LMFDB - Embedded newform 6003.2.a.q.1.11

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:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 3 x^{15} - 22 x^{14} + 68 x^{13} + 187 x^{12} - 597 x^{11} - 795 x^{10} + 2592 x^{9} + \cdots + 64 $ x^16 - 3x^15 - 22x^14 + 68x^13 + 187x^12 - 597x^11 - 795x^10 + 2592x^9 + 1860x^8 - 5877x^7 - 2496x^6 + 6612x^5 + 1842x^4 - 3011x^3 - 505x^2 + 336*x + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 667)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$-1.03000$ of defining polynomial
Character	$\chi$	$=$	6003.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.03000 q^{2} -0.939097 q^{4} -3.95819 q^{5} +4.48231 q^{7} -3.02727 q^{8} +O(q^{10})$	\(q+1.03000 q^{2} -0.939097 q^{4} -3.95819 q^{5} +4.48231 q^{7} -3.02727 q^{8} -4.07695 q^{10} -5.58245 q^{11} +1.54285 q^{13} +4.61678 q^{14} -1.23990 q^{16} +7.55730 q^{17} -3.20558 q^{19} +3.71713 q^{20} -5.74993 q^{22} -1.00000 q^{23} +10.6673 q^{25} +1.58914 q^{26} -4.20932 q^{28} +1.00000 q^{29} +4.20583 q^{31} +4.77745 q^{32} +7.78403 q^{34} -17.7419 q^{35} +4.87699 q^{37} -3.30176 q^{38} +11.9825 q^{40} +1.92475 q^{41} -11.4855 q^{43} +5.24246 q^{44} -1.03000 q^{46} +1.70768 q^{47} +13.0911 q^{49} +10.9873 q^{50} -1.44889 q^{52} -9.65100 q^{53} +22.0964 q^{55} -13.5692 q^{56} +1.03000 q^{58} +5.68257 q^{59} +0.740179 q^{61} +4.33201 q^{62} +7.40058 q^{64} -6.10691 q^{65} -10.1138 q^{67} -7.09704 q^{68} -18.2741 q^{70} -6.76766 q^{71} +6.55560 q^{73} +5.02331 q^{74} +3.01036 q^{76} -25.0223 q^{77} -10.6027 q^{79} +4.90777 q^{80} +1.98249 q^{82} +3.13550 q^{83} -29.9132 q^{85} -11.8301 q^{86} +16.8996 q^{88} -10.3943 q^{89} +6.91554 q^{91} +0.939097 q^{92} +1.75892 q^{94} +12.6883 q^{95} -8.85845 q^{97} +13.4838 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 3 q^{2} + 21 q^{4} - 16 q^{5} + q^{7} - 9 q^{8}+O(q^{10}) $ 16 * q - 3 * q^2 + 21 * q^4 - 16 * q^5 + q^7 - 9 * q^8	$ 16 q - 3 q^{2} + 21 q^{4} - 16 q^{5} + q^{7} - 9 q^{8} - 14 q^{10} - 4 q^{11} + 15 q^{13} - 8 q^{14} + 23 q^{16} - 20 q^{17} - 4 q^{19} - 25 q^{20} + 13 q^{22} - 16 q^{23} + 30 q^{25} - 25 q^{26} - 13 q^{28} + 16 q^{29} + 19 q^{32} - 23 q^{34} - 5 q^{35} + 5 q^{37} - 38 q^{38} - 20 q^{40} - 7 q^{41} - 17 q^{43} + 21 q^{44} + 3 q^{46} - 29 q^{47} + 31 q^{49} + 44 q^{50} + 20 q^{52} - 63 q^{53} + q^{55} + 19 q^{56} - 3 q^{58} - 11 q^{59} - 33 q^{62} + 29 q^{64} - 53 q^{65} - 13 q^{67} - 63 q^{68} - 46 q^{70} + 23 q^{71} - 38 q^{73} + 47 q^{74} - 56 q^{76} - 97 q^{77} - 27 q^{79} - 8 q^{80} + 9 q^{82} - 36 q^{83} + 6 q^{85} + 11 q^{86} - 24 q^{88} + 16 q^{89} - 47 q^{91} - 21 q^{92} + 37 q^{94} + 12 q^{95} - 30 q^{97} + 27 q^{98}+O(q^{100}) $ 16 * q - 3 * q^2 + 21 * q^4 - 16 * q^5 + q^7 - 9 * q^8 - 14 * q^10 - 4 * q^11 + 15 * q^13 - 8 * q^14 + 23 * q^16 - 20 * q^17 - 4 * q^19 - 25 * q^20 + 13 * q^22 - 16 * q^23 + 30 * q^25 - 25 * q^26 - 13 * q^28 + 16 * q^29 + 19 * q^32 - 23 * q^34 - 5 * q^35 + 5 * q^37 - 38 * q^38 - 20 * q^40 - 7 * q^41 - 17 * q^43 + 21 * q^44 + 3 * q^46 - 29 * q^47 + 31 * q^49 + 44 * q^50 + 20 * q^52 - 63 * q^53 + q^55 + 19 * q^56 - 3 * q^58 - 11 * q^59 - 33 * q^62 + 29 * q^64 - 53 * q^65 - 13 * q^67 - 63 * q^68 - 46 * q^70 + 23 * q^71 - 38 * q^73 + 47 * q^74 - 56 * q^76 - 97 * q^77 - 27 * q^79 - 8 * q^80 + 9 * q^82 - 36 * q^83 + 6 * q^85 + 11 * q^86 - 24 * q^88 + 16 * q^89 - 47 * q^91 - 21 * q^92 + 37 * q^94 + 12 * q^95 - 30 * q^97 + 27 * 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 $

$2$	1.03000	0.728321	0.364160	−	0.931336i	$-0.381356\pi$
$2$	1.03000	0.728321	0.364160	+	0.931336i	$0.381356\pi$
$3$	0	0
$3$	0	0
$4$	−0.939097	−0.469549
$5$	−3.95819	−1.77016	−0.885079	−	0.465441i	$-0.845896\pi$
$5$	−3.95819	−1.77016	−0.885079	+	0.465441i	$0.845896\pi$
$6$	0	0
$7$	4.48231	1.69415	0.847077	−	0.531471i	$-0.178360\pi$
$7$	4.48231	1.69415	0.847077	+	0.531471i	$0.178360\pi$
$8$	−3.02727	−1.07030
$9$	0	0
$10$	−4.07695	−1.28924
$11$	−5.58245	−1.68317	−0.841586	−	0.540124i	$-0.818377\pi$
$11$	−5.58245	−1.68317	−0.841586	+	0.540124i	$0.818377\pi$
$12$	0	0
$13$	1.54285	0.427910	0.213955	−	0.976844i	$-0.431365\pi$
$13$	1.54285	0.427910	0.213955	+	0.976844i	$0.431365\pi$
$14$	4.61678	1.23389
$15$	0	0
$16$	−1.23990	−0.309976
$17$	7.55730	1.83291	0.916457	−	0.400134i	$-0.131036\pi$
$17$	7.55730	1.83291	0.916457	+	0.400134i	$0.131036\pi$
$18$	0	0
$19$	−3.20558	−0.735412	−0.367706	−	0.929942i	$-0.619857\pi$
$19$	−3.20558	−0.735412	−0.367706	+	0.929942i	$0.619857\pi$
$20$	3.71713	0.831175
$21$	0	0
$22$	−5.74993	−1.22589
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	10.6673	2.13346
$26$	1.58914	0.311656
$27$	0	0
$28$	−4.20932	−0.795487
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	4.20583	0.755390	0.377695	−	0.925930i	$-0.376717\pi$
$31$	4.20583	0.755390	0.377695	+	0.925930i	$0.376717\pi$
$32$	4.77745	0.844541
$33$	0	0
$34$	7.78403	1.33495
$35$	−17.7419	−2.99892
$36$	0	0
$37$	4.87699	0.801773	0.400886	−	0.916128i	$-0.368702\pi$
$37$	4.87699	0.801773	0.400886	+	0.916128i	$0.368702\pi$
$38$	−3.30176	−0.535616
$39$	0	0
$40$	11.9825	1.89461
$41$	1.92475	0.300595	0.150298	−	0.988641i	$-0.451977\pi$
$41$	1.92475	0.300595	0.150298	+	0.988641i	$0.451977\pi$
$42$	0	0
