LMFDB - Embedded newform 6020.2.a.j.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6020,2,Mod(1,6020)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6020, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6020.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6020 = 2^{2} \cdot 5 \cdot 7 \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6020.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:	$48.0699420168$
Analytic rank:	$0$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - x^{12} - 28 x^{11} + 26 x^{10} + 286 x^{9} - 235 x^{8} - 1298 x^{7} + 895 x^{6} + 2571 x^{5} + \cdots - 216 $ x^13 - x^12 - 28x^11 + 26x^10 + 286x^9 - 235x^8 - 1298x^7 + 895x^6 + 2571x^5 - 1548x^4 - 1833x^3 + 1119x^2 + 342*x - 216
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$-3.05719$ of defining polynomial
Character	$\chi$	$=$	6020.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.05719 q^{3} -1.00000 q^{5} -1.00000 q^{7} +6.34642 q^{9} +O(q^{10})$	$q+3.05719 q^{3} -1.00000 q^{5} -1.00000 q^{7} +6.34642 q^{9} +3.45682 q^{11} +4.00012 q^{13} -3.05719 q^{15} +7.43356 q^{17} +3.50101 q^{19} -3.05719 q^{21} -4.81806 q^{23} +1.00000 q^{25} +10.2307 q^{27} -0.928454 q^{29} +2.35688 q^{31} +10.5682 q^{33} +1.00000 q^{35} -8.56397 q^{37} +12.2291 q^{39} -9.19942 q^{41} -1.00000 q^{43} -6.34642 q^{45} -6.81468 q^{47} +1.00000 q^{49} +22.7258 q^{51} +6.09556 q^{53} -3.45682 q^{55} +10.7033 q^{57} +3.86197 q^{59} +4.74079 q^{61} -6.34642 q^{63} -4.00012 q^{65} -5.67054 q^{67} -14.7297 q^{69} -9.38700 q^{71} +7.32410 q^{73} +3.05719 q^{75} -3.45682 q^{77} -1.69899 q^{79} +12.2378 q^{81} +4.92570 q^{83} -7.43356 q^{85} -2.83846 q^{87} +11.0437 q^{89} -4.00012 q^{91} +7.20542 q^{93} -3.50101 q^{95} +8.78581 q^{97} +21.9384 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - q^{3} - 13 q^{5} - 13 q^{7} + 18 q^{9}+O(q^{10}) $ 13 * q - q^3 - 13 * q^5 - 13 * q^7 + 18 * q^9	$ 13 q - q^{3} - 13 q^{5} - 13 q^{7} + 18 q^{9} + 6 q^{11} + q^{13} + q^{15} - 6 q^{17} + 2 q^{19} + q^{21} - 6 q^{23} + 13 q^{25} - q^{27} + 19 q^{29} - 24 q^{31} + 17 q^{33} + 13 q^{35} + 15 q^{39} + 4 q^{41} - 13 q^{43} - 18 q^{45} - 3 q^{47} + 13 q^{49} + 7 q^{51} - 5 q^{53} - 6 q^{55} + 6 q^{57} - 6 q^{59} + 3 q^{61} - 18 q^{63} - q^{65} - 2 q^{67} + 20 q^{69} + 18 q^{71} + 14 q^{73} - q^{75} - 6 q^{77} + 12 q^{79} + 37 q^{81} + 2 q^{83} + 6 q^{85} - 2 q^{87} + 17 q^{89} - q^{91} + 15 q^{93} - 2 q^{95} + 17 q^{97} + 80 q^{99}+O(q^{100}) $ 13 * q - q^3 - 13 * q^5 - 13 * q^7 + 18 * q^9 + 6 * q^11 + q^13 + q^15 - 6 * q^17 + 2 * q^19 + q^21 - 6 * q^23 + 13 * q^25 - q^27 + 19 * q^29 - 24 * q^31 + 17 * q^33 + 13 * q^35 + 15 * q^39 + 4 * q^41 - 13 * q^43 - 18 * q^45 - 3 * q^47 + 13 * q^49 + 7 * q^51 - 5 * q^53 - 6 * q^55 + 6 * q^57 - 6 * q^59 + 3 * q^61 - 18 * q^63 - q^65 - 2 * q^67 + 20 * q^69 + 18 * q^71 + 14 * q^73 - q^75 - 6 * q^77 + 12 * q^79 + 37 * q^81 + 2 * q^83 + 6 * q^85 - 2 * q^87 + 17 * q^89 - q^91 + 15 * q^93 - 2 * q^95 + 17 * q^97 + 80 * q^99

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	0
$2$	0	0
$3$	3.05719	1.76507	0.882535	−	0.470246i	$-0.155835\pi$
$3$	3.05719	1.76507	0.882535	+	0.470246i	$0.155835\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	6.34642	2.11547
$10$	0	0
$11$	3.45682	1.04227	0.521135	−	0.853474i	$-0.325509\pi$
$11$	3.45682	1.04227	0.521135	+	0.853474i	$0.325509\pi$
$12$	0	0
$13$	4.00012	1.10943	0.554716	−	0.832040i	$-0.312827\pi$
$13$	4.00012	1.10943	0.554716	+	0.832040i	$0.312827\pi$
$14$	0	0
$15$	−3.05719	−0.789364
$16$	0	0
$17$	7.43356	1.80290	0.901451	−	0.432881i	$-0.142503\pi$
$17$	7.43356	1.80290	0.901451	+	0.432881i	$0.142503\pi$
$18$	0	0
$19$	3.50101	0.803188	0.401594	−	0.915818i	$-0.368456\pi$
$19$	3.50101	0.803188	0.401594	+	0.915818i	$0.368456\pi$
$20$	0	0
$21$	−3.05719	−0.667134
$22$	0	0
$23$	−4.81806	−1.00463	−0.502317	−	0.864683i	$-0.667519\pi$
$23$	−4.81806	−1.00463	−0.502317	+	0.864683i	$0.667519\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	10.2307	1.96889
$28$	0	0
$29$	−0.928454	−0.172410	−0.0862048	−	0.996277i	$-0.527474\pi$
$29$	−0.928454	−0.172410	−0.0862048	+	0.996277i	$0.527474\pi$
$30$	0	0
$31$	2.35688	0.423308	0.211654	−	0.977345i	$-0.432115\pi$
$31$	2.35688	0.423308	0.211654	+	0.977345i	$0.432115\pi$
$32$	0	0
$33$	10.5682	1.83968
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	−8.56397	−1.40791	−0.703954	−	0.710246i	$-0.748584\pi$
$37$	−8.56397	−1.40791	−0.703954	+	0.710246i	$0.748584\pi$
$38$	0	0
$39$	12.2291	1.95823
$40$	0	0
$41$	−9.19942	−1.43671	−0.718354	−	0.695678i	$-0.755104\pi$
$41$	−9.19942	−1.43671	−0.718354	+	0.695678i	$0.755104\pi$
$42$	0	0
$43$	−1.00000	−0.152499
$43$	−1.00000	−0.152499
$44$	0	0
$45$	−6.34642	−0.946069
$46$	0	0
$47$	−6.81468	−0.994023	−0.497012	−	0.867744i	$-0.665569\pi$
