LMFDB - Embedded newform 432.4.c.d.431.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [432,4,Mod(431,432)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(432, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("432.431"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 432 = 2^{4} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	432.c (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,0,0,0,0,72,0,38] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$25.4888251225$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			431.2
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	432.431
Dual form			432.4.c.d.431.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+20.7846i q^{5} +29.4449i q^{7} +36.0000 q^{11} +19.0000 q^{13} +103.923i q^{17} +1.73205i q^{19} +108.000 q^{23} -307.000 q^{25} -207.846i q^{29} -93.5307i q^{31} -612.000 q^{35} -109.000 q^{37} -374.123i q^{41} +467.654i q^{43} +468.000 q^{47} -524.000 q^{49} +124.708i q^{53} +748.246i q^{55} -36.0000 q^{59} -145.000 q^{61} +394.908i q^{65} +313.501i q^{67} +576.000 q^{71} -809.000 q^{73} +1060.02i q^{77} -497.099i q^{79} -360.000 q^{83} -2160.00 q^{85} +270.200i q^{89} +559.452i q^{91} -36.0000 q^{95} +955.000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 72 q^{11} + 38 q^{13} + 216 q^{23} - 614 q^{25} - 1224 q^{35} - 218 q^{37} + 936 q^{47} - 1048 q^{49} - 72 q^{59} - 290 q^{61} + 1152 q^{71} - 1618 q^{73} - 720 q^{83} - 4320 q^{85} - 72 q^{95} + 1910 q^{97}+O(q^{100}) $ 2 * q + 72 * q^11 + 38 * q^13 + 216 * q^23 - 614 * q^25 - 1224 * q^35 - 218 * q^37 + 936 * q^47 - 1048 * q^49 - 72 * q^59 - 290 * q^61 + 1152 * q^71 - 1618 * q^73 - 720 * q^83 - 4320 * q^85 - 72 * q^95 + 1910 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/432\mathbb{Z}\right)^\times$.

$n$	$271$	$325$	$353$
$\chi(n)$	$-1$	$1$	$-1$

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
$5$	20.7846i	1.85903i	0.368782	+	0.929516i	$0.379775\pi$
$5$	20.7846i	1.85903i	−0.368782	+	0.929516i	$0.620225\pi$
$6$	0	0
$7$	29.4449i	1.58987i	0.606693	+	0.794937i	$0.292496\pi$
$7$	29.4449i	1.58987i	−0.606693	+	0.794937i	$0.707504\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	36.0000	0.986764	0.493382	−	0.869813i	$-0.335760\pi$
$11$	36.0000	0.986764	0.493382	+	0.869813i	$0.335760\pi$
$12$	0	0
$13$	19.0000	0.405358	0.202679	−	0.979245i	$-0.435035\pi$
$13$	19.0000	0.405358	0.202679	+	0.979245i	$0.435035\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	103.923i	1.48265i	0.671146	+	0.741325i	$0.265802\pi$
$17$	103.923i	1.48265i	−0.671146	+	0.741325i	$0.734198\pi$
$18$	0	0
$19$	1.73205i	0.0209137i	0.999945	+	0.0104568i	$0.00332857\pi$
$19$	1.73205i	0.0209137i	−0.999945	+	0.0104568i	$0.996671\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	108.000	0.979111	0.489556	−	0.871972i	$-0.337159\pi$
$23$	108.000	0.979111	0.489556	+	0.871972i	$0.337159\pi$
$24$	0	0
$25$	−307.000	−2.45600
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 207.846i	− 1.33090i	−0.746443	−	0.665449i	$-0.768240\pi$
$29$	− 207.846i	− 1.33090i	0.746443	−	0.665449i	$-0.231760\pi$
$30$	0	0
$31$	− 93.5307i	− 0.541891i	−0.962595	−	0.270945i	$-0.912664\pi$
$31$	− 93.5307i	− 0.541891i	0.962595	−	0.270945i	$-0.0873363\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−612.000	−2.95563
$36$	0	0
$37$	−109.000	−0.484311	−0.242155	−	0.970238i	$-0.577854\pi$
$37$	−109.000	−0.484311	−0.242155	+	0.970238i	$0.577854\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 374.123i	− 1.42508i	−0.701633	−	0.712539i	$-0.747545\pi$
$41$	− 374.123i	− 1.42508i	0.701633	−	0.712539i	$-0.252455\pi$
$42$	0	0
$43$	467.654i	1.65852i	0.558860	+	0.829262i	$0.311239\pi$
$43$	467.654i	1.65852i	−0.558860	+	0.829262i	$0.688761\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	468.000	1.45244	0.726221	−	0.687461i	$-0.241275\pi$
$47$	468.000	1.45244	0.726221	+	0.687461i	$0.241275\pi$
$48$	0	0
$49$	−524.000	−1.52770
$50$	0	0
$51$	0	0
$52$	0	0
$53$	124.708i	0.323206i	0.986856	+	0.161603i	$0.0516664\pi$
$53$	124.708i	0.323206i	−0.986856	+	0.161603i	$0.948334\pi$
$54$	0	0
$55$	748.246i	1.83443i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−36.0000	−0.0794373	−0.0397187	−	0.999211i	$-0.512646\pi$
$59$	−36.0000	−0.0794373	−0.0397187	+	0.999211i	$0.512646\pi$
$60$	0	0
$61$	−145.000	−0.304350	−0.152175	−	0.988354i	$-0.548628\pi$
$61$	−145.000	−0.304350	−0.152175	+	0.988354i	$0.548628\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	394.908i	0.753573i
$66$	0	0
$67$	313.501i	0.571646i	0.958282	+	0.285823i	$0.0922669\pi$
$67$	313.501i	0.571646i	−0.958282	+	0.285823i	$0.907733\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	576.000	0.962798	0.481399	−	0.876502i	$-0.340129\pi$
