LMFDB - Embedded newform 6036.2.a.h.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6036,2,Mod(1,6036)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6036, 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("6036.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6036 = 2^{2} \cdot 3 \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6036.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:	$48.1977026600$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Character	$\chi$	$=$	6036.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.97211 q^{5} +2.26181 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.97211 q^{5} +2.26181 q^{7} +1.00000 q^{9} +2.35709 q^{11} -0.135924 q^{13} +1.97211 q^{15} +2.74177 q^{17} +2.78972 q^{19} +2.26181 q^{21} -6.15147 q^{23} -1.11080 q^{25} +1.00000 q^{27} -0.635716 q^{29} +2.25306 q^{31} +2.35709 q^{33} +4.46053 q^{35} +6.70530 q^{37} -0.135924 q^{39} +8.99231 q^{41} -3.90784 q^{43} +1.97211 q^{45} -3.82047 q^{47} -1.88422 q^{49} +2.74177 q^{51} +12.4594 q^{53} +4.64843 q^{55} +2.78972 q^{57} +0.652656 q^{59} -6.69775 q^{61} +2.26181 q^{63} -0.268057 q^{65} +2.94969 q^{67} -6.15147 q^{69} -7.16191 q^{71} +4.18285 q^{73} -1.11080 q^{75} +5.33129 q^{77} +12.9329 q^{79} +1.00000 q^{81} +14.9066 q^{83} +5.40706 q^{85} -0.635716 q^{87} -15.2344 q^{89} -0.307434 q^{91} +2.25306 q^{93} +5.50162 q^{95} -11.9613 q^{97} +2.35709 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 24 q^{3} + 18 q^{5} + 9 q^{7} + 24 q^{9}+O(q^{10}) $ 24 * q + 24 * q^3 + 18 * q^5 + 9 * q^7 + 24 * q^9	$ 24 q + 24 q^{3} + 18 q^{5} + 9 q^{7} + 24 q^{9} + 9 q^{11} + 7 q^{13} + 18 q^{15} + 12 q^{17} - q^{19} + 9 q^{21} + 22 q^{23} + 48 q^{25} + 24 q^{27} + 44 q^{29} + 15 q^{31} + 9 q^{33} + 3 q^{35} + 18 q^{37} + 7 q^{39} + 37 q^{41} - 8 q^{43} + 18 q^{45} + 24 q^{47} + 63 q^{49} + 12 q^{51} + 59 q^{53} + 20 q^{55} - q^{57} + 44 q^{59} + 33 q^{61} + 9 q^{63} + 15 q^{65} + 13 q^{67} + 22 q^{69} + 47 q^{71} + 28 q^{73} + 48 q^{75} + 19 q^{77} + 12 q^{79} + 24 q^{81} + 17 q^{83} + 31 q^{85} + 44 q^{87} + 41 q^{89} - q^{91} + 15 q^{93} + 58 q^{95} + 51 q^{97} + 9 q^{99}+O(q^{100}) $ 24 * q + 24 * q^3 + 18 * q^5 + 9 * q^7 + 24 * q^9 + 9 * q^11 + 7 * q^13 + 18 * q^15 + 12 * q^17 - q^19 + 9 * q^21 + 22 * q^23 + 48 * q^25 + 24 * q^27 + 44 * q^29 + 15 * q^31 + 9 * q^33 + 3 * q^35 + 18 * q^37 + 7 * q^39 + 37 * q^41 - 8 * q^43 + 18 * q^45 + 24 * q^47 + 63 * q^49 + 12 * q^51 + 59 * q^53 + 20 * q^55 - q^57 + 44 * q^59 + 33 * q^61 + 9 * q^63 + 15 * q^65 + 13 * q^67 + 22 * q^69 + 47 * q^71 + 28 * q^73 + 48 * q^75 + 19 * q^77 + 12 * q^79 + 24 * q^81 + 17 * q^83 + 31 * q^85 + 44 * q^87 + 41 * q^89 - q^91 + 15 * q^93 + 58 * q^95 + 51 * q^97 + 9 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	1.97211	0.881952	0.440976	−	0.897519i	$-0.354632\pi$
$5$	1.97211	0.881952	0.440976	+	0.897519i	$0.354632\pi$
$6$	0	0
$7$	2.26181	0.854884	0.427442	−	0.904043i	$-0.359415\pi$
$7$	2.26181	0.854884	0.427442	+	0.904043i	$0.359415\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.35709	0.710690	0.355345	−	0.934735i	$-0.384363\pi$
$11$	2.35709	0.710690	0.355345	+	0.934735i	$0.384363\pi$
$12$	0	0
$13$	−0.135924	−0.0376986	−0.0188493	−	0.999822i	$-0.506000\pi$
$13$	−0.135924	−0.0376986	−0.0188493	+	0.999822i	$0.506000\pi$
$14$	0	0
$15$	1.97211	0.509196
$16$	0	0
$17$	2.74177	0.664976	0.332488	−	0.943107i	$-0.392112\pi$
$17$	2.74177	0.664976	0.332488	+	0.943107i	$0.392112\pi$
$18$	0	0
$19$	2.78972	0.640005	0.320003	−	0.947417i	$-0.396316\pi$
$19$	2.78972	0.640005	0.320003	+	0.947417i	$0.396316\pi$
$20$	0	0
$21$	2.26181	0.493567
$22$	0	0
$23$	−6.15147	−1.28267	−0.641336	−	0.767261i	$-0.721619\pi$
$23$	−6.15147	−1.28267	−0.641336	+	0.767261i	$0.721619\pi$
$24$	0	0
$25$	−1.11080	−0.222160
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−0.635716	−0.118050	−0.0590248	−	0.998257i	$-0.518799\pi$
$29$	−0.635716	−0.118050	−0.0590248	+	0.998257i	$0.518799\pi$
$30$	0	0
$31$	2.25306	0.404661	0.202330	−	0.979317i	$-0.435148\pi$
$31$	2.25306	0.404661	0.202330	+	0.979317i	$0.435148\pi$
$32$	0	0
$33$	2.35709	0.410317
$34$	0	0
$35$	4.46053	0.753967
$36$	0	0
$37$	6.70530	1.10234	0.551172	−	0.834391i	$-0.314181\pi$
$37$	6.70530	1.10234	0.551172	+	0.834391i	$0.314181\pi$
$38$	0	0
$39$	−0.135924	−0.0217653
$40$	0	0
$41$	8.99231	1.40436	0.702182	−	0.711998i	$-0.252209\pi$
$41$	8.99231	1.40436	0.702182	+	0.711998i	$0.252209\pi$
$42$	0	0
$43$	−3.90784	−0.595940	−0.297970	−	0.954575i	$-0.596310\pi$
$43$	−3.90784	−0.595940	−0.297970	+	0.954575i	$0.596310\pi$
$44$	0	0
$45$	1.97211	0.293984
$46$	0	0
$47$	−3.82047	−0.557273	−0.278636	−	0.960397i	$-0.589882\pi$
$47$	−3.82047	−0.557273	−0.278636	+	0.960397i	$0.589882\pi$
$48$	0	0
$49$	−1.88422	−0.269174
$50$	0	0
$51$	2.74177	0.383924
