LMFDB - Embedded newform 6027.2.a.m.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6027,2,Mod(1,6027)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6027, 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("6027.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6027 = 3 \cdot 7^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6027.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:	$48.1258372982$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 123)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	6027.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{2} -1.00000 q^{3} -0.585786 q^{5} -1.41421 q^{6} -2.82843 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.41421 q^{2} -1.00000 q^{3} -0.585786 q^{5} -1.41421 q^{6} -2.82843 q^{8} +1.00000 q^{9} -0.828427 q^{10} -0.414214 q^{11} +2.24264 q^{13} +0.585786 q^{15} -4.00000 q^{16} -2.41421 q^{17} +1.41421 q^{18} +2.58579 q^{19} -0.585786 q^{22} +1.41421 q^{23} +2.82843 q^{24} -4.65685 q^{25} +3.17157 q^{26} -1.00000 q^{27} +8.07107 q^{29} +0.828427 q^{30} +3.00000 q^{31} +0.414214 q^{33} -3.41421 q^{34} +7.48528 q^{37} +3.65685 q^{38} -2.24264 q^{39} +1.65685 q^{40} +1.00000 q^{41} -5.00000 q^{43} -0.585786 q^{45} +2.00000 q^{46} -7.58579 q^{47} +4.00000 q^{48} -6.58579 q^{50} +2.41421 q^{51} +1.17157 q^{53} -1.41421 q^{54} +0.242641 q^{55} -2.58579 q^{57} +11.4142 q^{58} -8.48528 q^{59} -6.65685 q^{61} +4.24264 q^{62} +8.00000 q^{64} -1.31371 q^{65} +0.585786 q^{66} -6.48528 q^{67} -1.41421 q^{69} -4.07107 q^{71} -2.82843 q^{72} -12.3137 q^{73} +10.5858 q^{74} +4.65685 q^{75} -3.17157 q^{78} +3.65685 q^{79} +2.34315 q^{80} +1.00000 q^{81} +1.41421 q^{82} +13.0711 q^{83} +1.41421 q^{85} -7.07107 q^{86} -8.07107 q^{87} +1.17157 q^{88} +0.343146 q^{89} -0.828427 q^{90} -3.00000 q^{93} -10.7279 q^{94} -1.51472 q^{95} -16.2426 q^{97} -0.414214 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 4 q^{5} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - 4 * q^5 + 2 * q^9	$ 2 q - 2 q^{3} - 4 q^{5} + 2 q^{9} + 4 q^{10} + 2 q^{11} - 4 q^{13} + 4 q^{15} - 8 q^{16} - 2 q^{17} + 8 q^{19} - 4 q^{22} + 2 q^{25} + 12 q^{26} - 2 q^{27} + 2 q^{29} - 4 q^{30} + 6 q^{31} - 2 q^{33} - 4 q^{34} - 2 q^{37} - 4 q^{38} + 4 q^{39} - 8 q^{40} + 2 q^{41} - 10 q^{43} - 4 q^{45} + 4 q^{46} - 18 q^{47} + 8 q^{48} - 16 q^{50} + 2 q^{51} + 8 q^{53} - 8 q^{55} - 8 q^{57} + 20 q^{58} - 2 q^{61} + 16 q^{64} + 20 q^{65} + 4 q^{66} + 4 q^{67} + 6 q^{71} - 2 q^{73} + 24 q^{74} - 2 q^{75} - 12 q^{78} - 4 q^{79} + 16 q^{80} + 2 q^{81} + 12 q^{83} - 2 q^{87} + 8 q^{88} + 12 q^{89} + 4 q^{90} - 6 q^{93} + 4 q^{94} - 20 q^{95} - 24 q^{97} + 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 4 * q^5 + 2 * q^9 + 4 * q^10 + 2 * q^11 - 4 * q^13 + 4 * q^15 - 8 * q^16 - 2 * q^17 + 8 * q^19 - 4 * q^22 + 2 * q^25 + 12 * q^26 - 2 * q^27 + 2 * q^29 - 4 * q^30 + 6 * q^31 - 2 * q^33 - 4 * q^34 - 2 * q^37 - 4 * q^38 + 4 * q^39 - 8 * q^40 + 2 * q^41 - 10 * q^43 - 4 * q^45 + 4 * q^46 - 18 * q^47 + 8 * q^48 - 16 * q^50 + 2 * q^51 + 8 * q^53 - 8 * q^55 - 8 * q^57 + 20 * q^58 - 2 * q^61 + 16 * q^64 + 20 * q^65 + 4 * q^66 + 4 * q^67 + 6 * q^71 - 2 * q^73 + 24 * q^74 - 2 * q^75 - 12 * q^78 - 4 * q^79 + 16 * q^80 + 2 * q^81 + 12 * q^83 - 2 * q^87 + 8 * q^88 + 12 * q^89 + 4 * q^90 - 6 * q^93 + 4 * q^94 - 20 * q^95 - 24 * q^97 + 2 * 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 $

$2$	1.41421	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$2$	1.41421	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−0.585786	−0.261972	−0.130986	−	0.991384i	$-0.541814\pi$
$5$	−0.585786	−0.261972	−0.130986	+	0.991384i	$0.541814\pi$
$6$	−1.41421	−0.577350
$7$	0	0
$7$	0	0
$8$	−2.82843	−1.00000
$9$	1.00000	0.333333
$10$	−0.828427	−0.261972
$11$	−0.414214	−0.124890	−0.0624450	−	0.998048i	$-0.519890\pi$
$11$	−0.414214	−0.124890	−0.0624450	+	0.998048i	$0.519890\pi$
$12$	0	0
$13$	2.24264	0.621997	0.310998	−	0.950410i	$-0.399337\pi$
$13$	2.24264	0.621997	0.310998	+	0.950410i	$0.399337\pi$
$14$	0	0
$15$	0.585786	0.151249
$16$	−4.00000	−1.00000
$17$	−2.41421	−0.585533	−0.292766	−	0.956184i	$-0.594576\pi$
$17$	−2.41421	−0.585533	−0.292766	+	0.956184i	$0.594576\pi$
$18$	1.41421	0.333333
$19$	2.58579	0.593220	0.296610	−	0.954999i	$-0.404144\pi$
$19$	2.58579	0.593220	0.296610	+	0.954999i	$0.404144\pi$
$20$	0	0
$21$	0	0
$22$	−0.585786	−0.124890
$23$	1.41421	0.294884	0.147442	−	0.989071i	$-0.452896\pi$
$23$	1.41421	0.294884	0.147442	+	0.989071i	$0.452896\pi$
$24$	2.82843	0.577350
$25$	−4.65685	−0.931371
$26$	3.17157	0.621997
$27$	−1.00000	−0.192450
$28$	0	0
$29$	8.07107	1.49876	0.749380	−	0.662140i	$-0.230352\pi$
$29$	8.07107	1.49876	0.749380	+	0.662140i	$0.230352\pi$
$30$	0.828427	0.151249
$31$	3.00000	0.538816	0.269408	−	0.963026i	$-0.413172\pi$
$31$	3.00000	0.538816	0.269408	+	0.963026i	$0.413172\pi$
$32$	0	0
$33$	0.414214	0.0721053
$34$	−3.41421	−0.585533
$35$	0	0
$36$	0	0
$37$	7.48528	1.23057	0.615286	−	0.788304i	$-0.289040\pi$
$37$	7.48528	1.23057	0.615286	+	0.788304i	$0.289040\pi$
$38$	3.65685	0.593220
$39$	−2.24264	−0.359110
$40$	1.65685	0.261972
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	−5.00000	−0.762493	−0.381246	−	0.924473i	$-0.624505\pi$
$43$	−5.00000	−0.762493	−0.381246	+	0.924473i	$0.624505\pi$
$44$	0	0
$45$	−0.585786	−0.0873239
$46$	2.00000	0.294884
$47$	−7.58579	−1.10650	−0.553250	−	0.833015i	$-0.686613\pi$
