LMFDB - Embedded newform 9065.2.a.bd.1.34

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			1.34
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)

$f(q)$	$=$	\(q+2.33502 q^{2} +1.99442 q^{3} +3.45231 q^{4} -1.00000 q^{5} +4.65702 q^{6} +3.39118 q^{8} +0.977724 q^{9} -2.33502 q^{10} -2.05753 q^{11} +6.88537 q^{12} -5.30258 q^{13} -1.99442 q^{15} +1.01384 q^{16} -3.88743 q^{17} +2.28300 q^{18} +0.886044 q^{19} -3.45231 q^{20} -4.80437 q^{22} -0.744489 q^{23} +6.76345 q^{24} +1.00000 q^{25} -12.3816 q^{26} -4.03327 q^{27} -8.44555 q^{29} -4.65702 q^{30} -0.531263 q^{31} -4.41502 q^{32} -4.10358 q^{33} -9.07723 q^{34} +3.37541 q^{36} -1.00000 q^{37} +2.06893 q^{38} -10.5756 q^{39} -3.39118 q^{40} +0.590897 q^{41} +6.51197 q^{43} -7.10323 q^{44} -0.977724 q^{45} -1.73840 q^{46} -6.64805 q^{47} +2.02203 q^{48} +2.33502 q^{50} -7.75319 q^{51} -18.3062 q^{52} +14.2942 q^{53} -9.41777 q^{54} +2.05753 q^{55} +1.76715 q^{57} -19.7205 q^{58} -9.13080 q^{59} -6.88537 q^{60} +5.01342 q^{61} -1.24051 q^{62} -12.3368 q^{64} +5.30258 q^{65} -9.58194 q^{66} +13.2968 q^{67} -13.4206 q^{68} -1.48483 q^{69} +0.554792 q^{71} +3.31564 q^{72} -3.04827 q^{73} -2.33502 q^{74} +1.99442 q^{75} +3.05890 q^{76} -24.6942 q^{78} -8.27660 q^{79} -1.01384 q^{80} -10.9772 q^{81} +1.37976 q^{82} -12.3856 q^{83} +3.88743 q^{85} +15.2056 q^{86} -16.8440 q^{87} -6.97745 q^{88} +12.8058 q^{89} -2.28300 q^{90} -2.57021 q^{92} -1.05956 q^{93} -15.5233 q^{94} -0.886044 q^{95} -8.80541 q^{96} +10.1253 q^{97} -2.01169 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.33502	1.65111	0.825554	−	0.564323i	$-0.190863\pi$
$2$	2.33502	1.65111	0.825554	+	0.564323i	$0.190863\pi$
$3$	1.99442	1.15148	0.575740	−	0.817633i	$-0.304714\pi$
$3$	1.99442	1.15148	0.575740	+	0.817633i	$0.304714\pi$
$4$	3.45231	1.72616
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	4.65702	1.90122
$7$	0	0
$7$	0	0
$8$	3.39118	1.19896
$9$	0.977724	0.325908
$10$	−2.33502	−0.738398
$11$	−2.05753	−0.620368	−0.310184	−	0.950677i	$-0.600391\pi$
$11$	−2.05753	−0.620368	−0.310184	+	0.950677i	$0.600391\pi$
$12$	6.88537	1.98764
$13$	−5.30258	−1.47067	−0.735336	−	0.677703i	$-0.762976\pi$
$13$	−5.30258	−1.47067	−0.735336	+	0.677703i	$0.762976\pi$
$14$	0	0
$15$	−1.99442	−0.514958
$16$	1.01384	0.253461
$17$	−3.88743	−0.942841	−0.471420	−	0.881909i	$-0.656259\pi$
$17$	−3.88743	−0.942841	−0.471420	+	0.881909i	$0.656259\pi$
$18$	2.28300	0.538109
$19$	0.886044	0.203272	0.101636	−	0.994822i	$-0.467592\pi$
$19$	0.886044	0.203272	0.101636	+	0.994822i	$0.467592\pi$
$20$	−3.45231	−0.771961
$21$	0	0
$22$	−4.80437	−1.02429
$23$	−0.744489	−0.155237	−0.0776184	−	0.996983i	$-0.524732\pi$
$23$	−0.744489	−0.155237	−0.0776184	+	0.996983i	$0.524732\pi$
$24$	6.76345	1.38058
$25$	1.00000	0.200000
$26$	−12.3816	−2.42824
$27$	−4.03327	−0.776204
$28$	0	0
$29$	−8.44555	−1.56830	−0.784150	−	0.620572i	$-0.786901\pi$
$29$	−8.44555	−1.56830	−0.784150	+	0.620572i	$0.786901\pi$
$30$	−4.65702	−0.850251
$31$	−0.531263	−0.0954176	−0.0477088	−	0.998861i	$-0.515192\pi$
$31$	−0.531263	−0.0954176	−0.0477088	+	0.998861i	$0.515192\pi$
$32$	−4.41502	−0.780472
$33$	−4.10358	−0.714342
$34$	−9.07723	−1.55673
$35$	0	0
$36$	3.37541	0.562568
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	2.06893	0.335625
$39$	−10.5756	−1.69345
$40$	−3.39118	−0.536193
$41$	0.590897	0.0922827	0.0461413	−	0.998935i	$-0.485308\pi$
$41$	0.590897	0.0922827	0.0461413	+	0.998935i	$0.485308\pi$
$42$	0	0
$43$	6.51197	0.993066	0.496533	−	0.868018i	$-0.334606\pi$
$43$	6.51197	0.993066	0.496533	+	0.868018i	$0.334606\pi$
$44$	−7.10323	−1.07085
$45$	−0.977724	−0.145750
$46$	−1.73840	−0.256313
$47$	−6.64805	−0.969717	−0.484859	−	0.874593i	$-0.661129\pi$
$47$	−6.64805	−0.969717	−0.484859	+	0.874593i	$0.661129\pi$
$48$	2.02203	0.291855
$49$	0	0
$50$	2.33502	0.330222
$51$	−7.75319	−1.08566
$52$	−18.3062	−2.53861
$53$	14.2942	1.96346	0.981732	−	0.190270i	$-0.0609363\pi$
$53$	14.2942	1.96346	0.981732	+	0.190270i	$0.0609363\pi$
$54$	−9.41777	−1.28160
$55$	2.05753	0.277437
$56$	0	0
$57$	1.76715	0.234064
$58$	−19.7205	−2.58943
$59$	−9.13080	−1.18873	−0.594364	−	0.804196i	$-0.702596\pi$
$59$	−9.13080	−1.18873	−0.594364	+	0.804196i	$0.702596\pi$
$60$	−6.88537	−0.888898
$61$	5.01342	0.641902	0.320951	−	0.947096i	$-0.395997\pi$
$61$	5.01342	0.641902	0.320951	+	0.947096i	$0.395997\pi$
$62$	−1.24051	−0.157545
$63$	0	0
$64$	−12.3368	−1.54210
$65$	5.30258	0.657704
$66$	−9.58194	−1.17946
$67$	13.2968	1.62447	0.812233	−	0.583333i	$-0.198252\pi$
$67$	13.2968	1.62447	0.812233	+	0.583333i	$0.198252\pi$
$68$	−13.4206	−1.62749
$69$	−1.48483	−0.178752
$70$	0	0
$71$	0.554792	0.0658417	0.0329208	−	0.999458i	$-0.489519\pi$
$71$	0.554792	0.0658417	0.0329208	+	0.999458i	$0.489519\pi$
$72$	3.31564	0.390752
$73$	−3.04827	−0.356773	−0.178387	−	0.983960i	$-0.557088\pi$
$73$	−3.04827	−0.356773	−0.178387	+	0.983960i	$0.557088\pi$
$74$	−2.33502	−0.271440
$75$	1.99442	0.230296
$76$	3.05890	0.350880
$77$	0	0
$78$	−24.6942	−2.79607
$79$	−8.27660	−0.931190	−0.465595	−	0.884998i	$-0.654160\pi$
$79$	−8.27660	−0.931190	−0.465595	+	0.884998i	$0.654160\pi$
$80$	−1.01384	−0.113351
$81$	−10.9772	−1.21969
$82$	1.37976	0.152369
