LMFDB - Embedded newform 432.8.a.q.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [432,8,Mod(1,432)] 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([0, 0, 0])) N = Newforms(chi, 8, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("432.1"); S:= CuspForms(chi, 8); N := Newforms(S);

Level:	$ N $	$=$	$ 432 = 2^{4} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	432.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,180,0,-700] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$134.950331009$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{65}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 16 $ x^2 - x - 16
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	no (minimal twist has level 27)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-3.53113$ of defining polynomial
Character	$\chi$	$=$	432.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+114.187 q^{5} -1438.40 q^{7} -5928.74 q^{11} -11447.2 q^{13} -20235.6 q^{17} +6354.94 q^{19} -75845.6 q^{23} -65086.4 q^{25} -74784.3 q^{29} +189363. q^{31} -164247. q^{35} -33407.2 q^{37} +141245. q^{41} +246197. q^{43} -335133. q^{47} +1.24547e6 q^{49} +1.65156e6 q^{53} -676983. q^{55} -2.04823e6 q^{59} -590469. q^{61} -1.30712e6 q^{65} -53575.5 q^{67} +4.95678e6 q^{71} +817542. q^{73} +8.52792e6 q^{77} -7.57257e6 q^{79} +1.01891e6 q^{83} -2.31064e6 q^{85} +1.37281e6 q^{89} +1.64658e7 q^{91} +725650. q^{95} -1.06023e7 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 180 q^{5} - 700 q^{7} - 10890 q^{11} - 5480 q^{13} + 16416 q^{17} - 16024 q^{19} - 24372 q^{23} - 138880 q^{25} - 143280 q^{29} + 38708 q^{31} - 115650 q^{35} + 455620 q^{37} + 731880 q^{41} + 1088840 q^{43}+ \cdots + 4098670 q^{97}+O(q^{100}) $ 2 * q + 180 * q^5 - 700 * q^7 - 10890 * q^11 - 5480 * q^13 + 16416 * q^17 - 16024 * q^19 - 24372 * q^23 - 138880 * q^25 - 143280 * q^29 + 38708 * q^31 - 115650 * q^35 + 455620 * q^37 + 731880 * q^41 + 1088840 * q^43 - 1561500 * q^47 + 967164 * q^49 + 2610468 * q^53 - 1003500 * q^55 - 1731960 * q^59 - 620192 * q^61 - 914400 * q^65 - 346600 * q^67 + 4242240 * q^71 - 3145190 * q^73 + 4864500 * q^77 - 10110616 * q^79 - 644202 * q^83 + 101520 * q^85 + 6021000 * q^89 + 20872000 * q^91 - 747180 * q^95 + 4098670 * 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
$5$	114.187	0.408527	0.204264	−	0.978916i	$-0.434520\pi$
$5$	114.187	0.408527	0.204264	+	0.978916i	$0.434520\pi$
$6$	0	0
$7$	−1438.40	−1.58503	−0.792516	−	0.609851i	$-0.791229\pi$
$7$	−1438.40	−1.58503	−0.792516	+	0.609851i	$0.791229\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−5928.74	−1.34304	−0.671518	−	0.740988i	$-0.734357\pi$
$11$	−5928.74	−1.34304	−0.671518	+	0.740988i	$0.734357\pi$
$12$	0	0
$13$	−11447.2	−1.44510	−0.722552	−	0.691317i	$-0.757031\pi$
$13$	−11447.2	−1.44510	−0.722552	+	0.691317i	$0.757031\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−20235.6	−0.998955	−0.499477	−	0.866327i	$-0.666475\pi$
$17$	−20235.6	−0.998955	−0.499477	+	0.866327i	$0.666475\pi$
$18$	0	0
$19$	6354.94	0.212556	0.106278	−	0.994336i	$-0.466107\pi$
$19$	6354.94	0.212556	0.106278	+	0.994336i	$0.466107\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−75845.6	−1.29982	−0.649910	−	0.760012i	$-0.725193\pi$
$23$	−75845.6	−1.29982	−0.649910	+	0.760012i	$0.725193\pi$
$24$	0	0
$25$	−65086.4	−0.833106
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−74784.3	−0.569400	−0.284700	−	0.958617i	$-0.591894\pi$
$29$	−74784.3	−0.569400	−0.284700	+	0.958617i	$0.591894\pi$
$30$	0	0
$31$	189363.	1.14164	0.570820	−	0.821076i	$-0.306626\pi$
$31$	189363.	1.14164	0.570820	+	0.821076i	$0.306626\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−164247.	−0.647528
$36$	0	0
$37$	−33407.2	−0.108426	−0.0542130	−	0.998529i	$-0.517265\pi$
$37$	−33407.2	−0.108426	−0.0542130	+	0.998529i	$0.517265\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	141245.	0.320058	0.160029	−	0.987112i	$-0.448841\pi$
$41$	141245.	0.320058	0.160029	+	0.987112i	$0.448841\pi$
$42$	0	0
$43$	246197.	0.472219	0.236109	−	0.971726i	$-0.424128\pi$
$43$	246197.	0.472219	0.236109	+	0.971726i	$0.424128\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−335133.	−0.470841	−0.235421	−	0.971894i	$-0.575647\pi$
$47$	−335133.	−0.470841	−0.235421	+	0.971894i	$0.575647\pi$
$48$	0	0
$49$	1.24547e6	1.51233
$50$	0	0
$51$	0	0
$52$	0	0
$53$	1.65156e6	1.52381	0.761904	−	0.647691i	$-0.224265\pi$
$53$	1.65156e6	1.52381	0.761904	+	0.647691i	$0.224265\pi$
$54$	0	0
$55$	−676983.	−0.548667
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−2.04823e6	−1.29836	−0.649182	−	0.760633i	$-0.724889\pi$
$59$	−2.04823e6	−1.29836	−0.649182	+	0.760633i	$0.724889\pi$
$60$	0	0
$61$	−590469.	−0.333076	−0.166538	−	0.986035i	$-0.553259\pi$
