	27.6900	1.22494
$512$	10.8759	0.480651
$513$	0	0
$514$	15.5843	0.687395
$515$	−60.5118	−2.66647
$516$	0	0
$517$	−1.76713	−0.0777182
$518$	18.6424	0.819101
$519$	0	0
$520$	−3.60581	−0.158125
$521$	20.4685	0.896743	0.448371	−	0.893847i	$-0.352004\pi$
$521$	20.4685	0.896743	0.448371	+	0.893847i	$0.352004\pi$
$522$	0	0
$523$	−16.5007	−0.721523	−0.360762	−	0.932658i	$-0.617483\pi$
$523$	−16.5007	−0.721523	−0.360762	+	0.932658i	$0.617483\pi$
$524$	7.62752	0.333210
$525$	0	0
$526$	−3.24916	−0.141670
$527$	3.86009	0.168148
$528$	0	0
$529$	1.00000	0.0434783
$530$	−2.72200	−0.118236
$531$	0	0
$532$	29.3325	1.27172
$533$	2.37605	0.102918
$534$	0	0
$535$	17.4185	0.753066
$536$	11.0090	0.475514
$537$	0	0
$538$	13.7811	0.594144
$539$	−3.29441	−0.141900
$540$	0	0
$541$	−7.89169	−0.339290	−0.169645	−	0.985505i	$-0.554262\pi$
$541$	−7.89169	−0.339290	−0.169645	+	0.985505i	$0.554262\pi$
$542$	−13.2481	−0.569055
$543$	0	0
$544$	−17.8288	−0.764402
$545$	−38.0375	−1.62935
$546$	0	0
$547$	−34.3963	−1.47068	−0.735340	−	0.677698i	$-0.762978\pi$
$547$	−34.3963	−1.47068	−0.735340	+	0.677698i	$0.762978\pi$
$548$	16.0620	0.686136
$549$	0	0
$550$	−8.11928	−0.346207
$551$	6.79263	0.289376
$552$	0	0
$553$	−7.26767	−0.309053
$554$	0.485348	0.0206205
$555$	0	0
$556$	19.9388	0.845592
$557$	−44.5893	−1.88931	−0.944655	−	0.328066i	$-0.893603\pi$
$557$	−44.5893	−1.88931	−0.944655	+	0.328066i	$0.893603\pi$
$558$	0	0
$559$	0.996652	0.0421539
$560$	−9.92866	−0.419562
$561$	0	0
$562$	18.2173	0.768452
$563$	1.52081	0.0640943	0.0320472	−	0.999486i	$-0.489797\pi$
$563$	1.52081	0.0640943	0.0320472	+	0.999486i	$0.489797\pi$
$564$	0	0
$565$	−20.4573	−0.860647
$566$	−21.8791	−0.919646
$567$	0	0
$568$	−17.9765	−0.754276
$569$	−2.96412	−0.124262	−0.0621311	−	0.998068i	$-0.519790\pi$
$569$	−2.96412	−0.124262	−0.0621311	+	0.998068i	$0.519790\pi$
$570$	0	0
$571$	22.2318	0.930371	0.465185	−	0.885213i	$-0.345988\pi$
$571$	22.2318	0.930371	0.465185	+	0.885213i	$0.345988\pi$
$572$	1.05006	0.0439051
$573$	0	0
$574$	−12.5897	−0.525483
$575$	−6.30168	−0.262798
$576$	0	0
$577$	1.31813	0.0548745	0.0274373	−	0.999624i	$-0.491265\pi$
$577$	1.31813	0.0548745	0.0274373	+	0.999624i	$0.491265\pi$
$578$	−5.75880	−0.239534
$579$	0	0
$580$	−4.86598	−0.202049
$581$	45.7012	1.89601
$582$	0	0
$583$	1.88797	0.0781917
$584$	23.7848	0.984223
$585$	0	0
$586$	−7.77107	−0.321020
$587$	−9.07620	−0.374615	−0.187307	−	0.982301i	$-0.559976\pi$
