LMFDB - Embedded newform 6028.2.a.f.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6028,2,Mod(1,6028)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6028, 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("6028.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6028 = 2^{2} \cdot 11 \cdot 137 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6028.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:	$48.1338223384$
Analytic rank:	$0$
Dimension:	$29$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Character	$\chi$	$=$	6028.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.14725 q^{3} +4.20688 q^{5} -0.775275 q^{7} -1.68381 q^{9} +O(q^{10})$	$q-1.14725 q^{3} +4.20688 q^{5} -0.775275 q^{7} -1.68381 q^{9} +1.00000 q^{11} +4.03596 q^{13} -4.82636 q^{15} -4.22872 q^{17} +2.69788 q^{19} +0.889438 q^{21} +6.08767 q^{23} +12.6978 q^{25} +5.37352 q^{27} -3.27567 q^{29} +3.97284 q^{31} -1.14725 q^{33} -3.26149 q^{35} -1.35413 q^{37} -4.63027 q^{39} -2.86060 q^{41} -4.82718 q^{43} -7.08357 q^{45} +3.30400 q^{47} -6.39895 q^{49} +4.85142 q^{51} +2.03493 q^{53} +4.20688 q^{55} -3.09515 q^{57} +2.41887 q^{59} -8.49867 q^{61} +1.30541 q^{63} +16.9788 q^{65} +10.7072 q^{67} -6.98411 q^{69} -0.849040 q^{71} -2.44878 q^{73} -14.5676 q^{75} -0.775275 q^{77} +12.8503 q^{79} -1.11338 q^{81} +8.06672 q^{83} -17.7897 q^{85} +3.75803 q^{87} +9.43566 q^{89} -3.12897 q^{91} -4.55786 q^{93} +11.3496 q^{95} -8.93235 q^{97} -1.68381 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 29 q + 14 q^{3} + 9 q^{5} + 14 q^{7} + 43 q^{9}+O(q^{10}) $ 29 * q + 14 * q^3 + 9 * q^5 + 14 * q^7 + 43 * q^9	$ 29 q + 14 q^{3} + 9 q^{5} + 14 q^{7} + 43 q^{9} + 29 q^{11} + 10 q^{15} + 29 q^{17} + 7 q^{19} + 2 q^{21} + 36 q^{23} + 36 q^{25} + 50 q^{27} + 9 q^{29} + 28 q^{31} + 14 q^{33} + 15 q^{35} + 25 q^{37} + 9 q^{39} + 19 q^{41} + 23 q^{43} + 5 q^{45} + 27 q^{47} + 27 q^{49} + 13 q^{51} + 4 q^{53} + 9 q^{55} + 14 q^{57} + 40 q^{59} + 20 q^{61} - 17 q^{63} + 9 q^{65} + 59 q^{67} + 30 q^{69} + 29 q^{71} - 5 q^{73} + 46 q^{75} + 14 q^{77} + 29 q^{79} + 61 q^{81} + 35 q^{83} - 57 q^{85} + 45 q^{87} + 39 q^{89} + 45 q^{91} - 8 q^{93} + q^{95} + 55 q^{97} + 43 q^{99}+O(q^{100}) $ 29 * q + 14 * q^3 + 9 * q^5 + 14 * q^7 + 43 * q^9 + 29 * q^11 + 10 * q^15 + 29 * q^17 + 7 * q^19 + 2 * q^21 + 36 * q^23 + 36 * q^25 + 50 * q^27 + 9 * q^29 + 28 * q^31 + 14 * q^33 + 15 * q^35 + 25 * q^37 + 9 * q^39 + 19 * q^41 + 23 * q^43 + 5 * q^45 + 27 * q^47 + 27 * q^49 + 13 * q^51 + 4 * q^53 + 9 * q^55 + 14 * q^57 + 40 * q^59 + 20 * q^61 - 17 * q^63 + 9 * q^65 + 59 * q^67 + 30 * q^69 + 29 * q^71 - 5 * q^73 + 46 * q^75 + 14 * q^77 + 29 * q^79 + 61 * q^81 + 35 * q^83 - 57 * q^85 + 45 * q^87 + 39 * q^89 + 45 * q^91 - 8 * q^93 + q^95 + 55 * q^97 + 43 * 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
$3$	−1.14725	−0.662368	−0.331184	−	0.943566i	$-0.607448\pi$
$3$	−1.14725	−0.662368	−0.331184	+	0.943566i	$0.607448\pi$
$4$	0	0
$5$	4.20688	1.88137	0.940686	−	0.339277i	$-0.110183\pi$
$5$	4.20688	1.88137	0.940686	+	0.339277i	$0.110183\pi$
$6$	0	0
$7$	−0.775275	−0.293026	−0.146513	−	0.989209i	$-0.546805\pi$
$7$	−0.775275	−0.293026	−0.146513	+	0.989209i	$0.546805\pi$
$8$	0	0
$9$	−1.68381	−0.561269
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	4.03596	1.11937	0.559686	−	0.828704i	$-0.310922\pi$
$13$	4.03596	1.11937	0.559686	+	0.828704i	$0.310922\pi$
$14$	0	0
$15$	−4.82636	−1.24616
$16$	0	0
$17$	−4.22872	−1.02561	−0.512807	−	0.858504i	$-0.671394\pi$
$17$	−4.22872	−1.02561	−0.512807	+	0.858504i	$0.671394\pi$
$18$	0	0
$19$	2.69788	0.618936	0.309468	−	0.950910i	$-0.399849\pi$
$19$	2.69788	0.618936	0.309468	+	0.950910i	$0.399849\pi$
$20$	0	0
$21$	0.889438	0.194091
$22$	0	0
$23$	6.08767	1.26937	0.634684	−	0.772772i	$-0.281130\pi$
$23$	6.08767	1.26937	0.634684	+	0.772772i	$0.281130\pi$
$24$	0	0
$25$	12.6978	2.53956
$26$	0	0
$27$	5.37352	1.03413
$28$	0	0
$29$	−3.27567	−0.608277	−0.304138	−	0.952628i	$-0.598369\pi$
$29$	−3.27567	−0.608277	−0.304138	+	0.952628i	$0.598369\pi$
$30$	0	0
$31$	3.97284	0.713543	0.356771	−	0.934192i	$-0.383878\pi$
$31$	3.97284	0.713543	0.356771	+	0.934192i	$0.383878\pi$
$32$	0	0
$33$	−1.14725	−0.199711
$34$	0	0
$35$	−3.26149	−0.551292
$36$	0	0
$37$	−1.35413	−0.222617	−0.111308	−	0.993786i	$-0.535504\pi$
$37$	−1.35413	−0.222617	−0.111308	+	0.993786i	$0.535504\pi$
$38$	0	0
$39$	−4.63027	−0.741437
$40$	0	0
$41$	−2.86060	−0.446751	−0.223376	−	0.974732i	$-0.571708\pi$
$41$	−2.86060	−0.446751	−0.223376	+	0.974732i	$0.571708\pi$
$42$	0	0
$43$	−4.82718	−0.736138	−0.368069	−	0.929798i	$-0.619981\pi$
$43$	−4.82718	−0.736138	−0.368069	+	0.929798i	$0.619981\pi$
$44$	0	0
$45$	−7.08357	−1.05596
$46$	0	0
$47$	3.30400	0.481938	0.240969	−	0.970533i	$-0.422535\pi$
$47$	3.30400	0.481938	0.240969	+	0.970533i	$0.422535\pi$
