LMFDB - Embedded newform 3888.2.a.ba.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:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 243)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ 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)

$f(q)$	$=$	$q-3.46410 q^{5} +1.00000 q^{7} -3.46410 q^{11} +5.00000 q^{13} +1.00000 q^{19} -6.92820 q^{23} +7.00000 q^{25} +3.46410 q^{29} -5.00000 q^{31} -3.46410 q^{35} -1.00000 q^{37} -3.46410 q^{41} +1.00000 q^{43} -3.46410 q^{47} -6.00000 q^{49} +10.3923 q^{53} +12.0000 q^{55} +3.46410 q^{59} +2.00000 q^{61} -17.3205 q^{65} -8.00000 q^{67} +10.3923 q^{71} +2.00000 q^{73} -3.46410 q^{77} +1.00000 q^{79} +6.92820 q^{83} -10.3923 q^{89} +5.00000 q^{91} -3.46410 q^{95} +17.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{7} + 10 q^{13} + 2 q^{19} + 14 q^{25} - 10 q^{31} - 2 q^{37} + 2 q^{43} - 12 q^{49} + 24 q^{55} + 4 q^{61} - 16 q^{67} + 4 q^{73} + 2 q^{79} + 10 q^{91} + 34 q^{97}+O(q^{100}) $ 2 * q + 2 * q^7 + 10 * q^13 + 2 * q^19 + 14 * q^25 - 10 * q^31 - 2 * q^37 + 2 * q^43 - 12 * q^49 + 24 * q^55 + 4 * q^61 - 16 * q^67 + 4 * q^73 + 2 * q^79 + 10 * q^91 + 34 * 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
$5$	−3.46410	−1.54919	−0.774597	−	0.632456i	$-0.782047\pi$
$5$	−3.46410	−1.54919	−0.774597	+	0.632456i	$0.782047\pi$
$6$	0	0
$7$	1.00000	0.377964	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−3.46410	−1.04447	−0.522233	−	0.852803i	$-0.674901\pi$
$11$	−3.46410	−1.04447	−0.522233	+	0.852803i	$0.674901\pi$
$12$	0	0
$13$	5.00000	1.38675	0.693375	−	0.720577i	$-0.256123\pi$
$13$	5.00000	1.38675	0.693375	+	0.720577i	$0.256123\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−6.92820	−1.44463	−0.722315	−	0.691564i	$-0.756922\pi$
$23$	−6.92820	−1.44463	−0.722315	+	0.691564i	$0.756922\pi$
$24$	0	0
$25$	7.00000	1.40000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	3.46410	0.643268	0.321634	−	0.946864i	$-0.395768\pi$
$29$	3.46410	0.643268	0.321634	+	0.946864i	$0.395768\pi$
$30$	0	0
$31$	−5.00000	−0.898027	−0.449013	−	0.893525i	$-0.648224\pi$
$31$	−5.00000	−0.898027	−0.449013	+	0.893525i	$0.648224\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.46410	−0.585540
$36$	0	0
$37$	−1.00000	−0.164399	−0.0821995	−	0.996616i	$-0.526194\pi$
$37$	−1.00000	−0.164399	−0.0821995	+	0.996616i	$0.526194\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−3.46410	−0.541002	−0.270501	−	0.962720i	$-0.587189\pi$
$41$	−3.46410	−0.541002	−0.270501	+	0.962720i	$0.587189\pi$
$42$	0	0
$43$	1.00000	0.152499	0.0762493	−	0.997089i	$-0.475706\pi$
$43$	1.00000	0.152499	0.0762493	+	0.997089i	$0.475706\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−3.46410	−0.505291	−0.252646	−	0.967559i	$-0.581301\pi$
$47$	−3.46410	−0.505291	−0.252646	+	0.967559i	$0.581301\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	0	0
$51$	0	0
$52$	0	0
$53$	10.3923	1.42749	0.713746	−	0.700404i	$-0.246997\pi$
$53$	10.3923	1.42749	0.713746	+	0.700404i	$0.246997\pi$
$54$	0	0
$55$	12.0000	1.61808
$56$	0	0
$57$	0	0
$58$	0	0
$59$	3.46410	0.450988	0.225494	−	0.974245i	$-0.427600\pi$
$59$	3.46410	0.450988	0.225494	+	0.974245i	$0.427600\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−17.3205	−2.14834
$66$	0	0
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	10.3923	1.23334	0.616670	−	0.787222i	$-0.288481\pi$
$71$	10.3923	1.23334	0.616670	+	0.787222i	$0.288481\pi$
$72$	0	0
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−3.46410	−0.394771
$78$	0	0
$79$	1.00000	0.112509	0.0562544	−	0.998416i	$-0.482084\pi$
$79$	1.00000	0.112509	0.0562544	+	0.998416i	$0.482084\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	6.92820	0.760469	0.380235	−	0.924890i	$-0.375843\pi$
$83$	6.92820	0.760469	0.380235	+	0.924890i	$0.375843\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−10.3923	−1.10158	−0.550791	−	0.834643i	$-0.685674\pi$
$89$	−10.3923	−1.10158	−0.550791	+	0.834643i	$0.685674\pi$
$90$	0	0
$91$	5.00000	0.524142
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−3.46410	−0.355409
$96$	0	0
$97$	17.0000	1.72609	0.863044	−	0.505128i	$-0.168555\pi$
$97$	17.0000	1.72609	0.863044	+	0.505128i	$0.168555\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	13.8564	1.37876	0.689382	−	0.724398i	$-0.257882\pi$
