LMFDB - Embedded newform 6003.2.a.u.1.15

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:	$22$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Character	$\chi$	$=$	6003.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.24737 q^{2} -0.444068 q^{4} +1.99313 q^{5} -4.26394 q^{7} -3.04866 q^{8} +O(q^{10})$	\(q+1.24737 q^{2} -0.444068 q^{4} +1.99313 q^{5} -4.26394 q^{7} -3.04866 q^{8} +2.48617 q^{10} +2.17945 q^{11} +0.563290 q^{13} -5.31871 q^{14} -2.91467 q^{16} +2.08203 q^{17} +7.02640 q^{19} -0.885084 q^{20} +2.71858 q^{22} -1.00000 q^{23} -1.02744 q^{25} +0.702630 q^{26} +1.89348 q^{28} -1.00000 q^{29} -5.22820 q^{31} +2.46165 q^{32} +2.59706 q^{34} -8.49858 q^{35} -3.21412 q^{37} +8.76453 q^{38} -6.07636 q^{40} -4.83996 q^{41} -12.2884 q^{43} -0.967826 q^{44} -1.24737 q^{46} +6.11511 q^{47} +11.1812 q^{49} -1.28160 q^{50} -0.250139 q^{52} -8.92910 q^{53} +4.34393 q^{55} +12.9993 q^{56} -1.24737 q^{58} -6.86887 q^{59} +1.96328 q^{61} -6.52150 q^{62} +8.89992 q^{64} +1.12271 q^{65} +13.0683 q^{67} -0.924563 q^{68} -10.6009 q^{70} -9.69199 q^{71} +7.14377 q^{73} -4.00920 q^{74} -3.12020 q^{76} -9.29306 q^{77} +11.3234 q^{79} -5.80930 q^{80} -6.03722 q^{82} -6.81709 q^{83} +4.14975 q^{85} -15.3282 q^{86} -6.64441 q^{88} -5.39987 q^{89} -2.40183 q^{91} +0.444068 q^{92} +7.62781 q^{94} +14.0045 q^{95} -3.65219 q^{97} +13.9471 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q + 3 q^{2} + 17 q^{4} - 6 q^{7} + 6 q^{8}+O(q^{10}) $ 22 * q + 3 * q^2 + 17 * q^4 - 6 * q^7 + 6 * q^8	$ 22 q + 3 q^{2} + 17 q^{4} - 6 q^{7} + 6 q^{8} - 12 q^{10} - 28 q^{13} + q^{14} + 3 q^{16} + 10 q^{17} - 8 q^{19} - 11 q^{22} - 22 q^{23} - 11 q^{26} - 21 q^{28} - 22 q^{29} - 18 q^{31} - 5 q^{32} - 33 q^{34} - 2 q^{35} - 28 q^{37} - 14 q^{38} - 30 q^{40} + 10 q^{41} - 14 q^{43} - 37 q^{44} - 3 q^{46} + 18 q^{47} + 2 q^{49} - 7 q^{50} - 57 q^{52} - 20 q^{53} - 42 q^{55} + 2 q^{56} - 3 q^{58} + 20 q^{59} - 38 q^{61} - 4 q^{62} - 24 q^{64} - 12 q^{65} - 50 q^{67} - 11 q^{68} - 48 q^{70} - 12 q^{71} - 46 q^{73} + 6 q^{74} - 16 q^{76} + 14 q^{77} - 20 q^{79} + 58 q^{80} - 42 q^{82} - 22 q^{83} - 66 q^{85} - 22 q^{86} - 68 q^{88} + 14 q^{89} - 16 q^{91} - 17 q^{92} - 27 q^{94} + 20 q^{95} - 48 q^{97} + 28 q^{98}+O(q^{100}) $ 22 * q + 3 * q^2 + 17 * q^4 - 6 * q^7 + 6 * q^8 - 12 * q^10 - 28 * q^13 + q^14 + 3 * q^16 + 10 * q^17 - 8 * q^19 - 11 * q^22 - 22 * q^23 - 11 * q^26 - 21 * q^28 - 22 * q^29 - 18 * q^31 - 5 * q^32 - 33 * q^34 - 2 * q^35 - 28 * q^37 - 14 * q^38 - 30 * q^40 + 10 * q^41 - 14 * q^43 - 37 * q^44 - 3 * q^46 + 18 * q^47 + 2 * q^49 - 7 * q^50 - 57 * q^52 - 20 * q^53 - 42 * q^55 + 2 * q^56 - 3 * q^58 + 20 * q^59 - 38 * q^61 - 4 * q^62 - 24 * q^64 - 12 * q^65 - 50 * q^67 - 11 * q^68 - 48 * q^70 - 12 * q^71 - 46 * q^73 + 6 * q^74 - 16 * q^76 + 14 * q^77 - 20 * q^79 + 58 * q^80 - 42 * q^82 - 22 * q^83 - 66 * q^85 - 22 * q^86 - 68 * q^88 + 14 * q^89 - 16 * q^91 - 17 * q^92 - 27 * q^94 + 20 * q^95 - 48 * q^97 + 28 * q^98

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 $

$2$	1.24737	0.882024	0.441012	−	0.897501i	$-0.354620\pi$
$2$	1.24737	0.882024	0.441012	+	0.897501i	$0.354620\pi$
$3$	0	0
$3$	0	0
$4$	−0.444068	−0.222034
$5$	1.99313	0.891354	0.445677	−	0.895194i	$-0.352963\pi$
$5$	1.99313	0.891354	0.445677	+	0.895194i	$0.352963\pi$
$6$	0	0
$7$	−4.26394	−1.61162	−0.805809	−	0.592175i	$-0.798269\pi$
$7$	−4.26394	−1.61162	−0.805809	+	0.592175i	$0.798269\pi$
$8$	−3.04866	−1.07786
$9$	0	0
$10$	2.48617	0.786195
$11$	2.17945	0.657130	0.328565	−	0.944481i	$-0.393435\pi$
$11$	2.17945	0.657130	0.328565	+	0.944481i	$0.393435\pi$
$12$	0	0
$13$	0.563290	0.156228	0.0781142	−	0.996944i	$-0.475110\pi$
$13$	0.563290	0.156228	0.0781142	+	0.996944i	$0.475110\pi$
$14$	−5.31871	−1.42149
$15$	0	0
$16$	−2.91467	−0.728667
$17$	2.08203	0.504966	0.252483	−	0.967601i	$-0.418753\pi$
$17$	2.08203	0.504966	0.252483	+	0.967601i	$0.418753\pi$
$18$	0	0
$19$	7.02640	1.61197	0.805984	−	0.591938i	$-0.201637\pi$
$19$	7.02640	1.61197	0.805984	+	0.591938i	$0.201637\pi$
$20$	−0.885084	−0.197911
$21$	0	0
$22$	2.71858	0.579604
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−1.02744	−0.205489
$26$	0.702630	0.137797
$27$	0	0
$28$	1.89348	0.357834
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−5.22820	−0.939012	−0.469506	−	0.882929i	$-0.655568\pi$
$31$	−5.22820	−0.939012	−0.469506	+	0.882929i	$0.655568\pi$
$32$	2.46165	0.435162
$33$	0	0
$34$	2.59706	0.445392
$35$	−8.49858	−1.43652
$36$	0	0
$37$	−3.21412	−0.528398	−0.264199	−	0.964468i	$-0.585108\pi$
$37$	−3.21412	−0.528398	−0.264199	+	0.964468i	$0.585108\pi$
$38$	8.76453	1.42179
$39$	0	0
$40$	−6.07636	−0.960757
$41$	−4.83996	−0.755875	−0.377937	−	0.925831i	$-0.623367\pi$
$41$	−4.83996	−0.755875	−0.377937	+	0.925831i	$0.623367\pi$
$42$	0	0
$43$	−12.2884	−1.87397	−0.936985	−	0.349368i	$-0.886396\pi$
$43$	−12.2884	−1.87397	−0.936985	+	0.349368i	$0.886396\pi$
$44$	−0.967826	−0.145905
$45$	0	0
$46$	−1.24737	−0.183915
$47$	6.11511	0.891981	0.445990	−	0.895038i	$-0.352852\pi$
$47$	6.11511	0.891981	0.445990	+	0.895038i	$0.352852\pi$
$48$	0	0
$49$	11.1812	1.59731
$50$	−1.28160	−0.181246
$51$	0	0
$52$	−0.250139	−0.0346880