$587$	−9.07620	−0.374615	−0.187307	+	0.982301i	$0.559976\pi$
$588$	0	0
$589$	−8.61980	−0.355173
$590$	1.08688	0.0447462
$591$	0	0
$592$	−8.32160	−0.342016
$593$	40.3794	1.65818	0.829092	−	0.559112i	$-0.188858\pi$
$593$	40.3794	1.65818	0.829092	+	0.559112i	$0.188858\pi$
$594$	0	0
$595$	30.5085	1.25073
$596$	3.11392	0.127551
$597$	0	0
$598$	−0.311125	−0.0127229
$599$	−13.4887	−0.551132	−0.275566	−	0.961282i	$-0.588865\pi$
$599$	−13.4887	−0.551132	−0.275566	+	0.961282i	$0.588865\pi$
$600$	0	0
$601$	−11.4881	−0.468608	−0.234304	−	0.972163i	$-0.575281\pi$
$601$	−11.4881	−0.468608	−0.234304	+	0.972163i	$0.575281\pi$
$602$	−5.28084	−0.215231
$603$	0	0
$604$	−26.5698	−1.08111
$605$	−26.8800	−1.09283
$606$	0	0
$607$	−3.94753	−0.160225	−0.0801126	−	0.996786i	$-0.525528\pi$
$607$	−3.94753	−0.160225	−0.0801126	+	0.996786i	$0.525528\pi$
$608$	39.8126	1.61462
$609$	0	0
$610$	−29.5849	−1.19786
$611$	0.426719	0.0172632
$612$	0	0
$613$	4.36193	0.176177	0.0880884	−	0.996113i	$-0.471924\pi$
$613$	4.36193	0.176177	0.0880884	+	0.996113i	$0.471924\pi$
$614$	−21.7906	−0.879396
$615$	0	0
$616$	−13.2516	−0.533923
$617$	−20.0408	−0.806813	−0.403407	−	0.915021i	$-0.632174\pi$
$617$	−20.0408	−0.806813	−0.403407	+	0.915021i	$0.632174\pi$
$618$	0	0
$619$	31.6150	1.27071	0.635357	−	0.772219i	$-0.280853\pi$
$619$	31.6150	1.27071	0.635357	+	0.772219i	$0.280853\pi$
$620$	6.17490	0.247990
$621$	0	0
$622$	10.5572	0.423306
$623$	35.3002	1.41427
$624$	0	0
$625$	−16.7973	−0.671891
$626$	−13.6039	−0.543721
$627$	0	0
$628$	0.790021	0.0315253
$629$	25.5704	1.01956
$630$	0	0
$631$	17.0088	0.677110	0.338555	−	0.940947i	$-0.390062\pi$
$631$	17.0088	0.677110	0.338555	+	0.940947i	$0.390062\pi$
$632$	−6.24269	−0.248321
$633$	0	0
$634$	−1.22343	−0.0485887
$635$	−9.01548	−0.357768
$636$	0	0
$637$	0.795520	0.0315196
$638$	−1.28843	−0.0510095
$639$	0	0
$640$	−33.4679	−1.32294
$641$	−17.9247	−0.707982	−0.353991	−	0.935249i	$-0.615176\pi$
$641$	−17.9247	−0.707982	−0.353991	+	0.935249i	$0.615176\pi$
$642$	0	0
$643$	19.9092	0.785143	0.392571	−	0.919722i	$-0.371586\pi$
$643$	19.9092	0.785143	0.392571	+	0.919722i	$0.371586\pi$
$644$	−4.31828	−0.170164
$645$	0	0
$646$	−15.3592	−0.604299
$647$	−43.0673	−1.69315	−0.846574	−	0.532270i	$-0.821339\pi$
$647$	−43.0673	−1.69315	−0.846574	+	0.532270i	$0.821339\pi$
$648$	0	0
$649$	−0.753857	−0.0295915
$650$	1.96061	0.0769015
$651$	0	0
$652$	0.788807	0.0308921
$653$	−1.82134	−0.0712744	−0.0356372	−	0.999365i	$-0.511346\pi$
