LMFDB - Embedded newform 3888.2.a.bk.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3888,2,Mod(1,3888)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3888, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3888.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3888 = 2^{4} \cdot 3^{5} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3888.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:	$31.0458363059$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 243)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.53209$ of defining polynomial
Character	$\chi$	$=$	3888.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.467911 q^{5} +3.22668 q^{7} +O(q^{10})$	$q+0.467911 q^{5} +3.22668 q^{7} +3.10607 q^{11} -2.18479 q^{13} +3.00000 q^{17} -0.0418891 q^{19} -6.10607 q^{23} -4.78106 q^{25} +6.57398 q^{29} +6.22668 q^{31} +1.50980 q^{35} +3.59627 q^{37} +7.70233 q^{41} +0.588526 q^{43} +9.66044 q^{47} +3.41147 q^{49} +4.95811 q^{53} +1.45336 q^{55} +8.53209 q^{59} -1.26857 q^{61} -1.02229 q^{65} -10.0077 q^{67} -11.8307 q^{71} -8.23442 q^{73} +10.0223 q^{77} -11.0496 q^{79} +1.50980 q^{83} +1.40373 q^{85} -15.8726 q^{89} -7.04963 q^{91} -0.0196004 q^{95} +18.6459 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 6 q^{5} + 3 q^{7}+O(q^{10}) $ 3 * q + 6 * q^5 + 3 * q^7	$ 3 q + 6 q^{5} + 3 q^{7} - 3 q^{11} - 3 q^{13} + 9 q^{17} + 3 q^{19} - 6 q^{23} + 3 q^{25} + 12 q^{29} + 12 q^{31} + 6 q^{35} - 3 q^{37} - 3 q^{41} + 12 q^{43} + 6 q^{47} + 18 q^{53} - 9 q^{55} + 21 q^{59} + 6 q^{61} + 3 q^{65} - 6 q^{67} + 9 q^{71} + 6 q^{73} + 24 q^{77} - 6 q^{79} + 6 q^{83} + 18 q^{85} + 6 q^{91} - 3 q^{95} + 15 q^{97}+O(q^{100}) $ 3 * q + 6 * q^5 + 3 * q^7 - 3 * q^11 - 3 * q^13 + 9 * q^17 + 3 * q^19 - 6 * q^23 + 3 * q^25 + 12 * q^29 + 12 * q^31 + 6 * q^35 - 3 * q^37 - 3 * q^41 + 12 * q^43 + 6 * q^47 + 18 * q^53 - 9 * q^55 + 21 * q^59 + 6 * q^61 + 3 * q^65 - 6 * q^67 + 9 * q^71 + 6 * q^73 + 24 * q^77 - 6 * q^79 + 6 * q^83 + 18 * q^85 + 6 * q^91 - 3 * q^95 + 15 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	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	$-0.291256\pi$
$7$	3.22668	1.21957	0.609786	+	0.792566i	$0.291256\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	3.10607	0.936514	0.468257	−	0.883592i	$-0.344882\pi$
$11$	3.10607	0.936514	0.468257	+	0.883592i	$0.344882\pi$
$12$	0	0
$13$	−2.18479	−0.605952	−0.302976	−	0.952998i	$-0.597980\pi$
$13$	−2.18479	−0.605952	−0.302976	+	0.952998i	$0.597980\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	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	$-0.501529\pi$
$19$	−0.0418891	−0.00961001	−0.00480501	+	0.999988i	$0.501529\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−6.10607	−1.27320	−0.636601	−	0.771193i	$-0.719660\pi$
$23$	−6.10607	−1.27320	−0.636601	+	0.771193i	$0.719660\pi$
$24$	0	0
$25$	−4.78106	−0.956212
$26$	0	0
$27$	0	0
$28$	0	0
$29$	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	$-0.311119\pi$
$31$	6.22668	1.11835	0.559173	+	0.829051i	$0.311119\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	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	0
$39$	0	0
$40$	0	0
$41$	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	$-0.485711\pi$
$43$	0.588526	0.0897494	0.0448747	+	0.998993i	$0.485711\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	9.66044	1.40912	0.704560	−	0.709644i	$-0.251144\pi$
$47$	9.66044	1.40912	0.704560	+	0.709644i	$0.251144\pi$
$48$	0	0
$49$	3.41147	0.487353
$50$	0	0
$51$	0	0
$52$	0	0
$53$	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
$56$	0	0
$57$	0	0
$58$	0	0
$59$	8.53209	1.11078	0.555392	−	0.831589i	$-0.312568\pi$
$59$	8.53209	1.11078	0.555392	+	0.831589i	$0.312568\pi$
$60$	0	0
$61$	−1.26857	−0.162424	−0.0812119	−	0.996697i	$-0.525879\pi$
$61$	−1.26857	−0.162424	−0.0812119	+	0.996697i	$0.525879\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.02229	−0.126799
$66$	0	0
