LMFDB - Embedded newform 243.2.a.f.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [243,2,Mod(1,243)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(243, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("243.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 243 = 3^{5} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	243.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:	$1.94036476912$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.53209$ of defining polynomial
Character	$\chi$	$=$	243.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.53209 q^{2} +4.41147 q^{4} +0.467911 q^{5} -3.22668 q^{7} +6.10607 q^{8} +1.18479 q^{10} -3.10607 q^{11} -2.18479 q^{13} -8.17024 q^{14} +6.63816 q^{16} +3.00000 q^{17} +0.0418891 q^{19} +2.06418 q^{20} -7.86484 q^{22} +6.10607 q^{23} -4.78106 q^{25} -5.53209 q^{26} -14.2344 q^{28} +6.57398 q^{29} -6.22668 q^{31} +4.59627 q^{32} +7.59627 q^{34} -1.50980 q^{35} +3.59627 q^{37} +0.106067 q^{38} +2.85710 q^{40} +7.70233 q^{41} -0.588526 q^{43} -13.7023 q^{44} +15.4611 q^{46} -9.66044 q^{47} +3.41147 q^{49} -12.1061 q^{50} -9.63816 q^{52} +4.95811 q^{53} -1.45336 q^{55} -19.7023 q^{56} +16.6459 q^{58} -8.53209 q^{59} -1.26857 q^{61} -15.7665 q^{62} -1.63816 q^{64} -1.02229 q^{65} +10.0077 q^{67} +13.2344 q^{68} -3.82295 q^{70} +11.8307 q^{71} -8.23442 q^{73} +9.10607 q^{74} +0.184793 q^{76} +10.0223 q^{77} +11.0496 q^{79} +3.10607 q^{80} +19.5030 q^{82} -1.50980 q^{83} +1.40373 q^{85} -1.49020 q^{86} -18.9659 q^{88} -15.8726 q^{89} +7.04963 q^{91} +26.9368 q^{92} -24.4611 q^{94} +0.0196004 q^{95} +18.6459 q^{97} +8.63816 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{4} + 6 q^{5} - 3 q^{7} + 6 q^{8} + 3 q^{11} - 3 q^{13} - 3 q^{14} + 3 q^{16} + 9 q^{17} - 3 q^{19} - 3 q^{20} + 6 q^{23} + 3 q^{25} - 12 q^{26} - 12 q^{28} + 12 q^{29} - 12 q^{31} + 9 q^{34}+ \cdots + 9 q^{98}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^4 + 6 * q^5 - 3 * q^7 + 6 * q^8 + 3 * q^11 - 3 * q^13 - 3 * q^14 + 3 * q^16 + 9 * q^17 - 3 * q^19 - 3 * q^20 + 6 * q^23 + 3 * q^25 - 12 * q^26 - 12 * q^28 + 12 * q^29 - 12 * q^31 + 9 * q^34 - 6 * q^35 - 3 * q^37 - 12 * q^38 + 9 * q^40 - 3 * q^41 - 12 * q^43 - 15 * q^44 + 9 * q^46 - 6 * q^47 - 24 * q^50 - 12 * q^52 + 18 * q^53 + 9 * q^55 - 33 * q^56 + 9 * q^58 - 21 * q^59 + 6 * q^61 - 12 * q^62 + 12 * q^64 + 3 * q^65 + 6 * q^67 + 9 * q^68 + 9 * q^70 - 9 * q^71 + 6 * q^73 + 15 * q^74 - 3 * q^76 + 24 * q^77 + 6 * q^79 - 3 * q^80 + 18 * q^82 - 6 * q^83 + 18 * q^85 - 3 * q^86 - 36 * q^88 - 6 * q^91 + 24 * q^92 - 36 * q^94 + 3 * q^95 + 15 * q^97 + 9 * q^98

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 $

$2$	2.53209	1.79046	0.895229	−	0.445607i	$-0.147012\pi$
$2$	2.53209	1.79046	0.895229	+	0.445607i	$0.147012\pi$
$3$	0	0
$3$	0	0
$4$	4.41147	2.20574
$5$	0.467911	0.209256	0.104628	−	0.994511i	$-0.466635\pi$
$5$	0.467911	0.209256	0.104628	+	0.994511i	$0.466635\pi$
$6$	0	0
$7$	−3.22668	−1.21957	−0.609786	−	0.792566i	$-0.708744\pi$
$7$	−3.22668	−1.21957	−0.609786	+	0.792566i	$0.708744\pi$
$8$	6.10607	2.15882
$9$	0	0
$10$	1.18479	0.374664
$11$	−3.10607	−0.936514	−0.468257	−	0.883592i	$-0.655118\pi$
$11$	−3.10607	−0.936514	−0.468257	+	0.883592i	$0.655118\pi$
$12$	0	0
$13$	−2.18479	−0.605952	−0.302976	−	0.952998i	$-0.597980\pi$
$13$	−2.18479	−0.605952	−0.302976	+	0.952998i	$0.597980\pi$
$14$	−8.17024	−2.18359
$15$	0	0
$16$	6.63816	1.65954
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	0	0
$19$	0.0418891	0.00961001	0.00480501	−	0.999988i	$-0.498471\pi$
$19$	0.0418891	0.00961001	0.00480501	+	0.999988i	$0.498471\pi$
$20$	2.06418	0.461564
$21$	0	0
$22$	−7.86484	−1.67679
$23$	6.10607	1.27320	0.636601	−	0.771193i	$-0.280340\pi$
$23$	6.10607	1.27320	0.636601	+	0.771193i	$0.280340\pi$
$24$	0	0
$25$	−4.78106	−0.956212
$26$	−5.53209	−1.08493
$27$	0	0
$28$	−14.2344	−2.69005
$29$	6.57398	1.22076	0.610379	−	0.792110i	$-0.291017\pi$
$29$	6.57398	1.22076	0.610379	+	0.792110i	$0.291017\pi$
$30$	0	0
$31$	−6.22668	−1.11835	−0.559173	−	0.829051i	$-0.688881\pi$
$31$	−6.22668	−1.11835	−0.559173	+	0.829051i	$0.688881\pi$
$32$	4.59627	0.812513
$33$	0	0
$34$	7.59627	1.30275
$35$	−1.50980	−0.255203
$36$	0	0
$37$	3.59627	0.591223	0.295611	−	0.955308i	$-0.404477\pi$
$37$	3.59627	0.591223	0.295611	+	0.955308i	$0.404477\pi$
$38$	0.106067	0.0172063
$39$	0	0
$40$	2.85710	0.451747
$41$	7.70233	1.20290	0.601451	−	0.798910i	$-0.294589\pi$
$41$	7.70233	1.20290	0.601451	+	0.798910i	$0.294589\pi$
$42$	0	0
$43$	−0.588526	−0.0897494	−0.0448747	−	0.998993i	$-0.514289\pi$
$43$	−0.588526	−0.0897494	−0.0448747	+	0.998993i	$0.514289\pi$
$44$	−13.7023	−2.06570
$45$	0	0
$46$	15.4611	2.27962
$47$	−9.66044	−1.40912	−0.704560	−	0.709644i	$-0.748856\pi$
$47$	−9.66044	−1.40912	−0.704560	+	0.709644i	$0.748856\pi$
$48$	0	0
$49$	3.41147	0.487353
$50$	−12.1061	−1.71206
$51$	0	0
$52$	−9.63816	−1.33657
$53$	4.95811	0.681049	0.340524	−	0.940236i	$-0.389395\pi$
$53$	4.95811	0.681049	0.340524	+	0.940236i	$0.389395\pi$
$54$	0	0
$55$	−1.45336	−0.195971
$56$	−19.7023	−2.63284
$57$	0	0
$58$	16.6459	2.18571
$59$	−8.53209	−1.11078	−0.555392	−	0.831589i	$-0.687432\pi$
$59$	−8.53209	−1.11078	−0.555392	+	0.831589i	$0.687432\pi$
$60$	0	0