$61$	−590469.	−0.333076	−0.166538	+	0.986035i	$0.553259\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.30712e6	−0.590364
$66$	0	0
$67$	−53575.5	−0.0217623	−0.0108811	−	0.999941i	$-0.503464\pi$
$67$	−53575.5	−0.0217623	−0.0108811	+	0.999941i	$0.503464\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	4.95678e6	1.64360	0.821798	−	0.569779i	$-0.192971\pi$
$71$	4.95678e6	1.64360	0.821798	+	0.569779i	$0.192971\pi$
$72$	0	0
$73$	817542.	0.245969	0.122984	−	0.992409i	$-0.460753\pi$
$73$	817542.	0.245969	0.122984	+	0.992409i	$0.460753\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	8.52792e6	2.12875
$78$	0	0
$79$	−7.57257e6	−1.72802	−0.864009	−	0.503476i	$-0.832054\pi$
$79$	−7.57257e6	−1.72802	−0.864009	+	0.503476i	$0.832054\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1.01891e6	0.195597	0.0977986	−	0.995206i	$-0.468820\pi$
$83$	1.01891e6	0.195597	0.0977986	+	0.995206i	$0.468820\pi$
$84$	0	0
$85$	−2.31064e6	−0.408100
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1.37281e6	0.206418	0.103209	−	0.994660i	$-0.467089\pi$
$89$	1.37281e6	0.206418	0.103209	+	0.994660i	$0.467089\pi$
$90$	0	0
$91$	1.64658e7	2.29054
$92$	0	0
$93$	0	0
$94$	0	0
$95$	725650.	0.0868350
$96$	0	0
$97$	−1.06023e7	−1.17950	−0.589750	−	0.807586i	$-0.700774\pi$
$97$	−1.06023e7	−1.17950	−0.589750	+	0.807586i	$0.700774\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1.04277e7	−1.00708	−0.503540	−	0.863972i	$-0.667969\pi$
$101$	−1.04277e7	−1.00708	−0.503540	+	0.863972i	$0.667969\pi$
$102$	0	0
$103$	−1.53315e6	−0.138246	−0.0691232	−	0.997608i	$-0.522020\pi$
$103$	−1.53315e6	−0.138246	−0.0691232	+	0.997608i	$0.522020\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−9.64859e6	−0.761414	−0.380707	−	0.924696i	$-0.624319\pi$
$107$	−9.64859e6	−0.761414	−0.380707	+	0.924696i	$0.624319\pi$
$108$	0	0
$109$	5.79409e6	0.428541	0.214270	−	0.976774i	$-0.431263\pi$
$109$	5.79409e6	0.428541	0.214270	+	0.976774i	$0.431263\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1.44843e7	−0.944327	−0.472164	−	0.881511i	$-0.656527\pi$
$113$	−1.44843e7	−0.944327	−0.472164	+	0.881511i	$0.656527\pi$
$114$	0	0
$115$	−8.66056e6	−0.531011
$116$	0	0
$117$	0	0
$118$	0	0
$119$	2.91070e7	1.58338
$120$	0	0
$121$	1.56627e7	0.803746
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.63528e7	−0.748873
$126$	0	0
$127$	1.65775e7	0.718136	0.359068	−	0.933311i	$-0.383095\pi$
$127$	1.65775e7	0.718136	0.359068	+	0.933311i	$0.383095\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1.44778e7	−0.562669	−0.281335	−	0.959610i	$-0.590777\pi$
$131$	−1.44778e7	−0.562669	−0.281335	+	0.959610i	$0.590777\pi$
$132$	0	0
$133$	−9.14098e6	−0.336909
$134$	0	0
$135$	0	0
$136$	0	0
$137$	4.21938e7	1.40193	0.700966	−	0.713194i	$-0.252752\pi$
$137$	4.21938e7	1.40193	0.700966	+	0.713194i	$0.252752\pi$
$138$	0	0
$139$	−4.79933e7	−1.51575	−0.757877	−	0.652398i	$-0.773763\pi$
$139$	−4.79933e7	−1.51575	−0.757877	+	0.652398i	$0.773763\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	6.78676e7	1.94083
$144$	0	0
$145$	−8.53938e6	−0.232615
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1.15668e7	0.286457	0.143229	−	0.989690i	$-0.454252\pi$
$149$	1.15668e7	0.286457	0.143229	+	0.989690i	$0.454252\pi$
$150$	0	0
$151$	−1.09933e7	−0.259843	−0.129921	−	0.991524i	$-0.541472\pi$
$151$	−1.09933e7	−0.259843	−0.129921	+	0.991524i	$0.541472\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2.16227e7	0.466390
$156$	0	0
$157$	2.48018e7	0.511487	0.255744	−	0.966745i	$-0.417680\pi$
$157$	2.48018e7	0.511487	0.255744	+	0.966745i	$0.417680\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1.09097e8	2.06025
$162$	0	0
$163$	−3.04106e6	−0.0550008	−0.0275004	−	0.999622i	$-0.508755\pi$
$163$	−3.04106e6	−0.0550008	−0.0275004	+	0.999622i	$0.508755\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.06849e8	1.77526	0.887629	−	0.460559i	$-0.152351\pi$
$167$	1.06849e8	1.77526	0.887629	+	0.460559i	$0.152351\pi$
$168$	0	0
$169$	6.82907e7	1.08832
$170$	0	0
$171$	0	0
$172$	0	0
$173$	9.94936e7	1.46094	0.730472	−	0.682942i	$-0.239300\pi$
$173$	9.94936e7	1.46094	0.730472	+	0.682942i	$0.239300\pi$
$174$	0	0
$175$	9.36206e7	1.32050
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−2.29173e7	−0.298661	−0.149330	−	0.988787i	$-0.547712\pi$
$179$	−2.29173e7	−0.298661	−0.149330	+	0.988787i	$0.547712\pi$
$180$	0	0
$181$	−1.35824e8	−1.70256	−0.851280	−	0.524712i	$-0.824173\pi$
$181$	−1.35824e8	−1.70256	−0.851280	+	0.524712i	$0.824173\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−3.81465e6	−0.0442949
$186$	0	0
$187$	1.19972e8	1.34163
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1.11919e8	−1.16221	−0.581107	−	0.813827i	$-0.697380\pi$
$191$	−1.11919e8	−1.16221	−0.581107	+	0.813827i	$0.697380\pi$
$192$	0	0
$193$	1.02898e8	1.03029	0.515143	−	0.857104i	$-0.327739\pi$
