LMFDB - Embedded newform 1386.2.a.q.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1386,2,Mod(1,1386)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1386, 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("1386.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1386.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:	$11.0672657201$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1304.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 11x - 2 $ x^3 - 11*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.40405$ of defining polynomial
Character	$\chi$	$=$	1386.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.09174 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -1.09174 q^{5} +1.00000 q^{7} -1.00000 q^{8} +1.09174 q^{10} -1.00000 q^{11} +6.80809 q^{13} -1.00000 q^{14} +1.00000 q^{16} -1.09174 q^{17} -3.89984 q^{19} -1.09174 q^{20} +1.00000 q^{22} +6.99158 q^{23} -3.80809 q^{25} -6.80809 q^{26} +1.00000 q^{28} -4.80809 q^{29} -1.27523 q^{31} -1.00000 q^{32} +1.09174 q^{34} -1.09174 q^{35} +4.18349 q^{37} +3.89984 q^{38} +1.09174 q^{40} +1.09174 q^{41} +10.6246 q^{43} -1.00000 q^{44} -6.99158 q^{46} +9.71635 q^{47} +1.00000 q^{49} +3.80809 q^{50} +6.80809 q^{52} -3.81651 q^{53} +1.09174 q^{55} -1.00000 q^{56} +4.80809 q^{58} -11.7997 q^{59} +11.1751 q^{61} +1.27523 q^{62} +1.00000 q^{64} -7.43270 q^{65} +8.99158 q^{67} -1.09174 q^{68} +1.09174 q^{70} +9.17507 q^{71} -5.09174 q^{73} -4.18349 q^{74} -3.89984 q^{76} -1.00000 q^{77} -4.99158 q^{79} -1.09174 q^{80} -1.09174 q^{82} +7.09174 q^{83} +1.19191 q^{85} -10.6246 q^{86} +1.00000 q^{88} +12.9916 q^{89} +6.80809 q^{91} +6.99158 q^{92} -9.71635 q^{94} +4.25763 q^{95} -7.61619 q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 3 q^{4} - 2 q^{5} + 3 q^{7} - 3 q^{8}+O(q^{10}) $ 3 * q - 3 * q^2 + 3 * q^4 - 2 * q^5 + 3 * q^7 - 3 * q^8	$ 3 q - 3 q^{2} + 3 q^{4} - 2 q^{5} + 3 q^{7} - 3 q^{8} + 2 q^{10} - 3 q^{11} - 3 q^{14} + 3 q^{16} - 2 q^{17} + 10 q^{19} - 2 q^{20} + 3 q^{22} - 2 q^{23} + 9 q^{25} + 3 q^{28} + 6 q^{29} - 3 q^{32} + 2 q^{34} - 2 q^{35} + 10 q^{37} - 10 q^{38} + 2 q^{40} + 2 q^{41} + 14 q^{43} - 3 q^{44} + 2 q^{46} + 10 q^{47} + 3 q^{49} - 9 q^{50} - 14 q^{53} + 2 q^{55} - 3 q^{56} - 6 q^{58} + 8 q^{59} + 8 q^{61} + 3 q^{64} + 16 q^{65} + 4 q^{67} - 2 q^{68} + 2 q^{70} + 2 q^{71} - 14 q^{73} - 10 q^{74} + 10 q^{76} - 3 q^{77} + 8 q^{79} - 2 q^{80} - 2 q^{82} + 20 q^{83} + 24 q^{85} - 14 q^{86} + 3 q^{88} + 16 q^{89} - 2 q^{92} - 10 q^{94} + 18 q^{97} - 3 q^{98}+O(q^{100}) $ 3 * q - 3 * q^2 + 3 * q^4 - 2 * q^5 + 3 * q^7 - 3 * q^8 + 2 * q^10 - 3 * q^11 - 3 * q^14 + 3 * q^16 - 2 * q^17 + 10 * q^19 - 2 * q^20 + 3 * q^22 - 2 * q^23 + 9 * q^25 + 3 * q^28 + 6 * q^29 - 3 * q^32 + 2 * q^34 - 2 * q^35 + 10 * q^37 - 10 * q^38 + 2 * q^40 + 2 * q^41 + 14 * q^43 - 3 * q^44 + 2 * q^46 + 10 * q^47 + 3 * q^49 - 9 * q^50 - 14 * q^53 + 2 * q^55 - 3 * q^56 - 6 * q^58 + 8 * q^59 + 8 * q^61 + 3 * q^64 + 16 * q^65 + 4 * q^67 - 2 * q^68 + 2 * q^70 + 2 * q^71 - 14 * q^73 - 10 * q^74 + 10 * q^76 - 3 * q^77 + 8 * q^79 - 2 * q^80 - 2 * q^82 + 20 * q^83 + 24 * q^85 - 14 * q^86 + 3 * q^88 + 16 * q^89 - 2 * q^92 - 10 * q^94 + 18 * q^97 - 3 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.09174	−0.488243	−0.244121	−	0.969745i	$-0.578500\pi$
$5$	−1.09174	−0.488243	−0.244121	+	0.969745i	$0.578500\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.00000	−0.353553
$9$	0	0
$10$	1.09174	0.345240
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	6.80809	1.88823	0.944113	−	0.329623i	$-0.106922\pi$
$13$	6.80809	1.88823	0.944113	+	0.329623i	$0.106922\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−1.09174	−0.264787	−0.132393	−	0.991197i	$-0.542266\pi$
$17$	−1.09174	−0.264787	−0.132393	+	0.991197i	$0.542266\pi$
$18$	0	0
$19$	−3.89984	−0.894684	−0.447342	−	0.894363i	$-0.647629\pi$
$19$	−3.89984	−0.894684	−0.447342	+	0.894363i	$0.647629\pi$
$20$	−1.09174	−0.244121
$21$	0	0
$22$	1.00000	0.213201
$23$	6.99158	1.45785	0.728923	−	0.684596i	$-0.240021\pi$
$23$	6.99158	1.45785	0.728923	+	0.684596i	$0.240021\pi$
$24$	0	0
$25$	−3.80809	−0.761619
$26$	−6.80809	−1.33518
$27$	0	0
$28$	1.00000	0.188982
$29$	−4.80809	−0.892841	−0.446420	−	0.894823i	$-0.647301\pi$
$29$	−4.80809	−0.892841	−0.446420	+	0.894823i	$0.647301\pi$
$30$	0	0
$31$	−1.27523	−0.229039	−0.114519	−	0.993421i	$-0.536533\pi$
$31$	−1.27523	−0.229039	−0.114519	+	0.993421i	$0.536533\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	1.09174	0.187233
$35$	−1.09174	−0.184538
$36$	0	0
$37$	4.18349	0.687761	0.343881	−	0.939013i	$-0.388258\pi$
$37$	4.18349	0.687761	0.343881	+	0.939013i	$0.388258\pi$
$38$	3.89984	0.632637
$39$	0	0
$40$	1.09174	0.172620
$41$	1.09174	0.170502	0.0852509	−	0.996360i	$-0.472831\pi$
$41$	1.09174	0.170502	0.0852509	+	0.996360i	$0.472831\pi$
$42$	0	0
$43$	10.6246	1.62024	0.810119	−	0.586266i	$-0.199403\pi$
$43$	10.6246	1.62024	0.810119	+	0.586266i	$0.199403\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	−6.99158	−1.03085
$47$	9.71635	1.41728	0.708638	−	0.705573i	$-0.249310\pi$
$47$	9.71635	1.41728	0.708638	+	0.705573i	$0.249310\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	3.80809	0.538546
$51$	0	0
$52$	6.80809	0.944113