$53$	−8.92910	−1.22651	−0.613253	−	0.789887i	$-0.710139\pi$
$53$	−8.92910	−1.22651	−0.613253	+	0.789887i	$0.710139\pi$
$54$	0	0
$55$	4.34393	0.585735
$56$	12.9993	1.73710
$57$	0	0
$58$	−1.24737	−0.163788
$59$	−6.86887	−0.894251	−0.447126	−	0.894471i	$-0.647552\pi$
$59$	−6.86887	−0.894251	−0.447126	+	0.894471i	$0.647552\pi$
$60$	0	0
$61$	1.96328	0.251372	0.125686	−	0.992070i	$-0.459887\pi$
$61$	1.96328	0.251372	0.125686	+	0.992070i	$0.459887\pi$
$62$	−6.52150	−0.828231
$63$	0	0
$64$	8.89992	1.11249
$65$	1.12271	0.139255
$66$	0	0
$67$	13.0683	1.59655	0.798276	−	0.602292i	$-0.205746\pi$
$67$	13.0683	1.59655	0.798276	+	0.602292i	$0.205746\pi$
$68$	−0.924563	−0.112120
$69$	0	0
$70$	−10.6009	−1.26705
$71$	−9.69199	−1.15023	−0.575114	−	0.818073i	$-0.695042\pi$
$71$	−9.69199	−1.15023	−0.575114	+	0.818073i	$0.695042\pi$
$72$	0	0
$73$	7.14377	0.836115	0.418057	−	0.908421i	$-0.362711\pi$
$73$	7.14377	0.836115	0.418057	+	0.908421i	$0.362711\pi$
$74$	−4.00920	−0.466060
$75$	0	0
$76$	−3.12020	−0.357912
$77$	−9.29306	−1.05904
$78$	0	0
$79$	11.3234	1.27399	0.636994	−	0.770869i	$-0.280178\pi$
$79$	11.3234	1.27399	0.636994	+	0.770869i	$0.280178\pi$
$80$	−5.80930	−0.649500
$81$	0	0
$82$	−6.03722	−0.666699
$83$	−6.81709	−0.748273	−0.374136	−	0.927374i	$-0.622061\pi$
$83$	−6.81709	−0.748273	−0.374136	+	0.927374i	$0.622061\pi$
$84$	0	0
$85$	4.14975	0.450103
$86$	−15.3282	−1.65289
$87$	0	0
$88$	−6.64441	−0.708296
$89$	−5.39987	−0.572385	−0.286192	−	0.958172i	$-0.592390\pi$
$89$	−5.39987	−0.572385	−0.286192	+	0.958172i	$0.592390\pi$
$90$	0	0
$91$	−2.40183	−0.251781
$92$	0.444068	0.0462973
$93$	0	0
$94$	7.62781	0.786748
$95$	14.0045	1.43683
$96$	0	0
$97$	−3.65219	−0.370824	−0.185412	−	0.982661i	$-0.559362\pi$
$97$	−3.65219	−0.370824	−0.185412	+	0.982661i	$0.559362\pi$
$98$	13.9471	1.40887
$99$	0	0
$100$	0.456255	0.0456255
$101$	−4.27417	−0.425296	−0.212648	−	0.977129i	$-0.568209\pi$
$101$	−4.27417	−0.425296	−0.212648	+	0.977129i	$0.568209\pi$
$102$	0	0
$103$	10.4801	1.03263	0.516316	−	0.856398i	$-0.327303\pi$
$103$	10.4801	1.03263	0.516316	+	0.856398i	$0.327303\pi$
$104$	−1.71728	−0.168393
$105$	0	0
$106$	−11.1379	−1.08181
$107$	−5.55127	−0.536661	−0.268331	−	0.963327i	$-0.586472\pi$
$107$	−5.55127	−0.536661	−0.268331	+	0.963327i	$0.586472\pi$
$108$	0	0
$109$	−16.4184	−1.57260	−0.786298	−	0.617848i	$-0.788005\pi$
$109$	−16.4184	−1.57260	−0.786298	+	0.617848i	$0.788005\pi$
$110$	5.41849	0.516632
$111$	0	0
$112$	12.4280	1.17433
$113$	7.72206	0.726430	0.363215	−	0.931705i	$-0.381679\pi$
$113$	7.72206	0.726430	0.363215	+	0.931705i	$0.381679\pi$
$114$	0	0
$115$	−1.99313	−0.185860
$116$	0.444068	0.0412307
$117$	0	0
$118$	−8.56803	−0.788751
$119$	−8.87765	−0.813813
$120$	0	0
$121$	−6.24998	−0.568180
$122$	2.44894	0.221717
$123$	0	0
$124$	2.32168	0.208493
$125$	−12.0135	−1.07452
$126$	0	0
$127$	−13.9951	−1.24186	−0.620932	−	0.783864i	$-0.713246\pi$
$127$	−13.9951	−1.24186	−0.620932	+	0.783864i	$0.713246\pi$
$128$	6.17820	0.546081
$129$	0	0
$130$	1.40043	0.122826
$131$	−10.9144	−0.953598	−0.476799	−	0.879012i	$-0.658203\pi$
$131$	−10.9144	−0.953598	−0.476799	+	0.879012i	$0.658203\pi$
$132$	0	0
$133$	−29.9602	−2.59788
$134$	16.3011	1.40820
$135$	0	0
$136$	−6.34739	−0.544284
$137$	−9.28667	−0.793414	−0.396707	−	0.917945i	$-0.629847\pi$
$137$	−9.28667	−0.793414	−0.396707	+	0.917945i	$0.629847\pi$
$138$	0	0
$139$	17.3020	1.46754	0.733768	−	0.679400i	$-0.237760\pi$
$139$	17.3020	1.46754	0.733768	+	0.679400i	$0.237760\pi$
$140$	3.77395	0.318957
$141$	0	0
$142$	−12.0895	−1.01453
$143$	1.22766	0.102662
$144$	0	0
$145$	−1.99313	−0.165520
$146$	8.91092	0.737473
$147$	0	0
$148$	1.42729	0.117322
$149$	−14.1617	−1.16017	−0.580085	−	0.814556i	$-0.696981\pi$
$149$	−14.1617	−1.16017	−0.580085	+	0.814556i	$0.696981\pi$
$150$	0	0
$151$	−13.2366	−1.07718	−0.538591	−	0.842567i	$-0.681043\pi$
$151$	−13.2366	−1.07718	−0.538591	+	0.842567i	$0.681043\pi$
$152$	−21.4211	−1.73748
$153$	0	0
$154$	−11.5919	−0.934101
$155$	−10.4205	−0.836992
$156$	0	0
$157$	−15.3147	−1.22225	−0.611124	−	0.791535i	$-0.709282\pi$
$157$	−15.3147	−1.22225	−0.611124	+	0.791535i	$0.709282\pi$
$158$	14.1245	1.12369
$159$	0	0
$160$	4.90637	0.387883
$161$	4.26394	0.336046
$162$	0	0
$163$	−14.1835	−1.11094	−0.555469	−	0.831537i	$-0.687461\pi$
$163$	−14.1835	−1.11094	−0.555469	+	0.831537i	$0.687461\pi$
$164$	2.14927	0.167830
$165$	0	0
$166$	−8.50343	−0.659994
$167$	−4.84408	−0.374846	−0.187423	−	0.982279i	$-0.560014\pi$
$167$	−4.84408	−0.374846	−0.187423	+	0.982279i	$0.560014\pi$
$168$	0	0
$169$	−12.6827	−0.975593
$170$	5.17627	0.397002
$171$	0	0
$172$	5.45691	0.416085
$173$	−20.4992	−1.55853	−0.779263	−	0.626697i	$-0.784407\pi$
$173$	−20.4992	−1.55853	−0.779263	+	0.626697i	$0.784407\pi$
$174$	0	0
$175$	4.38096	0.331169
$176$	−6.35238	−0.478829
$177$	0	0
$178$	−6.73563	−0.504857
$179$	−9.34936	−0.698804	−0.349402	−	0.936973i	$-0.613615\pi$
$179$	−9.34936	−0.698804	−0.349402	+	0.936973i	$0.613615\pi$
$180$	0	0
$181$	−4.92860	−0.366340	−0.183170	−	0.983081i	$-0.558636\pi$
$181$	−4.92860	−0.366340	−0.183170	+	0.983081i	$0.558636\pi$
$182$	−2.99598	−0.222076
$183$	0	0
$184$	3.04866	0.224750