$83$	−12.3856	−1.35950	−0.679751	−	0.733443i	$-0.737912\pi$
$83$	−12.3856	−1.35950	−0.679751	+	0.733443i	$0.737912\pi$
$84$	0	0
$85$	3.88743	0.421651
$86$	15.2056	1.63966
$87$	−16.8440	−1.80587
$88$	−6.97745	−0.743799
$89$	12.8058	1.35741	0.678706	−	0.734410i	$-0.262541\pi$
$89$	12.8058	1.35741	0.678706	+	0.734410i	$0.262541\pi$
$90$	−2.28300	−0.240650
$91$	0	0
$92$	−2.57021	−0.267963
$93$	−1.05956	−0.109872
$94$	−15.5233	−1.60111
$95$	−0.886044	−0.0909062
$96$	−8.80541	−0.898699
$97$	10.1253	1.02807	0.514036	−	0.857768i	$-0.328150\pi$
$97$	10.1253	1.02807	0.514036	+	0.857768i	$0.328150\pi$
$98$	0	0
$99$	−2.01169	−0.202183
$100$	3.45231	0.345231
$101$	−9.30816	−0.926197	−0.463098	−	0.886307i	$-0.653262\pi$
$101$	−9.30816	−0.926197	−0.463098	+	0.886307i	$0.653262\pi$
$102$	−18.1038	−1.79255
$103$	4.76681	0.469688	0.234844	−	0.972033i	$-0.424542\pi$
$103$	4.76681	0.469688	0.234844	+	0.972033i	$0.424542\pi$
$104$	−17.9820	−1.76328
$105$	0	0
$106$	33.3773	3.24189
$107$	7.09637	0.686032	0.343016	−	0.939329i	$-0.388551\pi$
$107$	7.09637	0.686032	0.343016	+	0.939329i	$0.388551\pi$
$108$	−13.9241	−1.33985
$109$	−17.1993	−1.64740	−0.823698	−	0.567029i	$-0.808093\pi$
$109$	−17.1993	−1.64740	−0.823698	+	0.567029i	$0.808093\pi$
$110$	4.80437	0.458078
$111$	−1.99442	−0.189302
$112$	0	0
$113$	4.83555	0.454890	0.227445	−	0.973791i	$-0.426963\pi$
$113$	4.83555	0.454890	0.227445	+	0.973791i	$0.426963\pi$
$114$	4.12632	0.386465
$115$	0.744489	0.0694240
$116$	−29.1567	−2.70713
$117$	−5.18446	−0.479304
$118$	−21.3206	−1.96272
$119$	0	0
$120$	−6.76345	−0.617416
$121$	−6.76658	−0.615144
$122$	11.7064	1.05985
$123$	1.17850	0.106262
$124$	−1.83409	−0.164706
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	4.76568	0.422886	0.211443	−	0.977390i	$-0.432184\pi$
$127$	4.76568	0.422886	0.211443	+	0.977390i	$0.432184\pi$
$128$	−19.9767	−1.76571
$129$	12.9876	1.14350
$130$	12.3816	1.08594
$131$	−4.84210	−0.423056	−0.211528	−	0.977372i	$-0.567844\pi$
$131$	−4.84210	−0.423056	−0.211528	+	0.977372i	$0.567844\pi$
$132$	−14.1669	−1.23307
$133$	0	0
$134$	31.0483	2.68217
$135$	4.03327	0.347129
$136$	−13.1830	−1.13043
$137$	21.4358	1.83138	0.915692	−	0.401880i	$-0.131643\pi$
$137$	21.4358	1.83138	0.915692	+	0.401880i	$0.131643\pi$
$138$	−3.46710	−0.295139
$139$	−10.3153	−0.874935	−0.437467	−	0.899234i	$-0.644124\pi$
$139$	−10.3153	−0.874935	−0.437467	+	0.899234i	$0.644124\pi$
$140$	0	0
$141$	−13.2590	−1.11661
$142$	1.29545	0.108712
$143$	10.9102	0.912358
$144$	0.991259	0.0826049
$145$	8.44555	0.701365
$146$	−7.11777	−0.589071
$147$	0	0
$148$	−3.45231	−0.283778
$149$	17.2915	1.41657	0.708286	−	0.705926i	$-0.249469\pi$
$149$	17.2915	1.41657	0.708286	+	0.705926i	$0.249469\pi$
$150$	4.65702	0.380244
$151$	−18.3698	−1.49491	−0.747457	−	0.664310i	$-0.768725\pi$
$151$	−18.3698	−1.49491	−0.747457	+	0.664310i	$0.768725\pi$
$152$	3.00474	0.243716
$153$	−3.80084	−0.307279
$154$	0	0
$155$	0.531263	0.0426721
$156$	−36.5103	−2.92316
$157$	−15.3019	−1.22122	−0.610612	−	0.791930i	$-0.709077\pi$
$157$	−15.3019	−1.22122	−0.610612	+	0.791930i	$0.709077\pi$
$158$	−19.3260	−1.53749
$159$	28.5087	2.26089
$160$	4.41502	0.349038
$161$	0	0
$162$	−25.6320	−2.01384
$163$	0.363267	0.0284533	0.0142266	−	0.999899i	$-0.495471\pi$
$163$	0.363267	0.0284533	0.0142266	+	0.999899i	$0.495471\pi$
$164$	2.03996	0.159294
$165$	4.10358	0.319463
$166$	−28.9207	−2.24468
$167$	−9.08619	−0.703111	−0.351555	−	0.936167i	$-0.614347\pi$
$167$	−9.08619	−0.703111	−0.351555	+	0.936167i	$0.614347\pi$
$168$	0	0
$169$	15.1174	1.16287
$170$	9.07723	0.696192
$171$	0.866307	0.0662481
$172$	22.4814	1.71419
$173$	22.3296	1.69769	0.848845	−	0.528641i	$-0.177298\pi$
$173$	22.3296	1.69769	0.848845	+	0.528641i	$0.177298\pi$
$174$	−39.3311	−2.98168
$175$	0	0
$176$	−2.08601	−0.157239
$177$	−18.2107	−1.36880
$178$	29.9018	2.24123
$179$	18.1549	1.35696	0.678480	−	0.734619i	$-0.262639\pi$
$179$	18.1549	1.35696	0.678480	+	0.734619i	$0.262639\pi$
$180$	−3.37541	−0.251588
$181$	−5.16766	−0.384109	−0.192055	−	0.981384i	$-0.561515\pi$
$181$	−5.16766	−0.384109	−0.192055	+	0.981384i	$0.561515\pi$
$182$	0	0
$183$	9.99887	0.739138
$184$	−2.52470	−0.186123
$185$	1.00000	0.0735215
$186$	−2.47410	−0.181410
$187$	7.99850	0.584908
$188$	−22.9511	−1.67388
$189$	0	0
$190$	−2.06893	−0.150096
$191$	−0.396923	−0.0287203	−0.0143602	−	0.999897i	$-0.504571\pi$
$191$	−0.396923	−0.0287203	−0.0143602	+	0.999897i	$0.504571\pi$
$192$	−24.6049	−1.77570
$193$	7.71941	0.555656	0.277828	−	0.960631i	$-0.410386\pi$
$193$	7.71941	0.555656	0.277828	+	0.960631i	$0.410386\pi$
$194$	23.6429	1.69746
$195$	10.5756	0.757334
$196$	0	0
$197$	−3.78696	−0.269810	−0.134905	−	0.990859i	$-0.543073\pi$
$197$	−3.78696	−0.269810	−0.134905	+	0.990859i	$0.543073\pi$
$198$	−4.69734	−0.333826
$199$	−14.3388	−1.01645	−0.508224	−	0.861225i	$-0.669698\pi$
$199$	−14.3388	−1.01645	−0.508224	+	0.861225i	$0.669698\pi$
$200$	3.39118	0.239793
$201$	26.5195	1.87054
$202$	−21.7347	−1.52925
$203$	0	0
$204$	−26.7664	−1.87402
$205$	−0.590897	−0.0412701
$206$	11.1306	0.775505
$207$	−0.727905	−0.0505929
$208$	−5.37599	−0.372758
$209$	−1.82306	−0.126104
$210$	0	0
$211$	26.5993	1.83117	0.915585	−	0.402124i	$-0.131728\pi$
$211$	26.5993	1.83117	0.915585	+	0.402124i	$0.131728\pi$