$47$	−7.58579	−1.10650	−0.553250	+	0.833015i	$0.686613\pi$
$48$	4.00000	0.577350
$49$	0	0
$50$	−6.58579	−0.931371
$51$	2.41421	0.338058
$52$	0	0
$53$	1.17157	0.160928	0.0804640	−	0.996758i	$-0.474360\pi$
$53$	1.17157	0.160928	0.0804640	+	0.996758i	$0.474360\pi$
$54$	−1.41421	−0.192450
$55$	0.242641	0.0327177
$56$	0	0
$57$	−2.58579	−0.342496
$58$	11.4142	1.49876
$59$	−8.48528	−1.10469	−0.552345	−	0.833616i	$-0.686267\pi$
$59$	−8.48528	−1.10469	−0.552345	+	0.833616i	$0.686267\pi$
$60$	0	0
$61$	−6.65685	−0.852323	−0.426161	−	0.904647i	$-0.640134\pi$
$61$	−6.65685	−0.852323	−0.426161	+	0.904647i	$0.640134\pi$
$62$	4.24264	0.538816
$63$	0	0
$64$	8.00000	1.00000
$65$	−1.31371	−0.162945
$66$	0.585786	0.0721053
$67$	−6.48528	−0.792303	−0.396152	−	0.918185i	$-0.629655\pi$
$67$	−6.48528	−0.792303	−0.396152	+	0.918185i	$0.629655\pi$
$68$	0	0
$69$	−1.41421	−0.170251
$70$	0	0
$71$	−4.07107	−0.483147	−0.241573	−	0.970383i	$-0.577663\pi$
$71$	−4.07107	−0.483147	−0.241573	+	0.970383i	$0.577663\pi$
$72$	−2.82843	−0.333333
$73$	−12.3137	−1.44121	−0.720605	−	0.693346i	$-0.756136\pi$
$73$	−12.3137	−1.44121	−0.720605	+	0.693346i	$0.756136\pi$
$74$	10.5858	1.23057
$75$	4.65685	0.537727
$76$	0	0
$77$	0	0
$78$	−3.17157	−0.359110
$79$	3.65685	0.411428	0.205714	−	0.978612i	$-0.434048\pi$
$79$	3.65685	0.411428	0.205714	+	0.978612i	$0.434048\pi$
$80$	2.34315	0.261972
$81$	1.00000	0.111111
$82$	1.41421	0.156174
$83$	13.0711	1.43474	0.717368	−	0.696694i	$-0.245347\pi$
$83$	13.0711	1.43474	0.717368	+	0.696694i	$0.245347\pi$
$84$	0	0
$85$	1.41421	0.153393
$86$	−7.07107	−0.762493
$87$	−8.07107	−0.865309
$88$	1.17157	0.124890
$89$	0.343146	0.0363734	0.0181867	−	0.999835i	$-0.494211\pi$
$89$	0.343146	0.0363734	0.0181867	+	0.999835i	$0.494211\pi$
$90$	−0.828427	−0.0873239
$91$	0	0
$92$	0	0
$93$	−3.00000	−0.311086
$94$	−10.7279	−1.10650
$95$	−1.51472	−0.155407
$96$	0	0
$97$	−16.2426	−1.64919	−0.824595	−	0.565723i	$-0.808597\pi$
$97$	−16.2426	−1.64919	−0.824595	+	0.565723i	$0.808597\pi$
$98$	0	0
$99$	−0.414214	−0.0416300
$100$	0	0
$101$	−2.41421	−0.240223	−0.120112	−	0.992760i	$-0.538325\pi$
$101$	−2.41421	−0.240223	−0.120112	+	0.992760i	$0.538325\pi$
$102$	3.41421	0.338058
$103$	16.3137	1.60744	0.803719	−	0.595009i	$-0.202852\pi$
$103$	16.3137	1.60744	0.803719	+	0.595009i	$0.202852\pi$
$104$	−6.34315	−0.621997
$105$	0	0
$106$	1.65685	0.160928
$107$	−11.6569	−1.12691	−0.563455	−	0.826147i	$-0.690528\pi$
$107$	−11.6569	−1.12691	−0.563455	+	0.826147i	$0.690528\pi$
$108$	0	0
$109$	−11.8995	−1.13976	−0.569882	−	0.821726i	$-0.693011\pi$
$109$	−11.8995	−1.13976	−0.569882	+	0.821726i	$0.693011\pi$
$110$	0.343146	0.0327177
$111$	−7.48528	−0.710471
$112$	0	0
$113$	−18.7279	−1.76177	−0.880887	−	0.473326i	$-0.843053\pi$
$113$	−18.7279	−1.76177	−0.880887	+	0.473326i	$0.843053\pi$
$114$	−3.65685	−0.342496
$115$	−0.828427	−0.0772512
$116$	0	0
$117$	2.24264	0.207332
$118$	−12.0000	−1.10469
$119$	0	0
$120$	−1.65685	−0.151249
$121$	−10.8284	−0.984402
$122$	−9.41421	−0.852323
$123$	−1.00000	−0.0901670
$124$	0	0
$125$	5.65685	0.505964
$126$	0	0
$127$	−13.3137	−1.18140	−0.590700	−	0.806891i	$-0.701148\pi$
$127$	−13.3137	−1.18140	−0.590700	+	0.806891i	$0.701148\pi$
$128$	11.3137	1.00000
$129$	5.00000	0.440225
$130$	−1.85786	−0.162945
$131$	13.4142	1.17201	0.586003	−	0.810309i	$-0.300701\pi$
$131$	13.4142	1.17201	0.586003	+	0.810309i	$0.300701\pi$
$132$	0	0
$133$	0	0
$134$	−9.17157	−0.792303
$135$	0.585786	0.0504165
$136$	6.82843	0.585533
$137$	18.0711	1.54392	0.771958	−	0.635674i	$-0.219278\pi$
$137$	18.0711	1.54392	0.771958	+	0.635674i	$0.219278\pi$
$138$	−2.00000	−0.170251
$139$	7.65685	0.649446	0.324723	−	0.945809i	$-0.394729\pi$
$139$	7.65685	0.649446	0.324723	+	0.945809i	$0.394729\pi$
$140$	0	0
$141$	7.58579	0.638838
$142$	−5.75736	−0.483147
$143$	−0.928932	−0.0776812
$144$	−4.00000	−0.333333
$145$	−4.72792	−0.392633
$146$	−17.4142	−1.44121
$147$	0	0
$148$	0	0
$149$	−5.31371	−0.435316	−0.217658	−	0.976025i	$-0.569842\pi$
$149$	−5.31371	−0.435316	−0.217658	+	0.976025i	$0.569842\pi$
$150$	6.58579	0.537727
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	−7.31371	−0.593220
$153$	−2.41421	−0.195178
$154$	0	0
$155$	−1.75736	−0.141154
$156$	0	0
$157$	−18.0000	−1.43656	−0.718278	−	0.695756i	$-0.755069\pi$
$157$	−18.0000	−1.43656	−0.718278	+	0.695756i	$0.755069\pi$
$158$	5.17157	0.411428
$159$	−1.17157	−0.0929118
$160$	0	0
$161$	0	0
$162$	1.41421	0.111111
$163$	7.48528	0.586292	0.293146	−	0.956068i	$-0.405298\pi$
$163$	7.48528	0.586292	0.293146	+	0.956068i	$0.405298\pi$
$164$	0	0
$165$	−0.242641	−0.0188896
$166$	18.4853	1.43474
$167$	−7.65685	−0.592505	−0.296253	−	0.955110i	$-0.595737\pi$
$167$	−7.65685	−0.592505	−0.296253	+	0.955110i	$0.595737\pi$
$168$	0	0
$169$	−7.97056	−0.613120
$170$	2.00000	0.153393
$171$	2.58579	0.197740
$172$	0	0
$173$	−5.07107	−0.385546	−0.192773	−	0.981243i	$-0.561748\pi$
$173$	−5.07107	−0.385546	−0.192773	+	0.981243i	$0.561748\pi$
$174$	−11.4142	−0.865309
$175$	0	0
$176$	1.65685	0.124890
$177$	8.48528	0.637793
$178$	0.485281	0.0363734
$179$	14.8995	1.11364	0.556820	−	0.830633i	$-0.312021\pi$
$179$	14.8995	1.11364	0.556820	+	0.830633i	$0.312021\pi$
$180$	0	0
$181$	−15.3137	−1.13826	−0.569129	−	0.822248i	$-0.692720\pi$
$181$	−15.3137	−1.13826	−0.569129	+	0.822248i	$0.692720\pi$
$182$	0	0
$183$	6.65685	0.492089
$184$	−4.00000	−0.294884
$185$	−4.38478	−0.322375
$186$	−4.24264	−0.311086
$187$	1.00000	0.0731272
$188$	0	0