$53$	−3.81651	−0.524238	−0.262119	−	0.965036i	$-0.584421\pi$
$53$	−3.81651	−0.524238	−0.262119	+	0.965036i	$0.584421\pi$
$54$	0	0
$55$	1.09174	0.147211
$56$	−1.00000	−0.133631
$57$	0	0
$58$	4.80809	0.631334
$59$	−11.7997	−1.53619	−0.768094	−	0.640338i	$-0.778794\pi$
$59$	−11.7997	−1.53619	−0.768094	+	0.640338i	$0.778794\pi$
$60$	0	0
$61$	11.1751	1.43082	0.715411	−	0.698704i	$-0.246240\pi$
$61$	11.1751	1.43082	0.715411	+	0.698704i	$0.246240\pi$
$62$	1.27523	0.161955
$63$	0	0
$64$	1.00000	0.125000
$65$	−7.43270	−0.921913
$66$	0	0
$67$	8.99158	1.09850	0.549248	−	0.835659i	$-0.314914\pi$
$67$	8.99158	1.09850	0.549248	+	0.835659i	$0.314914\pi$
$68$	−1.09174	−0.132393
$69$	0	0
$70$	1.09174	0.130488
$71$	9.17507	1.08888	0.544440	−	0.838800i	$-0.316742\pi$
$71$	9.17507	1.08888	0.544440	+	0.838800i	$0.316742\pi$
$72$	0	0
$73$	−5.09174	−0.595944	−0.297972	−	0.954575i	$-0.596310\pi$
$73$	−5.09174	−0.595944	−0.297972	+	0.954575i	$0.596310\pi$
$74$	−4.18349	−0.486321
$75$	0	0
$76$	−3.89984	−0.447342
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−4.99158	−0.561597	−0.280798	−	0.959767i	$-0.590599\pi$
$79$	−4.99158	−0.561597	−0.280798	+	0.959767i	$0.590599\pi$
$80$	−1.09174	−0.122061
$81$	0	0
$82$	−1.09174	−0.120563
$83$	7.09174	0.778420	0.389210	−	0.921149i	$-0.372748\pi$
$83$	7.09174	0.778420	0.389210	+	0.921149i	$0.372748\pi$
$84$	0	0
$85$	1.19191	0.129280
$86$	−10.6246	−1.14568
$87$	0	0
$88$	1.00000	0.106600
$89$	12.9916	1.37711	0.688553	−	0.725186i	$-0.258246\pi$
$89$	12.9916	1.37711	0.688553	+	0.725186i	$0.258246\pi$
$90$	0	0
$91$	6.80809	0.713682
$92$	6.99158	0.728923
$93$	0	0
$94$	−9.71635	−1.00216
$95$	4.25763	0.436823
$96$	0	0
$97$	−7.61619	−0.773307	−0.386653	−	0.922225i	$-0.626369\pi$
$97$	−7.61619	−0.773307	−0.386653	+	0.922225i	$0.626369\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	−3.80809	−0.380809
$101$	15.6162	1.55387	0.776934	−	0.629582i	$-0.216774\pi$
$101$	15.6162	1.55387	0.776934	+	0.629582i	$0.216774\pi$
$102$	0	0
$103$	3.09174	0.304639	0.152319	−	0.988331i	$-0.451326\pi$
$103$	3.09174	0.304639	0.152319	+	0.988331i	$0.451326\pi$
$104$	−6.80809	−0.667589
$105$	0	0
$106$	3.81651	0.370692
$107$	2.18349	0.211086	0.105543	−	0.994415i	$-0.466342\pi$
$107$	2.18349	0.211086	0.105543	+	0.994415i	$0.466342\pi$
$108$	0	0
$109$	8.00000	0.766261	0.383131	−	0.923694i	$-0.374846\pi$
$109$	8.00000	0.766261	0.383131	+	0.923694i	$0.374846\pi$
$110$	−1.09174	−0.104094
$111$	0	0
$112$	1.00000	0.0944911
$113$	−3.17507	−0.298686	−0.149343	−	0.988785i	$-0.547716\pi$
$113$	−3.17507	−0.298686	−0.149343	+	0.988785i	$0.547716\pi$
$114$	0	0
$115$	−7.63302	−0.711783
$116$	−4.80809	−0.446420
$117$	0	0
$118$	11.7997	1.08625
$119$	−1.09174	−0.100080
$120$	0	0
$121$	1.00000	0.0909091
$122$	−11.1751	−1.01174
$123$	0	0
$124$	−1.27523	−0.114519
$125$	9.61619	0.860098
$126$	0	0
$127$	−4.99158	−0.442931	−0.221466	−	0.975168i	$-0.571084\pi$
$127$	−4.99158	−0.442931	−0.221466	+	0.975168i	$0.571084\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	7.43270	0.651891
$131$	16.9083	1.47728	0.738641	−	0.674099i	$-0.235468\pi$
$131$	16.9083	1.47728	0.738641	+	0.674099i	$0.235468\pi$
$132$	0	0
$133$	−3.89984	−0.338159
$134$	−8.99158	−0.776754
$135$	0	0
$136$	1.09174	0.0936163
$137$	−15.1751	−1.29649	−0.648247	−	0.761430i	$-0.724498\pi$
$137$	−15.1751	−1.29649	−0.648247	+	0.761430i	$0.724498\pi$
$138$	0	0
$139$	−8.26682	−0.701182	−0.350591	−	0.936529i	$-0.614019\pi$
$139$	−8.26682	−0.701182	−0.350591	+	0.936529i	$0.614019\pi$
$140$	−1.09174	−0.0922692
$141$	0	0
$142$	−9.17507	−0.769955
$143$	−6.80809	−0.569321
$144$	0	0
$145$	5.24921	0.435923
$146$	5.09174	0.421396
$147$	0	0
$148$	4.18349	0.343881
$149$	15.6162	1.27933	0.639664	−	0.768655i	$-0.279074\pi$
$149$	15.6162	1.27933	0.639664	+	0.768655i	$0.279074\pi$
$150$	0	0
$151$	21.9832	1.78896	0.894482	−	0.447103i	$-0.147544\pi$
$151$	21.9832	1.78896	0.894482	+	0.447103i	$0.147544\pi$
$152$	3.89984	0.316319
$153$	0	0
$154$	1.00000	0.0805823
$155$	1.39223	0.111827
$156$	0	0
$157$	−14.9083	−1.18981	−0.594904	−	0.803797i	$-0.702810\pi$
$157$	−14.9083	−1.18981	−0.594904	+	0.803797i	$0.702810\pi$
$158$	4.99158	0.397109
$159$	0	0
$160$	1.09174	0.0863100
$161$	6.99158	0.551014
$162$	0	0
$163$	9.98317	0.781942	0.390971	−	0.920403i	$-0.372139\pi$
$163$	9.98317	0.781942	0.390971	+	0.920403i	$0.372139\pi$
$164$	1.09174	0.0852509
$165$	0	0
$166$	−7.09174	−0.550426
$167$	−7.43270	−0.575160	−0.287580	−	0.957757i	$-0.592851\pi$
$167$	−7.43270	−0.575160	−0.287580	+	0.957757i	$0.592851\pi$
$168$	0	0
$169$	33.3501	2.56540
$170$	−1.19191	−0.0914150
$171$	0	0
$172$	10.6246	0.810119
$173$	−5.79968	−0.440941	−0.220471	−	0.975394i	$-0.570759\pi$
$173$	−5.79968	−0.440941	−0.220471	+	0.975394i	$0.570759\pi$
$174$	0	0
$175$	−3.80809	−0.287865
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	−12.9916	−0.973760
$179$	−4.25763	−0.318230	−0.159115	−	0.987260i	$-0.550864\pi$
$179$	−4.25763	−0.318230	−0.159115	+	0.987260i	$0.550864\pi$
$180$	0	0
$181$	−5.09174	−0.378466	−0.189233	−	0.981932i	$-0.560600\pi$
$181$	−5.09174	−0.378466	−0.189233	+	0.981932i	$0.560600\pi$
$182$	−6.80809	−0.504650
$183$	0	0
$184$	−6.99158	−0.515426
$185$	−4.56730	−0.335795
$186$	0	0
$187$	1.09174	0.0798363