$193$	1.02898e8	1.03029	0.515143	+	0.857104i	$0.327739\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3.77994e7	0.352252	0.176126	−	0.984368i	$-0.443643\pi$
$197$	3.77994e7	0.352252	0.176126	+	0.984368i	$0.443643\pi$
$198$	0	0
$199$	49088.1	0.000441560 0	0.000220780	−	1.00000i	$-0.499930\pi$
$199$	49088.1	0.000441560 0	0.000220780		1.00000i	$0.499930\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1.07570e8	0.902517
$204$	0	0
$205$	1.61283e7	0.130752
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−3.76768e7	−0.285471
$210$	0	0
$211$	−1.88572e8	−1.38194	−0.690969	−	0.722885i	$-0.742816\pi$
$211$	−1.88572e8	−1.38194	−0.690969	+	0.722885i	$0.742816\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	2.81125e7	0.192914
$216$	0	0
$217$	−2.72380e8	−1.80953
$218$	0	0
$219$	0	0
$220$	0	0
$221$	2.31642e8	1.44359
$222$	0	0
$223$	−1.05676e8	−0.638127	−0.319064	−	0.947733i	$-0.603368\pi$
$223$	−1.05676e8	−0.638127	−0.319064	+	0.947733i	$0.603368\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4.59251e7	0.260591	0.130295	−	0.991475i	$-0.458407\pi$
$227$	4.59251e7	0.260591	0.130295	+	0.991475i	$0.458407\pi$
$228$	0	0
$229$	−1.22895e8	−0.676254	−0.338127	−	0.941100i	$-0.609793\pi$
$229$	−1.22895e8	−0.676254	−0.338127	+	0.941100i	$0.609793\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.68276e8	0.871519	0.435760	−	0.900063i	$-0.356480\pi$
$233$	1.68276e8	0.871519	0.435760	+	0.900063i	$0.356480\pi$
$234$	0	0
$235$	−3.82677e7	−0.192351
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−8.02527e7	−0.380248	−0.190124	−	0.981760i	$-0.560889\pi$
$239$	−8.02527e7	−0.380248	−0.190124	+	0.981760i	$0.560889\pi$
$240$	0	0
$241$	1.49328e8	0.687198	0.343599	−	0.939116i	$-0.388354\pi$
$241$	1.49328e8	0.687198	0.343599	+	0.939116i	$0.388354\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.42216e8	0.617826
$246$	0	0
$247$	−7.27466e7	−0.307166
$248$	0	0
$249$	0	0
$250$	0	0
$251$	4.54290e8	1.81332	0.906661	−	0.421861i	$-0.138623\pi$
$251$	4.54290e8	1.81332	0.906661	+	0.421861i	$0.138623\pi$
$252$	0	0
$253$	4.49668e8	1.74570
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−2.92144e7	−0.107357	−0.0536786	−	0.998558i	$-0.517095\pi$
$257$	−2.92144e7	−0.107357	−0.0536786	+	0.998558i	$0.517095\pi$
$258$	0	0
$259$	4.80530e7	0.171859
$260$	0	0
$261$	0	0
$262$	0	0
$263$	4.49406e8	1.52333	0.761664	−	0.647972i	$-0.224383\pi$
$263$	4.49406e8	1.52333	0.761664	+	0.647972i	$0.224383\pi$
$264$	0	0
$265$	1.88587e8	0.622516
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−1.94325e7	−0.0608689	−0.0304345	−	0.999537i	$-0.509689\pi$
$269$	−1.94325e7	−0.0608689	−0.0304345	+	0.999537i	$0.509689\pi$
$270$	0	0
$271$	2.13963e7	0.0653051	0.0326525	−	0.999467i	$-0.489605\pi$
$271$	2.13963e7	0.0653051	0.0326525	+	0.999467i	$0.489605\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	3.85880e8	1.11889
$276$	0	0
$277$	−1.85015e8	−0.523031	−0.261515	−	0.965199i	$-0.584222\pi$
$277$	−1.85015e8	−0.523031	−0.261515	+	0.965199i	$0.584222\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−3.62351e7	−0.0974221	−0.0487111	−	0.998813i	$-0.515511\pi$
$281$	−3.62351e7	−0.0974221	−0.0487111	+	0.998813i	$0.515511\pi$
$282$	0	0
$283$	3.96100e8	1.03885	0.519424	−	0.854517i	$-0.326147\pi$
$283$	3.96100e8	1.03885	0.519424	+	0.854517i	$0.326147\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2.03167e8	−0.507303
$288$	0	0
$289$	−857331.	−0.00208932
$290$	0	0
$291$	0	0
$292$	0	0
$293$	3.96468e8	0.920812	0.460406	−	0.887708i	$-0.347704\pi$
$293$	3.96468e8	0.920812	0.460406	+	0.887708i	$0.347704\pi$
$294$	0	0
$295$	−2.33881e8	−0.530417
$296$	0	0
$297$	0	0
$298$	0	0
$299$	8.68223e8	1.87837
$300$	0	0
$301$	−3.54131e8	−0.748482
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−6.74238e7	−0.136070
$306$	0	0
$307$	3.12263e8	0.615936	0.307968	−	0.951397i	$-0.400351\pi$
$307$	3.12263e8	0.615936	0.307968	+	0.951397i	$0.400351\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−2.82908e8	−0.533316	−0.266658	−	0.963791i	$-0.585919\pi$
$311$	−2.82908e8	−0.533316	−0.266658	+	0.963791i	$0.585919\pi$
$312$	0	0
$313$	−9.55802e8	−1.76182	−0.880912	−	0.473280i	$-0.843070\pi$
$313$	−9.55802e8	−1.76182	−0.880912	+	0.473280i	$0.843070\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	4.13052e8	0.728278	0.364139	−	0.931345i	$-0.381363\pi$
$317$	4.13052e8	0.728278	0.364139	+	0.931345i	$0.381363\pi$
$318$	0	0
$319$	4.43376e8	0.764724
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−1.28596e8	−0.212334
$324$	0	0
$325$	7.45059e8	1.20392
$326$	0	0
$327$	0	0
$328$	0	0
$329$	4.82057e8	0.746298
$330$	0	0
$331$	−2.28880e8	−0.346905	−0.173452	−	0.984842i	$-0.555492\pi$
$331$	−2.28880e8	−0.346905	−0.173452	+	0.984842i	$0.555492\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−6.11761e6	−0.00889048
