LMFDB - Embedded newform 6039.2.a.l.1.15

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			1.15
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)

$f(q)$	$=$	$q+1.45442 q^{2} +0.115325 q^{4} -3.82594 q^{5} +3.26042 q^{7} -2.74110 q^{8} +O(q^{10})$	\(q+1.45442 q^{2} +0.115325 q^{4} -3.82594 q^{5} +3.26042 q^{7} -2.74110 q^{8} -5.56450 q^{10} +1.00000 q^{11} +5.15443 q^{13} +4.74201 q^{14} -4.21735 q^{16} -1.81260 q^{17} -6.25538 q^{19} -0.441227 q^{20} +1.45442 q^{22} +3.26360 q^{23} +9.63778 q^{25} +7.49668 q^{26} +0.376009 q^{28} -9.11409 q^{29} +9.69614 q^{31} -0.651580 q^{32} -2.63628 q^{34} -12.4742 q^{35} -11.3749 q^{37} -9.09792 q^{38} +10.4873 q^{40} +1.74708 q^{41} +1.82661 q^{43} +0.115325 q^{44} +4.74662 q^{46} +3.66916 q^{47} +3.63034 q^{49} +14.0173 q^{50} +0.594436 q^{52} +6.16433 q^{53} -3.82594 q^{55} -8.93714 q^{56} -13.2557 q^{58} -8.95135 q^{59} +1.00000 q^{61} +14.1022 q^{62} +7.48703 q^{64} -19.7205 q^{65} +4.57270 q^{67} -0.209039 q^{68} -18.1426 q^{70} +5.54735 q^{71} -7.91540 q^{73} -16.5438 q^{74} -0.721404 q^{76} +3.26042 q^{77} +6.29660 q^{79} +16.1353 q^{80} +2.54098 q^{82} +2.81886 q^{83} +6.93490 q^{85} +2.65665 q^{86} -2.74110 q^{88} -5.92388 q^{89} +16.8056 q^{91} +0.376375 q^{92} +5.33648 q^{94} +23.9327 q^{95} +10.3200 q^{97} +5.28002 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.45442	1.02843	0.514214	−	0.857662i	$-0.328084\pi$
$2$	1.45442	1.02843	0.514214	+	0.857662i	$0.328084\pi$
$3$	0	0
$3$	0	0
$4$	0.115325	0.0576627
$5$	−3.82594	−1.71101	−0.855505	−	0.517794i	$-0.826753\pi$
$5$	−3.82594	−1.71101	−0.855505	+	0.517794i	$0.826753\pi$
$6$	0	0
$7$	3.26042	1.23232	0.616161	−	0.787620i	$-0.288687\pi$
$7$	3.26042	1.23232	0.616161	+	0.787620i	$0.288687\pi$
$8$	−2.74110	−0.969125
$9$	0	0
$10$	−5.56450	−1.75965
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	5.15443	1.42958	0.714791	−	0.699339i	$-0.246522\pi$
$13$	5.15443	1.42958	0.714791	+	0.699339i	$0.246522\pi$
$14$	4.74201	1.26735
$15$	0	0
$16$	−4.21735	−1.05434
$17$	−1.81260	−0.439621	−0.219810	−	0.975543i	$-0.570544\pi$
$17$	−1.81260	−0.439621	−0.219810	+	0.975543i	$0.570544\pi$
$18$	0	0
$19$	−6.25538	−1.43508	−0.717541	−	0.696516i	$-0.754732\pi$
$19$	−6.25538	−1.43508	−0.717541	+	0.696516i	$0.754732\pi$
$20$	−0.441227	−0.0986614
$21$	0	0
$22$	1.45442	0.310082
$23$	3.26360	0.680507	0.340253	−	0.940334i	$-0.389487\pi$
$23$	3.26360	0.680507	0.340253	+	0.940334i	$0.389487\pi$
$24$	0	0
$25$	9.63778	1.92756
$26$	7.49668	1.47022
$27$	0	0
$28$	0.376009	0.0710590
$29$	−9.11409	−1.69244	−0.846222	−	0.532830i	$-0.821128\pi$
$29$	−9.11409	−1.69244	−0.846222	+	0.532830i	$0.821128\pi$
$30$	0	0
$31$	9.69614	1.74148	0.870739	−	0.491745i	$-0.163641\pi$
$31$	9.69614	1.74148	0.870739	+	0.491745i	$0.163641\pi$
$32$	−0.651580	−0.115184
$33$	0	0
$34$	−2.63628	−0.452118
$35$	−12.4742	−2.10852
$36$	0	0
$37$	−11.3749	−1.87002	−0.935010	−	0.354622i	$-0.884610\pi$
$37$	−11.3749	−1.87002	−0.935010	+	0.354622i	$0.884610\pi$
$38$	−9.09792	−1.47588
$39$	0	0
$40$	10.4873	1.65818
$41$	1.74708	0.272848	0.136424	−	0.990651i	$-0.456439\pi$
$41$	1.74708	0.272848	0.136424	+	0.990651i	$0.456439\pi$
$42$	0	0
$43$	1.82661	0.278555	0.139278	−	0.990253i	$-0.455522\pi$
$43$	1.82661	0.278555	0.139278	+	0.990253i	$0.455522\pi$
$44$	0.115325	0.0173860
$45$	0	0
$46$	4.74662	0.699852
$47$	3.66916	0.535202	0.267601	−	0.963530i	$-0.413769\pi$
$47$	3.66916	0.535202	0.267601	+	0.963530i	$0.413769\pi$
$48$	0	0
$49$	3.63034	0.518620
$50$	14.0173	1.98235
$51$	0	0
$52$	0.594436	0.0824335
$53$	6.16433	0.846735	0.423368	−	0.905958i	$-0.360848\pi$
$53$	6.16433	0.846735	0.423368	+	0.905958i	$0.360848\pi$
$54$	0	0
$55$	−3.82594	−0.515889
$56$	−8.93714	−1.19428
$57$	0	0
$58$	−13.2557	−1.74056
$59$	−8.95135	−1.16537	−0.582683	−	0.812699i	$-0.697997\pi$
$59$	−8.95135	−1.16537	−0.582683	+	0.812699i	$0.697997\pi$
$60$	0	0
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	14.1022	1.79098
$63$	0	0
$64$	7.48703	0.935879
$65$	−19.7205	−2.44603
$66$	0	0
$67$	4.57270	0.558644	0.279322	−	0.960197i	$-0.409890\pi$
$67$	4.57270	0.558644	0.279322	+	0.960197i	$0.409890\pi$
$68$	−0.209039	−0.0253497
$69$	0	0
$70$	−18.1426	−2.16846
$71$	5.54735	0.658350	0.329175	−	0.944269i	$-0.393229\pi$
$71$	5.54735	0.658350	0.329175	+	0.944269i	$0.393229\pi$
$72$	0	0
$73$	−7.91540	−0.926427	−0.463214	−	0.886247i	$-0.653304\pi$
$73$	−7.91540	−0.926427	−0.463214	+	0.886247i	$0.653304\pi$
$74$	−16.5438	−1.92318
$75$	0	0
$76$	−0.721404	−0.0827507
$77$	3.26042	0.371559
$78$	0	0
$79$	6.29660	0.708422	0.354211	−	0.935165i	$-0.384749\pi$
$79$	6.29660	0.708422	0.354211	+	0.935165i	$0.384749\pi$
$80$	16.1353	1.80398
$81$	0	0
$82$	2.54098	0.280604
$83$	2.81886	0.309410	0.154705	−	0.987961i	$-0.450557\pi$
$83$	2.81886	0.309410	0.154705	+	0.987961i	$0.450557\pi$
