LMFDB - Embedded newform 6039.2.a.l.1.19

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			1.19
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)

$f(q)$	$=$	$q+2.55546 q^{2} +4.53040 q^{4} -4.10969 q^{5} -4.64385 q^{7} +6.46634 q^{8} +O(q^{10})$	\(q+2.55546 q^{2} +4.53040 q^{4} -4.10969 q^{5} -4.64385 q^{7} +6.46634 q^{8} -10.5022 q^{10} +1.00000 q^{11} +0.666644 q^{13} -11.8672 q^{14} +7.46372 q^{16} -4.42323 q^{17} +6.64507 q^{19} -18.6186 q^{20} +2.55546 q^{22} +1.40560 q^{23} +11.8896 q^{25} +1.70358 q^{26} -21.0385 q^{28} -5.66326 q^{29} +5.31632 q^{31} +6.14057 q^{32} -11.3034 q^{34} +19.0848 q^{35} +4.06782 q^{37} +16.9812 q^{38} -26.5747 q^{40} -6.02550 q^{41} +10.4418 q^{43} +4.53040 q^{44} +3.59197 q^{46} -1.84924 q^{47} +14.5654 q^{49} +30.3834 q^{50} +3.02016 q^{52} +11.3583 q^{53} -4.10969 q^{55} -30.0287 q^{56} -14.4723 q^{58} +2.44525 q^{59} +1.00000 q^{61} +13.5857 q^{62} +0.764583 q^{64} -2.73970 q^{65} +7.44418 q^{67} -20.0390 q^{68} +48.7706 q^{70} +3.85363 q^{71} -8.62822 q^{73} +10.3952 q^{74} +30.1048 q^{76} -4.64385 q^{77} -4.96078 q^{79} -30.6736 q^{80} -15.3980 q^{82} +12.6840 q^{83} +18.1781 q^{85} +26.6835 q^{86} +6.46634 q^{88} +0.290276 q^{89} -3.09579 q^{91} +6.36794 q^{92} -4.72566 q^{94} -27.3092 q^{95} +8.07144 q^{97} +37.2213 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.55546	1.80699	0.903493	−	0.428603i	$-0.140994\pi$
$2$	2.55546	1.80699	0.903493	+	0.428603i	$0.140994\pi$
$3$	0	0
$3$	0	0
$4$	4.53040	2.26520
$5$	−4.10969	−1.83791	−0.918956	−	0.394361i	$-0.870966\pi$
$5$	−4.10969	−1.83791	−0.918956	+	0.394361i	$0.870966\pi$
$6$	0	0
$7$	−4.64385	−1.75521	−0.877605	−	0.479384i	$-0.840860\pi$
$7$	−4.64385	−1.75521	−0.877605	+	0.479384i	$0.840860\pi$
$8$	6.46634	2.28620
$9$	0	0
$10$	−10.5022	−3.32108
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	0.666644	0.184894	0.0924468	−	0.995718i	$-0.470531\pi$
$13$	0.666644	0.184894	0.0924468	+	0.995718i	$0.470531\pi$
$14$	−11.8672	−3.17164
$15$	0	0
$16$	7.46372	1.86593
$17$	−4.42323	−1.07279	−0.536396	−	0.843967i	$-0.680215\pi$
$17$	−4.42323	−1.07279	−0.536396	+	0.843967i	$0.680215\pi$
$18$	0	0
$19$	6.64507	1.52448	0.762241	−	0.647293i	$-0.224099\pi$
$19$	6.64507	1.52448	0.762241	+	0.647293i	$0.224099\pi$
$20$	−18.6186	−4.16324
$21$	0	0
$22$	2.55546	0.544827
$23$	1.40560	0.293088	0.146544	−	0.989204i	$-0.453185\pi$
$23$	1.40560	0.293088	0.146544	+	0.989204i	$0.453185\pi$
$24$	0	0
$25$	11.8896	2.37792
$26$	1.70358	0.334100
$27$	0	0
$28$	−21.0385	−3.97590
$29$	−5.66326	−1.05164	−0.525821	−	0.850595i	$-0.676242\pi$
$29$	−5.66326	−1.05164	−0.525821	+	0.850595i	$0.676242\pi$
$30$	0	0
$31$	5.31632	0.954839	0.477420	−	0.878675i	$-0.341572\pi$
$31$	5.31632	0.954839	0.477420	+	0.878675i	$0.341572\pi$
$32$	6.14057	1.08551
$33$	0	0
$34$	−11.3034	−1.93852
$35$	19.0848	3.22592
$36$	0	0
$37$	4.06782	0.668745	0.334373	−	0.942441i	$-0.391476\pi$
$37$	4.06782	0.668745	0.334373	+	0.942441i	$0.391476\pi$
$38$	16.9812	2.75472
$39$	0	0
$40$	−26.5747	−4.20183
$41$	−6.02550	−0.941025	−0.470513	−	0.882393i	$-0.655931\pi$
$41$	−6.02550	−0.941025	−0.470513	+	0.882393i	$0.655931\pi$
$42$	0	0
$43$	10.4418	1.59235	0.796177	−	0.605064i	$-0.206853\pi$
$43$	10.4418	1.59235	0.796177	+	0.605064i	$0.206853\pi$
$44$	4.53040	0.682983
$45$	0	0
$46$	3.59197	0.529606
$47$	−1.84924	−0.269739	−0.134869	−	0.990863i	$-0.543062\pi$
$47$	−1.84924	−0.269739	−0.134869	+	0.990863i	$0.543062\pi$
$48$	0	0
$49$	14.5654	2.08077
$50$	30.3834	4.29686
$51$	0	0
$52$	3.02016	0.418821
$53$	11.3583	1.56018	0.780092	−	0.625665i	$-0.215172\pi$
$53$	11.3583	1.56018	0.780092	+	0.625665i	$0.215172\pi$
$54$	0	0
$55$	−4.10969	−0.554151
$56$	−30.0287	−4.01276
$57$	0	0
$58$	−14.4723	−1.90030
$59$	2.44525	0.318345	0.159172	−	0.987251i	$-0.449117\pi$
$59$	2.44525	0.318345	0.159172	+	0.987251i	$0.449117\pi$
$60$	0	0
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	13.5857	1.72538
$63$	0	0
$64$	0.764583	0.0955729
$65$	−2.73970	−0.339818
$66$	0	0
$67$	7.44418	0.909451	0.454725	−	0.890632i	$-0.349737\pi$
$67$	7.44418	0.909451	0.454725	+	0.890632i	$0.349737\pi$
$68$	−20.0390	−2.43009
$69$	0	0
$70$	48.7706	5.82920
$71$	3.85363	0.457342	0.228671	−	0.973504i	$-0.426562\pi$
$71$	3.85363	0.457342	0.228671	+	0.973504i	$0.426562\pi$
$72$	0	0
$73$	−8.62822	−1.00986	−0.504929	−	0.863161i	$-0.668481\pi$
$73$	−8.62822	−1.00986	−0.504929	+	0.863161i	$0.668481\pi$
$74$	10.3952	1.20841
$75$	0	0
$76$	30.1048	3.45326
$77$	−4.64385	−0.529216
$78$	0	0
$79$	−4.96078	−0.558131	−0.279065	−	0.960272i	$-0.590025\pi$
$79$	−4.96078	−0.558131	−0.279065	+	0.960272i	$0.590025\pi$
$80$	−30.6736	−3.42941
$81$	0	0
$82$	−15.3980	−1.70042
$83$	12.6840	1.39225	0.696123	−	0.717923i	$-0.254907\pi$
$83$	12.6840	1.39225	0.696123	+	0.717923i	$0.254907\pi$
