LMFDB - Embedded newform 6028.2.a.f.1.15

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			1.15
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)

$f(q)$	$=$	$q+0.785650 q^{3} +1.28190 q^{5} -3.55486 q^{7} -2.38275 q^{9} +O(q^{10})$	$q+0.785650 q^{3} +1.28190 q^{5} -3.55486 q^{7} -2.38275 q^{9} +1.00000 q^{11} -1.72689 q^{13} +1.00713 q^{15} +3.23783 q^{17} -0.00937882 q^{19} -2.79288 q^{21} -7.62869 q^{23} -3.35673 q^{25} -4.22896 q^{27} +2.21148 q^{29} +5.55268 q^{31} +0.785650 q^{33} -4.55698 q^{35} -0.632832 q^{37} -1.35673 q^{39} +8.90650 q^{41} +10.4595 q^{43} -3.05446 q^{45} +0.474700 q^{47} +5.63706 q^{49} +2.54380 q^{51} +3.85714 q^{53} +1.28190 q^{55} -0.00736847 q^{57} +11.3335 q^{59} -14.3119 q^{61} +8.47037 q^{63} -2.21370 q^{65} +11.4256 q^{67} -5.99348 q^{69} -2.33230 q^{71} -5.28565 q^{73} -2.63721 q^{75} -3.55486 q^{77} +7.27996 q^{79} +3.82578 q^{81} +4.81948 q^{83} +4.15057 q^{85} +1.73745 q^{87} +5.18596 q^{89} +6.13886 q^{91} +4.36247 q^{93} -0.0120227 q^{95} +14.0544 q^{97} -2.38275 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	0.785650	0.453595	0.226798	−	0.973942i	$-0.427174\pi$
$3$	0.785650	0.453595	0.226798	+	0.973942i	$0.427174\pi$
$4$	0	0
$5$	1.28190	0.573284	0.286642	−	0.958038i	$-0.407461\pi$
$5$	1.28190	0.573284	0.286642	+	0.958038i	$0.407461\pi$
$6$	0	0
$7$	−3.55486	−1.34361	−0.671806	−	0.740727i	$-0.734481\pi$
$7$	−3.55486	−1.34361	−0.671806	+	0.740727i	$0.734481\pi$
$8$	0	0
$9$	−2.38275	−0.794251
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−1.72689	−0.478954	−0.239477	−	0.970902i	$-0.576976\pi$
$13$	−1.72689	−0.478954	−0.239477	+	0.970902i	$0.576976\pi$
$14$	0	0
$15$	1.00713	0.260039
$16$	0	0
$17$	3.23783	0.785288	0.392644	−	0.919690i	$-0.371560\pi$
$17$	3.23783	0.785288	0.392644	+	0.919690i	$0.371560\pi$
$18$	0	0
$19$	−0.00937882	−0.00215165	−0.00107582	−	0.999999i	$-0.500342\pi$
$19$	−0.00937882	−0.00215165	−0.00107582	+	0.999999i	$0.500342\pi$
$20$	0	0
$21$	−2.79288	−0.609456
$22$	0	0
$23$	−7.62869	−1.59069	−0.795346	−	0.606156i	$-0.792711\pi$
$23$	−7.62869	−1.59069	−0.795346	+	0.606156i	$0.792711\pi$
$24$	0	0
$25$	−3.35673	−0.671346
$26$	0	0
$27$	−4.22896	−0.813864
$28$	0	0
$29$	2.21148	0.410662	0.205331	−	0.978693i	$-0.434173\pi$
$29$	2.21148	0.410662	0.205331	+	0.978693i	$0.434173\pi$
$30$	0	0
$31$	5.55268	0.997291	0.498646	−	0.866806i	$-0.333831\pi$
$31$	5.55268	0.997291	0.498646	+	0.866806i	$0.333831\pi$
$32$	0	0
$33$	0.785650	0.136764
$34$	0	0
$35$	−4.55698	−0.770271
$36$	0	0
$37$	−0.632832	−0.104037	−0.0520185	−	0.998646i	$-0.516565\pi$
$37$	−0.632832	−0.104037	−0.0520185	+	0.998646i	$0.516565\pi$
$38$	0	0
$39$	−1.35673	−0.217251
$40$	0	0
$41$	8.90650	1.39096	0.695481	−	0.718544i	$-0.255191\pi$
$41$	8.90650	1.39096	0.695481	+	0.718544i	$0.255191\pi$
$42$	0	0
$43$	10.4595	1.59506	0.797528	−	0.603282i	$-0.206141\pi$
$43$	10.4595	1.59506	0.797528	+	0.603282i	$0.206141\pi$
$44$	0	0
$45$	−3.05446	−0.455331
$46$	0	0
$47$	0.474700	0.0692421	0.0346210	−	0.999401i	$-0.488978\pi$
$47$	0.474700	0.0692421	0.0346210	+	0.999401i	$0.488978\pi$
$48$	0	0
$49$	5.63706	0.805294
$50$	0	0
$51$	2.54380	0.356203
$52$	0	0
$53$	3.85714	0.529818	0.264909	−	0.964273i	$-0.414658\pi$
$53$	3.85714	0.529818	0.264909	+	0.964273i	$0.414658\pi$
$54$	0	0
$55$	1.28190	0.172852
$56$	0	0
$57$	−0.00736847	−0.000975978 0
$58$	0	0
$59$	11.3335	1.47550	0.737749	−	0.675075i	$-0.235889\pi$
$59$	11.3335	1.47550	0.737749	+	0.675075i	$0.235889\pi$
$60$	0	0
$61$	−14.3119	−1.83245	−0.916223	−	0.400670i	$-0.868778\pi$
$61$	−14.3119	−1.83245	−0.916223	+	0.400670i	$0.868778\pi$
$62$	0	0
$63$	8.47037	1.06717
$64$	0	0
$65$	−2.21370	−0.274576
$66$	0	0
$67$	11.4256	1.39585	0.697927	−	0.716169i	$-0.254106\pi$
$67$	11.4256	1.39585	0.697927	+	0.716169i	$0.254106\pi$
$68$	0	0
$69$	−5.99348	−0.721530
$70$	0	0
$71$	−2.33230	−0.276793	−0.138396	−	0.990377i	$-0.544195\pi$
$71$	−2.33230	−0.276793	−0.138396	+	0.990377i	$0.544195\pi$
$72$	0	0
$73$	−5.28565	−0.618639	−0.309320	−	0.950958i	$-0.600101\pi$
$73$	−5.28565	−0.618639	−0.309320	+	0.950958i	$0.600101\pi$
$74$	0	0
$75$	−2.63721	−0.304519
$76$	0	0
$77$	−3.55486	−0.405114
$78$	0	0
$79$	7.27996	0.819060	0.409530	−	0.912297i	$-0.365693\pi$
$79$	7.27996	0.819060	0.409530	+	0.912297i	$0.365693\pi$
$80$	0	0
$81$	3.82578	0.425087
$82$	0	0
$83$	4.81948	0.529007	0.264503	−	0.964385i	$-0.414792\pi$
$83$	4.81948	0.529007	0.264503	+	0.964385i	$0.414792\pi$
$84$	0	0
$85$	4.15057	0.450193
