LMFDB - Embedded newform 9065.2.a.bd.1.20

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9065,2,Mod(1,9065)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9065.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9065, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 9065 = 5 \cdot 7^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9065.a (trivial)

Newform invariants

comment:select newform

sage:traces = [38,-2,-6,42,-38,-8,0,-6,48,2,34] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$72.3843894323$
Analytic rank:	$1$
Dimension:	$38$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Character	$\chi$	$=$	9065.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.0412902 q^{2} -1.58988 q^{3} -1.99830 q^{4} -1.00000 q^{5} +0.0656462 q^{6} +0.165090 q^{8} -0.472294 q^{9} +0.0412902 q^{10} +5.50113 q^{11} +3.17704 q^{12} +4.78243 q^{13} +1.58988 q^{15} +3.98977 q^{16} -5.25733 q^{17} +0.0195011 q^{18} +1.40799 q^{19} +1.99830 q^{20} -0.227143 q^{22} -5.30278 q^{23} -0.262473 q^{24} +1.00000 q^{25} -0.197467 q^{26} +5.52052 q^{27} -9.00328 q^{29} -0.0656462 q^{30} -7.92250 q^{31} -0.494919 q^{32} -8.74612 q^{33} +0.217076 q^{34} +0.943782 q^{36} -1.00000 q^{37} -0.0581363 q^{38} -7.60348 q^{39} -0.165090 q^{40} -0.195903 q^{41} +8.39361 q^{43} -10.9929 q^{44} +0.472294 q^{45} +0.218953 q^{46} -6.31043 q^{47} -6.34325 q^{48} -0.0412902 q^{50} +8.35850 q^{51} -9.55672 q^{52} +9.36408 q^{53} -0.227943 q^{54} -5.50113 q^{55} -2.23854 q^{57} +0.371747 q^{58} +9.60075 q^{59} -3.17704 q^{60} -7.49868 q^{61} +0.327121 q^{62} -7.95911 q^{64} -4.78243 q^{65} +0.361129 q^{66} -11.3096 q^{67} +10.5057 q^{68} +8.43076 q^{69} +16.1348 q^{71} -0.0779711 q^{72} +3.13716 q^{73} +0.0412902 q^{74} -1.58988 q^{75} -2.81359 q^{76} +0.313949 q^{78} +13.6088 q^{79} -3.98977 q^{80} -7.36006 q^{81} +0.00808886 q^{82} -5.52584 q^{83} +5.25733 q^{85} -0.346573 q^{86} +14.3141 q^{87} +0.908183 q^{88} -10.8370 q^{89} -0.0195011 q^{90} +10.5965 q^{92} +12.5958 q^{93} +0.260559 q^{94} -1.40799 q^{95} +0.786860 q^{96} +7.35709 q^{97} -2.59815 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 38 q - 2 q^{2} - 6 q^{3} + 42 q^{4} - 38 q^{5} - 8 q^{6} - 6 q^{8} + 48 q^{9} + 2 q^{10} + 34 q^{11} - 20 q^{12} - 22 q^{13} + 6 q^{15} + 46 q^{16} - 22 q^{17} - 36 q^{18} - 40 q^{19} - 42 q^{20} - 4 q^{22}+ \cdots + 92 q^{99}+O(q^{100}) $ 38 * q - 2 * q^2 - 6 * q^3 + 42 * q^4 - 38 * q^5 - 8 * q^6 - 6 * q^8 + 48 * q^9 + 2 * q^10 + 34 * q^11 - 20 * q^12 - 22 * q^13 + 6 * q^15 + 46 * q^16 - 22 * q^17 - 36 * q^18 - 40 * q^19 - 42 * q^20 - 4 * q^22 - 16 * q^23 - 24 * q^24 + 38 * q^25 - 20 * q^26 - 30 * q^27 + 26 * q^29 + 8 * q^30 - 52 * q^31 - 34 * q^32 - 38 * q^33 - 12 * q^34 + 74 * q^36 - 38 * q^37 - 16 * q^38 + 22 * q^39 + 6 * q^40 - 16 * q^41 + 72 * q^44 - 48 * q^45 + 12 * q^46 - 34 * q^47 - 34 * q^48 - 2 * q^50 + 46 * q^51 - 68 * q^52 + 4 * q^53 - 32 * q^54 - 34 * q^55 - 12 * q^57 - 40 * q^58 - 32 * q^59 + 20 * q^60 - 60 * q^61 + 38 * q^62 + 22 * q^64 + 22 * q^65 - 66 * q^66 - 20 * q^67 - 60 * q^68 - 24 * q^69 + 52 * q^71 - 78 * q^72 - 48 * q^73 + 2 * q^74 - 6 * q^75 - 86 * q^76 + 8 * q^78 + 22 * q^79 - 46 * q^80 + 58 * q^81 - 36 * q^82 - 44 * q^83 + 22 * q^85 + 20 * q^86 - 106 * q^87 - 66 * q^88 - 52 * q^89 + 36 * q^90 - 58 * q^92 - 48 * q^93 - 96 * q^94 + 40 * q^95 - 26 * q^96 - 38 * q^97 + 92 * q^99

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 $

$2$	−0.0412902	−0.0291966	−0.0145983	−	0.999893i	$-0.504647\pi$
$2$	−0.0412902	−0.0291966	−0.0145983	+	0.999893i	$0.504647\pi$
$3$	−1.58988	−0.917915	−0.458958	−	0.888458i	$-0.651777\pi$
$3$	−1.58988	−0.917915	−0.458958	+	0.888458i	$0.651777\pi$
$4$	−1.99830	−0.999148
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0.0656462	0.0268000
$7$	0	0
$7$	0	0
$8$	0.165090	0.0583682
$9$	−0.472294	−0.157431
$10$	0.0412902	0.0130571
$11$	5.50113	1.65865	0.829327	−	0.558764i	$-0.188724\pi$
$11$	5.50113	1.65865	0.829327	+	0.558764i	$0.188724\pi$
$12$	3.17704	0.917133
$13$	4.78243	1.32641	0.663204	−	0.748438i	$-0.269196\pi$
$13$	4.78243	1.32641	0.663204	+	0.748438i	$0.269196\pi$
$14$	0	0
$15$	1.58988	0.410504
$16$	3.98977	0.997443
$17$	−5.25733	−1.27509	−0.637545	−	0.770413i	$-0.720050\pi$
$17$	−5.25733	−1.27509	−0.637545	+	0.770413i	$0.720050\pi$
$18$	0.0195011	0.00459645
$19$	1.40799	0.323016	0.161508	−	0.986871i	$-0.448364\pi$
$19$	1.40799	0.323016	0.161508	+	0.986871i	$0.448364\pi$
$20$	1.99830	0.446832
$21$	0	0
$22$	−0.227143	−0.0484270
$23$	−5.30278	−1.10571	−0.552853	−	0.833279i	$-0.686461\pi$
$23$	−5.30278	−1.10571	−0.552853	+	0.833279i	$0.686461\pi$
$24$	−0.262473	−0.0535771
$25$	1.00000	0.200000
$26$	−0.197467	−0.0387266
$27$	5.52052	1.06242
$28$	0	0
$29$	−9.00328	−1.67187	−0.835934	−	0.548831i	$-0.815073\pi$
$29$	−9.00328	−1.67187	−0.835934	+	0.548831i	$0.815073\pi$
$30$	−0.0656462	−0.0119853
$31$	−7.92250	−1.42292	−0.711462	−	0.702725i	$-0.751966\pi$
$31$	−7.92250	−1.42292	−0.711462	+	0.702725i	$0.751966\pi$
$32$	−0.494919	−0.0874901
$33$	−8.74612	−1.52250
$34$	0.217076	0.0372282
$35$	0	0
$36$	0.943782	0.157297
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	−0.0581363	−0.00943095
$39$	−7.60348	−1.21753
$40$	−0.165090	−0.0261031
$41$	−0.195903	−0.0305949	−0.0152974	−	0.999883i	$-0.504870\pi$
$41$	−0.195903	−0.0305949	−0.0152974	+	0.999883i	$0.504870\pi$
$42$	0	0
$43$	8.39361	1.28001	0.640007	−	0.768369i	$-0.278932\pi$
$43$	8.39361	1.28001	0.640007	+	0.768369i	$0.278932\pi$
$44$	−10.9929	−1.65724
$45$	0.472294	0.0704054
$46$	0.218953	0.0322828
$47$	−6.31043	−0.920471	−0.460236	−	0.887797i	$-0.652235\pi$
$47$	−6.31043	−0.920471	−0.460236	+	0.887797i	$0.652235\pi$