$653$	−1.82134	−0.0712744	−0.0356372	+	0.999365i	$0.511346\pi$
$654$	0	0
$655$	−17.7156	−0.692205
$656$	5.61978	0.219416
$657$	0	0
$658$	−2.26100	−0.0881431
$659$	−26.8403	−1.04555	−0.522774	−	0.852471i	$-0.675103\pi$
$659$	−26.8403	−1.04555	−0.522774	+	0.852471i	$0.675103\pi$
$660$	0	0
$661$	4.26124	0.165743	0.0828715	−	0.996560i	$-0.473591\pi$
$661$	4.26124	0.165743	0.0828715	+	0.996560i	$0.473591\pi$
$662$	−10.7234	−0.416776
$663$	0	0
$664$	39.2558	1.52342
$665$	−68.1272	−2.64186
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	19.6009	0.758383
$669$	0	0
$670$	−10.7355	−0.414748
$671$	20.5200	0.792166
$672$	0	0
$673$	7.95990	0.306832	0.153416	−	0.988162i	$-0.450973\pi$
$673$	7.95990	0.306832	0.153416	+	0.988162i	$0.450973\pi$
$674$	−5.79337	−0.223152
$675$	0	0
$676$	18.5631	0.713965
$677$	−45.3855	−1.74431	−0.872154	−	0.489232i	$-0.837277\pi$
$677$	−45.3855	−1.74431	−0.872154	+	0.489232i	$0.837277\pi$
$678$	0	0
$679$	−13.0446	−0.500607
$680$	26.2058	1.00495
$681$	0	0
$682$	1.63501	0.0626078
$683$	−10.6867	−0.408914	−0.204457	−	0.978876i	$-0.565543\pi$
$683$	−10.6867	−0.408914	−0.204457	+	0.978876i	$0.565543\pi$
$684$	0	0
$685$	−37.3055	−1.42537
$686$	11.3088	0.431772
$687$	0	0
$688$	2.35726	0.0898697
$689$	−0.455899	−0.0173684
$690$	0	0
$691$	−28.8789	−1.09861	−0.549303	−	0.835623i	$-0.685107\pi$
$691$	−28.8789	−1.09861	−0.549303	+	0.835623i	$0.685107\pi$
$692$	−3.59528	−0.136672
$693$	0	0
$694$	20.4871	0.777678
$695$	−46.3096	−1.75662
$696$	0	0
$697$	−17.2683	−0.654085
$698$	−22.2896	−0.843675
$699$	0	0
$700$	27.2124	1.02853
$701$	39.1416	1.47836	0.739179	−	0.673510i	$-0.235214\pi$
$701$	39.1416	1.47836	0.739179	+	0.673510i	$0.235214\pi$
$702$	0	0
$703$	−57.1002	−2.15357
$704$	−4.12001	−0.155279
$705$	0	0
$706$	22.0714	0.830670
$707$	−21.5842	−0.811758
$708$	0	0
$709$	−38.0885	−1.43045	−0.715223	−	0.698897i	$-0.753675\pi$
$709$	−38.0885	−1.43045	−0.715223	+	0.698897i	$0.753675\pi$
$710$	17.5299	0.657886
$711$	0	0
$712$	30.3217	1.13635
$713$	1.26899	0.0475242
$714$	0	0
$715$	−2.43885	−0.0912078
$716$	−11.8647	−0.443406
$717$	0	0
$718$	−1.56033	−0.0582311
$719$	−9.52549	−0.355241	−0.177620	−	0.984099i	$-0.556840\pi$
$719$	−9.52549	−0.355241	−0.177620	+	0.984099i	$0.556840\pi$
$720$	0	0
$721$	53.7007	1.99992
$722$	20.1743	0.750810
$723$	0	0
$724$	11.8970	0.442148
$725$	6.30168	0.234038
$726$	0	0
$727$	−6.13864	−0.227669	−0.113835	−	0.993500i	$-0.536313\pi$
$727$	−6.13864	−0.227669	−0.113835	+	0.993500i	$0.536313\pi$
