LMFDB - Embedded newform 6039.2.a.l.1.8

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			1.8
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)

$f(q)$	$=$	$q-0.942064 q^{2} -1.11252 q^{4} -4.16220 q^{5} -0.914635 q^{7} +2.93219 q^{8} +O(q^{10})$	\(q-0.942064 q^{2} -1.11252 q^{4} -4.16220 q^{5} -0.914635 q^{7} +2.93219 q^{8} +3.92106 q^{10} +1.00000 q^{11} +6.85514 q^{13} +0.861645 q^{14} -0.537277 q^{16} +7.07723 q^{17} -0.758442 q^{19} +4.63051 q^{20} -0.942064 q^{22} -0.119595 q^{23} +12.3239 q^{25} -6.45798 q^{26} +1.01755 q^{28} +6.47853 q^{29} +8.12191 q^{31} -5.35823 q^{32} -6.66720 q^{34} +3.80690 q^{35} -3.59142 q^{37} +0.714500 q^{38} -12.2044 q^{40} +3.48107 q^{41} -5.70063 q^{43} -1.11252 q^{44} +0.112666 q^{46} +3.93114 q^{47} -6.16344 q^{49} -11.6099 q^{50} -7.62645 q^{52} -7.61747 q^{53} -4.16220 q^{55} -2.68188 q^{56} -6.10319 q^{58} +8.02724 q^{59} +1.00000 q^{61} -7.65135 q^{62} +6.12235 q^{64} -28.5325 q^{65} -9.13346 q^{67} -7.87353 q^{68} -3.58634 q^{70} -12.7273 q^{71} +10.0229 q^{73} +3.38335 q^{74} +0.843778 q^{76} -0.914635 q^{77} +6.32794 q^{79} +2.23625 q^{80} -3.27939 q^{82} +16.9673 q^{83} -29.4569 q^{85} +5.37035 q^{86} +2.93219 q^{88} +0.929850 q^{89} -6.26995 q^{91} +0.133051 q^{92} -3.70338 q^{94} +3.15679 q^{95} +15.9555 q^{97} +5.80636 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.942064	−0.666140	−0.333070	−	0.942902i	$-0.608084\pi$
$2$	−0.942064	−0.666140	−0.333070	+	0.942902i	$0.608084\pi$
$3$	0	0
$3$	0	0
$4$	−1.11252	−0.556258
$5$	−4.16220	−1.86139	−0.930696	−	0.365792i	$-0.880798\pi$
$5$	−4.16220	−1.86139	−0.930696	+	0.365792i	$0.880798\pi$
$6$	0	0
$7$	−0.914635	−0.345700	−0.172850	−	0.984948i	$-0.555298\pi$
$7$	−0.914635	−0.345700	−0.172850	+	0.984948i	$0.555298\pi$
$8$	2.93219	1.03669
$9$	0	0
$10$	3.92106	1.23995
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	6.85514	1.90127	0.950637	−	0.310306i	$-0.100431\pi$
$13$	6.85514	1.90127	0.950637	+	0.310306i	$0.100431\pi$
$14$	0.861645	0.230284
$15$	0	0
$16$	−0.537277	−0.134319
$17$	7.07723	1.71648	0.858241	−	0.513248i	$-0.171558\pi$
$17$	7.07723	1.71648	0.858241	+	0.513248i	$0.171558\pi$
$18$	0	0
$19$	−0.758442	−0.173998	−0.0869992	−	0.996208i	$-0.527728\pi$
$19$	−0.758442	−0.173998	−0.0869992	+	0.996208i	$0.527728\pi$
$20$	4.63051	1.03541
$21$	0	0
$22$	−0.942064	−0.200849
$23$	−0.119595	−0.0249372	−0.0124686	−	0.999922i	$-0.503969\pi$
$23$	−0.119595	−0.0249372	−0.0124686	+	0.999922i	$0.503969\pi$
$24$	0	0
$25$	12.3239	2.46478
$26$	−6.45798	−1.26651
$27$	0	0
$28$	1.01755	0.192298
$29$	6.47853	1.20303	0.601517	−	0.798860i	$-0.294563\pi$
$29$	6.47853	1.20303	0.601517	+	0.798860i	$0.294563\pi$
$30$	0	0
$31$	8.12191	1.45874	0.729369	−	0.684121i	$-0.239814\pi$
$31$	8.12191	1.45874	0.729369	+	0.684121i	$0.239814\pi$
$32$	−5.35823	−0.947210
$33$	0	0
$34$	−6.66720	−1.14342
$35$	3.80690	0.643483
$36$	0	0
$37$	−3.59142	−0.590426	−0.295213	−	0.955431i	$-0.595391\pi$
$37$	−3.59142	−0.590426	−0.295213	+	0.955431i	$0.595391\pi$
$38$	0.714500	0.115907
$39$	0	0
$40$	−12.2044	−1.92968
$41$	3.48107	0.543651	0.271826	−	0.962346i	$-0.412373\pi$
$41$	3.48107	0.543651	0.271826	+	0.962346i	$0.412373\pi$
$42$	0	0
$43$	−5.70063	−0.869338	−0.434669	−	0.900590i	$-0.643135\pi$
$43$	−5.70063	−0.869338	−0.434669	+	0.900590i	$0.643135\pi$
$44$	−1.11252	−0.167718
$45$	0	0
$46$	0.112666	0.0166117
$47$	3.93114	0.573415	0.286708	−	0.958018i	$-0.407439\pi$
$47$	3.93114	0.573415	0.286708	+	0.958018i	$0.407439\pi$
$48$	0	0
$49$	−6.16344	−0.880492
$50$	−11.6099	−1.64189
$51$	0	0
$52$	−7.62645	−1.05760
$53$	−7.61747	−1.04634	−0.523170	−	0.852228i	$-0.675251\pi$
$53$	−7.61747	−1.04634	−0.523170	+	0.852228i	$0.675251\pi$
$54$	0	0
$55$	−4.16220	−0.561231
$56$	−2.68188	−0.358382
$57$	0	0
$58$	−6.10319	−0.801388
$59$	8.02724	1.04506	0.522529	−	0.852621i	$-0.324989\pi$
$59$	8.02724	1.04506	0.522529	+	0.852621i	$0.324989\pi$
$60$	0	0
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	−7.65135	−0.971723
$63$	0	0
$64$	6.12235	0.765293
$65$	−28.5325	−3.53902
$66$	0	0
$67$	−9.13346	−1.11583	−0.557915	−	0.829898i	$-0.688399\pi$
$67$	−9.13346	−1.11583	−0.557915	+	0.829898i	$0.688399\pi$
$68$	−7.87353	−0.954806
$69$	0	0
$70$	−3.58634	−0.428650
$71$	−12.7273	−1.51046	−0.755229	−	0.655461i	$-0.772474\pi$
$71$	−12.7273	−1.51046	−0.755229	+	0.655461i	$0.772474\pi$
$72$	0	0
$73$	10.0229	1.17309	0.586545	−	0.809917i	$-0.300488\pi$
$73$	10.0229	1.17309	0.586545	+	0.809917i	$0.300488\pi$
$74$	3.38335	0.393306
$75$	0	0
$76$	0.843778	0.0967880
$77$	−0.914635	−0.104232
$78$	0	0
$79$	6.32794	0.711949	0.355975	−	0.934496i	$-0.384149\pi$
$79$	6.32794	0.711949	0.355975	+	0.934496i	$0.384149\pi$
$80$	2.23625	0.250021
$81$	0	0
$82$	−3.27939	−0.362148
$83$	16.9673	1.86240	0.931202	−	0.364504i	$-0.118762\pi$
