LMFDB - Embedded newform 6039.2.a.l.1.10

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			1.10
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)

$f(q)$	$=$	$q-0.543169 q^{2} -1.70497 q^{4} +0.382346 q^{5} -4.84691 q^{7} +2.01242 q^{8} +O(q^{10})$	\(q-0.543169 q^{2} -1.70497 q^{4} +0.382346 q^{5} -4.84691 q^{7} +2.01242 q^{8} -0.207678 q^{10} +1.00000 q^{11} +6.63881 q^{13} +2.63269 q^{14} +2.31685 q^{16} -4.99283 q^{17} +7.01598 q^{19} -0.651888 q^{20} -0.543169 q^{22} +7.26609 q^{23} -4.85381 q^{25} -3.60599 q^{26} +8.26382 q^{28} -1.46741 q^{29} -1.13218 q^{31} -5.28329 q^{32} +2.71195 q^{34} -1.85320 q^{35} -8.90479 q^{37} -3.81086 q^{38} +0.769442 q^{40} -2.75714 q^{41} -0.438418 q^{43} -1.70497 q^{44} -3.94671 q^{46} +9.41958 q^{47} +16.4925 q^{49} +2.63644 q^{50} -11.3190 q^{52} -8.67499 q^{53} +0.382346 q^{55} -9.75403 q^{56} +0.797050 q^{58} -9.48074 q^{59} +1.00000 q^{61} +0.614967 q^{62} -1.76398 q^{64} +2.53832 q^{65} +6.48857 q^{67} +8.51261 q^{68} +1.00660 q^{70} +3.31654 q^{71} +6.10854 q^{73} +4.83681 q^{74} -11.9620 q^{76} -4.84691 q^{77} -2.55910 q^{79} +0.885838 q^{80} +1.49759 q^{82} -14.7949 q^{83} -1.90899 q^{85} +0.238135 q^{86} +2.01242 q^{88} -10.4506 q^{89} -32.1777 q^{91} -12.3884 q^{92} -5.11642 q^{94} +2.68253 q^{95} +9.14364 q^{97} -8.95822 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 21 q + 32 q^{4} - 7 q^{5} + 5 q^{7} + 6 q^{8}+O(q^{10}) $ 21 * q + 32 * q^4 - 7 * q^5 + 5 * q^7 + 6 * q^8	$ 21 q + 32 q^{4} - 7 q^{5} + 5 q^{7} + 6 q^{8} + q^{10} + 21 q^{11} + 20 q^{13} - 17 q^{14} + 50 q^{16} - q^{17} + 15 q^{19} + 2 q^{20} - 11 q^{23} + 48 q^{25} + 5 q^{26} - 16 q^{28} + 9 q^{29} + 22 q^{31} - 3 q^{32} + 33 q^{34} + 39 q^{35} + 21 q^{37} - 11 q^{38} - 16 q^{40} - 7 q^{41} + 16 q^{43} + 32 q^{44} - 3 q^{46} - 5 q^{47} + 80 q^{49} + 33 q^{50} + 60 q^{52} - 9 q^{53} - 7 q^{55} - 44 q^{56} - 27 q^{58} - 13 q^{59} + 21 q^{61} + 23 q^{62} + 66 q^{64} - 25 q^{65} + 38 q^{67} + 74 q^{68} - 33 q^{70} - 12 q^{71} + 20 q^{73} + 12 q^{74} + 59 q^{76} + 5 q^{77} + q^{79} + 38 q^{80} + 7 q^{82} + 19 q^{83} + 38 q^{85} + 3 q^{86} + 6 q^{88} - 37 q^{89} + 24 q^{91} - 31 q^{92} - 64 q^{94} + 43 q^{95} + 68 q^{97} + 2 q^{98}+O(q^{100}) $ 21 * q + 32 * q^4 - 7 * q^5 + 5 * q^7 + 6 * q^8 + q^10 + 21 * q^11 + 20 * q^13 - 17 * q^14 + 50 * q^16 - q^17 + 15 * q^19 + 2 * q^20 - 11 * q^23 + 48 * q^25 + 5 * q^26 - 16 * q^28 + 9 * q^29 + 22 * q^31 - 3 * q^32 + 33 * q^34 + 39 * q^35 + 21 * q^37 - 11 * q^38 - 16 * q^40 - 7 * q^41 + 16 * q^43 + 32 * q^44 - 3 * q^46 - 5 * q^47 + 80 * q^49 + 33 * q^50 + 60 * q^52 - 9 * q^53 - 7 * q^55 - 44 * q^56 - 27 * q^58 - 13 * q^59 + 21 * q^61 + 23 * q^62 + 66 * q^64 - 25 * q^65 + 38 * q^67 + 74 * q^68 - 33 * q^70 - 12 * q^71 + 20 * q^73 + 12 * q^74 + 59 * q^76 + 5 * q^77 + q^79 + 38 * q^80 + 7 * q^82 + 19 * q^83 + 38 * q^85 + 3 * q^86 + 6 * q^88 - 37 * q^89 + 24 * q^91 - 31 * q^92 - 64 * q^94 + 43 * q^95 + 68 * q^97 + 2 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−0.543169	−0.384078	−0.192039	−	0.981387i	$-0.561510\pi$
$2$	−0.543169	−0.384078	−0.192039	+	0.981387i	$0.561510\pi$
$3$	0	0
$3$	0	0
$4$	−1.70497	−0.852484
$5$	0.382346	0.170990	0.0854952	−	0.996339i	$-0.472753\pi$
$5$	0.382346	0.170990	0.0854952	+	0.996339i	$0.472753\pi$
$6$	0	0
$7$	−4.84691	−1.83196	−0.915979	−	0.401225i	$-0.868585\pi$
$7$	−4.84691	−1.83196	−0.915979	+	0.401225i	$0.868585\pi$
$8$	2.01242	0.711499
$9$	0	0
$10$	−0.207678	−0.0656737
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	6.63881	1.84127	0.920637	−	0.390419i	$-0.127670\pi$
$13$	6.63881	1.84127	0.920637	+	0.390419i	$0.127670\pi$
$14$	2.63269	0.703616
$15$	0	0
$16$	2.31685	0.579212
$17$	−4.99283	−1.21094	−0.605469	−	0.795869i	$-0.707015\pi$
$17$	−4.99283	−1.21094	−0.605469	+	0.795869i	$0.707015\pi$
$18$	0	0
$19$	7.01598	1.60958	0.804789	−	0.593562i	$-0.202279\pi$
$19$	7.01598	1.60958	0.804789	+	0.593562i	$0.202279\pi$
$20$	−0.651888	−0.145766
$21$	0	0
$22$	−0.543169	−0.115804
$23$	7.26609	1.51508	0.757542	−	0.652786i	$-0.226400\pi$
$23$	7.26609	1.51508	0.757542	+	0.652786i	$0.226400\pi$
$24$	0	0
$25$	−4.85381	−0.970762
$26$	−3.60599	−0.707194
$27$	0	0
$28$	8.26382	1.56172
$29$	−1.46741	−0.272491	−0.136245	−	0.990675i	$-0.543504\pi$
$29$	−1.46741	−0.272491	−0.136245	+	0.990675i	$0.543504\pi$
$30$	0	0
$31$	−1.13218	−0.203346	−0.101673	−	0.994818i	$-0.532420\pi$
$31$	−1.13218	−0.203346	−0.101673	+	0.994818i	$0.532420\pi$
$32$	−5.28329	−0.933962
$33$	0	0
$34$	2.71195	0.465095
$35$	−1.85320	−0.313247
$36$	0	0
$37$	−8.90479	−1.46394	−0.731969	−	0.681337i	$-0.761399\pi$
$37$	−8.90479	−1.46394	−0.731969	+	0.681337i	$0.761399\pi$
$38$	−3.81086	−0.618204
$39$	0	0
$40$	0.769442	0.121659
$41$	−2.75714	−0.430594	−0.215297	−	0.976549i	$-0.569072\pi$
$41$	−2.75714	−0.430594	−0.215297	+	0.976549i	$0.569072\pi$
$42$	0	0
$43$	−0.438418	−0.0668582	−0.0334291	−	0.999441i	$-0.510643\pi$
$43$	−0.438418	−0.0668582	−0.0334291	+	0.999441i	$0.510643\pi$
$44$	−1.70497	−0.257034
$45$	0	0
$46$	−3.94671	−0.581911