$101$	13.8564	1.37876	0.689382	+	0.724398i	$0.257882\pi$
$102$	0	0
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	10.3923	1.00466	0.502331	−	0.864675i	$-0.332476\pi$
$107$	10.3923	1.00466	0.502331	+	0.864675i	$0.332476\pi$
$108$	0	0
$109$	17.0000	1.62830	0.814152	−	0.580651i	$-0.197202\pi$
$109$	17.0000	1.62830	0.814152	+	0.580651i	$0.197202\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	17.3205	1.62938	0.814688	−	0.579899i	$-0.196908\pi$
$113$	17.3205	1.62938	0.814688	+	0.579899i	$0.196908\pi$
$114$	0	0
$115$	24.0000	2.23801
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−6.92820	−0.619677
$126$	0	0
$127$	−17.0000	−1.50851	−0.754253	−	0.656584i	$-0.772001\pi$
$127$	−17.0000	−1.50851	−0.754253	+	0.656584i	$0.772001\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	3.46410	0.302660	0.151330	−	0.988483i	$-0.451644\pi$
$131$	3.46410	0.302660	0.151330	+	0.988483i	$0.451644\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−6.92820	−0.591916	−0.295958	−	0.955201i	$-0.595639\pi$
$137$	−6.92820	−0.591916	−0.295958	+	0.955201i	$0.595639\pi$
$138$	0	0
$139$	13.0000	1.10265	0.551323	−	0.834292i	$-0.314123\pi$
$139$	13.0000	1.10265	0.551323	+	0.834292i	$0.314123\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−17.3205	−1.44841
$144$	0	0
$145$	−12.0000	−0.996546
$146$	0	0
$147$	0	0
$148$	0	0
$149$	6.92820	0.567581	0.283790	−	0.958886i	$-0.408408\pi$
$149$	6.92820	0.567581	0.283790	+	0.958886i	$0.408408\pi$
$150$	0	0
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	17.3205	1.39122
$156$	0	0
$157$	−13.0000	−1.03751	−0.518756	−	0.854922i	$-0.673605\pi$
$157$	−13.0000	−1.03751	−0.518756	+	0.854922i	$0.673605\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−6.92820	−0.546019
$162$	0	0
$163$	1.00000	0.0783260	0.0391630	−	0.999233i	$-0.487531\pi$
$163$	1.00000	0.0783260	0.0391630	+	0.999233i	$0.487531\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	24.2487	1.87642	0.938211	−	0.346064i	$-0.112482\pi$
$167$	24.2487	1.87642	0.938211	+	0.346064i	$0.112482\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	0	0
$171$	0	0
$172$	0	0
$173$	13.8564	1.05348	0.526742	−	0.850026i	$-0.323414\pi$
$173$	13.8564	1.05348	0.526742	+	0.850026i	$0.323414\pi$
$174$	0	0
$175$	7.00000	0.529150
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−20.7846	−1.55351	−0.776757	−	0.629800i	$-0.783137\pi$
$179$	−20.7846	−1.55351	−0.776757	+	0.629800i	$0.783137\pi$
$180$	0	0
$181$	17.0000	1.26360	0.631800	−	0.775131i	$-0.282316\pi$
$181$	17.0000	1.26360	0.631800	+	0.775131i	$0.282316\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	3.46410	0.254686
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	6.92820	0.501307	0.250654	−	0.968077i	$-0.419354\pi$
$191$	6.92820	0.501307	0.250654	+	0.968077i	$0.419354\pi$
$192$	0	0
$193$	−10.0000	−0.719816	−0.359908	−	0.932988i	$-0.617192\pi$
$193$	−10.0000	−0.719816	−0.359908	+	0.932988i	$0.617192\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	10.3923	0.740421	0.370211	−	0.928948i	$-0.379286\pi$
$197$	10.3923	0.740421	0.370211	+	0.928948i	$0.379286\pi$
$198$	0	0
$199$	19.0000	1.34687	0.673437	−	0.739244i	$-0.264817\pi$
$199$	19.0000	1.34687	0.673437	+	0.739244i	$0.264817\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	3.46410	0.243132
$204$	0	0
$205$	12.0000	0.838116
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−3.46410	−0.239617
$210$	0	0
$211$	−5.00000	−0.344214	−0.172107	−	0.985078i	$-0.555058\pi$
$211$	−5.00000	−0.344214	−0.172107	+	0.985078i	$0.555058\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−3.46410	−0.236250
$216$	0	0
$217$	−5.00000	−0.339422
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	19.0000	1.27233	0.636167	−	0.771551i	$-0.280519\pi$
$223$	19.0000	1.27233	0.636167	+	0.771551i	$0.280519\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−13.8564	−0.919682	−0.459841	−	0.888001i	$-0.652094\pi$
$227$	−13.8564	−0.919682	−0.459841	+	0.888001i	$0.652094\pi$
$228$	0	0
$229$	5.00000	0.330409	0.165205	−	0.986259i	$-0.447172\pi$
$229$	5.00000	0.330409	0.165205	+	0.986259i	$0.447172\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	0	0
$235$	12.0000	0.782794
