LMFDB - Embedded newform 6028.2.a.d.1.19

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:	$1$
Dimension:	$27$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
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)

$f(q)$	$=$	$q+1.56304 q^{3} +2.78499 q^{5} -0.687006 q^{7} -0.556912 q^{9} +O(q^{10})$	$q+1.56304 q^{3} +2.78499 q^{5} -0.687006 q^{7} -0.556912 q^{9} -1.00000 q^{11} -2.44156 q^{13} +4.35304 q^{15} -2.53714 q^{17} -6.30320 q^{19} -1.07382 q^{21} +4.77356 q^{23} +2.75617 q^{25} -5.55959 q^{27} -9.75669 q^{29} +2.20324 q^{31} -1.56304 q^{33} -1.91330 q^{35} -1.86495 q^{37} -3.81625 q^{39} -6.35689 q^{41} +2.62902 q^{43} -1.55099 q^{45} +9.14926 q^{47} -6.52802 q^{49} -3.96564 q^{51} -6.15388 q^{53} -2.78499 q^{55} -9.85213 q^{57} -0.450577 q^{59} -4.86443 q^{61} +0.382602 q^{63} -6.79971 q^{65} +4.89098 q^{67} +7.46125 q^{69} -4.04697 q^{71} +12.0704 q^{73} +4.30799 q^{75} +0.687006 q^{77} -11.4400 q^{79} -7.01911 q^{81} -12.9546 q^{83} -7.06590 q^{85} -15.2501 q^{87} +17.2988 q^{89} +1.67736 q^{91} +3.44374 q^{93} -17.5543 q^{95} +8.69604 q^{97} +0.556912 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 27 q - 6 q^{3} + q^{5} - 14 q^{7} + 33 q^{9}+O(q^{10}) $ 27 * q - 6 * q^3 + q^5 - 14 * q^7 + 33 * q^9	$ 27 q - 6 q^{3} + q^{5} - 14 q^{7} + 33 q^{9} - 27 q^{11} - 6 q^{15} - 21 q^{17} - 3 q^{19} - 4 q^{21} - 44 q^{23} + 38 q^{25} - 18 q^{27} + q^{29} - 8 q^{31} + 6 q^{33} - 33 q^{35} + 11 q^{37} - 13 q^{39} - 19 q^{41} - 11 q^{43} + 17 q^{45} - 37 q^{47} + 41 q^{49} - 49 q^{51} - 12 q^{53} - q^{55} - 50 q^{57} - 14 q^{59} + 12 q^{61} - 53 q^{63} - 55 q^{65} - 5 q^{67} + 14 q^{69} - 67 q^{71} - 27 q^{73} - 70 q^{75} + 14 q^{77} - 31 q^{79} - 5 q^{81} - 55 q^{83} - 3 q^{85} - 31 q^{87} + 11 q^{89} - 11 q^{91} - 24 q^{93} - 47 q^{95} - q^{97} - 33 q^{99}+O(q^{100}) $ 27 * q - 6 * q^3 + q^5 - 14 * q^7 + 33 * q^9 - 27 * q^11 - 6 * q^15 - 21 * q^17 - 3 * q^19 - 4 * q^21 - 44 * q^23 + 38 * q^25 - 18 * q^27 + q^29 - 8 * q^31 + 6 * q^33 - 33 * q^35 + 11 * q^37 - 13 * q^39 - 19 * q^41 - 11 * q^43 + 17 * q^45 - 37 * q^47 + 41 * q^49 - 49 * q^51 - 12 * q^53 - q^55 - 50 * q^57 - 14 * q^59 + 12 * q^61 - 53 * q^63 - 55 * q^65 - 5 * q^67 + 14 * q^69 - 67 * q^71 - 27 * q^73 - 70 * q^75 + 14 * q^77 - 31 * q^79 - 5 * q^81 - 55 * q^83 - 3 * q^85 - 31 * q^87 + 11 * q^89 - 11 * q^91 - 24 * q^93 - 47 * q^95 - q^97 - 33 * 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
$3$	1.56304	0.902420	0.451210	−	0.892418i	$-0.350992\pi$
$3$	1.56304	0.902420	0.451210	+	0.892418i	$0.350992\pi$
$4$	0	0
$5$	2.78499	1.24549	0.622743	−	0.782427i	$-0.286018\pi$
$5$	2.78499	1.24549	0.622743	+	0.782427i	$0.286018\pi$
$6$	0	0
$7$	−0.687006	−0.259664	−0.129832	−	0.991536i	$-0.541444\pi$
$7$	−0.687006	−0.259664	−0.129832	+	0.991536i	$0.541444\pi$
$8$	0	0
$9$	−0.556912	−0.185637
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−2.44156	−0.677166	−0.338583	−	0.940936i	$-0.609948\pi$
$13$	−2.44156	−0.677166	−0.338583	+	0.940936i	$0.609948\pi$
$14$	0	0
$15$	4.35304	1.12395
$16$	0	0
$17$	−2.53714	−0.615346	−0.307673	−	0.951492i	$-0.599550\pi$
$17$	−2.53714	−0.615346	−0.307673	+	0.951492i	$0.599550\pi$
$18$	0	0
$19$	−6.30320	−1.44605	−0.723026	−	0.690821i	$-0.757249\pi$
$19$	−6.30320	−1.44605	−0.723026	+	0.690821i	$0.757249\pi$
$20$	0	0
$21$	−1.07382	−0.234326
$22$	0	0
$23$	4.77356	0.995356	0.497678	−	0.867362i	$-0.334186\pi$
$23$	4.77356	0.995356	0.497678	+	0.867362i	$0.334186\pi$
$24$	0	0
$25$	2.75617	0.551233
$26$	0	0
$27$	−5.55959	−1.06994
$28$	0	0
$29$	−9.75669	−1.81177	−0.905886	−	0.423522i	$-0.860793\pi$
$29$	−9.75669	−1.81177	−0.905886	+	0.423522i	$0.860793\pi$
$30$	0	0
$31$	2.20324	0.395713	0.197856	−	0.980231i	$-0.436602\pi$
$31$	2.20324	0.395713	0.197856	+	0.980231i	$0.436602\pi$
$32$	0	0
$33$	−1.56304	−0.272090
$34$	0	0
$35$	−1.91330	−0.323407
$36$	0	0
$37$	−1.86495	−0.306595	−0.153298	−	0.988180i	$-0.548989\pi$
$37$	−1.86495	−0.306595	−0.153298	+	0.988180i	$0.548989\pi$
$38$	0	0
$39$	−3.81625	−0.611089
$40$	0	0
$41$	−6.35689	−0.992779	−0.496389	−	0.868100i	$-0.665341\pi$
$41$	−6.35689	−0.992779	−0.496389	+	0.868100i	$0.665341\pi$
$42$	0	0
$43$	2.62902	0.400921	0.200461	−	0.979702i	$-0.435756\pi$
$43$	2.62902	0.400921	0.200461	+	0.979702i	$0.435756\pi$
$44$	0	0
$45$	−1.55099	−0.231208
$46$	0	0
$47$	9.14926	1.33456	0.667278	−	0.744809i	$-0.267459\pi$
$47$	9.14926	1.33456	0.667278	+	0.744809i	$0.267459\pi$
$48$	0	0
$49$	−6.52802	−0.932575
$50$	0	0
$51$	−3.96564	−0.555301
$52$	0	0
$53$	−6.15388	−0.845300	−0.422650	−	0.906293i	$-0.638900\pi$
$53$	−6.15388	−0.845300	−0.422650	+	0.906293i	$0.638900\pi$
$54$	0	0
$55$	−2.78499	−0.375528
$56$	0	0
$57$	−9.85213	−1.30495
$58$	0	0
$59$	−0.450577	−0.0586601	−0.0293301	−	0.999570i	$-0.509337\pi$