$48$	−6.34325	−0.915569
$49$	0	0
$50$	−0.0412902	−0.00583931
$51$	8.35850	1.17042
$52$	−9.55672	−1.32528
$53$	9.36408	1.28625	0.643127	−	0.765759i	$-0.277637\pi$
$53$	9.36408	1.28625	0.643127	+	0.765759i	$0.277637\pi$
$54$	−0.227943	−0.0310191
$55$	−5.50113	−0.741772
$56$	0	0
$57$	−2.23854	−0.296501
$58$	0.371747	0.0488128
$59$	9.60075	1.24991	0.624956	−	0.780660i	$-0.285117\pi$
$59$	9.60075	1.24991	0.624956	+	0.780660i	$0.285117\pi$
$60$	−3.17704	−0.410154
$61$	−7.49868	−0.960108	−0.480054	−	0.877239i	$-0.659383\pi$
$61$	−7.49868	−0.960108	−0.480054	+	0.877239i	$0.659383\pi$
$62$	0.327121	0.0415445
$63$	0	0
$64$	−7.95911	−0.994889
$65$	−4.78243	−0.593188
$66$	0.361129	0.0444519
$67$	−11.3096	−1.38169	−0.690847	−	0.723001i	$-0.742762\pi$
$67$	−11.3096	−1.38169	−0.690847	+	0.723001i	$0.742762\pi$
$68$	10.5057	1.27400
$69$	8.43076	1.01494
$70$	0	0
$71$	16.1348	1.91485	0.957423	−	0.288689i	$-0.0932192\pi$
$71$	16.1348	1.91485	0.957423	+	0.288689i	$0.0932192\pi$
$72$	−0.0779711	−0.00918898
$73$	3.13716	0.367177	0.183588	−	0.983003i	$-0.441229\pi$
$73$	3.13716	0.367177	0.183588	+	0.983003i	$0.441229\pi$
$74$	0.0412902	0.00479988
$75$	−1.58988	−0.183583
$76$	−2.81359	−0.322741
$77$	0	0
$78$	0.313949	0.0355477
$79$	13.6088	1.53111	0.765553	−	0.643373i	$-0.222466\pi$
$79$	13.6088	1.53111	0.765553	+	0.643373i	$0.222466\pi$
$80$	−3.98977	−0.446070
$81$	−7.36006	−0.817784
$82$	0.00808886	0.000893266 0
$83$	−5.52584	−0.606539	−0.303270	−	0.952905i	$-0.598078\pi$
$83$	−5.52584	−0.606539	−0.303270	+	0.952905i	$0.598078\pi$
$84$	0	0
$85$	5.25733	0.570237
$86$	−0.346573	−0.0373720
$87$	14.3141	1.53463
$88$	0.908183	0.0968127
$89$	−10.8370	−1.14871	−0.574357	−	0.818605i	$-0.694748\pi$
$89$	−10.8370	−1.14871	−0.574357	+	0.818605i	$0.694748\pi$
$90$	−0.0195011	−0.00205559
$91$	0	0
$92$	10.5965	1.10476
$93$	12.5958	1.30612
$94$	0.260559	0.0268746
$95$	−1.40799	−0.144457
$96$	0.786860	0.0803085
$97$	7.35709	0.746999	0.373499	−	0.927630i	$-0.378158\pi$
$97$	7.35709	0.746999	0.373499	+	0.927630i	$0.378158\pi$
$98$	0	0
$99$	−2.59815	−0.261124
$100$	−1.99830	−0.199830
$101$	5.43209	0.540513	0.270256	−	0.962788i	$-0.412892\pi$
$101$	5.43209	0.540513	0.270256	+	0.962788i	$0.412892\pi$
$102$	−0.345124	−0.0341724
$103$	−6.39829	−0.630442	−0.315221	−	0.949018i	$-0.602079\pi$
$103$	−6.39829	−0.630442	−0.315221	+	0.949018i	$0.602079\pi$
$104$	0.789533	0.0774201
$105$	0	0
$106$	−0.386644	−0.0375542
$107$	−0.925139	−0.0894366	−0.0447183	−	0.999000i	$-0.514239\pi$
$107$	−0.925139	−0.0894366	−0.0447183	+	0.999000i	$0.514239\pi$
$108$	−11.0316	−1.06152
$109$	2.26627	0.217069	0.108535	−	0.994093i	$-0.465384\pi$
$109$	2.26627	0.217069	0.108535	+	0.994093i	$0.465384\pi$
$110$	0.227143	0.0216572
$111$	1.58988	0.150904
$112$	0	0
$113$	13.4738	1.26751	0.633753	−	0.773535i	$-0.281513\pi$
$113$	13.4738	1.26751	0.633753	+	0.773535i	$0.281513\pi$
$114$	0.0924295	0.00865682
$115$	5.30278	0.494487
$116$	17.9912	1.67044
$117$	−2.25871	−0.208818
$118$	−0.396417	−0.0364931
$119$	0	0
$120$	0.262473	0.0239604
$121$	19.2624	1.75113
$122$	0.309622	0.0280318
$123$	0.311461	0.0280835
$124$	15.8315	1.42171
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	11.5910	1.02854	0.514268	−	0.857630i	$-0.328064\pi$
$127$	11.5910	1.02854	0.514268	+	0.857630i	$0.328064\pi$
$128$	1.31847	0.116537
$129$	−13.3448	−1.17494
$130$	0.197467	0.0173190
$131$	12.6123	1.10194	0.550970	−	0.834525i	$-0.314258\pi$
$131$	12.6123	1.10194	0.550970	+	0.834525i	$0.314258\pi$
$132$	17.4773	1.52121
$133$	0	0
$134$	0.466977	0.0403407
$135$	−5.52052	−0.475130
$136$	−0.867934	−0.0744247
$137$	−17.9243	−1.53138	−0.765690	−	0.643210i	$-0.777602\pi$
$137$	−17.9243	−1.53138	−0.765690	+	0.643210i	$0.777602\pi$
$138$	−0.348107	−0.0296329
$139$	−6.99249	−0.593095	−0.296548	−	0.955018i	$-0.595835\pi$
$139$	−6.99249	−0.593095	−0.296548	+	0.955018i	$0.595835\pi$
$140$	0	0
$141$	10.0328	0.844915
$142$	−0.666208	−0.0559069
$143$	26.3088	2.20005
$144$	−1.88435	−0.157029
$145$	9.00328	0.747682
$146$	−0.129534	−0.0107203
$147$	0	0
$148$	1.99830	0.164259
$149$	8.79235	0.720297	0.360149	−	0.932895i	$-0.382726\pi$
$149$	8.79235	0.720297	0.360149	+	0.932895i	$0.382726\pi$
$150$	0.0656462	0.00535999
$151$	−13.2328	−1.07687	−0.538437	−	0.842666i	$-0.680985\pi$
$151$	−13.2328	−1.07687	−0.538437	+	0.842666i	$0.680985\pi$
$152$	0.232446	0.0188539
$153$	2.48300	0.200739
$154$	0	0
$155$	7.92250	0.636351
$156$	15.1940	1.21649
$157$	−11.6787	−0.932060	−0.466030	−	0.884769i	$-0.654316\pi$
$157$	−11.6787	−0.932060	−0.466030	+	0.884769i	$0.654316\pi$
$158$	−0.561908	−0.0447030
$159$	−14.8877	−1.18067
$160$	0.494919	0.0391268
$161$	0	0
$162$	0.303898	0.0238765
$163$	−12.4831	−0.977751	−0.488876	−	0.872354i	$-0.662593\pi$
$163$	−12.4831	−0.977751	−0.488876	+	0.872354i	$0.662593\pi$
$164$	0.391472	0.0305688
$165$	8.74612	0.680884
$166$	0.228163	0.0177089
$167$	4.48947	0.347405	0.173703	−	0.984798i	$-0.444427\pi$
$167$	4.48947	0.347405	0.173703	+	0.984798i	$0.444427\pi$
$168$	0	0
$169$	9.87168	0.759360
$170$	−0.217076	−0.0166490
$171$	−0.664987	−0.0508528
$172$	−16.7729	−1.27892
$173$	11.4643	0.871613	0.435807	−	0.900040i	$-0.356463\pi$
$173$	11.4643	0.871613	0.435807	+	0.900040i	$0.356463\pi$
$174$	−0.591032	−0.0448060
$175$	0	0
$176$	21.9483	1.65441
$177$	−15.2640	−1.14731
$178$	0.447459	0.0335385
$179$	18.2813	1.36641	0.683204	−	0.730227i	$-0.260586\pi$
$179$	18.2813	1.36641	0.683204	+	0.730227i	$0.260586\pi$
$180$	−0.943782	−0.0703454
$181$	2.03170	0.151015	0.0755075	−	0.997145i	$-0.475942\pi$
$181$	2.03170	0.151015	0.0755075	+	0.997145i	$0.475942\pi$
$182$	0	0
$183$	11.9220	0.881298
$184$	−0.875437	−0.0645381