$84$	0	0
$85$	18.1781	1.97170
$86$	26.6835	2.87736
$87$	0	0
$88$	6.46634	0.689315
$89$	0.290276	0.0307692	0.0153846	−	0.999882i	$-0.495103\pi$
$89$	0.290276	0.0307692	0.0153846	+	0.999882i	$0.495103\pi$
$90$	0	0
$91$	−3.09579	−0.324527
$92$	6.36794	0.663903
$93$	0	0
$94$	−4.72566	−0.487414
$95$	−27.3092	−2.80186
$96$	0	0
$97$	8.07144	0.819531	0.409765	−	0.912191i	$-0.365611\pi$
$97$	8.07144	0.819531	0.409765	+	0.912191i	$0.365611\pi$
$98$	37.2213	3.75991
$99$	0	0
$100$	53.8646	5.38646
$101$	8.17458	0.813401	0.406701	−	0.913561i	$-0.366679\pi$
$101$	8.17458	0.813401	0.406701	+	0.913561i	$0.366679\pi$
$102$	0	0
$103$	−10.9265	−1.07662	−0.538309	−	0.842747i	$-0.680937\pi$
$103$	−10.9265	−1.07662	−0.538309	+	0.842747i	$0.680937\pi$
$104$	4.31075	0.422704
$105$	0	0
$106$	29.0258	2.81923
$107$	17.6392	1.70524	0.852622	−	0.522529i	$-0.175011\pi$
$107$	17.6392	1.70524	0.852622	+	0.522529i	$0.175011\pi$
$108$	0	0
$109$	9.09300	0.870951	0.435476	−	0.900200i	$-0.356580\pi$
$109$	9.09300	0.870951	0.435476	+	0.900200i	$0.356580\pi$
$110$	−10.5022	−1.00134
$111$	0	0
$112$	−34.6604	−3.27510
$113$	−1.42046	−0.133625	−0.0668127	−	0.997766i	$-0.521283\pi$
$113$	−1.42046	−0.133625	−0.0668127	+	0.997766i	$0.521283\pi$
$114$	0	0
$115$	−5.77659	−0.538670
$116$	−25.6568	−2.38218
$117$	0	0
$118$	6.24875	0.575244
$119$	20.5408	1.88298
$120$	0	0
$121$	1.00000	0.0909091
$122$	2.55546	0.231361
$123$	0	0
$124$	24.0850	2.16290
$125$	−28.3141	−2.53249
$126$	0	0
$127$	−14.4613	−1.28323	−0.641617	−	0.767025i	$-0.721736\pi$
$127$	−14.4613	−1.28323	−0.641617	+	0.767025i	$0.721736\pi$
$128$	−10.3273	−0.912811
$129$	0	0
$130$	−7.00121	−0.614047
$131$	5.53968	0.484004	0.242002	−	0.970276i	$-0.422196\pi$
$131$	5.53968	0.484004	0.242002	+	0.970276i	$0.422196\pi$
$132$	0	0
$133$	−30.8587	−2.67579
$134$	19.0233	1.64337
$135$	0	0
$136$	−28.6022	−2.45261
$137$	−16.9200	−1.44558	−0.722788	−	0.691070i	$-0.757140\pi$
$137$	−16.9200	−1.44558	−0.722788	+	0.691070i	$0.757140\pi$
$138$	0	0
$139$	6.73354	0.571132	0.285566	−	0.958359i	$-0.407818\pi$
$139$	6.73354	0.571132	0.285566	+	0.958359i	$0.407818\pi$
$140$	86.4618	7.30736
$141$	0	0
$142$	9.84782	0.826410
$143$	0.666644	0.0557475
$144$	0	0
$145$	23.2743	1.93282
$146$	−22.0491	−1.82480
$147$	0	0
$148$	18.4288	1.51484
$149$	−3.20949	−0.262932	−0.131466	−	0.991321i	$-0.541968\pi$
$149$	−3.20949	−0.262932	−0.131466	+	0.991321i	$0.541968\pi$
$150$	0	0
$151$	18.5000	1.50551	0.752753	−	0.658303i	$-0.228725\pi$
$151$	18.5000	1.50551	0.752753	+	0.658303i	$0.228725\pi$
$152$	42.9693	3.48527
$153$	0	0
$154$	−11.8672	−0.956286
$155$	−21.8484	−1.75491
$156$	0	0
$157$	2.76599	0.220750	0.110375	−	0.993890i	$-0.464795\pi$
$157$	2.76599	0.220750	0.110375	+	0.993890i	$0.464795\pi$
$158$	−12.6771	−1.00853
$159$	0	0
$160$	−25.2359	−1.99507
$161$	−6.52741	−0.514432
$162$	0	0
$163$	−3.05445	−0.239243	−0.119621	−	0.992820i	$-0.538168\pi$
$163$	−3.05445	−0.239243	−0.119621	+	0.992820i	$0.538168\pi$
$164$	−27.2979	−2.13161
$165$	0	0
$166$	32.4134	2.51577
$167$	9.05972	0.701062	0.350531	−	0.936551i	$-0.386001\pi$
$167$	9.05972	0.701062	0.350531	+	0.936551i	$0.386001\pi$
$168$	0	0
$169$	−12.5556	−0.965814
$170$	46.4536	3.56283
$171$	0	0
$172$	47.3053	3.60700
$173$	−7.33280	−0.557503	−0.278751	−	0.960363i	$-0.589921\pi$
$173$	−7.33280	−0.557503	−0.278751	+	0.960363i	$0.589921\pi$
$174$	0	0
$175$	−55.2135	−4.17375
$176$	7.46372	0.562599
$177$	0	0
$178$	0.741790	0.0555995
$179$	2.46630	0.184340	0.0921698	−	0.995743i	$-0.470620\pi$
$179$	2.46630	0.184340	0.0921698	+	0.995743i	$0.470620\pi$
$180$	0	0
$181$	−5.68151	−0.422303	−0.211152	−	0.977453i	$-0.567721\pi$
$181$	−5.68151	−0.422303	−0.211152	+	0.977453i	$0.567721\pi$
$182$	−7.91119	−0.586417
$183$	0	0
$184$	9.08911	0.670058
$185$	−16.7175	−1.22909
$186$	0	0
$187$	−4.42323	−0.323459
$188$	−8.37778	−0.611012
$189$	0	0
$190$	−69.7877	−5.06293
$191$	16.5871	1.20020	0.600101	−	0.799924i	$-0.295127\pi$
$191$	16.5871	1.20020	0.600101	+	0.799924i	$0.295127\pi$
$192$	0	0
$193$	−26.7259	−1.92377	−0.961886	−	0.273450i	$-0.911835\pi$
$193$	−26.7259	−1.92377	−0.961886	+	0.273450i	$0.911835\pi$
$194$	20.6263	1.48088
$195$	0	0
$196$	65.9869	4.71335
$197$	6.55487	0.467015	0.233508	−	0.972355i	$-0.424980\pi$
$197$	6.55487	0.467015	0.233508	+	0.972355i	$0.424980\pi$
$198$	0	0
$199$	0.207457	0.0147062	0.00735310	−	0.999973i	$-0.497659\pi$
$199$	0.207457	0.0147062	0.00735310	+	0.999973i	$0.497659\pi$
$200$	76.8822	5.43639
$201$	0	0
$202$	20.8899	1.46980
$203$	26.2993	1.84585
$204$	0	0
$205$	24.7630	1.72952
$206$	−27.9222	−1.94543
$207$	0	0
$208$	4.97564	0.344999
$209$	6.64507	0.459649
$210$	0	0
$211$	15.2259	1.04820	0.524098	−	0.851658i	$-0.324403\pi$
$211$	15.2259	1.04820	0.524098	+	0.851658i	$0.324403\pi$
