LMFDB - Embedded newform 6039.2.a.l.1.11

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:	$21$
Twist minimal:	no (minimal twist has level 671)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
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)

$f(q)$	$=$	$q-0.472572 q^{2} -1.77668 q^{4} +3.20957 q^{5} +4.06194 q^{7} +1.78475 q^{8} +O(q^{10})$	\(q-0.472572 q^{2} -1.77668 q^{4} +3.20957 q^{5} +4.06194 q^{7} +1.78475 q^{8} -1.51675 q^{10} +1.00000 q^{11} -6.29425 q^{13} -1.91956 q^{14} +2.70993 q^{16} -2.28683 q^{17} +4.41630 q^{19} -5.70236 q^{20} -0.472572 q^{22} -4.18874 q^{23} +5.30131 q^{25} +2.97449 q^{26} -7.21675 q^{28} -2.32215 q^{29} +4.52296 q^{31} -4.85014 q^{32} +1.08069 q^{34} +13.0371 q^{35} -8.64218 q^{37} -2.08702 q^{38} +5.72828 q^{40} -4.05541 q^{41} +11.3037 q^{43} -1.77668 q^{44} +1.97948 q^{46} +0.746581 q^{47} +9.49937 q^{49} -2.50525 q^{50} +11.1828 q^{52} +9.63873 q^{53} +3.20957 q^{55} +7.24956 q^{56} +1.09738 q^{58} -2.52363 q^{59} +1.00000 q^{61} -2.13743 q^{62} -3.12782 q^{64} -20.2018 q^{65} +10.5600 q^{67} +4.06296 q^{68} -6.16095 q^{70} +14.4115 q^{71} +5.36468 q^{73} +4.08405 q^{74} -7.84634 q^{76} +4.06194 q^{77} +10.4811 q^{79} +8.69769 q^{80} +1.91647 q^{82} +1.73345 q^{83} -7.33974 q^{85} -5.34181 q^{86} +1.78475 q^{88} +13.7096 q^{89} -25.5669 q^{91} +7.44203 q^{92} -0.352813 q^{94} +14.1744 q^{95} +0.00104411 q^{97} -4.48914 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 21 q + 32 q^{4} - 7 q^{5} + 5 q^{7} + 6 q^{8}+O(q^{10}) $ 21 * q + 32 * q^4 - 7 * q^5 + 5 * q^7 + 6 * q^8	$ 21 q + 32 q^{4} - 7 q^{5} + 5 q^{7} + 6 q^{8} + q^{10} + 21 q^{11} + 20 q^{13} - 17 q^{14} + 50 q^{16} - q^{17} + 15 q^{19} + 2 q^{20} - 11 q^{23} + 48 q^{25} + 5 q^{26} - 16 q^{28} + 9 q^{29} + 22 q^{31} - 3 q^{32} + 33 q^{34} + 39 q^{35} + 21 q^{37} - 11 q^{38} - 16 q^{40} - 7 q^{41} + 16 q^{43} + 32 q^{44} - 3 q^{46} - 5 q^{47} + 80 q^{49} + 33 q^{50} + 60 q^{52} - 9 q^{53} - 7 q^{55} - 44 q^{56} - 27 q^{58} - 13 q^{59} + 21 q^{61} + 23 q^{62} + 66 q^{64} - 25 q^{65} + 38 q^{67} + 74 q^{68} - 33 q^{70} - 12 q^{71} + 20 q^{73} + 12 q^{74} + 59 q^{76} + 5 q^{77} + q^{79} + 38 q^{80} + 7 q^{82} + 19 q^{83} + 38 q^{85} + 3 q^{86} + 6 q^{88} - 37 q^{89} + 24 q^{91} - 31 q^{92} - 64 q^{94} + 43 q^{95} + 68 q^{97} + 2 q^{98}+O(q^{100}) $ 21 * q + 32 * q^4 - 7 * q^5 + 5 * q^7 + 6 * q^8 + q^10 + 21 * q^11 + 20 * q^13 - 17 * q^14 + 50 * q^16 - q^17 + 15 * q^19 + 2 * q^20 - 11 * q^23 + 48 * q^25 + 5 * q^26 - 16 * q^28 + 9 * q^29 + 22 * q^31 - 3 * q^32 + 33 * q^34 + 39 * q^35 + 21 * q^37 - 11 * q^38 - 16 * q^40 - 7 * q^41 + 16 * q^43 + 32 * q^44 - 3 * q^46 - 5 * q^47 + 80 * q^49 + 33 * q^50 + 60 * q^52 - 9 * q^53 - 7 * q^55 - 44 * q^56 - 27 * q^58 - 13 * q^59 + 21 * q^61 + 23 * q^62 + 66 * q^64 - 25 * q^65 + 38 * q^67 + 74 * q^68 - 33 * q^70 - 12 * q^71 + 20 * q^73 + 12 * q^74 + 59 * q^76 + 5 * q^77 + q^79 + 38 * q^80 + 7 * q^82 + 19 * q^83 + 38 * q^85 + 3 * q^86 + 6 * q^88 - 37 * q^89 + 24 * q^91 - 31 * q^92 - 64 * q^94 + 43 * q^95 + 68 * q^97 + 2 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−0.472572	−0.334159	−0.167079	−	0.985943i	$-0.553434\pi$
$2$	−0.472572	−0.334159	−0.167079	+	0.985943i	$0.553434\pi$
$3$	0	0
$3$	0	0
$4$	−1.77668	−0.888338
$5$	3.20957	1.43536	0.717681	−	0.696372i	$-0.245204\pi$
$5$	3.20957	1.43536	0.717681	+	0.696372i	$0.245204\pi$
$6$	0	0
$7$	4.06194	1.53527	0.767635	−	0.640888i	$-0.221434\pi$
$7$	4.06194	1.53527	0.767635	+	0.640888i	$0.221434\pi$
$8$	1.78475	0.631005
$9$	0	0
$10$	−1.51675	−0.479639
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−6.29425	−1.74571	−0.872856	−	0.487978i	$-0.837734\pi$
$13$	−6.29425	−1.74571	−0.872856	+	0.487978i	$0.837734\pi$
$14$	−1.91956	−0.513024
$15$	0	0
$16$	2.70993	0.677482
$17$	−2.28683	−0.554638	−0.277319	−	0.960778i	$-0.589446\pi$
$17$	−2.28683	−0.554638	−0.277319	+	0.960778i	$0.589446\pi$
$18$	0	0
$19$	4.41630	1.01317	0.506585	−	0.862190i	$-0.330908\pi$
$19$	4.41630	1.01317	0.506585	+	0.862190i	$0.330908\pi$
$20$	−5.70236	−1.27509
$21$	0	0
$22$	−0.472572	−0.100753
$23$	−4.18874	−0.873413	−0.436706	−	0.899604i	$-0.643855\pi$
$23$	−4.18874	−0.873413	−0.436706	+	0.899604i	$0.643855\pi$
$24$	0	0
$25$	5.30131	1.06026
$26$	2.97449	0.583345
$27$	0	0
$28$	−7.21675	−1.36384
$29$	−2.32215	−0.431213	−0.215606	−	0.976480i	$-0.569173\pi$
$29$	−2.32215	−0.431213	−0.215606	+	0.976480i	$0.569173\pi$
$30$	0	0
$31$	4.52296	0.812348	0.406174	−	0.913796i	$-0.366863\pi$
$31$	4.52296	0.812348	0.406174	+	0.913796i	$0.366863\pi$
$32$	−4.85014	−0.857392
$33$	0	0
$34$	1.08069	0.185337
$35$	13.0371	2.20367
$36$	0	0
$37$	−8.64218	−1.42077	−0.710383	−	0.703816i	$-0.751478\pi$
$37$	−8.64218	−1.42077	−0.710383	+	0.703816i	$0.751478\pi$
$38$	−2.08702	−0.338560
$39$	0	0
$40$	5.72828	0.905720
$41$	−4.05541	−0.633349	−0.316674	−	0.948534i	$-0.602566\pi$
$41$	−4.05541	−0.633349	−0.316674	+	0.948534i	$0.602566\pi$
$42$	0	0
$43$	11.3037	1.72380	0.861898	−	0.507081i	$-0.169276\pi$
$43$	11.3037	1.72380	0.861898	+	0.507081i	$0.169276\pi$
$44$	−1.77668	−0.267844
$45$	0	0
$46$	1.97948	0.291859
$47$	0.746581	0.108900	0.0544500	−	0.998516i	$-0.482659\pi$
$47$	0.746581	0.108900	0.0544500	+	0.998516i	$0.482659\pi$
$48$	0	0