$185$	−6.40616	−0.470990
$186$	0	0
$187$	4.53768	0.331828
$188$	−2.71553	−0.198050
$189$	0	0
$190$	17.4688	1.26732
$191$	7.11425	0.514769	0.257385	−	0.966309i	$-0.417139\pi$
$191$	7.11425	0.514769	0.257385	+	0.966309i	$0.417139\pi$
$192$	0	0
$193$	−19.6914	−1.41742	−0.708709	−	0.705501i	$-0.750722\pi$
$193$	−19.6914	−1.41742	−0.708709	+	0.705501i	$0.750722\pi$
$194$	−4.55563	−0.327075
$195$	0	0
$196$	−4.96522	−0.354658
$197$	19.9579	1.42194	0.710971	−	0.703222i	$-0.248256\pi$
$197$	19.9579	1.42194	0.710971	+	0.703222i	$0.248256\pi$
$198$	0	0
$199$	7.39027	0.523882	0.261941	−	0.965084i	$-0.415637\pi$
$199$	7.39027	0.523882	0.261941	+	0.965084i	$0.415637\pi$
$200$	3.13232	0.221489
$201$	0	0
$202$	−5.33147	−0.375121
$203$	4.26394	0.299270
$204$	0	0
$205$	−9.64665	−0.673752
$206$	13.0725	0.910806
$207$	0	0
$208$	−1.64180	−0.113838
$209$	15.3137	1.05927
$210$	0	0
$211$	−12.6942	−0.873904	−0.436952	−	0.899485i	$-0.643942\pi$
$211$	−12.6942	−0.873904	−0.436952	+	0.899485i	$0.643942\pi$
$212$	3.96513	0.272326
$213$	0	0
$214$	−6.92448	−0.473348
$215$	−24.4924	−1.67037
$216$	0	0
$217$	22.2927	1.51333
$218$	−20.4798	−1.38707
$219$	0	0
$220$	−1.92900	−0.130053
$221$	1.17279	0.0788901
$222$	0	0
$223$	−10.5554	−0.706840	−0.353420	−	0.935465i	$-0.614981\pi$
$223$	−10.5554	−0.706840	−0.353420	+	0.935465i	$0.614981\pi$
$224$	−10.4963	−0.701315
$225$	0	0
$226$	9.63227	0.640729
$227$	8.54370	0.567065	0.283533	−	0.958963i	$-0.408494\pi$
$227$	8.54370	0.567065	0.283533	+	0.958963i	$0.408494\pi$
$228$	0	0
$229$	−0.974063	−0.0643679	−0.0321840	−	0.999482i	$-0.510246\pi$
$229$	−0.974063	−0.0643679	−0.0321840	+	0.999482i	$0.510246\pi$
$230$	−2.48617	−0.163933
$231$	0	0
$232$	3.04866	0.200154
$233$	21.7532	1.42510	0.712549	−	0.701622i	$-0.247541\pi$
$233$	21.7532	1.42510	0.712549	+	0.701622i	$0.247541\pi$
$234$	0	0
$235$	12.1882	0.795071
$236$	3.05025	0.198554
$237$	0	0
$238$	−11.0737	−0.717802
$239$	26.1349	1.69053	0.845264	−	0.534349i	$-0.179443\pi$
$239$	26.1349	1.69053	0.845264	+	0.534349i	$0.179443\pi$
$240$	0	0
$241$	21.9026	1.41087	0.705436	−	0.708773i	$-0.250751\pi$
$241$	21.9026	1.41087	0.705436	+	0.708773i	$0.250751\pi$
$242$	−7.79604	−0.501149
$243$	0	0
$244$	−0.871831	−0.0558133
$245$	22.2856	1.42377
$246$	0	0
$247$	3.95790	0.251835
$248$	15.9390	1.01213
$249$	0	0
$250$	−14.9852	−0.947749
$251$	25.5717	1.61407	0.807036	−	0.590502i	$-0.201070\pi$
$251$	25.5717	1.61407	0.807036	+	0.590502i	$0.201070\pi$
$252$	0	0
$253$	−2.17945	−0.137021
$254$	−17.4571	−1.09535
$255$	0	0
$256$	−10.0933	−0.630834
$257$	3.90624	0.243664	0.121832	−	0.992551i	$-0.461123\pi$
$257$	3.90624	0.243664	0.121832	+	0.992551i	$0.461123\pi$
$258$	0	0
$259$	13.7048	0.851577
$260$	−0.498559	−0.0309193
$261$	0	0
$262$	−13.6143	−0.841096
$263$	2.64870	0.163326	0.0816630	−	0.996660i	$-0.473977\pi$
$263$	2.64870	0.163326	0.0816630	+	0.996660i	$0.473977\pi$
$264$	0	0
$265$	−17.7968	−1.09325
$266$	−37.3714	−2.29139
$267$	0	0
$268$	−5.80323	−0.354489
$269$	−19.8887	−1.21264	−0.606319	−	0.795222i	$-0.707354\pi$
$269$	−19.8887	−1.21264	−0.606319	+	0.795222i	$0.707354\pi$
$270$	0	0
$271$	−9.52010	−0.578305	−0.289152	−	0.957283i	$-0.593373\pi$
$271$	−9.52010	−0.578305	−0.289152	+	0.957283i	$0.593373\pi$
$272$	−6.06842	−0.367952
$273$	0	0
$274$	−11.5839	−0.699810
$275$	−2.23926	−0.135033
$276$	0	0
$277$	−29.3962	−1.76625	−0.883124	−	0.469140i	$-0.844564\pi$
$277$	−29.3962	−1.76625	−0.883124	+	0.469140i	$0.844564\pi$
$278$	21.5820	1.29440
$279$	0	0
$280$	25.9093	1.54837
$281$	−27.3246	−1.63005	−0.815026	−	0.579425i	$-0.803277\pi$
$281$	−27.3246	−1.63005	−0.815026	+	0.579425i	$0.803277\pi$
$282$	0	0
$283$	11.2542	0.668990	0.334495	−	0.942398i	$-0.391434\pi$
$283$	11.2542	0.668990	0.334495	+	0.942398i	$0.391434\pi$
$284$	4.30390	0.255390
$285$	0	0
$286$	1.53135	0.0905506
$287$	20.6373	1.21818
$288$	0	0
$289$	−12.6652	−0.745009
$290$	−2.48617	−0.145993
$291$	0	0
$292$	−3.17232	−0.185646
$293$	12.4019	0.724526	0.362263	−	0.932076i	$-0.382004\pi$
$293$	12.4019	0.724526	0.362263	+	0.932076i	$0.382004\pi$
$294$	0	0
$295$	−13.6905	−0.797094
$296$	9.79876	0.569541
$297$	0	0
$298$	−17.6648	−1.02330
$299$	−0.563290	−0.0325759
$300$	0	0
$301$	52.3972	3.02013
$302$	−16.5110	−0.950101
$303$	0	0
$304$	−20.4796	−1.17459
$305$	3.91307	0.224062
$306$	0	0
$307$	2.27939	0.130092	0.0650460	−	0.997882i	$-0.479281\pi$
$307$	2.27939	0.130092	0.0650460	+	0.997882i	$0.479281\pi$
$308$	4.12675	0.235144
$309$	0	0
$310$	−12.9982	−0.738247
$311$	33.4135	1.89470	0.947352	−	0.320194i	$-0.103748\pi$
$311$	33.4135	1.89470	0.947352	+	0.320194i	$0.103748\pi$
$312$	0	0
$313$	3.95340	0.223459	0.111730	−	0.993739i	$-0.464361\pi$
$313$	3.95340	0.223459	0.111730	+	0.993739i	$0.464361\pi$
$314$	−19.1031	−1.07805
$315$	0	0
$316$	−5.02838	−0.282869
$317$	−33.7297	−1.89445	−0.947225	−	0.320570i	$-0.896125\pi$
$317$	−33.7297	−1.89445	−0.947225	+	0.320570i	$0.896125\pi$
$318$	0	0
$319$	−2.17945	−0.122026
$320$	17.7387	0.991622
$321$	0	0
$322$	5.31871	0.296400
$323$	14.6292	0.813989
$324$	0	0
$325$	−0.578748	−0.0321032
$326$	−17.6921	−0.979874
$327$	0	0
$328$	14.7554	0.814729
$329$	−26.0745	−1.43753
$330$	0	0
