LMFDB - Embedded newform 432.9.e.k.161.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [432,9,Mod(161,432)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(432, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("432.161");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 432 = 2^{4} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	432.e (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$175.987559546$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.6171673600.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 40x^{3} + 225x^{2} + 150x + 50 $ x^6 - 2x^5 + 2x^4 + 40x^3 + 225x^2 + 150*x + 50
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{9}\cdot 3^{21} $
Twist minimal:	no (minimal twist has level 27)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.2
Root			$3.41249 + 3.41249i$ of defining polynomial
Character	$\chi$	$=$	432.161
Dual form			432.9.e.k.161.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-799.282i q^{5} +4073.62 q^{7} +O(q^{10})$	$q-799.282i q^{5} +4073.62 q^{7} +24976.4i q^{11} -13577.8 q^{13} -100991. i q^{17} -18469.5 q^{19} -19397.6i q^{23} -248226. q^{25} -147325. i q^{29} -537585. q^{31} -3.25597e6i q^{35} -539398. q^{37} -155285. i q^{41} +5.25537e6 q^{43} -6.53898e6i q^{47} +1.08295e7 q^{49} -6.49732e6i q^{53} +1.99632e7 q^{55} +1.01924e7i q^{59} +1.22909e7 q^{61} +1.08524e7i q^{65} -3.31500e7 q^{67} -2.04368e7i q^{71} +1.03423e7 q^{73} +1.01744e8i q^{77} +2.57943e7 q^{79} -2.24588e7i q^{83} -8.07199e7 q^{85} +7.34429e6i q^{89} -5.53105e7 q^{91} +1.47623e7i q^{95} -1.55238e7 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 1698 q^{7}+O(q^{10}) $ 6 * q + 1698 * q^7	$ 6 q + 1698 q^{7} + 41844 q^{13} + 36384 q^{19} - 1646688 q^{25} - 2058474 q^{31} - 9395880 q^{37} + 2737284 q^{43} + 28900656 q^{49} + 26674542 q^{55} - 40180776 q^{61} - 111355284 q^{67} + 12821718 q^{73} - 21820404 q^{79} - 83844396 q^{85} - 156632964 q^{91} + 53735106 q^{97}+O(q^{100}) $ 6 * q + 1698 * q^7 + 41844 * q^13 + 36384 * q^19 - 1646688 * q^25 - 2058474 * q^31 - 9395880 * q^37 + 2737284 * q^43 + 28900656 * q^49 + 26674542 * q^55 - 40180776 * q^61 - 111355284 * q^67 + 12821718 * q^73 - 21820404 * q^79 - 83844396 * q^85 - 156632964 * q^91 + 53735106 * q^97

Display 100 coefficients 10 coefficients

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
$5$	− 799.282i	− 1.27885i	−0.768853	−	0.639425i	$-0.779172\pi$
$5$	− 799.282i	− 1.27885i	0.768853	−	0.639425i	$-0.220828\pi$
$6$	0	0
$7$	4073.62	1.69663	0.848316	−	0.529490i	$-0.177616\pi$
$7$	4073.62	1.69663	0.848316	+	0.529490i	$0.177616\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	24976.4i	1.70592i	0.521976	+	0.852960i	$0.325195\pi$
$11$	24976.4i	1.70592i	−0.521976	+	0.852960i	$0.674805\pi$
$12$	0	0
$13$	−13577.8	−0.475395	−0.237697	−	0.971339i	$-0.576393\pi$
$13$	−13577.8	−0.475395	−0.237697	+	0.971339i	$0.576393\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 100991.i	− 1.20916i	−0.796543	−	0.604582i	$-0.793340\pi$
$17$	− 100991.i	− 1.20916i	0.796543	−	0.604582i	$-0.206660\pi$
$18$	0	0
$19$	−18469.5	−0.141723	−0.0708615	−	0.997486i	$-0.522575\pi$
$19$	−18469.5	−0.141723	−0.0708615	+	0.997486i	$0.522575\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 19397.6i	− 0.0693164i	−0.999399	−	0.0346582i	$-0.988966\pi$
$23$	− 19397.6i	− 0.0693164i	0.999399	−	0.0346582i	$-0.0110343\pi$
$24$	0	0
$25$	−248226.	−0.635459
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 147325.i	− 0.208298i	−0.994562	−	0.104149i	$-0.966788\pi$
$29$	− 147325.i	− 0.208298i	0.994562	−	0.104149i	$-0.0332118\pi$
$30$	0	0
$31$	−537585.	−0.582104	−0.291052	−	0.956707i	$-0.594005\pi$
$31$	−537585.	−0.582104	−0.291052	+	0.956707i	$0.594005\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 3.25597e6i	− 2.16974i
$36$	0	0
$37$	−539398.	−0.287808	−0.143904	−	0.989592i	$-0.545966\pi$
$37$	−539398.	−0.287808	−0.143904	+	0.989592i	$0.545966\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 155285.i	− 0.0549535i	−0.999622	−	0.0274767i	$-0.991253\pi$
$41$	− 155285.i	− 0.0549535i	0.999622	−	0.0274767i	$-0.00874722\pi$
$42$	0	0
$43$	5.25537e6	1.53720	0.768598	−	0.639732i	$-0.220955\pi$
$43$	5.25537e6	1.53720	0.768598	+	0.639732i	$0.220955\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 6.53898e6i	− 1.34004i	−0.742342	−	0.670021i	$-0.766285\pi$
$47$	− 6.53898e6i	− 1.34004i	0.742342	−	0.670021i	$-0.233715\pi$
$48$	0	0
$49$	1.08295e7	1.87856
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 6.49732e6i	− 0.823438i	−0.911311	−	0.411719i	$-0.864929\pi$
$53$	− 6.49732e6i	− 0.823438i	0.911311	−	0.411719i	$-0.135071\pi$
$54$	0	0
$55$	1.99632e7	2.18162
$56$	0	0
$57$	0	0
$58$	0	0
$59$	1.01924e7i	0.841140i	0.907260	+	0.420570i	$0.138170\pi$