$47$	−6.81468	−0.994023	−0.497012	+	0.867744i	$0.665569\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	22.7258	3.18225
$52$	0	0
$53$	6.09556	0.837289	0.418645	−	0.908150i	$-0.362505\pi$
$53$	6.09556	0.837289	0.418645	+	0.908150i	$0.362505\pi$
$54$	0	0
$55$	−3.45682	−0.466118
$56$	0	0
$57$	10.7033	1.41768
$58$	0	0
$59$	3.86197	0.502786	0.251393	−	0.967885i	$-0.419111\pi$
$59$	3.86197	0.502786	0.251393	+	0.967885i	$0.419111\pi$
$60$	0	0
$61$	4.74079	0.606995	0.303498	−	0.952832i	$-0.401846\pi$
$61$	4.74079	0.606995	0.303498	+	0.952832i	$0.401846\pi$
$62$	0	0
$63$	−6.34642	−0.799574
$64$	0	0
$65$	−4.00012	−0.496153
$66$	0	0
$67$	−5.67054	−0.692767	−0.346383	−	0.938093i	$-0.612590\pi$
$67$	−5.67054	−0.692767	−0.346383	+	0.938093i	$0.612590\pi$
$68$	0	0
$69$	−14.7297	−1.77325
$70$	0	0
$71$	−9.38700	−1.11403	−0.557016	−	0.830502i	$-0.688054\pi$
$71$	−9.38700	−1.11403	−0.557016	+	0.830502i	$0.688054\pi$
$72$	0	0
$73$	7.32410	0.857221	0.428610	−	0.903489i	$-0.359003\pi$
$73$	7.32410	0.857221	0.428610	+	0.903489i	$0.359003\pi$
$74$	0	0
$75$	3.05719	0.353014
$76$	0	0
$77$	−3.45682	−0.393941
$78$	0	0
$79$	−1.69899	−0.191151	−0.0955755	−	0.995422i	$-0.530469\pi$
$79$	−1.69899	−0.191151	−0.0955755	+	0.995422i	$0.530469\pi$
$80$	0	0
$81$	12.2378	1.35976
$82$	0	0
$83$	4.92570	0.540665	0.270333	−	0.962767i	$-0.412866\pi$
$83$	4.92570	0.540665	0.270333	+	0.962767i	$0.412866\pi$
$84$	0	0
$85$	−7.43356	−0.806283
$86$	0	0
$87$	−2.83846	−0.304315
$88$	0	0
$89$	11.0437	1.17063	0.585317	−	0.810805i	$-0.300970\pi$
$89$	11.0437	1.17063	0.585317	+	0.810805i	$0.300970\pi$
$90$	0	0
$91$	−4.00012	−0.419326
$92$	0	0
$93$	7.20542	0.747168
$94$	0	0
$95$	−3.50101	−0.359196
$96$	0	0
$97$	8.78581	0.892064	0.446032	−	0.895017i	$-0.352837\pi$
$97$	8.78581	0.892064	0.446032	+	0.895017i	$0.352837\pi$
$98$	0	0
$99$	21.9384	2.20490
$100$	0	0
$101$	8.39648	0.835481	0.417740	−	0.908566i	$-0.362822\pi$
$101$	8.39648	0.835481	0.417740	+	0.908566i	$0.362822\pi$
$102$	0	0
$103$	7.90711	0.779111	0.389556	−	0.921003i	$-0.372629\pi$
$103$	7.90711	0.779111	0.389556	+	0.921003i	$0.372629\pi$
$104$	0	0
$105$	3.05719	0.298351
$106$	0	0
$107$	−16.1229	−1.55866	−0.779329	−	0.626615i	$-0.784440\pi$
$107$	−16.1229	−1.55866	−0.779329	+	0.626615i	$0.784440\pi$
$108$	0	0
$109$	0.171348	0.0164122	0.00820608	−	0.999966i	$-0.497388\pi$
$109$	0.171348	0.0164122	0.00820608	+	0.999966i	$0.497388\pi$
$110$	0	0
$111$	−26.1817	−2.48506
$112$	0	0
$113$	13.4199	1.26244	0.631221	−	0.775603i	$-0.282554\pi$
$113$	13.4199	1.26244	0.631221	+	0.775603i	$0.282554\pi$
$114$	0	0
$115$	4.81806	0.449286
$116$	0	0
$117$	25.3864	2.34698
$118$	0	0
$119$	−7.43356	−0.681433
$120$	0	0
$121$	0.949605	0.0863277
$122$	0	0
$123$	−28.1244	−2.53589
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−3.13609	−0.278283	−0.139141	−	0.990273i	$-0.544434\pi$
$127$	−3.13609	−0.278283	−0.139141	+	0.990273i	$0.544434\pi$
$128$	0	0
$129$	−3.05719	−0.269171
$130$	0	0
$131$	−1.70768	−0.149201	−0.0746004	−	0.997214i	$-0.523768\pi$
$131$	−1.70768	−0.149201	−0.0746004	+	0.997214i	$0.523768\pi$
$132$	0	0
$133$	−3.50101	−0.303576
$134$	0	0
$135$	−10.2307	−0.880515
$136$	0	0
$137$	−3.08969	−0.263970	−0.131985	−	0.991252i	$-0.542135\pi$
$137$	−3.08969	−0.263970	−0.131985	+	0.991252i	$0.542135\pi$
$138$	0	0
$139$	−9.38528	−0.796049	−0.398025	−	0.917375i	$-0.630304\pi$
$139$	−9.38528	−0.796049	−0.398025	+	0.917375i	$0.630304\pi$
$140$	0	0
$141$	−20.8338	−1.75452
$142$	0	0
$143$	13.8277	1.15633
$144$	0	0
$145$	0.928454	0.0771039
$146$	0	0
$147$	3.05719	0.252153
$148$	0	0
$149$	−2.39557	−0.196253	−0.0981263	−	0.995174i	$-0.531285\pi$
$149$	−2.39557	−0.196253	−0.0981263	+	0.995174i	$0.531285\pi$
$150$	0	0
$151$	−10.3862	−0.845221	−0.422610	−	0.906311i	$-0.638886\pi$
$151$	−10.3862	−0.845221	−0.422610	+	0.906311i	$0.638886\pi$
$152$	0	0
$153$	47.1765	3.81399
$154$	0	0
$155$	−2.35688	−0.189309
$156$	0	0
$157$	9.17653	0.732366	0.366183	−	0.930543i	$-0.380664\pi$
$157$	9.17653	0.732366	0.366183	+	0.930543i	$0.380664\pi$
$158$	0	0
$159$	18.6353	1.47787
$160$	0	0
$161$	4.81806	0.379716
$162$	0	0
$163$	−0.911780	−0.0714161	−0.0357081	−	0.999362i	$-0.511369\pi$
$163$	−0.911780	−0.0714161	−0.0357081	+	0.999362i	$0.511369\pi$
$164$	0	0
$165$	−10.5682	−0.822730
$166$	0	0
$167$	−5.98961	−0.463490	−0.231745	−	0.972777i	$-0.574444\pi$
$167$	−5.98961	−0.463490	−0.231745	+	0.972777i	$0.574444\pi$
$168$	0	0
$169$	3.00093	0.230841
$170$	0	0
$171$	22.2189	1.69912
$172$	0	0
$173$	−24.0515	−1.82860	−0.914302	−	0.405033i	$-0.867260\pi$
$173$	−24.0515	−1.82860	−0.914302	+	0.405033i	$0.867260\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	11.8068	0.887452
$178$	0	0
$179$	18.2473	1.36387	0.681933	−	0.731414i	$-0.261139\pi$
$179$	18.2473	1.36387	0.681933	+	0.731414i	$0.261139\pi$
$180$	0	0
