LMFDB - Embedded newform 6028.2.a.f.1.3

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			1.3
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)

$f(q)$	$=$	$q-2.65731 q^{3} +2.47175 q^{5} +0.711504 q^{7} +4.06129 q^{9} +O(q^{10})$	$q-2.65731 q^{3} +2.47175 q^{5} +0.711504 q^{7} +4.06129 q^{9} +1.00000 q^{11} -5.57958 q^{13} -6.56819 q^{15} -3.81904 q^{17} -2.65996 q^{19} -1.89068 q^{21} -3.08681 q^{23} +1.10953 q^{25} -2.82017 q^{27} -8.80699 q^{29} -2.60099 q^{31} -2.65731 q^{33} +1.75866 q^{35} +4.27018 q^{37} +14.8267 q^{39} +2.10972 q^{41} +8.51085 q^{43} +10.0385 q^{45} +4.18031 q^{47} -6.49376 q^{49} +10.1484 q^{51} +4.18768 q^{53} +2.47175 q^{55} +7.06832 q^{57} +5.22304 q^{59} +3.35127 q^{61} +2.88962 q^{63} -13.7913 q^{65} +2.39674 q^{67} +8.20260 q^{69} +1.66925 q^{71} +7.52899 q^{73} -2.94836 q^{75} +0.711504 q^{77} +1.11902 q^{79} -4.68981 q^{81} +0.280580 q^{83} -9.43970 q^{85} +23.4029 q^{87} +17.0319 q^{89} -3.96989 q^{91} +6.91164 q^{93} -6.57474 q^{95} -1.91091 q^{97} +4.06129 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.65731	−1.53420	−0.767099	−	0.641529i	$-0.778300\pi$
$3$	−2.65731	−1.53420	−0.767099	+	0.641529i	$0.778300\pi$
$4$	0	0
$5$	2.47175	1.10540	0.552699	−	0.833381i	$-0.313598\pi$
$5$	2.47175	1.10540	0.552699	+	0.833381i	$0.313598\pi$
$6$	0	0
$7$	0.711504	0.268923	0.134462	−	0.990919i	$-0.457070\pi$
$7$	0.711504	0.268923	0.134462	+	0.990919i	$0.457070\pi$
$8$	0	0
$9$	4.06129	1.35376
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−5.57958	−1.54750	−0.773749	−	0.633492i	$-0.781621\pi$
$13$	−5.57958	−1.54750	−0.773749	+	0.633492i	$0.781621\pi$
$14$	0	0
$15$	−6.56819	−1.69590
$16$	0	0
$17$	−3.81904	−0.926253	−0.463127	−	0.886292i	$-0.653272\pi$
$17$	−3.81904	−0.926253	−0.463127	+	0.886292i	$0.653272\pi$
$18$	0	0
$19$	−2.65996	−0.610236	−0.305118	−	0.952315i	$-0.598696\pi$
$19$	−2.65996	−0.610236	−0.305118	+	0.952315i	$0.598696\pi$
$20$	0	0
$21$	−1.89068	−0.412581
$22$	0	0
$23$	−3.08681	−0.643644	−0.321822	−	0.946800i	$-0.604295\pi$
$23$	−3.08681	−0.643644	−0.321822	+	0.946800i	$0.604295\pi$
$24$	0	0
$25$	1.10953	0.221906
$26$	0	0
$27$	−2.82017	−0.542741
$28$	0	0
$29$	−8.80699	−1.63542	−0.817708	−	0.575633i	$-0.804756\pi$
$29$	−8.80699	−1.63542	−0.817708	+	0.575633i	$0.804756\pi$
$30$	0	0
$31$	−2.60099	−0.467152	−0.233576	−	0.972339i	$-0.575043\pi$
$31$	−2.60099	−0.467152	−0.233576	+	0.972339i	$0.575043\pi$
$32$	0	0
$33$	−2.65731	−0.462578
$34$	0	0
$35$	1.75866	0.297267
$36$	0	0
$37$	4.27018	0.702014	0.351007	−	0.936373i	$-0.385839\pi$
$37$	4.27018	0.702014	0.351007	+	0.936373i	$0.385839\pi$
$38$	0	0
$39$	14.8267	2.37417
$40$	0	0
$41$	2.10972	0.329482	0.164741	−	0.986337i	$-0.447321\pi$
$41$	2.10972	0.329482	0.164741	+	0.986337i	$0.447321\pi$
$42$	0	0
$43$	8.51085	1.29789	0.648947	−	0.760834i	$-0.275210\pi$
$43$	8.51085	1.29789	0.648947	+	0.760834i	$0.275210\pi$
$44$	0	0
$45$	10.0385	1.49645
$46$	0	0
$47$	4.18031	0.609760	0.304880	−	0.952391i	$-0.401384\pi$
$47$	4.18031	0.609760	0.304880	+	0.952391i	$0.401384\pi$
$48$	0	0
$49$	−6.49376	−0.927680
$50$	0	0
$51$	10.1484	1.42106
$52$	0	0
$53$	4.18768	0.575222	0.287611	−	0.957747i	$-0.407139\pi$
$53$	4.18768	0.575222	0.287611	+	0.957747i	$0.407139\pi$
$54$	0	0
$55$	2.47175	0.333290
$56$	0	0
$57$	7.06832	0.936222
$58$	0	0
$59$	5.22304	0.679982	0.339991	−	0.940429i	$-0.389576\pi$
$59$	5.22304	0.679982	0.339991	+	0.940429i	$0.389576\pi$
$60$	0	0
$61$	3.35127	0.429086	0.214543	−	0.976715i	$-0.431174\pi$
$61$	3.35127	0.429086	0.214543	+	0.976715i	$0.431174\pi$
$62$	0	0
$63$	2.88962	0.364058
$64$	0	0
$65$	−13.7913	−1.71060
$66$	0	0
$67$	2.39674	0.292808	0.146404	−	0.989225i	$-0.453230\pi$
$67$	2.39674	0.292808	0.146404	+	0.989225i	$0.453230\pi$
$68$	0	0
$69$	8.20260	0.987477
$70$	0	0
$71$	1.66925	0.198103	0.0990517	−	0.995082i	$-0.468419\pi$
$71$	1.66925	0.198103	0.0990517	+	0.995082i	$0.468419\pi$
$72$	0	0
$73$	7.52899	0.881202	0.440601	−	0.897703i	$-0.354765\pi$
$73$	7.52899	0.881202	0.440601	+	0.897703i	$0.354765\pi$
$74$	0	0
$75$	−2.94836	−0.340448
$76$	0	0
$77$	0.711504	0.0810834
$78$	0	0
$79$	1.11902	0.125900	0.0629498	−	0.998017i	$-0.479949\pi$
$79$	1.11902	0.125900	0.0629498	+	0.998017i	$0.479949\pi$
$80$	0	0
$81$	−4.68981	−0.521090
$82$	0	0
$83$	0.280580	0.0307977	0.0153988	−	0.999881i	$-0.495098\pi$
$83$	0.280580	0.0307977	0.0153988	+	0.999881i	$0.495098\pi$
$84$	0	0
$85$	−9.43970	−1.02388
$86$	0	0