$48$	0	0
$49$	−6.39895	−0.914136
$50$	0	0
$51$	4.85142	0.679334
$52$	0	0
$53$	2.03493	0.279520	0.139760	−	0.990185i	$-0.455367\pi$
$53$	2.03493	0.279520	0.139760	+	0.990185i	$0.455367\pi$
$54$	0	0
$55$	4.20688	0.567255
$56$	0	0
$57$	−3.09515	−0.409963
$58$	0	0
$59$	2.41887	0.314910	0.157455	−	0.987526i	$-0.449671\pi$
$59$	2.41887	0.314910	0.157455	+	0.987526i	$0.449671\pi$
$60$	0	0
$61$	−8.49867	−1.08814	−0.544072	−	0.839039i	$-0.683118\pi$
$61$	−8.49867	−1.08814	−0.544072	+	0.839039i	$0.683118\pi$
$62$	0	0
$63$	1.30541	0.164466
$64$	0	0
$65$	16.9788	2.10596
$66$	0	0
$67$	10.7072	1.30809	0.654047	−	0.756454i	$-0.273070\pi$
$67$	10.7072	1.30809	0.654047	+	0.756454i	$0.273070\pi$
$68$	0	0
$69$	−6.98411	−0.840789
$70$	0	0
$71$	−0.849040	−0.100762	−0.0503812	−	0.998730i	$-0.516044\pi$
$71$	−0.849040	−0.100762	−0.0503812	+	0.998730i	$0.516044\pi$
$72$	0	0
$73$	−2.44878	−0.286608	−0.143304	−	0.989679i	$-0.545773\pi$
$73$	−2.44878	−0.286608	−0.143304	+	0.989679i	$0.545773\pi$
$74$	0	0
$75$	−14.5676	−1.68213
$76$	0	0
$77$	−0.775275	−0.0883507
$78$	0	0
$79$	12.8503	1.44577	0.722885	−	0.690968i	$-0.242816\pi$
$79$	12.8503	1.44577	0.722885	+	0.690968i	$0.242816\pi$
$80$	0	0
$81$	−1.11338	−0.123709
$82$	0	0
$83$	8.06672	0.885437	0.442719	−	0.896661i	$-0.354014\pi$
$83$	8.06672	0.885437	0.442719	+	0.896661i	$0.354014\pi$
$84$	0	0
$85$	−17.7897	−1.92956
$86$	0	0
$87$	3.75803	0.402903
$88$	0	0
$89$	9.43566	1.00018	0.500089	−	0.865974i	$-0.333301\pi$
$89$	9.43566	1.00018	0.500089	+	0.865974i	$0.333301\pi$
$90$	0	0
$91$	−3.12897	−0.328006
$92$	0	0
$93$	−4.55786	−0.472628
$94$	0	0
$95$	11.3496	1.16445
$96$	0	0
$97$	−8.93235	−0.906943	−0.453472	−	0.891271i	$-0.649815\pi$
$97$	−8.93235	−0.906943	−0.453472	+	0.891271i	$0.649815\pi$
$98$	0	0
$99$	−1.68381	−0.169229
$100$	0	0
$101$	5.64384	0.561583	0.280792	−	0.959769i	$-0.409403\pi$
$101$	5.64384	0.561583	0.280792	+	0.959769i	$0.409403\pi$
$102$	0	0
$103$	5.31641	0.523842	0.261921	−	0.965089i	$-0.415644\pi$
$103$	5.31641	0.523842	0.261921	+	0.965089i	$0.415644\pi$
$104$	0	0
$105$	3.74175	0.365158
$106$	0	0
$107$	−1.44357	−0.139555	−0.0697777	−	0.997563i	$-0.522229\pi$
$107$	−1.44357	−0.139555	−0.0697777	+	0.997563i	$0.522229\pi$
$108$	0	0
$109$	7.22240	0.691780	0.345890	−	0.938275i	$-0.387577\pi$
$109$	7.22240	0.691780	0.345890	+	0.938275i	$0.387577\pi$
$110$	0	0
$111$	1.55353	0.147454
$112$	0	0
$113$	−8.17874	−0.769391	−0.384696	−	0.923043i	$-0.625694\pi$
$113$	−8.17874	−0.769391	−0.384696	+	0.923043i	$0.625694\pi$
$114$	0	0
$115$	25.6101	2.38815
$116$	0	0
$117$	−6.79577	−0.628269
$118$	0	0
$119$	3.27842	0.300532
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	3.28184	0.295914
$124$	0	0
$125$	32.3838	2.89649
$126$	0	0
$127$	−13.7554	−1.22059	−0.610296	−	0.792174i	$-0.708949\pi$
$127$	−13.7554	−1.22059	−0.610296	+	0.792174i	$0.708949\pi$
$128$	0	0
$129$	5.53801	0.487594
$130$	0	0
$131$	5.78253	0.505222	0.252611	−	0.967568i	$-0.418711\pi$
$131$	5.78253	0.505222	0.252611	+	0.967568i	$0.418711\pi$
$132$	0	0
$133$	−2.09160	−0.181364
$134$	0	0
$135$	22.6057	1.94559
$136$	0	0
$137$	1.00000	0.0854358
$137$	1.00000	0.0854358
$138$	0	0
$139$	1.83102	0.155306	0.0776528	−	0.996980i	$-0.475257\pi$
$139$	1.83102	0.155306	0.0776528	+	0.996980i	$0.475257\pi$
$140$	0	0
$141$	−3.79053	−0.319220
$142$	0	0
$143$	4.03596	0.337504
$144$	0	0
$145$	−13.7803	−1.14440
$146$	0	0
$147$	7.34123	0.605494
$148$	0	0
$149$	8.32240	0.681798	0.340899	−	0.940100i	$-0.389269\pi$
$149$	8.32240	0.681798	0.340899	+	0.940100i	$0.389269\pi$
$150$	0	0
$151$	0.493357	0.0401489	0.0200744	−	0.999798i	$-0.493610\pi$
$151$	0.493357	0.0401489	0.0200744	+	0.999798i	$0.493610\pi$
$152$	0	0
$153$	7.12034	0.575645
$154$	0	0
$155$	16.7132	1.34244
$156$	0	0
$157$	7.10942	0.567394	0.283697	−	0.958914i	$-0.408439\pi$
$157$	7.10942	0.567394	0.283697	+	0.958914i	$0.408439\pi$
$158$	0	0
$159$	−2.33459	−0.185145
$160$	0	0
$161$	−4.71962	−0.371958
$162$	0	0
$163$	−1.25578	−0.0983606	−0.0491803	−	0.998790i	$-0.515661\pi$
$163$	−1.25578	−0.0983606	−0.0491803	+	0.998790i	$0.515661\pi$
$164$	0	0
$165$	−4.82636	−0.375732
$166$	0	0
$167$	−0.292854	−0.0226617	−0.0113308	−	0.999936i	$-0.503607\pi$
$167$	−0.292854	−0.0226617	−0.0113308	+	0.999936i	$0.503607\pi$
$168$	0	0
$169$	3.28894	0.252996
$170$	0	0
$171$	−4.54270	−0.347389
$172$	0	0
$173$	−23.0750	−1.75436	−0.877181	−	0.480161i	$-0.840578\pi$
$173$	−23.0750	−1.75436	−0.877181	+	0.480161i	$0.840578\pi$
$174$	0	0
$175$	−9.84430	−0.744159
$176$	0	0
$177$	−2.77506	−0.208586
$178$	0	0
$179$	13.4036	1.00183	0.500916	−	0.865496i	$-0.332997\pi$
$179$	13.4036	1.00183	0.500916	+	0.865496i	$0.332997\pi$
$180$	0	0
$181$	8.24159	0.612593	0.306296	−	0.951936i	$-0.400910\pi$
$181$	8.24159	0.612593	0.306296	+	0.951936i	$0.400910\pi$
$182$	0	0