$43$	−11.4855	−1.75152	−0.875760	−	0.482747i	$-0.839639\pi$
$43$	−11.4855	−1.75152	−0.875760	+	0.482747i	$0.839639\pi$
$44$	5.24246	0.790331
$45$	0	0
$46$	−1.03000	−0.151865
$47$	1.70768	0.249091	0.124546	−	0.992214i	$-0.460253\pi$
$47$	1.70768	0.249091	0.124546	+	0.992214i	$0.460253\pi$
$48$	0	0
$49$	13.0911	1.87016
$50$	10.9873	1.55384
$51$	0	0
$52$	−1.44889	−0.200925
$53$	−9.65100	−1.32567	−0.662833	−	0.748767i	$-0.730646\pi$
$53$	−9.65100	−1.32567	−0.662833	+	0.748767i	$0.730646\pi$
$54$	0	0
$55$	22.0964	2.97948
$56$	−13.5692	−1.81326
$57$	0	0
$58$	1.03000	0.135246
$59$	5.68257	0.739808	0.369904	−	0.929070i	$-0.379391\pi$
$59$	5.68257	0.739808	0.369904	+	0.929070i	$0.379391\pi$
$60$	0	0
$61$	0.740179	0.0947701	0.0473851	−	0.998877i	$-0.484911\pi$
$61$	0.740179	0.0947701	0.0473851	+	0.998877i	$0.484911\pi$
$62$	4.33201	0.550166
$63$	0	0
$64$	7.40058	0.925073
$65$	−6.10691	−0.757469
$66$	0	0
$67$	−10.1138	−1.23559	−0.617796	−	0.786339i	$-0.711974\pi$
$67$	−10.1138	−1.23559	−0.617796	+	0.786339i	$0.711974\pi$
$68$	−7.09704	−0.860642
$69$	0	0
$70$	−18.2741	−2.18418
$71$	−6.76766	−0.803174	−0.401587	−	0.915821i	$-0.631541\pi$
$71$	−6.76766	−0.803174	−0.401587	+	0.915821i	$0.631541\pi$
$72$	0	0
$73$	6.55560	0.767275	0.383637	−	0.923484i	$-0.374671\pi$
$73$	6.55560	0.767275	0.383637	+	0.923484i	$0.374671\pi$
$74$	5.02331	0.583948
$75$	0	0
$76$	3.01036	0.345311
$77$	−25.0223	−2.85155
$78$	0	0
$79$	−10.6027	−1.19290	−0.596451	−	0.802650i	$-0.703423\pi$
$79$	−10.6027	−1.19290	−0.596451	+	0.802650i	$0.703423\pi$
$80$	4.90777	0.548706
$81$	0	0
$82$	1.98249	0.218930
$83$	3.13550	0.344165	0.172083	−	0.985083i	$-0.444950\pi$
$83$	3.13550	0.344165	0.172083	+	0.985083i	$0.444950\pi$
$84$	0	0
$85$	−29.9132	−3.24455
$86$	−11.8301	−1.27567
$87$	0	0
$88$	16.8996	1.80150
$89$	−10.3943	−1.10180	−0.550899	−	0.834572i	$-0.685715\pi$
$89$	−10.3943	−1.10180	−0.550899	+	0.834572i	$0.685715\pi$
$90$	0	0
$91$	6.91554	0.724945
$92$	0.939097	0.0979076
$93$	0	0
$94$	1.75892	0.181418
$95$	12.6883	1.30179
$96$	0	0
$97$	−8.85845	−0.899439	−0.449720	−	0.893170i	$-0.648476\pi$
$97$	−8.85845	−0.899439	−0.449720	+	0.893170i	$0.648476\pi$
$98$	13.4838	1.36207
$99$	0	0
$100$	−10.0176	−1.00176
$101$	−3.50214	−0.348476	−0.174238	−	0.984704i	$-0.555746\pi$
$101$	−3.50214	−0.348476	−0.174238	+	0.984704i	$0.555746\pi$
$102$	0	0
$103$	10.2397	1.00895	0.504475	−	0.863426i	$-0.331686\pi$
$103$	10.2397	1.00895	0.504475	+	0.863426i	$0.331686\pi$
$104$	−4.67064	−0.457994
$105$	0	0
$106$	−9.94054	−0.965511
$107$	4.66882	0.451352	0.225676	−	0.974202i	$-0.427541\pi$
$107$	4.66882	0.451352	0.225676	+	0.974202i	$0.427541\pi$
$108$	0	0
$109$	−8.37166	−0.801859	−0.400930	−	0.916109i	$-0.631313\pi$
$109$	−8.37166	−0.801859	−0.400930	+	0.916109i	$0.631313\pi$
$110$	22.7593	2.17002
$111$	0	0
$112$	−5.55763	−0.525146
$113$	7.49063	0.704659	0.352330	−	0.935876i	$-0.385390\pi$
$113$	7.49063	0.704659	0.352330	+	0.935876i	$0.385390\pi$
$114$	0	0
$115$	3.95819	0.369104
$116$	−0.939097	−0.0871930
$117$	0	0
$118$	5.85306	0.538818
$119$	33.8741	3.10524
$120$	0	0
$121$	20.1637	1.83307
$122$	0.762385	0.0690231
$123$	0	0
$124$	−3.94969	−0.354692
$125$	−22.4323	−2.00640
$126$	0	0
$127$	−3.18394	−0.282529	−0.141265	−	0.989972i	$-0.545117\pi$
$127$	−3.18394	−0.282529	−0.141265	+	0.989972i	$0.545117\pi$
$128$	−1.93228	−0.170791
$129$	0	0
$130$	−6.29012	−0.551680
$131$	−1.43514	−0.125388	−0.0626942	−	0.998033i	$-0.519969\pi$
$131$	−1.43514	−0.125388	−0.0626942	+	0.998033i	$0.519969\pi$
$132$	0	0
$133$	−14.3684	−1.24590
$134$	−10.4172	−0.899907
$135$	0	0
$136$	−22.8780	−1.96177
$137$	−17.9076	−1.52995	−0.764973	−	0.644062i	$-0.777248\pi$
$137$	−17.9076	−1.52995	−0.764973	+	0.644062i	$0.777248\pi$
$138$	0	0
$139$	5.42185	0.459875	0.229938	−	0.973205i	$-0.426148\pi$
$139$	5.42185	0.459875	0.229938	+	0.973205i	$0.426148\pi$
$140$	16.6613	1.40814
$141$	0	0
$142$	−6.97070	−0.584968
$143$	−8.61289	−0.720246
$144$	0	0
$145$	−3.95819	−0.328710
$146$	6.75228	0.558822
$147$	0	0
$148$	−4.57997	−0.376471
$149$	−5.07035	−0.415380	−0.207690	−	0.978195i	$-0.566594\pi$
$149$	−5.07035	−0.415380	−0.207690	+	0.978195i	$0.566594\pi$
$150$	0	0
$151$	−16.4660	−1.33998	−0.669992	−	0.742368i	$-0.733703\pi$
$151$	−16.4660	−1.33998	−0.669992	+	0.742368i	$0.733703\pi$
$152$	9.70418	0.787113
$153$	0	0
$154$	−25.7730	−2.07684
$155$	−16.6475	−1.33716
$156$	0	0
$157$	−4.55430	−0.363472	−0.181736	−	0.983347i	$-0.558172\pi$
$157$	−4.55430	−0.363472	−0.181736	+	0.983347i	$0.558172\pi$
$158$	−10.9208	−0.868815
$159$	0	0
$160$	−18.9101	−1.49497
$161$	−4.48231	−0.353255
$162$	0	0
$163$	7.06221	0.553155	0.276578	−	0.960992i	$-0.410800\pi$
$163$	7.06221	0.553155	0.276578	+	0.960992i	$0.410800\pi$
$164$	−1.80753	−0.141144
$165$	0	0
$166$	3.22956	0.250663
$167$	−6.45941	−0.499844	−0.249922	−	0.968266i	$-0.580405\pi$
$167$	−6.45941	−0.499844	−0.249922	+	0.968266i	$0.580405\pi$
$168$	0	0
$169$	−10.6196	−0.816893
$170$	−30.8107	−2.36307
$171$	0	0
$172$	10.7860	0.822423
$173$	−11.4908	−0.873628	−0.436814	−	0.899552i	$-0.643893\pi$
$173$	−11.4908	−0.873628	−0.436814	+	0.899552i	$0.643893\pi$
$174$	0	0
$175$	47.8141	3.61441
$176$	6.92169	0.521742
$177$	0	0
$178$	−10.7062	−0.802463
$179$	9.25784	0.691964	0.345982	−	0.938241i	$-0.387546\pi$
