LMFDB - Embedded newform 6020.2.a.j.1.6

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			1.6
Root			$0.669559$ 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)

$f(q)$	$=$	$q-0.669559 q^{3} -1.00000 q^{5} -1.00000 q^{7} -2.55169 q^{9} +O(q^{10})$	$q-0.669559 q^{3} -1.00000 q^{5} -1.00000 q^{7} -2.55169 q^{9} -1.10944 q^{11} -4.93795 q^{13} +0.669559 q^{15} -7.84256 q^{17} -6.06606 q^{19} +0.669559 q^{21} -6.29690 q^{23} +1.00000 q^{25} +3.71718 q^{27} -6.73915 q^{29} -2.17417 q^{31} +0.742833 q^{33} +1.00000 q^{35} -0.816376 q^{37} +3.30625 q^{39} +3.61999 q^{41} -1.00000 q^{43} +2.55169 q^{45} +2.91400 q^{47} +1.00000 q^{49} +5.25105 q^{51} +0.776647 q^{53} +1.10944 q^{55} +4.06158 q^{57} -3.37582 q^{59} +11.6040 q^{61} +2.55169 q^{63} +4.93795 q^{65} +7.64051 q^{67} +4.21614 q^{69} +0.259538 q^{71} -11.0156 q^{73} -0.669559 q^{75} +1.10944 q^{77} +9.33277 q^{79} +5.16620 q^{81} -10.4111 q^{83} +7.84256 q^{85} +4.51226 q^{87} -0.837184 q^{89} +4.93795 q^{91} +1.45573 q^{93} +6.06606 q^{95} -3.83678 q^{97} +2.83094 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−0.669559	−0.386570	−0.193285	−	0.981143i	$-0.561914\pi$
$3$	−0.669559	−0.386570	−0.193285	+	0.981143i	$0.561914\pi$
$4$	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
$9$	−2.55169	−0.850564
$10$	0	0
$11$	−1.10944	−0.334508	−0.167254	−	0.985914i	$-0.553490\pi$
$11$	−1.10944	−0.334508	−0.167254	+	0.985914i	$0.553490\pi$
$12$	0	0
$13$	−4.93795	−1.36954	−0.684771	−	0.728759i	$-0.740098\pi$
$13$	−4.93795	−1.36954	−0.684771	+	0.728759i	$0.740098\pi$
$14$	0	0
$15$	0.669559	0.172879
$16$	0	0
$17$	−7.84256	−1.90210	−0.951050	−	0.309036i	$-0.899994\pi$
$17$	−7.84256	−1.90210	−0.951050	+	0.309036i	$0.899994\pi$
$18$	0	0
$19$	−6.06606	−1.39165	−0.695825	−	0.718211i	$-0.744961\pi$
$19$	−6.06606	−1.39165	−0.695825	+	0.718211i	$0.744961\pi$
$20$	0	0
$21$	0.669559	0.146110
$22$	0	0
$23$	−6.29690	−1.31299	−0.656497	−	0.754329i	$-0.727962\pi$
$23$	−6.29690	−1.31299	−0.656497	+	0.754329i	$0.727962\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	3.71718	0.715372
$28$	0	0
$29$	−6.73915	−1.25143	−0.625715	−	0.780052i	$-0.715192\pi$
$29$	−6.73915	−1.25143	−0.625715	+	0.780052i	$0.715192\pi$
$30$	0	0
$31$	−2.17417	−0.390492	−0.195246	−	0.980754i	$-0.562550\pi$
$31$	−2.17417	−0.390492	−0.195246	+	0.980754i	$0.562550\pi$
$32$	0	0
$33$	0.742833	0.129311
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	−0.816376	−0.134211	−0.0671057	−	0.997746i	$-0.521376\pi$
$37$	−0.816376	−0.134211	−0.0671057	+	0.997746i	$0.521376\pi$
$38$	0	0
$39$	3.30625	0.529423
$40$	0	0
$41$	3.61999	0.565347	0.282674	−	0.959216i	$-0.408779\pi$
$41$	3.61999	0.565347	0.282674	+	0.959216i	$0.408779\pi$
$42$	0	0
$43$	−1.00000	−0.152499
$43$	−1.00000	−0.152499
$44$	0	0
$45$	2.55169	0.380384
$46$	0	0
$47$	2.91400	0.425050	0.212525	−	0.977156i	$-0.431831\pi$
$47$	2.91400	0.425050	0.212525	+	0.977156i	$0.431831\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	5.25105	0.735295
$52$	0	0
$53$	0.776647	0.106681	0.0533403	−	0.998576i	$-0.483013\pi$
$53$	0.776647	0.106681	0.0533403	+	0.998576i	$0.483013\pi$
$54$	0	0
$55$	1.10944	0.149596
$56$	0	0
$57$	4.06158	0.537970
$58$	0	0
$59$	−3.37582	−0.439494	−0.219747	−	0.975557i	$-0.570523\pi$
$59$	−3.37582	−0.439494	−0.219747	+	0.975557i	$0.570523\pi$
$60$	0	0
$61$	11.6040	1.48574	0.742872	−	0.669433i	$-0.233463\pi$
$61$	11.6040	1.48574	0.742872	+	0.669433i	$0.233463\pi$
$62$	0	0
$63$	2.55169	0.321483
$64$	0	0
$65$	4.93795	0.612477
$66$	0	0
$67$	7.64051	0.933436	0.466718	−	0.884406i	$-0.345436\pi$
$67$	7.64051	0.933436	0.466718	+	0.884406i	$0.345436\pi$
$68$	0	0
$69$	4.21614	0.507564
$70$	0	0
$71$	0.259538	0.0308015	0.0154008	−	0.999881i	$-0.495098\pi$
$71$	0.259538	0.0308015	0.0154008	+	0.999881i	$0.495098\pi$
$72$	0	0
$73$	−11.0156	−1.28928	−0.644642	−	0.764485i	$-0.722994\pi$
$73$	−11.0156	−1.28928	−0.644642	+	0.764485i	$0.722994\pi$
$74$	0	0
$75$	−0.669559	−0.0773140
$76$	0	0
$77$	1.10944	0.126432
$78$	0	0
$79$	9.33277	1.05002	0.525009	−	0.851097i	$-0.324062\pi$
$79$	9.33277	1.05002	0.525009	+	0.851097i	$0.324062\pi$
$80$	0	0
$81$	5.16620	0.574022
$82$	0	0
$83$	−10.4111	−1.14276	−0.571381	−	0.820685i	$-0.693592\pi$
$83$	−10.4111	−1.14276	−0.571381	+	0.820685i	$0.693592\pi$
$84$	0	0
$85$	7.84256	0.850645
$86$	0	0
$87$	4.51226	0.483765
$88$	0	0
$89$	−0.837184	−0.0887414	−0.0443707	−	0.999015i	$-0.514128\pi$
$89$	−0.837184	−0.0887414	−0.0443707	+	0.999015i	$0.514128\pi$