$183$	9.75014	0.720751
$184$	0	0
$185$	−5.69664	−0.418825
$186$	0	0
$187$	−4.22872	−0.309234
$188$	0	0
$189$	−4.16595	−0.303029
$190$	0	0
$191$	−14.6226	−1.05805	−0.529027	−	0.848605i	$-0.677443\pi$
$191$	−14.6226	−1.05805	−0.529027	+	0.848605i	$0.677443\pi$
$192$	0	0
$193$	−24.6243	−1.77250	−0.886248	−	0.463211i	$-0.846697\pi$
$193$	−24.6243	−1.77250	−0.886248	+	0.463211i	$0.846697\pi$
$194$	0	0
$195$	−19.4790	−1.39492
$196$	0	0
$197$	15.0367	1.07132	0.535660	−	0.844434i	$-0.320063\pi$
$197$	15.0367	1.07132	0.535660	+	0.844434i	$0.320063\pi$
$198$	0	0
$199$	15.1162	1.07156	0.535781	−	0.844357i	$-0.320017\pi$
$199$	15.1162	1.07156	0.535781	+	0.844357i	$0.320017\pi$
$200$	0	0
$201$	−12.2839	−0.866440
$202$	0	0
$203$	2.53954	0.178241
$204$	0	0
$205$	−12.0342	−0.840506
$206$	0	0
$207$	−10.2505	−0.712456
$208$	0	0
$209$	2.69788	0.186616
$210$	0	0
$211$	1.65374	0.113848	0.0569242	−	0.998379i	$-0.481871\pi$
$211$	1.65374	0.113848	0.0569242	+	0.998379i	$0.481871\pi$
$212$	0	0
$213$	0.974065	0.0667418
$214$	0	0
$215$	−20.3074	−1.38495
$216$	0	0
$217$	−3.08004	−0.209087
$218$	0	0
$219$	2.80937	0.189840
$220$	0	0
$221$	−17.0669	−1.14805
$222$	0	0
$223$	−4.37312	−0.292846	−0.146423	−	0.989222i	$-0.546776\pi$
$223$	−4.37312	−0.292846	−0.146423	+	0.989222i	$0.546776\pi$
$224$	0	0
$225$	−21.3807	−1.42538
$226$	0	0
$227$	1.99900	0.132678	0.0663391	−	0.997797i	$-0.478868\pi$
$227$	1.99900	0.132678	0.0663391	+	0.997797i	$0.478868\pi$
$228$	0	0
$229$	17.6517	1.16646	0.583229	−	0.812308i	$-0.301789\pi$
$229$	17.6517	1.16646	0.583229	+	0.812308i	$0.301789\pi$
$230$	0	0
$231$	0.889438	0.0585207
$232$	0	0
$233$	−0.379930	−0.0248901	−0.0124450	−	0.999923i	$-0.503961\pi$
$233$	−0.379930	−0.0248901	−0.0124450	+	0.999923i	$0.503961\pi$
$234$	0	0
$235$	13.8995	0.906705
$236$	0	0
$237$	−14.7426	−0.957632
$238$	0	0
$239$	−12.9937	−0.840496	−0.420248	−	0.907409i	$-0.638057\pi$
$239$	−12.9937	−0.840496	−0.420248	+	0.907409i	$0.638057\pi$
$240$	0	0
$241$	1.47576	0.0950620	0.0475310	−	0.998870i	$-0.484865\pi$
$241$	1.47576	0.0950620	0.0475310	+	0.998870i	$0.484865\pi$
$242$	0	0
$243$	−14.8432	−0.952194
$244$	0	0
$245$	−26.9196	−1.71983
$246$	0	0
$247$	10.8885	0.692820
$248$	0	0
$249$	−9.25458	−0.586485
$250$	0	0
$251$	4.10748	0.259262	0.129631	−	0.991562i	$-0.458621\pi$
$251$	4.10748	0.259262	0.129631	+	0.991562i	$0.458621\pi$
$252$	0	0
$253$	6.08767	0.382729
$254$	0	0
$255$	20.4093	1.27808
$256$	0	0
$257$	−3.51726	−0.219401	−0.109700	−	0.993965i	$-0.534989\pi$
$257$	−3.51726	−0.219401	−0.109700	+	0.993965i	$0.534989\pi$
$258$	0	0
$259$	1.04982	0.0652326
$260$	0	0
$261$	5.51560	0.341407
$262$	0	0
$263$	8.91816	0.549917	0.274959	−	0.961456i	$-0.411336\pi$
$263$	8.91816	0.549917	0.274959	+	0.961456i	$0.411336\pi$
$264$	0	0
$265$	8.56072	0.525881
$266$	0	0
$267$	−10.8251	−0.662486
$268$	0	0
$269$	13.6688	0.833399	0.416699	−	0.909044i	$-0.363187\pi$
$269$	13.6688	0.833399	0.416699	+	0.909044i	$0.363187\pi$
$270$	0	0
$271$	−10.8029	−0.656228	−0.328114	−	0.944638i	$-0.606413\pi$
$271$	−10.8029	−0.656228	−0.328114	+	0.944638i	$0.606413\pi$
$272$	0	0
$273$	3.58973	0.217260
$274$	0	0
$275$	12.6978	0.765707
$276$	0	0
$277$	1.36310	0.0819005	0.0409503	−	0.999161i	$-0.486961\pi$
$277$	1.36310	0.0819005	0.0409503	+	0.999161i	$0.486961\pi$
$278$	0	0
$279$	−6.68949	−0.400489
$280$	0	0
$281$	−6.88831	−0.410922	−0.205461	−	0.978665i	$-0.565869\pi$
$281$	−6.88831	−0.410922	−0.205461	+	0.978665i	$0.565869\pi$
$282$	0	0
$283$	4.60663	0.273836	0.136918	−	0.990582i	$-0.456280\pi$
$283$	4.60663	0.273836	0.136918	+	0.990582i	$0.456280\pi$
$284$	0	0
$285$	−13.0209	−0.771293
$286$	0	0
$287$	2.21775	0.130910
$288$	0	0
$289$	0.882047	0.0518851
$290$	0	0
$291$	10.2477	0.600730
$292$	0	0
$293$	8.26261	0.482707	0.241353	−	0.970437i	$-0.422409\pi$
$293$	8.26261	0.482707	0.241353	+	0.970437i	$0.422409\pi$
$294$	0	0
$295$	10.1759	0.592462
$296$	0	0
$297$	5.37352	0.311803
$298$	0	0
$299$	24.5696	1.42090
$300$	0	0
$301$	3.74239	0.215708
$302$	0	0
$303$	−6.47493	−0.371975
$304$	0	0
$305$	−35.7529	−2.04720
$306$	0	0
$307$	23.9395	1.36630	0.683150	−	0.730278i	$-0.260610\pi$
$307$	23.9395	1.36630	0.683150	+	0.730278i	$0.260610\pi$
$308$	0	0
$309$	−6.09928	−0.346976
$310$	0	0
$311$	9.39137	0.532535	0.266268	−	0.963899i	$-0.414209\pi$
$311$	9.39137	0.532535	0.266268	+	0.963899i	$0.414209\pi$
$312$	0	0
$313$	22.7797	1.28758	0.643791	−	0.765201i	$-0.277361\pi$
$313$	22.7797	1.28758	0.643791	+	0.765201i	$0.277361\pi$
$314$	0	0
$315$	5.49171	0.309423
$316$	0	0
$317$	13.1582	0.739038	0.369519	−	0.929223i	$-0.379523\pi$
$317$	13.1582	0.739038	0.369519	+	0.929223i	$0.379523\pi$
$318$	0	0
$319$	−3.27567	−0.183402
$320$	0	0
$321$	1.65615	0.0924371
$322$	0	0
$323$	−11.4086	−0.634789
$324$	0	0
$325$	51.2478	2.84272
$326$	0	0
$327$	−8.28593	−0.458213