$49$	9.49937	1.35705
$50$	−2.50525	−0.354296
$51$	0	0
$52$	11.1828	1.55078
$53$	9.63873	1.32398	0.661991	−	0.749512i	$-0.269712\pi$
$53$	9.63873	1.32398	0.661991	+	0.749512i	$0.269712\pi$
$54$	0	0
$55$	3.20957	0.432778
$56$	7.24956	0.968763
$57$	0	0
$58$	1.09738	0.144094
$59$	−2.52363	−0.328549	−0.164275	−	0.986415i	$-0.552528\pi$
$59$	−2.52363	−0.328549	−0.164275	+	0.986415i	$0.552528\pi$
$60$	0	0
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	−2.13743	−0.271453
$63$	0	0
$64$	−3.12782	−0.390977
$65$	−20.2018	−2.50573
$66$	0	0
$67$	10.5600	1.29011	0.645056	−	0.764135i	$-0.276834\pi$
$67$	10.5600	1.29011	0.645056	+	0.764135i	$0.276834\pi$
$68$	4.06296	0.492706
$69$	0	0
$70$	−6.16095	−0.736375
$71$	14.4115	1.71033	0.855167	−	0.518353i	$-0.173455\pi$
$71$	14.4115	1.71033	0.855167	+	0.518353i	$0.173455\pi$
$72$	0	0
$73$	5.36468	0.627888	0.313944	−	0.949442i	$-0.398350\pi$
$73$	5.36468	0.627888	0.313944	+	0.949442i	$0.398350\pi$
$74$	4.08405	0.474761
$75$	0	0
$76$	−7.84634	−0.900037
$77$	4.06194	0.462901
$78$	0	0
$79$	10.4811	1.17921	0.589606	−	0.807691i	$-0.299283\pi$
$79$	10.4811	1.17921	0.589606	+	0.807691i	$0.299283\pi$
$80$	8.69769	0.972431
$81$	0	0
$82$	1.91647	0.211639
$83$	1.73345	0.190271	0.0951355	−	0.995464i	$-0.469672\pi$
$83$	1.73345	0.190271	0.0951355	+	0.995464i	$0.469672\pi$
$84$	0	0
$85$	−7.33974	−0.796106
$86$	−5.34181	−0.576022
$87$	0	0
$88$	1.78475	0.190255
$89$	13.7096	1.45321	0.726607	−	0.687053i	$-0.241096\pi$
$89$	13.7096	1.45321	0.726607	+	0.687053i	$0.241096\pi$
$90$	0	0
$91$	−25.5669	−2.68014
$92$	7.44203	0.775886
$93$	0	0
$94$	−0.352813	−0.0363899
$95$	14.1744	1.45426
$96$	0	0
$97$	0.00104411	0.000106013 0	5.30065e−5	−	1.00000i	$-0.499983\pi$
$97$	0.00104411	0.000106013 0	5.30065e−5		1.00000i	$0.499983\pi$
$98$	−4.48914	−0.453471
$99$	0	0
$100$	−9.41870	−0.941870
$101$	15.3172	1.52412	0.762059	−	0.647508i	$-0.224189\pi$
$101$	15.3172	1.52412	0.762059	+	0.647508i	$0.224189\pi$
$102$	0	0
$103$	−6.50221	−0.640682	−0.320341	−	0.947302i	$-0.603797\pi$
$103$	−6.50221	−0.640682	−0.320341	+	0.947302i	$0.603797\pi$
$104$	−11.2337	−1.10155
$105$	0	0
$106$	−4.55499	−0.442420
$107$	−8.21543	−0.794216	−0.397108	−	0.917772i	$-0.629986\pi$
$107$	−8.21543	−0.794216	−0.397108	+	0.917772i	$0.629986\pi$
$108$	0	0
$109$	12.7951	1.22555	0.612775	−	0.790258i	$-0.290053\pi$
$109$	12.7951	1.22555	0.612775	+	0.790258i	$0.290053\pi$
$110$	−1.51675	−0.144617
$111$	0	0
$112$	11.0076	1.04012
$113$	−13.0118	−1.22405	−0.612023	−	0.790840i	$-0.709644\pi$
$113$	−13.0118	−1.22405	−0.612023	+	0.790840i	$0.709644\pi$
$114$	0	0
$115$	−13.4440	−1.25366
$116$	4.12571	0.383063
$117$	0	0
$118$	1.19260	0.109788
$119$	−9.28898	−0.851519
$120$	0	0
$121$	1.00000	0.0909091
$122$	−0.472572	−0.0427847
$123$	0	0
$124$	−8.03584	−0.721640
$125$	0.967067	0.0864971
$126$	0	0
$127$	20.7171	1.83834	0.919172	−	0.393857i	$-0.128859\pi$
$127$	20.7171	1.83834	0.919172	+	0.393857i	$0.128859\pi$
$128$	11.1784	0.988040
$129$	0	0
$130$	9.54681	0.837311
$131$	1.48312	0.129581	0.0647905	−	0.997899i	$-0.479362\pi$
$131$	1.48312	0.129581	0.0647905	+	0.997899i	$0.479362\pi$
$132$	0	0
$133$	17.9388	1.55549
$134$	−4.99037	−0.431103
$135$	0	0
$136$	−4.08143	−0.349979
$137$	−13.6041	−1.16227	−0.581136	−	0.813806i	$-0.697392\pi$
$137$	−13.6041	−1.16227	−0.581136	+	0.813806i	$0.697392\pi$
$138$	0	0
$139$	−1.83421	−0.155575	−0.0777877	−	0.996970i	$-0.524786\pi$
$139$	−1.83421	−0.155575	−0.0777877	+	0.996970i	$0.524786\pi$
$140$	−23.1626	−1.95760
$141$	0	0
$142$	−6.81049	−0.571523
$143$	−6.29425	−0.526352
$144$	0	0
$145$	−7.45310	−0.618946
$146$	−2.53520	−0.209814
$147$	0	0
$148$	15.3543	1.26212
$149$	−20.6171	−1.68902	−0.844509	−	0.535542i	$-0.820107\pi$
$149$	−20.6171	−1.68902	−0.844509	+	0.535542i	$0.820107\pi$
$150$	0	0
$151$	13.2251	1.07624	0.538121	−	0.842867i	$-0.319134\pi$
$151$	13.2251	1.07624	0.538121	+	0.842867i	$0.319134\pi$
$152$	7.88200	0.639315
$153$	0	0
$154$	−1.91956	−0.154683
$155$	14.5167	1.16601
$156$	0	0
$157$	−23.4605	−1.87235	−0.936175	−	0.351534i	$-0.885660\pi$
$157$	−23.4605	−1.87235	−0.936175	+	0.351534i	$0.885660\pi$
$158$	−4.95306	−0.394044
$159$	0	0
$160$	−15.5668	−1.23067
$161$	−17.0144	−1.34092
$162$	0	0
$163$	5.47984	0.429214	0.214607	−	0.976700i	$-0.431153\pi$
$163$	5.47984	0.429214	0.214607	+	0.976700i	$0.431153\pi$
$164$	7.20515	0.562628
$165$	0	0
$166$	−0.819181	−0.0635808
$167$	7.23804	0.560096	0.280048	−	0.959986i	$-0.409650\pi$
$167$	7.23804	0.560096	0.280048	+	0.959986i	$0.409650\pi$
$168$	0	0
$169$	26.6176	2.04751
$170$	3.46855	0.266026
$171$	0	0
$172$	−20.0830	−1.53131
$173$	−5.48190	−0.416781	−0.208390	−	0.978046i	$-0.566822\pi$
$173$	−5.48190	−0.416781	−0.208390	+	0.978046i	$0.566822\pi$
$174$	0	0
$175$	21.5336	1.62779
$176$	2.70993	0.204268
$177$	0	0
$178$	−6.47877	−0.485605
$179$	13.5770	1.01479	0.507395	−	0.861713i	$-0.330608\pi$
$179$	13.5770	1.01479	0.507395	+	0.861713i	$0.330608\pi$
$180$	0	0
$181$	21.3974	1.59046	0.795230	−	0.606308i	$-0.207350\pi$
$181$	21.3974	1.59046	0.795230	+	0.606308i	$0.207350\pi$
$182$	12.0822	0.895592
$183$	0	0
$184$	−7.47586	−0.551128
$185$	−27.7376	−2.03931
$186$	0	0
$187$	−2.28683	−0.167230
$188$	−1.32643	−0.0967400
$189$	0	0