$59$	−0.450577	−0.0586601	−0.0293301	+	0.999570i	$0.509337\pi$
$60$	0	0
$61$	−4.86443	−0.622826	−0.311413	−	0.950275i	$-0.600802\pi$
$61$	−4.86443	−0.622826	−0.311413	+	0.950275i	$0.600802\pi$
$62$	0	0
$63$	0.382602	0.0482033
$64$	0	0
$65$	−6.79971	−0.843401
$66$	0	0
$67$	4.89098	0.597528	0.298764	−	0.954327i	$-0.403426\pi$
$67$	4.89098	0.597528	0.298764	+	0.954327i	$0.403426\pi$
$68$	0	0
$69$	7.46125	0.898229
$70$	0	0
$71$	−4.04697	−0.480287	−0.240143	−	0.970737i	$-0.577194\pi$
$71$	−4.04697	−0.480287	−0.240143	+	0.970737i	$0.577194\pi$
$72$	0	0
$73$	12.0704	1.41273	0.706366	−	0.707847i	$-0.250333\pi$
$73$	12.0704	1.41273	0.706366	+	0.707847i	$0.250333\pi$
$74$	0	0
$75$	4.30799	0.497444
$76$	0	0
$77$	0.687006	0.0782916
$78$	0	0
$79$	−11.4400	−1.28710	−0.643548	−	0.765406i	$-0.722538\pi$
$79$	−11.4400	−1.28710	−0.643548	+	0.765406i	$0.722538\pi$
$80$	0	0
$81$	−7.01911	−0.779902
$82$	0	0
$83$	−12.9546	−1.42195	−0.710973	−	0.703219i	$-0.751745\pi$
$83$	−12.9546	−1.42195	−0.710973	+	0.703219i	$0.751745\pi$
$84$	0	0
$85$	−7.06590	−0.766404
$86$	0	0
$87$	−15.2501	−1.63498
$88$	0	0
$89$	17.2988	1.83367	0.916833	−	0.399271i	$-0.130737\pi$
$89$	17.2988	1.83367	0.916833	+	0.399271i	$0.130737\pi$
$90$	0	0
$91$	1.67736	0.175836
$92$	0	0
$93$	3.44374	0.357099
$94$	0	0
$95$	−17.5543	−1.80104
$96$	0	0
$97$	8.69604	0.882949	0.441474	−	0.897274i	$-0.354456\pi$
$97$	8.69604	0.882949	0.441474	+	0.897274i	$0.354456\pi$
$98$	0	0
$99$	0.556912	0.0559717
$100$	0	0
$101$	12.0796	1.20196	0.600982	−	0.799262i	$-0.294776\pi$
$101$	12.0796	1.20196	0.600982	+	0.799262i	$0.294776\pi$
$102$	0	0
$103$	−13.0336	−1.28423	−0.642117	−	0.766607i	$-0.721944\pi$
$103$	−13.0336	−1.28423	−0.642117	+	0.766607i	$0.721944\pi$
$104$	0	0
$105$	−2.99057	−0.291850
$106$	0	0
$107$	9.74907	0.942479	0.471239	−	0.882005i	$-0.343807\pi$
$107$	9.74907	0.942479	0.471239	+	0.882005i	$0.343807\pi$
$108$	0	0
$109$	−19.3603	−1.85438	−0.927188	−	0.374596i	$-0.877781\pi$
$109$	−19.3603	−1.85438	−0.927188	+	0.374596i	$0.877781\pi$
$110$	0	0
$111$	−2.91498	−0.276678
$112$	0	0
$113$	−12.9830	−1.22134	−0.610670	−	0.791885i	$-0.709100\pi$
$113$	−12.9830	−1.22134	−0.610670	+	0.791885i	$0.709100\pi$
$114$	0	0
$115$	13.2943	1.23970
$116$	0	0
$117$	1.35973	0.125707
$118$	0	0
$119$	1.74303	0.159783
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−9.93605	−0.895904
$124$	0	0
$125$	−6.24905	−0.558932
$126$	0	0
$127$	−13.9971	−1.24204	−0.621021	−	0.783794i	$-0.713282\pi$
$127$	−13.9971	−1.24204	−0.621021	+	0.783794i	$0.713282\pi$
$128$	0	0
$129$	4.10925	0.361799
$130$	0	0
$131$	−3.42648	−0.299373	−0.149686	−	0.988734i	$-0.547826\pi$
$131$	−3.42648	−0.299373	−0.149686	+	0.988734i	$0.547826\pi$
$132$	0	0
$133$	4.33033	0.375487
$134$	0	0
$135$	−15.4834	−1.33260
$136$	0	0
$137$	1.00000	0.0854358
$137$	1.00000	0.0854358
$138$	0	0
$139$	20.2451	1.71717	0.858585	−	0.512671i	$-0.171344\pi$
$139$	20.2451	1.71717	0.858585	+	0.512671i	$0.171344\pi$
$140$	0	0
$141$	14.3006	1.20433
$142$	0	0
$143$	2.44156	0.204173
$144$	0	0
$145$	−27.1723	−2.25653
$146$	0	0
$147$	−10.2035	−0.841574
$148$	0	0
$149$	1.12036	0.0917834	0.0458917	−	0.998946i	$-0.485387\pi$
$149$	1.12036	0.0917834	0.0458917	+	0.998946i	$0.485387\pi$
$150$	0	0
$151$	−14.6759	−1.19431	−0.597153	−	0.802127i	$-0.703702\pi$
$151$	−14.6759	−1.19431	−0.597153	+	0.802127i	$0.703702\pi$
$152$	0	0
$153$	1.41296	0.114231
$154$	0	0
$155$	6.13599	0.492854
$156$	0	0
$157$	16.8530	1.34502	0.672510	−	0.740088i	$-0.265216\pi$
$157$	16.8530	1.34502	0.672510	+	0.740088i	$0.265216\pi$
$158$	0	0
$159$	−9.61874	−0.762816
$160$	0	0
$161$	−3.27946	−0.258458
$162$	0	0
$163$	−2.61301	−0.204667	−0.102333	−	0.994750i	$-0.532631\pi$
$163$	−2.61301	−0.204667	−0.102333	+	0.994750i	$0.532631\pi$
$164$	0	0
$165$	−4.35304	−0.338884
$166$	0	0
$167$	14.6282	1.13196	0.565980	−	0.824419i	$-0.308498\pi$
$167$	14.6282	1.13196	0.565980	+	0.824419i	$0.308498\pi$
$168$	0	0
$169$	−7.03879	−0.541446
$170$	0	0
$171$	3.51032	0.268441
$172$	0	0
$173$	21.9663	1.67007	0.835034	−	0.550199i	$-0.185448\pi$
$173$	21.9663	1.67007	0.835034	+	0.550199i	$0.185448\pi$
$174$	0	0
$175$	−1.89350	−0.143135
$176$	0	0
$177$	−0.704269	−0.0529361
$178$	0	0
$179$	−22.4947	−1.68133	−0.840666	−	0.541553i	$-0.817836\pi$
$179$	−22.4947	−1.68133	−0.840666	+	0.541553i	$0.817836\pi$
$180$	0	0
$181$	14.6351	1.08782	0.543910	−	0.839144i	$-0.316943\pi$
$181$	14.6351	1.08782	0.543910	+	0.839144i	$0.316943\pi$
$182$	0	0
$183$	−7.60329	−0.562051
$184$	0	0
$185$	−5.19386	−0.381860
$186$	0	0
$187$	2.53714	0.185534
$188$	0	0
$189$	3.81947	0.277826
$190$	0	0
$191$	−16.1420	−1.16800	−0.583999	−	0.811755i	$-0.698513\pi$
$191$	−16.1420	−1.16800	−0.583999	+	0.811755i	$0.698513\pi$
$192$	0	0