$328$	0	0
$329$	−2.56151	−0.141221
$330$	0	0
$331$	−10.0345	−0.551544	−0.275772	−	0.961223i	$-0.588933\pi$
$331$	−10.0345	−0.551544	−0.275772	+	0.961223i	$0.588933\pi$
$332$	0	0
$333$	2.28008	0.124948
$334$	0	0
$335$	45.0440	2.46101
$336$	0	0
$337$	−14.6007	−0.795350	−0.397675	−	0.917526i	$-0.630183\pi$
$337$	−14.6007	−0.795350	−0.397675	+	0.917526i	$0.630183\pi$
$338$	0	0
$339$	9.38310	0.509620
$340$	0	0
$341$	3.97284	0.215141
$342$	0	0
$343$	10.3879	0.560892
$344$	0	0
$345$	−29.3813	−1.58184
$346$	0	0
$347$	23.0927	1.23968	0.619840	−	0.784728i	$-0.287197\pi$
$347$	23.0927	1.23968	0.619840	+	0.784728i	$0.287197\pi$
$348$	0	0
$349$	18.8957	1.01146	0.505731	−	0.862691i	$-0.331223\pi$
$349$	18.8957	1.01146	0.505731	+	0.862691i	$0.331223\pi$
$350$	0	0
$351$	21.6873	1.15758
$352$	0	0
$353$	1.22846	0.0653843	0.0326921	−	0.999465i	$-0.489592\pi$
$353$	1.22846	0.0653843	0.0326921	+	0.999465i	$0.489592\pi$
$354$	0	0
$355$	−3.57181	−0.189572
$356$	0	0
$357$	−3.76118	−0.199063
$358$	0	0
$359$	11.7297	0.619070	0.309535	−	0.950888i	$-0.399827\pi$
$359$	11.7297	0.619070	0.309535	+	0.950888i	$0.399827\pi$
$360$	0	0
$361$	−11.7215	−0.616919
$362$	0	0
$363$	−1.14725	−0.0602153
$364$	0	0
$365$	−10.3017	−0.539216
$366$	0	0
$367$	−25.7480	−1.34404	−0.672018	−	0.740535i	$-0.734572\pi$
$367$	−25.7480	−1.34404	−0.672018	+	0.740535i	$0.734572\pi$
$368$	0	0
$369$	4.81670	0.250747
$370$	0	0
$371$	−1.57763	−0.0819066
$372$	0	0
$373$	34.7657	1.80010	0.900051	−	0.435784i	$-0.143529\pi$
$373$	34.7657	1.80010	0.900051	+	0.435784i	$0.143529\pi$
$374$	0	0
$375$	−37.1524	−1.91854
$376$	0	0
$377$	−13.2205	−0.680889
$378$	0	0
$379$	25.9542	1.33318	0.666590	−	0.745425i	$-0.267753\pi$
$379$	25.9542	1.33318	0.666590	+	0.745425i	$0.267753\pi$
$380$	0	0
$381$	15.7809	0.808481
$382$	0	0
$383$	−0.443423	−0.0226579	−0.0113289	−	0.999936i	$-0.503606\pi$
$383$	−0.443423	−0.0226579	−0.0113289	+	0.999936i	$0.503606\pi$
$384$	0	0
$385$	−3.26149	−0.166221
$386$	0	0
$387$	8.12804	0.413171
$388$	0	0
$389$	20.5133	1.04007	0.520034	−	0.854146i	$-0.325919\pi$
$389$	20.5133	1.04007	0.520034	+	0.854146i	$0.325919\pi$
$390$	0	0
$391$	−25.7430	−1.30188
$392$	0	0
$393$	−6.63403	−0.334643
$394$	0	0
$395$	54.0596	2.72003
$396$	0	0
$397$	−5.52029	−0.277056	−0.138528	−	0.990359i	$-0.544237\pi$
$397$	−5.52029	−0.277056	−0.138528	+	0.990359i	$0.544237\pi$
$398$	0	0
$399$	2.39959	0.120130
$400$	0	0
$401$	6.77246	0.338201	0.169100	−	0.985599i	$-0.445914\pi$
$401$	6.77246	0.338201	0.169100	+	0.985599i	$0.445914\pi$
$402$	0	0
$403$	16.0342	0.798720
$404$	0	0
$405$	−4.68385	−0.232742
$406$	0	0
$407$	−1.35413	−0.0671215
$408$	0	0
$409$	−9.27477	−0.458608	−0.229304	−	0.973355i	$-0.573645\pi$
$409$	−9.27477	−0.458608	−0.229304	+	0.973355i	$0.573645\pi$
$410$	0	0
$411$	−1.14725	−0.0565899
$412$	0	0
$413$	−1.87529	−0.0922768
$414$	0	0
$415$	33.9357	1.66584
$416$	0	0
$417$	−2.10065	−0.102869
$418$	0	0
$419$	−31.9614	−1.56142	−0.780708	−	0.624896i	$-0.785141\pi$
$419$	−31.9614	−1.56142	−0.780708	+	0.624896i	$0.785141\pi$
$420$	0	0
$421$	27.8046	1.35512	0.677558	−	0.735469i	$-0.263038\pi$
$421$	27.8046	1.35512	0.677558	+	0.735469i	$0.263038\pi$
$422$	0	0
$423$	−5.56330	−0.270497
$424$	0	0
$425$	−53.6955	−2.60461
$426$	0	0
$427$	6.58880	0.318855
$428$	0	0
$429$	−4.63027	−0.223552
$430$	0	0
$431$	−2.85285	−0.137417	−0.0687084	−	0.997637i	$-0.521888\pi$
$431$	−2.85285	−0.137417	−0.0687084	+	0.997637i	$0.521888\pi$
$432$	0	0
$433$	−18.7164	−0.899453	−0.449726	−	0.893166i	$-0.648478\pi$
$433$	−18.7164	−0.899453	−0.449726	+	0.893166i	$0.648478\pi$
$434$	0	0
$435$	15.8096	0.758011
$436$	0	0
$437$	16.4238	0.785657
$438$	0	0
$439$	10.0873	0.481442	0.240721	−	0.970594i	$-0.422616\pi$
$439$	10.0873	0.481442	0.240721	+	0.970594i	$0.422616\pi$
$440$	0	0
$441$	10.7746	0.513076
$442$	0	0
$443$	−26.9906	−1.28236	−0.641180	−	0.767391i	$-0.721555\pi$
$443$	−26.9906	−1.28236	−0.641180	+	0.767391i	$0.721555\pi$
$444$	0	0
$445$	39.6947	1.88171
$446$	0	0
$447$	−9.54792	−0.451601
$448$	0	0
$449$	−6.82866	−0.322264	−0.161132	−	0.986933i	$-0.551515\pi$
$449$	−6.82866	−0.322264	−0.161132	+	0.986933i	$0.551515\pi$
$450$	0	0
$451$	−2.86060	−0.134701
$452$	0	0
$453$	−0.566007	−0.0265933
$454$	0	0
$455$	−13.1632	−0.617101
$456$	0	0
$457$	34.0759	1.59400	0.797002	−	0.603977i	$-0.206418\pi$
$457$	34.0759	1.59400	0.797002	+	0.603977i	$0.206418\pi$
$458$	0	0
$459$	−22.7231	−1.06062
$460$	0	0
$461$	−35.3244	−1.64522	−0.822611	−	0.568604i	$-0.807484\pi$
$461$	−35.3244	−1.64522	−0.822611	+	0.568604i	$0.807484\pi$
$462$	0	0
$463$	8.96806	0.416781	0.208391	−	0.978046i	$-0.433177\pi$
$463$	8.96806	0.416781	0.208391	+	0.978046i	$0.433177\pi$
$464$	0	0
$465$	−19.1743	−0.889189
$466$	0	0
$467$	1.74718	0.0808497	0.0404249	−	0.999183i	$-0.487129\pi$