$86$	0	0
$87$	1.73745	0.186274
$88$	0	0
$89$	5.18596	0.549710	0.274855	−	0.961486i	$-0.411370\pi$
$89$	5.18596	0.549710	0.274855	+	0.961486i	$0.411370\pi$
$90$	0	0
$91$	6.13886	0.643528
$92$	0	0
$93$	4.36247	0.452367
$94$	0	0
$95$	−0.0120227	−0.00123351
$96$	0	0
$97$	14.0544	1.42701	0.713504	−	0.700651i	$-0.247107\pi$
$97$	14.0544	1.42701	0.713504	+	0.700651i	$0.247107\pi$
$98$	0	0
$99$	−2.38275	−0.239476
$100$	0	0
$101$	2.12758	0.211702	0.105851	−	0.994382i	$-0.466243\pi$
$101$	2.12758	0.211702	0.105851	+	0.994382i	$0.466243\pi$
$102$	0	0
$103$	12.3140	1.21334	0.606668	−	0.794955i	$-0.292506\pi$
$103$	12.3140	1.21334	0.606668	+	0.794955i	$0.292506\pi$
$104$	0	0
$105$	−3.58019	−0.349391
$106$	0	0
$107$	−0.0920398	−0.00889783	−0.00444891	−	0.999990i	$-0.501416\pi$
$107$	−0.0920398	−0.00889783	−0.00444891	+	0.999990i	$0.501416\pi$
$108$	0	0
$109$	−2.14287	−0.205250	−0.102625	−	0.994720i	$-0.532724\pi$
$109$	−2.14287	−0.205250	−0.102625	+	0.994720i	$0.532724\pi$
$110$	0	0
$111$	−0.497185	−0.0471907
$112$	0	0
$113$	−12.9992	−1.22286	−0.611429	−	0.791299i	$-0.709405\pi$
$113$	−12.9992	−1.22286	−0.611429	+	0.791299i	$0.709405\pi$
$114$	0	0
$115$	−9.77922	−0.911917
$116$	0	0
$117$	4.11476	0.380409
$118$	0	0
$119$	−11.5100	−1.05512
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	6.99739	0.630934
$124$	0	0
$125$	−10.7125	−0.958155
$126$	0	0
$127$	9.02177	0.800553	0.400276	−	0.916394i	$-0.368914\pi$
$127$	9.02177	0.800553	0.400276	+	0.916394i	$0.368914\pi$
$128$	0	0
$129$	8.21749	0.723510
$130$	0	0
$131$	−2.22519	−0.194416	−0.0972080	−	0.995264i	$-0.530991\pi$
$131$	−2.22519	−0.194416	−0.0972080	+	0.995264i	$0.530991\pi$
$132$	0	0
$133$	0.0333404	0.00289098
$134$	0	0
$135$	−5.42111	−0.466575
$136$	0	0
$137$	1.00000	0.0854358
$137$	1.00000	0.0854358
$138$	0	0
$139$	1.55124	0.131574	0.0657872	−	0.997834i	$-0.479044\pi$
$139$	1.55124	0.131574	0.0657872	+	0.997834i	$0.479044\pi$
$140$	0	0
$141$	0.372948	0.0314079
$142$	0	0
$143$	−1.72689	−0.144410
$144$	0	0
$145$	2.83490	0.235426
$146$	0	0
$147$	4.42875	0.365278
$148$	0	0
$149$	3.85647	0.315934	0.157967	−	0.987444i	$-0.449506\pi$
$149$	3.85647	0.315934	0.157967	+	0.987444i	$0.449506\pi$
$150$	0	0
$151$	−21.3596	−1.73822	−0.869110	−	0.494619i	$-0.835307\pi$
$151$	−21.3596	−1.73822	−0.869110	+	0.494619i	$0.835307\pi$
$152$	0	0
$153$	−7.71495	−0.623716
$154$	0	0
$155$	7.11799	0.571731
$156$	0	0
$157$	−3.91344	−0.312327	−0.156163	−	0.987731i	$-0.549913\pi$
$157$	−3.91344	−0.312327	−0.156163	+	0.987731i	$0.549913\pi$
$158$	0	0
$159$	3.03036	0.240323
$160$	0	0
$161$	27.1189	2.13727
$162$	0	0
$163$	23.9281	1.87420	0.937098	−	0.349068i	$-0.113502\pi$
$163$	23.9281	1.87420	0.937098	+	0.349068i	$0.113502\pi$
$164$	0	0
$165$	1.00713	0.0784046
$166$	0	0
$167$	−7.66575	−0.593194	−0.296597	−	0.955003i	$-0.595852\pi$
$167$	−7.66575	−0.593194	−0.296597	+	0.955003i	$0.595852\pi$
$168$	0	0
$169$	−10.0178	−0.770604
$170$	0	0
$171$	0.0223474	0.00170895
$172$	0	0
$173$	10.2807	0.781624	0.390812	−	0.920471i	$-0.372194\pi$
$173$	10.2807	0.781624	0.390812	+	0.920471i	$0.372194\pi$
$174$	0	0
$175$	11.9327	0.902029
$176$	0	0
$177$	8.90418	0.669279
$178$	0	0
$179$	1.74038	0.130082	0.0650409	−	0.997883i	$-0.479282\pi$
$179$	1.74038	0.130082	0.0650409	+	0.997883i	$0.479282\pi$
$180$	0	0
$181$	14.8656	1.10495	0.552476	−	0.833529i	$-0.313683\pi$
$181$	14.8656	1.10495	0.552476	+	0.833529i	$0.313683\pi$
$182$	0	0
$183$	−11.2441	−0.831188
$184$	0	0
$185$	−0.811229	−0.0596427
$186$	0	0
$187$	3.23783	0.236773
$188$	0	0
$189$	15.0334	1.09352
$190$	0	0
$191$	10.1691	0.735807	0.367904	−	0.929864i	$-0.380076\pi$
$191$	10.1691	0.735807	0.367904	+	0.929864i	$0.380076\pi$
$192$	0	0
$193$	13.8838	0.999376	0.499688	−	0.866206i	$-0.333448\pi$
$193$	13.8838	0.999376	0.499688	+	0.866206i	$0.333448\pi$
$194$	0	0
$195$	−1.73920	−0.124546
$196$	0	0
$197$	4.63426	0.330178	0.165089	−	0.986279i	$-0.447209\pi$
$197$	4.63426	0.330178	0.165089	+	0.986279i	$0.447209\pi$
$198$	0	0
$199$	16.0239	1.13590	0.567951	−	0.823062i	$-0.307736\pi$
$199$	16.0239	1.13590	0.567951	+	0.823062i	$0.307736\pi$
$200$	0	0
$201$	8.97649	0.633153
$202$	0	0
$203$	−7.86152	−0.551771
$204$	0	0
$205$	11.4173	0.797416
$206$	0	0
$207$	18.1773	1.26341
$208$	0	0
$209$	−0.00937882	−0.000648747 0
$210$	0	0
$211$	6.93910	0.477707	0.238854	−	0.971056i	$-0.423228\pi$
$211$	6.93910	0.477707	0.238854	+	0.971056i	$0.423228\pi$
$212$	0	0
$213$	−1.83237	−0.125552
$214$	0	0