$47$	9.41958	1.37399	0.686994	−	0.726664i	$-0.258930\pi$
$47$	9.41958	1.37399	0.686994	+	0.726664i	$0.258930\pi$
$48$	0	0
$49$	16.4925	2.35607
$50$	2.63644	0.372849
$51$	0	0
$52$	−11.3190	−1.56966
$53$	−8.67499	−1.19160	−0.595801	−	0.803132i	$-0.703165\pi$
$53$	−8.67499	−1.19160	−0.595801	+	0.803132i	$0.703165\pi$
$54$	0	0
$55$	0.382346	0.0515555
$56$	−9.75403	−1.30344
$57$	0	0
$58$	0.797050	0.104658
$59$	−9.48074	−1.23429	−0.617143	−	0.786851i	$-0.711710\pi$
$59$	−9.48074	−1.23429	−0.617143	+	0.786851i	$0.711710\pi$
$60$	0	0
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	0.614967	0.0781009
$63$	0	0
$64$	−1.76398	−0.220498
$65$	2.53832	0.314840
$66$	0	0
$67$	6.48857	0.792705	0.396353	−	0.918098i	$-0.370276\pi$
$67$	6.48857	0.792705	0.396353	+	0.918098i	$0.370276\pi$
$68$	8.51261	1.03231
$69$	0	0
$70$	1.00660	0.120312
$71$	3.31654	0.393601	0.196800	−	0.980444i	$-0.436945\pi$
$71$	3.31654	0.393601	0.196800	+	0.980444i	$0.436945\pi$
$72$	0	0
$73$	6.10854	0.714950	0.357475	−	0.933923i	$-0.383638\pi$
$73$	6.10854	0.714950	0.357475	+	0.933923i	$0.383638\pi$
$74$	4.83681	0.562267
$75$	0	0
$76$	−11.9620	−1.37214
$77$	−4.84691	−0.552356
$78$	0	0
$79$	−2.55910	−0.287922	−0.143961	−	0.989583i	$-0.545984\pi$
$79$	−2.55910	−0.287922	−0.143961	+	0.989583i	$0.545984\pi$
$80$	0.885838	0.0990397
$81$	0	0
$82$	1.49759	0.165382
$83$	−14.7949	−1.62395	−0.811974	−	0.583693i	$-0.801607\pi$
$83$	−14.7949	−1.62395	−0.811974	+	0.583693i	$0.801607\pi$
$84$	0	0
$85$	−1.90899	−0.207059
$86$	0.238135	0.0256788
$87$	0	0
$88$	2.01242	0.214525
$89$	−10.4506	−1.10776	−0.553879	−	0.832597i	$-0.686853\pi$
$89$	−10.4506	−1.10776	−0.553879	+	0.832597i	$0.686853\pi$
$90$	0	0
$91$	−32.1777	−3.37314
$92$	−12.3884	−1.29158
$93$	0	0
$94$	−5.11642	−0.527719
$95$	2.68253	0.275222
$96$	0	0
$97$	9.14364	0.928396	0.464198	−	0.885731i	$-0.346343\pi$
$97$	9.14364	0.928396	0.464198	+	0.885731i	$0.346343\pi$
$98$	−8.95822	−0.904917
$99$	0	0
$100$	8.27559	0.827559
$101$	−4.96054	−0.493592	−0.246796	−	0.969067i	$-0.579378\pi$
$101$	−4.96054	−0.493592	−0.246796	+	0.969067i	$0.579378\pi$
$102$	0	0
$103$	−11.2088	−1.10444	−0.552220	−	0.833698i	$-0.686219\pi$
$103$	−11.2088	−1.10444	−0.552220	+	0.833698i	$0.686219\pi$
$104$	13.3601	1.31007
$105$	0	0
$106$	4.71198	0.457668
$107$	−11.3950	−1.10160	−0.550798	−	0.834638i	$-0.685677\pi$
$107$	−11.3950	−1.10160	−0.550798	+	0.834638i	$0.685677\pi$
$108$	0	0
$109$	8.81005	0.843850	0.421925	−	0.906631i	$-0.361355\pi$
$109$	8.81005	0.843850	0.421925	+	0.906631i	$0.361355\pi$
$110$	−0.207678	−0.0198014
$111$	0	0
$112$	−11.2296	−1.06109
$113$	5.60704	0.527466	0.263733	−	0.964596i	$-0.415046\pi$
$113$	5.60704	0.527466	0.263733	+	0.964596i	$0.415046\pi$
$114$	0	0
$115$	2.77816	0.259065
$116$	2.50188	0.232294
$117$	0	0
$118$	5.14964	0.474063
$119$	24.1998	2.21839
$120$	0	0
$121$	1.00000	0.0909091
$122$	−0.543169	−0.0491762
$123$	0	0
$124$	1.93034	0.173349
$125$	−3.76757	−0.336981
$126$	0	0
$127$	9.96922	0.884625	0.442313	−	0.896861i	$-0.354158\pi$
$127$	9.96922	0.884625	0.442313	+	0.896861i	$0.354158\pi$
$128$	11.5247	1.01865
$129$	0	0
$130$	−1.37874	−0.120923
$131$	10.9236	0.954398	0.477199	−	0.878795i	$-0.341652\pi$
$131$	10.9236	0.954398	0.477199	+	0.878795i	$0.341652\pi$
$132$	0	0
$133$	−34.0058	−2.94868
$134$	−3.52439	−0.304461
$135$	0	0
$136$	−10.0477	−0.861582
$137$	20.4682	1.74871	0.874357	−	0.485283i	$-0.161283\pi$
$137$	20.4682	1.74871	0.874357	+	0.485283i	$0.161283\pi$
$138$	0	0
$139$	3.67439	0.311658	0.155829	−	0.987784i	$-0.450195\pi$
$139$	3.67439	0.311658	0.155829	+	0.987784i	$0.450195\pi$
$140$	3.15964	0.267038
$141$	0	0
$142$	−1.80144	−0.151174
$143$	6.63881	0.555165
$144$	0	0
$145$	−0.561057	−0.0465933
$146$	−3.31797	−0.274597
$147$	0	0
$148$	15.1824	1.24798
$149$	16.2675	1.33269	0.666343	−	0.745645i	$-0.267859\pi$
$149$	16.2675	1.33269	0.666343	+	0.745645i	$0.267859\pi$
$150$	0	0
$151$	8.76278	0.713105	0.356552	−	0.934275i	$-0.383952\pi$
$151$	8.76278	0.713105	0.356552	+	0.934275i	$0.383952\pi$
$152$	14.1191	1.14521
$153$	0	0
$154$	2.63269	0.212148
$155$	−0.432886	−0.0347702
$156$	0	0
$157$	9.66814	0.771601	0.385801	−	0.922582i	$-0.373925\pi$
$157$	9.66814	0.771601	0.385801	+	0.922582i	$0.373925\pi$
$158$	1.39002	0.110584
$159$	0	0
$160$	−2.02004	−0.159698
$161$	−35.2181	−2.77557
$162$	0	0
$163$	9.13632	0.715612	0.357806	−	0.933796i	$-0.383525\pi$
$163$	9.13632	0.715612	0.357806	+	0.933796i	$0.383525\pi$
$164$	4.70084	0.367074
$165$	0	0
$166$	8.03612	0.623724
$167$	0.383938	0.0297100	0.0148550	−	0.999890i	$-0.495271\pi$
$167$	0.383938	0.0297100	0.0148550	+	0.999890i	$0.495271\pi$
$168$	0	0
$169$	31.0738	2.39029
$170$	1.03690	0.0795268
$171$	0	0
$172$	0.747489	0.0569955
$173$	−7.59080	−0.577118	−0.288559	−	0.957462i	$-0.593176\pi$
$173$	−7.59080	−0.577118	−0.288559	+	0.957462i	$0.593176\pi$
$174$	0	0
$175$	23.5260	1.77840
$176$	2.31685	0.174639
$177$	0	0
$178$	5.67642	0.425466
$179$	−23.7154	−1.77257	−0.886287	−	0.463136i	$-0.846724\pi$
