LMFDB - Embedded newform 6039.2.a.o.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6039, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6039.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$48.2216577807$
Analytic rank:	$0$
Dimension:	$25$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
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)

$f(q)$	$=$	$q+0.226426 q^{2} -1.94873 q^{4} +3.14756 q^{5} -2.84661 q^{7} -0.894095 q^{8} +O(q^{10})$	\(q+0.226426 q^{2} -1.94873 q^{4} +3.14756 q^{5} -2.84661 q^{7} -0.894095 q^{8} +0.712690 q^{10} +1.00000 q^{11} +3.54011 q^{13} -0.644547 q^{14} +3.69502 q^{16} +6.41012 q^{17} +0.806922 q^{19} -6.13376 q^{20} +0.226426 q^{22} +4.07437 q^{23} +4.90716 q^{25} +0.801572 q^{26} +5.54729 q^{28} -0.826120 q^{29} +0.425981 q^{31} +2.62484 q^{32} +1.45142 q^{34} -8.95990 q^{35} +5.03556 q^{37} +0.182708 q^{38} -2.81422 q^{40} -9.88868 q^{41} +1.86637 q^{43} -1.94873 q^{44} +0.922542 q^{46} -7.37569 q^{47} +1.10322 q^{49} +1.11111 q^{50} -6.89872 q^{52} -0.550944 q^{53} +3.14756 q^{55} +2.54514 q^{56} -0.187055 q^{58} -4.11215 q^{59} -1.00000 q^{61} +0.0964531 q^{62} -6.79570 q^{64} +11.1427 q^{65} +3.12765 q^{67} -12.4916 q^{68} -2.02875 q^{70} -10.1748 q^{71} +5.68838 q^{73} +1.14018 q^{74} -1.57247 q^{76} -2.84661 q^{77} -9.00570 q^{79} +11.6303 q^{80} -2.23905 q^{82} -8.73064 q^{83} +20.1763 q^{85} +0.422594 q^{86} -0.894095 q^{88} +9.63618 q^{89} -10.0773 q^{91} -7.93985 q^{92} -1.67005 q^{94} +2.53984 q^{95} +18.5628 q^{97} +0.249797 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q + 5 q^{2} + 25 q^{4} + 4 q^{5} + 4 q^{7} + 15 q^{8}+O(q^{10}) $ 25 * q + 5 * q^2 + 25 * q^4 + 4 * q^5 + 4 * q^7 + 15 * q^8	$ 25 q + 5 q^{2} + 25 q^{4} + 4 q^{5} + 4 q^{7} + 15 q^{8} + 25 q^{11} + 4 q^{13} + 18 q^{14} + 21 q^{16} + 20 q^{17} + 14 q^{19} + 12 q^{20} + 5 q^{22} + 20 q^{23} + 13 q^{25} + 16 q^{26} - 14 q^{28} + 28 q^{29} - 12 q^{31} + 35 q^{32} + 6 q^{34} + 10 q^{35} - 8 q^{37} + 32 q^{38} + 24 q^{40} + 26 q^{41} + 18 q^{43} + 25 q^{44} + 4 q^{46} + 12 q^{47} + 23 q^{49} + 43 q^{50} + 22 q^{52} + 36 q^{53} + 4 q^{55} + 26 q^{56} - 20 q^{58} + 46 q^{59} - 25 q^{61} - 14 q^{62} - 13 q^{64} + 60 q^{65} - 20 q^{67} + 44 q^{68} - 20 q^{70} + 52 q^{71} + 6 q^{73} + 32 q^{74} + 4 q^{77} + 26 q^{79} + 52 q^{80} + 6 q^{82} + 38 q^{83} - 4 q^{85} + 34 q^{86} + 15 q^{88} + 82 q^{89} - 58 q^{91} + 36 q^{92} + 16 q^{94} + 30 q^{95} + 35 q^{98}+O(q^{100}) $ 25 * q + 5 * q^2 + 25 * q^4 + 4 * q^5 + 4 * q^7 + 15 * q^8 + 25 * q^11 + 4 * q^13 + 18 * q^14 + 21 * q^16 + 20 * q^17 + 14 * q^19 + 12 * q^20 + 5 * q^22 + 20 * q^23 + 13 * q^25 + 16 * q^26 - 14 * q^28 + 28 * q^29 - 12 * q^31 + 35 * q^32 + 6 * q^34 + 10 * q^35 - 8 * q^37 + 32 * q^38 + 24 * q^40 + 26 * q^41 + 18 * q^43 + 25 * q^44 + 4 * q^46 + 12 * q^47 + 23 * q^49 + 43 * q^50 + 22 * q^52 + 36 * q^53 + 4 * q^55 + 26 * q^56 - 20 * q^58 + 46 * q^59 - 25 * q^61 - 14 * q^62 - 13 * q^64 + 60 * q^65 - 20 * q^67 + 44 * q^68 - 20 * q^70 + 52 * q^71 + 6 * q^73 + 32 * q^74 + 4 * q^77 + 26 * q^79 + 52 * q^80 + 6 * q^82 + 38 * q^83 - 4 * q^85 + 34 * q^86 + 15 * q^88 + 82 * q^89 - 58 * q^91 + 36 * q^92 + 16 * q^94 + 30 * q^95 + 35 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0.226426	0.160107	0.0800536	−	0.996791i	$-0.474491\pi$
$2$	0.226426	0.160107	0.0800536	+	0.996791i	$0.474491\pi$
$3$	0	0
$3$	0	0
$4$	−1.94873	−0.974366
$5$	3.14756	1.40763	0.703817	−	0.710382i	$-0.251478\pi$
$5$	3.14756	1.40763	0.703817	+	0.710382i	$0.251478\pi$
$6$	0	0
$7$	−2.84661	−1.07592	−0.537960	−	0.842971i	$-0.680805\pi$
$7$	−2.84661	−1.07592	−0.537960	+	0.842971i	$0.680805\pi$
$8$	−0.894095	−0.316110
$9$	0	0
$10$	0.712690	0.225372
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	3.54011	0.981849	0.490924	−	0.871202i	$-0.336659\pi$
$13$	3.54011	0.981849	0.490924	+	0.871202i	$0.336659\pi$
$14$	−0.644547	−0.172263
$15$	0	0
$16$	3.69502	0.923754
$17$	6.41012	1.55468	0.777341	−	0.629079i	$-0.216568\pi$
$17$	6.41012	1.55468	0.777341	+	0.629079i	$0.216568\pi$
$18$	0	0
$19$	0.806922	0.185121	0.0925603	−	0.995707i	$-0.470495\pi$
$19$	0.806922	0.185121	0.0925603	+	0.995707i	$0.470495\pi$
$20$	−6.13376	−1.37155
$21$	0	0
$22$	0.226426	0.0482742
$23$	4.07437	0.849564	0.424782	−	0.905296i	$-0.360351\pi$
$23$	4.07437	0.849564	0.424782	+	0.905296i	$0.360351\pi$
$24$	0	0
$25$	4.90716	0.981432
$26$	0.801572	0.157201
$27$	0	0
$28$	5.54729	1.04834
$29$	−0.826120	−0.153407	−0.0767033	−	0.997054i	$-0.524439\pi$
$29$	−0.826120	−0.153407	−0.0767033	+	0.997054i	$0.524439\pi$
$30$	0	0
$31$	0.425981	0.0765084	0.0382542	−	0.999268i	$-0.487820\pi$
$31$	0.425981	0.0765084	0.0382542	+	0.999268i	$0.487820\pi$
$32$	2.62484	0.464010
$33$	0	0
$34$	1.45142	0.248916
$35$	−8.95990	−1.51450
$36$	0	0
$37$	5.03556	0.827841	0.413921	−	0.910313i	$-0.364159\pi$
$37$	5.03556	0.827841	0.413921	+	0.910313i	$0.364159\pi$
$38$	0.182708	0.0296392
$39$	0	0
$40$	−2.81422	−0.444968
$41$	−9.88868	−1.54435	−0.772176	−	0.635408i	$-0.780832\pi$
$41$	−9.88868	−1.54435	−0.772176	+	0.635408i	$0.780832\pi$
$42$	0	0
$43$	1.86637	0.284618	0.142309	−	0.989822i	$-0.454547\pi$
$43$	1.86637	0.284618	0.142309	+	0.989822i	$0.454547\pi$
$44$	−1.94873	−0.293782
$45$	0	0
$46$	0.922542	0.136021