$190$	−6.69843	−0.485955
$191$	6.13364	0.443815	0.221907	−	0.975068i	$-0.428772\pi$
$191$	6.13364	0.443815	0.221907	+	0.975068i	$0.428772\pi$
$192$	0	0
$193$	−15.0531	−1.08355	−0.541774	−	0.840524i	$-0.682247\pi$
$193$	−15.0531	−1.08355	−0.541774	+	0.840524i	$0.682247\pi$
$194$	−0.000493416 0	−3.54252e−5 0
$195$	0	0
$196$	−16.8773	−1.20552
$197$	−8.07480	−0.575306	−0.287653	−	0.957735i	$-0.592875\pi$
$197$	−8.07480	−0.575306	−0.287653	+	0.957735i	$0.592875\pi$
$198$	0	0
$199$	7.43252	0.526877	0.263439	−	0.964676i	$-0.415143\pi$
$199$	7.43252	0.526877	0.263439	+	0.964676i	$0.415143\pi$
$200$	9.46152	0.669030
$201$	0	0
$202$	−7.23848	−0.509298
$203$	−9.43245	−0.662028
$204$	0	0
$205$	−13.0161	−0.909084
$206$	3.07276	0.214089
$207$	0	0
$208$	−17.0570	−1.18269
$209$	4.41630	0.305482
$210$	0	0
$211$	26.4730	1.82247	0.911237	−	0.411883i	$-0.135129\pi$
$211$	26.4730	1.82247	0.911237	+	0.411883i	$0.135129\pi$
$212$	−17.1249	−1.17614
$213$	0	0
$214$	3.88238	0.265394
$215$	36.2799	2.47427
$216$	0	0
$217$	18.3720	1.24717
$218$	−6.04661	−0.409528
$219$	0	0
$220$	−5.70236	−0.384453
$221$	14.3939	0.968238
$222$	0	0
$223$	−2.84013	−0.190189	−0.0950946	−	0.995468i	$-0.530315\pi$
$223$	−2.84013	−0.190189	−0.0950946	+	0.995468i	$0.530315\pi$
$224$	−19.7010	−1.31633
$225$	0	0
$226$	6.14901	0.409026
$227$	−12.8755	−0.854576	−0.427288	−	0.904115i	$-0.640531\pi$
$227$	−12.8755	−0.854576	−0.427288	+	0.904115i	$0.640531\pi$
$228$	0	0
$229$	−11.1339	−0.735749	−0.367874	−	0.929876i	$-0.619914\pi$
$229$	−11.1339	−0.735749	−0.367874	+	0.929876i	$0.619914\pi$
$230$	6.35327	0.418923
$231$	0	0
$232$	−4.14446	−0.272097
$233$	−17.1736	−1.12508	−0.562539	−	0.826771i	$-0.690175\pi$
$233$	−17.1736	−1.12508	−0.562539	+	0.826771i	$0.690175\pi$
$234$	0	0
$235$	2.39620	0.156311
$236$	4.48368	0.291863
$237$	0	0
$238$	4.38971	0.284543
$239$	−10.9538	−0.708545	−0.354273	−	0.935142i	$-0.615272\pi$
$239$	−10.9538	−0.708545	−0.354273	+	0.935142i	$0.615272\pi$
$240$	0	0
$241$	7.88182	0.507713	0.253856	−	0.967242i	$-0.418301\pi$
$241$	7.88182	0.507713	0.253856	+	0.967242i	$0.418301\pi$
$242$	−0.472572	−0.0303781
$243$	0	0
$244$	−1.77668	−0.113740
$245$	30.4888	1.94786
$246$	0	0
$247$	−27.7973	−1.76870
$248$	8.07237	0.512596
$249$	0	0
$250$	−0.457009	−0.0289038
$251$	−1.64472	−0.103814	−0.0519068	−	0.998652i	$-0.516530\pi$
$251$	−1.64472	−0.103814	−0.0519068	+	0.998652i	$0.516530\pi$
$252$	0	0
$253$	−4.18874	−0.263344
$254$	−9.79031	−0.614299
$255$	0	0
$256$	0.973033	0.0608146
$257$	14.2756	0.890486	0.445243	−	0.895410i	$-0.353117\pi$
$257$	14.2756	0.890486	0.445243	+	0.895410i	$0.353117\pi$
$258$	0	0
$259$	−35.1040	−2.18126
$260$	35.8921	2.22593
$261$	0	0
$262$	−0.700882	−0.0433007
$263$	12.4281	0.766352	0.383176	−	0.923675i	$-0.374830\pi$
$263$	12.4281	0.766352	0.383176	+	0.923675i	$0.374830\pi$
$264$	0	0
$265$	30.9361	1.90039
$266$	−8.47736	−0.519780
$267$	0	0
$268$	−18.7617	−1.14606
$269$	−22.2727	−1.35799	−0.678996	−	0.734142i	$-0.737585\pi$
$269$	−22.2727	−1.35799	−0.678996	+	0.734142i	$0.737585\pi$
$270$	0	0
$271$	3.83646	0.233048	0.116524	−	0.993188i	$-0.462825\pi$
$271$	3.83646	0.233048	0.116524	+	0.993188i	$0.462825\pi$
$272$	−6.19715	−0.375757
$273$	0	0
$274$	6.42889	0.388384
$275$	5.30131	0.319681
$276$	0	0
$277$	−16.9872	−1.02066	−0.510331	−	0.859978i	$-0.670477\pi$
$277$	−16.9872	−1.02066	−0.510331	+	0.859978i	$0.670477\pi$
$278$	0.866794	0.0519869
$279$	0	0
$280$	23.2679	1.39052
$281$	−17.7829	−1.06084	−0.530418	−	0.847736i	$-0.677965\pi$
$281$	−17.7829	−1.06084	−0.530418	+	0.847736i	$0.677965\pi$
$282$	0	0
$283$	−10.1655	−0.604277	−0.302138	−	0.953264i	$-0.597700\pi$
$283$	−10.1655	−0.604277	−0.302138	+	0.953264i	$0.597700\pi$
$284$	−25.6046	−1.51935
$285$	0	0
$286$	2.97449	0.175885
$287$	−16.4728	−0.972361
$288$	0	0
$289$	−11.7704	−0.692376
$290$	3.52213	0.206826
$291$	0	0
$292$	−9.53129	−0.557776
$293$	−16.0638	−0.938458	−0.469229	−	0.883077i	$-0.655468\pi$
$293$	−16.0638	−0.938458	−0.469229	+	0.883077i	$0.655468\pi$
$294$	0	0
$295$	−8.09976	−0.471586
$296$	−15.4241	−0.896510
$297$	0	0
$298$	9.74306	0.564400
$299$	26.3650	1.52473
$300$	0	0
$301$	45.9149	2.64649
$302$	−6.24981	−0.359636
$303$	0	0
$304$	11.9679	0.686404
$305$	3.20957	0.183779
$306$	0	0
$307$	20.5144	1.17082	0.585409	−	0.810738i	$-0.300934\pi$
$307$	20.5144	1.17082	0.585409	+	0.810738i	$0.300934\pi$
$308$	−7.21675	−0.411213
$309$	0	0
$310$	−6.86021	−0.389634
$311$	−7.38695	−0.418876	−0.209438	−	0.977822i	$-0.567163\pi$
$311$	−7.38695	−0.418876	−0.209438	+	0.977822i	$0.567163\pi$
$312$	0	0
$313$	−18.4731	−1.04416	−0.522081	−	0.852896i	$-0.674844\pi$
$313$	−18.4731	−1.04416	−0.522081	+	0.852896i	$0.674844\pi$
$314$	11.0868	0.625662
$315$	0	0
$316$	−18.6214	−1.04754
$317$	27.1083	1.52256	0.761278	−	0.648426i	$-0.224572\pi$
$317$	27.1083	1.52256	0.761278	+	0.648426i	$0.224572\pi$
$318$	0	0
$319$	−2.32215	−0.130016
$320$	−10.0389	−0.561193
$321$	0	0
$322$	8.04054	0.448082
$323$	−10.0993	−0.561943
$324$	0	0
$325$	−33.3678	−1.85091
$326$	−2.58962	−0.143426
$327$	0	0
$328$	−7.23790	−0.399646
$329$	3.03257	0.167191
$330$	0	0
$331$	23.6583	1.30038	0.650188	−	0.759773i	$-0.274690\pi$
$331$	23.6583	1.30038	0.650188	+	0.759773i	$0.274690\pi$
$332$	−3.07978	−0.169025