$71$	576.000	0.962798	0.481399	+	0.876502i	$0.340129\pi$
$72$	0	0
$73$	−809.000	−1.29707	−0.648536	−	0.761184i	$-0.724618\pi$
$73$	−809.000	−1.29707	−0.648536	+	0.761184i	$0.724618\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1060.02i	1.56883i
$78$	0	0
$79$	− 497.099i	− 0.707949i	−0.935255	−	0.353974i	$-0.884830\pi$
$79$	− 497.099i	− 0.707949i	0.935255	−	0.353974i	$-0.115170\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−360.000	−0.476086	−0.238043	−	0.971255i	$-0.576506\pi$
$83$	−360.000	−0.476086	−0.238043	+	0.971255i	$0.576506\pi$
$84$	0	0
$85$	−2160.00	−2.75629
$86$	0	0
$87$	0	0
$88$	0	0
$89$	270.200i	0.321811i	0.986970	+	0.160905i	$0.0514414\pi$
$89$	270.200i	0.321811i	−0.986970	+	0.160905i	$0.948559\pi$
$90$	0	0
$91$	559.452i	0.644468i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−36.0000	−0.0388792
$96$	0	0
$97$	955.000	0.999645	0.499822	−	0.866128i	$-0.333399\pi$
$97$	955.000	0.999645	0.499822	+	0.866128i	$0.333399\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 249.415i	− 0.245720i	−0.992424	−	0.122860i	$-0.960793\pi$
$101$	− 249.415i	− 0.245720i	0.992424	−	0.122860i	$-0.0392067\pi$
$102$	0	0
$103$	− 1903.52i	− 1.82097i	−0.413544	−	0.910484i	$-0.635709\pi$
$103$	− 1903.52i	− 1.82097i	0.413544	−	0.910484i	$-0.364291\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1548.00	1.39861	0.699303	−	0.714826i	$-0.253494\pi$
$107$	1548.00	1.39861	0.699303	+	0.714826i	$0.253494\pi$
$108$	0	0
$109$	214.000	0.188050	0.0940251	−	0.995570i	$-0.470027\pi$
$109$	214.000	0.188050	0.0940251	+	0.995570i	$0.470027\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 1392.57i	− 1.15931i	−0.814863	−	0.579654i	$-0.803188\pi$
$113$	− 1392.57i	− 1.15931i	0.814863	−	0.579654i	$-0.196812\pi$
$114$	0	0
$115$	2244.74i	1.82020i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−3060.00	−2.35722
$120$	0	0
$121$	−35.0000	−0.0262960
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 3782.80i	− 2.70675i
$126$	0	0
$127$	2151.21i	1.50306i	0.659698	+	0.751531i	$0.270684\pi$
$127$	2151.21i	1.50306i	−0.659698	+	0.751531i	$0.729316\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−792.000	−0.528224	−0.264112	−	0.964492i	$-0.585079\pi$
$131$	−792.000	−0.528224	−0.264112	+	0.964492i	$0.585079\pi$
$132$	0	0
$133$	−51.0000	−0.0332501
$134$	0	0
$135$	0	0
$136$	0	0
$137$	394.908i	0.246272i	0.992390	+	0.123136i	$0.0392951\pi$
$137$	394.908i	0.246272i	−0.992390	+	0.123136i	$0.960705\pi$
$138$	0	0
$139$	− 1716.46i	− 1.04740i	−0.851903	−	0.523699i	$-0.824552\pi$
$139$	− 1716.46i	− 1.04740i	0.851903	−	0.523699i	$-0.175448\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	684.000	0.399993
$144$	0	0
$145$	4320.00	2.47418
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2161.60i	1.18849i	0.804284	+	0.594245i	$0.202549\pi$
$149$	2161.60i	1.18849i	−0.804284	+	0.594245i	$0.797451\pi$
$150$	0	0
$151$	− 1432.41i	− 0.771971i	−0.922505	−	0.385985i	$-0.873862\pi$
$151$	− 1432.41i	− 0.771971i	0.922505	−	0.385985i	$-0.126138\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1944.00	1.00739
$156$	0	0
$157$	1766.00	0.897721	0.448860	−	0.893602i	$-0.351830\pi$
$157$	1766.00	0.897721	0.448860	+	0.893602i	$0.351830\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	3180.05i	1.55666i
$162$	0	0
$163$	278.860i	0.134000i	0.997753	+	0.0670000i	$0.0213428\pi$
$163$	278.860i	0.134000i	−0.997753	+	0.0670000i	$0.978657\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2196.00	−1.01755	−0.508777	−	0.860898i	$-0.669902\pi$
$167$	−2196.00	−1.01755	−0.508777	+	0.860898i	$0.669902\pi$
$168$	0	0
$169$	−1836.00	−0.835685
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1787.48i	0.785545i	0.919636	+	0.392773i	$0.128484\pi$
$173$	1787.48i	0.785545i	−0.919636	+	0.392773i	$0.871516\pi$
$174$	0	0
$175$	− 9039.57i	− 3.90473i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	1152.00	0.481031	0.240515	−	0.970645i	$-0.422684\pi$
$179$	1152.00	0.481031	0.240515	+	0.970645i	$0.422684\pi$
$180$	0	0
$181$	2215.00	0.909611	0.454806	−	0.890591i	$-0.349709\pi$
$181$	2215.00	0.909611	0.454806	+	0.890591i	$0.349709\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 2265.52i	− 0.900349i
$186$	0	0
$187$	3741.23i	1.46303i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	4572.00	1.73203	0.866017	−	0.500015i	$-0.166672\pi$
$191$	4572.00	1.73203	0.866017	+	0.500015i	$0.166672\pi$
$192$	0	0
$193$	−937.000	−0.349465	−0.174732	−	0.984616i	$-0.555906\pi$
$193$	−937.000	−0.349465	−0.174732	+	0.984616i	$0.555906\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1849.83i	0.669010i	0.942394	+	0.334505i	$0.108569\pi$