$47$	−7.37569	−1.07586	−0.537928	−	0.842991i	$-0.680793\pi$
$47$	−7.37569	−1.07586	−0.537928	+	0.842991i	$0.680793\pi$
$48$	0	0
$49$	1.10322	0.157602
$50$	1.11111	0.157134
$51$	0	0
$52$	−6.89872	−0.956680
$53$	−0.550944	−0.0756780	−0.0378390	−	0.999284i	$-0.512047\pi$
$53$	−0.550944	−0.0756780	−0.0378390	+	0.999284i	$0.512047\pi$
$54$	0	0
$55$	3.14756	0.424417
$56$	2.54514	0.340109
$57$	0	0
$58$	−0.187055	−0.0245615
$59$	−4.11215	−0.535356	−0.267678	−	0.963508i	$-0.586256\pi$
$59$	−4.11215	−0.535356	−0.267678	+	0.963508i	$0.586256\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	0.0964531	0.0122496
$63$	0	0
$64$	−6.79570	−0.849463
$65$	11.1427	1.38208
$66$	0	0
$67$	3.12765	0.382104	0.191052	−	0.981580i	$-0.438810\pi$
$67$	3.12765	0.382104	0.191052	+	0.981580i	$0.438810\pi$
$68$	−12.4916	−1.51483
$69$	0	0
$70$	−2.02875	−0.242483
$71$	−10.1748	−1.20753	−0.603766	−	0.797162i	$-0.706334\pi$
$71$	−10.1748	−1.20753	−0.603766	+	0.797162i	$0.706334\pi$
$72$	0	0
$73$	5.68838	0.665774	0.332887	−	0.942967i	$-0.391977\pi$
$73$	5.68838	0.665774	0.332887	+	0.942967i	$0.391977\pi$
$74$	1.14018	0.132543
$75$	0	0
$76$	−1.57247	−0.180375
$77$	−2.84661	−0.324402
$78$	0	0
$79$	−9.00570	−1.01322	−0.506610	−	0.862175i	$-0.669102\pi$
$79$	−9.00570	−1.01322	−0.506610	+	0.862175i	$0.669102\pi$
$80$	11.6303	1.30031
$81$	0	0
$82$	−2.23905	−0.247262
$83$	−8.73064	−0.958312	−0.479156	−	0.877730i	$-0.659057\pi$
$83$	−8.73064	−0.958312	−0.479156	+	0.877730i	$0.659057\pi$
$84$	0	0
$85$	20.1763	2.18842
$86$	0.422594	0.0455695
$87$	0	0
$88$	−0.894095	−0.0953109
$89$	9.63618	1.02143	0.510717	−	0.859749i	$-0.329380\pi$
$89$	9.63618	1.02143	0.510717	+	0.859749i	$0.329380\pi$
$90$	0	0
$91$	−10.0773	−1.05639
$92$	−7.93985	−0.827786
$93$	0	0
$94$	−1.67005	−0.172252
$95$	2.53984	0.260582
$96$	0	0
$97$	18.5628	1.88477	0.942385	−	0.334529i	$-0.108577\pi$
$97$	18.5628	1.88477	0.942385	+	0.334529i	$0.108577\pi$
$98$	0.249797	0.0252333
$99$	0	0
$100$	−9.56274	−0.956274
$101$	5.06261	0.503749	0.251874	−	0.967760i	$-0.418953\pi$
$101$	5.06261	0.503749	0.251874	+	0.967760i	$0.418953\pi$
$102$	0	0
$103$	−17.5658	−1.73081	−0.865407	−	0.501069i	$-0.832940\pi$
$103$	−17.5658	−1.73081	−0.865407	+	0.501069i	$0.832940\pi$
$104$	−3.16519	−0.310373
$105$	0	0
$106$	−0.124748	−0.0121166
$107$	15.8560	1.53286	0.766429	−	0.642329i	$-0.222032\pi$
$107$	15.8560	1.53286	0.766429	+	0.642329i	$0.222032\pi$
$108$	0	0
$109$	12.2883	1.17700	0.588501	−	0.808497i	$-0.299718\pi$
$109$	12.2883	1.17700	0.588501	+	0.808497i	$0.299718\pi$
$110$	0.712690	0.0679523
$111$	0	0
$112$	−10.5183	−0.993885
$113$	16.2578	1.52940	0.764702	−	0.644385i	$-0.222886\pi$
$113$	16.2578	1.52940	0.764702	+	0.644385i	$0.222886\pi$
$114$	0	0
$115$	12.8243	1.19587
$116$	1.60989	0.149474
$117$	0	0
$118$	−0.931096	−0.0857144
$119$	−18.2471	−1.67271
$120$	0	0
$121$	1.00000	0.0909091
$122$	−0.226426	−0.0204996
$123$	0	0
$124$	−0.830122	−0.0745471
$125$	−0.292220	−0.0261369
$126$	0	0
$127$	18.1829	1.61347	0.806735	−	0.590913i	$-0.201232\pi$
$127$	18.1829	1.61347	0.806735	+	0.590913i	$0.201232\pi$
$128$	−6.78840	−0.600015
$129$	0	0
$130$	2.52300	0.221282
$131$	8.13584	0.710832	0.355416	−	0.934708i	$-0.384339\pi$
$131$	8.13584	0.710832	0.355416	+	0.934708i	$0.384339\pi$
$132$	0	0
$133$	−2.29700	−0.199175
$134$	0.708182	0.0611776
$135$	0	0
$136$	−5.73126	−0.491451
$137$	10.4806	0.895418	0.447709	−	0.894179i	$-0.352240\pi$
$137$	10.4806	0.895418	0.447709	+	0.894179i	$0.352240\pi$
$138$	0	0
$139$	8.05311	0.683056	0.341528	−	0.939872i	$-0.389056\pi$
$139$	8.05311	0.683056	0.341528	+	0.939872i	$0.389056\pi$
$140$	17.4604	1.47568
$141$	0	0
$142$	−2.30385	−0.193335
$143$	3.54011	0.296039
$144$	0	0
$145$	−2.60027	−0.215940
$146$	1.28800	0.106595
$147$	0	0
$148$	−9.81296	−0.806620
$149$	5.96964	0.489052	0.244526	−	0.969643i	$-0.421368\pi$
$149$	5.96964	0.489052	0.244526	+	0.969643i	$0.421368\pi$
$150$	0	0
$151$	15.8066	1.28632	0.643161	−	0.765731i	$-0.277623\pi$
$151$	15.8066	1.28632	0.643161	+	0.765731i	$0.277623\pi$
$152$	−0.721465	−0.0585186
$153$	0	0
$154$	−0.644547	−0.0519391
$155$	1.34080	0.107696
$156$	0	0
$157$	−18.7088	−1.49312	−0.746561	−	0.665316i	$-0.768297\pi$
$157$	−18.7088	−1.49312	−0.746561	+	0.665316i	$0.768297\pi$
$158$	−2.03912	−0.162224
$159$	0	0
$160$	8.26184	0.653156
$161$	−11.5982	−0.914062
$162$	0	0
$163$	20.9868	1.64381	0.821907	−	0.569621i	$-0.192910\pi$
$163$	20.9868	1.64381	0.821907	+	0.569621i	$0.192910\pi$
$164$	19.2704	1.50476
$165$	0	0
$166$	−1.97684	−0.153433
$167$	−3.23562	−0.250380	−0.125190	−	0.992133i	$-0.539954\pi$
$167$	−3.23562	−0.250380	−0.125190	+	0.992133i	$0.539954\pi$
$168$	0	0
$169$	−0.467646	−0.0359728
$170$	4.56843	0.350383
$171$	0	0
$172$	−3.63705	−0.277322
$173$	−0.398018	−0.0302607	−0.0151304	−	0.999886i	$-0.504816\pi$
$173$	−0.398018	−0.0302607	−0.0151304	+	0.999886i	$0.504816\pi$
$174$	0	0
$175$	−13.9688	−1.05594
$176$	3.69502	0.278522
$177$	0	0
$178$	2.18188	0.163539
$179$	−1.54070	−0.115157	−0.0575785	−	0.998341i	$-0.518338\pi$
$179$	−1.54070	−0.115157	−0.0575785	+	0.998341i	$0.518338\pi$
$180$	0	0
$181$	14.2718	1.06081	0.530406	−	0.847744i	$-0.322039\pi$
$181$	14.2718	1.06081	0.530406	+	0.847744i	$0.322039\pi$
$182$	−2.28177	−0.169136