$84$	0	0
$85$	6.93490	0.752196
$86$	2.65665	0.286474
$87$	0	0
$88$	−2.74110	−0.292202
$89$	−5.92388	−0.627930	−0.313965	−	0.949435i	$-0.601657\pi$
$89$	−5.92388	−0.627930	−0.313965	+	0.949435i	$0.601657\pi$
$90$	0	0
$91$	16.8056	1.76171
$92$	0.376375	0.0392398
$93$	0	0
$94$	5.33648	0.550416
$95$	23.9327	2.45544
$96$	0	0
$97$	10.3200	1.04784	0.523919	−	0.851768i	$-0.324469\pi$
$97$	10.3200	1.04784	0.523919	+	0.851768i	$0.324469\pi$
$98$	5.28002	0.533362
$99$	0	0
$100$	1.11148	0.111148
$101$	8.83521	0.879136	0.439568	−	0.898209i	$-0.355132\pi$
$101$	8.83521	0.879136	0.439568	+	0.898209i	$0.355132\pi$
$102$	0	0
$103$	17.0024	1.67529	0.837646	−	0.546213i	$-0.183931\pi$
$103$	17.0024	1.67529	0.837646	+	0.546213i	$0.183931\pi$
$104$	−14.1288	−1.38544
$105$	0	0
$106$	8.96549	0.870806
$107$	14.3578	1.38802	0.694010	−	0.719966i	$-0.255842\pi$
$107$	14.3578	1.38802	0.694010	+	0.719966i	$0.255842\pi$
$108$	0	0
$109$	12.6102	1.20784	0.603921	−	0.797044i	$-0.293604\pi$
$109$	12.6102	1.20784	0.603921	+	0.797044i	$0.293604\pi$
$110$	−5.56450	−0.530554
$111$	0	0
$112$	−13.7503	−1.29928
$113$	−1.44813	−0.136228	−0.0681142	−	0.997678i	$-0.521698\pi$
$113$	−1.44813	−0.136228	−0.0681142	+	0.997678i	$0.521698\pi$
$114$	0	0
$115$	−12.4863	−1.16435
$116$	−1.05109	−0.0975908
$117$	0	0
$118$	−13.0190	−1.19849
$119$	−5.90985	−0.541755
$120$	0	0
$121$	1.00000	0.0909091
$122$	1.45442	0.131677
$123$	0	0
$124$	1.11821	0.100418
$125$	−17.7439	−1.58706
$126$	0	0
$127$	1.73437	0.153901	0.0769503	−	0.997035i	$-0.475482\pi$
$127$	1.73437	0.153901	0.0769503	+	0.997035i	$0.475482\pi$
$128$	12.1924	1.07767
$129$	0	0
$130$	−28.6818	−2.51556
$131$	8.63290	0.754260	0.377130	−	0.926160i	$-0.376911\pi$
$131$	8.63290	0.754260	0.377130	+	0.926160i	$0.376911\pi$
$132$	0	0
$133$	−20.3952	−1.76848
$134$	6.65061	0.574525
$135$	0	0
$136$	4.96853	0.426048
$137$	6.05841	0.517605	0.258802	−	0.965930i	$-0.416672\pi$
$137$	6.05841	0.517605	0.258802	+	0.965930i	$0.416672\pi$
$138$	0	0
$139$	12.8486	1.08981	0.544903	−	0.838499i	$-0.316566\pi$
$139$	12.8486	1.08981	0.544903	+	0.838499i	$0.316566\pi$
$140$	−1.43859	−0.121583
$141$	0	0
$142$	8.06816	0.677065
$143$	5.15443	0.431035
$144$	0	0
$145$	34.8699	2.89579
$146$	−11.5123	−0.952763
$147$	0	0
$148$	−1.31181	−0.107830
$149$	6.70006	0.548891	0.274445	−	0.961603i	$-0.411506\pi$
$149$	6.70006	0.548891	0.274445	+	0.961603i	$0.411506\pi$
$150$	0	0
$151$	7.94403	0.646476	0.323238	−	0.946318i	$-0.395229\pi$
$151$	7.94403	0.646476	0.323238	+	0.946318i	$0.395229\pi$
$152$	17.1466	1.39077
$153$	0	0
$154$	4.74201	0.382122
$155$	−37.0968	−2.97969
$156$	0	0
$157$	−18.5035	−1.47674	−0.738369	−	0.674397i	$-0.764404\pi$
$157$	−18.5035	−1.47674	−0.738369	+	0.674397i	$0.764404\pi$
$158$	9.15787	0.728561
$159$	0	0
$160$	2.49290	0.197081
$161$	10.6407	0.838604
$162$	0	0
$163$	10.4832	0.821105	0.410552	−	0.911837i	$-0.365336\pi$
$163$	10.4832	0.821105	0.410552	+	0.911837i	$0.365336\pi$
$164$	0.201483	0.0157331
$165$	0	0
$166$	4.09980	0.318206
$167$	−5.69053	−0.440346	−0.220173	−	0.975461i	$-0.570662\pi$
$167$	−5.69053	−0.440346	−0.220173	+	0.975461i	$0.570662\pi$
$168$	0	0
$169$	13.5681	1.04370
$170$	10.0862	0.773579
$171$	0	0
$172$	0.210654	0.0160622
$173$	−2.76526	−0.210239	−0.105119	−	0.994460i	$-0.533522\pi$
$173$	−2.76526	−0.210239	−0.105119	+	0.994460i	$0.533522\pi$
$174$	0	0
$175$	31.4232	2.37537
$176$	−4.21735	−0.317895
$177$	0	0
$178$	−8.61578	−0.645780
$179$	−6.27826	−0.469259	−0.234630	−	0.972085i	$-0.575388\pi$
$179$	−6.27826	−0.469259	−0.234630	+	0.972085i	$0.575388\pi$
$180$	0	0
$181$	4.88002	0.362729	0.181365	−	0.983416i	$-0.441949\pi$
$181$	4.88002	0.362729	0.181365	+	0.983416i	$0.441949\pi$
$182$	24.4423	1.81179
$183$	0	0
$184$	−8.94584	−0.659496
$185$	43.5196	3.19962
$186$	0	0
$187$	−1.81260	−0.132551
$188$	0.423147	0.0308612
$189$	0	0
$190$	34.8081	2.52524
$191$	10.8614	0.785906	0.392953	−	0.919558i	$-0.371453\pi$
$191$	10.8614	0.785906	0.392953	+	0.919558i	$0.371453\pi$
$192$	0	0
$193$	17.1678	1.23576	0.617882	−	0.786271i	$-0.287991\pi$
$193$	17.1678	1.23576	0.617882	+	0.786271i	$0.287991\pi$
$194$	15.0096	1.07763
$195$	0	0
$196$	0.418670	0.0299050
$197$	1.25956	0.0897399	0.0448700	−	0.998993i	$-0.485713\pi$
$197$	1.25956	0.0897399	0.0448700	+	0.998993i	$0.485713\pi$
$198$	0	0
$199$	5.80007	0.411156	0.205578	−	0.978641i	$-0.434093\pi$
$199$	5.80007	0.411156	0.205578	+	0.978641i	$0.434093\pi$
$200$	−26.4181	−1.86804
$201$	0	0
$202$	12.8501	0.904128
$203$	−29.7158	−2.08564
$204$	0	0
$205$	−6.68421	−0.466846
$206$	24.7285	1.72292
$207$	0	0
$208$	−21.7380	−1.50726
$209$	−6.25538	−0.432694
$210$	0	0
$211$	−15.5879	−1.07312	−0.536559	−	0.843863i	$-0.680276\pi$
$211$	−15.5879	−1.07312	−0.536559	+	0.843863i	$0.680276\pi$
$212$	0.710903	0.0488250