$336$	0	0
$337$	9.78074e8	1.39209	0.696044	−	0.717999i	$-0.254942\pi$
$337$	9.78074e8	1.39209	0.696044	+	0.717999i	$0.254942\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−1.12268e9	−1.53326
$342$	0	0
$343$	−6.06895e8	−0.812053
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1.42099e8	0.182573	0.0912867	−	0.995825i	$-0.470902\pi$
$347$	1.42099e8	0.182573	0.0912867	+	0.995825i	$0.470902\pi$
$348$	0	0
$349$	−4.63334e8	−0.583452	−0.291726	−	0.956502i	$-0.594230\pi$
$349$	−4.63334e8	−0.583452	−0.291726	+	0.956502i	$0.594230\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−2.20636e8	−0.266971	−0.133486	−	0.991051i	$-0.542617\pi$
$353$	−2.20636e8	−0.266971	−0.133486	+	0.991051i	$0.542617\pi$
$354$	0	0
$355$	5.65998e8	0.671454
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−2.57110e8	−0.293284	−0.146642	−	0.989190i	$-0.546847\pi$
$359$	−2.57110e8	−0.293284	−0.146642	+	0.989190i	$0.546847\pi$
$360$	0	0
$361$	−8.53486e8	−0.954820
$362$	0	0
$363$	0	0
$364$	0	0
$365$	9.33525e7	0.100485
$366$	0	0
$367$	−7.22719e8	−0.763200	−0.381600	−	0.924328i	$-0.624627\pi$
$367$	−7.22719e8	−0.763200	−0.381600	+	0.924328i	$0.624627\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−2.37562e9	−2.41528
$372$	0	0
$373$	−5.89159e8	−0.587829	−0.293915	−	0.955832i	$-0.594958\pi$
$373$	−5.89159e8	−0.587829	−0.293915	+	0.955832i	$0.594958\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	8.56073e8	0.822842
$378$	0	0
$379$	−9.02541e8	−0.851589	−0.425794	−	0.904820i	$-0.640005\pi$
$379$	−9.02541e8	−0.851589	−0.425794	+	0.904820i	$0.640005\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−5.59375e8	−0.508754	−0.254377	−	0.967105i	$-0.581870\pi$
$383$	−5.59375e8	−0.508754	−0.254377	+	0.967105i	$0.581870\pi$
$384$	0	0
$385$	9.73776e8	0.869654
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−1.34140e9	−1.15541	−0.577705	−	0.816246i	$-0.696052\pi$
$389$	−1.34140e9	−1.15541	−0.577705	+	0.816246i	$0.696052\pi$
$390$	0	0
$391$	1.53478e9	1.29846
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−8.64687e8	−0.705942
$396$	0	0
$397$	2.24763e9	1.80285	0.901423	−	0.432939i	$-0.142523\pi$
$397$	2.24763e9	1.80285	0.901423	+	0.432939i	$0.142523\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−2.40503e9	−1.86258	−0.931289	−	0.364280i	$-0.881315\pi$
$401$	−2.40503e9	−1.86258	−0.931289	+	0.364280i	$0.881315\pi$
$402$	0	0
$403$	−2.16768e9	−1.64979
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.98062e8	0.145620
$408$	0	0
$409$	7.63994e8	0.552152	0.276076	−	0.961136i	$-0.410966\pi$
$409$	7.63994e8	0.552152	0.276076	+	0.961136i	$0.410966\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	2.94618e9	2.05795
$414$	0	0
$415$	1.16346e8	0.0799068
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−2.88611e8	−0.191675	−0.0958373	−	0.995397i	$-0.530553\pi$
$419$	−2.88611e8	−0.191675	−0.0958373	+	0.995397i	$0.530553\pi$
$420$	0	0
$421$	−1.55888e9	−1.01818	−0.509090	−	0.860713i	$-0.670018\pi$
$421$	−1.55888e9	−1.01818	−0.509090	+	0.860713i	$0.670018\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1.31706e9	0.832235
$426$	0	0
$427$	8.49334e8	0.527935
$428$	0	0
$429$	0	0
$430$	0	0
$431$	9.87526e8	0.594125	0.297063	−	0.954858i	$-0.403993\pi$
$431$	9.87526e8	0.594125	0.297063	+	0.954858i	$0.403993\pi$
$432$	0	0
$433$	−2.07048e9	−1.22564	−0.612820	−	0.790223i	$-0.709965\pi$
$433$	−2.07048e9	−1.22564	−0.612820	+	0.790223i	$0.709965\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−4.81994e8	−0.276285
$438$	0	0
$439$	3.10766e9	1.75311	0.876553	−	0.481306i	$-0.159837\pi$
$439$	3.10766e9	1.75311	0.876553	+	0.481306i	$0.159837\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−1.77729e9	−0.971280	−0.485640	−	0.874159i	$-0.661413\pi$
$443$	−1.77729e9	−0.971280	−0.485640	+	0.874159i	$0.661413\pi$
$444$	0	0
$445$	1.56757e8	0.0843271
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−3.38966e9	−1.76723	−0.883616	−	0.468212i	$-0.844898\pi$
$449$	−3.38966e9	−1.76723	−0.883616	+	0.468212i	$0.844898\pi$
$450$	0	0
$451$	−8.37404e8	−0.429850
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1.88017e9	0.935746
$456$	0	0
$457$	−2.09644e9	−1.02749	−0.513744	−	0.857944i	$-0.671742\pi$
$457$	−2.09644e9	−1.02749	−0.513744	+	0.857944i	$0.671742\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	1.78459e9	0.848371	0.424185	−	0.905575i	$-0.360560\pi$
$461$	1.78459e9	0.848371	0.424185	+	0.905575i	$0.360560\pi$
$462$	0	0
$463$	−2.29800e9	−1.07601	−0.538005	−	0.842942i	$-0.680822\pi$
$463$	−2.29800e9	−1.07601	−0.538005	+	0.842942i	$0.680822\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	8.59979e8	0.390732	0.195366	−	0.980730i	$-0.437411\pi$
$467$	8.59979e8	0.390732	0.195366	+	0.980730i	$0.437411\pi$
$468$	0	0