$215$	13.4080	0.914419
$216$	0	0
$217$	−19.7390	−1.33997
$218$	0	0
$219$	−4.15267	−0.280612
$220$	0	0
$221$	−5.59138	−0.376117
$222$	0	0
$223$	−8.10270	−0.542597	−0.271298	−	0.962495i	$-0.587453\pi$
$223$	−8.10270	−0.542597	−0.271298	+	0.962495i	$0.587453\pi$
$224$	0	0
$225$	7.99826	0.533217
$226$	0	0
$227$	−5.67945	−0.376958	−0.188479	−	0.982077i	$-0.560356\pi$
$227$	−5.67945	−0.376958	−0.188479	+	0.982077i	$0.560356\pi$
$228$	0	0
$229$	18.5770	1.22760	0.613802	−	0.789460i	$-0.289639\pi$
$229$	18.5770	1.22760	0.613802	+	0.789460i	$0.289639\pi$
$230$	0	0
$231$	−2.79288	−0.183758
$232$	0	0
$233$	−0.419629	−0.0274908	−0.0137454	−	0.999906i	$-0.504375\pi$
$233$	−0.419629	−0.0274908	−0.0137454	+	0.999906i	$0.504375\pi$
$234$	0	0
$235$	0.608518	0.0396953
$236$	0	0
$237$	5.71950	0.371522
$238$	0	0
$239$	−4.53704	−0.293477	−0.146738	−	0.989175i	$-0.546878\pi$
$239$	−4.53704	−0.293477	−0.146738	+	0.989175i	$0.546878\pi$
$240$	0	0
$241$	−9.31453	−0.600001	−0.300001	−	0.953939i	$-0.596987\pi$
$241$	−9.31453	−0.600001	−0.300001	+	0.953939i	$0.596987\pi$
$242$	0	0
$243$	15.6926	1.00668
$244$	0	0
$245$	7.22615	0.461662
$246$	0	0
$247$	0.0161962	0.00103054
$248$	0	0
$249$	3.78643	0.239955
$250$	0	0
$251$	−19.5411	−1.23342	−0.616710	−	0.787190i	$-0.711535\pi$
$251$	−19.5411	−1.23342	−0.616710	+	0.787190i	$0.711535\pi$
$252$	0	0
$253$	−7.62869	−0.479612
$254$	0	0
$255$	3.26090	0.204205
$256$	0	0
$257$	21.4623	1.33878	0.669391	−	0.742910i	$-0.266555\pi$
$257$	21.4623	1.33878	0.669391	+	0.742910i	$0.266555\pi$
$258$	0	0
$259$	2.24963	0.139785
$260$	0	0
$261$	−5.26942	−0.326169
$262$	0	0
$263$	12.0277	0.741663	0.370831	−	0.928700i	$-0.379073\pi$
$263$	12.0277	0.741663	0.370831	+	0.928700i	$0.379073\pi$
$264$	0	0
$265$	4.94447	0.303736
$266$	0	0
$267$	4.07435	0.249346
$268$	0	0
$269$	−16.3436	−0.996490	−0.498245	−	0.867036i	$-0.666022\pi$
$269$	−16.3436	−0.996490	−0.498245	+	0.867036i	$0.666022\pi$
$270$	0	0
$271$	1.58044	0.0960052	0.0480026	−	0.998847i	$-0.484714\pi$
$271$	1.58044	0.0960052	0.0480026	+	0.998847i	$0.484714\pi$
$272$	0	0
$273$	4.82300	0.291901
$274$	0	0
$275$	−3.35673	−0.202418
$276$	0	0
$277$	−12.6837	−0.762090	−0.381045	−	0.924556i	$-0.624436\pi$
$277$	−12.6837	−0.762090	−0.381045	+	0.924556i	$0.624436\pi$
$278$	0	0
$279$	−13.2307	−0.792100
$280$	0	0
$281$	−23.8709	−1.42402	−0.712010	−	0.702169i	$-0.752215\pi$
$281$	−23.8709	−1.42402	−0.712010	+	0.702169i	$0.752215\pi$
$282$	0	0
$283$	11.0629	0.657622	0.328811	−	0.944396i	$-0.393352\pi$
$283$	11.0629	0.657622	0.328811	+	0.944396i	$0.393352\pi$
$284$	0	0
$285$	−0.00944565	−0.000559512 0
$286$	0	0
$287$	−31.6614	−1.86891
$288$	0	0
$289$	−6.51648	−0.383322
$290$	0	0
$291$	11.0418	0.647284
$292$	0	0
$293$	7.85881	0.459117	0.229558	−	0.973295i	$-0.426272\pi$
$293$	7.85881	0.459117	0.229558	+	0.973295i	$0.426272\pi$
$294$	0	0
$295$	14.5285	0.845879
$296$	0	0
$297$	−4.22896	−0.245389
$298$	0	0
$299$	13.1739	0.761867
$300$	0	0
$301$	−37.1820	−2.14314
$302$	0	0
$303$	1.67153	0.0960271
$304$	0	0
$305$	−18.3464	−1.05051
$306$	0	0
$307$	17.5163	0.999707	0.499854	−	0.866110i	$-0.333387\pi$
$307$	17.5163	0.999707	0.499854	+	0.866110i	$0.333387\pi$
$308$	0	0
$309$	9.67451	0.550364
$310$	0	0
$311$	−8.77180	−0.497403	−0.248701	−	0.968580i	$-0.580004\pi$
$311$	−8.77180	−0.497403	−0.248701	+	0.968580i	$0.580004\pi$
$312$	0	0
$313$	6.22542	0.351881	0.175941	−	0.984401i	$-0.443703\pi$
$313$	6.22542	0.351881	0.175941	+	0.984401i	$0.443703\pi$
$314$	0	0
$315$	10.8582	0.611789
$316$	0	0
$317$	21.3519	1.19924	0.599621	−	0.800284i	$-0.295318\pi$
$317$	21.3519	1.19924	0.599621	+	0.800284i	$0.295318\pi$
$318$	0	0
$319$	2.21148	0.123819
$320$	0	0
$321$	−0.0723111	−0.00403601
$322$	0	0
$323$	−0.0303670	−0.00168967
$324$	0	0
$325$	5.79671	0.321543
$326$	0	0
$327$	−1.68354	−0.0931002
$328$	0	0
$329$	−1.68749	−0.0930345
$330$	0	0
$331$	10.4577	0.574806	0.287403	−	0.957810i	$-0.407208\pi$
$331$	10.4577	0.574806	0.287403	+	0.957810i	$0.407208\pi$
$332$	0	0
$333$	1.50788	0.0826315
$334$	0	0
$335$	14.6464	0.800221
$336$	0	0
$337$	−5.91803	−0.322376	−0.161188	−	0.986924i	$-0.551532\pi$
$337$	−5.91803	−0.322376	−0.161188	+	0.986924i	$0.551532\pi$
$338$	0	0
$339$	−10.2128	−0.554682
$340$	0	0
$341$	5.55268	0.300695
$342$	0	0
$343$	4.84507	0.261609
$344$	0	0
$345$	−7.68305	−0.413641
$346$	0	0
$347$	4.14688	0.222616	0.111308	−	0.993786i	$-0.464496\pi$
$347$	4.14688	0.222616	0.111308	+	0.993786i	$0.464496\pi$
$348$	0	0