$179$	9.25784	0.691964	0.345982	+	0.938241i	$0.387546\pi$
$180$	0	0
$181$	−23.0828	−1.71573	−0.857865	−	0.513874i	$-0.828210\pi$
$181$	−23.0828	−1.71573	−0.857865	+	0.513874i	$0.828210\pi$
$182$	7.12301	0.527993
$183$	0	0
$184$	3.02727	0.223174
$185$	−19.3041	−1.41926
$186$	0	0
$187$	−42.1882	−3.08511
$188$	−1.60368	−0.116960
$189$	0	0
$190$	13.0690	0.948124
$191$	4.08324	0.295453	0.147727	−	0.989028i	$-0.452804\pi$
$191$	4.08324	0.295453	0.147727	+	0.989028i	$0.452804\pi$
$192$	0	0
$193$	−13.0498	−0.939347	−0.469674	−	0.882840i	$-0.655628\pi$
$193$	−13.0498	−0.939347	−0.469674	+	0.882840i	$0.655628\pi$
$194$	−9.12421	−0.655080
$195$	0	0
$196$	−12.2938	−0.878129
$197$	−0.745096	−0.0530859	−0.0265430	−	0.999648i	$-0.508450\pi$
$197$	−0.745096	−0.0530859	−0.0265430	+	0.999648i	$0.508450\pi$
$198$	0	0
$199$	3.65472	0.259076	0.129538	−	0.991574i	$-0.458651\pi$
$199$	3.65472	0.259076	0.129538	+	0.991574i	$0.458651\pi$
$200$	−32.2928	−2.28345
$201$	0	0
$202$	−3.60721	−0.253802
$203$	4.48231	0.314596
$204$	0	0
$205$	−7.61853	−0.532101
$206$	10.5469	0.734839
$207$	0	0
$208$	−1.91299	−0.132642
$209$	17.8950	1.23782
$210$	0	0
$211$	−25.1728	−1.73297	−0.866483	−	0.499206i	$-0.833625\pi$
$211$	−25.1728	−1.73297	−0.866483	+	0.499206i	$0.833625\pi$
$212$	9.06322	0.622465
$213$	0	0
$214$	4.80890	0.328729
$215$	45.4618	3.10047
$216$	0	0
$217$	18.8518	1.27975
$218$	−8.62282	−0.584011
$219$	0	0
$220$	−20.7507	−1.39901
$221$	11.6598	0.784322
$222$	0	0
$223$	7.72283	0.517159	0.258579	−	0.965990i	$-0.416746\pi$
$223$	7.72283	0.517159	0.258579	+	0.965990i	$0.416746\pi$
$224$	21.4140	1.43078
$225$	0	0
$226$	7.71536	0.513218
$227$	1.94258	0.128934	0.0644669	−	0.997920i	$-0.479465\pi$
$227$	1.94258	0.128934	0.0644669	+	0.997920i	$0.479465\pi$
$228$	0	0
$229$	15.9480	1.05388	0.526938	−	0.849904i	$-0.323340\pi$
$229$	15.9480	1.05388	0.526938	+	0.849904i	$0.323340\pi$
$230$	4.07695	0.268826
$231$	0	0
$232$	−3.02727	−0.198750
$233$	−19.4550	−1.27454	−0.637268	−	0.770642i	$-0.719936\pi$
$233$	−19.4550	−1.27454	−0.637268	+	0.770642i	$0.719936\pi$
$234$	0	0
$235$	−6.75934	−0.440931
$236$	−5.33649	−0.347376
$237$	0	0
$238$	34.8904	2.26161
$239$	15.6778	1.01412	0.507058	−	0.861912i	$-0.330733\pi$
$239$	15.6778	1.01412	0.507058	+	0.861912i	$0.330733\pi$
$240$	0	0
$241$	−7.42619	−0.478363	−0.239181	−	0.970975i	$-0.576879\pi$
$241$	−7.42619	−0.478363	−0.239181	+	0.970975i	$0.576879\pi$
$242$	20.7687	1.33506
$243$	0	0
$244$	−0.695099	−0.0444992
$245$	−51.8171	−3.31047
$246$	0	0
$247$	−4.94574	−0.314690
$248$	−12.7322	−0.808496
$249$	0	0
$250$	−23.1053	−1.46131
$251$	10.3226	0.651554	0.325777	−	0.945447i	$-0.394374\pi$
$251$	10.3226	0.651554	0.325777	+	0.945447i	$0.394374\pi$
$252$	0	0
$253$	5.58245	0.350965
$254$	−3.27946	−0.205772
$255$	0	0
$256$	−16.7914	−1.04946
$257$	−29.9267	−1.86678	−0.933389	−	0.358866i	$-0.883164\pi$
$257$	−29.9267	−1.86678	−0.933389	+	0.358866i	$0.883164\pi$
$258$	0	0
$259$	21.8602	1.35833
$260$	5.73498	0.355668
$261$	0	0
$262$	−1.47819	−0.0913230
$263$	−3.49282	−0.215377	−0.107688	−	0.994185i	$-0.534345\pi$
$263$	−3.49282	−0.215377	−0.107688	+	0.994185i	$0.534345\pi$
$264$	0	0
$265$	38.2005	2.34664
$266$	−14.7995	−0.907415
$267$	0	0
$268$	9.49779	0.580170
$269$	−1.54881	−0.0944327	−0.0472164	−	0.998885i	$-0.515035\pi$
$269$	−1.54881	−0.0944327	−0.0472164	+	0.998885i	$0.515035\pi$
$270$	0	0
$271$	−10.7318	−0.651912	−0.325956	−	0.945385i	$-0.605686\pi$
$271$	−10.7318	−0.651912	−0.325956	+	0.945385i	$0.605686\pi$
$272$	−9.37031	−0.568159
$273$	0	0
$274$	−18.4448	−1.11429
$275$	−59.5497	−3.59098
$276$	0	0
$277$	−14.4071	−0.865637	−0.432818	−	0.901481i	$-0.642481\pi$
$277$	−14.4071	−0.865637	−0.432818	+	0.901481i	$0.642481\pi$
$278$	5.58451	0.334937
$279$	0	0
$280$	53.7094	3.20975
$281$	−13.9764	−0.833762	−0.416881	−	0.908961i	$-0.636877\pi$
$281$	−13.9764	−0.833762	−0.416881	+	0.908961i	$0.636877\pi$
$282$	0	0
$283$	21.1862	1.25939	0.629693	−	0.776844i	$-0.283181\pi$
$283$	21.1862	1.25939	0.629693	+	0.776844i	$0.283181\pi$
$284$	6.35549	0.377129
$285$	0	0
$286$	−8.87129	−0.524570
$287$	8.62731	0.509254
$288$	0	0
$289$	40.1127	2.35957
$290$	−4.07695	−0.239406
$291$	0	0
$292$	−6.15634	−0.360273
$293$	27.1475	1.58597	0.792987	−	0.609238i	$-0.208525\pi$
$293$	27.1475	1.58597	0.792987	+	0.609238i	$0.208525\pi$
$294$	0	0
$295$	−22.4927	−1.30958
$296$	−14.7640	−0.858140
$297$	0	0
$298$	−5.22247	−0.302530
$299$	−1.54285	−0.0892254
$300$	0	0
$301$	−51.4815	−2.96734
$302$	−16.9600	−0.975939
$303$	0	0
$304$	3.97461	0.227960
$305$	−2.92977	−0.167758
$306$	0	0
$307$	8.36373	0.477343	0.238672	−	0.971100i	$-0.423288\pi$
$307$	8.36373	0.477343	0.238672	+	0.971100i	$0.423288\pi$
$308$	23.4983	1.33894
$309$	0	0
$310$	−17.1470	−0.973882
$311$	27.9312	1.58383	0.791916	−	0.610630i	$-0.209084\pi$
$311$	27.9312	1.58383	0.791916	+	0.610630i	$0.209084\pi$
$312$	0	0
$313$	−4.19166	−0.236927	−0.118463	−	0.992958i	$-0.537797\pi$
$313$	−4.19166	−0.236927	−0.118463	+	0.992958i	$0.537797\pi$
$314$	−4.69093	−0.264725
$315$	0	0
$316$	9.95701	0.560125
$317$	3.35364	0.188359	0.0941796	−	0.995555i	$-0.469977\pi$
$317$	3.35364	0.188359	0.0941796	+	0.995555i	$0.469977\pi$
$318$	0	0
$319$	−5.58245	−0.312557
$320$	−29.2929	−1.63753