$67$	−10.0077	−1.22264	−0.611320	−	0.791383i	$-0.709361\pi$
$67$	−10.0077	−1.22264	−0.611320	+	0.791383i	$0.709361\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−11.8307	−1.40404	−0.702022	−	0.712155i	$-0.747719\pi$
$71$	−11.8307	−1.40404	−0.702022	+	0.712155i	$0.747719\pi$
$72$	0	0
$73$	−8.23442	−0.963766	−0.481883	−	0.876236i	$-0.660047\pi$
$73$	−8.23442	−0.963766	−0.481883	+	0.876236i	$0.660047\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	10.0223	1.14215
$78$	0	0
$79$	−11.0496	−1.24318	−0.621590	−	0.783343i	$-0.713513\pi$
$79$	−11.0496	−1.24318	−0.621590	+	0.783343i	$0.713513\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1.50980	0.165722	0.0828610	−	0.996561i	$-0.473594\pi$
$83$	1.50980	0.165722	0.0828610	+	0.996561i	$0.473594\pi$
$84$	0	0
$85$	1.40373	0.152256
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−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
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−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	$0.104491\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	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	$-0.504090\pi$
$103$	−0.260830	−0.0257003	−0.0128502	+	0.999917i	$0.504090\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−4.04189	−0.390744	−0.195372	−	0.980729i	$-0.562591\pi$
$107$	−4.04189	−0.390744	−0.195372	+	0.980729i	$0.562591\pi$
$108$	0	0
$109$	−5.40373	−0.517584	−0.258792	−	0.965933i	$-0.583324\pi$
$109$	−5.40373	−0.517584	−0.258792	+	0.965933i	$0.583324\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	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
$116$	0	0
$117$	0	0
$118$	0	0
$119$	9.68004	0.887368
$120$	0	0
$121$	−1.35235	−0.122941
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−4.57667	−0.409349
$126$	0	0
$127$	6.63816	0.589041	0.294521	−	0.955645i	$-0.404840\pi$
$127$	6.63816	0.589041	0.294521	+	0.955645i	$0.404840\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	12.6408	1.10444	0.552218	−	0.833700i	$-0.313782\pi$
$131$	12.6408	1.10444	0.552218	+	0.833700i	$0.313782\pi$
$132$	0	0
$133$	−0.135163	−0.0117201
$134$	0	0
$135$	0	0
$136$	0	0
$137$	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	$-0.397519\pi$
$139$	7.46110	0.632843	0.316421	+	0.948619i	$0.397519\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−6.78611	−0.567483
$144$	0	0
$145$	3.07604	0.255451
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−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	$-0.501751\pi$
$151$	−0.135163	−0.0109994	−0.00549969	+	0.999985i	$0.501751\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	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	$0.321530\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−19.7023	−1.55276
$162$	0	0
$163$	9.76382	0.764762	0.382381	−	0.924005i	$-0.375104\pi$
$163$	9.76382	0.764762	0.382381	+	0.924005i	$0.375104\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3.57398	0.276563	0.138281	−	0.990393i	$-0.455842\pi$
$167$	3.57398	0.276563	0.138281	+	0.990393i	$0.455842\pi$
$168$	0	0
$169$	−8.22668	−0.632822
$170$	0	0
$171$	0	0
$172$	0	0
$173$	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
$176$	0	0
$177$	0	0
$178$	0	0
$179$	5.08378	0.379979	0.189990	−	0.981786i	$-0.439155\pi$
$179$	5.08378	0.379979	0.189990	+	0.981786i	$0.439155\pi$
$180$	0	0
$181$	7.15064	0.531503	0.265752	−	0.964042i	$-0.414380\pi$
$181$	7.15064	0.531503	0.265752	+	0.964042i	$0.414380\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1.68273	0.123717
$186$	0	0
$187$	9.31820	0.681414
$188$	0	0
$189$	0	0
$190$	0	0
$191$	10.5098	0.760462	0.380231	−	0.924891i	$-0.375844\pi$
$191$	10.5098	0.760462	0.380231	+	0.924891i	$0.375844\pi$
$192$	0	0
$193$	−10.1429	−0.730102	−0.365051	−	0.930987i	$-0.618948\pi$
$193$	−10.1429	−0.730102	−0.365051	+	0.930987i	$0.618948\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	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	$-0.618669\pi$