$333$	0	0
$334$	−3.42049	−0.187161
$335$	33.8931	1.85178
$336$	0	0
$337$	28.0015	1.52534	0.762670	−	0.646788i	$-0.223888\pi$
$337$	28.0015	1.52534	0.762670	+	0.646788i	$0.223888\pi$
$338$	−12.5787	−0.684193
$339$	0	0
$340$	13.0403	0.707211
$341$	4.52296	0.244932
$342$	0	0
$343$	10.1523	0.548172
$344$	20.1743	1.08772
$345$	0	0
$346$	2.59059	0.139271
$347$	16.4975	0.885634	0.442817	−	0.896612i	$-0.353979\pi$
$347$	16.4975	0.885634	0.442817	+	0.896612i	$0.353979\pi$
$348$	0	0
$349$	−1.46317	−0.0783219	−0.0391610	−	0.999233i	$-0.512469\pi$
$349$	−1.46317	−0.0783219	−0.0391610	+	0.999233i	$0.512469\pi$
$350$	−10.1762	−0.543940
$351$	0	0
$352$	−4.85014	−0.258513
$353$	34.4299	1.83252	0.916259	−	0.400587i	$-0.131194\pi$
$353$	34.4299	1.83252	0.916259	+	0.400587i	$0.131194\pi$
$354$	0	0
$355$	46.2547	2.45495
$356$	−24.3575	−1.29095
$357$	0	0
$358$	−6.41610	−0.339101
$359$	19.9722	1.05409	0.527045	−	0.849837i	$-0.323300\pi$
$359$	19.9722	1.05409	0.527045	+	0.849837i	$0.323300\pi$
$360$	0	0
$361$	0.503742	0.0265127
$362$	−10.1118	−0.531466
$363$	0	0
$364$	45.4241	2.38087
$365$	17.2183	0.901246
$366$	0	0
$367$	9.55281	0.498653	0.249326	−	0.968420i	$-0.419791\pi$
$367$	9.55281	0.498653	0.249326	+	0.968420i	$0.419791\pi$
$368$	−11.3512	−0.591721
$369$	0	0
$370$	13.1080	0.681454
$371$	39.1520	2.03267
$372$	0	0
$373$	−6.77266	−0.350675	−0.175337	−	0.984508i	$-0.556102\pi$
$373$	−6.77266	−0.350675	−0.175337	+	0.984508i	$0.556102\pi$
$374$	1.08069	0.0558813
$375$	0	0
$376$	1.33246	0.0687164
$377$	14.6162	0.752773
$378$	0	0
$379$	0.317739	0.0163212	0.00816058	−	0.999967i	$-0.497402\pi$
$379$	0.317739	0.0163212	0.00816058	+	0.999967i	$0.497402\pi$
$380$	−25.1833	−1.29188
$381$	0	0
$382$	−2.89859	−0.148305
$383$	−24.0805	−1.23045	−0.615227	−	0.788350i	$-0.710936\pi$
$383$	−24.0805	−1.23045	−0.615227	+	0.788350i	$0.710936\pi$
$384$	0	0
$385$	13.0371	0.664430
$386$	7.11369	0.362078
$387$	0	0
$388$	−0.00185504	−9.41754e−5 0
$389$	2.26775	0.114980	0.0574898	−	0.998346i	$-0.481690\pi$
$389$	2.26775	0.114980	0.0574898	+	0.998346i	$0.481690\pi$
$390$	0	0
$391$	9.57895	0.484428
$392$	16.9540	0.856307
$393$	0	0
$394$	3.81593	0.192244
$395$	33.6396	1.69259
$396$	0	0
$397$	8.47515	0.425355	0.212678	−	0.977122i	$-0.431782\pi$
$397$	8.47515	0.425355	0.212678	+	0.977122i	$0.431782\pi$
$398$	−3.51240	−0.176061
$399$	0	0
$400$	14.3662	0.718308
$401$	1.69131	0.0844600	0.0422300	−	0.999108i	$-0.486554\pi$
$401$	1.69131	0.0844600	0.0422300	+	0.999108i	$0.486554\pi$
$402$	0	0
$403$	−28.4687	−1.41813
$404$	−27.2137	−1.35393
$405$	0	0
$406$	4.45751	0.221223
$407$	−8.64218	−0.428377
$408$	0	0
$409$	2.03412	0.100581	0.0502903	−	0.998735i	$-0.483985\pi$
$409$	2.03412	0.100581	0.0502903	+	0.998735i	$0.483985\pi$
$410$	6.15105	0.303779
$411$	0	0
$412$	11.5523	0.569142
$413$	−10.2508	−0.504411
$414$	0	0
$415$	5.56362	0.273108
$416$	30.5280	1.49676
$417$	0	0
$418$	−2.08702	−0.102080
$419$	17.6278	0.861174	0.430587	−	0.902549i	$-0.358307\pi$
$419$	17.6278	0.861174	0.430587	+	0.902549i	$0.358307\pi$
$420$	0	0
$421$	2.65032	0.129169	0.0645844	−	0.997912i	$-0.479428\pi$
$421$	2.65032	0.129169	0.0645844	+	0.997912i	$0.479428\pi$
$422$	−12.5104	−0.608996
$423$	0	0
$424$	17.2027	0.835439
$425$	−12.1232	−0.588062
$426$	0	0
$427$	4.06194	0.196571
$428$	14.5962	0.705532
$429$	0	0
$430$	−17.1449	−0.826800
$431$	27.8920	1.34351	0.671756	−	0.740773i	$-0.265541\pi$
$431$	27.8920	1.34351	0.671756	+	0.740773i	$0.265541\pi$
$432$	0	0
$433$	−24.3263	−1.16905	−0.584523	−	0.811377i	$-0.698718\pi$
$433$	−24.3263	−1.16905	−0.584523	+	0.811377i	$0.698718\pi$
$434$	−8.68210	−0.416754
$435$	0	0
$436$	−22.7328	−1.08870
$437$	−18.4988	−0.884915
$438$	0	0
$439$	20.9152	0.998226	0.499113	−	0.866537i	$-0.333659\pi$
$439$	20.9152	0.998226	0.499113	+	0.866537i	$0.333659\pi$
$440$	5.72828	0.273085
$441$	0	0
$442$	−6.80215	−0.323545
$443$	−28.9601	−1.37593	−0.687967	−	0.725742i	$-0.741497\pi$
$443$	−28.9601	−1.37593	−0.687967	+	0.725742i	$0.741497\pi$
$444$	0	0
$445$	44.0018	2.08589
$446$	1.34217	0.0635534
$447$	0	0
$448$	−12.7050	−0.600255
$449$	−23.4527	−1.10680	−0.553401	−	0.832915i	$-0.686670\pi$
$449$	−23.4527	−1.10680	−0.553401	+	0.832915i	$0.686670\pi$
$450$	0	0
$451$	−4.05541	−0.190962
$452$	23.1177	1.08737
$453$	0	0
$454$	6.08460	0.285564
$455$	−82.0586	−3.84697
$456$	0	0
$457$	21.3664	0.999476	0.499738	−	0.866177i	$-0.333430\pi$
$457$	21.3664	0.999476	0.499738	+	0.866177i	$0.333430\pi$
$458$	5.26157	0.245857
$459$	0	0
$460$	23.8857	1.11368
$461$	−28.7391	−1.33851	−0.669257	−	0.743031i	$-0.733388\pi$
$461$	−28.7391	−1.33851	−0.669257	+	0.743031i	$0.733388\pi$
$462$	0	0
$463$	4.67274	0.217161	0.108580	−	0.994088i	$-0.465370\pi$
$463$	4.67274	0.217161	0.108580	+	0.994088i	$0.465370\pi$
$464$	−6.29286	−0.292139
$465$	0	0
$466$	8.11575	0.375955
$467$	31.9660	1.47921	0.739605	−	0.673041i	$-0.235012\pi$
$467$	31.9660	1.47921	0.739605	+	0.673041i	$0.235012\pi$
$468$	0	0
$469$	42.8942	1.98067
$470$	−1.13238	−0.0522327
$471$	0	0
$472$	−4.50406	−0.207316
$473$	11.3037	0.519744
$474$	0	0
$475$	23.4122	1.07422
$476$	16.5035	0.756437
$477$	0	0
$478$	5.17648	0.236767
$479$	−28.9406	−1.32233	−0.661165	−	0.750241i	$-0.729938\pi$
