LMFDB - Embedded newform 3888.2.a.bd.1.3

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:	$1$
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			1.3
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)

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

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	$-0.533365\pi$
$5$	−0.467911	−0.209256	−0.104628	+	0.994511i	$0.533365\pi$
$6$	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	$-0.655118\pi$
$11$	−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	$0.597980\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	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	$-0.280340\pi$
$23$	6.10607	1.27320	0.636601	+	0.771193i	$0.280340\pi$
$24$	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	$-0.708983\pi$
$29$	−6.57398	−1.22076	−0.610379	+	0.792110i	$0.708983\pi$
$30$	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	$-0.705411\pi$
$41$	−7.70233	−1.20290	−0.601451	+	0.798910i	$0.705411\pi$
$42$	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	$-0.748856\pi$
$47$	−9.66044	−1.40912	−0.704560	+	0.709644i	$0.748856\pi$
$48$	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	$-0.610605\pi$
$53$	−4.95811	−0.681049	−0.340524	+	0.940236i	$0.610605\pi$
$54$	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	$-0.687432\pi$
$59$	−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	$0.525879\pi$
$62$	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	$-0.252281\pi$
$71$	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	$0.660047\pi$
$74$	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	$-0.526406\pi$
$83$	−1.50980	−0.165722	−0.0828610	+	0.996561i	$0.526406\pi$
$84$	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	$-0.181823\pi$
$89$	15.8726	1.68249	0.841245	+	0.540654i	$0.181823\pi$
$90$	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	$-0.649314\pi$
$101$	−9.08647	−0.904137	−0.452069	+	0.891983i	$0.649314\pi$
$102$	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	$-0.437409\pi$
$107$	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	$0.583324\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1.38413	−0.130208	−0.0651041	−	0.997878i	$-0.520738\pi$
$113$	−1.38413	−0.130208	−0.0651041	+	0.997878i	$0.520738\pi$
$114$	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	$-0.686218\pi$
$131$	−12.6408	−1.10444	−0.552218	+	0.833700i	$0.686218\pi$
$132$	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	$-0.649820\pi$
$137$	−10.6159	−0.906975	−0.453487	+	0.891263i	$0.649820\pi$
$138$	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	$-0.444249\pi$
$149$	4.25402	0.348503	0.174252	+	0.984701i	$0.444249\pi$
$150$	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	$-0.544158\pi$
$167$	−3.57398	−0.276563	−0.138281	+	0.990393i	$0.544158\pi$
$168$	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	$-0.752844\pi$
$173$	−18.7665	−1.42679	−0.713396	+	0.700761i	$0.752844\pi$
$174$	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	$-0.560845\pi$
$179$	−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	$0.414380\pi$
$182$	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	$-0.624156\pi$
$191$	−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	$0.618948\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−14.0838	−1.00343	−0.501714	−	0.865034i	$-0.667297\pi$
$197$	−14.0838	−1.00343	−0.501714	+	0.865034i	$0.667297\pi$
$198$	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	$-0.642248\pi$
$227$	−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	$0.878837\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	13.9145	0.911567	0.455784	−	0.890091i	$-0.349359\pi$
$233$	13.9145	0.911567	0.455784	+	0.890091i	$0.349359\pi$
$234$	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	$-0.661460\pi$
$239$	−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	$0.637217\pi$
$242$	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	$-0.491233\pi$
$251$	0.872578	0.0550766	0.0275383	+	0.999621i	$0.491233\pi$
$252$	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	$-0.545592\pi$
$257$	−4.57667	−0.285485	−0.142742	+	0.989760i	$0.545592\pi$
$258$	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	$-0.457699\pi$
$263$	4.29767	0.265005	0.132503	+	0.991183i	$0.457699\pi$
$264$	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	$-0.620525\pi$
$269$	−12.1257	−0.739315	−0.369657	+	0.929168i	$0.620525\pi$
$270$	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	$-0.792963\pi$
$281$	−26.6810	−1.59165	−0.795827	+	0.605524i	$0.792963\pi$
$282$	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	$-0.305522\pi$