$199$	−10.2763	−0.728468	−0.364234	+	0.931307i	$0.618669\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	21.2121	1.48880
$204$	0	0
$205$	3.60401	0.251715
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−0.130110	−0.00899991
$210$	0	0
$211$	7.14290	0.491738	0.245869	−	0.969303i	$-0.420927\pi$
$211$	7.14290	0.491738	0.245869	+	0.969303i	$0.420927\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0.275378	0.0187806
$216$	0	0
$217$	20.0915	1.36390
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−6.55438	−0.440895
$222$	0	0
$223$	10.3354	0.692112	0.346056	−	0.938214i	$-0.387521\pi$
$223$	10.3354	0.692112	0.346056	+	0.938214i	$0.387521\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	13.0223	0.864320	0.432160	−	0.901797i	$-0.357752\pi$
$227$	13.0223	0.864320	0.432160	+	0.901797i	$0.357752\pi$
$228$	0	0
$229$	−28.0993	−1.85685	−0.928426	−	0.371518i	$-0.878837\pi$
$229$	−28.0993	−1.85685	−0.928426	+	0.371518i	$0.878837\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−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
$236$	0	0
$237$	0	0
$238$	0	0
$239$	15.0196	0.971537	0.485769	−	0.874087i	$-0.338540\pi$
$239$	15.0196	0.971537	0.485769	+	0.874087i	$0.338540\pi$
$240$	0	0
$241$	−12.9736	−0.835703	−0.417851	−	0.908515i	$-0.637217\pi$
$241$	−12.9736	−0.835703	−0.417851	+	0.908515i	$0.637217\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.59627	0.101982
$246$	0	0
$247$	0.0915189	0.00582321
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−0.872578	−0.0550766	−0.0275383	−	0.999621i	$-0.508767\pi$
$251$	−0.872578	−0.0550766	−0.0275383	+	0.999621i	$0.508767\pi$
$252$	0	0
$253$	−18.9659	−1.19237
$254$	0	0
$255$	0	0
$256$	0	0
$257$	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
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−4.29767	−0.265005	−0.132503	−	0.991183i	$-0.542301\pi$
$263$	−4.29767	−0.265005	−0.132503	+	0.991183i	$0.542301\pi$
$264$	0	0
$265$	2.31996	0.142514
$266$	0	0
$267$	0	0
$268$	0	0
$269$	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	$-0.496907\pi$
$271$	0.319955	0.0194359	0.00971795	+	0.999953i	$0.496907\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−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	$0.798245\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	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	$-0.410917\pi$
$283$	9.29355	0.552444	0.276222	+	0.961094i	$0.410917\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	24.8530	1.46702
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−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
$296$	0	0
$297$	0	0
$298$	0	0
$299$	13.3405	0.771500
$300$	0	0
$301$	1.89899	0.109456
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−0.593578	−0.0339882
$306$	0	0
$307$	−28.3432	−1.61763	−0.808815	−	0.588063i	$-0.799891\pi$
$307$	−28.3432	−1.61763	−0.808815	+	0.588063i	$0.799891\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	2.04458	0.115937	0.0579687	−	0.998318i	$-0.481538\pi$
$311$	2.04458	0.115937	0.0579687	+	0.998318i	$0.481538\pi$
$312$	0	0
$313$	8.41147	0.475445	0.237722	−	0.971333i	$-0.423599\pi$
$313$	8.41147	0.475445	0.237722	+	0.971333i	$0.423599\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	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	0
$321$	0	0
$322$	0	0
$323$	−0.125667	−0.00699231
$324$	0	0
$325$	10.4456	0.579419
$326$	0	0
$327$	0	0
$328$	0	0
$329$	31.1712	1.71852
$330$	0	0
$331$	−31.0310	−1.70562	−0.852808	−	0.522225i	$-0.825102\pi$
$331$	−31.0310	−1.70562	−0.852808	+	0.522225i	$0.825102\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−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	$0.276308\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	19.3405	1.04735
$342$	0	0
$343$	−11.5790	−0.625209
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1.80840	0.0970800	0.0485400	−	0.998821i	$-0.484543\pi$
$347$	1.80840	0.0970800	0.0485400	+	0.998821i	$0.484543\pi$