$188$	9.71635	0.708638
$189$	0	0
$190$	−4.25763	−0.308881
$191$	24.2408	1.75400	0.877001	−	0.480488i	$-0.159541\pi$
$191$	24.2408	1.75400	0.877001	+	0.480488i	$0.159541\pi$
$192$	0	0
$193$	−5.63302	−0.405474	−0.202737	−	0.979233i	$-0.564984\pi$
$193$	−5.63302	−0.405474	−0.202737	+	0.979233i	$0.564984\pi$
$194$	7.61619	0.546810
$195$	0	0
$196$	1.00000	0.0714286
$197$	26.4243	1.88265	0.941326	−	0.337498i	$-0.109581\pi$
$197$	26.4243	1.88265	0.941326	+	0.337498i	$0.109581\pi$
$198$	0	0
$199$	−11.0917	−0.786273	−0.393136	−	0.919480i	$-0.628610\pi$
$199$	−11.0917	−0.786273	−0.393136	+	0.919480i	$0.628610\pi$
$200$	3.80809	0.269273
$201$	0	0
$202$	−15.6162	−1.09875
$203$	−4.80809	−0.337462
$204$	0	0
$205$	−1.19191	−0.0832463
$206$	−3.09174	−0.215412
$207$	0	0
$208$	6.80809	0.472056
$209$	3.89984	0.269757
$210$	0	0
$211$	−20.6078	−1.41870	−0.709349	−	0.704858i	$-0.751011\pi$
$211$	−20.6078	−1.41870	−0.709349	+	0.704858i	$0.751011\pi$
$212$	−3.81651	−0.262119
$213$	0	0
$214$	−2.18349	−0.149260
$215$	−11.5994	−0.791069
$216$	0	0
$217$	−1.27523	−0.0865685
$218$	−8.00000	−0.541828
$219$	0	0
$220$	1.09174	0.0736054
$221$	−7.43270	−0.499977
$222$	0	0
$223$	−5.64221	−0.377830	−0.188915	−	0.981993i	$-0.560497\pi$
$223$	−5.64221	−0.377830	−0.188915	+	0.981993i	$0.560497\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	3.17507	0.211203
$227$	11.4587	0.760542	0.380271	−	0.924875i	$-0.375831\pi$
$227$	11.4587	0.760542	0.380271	+	0.924875i	$0.375831\pi$
$228$	0	0
$229$	−12.5244	−0.827639	−0.413819	−	0.910359i	$-0.635805\pi$
$229$	−12.5244	−0.827639	−0.413819	+	0.910359i	$0.635805\pi$
$230$	7.63302	0.503307
$231$	0	0
$232$	4.80809	0.315667
$233$	−25.2324	−1.65303	−0.826514	−	0.562916i	$-0.809679\pi$
$233$	−25.2324	−1.65303	−0.826514	+	0.562916i	$0.809679\pi$
$234$	0	0
$235$	−10.6078	−0.691975
$236$	−11.7997	−0.768094
$237$	0	0
$238$	1.09174	0.0707673
$239$	13.9832	0.904496	0.452248	−	0.891892i	$-0.350622\pi$
$239$	13.9832	0.904496	0.452248	+	0.891892i	$0.350622\pi$
$240$	0	0
$241$	−7.27523	−0.468639	−0.234319	−	0.972160i	$-0.575286\pi$
$241$	−7.27523	−0.468639	−0.234319	+	0.972160i	$0.575286\pi$
$242$	−1.00000	−0.0642824
$243$	0	0
$244$	11.1751	0.715411
$245$	−1.09174	−0.0697490
$246$	0	0
$247$	−26.5505	−1.68937
$248$	1.27523	0.0809774
$249$	0	0
$250$	−9.61619	−0.608181
$251$	−28.1667	−1.77786	−0.888932	−	0.458040i	$-0.848552\pi$
$251$	−28.1667	−1.77786	−0.888932	+	0.458040i	$0.848552\pi$
$252$	0	0
$253$	−6.99158	−0.439557
$254$	4.99158	0.313200
$255$	0	0
$256$	1.00000	0.0625000
$257$	−8.62461	−0.537988	−0.268994	−	0.963142i	$-0.586691\pi$
$257$	−8.62461	−0.537988	−0.268994	+	0.963142i	$0.586691\pi$
$258$	0	0
$259$	4.18349	0.259949
$260$	−7.43270	−0.460956
$261$	0	0
$262$	−16.9083	−1.04460
$263$	22.6078	1.39405	0.697027	−	0.717044i	$-0.254506\pi$
$263$	22.6078	1.39405	0.697027	+	0.717044i	$0.254506\pi$
$264$	0	0
$265$	4.16665	0.255956
$266$	3.89984	0.239114
$267$	0	0
$268$	8.99158	0.549248
$269$	−22.9083	−1.39674	−0.698370	−	0.715736i	$-0.746091\pi$
$269$	−22.9083	−1.39674	−0.698370	+	0.715736i	$0.746091\pi$
$270$	0	0
$271$	−13.8165	−0.839293	−0.419647	−	0.907688i	$-0.637846\pi$
$271$	−13.8165	−0.839293	−0.419647	+	0.907688i	$0.637846\pi$
$272$	−1.09174	−0.0661967
$273$	0	0
$274$	15.1751	0.916760
$275$	3.80809	0.229637
$276$	0	0
$277$	17.6162	1.05845	0.529227	−	0.848480i	$-0.322482\pi$
$277$	17.6162	1.05845	0.529227	+	0.848480i	$0.322482\pi$
$278$	8.26682	0.495811
$279$	0	0
$280$	1.09174	0.0652442
$281$	19.9832	1.19210	0.596048	−	0.802949i	$-0.296737\pi$
$281$	19.9832	1.19210	0.596048	+	0.802949i	$0.296737\pi$
$282$	0	0
$283$	−23.5329	−1.39888	−0.699442	−	0.714690i	$-0.746568\pi$
$283$	−23.5329	−1.39888	−0.699442	+	0.714690i	$0.746568\pi$
$284$	9.17507	0.544440
$285$	0	0
$286$	6.80809	0.402571
$287$	1.09174	0.0644436
$288$	0	0
$289$	−15.8081	−0.929888
$290$	−5.24921	−0.308244
$291$	0	0
$292$	−5.09174	−0.297972
$293$	−3.81651	−0.222963	−0.111481	−	0.993767i	$-0.535560\pi$
$293$	−3.81651	−0.222963	−0.111481	+	0.993767i	$0.535560\pi$
$294$	0	0
$295$	12.8822	0.750033
$296$	−4.18349	−0.243160
$297$	0	0
$298$	−15.6162	−0.904621
$299$	47.5994	2.75274
$300$	0	0
$301$	10.6246	0.612392
$302$	−21.9832	−1.26499
$303$	0	0
$304$	−3.89984	−0.223671
$305$	−12.2003	−0.698588
$306$	0	0
$307$	29.5160	1.68457	0.842284	−	0.539034i	$-0.181210\pi$
$307$	29.5160	1.68457	0.842284	+	0.539034i	$0.181210\pi$
$308$	−1.00000	−0.0569803
$309$	0	0
$310$	−1.39223	−0.0790733
$311$	9.71635	0.550964	0.275482	−	0.961306i	$-0.411163\pi$
$311$	9.71635	0.550964	0.275482	+	0.961306i	$0.411163\pi$
$312$	0	0
$313$	−21.7997	−1.23219	−0.616095	−	0.787672i	$-0.711286\pi$
$313$	−21.7997	−1.23219	−0.616095	+	0.787672i	$0.711286\pi$
$314$	14.9083	0.841322
$315$	0	0
$316$	−4.99158	−0.280798
$317$	−29.7997	−1.67372	−0.836858	−	0.547420i	$-0.815610\pi$
$317$	−29.7997	−1.67372	−0.836858	+	0.547420i	$0.815610\pi$
$318$	0	0
$319$	4.80809	0.269202
$320$	−1.09174	−0.0610304
$321$	0	0
$322$	−6.99158	−0.389626
$323$	4.25763	0.236901
$324$	0	0
$325$	−25.9259	−1.43811
$326$	−9.98317	−0.552916
$327$	0	0
$328$	−1.09174	−0.0602815
$329$	9.71635	0.535680
$330$	0	0
$331$	−12.6246	−0.693911	−0.346956	−	0.937882i	$-0.612785\pi$
$331$	−12.6246	−0.693911	−0.346956	+	0.937882i	$0.612785\pi$