$83$	16.9673	1.86240	0.931202	+	0.364504i	$0.118762\pi$
$84$	0	0
$85$	−29.4569	−3.19505
$86$	5.37035	0.579100
$87$	0	0
$88$	2.93219	0.312572
$89$	0.929850	0.0985639	0.0492820	−	0.998785i	$-0.484307\pi$
$89$	0.929850	0.0985639	0.0492820	+	0.998785i	$0.484307\pi$
$90$	0	0
$91$	−6.26995	−0.657270
$92$	0.133051	0.0138715
$93$	0	0
$94$	−3.70338	−0.381975
$95$	3.15679	0.323879
$96$	0	0
$97$	15.9555	1.62003	0.810016	−	0.586408i	$-0.199458\pi$
$97$	15.9555	1.62003	0.810016	+	0.586408i	$0.199458\pi$
$98$	5.80636	0.586530
$99$	0	0
$100$	−13.7106	−1.37106
$101$	−3.89969	−0.388034	−0.194017	−	0.980998i	$-0.562152\pi$
$101$	−3.89969	−0.388034	−0.194017	+	0.980998i	$0.562152\pi$
$102$	0	0
$103$	−8.88441	−0.875407	−0.437703	−	0.899119i	$-0.644208\pi$
$103$	−8.88441	−0.875407	−0.437703	+	0.899119i	$0.644208\pi$
$104$	20.1006	1.97102
$105$	0	0
$106$	7.17615	0.697009
$107$	−1.43091	−0.138331	−0.0691657	−	0.997605i	$-0.522034\pi$
$107$	−1.43091	−0.138331	−0.0691657	+	0.997605i	$0.522034\pi$
$108$	0	0
$109$	5.58225	0.534682	0.267341	−	0.963602i	$-0.413855\pi$
$109$	5.58225	0.534682	0.267341	+	0.963602i	$0.413855\pi$
$110$	3.92106	0.373858
$111$	0	0
$112$	0.491412	0.0464341
$113$	0.911691	0.0857646	0.0428823	−	0.999080i	$-0.486346\pi$
$113$	0.911691	0.0857646	0.0428823	+	0.999080i	$0.486346\pi$
$114$	0	0
$115$	0.497777	0.0464179
$116$	−7.20747	−0.669197
$117$	0	0
$118$	−7.56218	−0.696155
$119$	−6.47309	−0.593387
$120$	0	0
$121$	1.00000	0.0909091
$122$	−0.942064	−0.0852904
$123$	0	0
$124$	−9.03575	−0.811434
$125$	−30.4836	−2.72654
$126$	0	0
$127$	2.04484	0.181450	0.0907250	−	0.995876i	$-0.471082\pi$
$127$	2.04484	0.181450	0.0907250	+	0.995876i	$0.471082\pi$
$128$	4.94882	0.437418
$129$	0	0
$130$	26.8794	2.35748
$131$	−2.29123	−0.200185	−0.100093	−	0.994978i	$-0.531914\pi$
$131$	−2.29123	−0.200185	−0.100093	+	0.994978i	$0.531914\pi$
$132$	0	0
$133$	0.693698	0.0601512
$134$	8.60431	0.743299
$135$	0	0
$136$	20.7518	1.77945
$137$	14.0851	1.20337	0.601686	−	0.798733i	$-0.294496\pi$
$137$	14.0851	1.20337	0.601686	+	0.798733i	$0.294496\pi$
$138$	0	0
$139$	−10.8222	−0.917926	−0.458963	−	0.888455i	$-0.651779\pi$
$139$	−10.8222	−0.917926	−0.458963	+	0.888455i	$0.651779\pi$
$140$	−4.23523	−0.357942
$141$	0	0
$142$	11.9900	1.00618
$143$	6.85514	0.573256
$144$	0	0
$145$	−26.9650	−2.23932
$146$	−9.44219	−0.781441
$147$	0	0
$148$	3.99551	0.328429
$149$	7.68545	0.629617	0.314808	−	0.949155i	$-0.398060\pi$
$149$	7.68545	0.629617	0.314808	+	0.949155i	$0.398060\pi$
$150$	0	0
$151$	−14.2789	−1.16200	−0.580998	−	0.813905i	$-0.697338\pi$
$151$	−14.2789	−1.16200	−0.580998	+	0.813905i	$0.697338\pi$
$152$	−2.22389	−0.180382
$153$	0	0
$154$	0.861645	0.0694333
$155$	−33.8050	−2.71528
$156$	0	0
$157$	13.5063	1.07792	0.538961	−	0.842331i	$-0.318817\pi$
$157$	13.5063	1.07792	0.538961	+	0.842331i	$0.318817\pi$
$158$	−5.96133	−0.474258
$159$	0	0
$160$	22.3020	1.76313
$161$	0.109385	0.00862078
$162$	0	0
$163$	−10.8779	−0.852024	−0.426012	−	0.904718i	$-0.640082\pi$
$163$	−10.8779	−0.852024	−0.426012	+	0.904718i	$0.640082\pi$
$164$	−3.87274	−0.302410
$165$	0	0
$166$	−15.9843	−1.24062
$167$	2.20692	0.170777	0.0853884	−	0.996348i	$-0.472787\pi$
$167$	2.20692	0.170777	0.0853884	+	0.996348i	$0.472787\pi$
$168$	0	0
$169$	33.9929	2.61484
$170$	27.7502	2.12835
$171$	0	0
$172$	6.34204	0.483576
$173$	−4.28336	−0.325658	−0.162829	−	0.986654i	$-0.552062\pi$
$173$	−4.28336	−0.325658	−0.162829	+	0.986654i	$0.552062\pi$
$174$	0	0
$175$	−11.2719	−0.852075
$176$	−0.537277	−0.0404988
$177$	0	0
$178$	−0.875978	−0.0656573
$179$	7.64364	0.571312	0.285656	−	0.958332i	$-0.407788\pi$
$179$	7.64364	0.571312	0.285656	+	0.958332i	$0.407788\pi$
$180$	0	0
$181$	11.9672	0.889515	0.444757	−	0.895651i	$-0.353290\pi$
$181$	11.9672	0.889515	0.444757	+	0.895651i	$0.353290\pi$
$182$	5.90670	0.437833
$183$	0	0
$184$	−0.350674	−0.0258520
$185$	14.9482	1.09901
$186$	0	0
$187$	7.07723	0.517539
$188$	−4.37345	−0.318967
$189$	0	0
$190$	−2.97389	−0.215749
$191$	0.835454	0.0604513	0.0302257	−	0.999543i	$-0.490377\pi$
$191$	0.835454	0.0604513	0.0302257	+	0.999543i	$0.490377\pi$
$192$	0	0
$193$	−10.7787	−0.775869	−0.387934	−	0.921687i	$-0.626811\pi$
$193$	−10.7787	−0.775869	−0.387934	+	0.921687i	$0.626811\pi$
$194$	−15.0311	−1.07917
$195$	0	0
$196$	6.85693	0.489780
$197$	−9.31428	−0.663615	−0.331808	−	0.943347i	$-0.607658\pi$
$197$	−9.31428	−0.663615	−0.331808	+	0.943347i	$0.607658\pi$
$198$	0	0
$199$	18.7278	1.32758	0.663788	−	0.747921i	$-0.268948\pi$
$199$	18.7278	1.32758	0.663788	+	0.747921i	$0.268948\pi$
$200$	36.1361	2.55520
$201$	0	0
$202$	3.67376	0.258485
$203$	−5.92550	−0.415888
$204$	0	0
$205$	−14.4889	−1.01195
$206$	8.36968	0.583143
$207$	0	0
$208$	−3.68311	−0.255378
$209$	−0.758442	−0.0524625
$210$	0	0
$211$	−21.8515	−1.50432	−0.752159	−	0.658982i	$-0.770987\pi$