$179$	−23.7154	−1.77257	−0.886287	+	0.463136i	$0.846724\pi$
$180$	0	0
$181$	0.0185149	0.00137620	0.000688100	−	1.00000i	$-0.499781\pi$
$181$	0.0185149	0.00137620	0.000688100		1.00000i	$0.499781\pi$
$182$	17.4779	1.29555
$183$	0	0
$184$	14.6224	1.07798
$185$	−3.40471	−0.250319
$186$	0	0
$187$	−4.99283	−0.365112
$188$	−16.0601	−1.17130
$189$	0	0
$190$	−1.45707	−0.105707
$191$	6.18153	0.447280	0.223640	−	0.974672i	$-0.428206\pi$
$191$	6.18153	0.447280	0.223640	+	0.974672i	$0.428206\pi$
$192$	0	0
$193$	−7.73036	−0.556443	−0.278222	−	0.960517i	$-0.589745\pi$
$193$	−7.73036	−0.556443	−0.278222	+	0.960517i	$0.589745\pi$
$194$	−4.96654	−0.356577
$195$	0	0
$196$	−28.1192	−2.00851
$197$	−0.334691	−0.0238457	−0.0119229	−	0.999929i	$-0.503795\pi$
$197$	−0.334691	−0.0238457	−0.0119229	+	0.999929i	$0.503795\pi$
$198$	0	0
$199$	22.7898	1.61553	0.807763	−	0.589507i	$-0.200678\pi$
$199$	22.7898	1.61553	0.807763	+	0.589507i	$0.200678\pi$
$200$	−9.76792	−0.690696
$201$	0	0
$202$	2.69441	0.189578
$203$	7.11239	0.499192
$204$	0	0
$205$	−1.05418	−0.0736273
$206$	6.08830	0.424192
$207$	0	0
$208$	15.3811	1.06649
$209$	7.01598	0.485306
$210$	0	0
$211$	3.08674	0.212500	0.106250	−	0.994339i	$-0.466116\pi$
$211$	3.08674	0.212500	0.106250	+	0.994339i	$0.466116\pi$
$212$	14.7906	1.01582
$213$	0	0
$214$	6.18941	0.423099
$215$	−0.167628	−0.0114321
$216$	0	0
$217$	5.48759	0.372522
$218$	−4.78535	−0.324105
$219$	0	0
$220$	−0.651888	−0.0439503
$221$	−33.1464	−2.22967
$222$	0	0
$223$	−13.6000	−0.910723	−0.455361	−	0.890307i	$-0.650490\pi$
$223$	−13.6000	−0.910723	−0.455361	+	0.890307i	$0.650490\pi$
$224$	25.6076	1.71098
$225$	0	0
$226$	−3.04557	−0.202588
$227$	−0.745926	−0.0495089	−0.0247544	−	0.999694i	$-0.507880\pi$
$227$	−0.745926	−0.0495089	−0.0247544	+	0.999694i	$0.507880\pi$
$228$	0	0
$229$	14.2486	0.941574	0.470787	−	0.882247i	$-0.343970\pi$
$229$	14.2486	0.941574	0.470787	+	0.882247i	$0.343970\pi$
$230$	−1.50901	−0.0995012
$231$	0	0
$232$	−2.95304	−0.193877
$233$	24.4434	1.60134	0.800670	−	0.599105i	$-0.204477\pi$
$233$	24.4434	1.60134	0.800670	+	0.599105i	$0.204477\pi$
$234$	0	0
$235$	3.60154	0.234939
$236$	16.1643	1.05221
$237$	0	0
$238$	−13.1446	−0.852036
$239$	−12.1455	−0.785629	−0.392814	−	0.919618i	$-0.628499\pi$
$239$	−12.1455	−0.785629	−0.392814	+	0.919618i	$0.628499\pi$
$240$	0	0
$241$	0.928093	0.0597837	0.0298918	−	0.999553i	$-0.490484\pi$
$241$	0.928093	0.0597837	0.0298918	+	0.999553i	$0.490484\pi$
$242$	−0.543169	−0.0349162
$243$	0	0
$244$	−1.70497	−0.109149
$245$	6.30585	0.402866
$246$	0	0
$247$	46.5778	2.96367
$248$	−2.27843	−0.144681
$249$	0	0
$250$	2.04642	0.129427
$251$	2.41807	0.152627	0.0763136	−	0.997084i	$-0.475685\pi$
$251$	2.41807	0.152627	0.0763136	+	0.997084i	$0.475685\pi$
$252$	0	0
$253$	7.26609	0.456815
$254$	−5.41497	−0.339765
$255$	0	0
$256$	−2.73190	−0.170744
$257$	6.27947	0.391703	0.195851	−	0.980634i	$-0.437253\pi$
$257$	6.27947	0.391703	0.195851	+	0.980634i	$0.437253\pi$
$258$	0	0
$259$	43.1607	2.68188
$260$	−4.32776	−0.268396
$261$	0	0
$262$	−5.93335	−0.366563
$263$	3.85746	0.237861	0.118931	−	0.992903i	$-0.462053\pi$
$263$	3.85746	0.237861	0.118931	+	0.992903i	$0.462053\pi$
$264$	0	0
$265$	−3.31685	−0.203752
$266$	18.4709	1.13252
$267$	0	0
$268$	−11.0628	−0.675768
$269$	3.12253	0.190384	0.0951921	−	0.995459i	$-0.469653\pi$
$269$	3.12253	0.190384	0.0951921	+	0.995459i	$0.469653\pi$
$270$	0	0
$271$	14.4266	0.876355	0.438177	−	0.898889i	$-0.355624\pi$
$271$	14.4266	0.876355	0.438177	+	0.898889i	$0.355624\pi$
$272$	−11.5676	−0.701391
$273$	0	0
$274$	−11.1177	−0.671643
$275$	−4.85381	−0.292696
$276$	0	0
$277$	5.31864	0.319566	0.159783	−	0.987152i	$-0.448921\pi$
$277$	5.31864	0.319566	0.159783	+	0.987152i	$0.448921\pi$
$278$	−1.99581	−0.119701
$279$	0	0
$280$	−3.72941	−0.222875
$281$	17.4272	1.03962	0.519809	−	0.854282i	$-0.326003\pi$
$281$	17.4272	1.03962	0.519809	+	0.854282i	$0.326003\pi$
$282$	0	0
$283$	0.559072	0.0332334	0.0166167	−	0.999862i	$-0.494711\pi$
$283$	0.559072	0.0332334	0.0166167	+	0.999862i	$0.494711\pi$
$284$	−5.65459	−0.335538
$285$	0	0
$286$	−3.60599	−0.213227
$287$	13.3636	0.788830
$288$	0	0
$289$	7.92833	0.466372
$290$	0.304749	0.0178955
$291$	0	0
$292$	−10.4149	−0.609483
$293$	1.57569	0.0920530	0.0460265	−	0.998940i	$-0.485344\pi$
$293$	1.57569	0.0920530	0.0460265	+	0.998940i	$0.485344\pi$
$294$	0	0
$295$	−3.62492	−0.211051
$296$	−17.9202	−1.04159
$297$	0	0
$298$	−8.83600	−0.511856
$299$	48.2382	2.78969
$300$	0	0
$301$	2.12497	0.122481
$302$	−4.75967	−0.273888
$303$	0	0
$304$	16.2550	0.932287
$305$	0.382346	0.0218931
$306$	0	0
$307$	−19.7291	−1.12600	−0.563001	−	0.826456i	$-0.690353\pi$
$307$	−19.7291	−1.12600	−0.563001	+	0.826456i	$0.690353\pi$
$308$	8.26382	0.470875
$309$	0	0
$310$	0.235130	0.0133545
$311$	8.10368	0.459518	0.229759	−	0.973248i	$-0.426206\pi$
$311$	8.10368	0.459518	0.229759	+	0.973248i	$0.426206\pi$
$312$	0	0
$313$	−3.29020	−0.185973	−0.0929865	−	0.995667i	$-0.529641\pi$
$313$	−3.29020	−0.185973	−0.0929865	+	0.995667i	$0.529641\pi$
$314$	−5.25143	−0.296355