$87$	23.4029	2.50905
$88$	0	0
$89$	17.0319	1.80538	0.902689	−	0.430294i	$-0.141590\pi$
$89$	17.0319	1.80538	0.902689	+	0.430294i	$0.141590\pi$
$90$	0	0
$91$	−3.96989	−0.416158
$92$	0	0
$93$	6.91164	0.716704
$94$	0	0
$95$	−6.57474	−0.674554
$96$	0	0
$97$	−1.91091	−0.194023	−0.0970117	−	0.995283i	$-0.530928\pi$
$97$	−1.91091	−0.194023	−0.0970117	+	0.995283i	$0.530928\pi$
$98$	0	0
$99$	4.06129	0.408175
$100$	0	0
$101$	6.09941	0.606913	0.303457	−	0.952845i	$-0.401859\pi$
$101$	6.09941	0.606913	0.303457	+	0.952845i	$0.401859\pi$
$102$	0	0
$103$	3.27552	0.322747	0.161373	−	0.986893i	$-0.448408\pi$
$103$	3.27552	0.322747	0.161373	+	0.986893i	$0.448408\pi$
$104$	0	0
$105$	−4.67329	−0.456067
$106$	0	0
$107$	5.97928	0.578039	0.289019	−	0.957323i	$-0.406671\pi$
$107$	5.97928	0.578039	0.289019	+	0.957323i	$0.406671\pi$
$108$	0	0
$109$	−12.6462	−1.21129	−0.605645	−	0.795735i	$-0.707085\pi$
$109$	−12.6462	−1.21129	−0.605645	+	0.795735i	$0.707085\pi$
$110$	0	0
$111$	−11.3472	−1.07703
$112$	0	0
$113$	−3.92533	−0.369264	−0.184632	−	0.982808i	$-0.559109\pi$
$113$	−3.92533	−0.369264	−0.184632	+	0.982808i	$0.559109\pi$
$114$	0	0
$115$	−7.62981	−0.711483
$116$	0	0
$117$	−22.6603	−2.09494
$118$	0	0
$119$	−2.71726	−0.249091
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−5.60617	−0.505491
$124$	0	0
$125$	−9.61625	−0.860104
$126$	0	0
$127$	10.8142	0.959601	0.479801	−	0.877378i	$-0.340709\pi$
$127$	10.8142	0.959601	0.479801	+	0.877378i	$0.340709\pi$
$128$	0	0
$129$	−22.6160	−1.99122
$130$	0	0
$131$	18.4831	1.61488	0.807439	−	0.589951i	$-0.200853\pi$
$131$	18.4831	1.61488	0.807439	+	0.589951i	$0.200853\pi$
$132$	0	0
$133$	−1.89257	−0.164106
$134$	0	0
$135$	−6.97074	−0.599945
$136$	0	0
$137$	1.00000	0.0854358
$137$	1.00000	0.0854358
$138$	0	0
$139$	−11.2378	−0.953180	−0.476590	−	0.879126i	$-0.658127\pi$
$139$	−11.2378	−0.953180	−0.476590	+	0.879126i	$0.658127\pi$
$140$	0	0
$141$	−11.1084	−0.935493
$142$	0	0
$143$	−5.57958	−0.466588
$144$	0	0
$145$	−21.7686	−1.80779
$146$	0	0
$147$	17.2559	1.42325
$148$	0	0
$149$	−6.62784	−0.542974	−0.271487	−	0.962442i	$-0.587515\pi$
$149$	−6.62784	−0.542974	−0.271487	+	0.962442i	$0.587515\pi$
$150$	0	0
$151$	4.19080	0.341042	0.170521	−	0.985354i	$-0.445455\pi$
$151$	4.19080	0.341042	0.170521	+	0.985354i	$0.445455\pi$
$152$	0	0
$153$	−15.5102	−1.25393
$154$	0	0
$155$	−6.42899	−0.516389
$156$	0	0
$157$	−5.46996	−0.436550	−0.218275	−	0.975887i	$-0.570043\pi$
$157$	−5.46996	−0.436550	−0.218275	+	0.975887i	$0.570043\pi$
$158$	0	0
$159$	−11.1280	−0.882505
$160$	0	0
$161$	−2.19627	−0.173091
$162$	0	0
$163$	17.6149	1.37970	0.689851	−	0.723951i	$-0.257676\pi$
$163$	17.6149	1.37970	0.689851	+	0.723951i	$0.257676\pi$
$164$	0	0
$165$	−6.56819	−0.511333
$166$	0	0
$167$	17.5025	1.35438	0.677192	−	0.735807i	$-0.263197\pi$
$167$	17.5025	1.35438	0.677192	+	0.735807i	$0.263197\pi$
$168$	0	0
$169$	18.1317	1.39475
$170$	0	0
$171$	−10.8028	−0.826114
$172$	0	0
$173$	−14.0053	−1.06480	−0.532401	−	0.846492i	$-0.678710\pi$
$173$	−14.0053	−1.06480	−0.532401	+	0.846492i	$0.678710\pi$
$174$	0	0
$175$	0.789435	0.0596757
$176$	0	0
$177$	−13.8792	−1.04323
$178$	0	0
$179$	−3.79430	−0.283599	−0.141800	−	0.989895i	$-0.545289\pi$
$179$	−3.79430	−0.283599	−0.141800	+	0.989895i	$0.545289\pi$
$180$	0	0
$181$	12.0492	0.895609	0.447804	−	0.894131i	$-0.352206\pi$
$181$	12.0492	0.895609	0.447804	+	0.894131i	$0.352206\pi$
$182$	0	0
$183$	−8.90536	−0.658303
$184$	0	0
$185$	10.5548	0.776005
$186$	0	0
$187$	−3.81904	−0.279276
$188$	0	0
$189$	−2.00656	−0.145956
$190$	0	0
$191$	15.4777	1.11992	0.559962	−	0.828518i	$-0.310816\pi$
$191$	15.4777	1.11992	0.559962	+	0.828518i	$0.310816\pi$
$192$	0	0
$193$	19.4162	1.39761	0.698805	−	0.715313i	$-0.253716\pi$
$193$	19.4162	1.39761	0.698805	+	0.715313i	$0.253716\pi$
$194$	0	0
$195$	36.6478	2.62440
$196$	0	0
$197$	−9.80109	−0.698299	−0.349149	−	0.937067i	$-0.613529\pi$
$197$	−9.80109	−0.698299	−0.349149	+	0.937067i	$0.613529\pi$
$198$	0	0
$199$	−7.04853	−0.499657	−0.249828	−	0.968290i	$-0.580374\pi$
$199$	−7.04853	−0.499657	−0.249828	+	0.968290i	$0.580374\pi$
$200$	0	0
$201$	−6.36887	−0.449226
$202$	0	0
$203$	−6.26620	−0.439801
$204$	0	0
$205$	5.21468	0.364209
$206$	0	0
$207$	−12.5364	−0.871341
$208$	0	0
$209$	−2.65996	−0.183993
$210$	0	0
$211$	10.5542	0.726584	0.363292	−	0.931675i	$-0.381653\pi$
$211$	10.5542	0.726584	0.363292	+	0.931675i	$0.381653\pi$
$212$	0	0
$213$	−4.43571	−0.303930
$214$	0	0