$213$	0	0
$214$	20.8822	1.42748
$215$	−6.98848	−0.476611
$216$	0	0
$217$	31.6135	2.14606
$218$	18.3405	1.24218
$219$	0	0
$220$	−0.441227	−0.0297475
$221$	−9.34293	−0.628474
$222$	0	0
$223$	7.31670	0.489962	0.244981	−	0.969528i	$-0.421218\pi$
$223$	7.31670	0.489962	0.244981	+	0.969528i	$0.421218\pi$
$224$	−2.12443	−0.141944
$225$	0	0
$226$	−2.10618	−0.140101
$227$	22.9379	1.52244	0.761220	−	0.648493i	$-0.224601\pi$
$227$	22.9379	1.52244	0.761220	+	0.648493i	$0.224601\pi$
$228$	0	0
$229$	13.4703	0.890144	0.445072	−	0.895495i	$-0.353178\pi$
$229$	13.4703	0.890144	0.445072	+	0.895495i	$0.353178\pi$
$230$	−18.1603	−1.19745
$231$	0	0
$232$	24.9826	1.64019
$233$	6.65333	0.435874	0.217937	−	0.975963i	$-0.430067\pi$
$233$	6.65333	0.435874	0.217937	+	0.975963i	$0.430067\pi$
$234$	0	0
$235$	−14.0380	−0.915735
$236$	−1.03232	−0.0671982
$237$	0	0
$238$	−8.59538	−0.557156
$239$	−3.00085	−0.194109	−0.0970545	−	0.995279i	$-0.530942\pi$
$239$	−3.00085	−0.194109	−0.0970545	+	0.995279i	$0.530942\pi$
$240$	0	0
$241$	−19.4575	−1.25337	−0.626683	−	0.779274i	$-0.715588\pi$
$241$	−19.4575	−1.25337	−0.626683	+	0.779274i	$0.715588\pi$
$242$	1.45442	0.0934934
$243$	0	0
$244$	0.115325	0.00738295
$245$	−13.8894	−0.887363
$246$	0	0
$247$	−32.2429	−2.05157
$248$	−26.5781	−1.68771
$249$	0	0
$250$	−25.8069	−1.63217
$251$	−21.7902	−1.37538	−0.687691	−	0.726003i	$-0.741376\pi$
$251$	−21.7902	−1.37538	−0.687691	+	0.726003i	$0.741376\pi$
$252$	0	0
$253$	3.26360	0.205180
$254$	2.52250	0.158276
$255$	0	0
$256$	2.75878	0.172424
$257$	5.38983	0.336208	0.168104	−	0.985769i	$-0.446235\pi$
$257$	5.38983	0.336208	0.168104	+	0.985769i	$0.446235\pi$
$258$	0	0
$259$	−37.0869	−2.30447
$260$	−2.27427	−0.141045
$261$	0	0
$262$	12.5558	0.775701
$263$	−6.98170	−0.430510	−0.215255	−	0.976558i	$-0.569058\pi$
$263$	−6.98170	−0.430510	−0.215255	+	0.976558i	$0.569058\pi$
$264$	0	0
$265$	−23.5843	−1.44877
$266$	−29.6630	−1.81876
$267$	0	0
$268$	0.527348	0.0322129
$269$	−15.9257	−0.971009	−0.485505	−	0.874234i	$-0.661364\pi$
$269$	−15.9257	−0.971009	−0.485505	+	0.874234i	$0.661364\pi$
$270$	0	0
$271$	11.8128	0.717579	0.358789	−	0.933419i	$-0.383190\pi$
$271$	11.8128	0.717579	0.358789	+	0.933419i	$0.383190\pi$
$272$	7.64439	0.463509
$273$	0	0
$274$	8.81144	0.532319
$275$	9.63778	0.581180
$276$	0	0
$277$	−18.2085	−1.09404	−0.547022	−	0.837118i	$-0.684239\pi$
$277$	−18.2085	−1.09404	−0.547022	+	0.837118i	$0.684239\pi$
$278$	18.6873	1.12079
$279$	0	0
$280$	34.1929	2.04342
$281$	−10.6699	−0.636515	−0.318258	−	0.948004i	$-0.603098\pi$
$281$	−10.6699	−0.636515	−0.318258	+	0.948004i	$0.603098\pi$
$282$	0	0
$283$	−2.57426	−0.153024	−0.0765119	−	0.997069i	$-0.524378\pi$
$283$	−2.57426	−0.153024	−0.0765119	+	0.997069i	$0.524378\pi$
$284$	0.639750	0.0379622
$285$	0	0
$286$	7.49668	0.443288
$287$	5.69621	0.336237
$288$	0	0
$289$	−13.7145	−0.806733
$290$	50.7154	2.97811
$291$	0	0
$292$	−0.912846	−0.0534203
$293$	28.0168	1.63676	0.818379	−	0.574679i	$-0.194873\pi$
$293$	28.0168	1.63676	0.818379	+	0.574679i	$0.194873\pi$
$294$	0	0
$295$	34.2473	1.99395
$296$	31.1797	1.81228
$297$	0	0
$298$	9.74468	0.564494
$299$	16.8220	0.972839
$300$	0	0
$301$	5.95551	0.343270
$302$	11.5539	0.664854
$303$	0	0
$304$	26.3811	1.51306
$305$	−3.82594	−0.219072
$306$	0	0
$307$	−20.1014	−1.14725	−0.573624	−	0.819119i	$-0.694463\pi$
$307$	−20.1014	−1.14725	−0.573624	+	0.819119i	$0.694463\pi$
$308$	0.376009	0.0214251
$309$	0	0
$310$	−53.9542	−3.06439
$311$	11.4362	0.648489	0.324245	−	0.945973i	$-0.394890\pi$
$311$	11.4362	0.648489	0.324245	+	0.945973i	$0.394890\pi$
$312$	0	0
$313$	22.3538	1.26351	0.631755	−	0.775168i	$-0.282335\pi$
$313$	22.3538	1.26351	0.631755	+	0.775168i	$0.282335\pi$
$314$	−26.9118	−1.51872
$315$	0	0
$316$	0.726157	0.0408495
$317$	−7.92764	−0.445261	−0.222630	−	0.974903i	$-0.571464\pi$
$317$	−7.92764	−0.445261	−0.222630	+	0.974903i	$0.571464\pi$
$318$	0	0
$319$	−9.11409	−0.510291
$320$	−28.6449	−1.60130
$321$	0	0
$322$	15.4760	0.862443
$323$	11.3385	0.630892
$324$	0	0
$325$	49.6773	2.75560
$326$	15.2469	0.844447
$327$	0	0
$328$	−4.78892	−0.264424
$329$	11.9630	0.659541
$330$	0	0
$331$	4.99164	0.274365	0.137183	−	0.990546i	$-0.456195\pi$
$331$	4.99164	0.274365	0.137183	+	0.990546i	$0.456195\pi$
$332$	0.325086	0.0178414
$333$	0	0
$334$	−8.27640	−0.452864
$335$	−17.4949	−0.955846
$336$	0	0
$337$	2.19574	0.119609	0.0598047	−	0.998210i	$-0.480952\pi$
$337$	2.19574	0.119609	0.0598047	+	0.998210i	$0.480952\pi$
$338$	19.7337	1.07337
$339$	0	0
$340$	0.799770	0.0433736
$341$	9.69614	0.525076
$342$	0	0
$343$	−10.9865	−0.593216
$344$	−5.00692	−0.269955
$345$	0	0
$346$	−4.02184	−0.216215
$347$	28.2253	1.51521	0.757606	−	0.652712i	$-0.226369\pi$
$347$	28.2253	1.51521	0.757606	+	0.652712i	$0.226369\pi$
$348$	0	0