$469$	7.70632e7	0.0344939
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−1.45964e9	−0.634207
$474$	0	0
$475$	−4.13620e8	−0.177082
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−3.06607e7	−0.0127470	−0.00637349	−	0.999980i	$-0.502029\pi$
$479$	−3.06607e7	−0.0127470	−0.00637349	+	0.999980i	$0.502029\pi$
$480$	0	0
$481$	3.82420e8	0.156687
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−1.21064e9	−0.481858
$486$	0	0
$487$	5.78717e8	0.227047	0.113523	−	0.993535i	$-0.463786\pi$
$487$	5.78717e8	0.227047	0.113523	+	0.993535i	$0.463786\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−3.29719e9	−1.25707	−0.628534	−	0.777782i	$-0.716345\pi$
$491$	−3.29719e9	−1.25707	−0.628534	+	0.777782i	$0.716345\pi$
$492$	0	0
$493$	1.51331e9	0.568805
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−7.12985e9	−2.60515
$498$	0	0
$499$	4.27181e9	1.53907	0.769537	−	0.638602i	$-0.220487\pi$
$499$	4.27181e9	1.53907	0.769537	+	0.638602i	$0.220487\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−1.85857e9	−0.651165	−0.325582	−	0.945514i	$-0.605560\pi$
$503$	−1.85857e9	−0.651165	−0.325582	+	0.945514i	$0.605560\pi$
$504$	0	0
$505$	−1.19071e9	−0.411419
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−2.12609e9	−0.714610	−0.357305	−	0.933988i	$-0.616304\pi$
$509$	−2.12609e9	−0.714610	−0.357305	+	0.933988i	$0.616304\pi$
$510$	0	0
$511$	−1.17596e9	−0.389868
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−1.75065e8	−0.0564774
$516$	0	0
$517$	1.98691e9	0.632357
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−5.97837e9	−1.85204	−0.926021	−	0.377472i	$-0.876793\pi$
$521$	−5.97837e9	−1.85204	−0.926021	+	0.377472i	$0.876793\pi$
$522$	0	0
$523$	−5.69436e8	−0.174056	−0.0870280	−	0.996206i	$-0.527737\pi$
$523$	−5.69436e8	−0.174056	−0.0870280	+	0.996206i	$0.527737\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−3.83188e9	−1.14045
$528$	0	0
$529$	2.34773e9	0.689530
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−1.61686e9	−0.462518
$534$	0	0
$535$	−1.10174e9	−0.311058
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−7.38403e9	−2.03111
$540$	0	0
$541$	−1.88830e9	−0.512722	−0.256361	−	0.966581i	$-0.582524\pi$
$541$	−1.88830e9	−0.512722	−0.256361	+	0.966581i	$0.582524\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	6.61608e8	0.175070
$546$	0	0
$547$	−5.88629e8	−0.153775	−0.0768875	−	0.997040i	$-0.524498\pi$
$547$	−5.88629e8	−0.153775	−0.0768875	+	0.997040i	$0.524498\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−4.75250e8	−0.121030
$552$	0	0
$553$	1.08924e10	2.73897
$554$	0	0
$555$	0	0
$556$	0	0
$557$	5.59795e9	1.37257	0.686287	−	0.727331i	$-0.259239\pi$
$557$	5.59795e9	1.37257	0.686287	+	0.727331i	$0.259239\pi$
$558$	0	0
$559$	−2.81828e9	−0.682405
$560$	0	0
$561$	0	0
$562$	0	0
$563$	3.06186e9	0.723113	0.361557	−	0.932350i	$-0.382245\pi$
$563$	3.06186e9	0.723113	0.361557	+	0.932350i	$0.382245\pi$
$564$	0	0
$565$	−1.65391e9	−0.385783
$566$	0	0
$567$	0	0
$568$	0	0
$569$	1.74572e9	0.397267	0.198633	−	0.980074i	$-0.436350\pi$
$569$	1.74572e9	0.397267	0.198633	+	0.980074i	$0.436350\pi$
$570$	0	0
$571$	5.28295e9	1.18755	0.593773	−	0.804633i	$-0.297638\pi$
$571$	5.28295e9	1.18755	0.593773	+	0.804633i	$0.297638\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	4.93651e9	1.08289
$576$	0	0
$577$	−6.38776e9	−1.38431	−0.692155	−	0.721749i	$-0.743338\pi$
$577$	−6.38776e9	−1.38431	−0.692155	+	0.721749i	$0.743338\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−1.46561e9	−0.310028
$582$	0	0
$583$	−9.79169e9	−2.04653
$584$	0	0
$585$	0	0
$586$	0	0
$587$	5.93488e9	1.21110	0.605548	−	0.795809i	$-0.292954\pi$
$587$	5.93488e9	1.21110	0.605548	+	0.795809i	$0.292954\pi$
$588$	0	0
$589$	1.20339e9	0.242663
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−2.18286e9	−0.429867	−0.214933	−	0.976629i	$-0.568953\pi$
$593$	−2.18286e9	−0.429867	−0.214933	+	0.976629i	$0.568953\pi$
$594$	0	0
$595$	3.32364e9	0.646852
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−8.51890e9	−1.61953	−0.809766	−	0.586752i	$-0.800406\pi$
$599$	−8.51890e9	−1.61953	−0.809766	+	0.586752i	$0.800406\pi$
$600$	0	0
$601$	−6.59540e9	−1.23931	−0.619656	−	0.784874i	$-0.712728\pi$
$601$	−6.59540e9	−1.23931	−0.619656	+	0.784874i	$0.712728\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	1.78848e9	0.328352
$606$	0	0
$607$	6.61480e9	1.20048	0.600242	−	0.799819i	$-0.295071\pi$
$607$	6.61480e9	1.20048	0.600242	+	0.799819i	$0.295071\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	3.83635e9	0.680414
$612$	0	0
$613$	−6.77351e9	−1.18769	−0.593843	−	0.804581i	$-0.702390\pi$
$613$	−6.77351e9	−1.18769	−0.593843	+	0.804581i	$0.702390\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1.09658e10	1.87950	0.939751	−	0.341860i	$-0.111057\pi$