$181$	16.1994	1.20409	0.602047	−	0.798461i	$-0.294352\pi$
$181$	16.1994	1.20409	0.602047	+	0.798461i	$0.294352\pi$
$182$	0	0
$183$	14.4935	1.07139
$184$	0	0
$185$	8.56397	0.629635
$186$	0	0
$187$	25.6965	1.87911
$188$	0	0
$189$	−10.2307	−0.744171
$190$	0	0
$191$	16.0637	1.16233	0.581165	−	0.813786i	$-0.302597\pi$
$191$	16.0637	1.16233	0.581165	+	0.813786i	$0.302597\pi$
$192$	0	0
$193$	−22.5384	−1.62235	−0.811175	−	0.584803i	$-0.801172\pi$
$193$	−22.5384	−1.62235	−0.811175	+	0.584803i	$0.801172\pi$
$194$	0	0
$195$	−12.2291	−0.875746
$196$	0	0
$197$	−4.71632	−0.336024	−0.168012	−	0.985785i	$-0.553735\pi$
$197$	−4.71632	−0.336024	−0.168012	+	0.985785i	$0.553735\pi$
$198$	0	0
$199$	27.4655	1.94698	0.973489	−	0.228732i	$-0.0734581\pi$
$199$	27.4655	1.94698	0.973489	+	0.228732i	$0.0734581\pi$
$200$	0	0
$201$	−17.3359	−1.22278
$202$	0	0
$203$	0.928454	0.0651647
$204$	0	0
$205$	9.19942	0.642515
$206$	0	0
$207$	−30.5774	−2.12528
$208$	0	0
$209$	12.1024	0.837139
$210$	0	0
$211$	5.49214	0.378095	0.189047	−	0.981968i	$-0.439460\pi$
$211$	5.49214	0.378095	0.189047	+	0.981968i	$0.439460\pi$
$212$	0	0
$213$	−28.6978	−1.96634
$214$	0	0
$215$	1.00000	0.0681994
$216$	0	0
$217$	−2.35688	−0.159995
$218$	0	0
$219$	22.3912	1.51306
$220$	0	0
$221$	29.7351	2.00020
$222$	0	0
$223$	−16.6677	−1.11615	−0.558075	−	0.829791i	$-0.688460\pi$
$223$	−16.6677	−1.11615	−0.558075	+	0.829791i	$0.688460\pi$
$224$	0	0
$225$	6.34642	0.423095
$226$	0	0
$227$	21.1000	1.40046	0.700228	−	0.713920i	$-0.253082\pi$
$227$	21.1000	1.40046	0.700228	+	0.713920i	$0.253082\pi$
$228$	0	0
$229$	−13.4045	−0.885793	−0.442897	−	0.896573i	$-0.646049\pi$
$229$	−13.4045	−0.885793	−0.442897	+	0.896573i	$0.646049\pi$
$230$	0	0
$231$	−10.5682	−0.695334
$232$	0	0
$233$	−21.6264	−1.41679	−0.708396	−	0.705815i	$-0.750581\pi$
$233$	−21.6264	−1.41679	−0.708396	+	0.705815i	$0.750581\pi$
$234$	0	0
$235$	6.81468	0.444541
$236$	0	0
$237$	−5.19413	−0.337395
$238$	0	0
$239$	−3.86686	−0.250127	−0.125063	−	0.992149i	$-0.539913\pi$
$239$	−3.86686	−0.250127	−0.125063	+	0.992149i	$0.539913\pi$
$240$	0	0
$241$	−0.253533	−0.0163315	−0.00816575	−	0.999967i	$-0.502599\pi$
$241$	−0.253533	−0.0163315	−0.00816575	+	0.999967i	$0.502599\pi$
$242$	0	0
$243$	6.72139	0.431177
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	14.0045	0.891083
$248$	0	0
$249$	15.0588	0.954313
$250$	0	0
$251$	−26.3619	−1.66395	−0.831975	−	0.554813i	$-0.812790\pi$
$251$	−26.3619	−1.66395	−0.831975	+	0.554813i	$0.812790\pi$
$252$	0	0
$253$	−16.6552	−1.04710
$254$	0	0
$255$	−22.7258	−1.42315
$256$	0	0
$257$	4.56382	0.284683	0.142342	−	0.989818i	$-0.454537\pi$
$257$	4.56382	0.284683	0.142342	+	0.989818i	$0.454537\pi$
$258$	0	0
$259$	8.56397	0.532139
$260$	0	0
$261$	−5.89236	−0.364728
$262$	0	0
$263$	1.84102	0.113522	0.0567610	−	0.998388i	$-0.481923\pi$
$263$	1.84102	0.113522	0.0567610	+	0.998388i	$0.481923\pi$
$264$	0	0
$265$	−6.09556	−0.374447
$266$	0	0
$267$	33.7628	2.06625
$268$	0	0
$269$	5.17629	0.315604	0.157802	−	0.987471i	$-0.449559\pi$
$269$	5.17629	0.315604	0.157802	+	0.987471i	$0.449559\pi$
$270$	0	0
$271$	−1.83633	−0.111549	−0.0557744	−	0.998443i	$-0.517763\pi$
$271$	−1.83633	−0.111549	−0.0557744	+	0.998443i	$0.517763\pi$
$272$	0	0
$273$	−12.2291	−0.740140
$274$	0	0
$275$	3.45682	0.208454
$276$	0	0
$277$	30.0638	1.80636	0.903178	−	0.429265i	$-0.141227\pi$
$277$	30.0638	1.80636	0.903178	+	0.429265i	$0.141227\pi$
$278$	0	0
$279$	14.9577	0.895496
$280$	0	0
$281$	18.7409	1.11799	0.558995	−	0.829171i	$-0.311187\pi$
$281$	18.7409	1.11799	0.558995	+	0.829171i	$0.311187\pi$
$282$	0	0
$283$	2.10341	0.125035	0.0625175	−	0.998044i	$-0.480087\pi$
$283$	2.10341	0.125035	0.0625175	+	0.998044i	$0.480087\pi$
$284$	0	0
$285$	−10.7033	−0.634007
$286$	0	0
$287$	9.19942	0.543024
$288$	0	0
$289$	38.2578	2.25046
$290$	0	0
$291$	26.8599	1.57456
$292$	0	0
$293$	4.09856	0.239440	0.119720	−	0.992808i	$-0.461800\pi$
$293$	4.09856	0.239440	0.119720	+	0.992808i	$0.461800\pi$
$294$	0	0
$295$	−3.86197	−0.224853
$296$	0	0
$297$	35.3655	2.05212
$298$	0	0
$299$	−19.2728	−1.11457
$300$	0	0
$301$	1.00000	0.0576390
$302$	0	0
$303$	25.6696	1.47468
$304$	0	0
$305$	−4.74079	−0.271457
$306$	0	0
$307$	−9.19968	−0.525054	−0.262527	−	0.964925i	$-0.584556\pi$
$307$	−9.19968	−0.525054	−0.262527	+	0.964925i	$0.584556\pi$
$308$	0	0
$309$	24.1736	1.37519
$310$	0	0
$311$	9.18401	0.520778	0.260389	−	0.965504i	$-0.416149\pi$
$311$	9.18401	0.520778	0.260389	+	0.965504i	$0.416149\pi$
$312$	0	0
$313$	11.0316	0.623545	0.311773	−	0.950157i	$-0.399077\pi$
$313$	11.0316	0.623545	0.311773	+	0.950157i	$0.399077\pi$
$314$	0	0
$315$	6.34642	0.357580
$316$	0	0
$317$	3.11441	0.174922	0.0874612	−	0.996168i	$-0.472125\pi$
$317$	3.11441	0.174922	0.0874612	+	0.996168i	$0.472125\pi$
$318$	0	0
$319$	−3.20950	−0.179697
$320$	0	0
$321$	−49.2908	−2.75114