$349$	−28.9465	−1.54947	−0.774735	−	0.632286i	$-0.782117\pi$
$349$	−28.9465	−1.54947	−0.774735	+	0.632286i	$0.782117\pi$
$350$	0	0
$351$	7.30296	0.389803
$352$	0	0
$353$	−13.0521	−0.694693	−0.347347	−	0.937737i	$-0.612917\pi$
$353$	−13.0521	−0.694693	−0.347347	+	0.937737i	$0.612917\pi$
$354$	0	0
$355$	−2.98977	−0.158681
$356$	0	0
$357$	−9.04286	−0.478599
$358$	0	0
$359$	19.1993	1.01330	0.506650	−	0.862152i	$-0.330884\pi$
$359$	19.1993	1.01330	0.506650	+	0.862152i	$0.330884\pi$
$360$	0	0
$361$	−18.9999	−0.999995
$362$	0	0
$363$	0.785650	0.0412359
$364$	0	0
$365$	−6.77569	−0.354656
$366$	0	0
$367$	−20.9327	−1.09268	−0.546339	−	0.837564i	$-0.683979\pi$
$367$	−20.9327	−1.09268	−0.546339	+	0.837564i	$0.683979\pi$
$368$	0	0
$369$	−21.2220	−1.10477
$370$	0	0
$371$	−13.7116	−0.711870
$372$	0	0
$373$	1.59689	0.0826840	0.0413420	−	0.999145i	$-0.486837\pi$
$373$	1.59689	0.0826840	0.0413420	+	0.999145i	$0.486837\pi$
$374$	0	0
$375$	−8.41628	−0.434615
$376$	0	0
$377$	−3.81899	−0.196688
$378$	0	0
$379$	18.1553	0.932576	0.466288	−	0.884633i	$-0.345591\pi$
$379$	18.1553	0.932576	0.466288	+	0.884633i	$0.345591\pi$
$380$	0	0
$381$	7.08795	0.363127
$382$	0	0
$383$	8.64941	0.441964	0.220982	−	0.975278i	$-0.429074\pi$
$383$	8.64941	0.441964	0.220982	+	0.975278i	$0.429074\pi$
$384$	0	0
$385$	−4.55698	−0.232245
$386$	0	0
$387$	−24.9224	−1.26687
$388$	0	0
$389$	−18.3103	−0.928367	−0.464184	−	0.885739i	$-0.653652\pi$
$389$	−18.3103	−0.928367	−0.464184	+	0.885739i	$0.653652\pi$
$390$	0	0
$391$	−24.7004	−1.24915
$392$	0	0
$393$	−1.74822	−0.0881861
$394$	0	0
$395$	9.33219	0.469553
$396$	0	0
$397$	27.8504	1.39777	0.698885	−	0.715234i	$-0.253680\pi$
$397$	27.8504	1.39777	0.698885	+	0.715234i	$0.253680\pi$
$398$	0	0
$399$	0.0261939	0.00131134
$400$	0	0
$401$	−17.3076	−0.864300	−0.432150	−	0.901802i	$-0.642245\pi$
$401$	−17.3076	−0.864300	−0.432150	+	0.901802i	$0.642245\pi$
$402$	0	0
$403$	−9.58888	−0.477656
$404$	0	0
$405$	4.90427	0.243695
$406$	0	0
$407$	−0.632832	−0.0313683
$408$	0	0
$409$	−34.5542	−1.70860	−0.854299	−	0.519783i	$-0.826013\pi$
$409$	−34.5542	−1.70860	−0.854299	+	0.519783i	$0.826013\pi$
$410$	0	0
$411$	0.785650	0.0387533
$412$	0	0
$413$	−40.2891	−1.98250
$414$	0	0
$415$	6.17810	0.303271
$416$	0	0
$417$	1.21873	0.0596815
$418$	0	0
$419$	4.80357	0.234670	0.117335	−	0.993092i	$-0.462565\pi$
$419$	4.80357	0.234670	0.117335	+	0.993092i	$0.462565\pi$
$420$	0	0
$421$	27.6121	1.34573	0.672865	−	0.739766i	$-0.265064\pi$
$421$	27.6121	1.34573	0.672865	+	0.739766i	$0.265064\pi$
$422$	0	0
$423$	−1.13109	−0.0549956
$424$	0	0
$425$	−10.8685	−0.527200
$426$	0	0
$427$	50.8767	2.46210
$428$	0	0
$429$	−1.35673	−0.0655037
$430$	0	0
$431$	−34.5764	−1.66549	−0.832744	−	0.553659i	$-0.813231\pi$
$431$	−34.5764	−1.66549	−0.832744	+	0.553659i	$0.813231\pi$
$432$	0	0
$433$	13.0894	0.629035	0.314517	−	0.949252i	$-0.398157\pi$
$433$	13.0894	0.629035	0.314517	+	0.949252i	$0.398157\pi$
$434$	0	0
$435$	2.22724	0.106788
$436$	0	0
$437$	0.0715481	0.00342261
$438$	0	0
$439$	7.16309	0.341875	0.170938	−	0.985282i	$-0.445320\pi$
$439$	7.16309	0.341875	0.170938	+	0.985282i	$0.445320\pi$
$440$	0	0
$441$	−13.4317	−0.639606
$442$	0	0
$443$	−17.4622	−0.829656	−0.414828	−	0.909900i	$-0.636158\pi$
$443$	−17.4622	−0.829656	−0.414828	+	0.909900i	$0.636158\pi$
$444$	0	0
$445$	6.64788	0.315140
$446$	0	0
$447$	3.02983	0.143306
$448$	0	0
$449$	21.5202	1.01560	0.507800	−	0.861475i	$-0.330459\pi$
$449$	21.5202	1.01560	0.507800	+	0.861475i	$0.330459\pi$
$450$	0	0
$451$	8.90650	0.419391
$452$	0	0
$453$	−16.7812	−0.788448
$454$	0	0
$455$	7.86942	0.368924
$456$	0	0
$457$	3.07741	0.143955	0.0719777	−	0.997406i	$-0.477069\pi$
$457$	3.07741	0.143955	0.0719777	+	0.997406i	$0.477069\pi$
$458$	0	0
$459$	−13.6926	−0.639118
$460$	0	0
$461$	−18.7593	−0.873709	−0.436854	−	0.899532i	$-0.643907\pi$
$461$	−18.7593	−0.873709	−0.436854	+	0.899532i	$0.643907\pi$
$462$	0	0
$463$	19.9192	0.925725	0.462863	−	0.886430i	$-0.346822\pi$
$463$	19.9192	0.925725	0.462863	+	0.886430i	$0.346822\pi$
$464$	0	0
$465$	5.59225	0.259334
$466$	0	0
$467$	25.6368	1.18633	0.593165	−	0.805081i	$-0.297878\pi$
$467$	25.6368	1.18633	0.593165	+	0.805081i	$0.297878\pi$
$468$	0	0
$469$	−40.6163	−1.87549
$470$	0	0
$471$	−3.07460	−0.141670
$472$	0	0
$473$	10.4595	0.480927
$474$	0	0
$475$	0.0314822	0.00144450
$476$	0	0
$477$	−9.19061	−0.420809
$478$	0	0
$479$	−40.8615	−1.86701	−0.933504	−	0.358566i	$-0.883266\pi$
$479$	−40.8615	−1.86701	−0.933504	+	0.358566i	$0.883266\pi$
