LMFDB - Embedded newform 6039.2.a.o.1.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6039,2,Mod(1,6039)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6039, 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("6039.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6039 = 3^{2} \cdot 11 \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6039.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:	$48.2216577807$
Analytic rank:	$0$
Dimension:	$25$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
Character	$\chi$	$=$	6039.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.56609 q^{2} +0.452630 q^{4} +1.36136 q^{5} +3.35990 q^{7} -2.42332 q^{8} +O(q^{10})$	\(q+1.56609 q^{2} +0.452630 q^{4} +1.36136 q^{5} +3.35990 q^{7} -2.42332 q^{8} +2.13201 q^{10} +1.00000 q^{11} +6.49192 q^{13} +5.26190 q^{14} -4.70039 q^{16} +3.12895 q^{17} -0.766546 q^{19} +0.616192 q^{20} +1.56609 q^{22} -0.696071 q^{23} -3.14670 q^{25} +10.1669 q^{26} +1.52079 q^{28} -1.69575 q^{29} -0.344190 q^{31} -2.51458 q^{32} +4.90022 q^{34} +4.57404 q^{35} +5.23948 q^{37} -1.20048 q^{38} -3.29901 q^{40} +8.27551 q^{41} -1.26225 q^{43} +0.452630 q^{44} -1.09011 q^{46} -2.80853 q^{47} +4.28895 q^{49} -4.92801 q^{50} +2.93844 q^{52} +13.4721 q^{53} +1.36136 q^{55} -8.14211 q^{56} -2.65570 q^{58} +7.04704 q^{59} -1.00000 q^{61} -0.539032 q^{62} +5.46272 q^{64} +8.83784 q^{65} -11.9010 q^{67} +1.41626 q^{68} +7.16334 q^{70} +1.74910 q^{71} -9.19391 q^{73} +8.20549 q^{74} -0.346962 q^{76} +3.35990 q^{77} -11.9855 q^{79} -6.39892 q^{80} +12.9602 q^{82} -1.29768 q^{83} +4.25963 q^{85} -1.97679 q^{86} -2.42332 q^{88} -1.41865 q^{89} +21.8122 q^{91} -0.315063 q^{92} -4.39840 q^{94} -1.04355 q^{95} -5.21258 q^{97} +6.71687 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q + 5 q^{2} + 25 q^{4} + 4 q^{5} + 4 q^{7} + 15 q^{8}+O(q^{10}) $ 25 * q + 5 * q^2 + 25 * q^4 + 4 * q^5 + 4 * q^7 + 15 * q^8	$ 25 q + 5 q^{2} + 25 q^{4} + 4 q^{5} + 4 q^{7} + 15 q^{8} + 25 q^{11} + 4 q^{13} + 18 q^{14} + 21 q^{16} + 20 q^{17} + 14 q^{19} + 12 q^{20} + 5 q^{22} + 20 q^{23} + 13 q^{25} + 16 q^{26} - 14 q^{28} + 28 q^{29} - 12 q^{31} + 35 q^{32} + 6 q^{34} + 10 q^{35} - 8 q^{37} + 32 q^{38} + 24 q^{40} + 26 q^{41} + 18 q^{43} + 25 q^{44} + 4 q^{46} + 12 q^{47} + 23 q^{49} + 43 q^{50} + 22 q^{52} + 36 q^{53} + 4 q^{55} + 26 q^{56} - 20 q^{58} + 46 q^{59} - 25 q^{61} - 14 q^{62} - 13 q^{64} + 60 q^{65} - 20 q^{67} + 44 q^{68} - 20 q^{70} + 52 q^{71} + 6 q^{73} + 32 q^{74} + 4 q^{77} + 26 q^{79} + 52 q^{80} + 6 q^{82} + 38 q^{83} - 4 q^{85} + 34 q^{86} + 15 q^{88} + 82 q^{89} - 58 q^{91} + 36 q^{92} + 16 q^{94} + 30 q^{95} + 35 q^{98}+O(q^{100}) $ 25 * q + 5 * q^2 + 25 * q^4 + 4 * q^5 + 4 * q^7 + 15 * q^8 + 25 * q^11 + 4 * q^13 + 18 * q^14 + 21 * q^16 + 20 * q^17 + 14 * q^19 + 12 * q^20 + 5 * q^22 + 20 * q^23 + 13 * q^25 + 16 * q^26 - 14 * q^28 + 28 * q^29 - 12 * q^31 + 35 * q^32 + 6 * q^34 + 10 * q^35 - 8 * q^37 + 32 * q^38 + 24 * q^40 + 26 * q^41 + 18 * q^43 + 25 * q^44 + 4 * q^46 + 12 * q^47 + 23 * q^49 + 43 * q^50 + 22 * q^52 + 36 * q^53 + 4 * q^55 + 26 * q^56 - 20 * q^58 + 46 * q^59 - 25 * q^61 - 14 * q^62 - 13 * q^64 + 60 * q^65 - 20 * q^67 + 44 * q^68 - 20 * q^70 + 52 * q^71 + 6 * q^73 + 32 * q^74 + 4 * q^77 + 26 * q^79 + 52 * q^80 + 6 * q^82 + 38 * q^83 - 4 * q^85 + 34 * q^86 + 15 * q^88 + 82 * q^89 - 58 * q^91 + 36 * q^92 + 16 * q^94 + 30 * q^95 + 35 * 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 $

$2$	1.56609	1.10739	0.553696	−	0.832719i	$-0.313217\pi$
$2$	1.56609	1.10739	0.553696	+	0.832719i	$0.313217\pi$
$3$	0	0
$3$	0	0
$4$	0.452630	0.226315
$5$	1.36136	0.608819	0.304409	−	0.952541i	$-0.401541\pi$
$5$	1.36136	0.608819	0.304409	+	0.952541i	$0.401541\pi$
$6$	0	0
$7$	3.35990	1.26992	0.634962	−	0.772543i	$-0.281016\pi$
$7$	3.35990	1.26992	0.634962	+	0.772543i	$0.281016\pi$
$8$	−2.42332	−0.856772
$9$	0	0
$10$	2.13201	0.674200
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	6.49192	1.80053	0.900267	−	0.435337i	$-0.143371\pi$
$13$	6.49192	1.80053	0.900267	+	0.435337i	$0.143371\pi$
$14$	5.26190	1.40630
$15$	0	0
$16$	−4.70039	−1.17510
$17$	3.12895	0.758883	0.379442	−	0.925216i	$-0.376116\pi$
$17$	3.12895	0.758883	0.379442	+	0.925216i	$0.376116\pi$
$18$	0	0
$19$	−0.766546	−0.175858	−0.0879289	−	0.996127i	$-0.528025\pi$
$19$	−0.766546	−0.175858	−0.0879289	+	0.996127i	$0.528025\pi$
$20$	0.616192	0.137785
$21$	0	0
$22$	1.56609	0.333891
$23$	−0.696071	−0.145141	−0.0725704	−	0.997363i	$-0.523120\pi$
$23$	−0.696071	−0.145141	−0.0725704	+	0.997363i	$0.523120\pi$
$24$	0	0
$25$	−3.14670	−0.629340
$26$	10.1669	1.99390
$27$	0	0
$28$	1.52079	0.287403
$29$	−1.69575	−0.314893	−0.157447	−	0.987527i	$-0.550326\pi$
$29$	−1.69575	−0.314893	−0.157447	+	0.987527i	$0.550326\pi$
$30$	0	0
$31$	−0.344190	−0.0618184	−0.0309092	−	0.999522i	$-0.509840\pi$
$31$	−0.344190	−0.0618184	−0.0309092	+	0.999522i	$0.509840\pi$
$32$	−2.51458	−0.444520
$33$	0	0
$34$	4.90022	0.840380
$35$	4.57404	0.773153
$36$	0	0
$37$	5.23948	0.861366	0.430683	−	0.902503i	$-0.358273\pi$
$37$	5.23948	0.861366	0.430683	+	0.902503i	$0.358273\pi$
$38$	−1.20048	−0.194743
$39$	0	0
$40$	−3.29901	−0.521619
$41$	8.27551	1.29242	0.646208	−	0.763161i	$-0.276354\pi$
$41$	8.27551	1.29242	0.646208	+	0.763161i	$0.276354\pi$