$322$	0	0
$323$	26.0250	1.44807
$324$	0	0
$325$	4.00012	0.221887
$326$	0	0
$327$	0.523844	0.0289686
$328$	0	0
$329$	6.81468	0.375705
$330$	0	0
$331$	−32.3922	−1.78044	−0.890219	−	0.455533i	$-0.849449\pi$
$331$	−32.3922	−1.78044	−0.890219	+	0.455533i	$0.849449\pi$
$332$	0	0
$333$	−54.3506	−2.97839
$334$	0	0
$335$	5.67054	0.309815
$336$	0	0
$337$	−6.86650	−0.374042	−0.187021	−	0.982356i	$-0.559883\pi$
$337$	−6.86650	−0.374042	−0.187021	+	0.982356i	$0.559883\pi$
$338$	0	0
$339$	41.0274	2.22830
$340$	0	0
$341$	8.14730	0.441201
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	14.7297	0.793022
$346$	0	0
$347$	10.8493	0.582419	0.291210	−	0.956659i	$-0.405942\pi$
$347$	10.8493	0.582419	0.291210	+	0.956659i	$0.405942\pi$
$348$	0	0
$349$	−9.30758	−0.498224	−0.249112	−	0.968475i	$-0.580139\pi$
$349$	−9.30758	−0.498224	−0.249112	+	0.968475i	$0.580139\pi$
$350$	0	0
$351$	40.9238	2.18435
$352$	0	0
$353$	0.530008	0.0282095	0.0141047	−	0.999901i	$-0.495510\pi$
$353$	0.530008	0.0282095	0.0141047	+	0.999901i	$0.495510\pi$
$354$	0	0
$355$	9.38700	0.498210
$356$	0	0
$357$	−22.7258	−1.20278
$358$	0	0
$359$	25.6881	1.35577	0.677883	−	0.735170i	$-0.262898\pi$
$359$	25.6881	1.35577	0.677883	+	0.735170i	$0.262898\pi$
$360$	0	0
$361$	−6.74290	−0.354889
$362$	0	0
$363$	2.90313	0.152375
$364$	0	0
$365$	−7.32410	−0.383361
$366$	0	0
$367$	23.3698	1.21989	0.609946	−	0.792443i	$-0.291191\pi$
$367$	23.3698	1.21989	0.609946	+	0.792443i	$0.291191\pi$
$368$	0	0
$369$	−58.3834	−3.03932
$370$	0	0
$371$	−6.09556	−0.316466
$372$	0	0
$373$	9.78042	0.506411	0.253206	−	0.967413i	$-0.418515\pi$
$373$	9.78042	0.506411	0.253206	+	0.967413i	$0.418515\pi$
$374$	0	0
$375$	−3.05719	−0.157873
$376$	0	0
$377$	−3.71393	−0.191277
$378$	0	0
$379$	26.5948	1.36608	0.683042	−	0.730379i	$-0.260657\pi$
$379$	26.5948	1.36608	0.683042	+	0.730379i	$0.260657\pi$
$380$	0	0
$381$	−9.58762	−0.491189
$382$	0	0
$383$	−11.8055	−0.603235	−0.301618	−	0.953429i	$-0.597527\pi$
$383$	−11.8055	−0.603235	−0.301618	+	0.953429i	$0.597527\pi$
$384$	0	0
$385$	3.45682	0.176176
$386$	0	0
$387$	−6.34642	−0.322607
$388$	0	0
$389$	−34.9280	−1.77092	−0.885460	−	0.464716i	$-0.846156\pi$
$389$	−34.9280	−1.77092	−0.885460	+	0.464716i	$0.846156\pi$
$390$	0	0
$391$	−35.8153	−1.81126
$392$	0	0
$393$	−5.22071	−0.263350
$394$	0	0
$395$	1.69899	0.0854853
$396$	0	0
$397$	20.5368	1.03071	0.515356	−	0.856976i	$-0.327660\pi$
$397$	20.5368	1.03071	0.515356	+	0.856976i	$0.327660\pi$
$398$	0	0
$399$	−10.7033	−0.535834
$400$	0	0
$401$	−1.99317	−0.0995339	−0.0497670	−	0.998761i	$-0.515848\pi$
$401$	−1.99317	−0.0995339	−0.0497670	+	0.998761i	$0.515848\pi$
$402$	0	0
$403$	9.42778	0.469631
$404$	0	0
$405$	−12.2378	−0.608102
$406$	0	0
$407$	−29.6041	−1.46742
$408$	0	0
$409$	14.4103	0.712545	0.356272	−	0.934382i	$-0.384048\pi$
$409$	14.4103	0.712545	0.356272	+	0.934382i	$0.384048\pi$
$410$	0	0
$411$	−9.44578	−0.465926
$412$	0	0
$413$	−3.86197	−0.190035
$414$	0	0
$415$	−4.92570	−0.241793
$416$	0	0
$417$	−28.6926	−1.40508
$418$	0	0
$419$	−22.3527	−1.09200	−0.546000	−	0.837785i	$-0.683850\pi$
$419$	−22.3527	−1.09200	−0.546000	+	0.837785i	$0.683850\pi$
$420$	0	0
$421$	−26.4509	−1.28914	−0.644568	−	0.764547i	$-0.722963\pi$
$421$	−26.4509	−1.28914	−0.644568	+	0.764547i	$0.722963\pi$
$422$	0	0
$423$	−43.2488	−2.10283
$424$	0	0
$425$	7.43356	0.360581
$426$	0	0
$427$	−4.74079	−0.229423
$428$	0	0
$429$	42.2739	2.04100
$430$	0	0
$431$	−5.69259	−0.274202	−0.137101	−	0.990557i	$-0.543779\pi$
$431$	−5.69259	−0.274202	−0.137101	+	0.990557i	$0.543779\pi$
$432$	0	0
$433$	−18.8461	−0.905685	−0.452843	−	0.891590i	$-0.649590\pi$
$433$	−18.8461	−0.905685	−0.452843	+	0.891590i	$0.649590\pi$
$434$	0	0
$435$	2.83846	0.136094
$436$	0	0
$437$	−16.8681	−0.806910
$438$	0	0
$439$	−12.0654	−0.575849	−0.287925	−	0.957653i	$-0.592965\pi$
$439$	−12.0654	−0.575849	−0.287925	+	0.957653i	$0.592965\pi$
$440$	0	0
$441$	6.34642	0.302211
$442$	0	0
$443$	−20.1618	−0.957914	−0.478957	−	0.877838i	$-0.658985\pi$
$443$	−20.1618	−0.957914	−0.478957	+	0.877838i	$0.658985\pi$
$444$	0	0
$445$	−11.0437	−0.523523
$446$	0	0
$447$	−7.32371	−0.346400
$448$	0	0
$449$	−12.8899	−0.608310	−0.304155	−	0.952623i	$-0.598374\pi$
$449$	−12.8899	−0.608310	−0.304155	+	0.952623i	$0.598374\pi$
$450$	0	0
$451$	−31.8007	−1.49744
$452$	0	0
$453$	−31.7527	−1.49187
$454$	0	0
$455$	4.00012	0.187528
$456$	0	0
$457$	−25.1684	−1.17733	−0.588663	−	0.808378i	$-0.700346\pi$
$457$	−25.1684	−1.17733	−0.588663	+	0.808378i	$0.700346\pi$
$458$	0	0
$459$	76.0502	3.54972
$460$	0	0
$461$	−1.78245	−0.0830171	−0.0415085	−	0.999138i	$-0.513216\pi$
$461$	−1.78245	−0.0830171	−0.0415085	+	0.999138i	$0.513216\pi$
$462$	0	0
$463$	−1.92420	−0.0894251	−0.0447126	−	0.999000i	$-0.514237\pi$
$463$	−1.92420	−0.0894251	−0.0447126	+	0.999000i	$0.514237\pi$
$464$	0	0