$480$	0	0
$481$	1.09283	0.0498289
$482$	0	0
$483$	21.3060	0.969457
$484$	0	0
$485$	18.0164	0.818081
$486$	0	0
$487$	24.8936	1.12804	0.564018	−	0.825762i	$-0.309255\pi$
$487$	24.8936	1.12804	0.564018	+	0.825762i	$0.309255\pi$
$488$	0	0
$489$	18.7991	0.850126
$490$	0	0
$491$	−9.36013	−0.422417	−0.211208	−	0.977441i	$-0.567740\pi$
$491$	−9.36013	−0.422417	−0.211208	+	0.977441i	$0.567740\pi$
$492$	0	0
$493$	7.16040	0.322488
$494$	0	0
$495$	−3.05446	−0.137288
$496$	0	0
$497$	8.29100	0.371902
$498$	0	0
$499$	17.9292	0.802620	0.401310	−	0.915942i	$-0.368555\pi$
$499$	17.9292	0.802620	0.401310	+	0.915942i	$0.368555\pi$
$500$	0	0
$501$	−6.02260	−0.269070
$502$	0	0
$503$	−28.0431	−1.25038	−0.625189	−	0.780473i	$-0.714978\pi$
$503$	−28.0431	−1.25038	−0.625189	+	0.780473i	$0.714978\pi$
$504$	0	0
$505$	2.72735	0.121365
$506$	0	0
$507$	−7.87052	−0.349542
$508$	0	0
$509$	43.3229	1.92025	0.960127	−	0.279564i	$-0.0901898\pi$
$509$	43.3229	1.92025	0.960127	+	0.279564i	$0.0901898\pi$
$510$	0	0
$511$	18.7898	0.831211
$512$	0	0
$513$	0.0396627	0.00175115
$514$	0	0
$515$	15.7854	0.695586
$516$	0	0
$517$	0.474700	0.0208773
$518$	0	0
$519$	8.07700	0.354541
$520$	0	0
$521$	−23.7845	−1.04202	−0.521008	−	0.853552i	$-0.674444\pi$
$521$	−23.7845	−1.04202	−0.521008	+	0.853552i	$0.674444\pi$
$522$	0	0
$523$	−3.92215	−0.171504	−0.0857518	−	0.996317i	$-0.527329\pi$
$523$	−3.92215	−0.171504	−0.0857518	+	0.996317i	$0.527329\pi$
$524$	0	0
$525$	9.37494	0.409156
$526$	0	0
$527$	17.9786	0.783161
$528$	0	0
$529$	35.1969	1.53030
$530$	0	0
$531$	−27.0050	−1.17192
$532$	0	0
$533$	−15.3806	−0.666206
$534$	0	0
$535$	−0.117986	−0.00510098
$536$	0	0
$537$	1.36733	0.0590045
$538$	0	0
$539$	5.63706	0.242805
$540$	0	0
$541$	29.8150	1.28185	0.640924	−	0.767605i	$-0.278551\pi$
$541$	29.8150	1.28185	0.640924	+	0.767605i	$0.278551\pi$
$542$	0	0
$543$	11.6792	0.501201
$544$	0	0
$545$	−2.74695	−0.117666
$546$	0	0
$547$	−18.6500	−0.797417	−0.398709	−	0.917078i	$-0.630542\pi$
$547$	−18.6500	−0.797417	−0.398709	+	0.917078i	$0.630542\pi$
$548$	0	0
$549$	34.1016	1.45542
$550$	0	0
$551$	−0.0207411	−0.000883601 0
$552$	0	0
$553$	−25.8793	−1.10050
$554$	0	0
$555$	−0.637342	−0.0270537
$556$	0	0
$557$	−0.411845	−0.0174504	−0.00872522	−	0.999962i	$-0.502777\pi$
$557$	−0.411845	−0.0174504	−0.00872522	+	0.999962i	$0.502777\pi$
$558$	0	0
$559$	−18.0624	−0.763957
$560$	0	0
$561$	2.54380	0.107399
$562$	0	0
$563$	2.24708	0.0947032	0.0473516	−	0.998878i	$-0.484922\pi$
$563$	2.24708	0.0947032	0.0473516	+	0.998878i	$0.484922\pi$
$564$	0	0
$565$	−16.6636	−0.701044
$566$	0	0
$567$	−13.6001	−0.571152
$568$	0	0
$569$	−3.24446	−0.136015	−0.0680075	−	0.997685i	$-0.521664\pi$
$569$	−3.24446	−0.136015	−0.0680075	+	0.997685i	$0.521664\pi$
$570$	0	0
$571$	−4.05280	−0.169604	−0.0848021	−	0.996398i	$-0.527026\pi$
$571$	−4.05280	−0.169604	−0.0848021	+	0.996398i	$0.527026\pi$
$572$	0	0
$573$	7.98932	0.333759
$574$	0	0
$575$	25.6074	1.06790
$576$	0	0
$577$	−3.64158	−0.151601	−0.0758006	−	0.997123i	$-0.524151\pi$
$577$	−3.64158	−0.151601	−0.0758006	+	0.997123i	$0.524151\pi$
$578$	0	0
$579$	10.9078	0.453312
$580$	0	0
$581$	−17.1326	−0.710780
$582$	0	0
$583$	3.85714	0.159746
$584$	0	0
$585$	5.27471	0.218083
$586$	0	0
$587$	−14.0711	−0.580775	−0.290388	−	0.956909i	$-0.593784\pi$
$587$	−14.0711	−0.580775	−0.290388	+	0.956909i	$0.593784\pi$
$588$	0	0
$589$	−0.0520776	−0.00214582
$590$	0	0
$591$	3.64091	0.149767
$592$	0	0
$593$	29.0575	1.19325	0.596623	−	0.802521i	$-0.296509\pi$
$593$	29.0575	1.19325	0.596623	+	0.802521i	$0.296509\pi$
$594$	0	0
$595$	−14.7547	−0.604885
$596$	0	0
$597$	12.5892	0.515240
$598$	0	0
$599$	−6.58190	−0.268929	−0.134465	−	0.990918i	$-0.542931\pi$
$599$	−6.58190	−0.268929	−0.134465	+	0.990918i	$0.542931\pi$
$600$	0	0
$601$	−11.8349	−0.482756	−0.241378	−	0.970431i	$-0.577599\pi$
$601$	−11.8349	−0.482756	−0.241378	+	0.970431i	$0.577599\pi$
$602$	0	0
$603$	−27.2243	−1.10866
$604$	0	0
$605$	1.28190	0.0521167
$606$	0	0
$607$	20.9446	0.850113	0.425057	−	0.905167i	$-0.360254\pi$
$607$	20.9446	0.850113	0.425057	+	0.905167i	$0.360254\pi$
$608$	0	0
$609$	−6.17640	−0.250281
$610$	0	0
$611$	−0.819755	−0.0331637
$612$	0	0
$613$	3.56277	0.143899	0.0719494	−	0.997408i	$-0.477078\pi$
$613$	3.56277	0.143899	0.0719494	+	0.997408i	$0.477078\pi$
$614$	0	0
$615$	8.96997	0.361704
$616$	0	0
$617$	23.1728	0.932901	0.466450	−	0.884547i	$-0.345533\pi$
$617$	23.1728	0.932901	0.466450	+	0.884547i	$0.345533\pi$