$212$	49.3482	3.38925
$213$	1.10649	0.0758154
$214$	16.5702	1.13271
$215$	−6.51197	−0.444113
$216$	−13.6776	−0.930640
$217$	0	0
$218$	−40.1607	−2.72003
$219$	−6.07954	−0.410817
$220$	7.10323	0.478900
$221$	20.6134	1.38661
$222$	−4.65702	−0.312558
$223$	−10.7716	−0.721319	−0.360659	−	0.932698i	$-0.617448\pi$
$223$	−10.7716	−0.721319	−0.360659	+	0.932698i	$0.617448\pi$
$224$	0	0
$225$	0.977724	0.0651816
$226$	11.2911	0.751072
$227$	−6.63335	−0.440271	−0.220136	−	0.975469i	$-0.570650\pi$
$227$	−6.63335	−0.440271	−0.220136	+	0.975469i	$0.570650\pi$
$228$	6.10075	0.404032
$229$	−13.8389	−0.914501	−0.457251	−	0.889338i	$-0.651166\pi$
$229$	−13.8389	−0.914501	−0.457251	+	0.889338i	$0.651166\pi$
$230$	1.73840	0.114627
$231$	0	0
$232$	−28.6404	−1.88033
$233$	−19.8586	−1.30098	−0.650489	−	0.759515i	$-0.725436\pi$
$233$	−19.8586	−1.30098	−0.650489	+	0.759515i	$0.725436\pi$
$234$	−12.1058	−0.791382
$235$	6.64805	0.433671
$236$	−31.5224	−2.05193
$237$	−16.5070	−1.07225
$238$	0	0
$239$	21.2568	1.37499	0.687495	−	0.726189i	$-0.258710\pi$
$239$	21.2568	1.37499	0.687495	+	0.726189i	$0.258710\pi$
$240$	−2.02203	−0.130522
$241$	3.62112	0.233257	0.116628	−	0.993176i	$-0.462791\pi$
$241$	3.62112	0.233257	0.116628	+	0.993176i	$0.462791\pi$
$242$	−15.8001	−1.01567
$243$	−9.79341	−0.628248
$244$	17.3079	1.10802
$245$	0	0
$246$	2.75182	0.175450
$247$	−4.69832	−0.298947
$248$	−1.80161	−0.114402
$249$	−24.7022	−1.56544
$250$	−2.33502	−0.147680
$251$	5.48077	0.345943	0.172971	−	0.984927i	$-0.444663\pi$
$251$	5.48077	0.345943	0.172971	+	0.984927i	$0.444663\pi$
$252$	0	0
$253$	1.53181	0.0963039
$254$	11.1280	0.698230
$255$	7.75319	0.485523
$256$	−21.9723	−1.37327
$257$	−4.27279	−0.266529	−0.133265	−	0.991080i	$-0.542546\pi$
$257$	−4.27279	−0.266529	−0.133265	+	0.991080i	$0.542546\pi$
$258$	30.3264	1.88804
$259$	0	0
$260$	18.3062	1.13530
$261$	−8.25741	−0.511121
$262$	−11.3064	−0.698512
$263$	−0.572948	−0.0353295	−0.0176647	−	0.999844i	$-0.505623\pi$
$263$	−0.572948	−0.0353295	−0.0176647	+	0.999844i	$0.505623\pi$
$264$	−13.9160	−0.856470
$265$	−14.2942	−0.878088
$266$	0	0
$267$	25.5402	1.56303
$268$	45.9048	2.80408
$269$	1.04702	0.0638380	0.0319190	−	0.999490i	$-0.489838\pi$
$269$	1.04702	0.0638380	0.0319190	+	0.999490i	$0.489838\pi$
$270$	9.41777	0.573147
$271$	−11.5962	−0.704418	−0.352209	−	0.935921i	$-0.614569\pi$
$271$	−11.5962	−0.704418	−0.352209	+	0.935921i	$0.614569\pi$
$272$	−3.94125	−0.238973
$273$	0	0
$274$	50.0530	3.02381
$275$	−2.05753	−0.124074
$276$	−5.12609	−0.308554
$277$	−8.63023	−0.518540	−0.259270	−	0.965805i	$-0.583482\pi$
$277$	−8.63023	−0.518540	−0.259270	+	0.965805i	$0.583482\pi$
$278$	−24.0865	−1.44461
$279$	−0.519428	−0.0310974
$280$	0	0
$281$	−8.90251	−0.531079	−0.265540	−	0.964100i	$-0.585550\pi$
$281$	−8.90251	−0.531079	−0.265540	+	0.964100i	$0.585550\pi$
$282$	−30.9601	−1.84364
$283$	−6.29501	−0.374200	−0.187100	−	0.982341i	$-0.559909\pi$
$283$	−6.29501	−0.374200	−0.187100	+	0.982341i	$0.559909\pi$
$284$	1.91532	0.113653
$285$	−1.76715	−0.104677
$286$	25.4755	1.50640
$287$	0	0
$288$	−4.31667	−0.254362
$289$	−1.88787	−0.111051
$290$	19.7205	1.15803
$291$	20.1942	1.18381
$292$	−10.5236	−0.615847
$293$	17.0831	0.998006	0.499003	−	0.866600i	$-0.333700\pi$
$293$	17.0831	0.998006	0.499003	+	0.866600i	$0.333700\pi$
$294$	0	0
$295$	9.13080	0.531616
$296$	−3.39118	−0.197108
$297$	8.29857	0.481532
$298$	40.3759	2.33891
$299$	3.94772	0.228302
$300$	6.88537	0.397527
$301$	0	0
$302$	−42.8938	−2.46826
$303$	−18.5644	−1.06650
$304$	0.898310	0.0515216
$305$	−5.01342	−0.287067
$306$	−8.87502	−0.507351
$307$	−6.14151	−0.350514	−0.175257	−	0.984523i	$-0.556076\pi$
$307$	−6.14151	−0.350514	−0.175257	+	0.984523i	$0.556076\pi$
$308$	0	0
$309$	9.50704	0.540836
$310$	1.24051	0.0704562
$311$	−4.88632	−0.277078	−0.138539	−	0.990357i	$-0.544241\pi$
$311$	−4.88632	−0.277078	−0.138539	+	0.990357i	$0.544241\pi$
$312$	−35.8637	−2.03038
$313$	−23.0340	−1.30196	−0.650980	−	0.759095i	$-0.725642\pi$
$313$	−23.0340	−1.30196	−0.650980	+	0.759095i	$0.725642\pi$
$314$	−35.7302	−2.01637
$315$	0	0
$316$	−28.5734	−1.60738
$317$	13.4715	0.756634	0.378317	−	0.925676i	$-0.376503\pi$
$317$	13.4715	0.756634	0.378317	+	0.925676i	$0.376503\pi$
$318$	66.5685	3.73297
$319$	17.3770	0.972923
$320$	12.3368	0.689650
$321$	14.1532	0.789953
$322$	0	0
$323$	−3.44444	−0.191654
$324$	−37.8968	−2.10538
$325$	−5.30258	−0.294134
$326$	0.848236	0.0469794
$327$	−34.3027	−1.89694
$328$	2.00384	0.110644
$329$	0	0
$330$	9.58194	0.527468
$331$	−34.0534	−1.87174	−0.935871	−	0.352344i	$-0.885385\pi$
$331$	−34.0534	−1.87174	−0.935871	+	0.352344i	$0.885385\pi$
$332$	−42.7591	−2.34671
$333$	−0.977724	−0.0535789
$334$	−21.2164	−1.16091
$335$	−13.2968	−0.726483
$336$	0	0
$337$	−9.60065	−0.522980	−0.261490	−	0.965206i	$-0.584214\pi$
$337$	−9.60065	−0.522980	−0.261490	+	0.965206i	$0.584214\pi$
$338$	35.2994	1.92003
$339$	9.64413	0.523797
$340$	13.4206	0.727836
$341$	1.09309	0.0591940
$342$	2.02284	0.109383
$343$	0	0
$344$	22.0833	1.19065
$345$	1.48483	0.0799404
$346$	52.1401	2.80307
$347$	−10.2333	−0.549354	−0.274677	−	0.961537i	$-0.588571\pi$
$347$	−10.2333	−0.549354	−0.274677	+	0.961537i	$0.588571\pi$
$348$	−58.1508	−3.11721