$197$	1849.83i	0.669010i	−0.942394	+	0.334505i	$0.891431\pi$
$198$	0	0
$199$	− 2527.06i	− 0.900194i	−0.892980	−	0.450097i	$-0.851389\pi$
$199$	− 2527.06i	− 0.900194i	0.892980	−	0.450097i	$-0.148611\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	6120.00	2.11596
$204$	0	0
$205$	7776.00	2.64927
$206$	0	0
$207$	0	0
$208$	0	0
$209$	62.3538i	0.0206369i
$210$	0	0
$211$	− 2277.65i	− 0.743126i	−0.928408	−	0.371563i	$-0.878822\pi$
$211$	− 2277.65i	− 0.743126i	0.928408	−	0.371563i	$-0.121178\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−9720.00	−3.08325
$216$	0	0
$217$	2754.00	0.861538
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1974.54i	0.601004i
$222$	0	0
$223$	− 1590.02i	− 0.477470i	−0.971085	−	0.238735i	$-0.923267\pi$
$223$	− 1590.02i	− 0.477470i	0.971085	−	0.238735i	$-0.0767327\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−3312.00	−0.968393	−0.484196	−	0.874959i	$-0.660888\pi$
$227$	−3312.00	−0.968393	−0.484196	+	0.874959i	$0.660888\pi$
$228$	0	0
$229$	−1010.00	−0.291453	−0.145726	−	0.989325i	$-0.546552\pi$
$229$	−1010.00	−0.291453	−0.145726	+	0.989325i	$0.546552\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	831.384i	0.233759i	0.993146	+	0.116879i	$0.0372891\pi$
$233$	831.384i	0.233759i	−0.993146	+	0.116879i	$0.962711\pi$
$234$	0	0
$235$	9727.20i	2.70014i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	2736.00	0.740490	0.370245	−	0.928934i	$-0.379274\pi$
$239$	2736.00	0.740490	0.370245	+	0.928934i	$0.379274\pi$
$240$	0	0
$241$	3007.00	0.803726	0.401863	−	0.915700i	$-0.368363\pi$
$241$	3007.00	0.803726	0.401863	+	0.915700i	$0.368363\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 10891.1i	− 2.84004i
$246$	0	0
$247$	32.9090i	0.00847752i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−5112.00	−1.28552	−0.642762	−	0.766066i	$-0.722212\pi$
$251$	−5112.00	−1.28552	−0.642762	+	0.766066i	$0.722212\pi$
$252$	0	0
$253$	3888.00	0.966152
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 6775.78i	− 1.64460i	−0.569056	−	0.822299i	$-0.692691\pi$
$257$	− 6775.78i	− 1.64460i	0.569056	−	0.822299i	$-0.307309\pi$
$258$	0	0
$259$	− 3209.49i	− 0.769992i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	3024.00	0.709003	0.354502	−	0.935055i	$-0.384651\pi$
$263$	3024.00	0.709003	0.354502	+	0.935055i	$0.384651\pi$
$264$	0	0
$265$	−2592.00	−0.600850
$266$	0	0
$267$	0	0
$268$	0	0
$269$	2016.11i	0.456967i	0.973548	+	0.228484i	$0.0733768\pi$
$269$	2016.11i	0.456967i	−0.973548	+	0.228484i	$0.926623\pi$
$270$	0	0
$271$	− 3178.31i	− 0.712431i	−0.934404	−	0.356215i	$-0.884067\pi$
$271$	− 3178.31i	− 0.712431i	0.934404	−	0.356215i	$-0.115933\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−11052.0	−2.42349
$276$	0	0
$277$	2734.00	0.593033	0.296516	−	0.955028i	$-0.404175\pi$
$277$	2734.00	0.593033	0.296516	+	0.955028i	$0.404175\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	1288.65i	0.273574i	0.990601	+	0.136787i	$0.0436775\pi$
$281$	1288.65i	0.273574i	−0.990601	+	0.136787i	$0.956322\pi$
$282$	0	0
$283$	− 2525.33i	− 0.530443i	−0.964188	−	0.265221i	$-0.914555\pi$
$283$	− 2525.33i	− 0.530443i	0.964188	−	0.265221i	$-0.0854450\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	11016.0	2.26569
$288$	0	0
$289$	−5887.00	−1.19825
$290$	0	0
$291$	0	0
$292$	0	0
$293$	3512.60i	0.700369i	0.936681	+	0.350185i	$0.113881\pi$
$293$	3512.60i	0.700369i	−0.936681	+	0.350185i	$0.886119\pi$
$294$	0	0
$295$	− 748.246i	− 0.147676i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2052.00	0.396890
$300$	0	0
$301$	−13770.0	−2.63684
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 3013.77i	− 0.565796i
$306$	0	0
$307$	6827.74i	1.26932i	0.772793	+	0.634658i	$0.218859\pi$
$307$	6827.74i	1.26932i	−0.772793	+	0.634658i	$0.781141\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	8676.00	1.58190	0.790950	−	0.611881i	$-0.209587\pi$
$311$	8676.00	1.58190	0.790950	+	0.611881i	$0.209587\pi$
$312$	0	0
$313$	−1333.00	−0.240721	−0.120360	−	0.992730i	$-0.538405\pi$
$313$	−1333.00	−0.240721	−0.120360	+	0.992730i	$0.538405\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	10433.9i	1.84866i	0.381596	+	0.924329i	$0.375375\pi$
$317$	10433.9i	1.84866i	−0.381596	+	0.924329i	$0.624625\pi$
$318$	0	0
$319$	− 7482.46i	− 1.31328i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−180.000	−0.0310076
$324$	0	0
$325$	−5833.00	−0.995559
$326$	0	0
$327$	0	0
$328$	0	0
$329$	13780.2i	2.30920i
$330$	0	0
$331$	7886.03i	1.30953i	0.755831	+	0.654766i	$0.227233\pi$
$331$	7886.03i	1.30953i	−0.755831	+	0.654766i	$0.772767\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−6516.00	−1.06271
$336$	0	0