$185$	1.00000	0.0735215
$186$	−0.520082	−0.0381343
$187$	−28.9213	−2.11493
$188$	12.6101	0.919687
$189$	0	0
$190$	0.0581363	0.00421765
$191$	0.614192	0.0444413	0.0222207	−	0.999753i	$-0.492926\pi$
$191$	0.614192	0.0444413	0.0222207	+	0.999753i	$0.492926\pi$
$192$	12.6540	0.913224
$193$	−7.79593	−0.561163	−0.280582	−	0.959830i	$-0.590527\pi$
$193$	−7.79593	−0.561163	−0.280582	+	0.959830i	$0.590527\pi$
$194$	−0.303775	−0.0218098
$195$	7.60348	0.544496
$196$	0	0
$197$	−4.93761	−0.351790	−0.175895	−	0.984409i	$-0.556282\pi$
$197$	−4.93761	−0.351790	−0.175895	+	0.984409i	$0.556282\pi$
$198$	0.107278	0.00762392
$199$	9.86235	0.699123	0.349562	−	0.936913i	$-0.386331\pi$
$199$	9.86235	0.699123	0.349562	+	0.936913i	$0.386331\pi$
$200$	0.165090	0.0116736
$201$	17.9809	1.26828
$202$	−0.224292	−0.0157811
$203$	0	0
$204$	−16.7028	−1.16943
$205$	0.195903	0.0136825
$206$	0.264186	0.0184067
$207$	2.50447	0.174073
$208$	19.0808	1.32302
$209$	7.74556	0.535772
$210$	0	0
$211$	−9.93956	−0.684267	−0.342134	−	0.939651i	$-0.611150\pi$
$211$	−9.93956	−0.684267	−0.342134	+	0.939651i	$0.611150\pi$
$212$	−18.7122	−1.28516
$213$	−25.6523	−1.75767
$214$	0.0381991	0.00261124
$215$	−8.39361	−0.572439
$216$	0.911384	0.0620118
$217$	0	0
$218$	−0.0935746	−0.00633767
$219$	−4.98769	−0.337037
$220$	10.9929	0.741140
$221$	−25.1428	−1.69129
$222$	−0.0656462	−0.00440589
$223$	12.5793	0.842372	0.421186	−	0.906974i	$-0.361614\pi$
$223$	12.5793	0.842372	0.421186	+	0.906974i	$0.361614\pi$
$224$	0	0
$225$	−0.472294	−0.0314863
$226$	−0.556334	−0.0370068
$227$	7.01840	0.465828	0.232914	−	0.972497i	$-0.425174\pi$
$227$	7.01840	0.465828	0.232914	+	0.972497i	$0.425174\pi$
$228$	4.47326	0.296249
$229$	−28.7647	−1.90083	−0.950413	−	0.310991i	$-0.899339\pi$
$229$	−28.7647	−1.90083	−0.950413	+	0.310991i	$0.899339\pi$
$230$	−0.218953	−0.0144373
$231$	0	0
$232$	−1.48635	−0.0975839
$233$	11.0795	0.725839	0.362920	−	0.931820i	$-0.381780\pi$
$233$	11.0795	0.725839	0.362920	+	0.931820i	$0.381780\pi$
$234$	0.0932627	0.00609677
$235$	6.31043	0.411647
$236$	−19.1851	−1.24885
$237$	−21.6362	−1.40543
$238$	0	0
$239$	3.76867	0.243775	0.121887	−	0.992544i	$-0.461105\pi$
$239$	3.76867	0.243775	0.121887	+	0.992544i	$0.461105\pi$
$240$	6.34325	0.409455
$241$	−24.9441	−1.60679	−0.803397	−	0.595444i	$-0.796976\pi$
$241$	−24.9441	−1.60679	−0.803397	+	0.595444i	$0.796976\pi$
$242$	−0.795350	−0.0511270
$243$	−4.85997	−0.311767
$244$	14.9846	0.959290
$245$	0	0
$246$	−0.0128603	−0.000819942 0
$247$	6.73364	0.428451
$248$	−1.30793	−0.0830535
$249$	8.78540	0.556752
$250$	0.0412902	0.00261142
$251$	−1.49287	−0.0942291	−0.0471146	−	0.998889i	$-0.515003\pi$
$251$	−1.49287	−0.0942291	−0.0471146	+	0.998889i	$0.515003\pi$
$252$	0	0
$253$	−29.1713	−1.83398
$254$	−0.478595	−0.0300297
$255$	−8.35850	−0.523430
$256$	15.8638	0.991487
$257$	−15.7647	−0.983376	−0.491688	−	0.870771i	$-0.663620\pi$
$257$	−15.7647	−0.983376	−0.491688	+	0.870771i	$0.663620\pi$
$258$	0.551009	0.0343043
$259$	0	0
$260$	9.55672	0.592682
$261$	4.25219	0.263204
$262$	−0.520763	−0.0321728
$263$	2.76898	0.170742	0.0853712	−	0.996349i	$-0.472792\pi$
$263$	2.76898	0.170742	0.0853712	+	0.996349i	$0.472792\pi$
$264$	−1.44390	−0.0888658
$265$	−9.36408	−0.575231
$266$	0	0
$267$	17.2294	1.05442
$268$	22.6000	1.38052
$269$	−25.9399	−1.58159	−0.790793	−	0.612084i	$-0.790332\pi$
$269$	−25.9399	−1.58159	−0.790793	+	0.612084i	$0.790332\pi$
$270$	0.227943	0.0138722
$271$	2.35321	0.142947	0.0714737	−	0.997442i	$-0.477230\pi$
$271$	2.35321	0.142947	0.0714737	+	0.997442i	$0.477230\pi$
$272$	−20.9756	−1.27183
$273$	0	0
$274$	0.740099	0.0447110
$275$	5.50113	0.331731
$276$	−16.8471	−1.01408
$277$	5.92183	0.355808	0.177904	−	0.984048i	$-0.443068\pi$
$277$	5.92183	0.355808	0.177904	+	0.984048i	$0.443068\pi$
$278$	0.288721	0.0173163
$279$	3.74175	0.224013
$280$	0	0
$281$	32.8398	1.95906	0.979528	−	0.201307i	$-0.0645188\pi$
$281$	32.8398	1.95906	0.979528	+	0.201307i	$0.0645188\pi$
$282$	−0.414256	−0.0246686
$283$	−11.4540	−0.680872	−0.340436	−	0.940268i	$-0.610575\pi$
$283$	−11.4540	−0.680872	−0.340436	+	0.940268i	$0.610575\pi$
$284$	−32.2421	−1.91321
$285$	2.23854	0.132599
$286$	−1.08629	−0.0642339
$287$	0	0
$288$	0.233747	0.0137737
$289$	10.6395	0.625854
$290$	−0.371747	−0.0218297
$291$	−11.6969	−0.685682
$292$	−6.26897	−0.366864
$293$	−12.8139	−0.748595	−0.374298	−	0.927309i	$-0.622116\pi$
$293$	−12.8139	−0.748595	−0.374298	+	0.927309i	$0.622116\pi$
$294$	0	0
$295$	−9.60075	−0.558977
$296$	−0.165090	−0.00959568
$297$	30.3691	1.76219
$298$	−0.363037	−0.0210302
$299$	−25.3602	−1.46662
$300$	3.17704	0.183427
$301$	0	0
$302$	0.546386	0.0314410
$303$	−8.63635	−0.496145
$304$	5.61758	0.322190
$305$	7.49868	0.429373
$306$	−0.102524	−0.00586089
$307$	−8.99752	−0.513515	−0.256758	−	0.966476i	$-0.582654\pi$
$307$	−8.99752	−0.513515	−0.256758	+	0.966476i	$0.582654\pi$
$308$	0	0
$309$	10.1725	0.578692
$310$	−0.327121	−0.0185792
$311$	−0.140522	−0.00796829	−0.00398414	−	0.999992i	$-0.501268\pi$
$311$	−0.140522	−0.00796829	−0.00398414	+	0.999992i	$0.501268\pi$
$312$	−1.25526	−0.0710651
$313$	−13.5231	−0.764369	−0.382185	−	0.924086i	$-0.624828\pi$
$313$	−13.5231	−0.764369	−0.382185	+	0.924086i	$0.624828\pi$
$314$	0.482215	0.0272129
$315$	0	0
$316$	−27.1943	−1.52980
$317$	−30.6688	−1.72253	−0.861266	−	0.508154i	$-0.830328\pi$
$317$	−30.6688	−1.72253	−0.861266	+	0.508154i	$0.830328\pi$
$318$	0.614716	0.0344716
$319$	−49.5282	−2.77305
$320$	7.95911	0.444928
$321$	1.47086	0.0820952
$322$	0	0
$323$	−7.40229	−0.411874
$324$	14.7076	0.817087
$325$	4.78243	0.265282
$326$	0.515429	0.0285470
$327$	−3.60309	−0.199251
$328$	−0.0323417	−0.00178577
$329$	0	0