$315$	0	0
$316$	4.36319	0.245448
$317$	9.56197	0.537054	0.268527	−	0.963272i	$-0.413463\pi$
$317$	9.56197	0.537054	0.268527	+	0.963272i	$0.413463\pi$
$318$	0	0
$319$	−1.46741	−0.0821590
$320$	−0.674452	−0.0377030
$321$	0	0
$322$	19.1294	1.06604
$323$	−35.0296	−1.94910
$324$	0	0
$325$	−32.2235	−1.78744
$326$	−4.96257	−0.274851
$327$	0	0
$328$	−5.54854	−0.306367
$329$	−45.6558	−2.51709
$330$	0	0
$331$	−31.5705	−1.73527	−0.867636	−	0.497200i	$-0.834361\pi$
$331$	−31.5705	−1.73527	−0.867636	+	0.497200i	$0.834361\pi$
$332$	25.2248	1.38439
$333$	0	0
$334$	−0.208543	−0.0114110
$335$	2.48088	0.135545
$336$	0	0
$337$	2.48834	0.135548	0.0677742	−	0.997701i	$-0.478410\pi$
$337$	2.48834	0.135548	0.0677742	+	0.997701i	$0.478410\pi$
$338$	−16.8783	−0.918060
$339$	0	0
$340$	3.25476	0.176514
$341$	−1.13218	−0.0613112
$342$	0	0
$343$	−46.0094	−2.48427
$344$	−0.882283	−0.0475695
$345$	0	0
$346$	4.12309	0.221659
$347$	−0.0808406	−0.00433975	−0.00216987	−	0.999998i	$-0.500691\pi$
$347$	−0.0808406	−0.00433975	−0.00216987	+	0.999998i	$0.500691\pi$
$348$	0	0
$349$	24.9294	1.33444	0.667219	−	0.744862i	$-0.267485\pi$
$349$	24.9294	1.33444	0.667219	+	0.744862i	$0.267485\pi$
$350$	−12.7786	−0.683044
$351$	0	0
$352$	−5.28329	−0.281600
$353$	−24.0803	−1.28167	−0.640833	−	0.767680i	$-0.721411\pi$
$353$	−24.0803	−1.28167	−0.640833	+	0.767680i	$0.721411\pi$
$354$	0	0
$355$	1.26807	0.0673019
$356$	17.8179	0.944345
$357$	0	0
$358$	12.8815	0.680808
$359$	−7.77746	−0.410478	−0.205239	−	0.978712i	$-0.565797\pi$
$359$	−7.77746	−0.410478	−0.205239	+	0.978712i	$0.565797\pi$
$360$	0	0
$361$	30.2240	1.59074
$362$	−0.0100567	−0.000528568 0
$363$	0	0
$364$	54.8619	2.87555
$365$	2.33557	0.122250
$366$	0	0
$367$	−0.661468	−0.0345283	−0.0172642	−	0.999851i	$-0.505496\pi$
$367$	−0.661468	−0.0345283	−0.0172642	+	0.999851i	$0.505496\pi$
$368$	16.8344	0.877556
$369$	0	0
$370$	1.84933	0.0961423
$371$	42.0469	2.18297
$372$	0	0
$373$	16.7512	0.867346	0.433673	−	0.901070i	$-0.357217\pi$
$373$	16.7512	0.867346	0.433673	+	0.901070i	$0.357217\pi$
$374$	2.71195	0.140232
$375$	0	0
$376$	18.9562	0.977590
$377$	−9.74184	−0.501730
$378$	0	0
$379$	14.6827	0.754200	0.377100	−	0.926173i	$-0.376921\pi$
$379$	14.6827	0.754200	0.377100	+	0.926173i	$0.376921\pi$
$380$	−4.57363	−0.234622
$381$	0	0
$382$	−3.35761	−0.171790
$383$	19.4800	0.995384	0.497692	−	0.867354i	$-0.334181\pi$
$383$	19.4800	0.995384	0.497692	+	0.867354i	$0.334181\pi$
$384$	0	0
$385$	−1.85320	−0.0944476
$386$	4.19889	0.213718
$387$	0	0
$388$	−15.5896	−0.791443
$389$	2.50398	0.126957	0.0634784	−	0.997983i	$-0.479781\pi$
$389$	2.50398	0.126957	0.0634784	+	0.997983i	$0.479781\pi$
$390$	0	0
$391$	−36.2783	−1.83467
$392$	33.1899	1.67634
$393$	0	0
$394$	0.181794	0.00915862
$395$	−0.978463	−0.0492318
$396$	0	0
$397$	−20.8712	−1.04749	−0.523747	−	0.851874i	$-0.675466\pi$
$397$	−20.8712	−1.04749	−0.523747	+	0.851874i	$0.675466\pi$
$398$	−12.3787	−0.620489
$399$	0	0
$400$	−11.2456	−0.562278
$401$	−16.2136	−0.809669	−0.404834	−	0.914390i	$-0.632671\pi$
$401$	−16.2136	−0.809669	−0.404834	+	0.914390i	$0.632671\pi$
$402$	0	0
$403$	−7.51635	−0.374416
$404$	8.45756	0.420779
$405$	0	0
$406$	−3.86323	−0.191729
$407$	−8.90479	−0.441394
$408$	0	0
$409$	26.6015	1.31536	0.657680	−	0.753298i	$-0.271538\pi$
$409$	26.6015	1.31536	0.657680	+	0.753298i	$0.271538\pi$
$410$	0.572599	0.0282787
$411$	0	0
$412$	19.1107	0.941518
$413$	45.9523	2.26116
$414$	0	0
$415$	−5.65676	−0.277680
$416$	−35.0747	−1.71968
$417$	0	0
$418$	−3.81086	−0.186395
$419$	−2.09758	−0.102474	−0.0512368	−	0.998687i	$-0.516316\pi$
$419$	−2.09758	−0.102474	−0.0512368	+	0.998687i	$0.516316\pi$
$420$	0	0
$421$	−1.48575	−0.0724108	−0.0362054	−	0.999344i	$-0.511527\pi$
$421$	−1.48575	−0.0724108	−0.0362054	+	0.999344i	$0.511527\pi$
$422$	−1.67662	−0.0816168
$423$	0	0
$424$	−17.4578	−0.847823
$425$	24.2342	1.17553
$426$	0	0
$427$	−4.84691	−0.234558
$428$	19.4281	0.939093
$429$	0	0
$430$	0.0910501	0.00439082
$431$	−34.7087	−1.67186	−0.835928	−	0.548838i	$-0.815070\pi$
$431$	−34.7087	−1.67186	−0.835928	+	0.548838i	$0.815070\pi$
$432$	0	0
$433$	31.4962	1.51361	0.756806	−	0.653639i	$-0.226759\pi$
$433$	31.4962	1.51361	0.756806	+	0.653639i	$0.226759\pi$
$434$	−2.98069	−0.143078
$435$	0	0
$436$	−15.0209	−0.719369
$437$	50.9788	2.43865
$438$	0	0
$439$	35.4239	1.69069	0.845344	−	0.534222i	$-0.179395\pi$
$439$	35.4239	1.69069	0.845344	+	0.534222i	$0.179395\pi$
$440$	0.769442	0.0366817
$441$	0	0
$442$	18.0041	0.856368
$443$	−8.39721	−0.398964	−0.199482	−	0.979902i	$-0.563926\pi$
$443$	−8.39721	−0.398964	−0.199482	+	0.979902i	$0.563926\pi$
$444$	0	0
$445$	−3.99573	−0.189416
$446$	7.38709	0.349789
$447$	0	0
$448$	8.54986	0.403943
$449$	2.16357	0.102105	0.0510525	−	0.998696i	$-0.483742\pi$
$449$	2.16357	0.102105	0.0510525	+	0.998696i	$0.483742\pi$
$450$	0	0
$451$	−2.75714	−0.129829
$452$	−9.55982	−0.449656
$453$	0	0
$454$	0.405164	0.0190153
$455$	−12.3030	−0.576774
$456$	0	0
$457$	10.3573	0.484496	0.242248	−	0.970214i	$-0.422115\pi$
