LMFDB - Embedded newform 6039.2.a.m.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:	$1$
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)

$f(q)$	$=$	$q+0.870092 q^{2} -1.24294 q^{4} -3.50217 q^{5} -4.57282 q^{7} -2.82166 q^{8} +O(q^{10})$	\(q+0.870092 q^{2} -1.24294 q^{4} -3.50217 q^{5} -4.57282 q^{7} -2.82166 q^{8} -3.04721 q^{10} +1.00000 q^{11} +5.30244 q^{13} -3.97878 q^{14} +0.0307781 q^{16} +2.88563 q^{17} -3.82706 q^{19} +4.35299 q^{20} +0.870092 q^{22} +7.78717 q^{23} +7.26521 q^{25} +4.61361 q^{26} +5.68374 q^{28} -0.514252 q^{29} -5.54925 q^{31} +5.67009 q^{32} +2.51076 q^{34} +16.0148 q^{35} +0.537557 q^{37} -3.32989 q^{38} +9.88193 q^{40} +3.66679 q^{41} +11.1035 q^{43} -1.24294 q^{44} +6.77556 q^{46} -12.3217 q^{47} +13.9107 q^{49} +6.32140 q^{50} -6.59061 q^{52} -9.47756 q^{53} -3.50217 q^{55} +12.9029 q^{56} -0.447446 q^{58} -1.31206 q^{59} +1.00000 q^{61} -4.82836 q^{62} +4.87195 q^{64} -18.5701 q^{65} -12.7269 q^{67} -3.58666 q^{68} +13.9344 q^{70} +5.58054 q^{71} +8.67013 q^{73} +0.467724 q^{74} +4.75680 q^{76} -4.57282 q^{77} +9.71332 q^{79} -0.107790 q^{80} +3.19045 q^{82} +10.8261 q^{83} -10.1060 q^{85} +9.66107 q^{86} -2.82166 q^{88} -10.0220 q^{89} -24.2471 q^{91} -9.67898 q^{92} -10.7210 q^{94} +13.4030 q^{95} -16.0811 q^{97} +12.1036 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q - 5 q^{2} + 25 q^{4} - 12 q^{5} - 4 q^{7} - 15 q^{8}+O(q^{10}) $ 25 * q - 5 * q^2 + 25 * q^4 - 12 * q^5 - 4 * q^7 - 15 * q^8	$ 25 q - 5 q^{2} + 25 q^{4} - 12 q^{5} - 4 q^{7} - 15 q^{8} - 12 q^{10} + 25 q^{11} - 4 q^{13} - 14 q^{14} + 21 q^{16} - 16 q^{17} - 18 q^{19} - 28 q^{20} - 5 q^{22} - 8 q^{23} + 29 q^{25} - 16 q^{26} + 18 q^{28} - 28 q^{29} - 8 q^{31} - 35 q^{32} + 6 q^{34} - 22 q^{35} + 4 q^{37} + 4 q^{38} - 12 q^{40} - 58 q^{41} - 26 q^{43} + 25 q^{44} + 8 q^{46} - 20 q^{47} + 23 q^{49} - 27 q^{50} - 2 q^{52} - 36 q^{53} - 12 q^{55} - 70 q^{56} + 12 q^{58} - 18 q^{59} + 25 q^{61} - 42 q^{62} + 35 q^{64} - 76 q^{65} - 8 q^{67} - 28 q^{68} + 76 q^{70} - 24 q^{71} + 2 q^{73} - 40 q^{74} - 64 q^{76} - 4 q^{77} - 22 q^{79} - 36 q^{80} + 30 q^{82} - 14 q^{83} - 70 q^{86} - 15 q^{88} - 82 q^{89} - 6 q^{91} - 48 q^{92} - 16 q^{94} - 34 q^{95} + 16 q^{97} - 35 q^{98}+O(q^{100}) $ 25 * q - 5 * q^2 + 25 * q^4 - 12 * q^5 - 4 * q^7 - 15 * q^8 - 12 * q^10 + 25 * q^11 - 4 * q^13 - 14 * q^14 + 21 * q^16 - 16 * q^17 - 18 * q^19 - 28 * q^20 - 5 * q^22 - 8 * q^23 + 29 * q^25 - 16 * q^26 + 18 * q^28 - 28 * q^29 - 8 * q^31 - 35 * q^32 + 6 * q^34 - 22 * q^35 + 4 * q^37 + 4 * q^38 - 12 * q^40 - 58 * q^41 - 26 * q^43 + 25 * q^44 + 8 * q^46 - 20 * q^47 + 23 * q^49 - 27 * q^50 - 2 * q^52 - 36 * q^53 - 12 * q^55 - 70 * q^56 + 12 * q^58 - 18 * q^59 + 25 * q^61 - 42 * q^62 + 35 * q^64 - 76 * q^65 - 8 * q^67 - 28 * q^68 + 76 * q^70 - 24 * q^71 + 2 * q^73 - 40 * q^74 - 64 * q^76 - 4 * q^77 - 22 * q^79 - 36 * q^80 + 30 * q^82 - 14 * q^83 - 70 * q^86 - 15 * q^88 - 82 * q^89 - 6 * q^91 - 48 * q^92 - 16 * q^94 - 34 * q^95 + 16 * q^97 - 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 $

$2$	0.870092	0.615248	0.307624	−	0.951508i	$-0.400466\pi$
$2$	0.870092	0.615248	0.307624	+	0.951508i	$0.400466\pi$
$3$	0	0
$3$	0	0
$4$	−1.24294	−0.621470
$5$	−3.50217	−1.56622	−0.783109	−	0.621884i	$-0.786368\pi$
$5$	−3.50217	−1.56622	−0.783109	+	0.621884i	$0.786368\pi$
$6$	0	0
$7$	−4.57282	−1.72836	−0.864182	−	0.503179i	$-0.832164\pi$
$7$	−4.57282	−1.72836	−0.864182	+	0.503179i	$0.832164\pi$
$8$	−2.82166	−0.997606
$9$	0	0
$10$	−3.04721	−0.963613
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	5.30244	1.47063	0.735316	−	0.677724i	$-0.237034\pi$
$13$	5.30244	1.47063	0.735316	+	0.677724i	$0.237034\pi$
$14$	−3.97878	−1.06337
$15$	0	0
$16$	0.0307781	0.00769452
$17$	2.88563	0.699867	0.349934	−	0.936775i	$-0.386204\pi$
$17$	2.88563	0.699867	0.349934	+	0.936775i	$0.386204\pi$
$18$	0	0
$19$	−3.82706	−0.877988	−0.438994	−	0.898490i	$-0.644665\pi$
$19$	−3.82706	−0.877988	−0.438994	+	0.898490i	$0.644665\pi$
$20$	4.35299	0.973358
$21$	0	0
$22$	0.870092	0.185504
$23$	7.78717	1.62374	0.811869	−	0.583840i	$-0.198450\pi$
$23$	7.78717	1.62374	0.811869	+	0.583840i	$0.198450\pi$
$24$	0	0
$25$	7.26521	1.45304
$26$	4.61361	0.904804
$27$	0	0
$28$	5.68374	1.07413
$29$	−0.514252	−0.0954941	−0.0477471	−	0.998859i	$-0.515204\pi$
$29$	−0.514252	−0.0954941	−0.0477471	+	0.998859i	$0.515204\pi$
$30$	0	0
$31$	−5.54925	−0.996675	−0.498338	−	0.866983i	$-0.666056\pi$
$31$	−5.54925	−0.996675	−0.498338	+	0.866983i	$0.666056\pi$
$32$	5.67009	1.00234
$33$	0	0
$34$	2.51076	0.430592
$35$	16.0148	2.70700
$36$	0	0
$37$	0.537557	0.0883738	0.0441869	−	0.999023i	$-0.485930\pi$
$37$	0.537557	0.0883738	0.0441869	+	0.999023i	$0.485930\pi$
$38$	−3.32989	−0.540180
$39$	0	0
$40$	9.88193	1.56247
$41$	3.66679	0.572657	0.286329	−	0.958132i	$-0.407565\pi$
$41$	3.66679	0.572657	0.286329	+	0.958132i	$0.407565\pi$
$42$	0	0
$43$	11.1035	1.69327	0.846634	−	0.532175i	$-0.178625\pi$
$43$	11.1035	1.69327	0.846634	+	0.532175i	$0.178625\pi$
$44$	−1.24294	−0.187380
$45$	0	0
$46$	6.77556	0.999001
$47$	−12.3217	−1.79730	−0.898652	−	0.438663i	$-0.855452\pi$
$47$	−12.3217	−1.79730	−0.898652	+	0.438663i	$0.855452\pi$
$48$	0	0