$331$	−16.8294	−0.925028	−0.462514	−	0.886612i	$-0.653052\pi$
$331$	−16.8294	−0.925028	−0.462514	+	0.886612i	$0.653052\pi$
$332$	3.02725	0.166142
$333$	0	0
$334$	−6.04236	−0.330623
$335$	26.0469	1.42309
$336$	0	0
$337$	−12.4562	−0.678532	−0.339266	−	0.940690i	$-0.610179\pi$
$337$	−12.4562	−0.678532	−0.339266	+	0.940690i	$0.610179\pi$
$338$	−15.8200	−0.860496
$339$	0	0
$340$	−1.84277	−0.0999383
$341$	−11.3946	−0.617053
$342$	0	0
$343$	−17.8284	−0.962644
$344$	37.4633	2.01988
$345$	0	0
$346$	−25.5701	−1.37466
$347$	−23.6920	−1.27185	−0.635926	−	0.771750i	$-0.719382\pi$
$347$	−23.6920	−1.27185	−0.635926	+	0.771750i	$0.719382\pi$
$348$	0	0
$349$	−15.4566	−0.827374	−0.413687	−	0.910419i	$-0.635759\pi$
$349$	−15.4566	−0.827374	−0.413687	+	0.910419i	$0.635759\pi$
$350$	5.46467	0.292099
$351$	0	0
$352$	5.36504	0.285958
$353$	25.4882	1.35660	0.678300	−	0.734785i	$-0.262717\pi$
$353$	25.4882	1.35660	0.678300	+	0.734785i	$0.262717\pi$
$354$	0	0
$355$	−19.3174	−1.02526
$356$	2.39791	0.127089
$357$	0	0
$358$	−11.6621	−0.616362
$359$	−18.4212	−0.972232	−0.486116	−	0.873894i	$-0.661587\pi$
$359$	−18.4212	−0.972232	−0.486116	+	0.873894i	$0.661587\pi$
$360$	0	0
$361$	30.3704	1.59844
$362$	−6.14779	−0.323121
$363$	0	0
$364$	1.06658	0.0559039
$365$	14.2384	0.745274
$366$	0	0
$367$	26.3389	1.37488	0.687440	−	0.726241i	$-0.258734\pi$
$367$	26.3389	1.37488	0.687440	+	0.726241i	$0.258734\pi$
$368$	2.91467	0.151938
$369$	0	0
$370$	−7.99085	−0.415424
$371$	38.0732	1.97666
$372$	0	0
$373$	37.3063	1.93165	0.965824	−	0.259198i	$-0.0834582\pi$
$373$	37.3063	1.93165	0.965824	+	0.259198i	$0.0834582\pi$
$374$	5.66017	0.292681
$375$	0	0
$376$	−18.6429	−0.961433
$377$	−0.563290	−0.0290109
$378$	0	0
$379$	11.7605	0.604099	0.302049	−	0.953292i	$-0.402329\pi$
$379$	11.7605	0.604099	0.302049	+	0.953292i	$0.402329\pi$
$380$	−6.21896	−0.319026
$381$	0	0
$382$	8.87411	0.454039
$383$	29.6295	1.51400	0.756998	−	0.653417i	$-0.226665\pi$
$383$	29.6295	1.51400	0.756998	+	0.653417i	$0.226665\pi$
$384$	0	0
$385$	−18.5223	−0.943982
$386$	−24.5625	−1.25020
$387$	0	0
$388$	1.62182	0.0823355
$389$	29.2796	1.48454	0.742268	−	0.670103i	$-0.233750\pi$
$389$	29.2796	1.48454	0.742268	+	0.670103i	$0.233750\pi$
$390$	0	0
$391$	−2.08203	−0.105293
$392$	−34.0877	−1.72169
$393$	0	0
$394$	24.8949	1.25419
$395$	22.5691	1.13557
$396$	0	0
$397$	−34.7591	−1.74451	−0.872256	−	0.489050i	$-0.837344\pi$
$397$	−34.7591	−1.74451	−0.872256	+	0.489050i	$0.837344\pi$
$398$	9.21840	0.462077
$399$	0	0
$400$	2.99465	0.149733
$401$	16.4277	0.820361	0.410181	−	0.912004i	$-0.365466\pi$
$401$	16.4277	0.820361	0.410181	+	0.912004i	$0.365466\pi$
$402$	0	0
$403$	−2.94499	−0.146700
$404$	1.89802	0.0944301
$405$	0	0
$406$	5.31871	0.263963
$407$	−7.00503	−0.347226
$408$	0	0
$409$	25.1263	1.24242	0.621208	−	0.783646i	$-0.286642\pi$
$409$	25.1263	1.24242	0.621208	+	0.783646i	$0.286642\pi$
$410$	−12.0329	−0.594265
$411$	0	0
$412$	−4.65387	−0.229280
$413$	29.2885	1.44119
$414$	0	0
$415$	−13.5873	−0.666976
$416$	1.38662	0.0679846
$417$	0	0
$418$	19.1019	0.934303
$419$	−34.9848	−1.70912	−0.854560	−	0.519353i	$-0.826173\pi$
$419$	−34.9848	−1.70912	−0.854560	+	0.519353i	$0.826173\pi$
$420$	0	0
$421$	10.9782	0.535042	0.267521	−	0.963552i	$-0.413795\pi$
$421$	10.9782	0.535042	0.267521	+	0.963552i	$0.413795\pi$
$422$	−15.8344	−0.770804
$423$	0	0
$424$	27.2218	1.32201
$425$	−2.13917	−0.103765
$426$	0	0
$427$	−8.37132	−0.405117
$428$	2.46514	0.119157
$429$	0	0
$430$	−30.5511	−1.47331
$431$	−26.9852	−1.29983	−0.649915	−	0.760007i	$-0.725195\pi$
$431$	−26.9852	−1.29983	−0.649915	+	0.760007i	$0.725195\pi$
$432$	0	0
$433$	−5.30372	−0.254881	−0.127440	−	0.991846i	$-0.540676\pi$
$433$	−5.30372	−0.254881	−0.127440	+	0.991846i	$0.540676\pi$
$434$	27.8073	1.33479
$435$	0	0
$436$	7.29088	0.349170
$437$	−7.02640	−0.336118
$438$	0	0
$439$	16.0713	0.767040	0.383520	−	0.923533i	$-0.374712\pi$
$439$	16.0713	0.767040	0.383520	+	0.923533i	$0.374712\pi$
$440$	−13.2431	−0.631342
$441$	0	0
$442$	1.46290	0.0695829
$443$	−28.9967	−1.37767	−0.688837	−	0.724916i	$-0.741879\pi$
$443$	−28.9967	−1.37767	−0.688837	+	0.724916i	$0.741879\pi$
$444$	0	0
$445$	−10.7626	−0.510197
$446$	−13.1665	−0.623450
$447$	0	0
$448$	−37.9487	−1.79291
$449$	16.5954	0.783186	0.391593	−	0.920139i	$-0.371924\pi$
$449$	16.5954	0.783186	0.391593	+	0.920139i	$0.371924\pi$
$450$	0	0
$451$	−10.5485	−0.496708
$452$	−3.42912	−0.161292
$453$	0	0
$454$	10.6572	0.500165
$455$	−4.78716	−0.224426
$456$	0	0
$457$	−15.1292	−0.707713	−0.353857	−	0.935300i	$-0.615130\pi$
$457$	−15.1292	−0.707713	−0.353857	+	0.935300i	$0.615130\pi$
$458$	−1.21502	−0.0567740
$459$	0	0
$460$	0.885084	0.0412673
$461$	34.1418	1.59014	0.795071	−	0.606516i	$-0.207433\pi$
$461$	34.1418	1.59014	0.795071	+	0.606516i	$0.207433\pi$
$462$	0	0
$463$	0.401403	0.0186548	0.00932740	−	0.999956i	$-0.497031\pi$
$463$	0.401403	0.0186548	0.00932740	+	0.999956i	$0.497031\pi$
$464$	2.91467	0.135310
$465$	0	0
$466$	27.1343	1.25697
$467$	−28.5366	−1.32052	−0.660258	−	0.751039i	$-0.729553\pi$
$467$	−28.5366	−1.32052	−0.660258	+	0.751039i	$0.729553\pi$
$468$	0	0
$469$	−55.7226	−2.57303
$470$	15.2032	0.701271
$471$	0	0
$472$	20.9408	0.963880