$457$	10.3573	0.484496	0.242248	+	0.970214i	$0.422115\pi$
$458$	−7.73940	−0.361638
$459$	0	0
$460$	−4.73667	−0.220849
$461$	3.38383	0.157601	0.0788004	−	0.996890i	$-0.474891\pi$
$461$	3.38383	0.157601	0.0788004	+	0.996890i	$0.474891\pi$
$462$	0	0
$463$	−11.3541	−0.527668	−0.263834	−	0.964568i	$-0.584987\pi$
$463$	−11.3541	−0.527668	−0.263834	+	0.964568i	$0.584987\pi$
$464$	−3.39976	−0.157830
$465$	0	0
$466$	−13.2769	−0.615040
$467$	24.1374	1.11694	0.558472	−	0.829523i	$-0.311388\pi$
$467$	24.1374	1.11694	0.558472	+	0.829523i	$0.311388\pi$
$468$	0	0
$469$	−31.4495	−1.45220
$470$	−1.95624	−0.0902348
$471$	0	0
$472$	−19.0793	−0.878194
$473$	−0.438418	−0.0201585
$474$	0	0
$475$	−34.0543	−1.56252
$476$	−41.2598	−1.89114
$477$	0	0
$478$	6.59707	0.301743
$479$	−2.16493	−0.0989181	−0.0494591	−	0.998776i	$-0.515750\pi$
$479$	−2.16493	−0.0989181	−0.0494591	+	0.998776i	$0.515750\pi$
$480$	0	0
$481$	−59.1172	−2.69551
$482$	−0.504111	−0.0229616
$483$	0	0
$484$	−1.70497	−0.0774985
$485$	3.49603	0.158747
$486$	0	0
$487$	−42.2450	−1.91430	−0.957151	−	0.289590i	$-0.906481\pi$
$487$	−42.2450	−1.91430	−0.957151	+	0.289590i	$0.906481\pi$
$488$	2.01242	0.0910981
$489$	0	0
$490$	−3.42514	−0.154732
$491$	−6.16680	−0.278304	−0.139152	−	0.990271i	$-0.544438\pi$
$491$	−6.16680	−0.278304	−0.139152	+	0.990271i	$0.544438\pi$
$492$	0	0
$493$	7.32651	0.329969
$494$	−25.2996	−1.13828
$495$	0	0
$496$	−2.62310	−0.117781
$497$	−16.0750	−0.721061
$498$	0	0
$499$	31.3774	1.40465	0.702323	−	0.711859i	$-0.252146\pi$
$499$	31.3774	1.40465	0.702323	+	0.711859i	$0.252146\pi$
$500$	6.42358	0.287271
$501$	0	0
$502$	−1.31342	−0.0586208
$503$	0.489122	0.0218089	0.0109044	−	0.999941i	$-0.496529\pi$
$503$	0.489122	0.0218089	0.0109044	+	0.999941i	$0.496529\pi$
$504$	0	0
$505$	−1.89664	−0.0843995
$506$	−3.94671	−0.175453
$507$	0	0
$508$	−16.9972	−0.754129
$509$	13.6789	0.606306	0.303153	−	0.952942i	$-0.401961\pi$
$509$	13.6789	0.606306	0.303153	+	0.952942i	$0.401961\pi$
$510$	0	0
$511$	−29.6075	−1.30976
$512$	−21.5655	−0.953071
$513$	0	0
$514$	−3.41081	−0.150445
$515$	−4.28566	−0.188849
$516$	0	0
$517$	9.41958	0.414273
$518$	−23.4436	−1.03005
$519$	0	0
$520$	5.10818	0.224008
$521$	−21.1576	−0.926933	−0.463466	−	0.886114i	$-0.653395\pi$
$521$	−21.1576	−0.926933	−0.463466	+	0.886114i	$0.653395\pi$
$522$	0	0
$523$	27.5811	1.20604	0.603019	−	0.797727i	$-0.293964\pi$
$523$	27.5811	1.20604	0.603019	+	0.797727i	$0.293964\pi$
$524$	−18.6244	−0.813609
$525$	0	0
$526$	−2.09525	−0.0913573
$527$	5.65280	0.246240
$528$	0	0
$529$	29.7961	1.29548
$530$	1.80161	0.0782569
$531$	0	0
$532$	57.9788	2.51370
$533$	−18.3042	−0.792841
$534$	0	0
$535$	−4.35683	−0.188362
$536$	13.0578	0.564009
$537$	0	0
$538$	−1.69606	−0.0731224
$539$	16.4925	0.710383
$540$	0	0
$541$	18.8555	0.810662	0.405331	−	0.914170i	$-0.367156\pi$
$541$	18.8555	0.810662	0.405331	+	0.914170i	$0.367156\pi$
$542$	−7.83609	−0.336589
$543$	0	0
$544$	26.3785	1.13097
$545$	3.36849	0.144290
$546$	0	0
$547$	−11.5604	−0.494289	−0.247144	−	0.968979i	$-0.579492\pi$
$547$	−11.5604	−0.494289	−0.247144	+	0.968979i	$0.579492\pi$
$548$	−34.8976	−1.49075
$549$	0	0
$550$	2.63644	0.112418
$551$	−10.2953	−0.438595
$552$	0	0
$553$	12.4037	0.527460
$554$	−2.88892	−0.122738
$555$	0	0
$556$	−6.26472	−0.265683
$557$	−1.65640	−0.0701838	−0.0350919	−	0.999384i	$-0.511172\pi$
$557$	−1.65640	−0.0701838	−0.0350919	+	0.999384i	$0.511172\pi$
$558$	0	0
$559$	−2.91058	−0.123104
$560$	−4.29358	−0.181437
$561$	0	0
$562$	−9.46590	−0.399295
$563$	−8.71228	−0.367179	−0.183589	−	0.983003i	$-0.558772\pi$
$563$	−8.71228	−0.367179	−0.183589	+	0.983003i	$0.558772\pi$
$564$	0	0
$565$	2.14383	0.0901916
$566$	−0.303670	−0.0127642
$567$	0	0
$568$	6.67428	0.280047
$569$	32.6564	1.36903	0.684514	−	0.729000i	$-0.260015\pi$
$569$	32.6564	1.36903	0.684514	+	0.729000i	$0.260015\pi$
$570$	0	0
$571$	−27.7851	−1.16277	−0.581385	−	0.813628i	$-0.697489\pi$
$571$	−27.7851	−1.16277	−0.581385	+	0.813628i	$0.697489\pi$
$572$	−11.3190	−0.473269
$573$	0	0
$574$	−7.25870	−0.302972
$575$	−35.2682	−1.47079
$576$	0	0
$577$	18.0541	0.751600	0.375800	−	0.926701i	$-0.377368\pi$
$577$	18.0541	0.751600	0.375800	+	0.926701i	$0.377368\pi$
$578$	−4.30642	−0.179123
$579$	0	0
$580$	0.956584	0.0397200
$581$	71.7094	2.97501
$582$	0	0
$583$	−8.67499	−0.359281
$584$	12.2930	0.508686
$585$	0	0
$586$	−0.855868	−0.0353556
$587$	−30.6820	−1.26638	−0.633191	−	0.773996i	$-0.718255\pi$
$587$	−30.6820	−1.26638	−0.633191	+	0.773996i	$0.718255\pi$
$588$	0	0
$589$	−7.94338	−0.327301
$590$	1.96894	0.0810602
$591$	0	0
$592$	−20.6311	−0.847932
$593$	25.9352	1.06503	0.532515	−	0.846420i	$-0.321247\pi$
$593$	25.9352	1.06503	0.532515	+	0.846420i	$0.321247\pi$
$594$	0	0
$595$	9.25269	0.379323
$596$	−27.7356	−1.13609
$597$	0	0
$598$	−26.2015	−1.07146
$599$	−0.490442	−0.0200389	−0.0100195	−	0.999950i	$-0.503189\pi$
$599$	−0.490442	−0.0200389	−0.0100195	+	0.999950i	$0.503189\pi$
$600$	0	0
$601$	32.3923	1.32131	0.660654	−	0.750690i	$-0.270279\pi$
$601$	32.3923	1.32131	0.660654	+	0.750690i	$0.270279\pi$