$211$	−21.8515	−1.50432	−0.752159	+	0.658982i	$0.770987\pi$
$212$	8.47456	0.582035
$213$	0	0
$214$	1.34801	0.0921481
$215$	23.7272	1.61818
$216$	0	0
$217$	−7.42858	−0.504285
$218$	−5.25883	−0.356173
$219$	0	0
$220$	4.63051	0.312189
$221$	48.5154	3.26350
$222$	0	0
$223$	5.38796	0.360805	0.180402	−	0.983593i	$-0.442260\pi$
$223$	5.38796	0.360805	0.180402	+	0.983593i	$0.442260\pi$
$224$	4.90082	0.327450
$225$	0	0
$226$	−0.858871	−0.0571312
$227$	−6.06937	−0.402838	−0.201419	−	0.979505i	$-0.564555\pi$
$227$	−6.06937	−0.402838	−0.201419	+	0.979505i	$0.564555\pi$
$228$	0	0
$229$	3.78348	0.250019	0.125010	−	0.992156i	$-0.460104\pi$
$229$	3.78348	0.250019	0.125010	+	0.992156i	$0.460104\pi$
$230$	−0.468937	−0.0309208
$231$	0	0
$232$	18.9963	1.24717
$233$	−17.0813	−1.11903	−0.559515	−	0.828820i	$-0.689013\pi$
$233$	−17.0813	−1.11903	−0.559515	+	0.828820i	$0.689013\pi$
$234$	0	0
$235$	−16.3622	−1.06735
$236$	−8.93044	−0.581322
$237$	0	0
$238$	6.09806	0.395279
$239$	−5.52756	−0.357548	−0.178774	−	0.983890i	$-0.557213\pi$
$239$	−5.52756	−0.357548	−0.178774	+	0.983890i	$0.557213\pi$
$240$	0	0
$241$	18.4791	1.19034	0.595172	−	0.803598i	$-0.297084\pi$
$241$	18.4791	1.19034	0.595172	+	0.803598i	$0.297084\pi$
$242$	−0.942064	−0.0605582
$243$	0	0
$244$	−1.11252	−0.0712215
$245$	25.6535	1.63894
$246$	0	0
$247$	−5.19922	−0.330819
$248$	23.8150	1.51225
$249$	0	0
$250$	28.7175	1.81626
$251$	−14.0522	−0.886966	−0.443483	−	0.896283i	$-0.646257\pi$
$251$	−14.0522	−0.886966	−0.443483	+	0.896283i	$0.646257\pi$
$252$	0	0
$253$	−0.119595	−0.00751885
$254$	−1.92637	−0.120871
$255$	0	0
$256$	−16.9068	−1.05667
$257$	18.0106	1.12347	0.561734	−	0.827318i	$-0.310134\pi$
$257$	18.0106	1.12347	0.561734	+	0.827318i	$0.310134\pi$
$258$	0	0
$259$	3.28484	0.204110
$260$	31.7428	1.96861
$261$	0	0
$262$	2.15848	0.133351
$263$	−8.90385	−0.549035	−0.274517	−	0.961582i	$-0.588518\pi$
$263$	−8.90385	−0.549035	−0.274517	+	0.961582i	$0.588518\pi$
$264$	0	0
$265$	31.7055	1.94765
$266$	−0.653507	−0.0400691
$267$	0	0
$268$	10.1611	0.620689
$269$	−22.2434	−1.35620	−0.678102	−	0.734968i	$-0.737197\pi$
$269$	−22.2434	−1.35620	−0.678102	+	0.734968i	$0.737197\pi$
$270$	0	0
$271$	−15.7517	−0.956850	−0.478425	−	0.878128i	$-0.658792\pi$
$271$	−15.7517	−0.956850	−0.478425	+	0.878128i	$0.658792\pi$
$272$	−3.80243	−0.230556
$273$	0	0
$274$	−13.2691	−0.801614
$275$	12.3239	0.743160
$276$	0	0
$277$	−8.81685	−0.529753	−0.264876	−	0.964282i	$-0.585331\pi$
$277$	−8.81685	−0.529753	−0.264876	+	0.964282i	$0.585331\pi$
$278$	10.1952	0.611467
$279$	0	0
$280$	11.1625	0.667089
$281$	−5.42753	−0.323779	−0.161890	−	0.986809i	$-0.551759\pi$
$281$	−5.42753	−0.323779	−0.161890	+	0.986809i	$0.551759\pi$
$282$	0	0
$283$	25.2993	1.50389	0.751943	−	0.659228i	$-0.229117\pi$
$283$	25.2993	1.50389	0.751943	+	0.659228i	$0.229117\pi$
$284$	14.1594	0.840204
$285$	0	0
$286$	−6.45798	−0.381868
$287$	−3.18391	−0.187940
$288$	0	0
$289$	33.0872	1.94631
$290$	25.4027	1.49170
$291$	0	0
$292$	−11.1506	−0.652540
$293$	−30.7679	−1.79748	−0.898739	−	0.438485i	$-0.855515\pi$
$293$	−30.7679	−1.79748	−0.898739	+	0.438485i	$0.855515\pi$
$294$	0	0
$295$	−33.4110	−1.94526
$296$	−10.5307	−0.612086
$297$	0	0
$298$	−7.24018	−0.419413
$299$	−0.819837	−0.0474124
$300$	0	0
$301$	5.21400	0.300530
$302$	13.4516	0.774052
$303$	0	0
$304$	0.407493	0.0233713
$305$	−4.16220	−0.238327
$306$	0	0
$307$	5.68781	0.324620	0.162310	−	0.986740i	$-0.448105\pi$
$307$	5.68781	0.324620	0.162310	+	0.986740i	$0.448105\pi$
$308$	1.01755	0.0579801
$309$	0	0
$310$	31.8465	1.80876
$311$	−4.03491	−0.228799	−0.114399	−	0.993435i	$-0.536494\pi$
$311$	−4.03491	−0.228799	−0.114399	+	0.993435i	$0.536494\pi$
$312$	0	0
$313$	21.9128	1.23859	0.619294	−	0.785159i	$-0.287419\pi$
$313$	21.9128	1.23859	0.619294	+	0.785159i	$0.287419\pi$
$314$	−12.7238	−0.718046
$315$	0	0
$316$	−7.03994	−0.396027
$317$	25.9100	1.45525	0.727625	−	0.685975i	$-0.240624\pi$
$317$	25.9100	1.45525	0.727625	+	0.685975i	$0.240624\pi$
$318$	0	0
$319$	6.47853	0.362728
$320$	−25.4824	−1.42451
$321$	0	0
$322$	−0.103048	−0.00574264
$323$	−5.36767	−0.298665
$324$	0	0
$325$	84.4822	4.68623
$326$	10.2477	0.567567
$327$	0	0
$328$	10.2071	0.563595
$329$	−3.59556	−0.198230
$330$	0	0
$331$	−4.18807	−0.230197	−0.115099	−	0.993354i	$-0.536718\pi$
$331$	−4.18807	−0.230197	−0.115099	+	0.993354i	$0.536718\pi$
$332$	−18.8764	−1.03598
$333$	0	0
$334$	−2.07906	−0.113761
$335$	38.0153	2.07700
$336$	0	0
$337$	31.1255	1.69551	0.847757	−	0.530385i	$-0.177953\pi$
$337$	31.1255	1.69551	0.847757	+	0.530385i	$0.177953\pi$
$338$	−32.0235	−1.74185
$339$	0	0
$340$	32.7712	1.77727
$341$	8.12191	0.439826
$342$	0	0
$343$	12.0398	0.650085
$344$	−16.7153	−0.901229
$345$	0	0
$346$	4.03520	0.216934
$347$	−15.5977	−0.837326	−0.418663	−	0.908142i	$-0.637501\pi$