$728$	3.19995	0.118598
$729$	0	0
$730$	−23.1940	−0.858448
$731$	−7.24333	−0.267904
$732$	0	0
$733$	37.0637	1.36898	0.684489	−	0.729023i	$-0.260025\pi$
$733$	37.0637	1.36898	0.684489	+	0.729023i	$0.260025\pi$
$734$	14.4892	0.534806
$735$	0	0
$736$	−5.86115	−0.216045
$737$	7.44609	0.274280
$738$	0	0
$739$	27.6507	1.01715	0.508573	−	0.861019i	$-0.330173\pi$
$739$	27.6507	1.01715	0.508573	+	0.861019i	$0.330173\pi$
$740$	40.9044	1.50367
$741$	0	0
$742$	2.41562	0.0886801
$743$	23.5165	0.862738	0.431369	−	0.902176i	$-0.358031\pi$
$743$	23.5165	0.862738	0.431369	+	0.902176i	$0.358031\pi$
$744$	0	0
$745$	−7.23236	−0.264973
$746$	−24.8508	−0.909852
$747$	0	0
$748$	−7.63146	−0.279034
$749$	−15.4579	−0.564819
$750$	0	0
$751$	−39.7925	−1.45205	−0.726025	−	0.687669i	$-0.758634\pi$
$751$	−39.7925	−1.45205	−0.726025	+	0.687669i	$0.758634\pi$
$752$	1.00927	0.0368042
$753$	0	0
$754$	0.311125	0.0113305
$755$	61.7108	2.24588
$756$	0	0
$757$	19.1170	0.694818	0.347409	−	0.937714i	$-0.387062\pi$
$757$	19.1170	0.694818	0.347409	+	0.937714i	$0.387062\pi$
$758$	17.4554	0.634010
$759$	0	0
$760$	−58.5190	−2.12271
$761$	23.8344	0.863996	0.431998	−	0.901874i	$-0.357809\pi$
$761$	23.8344	0.863996	0.431998	+	0.901874i	$0.357809\pi$
$762$	0	0
$763$	33.7561	1.22205
$764$	−20.3958	−0.737894
$765$	0	0
$766$	−16.3605	−0.591129
$767$	0.182038	0.00657303
$768$	0	0
$769$	36.2669	1.30782	0.653909	−	0.756573i	$-0.273128\pi$
$769$	36.2669	1.30782	0.653909	+	0.756573i	$0.273128\pi$
$770$	12.9224	0.465692
$771$	0	0
$772$	−0.394706	−0.0142058
$773$	−31.7295	−1.14123	−0.570616	−	0.821217i	$-0.693296\pi$
$773$	−31.7295	−1.14123	−0.570616	+	0.821217i	$0.693296\pi$
$774$	0	0
$775$	−7.99678	−0.287253
$776$	−11.2049	−0.402233
$777$	0	0
$778$	−10.9810	−0.393689
$779$	38.5611	1.38160
$780$	0	0
$781$	−12.1587	−0.435072
$782$	2.26116	0.0808587
$783$	0	0
$784$	1.88155	0.0671981
$785$	−1.83489	−0.0654902
$786$	0	0
$787$	−4.23836	−0.151081	−0.0755405	−	0.997143i	$-0.524068\pi$
$787$	−4.23836	−0.151081	−0.0755405	+	0.997143i	$0.524068\pi$
$788$	34.1505	1.21656
$789$	0	0
$790$	6.08761	0.216588
$791$	18.1547	0.645507
$792$	0	0
$793$	−4.95509	−0.175960
$794$	16.1859	0.574418
$795$	0	0
$796$	−5.08113	−0.180096
$797$	24.4161	0.864863	0.432432	−	0.901667i	$-0.357656\pi$
$797$	24.4161	0.864863	0.432432	+	0.901667i	$0.357656\pi$
$798$	0	0
$799$	−3.10125	−0.109714
$800$	36.9351	1.30585
$801$	0	0
$802$	1.76657	0.0623799
$803$	16.0873	0.567707