$293$	19.6391	1.14733	0.573664	+	0.819091i	$0.305522\pi$
$294$	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	$-0.518462\pi$
$311$	−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	$0.423599\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−31.1284	−1.74834	−0.874171	−	0.485618i	$-0.838595\pi$
$317$	−31.1284	−1.74834	−0.874171	+	0.485618i	$0.838595\pi$
$318$	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	$-0.515457\pi$
$347$	−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	$0.366071\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	32.3824	1.72354	0.861770	−	0.507299i	$-0.169356\pi$
$353$	32.3824	1.72354	0.861770	+	0.507299i	$0.169356\pi$
$354$	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	$-0.516088\pi$
$359$	−1.91447	−0.101042	−0.0505209	+	0.998723i	$0.516088\pi$
$360$	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	$-0.576186\pi$
$383$	−9.27900	−0.474135	−0.237067	+	0.971493i	$0.576186\pi$
$384$	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	$-0.367245\pi$
$389$	15.9786	0.810149	0.405075	+	0.914284i	$0.367245\pi$
$390$	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	$-0.509125\pi$
$401$	−1.14796	−0.0573262	−0.0286631	+	0.999589i	$0.509125\pi$
$402$	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	$-0.166744\pi$
$419$	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	$0.427889\pi$
$422$	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	$-0.410037\pi$
$431$	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	$0.453445\pi$
$434$	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	$-0.237714\pi$
$443$	30.8922	1.46773	0.733866	+	0.679294i	$0.237714\pi$
$444$	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	$-0.126392\pi$
$449$	39.0820	1.84439	0.922197	+	0.386720i	$0.126392\pi$
$450$	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	$-0.709229\pi$
$461$	−26.2371	−1.22198	−0.610992	+	0.791637i	$0.709229\pi$
$462$	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	$-0.214806\pi$
$467$	33.7469	1.56162	0.780810	+	0.624768i	$0.214806\pi$
$468$	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	$-0.418190\pi$
$479$	11.1266	0.508387	0.254194	+	0.967153i	$0.418190\pi$
$480$	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	$-0.331822\pi$
$491$	22.3405	1.00821	0.504106	+	0.863642i	$0.331822\pi$
$492$	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	$-0.311461\pi$
$503$	25.0419	1.11656	0.558281	+	0.829652i	$0.311461\pi$
$504$	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	$-0.631095\pi$
$509$	−18.0624	−0.800603	−0.400301	+	0.916384i	$0.631095\pi$
$510$	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	$-0.307476\pi$
$521$	25.9581	1.13725	0.568623	+	0.822598i	$0.307476\pi$
$522$	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	$-0.691419\pi$
$557$	−26.7050	−1.13153	−0.565764	+	0.824567i	$0.691419\pi$
$558$	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	$-0.225236\pi$
$563$	36.0624	1.51985	0.759925	+	0.650011i	$0.225236\pi$
$564$	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	$-0.439268\pi$
$569$	9.04727	0.379281	0.189641	+	0.981854i	$0.439268\pi$
$570$	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	$-0.358242\pi$
$587$	20.8735	0.861542	0.430771	+	0.902461i	$0.358242\pi$
$588$	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	$-0.603933\pi$
$593$	−15.6212	−0.641488	−0.320744	+	0.947166i	$0.603933\pi$
$594$	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	$-0.502915\pi$
$599$	−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	$0.617371\pi$
$602$	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	$-0.591072\pi$
$617$	−14.0205	−0.564445	−0.282223	+	0.959349i	$0.591072\pi$
$618$	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	$-0.293315\pi$
$641$	30.6168	1.20929	0.604645	+	0.796495i	$0.293315\pi$
$642$	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	$-0.581429\pi$
$647$	−12.8726	−0.506073	−0.253037	+	0.967457i	$0.581429\pi$
$648$	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	$-0.428913\pi$
$653$	11.3191	0.442952	0.221476	+	0.975166i	$0.428913\pi$
$654$	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	$-0.586424\pi$
$659$	−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	$0.628959\pi$
$662$	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	$-0.477231\pi$
$677$	3.71925	0.142942	0.0714711	+	0.997443i	$0.477231\pi$
$678$	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	$-0.636590\pi$
$683$	−21.7469	−0.832122	−0.416061	+	0.909337i	$0.636590\pi$
$684$	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	$-0.645262\pi$