$349$	−33.7956	−1.80904	−0.904519	−	0.426434i	$-0.859770\pi$
$349$	−33.7956	−1.80904	−0.904519	+	0.426434i	$0.859770\pi$
$350$	0	0
$351$	21.3868	1.14154
$352$	9.08402	0.484180
$353$	3.65202	0.194377	0.0971887	−	0.995266i	$-0.469015\pi$
$353$	3.65202	0.194377	0.0971887	+	0.995266i	$0.469015\pi$
$354$	−42.5223	−2.26003
$355$	−0.554792	−0.0294453
$356$	44.2097	2.34311
$357$	0	0
$358$	42.3920	2.24049
$359$	10.4508	0.551573	0.275786	−	0.961219i	$-0.411062\pi$
$359$	10.4508	0.551573	0.275786	+	0.961219i	$0.411062\pi$
$360$	−3.31564	−0.174749
$361$	−18.2149	−0.958680
$362$	−12.0666	−0.634205
$363$	−13.4954	−0.708326
$364$	0	0
$365$	3.04827	0.159554
$366$	23.3476	1.22040
$367$	−3.39084	−0.177000	−0.0885002	−	0.996076i	$-0.528207\pi$
$367$	−3.39084	−0.177000	−0.0885002	+	0.996076i	$0.528207\pi$
$368$	−0.754796	−0.0393464
$369$	0.577734	0.0300756
$370$	2.33502	0.121392
$371$	0	0
$372$	−3.65794	−0.189656
$373$	−12.7351	−0.659398	−0.329699	−	0.944086i	$-0.606947\pi$
$373$	−12.7351	−0.659398	−0.329699	+	0.944086i	$0.606947\pi$
$374$	18.6767	0.965747
$375$	−1.99442	−0.102992
$376$	−22.5447	−1.16266
$377$	44.7832	2.30645
$378$	0	0
$379$	13.7772	0.707686	0.353843	−	0.935305i	$-0.384875\pi$
$379$	13.7772	0.707686	0.353843	+	0.935305i	$0.384875\pi$
$380$	−3.05890	−0.156918
$381$	9.50478	0.486945
$382$	−0.926822	−0.0474203
$383$	2.08143	0.106356	0.0531781	−	0.998585i	$-0.483065\pi$
$383$	2.08143	0.106356	0.0531781	+	0.998585i	$0.483065\pi$
$384$	−39.8420	−2.03318
$385$	0	0
$386$	18.0250	0.917447
$387$	6.36691	0.323648
$388$	34.9559	1.77462
$389$	4.78908	0.242816	0.121408	−	0.992603i	$-0.461259\pi$
$389$	4.78908	0.242816	0.121408	+	0.992603i	$0.461259\pi$
$390$	24.6942	1.25044
$391$	2.89415	0.146364
$392$	0	0
$393$	−9.65720	−0.487141
$394$	−8.84262	−0.445485
$395$	8.27660	0.416441
$396$	−6.94500	−0.348999
$397$	−6.19533	−0.310935	−0.155467	−	0.987841i	$-0.549688\pi$
$397$	−6.19533	−0.310935	−0.155467	+	0.987841i	$0.549688\pi$
$398$	−33.4813	−1.67826
$399$	0	0
$400$	1.01384	0.0506922
$401$	−20.3734	−1.01740	−0.508699	−	0.860945i	$-0.669873\pi$
$401$	−20.3734	−1.01740	−0.508699	+	0.860945i	$0.669873\pi$
$402$	61.9235	3.08847
$403$	2.81706	0.140328
$404$	−32.1347	−1.59876
$405$	10.9772	0.545463
$406$	0	0
$407$	2.05753	0.101988
$408$	−26.2925	−1.30167
$409$	−14.7756	−0.730606	−0.365303	−	0.930889i	$-0.619035\pi$
$409$	−14.7756	−0.730606	−0.365303	+	0.930889i	$0.619035\pi$
$410$	−1.37976	−0.0681413
$411$	42.7521	2.10880
$412$	16.4565	0.810755
$413$	0	0
$414$	−1.69967	−0.0835343
$415$	12.3856	0.607987
$416$	23.4110	1.14782
$417$	−20.5731	−1.00747
$418$	−4.25688	−0.208211
$419$	36.4864	1.78248	0.891238	−	0.453537i	$-0.149838\pi$
$419$	36.4864	1.78248	0.891238	+	0.453537i	$0.149838\pi$
$420$	0	0
$421$	−24.2906	−1.18385	−0.591925	−	0.805993i	$-0.701632\pi$
$421$	−24.2906	−1.18385	−0.591925	+	0.805993i	$0.701632\pi$
$422$	62.1098	3.02346
$423$	−6.49995	−0.316039
$424$	48.4743	2.35412
$425$	−3.88743	−0.188568
$426$	2.58367	0.125179
$427$	0	0
$428$	24.4989	1.18420
$429$	21.7596	1.05056
$430$	−15.2056	−0.733278
$431$	5.54323	0.267008	0.133504	−	0.991048i	$-0.457377\pi$
$431$	5.54323	0.267008	0.133504	+	0.991048i	$0.457377\pi$
$432$	−4.08911	−0.196737
$433$	14.7697	0.709786	0.354893	−	0.934907i	$-0.384517\pi$
$433$	14.7697	0.709786	0.354893	+	0.934907i	$0.384517\pi$
$434$	0	0
$435$	16.8440	0.807608
$436$	−59.3774	−2.84366
$437$	−0.659651	−0.0315554
$438$	−14.1958	−0.678304
$439$	−3.80842	−0.181766	−0.0908830	−	0.995862i	$-0.528969\pi$
$439$	−3.80842	−0.181766	−0.0908830	+	0.995862i	$0.528969\pi$
$440$	6.97745	0.332637
$441$	0	0
$442$	48.1327	2.28944
$443$	−1.29336	−0.0614496	−0.0307248	−	0.999528i	$-0.509782\pi$
$443$	−1.29336	−0.0614496	−0.0307248	+	0.999528i	$0.509782\pi$
$444$	−6.88537	−0.326765
$445$	−12.8058	−0.607053
$446$	−25.1519	−1.19097
$447$	34.4865	1.63116
$448$	0	0
$449$	23.3359	1.10129	0.550645	−	0.834740i	$-0.314382\pi$
$449$	23.3359	1.10129	0.550645	+	0.834740i	$0.314382\pi$
$450$	2.28300	0.107622
$451$	−1.21579	−0.0572492
$452$	16.6938	0.785212
$453$	−36.6372	−1.72136
$454$	−15.4890	−0.726935
$455$	0	0
$456$	5.99272	0.280635
$457$	−0.838110	−0.0392051	−0.0196026	−	0.999808i	$-0.506240\pi$
$457$	−0.838110	−0.0392051	−0.0196026	+	0.999808i	$0.506240\pi$
$458$	−32.3141	−1.50994
$459$	15.6791	0.731837
$460$	2.57021	0.119837
$461$	26.5278	1.23552	0.617761	−	0.786366i	$-0.288040\pi$
$461$	26.5278	1.23552	0.617761	+	0.786366i	$0.288040\pi$
$462$	0	0
$463$	14.9420	0.694415	0.347207	−	0.937788i	$-0.387130\pi$
$463$	14.9420	0.694415	0.347207	+	0.937788i	$0.387130\pi$
$464$	−8.56246	−0.397502
$465$	1.05956	0.0491360
$466$	−46.3701	−2.14806
$467$	−21.1416	−0.978315	−0.489158	−	0.872195i	$-0.662696\pi$
$467$	−21.1416	−0.978315	−0.489158	+	0.872195i	$0.662696\pi$
$468$	−17.8984	−0.827353
$469$	0	0
$470$	15.5233	0.716037
$471$	−30.5185	−1.40622
$472$	−30.9642	−1.42524
$473$	−13.3986	−0.616067
$474$	−38.5442	−1.77040
$475$	0.886044	0.0406545
$476$	0	0
$477$	13.9758	0.639908
$478$	49.6351	2.27026
$479$	−20.1255	−0.919557	−0.459779	−	0.888034i	$-0.652071\pi$
$479$	−20.1255	−0.919557	−0.459779	+	0.888034i	$0.652071\pi$
$480$	8.80541	0.401910
$481$	5.30258	0.241777
$482$	8.45538	0.385132
$483$	0	0