$332$	7.09174	0.389210
$333$	0	0
$334$	7.43270	0.406699
$335$	−9.81651	−0.536333
$336$	0	0
$337$	−2.56730	−0.139850	−0.0699249	−	0.997552i	$-0.522276\pi$
$337$	−2.56730	−0.139850	−0.0699249	+	0.997552i	$0.522276\pi$
$338$	−33.3501	−1.81401
$339$	0	0
$340$	1.19191	0.0646402
$341$	1.27523	0.0690578
$342$	0	0
$343$	1.00000	0.0539949
$344$	−10.6246	−0.572840
$345$	0	0
$346$	5.79968	0.311793
$347$	2.18349	0.117216	0.0586079	−	0.998281i	$-0.481334\pi$
$347$	2.18349	0.117216	0.0586079	+	0.998281i	$0.481334\pi$
$348$	0	0
$349$	7.00842	0.375152	0.187576	−	0.982250i	$-0.439937\pi$
$349$	7.00842	0.375152	0.187576	+	0.982250i	$0.439937\pi$
$350$	3.80809	0.203551
$351$	0	0
$352$	1.00000	0.0533002
$353$	−8.62461	−0.459041	−0.229521	−	0.973304i	$-0.573716\pi$
$353$	−8.62461	−0.459041	−0.229521	+	0.973304i	$0.573716\pi$
$354$	0	0
$355$	−10.0168	−0.531638
$356$	12.9916	0.688553
$357$	0	0
$358$	4.25763	0.225023
$359$	−26.9747	−1.42367	−0.711836	−	0.702345i	$-0.752136\pi$
$359$	−26.9747	−1.42367	−0.711836	+	0.702345i	$0.752136\pi$
$360$	0	0
$361$	−3.79126	−0.199540
$362$	5.09174	0.267616
$363$	0	0
$364$	6.80809	0.356841
$365$	5.55888	0.290965
$366$	0	0
$367$	−6.72477	−0.351030	−0.175515	−	0.984477i	$-0.556159\pi$
$367$	−6.72477	−0.351030	−0.175515	+	0.984477i	$0.556159\pi$
$368$	6.99158	0.364461
$369$	0	0
$370$	4.56730	0.237443
$371$	−3.81651	−0.198143
$372$	0	0
$373$	−10.4411	−0.540620	−0.270310	−	0.962773i	$-0.587126\pi$
$373$	−10.4411	−0.540620	−0.270310	+	0.962773i	$0.587126\pi$
$374$	−1.09174	−0.0564528
$375$	0	0
$376$	−9.71635	−0.501082
$377$	−32.7340	−1.68588
$378$	0	0
$379$	14.3501	0.737117	0.368559	−	0.929604i	$-0.379851\pi$
$379$	14.3501	0.737117	0.368559	+	0.929604i	$0.379851\pi$
$380$	4.25763	0.218412
$381$	0	0
$382$	−24.2408	−1.24027
$383$	−9.91667	−0.506718	−0.253359	−	0.967372i	$-0.581535\pi$
$383$	−9.91667	−0.506718	−0.253359	+	0.967372i	$0.581535\pi$
$384$	0	0
$385$	1.09174	0.0556405
$386$	5.63302	0.286713
$387$	0	0
$388$	−7.61619	−0.386653
$389$	−33.9663	−1.72216	−0.861081	−	0.508468i	$-0.830212\pi$
$389$	−33.9663	−1.72216	−0.861081	+	0.508468i	$0.830212\pi$
$390$	0	0
$391$	−7.63302	−0.386019
$392$	−1.00000	−0.0505076
$393$	0	0
$394$	−26.4243	−1.33124
$395$	5.44953	0.274196
$396$	0	0
$397$	−26.7079	−1.34043	−0.670216	−	0.742166i	$-0.733799\pi$
$397$	−26.7079	−1.34043	−0.670216	+	0.742166i	$0.733799\pi$
$398$	11.0917	0.555979
$399$	0	0
$400$	−3.80809	−0.190405
$401$	−15.1751	−0.757807	−0.378903	−	0.925436i	$-0.623699\pi$
$401$	−15.1751	−0.757807	−0.378903	+	0.925436i	$0.623699\pi$
$402$	0	0
$403$	−8.68191	−0.432477
$404$	15.6162	0.776934
$405$	0	0
$406$	4.80809	0.238622
$407$	−4.18349	−0.207368
$408$	0	0
$409$	−12.7248	−0.629199	−0.314600	−	0.949224i	$-0.601870\pi$
$409$	−12.7248	−0.629199	−0.314600	+	0.949224i	$0.601870\pi$
$410$	1.19191	0.0588640
$411$	0	0
$412$	3.09174	0.152319
$413$	−11.7997	−0.580624
$414$	0	0
$415$	−7.74237	−0.380058
$416$	−6.80809	−0.333794
$417$	0	0
$418$	−3.89984	−0.190747
$419$	31.2324	1.52580	0.762901	−	0.646516i	$-0.223775\pi$
$419$	31.2324	1.52580	0.762901	+	0.646516i	$0.223775\pi$
$420$	0	0
$421$	18.3670	0.895152	0.447576	−	0.894246i	$-0.352287\pi$
$421$	18.3670	0.895152	0.447576	+	0.894246i	$0.352287\pi$
$422$	20.6078	1.00317
$423$	0	0
$424$	3.81651	0.185346
$425$	4.15747	0.201667
$426$	0	0
$427$	11.1751	0.540800
$428$	2.18349	0.105543
$429$	0	0
$430$	11.5994	0.559371
$431$	−3.37539	−0.162587	−0.0812935	−	0.996690i	$-0.525905\pi$
$431$	−3.37539	−0.162587	−0.0812935	+	0.996690i	$0.525905\pi$
$432$	0	0
$433$	5.26604	0.253070	0.126535	−	0.991962i	$-0.459614\pi$
$433$	5.26604	0.253070	0.126535	+	0.991962i	$0.459614\pi$
$434$	1.27523	0.0612132
$435$	0	0
$436$	8.00000	0.383131
$437$	−27.2660	−1.30431
$438$	0	0
$439$	−3.79968	−0.181349	−0.0906743	−	0.995881i	$-0.528902\pi$
$439$	−3.79968	−0.181349	−0.0906743	+	0.995881i	$0.528902\pi$
$440$	−1.09174	−0.0520469
$441$	0	0
$442$	7.43270	0.353537
$443$	20.6246	0.979905	0.489952	−	0.871749i	$-0.337014\pi$
$443$	20.6246	0.979905	0.489952	+	0.871749i	$0.337014\pi$
$444$	0	0
$445$	−14.1835	−0.672362
$446$	5.64221	0.267166
$447$	0	0
$448$	1.00000	0.0472456
$449$	−27.1751	−1.28247	−0.641235	−	0.767344i	$-0.721578\pi$
$449$	−27.1751	−1.28247	−0.641235	+	0.767344i	$0.721578\pi$
$450$	0	0
$451$	−1.09174	−0.0514082
$452$	−3.17507	−0.149343
$453$	0	0
$454$	−11.4587	−0.537784
$455$	−7.43270	−0.348450
$456$	0	0
$457$	−15.2492	−0.713328	−0.356664	−	0.934233i	$-0.616086\pi$
$457$	−15.2492	−0.713328	−0.356664	+	0.934233i	$0.616086\pi$
$458$	12.5244	0.585229
$459$	0	0
$460$	−7.63302	−0.355891
$461$	−13.2324	−0.616293	−0.308147	−	0.951339i	$-0.599709\pi$
$461$	−13.2324	−0.616293	−0.308147	+	0.951339i	$0.599709\pi$
$462$	0	0
$463$	3.63302	0.168841	0.0844204	−	0.996430i	$-0.473096\pi$
$463$	3.63302	0.168841	0.0844204	+	0.996430i	$0.473096\pi$
$464$	−4.80809	−0.223210
$465$	0	0
$466$	25.2324	1.16887
$467$	13.9832	0.647064	0.323532	−	0.946217i	$-0.395130\pi$
$467$	13.9832	0.647064	0.323532	+	0.946217i	$0.395130\pi$
$468$	0	0
$469$	8.99158	0.415193
$470$	10.6078	0.489300
$471$	0	0
$472$	11.7997	0.543124
$473$	−10.6246	−0.488520
$474$	0	0
$475$	14.8510	0.681408
$476$	−1.09174	−0.0500400
$477$	0	0
$478$	−13.9832	−0.639575