$349$	−7.90778	−0.423294	−0.211647	−	0.977346i	$-0.567883\pi$
$349$	−7.90778	−0.423294	−0.211647	+	0.977346i	$0.567883\pi$
$350$	45.7024	2.44290
$351$	0	0
$352$	−0.651580	−0.0347294
$353$	3.84148	0.204461	0.102231	−	0.994761i	$-0.467402\pi$
$353$	3.84148	0.204461	0.102231	+	0.994761i	$0.467402\pi$
$354$	0	0
$355$	−21.2238	−1.12644
$356$	−0.683173	−0.0362081
$357$	0	0
$358$	−9.13120	−0.482599
$359$	16.0984	0.849641	0.424821	−	0.905278i	$-0.360337\pi$
$359$	16.0984	0.849641	0.424821	+	0.905278i	$0.360337\pi$
$360$	0	0
$361$	20.1297	1.05946
$362$	7.09758	0.373041
$363$	0	0
$364$	1.93811	0.101585
$365$	30.2838	1.58513
$366$	0	0
$367$	26.0929	1.36204	0.681020	−	0.732264i	$-0.261536\pi$
$367$	26.0929	1.36204	0.681020	+	0.732264i	$0.261536\pi$
$368$	−13.7637	−0.717484
$369$	0	0
$370$	63.2956	3.29058
$371$	20.0983	1.04345
$372$	0	0
$373$	7.64383	0.395782	0.197891	−	0.980224i	$-0.436591\pi$
$373$	7.64383	0.395782	0.197891	+	0.980224i	$0.436591\pi$
$374$	−2.63628	−0.136319
$375$	0	0
$376$	−10.0575	−0.518677
$377$	−46.9779	−2.41949
$378$	0	0
$379$	−9.14079	−0.469531	−0.234765	−	0.972052i	$-0.575432\pi$
$379$	−9.14079	−0.469531	−0.234765	+	0.972052i	$0.575432\pi$
$380$	2.76004	0.141587
$381$	0	0
$382$	15.7971	0.808248
$383$	16.8609	0.861550	0.430775	−	0.902459i	$-0.358240\pi$
$383$	16.8609	0.861550	0.430775	+	0.902459i	$0.358240\pi$
$384$	0	0
$385$	−12.4742	−0.635742
$386$	24.9691	1.27089
$387$	0	0
$388$	1.19016	0.0604212
$389$	−12.9794	−0.658083	−0.329041	−	0.944315i	$-0.606726\pi$
$389$	−12.9794	−0.658083	−0.329041	+	0.944315i	$0.606726\pi$
$390$	0	0
$391$	−5.91560	−0.299165
$392$	−9.95112	−0.502607
$393$	0	0
$394$	1.83192	0.0922910
$395$	−24.0904	−1.21212
$396$	0	0
$397$	−0.913635	−0.0458540	−0.0229270	−	0.999737i	$-0.507299\pi$
$397$	−0.913635	−0.0458540	−0.0229270	+	0.999737i	$0.507299\pi$
$398$	8.43571	0.422844
$399$	0	0
$400$	−40.6459	−2.03230
$401$	−36.2175	−1.80862	−0.904309	−	0.426879i	$-0.859613\pi$
$401$	−36.2175	−1.80862	−0.904309	+	0.426879i	$0.859613\pi$
$402$	0	0
$403$	49.9781	2.48959
$404$	1.01892	0.0506933
$405$	0	0
$406$	−43.2191	−2.14493
$407$	−11.3749	−0.563832
$408$	0	0
$409$	3.86996	0.191357	0.0956787	−	0.995412i	$-0.469498\pi$
$409$	3.86996	0.191357	0.0956787	+	0.995412i	$0.469498\pi$
$410$	−9.72163	−0.480117
$411$	0	0
$412$	1.96080	0.0966018
$413$	−29.1852	−1.43611
$414$	0	0
$415$	−10.7848	−0.529404
$416$	−3.35852	−0.164665
$417$	0	0
$418$	−9.09792	−0.444994
$419$	−5.40560	−0.264081	−0.132040	−	0.991244i	$-0.542153\pi$
$419$	−5.40560	−0.264081	−0.132040	+	0.991244i	$0.542153\pi$
$420$	0	0
$421$	20.8768	1.01748	0.508738	−	0.860922i	$-0.330112\pi$
$421$	20.8768	1.01748	0.508738	+	0.860922i	$0.330112\pi$
$422$	−22.6713	−1.10362
$423$	0	0
$424$	−16.8970	−0.820593
$425$	−17.4695	−0.847394
$426$	0	0
$427$	3.26042	0.157783
$428$	1.65582	0.0800369
$429$	0	0
$430$	−10.1642	−0.490159
$431$	24.5720	1.18359	0.591796	−	0.806087i	$-0.298419\pi$
$431$	24.5720	1.18359	0.591796	+	0.806087i	$0.298419\pi$
$432$	0	0
$433$	−39.3999	−1.89344	−0.946719	−	0.322061i	$-0.895624\pi$
$433$	−39.3999	−1.89344	−0.946719	+	0.322061i	$0.895624\pi$
$434$	45.9792	2.20707
$435$	0	0
$436$	1.45428	0.0696474
$437$	−20.4150	−0.976583
$438$	0	0
$439$	8.87304	0.423487	0.211744	−	0.977325i	$-0.432086\pi$
$439$	8.87304	0.423487	0.211744	+	0.977325i	$0.432086\pi$
$440$	10.4873	0.499961
$441$	0	0
$442$	−13.5885	−0.646340
$443$	0.994527	0.0472514	0.0236257	−	0.999721i	$-0.492479\pi$
$443$	0.994527	0.0472514	0.0236257	+	0.999721i	$0.492479\pi$
$444$	0	0
$445$	22.6644	1.07439
$446$	10.6415	0.503891
$447$	0	0
$448$	24.4109	1.15331
$449$	8.22095	0.387971	0.193985	−	0.981004i	$-0.437859\pi$
$449$	8.22095	0.387971	0.193985	+	0.981004i	$0.437859\pi$
$450$	0	0
$451$	1.74708	0.0822668
$452$	−0.167006	−0.00785530
$453$	0	0
$454$	33.3612	1.56572
$455$	−64.2971	−3.01430
$456$	0	0
$457$	−39.5336	−1.84931	−0.924653	−	0.380811i	$-0.875645\pi$
$457$	−39.5336	−1.84931	−0.924653	+	0.380811i	$0.875645\pi$
$458$	19.5915	0.915449
$459$	0	0
$460$	−1.43999	−0.0671398
$461$	−3.35525	−0.156270	−0.0781349	−	0.996943i	$-0.524896\pi$
$461$	−3.35525	−0.156270	−0.0781349	+	0.996943i	$0.524896\pi$
$462$	0	0
$463$	3.96701	0.184363	0.0921814	−	0.995742i	$-0.470616\pi$
$463$	3.96701	0.184363	0.0921814	+	0.995742i	$0.470616\pi$
$464$	38.4373	1.78441
$465$	0	0
$466$	9.67670	0.448265
$467$	27.4883	1.27201	0.636004	−	0.771686i	$-0.280586\pi$
$467$	27.4883	1.27201	0.636004	+	0.771686i	$0.280586\pi$
$468$	0	0
$469$	14.9089	0.688430
$470$	−20.4170	−0.941767
$471$	0	0
$472$	24.5366	1.12939
$473$	1.82661	0.0839875
$474$	0	0
$475$	−60.2880	−2.76620
$476$	−0.681555	−0.0312390
$477$	0	0
$478$	−4.36449	−0.199627
$479$	11.9418	0.545635	0.272817	−	0.962066i	$-0.412045\pi$
$479$	11.9418	0.545635	0.272817	+	0.962066i	$0.412045\pi$
$480$	0	0
$481$	−58.6310	−2.67334