$59$	1.01924e7i	0.841140i	−0.907260	+	0.420570i	$0.861830\pi$
$60$	0	0
$61$	1.22909e7	0.887693	0.443847	−	0.896103i	$-0.353613\pi$
$61$	1.22909e7	0.887693	0.443847	+	0.896103i	$0.353613\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.08524e7i	0.607959i
$66$	0	0
$67$	−3.31500e7	−1.64507	−0.822536	−	0.568713i	$-0.807441\pi$
$67$	−3.31500e7	−1.64507	−0.822536	+	0.568713i	$0.807441\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 2.04368e7i	− 0.804231i	−0.915589	−	0.402115i	$-0.868275\pi$
$71$	− 2.04368e7i	− 0.804231i	0.915589	−	0.402115i	$-0.131725\pi$
$72$	0	0
$73$	1.03423e7	0.364189	0.182094	−	0.983281i	$-0.441712\pi$
$73$	1.03423e7	0.364189	0.182094	+	0.983281i	$0.441712\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.01744e8i	2.89432i
$78$	0	0
$79$	2.57943e7	0.662239	0.331120	−	0.943589i	$-0.392574\pi$
$79$	2.57943e7	0.662239	0.331120	+	0.943589i	$0.392574\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 2.24588e7i	− 0.473233i	−0.971603	−	0.236616i	$-0.923962\pi$
$83$	− 2.24588e7i	− 0.473233i	0.971603	−	0.236616i	$-0.0760385\pi$
$84$	0	0
$85$	−8.07199e7	−1.54634
$86$	0	0
$87$	0	0
$88$	0	0
$89$	7.34429e6i	0.117055i	0.998286	+	0.0585275i	$0.0186405\pi$
$89$	7.34429e6i	0.117055i	−0.998286	+	0.0585275i	$0.981359\pi$
$90$	0	0
$91$	−5.53105e7	−0.806570
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.47623e7i	0.181243i
$96$	0	0
$97$	−1.55238e7	−0.175352	−0.0876762	−	0.996149i	$-0.527944\pi$
$97$	−1.55238e7	−0.175352	−0.0876762	+	0.996149i	$0.527944\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 1.15333e8i	− 1.10833i	−0.832407	−	0.554164i	$-0.813038\pi$
$101$	− 1.15333e8i	− 1.10833i	0.832407	−	0.554164i	$-0.186962\pi$
$102$	0	0
$103$	3.92352e6	0.0348600	0.0174300	−	0.999848i	$-0.494452\pi$
$103$	3.92352e6	0.0348600	0.0174300	+	0.999848i	$0.494452\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.72844e8i	1.31862i	0.751872	+	0.659309i	$0.229151\pi$
$107$	1.72844e8i	1.31862i	−0.751872	+	0.659309i	$0.770849\pi$
$108$	0	0
$109$	2.05642e8	1.45682	0.728409	−	0.685142i	$-0.240260\pi$
$109$	2.05642e8	1.45682	0.728409	+	0.685142i	$0.240260\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 1.28415e8i	− 0.787595i	−0.919197	−	0.393797i	$-0.871161\pi$
$113$	− 1.28415e8i	− 0.787595i	0.919197	−	0.393797i	$-0.128839\pi$
$114$	0	0
$115$	−1.55041e7	−0.0886454
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 4.11397e8i	− 2.05151i
$120$	0	0
$121$	−4.09461e8	−1.91017
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 1.13817e8i	− 0.466194i
$126$	0	0
$127$	1.68860e8	0.649101	0.324550	−	0.945868i	$-0.394787\pi$
$127$	1.68860e8	0.649101	0.324550	+	0.945868i	$0.394787\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	7.46165e7i	0.253367i	0.991943	+	0.126683i	$0.0404332\pi$
$131$	7.46165e7i	0.253367i	−0.991943	+	0.126683i	$0.959567\pi$
$132$	0	0
$133$	−7.52376e7	−0.240452
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 2.90045e8i	− 0.823347i	−0.911331	−	0.411674i	$-0.864944\pi$
$137$	− 2.90045e8i	− 0.823347i	0.911331	−	0.411674i	$-0.135056\pi$
$138$	0	0
$139$	−6.95071e8	−1.86196	−0.930979	−	0.365072i	$-0.881044\pi$
$139$	−6.95071e8	−1.86196	−0.930979	+	0.365072i	$0.881044\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 3.39123e8i	− 0.810986i
$144$	0	0
$145$	−1.17754e8	−0.266382
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 3.43900e8i	− 0.697730i	−0.937173	−	0.348865i	$-0.886567\pi$
$149$	− 3.43900e8i	− 0.697730i	0.937173	−	0.348865i	$-0.113433\pi$
$150$	0	0
$151$	3.63618e8	0.699419	0.349710	−	0.936858i	$-0.386280\pi$
$151$	3.63618e8	0.699419	0.349710	+	0.936858i	$0.386280\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	4.29682e8i	0.744424i
$156$	0	0
$157$	9.40287e8	1.54761	0.773806	−	0.633423i	$-0.218351\pi$
$157$	9.40287e8	1.54761	0.773806	+	0.633423i	$0.218351\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 7.90183e7i	− 0.117605i
$162$	0	0
$163$	3.84041e8	0.544035	0.272018	−	0.962292i	$-0.412309\pi$
$163$	3.84041e8	0.544035	0.272018	+	0.962292i	$0.412309\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 1.35165e8i	− 0.173780i	−0.996218	−	0.0868899i	$-0.972307\pi$
$167$	− 1.35165e8i	− 0.173780i	0.996218	−	0.0868899i	$-0.0276929\pi$
$168$	0	0
$169$	−6.31375e8	−0.774000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 1.16873e9i	− 1.30475i	−0.757895	−	0.652377i	$-0.773772\pi$
$173$	− 1.16873e9i	− 1.30475i	0.757895	−	0.652377i	$-0.226228\pi$
$174$	0	0
$175$	−1.01118e9	−1.07814
$176$	0	0
$177$	0	0
$178$	0	0
$179$	1.63803e9i	1.59555i	0.602957	+	0.797773i	$0.293989\pi$
$179$	1.63803e9i	1.59555i	−0.602957	+	0.797773i	$0.706011\pi$
$180$	0	0
$181$	1.06719e9	0.994327	0.497163	−	0.867657i	$-0.334375\pi$