$42$	0	0
$43$	−1.26225	−0.192491	−0.0962456	−	0.995358i	$-0.530683\pi$
$43$	−1.26225	−0.192491	−0.0962456	+	0.995358i	$0.530683\pi$
$44$	0.452630	0.0682365
$45$	0	0
$46$	−1.09011	−0.160728
$47$	−2.80853	−0.409666	−0.204833	−	0.978797i	$-0.565665\pi$
$47$	−2.80853	−0.409666	−0.204833	+	0.978797i	$0.565665\pi$
$48$	0	0
$49$	4.28895	0.612707
$50$	−4.92801	−0.696925
$51$	0	0
$52$	2.93844	0.407488
$53$	13.4721	1.85054	0.925270	−	0.379309i	$-0.123838\pi$
$53$	13.4721	1.85054	0.925270	+	0.379309i	$0.123838\pi$
$54$	0	0
$55$	1.36136	0.183566
$56$	−8.14211	−1.08804
$57$	0	0
$58$	−2.65570	−0.348710
$59$	7.04704	0.917446	0.458723	−	0.888579i	$-0.348307\pi$
$59$	7.04704	0.917446	0.458723	+	0.888579i	$0.348307\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	−0.539032	−0.0684571
$63$	0	0
$64$	5.46272	0.682840
$65$	8.83784	1.09620
$66$	0	0
$67$	−11.9010	−1.45393	−0.726967	−	0.686672i	$-0.759071\pi$
$67$	−11.9010	−1.45393	−0.726967	+	0.686672i	$0.759071\pi$
$68$	1.41626	0.171747
$69$	0	0
$70$	7.16334	0.856183
$71$	1.74910	0.207580	0.103790	−	0.994599i	$-0.466903\pi$
$71$	1.74910	0.207580	0.103790	+	0.994599i	$0.466903\pi$
$72$	0	0
$73$	−9.19391	−1.07607	−0.538033	−	0.842924i	$-0.680832\pi$
$73$	−9.19391	−1.07607	−0.538033	+	0.842924i	$0.680832\pi$
$74$	8.20549	0.953868
$75$	0	0
$76$	−0.346962	−0.0397993
$77$	3.35990	0.382896
$78$	0	0
$79$	−11.9855	−1.34847	−0.674236	−	0.738516i	$-0.735527\pi$
$79$	−11.9855	−1.34847	−0.674236	+	0.738516i	$0.735527\pi$
$80$	−6.39892	−0.715421
$81$	0	0
$82$	12.9602	1.43121
$83$	−1.29768	−0.142439	−0.0712196	−	0.997461i	$-0.522689\pi$
$83$	−1.29768	−0.142439	−0.0712196	+	0.997461i	$0.522689\pi$
$84$	0	0
$85$	4.25963	0.462022
$86$	−1.97679	−0.213163
$87$	0	0
$88$	−2.42332	−0.258326
$89$	−1.41865	−0.150377	−0.0751885	−	0.997169i	$-0.523956\pi$
$89$	−1.41865	−0.150377	−0.0751885	+	0.997169i	$0.523956\pi$
$90$	0	0
$91$	21.8122	2.28654
$92$	−0.315063	−0.0328476
$93$	0	0
$94$	−4.39840	−0.453660
$95$	−1.04355	−0.107065
$96$	0	0
$97$	−5.21258	−0.529257	−0.264628	−	0.964350i	$-0.585249\pi$
$97$	−5.21258	−0.529257	−0.264628	+	0.964350i	$0.585249\pi$
$98$	6.71687	0.678506
$99$	0	0
$100$	−1.42429	−0.142429
$101$	11.6285	1.15708	0.578539	−	0.815655i	$-0.303623\pi$
$101$	11.6285	1.15708	0.578539	+	0.815655i	$0.303623\pi$
$102$	0	0
$103$	−4.41364	−0.434889	−0.217445	−	0.976073i	$-0.569772\pi$
$103$	−4.41364	−0.434889	−0.217445	+	0.976073i	$0.569772\pi$
$104$	−15.7320	−1.54265
$105$	0	0
$106$	21.0985	2.04927
$107$	4.86398	0.470219	0.235110	−	0.971969i	$-0.424455\pi$
$107$	4.86398	0.470219	0.235110	+	0.971969i	$0.424455\pi$
$108$	0	0
$109$	5.03958	0.482704	0.241352	−	0.970438i	$-0.422409\pi$
$109$	5.03958	0.482704	0.241352	+	0.970438i	$0.422409\pi$
$110$	2.13201	0.203279
$111$	0	0
$112$	−15.7928	−1.49228
$113$	−5.27857	−0.496566	−0.248283	−	0.968688i	$-0.579866\pi$
$113$	−5.27857	−0.496566	−0.248283	+	0.968688i	$0.579866\pi$
$114$	0	0
$115$	−0.947603	−0.0883645
$116$	−0.767549	−0.0712651
$117$	0	0
$118$	11.0363	1.01597
$119$	10.5130	0.963724
$120$	0	0
$121$	1.00000	0.0909091
$122$	−1.56609	−0.141787
$123$	0	0
$124$	−0.155791	−0.0139904
$125$	−11.0906	−0.991972
$126$	0	0
$127$	−7.24882	−0.643229	−0.321615	−	0.946871i	$-0.604226\pi$
$127$	−7.24882	−0.643229	−0.321615	+	0.946871i	$0.604226\pi$
$128$	13.5843	1.20069
$129$	0	0
$130$	13.8408	1.21392
$131$	−0.0866722	−0.00757259	−0.00378629	−	0.999993i	$-0.501205\pi$
$131$	−0.0866722	−0.00757259	−0.00378629	+	0.999993i	$0.501205\pi$
$132$	0	0
$133$	−2.57552	−0.223326
$134$	−18.6380	−1.61007
$135$	0	0
$136$	−7.58245	−0.650190
$137$	−19.1955	−1.63999	−0.819993	−	0.572374i	$-0.806023\pi$
$137$	−19.1955	−1.63999	−0.819993	+	0.572374i	$0.806023\pi$
$138$	0	0
$139$	14.1692	1.20182	0.600910	−	0.799317i	$-0.294805\pi$
$139$	14.1692	1.20182	0.600910	+	0.799317i	$0.294805\pi$
$140$	2.07035	0.174976
$141$	0	0
$142$	2.73925	0.229873
$143$	6.49192	0.542882
$144$	0	0
$145$	−2.30853	−0.191713
$146$	−14.3985	−1.19163
$147$	0	0
$148$	2.37155	0.194940
$149$	1.84097	0.150818	0.0754090	−	0.997153i	$-0.475974\pi$
$149$	1.84097	0.150818	0.0754090	+	0.997153i	$0.475974\pi$
$150$	0	0
$151$	1.00745	0.0819850	0.0409925	−	0.999159i	$-0.486948\pi$
$151$	1.00745	0.0819850	0.0409925	+	0.999159i	$0.486948\pi$
$152$	1.85758	0.150670
$153$	0	0
$154$	5.26190	0.424016
$155$	−0.468567	−0.0376362
$156$	0	0
$157$	22.8792	1.82596	0.912979	−	0.408007i	$-0.133776\pi$
$157$	22.8792	1.82596	0.912979	+	0.408007i	$0.133776\pi$
$158$	−18.7703	−1.49329
$159$	0	0
$160$	−3.42325	−0.270632
$161$	−2.33873	−0.184318
$162$	0	0
$163$	7.76465	0.608174	0.304087	−	0.952644i	$-0.401649\pi$
$163$	7.76465	0.608174	0.304087	+	0.952644i	$0.401649\pi$
$164$	3.74574	0.292493
$165$	0	0
$166$	−2.03228	−0.157736
$167$	−9.83034	−0.760695	−0.380347	−	0.924844i	$-0.624196\pi$
$167$	−9.83034	−0.760695	−0.380347	+	0.924844i	$0.624196\pi$
$168$	0	0
$169$	29.1450	2.24193
$170$	6.67096	0.511639
$171$	0	0
$172$	−0.571332	−0.0435637
$173$	15.9889	1.21561	0.607807	−	0.794085i	$-0.292049\pi$
$173$	15.9889	1.21561	0.607807	+	0.794085i	$0.292049\pi$
$174$	0	0
$175$	−10.5726	−0.799214
$176$	−4.70039	−0.354305
$177$	0	0
$178$	−2.22174	−0.166526
$179$	15.4714	1.15639	0.578194	−	0.815899i	$-0.303757\pi$
$179$	15.4714	1.15639	0.578194	+	0.815899i	$0.303757\pi$