$321$	0	0
$322$	−4.61678	−0.257283
$323$	−24.2256	−1.34795
$324$	0	0
$325$	16.4581	0.912929
$326$	7.27409	0.402875
$327$	0	0
$328$	−5.82674	−0.321728
$329$	7.65436	0.421999
$330$	0	0
$331$	26.3713	1.44950	0.724748	−	0.689014i	$-0.241956\pi$
$331$	26.3713	1.44950	0.724748	+	0.689014i	$0.241956\pi$
$332$	−2.94453	−0.161602
$333$	0	0
$334$	−6.65320	−0.364047
$335$	40.0322	2.18719
$336$	0	0
$337$	−29.2668	−1.59426	−0.797132	−	0.603806i	$-0.793650\pi$
$337$	−29.2668	−1.59426	−0.797132	+	0.603806i	$0.793650\pi$
$338$	−10.9382	−0.594960
$339$	0	0
$340$	28.0914	1.52347
$341$	−23.4788	−1.27145
$342$	0	0
$343$	27.3022	1.47418
$344$	34.7697	1.87466
$345$	0	0
$346$	−11.8355	−0.636281
$347$	−28.4110	−1.52518	−0.762592	−	0.646880i	$-0.776074\pi$
$347$	−28.4110	−1.52518	−0.762592	+	0.646880i	$0.776074\pi$
$348$	0	0
$349$	−9.09625	−0.486911	−0.243456	−	0.969912i	$-0.578281\pi$
$349$	−9.09625	−0.486911	−0.243456	+	0.969912i	$0.578281\pi$
$350$	49.2486	2.63245
$351$	0	0
$352$	−26.6698	−1.42151
$353$	13.8729	0.738379	0.369190	−	0.929354i	$-0.379635\pi$
$353$	13.8729	0.738379	0.369190	+	0.929354i	$0.379635\pi$
$354$	0	0
$355$	26.7877	1.42175
$356$	9.76130	0.517348
$357$	0	0
$358$	9.53559	0.503972
$359$	8.10850	0.427950	0.213975	−	0.976839i	$-0.431359\pi$
$359$	8.10850	0.427950	0.213975	+	0.976839i	$0.431359\pi$
$360$	0	0
$361$	−8.72423	−0.459170
$362$	−23.7753	−1.24960
$363$	0	0
$364$	−6.49436	−0.340397
$365$	−25.9483	−1.35820
$366$	0	0
$367$	−26.5897	−1.38797	−0.693987	−	0.719988i	$-0.744147\pi$
$367$	−26.5897	−1.38797	−0.693987	+	0.719988i	$0.744147\pi$
$368$	1.23990	0.0646344
$369$	0	0
$370$	−19.8832	−1.03368
$371$	−43.2587	−2.24588
$372$	0	0
$373$	−16.3583	−0.846999	−0.423500	−	0.905896i	$-0.639199\pi$
$373$	−16.3583	−0.846999	−0.423500	+	0.905896i	$0.639199\pi$
$374$	−43.4539	−2.24695
$375$	0	0
$376$	−5.16963	−0.266603
$377$	1.54285	0.0794609
$378$	0	0
$379$	−10.4643	−0.537513	−0.268756	−	0.963208i	$-0.586613\pi$
$379$	−10.4643	−0.537513	−0.268756	+	0.963208i	$0.586613\pi$
$380$	−11.9156	−0.611256
$381$	0	0
$382$	4.20575	0.215185
$383$	−7.07450	−0.361490	−0.180745	−	0.983530i	$-0.557851\pi$
$383$	−7.07450	−0.361490	−0.180745	+	0.983530i	$0.557851\pi$
$384$	0	0
$385$	99.0430	5.04770
$386$	−13.4413	−0.684146
$387$	0	0
$388$	8.31894	0.422330
$389$	2.37583	0.120459	0.0602297	−	0.998185i	$-0.480817\pi$
$389$	2.37583	0.120459	0.0602297	+	0.998185i	$0.480817\pi$
$390$	0	0
$391$	−7.55730	−0.382189
$392$	−39.6303	−2.00163
$393$	0	0
$394$	−0.767450	−0.0386636
$395$	41.9677	2.11163
$396$	0	0
$397$	32.4735	1.62980	0.814899	−	0.579603i	$-0.196792\pi$
$397$	32.4735	1.62980	0.814899	+	0.579603i	$0.196792\pi$
$398$	3.76437	0.188691
$399$	0	0
$400$	−13.2264	−0.661321
$401$	−9.72352	−0.485569	−0.242785	−	0.970080i	$-0.578061\pi$
$401$	−9.72352	−0.485569	−0.242785	+	0.970080i	$0.578061\pi$
$402$	0	0
$403$	6.48898	0.323239
$404$	3.28885	0.163626
$405$	0	0
$406$	4.61678	0.229127
$407$	−27.2256	−1.34952
$408$	0	0
$409$	−26.8132	−1.32583	−0.662915	−	0.748695i	$-0.730681\pi$
$409$	−26.8132	−1.32583	−0.662915	+	0.748695i	$0.730681\pi$
$410$	−7.84709	−0.387540
$411$	0	0
$412$	−9.61609	−0.473751
$413$	25.4710	1.25335
$414$	0	0
$415$	−12.4109	−0.609227
$416$	7.37089	0.361388
$417$	0	0
$418$	18.4319	0.901533
$419$	28.6227	1.39831	0.699156	−	0.714969i	$-0.253559\pi$
$419$	28.6227	1.39831	0.699156	+	0.714969i	$0.253559\pi$
$420$	0	0
$421$	−12.4491	−0.606732	−0.303366	−	0.952874i	$-0.598110\pi$
$421$	−12.4491	−0.606732	−0.303366	+	0.952874i	$0.598110\pi$
$422$	−25.9280	−1.26216
$423$	0	0
$424$	29.2162	1.41886
$425$	80.6160	3.91045
$426$	0	0
$427$	3.31771	0.160555
$428$	−4.38448	−0.211932
$429$	0	0
$430$	46.8257	2.25813
$431$	23.2274	1.11883	0.559413	−	0.828889i	$-0.311027\pi$
$431$	23.2274	1.11883	0.559413	+	0.828889i	$0.311027\pi$
$432$	0	0
$433$	−12.9335	−0.621544	−0.310772	−	0.950484i	$-0.600588\pi$
$433$	−12.9335	−0.621544	−0.310772	+	0.950484i	$0.600588\pi$
$434$	19.4174	0.932066
$435$	0	0
$436$	7.86180	0.376512
$437$	3.20558	0.153344
$438$	0	0
$439$	−15.4117	−0.735560	−0.367780	−	0.929913i	$-0.619882\pi$
$439$	−15.4117	−0.735560	−0.367780	+	0.929913i	$0.619882\pi$
$440$	−66.8919	−3.18895
$441$	0	0
$442$	12.0096	0.571238
$443$	−20.3157	−0.965226	−0.482613	−	0.875834i	$-0.660312\pi$
$443$	−20.3157	−0.965226	−0.482613	+	0.875834i	$0.660312\pi$
$444$	0	0
$445$	41.1428	1.95036
$446$	7.95453	0.376658
$447$	0	0
$448$	33.1717	1.56722
$449$	6.05351	0.285683	0.142841	−	0.989746i	$-0.454376\pi$
$449$	6.05351	0.285683	0.142841	+	0.989746i	$0.454376\pi$
$450$	0	0
$451$	−10.7448	−0.505953
$452$	−7.03443	−0.330872
$453$	0	0
$454$	2.00086	0.0939052
$455$	−27.3730	−1.28327
$456$	0	0
$457$	28.1224	1.31551	0.657754	−	0.753233i	$-0.271507\pi$
$457$	28.1224	1.31551	0.657754	+	0.753233i	$0.271507\pi$
$458$	16.4265	0.767560
$459$	0	0
$460$	−3.71713	−0.173312
$461$	−16.8087	−0.782857	−0.391429	−	0.920208i	$-0.628019\pi$
$461$	−16.8087	−0.782857	−0.391429	+	0.920208i	$0.628019\pi$
$462$	0	0
$463$	−21.4247	−0.995691	−0.497845	−	0.867266i	$-0.665875\pi$
$463$	−21.4247	−0.995691	−0.497845	+	0.867266i	$0.665875\pi$
$464$	−1.23990	−0.0575610
$465$	0	0
$466$	−20.0386	−0.928271
$467$	11.7990	0.545993	0.272997	−	0.962015i	$-0.411985\pi$