$484$	−23.3604	−1.06183
$485$	−10.1253	−0.459768
$486$	−22.8678	−1.03730
$487$	22.2052	1.00621	0.503106	−	0.864225i	$-0.332190\pi$
$487$	22.2052	1.00621	0.503106	+	0.864225i	$0.332190\pi$
$488$	17.0014	0.769617
$489$	0.724508	0.0327634
$490$	0	0
$491$	−11.5432	−0.520938	−0.260469	−	0.965482i	$-0.583877\pi$
$491$	−11.5432	−0.520938	−0.260469	+	0.965482i	$0.583877\pi$
$492$	4.06855	0.183424
$493$	32.8315	1.47866
$494$	−10.9707	−0.493594
$495$	2.01169	0.0904189
$496$	−0.538617	−0.0241846
$497$	0	0
$498$	−57.6802	−2.58471
$499$	32.6808	1.46300	0.731498	−	0.681844i	$-0.238822\pi$
$499$	32.6808	1.46300	0.731498	+	0.681844i	$0.238822\pi$
$500$	−3.45231	−0.154392
$501$	−18.1217	−0.809618
$502$	12.7977	0.571189
$503$	28.6283	1.27647	0.638237	−	0.769840i	$-0.279664\pi$
$503$	28.6283	1.27647	0.638237	+	0.769840i	$0.279664\pi$
$504$	0	0
$505$	9.30816	0.414208
$506$	3.57680	0.159008
$507$	30.1504	1.33903
$508$	16.4526	0.729967
$509$	9.28117	0.411381	0.205690	−	0.978617i	$-0.434056\pi$
$509$	9.28117	0.411381	0.205690	+	0.978617i	$0.434056\pi$
$510$	18.1038	0.801651
$511$	0	0
$512$	−11.3524	−0.501709
$513$	−3.57366	−0.157781
$514$	−9.97705	−0.440069
$515$	−4.76681	−0.210051
$516$	44.8374	1.97386
$517$	13.6785	0.601582
$518$	0	0
$519$	44.5347	1.95486
$520$	17.9820	0.788563
$521$	4.30099	0.188430	0.0942148	−	0.995552i	$-0.469966\pi$
$521$	4.30099	0.188430	0.0942148	+	0.995552i	$0.469966\pi$
$522$	−19.2812	−0.843916
$523$	3.36743	0.147247	0.0736237	−	0.997286i	$-0.476544\pi$
$523$	3.36743	0.147247	0.0736237	+	0.997286i	$0.476544\pi$
$524$	−16.7165	−0.730262
$525$	0	0
$526$	−1.33784	−0.0583328
$527$	2.06525	0.0899636
$528$	−4.16039	−0.181058
$529$	−22.4457	−0.975902
$530$	−33.3773	−1.44982
$531$	−8.92740	−0.387416
$532$	0	0
$533$	−3.13328	−0.135717
$534$	59.6368	2.58074
$535$	−7.09637	−0.306803
$536$	45.0919	1.94768
$537$	36.2085	1.56251
$538$	2.44482	0.105403
$539$	0	0
$540$	13.9241	0.599199
$541$	−33.6705	−1.44761	−0.723803	−	0.690006i	$-0.757608\pi$
$541$	−33.6705	−1.44761	−0.723803	+	0.690006i	$0.757608\pi$
$542$	−27.0773	−1.16307
$543$	−10.3065	−0.442294
$544$	17.1631	0.735861
$545$	17.1993	0.736738
$546$	0	0
$547$	−21.7800	−0.931247	−0.465623	−	0.884983i	$-0.654170\pi$
$547$	−21.7800	−0.931247	−0.465623	+	0.884983i	$0.654170\pi$
$548$	74.0031	3.16126
$549$	4.90174	0.209201
$550$	−4.80437	−0.204859
$551$	−7.48313	−0.318792
$552$	−5.03532	−0.214317
$553$	0	0
$554$	−20.1518	−0.856166
$555$	1.99442	0.0846585
$556$	−35.6118	−1.51027
$557$	−38.3353	−1.62432	−0.812159	−	0.583437i	$-0.801708\pi$
$557$	−38.3353	−1.62432	−0.812159	+	0.583437i	$0.801708\pi$
$558$	−1.21287	−0.0513451
$559$	−34.5303	−1.46047
$560$	0	0
$561$	15.9524	0.673511
$562$	−20.7875	−0.876869
$563$	1.76908	0.0745576	0.0372788	−	0.999305i	$-0.488131\pi$
$563$	1.76908	0.0745576	0.0372788	+	0.999305i	$0.488131\pi$
$564$	−45.7743	−1.92745
$565$	−4.83555	−0.203433
$566$	−14.6990	−0.617844
$567$	0	0
$568$	1.88140	0.0789418
$569$	−31.8451	−1.33502	−0.667509	−	0.744602i	$-0.732639\pi$
$569$	−31.8451	−1.33502	−0.667509	+	0.744602i	$0.732639\pi$
$570$	−4.12632	−0.172833
$571$	31.5415	1.31997	0.659985	−	0.751279i	$-0.270563\pi$
$571$	31.5415	1.31997	0.659985	+	0.751279i	$0.270563\pi$
$572$	37.6655	1.57487
$573$	−0.791632	−0.0330709
$574$	0	0
$575$	−0.744489	−0.0310474
$576$	−12.0620	−0.502584
$577$	−4.38750	−0.182654	−0.0913270	−	0.995821i	$-0.529111\pi$
$577$	−4.38750	−0.182654	−0.0913270	+	0.995821i	$0.529111\pi$
$578$	−4.40821	−0.183357
$579$	15.3958	0.639827
$580$	29.1567	1.21067
$581$	0	0
$582$	47.1539	1.95459
$583$	−29.4108	−1.21807
$584$	−10.3372	−0.427758
$585$	5.18446	0.214351
$586$	39.8894	1.64782
$587$	−35.7210	−1.47436	−0.737182	−	0.675695i	$-0.763844\pi$
$587$	−35.7210	−1.47436	−0.737182	+	0.675695i	$0.763844\pi$
$588$	0	0
$589$	−0.470722	−0.0193958
$590$	21.3206	0.877755
$591$	−7.55280	−0.310681
$592$	−1.01384	−0.0416687
$593$	−37.5667	−1.54268	−0.771340	−	0.636423i	$-0.780413\pi$
$593$	−37.5667	−1.54268	−0.771340	+	0.636423i	$0.780413\pi$
$594$	19.3773	0.795061
$595$	0	0
$596$	59.6956	2.44523
$597$	−28.5976	−1.17042
$598$	9.21799	0.376952
$599$	−47.2585	−1.93093	−0.965465	−	0.260532i	$-0.916102\pi$
$599$	−47.2585	−1.93093	−0.965465	+	0.260532i	$0.916102\pi$
$600$	6.76345	0.276117
$601$	32.4297	1.32283	0.661417	−	0.750019i	$-0.269955\pi$
$601$	32.4297	1.32283	0.661417	+	0.750019i	$0.269955\pi$
$602$	0	0
$603$	13.0006	0.529426
$604$	−63.4183	−2.58046
$605$	6.76658	0.275101
$606$	−43.3483	−1.76090
$607$	−12.8532	−0.521694	−0.260847	−	0.965380i	$-0.584002\pi$
$607$	−12.8532	−0.521694	−0.260847	+	0.965380i	$0.584002\pi$
$608$	−3.91190	−0.158649
$609$	0	0
$610$	−11.7064	−0.473979
$611$	35.2518	1.42614
$612$	−13.1217	−0.530412
$613$	−7.64274	−0.308687	−0.154344	−	0.988017i	$-0.549326\pi$
$613$	−7.64274	−0.308687	−0.154344	+	0.988017i	$0.549326\pi$
$614$	−14.3405	−0.578737
$615$	−1.17850	−0.0475217
$616$	0	0
$617$	−16.8708	−0.679192	−0.339596	−	0.940571i	$-0.610290\pi$
$617$	−16.8708	−0.679192	−0.339596	+	0.940571i	$0.610290\pi$
$618$	22.1991	0.892979
$619$	34.4221	1.38354	0.691770	−	0.722118i	$-0.256831\pi$
$619$	34.4221	1.38354	0.691770	+	0.722118i	$0.256831\pi$
$620$	1.83409	0.0736587
$621$	3.00273	0.120495