$482$	−28.2993	−1.28900
$483$	0	0
$484$	0.115325	0.00524206
$485$	−39.4837	−1.79286
$486$	0	0
$487$	34.5789	1.56692	0.783460	−	0.621442i	$-0.213453\pi$
$487$	34.5789	1.56692	0.783460	+	0.621442i	$0.213453\pi$
$488$	−2.74110	−0.124084
$489$	0	0
$490$	−20.2010	−0.912589
$491$	22.9030	1.03360	0.516799	−	0.856106i	$-0.327123\pi$
$491$	22.9030	1.03360	0.516799	+	0.856106i	$0.327123\pi$
$492$	0	0
$493$	16.5202	0.744034
$494$	−46.8946	−2.10989
$495$	0	0
$496$	−40.8920	−1.83611
$497$	18.0867	0.811299
$498$	0	0
$499$	−23.7634	−1.06380	−0.531898	−	0.846808i	$-0.678521\pi$
$499$	−23.7634	−1.06380	−0.531898	+	0.846808i	$0.678521\pi$
$500$	−2.04632	−0.0915141
$501$	0	0
$502$	−31.6920	−1.41448
$503$	−19.3269	−0.861744	−0.430872	−	0.902413i	$-0.641794\pi$
$503$	−19.3269	−0.861744	−0.430872	+	0.902413i	$0.641794\pi$
$504$	0	0
$505$	−33.8029	−1.50421
$506$	4.74662	0.211013
$507$	0	0
$508$	0.200017	0.00887432
$509$	−16.3288	−0.723763	−0.361881	−	0.932224i	$-0.617865\pi$
$509$	−16.3288	−0.723763	−0.361881	+	0.932224i	$0.617865\pi$
$510$	0	0
$511$	−25.8075	−1.14166
$512$	−20.3724	−0.900342
$513$	0	0
$514$	7.83906	0.345766
$515$	−65.0499	−2.86644
$516$	0	0
$517$	3.66916	0.161369
$518$	−53.9398	−2.36998
$519$	0	0
$520$	54.0559	2.37051
$521$	0.718142	0.0314624	0.0157312	−	0.999876i	$-0.494992\pi$
$521$	0.718142	0.0314624	0.0157312	+	0.999876i	$0.494992\pi$
$522$	0	0
$523$	6.65017	0.290792	0.145396	−	0.989374i	$-0.453554\pi$
$523$	6.65017	0.290792	0.145396	+	0.989374i	$0.453554\pi$
$524$	0.995592	0.0434926
$525$	0	0
$526$	−10.1543	−0.442749
$527$	−17.5753	−0.765591
$528$	0	0
$529$	−12.3489	−0.536911
$530$	−34.3014	−1.48996
$531$	0	0
$532$	−2.35208	−0.101976
$533$	9.00519	0.390058
$534$	0	0
$535$	−54.9320	−2.37492
$536$	−12.5342	−0.541396
$537$	0	0
$538$	−23.1626	−0.998612
$539$	3.63034	0.156370
$540$	0	0
$541$	−14.3220	−0.615750	−0.307875	−	0.951427i	$-0.599618\pi$
$541$	−14.3220	−0.615750	−0.307875	+	0.951427i	$0.599618\pi$
$542$	17.1808	0.737978
$543$	0	0
$544$	1.18106	0.0506374
$545$	−48.2459	−2.06663
$546$	0	0
$547$	−6.09736	−0.260704	−0.130352	−	0.991468i	$-0.541611\pi$
$547$	−6.09736	−0.260704	−0.130352	+	0.991468i	$0.541611\pi$
$548$	0.698688	0.0298465
$549$	0	0
$550$	14.0173	0.597702
$551$	57.0121	2.42880
$552$	0	0
$553$	20.5295	0.873005
$554$	−26.4828	−1.12514
$555$	0	0
$556$	1.48177	0.0628412
$557$	−33.7124	−1.42844	−0.714219	−	0.699922i	$-0.753218\pi$
$557$	−33.7124	−1.42844	−0.714219	+	0.699922i	$0.753218\pi$
$558$	0	0
$559$	9.41512	0.398217
$560$	52.6079	2.22309
$561$	0	0
$562$	−15.5185	−0.654610
$563$	17.1856	0.724285	0.362142	−	0.932123i	$-0.382045\pi$
$563$	17.1856	0.724285	0.362142	+	0.932123i	$0.382045\pi$
$564$	0	0
$565$	5.54044	0.233088
$566$	−3.74404	−0.157374
$567$	0	0
$568$	−15.2059	−0.638023
$569$	30.9151	1.29603	0.648015	−	0.761628i	$-0.275599\pi$
$569$	30.9151	1.29603	0.648015	+	0.761628i	$0.275599\pi$
$570$	0	0
$571$	−13.0403	−0.545718	−0.272859	−	0.962054i	$-0.587969\pi$
$571$	−13.0403	−0.545718	−0.272859	+	0.962054i	$0.587969\pi$
$572$	0.594436	0.0248546
$573$	0	0
$574$	8.28466	0.345795
$575$	31.4538	1.31171
$576$	0	0
$577$	−14.3202	−0.596157	−0.298078	−	0.954541i	$-0.596346\pi$
$577$	−14.3202	−0.596157	−0.298078	+	0.954541i	$0.596346\pi$
$578$	−19.9465	−0.829667
$579$	0	0
$580$	4.02139	0.166979
$581$	9.19067	0.381293
$582$	0	0
$583$	6.16433	0.255300
$584$	21.6969	0.897824
$585$	0	0
$586$	40.7480	1.68329
$587$	32.3428	1.33493	0.667466	−	0.744640i	$-0.267379\pi$
$587$	32.3428	1.33493	0.667466	+	0.744640i	$0.267379\pi$
$588$	0	0
$589$	−60.6530	−2.49917
$590$	49.8098	2.05064
$591$	0	0
$592$	47.9719	1.97163
$593$	−7.57405	−0.311029	−0.155514	−	0.987834i	$-0.549704\pi$
$593$	−7.57405	−0.311029	−0.155514	+	0.987834i	$0.549704\pi$
$594$	0	0
$595$	22.6107	0.926948
$596$	0.772687	0.0316505
$597$	0	0
$598$	24.4661	1.00049
$599$	37.0706	1.51466	0.757332	−	0.653030i	$-0.226502\pi$
$599$	37.0706	1.51466	0.757332	+	0.653030i	$0.226502\pi$
$600$	0	0
$601$	−34.1013	−1.39102	−0.695511	−	0.718515i	$-0.744822\pi$
$601$	−34.1013	−1.39102	−0.695511	+	0.718515i	$0.744822\pi$
$602$	8.66178	0.353028
$603$	0	0
$604$	0.916149	0.0372776
$605$	−3.82594	−0.155546
$606$	0	0
$607$	26.0356	1.05675	0.528375	−	0.849011i	$-0.322801\pi$
$607$	26.0356	1.05675	0.528375	+	0.849011i	$0.322801\pi$
$608$	4.07588	0.165299
$609$	0	0
$610$	−5.56450	−0.225300
$611$	18.9124	0.765114
$612$	0	0
$613$	30.6659	1.23859	0.619293	−	0.785160i	$-0.287419\pi$
$613$	30.6659	1.23859	0.619293	+	0.785160i	$0.287419\pi$
$614$	−29.2358	−1.17986
$615$	0	0
$616$	−8.93714	−0.360088
$617$	−15.6936	−0.631801	−0.315901	−	0.948792i	$-0.602307\pi$
$617$	−15.6936	−0.631801	−0.315901	+	0.948792i	$0.602307\pi$
$618$	0	0
$619$	15.1794	0.610111	0.305055	−	0.952335i	$-0.401325\pi$
$619$	15.1794	0.610111	0.305055	+	0.952335i	$0.401325\pi$