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−6.92820	−0.448148	−0.224074	−	0.974572i	$-0.571936\pi$
$239$	−6.92820	−0.448148	−0.224074	+	0.974572i	$0.571936\pi$
$240$	0	0
$241$	−19.0000	−1.22390	−0.611949	−	0.790897i	$-0.709614\pi$
$241$	−19.0000	−1.22390	−0.611949	+	0.790897i	$0.709614\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	20.7846	1.32788
$246$	0	0
$247$	5.00000	0.318142
$248$	0	0
$249$	0	0
$250$	0	0
$251$	20.7846	1.31191	0.655956	−	0.754799i	$-0.272265\pi$
$251$	20.7846	1.31191	0.655956	+	0.754799i	$0.272265\pi$
$252$	0	0
$253$	24.0000	1.50887
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−3.46410	−0.216085	−0.108042	−	0.994146i	$-0.534458\pi$
$257$	−3.46410	−0.216085	−0.108042	+	0.994146i	$0.534458\pi$
$258$	0	0
$259$	−1.00000	−0.0621370
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−13.8564	−0.854423	−0.427211	−	0.904152i	$-0.640504\pi$
$263$	−13.8564	−0.854423	−0.427211	+	0.904152i	$0.640504\pi$
$264$	0	0
$265$	−36.0000	−2.21146
$266$	0	0
$267$	0	0
$268$	0	0
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	16.0000	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−24.2487	−1.46225
$276$	0	0
$277$	17.0000	1.02143	0.510716	−	0.859750i	$-0.329381\pi$
$277$	17.0000	1.02143	0.510716	+	0.859750i	$0.329381\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	13.8564	0.826604	0.413302	−	0.910594i	$-0.364375\pi$
$281$	13.8564	0.826604	0.413302	+	0.910594i	$0.364375\pi$
$282$	0	0
$283$	13.0000	0.772770	0.386385	−	0.922338i	$-0.373724\pi$
$283$	13.0000	0.772770	0.386385	+	0.922338i	$0.373724\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−3.46410	−0.204479
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−13.8564	−0.809500	−0.404750	−	0.914427i	$-0.632641\pi$
$293$	−13.8564	−0.809500	−0.404750	+	0.914427i	$0.632641\pi$
$294$	0	0
$295$	−12.0000	−0.698667
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−34.6410	−2.00334
$300$	0	0
$301$	1.00000	0.0576390
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−6.92820	−0.396708
$306$	0	0
$307$	−20.0000	−1.14146	−0.570730	−	0.821138i	$-0.693340\pi$
$307$	−20.0000	−1.14146	−0.570730	+	0.821138i	$0.693340\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	13.8564	0.785725	0.392862	−	0.919597i	$-0.371485\pi$
$311$	13.8564	0.785725	0.392862	+	0.919597i	$0.371485\pi$
$312$	0	0
$313$	−1.00000	−0.0565233	−0.0282617	−	0.999601i	$-0.508997\pi$
$313$	−1.00000	−0.0565233	−0.0282617	+	0.999601i	$0.508997\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−27.7128	−1.55651	−0.778253	−	0.627950i	$-0.783894\pi$
$317$	−27.7128	−1.55651	−0.778253	+	0.627950i	$0.783894\pi$
$318$	0	0
$319$	−12.0000	−0.671871
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	35.0000	1.94145
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−3.46410	−0.190982
$330$	0	0
$331$	19.0000	1.04433	0.522167	−	0.852843i	$-0.325124\pi$
$331$	19.0000	1.04433	0.522167	+	0.852843i	$0.325124\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	27.7128	1.51411
$336$	0	0
$337$	5.00000	0.272367	0.136184	−	0.990684i	$-0.456516\pi$
$337$	5.00000	0.272367	0.136184	+	0.990684i	$0.456516\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	17.3205	0.937958
$342$	0	0
$343$	−13.0000	−0.701934
$344$	0	0
$345$	0	0
$346$	0	0
$347$	24.2487	1.30174	0.650870	−	0.759190i	$-0.274404\pi$
$347$	24.2487	1.30174	0.650870	+	0.759190i	$0.274404\pi$
$348$	0	0
$349$	−1.00000	−0.0535288	−0.0267644	−	0.999642i	$-0.508520\pi$
$349$	−1.00000	−0.0535288	−0.0267644	+	0.999642i	$0.508520\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−17.3205	−0.921878	−0.460939	−	0.887432i	$-0.652487\pi$
$353$	−17.3205	−0.921878	−0.460939	+	0.887432i	$0.652487\pi$
$354$	0	0
$355$	−36.0000	−1.91068
$356$	0	0
$357$	0	0
$358$	0	0
$359$	31.1769	1.64545	0.822727	−	0.568436i	$-0.192451\pi$
$359$	31.1769	1.64545	0.822727	+	0.568436i	$0.192451\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−6.92820	−0.362639
$366$	0	0
$367$	16.0000	0.835193	0.417597	−	0.908633i	$-0.362873\pi$
$367$	16.0000	0.835193	0.417597	+	0.908633i	$0.362873\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	10.3923	0.539542
$372$	0	0
$373$	23.0000	1.19089	0.595447	−	0.803394i	$-0.296975\pi$
$373$	23.0000	1.19089	0.595447	+	0.803394i	$0.296975\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	17.3205	0.892052