$479$	−28.9406	−1.32233	−0.661165	+	0.750241i	$0.729938\pi$
$480$	0	0
$481$	54.3960	2.48025
$482$	−3.72473	−0.169657
$483$	0	0
$484$	−1.77668	−0.0807580
$485$	0.00335113	0.000152167 0
$486$	0	0
$487$	−16.6396	−0.754014	−0.377007	−	0.926210i	$-0.623047\pi$
$487$	−16.6396	−0.754014	−0.377007	+	0.926210i	$0.623047\pi$
$488$	1.78475	0.0807919
$489$	0	0
$490$	−14.4082	−0.650895
$491$	0.638708	0.0288245	0.0144122	−	0.999896i	$-0.495412\pi$
$491$	0.638708	0.0288245	0.0144122	+	0.999896i	$0.495412\pi$
$492$	0	0
$493$	5.31037	0.239167
$494$	13.1362	0.591027
$495$	0	0
$496$	12.2569	0.550351
$497$	58.5388	2.62582
$498$	0	0
$499$	−15.9228	−0.712803	−0.356401	−	0.934333i	$-0.615996\pi$
$499$	−15.9228	−0.712803	−0.356401	+	0.934333i	$0.615996\pi$
$500$	−1.71816	−0.0768386
$501$	0	0
$502$	0.777248	0.0346903
$503$	10.8102	0.482001	0.241001	−	0.970525i	$-0.422524\pi$
$503$	10.8102	0.482001	0.241001	+	0.970525i	$0.422524\pi$
$504$	0	0
$505$	49.1615	2.18766
$506$	1.97948	0.0879987
$507$	0	0
$508$	−36.8075	−1.63307
$509$	26.4197	1.17103	0.585516	−	0.810661i	$-0.300892\pi$
$509$	26.4197	1.17103	0.585516	+	0.810661i	$0.300892\pi$
$510$	0	0
$511$	21.7910	0.963977
$512$	−22.8166	−1.00836
$513$	0	0
$514$	−6.74624	−0.297564
$515$	−20.8693	−0.919610
$516$	0	0
$517$	0.746581	0.0328346
$518$	16.5892	0.728887
$519$	0	0
$520$	−36.0552	−1.58113
$521$	4.91123	0.215165	0.107583	−	0.994196i	$-0.465689\pi$
$521$	4.91123	0.215165	0.107583	+	0.994196i	$0.465689\pi$
$522$	0	0
$523$	−0.691542	−0.0302390	−0.0151195	−	0.999886i	$-0.504813\pi$
$523$	−0.691542	−0.0302390	−0.0151195	+	0.999886i	$0.504813\pi$
$524$	−2.63503	−0.115112
$525$	0	0
$526$	−5.87319	−0.256083
$527$	−10.3433	−0.450559
$528$	0	0
$529$	−5.45445	−0.237150
$530$	−14.6195	−0.635033
$531$	0	0
$532$	−31.8714	−1.38180
$533$	25.5258	1.10564
$534$	0	0
$535$	−26.3680	−1.13999
$536$	18.8470	0.814067
$537$	0	0
$538$	10.5255	0.453785
$539$	9.49937	0.409167
$540$	0	0
$541$	29.4372	1.26560	0.632801	−	0.774314i	$-0.281905\pi$
$541$	29.4372	1.26560	0.632801	+	0.774314i	$0.281905\pi$
$542$	−1.81300	−0.0778751
$543$	0	0
$544$	11.0915	0.475542
$545$	41.0668	1.75911
$546$	0	0
$547$	−12.2980	−0.525826	−0.262913	−	0.964820i	$-0.584683\pi$
$547$	−12.2980	−0.525826	−0.262913	+	0.964820i	$0.584683\pi$
$548$	24.1700	1.03249
$549$	0	0
$550$	−2.50525	−0.106824
$551$	−10.2553	−0.436892
$552$	0	0
$553$	42.5735	1.81041
$554$	8.02767	0.341063
$555$	0	0
$556$	3.25879	0.138203
$557$	0.790508	0.0334949	0.0167474	−	0.999860i	$-0.494669\pi$
$557$	0.790508	0.0334949	0.0167474	+	0.999860i	$0.494669\pi$
$558$	0	0
$559$	−71.1483	−3.00925
$560$	35.3295	1.49294
$561$	0	0
$562$	8.40369	0.354488
$563$	−13.6547	−0.575479	−0.287739	−	0.957709i	$-0.592904\pi$
$563$	−13.6547	−0.575479	−0.287739	+	0.957709i	$0.592904\pi$
$564$	0	0
$565$	−41.7622	−1.75695
$566$	4.80393	0.201924
$567$	0	0
$568$	25.7210	1.07923
$569$	−4.57231	−0.191681	−0.0958406	−	0.995397i	$-0.530554\pi$
$569$	−4.57231	−0.191681	−0.0958406	+	0.995397i	$0.530554\pi$
$570$	0	0
$571$	0.856002	0.0358226	0.0179113	−	0.999840i	$-0.494298\pi$
$571$	0.856002	0.0358226	0.0179113	+	0.999840i	$0.494298\pi$
$572$	11.1828	0.467578
$573$	0	0
$574$	7.78460	0.324923
$575$	−22.2058	−0.926046
$576$	0	0
$577$	−5.43867	−0.226415	−0.113207	−	0.993571i	$-0.536112\pi$
$577$	−5.43867	−0.226415	−0.113207	+	0.993571i	$0.536112\pi$
$578$	5.56236	0.231364
$579$	0	0
$580$	13.2417	0.549833
$581$	7.04118	0.292117
$582$	0	0
$583$	9.63873	0.399195
$584$	9.57461	0.396200
$585$	0	0
$586$	7.59131	0.313594
$587$	−8.73008	−0.360329	−0.180165	−	0.983636i	$-0.557663\pi$
$587$	−8.73008	−0.360329	−0.180165	+	0.983636i	$0.557663\pi$
$588$	0	0
$589$	19.9748	0.823047
$590$	3.82772	0.157585
$591$	0	0
$592$	−23.4197	−0.962543
$593$	17.6920	0.726522	0.363261	−	0.931687i	$-0.381663\pi$
$593$	17.6920	0.726522	0.363261	+	0.931687i	$0.381663\pi$
$594$	0	0
$595$	−29.8136	−1.22224
$596$	36.6299	1.50042
$597$	0	0
$598$	−12.4594	−0.509501
$599$	−41.7205	−1.70465	−0.852327	−	0.523009i	$-0.824810\pi$
$599$	−41.7205	−1.70465	−0.852327	+	0.523009i	$0.824810\pi$
$600$	0	0
$601$	29.0422	1.18466	0.592328	−	0.805697i	$-0.298209\pi$
$601$	29.0422	1.18466	0.592328	+	0.805697i	$0.298209\pi$
$602$	−21.6981	−0.884349
$603$	0	0
$604$	−23.4967	−0.956067
$605$	3.20957	0.130487
$606$	0	0
$607$	−10.3868	−0.421589	−0.210795	−	0.977530i	$-0.567605\pi$
$607$	−10.3868	−0.421589	−0.210795	+	0.977530i	$0.567605\pi$
$608$	−21.4197	−0.868683
$609$	0	0
$610$	−1.51675	−0.0614114
$611$	−4.69917	−0.190108
$612$	0	0
$613$	−13.7211	−0.554191	−0.277095	−	0.960842i	$-0.589372\pi$
$613$	−13.7211	−0.554191	−0.277095	+	0.960842i	$0.589372\pi$
$614$	−9.69452	−0.391239
$615$	0	0
$616$	7.24956	0.292093
$617$	27.1311	1.09226	0.546129	−	0.837701i	$-0.316101\pi$
$617$	27.1311	1.09226	0.546129	+	0.837701i	$0.316101\pi$
$618$	0	0
$619$	17.0390	0.684854	0.342427	−	0.939544i	$-0.388751\pi$
$619$	17.0390	0.684854	0.342427	+	0.939544i	$0.388751\pi$
$620$	−25.7916	−1.03581
$621$	0	0
$622$	3.49087	0.139971
$623$	55.6876	2.23108
$624$	0	0
$625$	−23.4027	−0.936107
$626$	8.72988	0.348916
$627$	0	0
$628$	41.6817	1.66328
$629$	19.7632	0.788011
$630$	0	0
$631$	−4.70714	−0.187388	−0.0936942	−	0.995601i	$-0.529868\pi$
$631$	−4.70714	−0.187388	−0.0936942	+	0.995601i	$0.529868\pi$