$181$	1.06719e9	0.994327	0.497163	+	0.867657i	$0.334375\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	4.31131e8i	0.368063i
$186$	0	0
$187$	2.52238e9	2.06274
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 2.24410e9i	− 1.68620i	−0.537759	−	0.843099i	$-0.680729\pi$
$191$	− 2.24410e9i	− 1.68620i	0.537759	−	0.843099i	$-0.319271\pi$
$192$	0	0
$193$	1.06657e9	0.768704	0.384352	−	0.923187i	$-0.374425\pi$
$193$	1.06657e9	0.768704	0.384352	+	0.923187i	$0.374425\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1.28343e9i	0.852133i	0.904692	+	0.426067i	$0.140101\pi$
$197$	1.28343e9i	0.852133i	−0.904692	+	0.426067i	$0.859899\pi$
$198$	0	0
$199$	−4.60815e8	−0.293843	−0.146921	−	0.989148i	$-0.546936\pi$
$199$	−4.60815e8	−0.293843	−0.146921	+	0.989148i	$0.546936\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 6.00145e8i	− 0.353405i
$204$	0	0
$205$	−1.24117e8	−0.0702773
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 4.61301e8i	− 0.241768i
$210$	0	0
$211$	3.00503e9	1.51607	0.758036	−	0.652213i	$-0.226159\pi$
$211$	3.00503e9	1.51607	0.758036	+	0.652213i	$0.226159\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 4.20052e9i	− 1.96584i
$216$	0	0
$217$	−2.18992e9	−0.987617
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1.37122e9i	0.574830i
$222$	0	0
$223$	−1.65731e9	−0.670170	−0.335085	−	0.942188i	$-0.608765\pi$
$223$	−1.65731e9	−0.670170	−0.335085	+	0.942188i	$0.608765\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 3.86497e9i	− 1.45560i	−0.685789	−	0.727800i	$-0.740543\pi$
$227$	− 3.86497e9i	− 1.45560i	0.685789	−	0.727800i	$-0.259457\pi$
$228$	0	0
$229$	2.11385e9	0.768656	0.384328	−	0.923197i	$-0.374433\pi$
$229$	2.11385e9	0.768656	0.384328	+	0.923197i	$0.374433\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 3.35951e9i	− 1.13986i	−0.821693	−	0.569930i	$-0.806970\pi$
$233$	− 3.35951e9i	− 1.13986i	0.821693	−	0.569930i	$-0.193030\pi$
$234$	0	0
$235$	−5.22649e9	−1.71371
$236$	0	0
$237$	0	0
$238$	0	0
$239$	5.99631e8i	0.183778i	0.995769	+	0.0918888i	$0.0292904\pi$
$239$	5.99631e8i	0.183778i	−0.995769	+	0.0918888i	$0.970710\pi$
$240$	0	0
$241$	−4.04195e9	−1.19818	−0.599091	−	0.800681i	$-0.704471\pi$
$241$	−4.04195e9	−1.19818	−0.599091	+	0.800681i	$0.704471\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 8.65586e9i	− 2.40240i
$246$	0	0
$247$	2.50774e8	0.0673744
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 5.86397e9i	− 1.47740i	−0.674037	−	0.738698i	$-0.735441\pi$
$251$	− 5.86397e9i	− 1.47740i	0.674037	−	0.738698i	$-0.264559\pi$
$252$	0	0
$253$	4.84482e8	0.118248
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 6.71902e9i	− 1.54019i	−0.637931	−	0.770094i	$-0.720209\pi$
$257$	− 6.71902e9i	− 1.54019i	0.637931	−	0.770094i	$-0.279791\pi$
$258$	0	0
$259$	−2.19730e9	−0.488304
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 2.13401e9i	− 0.446040i	−0.974814	−	0.223020i	$-0.928408\pi$
$263$	− 2.13401e9i	− 0.446040i	0.974814	−	0.223020i	$-0.0715915\pi$
$264$	0	0
$265$	−5.19319e9	−1.05305
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 2.66584e9i	− 0.509125i	−0.967056	−	0.254562i	$-0.918069\pi$
$269$	− 2.66584e9i	− 0.509125i	0.967056	−	0.254562i	$-0.0819314\pi$
$270$	0	0
$271$	−4.61166e9	−0.855028	−0.427514	−	0.904009i	$-0.640611\pi$
$271$	−4.61166e9	−0.855028	−0.427514	+	0.904009i	$0.640611\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 6.19979e9i	− 1.08404i
$276$	0	0
$277$	−3.18011e9	−0.540160	−0.270080	−	0.962838i	$-0.587050\pi$
$277$	−3.18011e9	−0.540160	−0.270080	+	0.962838i	$0.587050\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	7.42091e9i	1.19023i	0.803640	+	0.595116i	$0.202894\pi$
$281$	7.42091e9i	1.19023i	−0.803640	+	0.595116i	$0.797106\pi$
$282$	0	0
$283$	−1.31058e9	−0.204323	−0.102162	−	0.994768i	$-0.532576\pi$
$283$	−1.31058e9	−0.204323	−0.102162	+	0.994768i	$0.532576\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 6.32573e8i	− 0.0932359i
$288$	0	0
$289$	−3.22333e9	−0.462076
$290$	0	0
$291$	0	0
$292$	0	0
$293$	3.53913e9i	0.480204i	0.970748	+	0.240102i	$0.0771809\pi$
$293$	3.53913e9i	0.480204i	−0.970748	+	0.240102i	$0.922819\pi$
$294$	0	0
$295$	8.14659e9	1.07569
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2.63376e8i	0.0329527i
$300$	0	0
$301$	2.14084e10	2.60806
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 9.82386e9i	− 1.13523i
$306$	0	0
$307$	5.34860e9	0.602125	0.301063	−	0.953604i	$-0.402659\pi$
$307$	5.34860e9	0.602125	0.301063	+	0.953604i	$0.402659\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	1.98105e9i	0.211765i	0.994379	+	0.105882i	$0.0337667\pi$
$311$	1.98105e9i	0.211765i	−0.994379	+	0.105882i	$0.966233\pi$
$312$	0	0
$313$	5.46022e9	0.568896	0.284448	−	0.958691i	$-0.408190\pi$
$313$	5.46022e9	0.568896	0.284448	+	0.958691i	$0.408190\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 8.79043e8i	− 0.0870508i	−0.999052	−	0.0435254i	$-0.986141\pi$