$804$	0	0
$805$	10.0296	0.353496
$806$	−0.394816	−0.0139068
$807$	0	0
$808$	−18.5401	−0.652239
$809$	−27.0005	−0.949287	−0.474644	−	0.880178i	$-0.657423\pi$
$809$	−27.0005	−0.949287	−0.474644	+	0.880178i	$0.657423\pi$
$810$	0	0
$811$	26.8385	0.942428	0.471214	−	0.882019i	$-0.343816\pi$
$811$	26.8385	0.942428	0.471214	+	0.882019i	$0.343816\pi$
$812$	4.31828	0.151542
$813$	0	0
$814$	10.8308	0.379620
$815$	−1.83207	−0.0641748
$816$	0	0
$817$	16.1748	0.565883
$818$	0.106457	0.00372217
$819$	0	0
$820$	−27.6237	−0.964662
$821$	53.0026	1.84980	0.924901	−	0.380208i	$-0.124148\pi$
$821$	53.0026	1.84980	0.924901	+	0.380208i	$0.124148\pi$
$822$	0	0
$823$	−5.65489	−0.197117	−0.0985585	−	0.995131i	$-0.531423\pi$
$823$	−5.65489	−0.197117	−0.0985585	+	0.995131i	$0.531423\pi$
$824$	46.1271	1.60691
$825$	0	0
$826$	−0.964545	−0.0335608
$827$	39.2613	1.36525	0.682624	−	0.730770i	$-0.260839\pi$
$827$	39.2613	1.36525	0.682624	+	0.730770i	$0.260839\pi$
$828$	0	0
$829$	30.1229	1.04621	0.523106	−	0.852268i	$-0.324773\pi$
$829$	30.1229	1.04621	0.523106	+	0.852268i	$0.324773\pi$
$830$	−38.2807	−1.32874
$831$	0	0
$832$	0.994883	0.0344914
$833$	−5.78157	−0.200320
$834$	0	0
$835$	−45.5249	−1.57546
$836$	17.0415	0.589391
$837$	0	0
$838$	1.38058	0.0476913
$839$	−2.39338	−0.0826286	−0.0413143	−	0.999146i	$-0.513155\pi$
$839$	−2.39338	−0.0826286	−0.0413143	+	0.999146i	$0.513155\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	12.4852	0.430269
$843$	0	0
$844$	0.199963	0.00688301
$845$	−43.1144	−1.48318
$846$	0	0
$847$	23.8544	0.819648
$848$	−1.07828	−0.0370284
$849$	0	0
$850$	−14.2491	−0.488739
$851$	8.40619	0.288161
$852$	0	0
$853$	20.8704	0.714588	0.357294	−	0.933992i	$-0.383699\pi$
$853$	20.8704	0.714588	0.357294	+	0.933992i	$0.383699\pi$
$854$	26.2549	0.898425
$855$	0	0
$856$	−13.2778	−0.453826
$857$	−45.3139	−1.54789	−0.773946	−	0.633251i	$-0.781720\pi$
$857$	−45.3139	−1.54789	−0.773946	+	0.633251i	$0.781720\pi$
$858$	0	0
$859$	−35.5885	−1.21427	−0.607133	−	0.794600i	$-0.707680\pi$
$859$	−35.5885	−1.21427	−0.607133	+	0.794600i	$0.707680\pi$
$860$	−11.5870	−0.395113
$861$	0	0
$862$	−11.5094	−0.392013
$863$	20.3394	0.692362	0.346181	−	0.938168i	$-0.387478\pi$
$863$	20.3394	0.692362	0.346181	+	0.938168i	$0.387478\pi$
$864$	0	0
$865$	8.35036	0.283921
$866$	−8.05689	−0.273784
$867$	0	0
$868$	−5.47986	−0.185999
$869$	−4.22235	−0.143233
$870$	0	0
$871$	−1.79805	−0.0609246
$872$	28.9954	0.981907
$873$	0	0