$348$	0	0
$349$	15.2608	0.816893	0.408447	−	0.912782i	$-0.366071\pi$
$349$	15.2608	0.816893	0.408447	+	0.912782i	$0.366071\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−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
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1.91447	0.101042	0.0505209	−	0.998723i	$-0.483912\pi$
$359$	1.91447	0.101042	0.0505209	+	0.998723i	$0.483912\pi$
$360$	0	0
$361$	−18.9982	−0.999908
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−3.85298	−0.201674
$366$	0	0
$367$	26.8648	1.40233	0.701167	−	0.712998i	$-0.252663\pi$
$367$	26.8648	1.40233	0.701167	+	0.712998i	$0.252663\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	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	$0.374478\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−14.3628	−0.739721
$378$	0	0
$379$	33.7870	1.73552	0.867762	−	0.496980i	$-0.165558\pi$
$379$	33.7870	1.73552	0.867762	+	0.496980i	$0.165558\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	9.27900	0.474135	0.237067	−	0.971493i	$-0.423814\pi$
$383$	9.27900	0.474135	0.237067	+	0.971493i	$0.423814\pi$
$384$	0	0
$385$	4.68954	0.239001
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−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
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−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	$0.335357\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	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
$404$	0	0
$405$	0	0
$406$	0	0
$407$	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	$0.524428\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	27.5303	1.35468
$414$	0	0
$415$	0.706452	0.0346784
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−35.4492	−1.73181	−0.865904	−	0.500209i	$-0.833256\pi$
$419$	−35.4492	−1.73181	−0.865904	+	0.500209i	$0.833256\pi$
$420$	0	0
$421$	9.21719	0.449218	0.224609	−	0.974449i	$-0.427889\pi$
$421$	9.21719	0.449218	0.224609	+	0.974449i	$0.427889\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−14.3432	−0.695746
$426$	0	0
$427$	−4.09327	−0.198087
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−11.5794	−0.557758	−0.278879	−	0.960326i	$-0.589963\pi$
$431$	−11.5794	−0.557758	−0.278879	+	0.960326i	$0.589963\pi$
$432$	0	0
$433$	6.06511	0.291471	0.145735	−	0.989324i	$-0.453445\pi$
$433$	6.06511	0.291471	0.145735	+	0.989324i	$0.453445\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0.255777	0.0122355
$438$	0	0
$439$	−29.0060	−1.38438	−0.692190	−	0.721715i	$-0.743354\pi$
$439$	−29.0060	−1.38438	−0.692190	+	0.721715i	$0.743354\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−30.8922	−1.46773	−0.733866	−	0.679294i	$-0.762286\pi$
$443$	−30.8922	−1.46773	−0.733866	+	0.679294i	$0.762286\pi$
$444$	0	0
$445$	−7.42696	−0.352071
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−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
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−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	$0.511205\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	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	$-0.550182\pi$
$463$	−6.75641	−0.313997	−0.156998	+	0.987599i	$0.550182\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−33.7469	−1.56162	−0.780810	−	0.624768i	$-0.785194\pi$
$467$	−33.7469	−1.56162	−0.780810	+	0.624768i	$0.785194\pi$
$468$	0	0
$469$	−32.2918	−1.49110
$470$	0	0
$471$	0	0
$472$	0	0
$473$	1.82800	0.0840516
$474$	0	0
$475$	0.200274	0.00918921
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−11.1266	−0.508387	−0.254194	−	0.967153i	$-0.581810\pi$
$479$	−11.1266	−0.508387	−0.254194	+	0.967153i	$0.581810\pi$
$480$	0	0
$481$	−7.85710	−0.358253
$482$	0	0
$483$	0	0
$484$	0	0
$485$	8.72462	0.396165
$486$	0	0
$487$	4.74691	0.215103	0.107552	−	0.994200i	$-0.465699\pi$
$487$	4.74691	0.215103	0.107552	+	0.994200i	$0.465699\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−22.3405	−1.00821	−0.504106	−	0.863642i	$-0.668178\pi$