$479$	16.3670	0.747826	0.373913	−	0.927464i	$-0.378016\pi$
$479$	16.3670	0.747826	0.373913	+	0.927464i	$0.378016\pi$
$480$	0	0
$481$	28.4816	1.29865
$482$	7.27523	0.331378
$483$	0	0
$484$	1.00000	0.0454545
$485$	8.31493	0.377562
$486$	0	0
$487$	24.1667	1.09510	0.547548	−	0.836774i	$-0.315561\pi$
$487$	24.1667	1.09510	0.547548	+	0.836774i	$0.315561\pi$
$488$	−11.1751	−0.505872
$489$	0	0
$490$	1.09174	0.0493200
$491$	7.63302	0.344473	0.172237	−	0.985056i	$-0.444901\pi$
$491$	7.63302	0.344473	0.172237	+	0.985056i	$0.444901\pi$
$492$	0	0
$493$	5.24921	0.236413
$494$	26.5505	1.19456
$495$	0	0
$496$	−1.27523	−0.0572597
$497$	9.17507	0.411558
$498$	0	0
$499$	12.2576	0.548727	0.274363	−	0.961626i	$-0.411533\pi$
$499$	12.2576	0.548727	0.274363	+	0.961626i	$0.411533\pi$
$500$	9.61619	0.430049
$501$	0	0
$502$	28.1667	1.25714
$503$	−31.4327	−1.40151	−0.700757	−	0.713400i	$-0.747154\pi$
$503$	−31.4327	−1.40151	−0.700757	+	0.713400i	$0.747154\pi$
$504$	0	0
$505$	−17.0489	−0.758666
$506$	6.99158	0.310814
$507$	0	0
$508$	−4.99158	−0.221466
$509$	−37.7737	−1.67429	−0.837144	−	0.546983i	$-0.815776\pi$
$509$	−37.7737	−1.67429	−0.837144	+	0.546983i	$0.815776\pi$
$510$	0	0
$511$	−5.09174	−0.225246
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	8.62461	0.380415
$515$	−3.37539	−0.148738
$516$	0	0
$517$	−9.71635	−0.427325
$518$	−4.18349	−0.183812
$519$	0	0
$520$	7.43270	0.325945
$521$	−4.25763	−0.186530	−0.0932650	−	0.995641i	$-0.529730\pi$
$521$	−4.25763	−0.186530	−0.0932650	+	0.995641i	$0.529730\pi$
$522$	0	0
$523$	4.83412	0.211381	0.105691	−	0.994399i	$-0.466295\pi$
$523$	4.83412	0.211381	0.105691	+	0.994399i	$0.466295\pi$
$524$	16.9083	0.738641
$525$	0	0
$526$	−22.6078	−0.985746
$527$	1.39223	0.0606465
$528$	0	0
$529$	25.8822	1.12531
$530$	−4.16665	−0.180988
$531$	0	0
$532$	−3.89984	−0.169079
$533$	7.43270	0.321946
$534$	0	0
$535$	−2.38381	−0.103061
$536$	−8.99158	−0.388377
$537$	0	0
$538$	22.9083	0.987645
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−20.0573	−0.862331	−0.431165	−	0.902273i	$-0.641897\pi$
$541$	−20.0573	−0.862331	−0.431165	+	0.902273i	$0.641897\pi$
$542$	13.8165	0.593470
$543$	0	0
$544$	1.09174	0.0468082
$545$	−8.73396	−0.374122
$546$	0	0
$547$	25.9090	1.10779	0.553895	−	0.832587i	$-0.313141\pi$
$547$	25.9090	1.10779	0.553895	+	0.832587i	$0.313141\pi$
$548$	−15.1751	−0.648247
$549$	0	0
$550$	−3.80809	−0.162378
$551$	18.7508	0.798811
$552$	0	0
$553$	−4.99158	−0.212264
$554$	−17.6162	−0.748440
$555$	0	0
$556$	−8.26682	−0.350591
$557$	1.54205	0.0653387	0.0326694	−	0.999466i	$-0.489599\pi$
$557$	1.54205	0.0653387	0.0326694	+	0.999466i	$0.489599\pi$
$558$	0	0
$559$	72.3333	3.05937
$560$	−1.09174	−0.0461346
$561$	0	0
$562$	−19.9832	−0.842939
$563$	31.0917	1.31036	0.655180	−	0.755472i	$-0.272593\pi$
$563$	31.0917	1.31036	0.655180	+	0.755472i	$0.272593\pi$
$564$	0	0
$565$	3.46637	0.145831
$566$	23.5329	0.989160
$567$	0	0
$568$	−9.17507	−0.384977
$569$	−16.0168	−0.671461	−0.335730	−	0.941958i	$-0.608983\pi$
$569$	−16.0168	−0.671461	−0.335730	+	0.941958i	$0.608983\pi$
$570$	0	0
$571$	12.8081	0.536002	0.268001	−	0.963419i	$-0.413637\pi$
$571$	12.8081	0.536002	0.268001	+	0.963419i	$0.413637\pi$
$572$	−6.80809	−0.284661
$573$	0	0
$574$	−1.09174	−0.0455685
$575$	−26.6246	−1.11032
$576$	0	0
$577$	−21.7997	−0.907532	−0.453766	−	0.891121i	$-0.649920\pi$
$577$	−21.7997	−0.907532	−0.453766	+	0.891121i	$0.649920\pi$
$578$	15.8081	0.657530
$579$	0	0
$580$	5.24921	0.217962
$581$	7.09174	0.294215
$582$	0	0
$583$	3.81651	0.158064
$584$	5.09174	0.210698
$585$	0	0
$586$	3.81651	0.157659
$587$	−9.81651	−0.405171	−0.202585	−	0.979265i	$-0.564934\pi$
$587$	−9.81651	−0.405171	−0.202585	+	0.979265i	$0.564934\pi$
$588$	0	0
$589$	4.97320	0.204917
$590$	−12.8822	−0.530353
$591$	0	0
$592$	4.18349	0.171940
$593$	9.82570	0.403493	0.201747	−	0.979438i	$-0.435338\pi$
$593$	9.82570	0.403493	0.201747	+	0.979438i	$0.435338\pi$
$594$	0	0
$595$	1.19191	0.0488634
$596$	15.6162	0.639664
$597$	0	0
$598$	−47.5994	−1.94648
$599$	−2.82493	−0.115423	−0.0577117	−	0.998333i	$-0.518380\pi$
$599$	−2.82493	−0.115423	−0.0577117	+	0.998333i	$0.518380\pi$
$600$	0	0
$601$	39.2416	1.60070	0.800348	−	0.599535i	$-0.204648\pi$
$601$	39.2416	1.60070	0.800348	+	0.599535i	$0.204648\pi$
$602$	−10.6246	−0.433027
$603$	0	0
$604$	21.9832	0.894482
$605$	−1.09174	−0.0443857
$606$	0	0
$607$	−16.8822	−0.685229	−0.342614	−	0.939476i	$-0.611312\pi$
$607$	−16.8822	−0.685229	−0.342614	+	0.939476i	$0.611312\pi$
$608$	3.89984	0.158159
$609$	0	0
$610$	12.2003	0.493977
$611$	66.1498	2.67614
$612$	0	0
$613$	44.7913	1.80910	0.904551	−	0.426365i	$-0.140206\pi$
$613$	44.7913	1.80910	0.904551	+	0.426365i	$0.140206\pi$
$614$	−29.5160	−1.19117
$615$	0	0
$616$	1.00000	0.0402911
$617$	37.9832	1.52914	0.764572	−	0.644538i	$-0.222950\pi$
$617$	37.9832	1.52914	0.764572	+	0.644538i	$0.222950\pi$
$618$	0	0
$619$	9.98317	0.401257	0.200629	−	0.979667i	$-0.435702\pi$
$619$	9.98317	0.401257	0.200629	+	0.979667i	$0.435702\pi$
$620$	1.39223	0.0559133
$621$	0	0
$622$	−9.71635	−0.389590
$623$	12.9916	0.520497
$624$	0	0
$625$	8.54205	0.341682
$626$	21.7997	0.871290
$627$	0	0
$628$	−14.9083	−0.594904
$629$	−4.56730	−0.182110
$630$	0	0
$631$	−1.41586	−0.0563647	−0.0281823	−	0.999603i	$-0.508972\pi$