$701$	−23.3351	−0.881355	−0.440678	+	0.897665i	$0.645262\pi$
$702$	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	$-0.601626\pi$
$719$	−16.8324	−0.627744	−0.313872	+	0.949465i	$0.601626\pi$
$720$	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	$-0.196549\pi$
$743$	44.4492	1.63068	0.815342	+	0.578979i	$0.196549\pi$
$744$	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	$-0.264505\pi$
$761$	37.1952	1.34833	0.674163	+	0.738583i	$0.264505\pi$
$762$	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	$-0.898949\pi$
$773$	−52.8272	−1.90006	−0.950031	+	0.312156i	$0.898949\pi$
$774$	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	$-0.403130\pi$
$797$	16.9189	0.599299	0.299649	+	0.954049i	$0.403130\pi$
$798$	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	$-0.289655\pi$
$809$	34.9145	1.22753	0.613764	+	0.789490i	$0.289655\pi$
$810$	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	$-0.750924\pi$
$821$	−40.6391	−1.41831	−0.709157	+	0.705051i	$0.750924\pi$
$822$	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	$-0.726944\pi$
$827$	−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	$0.708097\pi$
$830$	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	$-0.348061\pi$
$839$	26.6141	0.918821	0.459411	+	0.888224i	$0.348061\pi$
$840$	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	$-0.446305\pi$
$857$	9.82976	0.335778	0.167889	+	0.985806i	$0.446305\pi$
$858$	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	$-0.535948\pi$
$863$	−6.62124	−0.225390	−0.112695	+	0.993630i	$0.535948\pi$
$864$	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	$-0.549809\pi$
$881$	−9.25133	−0.311685	−0.155843	+	0.987782i	$0.549809\pi$
$882$	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	$-0.396781\pi$
$887$	18.9786	0.637241	0.318620	+	0.947882i	$0.396781\pi$
$888$	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	$-0.534526\pi$
$911$	−6.53478	−0.216507	−0.108253	+	0.994123i	$0.534526\pi$
$912$	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	$-0.273188\pi$
$929$	39.8530	1.30753	0.653767	+	0.756696i	$0.273188\pi$
$930$	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	$-0.159526\pi$
$941$	53.8066	1.75405	0.877023	+	0.480448i	$0.159526\pi$
$942$	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	$-0.259263\pi$
$947$	42.2354	1.37246	0.686232	+	0.727382i	$0.259263\pi$
$948$	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	$-0.630321\pi$
$953$	−24.5776	−0.796147	−0.398073	+	0.917354i	$0.630321\pi$
$954$	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	$-0.523616\pi$
$971$	−4.61949	−0.148246	−0.0741232	+	0.997249i	$0.523616\pi$
$972$	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	$-0.442244\pi$
$977$	11.2808	0.360903	0.180452	+	0.983584i	$0.442244\pi$
$978$	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	$-0.587729\pi$
$983$	−17.0642	−0.544263	−0.272131	+	0.962260i	$0.587729\pi$
$984$	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.bd.1.3		3
3.2	odd	2		3888.2.a.bk.1.1		3
4.3	odd	2		243.2.a.e.1.1	✓	3
12.11	even	2		243.2.a.f.1.3	yes	3
20.19	odd	2		6075.2.a.bv.1.3		3
36.7	odd	6		243.2.c.f.82.3		6
36.11	even	6		243.2.c.e.82.1		6
36.23	even	6		243.2.c.e.163.1		6
36.31	odd	6		243.2.c.f.163.3		6
60.59	even	2		6075.2.a.bq.1.1		3
108.7	odd	18		729.2.e.a.325.1		6
108.11	even	18		729.2.e.b.82.1		6
108.23	even	18		729.2.e.i.406.1		6
108.31	odd	18		729.2.e.a.406.1		6
108.43	odd	18		729.2.e.g.82.1		6
108.47	even	18		729.2.e.i.325.1		6
108.59	even	18		729.2.e.b.649.1		6
108.67	odd	18		729.2.e.h.163.1		6
108.79	odd	18		729.2.e.h.568.1		6
108.83	even	18		729.2.e.c.568.1		6
108.95	even	18		729.2.e.c.163.1		6
108.103	odd	18		729.2.e.g.649.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
243.2.a.e.1.1	✓	3	4.3	odd	2
243.2.a.f.1.3	yes	3	12.11	even	2
243.2.c.e.82.1		6	36.11	even	6
243.2.c.e.163.1		6	36.23	even	6
243.2.c.f.82.3		6	36.7	odd	6
243.2.c.f.163.3		6	36.31	odd	6
729.2.e.a.325.1		6	108.7	odd	18
729.2.e.a.406.1		6	108.31	odd	18
729.2.e.b.82.1		6	108.11	even	18
729.2.e.b.649.1		6	108.59	even	18
729.2.e.c.163.1		6	108.95	even	18
729.2.e.c.568.1		6	108.83	even	18
729.2.e.g.82.1		6	108.43	odd	18
729.2.e.g.649.1		6	108.103	odd	18
729.2.e.h.163.1		6	108.67	odd	18
729.2.e.h.568.1		6	108.79	odd	18
729.2.e.i.325.1		6	108.47	even	18
729.2.e.i.406.1		6	108.23	even	18
3888.2.a.bd.1.3		3	1.1	even	1	trivial
3888.2.a.bk.1.1		3	3.2	odd	2
6075.2.a.bq.1.1		3	60.59	even	2
6075.2.a.bv.1.3		3	20.19	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:	\( N \)	\(=\)	\( 3888 = 2^{4} \cdot 3^{5} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3888.a (trivial)

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