$212$	51.4577	3.53413
$213$	0	0
$214$	45.0763	3.08135
$215$	−42.9124	−2.92660
$216$	0	0
$217$	−24.6882	−1.67594
$218$	23.2368	1.57380
$219$	0	0
$220$	−18.6186	−1.25526
$221$	−2.94872	−0.198352
$222$	0	0
$223$	−11.7887	−0.789428	−0.394714	−	0.918804i	$-0.629156\pi$
$223$	−11.7887	−0.789428	−0.394714	+	0.918804i	$0.629156\pi$
$224$	−28.5159	−1.90530
$225$	0	0
$226$	−3.62993	−0.241459
$227$	−1.77042	−0.117507	−0.0587535	−	0.998273i	$-0.518713\pi$
$227$	−1.77042	−0.117507	−0.0587535	+	0.998273i	$0.518713\pi$
$228$	0	0
$229$	−15.2622	−1.00856	−0.504279	−	0.863541i	$-0.668242\pi$
$229$	−15.2622	−1.00856	−0.504279	+	0.863541i	$0.668242\pi$
$230$	−14.7619	−0.973370
$231$	0	0
$232$	−36.6206	−2.40426
$233$	12.7162	0.833064	0.416532	−	0.909121i	$-0.363245\pi$
$233$	12.7162	0.833064	0.416532	+	0.909121i	$0.363245\pi$
$234$	0	0
$235$	7.59980	0.495756
$236$	11.0780	0.721114
$237$	0	0
$238$	52.4914	3.40251
$239$	−2.15176	−0.139186	−0.0695930	−	0.997575i	$-0.522170\pi$
$239$	−2.15176	−0.139186	−0.0695930	+	0.997575i	$0.522170\pi$
$240$	0	0
$241$	15.9652	1.02841	0.514205	−	0.857667i	$-0.328087\pi$
$241$	15.9652	1.02841	0.514205	+	0.857667i	$0.328087\pi$
$242$	2.55546	0.164271
$243$	0	0
$244$	4.53040	0.290029
$245$	−59.8592	−3.82426
$246$	0	0
$247$	4.42989	0.281867
$248$	34.3771	2.18295
$249$	0	0
$250$	−72.3557	−4.57617
$251$	1.05335	0.0664867	0.0332433	−	0.999447i	$-0.489416\pi$
$251$	1.05335	0.0664867	0.0332433	+	0.999447i	$0.489416\pi$
$252$	0	0
$253$	1.40560	0.0883694
$254$	−36.9554	−2.31879
$255$	0	0
$256$	−27.9202	−1.74501
$257$	−2.08120	−0.129822	−0.0649109	−	0.997891i	$-0.520676\pi$
$257$	−2.08120	−0.129822	−0.0649109	+	0.997891i	$0.520676\pi$
$258$	0	0
$259$	−18.8903	−1.17379
$260$	−12.4119	−0.769756
$261$	0	0
$262$	14.1565	0.874589
$263$	10.6735	0.658155	0.329077	−	0.944303i	$-0.393262\pi$
$263$	10.6735	0.658155	0.329077	+	0.944303i	$0.393262\pi$
$264$	0	0
$265$	−46.6792	−2.86748
$266$	−78.8583	−4.83511
$267$	0	0
$268$	33.7251	2.06009
$269$	−14.9995	−0.914535	−0.457267	−	0.889329i	$-0.651172\pi$
$269$	−14.9995	−0.914535	−0.457267	+	0.889329i	$0.651172\pi$
$270$	0	0
$271$	14.3260	0.870244	0.435122	−	0.900372i	$-0.356705\pi$
$271$	14.3260	0.870244	0.435122	+	0.900372i	$0.356705\pi$
$272$	−33.0138	−2.00175
$273$	0	0
$274$	−43.2386	−2.61214
$275$	11.8896	0.716969
$276$	0	0
$277$	23.2753	1.39848	0.699238	−	0.714889i	$-0.253523\pi$
$277$	23.2753	1.39848	0.699238	+	0.714889i	$0.253523\pi$
$278$	17.2073	1.03203
$279$	0	0
$280$	123.409	7.37510
$281$	29.5558	1.76315	0.881576	−	0.472042i	$-0.156483\pi$
$281$	29.5558	1.76315	0.881576	+	0.472042i	$0.156483\pi$
$282$	0	0
$283$	14.7770	0.878402	0.439201	−	0.898389i	$-0.355262\pi$
$283$	14.7770	0.878402	0.439201	+	0.898389i	$0.355262\pi$
$284$	17.4585	1.03597
$285$	0	0
$286$	1.70358	0.100735
$287$	27.9815	1.65170
$288$	0	0
$289$	2.56499	0.150882
$290$	59.4766	3.49259
$291$	0	0
$292$	−39.0893	−2.28753
$293$	−17.3480	−1.01348	−0.506741	−	0.862099i	$-0.669150\pi$
$293$	−17.3480	−1.01348	−0.506741	+	0.862099i	$0.669150\pi$
$294$	0	0
$295$	−10.0492	−0.585089
$296$	26.3039	1.52888
$297$	0	0
$298$	−8.20174	−0.475114
$299$	0.937036	0.0541902
$300$	0	0
$301$	−48.4900	−2.79492
$302$	47.2760	2.72043
$303$	0	0
$304$	49.5969	2.84458
$305$	−4.10969	−0.235320
$306$	0	0
$307$	21.3713	1.21972	0.609861	−	0.792508i	$-0.291225\pi$
$307$	21.3713	1.21972	0.609861	+	0.792508i	$0.291225\pi$
$308$	−21.0385	−1.19878
$309$	0	0
$310$	−55.8329	−3.17110
$311$	−17.5491	−0.995121	−0.497560	−	0.867429i	$-0.665771\pi$
$311$	−17.5491	−0.995121	−0.497560	+	0.867429i	$0.665771\pi$
$312$	0	0
$313$	−6.04955	−0.341941	−0.170970	−	0.985276i	$-0.554690\pi$
$313$	−6.04955	−0.341941	−0.170970	+	0.985276i	$0.554690\pi$
$314$	7.06839	0.398892
$315$	0	0
$316$	−22.4743	−1.26428
$317$	−4.99781	−0.280705	−0.140353	−	0.990102i	$-0.544824\pi$
$317$	−4.99781	−0.280705	−0.140353	+	0.990102i	$0.544824\pi$
$318$	0	0
$319$	−5.66326	−0.317082
$320$	−3.14220	−0.175655
$321$	0	0
$322$	−16.6806	−0.929571
$323$	−29.3927	−1.63545
$324$	0	0
$325$	7.92612	0.439662
$326$	−7.80554	−0.432309
$327$	0	0
$328$	−38.9630	−2.15137
$329$	8.58758	0.473449
$330$	0	0
$331$	19.8112	1.08892	0.544462	−	0.838786i	$-0.316734\pi$
$331$	19.8112	1.08892	0.544462	+	0.838786i	$0.316734\pi$
$332$	57.4634	3.15371
$333$	0	0
$334$	23.1518	1.26681
$335$	−30.5933	−1.67149
$336$	0	0
$337$	−8.12725	−0.442720	−0.221360	−	0.975192i	$-0.571050\pi$
$337$	−8.12725	−0.442720	−0.221360	+	0.975192i	$0.571050\pi$
$338$	−32.0854	−1.74521
$339$	0	0
$340$	82.3542	4.46628
$341$	5.31632	0.287895
$342$	0	0
$343$	−35.1324	−1.89697
$344$	67.5200	3.64044
$345$	0	0
$346$	−18.7387	−1.00740
$347$	17.6978	0.950067	0.475034	−	0.879968i	$-0.342436\pi$
$347$	17.6978	0.950067	0.475034	+	0.879968i	$0.342436\pi$