$215$	21.0367	1.43469
$216$	0	0
$217$	−1.85062	−0.125628
$218$	0	0
$219$	−20.0068	−1.35194
$220$	0	0
$221$	21.3087	1.43338
$222$	0	0
$223$	20.7595	1.39016	0.695081	−	0.718932i	$-0.255369\pi$
$223$	20.7595	1.39016	0.695081	+	0.718932i	$0.255369\pi$
$224$	0	0
$225$	4.50612	0.300408
$226$	0	0
$227$	−9.76123	−0.647876	−0.323938	−	0.946078i	$-0.605007\pi$
$227$	−9.76123	−0.647876	−0.323938	+	0.946078i	$0.605007\pi$
$228$	0	0
$229$	−8.63152	−0.570387	−0.285193	−	0.958470i	$-0.592058\pi$
$229$	−8.63152	−0.570387	−0.285193	+	0.958470i	$0.592058\pi$
$230$	0	0
$231$	−1.89068	−0.124398
$232$	0	0
$233$	−26.9983	−1.76871	−0.884357	−	0.466811i	$-0.845403\pi$
$233$	−26.9983	−1.76871	−0.884357	+	0.466811i	$0.845403\pi$
$234$	0	0
$235$	10.3327	0.674028
$236$	0	0
$237$	−2.97358	−0.193155
$238$	0	0
$239$	−8.49756	−0.549661	−0.274831	−	0.961493i	$-0.588622\pi$
$239$	−8.49756	−0.549661	−0.274831	+	0.961493i	$0.588622\pi$
$240$	0	0
$241$	16.4579	1.06014	0.530072	−	0.847953i	$-0.322165\pi$
$241$	16.4579	1.06014	0.530072	+	0.847953i	$0.322165\pi$
$242$	0	0
$243$	20.9228	1.34220
$244$	0	0
$245$	−16.0509	−1.02546
$246$	0	0
$247$	14.8414	0.944338
$248$	0	0
$249$	−0.745588	−0.0472497
$250$	0	0
$251$	−19.6348	−1.23934	−0.619669	−	0.784863i	$-0.712733\pi$
$251$	−19.6348	−1.23934	−0.619669	+	0.784863i	$0.712733\pi$
$252$	0	0
$253$	−3.08681	−0.194066
$254$	0	0
$255$	25.0842	1.57083
$256$	0	0
$257$	22.3772	1.39585	0.697925	−	0.716171i	$-0.254107\pi$
$257$	22.3772	1.39585	0.697925	+	0.716171i	$0.254107\pi$
$258$	0	0
$259$	3.03825	0.188788
$260$	0	0
$261$	−35.7677	−2.21396
$262$	0	0
$263$	−7.45887	−0.459934	−0.229967	−	0.973198i	$-0.573862\pi$
$263$	−7.45887	−0.459934	−0.229967	+	0.973198i	$0.573862\pi$
$264$	0	0
$265$	10.3509	0.635850
$266$	0	0
$267$	−45.2590	−2.76981
$268$	0	0
$269$	19.5587	1.19252	0.596258	−	0.802793i	$-0.296654\pi$
$269$	19.5587	1.19252	0.596258	+	0.802793i	$0.296654\pi$
$270$	0	0
$271$	−30.1524	−1.83163	−0.915814	−	0.401603i	$-0.868453\pi$
$271$	−30.1524	−1.83163	−0.915814	+	0.401603i	$0.868453\pi$
$272$	0	0
$273$	10.5492	0.638468
$274$	0	0
$275$	1.10953	0.0669072
$276$	0	0
$277$	15.3487	0.922214	0.461107	−	0.887345i	$-0.347452\pi$
$277$	15.3487	0.922214	0.461107	+	0.887345i	$0.347452\pi$
$278$	0	0
$279$	−10.5634	−0.632413
$280$	0	0
$281$	−6.46742	−0.385814	−0.192907	−	0.981217i	$-0.561792\pi$
$281$	−6.46742	−0.385814	−0.192907	+	0.981217i	$0.561792\pi$
$282$	0	0
$283$	16.2626	0.966712	0.483356	−	0.875424i	$-0.339418\pi$
$283$	16.2626	0.966712	0.483356	+	0.875424i	$0.339418\pi$
$284$	0	0
$285$	17.4711	1.03490
$286$	0	0
$287$	1.50107	0.0886054
$288$	0	0
$289$	−2.41493	−0.142055
$290$	0	0
$291$	5.07788	0.297670
$292$	0	0
$293$	−26.3099	−1.53704	−0.768520	−	0.639826i	$-0.779006\pi$
$293$	−26.3099	−1.53704	−0.768520	+	0.639826i	$0.779006\pi$
$294$	0	0
$295$	12.9100	0.751651
$296$	0	0
$297$	−2.82017	−0.163643
$298$	0	0
$299$	17.2231	0.996038
$300$	0	0
$301$	6.05550	0.349033
$302$	0	0
$303$	−16.2080	−0.931125
$304$	0	0
$305$	8.28349	0.474311
$306$	0	0
$307$	29.2456	1.66914	0.834568	−	0.550905i	$-0.185717\pi$
$307$	29.2456	1.66914	0.834568	+	0.550905i	$0.185717\pi$
$308$	0	0
$309$	−8.70407	−0.495157
$310$	0	0
$311$	20.9826	1.18981	0.594907	−	0.803794i	$-0.297189\pi$
$311$	20.9826	1.18981	0.594907	+	0.803794i	$0.297189\pi$
$312$	0	0
$313$	27.2457	1.54002	0.770010	−	0.638032i	$-0.220251\pi$
$313$	27.2457	1.54002	0.770010	+	0.638032i	$0.220251\pi$
$314$	0	0
$315$	7.14241	0.402429
$316$	0	0
$317$	−10.4658	−0.587818	−0.293909	−	0.955833i	$-0.594956\pi$
$317$	−10.4658	−0.587818	−0.293909	+	0.955833i	$0.594956\pi$
$318$	0	0
$319$	−8.80699	−0.493097
$320$	0	0
$321$	−15.8888	−0.886826
$322$	0	0
$323$	10.1585	0.565233
$324$	0	0
$325$	−6.19072	−0.343399
$326$	0	0
$327$	33.6049	1.85836
$328$	0	0
$329$	2.97430	0.163979
$330$	0	0
$331$	−4.92248	−0.270564	−0.135282	−	0.990807i	$-0.543194\pi$
$331$	−4.92248	−0.270564	−0.135282	+	0.990807i	$0.543194\pi$
$332$	0	0
$333$	17.3424	0.950360
$334$	0	0
$335$	5.92413	0.323670
$336$	0	0
$337$	−20.3843	−1.11040	−0.555201	−	0.831716i	$-0.687359\pi$
$337$	−20.3843	−1.11040	−0.555201	+	0.831716i	$0.687359\pi$
$338$	0	0
$339$	10.4308	0.566524
$340$	0	0
$341$	−2.60099	−0.140852
$342$	0	0
$343$	−9.60086	−0.518398
$344$	0	0
$345$	20.2747	1.09156
$346$	0	0
$347$	14.0829	0.756009	0.378004	−	0.925804i	$-0.376610\pi$
$347$	14.0829	0.756009	0.378004	+	0.925804i	$0.376610\pi$
$348$	0	0