$602$	−1.15422	−0.0470425
$603$	0	0
$604$	−14.9403	−0.607910
$605$	0.382346	0.0155446
$606$	0	0
$607$	8.19359	0.332567	0.166284	−	0.986078i	$-0.446823\pi$
$607$	8.19359	0.332567	0.166284	+	0.986078i	$0.446823\pi$
$608$	−37.0675	−1.50328
$609$	0	0
$610$	−0.207678	−0.00840865
$611$	62.5348	2.52989
$612$	0	0
$613$	−12.4348	−0.502236	−0.251118	−	0.967957i	$-0.580798\pi$
$613$	−12.4348	−0.502236	−0.251118	+	0.967957i	$0.580798\pi$
$614$	10.7163	0.432473
$615$	0	0
$616$	−9.75403	−0.393001
$617$	−3.32280	−0.133771	−0.0668854	−	0.997761i	$-0.521306\pi$
$617$	−3.32280	−0.133771	−0.0668854	+	0.997761i	$0.521306\pi$
$618$	0	0
$619$	8.67691	0.348755	0.174377	−	0.984679i	$-0.444209\pi$
$619$	8.67691	0.348755	0.174377	+	0.984679i	$0.444209\pi$
$620$	0.738056	0.0296411
$621$	0	0
$622$	−4.40167	−0.176491
$623$	50.6529	2.02937
$624$	0	0
$625$	22.8285	0.913142
$626$	1.78713	0.0714282
$627$	0	0
$628$	−16.4839	−0.657778
$629$	44.4601	1.77274
$630$	0	0
$631$	−11.6248	−0.462777	−0.231388	−	0.972861i	$-0.574327\pi$
$631$	−11.6248	−0.462777	−0.231388	+	0.972861i	$0.574327\pi$
$632$	−5.15000	−0.204856
$633$	0	0
$634$	−5.19377	−0.206271
$635$	3.81169	0.151262
$636$	0	0
$637$	109.491	4.33818
$638$	0.797050	0.0315555
$639$	0	0
$640$	4.40643	0.174179
$641$	49.6826	1.96235	0.981173	−	0.193131i	$-0.0618643\pi$
$641$	49.6826	1.96235	0.981173	+	0.193131i	$0.0618643\pi$
$642$	0	0
$643$	−11.9840	−0.472604	−0.236302	−	0.971680i	$-0.575935\pi$
$643$	−11.9840	−0.472604	−0.236302	+	0.971680i	$0.575935\pi$
$644$	60.0457	2.36613
$645$	0	0
$646$	19.0270	0.748607
$647$	8.19183	0.322054	0.161027	−	0.986950i	$-0.448519\pi$
$647$	8.19183	0.322054	0.161027	+	0.986950i	$0.448519\pi$
$648$	0	0
$649$	−9.48074	−0.372151
$650$	17.5028	0.686517
$651$	0	0
$652$	−15.5771	−0.610048
$653$	−43.5568	−1.70451	−0.852255	−	0.523127i	$-0.824765\pi$
$653$	−43.5568	−1.70451	−0.852255	+	0.523127i	$0.824765\pi$
$654$	0	0
$655$	4.17659	0.163193
$656$	−6.38789	−0.249405
$657$	0	0
$658$	24.7988	0.966759
$659$	41.1666	1.60362	0.801811	−	0.597578i	$-0.203870\pi$
$659$	41.1666	1.60362	0.801811	+	0.597578i	$0.203870\pi$
$660$	0	0
$661$	−10.8706	−0.422819	−0.211409	−	0.977398i	$-0.567805\pi$
$661$	−10.8706	−0.422819	−0.211409	+	0.977398i	$0.567805\pi$
$662$	17.1481	0.666480
$663$	0	0
$664$	−29.7736	−1.15544
$665$	−13.0020	−0.504196
$666$	0	0
$667$	−10.6623	−0.412846
$668$	−0.654603	−0.0253273
$669$	0	0
$670$	−1.34754	−0.0520599
$671$	1.00000	0.0386046
$672$	0	0
$673$	0.548928	0.0211596	0.0105798	−	0.999944i	$-0.496632\pi$
$673$	0.548928	0.0211596	0.0105798	+	0.999944i	$0.496632\pi$
$674$	−1.35159	−0.0520612
$675$	0	0
$676$	−52.9798	−2.03769
$677$	44.4999	1.71027	0.855135	−	0.518405i	$-0.173474\pi$
$677$	44.4999	1.71027	0.855135	+	0.518405i	$0.173474\pi$
$678$	0	0
$679$	−44.3184	−1.70078
$680$	−3.84169	−0.147322
$681$	0	0
$682$	0.614967	0.0235483
$683$	−17.8878	−0.684456	−0.342228	−	0.939617i	$-0.611181\pi$
$683$	−17.8878	−0.684456	−0.342228	+	0.939617i	$0.611181\pi$
$684$	0	0
$685$	7.82593	0.299013
$686$	24.9908	0.954155
$687$	0	0
$688$	−1.01575	−0.0387251
$689$	−57.5916	−2.19407
$690$	0	0
$691$	20.7703	0.790140	0.395070	−	0.918651i	$-0.370720\pi$
$691$	20.7703	0.790140	0.395070	+	0.918651i	$0.370720\pi$
$692$	12.9421	0.491984
$693$	0	0
$694$	0.0439101	0.00166680
$695$	1.40489	0.0532905
$696$	0	0
$697$	13.7659	0.521422
$698$	−13.5409	−0.512529
$699$	0	0
$700$	−40.1110	−1.51605
$701$	44.6060	1.68474	0.842372	−	0.538896i	$-0.181158\pi$
$701$	44.6060	1.68474	0.842372	+	0.538896i	$0.181158\pi$
$702$	0	0
$703$	−62.4759	−2.35632
$704$	−1.76398	−0.0664826
$705$	0	0
$706$	13.0797	0.492260
$707$	24.0433	0.904240
$708$	0	0
$709$	28.2143	1.05961	0.529806	−	0.848119i	$-0.322265\pi$
$709$	28.2143	1.05961	0.529806	+	0.848119i	$0.322265\pi$
$710$	−0.688774	−0.0258492
$711$	0	0
$712$	−21.0310	−0.788168
$713$	−8.22655	−0.308087
$714$	0	0
$715$	2.53832	0.0949279
$716$	40.4341	1.51109
$717$	0	0
$718$	4.22447	0.157656
$719$	−46.9913	−1.75248	−0.876241	−	0.481874i	$-0.839956\pi$
$719$	−46.9913	−1.75248	−0.876241	+	0.481874i	$0.839956\pi$
$720$	0	0
$721$	54.3282	2.02329
$722$	−16.4168	−0.610968
$723$	0	0
$724$	−0.0315672	−0.00117319
$725$	7.12252	0.264524
$726$	0	0
$727$	32.3241	1.19883	0.599417	−	0.800437i	$-0.295399\pi$
$727$	32.3241	1.19883	0.599417	+	0.800437i	$0.295399\pi$
$728$	−64.7551	−2.39999
$729$	0	0
$730$	−1.26861	−0.0469534
$731$	2.18895	0.0809612
$732$	0	0
$733$	36.5957	1.35169	0.675846	−	0.737043i	$-0.263779\pi$
$733$	36.5957	1.35169	0.675846	+	0.737043i	$0.263779\pi$
$734$	0.359289	0.0132616
$735$	0	0
$736$	−38.3888	−1.41503
$737$	6.48857	0.239010
$738$	0	0
$739$	−39.6416	−1.45824	−0.729120	−	0.684386i	$-0.760070\pi$
$739$	−39.6416	−1.45824	−0.729120	+	0.684386i	$0.760070\pi$
$740$	5.80492	0.213393
$741$	0	0
$742$	−22.8386	−0.838430
$743$	2.00195	0.0734442	0.0367221	−	0.999326i	$-0.488308\pi$
$743$	2.00195	0.0734442	0.0367221	+	0.999326i	$0.488308\pi$
$744$	0	0
$745$	6.21981	0.227876
$746$	−9.09874	−0.333129
$747$	0	0
$748$	8.51261	0.311252
$749$	55.2305	2.01808