$90$	0	0
$91$	4.93795	0.517638
$92$	0	0
$93$	1.45573	0.150952
$94$	0	0
$95$	6.06606	0.622365
$96$	0	0
$97$	−3.83678	−0.389566	−0.194783	−	0.980846i	$-0.562400\pi$
$97$	−3.83678	−0.389566	−0.194783	+	0.980846i	$0.562400\pi$
$98$	0	0
$99$	2.83094	0.284520
$100$	0	0
$101$	−11.4927	−1.14357	−0.571786	−	0.820403i	$-0.693749\pi$
$101$	−11.4927	−1.14357	−0.571786	+	0.820403i	$0.693749\pi$
$102$	0	0
$103$	4.71928	0.465004	0.232502	−	0.972596i	$-0.425309\pi$
$103$	4.71928	0.465004	0.232502	+	0.972596i	$0.425309\pi$
$104$	0	0
$105$	−0.669559	−0.0653422
$106$	0	0
$107$	11.6617	1.12737	0.563687	−	0.825988i	$-0.309382\pi$
$107$	11.6617	1.12737	0.563687	+	0.825988i	$0.309382\pi$
$108$	0	0
$109$	−9.12274	−0.873800	−0.436900	−	0.899510i	$-0.643924\pi$
$109$	−9.12274	−0.873800	−0.436900	+	0.899510i	$0.643924\pi$
$110$	0	0
$111$	0.546612	0.0518821
$112$	0	0
$113$	1.77032	0.166538	0.0832689	−	0.996527i	$-0.473464\pi$
$113$	1.77032	0.166538	0.0832689	+	0.996527i	$0.473464\pi$
$114$	0	0
$115$	6.29690	0.587189
$116$	0	0
$117$	12.6001	1.16488
$118$	0	0
$119$	7.84256	0.718926
$120$	0	0
$121$	−9.76915	−0.888105
$122$	0	0
$123$	−2.42379	−0.218546
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−11.0246	−0.978271	−0.489136	−	0.872208i	$-0.662688\pi$
$127$	−11.0246	−0.978271	−0.489136	+	0.872208i	$0.662688\pi$
$128$	0	0
$129$	0.669559	0.0589514
$130$	0	0
$131$	−2.14594	−0.187492	−0.0937458	−	0.995596i	$-0.529884\pi$
$131$	−2.14594	−0.187492	−0.0937458	+	0.995596i	$0.529884\pi$
$132$	0	0
$133$	6.06606	0.525994
$134$	0	0
$135$	−3.71718	−0.319924
$136$	0	0
$137$	−13.8398	−1.18242	−0.591208	−	0.806519i	$-0.701349\pi$
$137$	−13.8398	−1.18242	−0.591208	+	0.806519i	$0.701349\pi$
$138$	0	0
$139$	8.00439	0.678923	0.339462	−	0.940620i	$-0.389755\pi$
$139$	8.00439	0.678923	0.339462	+	0.940620i	$0.389755\pi$
$140$	0	0
$141$	−1.95109	−0.164312
$142$	0	0
$143$	5.47834	0.458122
$144$	0	0
$145$	6.73915	0.559656
$146$	0	0
$147$	−0.669559	−0.0552243
$148$	0	0
$149$	18.1724	1.48874	0.744372	−	0.667766i	$-0.232749\pi$
$149$	18.1724	1.48874	0.744372	+	0.667766i	$0.232749\pi$
$150$	0	0
$151$	−8.97991	−0.730775	−0.365387	−	0.930856i	$-0.619063\pi$
$151$	−8.97991	−0.730775	−0.365387	+	0.930856i	$0.619063\pi$
$152$	0	0
$153$	20.0118	1.61786
$154$	0	0
$155$	2.17417	0.174633
$156$	0	0
$157$	−17.0794	−1.36308	−0.681541	−	0.731780i	$-0.738690\pi$
$157$	−17.0794	−1.36308	−0.681541	+	0.731780i	$0.738690\pi$
$158$	0	0
$159$	−0.520011	−0.0412395
$160$	0	0
$161$	6.29690	0.496265
$162$	0	0
$163$	−1.73122	−0.135600	−0.0677998	−	0.997699i	$-0.521598\pi$
$163$	−1.73122	−0.135600	−0.0677998	+	0.997699i	$0.521598\pi$
$164$	0	0
$165$	−0.742833	−0.0578295
$166$	0	0
$167$	−1.54701	−0.119711	−0.0598556	−	0.998207i	$-0.519064\pi$
$167$	−1.54701	−0.119711	−0.0598556	+	0.998207i	$0.519064\pi$
$168$	0	0
$169$	11.3834	0.875643
$170$	0	0
$171$	15.4787	1.18369
$172$	0	0
$173$	−22.9973	−1.74845	−0.874225	−	0.485521i	$-0.838630\pi$
$173$	−22.9973	−1.74845	−0.874225	+	0.485521i	$0.838630\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	2.26031	0.169895
$178$	0	0
$179$	−23.3522	−1.74543	−0.872713	−	0.488233i	$-0.837641\pi$
$179$	−23.3522	−1.74543	−0.872713	+	0.488233i	$0.837641\pi$
$180$	0	0
$181$	15.3821	1.14334	0.571670	−	0.820484i	$-0.306296\pi$
$181$	15.3821	1.14334	0.571670	+	0.820484i	$0.306296\pi$
$182$	0	0
$183$	−7.76958	−0.574344
$184$	0	0
$185$	0.816376	0.0600211
$186$	0	0
$187$	8.70082	0.636267
$188$	0	0
$189$	−3.71718	−0.270385
$190$	0	0
$191$	12.4514	0.900954	0.450477	−	0.892788i	$-0.351254\pi$
$191$	12.4514	0.900954	0.450477	+	0.892788i	$0.351254\pi$
$192$	0	0
$193$	8.08445	0.581931	0.290966	−	0.956734i	$-0.406023\pi$
$193$	8.08445	0.581931	0.290966	+	0.956734i	$0.406023\pi$
$194$	0	0
$195$	−3.30625	−0.236765
$196$	0	0
$197$	6.27574	0.447128	0.223564	−	0.974689i	$-0.428231\pi$
$197$	6.27574	0.447128	0.223564	+	0.974689i	$0.428231\pi$
$198$	0	0
$199$	−17.9267	−1.27079	−0.635394	−	0.772188i	$-0.719162\pi$
$199$	−17.9267	−1.27079	−0.635394	+	0.772188i	$0.719162\pi$
$200$	0	0
$201$	−5.11577	−0.360838
$202$	0	0
$203$	6.73915	0.472996
$204$	0	0
$205$	−3.61999	−0.252831
$206$	0	0
$207$	16.0677	1.11678
$208$	0	0
$209$	6.72991	0.465518
$210$	0	0
$211$	5.68555	0.391409	0.195705	−	0.980663i	$-0.437301\pi$
$211$	5.68555	0.391409	0.195705	+	0.980663i	$0.437301\pi$
$212$	0	0
$213$	−0.173776	−0.0119069
$214$	0	0
$215$	1.00000	0.0681994
$216$	0	0
$217$	2.17417	0.147592
$218$	0	0