$349$	3.67666	0.196807	0.0984037	−	0.995147i	$-0.468626\pi$
$349$	3.67666	0.196807	0.0984037	+	0.995147i	$0.468626\pi$
$350$	0	0
$351$	15.7354	0.839891
$352$	0	0
$353$	29.2177	1.55510	0.777552	−	0.628819i	$-0.216461\pi$
$353$	29.2177	1.55510	0.777552	+	0.628819i	$0.216461\pi$
$354$	0	0
$355$	4.12596	0.218983
$356$	0	0
$357$	7.22060	0.382155
$358$	0	0
$359$	−32.1254	−1.69552	−0.847758	−	0.530383i	$-0.822048\pi$
$359$	−32.1254	−1.69552	−0.847758	+	0.530383i	$0.822048\pi$
$360$	0	0
$361$	−11.9246	−0.627612
$362$	0	0
$363$	−2.65731	−0.139473
$364$	0	0
$365$	18.6098	0.974079
$366$	0	0
$367$	34.3966	1.79549	0.897743	−	0.440520i	$-0.145206\pi$
$367$	34.3966	1.79549	0.897743	+	0.440520i	$0.145206\pi$
$368$	0	0
$369$	8.56816	0.446041
$370$	0	0
$371$	2.97955	0.154691
$372$	0	0
$373$	18.9900	0.983265	0.491633	−	0.870803i	$-0.336400\pi$
$373$	18.9900	0.983265	0.491633	+	0.870803i	$0.336400\pi$
$374$	0	0
$375$	25.5534	1.31957
$376$	0	0
$377$	49.1393	2.53080
$378$	0	0
$379$	16.0400	0.823918	0.411959	−	0.911202i	$-0.364845\pi$
$379$	16.0400	0.823918	0.411959	+	0.911202i	$0.364845\pi$
$380$	0	0
$381$	−28.7365	−1.47222
$382$	0	0
$383$	−6.65535	−0.340072	−0.170036	−	0.985438i	$-0.554388\pi$
$383$	−6.65535	−0.340072	−0.170036	+	0.985438i	$0.554388\pi$
$384$	0	0
$385$	1.75866	0.0896294
$386$	0	0
$387$	34.5650	1.75704
$388$	0	0
$389$	−24.8221	−1.25853	−0.629266	−	0.777190i	$-0.716644\pi$
$389$	−24.8221	−1.25853	−0.629266	+	0.777190i	$0.716644\pi$
$390$	0	0
$391$	11.7886	0.596177
$392$	0	0
$393$	−49.1154	−2.47754
$394$	0	0
$395$	2.76593	0.139169
$396$	0	0
$397$	−27.4645	−1.37840	−0.689201	−	0.724570i	$-0.742038\pi$
$397$	−27.4645	−1.37840	−0.689201	+	0.724570i	$0.742038\pi$
$398$	0	0
$399$	5.02914	0.251772
$400$	0	0
$401$	17.4697	0.872393	0.436196	−	0.899851i	$-0.356325\pi$
$401$	17.4697	0.872393	0.436196	+	0.899851i	$0.356325\pi$
$402$	0	0
$403$	14.5125	0.722917
$404$	0	0
$405$	−11.5920	−0.576012
$406$	0	0
$407$	4.27018	0.211665
$408$	0	0
$409$	16.9665	0.838941	0.419471	−	0.907769i	$-0.362216\pi$
$409$	16.9665	0.838941	0.419471	+	0.907769i	$0.362216\pi$
$410$	0	0
$411$	−2.65731	−0.131075
$412$	0	0
$413$	3.71621	0.182863
$414$	0	0
$415$	0.693523	0.0340437
$416$	0	0
$417$	29.8624	1.46237
$418$	0	0
$419$	−13.8333	−0.675803	−0.337901	−	0.941181i	$-0.609717\pi$
$419$	−13.8333	−0.675803	−0.337901	+	0.941181i	$0.609717\pi$
$420$	0	0
$421$	−25.7319	−1.25410	−0.627049	−	0.778980i	$-0.715737\pi$
$421$	−25.7319	−1.25410	−0.627049	+	0.778980i	$0.715737\pi$
$422$	0	0
$423$	16.9774	0.825470
$424$	0	0
$425$	−4.23734	−0.205541
$426$	0	0
$427$	2.38444	0.115391
$428$	0	0
$429$	14.8267	0.715838
$430$	0	0
$431$	31.8249	1.53295	0.766476	−	0.642273i	$-0.222008\pi$
$431$	31.8249	1.53295	0.766476	+	0.642273i	$0.222008\pi$
$432$	0	0
$433$	20.0948	0.965692	0.482846	−	0.875705i	$-0.339603\pi$
$433$	20.0948	0.965692	0.482846	+	0.875705i	$0.339603\pi$
$434$	0	0
$435$	57.8460	2.77350
$436$	0	0
$437$	8.21077	0.392774
$438$	0	0
$439$	−11.8603	−0.566060	−0.283030	−	0.959111i	$-0.591340\pi$
$439$	−11.8603	−0.566060	−0.283030	+	0.959111i	$0.591340\pi$
$440$	0	0
$441$	−26.3730	−1.25586
$442$	0	0
$443$	33.4948	1.59139	0.795693	−	0.605700i	$-0.207107\pi$
$443$	33.4948	1.59139	0.795693	+	0.605700i	$0.207107\pi$
$444$	0	0
$445$	42.0985	1.99566
$446$	0	0
$447$	17.6122	0.833029
$448$	0	0
$449$	1.73557	0.0819066	0.0409533	−	0.999161i	$-0.486961\pi$
$449$	1.73557	0.0819066	0.0409533	+	0.999161i	$0.486961\pi$
$450$	0	0
$451$	2.10972	0.0993427
$452$	0	0
$453$	−11.1362	−0.523226
$454$	0	0
$455$	−9.81257	−0.460020
$456$	0	0
$457$	40.4309	1.89128	0.945639	−	0.325218i	$-0.105438\pi$
$457$	40.4309	1.89128	0.945639	+	0.325218i	$0.105438\pi$
$458$	0	0
$459$	10.7703	0.502716
$460$	0	0
$461$	−12.2764	−0.571767	−0.285883	−	0.958264i	$-0.592287\pi$
$461$	−12.2764	−0.571767	−0.285883	+	0.958264i	$0.592287\pi$
$462$	0	0
$463$	3.23524	0.150354	0.0751771	−	0.997170i	$-0.476048\pi$
$463$	3.23524	0.150354	0.0751771	+	0.997170i	$0.476048\pi$
$464$	0	0
$465$	17.0838	0.792243
$466$	0	0
$467$	−9.35582	−0.432936	−0.216468	−	0.976290i	$-0.569454\pi$
$467$	−9.35582	−0.432936	−0.216468	+	0.976290i	$0.569454\pi$
$468$	0	0
$469$	1.70529	0.0787429
$470$	0	0
$471$	14.5354	0.669754
$472$	0	0
$473$	8.51085	0.391329
$474$	0	0
$475$	−2.95130	−0.135415
$476$	0	0
$477$	17.0074	0.778715
$478$	0	0
$479$	33.4910	1.53024	0.765121	−	0.643887i	$-0.222679\pi$
$479$	33.4910	1.53024	0.765121	+	0.643887i	$0.222679\pi$
$480$	0	0