$52$	0	0
$53$	12.4594	1.71143	0.855714	−	0.517449i	$-0.173118\pi$
$53$	12.4594	1.71143	0.855714	+	0.517449i	$0.173118\pi$
$54$	0	0
$55$	4.64843	0.626795
$56$	0	0
$57$	2.78972	0.369507
$58$	0	0
$59$	0.652656	0.0849686	0.0424843	−	0.999097i	$-0.486473\pi$
$59$	0.652656	0.0849686	0.0424843	+	0.999097i	$0.486473\pi$
$60$	0	0
$61$	−6.69775	−0.857559	−0.428780	−	0.903409i	$-0.641056\pi$
$61$	−6.69775	−0.857559	−0.428780	+	0.903409i	$0.641056\pi$
$62$	0	0
$63$	2.26181	0.284961
$64$	0	0
$65$	−0.268057	−0.0332483
$66$	0	0
$67$	2.94969	0.360362	0.180181	−	0.983633i	$-0.442332\pi$
$67$	2.94969	0.360362	0.180181	+	0.983633i	$0.442332\pi$
$68$	0	0
$69$	−6.15147	−0.740551
$70$	0	0
$71$	−7.16191	−0.849963	−0.424981	−	0.905202i	$-0.639719\pi$
$71$	−7.16191	−0.849963	−0.424981	+	0.905202i	$0.639719\pi$
$72$	0	0
$73$	4.18285	0.489565	0.244783	−	0.969578i	$-0.421283\pi$
$73$	4.18285	0.489565	0.244783	+	0.969578i	$0.421283\pi$
$74$	0	0
$75$	−1.11080	−0.128264
$76$	0	0
$77$	5.33129	0.607557
$78$	0	0
$79$	12.9329	1.45506	0.727530	−	0.686076i	$-0.240668\pi$
$79$	12.9329	1.45506	0.727530	+	0.686076i	$0.240668\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	14.9066	1.63622	0.818108	−	0.575065i	$-0.195023\pi$
$83$	14.9066	1.63622	0.818108	+	0.575065i	$0.195023\pi$
$84$	0	0
$85$	5.40706	0.586477
$86$	0	0
$87$	−0.635716	−0.0681559
$88$	0	0
$89$	−15.2344	−1.61484	−0.807422	−	0.589974i	$-0.799138\pi$
$89$	−15.2344	−1.61484	−0.807422	+	0.589974i	$0.799138\pi$
$90$	0	0
$91$	−0.307434	−0.0322279
$92$	0	0
$93$	2.25306	0.233631
$94$	0	0
$95$	5.50162	0.564454
$96$	0	0
$97$	−11.9613	−1.21448	−0.607242	−	0.794517i	$-0.707724\pi$
$97$	−11.9613	−1.21448	−0.607242	+	0.794517i	$0.707724\pi$
$98$	0	0
$99$	2.35709	0.236897
$100$	0	0
$101$	6.57789	0.654524	0.327262	−	0.944934i	$-0.393874\pi$
$101$	6.57789	0.654524	0.327262	+	0.944934i	$0.393874\pi$
$102$	0	0
$103$	6.41855	0.632439	0.316219	−	0.948686i	$-0.397586\pi$
$103$	6.41855	0.632439	0.316219	+	0.948686i	$0.397586\pi$
$104$	0	0
$105$	4.46053	0.435303
$106$	0	0
$107$	−14.4147	−1.39352	−0.696759	−	0.717305i	$-0.745375\pi$
$107$	−14.4147	−1.39352	−0.696759	+	0.717305i	$0.745375\pi$
$108$	0	0
$109$	0.895179	0.0857426	0.0428713	−	0.999081i	$-0.486349\pi$
$109$	0.895179	0.0857426	0.0428713	+	0.999081i	$0.486349\pi$
$110$	0	0
$111$	6.70530	0.636439
$112$	0	0
$113$	12.4200	1.16838	0.584189	−	0.811618i	$-0.301413\pi$
$113$	12.4200	1.16838	0.584189	+	0.811618i	$0.301413\pi$
$114$	0	0
$115$	−12.1314	−1.13125
$116$	0	0
$117$	−0.135924	−0.0125662
$118$	0	0
$119$	6.20136	0.568477
$120$	0	0
$121$	−5.44412	−0.494920
$122$	0	0
$123$	8.99231	0.810810
$124$	0	0
$125$	−12.0511	−1.07789
$126$	0	0
$127$	−17.7387	−1.57406	−0.787028	−	0.616917i	$-0.788381\pi$
$127$	−17.7387	−1.57406	−0.787028	+	0.616917i	$0.788381\pi$
$128$	0	0
$129$	−3.90784	−0.344066
$130$	0	0
$131$	−5.34555	−0.467043	−0.233521	−	0.972352i	$-0.575025\pi$
$131$	−5.34555	−0.467043	−0.233521	+	0.972352i	$0.575025\pi$
$132$	0	0
$133$	6.30981	0.547130
$134$	0	0
$135$	1.97211	0.169732
$136$	0	0
$137$	8.08419	0.690679	0.345340	−	0.938478i	$-0.387764\pi$
$137$	8.08419	0.690679	0.345340	+	0.938478i	$0.387764\pi$
$138$	0	0
$139$	4.59643	0.389864	0.194932	−	0.980817i	$-0.437551\pi$
$139$	4.59643	0.389864	0.194932	+	0.980817i	$0.437551\pi$
$140$	0	0
$141$	−3.82047	−0.321742
$142$	0	0
$143$	−0.320386	−0.0267920
$144$	0	0
$145$	−1.25370	−0.104114
$146$	0	0
$147$	−1.88422	−0.155408
$148$	0	0
$149$	−16.0490	−1.31478	−0.657392	−	0.753549i	$-0.728341\pi$
$149$	−16.0490	−1.31478	−0.657392	+	0.753549i	$0.728341\pi$
$150$	0	0
$151$	21.5173	1.75105	0.875527	−	0.483169i	$-0.160514\pi$
$151$	21.5173	1.75105	0.875527	+	0.483169i	$0.160514\pi$
$152$	0	0
$153$	2.74177	0.221659
$154$	0	0
$155$	4.44327	0.356892
$156$	0	0
$157$	−6.04399	−0.482363	−0.241181	−	0.970480i	$-0.577535\pi$
$157$	−6.04399	−0.482363	−0.241181	+	0.970480i	$0.577535\pi$
$158$	0	0
$159$	12.4594	0.988094
$160$	0	0
$161$	−13.9135	−1.09653
$162$	0	0
$163$	−5.64752	−0.442348	−0.221174	−	0.975234i	$-0.570989\pi$
$163$	−5.64752	−0.442348	−0.221174	+	0.975234i	$0.570989\pi$
$164$	0	0
$165$	4.64843	0.361880
$166$	0	0
$167$	−15.6277	−1.20931	−0.604654	−	0.796488i	$-0.706689\pi$
$167$	−15.6277	−1.20931	−0.604654	+	0.796488i	$0.706689\pi$
$168$	0	0
$169$	−12.9815	−0.998579
$170$	0	0
$171$	2.78972	0.213335
$172$	0	0
$173$	2.98247	0.226753	0.113376	−	0.993552i	$-0.463833\pi$
$173$	2.98247	0.226753	0.113376	+	0.993552i	$0.463833\pi$
$174$	0	0
$175$	−2.51242	−0.189921
$176$	0	0
$177$	0.652656	0.0490566
$178$	0	0
$179$	19.5554	1.46164	0.730818	−	0.682572i	$-0.239139\pi$
$179$	19.5554	1.46164	0.730818	+	0.682572i	$0.239139\pi$
$180$	0	0
$181$	10.5388	0.783346	0.391673	−	0.920104i	$-0.371896\pi$
$181$	10.5388	0.783346	0.391673	+	0.920104i	$0.371896\pi$
$182$	0	0
$183$	−6.69775	−0.495112