$49$	13.9107	1.98724
$50$	6.32140	0.893981
$51$	0	0
$52$	−6.59061	−0.913954
$53$	−9.47756	−1.30184	−0.650921	−	0.759145i	$-0.725617\pi$
$53$	−9.47756	−1.30184	−0.650921	+	0.759145i	$0.725617\pi$
$54$	0	0
$55$	−3.50217	−0.472233
$56$	12.9029	1.72423
$57$	0	0
$58$	−0.447446	−0.0587526
$59$	−1.31206	−0.170816	−0.0854078	−	0.996346i	$-0.527219\pi$
$59$	−1.31206	−0.170816	−0.0854078	+	0.996346i	$0.527219\pi$
$60$	0	0
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	−4.82836	−0.613202
$63$	0	0
$64$	4.87195	0.608993
$65$	−18.5701	−2.30333
$66$	0	0
$67$	−12.7269	−1.55484	−0.777420	−	0.628981i	$-0.783472\pi$
$67$	−12.7269	−1.55484	−0.777420	+	0.628981i	$0.783472\pi$
$68$	−3.58666	−0.434946
$69$	0	0
$70$	13.9344	1.66547
$71$	5.58054	0.662288	0.331144	−	0.943580i	$-0.392565\pi$
$71$	5.58054	0.662288	0.331144	+	0.943580i	$0.392565\pi$
$72$	0	0
$73$	8.67013	1.01476	0.507381	−	0.861722i	$-0.330614\pi$
$73$	8.67013	1.01476	0.507381	+	0.861722i	$0.330614\pi$
$74$	0.467724	0.0543718
$75$	0	0
$76$	4.75680	0.545643
$77$	−4.57282	−0.521122
$78$	0	0
$79$	9.71332	1.09283	0.546417	−	0.837513i	$-0.315991\pi$
$79$	9.71332	1.09283	0.546417	+	0.837513i	$0.315991\pi$
$80$	−0.107790	−0.0120513
$81$	0	0
$82$	3.19045	0.352326
$83$	10.8261	1.18832	0.594158	−	0.804349i	$-0.297486\pi$
$83$	10.8261	1.18832	0.594158	+	0.804349i	$0.297486\pi$
$84$	0	0
$85$	−10.1060	−1.09615
$86$	9.66107	1.04178
$87$	0	0
$88$	−2.82166	−0.300790
$89$	−10.0220	−1.06233	−0.531165	−	0.847268i	$-0.678246\pi$
$89$	−10.0220	−1.06233	−0.531165	+	0.847268i	$0.678246\pi$
$90$	0	0
$91$	−24.2471	−2.54179
$92$	−9.67898	−1.00910
$93$	0	0
$94$	−10.7210	−1.10579
$95$	13.4030	1.37512
$96$	0	0
$97$	−16.0811	−1.63279	−0.816394	−	0.577495i	$-0.804030\pi$
$97$	−16.0811	−1.63279	−0.816394	+	0.577495i	$0.804030\pi$
$98$	12.1036	1.22265
$99$	0	0
$100$	−9.03022	−0.903022
$101$	13.7526	1.36844	0.684219	−	0.729277i	$-0.260143\pi$
$101$	13.7526	1.36844	0.684219	+	0.729277i	$0.260143\pi$
$102$	0	0
$103$	10.8156	1.06569	0.532846	−	0.846213i	$-0.321123\pi$
$103$	10.8156	1.06569	0.532846	+	0.846213i	$0.321123\pi$
$104$	−14.9617	−1.46711
$105$	0	0
$106$	−8.24635	−0.800956
$107$	−7.71805	−0.746132	−0.373066	−	0.927805i	$-0.621694\pi$
$107$	−7.71805	−0.746132	−0.373066	+	0.927805i	$0.621694\pi$
$108$	0	0
$109$	−7.18154	−0.687866	−0.343933	−	0.938994i	$-0.611759\pi$
$109$	−7.18154	−0.687866	−0.343933	+	0.938994i	$0.611759\pi$
$110$	−3.04721	−0.290540
$111$	0	0
$112$	−0.140743	−0.0132989
$113$	−20.0416	−1.88536	−0.942680	−	0.333699i	$-0.891703\pi$
$113$	−20.0416	−1.88536	−0.942680	+	0.333699i	$0.891703\pi$
$114$	0	0
$115$	−27.2720	−2.54313
$116$	0.639184	0.0593467
$117$	0	0
$118$	−1.14161	−0.105094
$119$	−13.1955	−1.20963
$120$	0	0
$121$	1.00000	0.0909091
$122$	0.870092	0.0787744
$123$	0	0
$124$	6.89739	0.619403
$125$	−7.93315	−0.709563
$126$	0	0
$127$	−1.48042	−0.131366	−0.0656832	−	0.997841i	$-0.520923\pi$
$127$	−1.48042	−0.131366	−0.0656832	+	0.997841i	$0.520923\pi$
$128$	−7.10114	−0.627658
$129$	0	0
$130$	−16.1577	−1.41712
$131$	1.69159	0.147795	0.0738974	−	0.997266i	$-0.476456\pi$
$131$	1.69159	0.147795	0.0738974	+	0.997266i	$0.476456\pi$
$132$	0	0
$133$	17.5005	1.51748
$134$	−11.0736	−0.956613
$135$	0	0
$136$	−8.14225	−0.698192
$137$	−4.48975	−0.383585	−0.191792	−	0.981436i	$-0.561430\pi$
$137$	−4.48975	−0.383585	−0.191792	+	0.981436i	$0.561430\pi$
$138$	0	0
$139$	−5.71957	−0.485127	−0.242564	−	0.970135i	$-0.577988\pi$
$139$	−5.71957	−0.485127	−0.242564	+	0.970135i	$0.577988\pi$
$140$	−19.9054	−1.68232
$141$	0	0
$142$	4.85558	0.407471
$143$	5.30244	0.443412
$144$	0	0
$145$	1.80100	0.149565
$146$	7.54382	0.624331
$147$	0	0
$148$	−0.668151	−0.0549217
$149$	−1.93753	−0.158728	−0.0793642	−	0.996846i	$-0.525289\pi$
$149$	−1.93753	−0.158728	−0.0793642	+	0.996846i	$0.525289\pi$
$150$	0	0
$151$	14.2677	1.16109	0.580545	−	0.814228i	$-0.302839\pi$
$151$	14.2677	1.16109	0.580545	+	0.814228i	$0.302839\pi$
$152$	10.7986	0.875886
$153$	0	0
$154$	−3.97878	−0.320619
$155$	19.4344	1.56101
$156$	0	0
$157$	11.3609	0.906696	0.453348	−	0.891334i	$-0.350229\pi$
$157$	11.3609	0.906696	0.453348	+	0.891334i	$0.350229\pi$
$158$	8.45148	0.672364
$159$	0	0
$160$	−19.8576	−1.56988
$161$	−35.6094	−2.80641
$162$	0	0
$163$	15.6565	1.22631	0.613156	−	0.789962i	$-0.289900\pi$
$163$	15.6565	1.22631	0.613156	+	0.789962i	$0.289900\pi$
$164$	−4.55760	−0.355889
$165$	0	0
$166$	9.41968	0.731109
$167$	22.1417	1.71338	0.856689	−	0.515833i	$-0.172517\pi$
$167$	22.1417	1.71338	0.856689	+	0.515833i	$0.172517\pi$
$168$	0	0
$169$	15.1159	1.16276
$170$	−8.79312	−0.674401
$171$	0	0
$172$	−13.8010	−1.05232
$173$	0.155831	0.0118476	0.00592380	−	0.999982i	$-0.498114\pi$
$173$	0.155831	0.0118476	0.00592380	+	0.999982i	$0.498114\pi$
$174$	0	0
$175$	−33.2225	−2.51139
$176$	0.0307781	0.00231999
$177$	0	0
$178$	−8.72006	−0.653596
$179$	−14.3356	−1.07149	−0.535746	−	0.844379i	$-0.679970\pi$
$179$	−14.3356	−1.07149	−0.535746	+	0.844379i	$0.679970\pi$
$180$	0	0
$181$	−1.69699	−0.126136	−0.0630682	−	0.998009i	$-0.520089\pi$
$181$	−1.69699	−0.126136	−0.0630682	+	0.998009i	$0.520089\pi$
$182$	−21.0972	−1.56383
$183$	0	0
$184$	−21.9727	−1.61985
$185$	−1.88262	−0.138413
$186$	0	0