$348$	0	0
$349$	−31.7603	−1.70009	−0.850044	−	0.526712i	$-0.823425\pi$
$349$	−31.7603	−1.70009	−0.850044	+	0.526712i	$0.823425\pi$
$350$	−141.096	−7.54190
$351$	0	0
$352$	6.14057	0.327294
$353$	5.73302	0.305138	0.152569	−	0.988293i	$-0.451245\pi$
$353$	5.73302	0.305138	0.152569	+	0.988293i	$0.451245\pi$
$354$	0	0
$355$	−15.8372	−0.840554
$356$	1.31507	0.0696983
$357$	0	0
$358$	6.30253	0.333099
$359$	4.65129	0.245486	0.122743	−	0.992438i	$-0.460831\pi$
$359$	4.65129	0.245486	0.122743	+	0.992438i	$0.460831\pi$
$360$	0	0
$361$	25.1569	1.32405
$362$	−14.5189	−0.763096
$363$	0	0
$364$	−14.0252	−0.735119
$365$	35.4594	1.85603
$366$	0	0
$367$	−16.4324	−0.857764	−0.428882	−	0.903360i	$-0.641092\pi$
$367$	−16.4324	−0.857764	−0.428882	+	0.903360i	$0.641092\pi$
$368$	10.4910	0.546882
$369$	0	0
$370$	−42.7210	−2.22096
$371$	−52.7463	−2.73845
$372$	0	0
$373$	−8.15729	−0.422368	−0.211184	−	0.977446i	$-0.567732\pi$
$373$	−8.15729	−0.422368	−0.211184	+	0.977446i	$0.567732\pi$
$374$	−11.3034	−0.584486
$375$	0	0
$376$	−11.9578	−0.616676
$377$	−3.77538	−0.194442
$378$	0	0
$379$	17.5944	0.903764	0.451882	−	0.892078i	$-0.350753\pi$
$379$	17.5944	0.903764	0.451882	+	0.892078i	$0.350753\pi$
$380$	−123.722	−6.34678
$381$	0	0
$382$	42.3878	2.16875
$383$	−10.7199	−0.547762	−0.273881	−	0.961764i	$-0.588307\pi$
$383$	−10.7199	−0.547762	−0.273881	+	0.961764i	$0.588307\pi$
$384$	0	0
$385$	19.0848	0.972652
$386$	−68.2971	−3.47623
$387$	0	0
$388$	36.5669	1.85640
$389$	13.7433	0.696811	0.348406	−	0.937344i	$-0.386723\pi$
$389$	13.7433	0.696811	0.348406	+	0.937344i	$0.386723\pi$
$390$	0	0
$391$	−6.21731	−0.314423
$392$	94.1846	4.75704
$393$	0	0
$394$	16.7507	0.843890
$395$	20.3873	1.02580
$396$	0	0
$397$	27.5986	1.38513	0.692567	−	0.721353i	$-0.256480\pi$
$397$	27.5986	1.38513	0.692567	+	0.721353i	$0.256480\pi$
$398$	0.530148	0.0265739
$399$	0	0
$400$	88.7405	4.43703
$401$	−12.7201	−0.635210	−0.317605	−	0.948223i	$-0.602879\pi$
$401$	−12.7201	−0.635210	−0.317605	+	0.948223i	$0.602879\pi$
$402$	0	0
$403$	3.54409	0.176544
$404$	37.0341	1.84252
$405$	0	0
$406$	67.2070	3.33543
$407$	4.06782	0.201634
$408$	0	0
$409$	−18.4882	−0.914180	−0.457090	−	0.889420i	$-0.651108\pi$
$409$	−18.4882	−0.914180	−0.457090	+	0.889420i	$0.651108\pi$
$410$	63.2809	3.12522
$411$	0	0
$412$	−49.5013	−2.43876
$413$	−11.3554	−0.558762
$414$	0	0
$415$	−52.1272	−2.55882
$416$	4.09357	0.200704
$417$	0	0
$418$	16.9812	0.830579
$419$	−17.5999	−0.859810	−0.429905	−	0.902874i	$-0.641453\pi$
$419$	−17.5999	−0.859810	−0.429905	+	0.902874i	$0.641453\pi$
$420$	0	0
$421$	30.1765	1.47071	0.735356	−	0.677681i	$-0.237015\pi$
$421$	30.1765	1.47071	0.735356	+	0.677681i	$0.237015\pi$
$422$	38.9093	1.89407
$423$	0	0
$424$	73.4467	3.56689
$425$	−52.5904	−2.55101
$426$	0	0
$427$	−4.64385	−0.224732
$428$	79.9125	3.86272
$429$	0	0
$430$	−109.661	−5.28833
$431$	25.7244	1.23910	0.619550	−	0.784957i	$-0.287315\pi$
$431$	25.7244	1.23910	0.619550	+	0.784957i	$0.287315\pi$
$432$	0	0
$433$	4.10568	0.197306	0.0986532	−	0.995122i	$-0.468547\pi$
$433$	4.10568	0.197306	0.0986532	+	0.995122i	$0.468547\pi$
$434$	−63.0898	−3.02841
$435$	0	0
$436$	41.1949	1.97288
$437$	9.34032	0.446808
$438$	0	0
$439$	−25.9632	−1.23915	−0.619577	−	0.784936i	$-0.712696\pi$
$439$	−25.9632	−1.23915	−0.619577	+	0.784936i	$0.712696\pi$
$440$	−26.5747	−1.26690
$441$	0	0
$442$	−7.53535	−0.358420
$443$	−31.5884	−1.50081	−0.750405	−	0.660979i	$-0.770141\pi$
$443$	−31.5884	−1.50081	−0.750405	+	0.660979i	$0.770141\pi$
$444$	0	0
$445$	−1.19295	−0.0565510
$446$	−30.1255	−1.42649
$447$	0	0
$448$	−3.55061	−0.167751
$449$	9.68232	0.456937	0.228468	−	0.973551i	$-0.426628\pi$
$449$	9.68232	0.456937	0.228468	+	0.973551i	$0.426628\pi$
$450$	0	0
$451$	−6.02550	−0.283730
$452$	−6.43524	−0.302688
$453$	0	0
$454$	−4.52425	−0.212334
$455$	12.7228	0.596453
$456$	0	0
$457$	24.2305	1.13345	0.566727	−	0.823906i	$-0.308210\pi$
$457$	24.2305	1.13345	0.566727	+	0.823906i	$0.308210\pi$
$458$	−39.0021	−1.82245
$459$	0	0
$460$	−26.1703	−1.22020
$461$	19.6817	0.916667	0.458334	−	0.888780i	$-0.348447\pi$
$461$	19.6817	0.916667	0.458334	+	0.888780i	$0.348447\pi$
$462$	0	0
$463$	−29.3838	−1.36558	−0.682790	−	0.730615i	$-0.739234\pi$
$463$	−29.3838	−1.36558	−0.682790	+	0.730615i	$0.739234\pi$
$464$	−42.2690	−1.96229
$465$	0	0
$466$	32.4957	1.50534
$467$	1.53420	0.0709943	0.0354971	−	0.999370i	$-0.488699\pi$
$467$	1.53420	0.0709943	0.0354971	+	0.999370i	$0.488699\pi$
$468$	0	0
$469$	−34.5696	−1.59628
$470$	19.4210	0.895824
$471$	0	0
$472$	15.8118	0.727799
$473$	10.4418	0.480113
$474$	0	0
$475$	79.0071	3.62509
$476$	93.0582	4.26532
$477$	0	0
$478$	−5.49876	−0.251507
$479$	−19.5431	−0.892949	−0.446474	−	0.894796i	$-0.647321\pi$
$479$	−19.5431	−0.892949	−0.446474	+	0.894796i	$0.647321\pi$