$337$	6931.00	1.12034	0.560172	−	0.828377i	$-0.310735\pi$
$337$	6931.00	1.12034	0.560172	+	0.828377i	$0.310735\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 3367.11i	− 0.534719i
$342$	0	0
$343$	− 5329.52i	− 0.838971i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−1296.00	−0.200498	−0.100249	−	0.994962i	$-0.531964\pi$
$347$	−1296.00	−0.200498	−0.100249	+	0.994962i	$0.531964\pi$
$348$	0	0
$349$	−9757.00	−1.49650	−0.748252	−	0.663414i	$-0.769107\pi$
$349$	−9757.00	−1.49650	−0.748252	+	0.663414i	$0.769107\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	8729.54i	1.31622i	0.752921	+	0.658111i	$0.228644\pi$
$353$	8729.54i	1.31622i	−0.752921	+	0.658111i	$0.771356\pi$
$354$	0	0
$355$	11971.9i	1.78987i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	8388.00	1.23315	0.616576	−	0.787295i	$-0.288519\pi$
$359$	8388.00	1.23315	0.616576	+	0.787295i	$0.288519\pi$
$360$	0	0
$361$	6856.00	0.999563
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 16814.7i	− 2.41130i
$366$	0	0
$367$	8821.33i	1.25469i	0.778743	+	0.627343i	$0.215858\pi$
$367$	8821.33i	1.25469i	−0.778743	+	0.627343i	$0.784142\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−3672.00	−0.513856
$372$	0	0
$373$	−181.000	−0.0251255	−0.0125628	−	0.999921i	$-0.503999\pi$
$373$	−181.000	−0.0251255	−0.0125628	+	0.999921i	$0.503999\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 3949.08i	− 0.539490i
$378$	0	0
$379$	10664.2i	1.44534i	0.691192	+	0.722672i	$0.257086\pi$
$379$	10664.2i	1.44534i	−0.691192	+	0.722672i	$0.742914\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−11664.0	−1.55614	−0.778071	−	0.628176i	$-0.783802\pi$
$383$	−11664.0	−1.55614	−0.778071	+	0.628176i	$0.783802\pi$
$384$	0	0
$385$	−22032.0	−2.91651
$386$	0	0
$387$	0	0
$388$	0	0
$389$	5549.49i	0.723317i	0.932311	+	0.361659i	$0.117789\pi$
$389$	5549.49i	0.723317i	−0.932311	+	0.361659i	$0.882211\pi$
$390$	0	0
$391$	11223.7i	1.45168i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	10332.0	1.31610
$396$	0	0
$397$	−11338.0	−1.43334	−0.716672	−	0.697410i	$-0.754336\pi$
$397$	−11338.0	−1.43334	−0.716672	+	0.697410i	$0.754336\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2826.71i	0.352017i	0.984389	+	0.176009i	$0.0563187\pi$
$401$	2826.71i	0.352017i	−0.984389	+	0.176009i	$0.943681\pi$
$402$	0	0
$403$	− 1777.08i	− 0.219660i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−3924.00	−0.477900
$408$	0	0
$409$	6607.00	0.798766	0.399383	−	0.916784i	$-0.369224\pi$
$409$	6607.00	0.798766	0.399383	+	0.916784i	$0.369224\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 1060.02i	− 0.126295i
$414$	0	0
$415$	− 7482.46i	− 0.885059i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	8892.00	1.03676	0.518380	−	0.855150i	$-0.326535\pi$
$419$	8892.00	1.03676	0.518380	+	0.855150i	$0.326535\pi$
$420$	0	0
$421$	5707.00	0.660670	0.330335	−	0.943864i	$-0.392838\pi$
$421$	5707.00	0.660670	0.330335	+	0.943864i	$0.392838\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 31904.4i	− 3.64139i
$426$	0	0
$427$	− 4269.51i	− 0.483878i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−6588.00	−0.736271	−0.368135	−	0.929772i	$-0.620004\pi$
$431$	−6588.00	−0.736271	−0.368135	+	0.929772i	$0.620004\pi$
$432$	0	0
$433$	−4750.00	−0.527183	−0.263592	−	0.964634i	$-0.584907\pi$
$433$	−4750.00	−0.527183	−0.263592	+	0.964634i	$0.584907\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	187.061i	0.0204768i
$438$	0	0
$439$	1028.84i	0.111854i	0.998435	+	0.0559268i	$0.0178114\pi$
$439$	1028.84i	0.111854i	−0.998435	+	0.0559268i	$0.982189\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−10152.0	−1.08879	−0.544397	−	0.838827i	$-0.683242\pi$
$443$	−10152.0	−1.08879	−0.544397	+	0.838827i	$0.683242\pi$
$444$	0	0
$445$	−5616.00	−0.598256
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 15276.7i	− 1.60568i	−0.596193	−	0.802841i	$-0.703321\pi$
$449$	− 15276.7i	− 1.60568i	0.596193	−	0.802841i	$-0.296679\pi$
$450$	0	0
$451$	− 13468.4i	− 1.40622i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−11628.0	−1.19809
$456$	0	0
$457$	−13142.0	−1.34520	−0.672600	−	0.740006i	$-0.734823\pi$
$457$	−13142.0	−1.34520	−0.672600	+	0.740006i	$0.734823\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	6048.32i	0.611059i	0.952183	+	0.305530i	$0.0988335\pi$
$461$	6048.32i	0.611059i	−0.952183	+	0.305530i	$0.901166\pi$
$462$	0	0
$463$	2398.89i	0.240790i	0.992726	+	0.120395i	$0.0384162\pi$
$463$	2398.89i	0.240790i	−0.992726	+	0.120395i	$0.961584\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−4500.00	−0.445900	−0.222950	−	0.974830i	$-0.571569\pi$
$467$	−4500.00	−0.445900	−0.222950	+	0.974830i	$0.571569\pi$
$468$	0	0
$469$	−9231.00	−0.908844