$61$	−1.26857	−0.162424	−0.0812119	−	0.996697i	$-0.525879\pi$
$61$	−1.26857	−0.162424	−0.0812119	+	0.996697i	$0.525879\pi$
$62$	−15.7665	−2.00235
$63$	0	0
$64$	−1.63816	−0.204769
$65$	−1.02229	−0.126799
$66$	0	0
$67$	10.0077	1.22264	0.611320	−	0.791383i	$-0.290639\pi$
$67$	10.0077	1.22264	0.611320	+	0.791383i	$0.290639\pi$
$68$	13.2344	1.60491
$69$	0	0
$70$	−3.82295	−0.456930
$71$	11.8307	1.40404	0.702022	−	0.712155i	$-0.252281\pi$
$71$	11.8307	1.40404	0.702022	+	0.712155i	$0.252281\pi$
$72$	0	0
$73$	−8.23442	−0.963766	−0.481883	−	0.876236i	$-0.660047\pi$
$73$	−8.23442	−0.963766	−0.481883	+	0.876236i	$0.660047\pi$
$74$	9.10607	1.05856
$75$	0	0
$76$	0.184793	0.0211972
$77$	10.0223	1.14215
$78$	0	0
$79$	11.0496	1.24318	0.621590	−	0.783343i	$-0.286487\pi$
$79$	11.0496	1.24318	0.621590	+	0.783343i	$0.286487\pi$
$80$	3.10607	0.347269
$81$	0	0
$82$	19.5030	2.15375
$83$	−1.50980	−0.165722	−0.0828610	−	0.996561i	$-0.526406\pi$
$83$	−1.50980	−0.165722	−0.0828610	+	0.996561i	$0.526406\pi$
$84$	0	0
$85$	1.40373	0.152256
$86$	−1.49020	−0.160692
$87$	0	0
$88$	−18.9659	−2.02177
$89$	−15.8726	−1.68249	−0.841245	−	0.540654i	$-0.818177\pi$
$89$	−15.8726	−1.68249	−0.841245	+	0.540654i	$0.818177\pi$
$90$	0	0
$91$	7.04963	0.739002
$92$	26.9368	2.80835
$93$	0	0
$94$	−24.4611	−2.52297
$95$	0.0196004	0.00201095
$96$	0	0
$97$	18.6459	1.89320	0.946602	−	0.322405i	$-0.104491\pi$
$97$	18.6459	1.89320	0.946602	+	0.322405i	$0.104491\pi$
$98$	8.63816	0.872585
$99$	0	0
$100$	−21.0915	−2.10915
$101$	9.08647	0.904137	0.452069	−	0.891983i	$-0.350686\pi$
$101$	9.08647	0.904137	0.452069	+	0.891983i	$0.350686\pi$
$102$	0	0
$103$	0.260830	0.0257003	0.0128502	−	0.999917i	$-0.495910\pi$
$103$	0.260830	0.0257003	0.0128502	+	0.999917i	$0.495910\pi$
$104$	−13.3405	−1.30814
$105$	0	0
$106$	12.5544	1.21939
$107$	4.04189	0.390744	0.195372	−	0.980729i	$-0.437409\pi$
$107$	4.04189	0.390744	0.195372	+	0.980729i	$0.437409\pi$
$108$	0	0
$109$	−5.40373	−0.517584	−0.258792	−	0.965933i	$-0.583324\pi$
$109$	−5.40373	−0.517584	−0.258792	+	0.965933i	$0.583324\pi$
$110$	−3.68004	−0.350879
$111$	0	0
$112$	−21.4192	−2.02393
$113$	1.38413	0.130208	0.0651041	−	0.997878i	$-0.479262\pi$
$113$	1.38413	0.130208	0.0651041	+	0.997878i	$0.479262\pi$
$114$	0	0
$115$	2.85710	0.266426
$116$	29.0009	2.69267
$117$	0	0
$118$	−21.6040	−1.98881
$119$	−9.68004	−0.887368
$120$	0	0
$121$	−1.35235	−0.122941
$122$	−3.21213	−0.290813
$123$	0	0
$124$	−27.4688	−2.46678
$125$	−4.57667	−0.409349
$126$	0	0
$127$	−6.63816	−0.589041	−0.294521	−	0.955645i	$-0.595160\pi$
$127$	−6.63816	−0.589041	−0.294521	+	0.955645i	$0.595160\pi$
$128$	−13.3405	−1.17914
$129$	0	0
$130$	−2.58853	−0.227029
$131$	−12.6408	−1.10444	−0.552218	−	0.833700i	$-0.686218\pi$
$131$	−12.6408	−1.10444	−0.552218	+	0.833700i	$0.686218\pi$
$132$	0	0
$133$	−0.135163	−0.0117201
$134$	25.3405	2.18908
$135$	0	0
$136$	18.3182	1.57077
$137$	10.6159	0.906975	0.453487	−	0.891263i	$-0.350180\pi$
$137$	10.6159	0.906975	0.453487	+	0.891263i	$0.350180\pi$
$138$	0	0
$139$	−7.46110	−0.632843	−0.316421	−	0.948619i	$-0.602481\pi$
$139$	−7.46110	−0.632843	−0.316421	+	0.948619i	$0.602481\pi$
$140$	−6.66044	−0.562910
$141$	0	0
$142$	29.9564	2.51388
$143$	6.78611	0.567483
$144$	0	0
$145$	3.07604	0.255451
$146$	−20.8503	−1.72558
$147$	0	0
$148$	15.8648	1.30408
$149$	−4.25402	−0.348503	−0.174252	−	0.984701i	$-0.555751\pi$
$149$	−4.25402	−0.348503	−0.174252	+	0.984701i	$0.555751\pi$
$150$	0	0
$151$	0.135163	0.0109994	0.00549969	−	0.999985i	$-0.498249\pi$
$151$	0.135163	0.0109994	0.00549969	+	0.999985i	$0.498249\pi$
$152$	0.255777	0.0207463
$153$	0	0
$154$	25.3773	2.04496
$155$	−2.91353	−0.234021
$156$	0	0
$157$	13.3259	1.06353	0.531763	−	0.846893i	$-0.321530\pi$
$157$	13.3259	1.06353	0.531763	+	0.846893i	$0.321530\pi$
$158$	27.9786	2.22586
$159$	0	0
$160$	2.15064	0.170023
$161$	−19.7023	−1.55276
$162$	0	0
$163$	−9.76382	−0.764762	−0.382381	−	0.924005i	$-0.624896\pi$
$163$	−9.76382	−0.764762	−0.382381	+	0.924005i	$0.624896\pi$
$164$	33.9786	2.65329
$165$	0	0
$166$	−3.82295	−0.296718
$167$	−3.57398	−0.276563	−0.138281	−	0.990393i	$-0.544158\pi$
$167$	−3.57398	−0.276563	−0.138281	+	0.990393i	$0.544158\pi$
$168$	0	0
$169$	−8.22668	−0.632822
$170$	3.55438	0.272608
$171$	0	0
$172$	−2.59627	−0.197963
$173$	18.7665	1.42679	0.713396	−	0.700761i	$-0.247156\pi$
$173$	18.7665	1.42679	0.713396	+	0.700761i	$0.247156\pi$
$174$	0	0
$175$	15.4270	1.16617
$176$	−20.6186	−1.55418
$177$	0	0
$178$	−40.1908	−3.01243
$179$	−5.08378	−0.379979	−0.189990	−	0.981786i	$-0.560845\pi$
$179$	−5.08378	−0.379979	−0.189990	+	0.981786i	$0.560845\pi$
$180$	0	0
$181$	7.15064	0.531503	0.265752	−	0.964042i	$-0.414380\pi$
$181$	7.15064	0.531503	0.265752	+	0.964042i	$0.414380\pi$
$182$	17.8503	1.32315
$183$	0	0
$184$	37.2841	2.74862
$185$	1.68273	0.123717
$186$	0	0
$187$	−9.31820	−0.681414
$188$	−42.6168	−3.10815
$189$	0	0
$190$	0.0496299	0.00360053
$191$	−10.5098	−0.760462	−0.380231	−	0.924891i	$-0.624156\pi$
$191$	−10.5098	−0.760462	−0.380231	+	0.924891i	$0.624156\pi$
$192$	0	0
$193$	−10.1429	−0.730102	−0.365051	−	0.930987i	$-0.618948\pi$
$193$	−10.1429	−0.730102	−0.365051	+	0.930987i	$0.618948\pi$
$194$	47.2131	3.38970
$195$	0	0
$196$	15.0496	1.07497