$378$	0	0
$379$	19.0000	0.975964	0.487982	−	0.872854i	$-0.337733\pi$
$379$	19.0000	0.975964	0.487982	+	0.872854i	$0.337733\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−17.3205	−0.885037	−0.442518	−	0.896759i	$-0.645915\pi$
$383$	−17.3205	−0.885037	−0.442518	+	0.896759i	$0.645915\pi$
$384$	0	0
$385$	12.0000	0.611577
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−6.92820	−0.351274	−0.175637	−	0.984455i	$-0.556198\pi$
$389$	−6.92820	−0.351274	−0.175637	+	0.984455i	$0.556198\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−3.46410	−0.174298
$396$	0	0
$397$	−1.00000	−0.0501886	−0.0250943	−	0.999685i	$-0.507989\pi$
$397$	−1.00000	−0.0501886	−0.0250943	+	0.999685i	$0.507989\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−3.46410	−0.172989	−0.0864945	−	0.996252i	$-0.527566\pi$
$401$	−3.46410	−0.172989	−0.0864945	+	0.996252i	$0.527566\pi$
$402$	0	0
$403$	−25.0000	−1.24534
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3.46410	0.171709
$408$	0	0
$409$	5.00000	0.247234	0.123617	−	0.992330i	$-0.460551\pi$
$409$	5.00000	0.247234	0.123617	+	0.992330i	$0.460551\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	3.46410	0.170457
$414$	0	0
$415$	−24.0000	−1.17811
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−38.1051	−1.86156	−0.930778	−	0.365584i	$-0.880869\pi$
$419$	−38.1051	−1.86156	−0.930778	+	0.365584i	$0.880869\pi$
$420$	0	0
$421$	−19.0000	−0.926003	−0.463002	−	0.886357i	$-0.653228\pi$
$421$	−19.0000	−0.926003	−0.463002	+	0.886357i	$0.653228\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	2.00000	0.0967868
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−20.7846	−1.00116	−0.500580	−	0.865690i	$-0.666880\pi$
$431$	−20.7846	−1.00116	−0.500580	+	0.865690i	$0.666880\pi$
$432$	0	0
$433$	−1.00000	−0.0480569	−0.0240285	−	0.999711i	$-0.507649\pi$
$433$	−1.00000	−0.0480569	−0.0240285	+	0.999711i	$0.507649\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−6.92820	−0.331421
$438$	0	0
$439$	−20.0000	−0.954548	−0.477274	−	0.878755i	$-0.658375\pi$
$439$	−20.0000	−0.954548	−0.477274	+	0.878755i	$0.658375\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−13.8564	−0.658338	−0.329169	−	0.944271i	$-0.606769\pi$
$443$	−13.8564	−0.658338	−0.329169	+	0.944271i	$0.606769\pi$
$444$	0	0
$445$	36.0000	1.70656
$446$	0	0
$447$	0	0
$448$	0	0
$449$	10.3923	0.490443	0.245222	−	0.969467i	$-0.421139\pi$
$449$	10.3923	0.490443	0.245222	+	0.969467i	$0.421139\pi$
$450$	0	0
$451$	12.0000	0.565058
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−17.3205	−0.811998
$456$	0	0
$457$	17.0000	0.795226	0.397613	−	0.917553i	$-0.369839\pi$
$457$	17.0000	0.795226	0.397613	+	0.917553i	$0.369839\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−27.7128	−1.29071	−0.645357	−	0.763881i	$-0.723291\pi$
$461$	−27.7128	−1.29071	−0.645357	+	0.763881i	$0.723291\pi$
$462$	0	0
$463$	31.0000	1.44069	0.720346	−	0.693615i	$-0.243983\pi$
$463$	31.0000	1.44069	0.720346	+	0.693615i	$0.243983\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−10.3923	−0.480899	−0.240449	−	0.970662i	$-0.577295\pi$
$467$	−10.3923	−0.480899	−0.240449	+	0.970662i	$0.577295\pi$
$468$	0	0
$469$	−8.00000	−0.369406
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−3.46410	−0.159280
$474$	0	0
$475$	7.00000	0.321182
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−13.8564	−0.633115	−0.316558	−	0.948573i	$-0.602527\pi$
$479$	−13.8564	−0.633115	−0.316558	+	0.948573i	$0.602527\pi$
$480$	0	0
$481$	−5.00000	−0.227980
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−58.8897	−2.67404
$486$	0	0
$487$	19.0000	0.860972	0.430486	−	0.902597i	$-0.358342\pi$
$487$	19.0000	0.860972	0.430486	+	0.902597i	$0.358342\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−38.1051	−1.71966	−0.859830	−	0.510581i	$-0.829431\pi$
$491$	−38.1051	−1.71966	−0.859830	+	0.510581i	$0.829431\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	10.3923	0.466159
$498$	0	0
$499$	28.0000	1.25345	0.626726	−	0.779240i	$-0.284395\pi$
$499$	28.0000	1.25345	0.626726	+	0.779240i	$0.284395\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	41.5692	1.85348	0.926740	−	0.375703i	$-0.122599\pi$
$503$	41.5692	1.85348	0.926740	+	0.375703i	$0.122599\pi$