$467$	11.7990	0.545993	0.272997	+	0.962015i	$0.411985\pi$
$468$	0	0
$469$	−45.3330	−2.09328
$470$	−6.96213	−0.321139
$471$	0	0
$472$	−17.2027	−0.791819
$473$	64.1171	2.94811
$474$	0	0
$475$	−34.1949	−1.56897
$476$	−31.8111	−1.45806
$477$	0	0
$478$	16.1482	0.738602
$479$	−30.3535	−1.38689	−0.693444	−	0.720510i	$-0.743908\pi$
$479$	−30.3535	−1.38689	−0.693444	+	0.720510i	$0.743908\pi$
$480$	0	0
$481$	7.52448	0.343087
$482$	−7.64898	−0.348402
$483$	0	0
$484$	−18.9357	−0.860713
$485$	35.0635	1.59215
$486$	0	0
$487$	8.43053	0.382024	0.191012	−	0.981588i	$-0.438823\pi$
$487$	8.43053	0.382024	0.191012	+	0.981588i	$0.438823\pi$
$488$	−2.24072	−0.101433
$489$	0	0
$490$	−53.3717	−2.41109
$491$	−36.0960	−1.62899	−0.814495	−	0.580170i	$-0.802986\pi$
$491$	−36.0960	−1.62899	−0.814495	+	0.580170i	$0.802986\pi$
$492$	0	0
$493$	7.55730	0.340364
$494$	−5.09412	−0.229195
$495$	0	0
$496$	−5.21482	−0.234152
$497$	−30.3348	−1.36070
$498$	0	0
$499$	−22.1018	−0.989413	−0.494707	−	0.869060i	$-0.664724\pi$
$499$	−22.1018	−0.989413	−0.494707	+	0.869060i	$0.664724\pi$
$500$	21.0661	0.942104
$501$	0	0
$502$	10.6323	0.474541
$503$	9.65157	0.430342	0.215171	−	0.976576i	$-0.430969\pi$
$503$	9.65157	0.430342	0.215171	+	0.976576i	$0.430969\pi$
$504$	0	0
$505$	13.8621	0.616858
$506$	5.74993	0.255616
$507$	0	0
$508$	2.99003	0.132661
$509$	28.3884	1.25829	0.629146	−	0.777287i	$-0.283405\pi$
$509$	28.3884	1.25829	0.629146	+	0.777287i	$0.283405\pi$
$510$	0	0
$511$	29.3842	1.29988
$512$	−13.4306	−0.593555
$513$	0	0
$514$	−30.8246	−1.35961
$515$	−40.5308	−1.78600
$516$	0	0
$517$	−9.53305	−0.419263
$518$	22.5160	0.989298
$519$	0	0
$520$	18.4873	0.810721
$521$	−2.27334	−0.0995970	−0.0497985	−	0.998759i	$-0.515858\pi$
$521$	−2.27334	−0.0995970	−0.0497985	+	0.998759i	$0.515858\pi$
$522$	0	0
$523$	−10.3759	−0.453705	−0.226853	−	0.973929i	$-0.572844\pi$
$523$	−10.3759	−0.453705	−0.226853	+	0.973929i	$0.572844\pi$
$524$	1.34773	0.0588759
$525$	0	0
$526$	−3.59761	−0.156863
$527$	31.7847	1.38456
$528$	0	0
$529$	1.00000	0.0434783
$530$	39.3466	1.70911
$531$	0	0
$532$	13.4933	0.585011
$533$	2.96960	0.128628
$534$	0	0
$535$	−18.4801	−0.798965
$536$	30.6171	1.32246
$537$	0	0
$538$	−1.59528	−0.0687773
$539$	−73.0804	−3.14779
$540$	0	0
$541$	−23.0163	−0.989546	−0.494773	−	0.869022i	$-0.664749\pi$
$541$	−23.0163	−0.989546	−0.494773	+	0.869022i	$0.664749\pi$
$542$	−11.0538	−0.474801
$543$	0	0
$544$	36.1046	1.54797
$545$	33.1366	1.41942
$546$	0	0
$547$	28.0291	1.19844	0.599219	−	0.800585i	$-0.295478\pi$
$547$	28.0291	1.19844	0.599219	+	0.800585i	$0.295478\pi$
$548$	16.8169	0.718384
$549$	0	0
$550$	−61.3362	−2.61539
$551$	−3.20558	−0.136562
$552$	0	0
$553$	−47.5248	−2.02096
$554$	−14.8393	−0.630462
$555$	0	0
$556$	−5.09164	−0.215934
$557$	4.79213	0.203049	0.101525	−	0.994833i	$-0.467628\pi$
$557$	4.79213	0.203049	0.101525	+	0.994833i	$0.467628\pi$
$558$	0	0
$559$	−17.7204	−0.749493
$560$	21.9982	0.929592
$561$	0	0
$562$	−14.3957	−0.607246
$563$	4.03874	0.170213	0.0851063	−	0.996372i	$-0.472877\pi$
$563$	4.03874	0.170213	0.0851063	+	0.996372i	$0.472877\pi$
$564$	0	0
$565$	−29.6494	−1.24736
$566$	21.8218	0.917237
$567$	0	0
$568$	20.4876	0.859640
$569$	5.07151	0.212609	0.106304	−	0.994334i	$-0.466098\pi$
$569$	5.07151	0.212609	0.106304	+	0.994334i	$0.466098\pi$
$570$	0	0
$571$	−18.6566	−0.780755	−0.390377	−	0.920655i	$-0.627655\pi$
$571$	−18.6566	−0.780755	−0.390377	+	0.920655i	$0.627655\pi$
$572$	8.08834	0.338190
$573$	0	0
$574$	8.88615	0.370901
$575$	−10.6673	−0.444857
$576$	0	0
$577$	−5.09852	−0.212254	−0.106127	−	0.994353i	$-0.533845\pi$
$577$	−5.09852	−0.212254	−0.106127	+	0.994353i	$0.533845\pi$
$578$	41.3162	1.71853
$579$	0	0
$580$	3.71713	0.154345
$581$	14.0543	0.583069
$582$	0	0
$583$	53.8762	2.23132
$584$	−19.8456	−0.821217
$585$	0	0
$586$	27.9620	1.15510
$587$	16.7995	0.693392	0.346696	−	0.937978i	$-0.387304\pi$
$587$	16.7995	0.693392	0.346696	+	0.937978i	$0.387304\pi$
$588$	0	0
$589$	−13.4822	−0.555523
$590$	−23.1675	−0.953793
$591$	0	0
$592$	−6.04700	−0.248530
$593$	−17.8837	−0.734395	−0.367197	−	0.930143i	$-0.619683\pi$
$593$	−17.8837	−0.734395	−0.367197	+	0.930143i	$0.619683\pi$
$594$	0	0
$595$	−134.080	−5.49676
$596$	4.76155	0.195041
$597$	0	0
$598$	−1.58914	−0.0649848
$599$	7.25904	0.296596	0.148298	−	0.988943i	$-0.452620\pi$
$599$	7.25904	0.296596	0.148298	+	0.988943i	$0.452620\pi$
$600$	0	0
$601$	−1.93195	−0.0788059	−0.0394030	−	0.999223i	$-0.512546\pi$
$601$	−1.93195	−0.0788059	−0.0394030	+	0.999223i	$0.512546\pi$
$602$	−53.0260	−2.16118
$603$	0	0
$604$	15.4632	0.629188
$605$	−79.8119	−3.24482
$606$	0	0
$607$	28.8068	1.16923	0.584615	−	0.811311i	$-0.301246\pi$
$607$	28.8068	1.16923	0.584615	+	0.811311i	$0.301246\pi$
$608$	−15.3145	−0.621085
$609$	0	0
$610$	−3.01767	−0.122182
$611$	2.63470	0.106589
$612$	0	0
$613$	13.6370	0.550794	0.275397	−	0.961331i	$-0.411191\pi$
$613$	13.6370	0.550794	0.275397	+	0.961331i	$0.411191\pi$
$614$	8.61465	0.347659
$615$	0	0
$616$	75.7492	3.05202
$617$	−5.82175	−0.234375	−0.117187	−	0.993110i	$-0.537388\pi$
$617$	−5.82175	−0.234375	−0.117187	+	0.993110i	$0.537388\pi$
$618$	0	0
$619$	28.8940	1.16135	0.580674	−	0.814136i	$-0.302789\pi$
$619$	28.8940	1.16135	0.580674	+	0.814136i	$0.302789\pi$
$620$	15.6336	0.627862