$187$	2.88563	0.211018
$188$	15.3151	1.11697
$189$	0	0
$190$	11.6619	0.846040
$191$	−14.7328	−1.06603	−0.533015	−	0.846106i	$-0.678941\pi$
$191$	−14.7328	−1.06603	−0.533015	+	0.846106i	$0.678941\pi$
$192$	0	0
$193$	−16.6265	−1.19680	−0.598400	−	0.801197i	$-0.704197\pi$
$193$	−16.6265	−1.19680	−0.598400	+	0.801197i	$0.704197\pi$
$194$	−13.9920	−1.00457
$195$	0	0
$196$	−17.2902	−1.23501
$197$	−5.11833	−0.364666	−0.182333	−	0.983237i	$-0.558365\pi$
$197$	−5.11833	−0.364666	−0.182333	+	0.983237i	$0.558365\pi$
$198$	0	0
$199$	20.3100	1.43974	0.719868	−	0.694111i	$-0.244202\pi$
$199$	20.3100	1.43974	0.719868	+	0.694111i	$0.244202\pi$
$200$	−20.4999	−1.44956
$201$	0	0
$202$	11.9661	0.841928
$203$	2.35158	0.165049
$204$	0	0
$205$	−12.8417	−0.896906
$206$	9.41055	0.655664
$207$	0	0
$208$	0.163199	0.0113158
$209$	−3.82706	−0.264723
$210$	0	0
$211$	−24.0523	−1.65583	−0.827913	−	0.560857i	$-0.810472\pi$
$211$	−24.0523	−1.65583	−0.827913	+	0.560857i	$0.810472\pi$
$212$	11.7800	0.809056
$213$	0	0
$214$	−6.71542	−0.459056
$215$	−38.8864	−2.65203
$216$	0	0
$217$	25.3757	1.72262
$218$	−6.24860	−0.423209
$219$	0	0
$220$	4.35299	0.293478
$221$	15.3009	1.02925
$222$	0	0
$223$	13.4767	0.902470	0.451235	−	0.892405i	$-0.350984\pi$
$223$	13.4767	0.902470	0.451235	+	0.892405i	$0.350984\pi$
$224$	−25.9283	−1.73241
$225$	0	0
$226$	−17.4381	−1.15996
$227$	−8.86033	−0.588081	−0.294040	−	0.955793i	$-0.595000\pi$
$227$	−8.86033	−0.588081	−0.294040	+	0.955793i	$0.595000\pi$
$228$	0	0
$229$	−27.4861	−1.81633	−0.908167	−	0.418608i	$-0.862518\pi$
$229$	−27.4861	−1.81633	−0.908167	+	0.418608i	$0.862518\pi$
$230$	−23.7292	−1.56465
$231$	0	0
$232$	1.45104	0.0952655
$233$	−5.12885	−0.336002	−0.168001	−	0.985787i	$-0.553731\pi$
$233$	−5.12885	−0.336002	−0.168001	+	0.985787i	$0.553731\pi$
$234$	0	0
$235$	43.1527	2.81497
$236$	1.63081	0.106157
$237$	0	0
$238$	−11.4813	−0.744220
$239$	−16.8507	−1.08998	−0.544991	−	0.838442i	$-0.683467\pi$
$239$	−16.8507	−1.08998	−0.544991	+	0.838442i	$0.683467\pi$
$240$	0	0
$241$	5.37801	0.346428	0.173214	−	0.984884i	$-0.444585\pi$
$241$	5.37801	0.346428	0.173214	+	0.984884i	$0.444585\pi$
$242$	0.870092	0.0559316
$243$	0	0
$244$	−1.24294	−0.0795711
$245$	−48.7177	−3.11246
$246$	0	0
$247$	−20.2928	−1.29120
$248$	15.6581	0.994289
$249$	0	0
$250$	−6.90257	−0.436557
$251$	7.33537	0.463005	0.231502	−	0.972834i	$-0.425636\pi$
$251$	7.33537	0.463005	0.231502	+	0.972834i	$0.425636\pi$
$252$	0	0
$253$	7.78717	0.489575
$254$	−1.28811	−0.0808229
$255$	0	0
$256$	−15.9225	−0.995159
$257$	−24.3935	−1.52163	−0.760813	−	0.648971i	$-0.775200\pi$
$257$	−24.3935	−1.52163	−0.760813	+	0.648971i	$0.775200\pi$
$258$	0	0
$259$	−2.45815	−0.152742
$260$	23.0815	1.43145
$261$	0	0
$262$	1.47184	0.0909304
$263$	11.8596	0.731295	0.365648	−	0.930753i	$-0.380848\pi$
$263$	11.8596	0.731295	0.365648	+	0.930753i	$0.380848\pi$
$264$	0	0
$265$	33.1920	2.03897
$266$	15.2270	0.933628
$267$	0	0
$268$	15.8188	0.966286
$269$	18.8085	1.14678	0.573388	−	0.819284i	$-0.305629\pi$
$269$	18.8085	1.14678	0.573388	+	0.819284i	$0.305629\pi$
$270$	0	0
$271$	27.3588	1.66193	0.830965	−	0.556324i	$-0.187789\pi$
$271$	27.3588	1.66193	0.830965	+	0.556324i	$0.187789\pi$
$272$	0.0888141	0.00538514
$273$	0	0
$274$	−3.90649	−0.236000
$275$	7.26521	0.438109
$276$	0	0
$277$	21.0441	1.26442	0.632210	−	0.774797i	$-0.282148\pi$
$277$	21.0441	1.26442	0.632210	+	0.774797i	$0.282148\pi$
$278$	−4.97655	−0.298474
$279$	0	0
$280$	−45.1883	−2.70052
$281$	−24.9123	−1.48614	−0.743071	−	0.669213i	$-0.766632\pi$
$281$	−24.9123	−1.48614	−0.743071	+	0.669213i	$0.766632\pi$
$282$	0	0
$283$	−6.52769	−0.388031	−0.194015	−	0.980998i	$-0.562151\pi$
$283$	−6.52769	−0.388031	−0.194015	+	0.980998i	$0.562151\pi$
$284$	−6.93627	−0.411592
$285$	0	0
$286$	4.61361	0.272809
$287$	−16.7676	−0.989760
$288$	0	0
$289$	−8.67316	−0.510186
$290$	1.56703	0.0920194
$291$	0	0
$292$	−10.7765	−0.630644
$293$	−13.5176	−0.789706	−0.394853	−	0.918744i	$-0.629205\pi$
$293$	−13.5176	−0.789706	−0.394853	+	0.918744i	$0.629205\pi$
$294$	0	0
$295$	4.59506	0.267535
$296$	−1.51680	−0.0881623
$297$	0	0
$298$	−1.68583	−0.0976574
$299$	41.2910	2.38792
$300$	0	0
$301$	−50.7744	−2.92659
$302$	12.4142	0.714359
$303$	0	0
$304$	−0.117790	−0.00675569
$305$	−3.50217	−0.200534
$306$	0	0
$307$	−3.29228	−0.187900	−0.0939501	−	0.995577i	$-0.529949\pi$
$307$	−3.29228	−0.187900	−0.0939501	+	0.995577i	$0.529949\pi$
$308$	5.68374	0.323861
$309$	0	0
$310$	16.9097	0.960409
$311$	−0.373664	−0.0211885	−0.0105943	−	0.999944i	$-0.503372\pi$
$311$	−0.373664	−0.0211885	−0.0105943	+	0.999944i	$0.503372\pi$
$312$	0	0
$313$	−26.7710	−1.51319	−0.756594	−	0.653885i	$-0.773138\pi$
$313$	−26.7710	−1.51319	−0.756594	+	0.653885i	$0.773138\pi$
$314$	9.88500	0.557843
$315$	0	0
$316$	−12.0731	−0.679163
$317$	−21.6784	−1.21758	−0.608789	−	0.793332i	$-0.708344\pi$
$317$	−21.6784	−1.21758	−0.608789	+	0.793332i	$0.708344\pi$
$318$	0	0
$319$	−0.514252	−0.0287926
$320$	−17.0624	−0.953817
$321$	0	0
$322$	−30.9834	−1.72664
$323$	−11.0435	−0.614475
$324$	0	0
$325$	38.5233	2.13689
$326$	13.6226	0.754486
$327$	0	0
$328$	−10.3464	−0.571286
$329$	56.3449	3.10640
$330$	0	0
$331$	31.4549	1.72892	0.864458	−	0.502706i	$-0.167662\pi$