$470$	0	0
$471$	0	0
$472$	0	0
$473$	16835.5i	1.63657i
$474$	0	0
$475$	− 531.740i	− 0.0513640i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	2232.00	0.212907	0.106454	−	0.994318i	$-0.466050\pi$
$479$	2232.00	0.212907	0.106454	+	0.994318i	$0.466050\pi$
$480$	0	0
$481$	−2071.00	−0.196319
$482$	0	0
$483$	0	0
$484$	0	0
$485$	19849.3i	1.85837i
$486$	0	0
$487$	− 4584.74i	− 0.426600i	−0.976987	−	0.213300i	$-0.931579\pi$
$487$	− 4584.74i	− 0.426600i	0.976987	−	0.213300i	$-0.0684212\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	9108.00	0.837145	0.418573	−	0.908183i	$-0.362531\pi$
$491$	9108.00	0.837145	0.418573	+	0.908183i	$0.362531\pi$
$492$	0	0
$493$	21600.0	1.97326
$494$	0	0
$495$	0	0
$496$	0	0
$497$	16960.2i	1.53073i
$498$	0	0
$499$	16554.9i	1.48517i	0.669751	+	0.742586i	$0.266401\pi$
$499$	16554.9i	1.48517i	−0.669751	+	0.742586i	$0.733599\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−252.000	−0.0223382	−0.0111691	−	0.999938i	$-0.503555\pi$
$503$	−252.000	−0.0223382	−0.0111691	+	0.999938i	$0.503555\pi$
$504$	0	0
$505$	5184.00	0.456802
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 15650.8i	− 1.36289i	−0.731870	−	0.681444i	$-0.761352\pi$
$509$	− 15650.8i	− 1.36289i	0.731870	−	0.681444i	$-0.238648\pi$
$510$	0	0
$511$	− 23820.9i	− 2.06218i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	39564.0	3.38524
$516$	0	0
$517$	16848.0	1.43322
$518$	0	0
$519$	0	0
$520$	0	0
$521$	17521.4i	1.47337i	0.676234	+	0.736687i	$0.263611\pi$
$521$	17521.4i	1.47337i	−0.676234	+	0.736687i	$0.736389\pi$
$522$	0	0
$523$	2059.41i	0.172183i	0.996287	+	0.0860914i	$0.0274377\pi$
$523$	2059.41i	0.172183i	−0.996287	+	0.0860914i	$0.972562\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	9720.00	0.803434
$528$	0	0
$529$	−503.000	−0.0413413
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 7108.34i	− 0.577666i
$534$	0	0
$535$	32174.6i	2.60005i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−18864.0	−1.50748
$540$	0	0
$541$	2971.00	0.236106	0.118053	−	0.993007i	$-0.462335\pi$
$541$	2971.00	0.236106	0.118053	+	0.993007i	$0.462335\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	4447.91i	0.349592i
$546$	0	0
$547$	− 22043.8i	− 1.72308i	−0.507689	−	0.861540i	$-0.669500\pi$
$547$	− 22043.8i	− 1.72308i	0.507689	−	0.861540i	$-0.330500\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	360.000	0.0278340
$552$	0	0
$553$	14637.0	1.12555
$554$	0	0
$555$	0	0
$556$	0	0
$557$	16814.7i	1.27911i	0.768746	+	0.639555i	$0.220881\pi$
$557$	16814.7i	1.27911i	−0.768746	+	0.639555i	$0.779119\pi$
$558$	0	0
$559$	8885.42i	0.672296i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	13608.0	1.01867	0.509333	−	0.860570i	$-0.329892\pi$
$563$	13608.0	1.01867	0.509333	+	0.860570i	$0.329892\pi$
$564$	0	0
$565$	28944.0	2.15519
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 7752.66i	− 0.571192i	−0.958350	−	0.285596i	$-0.907808\pi$
$569$	− 7752.66i	− 0.571192i	0.958350	−	0.285596i	$-0.0921916\pi$
$570$	0	0
$571$	16587.9i	1.21573i	0.794042	+	0.607863i	$0.207973\pi$
$571$	16587.9i	1.21573i	−0.794042	+	0.607863i	$0.792027\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−33156.0	−2.40470
$576$	0	0
$577$	−10477.0	−0.755915	−0.377958	−	0.925823i	$-0.623374\pi$
$577$	−10477.0	−0.755915	−0.377958	+	0.925823i	$0.623374\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 10600.2i	− 0.756916i
$582$	0	0
$583$	4489.48i	0.318928i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−828.000	−0.0582201	−0.0291101	−	0.999576i	$-0.509267\pi$
$587$	−828.000	−0.0582201	−0.0291101	+	0.999576i	$0.509267\pi$
$588$	0	0
$589$	162.000	0.0113329
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 16752.4i	− 1.16010i	−0.814581	−	0.580049i	$-0.803033\pi$
$593$	− 16752.4i	− 1.16010i	0.814581	−	0.580049i	$-0.196967\pi$
$594$	0	0
$595$	− 63600.9i	− 4.38216i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	11016.0	0.751422	0.375711	−	0.926737i	$-0.377399\pi$
$599$	11016.0	0.751422	0.375711	+	0.926737i	$0.377399\pi$
$600$	0	0
$601$	7418.00	0.503472	0.251736	−	0.967796i	$-0.418999\pi$
$601$	7418.00	0.503472	0.251736	+	0.967796i	$0.418999\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 727.461i	− 0.0488851i
$606$	0	0
$607$	− 18926.1i	− 1.26555i	−0.774337	−	0.632774i	$-0.781916\pi$
$607$	− 18926.1i	− 1.26555i	0.774337	−	0.632774i	$-0.218084\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	8892.00	0.588759
$612$	0	0
$613$	12527.0	0.825385	0.412692	−	0.910870i	$-0.364588\pi$
$613$	12527.0	0.825385	0.412692	+	0.910870i	$0.364588\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 11701.7i	− 0.763524i	−0.924261	−	0.381762i	$-0.875317\pi$