$620$	−4.27820	−0.171817
$621$	0	0
$622$	16.6330	0.666924
$623$	−19.3143	−0.773812
$624$	0	0
$625$	19.6979	0.787917
$626$	32.5117	1.29943
$627$	0	0
$628$	−2.13392	−0.0851527
$629$	20.6182	0.822100
$630$	0	0
$631$	31.6400	1.25957	0.629784	−	0.776771i	$-0.283144\pi$
$631$	31.6400	1.25957	0.629784	+	0.776771i	$0.283144\pi$
$632$	−17.2596	−0.686550
$633$	0	0
$634$	−11.5301	−0.457918
$635$	−6.63559	−0.263326
$636$	0	0
$637$	18.7123	0.741409
$638$	−13.2557	−0.524797
$639$	0	0
$640$	−46.6474	−1.84390
$641$	−5.81111	−0.229525	−0.114762	−	0.993393i	$-0.536611\pi$
$641$	−5.81111	−0.229525	−0.114762	+	0.993393i	$0.536611\pi$
$642$	0	0
$643$	−49.2787	−1.94336	−0.971680	−	0.236300i	$-0.924065\pi$
$643$	−49.2787	−1.94336	−0.971680	+	0.236300i	$0.924065\pi$
$644$	1.22714	0.0483561
$645$	0	0
$646$	16.4909	0.648827
$647$	−0.304638	−0.0119766	−0.00598828	−	0.999982i	$-0.501906\pi$
$647$	−0.304638	−0.0119766	−0.00598828	+	0.999982i	$0.501906\pi$
$648$	0	0
$649$	−8.95135	−0.351371
$650$	72.2514	2.83393
$651$	0	0
$652$	1.20897	0.0473471
$653$	−3.17370	−0.124196	−0.0620981	−	0.998070i	$-0.519779\pi$
$653$	−3.17370	−0.124196	−0.0620981	+	0.998070i	$0.519779\pi$
$654$	0	0
$655$	−33.0289	−1.29055
$656$	−7.36805	−0.287674
$657$	0	0
$658$	17.3992	0.678290
$659$	−33.4416	−1.30270	−0.651349	−	0.758778i	$-0.725797\pi$
$659$	−33.4416	−1.30270	−0.651349	+	0.758778i	$0.725797\pi$
$660$	0	0
$661$	−18.6941	−0.727118	−0.363559	−	0.931571i	$-0.618438\pi$
$661$	−18.6941	−0.727118	−0.363559	+	0.931571i	$0.618438\pi$
$662$	7.25991	0.282165
$663$	0	0
$664$	−7.72678	−0.299857
$665$	78.0306	3.02590
$666$	0	0
$667$	−29.7447	−1.15172
$668$	−0.656262	−0.0253916
$669$	0	0
$670$	−25.4448	−0.983018
$671$	1.00000	0.0386046
$672$	0	0
$673$	8.20391	0.316237	0.158119	−	0.987420i	$-0.449457\pi$
$673$	8.20391	0.316237	0.158119	+	0.987420i	$0.449457\pi$
$674$	3.19351	0.123010
$675$	0	0
$676$	1.56475	0.0601826
$677$	31.6490	1.21637	0.608185	−	0.793795i	$-0.291898\pi$
$677$	31.6490	1.21637	0.608185	+	0.793795i	$0.291898\pi$
$678$	0	0
$679$	33.6476	1.29128
$680$	−19.0093	−0.728972
$681$	0	0
$682$	14.1022	0.540002
$683$	34.5524	1.32211	0.661056	−	0.750337i	$-0.270109\pi$
$683$	34.5524	1.32211	0.661056	+	0.750337i	$0.270109\pi$
$684$	0	0
$685$	−23.1791	−0.885627
$686$	−15.9790	−0.610080
$687$	0	0
$688$	−7.70345	−0.293691
$689$	31.7736	1.21048
$690$	0	0
$691$	−29.6034	−1.12617	−0.563083	−	0.826401i	$-0.690385\pi$
$691$	−29.6034	−1.12617	−0.563083	+	0.826401i	$0.690385\pi$
$692$	−0.318904	−0.0121229
$693$	0	0
$694$	41.0513	1.55829
$695$	−49.1580	−1.86467
$696$	0	0
$697$	−3.16676	−0.119950
$698$	−11.5012	−0.435327
$699$	0	0
$700$	3.62389	0.136970
$701$	−22.7553	−0.859455	−0.429728	−	0.902959i	$-0.641390\pi$
$701$	−22.7553	−0.859455	−0.429728	+	0.902959i	$0.641390\pi$
$702$	0	0
$703$	71.1542	2.68363
$704$	7.48703	0.282178
$705$	0	0
$706$	5.58710	0.210273
$707$	28.8065	1.08338
$708$	0	0
$709$	2.14817	0.0806762	0.0403381	−	0.999186i	$-0.487157\pi$
$709$	2.14817	0.0806762	0.0403381	+	0.999186i	$0.487157\pi$
$710$	−30.8683	−1.15846
$711$	0	0
$712$	16.2379	0.608543
$713$	31.6443	1.18509
$714$	0	0
$715$	−19.7205	−0.737505
$716$	−0.724043	−0.0270587
$717$	0	0
$718$	23.4138	0.873794
$719$	−21.9828	−0.819820	−0.409910	−	0.912126i	$-0.634440\pi$
$719$	−21.9828	−0.819820	−0.409910	+	0.912126i	$0.634440\pi$
$720$	0	0
$721$	55.4348	2.06450
$722$	29.2770	1.08958
$723$	0	0
$724$	0.562790	0.0209159
$725$	−87.8396	−3.26228
$726$	0	0
$727$	48.6664	1.80494	0.902468	−	0.430757i	$-0.141753\pi$
$727$	48.6664	1.80494	0.902468	+	0.430757i	$0.141753\pi$
$728$	−46.0658	−1.70731
$729$	0	0
$730$	44.0452	1.63019
$731$	−3.31092	−0.122459
$732$	0	0
$733$	5.69436	0.210326	0.105163	−	0.994455i	$-0.466464\pi$
$733$	5.69436	0.210326	0.105163	+	0.994455i	$0.466464\pi$
$734$	37.9500	1.40076
$735$	0	0
$736$	−2.12649	−0.0783836
$737$	4.57270	0.168438
$738$	0	0
$739$	33.6899	1.23930	0.619652	−	0.784877i	$-0.287274\pi$
$739$	33.6899	1.23930	0.619652	+	0.784877i	$0.287274\pi$
$740$	5.01891	0.184499
$741$	0	0
$742$	29.2313	1.07311
$743$	−25.4613	−0.934083	−0.467042	−	0.884235i	$-0.654680\pi$
$743$	−25.4613	−0.934083	−0.467042	+	0.884235i	$0.654680\pi$
$744$	0	0
$745$	−25.6340	−0.939157
$746$	11.1173	0.407033
$747$	0	0
$748$	−0.209039	−0.00764323
$749$	46.8124	1.71049
$750$	0	0
$751$	11.6876	0.426485	0.213243	−	0.976999i	$-0.431598\pi$
$751$	11.6876	0.426485	0.213243	+	0.976999i	$0.431598\pi$
$752$	−15.4741	−0.564283
$753$	0	0
$754$	−68.3254	−2.48827
$755$	−30.3934	−1.10613
$756$	0	0
$757$	−32.0082	−1.16336	−0.581678	−	0.813419i	$-0.697604\pi$
$757$	−32.0082	−1.16336	−0.581678	+	0.813419i	$0.697604\pi$
$758$	−13.2945	−0.482878
$759$	0	0
$760$	−65.6019	−2.37963
$761$	−36.3035	−1.31600	−0.658001	−	0.753017i	$-0.728598\pi$