$317$	− 8.79043e8i	− 0.0870508i	0.999052	−	0.0435254i	$-0.0138590\pi$
$318$	0	0
$319$	3.67965e9	0.355339
$320$	0	0
$321$	0	0
$322$	0	0
$323$	1.86524e9i	0.171366i
$324$	0	0
$325$	3.37035e9	0.302094
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 2.66373e10i	− 2.27356i
$330$	0	0
$331$	−4.89577e8	−0.0407858	−0.0203929	−	0.999792i	$-0.506492\pi$
$331$	−4.89577e8	−0.0407858	−0.0203929	+	0.999792i	$0.506492\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	2.64962e10i	2.10380i
$336$	0	0
$337$	−2.24281e9	−0.173889	−0.0869447	−	0.996213i	$-0.527710\pi$
$337$	−2.24281e9	−0.173889	−0.0869447	+	0.996213i	$0.527710\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 1.34269e10i	− 0.993023i
$342$	0	0
$343$	2.06318e10	1.49060
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 2.81689e10i	− 1.94290i	−0.237235	−	0.971452i	$-0.576241\pi$
$347$	− 2.81689e10i	− 1.94290i	0.237235	−	0.971452i	$-0.423759\pi$
$348$	0	0
$349$	−1.06312e10	−0.716608	−0.358304	−	0.933605i	$-0.616645\pi$
$349$	−1.06312e10	−0.716608	−0.358304	+	0.933605i	$0.616645\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	1.08514e10i	0.698855i	0.936963	+	0.349428i	$0.113624\pi$
$353$	1.08514e10i	0.698855i	−0.936963	+	0.349428i	$0.886376\pi$
$354$	0	0
$355$	−1.63348e10	−1.02849
$356$	0	0
$357$	0	0
$358$	0	0
$359$	5.67830e9i	0.341854i	0.985284	+	0.170927i	$0.0546762\pi$
$359$	5.67830e9i	0.341854i	−0.985284	+	0.170927i	$0.945324\pi$
$360$	0	0
$361$	−1.66424e10	−0.979915
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 8.26642e9i	− 0.465743i
$366$	0	0
$367$	1.14936e9	0.0633563	0.0316782	−	0.999498i	$-0.489915\pi$
$367$	1.14936e9	0.0633563	0.0316782	+	0.999498i	$0.489915\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 2.64676e10i	− 1.39707i
$372$	0	0
$373$	−3.70986e10	−1.91656	−0.958280	−	0.285830i	$-0.907731\pi$
$373$	−3.70986e10	−1.91656	−0.958280	+	0.285830i	$0.907731\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.00034e9i	0.0990236i
$378$	0	0
$379$	−1.48862e10	−0.721482	−0.360741	−	0.932666i	$-0.617476\pi$
$379$	−1.48862e10	−0.721482	−0.360741	+	0.932666i	$0.617476\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 3.75932e10i	− 1.74709i	−0.486746	−	0.873543i	$-0.661816\pi$
$383$	− 3.75932e10i	− 1.74709i	0.486746	−	0.873543i	$-0.338184\pi$
$384$	0	0
$385$	8.13223e10	3.70140
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 2.12564e10i	− 0.928305i	−0.885755	−	0.464152i	$-0.846359\pi$
$389$	− 2.12564e10i	− 0.928305i	0.885755	−	0.464152i	$-0.153641\pi$
$390$	0	0
$391$	−1.95897e9	−0.0838149
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 2.06169e10i	− 0.846905i
$396$	0	0
$397$	2.52477e10	1.01639	0.508194	−	0.861243i	$-0.330313\pi$
$397$	2.52477e10	1.01639	0.508194	+	0.861243i	$0.330313\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 1.18250e10i	− 0.457325i	−0.973506	−	0.228662i	$-0.926565\pi$
$401$	− 1.18250e10i	− 0.457325i	0.973506	−	0.228662i	$-0.0734352\pi$
$402$	0	0
$403$	7.29920e9	0.276729
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 1.34722e10i	− 0.490977i
$408$	0	0
$409$	−4.70183e10	−1.68025	−0.840125	−	0.542393i	$-0.817518\pi$
$409$	−4.70183e10	−1.68025	−0.840125	+	0.542393i	$0.817518\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	4.15199e10i	1.42711i
$414$	0	0
$415$	−1.79509e10	−0.605194
$416$	0	0
$417$	0	0
$418$	0	0
$419$	4.39477e10i	1.42587i	0.701231	+	0.712935i	$0.252634\pi$
$419$	4.39477e10i	1.42587i	−0.701231	+	0.712935i	$0.747366\pi$
$420$	0	0
$421$	−3.94543e10	−1.25593	−0.627966	−	0.778241i	$-0.716112\pi$
$421$	−3.94543e10	−1.25593	−0.627966	+	0.778241i	$0.716112\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	2.50685e10i	0.768374i
$426$	0	0
$427$	5.00682e10	1.50609
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 2.68297e10i	− 0.777511i	−0.921341	−	0.388755i	$-0.872905\pi$
$431$	− 2.68297e10i	− 0.777511i	0.921341	−	0.388755i	$-0.127095\pi$
$432$	0	0
$433$	3.42328e10	0.973846	0.486923	−	0.873445i	$-0.338119\pi$
$433$	3.42328e10	0.973846	0.486923	+	0.873445i	$0.338119\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3.58263e8i	0.00982373i
$438$	0	0
$439$	−5.21052e10	−1.40289	−0.701444	−	0.712724i	$-0.747461\pi$
$439$	−5.21052e10	−1.40289	−0.701444	+	0.712724i	$0.747461\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	1.08565e10i	0.281887i	0.990018	+	0.140943i	$0.0450136\pi$
$443$	1.08565e10i	0.281887i	−0.990018	+	0.140943i	$0.954986\pi$
$444$	0	0
$445$	5.87016e9	0.149696
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 4.63261e10i	− 1.13983i	−0.821704	−	0.569915i	$-0.806976\pi$
$449$	− 4.63261e10i	− 1.13983i	0.821704	−	0.569915i	$-0.193024\pi$
$450$	0	0
$451$	3.87847e9	0.0937463
$452$	0	0