$617$	− 11701.7i	− 0.763524i	0.924261	−	0.381762i	$-0.124683\pi$
$618$	0	0
$619$	2121.76i	0.137772i	0.997625	+	0.0688860i	$0.0219445\pi$
$619$	2121.76i	0.137772i	−0.997625	+	0.0688860i	$0.978056\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−7956.00	−0.511638
$624$	0	0
$625$	40249.0	2.57594
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 11327.6i	− 0.718063i
$630$	0	0
$631$	5426.52i	0.342355i	0.985240	+	0.171178i	$0.0547572\pi$
$631$	5426.52i	0.342355i	−0.985240	+	0.171178i	$0.945243\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−44712.0	−2.79424
$636$	0	0
$637$	−9956.00	−0.619264
$638$	0	0
$639$	0	0
$640$	0	0
$641$	11306.8i	0.696713i	0.937362	+	0.348356i	$0.113260\pi$
$641$	11306.8i	0.696713i	−0.937362	+	0.348356i	$0.886740\pi$
$642$	0	0
$643$	− 14123.1i	− 0.866193i	−0.901347	−	0.433097i	$-0.857421\pi$
$643$	− 14123.1i	− 0.866193i	0.901347	−	0.433097i	$-0.142579\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	13464.0	0.818121	0.409061	−	0.912507i	$-0.365856\pi$
$647$	13464.0	0.818121	0.409061	+	0.912507i	$0.365856\pi$
$648$	0	0
$649$	−1296.00	−0.0783859
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 3865.94i	− 0.231678i	−0.993268	−	0.115839i	$-0.963044\pi$
$653$	− 3865.94i	− 0.231678i	0.993268	−	0.115839i	$-0.0369557\pi$
$654$	0	0
$655$	− 16461.4i	− 0.981986i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−23400.0	−1.38321	−0.691604	−	0.722277i	$-0.743096\pi$
$659$	−23400.0	−1.38321	−0.691604	+	0.722277i	$0.743096\pi$
$660$	0	0
$661$	−15409.0	−0.906718	−0.453359	−	0.891328i	$-0.649774\pi$
$661$	−15409.0	−0.906718	−0.453359	+	0.891328i	$0.649774\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 1060.02i	− 0.0618130i
$666$	0	0
$667$	− 22447.4i	− 1.30310i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−5220.00	−0.300322
$672$	0	0
$673$	−19693.0	−1.12795	−0.563974	−	0.825792i	$-0.690728\pi$
$673$	−19693.0	−1.12795	−0.563974	+	0.825792i	$0.690728\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 25876.8i	− 1.46902i	−0.678597	−	0.734511i	$-0.737412\pi$
$677$	− 25876.8i	− 1.46902i	0.678597	−	0.734511i	$-0.262588\pi$
$678$	0	0
$679$	28119.8i	1.58931i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	17856.0	1.00035	0.500176	−	0.865924i	$-0.333269\pi$
$683$	17856.0	1.00035	0.500176	+	0.865924i	$0.333269\pi$
$684$	0	0
$685$	−8208.00	−0.457827
$686$	0	0
$687$	0	0
$688$	0	0
$689$	2369.45i	0.131014i
$690$	0	0
$691$	− 2899.45i	− 0.159624i	−0.996810	−	0.0798122i	$-0.974568\pi$
$691$	− 2899.45i	− 0.159624i	0.996810	−	0.0798122i	$-0.0254320\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	35676.0	1.94715
$696$	0	0
$697$	38880.0	2.11289
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 27414.9i	− 1.47710i	−0.674199	−	0.738550i	$-0.735511\pi$
$701$	− 27414.9i	− 1.47710i	0.674199	−	0.738550i	$-0.264489\pi$
$702$	0	0
$703$	− 188.794i	− 0.0101287i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	7344.00	0.390664
$708$	0	0
$709$	1603.00	0.0849110	0.0424555	−	0.999098i	$-0.486482\pi$
$709$	1603.00	0.0849110	0.0424555	+	0.999098i	$0.486482\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 10101.3i	− 0.530571i
$714$	0	0
$715$	14216.7i	0.743599i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	16344.0	0.847745	0.423872	−	0.905722i	$-0.360671\pi$
$719$	16344.0	0.847745	0.423872	+	0.905722i	$0.360671\pi$
$720$	0	0
$721$	56049.0	2.89511
$722$	0	0
$723$	0	0
$724$	0	0
$725$	63808.8i	3.26869i
$726$	0	0
$727$	4021.82i	0.205173i	0.994724	+	0.102587i	$0.0327119\pi$
$727$	4021.82i	0.205173i	−0.994724	+	0.102587i	$0.967288\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−48600.0	−2.45901
$732$	0	0
$733$	38086.0	1.91915	0.959576	−	0.281449i	$-0.0908152\pi$
$733$	38086.0	1.91915	0.959576	+	0.281449i	$0.0908152\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	11286.0i	0.564080i
$738$	0	0
$739$	7575.99i	0.377114i	0.982062	+	0.188557i	$0.0603810\pi$
$739$	7575.99i	0.377114i	−0.982062	+	0.188557i	$0.939619\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	18036.0	0.890547	0.445274	−	0.895395i	$-0.353106\pi$
$743$	18036.0	0.890547	0.445274	+	0.895395i	$0.353106\pi$
$744$	0	0
$745$	−44928.0	−2.20944
$746$	0	0
$747$	0	0
$748$	0	0
$749$	45580.6i	2.22361i
$750$	0	0
$751$	9853.64i	0.478781i	0.970923	+	0.239390i	$0.0769476\pi$
$751$	9853.64i	0.478781i	−0.970923	+	0.239390i	$0.923052\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	29772.0	1.43512
$756$	0	0
$757$	−40661.0	−1.95224	−0.976122	−	0.217222i	$-0.930301\pi$
$757$	−40661.0	−1.95224	−0.976122	+	0.217222i	$0.930301\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 9706.41i	− 0.462362i	−0.972911	−	0.231181i	$-0.925741\pi$