$197$	14.0838	1.00343	0.501714	−	0.865034i	$-0.332703\pi$
$197$	14.0838	1.00343	0.501714	+	0.865034i	$0.332703\pi$
$198$	0	0
$199$	10.2763	0.728468	0.364234	−	0.931307i	$-0.381331\pi$
$199$	10.2763	0.728468	0.364234	+	0.931307i	$0.381331\pi$
$200$	−29.1935	−2.06429
$201$	0	0
$202$	23.0077	1.61882
$203$	−21.2121	−1.48880
$204$	0	0
$205$	3.60401	0.251715
$206$	0.660444	0.0460153
$207$	0	0
$208$	−14.5030	−1.00560
$209$	−0.130110	−0.00899991
$210$	0	0
$211$	−7.14290	−0.491738	−0.245869	−	0.969303i	$-0.579073\pi$
$211$	−7.14290	−0.491738	−0.245869	+	0.969303i	$0.579073\pi$
$212$	21.8726	1.50221
$213$	0	0
$214$	10.2344	0.699611
$215$	−0.275378	−0.0187806
$216$	0	0
$217$	20.0915	1.36390
$218$	−13.6827	−0.926712
$219$	0	0
$220$	−6.41147	−0.432261
$221$	−6.55438	−0.440895
$222$	0	0
$223$	−10.3354	−0.692112	−0.346056	−	0.938214i	$-0.612479\pi$
$223$	−10.3354	−0.692112	−0.346056	+	0.938214i	$0.612479\pi$
$224$	−14.8307	−0.990917
$225$	0	0
$226$	3.50475	0.233132
$227$	−13.0223	−0.864320	−0.432160	−	0.901797i	$-0.642248\pi$
$227$	−13.0223	−0.864320	−0.432160	+	0.901797i	$0.642248\pi$
$228$	0	0
$229$	−28.0993	−1.85685	−0.928426	−	0.371518i	$-0.878837\pi$
$229$	−28.0993	−1.85685	−0.928426	+	0.371518i	$0.878837\pi$
$230$	7.23442	0.477024
$231$	0	0
$232$	40.1411	2.63540
$233$	−13.9145	−0.911567	−0.455784	−	0.890091i	$-0.650641\pi$
$233$	−13.9145	−0.911567	−0.455784	+	0.890091i	$0.650641\pi$
$234$	0	0
$235$	−4.52023	−0.294867
$236$	−37.6391	−2.45010
$237$	0	0
$238$	−24.5107	−1.58879
$239$	−15.0196	−0.971537	−0.485769	−	0.874087i	$-0.661460\pi$
$239$	−15.0196	−0.971537	−0.485769	+	0.874087i	$0.661460\pi$
$240$	0	0
$241$	−12.9736	−0.835703	−0.417851	−	0.908515i	$-0.637217\pi$
$241$	−12.9736	−0.835703	−0.417851	+	0.908515i	$0.637217\pi$
$242$	−3.42427	−0.220120
$243$	0	0
$244$	−5.59627	−0.358264
$245$	1.59627	0.101982
$246$	0	0
$247$	−0.0915189	−0.00582321
$248$	−38.0205	−2.41431
$249$	0	0
$250$	−11.5885	−0.732923
$251$	0.872578	0.0550766	0.0275383	−	0.999621i	$-0.491233\pi$
$251$	0.872578	0.0550766	0.0275383	+	0.999621i	$0.491233\pi$
$252$	0	0
$253$	−18.9659	−1.19237
$254$	−16.8084	−1.05465
$255$	0	0
$256$	−30.5030	−1.90644
$257$	4.57667	0.285485	0.142742	−	0.989760i	$-0.454408\pi$
$257$	4.57667	0.285485	0.142742	+	0.989760i	$0.454408\pi$
$258$	0	0
$259$	−11.6040	−0.721038
$260$	−4.50980	−0.279686
$261$	0	0
$262$	−32.0077	−1.97744
$263$	4.29767	0.265005	0.132503	−	0.991183i	$-0.457699\pi$
$263$	4.29767	0.265005	0.132503	+	0.991183i	$0.457699\pi$
$264$	0	0
$265$	2.31996	0.142514
$266$	−0.342244	−0.0209843
$267$	0	0
$268$	44.1489	2.69682
$269$	12.1257	0.739315	0.369657	−	0.929168i	$-0.379475\pi$
$269$	12.1257	0.739315	0.369657	+	0.929168i	$0.379475\pi$
$270$	0	0
$271$	−0.319955	−0.0194359	−0.00971795	−	0.999953i	$-0.503093\pi$
$271$	−0.319955	−0.0194359	−0.00971795	+	0.999953i	$0.503093\pi$
$272$	19.9145	1.20749
$273$	0	0
$274$	26.8803	1.62390
$275$	14.8503	0.895506
$276$	0	0
$277$	−26.8212	−1.61153	−0.805765	−	0.592236i	$-0.798245\pi$
$277$	−26.8212	−1.61153	−0.805765	+	0.592236i	$0.798245\pi$
$278$	−18.8922	−1.13308
$279$	0	0
$280$	−9.21894	−0.550937
$281$	26.6810	1.59165	0.795827	−	0.605524i	$-0.207037\pi$
$281$	26.6810	1.59165	0.795827	+	0.605524i	$0.207037\pi$
$282$	0	0
$283$	−9.29355	−0.552444	−0.276222	−	0.961094i	$-0.589083\pi$
$283$	−9.29355	−0.552444	−0.276222	+	0.961094i	$0.589083\pi$
$284$	52.1908	3.09695
$285$	0	0
$286$	17.1830	1.01605
$287$	−24.8530	−1.46702
$288$	0	0
$289$	−8.00000	−0.470588
$290$	7.78880	0.457374
$291$	0	0
$292$	−36.3259	−2.12581
$293$	−19.6391	−1.14733	−0.573664	−	0.819091i	$-0.694478\pi$
$293$	−19.6391	−1.14733	−0.573664	+	0.819091i	$0.694478\pi$
$294$	0	0
$295$	−3.99226	−0.232438
$296$	21.9590	1.27634
$297$	0	0
$298$	−10.7716	−0.623980
$299$	−13.3405	−0.771500
$300$	0	0
$301$	1.89899	0.109456
$302$	0.342244	0.0196939
$303$	0	0
$304$	0.278066	0.0159482
$305$	−0.593578	−0.0339882
$306$	0	0
$307$	28.3432	1.61763	0.808815	−	0.588063i	$-0.200109\pi$
$307$	28.3432	1.61763	0.808815	+	0.588063i	$0.200109\pi$
$308$	44.2131	2.51927
$309$	0	0
$310$	−7.37733	−0.419004
$311$	−2.04458	−0.115937	−0.0579687	−	0.998318i	$-0.518462\pi$
$311$	−2.04458	−0.115937	−0.0579687	+	0.998318i	$0.518462\pi$
$312$	0	0
$313$	8.41147	0.475445	0.237722	−	0.971333i	$-0.423599\pi$
$313$	8.41147	0.475445	0.237722	+	0.971333i	$0.423599\pi$
$314$	33.7425	1.90420
$315$	0	0
$316$	48.7452	2.74213
$317$	31.1284	1.74834	0.874171	−	0.485618i	$-0.161405\pi$
$317$	31.1284	1.74834	0.874171	+	0.485618i	$0.161405\pi$
$318$	0	0
$319$	−20.4192	−1.14326
$320$	−0.766511	−0.0428493
$321$	0	0
$322$	−49.8881	−2.78015
$323$	0.125667	0.00699231
$324$	0	0
$325$	10.4456	0.579419
$326$	−24.7229	−1.36927
$327$	0	0
$328$	47.0310	2.59685
$329$	31.1712	1.71852
$330$	0	0
$331$	31.0310	1.70562	0.852808	−	0.522225i	$-0.174898\pi$
$331$	31.0310	1.70562	0.852808	+	0.522225i	$0.174898\pi$
$332$	−6.66044	−0.365539
$333$	0	0
$334$	−9.04963	−0.495174
$335$	4.68273	0.255845
$336$	0	0
$337$	23.7297	1.29264	0.646319	−	0.763067i	$-0.276308\pi$
$337$	23.7297	1.29264	0.646319	+	0.763067i	$0.276308\pi$
$338$	−20.8307	−1.13304
$339$	0	0
$340$	6.19253	0.335837
$341$	19.3405	1.04735
$342$	0	0
$343$	11.5790	0.625209
$344$	−3.59358	−0.193753
$345$	0	0