$622$	−11.4097	−0.457486
$623$	0	0
$624$	−10.7220	−0.429223
$625$	1.00000	0.0400000
$626$	−53.7849	−2.14968
$627$	−3.63595	−0.145206
$628$	−52.8269	−2.10802
$629$	3.88743	0.155002
$630$	0	0
$631$	40.0624	1.59486	0.797430	−	0.603411i	$-0.206192\pi$
$631$	40.0624	1.59486	0.797430	+	0.603411i	$0.206192\pi$
$632$	−28.0674	−1.11646
$633$	53.0502	2.10856
$634$	31.4562	1.24928
$635$	−4.76568	−0.189120
$636$	98.4211	3.90265
$637$	0	0
$638$	40.5755	1.60640
$639$	0.542433	0.0214583
$640$	19.9767	0.789649
$641$	7.28133	0.287595	0.143798	−	0.989607i	$-0.454069\pi$
$641$	7.28133	0.287595	0.143798	+	0.989607i	$0.454069\pi$
$642$	33.0479	1.30430
$643$	8.94393	0.352714	0.176357	−	0.984326i	$-0.443569\pi$
$643$	8.94393	0.352714	0.176357	+	0.984326i	$0.443569\pi$
$644$	0	0
$645$	−12.9876	−0.511387
$646$	−8.04283	−0.316441
$647$	−8.22738	−0.323452	−0.161726	−	0.986836i	$-0.551706\pi$
$647$	−8.22738	−0.323452	−0.161726	+	0.986836i	$0.551706\pi$
$648$	−37.2258	−1.46237
$649$	18.7869	0.737449
$650$	−12.3816	−0.485647
$651$	0	0
$652$	1.25411	0.0491148
$653$	−9.37665	−0.366937	−0.183468	−	0.983026i	$-0.558732\pi$
$653$	−9.37665	−0.366937	−0.183468	+	0.983026i	$0.558732\pi$
$654$	−80.0975	−3.13206
$655$	4.84210	0.189197
$656$	0.599077	0.0233900
$657$	−2.98037	−0.116275
$658$	0	0
$659$	−23.2317	−0.904980	−0.452490	−	0.891769i	$-0.649464\pi$
$659$	−23.2317	−0.904980	−0.452490	+	0.891769i	$0.649464\pi$
$660$	14.1669	0.551444
$661$	3.21035	0.124868	0.0624341	−	0.998049i	$-0.480114\pi$
$661$	3.21035	0.124868	0.0624341	+	0.998049i	$0.480114\pi$
$662$	−79.5152	−3.09045
$663$	41.1119	1.59665
$664$	−42.0020	−1.62999
$665$	0	0
$666$	−2.28300	−0.0884646
$667$	6.28762	0.243458
$668$	−31.3684	−1.21368
$669$	−21.4831	−0.830584
$670$	−31.0483	−1.19950
$671$	−10.3152	−0.398216
$672$	0	0
$673$	45.5553	1.75603	0.878014	−	0.478635i	$-0.158868\pi$
$673$	45.5553	1.75603	0.878014	+	0.478635i	$0.158868\pi$
$674$	−22.4177	−0.863497
$675$	−4.03327	−0.155241
$676$	52.1899	2.00730
$677$	−15.4647	−0.594357	−0.297179	−	0.954822i	$-0.596046\pi$
$677$	−15.4647	−0.594357	−0.297179	+	0.954822i	$0.596046\pi$
$678$	22.5192	0.864845
$679$	0	0
$680$	13.1830	0.505544
$681$	−13.2297	−0.506964
$682$	2.55238	0.0977357
$683$	−47.6098	−1.82174	−0.910869	−	0.412696i	$-0.864587\pi$
$683$	−47.6098	−1.82174	−0.910869	+	0.412696i	$0.864587\pi$
$684$	2.99076	0.114355
$685$	−21.4358	−0.819020
$686$	0	0
$687$	−27.6006	−1.05303
$688$	6.60212	0.251703
$689$	−75.7963	−2.88761
$690$	3.46710	0.131990
$691$	21.4983	0.817835	0.408918	−	0.912571i	$-0.365906\pi$
$691$	21.4983	0.817835	0.408918	+	0.912571i	$0.365906\pi$
$692$	77.0889	2.93048
$693$	0	0
$694$	−23.8950	−0.907042
$695$	10.3153	0.391283
$696$	−57.1210	−2.16517
$697$	−2.29707	−0.0870079
$698$	−78.9134	−2.98692
$699$	−39.6064	−1.49805
$700$	0	0
$701$	40.9424	1.54637	0.773186	−	0.634180i	$-0.218662\pi$
$701$	40.9424	1.54637	0.773186	+	0.634180i	$0.218662\pi$
$702$	49.9385	1.88481
$703$	−0.886044	−0.0334178
$704$	25.3834	0.956672
$705$	13.2590	0.499364
$706$	8.52754	0.320938
$707$	0	0
$708$	−62.8690	−2.36276
$709$	1.63237	0.0613048	0.0306524	−	0.999530i	$-0.490242\pi$
$709$	1.63237	0.0613048	0.0306524	+	0.999530i	$0.490242\pi$
$710$	−1.29545	−0.0486174
$711$	−8.09223	−0.303482
$712$	43.4268	1.62749
$713$	0.395519	0.0148123
$714$	0	0
$715$	−10.9102	−0.408019
$716$	62.6764	2.34233
$717$	42.3951	1.58328
$718$	24.4028	0.910706
$719$	35.4624	1.32252	0.661262	−	0.750155i	$-0.270021\pi$
$719$	35.4624	1.32252	0.661262	+	0.750155i	$0.270021\pi$
$720$	−0.991259	−0.0369420
$721$	0	0
$722$	−42.5322	−1.58288
$723$	7.22204	0.268591
$724$	−17.8404	−0.663032
$725$	−8.44555	−0.313660
$726$	−31.5121	−1.16952
$727$	−35.3049	−1.30939	−0.654693	−	0.755895i	$-0.727202\pi$
$727$	−35.3049	−1.30939	−0.654693	+	0.755895i	$0.727202\pi$
$728$	0	0
$729$	13.3995	0.496277
$730$	7.11777	0.263441
$731$	−25.3149	−0.936304
$732$	34.5192	1.27587
$733$	−42.6297	−1.57456	−0.787281	−	0.616594i	$-0.788512\pi$
$733$	−42.6297	−1.57456	−0.787281	+	0.616594i	$0.788512\pi$
$734$	−7.91767	−0.292247
$735$	0	0
$736$	3.28693	0.121158
$737$	−27.3586	−1.00777
$738$	1.34902	0.0496581
$739$	−17.5915	−0.647114	−0.323557	−	0.946209i	$-0.604879\pi$
$739$	−17.5915	−0.647114	−0.323557	+	0.946209i	$0.604879\pi$
$740$	3.45231	0.126910
$741$	−9.37044	−0.344232
$742$	0	0
$743$	14.5952	0.535445	0.267723	−	0.963496i	$-0.413729\pi$
$743$	14.5952	0.535445	0.267723	+	0.963496i	$0.413729\pi$
$744$	−3.59317	−0.131732
$745$	−17.2915	−0.633510
$746$	−29.7367	−1.08874
$747$	−12.1097	−0.443072
$748$	27.6133	1.00964
$749$	0	0
$750$	−4.65702	−0.170050
$751$	1.34737	0.0491662	0.0245831	−	0.999698i	$-0.492174\pi$
$751$	1.34737	0.0491662	0.0245831	+	0.999698i	$0.492174\pi$
$752$	−6.74008	−0.245785
$753$	10.9310	0.398347
$754$	104.570	3.80820
$755$	18.3698	0.668546
$756$	0	0
$757$	1.51676	0.0551277	0.0275639	−	0.999620i	$-0.491225\pi$
$757$	1.51676	0.0551277	0.0275639	+	0.999620i	$0.491225\pi$
$758$	32.1700	1.16847
$759$	3.05507	0.110892
$760$	−3.00474	−0.108993
$761$	18.3985	0.666944	0.333472	−	0.942760i	$-0.391780\pi$
$761$	18.3985	0.666944	0.333472	+	0.942760i	$0.391780\pi$
$762$	22.1938	0.803998
$763$	0	0
$764$	−1.37030	−0.0495758
$765$	3.80084	0.137419
$766$	4.86018	0.175605