$481$	−23.8258	−1.08636
$482$	0	0
$483$	5.83618	0.265555
$484$	0	0
$485$	−4.72328	−0.214473
$486$	0	0
$487$	−2.39912	−0.108714	−0.0543572	−	0.998522i	$-0.517311\pi$
$487$	−2.39912	−0.108714	−0.0543572	+	0.998522i	$0.517311\pi$
$488$	0	0
$489$	−46.8081	−2.11674
$490$	0	0
$491$	−15.2184	−0.686796	−0.343398	−	0.939190i	$-0.611578\pi$
$491$	−15.2184	−0.686796	−0.343398	+	0.939190i	$0.611578\pi$
$492$	0	0
$493$	33.6342	1.51481
$494$	0	0
$495$	10.0385	0.451196
$496$	0	0
$497$	1.18768	0.0532746
$498$	0	0
$499$	12.1494	0.543881	0.271941	−	0.962314i	$-0.412335\pi$
$499$	12.1494	0.543881	0.271941	+	0.962314i	$0.412335\pi$
$500$	0	0
$501$	−46.5095	−2.07789
$502$	0	0
$503$	−17.2600	−0.769585	−0.384793	−	0.923003i	$-0.625727\pi$
$503$	−17.2600	−0.769585	−0.384793	+	0.923003i	$0.625727\pi$
$504$	0	0
$505$	15.0762	0.670881
$506$	0	0
$507$	−48.1816	−2.13982
$508$	0	0
$509$	2.28420	0.101245	0.0506226	−	0.998718i	$-0.483879\pi$
$509$	2.28420	0.101245	0.0506226	+	0.998718i	$0.483879\pi$
$510$	0	0
$511$	5.35690	0.236975
$512$	0	0
$513$	7.50152	0.331200
$514$	0	0
$515$	8.09626	0.356764
$516$	0	0
$517$	4.18031	0.183850
$518$	0	0
$519$	37.2164	1.63362
$520$	0	0
$521$	37.1173	1.62614	0.813070	−	0.582166i	$-0.197795\pi$
$521$	37.1173	1.62614	0.813070	+	0.582166i	$0.197795\pi$
$522$	0	0
$523$	−1.32277	−0.0578408	−0.0289204	−	0.999582i	$-0.509207\pi$
$523$	−1.32277	−0.0578408	−0.0289204	+	0.999582i	$0.509207\pi$
$524$	0	0
$525$	−2.09777	−0.0915543
$526$	0	0
$527$	9.93329	0.432701
$528$	0	0
$529$	−13.4716	−0.585723
$530$	0	0
$531$	21.2123	0.920534
$532$	0	0
$533$	−11.7713	−0.509873
$534$	0	0
$535$	14.7793	0.638963
$536$	0	0
$537$	10.0826	0.435097
$538$	0	0
$539$	−6.49376	−0.279706
$540$	0	0
$541$	−12.9518	−0.556843	−0.278421	−	0.960459i	$-0.589811\pi$
$541$	−12.9518	−0.556843	−0.278421	+	0.960459i	$0.589811\pi$
$542$	0	0
$543$	−32.0184	−1.37404
$544$	0	0
$545$	−31.2583	−1.33896
$546$	0	0
$547$	−15.5762	−0.665992	−0.332996	−	0.942928i	$-0.608060\pi$
$547$	−15.5762	−0.665992	−0.332996	+	0.942928i	$0.608060\pi$
$548$	0	0
$549$	13.6105	0.580881
$550$	0	0
$551$	23.4262	0.997989
$552$	0	0
$553$	0.796187	0.0338573
$554$	0	0
$555$	−28.0474	−1.19055
$556$	0	0
$557$	−30.6738	−1.29969	−0.649845	−	0.760067i	$-0.725166\pi$
$557$	−30.6738	−1.29969	−0.649845	+	0.760067i	$0.725166\pi$
$558$	0	0
$559$	−47.4870	−2.00849
$560$	0	0
$561$	10.1484	0.428464
$562$	0	0
$563$	12.7733	0.538329	0.269164	−	0.963094i	$-0.413253\pi$
$563$	12.7733	0.538329	0.269164	+	0.963094i	$0.413253\pi$
$564$	0	0
$565$	−9.70243	−0.408184
$566$	0	0
$567$	−3.33682	−0.140133
$568$	0	0
$569$	23.0493	0.966278	0.483139	−	0.875544i	$-0.339497\pi$
$569$	23.0493	0.966278	0.483139	+	0.875544i	$0.339497\pi$
$570$	0	0
$571$	−36.4277	−1.52445	−0.762227	−	0.647310i	$-0.775894\pi$
$571$	−36.4277	−1.52445	−0.762227	+	0.647310i	$0.775894\pi$
$572$	0	0
$573$	−41.1289	−1.71819
$574$	0	0
$575$	−3.42491	−0.142828
$576$	0	0
$577$	16.4183	0.683504	0.341752	−	0.939790i	$-0.388980\pi$
$577$	16.4183	0.683504	0.341752	+	0.939790i	$0.388980\pi$
$578$	0	0
$579$	−51.5949	−2.14421
$580$	0	0
$581$	0.199634	0.00828221
$582$	0	0
$583$	4.18768	0.173436
$584$	0	0
$585$	−56.0105	−2.31575
$586$	0	0
$587$	14.4233	0.595314	0.297657	−	0.954673i	$-0.403795\pi$
$587$	14.4233	0.595314	0.297657	+	0.954673i	$0.403795\pi$
$588$	0	0
$589$	6.91852	0.285073
$590$	0	0
$591$	26.0445	1.07133
$592$	0	0
$593$	−29.7137	−1.22020	−0.610098	−	0.792326i	$-0.708870\pi$
$593$	−29.7137	−1.22020	−0.610098	+	0.792326i	$0.708870\pi$
$594$	0	0
$595$	−6.71638	−0.275345
$596$	0	0
$597$	18.7301	0.766572
$598$	0	0
$599$	22.4518	0.917357	0.458679	−	0.888602i	$-0.348323\pi$
$599$	22.4518	0.917357	0.458679	+	0.888602i	$0.348323\pi$
$600$	0	0
$601$	1.06243	0.0433375	0.0216687	−	0.999765i	$-0.493102\pi$
$601$	1.06243	0.0433375	0.0216687	+	0.999765i	$0.493102\pi$
$602$	0	0
$603$	9.73384	0.396393
$604$	0	0
$605$	2.47175	0.100491
$606$	0	0
$607$	−19.7549	−0.801825	−0.400913	−	0.916116i	$-0.631307\pi$
$607$	−19.7549	−0.801825	−0.400913	+	0.916116i	$0.631307\pi$
$608$	0	0
$609$	16.6512	0.674742
$610$	0	0
$611$	−23.3244	−0.943603
$612$	0	0
$613$	−13.3196	−0.537973	−0.268987	−	0.963144i	$-0.586689\pi$
$613$	−13.3196	−0.537973	−0.268987	+	0.963144i	$0.586689\pi$
$614$	0	0
$615$	−13.8570	−0.558769
$616$	0	0
$617$	−31.3466	−1.26197	−0.630984	−	0.775796i	$-0.717349\pi$
$617$	−31.3466	−1.26197	−0.630984	+	0.775796i	$0.717349\pi$
$618$	0	0