$331$	31.4549	1.72892	0.864458	+	0.502706i	$0.167662\pi$
$332$	−13.4561	−0.738502
$333$	0	0
$334$	19.2653	1.05415
$335$	44.5719	2.43522
$336$	0	0
$337$	3.07985	0.167770	0.0838850	−	0.996475i	$-0.473267\pi$
$337$	3.07985	0.167770	0.0838850	+	0.996475i	$0.473267\pi$
$338$	13.1522	0.715386
$339$	0	0
$340$	12.5611	0.681221
$341$	−5.54925	−0.300509
$342$	0	0
$343$	−31.6014	−1.70632
$344$	−31.3303	−1.68922
$345$	0	0
$346$	0.135587	0.00728921
$347$	13.7788	0.739683	0.369841	−	0.929095i	$-0.379412\pi$
$347$	13.7788	0.739683	0.369841	+	0.929095i	$0.379412\pi$
$348$	0	0
$349$	22.0160	1.17849	0.589245	−	0.807954i	$-0.299425\pi$
$349$	22.0160	1.17849	0.589245	+	0.807954i	$0.299425\pi$
$350$	−28.9066	−1.54513
$351$	0	0
$352$	5.67009	0.302217
$353$	−23.8985	−1.27199	−0.635995	−	0.771693i	$-0.719410\pi$
$353$	−23.8985	−1.27199	−0.635995	+	0.771693i	$0.719410\pi$
$354$	0	0
$355$	−19.5440	−1.03729
$356$	12.4567	0.660206
$357$	0	0
$358$	−12.4733	−0.659234
$359$	6.83651	0.360817	0.180409	−	0.983592i	$-0.442258\pi$
$359$	6.83651	0.360817	0.180409	+	0.983592i	$0.442258\pi$
$360$	0	0
$361$	−4.35362	−0.229138
$362$	−1.47654	−0.0776052
$363$	0	0
$364$	30.1377	1.57964
$365$	−30.3643	−1.58934
$366$	0	0
$367$	−30.3251	−1.58296	−0.791478	−	0.611198i	$-0.790688\pi$
$367$	−30.3251	−1.58296	−0.791478	+	0.611198i	$0.790688\pi$
$368$	0.239674	0.0124939
$369$	0	0
$370$	−1.63805	−0.0851582
$371$	43.3392	2.25006
$372$	0	0
$373$	−15.9526	−0.825996	−0.412998	−	0.910732i	$-0.635518\pi$
$373$	−15.9526	−0.825996	−0.412998	+	0.910732i	$0.635518\pi$
$374$	2.51076	0.129828
$375$	0	0
$376$	34.7676	1.79300
$377$	−2.72679	−0.140437
$378$	0	0
$379$	30.7458	1.57931	0.789654	−	0.613553i	$-0.210260\pi$
$379$	30.7458	1.57931	0.789654	+	0.613553i	$0.210260\pi$
$380$	−16.6591	−0.854596
$381$	0	0
$382$	−12.8189	−0.655873
$383$	−1.07524	−0.0549422	−0.0274711	−	0.999623i	$-0.508745\pi$
$383$	−1.07524	−0.0549422	−0.0274711	+	0.999623i	$0.508745\pi$
$384$	0	0
$385$	16.0148	0.816190
$386$	−14.4666	−0.736329
$387$	0	0
$388$	19.9878	1.01473
$389$	−5.40054	−0.273818	−0.136909	−	0.990584i	$-0.543717\pi$
$389$	−5.40054	−0.273818	−0.136909	+	0.990584i	$0.543717\pi$
$390$	0	0
$391$	22.4709	1.13640
$392$	−39.2512	−1.98249
$393$	0	0
$394$	−4.45342	−0.224360
$395$	−34.0177	−1.71162
$396$	0	0
$397$	19.7594	0.991694	0.495847	−	0.868410i	$-0.334858\pi$
$397$	19.7594	0.991694	0.495847	+	0.868410i	$0.334858\pi$
$398$	17.6715	0.885795
$399$	0	0
$400$	0.223609	0.0111805
$401$	−38.5327	−1.92423	−0.962116	−	0.272639i	$-0.912104\pi$
$401$	−38.5327	−1.92423	−0.962116	+	0.272639i	$0.912104\pi$
$402$	0	0
$403$	−29.4246	−1.46574
$404$	−17.0937	−0.850442
$405$	0	0
$406$	2.04609	0.101546
$407$	0.537557	0.0266457
$408$	0	0
$409$	−29.9708	−1.48196	−0.740980	−	0.671527i	$-0.765639\pi$
$409$	−29.9708	−1.48196	−0.740980	+	0.671527i	$0.765639\pi$
$410$	−11.1735	−0.551820
$411$	0	0
$412$	−13.4431	−0.662295
$413$	5.99981	0.295232
$414$	0	0
$415$	−37.9148	−1.86116
$416$	30.0653	1.47407
$417$	0	0
$418$	−3.32989	−0.162870
$419$	−5.98139	−0.292210	−0.146105	−	0.989269i	$-0.546674\pi$
$419$	−5.98139	−0.292210	−0.146105	+	0.989269i	$0.546674\pi$
$420$	0	0
$421$	22.9831	1.12013	0.560064	−	0.828449i	$-0.310777\pi$
$421$	22.9831	1.12013	0.560064	+	0.828449i	$0.310777\pi$
$422$	−20.9277	−1.01874
$423$	0	0
$424$	26.7424	1.29873
$425$	20.9647	1.01694
$426$	0	0
$427$	−4.57282	−0.221294
$428$	9.59307	0.463699
$429$	0	0
$430$	−33.8347	−1.63166
$431$	−31.4780	−1.51624	−0.758121	−	0.652114i	$-0.773882\pi$
$431$	−31.4780	−1.51624	−0.758121	+	0.652114i	$0.773882\pi$
$432$	0	0
$433$	11.8526	0.569598	0.284799	−	0.958587i	$-0.408073\pi$
$433$	11.8526	0.569598	0.284799	+	0.958587i	$0.408073\pi$
$434$	22.0792	1.05984
$435$	0	0
$436$	8.92622	0.427488
$437$	−29.8020	−1.42562
$438$	0	0
$439$	10.6629	0.508913	0.254456	−	0.967084i	$-0.418103\pi$
$439$	10.6629	0.508913	0.254456	+	0.967084i	$0.418103\pi$
$440$	9.88193	0.471102
$441$	0	0
$442$	13.3132	0.633242
$443$	−26.2094	−1.24524	−0.622622	−	0.782523i	$-0.713933\pi$
$443$	−26.2094	−1.24524	−0.622622	+	0.782523i	$0.713933\pi$
$444$	0	0
$445$	35.0988	1.66384
$446$	11.7260	0.555243
$447$	0	0
$448$	−22.2785	−1.05256
$449$	17.2898	0.815958	0.407979	−	0.912991i	$-0.366234\pi$
$449$	17.2898	0.815958	0.407979	+	0.912991i	$0.366234\pi$
$450$	0	0
$451$	3.66679	0.172663
$452$	24.9106	1.17169
$453$	0	0
$454$	−7.70931	−0.361816
$455$	84.9176	3.98100
$456$	0	0
$457$	−7.35976	−0.344275	−0.172138	−	0.985073i	$-0.555067\pi$
$457$	−7.35976	−0.344275	−0.172138	+	0.985073i	$0.555067\pi$
$458$	−23.9155	−1.11750
$459$	0	0
$460$	33.8975	1.58048
$461$	20.2166	0.941581	0.470790	−	0.882245i	$-0.343969\pi$
$461$	20.2166	0.941581	0.470790	+	0.882245i	$0.343969\pi$
$462$	0	0
$463$	22.7126	1.05554	0.527772	−	0.849386i	$-0.323028\pi$
$463$	22.7126	1.05554	0.527772	+	0.849386i	$0.323028\pi$
$464$	−0.0158277	−0.000734782 0
$465$	0	0
$466$	−4.46257	−0.206725
$467$	10.3822	0.480431	0.240215	−	0.970720i	$-0.422782\pi$
$467$	10.3822	0.480431	0.240215	+	0.970720i	$0.422782\pi$
$468$	0	0
$469$	58.1979	2.68733
$470$	37.5468	1.73191
$471$	0	0
$472$	3.70218	0.170407
$473$	11.1035	0.510540
$474$	0	0
$475$	−27.8044	−1.27575
$476$	16.4012	0.751746
$477$	0	0