$180$	0	0
$181$	4.75922	0.353750	0.176875	−	0.984233i	$-0.443401\pi$
$181$	4.75922	0.353750	0.176875	+	0.984233i	$0.443401\pi$
$182$	34.1598	2.53210
$183$	0	0
$184$	1.68680	0.124353
$185$	7.13282	0.524415
$186$	0	0
$187$	3.12895	0.228812
$188$	−1.27122	−0.0927135
$189$	0	0
$190$	−1.63428	−0.118563
$191$	−20.8442	−1.50823	−0.754116	−	0.656742i	$-0.771934\pi$
$191$	−20.8442	−1.50823	−0.754116	+	0.656742i	$0.771934\pi$
$192$	0	0
$193$	−10.9933	−0.791316	−0.395658	−	0.918398i	$-0.629483\pi$
$193$	−10.9933	−0.791316	−0.395658	+	0.918398i	$0.629483\pi$
$194$	−8.16335	−0.586094
$195$	0	0
$196$	1.94131	0.138665
$197$	9.78826	0.697385	0.348692	−	0.937237i	$-0.386626\pi$
$197$	9.78826	0.697385	0.348692	+	0.937237i	$0.386626\pi$
$198$	0	0
$199$	−11.5773	−0.820695	−0.410347	−	0.911929i	$-0.634593\pi$
$199$	−11.5773	−0.820695	−0.410347	+	0.911929i	$0.634593\pi$
$200$	7.62545	0.539201
$201$	0	0
$202$	18.2112	1.28134
$203$	−5.69756	−0.399891
$204$	0	0
$205$	11.2659	0.786847
$206$	−6.91215	−0.481592
$207$	0	0
$208$	−30.5145	−2.11580
$209$	−0.766546	−0.0530231
$210$	0	0
$211$	−11.0359	−0.759746	−0.379873	−	0.925039i	$-0.624032\pi$
$211$	−11.0359	−0.759746	−0.379873	+	0.925039i	$0.624032\pi$
$212$	6.09789	0.418805
$213$	0	0
$214$	7.61742	0.520716
$215$	−1.71838	−0.117192
$216$	0	0
$217$	−1.15645	−0.0785046
$218$	7.89242	0.534542
$219$	0	0
$220$	0.616192	0.0415437
$221$	20.3129	1.36640
$222$	0	0
$223$	7.01627	0.469844	0.234922	−	0.972014i	$-0.424517\pi$
$223$	7.01627	0.469844	0.234922	+	0.972014i	$0.424517\pi$
$224$	−8.44875	−0.564506
$225$	0	0
$226$	−8.26670	−0.549893
$227$	1.47103	0.0976358	0.0488179	−	0.998808i	$-0.484455\pi$
$227$	1.47103	0.0976358	0.0488179	+	0.998808i	$0.484455\pi$
$228$	0	0
$229$	7.81248	0.516263	0.258132	−	0.966110i	$-0.416893\pi$
$229$	7.81248	0.516263	0.258132	+	0.966110i	$0.416893\pi$
$230$	−1.48403	−0.0978540
$231$	0	0
$232$	4.10935	0.269792
$233$	−6.56622	−0.430167	−0.215084	−	0.976596i	$-0.569002\pi$
$233$	−6.56622	−0.430167	−0.215084	+	0.976596i	$0.569002\pi$
$234$	0	0
$235$	−3.82342	−0.249412
$236$	3.18970	0.207632
$237$	0	0
$238$	16.4643	1.06722
$239$	−11.6820	−0.755647	−0.377823	−	0.925878i	$-0.623327\pi$
$239$	−11.6820	−0.755647	−0.377823	+	0.925878i	$0.623327\pi$
$240$	0	0
$241$	23.4303	1.50928	0.754638	−	0.656141i	$-0.227813\pi$
$241$	23.4303	1.50928	0.754638	+	0.656141i	$0.227813\pi$
$242$	1.56609	0.100672
$243$	0	0
$244$	−0.452630	−0.0289767
$245$	5.83880	0.373027
$246$	0	0
$247$	−4.97636	−0.316638
$248$	0.834082	0.0529642
$249$	0	0
$250$	−17.3688	−1.09850
$251$	17.2071	1.08610	0.543051	−	0.839699i	$-0.317269\pi$
$251$	17.2071	1.08610	0.543051	+	0.839699i	$0.317269\pi$
$252$	0	0
$253$	−0.696071	−0.0437616
$254$	−11.3523	−0.712306
$255$	0	0
$256$	10.3487	0.646794
$257$	21.5802	1.34614	0.673068	−	0.739581i	$-0.264976\pi$
$257$	21.5802	1.34614	0.673068	+	0.739581i	$0.264976\pi$
$258$	0	0
$259$	17.6041	1.09387
$260$	4.00027	0.248086
$261$	0	0
$262$	−0.135736	−0.00838582
$263$	11.4225	0.704339	0.352169	−	0.935936i	$-0.385444\pi$
$263$	11.4225	0.704339	0.352169	+	0.935936i	$0.385444\pi$
$264$	0	0
$265$	18.3404	1.12664
$266$	−4.03349	−0.247309
$267$	0	0
$268$	−5.38673	−0.329047
$269$	19.5550	1.19229	0.596144	−	0.802877i	$-0.296699\pi$
$269$	19.5550	1.19229	0.596144	+	0.802877i	$0.296699\pi$
$270$	0	0
$271$	22.7236	1.38036	0.690179	−	0.723639i	$-0.257532\pi$
$271$	22.7236	1.38036	0.690179	+	0.723639i	$0.257532\pi$
$272$	−14.7073	−0.891761
$273$	0	0
$274$	−30.0619	−1.81611
$275$	−3.14670	−0.189753
$276$	0	0
$277$	−22.6998	−1.36390	−0.681950	−	0.731398i	$-0.738868\pi$
$277$	−22.6998	−1.36390	−0.681950	+	0.731398i	$0.738868\pi$
$278$	22.1903	1.33088
$279$	0	0
$280$	−11.0843	−0.662416
$281$	−14.5156	−0.865929	−0.432965	−	0.901411i	$-0.642532\pi$
$281$	−14.5156	−0.865929	−0.432965	+	0.901411i	$0.642532\pi$
$282$	0	0
$283$	8.44305	0.501888	0.250944	−	0.968002i	$-0.419259\pi$
$283$	8.44305	0.501888	0.250944	+	0.968002i	$0.419259\pi$
$284$	0.791697	0.0469785
$285$	0	0
$286$	10.1669	0.601182
$287$	27.8049	1.64127
$288$	0	0
$289$	−7.20964	−0.424097
$290$	−3.61536	−0.212301
$291$	0	0
$292$	−4.16144	−0.243530
$293$	33.7697	1.97284	0.986422	−	0.164230i	$-0.0525140\pi$
$293$	33.7697	1.97284	0.986422	+	0.164230i	$0.0525140\pi$
$294$	0	0
$295$	9.59355	0.558558
$296$	−12.6969	−0.737994
$297$	0	0
$298$	2.88312	0.167015
$299$	−4.51884	−0.261331
$300$	0	0
$301$	−4.24104	−0.244449
$302$	1.57775	0.0907895
$303$	0	0
$304$	3.60306	0.206650
$305$	−1.36136	−0.0779512
$306$	0	0
$307$	−28.9927	−1.65470	−0.827352	−	0.561684i	$-0.810154\pi$
$307$	−28.9927	−1.65470	−0.827352	+	0.561684i	$0.810154\pi$
$308$	1.52079	0.0866552
$309$	0	0
$310$	−0.733816	−0.0416780
$311$	−11.0448	−0.626293	−0.313146	−	0.949705i	$-0.601383\pi$
$311$	−11.0448	−0.626293	−0.313146	+	0.949705i	$0.601383\pi$
$312$	0	0
$313$	5.07426	0.286814	0.143407	−	0.989664i	$-0.454194\pi$
$313$	5.07426	0.286814	0.143407	+	0.989664i	$0.454194\pi$
$314$	35.8308	2.02205
$315$	0	0
$316$	−5.42499	−0.305179
$317$	10.8321	0.608391	0.304195	−	0.952610i	$-0.401612\pi$
$317$	10.8321	0.608391	0.304195	+	0.952610i	$0.401612\pi$
$318$	0	0
$319$	−1.69575	−0.0949439
$320$	7.43672	0.415725
$321$	0	0
$322$	−3.66266	−0.204112
$323$	−2.39849	−0.133455
$324$	0	0
$325$	−20.4281	−1.13315
$326$	12.1601	0.673487