$193$	−8.70023	−0.626256	−0.313128	−	0.949711i	$-0.601377\pi$
$193$	−8.70023	−0.626256	−0.313128	+	0.949711i	$0.601377\pi$
$194$	0	0
$195$	−10.6282	−0.761102
$196$	0	0
$197$	11.2196	0.799363	0.399681	−	0.916654i	$-0.369121\pi$
$197$	11.2196	0.799363	0.399681	+	0.916654i	$0.369121\pi$
$198$	0	0
$199$	−7.69556	−0.545524	−0.272762	−	0.962082i	$-0.587937\pi$
$199$	−7.69556	−0.545524	−0.272762	+	0.962082i	$0.587937\pi$
$200$	0	0
$201$	7.64478	0.539221
$202$	0	0
$203$	6.70290	0.470451
$204$	0	0
$205$	−17.7039	−1.23649
$206$	0	0
$207$	−2.65845	−0.184775
$208$	0	0
$209$	6.30320	0.436001
$210$	0	0
$211$	3.28568	0.226195	0.113098	−	0.993584i	$-0.463923\pi$
$211$	3.28568	0.226195	0.113098	+	0.993584i	$0.463923\pi$
$212$	0	0
$213$	−6.32556	−0.433421
$214$	0	0
$215$	7.32178	0.499341
$216$	0	0
$217$	−1.51364	−0.102752
$218$	0	0
$219$	18.8665	1.27488
$220$	0	0
$221$	6.19457	0.416692
$222$	0	0
$223$	1.70732	0.114330	0.0571652	−	0.998365i	$-0.481794\pi$
$223$	1.70732	0.114330	0.0571652	+	0.998365i	$0.481794\pi$
$224$	0	0
$225$	−1.53494	−0.102329
$226$	0	0
$227$	−8.52535	−0.565847	−0.282924	−	0.959142i	$-0.591304\pi$
$227$	−8.52535	−0.565847	−0.282924	+	0.959142i	$0.591304\pi$
$228$	0	0
$229$	−8.65310	−0.571813	−0.285906	−	0.958258i	$-0.592295\pi$
$229$	−8.65310	−0.571813	−0.285906	+	0.958258i	$0.592295\pi$
$230$	0	0
$231$	1.07382	0.0706519
$232$	0	0
$233$	28.1986	1.84735	0.923677	−	0.383171i	$-0.125168\pi$
$233$	28.1986	1.84735	0.923677	+	0.383171i	$0.125168\pi$
$234$	0	0
$235$	25.4806	1.66217
$236$	0	0
$237$	−17.8811	−1.16150
$238$	0	0
$239$	−24.6230	−1.59273	−0.796365	−	0.604816i	$-0.793246\pi$
$239$	−24.6230	−1.59273	−0.796365	+	0.604816i	$0.793246\pi$
$240$	0	0
$241$	−2.41161	−0.155345	−0.0776727	−	0.996979i	$-0.524749\pi$
$241$	−2.41161	−0.155345	−0.0776727	+	0.996979i	$0.524749\pi$
$242$	0	0
$243$	5.70762	0.366144
$244$	0	0
$245$	−18.1805	−1.16151
$246$	0	0
$247$	15.3896	0.979218
$248$	0	0
$249$	−20.2485	−1.28319
$250$	0	0
$251$	12.9349	0.816443	0.408222	−	0.912883i	$-0.366149\pi$
$251$	12.9349	0.816443	0.408222	+	0.912883i	$0.366149\pi$
$252$	0	0
$253$	−4.77356	−0.300111
$254$	0	0
$255$	−11.0443	−0.691619
$256$	0	0
$257$	25.4147	1.58533	0.792663	−	0.609660i	$-0.208694\pi$
$257$	25.4147	1.58533	0.792663	+	0.609660i	$0.208694\pi$
$258$	0	0
$259$	1.28123	0.0796117
$260$	0	0
$261$	5.43361	0.336332
$262$	0	0
$263$	−3.62386	−0.223457	−0.111728	−	0.993739i	$-0.535639\pi$
$263$	−3.62386	−0.223457	−0.111728	+	0.993739i	$0.535639\pi$
$264$	0	0
$265$	−17.1385	−1.05281
$266$	0	0
$267$	27.0386	1.65474
$268$	0	0
$269$	−13.0612	−0.796358	−0.398179	−	0.917308i	$-0.630358\pi$
$269$	−13.0612	−0.796358	−0.398179	+	0.917308i	$0.630358\pi$
$270$	0	0
$271$	−11.3688	−0.690608	−0.345304	−	0.938491i	$-0.612224\pi$
$271$	−11.3688	−0.690608	−0.345304	+	0.938491i	$0.612224\pi$
$272$	0	0
$273$	2.62179	0.158678
$274$	0	0
$275$	−2.75617	−0.166203
$276$	0	0
$277$	−15.2283	−0.914982	−0.457491	−	0.889214i	$-0.651252\pi$
$277$	−15.2283	−0.914982	−0.457491	+	0.889214i	$0.651252\pi$
$278$	0	0
$279$	−1.22701	−0.0734590
$280$	0	0
$281$	12.9635	0.773340	0.386670	−	0.922218i	$-0.373625\pi$
$281$	12.9635	0.773340	0.386670	+	0.922218i	$0.373625\pi$
$282$	0	0
$283$	1.69691	0.100871	0.0504354	−	0.998727i	$-0.483939\pi$
$283$	1.69691	0.100871	0.0504354	+	0.998727i	$0.483939\pi$
$284$	0	0
$285$	−27.4381	−1.62529
$286$	0	0
$287$	4.36722	0.257789
$288$	0	0
$289$	−10.5629	−0.621349
$290$	0	0
$291$	13.5922	0.796791
$292$	0	0
$293$	23.4231	1.36839	0.684197	−	0.729298i	$-0.260153\pi$
$293$	23.4231	1.36839	0.684197	+	0.729298i	$0.260153\pi$
$294$	0	0
$295$	−1.25485	−0.0730603
$296$	0	0
$297$	5.55959	0.322600
$298$	0	0
$299$	−11.6549	−0.674021
$300$	0	0
$301$	−1.80615	−0.104105
$302$	0	0
$303$	18.8809	1.08468
$304$	0	0
$305$	−13.5474	−0.775721
$306$	0	0
$307$	15.9323	0.909306	0.454653	−	0.890669i	$-0.349763\pi$
$307$	15.9323	0.909306	0.454653	+	0.890669i	$0.349763\pi$
$308$	0	0
$309$	−20.3719	−1.15892
$310$	0	0
$311$	16.6050	0.941584	0.470792	−	0.882244i	$-0.343968\pi$
$311$	16.6050	0.941584	0.470792	+	0.882244i	$0.343968\pi$
$312$	0	0
$313$	14.0647	0.794985	0.397493	−	0.917605i	$-0.369880\pi$
$313$	14.0647	0.794985	0.397493	+	0.917605i	$0.369880\pi$
$314$	0	0
$315$	1.06554	0.0600365
$316$	0	0
$317$	31.5460	1.77180	0.885901	−	0.463874i	$-0.153541\pi$
$317$	31.5460	1.77180	0.885901	+	0.463874i	$0.153541\pi$
$318$	0	0
$319$	9.75669	0.546270
$320$	0	0
$321$	15.2382	0.850512
$322$	0	0
$323$	15.9921	0.889823
$324$	0	0
$325$	−6.72934	−0.373277
$326$	0	0
$327$	−30.2608	−1.67343
$328$	0	0
$329$	−6.28559	−0.346536
$330$	0	0
$331$	−5.95383	−0.327252	−0.163626	−	0.986522i	$-0.552319\pi$
$331$	−5.95383	−0.327252	−0.163626	+	0.986522i	$0.552319\pi$
$332$	0	0
$333$	1.03861	0.0569155