$467$	1.74718	0.0808497	0.0404249	+	0.999183i	$0.487129\pi$
$468$	0	0
$469$	−8.30103	−0.383306
$470$	0	0
$471$	−8.15632	−0.375823
$472$	0	0
$473$	−4.82718	−0.221954
$474$	0	0
$475$	34.2572	1.57183
$476$	0	0
$477$	−3.42643	−0.156886
$478$	0	0
$479$	12.5314	0.572573	0.286286	−	0.958144i	$-0.407579\pi$
$479$	12.5314	0.572573	0.286286	+	0.958144i	$0.407579\pi$
$480$	0	0
$481$	−5.46519	−0.249191
$482$	0	0
$483$	5.41461	0.246373
$484$	0	0
$485$	−37.5773	−1.70630
$486$	0	0
$487$	30.2674	1.37155	0.685774	−	0.727814i	$-0.259464\pi$
$487$	30.2674	1.37155	0.685774	+	0.727814i	$0.259464\pi$
$488$	0	0
$489$	1.44070	0.0651509
$490$	0	0
$491$	−24.5345	−1.10722	−0.553612	−	0.832775i	$-0.686751\pi$
$491$	−24.5345	−1.10722	−0.553612	+	0.832775i	$0.686751\pi$
$492$	0	0
$493$	13.8519	0.623858
$494$	0	0
$495$	−7.08357	−0.318383
$496$	0	0
$497$	0.658239	0.0295261
$498$	0	0
$499$	−17.7465	−0.794443	−0.397221	−	0.917723i	$-0.630025\pi$
$499$	−17.7465	−0.794443	−0.397221	+	0.917723i	$0.630025\pi$
$500$	0	0
$501$	0.335978	0.0150104
$502$	0	0
$503$	24.9771	1.11367	0.556837	−	0.830622i	$-0.312015\pi$
$503$	24.9771	1.11367	0.556837	+	0.830622i	$0.312015\pi$
$504$	0	0
$505$	23.7430	1.05655
$506$	0	0
$507$	−3.77326	−0.167576
$508$	0	0
$509$	−18.7533	−0.831224	−0.415612	−	0.909542i	$-0.636433\pi$
$509$	−18.7533	−0.831224	−0.415612	+	0.909542i	$0.636433\pi$
$510$	0	0
$511$	1.89847	0.0839835
$512$	0	0
$513$	14.4971	0.640063
$514$	0	0
$515$	22.3655	0.985541
$516$	0	0
$517$	3.30400	0.145310
$518$	0	0
$519$	26.4729	1.16203
$520$	0	0
$521$	−10.4475	−0.457713	−0.228857	−	0.973460i	$-0.573499\pi$
$521$	−10.4475	−0.457713	−0.228857	+	0.973460i	$0.573499\pi$
$522$	0	0
$523$	−41.5581	−1.81721	−0.908605	−	0.417656i	$-0.862852\pi$
$523$	−41.5581	−1.81721	−0.908605	+	0.417656i	$0.862852\pi$
$524$	0	0
$525$	11.2939	0.492907
$526$	0	0
$527$	−16.8000	−0.731820
$528$	0	0
$529$	14.0598	0.611295
$530$	0	0
$531$	−4.07290	−0.176749
$532$	0	0
$533$	−11.5453	−0.500081
$534$	0	0
$535$	−6.07293	−0.262556
$536$	0	0
$537$	−15.3773	−0.663581
$538$	0	0
$539$	−6.39895	−0.275622
$540$	0	0
$541$	4.31934	0.185703	0.0928516	−	0.995680i	$-0.470402\pi$
$541$	4.31934	0.185703	0.0928516	+	0.995680i	$0.470402\pi$
$542$	0	0
$543$	−9.45521	−0.405762
$544$	0	0
$545$	30.3838	1.30150
$546$	0	0
$547$	14.2013	0.607203	0.303601	−	0.952799i	$-0.401811\pi$
$547$	14.2013	0.607203	0.303601	+	0.952799i	$0.401811\pi$
$548$	0	0
$549$	14.3101	0.610741
$550$	0	0
$551$	−8.83736	−0.376484
$552$	0	0
$553$	−9.96250	−0.423649
$554$	0	0
$555$	6.53550	0.277416
$556$	0	0
$557$	−28.4366	−1.20490	−0.602449	−	0.798157i	$-0.705808\pi$
$557$	−28.4366	−1.20490	−0.602449	+	0.798157i	$0.705808\pi$
$558$	0	0
$559$	−19.4823	−0.824013
$560$	0	0
$561$	4.85142	0.204827
$562$	0	0
$563$	18.4269	0.776600	0.388300	−	0.921533i	$-0.373063\pi$
$563$	18.4269	0.776600	0.388300	+	0.921533i	$0.373063\pi$
$564$	0	0
$565$	−34.4070	−1.44751
$566$	0	0
$567$	0.863174	0.0362499
$568$	0	0
$569$	−26.6552	−1.11744	−0.558722	−	0.829355i	$-0.688708\pi$
$569$	−26.6552	−1.11744	−0.558722	+	0.829355i	$0.688708\pi$
$570$	0	0
$571$	−6.32175	−0.264557	−0.132278	−	0.991213i	$-0.542229\pi$
$571$	−6.32175	−0.264557	−0.132278	+	0.991213i	$0.542229\pi$
$572$	0	0
$573$	16.7758	0.700821
$574$	0	0
$575$	77.3002	3.22364
$576$	0	0
$577$	−5.16419	−0.214988	−0.107494	−	0.994206i	$-0.534283\pi$
$577$	−5.16419	−0.214988	−0.107494	+	0.994206i	$0.534283\pi$
$578$	0	0
$579$	28.2504	1.17404
$580$	0	0
$581$	−6.25392	−0.259456
$582$	0	0
$583$	2.03493	0.0842784
$584$	0	0
$585$	−28.5890	−1.18201
$586$	0	0
$587$	3.23190	0.133395	0.0666975	−	0.997773i	$-0.478754\pi$
$587$	3.23190	0.133395	0.0666975	+	0.997773i	$0.478754\pi$
$588$	0	0
$589$	10.7182	0.441637
$590$	0	0
$591$	−17.2509	−0.709608
$592$	0	0
$593$	27.0283	1.10992	0.554961	−	0.831877i	$-0.312733\pi$
$593$	27.0283	1.10992	0.554961	+	0.831877i	$0.312733\pi$
$594$	0	0
$595$	13.7919	0.565413
$596$	0	0
$597$	−17.3422	−0.709768
$598$	0	0
$599$	−28.4449	−1.16223	−0.581113	−	0.813823i	$-0.697382\pi$
$599$	−28.4449	−1.16223	−0.581113	+	0.813823i	$0.697382\pi$
$600$	0	0
$601$	−35.0557	−1.42995	−0.714976	−	0.699149i	$-0.753562\pi$
$601$	−35.0557	−1.42995	−0.714976	+	0.699149i	$0.753562\pi$
$602$	0	0
$603$	−18.0289	−0.734193
$604$	0	0
$605$	4.20688	0.171034
$606$	0	0
$607$	28.9387	1.17459	0.587293	−	0.809374i	$-0.300194\pi$
$607$	28.9387	1.17459	0.587293	+	0.809374i	$0.300194\pi$
$608$	0	0
$609$	−2.91351	−0.118061
$610$	0	0
$611$	13.3348	0.539468
$612$	0	0
$613$	32.0539	1.29464	0.647321	−	0.762217i	$-0.275889\pi$
$613$	32.0539	1.29464	0.647321	+	0.762217i	$0.275889\pi$
$614$	0	0
$615$	13.8063	0.556724
$616$	0	0
$617$	5.15849	0.207673	0.103837	−	0.994594i	$-0.466888\pi$
$617$	5.15849	0.207673	0.103837	+	0.994594i	$0.466888\pi$
$618$	0	0
$619$	−40.1239	−1.61272	−0.806358	−	0.591428i	$-0.798564\pi$