$346$	47.5185	2.55461
$347$	−1.80840	−0.0970800	−0.0485400	−	0.998821i	$-0.515457\pi$
$347$	−1.80840	−0.0970800	−0.0485400	+	0.998821i	$0.515457\pi$
$348$	0	0
$349$	15.2608	0.816893	0.408447	−	0.912782i	$-0.366071\pi$
$349$	15.2608	0.816893	0.408447	+	0.912782i	$0.366071\pi$
$350$	39.0624	2.08797
$351$	0	0
$352$	−14.2763	−0.760930
$353$	−32.3824	−1.72354	−0.861770	−	0.507299i	$-0.830644\pi$
$353$	−32.3824	−1.72354	−0.861770	+	0.507299i	$0.830644\pi$
$354$	0	0
$355$	5.53571	0.293805
$356$	−70.0215	−3.71113
$357$	0	0
$358$	−12.8726	−0.680337
$359$	−1.91447	−0.101042	−0.0505209	−	0.998723i	$-0.516088\pi$
$359$	−1.91447	−0.101042	−0.0505209	+	0.998723i	$0.516088\pi$
$360$	0	0
$361$	−18.9982	−0.999908
$362$	18.1061	0.951634
$363$	0	0
$364$	31.0993	1.63004
$365$	−3.85298	−0.201674
$366$	0	0
$367$	−26.8648	−1.40233	−0.701167	−	0.712998i	$-0.747337\pi$
$367$	−26.8648	−1.40233	−0.701167	+	0.712998i	$0.747337\pi$
$368$	40.5330	2.11293
$369$	0	0
$370$	4.26083	0.221510
$371$	−15.9982	−0.830588
$372$	0	0
$373$	14.8402	0.768396	0.384198	−	0.923251i	$-0.374478\pi$
$373$	14.8402	0.768396	0.384198	+	0.923251i	$0.374478\pi$
$374$	−23.5945	−1.22004
$375$	0	0
$376$	−58.9873	−3.04204
$377$	−14.3628	−0.739721
$378$	0	0
$379$	−33.7870	−1.73552	−0.867762	−	0.496980i	$-0.834442\pi$
$379$	−33.7870	−1.73552	−0.867762	+	0.496980i	$0.834442\pi$
$380$	0.0864665	0.00443564
$381$	0	0
$382$	−26.6117	−1.36158
$383$	−9.27900	−0.474135	−0.237067	−	0.971493i	$-0.576186\pi$
$383$	−9.27900	−0.474135	−0.237067	+	0.971493i	$0.576186\pi$
$384$	0	0
$385$	4.68954	0.239001
$386$	−25.6827	−1.30722
$387$	0	0
$388$	82.2559	4.17591
$389$	−15.9786	−0.810149	−0.405075	−	0.914284i	$-0.632755\pi$
$389$	−15.9786	−0.810149	−0.405075	+	0.914284i	$0.632755\pi$
$390$	0	0
$391$	18.3182	0.926391
$392$	20.8307	1.05211
$393$	0	0
$394$	35.6614	1.79659
$395$	5.17024	0.260143
$396$	0	0
$397$	19.7050	0.988967	0.494483	−	0.869187i	$-0.335357\pi$
$397$	19.7050	0.988967	0.494483	+	0.869187i	$0.335357\pi$
$398$	26.0205	1.30429
$399$	0	0
$400$	−31.7374	−1.58687
$401$	1.14796	0.0573262	0.0286631	−	0.999589i	$-0.490875\pi$
$401$	1.14796	0.0573262	0.0286631	+	0.999589i	$0.490875\pi$
$402$	0	0
$403$	13.6040	0.677664
$404$	40.0847	1.99429
$405$	0	0
$406$	−53.7110	−2.66563
$407$	−11.1702	−0.553688
$408$	0	0
$409$	−3.10101	−0.153335	−0.0766676	−	0.997057i	$-0.524428\pi$
$409$	−3.10101	−0.153335	−0.0766676	+	0.997057i	$0.524428\pi$
$410$	9.12567	0.450685
$411$	0	0
$412$	1.15064	0.0566882
$413$	27.5303	1.35468
$414$	0	0
$415$	−0.706452	−0.0346784
$416$	−10.0419	−0.492344
$417$	0	0
$418$	−0.329451	−0.0161140
$419$	35.4492	1.73181	0.865904	−	0.500209i	$-0.166744\pi$
$419$	35.4492	1.73181	0.865904	+	0.500209i	$0.166744\pi$
$420$	0	0
$421$	9.21719	0.449218	0.224609	−	0.974449i	$-0.427889\pi$
$421$	9.21719	0.449218	0.224609	+	0.974449i	$0.427889\pi$
$422$	−18.0865	−0.880435
$423$	0	0
$424$	30.2746	1.47026
$425$	−14.3432	−0.695746
$426$	0	0
$427$	4.09327	0.198087
$428$	17.8307	0.861879
$429$	0	0
$430$	−0.697281	−0.0336259
$431$	11.5794	0.557758	0.278879	−	0.960326i	$-0.410037\pi$
$431$	11.5794	0.557758	0.278879	+	0.960326i	$0.410037\pi$
$432$	0	0
$433$	6.06511	0.291471	0.145735	−	0.989324i	$-0.453445\pi$
$433$	6.06511	0.291471	0.145735	+	0.989324i	$0.453445\pi$
$434$	50.8735	2.44201
$435$	0	0
$436$	−23.8384	−1.14165
$437$	0.255777	0.0122355
$438$	0	0
$439$	29.0060	1.38438	0.692190	−	0.721715i	$-0.256646\pi$
$439$	29.0060	1.38438	0.692190	+	0.721715i	$0.256646\pi$
$440$	−8.87433	−0.423067
$441$	0	0
$442$	−16.5963	−0.789404
$443$	30.8922	1.46773	0.733866	−	0.679294i	$-0.237714\pi$
$443$	30.8922	1.46773	0.733866	+	0.679294i	$0.237714\pi$
$444$	0	0
$445$	−7.42696	−0.352071
$446$	−26.1702	−1.23920
$447$	0	0
$448$	5.28581	0.249731
$449$	−39.0820	−1.84439	−0.922197	−	0.386720i	$-0.873608\pi$
$449$	−39.0820	−1.84439	−0.922197	+	0.386720i	$0.873608\pi$
$450$	0	0
$451$	−23.9240	−1.12654
$452$	6.10607	0.287205
$453$	0	0
$454$	−32.9736	−1.54753
$455$	3.29860	0.154641
$456$	0	0
$457$	−1.50475	−0.0703891	−0.0351946	−	0.999380i	$-0.511205\pi$
$457$	−1.50475	−0.0703891	−0.0351946	+	0.999380i	$0.511205\pi$
$458$	−71.1498	−3.32461
$459$	0	0
$460$	12.6040	0.587665
$461$	26.2371	1.22198	0.610992	−	0.791637i	$-0.290771\pi$
$461$	26.2371	1.22198	0.610992	+	0.791637i	$0.290771\pi$
$462$	0	0
$463$	6.75641	0.313997	0.156998	−	0.987599i	$-0.449818\pi$
$463$	6.75641	0.313997	0.156998	+	0.987599i	$0.449818\pi$
$464$	43.6391	2.02589
$465$	0	0
$466$	−35.2327	−1.63212
$467$	33.7469	1.56162	0.780810	−	0.624768i	$-0.214806\pi$
$467$	33.7469	1.56162	0.780810	+	0.624768i	$0.214806\pi$
$468$	0	0
$469$	−32.2918	−1.49110
$470$	−11.4456	−0.527947
$471$	0	0
$472$	−52.0975	−2.39798
$473$	1.82800	0.0840516
$474$	0	0
$475$	−0.200274	−0.00918921
$476$	−42.7033	−1.95730
$477$	0	0
$478$	−38.0310	−1.73950
$479$	11.1266	0.508387	0.254194	−	0.967153i	$-0.418190\pi$
$479$	11.1266	0.508387	0.254194	+	0.967153i	$0.418190\pi$
$480$	0	0
$481$	−7.85710	−0.358253
$482$	−32.8503	−1.49629
$483$	0	0
$484$	−5.96585	−0.271175
$485$	8.72462	0.396165
$486$	0	0
$487$	−4.74691	−0.215103	−0.107552	−	0.994200i	$-0.534301\pi$
$487$	−4.74691	−0.215103	−0.107552	+	0.994200i	$0.534301\pi$
$488$	−7.74598	−0.350644
$489$	0	0