$334$	0	0
$335$	13.6213	0.744212
$336$	0	0
$337$	25.4394	1.38577	0.692887	−	0.721046i	$-0.256338\pi$
$337$	25.4394	1.38577	0.692887	+	0.721046i	$0.256338\pi$
$338$	0	0
$339$	−20.2930	−1.10216
$340$	0	0
$341$	−2.20324	−0.119312
$342$	0	0
$343$	9.29383	0.501820
$344$	0	0
$345$	20.7795	1.11873
$346$	0	0
$347$	−21.9391	−1.17775	−0.588876	−	0.808223i	$-0.700429\pi$
$347$	−21.9391	−1.17775	−0.588876	+	0.808223i	$0.700429\pi$
$348$	0	0
$349$	4.47977	0.239797	0.119898	−	0.992786i	$-0.461743\pi$
$349$	4.47977	0.239797	0.119898	+	0.992786i	$0.461743\pi$
$350$	0	0
$351$	13.5741	0.724530
$352$	0	0
$353$	−31.9567	−1.70089	−0.850443	−	0.526068i	$-0.823666\pi$
$353$	−31.9567	−1.70089	−0.850443	+	0.526068i	$0.823666\pi$
$354$	0	0
$355$	−11.2708	−0.598190
$356$	0	0
$357$	2.72442	0.144192
$358$	0	0
$359$	−27.7825	−1.46630	−0.733152	−	0.680065i	$-0.761952\pi$
$359$	−27.7825	−1.46630	−0.733152	+	0.680065i	$0.761952\pi$
$360$	0	0
$361$	20.7303	1.09107
$362$	0	0
$363$	1.56304	0.0820382
$364$	0	0
$365$	33.6159	1.75954
$366$	0	0
$367$	−0.104167	−0.00543746	−0.00271873	−	0.999996i	$-0.500865\pi$
$367$	−0.104167	−0.00543746	−0.00271873	+	0.999996i	$0.500865\pi$
$368$	0	0
$369$	3.54022	0.184297
$370$	0	0
$371$	4.22775	0.219494
$372$	0	0
$373$	4.45518	0.230680	0.115340	−	0.993326i	$-0.463204\pi$
$373$	4.45518	0.230680	0.115340	+	0.993326i	$0.463204\pi$
$374$	0	0
$375$	−9.76751	−0.504392
$376$	0	0
$377$	23.8215	1.22687
$378$	0	0
$379$	−23.4074	−1.20236	−0.601179	−	0.799114i	$-0.705302\pi$
$379$	−23.4074	−1.20236	−0.601179	+	0.799114i	$0.705302\pi$
$380$	0	0
$381$	−21.8780	−1.12084
$382$	0	0
$383$	−4.64207	−0.237198	−0.118599	−	0.992942i	$-0.537840\pi$
$383$	−4.64207	−0.237198	−0.118599	+	0.992942i	$0.537840\pi$
$384$	0	0
$385$	1.91330	0.0975110
$386$	0	0
$387$	−1.46413	−0.0744259
$388$	0	0
$389$	14.0017	0.709913	0.354957	−	0.934883i	$-0.384496\pi$
$389$	14.0017	0.709913	0.354957	+	0.934883i	$0.384496\pi$
$390$	0	0
$391$	−12.1112	−0.612488
$392$	0	0
$393$	−5.35571	−0.270160
$394$	0	0
$395$	−31.8602	−1.60306
$396$	0	0
$397$	24.3350	1.22134	0.610670	−	0.791886i	$-0.290900\pi$
$397$	24.3350	1.22134	0.610670	+	0.791886i	$0.290900\pi$
$398$	0	0
$399$	6.76848	0.338848
$400$	0	0
$401$	3.89164	0.194339	0.0971696	−	0.995268i	$-0.469021\pi$
$401$	3.89164	0.194339	0.0971696	+	0.995268i	$0.469021\pi$
$402$	0	0
$403$	−5.37933	−0.267963
$404$	0	0
$405$	−19.5482	−0.971356
$406$	0	0
$407$	1.86495	0.0924420
$408$	0	0
$409$	−19.4630	−0.962382	−0.481191	−	0.876616i	$-0.659796\pi$
$409$	−19.4630	−0.962382	−0.481191	+	0.876616i	$0.659796\pi$
$410$	0	0
$411$	1.56304	0.0770990
$412$	0	0
$413$	0.309549	0.0152319
$414$	0	0
$415$	−36.0783	−1.77101
$416$	0	0
$417$	31.6439	1.54961
$418$	0	0
$419$	26.9424	1.31622	0.658111	−	0.752921i	$-0.271356\pi$
$419$	26.9424	1.31622	0.658111	+	0.752921i	$0.271356\pi$
$420$	0	0
$421$	−5.49987	−0.268047	−0.134024	−	0.990978i	$-0.542790\pi$
$421$	−5.49987	−0.268047	−0.134024	+	0.990978i	$0.542790\pi$
$422$	0	0
$423$	−5.09533	−0.247743
$424$	0	0
$425$	−6.99277	−0.339199
$426$	0	0
$427$	3.34189	0.161725
$428$	0	0
$429$	3.81625	0.184250
$430$	0	0
$431$	13.5405	0.652221	0.326110	−	0.945332i	$-0.394262\pi$
$431$	13.5405	0.652221	0.326110	+	0.945332i	$0.394262\pi$
$432$	0	0
$433$	−28.0252	−1.34681	−0.673404	−	0.739275i	$-0.735168\pi$
$433$	−28.0252	−1.34681	−0.673404	+	0.739275i	$0.735168\pi$
$434$	0	0
$435$	−42.4713	−2.03634
$436$	0	0
$437$	−30.0887	−1.43934
$438$	0	0
$439$	10.2042	0.487019	0.243510	−	0.969898i	$-0.421701\pi$
$439$	10.2042	0.487019	0.243510	+	0.969898i	$0.421701\pi$
$440$	0	0
$441$	3.63553	0.173121
$442$	0	0
$443$	−9.44280	−0.448641	−0.224320	−	0.974515i	$-0.572016\pi$
$443$	−9.44280	−0.448641	−0.224320	+	0.974515i	$0.572016\pi$
$444$	0	0
$445$	48.1769	2.28380
$446$	0	0
$447$	1.75116	0.0828272
$448$	0	0
$449$	17.7950	0.839797	0.419898	−	0.907571i	$-0.362066\pi$
$449$	17.7950	0.839797	0.419898	+	0.907571i	$0.362066\pi$
$450$	0	0
$451$	6.35689	0.299334
$452$	0	0
$453$	−22.9390	−1.07777
$454$	0	0
$455$	4.67144	0.219001
$456$	0	0
$457$	−17.0792	−0.798934	−0.399467	−	0.916748i	$-0.630805\pi$
$457$	−17.0792	−0.798934	−0.399467	+	0.916748i	$0.630805\pi$
$458$	0	0
$459$	14.1054	0.658385
$460$	0	0
$461$	−13.8111	−0.643247	−0.321624	−	0.946868i	$-0.604229\pi$
$461$	−13.8111	−0.643247	−0.321624	+	0.946868i	$0.604229\pi$
$462$	0	0
$463$	−27.1096	−1.25989	−0.629946	−	0.776639i	$-0.716923\pi$
$463$	−27.1096	−1.25989	−0.629946	+	0.776639i	$0.716923\pi$
$464$	0	0
$465$	9.59078	0.444762
$466$	0	0
$467$	−19.3189	−0.893974	−0.446987	−	0.894540i	$-0.647503\pi$
$467$	−19.3189	−0.893974	−0.446987	+	0.894540i	$0.647503\pi$
$468$	0	0
$469$	−3.36013	−0.155156
$470$	0	0
$471$	26.3420	1.21377
$472$	0	0
$473$	−2.62902	−0.120882
$474$	0	0