$189$	0	0
$190$	−2.14214	−0.155407
$191$	13.1716	0.953062	0.476531	−	0.879158i	$-0.341894\pi$
$191$	13.1716	0.953062	0.476531	+	0.879158i	$0.341894\pi$
$192$	−8.00000	−0.577350
$193$	3.41421	0.245760	0.122880	−	0.992422i	$-0.460787\pi$
$193$	3.41421	0.245760	0.122880	+	0.992422i	$0.460787\pi$
$194$	−22.9706	−1.64919
$195$	1.31371	0.0940766
$196$	0	0
$197$	13.7574	0.980171	0.490086	−	0.871674i	$-0.336965\pi$
$197$	13.7574	0.980171	0.490086	+	0.871674i	$0.336965\pi$
$198$	−0.585786	−0.0416300
$199$	7.75736	0.549905	0.274952	−	0.961458i	$-0.411338\pi$
$199$	7.75736	0.549905	0.274952	+	0.961458i	$0.411338\pi$
$200$	13.1716	0.931371
$201$	6.48528	0.457436
$202$	−3.41421	−0.240223
$203$	0	0
$204$	0	0
$205$	−0.585786	−0.0409131
$206$	23.0711	1.60744
$207$	1.41421	0.0982946
$208$	−8.97056	−0.621997
$209$	−1.07107	−0.0740873
$210$	0	0
$211$	−7.51472	−0.517335	−0.258667	−	0.965966i	$-0.583283\pi$
$211$	−7.51472	−0.517335	−0.258667	+	0.965966i	$0.583283\pi$
$212$	0	0
$213$	4.07107	0.278945
$214$	−16.4853	−1.12691
$215$	2.92893	0.199752
$216$	2.82843	0.192450
$217$	0	0
$218$	−16.8284	−1.13976
$219$	12.3137	0.832083
$220$	0	0
$221$	−5.41421	−0.364199
$222$	−10.5858	−0.710471
$223$	−21.6569	−1.45025	−0.725125	−	0.688617i	$-0.758218\pi$
$223$	−21.6569	−1.45025	−0.725125	+	0.688617i	$0.758218\pi$
$224$	0	0
$225$	−4.65685	−0.310457
$226$	−26.4853	−1.76177
$227$	−20.8995	−1.38715	−0.693574	−	0.720385i	$-0.743965\pi$
$227$	−20.8995	−1.38715	−0.693574	+	0.720385i	$0.743965\pi$
$228$	0	0
$229$	−8.24264	−0.544689	−0.272345	−	0.962200i	$-0.587799\pi$
$229$	−8.24264	−0.544689	−0.272345	+	0.962200i	$0.587799\pi$
$230$	−1.17157	−0.0772512
$231$	0	0
$232$	−22.8284	−1.49876
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	3.17157	0.207332
$235$	4.44365	0.289872
$236$	0	0
$237$	−3.65685	−0.237538
$238$	0	0
$239$	−10.1421	−0.656040	−0.328020	−	0.944671i	$-0.606381\pi$
$239$	−10.1421	−0.656040	−0.328020	+	0.944671i	$0.606381\pi$
$240$	−2.34315	−0.151249
$241$	7.48528	0.482169	0.241085	−	0.970504i	$-0.422497\pi$
$241$	7.48528	0.482169	0.241085	+	0.970504i	$0.422497\pi$
$242$	−15.3137	−0.984402
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	−1.41421	−0.0901670
$247$	5.79899	0.368981
$248$	−8.48528	−0.538816
$249$	−13.0711	−0.828345
$250$	8.00000	0.505964
$251$	−3.17157	−0.200188	−0.100094	−	0.994978i	$-0.531914\pi$
$251$	−3.17157	−0.200188	−0.100094	+	0.994978i	$0.531914\pi$
$252$	0	0
$253$	−0.585786	−0.0368281
$254$	−18.8284	−1.18140
$255$	−1.41421	−0.0885615
$256$	0	0
$257$	24.8995	1.55319	0.776594	−	0.630002i	$-0.216946\pi$
$257$	24.8995	1.55319	0.776594	+	0.630002i	$0.216946\pi$
$258$	7.07107	0.440225
$259$	0	0
$260$	0	0
$261$	8.07107	0.499587
$262$	18.9706	1.17201
$263$	−0.414214	−0.0255415	−0.0127708	−	0.999918i	$-0.504065\pi$
$263$	−0.414214	−0.0255415	−0.0127708	+	0.999918i	$0.504065\pi$
$264$	−1.17157	−0.0721053
$265$	−0.686292	−0.0421586
$266$	0	0
$267$	−0.343146	−0.0210002
$268$	0	0
$269$	−8.48528	−0.517357	−0.258678	−	0.965964i	$-0.583287\pi$
$269$	−8.48528	−0.517357	−0.258678	+	0.965964i	$0.583287\pi$
$270$	0.828427	0.0504165
$271$	20.7990	1.26345	0.631724	−	0.775193i	$-0.282347\pi$
$271$	20.7990	1.26345	0.631724	+	0.775193i	$0.282347\pi$
$272$	9.65685	0.585533
$273$	0	0
$274$	25.5563	1.54392
$275$	1.92893	0.116319
$276$	0	0
$277$	29.2843	1.75952	0.879761	−	0.475417i	$-0.157703\pi$
$277$	29.2843	1.75952	0.879761	+	0.475417i	$0.157703\pi$
$278$	10.8284	0.649446
$279$	3.00000	0.179605
$280$	0	0
$281$	−19.3848	−1.15640	−0.578199	−	0.815895i	$-0.696244\pi$
$281$	−19.3848	−1.15640	−0.578199	+	0.815895i	$0.696244\pi$
$282$	10.7279	0.638838
$283$	−21.8284	−1.29757	−0.648783	−	0.760974i	$-0.724722\pi$
$283$	−21.8284	−1.29757	−0.648783	+	0.760974i	$0.724722\pi$
$284$	0	0
$285$	1.51472	0.0897242
$286$	−1.31371	−0.0776812
$287$	0	0
$288$	0	0
$289$	−11.1716	−0.657151
$290$	−6.68629	−0.392633
$291$	16.2426	0.952160
$292$	0	0
$293$	−12.4142	−0.725246	−0.362623	−	0.931936i	$-0.618119\pi$
$293$	−12.4142	−0.725246	−0.362623	+	0.931936i	$0.618119\pi$
$294$	0	0
$295$	4.97056	0.289397
$296$	−21.1716	−1.23057
$297$	0.414214	0.0240351
$298$	−7.51472	−0.435316
$299$	3.17157	0.183417
$300$	0	0
$301$	0	0
$302$	−22.6274	−1.30206
$303$	2.41421	0.138693
$304$	−10.3431	−0.593220
$305$	3.89949	0.223284
$306$	−3.41421	−0.195178
$307$	−22.3137	−1.27351	−0.636755	−	0.771066i	$-0.719724\pi$
$307$	−22.3137	−1.27351	−0.636755	+	0.771066i	$0.719724\pi$
$308$	0	0
$309$	−16.3137	−0.928054
$310$	−2.48528	−0.141154
$311$	−21.1716	−1.20053	−0.600265	−	0.799801i	$-0.704938\pi$
$311$	−21.1716	−1.20053	−0.600265	+	0.799801i	$0.704938\pi$
$312$	6.34315	0.359110
$313$	−24.8701	−1.40574	−0.702869	−	0.711319i	$-0.748098\pi$
$313$	−24.8701	−1.40574	−0.702869	+	0.711319i	$0.748098\pi$
$314$	−25.4558	−1.43656
$315$	0	0
$316$	0	0
$317$	2.07107	0.116323	0.0581614	−	0.998307i	$-0.481476\pi$
$317$	2.07107	0.116323	0.0581614	+	0.998307i	$0.481476\pi$
$318$	−1.65685	−0.0929118
$319$	−3.34315	−0.187180
$320$	−4.68629	−0.261972
$321$	11.6569	0.650622
$322$	0	0
$323$	−6.24264	−0.347350
$324$	0	0
$325$	−10.4437	−0.579310
$326$	10.5858	0.586292
$327$	11.8995	0.658044
$328$	−2.82843	−0.156174
$329$	0	0
$330$	−0.343146	−0.0188896
$331$	7.75736	0.426383	0.213192	−	0.977010i	$-0.431614\pi$
$331$	7.75736	0.426383	0.213192	+	0.977010i	$0.431614\pi$
$332$	0	0
$333$	7.48528	0.410191
$334$	−10.8284	−0.592505
$335$	3.79899	0.207561
$336$	0	0