$504$	0	0
$505$	−48.0000	−2.13597
$506$	0	0
$507$	0	0
$508$	0	0
$509$	27.7128	1.22835	0.614174	−	0.789170i	$-0.289489\pi$
$509$	27.7128	1.22835	0.614174	+	0.789170i	$0.289489\pi$
$510$	0	0
$511$	2.00000	0.0884748
$512$	0	0
$513$	0	0
$514$	0	0
$515$	27.7128	1.22117
$516$	0	0
$517$	12.0000	0.527759
$518$	0	0
$519$	0	0
$520$	0	0
$521$	20.7846	0.910590	0.455295	−	0.890341i	$-0.349534\pi$
$521$	20.7846	0.910590	0.455295	+	0.890341i	$0.349534\pi$
$522$	0	0
$523$	−20.0000	−0.874539	−0.437269	−	0.899331i	$-0.644054\pi$
$523$	−20.0000	−0.874539	−0.437269	+	0.899331i	$0.644054\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	25.0000	1.08696
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−17.3205	−0.750234
$534$	0	0
$535$	−36.0000	−1.55642
$536$	0	0
$537$	0	0
$538$	0	0
$539$	20.7846	0.895257
$540$	0	0
$541$	17.0000	0.730887	0.365444	−	0.930834i	$-0.380917\pi$
$541$	17.0000	0.730887	0.365444	+	0.930834i	$0.380917\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−58.8897	−2.52256
$546$	0	0
$547$	−20.0000	−0.855138	−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000	−0.855138	−0.427569	+	0.903983i	$0.640630\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	3.46410	0.147576
$552$	0	0
$553$	1.00000	0.0425243
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−10.3923	−0.440336	−0.220168	−	0.975462i	$-0.570661\pi$
$557$	−10.3923	−0.440336	−0.220168	+	0.975462i	$0.570661\pi$
$558$	0	0
$559$	5.00000	0.211477
$560$	0	0
$561$	0	0
$562$	0	0
$563$	34.6410	1.45994	0.729972	−	0.683477i	$-0.239533\pi$
$563$	34.6410	1.45994	0.729972	+	0.683477i	$0.239533\pi$
$564$	0	0
$565$	−60.0000	−2.52422
$566$	0	0
$567$	0	0
$568$	0	0
$569$	24.2487	1.01656	0.508279	−	0.861192i	$-0.330282\pi$
$569$	24.2487	1.01656	0.508279	+	0.861192i	$0.330282\pi$
$570$	0	0
$571$	−41.0000	−1.71580	−0.857898	−	0.513820i	$-0.828230\pi$
$571$	−41.0000	−1.71580	−0.857898	+	0.513820i	$0.828230\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−48.4974	−2.02248
$576$	0	0
$577$	−34.0000	−1.41544	−0.707719	−	0.706494i	$-0.750276\pi$
$577$	−34.0000	−1.41544	−0.707719	+	0.706494i	$0.750276\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	6.92820	0.287430
$582$	0	0
$583$	−36.0000	−1.49097
$584$	0	0
$585$	0	0
$586$	0	0
$587$	6.92820	0.285958	0.142979	−	0.989726i	$-0.454332\pi$
$587$	6.92820	0.285958	0.142979	+	0.989726i	$0.454332\pi$
$588$	0	0
$589$	−5.00000	−0.206021
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−20.7846	−0.853522	−0.426761	−	0.904365i	$-0.640345\pi$
$593$	−20.7846	−0.853522	−0.426761	+	0.904365i	$0.640345\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	24.2487	0.990775	0.495388	−	0.868672i	$-0.335026\pi$
$599$	24.2487	0.990775	0.495388	+	0.868672i	$0.335026\pi$
$600$	0	0
$601$	35.0000	1.42768	0.713840	−	0.700309i	$-0.246954\pi$
$601$	35.0000	1.42768	0.713840	+	0.700309i	$0.246954\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−3.46410	−0.140836
$606$	0	0
$607$	13.0000	0.527654	0.263827	−	0.964570i	$-0.415015\pi$
$607$	13.0000	0.527654	0.263827	+	0.964570i	$0.415015\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−17.3205	−0.700713
$612$	0	0
$613$	2.00000	0.0807792	0.0403896	−	0.999184i	$-0.487140\pi$
$613$	2.00000	0.0807792	0.0403896	+	0.999184i	$0.487140\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	6.92820	0.278919	0.139459	−	0.990228i	$-0.455464\pi$
$617$	6.92820	0.278919	0.139459	+	0.990228i	$0.455464\pi$
$618$	0	0
$619$	−20.0000	−0.803868	−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000	−0.803868	−0.401934	+	0.915669i	$0.631662\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−10.3923	−0.416359
$624$	0	0
$625$	−11.0000	−0.440000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−17.0000	−0.676759	−0.338380	−	0.941010i	$-0.609879\pi$
$631$	−17.0000	−0.676759	−0.338380	+	0.941010i	$0.609879\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	58.8897	2.33697
$636$	0	0
$637$	−30.0000	−1.18864
$638$	0	0
$639$	0	0
$640$	0	0
$641$	45.0333	1.77871	0.889355	−	0.457218i	$-0.151154\pi$
$641$	45.0333	1.77871	0.889355	+	0.457218i	$0.151154\pi$
$642$	0	0
$643$	−44.0000	−1.73519	−0.867595	−	0.497271i	$-0.834335\pi$