$184$	0	0
$185$	13.2236	0.972216
$186$	0	0
$187$	6.46260	0.472592
$188$	0	0
$189$	2.26181	0.164522
$190$	0	0
$191$	−16.7072	−1.20889	−0.604445	−	0.796647i	$-0.706605\pi$
$191$	−16.7072	−1.20889	−0.604445	+	0.796647i	$0.706605\pi$
$192$	0	0
$193$	−16.6778	−1.20049	−0.600247	−	0.799815i	$-0.704931\pi$
$193$	−16.6778	−1.20049	−0.600247	+	0.799815i	$0.704931\pi$
$194$	0	0
$195$	−0.268057	−0.0191959
$196$	0	0
$197$	22.8157	1.62555	0.812776	−	0.582576i	$-0.197955\pi$
$197$	22.8157	1.62555	0.812776	+	0.582576i	$0.197955\pi$
$198$	0	0
$199$	13.1791	0.934244	0.467122	−	0.884193i	$-0.345291\pi$
$199$	13.1791	0.934244	0.467122	+	0.884193i	$0.345291\pi$
$200$	0	0
$201$	2.94969	0.208055
$202$	0	0
$203$	−1.43787	−0.100919
$204$	0	0
$205$	17.7338	1.23858
$206$	0	0
$207$	−6.15147	−0.427557
$208$	0	0
$209$	6.57562	0.454845
$210$	0	0
$211$	−2.65663	−0.182890	−0.0914449	−	0.995810i	$-0.529149\pi$
$211$	−2.65663	−0.182890	−0.0914449	+	0.995810i	$0.529149\pi$
$212$	0	0
$213$	−7.16191	−0.490726
$214$	0	0
$215$	−7.70667	−0.525591
$216$	0	0
$217$	5.09599	0.345938
$218$	0	0
$219$	4.18285	0.282651
$220$	0	0
$221$	−0.372672	−0.0250687
$222$	0	0
$223$	19.7011	1.31928	0.659642	−	0.751580i	$-0.270708\pi$
$223$	19.7011	1.31928	0.659642	+	0.751580i	$0.270708\pi$
$224$	0	0
$225$	−1.11080	−0.0740533
$226$	0	0
$227$	15.0651	0.999908	0.499954	−	0.866052i	$-0.333350\pi$
$227$	15.0651	0.999908	0.499954	+	0.866052i	$0.333350\pi$
$228$	0	0
$229$	13.0249	0.860710	0.430355	−	0.902660i	$-0.358388\pi$
$229$	13.0249	0.860710	0.430355	+	0.902660i	$0.358388\pi$
$230$	0	0
$231$	5.33129	0.350773
$232$	0	0
$233$	−27.4712	−1.79970	−0.899848	−	0.436204i	$-0.856323\pi$
$233$	−27.4712	−1.79970	−0.899848	+	0.436204i	$0.856323\pi$
$234$	0	0
$235$	−7.53437	−0.491488
$236$	0	0
$237$	12.9329	0.840079
$238$	0	0
$239$	11.4011	0.737479	0.368739	−	0.929533i	$-0.379790\pi$
$239$	11.4011	0.737479	0.368739	+	0.929533i	$0.379790\pi$
$240$	0	0
$241$	−16.5462	−1.06583	−0.532916	−	0.846168i	$-0.678904\pi$
$241$	−16.5462	−1.06583	−0.532916	+	0.846168i	$0.678904\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−3.71588	−0.237399
$246$	0	0
$247$	−0.379190	−0.0241273
$248$	0	0
$249$	14.9066	0.944670
$250$	0	0
$251$	−12.2948	−0.776040	−0.388020	−	0.921651i	$-0.626841\pi$
$251$	−12.2948	−0.776040	−0.388020	+	0.921651i	$0.626841\pi$
$252$	0	0
$253$	−14.4996	−0.911581
$254$	0	0
$255$	5.40706	0.338603
$256$	0	0
$257$	24.1347	1.50548	0.752742	−	0.658315i	$-0.228731\pi$
$257$	24.1347	1.50548	0.752742	+	0.658315i	$0.228731\pi$
$258$	0	0
$259$	15.1661	0.942377
$260$	0	0
$261$	−0.635716	−0.0393498
$262$	0	0
$263$	−18.7802	−1.15804	−0.579018	−	0.815315i	$-0.696564\pi$
$263$	−18.7802	−1.15804	−0.579018	+	0.815315i	$0.696564\pi$
$264$	0	0
$265$	24.5712	1.50940
$266$	0	0
$267$	−15.2344	−0.932331
$268$	0	0
$269$	3.46521	0.211277	0.105639	−	0.994405i	$-0.466311\pi$
$269$	3.46521	0.211277	0.105639	+	0.994405i	$0.466311\pi$
$270$	0	0
$271$	−28.9756	−1.76014	−0.880071	−	0.474841i	$-0.842505\pi$
$271$	−28.9756	−1.76014	−0.880071	+	0.474841i	$0.842505\pi$
$272$	0	0
$273$	−0.307434	−0.0186068
$274$	0	0
$275$	−2.61826	−0.157887
$276$	0	0
$277$	4.29373	0.257985	0.128993	−	0.991646i	$-0.458826\pi$
$277$	4.29373	0.257985	0.128993	+	0.991646i	$0.458826\pi$
$278$	0	0
$279$	2.25306	0.134887
$280$	0	0
$281$	6.85543	0.408961	0.204480	−	0.978871i	$-0.434450\pi$
$281$	6.85543	0.408961	0.204480	+	0.978871i	$0.434450\pi$
$282$	0	0
$283$	−12.9716	−0.771082	−0.385541	−	0.922691i	$-0.625985\pi$
$283$	−12.9716	−0.771082	−0.385541	+	0.922691i	$0.625985\pi$
$284$	0	0
$285$	5.50162	0.325888
$286$	0	0
$287$	20.3389	1.20057
$288$	0	0
$289$	−9.48271	−0.557807
$290$	0	0
$291$	−11.9613	−0.701183
$292$	0	0
$293$	19.0626	1.11365	0.556825	−	0.830630i	$-0.312019\pi$
$293$	19.0626	1.11365	0.556825	+	0.830630i	$0.312019\pi$
$294$	0	0
$295$	1.28711	0.0749383
$296$	0	0
$297$	2.35709	0.136772
$298$	0	0
$299$	0.836134	0.0483549
$300$	0	0
$301$	−8.83879	−0.509459
$302$	0	0
$303$	6.57789	0.377890
$304$	0	0
$305$	−13.2087	−0.756327
$306$	0	0
$307$	−20.3283	−1.16020	−0.580099	−	0.814546i	$-0.696986\pi$
$307$	−20.3283	−1.16020	−0.580099	+	0.814546i	$0.696986\pi$
$308$	0	0
$309$	6.41855	0.365139
$310$	0	0
$311$	−3.07074	−0.174126	−0.0870629	−	0.996203i	$-0.527748\pi$
$311$	−3.07074	−0.174126	−0.0870629	+	0.996203i	$0.527748\pi$
$312$	0	0
$313$	16.3493	0.924114	0.462057	−	0.886850i	$-0.347111\pi$
$313$	16.3493	0.924114	0.462057	+	0.886850i	$0.347111\pi$
$314$	0	0
$315$	4.46053	0.251322
$316$	0	0
$317$	6.67789	0.375068	0.187534	−	0.982258i	$-0.439951\pi$
$317$	6.67789	0.375068	0.187534	+	0.982258i	$0.439951\pi$
$318$	0	0
$319$	−1.49844	−0.0838966
$320$	0	0
$321$	−14.4147	−0.804548
$322$	0	0
$323$	7.64876	0.425588
$324$	0	0
$325$	0.150984	0.00837511
$326$	0	0
$327$	0.895179	0.0495035