$761$	− 9706.41i	− 0.462362i	0.972911	−	0.231181i	$-0.0742589\pi$
$762$	0	0
$763$	6301.20i	0.298976i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−684.000	−0.0322005
$768$	0	0
$769$	−5041.00	−0.236389	−0.118194	−	0.992990i	$-0.537711\pi$
$769$	−5041.00	−0.236389	−0.118194	+	0.992990i	$0.537711\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	13011.2i	0.605406i	0.953085	+	0.302703i	$0.0978891\pi$
$773$	13011.2i	0.605406i	−0.953085	+	0.302703i	$0.902111\pi$
$774$	0	0
$775$	28713.9i	1.33088i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	648.000	0.0298036
$780$	0	0
$781$	20736.0	0.950054
$782$	0	0
$783$	0	0
$784$	0	0
$785$	36705.6i	1.66889i
$786$	0	0
$787$	− 26062.2i	− 1.18045i	−0.807238	−	0.590226i	$-0.799039\pi$
$787$	− 26062.2i	− 1.18045i	0.807238	−	0.590226i	$-0.200961\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	41004.0	1.84315
$792$	0	0
$793$	−2755.00	−0.123371
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1163.94i	0.0517300i	0.999665	+	0.0258650i	$0.00823400\pi$
$797$	1163.94i	0.0517300i	−0.999665	+	0.0258650i	$0.991766\pi$
$798$	0	0
$799$	48636.0i	2.15346i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−29124.0	−1.27990
$804$	0	0
$805$	−66096.0	−2.89389
$806$	0	0
$807$	0	0
$808$	0	0
$809$	15796.3i	0.686488i	0.939246	+	0.343244i	$0.111526\pi$
$809$	15796.3i	0.686488i	−0.939246	+	0.343244i	$0.888474\pi$
$810$	0	0
$811$	− 24972.7i	− 1.08127i	−0.841257	−	0.540635i	$-0.818184\pi$
$811$	− 24972.7i	− 1.08127i	0.841257	−	0.540635i	$-0.181816\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−5796.00	−0.249110
$816$	0	0
$817$	−810.000	−0.0346858
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 22676.0i	− 0.963944i	−0.876187	−	0.481972i	$-0.839921\pi$
$821$	− 22676.0i	− 0.963944i	0.876187	−	0.481972i	$-0.160079\pi$
$822$	0	0
$823$	11911.3i	0.504499i	0.967662	+	0.252249i	$0.0811703\pi$
$823$	11911.3i	0.504499i	−0.967662	+	0.252249i	$0.918830\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−43956.0	−1.84825	−0.924123	−	0.382094i	$-0.875203\pi$
$827$	−43956.0	−1.84825	−0.924123	+	0.382094i	$0.875203\pi$
$828$	0	0
$829$	−17225.0	−0.721651	−0.360826	−	0.932633i	$-0.617505\pi$
$829$	−17225.0	−0.721651	−0.360826	+	0.932633i	$0.617505\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 54455.7i	− 2.26504i
$834$	0	0
$835$	− 45643.0i	− 1.89167i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−22680.0	−0.933255	−0.466627	−	0.884454i	$-0.654531\pi$
$839$	−22680.0	−0.933255	−0.466627	+	0.884454i	$0.654531\pi$
$840$	0	0
$841$	−18811.0	−0.771290
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 38160.5i	− 1.55357i
$846$	0	0
$847$	− 1030.57i	− 0.0418073i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−11772.0	−0.474194
$852$	0	0
$853$	35819.0	1.43777	0.718886	−	0.695128i	$-0.244652\pi$
$853$	35819.0	1.43777	0.718886	+	0.695128i	$0.244652\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 7399.32i	− 0.294931i	−0.989067	−	0.147466i	$-0.952888\pi$
$857$	− 7399.32i	− 0.294931i	0.989067	−	0.147466i	$-0.0471116\pi$
$858$	0	0
$859$	− 8263.61i	− 0.328232i	−0.986441	−	0.164116i	$-0.947523\pi$
$859$	− 8263.61i	− 0.328232i	0.986441	−	0.164116i	$-0.0524771\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	4932.00	0.194539	0.0972695	−	0.995258i	$-0.468989\pi$
$863$	4932.00	0.194539	0.0972695	+	0.995258i	$0.468989\pi$
$864$	0	0
$865$	−37152.0	−1.46035
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 17895.5i	− 0.698579i
$870$	0	0
$871$	5956.52i	0.231721i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	111384.	4.30339
$876$	0	0
$877$	−35821.0	−1.37924	−0.689618	−	0.724174i	$-0.742221\pi$
$877$	−35821.0	−1.37924	−0.689618	+	0.724174i	$0.742221\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 21886.2i	− 0.836963i	−0.908225	−	0.418482i	$-0.862562\pi$
$881$	− 21886.2i	− 0.836963i	0.908225	−	0.418482i	$-0.137438\pi$
$882$	0	0
$883$	652.983i	0.0248863i	0.999923	+	0.0124432i	$0.00396089\pi$
$883$	652.983i	0.0248863i	−0.999923	+	0.0124432i	$0.996039\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−26856.0	−1.01661	−0.508307	−	0.861176i	$-0.669728\pi$
$887$	−26856.0	−1.01661	−0.508307	+	0.861176i	$0.669728\pi$
$888$	0	0
$889$	−63342.0	−2.38968
$890$	0	0
$891$	0	0
$892$	0	0
$893$	810.600i	0.0303759i
$894$	0	0
$895$	23943.9i	0.894252i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−19440.0	−0.721202
$900$	0	0
$901$	−12960.0	−0.479201
$902$	0	0
$903$	0	0
$904$	0	0
$905$	46037.9i	1.69100i
$906$	0	0
$907$	19144.4i	0.700858i	0.936589	+	0.350429i	$0.113964\pi$
$907$	19144.4i	0.700858i	−0.936589	+	0.350429i	$0.886036\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−32904.0	−1.19666	−0.598330	−	0.801249i	$-0.704169\pi$