$478$	−14.6617	−0.670610
$479$	−2.88453	−0.131797	−0.0658987	−	0.997826i	$-0.520991\pi$
$479$	−2.88453	−0.131797	−0.0658987	+	0.997826i	$0.520991\pi$
$480$	0	0
$481$	2.85036	0.129965
$482$	4.67936	0.213139
$483$	0	0
$484$	−1.24294	−0.0564973
$485$	56.3188	2.55730
$486$	0	0
$487$	21.2778	0.964188	0.482094	−	0.876119i	$-0.339876\pi$
$487$	21.2778	0.964188	0.482094	+	0.876119i	$0.339876\pi$
$488$	−2.82166	−0.127730
$489$	0	0
$490$	−42.3889	−1.91493
$491$	−1.75605	−0.0792493	−0.0396246	−	0.999215i	$-0.512616\pi$
$491$	−1.75605	−0.0792493	−0.0396246	+	0.999215i	$0.512616\pi$
$492$	0	0
$493$	−1.48394	−0.0668332
$494$	−17.6566	−0.794406
$495$	0	0
$496$	−0.170795	−0.00766894
$497$	−25.5188	−1.14467
$498$	0	0
$499$	13.7801	0.616883	0.308442	−	0.951243i	$-0.400193\pi$
$499$	13.7801	0.616883	0.308442	+	0.951243i	$0.400193\pi$
$500$	9.86043	0.440972
$501$	0	0
$502$	6.38245	0.284863
$503$	42.4947	1.89474	0.947372	−	0.320136i	$-0.103729\pi$
$503$	42.4947	1.89474	0.947372	+	0.320136i	$0.103729\pi$
$504$	0	0
$505$	−48.1641	−2.14327
$506$	6.77556	0.301210
$507$	0	0
$508$	1.84008	0.0816403
$509$	−1.49009	−0.0660469	−0.0330235	−	0.999455i	$-0.510514\pi$
$509$	−1.49009	−0.0660469	−0.0330235	+	0.999455i	$0.510514\pi$
$510$	0	0
$511$	−39.6470	−1.75388
$512$	0.348205	0.0153886
$513$	0	0
$514$	−21.2246	−0.936178
$515$	−37.8780	−1.66911
$516$	0	0
$517$	−12.3217	−0.541907
$518$	−2.13882	−0.0939743
$519$	0	0
$520$	52.3983	2.29782
$521$	−23.3806	−1.02432	−0.512162	−	0.858889i	$-0.671155\pi$
$521$	−23.3806	−1.02432	−0.512162	+	0.858889i	$0.671155\pi$
$522$	0	0
$523$	−2.58305	−0.112949	−0.0564744	−	0.998404i	$-0.517986\pi$
$523$	−2.58305	−0.112949	−0.0564744	+	0.998404i	$0.517986\pi$
$524$	−2.10254	−0.0918500
$525$	0	0
$526$	10.3190	0.449928
$527$	−16.0131	−0.697540
$528$	0	0
$529$	37.6400	1.63652
$530$	28.8801	1.25447
$531$	0	0
$532$	−21.7520	−0.943070
$533$	19.4430	0.842168
$534$	0	0
$535$	27.0299	1.16861
$536$	35.9110	1.55112
$537$	0	0
$538$	16.3652	0.705552
$539$	13.9107	0.599177
$540$	0	0
$541$	−30.8852	−1.32786	−0.663929	−	0.747795i	$-0.731112\pi$
$541$	−30.8852	−1.32786	−0.663929	+	0.747795i	$0.731112\pi$
$542$	23.8047	1.02250
$543$	0	0
$544$	16.3618	0.701505
$545$	25.1510	1.07735
$546$	0	0
$547$	−22.9234	−0.980132	−0.490066	−	0.871685i	$-0.663027\pi$
$547$	−22.9234	−0.980132	−0.490066	+	0.871685i	$0.663027\pi$
$548$	5.58048	0.238386
$549$	0	0
$550$	6.32140	0.269545
$551$	1.96807	0.0838426
$552$	0	0
$553$	−44.4173	−1.88882
$554$	18.3103	0.777932
$555$	0	0
$556$	7.10908	0.301492
$557$	−27.3896	−1.16053	−0.580267	−	0.814426i	$-0.697052\pi$
$557$	−27.3896	−1.16053	−0.580267	+	0.814426i	$0.697052\pi$
$558$	0	0
$559$	58.8757	2.49018
$560$	0.492905	0.0208290
$561$	0	0
$562$	−21.6760	−0.914346
$563$	−5.45310	−0.229821	−0.114910	−	0.993376i	$-0.536658\pi$
$563$	−5.45310	−0.229821	−0.114910	+	0.993376i	$0.536658\pi$
$564$	0	0
$565$	70.1893	2.95289
$566$	−5.67969	−0.238735
$567$	0	0
$568$	−15.7464	−0.660702
$569$	−10.0014	−0.419279	−0.209639	−	0.977779i	$-0.567229\pi$
$569$	−10.0014	−0.419279	−0.209639	+	0.977779i	$0.567229\pi$
$570$	0	0
$571$	16.0971	0.673641	0.336821	−	0.941569i	$-0.390648\pi$
$571$	16.0971	0.673641	0.336821	+	0.941569i	$0.390648\pi$
$572$	−6.59061	−0.275567
$573$	0	0
$574$	−14.5894	−0.608948
$575$	56.5754	2.35936
$576$	0	0
$577$	15.5141	0.645861	0.322931	−	0.946423i	$-0.395332\pi$
$577$	15.5141	0.645861	0.322931	+	0.946423i	$0.395332\pi$
$578$	−7.54645	−0.313891
$579$	0	0
$580$	−2.23853	−0.0929499
$581$	−49.5057	−2.05384
$582$	0	0
$583$	−9.47756	−0.392520
$584$	−24.4641	−1.01233
$585$	0	0
$586$	−11.7616	−0.485865
$587$	0.904991	0.0373530	0.0186765	−	0.999826i	$-0.494055\pi$
$587$	0.904991	0.0373530	0.0186765	+	0.999826i	$0.494055\pi$
$588$	0	0
$589$	21.2373	0.875068
$590$	3.99812	0.164600
$591$	0	0
$592$	0.0165450	0.000679994 0
$593$	8.62689	0.354264	0.177132	−	0.984187i	$-0.443318\pi$
$593$	8.62689	0.354264	0.177132	+	0.984187i	$0.443318\pi$
$594$	0	0
$595$	46.2128	1.89454
$596$	2.40823	0.0986449
$597$	0	0
$598$	35.9270	1.46916
$599$	−34.8513	−1.42399	−0.711993	−	0.702187i	$-0.752207\pi$
$599$	−34.8513	−1.42399	−0.711993	+	0.702187i	$0.752207\pi$
$600$	0	0
$601$	8.37025	0.341430	0.170715	−	0.985320i	$-0.445392\pi$
$601$	8.37025	0.341430	0.170715	+	0.985320i	$0.445392\pi$
$602$	−44.1784	−1.80058
$603$	0	0
$604$	−17.7339	−0.721583
$605$	−3.50217	−0.142384
$606$	0	0
$607$	6.18029	0.250850	0.125425	−	0.992103i	$-0.459970\pi$
$607$	6.18029	0.250850	0.125425	+	0.992103i	$0.459970\pi$
$608$	−21.6998	−0.880042
$609$	0	0
$610$	−3.04721	−0.123378
$611$	−65.3350	−2.64317
$612$	0	0
$613$	12.2836	0.496130	0.248065	−	0.968743i	$-0.420205\pi$
$613$	12.2836	0.496130	0.248065	+	0.968743i	$0.420205\pi$
$614$	−2.86458	−0.115605
$615$	0	0
$616$	12.9029	0.519874
$617$	−5.01569	−0.201924	−0.100962	−	0.994890i	$-0.532192\pi$
$617$	−5.01569	−0.201924	−0.100962	+	0.994890i	$0.532192\pi$
$618$	0	0
$619$	6.36280	0.255742	0.127871	−	0.991791i	$-0.459186\pi$
$619$	6.36280	0.255742	0.127871	+	0.991791i	$0.459186\pi$
$620$	−24.1558	−0.970121
$621$	0	0
$622$	−0.325122	−0.0130362
$623$	45.8288	1.83609
$624$	0	0
$625$	−8.54278	−0.341711
$626$	−23.2933	−0.930986
$627$	0	0
$628$	−14.1209	−0.563484