$621$	0	0
$622$	28.7692	1.15354
$623$	−46.5907	−1.86662
$624$	0	0
$625$	35.4548	1.41819
$626$	−4.31742	−0.172559
$627$	0	0
$628$	4.27693	0.170668
$629$	36.8569	1.46958
$630$	0	0
$631$	−33.9972	−1.35341	−0.676703	−	0.736256i	$-0.736592\pi$
$631$	−33.9972	−1.35341	−0.676703	+	0.736256i	$0.736592\pi$
$632$	32.0974	1.27677
$633$	0	0
$634$	3.45425	0.137186
$635$	12.6027	0.500121
$636$	0	0
$637$	20.1976	0.800259
$638$	−5.74993	−0.227642
$639$	0	0
$640$	7.64836	0.302328
$641$	5.20966	0.205769	0.102885	−	0.994693i	$-0.467193\pi$
$641$	5.20966	0.205769	0.102885	+	0.994693i	$0.467193\pi$
$642$	0	0
$643$	−9.97538	−0.393391	−0.196695	−	0.980465i	$-0.563021\pi$
$643$	−9.97538	−0.393391	−0.196695	+	0.980465i	$0.563021\pi$
$644$	4.20932	0.165871
$645$	0	0
$646$	−24.9524	−0.981737
$647$	9.14442	0.359504	0.179752	−	0.983712i	$-0.442470\pi$
$647$	9.14442	0.359504	0.179752	+	0.983712i	$0.442470\pi$
$648$	0	0
$649$	−31.7227	−1.24522
$650$	16.9518	0.664906
$651$	0	0
$652$	−6.63210	−0.259733
$653$	2.87403	0.112470	0.0562348	−	0.998418i	$-0.482090\pi$
$653$	2.87403	0.112470	0.0562348	+	0.998418i	$0.482090\pi$
$654$	0	0
$655$	5.68055	0.221957
$656$	−2.38650	−0.0931772
$657$	0	0
$658$	7.88401	0.307351
$659$	−11.8713	−0.462439	−0.231219	−	0.972902i	$-0.574271\pi$
$659$	−11.8713	−0.462439	−0.231219	+	0.972902i	$0.574271\pi$
$660$	0	0
$661$	−9.74878	−0.379184	−0.189592	−	0.981863i	$-0.560716\pi$
$661$	−9.74878	−0.379184	−0.189592	+	0.981863i	$0.560716\pi$
$662$	27.1625	1.05570
$663$	0	0
$664$	−9.49200	−0.368361
$665$	56.8730	2.20544
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	6.06602	0.234701
$669$	0	0
$670$	41.2332	1.59298
$671$	−4.13201	−0.159514
$672$	0	0
$673$	−17.7848	−0.685554	−0.342777	−	0.939417i	$-0.611368\pi$
$673$	−17.7848	−0.685554	−0.342777	+	0.939417i	$0.611368\pi$
$674$	−30.1448	−1.16114
$675$	0	0
$676$	9.97284	0.383571
$677$	−7.08914	−0.272458	−0.136229	−	0.990677i	$-0.543498\pi$
$677$	−7.08914	−0.272458	−0.136229	+	0.990677i	$0.543498\pi$
$678$	0	0
$679$	−39.7063	−1.52379
$680$	90.5556	3.47265
$681$	0	0
$682$	−24.1832	−0.926024
$683$	−2.72661	−0.104331	−0.0521654	−	0.998638i	$-0.516612\pi$
$683$	−2.72661	−0.104331	−0.0521654	+	0.998638i	$0.516612\pi$
$684$	0	0
$685$	70.8816	2.70825
$686$	28.1213	1.07368
$687$	0	0
$688$	14.2409	0.542928
$689$	−14.8901	−0.567266
$690$	0	0
$691$	−36.0905	−1.37295	−0.686473	−	0.727155i	$-0.740842\pi$
$691$	−36.0905	−1.37295	−0.686473	+	0.727155i	$0.740842\pi$
$692$	10.7910	0.410211
$693$	0	0
$694$	−29.2634	−1.11082
$695$	−21.4607	−0.814052
$696$	0	0
$697$	14.5459	0.550965
$698$	−9.36915	−0.354628
$699$	0	0
$700$	−44.9021	−1.69714
$701$	−9.17628	−0.346583	−0.173292	−	0.984871i	$-0.555440\pi$
$701$	−9.17628	−0.346583	−0.173292	+	0.984871i	$0.555440\pi$
$702$	0	0
$703$	−15.6336	−0.589633
$704$	−41.3134	−1.55706
$705$	0	0
$706$	14.2891	0.537777
$707$	−15.6977	−0.590372
$708$	0	0
$709$	−11.9213	−0.447713	−0.223856	−	0.974622i	$-0.571865\pi$
$709$	−11.9213	−0.447713	−0.223856	+	0.974622i	$0.571865\pi$
$710$	27.5914	1.03549
$711$	0	0
$712$	31.4665	1.17926
$713$	−4.20583	−0.157510
$714$	0	0
$715$	34.0915	1.27495
$716$	−8.69401	−0.324911
$717$	0	0
$718$	8.35177	0.311685
$719$	46.8342	1.74662	0.873310	−	0.487166i	$-0.161969\pi$
$719$	46.8342	1.74662	0.873310	+	0.487166i	$0.161969\pi$
$720$	0	0
$721$	45.8976	1.70932
$722$	−8.98597	−0.334423
$723$	0	0
$724$	21.6770	0.805619
$725$	10.6673	0.396174
$726$	0	0
$727$	0.568609	0.0210886	0.0105443	−	0.999944i	$-0.496644\pi$
$727$	0.568609	0.0210886	0.0105443	+	0.999944i	$0.496644\pi$
$728$	−20.9352	−0.775911
$729$	0	0
$730$	−26.7268	−0.989204
$731$	−86.7992	−3.21038
$732$	0	0
$733$	43.9920	1.62488	0.812441	−	0.583043i	$-0.198138\pi$
$733$	43.9920	1.62488	0.812441	+	0.583043i	$0.198138\pi$
$734$	−27.3875	−1.01089
$735$	0	0
$736$	−4.77745	−0.176099
$737$	56.4595	2.07971
$738$	0	0
$739$	30.8281	1.13403	0.567015	−	0.823708i	$-0.308098\pi$
$739$	30.8281	1.13403	0.567015	+	0.823708i	$0.308098\pi$
$740$	18.1284	0.666414
$741$	0	0
$742$	−44.5566	−1.63572
$743$	21.2479	0.779510	0.389755	−	0.920919i	$-0.372560\pi$
$743$	21.2479	0.779510	0.389755	+	0.920919i	$0.372560\pi$
$744$	0	0
$745$	20.0694	0.735288
$746$	−16.8490	−0.616887
$747$	0	0
$748$	39.6188	1.44861
$749$	20.9271	0.764660
$750$	0	0
$751$	3.75923	0.137176	0.0685882	−	0.997645i	$-0.478151\pi$
$751$	3.75923	0.137176	0.0685882	+	0.997645i	$0.478151\pi$
$752$	−2.11736	−0.0772122
$753$	0	0
$754$	1.58914	0.0578731
$755$	65.1757	2.37198
$756$	0	0
$757$	17.0669	0.620307	0.310154	−	0.950686i	$-0.399619\pi$
$757$	17.0669	0.620307	0.310154	+	0.950686i	$0.399619\pi$
$758$	−10.7782	−0.391482
$759$	0	0
$760$	−38.4110	−1.39331
$761$	−11.1752	−0.405101	−0.202551	−	0.979272i	$-0.564923\pi$
$761$	−11.1752	−0.405101	−0.202551	+	0.979272i	$0.564923\pi$
$762$	0	0
$763$	−37.5243	−1.35847
$764$	−3.83456	−0.138730
$765$	0	0
$766$	−7.28675	−0.263281
$767$	8.76737	0.316571
$768$	0	0
$769$	−46.6242	−1.68131	−0.840656	−	0.541569i	$-0.817831\pi$
$769$	−46.6242	−1.68131	−0.840656	+	0.541569i	$0.817831\pi$
$770$	102.014	3.67634
$771$	0	0
$772$	12.2551	0.441069
$773$	−37.9835	−1.36617	−0.683086	−	0.730338i	$-0.739362\pi$
$773$	−37.9835	−1.36617	−0.683086	+	0.730338i	$0.739362\pi$
$774$	0	0
$775$	44.8649	1.61159