$619$	−40.1239	−1.61272	−0.806358	+	0.591428i	$0.798564\pi$
$620$	0	0
$621$	32.7122	1.31270
$622$	0	0
$623$	−7.31523	−0.293079
$624$	0	0
$625$	72.7455	2.90982
$626$	0	0
$627$	−3.09515	−0.123609
$628$	0	0
$629$	5.72621	0.228319
$630$	0	0
$631$	14.5287	0.578378	0.289189	−	0.957272i	$-0.406614\pi$
$631$	14.5287	0.578378	0.289189	+	0.957272i	$0.406614\pi$
$632$	0	0
$633$	−1.89726	−0.0754095
$634$	0	0
$635$	−57.8672	−2.29639
$636$	0	0
$637$	−25.8259	−1.02326
$638$	0	0
$639$	1.42962	0.0565548
$640$	0	0
$641$	−22.2960	−0.880640	−0.440320	−	0.897841i	$-0.645135\pi$
$641$	−22.2960	−0.880640	−0.440320	+	0.897841i	$0.645135\pi$
$642$	0	0
$643$	21.7777	0.858827	0.429413	−	0.903108i	$-0.358720\pi$
$643$	21.7777	0.858827	0.429413	+	0.903108i	$0.358720\pi$
$644$	0	0
$645$	23.2977	0.917347
$646$	0	0
$647$	12.0068	0.472038	0.236019	−	0.971748i	$-0.424157\pi$
$647$	12.0068	0.472038	0.236019	+	0.971748i	$0.424157\pi$
$648$	0	0
$649$	2.41887	0.0949488
$650$	0	0
$651$	3.53359	0.138492
$652$	0	0
$653$	37.3454	1.46144	0.730720	−	0.682677i	$-0.239185\pi$
$653$	37.3454	1.46144	0.730720	+	0.682677i	$0.239185\pi$
$654$	0	0
$655$	24.3264	0.950510
$656$	0	0
$657$	4.12326	0.160864
$658$	0	0
$659$	−7.19220	−0.280168	−0.140084	−	0.990140i	$-0.544737\pi$
$659$	−7.19220	−0.280168	−0.140084	+	0.990140i	$0.544737\pi$
$660$	0	0
$661$	−49.5424	−1.92697	−0.963487	−	0.267754i	$-0.913718\pi$
$661$	−49.5424	−1.92697	−0.963487	+	0.267754i	$0.913718\pi$
$662$	0	0
$663$	19.5801	0.760428
$664$	0	0
$665$	−8.79909	−0.341214
$666$	0	0
$667$	−19.9412	−0.772127
$668$	0	0
$669$	5.01708	0.193972
$670$	0	0
$671$	−8.49867	−0.328088
$672$	0	0
$673$	13.5273	0.521437	0.260719	−	0.965415i	$-0.416041\pi$
$673$	13.5273	0.521437	0.260719	+	0.965415i	$0.416041\pi$
$674$	0	0
$675$	68.2320	2.62625
$676$	0	0
$677$	−17.1814	−0.660334	−0.330167	−	0.943923i	$-0.607105\pi$
$677$	−17.1814	−0.660334	−0.330167	+	0.943923i	$0.607105\pi$
$678$	0	0
$679$	6.92503	0.265758
$680$	0	0
$681$	−2.29336	−0.0878818
$682$	0	0
$683$	49.5126	1.89455	0.947273	−	0.320427i	$-0.103826\pi$
$683$	49.5126	1.89455	0.947273	+	0.320427i	$0.103826\pi$
$684$	0	0
$685$	4.20688	0.160737
$686$	0	0
$687$	−20.2510	−0.772624
$688$	0	0
$689$	8.21290	0.312887
$690$	0	0
$691$	24.5815	0.935124	0.467562	−	0.883960i	$-0.345133\pi$
$691$	24.5815	0.935124	0.467562	+	0.883960i	$0.345133\pi$
$692$	0	0
$693$	1.30541	0.0495885
$694$	0	0
$695$	7.70290	0.292188
$696$	0	0
$697$	12.0967	0.458194
$698$	0	0
$699$	0.435877	0.0164864
$700$	0	0
$701$	10.5722	0.399306	0.199653	−	0.979867i	$-0.436019\pi$
$701$	10.5722	0.399306	0.199653	+	0.979867i	$0.436019\pi$
$702$	0	0
$703$	−3.65326	−0.137785
$704$	0	0
$705$	−15.9463	−0.600572
$706$	0	0
$707$	−4.37553	−0.164559
$708$	0	0
$709$	41.7562	1.56819	0.784094	−	0.620642i	$-0.213128\pi$
$709$	41.7562	1.56819	0.784094	+	0.620642i	$0.213128\pi$
$710$	0	0
$711$	−21.6374	−0.811466
$712$	0	0
$713$	24.1853	0.905748
$714$	0	0
$715$	16.9788	0.634970
$716$	0	0
$717$	14.9071	0.556717
$718$	0	0
$719$	49.5329	1.84727	0.923633	−	0.383277i	$-0.125205\pi$
$719$	49.5329	1.84727	0.923633	+	0.383277i	$0.125205\pi$
$720$	0	0
$721$	−4.12168	−0.153499
$722$	0	0
$723$	−1.69307	−0.0629660
$724$	0	0
$725$	−41.5939	−1.54476
$726$	0	0
$727$	−34.0337	−1.26224	−0.631121	−	0.775684i	$-0.717405\pi$
$727$	−34.0337	−1.26224	−0.631121	+	0.775684i	$0.717405\pi$
$728$	0	0
$729$	20.3691	0.754411
$730$	0	0
$731$	20.4128	0.754994
$732$	0	0
$733$	20.2668	0.748572	0.374286	−	0.927313i	$-0.377888\pi$
$733$	20.2668	0.748572	0.374286	+	0.927313i	$0.377888\pi$
$734$	0	0
$735$	30.8836	1.13916
$736$	0	0
$737$	10.7072	0.394405
$738$	0	0
$739$	−49.3477	−1.81529	−0.907643	−	0.419743i	$-0.862120\pi$
$739$	−49.3477	−1.81529	−0.907643	+	0.419743i	$0.862120\pi$
$740$	0	0
$741$	−12.4919	−0.458902
$742$	0	0
$743$	−20.7580	−0.761538	−0.380769	−	0.924670i	$-0.624341\pi$
$743$	−20.7580	−0.761538	−0.380769	+	0.924670i	$0.624341\pi$
$744$	0	0
$745$	35.0113	1.28272
$746$	0	0
$747$	−13.5828	−0.496968
$748$	0	0
$749$	1.11917	0.0408934
$750$	0	0
$751$	−26.7720	−0.976923	−0.488462	−	0.872585i	$-0.662442\pi$
$751$	−26.7720	−0.976923	−0.488462	+	0.872585i	$0.662442\pi$
$752$	0	0
$753$	−4.71232	−0.171727
$754$	0	0
$755$	2.07549	0.0755350
$756$	0	0
$757$	50.8946	1.84979	0.924897	−	0.380217i	$-0.124151\pi$
$757$	50.8946	1.84979	0.924897	+	0.380217i	$0.124151\pi$
$758$	0	0
$759$	−6.98411	−0.253507
$760$	0	0
$761$	−43.5722	−1.57949	−0.789746	−	0.613434i	$-0.789788\pi$
$761$	−43.5722	−1.57949	−0.789746	+	0.613434i	$0.789788\pi$
$762$	0	0
$763$	−5.59934	−0.202710
$764$	0	0
$765$	29.9544	1.08300
$766$	0	0
$767$	9.76244	0.352501
$768$	0	0
$769$	18.1068	0.652947	0.326474	−	0.945206i	$-0.394140\pi$
$769$	18.1068	0.652947	0.326474	+	0.945206i	$0.394140\pi$
$770$	0	0
$771$	4.03520	0.145324
$772$	0	0