$327$	0	0
$328$	−20.0542	−1.10731
$329$	−9.43638	−0.520244
$330$	0	0
$331$	20.2052	1.11058	0.555289	−	0.831657i	$-0.312608\pi$
$331$	20.2052	1.11058	0.555289	+	0.831657i	$0.312608\pi$
$332$	−0.587370	−0.0322361
$333$	0	0
$334$	−15.3952	−0.842386
$335$	−16.2015	−0.885182
$336$	0	0
$337$	−12.1303	−0.660782	−0.330391	−	0.943844i	$-0.607181\pi$
$337$	−12.1303	−0.660782	−0.330391	+	0.943844i	$0.607181\pi$
$338$	45.6437	2.48269
$339$	0	0
$340$	1.92804	0.104563
$341$	−0.344190	−0.0186389
$342$	0	0
$343$	−9.10888	−0.491833
$344$	3.05883	0.164921
$345$	0	0
$346$	25.0400	1.34616
$347$	−18.8939	−1.01428	−0.507139	−	0.861864i	$-0.669297\pi$
$347$	−18.8939	−1.01428	−0.507139	+	0.861864i	$0.669297\pi$
$348$	0	0
$349$	−33.9066	−1.81498	−0.907490	−	0.420075i	$-0.862004\pi$
$349$	−33.9066	−1.81498	−0.907490	+	0.420075i	$0.862004\pi$
$350$	−16.5576	−0.885042
$351$	0	0
$352$	−2.51458	−0.134028
$353$	17.7095	0.942579	0.471290	−	0.881978i	$-0.343789\pi$
$353$	17.7095	0.942579	0.471290	+	0.881978i	$0.343789\pi$
$354$	0	0
$355$	2.38116	0.126379
$356$	−0.642125	−0.0340326
$357$	0	0
$358$	24.2296	1.28057
$359$	11.8177	0.623713	0.311857	−	0.950129i	$-0.399049\pi$
$359$	11.8177	0.623713	0.311857	+	0.950129i	$0.399049\pi$
$360$	0	0
$361$	−18.4124	−0.969074
$362$	7.45336	0.391740
$363$	0	0
$364$	9.87287	0.517479
$365$	−12.5162	−0.655129
$366$	0	0
$367$	7.22914	0.377358	0.188679	−	0.982039i	$-0.439579\pi$
$367$	7.22914	0.377358	0.188679	+	0.982039i	$0.439579\pi$
$368$	3.27180	0.170555
$369$	0	0
$370$	11.1706	0.580733
$371$	45.2651	2.35005
$372$	0	0
$373$	−18.2438	−0.944631	−0.472315	−	0.881430i	$-0.656582\pi$
$373$	−18.2438	−0.944631	−0.472315	+	0.881430i	$0.656582\pi$
$374$	4.90022	0.253384
$375$	0	0
$376$	6.80595	0.350990
$377$	−11.0087	−0.566977
$378$	0	0
$379$	−30.8681	−1.58559	−0.792794	−	0.609490i	$-0.791374\pi$
$379$	−30.8681	−1.58559	−0.792794	+	0.609490i	$0.791374\pi$
$380$	−0.472340	−0.0242305
$381$	0	0
$382$	−32.6438	−1.67020
$383$	31.8273	1.62630	0.813148	−	0.582056i	$-0.197752\pi$
$383$	31.8273	1.62630	0.813148	+	0.582056i	$0.197752\pi$
$384$	0	0
$385$	4.57404	0.233114
$386$	−17.2165	−0.876296
$387$	0	0
$388$	−2.35937	−0.119779
$389$	−23.8104	−1.20724	−0.603618	−	0.797273i	$-0.706275\pi$
$389$	−23.8104	−1.20724	−0.603618	+	0.797273i	$0.706275\pi$
$390$	0	0
$391$	−2.17798	−0.110145
$392$	−10.3935	−0.524950
$393$	0	0
$394$	15.3293	0.772278
$395$	−16.3166	−0.820975
$396$	0	0
$397$	0.237691	0.0119294	0.00596470	−	0.999982i	$-0.498101\pi$
$397$	0.237691	0.0119294	0.00596470	+	0.999982i	$0.498101\pi$
$398$	−18.1311	−0.908830
$399$	0	0
$400$	14.7907	0.739535
$401$	−7.74372	−0.386703	−0.193352	−	0.981130i	$-0.561936\pi$
$401$	−7.74372	−0.386703	−0.193352	+	0.981130i	$0.561936\pi$
$402$	0	0
$403$	−2.23445	−0.111306
$404$	5.26340	0.261864
$405$	0	0
$406$	−8.92288	−0.442835
$407$	5.23948	0.259711
$408$	0	0
$409$	−26.4977	−1.31023	−0.655114	−	0.755530i	$-0.727380\pi$
$409$	−26.4977	−1.31023	−0.655114	+	0.755530i	$0.727380\pi$
$410$	17.6434	0.871348
$411$	0	0
$412$	−1.99775	−0.0984219
$413$	23.6774	1.16509
$414$	0	0
$415$	−1.76661	−0.0867196
$416$	−16.3245	−0.800373
$417$	0	0
$418$	−1.20048	−0.0587173
$419$	4.24640	0.207450	0.103725	−	0.994606i	$-0.466924\pi$
$419$	4.24640	0.207450	0.103725	+	0.994606i	$0.466924\pi$
$420$	0	0
$421$	−4.39859	−0.214374	−0.107187	−	0.994239i	$-0.534184\pi$
$421$	−4.39859	−0.214374	−0.107187	+	0.994239i	$0.534184\pi$
$422$	−17.2832	−0.841335
$423$	0	0
$424$	−32.6473	−1.58549
$425$	−9.84588	−0.477595
$426$	0	0
$427$	−3.35990	−0.162597
$428$	2.20159	0.106418
$429$	0	0
$430$	−2.69113	−0.129778
$431$	−20.3462	−0.980043	−0.490022	−	0.871710i	$-0.663011\pi$
$431$	−20.3462	−0.980043	−0.490022	+	0.871710i	$0.663011\pi$
$432$	0	0
$433$	−6.50313	−0.312521	−0.156260	−	0.987716i	$-0.549944\pi$
$433$	−6.50313	−0.312521	−0.156260	+	0.987716i	$0.549944\pi$
$434$	−1.81109	−0.0869353
$435$	0	0
$436$	2.28107	0.109243
$437$	0.533571	0.0255241
$438$	0	0
$439$	8.54133	0.407655	0.203828	−	0.979007i	$-0.434662\pi$
$439$	8.54133	0.407655	0.203828	+	0.979007i	$0.434662\pi$
$440$	−3.29901	−0.157274
$441$	0	0
$442$	31.8118	1.51313
$443$	−7.02111	−0.333583	−0.166791	−	0.985992i	$-0.553341\pi$
$443$	−7.02111	−0.333583	−0.166791	+	0.985992i	$0.553341\pi$
$444$	0	0
$445$	−1.93130	−0.0915523
$446$	10.9881	0.520301
$447$	0	0
$448$	18.3542	0.867154
$449$	−6.70573	−0.316463	−0.158232	−	0.987402i	$-0.550579\pi$
$449$	−6.70573	−0.316463	−0.158232	+	0.987402i	$0.550579\pi$
$450$	0	0
$451$	8.27551	0.389678
$452$	−2.38924	−0.112380
$453$	0	0
$454$	2.30376	0.108121
$455$	29.6943	1.39209
$456$	0	0
$457$	3.04900	0.142626	0.0713132	−	0.997454i	$-0.477281\pi$
$457$	3.04900	0.142626	0.0713132	+	0.997454i	$0.477281\pi$
$458$	12.2350	0.571705
$459$	0	0
$460$	−0.428914	−0.0199982
$461$	−25.0006	−1.16439	−0.582197	−	0.813048i	$-0.697807\pi$
$461$	−25.0006	−1.16439	−0.582197	+	0.813048i	$0.697807\pi$
$462$	0	0
$463$	1.57111	0.0730158	0.0365079	−	0.999333i	$-0.488377\pi$
$463$	1.57111	0.0730158	0.0365079	+	0.999333i	$0.488377\pi$
$464$	7.97069	0.370030
$465$	0	0
$466$	−10.2833	−0.476363
$467$	−4.67308	−0.216245	−0.108122	−	0.994138i	$-0.534484\pi$
$467$	−4.67308	−0.216245	−0.108122	+	0.994138i	$0.534484\pi$
$468$	0	0
$469$	−39.9861	−1.84639
$470$	−5.98780	−0.276197