$643$	−44.0000	−1.73519	−0.867595	+	0.497271i	$0.834335\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−20.7846	−0.817127	−0.408564	−	0.912730i	$-0.633970\pi$
$647$	−20.7846	−0.817127	−0.408564	+	0.912730i	$0.633970\pi$
$648$	0	0
$649$	−12.0000	−0.471041
$650$	0	0
$651$	0	0
$652$	0	0
$653$	17.3205	0.677804	0.338902	−	0.940822i	$-0.389945\pi$
$653$	17.3205	0.677804	0.338902	+	0.940822i	$0.389945\pi$
$654$	0	0
$655$	−12.0000	−0.468879
$656$	0	0
$657$	0	0
$658$	0	0
$659$	27.7128	1.07954	0.539769	−	0.841813i	$-0.318512\pi$
$659$	27.7128	1.07954	0.539769	+	0.841813i	$0.318512\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−3.46410	−0.134332
$666$	0	0
$667$	−24.0000	−0.929284
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−6.92820	−0.267460
$672$	0	0
$673$	−37.0000	−1.42625	−0.713123	−	0.701039i	$-0.752720\pi$
$673$	−37.0000	−1.42625	−0.713123	+	0.701039i	$0.752720\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−17.3205	−0.665681	−0.332841	−	0.942983i	$-0.608007\pi$
$677$	−17.3205	−0.665681	−0.332841	+	0.942983i	$0.608007\pi$
$678$	0	0
$679$	17.0000	0.652400
$680$	0	0
$681$	0	0
$682$	0	0
$683$	41.5692	1.59060	0.795301	−	0.606215i	$-0.207313\pi$
$683$	41.5692	1.59060	0.795301	+	0.606215i	$0.207313\pi$
$684$	0	0
$685$	24.0000	0.916993
$686$	0	0
$687$	0	0
$688$	0	0
$689$	51.9615	1.97958
$690$	0	0
$691$	−17.0000	−0.646710	−0.323355	−	0.946278i	$-0.604811\pi$
$691$	−17.0000	−0.646710	−0.323355	+	0.946278i	$0.604811\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−45.0333	−1.70821
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	41.5692	1.57005	0.785024	−	0.619466i	$-0.212651\pi$
$701$	41.5692	1.57005	0.785024	+	0.619466i	$0.212651\pi$
$702$	0	0
$703$	−1.00000	−0.0377157
$704$	0	0
$705$	0	0
$706$	0	0
$707$	13.8564	0.521124
$708$	0	0
$709$	−19.0000	−0.713560	−0.356780	−	0.934188i	$-0.616125\pi$
$709$	−19.0000	−0.713560	−0.356780	+	0.934188i	$0.616125\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	34.6410	1.29732
$714$	0	0
$715$	60.0000	2.24387
$716$	0	0
$717$	0	0
$718$	0	0
$719$	10.3923	0.387568	0.193784	−	0.981044i	$-0.437924\pi$
$719$	10.3923	0.387568	0.193784	+	0.981044i	$0.437924\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	0	0
$723$	0	0
$724$	0	0
$725$	24.2487	0.900575
$726$	0	0
$727$	16.0000	0.593407	0.296704	−	0.954970i	$-0.404113\pi$
$727$	16.0000	0.593407	0.296704	+	0.954970i	$0.404113\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	41.0000	1.51437	0.757185	−	0.653201i	$-0.226574\pi$
$733$	41.0000	1.51437	0.757185	+	0.653201i	$0.226574\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	27.7128	1.02081
$738$	0	0
$739$	19.0000	0.698926	0.349463	−	0.936950i	$-0.386364\pi$
$739$	19.0000	0.698926	0.349463	+	0.936950i	$0.386364\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−6.92820	−0.254171	−0.127086	−	0.991892i	$-0.540562\pi$
$743$	−6.92820	−0.254171	−0.127086	+	0.991892i	$0.540562\pi$
$744$	0	0
$745$	−24.0000	−0.879292
$746$	0	0
$747$	0	0
$748$	0	0
$749$	10.3923	0.379727
$750$	0	0
$751$	−5.00000	−0.182453	−0.0912263	−	0.995830i	$-0.529079\pi$
$751$	−5.00000	−0.182453	−0.0912263	+	0.995830i	$0.529079\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−55.4256	−2.01715
$756$	0	0
$757$	38.0000	1.38113	0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.0000	1.38113	0.690567	+	0.723269i	$0.257361\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	27.7128	1.00459	0.502294	−	0.864697i	$-0.332489\pi$
$761$	27.7128	1.00459	0.502294	+	0.864697i	$0.332489\pi$
$762$	0	0
$763$	17.0000	0.615441
$764$	0	0
$765$	0	0
$766$	0	0
$767$	17.3205	0.625407
$768$	0	0
$769$	−13.0000	−0.468792	−0.234396	−	0.972141i	$-0.575311\pi$
$769$	−13.0000	−0.468792	−0.234396	+	0.972141i	$0.575311\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−20.7846	−0.747570	−0.373785	−	0.927515i	$-0.621940\pi$
$773$	−20.7846	−0.747570	−0.373785	+	0.927515i	$0.621940\pi$
$774$	0	0
$775$	−35.0000	−1.25724
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−3.46410	−0.124114
$780$	0	0
$781$	−36.0000	−1.28818
$782$	0	0
$783$	0	0
$784$	0	0
$785$	45.0333	1.60731
$786$	0	0
$787$	28.0000	0.998092	0.499046	−	0.866575i	$-0.333684\pi$
$787$	28.0000	0.998092	0.499046	+	0.866575i	$0.333684\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	17.3205	0.615846
$792$	0	0
$793$	10.0000	0.355110