$490$	4.04189	0.182594
$491$	22.3405	1.00821	0.504106	−	0.863642i	$-0.331822\pi$
$491$	22.3405	1.00821	0.504106	+	0.863642i	$0.331822\pi$
$492$	0	0
$493$	19.7219	0.888231
$494$	−0.231734	−0.0104262
$495$	0	0
$496$	−41.3337	−1.85594
$497$	−38.1739	−1.71233
$498$	0	0
$499$	−10.4611	−0.468303	−0.234152	−	0.972200i	$-0.575231\pi$
$499$	−10.4611	−0.468303	−0.234152	+	0.972200i	$0.575231\pi$
$500$	−20.1898	−0.902917
$501$	0	0
$502$	2.20945	0.0986124
$503$	25.0419	1.11656	0.558281	−	0.829652i	$-0.311461\pi$
$503$	25.0419	1.11656	0.558281	+	0.829652i	$0.311461\pi$
$504$	0	0
$505$	4.25166	0.189196
$506$	−48.0232	−2.13489
$507$	0	0
$508$	−29.2841	−1.29927
$509$	18.0624	0.800603	0.400301	−	0.916384i	$-0.368905\pi$
$509$	18.0624	0.800603	0.400301	+	0.916384i	$0.368905\pi$
$510$	0	0
$511$	26.5699	1.17538
$512$	−50.5553	−2.23425
$513$	0	0
$514$	11.5885	0.511148
$515$	0.122045	0.00537795
$516$	0	0
$517$	30.0060	1.31966
$518$	−29.3824	−1.29099
$519$	0	0
$520$	−6.24216	−0.273737
$521$	−25.9581	−1.13725	−0.568623	−	0.822598i	$-0.692524\pi$
$521$	−25.9581	−1.13725	−0.568623	+	0.822598i	$0.692524\pi$
$522$	0	0
$523$	25.5945	1.11917	0.559585	−	0.828773i	$-0.310961\pi$
$523$	25.5945	1.11917	0.559585	+	0.828773i	$0.310961\pi$
$524$	−55.7648	−2.43609
$525$	0	0
$526$	10.8821	0.474481
$527$	−18.6800	−0.813716
$528$	0	0
$529$	14.2841	0.621046
$530$	5.87433	0.255165
$531$	0	0
$532$	−0.596267	−0.0258514
$533$	−16.8280	−0.728902
$534$	0	0
$535$	1.89124	0.0817656
$536$	61.1079	2.63946
$537$	0	0
$538$	30.7033	1.32371
$539$	−10.5963	−0.456414
$540$	0	0
$541$	27.8476	1.19726	0.598631	−	0.801025i	$-0.295712\pi$
$541$	27.8476	1.19726	0.598631	+	0.801025i	$0.295712\pi$
$542$	−0.810155	−0.0347991
$543$	0	0
$544$	13.7888	0.591190
$545$	−2.52847	−0.108308
$546$	0	0
$547$	−5.90848	−0.252628	−0.126314	−	0.991990i	$-0.540315\pi$
$547$	−5.90848	−0.252628	−0.126314	+	0.991990i	$0.540315\pi$
$548$	46.8316	2.00055
$549$	0	0
$550$	37.6023	1.60337
$551$	0.275378	0.0117315
$552$	0	0
$553$	−35.6536	−1.51615
$554$	−67.9136	−2.88537
$555$	0	0
$556$	−32.9145	−1.39588
$557$	26.7050	1.13153	0.565764	−	0.824567i	$-0.308581\pi$
$557$	26.7050	1.13153	0.565764	+	0.824567i	$0.308581\pi$
$558$	0	0
$559$	1.28581	0.0543838
$560$	−10.0223	−0.423519
$561$	0	0
$562$	67.5586	2.84979
$563$	36.0624	1.51985	0.759925	−	0.650011i	$-0.225236\pi$
$563$	36.0624	1.51985	0.759925	+	0.650011i	$0.225236\pi$
$564$	0	0
$565$	0.647651	0.0272469
$566$	−23.5321	−0.989127
$567$	0	0
$568$	72.2390	3.03108
$569$	−9.04727	−0.379281	−0.189641	−	0.981854i	$-0.560732\pi$
$569$	−9.04727	−0.379281	−0.189641	+	0.981854i	$0.560732\pi$
$570$	0	0
$571$	30.5790	1.27969	0.639846	−	0.768503i	$-0.278998\pi$
$571$	30.5790	1.27969	0.639846	+	0.768503i	$0.278998\pi$
$572$	29.9368	1.25172
$573$	0	0
$574$	−62.9299	−2.62665
$575$	−29.1935	−1.21745
$576$	0	0
$577$	−25.1489	−1.04696	−0.523481	−	0.852037i	$-0.675367\pi$
$577$	−25.1489	−1.04696	−0.523481	+	0.852037i	$0.675367\pi$
$578$	−20.2567	−0.842568
$579$	0	0
$580$	13.5699	0.563458
$581$	4.87164	0.202110
$582$	0	0
$583$	−15.4002	−0.637812
$584$	−50.2799	−2.08060
$585$	0	0
$586$	−49.7279	−2.05424
$587$	20.8735	0.861542	0.430771	−	0.902461i	$-0.358242\pi$
$587$	20.8735	0.861542	0.430771	+	0.902461i	$0.358242\pi$
$588$	0	0
$589$	−0.260830	−0.0107473
$590$	−10.1088	−0.416171
$591$	0	0
$592$	23.8726	0.981157
$593$	15.6212	0.641488	0.320744	−	0.947166i	$-0.396067\pi$
$593$	15.6212	0.641488	0.320744	+	0.947166i	$0.396067\pi$
$594$	0	0
$595$	−4.52940	−0.185687
$596$	−18.7665	−0.768706
$597$	0	0
$598$	−33.7793	−1.38134
$599$	−0.448311	−0.0183175	−0.00915874	−	0.999958i	$-0.502915\pi$
$599$	−0.448311	−0.0183175	−0.00915874	+	0.999958i	$0.502915\pi$
$600$	0	0
$601$	−17.6723	−0.720868	−0.360434	−	0.932785i	$-0.617371\pi$
$601$	−17.6723	−0.720868	−0.360434	+	0.932785i	$0.617371\pi$
$602$	4.80840	0.195976
$603$	0	0
$604$	0.596267	0.0242617
$605$	−0.632779	−0.0257261
$606$	0	0
$607$	26.1985	1.06337	0.531683	−	0.846944i	$-0.321560\pi$
$607$	26.1985	1.06337	0.531683	+	0.846944i	$0.321560\pi$
$608$	0.192533	0.00780826
$609$	0	0
$610$	−1.50299	−0.0608544
$611$	21.1061	0.853860
$612$	0	0
$613$	14.5544	0.587846	0.293923	−	0.955829i	$-0.405039\pi$
$613$	14.5544	0.587846	0.293923	+	0.955829i	$0.405039\pi$
$614$	71.7674	2.89630
$615$	0	0
$616$	61.1968	2.46569
$617$	14.0205	0.564445	0.282223	−	0.959349i	$-0.408928\pi$
$617$	14.0205	0.564445	0.282223	+	0.959349i	$0.408928\pi$
$618$	0	0
$619$	−31.7124	−1.27463	−0.637315	−	0.770603i	$-0.719955\pi$
$619$	−31.7124	−1.27463	−0.637315	+	0.770603i	$0.719955\pi$
$620$	−12.8530	−0.516188
$621$	0	0
$622$	−5.17705	−0.207581
$623$	51.2158	2.05192
$624$	0	0
$625$	21.7638	0.870553
$626$	21.2986	0.851263
$627$	0	0
$628$	58.7870	2.34586
$629$	10.7888	0.430178
$630$	0	0
$631$	−38.5758	−1.53568	−0.767840	−	0.640642i	$-0.778668\pi$
$631$	−38.5758	−1.53568	−0.767840	+	0.640642i	$0.778668\pi$
$632$	67.4698	2.68380
$633$	0	0
$634$	78.8198	3.13033
$635$	−3.10607	−0.123261
$636$	0	0
$637$	−7.45336	−0.295313
$638$	−51.7033	−2.04695
$639$	0	0
$640$	−6.24216	−0.246743
$641$	−30.6168	−1.20929	−0.604645	−	0.796495i	$-0.706685\pi$
$641$	−30.6168	−1.20929	−0.604645	+	0.796495i	$0.706685\pi$
$642$	0	0
$643$	−34.2249	−1.34970	−0.674850	−	0.737955i	$-0.735792\pi$