$631$	−1.41586	−0.0563647	−0.0281823	+	0.999603i	$0.508972\pi$
$632$	4.99158	0.198555
$633$	0	0
$634$	29.7997	1.18350
$635$	5.44953	0.216258
$636$	0	0
$637$	6.80809	0.269747
$638$	−4.80809	−0.190354
$639$	0	0
$640$	1.09174	0.0431550
$641$	33.2155	1.31194	0.655968	−	0.754789i	$-0.272261\pi$
$641$	33.2155	1.31194	0.655968	+	0.754789i	$0.272261\pi$
$642$	0	0
$643$	−27.7997	−1.09631	−0.548156	−	0.836376i	$-0.684670\pi$
$643$	−27.7997	−1.09631	−0.548156	+	0.836376i	$0.684670\pi$
$644$	6.99158	0.275507
$645$	0	0
$646$	−4.25763	−0.167514
$647$	−14.2837	−0.561548	−0.280774	−	0.959774i	$-0.590591\pi$
$647$	−14.2837	−0.561548	−0.280774	+	0.959774i	$0.590591\pi$
$648$	0	0
$649$	11.7997	0.463178
$650$	25.9259	1.01690
$651$	0	0
$652$	9.98317	0.390971
$653$	37.2324	1.45702	0.728508	−	0.685038i	$-0.240214\pi$
$653$	37.2324	1.45702	0.728508	+	0.685038i	$0.240214\pi$
$654$	0	0
$655$	−18.4595	−0.721272
$656$	1.09174	0.0426255
$657$	0	0
$658$	−9.71635	−0.378783
$659$	−23.3990	−0.911497	−0.455748	−	0.890109i	$-0.650628\pi$
$659$	−23.3990	−0.911497	−0.455748	+	0.890109i	$0.650628\pi$
$660$	0	0
$661$	−36.1238	−1.40505	−0.702526	−	0.711658i	$-0.747945\pi$
$661$	−36.1238	−1.40505	−0.702526	+	0.711658i	$0.747945\pi$
$662$	12.6246	0.490669
$663$	0	0
$664$	−7.09174	−0.275213
$665$	4.25763	0.165104
$666$	0	0
$667$	−33.6162	−1.30162
$668$	−7.43270	−0.287580
$669$	0	0
$670$	9.81651	0.379245
$671$	−11.1751	−0.431409
$672$	0	0
$673$	30.1667	1.16284	0.581420	−	0.813604i	$-0.302498\pi$
$673$	30.1667	1.16284	0.581420	+	0.813604i	$0.302498\pi$
$674$	2.56730	0.0988887
$675$	0	0
$676$	33.3501	1.28270
$677$	−2.93428	−0.112773	−0.0563867	−	0.998409i	$-0.517958\pi$
$677$	−2.93428	−0.112773	−0.0563867	+	0.998409i	$0.517958\pi$
$678$	0	0
$679$	−7.61619	−0.292282
$680$	−1.19191	−0.0457075
$681$	0	0
$682$	−1.27523	−0.0488312
$683$	17.3586	0.664207	0.332103	−	0.943243i	$-0.392242\pi$
$683$	17.3586	0.664207	0.332103	+	0.943243i	$0.392242\pi$
$684$	0	0
$685$	16.5673	0.633004
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	10.6246	0.405059
$689$	−25.9832	−0.989880
$690$	0	0
$691$	50.1498	1.90779	0.953895	−	0.300142i	$-0.0970341\pi$
$691$	50.1498	1.90779	0.953895	+	0.300142i	$0.0970341\pi$
$692$	−5.79968	−0.220471
$693$	0	0
$694$	−2.18349	−0.0828841
$695$	9.02525	0.342347
$696$	0	0
$697$	−1.19191	−0.0451467
$698$	−7.00842	−0.265272
$699$	0	0
$700$	−3.80809	−0.143932
$701$	−43.5825	−1.64609	−0.823045	−	0.567977i	$-0.807726\pi$
$701$	−43.5825	−1.64609	−0.823045	+	0.567977i	$0.807726\pi$
$702$	0	0
$703$	−16.3149	−0.615329
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	8.62461	0.324591
$707$	15.6162	0.587307
$708$	0	0
$709$	41.7660	1.56856	0.784278	−	0.620410i	$-0.213034\pi$
$709$	41.7660	1.56856	0.784278	+	0.620410i	$0.213034\pi$
$710$	10.0168	0.375925
$711$	0	0
$712$	−12.9916	−0.486880
$713$	−8.91590	−0.333903
$714$	0	0
$715$	7.43270	0.277967
$716$	−4.25763	−0.159115
$717$	0	0
$718$	26.9747	1.00669
$719$	−28.2668	−1.05417	−0.527087	−	0.849811i	$-0.676716\pi$
$719$	−28.2668	−1.05417	−0.527087	+	0.849811i	$0.676716\pi$
$720$	0	0
$721$	3.09174	0.115143
$722$	3.79126	0.141096
$723$	0	0
$724$	−5.09174	−0.189233
$725$	18.3097	0.680004
$726$	0	0
$727$	−29.2416	−1.08451	−0.542255	−	0.840214i	$-0.682429\pi$
$727$	−29.2416	−1.08451	−0.542255	+	0.840214i	$0.682429\pi$
$728$	−6.80809	−0.252325
$729$	0	0
$730$	−5.55888	−0.205744
$731$	−11.5994	−0.429018
$732$	0	0
$733$	−36.4243	−1.34536	−0.672681	−	0.739933i	$-0.734857\pi$
$733$	−36.4243	−1.34536	−0.672681	+	0.739933i	$0.734857\pi$
$734$	6.72477	0.248216
$735$	0	0
$736$	−6.99158	−0.257713
$737$	−8.99158	−0.331209
$738$	0	0
$739$	46.4243	1.70774	0.853872	−	0.520482i	$-0.174248\pi$
$739$	46.4243	1.70774	0.853872	+	0.520482i	$0.174248\pi$
$740$	−4.56730	−0.167897
$741$	0	0
$742$	3.81651	0.140109
$743$	−4.36698	−0.160209	−0.0801044	−	0.996786i	$-0.525525\pi$
$743$	−4.36698	−0.160209	−0.0801044	+	0.996786i	$0.525525\pi$
$744$	0	0
$745$	−17.0489	−0.624623
$746$	10.4411	0.382276
$747$	0	0
$748$	1.09174	0.0399181
$749$	2.18349	0.0797830
$750$	0	0
$751$	−34.1498	−1.24614	−0.623072	−	0.782164i	$-0.714116\pi$
$751$	−34.1498	−1.24614	−0.623072	+	0.782164i	$0.714116\pi$
$752$	9.71635	0.354319
$753$	0	0
$754$	32.7340	1.19210
$755$	−24.0000	−0.873449
$756$	0	0
$757$	−6.93428	−0.252031	−0.126015	−	0.992028i	$-0.540219\pi$
$757$	−6.93428	−0.252031	−0.126015	+	0.992028i	$0.540219\pi$
$758$	−14.3501	−0.521221
$759$	0	0
$760$	−4.25763	−0.154440
$761$	40.1575	1.45571	0.727853	−	0.685733i	$-0.240518\pi$
$761$	40.1575	1.45571	0.727853	+	0.685733i	$0.240518\pi$
$762$	0	0
$763$	8.00000	0.289619
$764$	24.2408	0.877001
$765$	0	0
$766$	9.91667	0.358304
$767$	−80.3333	−2.90067
$768$	0	0
$769$	−10.1406	−0.365681	−0.182840	−	0.983143i	$-0.558529\pi$
$769$	−10.1406	−0.365681	−0.182840	+	0.983143i	$0.558529\pi$
$770$	−1.09174	−0.0393437
$771$	0	0
$772$	−5.63302	−0.202737
$773$	2.17430	0.0782041	0.0391021	−	0.999235i	$-0.487550\pi$
$773$	2.17430	0.0782041	0.0391021	+	0.999235i	$0.487550\pi$
$774$	0	0
$775$	4.85621	0.174440
$776$	7.61619	0.273405
$777$	0	0
$778$	33.9663	1.21775
$779$	−4.25763	−0.152545
$780$	0	0
$781$	−9.17507	−0.328310
$782$	7.63302	0.272956
$783$	0	0
$784$	1.00000	0.0357143