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−13.8564	−0.490819	−0.245410	−	0.969419i	$-0.578922\pi$
$797$	−13.8564	−0.490819	−0.245410	+	0.969419i	$0.578922\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−6.92820	−0.244491
$804$	0	0
$805$	24.0000	0.845889
$806$	0	0
$807$	0	0
$808$	0	0
$809$	31.1769	1.09612	0.548061	−	0.836438i	$-0.315366\pi$
$809$	31.1769	1.09612	0.548061	+	0.836438i	$0.315366\pi$
$810$	0	0
$811$	−35.0000	−1.22902	−0.614508	−	0.788911i	$-0.710645\pi$
$811$	−35.0000	−1.22902	−0.614508	+	0.788911i	$0.710645\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−3.46410	−0.121342
$816$	0	0
$817$	1.00000	0.0349856
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−38.1051	−1.32988	−0.664939	−	0.746898i	$-0.731542\pi$
$821$	−38.1051	−1.32988	−0.664939	+	0.746898i	$0.731542\pi$
$822$	0	0
$823$	−23.0000	−0.801730	−0.400865	−	0.916137i	$-0.631290\pi$
$823$	−23.0000	−0.801730	−0.400865	+	0.916137i	$0.631290\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−10.3923	−0.361376	−0.180688	−	0.983540i	$-0.557832\pi$
$827$	−10.3923	−0.361376	−0.180688	+	0.983540i	$0.557832\pi$
$828$	0	0
$829$	−19.0000	−0.659897	−0.329949	−	0.943999i	$-0.607031\pi$
$829$	−19.0000	−0.659897	−0.329949	+	0.943999i	$0.607031\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−84.0000	−2.90694
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−3.46410	−0.119594	−0.0597970	−	0.998211i	$-0.519045\pi$
$839$	−3.46410	−0.119594	−0.0597970	+	0.998211i	$0.519045\pi$
$840$	0	0
$841$	−17.0000	−0.586207
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−41.5692	−1.43002
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	0	0
$850$	0	0
$851$	6.92820	0.237496
$852$	0	0
$853$	2.00000	0.0684787	0.0342393	−	0.999414i	$-0.489099\pi$
$853$	2.00000	0.0684787	0.0342393	+	0.999414i	$0.489099\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−38.1051	−1.30165	−0.650823	−	0.759229i	$-0.725576\pi$
$857$	−38.1051	−1.30165	−0.650823	+	0.759229i	$0.725576\pi$
$858$	0	0
$859$	49.0000	1.67186	0.835929	−	0.548837i	$-0.184929\pi$
$859$	49.0000	1.67186	0.835929	+	0.548837i	$0.184929\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	10.3923	0.353758	0.176879	−	0.984233i	$-0.443400\pi$
$863$	10.3923	0.353758	0.176879	+	0.984233i	$0.443400\pi$
$864$	0	0
$865$	−48.0000	−1.63205
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−3.46410	−0.117512
$870$	0	0
$871$	−40.0000	−1.35535
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−6.92820	−0.234216
$876$	0	0
$877$	23.0000	0.776655	0.388327	−	0.921521i	$-0.373053\pi$
$877$	23.0000	0.776655	0.388327	+	0.921521i	$0.373053\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	0	0
$883$	19.0000	0.639401	0.319700	−	0.947519i	$-0.396418\pi$
$883$	19.0000	0.639401	0.319700	+	0.947519i	$0.396418\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	13.8564	0.465253	0.232626	−	0.972566i	$-0.425268\pi$
$887$	13.8564	0.465253	0.232626	+	0.972566i	$0.425268\pi$
$888$	0	0
$889$	−17.0000	−0.570162
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−3.46410	−0.115922
$894$	0	0
$895$	72.0000	2.40669
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−17.3205	−0.577671
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−58.8897	−1.95756
$906$	0	0
$907$	37.0000	1.22856	0.614282	−	0.789086i	$-0.289446\pi$
$907$	37.0000	1.22856	0.614282	+	0.789086i	$0.289446\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	27.7128	0.918166	0.459083	−	0.888393i	$-0.348178\pi$
$911$	27.7128	0.918166	0.459083	+	0.888393i	$0.348178\pi$
$912$	0	0
$913$	−24.0000	−0.794284
$914$	0	0
$915$	0	0
$916$	0	0
$917$	3.46410	0.114395
$918$	0	0
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	51.9615	1.71033
$924$	0	0
$925$	−7.00000	−0.230159
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−6.92820	−0.227307	−0.113653	−	0.993520i	$-0.536255\pi$
$929$	−6.92820	−0.227307	−0.113653	+	0.993520i	$0.536255\pi$
$930$	0	0
$931$	−6.00000	−0.196642
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−37.0000	−1.20874	−0.604369	−	0.796705i	$-0.706575\pi$
$937$	−37.0000	−1.20874	−0.604369	+	0.796705i	$0.706575\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−3.46410	−0.112926	−0.0564632	−	0.998405i	$-0.517982\pi$