$480$	0	0
$481$	2.71179	0.123647
$482$	40.7985	1.85832
$483$	0	0
$484$	4.53040	0.205927
$485$	−33.1712	−1.50622
$486$	0	0
$487$	15.7254	0.712585	0.356292	−	0.934375i	$-0.384041\pi$
$487$	15.7254	0.712585	0.356292	+	0.934375i	$0.384041\pi$
$488$	6.46634	0.292718
$489$	0	0
$490$	−152.968	−6.91039
$491$	22.1504	0.999635	0.499817	−	0.866131i	$-0.333400\pi$
$491$	22.1504	0.999635	0.499817	+	0.866131i	$0.333400\pi$
$492$	0	0
$493$	25.0499	1.12819
$494$	11.3204	0.509330
$495$	0	0
$496$	39.6795	1.78166
$497$	−17.8957	−0.802731
$498$	0	0
$499$	−11.7623	−0.526553	−0.263276	−	0.964720i	$-0.584803\pi$
$499$	−11.7623	−0.526553	−0.263276	+	0.964720i	$0.584803\pi$
$500$	−128.274	−5.73660
$501$	0	0
$502$	2.69179	0.120140
$503$	28.1289	1.25421	0.627103	−	0.778936i	$-0.284240\pi$
$503$	28.1289	1.25421	0.627103	+	0.778936i	$0.284240\pi$
$504$	0	0
$505$	−33.5950	−1.49496
$506$	3.59197	0.159682
$507$	0	0
$508$	−65.5155	−2.90678
$509$	6.45547	0.286134	0.143067	−	0.989713i	$-0.454304\pi$
$509$	6.45547	0.286134	0.143067	+	0.989713i	$0.454304\pi$
$510$	0	0
$511$	40.0682	1.77251
$512$	−50.6944	−2.24040
$513$	0	0
$514$	−5.31844	−0.234586
$515$	44.9045	1.97873
$516$	0	0
$517$	−1.84924	−0.0813293
$518$	−48.2736	−2.12102
$519$	0	0
$520$	−17.7159	−0.776892
$521$	27.5801	1.20830	0.604152	−	0.796869i	$-0.293512\pi$
$521$	27.5801	1.20830	0.604152	+	0.796869i	$0.293512\pi$
$522$	0	0
$523$	−26.3856	−1.15376	−0.576882	−	0.816828i	$-0.695731\pi$
$523$	−26.3856	−1.15376	−0.576882	+	0.816828i	$0.695731\pi$
$524$	25.0970	1.09637
$525$	0	0
$526$	27.2757	1.18928
$527$	−23.5153	−1.02434
$528$	0	0
$529$	−21.0243	−0.914099
$530$	−119.287	−5.18150
$531$	0	0
$532$	−139.802	−6.06120
$533$	−4.01686	−0.173990
$534$	0	0
$535$	−72.4916	−3.13409
$536$	48.1366	2.07918
$537$	0	0
$538$	−38.3306	−1.65255
$539$	14.5654	0.627374
$540$	0	0
$541$	12.1153	0.520877	0.260438	−	0.965491i	$-0.416133\pi$
$541$	12.1153	0.520877	0.260438	+	0.965491i	$0.416133\pi$
$542$	36.6096	1.57252
$543$	0	0
$544$	−27.1612	−1.16453
$545$	−37.3695	−1.60073
$546$	0	0
$547$	−21.7847	−0.931448	−0.465724	−	0.884930i	$-0.654206\pi$
$547$	−21.7847	−0.931448	−0.465724	+	0.884930i	$0.654206\pi$
$548$	−76.6545	−3.27452
$549$	0	0
$550$	30.3834	1.29555
$551$	−37.6328	−1.60321
$552$	0	0
$553$	23.0371	0.979638
$554$	59.4792	2.52703
$555$	0	0
$556$	30.5056	1.29373
$557$	38.1464	1.61631	0.808157	−	0.588968i	$-0.200465\pi$
$557$	38.1464	1.61631	0.808157	+	0.588968i	$0.200465\pi$
$558$	0	0
$559$	6.96093	0.294416
$560$	142.444	6.01934
$561$	0	0
$562$	75.5288	3.18599
$563$	12.7597	0.537756	0.268878	−	0.963174i	$-0.413347\pi$
$563$	12.7597	0.537756	0.268878	+	0.963174i	$0.413347\pi$
$564$	0	0
$565$	5.83765	0.245592
$566$	37.7621	1.58726
$567$	0	0
$568$	24.9189	1.04557
$569$	−45.2759	−1.89806	−0.949032	−	0.315179i	$-0.897936\pi$
$569$	−45.2759	−1.89806	−0.949032	+	0.315179i	$0.897936\pi$
$570$	0	0
$571$	−1.77216	−0.0741627	−0.0370813	−	0.999312i	$-0.511806\pi$
$571$	−1.77216	−0.0741627	−0.0370813	+	0.999312i	$0.511806\pi$
$572$	3.02016	0.126279
$573$	0	0
$574$	71.5058	2.98459
$575$	16.7120	0.696940
$576$	0	0
$577$	1.31236	0.0546341	0.0273170	−	0.999627i	$-0.491304\pi$
$577$	1.31236	0.0546341	0.0273170	+	0.999627i	$0.491304\pi$
$578$	6.55475	0.272642
$579$	0	0
$580$	105.442	4.37823
$581$	−58.9024	−2.44369
$582$	0	0
$583$	11.3583	0.470413
$584$	−55.7931	−2.30873
$585$	0	0
$586$	−44.3322	−1.83135
$587$	21.1079	0.871218	0.435609	−	0.900136i	$-0.356533\pi$
$587$	21.1079	0.871218	0.435609	+	0.900136i	$0.356533\pi$
$588$	0	0
$589$	35.3273	1.45564
$590$	−25.6805	−1.05725
$591$	0	0
$592$	30.3610	1.24783
$593$	−23.0846	−0.947972	−0.473986	−	0.880532i	$-0.657185\pi$
$593$	−23.0846	−0.947972	−0.473986	+	0.880532i	$0.657185\pi$
$594$	0	0
$595$	−84.4166	−3.46074
$596$	−14.5403	−0.595593
$597$	0	0
$598$	2.39456	0.0979209
$599$	31.8521	1.30144	0.650721	−	0.759317i	$-0.274467\pi$
$599$	31.8521	1.30144	0.650721	+	0.759317i	$0.274467\pi$
$600$	0	0
$601$	−15.8717	−0.647421	−0.323711	−	0.946156i	$-0.604930\pi$
$601$	−15.8717	−0.647421	−0.323711	+	0.946156i	$0.604930\pi$
$602$	−123.914	−5.05038
$603$	0	0
$604$	83.8123	3.41027
$605$	−4.10969	−0.167083
$606$	0	0
$607$	38.9427	1.58063	0.790317	−	0.612698i	$-0.209916\pi$
$607$	38.9427	1.58063	0.790317	+	0.612698i	$0.209916\pi$
$608$	40.8045	1.65484
$609$	0	0
$610$	−10.5022	−0.425221
$611$	−1.23278	−0.0498730
$612$	0	0
$613$	−9.68199	−0.391052	−0.195526	−	0.980699i	$-0.562641\pi$
$613$	−9.68199	−0.391052	−0.195526	+	0.980699i	$0.562641\pi$
$614$	54.6135	2.20402
$615$	0	0
$616$	−30.0287	−1.20989
$617$	21.1567	0.851736	0.425868	−	0.904785i	$-0.359969\pi$
$617$	21.1567	0.851736	0.425868	+	0.904785i	$0.359969\pi$
$618$	0	0
$619$	15.5848	0.626405	0.313202	−	0.949686i	$-0.398598\pi$