$773$	5.59520	0.201245	0.100623	−	0.994925i	$-0.467916\pi$
$773$	5.59520	0.201245	0.100623	+	0.994925i	$0.467916\pi$
$774$	0	0
$775$	50.4464	1.81209
$776$	0	0
$777$	−1.20441	−0.0432080
$778$	0	0
$779$	−7.71756	−0.276510
$780$	0	0
$781$	−0.849040	−0.0303810
$782$	0	0
$783$	−17.6019	−0.629040
$784$	0	0
$785$	29.9085	1.06748
$786$	0	0
$787$	−16.0785	−0.573138	−0.286569	−	0.958060i	$-0.592515\pi$
$787$	−16.0785	−0.573138	−0.286569	+	0.958060i	$0.592515\pi$
$788$	0	0
$789$	−10.2314	−0.364247
$790$	0	0
$791$	6.34077	0.225452
$792$	0	0
$793$	−34.3003	−1.21804
$794$	0	0
$795$	−9.82133	−0.348327
$796$	0	0
$797$	1.78558	0.0632486	0.0316243	−	0.999500i	$-0.489932\pi$
$797$	1.78558	0.0632486	0.0316243	+	0.999500i	$0.489932\pi$
$798$	0	0
$799$	−13.9717	−0.494283
$800$	0	0
$801$	−15.8878	−0.561369
$802$	0	0
$803$	−2.44878	−0.0864154
$804$	0	0
$805$	−19.8549	−0.699792
$806$	0	0
$807$	−15.6815	−0.552016
$808$	0	0
$809$	9.49027	0.333660	0.166830	−	0.985986i	$-0.446647\pi$
$809$	9.49027	0.333660	0.166830	+	0.985986i	$0.446647\pi$
$810$	0	0
$811$	49.2079	1.72792	0.863962	−	0.503558i	$-0.167976\pi$
$811$	49.2079	1.72792	0.863962	+	0.503558i	$0.167976\pi$
$812$	0	0
$813$	12.3937	0.434664
$814$	0	0
$815$	−5.28293	−0.185053
$816$	0	0
$817$	−13.0231	−0.455622
$818$	0	0
$819$	5.26859	0.184099
$820$	0	0
$821$	−31.1705	−1.08786	−0.543930	−	0.839131i	$-0.683064\pi$
$821$	−31.1705	−1.08786	−0.543930	+	0.839131i	$0.683064\pi$
$822$	0	0
$823$	−18.8058	−0.655528	−0.327764	−	0.944760i	$-0.606295\pi$
$823$	−18.8058	−0.655528	−0.327764	+	0.944760i	$0.606295\pi$
$824$	0	0
$825$	−14.5676	−0.507180
$826$	0	0
$827$	47.5942	1.65501	0.827506	−	0.561457i	$-0.189759\pi$
$827$	47.5942	1.65501	0.827506	+	0.561457i	$0.189759\pi$
$828$	0	0
$829$	23.9305	0.831141	0.415571	−	0.909561i	$-0.363582\pi$
$829$	23.9305	0.831141	0.415571	+	0.909561i	$0.363582\pi$
$830$	0	0
$831$	−1.56382	−0.0542483
$832$	0	0
$833$	27.0593	0.937551
$834$	0	0
$835$	−1.23200	−0.0426351
$836$	0	0
$837$	21.3481	0.737899
$838$	0	0
$839$	−20.2122	−0.697803	−0.348902	−	0.937159i	$-0.613445\pi$
$839$	−20.2122	−0.697803	−0.348902	+	0.937159i	$0.613445\pi$
$840$	0	0
$841$	−18.2700	−0.629999
$842$	0	0
$843$	7.90264	0.272182
$844$	0	0
$845$	13.8362	0.475979
$846$	0	0
$847$	−0.775275	−0.0266387
$848$	0	0
$849$	−5.28498	−0.181380
$850$	0	0
$851$	−8.24347	−0.282583
$852$	0	0
$853$	−14.5201	−0.497159	−0.248579	−	0.968612i	$-0.579964\pi$
$853$	−14.5201	−0.497159	−0.248579	+	0.968612i	$0.579964\pi$
$854$	0	0
$855$	−19.1106	−0.653569
$856$	0	0
$857$	−45.1408	−1.54198	−0.770991	−	0.636846i	$-0.780239\pi$
$857$	−45.1408	−1.54198	−0.770991	+	0.636846i	$0.780239\pi$
$858$	0	0
$859$	−47.1760	−1.60962	−0.804812	−	0.593530i	$-0.797734\pi$
$859$	−47.1760	−1.60962	−0.804812	+	0.593530i	$0.797734\pi$
$860$	0	0
$861$	−2.54433	−0.0867105
$862$	0	0
$863$	14.4344	0.491352	0.245676	−	0.969352i	$-0.420990\pi$
$863$	14.4344	0.491352	0.245676	+	0.969352i	$0.420990\pi$
$864$	0	0
$865$	−97.0738	−3.30061
$866$	0	0
$867$	−1.01193	−0.0343670
$868$	0	0
$869$	12.8503	0.435916
$870$	0	0
$871$	43.2139	1.46425
$872$	0	0
$873$	15.0404	0.509039
$874$	0	0
$875$	−25.1063	−0.848748
$876$	0	0
$877$	15.3418	0.518056	0.259028	−	0.965870i	$-0.416598\pi$
$877$	15.3418	0.518056	0.259028	+	0.965870i	$0.416598\pi$
$878$	0	0
$879$	−9.47931	−0.319729
$880$	0	0
$881$	21.2563	0.716144	0.358072	−	0.933694i	$-0.383434\pi$
$881$	21.2563	0.716144	0.358072	+	0.933694i	$0.383434\pi$
$882$	0	0
$883$	−49.9997	−1.68262	−0.841312	−	0.540550i	$-0.818216\pi$
$883$	−49.9997	−1.68262	−0.841312	+	0.540550i	$0.818216\pi$
$884$	0	0
$885$	−11.6743	−0.392428
$886$	0	0
$887$	24.2268	0.813455	0.406728	−	0.913549i	$-0.366670\pi$
$887$	24.2268	0.813455	0.406728	+	0.913549i	$0.366670\pi$
$888$	0	0
$889$	10.6642	0.357665
$890$	0	0
$891$	−1.11338	−0.0372996
$892$	0	0
$893$	8.91379	0.298289
$894$	0	0
$895$	56.3873	1.88482
$896$	0	0
$897$	−28.1876	−0.941156
$898$	0	0
$899$	−13.0137	−0.434032
$900$	0	0
$901$	−8.60516	−0.286679
$902$	0	0
$903$	−4.29348	−0.142878
$904$	0	0
$905$	34.6714	1.15252
$906$	0	0
$907$	−19.5906	−0.650495	−0.325248	−	0.945629i	$-0.605448\pi$
$907$	−19.5906	−0.650495	−0.325248	+	0.945629i	$0.605448\pi$
$908$	0	0
$909$	−9.50314	−0.315199
$910$	0	0
$911$	37.8521	1.25410	0.627049	−	0.778980i	$-0.284263\pi$
$911$	37.8521	1.25410	0.627049	+	0.778980i	$0.284263\pi$
$912$	0	0
$913$	8.06672	0.266969
$914$	0	0
$915$	41.0177	1.35600
$916$	0	0
$917$	−4.48305	−0.148043
$918$	0	0
$919$	−5.31491	−0.175323	−0.0876614	−	0.996150i	$-0.527939\pi$
$919$	−5.31491	−0.175323	−0.0876614	+	0.996150i	$0.527939\pi$
$920$	0	0
$921$	−27.4647	−0.904993
$922$	0	0
$923$	−3.42669	−0.112791
$924$	0	0
$925$	−17.1944	−0.565350
$926$	0	0
$927$	−8.95181	−0.294016
$928$	0	0
$929$	−24.1841	−0.793455	−0.396728	−	0.917936i	$-0.629854\pi$