$473$	−26.7821	−1.23144
$474$	0	0
$475$	−7.21923	−0.331241
$476$	3.94228	0.180694
$477$	0	0
$478$	32.5999	1.49109
$479$	4.20402	0.192087	0.0960434	−	0.995377i	$-0.469381\pi$
$479$	4.20402	0.192087	0.0960434	+	0.995377i	$0.469381\pi$
$480$	0	0
$481$	−1.81048	−0.0825509
$482$	27.3207	1.24442
$483$	0	0
$484$	2.77542	0.126155
$485$	−7.27928	−0.330535
$486$	0	0
$487$	11.7731	0.533492	0.266746	−	0.963767i	$-0.414051\pi$
$487$	11.7731	0.533492	0.266746	+	0.963767i	$0.414051\pi$
$488$	−5.98537	−0.270945
$489$	0	0
$490$	27.7983	1.25580
$491$	23.4405	1.05785	0.528927	−	0.848667i	$-0.322595\pi$
$491$	23.4405	1.05785	0.528927	+	0.848667i	$0.322595\pi$
$492$	0	0
$493$	−2.08203	−0.0937699
$494$	4.93697	0.222125
$495$	0	0
$496$	15.2385	0.684227
$497$	41.3261	1.85373
$498$	0	0
$499$	9.65443	0.432192	0.216096	−	0.976372i	$-0.430668\pi$
$499$	9.65443	0.432192	0.216096	+	0.976372i	$0.430668\pi$
$500$	5.33480	0.238579
$501$	0	0
$502$	31.8974	1.42365
$503$	11.2210	0.500320	0.250160	−	0.968204i	$-0.419517\pi$
$503$	11.2210	0.500320	0.250160	+	0.968204i	$0.419517\pi$
$504$	0	0
$505$	−8.51896	−0.379089
$506$	−2.71858	−0.120856
$507$	0	0
$508$	6.21478	0.275736
$509$	9.20808	0.408141	0.204070	−	0.978956i	$-0.434583\pi$
$509$	9.20808	0.408141	0.204070	+	0.978956i	$0.434583\pi$
$510$	0	0
$511$	−30.4606	−1.34750
$512$	−24.9465	−1.10249
$513$	0	0
$514$	4.87252	0.214918
$515$	20.8881	0.920441
$516$	0	0
$517$	13.3276	0.586147
$518$	17.0950	0.751111
$519$	0	0
$520$	−3.42275	−0.150098
$521$	20.6804	0.906023	0.453011	−	0.891505i	$-0.350350\pi$
$521$	20.6804	0.906023	0.453011	+	0.891505i	$0.350350\pi$
$522$	0	0
$523$	7.40668	0.323872	0.161936	−	0.986801i	$-0.448226\pi$
$523$	7.40668	0.323872	0.161936	+	0.986801i	$0.448226\pi$
$524$	4.84675	0.211731
$525$	0	0
$526$	3.30391	0.144057
$527$	−10.8853	−0.474170
$528$	0	0
$529$	1.00000	0.0434783
$530$	−22.1992	−0.964273
$531$	0	0
$532$	13.3044	0.576817
$533$	−2.72630	−0.118089
$534$	0	0
$535$	−11.0644	−0.478355
$536$	−39.8409	−1.72086
$537$	0	0
$538$	−24.8086	−1.06957
$539$	24.3689	1.04964
$540$	0	0
$541$	17.4208	0.748979	0.374490	−	0.927231i	$-0.377818\pi$
$541$	17.4208	0.748979	0.374490	+	0.927231i	$0.377818\pi$
$542$	−11.8751	−0.510079
$543$	0	0
$544$	5.12522	0.219742
$545$	−32.7239	−1.40174
$546$	0	0
$547$	12.2981	0.525828	0.262914	−	0.964819i	$-0.415316\pi$
$547$	12.2981	0.525828	0.262914	+	0.964819i	$0.415316\pi$
$548$	4.12391	0.176165
$549$	0	0
$550$	−2.79319	−0.119102
$551$	−7.02640	−0.299335
$552$	0	0
$553$	−48.2825	−2.05318
$554$	−36.6680	−1.55787
$555$	0	0
$556$	−7.68326	−0.325843
$557$	27.2236	1.15350	0.576751	−	0.816920i	$-0.304320\pi$
$557$	27.2236	1.15350	0.576751	+	0.816920i	$0.304320\pi$
$558$	0	0
$559$	−6.92195	−0.292767
$560$	24.7705	1.04675
$561$	0	0
$562$	−34.0839	−1.43774
$563$	19.9191	0.839491	0.419745	−	0.907642i	$-0.362119\pi$
$563$	19.9191	0.839491	0.419745	+	0.907642i	$0.362119\pi$
$564$	0	0
$565$	15.3911	0.647506
$566$	14.0381	0.590065
$567$	0	0
$568$	29.5476	1.23979
$569$	0.396105	0.0166056	0.00830280	−	0.999966i	$-0.497357\pi$
$569$	0.396105	0.0166056	0.00830280	+	0.999966i	$0.497357\pi$
$570$	0	0
$571$	−13.3273	−0.557729	−0.278864	−	0.960331i	$-0.589958\pi$
$571$	−13.3273	−0.557729	−0.278864	+	0.960331i	$0.589958\pi$
$572$	−0.545166	−0.0227945
$573$	0	0
$574$	25.7424	1.07447
$575$	1.02744	0.0428473
$576$	0	0
$577$	42.4841	1.76864	0.884319	−	0.466884i	$-0.154623\pi$
$577$	42.4841	1.76864	0.884319	+	0.466884i	$0.154623\pi$
$578$	−15.7981	−0.657116
$579$	0	0
$580$	0.885084	0.0367511
$581$	29.0677	1.20593
$582$	0	0
$583$	−19.4605	−0.805974
$584$	−21.7789	−0.901217
$585$	0	0
$586$	15.4697	0.639049
$587$	39.3902	1.62581	0.812904	−	0.582397i	$-0.197885\pi$
$587$	39.3902	1.62581	0.812904	+	0.582397i	$0.197885\pi$
$588$	0	0
$589$	−36.7354	−1.51366
$590$	−17.0772	−0.703056
$591$	0	0
$592$	9.36810	0.385026
$593$	−36.5618	−1.50141	−0.750707	−	0.660635i	$-0.770287\pi$
$593$	−36.5618	−1.50141	−0.750707	+	0.660635i	$0.770287\pi$
$594$	0	0
$595$	−17.6943	−0.725395
$596$	6.28875	0.257597
$597$	0	0
$598$	−0.702630	−0.0287327
$599$	8.57473	0.350354	0.175177	−	0.984537i	$-0.443950\pi$
$599$	8.57473	0.350354	0.175177	+	0.984537i	$0.443950\pi$
$600$	0	0
$601$	−6.97815	−0.284645	−0.142322	−	0.989820i	$-0.545457\pi$
$601$	−6.97815	−0.284645	−0.142322	+	0.989820i	$0.545457\pi$
$602$	65.3587	2.66382
$603$	0	0
$604$	5.87797	0.239171
$605$	−12.4570	−0.506450
$606$	0	0
$607$	6.33213	0.257013	0.128507	−	0.991709i	$-0.458982\pi$
$607$	6.33213	0.257013	0.128507	+	0.991709i	$0.458982\pi$
$608$	17.2965	0.701466
$609$	0	0
$610$	4.88105	0.197628
$611$	3.44458	0.139353
$612$	0	0
$613$	−26.5212	−1.07118	−0.535590	−	0.844478i	$-0.679911\pi$
$613$	−26.5212	−1.07118	−0.535590	+	0.844478i	$0.679911\pi$
$614$	2.84325	0.114744
$615$	0	0
$616$	28.3314	1.14150
$617$	−15.4097	−0.620373	−0.310186	−	0.950676i	$-0.600391\pi$
$617$	−15.4097	−0.620373	−0.310186	+	0.950676i	$0.600391\pi$
$618$	0	0
$619$	15.1345	0.608306	0.304153	−	0.952623i	$-0.401627\pi$
$619$	15.1345	0.608306	0.304153	+	0.952623i	$0.401627\pi$
$620$	4.62740	0.185841
$621$	0	0
$622$	41.6790	1.67117
$623$	23.0247	0.922466
$624$	0	0
$625$	−18.8071	−0.752286
$626$	4.93135	0.197096