$761$	−36.3035	−1.31600	−0.658001	+	0.753017i	$0.728598\pi$
$762$	0	0
$763$	41.1147	1.48845
$764$	1.25260	0.0453175
$765$	0	0
$766$	24.5227	0.886042
$767$	−46.1391	−1.66599
$768$	0	0
$769$	−30.3964	−1.09612	−0.548062	−	0.836438i	$-0.684634\pi$
$769$	−30.3964	−1.09612	−0.548062	+	0.836438i	$0.684634\pi$
$770$	−18.1426	−0.653814
$771$	0	0
$772$	1.97988	0.0712574
$773$	8.45088	0.303957	0.151979	−	0.988384i	$-0.451436\pi$
$773$	8.45088	0.303957	0.151979	+	0.988384i	$0.451436\pi$
$774$	0	0
$775$	93.4493	3.35680
$776$	−28.2882	−1.01549
$777$	0	0
$778$	−18.8775	−0.676791
$779$	−10.9286	−0.391559
$780$	0	0
$781$	5.54735	0.198500
$782$	−8.60375	−0.307669
$783$	0	0
$784$	−15.3104	−0.546800
$785$	70.7931	2.52672
$786$	0	0
$787$	−24.9875	−0.890707	−0.445354	−	0.895355i	$-0.646922\pi$
$787$	−24.9875	−0.890707	−0.445354	+	0.895355i	$0.646922\pi$
$788$	0.145259	0.00517464
$789$	0	0
$790$	−35.0374	−1.24658
$791$	−4.72151	−0.167877
$792$	0	0
$793$	5.15443	0.183039
$794$	−1.32881	−0.0471576
$795$	0	0
$796$	0.668895	0.0237083
$797$	18.0752	0.640257	0.320128	−	0.947374i	$-0.396274\pi$
$797$	18.0752	0.640257	0.320128	+	0.947374i	$0.396274\pi$
$798$	0	0
$799$	−6.65073	−0.235286
$800$	−6.27979	−0.222024
$801$	0	0
$802$	−52.6754	−1.86003
$803$	−7.91540	−0.279328
$804$	0	0
$805$	−40.7106	−1.43486
$806$	72.6889	2.56036
$807$	0	0
$808$	−24.2182	−0.851993
$809$	−30.1014	−1.05831	−0.529154	−	0.848526i	$-0.677491\pi$
$809$	−30.1014	−1.05831	−0.529154	+	0.848526i	$0.677491\pi$
$810$	0	0
$811$	−12.8525	−0.451311	−0.225656	−	0.974207i	$-0.572452\pi$
$811$	−12.8525	−0.451311	−0.225656	+	0.974207i	$0.572452\pi$
$812$	−3.42698	−0.120263
$813$	0	0
$814$	−16.5438	−0.579860
$815$	−40.1079	−1.40492
$816$	0	0
$817$	−11.4261	−0.399749
$818$	5.62853	0.196797
$819$	0	0
$820$	−0.770859	−0.0269196
$821$	−32.8927	−1.14796	−0.573981	−	0.818869i	$-0.694602\pi$
$821$	−32.8927	−1.14796	−0.573981	+	0.818869i	$0.694602\pi$
$822$	0	0
$823$	32.1184	1.11958	0.559789	−	0.828635i	$-0.310882\pi$
$823$	32.1184	1.11958	0.559789	+	0.828635i	$0.310882\pi$
$824$	−46.6052	−1.62357
$825$	0	0
$826$	−42.4474	−1.47693
$827$	7.66041	0.266378	0.133189	−	0.991091i	$-0.457478\pi$
$827$	7.66041	0.266378	0.133189	+	0.991091i	$0.457478\pi$
$828$	0	0
$829$	−27.0956	−0.941068	−0.470534	−	0.882382i	$-0.655939\pi$
$829$	−27.0956	−0.941068	−0.470534	+	0.882382i	$0.655939\pi$
$830$	−15.6856	−0.544454
$831$	0	0
$832$	38.5914	1.33791
$833$	−6.58036	−0.227996
$834$	0	0
$835$	21.7716	0.753437
$836$	−0.721404	−0.0249503
$837$	0	0
$838$	−7.86199	−0.271588
$839$	−57.2281	−1.97573	−0.987866	−	0.155307i	$-0.950363\pi$
$839$	−57.2281	−1.97573	−0.987866	+	0.155307i	$0.950363\pi$
$840$	0	0
$841$	54.0666	1.86437
$842$	30.3636	1.04640
$843$	0	0
$844$	−1.79768	−0.0618788
$845$	−51.9108	−1.78578
$846$	0	0
$847$	3.26042	0.112029
$848$	−25.9971	−0.892745
$849$	0	0
$850$	−25.4079	−0.871483
$851$	−37.1230	−1.27256
$852$	0	0
$853$	2.91211	0.0997089	0.0498544	−	0.998756i	$-0.484124\pi$
$853$	2.91211	0.0997089	0.0498544	+	0.998756i	$0.484124\pi$
$854$	4.74201	0.162268
$855$	0	0
$856$	−39.3561	−1.34516
$857$	11.9479	0.408133	0.204066	−	0.978957i	$-0.434584\pi$
$857$	11.9479	0.408133	0.204066	+	0.978957i	$0.434584\pi$
$858$	0	0
$859$	−28.3977	−0.968917	−0.484459	−	0.874814i	$-0.660983\pi$
$859$	−28.3977	−0.968917	−0.484459	+	0.874814i	$0.660983\pi$
$860$	−0.805949	−0.0274826
$861$	0	0
$862$	35.7379	1.21724
$863$	27.3821	0.932097	0.466049	−	0.884759i	$-0.345677\pi$
$863$	27.3821	0.932097	0.466049	+	0.884759i	$0.345677\pi$
$864$	0	0
$865$	10.5797	0.359721
$866$	−57.3038	−1.94726
$867$	0	0
$868$	3.64584	0.123748
$869$	6.29660	0.213597
$870$	0	0
$871$	23.5697	0.798627
$872$	−34.5659	−1.17055
$873$	0	0
$874$	−29.6919	−1.00434
$875$	−57.8524	−1.95577
$876$	0	0
$877$	−11.0166	−0.372004	−0.186002	−	0.982549i	$-0.559553\pi$
$877$	−11.0166	−0.372004	−0.186002	+	0.982549i	$0.559553\pi$
$878$	12.9051	0.435526
$879$	0	0
$880$	16.1353	0.543921
$881$	26.1108	0.879695	0.439848	−	0.898072i	$-0.355032\pi$
$881$	26.1108	0.879695	0.439848	+	0.898072i	$0.355032\pi$
$882$	0	0
$883$	8.12745	0.273511	0.136755	−	0.990605i	$-0.456333\pi$
$883$	8.12745	0.273511	0.136755	+	0.990605i	$0.456333\pi$
$884$	−1.07748	−0.0362395
$885$	0	0
$886$	1.44646	0.0485946
$887$	−32.6845	−1.09744	−0.548720	−	0.836006i	$-0.684885\pi$
$887$	−32.6845	−1.09744	−0.548720	+	0.836006i	$0.684885\pi$
$888$	0	0
$889$	5.65478	0.189655
$890$	32.9634	1.10494
$891$	0	0
$892$	0.843801	0.0282525
$893$	−22.9520	−0.768058
$894$	0	0
$895$	24.0202	0.802907
$896$	39.7524	1.32803
$897$	0	0
$898$	11.9567	0.399000
$899$	−88.3715	−2.94736
$900$	0	0
$901$	−11.1735	−0.372243
$902$	2.54098	0.0846054
$903$	0	0
$904$	3.96946	0.132022
$905$	−18.6707	−0.620634
$906$	0	0
$907$	−37.3623	−1.24060	−0.620298	−	0.784366i	$-0.712988\pi$