$219$	7.37562	0.498398
$220$	0	0
$221$	38.7262	2.60500
$222$	0	0
$223$	−16.6215	−1.11306	−0.556528	−	0.830829i	$-0.687867\pi$
$223$	−16.6215	−1.11306	−0.556528	+	0.830829i	$0.687867\pi$
$224$	0	0
$225$	−2.55169	−0.170113
$226$	0	0
$227$	−9.89425	−0.656704	−0.328352	−	0.944555i	$-0.606493\pi$
$227$	−9.89425	−0.656704	−0.328352	+	0.944555i	$0.606493\pi$
$228$	0	0
$229$	−17.9917	−1.18892	−0.594461	−	0.804124i	$-0.702635\pi$
$229$	−17.9917	−1.18892	−0.594461	+	0.804124i	$0.702635\pi$
$230$	0	0
$231$	−0.742833	−0.0488748
$232$	0	0
$233$	−5.01365	−0.328455	−0.164227	−	0.986422i	$-0.552513\pi$
$233$	−5.01365	−0.328455	−0.164227	+	0.986422i	$0.552513\pi$
$234$	0	0
$235$	−2.91400	−0.190088
$236$	0	0
$237$	−6.24884	−0.405906
$238$	0	0
$239$	−5.05926	−0.327257	−0.163628	−	0.986522i	$-0.552320\pi$
$239$	−5.05926	−0.327257	−0.163628	+	0.986522i	$0.552320\pi$
$240$	0	0
$241$	14.1613	0.912208	0.456104	−	0.889926i	$-0.349244\pi$
$241$	14.1613	0.912208	0.456104	+	0.889926i	$0.349244\pi$
$242$	0	0
$243$	−14.6106	−0.937272
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	29.9539	1.90592
$248$	0	0
$249$	6.97081	0.441757
$250$	0	0
$251$	−4.12661	−0.260469	−0.130235	−	0.991483i	$-0.541573\pi$
$251$	−4.12661	−0.260469	−0.130235	+	0.991483i	$0.541573\pi$
$252$	0	0
$253$	6.98601	0.439207
$254$	0	0
$255$	−5.25105	−0.328834
$256$	0	0
$257$	−13.6208	−0.849645	−0.424822	−	0.905277i	$-0.639663\pi$
$257$	−13.6208	−0.849645	−0.424822	+	0.905277i	$0.639663\pi$
$258$	0	0
$259$	0.816376	0.0507271
$260$	0	0
$261$	17.1962	1.06442
$262$	0	0
$263$	−8.32507	−0.513346	−0.256673	−	0.966498i	$-0.582626\pi$
$263$	−8.32507	−0.513346	−0.256673	+	0.966498i	$0.582626\pi$
$264$	0	0
$265$	−0.776647	−0.0477091
$266$	0	0
$267$	0.560544	0.0343047
$268$	0	0
$269$	16.8961	1.03017	0.515087	−	0.857138i	$-0.327760\pi$
$269$	16.8961	1.03017	0.515087	+	0.857138i	$0.327760\pi$
$270$	0	0
$271$	−18.3749	−1.11620	−0.558099	−	0.829775i	$-0.688469\pi$
$271$	−18.3749	−1.11620	−0.558099	+	0.829775i	$0.688469\pi$
$272$	0	0
$273$	−3.30625	−0.200103
$274$	0	0
$275$	−1.10944	−0.0669015
$276$	0	0
$277$	18.7058	1.12392	0.561960	−	0.827164i	$-0.310047\pi$
$277$	18.7058	1.12392	0.561960	+	0.827164i	$0.310047\pi$
$278$	0	0
$279$	5.54780	0.332138
$280$	0	0
$281$	2.01693	0.120320	0.0601599	−	0.998189i	$-0.480839\pi$
$281$	2.01693	0.120320	0.0601599	+	0.998189i	$0.480839\pi$
$282$	0	0
$283$	−2.14265	−0.127367	−0.0636836	−	0.997970i	$-0.520285\pi$
$283$	−2.14265	−0.127367	−0.0636836	+	0.997970i	$0.520285\pi$
$284$	0	0
$285$	−4.06158	−0.240587
$286$	0	0
$287$	−3.61999	−0.213681
$288$	0	0
$289$	44.5058	2.61799
$290$	0	0
$291$	2.56895	0.150595
$292$	0	0
$293$	15.6204	0.912553	0.456277	−	0.889838i	$-0.349183\pi$
$293$	15.6204	0.912553	0.456277	+	0.889838i	$0.349183\pi$
$294$	0	0
$295$	3.37582	0.196548
$296$	0	0
$297$	−4.12398	−0.239298
$298$	0	0
$299$	31.0938	1.79820
$300$	0	0
$301$	1.00000	0.0576390
$302$	0	0
$303$	7.69507	0.442070
$304$	0	0
$305$	−11.6040	−0.664445
$306$	0	0
$307$	−1.01362	−0.0578505	−0.0289253	−	0.999582i	$-0.509208\pi$
$307$	−1.01362	−0.0578505	−0.0289253	+	0.999582i	$0.509208\pi$
$308$	0	0
$309$	−3.15983	−0.179757
$310$	0	0
$311$	−15.0610	−0.854033	−0.427016	−	0.904244i	$-0.640435\pi$
$311$	−15.0610	−0.854033	−0.427016	+	0.904244i	$0.640435\pi$
$312$	0	0
$313$	1.78077	0.100655	0.0503276	−	0.998733i	$-0.483973\pi$
$313$	1.78077	0.100655	0.0503276	+	0.998733i	$0.483973\pi$
$314$	0	0
$315$	−2.55169	−0.143772
$316$	0	0
$317$	−18.3370	−1.02991	−0.514954	−	0.857218i	$-0.672191\pi$
$317$	−18.3370	−1.02991	−0.514954	+	0.857218i	$0.672191\pi$
$318$	0	0
$319$	7.47666	0.418613
$320$	0	0
$321$	−7.80816	−0.435809
$322$	0	0
$323$	47.5735	2.64706
$324$	0	0
$325$	−4.93795	−0.273908
$326$	0	0
$327$	6.10821	0.337785
$328$	0	0
$329$	−2.91400	−0.160654
$330$	0	0
$331$	−10.2633	−0.564119	−0.282060	−	0.959397i	$-0.591018\pi$
$331$	−10.2633	−0.564119	−0.282060	+	0.959397i	$0.591018\pi$
$332$	0	0
$333$	2.08314	0.114155
$334$	0	0
$335$	−7.64051	−0.417445
$336$	0	0
$337$	8.74487	0.476364	0.238182	−	0.971221i	$-0.423449\pi$
$337$	8.74487	0.476364	0.238182	+	0.971221i	$0.423449\pi$
$338$	0	0
$339$	−1.18533	−0.0643785
$340$	0	0
$341$	2.41210	0.130622
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−4.21614	−0.226989
$346$	0	0
$347$	5.14910	0.276418	0.138209	−	0.990403i	$-0.455865\pi$
$347$	5.14910	0.276418	0.138209	+	0.990403i	$0.455865\pi$
$348$	0	0
$349$	−4.46265	−0.238880	−0.119440	−	0.992841i	$-0.538110\pi$