$874$	−5.04928	−0.170795
$875$	−13.0553	−0.441348
$876$	0	0
$877$	−47.6368	−1.60858	−0.804290	−	0.594238i	$-0.797454\pi$
$877$	−47.6368	−1.60858	−0.804290	+	0.594238i	$0.797454\pi$
$878$	9.70718	0.327601
$879$	0	0
$880$	−5.76832	−0.194450
$881$	−22.4664	−0.756911	−0.378455	−	0.925620i	$-0.623545\pi$
$881$	−22.4664	−0.756911	−0.378455	+	0.925620i	$0.623545\pi$
$882$	0	0
$883$	47.1244	1.58586	0.792931	−	0.609311i	$-0.208554\pi$
$883$	47.1244	1.58586	0.792931	+	0.609311i	$0.208554\pi$
$884$	1.84281	0.0619805
$885$	0	0
$886$	10.3709	0.348417
$887$	−24.4228	−0.820036	−0.410018	−	0.912077i	$-0.634478\pi$
$887$	−24.4228	−0.820036	−0.410018	+	0.912077i	$0.634478\pi$
$888$	0	0
$889$	8.00071	0.268335
$890$	−29.5684	−0.991136
$891$	0	0
$892$	−28.5978	−0.957524
$893$	6.92526	0.231745
$894$	0	0
$895$	27.5569	0.921126
$896$	29.7008	0.992235
$897$	0	0
$898$	−0.0806501	−0.00269133
$899$	−1.26899	−0.0423233
$900$	0	0
$901$	3.31332	0.110383
$902$	−7.31431	−0.243540
$903$	0	0
$904$	15.5943	0.518658
$905$	−27.6318	−0.918512
$906$	0	0
$907$	−38.1574	−1.26700	−0.633498	−	0.773744i	$-0.718381\pi$
$907$	−38.1574	−1.26700	−0.633498	+	0.773744i	$0.718381\pi$
$908$	−30.8138	−1.02259
$909$	0	0
$910$	−3.12046	−0.103442
$911$	21.7734	0.721386	0.360693	−	0.932685i	$-0.382540\pi$
$911$	21.7734	0.721386	0.360693	+	0.932685i	$0.382540\pi$
$912$	0	0
$913$	26.5513	0.878721
$914$	21.2818	0.703940
$915$	0	0
$916$	14.4501	0.477444
$917$	15.7216	0.519171
$918$	0	0
$919$	18.5974	0.613472	0.306736	−	0.951795i	$-0.400763\pi$
$919$	18.5974	0.613472	0.306736	+	0.951795i	$0.400763\pi$
$920$	8.61508	0.284031
$921$	0	0
$922$	−20.3434	−0.669973
$923$	2.93603	0.0966406
$924$	0	0
$925$	−52.9731	−1.74174
$926$	−6.87100	−0.225795
$927$	0	0
$928$	5.86115	0.192402
$929$	8.38393	0.275068	0.137534	−	0.990497i	$-0.456082\pi$
$929$	8.38393	0.275068	0.137534	+	0.990497i	$0.456082\pi$
$930$	0	0
$931$	12.9106	0.423127
$932$	−17.0233	−0.557617
$933$	0	0
$934$	−20.1769	−0.660209
$935$	17.7247	0.579661
$936$	0	0
$937$	46.3141	1.51301	0.756507	−	0.653985i	$-0.226904\pi$
$937$	46.3141	1.51301	0.756507	+	0.653985i	$0.226904\pi$
$938$	9.52711	0.311071
$939$	0	0
$940$	−4.96099	−0.161810
$941$	−38.1202	−1.24268	−0.621342	−	0.783540i	$-0.713412\pi$
$941$	−38.1202	−1.24268	−0.621342	+	0.783540i	$0.713412\pi$
$942$	0	0
$943$	−5.67691	−0.184865
$944$	0.430553	0.0140133
$945$	0	0
$946$	−3.06804	−0.0997507
$947$	25.5915	0.831612	0.415806	−	0.909453i	$-0.363500\pi$