$776$	26.8170	0.962672
$777$	0	0
$778$	2.44711	0.0877332
$779$	−6.16994	−0.221061
$780$	0	0
$781$	37.7801	1.35188
$782$	−7.78403	−0.278356
$783$	0	0
$784$	−16.2317	−0.579703
$785$	18.0268	0.643404
$786$	0	0
$787$	28.3953	1.01218	0.506092	−	0.862480i	$-0.331090\pi$
$787$	28.3953	1.01218	0.506092	+	0.862480i	$0.331090\pi$
$788$	0.699718	0.0249264
$789$	0	0
$790$	43.2268	1.53794
$791$	33.5753	1.19380
$792$	0	0
$793$	1.14199	0.0405531
$794$	33.4477	1.18702
$795$	0	0
$796$	−3.43214	−0.121649
$797$	−29.7031	−1.05214	−0.526068	−	0.850442i	$-0.676334\pi$
$797$	−29.7031	−1.05214	−0.526068	+	0.850442i	$0.676334\pi$
$798$	0	0
$799$	12.9055	0.456563
$800$	50.9625	1.80180
$801$	0	0
$802$	−10.0152	−0.353650
$803$	−36.5963	−1.29146
$804$	0	0
$805$	17.7419	0.625318
$806$	6.68366	0.235422
$807$	0	0
$808$	10.6019	0.372975
$809$	−30.6595	−1.07793	−0.538965	−	0.842328i	$-0.681184\pi$
$809$	−30.6595	−1.07793	−0.538965	+	0.842328i	$0.681184\pi$
$810$	0	0
$811$	18.4219	0.646881	0.323441	−	0.946248i	$-0.395160\pi$
$811$	18.4219	0.646881	0.323441	+	0.946248i	$0.395160\pi$
$812$	−4.20932	−0.147718
$813$	0	0
$814$	−28.0424	−0.982885
$815$	−27.9536	−0.979172
$816$	0	0
$817$	36.8177	1.28809
$818$	−27.6177	−0.965629
$819$	0	0
$820$	7.15454	0.249847
$821$	46.7160	1.63040	0.815199	−	0.579181i	$-0.196627\pi$
$821$	46.7160	1.63040	0.815199	+	0.579181i	$0.196627\pi$
$822$	0	0
$823$	9.15823	0.319236	0.159618	−	0.987179i	$-0.448974\pi$
$823$	9.15823	0.319236	0.159618	+	0.987179i	$0.448974\pi$
$824$	−30.9984	−1.07988
$825$	0	0
$826$	26.2352	0.912840
$827$	56.5427	1.96618	0.983090	−	0.183121i	$-0.0586199\pi$
$827$	56.5427	1.96618	0.983090	+	0.183121i	$0.0586199\pi$
$828$	0	0
$829$	−3.31793	−0.115236	−0.0576182	−	0.998339i	$-0.518351\pi$
$829$	−3.31793	−0.115236	−0.0576182	+	0.998339i	$0.518351\pi$
$830$	−12.7832	−0.443713
$831$	0	0
$832$	11.4180	0.395848
$833$	98.9333	3.42784
$834$	0	0
$835$	25.5676	0.884804
$836$	−16.8051	−0.581218
$837$	0	0
$838$	29.4815	1.01842
$839$	−39.7713	−1.37306	−0.686530	−	0.727102i	$-0.740867\pi$
$839$	−39.7713	−1.37306	−0.686530	+	0.727102i	$0.740867\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−12.8226	−0.441896
$843$	0	0
$844$	23.6397	0.813712
$845$	42.0345	1.44603
$846$	0	0
$847$	90.3800	3.10549
$848$	11.9663	0.410924
$849$	0	0
$850$	83.0346	2.84806
$851$	−4.87699	−0.167181
$852$	0	0
$853$	20.8747	0.714737	0.357369	−	0.933963i	$-0.383674\pi$
$853$	20.8747	0.714737	0.357369	+	0.933963i	$0.383674\pi$
$854$	3.41724	0.116936
$855$	0	0
$856$	−14.1338	−0.483084
$857$	43.5186	1.48657	0.743283	−	0.668977i	$-0.233268\pi$
$857$	43.5186	1.48657	0.743283	+	0.668977i	$0.233268\pi$
$858$	0	0
$859$	−21.7817	−0.743181	−0.371591	−	0.928397i	$-0.621187\pi$
$859$	−21.7817	−0.743181	−0.371591	+	0.928397i	$0.621187\pi$
$860$	−42.6930	−1.45582
$861$	0	0
$862$	23.9243	0.814864
$863$	40.8871	1.39181	0.695906	−	0.718133i	$-0.255003\pi$
$863$	40.8871	1.39181	0.695906	+	0.718133i	$0.255003\pi$
$864$	0	0
$865$	45.4827	1.54646
$866$	−13.3215	−0.452684
$867$	0	0
$868$	−17.7037	−0.600903
$869$	59.1893	2.00786
$870$	0	0
$871$	−15.6040	−0.528722
$872$	25.3433	0.858232
$873$	0	0
$874$	3.30176	0.111684
$875$	−100.548	−3.39916
$876$	0	0
$877$	31.0562	1.04869	0.524347	−	0.851505i	$-0.324309\pi$
$877$	31.0562	1.04869	0.524347	+	0.851505i	$0.324309\pi$
$878$	−15.8741	−0.535724
$879$	0	0
$880$	−27.3974	−0.923566
$881$	−49.5220	−1.66844	−0.834219	−	0.551433i	$-0.814081\pi$
$881$	−49.5220	−1.66844	−0.834219	+	0.551433i	$0.814081\pi$
$882$	0	0
$883$	15.7559	0.530229	0.265114	−	0.964217i	$-0.414590\pi$
$883$	15.7559	0.530229	0.265114	+	0.964217i	$0.414590\pi$
$884$	−10.9497	−0.368277
$885$	0	0
$886$	−20.9252	−0.702994
$887$	12.3349	0.414165	0.207082	−	0.978323i	$-0.433603\pi$
$887$	12.3349	0.414165	0.207082	+	0.978323i	$0.433603\pi$
$888$	0	0
$889$	−14.2714	−0.478648
$890$	42.3772	1.42049
$891$	0	0
$892$	−7.25249	−0.242831
$893$	−5.47412	−0.183185
$894$	0	0
$895$	−36.6443	−1.22489
$896$	−8.66109	−0.289347
$897$	0	0
$898$	6.23513	0.208069
$899$	4.20583	0.140272
$900$	0	0
$901$	−72.9354	−2.42983
$902$	−11.0672	−0.368496
$903$	0	0
$904$	−22.6762	−0.754199
$905$	91.3662	3.03712
$906$	0	0
$907$	−12.4717	−0.414117	−0.207058	−	0.978329i	$-0.566389\pi$
$907$	−12.4717	−0.414117	−0.207058	+	0.978329i	$0.566389\pi$
$908$	−1.82427	−0.0605407
$909$	0	0
$910$	−28.1943	−0.934631
$911$	−34.3809	−1.13909	−0.569545	−	0.821960i	$-0.692881\pi$
$911$	−34.3809	−1.13909	−0.569545	+	0.821960i	$0.692881\pi$
$912$	0	0
$913$	−17.5037	−0.579289
$914$	28.9661	0.958112
$915$	0	0
$916$	−14.9768	−0.494846
$917$	−6.43272	−0.212427
$918$	0	0
$919$	−15.5788	−0.513898	−0.256949	−	0.966425i	$-0.582717\pi$
$919$	−15.5788	−0.513898	−0.256949	+	0.966425i	$0.582717\pi$
$920$	−11.9825	−0.395053
$921$	0	0
$922$	−17.3129	−0.570171
$923$	−10.4415	−0.343686
$924$	0	0
$925$	52.0244	1.71055
$926$	−22.0675	−0.725182
$927$	0	0
$928$	4.77745	0.156827
$929$	−43.9844	−1.44308	−0.721541	−	0.692371i	$-0.756566\pi$
$929$	−43.9844	−1.44308	−0.721541	+	0.692371i	$0.756566\pi$
$930$	0	0
$931$	−41.9646	−1.37533
$932$	18.2701	0.598457
$933$	0	0
$934$	12.1530	0.397658
$935$	166.989	5.46113
$936$	0	0
$937$	56.3144	1.83971	0.919855	−	0.392259i	$-0.128306\pi$