$337$	−13.9706	−0.761025	−0.380513	−	0.924776i	$-0.624252\pi$
$337$	−13.9706	−0.761025	−0.380513	+	0.924776i	$0.624252\pi$
$338$	−11.2721	−0.613120
$339$	18.7279	1.01716
$340$	0	0
$341$	−1.24264	−0.0672928
$342$	3.65685	0.197740
$343$	0	0
$344$	14.1421	0.762493
$345$	0.828427	0.0446010
$346$	−7.17157	−0.385546
$347$	−18.8995	−1.01458	−0.507289	−	0.861776i	$-0.669352\pi$
$347$	−18.8995	−1.01458	−0.507289	+	0.861776i	$0.669352\pi$
$348$	0	0
$349$	19.1421	1.02466	0.512328	−	0.858790i	$-0.328783\pi$
$349$	19.1421	1.02466	0.512328	+	0.858790i	$0.328783\pi$
$350$	0	0
$351$	−2.24264	−0.119703
$352$	0	0
$353$	1.02944	0.0547914	0.0273957	−	0.999625i	$-0.491279\pi$
$353$	1.02944	0.0547914	0.0273957	+	0.999625i	$0.491279\pi$
$354$	12.0000	0.637793
$355$	2.38478	0.126571
$356$	0	0
$357$	0	0
$358$	21.0711	1.11364
$359$	7.75736	0.409418	0.204709	−	0.978823i	$-0.434375\pi$
$359$	7.75736	0.409418	0.204709	+	0.978823i	$0.434375\pi$
$360$	1.65685	0.0873239
$361$	−12.3137	−0.648090
$362$	−21.6569	−1.13826
$363$	10.8284	0.568345
$364$	0	0
$365$	7.21320	0.377556
$366$	9.41421	0.492089
$367$	−4.65685	−0.243086	−0.121543	−	0.992586i	$-0.538784\pi$
$367$	−4.65685	−0.243086	−0.121543	+	0.992586i	$0.538784\pi$
$368$	−5.65685	−0.294884
$369$	1.00000	0.0520579
$370$	−6.20101	−0.322375
$371$	0	0
$372$	0	0
$373$	4.85786	0.251531	0.125765	−	0.992060i	$-0.459861\pi$
$373$	4.85786	0.251531	0.125765	+	0.992060i	$0.459861\pi$
$374$	1.41421	0.0731272
$375$	−5.65685	−0.292119
$376$	21.4558	1.10650
$377$	18.1005	0.932223
$378$	0	0
$379$	−6.97056	−0.358054	−0.179027	−	0.983844i	$-0.557295\pi$
$379$	−6.97056	−0.358054	−0.179027	+	0.983844i	$0.557295\pi$
$380$	0	0
$381$	13.3137	0.682082
$382$	18.6274	0.953062
$383$	−36.8995	−1.88548	−0.942738	−	0.333534i	$-0.891759\pi$
$383$	−36.8995	−1.88548	−0.942738	+	0.333534i	$0.891759\pi$
$384$	−11.3137	−0.577350
$385$	0	0
$386$	4.82843	0.245760
$387$	−5.00000	−0.254164
$388$	0	0
$389$	−13.7574	−0.697526	−0.348763	−	0.937211i	$-0.613398\pi$
$389$	−13.7574	−0.697526	−0.348763	+	0.937211i	$0.613398\pi$
$390$	1.85786	0.0940766
$391$	−3.41421	−0.172664
$392$	0	0
$393$	−13.4142	−0.676658
$394$	19.4558	0.980171
$395$	−2.14214	−0.107783
$396$	0	0
$397$	34.5269	1.73286	0.866428	−	0.499302i	$-0.166410\pi$
$397$	34.5269	1.73286	0.866428	+	0.499302i	$0.166410\pi$
$398$	10.9706	0.549905
$399$	0	0
$400$	18.6274	0.931371
$401$	−34.6274	−1.72921	−0.864605	−	0.502452i	$-0.832432\pi$
$401$	−34.6274	−1.72921	−0.864605	+	0.502452i	$0.832432\pi$
$402$	9.17157	0.457436
$403$	6.72792	0.335142
$404$	0	0
$405$	−0.585786	−0.0291080
$406$	0	0
$407$	−3.10051	−0.153686
$408$	−6.82843	−0.338058
$409$	−21.8284	−1.07935	−0.539673	−	0.841875i	$-0.681452\pi$
$409$	−21.8284	−1.07935	−0.539673	+	0.841875i	$0.681452\pi$
$410$	−0.828427	−0.0409131
$411$	−18.0711	−0.891380
$412$	0	0
$413$	0	0
$414$	2.00000	0.0982946
$415$	−7.65685	−0.375860
$416$	0	0
$417$	−7.65685	−0.374958
$418$	−1.51472	−0.0740873
$419$	−0.928932	−0.0453813	−0.0226907	−	0.999743i	$-0.507223\pi$
$419$	−0.928932	−0.0453813	−0.0226907	+	0.999743i	$0.507223\pi$
$420$	0	0
$421$	24.9706	1.21699	0.608495	−	0.793557i	$-0.291773\pi$
$421$	24.9706	1.21699	0.608495	+	0.793557i	$0.291773\pi$
$422$	−10.6274	−0.517335
$423$	−7.58579	−0.368834
$424$	−3.31371	−0.160928
$425$	11.2426	0.545348
$426$	5.75736	0.278945
$427$	0	0
$428$	0	0
$429$	0.928932	0.0448493
$430$	4.14214	0.199752
$431$	39.0711	1.88199	0.940994	−	0.338424i	$-0.109894\pi$
$431$	39.0711	1.88199	0.940994	+	0.338424i	$0.109894\pi$
$432$	4.00000	0.192450
$433$	11.4853	0.551947	0.275974	−	0.961165i	$-0.411000\pi$
$433$	11.4853	0.551947	0.275974	+	0.961165i	$0.411000\pi$
$434$	0	0
$435$	4.72792	0.226687
$436$	0	0
$437$	3.65685	0.174931
$438$	17.4142	0.832083
$439$	30.9706	1.47814	0.739072	−	0.673626i	$-0.235264\pi$
$439$	30.9706	1.47814	0.739072	+	0.673626i	$0.235264\pi$
$440$	−0.686292	−0.0327177
$441$	0	0
$442$	−7.65685	−0.364199
$443$	20.2843	0.963735	0.481867	−	0.876244i	$-0.339959\pi$
$443$	20.2843	0.963735	0.481867	+	0.876244i	$0.339959\pi$
$444$	0	0
$445$	−0.201010	−0.00952879
$446$	−30.6274	−1.45025
$447$	5.31371	0.251330
$448$	0	0
$449$	20.4853	0.966760	0.483380	−	0.875411i	$-0.339409\pi$
$449$	20.4853	0.966760	0.483380	+	0.875411i	$0.339409\pi$
$450$	−6.58579	−0.310457
$451$	−0.414214	−0.0195046
$452$	0	0
$453$	16.0000	0.751746
$454$	−29.5563	−1.38715
$455$	0	0
$456$	7.31371	0.342496
$457$	−4.00000	−0.187112	−0.0935561	−	0.995614i	$-0.529823\pi$
$457$	−4.00000	−0.187112	−0.0935561	+	0.995614i	$0.529823\pi$
$458$	−11.6569	−0.544689
$459$	2.41421	0.112686
$460$	0	0
$461$	16.6274	0.774416	0.387208	−	0.921992i	$-0.373440\pi$
$461$	16.6274	0.774416	0.387208	+	0.921992i	$0.373440\pi$
$462$	0	0
$463$	5.55635	0.258225	0.129113	−	0.991630i	$-0.458787\pi$
$463$	5.55635	0.258225	0.129113	+	0.991630i	$0.458787\pi$
$464$	−32.2843	−1.49876
$465$	1.75736	0.0814956
$466$	−25.4558	−1.17922
$467$	−21.3137	−0.986281	−0.493140	−	0.869950i	$-0.664151\pi$
$467$	−21.3137	−0.986281	−0.493140	+	0.869950i	$0.664151\pi$
$468$	0	0
$469$	0	0
$470$	6.28427	0.289872
$471$	18.0000	0.829396
$472$	24.0000	1.10469
$473$	2.07107	0.0952278
$474$	−5.17157	−0.237538
$475$	−12.0416	−0.552508
$476$	0	0
$477$	1.17157	0.0536426
$478$	−14.3431	−0.656040
$479$	12.8995	0.589393	0.294696	−	0.955591i	$-0.404781\pi$
$479$	12.8995	0.589393	0.294696	+	0.955591i	$0.404781\pi$
$480$	0	0
$481$	16.7868	0.765412
$482$	10.5858	0.482169
$483$	0	0