$183$	0	0
$184$	−3.64287	−0.268556
$185$	15.8498	1.16530
$186$	0	0
$187$	6.41012	0.468754
$188$	14.3732	1.04828
$189$	0	0
$190$	0.575085	0.0417211
$191$	−6.40614	−0.463532	−0.231766	−	0.972772i	$-0.574450\pi$
$191$	−6.40614	−0.463532	−0.231766	+	0.972772i	$0.574450\pi$
$192$	0	0
$193$	2.95024	0.212363	0.106182	−	0.994347i	$-0.466138\pi$
$193$	2.95024	0.212363	0.106182	+	0.994347i	$0.466138\pi$
$194$	4.20311	0.301766
$195$	0	0
$196$	−2.14987	−0.153562
$197$	20.0123	1.42582	0.712910	−	0.701255i	$-0.247377\pi$
$197$	20.0123	1.42582	0.712910	+	0.701255i	$0.247377\pi$
$198$	0	0
$199$	−16.2136	−1.14935	−0.574677	−	0.818380i	$-0.694872\pi$
$199$	−16.2136	−1.14935	−0.574677	+	0.818380i	$0.694872\pi$
$200$	−4.38747	−0.310241
$201$	0	0
$202$	1.14631	0.0806539
$203$	2.35165	0.165053
$204$	0	0
$205$	−31.1253	−2.17388
$206$	−3.97736	−0.277116
$207$	0	0
$208$	13.0808	0.906987
$209$	0.806922	0.0558160
$210$	0	0
$211$	−6.36140	−0.437937	−0.218968	−	0.975732i	$-0.570269\pi$
$211$	−6.36140	−0.437937	−0.218968	+	0.975732i	$0.570269\pi$
$212$	1.07364	0.0737380
$213$	0	0
$214$	3.59021	0.245422
$215$	5.87451	0.400638
$216$	0	0
$217$	−1.21260	−0.0823168
$218$	2.78238	0.188447
$219$	0	0
$220$	−6.13376	−0.413538
$221$	22.6925	1.52646
$222$	0	0
$223$	17.9918	1.20482	0.602411	−	0.798186i	$-0.294207\pi$
$223$	17.9918	1.20482	0.602411	+	0.798186i	$0.294207\pi$
$224$	−7.47190	−0.499237
$225$	0	0
$226$	3.68118	0.244869
$227$	16.7803	1.11375	0.556873	−	0.830598i	$-0.312001\pi$
$227$	16.7803	1.11375	0.556873	+	0.830598i	$0.312001\pi$
$228$	0	0
$229$	−27.6343	−1.82613	−0.913063	−	0.407818i	$-0.866290\pi$
$229$	−27.6343	−1.82613	−0.913063	+	0.407818i	$0.866290\pi$
$230$	2.90376	0.191468
$231$	0	0
$232$	0.738630	0.0484934
$233$	1.36019	0.0891090	0.0445545	−	0.999007i	$-0.485813\pi$
$233$	1.36019	0.0891090	0.0445545	+	0.999007i	$0.485813\pi$
$234$	0	0
$235$	−23.2155	−1.51441
$236$	8.01347	0.521632
$237$	0	0
$238$	−4.13163	−0.267814
$239$	22.8575	1.47853	0.739264	−	0.673416i	$-0.235174\pi$
$239$	22.8575	1.47853	0.739264	+	0.673416i	$0.235174\pi$
$240$	0	0
$241$	−15.2565	−0.982757	−0.491379	−	0.870946i	$-0.663507\pi$
$241$	−15.2565	−0.982757	−0.491379	+	0.870946i	$0.663507\pi$
$242$	0.226426	0.0145552
$243$	0	0
$244$	1.94873	0.124755
$245$	3.47245	0.221846
$246$	0	0
$247$	2.85659	0.181761
$248$	−0.380867	−0.0241851
$249$	0	0
$250$	−0.0661661	−0.00418471
$251$	28.2558	1.78349	0.891745	−	0.452537i	$-0.149481\pi$
$251$	28.2558	1.78349	0.891745	+	0.452537i	$0.149481\pi$
$252$	0	0
$253$	4.07437	0.256153
$254$	4.11708	0.258328
$255$	0	0
$256$	12.0543	0.753396
$257$	−4.14060	−0.258284	−0.129142	−	0.991626i	$-0.541222\pi$
$257$	−4.14060	−0.258284	−0.129142	+	0.991626i	$0.541222\pi$
$258$	0	0
$259$	−14.3343	−0.890691
$260$	−21.7142	−1.34665
$261$	0	0
$262$	1.84217	0.113809
$263$	−6.91648	−0.426488	−0.213244	−	0.976999i	$-0.568403\pi$
$263$	−6.91648	−0.426488	−0.213244	+	0.976999i	$0.568403\pi$
$264$	0	0
$265$	−1.73413	−0.106527
$266$	−0.520100	−0.0318894
$267$	0	0
$268$	−6.09496	−0.372309
$269$	15.0751	0.919144	0.459572	−	0.888141i	$-0.348003\pi$
$269$	15.0751	0.919144	0.459572	+	0.888141i	$0.348003\pi$
$270$	0	0
$271$	−18.3019	−1.11176	−0.555879	−	0.831263i	$-0.687618\pi$
$271$	−18.3019	−1.11176	−0.555879	+	0.831263i	$0.687618\pi$
$272$	23.6855	1.43614
$273$	0	0
$274$	2.37308	0.143363
$275$	4.90716	0.295913
$276$	0	0
$277$	11.7347	0.705072	0.352536	−	0.935798i	$-0.385319\pi$
$277$	11.7347	0.705072	0.352536	+	0.935798i	$0.385319\pi$
$278$	1.82343	0.109362
$279$	0	0
$280$	8.01101	0.478749
$281$	−21.2986	−1.27057	−0.635283	−	0.772280i	$-0.719116\pi$
$281$	−21.2986	−1.27057	−0.635283	+	0.772280i	$0.719116\pi$
$282$	0	0
$283$	−8.04474	−0.478210	−0.239105	−	0.970994i	$-0.576854\pi$
$283$	−8.04474	−0.478210	−0.239105	+	0.970994i	$0.576854\pi$
$284$	19.8280	1.17658
$285$	0	0
$286$	0.801572	0.0473979
$287$	28.1493	1.66160
$288$	0	0
$289$	24.0896	1.41704
$290$	−0.588767	−0.0345736
$291$	0	0
$292$	−11.0851	−0.648708
$293$	−22.1201	−1.29227	−0.646134	−	0.763224i	$-0.723615\pi$
$293$	−22.1201	−1.29227	−0.646134	+	0.763224i	$0.723615\pi$
$294$	0	0
$295$	−12.9432	−0.753585
$296$	−4.50227	−0.261689
$297$	0	0
$298$	1.35168	0.0783008
$299$	14.4237	0.834144
$300$	0	0
$301$	−5.31283	−0.306226
$302$	3.57902	0.205950
$303$	0	0
$304$	2.98159	0.171006
$305$	−3.14756	−0.180229
$306$	0	0
$307$	−16.5587	−0.945057	−0.472529	−	0.881315i	$-0.656659\pi$
$307$	−16.5587	−0.945057	−0.472529	+	0.881315i	$0.656659\pi$
$308$	5.54729	0.316086
$309$	0	0
$310$	0.303592	0.0172429
$311$	−33.8654	−1.92033	−0.960164	−	0.279436i	$-0.909852\pi$
$311$	−33.8654	−1.92033	−0.960164	+	0.279436i	$0.909852\pi$
$312$	0	0
$313$	29.5112	1.66807	0.834037	−	0.551708i	$-0.186024\pi$
$313$	29.5112	1.66807	0.834037	+	0.551708i	$0.186024\pi$
$314$	−4.23615	−0.239060
$315$	0	0
$316$	17.5497	0.987248
$317$	23.8779	1.34112	0.670560	−	0.741856i	$-0.266054\pi$
$317$	23.8779	1.34112	0.670560	+	0.741856i	$0.266054\pi$
$318$	0	0
$319$	−0.826120	−0.0462538
$320$	−21.3899	−1.19573
$321$	0	0
$322$	−2.62612	−0.146348
$323$	5.17247	0.287804
$324$	0	0
$325$	17.3719	0.963618
$326$	4.75196	0.263187
$327$	0	0
$328$	8.84142	0.488186
$329$	20.9958	1.15753
$330$	0	0
$331$	0.956704	0.0525852	0.0262926	−	0.999654i	$-0.491630\pi$