$618$	0	0
$619$	18.6751	0.750615	0.375308	−	0.926900i	$-0.377537\pi$
$619$	18.6751	0.750615	0.375308	+	0.926900i	$0.377537\pi$
$620$	0	0
$621$	32.2614	1.29461
$622$	0	0
$623$	−18.4354	−0.738598
$624$	0	0
$625$	3.05128	0.122051
$626$	0	0
$627$	−0.00736847	−0.000294268 0
$628$	0	0
$629$	−2.04900	−0.0816991
$630$	0	0
$631$	−0.180505	−0.00718580	−0.00359290	−	0.999994i	$-0.501144\pi$
$631$	−0.180505	−0.00718580	−0.00359290	+	0.999994i	$0.501144\pi$
$632$	0	0
$633$	5.45170	0.216686
$634$	0	0
$635$	11.5650	0.458944
$636$	0	0
$637$	−9.73459	−0.385698
$638$	0	0
$639$	5.55729	0.219843
$640$	0	0
$641$	20.1691	0.796630	0.398315	−	0.917249i	$-0.369595\pi$
$641$	20.1691	0.796630	0.398315	+	0.917249i	$0.369595\pi$
$642$	0	0
$643$	26.1824	1.03253	0.516267	−	0.856428i	$-0.327321\pi$
$643$	26.1824	1.03253	0.516267	+	0.856428i	$0.327321\pi$
$644$	0	0
$645$	10.5340	0.414776
$646$	0	0
$647$	30.9941	1.21850	0.609252	−	0.792977i	$-0.291470\pi$
$647$	30.9941	1.21850	0.609252	+	0.792977i	$0.291470\pi$
$648$	0	0
$649$	11.3335	0.444880
$650$	0	0
$651$	−15.5080	−0.607805
$652$	0	0
$653$	−21.4034	−0.837580	−0.418790	−	0.908083i	$-0.637546\pi$
$653$	−21.4034	−0.837580	−0.418790	+	0.908083i	$0.637546\pi$
$654$	0	0
$655$	−2.85248	−0.111455
$656$	0	0
$657$	12.5944	0.491355
$658$	0	0
$659$	49.8468	1.94176	0.970879	−	0.239572i	$-0.0770071\pi$
$659$	49.8468	1.94176	0.970879	+	0.239572i	$0.0770071\pi$
$660$	0	0
$661$	−5.68838	−0.221252	−0.110626	−	0.993862i	$-0.535286\pi$
$661$	−5.68838	−0.221252	−0.110626	+	0.993862i	$0.535286\pi$
$662$	0	0
$663$	−4.39286	−0.170605
$664$	0	0
$665$	0.0427391	0.00165735
$666$	0	0
$667$	−16.8707	−0.653237
$668$	0	0
$669$	−6.36588	−0.246119
$670$	0	0
$671$	−14.3119	−0.552503
$672$	0	0
$673$	−4.64103	−0.178899	−0.0894493	−	0.995991i	$-0.528511\pi$
$673$	−4.64103	−0.178899	−0.0894493	+	0.995991i	$0.528511\pi$
$674$	0	0
$675$	14.1955	0.546384
$676$	0	0
$677$	−22.4035	−0.861036	−0.430518	−	0.902582i	$-0.641669\pi$
$677$	−22.4035	−0.861036	−0.430518	+	0.902582i	$0.641669\pi$
$678$	0	0
$679$	−49.9615	−1.91735
$680$	0	0
$681$	−4.46206	−0.170986
$682$	0	0
$683$	21.3668	0.817578	0.408789	−	0.912629i	$-0.365951\pi$
$683$	21.3668	0.817578	0.408789	+	0.912629i	$0.365951\pi$
$684$	0	0
$685$	1.28190	0.0489789
$686$	0	0
$687$	14.5950	0.556835
$688$	0	0
$689$	−6.66085	−0.253758
$690$	0	0
$691$	17.0973	0.650411	0.325206	−	0.945643i	$-0.394567\pi$
$691$	17.0973	0.650411	0.325206	+	0.945643i	$0.394567\pi$
$692$	0	0
$693$	8.47037	0.321763
$694$	0	0
$695$	1.98854	0.0754294
$696$	0	0
$697$	28.8377	1.09231
$698$	0	0
$699$	−0.329682	−0.0124697
$700$	0	0
$701$	41.3790	1.56286	0.781431	−	0.623991i	$-0.214490\pi$
$701$	41.3790	1.56286	0.781431	+	0.623991i	$0.214490\pi$
$702$	0	0
$703$	0.00593522	0.000223851 0
$704$	0	0
$705$	0.478082	0.0180056
$706$	0	0
$707$	−7.56326	−0.284446
$708$	0	0
$709$	−30.4516	−1.14363	−0.571816	−	0.820382i	$-0.693761\pi$
$709$	−30.4516	−1.14363	−0.571816	+	0.820382i	$0.693761\pi$
$710$	0	0
$711$	−17.3464	−0.650539
$712$	0	0
$713$	−42.3597	−1.58638
$714$	0	0
$715$	−2.21370	−0.0827878
$716$	0	0
$717$	−3.56452	−0.133120
$718$	0	0
$719$	44.2279	1.64942	0.824711	−	0.565554i	$-0.191338\pi$
$719$	44.2279	1.64942	0.824711	+	0.565554i	$0.191338\pi$
$720$	0	0
$721$	−43.7747	−1.63025
$722$	0	0
$723$	−7.31796	−0.272158
$724$	0	0
$725$	−7.42335	−0.275696
$726$	0	0
$727$	−28.2729	−1.04858	−0.524292	−	0.851538i	$-0.675670\pi$
$727$	−28.2729	−1.04858	−0.524292	+	0.851538i	$0.675670\pi$
$728$	0	0
$729$	0.851558	0.0315392
$730$	0	0
$731$	33.8660	1.25258
$732$	0	0
$733$	−14.0834	−0.520181	−0.260090	−	0.965584i	$-0.583752\pi$
$733$	−14.0834	−0.520181	−0.260090	+	0.965584i	$0.583752\pi$
$734$	0	0
$735$	5.67723	0.209408
$736$	0	0
$737$	11.4256	0.420866
$738$	0	0
$739$	36.7252	1.35096	0.675479	−	0.737380i	$-0.263937\pi$
$739$	36.7252	1.35096	0.675479	+	0.737380i	$0.263937\pi$
$740$	0	0
$741$	0.0127246	0.000467448 0
$742$	0	0
$743$	−7.77611	−0.285278	−0.142639	−	0.989775i	$-0.545559\pi$
$743$	−7.77611	−0.285278	−0.142639	+	0.989775i	$0.545559\pi$
$744$	0	0
$745$	4.94361	0.181120
$746$	0	0
$747$	−11.4836	−0.420165
$748$	0	0
$749$	0.327189	0.0119552
$750$	0	0
$751$	−37.5777	−1.37123	−0.685615	−	0.727964i	$-0.740467\pi$
$751$	−37.5777	−1.37123	−0.685615	+	0.727964i	$0.740467\pi$
$752$	0	0
$753$	−15.3524	−0.559474
$754$	0	0
$755$	−27.3809	−0.996493
$756$	0	0
$757$	20.2412	0.735678	0.367839	−	0.929890i	$-0.380098\pi$
$757$	20.2412	0.735678	0.367839	+	0.929890i	$0.380098\pi$