$627$	0	0
$628$	6.80078	0.271380
$629$	−6.69190	−0.266823
$630$	0	0
$631$	12.6845	0.504961	0.252480	−	0.967602i	$-0.418754\pi$
$631$	12.6845	0.504961	0.252480	+	0.967602i	$0.418754\pi$
$632$	−34.5213	−1.37318
$633$	0	0
$634$	−42.0734	−1.67095
$635$	−27.8940	−1.10694
$636$	0	0
$637$	6.29826	0.249546
$638$	−2.71858	−0.107630
$639$	0	0
$640$	12.3139	0.486751
$641$	23.3330	0.921597	0.460799	−	0.887505i	$-0.347563\pi$
$641$	23.3330	0.921597	0.460799	+	0.887505i	$0.347563\pi$
$642$	0	0
$643$	−2.79864	−0.110368	−0.0551838	−	0.998476i	$-0.517574\pi$
$643$	−2.79864	−0.110368	−0.0551838	+	0.998476i	$0.517574\pi$
$644$	−1.89348	−0.0746136
$645$	0	0
$646$	18.2480	0.717958
$647$	−5.87120	−0.230821	−0.115410	−	0.993318i	$-0.536818\pi$
$647$	−5.87120	−0.230821	−0.115410	+	0.993318i	$0.536818\pi$
$648$	0	0
$649$	−14.9704	−0.587639
$650$	−0.721913	−0.0283157
$651$	0	0
$652$	6.29844	0.246666
$653$	−39.3801	−1.54106	−0.770532	−	0.637402i	$-0.780009\pi$
$653$	−39.3801	−1.54106	−0.770532	+	0.637402i	$0.780009\pi$
$654$	0	0
$655$	−21.7538	−0.849993
$656$	14.1069	0.550781
$657$	0	0
$658$	−32.5245	−1.26794
$659$	−12.6740	−0.493707	−0.246854	−	0.969053i	$-0.579397\pi$
$659$	−12.6740	−0.493707	−0.246854	+	0.969053i	$0.579397\pi$
$660$	0	0
$661$	42.7161	1.66146	0.830731	−	0.556674i	$-0.187923\pi$
$661$	42.7161	1.66146	0.830731	+	0.556674i	$0.187923\pi$
$662$	−20.9925	−0.815896
$663$	0	0
$664$	20.7830	0.806536
$665$	−59.7145	−2.31563
$666$	0	0
$667$	1.00000	0.0387202
$668$	2.15110	0.0832287
$669$	0	0
$670$	32.4901	1.25520
$671$	4.27888	0.165184
$672$	0	0
$673$	18.6162	0.717602	0.358801	−	0.933414i	$-0.383186\pi$
$673$	18.6162	0.717602	0.358801	+	0.933414i	$0.383186\pi$
$674$	−15.5375	−0.598482
$675$	0	0
$676$	5.63198	0.216615
$677$	14.3352	0.550948	0.275474	−	0.961309i	$-0.411165\pi$
$677$	14.3352	0.550948	0.275474	+	0.961309i	$0.411165\pi$
$678$	0	0
$679$	15.5727	0.597626
$680$	−12.6512	−0.485150
$681$	0	0
$682$	−14.2133	−0.544255
$683$	28.8242	1.10293	0.551464	−	0.834199i	$-0.314070\pi$
$683$	28.8242	1.10293	0.551464	+	0.834199i	$0.314070\pi$
$684$	0	0
$685$	−18.5095	−0.707212
$686$	−22.2386	−0.849075
$687$	0	0
$688$	35.8167	1.36550
$689$	−5.02967	−0.191615
$690$	0	0
$691$	39.8007	1.51409	0.757045	−	0.653362i	$-0.226642\pi$
$691$	39.8007	1.51409	0.757045	+	0.653362i	$0.226642\pi$
$692$	9.10305	0.346046
$693$	0	0
$694$	−29.5527	−1.12180
$695$	34.4851	1.30809
$696$	0	0
$697$	−10.0769	−0.381691
$698$	−19.2801	−0.729763
$699$	0	0
$700$	−1.94544	−0.0735309
$701$	−46.5650	−1.75874	−0.879368	−	0.476144i	$-0.842034\pi$
$701$	−46.5650	−1.75874	−0.879368	+	0.476144i	$0.842034\pi$
$702$	0	0
$703$	−22.5837	−0.851761
$704$	19.3970	0.731050
$705$	0	0
$706$	31.7932	1.19655
$707$	18.2248	0.685415
$708$	0	0
$709$	−11.8273	−0.444183	−0.222092	−	0.975026i	$-0.571288\pi$
$709$	−11.8273	−0.444183	−0.222092	+	0.975026i	$0.571288\pi$
$710$	−24.0959	−0.904303
$711$	0	0
$712$	16.4623	0.616952
$713$	5.22820	0.195798
$714$	0	0
$715$	2.44689	0.0915085
$716$	4.15175	0.155158
$717$	0	0
$718$	−22.9780	−0.857532
$719$	43.6512	1.62792	0.813958	−	0.580923i	$-0.197308\pi$
$719$	43.6512	1.62792	0.813958	+	0.580923i	$0.197308\pi$
$720$	0	0
$721$	−44.6864	−1.66421
$722$	37.8831	1.40986
$723$	0	0
$724$	2.18864	0.0813400
$725$	1.02744	0.0381583
$726$	0	0
$727$	−26.7061	−0.990473	−0.495236	−	0.868758i	$-0.664919\pi$
$727$	−26.7061	−0.990473	−0.495236	+	0.868758i	$0.664919\pi$
$728$	7.32237	0.271385
$729$	0	0
$730$	17.7606	0.657350
$731$	−25.5849	−0.946292
$732$	0	0
$733$	−1.08710	−0.0401531	−0.0200765	−	0.999798i	$-0.506391\pi$
$733$	−1.08710	−0.0401531	−0.0200765	+	0.999798i	$0.506391\pi$
$734$	32.8544	1.21268
$735$	0	0
$736$	−2.46165	−0.0907375
$737$	28.4818	1.04914
$738$	0	0
$739$	6.75646	0.248540	0.124270	−	0.992248i	$-0.460341\pi$
$739$	6.75646	0.248540	0.124270	+	0.992248i	$0.460341\pi$
$740$	2.84477	0.104576
$741$	0	0
$742$	47.4913	1.74346
$743$	−5.56646	−0.204214	−0.102107	−	0.994773i	$-0.532558\pi$
$743$	−5.56646	−0.204214	−0.102107	+	0.994773i	$0.532558\pi$
$744$	0	0
$745$	−28.2260	−1.03412
$746$	46.5348	1.70376
$747$	0	0
$748$	−2.01504	−0.0736772
$749$	23.6703	0.864893
$750$	0	0
$751$	−33.9034	−1.23715	−0.618577	−	0.785724i	$-0.712291\pi$
$751$	−33.9034	−1.23715	−0.618577	+	0.785724i	$0.712291\pi$
$752$	−17.8235	−0.649957
$753$	0	0
$754$	−0.702630	−0.0255883
$755$	−26.3823	−0.960151
$756$	0	0
$757$	−33.6145	−1.22174	−0.610869	−	0.791732i	$-0.709180\pi$
$757$	−33.6145	−1.22174	−0.610869	+	0.791732i	$0.709180\pi$
$758$	14.6697	0.532829
$759$	0	0
$760$	−42.6950	−1.54871
$761$	46.1969	1.67464	0.837319	−	0.546715i	$-0.184122\pi$
$761$	46.1969	1.67464	0.837319	+	0.546715i	$0.184122\pi$
$762$	0	0
$763$	70.0070	2.53442
$764$	−3.15921	−0.114296
$765$	0	0
$766$	36.9589	1.33538
$767$	−3.86916	−0.139707
$768$	0	0
$769$	−39.8972	−1.43873	−0.719364	−	0.694633i	$-0.755567\pi$
$769$	−39.8972	−1.43873	−0.719364	+	0.694633i	$0.755567\pi$
$770$	−23.1041	−0.832614
$771$	0	0
$772$	8.74432	0.314715
$773$	31.5932	1.13633	0.568164	−	0.822916i	$-0.307654\pi$
$773$	31.5932	1.13633	0.568164	+	0.822916i	$0.307654\pi$
$774$	0	0
$775$	5.37168	0.192956
$776$	11.1343	0.399697
$777$	0	0