$330$	−0.361129	−0.0198795
$331$	−2.82312	−0.155173	−0.0775863	−	0.996986i	$-0.524721\pi$
$331$	−2.82312	−0.155173	−0.0775863	+	0.996986i	$0.524721\pi$
$332$	11.0423	0.606022
$333$	0.472294	0.0258815
$334$	−0.185371	−0.0101430
$335$	11.3096	0.617912
$336$	0	0
$337$	7.89171	0.429889	0.214944	−	0.976626i	$-0.431043\pi$
$337$	7.89171	0.429889	0.214944	+	0.976626i	$0.431043\pi$
$338$	−0.407603	−0.0221707
$339$	−21.4216	−1.16346
$340$	−10.5057	−0.569751
$341$	−43.5827	−2.36014
$342$	0.0274574	0.00148473
$343$	0	0
$344$	1.38570	0.0747121
$345$	−8.43076	−0.453897
$346$	−0.473362	−0.0254481
$347$	18.9912	1.01950	0.509751	−	0.860322i	$-0.329737\pi$
$347$	18.9912	1.01950	0.509751	+	0.860322i	$0.329737\pi$
$348$	−28.6038	−1.53332
$349$	10.5651	0.565540	0.282770	−	0.959188i	$-0.408747\pi$
$349$	10.5651	0.565540	0.282770	+	0.959188i	$0.408747\pi$
$350$	0	0
$351$	26.4015	1.40921
$352$	−2.72261	−0.145116
$353$	−24.3520	−1.29613	−0.648063	−	0.761587i	$-0.724420\pi$
$353$	−24.3520	−1.29613	−0.648063	+	0.761587i	$0.724420\pi$
$354$	0.630253	0.0334976
$355$	−16.1348	−0.856345
$356$	21.6554	1.14774
$357$	0	0
$358$	−0.754838	−0.0398944
$359$	36.2736	1.91445	0.957225	−	0.289346i	$-0.0934378\pi$
$359$	36.2736	1.91445	0.957225	+	0.289346i	$0.0934378\pi$
$360$	0.0779711	0.00410944
$361$	−17.0176	−0.895661
$362$	−0.0838892	−0.00440912
$363$	−30.6249	−1.60739
$364$	0	0
$365$	−3.13716	−0.164206
$366$	−0.492260	−0.0257309
$367$	−32.2560	−1.68375	−0.841876	−	0.539671i	$-0.818549\pi$
$367$	−32.2560	−1.68375	−0.841876	+	0.539671i	$0.818549\pi$
$368$	−21.1569	−1.10288
$369$	0.0925237	0.00481659
$370$	−0.0412902	−0.00214657
$371$	0	0
$372$	−25.1701	−1.30501
$373$	−22.4092	−1.16030	−0.580151	−	0.814509i	$-0.697006\pi$
$373$	−22.4092	−1.16030	−0.580151	+	0.814509i	$0.697006\pi$
$374$	1.19416	0.0617487
$375$	1.58988	0.0821009
$376$	−1.04179	−0.0537263
$377$	−43.0576	−2.21758
$378$	0	0
$379$	−19.7775	−1.01590	−0.507950	−	0.861387i	$-0.669597\pi$
$379$	−19.7775	−1.01590	−0.507950	+	0.861387i	$0.669597\pi$
$380$	2.81359	0.144334
$381$	−18.4283	−0.944109
$382$	−0.0253601	−0.00129753
$383$	13.0748	0.668089	0.334044	−	0.942557i	$-0.391586\pi$
$383$	13.0748	0.668089	0.334044	+	0.942557i	$0.391586\pi$
$384$	−2.09621	−0.106972
$385$	0	0
$386$	0.321895	0.0163840
$387$	−3.96425	−0.201514
$388$	−14.7016	−0.746362
$389$	13.8368	0.701552	0.350776	−	0.936459i	$-0.385918\pi$
$389$	13.8368	0.701552	0.350776	+	0.936459i	$0.385918\pi$
$390$	−0.313949	−0.0158974
$391$	27.8784	1.40987
$392$	0	0
$393$	−20.0519	−1.01149
$394$	0.203875	0.0102711
$395$	−13.6088	−0.684731
$396$	5.19187	0.260901
$397$	14.6229	0.733904	0.366952	−	0.930240i	$-0.380401\pi$
$397$	14.6229	0.733904	0.366952	+	0.930240i	$0.380401\pi$
$398$	−0.407218	−0.0204120
$399$	0	0
$400$	3.98977	0.199489
$401$	18.7563	0.936644	0.468322	−	0.883558i	$-0.344859\pi$
$401$	18.7563	0.936644	0.468322	+	0.883558i	$0.344859\pi$
$402$	−0.742436	−0.0370293
$403$	−37.8888	−1.88738
$404$	−10.8549	−0.540052
$405$	7.36006	0.365724
$406$	0	0
$407$	−5.50113	−0.272681
$408$	1.37991	0.0683156
$409$	30.4024	1.50330	0.751650	−	0.659562i	$-0.229258\pi$
$409$	30.4024	1.50330	0.751650	+	0.659562i	$0.229258\pi$
$410$	−0.00808886	−0.000399480 0
$411$	28.4975	1.40568
$412$	12.7857	0.629904
$413$	0	0
$414$	−0.103410	−0.00508232
$415$	5.52584	0.271253
$416$	−2.36692	−0.116048
$417$	11.1172	0.544411
$418$	−0.319815	−0.0156427
$419$	2.58014	0.126048	0.0630241	−	0.998012i	$-0.479925\pi$
$419$	2.58014	0.126048	0.0630241	+	0.998012i	$0.479925\pi$
$420$	0	0
$421$	−2.92375	−0.142495	−0.0712475	−	0.997459i	$-0.522698\pi$
$421$	−2.92375	−0.142495	−0.0712475	+	0.997459i	$0.522698\pi$
$422$	0.410406	0.0199782
$423$	2.98038	0.144911
$424$	1.54592	0.0750764
$425$	−5.25733	−0.255018
$426$	1.05919	0.0513178
$427$	0	0
$428$	1.84870	0.0893603
$429$	−41.8277	−2.01946
$430$	0.346573	0.0167133
$431$	38.8482	1.87125	0.935627	−	0.352990i	$-0.114835\pi$
$431$	38.8482	1.87125	0.935627	+	0.352990i	$0.114835\pi$
$432$	22.0256	1.05971
$433$	−27.7134	−1.33182	−0.665911	−	0.746032i	$-0.731957\pi$
$433$	−27.7134	−1.33182	−0.665911	+	0.746032i	$0.731957\pi$
$434$	0	0
$435$	−14.3141	−0.686309
$436$	−4.52867	−0.216884
$437$	−7.46628	−0.357161
$438$	0.205943	0.00984032
$439$	−17.2384	−0.822743	−0.411372	−	0.911468i	$-0.634950\pi$
$439$	−17.2384	−0.822743	−0.411372	+	0.911468i	$0.634950\pi$
$440$	−0.908183	−0.0432959
$441$	0	0
$442$	1.03815	0.0493798
$443$	24.3921	1.15890	0.579452	−	0.815007i	$-0.303267\pi$
$443$	24.3921	1.15890	0.579452	+	0.815007i	$0.303267\pi$
$444$	−3.17704	−0.150776
$445$	10.8370	0.513721
$446$	−0.519401	−0.0245944
$447$	−13.9787	−0.661172
$448$	0	0
$449$	−25.2738	−1.19275	−0.596373	−	0.802707i	$-0.703392\pi$
$449$	−25.2738	−1.19275	−0.596373	+	0.802707i	$0.703392\pi$
$450$	0.0195011	0.000919290 0
$451$	−1.07769	−0.0507463
$452$	−26.9246	−1.26643
$453$	21.0386	0.988479
$454$	−0.289791	−0.0136006
$455$	0	0
$456$	−0.369561	−0.0173063
$457$	20.0738	0.939013	0.469507	−	0.882929i	$-0.344432\pi$
$457$	20.0738	0.939013	0.469507	+	0.882929i	$0.344432\pi$
$458$	1.18770	0.0554975
$459$	−29.0232	−1.35469
$460$	−10.5965	−0.494065
$461$	−22.9576	−1.06924	−0.534622	−	0.845091i	$-0.679546\pi$
$461$	−22.9576	−1.06924	−0.534622	+	0.845091i	$0.679546\pi$
$462$	0	0
$463$	8.13863	0.378234	0.189117	−	0.981955i	$-0.439437\pi$
$463$	8.13863	0.378234	0.189117	+	0.981955i	$0.439437\pi$
$464$	−35.9211	−1.66759
$465$	−12.5958	−0.584116
$466$	−0.457472	−0.0211920
$467$	17.5697	0.813028	0.406514	−	0.913645i	$-0.366744\pi$
$467$	17.5697	0.813028	0.406514	+	0.913645i	$0.366744\pi$
$468$	4.51358	0.208640
$469$	0	0
$470$	−0.260559	−0.0120187
$471$	18.5677	0.855552
$472$	1.58499	0.0729551