$453$	0	0
$454$	0	0
$455$	4.42087e10i	1.03148i
$456$	0	0
$457$	3.90812e10	0.895990	0.447995	−	0.894036i	$-0.352138\pi$
$457$	3.90812e10	0.895990	0.447995	+	0.894036i	$0.352138\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	3.82980e10i	0.847955i	0.905673	+	0.423977i	$0.139366\pi$
$461$	3.82980e10i	0.847955i	−0.905673	+	0.423977i	$0.860634\pi$
$462$	0	0
$463$	−1.82334e10	−0.396774	−0.198387	−	0.980124i	$-0.563570\pi$
$463$	−1.82334e10	−0.396774	−0.198387	+	0.980124i	$0.563570\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	4.62840e10i	0.973114i	0.873649	+	0.486557i	$0.161747\pi$
$467$	4.62840e10i	0.973114i	−0.873649	+	0.486557i	$0.838253\pi$
$468$	0	0
$469$	−1.35041e11	−2.79108
$470$	0	0
$471$	0	0
$472$	0	0
$473$	1.31260e11i	2.62234i
$474$	0	0
$475$	4.58461e9	0.0900591
$476$	0	0
$477$	0	0
$478$	0	0
$479$	3.28490e10i	0.623994i	0.950083	+	0.311997i	$0.100998\pi$
$479$	3.28490e10i	0.623994i	−0.950083	+	0.311997i	$0.899002\pi$
$480$	0	0
$481$	7.32381e9	0.136822
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1.24079e10i	0.224250i
$486$	0	0
$487$	−5.84968e10	−1.03996	−0.519980	−	0.854179i	$-0.674060\pi$
$487$	−5.84968e10	−1.03996	−0.519980	+	0.854179i	$0.674060\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	4.07576e10i	0.701266i	0.936513	+	0.350633i	$0.114034\pi$
$491$	4.07576e10i	0.701266i	−0.936513	+	0.350633i	$0.885966\pi$
$492$	0	0
$493$	−1.48784e10	−0.251866
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 8.32519e10i	− 1.36448i
$498$	0	0
$499$	−3.26104e10	−0.525962	−0.262981	−	0.964801i	$-0.584706\pi$
$499$	−3.26104e10	−0.525962	−0.262981	+	0.964801i	$0.584706\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 9.61730e10i	− 1.50238i	−0.660083	−	0.751192i	$-0.729479\pi$
$503$	− 9.61730e10i	− 1.50238i	0.660083	−	0.751192i	$-0.270521\pi$
$504$	0	0
$505$	−9.21837e10	−1.41739
$506$	0	0
$507$	0	0
$508$	0	0
$509$	1.12103e11i	1.67011i	0.550167	+	0.835055i	$0.314564\pi$
$509$	1.12103e11i	1.67011i	−0.550167	+	0.835055i	$0.685436\pi$
$510$	0	0
$511$	4.21306e10	0.617895
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 3.13600e9i	− 0.0445807i
$516$	0	0
$517$	1.63320e11	2.28601
$518$	0	0
$519$	0	0
$520$	0	0
$521$	8.96027e10i	1.21610i	0.793898	+	0.608051i	$0.208049\pi$
$521$	8.96027e10i	1.21610i	−0.793898	+	0.608051i	$0.791951\pi$
$522$	0	0
$523$	7.63047e10	1.01987	0.509935	−	0.860213i	$-0.329670\pi$
$523$	7.63047e10	1.01987	0.509935	+	0.860213i	$0.329670\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5.42910e10i	0.703859i
$528$	0	0
$529$	7.79347e10	0.995195
$530$	0	0
$531$	0	0
$532$	0	0
$533$	2.10843e9i	0.0261246i
$534$	0	0
$535$	1.38151e11	1.68632
$536$	0	0
$537$	0	0
$538$	0	0
$539$	2.70483e11i	3.20468i
$540$	0	0
$541$	5.33718e10	0.623050	0.311525	−	0.950238i	$-0.399160\pi$
$541$	5.33718e10	0.623050	0.311525	+	0.950238i	$0.399160\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 1.64366e11i	− 1.86305i
$546$	0	0
$547$	−2.29943e9	−0.0256844	−0.0128422	−	0.999918i	$-0.504088\pi$
$547$	−2.29943e9	−0.0256844	−0.0128422	+	0.999918i	$0.504088\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	2.72102e9i	0.0295206i
$552$	0	0
$553$	1.05076e11	1.12358
$554$	0	0
$555$	0	0
$556$	0	0
$557$	8.58190e9i	0.0891585i	0.999006	+	0.0445793i	$0.0141947\pi$
$557$	8.58190e9i	0.0891585i	−0.999006	+	0.0445793i	$0.985805\pi$
$558$	0	0
$559$	−7.13561e10	−0.730775
$560$	0	0
$561$	0	0
$562$	0	0
$563$	3.97634e10i	0.395776i	0.980225	+	0.197888i	$0.0634083\pi$
$563$	3.97634e10i	0.395776i	−0.980225	+	0.197888i	$0.936592\pi$
$564$	0	0
$565$	−1.02640e11	−1.00722
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 1.60448e11i	− 1.53068i	−0.643625	−	0.765341i	$-0.722570\pi$
$569$	− 1.60448e11i	− 1.53068i	0.643625	−	0.765341i	$-0.277430\pi$
$570$	0	0
$571$	1.12801e11	1.06113	0.530566	−	0.847643i	$-0.321979\pi$
$571$	1.12801e11	1.06113	0.530566	+	0.847643i	$0.321979\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	4.81499e9i	0.0440478i
$576$	0	0
$577$	−9.32981e10	−0.841724	−0.420862	−	0.907125i	$-0.638272\pi$
$577$	−9.32981e10	−0.841724	−0.420862	+	0.907125i	$0.638272\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 9.14887e10i	− 0.802903i
$582$	0	0
$583$	1.62280e11	1.40472
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 5.67306e10i	− 0.477821i	−0.971042	−	0.238910i	$-0.923210\pi$
$587$	− 5.67306e10i	− 0.477821i	0.971042	−	0.238910i	$-0.0767902\pi$
$588$	0	0
$589$	9.92892e9	0.0824975
$590$	0	0
$591$	0	0
$592$	0	0
$593$	1.06286e11i	0.859521i	0.902943	+	0.429761i	$0.141402\pi$
$593$	1.06286e11i	0.859521i	−0.902943	+	0.429761i	$0.858598\pi$
$594$	0	0
$595$	−3.28822e11	−2.62357
$596$	0	0
$597$	0	0
$598$	0	0
$599$	3.06254e10i	0.237889i	0.992901	+	0.118945i	$0.0379511\pi$
$599$	3.06254e10i	0.237889i	−0.992901	+	0.118945i	$0.962049\pi$