$484$	0	0
$485$	9.51472	0.432041
$486$	−1.41421	−0.0641500
$487$	−31.9706	−1.44872	−0.724362	−	0.689420i	$-0.757866\pi$
$487$	−31.9706	−1.44872	−0.724362	+	0.689420i	$0.757866\pi$
$488$	18.8284	0.852323
$489$	−7.48528	−0.338496
$490$	0	0
$491$	1.07107	0.0483366	0.0241683	−	0.999708i	$-0.492306\pi$
$491$	1.07107	0.0483366	0.0241683	+	0.999708i	$0.492306\pi$
$492$	0	0
$493$	−19.4853	−0.877573
$494$	8.20101	0.368981
$495$	0.242641	0.0109059
$496$	−12.0000	−0.538816
$497$	0	0
$498$	−18.4853	−0.828345
$499$	−36.4853	−1.63331	−0.816653	−	0.577129i	$-0.804173\pi$
$499$	−36.4853	−1.63331	−0.816653	+	0.577129i	$0.804173\pi$
$500$	0	0
$501$	7.65685	0.342083
$502$	−4.48528	−0.200188
$503$	35.3848	1.57773	0.788865	−	0.614567i	$-0.210669\pi$
$503$	35.3848	1.57773	0.788865	+	0.614567i	$0.210669\pi$
$504$	0	0
$505$	1.41421	0.0629317
$506$	−0.828427	−0.0368281
$507$	7.97056	0.353985
$508$	0	0
$509$	33.8701	1.50126	0.750632	−	0.660721i	$-0.229749\pi$
$509$	33.8701	1.50126	0.750632	+	0.660721i	$0.229749\pi$
$510$	−2.00000	−0.0885615
$511$	0	0
$512$	−22.6274	−1.00000
$513$	−2.58579	−0.114165
$514$	35.2132	1.55319
$515$	−9.55635	−0.421103
$516$	0	0
$517$	3.14214	0.138191
$518$	0	0
$519$	5.07107	0.222595
$520$	3.71573	0.162945
$521$	−22.7574	−0.997018	−0.498509	−	0.866885i	$-0.666119\pi$
$521$	−22.7574	−0.997018	−0.498509	+	0.866885i	$0.666119\pi$
$522$	11.4142	0.499587
$523$	−34.0000	−1.48672	−0.743358	−	0.668894i	$-0.766768\pi$
$523$	−34.0000	−1.48672	−0.743358	+	0.668894i	$0.766768\pi$
$524$	0	0
$525$	0	0
$526$	−0.585786	−0.0255415
$527$	−7.24264	−0.315494
$528$	−1.65685	−0.0721053
$529$	−21.0000	−0.913043
$530$	−0.970563	−0.0421586
$531$	−8.48528	−0.368230
$532$	0	0
$533$	2.24264	0.0971396
$534$	−0.485281	−0.0210002
$535$	6.82843	0.295219
$536$	18.3431	0.792303
$537$	−14.8995	−0.642961
$538$	−12.0000	−0.517357
$539$	0	0
$540$	0	0
$541$	18.9706	0.815608	0.407804	−	0.913069i	$-0.366295\pi$
$541$	18.9706	0.815608	0.407804	+	0.913069i	$0.366295\pi$
$542$	29.4142	1.26345
$543$	15.3137	0.657174
$544$	0	0
$545$	6.97056	0.298586
$546$	0	0
$547$	−7.51472	−0.321306	−0.160653	−	0.987011i	$-0.551360\pi$
$547$	−7.51472	−0.321306	−0.160653	+	0.987011i	$0.551360\pi$
$548$	0	0
$549$	−6.65685	−0.284108
$550$	2.72792	0.116319
$551$	20.8701	0.889094
$552$	4.00000	0.170251
$553$	0	0
$554$	41.4142	1.75952
$555$	4.38478	0.186123
$556$	0	0
$557$	39.7279	1.68333	0.841663	−	0.540003i	$-0.181577\pi$
$557$	39.7279	1.68333	0.841663	+	0.540003i	$0.181577\pi$
$558$	4.24264	0.179605
$559$	−11.2132	−0.474268
$560$	0	0
$561$	−1.00000	−0.0422200
$562$	−27.4142	−1.15640
$563$	12.8995	0.543649	0.271824	−	0.962347i	$-0.412373\pi$
$563$	12.8995	0.543649	0.271824	+	0.962347i	$0.412373\pi$
$564$	0	0
$565$	10.9706	0.461535
$566$	−30.8701	−1.29757
$567$	0	0
$568$	11.5147	0.483147
$569$	−35.5563	−1.49060	−0.745300	−	0.666729i	$-0.767694\pi$
$569$	−35.5563	−1.49060	−0.745300	+	0.666729i	$0.767694\pi$
$570$	2.14214	0.0897242
$571$	42.6274	1.78390	0.891951	−	0.452132i	$-0.149336\pi$
$571$	42.6274	1.78390	0.891951	+	0.452132i	$0.149336\pi$
$572$	0	0
$573$	−13.1716	−0.550250
$574$	0	0
$575$	−6.58579	−0.274646
$576$	8.00000	0.333333
$577$	19.5147	0.812408	0.406204	−	0.913782i	$-0.366852\pi$
$577$	19.5147	0.812408	0.406204	+	0.913782i	$0.366852\pi$
$578$	−15.7990	−0.657151
$579$	−3.41421	−0.141890
$580$	0	0
$581$	0	0
$582$	22.9706	0.952160
$583$	−0.485281	−0.0200983
$584$	34.8284	1.44121
$585$	−1.31371	−0.0543152
$586$	−17.5563	−0.725246
$587$	−7.38478	−0.304802	−0.152401	−	0.988319i	$-0.548701\pi$
$587$	−7.38478	−0.304802	−0.152401	+	0.988319i	$0.548701\pi$
$588$	0	0
$589$	7.75736	0.319636
$590$	7.02944	0.289397
$591$	−13.7574	−0.565902
$592$	−29.9411	−1.23057
$593$	−4.41421	−0.181270	−0.0906350	−	0.995884i	$-0.528890\pi$
$593$	−4.41421	−0.181270	−0.0906350	+	0.995884i	$0.528890\pi$
$594$	0.585786	0.0240351
$595$	0	0
$596$	0	0
$597$	−7.75736	−0.317488
$598$	4.48528	0.183417
$599$	35.3553	1.44458	0.722290	−	0.691590i	$-0.243090\pi$
$599$	35.3553	1.44458	0.722290	+	0.691590i	$0.243090\pi$
$600$	−13.1716	−0.537727
$601$	9.51472	0.388113	0.194057	−	0.980990i	$-0.437835\pi$
$601$	9.51472	0.388113	0.194057	+	0.980990i	$0.437835\pi$
$602$	0	0
$603$	−6.48528	−0.264101
$604$	0	0
$605$	6.34315	0.257886
$606$	3.41421	0.138693
$607$	17.5147	0.710900	0.355450	−	0.934695i	$-0.384328\pi$
$607$	17.5147	0.710900	0.355450	+	0.934695i	$0.384328\pi$
$608$	0	0
$609$	0	0
$610$	5.51472	0.223284
$611$	−17.0122	−0.688240
$612$	0	0
$613$	−3.31371	−0.133839	−0.0669197	−	0.997758i	$-0.521317\pi$
$613$	−3.31371	−0.133839	−0.0669197	+	0.997758i	$0.521317\pi$
$614$	−31.5563	−1.27351
$615$	0.585786	0.0236212
$616$	0	0
$617$	27.2132	1.09556	0.547781	−	0.836622i	$-0.315473\pi$
$617$	27.2132	1.09556	0.547781	+	0.836622i	$0.315473\pi$
$618$	−23.0711	−0.928054
$619$	−12.3137	−0.494930	−0.247465	−	0.968897i	$-0.579597\pi$
$619$	−12.3137	−0.494930	−0.247465	+	0.968897i	$0.579597\pi$
$620$	0	0
$621$	−1.41421	−0.0567504
$622$	−29.9411	−1.20053
$623$	0	0
$624$	8.97056	0.359110
$625$	19.9706	0.798823
$626$	−35.1716	−1.40574
$627$	1.07107	0.0427743
$628$	0	0
$629$	−18.0711	−0.720541
$630$	0	0
$631$	22.4558	0.893953	0.446977	−	0.894546i	$-0.352501\pi$
$631$	22.4558	0.893953	0.446977	+	0.894546i	$0.352501\pi$
$632$	−10.3431	−0.411428
$633$	7.51472	0.298683
$634$	2.92893	0.116323
$635$	7.79899	0.309493
$636$	0	0
$637$	0	0
$638$	−4.72792	−0.187180
$639$	−4.07107	−0.161049