$617$	1.09658e10	1.87950	0.939751	+	0.341860i	$0.111057\pi$
$618$	0	0
$619$	−5.57764e9	−0.945221	−0.472610	−	0.881271i	$-0.656688\pi$
$619$	−5.57764e9	−0.945221	−0.472610	+	0.881271i	$0.656688\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−1.97466e9	−0.327178
$624$	0	0
$625$	3.21759e9	0.527171
$626$	0	0
$627$	0	0
$628$	0	0
$629$	6.76015e8	0.108313
$630$	0	0
$631$	6.36080e9	1.00788	0.503940	−	0.863739i	$-0.331883\pi$
$631$	6.36080e9	1.00788	0.503940	+	0.863739i	$0.331883\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	1.89293e9	0.293378
$636$	0	0
$637$	−1.42571e10	−2.18547
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−7.54816e8	−0.113198	−0.0565989	−	0.998397i	$-0.518026\pi$
$641$	−7.54816e8	−0.113198	−0.0565989	+	0.998397i	$0.518026\pi$
$642$	0	0
$643$	−1.04241e10	−1.54633	−0.773164	−	0.634206i	$-0.781327\pi$
$643$	−1.04241e10	−1.54633	−0.773164	+	0.634206i	$0.781327\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−8.26499e9	−1.19971	−0.599857	−	0.800107i	$-0.704776\pi$
$647$	−8.26499e9	−1.19971	−0.599857	+	0.800107i	$0.704776\pi$
$648$	0	0
$649$	1.21434e10	1.74375
$650$	0	0
$651$	0	0
$652$	0	0
$653$	9.12709e9	1.28273	0.641366	−	0.767235i	$-0.278368\pi$
$653$	9.12709e9	1.28273	0.641366	+	0.767235i	$0.278368\pi$
$654$	0	0
$655$	−1.65317e9	−0.229866
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−1.50605e9	−0.204994	−0.102497	−	0.994733i	$-0.532683\pi$
$659$	−1.50605e9	−0.204994	−0.102497	+	0.994733i	$0.532683\pi$
$660$	0	0
$661$	9.79847e9	1.31963	0.659816	−	0.751427i	$-0.270634\pi$
$661$	9.79847e9	1.31963	0.659816	+	0.751427i	$0.270634\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1.04378e9	−0.137636
$666$	0	0
$667$	5.67206e9	0.740117
$668$	0	0
$669$	0	0
$670$	0	0
$671$	3.50073e9	0.447332
$672$	0	0
$673$	−9.18829e9	−1.16194	−0.580968	−	0.813927i	$-0.697326\pi$
$673$	−9.18829e9	−1.16194	−0.580968	+	0.813927i	$0.697326\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.01615e10	1.25862	0.629312	−	0.777153i	$-0.283337\pi$
$677$	1.01615e10	1.25862	0.629312	+	0.777153i	$0.283337\pi$
$678$	0	0
$679$	1.52504e10	1.86955
$680$	0	0
$681$	0	0
$682$	0	0
$683$	9.85009e9	1.18295	0.591477	−	0.806322i	$-0.298545\pi$
$683$	9.85009e9	1.18295	0.591477	+	0.806322i	$0.298545\pi$
$684$	0	0
$685$	4.81798e9	0.572727
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−1.89059e10	−2.20206
$690$	0	0
$691$	−3.54327e9	−0.408536	−0.204268	−	0.978915i	$-0.565481\pi$
$691$	−3.54327e9	−0.408536	−0.204268	+	0.978915i	$0.565481\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−5.48020e9	−0.619226
$696$	0	0
$697$	−2.85818e9	−0.319724
$698$	0	0
$699$	0	0
$700$	0	0
$701$	1.22196e10	1.33982	0.669908	−	0.742444i	$-0.266334\pi$
$701$	1.22196e10	1.33982	0.669908	+	0.742444i	$0.266334\pi$
$702$	0	0
$703$	−2.12301e8	−0.0230466
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.49992e10	1.59625
$708$	0	0
$709$	1.01823e10	1.07297	0.536483	−	0.843911i	$-0.319753\pi$
$709$	1.01823e10	1.07297	0.536483	+	0.843911i	$0.319753\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−1.43623e10	−1.48392
$714$	0	0
$715$	7.74959e9	0.792880
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−2.32175e9	−0.232951	−0.116476	−	0.993194i	$-0.537160\pi$
$719$	−2.32175e9	−0.232951	−0.116476	+	0.993194i	$0.537160\pi$
$720$	0	0
$721$	2.20529e9	0.219125
$722$	0	0
$723$	0	0
$724$	0	0
$725$	4.86744e9	0.474370
$726$	0	0
$727$	−1.20892e10	−1.16688	−0.583439	−	0.812157i	$-0.698293\pi$
$727$	−1.20892e10	−1.16688	−0.583439	+	0.812157i	$0.698293\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−4.98196e9	−0.471725
$732$	0	0
$733$	1.35185e10	1.26784	0.633919	−	0.773399i	$-0.281445\pi$
$733$	1.35185e10	1.26784	0.633919	+	0.773399i	$0.281445\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3.17635e8	0.0292275
$738$	0	0
$739$	−9.64611e9	−0.879218	−0.439609	−	0.898189i	$-0.644883\pi$
$739$	−9.64611e9	−0.879218	−0.439609	+	0.898189i	$0.644883\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1.26934e10	1.13532	0.567660	−	0.823263i	$-0.307849\pi$
$743$	1.26934e10	1.13532	0.567660	+	0.823263i	$0.307849\pi$
$744$	0	0
$745$	1.32077e9	0.117025
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1.38786e10	1.20686
$750$	0	0
$751$	1.83866e9	0.158403	0.0792013	−	0.996859i	$-0.474763\pi$
$751$	1.83866e9	0.158403	0.0792013	+	0.996859i	$0.474763\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−1.25529e9	−0.106153
$756$	0	0
$757$	−3.69951e9	−0.309962	−0.154981	−	0.987917i	$-0.549532\pi$
$757$	−3.69951e9	−0.309962	−0.154981	+	0.987917i	$0.549532\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−9.10481e9	−0.748901	−0.374451	−	0.927247i	$-0.622169\pi$
$761$	−9.10481e9	−0.748901	−0.374451	+	0.927247i	$0.622169\pi$
$762$	0	0
$763$	−8.33424e9	−0.679251