$632$	18.7061	0.744088
$633$	0	0
$634$	−12.8106	−0.508776
$635$	66.4928	2.63869
$636$	0	0
$637$	−59.7914	−2.36902
$638$	1.09738	0.0434459
$639$	0	0
$640$	35.8778	1.41819
$641$	−29.1072	−1.14967	−0.574833	−	0.818270i	$-0.694933\pi$
$641$	−29.1072	−1.14967	−0.574833	+	0.818270i	$0.694933\pi$
$642$	0	0
$643$	6.29208	0.248135	0.124068	−	0.992274i	$-0.460406\pi$
$643$	6.29208	0.248135	0.124068	+	0.992274i	$0.460406\pi$
$644$	30.2291	1.19119
$645$	0	0
$646$	4.77267	0.187778
$647$	1.36523	0.0536729	0.0268365	−	0.999640i	$-0.491457\pi$
$647$	1.36523	0.0536729	0.0268365	+	0.999640i	$0.491457\pi$
$648$	0	0
$649$	−2.52363	−0.0990613
$650$	15.7687	0.618498
$651$	0	0
$652$	−9.73589	−0.381287
$653$	28.7882	1.12657	0.563285	−	0.826263i	$-0.309537\pi$
$653$	28.7882	1.12657	0.563285	+	0.826263i	$0.309537\pi$
$654$	0	0
$655$	4.76018	0.185996
$656$	−10.9899	−0.429082
$657$	0	0
$658$	−1.43311	−0.0558683
$659$	−37.0741	−1.44420	−0.722101	−	0.691788i	$-0.756823\pi$
$659$	−37.0741	−1.44420	−0.722101	+	0.691788i	$0.756823\pi$
$660$	0	0
$661$	−14.6348	−0.569227	−0.284614	−	0.958642i	$-0.591865\pi$
$661$	−14.6348	−0.569227	−0.284614	+	0.958642i	$0.591865\pi$
$662$	−11.1802	−0.434532
$663$	0	0
$664$	3.09378	0.120062
$665$	57.5756	2.23269
$666$	0	0
$667$	9.72689	0.376627
$668$	−12.8596	−0.497555
$669$	0	0
$670$	−16.0169	−0.618788
$671$	1.00000	0.0386046
$672$	0	0
$673$	−32.0014	−1.23356	−0.616782	−	0.787134i	$-0.711564\pi$
$673$	−32.0014	−1.23356	−0.616782	+	0.787134i	$0.711564\pi$
$674$	−13.2327	−0.509706
$675$	0	0
$676$	−47.2909	−1.81888
$677$	−11.1250	−0.427569	−0.213785	−	0.976881i	$-0.568579\pi$
$677$	−11.1250	−0.427569	−0.213785	+	0.976881i	$0.568579\pi$
$678$	0	0
$679$	0.00424110	0.000162759 0
$680$	−13.0996	−0.502347
$681$	0	0
$682$	−2.13743	−0.0818463
$683$	−10.0372	−0.384063	−0.192031	−	0.981389i	$-0.561508\pi$
$683$	−10.0372	−0.384063	−0.192031	+	0.981389i	$0.561508\pi$
$684$	0	0
$685$	−43.6631	−1.66828
$686$	−4.79769	−0.183177
$687$	0	0
$688$	30.6322	1.16784
$689$	−60.6686	−2.31129
$690$	0	0
$691$	−47.0727	−1.79073	−0.895365	−	0.445332i	$-0.853086\pi$
$691$	−47.0727	−1.79073	−0.895365	+	0.445332i	$0.853086\pi$
$692$	9.73955	0.370242
$693$	0	0
$694$	−7.79627	−0.295942
$695$	−5.88700	−0.223307
$696$	0	0
$697$	9.27404	0.351279
$698$	0.691455	0.0261720
$699$	0	0
$700$	−38.2582	−1.44602
$701$	−3.00915	−0.113654	−0.0568271	−	0.998384i	$-0.518098\pi$
$701$	−3.00915	−0.113654	−0.0568271	+	0.998384i	$0.518098\pi$
$702$	0	0
$703$	−38.1665	−1.43948
$704$	−3.12782	−0.117884
$705$	0	0
$706$	−16.2706	−0.612352
$707$	62.2176	2.33993
$708$	0	0
$709$	−5.46583	−0.205274	−0.102637	−	0.994719i	$-0.532728\pi$
$709$	−5.46583	−0.205274	−0.102637	+	0.994719i	$0.532728\pi$
$710$	−21.8587	−0.820342
$711$	0	0
$712$	24.4682	0.916986
$713$	−18.9455	−0.709515
$714$	0	0
$715$	−20.2018	−0.755505
$716$	−24.1219	−0.901477
$717$	0	0
$718$	−9.43829	−0.352234
$719$	−8.76833	−0.327003	−0.163502	−	0.986543i	$-0.552279\pi$
$719$	−8.76833	−0.327003	−0.163502	+	0.986543i	$0.552279\pi$
$720$	0	0
$721$	−26.4116	−0.983619
$722$	−0.238054	−0.00885946
$723$	0	0
$724$	−38.0163	−1.41286
$725$	−12.3104	−0.457198
$726$	0	0
$727$	−42.9839	−1.59418	−0.797092	−	0.603858i	$-0.793629\pi$
$727$	−42.9839	−1.59418	−0.797092	+	0.603858i	$0.793629\pi$
$728$	−45.6305	−1.69118
$729$	0	0
$730$	−8.13688	−0.301159
$731$	−25.8496	−0.956084
$732$	0	0
$733$	−20.6856	−0.764042	−0.382021	−	0.924154i	$-0.624772\pi$
$733$	−20.6856	−0.764042	−0.382021	+	0.924154i	$0.624772\pi$
$734$	−4.51439	−0.166629
$735$	0	0
$736$	20.3160	0.748857
$737$	10.5600	0.388984
$738$	0	0
$739$	31.1414	1.14555	0.572777	−	0.819711i	$-0.305866\pi$
$739$	31.1414	1.14555	0.572777	+	0.819711i	$0.305866\pi$
$740$	49.2808	1.81160
$741$	0	0
$742$	−18.5021	−0.679234
$743$	−0.409099	−0.0150084	−0.00750420	−	0.999972i	$-0.502389\pi$
$743$	−0.409099	−0.0150084	−0.00750420	+	0.999972i	$0.502389\pi$
$744$	0	0
$745$	−66.1719	−2.42435
$746$	3.20057	0.117181
$747$	0	0
$748$	4.06296	0.148556
$749$	−33.3706	−1.21933
$750$	0	0
$751$	47.1958	1.72220	0.861100	−	0.508436i	$-0.169776\pi$
$751$	47.1958	1.72220	0.861100	+	0.508436i	$0.169776\pi$
$752$	2.02318	0.0737778
$753$	0	0
$754$	−6.90721	−0.251546
$755$	42.4468	1.54480
$756$	0	0
$757$	−21.2920	−0.773871	−0.386935	−	0.922107i	$-0.626466\pi$
$757$	−21.2920	−0.773871	−0.386935	+	0.922107i	$0.626466\pi$
$758$	−0.150155	−0.00545386
$759$	0	0
$760$	25.2978	0.917648
$761$	40.4700	1.46703	0.733517	−	0.679671i	$-0.237877\pi$
$761$	40.4700	1.46703	0.733517	+	0.679671i	$0.237877\pi$
$762$	0	0
$763$	51.9730	1.88155
$764$	−10.8975	−0.394257
$765$	0	0
$766$	11.3798	0.411167
$767$	15.8844	0.573552
$768$	0	0
$769$	−28.0251	−1.01061	−0.505306	−	0.862940i	$-0.668620\pi$
$769$	−28.0251	−1.01061	−0.505306	+	0.862940i	$0.668620\pi$
$770$	−6.16095	−0.222025
$771$	0	0
$772$	26.7446	0.962557
$773$	−10.8737	−0.391100	−0.195550	−	0.980694i	$-0.562649\pi$
$773$	−10.8737	−0.391100	−0.195550	+	0.980694i	$0.562649\pi$
$774$	0	0
$775$	23.9776	0.861302
$776$	0.00186347	6.68948e−5 0
$777$	0	0
$778$	−1.07168	−0.0384215
$779$	−17.9099	−0.641690
$780$	0	0
$781$	14.4115	0.515685
$782$	−4.52674	−0.161876
$783$	0	0
$784$	25.7426	0.919379
$785$	−75.2979	−2.68750
$786$	0	0