$929$	−24.1841	−0.793455	−0.396728	+	0.917936i	$0.629854\pi$
$930$	0	0
$931$	−17.2636	−0.565791
$932$	0	0
$933$	−10.7743	−0.352734
$934$	0	0
$935$	−17.7897	−0.581785
$936$	0	0
$937$	−39.8202	−1.30087	−0.650435	−	0.759562i	$-0.725414\pi$
$937$	−39.8202	−1.30087	−0.650435	+	0.759562i	$0.725414\pi$
$938$	0	0
$939$	−26.1341	−0.852853
$940$	0	0
$941$	6.50585	0.212085	0.106042	−	0.994362i	$-0.466182\pi$
$941$	6.50585	0.212085	0.106042	+	0.994362i	$0.466182\pi$
$942$	0	0
$943$	−17.4144	−0.567092
$944$	0	0
$945$	−17.5257	−0.570110
$946$	0	0
$947$	18.1216	0.588874	0.294437	−	0.955671i	$-0.404868\pi$
$947$	18.1216	0.588874	0.294437	+	0.955671i	$0.404868\pi$
$948$	0	0
$949$	−9.88315	−0.320821
$950$	0	0
$951$	−15.0958	−0.489515
$952$	0	0
$953$	35.4380	1.14795	0.573974	−	0.818874i	$-0.305401\pi$
$953$	35.4380	1.14795	0.573974	+	0.818874i	$0.305401\pi$
$954$	0	0
$955$	−61.5155	−1.99059
$956$	0	0
$957$	3.75803	0.121480
$958$	0	0
$959$	−0.775275	−0.0250349
$960$	0	0
$961$	−15.2166	−0.490857
$962$	0	0
$963$	2.43070	0.0783281
$964$	0	0
$965$	−103.591	−3.33473
$966$	0	0
$967$	−14.4902	−0.465973	−0.232986	−	0.972480i	$-0.574850\pi$
$967$	−14.4902	−0.465973	−0.232986	+	0.972480i	$0.574850\pi$
$968$	0	0
$969$	13.0885	0.420464
$970$	0	0
$971$	−39.0968	−1.25468	−0.627338	−	0.778747i	$-0.715856\pi$
$971$	−39.0968	−1.25468	−0.627338	+	0.778747i	$0.715856\pi$
$972$	0	0
$973$	−1.41955	−0.0455086
$974$	0	0
$975$	−58.7943	−1.88293
$976$	0	0
$977$	−5.42114	−0.173438	−0.0867188	−	0.996233i	$-0.527638\pi$
$977$	−5.42114	−0.173438	−0.0867188	+	0.996233i	$0.527638\pi$
$978$	0	0
$979$	9.43566	0.301565
$980$	0	0
$981$	−12.1611	−0.388275
$982$	0	0
$983$	−43.6847	−1.39333	−0.696663	−	0.717399i	$-0.745333\pi$
$983$	−43.6847	−1.39333	−0.696663	+	0.717399i	$0.745333\pi$
$984$	0	0
$985$	63.2575	2.01555
$986$	0	0
$987$	2.93870	0.0935399
$988$	0	0
$989$	−29.3863	−0.934430
$990$	0	0
$991$	−16.2376	−0.515804	−0.257902	−	0.966171i	$-0.583031\pi$
$991$	−16.2376	−0.515804	−0.257902	+	0.966171i	$0.583031\pi$
$992$	0	0
$993$	11.5121	0.365325
$994$	0	0
$995$	63.5922	2.01601
$996$	0	0
$997$	−23.1146	−0.732047	−0.366024	−	0.930606i	$-0.619281\pi$
$997$	−23.1146	−0.732047	−0.366024	+	0.930606i	$0.619281\pi$
$998$	0	0
$999$	−7.27642	−0.230216

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6028.2.a.f.1.9	✓	29

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6028.2.a.f.1.9	✓	29	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6028 = 2^{2} \cdot 11 \cdot 137 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6028.a (trivial)

\(f(q)\)	\(=\)	\(q-1.14725 q^{3} +4.20688 q^{5} -0.775275 q^{7} -1.68381 q^{9} +O(q^{10})\)	\(q-1.14725 q^{3} +4.20688 q^{5} -0.775275 q^{7} -1.68381 q^{9} +1.00000 q^{11} +4.03596 q^{13} -4.82636 q^{15} -4.22872 q^{17} +2.69788 q^{19} +0.889438 q^{21} +6.08767 q^{23} +12.6978 q^{25} +5.37352 q^{27} -3.27567 q^{29} +3.97284 q^{31} -1.14725 q^{33} -3.26149 q^{35} -1.35413 q^{37} -4.63027 q^{39} -2.86060 q^{41} -4.82718 q^{43} -7.08357 q^{45} +3.30400 q^{47} -6.39895 q^{49} +4.85142 q^{51} +2.03493 q^{53} +4.20688 q^{55} -3.09515 q^{57} +2.41887 q^{59} -8.49867 q^{61} +1.30541 q^{63} +16.9788 q^{65} +10.7072 q^{67} -6.98411 q^{69} -0.849040 q^{71} -2.44878 q^{73} -14.5676 q^{75} -0.775275 q^{77} +12.8503 q^{79} -1.11338 q^{81} +8.06672 q^{83} -17.7897 q^{85} +3.75803 q^{87} +9.43566 q^{89} -3.12897 q^{91} -4.55786 q^{93} +11.3496 q^{95} -8.93235 q^{97} -1.68381 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 29 q + 14 q^{3} + 9 q^{5} + 14 q^{7} + 43 q^{9}+O(q^{10}) \) 29 * q + 14 * q^3 + 9 * q^5 + 14 * q^7 + 43 * q^9	\( 29 q + 14 q^{3} + 9 q^{5} + 14 q^{7} + 43 q^{9} + 29 q^{11} + 10 q^{15} + 29 q^{17} + 7 q^{19} + 2 q^{21} + 36 q^{23} + 36 q^{25} + 50 q^{27} + 9 q^{29} + 28 q^{31} + 14 q^{33} + 15 q^{35} + 25 q^{37} + 9 q^{39} + 19 q^{41} + 23 q^{43} + 5 q^{45} + 27 q^{47} + 27 q^{49} + 13 q^{51} + 4 q^{53} + 9 q^{55} + 14 q^{57} + 40 q^{59} + 20 q^{61} - 17 q^{63} + 9 q^{65} + 59 q^{67} + 30 q^{69} + 29 q^{71} - 5 q^{73} + 46 q^{75} + 14 q^{77} + 29 q^{79} + 61 q^{81} + 35 q^{83} - 57 q^{85} + 45 q^{87} + 39 q^{89} + 45 q^{91} - 8 q^{93} + q^{95} + 55 q^{97} + 43 q^{99}+O(q^{100}) \) 29 * q + 14 * q^3 + 9 * q^5 + 14 * q^7 + 43 * q^9 + 29 * q^11 + 10 * q^15 + 29 * q^17 + 7 * q^19 + 2 * q^21 + 36 * q^23 + 36 * q^25 + 50 * q^27 + 9 * q^29 + 28 * q^31 + 14 * q^33 + 15 * q^35 + 25 * q^37 + 9 * q^39 + 19 * q^41 + 23 * q^43 + 5 * q^45 + 27 * q^47 + 27 * q^49 + 13 * q^51 + 4 * q^53 + 9 * q^55 + 14 * q^57 + 40 * q^59 + 20 * q^61 - 17 * q^63 + 9 * q^65 + 59 * q^67 + 30 * q^69 + 29 * q^71 - 5 * q^73 + 46 * q^75 + 14 * q^77 + 29 * q^79 + 61 * q^81 + 35 * q^83 - 57 * q^85 + 45 * q^87 + 39 * q^89 + 45 * q^91 - 8 * q^93 + q^95 + 55 * q^97 + 43 * q^99

Introduction

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