$491$	−22.3405	−1.00821	−0.504106	+	0.863642i	$0.668178\pi$
$492$	0	0
$493$	19.7219	0.888231
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−38.1739	−1.71233
$498$	0	0
$499$	10.4611	0.468303	0.234152	−	0.972200i	$-0.424769\pi$
$499$	10.4611	0.468303	0.234152	+	0.972200i	$0.424769\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−25.0419	−1.11656	−0.558281	−	0.829652i	$-0.688539\pi$
$503$	−25.0419	−1.11656	−0.558281	+	0.829652i	$0.688539\pi$
$504$	0	0
$505$	4.25166	0.189196
$506$	0	0
$507$	0	0
$508$	0	0
$509$	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
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−0.122045	−0.00537795
$516$	0	0
$517$	30.0060	1.31966
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−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	$-0.689039\pi$
$523$	−25.5945	−1.11917	−0.559585	+	0.828773i	$0.689039\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	18.6800	0.813716
$528$	0	0
$529$	14.2841	0.621046
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−16.8280	−0.728902
$534$	0	0
$535$	−1.89124	−0.0817656
$536$	0	0
$537$	0	0
$538$	0	0
$539$	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	0
$543$	0	0
$544$	0	0
$545$	−2.52847	−0.108308
$546$	0	0
$547$	5.90848	0.252628	0.126314	−	0.991990i	$-0.459685\pi$
$547$	5.90848	0.252628	0.126314	+	0.991990i	$0.459685\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−0.275378	−0.0117315
$552$	0	0
$553$	−35.6536	−1.51615
$554$	0	0
$555$	0	0
$556$	0	0
$557$	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
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−36.0624	−1.51985	−0.759925	−	0.650011i	$-0.774764\pi$
$563$	−36.0624	−1.51985	−0.759925	+	0.650011i	$0.774764\pi$
$564$	0	0
$565$	0.647651	0.0272469
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−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	$-0.721002\pi$
$571$	−30.5790	−1.27969	−0.639846	+	0.768503i	$0.721002\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	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	$0.675367\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	4.87164	0.202110
$582$	0	0
$583$	15.4002	0.637812
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−20.8735	−0.861542	−0.430771	−	0.902461i	$-0.641758\pi$
$587$	−20.8735	−0.861542	−0.430771	+	0.902461i	$0.641758\pi$
$588$	0	0
$589$	−0.260830	−0.0107473
$590$	0	0
$591$	0	0
$592$	0	0
$593$	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
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0.448311	0.0183175	0.00915874	−	0.999958i	$-0.497085\pi$
$599$	0.448311	0.0183175	0.00915874	+	0.999958i	$0.497085\pi$
$600$	0	0
$601$	−17.6723	−0.720868	−0.360434	−	0.932785i	$-0.617371\pi$
$601$	−17.6723	−0.720868	−0.360434	+	0.932785i	$0.617371\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−0.632779	−0.0257261
$606$	0	0
$607$	−26.1985	−1.06337	−0.531683	−	0.846944i	$-0.678440\pi$
$607$	−26.1985	−1.06337	−0.531683	+	0.846944i	$0.678440\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−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	$0.405039\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	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	$-0.280045\pi$
$619$	31.7124	1.27463	0.637315	+	0.770603i	$0.280045\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−51.2158	−2.05192
$624$	0	0
$625$	21.7638	0.870553
$626$	0	0
$627$	0	0
$628$	0	0
$629$	10.7888	0.430178
$630$	0	0
$631$	38.5758	1.53568	0.767840	−	0.640642i	$-0.221332\pi$
$631$	38.5758	1.53568	0.767840	+	0.640642i	$0.221332\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	3.10607	0.123261
$636$	0	0
$637$	−7.45336	−0.295313
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−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	$-0.264208\pi$
$643$	34.2249	1.34970	0.674850	+	0.737955i	$0.264208\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	12.8726	0.506073	0.253037	−	0.967457i	$-0.418571\pi$