$349$	−4.46265	−0.238880	−0.119440	+	0.992841i	$0.538110\pi$
$350$	0	0
$351$	−18.3553	−0.979732
$352$	0	0
$353$	−1.32068	−0.0702927	−0.0351463	−	0.999382i	$-0.511190\pi$
$353$	−1.32068	−0.0702927	−0.0351463	+	0.999382i	$0.511190\pi$
$354$	0	0
$355$	−0.259538	−0.0137749
$356$	0	0
$357$	−5.25105	−0.277915
$358$	0	0
$359$	19.2082	1.01377	0.506885	−	0.862014i	$-0.330797\pi$
$359$	19.2082	1.01377	0.506885	+	0.862014i	$0.330797\pi$
$360$	0	0
$361$	17.7971	0.936690
$362$	0	0
$363$	6.54102	0.343314
$364$	0	0
$365$	11.0156	0.576585
$366$	0	0
$367$	−7.04601	−0.367799	−0.183899	−	0.982945i	$-0.558872\pi$
$367$	−7.04601	−0.367799	−0.183899	+	0.982945i	$0.558872\pi$
$368$	0	0
$369$	−9.23709	−0.480864
$370$	0	0
$371$	−0.776647	−0.0403215
$372$	0	0
$373$	−15.0174	−0.777571	−0.388786	−	0.921328i	$-0.627105\pi$
$373$	−15.0174	−0.777571	−0.388786	+	0.921328i	$0.627105\pi$
$374$	0	0
$375$	0.669559	0.0345759
$376$	0	0
$377$	33.2776	1.71388
$378$	0	0
$379$	−11.3619	−0.583622	−0.291811	−	0.956476i	$-0.594258\pi$
$379$	−11.3619	−0.583622	−0.291811	+	0.956476i	$0.594258\pi$
$380$	0	0
$381$	7.38159	0.378170
$382$	0	0
$383$	−31.4337	−1.60619	−0.803093	−	0.595854i	$-0.796814\pi$
$383$	−31.4337	−1.60619	−0.803093	+	0.595854i	$0.796814\pi$
$384$	0	0
$385$	−1.10944	−0.0565421
$386$	0	0
$387$	2.55169	0.129710
$388$	0	0
$389$	−37.6352	−1.90818	−0.954091	−	0.299516i	$-0.903175\pi$
$389$	−37.6352	−1.90818	−0.954091	+	0.299516i	$0.903175\pi$
$390$	0	0
$391$	49.3838	2.49745
$392$	0	0
$393$	1.43683	0.0724786
$394$	0	0
$395$	−9.33277	−0.469583
$396$	0	0
$397$	33.6235	1.68752	0.843758	−	0.536724i	$-0.180339\pi$
$397$	33.6235	1.68752	0.843758	+	0.536724i	$0.180339\pi$
$398$	0	0
$399$	−4.06158	−0.203334
$400$	0	0
$401$	5.24892	0.262118	0.131059	−	0.991375i	$-0.458162\pi$
$401$	5.24892	0.262118	0.131059	+	0.991375i	$0.458162\pi$
$402$	0	0
$403$	10.7359	0.534794
$404$	0	0
$405$	−5.16620	−0.256711
$406$	0	0
$407$	0.905717	0.0448947
$408$	0	0
$409$	25.8924	1.28029	0.640147	−	0.768252i	$-0.278873\pi$
$409$	25.8924	1.28029	0.640147	+	0.768252i	$0.278873\pi$
$410$	0	0
$411$	9.26657	0.457086
$412$	0	0
$413$	3.37582	0.166113
$414$	0	0
$415$	10.4111	0.511059
$416$	0	0
$417$	−5.35941	−0.262451
$418$	0	0
$419$	2.87407	0.140407	0.0702037	−	0.997533i	$-0.477635\pi$
$419$	2.87407	0.140407	0.0702037	+	0.997533i	$0.477635\pi$
$420$	0	0
$421$	1.00267	0.0488674	0.0244337	−	0.999701i	$-0.492222\pi$
$421$	1.00267	0.0488674	0.0244337	+	0.999701i	$0.492222\pi$
$422$	0	0
$423$	−7.43562	−0.361532
$424$	0	0
$425$	−7.84256	−0.380420
$426$	0	0
$427$	−11.6040	−0.561559
$428$	0	0
$429$	−3.66807	−0.177096
$430$	0	0
$431$	−21.1784	−1.02013	−0.510063	−	0.860137i	$-0.670378\pi$
$431$	−21.1784	−1.02013	−0.510063	+	0.860137i	$0.670378\pi$
$432$	0	0
$433$	20.1337	0.967565	0.483783	−	0.875188i	$-0.339263\pi$
$433$	20.1337	0.967565	0.483783	+	0.875188i	$0.339263\pi$
$434$	0	0
$435$	−4.51226	−0.216346
$436$	0	0
$437$	38.1974	1.82723
$438$	0	0
$439$	−34.6902	−1.65567	−0.827837	−	0.560968i	$-0.810429\pi$
$439$	−34.6902	−1.65567	−0.827837	+	0.560968i	$0.810429\pi$
$440$	0	0
$441$	−2.55169	−0.121509
$442$	0	0
$443$	3.37725	0.160458	0.0802291	−	0.996776i	$-0.474435\pi$
$443$	3.37725	0.160458	0.0802291	+	0.996776i	$0.474435\pi$
$444$	0	0
$445$	0.837184	0.0396863
$446$	0	0
$447$	−12.1675	−0.575503
$448$	0	0
$449$	13.2680	0.626157	0.313079	−	0.949727i	$-0.398640\pi$
$449$	13.2680	0.626157	0.313079	+	0.949727i	$0.398640\pi$
$450$	0	0
$451$	−4.01615	−0.189113
$452$	0	0
$453$	6.01258	0.282495
$454$	0	0
$455$	−4.93795	−0.231495
$456$	0	0
$457$	−19.4561	−0.910120	−0.455060	−	0.890461i	$-0.650382\pi$
$457$	−19.4561	−0.910120	−0.455060	+	0.890461i	$0.650382\pi$
$458$	0	0
$459$	−29.1522	−1.36071
$460$	0	0
$461$	−6.16267	−0.287024	−0.143512	−	0.989649i	$-0.545840\pi$
$461$	−6.16267	−0.287024	−0.143512	+	0.989649i	$0.545840\pi$
$462$	0	0
$463$	23.3817	1.08664	0.543320	−	0.839526i	$-0.317167\pi$
$463$	23.3817	1.08664	0.543320	+	0.839526i	$0.317167\pi$
$464$	0	0
$465$	−1.45573	−0.0675079
$466$	0	0
$467$	21.1531	0.978848	0.489424	−	0.872046i	$-0.337207\pi$
$467$	21.1531	0.978848	0.489424	+	0.872046i	$0.337207\pi$
$468$	0	0
$469$	−7.64051	−0.352806
$470$	0	0
$471$	11.4356	0.526927
$472$	0	0
$473$	1.10944	0.0510119
$474$	0	0
$475$	−6.06606	−0.278330
$476$	0	0
$477$	−1.98176	−0.0907387
$478$	0	0
$479$	20.6357	0.942868	0.471434	−	0.881901i	$-0.343737\pi$
$479$	20.6357	0.942868	0.471434	+	0.881901i	$0.343737\pi$
$480$	0	0