$643$	−34.2249	−1.34970	−0.674850	+	0.737955i	$0.735792\pi$
$644$	−86.9163	−3.42498
$645$	0	0
$646$	0.318201	0.0125194
$647$	−12.8726	−0.506073	−0.253037	−	0.967457i	$-0.581429\pi$
$647$	−12.8726	−0.506073	−0.253037	+	0.967457i	$0.581429\pi$
$648$	0	0
$649$	26.5012	1.04026
$650$	26.4492	1.03742
$651$	0	0
$652$	−43.0729	−1.68686
$653$	−11.3191	−0.442952	−0.221476	−	0.975166i	$-0.571087\pi$
$653$	−11.3191	−0.442952	−0.221476	+	0.975166i	$0.571087\pi$
$654$	0	0
$655$	−5.91479	−0.231110
$656$	51.1293	1.99626
$657$	0	0
$658$	78.9282	3.07694
$659$	−13.7692	−0.536372	−0.268186	−	0.963367i	$-0.586424\pi$
$659$	−13.7692	−0.536372	−0.268186	+	0.963367i	$0.586424\pi$
$660$	0	0
$661$	−20.2668	−0.788288	−0.394144	−	0.919049i	$-0.628959\pi$
$661$	−20.2668	−0.788288	−0.394144	+	0.919049i	$0.628959\pi$
$662$	78.5732	3.05383
$663$	0	0
$664$	−9.21894	−0.357764
$665$	−0.0632441	−0.00245250
$666$	0	0
$667$	40.1411	1.55427
$668$	−15.7665	−0.610025
$669$	0	0
$670$	11.8571	0.458080
$671$	3.94027	0.152112
$672$	0	0
$673$	−30.5449	−1.17742	−0.588709	−	0.808345i	$-0.700364\pi$
$673$	−30.5449	−1.17742	−0.588709	+	0.808345i	$0.700364\pi$
$674$	60.0856	2.31441
$675$	0	0
$676$	−36.2918	−1.39584
$677$	−3.71925	−0.142942	−0.0714711	−	0.997443i	$-0.522769\pi$
$677$	−3.71925	−0.142942	−0.0714711	+	0.997443i	$0.522769\pi$
$678$	0	0
$679$	−60.1644	−2.30890
$680$	8.57129	0.328694
$681$	0	0
$682$	48.9718	1.87523
$683$	−21.7469	−0.832122	−0.416061	−	0.909337i	$-0.636590\pi$
$683$	−21.7469	−0.832122	−0.416061	+	0.909337i	$0.636590\pi$
$684$	0	0
$685$	4.96728	0.189790
$686$	29.3191	1.11941
$687$	0	0
$688$	−3.90673	−0.148943
$689$	−10.8324	−0.412683
$690$	0	0
$691$	−37.5948	−1.43017	−0.715087	−	0.699035i	$-0.753613\pi$
$691$	−37.5948	−1.43017	−0.715087	+	0.699035i	$0.753613\pi$
$692$	82.7880	3.14713
$693$	0	0
$694$	−4.57903	−0.173818
$695$	−3.49113	−0.132426
$696$	0	0
$697$	23.1070	0.875240
$698$	38.6418	1.46261
$699$	0	0
$700$	68.0556	2.57226
$701$	23.3351	0.881355	0.440678	−	0.897665i	$-0.354738\pi$
$701$	23.3351	0.881355	0.440678	+	0.897665i	$0.354738\pi$
$702$	0	0
$703$	0.150644	0.00568166
$704$	5.08822	0.191770
$705$	0	0
$706$	−81.9951	−3.08592
$707$	−29.3191	−1.10266
$708$	0	0
$709$	6.57903	0.247081	0.123540	−	0.992340i	$-0.460575\pi$
$709$	6.57903	0.247081	0.123540	+	0.992340i	$0.460575\pi$
$710$	14.0169	0.526045
$711$	0	0
$712$	−96.9190	−3.63219
$713$	−38.0205	−1.42388
$714$	0	0
$715$	3.17530	0.118749
$716$	−22.4270	−0.838135
$717$	0	0
$718$	−4.84760	−0.180911
$719$	−16.8324	−0.627744	−0.313872	−	0.949465i	$-0.601626\pi$
$719$	−16.8324	−0.627744	−0.313872	+	0.949465i	$0.601626\pi$
$720$	0	0
$721$	−0.841615	−0.0313434
$722$	−48.1052	−1.79029
$723$	0	0
$724$	31.5449	1.17236
$725$	−31.4306	−1.16730
$726$	0	0
$727$	25.5621	0.948046	0.474023	−	0.880512i	$-0.342801\pi$
$727$	25.5621	0.948046	0.474023	+	0.880512i	$0.342801\pi$
$728$	43.0455	1.59537
$729$	0	0
$730$	−9.75608	−0.361089
$731$	−1.76558	−0.0653022
$732$	0	0
$733$	−14.7980	−0.546576	−0.273288	−	0.961932i	$-0.588111\pi$
$733$	−14.7980	−0.546576	−0.273288	+	0.961932i	$0.588111\pi$
$734$	−68.0242	−2.51082
$735$	0	0
$736$	28.0651	1.03449
$737$	−31.0847	−1.14502
$738$	0	0
$739$	9.19078	0.338088	0.169044	−	0.985608i	$-0.445932\pi$
$739$	9.19078	0.338088	0.169044	+	0.985608i	$0.445932\pi$
$740$	7.42333	0.272887
$741$	0	0
$742$	−40.5090	−1.48713
$743$	44.4492	1.63068	0.815342	−	0.578979i	$-0.196549\pi$
$743$	44.4492	1.63068	0.815342	+	0.578979i	$0.196549\pi$
$744$	0	0
$745$	−1.99050	−0.0729264
$746$	37.5767	1.37578
$747$	0	0
$748$	−41.1070	−1.50302
$749$	−13.0419	−0.476540
$750$	0	0
$751$	−35.8060	−1.30658	−0.653290	−	0.757107i	$-0.726612\pi$
$751$	−35.8060	−1.30658	−0.653290	+	0.757107i	$0.726612\pi$
$752$	−64.1275	−2.33849
$753$	0	0
$754$	−36.3678	−1.32444
$755$	0.0632441	0.00230169
$756$	0	0
$757$	−45.8976	−1.66818	−0.834088	−	0.551632i	$-0.814005\pi$
$757$	−45.8976	−1.66818	−0.834088	+	0.551632i	$0.814005\pi$
$758$	−85.5518	−3.10738
$759$	0	0
$760$	0.119681	0.00434129
$761$	−37.1952	−1.34833	−0.674163	−	0.738583i	$-0.735495\pi$
$761$	−37.1952	−1.34833	−0.674163	+	0.738583i	$0.735495\pi$
$762$	0	0
$763$	17.4361	0.631230
$764$	−46.3637	−1.67738
$765$	0	0
$766$	−23.4953	−0.848918
$767$	18.6408	0.673082
$768$	0	0
$769$	−38.3337	−1.38235	−0.691174	−	0.722688i	$-0.742906\pi$
$769$	−38.3337	−1.38235	−0.691174	+	0.722688i	$0.742906\pi$
$770$	11.8743	0.427921
$771$	0	0
$772$	−44.7452	−1.61041
$773$	52.8272	1.90006	0.950031	−	0.312156i	$-0.101051\pi$
$773$	52.8272	1.90006	0.950031	+	0.312156i	$0.101051\pi$
$774$	0	0
$775$	29.7701	1.06937
$776$	113.853	4.08709
$777$	0	0
$778$	−40.4593	−1.45054
$779$	0.322644	0.0115599
$780$	0	0
$781$	−36.7469	−1.31491
$782$	46.3833	1.65866
$783$	0	0
$784$	22.6459	0.808782
$785$	6.23536	0.222549
$786$	0	0
$787$	45.2404	1.61265	0.806323	−	0.591475i	$-0.201454\pi$
$787$	45.2404	1.61265	0.806323	+	0.591475i	$0.201454\pi$
$788$	62.1302	2.21330
$789$	0	0
$790$	13.0915	0.465775
$791$	−4.46616	−0.158798
$792$	0	0
$793$	2.77156	0.0984211
$794$	49.8949	1.77070
$795$	0	0
$796$	45.3337	1.60681
$797$	−16.9189	−0.599299	−0.299649	−	0.954049i	$-0.596870\pi$
$797$	−16.9189	−0.599299	−0.299649	+	0.954049i	$0.596870\pi$
$798$	0	0