$647$	12.8726	0.506073	0.253037	+	0.967457i	$0.418571\pi$
$648$	0	0
$649$	26.5012	1.04026
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−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
$656$	0	0
$657$	0	0
$658$	0	0
$659$	13.7692	0.536372	0.268186	−	0.963367i	$-0.413576\pi$
$659$	13.7692	0.536372	0.268186	+	0.963367i	$0.413576\pi$
$660$	0	0
$661$	−20.2668	−0.788288	−0.394144	−	0.919049i	$-0.628959\pi$
$661$	−20.2668	−0.788288	−0.394144	+	0.919049i	$0.628959\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−0.0632441	−0.00245250
$666$	0	0
$667$	−40.1411	−1.55427
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−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	$0.700364\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−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
$680$	0	0
$681$	0	0
$682$	0	0
$683$	21.7469	0.832122	0.416061	−	0.909337i	$-0.363410\pi$
$683$	21.7469	0.832122	0.416061	+	0.909337i	$0.363410\pi$
$684$	0	0
$685$	4.96728	0.189790
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−10.8324	−0.412683
$690$	0	0
$691$	37.5948	1.43017	0.715087	−	0.699035i	$-0.246387\pi$
$691$	37.5948	1.43017	0.715087	+	0.699035i	$0.246387\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	3.49113	0.132426
$696$	0	0
$697$	23.1070	0.875240
$698$	0	0
$699$	0	0
$700$	0	0
$701$	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
$704$	0	0
$705$	0	0
$706$	0	0
$707$	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	$0.460575\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−38.0205	−1.42388
$714$	0	0
$715$	−3.17530	−0.118749
$716$	0	0
$717$	0	0
$718$	0	0
$719$	16.8324	0.627744	0.313872	−	0.949465i	$-0.398374\pi$
$719$	16.8324	0.627744	0.313872	+	0.949465i	$0.398374\pi$
$720$	0	0
$721$	−0.841615	−0.0313434
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−31.4306	−1.16730
$726$	0	0
$727$	−25.5621	−0.948046	−0.474023	−	0.880512i	$-0.657199\pi$
$727$	−25.5621	−0.948046	−0.474023	+	0.880512i	$0.657199\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	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	$0.588111\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−31.0847	−1.14502
$738$	0	0
$739$	−9.19078	−0.338088	−0.169044	−	0.985608i	$-0.554068\pi$
$739$	−9.19078	−0.338088	−0.169044	+	0.985608i	$0.554068\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−44.4492	−1.63068	−0.815342	−	0.578979i	$-0.803451\pi$
$743$	−44.4492	−1.63068	−0.815342	+	0.578979i	$0.803451\pi$
$744$	0	0
$745$	−1.99050	−0.0729264
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−13.0419	−0.476540
$750$	0	0
$751$	35.8060	1.30658	0.653290	−	0.757107i	$-0.273388\pi$
$751$	35.8060	1.30658	0.653290	+	0.757107i	$0.273388\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−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	$0.814005\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−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
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−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	$0.742906\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	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
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−0.322644	−0.0115599
$780$	0	0
$781$	−36.7469	−1.31491
$782$	0	0
$783$	0	0
$784$	0	0
$785$	6.23536	0.222549
$786$	0	0
$787$	−45.2404	−1.61265	−0.806323	−	0.591475i	$-0.798546\pi$
$787$	−45.2404	−1.61265	−0.806323	+	0.591475i	$0.798546\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	4.46616	0.158798
$792$	0	0
$793$	2.77156	0.0984211
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−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
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−25.5767	−0.902581
$804$	0	0
$805$	−9.21894	−0.324925
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−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	$-0.602598\pi$