$481$	4.03122	0.183808
$482$	0	0
$483$	−4.21614	−0.191841
$484$	0	0
$485$	3.83678	0.174219
$486$	0	0
$487$	−16.9168	−0.766571	−0.383286	−	0.923630i	$-0.625208\pi$
$487$	−16.9168	−0.766571	−0.383286	+	0.923630i	$0.625208\pi$
$488$	0	0
$489$	1.15915	0.0524187
$490$	0	0
$491$	−32.0969	−1.44851	−0.724257	−	0.689530i	$-0.757817\pi$
$491$	−32.0969	−1.44851	−0.724257	+	0.689530i	$0.757817\pi$
$492$	0	0
$493$	52.8522	2.38034
$494$	0	0
$495$	−2.83094	−0.127241
$496$	0	0
$497$	−0.259538	−0.0116419
$498$	0	0
$499$	−28.6627	−1.28312	−0.641560	−	0.767073i	$-0.721713\pi$
$499$	−28.6627	−1.28312	−0.641560	+	0.767073i	$0.721713\pi$
$500$	0	0
$501$	1.03581	0.0462767
$502$	0	0
$503$	−30.9721	−1.38098	−0.690488	−	0.723343i	$-0.742604\pi$
$503$	−30.9721	−1.38098	−0.690488	+	0.723343i	$0.742604\pi$
$504$	0	0
$505$	11.4927	0.511421
$506$	0	0
$507$	−7.62182	−0.338497
$508$	0	0
$509$	41.5970	1.84376	0.921878	−	0.387480i	$-0.126654\pi$
$509$	41.5970	1.84376	0.921878	+	0.387480i	$0.126654\pi$
$510$	0	0
$511$	11.0156	0.487304
$512$	0	0
$513$	−22.5487	−0.995548
$514$	0	0
$515$	−4.71928	−0.207956
$516$	0	0
$517$	−3.23289	−0.142183
$518$	0	0
$519$	15.3980	0.675898
$520$	0	0
$521$	−9.24134	−0.404870	−0.202435	−	0.979296i	$-0.564886\pi$
$521$	−9.24134	−0.404870	−0.202435	+	0.979296i	$0.564886\pi$
$522$	0	0
$523$	−31.1581	−1.36245	−0.681224	−	0.732075i	$-0.738552\pi$
$523$	−31.1581	−1.36245	−0.681224	+	0.732075i	$0.738552\pi$
$524$	0	0
$525$	0.669559	0.0292219
$526$	0	0
$527$	17.0510	0.742754
$528$	0	0
$529$	16.6509	0.723953
$530$	0	0
$531$	8.61405	0.373818
$532$	0	0
$533$	−17.8753	−0.774266
$534$	0	0
$535$	−11.6617	−0.504177
$536$	0	0
$537$	15.6357	0.674729
$538$	0	0
$539$	−1.10944	−0.0477868
$540$	0	0
$541$	27.7809	1.19440	0.597198	−	0.802094i	$-0.296281\pi$
$541$	27.7809	1.19440	0.597198	+	0.802094i	$0.296281\pi$
$542$	0	0
$543$	−10.2992	−0.441980
$544$	0	0
$545$	9.12274	0.390775
$546$	0	0
$547$	−13.1127	−0.560658	−0.280329	−	0.959904i	$-0.590444\pi$
$547$	−13.1127	−0.560658	−0.280329	+	0.959904i	$0.590444\pi$
$548$	0	0
$549$	−29.6099	−1.26372
$550$	0	0
$551$	40.8801	1.74155
$552$	0	0
$553$	−9.33277	−0.396870
$554$	0	0
$555$	−0.546612	−0.0232024
$556$	0	0
$557$	9.01258	0.381875	0.190938	−	0.981602i	$-0.438847\pi$
$557$	9.01258	0.381875	0.190938	+	0.981602i	$0.438847\pi$
$558$	0	0
$559$	4.93795	0.208853
$560$	0	0
$561$	−5.82571	−0.245962
$562$	0	0
$563$	14.0023	0.590128	0.295064	−	0.955478i	$-0.404659\pi$
$563$	14.0023	0.590128	0.295064	+	0.955478i	$0.404659\pi$
$564$	0	0
$565$	−1.77032	−0.0744780
$566$	0	0
$567$	−5.16620	−0.216960
$568$	0	0
$569$	−9.72181	−0.407559	−0.203780	−	0.979017i	$-0.565323\pi$
$569$	−9.72181	−0.407559	−0.203780	+	0.979017i	$0.565323\pi$
$570$	0	0
$571$	26.7499	1.11945	0.559725	−	0.828679i	$-0.310907\pi$
$571$	26.7499	1.11945	0.559725	+	0.828679i	$0.310907\pi$
$572$	0	0
$573$	−8.33696	−0.348282
$574$	0	0
$575$	−6.29690	−0.262599
$576$	0	0
$577$	−34.1184	−1.42037	−0.710184	−	0.704016i	$-0.751388\pi$
$577$	−34.1184	−1.42037	−0.710184	+	0.704016i	$0.751388\pi$
$578$	0	0
$579$	−5.41301	−0.224957
$580$	0	0
$581$	10.4111	0.431923
$582$	0	0
$583$	−0.861641	−0.0356855
$584$	0	0
$585$	−12.6001	−0.520951
$586$	0	0
$587$	−25.5388	−1.05410	−0.527050	−	0.849835i	$-0.676702\pi$
$587$	−25.5388	−1.05410	−0.527050	+	0.849835i	$0.676702\pi$
$588$	0	0
$589$	13.1886	0.543428
$590$	0	0
$591$	−4.20198	−0.172846
$592$	0	0
$593$	−1.10692	−0.0454557	−0.0227279	−	0.999742i	$-0.507235\pi$
$593$	−1.10692	−0.0454557	−0.0227279	+	0.999742i	$0.507235\pi$
$594$	0	0
$595$	−7.84256	−0.321514
$596$	0	0
$597$	12.0029	0.491248
$598$	0	0
$599$	−4.86167	−0.198643	−0.0993213	−	0.995055i	$-0.531667\pi$
$599$	−4.86167	−0.198643	−0.0993213	+	0.995055i	$0.531667\pi$
$600$	0	0
$601$	45.5320	1.85729	0.928645	−	0.370970i	$-0.120975\pi$
$601$	45.5320	1.85729	0.928645	+	0.370970i	$0.120975\pi$
$602$	0	0
$603$	−19.4962	−0.793947
$604$	0	0
$605$	9.76915	0.397172
$606$	0	0
$607$	−21.9730	−0.891857	−0.445929	−	0.895068i	$-0.647127\pi$
$607$	−21.9730	−0.891857	−0.445929	+	0.895068i	$0.647127\pi$
$608$	0	0
$609$	−4.51226	−0.182846
$610$	0	0
$611$	−14.3892	−0.582123
$612$	0	0
$613$	19.5676	0.790329	0.395164	−	0.918610i	$-0.370688\pi$
$613$	19.5676	0.790329	0.395164	+	0.918610i	$0.370688\pi$
$614$	0	0
$615$	2.42379	0.0977368
$616$	0	0
$617$	−44.0977	−1.77531	−0.887654	−	0.460511i	$-0.847666\pi$
$617$	−44.0977	−1.77531	−0.887654	+	0.460511i	$0.847666\pi$
$618$	0	0