$465$	−7.20542	−0.334144
$466$	0	0
$467$	−26.5132	−1.22688	−0.613442	−	0.789740i	$-0.710215\pi$
$467$	−26.5132	−1.22688	−0.613442	+	0.789740i	$0.710215\pi$
$468$	0	0
$469$	5.67054	0.261841
$470$	0	0
$471$	28.0544	1.29268
$472$	0	0
$473$	−3.45682	−0.158945
$474$	0	0
$475$	3.50101	0.160638
$476$	0	0
$477$	38.6850	1.77126
$478$	0	0
$479$	−9.79924	−0.447739	−0.223869	−	0.974619i	$-0.571869\pi$
$479$	−9.79924	−0.447739	−0.223869	+	0.974619i	$0.571869\pi$
$480$	0	0
$481$	−34.2569	−1.56198
$482$	0	0
$483$	14.7297	0.670226
$484$	0	0
$485$	−8.78581	−0.398943
$486$	0	0
$487$	21.7334	0.984833	0.492417	−	0.870360i	$-0.336114\pi$
$487$	21.7334	0.984833	0.492417	+	0.870360i	$0.336114\pi$
$488$	0	0
$489$	−2.78749	−0.126055
$490$	0	0
$491$	8.82113	0.398092	0.199046	−	0.979990i	$-0.436216\pi$
$491$	8.82113	0.398092	0.199046	+	0.979990i	$0.436216\pi$
$492$	0	0
$493$	−6.90172	−0.310838
$494$	0	0
$495$	−21.9384	−0.986060
$496$	0	0
$497$	9.38700	0.421064
$498$	0	0
$499$	−15.7526	−0.705183	−0.352592	−	0.935777i	$-0.614700\pi$
$499$	−15.7526	−0.705183	−0.352592	+	0.935777i	$0.614700\pi$
$500$	0	0
$501$	−18.3114	−0.818093
$502$	0	0
$503$	12.8274	0.571948	0.285974	−	0.958237i	$-0.407683\pi$
$503$	12.8274	0.571948	0.285974	+	0.958237i	$0.407683\pi$
$504$	0	0
$505$	−8.39648	−0.373638
$506$	0	0
$507$	9.17442	0.407450
$508$	0	0
$509$	14.4492	0.640449	0.320224	−	0.947342i	$-0.396242\pi$
$509$	14.4492	0.640449	0.320224	+	0.947342i	$0.396242\pi$
$510$	0	0
$511$	−7.32410	−0.323999
$512$	0	0
$513$	35.8177	1.58139
$514$	0	0
$515$	−7.90711	−0.348429
$516$	0	0
$517$	−23.5571	−1.03604
$518$	0	0
$519$	−73.5301	−3.22761
$520$	0	0
$521$	−29.7728	−1.30437	−0.652184	−	0.758061i	$-0.726147\pi$
$521$	−29.7728	−1.30437	−0.652184	+	0.758061i	$0.726147\pi$
$522$	0	0
$523$	6.12913	0.268008	0.134004	−	0.990981i	$-0.457217\pi$
$523$	6.12913	0.268008	0.134004	+	0.990981i	$0.457217\pi$
$524$	0	0
$525$	−3.05719	−0.133427
$526$	0	0
$527$	17.5200	0.763182
$528$	0	0
$529$	0.213678	0.00929033
$530$	0	0
$531$	24.5097	1.06363
$532$	0	0
$533$	−36.7987	−1.59393
$534$	0	0
$535$	16.1229	0.697053
$536$	0	0
$537$	55.7855	2.40732
$538$	0	0
$539$	3.45682	0.148896
$540$	0	0
$541$	−14.3705	−0.617837	−0.308919	−	0.951088i	$-0.599967\pi$
$541$	−14.3705	−0.617837	−0.308919	+	0.951088i	$0.599967\pi$
$542$	0	0
$543$	49.5247	2.12531
$544$	0	0
$545$	−0.171348	−0.00733974
$546$	0	0
$547$	15.4280	0.659655	0.329827	−	0.944041i	$-0.393009\pi$
$547$	15.4280	0.659655	0.329827	+	0.944041i	$0.393009\pi$
$548$	0	0
$549$	30.0870	1.28408
$550$	0	0
$551$	−3.25053	−0.138477
$552$	0	0
$553$	1.69899	0.0722483
$554$	0	0
$555$	26.1817	1.11135
$556$	0	0
$557$	−29.0130	−1.22932	−0.614660	−	0.788792i	$-0.710707\pi$
$557$	−29.0130	−1.22932	−0.614660	+	0.788792i	$0.710707\pi$
$558$	0	0
$559$	−4.00012	−0.169187
$560$	0	0
$561$	78.5590	3.31677
$562$	0	0
$563$	−21.6764	−0.913551	−0.456775	−	0.889582i	$-0.650996\pi$
$563$	−21.6764	−0.913551	−0.456775	+	0.889582i	$0.650996\pi$
$564$	0	0
$565$	−13.4199	−0.564581
$566$	0	0
$567$	−12.2378	−0.513940
$568$	0	0
$569$	38.1547	1.59953	0.799763	−	0.600316i	$-0.204958\pi$
$569$	38.1547	1.59953	0.799763	+	0.600316i	$0.204958\pi$
$570$	0	0
$571$	−42.5455	−1.78047	−0.890237	−	0.455498i	$-0.849461\pi$
$571$	−42.5455	−1.78047	−0.890237	+	0.455498i	$0.849461\pi$
$572$	0	0
$573$	49.1099	2.05159
$574$	0	0
$575$	−4.81806	−0.200927
$576$	0	0
$577$	23.0368	0.959034	0.479517	−	0.877533i	$-0.340812\pi$
$577$	23.0368	0.959034	0.479517	+	0.877533i	$0.340812\pi$
$578$	0	0
$579$	−68.9043	−2.86356
$580$	0	0
$581$	−4.92570	−0.204352
$582$	0	0
$583$	21.0712	0.872682
$584$	0	0
$585$	−25.3864	−1.04960
$586$	0	0
$587$	40.6843	1.67922	0.839610	−	0.543189i	$-0.182783\pi$
$587$	40.6843	1.67922	0.839610	+	0.543189i	$0.182783\pi$
$588$	0	0
$589$	8.25146	0.339995
$590$	0	0
$591$	−14.4187	−0.593106
$592$	0	0
$593$	−30.5938	−1.25634	−0.628169	−	0.778077i	$-0.716195\pi$
$593$	−30.5938	−1.25634	−0.628169	+	0.778077i	$0.716195\pi$
$594$	0	0
$595$	7.43356	0.304746
$596$	0	0
$597$	83.9673	3.43655
$598$	0	0
$599$	−20.1108	−0.821705	−0.410853	−	0.911702i	$-0.634769\pi$
$599$	−20.1108	−0.821705	−0.410853	+	0.911702i	$0.634769\pi$
$600$	0	0
$601$	25.1188	1.02462	0.512309	−	0.858801i	$-0.328790\pi$
$601$	25.1188	1.02462	0.512309	+	0.858801i	$0.328790\pi$
$602$	0	0
$603$	−35.9877	−1.46553
$604$	0	0
$605$	−0.949605	−0.0386069
$606$	0	0
$607$	−4.24456	−0.172282	−0.0861408	−	0.996283i	$-0.527453\pi$
$607$	−4.24456	−0.172282	−0.0861408	+	0.996283i	$0.527453\pi$
$608$	0	0
$609$	2.83846	0.115020
$610$	0	0
$611$	−27.2595	−1.10280
$612$	0	0
$613$	−19.0986	−0.771386	−0.385693	−	0.922627i	$-0.626038\pi$
$613$	−19.0986	−0.771386	−0.385693	+	0.922627i	$0.626038\pi$
$614$	0	0
$615$	28.1244	1.13408
$616$	0	0
$617$	−15.2984	−0.615892	−0.307946	−	0.951404i	$-0.599642\pi$