$473$	46.1743	2.12310
$474$	0.893364	0.0410336
$475$	1.40799	0.0646032
$476$	0	0
$477$	−4.42260	−0.202497
$478$	−0.155609	−0.00711739
$479$	−28.8315	−1.31734	−0.658672	−	0.752430i	$-0.728881\pi$
$479$	−28.8315	−1.31734	−0.658672	+	0.752430i	$0.728881\pi$
$480$	−0.786860	−0.0359151
$481$	−4.78243	−0.218060
$482$	1.02995	0.0469128
$483$	0	0
$484$	−38.4921	−1.74964
$485$	−7.35709	−0.334068
$486$	0.200669	0.00910253
$487$	10.4111	0.471770	0.235885	−	0.971781i	$-0.424201\pi$
$487$	10.4111	0.471770	0.235885	+	0.971781i	$0.424201\pi$
$488$	−1.23796	−0.0560398
$489$	19.8466	0.897493
$490$	0	0
$491$	−29.0040	−1.30893	−0.654466	−	0.756091i	$-0.727106\pi$
$491$	−29.0040	−1.30893	−0.654466	+	0.756091i	$0.727106\pi$
$492$	−0.622392	−0.0280596
$493$	47.3332	2.13178
$494$	−0.278033	−0.0125093
$495$	2.59815	0.116778
$496$	−31.6090	−1.41929
$497$	0	0
$498$	−0.362750	−0.0162552
$499$	−6.97740	−0.312351	−0.156176	−	0.987729i	$-0.549917\pi$
$499$	−6.97740	−0.312351	−0.156176	+	0.987729i	$0.549917\pi$
$500$	1.99830	0.0893665
$501$	−7.13770	−0.318889
$502$	0.0616408	0.00275117
$503$	−20.2329	−0.902138	−0.451069	−	0.892489i	$-0.648957\pi$
$503$	−20.2329	−0.902138	−0.451069	+	0.892489i	$0.648957\pi$
$504$	0	0
$505$	−5.43209	−0.241725
$506$	1.20449	0.0535460
$507$	−15.6947	−0.697028
$508$	−23.1623	−1.02766
$509$	13.5623	0.601138	0.300569	−	0.953760i	$-0.402823\pi$
$509$	13.5623	0.601138	0.300569	+	0.953760i	$0.402823\pi$
$510$	0.345124	0.0152823
$511$	0	0
$512$	−3.29196	−0.145485
$513$	7.77286	0.343180
$514$	0.650928	0.0287112
$515$	6.39829	0.281942
$516$	26.6668	1.17394
$517$	−34.7145	−1.52674
$518$	0	0
$519$	−18.2268	−0.800067
$520$	−0.789533	−0.0346233
$521$	−6.77322	−0.296740	−0.148370	−	0.988932i	$-0.547403\pi$
$521$	−6.77322	−0.296740	−0.148370	+	0.988932i	$0.547403\pi$
$522$	−0.175574	−0.00768465
$523$	−22.9436	−1.00325	−0.501626	−	0.865084i	$-0.667265\pi$
$523$	−22.9436	−1.00325	−0.501626	+	0.865084i	$0.667265\pi$
$524$	−25.2030	−1.10100
$525$	0	0
$526$	−0.114331	−0.00498509
$527$	41.6512	1.81435
$528$	−34.8950	−1.51861
$529$	5.11945	0.222585
$530$	0.386644	0.0167948
$531$	−4.53438	−0.196775
$532$	0	0
$533$	−0.936893	−0.0405813
$534$	−0.711405	−0.0307855
$535$	0.925139	0.0399972
$536$	−1.86711	−0.0806470
$537$	−29.0650	−1.25425
$538$	1.07106	0.0461769
$539$	0	0
$540$	11.0316	0.474725
$541$	9.06056	0.389544	0.194772	−	0.980849i	$-0.437603\pi$
$541$	9.06056	0.389544	0.194772	+	0.980849i	$0.437603\pi$
$542$	−0.0971644	−0.00417357
$543$	−3.23015	−0.138619
$544$	2.60195	0.111558
$545$	−2.26627	−0.0970763
$546$	0	0
$547$	−37.9458	−1.62245	−0.811223	−	0.584737i	$-0.801198\pi$
$547$	−37.9458	−1.62245	−0.811223	+	0.584737i	$0.801198\pi$
$548$	35.8181	1.53008
$549$	3.54158	0.151151
$550$	−0.227143	−0.00968539
$551$	−12.6766	−0.540040
$552$	1.39184	0.0592405
$553$	0	0
$554$	−0.244513	−0.0103884
$555$	−1.58988	−0.0674865
$556$	13.9731	0.592590
$557$	7.29853	0.309249	0.154624	−	0.987973i	$-0.450583\pi$
$557$	7.29853	0.309249	0.154624	+	0.987973i	$0.450583\pi$
$558$	−0.154497	−0.00654040
$559$	40.1419	1.69782
$560$	0	0
$561$	45.9812	1.94133
$562$	−1.35596	−0.0571977
$563$	4.18085	0.176202	0.0881010	−	0.996112i	$-0.471920\pi$
$563$	4.18085	0.176202	0.0881010	+	0.996112i	$0.471920\pi$
$564$	−20.0485	−0.844195
$565$	−13.4738	−0.566846
$566$	0.472939	0.0198791
$567$	0	0
$568$	2.66369	0.111766
$569$	6.43413	0.269733	0.134866	−	0.990864i	$-0.456939\pi$
$569$	6.43413	0.269733	0.134866	+	0.990864i	$0.456939\pi$
$570$	−0.0924295	−0.00387145
$571$	3.94177	0.164958	0.0824790	−	0.996593i	$-0.473716\pi$
$571$	3.94177	0.164958	0.0824790	+	0.996593i	$0.473716\pi$
$572$	−52.5728	−2.19818
$573$	−0.976489	−0.0407934
$574$	0	0
$575$	−5.30278	−0.221141
$576$	3.75904	0.156627
$577$	−23.1440	−0.963495	−0.481748	−	0.876310i	$-0.659998\pi$
$577$	−23.1440	−0.963495	−0.481748	+	0.876310i	$0.659998\pi$
$578$	−0.439307	−0.0182728
$579$	12.3946	0.515101
$580$	−17.9912	−0.747044
$581$	0	0
$582$	0.482965	0.0200195
$583$	51.5130	2.13345
$584$	0.517914	0.0214314
$585$	2.25871	0.0933863
$586$	0.529088	0.0218564
$587$	−34.0634	−1.40595	−0.702973	−	0.711217i	$-0.748144\pi$
$587$	−34.0634	−1.40595	−0.702973	+	0.711217i	$0.748144\pi$
$588$	0	0
$589$	−11.1548	−0.459627
$590$	0.396417	0.0163202
$591$	7.85019	0.322914
$592$	−3.98977	−0.163979
$593$	17.1113	0.702676	0.351338	−	0.936249i	$-0.385727\pi$
$593$	17.1113	0.702676	0.351338	+	0.936249i	$0.385727\pi$
$594$	−1.25394	−0.0514500
$595$	0	0
$596$	−17.5697	−0.719683
$597$	−15.6799	−0.641736
$598$	1.04713	0.0428202
$599$	−25.5262	−1.04297	−0.521486	−	0.853260i	$-0.674622\pi$
$599$	−25.5262	−1.04297	−0.521486	+	0.853260i	$0.674622\pi$
$600$	−0.262473	−0.0107154
$601$	24.0530	0.981141	0.490571	−	0.871401i	$-0.336788\pi$
$601$	24.0530	0.981141	0.490571	+	0.871401i	$0.336788\pi$
$602$	0	0
$603$	5.34148	0.217522
$604$	26.4431	1.07596
$605$	−19.2624	−0.783130
$606$	0.356596	0.0144857
$607$	−16.3270	−0.662692	−0.331346	−	0.943509i	$-0.607503\pi$
$607$	−16.3270	−0.662692	−0.331346	+	0.943509i	$0.607503\pi$
$608$	−0.696843	−0.0282607
$609$	0	0
$610$	−0.309622	−0.0125362
$611$	−30.1792	−1.22092
$612$	−4.96177	−0.200568
$613$	−40.9135	−1.65248	−0.826239	−	0.563319i	$-0.809524\pi$
$613$	−40.9135	−1.65248	−0.826239	+	0.563319i	$0.809524\pi$
$614$	0.371509	0.0149929
$615$	−0.311461	−0.0125593
$616$	0	0
$617$	17.3649	0.699086	0.349543	−	0.936920i	$-0.386337\pi$
$617$	17.3649	0.699086	0.349543	+	0.936920i	$0.386337\pi$
$618$	−0.420023	−0.0168958
$619$	37.5432	1.50899	0.754495	−	0.656306i	$-0.227882\pi$
$619$	37.5432	1.50899	0.754495	+	0.656306i	$0.227882\pi$
$620$	−15.8315	−0.635808
$621$	−29.2741	−1.17473
$622$	0.00580219	0.000232647 0