$328$	0	0
$329$	−8.64118	−0.476403
$330$	0	0
$331$	−17.0109	−0.935005	−0.467502	−	0.883992i	$-0.654846\pi$
$331$	−17.0109	−0.935005	−0.467502	+	0.883992i	$0.654846\pi$
$332$	0	0
$333$	6.70530	0.367448
$334$	0	0
$335$	5.81711	0.317823
$336$	0	0
$337$	22.6445	1.23353	0.616763	−	0.787149i	$-0.288444\pi$
$337$	22.6445	1.23353	0.616763	+	0.787149i	$0.288444\pi$
$338$	0	0
$339$	12.4200	0.674563
$340$	0	0
$341$	5.31066	0.287588
$342$	0	0
$343$	−20.0944	−1.08500
$344$	0	0
$345$	−12.1314	−0.653130
$346$	0	0
$347$	−8.58140	−0.460674	−0.230337	−	0.973111i	$-0.573983\pi$
$347$	−8.58140	−0.460674	−0.230337	+	0.973111i	$0.573983\pi$
$348$	0	0
$349$	0.0604467	0.00323564	0.00161782	−	0.999999i	$-0.499485\pi$
$349$	0.0604467	0.00323564	0.00161782	+	0.999999i	$0.499485\pi$
$350$	0	0
$351$	−0.135924	−0.00725509
$352$	0	0
$353$	5.02319	0.267358	0.133679	−	0.991025i	$-0.457321\pi$
$353$	5.02319	0.267358	0.133679	+	0.991025i	$0.457321\pi$
$354$	0	0
$355$	−14.1240	−0.749627
$356$	0	0
$357$	6.20136	0.328211
$358$	0	0
$359$	28.5827	1.50854	0.754269	−	0.656566i	$-0.227992\pi$
$359$	28.5827	1.50854	0.754269	+	0.656566i	$0.227992\pi$
$360$	0	0
$361$	−11.2175	−0.590393
$362$	0	0
$363$	−5.44412	−0.285742
$364$	0	0
$365$	8.24902	0.431774
$366$	0	0
$367$	−19.5735	−1.02173	−0.510863	−	0.859662i	$-0.670674\pi$
$367$	−19.5735	−1.02173	−0.510863	+	0.859662i	$0.670674\pi$
$368$	0	0
$369$	8.99231	0.468121
$370$	0	0
$371$	28.1808	1.46307
$372$	0	0
$373$	−18.3826	−0.951814	−0.475907	−	0.879496i	$-0.657880\pi$
$373$	−18.3826	−0.951814	−0.475907	+	0.879496i	$0.657880\pi$
$374$	0	0
$375$	−12.0511	−0.622318
$376$	0	0
$377$	0.0864092	0.00445030
$378$	0	0
$379$	−14.0059	−0.719436	−0.359718	−	0.933061i	$-0.617127\pi$
$379$	−14.0059	−0.719436	−0.359718	+	0.933061i	$0.617127\pi$
$380$	0	0
$381$	−17.7387	−0.908782
$382$	0	0
$383$	−33.7161	−1.72281	−0.861406	−	0.507916i	$-0.830416\pi$
$383$	−33.7161	−1.72281	−0.861406	+	0.507916i	$0.830416\pi$
$384$	0	0
$385$	10.5139	0.535837
$386$	0	0
$387$	−3.90784	−0.198647
$388$	0	0
$389$	24.4149	1.23788	0.618942	−	0.785436i	$-0.287561\pi$
$389$	24.4149	1.23788	0.618942	+	0.785436i	$0.287561\pi$
$390$	0	0
$391$	−16.8659	−0.852946
$392$	0	0
$393$	−5.34555	−0.269647
$394$	0	0
$395$	25.5050	1.28329
$396$	0	0
$397$	−9.99782	−0.501776	−0.250888	−	0.968016i	$-0.580723\pi$
$397$	−9.99782	−0.501776	−0.250888	+	0.968016i	$0.580723\pi$
$398$	0	0
$399$	6.30981	0.315886
$400$	0	0
$401$	10.6859	0.533628	0.266814	−	0.963748i	$-0.414029\pi$
$401$	10.6859	0.533628	0.266814	+	0.963748i	$0.414029\pi$
$402$	0	0
$403$	−0.306245	−0.0152551
$404$	0	0
$405$	1.97211	0.0979947
$406$	0	0
$407$	15.8050	0.783425
$408$	0	0
$409$	36.3400	1.79690	0.898448	−	0.439079i	$-0.144695\pi$
$409$	36.3400	1.79690	0.898448	+	0.439079i	$0.144695\pi$
$410$	0	0
$411$	8.08419	0.398764
$412$	0	0
$413$	1.47618	0.0726383
$414$	0	0
$415$	29.3975	1.44306
$416$	0	0
$417$	4.59643	0.225088
$418$	0	0
$419$	−29.7299	−1.45240	−0.726201	−	0.687482i	$-0.758716\pi$
$419$	−29.7299	−1.45240	−0.726201	+	0.687482i	$0.758716\pi$
$420$	0	0
$421$	−20.2790	−0.988338	−0.494169	−	0.869366i	$-0.664528\pi$
$421$	−20.2790	−0.988338	−0.494169	+	0.869366i	$0.664528\pi$
$422$	0	0
$423$	−3.82047	−0.185758
$424$	0	0
$425$	−3.04555	−0.147731
$426$	0	0
$427$	−15.1490	−0.733113
$428$	0	0
$429$	−0.320386	−0.0154684
$430$	0	0
$431$	15.1409	0.729313	0.364657	−	0.931142i	$-0.381186\pi$
$431$	15.1409	0.729313	0.364657	+	0.931142i	$0.381186\pi$
$432$	0	0
$433$	−35.2209	−1.69261	−0.846304	−	0.532700i	$-0.821177\pi$
$433$	−35.2209	−1.69261	−0.846304	+	0.532700i	$0.821177\pi$
$434$	0	0
$435$	−1.25370	−0.0601103
$436$	0	0
$437$	−17.1609	−0.820916
$438$	0	0
$439$	26.9316	1.28538	0.642689	−	0.766128i	$-0.277819\pi$
$439$	26.9316	1.28538	0.642689	+	0.766128i	$0.277819\pi$
$440$	0	0
$441$	−1.88422	−0.0897246
$442$	0	0
$443$	−3.65511	−0.173659	−0.0868296	−	0.996223i	$-0.527674\pi$
$443$	−3.65511	−0.173659	−0.0868296	+	0.996223i	$0.527674\pi$
$444$	0	0
$445$	−30.0439	−1.42422
$446$	0	0
$447$	−16.0490	−0.759091
$448$	0	0
$449$	18.6432	0.879827	0.439914	−	0.898040i	$-0.355009\pi$
$449$	18.6432	0.879827	0.439914	+	0.898040i	$0.355009\pi$
$450$	0	0
$451$	21.1957	0.998067
$452$	0	0
$453$	21.5173	1.01097
$454$	0	0
$455$	−0.606293	−0.0284235
$456$	0	0
$457$	8.88808	0.415767	0.207883	−	0.978154i	$-0.433343\pi$
$457$	8.88808	0.415767	0.207883	+	0.978154i	$0.433343\pi$
$458$	0	0
$459$	2.74177	0.127975
$460$	0	0
$461$	29.4288	1.37063	0.685317	−	0.728245i	$-0.259664\pi$
$461$	29.4288	1.37063	0.685317	+	0.728245i	$0.259664\pi$
$462$	0	0
$463$	11.3261	0.526370	0.263185	−	0.964745i	$-0.415227\pi$
$463$	11.3261	0.526370	0.263185	+	0.964745i	$0.415227\pi$
$464$	0	0
$465$	4.44327	0.206052
$466$	0	0
$467$	−22.8754	−1.05855	−0.529273	−	0.848452i	$-0.677535\pi$