$475$	−17.3727	−0.797112
$476$	0	0
$477$	3.42717	0.156919
$478$	0	0
$479$	−6.20728	−0.283618	−0.141809	−	0.989894i	$-0.545292\pi$
$479$	−6.20728	−0.283618	−0.141809	+	0.989894i	$0.545292\pi$
$480$	0	0
$481$	4.55337	0.207616
$482$	0	0
$483$	−5.12593	−0.233238
$484$	0	0
$485$	24.2184	1.09970
$486$	0	0
$487$	−6.97901	−0.316249	−0.158125	−	0.987419i	$-0.550545\pi$
$487$	−6.97901	−0.316249	−0.158125	+	0.987419i	$0.550545\pi$
$488$	0	0
$489$	−4.08424	−0.184696
$490$	0	0
$491$	−12.9881	−0.586146	−0.293073	−	0.956090i	$-0.594678\pi$
$491$	−12.9881	−0.586146	−0.293073	+	0.956090i	$0.594678\pi$
$492$	0	0
$493$	24.7540	1.11487
$494$	0	0
$495$	1.55099	0.0697120
$496$	0	0
$497$	2.78029	0.124713
$498$	0	0
$499$	−2.41143	−0.107951	−0.0539753	−	0.998542i	$-0.517189\pi$
$499$	−2.41143	−0.107951	−0.0539753	+	0.998542i	$0.517189\pi$
$500$	0	0
$501$	22.8644	1.02150
$502$	0	0
$503$	−2.72397	−0.121456	−0.0607278	−	0.998154i	$-0.519342\pi$
$503$	−2.72397	−0.121456	−0.0607278	+	0.998154i	$0.519342\pi$
$504$	0	0
$505$	33.6415	1.49703
$506$	0	0
$507$	−11.0019	−0.488612
$508$	0	0
$509$	10.9925	0.487235	0.243618	−	0.969871i	$-0.421666\pi$
$509$	10.9925	0.487235	0.243618	+	0.969871i	$0.421666\pi$
$510$	0	0
$511$	−8.29243	−0.366836
$512$	0	0
$513$	35.0432	1.54719
$514$	0	0
$515$	−36.2983	−1.59949
$516$	0	0
$517$	−9.14926	−0.402384
$518$	0	0
$519$	34.3342	1.50710
$520$	0	0
$521$	−32.4531	−1.42179	−0.710897	−	0.703296i	$-0.751711\pi$
$521$	−32.4531	−1.42179	−0.710897	+	0.703296i	$0.751711\pi$
$522$	0	0
$523$	16.6535	0.728205	0.364102	−	0.931359i	$-0.381376\pi$
$523$	16.6535	0.728205	0.364102	+	0.931359i	$0.381376\pi$
$524$	0	0
$525$	−2.95962	−0.129168
$526$	0	0
$527$	−5.58991	−0.243500
$528$	0	0
$529$	−0.213145	−0.00926718
$530$	0	0
$531$	0.250932	0.0108895
$532$	0	0
$533$	15.5207	0.672276
$534$	0	0
$535$	27.1511	1.17384
$536$	0	0
$537$	−35.1601	−1.51727
$538$	0	0
$539$	6.52802	0.281182
$540$	0	0
$541$	37.8445	1.62706	0.813532	−	0.581521i	$-0.197542\pi$
$541$	37.8445	1.62706	0.813532	+	0.581521i	$0.197542\pi$
$542$	0	0
$543$	22.8752	0.981671
$544$	0	0
$545$	−53.9181	−2.30960
$546$	0	0
$547$	−9.28831	−0.397139	−0.198570	−	0.980087i	$-0.563630\pi$
$547$	−9.28831	−0.397139	−0.198570	+	0.980087i	$0.563630\pi$
$548$	0	0
$549$	2.70906	0.115620
$550$	0	0
$551$	61.4983	2.61992
$552$	0	0
$553$	7.85932	0.334212
$554$	0	0
$555$	−8.11820	−0.344598
$556$	0	0
$557$	21.9471	0.929929	0.464965	−	0.885329i	$-0.346067\pi$
$557$	21.9471	0.929929	0.464965	+	0.885329i	$0.346067\pi$
$558$	0	0
$559$	−6.41889	−0.271490
$560$	0	0
$561$	3.96564	0.167430
$562$	0	0
$563$	19.2659	0.811961	0.405981	−	0.913882i	$-0.366930\pi$
$563$	19.2659	0.811961	0.405981	+	0.913882i	$0.366930\pi$
$564$	0	0
$565$	−36.1576	−1.52116
$566$	0	0
$567$	4.82217	0.202512
$568$	0	0
$569$	37.3490	1.56575	0.782877	−	0.622177i	$-0.213752\pi$
$569$	37.3490	1.56575	0.782877	+	0.622177i	$0.213752\pi$
$570$	0	0
$571$	15.1184	0.632687	0.316344	−	0.948645i	$-0.397545\pi$
$571$	15.1184	0.632687	0.316344	+	0.948645i	$0.397545\pi$
$572$	0	0
$573$	−25.2306	−1.05402
$574$	0	0
$575$	13.1567	0.548673
$576$	0	0
$577$	−17.3388	−0.721826	−0.360913	−	0.932600i	$-0.617535\pi$
$577$	−17.3388	−0.721826	−0.360913	+	0.932600i	$0.617535\pi$
$578$	0	0
$579$	−13.5988	−0.565146
$580$	0	0
$581$	8.89985	0.369228
$582$	0	0
$583$	6.15388	0.254868
$584$	0	0
$585$	3.78684	0.156567
$586$	0	0
$587$	13.8006	0.569613	0.284807	−	0.958585i	$-0.408071\pi$
$587$	13.8006	0.569613	0.284807	+	0.958585i	$0.408071\pi$
$588$	0	0
$589$	−13.8874	−0.572221
$590$	0	0
$591$	17.5367	0.721361
$592$	0	0
$593$	−28.9323	−1.18811	−0.594054	−	0.804425i	$-0.702474\pi$
$593$	−28.9323	−1.18811	−0.594054	+	0.804425i	$0.702474\pi$
$594$	0	0
$595$	4.85431	0.199008
$596$	0	0
$597$	−12.0284	−0.492292
$598$	0	0
$599$	−24.6801	−1.00840	−0.504201	−	0.863586i	$-0.668213\pi$
$599$	−24.6801	−1.00840	−0.504201	+	0.863586i	$0.668213\pi$
$600$	0	0
$601$	−14.2597	−0.581665	−0.290833	−	0.956774i	$-0.593932\pi$
$601$	−14.2597	−0.581665	−0.290833	+	0.956774i	$0.593932\pi$
$602$	0	0
$603$	−2.72384	−0.110923
$604$	0	0
$605$	2.78499	0.113226
$606$	0	0
$607$	41.5318	1.68573	0.842863	−	0.538129i	$-0.180869\pi$
$607$	41.5318	1.68573	0.842863	+	0.538129i	$0.180869\pi$
$608$	0	0
$609$	10.4769	0.424545
$610$	0	0
$611$	−22.3384	−0.903716
$612$	0	0
$613$	−31.7210	−1.28120	−0.640599	−	0.767876i	$-0.721314\pi$
$613$	−31.7210	−1.28120	−0.640599	+	0.767876i	$0.721314\pi$
$614$	0	0
$615$	−27.6718	−1.11583
$616$	0	0
$617$	−40.0058	−1.61057	−0.805286	−	0.592887i	$-0.797988\pi$
$617$	−40.0058	−1.61057	−0.805286	+	0.592887i	$0.797988\pi$
$618$	0	0
$619$	−13.9581	−0.561024	−0.280512	−	0.959851i	$-0.590504\pi$
$619$	−13.9581	−0.561024	−0.280512	+	0.959851i	$0.590504\pi$
$620$	0	0
$621$	−26.5390	−1.06497