$758$	0	0
$759$	−5.99348	−0.217549
$760$	0	0
$761$	−28.1714	−1.02121	−0.510607	−	0.859814i	$-0.670579\pi$
$761$	−28.1714	−1.02121	−0.510607	+	0.859814i	$0.670579\pi$
$762$	0	0
$763$	7.61761	0.275776
$764$	0	0
$765$	−9.88980	−0.357566
$766$	0	0
$767$	−19.5718	−0.706695
$768$	0	0
$769$	−41.0260	−1.47944	−0.739718	−	0.672917i	$-0.765041\pi$
$769$	−41.0260	−1.47944	−0.739718	+	0.672917i	$0.765041\pi$
$770$	0	0
$771$	16.8619	0.607265
$772$	0	0
$773$	21.6641	0.779203	0.389601	−	0.920984i	$-0.372613\pi$
$773$	21.6641	0.779203	0.389601	+	0.920984i	$0.372613\pi$
$774$	0	0
$775$	−18.6389	−0.669527
$776$	0	0
$777$	1.76742	0.0634060
$778$	0	0
$779$	−0.0835325	−0.00299286
$780$	0	0
$781$	−2.33230	−0.0834562
$782$	0	0
$783$	−9.35227	−0.334223
$784$	0	0
$785$	−5.01665	−0.179052
$786$	0	0
$787$	−24.9148	−0.888115	−0.444058	−	0.895998i	$-0.646461\pi$
$787$	−24.9148	−0.888115	−0.444058	+	0.895998i	$0.646461\pi$
$788$	0	0
$789$	9.44960	0.336415
$790$	0	0
$791$	46.2102	1.64305
$792$	0	0
$793$	24.7150	0.877656
$794$	0	0
$795$	3.88462	0.137773
$796$	0	0
$797$	−44.8311	−1.58800	−0.794000	−	0.607918i	$-0.792005\pi$
$797$	−44.8311	−1.58800	−0.794000	+	0.607918i	$0.792005\pi$
$798$	0	0
$799$	1.53700	0.0543750
$800$	0	0
$801$	−12.3569	−0.436608
$802$	0	0
$803$	−5.28565	−0.186527
$804$	0	0
$805$	34.7638	1.22526
$806$	0	0
$807$	−12.8404	−0.452003
$808$	0	0
$809$	12.4682	0.438357	0.219179	−	0.975685i	$-0.429662\pi$
$809$	12.4682	0.438357	0.219179	+	0.975685i	$0.429662\pi$
$810$	0	0
$811$	12.1238	0.425723	0.212861	−	0.977082i	$-0.431722\pi$
$811$	12.1238	0.425723	0.212861	+	0.977082i	$0.431722\pi$
$812$	0	0
$813$	1.24168	0.0435475
$814$	0	0
$815$	30.6735	1.07445
$816$	0	0
$817$	−0.0980976	−0.00343200
$818$	0	0
$819$	−14.6274	−0.511123
$820$	0	0
$821$	22.5794	0.788025	0.394013	−	0.919105i	$-0.371087\pi$
$821$	22.5794	0.788025	0.394013	+	0.919105i	$0.371087\pi$
$822$	0	0
$823$	17.3132	0.603501	0.301751	−	0.953387i	$-0.402429\pi$
$823$	17.3132	0.603501	0.301751	+	0.953387i	$0.402429\pi$
$824$	0	0
$825$	−2.63721	−0.0918160
$826$	0	0
$827$	−14.5039	−0.504349	−0.252175	−	0.967682i	$-0.581146\pi$
$827$	−14.5039	−0.504349	−0.252175	+	0.967682i	$0.581146\pi$
$828$	0	0
$829$	−52.9807	−1.84010	−0.920048	−	0.391806i	$-0.871850\pi$
$829$	−52.9807	−1.84010	−0.920048	+	0.391806i	$0.871850\pi$
$830$	0	0
$831$	−9.96496	−0.345681
$832$	0	0
$833$	18.2518	0.632388
$834$	0	0
$835$	−9.82674	−0.340068
$836$	0	0
$837$	−23.4821	−0.811659
$838$	0	0
$839$	45.7698	1.58015	0.790074	−	0.613011i	$-0.210042\pi$
$839$	45.7698	1.58015	0.790074	+	0.613011i	$0.210042\pi$
$840$	0	0
$841$	−24.1093	−0.831357
$842$	0	0
$843$	−18.7542	−0.645929
$844$	0	0
$845$	−12.8419	−0.441774
$846$	0	0
$847$	−3.55486	−0.122147
$848$	0	0
$849$	8.69158	0.298294
$850$	0	0
$851$	4.82768	0.165491
$852$	0	0
$853$	25.0087	0.856282	0.428141	−	0.903712i	$-0.359169\pi$
$853$	25.0087	0.856282	0.428141	+	0.903712i	$0.359169\pi$
$854$	0	0
$855$	0.0286472	0.000979713 0
$856$	0	0
$857$	25.6762	0.877082	0.438541	−	0.898711i	$-0.355495\pi$
$857$	25.6762	0.877082	0.438541	+	0.898711i	$0.355495\pi$
$858$	0	0
$859$	22.1460	0.755612	0.377806	−	0.925885i	$-0.376679\pi$
$859$	22.1460	0.755612	0.377806	+	0.925885i	$0.376679\pi$
$860$	0	0
$861$	−24.8748	−0.847730
$862$	0	0
$863$	−0.668143	−0.0227439	−0.0113719	−	0.999935i	$-0.503620\pi$
$863$	−0.668143	−0.0227439	−0.0113719	+	0.999935i	$0.503620\pi$
$864$	0	0
$865$	13.1788	0.448092
$866$	0	0
$867$	−5.11967	−0.173873
$868$	0	0
$869$	7.27996	0.246956
$870$	0	0
$871$	−19.7307	−0.668550
$872$	0	0
$873$	−33.4882	−1.13340
$874$	0	0
$875$	38.0815	1.28739
$876$	0	0
$877$	−31.2176	−1.05414	−0.527072	−	0.849821i	$-0.676710\pi$
$877$	−31.2176	−1.05414	−0.527072	+	0.849821i	$0.676710\pi$
$878$	0	0
$879$	6.17427	0.208253
$880$	0	0
$881$	−22.4350	−0.755855	−0.377928	−	0.925835i	$-0.623363\pi$
$881$	−22.4350	−0.755855	−0.377928	+	0.925835i	$0.623363\pi$
$882$	0	0
$883$	−19.6092	−0.659903	−0.329952	−	0.943998i	$-0.607032\pi$
$883$	−19.6092	−0.659903	−0.329952	+	0.943998i	$0.607032\pi$
$884$	0	0
$885$	11.4143	0.383687
$886$	0	0
$887$	−17.9881	−0.603981	−0.301991	−	0.953311i	$-0.597651\pi$
$887$	−17.9881	−0.603981	−0.301991	+	0.953311i	$0.597651\pi$
$888$	0	0
$889$	−32.0712	−1.07563
$890$	0	0
$891$	3.82578	0.128168
$892$	0	0
$893$	−0.00445212	−0.000148985 0
$894$	0	0
$895$	2.23099	0.0745738
$896$	0	0
$897$	10.3501	0.345579
$898$	0	0
$899$	12.2797	0.409550
$900$	0	0