$331$	0.956704	0.0525852	0.0262926	+	0.999654i	$0.491630\pi$
$332$	17.0137	0.933746
$333$	0	0
$334$	−0.732628	−0.0400876
$335$	9.84449	0.537862
$336$	0	0
$337$	15.0654	0.820666	0.410333	−	0.911936i	$-0.365412\pi$
$337$	15.0654	0.820666	0.410333	+	0.911936i	$0.365412\pi$
$338$	−0.105887	−0.00575951
$339$	0	0
$340$	−39.3181	−2.13232
$341$	0.425981	0.0230681
$342$	0	0
$343$	16.7859	0.906352
$344$	−1.66871	−0.0899708
$345$	0	0
$346$	−0.0901216	−0.00484497
$347$	21.4639	1.15224	0.576121	−	0.817365i	$-0.304566\pi$
$347$	21.4639	1.15224	0.576121	+	0.817365i	$0.304566\pi$
$348$	0	0
$349$	−24.0720	−1.28854	−0.644271	−	0.764797i	$-0.722839\pi$
$349$	−24.0720	−1.28854	−0.644271	+	0.764797i	$0.722839\pi$
$350$	−3.16290	−0.169064
$351$	0	0
$352$	2.62484	0.139904
$353$	−14.7507	−0.785099	−0.392549	−	0.919731i	$-0.628407\pi$
$353$	−14.7507	−0.785099	−0.392549	+	0.919731i	$0.628407\pi$
$354$	0	0
$355$	−32.0260	−1.69976
$356$	−18.7783	−0.995249
$357$	0	0
$358$	−0.348853	−0.0184375
$359$	23.0106	1.21445	0.607227	−	0.794529i	$-0.292282\pi$
$359$	23.0106	1.21445	0.607227	+	0.794529i	$0.292282\pi$
$360$	0	0
$361$	−18.3489	−0.965730
$362$	3.23150	0.169844
$363$	0	0
$364$	19.6380	1.02931
$365$	17.9045	0.937166
$366$	0	0
$367$	−2.19366	−0.114508	−0.0572542	−	0.998360i	$-0.518235\pi$
$367$	−2.19366	−0.114508	−0.0572542	+	0.998360i	$0.518235\pi$
$368$	15.0549	0.784788
$369$	0	0
$370$	3.58880	0.186573
$371$	1.56832	0.0814234
$372$	0	0
$373$	−2.58790	−0.133996	−0.0669982	−	0.997753i	$-0.521342\pi$
$373$	−2.58790	−0.133996	−0.0669982	+	0.997753i	$0.521342\pi$
$374$	1.45142	0.0750510
$375$	0	0
$376$	6.59457	0.340089
$377$	−2.92455	−0.150622
$378$	0	0
$379$	7.49512	0.384998	0.192499	−	0.981297i	$-0.438341\pi$
$379$	7.49512	0.384998	0.192499	+	0.981297i	$0.438341\pi$
$380$	−4.94946	−0.253902
$381$	0	0
$382$	−1.45052	−0.0742149
$383$	−19.6883	−1.00602	−0.503012	−	0.864279i	$-0.667775\pi$
$383$	−19.6883	−1.00602	−0.503012	+	0.864279i	$0.667775\pi$
$384$	0	0
$385$	−8.95990	−0.456639
$386$	0.668012	0.0340009
$387$	0	0
$388$	−36.1740	−1.83646
$389$	−1.86633	−0.0946265	−0.0473132	−	0.998880i	$-0.515066\pi$
$389$	−1.86633	−0.0946265	−0.0473132	+	0.998880i	$0.515066\pi$
$390$	0	0
$391$	26.1172	1.32080
$392$	−0.986381	−0.0498197
$393$	0	0
$394$	4.53131	0.228284
$395$	−28.3460	−1.42624
$396$	0	0
$397$	−30.3910	−1.52528	−0.762639	−	0.646824i	$-0.776097\pi$
$397$	−30.3910	−1.52528	−0.762639	+	0.646824i	$0.776097\pi$
$398$	−3.67119	−0.184020
$399$	0	0
$400$	18.1320	0.906602
$401$	−4.27044	−0.213256	−0.106628	−	0.994299i	$-0.534005\pi$
$401$	−4.27044	−0.213256	−0.106628	+	0.994299i	$0.534005\pi$
$402$	0	0
$403$	1.50802	0.0751197
$404$	−9.86568	−0.490836
$405$	0	0
$406$	0.532473	0.0264262
$407$	5.03556	0.249604
$408$	0	0
$409$	−26.0156	−1.28639	−0.643195	−	0.765703i	$-0.722391\pi$
$409$	−26.0156	−1.28639	−0.643195	+	0.765703i	$0.722391\pi$
$410$	−7.04756	−0.348054
$411$	0	0
$412$	34.2311	1.68645
$413$	11.7057	0.576000
$414$	0	0
$415$	−27.4802	−1.34895
$416$	9.29220	0.455588
$417$	0	0
$418$	0.182708	0.00893655
$419$	11.0956	0.542056	0.271028	−	0.962571i	$-0.412636\pi$
$419$	11.0956	0.542056	0.271028	+	0.962571i	$0.412636\pi$
$420$	0	0
$421$	3.78576	0.184506	0.0922532	−	0.995736i	$-0.470593\pi$
$421$	3.78576	0.184506	0.0922532	+	0.995736i	$0.470593\pi$
$422$	−1.44039	−0.0701169
$423$	0	0
$424$	0.492596	0.0239226
$425$	31.4555	1.52582
$426$	0	0
$427$	2.84661	0.137757
$428$	−30.8991	−1.49356
$429$	0	0
$430$	1.33014	0.0641451
$431$	2.55580	0.123109	0.0615544	−	0.998104i	$-0.480394\pi$
$431$	2.55580	0.123109	0.0615544	+	0.998104i	$0.480394\pi$
$432$	0	0
$433$	−32.6685	−1.56995	−0.784975	−	0.619528i	$-0.787324\pi$
$433$	−32.6685	−1.56995	−0.784975	+	0.619528i	$0.787324\pi$
$434$	−0.274565	−0.0131795
$435$	0	0
$436$	−23.9465	−1.14683
$437$	3.28770	0.157272
$438$	0	0
$439$	−13.0101	−0.620937	−0.310469	−	0.950584i	$-0.600486\pi$
$439$	−13.0101	−0.620937	−0.310469	+	0.950584i	$0.600486\pi$
$440$	−2.81422	−0.134163
$441$	0	0
$442$	5.13817	0.244398
$443$	−16.5384	−0.785763	−0.392882	−	0.919589i	$-0.628522\pi$
$443$	−16.5384	−0.785763	−0.392882	+	0.919589i	$0.628522\pi$
$444$	0	0
$445$	30.3305	1.43780
$446$	4.07382	0.192901
$447$	0	0
$448$	19.3447	0.913953
$449$	0.241989	0.0114202	0.00571008	−	0.999984i	$-0.498182\pi$
$449$	0.241989	0.0114202	0.00571008	+	0.999984i	$0.498182\pi$
$450$	0	0
$451$	−9.88868	−0.465640
$452$	−31.6820	−1.49020
$453$	0	0
$454$	3.79949	0.178319
$455$	−31.7190	−1.48701
$456$	0	0
$457$	32.8406	1.53622	0.768110	−	0.640318i	$-0.221198\pi$
$457$	32.8406	1.53622	0.768110	+	0.640318i	$0.221198\pi$
$458$	−6.25712	−0.292376
$459$	0	0
$460$	−24.9912	−1.16522
$461$	−3.59982	−0.167661	−0.0838303	−	0.996480i	$-0.526715\pi$
$461$	−3.59982	−0.167661	−0.0838303	+	0.996480i	$0.526715\pi$
$462$	0	0
$463$	13.8343	0.642937	0.321468	−	0.946920i	$-0.395824\pi$
$463$	13.8343	0.642937	0.321468	+	0.946920i	$0.395824\pi$
$464$	−3.05253	−0.141710
$465$	0	0
$466$	0.307982	0.0142670
$467$	−20.1841	−0.934010	−0.467005	−	0.884255i	$-0.654667\pi$
$467$	−20.1841	−0.934010	−0.467005	+	0.884255i	$0.654667\pi$
$468$	0	0
$469$	−8.90323	−0.411113
$470$	−5.25658	−0.242468
$471$	0	0
$472$	3.67665	0.169232
$473$	1.86637	0.0858156
$474$	0	0