$467$	−22.8754	−1.05855	−0.529273	+	0.848452i	$0.677535\pi$
$468$	0	0
$469$	6.67164	0.308068
$470$	0	0
$471$	−6.04399	−0.278492
$472$	0	0
$473$	−9.21114	−0.423529
$474$	0	0
$475$	−3.09882	−0.142183
$476$	0	0
$477$	12.4594	0.570476
$478$	0	0
$479$	−10.3119	−0.471162	−0.235581	−	0.971855i	$-0.575699\pi$
$479$	−10.3119	−0.471162	−0.235581	+	0.971855i	$0.575699\pi$
$480$	0	0
$481$	−0.911412	−0.0415568
$482$	0	0
$483$	−13.9135	−0.633085
$484$	0	0
$485$	−23.5889	−1.07112
$486$	0	0
$487$	22.1575	1.00405	0.502026	−	0.864853i	$-0.332588\pi$
$487$	22.1575	1.00405	0.502026	+	0.864853i	$0.332588\pi$
$488$	0	0
$489$	−5.64752	−0.255390
$490$	0	0
$491$	27.6414	1.24744	0.623720	−	0.781648i	$-0.285621\pi$
$491$	27.6414	1.24744	0.623720	+	0.781648i	$0.285621\pi$
$492$	0	0
$493$	−1.74299	−0.0785001
$494$	0	0
$495$	4.64843	0.208932
$496$	0	0
$497$	−16.1989	−0.726619
$498$	0	0
$499$	44.4915	1.99172	0.995858	−	0.0909274i	$-0.0289831\pi$
$499$	44.4915	1.99172	0.995858	+	0.0909274i	$0.0289831\pi$
$500$	0	0
$501$	−15.6277	−0.698195
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	12.9723	0.577259
$506$	0	0
$507$	−12.9815	−0.576530
$508$	0	0
$509$	−0.408808	−0.0181201	−0.00906006	−	0.999959i	$-0.502884\pi$
$509$	−0.408808	−0.0181201	−0.00906006	+	0.999959i	$0.502884\pi$
$510$	0	0
$511$	9.46081	0.418522
$512$	0	0
$513$	2.78972	0.123169
$514$	0	0
$515$	12.6581	0.557781
$516$	0	0
$517$	−9.00520	−0.396048
$518$	0	0
$519$	2.98247	0.130916
$520$	0	0
$521$	30.7249	1.34608	0.673041	−	0.739606i	$-0.264988\pi$
$521$	30.7249	1.34608	0.673041	+	0.739606i	$0.264988\pi$
$522$	0	0
$523$	8.83156	0.386177	0.193089	−	0.981181i	$-0.438150\pi$
$523$	8.83156	0.386177	0.193089	+	0.981181i	$0.438150\pi$
$524$	0	0
$525$	−2.51242	−0.109651
$526$	0	0
$527$	6.17736	0.269090
$528$	0	0
$529$	14.8406	0.645245
$530$	0	0
$531$	0.652656	0.0283229
$532$	0	0
$533$	−1.22227	−0.0529425
$534$	0	0
$535$	−28.4272	−1.22902
$536$	0	0
$537$	19.5554	0.843876
$538$	0	0
$539$	−4.44127	−0.191299
$540$	0	0
$541$	−2.09455	−0.0900516	−0.0450258	−	0.998986i	$-0.514337\pi$
$541$	−2.09455	−0.0900516	−0.0450258	+	0.998986i	$0.514337\pi$
$542$	0	0
$543$	10.5388	0.452265
$544$	0	0
$545$	1.76539	0.0756209
$546$	0	0
$547$	14.2353	0.608657	0.304329	−	0.952567i	$-0.401568\pi$
$547$	14.2353	0.608657	0.304329	+	0.952567i	$0.401568\pi$
$548$	0	0
$549$	−6.69775	−0.285853
$550$	0	0
$551$	−1.77347	−0.0755523
$552$	0	0
$553$	29.2517	1.24391
$554$	0	0
$555$	13.2236	0.561309
$556$	0	0
$557$	7.48707	0.317237	0.158619	−	0.987340i	$-0.449296\pi$
$557$	7.48707	0.317237	0.158619	+	0.987340i	$0.449296\pi$
$558$	0	0
$559$	0.531170	0.0224661
$560$	0	0
$561$	6.46260	0.272851
$562$	0	0
$563$	34.2128	1.44190	0.720949	−	0.692988i	$-0.243706\pi$
$563$	34.2128	1.44190	0.720949	+	0.692988i	$0.243706\pi$
$564$	0	0
$565$	24.4936	1.03045
$566$	0	0
$567$	2.26181	0.0949871
$568$	0	0
$569$	36.4969	1.53003	0.765015	−	0.644013i	$-0.222732\pi$
$569$	36.4969	1.53003	0.765015	+	0.644013i	$0.222732\pi$
$570$	0	0
$571$	−40.6264	−1.70016	−0.850081	−	0.526652i	$-0.823447\pi$
$571$	−40.6264	−1.70016	−0.850081	+	0.526652i	$0.823447\pi$
$572$	0	0
$573$	−16.7072	−0.697953
$574$	0	0
$575$	6.83305	0.284958
$576$	0	0
$577$	−27.7189	−1.15395	−0.576977	−	0.816761i	$-0.695768\pi$
$577$	−27.7189	−1.15395	−0.576977	+	0.816761i	$0.695768\pi$
$578$	0	0
$579$	−16.6778	−0.693105
$580$	0	0
$581$	33.7160	1.39877
$582$	0	0
$583$	29.3679	1.21629
$584$	0	0
$585$	−0.268057	−0.0110828
$586$	0	0
$587$	4.09130	0.168866	0.0844330	−	0.996429i	$-0.473092\pi$
$587$	4.09130	0.168866	0.0844330	+	0.996429i	$0.473092\pi$
$588$	0	0
$589$	6.28539	0.258985
$590$	0	0
$591$	22.8157	0.938513
$592$	0	0
$593$	25.9818	1.06695	0.533473	−	0.845817i	$-0.320887\pi$
$593$	25.9818	1.06695	0.533473	+	0.845817i	$0.320887\pi$
$594$	0	0
$595$	12.2297	0.501370
$596$	0	0
$597$	13.1791	0.539386
$598$	0	0
$599$	−21.3515	−0.872400	−0.436200	−	0.899850i	$-0.643676\pi$
$599$	−21.3515	−0.872400	−0.436200	+	0.899850i	$0.643676\pi$
$600$	0	0
$601$	−4.05490	−0.165403	−0.0827014	−	0.996574i	$-0.526355\pi$
$601$	−4.05490	−0.165403	−0.0827014	+	0.996574i	$0.526355\pi$
$602$	0	0
$603$	2.94969	0.120121
$604$	0	0
$605$	−10.7364	−0.436496
$606$	0	0
$607$	−48.2654	−1.95903	−0.979516	−	0.201368i	$-0.935461\pi$
$607$	−48.2654	−1.95903	−0.979516	+	0.201368i	$0.935461\pi$
$608$	0	0
$609$	−1.43787	−0.0582654
$610$	0	0
$611$	0.519294	0.0210084
$612$	0	0
$613$	45.3081	1.82998	0.914989	−	0.403479i	$-0.132199\pi$
$613$	45.3081	1.82998	0.914989	+	0.403479i	$0.132199\pi$
$614$	0	0
$615$	17.7338	0.715096
$616$	0	0
$617$	−43.3287	−1.74435	−0.872174	−	0.489195i	$-0.837291\pi$
$617$	−43.3287	−1.74435	−0.872174	+	0.489195i	$0.837291\pi$
$618$	0	0
$619$	−29.3462	−1.17952	−0.589762	−	0.807577i	$-0.700778\pi$
$619$	−29.3462	−1.17952	−0.589762	+	0.807577i	$0.700778\pi$
$620$	0	0