$640$	−6.62742	−0.261972
$641$	22.5563	0.890922	0.445461	−	0.895301i	$-0.353040\pi$
$641$	22.5563	0.890922	0.445461	+	0.895301i	$0.353040\pi$
$642$	16.4853	0.650622
$643$	13.4558	0.530647	0.265323	−	0.964159i	$-0.414521\pi$
$643$	13.4558	0.530647	0.265323	+	0.964159i	$0.414521\pi$
$644$	0	0
$645$	−2.92893	−0.115327
$646$	−8.82843	−0.347350
$647$	−7.55635	−0.297071	−0.148535	−	0.988907i	$-0.547456\pi$
$647$	−7.55635	−0.297071	−0.148535	+	0.988907i	$0.547456\pi$
$648$	−2.82843	−0.111111
$649$	3.51472	0.137965
$650$	−14.7696	−0.579310
$651$	0	0
$652$	0	0
$653$	12.8995	0.504796	0.252398	−	0.967623i	$-0.418781\pi$
$653$	12.8995	0.504796	0.252398	+	0.967623i	$0.418781\pi$
$654$	16.8284	0.658044
$655$	−7.85786	−0.307032
$656$	−4.00000	−0.156174
$657$	−12.3137	−0.480404
$658$	0	0
$659$	−29.1716	−1.13636	−0.568182	−	0.822903i	$-0.692353\pi$
$659$	−29.1716	−1.13636	−0.568182	+	0.822903i	$0.692353\pi$
$660$	0	0
$661$	37.4558	1.45686	0.728432	−	0.685118i	$-0.240250\pi$
$661$	37.4558	1.45686	0.728432	+	0.685118i	$0.240250\pi$
$662$	10.9706	0.426383
$663$	5.41421	0.210271
$664$	−36.9706	−1.43474
$665$	0	0
$666$	10.5858	0.410191
$667$	11.4142	0.441960
$668$	0	0
$669$	21.6569	0.837302
$670$	5.37258	0.207561
$671$	2.75736	0.106447
$672$	0	0
$673$	−17.7990	−0.686101	−0.343050	−	0.939317i	$-0.611460\pi$
$673$	−17.7990	−0.686101	−0.343050	+	0.939317i	$0.611460\pi$
$674$	−19.7574	−0.761025
$675$	4.65685	0.179242
$676$	0	0
$677$	−2.44365	−0.0939171	−0.0469586	−	0.998897i	$-0.514953\pi$
$677$	−2.44365	−0.0939171	−0.0469586	+	0.998897i	$0.514953\pi$
$678$	26.4853	1.01716
$679$	0	0
$680$	−4.00000	−0.153393
$681$	20.8995	0.800870
$682$	−1.75736	−0.0672928
$683$	50.2843	1.92407	0.962037	−	0.272919i	$-0.0879890\pi$
$683$	50.2843	1.92407	0.962037	+	0.272919i	$0.0879890\pi$
$684$	0	0
$685$	−10.5858	−0.404462
$686$	0	0
$687$	8.24264	0.314476
$688$	20.0000	0.762493
$689$	2.62742	0.100097
$690$	1.17157	0.0446010
$691$	14.7279	0.560277	0.280138	−	0.959960i	$-0.409620\pi$
$691$	14.7279	0.560277	0.280138	+	0.959960i	$0.409620\pi$
$692$	0	0
$693$	0	0
$694$	−26.7279	−1.01458
$695$	−4.48528	−0.170136
$696$	22.8284	0.865309
$697$	−2.41421	−0.0914449
$698$	27.0711	1.02466
$699$	18.0000	0.680823
$700$	0	0
$701$	−6.20101	−0.234209	−0.117104	−	0.993120i	$-0.537361\pi$
$701$	−6.20101	−0.234209	−0.117104	+	0.993120i	$0.537361\pi$
$702$	−3.17157	−0.119703
$703$	19.3553	0.730000
$704$	−3.31371	−0.124890
$705$	−4.44365	−0.167358
$706$	1.45584	0.0547914
$707$	0	0
$708$	0	0
$709$	−13.5147	−0.507556	−0.253778	−	0.967263i	$-0.581673\pi$
$709$	−13.5147	−0.507556	−0.253778	+	0.967263i	$0.581673\pi$
$710$	3.37258	0.126571
$711$	3.65685	0.137143
$712$	−0.970563	−0.0363734
$713$	4.24264	0.158888
$714$	0	0
$715$	0.544156	0.0203503
$716$	0	0
$717$	10.1421	0.378765
$718$	10.9706	0.409418
$719$	−9.44365	−0.352189	−0.176094	−	0.984373i	$-0.556346\pi$
$719$	−9.44365	−0.352189	−0.176094	+	0.984373i	$0.556346\pi$
$720$	2.34315	0.0873239
$721$	0	0
$722$	−17.4142	−0.648090
$723$	−7.48528	−0.278381
$724$	0	0
$725$	−37.5858	−1.39590
$726$	15.3137	0.568345
$727$	7.27208	0.269706	0.134853	−	0.990866i	$-0.456944\pi$
$727$	7.27208	0.269706	0.134853	+	0.990866i	$0.456944\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	10.2010	0.377556
$731$	12.0711	0.446465
$732$	0	0
$733$	2.85786	0.105558	0.0527788	−	0.998606i	$-0.483192\pi$
$733$	2.85786	0.105558	0.0527788	+	0.998606i	$0.483192\pi$
$734$	−6.58579	−0.243086
$735$	0	0
$736$	0	0
$737$	2.68629	0.0989508
$738$	1.41421	0.0520579
$739$	11.3431	0.417264	0.208632	−	0.977994i	$-0.433099\pi$
$739$	11.3431	0.417264	0.208632	+	0.977994i	$0.433099\pi$
$740$	0	0
$741$	−5.79899	−0.213031
$742$	0	0
$743$	−28.6274	−1.05024	−0.525119	−	0.851029i	$-0.675979\pi$
$743$	−28.6274	−1.05024	−0.525119	+	0.851029i	$0.675979\pi$
$744$	8.48528	0.311086
$745$	3.11270	0.114040
$746$	6.87006	0.251531
$747$	13.0711	0.478245
$748$	0	0
$749$	0	0
$750$	−8.00000	−0.292119
$751$	1.02944	0.0375647	0.0187823	−	0.999824i	$-0.494021\pi$
$751$	1.02944	0.0375647	0.0187823	+	0.999824i	$0.494021\pi$
$752$	30.3431	1.10650
$753$	3.17157	0.115579
$754$	25.5980	0.932223
$755$	9.37258	0.341103
$756$	0	0
$757$	−0.242641	−0.00881893	−0.00440946	−	0.999990i	$-0.501404\pi$
$757$	−0.242641	−0.00881893	−0.00440946	+	0.999990i	$0.501404\pi$
$758$	−9.85786	−0.358054
$759$	0.585786	0.0212627
$760$	4.28427	0.155407
$761$	−9.17157	−0.332469	−0.166235	−	0.986086i	$-0.553161\pi$
$761$	−9.17157	−0.332469	−0.166235	+	0.986086i	$0.553161\pi$
$762$	18.8284	0.682082
$763$	0	0
$764$	0	0
$765$	1.41421	0.0511310
$766$	−52.1838	−1.88548
$767$	−19.0294	−0.687113
$768$	0	0
$769$	16.5147	0.595536	0.297768	−	0.954638i	$-0.403758\pi$
$769$	16.5147	0.595536	0.297768	+	0.954638i	$0.403758\pi$
$770$	0	0
$771$	−24.8995	−0.896733
$772$	0	0
$773$	−47.5858	−1.71154	−0.855771	−	0.517355i	$-0.826917\pi$
$773$	−47.5858	−1.71154	−0.855771	+	0.517355i	$0.826917\pi$
$774$	−7.07107	−0.254164
$775$	−13.9706	−0.501837
$776$	45.9411	1.64919
$777$	0	0
$778$	−19.4558	−0.697526
$779$	2.58579	0.0926454
$780$	0	0
$781$	1.68629	0.0603403
$782$	−4.82843	−0.172664
$783$	−8.07107	−0.288436
$784$	0	0
$785$	10.5442	0.376337
$786$	−18.9706	−0.676658
$787$	−19.9706	−0.711874	−0.355937	−	0.934510i	$-0.615838\pi$
$787$	−19.9706	−0.711874	−0.355937	+	0.934510i	$0.615838\pi$
$788$	0	0
$789$	0.414214	0.0147464
$790$	−3.02944	−0.107783
$791$	0	0
$792$	1.17157	0.0416300
$793$	−14.9289	−0.530142