$623$	0	0
$624$	−30.3362	−1.21442
$625$	1.00000	0.0400000
$626$	0.558370	0.0223169
$627$	−12.3145	−0.491793
$628$	23.3374	0.931266
$629$	5.25733	0.209623
$630$	0	0
$631$	−38.1269	−1.51781	−0.758904	−	0.651202i	$-0.774265\pi$
$631$	−38.1269	−1.51781	−0.758904	+	0.651202i	$0.774265\pi$
$632$	2.24667	0.0893679
$633$	15.8027	0.628100
$634$	1.26632	0.0502920
$635$	−11.5910	−0.459975
$636$	29.7501	1.17967
$637$	0	0
$638$	2.04503	0.0809635
$639$	−7.62036	−0.301457
$640$	−1.31847	−0.0521171
$641$	−1.86184	−0.0735384	−0.0367692	−	0.999324i	$-0.511707\pi$
$641$	−1.86184	−0.0735384	−0.0367692	+	0.999324i	$0.511707\pi$
$642$	−0.0607319	−0.00239690
$643$	16.7644	0.661124	0.330562	−	0.943784i	$-0.392762\pi$
$643$	16.7644	0.661124	0.330562	+	0.943784i	$0.392762\pi$
$644$	0	0
$645$	13.3448	0.525451
$646$	0.305642	0.0120253
$647$	22.4902	0.884180	0.442090	−	0.896971i	$-0.354237\pi$
$647$	22.4902	0.884180	0.442090	+	0.896971i	$0.354237\pi$
$648$	−1.21507	−0.0477326
$649$	52.8150	2.07317
$650$	−0.197467	−0.00774531
$651$	0	0
$652$	24.9449	0.976918
$653$	−0.117286	−0.00458974	−0.00229487	−	0.999997i	$-0.500730\pi$
$653$	−0.117286	−0.00458974	−0.00229487	+	0.999997i	$0.500730\pi$
$654$	0.148772	0.00581744
$655$	−12.6123	−0.492802
$656$	−0.781608	−0.0305167
$657$	−1.48166	−0.0578051
$658$	0	0
$659$	−4.29295	−0.167230	−0.0836149	−	0.996498i	$-0.526647\pi$
$659$	−4.29295	−0.167230	−0.0836149	+	0.996498i	$0.526647\pi$
$660$	−17.4773	−0.680304
$661$	28.3007	1.10077	0.550384	−	0.834911i	$-0.314481\pi$
$661$	28.3007	1.10077	0.550384	+	0.834911i	$0.314481\pi$
$662$	0.116567	0.00453050
$663$	39.9740	1.55246
$664$	−0.912262	−0.0354026
$665$	0	0
$666$	−0.0195011	−0.000755652 0
$667$	47.7424	1.84859
$668$	−8.97128	−0.347109
$669$	−19.9995	−0.773227
$670$	−0.466977	−0.0180409
$671$	−41.2512	−1.59249
$672$	0	0
$673$	20.9805	0.808739	0.404369	−	0.914596i	$-0.367491\pi$
$673$	20.9805	0.808739	0.404369	+	0.914596i	$0.367491\pi$
$674$	−0.325850	−0.0125513
$675$	5.52052	0.212485
$676$	−19.7265	−0.758713
$677$	−17.5349	−0.673921	−0.336961	−	0.941519i	$-0.609399\pi$
$677$	−17.5349	−0.673921	−0.336961	+	0.941519i	$0.609399\pi$
$678$	0.884503	0.0339691
$679$	0	0
$680$	0.867934	0.0332837
$681$	−11.1584	−0.427590
$682$	1.79954	0.0689079
$683$	−48.4807	−1.85506	−0.927531	−	0.373746i	$-0.878073\pi$
$683$	−48.4807	−1.85506	−0.927531	+	0.373746i	$0.878073\pi$
$684$	1.32884	0.0508095
$685$	17.9243	0.684854
$686$	0	0
$687$	45.7323	1.74480
$688$	33.4886	1.27674
$689$	44.7831	1.70610
$690$	0.348107	0.0132522
$691$	12.9001	0.490742	0.245371	−	0.969429i	$-0.421090\pi$
$691$	12.9001	0.490742	0.245371	+	0.969429i	$0.421090\pi$
$692$	−22.9090	−0.870870
$693$	0	0
$694$	−0.784151	−0.0297660
$695$	6.99249	0.265240
$696$	2.36312	0.0895738
$697$	1.02993	0.0390112
$698$	−0.436237	−0.0165118
$699$	−17.6150	−0.666259
$700$	0	0
$701$	7.86699	0.297132	0.148566	−	0.988902i	$-0.452534\pi$
$701$	7.86699	0.297132	0.148566	+	0.988902i	$0.452534\pi$
$702$	−1.09012	−0.0411440
$703$	−1.40799	−0.0531035
$704$	−43.7841	−1.65018
$705$	−10.0328	−0.377857
$706$	1.00550	0.0378424
$707$	0	0
$708$	30.5020	1.14633
$709$	13.0037	0.488365	0.244183	−	0.969729i	$-0.421480\pi$
$709$	13.0037	0.488365	0.244183	+	0.969729i	$0.421480\pi$
$710$	0.666208	0.0250023
$711$	−6.42733	−0.241044
$712$	−1.78907	−0.0670484
$713$	42.0113	1.57333
$714$	0	0
$715$	−26.3088	−0.983893
$716$	−36.5314	−1.36524
$717$	−5.99172	−0.223765
$718$	−1.49774	−0.0558953
$719$	−40.8141	−1.52211	−0.761055	−	0.648687i	$-0.775318\pi$
$719$	−40.8141	−1.52211	−0.761055	+	0.648687i	$0.775318\pi$
$720$	1.88435	0.0702254
$721$	0	0
$722$	0.702657	0.0261502
$723$	39.6581	1.47490
$724$	−4.05993	−0.150886
$725$	−9.00328	−0.334373
$726$	1.26451	0.0469303
$727$	−42.2934	−1.56858	−0.784288	−	0.620397i	$-0.786971\pi$
$727$	−42.2934	−1.56858	−0.784288	+	0.620397i	$0.786971\pi$
$728$	0	0
$729$	29.8069	1.10396
$730$	0.129534	0.00479426
$731$	−44.1280	−1.63213
$732$	−23.8236	−0.880547
$733$	51.6129	1.90637	0.953183	−	0.302394i	$-0.0977859\pi$
$733$	51.6129	1.90637	0.953183	+	0.302394i	$0.0977859\pi$
$734$	1.33186	0.0491598
$735$	0	0
$736$	2.62444	0.0967383
$737$	−62.2159	−2.29175
$738$	−0.00382032	−0.000140628 0
$739$	6.37583	0.234539	0.117269	−	0.993100i	$-0.462586\pi$
$739$	6.37583	0.234539	0.117269	+	0.993100i	$0.462586\pi$
$740$	−1.99830	−0.0734588
$741$	−10.7057	−0.393282
$742$	0	0
$743$	15.1764	0.556767	0.278384	−	0.960470i	$-0.410201\pi$
$743$	15.1764	0.556767	0.278384	+	0.960470i	$0.410201\pi$
$744$	2.07944	0.0762361
$745$	−8.79235	−0.322127
$746$	0.925277	0.0338768
$747$	2.60982	0.0954883
$748$	57.7932	2.11313
$749$	0	0
$750$	−0.0656462	−0.00239706
$751$	−22.7746	−0.831056	−0.415528	−	0.909580i	$-0.636403\pi$
$751$	−22.7746	−0.831056	−0.415528	+	0.909580i	$0.636403\pi$
$752$	−25.1772	−0.918118
$753$	2.37348	0.0864944
$754$	1.77786	0.0647457
$755$	13.2328	0.481593
$756$	0	0
$757$	2.88003	0.104677	0.0523383	−	0.998629i	$-0.483333\pi$
$757$	2.88003	0.104677	0.0523383	+	0.998629i	$0.483333\pi$
$758$	0.816615	0.0296608
$759$	46.3787	1.68344
$760$	−0.232446	−0.00843171
$761$	−11.3790	−0.412489	−0.206244	−	0.978501i	$-0.566124\pi$
$761$	−11.3790	−0.412489	−0.206244	+	0.978501i	$0.566124\pi$
$762$	0.760906	0.0275647
$763$	0	0
$764$	−1.22734	−0.0444035
$765$	−2.48300	−0.0897732
$766$	−0.539859	−0.0195059
$767$	45.9150	1.65789
$768$	−25.2215	−0.910101
$769$	−4.32930	−0.156119	−0.0780594	−	0.996949i	$-0.524872\pi$
$769$	−4.32930	−0.156119	−0.0780594	+	0.996949i	$0.524872\pi$
$770$	0	0
$771$	25.0640	0.902656
$772$	15.5786	0.560685
$773$	43.0317	1.54774	0.773872	−	0.633343i	$-0.218318\pi$
$773$	43.0317	1.54774	0.773872	+	0.633343i	$0.218318\pi$