$621$	−6.15147	−0.246850
$622$	0	0
$623$	−34.4573	−1.38050
$624$	0	0
$625$	−18.2121	−0.728485
$626$	0	0
$627$	6.57562	0.262605
$628$	0	0
$629$	18.3844	0.733033
$630$	0	0
$631$	−22.5155	−0.896329	−0.448165	−	0.893951i	$-0.647922\pi$
$631$	−22.5155	−0.896329	−0.448165	+	0.893951i	$0.647922\pi$
$632$	0	0
$633$	−2.65663	−0.105592
$634$	0	0
$635$	−34.9826	−1.38824
$636$	0	0
$637$	0.256111	0.0101475
$638$	0	0
$639$	−7.16191	−0.283321
$640$	0	0
$641$	31.5511	1.24620	0.623098	−	0.782144i	$-0.285874\pi$
$641$	31.5511	1.24620	0.623098	+	0.782144i	$0.285874\pi$
$642$	0	0
$643$	−42.8468	−1.68971	−0.844856	−	0.534993i	$-0.820314\pi$
$643$	−42.8468	−1.68971	−0.844856	+	0.534993i	$0.820314\pi$
$644$	0	0
$645$	−7.70667	−0.303450
$646$	0	0
$647$	−42.6587	−1.67709	−0.838543	−	0.544835i	$-0.816592\pi$
$647$	−42.6587	−1.67709	−0.838543	+	0.544835i	$0.816592\pi$
$648$	0	0
$649$	1.53837	0.0603863
$650$	0	0
$651$	5.09599	0.199727
$652$	0	0
$653$	2.69408	0.105428	0.0527138	−	0.998610i	$-0.483213\pi$
$653$	2.69408	0.105428	0.0527138	+	0.998610i	$0.483213\pi$
$654$	0	0
$655$	−10.5420	−0.411909
$656$	0	0
$657$	4.18285	0.163188
$658$	0	0
$659$	5.92483	0.230799	0.115399	−	0.993319i	$-0.463185\pi$
$659$	5.92483	0.230799	0.115399	+	0.993319i	$0.463185\pi$
$660$	0	0
$661$	−37.8809	−1.47339	−0.736697	−	0.676222i	$-0.763616\pi$
$661$	−37.8809	−1.47339	−0.736697	+	0.676222i	$0.763616\pi$
$662$	0	0
$663$	−0.372672	−0.0144734
$664$	0	0
$665$	12.4436	0.482543
$666$	0	0
$667$	3.91059	0.151419
$668$	0	0
$669$	19.7011	0.761689
$670$	0	0
$671$	−15.7872	−0.609459
$672$	0	0
$673$	−46.3664	−1.78729	−0.893647	−	0.448771i	$-0.851862\pi$
$673$	−46.3664	−1.78729	−0.893647	+	0.448771i	$0.851862\pi$
$674$	0	0
$675$	−1.11080	−0.0427547
$676$	0	0
$677$	−48.6272	−1.86890	−0.934448	−	0.356100i	$-0.884106\pi$
$677$	−48.6272	−1.86890	−0.934448	+	0.356100i	$0.884106\pi$
$678$	0	0
$679$	−27.0541	−1.03824
$680$	0	0
$681$	15.0651	0.577297
$682$	0	0
$683$	−16.4290	−0.628637	−0.314319	−	0.949318i	$-0.601776\pi$
$683$	−16.4290	−0.628637	−0.314319	+	0.949318i	$0.601776\pi$
$684$	0	0
$685$	15.9429	0.609146
$686$	0	0
$687$	13.0249	0.496931
$688$	0	0
$689$	−1.69353	−0.0645184
$690$	0	0
$691$	2.20585	0.0839144	0.0419572	−	0.999119i	$-0.486641\pi$
$691$	2.20585	0.0839144	0.0419572	+	0.999119i	$0.486641\pi$
$692$	0	0
$693$	5.33129	0.202519
$694$	0	0
$695$	9.06464	0.343841
$696$	0	0
$697$	24.6548	0.933868
$698$	0	0
$699$	−27.4712	−1.03905
$700$	0	0
$701$	−26.7331	−1.00969	−0.504847	−	0.863209i	$-0.668451\pi$
$701$	−26.7331	−1.00969	−0.504847	+	0.863209i	$0.668451\pi$
$702$	0	0
$703$	18.7059	0.705506
$704$	0	0
$705$	−7.53437	−0.283761
$706$	0	0
$707$	14.8779	0.559542
$708$	0	0
$709$	−32.0410	−1.20333	−0.601663	−	0.798750i	$-0.705495\pi$
$709$	−32.0410	−1.20333	−0.601663	+	0.798750i	$0.705495\pi$
$710$	0	0
$711$	12.9329	0.485020
$712$	0	0
$713$	−13.8596	−0.519047
$714$	0	0
$715$	−0.631834	−0.0236293
$716$	0	0
$717$	11.4011	0.425783
$718$	0	0
$719$	−40.8108	−1.52198	−0.760992	−	0.648761i	$-0.775288\pi$
$719$	−40.8108	−1.52198	−0.760992	+	0.648761i	$0.775288\pi$
$720$	0	0
$721$	14.5175	0.540662
$722$	0	0
$723$	−16.5462	−0.615359
$724$	0	0
$725$	0.706153	0.0262259
$726$	0	0
$727$	−17.2819	−0.640949	−0.320474	−	0.947257i	$-0.603842\pi$
$727$	−17.2819	−0.640949	−0.320474	+	0.947257i	$0.603842\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−10.7144	−0.396286
$732$	0	0
$733$	45.7335	1.68921	0.844603	−	0.535392i	$-0.179836\pi$
$733$	45.7335	1.68921	0.844603	+	0.535392i	$0.179836\pi$
$734$	0	0
$735$	−3.71588	−0.137062
$736$	0	0
$737$	6.95270	0.256106
$738$	0	0
$739$	−23.7741	−0.874543	−0.437271	−	0.899330i	$-0.644055\pi$
$739$	−23.7741	−0.874543	−0.437271	+	0.899330i	$0.644055\pi$
$740$	0	0
$741$	−0.379190	−0.0139299
$742$	0	0
$743$	30.7677	1.12876	0.564378	−	0.825517i	$-0.309116\pi$
$743$	30.7677	1.12876	0.564378	+	0.825517i	$0.309116\pi$
$744$	0	0
$745$	−31.6503	−1.15958
$746$	0	0
$747$	14.9066	0.545405
$748$	0	0
$749$	−32.6032	−1.19130
$750$	0	0
$751$	28.3820	1.03567	0.517836	−	0.855480i	$-0.326738\pi$
$751$	28.3820	1.03567	0.517836	+	0.855480i	$0.326738\pi$
$752$	0	0
$753$	−12.2948	−0.448047
$754$	0	0
$755$	42.4344	1.54435
$756$	0	0
$757$	32.9544	1.19775	0.598874	−	0.800844i	$-0.295615\pi$
$757$	32.9544	1.19775	0.598874	+	0.800844i	$0.295615\pi$
$758$	0	0
$759$	−14.4996	−0.526302
$760$	0	0
$761$	−17.3393	−0.628548	−0.314274	−	0.949332i	$-0.601761\pi$
$761$	−17.3393	−0.628548	−0.314274	+	0.949332i	$0.601761\pi$
$762$	0	0
$763$	2.02472	0.0732999
$764$	0	0
$765$	5.40706	0.195492
$766$	0	0
$767$	−0.0887117	−0.00320319
$768$	0	0
$769$	8.40726	0.303174	0.151587	−	0.988444i	$-0.451562\pi$
$769$	8.40726	0.303174	0.151587	+	0.988444i	$0.451562\pi$
$770$	0	0
$771$	24.1347	0.869192
$772$	0	0
$773$	48.3344	1.73847	0.869234	−	0.494401i	$-0.164613\pi$