$794$	48.8284	1.73286
$795$	0.686292	0.0243403
$796$	0	0
$797$	15.8579	0.561714	0.280857	−	0.959750i	$-0.409381\pi$
$797$	15.8579	0.561714	0.280857	+	0.959750i	$0.409381\pi$
$798$	0	0
$799$	18.3137	0.647892
$800$	0	0
$801$	0.343146	0.0121245
$802$	−48.9706	−1.72921
$803$	5.10051	0.179993
$804$	0	0
$805$	0	0
$806$	9.51472	0.335142
$807$	8.48528	0.298696
$808$	6.82843	0.240223
$809$	−37.1127	−1.30481	−0.652406	−	0.757869i	$-0.726240\pi$
$809$	−37.1127	−1.30481	−0.652406	+	0.757869i	$0.726240\pi$
$810$	−0.828427	−0.0291080
$811$	−15.9706	−0.560802	−0.280401	−	0.959883i	$-0.590467\pi$
$811$	−15.9706	−0.560802	−0.280401	+	0.959883i	$0.590467\pi$
$812$	0	0
$813$	−20.7990	−0.729452
$814$	−4.38478	−0.153686
$815$	−4.38478	−0.153592
$816$	−9.65685	−0.338058
$817$	−12.9289	−0.452326
$818$	−30.8701	−1.07935
$819$	0	0
$820$	0	0
$821$	14.8701	0.518969	0.259484	−	0.965747i	$-0.416447\pi$
$821$	14.8701	0.518969	0.259484	+	0.965747i	$0.416447\pi$
$822$	−25.5563	−0.891380
$823$	−52.4264	−1.82747	−0.913735	−	0.406311i	$-0.866815\pi$
$823$	−52.4264	−1.82747	−0.913735	+	0.406311i	$0.866815\pi$
$824$	−46.1421	−1.60744
$825$	−1.92893	−0.0671568
$826$	0	0
$827$	22.0711	0.767486	0.383743	−	0.923440i	$-0.374635\pi$
$827$	22.0711	0.767486	0.383743	+	0.923440i	$0.374635\pi$
$828$	0	0
$829$	−42.9411	−1.49141	−0.745703	−	0.666278i	$-0.767886\pi$
$829$	−42.9411	−1.49141	−0.745703	+	0.666278i	$0.767886\pi$
$830$	−10.8284	−0.375860
$831$	−29.2843	−1.01586
$832$	17.9411	0.621997
$833$	0	0
$834$	−10.8284	−0.374958
$835$	4.48528	0.155220
$836$	0	0
$837$	−3.00000	−0.103695
$838$	−1.31371	−0.0453813
$839$	17.1716	0.592829	0.296414	−	0.955059i	$-0.404209\pi$
$839$	17.1716	0.592829	0.296414	+	0.955059i	$0.404209\pi$
$840$	0	0
$841$	36.1421	1.24628
$842$	35.3137	1.21699
$843$	19.3848	0.667647
$844$	0	0
$845$	4.66905	0.160620
$846$	−10.7279	−0.368834
$847$	0	0
$848$	−4.68629	−0.160928
$849$	21.8284	0.749150
$850$	15.8995	0.545348
$851$	10.5858	0.362876
$852$	0	0
$853$	0	0		−	1.00000i	$-0.5\pi$
$853$	0	0			1.00000i	$0.5\pi$
$854$	0	0
$855$	−1.51472	−0.0518023
$856$	32.9706	1.12691
$857$	−12.2010	−0.416779	−0.208389	−	0.978046i	$-0.566822\pi$
$857$	−12.2010	−0.416779	−0.208389	+	0.978046i	$0.566822\pi$
$858$	1.31371	0.0448493
$859$	−5.34315	−0.182306	−0.0911529	−	0.995837i	$-0.529055\pi$
$859$	−5.34315	−0.182306	−0.0911529	+	0.995837i	$0.529055\pi$
$860$	0	0
$861$	0	0
$862$	55.2548	1.88199
$863$	10.9289	0.372025	0.186013	−	0.982547i	$-0.440443\pi$
$863$	10.9289	0.372025	0.186013	+	0.982547i	$0.440443\pi$
$864$	0	0
$865$	2.97056	0.101002
$866$	16.2426	0.551947
$867$	11.1716	0.379407
$868$	0	0
$869$	−1.51472	−0.0513833
$870$	6.68629	0.226687
$871$	−14.5442	−0.492810
$872$	33.6569	1.13976
$873$	−16.2426	−0.549730
$874$	5.17157	0.174931
$875$	0	0
$876$	0	0
$877$	1.14214	0.0385672	0.0192836	−	0.999814i	$-0.493861\pi$
$877$	1.14214	0.0385672	0.0192836	+	0.999814i	$0.493861\pi$
$878$	43.7990	1.47814
$879$	12.4142	0.418721
$880$	−0.970563	−0.0327177
$881$	−2.62742	−0.0885199	−0.0442600	−	0.999020i	$-0.514093\pi$
$881$	−2.62742	−0.0885199	−0.0442600	+	0.999020i	$0.514093\pi$
$882$	0	0
$883$	−6.58579	−0.221629	−0.110815	−	0.993841i	$-0.535346\pi$
$883$	−6.58579	−0.221629	−0.110815	+	0.993841i	$0.535346\pi$
$884$	0	0
$885$	−4.97056	−0.167084
$886$	28.6863	0.963735
$887$	46.0122	1.54494	0.772469	−	0.635052i	$-0.219021\pi$
$887$	46.0122	1.54494	0.772469	+	0.635052i	$0.219021\pi$
$888$	21.1716	0.710471
$889$	0	0
$890$	−0.284271	−0.00952879
$891$	−0.414214	−0.0138767
$892$	0	0
$893$	−19.6152	−0.656398
$894$	7.51472	0.251330
$895$	−8.72792	−0.291742
$896$	0	0
$897$	−3.17157	−0.105896
$898$	28.9706	0.966760
$899$	24.2132	0.807556
$900$	0	0
$901$	−2.82843	−0.0942286
$902$	−0.585786	−0.0195046
$903$	0	0
$904$	52.9706	1.76177
$905$	8.97056	0.298192
$906$	22.6274	0.751746
$907$	4.34315	0.144212	0.0721059	−	0.997397i	$-0.477028\pi$
$907$	4.34315	0.144212	0.0721059	+	0.997397i	$0.477028\pi$
$908$	0	0
$909$	−2.41421	−0.0800744
$910$	0	0
$911$	53.8995	1.78577	0.892885	−	0.450285i	$-0.148678\pi$
$911$	53.8995	1.78577	0.892885	+	0.450285i	$0.148678\pi$
$912$	10.3431	0.342496
$913$	−5.41421	−0.179184
$914$	−5.65685	−0.187112
$915$	−3.89949	−0.128913
$916$	0	0
$917$	0	0
$918$	3.41421	0.112686
$919$	−12.5858	−0.415167	−0.207583	−	0.978217i	$-0.566560\pi$
$919$	−12.5858	−0.415167	−0.207583	+	0.978217i	$0.566560\pi$
$920$	2.34315	0.0772512
$921$	22.3137	0.735262
$922$	23.5147	0.774416
$923$	−9.12994	−0.300516
$924$	0	0
$925$	−34.8579	−1.14612
$926$	7.85786	0.258225
$927$	16.3137	0.535812
$928$	0	0
$929$	−45.8701	−1.50495	−0.752474	−	0.658622i	$-0.771140\pi$
$929$	−45.8701	−1.50495	−0.752474	+	0.658622i	$0.771140\pi$
$930$	2.48528	0.0814956
$931$	0	0
$932$	0	0
$933$	21.1716	0.693126
$934$	−30.1421	−0.986281
$935$	−0.585786	−0.0191573
$936$	−6.34315	−0.207332
$937$	−19.0294	−0.621665	−0.310832	−	0.950465i	$-0.600608\pi$
$937$	−19.0294	−0.621665	−0.310832	+	0.950465i	$0.600608\pi$
$938$	0	0
$939$	24.8701	0.811604
$940$	0	0
$941$	7.37258	0.240339	0.120170	−	0.992753i	$-0.461656\pi$
$941$	7.37258	0.240339	0.120170	+	0.992753i	$0.461656\pi$
$942$	25.4558	0.829396
$943$	1.41421	0.0460531
$944$	33.9411	1.10469
$945$	0	0
$946$	2.92893	0.0952278
$947$	−57.5980	−1.87168	−0.935841	−	0.352421i	$-0.885358\pi$
$947$	−57.5980	−1.87168	−0.935841	+	0.352421i	$0.885358\pi$
$948$	0	0
$949$	−27.6152	−0.896428
$950$	−17.0294	−0.552508
$951$	−2.07107	−0.0671590
$952$	0	0