$774$	0.163684	0.00588352
$775$	−7.92250	−0.284585
$776$	1.21458	0.0436010
$777$	0	0
$778$	−0.571322	−0.0204829
$779$	−0.275830	−0.00988264
$780$	−15.1940	−0.544032
$781$	88.7596	3.17607
$782$	−1.15111	−0.0411634
$783$	−49.7028	−1.77623
$784$	0	0
$785$	11.6787	0.416830
$786$	0.827948	0.0295319
$787$	40.7462	1.45244	0.726222	−	0.687460i	$-0.241274\pi$
$787$	40.7462	1.45244	0.726222	+	0.687460i	$0.241274\pi$
$788$	9.86680	0.351490
$789$	−4.40233	−0.156727
$790$	0.561908	0.0199918
$791$	0	0
$792$	−0.428929	−0.0152413
$793$	−35.8620	−1.27350
$794$	−0.603783	−0.0214275
$795$	14.8877	0.528013
$796$	−19.7079	−0.698527
$797$	−48.3137	−1.71136	−0.855680	−	0.517505i	$-0.826861\pi$
$797$	−48.3137	−1.71136	−0.855680	+	0.517505i	$0.826861\pi$
$798$	0	0
$799$	33.1760	1.17368
$800$	−0.494919	−0.0174980
$801$	5.11822	0.180844
$802$	−0.774450	−0.0273468
$803$	17.2579	0.609019
$804$	−35.9312	−1.26720
$805$	0	0
$806$	1.56444	0.0551049
$807$	41.2413	1.45176
$808$	0.896785	0.0315488
$809$	12.7933	0.449790	0.224895	−	0.974383i	$-0.427796\pi$
$809$	12.7933	0.449790	0.224895	+	0.974383i	$0.427796\pi$
$810$	−0.303898	−0.0106779
$811$	42.5579	1.49441	0.747206	−	0.664593i	$-0.231395\pi$
$811$	42.5579	1.49441	0.747206	+	0.664593i	$0.231395\pi$
$812$	0	0
$813$	−3.74131	−0.131214
$814$	0.227143	0.00796134
$815$	12.4831	0.437264
$816$	33.3485	1.16743
$817$	11.8182	0.413465
$818$	−1.25532	−0.0438912
$819$	0	0
$820$	−0.391472	−0.0136708
$821$	−49.3419	−1.72205	−0.861023	−	0.508566i	$-0.830176\pi$
$821$	−49.3419	−1.72205	−0.861023	+	0.508566i	$0.830176\pi$
$822$	−1.17667	−0.0410409
$823$	37.9589	1.32316	0.661582	−	0.749873i	$-0.269885\pi$
$823$	37.9589	1.32316	0.661582	+	0.749873i	$0.269885\pi$
$824$	−1.05629	−0.0367978
$825$	−8.74612	−0.304501
$826$	0	0
$827$	−43.7194	−1.52027	−0.760136	−	0.649764i	$-0.774868\pi$
$827$	−43.7194	−1.52027	−0.760136	+	0.649764i	$0.774868\pi$
$828$	−5.00467	−0.173924
$829$	−36.4725	−1.26674	−0.633371	−	0.773848i	$-0.718329\pi$
$829$	−36.4725	−1.26674	−0.633371	+	0.773848i	$0.718329\pi$
$830$	−0.228163	−0.00791964
$831$	−9.41497	−0.326602
$832$	−38.0639	−1.31963
$833$	0	0
$834$	−0.459031	−0.0158949
$835$	−4.48947	−0.155364
$836$	−15.4779	−0.535315
$837$	−43.7363	−1.51175
$838$	−0.106535	−0.00368017
$839$	−36.5934	−1.26334	−0.631671	−	0.775236i	$-0.717631\pi$
$839$	−36.5934	−1.26334	−0.631671	+	0.775236i	$0.717631\pi$
$840$	0	0
$841$	52.0591	1.79514
$842$	0.120722	0.00416036
$843$	−52.2112	−1.79825
$844$	19.8622	0.683684
$845$	−9.87168	−0.339596
$846$	−0.123060	−0.00423090
$847$	0	0
$848$	37.3605	1.28297
$849$	18.2105	0.624983
$850$	0.217076	0.00744564
$851$	5.30278	0.181777
$852$	51.2609	1.75617
$853$	−24.3273	−0.832952	−0.416476	−	0.909147i	$-0.636735\pi$
$853$	−24.3273	−0.832952	−0.416476	+	0.909147i	$0.636735\pi$
$854$	0	0
$855$	0.664987	0.0227421
$856$	−0.152731	−0.00522025
$857$	21.4304	0.732048	0.366024	−	0.930605i	$-0.380719\pi$
$857$	21.4304	0.732048	0.366024	+	0.930605i	$0.380719\pi$
$858$	1.72707	0.0589613
$859$	22.9170	0.781917	0.390959	−	0.920408i	$-0.372144\pi$
$859$	22.9170	0.781917	0.390959	+	0.920408i	$0.372144\pi$
$860$	16.7729	0.571951
$861$	0	0
$862$	−1.60405	−0.0546342
$863$	−27.8563	−0.948240	−0.474120	−	0.880460i	$-0.657234\pi$
$863$	−27.8563	−0.948240	−0.474120	+	0.880460i	$0.657234\pi$
$864$	−2.73221	−0.0929516
$865$	−11.4643	−0.389797
$866$	1.14429	0.0388846
$867$	−16.9155	−0.574481
$868$	0	0
$869$	74.8636	2.53957
$870$	0.591032	0.0200378
$871$	−54.0877	−1.83269
$872$	0.374139	0.0126699
$873$	−3.47471	−0.117601
$874$	0.308284	0.0104279
$875$	0	0
$876$	9.96688	0.336750
$877$	−13.5344	−0.457023	−0.228511	−	0.973541i	$-0.573386\pi$
$877$	−13.5344	−0.457023	−0.228511	+	0.973541i	$0.573386\pi$
$878$	0.711776	0.0240213
$879$	20.3725	0.687147
$880$	−21.9483	−0.739876
$881$	8.99878	0.303177	0.151588	−	0.988444i	$-0.451561\pi$
$881$	8.99878	0.303177	0.151588	+	0.988444i	$0.451561\pi$
$882$	0	0
$883$	4.55777	0.153381	0.0766906	−	0.997055i	$-0.475565\pi$
$883$	4.55777	0.153381	0.0766906	+	0.997055i	$0.475565\pi$
$884$	50.2428	1.68985
$885$	15.2640	0.513094
$886$	−1.00715	−0.0338360
$887$	15.7142	0.527630	0.263815	−	0.964573i	$-0.415019\pi$
$887$	15.7142	0.527630	0.263815	+	0.964573i	$0.415019\pi$
$888$	0.262473	0.00880802
$889$	0	0
$890$	−0.447459	−0.0149989
$891$	−40.4886	−1.35642
$892$	−25.1372	−0.841654
$893$	−8.88506	−0.297327
$894$	0.577184	0.0193039
$895$	−18.2813	−0.611077
$896$	0	0
$897$	40.3196	1.34623
$898$	1.04356	0.0348241
$899$	71.3285	2.37894
$900$	0.943782	0.0314594
$901$	−49.2300	−1.64009
$902$	0.0444979	0.00148162
$903$	0	0
$904$	2.22439	0.0739821
$905$	−2.03170	−0.0675360
$906$	−0.868687	−0.0288602
$907$	−35.2057	−1.16899	−0.584494	−	0.811398i	$-0.698707\pi$
$907$	−35.2057	−1.16899	−0.584494	+	0.811398i	$0.698707\pi$
$908$	−14.0248	−0.465431
$909$	−2.56554	−0.0850936
$910$	0	0
$911$	−38.9632	−1.29091	−0.645455	−	0.763799i	$-0.723332\pi$
$911$	−38.9632	−1.29091	−0.645455	+	0.763799i	$0.723332\pi$
$912$	−8.93125	−0.295743
$913$	−30.3984	−1.00604
$914$	−0.828851	−0.0274159
$915$	−11.9220	−0.394128
$916$	57.4804	1.89921
$917$	0	0
$918$	1.19837	0.0395522
$919$	7.70471	0.254155	0.127077	−	0.991893i	$-0.459440\pi$
$919$	7.70471	0.254155	0.127077	+	0.991893i	$0.459440\pi$
$920$	0.875437	0.0288623
$921$	14.3049	0.471364
$922$	0.947925	0.0312182
$923$	77.1635	2.53987
$924$	0	0
$925$	−1.00000	−0.0328798
$926$	−0.336045	−0.0110431
$927$	3.02187	0.0992512
$928$	4.45589	0.146272
$929$	−32.6555	−1.07139	−0.535697	−	0.844410i	$-0.679951\pi$
$929$	−32.6555	−1.07139	−0.535697	+	0.844410i	$0.679951\pi$
$930$	0.520082	0.0170542
$931$	0	0
$932$	−22.1400	−0.725220
$933$	0.223413	0.00731421