$773$	48.3344	1.73847	0.869234	+	0.494401i	$0.164613\pi$
$774$	0	0
$775$	−2.50269	−0.0898994
$776$	0	0
$777$	15.1661	0.544081
$778$	0	0
$779$	25.0860	0.898800
$780$	0	0
$781$	−16.8813	−0.604060
$782$	0	0
$783$	−0.635716	−0.0227186
$784$	0	0
$785$	−11.9194	−0.425421
$786$	0	0
$787$	−1.78651	−0.0636822	−0.0318411	−	0.999493i	$-0.510137\pi$
$787$	−1.78651	−0.0636822	−0.0318411	+	0.999493i	$0.510137\pi$
$788$	0	0
$789$	−18.7802	−0.668592
$790$	0	0
$791$	28.0917	0.998827
$792$	0	0
$793$	0.910386	0.0323288
$794$	0	0
$795$	24.5712	0.871452
$796$	0	0
$797$	20.6857	0.732725	0.366363	−	0.930472i	$-0.380603\pi$
$797$	20.6857	0.732725	0.366363	+	0.930472i	$0.380603\pi$
$798$	0	0
$799$	−10.4748	−0.370573
$800$	0	0
$801$	−15.2344	−0.538282
$802$	0	0
$803$	9.85936	0.347929
$804$	0	0
$805$	−27.4388	−0.967091
$806$	0	0
$807$	3.46521	0.121981
$808$	0	0
$809$	1.90278	0.0668982	0.0334491	−	0.999440i	$-0.489351\pi$
$809$	1.90278	0.0668982	0.0334491	+	0.999440i	$0.489351\pi$
$810$	0	0
$811$	45.4644	1.59647	0.798235	−	0.602346i	$-0.205767\pi$
$811$	45.4644	1.59647	0.798235	+	0.602346i	$0.205767\pi$
$812$	0	0
$813$	−28.9756	−1.01622
$814$	0	0
$815$	−11.1375	−0.390130
$816$	0	0
$817$	−10.9018	−0.381405
$818$	0	0
$819$	−0.307434	−0.0107426
$820$	0	0
$821$	30.4415	1.06242	0.531208	−	0.847242i	$-0.321738\pi$
$821$	30.4415	1.06242	0.531208	+	0.847242i	$0.321738\pi$
$822$	0	0
$823$	−6.18586	−0.215626	−0.107813	−	0.994171i	$-0.534385\pi$
$823$	−6.18586	−0.215626	−0.107813	+	0.994171i	$0.534385\pi$
$824$	0	0
$825$	−2.61826	−0.0911559
$826$	0	0
$827$	10.0670	0.350065	0.175033	−	0.984563i	$-0.443997\pi$
$827$	10.0670	0.350065	0.175033	+	0.984563i	$0.443997\pi$
$828$	0	0
$829$	0.568021	0.0197282	0.00986410	−	0.999951i	$-0.496860\pi$
$829$	0.568021	0.0197282	0.00986410	+	0.999951i	$0.496860\pi$
$830$	0	0
$831$	4.29373	0.148948
$832$	0	0
$833$	−5.16609	−0.178994
$834$	0	0
$835$	−30.8195	−1.06655
$836$	0	0
$837$	2.25306	0.0778770
$838$	0	0
$839$	−40.5328	−1.39935	−0.699674	−	0.714462i	$-0.746671\pi$
$839$	−40.5328	−1.39935	−0.699674	+	0.714462i	$0.746671\pi$
$840$	0	0
$841$	−28.5959	−0.986064
$842$	0	0
$843$	6.85543	0.236114
$844$	0	0
$845$	−25.6009	−0.880699
$846$	0	0
$847$	−12.3136	−0.423099
$848$	0	0
$849$	−12.9716	−0.445185
$850$	0	0
$851$	−41.2475	−1.41395
$852$	0	0
$853$	7.64327	0.261700	0.130850	−	0.991402i	$-0.458229\pi$
$853$	7.64327	0.261700	0.130850	+	0.991402i	$0.458229\pi$
$854$	0	0
$855$	5.50162	0.188151
$856$	0	0
$857$	35.3521	1.20761	0.603803	−	0.797133i	$-0.293651\pi$
$857$	35.3521	1.20761	0.603803	+	0.797133i	$0.293651\pi$
$858$	0	0
$859$	−21.0370	−0.717774	−0.358887	−	0.933381i	$-0.616844\pi$
$859$	−21.0370	−0.717774	−0.358887	+	0.933381i	$0.616844\pi$
$860$	0	0
$861$	20.3389	0.693148
$862$	0	0
$863$	19.1286	0.651145	0.325573	−	0.945517i	$-0.394443\pi$
$863$	19.1286	0.651145	0.325573	+	0.945517i	$0.394443\pi$
$864$	0	0
$865$	5.88174	0.199985
$866$	0	0
$867$	−9.48271	−0.322050
$868$	0	0
$869$	30.4839	1.03410
$870$	0	0
$871$	−0.400934	−0.0135851
$872$	0	0
$873$	−11.9613	−0.404828
$874$	0	0
$875$	−27.2574	−0.921468
$876$	0	0
$877$	21.3434	0.720717	0.360358	−	0.932814i	$-0.382654\pi$
$877$	21.3434	0.720717	0.360358	+	0.932814i	$0.382654\pi$
$878$	0	0
$879$	19.0626	0.642966
$880$	0	0
$881$	29.1851	0.983271	0.491635	−	0.870801i	$-0.336399\pi$
$881$	29.1851	0.983271	0.491635	+	0.870801i	$0.336399\pi$
$882$	0	0
$883$	13.8245	0.465230	0.232615	−	0.972569i	$-0.425272\pi$
$883$	13.8245	0.465230	0.232615	+	0.972569i	$0.425272\pi$
$884$	0	0
$885$	1.28711	0.0432656
$886$	0	0
$887$	−3.72753	−0.125158	−0.0625791	−	0.998040i	$-0.519933\pi$
$887$	−3.72753	−0.125158	−0.0625791	+	0.998040i	$0.519933\pi$
$888$	0	0
$889$	−40.1216	−1.34564
$890$	0	0
$891$	2.35709	0.0789655
$892$	0	0
$893$	−10.6580	−0.356658
$894$	0	0
$895$	38.5652	1.28909
$896$	0	0
$897$	0.836134	0.0279177
$898$	0	0
$899$	−1.43230	−0.0477700
$900$	0	0
$901$	34.1607	1.13806
$902$	0	0
$903$	−8.83879	−0.294137
$904$	0	0
$905$	20.7837	0.690874
$906$	0	0
$907$	−54.1355	−1.79754	−0.898770	−	0.438420i	$-0.855538\pi$
$907$	−54.1355	−1.79754	−0.898770	+	0.438420i	$0.855538\pi$
$908$	0	0
$909$	6.57789	0.218175
$910$	0	0
$911$	−36.5982	−1.21255	−0.606276	−	0.795254i	$-0.707337\pi$
$911$	−36.5982	−1.21255	−0.606276	+	0.795254i	$0.707337\pi$
$912$	0	0
$913$	35.1363	1.16284
$914$	0	0
$915$	−13.2087	−0.436665
$916$	0	0
$917$	−12.0906	−0.399267
$918$	0	0
$919$	−26.9748	−0.889815	−0.444908	−	0.895576i	$-0.646764\pi$
$919$	−26.9748	−0.889815	−0.444908	+	0.895576i	$0.646764\pi$
$920$	0	0
$921$	−20.3283	−0.669840
$922$	0	0
$923$	0.973477	0.0320424
$924$	0	0
$925$	−7.44824	−0.244897
$926$	0	0
$927$	6.41855	0.210813
$928$	0	0
$929$	1.97615	0.0648353	0.0324177	−	0.999474i	$-0.489679\pi$
$929$	1.97615	0.0648353	0.0324177	+	0.999474i	$0.489679\pi$