$787$	13.1203	0.467690	0.233845	−	0.972274i	$-0.424869\pi$
$787$	13.1203	0.467690	0.233845	+	0.972274i	$0.424869\pi$
$788$	14.3463	0.511066
$789$	0	0
$790$	−15.8972	−0.565596
$791$	−52.8531	−1.87924
$792$	0	0
$793$	−6.29425	−0.223515
$794$	−4.00512	−0.142136
$795$	0	0
$796$	−13.2052	−0.468045
$797$	−15.1688	−0.537307	−0.268653	−	0.963237i	$-0.586579\pi$
$797$	−15.1688	−0.537307	−0.268653	+	0.963237i	$0.586579\pi$
$798$	0	0
$799$	−1.70731	−0.0604001
$800$	−25.7121	−0.909059
$801$	0	0
$802$	−0.799266	−0.0282231
$803$	5.36468	0.189315
$804$	0	0
$805$	−54.6089	−1.92471
$806$	13.4535	0.473879
$807$	0	0
$808$	27.3374	0.961726
$809$	34.4544	1.21135	0.605676	−	0.795712i	$-0.292903\pi$
$809$	34.4544	1.21135	0.605676	+	0.795712i	$0.292903\pi$
$810$	0	0
$811$	−21.5549	−0.756896	−0.378448	−	0.925623i	$-0.623542\pi$
$811$	−21.5549	−0.756896	−0.378448	+	0.925623i	$0.623542\pi$
$812$	16.7584	0.588104
$813$	0	0
$814$	4.08405	0.143146
$815$	17.5879	0.616077
$816$	0	0
$817$	49.9205	1.74650
$818$	−0.961267	−0.0336099
$819$	0	0
$820$	23.1254	0.807574
$821$	16.6186	0.579994	0.289997	−	0.957027i	$-0.406346\pi$
$821$	16.6186	0.579994	0.289997	+	0.957027i	$0.406346\pi$
$822$	0	0
$823$	−18.1054	−0.631113	−0.315557	−	0.948907i	$-0.602191\pi$
$823$	−18.1054	−0.631113	−0.315557	+	0.948907i	$0.602191\pi$
$824$	−11.6048	−0.404273
$825$	0	0
$826$	4.84426	0.168554
$827$	−16.4273	−0.571234	−0.285617	−	0.958344i	$-0.592198\pi$
$827$	−16.4273	−0.571234	−0.285617	+	0.958344i	$0.592198\pi$
$828$	0	0
$829$	−31.9926	−1.11115	−0.555575	−	0.831467i	$-0.687502\pi$
$829$	−31.9926	−1.11115	−0.555575	+	0.831467i	$0.687502\pi$
$830$	−2.62921	−0.0912613
$831$	0	0
$832$	19.6873	0.682533
$833$	−21.7235	−0.752673
$834$	0	0
$835$	23.2310	0.803940
$836$	−7.84634	−0.271371
$837$	0	0
$838$	−8.33041	−0.287769
$839$	−33.0036	−1.13941	−0.569706	−	0.821849i	$-0.692943\pi$
$839$	−33.0036	−1.13941	−0.569706	+	0.821849i	$0.692943\pi$
$840$	0	0
$841$	−23.6076	−0.814056
$842$	−1.25247	−0.0431629
$843$	0	0
$844$	−47.0339	−1.61897
$845$	85.4310	2.93891
$846$	0	0
$847$	4.06194	0.139570
$848$	26.1203	0.896973
$849$	0	0
$850$	5.72909	0.196506
$851$	36.1998	1.24091
$852$	0	0
$853$	10.9871	0.376190	0.188095	−	0.982151i	$-0.439769\pi$
$853$	10.9871	0.376190	0.188095	+	0.982151i	$0.439769\pi$
$854$	−1.91956	−0.0656860
$855$	0	0
$856$	−14.6625	−0.501154
$857$	−15.7703	−0.538705	−0.269352	−	0.963042i	$-0.586810\pi$
$857$	−15.7703	−0.538705	−0.269352	+	0.963042i	$0.586810\pi$
$858$	0	0
$859$	27.0638	0.923403	0.461702	−	0.887035i	$-0.347239\pi$
$859$	27.0638	0.923403	0.461702	+	0.887035i	$0.347239\pi$
$860$	−64.4577	−2.19799
$861$	0	0
$862$	−13.1810	−0.448946
$863$	6.40423	0.218002	0.109001	−	0.994042i	$-0.465235\pi$
$863$	6.40423	0.218002	0.109001	+	0.994042i	$0.465235\pi$
$864$	0	0
$865$	−17.5945	−0.598231
$866$	11.4959	0.390647
$867$	0	0
$868$	−32.6411	−1.10791
$869$	10.4811	0.355546
$870$	0	0
$871$	−66.4675	−2.25216
$872$	22.8361	0.773328
$873$	0	0
$874$	8.74199	0.295702
$875$	3.92817	0.132796
$876$	0	0
$877$	−18.4953	−0.624543	−0.312272	−	0.949993i	$-0.601090\pi$
$877$	−18.4953	−0.624543	−0.312272	+	0.949993i	$0.601090\pi$
$878$	−9.88392	−0.333566
$879$	0	0
$880$	8.69769	0.293199
$881$	−17.1991	−0.579454	−0.289727	−	0.957109i	$-0.593564\pi$
$881$	−17.1991	−0.579454	−0.289727	+	0.957109i	$0.593564\pi$
$882$	0	0
$883$	21.1699	0.712422	0.356211	−	0.934406i	$-0.384068\pi$
$883$	21.1699	0.712422	0.356211	+	0.934406i	$0.384068\pi$
$884$	−25.5733	−0.860123
$885$	0	0
$886$	13.6857	0.459780
$887$	−41.1741	−1.38249	−0.691245	−	0.722620i	$-0.742937\pi$
$887$	−41.1741	−1.38249	−0.691245	+	0.722620i	$0.742937\pi$
$888$	0	0
$889$	84.1516	2.82235
$890$	−20.7940	−0.697018
$891$	0	0
$892$	5.04599	0.168952
$893$	3.29713	0.110334
$894$	0	0
$895$	43.5762	1.45659
$896$	45.4060	1.51691
$897$	0	0
$898$	11.0831	0.369847
$899$	−10.5030	−0.350295
$900$	0	0
$901$	−22.0422	−0.734331
$902$	1.91647	0.0638116
$903$	0	0
$904$	−23.2228	−0.772379
$905$	68.6765	2.28288
$906$	0	0
$907$	35.5805	1.18143	0.590715	−	0.806880i	$-0.298846\pi$
$907$	35.5805	1.18143	0.590715	+	0.806880i	$0.298846\pi$
$908$	22.8756	0.759152
$909$	0	0
$910$	38.7786	1.28550
$911$	−8.92521	−0.295705	−0.147853	−	0.989009i	$-0.547236\pi$
$911$	−8.92521	−0.295705	−0.147853	+	0.989009i	$0.547236\pi$
$912$	0	0
$913$	1.73345	0.0573689
$914$	−10.0971	−0.333984
$915$	0	0
$916$	19.7813	0.653593
$917$	6.02436	0.198942
$918$	0	0
$919$	−43.8505	−1.44649	−0.723247	−	0.690590i	$-0.757351\pi$
$919$	−43.8505	−1.44649	−0.723247	+	0.690590i	$0.757351\pi$
$920$	−23.9943	−0.791067
$921$	0	0
$922$	13.5813	0.447276
$923$	−90.7098	−2.98575
$924$	0	0
$925$	−45.8148	−1.50638
$926$	−2.20821	−0.0725662
$927$	0	0
$928$	11.2628	0.369718
$929$	−16.9397	−0.555772	−0.277886	−	0.960614i	$-0.589634\pi$
$929$	−16.9397	−0.555772	−0.277886	+	0.960614i	$0.589634\pi$
$930$	0	0
$931$	41.9521	1.37492
$932$	30.5119	0.999450
$933$	0	0
$934$	−15.1062	−0.494291
$935$	−7.33974	−0.240035
$936$	0	0
$937$	−14.7259	−0.481076	−0.240538	−	0.970640i	$-0.577324\pi$
$937$	−14.7259	−0.481076	−0.240538	+	0.970640i	$0.577324\pi$
$938$	−20.2706	−0.661859
$939$	0	0
$940$	−4.25727	−0.138857
$941$	−21.5676	−0.703083	−0.351542	−	0.936172i	$-0.614342\pi$