$750$	0	0
$751$	35.5941	1.29885	0.649423	−	0.760427i	$-0.275010\pi$
$751$	35.5941	1.29885	0.649423	+	0.760427i	$0.275010\pi$
$752$	21.8238	0.795830
$753$	0	0
$754$	5.29146	0.192704
$755$	3.35041	0.121934
$756$	0	0
$757$	−34.5632	−1.25622	−0.628110	−	0.778124i	$-0.716171\pi$
$757$	−34.5632	−1.25622	−0.628110	+	0.778124i	$0.716171\pi$
$758$	−7.97519	−0.289672
$759$	0	0
$760$	5.39839	0.195820
$761$	−23.6216	−0.856282	−0.428141	−	0.903712i	$-0.640831\pi$
$761$	−23.6216	−0.856282	−0.428141	+	0.903712i	$0.640831\pi$
$762$	0	0
$763$	−42.7015	−1.54590
$764$	−10.5393	−0.381299
$765$	0	0
$766$	−10.5810	−0.382305
$767$	−62.9408	−2.27266
$768$	0	0
$769$	43.1853	1.55730	0.778651	−	0.627457i	$-0.215904\pi$
$769$	43.1853	1.55730	0.778651	+	0.627457i	$0.215904\pi$
$770$	1.00660	0.0362753
$771$	0	0
$772$	13.1800	0.474359
$773$	−15.2209	−0.547457	−0.273729	−	0.961807i	$-0.588257\pi$
$773$	−15.2209	−0.547457	−0.273729	+	0.961807i	$0.588257\pi$
$774$	0	0
$775$	5.49541	0.197401
$776$	18.4009	0.660553
$777$	0	0
$778$	−1.36008	−0.0487614
$779$	−19.3441	−0.693073
$780$	0	0
$781$	3.31654	0.118675
$782$	19.7053	0.704659
$783$	0	0
$784$	38.2107	1.36467
$785$	3.69657	0.131936
$786$	0	0
$787$	−42.0410	−1.49860	−0.749300	−	0.662231i	$-0.769610\pi$
$787$	−42.0410	−1.49860	−0.749300	+	0.662231i	$0.769610\pi$
$788$	0.570637	0.0203281
$789$	0	0
$790$	0.531470	0.0189089
$791$	−27.1768	−0.966296
$792$	0	0
$793$	6.63881	0.235751
$794$	11.3366	0.402320
$795$	0	0
$796$	−38.8559	−1.37721
$797$	−19.2573	−0.682128	−0.341064	−	0.940040i	$-0.610787\pi$
$797$	−19.2573	−0.682128	−0.341064	+	0.940040i	$0.610787\pi$
$798$	0	0
$799$	−47.0303	−1.66381
$800$	25.6441	0.906655
$801$	0	0
$802$	8.80672	0.310976
$803$	6.10854	0.215566
$804$	0	0
$805$	−13.4655	−0.474596
$806$	4.08265	0.143805
$807$	0	0
$808$	−9.98270	−0.351190
$809$	38.0646	1.33828	0.669139	−	0.743137i	$-0.266663\pi$
$809$	38.0646	1.33828	0.669139	+	0.743137i	$0.266663\pi$
$810$	0	0
$811$	45.5995	1.60121	0.800607	−	0.599190i	$-0.204511\pi$
$811$	45.5995	1.60121	0.800607	+	0.599190i	$0.204511\pi$
$812$	−12.1264	−0.425553
$813$	0	0
$814$	4.83681	0.169530
$815$	3.49324	0.122363
$816$	0	0
$817$	−3.07594	−0.107613
$818$	−14.4491	−0.505201
$819$	0	0
$820$	1.79735	0.0627661
$821$	−28.3822	−0.990544	−0.495272	−	0.868738i	$-0.664932\pi$
$821$	−28.3822	−0.990544	−0.495272	+	0.868738i	$0.664932\pi$
$822$	0	0
$823$	5.50235	0.191800	0.0958999	−	0.995391i	$-0.469427\pi$
$823$	5.50235	0.191800	0.0958999	+	0.995391i	$0.469427\pi$
$824$	−22.5569	−0.785808
$825$	0	0
$826$	−24.9598	−0.868464
$827$	20.3520	0.707707	0.353854	−	0.935301i	$-0.384871\pi$
$827$	20.3520	0.707707	0.353854	+	0.935301i	$0.384871\pi$
$828$	0	0
$829$	4.27755	0.148565	0.0742827	−	0.997237i	$-0.476333\pi$
$829$	4.27755	0.148565	0.0742827	+	0.997237i	$0.476333\pi$
$830$	3.07258	0.106651
$831$	0	0
$832$	−11.7107	−0.405997
$833$	−82.3443	−2.85306
$834$	0	0
$835$	0.146797	0.00508013
$836$	−11.9620	−0.413715
$837$	0	0
$838$	1.13934	0.0393579
$839$	42.2443	1.45844	0.729218	−	0.684281i	$-0.239884\pi$
$839$	42.2443	1.45844	0.729218	+	0.684281i	$0.239884\pi$
$840$	0	0
$841$	−26.8467	−0.925749
$842$	0.807011	0.0278114
$843$	0	0
$844$	−5.26280	−0.181153
$845$	11.8809	0.408717
$846$	0	0
$847$	−4.84691	−0.166542
$848$	−20.0986	−0.690190
$849$	0	0
$850$	−13.1633	−0.451497
$851$	−64.7030	−2.21799
$852$	0	0
$853$	34.5772	1.18390	0.591950	−	0.805975i	$-0.298358\pi$
$853$	34.5772	1.18390	0.591950	+	0.805975i	$0.298358\pi$
$854$	2.63269	0.0900888
$855$	0	0
$856$	−22.9316	−0.783785
$857$	11.1524	0.380959	0.190480	−	0.981691i	$-0.438996\pi$
$857$	11.1524	0.380959	0.190480	+	0.981691i	$0.438996\pi$
$858$	0	0
$859$	16.8745	0.575751	0.287876	−	0.957668i	$-0.407051\pi$
$859$	16.8745	0.575751	0.287876	+	0.957668i	$0.407051\pi$
$860$	0.285800	0.00974568
$861$	0	0
$862$	18.8527	0.642124
$863$	45.4420	1.54686	0.773432	−	0.633879i	$-0.218538\pi$
$863$	45.4420	1.54686	0.773432	+	0.633879i	$0.218538\pi$
$864$	0	0
$865$	−2.90231	−0.0986816
$866$	−17.1078	−0.581346
$867$	0	0
$868$	−9.35616	−0.317569
$869$	−2.55910	−0.0868116
$870$	0	0
$871$	43.0764	1.45959
$872$	17.7296	0.600399
$873$	0	0
$874$	−27.6901	−0.936631
$875$	18.2610	0.617336
$876$	0	0
$877$	24.3604	0.822593	0.411296	−	0.911502i	$-0.365076\pi$
$877$	24.3604	0.822593	0.411296	+	0.911502i	$0.365076\pi$
$878$	−19.2411	−0.649357
$879$	0	0
$880$	0.885838	0.0298616
$881$	32.4348	1.09276	0.546379	−	0.837538i	$-0.316006\pi$
$881$	32.4348	1.09276	0.546379	+	0.837538i	$0.316006\pi$
$882$	0	0
$883$	−26.0372	−0.876220	−0.438110	−	0.898921i	$-0.644352\pi$
$883$	−26.0372	−0.876220	−0.438110	+	0.898921i	$0.644352\pi$
$884$	56.5136	1.90076
$885$	0	0
$886$	4.56110	0.153233
$887$	14.7623	0.495668	0.247834	−	0.968803i	$-0.420281\pi$
$887$	14.7623	0.495668	0.247834	+	0.968803i	$0.420281\pi$
$888$	0	0
$889$	−48.3199	−1.62060
$890$	2.17036	0.0727505
$891$	0	0
$892$	23.1875	0.776376
$893$	66.0876	2.21154
$894$	0	0
$895$	−9.06750	−0.303093
$896$	−55.8592	−1.86613
$897$	0	0
$898$	−1.17518	−0.0392163