$811$	−18.0419	−0.633536	−0.316768	+	0.948503i	$0.602598\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	4.56860	0.160031
$816$	0	0
$817$	−0.0246528	−0.000862492 0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	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	$-0.650009\pi$
$823$	−26.0496	−0.908033	−0.454017	+	0.890993i	$0.650009\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	37.6195	1.30816	0.654079	−	0.756426i	$-0.273056\pi$
$827$	37.6195	1.30816	0.654079	+	0.756426i	$0.273056\pi$
$828$	0	0
$829$	−35.0215	−1.21635	−0.608173	−	0.793805i	$-0.708097\pi$
$829$	−35.0215	−1.21635	−0.608173	+	0.793805i	$0.708097\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	10.2344	0.354602
$834$	0	0
$835$	1.67230	0.0578725
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−26.6141	−0.918821	−0.459411	−	0.888224i	$-0.651939\pi$
$839$	−26.6141	−0.918821	−0.459411	+	0.888224i	$0.651939\pi$
$840$	0	0
$841$	14.2172	0.490248
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−3.84936	−0.132422
$846$	0	0
$847$	−4.36360	−0.149935
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−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	$0.486617\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−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	$-0.548061\pi$
$859$	−8.81696	−0.300831	−0.150415	+	0.988623i	$0.548061\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	6.62124	0.225390	0.112695	−	0.993630i	$-0.464052\pi$
$863$	6.62124	0.225390	0.112695	+	0.993630i	$0.464052\pi$
$864$	0	0
$865$	8.78106	0.298565
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−34.3209	−1.16426
$870$	0	0
$871$	21.8648	0.740862
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−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	$0.521923\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	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	$-0.280780\pi$
$883$	37.7701	1.27107	0.635533	+	0.772074i	$0.280780\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−18.9786	−0.637241	−0.318620	−	0.947882i	$-0.603219\pi$
$887$	−18.9786	−0.637241	−0.318620	+	0.947882i	$0.603219\pi$
$888$	0	0
$889$	21.4192	0.718377
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−0.404667	−0.0135417
$894$	0	0
$895$	2.37876	0.0795131
$896$	0	0
$897$	0	0
$898$	0	0
$899$	40.9341	1.36523
$900$	0	0
$901$	14.8743	0.495536
$902$	0	0
$903$	0	0
$904$	0	0
$905$	3.34587	0.111220
$906$	0	0
$907$	−9.64590	−0.320287	−0.160143	−	0.987094i	$-0.551196\pi$
$907$	−9.64590	−0.320287	−0.160143	+	0.987094i	$0.551196\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	6.53478	0.216507	0.108253	−	0.994123i	$-0.465474\pi$
$911$	6.53478	0.216507	0.108253	+	0.994123i	$0.465474\pi$
$912$	0	0
$913$	4.68954	0.155201
$914$	0	0
$915$	0	0
$916$	0	0
$917$	40.7880	1.34694
$918$	0	0
$919$	−3.89124	−0.128360	−0.0641802	−	0.997938i	$-0.520443\pi$
$919$	−3.89124	−0.128360	−0.0641802	+	0.997938i	$0.520443\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	25.8476	0.850784
$924$	0	0
$925$	−17.1940	−0.565334
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−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
$932$	0	0
$933$	0	0
$934$	0	0
$935$	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	$0.318388\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−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
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−42.2354	−1.37246	−0.686232	−	0.727382i	$-0.740737\pi$
$947$	−42.2354	−1.37246	−0.686232	+	0.727382i	$0.740737\pi$
$948$	0	0
$949$	17.9905	0.583996
$950$	0	0
$951$	0	0
$952$	0	0
$953$	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
$956$	0	0
$957$	0	0
$958$	0	0
$959$	34.2540	1.10612
$960$	0	0
$961$	7.77156	0.250696
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−4.74598	−0.152778
$966$	0	0
$967$	−4.47203	−0.143811	−0.0719054	−	0.997411i	$-0.522908\pi$