$785$	16.2760	0.580916
$786$	0	0
$787$	−8.66746	−0.308962	−0.154481	−	0.987996i	$-0.549370\pi$
$787$	−8.66746	−0.308962	−0.154481	+	0.987996i	$0.549370\pi$
$788$	26.4243	0.941326
$789$	0	0
$790$	−5.44953	−0.193886
$791$	−3.17507	−0.112893
$792$	0	0
$793$	76.0809	2.70171
$794$	26.7079	0.947829
$795$	0	0
$796$	−11.0917	−0.393136
$797$	−3.47556	−0.123111	−0.0615553	−	0.998104i	$-0.519606\pi$
$797$	−3.47556	−0.123111	−0.0615553	+	0.998104i	$0.519606\pi$
$798$	0	0
$799$	−10.6078	−0.375276
$800$	3.80809	0.134636
$801$	0	0
$802$	15.1751	0.535850
$803$	5.09174	0.179684
$804$	0	0
$805$	−7.63302	−0.269029
$806$	8.68191	0.305807
$807$	0	0
$808$	−15.6162	−0.549376
$809$	−10.1667	−0.357441	−0.178720	−	0.983900i	$-0.557196\pi$
$809$	−10.1667	−0.357441	−0.178720	+	0.983900i	$0.557196\pi$
$810$	0	0
$811$	−45.8493	−1.60999	−0.804994	−	0.593283i	$-0.797832\pi$
$811$	−45.8493	−1.60999	−0.804994	+	0.593283i	$0.797832\pi$
$812$	−4.80809	−0.168731
$813$	0	0
$814$	4.18349	0.146631
$815$	−10.8991	−0.381778
$816$	0	0
$817$	−41.4342	−1.44960
$818$	12.7248	0.444911
$819$	0	0
$820$	−1.19191	−0.0416232
$821$	18.0000	0.628204	0.314102	−	0.949389i	$-0.398297\pi$
$821$	18.0000	0.628204	0.314102	+	0.949389i	$0.398297\pi$
$822$	0	0
$823$	−1.41586	−0.0493539	−0.0246770	−	0.999695i	$-0.507856\pi$
$823$	−1.41586	−0.0493539	−0.0246770	+	0.999695i	$0.507856\pi$
$824$	−3.09174	−0.107706
$825$	0	0
$826$	11.7997	0.410563
$827$	16.3670	0.569136	0.284568	−	0.958656i	$-0.408150\pi$
$827$	16.3670	0.569136	0.284568	+	0.958656i	$0.408150\pi$
$828$	0	0
$829$	52.1238	1.81033	0.905167	−	0.425056i	$-0.139746\pi$
$829$	52.1238	1.81033	0.905167	+	0.425056i	$0.139746\pi$
$830$	7.74237	0.268742
$831$	0	0
$832$	6.80809	0.236028
$833$	−1.09174	−0.0378267
$834$	0	0
$835$	8.11461	0.280818
$836$	3.89984	0.134879
$837$	0	0
$838$	−31.2324	−1.07890
$839$	−28.2668	−0.975879	−0.487939	−	0.872877i	$-0.662251\pi$
$839$	−28.2668	−0.975879	−0.487939	+	0.872877i	$0.662251\pi$
$840$	0	0
$841$	−5.88223	−0.202836
$842$	−18.3670	−0.632968
$843$	0	0
$844$	−20.6078	−0.709349
$845$	−36.4098	−1.25254
$846$	0	0
$847$	1.00000	0.0343604
$848$	−3.81651	−0.131060
$849$	0	0
$850$	−4.15747	−0.142600
$851$	29.2492	1.00265
$852$	0	0
$853$	17.5252	0.600052	0.300026	−	0.953931i	$-0.403005\pi$
$853$	17.5252	0.600052	0.300026	+	0.953931i	$0.403005\pi$
$854$	−11.1751	−0.382403
$855$	0	0
$856$	−2.18349	−0.0746301
$857$	3.95714	0.135173	0.0675867	−	0.997713i	$-0.478470\pi$
$857$	3.95714	0.135173	0.0675867	+	0.997713i	$0.478470\pi$
$858$	0	0
$859$	53.4159	1.82253	0.911263	−	0.411825i	$-0.135109\pi$
$859$	53.4159	1.82253	0.911263	+	0.411825i	$0.135109\pi$
$860$	−11.5994	−0.395535
$861$	0	0
$862$	3.37539	0.114966
$863$	6.10935	0.207965	0.103982	−	0.994579i	$-0.466841\pi$
$863$	6.10935	0.207965	0.103982	+	0.994579i	$0.466841\pi$
$864$	0	0
$865$	6.33177	0.215286
$866$	−5.26604	−0.178947
$867$	0	0
$868$	−1.27523	−0.0432842
$869$	4.99158	0.169328
$870$	0	0
$871$	61.2155	2.07421
$872$	−8.00000	−0.270914
$873$	0	0
$874$	27.2660	0.922288
$875$	9.61619	0.325086
$876$	0	0
$877$	0.366978	0.0123920	0.00619598	−	0.999981i	$-0.498028\pi$
$877$	0.366978	0.0123920	0.00619598	+	0.999981i	$0.498028\pi$
$878$	3.79968	0.128233
$879$	0	0
$880$	1.09174	0.0368027
$881$	−36.3906	−1.22603	−0.613015	−	0.790071i	$-0.710044\pi$
$881$	−36.3906	−1.22603	−0.613015	+	0.790071i	$0.710044\pi$
$882$	0	0
$883$	23.0841	0.776842	0.388421	−	0.921482i	$-0.373021\pi$
$883$	23.0841	0.776842	0.388421	+	0.921482i	$0.373021\pi$
$884$	−7.43270	−0.249989
$885$	0	0
$886$	−20.6246	−0.692897
$887$	−29.2492	−0.982092	−0.491046	−	0.871134i	$-0.663385\pi$
$887$	−29.2492	−0.982092	−0.491046	+	0.871134i	$0.663385\pi$
$888$	0	0
$889$	−4.99158	−0.167412
$890$	14.1835	0.475432
$891$	0	0
$892$	−5.64221	−0.188915
$893$	−37.8922	−1.26801
$894$	0	0
$895$	4.64824	0.155374
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	27.1751	0.906844
$899$	6.13144	0.204495
$900$	0	0
$901$	4.16665	0.138811
$902$	1.09174	0.0363511
$903$	0	0
$904$	3.17507	0.105601
$905$	5.55888	0.184784
$906$	0	0
$907$	−24.6246	−0.817647	−0.408823	−	0.912614i	$-0.634061\pi$
$907$	−24.6246	−0.817647	−0.408823	+	0.912614i	$0.634061\pi$
$908$	11.4587	0.380271
$909$	0	0
$910$	7.43270	0.246392
$911$	−17.8906	−0.592744	−0.296372	−	0.955073i	$-0.595777\pi$
$911$	−17.8906	−0.592744	−0.296372	+	0.955073i	$0.595777\pi$
$912$	0	0
$913$	−7.09174	−0.234702
$914$	15.2492	0.504399
$915$	0	0
$916$	−12.5244	−0.413819
$917$	16.9083	0.558360
$918$	0	0
$919$	−1.61619	−0.0533131	−0.0266566	−	0.999645i	$-0.508486\pi$
$919$	−1.61619	−0.0533131	−0.0266566	+	0.999645i	$0.508486\pi$
$920$	7.63302	0.251653
$921$	0	0
$922$	13.2324	0.435785
$923$	62.4648	2.05605
$924$	0	0
$925$	−15.9311	−0.523812
$926$	−3.63302	−0.119389
$927$	0	0
$928$	4.80809	0.157833
$929$	−38.9747	−1.27872	−0.639360	−	0.768908i	$-0.720801\pi$
$929$	−38.9747	−1.27872	−0.639360	+	0.768908i	$0.720801\pi$
$930$	0	0
$931$	−3.89984	−0.127812
$932$	−25.2324	−0.826514
$933$	0	0
$934$	−13.9832	−0.457543
$935$	−1.19191	−0.0389795
$936$	0	0
$937$	4.92509	0.160896	0.0804478	−	0.996759i	$-0.474365\pi$
$937$	4.92509	0.160896	0.0804478	+	0.996759i	$0.474365\pi$
$938$	−8.99158	−0.293586
$939$	0	0
$940$	−10.6078	−0.345987