$471$	0	0
$472$	−17.0772	−0.786042
$473$	−1.26225	−0.0580383
$474$	0	0
$475$	2.41209	0.110674
$476$	4.75849	0.218105
$477$	0	0
$478$	−18.2951	−0.836796
$479$	22.7806	1.04087	0.520437	−	0.853900i	$-0.325769\pi$
$479$	22.7806	1.04087	0.520437	+	0.853900i	$0.325769\pi$
$480$	0	0
$481$	34.0143	1.55092
$482$	36.6938	1.67136
$483$	0	0
$484$	0.452630	0.0205741
$485$	−7.09619	−0.322221
$486$	0	0
$487$	−32.7483	−1.48397	−0.741983	−	0.670419i	$-0.766114\pi$
$487$	−32.7483	−1.48397	−0.741983	+	0.670419i	$0.766114\pi$
$488$	2.42332	0.109698
$489$	0	0
$490$	9.14407	0.413087
$491$	−39.7785	−1.79518	−0.897590	−	0.440831i	$-0.854684\pi$
$491$	−39.7785	−1.79518	−0.897590	+	0.440831i	$0.854684\pi$
$492$	0	0
$493$	−5.30593	−0.238967
$494$	−7.79341	−0.350642
$495$	0	0
$496$	1.61783	0.0726425
$497$	5.87682	0.263611
$498$	0	0
$499$	0.606793	0.0271638	0.0135819	−	0.999908i	$-0.495677\pi$
$499$	0.606793	0.0271638	0.0135819	+	0.999908i	$0.495677\pi$
$500$	−5.01993	−0.224498
$501$	0	0
$502$	26.9478	1.20274
$503$	17.8924	0.797783	0.398891	−	0.916998i	$-0.369395\pi$
$503$	17.8924	0.797783	0.398891	+	0.916998i	$0.369395\pi$
$504$	0	0
$505$	15.8306	0.704450
$506$	−1.09011	−0.0484612
$507$	0	0
$508$	−3.28104	−0.145572
$509$	−34.3179	−1.52112	−0.760558	−	0.649270i	$-0.775074\pi$
$509$	−34.3179	−1.52112	−0.760558	+	0.649270i	$0.775074\pi$
$510$	0	0
$511$	−30.8906	−1.36652
$512$	−10.9615	−0.484436
$513$	0	0
$514$	33.7965	1.49070
$515$	−6.00855	−0.264769
$516$	0	0
$517$	−2.80853	−0.123519
$518$	27.5696	1.21134
$519$	0	0
$520$	−21.4169	−0.939193
$521$	−14.6255	−0.640756	−0.320378	−	0.947290i	$-0.603810\pi$
$521$	−14.6255	−0.640756	−0.320378	+	0.947290i	$0.603810\pi$
$522$	0	0
$523$	−13.0719	−0.571593	−0.285797	−	0.958290i	$-0.592258\pi$
$523$	−13.0719	−0.571593	−0.285797	+	0.958290i	$0.592258\pi$
$524$	−0.0392305	−0.00171379
$525$	0	0
$526$	17.8886	0.779978
$527$	−1.07696	−0.0469129
$528$	0	0
$529$	−22.5155	−0.978934
$530$	28.7227	1.24763
$531$	0	0
$532$	−1.16576	−0.0505420
$533$	53.7239	2.32704
$534$	0	0
$535$	6.62163	0.286278
$536$	28.8398	1.24569
$537$	0	0
$538$	30.6248	1.32033
$539$	4.28895	0.184738
$540$	0	0
$541$	−29.1984	−1.25534	−0.627669	−	0.778480i	$-0.715991\pi$
$541$	−29.1984	−1.25534	−0.627669	+	0.778480i	$0.715991\pi$
$542$	35.5871	1.52860
$543$	0	0
$544$	−7.86802	−0.337338
$545$	6.86068	0.293879
$546$	0	0
$547$	−34.5379	−1.47673	−0.738366	−	0.674400i	$-0.764402\pi$
$547$	−34.5379	−1.47673	−0.738366	+	0.674400i	$0.764402\pi$
$548$	−8.68848	−0.371153
$549$	0	0
$550$	−4.92801	−0.210131
$551$	1.29987	0.0553764
$552$	0	0
$553$	−40.2700	−1.71246
$554$	−35.5499	−1.51037
$555$	0	0
$556$	6.41342	0.271990
$557$	−33.0302	−1.39953	−0.699767	−	0.714371i	$-0.746713\pi$
$557$	−33.0302	−1.39953	−0.699767	+	0.714371i	$0.746713\pi$
$558$	0	0
$559$	−8.19442	−0.346587
$560$	−21.4997	−0.908530
$561$	0	0
$562$	−22.7327	−0.958922
$563$	−20.8159	−0.877285	−0.438642	−	0.898662i	$-0.644541\pi$
$563$	−20.8159	−0.877285	−0.438642	+	0.898662i	$0.644541\pi$
$564$	0	0
$565$	−7.18603	−0.302319
$566$	13.2226	0.555786
$567$	0	0
$568$	−4.23863	−0.177849
$569$	−5.73465	−0.240409	−0.120204	−	0.992749i	$-0.538355\pi$
$569$	−5.73465	−0.240409	−0.120204	+	0.992749i	$0.538355\pi$
$570$	0	0
$571$	3.24657	0.135865	0.0679323	−	0.997690i	$-0.478360\pi$
$571$	3.24657	0.135865	0.0679323	+	0.997690i	$0.478360\pi$
$572$	2.93844	0.122862
$573$	0	0
$574$	43.5449	1.81753
$575$	2.19033	0.0913429
$576$	0	0
$577$	21.8389	0.909165	0.454583	−	0.890705i	$-0.349788\pi$
$577$	21.8389	0.909165	0.454583	+	0.890705i	$0.349788\pi$
$578$	−11.2909	−0.469641
$579$	0	0
$580$	−1.04491	−0.0433875
$581$	−4.36009	−0.180887
$582$	0	0
$583$	13.4721	0.557959
$584$	22.2797	0.921943
$585$	0	0
$586$	52.8862	2.18471
$587$	5.80443	0.239574	0.119787	−	0.992800i	$-0.461779\pi$
$587$	5.80443	0.239574	0.119787	+	0.992800i	$0.461779\pi$
$588$	0	0
$589$	0.263838	0.0108712
$590$	15.0243	0.618542
$591$	0	0
$592$	−24.6276	−1.01219
$593$	−27.9671	−1.14847	−0.574235	−	0.818690i	$-0.694701\pi$
$593$	−27.9671	−1.14847	−0.574235	+	0.818690i	$0.694701\pi$
$594$	0	0
$595$	14.3120	0.586733
$596$	0.833278	0.0341324
$597$	0	0
$598$	−7.07690	−0.289396
$599$	1.61130	0.0658358	0.0329179	−	0.999458i	$-0.489520\pi$
$599$	1.61130	0.0658358	0.0329179	+	0.999458i	$0.489520\pi$
$600$	0	0
$601$	−15.5337	−0.633632	−0.316816	−	0.948487i	$-0.602614\pi$
$601$	−15.5337	−0.633632	−0.316816	+	0.948487i	$0.602614\pi$
$602$	−6.64183	−0.270701
$603$	0	0
$604$	0.456002	0.0185544
$605$	1.36136	0.0553471
$606$	0	0
$607$	−35.6663	−1.44765	−0.723826	−	0.689982i	$-0.757618\pi$
$607$	−35.6663	−1.44765	−0.723826	+	0.689982i	$0.757618\pi$
$608$	1.92754	0.0781722
$609$	0	0
$610$	−2.13201	−0.0863225
$611$	−18.2327	−0.737618
$612$	0	0
$613$	42.4297	1.71372	0.856859	−	0.515550i	$-0.172412\pi$
$613$	42.4297	1.71372	0.856859	+	0.515550i	$0.172412\pi$
$614$	−45.4052	−1.83240
$615$	0	0
$616$	−8.14211	−0.328055
$617$	7.26110	0.292321	0.146160	−	0.989261i	$-0.453308\pi$
$617$	7.26110	0.292321	0.146160	+	0.989261i	$0.453308\pi$
$618$	0	0
$619$	15.5594	0.625385	0.312692	−	0.949854i	$-0.398769\pi$
$619$	15.5594	0.625385	0.312692	+	0.949854i	$0.398769\pi$
$620$	−0.212087	−0.00851763
$621$	0	0
$622$	−17.2971	−0.693551
$623$	−4.76654	−0.190967
$624$	0	0
$625$	0.635214	0.0254086