$619$	−13.9503	−0.560712	−0.280356	−	0.959896i	$-0.590452\pi$
$619$	−13.9503	−0.560712	−0.280356	+	0.959896i	$0.590452\pi$
$620$	0	0
$621$	8.70531	0.349332
$622$	0	0
$623$	12.1183	0.485508
$624$	0	0
$625$	−29.3166	−1.17266
$626$	0	0
$627$	7.06832	0.282282
$628$	0	0
$629$	−16.3080	−0.650243
$630$	0	0
$631$	12.2997	0.489644	0.244822	−	0.969568i	$-0.421271\pi$
$631$	12.2997	0.489644	0.244822	+	0.969568i	$0.421271\pi$
$632$	0	0
$633$	−28.0459	−1.11472
$634$	0	0
$635$	26.7299	1.06074
$636$	0	0
$637$	36.2325	1.43558
$638$	0	0
$639$	6.77930	0.268185
$640$	0	0
$641$	20.4461	0.807571	0.403786	−	0.914854i	$-0.367694\pi$
$641$	20.4461	0.807571	0.403786	+	0.914854i	$0.367694\pi$
$642$	0	0
$643$	−11.4844	−0.452899	−0.226450	−	0.974023i	$-0.572712\pi$
$643$	−11.4844	−0.452899	−0.226450	+	0.974023i	$0.572712\pi$
$644$	0	0
$645$	−55.9009	−2.20110
$646$	0	0
$647$	37.2362	1.46391	0.731953	−	0.681355i	$-0.238609\pi$
$647$	37.2362	1.46391	0.731953	+	0.681355i	$0.238609\pi$
$648$	0	0
$649$	5.22304	0.205022
$650$	0	0
$651$	4.91766	0.192738
$652$	0	0
$653$	−2.93735	−0.114947	−0.0574737	−	0.998347i	$-0.518305\pi$
$653$	−2.93735	−0.114947	−0.0574737	+	0.998347i	$0.518305\pi$
$654$	0	0
$655$	45.6856	1.78508
$656$	0	0
$657$	30.5774	1.19294
$658$	0	0
$659$	−22.4083	−0.872903	−0.436452	−	0.899728i	$-0.643765\pi$
$659$	−22.4083	−0.872903	−0.436452	+	0.899728i	$0.643765\pi$
$660$	0	0
$661$	−10.4848	−0.407811	−0.203906	−	0.978991i	$-0.565364\pi$
$661$	−10.4848	−0.407811	−0.203906	+	0.978991i	$0.565364\pi$
$662$	0	0
$663$	−56.6237	−2.19908
$664$	0	0
$665$	−4.67795	−0.181403
$666$	0	0
$667$	27.1855	1.05263
$668$	0	0
$669$	−55.1645	−2.13278
$670$	0	0
$671$	3.35127	0.129374
$672$	0	0
$673$	−9.38322	−0.361697	−0.180848	−	0.983511i	$-0.557884\pi$
$673$	−9.38322	−0.361697	−0.180848	+	0.983511i	$0.557884\pi$
$674$	0	0
$675$	−3.12906	−0.120438
$676$	0	0
$677$	37.2336	1.43100	0.715501	−	0.698612i	$-0.246198\pi$
$677$	37.2336	1.43100	0.715501	+	0.698612i	$0.246198\pi$
$678$	0	0
$679$	−1.35962	−0.0521774
$680$	0	0
$681$	25.9386	0.993969
$682$	0	0
$683$	−15.1323	−0.579020	−0.289510	−	0.957175i	$-0.593492\pi$
$683$	−15.1323	−0.579020	−0.289510	+	0.957175i	$0.593492\pi$
$684$	0	0
$685$	2.47175	0.0944406
$686$	0	0
$687$	22.9366	0.875086
$688$	0	0
$689$	−23.3655	−0.890156
$690$	0	0
$691$	20.1532	0.766665	0.383333	−	0.923610i	$-0.374776\pi$
$691$	20.1532	0.766665	0.383333	+	0.923610i	$0.374776\pi$
$692$	0	0
$693$	2.88962	0.109768
$694$	0	0
$695$	−27.7771	−1.05364
$696$	0	0
$697$	−8.05709	−0.305184
$698$	0	0
$699$	71.7427	2.71356
$700$	0	0
$701$	15.3769	0.580776	0.290388	−	0.956909i	$-0.406216\pi$
$701$	15.3769	0.580776	0.290388	+	0.956909i	$0.406216\pi$
$702$	0	0
$703$	−11.3585	−0.428394
$704$	0	0
$705$	−27.4571	−1.03409
$706$	0	0
$707$	4.33975	0.163213
$708$	0	0
$709$	2.81603	0.105758	0.0528792	−	0.998601i	$-0.483160\pi$
$709$	2.81603	0.105758	0.0528792	+	0.998601i	$0.483160\pi$
$710$	0	0
$711$	4.54466	0.170438
$712$	0	0
$713$	8.02876	0.300680
$714$	0	0
$715$	−13.7913	−0.515766
$716$	0	0
$717$	22.5806	0.843289
$718$	0	0
$719$	−38.3344	−1.42963	−0.714816	−	0.699313i	$-0.753490\pi$
$719$	−38.3344	−1.42963	−0.714816	+	0.699313i	$0.753490\pi$
$720$	0	0
$721$	2.33055	0.0867941
$722$	0	0
$723$	−43.7336	−1.62647
$724$	0	0
$725$	−9.77162	−0.362909
$726$	0	0
$727$	6.39633	0.237227	0.118613	−	0.992941i	$-0.462155\pi$
$727$	6.39633	0.237227	0.118613	+	0.992941i	$0.462155\pi$
$728$	0	0
$729$	−41.5288	−1.53810
$730$	0	0
$731$	−32.5033	−1.20218
$732$	0	0
$733$	−13.6114	−0.502749	−0.251374	−	0.967890i	$-0.580883\pi$
$733$	−13.6114	−0.502749	−0.251374	+	0.967890i	$0.580883\pi$
$734$	0	0
$735$	42.6523	1.57325
$736$	0	0
$737$	2.39674	0.0882850
$738$	0	0
$739$	2.56509	0.0943585	0.0471792	−	0.998886i	$-0.484977\pi$
$739$	2.56509	0.0943585	0.0471792	+	0.998886i	$0.484977\pi$
$740$	0	0
$741$	−39.4383	−1.44880
$742$	0	0
$743$	52.5871	1.92924	0.964618	−	0.263653i	$-0.0849273\pi$
$743$	52.5871	1.92924	0.964618	+	0.263653i	$0.0849273\pi$
$744$	0	0
$745$	−16.3823	−0.600203
$746$	0	0
$747$	1.13952	0.0416927
$748$	0	0
$749$	4.25428	0.155448
$750$	0	0
$751$	21.1179	0.770603	0.385301	−	0.922791i	$-0.374097\pi$
$751$	21.1179	0.770603	0.385301	+	0.922791i	$0.374097\pi$
$752$	0	0
$753$	52.1757	1.90139
$754$	0	0
$755$	10.3586	0.376988
$756$	0	0
$757$	40.6182	1.47629	0.738146	−	0.674641i	$-0.235701\pi$
$757$	40.6182	1.47629	0.738146	+	0.674641i	$0.235701\pi$
$758$	0	0
$759$	8.20260	0.297735
$760$	0	0