$475$	3.95970	0.181683
$476$	35.5588	1.62983
$477$	0	0
$478$	5.17553	0.236723
$479$	−41.0123	−1.87390	−0.936950	−	0.349464i	$-0.886364\pi$
$479$	−41.0123	−1.87390	−0.936950	+	0.349464i	$0.886364\pi$
$480$	0	0
$481$	17.8264	0.812815
$482$	−3.45446	−0.157347
$483$	0	0
$484$	−1.94873	−0.0885787
$485$	58.4277	2.65307
$486$	0	0
$487$	20.7931	0.942223	0.471112	−	0.882074i	$-0.343853\pi$
$487$	20.7931	0.942223	0.471112	+	0.882074i	$0.343853\pi$
$488$	0.894095	0.0404738
$489$	0	0
$490$	0.786252	0.0355192
$491$	−7.31240	−0.330004	−0.165002	−	0.986293i	$-0.552763\pi$
$491$	−7.31240	−0.330004	−0.165002	+	0.986293i	$0.552763\pi$
$492$	0	0
$493$	−5.29553	−0.238499
$494$	0.646806	0.0291012
$495$	0	0
$496$	1.57401	0.0706749
$497$	28.9639	1.29921
$498$	0	0
$499$	−22.5268	−1.00844	−0.504219	−	0.863576i	$-0.668220\pi$
$499$	−22.5268	−1.00844	−0.504219	+	0.863576i	$0.668220\pi$
$500$	0.569458	0.0254669
$501$	0	0
$502$	6.39785	0.285550
$503$	28.2970	1.26170	0.630850	−	0.775905i	$-0.282706\pi$
$503$	28.2970	1.26170	0.630850	+	0.775905i	$0.282706\pi$
$504$	0	0
$505$	15.9349	0.709094
$506$	0.922542	0.0410120
$507$	0	0
$508$	−35.4336	−1.57211
$509$	8.78335	0.389315	0.194658	−	0.980871i	$-0.437640\pi$
$509$	8.78335	0.389315	0.194658	+	0.980871i	$0.437640\pi$
$510$	0	0
$511$	−16.1926	−0.716319
$512$	16.3062	0.720639
$513$	0	0
$514$	−0.937540	−0.0413531
$515$	−55.2896	−2.43635
$516$	0	0
$517$	−7.37569	−0.324383
$518$	−3.24566	−0.142606
$519$	0	0
$520$	−9.96264	−0.436891
$521$	30.7107	1.34546	0.672730	−	0.739888i	$-0.265122\pi$
$521$	30.7107	1.34546	0.672730	+	0.739888i	$0.265122\pi$
$522$	0	0
$523$	−35.4848	−1.55164	−0.775820	−	0.630954i	$-0.782664\pi$
$523$	−35.4848	−1.55164	−0.775820	+	0.630954i	$0.782664\pi$
$524$	−15.8546	−0.692610
$525$	0	0
$526$	−1.56607	−0.0682839
$527$	2.73059	0.118946
$528$	0	0
$529$	−6.39954	−0.278241
$530$	−0.392652	−0.0170557
$531$	0	0
$532$	4.47623	0.194069
$533$	−35.0070	−1.51632
$534$	0	0
$535$	49.9078	2.15770
$536$	−2.79642	−0.120787
$537$	0	0
$538$	3.41339	0.147162
$539$	1.10322	0.0475189
$540$	0	0
$541$	15.0212	0.645813	0.322907	−	0.946431i	$-0.395340\pi$
$541$	15.0212	0.645813	0.322907	+	0.946431i	$0.395340\pi$
$542$	−4.14402	−0.178001
$543$	0	0
$544$	16.8255	0.721388
$545$	38.6781	1.65679
$546$	0	0
$547$	15.6137	0.667592	0.333796	−	0.942645i	$-0.391670\pi$
$547$	15.6137	0.667592	0.333796	+	0.942645i	$0.391670\pi$
$548$	−20.4239	−0.872464
$549$	0	0
$550$	1.11111	0.0473778
$551$	−0.666615	−0.0283987
$552$	0	0
$553$	25.6358	1.09014
$554$	2.65705	0.112887
$555$	0	0
$556$	−15.6934	−0.665546
$557$	−1.93285	−0.0818976	−0.0409488	−	0.999161i	$-0.513038\pi$
$557$	−1.93285	−0.0818976	−0.0409488	+	0.999161i	$0.513038\pi$
$558$	0	0
$559$	6.60714	0.279452
$560$	−33.1070	−1.39903
$561$	0	0
$562$	−4.82255	−0.203427
$563$	−27.2126	−1.14688	−0.573438	−	0.819249i	$-0.694391\pi$
$563$	−27.2126	−1.14688	−0.573438	+	0.819249i	$0.694391\pi$
$564$	0	0
$565$	51.1724	2.15284
$566$	−1.82154	−0.0765649
$567$	0	0
$568$	9.09728	0.381713
$569$	31.4184	1.31713	0.658564	−	0.752525i	$-0.271164\pi$
$569$	31.4184	1.31713	0.658564	+	0.752525i	$0.271164\pi$
$570$	0	0
$571$	36.4671	1.52610	0.763051	−	0.646338i	$-0.223700\pi$
$571$	36.4671	1.52610	0.763051	+	0.646338i	$0.223700\pi$
$572$	−6.89872	−0.288450
$573$	0	0
$574$	6.37372	0.266034
$575$	19.9936	0.833789
$576$	0	0
$577$	22.6555	0.943163	0.471581	−	0.881823i	$-0.343683\pi$
$577$	22.6555	0.943163	0.471581	+	0.881823i	$0.343683\pi$
$578$	5.45452	0.226878
$579$	0	0
$580$	5.06722	0.210405
$581$	24.8528	1.03107
$582$	0	0
$583$	−0.550944	−0.0228178
$584$	−5.08595	−0.210458
$585$	0	0
$586$	−5.00855	−0.206901
$587$	−5.99148	−0.247295	−0.123647	−	0.992326i	$-0.539459\pi$
$587$	−5.99148	−0.247295	−0.123647	+	0.992326i	$0.539459\pi$
$588$	0	0
$589$	0.343733	0.0141633
$590$	−2.93069	−0.120654
$591$	0	0
$592$	18.6065	0.764722
$593$	−9.22080	−0.378653	−0.189326	−	0.981914i	$-0.560630\pi$
$593$	−9.22080	−0.378653	−0.189326	+	0.981914i	$0.560630\pi$
$594$	0	0
$595$	−57.4341	−2.35457
$596$	−11.6332	−0.476516
$597$	0	0
$598$	3.26590	0.133552
$599$	−3.17645	−0.129786	−0.0648931	−	0.997892i	$-0.520671\pi$
$599$	−3.17645	−0.129786	−0.0648931	+	0.997892i	$0.520671\pi$
$600$	0	0
$601$	20.1398	0.821521	0.410760	−	0.911743i	$-0.365263\pi$
$601$	20.1398	0.821521	0.410760	+	0.911743i	$0.365263\pi$
$602$	−1.20296	−0.0490291
$603$	0	0
$604$	−30.8028	−1.25335
$605$	3.14756	0.127967
$606$	0	0
$607$	13.3532	0.541991	0.270996	−	0.962581i	$-0.412647\pi$
$607$	13.3532	0.541991	0.270996	+	0.962581i	$0.412647\pi$
$608$	2.11804	0.0858979
$609$	0	0
$610$	−0.712690	−0.0288560
$611$	−26.1107	−1.05633
$612$	0	0
$613$	30.3416	1.22549	0.612743	−	0.790282i	$-0.290066\pi$
$613$	30.3416	1.22549	0.612743	+	0.790282i	$0.290066\pi$
$614$	−3.74933	−0.151311
$615$	0	0
$616$	2.54514	0.102547
$617$	11.7675	0.473742	0.236871	−	0.971541i	$-0.423878\pi$
$617$	11.7675	0.473742	0.236871	+	0.971541i	$0.423878\pi$
$618$	0	0
$619$	40.9785	1.64706	0.823532	−	0.567270i	$-0.192000\pi$
$619$	40.9785	1.64706	0.823532	+	0.567270i	$0.192000\pi$
$620$	−2.61286	−0.104935
$621$	0	0
$622$	−7.66799	−0.307459
$623$	−27.4305	−1.09898
$624$	0	0
$625$	−25.4556	−1.01822
$626$	6.68211	0.267071
$627$	0	0
$628$	36.4584	1.45485
$629$	32.2786	1.28703