$947$	25.5915	0.831612	0.415806	+	0.909453i	$0.363500\pi$
$948$	0	0
$949$	−3.88469	−0.126102
$950$	31.8189	1.03234
$951$	0	0
$952$	−23.2561	−0.753736
$953$	9.53149	0.308755	0.154378	−	0.988012i	$-0.450663\pi$
$953$	9.53149	0.308755	0.154378	+	0.988012i	$0.450663\pi$
$954$	0	0
$955$	47.3710	1.53289
$956$	−21.2815	−0.688294
$957$	0	0
$958$	11.4097	0.368629
$959$	33.1064	1.06906
$960$	0	0
$961$	−29.3897	−0.948053
$962$	−2.61538	−0.0843232
$963$	0	0
$964$	9.38197	0.302173
$965$	0.916739	0.0295109
$966$	0	0
$967$	57.4719	1.84817	0.924086	−	0.382184i	$-0.124828\pi$
$967$	57.4719	1.84817	0.924086	+	0.382184i	$0.124828\pi$
$968$	20.4902	0.658579
$969$	0	0
$970$	10.9266	0.350831
$971$	−2.22999	−0.0715636	−0.0357818	−	0.999360i	$-0.511392\pi$
$971$	−2.22999	−0.0715636	−0.0357818	+	0.999360i	$0.511392\pi$
$972$	0	0
$973$	41.0970	1.31751
$974$	−17.3197	−0.554960
$975$	0	0
$976$	−11.7197	−0.375137
$977$	−16.0703	−0.514133	−0.257067	−	0.966394i	$-0.582756\pi$
$977$	−16.0703	−0.514133	−0.257067	+	0.966394i	$0.582756\pi$
$978$	0	0
$979$	20.5086	0.655456
$980$	−9.24863	−0.295437
$981$	0	0
$982$	23.7598	0.758207
$983$	43.1593	1.37657	0.688284	−	0.725441i	$-0.258364\pi$
$983$	43.1593	1.37657	0.688284	+	0.725441i	$0.258364\pi$
$984$	0	0
$985$	−79.3176	−2.52727
$986$	−2.26116	−0.0720098
$987$	0	0
$988$	−4.11510	−0.130919
$989$	−2.38122	−0.0757184
$990$	0	0
$991$	28.6889	0.911334	0.455667	−	0.890150i	$-0.349401\pi$
$991$	28.6889	0.911334	0.455667	+	0.890150i	$0.349401\pi$
$992$	−7.43776	−0.236149
$993$	0	0
$994$	−15.5568	−0.493431
$995$	11.8014	0.374129
$996$	0	0
$997$	51.9493	1.64525	0.822625	−	0.568584i	$-0.192509\pi$
$997$	51.9493	1.64525	0.822625	+	0.568584i	$0.192509\pi$
$998$	15.2135	0.481575
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.j.1.4		7
3.2	odd	2		2001.2.a.i.1.4	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.2.a.i.1.4	✓	7	3.2	odd	2
6003.2.a.j.1.4		7	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)

\(f(q)\)	\(=\)	\(q+0.743347 q^{2} -1.44743 q^{4} +3.36180 q^{5} -2.98340 q^{7} -2.56264 q^{8} +O(q^{10})\)	\(q+0.743347 q^{2} -1.44743 q^{4} +3.36180 q^{5} -2.98340 q^{7} -2.56264 q^{8} +2.49898 q^{10} -1.73328 q^{11} +0.418547 q^{13} -2.21770 q^{14} +0.989937 q^{16} -3.04186 q^{17} +6.79263 q^{19} -4.86598 q^{20} -1.28843 q^{22} -1.00000 q^{23} +6.30168 q^{25} +0.311125 q^{26} +4.31828 q^{28} +1.00000 q^{29} -1.26899 q^{31} +5.86115 q^{32} -2.26116 q^{34} -10.0296 q^{35} -8.40619 