$907$	−37.3623	−1.24060	−0.620298	+	0.784366i	$0.712988\pi$
$908$	2.64532	0.0877880
$909$	0	0
$910$	−93.5148	−3.09998
$911$	0.782097	0.0259120	0.0129560	−	0.999916i	$-0.495876\pi$
$911$	0.782097	0.0259120	0.0129560	+	0.999916i	$0.495876\pi$
$912$	0	0
$913$	2.81886	0.0932907
$914$	−57.4984	−1.90188
$915$	0	0
$916$	1.55347	0.0513281
$917$	28.1469	0.929491
$918$	0	0
$919$	3.64261	0.120159	0.0600793	−	0.998194i	$-0.480865\pi$
$919$	3.64261	0.120159	0.0600793	+	0.998194i	$0.480865\pi$
$920$	34.2262	1.12840
$921$	0	0
$922$	−4.87994	−0.160712
$923$	28.5934	0.941164
$924$	0	0
$925$	−109.629	−3.60457
$926$	5.76969	0.189604
$927$	0	0
$928$	5.93856	0.194943
$929$	6.85276	0.224832	0.112416	−	0.993661i	$-0.464141\pi$
$929$	6.85276	0.224832	0.112416	+	0.993661i	$0.464141\pi$
$930$	0	0
$931$	−22.7091	−0.744262
$932$	0.767297	0.0251337
$933$	0	0
$934$	39.9795	1.30817
$935$	6.93490	0.226796
$936$	0	0
$937$	−24.9476	−0.815002	−0.407501	−	0.913205i	$-0.633600\pi$
$937$	−24.9476	−0.815002	−0.407501	+	0.913205i	$0.633600\pi$
$938$	21.6838	0.708000
$939$	0	0
$940$	−1.61893	−0.0528038
$941$	47.1630	1.53747	0.768735	−	0.639568i	$-0.220887\pi$
$941$	47.1630	1.53747	0.768735	+	0.639568i	$0.220887\pi$
$942$	0	0
$943$	5.70176	0.185675
$944$	37.7510	1.22869
$945$	0	0
$946$	2.65665	0.0863750
$947$	49.6803	1.61439	0.807196	−	0.590283i	$-0.200984\pi$
$947$	49.6803	1.61439	0.807196	+	0.590283i	$0.200984\pi$
$948$	0	0
$949$	−40.7993	−1.32440
$950$	−87.6838	−2.84484
$951$	0	0
$952$	16.1995	0.525028
$953$	31.7621	1.02887	0.514437	−	0.857528i	$-0.328001\pi$
$953$	31.7621	1.02887	0.514437	+	0.857528i	$0.328001\pi$
$954$	0	0
$955$	−41.5552	−1.34469
$956$	−0.346074	−0.0111928
$957$	0	0
$958$	17.3683	0.561146
$959$	19.7529	0.637856
$960$	0	0
$961$	63.0152	2.03275
$962$	−85.2739	−2.74934
$963$	0	0
$964$	−2.24394	−0.0722725
$965$	−65.6828	−2.11440
$966$	0	0
$967$	−9.66371	−0.310764	−0.155382	−	0.987854i	$-0.549661\pi$
$967$	−9.66371	−0.310764	−0.155382	+	0.987854i	$0.549661\pi$
$968$	−2.74110	−0.0881023
$969$	0	0
$970$	−57.4257	−1.84383
$971$	−49.0323	−1.57352	−0.786760	−	0.617259i	$-0.788243\pi$
$971$	−49.0323	−1.57352	−0.786760	+	0.617259i	$0.788243\pi$
$972$	0	0
$973$	41.8919	1.34299
$974$	50.2921	1.61146
$975$	0	0
$976$	−4.21735	−0.134994
$977$	−26.3062	−0.841610	−0.420805	−	0.907151i	$-0.638252\pi$
$977$	−26.3062	−0.841610	−0.420805	+	0.907151i	$0.638252\pi$
$978$	0	0
$979$	−5.92388	−0.189328
$980$	−1.60180	−0.0511677
$981$	0	0
$982$	33.3105	1.06298
$983$	10.5887	0.337728	0.168864	−	0.985639i	$-0.445990\pi$
$983$	10.5887	0.337728	0.168864	+	0.985639i	$0.445990\pi$
$984$	0	0
$985$	−4.81899	−0.153546
$986$	24.0273	0.765185
$987$	0	0
$988$	−3.71842	−0.118299
$989$	5.96131	0.189559
$990$	0	0
$991$	−14.3949	−0.457270	−0.228635	−	0.973512i	$-0.573426\pi$
$991$	−14.3949	−0.457270	−0.228635	+	0.973512i	$0.573426\pi$
$992$	−6.31782	−0.200591
$993$	0	0
$994$	26.3056	0.834362
$995$	−22.1907	−0.703492
$996$	0	0
$997$	−56.8098	−1.79919	−0.899593	−	0.436730i	$-0.856137\pi$
$997$	−56.8098	−1.79919	−0.899593	+	0.436730i	$0.856137\pi$
$998$	−34.5619	−1.09404
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.l.1.15		21
3.2	odd	2		671.2.a.d.1.7	✓	21
33.32	even	2		7381.2.a.j.1.15		21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
671.2.a.d.1.7	✓	21	3.2	odd	2
6039.2.a.l.1.15		21	1.1	even	1	trivial
7381.2.a.j.1.15		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)

\(f(q)\)	\(=\)	\(q+1.45442 q^{2} +0.115325 q^{4} -3.82594 q^{5} +3.26042 q^{7} -2.74110 q^{8} +O(q^{10})\)	\(q+1.45442 q^{2} +0.115325 q^{4} -3.82594 q^{5} +3.26042 q^{7} -2.74110 q^{8} -5.56450 q^{10} +1.00000 q^{11} +5.15443 q^{13} +4.74201 q^{14} -4.21735 q^{16} -1.81260 q^{17} -6.25538 q^{19} -0.441227 q^{20} +1.45442 q^{22} +3.26360 q^{23} +9.63778 q^{25} +7.49668 q^{26} +0.376009 q^{28} -9.11409 q^{29} +9.69614 q^{31} -0.651580 q^{32} -2.63628 q^{34} -12.4742 q^{35} -11.3749 q^{37} -9.09792 q^{38} +10.4873 q^{40} +1.74708 q^{41} +1.82661 q^{43} +0.115325 q^{44} +4.74662 q^{46} +3.66916 q^{47} +3.63034 q^{49} +14.0173 q^{50} +0.594436 q^{52} +6.16433 q^{53} -3.82594 q^{55} -8.93714 q^{56} -13.2557 q^{58} -8.95135 q^{59} +1.00000 q^{61} +14.1022 q^{62} +7.48703 q^{64} -19.7205 q^{65} +4.57270 q^{67} -0.209039 q^{68} -18.1426 q^{70} +5.54735 q^{71} -7.91540 q^{73} -16.5438 q^{74} -0.721404 q^{76} +3.26042 q^{77} +6.29660 q^{79} +16.1353 q^{80} +2.54098 q^{82} +2.81886 q^{83} +6.93490 q^{85} +2.65665 q^{86} -2.74110 q^{88} -5.92388 q^{89} +16.8056 q^{91} +0.376375 q^{92} +5.33648 q^{94} +23.9327 q^{95} +10.3200 q^{97} +5.28002 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q + 32 q^{4} - 7 q^{5} + 5 q^{7} + 6 q^{8}+O(q^{10}) \) 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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