$630$	0	0
$631$	−17.0008	−0.676789	−0.338395	−	0.941004i	$-0.609884\pi$
$631$	−17.0008	−0.676789	−0.338395	+	0.941004i	$0.609884\pi$
$632$	8.05196	0.320290
$633$	0	0
$634$	5.40659	0.214723
$635$	57.2318	2.27117
$636$	0	0
$637$	3.90550	0.154742
$638$	−0.187055	−0.00740558
$639$	0	0
$640$	−21.3669	−0.844602
$641$	−15.1781	−0.599500	−0.299750	−	0.954018i	$-0.596903\pi$
$641$	−15.1781	−0.599500	−0.299750	+	0.954018i	$0.596903\pi$
$642$	0	0
$643$	5.58966	0.220435	0.110217	−	0.993908i	$-0.464845\pi$
$643$	5.58966	0.220435	0.110217	+	0.993908i	$0.464845\pi$
$644$	22.6017	0.890631
$645$	0	0
$646$	1.17118	0.0460795
$647$	−24.7711	−0.973852	−0.486926	−	0.873443i	$-0.661882\pi$
$647$	−24.7711	−0.973852	−0.486926	+	0.873443i	$0.661882\pi$
$648$	0	0
$649$	−4.11215	−0.161416
$650$	3.93344	0.154282
$651$	0	0
$652$	−40.8977	−1.60168
$653$	14.8153	0.579768	0.289884	−	0.957062i	$-0.406383\pi$
$653$	14.8153	0.579768	0.289884	+	0.957062i	$0.406383\pi$
$654$	0	0
$655$	25.6081	1.00059
$656$	−36.5388	−1.42660
$657$	0	0
$658$	4.75398	0.185330
$659$	13.0832	0.509648	0.254824	−	0.966987i	$-0.417982\pi$
$659$	13.0832	0.509648	0.254824	+	0.966987i	$0.417982\pi$
$660$	0	0
$661$	−10.6987	−0.416132	−0.208066	−	0.978115i	$-0.566717\pi$
$661$	−10.6987	−0.416132	−0.208066	+	0.978115i	$0.566717\pi$
$662$	0.216622	0.00841927
$663$	0	0
$664$	7.80602	0.302932
$665$	−7.22994	−0.280365
$666$	0	0
$667$	−3.36592	−0.130329
$668$	6.30535	0.243961
$669$	0	0
$670$	2.22905	0.0861156
$671$	−1.00000	−0.0386046
$672$	0	0
$673$	−17.6860	−0.681747	−0.340873	−	0.940109i	$-0.610723\pi$
$673$	−17.6860	−0.681747	−0.340873	+	0.940109i	$0.610723\pi$
$674$	3.41120	0.131395
$675$	0	0
$676$	0.911317	0.0350507
$677$	−14.8409	−0.570383	−0.285191	−	0.958471i	$-0.592057\pi$
$677$	−14.8409	−0.570383	−0.285191	+	0.958471i	$0.592057\pi$
$678$	0	0
$679$	−52.8413	−2.02786
$680$	−18.0395	−0.691783
$681$	0	0
$682$	0.0964531	0.00369338
$683$	−12.0046	−0.459343	−0.229671	−	0.973268i	$-0.573765\pi$
$683$	−12.0046	−0.459343	−0.229671	+	0.973268i	$0.573765\pi$
$684$	0	0
$685$	32.9883	1.26042
$686$	3.80076	0.145114
$687$	0	0
$688$	6.89626	0.262917
$689$	−1.95040	−0.0743043
$690$	0	0
$691$	−11.2959	−0.429716	−0.214858	−	0.976645i	$-0.568929\pi$
$691$	−11.2959	−0.429716	−0.214858	+	0.976645i	$0.568929\pi$
$692$	0.775630	0.0294850
$693$	0	0
$694$	4.85998	0.184482
$695$	25.3477	0.961493
$696$	0	0
$697$	−63.3876	−2.40098
$698$	−5.45052	−0.206305
$699$	0	0
$700$	27.2214	1.02887
$701$	49.1999	1.85825	0.929127	−	0.369760i	$-0.120560\pi$
$701$	49.1999	1.85825	0.929127	+	0.369760i	$0.120560\pi$
$702$	0	0
$703$	4.06331	0.153251
$704$	−6.79570	−0.256123
$705$	0	0
$706$	−3.33993	−0.125700
$707$	−14.4113	−0.541993
$708$	0	0
$709$	37.2184	1.39777	0.698884	−	0.715235i	$-0.253680\pi$
$709$	37.2184	1.39777	0.698884	+	0.715235i	$0.253680\pi$
$710$	−7.25151	−0.272144
$711$	0	0
$712$	−8.61566	−0.322886
$713$	1.73560	0.0649988
$714$	0	0
$715$	11.1427	0.416714
$716$	3.00240	0.112205
$717$	0	0
$718$	5.21020	0.194443
$719$	22.7760	0.849400	0.424700	−	0.905334i	$-0.360380\pi$
$719$	22.7760	0.849400	0.424700	+	0.905334i	$0.360380\pi$
$720$	0	0
$721$	50.0032	1.86222
$722$	−4.15466	−0.154620
$723$	0	0
$724$	−27.8119	−1.03362
$725$	−4.05390	−0.150558
$726$	0	0
$727$	11.6536	0.432209	0.216105	−	0.976370i	$-0.430665\pi$
$727$	11.6536	0.432209	0.216105	+	0.976370i	$0.430665\pi$
$728$	9.01008	0.333936
$729$	0	0
$730$	4.05405	0.150047
$731$	11.9636	0.442491
$732$	0	0
$733$	48.3850	1.78714	0.893569	−	0.448925i	$-0.148193\pi$
$733$	48.3850	1.78714	0.893569	+	0.448925i	$0.148193\pi$
$734$	−0.496702	−0.0183336
$735$	0	0
$736$	10.6946	0.394206
$737$	3.12765	0.115209
$738$	0	0
$739$	−23.8439	−0.877111	−0.438555	−	0.898704i	$-0.644510\pi$
$739$	−23.8439	−0.877111	−0.438555	+	0.898704i	$0.644510\pi$
$740$	−30.8869	−1.13543
$741$	0	0
$742$	0.355109	0.0130365
$743$	19.9222	0.730876	0.365438	−	0.930836i	$-0.380919\pi$
$743$	19.9222	0.730876	0.365438	+	0.930836i	$0.380919\pi$
$744$	0	0
$745$	18.7898	0.688406
$746$	−0.585968	−0.0214538
$747$	0	0
$748$	−12.4916	−0.456738
$749$	−45.1360	−1.64923
$750$	0	0
$751$	−0.875744	−0.0319563	−0.0159782	−	0.999872i	$-0.505086\pi$
$751$	−0.875744	−0.0319563	−0.0159782	+	0.999872i	$0.505086\pi$
$752$	−27.2533	−0.993826
$753$	0	0
$754$	−0.662194	−0.0241157
$755$	49.7522	1.81067
$756$	0	0
$757$	34.4888	1.25352	0.626759	−	0.779213i	$-0.284381\pi$
$757$	34.4888	1.25352	0.626759	+	0.779213i	$0.284381\pi$
$758$	1.69709	0.0616410
$759$	0	0
$760$	−2.27086	−0.0823727
$761$	40.4519	1.46638	0.733190	−	0.680024i	$-0.238031\pi$
$761$	40.4519	1.46638	0.733190	+	0.680024i	$0.238031\pi$
$762$	0	0
$763$	−34.9799	−1.26636
$764$	12.4838	0.451650
$765$	0	0
$766$	−4.45794	−0.161072
$767$	−14.5574	−0.525638
$768$	0	0
$769$	7.12060	0.256775	0.128388	−	0.991724i	$-0.459020\pi$
$769$	7.12060	0.256775	0.128388	+	0.991724i	$0.459020\pi$
$770$	−2.02875	−0.0731112
$771$	0	0
$772$	−5.74923	−0.206919
$773$	10.2170	0.367480	0.183740	−	0.982975i	$-0.441179\pi$
$773$	10.2170	0.367480	0.183740	+	0.982975i	$0.441179\pi$
$774$	0	0
$775$	2.09036	0.0750878
$776$	−16.5969	−0.595796
$777$	0	0
$778$	−0.422585	−0.0151504
$779$	−7.97940	−0.285892
$780$	0	0
$781$	−10.1748	−0.364085
$782$	5.91361	0.211470
$783$	0	0