$619$	15.5848	0.626405	0.313202	+	0.949686i	$0.398598\pi$
$620$	−98.9822	−3.97522
$621$	0	0
$622$	−44.8462	−1.79817
$623$	−1.34800	−0.0540064
$624$	0	0
$625$	56.9144	2.27657
$626$	−15.4594	−0.617883
$627$	0	0
$628$	12.5310	0.500043
$629$	−17.9929	−0.717424
$630$	0	0
$631$	−36.5302	−1.45425	−0.727123	−	0.686507i	$-0.759143\pi$
$631$	−36.5302	−1.45425	−0.727123	+	0.686507i	$0.759143\pi$
$632$	−32.0781	−1.27600
$633$	0	0
$634$	−12.7717	−0.507230
$635$	59.4316	2.35847
$636$	0	0
$637$	9.70991	0.384720
$638$	−14.4723	−0.572962
$639$	0	0
$640$	42.4420	1.67767
$641$	21.4341	0.846597	0.423298	−	0.905990i	$-0.360872\pi$
$641$	21.4341	0.846597	0.423298	+	0.905990i	$0.360872\pi$
$642$	0	0
$643$	21.0844	0.831488	0.415744	−	0.909482i	$-0.363521\pi$
$643$	21.0844	0.831488	0.415744	+	0.909482i	$0.363521\pi$
$644$	−29.5718	−1.16529
$645$	0	0
$646$	−75.1120	−2.95524
$647$	−21.5720	−0.848083	−0.424041	−	0.905643i	$-0.639389\pi$
$647$	−21.5720	−0.848083	−0.424041	+	0.905643i	$0.639389\pi$
$648$	0	0
$649$	2.44525	0.0959845
$650$	20.2549	0.794463
$651$	0	0
$652$	−13.8379	−0.541933
$653$	12.2422	0.479074	0.239537	−	0.970887i	$-0.423004\pi$
$653$	12.2422	0.479074	0.239537	+	0.970887i	$0.423004\pi$
$654$	0	0
$655$	−22.7664	−0.889557
$656$	−44.9726	−1.75589
$657$	0	0
$658$	21.9453	0.855515
$659$	10.3699	0.403952	0.201976	−	0.979390i	$-0.435264\pi$
$659$	10.3699	0.403952	0.201976	+	0.979390i	$0.435264\pi$
$660$	0	0
$661$	−15.9771	−0.621437	−0.310719	−	0.950502i	$-0.600570\pi$
$661$	−15.9771	−0.621437	−0.310719	+	0.950502i	$0.600570\pi$
$662$	50.6269	1.96767
$663$	0	0
$664$	82.0189	3.18295
$665$	126.820	4.91786
$666$	0	0
$667$	−7.96029	−0.308224
$668$	41.0441	1.58804
$669$	0	0
$670$	−78.1801	−3.02036
$671$	1.00000	0.0386046
$672$	0	0
$673$	7.29351	0.281144	0.140572	−	0.990070i	$-0.455106\pi$
$673$	7.29351	0.281144	0.140572	+	0.990070i	$0.455106\pi$
$674$	−20.7689	−0.799988
$675$	0	0
$676$	−56.8818	−2.18776
$677$	−1.80650	−0.0694294	−0.0347147	−	0.999397i	$-0.511052\pi$
$677$	−1.80650	−0.0694294	−0.0347147	+	0.999397i	$0.511052\pi$
$678$	0	0
$679$	−37.4826	−1.43845
$680$	117.546	4.50769
$681$	0	0
$682$	13.5857	0.520222
$683$	−28.5417	−1.09212	−0.546059	−	0.837747i	$-0.683872\pi$
$683$	−28.5417	−1.09212	−0.546059	+	0.837747i	$0.683872\pi$
$684$	0	0
$685$	69.5362	2.65684
$686$	−89.7796	−3.42780
$687$	0	0
$688$	77.9343	2.97122
$689$	7.57194	0.288468
$690$	0	0
$691$	23.4829	0.893330	0.446665	−	0.894701i	$-0.352612\pi$
$691$	23.4829	0.893330	0.446665	+	0.894701i	$0.352612\pi$
$692$	−33.2205	−1.26285
$693$	0	0
$694$	45.2261	1.71676
$695$	−27.6728	−1.04969
$696$	0	0
$697$	26.6522	1.00952
$698$	−81.1622	−3.07203
$699$	0	0
$700$	−250.139	−9.45437
$701$	11.2981	0.426722	0.213361	−	0.976973i	$-0.431559\pi$
$701$	11.2981	0.426722	0.213361	+	0.976973i	$0.431559\pi$
$702$	0	0
$703$	27.0309	1.01949
$704$	0.764583	0.0288163
$705$	0	0
$706$	14.6505	0.551380
$707$	−37.9615	−1.42769
$708$	0	0
$709$	−18.1320	−0.680961	−0.340481	−	0.940252i	$-0.610590\pi$
$709$	−18.1320	−0.680961	−0.340481	+	0.940252i	$0.610590\pi$
$710$	−40.4715	−1.51887
$711$	0	0
$712$	1.87702	0.0703445
$713$	7.47263	0.279852
$714$	0	0
$715$	−2.73970	−0.102459
$716$	11.1733	0.417566
$717$	0	0
$718$	11.8862	0.443590
$719$	−41.2774	−1.53939	−0.769694	−	0.638414i	$-0.779591\pi$
$719$	−41.2774	−1.53939	−0.769694	+	0.638414i	$0.779591\pi$
$720$	0	0
$721$	50.7410	1.88969
$722$	64.2876	2.39254
$723$	0	0
$724$	−25.7395	−0.956601
$725$	−67.3338	−2.50072
$726$	0	0
$727$	−24.7151	−0.916634	−0.458317	−	0.888789i	$-0.651547\pi$
$727$	−24.7151	−0.916634	−0.458317	+	0.888789i	$0.651547\pi$
$728$	−20.0185	−0.741934
$729$	0	0
$730$	90.6151	3.35382
$731$	−46.1863	−1.70826
$732$	0	0
$733$	28.0872	1.03742	0.518712	−	0.854949i	$-0.326412\pi$
$733$	28.0872	1.03742	0.518712	+	0.854949i	$0.326412\pi$
$734$	−41.9924	−1.54997
$735$	0	0
$736$	8.63120	0.318150
$737$	7.44418	0.274210
$738$	0	0
$739$	36.3266	1.33630	0.668148	−	0.744029i	$-0.267087\pi$
$739$	36.3266	1.33630	0.668148	+	0.744029i	$0.267087\pi$
$740$	−75.7369	−2.78414
$741$	0	0
$742$	−134.791	−4.94834
$743$	−37.8239	−1.38762	−0.693812	−	0.720156i	$-0.744070\pi$
$743$	−37.8239	−1.38762	−0.693812	+	0.720156i	$0.744070\pi$
$744$	0	0
$745$	13.1900	0.483245
$746$	−20.8457	−0.763214
$747$	0	0
$748$	−20.0390	−0.732699
$749$	−81.9137	−2.99306
$750$	0	0
$751$	−11.3714	−0.414948	−0.207474	−	0.978241i	$-0.566524\pi$
$751$	−11.3714	−0.414948	−0.207474	+	0.978241i	$0.566524\pi$
$752$	−13.8022	−0.503314
$753$	0	0
$754$	−9.64784	−0.351354
$755$	−76.0292	−2.76699
$756$	0	0
$757$	−42.1631	−1.53244	−0.766222	−	0.642576i	$-0.777866\pi$
$757$	−42.1631	−1.53244	−0.766222	+	0.642576i	$0.777866\pi$
$758$	44.9619	1.63309
$759$	0	0
$760$	−176.591	−6.40562
$761$	16.4116	0.594920	0.297460	−	0.954734i	$-0.403861\pi$