$778$	36.5225	1.30940
$779$	−34.0075	−1.21845
$780$	0	0
$781$	−21.1232	−0.755849
$782$	−2.59706	−0.0928707
$783$	0	0
$784$	−32.5895	−1.16391
$785$	−30.5242	−1.08945
$786$	0	0
$787$	11.2307	0.400333	0.200166	−	0.979762i	$-0.435852\pi$
$787$	11.2307	0.400333	0.200166	+	0.979762i	$0.435852\pi$
$788$	−8.86267	−0.315719
$789$	0	0
$790$	28.1520	1.00160
$791$	−32.9264	−1.17073
$792$	0	0
$793$	1.10590	0.0392715
$794$	−43.3575	−1.53870
$795$	0	0
$796$	−3.28178	−0.116320
$797$	−39.7919	−1.40950	−0.704750	−	0.709456i	$-0.748941\pi$
$797$	−39.7919	−1.40950	−0.704750	+	0.709456i	$0.748941\pi$
$798$	0	0
$799$	12.7318	0.450420
$800$	−2.52920	−0.0894207
$801$	0	0
$802$	20.4914	0.723578
$803$	15.5695	0.549436
$804$	0	0
$805$	8.49858	0.299536
$806$	−3.67349	−0.129393
$807$	0	0
$808$	13.0305	0.458411
$809$	18.6090	0.654259	0.327129	−	0.944980i	$-0.393919\pi$
$809$	18.6090	0.654259	0.327129	+	0.944980i	$0.393919\pi$
$810$	0	0
$811$	7.44117	0.261295	0.130647	−	0.991429i	$-0.458294\pi$
$811$	7.44117	0.261295	0.130647	+	0.991429i	$0.458294\pi$
$812$	−1.89348	−0.0664482
$813$	0	0
$814$	−8.73786	−0.306262
$815$	−28.2695	−0.990239
$816$	0	0
$817$	−86.3436	−3.02078
$818$	31.3418	1.09584
$819$	0	0
$820$	4.28377	0.149596
$821$	−19.1494	−0.668318	−0.334159	−	0.942517i	$-0.608452\pi$
$821$	−19.1494	−0.668318	−0.334159	+	0.942517i	$0.608452\pi$
$822$	0	0
$823$	−52.7502	−1.83876	−0.919379	−	0.393373i	$-0.871308\pi$
$823$	−52.7502	−1.83876	−0.919379	+	0.393373i	$0.871308\pi$
$824$	−31.9502	−1.11304
$825$	0	0
$826$	36.5336	1.27117
$827$	−3.02517	−0.105195	−0.0525977	−	0.998616i	$-0.516750\pi$
$827$	−3.02517	−0.105195	−0.0525977	+	0.998616i	$0.516750\pi$
$828$	0	0
$829$	−28.9116	−1.00414	−0.502070	−	0.864827i	$-0.667428\pi$
$829$	−28.9116	−1.00414	−0.502070	+	0.864827i	$0.667428\pi$
$830$	−16.9484	−0.588288
$831$	0	0
$832$	5.01323	0.173802
$833$	23.2796	0.806590
$834$	0	0
$835$	−9.65488	−0.334121
$836$	−6.80033	−0.235194
$837$	0	0
$838$	−43.6390	−1.50748
$839$	−15.5459	−0.536702	−0.268351	−	0.963321i	$-0.586479\pi$
$839$	−15.5459	−0.536702	−0.268351	+	0.963321i	$0.586479\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	13.6938	0.471920
$843$	0	0
$844$	5.63709	0.194036
$845$	−25.2782	−0.869598
$846$	0	0
$847$	26.6496	0.915690
$848$	26.0253	0.893714
$849$	0	0
$850$	−2.66833	−0.0915230
$851$	3.21412	0.110179
$852$	0	0
$853$	−25.6274	−0.877466	−0.438733	−	0.898617i	$-0.644573\pi$
$853$	−25.6274	−0.877466	−0.438733	+	0.898617i	$0.644573\pi$
$854$	−10.4421	−0.357322
$855$	0	0
$856$	16.9239	0.578447
$857$	−30.2135	−1.03207	−0.516037	−	0.856566i	$-0.672593\pi$
$857$	−30.2135	−1.03207	−0.516037	+	0.856566i	$0.672593\pi$
$858$	0	0
$859$	−13.0782	−0.446221	−0.223111	−	0.974793i	$-0.571621\pi$
$859$	−13.0782	−0.446221	−0.223111	+	0.974793i	$0.571621\pi$
$860$	10.8763	0.370879
$861$	0	0
$862$	−33.6605	−1.14648
$863$	5.97393	0.203355	0.101678	−	0.994817i	$-0.467579\pi$
$863$	5.97393	0.203355	0.101678	+	0.994817i	$0.467579\pi$
$864$	0	0
$865$	−40.8576	−1.38920
$866$	−6.61570	−0.224811
$867$	0	0
$868$	−9.89950	−0.336011
$869$	24.6789	0.837175
$870$	0	0
$871$	7.36126	0.249427
$872$	50.0540	1.69504
$873$	0	0
$874$	−8.76453	−0.296465
$875$	51.2247	1.73171
$876$	0	0
$877$	12.7561	0.430743	0.215372	−	0.976532i	$-0.430904\pi$
$877$	12.7561	0.430743	0.215372	+	0.976532i	$0.430904\pi$
$878$	20.0468	0.676547
$879$	0	0
$880$	−12.6611	−0.426806
$881$	−17.1776	−0.578728	−0.289364	−	0.957219i	$-0.593444\pi$
$881$	−17.1776	−0.578728	−0.289364	+	0.957219i	$0.593444\pi$
$882$	0	0
$883$	8.39238	0.282426	0.141213	−	0.989979i	$-0.454900\pi$
$883$	8.39238	0.282426	0.141213	+	0.989979i	$0.454900\pi$
$884$	−0.520796	−0.0175163
$885$	0	0
$886$	−36.1696	−1.21514
$887$	13.0653	0.438691	0.219346	−	0.975647i	$-0.429608\pi$
$887$	13.0653	0.438691	0.219346	+	0.975647i	$0.429608\pi$
$888$	0	0
$889$	59.6743	2.00141
$890$	−13.4250	−0.450006
$891$	0	0
$892$	4.68731	0.156943
$893$	42.9673	1.43784
$894$	0	0
$895$	−18.6345	−0.622882
$896$	−26.3435	−0.880074
$897$	0	0
$898$	20.7006	0.690789
$899$	5.22820	0.174370
$900$	0	0
$901$	−18.5906	−0.619344
$902$	−13.1578	−0.438108
$903$	0	0
$904$	−23.5419	−0.782992
$905$	−9.82334	−0.326539
$906$	0	0
$907$	−12.1824	−0.404510	−0.202255	−	0.979333i	$-0.564827\pi$
$907$	−12.1824	−0.404510	−0.202255	+	0.979333i	$0.564827\pi$
$908$	−3.79399	−0.125908
$909$	0	0
$910$	−5.97136	−0.197949
$911$	−48.5411	−1.60824	−0.804119	−	0.594468i	$-0.797363\pi$
$911$	−48.5411	−1.60824	−0.804119	+	0.594468i	$0.797363\pi$
$912$	0	0
$913$	−14.8575	−0.491712
$914$	−18.8717	−0.624220
$915$	0	0
$916$	0.432550	0.0142919
$917$	46.5385	1.53684
$918$	0	0
$919$	−47.2023	−1.55706	−0.778531	−	0.627606i	$-0.784035\pi$
$919$	−47.2023	−1.55706	−0.778531	+	0.627606i	$0.784035\pi$
$920$	6.07636	0.200332
$921$	0	0
$922$	42.5875	1.40254
$923$	−5.45940	−0.179698
$924$	0	0
$925$	3.30233	0.108580
$926$	0.500699	0.0164540
$927$	0	0
$928$	−2.46165	−0.0808075
$929$	12.1402	0.398306	0.199153	−	0.979968i	$-0.436181\pi$
$929$	12.1402	0.398306	0.199153	+	0.979968i	$0.436181\pi$
$930$	0	0
$931$	78.5637	2.57482
$932$	−9.65989	−0.316420
$933$	0	0
$934$	−35.5957	−1.16473
$935$	9.04418	0.295776
$936$	0	0