$799$	−28.9813	−1.02529
$800$	−21.9750	−0.776934
$801$	0	0
$802$	2.90673	0.102640
$803$	25.5767	0.902581
$804$	0	0
$805$	−9.21894	−0.324925
$806$	34.4466	1.21333
$807$	0	0
$808$	55.4826	1.95187
$809$	−34.9145	−1.22753	−0.613764	−	0.789490i	$-0.710345\pi$
$809$	−34.9145	−1.22753	−0.613764	+	0.789490i	$0.710345\pi$
$810$	0	0
$811$	18.0419	0.633536	0.316768	−	0.948503i	$-0.397402\pi$
$811$	18.0419	0.633536	0.316768	+	0.948503i	$0.397402\pi$
$812$	−93.5768	−3.28390
$813$	0	0
$814$	−28.2841	−0.991356
$815$	−4.56860	−0.160031
$816$	0	0
$817$	−0.0246528	−0.000862492 0
$818$	−7.85204	−0.274540
$819$	0	0
$820$	15.8990	0.555217
$821$	40.6391	1.41831	0.709157	−	0.705051i	$-0.249076\pi$
$821$	40.6391	1.41831	0.709157	+	0.705051i	$0.249076\pi$
$822$	0	0
$823$	26.0496	0.908033	0.454017	−	0.890993i	$-0.349991\pi$
$823$	26.0496	0.908033	0.454017	+	0.890993i	$0.349991\pi$
$824$	1.59264	0.0554824
$825$	0	0
$826$	69.7093	2.42550
$827$	−37.6195	−1.30816	−0.654079	−	0.756426i	$-0.726944\pi$
$827$	−37.6195	−1.30816	−0.654079	+	0.756426i	$0.726944\pi$
$828$	0	0
$829$	−35.0215	−1.21635	−0.608173	−	0.793805i	$-0.708097\pi$
$829$	−35.0215	−1.21635	−0.608173	+	0.793805i	$0.708097\pi$
$830$	−1.78880	−0.0620902
$831$	0	0
$832$	3.57903	0.124081
$833$	10.2344	0.354602
$834$	0	0
$835$	−1.67230	−0.0578725
$836$	−0.573978	−0.0198514
$837$	0	0
$838$	89.7606	3.10073
$839$	26.6141	0.918821	0.459411	−	0.888224i	$-0.348061\pi$
$839$	26.6141	0.918821	0.459411	+	0.888224i	$0.348061\pi$
$840$	0	0
$841$	14.2172	0.490248
$842$	23.3387	0.804306
$843$	0	0
$844$	−31.5107	−1.08464
$845$	−3.84936	−0.132422
$846$	0	0
$847$	4.36360	0.149935
$848$	32.9127	1.13023
$849$	0	0
$850$	−36.3182	−1.24570
$851$	21.9590	0.752746
$852$	0	0
$853$	2.45512	0.0840616	0.0420308	−	0.999116i	$-0.486617\pi$
$853$	2.45512	0.0840616	0.0420308	+	0.999116i	$0.486617\pi$
$854$	10.3645	0.354667
$855$	0	0
$856$	24.6800	0.843547
$857$	−9.82976	−0.335778	−0.167889	−	0.985806i	$-0.553695\pi$
$857$	−9.82976	−0.335778	−0.167889	+	0.985806i	$0.553695\pi$
$858$	0	0
$859$	8.81696	0.300831	0.150415	−	0.988623i	$-0.451939\pi$
$859$	8.81696	0.300831	0.150415	+	0.988623i	$0.451939\pi$
$860$	−1.21482	−0.0414251
$861$	0	0
$862$	29.3200	0.998642
$863$	−6.62124	−0.225390	−0.112695	−	0.993630i	$-0.535948\pi$
$863$	−6.62124	−0.225390	−0.112695	+	0.993630i	$0.535948\pi$
$864$	0	0
$865$	8.78106	0.298565
$866$	15.3574	0.521866
$867$	0	0
$868$	88.6332	3.00841
$869$	−34.3209	−1.16426
$870$	0	0
$871$	−21.8648	−0.740862
$872$	−32.9956	−1.11737
$873$	0	0
$874$	0.647651	0.0219071
$875$	14.7674	0.499231
$876$	0	0
$877$	−4.07604	−0.137638	−0.0688190	−	0.997629i	$-0.521923\pi$
$877$	−4.07604	−0.137638	−0.0688190	+	0.997629i	$0.521923\pi$
$878$	73.4457	2.47867
$879$	0	0
$880$	−9.64765	−0.325222
$881$	9.25133	0.311685	0.155843	−	0.987782i	$-0.450191\pi$
$881$	9.25133	0.311685	0.155843	+	0.987782i	$0.450191\pi$
$882$	0	0
$883$	−37.7701	−1.27107	−0.635533	−	0.772074i	$-0.719220\pi$
$883$	−37.7701	−1.27107	−0.635533	+	0.772074i	$0.719220\pi$
$884$	−28.9145	−0.972499
$885$	0	0
$886$	78.2217	2.62791
$887$	18.9786	0.637241	0.318620	−	0.947882i	$-0.396781\pi$
$887$	18.9786	0.637241	0.318620	+	0.947882i	$0.396781\pi$
$888$	0	0
$889$	21.4192	0.718377
$890$	−18.8057	−0.630369
$891$	0	0
$892$	−45.5945	−1.52662
$893$	−0.404667	−0.0135417
$894$	0	0
$895$	−2.37876	−0.0795131
$896$	43.0455	1.43805
$897$	0	0
$898$	−98.9592	−3.30231
$899$	−40.9341	−1.36523
$900$	0	0
$901$	14.8743	0.495536
$902$	−60.5776	−2.01701
$903$	0	0
$904$	8.45161	0.281096
$905$	3.34587	0.111220
$906$	0	0
$907$	9.64590	0.320287	0.160143	−	0.987094i	$-0.448804\pi$
$907$	9.64590	0.320287	0.160143	+	0.987094i	$0.448804\pi$
$908$	−57.4475	−1.90646
$909$	0	0
$910$	8.35235	0.276878
$911$	−6.53478	−0.216507	−0.108253	−	0.994123i	$-0.534526\pi$
$911$	−6.53478	−0.216507	−0.108253	+	0.994123i	$0.534526\pi$
$912$	0	0
$913$	4.68954	0.155201
$914$	−3.81016	−0.126029
$915$	0	0
$916$	−123.959	−4.09573
$917$	40.7880	1.34694
$918$	0	0
$919$	3.89124	0.128360	0.0641802	−	0.997938i	$-0.479557\pi$
$919$	3.89124	0.128360	0.0641802	+	0.997938i	$0.479557\pi$
$920$	17.4456	0.575165
$921$	0	0
$922$	66.4347	2.18791
$923$	−25.8476	−0.850784
$924$	0	0
$925$	−17.1940	−0.565334
$926$	17.1078	0.562198
$927$	0	0
$928$	30.2158	0.991881
$929$	−39.8530	−1.30753	−0.653767	−	0.756696i	$-0.726812\pi$
$929$	−39.8530	−1.30753	−0.653767	+	0.756696i	$0.726812\pi$
$930$	0	0
$931$	0.142903	0.00468347
$932$	−61.3833	−2.01068
$933$	0	0
$934$	85.4502	2.79602
$935$	−4.36009	−0.142590
$936$	0	0
$937$	33.0651	1.08019	0.540095	−	0.841604i	$-0.318388\pi$
$937$	33.0651	1.08019	0.540095	+	0.841604i	$0.318388\pi$
$938$	−81.7657	−2.66974
$939$	0	0
$940$	−19.9409	−0.650400
$941$	−53.8066	−1.75405	−0.877023	−	0.480448i	$-0.840474\pi$
$941$	−53.8066	−1.75405	−0.877023	+	0.480448i	$0.840474\pi$
$942$	0	0
$943$	47.0310	1.53154
$944$	−56.6373	−1.84339
$945$	0	0
$946$	4.62866	0.150491
$947$	42.2354	1.37246	0.686232	−	0.727382i	$-0.259263\pi$
$947$	42.2354	1.37246	0.686232	+	0.727382i	$0.259263\pi$
$948$	0	0
$949$	17.9905	0.583996
$950$	−0.507112	−0.0164529
$951$	0	0
$952$	−59.1070	−1.91567
$953$	24.5776	0.796147	0.398073	−	0.917354i	$-0.369679\pi$
$953$	24.5776	0.796147	0.398073	+	0.917354i	$0.369679\pi$