$617$	−15.2984	−0.615892	−0.307946	+	0.951404i	$0.599642\pi$
$618$	0	0
$619$	−21.0897	−0.847668	−0.423834	−	0.905740i	$-0.639316\pi$
$619$	−21.0897	−0.847668	−0.423834	+	0.905740i	$0.639316\pi$
$620$	0	0
$621$	−49.2919	−1.97802
$622$	0	0
$623$	−11.0437	−0.442458
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	36.9993	1.47761
$628$	0	0
$629$	−63.6607	−2.53832
$630$	0	0
$631$	−37.1680	−1.47963	−0.739817	−	0.672809i	$-0.765088\pi$
$631$	−37.1680	−1.47963	−0.739817	+	0.672809i	$0.765088\pi$
$632$	0	0
$633$	16.7905	0.667364
$634$	0	0
$635$	3.13609	0.124452
$636$	0	0
$637$	4.00012	0.158490
$638$	0	0
$639$	−59.5738	−2.35670
$640$	0	0
$641$	3.91835	0.154766	0.0773828	−	0.997001i	$-0.475344\pi$
$641$	3.91835	0.154766	0.0773828	+	0.997001i	$0.475344\pi$
$642$	0	0
$643$	−30.4745	−1.20180	−0.600898	−	0.799326i	$-0.705190\pi$
$643$	−30.4745	−1.20180	−0.600898	+	0.799326i	$0.705190\pi$
$644$	0	0
$645$	3.05719	0.120377
$646$	0	0
$647$	−27.3862	−1.07666	−0.538332	−	0.842733i	$-0.680945\pi$
$647$	−27.3862	−1.07666	−0.538332	+	0.842733i	$0.680945\pi$
$648$	0	0
$649$	13.3501	0.524039
$650$	0	0
$651$	−7.20542	−0.282403
$652$	0	0
$653$	−34.4678	−1.34883	−0.674414	−	0.738353i	$-0.735604\pi$
$653$	−34.4678	−1.34883	−0.674414	+	0.738353i	$0.735604\pi$
$654$	0	0
$655$	1.70768	0.0667246
$656$	0	0
$657$	46.4818	1.81343
$658$	0	0
$659$	−32.8701	−1.28044	−0.640219	−	0.768192i	$-0.721156\pi$
$659$	−32.8701	−1.28044	−0.640219	+	0.768192i	$0.721156\pi$
$660$	0	0
$661$	−35.4998	−1.38078	−0.690391	−	0.723436i	$-0.742562\pi$
$661$	−35.4998	−1.38078	−0.690391	+	0.723436i	$0.742562\pi$
$662$	0	0
$663$	90.9059	3.53049
$664$	0	0
$665$	3.50101	0.135764
$666$	0	0
$667$	4.47335	0.173209
$668$	0	0
$669$	−50.9563	−1.97008
$670$	0	0
$671$	16.3880	0.632653
$672$	0	0
$673$	−27.2696	−1.05116	−0.525582	−	0.850743i	$-0.676153\pi$
$673$	−27.2696	−1.05116	−0.525582	+	0.850743i	$0.676153\pi$
$674$	0	0
$675$	10.2307	0.393778
$676$	0	0
$677$	−34.3054	−1.31846	−0.659231	−	0.751941i	$-0.729118\pi$
$677$	−34.3054	−1.31846	−0.659231	+	0.751941i	$0.729118\pi$
$678$	0	0
$679$	−8.78581	−0.337168
$680$	0	0
$681$	64.5067	2.47190
$682$	0	0
$683$	28.3957	1.08653	0.543265	−	0.839561i	$-0.317188\pi$
$683$	28.3957	1.08653	0.543265	+	0.839561i	$0.317188\pi$
$684$	0	0
$685$	3.08969	0.118051
$686$	0	0
$687$	−40.9801	−1.56349
$688$	0	0
$689$	24.3829	0.928916
$690$	0	0
$691$	45.7628	1.74090	0.870450	−	0.492257i	$-0.163828\pi$
$691$	45.7628	1.74090	0.870450	+	0.492257i	$0.163828\pi$
$692$	0	0
$693$	−21.9384	−0.833373
$694$	0	0
$695$	9.38528	0.356004
$696$	0	0
$697$	−68.3844	−2.59024
$698$	0	0
$699$	−66.1160	−2.50074
$700$	0	0
$701$	0.905291	0.0341924	0.0170962	−	0.999854i	$-0.494558\pi$
$701$	0.905291	0.0341924	0.0170962	+	0.999854i	$0.494558\pi$
$702$	0	0
$703$	−29.9826	−1.13081
$704$	0	0
$705$	20.8338	0.784646
$706$	0	0
$707$	−8.39648	−0.315782
$708$	0	0
$709$	24.3041	0.912758	0.456379	−	0.889785i	$-0.349146\pi$
$709$	24.3041	0.912758	0.456379	+	0.889785i	$0.349146\pi$
$710$	0	0
$711$	−10.7825	−0.404375
$712$	0	0
$713$	−11.3556	−0.425269
$714$	0	0
$715$	−13.8277	−0.517126
$716$	0	0
$717$	−11.8217	−0.441491
$718$	0	0
$719$	−38.0341	−1.41843	−0.709217	−	0.704991i	$-0.750951\pi$
$719$	−38.0341	−1.41843	−0.709217	+	0.704991i	$0.750951\pi$
$720$	0	0
$721$	−7.90711	−0.294476
$722$	0	0
$723$	−0.775099	−0.0288263
$724$	0	0
$725$	−0.928454	−0.0344819
$726$	0	0
$727$	38.1140	1.41357	0.706785	−	0.707428i	$-0.250145\pi$
$727$	38.1140	1.41357	0.706785	+	0.707428i	$0.250145\pi$
$728$	0	0
$729$	−16.1649	−0.598700
$730$	0	0
$731$	−7.43356	−0.274940
$732$	0	0
$733$	16.6792	0.616060	0.308030	−	0.951377i	$-0.400330\pi$
$733$	16.6792	0.616060	0.308030	+	0.951377i	$0.400330\pi$
$734$	0	0
$735$	−3.05719	−0.112766
$736$	0	0
$737$	−19.6020	−0.722050
$738$	0	0
$739$	45.3905	1.66972	0.834858	−	0.550465i	$-0.185550\pi$
$739$	45.3905	1.66972	0.834858	+	0.550465i	$0.185550\pi$
$740$	0	0
$741$	42.8143	1.57282
$742$	0	0
$743$	3.42186	0.125536	0.0627680	−	0.998028i	$-0.480007\pi$
$743$	3.42186	0.125536	0.0627680	+	0.998028i	$0.480007\pi$
$744$	0	0
$745$	2.39557	0.0877668
$746$	0	0
$747$	31.2606	1.14376
$748$	0	0
$749$	16.1229	0.589118
$750$	0	0
$751$	41.1180	1.50042	0.750209	−	0.661201i	$-0.229953\pi$
$751$	41.1180	1.50042	0.750209	+	0.661201i	$0.229953\pi$
$752$	0	0
$753$	−80.5935	−2.93699
$754$	0	0
$755$	10.3862	0.377994
$756$	0	0
$757$	39.6972	1.44282	0.721410	−	0.692508i	$-0.243494\pi$
$757$	39.6972	1.44282	0.721410	+	0.692508i	$0.243494\pi$
$758$	0	0
$759$	−50.9180	−1.84821
$760$	0	0
$761$	25.2836	0.916531	0.458266	−	0.888815i	$-0.348471\pi$
$761$	25.2836	0.916531	0.458266	+	0.888815i	$0.348471\pi$
$762$	0	0
$763$	−0.171348	−0.00620322
$764$	0	0
$765$	−47.1765	−1.70567
$766$	0	0
$767$	15.4483	0.557807
$768$	0	0
$769$	6.23308	0.224771	0.112385	−	0.993665i	$-0.464151\pi$