$761$	18.9401	0.686579	0.343290	−	0.939230i	$-0.388459\pi$
$761$	18.9401	0.686579	0.343290	+	0.939230i	$0.388459\pi$
$762$	0	0
$763$	−8.99784	−0.325744
$764$	0	0
$765$	−38.3373	−1.38609
$766$	0	0
$767$	−29.1424	−1.05227
$768$	0	0
$769$	−11.3276	−0.408483	−0.204242	−	0.978921i	$-0.565473\pi$
$769$	−11.3276	−0.408483	−0.204242	+	0.978921i	$0.565473\pi$
$770$	0	0
$771$	−59.4630	−2.14151
$772$	0	0
$773$	31.5447	1.13458	0.567292	−	0.823516i	$-0.307991\pi$
$773$	31.5447	1.13458	0.567292	+	0.823516i	$0.307991\pi$
$774$	0	0
$775$	−2.88588	−0.103664
$776$	0	0
$777$	−8.07357	−0.289638
$778$	0	0
$779$	−5.61175	−0.201062
$780$	0	0
$781$	1.66925	0.0597304
$782$	0	0
$783$	24.8372	0.887608
$784$	0	0
$785$	−13.5203	−0.482562
$786$	0	0
$787$	19.7103	0.702597	0.351298	−	0.936264i	$-0.385740\pi$
$787$	19.7103	0.702597	0.351298	+	0.936264i	$0.385740\pi$
$788$	0	0
$789$	19.8205	0.705629
$790$	0	0
$791$	−2.79289	−0.0993037
$792$	0	0
$793$	−18.6987	−0.664010
$794$	0	0
$795$	−27.5055	−0.975520
$796$	0	0
$797$	28.3533	1.00433	0.502163	−	0.864773i	$-0.332538\pi$
$797$	28.3533	1.00433	0.502163	+	0.864773i	$0.332538\pi$
$798$	0	0
$799$	−15.9648	−0.564792
$800$	0	0
$801$	69.1714	2.44405
$802$	0	0
$803$	7.52899	0.265692
$804$	0	0
$805$	−5.42863	−0.191334
$806$	0	0
$807$	−51.9735	−1.82955
$808$	0	0
$809$	−17.0074	−0.597949	−0.298975	−	0.954261i	$-0.596645\pi$
$809$	−17.0074	−0.597949	−0.298975	+	0.954261i	$0.596645\pi$
$810$	0	0
$811$	20.5447	0.721420	0.360710	−	0.932678i	$-0.382534\pi$
$811$	20.5447	0.721420	0.360710	+	0.932678i	$0.382534\pi$
$812$	0	0
$813$	80.1242	2.81008
$814$	0	0
$815$	43.5395	1.52512
$816$	0	0
$817$	−22.6385	−0.792021
$818$	0	0
$819$	−16.1229	−0.563379
$820$	0	0
$821$	−18.7653	−0.654915	−0.327457	−	0.944866i	$-0.606192\pi$
$821$	−18.7653	−0.654915	−0.327457	+	0.944866i	$0.606192\pi$
$822$	0	0
$823$	−17.7731	−0.619532	−0.309766	−	0.950813i	$-0.600251\pi$
$823$	−17.7731	−0.619532	−0.309766	+	0.950813i	$0.600251\pi$
$824$	0	0
$825$	−2.94836	−0.102649
$826$	0	0
$827$	−1.89797	−0.0659988	−0.0329994	−	0.999455i	$-0.510506\pi$
$827$	−1.89797	−0.0659988	−0.0329994	+	0.999455i	$0.510506\pi$
$828$	0	0
$829$	−31.0432	−1.07818	−0.539088	−	0.842250i	$-0.681231\pi$
$829$	−31.0432	−1.07818	−0.539088	+	0.842250i	$0.681231\pi$
$830$	0	0
$831$	−40.7862	−1.41486
$832$	0	0
$833$	24.7999	0.859267
$834$	0	0
$835$	43.2617	1.49713
$836$	0	0
$837$	7.33523	0.253543
$838$	0	0
$839$	−16.7979	−0.579928	−0.289964	−	0.957038i	$-0.593643\pi$
$839$	−16.7979	−0.579928	−0.289964	+	0.957038i	$0.593643\pi$
$840$	0	0
$841$	48.5630	1.67459
$842$	0	0
$843$	17.1859	0.591915
$844$	0	0
$845$	44.8171	1.54175
$846$	0	0
$847$	0.711504	0.0244476
$848$	0	0
$849$	−43.2148	−1.48313
$850$	0	0
$851$	−13.1812	−0.451847
$852$	0	0
$853$	39.6254	1.35675	0.678374	−	0.734717i	$-0.262685\pi$
$853$	39.6254	1.35675	0.678374	+	0.734717i	$0.262685\pi$
$854$	0	0
$855$	−26.7019	−0.913185
$856$	0	0
$857$	−5.47139	−0.186899	−0.0934495	−	0.995624i	$-0.529789\pi$
$857$	−5.47139	−0.186899	−0.0934495	+	0.995624i	$0.529789\pi$
$858$	0	0
$859$	−7.11536	−0.242773	−0.121387	−	0.992605i	$-0.538734\pi$
$859$	−7.11536	−0.242773	−0.121387	+	0.992605i	$0.538734\pi$
$860$	0	0
$861$	−3.98881	−0.135938
$862$	0	0
$863$	34.6335	1.17894	0.589470	−	0.807791i	$-0.299337\pi$
$863$	34.6335	1.17894	0.589470	+	0.807791i	$0.299337\pi$
$864$	0	0
$865$	−34.6175	−1.17703
$866$	0	0
$867$	6.41722	0.217940
$868$	0	0
$869$	1.11902	0.0379602
$870$	0	0
$871$	−13.3728	−0.453120
$872$	0	0
$873$	−7.76075	−0.262662
$874$	0	0
$875$	−6.84200	−0.231302
$876$	0	0
$877$	−10.7598	−0.363334	−0.181667	−	0.983360i	$-0.558149\pi$
$877$	−10.7598	−0.363334	−0.181667	+	0.983360i	$0.558149\pi$
$878$	0	0
$879$	69.9135	2.35812
$880$	0	0
$881$	15.4575	0.520775	0.260387	−	0.965504i	$-0.416150\pi$
$881$	15.4575	0.520775	0.260387	+	0.965504i	$0.416150\pi$
$882$	0	0
$883$	−36.3293	−1.22258	−0.611289	−	0.791407i	$-0.709349\pi$
$883$	−36.3293	−1.22258	−0.611289	+	0.791407i	$0.709349\pi$
$884$	0	0
$885$	−34.3059	−1.15318
$886$	0	0
$887$	−36.9254	−1.23983	−0.619917	−	0.784668i	$-0.712834\pi$
$887$	−36.9254	−1.23983	−0.619917	+	0.784668i	$0.712834\pi$
$888$	0	0
$889$	7.69431	0.258059
$890$	0	0
$891$	−4.68981	−0.157115
$892$	0	0
$893$	−11.1194	−0.372097
$894$	0	0
$895$	−9.37855	−0.313490
$896$	0	0
$897$	−45.7671	−1.52812
$898$	0	0
$899$	22.9069	0.763988
$900$	0	0
$901$	−15.9929	−0.532802
$902$	0	0
$903$	−16.0913	−0.535486
$904$	0	0
$905$	29.7825	0.990005