$629$	1.55119	0.0618499
$630$	0	0
$631$	34.1120	1.35798	0.678988	−	0.734149i	$-0.262419\pi$
$631$	34.1120	1.35798	0.678988	+	0.734149i	$0.262419\pi$
$632$	−27.4076	−1.09022
$633$	0	0
$634$	−18.8622	−0.749112
$635$	5.18470	0.205749
$636$	0	0
$637$	73.7607	2.92251
$638$	−0.447446	−0.0177146
$639$	0	0
$640$	24.8694	0.983050
$641$	26.1153	1.03149	0.515747	−	0.856741i	$-0.327515\pi$
$641$	26.1153	1.03149	0.515747	+	0.856741i	$0.327515\pi$
$642$	0	0
$643$	−41.4427	−1.63434	−0.817170	−	0.576397i	$-0.804458\pi$
$643$	−41.4427	−1.63434	−0.817170	+	0.576397i	$0.804458\pi$
$644$	44.2603	1.74410
$645$	0	0
$646$	−9.60883	−0.378054
$647$	−34.1702	−1.34337	−0.671684	−	0.740837i	$-0.734429\pi$
$647$	−34.1702	−1.34337	−0.671684	+	0.740837i	$0.734429\pi$
$648$	0	0
$649$	−1.31206	−0.0515028
$650$	33.5189	1.31472
$651$	0	0
$652$	−19.4601	−0.762116
$653$	18.9094	0.739982	0.369991	−	0.929035i	$-0.379361\pi$
$653$	18.9094	0.739982	0.369991	+	0.929035i	$0.379361\pi$
$654$	0	0
$655$	−5.92423	−0.231479
$656$	0.112857	0.00440632
$657$	0	0
$658$	49.0253	1.91120
$659$	−47.3387	−1.84405	−0.922026	−	0.387127i	$-0.873467\pi$
$659$	−47.3387	−1.84405	−0.922026	+	0.387127i	$0.873467\pi$
$660$	0	0
$661$	−5.04450	−0.196208	−0.0981042	−	0.995176i	$-0.531278\pi$
$661$	−5.04450	−0.196208	−0.0981042	+	0.995176i	$0.531278\pi$
$662$	27.3686	1.06371
$663$	0	0
$664$	−30.5474	−1.18547
$665$	−61.2896	−2.37671
$666$	0	0
$667$	−4.00457	−0.155057
$668$	−27.5208	−1.06481
$669$	0	0
$670$	38.7816	1.49826
$671$	1.00000	0.0386046
$672$	0	0
$673$	18.8450	0.726423	0.363212	−	0.931707i	$-0.381680\pi$
$673$	18.8450	0.726423	0.363212	+	0.931707i	$0.381680\pi$
$674$	2.67975	0.103220
$675$	0	0
$676$	−18.7881	−0.722620
$677$	7.78484	0.299196	0.149598	−	0.988747i	$-0.452202\pi$
$677$	7.78484	0.299196	0.149598	+	0.988747i	$0.452202\pi$
$678$	0	0
$679$	73.5360	2.82205
$680$	28.5155	1.09352
$681$	0	0
$682$	−4.82836	−0.184887
$683$	−16.6104	−0.635581	−0.317790	−	0.948161i	$-0.602941\pi$
$683$	−16.6104	−0.635581	−0.317790	+	0.948161i	$0.602941\pi$
$684$	0	0
$685$	15.7239	0.600778
$686$	−27.4962	−1.04981
$687$	0	0
$688$	0.341745	0.0130289
$689$	−50.2542	−1.91453
$690$	0	0
$691$	44.9915	1.71156	0.855779	−	0.517342i	$-0.173078\pi$
$691$	44.9915	1.71156	0.855779	+	0.517342i	$0.173078\pi$
$692$	−0.193688	−0.00736292
$693$	0	0
$694$	11.9888	0.455088
$695$	20.0309	0.759816
$696$	0	0
$697$	10.5810	0.400784
$698$	19.1560	0.725064
$699$	0	0
$700$	41.2936	1.56075
$701$	−16.9502	−0.640198	−0.320099	−	0.947384i	$-0.603716\pi$
$701$	−16.9502	−0.640198	−0.320099	+	0.947384i	$0.603716\pi$
$702$	0	0
$703$	−2.05726	−0.0775911
$704$	4.87195	0.183618
$705$	0	0
$706$	−20.7939	−0.782589
$707$	−62.8883	−2.36516
$708$	0	0
$709$	−9.89737	−0.371704	−0.185852	−	0.982578i	$-0.559504\pi$
$709$	−9.89737	−0.371704	−0.185852	+	0.982578i	$0.559504\pi$
$710$	−17.0051	−0.638189
$711$	0	0
$712$	28.2786	1.05979
$713$	−43.2130	−1.61834
$714$	0	0
$715$	−18.5701	−0.694481
$716$	17.8183	0.665900
$717$	0	0
$718$	5.94840	0.221992
$719$	−0.754196	−0.0281268	−0.0140634	−	0.999901i	$-0.504477\pi$
$719$	−0.754196	−0.0281268	−0.0140634	+	0.999901i	$0.504477\pi$
$720$	0	0
$721$	−49.4577	−1.84190
$722$	−3.78805	−0.140977
$723$	0	0
$724$	2.10926	0.0783900
$725$	−3.73614	−0.138757
$726$	0	0
$727$	10.8101	0.400924	0.200462	−	0.979701i	$-0.435756\pi$
$727$	10.8101	0.400924	0.200462	+	0.979701i	$0.435756\pi$
$728$	68.4170	2.53570
$729$	0	0
$730$	−26.4197	−0.977838
$731$	32.0406	1.18506
$732$	0	0
$733$	−28.1476	−1.03965	−0.519827	−	0.854272i	$-0.674004\pi$
$733$	−28.1476	−1.03965	−0.519827	+	0.854272i	$0.674004\pi$
$734$	−26.3856	−0.973910
$735$	0	0
$736$	44.1540	1.62754
$737$	−12.7269	−0.468802
$738$	0	0
$739$	−34.1542	−1.25638	−0.628190	−	0.778060i	$-0.716204\pi$
$739$	−34.1542	−1.25638	−0.628190	+	0.778060i	$0.716204\pi$
$740$	2.33998	0.0860193
$741$	0	0
$742$	37.7091	1.38434
$743$	1.97360	0.0724043	0.0362021	−	0.999344i	$-0.488474\pi$
$743$	1.97360	0.0724043	0.0362021	+	0.999344i	$0.488474\pi$
$744$	0	0
$745$	6.78556	0.248604
$746$	−13.8803	−0.508192
$747$	0	0
$748$	−3.58666	−0.131141
$749$	35.2933	1.28959
$750$	0	0
$751$	16.1139	0.588005	0.294003	−	0.955805i	$-0.405013\pi$
$751$	16.1139	0.588005	0.294003	+	0.955805i	$0.405013\pi$
$752$	−0.379238	−0.0138294
$753$	0	0
$754$	−2.37256	−0.0864034
$755$	−49.9680	−1.81852
$756$	0	0
$757$	−27.9699	−1.01658	−0.508292	−	0.861185i	$-0.669723\pi$
$757$	−27.9699	−1.01658	−0.508292	+	0.861185i	$0.669723\pi$
$758$	26.7517	0.971666
$759$	0	0
$760$	−37.8187	−1.37183
$761$	−24.3226	−0.881693	−0.440846	−	0.897583i	$-0.645322\pi$
$761$	−24.3226	−0.881693	−0.440846	+	0.897583i	$0.645322\pi$
$762$	0	0
$763$	32.8399	1.18888
$764$	18.3120	0.662506
$765$	0	0
$766$	−0.935558	−0.0338031
$767$	−6.95712	−0.251207
$768$	0	0
$769$	−17.4043	−0.627616	−0.313808	−	0.949487i	$-0.601605\pi$
$769$	−17.4043	−0.627616	−0.313808	+	0.949487i	$0.601605\pi$
$770$	13.9344	0.502160
$771$	0	0
$772$	20.6657	0.743776
$773$	−50.6239	−1.82082	−0.910408	−	0.413712i	$-0.864232\pi$
$773$	−50.6239	−1.82082	−0.910408	+	0.413712i	$0.864232\pi$
$774$	0	0
$775$	−40.3165	−1.44821
$776$	45.3753	1.62888
$777$	0	0
$778$	−4.69897	−0.168466
$779$	−14.0330	−0.502786
$780$	0	0