$769$	6.23308	0.224771	0.112385	+	0.993665i	$0.464151\pi$
$770$	0	0
$771$	13.9525	0.502486
$772$	0	0
$773$	16.0169	0.576087	0.288043	−	0.957617i	$-0.406995\pi$
$773$	16.0169	0.576087	0.288043	+	0.957617i	$0.406995\pi$
$774$	0	0
$775$	2.35688	0.0846615
$776$	0	0
$777$	26.1817	0.939263
$778$	0	0
$779$	−32.2073	−1.15395
$780$	0	0
$781$	−32.4492	−1.16112
$782$	0	0
$783$	−9.49870	−0.339456
$784$	0	0
$785$	−9.17653	−0.327524
$786$	0	0
$787$	−11.9747	−0.426852	−0.213426	−	0.976959i	$-0.568462\pi$
$787$	−11.9747	−0.426852	−0.213426	+	0.976959i	$0.568462\pi$
$788$	0	0
$789$	5.62835	0.200374
$790$	0	0
$791$	−13.4199	−0.477158
$792$	0	0
$793$	18.9637	0.673421
$794$	0	0
$795$	−18.6353	−0.660926
$796$	0	0
$797$	37.8238	1.33979	0.669894	−	0.742457i	$-0.266340\pi$
$797$	37.8238	1.33979	0.669894	+	0.742457i	$0.266340\pi$
$798$	0	0
$799$	−50.6573	−1.79213
$800$	0	0
$801$	70.0882	2.47644
$802$	0	0
$803$	25.3181	0.893456
$804$	0	0
$805$	−4.81806	−0.169814
$806$	0	0
$807$	15.8249	0.557063
$808$	0	0
$809$	13.1750	0.463208	0.231604	−	0.972810i	$-0.425603\pi$
$809$	13.1750	0.463208	0.231604	+	0.972810i	$0.425603\pi$
$810$	0	0
$811$	−10.6411	−0.373658	−0.186829	−	0.982392i	$-0.559821\pi$
$811$	−10.6411	−0.373658	−0.186829	+	0.982392i	$0.559821\pi$
$812$	0	0
$813$	−5.61400	−0.196892
$814$	0	0
$815$	0.911780	0.0319383
$816$	0	0
$817$	−3.50101	−0.122485
$818$	0	0
$819$	−25.3864	−0.887074
$820$	0	0
$821$	−43.2313	−1.50878	−0.754391	−	0.656426i	$-0.772068\pi$
$821$	−43.2313	−1.50878	−0.754391	+	0.656426i	$0.772068\pi$
$822$	0	0
$823$	−46.2336	−1.61160	−0.805800	−	0.592187i	$-0.798265\pi$
$823$	−46.2336	−1.61160	−0.805800	+	0.592187i	$0.798265\pi$
$824$	0	0
$825$	10.5682	0.367936
$826$	0	0
$827$	−26.0245	−0.904962	−0.452481	−	0.891774i	$-0.649461\pi$
$827$	−26.0245	−0.904962	−0.452481	+	0.891774i	$0.649461\pi$
$828$	0	0
$829$	26.3539	0.915307	0.457654	−	0.889131i	$-0.348690\pi$
$829$	26.3539	0.915307	0.457654	+	0.889131i	$0.348690\pi$
$830$	0	0
$831$	91.9107	3.18835
$832$	0	0
$833$	7.43356	0.257558
$834$	0	0
$835$	5.98961	0.207279
$836$	0	0
$837$	24.1124	0.833446
$838$	0	0
$839$	−39.2069	−1.35357	−0.676786	−	0.736180i	$-0.736628\pi$
$839$	−39.2069	−1.35357	−0.676786	+	0.736180i	$0.736628\pi$
$840$	0	0
$841$	−28.1380	−0.970275
$842$	0	0
$843$	57.2946	1.97333
$844$	0	0
$845$	−3.00093	−0.103235
$846$	0	0
$847$	−0.949605	−0.0326288
$848$	0	0
$849$	6.43054	0.220696
$850$	0	0
$851$	41.2617	1.41443
$852$	0	0
$853$	26.2080	0.897345	0.448672	−	0.893696i	$-0.351897\pi$
$853$	26.2080	0.897345	0.448672	+	0.893696i	$0.351897\pi$
$854$	0	0
$855$	−22.2189	−0.759871
$856$	0	0
$857$	14.1033	0.481760	0.240880	−	0.970555i	$-0.422564\pi$
$857$	14.1033	0.481760	0.240880	+	0.970555i	$0.422564\pi$
$858$	0	0
$859$	19.4392	0.663255	0.331628	−	0.943410i	$-0.392402\pi$
$859$	19.4392	0.663255	0.331628	+	0.943410i	$0.392402\pi$
$860$	0	0
$861$	28.1244	0.958476
$862$	0	0
$863$	19.0871	0.649733	0.324867	−	0.945760i	$-0.394681\pi$
$863$	19.0871	0.649733	0.324867	+	0.945760i	$0.394681\pi$
$864$	0	0
$865$	24.0515	0.817776
$866$	0	0
$867$	116.961	3.97222
$868$	0	0
$869$	−5.87309	−0.199231
$870$	0	0
$871$	−22.6828	−0.768578
$872$	0	0
$873$	55.7584	1.88714
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	−2.71727	−0.0917558	−0.0458779	−	0.998947i	$-0.514609\pi$
$877$	−2.71727	−0.0917558	−0.0458779	+	0.998947i	$0.514609\pi$
$878$	0	0
$879$	12.5301	0.422629
$880$	0	0
$881$	28.4673	0.959087	0.479544	−	0.877518i	$-0.340802\pi$
$881$	28.4673	0.959087	0.479544	+	0.877518i	$0.340802\pi$
$882$	0	0
$883$	−13.8378	−0.465678	−0.232839	−	0.972515i	$-0.574802\pi$
$883$	−13.8378	−0.465678	−0.232839	+	0.972515i	$0.574802\pi$
$884$	0	0
$885$	−11.8068	−0.396881
$886$	0	0
$887$	−40.2310	−1.35082	−0.675412	−	0.737441i	$-0.736034\pi$
$887$	−40.2310	−1.35082	−0.675412	+	0.737441i	$0.736034\pi$
$888$	0	0
$889$	3.13609	0.105181
$890$	0	0
$891$	42.3039	1.41724
$892$	0	0
$893$	−23.8583	−0.798387
$894$	0	0
$895$	−18.2473	−0.609940
$896$	0	0
$897$	−58.9206	−1.96730
$898$	0	0
$899$	−2.18825	−0.0729823
$900$	0	0
$901$	45.3117	1.50955
$902$	0	0
$903$	3.05719	0.101737
$904$	0	0
$905$	−16.1994	−0.538487
$906$	0	0
$907$	−17.7217	−0.588440	−0.294220	−	0.955738i	$-0.595060\pi$
$907$	−17.7217	−0.588440	−0.294220	+	0.955738i	$0.595060\pi$
$908$	0	0
$909$	53.2876	1.76744
$910$	0	0
$911$	42.9929	1.42442	0.712210	−	0.701967i	$-0.247695\pi$
$911$	42.9929	1.42442	0.712210	+	0.701967i	$0.247695\pi$
$912$	0	0
$913$	17.0272	0.563520
$914$	0	0
$915$	−14.4935	−0.479140
$916$	0	0
$917$	1.70768	0.0563926
$918$	0	0
$919$	41.5485	1.37056	0.685279	−	0.728281i	$-0.259680\pi$
$919$	41.5485	1.37056	0.685279	+	0.728281i	$0.259680\pi$
$920$	0	0
$921$	−28.1252	−0.926757
$922$	0	0
$923$	−37.5491	−1.23594
$924$	0	0
$925$	−8.56397	−0.281582
$926$	0	0
$927$	50.1819	1.64819
$928$	0	0