$600$	0	0
$601$	2.15827e11	1.65428	0.827139	−	0.561997i	$-0.189967\pi$
$601$	2.15827e11	1.65428	0.827139	+	0.561997i	$0.189967\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	3.27275e11i	2.44282i
$606$	0	0
$607$	7.82220e9	0.0576201	0.0288100	−	0.999585i	$-0.490828\pi$
$607$	7.82220e9	0.0576201	0.0288100	+	0.999585i	$0.490828\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	8.87846e10i	0.637049i
$612$	0	0
$613$	8.21846e10	0.582034	0.291017	−	0.956718i	$-0.406006\pi$
$613$	8.21846e10	0.582034	0.291017	+	0.956718i	$0.406006\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1.85390e11i	1.27922i	0.768699	+	0.639610i	$0.220904\pi$
$617$	1.85390e11i	1.27922i	−0.768699	+	0.639610i	$0.779096\pi$
$618$	0	0
$619$	−5.59724e9	−0.0381251	−0.0190626	−	0.999818i	$-0.506068\pi$
$619$	−5.59724e9	−0.0381251	−0.0190626	+	0.999818i	$0.506068\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	2.99178e10i	0.198599i
$624$	0	0
$625$	−1.87935e11	−1.23165
$626$	0	0
$627$	0	0
$628$	0	0
$629$	5.44741e10i	0.348006i
$630$	0	0
$631$	−1.47220e11	−0.928646	−0.464323	−	0.885666i	$-0.653702\pi$
$631$	−1.47220e11	−0.928646	−0.464323	+	0.885666i	$0.653702\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 1.34967e11i	− 0.830103i
$636$	0	0
$637$	−1.47041e11	−0.893059
$638$	0	0
$639$	0	0
$640$	0	0
$641$	6.78345e10i	0.401808i	0.979611	+	0.200904i	$0.0643879\pi$
$641$	6.78345e10i	0.401808i	−0.979611	+	0.200904i	$0.935612\pi$
$642$	0	0
$643$	−8.05136e10	−0.471005	−0.235502	−	0.971874i	$-0.575674\pi$
$643$	−8.05136e10	−0.471005	−0.235502	+	0.971874i	$0.575674\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 3.52374e10i	− 0.201088i	−0.994933	−	0.100544i	$-0.967942\pi$
$647$	− 3.52374e10i	− 0.201088i	0.994933	−	0.100544i	$-0.0320584\pi$
$648$	0	0
$649$	−2.54569e11	−1.43492
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1.55714e11i	0.856395i	0.903685	+	0.428197i	$0.140851\pi$
$653$	1.55714e11i	0.856395i	−0.903685	+	0.428197i	$0.859149\pi$
$654$	0	0
$655$	5.96396e10	0.324018
$656$	0	0
$657$	0	0
$658$	0	0
$659$	3.96051e10i	0.209995i	0.994472	+	0.104998i	$0.0334835\pi$
$659$	3.96051e10i	0.209995i	−0.994472	+	0.104998i	$0.966517\pi$
$660$	0	0
$661$	3.14848e10	0.164928	0.0824642	−	0.996594i	$-0.473721\pi$
$661$	3.14848e10	0.164928	0.0824642	+	0.996594i	$0.473721\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	6.01360e10i	0.307502i
$666$	0	0
$667$	−2.85775e9	−0.0144385
$668$	0	0
$669$	0	0
$670$	0	0
$671$	3.06981e11i	1.51433i
$672$	0	0
$673$	−9.28635e10	−0.452673	−0.226337	−	0.974049i	$-0.572675\pi$
$673$	−9.28635e10	−0.452673	−0.226337	+	0.974049i	$0.572675\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 1.67632e10i	− 0.0798001i	−0.999204	−	0.0399000i	$-0.987296\pi$
$677$	− 1.67632e10i	− 0.0798001i	0.999204	−	0.0399000i	$-0.0127040\pi$
$678$	0	0
$679$	−6.32381e10	−0.297509
$680$	0	0
$681$	0	0
$682$	0	0
$683$	2.34709e11i	1.07857i	0.842124	+	0.539283i	$0.181305\pi$
$683$	2.34709e11i	1.07857i	−0.842124	+	0.539283i	$0.818695\pi$
$684$	0	0
$685$	−2.31828e11	−1.05294
$686$	0	0
$687$	0	0
$688$	0	0
$689$	8.82190e10i	0.391458i
$690$	0	0
$691$	1.45892e11	0.639909	0.319955	−	0.947433i	$-0.396332\pi$
$691$	1.45892e11	0.639909	0.319955	+	0.947433i	$0.396332\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	5.55558e11i	2.38117i
$696$	0	0
$697$	−1.56824e10	−0.0664477
$698$	0	0
$699$	0	0
$700$	0	0
$701$	1.84146e11i	0.762590i	0.924453	+	0.381295i	$0.124522\pi$
$701$	1.84146e11i	0.762590i	−0.924453	+	0.381295i	$0.875478\pi$
$702$	0	0
$703$	9.96240e9	0.0407890
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 4.69823e11i	− 1.88043i
$708$	0	0
$709$	3.69970e11	1.46414	0.732068	−	0.681232i	$-0.238555\pi$
$709$	3.69970e11	1.46414	0.732068	+	0.681232i	$0.238555\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	1.04279e10i	0.0403494i
$714$	0	0
$715$	−2.71055e11	−1.03713
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 3.88643e11i	− 1.45424i	−0.686511	−	0.727120i	$-0.740859\pi$
$719$	− 3.88643e11i	− 1.45424i	0.686511	−	0.727120i	$-0.259141\pi$
$720$	0	0
$721$	1.59829e10	0.0591446
$722$	0	0
$723$	0	0
$724$	0	0
$725$	3.65699e10i	0.132365i
$726$	0	0
$727$	4.71080e11	1.68639	0.843193	−	0.537611i	$-0.180673\pi$
$727$	4.71080e11	1.68639	0.843193	+	0.537611i	$0.180673\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 5.30742e11i	− 1.85872i
$732$	0	0
$733$	4.73027e11	1.63859	0.819294	−	0.573373i	$-0.194365\pi$
$733$	4.73027e11	1.63859	0.819294	+	0.573373i	$0.194365\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 8.27968e11i	− 2.80636i
$738$	0	0
$739$	−2.68388e11	−0.899883	−0.449941	−	0.893058i	$-0.648555\pi$
$739$	−2.68388e11	−0.899883	−0.449941	+	0.893058i	$0.648555\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	2.56982e11i	0.843231i	0.906775	+	0.421616i	$0.138537\pi$