$901$	12.4887	0.416060
$902$	0	0
$903$	−29.2121	−0.972116
$904$	0	0
$905$	19.0563	0.633451
$906$	0	0
$907$	15.5059	0.514864	0.257432	−	0.966296i	$-0.417124\pi$
$907$	15.5059	0.514864	0.257432	+	0.966296i	$0.417124\pi$
$908$	0	0
$909$	−5.06950	−0.168145
$910$	0	0
$911$	−4.73245	−0.156793	−0.0783965	−	0.996922i	$-0.524980\pi$
$911$	−4.73245	−0.156793	−0.0783965	+	0.996922i	$0.524980\pi$
$912$	0	0
$913$	4.81948	0.159502
$914$	0	0
$915$	−14.4138	−0.476507
$916$	0	0
$917$	7.91025	0.261220
$918$	0	0
$919$	−49.3820	−1.62896	−0.814481	−	0.580190i	$-0.802978\pi$
$919$	−49.3820	−1.62896	−0.814481	+	0.580190i	$0.802978\pi$
$920$	0	0
$921$	13.7617	0.453462
$922$	0	0
$923$	4.02762	0.132571
$924$	0	0
$925$	2.12425	0.0698448
$926$	0	0
$927$	−29.3413	−0.963694
$928$	0	0
$929$	6.92738	0.227280	0.113640	−	0.993522i	$-0.463749\pi$
$929$	6.92738	0.227280	0.113640	+	0.993522i	$0.463749\pi$
$930$	0	0
$931$	−0.0528690	−0.00173271
$932$	0	0
$933$	−6.89156	−0.225620
$934$	0	0
$935$	4.15057	0.135738
$936$	0	0
$937$	−4.05054	−0.132325	−0.0661627	−	0.997809i	$-0.521076\pi$
$937$	−4.05054	−0.132325	−0.0661627	+	0.997809i	$0.521076\pi$
$938$	0	0
$939$	4.89100	0.159612
$940$	0	0
$941$	24.3281	0.793074	0.396537	−	0.918019i	$-0.370212\pi$
$941$	24.3281	0.793074	0.396537	+	0.918019i	$0.370212\pi$
$942$	0	0
$943$	−67.9449	−2.21259
$944$	0	0
$945$	19.2713	0.626896
$946$	0	0
$947$	35.8640	1.16542	0.582712	−	0.812679i	$-0.301992\pi$
$947$	35.8640	1.16542	0.582712	+	0.812679i	$0.301992\pi$
$948$	0	0
$949$	9.12775	0.296299
$950$	0	0
$951$	16.7751	0.543971
$952$	0	0
$953$	16.8227	0.544939	0.272470	−	0.962164i	$-0.412160\pi$
$953$	16.8227	0.544939	0.272470	+	0.962164i	$0.412160\pi$
$954$	0	0
$955$	13.0357	0.421826
$956$	0	0
$957$	1.73745	0.0561638
$958$	0	0
$959$	−3.55486	−0.114793
$960$	0	0
$961$	−0.167710	−0.00540999
$962$	0	0
$963$	0.219308	0.00706711
$964$	0	0
$965$	17.7976	0.572926
$966$	0	0
$967$	−24.1211	−0.775683	−0.387842	−	0.921726i	$-0.626779\pi$
$967$	−24.1211	−0.775683	−0.387842	+	0.921726i	$0.626779\pi$
$968$	0	0
$969$	−0.0238578	−0.000766424 0
$970$	0	0
$971$	49.1252	1.57650	0.788252	−	0.615353i	$-0.210987\pi$
$971$	49.1252	1.57650	0.788252	+	0.615353i	$0.210987\pi$
$972$	0	0
$973$	−5.51444	−0.176785
$974$	0	0
$975$	4.55418	0.145851
$976$	0	0
$977$	4.03424	0.129067	0.0645334	−	0.997916i	$-0.479444\pi$
$977$	4.03424	0.129067	0.0645334	+	0.997916i	$0.479444\pi$
$978$	0	0
$979$	5.18596	0.165744
$980$	0	0
$981$	5.10593	0.163020
$982$	0	0
$983$	18.8970	0.602722	0.301361	−	0.953510i	$-0.402559\pi$
$983$	18.8970	0.602722	0.301361	+	0.953510i	$0.402559\pi$
$984$	0	0
$985$	5.94067	0.189285
$986$	0	0
$987$	−1.32578	−0.0422000
$988$	0	0
$989$	−79.7921	−2.53724
$990$	0	0
$991$	−29.5607	−0.939027	−0.469514	−	0.882925i	$-0.655571\pi$
$991$	−29.5607	−0.939027	−0.469514	+	0.882925i	$0.655571\pi$
$992$	0	0
$993$	8.21608	0.260729
$994$	0	0
$995$	20.5410	0.651194
$996$	0	0
$997$	−12.7298	−0.403156	−0.201578	−	0.979472i	$-0.564607\pi$
$997$	−12.7298	−0.403156	−0.201578	+	0.979472i	$0.564607\pi$
$998$	0	0
$999$	2.67622	0.0846720

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6028.2.a.f.1.15	✓	29

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6028.2.a.f.1.15	✓	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)

\(f(q)\)	\(=\)	\(q+0.785650 q^{3} +1.28190 q^{5} -3.55486 q^{7} -2.38275 q^{9} +O(q^{10})\)	\(q+0.785650 q^{3} +1.28190 q^{5} -3.55486 q^{7} -2.38275 q^{9} +1.00000 q^{11} -1.72689 q^{13} +1.00713 q^{15} +3.23783 q^{17} -0.00937882 q^{19} -2.79288 q^{21} -7.62869 q^{23} -3.35673 q^{25} -4.22896 q^{27} +2.21148 q^{29} +5.55268 q^{31} +0.785650 q^{33} -4.55698 q^{35} -0.632832 q^{37} -1.35673 q^{39} +8.90650 q^{41} +10.4595 q^{43} -3.05446 q^{45} +0.474700 q^{47} +5.63706 q^{49} +2.54380 q^{51} +3.85714 q^{53} +1.28190 q^{55} -0.00736847 q^{57} +11.3335 q^{59} -14.3119 q^{61} +8.47037 q^{63} -2.21370 q^{65} +11.4256 q^{67} -5.99348 q^{69} -2.33230 q^{71} -5.28565 q^{73} -2.63721 q^{75} -3.55486 q^{77} +7.27996 q^{79} +3.82578 q^{81} +4.81948 q^{83} +4.15057 q^{85} +1.73745 q^{87} +5.18596 q^{89} +6.13886 q^{91} +4.36247 q^{93} -0.0120227 q^{95} +14.0544 q^{97} -2.38275 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 29 q + 14 q^{3} + 9 q^{5} + 14 q^{7} + 43 q^{9}+O(q^{10}) \) 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

L-functions

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Database

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