$784$	4.07640	0.145586
$785$	−58.8871	−2.10177
$786$	0	0
$787$	13.1987	0.470482	0.235241	−	0.971937i	$-0.424412\pi$
$787$	13.1987	0.470482	0.235241	+	0.971937i	$0.424412\pi$
$788$	−38.9987	−1.38927
$789$	0	0
$790$	−6.41828	−0.228352
$791$	−46.2796	−1.64551
$792$	0	0
$793$	−3.54011	−0.125713
$794$	−6.88130	−0.244208
$795$	0	0
$796$	31.5960	1.11989
$797$	−15.6414	−0.554048	−0.277024	−	0.960863i	$-0.589348\pi$
$797$	−15.6414	−0.554048	−0.277024	+	0.960863i	$0.589348\pi$
$798$	0	0
$799$	−47.2791	−1.67261
$800$	12.8805	0.455394
$801$	0	0
$802$	−0.966939	−0.0341438
$803$	5.68838	0.200739
$804$	0	0
$805$	−36.5059	−1.28666
$806$	0.341454	0.0120272
$807$	0	0
$808$	−4.52646	−0.159240
$809$	−35.6124	−1.25206	−0.626032	−	0.779797i	$-0.715322\pi$
$809$	−35.6124	−1.25206	−0.626032	+	0.779797i	$0.715322\pi$
$810$	0	0
$811$	25.8284	0.906958	0.453479	−	0.891267i	$-0.350183\pi$
$811$	25.8284	0.906958	0.453479	+	0.891267i	$0.350183\pi$
$812$	−4.58272	−0.160822
$813$	0	0
$814$	1.14018	0.0399634
$815$	66.0574	2.31389
$816$	0	0
$817$	1.50601	0.0526887
$818$	−5.89061	−0.205960
$819$	0	0
$820$	60.6548	2.11816
$821$	−28.3972	−0.991069	−0.495535	−	0.868588i	$-0.665028\pi$
$821$	−28.3972	−0.991069	−0.495535	+	0.868588i	$0.665028\pi$
$822$	0	0
$823$	−1.95691	−0.0682136	−0.0341068	−	0.999418i	$-0.510859\pi$
$823$	−1.95691	−0.0682136	−0.0341068	+	0.999418i	$0.510859\pi$
$824$	15.7055	0.547128
$825$	0	0
$826$	2.65047	0.0922217
$827$	12.4574	0.433186	0.216593	−	0.976262i	$-0.430505\pi$
$827$	12.4574	0.433186	0.216593	+	0.976262i	$0.430505\pi$
$828$	0	0
$829$	10.1274	0.351740	0.175870	−	0.984413i	$-0.443726\pi$
$829$	10.1274	0.351740	0.175870	+	0.984413i	$0.443726\pi$
$830$	−6.22224	−0.215977
$831$	0	0
$832$	−24.0575	−0.834044
$833$	7.07175	0.245022
$834$	0	0
$835$	−10.1843	−0.352443
$836$	−1.57247	−0.0543852
$837$	0	0
$838$	2.51234	0.0867872
$839$	−44.0624	−1.52120	−0.760602	−	0.649219i	$-0.775096\pi$
$839$	−44.0624	−1.52120	−0.760602	+	0.649219i	$0.775096\pi$
$840$	0	0
$841$	−28.3175	−0.976466
$842$	0.857193	0.0295408
$843$	0	0
$844$	12.3967	0.426711
$845$	−1.47195	−0.0506365
$846$	0	0
$847$	−2.84661	−0.0978108
$848$	−2.03575	−0.0699078
$849$	0	0
$850$	7.12234	0.244294
$851$	20.5167	0.703304
$852$	0	0
$853$	−16.0696	−0.550211	−0.275106	−	0.961414i	$-0.588713\pi$
$853$	−16.0696	−0.550211	−0.275106	+	0.961414i	$0.588713\pi$
$854$	0.644547	0.0220560
$855$	0	0
$856$	−14.1768	−0.484552
$857$	32.9166	1.12441	0.562205	−	0.826998i	$-0.309953\pi$
$857$	32.9166	1.12441	0.562205	+	0.826998i	$0.309953\pi$
$858$	0	0
$859$	−17.5315	−0.598166	−0.299083	−	0.954227i	$-0.596681\pi$
$859$	−17.5315	−0.598166	−0.299083	+	0.954227i	$0.596681\pi$
$860$	−11.4478	−0.390368
$861$	0	0
$862$	0.578700	0.0197106
$863$	−48.1147	−1.63784	−0.818921	−	0.573906i	$-0.805427\pi$
$863$	−48.1147	−1.63784	−0.818921	+	0.573906i	$0.805427\pi$
$864$	0	0
$865$	−1.25279	−0.0425960
$866$	−7.39700	−0.251360
$867$	0	0
$868$	2.36304	0.0802067
$869$	−9.00570	−0.305498
$870$	0	0
$871$	11.0722	0.375168
$872$	−10.9869	−0.372062
$873$	0	0
$874$	0.744420	0.0251804
$875$	0.831837	0.0281212
$876$	0	0
$877$	−42.6878	−1.44146	−0.720732	−	0.693214i	$-0.756194\pi$
$877$	−42.6878	−1.44146	−0.720732	+	0.693214i	$0.756194\pi$
$878$	−2.94582	−0.0994166
$879$	0	0
$880$	11.6303	0.392057
$881$	−13.3091	−0.448394	−0.224197	−	0.974544i	$-0.571976\pi$
$881$	−13.3091	−0.448394	−0.224197	+	0.974544i	$0.571976\pi$
$882$	0	0
$883$	−47.9163	−1.61251	−0.806256	−	0.591567i	$-0.798510\pi$
$883$	−47.9163	−1.61251	−0.806256	+	0.591567i	$0.798510\pi$
$884$	−44.2216	−1.48733
$885$	0	0
$886$	−3.74472	−0.125806
$887$	34.4170	1.15561	0.577804	−	0.816175i	$-0.303910\pi$
$887$	34.4170	1.15561	0.577804	+	0.816175i	$0.303910\pi$
$888$	0	0
$889$	−51.7597	−1.73596
$890$	6.86761	0.230203
$891$	0	0
$892$	−35.0613	−1.17394
$893$	−5.95161	−0.199163
$894$	0	0
$895$	−4.84944	−0.162099
$896$	19.3240	0.645568
$897$	0	0
$898$	0.0547926	0.00182845
$899$	−0.351911	−0.0117369
$900$	0	0
$901$	−3.53162	−0.117655
$902$	−2.23905	−0.0745523
$903$	0	0
$904$	−14.5360	−0.483460
$905$	44.9213	1.49324
$906$	0	0
$907$	−2.63661	−0.0875473	−0.0437737	−	0.999041i	$-0.513938\pi$
$907$	−2.63661	−0.0875473	−0.0437737	+	0.999041i	$0.513938\pi$
$908$	−32.7002	−1.08519
$909$	0	0
$910$	−7.18201	−0.238081
$911$	6.75444	0.223785	0.111892	−	0.993720i	$-0.464309\pi$
$911$	6.75444	0.223785	0.111892	+	0.993720i	$0.464309\pi$
$912$	0	0
$913$	−8.73064	−0.288942
$914$	7.43597	0.245960
$915$	0	0
$916$	53.8519	1.77932
$917$	−23.1596	−0.764798
$918$	0	0
$919$	−30.1005	−0.992924	−0.496462	−	0.868058i	$-0.665368\pi$
$919$	−30.1005	−0.992924	−0.496462	+	0.868058i	$0.665368\pi$
$920$	−11.4662	−0.378028
$921$	0	0
$922$	−0.815093	−0.0268437
$923$	−36.0200	−1.18561
$924$	0	0
$925$	24.7103	0.812470
$926$	3.13245	0.102939
$927$	0	0
$928$	−2.16843	−0.0711822
$929$	−16.8486	−0.552786	−0.276393	−	0.961045i	$-0.589139\pi$
$929$	−16.8486	−0.552786	−0.276393	+	0.961045i	$0.589139\pi$
$930$	0	0
$931$	0.890210	0.0291755
$932$	−2.65065	−0.0868248
$933$	0	0
$934$	−4.57021	−0.149542
$935$	20.1763	0.659834
$936$	0	0
$937$	−9.84900	−0.321753	−0.160876	−	0.986975i	$-0.551432\pi$
$937$	−9.84900	−0.321753	−0.160876	+	0.986975i	$0.551432\pi$
$938$	−2.01592	−0.0658222
$939$	0	0