$743$	2.56982e11i	0.843231i	−0.906775	+	0.421616i	$0.861463\pi$
$744$	0	0
$745$	−2.74873e11	−0.892293
$746$	0	0
$747$	0	0
$748$	0	0
$749$	7.04100e11i	2.23721i
$750$	0	0
$751$	−4.51336e11	−1.41886	−0.709431	−	0.704775i	$-0.751048\pi$
$751$	−4.51336e11	−1.41886	−0.709431	+	0.704775i	$0.751048\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 2.90633e11i	− 0.894452i
$756$	0	0
$757$	1.49853e11	0.456334	0.228167	−	0.973622i	$-0.426727\pi$
$757$	1.49853e11	0.456334	0.228167	+	0.973622i	$0.426727\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 7.38110e10i	− 0.220081i	−0.993927	−	0.110040i	$-0.964902\pi$
$761$	− 7.38110e10i	− 0.220081i	0.993927	−	0.110040i	$-0.0350980\pi$
$762$	0	0
$763$	8.37706e11	2.47169
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 1.38390e11i	− 0.399873i
$768$	0	0
$769$	−5.20453e10	−0.148825	−0.0744126	−	0.997228i	$-0.523708\pi$
$769$	−5.20453e10	−0.148825	−0.0744126	+	0.997228i	$0.523708\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 2.09052e11i	− 0.585512i	−0.956187	−	0.292756i	$-0.905428\pi$
$773$	− 2.09052e11i	− 0.585512i	0.956187	−	0.292756i	$-0.0945724\pi$
$774$	0	0
$775$	1.33443e11	0.369903
$776$	0	0
$777$	0	0
$778$	0	0
$779$	2.86804e9i	0.00778817i
$780$	0	0
$781$	5.10439e11	1.37195
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 7.51555e11i	− 1.97916i
$786$	0	0
$787$	3.37714e11	0.880340	0.440170	−	0.897914i	$-0.354918\pi$
$787$	3.37714e11	0.880340	0.440170	+	0.897914i	$0.354918\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 5.23114e11i	− 1.33626i
$792$	0	0
$793$	−1.66882e11	−0.422005
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 5.49760e11i	− 1.36251i	−0.732046	−	0.681255i	$-0.761434\pi$
$797$	− 5.49760e11i	− 1.36251i	0.732046	−	0.681255i	$-0.238566\pi$
$798$	0	0
$799$	−6.60375e11	−1.62033
$800$	0	0
$801$	0	0
$802$	0	0
$803$	2.58314e11i	0.621277i
$804$	0	0
$805$	−6.31579e10	−0.150399
$806$	0	0
$807$	0	0
$808$	0	0
$809$	2.45301e11i	0.572671i	0.958129	+	0.286336i	$0.0924372\pi$
$809$	2.45301e11i	0.572671i	−0.958129	+	0.286336i	$0.907563\pi$
$810$	0	0
$811$	3.09248e11	0.714864	0.357432	−	0.933939i	$-0.383652\pi$
$811$	3.09248e11	0.714864	0.357432	+	0.933939i	$0.383652\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 3.06957e11i	− 0.695740i
$816$	0	0
$817$	−9.70639e10	−0.217856
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 4.13727e11i	− 0.910628i	−0.890331	−	0.455314i	$-0.849527\pi$
$821$	− 4.13727e11i	− 0.910628i	0.890331	−	0.455314i	$-0.150473\pi$
$822$	0	0
$823$	4.28713e11	0.934474	0.467237	−	0.884132i	$-0.345249\pi$
$823$	4.28713e11	0.934474	0.467237	+	0.884132i	$0.345249\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	8.95971e10i	0.191545i	0.995403	+	0.0957727i	$0.0305322\pi$
$827$	8.95971e10i	0.191545i	−0.995403	+	0.0957727i	$0.969468\pi$
$828$	0	0
$829$	−4.41213e11	−0.934179	−0.467090	−	0.884210i	$-0.654697\pi$
$829$	−4.41213e11	−0.934179	−0.467090	+	0.884210i	$0.654697\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 1.09368e12i	− 2.27149i
$834$	0	0
$835$	−1.08035e11	−0.222239
$836$	0	0
$837$	0	0
$838$	0	0
$839$	6.91542e11i	1.39563i	0.716277	+	0.697816i	$0.245845\pi$
$839$	6.91542e11i	1.39563i	−0.716277	+	0.697816i	$0.754155\pi$
$840$	0	0
$841$	4.78542e11	0.956612
$842$	0	0
$843$	0	0
$844$	0	0
$845$	5.04647e11i	0.989830i
$846$	0	0
$847$	−1.66799e12	−3.24085
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1.04630e10i	0.0199498i
$852$	0	0
$853$	7.98823e11	1.50888	0.754440	−	0.656369i	$-0.227909\pi$
$853$	7.98823e11	1.50888	0.754440	+	0.656369i	$0.227909\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 2.04186e11i	− 0.378532i	−0.981926	−	0.189266i	$-0.939389\pi$
$857$	− 2.04186e11i	− 0.378532i	0.981926	−	0.189266i	$-0.0606108\pi$
$858$	0	0
$859$	5.69484e11	1.04594	0.522972	−	0.852350i	$-0.324823\pi$
$859$	5.69484e11	1.04594	0.522972	+	0.852350i	$0.324823\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	9.98074e11i	1.79937i	0.436544	+	0.899683i	$0.356202\pi$
$863$	9.98074e11i	1.79937i	−0.436544	+	0.899683i	$0.643798\pi$
$864$	0	0
$865$	−9.34142e11	−1.66858
$866$	0	0
$867$	0	0
$868$	0	0
$869$	6.44248e11i	1.12973i
$870$	0	0
$871$	4.50103e11	0.782059
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 4.63646e11i	− 0.790959i
$876$	0	0
$877$	−9.52277e11	−1.60977	−0.804887	−	0.593428i	$-0.797774\pi$
$877$	−9.52277e11	−1.60977	−0.804887	+	0.593428i	$0.797774\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 4.61083e11i	− 0.765378i	−0.923877	−	0.382689i	$-0.874998\pi$
$881$	− 4.61083e11i	− 0.765378i	0.923877	−	0.382689i	$-0.125002\pi$
$882$	0	0
$883$	−7.93717e11	−1.30564	−0.652819	−	0.757514i	$-0.726414\pi$