$934$	−0.725455	−0.0237376
$935$	28.9213	0.945826
$936$	−0.372892	−0.0121883
$937$	7.29911	0.238452	0.119226	−	0.992867i	$-0.461959\pi$
$937$	7.29911	0.238452	0.119226	+	0.992867i	$0.461959\pi$
$938$	0	0
$939$	21.5000	0.701626
$940$	−12.6101	−0.411296
$941$	−10.4128	−0.339448	−0.169724	−	0.985492i	$-0.554288\pi$
$941$	−10.4128	−0.339448	−0.169724	+	0.985492i	$0.554288\pi$
$942$	−0.766661	−0.0249792
$943$	1.03883	0.0338289
$944$	38.3048	1.24672
$945$	0	0
$946$	−1.90655	−0.0619872
$947$	−18.1364	−0.589353	−0.294676	−	0.955597i	$-0.595212\pi$
$947$	−18.1364	−0.589353	−0.294676	+	0.955597i	$0.595212\pi$
$948$	43.2356	1.40423
$949$	15.0033	0.487026
$950$	−0.0581363	−0.00188619
$951$	48.7596	1.58114
$952$	0	0
$953$	−40.0089	−1.29602	−0.648008	−	0.761633i	$-0.724398\pi$
$953$	−40.0089	−1.29602	−0.648008	+	0.761633i	$0.724398\pi$
$954$	0.182610	0.00591221
$955$	−0.614192	−0.0198748
$956$	−7.53091	−0.243567
$957$	78.7438	2.54542
$958$	1.19046	0.0384619
$959$	0	0
$960$	−12.6540	−0.408406
$961$	31.7660	1.02471
$962$	0.197467	0.00636661
$963$	0.436937	0.0140801
$964$	49.8458	1.60542
$965$	7.79593	0.250960
$966$	0	0
$967$	−28.4278	−0.914176	−0.457088	−	0.889421i	$-0.651108\pi$
$967$	−28.4278	−0.914176	−0.457088	+	0.889421i	$0.651108\pi$
$968$	3.18004	0.102210
$969$	11.7687	0.378066
$970$	0.303775	0.00975364
$971$	−32.4048	−1.03992	−0.519959	−	0.854191i	$-0.674053\pi$
$971$	−32.4048	−1.03992	−0.519959	+	0.854191i	$0.674053\pi$
$972$	9.71166	0.311502
$973$	0	0
$974$	−0.429874	−0.0137741
$975$	−7.60348	−0.243506
$976$	−29.9180	−0.957653
$977$	−45.0477	−1.44120	−0.720602	−	0.693349i	$-0.756134\pi$
$977$	−45.0477	−1.44120	−0.720602	+	0.693349i	$0.756134\pi$
$978$	−0.819468	−0.0262037
$979$	−59.6155	−1.90532
$980$	0	0
$981$	−1.07034	−0.0341735
$982$	1.19758	0.0382163
$983$	−3.97656	−0.126833	−0.0634163	−	0.997987i	$-0.520200\pi$
$983$	−3.97656	−0.126833	−0.0634163	+	0.997987i	$0.520200\pi$
$984$	0.0514192	0.00163919
$985$	4.93761	0.157325
$986$	−1.95440	−0.0622406
$987$	0	0
$988$	−13.4558	−0.428086
$989$	−44.5094	−1.41532
$990$	−0.107278	−0.00340952
$991$	−55.2742	−1.75584	−0.877922	−	0.478804i	$-0.841070\pi$
$991$	−55.2742	−1.75584	−0.877922	+	0.478804i	$0.841070\pi$
$992$	3.92100	0.124492
$993$	4.48841	0.142435
$994$	0	0
$995$	−9.86235	−0.312658
$996$	−17.5558	−0.556277
$997$	−54.3278	−1.72058	−0.860289	−	0.509807i	$-0.829717\pi$
$997$	−54.3278	−1.72058	−0.860289	+	0.509807i	$0.829717\pi$
$998$	0.288098	0.00911958
$999$	−5.52052	−0.174661

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9065.2.a.bd.1.20	✓	38
7.6	odd	2		9065.2.a.be.1.20	yes	38

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9065.2.a.bd.1.20	✓	38	1.1	even	1	trivial
9065.2.a.be.1.20	yes	38	7.6	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 9065 = 5 \cdot 7^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9065.a (trivial)

\(f(q)\)	\(=\)	\(q-0.0412902 q^{2} -1.58988 q^{3} -1.99830 q^{4} -1.00000 q^{5} +0.0656462 q^{6} +0.165090 q^{8} -0.472294 q^{9} +0.0412902 q^{10} +5.50113 q^{11} +3.17704 q^{12} +4.78243 q^{13} +1.58988 q^{15} +3.98977 q^{16} -5.25733 q^{17} +0.0195011 q^{18} +1.40799 q^{19} +1.99830 q^{20} -0.227143 q^{22} -5.30278 q^{23} -0.262473 q^{24} +1.00000 q^{25} -0.197467 q^{26} +5.52052 q^{27} -9.00328 q^{29} -0.0656462 q^{30} -7.92250 q^{31} -0.494919 q^{32} -8.74612 q^{33} +0.217076 q^{34} +0.943782 q^{36} -1.00000 q^{37} -0.0581363 q^{38} -7.60348 q^{39} -0.165090 q^{40} -0.195903 q^{41} +8.39361 q^{43} -10.9929 q^{44} +0.472294 q^{45} +0.218953 q^{46} -6.31043 q^{47} -6.34325 q^{48} -0.0412902 q^{50} +8.35850 q^{51} -9.55672 q^{52} +9.36408 q^{53} -0.227943 q^{54} -5.50113 q^{55} -2.23854 q^{57} +0.371747 q^{58} +9.60075 q^{59} -3.17704 q^{60} -7.49868 q^{61} +0.327121 q^{62} -7.95911 q^{64} -4.78243 q^{65} +0.361129 q^{66} -11.3096 q^{67} +10.5057 q^{68} +8.43076 q^{69} +16.1348 q^{71} -0.0779711 q^{72} +3.13716 q^{73} +0.0412902 q^{74} -1.58988 q^{75} -2.81359 q^{76} +0.313949 q^{78} +13.6088 q^{79} -3.98977 q^{80} -7.36006 q^{81} +0.00808886 q^{82} -5.52584 q^{83} +5.25733 q^{85} -0.346573 q^{86} +14.3141 q^{87} +0.908183 q^{88} -10.8370 q^{89} -0.0195011 q^{90} +10.5965 q^{92} +12.5958 q^{93} +0.260559 q^{94} -1.40799 q^{95} +0.786860 q^{96} +7.35709 q^{97} -2.59815 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 38 q - 2 q^{2} - 6 q^{3} + 42 q^{4} - 38 q^{5} - 8 q^{6} - 6 q^{8} + 48 q^{9} + 2 q^{10} + 34 q^{11} - 20 q^{12} - 22 q^{13} + 6 q^{15} + 46 q^{16} - 22 q^{17} - 36 q^{18} - 40 q^{19} - 42 q^{20} - 4 q^{22}+ \cdots + 92 q^{99}+O(q^{100}) \) 38 * q - 2 * q^2 - 6 * q^3 + 42 * q^4 - 38 * q^5 - 8 * q^6 - 6 * q^8 + 48 * q^9 + 2 * q^10 + 34 * q^11 - 20 * q^12 - 22 * q^13 + 6 * q^15 + 46 * q^16 - 22 * q^17 - 36 * q^18 - 40 * q^19 - 42 * q^20 - 4 * q^22 - 16 * q^23 - 24 * q^24 + 38 * q^25 - 20 * q^26 - 30 * q^27 + 26 * q^29 + 8 * q^30 - 52 * q^31 - 34 * q^32 - 38 * q^33 - 12 * q^34 + 74 * q^36 - 38 * q^37 - 16 * q^38 + 22 * q^39 + 6 * q^40 - 16 * q^41 + 72 * q^44 - 48 * q^45 + 12 * q^46 - 34 * q^47 - 34 * q^48 - 2 * q^50 + 46 * q^51 - 68 * q^52 + 4 * q^53 - 32 * q^54 - 34 * q^55 - 12 * q^57 - 40 * q^58 - 32 * q^59 + 20 * q^60 - 60 * q^61 + 38 * q^62 + 22 * q^64 + 22 * q^65 - 66 * q^66 - 20 * q^67 - 60 * q^68 - 24 * q^69 + 52 * q^71 - 78 * q^72 - 48 * q^73 + 2 * q^74 - 6 * q^75 - 86 * q^76 + 8 * q^78 + 22 * q^79 - 46 * q^80 + 58 * q^81 - 36 * q^82 - 44 * q^83 + 22 * q^85 + 20 * q^86 - 106 * q^87 - 66 * q^88 - 52 * q^89 + 36 * q^90 - 58 * q^92 - 48 * q^93 - 96 * q^94 + 40 * q^95 - 26 * q^96 - 38 * q^97 + 92 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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