$899$	1.66137	0.0554099
$900$	0	0
$901$	43.3127	1.44296
$902$	1.49759	0.0498644
$903$	0	0
$904$	11.2837	0.375292
$905$	0.00707908	0.000235317 0
$906$	0	0
$907$	42.1811	1.40060	0.700300	−	0.713849i	$-0.253050\pi$
$907$	42.1811	1.40060	0.700300	+	0.713849i	$0.253050\pi$
$908$	1.27178	0.0422055
$909$	0	0
$910$	6.68262	0.221527
$911$	−57.2649	−1.89727	−0.948635	−	0.316373i	$-0.897535\pi$
$911$	−57.2649	−1.89727	−0.948635	+	0.316373i	$0.897535\pi$
$912$	0	0
$913$	−14.7949	−0.489639
$914$	−5.62578	−0.186084
$915$	0	0
$916$	−24.2934	−0.802677
$917$	−52.9456	−1.74842
$918$	0	0
$919$	1.86371	0.0614780	0.0307390	−	0.999527i	$-0.490214\pi$
$919$	1.86371	0.0614780	0.0307390	+	0.999527i	$0.490214\pi$
$920$	5.59083	0.184324
$921$	0	0
$922$	−1.83799	−0.0605310
$923$	22.0179	0.724727
$924$	0	0
$925$	43.2222	1.42114
$926$	6.16717	0.202666
$927$	0	0
$928$	7.75273	0.254496
$929$	36.0486	1.18272	0.591358	−	0.806409i	$-0.298592\pi$
$929$	36.0486	1.18272	0.591358	+	0.806409i	$0.298592\pi$
$930$	0	0
$931$	115.711	3.79228
$932$	−41.6752	−1.36512
$933$	0	0
$934$	−13.1107	−0.428994
$935$	−1.90899	−0.0624306
$936$	0	0
$937$	−16.1499	−0.527596	−0.263798	−	0.964578i	$-0.584975\pi$
$937$	−16.1499	−0.527596	−0.263798	+	0.964578i	$0.584975\pi$
$938$	17.0824	0.557760
$939$	0	0
$940$	−6.14051	−0.200281
$941$	−54.6806	−1.78254	−0.891268	−	0.453476i	$-0.850184\pi$
$941$	−54.6806	−1.78254	−0.891268	+	0.453476i	$0.850184\pi$
$942$	0	0
$943$	−20.0337	−0.652386
$944$	−21.9654	−0.714914
$945$	0	0
$946$	0.238135	0.00774244
$947$	−30.3263	−0.985472	−0.492736	−	0.870179i	$-0.664003\pi$
$947$	−30.3263	−0.985472	−0.492736	+	0.870179i	$0.664003\pi$
$948$	0	0
$949$	40.5534	1.31642
$950$	18.4972	0.600129
$951$	0	0
$952$	48.7002	1.57838
$953$	46.0196	1.49072	0.745361	−	0.666662i	$-0.232277\pi$
$953$	46.0196	1.49072	0.745361	+	0.666662i	$0.232277\pi$
$954$	0	0
$955$	2.36348	0.0764805
$956$	20.7077	0.669736
$957$	0	0
$958$	1.17592	0.0379923
$959$	−99.2074	−3.20357
$960$	0	0
$961$	−29.7182	−0.958650
$962$	32.1106	1.03529
$963$	0	0
$964$	−1.58237	−0.0509646
$965$	−2.95567	−0.0951465
$966$	0	0
$967$	55.9407	1.79893	0.899466	−	0.436991i	$-0.143956\pi$
$967$	55.9407	1.79893	0.899466	+	0.436991i	$0.143956\pi$
$968$	2.01242	0.0646817
$969$	0	0
$970$	−1.89894	−0.0609712
$971$	−0.422452	−0.0135571	−0.00677856	−	0.999977i	$-0.502158\pi$
$971$	−0.422452	−0.0135571	−0.00677856	+	0.999977i	$0.502158\pi$
$972$	0	0
$973$	−17.8094	−0.570944
$974$	22.9461	0.735242
$975$	0	0
$976$	2.31685	0.0741606
$977$	20.1876	0.645858	0.322929	−	0.946423i	$-0.395333\pi$
$977$	20.1876	0.645858	0.322929	+	0.946423i	$0.395333\pi$
$978$	0	0
$979$	−10.4506	−0.334001
$980$	−10.7513	−0.343437
$981$	0	0
$982$	3.34962	0.106891
$983$	25.6609	0.818457	0.409228	−	0.912432i	$-0.365798\pi$
$983$	25.6609	0.818457	0.409228	+	0.912432i	$0.365798\pi$
$984$	0	0
$985$	−0.127968	−0.00407739
$986$	−3.97953	−0.126734
$987$	0	0
$988$	−79.4136	−2.52648
$989$	−3.18559	−0.101296
$990$	0	0
$991$	−47.8468	−1.51991	−0.759953	−	0.649978i	$-0.774778\pi$
$991$	−47.8468	−1.51991	−0.759953	+	0.649978i	$0.774778\pi$
$992$	5.98165	0.189918
$993$	0	0
$994$	8.73142	0.276944
$995$	8.71359	0.276239
$996$	0	0
$997$	−25.9634	−0.822268	−0.411134	−	0.911575i	$-0.634867\pi$
$997$	−25.9634	−0.822268	−0.411134	+	0.911575i	$0.634867\pi$
$998$	−17.0432	−0.539494
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.l.1.10		21
3.2	odd	2		671.2.a.d.1.12	✓	21
33.32	even	2		7381.2.a.j.1.10		21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
671.2.a.d.1.12	✓	21	3.2	odd	2
6039.2.a.l.1.10		21	1.1	even	1	trivial
7381.2.a.j.1.10		21	33.32	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-0.543169 q^{2} -1.70497 q^{4} +0.382346 q^{5} -4.84691 q^{7} +2.01242 q^{8} +O(q^{10})\)	\(q-0.543169 q^{2} -1.70497 q^{4} +0.382346 q^{5} -4.84691 q^{7} +2.01242 q^{8} -0.207678 q^{10} +1.00000 q^{11} +6.63881 q^{13} +2.63269 q^{14} +2.31685 q^{16} -4.99283 q^{17} +7.01598 q^{19} -0.651888 q^{20} -0.543169 q^{22} +7.26609 q^{23} -4.85381 q^{25} -3.60599 q^{26} +8.26382 q^{28} -1.46741 q^{29} -1.13218 q^{31} -5.28329 q^{32} +2.71195 q^{34} -1.85320 q^{35} -8.90479 q^{37} -3.81086 q^{38} +0.769442 q^{40} -2.75714 q^{41} -0.438418 q^{43} -1.70497 q^{44} -3.94671 q^{46} +9.41958 q^{47} +16.4925 q^{49} +2.63644 q^{50} -11.3190 q^{52} -8.67499 q^{53} +0.382346 q^{55} -9.75403 q^{56} +0.797050 q^{58} -9.48074 q^{59} +1.00000 q^{61} +0.614967 q^{62} -1.76398 q^{64} +2.53832 q^{65} +6.48857 q^{67} +8.51261 q^{68} +1.00660 q^{70} +3.31654 q^{71} +6.10854 q^{73} +4.83681 q^{74} -11.9620 q^{76} -4.84691 q^{77} -2.55910 q^{79} +0.885838 q^{80} +1.49759 q^{82} -14.7949 q^{83} -1.90899 q^{85} +0.238135 q^{86} +2.01242 q^{88} -10.4506 q^{89} -32.1777 q^{91} -12.3884 q^{92} -5.11642 q^{94} +2.68253 q^{95} +9.14364 q^{97} -8.95822 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q + 32 q^{4} - 7 q^{5} + 5 q^{7} + 6 q^{8}+O(q^{10}) \) 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

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