$954$	0	0
$955$	−4.91765	−0.159131
$956$	−66.2586	−2.14296
$957$	0	0
$958$	28.1735	0.910246
$959$	−34.2540	−1.10612
$960$	0	0
$961$	7.77156	0.250696
$962$	−19.8949	−0.641436
$963$	0	0
$964$	−57.2327	−1.84334
$965$	−4.74598	−0.152778
$966$	0	0
$967$	4.47203	0.143811	0.0719054	−	0.997411i	$-0.477092\pi$
$967$	4.47203	0.143811	0.0719054	+	0.997411i	$0.477092\pi$
$968$	−8.25753	−0.265407
$969$	0	0
$970$	22.0915	0.709316
$971$	−4.61949	−0.148246	−0.0741232	−	0.997249i	$-0.523616\pi$
$971$	−4.61949	−0.148246	−0.0741232	+	0.997249i	$0.523616\pi$
$972$	0	0
$973$	24.0746	0.771796
$974$	−12.0196	−0.385133
$975$	0	0
$976$	−8.42097	−0.269549
$977$	−11.2808	−0.360903	−0.180452	−	0.983584i	$-0.557756\pi$
$977$	−11.2808	−0.360903	−0.180452	+	0.983584i	$0.557756\pi$
$978$	0	0
$979$	49.3013	1.57568
$980$	7.04189	0.224945
$981$	0	0
$982$	56.5681	1.80516
$983$	−17.0642	−0.544263	−0.272131	−	0.962260i	$-0.587729\pi$
$983$	−17.0642	−0.544263	−0.272131	+	0.962260i	$0.587729\pi$
$984$	0	0
$985$	6.58996	0.209973
$986$	49.9377	1.59034
$987$	0	0
$988$	−0.403733	−0.0128445
$989$	−3.59358	−0.114269
$990$	0	0
$991$	2.00000	0.0635321	0.0317660	−	0.999495i	$-0.489887\pi$
$991$	2.00000	0.0635321	0.0317660	+	0.999495i	$0.489887\pi$
$992$	−28.6195	−0.908670
$993$	0	0
$994$	−96.6596	−3.06586
$995$	4.80840	0.152437
$996$	0	0
$997$	22.3847	0.708932	0.354466	−	0.935069i	$-0.384663\pi$
$997$	22.3847	0.708932	0.354466	+	0.935069i	$0.384663\pi$
$998$	−26.4884	−0.838477
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	243.2.a.f.1.3	yes	3
3.2	odd	2		243.2.a.e.1.1	✓	3
4.3	odd	2		3888.2.a.bk.1.1		3
5.4	even	2		6075.2.a.bq.1.1		3
9.2	odd	6		243.2.c.f.82.3		6
9.4	even	3		243.2.c.e.163.1		6
9.5	odd	6		243.2.c.f.163.3		6
9.7	even	3		243.2.c.e.82.1		6
12.11	even	2		3888.2.a.bd.1.3		3
15.14	odd	2		6075.2.a.bv.1.3		3
27.2	odd	18		729.2.e.h.568.1		6
27.4	even	9		729.2.e.i.406.1		6
27.5	odd	18		729.2.e.g.649.1		6
27.7	even	9		729.2.e.i.325.1		6
27.11	odd	18		729.2.e.g.82.1		6
27.13	even	9		729.2.e.c.163.1		6
27.14	odd	18		729.2.e.h.163.1		6
27.16	even	9		729.2.e.b.82.1		6
27.20	odd	18		729.2.e.a.325.1		6
27.22	even	9		729.2.e.b.649.1		6
27.23	odd	18		729.2.e.a.406.1		6
27.25	even	9		729.2.e.c.568.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
243.2.a.e.1.1	✓	3	3.2	odd	2
243.2.a.f.1.3	yes	3	1.1	even	1	trivial
243.2.c.e.82.1		6	9.7	even	3
243.2.c.e.163.1		6	9.4	even	3
243.2.c.f.82.3		6	9.2	odd	6
243.2.c.f.163.3		6	9.5	odd	6
729.2.e.a.325.1		6	27.20	odd	18
729.2.e.a.406.1		6	27.23	odd	18
729.2.e.b.82.1		6	27.16	even	9
729.2.e.b.649.1		6	27.22	even	9
729.2.e.c.163.1		6	27.13	even	9
729.2.e.c.568.1		6	27.25	even	9
729.2.e.g.82.1		6	27.11	odd	18
729.2.e.g.649.1		6	27.5	odd	18
729.2.e.h.163.1		6	27.14	odd	18
729.2.e.h.568.1		6	27.2	odd	18
729.2.e.i.325.1		6	27.7	even	9
729.2.e.i.406.1		6	27.4	even	9
3888.2.a.bd.1.3		3	12.11	even	2
3888.2.a.bk.1.1		3	4.3	odd	2
6075.2.a.bq.1.1		3	5.4	even	2
6075.2.a.bv.1.3		3	15.14	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 243 = 3^{5} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	243.a (trivial)

\(f(q)\)	\(=\)	\(q+2.53209 q^{2} +4.41147 q^{4} +0.467911 q^{5} -3.22668 q^{7} +6.10607 q^{8} +1.18479 q^{10} -3.10607 q^{11} -2.18479 q^{13} -8.17024 q^{14} +6.63816 q^{16} +3.00000 q^{17} +0.0418891 q^{19} +2.06418 q^{20} -7.86484 q^{22} +6.10607 q^{23} -4.78106 q^{25} -5.53209 q^{26} -14.2344 q^{28} +6.57398 q^{29} -6.22668 q^{31} +4.59627 q^{32} +7.59627 q^{34} -1.50980 q^{35} +3.59627 q^{37} +0.106067 q^{38} +2.85710 q^{40} +7.70233 q^{41} -0.588526 q^{43} -13.7023 q^{44} +15.4611 q^{46} -9.66044 q^{47} +3.41147 q^{49} -12.1061 q^{50} -9.63816 q^{52} +4.95811 q^{53} -1.45336 q^{55} -19.7023 q^{56} +16.6459 q^{58} -8.53209 q^{59} -1.26857 q^{61} -15.7665 q^{62} -1.63816 q^{64} -1.02229 q^{65} +10.0077 q^{67} +13.2344 q^{68} -3.82295 q^{70} +11.8307 q^{71} -8.23442 q^{73} +9.10607 q^{74} +0.184793 q^{76} +10.0223 q^{77} +11.0496 q^{79} +3.10607 q^{80} +19.5030 q^{82} -1.50980 q^{83} +1.40373 q^{85} -1.49020 q^{86} -18.9659 q^{88} -15.8726 q^{89} +7.04963 q^{91} +26.9368 q^{92} -24.4611 q^{94} +0.0196004 q^{95} +18.6459 q^{97} +8.63816 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} + 6 q^{5} - 3 q^{7} + 6 q^{8} + 3 q^{11} - 3 q^{13} - 3 q^{14} + 3 q^{16} + 9 q^{17} - 3 q^{19} - 3 q^{20} + 6 q^{23} + 3 q^{25} - 12 q^{26} - 12 q^{28} + 12 q^{29} - 12 q^{31} + 9 q^{34}+ \cdots + 9 q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^4 + 6 * q^5 - 3 * q^7 + 6 * q^8 + 3 * q^11 - 3 * q^13 - 3 * q^14 + 3 * q^16 + 9 * q^17 - 3 * q^19 - 3 * q^20 + 6 * q^23 + 3 * q^25 - 12 * q^26 - 12 * q^28 + 12 * q^29 - 12 * q^31 + 9 * q^34 - 6 * q^35 - 3 * q^37 - 12 * q^38 + 9 * q^40 - 3 * q^41 - 12 * q^43 - 15 * q^44 + 9 * q^46 - 6 * q^47 - 24 * q^50 - 12 * q^52 + 18 * q^53 + 9 * q^55 - 33 * q^56 + 9 * q^58 - 21 * q^59 + 6 * q^61 - 12 * q^62 + 12 * q^64 + 3 * q^65 + 6 * q^67 + 9 * q^68 + 9 * q^70 - 9 * q^71 + 6 * q^73 + 15 * q^74 - 3 * q^76 + 24 * q^77 + 6 * q^79 - 3 * q^80 + 18 * q^82 - 6 * q^83 + 18 * q^85 - 3 * q^86 - 36 * q^88 - 6 * q^91 + 24 * q^92 - 36 * q^94 + 3 * q^95 + 15 * q^97 + 9 * q^98

Introduction

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