$941$	−45.7660	−1.49193	−0.745965	−	0.665986i	$-0.768011\pi$
$941$	−45.7660	−1.49193	−0.745965	+	0.665986i	$0.768011\pi$
$942$	0	0
$943$	7.63302	0.248565
$944$	−11.7997	−0.384047
$945$	0	0
$946$	10.6246	0.345436
$947$	−25.5825	−0.831320	−0.415660	−	0.909520i	$-0.636449\pi$
$947$	−25.5825	−0.831320	−0.415660	+	0.909520i	$0.636449\pi$
$948$	0	0
$949$	−34.6651	−1.12528
$950$	−14.8510	−0.481828
$951$	0	0
$952$	1.09174	0.0353836
$953$	−32.1835	−1.04253	−0.521263	−	0.853396i	$-0.674539\pi$
$953$	−32.1835	−1.04253	−0.521263	+	0.853396i	$0.674539\pi$
$954$	0	0
$955$	−26.4648	−0.856379
$956$	13.9832	0.452248
$957$	0	0
$958$	−16.3670	−0.528793
$959$	−15.1751	−0.490029
$960$	0	0
$961$	−29.3738	−0.947541
$962$	−28.4816	−0.918283
$963$	0	0
$964$	−7.27523	−0.234319
$965$	6.14982	0.197970
$966$	0	0
$967$	2.35014	0.0755755	0.0377878	−	0.999286i	$-0.487969\pi$
$967$	2.35014	0.0755755	0.0377878	+	0.999286i	$0.487969\pi$
$968$	−1.00000	−0.0321412
$969$	0	0
$970$	−8.31493	−0.266976
$971$	1.98317	0.0636428	0.0318214	−	0.999494i	$-0.489869\pi$
$971$	1.98317	0.0636428	0.0318214	+	0.999494i	$0.489869\pi$
$972$	0	0
$973$	−8.26682	−0.265022
$974$	−24.1667	−0.774350
$975$	0	0
$976$	11.1751	0.357705
$977$	−1.98317	−0.0634471	−0.0317235	−	0.999497i	$-0.510100\pi$
$977$	−1.98317	−0.0634471	−0.0317235	+	0.999497i	$0.510100\pi$
$978$	0	0
$979$	−12.9916	−0.415213
$980$	−1.09174	−0.0348745
$981$	0	0
$982$	−7.63302	−0.243580
$983$	48.5817	1.54952	0.774759	−	0.632257i	$-0.217871\pi$
$983$	48.5817	1.54952	0.774759	+	0.632257i	$0.217871\pi$
$984$	0	0
$985$	−28.8486	−0.919192
$986$	−5.24921	−0.167169
$987$	0	0
$988$	−26.5505	−0.844683
$989$	74.2828	2.36206
$990$	0	0
$991$	14.3501	0.455847	0.227924	−	0.973679i	$-0.426806\pi$
$991$	14.3501	0.455847	0.227924	+	0.973679i	$0.426806\pi$
$992$	1.27523	0.0404887
$993$	0	0
$994$	−9.17507	−0.291016
$995$	12.1094	0.383892
$996$	0	0
$997$	−36.6246	−1.15991	−0.579956	−	0.814647i	$-0.696930\pi$
$997$	−36.6246	−1.15991	−0.579956	+	0.814647i	$0.696930\pi$
$998$	−12.2576	−0.388008
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1386.2.a.q.1.2	✓	3
3.2	odd	2		1386.2.a.r.1.2	yes	3
7.6	odd	2		9702.2.a.dx.1.2		3
21.20	even	2		9702.2.a.dy.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1386.2.a.q.1.2	✓	3	1.1	even	1	trivial
1386.2.a.r.1.2	yes	3	3.2	odd	2
9702.2.a.dx.1.2		3	7.6	odd	2
9702.2.a.dy.1.2		3	21.20	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:	\( N \)	\(=\)	\( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1386.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.09174 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.09174 q^{5} +1.00000 q^{7} -1.00000 q^{8} +1.09174 q^{10} -1.00000 q^{11} +6.80809 q^{13} -1.00000 q^{14} +1.00000 q^{16} -1.09174 q^{17} -3.89984 q^{19} -1.09174 q^{20} +1.00000 q^{22} +6.99158 q^{23} -3.80809 q^{25} -6.80809 q^{26} +1.00000 q^{28} -4.80809 q^{29} -1.27523 q^{31} -1.00000 q^{32} +1.09174 q^{34} -1.09174 q^{35} +4.18349 q^{37} +3.89984 q^{38} +1.09174 q^{40} +1.09174 q^{41} +10.6246 q^{43} -1.00000 q^{44} -6.99158 q^{46} +9.71635 q^{47} +1.00000 q^{49} +3.80809 q^{50} +6.80809 q^{52} -3.81651 q^{53} +1.09174 q^{55} -1.00000 q^{56} +4.80809 q^{58} -11.7997 q^{59} +11.1751 q^{61} +1.27523 q^{62} +1.00000 q^{64} -7.43270 q^{65} +8.99158 q^{67} -1.09174 q^{68} +1.09174 q^{70} +9.17507 q^{71} -5.09174 q^{73} -4.18349 q^{74} -3.89984 q^{76} -1.00000 q^{77} -4.99158 q^{79} -1.09174 q^{80} -1.09174 q^{82} +7.09174 q^{83} +1.19191 q^{85} -10.6246 q^{86} +1.00000 q^{88} +12.9916 q^{89} +6.80809 q^{91} +6.99158 q^{92} -9.71635 q^{94} +4.25763 q^{95} -7.61619 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{4} - 2 q^{5} + 3 q^{7} - 3 q^{8}+O(q^{10}) \) 3 * q - 3 * q^2 + 3 * q^4 - 2 * q^5 + 3 * q^7 - 3 * q^8	\( 3 q - 3 q^{2} + 3 q^{4} - 2 q^{5} + 3 q^{7} - 3 q^{8} + 2 q^{10} - 3 q^{11} - 3 q^{14} + 3 q^{16} - 2 q^{17} + 10 q^{19} - 2 q^{20} + 3 q^{22} - 2 q^{23} + 9 q^{25} + 3 q^{28} + 6 q^{29} - 3 q^{32} + 2 q^{34} - 2 q^{35} + 10 q^{37} - 10 q^{38} + 2 q^{40} + 2 q^{41} + 14 q^{43} - 3 q^{44} + 2 q^{46} + 10 q^{47} + 3 q^{49} - 9 q^{50} - 14 q^{53} + 2 q^{55} - 3 q^{56} - 6 q^{58} + 8 q^{59} + 8 q^{61} + 3 q^{64} + 16 q^{65} + 4 q^{67} - 2 q^{68} + 2 q^{70} + 2 q^{71} - 14 q^{73} - 10 q^{74} + 10 q^{76} - 3 q^{77} + 8 q^{79} - 2 q^{80} - 2 q^{82} + 20 q^{83} + 24 q^{85} - 14 q^{86} + 3 q^{88} + 16 q^{89} - 2 q^{92} - 10 q^{94} + 18 q^{97} - 3 q^{98}+O(q^{100}) \) 3 * q - 3 * q^2 + 3 * q^4 - 2 * q^5 + 3 * q^7 - 3 * q^8 + 2 * q^10 - 3 * q^11 - 3 * q^14 + 3 * q^16 - 2 * q^17 + 10 * q^19 - 2 * q^20 + 3 * q^22 - 2 * q^23 + 9 * q^25 + 3 * q^28 + 6 * q^29 - 3 * q^32 + 2 * q^34 - 2 * q^35 + 10 * q^37 - 10 * q^38 + 2 * q^40 + 2 * q^41 + 14 * q^43 - 3 * q^44 + 2 * q^46 + 10 * q^47 + 3 * q^49 - 9 * q^50 - 14 * q^53 + 2 * q^55 - 3 * q^56 - 6 * q^58 + 8 * q^59 + 8 * q^61 + 3 * q^64 + 16 * q^65 + 4 * q^67 - 2 * q^68 + 2 * q^70 + 2 * q^71 - 14 * q^73 - 10 * q^74 + 10 * q^76 - 3 * q^77 + 8 * q^79 - 2 * q^80 - 2 * q^82 + 20 * q^83 + 24 * q^85 - 14 * q^86 + 3 * q^88 + 16 * q^89 - 2 * q^92 - 10 * q^94 + 18 * q^97 - 3 * q^98

Introduction

L-functions

Modular forms

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Database

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