$781$	5.58054	0.199687
$782$	19.5517	0.699168
$783$	0	0
$784$	0.428145	0.0152909
$785$	−39.7877	−1.42008
$786$	0	0
$787$	−16.9860	−0.605487	−0.302743	−	0.953072i	$-0.597902\pi$
$787$	−16.9860	−0.605487	−0.302743	+	0.953072i	$0.597902\pi$
$788$	6.36178	0.226629
$789$	0	0
$790$	−29.5985	−1.05307
$791$	91.6469	3.25859
$792$	0	0
$793$	5.30244	0.188295
$794$	17.1925	0.610138
$795$	0	0
$796$	−25.2441	−0.894752
$797$	10.7741	0.381637	0.190819	−	0.981625i	$-0.438886\pi$
$797$	10.7741	0.381637	0.190819	+	0.981625i	$0.438886\pi$
$798$	0	0
$799$	−35.5558	−1.25787
$800$	41.1944	1.45644
$801$	0	0
$802$	−33.5270	−1.18388
$803$	8.67013	0.305962
$804$	0	0
$805$	124.710	4.39545
$806$	−25.6021	−0.901795
$807$	0	0
$808$	−38.8052	−1.36516
$809$	−33.7476	−1.18650	−0.593251	−	0.805018i	$-0.702156\pi$
$809$	−33.7476	−1.18650	−0.593251	+	0.805018i	$0.702156\pi$
$810$	0	0
$811$	−31.8594	−1.11873	−0.559367	−	0.828920i	$-0.688956\pi$
$811$	−31.8594	−1.11873	−0.559367	+	0.828920i	$0.688956\pi$
$812$	−2.92287	−0.102573
$813$	0	0
$814$	0.467724	0.0163937
$815$	−54.8318	−1.92067
$816$	0	0
$817$	−42.4938	−1.48667
$818$	−26.0773	−0.911773
$819$	0	0
$820$	15.9615	0.557400
$821$	−1.90272	−0.0664054	−0.0332027	−	0.999449i	$-0.510571\pi$
$821$	−1.90272	−0.0664054	−0.0332027	+	0.999449i	$0.510571\pi$
$822$	0	0
$823$	23.6587	0.824689	0.412344	−	0.911028i	$-0.364710\pi$
$823$	23.6587	0.824689	0.412344	+	0.911028i	$0.364710\pi$
$824$	−30.5179	−1.06314
$825$	0	0
$826$	5.22039	0.181641
$827$	−13.2524	−0.460833	−0.230416	−	0.973092i	$-0.574009\pi$
$827$	−13.2524	−0.460833	−0.230416	+	0.973092i	$0.574009\pi$
$828$	0	0
$829$	−23.6808	−0.822470	−0.411235	−	0.911529i	$-0.634902\pi$
$829$	−23.6808	−0.822470	−0.411235	+	0.911529i	$0.634902\pi$
$830$	−32.9893	−1.14508
$831$	0	0
$832$	25.8332	0.895605
$833$	40.1411	1.39081
$834$	0	0
$835$	−77.5442	−2.68353
$836$	4.75680	0.164517
$837$	0	0
$838$	−5.20436	−0.179782
$839$	39.5598	1.36576	0.682878	−	0.730532i	$-0.260728\pi$
$839$	39.5598	1.36576	0.682878	+	0.730532i	$0.260728\pi$
$840$	0	0
$841$	−28.7355	−0.990881
$842$	19.9974	0.689157
$843$	0	0
$844$	29.8955	1.02905
$845$	−52.9384	−1.82114
$846$	0	0
$847$	−4.57282	−0.157124
$848$	−0.291701	−0.0100171
$849$	0	0
$850$	18.2412	0.625668
$851$	4.18605	0.143496
$852$	0	0
$853$	−12.8148	−0.438771	−0.219386	−	0.975638i	$-0.570405\pi$
$853$	−12.8148	−0.438771	−0.219386	+	0.975638i	$0.570405\pi$
$854$	−3.97878	−0.136151
$855$	0	0
$856$	21.7777	0.744346
$857$	43.8073	1.49643	0.748215	−	0.663456i	$-0.230911\pi$
$857$	43.8073	1.49643	0.748215	+	0.663456i	$0.230911\pi$
$858$	0	0
$859$	−53.5322	−1.82650	−0.913248	−	0.407404i	$-0.866434\pi$
$859$	−53.5322	−1.82650	−0.913248	+	0.407404i	$0.866434\pi$
$860$	48.3334	1.64816
$861$	0	0
$862$	−27.3887	−0.932864
$863$	−37.9235	−1.29093	−0.645465	−	0.763790i	$-0.723336\pi$
$863$	−37.9235	−1.29093	−0.645465	+	0.763790i	$0.723336\pi$
$864$	0	0
$865$	−0.545746	−0.0185559
$866$	10.3128	0.350444
$867$	0	0
$868$	−31.5405	−1.07055
$869$	9.71332	0.329502
$870$	0	0
$871$	−67.4837	−2.28660
$872$	20.2638	0.686220
$873$	0	0
$874$	−25.9305	−0.877111
$875$	36.2769	1.22638
$876$	0	0
$877$	−37.4073	−1.26315	−0.631577	−	0.775313i	$-0.717592\pi$
$877$	−37.4073	−1.26315	−0.631577	+	0.775313i	$0.717592\pi$
$878$	9.27771	0.313108
$879$	0	0
$880$	−0.107790	−0.00363361
$881$	15.4415	0.520238	0.260119	−	0.965577i	$-0.416238\pi$
$881$	15.4415	0.520238	0.260119	+	0.965577i	$0.416238\pi$
$882$	0	0
$883$	13.7234	0.461831	0.230915	−	0.972974i	$-0.425828\pi$
$883$	13.7234	0.461831	0.230915	+	0.972974i	$0.425828\pi$
$884$	−19.0180	−0.639646
$885$	0	0
$886$	−22.8046	−0.766134
$887$	11.1814	0.375436	0.187718	−	0.982223i	$-0.439891\pi$
$887$	11.1814	0.375436	0.187718	+	0.982223i	$0.439891\pi$
$888$	0	0
$889$	6.76972	0.227049
$890$	30.5392	1.02368
$891$	0	0
$892$	−16.7508	−0.560858
$893$	47.1558	1.57801
$894$	0	0
$895$	50.2057	1.67819
$896$	32.4723	1.08482
$897$	0	0
$898$	15.0437	0.502016
$899$	2.85371	0.0951766
$900$	0	0
$901$	−27.3487	−0.911117
$902$	3.19045	0.106230
$903$	0	0
$904$	56.5506	1.88085
$905$	5.94316	0.197557
$906$	0	0
$907$	−30.2905	−1.00578	−0.502890	−	0.864350i	$-0.667730\pi$
$907$	−30.2905	−1.00578	−0.502890	+	0.864350i	$0.667730\pi$
$908$	11.0129	0.365475
$909$	0	0
$910$	73.8861	2.44930
$911$	33.8203	1.12052	0.560259	−	0.828318i	$-0.310702\pi$
$911$	33.8203	1.12052	0.560259	+	0.828318i	$0.310702\pi$
$912$	0	0
$913$	10.8261	0.358291
$914$	−6.40367	−0.211815
$915$	0	0
$916$	34.1636	1.12880
$917$	−7.73533	−0.255443
$918$	0	0
$919$	−44.1324	−1.45579	−0.727897	−	0.685687i	$-0.759502\pi$
$919$	−44.1324	−1.45579	−0.727897	+	0.685687i	$0.759502\pi$
$920$	76.9523	2.53704
$921$	0	0
$922$	17.5903	0.579306
$923$	29.5905	0.973982
$924$	0	0
$925$	3.90546	0.128411
$926$	19.7620	0.649421
$927$	0	0
$928$	−2.91585	−0.0957176
$929$	1.07558	0.0352885	0.0176443	−	0.999844i	$-0.494383\pi$
$929$	1.07558	0.0352885	0.0176443	+	0.999844i	$0.494383\pi$
$930$	0	0
$931$	−53.2371	−1.74478
$932$	6.37485	0.208815
$933$	0	0
$934$	9.03347	0.295584
$935$	−10.1060	−0.330500
$936$	0	0
$937$	−23.3023	−0.761254	−0.380627	−	0.924729i	$-0.624292\pi$
$937$	−23.3023	−0.761254	−0.380627	+	0.924729i	$0.624292\pi$
$938$	50.6376	1.65338