$347$	−15.5977	−0.837326	−0.418663	+	0.908142i	$0.637501\pi$
$348$	0	0
$349$	−6.70862	−0.359104	−0.179552	−	0.983748i	$-0.557465\pi$
$349$	−6.70862	−0.359104	−0.179552	+	0.983748i	$0.557465\pi$
$350$	10.6188	0.567601
$351$	0	0
$352$	−5.35823	−0.285595
$353$	1.11688	0.0594453	0.0297227	−	0.999558i	$-0.490538\pi$
$353$	1.11688	0.0594453	0.0297227	+	0.999558i	$0.490538\pi$
$354$	0	0
$355$	52.9738	2.81156
$356$	−1.03447	−0.0548270
$357$	0	0
$358$	−7.20079	−0.380574
$359$	−27.4685	−1.44973	−0.724866	−	0.688890i	$-0.758098\pi$
$359$	−27.4685	−1.44973	−0.724866	+	0.688890i	$0.758098\pi$
$360$	0	0
$361$	−18.4248	−0.969725
$362$	−11.2739	−0.592541
$363$	0	0
$364$	6.97542	0.365611
$365$	−41.7172	−2.18358
$366$	0	0
$367$	37.7724	1.97170	0.985852	−	0.167620i	$-0.0536082\pi$
$367$	37.7724	1.97170	0.985852	+	0.167620i	$0.0536082\pi$
$368$	0.0642554	0.00334954
$369$	0	0
$370$	−14.0822	−0.732097
$371$	6.96721	0.361720
$372$	0	0
$373$	−11.1892	−0.579353	−0.289676	−	0.957125i	$-0.593548\pi$
$373$	−11.1892	−0.579353	−0.289676	+	0.957125i	$0.593548\pi$
$374$	−6.66720	−0.344753
$375$	0	0
$376$	11.5268	0.594451
$377$	44.4112	2.28730
$378$	0	0
$379$	22.3776	1.14946	0.574730	−	0.818343i	$-0.305107\pi$
$379$	22.3776	1.14946	0.574730	+	0.818343i	$0.305107\pi$
$380$	−3.51198	−0.180161
$381$	0	0
$382$	−0.787051	−0.0402690
$383$	−18.5532	−0.948022	−0.474011	−	0.880519i	$-0.657194\pi$
$383$	−18.5532	−0.948022	−0.474011	+	0.880519i	$0.657194\pi$
$384$	0	0
$385$	3.80690	0.194017
$386$	10.1542	0.516837
$387$	0	0
$388$	−17.7507	−0.901156
$389$	18.3647	0.931126	0.465563	−	0.885015i	$-0.345852\pi$
$389$	18.3647	0.931126	0.465563	+	0.885015i	$0.345852\pi$
$390$	0	0
$391$	−0.846399	−0.0428042
$392$	−18.0724	−0.912793
$393$	0	0
$394$	8.77465	0.442060
$395$	−26.3382	−1.32522
$396$	0	0
$397$	−1.08826	−0.0546182	−0.0273091	−	0.999627i	$-0.508694\pi$
$397$	−1.08826	−0.0546182	−0.0273091	+	0.999627i	$0.508694\pi$
$398$	−17.6427	−0.884351
$399$	0	0
$400$	−6.62136	−0.331068
$401$	6.73481	0.336320	0.168160	−	0.985760i	$-0.446217\pi$
$401$	6.73481	0.336320	0.168160	+	0.985760i	$0.446217\pi$
$402$	0	0
$403$	55.6768	2.77346
$404$	4.33847	0.215847
$405$	0	0
$406$	5.58220	0.277040
$407$	−3.59142	−0.178020
$408$	0	0
$409$	−30.6490	−1.51550	−0.757748	−	0.652547i	$-0.773700\pi$
$409$	−30.6490	−1.51550	−0.757748	+	0.652547i	$0.773700\pi$
$410$	13.6495	0.674099
$411$	0	0
$412$	9.88405	0.486952
$413$	−7.34200	−0.361276
$414$	0	0
$415$	−70.6213	−3.46666
$416$	−36.7314	−1.80090
$417$	0	0
$418$	0.714500	0.0349474
$419$	17.2551	0.842966	0.421483	−	0.906836i	$-0.361510\pi$
$419$	17.2551	0.842966	0.421483	+	0.906836i	$0.361510\pi$
$420$	0	0
$421$	−5.65725	−0.275717	−0.137859	−	0.990452i	$-0.544022\pi$
$421$	−5.65725	−0.275717	−0.137859	+	0.990452i	$0.544022\pi$
$422$	20.5855	1.00209
$423$	0	0
$424$	−22.3359	−1.08473
$425$	87.2192	4.23075
$426$	0	0
$427$	−0.914635	−0.0442623
$428$	1.59191	0.0769480
$429$	0	0
$430$	−22.3525	−1.07793
$431$	11.6170	0.559571	0.279786	−	0.960062i	$-0.409737\pi$
$431$	11.6170	0.559571	0.279786	+	0.960062i	$0.409737\pi$
$432$	0	0
$433$	2.70697	0.130089	0.0650444	−	0.997882i	$-0.479281\pi$
$433$	2.70697	0.130089	0.0650444	+	0.997882i	$0.479281\pi$
$434$	6.99820	0.335924
$435$	0	0
$436$	−6.21034	−0.297421
$437$	0.0907055	0.00433903
$438$	0	0
$439$	11.8107	0.563694	0.281847	−	0.959459i	$-0.409053\pi$
$439$	11.8107	0.563694	0.281847	+	0.959459i	$0.409053\pi$
$440$	−12.2044	−0.581820
$441$	0	0
$442$	−45.7046	−2.17395
$443$	−1.36325	−0.0647702	−0.0323851	−	0.999475i	$-0.510310\pi$
$443$	−1.36325	−0.0647702	−0.0323851	+	0.999475i	$0.510310\pi$
$444$	0	0
$445$	−3.87022	−0.183466
$446$	−5.07581	−0.240346
$447$	0	0
$448$	−5.59971	−0.264562
$449$	−22.8380	−1.07779	−0.538895	−	0.842373i	$-0.681158\pi$
$449$	−22.8380	−1.07779	−0.538895	+	0.842373i	$0.681158\pi$
$450$	0	0
$451$	3.48107	0.163917
$452$	−1.01427	−0.0477073
$453$	0	0
$454$	5.71774	0.268347
$455$	26.0968	1.22344
$456$	0	0
$457$	3.92443	0.183577	0.0917886	−	0.995779i	$-0.470742\pi$
$457$	3.92443	0.183577	0.0917886	+	0.995779i	$0.470742\pi$
$458$	−3.56428	−0.166548
$459$	0	0
$460$	−0.553784	−0.0258203
$461$	18.0655	0.841396	0.420698	−	0.907201i	$-0.361785\pi$
$461$	18.0655	0.841396	0.420698	+	0.907201i	$0.361785\pi$
$462$	0	0
$463$	−26.8804	−1.24924	−0.624618	−	0.780930i	$-0.714745\pi$
$463$	−26.8804	−1.24924	−0.624618	+	0.780930i	$0.714745\pi$
$464$	−3.48077	−0.161591
$465$	0	0
$466$	16.0916	0.745431
$467$	−28.9914	−1.34156	−0.670781	−	0.741655i	$-0.734041\pi$
$467$	−28.9914	−1.34156	−0.670781	+	0.741655i	$0.734041\pi$
$468$	0	0
$469$	8.35379	0.385742
$470$	15.4142	0.711005
$471$	0	0
$472$	23.5374	1.08340
$473$	−5.70063	−0.262115
$474$	0	0
$475$	−9.34697	−0.428869
$476$	7.20141	0.330076
$477$	0	0
$478$	5.20731	0.238177
$479$	12.0538	0.550752	0.275376	−	0.961337i	$-0.411198\pi$