$953$	17.6985	0.573310	0.286655	−	0.958034i	$-0.407457\pi$
$953$	17.6985	0.573310	0.286655	+	0.958034i	$0.407457\pi$
$954$	1.65685	0.0536426
$955$	−7.71573	−0.249675
$956$	0	0
$957$	3.34315	0.108069
$958$	18.2426	0.589393
$959$	0	0
$960$	4.68629	0.151249
$961$	−22.0000	−0.709677
$962$	23.7401	0.765412
$963$	−11.6569	−0.375637
$964$	0	0
$965$	−2.00000	−0.0643823
$966$	0	0
$967$	−34.9706	−1.12458	−0.562289	−	0.826941i	$-0.690079\pi$
$967$	−34.9706	−1.12458	−0.562289	+	0.826941i	$0.690079\pi$
$968$	30.6274	0.984402
$969$	6.24264	0.200543
$970$	13.4558	0.432041
$971$	−9.44365	−0.303061	−0.151531	−	0.988453i	$-0.548420\pi$
$971$	−9.44365	−0.303061	−0.151531	+	0.988453i	$0.548420\pi$
$972$	0	0
$973$	0	0
$974$	−45.2132	−1.44872
$975$	10.4437	0.334465
$976$	26.6274	0.852323
$977$	10.4142	0.333180	0.166590	−	0.986026i	$-0.446724\pi$
$977$	10.4142	0.333180	0.166590	+	0.986026i	$0.446724\pi$
$978$	−10.5858	−0.338496
$979$	−0.142136	−0.00454267
$980$	0	0
$981$	−11.8995	−0.379922
$982$	1.51472	0.0483366
$983$	37.0711	1.18238	0.591192	−	0.806531i	$-0.298658\pi$
$983$	37.0711	1.18238	0.591192	+	0.806531i	$0.298658\pi$
$984$	2.82843	0.0901670
$985$	−8.05887	−0.256777
$986$	−27.5563	−0.877573
$987$	0	0
$988$	0	0
$989$	−7.07107	−0.224847
$990$	0.343146	0.0109059
$991$	−11.5147	−0.365777	−0.182889	−	0.983134i	$-0.558545\pi$
$991$	−11.5147	−0.365777	−0.182889	+	0.983134i	$0.558545\pi$
$992$	0	0
$993$	−7.75736	−0.246172
$994$	0	0
$995$	−4.54416	−0.144059
$996$	0	0
$997$	−12.5858	−0.398596	−0.199298	−	0.979939i	$-0.563866\pi$
$997$	−12.5858	−0.398596	−0.199298	+	0.979939i	$0.563866\pi$
$998$	−51.5980	−1.63331
$999$	−7.48528	−0.236824

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6027.2.a.m.1.2		2
7.6	odd	2		123.2.a.c.1.2	✓	2
21.20	even	2		369.2.a.c.1.1		2
28.27	even	2		1968.2.a.r.1.1		2
35.34	odd	2		3075.2.a.p.1.1		2
56.13	odd	2		7872.2.a.bk.1.2		2
56.27	even	2		7872.2.a.bo.1.2		2
84.83	odd	2		5904.2.a.w.1.2		2
105.104	even	2		9225.2.a.bm.1.2		2
287.286	odd	2		5043.2.a.e.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
123.2.a.c.1.2	✓	2	7.6	odd	2
369.2.a.c.1.1		2	21.20	even	2
1968.2.a.r.1.1		2	28.27	even	2
3075.2.a.p.1.1		2	35.34	odd	2
5043.2.a.e.1.2		2	287.286	odd	2
5904.2.a.w.1.2		2	84.83	odd	2
6027.2.a.m.1.2		2	1.1	even	1	trivial
7872.2.a.bk.1.2		2	56.13	odd	2
7872.2.a.bo.1.2		2	56.27	even	2
9225.2.a.bm.1.2		2	105.104	even	2

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:	\( N \)	\(=\)	\( 6027 = 3 \cdot 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6027.a (trivial)

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} -1.00000 q^{3} -0.585786 q^{5} -1.41421 q^{6} -2.82843 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.41421 q^{2} -1.00000 q^{3} -0.585786 q^{5} -1.41421 q^{6} -2.82843 q^{8} +1.00000 q^{9} -0.828427 q^{10} -0.414214 q^{11} +2.24264 q^{13} +0.585786 q^{15} -4.00000 q^{16} -2.41421 q^{17} +1.41421 q^{18} +2.58579 q^{19} -0.585786 q^{22} +1.41421 q^{23} +2.82843 q^{24} -4.65685 q^{25} +3.17157 q^{26} -1.00000 q^{27} +8.07107 q^{29} +0.828427 q^{30} +3.00000 q^{31} +0.414214 q^{33} -3.41421 q^{34} +7.48528 q^{37} +3.65685 q^{38} -2.24264 q^{39} +1.65685 q^{40} +1.00000 q^{41} -5.00000 q^{43} -0.585786 q^{45} +2.00000 q^{46} -7.58579 q^{47} +4.00000 q^{48} -6.58579 q^{50} +2.41421 q^{51} +1.17157 q^{53} -1.41421 q^{54} +0.242641 q^{55} -2.58579 q^{57} +11.4142 q^{58} -8.48528 q^{59} -6.65685 q^{61} +4.24264 q^{62} +8.00000 q^{64} -1.31371 q^{65} +0.585786 q^{66} -6.48528 q^{67} -1.41421 q^{69} -4.07107 q^{71} -2.82843 q^{72} -12.3137 q^{73} +10.5858 q^{74} +4.65685 q^{75} -3.17157 q^{78} +3.65685 q^{79} +2.34315 q^{80} +1.00000 q^{81} +1.41421 q^{82} +13.0711 q^{83} +1.41421 q^{85} -7.07107 q^{86} -8.07107 q^{87} +1.17157 q^{88} +0.343146 q^{89} -0.828427 q^{90} -3.00000 q^{93} -10.7279 q^{94} -1.51472 q^{95} -16.2426 q^{97} -0.414214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 4 q^{5} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 4 * q^5 + 2 * q^9	\( 2 q - 2 q^{3} - 4 q^{5} + 2 q^{9} + 4 q^{10} + 2 q^{11} - 4 q^{13} + 4 q^{15} - 8 q^{16} - 2 q^{17} + 8 q^{19} - 4 q^{22} + 2 q^{25} + 12 q^{26} - 2 q^{27} + 2 q^{29} - 4 q^{30} + 6 q^{31} - 2 q^{33} - 4 q^{34} - 2 q^{37} - 4 q^{38} + 4 q^{39} - 8 q^{40} + 2 q^{41} - 10 q^{43} - 4 q^{45} + 4 q^{46} - 18 q^{47} + 8 q^{48} - 16 q^{50} + 2 q^{51} + 8 q^{53} - 8 q^{55} - 8 q^{57} + 20 q^{58} - 2 q^{61} + 16 q^{64} + 20 q^{65} + 4 q^{66} + 4 q^{67} + 6 q^{71} - 2 q^{73} + 24 q^{74} - 2 q^{75} - 12 q^{78} - 4 q^{79} + 16 q^{80} + 2 q^{81} + 12 q^{83} - 2 q^{87} + 8 q^{88} + 12 q^{89} + 4 q^{90} - 6 q^{93} + 4 q^{94} - 20 q^{95} - 24 q^{97} + 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 4 * q^5 + 2 * q^9 + 4 * q^10 + 2 * q^11 - 4 * q^13 + 4 * q^15 - 8 * q^16 - 2 * q^17 + 8 * q^19 - 4 * q^22 + 2 * q^25 + 12 * q^26 - 2 * q^27 + 2 * q^29 - 4 * q^30 + 6 * q^31 - 2 * q^33 - 4 * q^34 - 2 * q^37 - 4 * q^38 + 4 * q^39 - 8 * q^40 + 2 * q^41 - 10 * q^43 - 4 * q^45 + 4 * q^46 - 18 * q^47 + 8 * q^48 - 16 * q^50 + 2 * q^51 + 8 * q^53 - 8 * q^55 - 8 * q^57 + 20 * q^58 - 2 * q^61 + 16 * q^64 + 20 * q^65 + 4 * q^66 + 4 * q^67 + 6 * q^71 - 2 * q^73 + 24 * q^74 - 2 * q^75 - 12 * q^78 - 4 * q^79 + 16 * q^80 + 2 * q^81 + 12 * q^83 - 2 * q^87 + 8 * q^88 + 12 * q^89 + 4 * q^90 - 6 * q^93 + 4 * q^94 - 20 * q^95 - 24 * q^97 + 2 * q^99

Introduction

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