$941$	−21.5676	−0.703083	−0.351542	+	0.936172i	$0.614342\pi$
$942$	0	0
$943$	16.9871	0.553175
$944$	−6.83886	−0.222586
$945$	0	0
$946$	−5.34181	−0.173677
$947$	0.356161	0.0115737	0.00578685	−	0.999983i	$-0.498158\pi$
$947$	0.356161	0.0115737	0.00578685	+	0.999983i	$0.498158\pi$
$948$	0	0
$949$	−33.7666	−1.09611
$950$	−11.0639	−0.358962
$951$	0	0
$952$	−16.5785	−0.537313
$953$	−0.771616	−0.0249951	−0.0124975	−	0.999922i	$-0.503978\pi$
$953$	−0.771616	−0.0249951	−0.0124975	+	0.999922i	$0.503978\pi$
$954$	0	0
$955$	19.6863	0.637034
$956$	19.4614	0.629428
$957$	0	0
$958$	13.6765	0.441868
$959$	−55.2589	−1.78440
$960$	0	0
$961$	−10.5428	−0.340090
$962$	−25.7061	−0.828796
$963$	0	0
$964$	−14.0034	−0.451020
$965$	−48.3140	−1.55528
$966$	0	0
$967$	39.1087	1.25765	0.628826	−	0.777546i	$-0.283536\pi$
$967$	39.1087	1.25765	0.628826	+	0.777546i	$0.283536\pi$
$968$	1.78475	0.0573641
$969$	0	0
$970$	−0.00158365	−5.08480e−5 0
$971$	−62.2015	−1.99614	−0.998071	−	0.0620867i	$-0.980224\pi$
$971$	−62.2015	−1.99614	−0.998071	+	0.0620867i	$0.980224\pi$
$972$	0	0
$973$	−7.45044	−0.238850
$974$	7.86343	0.251960
$975$	0	0
$976$	2.70993	0.0867427
$977$	−33.6565	−1.07677	−0.538383	−	0.842700i	$-0.680964\pi$
$977$	−33.6565	−1.07677	−0.538383	+	0.842700i	$0.680964\pi$
$978$	0	0
$979$	13.7096	0.438161
$980$	−54.1688	−1.73036
$981$	0	0
$982$	−0.301836	−0.00963196
$983$	31.7228	1.01180	0.505901	−	0.862592i	$-0.331160\pi$
$983$	31.7228	1.01180	0.505901	+	0.862592i	$0.331160\pi$
$984$	0	0
$985$	−25.9166	−0.825772
$986$	−2.50953	−0.0799198
$987$	0	0
$988$	49.3868	1.57120
$989$	−47.3482	−1.50559
$990$	0	0
$991$	−14.2004	−0.451089	−0.225545	−	0.974233i	$-0.572416\pi$
$991$	−14.2004	−0.451089	−0.225545	+	0.974233i	$0.572416\pi$
$992$	−21.9370	−0.696501
$993$	0	0
$994$	−27.6638	−0.877442
$995$	23.8551	0.756259
$996$	0	0
$997$	−15.9252	−0.504357	−0.252179	−	0.967681i	$-0.581147\pi$
$997$	−15.9252	−0.504357	−0.252179	+	0.967681i	$0.581147\pi$
$998$	7.52468	0.238189
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.l.1.11		21
3.2	odd	2		671.2.a.d.1.11	✓	21
33.32	even	2		7381.2.a.j.1.11		21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
671.2.a.d.1.11	✓	21	3.2	odd	2
6039.2.a.l.1.11		21	1.1	even	1	trivial
7381.2.a.j.1.11		21	33.32	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6039 = 3^{2} \cdot 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6039.a (trivial)

\(f(q)\)	\(=\)	\(q-0.472572 q^{2} -1.77668 q^{4} +3.20957 q^{5} +4.06194 q^{7} +1.78475 q^{8} +O(q^{10})\)	\(q-0.472572 q^{2} -1.77668 q^{4} +3.20957 q^{5} +4.06194 q^{7} +1.78475 q^{8} -1.51675 q^{10} +1.00000 q^{11} -6.29425 q^{13} -1.91956 q^{14} +2.70993 q^{16} -2.28683 q^{17} +4.41630 q^{19} -5.70236 q^{20} -0.472572 q^{22} -4.18874 q^{23} +5.30131 q^{25} +2.97449 q^{26} -7.21675 q^{28} -2.32215 q^{29} +4.52296 q^{31} -4.85014 q^{32} +1.08069 q^{34} +13.0371 q^{35} -8.64218 q^{37} -2.08702 q^{38} +5.72828 q^{40} -4.05541 q^{41} +11.3037 q^{43} -1.77668 q^{44} +1.97948 q^{46} +0.746581 q^{47} +9.49937 q^{49} -2.50525 q^{50} +11.1828 q^{52} +9.63873 q^{53} +3.20957 q^{55} +7.24956 q^{56} +1.09738 q^{58} -2.52363 q^{59} +1.00000 q^{61} -2.13743 q^{62} -3.12782 q^{64} -20.2018 q^{65} +10.5600 q^{67} +4.06296 q^{68} -6.16095 q^{70} +14.4115 q^{71} +5.36468 q^{73} +4.08405 q^{74} -7.84634 q^{76} +4.06194 q^{77} +10.4811 q^{79} +8.69769 q^{80} +1.91647 q^{82} +1.73345 q^{83} -7.33974 q^{85} -5.34181 q^{86} +1.78475 q^{88} +13.7096 q^{89} -25.5669 q^{91} +7.44203 q^{92} -0.352813 q^{94} +14.1744 q^{95} +0.00104411 q^{97} -4.48914 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q + 32 q^{4} - 7 q^{5} + 5 q^{7} + 6 q^{8}+O(q^{10}) \) 21 * q + 32 * q^4 - 7 * q^5 + 5 * q^7 + 6 * q^8	\( 21 q + 32 q^{4} - 7 q^{5} + 5 q^{7} + 6 q^{8} + q^{10} + 21 q^{11} + 20 q^{13} - 17 q^{14} + 50 q^{16} - q^{17} + 15 q^{19} + 2 q^{20} - 11 q^{23} + 48 q^{25} + 5 q^{26} - 16 q^{28} + 9 q^{29} + 22 q^{31} - 3 q^{32} + 33 q^{34} + 39 q^{35} + 21 q^{37} - 11 q^{38} - 16 q^{40} - 7 q^{41} + 16 q^{43} + 32 q^{44} - 3 q^{46} - 5 q^{47} + 80 q^{49} + 33 q^{50} + 60 q^{52} - 9 q^{53} - 7 q^{55} - 44 q^{56} - 27 q^{58} - 13 q^{59} + 21 q^{61} + 23 q^{62} + 66 q^{64} - 25 q^{65} + 38 q^{67} + 74 q^{68} - 33 q^{70} - 12 q^{71} + 20 q^{73} + 12 q^{74} + 59 q^{76} + 5 q^{77} + q^{79} + 38 q^{80} + 7 q^{82} + 19 q^{83} + 38 q^{85} + 3 q^{86} + 6 q^{88} - 37 q^{89} + 24 q^{91} - 31 q^{92} - 64 q^{94} + 43 q^{95} + 68 q^{97} + 2 q^{98}+O(q^{100}) \) 21 * q + 32 * q^4 - 7 * q^5 + 5 * q^7 + 6 * q^8 + q^10 + 21 * q^11 + 20 * q^13 - 17 * q^14 + 50 * q^16 - q^17 + 15 * q^19 + 2 * q^20 - 11 * q^23 + 48 * q^25 + 5 * q^26 - 16 * q^28 + 9 * q^29 + 22 * q^31 - 3 * q^32 + 33 * q^34 + 39 * q^35 + 21 * q^37 - 11 * q^38 - 16 * q^40 - 7 * q^41 + 16 * q^43 + 32 * q^44 - 3 * q^46 - 5 * q^47 + 80 * q^49 + 33 * q^50 + 60 * q^52 - 9 * q^53 - 7 * q^55 - 44 * q^56 - 27 * q^58 - 13 * q^59 + 21 * q^61 + 23 * q^62 + 66 * q^64 - 25 * q^65 + 38 * q^67 + 74 * q^68 - 33 * q^70 - 12 * q^71 + 20 * q^73 + 12 * q^74 + 59 * q^76 + 5 * q^77 + q^79 + 38 * q^80 + 7 * q^82 + 19 * q^83 + 38 * q^85 + 3 * q^86 + 6 * q^88 - 37 * q^89 + 24 * q^91 - 31 * q^92 - 64 * q^94 + 43 * q^95 + 68 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more