$626$	7.94674	0.317616
$627$	0	0
$628$	10.3558	0.413242
$629$	16.3941	0.653676
$630$	0	0
$631$	−5.24432	−0.208773	−0.104387	−	0.994537i	$-0.533288\pi$
$631$	−5.24432	−0.208773	−0.104387	+	0.994537i	$0.533288\pi$
$632$	29.0446	1.15533
$633$	0	0
$634$	16.9640	0.673726
$635$	−9.86826	−0.391610
$636$	0	0
$637$	27.8435	1.10320
$638$	−2.65570	−0.105140
$639$	0	0
$640$	18.4931	0.731002
$641$	−2.17971	−0.0860935	−0.0430468	−	0.999073i	$-0.513706\pi$
$641$	−2.17971	−0.0860935	−0.0430468	+	0.999073i	$0.513706\pi$
$642$	0	0
$643$	−4.84766	−0.191173	−0.0955865	−	0.995421i	$-0.530473\pi$
$643$	−4.84766	−0.191173	−0.0955865	+	0.995421i	$0.530473\pi$
$644$	−1.05858	−0.0417139
$645$	0	0
$646$	−3.75624	−0.147787
$647$	40.2387	1.58195	0.790973	−	0.611851i	$-0.209575\pi$
$647$	40.2387	1.58195	0.790973	+	0.611851i	$0.209575\pi$
$648$	0	0
$649$	7.04704	0.276620
$650$	−31.9922	−1.25484
$651$	0	0
$652$	3.51451	0.137639
$653$	6.85500	0.268257	0.134129	−	0.990964i	$-0.457177\pi$
$653$	6.85500	0.268257	0.134129	+	0.990964i	$0.457177\pi$
$654$	0	0
$655$	−0.117992	−0.00461033
$656$	−38.8981	−1.51871
$657$	0	0
$658$	−14.7782	−0.576114
$659$	−26.8314	−1.04520	−0.522601	−	0.852577i	$-0.675038\pi$
$659$	−26.8314	−1.04520	−0.522601	+	0.852577i	$0.675038\pi$
$660$	0	0
$661$	30.6381	1.19168	0.595842	−	0.803102i	$-0.296819\pi$
$661$	30.6381	1.19168	0.595842	+	0.803102i	$0.296819\pi$
$662$	31.6431	1.22984
$663$	0	0
$664$	3.14470	0.122038
$665$	−3.50621	−0.135965
$666$	0	0
$667$	1.18036	0.0457039
$668$	−4.44951	−0.172157
$669$	0	0
$670$	−25.3730	−0.980243
$671$	−1.00000	−0.0386046
$672$	0	0
$673$	31.9283	1.23075	0.615373	−	0.788236i	$-0.289005\pi$
$673$	31.9283	1.23075	0.615373	+	0.788236i	$0.289005\pi$
$674$	−18.9972	−0.731744
$675$	0	0
$676$	13.1919	0.507381
$677$	−23.5546	−0.905275	−0.452638	−	0.891695i	$-0.649517\pi$
$677$	−23.5546	−0.905275	−0.452638	+	0.891695i	$0.649517\pi$
$678$	0	0
$679$	−17.5137	−0.672116
$680$	−10.3224	−0.395848
$681$	0	0
$682$	−0.539032	−0.0206406
$683$	−17.5832	−0.672802	−0.336401	−	0.941719i	$-0.609210\pi$
$683$	−17.5832	−0.672802	−0.336401	+	0.941719i	$0.609210\pi$
$684$	0	0
$685$	−26.1320	−0.998454
$686$	−14.2653	−0.544652
$687$	0	0
$688$	5.93306	0.226196
$689$	87.4600	3.33196
$690$	0	0
$691$	−17.1914	−0.653992	−0.326996	−	0.945026i	$-0.606036\pi$
$691$	−17.1914	−0.653992	−0.326996	+	0.945026i	$0.606036\pi$
$692$	7.23706	0.275112
$693$	0	0
$694$	−29.5895	−1.12320
$695$	19.2894	0.731690
$696$	0	0
$697$	25.8937	0.980793
$698$	−53.1007	−2.00989
$699$	0	0
$700$	−4.78548	−0.180874
$701$	38.0758	1.43810	0.719052	−	0.694956i	$-0.244576\pi$
$701$	38.0758	1.43810	0.719052	+	0.694956i	$0.244576\pi$
$702$	0	0
$703$	−4.01631	−0.151478
$704$	5.46272	0.205884
$705$	0	0
$706$	27.7346	1.04380
$707$	39.0706	1.46940
$708$	0	0
$709$	7.43128	0.279088	0.139544	−	0.990216i	$-0.455436\pi$
$709$	7.43128	0.279088	0.139544	+	0.990216i	$0.455436\pi$
$710$	3.72910	0.139951
$711$	0	0
$712$	3.43785	0.128839
$713$	0.239581	0.00897237
$714$	0	0
$715$	8.83784	0.330516
$716$	7.00283	0.261708
$717$	0	0
$718$	18.5075	0.690694
$719$	22.8101	0.850671	0.425336	−	0.905036i	$-0.360156\pi$
$719$	22.8101	0.850671	0.425336	+	0.905036i	$0.360156\pi$
$720$	0	0
$721$	−14.8294	−0.552276
$722$	−28.8354	−1.07314
$723$	0	0
$724$	2.15417	0.0800590
$725$	5.33602	0.198175
$726$	0	0
$727$	−16.7830	−0.622447	−0.311223	−	0.950337i	$-0.600739\pi$
$727$	−16.7830	−0.622447	−0.311223	+	0.950337i	$0.600739\pi$
$728$	−52.8579	−1.95904
$729$	0	0
$730$	−19.6015	−0.725484
$731$	−3.94952	−0.146078
$732$	0	0
$733$	12.5004	0.461713	0.230857	−	0.972988i	$-0.425847\pi$
$733$	12.5004	0.461713	0.230857	+	0.972988i	$0.425847\pi$
$734$	11.3215	0.417883
$735$	0	0
$736$	1.75033	0.0645180
$737$	−11.9010	−0.438378
$738$	0	0
$739$	−30.0947	−1.10705	−0.553526	−	0.832832i	$-0.686718\pi$
$739$	−30.0947	−1.10705	−0.553526	+	0.832832i	$0.686718\pi$
$740$	3.22853	0.118683
$741$	0	0
$742$	70.8891	2.60242
$743$	−38.8598	−1.42563	−0.712813	−	0.701354i	$-0.752579\pi$
$743$	−38.8598	−1.42563	−0.712813	+	0.701354i	$0.752579\pi$
$744$	0	0
$745$	2.50622	0.0918209
$746$	−28.5715	−1.04608
$747$	0	0
$748$	1.41626	0.0517836
$749$	16.3425	0.597142
$750$	0	0
$751$	30.7717	1.12288	0.561438	−	0.827519i	$-0.310248\pi$
$751$	30.7717	1.12288	0.561438	+	0.827519i	$0.310248\pi$
$752$	13.2012	0.481397
$753$	0	0
$754$	−17.2406	−0.627865
$755$	1.37150	0.0499140
$756$	0	0
$757$	−22.9285	−0.833351	−0.416676	−	0.909055i	$-0.636805\pi$
$757$	−22.9285	−0.833351	−0.416676	+	0.909055i	$0.636805\pi$
$758$	−48.3421	−1.75587
$759$	0	0
$760$	2.52884	0.0917307
$761$	12.8001	0.464001	0.232001	−	0.972716i	$-0.425473\pi$
$761$	12.8001	0.464001	0.232001	+	0.972716i	$0.425473\pi$
$762$	0	0
$763$	16.9325	0.612998
$764$	−9.43470	−0.341335
$765$	0	0
$766$	49.8443	1.80095
$767$	45.7488	1.65189
$768$	0	0
$769$	30.0017	1.08189	0.540945	−	0.841058i	$-0.318067\pi$
$769$	30.0017	1.08189	0.540945	+	0.841058i	$0.318067\pi$
$770$	7.16334	0.258149
$771$	0	0
$772$	−4.97590	−0.179087
$773$	25.4863	0.916680	0.458340	−	0.888777i	$-0.348444\pi$
$773$	25.4863	0.916680	0.458340	+	0.888777i	$0.348444\pi$
$774$	0	0
$775$	1.08306	0.0389048
$776$	12.6317	0.453452
$777$	0	0
$778$	−37.2892	−1.33688
$779$	−6.34356	−0.227282
$780$	0	0
$781$	1.74910	0.0625878
$782$	−3.41090	−0.121974