$940$	45.2407	1.47559
$941$	−8.53936	−0.278375	−0.139188	−	0.990266i	$-0.544449\pi$
$941$	−8.53936	−0.278375	−0.139188	+	0.990266i	$0.544449\pi$
$942$	0	0
$943$	−40.2901	−1.31203
$944$	−15.1944	−0.494537
$945$	0	0
$946$	0.422594	0.0137397
$947$	21.1687	0.687889	0.343945	−	0.938990i	$-0.388237\pi$
$947$	21.1687	0.687889	0.343945	+	0.938990i	$0.388237\pi$
$948$	0	0
$949$	20.1375	0.653690
$950$	0.896578	0.0290888
$951$	0	0
$952$	16.3147	0.528762
$953$	23.7144	0.768185	0.384093	−	0.923295i	$-0.374514\pi$
$953$	23.7144	0.768185	0.384093	+	0.923295i	$0.374514\pi$
$954$	0	0
$955$	−20.1637	−0.652483
$956$	−44.5431	−1.44063
$957$	0	0
$958$	−9.28625	−0.300025
$959$	−29.8342	−0.963397
$960$	0	0
$961$	−30.8185	−0.994146
$962$	4.03636	0.130138
$963$	0	0
$964$	29.7308	0.957565
$965$	9.28608	0.298930
$966$	0	0
$967$	22.6678	0.728949	0.364474	−	0.931213i	$-0.381249\pi$
$967$	22.6678	0.728949	0.364474	+	0.931213i	$0.381249\pi$
$968$	−0.894095	−0.0287373
$969$	0	0
$970$	13.2296	0.424775
$971$	−19.6399	−0.630273	−0.315137	−	0.949046i	$-0.602050\pi$
$971$	−19.6399	−0.630273	−0.315137	+	0.949046i	$0.602050\pi$
$972$	0	0
$973$	−22.9241	−0.734913
$974$	4.70809	0.150857
$975$	0	0
$976$	−3.69502	−0.118275
$977$	2.10212	0.0672527	0.0336263	−	0.999434i	$-0.489294\pi$
$977$	2.10212	0.0672527	0.0336263	+	0.999434i	$0.489294\pi$
$978$	0	0
$979$	9.63618	0.307974
$980$	−6.76686	−0.216160
$981$	0	0
$982$	−1.65572	−0.0528360
$983$	40.6024	1.29502	0.647508	−	0.762059i	$-0.275811\pi$
$983$	40.6024	1.29502	0.647508	+	0.762059i	$0.275811\pi$
$984$	0	0
$985$	62.9901	2.00703
$986$	−1.19904	−0.0381854
$987$	0	0
$988$	−5.56673	−0.177101
$989$	7.60426	0.241801
$990$	0	0
$991$	53.1949	1.68979	0.844896	−	0.534931i	$-0.179662\pi$
$991$	53.1949	1.68979	0.844896	+	0.534931i	$0.179662\pi$
$992$	1.11813	0.0355007
$993$	0	0
$994$	6.55817	0.208013
$995$	−51.0335	−1.61787
$996$	0	0
$997$	25.9868	0.823010	0.411505	−	0.911408i	$-0.365003\pi$
$997$	25.9868	0.823010	0.411505	+	0.911408i	$0.365003\pi$
$998$	−5.10065	−0.161458
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.o.1.14	yes	25
3.2	odd	2		6039.2.a.n.1.12	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6039.2.a.n.1.12	✓	25	3.2	odd	2
6039.2.a.o.1.14	yes	25	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+0.226426 q^{2} -1.94873 q^{4} +3.14756 q^{5} -2.84661 q^{7} -0.894095 q^{8} +O(q^{10})\)	\(q+0.226426 q^{2} -1.94873 q^{4} +3.14756 q^{5} -2.84661 q^{7} -0.894095 q^{8} +0.712690 q^{10} +1.00000 q^{11} +3.54011 q^{13} -0.644547 q^{14} +3.69502 q^{16} +6.41012 q^{17} +0.806922 q^{19} -6.13376 q^{20} +0.226426 q^{22} +4.07437 q^{23} +4.90716 q^{25} +0.801572 q^{26} +5.54729 q^{28} -0.826120 q^{29} +0.425981 q^{31} +2.62484 q^{32} +1.45142 q^{34} -8.95990 q^{35} +5.03556 q^{37} +0.182708 q^{38} -2.81422 q^{40} -9.88868 q^{41} +1.86637 q^{43} -1.94873 q^{44} +0.922542 q^{46} -7.37569 q^{47} +1.10322 q^{49} +1.11111 q^{50} -6.89872 q^{52} -0.550944 q^{53} +3.14756 q^{55} +2.54514 q^{56} -0.187055 q^{58} -4.11215 q^{59} -1.00000 q^{61} +0.0964531 q^{62} -6.79570 q^{64} +11.1427 q^{65} +3.12765 q^{67} -12.4916 q^{68} -2.02875 q^{70} -10.1748 q^{71} +5.68838 q^{73} +1.14018 q^{74} -1.57247 q^{76} -2.84661 q^{77} -9.00570 q^{79} +11.6303 q^{80} -2.23905 q^{82} -8.73064 q^{83} +20.1763 q^{85} +0.422594 q^{86} -0.894095 q^{88} +9.63618 q^{89} -10.0773 q^{91} -7.93985 q^{92} -1.67005 q^{94} +2.53984 q^{95} +18.5628 q^{97} +0.249797 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q + 5 q^{2} + 25 q^{4} + 4 q^{5} + 4 q^{7} + 15 q^{8}+O(q^{10}) \) 25 * q + 5 * q^2 + 25 * q^4 + 4 * q^5 + 4 * q^7 + 15 * q^8	\( 25 q + 5 q^{2} + 25 q^{4} + 4 q^{5} + 4 q^{7} + 15 q^{8} + 25 q^{11} + 4 q^{13} + 18 q^{14} + 21 q^{16} + 20 q^{17} + 14 q^{19} + 12 q^{20} + 5 q^{22} + 20 q^{23} + 13 q^{25} + 16 q^{26} - 14 q^{28} + 28 q^{29} - 12 q^{31} + 35 q^{32} + 6 q^{34} + 10 q^{35} - 8 q^{37} + 32 q^{38} + 24 q^{40} + 26 q^{41} + 18 q^{43} + 25 q^{44} + 4 q^{46} + 12 q^{47} + 23 q^{49} + 43 q^{50} + 22 q^{52} + 36 q^{53} + 4 q^{55} + 26 q^{56} - 20 q^{58} + 46 q^{59} - 25 q^{61} - 14 q^{62} - 13 q^{64} + 60 q^{65} - 20 q^{67} + 44 q^{68} - 20 q^{70} + 52 q^{71} + 6 q^{73} + 32 q^{74} + 4 q^{77} + 26 q^{79} + 52 q^{80} + 6 q^{82} + 38 q^{83} - 4 q^{85} + 34 q^{86} + 15 q^{88} + 82 q^{89} - 58 q^{91} + 36 q^{92} + 16 q^{94} + 30 q^{95} + 35 q^{98}+O(q^{100}) \) 25 * q + 5 * q^2 + 25 * q^4 + 4 * q^5 + 4 * q^7 + 15 * q^8 + 25 * q^11 + 4 * q^13 + 18 * q^14 + 21 * q^16 + 20 * q^17 + 14 * q^19 + 12 * q^20 + 5 * q^22 + 20 * q^23 + 13 * q^25 + 16 * q^26 - 14 * q^28 + 28 * q^29 - 12 * q^31 + 35 * q^32 + 6 * q^34 + 10 * q^35 - 8 * q^37 + 32 * q^38 + 24 * q^40 + 26 * q^41 + 18 * q^43 + 25 * q^44 + 4 * q^46 + 12 * q^47 + 23 * q^49 + 43 * q^50 + 22 * q^52 + 36 * q^53 + 4 * q^55 + 26 * q^56 - 20 * q^58 + 46 * q^59 - 25 * q^61 - 14 * q^62 - 13 * q^64 + 60 * q^65 - 20 * q^67 + 44 * q^68 - 20 * q^70 + 52 * q^71 + 6 * q^73 + 32 * q^74 + 4 * q^77 + 26 * q^79 + 52 * q^80 + 6 * q^82 + 38 * q^83 - 4 * q^85 + 34 * q^86 + 15 * q^88 + 82 * q^89 - 58 * q^91 + 36 * q^92 + 16 * q^94 + 30 * q^95 + 35 * q^98

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

Properties

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