$930$	0	0
$931$	−5.25644	−0.172273
$932$	0	0
$933$	−3.07074	−0.100532
$934$	0	0
$935$	12.7449	0.416804
$936$	0	0
$937$	−9.58943	−0.313273	−0.156637	−	0.987656i	$-0.550065\pi$
$937$	−9.58943	−0.313273	−0.156637	+	0.987656i	$0.550065\pi$
$938$	0	0
$939$	16.3493	0.533538
$940$	0	0
$941$	−20.2814	−0.661153	−0.330577	−	0.943779i	$-0.607243\pi$
$941$	−20.2814	−0.661153	−0.330577	+	0.943779i	$0.607243\pi$
$942$	0	0
$943$	−55.3160	−1.80134
$944$	0	0
$945$	4.46053	0.145101
$946$	0	0
$947$	14.6480	0.475996	0.237998	−	0.971266i	$-0.423509\pi$
$947$	14.6480	0.475996	0.237998	+	0.971266i	$0.423509\pi$
$948$	0	0
$949$	−0.568550	−0.0184559
$950$	0	0
$951$	6.67789	0.216545
$952$	0	0
$953$	51.4003	1.66502	0.832509	−	0.554011i	$-0.186903\pi$
$953$	51.4003	1.66502	0.832509	+	0.554011i	$0.186903\pi$
$954$	0	0
$955$	−32.9484	−1.06618
$956$	0	0
$957$	−1.49844	−0.0484377
$958$	0	0
$959$	18.2849	0.590450
$960$	0	0
$961$	−25.9237	−0.836250
$962$	0	0
$963$	−14.4147	−0.464506
$964$	0	0
$965$	−32.8904	−1.05878
$966$	0	0
$967$	−5.37674	−0.172904	−0.0864522	−	0.996256i	$-0.527553\pi$
$967$	−5.37674	−0.172904	−0.0864522	+	0.996256i	$0.527553\pi$
$968$	0	0
$969$	7.64876	0.245713
$970$	0	0
$971$	−58.0108	−1.86166	−0.930828	−	0.365457i	$-0.880913\pi$
$971$	−58.0108	−1.86166	−0.930828	+	0.365457i	$0.880913\pi$
$972$	0	0
$973$	10.3962	0.333288
$974$	0	0
$975$	0.150984	0.00483537
$976$	0	0
$977$	−15.2133	−0.486716	−0.243358	−	0.969936i	$-0.578249\pi$
$977$	−15.2133	−0.486716	−0.243358	+	0.969936i	$0.578249\pi$
$978$	0	0
$979$	−35.9089	−1.14765
$980$	0	0
$981$	0.895179	0.0285809
$982$	0	0
$983$	30.5998	0.975982	0.487991	−	0.872849i	$-0.337730\pi$
$983$	30.5998	0.975982	0.487991	+	0.872849i	$0.337730\pi$
$984$	0	0
$985$	44.9950	1.43366
$986$	0	0
$987$	−8.64118	−0.275052
$988$	0	0
$989$	24.0390	0.764395
$990$	0	0
$991$	−55.4345	−1.76093	−0.880467	−	0.474107i	$-0.842771\pi$
$991$	−55.4345	−1.76093	−0.880467	+	0.474107i	$0.842771\pi$
$992$	0	0
$993$	−17.0109	−0.539825
$994$	0	0
$995$	25.9906	0.823959
$996$	0	0
$997$	12.7798	0.404739	0.202370	−	0.979309i	$-0.435136\pi$
$997$	12.7798	0.404739	0.202370	+	0.979309i	$0.435136\pi$
$998$	0	0
$999$	6.70530	0.212146

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6036.2.a.h.1.16	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6036.2.a.h.1.16	✓	24	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:	\( N \)	\(=\)	\( 6036 = 2^{2} \cdot 3 \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6036.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.97211 q^{5} +2.26181 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.97211 q^{5} +2.26181 q^{7} +1.00000 q^{9} +2.35709 q^{11} -0.135924 q^{13} +1.97211 q^{15} +2.74177 q^{17} +2.78972 q^{19} +2.26181 q^{21} -6.15147 q^{23} -1.11080 q^{25} +1.00000 q^{27} -0.635716 q^{29} +2.25306 q^{31} +2.35709 q^{33} +4.46053 q^{35} +6.70530 q^{37} -0.135924 q^{39} +8.99231 q^{41} -3.90784 q^{43} +1.97211 q^{45} -3.82047 q^{47} -1.88422 q^{49} +2.74177 q^{51} +12.4594 q^{53} +4.64843 q^{55} +2.78972 q^{57} +0.652656 q^{59} -6.69775 q^{61} +2.26181 q^{63} -0.268057 q^{65} +2.94969 q^{67} -6.15147 q^{69} -7.16191 q^{71} +4.18285 q^{73} -1.11080 q^{75} +5.33129 q^{77} +12.9329 q^{79} +1.00000 q^{81} +14.9066 q^{83} +5.40706 q^{85} -0.635716 q^{87} -15.2344 q^{89} -0.307434 q^{91} +2.25306 q^{93} +5.50162 q^{95} -11.9613 q^{97} +2.35709 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 24 q^{3} + 18 q^{5} + 9 q^{7} + 24 q^{9}+O(q^{10}) \) 24 * q + 24 * q^3 + 18 * q^5 + 9 * q^7 + 24 * q^9	\( 24 q + 24 q^{3} + 18 q^{5} + 9 q^{7} + 24 q^{9} + 9 q^{11} + 7 q^{13} + 18 q^{15} + 12 q^{17} - q^{19} + 9 q^{21} + 22 q^{23} + 48 q^{25} + 24 q^{27} + 44 q^{29} + 15 q^{31} + 9 q^{33} + 3 q^{35} + 18 q^{37} + 7 q^{39} + 37 q^{41} - 8 q^{43} + 18 q^{45} + 24 q^{47} + 63 q^{49} + 12 q^{51} + 59 q^{53} + 20 q^{55} - q^{57} + 44 q^{59} + 33 q^{61} + 9 q^{63} + 15 q^{65} + 13 q^{67} + 22 q^{69} + 47 q^{71} + 28 q^{73} + 48 q^{75} + 19 q^{77} + 12 q^{79} + 24 q^{81} + 17 q^{83} + 31 q^{85} + 44 q^{87} + 41 q^{89} - q^{91} + 15 q^{93} + 58 q^{95} + 51 q^{97} + 9 q^{99}+O(q^{100}) \) 24 * q + 24 * q^3 + 18 * q^5 + 9 * q^7 + 24 * q^9 + 9 * q^11 + 7 * q^13 + 18 * q^15 + 12 * q^17 - q^19 + 9 * q^21 + 22 * q^23 + 48 * q^25 + 24 * q^27 + 44 * q^29 + 15 * q^31 + 9 * q^33 + 3 * q^35 + 18 * q^37 + 7 * q^39 + 37 * q^41 - 8 * q^43 + 18 * q^45 + 24 * q^47 + 63 * q^49 + 12 * q^51 + 59 * q^53 + 20 * q^55 - q^57 + 44 * q^59 + 33 * q^61 + 9 * q^63 + 15 * q^65 + 13 * q^67 + 22 * q^69 + 47 * q^71 + 28 * q^73 + 48 * q^75 + 19 * q^77 + 12 * q^79 + 24 * q^81 + 17 * q^83 + 31 * q^85 + 44 * q^87 + 41 * q^89 - q^91 + 15 * q^93 + 58 * q^95 + 51 * q^97 + 9 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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