$619$	−17.9229	−0.720384	−0.360192	−	0.932878i	$-0.617289\pi$
$619$	−17.9229	−0.720384	−0.360192	+	0.932878i	$0.617289\pi$
$620$	0	0
$621$	−23.4067	−0.939279
$622$	0	0
$623$	0.837184	0.0335411
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−4.50607	−0.179955
$628$	0	0
$629$	6.40248	0.255284
$630$	0	0
$631$	7.31866	0.291351	0.145676	−	0.989332i	$-0.453464\pi$
$631$	7.31866	0.291351	0.145676	+	0.989332i	$0.453464\pi$
$632$	0	0
$633$	−3.80681	−0.151307
$634$	0	0
$635$	11.0246	0.437496
$636$	0	0
$637$	−4.93795	−0.195649
$638$	0	0
$639$	−0.662261	−0.0261987
$640$	0	0
$641$	−2.09711	−0.0828310	−0.0414155	−	0.999142i	$-0.513187\pi$
$641$	−2.09711	−0.0828310	−0.0414155	+	0.999142i	$0.513187\pi$
$642$	0	0
$643$	−38.7112	−1.52662	−0.763310	−	0.646032i	$-0.776427\pi$
$643$	−38.7112	−1.52662	−0.763310	+	0.646032i	$0.776427\pi$
$644$	0	0
$645$	−0.669559	−0.0263638
$646$	0	0
$647$	22.1643	0.871370	0.435685	−	0.900099i	$-0.356506\pi$
$647$	22.1643	0.871370	0.435685	+	0.900099i	$0.356506\pi$
$648$	0	0
$649$	3.74526	0.147014
$650$	0	0
$651$	−1.45573	−0.0570546
$652$	0	0
$653$	5.52449	0.216190	0.108095	−	0.994141i	$-0.465525\pi$
$653$	5.52449	0.216190	0.108095	+	0.994141i	$0.465525\pi$
$654$	0	0
$655$	2.14594	0.0838488
$656$	0	0
$657$	28.1085	1.09662
$658$	0	0
$659$	18.7721	0.731258	0.365629	−	0.930761i	$-0.380854\pi$
$659$	18.7721	0.731258	0.365629	+	0.930761i	$0.380854\pi$
$660$	0	0
$661$	−17.5389	−0.682183	−0.341092	−	0.940030i	$-0.610797\pi$
$661$	−17.5389	−0.682183	−0.341092	+	0.940030i	$0.610797\pi$
$662$	0	0
$663$	−25.9295	−1.00702
$664$	0	0
$665$	−6.06606	−0.235232
$666$	0	0
$667$	42.4357	1.64312
$668$	0	0
$669$	11.1291	0.430274
$670$	0	0
$671$	−12.8739	−0.496993
$672$	0	0
$673$	5.82865	0.224678	0.112339	−	0.993670i	$-0.464166\pi$
$673$	5.82865	0.224678	0.112339	+	0.993670i	$0.464166\pi$
$674$	0	0
$675$	3.71718	0.143074
$676$	0	0
$677$	41.8819	1.60965	0.804827	−	0.593510i	$-0.202258\pi$
$677$	41.8819	1.60965	0.804827	+	0.593510i	$0.202258\pi$
$678$	0	0
$679$	3.83678	0.147242
$680$	0	0
$681$	6.62478	0.253862
$682$	0	0
$683$	−3.06653	−0.117338	−0.0586688	−	0.998278i	$-0.518686\pi$
$683$	−3.06653	−0.117338	−0.0586688	+	0.998278i	$0.518686\pi$
$684$	0	0
$685$	13.8398	0.528792
$686$	0	0
$687$	12.0465	0.459601
$688$	0	0
$689$	−3.83504	−0.146104
$690$	0	0
$691$	10.1542	0.386285	0.193142	−	0.981171i	$-0.438132\pi$
$691$	10.1542	0.386285	0.193142	+	0.981171i	$0.438132\pi$
$692$	0	0
$693$	−2.83094	−0.107539
$694$	0	0
$695$	−8.00439	−0.303624
$696$	0	0
$697$	−28.3900	−1.07535
$698$	0	0
$699$	3.35693	0.126971
$700$	0	0
$701$	22.7469	0.859137	0.429569	−	0.903034i	$-0.358666\pi$
$701$	22.7469	0.859137	0.429569	+	0.903034i	$0.358666\pi$
$702$	0	0
$703$	4.95219	0.186775
$704$	0	0
$705$	1.95109	0.0734824
$706$	0	0
$707$	11.4927	0.432229
$708$	0	0
$709$	33.7371	1.26702	0.633512	−	0.773733i	$-0.281613\pi$
$709$	33.7371	1.26702	0.633512	+	0.773733i	$0.281613\pi$
$710$	0	0
$711$	−23.8143	−0.893108
$712$	0	0
$713$	13.6905	0.512713
$714$	0	0
$715$	−5.47834	−0.204878
$716$	0	0
$717$	3.38747	0.126508
$718$	0	0
$719$	−15.4523	−0.576275	−0.288138	−	0.957589i	$-0.593036\pi$
$719$	−15.4523	−0.576275	−0.288138	+	0.957589i	$0.593036\pi$
$720$	0	0
$721$	−4.71928	−0.175755
$722$	0	0
$723$	−9.48181	−0.352632
$724$	0	0
$725$	−6.73915	−0.250286
$726$	0	0
$727$	−15.8604	−0.588230	−0.294115	−	0.955770i	$-0.595025\pi$
$727$	−15.8604	−0.588230	−0.294115	+	0.955770i	$0.595025\pi$
$728$	0	0
$729$	−5.71594	−0.211701
$730$	0	0
$731$	7.84256	0.290068
$732$	0	0
$733$	−17.7300	−0.654871	−0.327435	−	0.944874i	$-0.606184\pi$
$733$	−17.7300	−0.654871	−0.327435	+	0.944874i	$0.606184\pi$
$734$	0	0
$735$	0.669559	0.0246970
$736$	0	0
$737$	−8.47666	−0.312242
$738$	0	0
$739$	−2.34732	−0.0863475	−0.0431737	−	0.999068i	$-0.513747\pi$
$739$	−2.34732	−0.0863475	−0.0431737	+	0.999068i	$0.513747\pi$
$740$	0	0
$741$	−20.0559	−0.736772
$742$	0	0
$743$	32.9350	1.20827	0.604135	−	0.796882i	$-0.293519\pi$
$743$	32.9350	1.20827	0.604135	+	0.796882i	$0.293519\pi$
$744$	0	0
$745$	−18.1724	−0.665786
$746$	0	0
$747$	26.5658	0.971992
$748$	0	0
$749$	−11.6617	−0.426107
$750$	0	0
$751$	38.5947	1.40834	0.704170	−	0.710031i	$-0.251319\pi$
$751$	38.5947	1.40834	0.704170	+	0.710031i	$0.251319\pi$
$752$	0	0
$753$	2.76301	0.100690
$754$	0	0
$755$	8.97991	0.326812
$756$	0	0
$757$	50.2783	1.82739	0.913697	−	0.406395i	$-0.133214\pi$
$757$	50.2783	1.82739	0.913697	+	0.406395i	$0.133214\pi$
$758$	0	0
$759$	−4.67754	−0.169784