$622$	0	0
$623$	−11.8844	−0.476137
$624$	0	0
$625$	−31.1844	−1.24738
$626$	0	0
$627$	9.85213	0.393456
$628$	0	0
$629$	4.73162	0.188662
$630$	0	0
$631$	−1.98410	−0.0789859	−0.0394929	−	0.999220i	$-0.512574\pi$
$631$	−1.98410	−0.0789859	−0.0394929	+	0.999220i	$0.512574\pi$
$632$	0	0
$633$	5.13564	0.204123
$634$	0	0
$635$	−38.9818	−1.54694
$636$	0	0
$637$	15.9385	0.631508
$638$	0	0
$639$	2.25380	0.0891591
$640$	0	0
$641$	−36.4755	−1.44070	−0.720348	−	0.693613i	$-0.756018\pi$
$641$	−36.4755	−1.44070	−0.720348	+	0.693613i	$0.756018\pi$
$642$	0	0
$643$	−40.9148	−1.61352	−0.806762	−	0.590877i	$-0.798782\pi$
$643$	−40.9148	−1.61352	−0.806762	+	0.590877i	$0.798782\pi$
$644$	0	0
$645$	11.4442	0.450616
$646$	0	0
$647$	15.2007	0.597602	0.298801	−	0.954316i	$-0.403413\pi$
$647$	15.2007	0.597602	0.298801	+	0.954316i	$0.403413\pi$
$648$	0	0
$649$	0.450577	0.0176867
$650$	0	0
$651$	−2.36587	−0.0927258
$652$	0	0
$653$	34.6832	1.35726	0.678628	−	0.734482i	$-0.262575\pi$
$653$	34.6832	1.35726	0.678628	+	0.734482i	$0.262575\pi$
$654$	0	0
$655$	−9.54270	−0.372864
$656$	0	0
$657$	−6.72214	−0.262256
$658$	0	0
$659$	−21.9321	−0.854354	−0.427177	−	0.904168i	$-0.640492\pi$
$659$	−21.9321	−0.854354	−0.427177	+	0.904168i	$0.640492\pi$
$660$	0	0
$661$	−21.7835	−0.847282	−0.423641	−	0.905830i	$-0.639248\pi$
$661$	−21.7835	−0.847282	−0.423641	+	0.905830i	$0.639248\pi$
$662$	0	0
$663$	9.68234	0.376031
$664$	0	0
$665$	12.0599	0.467664
$666$	0	0
$667$	−46.5741	−1.80336
$668$	0	0
$669$	2.66860	0.103174
$670$	0	0
$671$	4.86443	0.187789
$672$	0	0
$673$	−4.75355	−0.183236	−0.0916180	−	0.995794i	$-0.529204\pi$
$673$	−4.75355	−0.183236	−0.0916180	+	0.995794i	$0.529204\pi$
$674$	0	0
$675$	−15.3232	−0.589788
$676$	0	0
$677$	−1.31848	−0.0506733	−0.0253367	−	0.999679i	$-0.508066\pi$
$677$	−1.31848	−0.0506733	−0.0253367	+	0.999679i	$0.508066\pi$
$678$	0	0
$679$	−5.97423	−0.229270
$680$	0	0
$681$	−13.3254	−0.510632
$682$	0	0
$683$	−13.5120	−0.517024	−0.258512	−	0.966008i	$-0.583232\pi$
$683$	−13.5120	−0.517024	−0.258512	+	0.966008i	$0.583232\pi$
$684$	0	0
$685$	2.78499	0.106409
$686$	0	0
$687$	−13.5251	−0.516015
$688$	0	0
$689$	15.0250	0.572409
$690$	0	0
$691$	25.9273	0.986322	0.493161	−	0.869938i	$-0.335841\pi$
$691$	25.9273	0.986322	0.493161	+	0.869938i	$0.335841\pi$
$692$	0	0
$693$	−0.382602	−0.0145338
$694$	0	0
$695$	56.3825	2.13871
$696$	0	0
$697$	16.1283	0.610902
$698$	0	0
$699$	44.0756	1.66709
$700$	0	0
$701$	−7.11998	−0.268918	−0.134459	−	0.990919i	$-0.542930\pi$
$701$	−7.11998	−0.268918	−0.134459	+	0.990919i	$0.542930\pi$
$702$	0	0
$703$	11.7551	0.443353
$704$	0	0
$705$	39.8271	1.49998
$706$	0	0
$707$	−8.29875	−0.312107
$708$	0	0
$709$	−40.9864	−1.53928	−0.769638	−	0.638480i	$-0.779564\pi$
$709$	−40.9864	−1.53928	−0.769638	+	0.638480i	$0.779564\pi$
$710$	0	0
$711$	6.37105	0.238933
$712$	0	0
$713$	10.5173	0.393875
$714$	0	0
$715$	6.79971	0.254295
$716$	0	0
$717$	−38.4867	−1.43731
$718$	0	0
$719$	−49.3228	−1.83943	−0.919715	−	0.392587i	$-0.871580\pi$
$719$	−49.3228	−1.83943	−0.919715	+	0.392587i	$0.871580\pi$
$720$	0	0
$721$	8.95413	0.333469
$722$	0	0
$723$	−3.76944	−0.140187
$724$	0	0
$725$	−26.8911	−0.998709
$726$	0	0
$727$	39.6950	1.47221	0.736103	−	0.676869i	$-0.236664\pi$
$727$	39.6950	1.47221	0.736103	+	0.676869i	$0.236664\pi$
$728$	0	0
$729$	29.9786	1.11032
$730$	0	0
$731$	−6.67017	−0.246705
$732$	0	0
$733$	−8.36063	−0.308807	−0.154404	−	0.988008i	$-0.549346\pi$
$733$	−8.36063	−0.308807	−0.154404	+	0.988008i	$0.549346\pi$
$734$	0	0
$735$	−28.4168	−1.04817
$736$	0	0
$737$	−4.89098	−0.180161
$738$	0	0
$739$	22.7622	0.837320	0.418660	−	0.908143i	$-0.362500\pi$
$739$	22.7622	0.837320	0.418660	+	0.908143i	$0.362500\pi$
$740$	0	0
$741$	24.0546	0.883666
$742$	0	0
$743$	−31.4701	−1.15453	−0.577263	−	0.816558i	$-0.695879\pi$
$743$	−31.4701	−1.15453	−0.577263	+	0.816558i	$0.695879\pi$
$744$	0	0
$745$	3.12019	0.114315
$746$	0	0
$747$	7.21454	0.263966
$748$	0	0
$749$	−6.69767	−0.244728
$750$	0	0
$751$	9.37869	0.342233	0.171117	−	0.985251i	$-0.445263\pi$
$751$	9.37869	0.342233	0.171117	+	0.985251i	$0.445263\pi$
$752$	0	0
$753$	20.2177	0.736775
$754$	0	0
$755$	−40.8722	−1.48749
$756$	0	0
$757$	7.76616	0.282266	0.141133	−	0.989991i	$-0.454925\pi$
$757$	7.76616	0.282266	0.141133	+	0.989991i	$0.454925\pi$
$758$	0	0
$759$	−7.46125	−0.270826
$760$	0	0
$761$	−48.2867	−1.75039	−0.875196	−	0.483769i	$-0.839268\pi$
$761$	−48.2867	−1.75039	−0.875196	+	0.483769i	$0.839268\pi$
$762$	0	0
$763$	13.3006	0.481514
$764$	0	0
$765$	3.93508	0.142273
$766$	0	0
$767$	1.10011	0.0397226
$768$	0	0
$769$	2.55252	0.0920462	0.0460231	−	0.998940i	$-0.485345\pi$
$769$	2.55252	0.0920462	0.0460231	+	0.998940i	$0.485345\pi$
$770$	0	0
$771$	39.7241	1.43063
$772$	0	0