q^{37} +5.04928 q^{38} -8.61508 q^{40} +5.67691 q^{41} +2.38122 q^{43} +2.50882 q^{44} -0.743347 q^{46} +1.01953 q^{47} +1.90067 q^{49} +4.68433 q^{50} -0.605819 q^{52} -1.08924 q^{53} -5.82695 q^{55} +7.64538 q^{56} +0.743347 q^{58} +0.434930 q^{59} -11.8388 q^{61} -0.943302 q^{62} +2.37699 q^{64} +1.40707 q^{65} -4.29594 q^{67} +4.40289 q^{68} -7.45546 q^{70} +7.01482 q^{71} -9.28137 q^{73} -6.24872 q^{74} -9.83189 q^{76} +5.17108 q^{77} +2.43604 q^{79} +3.32797 q^{80} +4.21991 q^{82} -15.3185 q^{83} -10.2261 q^{85} +1.77007 q^{86} +4.44179 q^{88} -11.8322 q^{89} -1.24869 q^{91} +1.44743 q^{92} +0.757862 q^{94} +22.8354 q^{95} +4.37241 q^{97} +1.41286 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 3 q^{2} + 5 q^{4} + 3 q^{5} - 5 q^{7} + 6 q^{8}+O(q^{10}) \) 7 * q + 3 * q^2 + 5 * q^4 + 3 * q^5 - 5 * q^7 + 6 * q^8	\( 7 q + 3 q^{2} + 5 q^{4} + 3 q^{5} - 5 q^{7} + 6 q^{8} + 3 q^{10} + 4 q^{11} - 18 q^{13} + 2 q^{14} - 7 q^{16} + 3 q^{17} - 4 q^{19} + 2 q^{20} - 26 q^{22} - 7 q^{23} - 8 q^{25} + 7 q^{26} - 6 q^{28} + 7 q^{29} - 22 q^{31} - 5 q^{32} + 9 q^{34} - 3 q^{35} - 25 q^{37} - 14 q^{38} - 10 q^{40} + 13 q^{41} - 2 q^{43} - 4 q^{44} - 3 q^{46} + 25 q^{47} - 8 q^{49} - 19 q^{50} - 12 q^{52} + 5 q^{53} - 15 q^{55} - 18 q^{56} + 3 q^{58} - 11 q^{59} - 33 q^{61} - 28 q^{62} - 14 q^{64} + 2 q^{65} + 8 q^{67} - 12 q^{68} - 22 q^{70} + 6 q^{71} + 15 q^{73} - 34 q^{74} - 28 q^{76} + q^{77} - 15 q^{79} + 12 q^{80} - 14 q^{82} - 21 q^{83} - 28 q^{85} + 12 q^{86} - 13 q^{88} - 8 q^{89} + 6 q^{91} - 5 q^{92} - 35 q^{94} + 25 q^{95} + 13 q^{97} - q^{98}+O(q^{100}) \) 7 * q + 3 * q^2 + 5 * q^4 + 3 * q^5 - 5 * q^7 + 6 * q^8 + 3 * q^10 + 4 * q^11 - 18 * q^13 + 2 * q^14 - 7 * q^16 + 3 * q^17 - 4 * q^19 + 2 * q^20 - 26 * q^22 - 7 * q^23 - 8 * q^25 + 7 * q^26 - 6 * q^28 + 7 * q^29 - 22 * q^31 - 5 * q^32 + 9 * q^34 - 3 * q^35 - 25 * q^37 - 14 * q^38 - 10 * q^40 + 13 * q^41 - 2 * q^43 - 4 * q^44 - 3 * q^46 + 25 * q^47 - 8 * q^49 - 19 * q^50 - 12 * q^52 + 5 * q^53 - 15 * q^55 - 18 * q^56 + 3 * q^58 - 11 * q^59 - 33 * q^61 - 28 * q^62 - 14 * q^64 + 2 * q^65 + 8 * q^67 - 12 * q^68 - 22 * q^70 + 6 * q^71 + 15 * q^73 - 34 * q^74 - 28 * q^76 + q^77 - 15 * q^79 + 12 * q^80 - 14 * q^82 - 21 * q^83 - 28 * q^85 + 12 * q^86 - 13 * q^88 - 8 * q^89 + 6 * q^91 - 5 * q^92 - 35 * q^94 + 25 * q^95 + 13 * q^97 - q^98

Introduction

L-functions

Modular forms

Varieties

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