$760$	0	0
$761$	34.4317	1.24815	0.624075	−	0.781365i	$-0.285476\pi$
$761$	34.4317	1.24815	0.624075	+	0.781365i	$0.285476\pi$
$762$	0	0
$763$	9.12274	0.330265
$764$	0	0
$765$	−20.0118	−0.723528
$766$	0	0
$767$	16.6696	0.601905
$768$	0	0
$769$	26.7812	0.965756	0.482878	−	0.875688i	$-0.339591\pi$
$769$	26.7812	0.965756	0.482878	+	0.875688i	$0.339591\pi$
$770$	0	0
$771$	9.11995	0.328447
$772$	0	0
$773$	−5.40578	−0.194433	−0.0972163	−	0.995263i	$-0.530994\pi$
$773$	−5.40578	−0.194433	−0.0972163	+	0.995263i	$0.530994\pi$
$774$	0	0
$775$	−2.17417	−0.0780983
$776$	0	0
$777$	−0.546612	−0.0196096
$778$	0	0
$779$	−21.9591	−0.786765
$780$	0	0
$781$	−0.287941	−0.0103033
$782$	0	0
$783$	−25.0507	−0.895237
$784$	0	0
$785$	17.0794	0.609589
$786$	0	0
$787$	−37.8528	−1.34931	−0.674653	−	0.738135i	$-0.735707\pi$
$787$	−37.8528	−1.34931	−0.674653	+	0.738135i	$0.735707\pi$
$788$	0	0
$789$	5.57412	0.198444
$790$	0	0
$791$	−1.77032	−0.0629454
$792$	0	0
$793$	−57.3001	−2.03479
$794$	0	0
$795$	0.520011	0.0184429
$796$	0	0
$797$	−43.2403	−1.53165	−0.765825	−	0.643049i	$-0.777669\pi$
$797$	−43.2403	−1.53165	−0.765825	+	0.643049i	$0.777669\pi$
$798$	0	0
$799$	−22.8532	−0.808488
$800$	0	0
$801$	2.13624	0.0754802
$802$	0	0
$803$	12.2212	0.431275
$804$	0	0
$805$	−6.29690	−0.221936
$806$	0	0
$807$	−11.3129	−0.398234
$808$	0	0
$809$	−42.2924	−1.48692	−0.743460	−	0.668780i	$-0.766817\pi$
$809$	−42.2924	−1.48692	−0.743460	+	0.668780i	$0.766817\pi$
$810$	0	0
$811$	41.3214	1.45099	0.725495	−	0.688227i	$-0.241611\pi$
$811$	41.3214	1.45099	0.725495	+	0.688227i	$0.241611\pi$
$812$	0	0
$813$	12.3031	0.431488
$814$	0	0
$815$	1.73122	0.0606420
$816$	0	0
$817$	6.06606	0.212225
$818$	0	0
$819$	−12.6001	−0.440284
$820$	0	0
$821$	25.3954	0.886307	0.443153	−	0.896446i	$-0.353860\pi$
$821$	25.3954	0.886307	0.443153	+	0.896446i	$0.353860\pi$
$822$	0	0
$823$	−27.7476	−0.967220	−0.483610	−	0.875284i	$-0.660675\pi$
$823$	−27.7476	−0.967220	−0.483610	+	0.875284i	$0.660675\pi$
$824$	0	0
$825$	0.742833	0.0258621
$826$	0	0
$827$	24.1463	0.839650	0.419825	−	0.907605i	$-0.362091\pi$
$827$	24.1463	0.839650	0.419825	+	0.907605i	$0.362091\pi$
$828$	0	0
$829$	−23.2394	−0.807138	−0.403569	−	0.914949i	$-0.632230\pi$
$829$	−23.2394	−0.807138	−0.403569	+	0.914949i	$0.632230\pi$
$830$	0	0
$831$	−12.5246	−0.434474
$832$	0	0
$833$	−7.84256	−0.271729
$834$	0	0
$835$	1.54701	0.0535365
$836$	0	0
$837$	−8.08177	−0.279347
$838$	0	0
$839$	−36.9185	−1.27457	−0.637283	−	0.770629i	$-0.719942\pi$
$839$	−36.9185	−1.27457	−0.637283	+	0.770629i	$0.719942\pi$
$840$	0	0
$841$	16.4162	0.566075
$842$	0	0
$843$	−1.35045	−0.0465120
$844$	0	0
$845$	−11.3834	−0.391599
$846$	0	0
$847$	9.76915	0.335672
$848$	0	0
$849$	1.43463	0.0492363
$850$	0	0
$851$	5.14063	0.176219
$852$	0	0
$853$	−56.4658	−1.93335	−0.966677	−	0.256001i	$-0.917595\pi$
$853$	−56.4658	−1.93335	−0.966677	+	0.256001i	$0.917595\pi$
$854$	0	0
$855$	−15.4787	−0.529361
$856$	0	0
$857$	−9.36288	−0.319830	−0.159915	−	0.987131i	$-0.551122\pi$
$857$	−9.36288	−0.319830	−0.159915	+	0.987131i	$0.551122\pi$
$858$	0	0
$859$	−3.25944	−0.111211	−0.0556053	−	0.998453i	$-0.517709\pi$
$859$	−3.25944	−0.111211	−0.0556053	+	0.998453i	$0.517709\pi$
$860$	0	0
$861$	2.42379	0.0826027
$862$	0	0
$863$	−20.7206	−0.705336	−0.352668	−	0.935748i	$-0.614726\pi$
$863$	−20.7206	−0.705336	−0.352668	+	0.935748i	$0.614726\pi$
$864$	0	0
$865$	22.9973	0.781931
$866$	0	0
$867$	−29.7992	−1.01203
$868$	0	0
$869$	−10.3541	−0.351239
$870$	0	0
$871$	−37.7284	−1.27838
$872$	0	0
$873$	9.79028	0.331351
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	41.3163	1.39515	0.697576	−	0.716510i	$-0.254262\pi$
$877$	41.3163	1.39515	0.697576	+	0.716510i	$0.254262\pi$
$878$	0	0
$879$	−10.4588	−0.352766
$880$	0	0
$881$	0.795227	0.0267919	0.0133959	−	0.999910i	$-0.495736\pi$
$881$	0.795227	0.0267919	0.0133959	+	0.999910i	$0.495736\pi$
$882$	0	0
$883$	−26.8395	−0.903220	−0.451610	−	0.892215i	$-0.649150\pi$
$883$	−26.8395	−0.903220	−0.451610	+	0.892215i	$0.649150\pi$
$884$	0	0
$885$	−2.26031	−0.0759795
$886$	0	0
$887$	−35.8175	−1.20263	−0.601317	−	0.799010i	$-0.705357\pi$
$887$	−35.8175	−1.20263	−0.601317	+	0.799010i	$0.705357\pi$
$888$	0	0
$889$	11.0246	0.369752
$890$	0	0
$891$	−5.73157	−0.192015
$892$	0	0
$893$	−17.6765	−0.591521
$894$	0	0
$895$	23.3522	0.780578
$896$	0	0
$897$	−20.8191	−0.695129
$898$	0	0
$899$	14.6520	0.488673
$900$	0	0
$901$	−6.09090	−0.202917
$902$	0	0
$903$	−0.669559	−0.0222815
$904$	0	0