$929$	−7.97077	−0.261513	−0.130756	−	0.991415i	$-0.541741\pi$
$929$	−7.97077	−0.261513	−0.130756	+	0.991415i	$0.541741\pi$
$930$	0	0
$931$	3.50101	0.114741
$932$	0	0
$933$	28.0773	0.919209
$934$	0	0
$935$	−25.6965	−0.840364
$936$	0	0
$937$	−15.8569	−0.518021	−0.259011	−	0.965874i	$-0.583396\pi$
$937$	−15.8569	−0.518021	−0.259011	+	0.965874i	$0.583396\pi$
$938$	0	0
$939$	33.7259	1.10060
$940$	0	0
$941$	−55.3359	−1.80390	−0.901950	−	0.431840i	$-0.857864\pi$
$941$	−55.3359	−1.80390	−0.901950	+	0.431840i	$0.857864\pi$
$942$	0	0
$943$	44.3233	1.44337
$944$	0	0
$945$	10.2307	0.332803
$946$	0	0
$947$	21.3507	0.693806	0.346903	−	0.937901i	$-0.387233\pi$
$947$	21.3507	0.693806	0.346903	+	0.937901i	$0.387233\pi$
$948$	0	0
$949$	29.2972	0.951029
$950$	0	0
$951$	9.52134	0.308750
$952$	0	0
$953$	−45.4882	−1.47351	−0.736754	−	0.676160i	$-0.763643\pi$
$953$	−45.4882	−1.47351	−0.736754	+	0.676160i	$0.763643\pi$
$954$	0	0
$955$	−16.0637	−0.519810
$956$	0	0
$957$	−9.81206	−0.317179
$958$	0	0
$959$	3.08969	0.0997714
$960$	0	0
$961$	−25.4451	−0.820811
$962$	0	0
$963$	−102.323	−3.29730
$964$	0	0
$965$	22.5384	0.725537
$966$	0	0
$967$	−54.6529	−1.75752	−0.878760	−	0.477264i	$-0.841629\pi$
$967$	−54.6529	−1.75752	−0.878760	+	0.477264i	$0.841629\pi$
$968$	0	0
$969$	79.5634	2.55594
$970$	0	0
$971$	−50.4839	−1.62011	−0.810053	−	0.586357i	$-0.800562\pi$
$971$	−50.4839	−1.62011	−0.810053	+	0.586357i	$0.800562\pi$
$972$	0	0
$973$	9.38528	0.300878
$974$	0	0
$975$	12.2291	0.391645
$976$	0	0
$977$	−51.5098	−1.64794	−0.823972	−	0.566630i	$-0.808247\pi$
$977$	−51.5098	−1.64794	−0.823972	+	0.566630i	$0.808247\pi$
$978$	0	0
$979$	38.1762	1.22012
$980$	0	0
$981$	1.08745	0.0347195
$982$	0	0
$983$	10.1403	0.323426	0.161713	−	0.986838i	$-0.448298\pi$
$983$	10.1403	0.323426	0.161713	+	0.986838i	$0.448298\pi$
$984$	0	0
$985$	4.71632	0.150274
$986$	0	0
$987$	20.8338	0.663147
$988$	0	0
$989$	4.81806	0.153205
$990$	0	0
$991$	−31.9814	−1.01592	−0.507962	−	0.861380i	$-0.669601\pi$
$991$	−31.9814	−1.01592	−0.507962	+	0.861380i	$0.669601\pi$
$992$	0	0
$993$	−99.0293	−3.14260
$994$	0	0
$995$	−27.4655	−0.870715
$996$	0	0
$997$	15.3965	0.487614	0.243807	−	0.969824i	$-0.421604\pi$
$997$	15.3965	0.487614	0.243807	+	0.969824i	$0.421604\pi$
$998$	0	0
$999$	−87.6150	−2.77202

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6020.2.a.j.1.12	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6020.2.a.j.1.12	✓	13	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 6020 = 2^{2} \cdot 5 \cdot 7 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6020.a (trivial)

\(f(q)\)	\(=\)	\(q+3.05719 q^{3} -1.00000 q^{5} -1.00000 q^{7} +6.34642 q^{9} +O(q^{10})\)	\(q+3.05719 q^{3} -1.00000 q^{5} -1.00000 q^{7} +6.34642 q^{9} +3.45682 q^{11} +4.00012 q^{13} -3.05719 q^{15} +7.43356 q^{17} +3.50101 q^{19} -3.05719 q^{21} -4.81806 q^{23} +1.00000 q^{25} +10.2307 q^{27} -0.928454 q^{29} +2.35688 q^{31} +10.5682 q^{33} +1.00000 q^{35} -8.56397 q^{37} +12.2291 q^{39} -9.19942 q^{41} -1.00000 q^{43} -6.34642 q^{45} -6.81468 q^{47} +1.00000 q^{49} +22.7258 q^{51} +6.09556 q^{53} -3.45682 q^{55} +10.7033 q^{57} +3.86197 q^{59} +4.74079 q^{61} -6.34642 q^{63} -4.00012 q^{65} -5.67054 q^{67} -14.7297 q^{69} -9.38700 q^{71} +7.32410 q^{73} +3.05719 q^{75} -3.45682 q^{77} -1.69899 q^{79} +12.2378 q^{81} +4.92570 q^{83} -7.43356 q^{85} -2.83846 q^{87} +11.0437 q^{89} -4.00012 q^{91} +7.20542 q^{93} -3.50101 q^{95} +8.78581 q^{97} +21.9384 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - q^{3} - 13 q^{5} - 13 q^{7} + 18 q^{9}+O(q^{10}) \) 13 * q - q^3 - 13 * q^5 - 13 * q^7 + 18 * q^9	\( 13 q - q^{3} - 13 q^{5} - 13 q^{7} + 18 q^{9} + 6 q^{11} + q^{13} + q^{15} - 6 q^{17} + 2 q^{19} + q^{21} - 6 q^{23} + 13 q^{25} - q^{27} + 19 q^{29} - 24 q^{31} + 17 q^{33} + 13 q^{35} + 15 q^{39} + 4 q^{41} - 13 q^{43} - 18 q^{45} - 3 q^{47} + 13 q^{49} + 7 q^{51} - 5 q^{53} - 6 q^{55} + 6 q^{57} - 6 q^{59} + 3 q^{61} - 18 q^{63} - q^{65} - 2 q^{67} + 20 q^{69} + 18 q^{71} + 14 q^{73} - q^{75} - 6 q^{77} + 12 q^{79} + 37 q^{81} + 2 q^{83} + 6 q^{85} - 2 q^{87} + 17 q^{89} - q^{91} + 15 q^{93} - 2 q^{95} + 17 q^{97} + 80 q^{99}+O(q^{100}) \) 13 * q - q^3 - 13 * q^5 - 13 * q^7 + 18 * q^9 + 6 * q^11 + q^13 + q^15 - 6 * q^17 + 2 * q^19 + q^21 - 6 * q^23 + 13 * q^25 - q^27 + 19 * q^29 - 24 * q^31 + 17 * q^33 + 13 * q^35 + 15 * q^39 + 4 * q^41 - 13 * q^43 - 18 * q^45 - 3 * q^47 + 13 * q^49 + 7 * q^51 - 5 * q^53 - 6 * q^55 + 6 * q^57 - 6 * q^59 + 3 * q^61 - 18 * q^63 - q^65 - 2 * q^67 + 20 * q^69 + 18 * q^71 + 14 * q^73 - q^75 - 6 * q^77 + 12 * q^79 + 37 * q^81 + 2 * q^83 + 6 * q^85 - 2 * q^87 + 17 * q^89 - q^91 + 15 * q^93 - 2 * q^95 + 17 * q^97 + 80 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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