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2.34466e10	1.87627
$768$	0	0
$769$	−4.87498e9	−0.386572	−0.193286	−	0.981142i	$-0.561915\pi$
$769$	−4.87498e9	−0.386572	−0.193286	+	0.981142i	$0.561915\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−1.86703e9	−0.145386	−0.0726930	−	0.997354i	$-0.523159\pi$
$773$	−1.86703e9	−0.145386	−0.0726930	+	0.997354i	$0.523159\pi$
$774$	0	0
$775$	−1.23249e10	−0.951106
$776$	0	0
$777$	0	0
$778$	0	0
$779$	8.97603e8	0.0680304
$780$	0	0
$781$	−2.93874e10	−2.20741
$782$	0	0
$783$	0	0
$784$	0	0
$785$	2.83204e9	0.208956
$786$	0	0
$787$	6.76549e8	0.0494752	0.0247376	−	0.999694i	$-0.492125\pi$
$787$	6.76549e8	0.0494752	0.0247376	+	0.999694i	$0.492125\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	2.08343e10	1.49679
$792$	0	0
$793$	6.75924e9	0.481329
$794$	0	0
$795$	0	0
$796$	0	0
$797$	2.78422e10	1.94805	0.974025	−	0.226440i	$-0.0727088\pi$
$797$	2.78422e10	1.94805	0.974025	+	0.226440i	$0.0727088\pi$
$798$	0	0
$799$	6.78163e9	0.470349
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−4.84699e9	−0.330345
$804$	0	0
$805$	1.24574e10	0.841670
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−5.64823e9	−0.375053	−0.187527	−	0.982260i	$-0.560047\pi$
$809$	−5.64823e9	−0.375053	−0.187527	+	0.982260i	$0.560047\pi$
$810$	0	0
$811$	7.40101e9	0.487212	0.243606	−	0.969874i	$-0.421670\pi$
$811$	7.40101e9	0.487212	0.243606	+	0.969874i	$0.421670\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−3.47249e8	−0.0224693
$816$	0	0
$817$	1.56457e9	0.100373
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1.52865e10	0.964066	0.482033	−	0.876153i	$-0.339899\pi$
$821$	1.52865e10	0.964066	0.482033	+	0.876153i	$0.339899\pi$
$822$	0	0
$823$	2.97997e10	1.86343	0.931714	−	0.363194i	$-0.118314\pi$
$823$	2.97997e10	1.86343	0.931714	+	0.363194i	$0.118314\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	6.76407e9	0.415852	0.207926	−	0.978145i	$-0.433329\pi$
$827$	6.76407e9	0.415852	0.207926	+	0.978145i	$0.433329\pi$
$828$	0	0
$829$	−2.81440e10	−1.71571	−0.857857	−	0.513888i	$-0.828205\pi$
$829$	−2.81440e10	−1.71571	−0.857857	+	0.513888i	$0.828205\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−2.52028e10	−1.51075
$834$	0	0
$835$	1.22007e10	0.725241
$836$	0	0
$837$	0	0
$838$	0	0
$839$	8.02651e9	0.469202	0.234601	−	0.972092i	$-0.424622\pi$
$839$	8.02651e9	0.469202	0.234601	+	0.972092i	$0.424622\pi$
$840$	0	0
$841$	−1.16572e10	−0.675784
$842$	0	0
$843$	0	0
$844$	0	0
$845$	7.79790e9	0.444610
$846$	0	0
$847$	−2.25294e10	−1.27396
$848$	0	0
$849$	0	0
$850$	0	0
$851$	2.53378e9	0.140934
$852$	0	0
$853$	3.45404e10	1.90549	0.952743	−	0.303779i	$-0.0982483\pi$
$853$	3.45404e10	1.90549	0.952743	+	0.303779i	$0.0982483\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1.82602e10	0.990995	0.495498	−	0.868609i	$-0.334986\pi$
$857$	1.82602e10	0.990995	0.495498	+	0.868609i	$0.334986\pi$
$858$	0	0
$859$	2.17949e10	1.17322	0.586608	−	0.809871i	$-0.300463\pi$
$859$	2.17949e10	1.17322	0.586608	+	0.809871i	$0.300463\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−1.25884e10	−0.666702	−0.333351	−	0.942803i	$-0.608179\pi$
$863$	−1.25884e10	−0.666702	−0.333351	+	0.942803i	$0.608179\pi$
$864$	0	0
$865$	1.13609e10	0.596835
$866$	0	0
$867$	0	0
$868$	0	0
$869$	4.48958e10	2.32079
$870$	0	0
$871$	6.13291e8	0.0314487
$872$	0	0
$873$	0	0
$874$	0	0
$875$	2.35220e10	1.18699
$876$	0	0
$877$	1.83497e10	0.918607	0.459304	−	0.888279i	$-0.348099\pi$
$877$	1.83497e10	0.918607	0.459304	+	0.888279i	$0.348099\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1.25257e10	0.617146	0.308573	−	0.951201i	$-0.400149\pi$
$881$	1.25257e10	0.617146	0.308573	+	0.951201i	$0.400149\pi$
$882$	0	0
$883$	−2.79816e10	−1.36776	−0.683880	−	0.729594i	$-0.739709\pi$
$883$	−2.79816e10	−1.36776	−0.683880	+	0.729594i	$0.739709\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	2.74565e10	1.32103	0.660515	−	0.750813i	$-0.270338\pi$
$887$	2.74565e10	1.32103	0.660515	+	0.750813i	$0.270338\pi$
$888$	0	0
$889$	−2.38452e10	−1.13827
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−2.12975e9	−0.100080
$894$	0	0
$895$	−2.61686e9	−0.122011
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−1.41614e10	−0.650049
$900$	0	0
$901$	−3.34205e10	−1.52221
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1.55093e10	−0.695541
$906$	0	0
$907$	−3.16369e9	−0.140789	−0.0703944	−	0.997519i	$-0.522426\pi$
$907$	−3.16369e9	−0.140789	−0.0703944	+	0.997519i	$0.522426\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1.72740e10	0.756969	0.378485	−	0.925608i	$-0.376445\pi$
$911$	1.72740e10	0.756969	0.378485	+	0.925608i	$0.376445\pi$
$912$	0	0
$913$	−6.04085e9	−0.262694
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2.08249e10	0.891849
$918$	0	0
$919$	−9.42182e9	−0.400434	−0.200217	−	0.979752i	$-0.564165\pi$