$941$	−3.46410	−0.112926	−0.0564632	+	0.998405i	$0.517982\pi$
$942$	0	0
$943$	24.0000	0.781548
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−13.8564	−0.450273	−0.225136	−	0.974327i	$-0.572283\pi$
$947$	−13.8564	−0.450273	−0.225136	+	0.974327i	$0.572283\pi$
$948$	0	0
$949$	10.0000	0.324614
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	0	0
$955$	−24.0000	−0.776622
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−6.92820	−0.223723
$960$	0	0
$961$	−6.00000	−0.193548
$962$	0	0
$963$	0	0
$964$	0	0
$965$	34.6410	1.11513
$966$	0	0
$967$	−8.00000	−0.257263	−0.128631	−	0.991692i	$-0.541058\pi$
$967$	−8.00000	−0.257263	−0.128631	+	0.991692i	$0.541058\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	51.9615	1.66752	0.833762	−	0.552124i	$-0.186182\pi$
$971$	51.9615	1.66752	0.833762	+	0.552124i	$0.186182\pi$
$972$	0	0
$973$	13.0000	0.416761
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−3.46410	−0.110826	−0.0554132	−	0.998464i	$-0.517648\pi$
$977$	−3.46410	−0.110826	−0.0554132	+	0.998464i	$0.517648\pi$
$978$	0	0
$979$	36.0000	1.15056
$980$	0	0
$981$	0	0
$982$	0	0
$983$	6.92820	0.220975	0.110488	−	0.993877i	$-0.464759\pi$
$983$	6.92820	0.220975	0.110488	+	0.993877i	$0.464759\pi$
$984$	0	0
$985$	−36.0000	−1.14706
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−6.92820	−0.220304
$990$	0	0
$991$	−20.0000	−0.635321	−0.317660	−	0.948205i	$-0.602897\pi$
$991$	−20.0000	−0.635321	−0.317660	+	0.948205i	$0.602897\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−65.8179	−2.08657
$996$	0	0
$997$	2.00000	0.0633406	0.0316703	−	0.999498i	$-0.489917\pi$
$997$	2.00000	0.0633406	0.0316703	+	0.999498i	$0.489917\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3888.2.a.ba.1.1		2
3.2	odd	2	inner	3888.2.a.ba.1.2		2
4.3	odd	2		243.2.a.c.1.1	✓	2
12.11	even	2		243.2.a.c.1.2	yes	2
20.19	odd	2		6075.2.a.bm.1.2		2
36.7	odd	6		243.2.c.d.82.2		4
36.11	even	6		243.2.c.d.82.1		4
36.23	even	6		243.2.c.d.163.1		4
36.31	odd	6		243.2.c.d.163.2		4
60.59	even	2		6075.2.a.bm.1.1		2
108.7	odd	18		729.2.e.n.325.1		12
108.11	even	18		729.2.e.n.82.2		12
108.23	even	18		729.2.e.n.406.2		12
108.31	odd	18		729.2.e.n.406.1		12
108.43	odd	18		729.2.e.n.82.1		12
108.47	even	18		729.2.e.n.325.2		12
108.59	even	18		729.2.e.n.649.2		12
108.67	odd	18		729.2.e.n.163.2		12
108.79	odd	18		729.2.e.n.568.2		12
108.83	even	18		729.2.e.n.568.1		12
108.95	even	18		729.2.e.n.163.1		12
108.103	odd	18		729.2.e.n.649.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
243.2.a.c.1.1	✓	2	4.3	odd	2
243.2.a.c.1.2	yes	2	12.11	even	2
243.2.c.d.82.1		4	36.11	even	6
243.2.c.d.82.2		4	36.7	odd	6
243.2.c.d.163.1		4	36.23	even	6
243.2.c.d.163.2		4	36.31	odd	6
729.2.e.n.82.1		12	108.43	odd	18
729.2.e.n.82.2		12	108.11	even	18
729.2.e.n.163.1		12	108.95	even	18
729.2.e.n.163.2		12	108.67	odd	18
729.2.e.n.325.1		12	108.7	odd	18
729.2.e.n.325.2		12	108.47	even	18
729.2.e.n.406.1		12	108.31	odd	18
729.2.e.n.406.2		12	108.23	even	18
729.2.e.n.568.1		12	108.83	even	18
729.2.e.n.568.2		12	108.79	odd	18
729.2.e.n.649.1		12	108.103	odd	18
729.2.e.n.649.2		12	108.59	even	18
3888.2.a.ba.1.1		2	1.1	even	1	trivial
3888.2.a.ba.1.2		2	3.2	odd	2	inner
6075.2.a.bm.1.1		2	60.59	even	2
6075.2.a.bm.1.2		2	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)

\(f(q)\)	\(=\)	\(q-3.46410 q^{5} +1.00000 q^{7} -3.46410 q^{11} +5.00000 q^{13} +1.00000 q^{19} -6.92820 q^{23} +7.00000 q^{25} +3.46410 q^{29} -5.00000 q^{31} -3.46410 q^{35} -1.00000 q^{37} -3.46410 q^{41} +1.00000 q^{43} -3.46410 q^{47} -6.00000 q^{49} +10.3923 q^{53} +12.0000 q^{55} +3.46410 q^{59} +2.00000 q^{61} -17.3205 q^{65} -8.00000 q^{67} +10.3923 q^{71} +2.00000 q^{73} -3.46410 q^{77} +1.00000 q^{79} +6.92820 q^{83} -10.3923 q^{89} +5.00000 q^{91} -3.46410 q^{95} +17.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{7} + 10 q^{13} + 2 q^{19} + 14 q^{25} - 10 q^{31} - 2 q^{37} + 2 q^{43} - 12 q^{49} + 24 q^{55} + 4 q^{61} - 16 q^{67} + 4 q^{73} + 2 q^{79} + 10 q^{91} + 34 q^{97}+O(q^{100}) \) 2 * q + 2 * q^7 + 10 * q^13 + 2 * q^19 + 14 * q^25 - 10 * q^31 - 2 * q^37 + 2 * q^43 - 12 * q^49 + 24 * q^55 + 4 * q^61 - 16 * q^67 + 4 * q^73 + 2 * q^79 + 10 * q^91 + 34 * q^97

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