$939$	0	0
$940$	−53.6362	−1.74942
$941$	−17.6387	−0.575006	−0.287503	−	0.957780i	$-0.592825\pi$
$941$	−17.6387	−0.575006	−0.287503	+	0.957780i	$0.592825\pi$
$942$	0	0
$943$	28.5540	0.929845
$944$	−0.0403827	−0.00131434
$945$	0	0
$946$	9.66107	0.314109
$947$	−4.32732	−0.140619	−0.0703095	−	0.997525i	$-0.522399\pi$
$947$	−4.32732	−0.140619	−0.0703095	+	0.997525i	$0.522399\pi$
$948$	0	0
$949$	45.9729	1.49234
$950$	−24.1924	−0.784904
$951$	0	0
$952$	37.2330	1.20673
$953$	3.37233	0.109240	0.0546202	−	0.998507i	$-0.482605\pi$
$953$	3.37233	0.109240	0.0546202	+	0.998507i	$0.482605\pi$
$954$	0	0
$955$	51.5969	1.66964
$956$	20.9444	0.677391
$957$	0	0
$958$	−2.50980	−0.0810881
$959$	20.5308	0.662974
$960$	0	0
$961$	−0.205803	−0.00663879
$962$	2.48008	0.0799609
$963$	0	0
$964$	−6.68454	−0.215295
$965$	58.2288	1.87445
$966$	0	0
$967$	−35.3973	−1.13830	−0.569150	−	0.822234i	$-0.692728\pi$
$967$	−35.3973	−1.13830	−0.569150	+	0.822234i	$0.692728\pi$
$968$	−2.82166	−0.0906915
$969$	0	0
$970$	49.0025	1.57338
$971$	25.4174	0.815684	0.407842	−	0.913052i	$-0.366281\pi$
$971$	25.4174	0.815684	0.407842	+	0.913052i	$0.366281\pi$
$972$	0	0
$973$	26.1546	0.838477
$974$	18.5136	0.593215
$975$	0	0
$976$	0.0307781	0.000985183 0
$977$	−34.8842	−1.11604	−0.558022	−	0.829826i	$-0.688439\pi$
$977$	−34.8842	−1.11604	−0.558022	+	0.829826i	$0.688439\pi$
$978$	0	0
$979$	−10.0220	−0.320305
$980$	60.5531	1.93430
$981$	0	0
$982$	−1.52792	−0.0487580
$983$	−18.0698	−0.576336	−0.288168	−	0.957580i	$-0.593046\pi$
$983$	−18.0698	−0.576336	−0.288168	+	0.957580i	$0.593046\pi$
$984$	0	0
$985$	17.9253	0.571147
$986$	−1.29116	−0.0411190
$987$	0	0
$988$	25.2227	0.802440
$989$	86.4649	2.74942
$990$	0	0
$991$	−40.6820	−1.29231	−0.646153	−	0.763208i	$-0.723623\pi$
$991$	−40.6820	−1.29231	−0.646153	+	0.763208i	$0.723623\pi$
$992$	−31.4648	−0.999008
$993$	0	0
$994$	−22.2037	−0.704259
$995$	−71.1290	−2.25494
$996$	0	0
$997$	4.06368	0.128698	0.0643491	−	0.997927i	$-0.479503\pi$
$997$	4.06368	0.128698	0.0643491	+	0.997927i	$0.479503\pi$
$998$	11.9900	0.379536
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6039.2.a.m.1.17	✓	25	1.1	even	1	trivial
6039.2.a.p.1.9	yes	25	3.2	odd	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)

\(f(q)\)	\(=\)	\(q+0.870092 q^{2} -1.24294 q^{4} -3.50217 q^{5} -4.57282 q^{7} -2.82166 q^{8} +O(q^{10})\)	\(q+0.870092 q^{2} -1.24294 q^{4} -3.50217 q^{5} -4.57282 q^{7} -2.82166 q^{8} -3.04721 q^{10} +1.00000 q^{11} +5.30244 q^{13} -3.97878 q^{14} +0.0307781 q^{16} +2.88563 q^{17} -3.82706 q^{19} +4.35299 q^{20} +0.870092 q^{22} +7.78717 q^{23} +7.26521 q^{25} +4.61361 q^{26} +5.68374 q^{28} -0.514252 q^{29} -5.54925 q^{31} +5.67009 q^{32} +2.51076 q^{34} +16.0148 q^{35} +0.537557 q^{37} -3.32989 q^{38} +9.88193 q^{40} +3.66679 q^{41} +11.1035 q^{43} -1.24294 q^{44} +6.77556 q^{46} -12.3217 q^{47} +13.9107 q^{49} +6.32140 q^{50} -6.59061 q^{52} -9.47756 q^{53} -3.50217 q^{55} +12.9029 q^{56} -0.447446 q^{58} -1.31206 q^{59} +1.00000 q^{61} -4.82836 q^{62} +4.87195 q^{64} -18.5701 q^{65} -12.7269 q^{67} -3.58666 q^{68} +13.9344 q^{70} +5.58054 q^{71} +8.67013 q^{73} +0.467724 q^{74} +4.75680 q^{76} -4.57282 q^{77} +9.71332 q^{79} -0.107790 q^{80} +3.19045 q^{82} +10.8261 q^{83} -10.1060 q^{85} +9.66107 q^{86} -2.82166 q^{88} -10.0220 q^{89} -24.2471 q^{91} -9.67898 q^{92} -10.7210 q^{94} +13.4030 q^{95} -16.0811 q^{97} +12.1036 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q - 5 q^{2} + 25 q^{4} - 12 q^{5} - 4 q^{7} - 15 q^{8}+O(q^{10}) \) 25 * q - 5 * q^2 + 25 * q^4 - 12 * q^5 - 4 * q^7 - 15 * q^8	\( 25 q - 5 q^{2} + 25 q^{4} - 12 q^{5} - 4 q^{7} - 15 q^{8} - 12 q^{10} + 25 q^{11} - 4 q^{13} - 14 q^{14} + 21 q^{16} - 16 q^{17} - 18 q^{19} - 28 q^{20} - 5 q^{22} - 8 q^{23} + 29 q^{25} - 16 q^{26} + 18 q^{28} - 28 q^{29} - 8 q^{31} - 35 q^{32} + 6 q^{34} - 22 q^{35} + 4 q^{37} + 4 q^{38} - 12 q^{40} - 58 q^{41} - 26 q^{43} + 25 q^{44} + 8 q^{46} - 20 q^{47} + 23 q^{49} - 27 q^{50} - 2 q^{52} - 36 q^{53} - 12 q^{55} - 70 q^{56} + 12 q^{58} - 18 q^{59} + 25 q^{61} - 42 q^{62} + 35 q^{64} - 76 q^{65} - 8 q^{67} - 28 q^{68} + 76 q^{70} - 24 q^{71} + 2 q^{73} - 40 q^{74} - 64 q^{76} - 4 q^{77} - 22 q^{79} - 36 q^{80} + 30 q^{82} - 14 q^{83} - 70 q^{86} - 15 q^{88} - 82 q^{89} - 6 q^{91} - 48 q^{92} - 16 q^{94} - 34 q^{95} + 16 q^{97} - 35 q^{98}+O(q^{100}) \) 25 * q - 5 * q^2 + 25 * q^4 - 12 * q^5 - 4 * q^7 - 15 * q^8 - 12 * q^10 + 25 * q^11 - 4 * q^13 - 14 * q^14 + 21 * q^16 - 16 * q^17 - 18 * q^19 - 28 * q^20 - 5 * q^22 - 8 * q^23 + 29 * q^25 - 16 * q^26 + 18 * q^28 - 28 * q^29 - 8 * q^31 - 35 * q^32 + 6 * q^34 - 22 * q^35 + 4 * q^37 + 4 * q^38 - 12 * q^40 - 58 * q^41 - 26 * q^43 + 25 * q^44 + 8 * q^46 - 20 * q^47 + 23 * q^49 - 27 * q^50 - 2 * q^52 - 36 * q^53 - 12 * q^55 - 70 * q^56 + 12 * q^58 - 18 * q^59 + 25 * q^61 - 42 * q^62 + 35 * q^64 - 76 * q^65 - 8 * q^67 - 28 * q^68 + 76 * q^70 - 24 * q^71 + 2 * q^73 - 40 * q^74 - 64 * q^76 - 4 * q^77 - 22 * q^79 - 36 * q^80 + 30 * q^82 - 14 * q^83 - 70 * q^86 - 15 * q^88 - 82 * q^89 - 6 * q^91 - 48 * q^92 - 16 * q^94 - 34 * q^95 + 16 * q^97 - 35 * q^98

Introduction

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