$773$	−21.9459	−0.789341	−0.394670	−	0.918823i	$-0.629141\pi$
$773$	−21.9459	−0.789341	−0.394670	+	0.918823i	$0.629141\pi$
$774$	0	0
$775$	6.07248	0.218130
$776$	0	0
$777$	2.00261	0.0718432
$778$	0	0
$779$	40.0687	1.43561
$780$	0	0
$781$	4.04697	0.144812
$782$	0	0
$783$	54.2432	1.93849
$784$	0	0
$785$	46.9356	1.67520
$786$	0	0
$787$	9.36651	0.333880	0.166940	−	0.985967i	$-0.446611\pi$
$787$	9.36651	0.333880	0.166940	+	0.985967i	$0.446611\pi$
$788$	0	0
$789$	−5.66423	−0.201652
$790$	0	0
$791$	8.91942	0.317138
$792$	0	0
$793$	11.8768	0.421757
$794$	0	0
$795$	−26.7881	−0.950076
$796$	0	0
$797$	26.0103	0.921332	0.460666	−	0.887573i	$-0.347611\pi$
$797$	26.0103	0.921332	0.460666	+	0.887573i	$0.347611\pi$
$798$	0	0
$799$	−23.2129	−0.821214
$800$	0	0
$801$	−9.63389	−0.340397
$802$	0	0
$803$	−12.0704	−0.425955
$804$	0	0
$805$	−9.13327	−0.321905
$806$	0	0
$807$	−20.4152	−0.718650
$808$	0	0
$809$	−44.0342	−1.54816	−0.774081	−	0.633087i	$-0.781788\pi$
$809$	−44.0342	−1.54816	−0.774081	+	0.633087i	$0.781788\pi$
$810$	0	0
$811$	14.0662	0.493932	0.246966	−	0.969024i	$-0.420566\pi$
$811$	14.0662	0.493932	0.246966	+	0.969024i	$0.420566\pi$
$812$	0	0
$813$	−17.7699	−0.623219
$814$	0	0
$815$	−7.27721	−0.254909
$816$	0	0
$817$	−16.5712	−0.579753
$818$	0	0
$819$	−0.934144	−0.0326416
$820$	0	0
$821$	33.5129	1.16961	0.584803	−	0.811175i	$-0.301172\pi$
$821$	33.5129	1.16961	0.584803	+	0.811175i	$0.301172\pi$
$822$	0	0
$823$	−13.6419	−0.475528	−0.237764	−	0.971323i	$-0.576414\pi$
$823$	−13.6419	−0.475528	−0.237764	+	0.971323i	$0.576414\pi$
$824$	0	0
$825$	−4.30799	−0.149985
$826$	0	0
$827$	−38.3902	−1.33496	−0.667480	−	0.744628i	$-0.732627\pi$
$827$	−38.3902	−1.33496	−0.667480	+	0.744628i	$0.732627\pi$
$828$	0	0
$829$	−2.40083	−0.0833843	−0.0416922	−	0.999131i	$-0.513275\pi$
$829$	−2.40083	−0.0833843	−0.0416922	+	0.999131i	$0.513275\pi$
$830$	0	0
$831$	−23.8025	−0.825699
$832$	0	0
$833$	16.5625	0.573856
$834$	0	0
$835$	40.7393	1.40984
$836$	0	0
$837$	−12.2491	−0.423390
$838$	0	0
$839$	16.6563	0.575038	0.287519	−	0.957775i	$-0.407169\pi$
$839$	16.6563	0.575038	0.287519	+	0.957775i	$0.407169\pi$
$840$	0	0
$841$	66.1929	2.28252
$842$	0	0
$843$	20.2625	0.697878
$844$	0	0
$845$	−19.6030	−0.674363
$846$	0	0
$847$	−0.687006	−0.0236058
$848$	0	0
$849$	2.65233	0.0910278
$850$	0	0
$851$	−8.90243	−0.305171
$852$	0	0
$853$	−11.6584	−0.399175	−0.199587	−	0.979880i	$-0.563960\pi$
$853$	−11.6584	−0.399175	−0.199587	+	0.979880i	$0.563960\pi$
$854$	0	0
$855$	9.77622	0.334340
$856$	0	0
$857$	45.5257	1.55513	0.777565	−	0.628803i	$-0.216455\pi$
$857$	45.5257	1.55513	0.777565	+	0.628803i	$0.216455\pi$
$858$	0	0
$859$	−21.4911	−0.733268	−0.366634	−	0.930365i	$-0.619490\pi$
$859$	−21.4911	−0.733268	−0.366634	+	0.930365i	$0.619490\pi$
$860$	0	0
$861$	6.82613	0.232634
$862$	0	0
$863$	−7.37563	−0.251069	−0.125535	−	0.992089i	$-0.540065\pi$
$863$	−7.37563	−0.251069	−0.125535	+	0.992089i	$0.540065\pi$
$864$	0	0
$865$	61.1759	2.08004
$866$	0	0
$867$	−16.5103	−0.560718
$868$	0	0
$869$	11.4400	0.388074
$870$	0	0
$871$	−11.9416	−0.404626
$872$	0	0
$873$	−4.84293	−0.163908
$874$	0	0
$875$	4.29314	0.145134
$876$	0	0
$877$	−53.3061	−1.80002	−0.900009	−	0.435872i	$-0.856440\pi$
$877$	−53.3061	−1.80002	−0.900009	+	0.435872i	$0.856440\pi$
$878$	0	0
$879$	36.6112	1.23487
$880$	0	0
$881$	−12.7628	−0.429990	−0.214995	−	0.976615i	$-0.568974\pi$
$881$	−12.7628	−0.429990	−0.214995	+	0.976615i	$0.568974\pi$
$882$	0	0
$883$	−40.8922	−1.37613	−0.688066	−	0.725648i	$-0.741540\pi$
$883$	−40.8922	−1.37613	−0.688066	+	0.725648i	$0.741540\pi$
$884$	0	0
$885$	−1.96138	−0.0659311
$886$	0	0
$887$	−12.8105	−0.430133	−0.215066	−	0.976599i	$-0.568997\pi$
$887$	−12.8105	−0.430133	−0.215066	+	0.976599i	$0.568997\pi$
$888$	0	0
$889$	9.61609	0.322513
$890$	0	0
$891$	7.01911	0.235149
$892$	0	0
$893$	−57.6696	−1.92984
$894$	0	0
$895$	−62.6475	−2.09408
$896$	0	0
$897$	−18.2171	−0.608251
$898$	0	0
$899$	−21.4963	−0.716941
$900$	0	0
$901$	15.6132	0.520152
$902$	0	0
$903$	−2.82308	−0.0939462
$904$	0	0
$905$	40.7587	1.35486
$906$	0	0
$907$	−14.9708	−0.497097	−0.248548	−	0.968620i	$-0.579953\pi$
$907$	−14.9708	−0.497097	−0.248548	+	0.968620i	$0.579953\pi$
$908$	0	0
$909$	−6.72727	−0.223129
$910$	0	0
$911$	−4.68084	−0.155083	−0.0775416	−	0.996989i	$-0.524707\pi$
$911$	−4.68084	−0.155083	−0.0775416	+	0.996989i	$0.524707\pi$
$912$	0	0
$913$	12.9546	0.428733
$914$	0	0
$915$	−21.1751	−0.700026
$916$	0	0
$917$	2.35401	0.0777363
$918$	0	0
$919$	30.9487	1.02090	0.510452	−	0.859906i	$-0.329478\pi$
$919$	30.9487	1.02090	0.510452	+	0.859906i	$0.329478\pi$
$920$	0	0
$921$	24.9028	0.820577
$922$	0	0
$923$	9.88091	0.325234
$924$	0	0
$925$	−5.14010	−0.169006
$926$	0	0
$927$	7.25854	0.238402
$928$	0	0