$761$	16.4116	0.594920	0.297460	+	0.954734i	$0.403861\pi$
$762$	0	0
$763$	−42.2265	−1.52870
$764$	75.1463	2.71870
$765$	0	0
$766$	−27.3944	−0.989799
$767$	1.63011	0.0588599
$768$	0	0
$769$	9.09945	0.328134	0.164067	−	0.986449i	$-0.447539\pi$
$769$	9.09945	0.328134	0.164067	+	0.986449i	$0.447539\pi$
$770$	48.7706	1.75757
$771$	0	0
$772$	−121.079	−4.35773
$773$	−7.14478	−0.256980	−0.128490	−	0.991711i	$-0.541013\pi$
$773$	−7.14478	−0.256980	−0.128490	+	0.991711i	$0.541013\pi$
$774$	0	0
$775$	63.2088	2.27053
$776$	52.1927	1.87361
$777$	0	0
$778$	35.1204	1.25913
$779$	−40.0399	−1.43458
$780$	0	0
$781$	3.85363	0.137894
$782$	−15.8881	−0.568157
$783$	0	0
$784$	108.712	3.88256
$785$	−11.3674	−0.405719
$786$	0	0
$787$	45.7797	1.63187	0.815936	−	0.578143i	$-0.196222\pi$
$787$	45.7797	1.63187	0.815936	+	0.578143i	$0.196222\pi$
$788$	29.6962	1.05788
$789$	0	0
$790$	52.0990	1.85360
$791$	6.59640	0.234541
$792$	0	0
$793$	0.666644	0.0236732
$794$	70.5273	2.50292
$795$	0	0
$796$	0.939861	0.0333125
$797$	−48.6021	−1.72157	−0.860787	−	0.508965i	$-0.830028\pi$
$797$	−48.6021	−1.72157	−0.860787	+	0.508965i	$0.830028\pi$
$798$	0	0
$799$	8.17960	0.289374
$800$	73.0089	2.58125
$801$	0	0
$802$	−32.5057	−1.14782
$803$	−8.62822	−0.304483
$804$	0	0
$805$	26.8256	0.945480
$806$	9.05680	0.319012
$807$	0	0
$808$	52.8597	1.85960
$809$	34.3381	1.20726	0.603632	−	0.797263i	$-0.293720\pi$
$809$	34.3381	1.20726	0.603632	+	0.797263i	$0.293720\pi$
$810$	0	0
$811$	10.0089	0.351461	0.175731	−	0.984438i	$-0.443771\pi$
$811$	10.0089	0.351461	0.175731	+	0.984438i	$0.443771\pi$
$812$	119.147	4.18122
$813$	0	0
$814$	10.3952	0.364350
$815$	12.5529	0.439707
$816$	0	0
$817$	69.3862	2.42752
$818$	−47.2458	−1.65191
$819$	0	0
$820$	112.186	3.91771
$821$	1.61559	0.0563846	0.0281923	−	0.999603i	$-0.491025\pi$
$821$	1.61559	0.0563846	0.0281923	+	0.999603i	$0.491025\pi$
$822$	0	0
$823$	−47.3899	−1.65191	−0.825953	−	0.563739i	$-0.809363\pi$
$823$	−47.3899	−1.65191	−0.825953	+	0.563739i	$0.809363\pi$
$824$	−70.6544	−2.46136
$825$	0	0
$826$	−29.0183	−1.00968
$827$	27.5468	0.957896	0.478948	−	0.877843i	$-0.341018\pi$
$827$	27.5468	0.957896	0.478948	+	0.877843i	$0.341018\pi$
$828$	0	0
$829$	42.1510	1.46396	0.731982	−	0.681324i	$-0.238596\pi$
$829$	42.1510	1.46396	0.731982	+	0.681324i	$0.238596\pi$
$830$	−133.209	−4.62376
$831$	0	0
$832$	0.509705	0.0176708
$833$	−64.4260	−2.23223
$834$	0	0
$835$	−37.2327	−1.28849
$836$	30.1048	1.04120
$837$	0	0
$838$	−44.9758	−1.55366
$839$	4.16582	0.143820	0.0719101	−	0.997411i	$-0.477091\pi$
$839$	4.16582	0.143820	0.0719101	+	0.997411i	$0.477091\pi$
$840$	0	0
$841$	3.07253	0.105949
$842$	77.1150	2.65756
$843$	0	0
$844$	68.9795	2.37437
$845$	51.5996	1.77508
$846$	0	0
$847$	−4.64385	−0.159565
$848$	84.7752	2.91119
$849$	0	0
$850$	−134.393	−4.60964
$851$	5.71773	0.196001
$852$	0	0
$853$	−49.3442	−1.68951	−0.844756	−	0.535152i	$-0.820254\pi$
$853$	−49.3442	−1.68951	−0.844756	+	0.535152i	$0.820254\pi$
$854$	−11.8672	−0.406087
$855$	0	0
$856$	114.061	3.89852
$857$	−19.4233	−0.663486	−0.331743	−	0.943370i	$-0.607637\pi$
$857$	−19.4233	−0.663486	−0.331743	+	0.943370i	$0.607637\pi$
$858$	0	0
$859$	31.7807	1.08434	0.542171	−	0.840268i	$-0.317603\pi$
$859$	31.7807	1.08434	0.542171	+	0.840268i	$0.317603\pi$
$860$	−194.410	−6.62934
$861$	0	0
$862$	65.7378	2.23904
$863$	5.84413	0.198936	0.0994682	−	0.995041i	$-0.468286\pi$
$863$	5.84413	0.198936	0.0994682	+	0.995041i	$0.468286\pi$
$864$	0	0
$865$	30.1356	1.02464
$866$	10.4919	0.356530
$867$	0	0
$868$	−111.847	−3.79635
$869$	−4.96078	−0.168283
$870$	0	0
$871$	4.96261	0.168152
$872$	58.7985	1.99117
$873$	0	0
$874$	23.8689	0.807376
$875$	131.486	4.44505
$876$	0	0
$877$	−14.8060	−0.499962	−0.249981	−	0.968251i	$-0.580424\pi$
$877$	−14.8060	−0.499962	−0.249981	+	0.968251i	$0.580424\pi$
$878$	−66.3479	−2.23913
$879$	0	0
$880$	−30.6736	−1.03401
$881$	41.4772	1.39740	0.698701	−	0.715414i	$-0.253762\pi$
$881$	41.4772	1.39740	0.698701	+	0.715414i	$0.253762\pi$
$882$	0	0
$883$	−33.2315	−1.11833	−0.559164	−	0.829057i	$-0.688878\pi$
$883$	−33.2315	−1.11833	−0.559164	+	0.829057i	$0.688878\pi$
$884$	−13.3589	−0.449308
$885$	0	0
$886$	−80.7230	−2.71194
$887$	16.9919	0.570533	0.285266	−	0.958448i	$-0.407918\pi$
$887$	16.9919	0.570533	0.285266	+	0.958448i	$0.407918\pi$
$888$	0	0
$889$	67.1562	2.25235
$890$	−3.04853	−0.102187
$891$	0	0
$892$	−53.4074	−1.78821
$893$	−12.2883	−0.411212
$894$	0	0
$895$	−10.1357	−0.338800
$896$	47.9584	1.60218
$897$	0	0
$898$	24.7428	0.825678
$899$	−30.1077	−1.00415
$900$	0	0
$901$	−50.2404	−1.67375
$902$	−15.3980	−0.512696
$903$	0	0
$904$	−9.18517	−0.305494
$905$	23.3493	0.776156
$906$	0	0
$907$	5.14525	0.170845	0.0854226	−	0.996345i	$-0.472776\pi$
$907$	5.14525	0.170845	0.0854226	+	0.996345i	$0.472776\pi$