$905$	−15.3821	−0.511317
$906$	0	0
$907$	39.1140	1.29876	0.649380	−	0.760464i	$-0.275029\pi$
$907$	39.1140	1.29876	0.649380	+	0.760464i	$0.275029\pi$
$908$	0	0
$909$	29.3259	0.972680
$910$	0	0
$911$	−7.66788	−0.254048	−0.127024	−	0.991900i	$-0.540543\pi$
$911$	−7.66788	−0.254048	−0.127024	+	0.991900i	$0.540543\pi$
$912$	0	0
$913$	11.5504	0.382263
$914$	0	0
$915$	7.76958	0.256854
$916$	0	0
$917$	2.14594	0.0708652
$918$	0	0
$919$	45.1283	1.48865	0.744323	−	0.667820i	$-0.232772\pi$
$919$	45.1283	1.48865	0.744323	+	0.667820i	$0.232772\pi$
$920$	0	0
$921$	0.678680	0.0223633
$922$	0	0
$923$	−1.28159	−0.0421839
$924$	0	0
$925$	−0.816376	−0.0268423
$926$	0	0
$927$	−12.0421	−0.395516
$928$	0	0
$929$	22.4761	0.737417	0.368708	−	0.929545i	$-0.379800\pi$
$929$	22.4761	0.737417	0.368708	+	0.929545i	$0.379800\pi$
$930$	0	0
$931$	−6.06606	−0.198807
$932$	0	0
$933$	10.0842	0.330143
$934$	0	0
$935$	−8.70082	−0.284547
$936$	0	0
$937$	40.5300	1.32406	0.662029	−	0.749478i	$-0.269696\pi$
$937$	40.5300	1.32406	0.662029	+	0.749478i	$0.269696\pi$
$938$	0	0
$939$	−1.19233	−0.0389103
$940$	0	0
$941$	−58.4823	−1.90647	−0.953234	−	0.302234i	$-0.902267\pi$
$941$	−58.4823	−1.90647	−0.953234	+	0.302234i	$0.902267\pi$
$942$	0	0
$943$	−22.7947	−0.742297
$944$	0	0
$945$	3.71718	0.120920
$946$	0	0
$947$	−16.1359	−0.524345	−0.262173	−	0.965021i	$-0.584439\pi$
$947$	−16.1359	−0.524345	−0.262173	+	0.965021i	$0.584439\pi$
$948$	0	0
$949$	54.3947	1.76573
$950$	0	0
$951$	12.2777	0.398131
$952$	0	0
$953$	−51.7220	−1.67544	−0.837720	−	0.546100i	$-0.816112\pi$
$953$	−51.7220	−1.67544	−0.837720	+	0.546100i	$0.816112\pi$
$954$	0	0
$955$	−12.4514	−0.402919
$956$	0	0
$957$	−5.00606	−0.161823
$958$	0	0
$959$	13.8398	0.446911
$960$	0	0
$961$	−26.2730	−0.847516
$962$	0	0
$963$	−29.7569	−0.958904
$964$	0	0
$965$	−8.08445	−0.260248
$966$	0	0
$967$	20.6862	0.665224	0.332612	−	0.943064i	$-0.392070\pi$
$967$	20.6862	0.665224	0.332612	+	0.943064i	$0.392070\pi$
$968$	0	0
$969$	−31.8532	−1.02327
$970$	0	0
$971$	27.2312	0.873889	0.436945	−	0.899488i	$-0.356061\pi$
$971$	27.2312	0.873889	0.436945	+	0.899488i	$0.356061\pi$
$972$	0	0
$973$	−8.00439	−0.256609
$974$	0	0
$975$	3.30625	0.105885
$976$	0	0
$977$	−25.5112	−0.816175	−0.408087	−	0.912943i	$-0.633804\pi$
$977$	−25.5112	−0.816175	−0.408087	+	0.912943i	$0.633804\pi$
$978$	0	0
$979$	0.928803	0.0296847
$980$	0	0
$981$	23.2784	0.743222
$982$	0	0
$983$	−38.7437	−1.23573	−0.617866	−	0.786283i	$-0.712003\pi$
$983$	−38.7437	−1.23573	−0.617866	+	0.786283i	$0.712003\pi$
$984$	0	0
$985$	−6.27574	−0.199962
$986$	0	0
$987$	1.95109	0.0621039
$988$	0	0
$989$	6.29690	0.200230
$990$	0	0
$991$	42.3118	1.34408	0.672039	−	0.740515i	$-0.265419\pi$
$991$	42.3118	1.34408	0.672039	+	0.740515i	$0.265419\pi$
$992$	0	0
$993$	6.87185	0.218072
$994$	0	0
$995$	17.9267	0.568313
$996$	0	0
$997$	−14.6936	−0.465350	−0.232675	−	0.972555i	$-0.574748\pi$
$997$	−14.6936	−0.465350	−0.232675	+	0.972555i	$0.574748\pi$
$998$	0	0
$999$	−3.03462	−0.0960111

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6020.2.a.j.1.6	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6020.2.a.j.1.6	✓	13	1.1	even	1	trivial

Overview	Random
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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)

\(f(q)\)	\(=\)	\(q-0.669559 q^{3} -1.00000 q^{5} -1.00000 q^{7} -2.55169 q^{9} +O(q^{10})\)	\(q-0.669559 q^{3} -1.00000 q^{5} -1.00000 q^{7} -2.55169 q^{9} -1.10944 q^{11} -4.93795 q^{13} +0.669559 q^{15} -7.84256 q^{17} -6.06606 q^{19} +0.669559 q^{21} -6.29690 q^{23} +1.00000 q^{25} +3.71718 q^{27} -6.73915 q^{29} -2.17417 q^{31} +0.742833 q^{33} +1.00000 q^{35} -0.816376 q^{37} +3.30625 q^{39} +3.61999 q^{41} -1.00000 q^{43} +2.55169 q^{45} +2.91400 q^{47} +1.00000 q^{49} +5.25105 q^{51} +0.776647 q^{53} +1.10944 q^{55} +4.06158 q^{57} -3.37582 q^{59} +11.6040 q^{61} +2.55169 q^{63} +4.93795 q^{65} +7.64051 q^{67} +4.21614 q^{69} +0.259538 q^{71} -11.0156 q^{73} -0.669559 q^{75} +1.10944 q^{77} +9.33277 q^{79} +5.16620 q^{81} -10.4111 q^{83} +7.84256 q^{85} +4.51226 q^{87} -0.837184 q^{89} +4.93795 q^{91} +1.45573 q^{93} +6.06606 q^{95} -3.83678 q^{97} +2.83094 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - q^{3} - 13 q^{5} - 13 q^{7} + 18 q^{9}+O(q^{10}) \) 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

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Database

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