$479$	12.0538	0.550752	0.275376	+	0.961337i	$0.411198\pi$
$480$	0	0
$481$	−24.6197	−1.12256
$482$	−17.4085	−0.792935
$483$	0	0
$484$	−1.11252	−0.0505689
$485$	−66.4099	−3.01552
$486$	0	0
$487$	2.40506	0.108984	0.0544918	−	0.998514i	$-0.482646\pi$
$487$	2.40506	0.108984	0.0544918	+	0.998514i	$0.482646\pi$
$488$	2.93219	0.132734
$489$	0	0
$490$	−24.1672	−1.09176
$491$	3.18966	0.143947	0.0719737	−	0.997407i	$-0.477070\pi$
$491$	3.18966	0.143947	0.0719737	+	0.997407i	$0.477070\pi$
$492$	0	0
$493$	45.8501	2.06498
$494$	4.89800	0.220371
$495$	0	0
$496$	−4.36371	−0.195936
$497$	11.6409	0.522165
$498$	0	0
$499$	−31.1467	−1.39432	−0.697160	−	0.716916i	$-0.745553\pi$
$499$	−31.1467	−1.39432	−0.697160	+	0.716916i	$0.745553\pi$
$500$	33.9135	1.51666
$501$	0	0
$502$	13.2381	0.590844
$503$	0.912851	0.0407020	0.0203510	−	0.999793i	$-0.493522\pi$
$503$	0.912851	0.0407020	0.0203510	+	0.999793i	$0.493522\pi$
$504$	0	0
$505$	16.2313	0.722284
$506$	0.112666	0.00500860
$507$	0	0
$508$	−2.27491	−0.100933
$509$	−15.5630	−0.689820	−0.344910	−	0.938636i	$-0.612090\pi$
$509$	−15.5630	−0.689820	−0.344910	+	0.938636i	$0.612090\pi$
$510$	0	0
$511$	−9.16728	−0.405537
$512$	6.02965	0.266475
$513$	0	0
$514$	−16.9671	−0.748387
$515$	36.9787	1.62948
$516$	0	0
$517$	3.93114	0.172891
$518$	−3.09453	−0.135966
$519$	0	0
$520$	−83.6626	−3.66885
$521$	−20.7782	−0.910309	−0.455155	−	0.890412i	$-0.650416\pi$
$521$	−20.7782	−0.910309	−0.455155	+	0.890412i	$0.650416\pi$
$522$	0	0
$523$	−23.9406	−1.04685	−0.523426	−	0.852071i	$-0.675346\pi$
$523$	−23.9406	−1.04685	−0.523426	+	0.852071i	$0.675346\pi$
$524$	2.54903	0.111355
$525$	0	0
$526$	8.38799	0.365734
$527$	57.4806	2.50390
$528$	0	0
$529$	−22.9857	−0.999378
$530$	−29.8686	−1.29741
$531$	0	0
$532$	−0.771750	−0.0334596
$533$	23.8632	1.03363
$534$	0	0
$535$	5.95574	0.257489
$536$	−26.7810	−1.15676
$537$	0	0
$538$	20.9547	0.903421
$539$	−6.16344	−0.265478
$540$	0	0
$541$	32.2705	1.38742	0.693709	−	0.720256i	$-0.255975\pi$
$541$	32.2705	1.38742	0.693709	+	0.720256i	$0.255975\pi$
$542$	14.8391	0.637396
$543$	0	0
$544$	−37.9214	−1.62587
$545$	−23.2344	−0.995254
$546$	0	0
$547$	5.16940	0.221028	0.110514	−	0.993875i	$-0.464750\pi$
$547$	5.16940	0.221028	0.110514	+	0.993875i	$0.464750\pi$
$548$	−15.6699	−0.669385
$549$	0	0
$550$	−11.6099	−0.495049
$551$	−4.91359	−0.209326
$552$	0	0
$553$	−5.78776	−0.246121
$554$	8.30603	0.352889
$555$	0	0
$556$	12.0399	0.510604
$557$	−28.5616	−1.21019	−0.605096	−	0.796153i	$-0.706865\pi$
$557$	−28.5616	−1.21019	−0.605096	+	0.796153i	$0.706865\pi$
$558$	0	0
$559$	−39.0786	−1.65285
$560$	−2.04536	−0.0864321
$561$	0	0
$562$	5.11308	0.215682
$563$	41.5103	1.74945	0.874724	−	0.484621i	$-0.161042\pi$
$563$	41.5103	1.74945	0.874724	+	0.484621i	$0.161042\pi$
$564$	0	0
$565$	−3.79464	−0.159642
$566$	−23.8335	−1.00180
$567$	0	0
$568$	−37.3190	−1.56587
$569$	13.2623	0.555985	0.277992	−	0.960583i	$-0.410331\pi$
$569$	13.2623	0.555985	0.277992	+	0.960583i	$0.410331\pi$
$570$	0	0
$571$	−1.52594	−0.0638588	−0.0319294	−	0.999490i	$-0.510165\pi$
$571$	−1.52594	−0.0638588	−0.0319294	+	0.999490i	$0.510165\pi$
$572$	−7.62645	−0.318878
$573$	0	0
$574$	2.99944	0.125194
$575$	−1.47387	−0.0614648
$576$	0	0
$577$	−13.1176	−0.546091	−0.273046	−	0.962001i	$-0.588031\pi$
$577$	−13.1176	−0.546091	−0.273046	+	0.962001i	$0.588031\pi$
$578$	−31.1703	−1.29651
$579$	0	0
$580$	29.9989	1.24564
$581$	−15.5189	−0.643832
$582$	0	0
$583$	−7.61747	−0.315484
$584$	29.3890	1.21612
$585$	0	0
$586$	28.9853	1.19737
$587$	12.6007	0.520088	0.260044	−	0.965597i	$-0.416263\pi$
$587$	12.6007	0.520088	0.260044	+	0.965597i	$0.416263\pi$
$588$	0	0
$589$	−6.15999	−0.253818
$590$	31.4753	1.29582
$591$	0	0
$592$	1.92959	0.0793055
$593$	15.0972	0.619967	0.309984	−	0.950742i	$-0.399676\pi$
$593$	15.0972	0.619967	0.309984	+	0.950742i	$0.399676\pi$
$594$	0	0
$595$	26.9423	1.10453
$596$	−8.55018	−0.350229
$597$	0	0
$598$	0.772339	0.0315833
$599$	37.1076	1.51618	0.758089	−	0.652151i	$-0.226133\pi$
$599$	37.1076	1.51618	0.758089	+	0.652151i	$0.226133\pi$
$600$	0	0
$601$	0.977214	0.0398614	0.0199307	−	0.999801i	$-0.493655\pi$
$601$	0.977214	0.0398614	0.0199307	+	0.999801i	$0.493655\pi$
$602$	−4.91192	−0.200195
$603$	0	0
$604$	15.8854	0.646370
$605$	−4.16220	−0.169218
$606$	0	0
$607$	−12.9781	−0.526763	−0.263382	−	0.964692i	$-0.584838\pi$
$607$	−12.9781	−0.526763	−0.263382	+	0.964692i	$0.584838\pi$
$608$	4.06390	0.164813
$609$	0	0
$610$	3.92106	0.158759
$611$	26.9485	1.09022
$612$	0	0
$613$	31.6608	1.27877	0.639383	−	0.768888i	$-0.279190\pi$
$613$	31.6608	1.27877	0.639383	+	0.768888i	$0.279190\pi$
$614$	−5.35828	−0.216243
$615$	0	0
$616$	−2.68188	−0.108056
$617$	−2.15571	−0.0867857	−0.0433928	−	0.999058i	$-0.513817\pi$
$617$	−2.15571	−0.0867857	−0.0433928	+	0.999058i	$0.513817\pi$
$618$	0	0