$767$	48.4168	1.74823
$768$	−43.8221	−1.58129
$769$	−30.1956	−1.08888	−0.544441	−	0.838799i	$-0.683258\pi$
$769$	−30.1956	−1.08888	−0.544441	+	0.838799i	$0.683258\pi$
$770$	0	0
$771$	−8.52175	−0.306903
$772$	26.6498	0.959149
$773$	3.11757	0.112131	0.0560656	−	0.998427i	$-0.482144\pi$
$773$	3.11757	0.112131	0.0560656	+	0.998427i	$0.482144\pi$
$774$	14.8669	0.534378
$775$	−0.531263	−0.0190835
$776$	34.3369	1.23262
$777$	0	0
$778$	11.1826	0.400915
$779$	0.523561	0.0187585
$780$	36.5103	1.30728
$781$	−1.14150	−0.0408461
$782$	6.75790	0.241662
$783$	34.0632	1.21732
$784$	0	0
$785$	15.3019	0.546148
$786$	−22.5498	−0.804323
$787$	−2.06553	−0.0736282	−0.0368141	−	0.999322i	$-0.511721\pi$
$787$	−2.06553	−0.0736282	−0.0368141	+	0.999322i	$0.511721\pi$
$788$	−13.0738	−0.465734
$789$	−1.14270	−0.0406812
$790$	19.3260	0.687589
$791$	0	0
$792$	−6.82202	−0.242410
$793$	−26.5840	−0.944027
$794$	−14.4662	−0.513386
$795$	−28.5087	−1.01110
$796$	−49.5019	−1.75455
$797$	10.3454	0.366452	0.183226	−	0.983071i	$-0.441346\pi$
$797$	10.3454	0.366452	0.183226	+	0.983071i	$0.441346\pi$
$798$	0	0
$799$	25.8438	0.914289
$800$	−4.41502	−0.156094
$801$	12.5205	0.442392
$802$	−47.5722	−1.67983
$803$	6.27190	0.221331
$804$	91.5536	3.22885
$805$	0	0
$806$	6.57790	0.231697
$807$	2.08820	0.0735083
$808$	−31.5657	−1.11048
$809$	16.6790	0.586401	0.293200	−	0.956051i	$-0.405280\pi$
$809$	16.6790	0.586401	0.293200	+	0.956051i	$0.405280\pi$
$810$	25.6320	0.900618
$811$	55.7573	1.95790	0.978951	−	0.204093i	$-0.0654246\pi$
$811$	55.7573	1.95790	0.978951	+	0.204093i	$0.0654246\pi$
$812$	0	0
$813$	−23.1277	−0.811123
$814$	4.80437	0.168393
$815$	−0.363267	−0.0127247
$816$	−7.86052	−0.275173
$817$	5.76990	0.201863
$818$	−34.5013	−1.20631
$819$	0	0
$820$	−2.03996	−0.0712386
$821$	−28.6230	−0.998949	−0.499475	−	0.866329i	$-0.666474\pi$
$821$	−28.6230	−0.998949	−0.499475	+	0.866329i	$0.666474\pi$
$822$	99.8269	3.48186
$823$	−37.4904	−1.30683	−0.653416	−	0.756999i	$-0.726665\pi$
$823$	−37.4904	−1.30683	−0.653416	+	0.756999i	$0.726665\pi$
$824$	16.1651	0.563138
$825$	−4.10358	−0.142868
$826$	0	0
$827$	−53.9684	−1.87666	−0.938332	−	0.345736i	$-0.887629\pi$
$827$	−53.9684	−1.87666	−0.938332	+	0.345736i	$0.887629\pi$
$828$	−2.51296	−0.0873313
$829$	−23.8717	−0.829100	−0.414550	−	0.910027i	$-0.636061\pi$
$829$	−23.8717	−0.829100	−0.414550	+	0.910027i	$0.636061\pi$
$830$	28.9207	1.00385
$831$	−17.2123	−0.597089
$832$	65.4171	2.26793
$833$	0	0
$834$	−48.0387	−1.66344
$835$	9.08619	0.314441
$836$	−6.29378	−0.217675
$837$	2.14273	0.0740635
$838$	85.1963	2.94306
$839$	55.4905	1.91574	0.957872	−	0.287197i	$-0.0927233\pi$
$839$	55.4905	1.91574	0.957872	+	0.287197i	$0.0927233\pi$
$840$	0	0
$841$	42.3273	1.45956
$842$	−56.7189	−1.95466
$843$	−17.7554	−0.611528
$844$	91.8291	3.16089
$845$	−15.1174	−0.520053
$846$	−15.1775	−0.521814
$847$	0	0
$848$	14.4921	0.497661
$849$	−12.5549	−0.430884
$850$	−9.07723	−0.311346
$851$	0.744489	0.0255208
$852$	3.81995	0.130869
$853$	17.8941	0.612681	0.306341	−	0.951922i	$-0.400895\pi$
$853$	17.8941	0.612681	0.306341	+	0.951922i	$0.400895\pi$
$854$	0	0
$855$	−0.866307	−0.0296271
$856$	24.0651	0.822528
$857$	15.1126	0.516237	0.258118	−	0.966113i	$-0.416898\pi$
$857$	15.1126	0.516237	0.258118	+	0.966113i	$0.416898\pi$
$858$	50.8090	1.73459
$859$	−39.2354	−1.33870	−0.669348	−	0.742949i	$-0.733426\pi$
$859$	−39.2354	−1.33870	−0.669348	+	0.742949i	$0.733426\pi$
$860$	−22.4814	−0.766608
$861$	0	0
$862$	12.9435	0.440859
$863$	38.8554	1.32265	0.661327	−	0.750098i	$-0.269994\pi$
$863$	38.8554	1.32265	0.661327	+	0.750098i	$0.269994\pi$
$864$	17.8070	0.605806
$865$	−22.3296	−0.759230
$866$	34.4875	1.17193
$867$	−3.76521	−0.127873
$868$	0	0
$869$	17.0293	0.577680
$870$	39.3311	1.33345
$871$	−70.5075	−2.38906
$872$	−58.3260	−1.97517
$873$	9.89979	0.335057
$874$	−1.54030	−0.0521013
$875$	0	0
$876$	−20.9885	−0.709135
$877$	11.1573	0.376755	0.188377	−	0.982097i	$-0.439677\pi$
$877$	11.1573	0.376755	0.188377	+	0.982097i	$0.439677\pi$
$878$	−8.89274	−0.300115
$879$	34.0709	1.14918
$880$	2.08601	0.0703194
$881$	−35.5972	−1.19930	−0.599651	−	0.800262i	$-0.704694\pi$
$881$	−35.5972	−1.19930	−0.599651	+	0.800262i	$0.704694\pi$
$882$	0	0
$883$	−17.2704	−0.581196	−0.290598	−	0.956845i	$-0.593854\pi$
$883$	−17.2704	−0.581196	−0.290598	+	0.956845i	$0.593854\pi$
$884$	71.1640	2.39351
$885$	18.2107	0.612145
$886$	−3.02003	−0.101460
$887$	−8.22033	−0.276012	−0.138006	−	0.990431i	$-0.544069\pi$
$887$	−8.22033	−0.276012	−0.138006	+	0.990431i	$0.544069\pi$
$888$	−6.76345	−0.226966
$889$	0	0
$890$	−29.9018	−1.00231
$891$	22.5860	0.756658
$892$	−37.1869	−1.24511
$893$	−5.89046	−0.197117
$894$	80.5266	2.69321
$895$	−18.1549	−0.606851
$896$	0	0
$897$	7.87342	0.262886
$898$	54.4898	1.81835
$899$	4.48681	0.149643
$900$	3.37541	0.112514
$901$	−55.5679	−1.85123
$902$	−2.83889	−0.0945246
$903$	0	0
$904$	16.3982	0.545396
$905$	5.16766	0.171779
$906$	−85.5485	−2.84216
$907$	47.2081	1.56752	0.783759	−	0.621065i	$-0.213300\pi$
$907$	47.2081	1.56752	0.783759	+	0.621065i	$0.213300\pi$
$908$	−22.9004	−0.759977
$909$	−9.10081	−0.301855
$910$	0	0
$911$	−29.1565	−0.965997	−0.482998	−	0.875621i	$-0.660452\pi$
$911$	−29.1565	−0.965997	−0.482998	+	0.875621i	$0.660452\pi$