$919$	−9.42182e9	−0.400434	−0.200217	+	0.979752i	$0.564165\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−5.67414e10	−2.37517
$924$	0	0
$925$	2.17435e9	0.0903303
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−2.32592e10	−0.951788	−0.475894	−	0.879503i	$-0.657875\pi$
$929$	−2.32592e10	−0.951788	−0.475894	+	0.879503i	$0.657875\pi$
$930$	0	0
$931$	7.91486e9	0.321454
$932$	0	0
$933$	0	0
$934$	0	0
$935$	1.36992e10	0.548093
$936$	0	0
$937$	2.57549e10	1.02275	0.511377	−	0.859357i	$-0.329136\pi$
$937$	2.57549e10	1.02275	0.511377	+	0.859357i	$0.329136\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−2.13825e10	−0.836557	−0.418278	−	0.908319i	$-0.637366\pi$
$941$	−2.13825e10	−0.836557	−0.418278	+	0.908319i	$0.637366\pi$
$942$	0	0
$943$	−1.07128e10	−0.416018
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−3.90679e10	−1.49484	−0.747420	−	0.664351i	$-0.768708\pi$
$947$	−3.90679e10	−1.49484	−0.747420	+	0.664351i	$0.768708\pi$
$948$	0	0
$949$	−9.35860e9	−0.355450
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−1.44692e10	−0.541527	−0.270763	−	0.962646i	$-0.587276\pi$
$953$	−1.44692e10	−0.541527	−0.270763	+	0.962646i	$0.587276\pi$
$954$	0	0
$955$	−1.27796e10	−0.474796
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−6.06918e10	−2.22211
$960$	0	0
$961$	8.34567e9	0.303340
$962$	0	0
$963$	0	0
$964$	0	0
$965$	1.17496e10	0.420900
$966$	0	0
$967$	−3.68538e10	−1.31066	−0.655329	−	0.755343i	$-0.727470\pi$
$967$	−3.68538e10	−1.31066	−0.655329	+	0.755343i	$0.727470\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1.36704e10	0.479198	0.239599	−	0.970872i	$-0.422984\pi$
$971$	1.36704e10	0.479198	0.239599	+	0.970872i	$0.422984\pi$
$972$	0	0
$973$	6.90337e10	2.40252
$974$	0	0
$975$	0	0
$976$	0	0
$977$	3.51417e10	1.20557	0.602785	−	0.797904i	$-0.294058\pi$
$977$	3.51417e10	1.20557	0.602785	+	0.797904i	$0.294058\pi$
$978$	0	0
$979$	−8.13905e9	−0.277226
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−3.29903e10	−1.10777	−0.553885	−	0.832593i	$-0.686855\pi$
$983$	−3.29903e10	−1.10777	−0.553885	+	0.832593i	$0.686855\pi$
$984$	0	0
$985$	4.31620e9	0.143905
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1.86730e10	−0.613799
$990$	0	0
$991$	2.08796e10	0.681496	0.340748	−	0.940155i	$-0.389320\pi$
$991$	2.08796e10	0.681496	0.340748	+	0.940155i	$0.389320\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	5.60521e6	0.000180389 0
$996$	0	0
$997$	−3.29887e10	−1.05422	−0.527111	−	0.849797i	$-0.676725\pi$
$997$	−3.29887e10	−1.05422	−0.527111	+	0.849797i	$0.676725\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	432.8.a.q.1.2		2
3.2	odd	2		432.8.a.j.1.1		2
4.3	odd	2		27.8.a.e.1.2	yes	2
12.11	even	2		27.8.a.b.1.1	✓	2
36.7	odd	6		81.8.c.d.28.1		4
36.11	even	6		81.8.c.h.28.2		4
36.23	even	6		81.8.c.h.55.2		4
36.31	odd	6		81.8.c.d.55.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.8.a.b.1.1	✓	2	12.11	even	2
27.8.a.e.1.2	yes	2	4.3	odd	2
81.8.c.d.28.1		4	36.7	odd	6
81.8.c.d.55.1		4	36.31	odd	6
81.8.c.h.28.2		4	36.11	even	6
81.8.c.h.55.2		4	36.23	even	6
432.8.a.j.1.1		2	3.2	odd	2
432.8.a.q.1.2		2	1.1	even	1	trivial

Overview	Random
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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:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	432.a (trivial)

\(f(q)\)	\(=\)	\(q+114.187 q^{5} -1438.40 q^{7} -5928.74 q^{11} -11447.2 q^{13} -20235.6 q^{17} +6354.94 q^{19} -75845.6 q^{23} -65086.4 q^{25} -74784.3 q^{29} +189363. q^{31} -164247. q^{35} -33407.2 q^{37} +141245. q^{41} +246197. q^{43} -335133. q^{47} +1.24547e6 q^{49} +1.65156e6 q^{53} -676983. q^{55} -2.04823e6 q^{59} -590469. q^{61} -1.30712e6 q^{65} -53575.5 q^{67} +4.95678e6 q^{71} +817542. q^{73} +8.52792e6 q^{77} -7.57257e6 q^{79} +1.01891e6 q^{83} -2.31064e6 q^{85} +1.37281e6 q^{89} +1.64658e7 q^{91} +725650. q^{95} -1.06023e7 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 180 q^{5} - 700 q^{7} - 10890 q^{11} - 5480 q^{13} + 16416 q^{17} - 16024 q^{19} - 24372 q^{23} - 138880 q^{25} - 143280 q^{29} + 38708 q^{31} - 115650 q^{35} + 455620 q^{37} + 731880 q^{41} + 1088840 q^{43}+ \cdots + 4098670 q^{97}+O(q^{100}) \) 2 * q + 180 * q^5 - 700 * q^7 - 10890 * q^11 - 5480 * q^13 + 16416 * q^17 - 16024 * q^19 - 24372 * q^23 - 138880 * q^25 - 143280 * q^29 + 38708 * q^31 - 115650 * q^35 + 455620 * q^37 + 731880 * q^41 + 1088840 * q^43 - 1561500 * q^47 + 967164 * q^49 + 2610468 * q^53 - 1003500 * q^55 - 1731960 * q^59 - 620192 * q^61 - 914400 * q^65 - 346600 * q^67 + 4242240 * q^71 - 3145190 * q^73 + 4864500 * q^77 - 10110616 * q^79 - 644202 * q^83 + 101520 * q^85 + 6021000 * q^89 + 20872000 * q^91 - 747180 * q^95 + 4098670 * q^97

Introduction

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