$911$	−32904.0	−1.19666	−0.598330	+	0.801249i	$0.704169\pi$
$912$	0	0
$913$	−12960.0	−0.469785
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 23320.3i	− 0.839809i
$918$	0	0
$919$	50413.1i	1.80955i	0.425893	+	0.904774i	$0.359960\pi$
$919$	50413.1i	1.80955i	−0.425893	+	0.904774i	$0.640040\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	10944.0	0.390278
$924$	0	0
$925$	33463.0	1.18947
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 36248.4i	− 1.28016i	−0.768308	−	0.640081i	$-0.778901\pi$
$929$	− 36248.4i	− 1.28016i	0.768308	−	0.640081i	$-0.221099\pi$
$930$	0	0
$931$	− 907.595i	− 0.0319497i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−77760.0	−2.71981
$936$	0	0
$937$	−11989.0	−0.417997	−0.208999	−	0.977916i	$-0.567020\pi$
$937$	−11989.0	−0.417997	−0.208999	+	0.977916i	$0.567020\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	10454.7i	0.362181i	0.983466	+	0.181090i	$0.0579627\pi$
$941$	10454.7i	0.362181i	−0.983466	+	0.181090i	$0.942037\pi$
$942$	0	0
$943$	− 40405.3i	− 1.39531i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−3780.00	−0.129708	−0.0648540	−	0.997895i	$-0.520658\pi$
$947$	−3780.00	−0.129708	−0.0648540	+	0.997895i	$0.520658\pi$
$948$	0	0
$949$	−15371.0	−0.525778
$950$	0	0
$951$	0	0
$952$	0	0
$953$	31197.7i	1.06043i	0.847862	+	0.530216i	$0.177889\pi$
$953$	31197.7i	1.06043i	−0.847862	+	0.530216i	$0.822111\pi$
$954$	0	0
$955$	95027.2i	3.21990i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−11628.0	−0.391541
$960$	0	0
$961$	21043.0	0.706354
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 19475.2i	− 0.649667i
$966$	0	0
$967$	− 6607.77i	− 0.219743i	−0.993946	−	0.109872i	$-0.964956\pi$
$967$	− 6607.77i	− 0.219743i	0.993946	−	0.109872i	$-0.0350440\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	33012.0	1.09105	0.545523	−	0.838096i	$-0.316331\pi$
$971$	33012.0	1.09105	0.545523	+	0.838096i	$0.316331\pi$
$972$	0	0
$973$	50541.0	1.66523
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 57240.8i	− 1.87441i	−0.348783	−	0.937204i	$-0.613405\pi$
$977$	− 57240.8i	− 1.87441i	0.348783	−	0.937204i	$-0.386595\pi$
$978$	0	0
$979$	9727.20i	0.317551i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	15084.0	0.489425	0.244713	−	0.969596i	$-0.421306\pi$
$983$	15084.0	0.489425	0.244713	+	0.969596i	$0.421306\pi$
$984$	0	0
$985$	−38448.0	−1.24371
$986$	0	0
$987$	0	0
$988$	0	0
$989$	50506.6i	1.62388i
$990$	0	0
$991$	− 24600.3i	− 0.788552i	−0.918992	−	0.394276i	$-0.870995\pi$
$991$	− 24600.3i	− 0.788552i	0.918992	−	0.394276i	$-0.129005\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	52524.0	1.67349
$996$	0	0
$997$	3422.00	0.108702	0.0543510	−	0.998522i	$-0.482691\pi$
$997$	3422.00	0.108702	0.0543510	+	0.998522i	$0.482691\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	432.4.c.d.431.2	yes	2
3.2	odd	2		432.4.c.a.431.1	✓	2
4.3	odd	2		432.4.c.a.431.2	yes	2
8.3	odd	2		1728.4.c.d.1727.1		2
8.5	even	2		1728.4.c.a.1727.1		2
12.11	even	2	inner	432.4.c.d.431.1	yes	2
24.5	odd	2		1728.4.c.d.1727.2		2
24.11	even	2		1728.4.c.a.1727.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
432.4.c.a.431.1	✓	2	3.2	odd	2
432.4.c.a.431.2	yes	2	4.3	odd	2
432.4.c.d.431.1	yes	2	12.11	even	2	inner
432.4.c.d.431.2	yes	2	1.1	even	1	trivial
1728.4.c.a.1727.1		2	8.5	even	2
1728.4.c.a.1727.2		2	24.11	even	2
1728.4.c.d.1727.1		2	8.3	odd	2
1728.4.c.d.1727.2		2	24.5	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 432 = 2^{4} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	432.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+20.7846i q^{5} +29.4449i q^{7} +36.0000 q^{11} +19.0000 q^{13} +103.923i q^{17} +1.73205i q^{19} +108.000 q^{23} -307.000 q^{25} -207.846i q^{29} -93.5307i q^{31} -612.000 q^{35} -109.000 q^{37} -374.123i q^{41} +467.654i q^{43} +468.000 q^{47} -524.000 q^{49} +124.708i q^{53} +748.246i q^{55} -36.0000 q^{59} -145.000 q^{61} +394.908i q^{65} +313.501i q^{67} +576.000 q^{71} -809.000 q^{73} +1060.02i q^{77} -497.099i q^{79} -360.000 q^{83} -2160.00 q^{85} +270.200i q^{89} +559.452i q^{91} -36.0000 q^{95} +955.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 72 q^{11} + 38 q^{13} + 216 q^{23} - 614 q^{25} - 1224 q^{35} - 218 q^{37} + 936 q^{47} - 1048 q^{49} - 72 q^{59} - 290 q^{61} + 1152 q^{71} - 1618 q^{73} - 720 q^{83} - 4320 q^{85} - 72 q^{95} + 1910 q^{97}+O(q^{100}) \) 2 * q + 72 * q^11 + 38 * q^13 + 216 * q^23 - 614 * q^25 - 1224 * q^35 - 218 * q^37 + 936 * q^47 - 1048 * q^49 - 72 * q^59 - 290 * q^61 + 1152 * q^71 - 1618 * q^73 - 720 * q^83 - 4320 * q^85 - 72 * q^95 + 1910 * q^97

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