$967$	−4.47203	−0.143811	−0.0719054	+	0.997411i	$0.522908\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	4.61949	0.148246	0.0741232	−	0.997249i	$-0.476384\pi$
$971$	4.61949	0.148246	0.0741232	+	0.997249i	$0.476384\pi$
$972$	0	0
$973$	24.0746	0.771796
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−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
$980$	0	0
$981$	0	0
$982$	0	0
$983$	17.0642	0.544263	0.272131	−	0.962260i	$-0.412271\pi$
$983$	17.0642	0.544263	0.272131	+	0.962260i	$0.412271\pi$
$984$	0	0
$985$	6.58996	0.209973
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−3.59358	−0.114269
$990$	0	0
$991$	−2.00000	−0.0635321	−0.0317660	−	0.999495i	$-0.510113\pi$
$991$	−2.00000	−0.0635321	−0.0317660	+	0.999495i	$0.510113\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−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	$0.384663\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3888.2.a.bk.1.1		3
3.2	odd	2		3888.2.a.bd.1.3		3
4.3	odd	2		243.2.a.f.1.3	yes	3
12.11	even	2		243.2.a.e.1.1	✓	3
20.19	odd	2		6075.2.a.bq.1.1		3
36.7	odd	6		243.2.c.e.82.1		6
36.11	even	6		243.2.c.f.82.3		6
36.23	even	6		243.2.c.f.163.3		6
36.31	odd	6		243.2.c.e.163.1		6
60.59	even	2		6075.2.a.bv.1.3		3
108.7	odd	18		729.2.e.i.325.1		6
108.11	even	18		729.2.e.g.82.1		6
108.23	even	18		729.2.e.a.406.1		6
108.31	odd	18		729.2.e.i.406.1		6
108.43	odd	18		729.2.e.b.82.1		6
108.47	even	18		729.2.e.a.325.1		6
108.59	even	18		729.2.e.g.649.1		6
108.67	odd	18		729.2.e.c.163.1		6
108.79	odd	18		729.2.e.c.568.1		6
108.83	even	18		729.2.e.h.568.1		6
108.95	even	18		729.2.e.h.163.1		6
108.103	odd	18		729.2.e.b.649.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
243.2.a.e.1.1	✓	3	12.11	even	2
243.2.a.f.1.3	yes	3	4.3	odd	2
243.2.c.e.82.1		6	36.7	odd	6
243.2.c.e.163.1		6	36.31	odd	6
243.2.c.f.82.3		6	36.11	even	6
243.2.c.f.163.3		6	36.23	even	6
729.2.e.a.325.1		6	108.47	even	18
729.2.e.a.406.1		6	108.23	even	18
729.2.e.b.82.1		6	108.43	odd	18
729.2.e.b.649.1		6	108.103	odd	18
729.2.e.c.163.1		6	108.67	odd	18
729.2.e.c.568.1		6	108.79	odd	18
729.2.e.g.82.1		6	108.11	even	18
729.2.e.g.649.1		6	108.59	even	18
729.2.e.h.163.1		6	108.95	even	18
729.2.e.h.568.1		6	108.83	even	18
729.2.e.i.325.1		6	108.7	odd	18
729.2.e.i.406.1		6	108.31	odd	18
3888.2.a.bd.1.3		3	3.2	odd	2
3888.2.a.bk.1.1		3	1.1	even	1	trivial
6075.2.a.bq.1.1		3	20.19	odd	2
6075.2.a.bv.1.3		3	60.59	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3888 = 2^{4} \cdot 3^{5} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3888.a (trivial)

\(f(q)\)	\(=\)	\(q+0.467911 q^{5} +3.22668 q^{7} +O(q^{10})\)	\(q+0.467911 q^{5} +3.22668 q^{7} +3.10607 q^{11} -2.18479 q^{13} +3.00000 q^{17} -0.0418891 q^{19} -6.10607 q^{23} -4.78106 q^{25} +6.57398 q^{29} +6.22668 q^{31} +1.50980 q^{35} +3.59627 q^{37} +7.70233 q^{41} +0.588526 q^{43} +9.66044 q^{47} +3.41147 q^{49} +4.95811 q^{53} +1.45336 q^{55} +8.53209 q^{59} -1.26857 q^{61} -1.02229 q^{65} -10.0077 q^{67} -11.8307 q^{71} -8.23442 q^{73} +10.0223 q^{77} -11.0496 q^{79} +1.50980 q^{83} +1.40373 q^{85} -15.8726 q^{89} -7.04963 q^{91} -0.0196004 q^{95} +18.6459 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 6 q^{5} + 3 q^{7}+O(q^{10}) \) 3 * q + 6 * q^5 + 3 * q^7	\( 3 q + 6 q^{5} + 3 q^{7} - 3 q^{11} - 3 q^{13} + 9 q^{17} + 3 q^{19} - 6 q^{23} + 3 q^{25} + 12 q^{29} + 12 q^{31} + 6 q^{35} - 3 q^{37} - 3 q^{41} + 12 q^{43} + 6 q^{47} + 18 q^{53} - 9 q^{55} + 21 q^{59} + 6 q^{61} + 3 q^{65} - 6 q^{67} + 9 q^{71} + 6 q^{73} + 24 q^{77} - 6 q^{79} + 6 q^{83} + 18 q^{85} + 6 q^{91} - 3 q^{95} + 15 q^{97}+O(q^{100}) \) 3 * q + 6 * q^5 + 3 * q^7 - 3 * q^11 - 3 * q^13 + 9 * q^17 + 3 * q^19 - 6 * q^23 + 3 * q^25 + 12 * q^29 + 12 * q^31 + 6 * q^35 - 3 * q^37 - 3 * q^41 + 12 * q^43 + 6 * q^47 + 18 * q^53 - 9 * q^55 + 21 * q^59 + 6 * q^61 + 3 * q^65 - 6 * q^67 + 9 * q^71 + 6 * q^73 + 24 * q^77 - 6 * q^79 + 6 * q^83 + 18 * q^85 + 6 * q^91 - 3 * q^95 + 15 * q^97

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