$619$	−3.90531	−0.156968	−0.0784839	−	0.996915i	$-0.525008\pi$
$619$	−3.90531	−0.156968	−0.0784839	+	0.996915i	$0.525008\pi$
$620$	37.6086	1.51040
$621$	0	0
$622$	3.80114	0.152412
$623$	−0.850474	−0.0340735
$624$	0	0
$625$	65.2594	2.61037
$626$	−20.6433	−0.825072
$627$	0	0
$628$	−15.0260	−0.599602
$629$	−25.4173	−1.01345
$630$	0	0
$631$	−28.4071	−1.13087	−0.565434	−	0.824793i	$-0.691291\pi$
$631$	−28.4071	−1.13087	−0.565434	+	0.824793i	$0.691291\pi$
$632$	18.5547	0.738067
$633$	0	0
$634$	−24.4089	−0.969400
$635$	−8.51103	−0.337750
$636$	0	0
$637$	−42.2513	−1.67406
$638$	−6.10319	−0.241628
$639$	0	0
$640$	−20.5980	−0.814206
$641$	−27.3714	−1.08111	−0.540553	−	0.841310i	$-0.681785\pi$
$641$	−27.3714	−1.08111	−0.540553	+	0.841310i	$0.681785\pi$
$642$	0	0
$643$	45.1938	1.78227	0.891135	−	0.453738i	$-0.149910\pi$
$643$	45.1938	1.78227	0.891135	+	0.453738i	$0.149910\pi$
$644$	−0.121693	−0.00479538
$645$	0	0
$646$	5.05669	0.198953
$647$	28.4332	1.11783	0.558913	−	0.829226i	$-0.311219\pi$
$647$	28.4332	1.11783	0.558913	+	0.829226i	$0.311219\pi$
$648$	0	0
$649$	8.02724	0.315097
$650$	−79.5876	−3.12168
$651$	0	0
$652$	12.1018	0.473945
$653$	26.0993	1.02134	0.510672	−	0.859776i	$-0.329397\pi$
$653$	26.0993	1.02134	0.510672	+	0.859776i	$0.329397\pi$
$654$	0	0
$655$	9.53654	0.372624
$656$	−1.87030	−0.0730228
$657$	0	0
$658$	3.38724	0.132049
$659$	14.9194	0.581178	0.290589	−	0.956848i	$-0.406149\pi$
$659$	14.9194	0.581178	0.290589	+	0.956848i	$0.406149\pi$
$660$	0	0
$661$	8.03694	0.312601	0.156300	−	0.987710i	$-0.450043\pi$
$661$	8.03694	0.312601	0.156300	+	0.987710i	$0.450043\pi$
$662$	3.94543	0.153343
$663$	0	0
$664$	49.7513	1.93073
$665$	−2.88731	−0.111965
$666$	0	0
$667$	−0.774797	−0.0300003
$668$	−2.45524	−0.0949959
$669$	0	0
$670$	−35.8129	−1.38357
$671$	1.00000	0.0386046
$672$	0	0
$673$	4.23492	0.163244	0.0816220	−	0.996663i	$-0.473990\pi$
$673$	4.23492	0.163244	0.0816220	+	0.996663i	$0.473990\pi$
$674$	−29.3222	−1.12945
$675$	0	0
$676$	−37.8177	−1.45453
$677$	−9.81344	−0.377161	−0.188580	−	0.982058i	$-0.560389\pi$
$677$	−9.81344	−0.377161	−0.188580	+	0.982058i	$0.560389\pi$
$678$	0	0
$679$	−14.5934	−0.560045
$680$	−86.3731	−3.31226
$681$	0	0
$682$	−7.65135	−0.292985
$683$	−13.0230	−0.498312	−0.249156	−	0.968463i	$-0.580153\pi$
$683$	−13.0230	−0.498312	−0.249156	+	0.968463i	$0.580153\pi$
$684$	0	0
$685$	−58.6251	−2.23995
$686$	−11.3422	−0.433048
$687$	0	0
$688$	3.06281	0.116769
$689$	−52.2188	−1.98938
$690$	0	0
$691$	−15.1399	−0.575950	−0.287975	−	0.957638i	$-0.592982\pi$
$691$	−15.1399	−0.575950	−0.287975	+	0.957638i	$0.592982\pi$
$692$	4.76531	0.181150
$693$	0	0
$694$	14.6940	0.557776
$695$	45.0441	1.70862
$696$	0	0
$697$	24.6363	0.933168
$698$	6.31995	0.239214
$699$	0	0
$700$	12.5402	0.473973
$701$	−21.3905	−0.807909	−0.403954	−	0.914779i	$-0.632365\pi$
$701$	−21.3905	−0.807909	−0.403954	+	0.914779i	$0.632365\pi$
$702$	0	0
$703$	2.72388	0.102733
$704$	6.12235	0.230745
$705$	0	0
$706$	−1.05217	−0.0395989
$707$	3.56680	0.134143
$708$	0	0
$709$	−15.7681	−0.592182	−0.296091	−	0.955160i	$-0.595683\pi$
$709$	−15.7681	−0.592182	−0.296091	+	0.955160i	$0.595683\pi$
$710$	−49.9047	−1.87289
$711$	0	0
$712$	2.72650	0.102180
$713$	−0.971336	−0.0363768
$714$	0	0
$715$	−28.5325	−1.06705
$716$	−8.50367	−0.317797
$717$	0	0
$718$	25.8771	0.965723
$719$	25.7622	0.960767	0.480383	−	0.877059i	$-0.340498\pi$
$719$	25.7622	0.960767	0.480383	+	0.877059i	$0.340498\pi$
$720$	0	0
$721$	8.12600	0.302628
$722$	17.3573	0.645972
$723$	0	0
$724$	−13.3137	−0.494800
$725$	79.8409	2.96522
$726$	0	0
$727$	5.90292	0.218927	0.109464	−	0.993991i	$-0.465087\pi$
$727$	5.90292	0.218927	0.109464	+	0.993991i	$0.465087\pi$
$728$	−18.3847	−0.681382
$729$	0	0
$730$	39.3003	1.45457
$731$	−40.3447	−1.49220
$732$	0	0
$733$	−19.2956	−0.712700	−0.356350	−	0.934353i	$-0.615979\pi$
$733$	−19.2956	−0.712700	−0.356350	+	0.934353i	$0.615979\pi$
$734$	−35.5840	−1.31343
$735$	0	0
$736$	0.640815	0.0236208
$737$	−9.13346	−0.336435
$738$	0	0
$739$	29.4469	1.08322	0.541612	−	0.840629i	$-0.317814\pi$
$739$	29.4469	1.08322	0.541612	+	0.840629i	$0.317814\pi$
$740$	−16.6301	−0.611336
$741$	0	0
$742$	−6.56356	−0.240956
$743$	−20.4062	−0.748630	−0.374315	−	0.927302i	$-0.622122\pi$
$743$	−20.4062	−0.748630	−0.374315	+	0.927302i	$0.622122\pi$
$744$	0	0
$745$	−31.9884	−1.17196
$746$	10.5409	0.385930
$747$	0	0
$748$	−7.87353	−0.287885
$749$	1.30876	0.0478212
$750$	0	0
$751$	−52.1463	−1.90284	−0.951422	−	0.307890i	$-0.900377\pi$
$751$	−52.1463	−1.90284	−0.951422	+	0.307890i	$0.900377\pi$
$752$	−2.11211	−0.0770207
$753$	0	0
$754$	−41.8382	−1.52366
$755$	59.4315	2.16293
$756$	0	0
$757$	−16.2427	−0.590350	−0.295175	−	0.955443i	$-0.595378\pi$
$757$	−16.2427	−0.590350	−0.295175	+	0.955443i	$0.595378\pi$
$758$	−21.0811	−0.765701
$759$	0	0
$760$	9.25629	0.335761