$783$	0	0
$784$	−20.1597	−0.719989
$785$	31.1468	1.11168
$786$	0	0
$787$	17.9205	0.638796	0.319398	−	0.947621i	$-0.396519\pi$
$787$	17.9205	0.638796	0.319398	+	0.947621i	$0.396519\pi$
$788$	4.43046	0.157829
$789$	0	0
$790$	−25.5531	−0.909140
$791$	−17.7355	−0.630601
$792$	0	0
$793$	−6.49192	−0.230535
$794$	0.372246	0.0132105
$795$	0	0
$796$	−5.24024	−0.185736
$797$	−2.75839	−0.0977072	−0.0488536	−	0.998806i	$-0.515557\pi$
$797$	−2.75839	−0.0977072	−0.0488536	+	0.998806i	$0.515557\pi$
$798$	0	0
$799$	−8.78775	−0.310888
$800$	7.91264	0.279754
$801$	0	0
$802$	−12.1273	−0.428232
$803$	−9.19391	−0.324446
$804$	0	0
$805$	−3.18386	−0.112216
$806$	−3.49935	−0.123259
$807$	0	0
$808$	−28.1795	−0.991351
$809$	−3.91072	−0.137494	−0.0687468	−	0.997634i	$-0.521900\pi$
$809$	−3.91072	−0.137494	−0.0687468	+	0.997634i	$0.521900\pi$
$810$	0	0
$811$	−54.0477	−1.89787	−0.948936	−	0.315468i	$-0.897839\pi$
$811$	−54.0477	−1.89787	−0.948936	+	0.315468i	$0.897839\pi$
$812$	−2.57889	−0.0905013
$813$	0	0
$814$	8.20549	0.287602
$815$	10.5705	0.370268
$816$	0	0
$817$	0.967573	0.0338511
$818$	−41.4978	−1.45094
$819$	0	0
$820$	5.09930	0.178075
$821$	10.4632	0.365169	0.182585	−	0.983190i	$-0.441554\pi$
$821$	10.4632	0.365169	0.182585	+	0.983190i	$0.441554\pi$
$822$	0	0
$823$	−41.4188	−1.44377	−0.721884	−	0.692015i	$-0.756723\pi$
$823$	−41.4188	−1.44377	−0.721884	+	0.692015i	$0.756723\pi$
$824$	10.6957	0.372601
$825$	0	0
$826$	37.0808	1.29021
$827$	25.7169	0.894264	0.447132	−	0.894468i	$-0.352445\pi$
$827$	25.7169	0.894264	0.447132	+	0.894468i	$0.352445\pi$
$828$	0	0
$829$	−56.0823	−1.94782	−0.973909	−	0.226938i	$-0.927129\pi$
$829$	−56.0823	−1.94782	−0.973909	+	0.226938i	$0.927129\pi$
$830$	−2.76667	−0.0960325
$831$	0	0
$832$	35.4635	1.22948
$833$	13.4199	0.464973
$834$	0	0
$835$	−13.3826	−0.463125
$836$	−0.346962	−0.0119999
$837$	0	0
$838$	6.65023	0.229728
$839$	−18.3203	−0.632486	−0.316243	−	0.948678i	$-0.602422\pi$
$839$	−18.3203	−0.632486	−0.316243	+	0.948678i	$0.602422\pi$
$840$	0	0
$841$	−26.1244	−0.900842
$842$	−6.88857	−0.237396
$843$	0	0
$844$	−4.99520	−0.171942
$845$	39.6769	1.36493
$846$	0	0
$847$	3.35990	0.115448
$848$	−63.3242	−2.17456
$849$	0	0
$850$	−15.4195	−0.528885
$851$	−3.64705	−0.125019
$852$	0	0
$853$	47.0478	1.61088	0.805442	−	0.592674i	$-0.201928\pi$
$853$	47.0478	1.61088	0.805442	+	0.592674i	$0.201928\pi$
$854$	−5.26190	−0.180059
$855$	0	0
$856$	−11.7870	−0.402870
$857$	−55.0112	−1.87915	−0.939573	−	0.342350i	$-0.888777\pi$
$857$	−55.0112	−1.87915	−0.939573	+	0.342350i	$0.888777\pi$
$858$	0	0
$859$	45.6370	1.55711	0.778557	−	0.627574i	$-0.215952\pi$
$859$	45.6370	1.55711	0.778557	+	0.627574i	$0.215952\pi$
$860$	−0.777789	−0.0265224
$861$	0	0
$862$	−31.8640	−1.08529
$863$	53.4991	1.82113	0.910565	−	0.413365i	$-0.135647\pi$
$863$	53.4991	1.82113	0.910565	+	0.413365i	$0.135647\pi$
$864$	0	0
$865$	21.7666	0.740088
$866$	−10.1845	−0.346082
$867$	0	0
$868$	−0.523442	−0.0177668
$869$	−11.9855	−0.406580
$870$	0	0
$871$	−77.2601	−2.61786
$872$	−12.2125	−0.413567
$873$	0	0
$874$	0.835618	0.0282652
$875$	−37.2633	−1.25973
$876$	0	0
$877$	−7.59394	−0.256429	−0.128214	−	0.991746i	$-0.540925\pi$
$877$	−7.59394	−0.256429	−0.128214	+	0.991746i	$0.540925\pi$
$878$	13.3765	0.451434
$879$	0	0
$880$	−6.39892	−0.215707
$881$	−18.4709	−0.622300	−0.311150	−	0.950361i	$-0.600714\pi$
$881$	−18.4709	−0.622300	−0.311150	+	0.950361i	$0.600714\pi$
$882$	0	0
$883$	43.3310	1.45820	0.729102	−	0.684405i	$-0.239938\pi$
$883$	43.3310	1.45820	0.729102	+	0.684405i	$0.239938\pi$
$884$	9.19424	0.309236
$885$	0	0
$886$	−10.9957	−0.369407
$887$	33.1903	1.11442	0.557211	−	0.830371i	$-0.311872\pi$
$887$	33.1903	1.11442	0.557211	+	0.830371i	$0.311872\pi$
$888$	0	0
$889$	−24.3553	−0.816852
$890$	−3.02458	−0.101384
$891$	0	0
$892$	3.17577	0.106333
$893$	2.15287	0.0720429
$894$	0	0
$895$	21.0622	0.704031
$896$	45.6418	1.52478
$897$	0	0
$898$	−10.5018	−0.350448
$899$	0.583661	0.0194662
$900$	0	0
$901$	42.1537	1.40434
$902$	12.9602	0.431526
$903$	0	0
$904$	12.7916	0.425444
$905$	6.47901	0.215370
$906$	0	0
$907$	−35.1466	−1.16702	−0.583511	−	0.812105i	$-0.698322\pi$
$907$	−35.1466	−1.16702	−0.583511	+	0.812105i	$0.698322\pi$
$908$	0.665833	0.0220964
$909$	0	0
$910$	46.5038	1.54159
$911$	−2.35749	−0.0781070	−0.0390535	−	0.999237i	$-0.512434\pi$
$911$	−2.35749	−0.0781070	−0.0390535	+	0.999237i	$0.512434\pi$
$912$	0	0
$913$	−1.29768	−0.0429470
$914$	4.77501	0.157943
$915$	0	0
$916$	3.53616	0.116838
$917$	−0.291210	−0.00961661
$918$	0	0
$919$	−54.1630	−1.78667	−0.893335	−	0.449390i	$-0.851641\pi$
$919$	−54.1630	−1.78667	−0.893335	+	0.449390i	$0.851641\pi$
$920$	2.29634	0.0757082
$921$	0	0
$922$	−39.1531	−1.28944
$923$	11.3550	0.373756
$924$	0	0
$925$	−16.4871	−0.542092
$926$	2.46050	0.0808570
$927$	0	0
$928$	4.26411	0.139976
$929$	3.87710	0.127203	0.0636017	−	0.997975i	$-0.479741\pi$
$929$	3.87710	0.127203	0.0636017	+	0.997975i	$0.479741\pi$
$930$	0	0
$931$	−3.28768	−0.107749
$932$	−2.97207	−0.0973533
$933$	0	0
$934$	−7.31846	−0.239467
$935$	4.25963	0.139305
$936$	0	0
$937$	5.02820	0.164264	0.0821321	−	0.996621i	$-0.473827\pi$
$937$	5.02820	0.164264	0.0821321	+	0.996621i	$0.473827\pi$
$938$	−62.6217	−2.04467
$939$	0	0
$940$	−1.73059	−0.0564457
$941$	−19.4123	−0.632824	−0.316412	−	0.948622i	$-0.602478\pi$