$912$	1.79161	0.0593262
$913$	25.4838	0.843391
$914$	−1.95700	−0.0647319
$915$	−9.99887	−0.330553
$916$	−47.7763	−1.57857
$917$	0	0
$918$	36.6110	1.20834
$919$	−12.0346	−0.396984	−0.198492	−	0.980102i	$-0.563604\pi$
$919$	−12.0346	−0.396984	−0.198492	+	0.980102i	$0.563604\pi$
$920$	2.52470	0.0832368
$921$	−12.2488	−0.403611
$922$	61.9429	2.03998
$923$	−2.94183	−0.0968315
$924$	0	0
$925$	−1.00000	−0.0328798
$926$	34.8899	1.14655
$927$	4.66062	0.153075
$928$	37.2872	1.22401
$929$	−9.35722	−0.307000	−0.153500	−	0.988149i	$-0.549055\pi$
$929$	−9.35722	−0.307000	−0.153500	+	0.988149i	$0.549055\pi$
$930$	2.47410	0.0811289
$931$	0	0
$932$	−68.5580	−2.24569
$933$	−9.74540	−0.319050
$934$	−49.3660	−1.61530
$935$	−7.99850	−0.261579
$936$	−17.5814	−0.574667
$937$	−6.94565	−0.226905	−0.113452	−	0.993543i	$-0.536191\pi$
$937$	−6.94565	−0.226905	−0.113452	+	0.993543i	$0.536191\pi$
$938$	0	0
$939$	−45.9396	−1.49918
$940$	22.9511	0.748584
$941$	31.4542	1.02538	0.512688	−	0.858575i	$-0.328650\pi$
$941$	31.4542	1.02538	0.512688	+	0.858575i	$0.328650\pi$
$942$	−71.2612	−2.32181
$943$	−0.439917	−0.0143257
$944$	−9.25720	−0.301296
$945$	0	0
$946$	−31.2859	−1.01719
$947$	−2.62121	−0.0851778	−0.0425889	−	0.999093i	$-0.513561\pi$
$947$	−2.62121	−0.0851778	−0.0425889	+	0.999093i	$0.513561\pi$
$948$	−56.9875	−1.85087
$949$	16.1637	0.524696
$950$	2.06893	0.0671250
$951$	26.8679	0.871250
$952$	0	0
$953$	−56.7897	−1.83960	−0.919799	−	0.392390i	$-0.871648\pi$
$953$	−56.7897	−1.83960	−0.919799	+	0.392390i	$0.871648\pi$
$954$	32.6338	1.05656
$955$	0.396923	0.0128441
$956$	73.3853	2.37345
$957$	34.6570	1.12030
$958$	−46.9934	−1.51829
$959$	0	0
$960$	24.6049	0.794119
$961$	−30.7178	−0.990895
$962$	12.3816	0.399200
$963$	6.93829	0.223583
$964$	12.5012	0.402638
$965$	−7.71941	−0.248497
$966$	0	0
$967$	2.20757	0.0709906	0.0354953	−	0.999370i	$-0.488699\pi$
$967$	2.20757	0.0709906	0.0354953	+	0.999370i	$0.488699\pi$
$968$	−22.9467	−0.737535
$969$	−6.86967	−0.220685
$970$	−23.6429	−0.759127
$971$	43.7537	1.40412	0.702062	−	0.712116i	$-0.252263\pi$
$971$	43.7537	1.40412	0.702062	+	0.712116i	$0.252263\pi$
$972$	−33.8099	−1.08445
$973$	0	0
$974$	51.8495	1.66137
$975$	−10.5756	−0.338690
$976$	5.08282	0.162697
$977$	3.79551	0.121429	0.0607146	−	0.998155i	$-0.480662\pi$
$977$	3.79551	0.121429	0.0607146	+	0.998155i	$0.480662\pi$
$978$	1.69174	0.0540959
$979$	−26.3483	−0.842095
$980$	0	0
$981$	−16.8162	−0.536899
$982$	−26.9536	−0.860124
$983$	0.344706	0.0109944	0.00549721	−	0.999985i	$-0.498250\pi$
$983$	0.344706	0.0109944	0.00549721	+	0.999985i	$0.498250\pi$
$984$	3.99650	0.127404
$985$	3.78696	0.120663
$986$	76.6622	2.44142
$987$	0	0
$988$	−16.2201	−0.516030
$989$	−4.84809	−0.154160
$990$	4.69734	0.149291
$991$	3.79464	0.120541	0.0602704	−	0.998182i	$-0.480804\pi$
$991$	3.79464	0.120541	0.0602704	+	0.998182i	$0.480804\pi$
$992$	2.34553	0.0744708
$993$	−67.9168	−2.15527
$994$	0	0
$995$	14.3388	0.454569
$996$	−85.2798	−2.70219
$997$	−45.8738	−1.45284	−0.726419	−	0.687252i	$-0.758817\pi$
$997$	−45.8738	−1.45284	−0.726419	+	0.687252i	$0.758817\pi$
$998$	76.3104	2.41556
$999$	4.03327	0.127607

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9065.2.a.bd.1.34	✓	38
7.6	odd	2		9065.2.a.be.1.34	yes	38

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9065.2.a.bd.1.34	✓	38	1.1	even	1	trivial
9065.2.a.be.1.34	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)

\(f(q)\)	\(=\)	\(q+2.33502 q^{2} +1.99442 q^{3} +3.45231 q^{4} -1.00000 q^{5} +4.65702 q^{6} +3.39118 q^{8} +0.977724 q^{9} -2.33502 q^{10} -2.05753 q^{11} +6.88537 q^{12} -5.30258 q^{13} -1.99442 q^{15} +1.01384 q^{16} -3.88743 q^{17} +2.28300 q^{18} +0.886044 q^{19} -3.45231 q^{20} -4.80437 q^{22} -0.744489 q^{23} +6.76345 q^{24} +1.00000 q^{25} -12.3816 q^{26} -4.03327 q^{27} -8.44555 q^{29} -4.65702 q^{30} -0.531263 q^{31} -4.41502 q^{32} -4.10358 q^{33} -9.07723 q^{34} +3.37541 q^{36} -1.00000 q^{37} +2.06893 q^{38} -10.5756 q^{39} -3.39118 q^{40} +0.590897 q^{41} +6.51197 q^{43} -7.10323 q^{44} -0.977724 q^{45} -1.73840 q^{46} -6.64805 q^{47} +2.02203 q^{48} +2.33502 q^{50} -7.75319 q^{51} -18.3062 q^{52} +14.2942 q^{53} -9.41777 q^{54} +2.05753 q^{55} +1.76715 q^{57} -19.7205 q^{58} -9.13080 q^{59} -6.88537 q^{60} +5.01342 q^{61} -1.24051 q^{62} -12.3368 q^{64} +5.30258 q^{65} -9.58194 q^{66} +13.2968 q^{67} -13.4206 q^{68} -1.48483 q^{69} +0.554792 q^{71} +3.31564 q^{72} -3.04827 q^{73} -2.33502 q^{74} +1.99442 q^{75} +3.05890 q^{76} -24.6942 q^{78} -8.27660 q^{79} -1.01384 q^{80} -10.9772 q^{81} +1.37976 q^{82} -12.3856 q^{83} +3.88743 q^{85} +15.2056 q^{86} -16.8440 q^{87} -6.97745 q^{88} +12.8058 q^{89} -2.28300 q^{90} -2.57021 q^{92} -1.05956 q^{93} -15.5233 q^{94} -0.886044 q^{95} -8.80541 q^{96} +10.1253 q^{97} -2.01169 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 38 q - 2 q^{2} - 6 q^{3} + 42 q^{4} - 38 q^{5} - 8 q^{6} - 6 q^{8} + 48 q^{9} + 2 q^{10} + 34 q^{11} - 20 q^{12} - 22 q^{13} + 6 q^{15} + 46 q^{16} - 22 q^{17} - 36 q^{18} - 40 q^{19} - 42 q^{20} - 4 q^{22}+ \cdots + 92 q^{99}+O(q^{100}) \) 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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