$761$	4.13096	0.149747	0.0748737	−	0.997193i	$-0.476145\pi$
$761$	4.13096	0.149747	0.0748737	+	0.997193i	$0.476145\pi$
$762$	0	0
$763$	−5.10572	−0.184839
$764$	−0.929456	−0.0336265
$765$	0	0
$766$	17.4783	0.631515
$767$	55.0279	1.98694
$768$	0	0
$769$	39.8109	1.43562	0.717809	−	0.696240i	$-0.245145\pi$
$769$	39.8109	1.43562	0.717809	+	0.696240i	$0.245145\pi$
$770$	−3.58634	−0.129243
$771$	0	0
$772$	11.9915	0.431583
$773$	44.1442	1.58776	0.793879	−	0.608076i	$-0.208059\pi$
$773$	44.1442	1.58776	0.793879	+	0.608076i	$0.208059\pi$
$774$	0	0
$775$	100.094	3.59547
$776$	46.7844	1.67946
$777$	0	0
$778$	−17.3007	−0.620260
$779$	−2.64019	−0.0945945
$780$	0	0
$781$	−12.7273	−0.455420
$782$	0.797361	0.0285136
$783$	0	0
$784$	3.31147	0.118267
$785$	−56.2160	−2.00644
$786$	0	0
$787$	6.00401	0.214020	0.107010	−	0.994258i	$-0.465872\pi$
$787$	6.00401	0.214020	0.107010	+	0.994258i	$0.465872\pi$
$788$	10.3623	0.369141
$789$	0	0
$790$	24.8122	0.882780
$791$	−0.833864	−0.0296488
$792$	0	0
$793$	6.85514	0.243433
$794$	1.02521	0.0363834
$795$	0	0
$796$	−20.8349	−0.738474
$797$	29.5364	1.04623	0.523116	−	0.852261i	$-0.324769\pi$
$797$	29.5364	1.04623	0.523116	+	0.852261i	$0.324769\pi$
$798$	0	0
$799$	27.8216	0.984257
$800$	−66.0344	−2.33467
$801$	0	0
$802$	−6.34462	−0.224036
$803$	10.0229	0.353700
$804$	0	0
$805$	−0.455284	−0.0160467
$806$	−52.4511	−1.84751
$807$	0	0
$808$	−11.4346	−0.402269
$809$	28.0765	0.987119	0.493559	−	0.869712i	$-0.335696\pi$
$809$	28.0765	0.987119	0.493559	+	0.869712i	$0.335696\pi$
$810$	0	0
$811$	−25.8877	−0.909039	−0.454520	−	0.890737i	$-0.650189\pi$
$811$	−25.8877	−0.909039	−0.454520	+	0.890737i	$0.650189\pi$
$812$	6.59221	0.231341
$813$	0	0
$814$	3.38335	0.118586
$815$	45.2761	1.58595
$816$	0	0
$817$	4.32359	0.151263
$818$	28.8733	1.00953
$819$	0	0
$820$	16.1191	0.562905
$821$	29.0400	1.01350	0.506752	−	0.862092i	$-0.330846\pi$
$821$	29.0400	1.01350	0.506752	+	0.862092i	$0.330846\pi$
$822$	0	0
$823$	−44.3381	−1.54553	−0.772764	−	0.634694i	$-0.781126\pi$
$823$	−44.3381	−1.54553	−0.772764	+	0.634694i	$0.781126\pi$
$824$	−26.0508	−0.907521
$825$	0	0
$826$	6.91663	0.240660
$827$	36.0307	1.25291	0.626455	−	0.779457i	$-0.284505\pi$
$827$	36.0307	1.25291	0.626455	+	0.779457i	$0.284505\pi$
$828$	0	0
$829$	7.27308	0.252604	0.126302	−	0.991992i	$-0.459689\pi$
$829$	7.27308	0.252604	0.126302	+	0.991992i	$0.459689\pi$
$830$	66.5298	2.30928
$831$	0	0
$832$	41.9695	1.45503
$833$	−43.6201	−1.51135
$834$	0	0
$835$	−9.18565	−0.317883
$836$	0.843778	0.0291827
$837$	0	0
$838$	−16.2554	−0.561533
$839$	33.4149	1.15361	0.576806	−	0.816881i	$-0.304299\pi$
$839$	33.4149	1.15361	0.576806	+	0.816881i	$0.304299\pi$
$840$	0	0
$841$	12.9714	0.447290
$842$	5.32949	0.183666
$843$	0	0
$844$	24.3101	0.836788
$845$	−141.485	−4.86725
$846$	0	0
$847$	−0.914635	−0.0314272
$848$	4.09269	0.140544
$849$	0	0
$850$	−82.1661	−2.81827
$851$	0.429514	0.0147236
$852$	0	0
$853$	10.4507	0.357824	0.178912	−	0.983865i	$-0.442742\pi$
$853$	10.4507	0.357824	0.178912	+	0.983865i	$0.442742\pi$
$854$	0.861645	0.0294849
$855$	0	0
$856$	−4.19570	−0.143406
$857$	4.06247	0.138771	0.0693857	−	0.997590i	$-0.477896\pi$
$857$	4.06247	0.138771	0.0693857	+	0.997590i	$0.477896\pi$
$858$	0	0
$859$	−33.1960	−1.13263	−0.566317	−	0.824187i	$-0.691632\pi$
$859$	−33.1960	−1.13263	−0.566317	+	0.824187i	$0.691632\pi$
$860$	−26.3968	−0.900125
$861$	0	0
$862$	−10.9440	−0.372753
$863$	38.9594	1.32619	0.663097	−	0.748533i	$-0.269242\pi$
$863$	38.9594	1.32619	0.663097	+	0.748533i	$0.269242\pi$
$864$	0	0
$865$	17.8282	0.606178
$866$	−2.55014	−0.0866574
$867$	0	0
$868$	8.26442	0.280513
$869$	6.32794	0.214661
$870$	0	0
$871$	−62.6112	−2.12150
$872$	16.3682	0.554297
$873$	0	0
$874$	−0.0854504	−0.00289040
$875$	27.8814	0.942563
$876$	0	0
$877$	−45.7454	−1.54471	−0.772356	−	0.635190i	$-0.780922\pi$
$877$	−45.7454	−1.54471	−0.772356	+	0.635190i	$0.780922\pi$
$878$	−11.1264	−0.375499
$879$	0	0
$880$	2.23625	0.0753841
$881$	−32.9922	−1.11154	−0.555768	−	0.831337i	$-0.687576\pi$
$881$	−32.9922	−1.11154	−0.555768	+	0.831337i	$0.687576\pi$
$882$	0	0
$883$	33.9586	1.14280	0.571399	−	0.820672i	$-0.306401\pi$
$883$	33.9586	1.14280	0.571399	+	0.820672i	$0.306401\pi$
$884$	−53.9742	−1.81535
$885$	0	0
$886$	1.28427	0.0431460
$887$	−4.70806	−0.158081	−0.0790406	−	0.996871i	$-0.525186\pi$
$887$	−4.70806	−0.158081	−0.0790406	+	0.996871i	$0.525186\pi$
$888$	0	0
$889$	−1.87028	−0.0627272
$890$	3.64600	0.122214
$891$	0	0
$892$	−5.99420	−0.200701
$893$	−2.98154	−0.0997734
$894$	0	0
$895$	−31.8144	−1.06344
$896$	−4.52636	−0.151215
$897$	0	0
$898$	21.5148	0.717959
$899$	52.6180	1.75491
$900$	0	0
$901$	−53.9106	−1.79602
$902$	−3.27939	−0.109192
$903$	0	0
$904$	2.67325	0.0889109
$905$	−49.8099	−1.65574
$906$	0	0
$907$	−9.21665	−0.306034	−0.153017	−	0.988224i	$-0.548899\pi$