$941$	−19.4123	−0.632824	−0.316412	+	0.948622i	$0.602478\pi$
$942$	0	0
$943$	−5.76034	−0.187583
$944$	−33.1238	−1.07809
$945$	0	0
$946$	−1.97679	−0.0642711
$947$	−57.0998	−1.85549	−0.927746	−	0.373211i	$-0.878257\pi$
$947$	−57.0998	−1.85549	−0.927746	+	0.373211i	$0.878257\pi$
$948$	0	0
$949$	−59.6861	−1.93749
$950$	3.77754	0.122560
$951$	0	0
$952$	−25.4763	−0.825691
$953$	−25.4030	−0.822884	−0.411442	−	0.911436i	$-0.634975\pi$
$953$	−25.4030	−0.822884	−0.411442	+	0.911436i	$0.634975\pi$
$954$	0	0
$955$	−28.3764	−0.918239
$956$	−5.28763	−0.171014
$957$	0	0
$958$	35.6765	1.15265
$959$	−64.4951	−2.08266
$960$	0	0
$961$	−30.8815	−0.996178
$962$	53.2694	1.71747
$963$	0	0
$964$	10.6052	0.341572
$965$	−14.9659	−0.481768
$966$	0	0
$967$	30.7013	0.987287	0.493644	−	0.869664i	$-0.335665\pi$
$967$	30.7013	0.987287	0.493644	+	0.869664i	$0.335665\pi$
$968$	−2.42332	−0.0778883
$969$	0	0
$970$	−11.1133	−0.356825
$971$	48.6547	1.56140	0.780701	−	0.624904i	$-0.214862\pi$
$971$	48.6547	1.56140	0.780701	+	0.624904i	$0.214862\pi$
$972$	0	0
$973$	47.6073	1.52622
$974$	−51.2866	−1.64333
$975$	0	0
$976$	4.70039	0.150456
$977$	50.1834	1.60551	0.802755	−	0.596309i	$-0.203367\pi$
$977$	50.1834	1.60551	0.802755	+	0.596309i	$0.203367\pi$
$978$	0	0
$979$	−1.41865	−0.0453404
$980$	2.64282	0.0844217
$981$	0	0
$982$	−62.2967	−1.98797
$983$	7.93819	0.253189	0.126594	−	0.991955i	$-0.459595\pi$
$983$	7.93819	0.253189	0.126594	+	0.991955i	$0.459595\pi$
$984$	0	0
$985$	13.3253	0.424581
$986$	−8.30956	−0.264630
$987$	0	0
$988$	−2.25245	−0.0716599
$989$	0.878616	0.0279384
$990$	0	0
$991$	−39.0515	−1.24051	−0.620256	−	0.784400i	$-0.712971\pi$
$991$	−39.0515	−1.24051	−0.620256	+	0.784400i	$0.712971\pi$
$992$	0.865494	0.0274795
$993$	0	0
$994$	9.20361	0.291921
$995$	−15.7609	−0.499654
$996$	0	0
$997$	−24.2028	−0.766512	−0.383256	−	0.923642i	$-0.625197\pi$
$997$	−24.2028	−0.766512	−0.383256	+	0.923642i	$0.625197\pi$
$998$	0.950290	0.0300809
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.o.1.17	yes	25
3.2	odd	2		6039.2.a.n.1.9	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6039.2.a.n.1.9	✓	25	3.2	odd	2
6039.2.a.o.1.17	yes	25	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:	\( N \)	\(=\)	\( 6039 = 3^{2} \cdot 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6039.a (trivial)

\(f(q)\)	\(=\)	\(q+1.56609 q^{2} +0.452630 q^{4} +1.36136 q^{5} +3.35990 q^{7} -2.42332 q^{8} +O(q^{10})\)	\(q+1.56609 q^{2} +0.452630 q^{4} +1.36136 q^{5} +3.35990 q^{7} -2.42332 q^{8} +2.13201 q^{10} +1.00000 q^{11} +6.49192 q^{13} +5.26190 q^{14} -4.70039 q^{16} +3.12895 q^{17} -0.766546 q^{19} +0.616192 q^{20} +1.56609 q^{22} -0.696071 q^{23} -3.14670 q^{25} +10.1669 q^{26} +1.52079 q^{28} -1.69575 q^{29} -0.344190 q^{31} -2.51458 q^{32} +4.90022 q^{34} +4.57404 q^{35} +5.23948 q^{37} -1.20048 q^{38} -3.29901 q^{40} +8.27551 q^{41} -1.26225 q^{43} +0.452630 q^{44} -1.09011 q^{46} -2.80853 q^{47} +4.28895 q^{49} -4.92801 q^{50} +2.93844 q^{52} +13.4721 q^{53} +1.36136 q^{55} -8.14211 q^{56} -2.65570 q^{58} +7.04704 q^{59} -1.00000 q^{61} -0.539032 q^{62} +5.46272 q^{64} +8.83784 q^{65} -11.9010 q^{67} +1.41626 q^{68} +7.16334 q^{70} +1.74910 q^{71} -9.19391 q^{73} +8.20549 q^{74} -0.346962 q^{76} +3.35990 q^{77} -11.9855 q^{79} -6.39892 q^{80} +12.9602 q^{82} -1.29768 q^{83} +4.25963 q^{85} -1.97679 q^{86} -2.42332 q^{88} -1.41865 q^{89} +21.8122 q^{91} -0.315063 q^{92} -4.39840 q^{94} -1.04355 q^{95} -5.21258 q^{97} +6.71687 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q + 5 q^{2} + 25 q^{4} + 4 q^{5} + 4 q^{7} + 15 q^{8}+O(q^{10}) \) 25 * q + 5 * q^2 + 25 * q^4 + 4 * q^5 + 4 * q^7 + 15 * q^8	\( 25 q + 5 q^{2} + 25 q^{4} + 4 q^{5} + 4 q^{7} + 15 q^{8} + 25 q^{11} + 4 q^{13} + 18 q^{14} + 21 q^{16} + 20 q^{17} + 14 q^{19} + 12 q^{20} + 5 q^{22} + 20 q^{23} + 13 q^{25} + 16 q^{26} - 14 q^{28} + 28 q^{29} - 12 q^{31} + 35 q^{32} + 6 q^{34} + 10 q^{35} - 8 q^{37} + 32 q^{38} + 24 q^{40} + 26 q^{41} + 18 q^{43} + 25 q^{44} + 4 q^{46} + 12 q^{47} + 23 q^{49} + 43 q^{50} + 22 q^{52} + 36 q^{53} + 4 q^{55} + 26 q^{56} - 20 q^{58} + 46 q^{59} - 25 q^{61} - 14 q^{62} - 13 q^{64} + 60 q^{65} - 20 q^{67} + 44 q^{68} - 20 q^{70} + 52 q^{71} + 6 q^{73} + 32 q^{74} + 4 q^{77} + 26 q^{79} + 52 q^{80} + 6 q^{82} + 38 q^{83} - 4 q^{85} + 34 q^{86} + 15 q^{88} + 82 q^{89} - 58 q^{91} + 36 q^{92} + 16 q^{94} + 30 q^{95} + 35 q^{98}+O(q^{100}) \) 25 * q + 5 * q^2 + 25 * q^4 + 4 * q^5 + 4 * q^7 + 15 * q^8 + 25 * q^11 + 4 * q^13 + 18 * q^14 + 21 * q^16 + 20 * q^17 + 14 * q^19 + 12 * q^20 + 5 * q^22 + 20 * q^23 + 13 * q^25 + 16 * q^26 - 14 * q^28 + 28 * q^29 - 12 * q^31 + 35 * q^32 + 6 * q^34 + 10 * q^35 - 8 * q^37 + 32 * q^38 + 24 * q^40 + 26 * q^41 + 18 * q^43 + 25 * q^44 + 4 * q^46 + 12 * q^47 + 23 * q^49 + 43 * q^50 + 22 * q^52 + 36 * q^53 + 4 * q^55 + 26 * q^56 - 20 * q^58 + 46 * q^59 - 25 * q^61 - 14 * q^62 - 13 * q^64 + 60 * q^65 - 20 * q^67 + 44 * q^68 - 20 * q^70 + 52 * q^71 + 6 * q^73 + 32 * q^74 + 4 * q^77 + 26 * q^79 + 52 * q^80 + 6 * q^82 + 38 * q^83 - 4 * q^85 + 34 * q^86 + 15 * q^88 + 82 * q^89 - 58 * q^91 + 36 * q^92 + 16 * q^94 + 30 * q^95 + 35 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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