$937$	−40.3140	−1.31700	−0.658501	−	0.752580i	$-0.728809\pi$
$937$	−40.3140	−1.31700	−0.658501	+	0.752580i	$0.728809\pi$
$938$	−69.5068	−2.26948
$939$	0	0
$940$	−5.41239	−0.176533
$941$	57.8357	1.88539	0.942695	−	0.333654i	$-0.108282\pi$
$941$	57.8357	1.88539	0.942695	+	0.333654i	$0.108282\pi$
$942$	0	0
$943$	4.83996	0.157611
$944$	20.0205	0.651611
$945$	0	0
$946$	−33.4072	−1.08616
$947$	−44.4178	−1.44338	−0.721692	−	0.692214i	$-0.756635\pi$
$947$	−44.4178	−1.44338	−0.721692	+	0.692214i	$0.756635\pi$
$948$	0	0
$949$	4.02401	0.130625
$950$	−9.00505	−0.292162
$951$	0	0
$952$	27.0649	0.877179
$953$	15.0278	0.486798	0.243399	−	0.969926i	$-0.421737\pi$
$953$	15.0278	0.486798	0.243399	+	0.969926i	$0.421737\pi$
$954$	0	0
$955$	14.1796	0.458842
$956$	−11.6057	−0.375355
$957$	0	0
$958$	5.24397	0.169425
$959$	39.5978	1.27868
$960$	0	0
$961$	−3.66593	−0.118256
$962$	−2.25834	−0.0728118
$963$	0	0
$964$	−9.72626	−0.313262
$965$	−39.2475	−1.26342
$966$	0	0
$967$	−39.8659	−1.28200	−0.641001	−	0.767540i	$-0.721481\pi$
$967$	−39.8659	−1.28200	−0.641001	+	0.767540i	$0.721481\pi$
$968$	19.0541	0.612421
$969$	0	0
$970$	−9.07995	−0.291540
$971$	−11.9856	−0.384635	−0.192318	−	0.981333i	$-0.561600\pi$
$971$	−11.9856	−0.384635	−0.192318	+	0.981333i	$0.561600\pi$
$972$	0	0
$973$	−73.7747	−2.36511
$974$	14.6855	0.470553
$975$	0	0
$976$	−5.72231	−0.183167
$977$	18.9625	0.606664	0.303332	−	0.952885i	$-0.401901\pi$
$977$	18.9625	0.606664	0.303332	+	0.952885i	$0.401901\pi$
$978$	0	0
$979$	−11.7688	−0.376131
$980$	−9.89631	−0.316126
$981$	0	0
$982$	29.2389	0.933052
$983$	−34.5151	−1.10086	−0.550431	−	0.834881i	$-0.685536\pi$
$983$	−34.5151	−1.10086	−0.550431	+	0.834881i	$0.685536\pi$
$984$	0	0
$985$	39.7786	1.26745
$986$	−2.59706	−0.0827073
$987$	0	0
$988$	−1.75758	−0.0559160
$989$	12.2884	0.390750
$990$	0	0
$991$	55.9207	1.77638	0.888191	−	0.459475i	$-0.151963\pi$
$991$	55.9207	1.77638	0.888191	+	0.459475i	$0.151963\pi$
$992$	−12.8700	−0.408622
$993$	0	0
$994$	51.5489	1.63503
$995$	14.7297	0.466964
$996$	0	0
$997$	32.5201	1.02992	0.514961	−	0.857214i	$-0.327806\pi$
$997$	32.5201	1.02992	0.514961	+	0.857214i	$0.327806\pi$
$998$	12.0427	0.381203
$999$	0	0

Twists

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

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

\(f(q)\)	\(=\)	\(q+1.24737 q^{2} -0.444068 q^{4} +1.99313 q^{5} -4.26394 q^{7} -3.04866 q^{8} +O(q^{10})\)	\(q+1.24737 q^{2} -0.444068 q^{4} +1.99313 q^{5} -4.26394 q^{7} -3.04866 q^{8} +2.48617 q^{10} +2.17945 q^{11} +0.563290 q^{13} -5.31871 q^{14} -2.91467 q^{16} +2.08203 q^{17} +7.02640 q^{19} -0.885084 q^{20} +2.71858 q^{22} -1.00000 q^{23} -1.02744 q^{25} +0.702630 q^{26} +1.89348 q^{28} -1.00000 q^{29} -5.22820 q^{31} +2.46165 q^{32} +2.59706 q^{34} -8.49858 q^{35} -3.21412 q^{37} +8.76453 q^{38} -6.07636 q^{40} -4.83996 q^{41} -12.2884 q^{43} -0.967826 q^{44} -1.24737 q^{46} +6.11511 q^{47} +11.1812 q^{49} -1.28160 q^{50} -0.250139 q^{52} -8.92910 q^{53} +4.34393 q^{55} +12.9993 q^{56} -1.24737 q^{58} -6.86887 q^{59} +1.96328 q^{61} -6.52150 q^{62} +8.89992 q^{64} +1.12271 q^{65} +13.0683 q^{67} -0.924563 q^{68} -10.6009 q^{70} -9.69199 q^{71} +7.14377 q^{73} -4.00920 q^{74} -3.12020 q^{76} -9.29306 q^{77} +11.3234 q^{79} -5.80930 q^{80} -6.03722 q^{82} -6.81709 q^{83} +4.14975 q^{85} -15.3282 q^{86} -6.64441 q^{88} -5.39987 q^{89} -2.40183 q^{91} +0.444068 q^{92} +7.62781 q^{94} +14.0045 q^{95} -3.65219 q^{97} +13.9471 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 3 q^{2} + 17 q^{4} - 6 q^{7} + 6 q^{8}+O(q^{10}) \) 22 * q + 3 * q^2 + 17 * q^4 - 6 * q^7 + 6 * q^8	\( 22 q + 3 q^{2} + 17 q^{4} - 6 q^{7} + 6 q^{8} - 12 q^{10} - 28 q^{13} + q^{14} + 3 q^{16} + 10 q^{17} - 8 q^{19} - 11 q^{22} - 22 q^{23} - 11 q^{26} - 21 q^{28} - 22 q^{29} - 18 q^{31} - 5 q^{32} - 33 q^{34} - 2 q^{35} - 28 q^{37} - 14 q^{38} - 30 q^{40} + 10 q^{41} - 14 q^{43} - 37 q^{44} - 3 q^{46} + 18 q^{47} + 2 q^{49} - 7 q^{50} - 57 q^{52} - 20 q^{53} - 42 q^{55} + 2 q^{56} - 3 q^{58} + 20 q^{59} - 38 q^{61} - 4 q^{62} - 24 q^{64} - 12 q^{65} - 50 q^{67} - 11 q^{68} - 48 q^{70} - 12 q^{71} - 46 q^{73} + 6 q^{74} - 16 q^{76} + 14 q^{77} - 20 q^{79} + 58 q^{80} - 42 q^{82} - 22 q^{83} - 66 q^{85} - 22 q^{86} - 68 q^{88} + 14 q^{89} - 16 q^{91} - 17 q^{92} - 27 q^{94} + 20 q^{95} - 48 q^{97} + 28 q^{98}+O(q^{100}) \) 22 * q + 3 * q^2 + 17 * q^4 - 6 * q^7 + 6 * q^8 - 12 * q^10 - 28 * q^13 + q^14 + 3 * q^16 + 10 * q^17 - 8 * q^19 - 11 * q^22 - 22 * q^23 - 11 * q^26 - 21 * q^28 - 22 * q^29 - 18 * q^31 - 5 * q^32 - 33 * q^34 - 2 * q^35 - 28 * q^37 - 14 * q^38 - 30 * q^40 + 10 * q^41 - 14 * q^43 - 37 * q^44 - 3 * q^46 + 18 * q^47 + 2 * q^49 - 7 * q^50 - 57 * q^52 - 20 * q^53 - 42 * q^55 + 2 * q^56 - 3 * q^58 + 20 * q^59 - 38 * q^61 - 4 * q^62 - 24 * q^64 - 12 * q^65 - 50 * q^67 - 11 * q^68 - 48 * q^70 - 12 * q^71 - 46 * q^73 + 6 * q^74 - 16 * q^76 + 14 * q^77 - 20 * q^79 + 58 * q^80 - 42 * q^82 - 22 * q^83 - 66 * q^85 - 22 * q^86 - 68 * q^88 + 14 * q^89 - 16 * q^91 - 17 * q^92 - 27 * q^94 + 20 * q^95 - 48 * q^97 + 28 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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