$907$	−9.21665	−0.306034	−0.153017	+	0.988224i	$0.548899\pi$
$908$	6.75227	0.224082
$909$	0	0
$910$	−24.5849	−0.814980
$911$	−34.1955	−1.13295	−0.566474	−	0.824080i	$-0.691693\pi$
$911$	−34.1955	−1.13295	−0.566474	+	0.824080i	$0.691693\pi$
$912$	0	0
$913$	16.9673	0.561536
$914$	−3.69707	−0.122288
$915$	0	0
$916$	−4.20918	−0.139075
$917$	2.09564	0.0692040
$918$	0	0
$919$	51.2959	1.69210	0.846048	−	0.533107i	$-0.178976\pi$
$919$	51.2959	1.69210	0.846048	+	0.533107i	$0.178976\pi$
$920$	1.45957	0.0481208
$921$	0	0
$922$	−17.0189	−0.560487
$923$	−87.2477	−2.87179
$924$	0	0
$925$	−44.2604	−1.45527
$926$	25.3230	0.832166
$927$	0	0
$928$	−34.7135	−1.13953
$929$	11.3176	0.371317	0.185658	−	0.982614i	$-0.440558\pi$
$929$	11.3176	0.371317	0.185658	+	0.982614i	$0.440558\pi$
$930$	0	0
$931$	4.67461	0.153204
$932$	19.0032	0.622470
$933$	0	0
$934$	27.3117	0.893668
$935$	−29.4569	−0.963343
$936$	0	0
$937$	−7.92226	−0.258809	−0.129404	−	0.991592i	$-0.541307\pi$
$937$	−7.92226	−0.258809	−0.129404	+	0.991592i	$0.541307\pi$
$938$	−7.86980	−0.256958
$939$	0	0
$940$	18.2032	0.593723
$941$	19.8214	0.646161	0.323080	−	0.946372i	$-0.395282\pi$
$941$	19.8214	0.646161	0.323080	+	0.946372i	$0.395282\pi$
$942$	0	0
$943$	−0.416317	−0.0135571
$944$	−4.31285	−0.140371
$945$	0	0
$946$	5.37035	0.174605
$947$	3.76239	0.122261	0.0611306	−	0.998130i	$-0.480529\pi$
$947$	3.76239	0.122261	0.0611306	+	0.998130i	$0.480529\pi$
$948$	0	0
$949$	68.7082	2.23036
$950$	8.80544	0.285686
$951$	0	0
$952$	−18.9803	−0.615155
$953$	22.2293	0.720077	0.360039	−	0.932937i	$-0.382764\pi$
$953$	22.2293	0.720077	0.360039	+	0.932937i	$0.382764\pi$
$954$	0	0
$955$	−3.47733	−0.112524
$956$	6.14950	0.198889
$957$	0	0
$958$	−11.3554	−0.366878
$959$	−12.8827	−0.416005
$960$	0	0
$961$	34.9654	1.12791
$962$	23.1933	0.747782
$963$	0	0
$964$	−20.5583	−0.662138
$965$	44.8632	1.44420
$966$	0	0
$967$	6.82929	0.219615	0.109808	−	0.993953i	$-0.464977\pi$
$967$	6.82929	0.219615	0.109808	+	0.993953i	$0.464977\pi$
$968$	2.93219	0.0942441
$969$	0	0
$970$	62.5623	2.00875
$971$	35.6329	1.14351	0.571757	−	0.820423i	$-0.306262\pi$
$971$	35.6329	1.14351	0.571757	+	0.820423i	$0.306262\pi$
$972$	0	0
$973$	9.89835	0.317327
$974$	−2.26572	−0.0725984
$975$	0	0
$976$	−0.537277	−0.0171978
$977$	46.8490	1.49883	0.749417	−	0.662099i	$-0.230334\pi$
$977$	46.8490	1.49883	0.749417	+	0.662099i	$0.230334\pi$
$978$	0	0
$979$	0.929850	0.0297181
$980$	−28.5399	−0.911674
$981$	0	0
$982$	−3.00487	−0.0958891
$983$	20.4677	0.652818	0.326409	−	0.945229i	$-0.394161\pi$
$983$	20.4677	0.652818	0.326409	+	0.945229i	$0.394161\pi$
$984$	0	0
$985$	38.7679	1.23525
$986$	−43.1937	−1.37557
$987$	0	0
$988$	5.78422	0.184020
$989$	0.681764	0.0216788
$990$	0	0
$991$	56.8308	1.80529	0.902646	−	0.430384i	$-0.141622\pi$
$991$	56.8308	1.80529	0.902646	+	0.430384i	$0.141622\pi$
$992$	−43.5190	−1.38173
$993$	0	0
$994$	−10.9665	−0.347835
$995$	−77.9487	−2.47114
$996$	0	0
$997$	−9.65434	−0.305756	−0.152878	−	0.988245i	$-0.548854\pi$
$997$	−9.65434	−0.305756	−0.152878	+	0.988245i	$0.548854\pi$
$998$	29.3422	0.928812
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.l.1.8		21
3.2	odd	2		671.2.a.d.1.14	✓	21
33.32	even	2		7381.2.a.j.1.8		21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
671.2.a.d.1.14	✓	21	3.2	odd	2
6039.2.a.l.1.8		21	1.1	even	1	trivial
7381.2.a.j.1.8		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)

\(f(q)\)	\(=\)	\(q-0.942064 q^{2} -1.11252 q^{4} -4.16220 q^{5} -0.914635 q^{7} +2.93219 q^{8} +O(q^{10})\)	\(q-0.942064 q^{2} -1.11252 q^{4} -4.16220 q^{5} -0.914635 q^{7} +2.93219 q^{8} +3.92106 q^{10} +1.00000 q^{11} +6.85514 q^{13} +0.861645 q^{14} -0.537277 q^{16} +7.07723 q^{17} -0.758442 q^{19} +4.63051 q^{20} -0.942064 q^{22} -0.119595 q^{23} +12.3239 q^{25} -6.45798 q^{26} +1.01755 q^{28} +6.47853 q^{29} +8.12191 q^{31} -5.35823 q^{32} -6.66720 q^{34} +3.80690 q^{35} -3.59142 q^{37} +0.714500 q^{38} -12.2044 q^{40} +3.48107 q^{41} -5.70063 q^{43} -1.11252 q^{44} +0.112666 q^{46} +3.93114 q^{47} -6.16344 q^{49} -11.6099 q^{50} -7.62645 q^{52} -7.61747 q^{53} -4.16220 q^{55} -2.68188 q^{56} -6.10319 q^{58} +8.02724 q^{59} +1.00000 q^{61} -7.65135 q^{62} +6.12235 q^{64} -28.5325 q^{65} -9.13346 q^{67} -7.87353 q^{68} -3.58634 q^{70} -12.7273 q^{71} +10.0229 q^{73} +3.38335 q^{74} +0.843778 q^{76} -0.914635 q^{77} +6.32794 q^{79} +2.23625 q^{80} -3.27939 q^{82} +16.9673 q^{83} -29.4569 q^{85} +5.37035 q^{86} +2.93219 q^{88} +0.929850 q^{89} -6.26995 q^{91} +0.133051 q^{92} -3.70338 q^{94} +3.15679 q^{95} +15.9555 q^{97} +5.80636 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q + 32 q^{4} - 7 q^{5} + 5 q^{7} + 6 q^{8}+O(q^{10}) \) 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

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