$937$	56.3144	1.83971	0.919855	+	0.392259i	$0.128306\pi$
$938$	−46.6930	−1.52458
$939$	0	0
$940$	6.34768	0.207038
$941$	16.2465	0.529622	0.264811	−	0.964300i	$-0.414690\pi$
$941$	16.2465	0.529622	0.264811	+	0.964300i	$0.414690\pi$
$942$	0	0
$943$	−1.92475	−0.0626784
$944$	−7.04584	−0.229322
$945$	0	0
$946$	66.0407	2.14717
$947$	−36.2464	−1.17785	−0.588926	−	0.808187i	$-0.700449\pi$
$947$	−36.2464	−1.17785	−0.588926	+	0.808187i	$0.700449\pi$
$948$	0	0
$949$	10.1143	0.328325
$950$	−35.2208	−1.14271
$951$	0	0
$952$	−102.546	−3.32354
$953$	53.9400	1.74729	0.873643	−	0.486567i	$-0.161751\pi$
$953$	53.9400	1.74729	0.873643	+	0.486567i	$0.161751\pi$
$954$	0	0
$955$	−16.1623	−0.522999
$956$	−14.7230	−0.476177
$957$	0	0
$958$	−31.2642	−1.01010
$959$	−80.2672	−2.59196
$960$	0	0
$961$	−13.3110	−0.429386
$962$	7.75022	0.249877
$963$	0	0
$964$	6.97391	0.224614
$965$	51.6538	1.66279
$966$	0	0
$967$	62.0485	1.99535	0.997673	−	0.0681872i	$-0.0217215\pi$
$967$	62.0485	1.99535	0.997673	+	0.0681872i	$0.0217215\pi$
$968$	−61.0411	−1.96194
$969$	0	0
$970$	36.1154	1.15960
$971$	24.3236	0.780583	0.390291	−	0.920691i	$-0.372374\pi$
$971$	24.3236	0.780583	0.390291	+	0.920691i	$0.372374\pi$
$972$	0	0
$973$	24.3024	0.779100
$974$	8.68345	0.278236
$975$	0	0
$976$	−0.917749	−0.0293764
$977$	−34.3435	−1.09875	−0.549373	−	0.835577i	$-0.685133\pi$
$977$	−34.3435	−1.09875	−0.549373	+	0.835577i	$0.685133\pi$
$978$	0	0
$979$	58.0259	1.85452
$980$	48.6613	1.55443
$981$	0	0
$982$	−37.1790	−1.18643
$983$	52.1361	1.66288	0.831442	−	0.555612i	$-0.187516\pi$
$983$	52.1361	1.66288	0.831442	+	0.555612i	$0.187516\pi$
$984$	0	0
$985$	2.94924	0.0939705
$986$	7.78403	0.247894
$987$	0	0
$988$	4.64453	0.147762
$989$	11.4855	0.365217
$990$	0	0
$991$	21.4342	0.680880	0.340440	−	0.940266i	$-0.389424\pi$
$991$	21.4342	0.680880	0.340440	+	0.940266i	$0.389424\pi$
$992$	20.0931	0.637958
$993$	0	0
$994$	−31.2449	−0.991026
$995$	−14.4661	−0.458606
$996$	0	0
$997$	−20.0157	−0.633904	−0.316952	−	0.948442i	$-0.602659\pi$
$997$	−20.0157	−0.633904	−0.316952	+	0.948442i	$0.602659\pi$
$998$	−22.7649	−0.720610
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.q.1.11		16
3.2	odd	2		667.2.a.d.1.6	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
667.2.a.d.1.6	✓	16	3.2	odd	2
6003.2.a.q.1.11		16	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+1.03000 q^{2} -0.939097 q^{4} -3.95819 q^{5} +4.48231 q^{7} -3.02727 q^{8} +O(q^{10})\)	\(q+1.03000 q^{2} -0.939097 q^{4} -3.95819 q^{5} +4.48231 q^{7} -3.02727 q^{8} -4.07695 q^{10} -5.58245 q^{11} +1.54285 q^{13} +4.61678 q^{14} -1.23990 q^{16} +7.55730 q^{17} -3.20558 q^{19} +3.71713 q^{20} -5.74993 q^{22} -1.00000 q^{23} +10.6673 q^{25} +1.58914 q^{26} -4.20932 q^{28} +1.00000 q^{29} +4.20583 q^{31} +4.77745 q^{32} +7.78403 q^{34} -17.7419 q^{35} +4.87699 q^{37} -3.30176 q^{38} +11.9825 q^{40} +1.92475 q^{41} -11.4855 q^{43} +5.24246 q^{44} -1.03000 q^{46} +1.70768 q^{47} +13.0911 q^{49} +10.9873 q^{50} -1.44889 q^{52} -9.65100 q^{53} +22.0964 q^{55} -13.5692 q^{56} +1.03000 q^{58} +5.68257 q^{59} +0.740179 q^{61} +4.33201 q^{62} +7.40058 q^{64} -6.10691 q^{65} -10.1138 q^{67} -7.09704 q^{68} -18.2741 q^{70} -6.76766 q^{71} +6.55560 q^{73} +5.02331 q^{74} +3.01036 q^{76} -25.0223 q^{77} -10.6027 q^{79} +4.90777 q^{80} +1.98249 q^{82} +3.13550 q^{83} -29.9132 q^{85} -11.8301 q^{86} +16.8996 q^{88} -10.3943 q^{89} +6.91554 q^{91} +0.939097 q^{92} +1.75892 q^{94} +12.6883 q^{95} -8.85845 q^{97} +13.4838 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 3 q^{2} + 21 q^{4} - 16 q^{5} + q^{7} - 9 q^{8}+O(q^{10}) \) 16 * q - 3 * q^2 + 21 * q^4 - 16 * q^5 + q^7 - 9 * q^8	\( 16 q - 3 q^{2} + 21 q^{4} - 16 q^{5} + q^{7} - 9 q^{8} - 14 q^{10} - 4 q^{11} + 15 q^{13} - 8 q^{14} + 23 q^{16} - 20 q^{17} - 4 q^{19} - 25 q^{20} + 13 q^{22} - 16 q^{23} + 30 q^{25} - 25 q^{26} - 13 q^{28} + 16 q^{29} + 19 q^{32} - 23 q^{34} - 5 q^{35} + 5 q^{37} - 38 q^{38} - 20 q^{40} - 7 q^{41} - 17 q^{43} + 21 q^{44} + 3 q^{46} - 29 q^{47} + 31 q^{49} + 44 q^{50} + 20 q^{52} - 63 q^{53} + q^{55} + 19 q^{56} - 3 q^{58} - 11 q^{59} - 33 q^{62} + 29 q^{64} - 53 q^{65} - 13 q^{67} - 63 q^{68} - 46 q^{70} + 23 q^{71} - 38 q^{73} + 47 q^{74} - 56 q^{76} - 97 q^{77} - 27 q^{79} - 8 q^{80} + 9 q^{82} - 36 q^{83} + 6 q^{85} + 11 q^{86} - 24 q^{88} + 16 q^{89} - 47 q^{91} - 21 q^{92} + 37 q^{94} + 12 q^{95} - 30 q^{97} + 27 q^{98}+O(q^{100}) \) 16 * q - 3 * q^2 + 21 * q^4 - 16 * q^5 + q^7 - 9 * q^8 - 14 * q^10 - 4 * q^11 + 15 * q^13 - 8 * q^14 + 23 * q^16 - 20 * q^17 - 4 * q^19 - 25 * q^20 + 13 * q^22 - 16 * q^23 + 30 * q^25 - 25 * q^26 - 13 * q^28 + 16 * q^29 + 19 * q^32 - 23 * q^34 - 5 * q^35 + 5 * q^37 - 38 * q^38 - 20 * q^40 - 7 * q^41 - 17 * q^43 + 21 * q^44 + 3 * q^46 - 29 * q^47 + 31 * q^49 + 44 * q^50 + 20 * q^52 - 63 * q^53 + q^55 + 19 * q^56 - 3 * q^58 - 11 * q^59 - 33 * q^62 + 29 * q^64 - 53 * q^65 - 13 * q^67 - 63 * q^68 - 46 * q^70 + 23 * q^71 - 38 * q^73 + 47 * q^74 - 56 * q^76 - 97 * q^77 - 27 * q^79 - 8 * q^80 + 9 * q^82 - 36 * q^83 + 6 * q^85 + 11 * q^86 - 24 * q^88 + 16 * q^89 - 47 * q^91 - 21 * q^92 + 37 * q^94 + 12 * q^95 - 30 * q^97 + 27 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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