$929$	48.2071	1.58162	0.790811	−	0.612060i	$-0.209659\pi$
$929$	48.2071	1.58162	0.790811	+	0.612060i	$0.209659\pi$
$930$	0	0
$931$	41.1474	1.34855
$932$	0	0
$933$	25.9543	0.849705
$934$	0	0
$935$	7.06590	0.231080
$936$	0	0
$937$	26.8952	0.878629	0.439315	−	0.898333i	$-0.355221\pi$
$937$	26.8952	0.878629	0.439315	+	0.898333i	$0.355221\pi$
$938$	0	0
$939$	21.9837	0.717411
$940$	0	0
$941$	−47.5030	−1.54855	−0.774277	−	0.632847i	$-0.781886\pi$
$941$	−47.5030	−1.54855	−0.774277	+	0.632847i	$0.781886\pi$
$942$	0	0
$943$	−30.3450	−0.988168
$944$	0	0
$945$	10.6372	0.346028
$946$	0	0
$947$	58.1636	1.89006	0.945032	−	0.326978i	$-0.106030\pi$
$947$	58.1636	1.89006	0.945032	+	0.326978i	$0.106030\pi$
$948$	0	0
$949$	−29.4706	−0.956655
$950$	0	0
$951$	49.3077	1.59891
$952$	0	0
$953$	−36.5509	−1.18400	−0.592000	−	0.805938i	$-0.701662\pi$
$953$	−36.5509	−1.18400	−0.592000	+	0.805938i	$0.701662\pi$
$954$	0	0
$955$	−44.9554	−1.45472
$956$	0	0
$957$	15.2501	0.492965
$958$	0	0
$959$	−0.687006	−0.0221846
$960$	0	0
$961$	−26.1458	−0.843411
$962$	0	0
$963$	−5.42937	−0.174959
$964$	0	0
$965$	−24.2300	−0.779993
$966$	0	0
$967$	54.3373	1.74737	0.873684	−	0.486493i	$-0.161724\pi$
$967$	54.3373	1.74737	0.873684	+	0.486493i	$0.161724\pi$
$968$	0	0
$969$	24.9962	0.802994
$970$	0	0
$971$	−28.4619	−0.913387	−0.456693	−	0.889624i	$-0.650966\pi$
$971$	−28.4619	−0.913387	−0.456693	+	0.889624i	$0.650966\pi$
$972$	0	0
$973$	−13.9085	−0.445887
$974$	0	0
$975$	−10.5182	−0.336853
$976$	0	0
$977$	−3.09103	−0.0988907	−0.0494453	−	0.998777i	$-0.515745\pi$
$977$	−3.09103	−0.0988907	−0.0494453	+	0.998777i	$0.515745\pi$
$978$	0	0
$979$	−17.2988	−0.552871
$980$	0	0
$981$	10.7820	0.344241
$982$	0	0
$983$	−11.2587	−0.359098	−0.179549	−	0.983749i	$-0.557464\pi$
$983$	−11.2587	−0.359098	−0.179549	+	0.983749i	$0.557464\pi$
$984$	0	0
$985$	31.2464	0.995595
$986$	0	0
$987$	−9.82462	−0.312721
$988$	0	0
$989$	12.5498	0.399059
$990$	0	0
$991$	42.3313	1.34470	0.672350	−	0.740234i	$-0.265285\pi$
$991$	42.3313	1.34470	0.672350	+	0.740234i	$0.265285\pi$
$992$	0	0
$993$	−9.30606	−0.295319
$994$	0	0
$995$	−21.4320	−0.679442
$996$	0	0
$997$	48.4549	1.53458	0.767291	−	0.641300i	$-0.221604\pi$
$997$	48.4549	1.53458	0.767291	+	0.641300i	$0.221604\pi$
$998$	0	0
$999$	10.3683	0.328040

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6028.2.a.d.1.19	✓	27

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6028.2.a.d.1.19	✓	27	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)

\(f(q)\)	\(=\)	\(q+1.56304 q^{3} +2.78499 q^{5} -0.687006 q^{7} -0.556912 q^{9} +O(q^{10})\)	\(q+1.56304 q^{3} +2.78499 q^{5} -0.687006 q^{7} -0.556912 q^{9} -1.00000 q^{11} -2.44156 q^{13} +4.35304 q^{15} -2.53714 q^{17} -6.30320 q^{19} -1.07382 q^{21} +4.77356 q^{23} +2.75617 q^{25} -5.55959 q^{27} -9.75669 q^{29} +2.20324 q^{31} -1.56304 q^{33} -1.91330 q^{35} -1.86495 q^{37} -3.81625 q^{39} -6.35689 q^{41} +2.62902 q^{43} -1.55099 q^{45} +9.14926 q^{47} -6.52802 q^{49} -3.96564 q^{51} -6.15388 q^{53} -2.78499 q^{55} -9.85213 q^{57} -0.450577 q^{59} -4.86443 q^{61} +0.382602 q^{63} -6.79971 q^{65} +4.89098 q^{67} +7.46125 q^{69} -4.04697 q^{71} +12.0704 q^{73} +4.30799 q^{75} +0.687006 q^{77} -11.4400 q^{79} -7.01911 q^{81} -12.9546 q^{83} -7.06590 q^{85} -15.2501 q^{87} +17.2988 q^{89} +1.67736 q^{91} +3.44374 q^{93} -17.5543 q^{95} +8.69604 q^{97} +0.556912 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 27 q - 6 q^{3} + q^{5} - 14 q^{7} + 33 q^{9}+O(q^{10}) \) 27 * q - 6 * q^3 + q^5 - 14 * q^7 + 33 * q^9	\( 27 q - 6 q^{3} + q^{5} - 14 q^{7} + 33 q^{9} - 27 q^{11} - 6 q^{15} - 21 q^{17} - 3 q^{19} - 4 q^{21} - 44 q^{23} + 38 q^{25} - 18 q^{27} + q^{29} - 8 q^{31} + 6 q^{33} - 33 q^{35} + 11 q^{37} - 13 q^{39} - 19 q^{41} - 11 q^{43} + 17 q^{45} - 37 q^{47} + 41 q^{49} - 49 q^{51} - 12 q^{53} - q^{55} - 50 q^{57} - 14 q^{59} + 12 q^{61} - 53 q^{63} - 55 q^{65} - 5 q^{67} + 14 q^{69} - 67 q^{71} - 27 q^{73} - 70 q^{75} + 14 q^{77} - 31 q^{79} - 5 q^{81} - 55 q^{83} - 3 q^{85} - 31 q^{87} + 11 q^{89} - 11 q^{91} - 24 q^{93} - 47 q^{95} - q^{97} - 33 q^{99}+O(q^{100}) \) 27 * q - 6 * q^3 + q^5 - 14 * q^7 + 33 * q^9 - 27 * q^11 - 6 * q^15 - 21 * q^17 - 3 * q^19 - 4 * q^21 - 44 * q^23 + 38 * q^25 - 18 * q^27 + q^29 - 8 * q^31 + 6 * q^33 - 33 * q^35 + 11 * q^37 - 13 * q^39 - 19 * q^41 - 11 * q^43 + 17 * q^45 - 37 * q^47 + 41 * q^49 - 49 * q^51 - 12 * q^53 - q^55 - 50 * q^57 - 14 * q^59 + 12 * q^61 - 53 * q^63 - 55 * q^65 - 5 * q^67 + 14 * q^69 - 67 * q^71 - 27 * q^73 - 70 * q^75 + 14 * q^77 - 31 * q^79 - 5 * q^81 - 55 * q^83 - 3 * q^85 - 31 * q^87 + 11 * q^89 - 11 * q^91 - 24 * q^93 - 47 * q^95 - q^97 - 33 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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