$908$	−8.02072	−0.266177
$909$	0	0
$910$	32.5126	1.07778
$911$	23.1471	0.766899	0.383449	−	0.923562i	$-0.374736\pi$
$911$	23.1471	0.766899	0.383449	+	0.923562i	$0.374736\pi$
$912$	0	0
$913$	12.6840	0.419778
$914$	61.9201	2.04814
$915$	0	0
$916$	−69.1441	−2.28458
$917$	−25.7255	−0.849529
$918$	0	0
$919$	55.6699	1.83638	0.918190	−	0.396141i	$-0.129651\pi$
$919$	55.6699	1.83638	0.918190	+	0.396141i	$0.129651\pi$
$920$	−37.3535	−1.23151
$921$	0	0
$922$	50.2958	1.65640
$923$	2.56900	0.0845596
$924$	0	0
$925$	48.3647	1.59022
$926$	−75.0892	−2.46758
$927$	0	0
$928$	−34.7757	−1.14157
$929$	26.0308	0.854044	0.427022	−	0.904241i	$-0.359563\pi$
$929$	26.0308	0.854044	0.427022	+	0.904241i	$0.359563\pi$
$930$	0	0
$931$	96.7878	3.17209
$932$	57.6093	1.88706
$933$	0	0
$934$	3.92059	0.128286
$935$	18.1781	0.594489
$936$	0	0
$937$	−36.2353	−1.18376	−0.591878	−	0.806027i	$-0.701613\pi$
$937$	−36.2353	−1.18376	−0.591878	+	0.806027i	$0.701613\pi$
$938$	−88.3415	−2.88445
$939$	0	0
$940$	34.4301	1.12299
$941$	5.78164	0.188476	0.0942380	−	0.995550i	$-0.469959\pi$
$941$	5.78164	0.188476	0.0942380	+	0.995550i	$0.469959\pi$
$942$	0	0
$943$	−8.46945	−0.275803
$944$	18.2507	0.594009
$945$	0	0
$946$	26.6835	0.867557
$947$	−5.19982	−0.168972	−0.0844858	−	0.996425i	$-0.526925\pi$
$947$	−5.19982	−0.168972	−0.0844858	+	0.996425i	$0.526925\pi$
$948$	0	0
$949$	−5.75195	−0.186716
$950$	201.900	6.55050
$951$	0	0
$952$	132.824	4.30486
$953$	5.45893	0.176832	0.0884160	−	0.996084i	$-0.471820\pi$
$953$	5.45893	0.176832	0.0884160	+	0.996084i	$0.471820\pi$
$954$	0	0
$955$	−68.1681	−2.20587
$956$	−9.74835	−0.315284
$957$	0	0
$958$	−49.9418	−1.61355
$959$	78.5741	2.53729
$960$	0	0
$961$	−2.73675	−0.0882824
$962$	6.92987	0.223428
$963$	0	0
$964$	72.3288	2.32955
$965$	109.835	3.53572
$966$	0	0
$967$	52.8290	1.69887	0.849433	−	0.527697i	$-0.176944\pi$
$967$	52.8290	1.69887	0.849433	+	0.527697i	$0.176944\pi$
$968$	6.46634	0.207836
$969$	0	0
$970$	−84.7677	−2.72173
$971$	47.3848	1.52065	0.760326	−	0.649542i	$-0.225039\pi$
$971$	47.3848	1.52065	0.760326	+	0.649542i	$0.225039\pi$
$972$	0	0
$973$	−31.2696	−1.00246
$974$	40.1856	1.28763
$975$	0	0
$976$	7.46372	0.238908
$977$	−49.1774	−1.57332	−0.786662	−	0.617384i	$-0.788192\pi$
$977$	−49.1774	−1.57332	−0.786662	+	0.617384i	$0.788192\pi$
$978$	0	0
$979$	0.290276	0.00927726
$980$	−271.186	−8.66272
$981$	0	0
$982$	56.6046	1.80633
$983$	30.5252	0.973601	0.486801	−	0.873513i	$-0.338164\pi$
$983$	30.5252	0.973601	0.486801	+	0.873513i	$0.338164\pi$
$984$	0	0
$985$	−26.9385	−0.858332
$986$	64.0142	2.03863
$987$	0	0
$988$	20.0692	0.638486
$989$	14.6770	0.466700
$990$	0	0
$991$	−31.0393	−0.985997	−0.492999	−	0.870030i	$-0.664099\pi$
$991$	−31.0393	−0.985997	−0.492999	+	0.870030i	$0.664099\pi$
$992$	32.6452	1.03649
$993$	0	0
$994$	−45.7318	−1.45052
$995$	−0.852583	−0.0270287
$996$	0	0
$997$	27.6505	0.875700	0.437850	−	0.899048i	$-0.355740\pi$
$997$	27.6505	0.875700	0.437850	+	0.899048i	$0.355740\pi$
$998$	−30.0581	−0.951473
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.l.1.19		21
3.2	odd	2		671.2.a.d.1.3	✓	21
33.32	even	2		7381.2.a.j.1.19		21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
671.2.a.d.1.3	✓	21	3.2	odd	2
6039.2.a.l.1.19		21	1.1	even	1	trivial
7381.2.a.j.1.19		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)

\(f(q)\)	\(=\)	\(q+2.55546 q^{2} +4.53040 q^{4} -4.10969 q^{5} -4.64385 q^{7} +6.46634 q^{8} +O(q^{10})\)	\(q+2.55546 q^{2} +4.53040 q^{4} -4.10969 q^{5} -4.64385 q^{7} +6.46634 q^{8} -10.5022 q^{10} +1.00000 q^{11} +0.666644 q^{13} -11.8672 q^{14} +7.46372 q^{16} -4.42323 q^{17} +6.64507 q^{19} -18.6186 q^{20} +2.55546 q^{22} +1.40560 q^{23} +11.8896 q^{25} +1.70358 q^{26} -21.0385 q^{28} -5.66326 q^{29} +5.31632 q^{31} +6.14057 q^{32} -11.3034 q^{34} +19.0848 q^{35} +4.06782 q^{37} +16.9812 q^{38} -26.5747 q^{40} -6.02550 q^{41} +10.4418 q^{43} +4.53040 q^{44} +3.59197 q^{46} -1.84924 q^{47} +14.5654 q^{49} +30.3834 q^{50} +3.02016 q^{52} +11.3583 q^{53} -4.10969 q^{55} -30.0287 q^{56} -14.4723 q^{58} +2.44525 q^{59} +1.00000 q^{61} +13.5857 q^{62} +0.764583 q^{64} -2.73970 q^{65} +7.44418 q^{67} -20.0390 q^{68} +48.7706 q^{70} +3.85363 q^{71} -8.62822 q^{73} +10.3952 q^{74} +30.1048 q^{76} -4.64385 q^{77} -4.96078 q^{79} -30.6736 q^{80} -15.3980 q^{82} +12.6840 q^{83} +18.1781 q^{85} +26.6835 q^{86} +6.46634 q^{88} +0.290276 q^{89} -3.09579 q^{91} +6.36794 q^{92} -4.72566 q^{94} -27.3092 q^{95} +8.07144 q^{97} +37.2213 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q + 32 q^{4} - 7 q^{5} + 5 q^{7} + 6 q^{8}+O(q^{10}) \) 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

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Database

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