$906$	0	0
$907$	−27.1558	−0.901695	−0.450847	−	0.892601i	$-0.648878\pi$
$907$	−27.1558	−0.901695	−0.450847	+	0.892601i	$0.648878\pi$
$908$	0	0
$909$	24.7714	0.821617
$910$	0	0
$911$	−3.98541	−0.132042	−0.0660212	−	0.997818i	$-0.521030\pi$
$911$	−3.98541	−0.132042	−0.0660212	+	0.997818i	$0.521030\pi$
$912$	0	0
$913$	0.280580	0.00928585
$914$	0	0
$915$	−22.0118	−0.727687
$916$	0	0
$917$	13.1508	0.434278
$918$	0	0
$919$	5.94971	0.196263	0.0981314	−	0.995173i	$-0.468713\pi$
$919$	5.94971	0.196263	0.0981314	+	0.995173i	$0.468713\pi$
$920$	0	0
$921$	−77.7146	−2.56078
$922$	0	0
$923$	−9.31372	−0.306565
$924$	0	0
$925$	4.73790	0.155781
$926$	0	0
$927$	13.3028	0.436922
$928$	0	0
$929$	−47.9201	−1.57221	−0.786104	−	0.618094i	$-0.787905\pi$
$929$	−47.9201	−1.57221	−0.786104	+	0.618094i	$0.787905\pi$
$930$	0	0
$931$	17.2731	0.566104
$932$	0	0
$933$	−55.7573	−1.82541
$934$	0	0
$935$	−9.43970	−0.308711
$936$	0	0
$937$	−1.28111	−0.0418521	−0.0209261	−	0.999781i	$-0.506661\pi$
$937$	−1.28111	−0.0418521	−0.0209261	+	0.999781i	$0.506661\pi$
$938$	0	0
$939$	−72.4003	−2.36269
$940$	0	0
$941$	19.7459	0.643699	0.321849	−	0.946791i	$-0.395696\pi$
$941$	19.7459	0.643699	0.321849	+	0.946791i	$0.395696\pi$
$942$	0	0
$943$	−6.51229	−0.212069
$944$	0	0
$945$	−4.95970	−0.161339
$946$	0	0
$947$	−26.3106	−0.854979	−0.427489	−	0.904020i	$-0.640602\pi$
$947$	−26.3106	−0.854979	−0.427489	+	0.904020i	$0.640602\pi$
$948$	0	0
$949$	−42.0086	−1.36366
$950$	0	0
$951$	27.8109	0.901829
$952$	0	0
$953$	37.7105	1.22156	0.610782	−	0.791799i	$-0.290855\pi$
$953$	37.7105	1.22156	0.610782	+	0.791799i	$0.290855\pi$
$954$	0	0
$955$	38.2569	1.23796
$956$	0	0
$957$	23.4029	0.756508
$958$	0	0
$959$	0.711504	0.0229757
$960$	0	0
$961$	−24.2348	−0.781769
$962$	0	0
$963$	24.2836	0.782527
$964$	0	0
$965$	47.9919	1.54492
$966$	0	0
$967$	−31.3828	−1.00920	−0.504601	−	0.863353i	$-0.668360\pi$
$967$	−31.3828	−1.00920	−0.504601	+	0.863353i	$0.668360\pi$
$968$	0	0
$969$	−26.9942	−0.867179
$970$	0	0
$971$	−7.04947	−0.226228	−0.113114	−	0.993582i	$-0.536083\pi$
$971$	−7.04947	−0.226228	−0.113114	+	0.993582i	$0.536083\pi$
$972$	0	0
$973$	−7.99576	−0.256332
$974$	0	0
$975$	16.4506	0.526842
$976$	0	0
$977$	13.6624	0.437099	0.218550	−	0.975826i	$-0.429867\pi$
$977$	13.6624	0.437099	0.218550	+	0.975826i	$0.429867\pi$
$978$	0	0
$979$	17.0319	0.544342
$980$	0	0
$981$	−51.3600	−1.63980
$982$	0	0
$983$	−15.3481	−0.489527	−0.244764	−	0.969583i	$-0.578710\pi$
$983$	−15.3481	−0.489527	−0.244764	+	0.969583i	$0.578710\pi$
$984$	0	0
$985$	−24.2258	−0.771898
$986$	0	0
$987$	−7.90364	−0.251576
$988$	0	0
$989$	−26.2714	−0.835381
$990$	0	0
$991$	−8.92020	−0.283359	−0.141680	−	0.989913i	$-0.545250\pi$
$991$	−8.92020	−0.283359	−0.141680	+	0.989913i	$0.545250\pi$
$992$	0	0
$993$	13.0805	0.415098
$994$	0	0
$995$	−17.4222	−0.552320
$996$	0	0
$997$	−11.4759	−0.363445	−0.181722	−	0.983350i	$-0.558167\pi$
$997$	−11.4759	−0.363445	−0.181722	+	0.983350i	$0.558167\pi$
$998$	0	0
$999$	−12.0426	−0.381012

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6028.2.a.f.1.3	✓	29

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6028.2.a.f.1.3	✓	29	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:	\( N \)	\(=\)	\( 6028 = 2^{2} \cdot 11 \cdot 137 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6028.a (trivial)

\(f(q)\)	\(=\)	\(q-2.65731 q^{3} +2.47175 q^{5} +0.711504 q^{7} +4.06129 q^{9} +O(q^{10})\)	\(q-2.65731 q^{3} +2.47175 q^{5} +0.711504 q^{7} +4.06129 q^{9} +1.00000 q^{11} -5.57958 q^{13} -6.56819 q^{15} -3.81904 q^{17} -2.65996 q^{19} -1.89068 q^{21} -3.08681 q^{23} +1.10953 q^{25} -2.82017 q^{27} -8.80699 q^{29} -2.60099 q^{31} -2.65731 q^{33} +1.75866 q^{35} +4.27018 q^{37} +14.8267 q^{39} +2.10972 q^{41} +8.51085 q^{43} +10.0385 q^{45} +4.18031 q^{47} -6.49376 q^{49} +10.1484 q^{51} +4.18768 q^{53} +2.47175 q^{55} +7.06832 q^{57} +5.22304 q^{59} +3.35127 q^{61} +2.88962 q^{63} -13.7913 q^{65} +2.39674 q^{67} +8.20260 q^{69} +1.66925 q^{71} +7.52899 q^{73} -2.94836 q^{75} +0.711504 q^{77} +1.11902 q^{79} -4.68981 q^{81} +0.280580 q^{83} -9.43970 q^{85} +23.4029 q^{87} +17.0319 q^{89} -3.96989 q^{91} +6.91164 q^{93} -6.57474 q^{95} -1.91091 q^{97} +4.06129 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 29 q + 14 q^{3} + 9 q^{5} + 14 q^{7} + 43 q^{9}+O(q^{10}) \) 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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