$883$	−7.93717e11	−1.30564	−0.652819	+	0.757514i	$0.726414\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 8.56491e11i	− 1.38366i	−0.722062	−	0.691828i	$-0.756806\pi$
$887$	− 8.56491e11i	− 1.38366i	0.722062	−	0.691828i	$-0.243194\pi$
$888$	0	0
$889$	6.87871e11	1.10129
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1.20772e11i	0.189915i
$894$	0	0
$895$	1.30925e12	2.04047
$896$	0	0
$897$	0	0
$898$	0	0
$899$	7.91997e10i	0.121251i
$900$	0	0
$901$	−6.56168e11	−0.995670
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 8.52989e11i	− 1.27160i
$906$	0	0
$907$	−2.84798e11	−0.420831	−0.210416	−	0.977612i	$-0.567482\pi$
$907$	−2.84798e11	−0.420831	−0.210416	+	0.977612i	$0.567482\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 6.37115e11i	− 0.925006i	−0.886618	−	0.462503i	$-0.846952\pi$
$911$	− 6.37115e11i	− 0.925006i	0.886618	−	0.462503i	$-0.153048\pi$
$912$	0	0
$913$	5.60941e11	0.807298
$914$	0	0
$915$	0	0
$916$	0	0
$917$	3.03959e11i	0.429871i
$918$	0	0
$919$	9.37584e11	1.31446	0.657231	−	0.753689i	$-0.271728\pi$
$919$	9.37584e11	1.31446	0.657231	+	0.753689i	$0.271728\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	2.77486e11i	0.382327i
$924$	0	0
$925$	1.33893e11	0.182890
$926$	0	0
$927$	0	0
$928$	0	0
$929$	9.26841e11i	1.24435i	0.782878	+	0.622175i	$0.213751\pi$
$929$	9.26841e11i	1.24435i	−0.782878	+	0.622175i	$0.786249\pi$
$930$	0	0
$931$	−2.00016e11	−0.266236
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 2.01609e12i	− 2.63793i
$936$	0	0
$937$	8.85787e11	1.14913	0.574567	−	0.818457i	$-0.305170\pi$
$937$	8.85787e11	1.14913	0.574567	+	0.818457i	$0.305170\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1.34583e12i	1.71645i	0.513271	+	0.858227i	$0.328434\pi$
$941$	1.34583e12i	1.71645i	−0.513271	+	0.858227i	$0.671566\pi$
$942$	0	0
$943$	−3.01216e9	−0.00380918
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 1.91796e11i	− 0.238473i	−0.992866	−	0.119236i	$-0.961955\pi$
$947$	− 1.91796e11i	− 0.238473i	0.992866	−	0.119236i	$-0.0380446\pi$
$948$	0	0
$949$	−1.40425e11	−0.173133
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 1.35503e12i	− 1.64277i	−0.570373	−	0.821386i	$-0.693201\pi$
$953$	− 1.35503e12i	− 1.64277i	0.570373	−	0.821386i	$-0.306799\pi$
$954$	0	0
$955$	−1.79367e12	−2.15640
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 1.18153e12i	− 1.39692i
$960$	0	0
$961$	−5.63893e11	−0.661155
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 8.52488e11i	− 0.983058i
$966$	0	0
$967$	−2.42676e11	−0.277537	−0.138769	−	0.990325i	$-0.544314\pi$
$967$	−2.42676e11	−0.277537	−0.138769	+	0.990325i	$0.544314\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	9.24884e11i	1.04042i	0.854037	+	0.520212i	$0.174147\pi$
$971$	9.24884e11i	1.04042i	−0.854037	+	0.520212i	$0.825853\pi$
$972$	0	0
$973$	−2.83145e12	−3.15906
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 1.39429e12i	− 1.53029i	−0.643858	−	0.765145i	$-0.722667\pi$
$977$	− 1.39429e12i	− 1.53029i	0.643858	−	0.765145i	$-0.277333\pi$
$978$	0	0
$979$	−1.83434e11	−0.199687
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 5.78787e11i	− 0.619875i	−0.950757	−	0.309938i	$-0.899692\pi$
$983$	− 5.78787e11i	− 0.619875i	0.950757	−	0.309938i	$-0.100308\pi$
$984$	0	0
$985$	1.02582e12	1.08975
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 1.01941e11i	− 0.106553i
$990$	0	0
$991$	−7.03550e11	−0.729458	−0.364729	−	0.931114i	$-0.618838\pi$
$991$	−7.03550e11	−0.729458	−0.364729	+	0.931114i	$0.618838\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	3.68321e11i	0.375781i
$996$	0	0
$997$	8.56148e11	0.866499	0.433249	−	0.901274i	$-0.357367\pi$
$997$	8.56148e11	0.866499	0.433249	+	0.901274i	$0.357367\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	432.9.e.k.161.2		6
3.2	odd	2	inner	432.9.e.k.161.5		6
4.3	odd	2		27.9.b.d.26.4	yes	6
12.11	even	2		27.9.b.d.26.3	✓	6
36.7	odd	6		81.9.d.f.53.4		12
36.11	even	6		81.9.d.f.53.3		12
36.23	even	6		81.9.d.f.26.4		12
36.31	odd	6		81.9.d.f.26.3		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.9.b.d.26.3	✓	6	12.11	even	2
27.9.b.d.26.4	yes	6	4.3	odd	2
81.9.d.f.26.3		12	36.31	odd	6
81.9.d.f.26.4		12	36.23	even	6
81.9.d.f.53.3		12	36.11	even	6
81.9.d.f.53.4		12	36.7	odd	6
432.9.e.k.161.2		6	1.1	even	1	trivial
432.9.e.k.161.5		6	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 432 = 2^{4} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	432.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-799.282i q^{5} +4073.62 q^{7} +O(q^{10})\)	\(q-799.282i q^{5} +4073.62 q^{7} +24976.4i q^{11} -13577.8 q^{13} -100991. i q^{17} -18469.5 q^{19} -19397.6i q^{23} -248226. q^{25} -147325. i q^{29} -537585. q^{31} -3.25597e6i q^{35} -539398. q^{37} -155285. i q^{41} +5.25537e6 q^{43} -6.53898e6i q^{47} +1.08295e7 q^{49} -6.49732e6i q^{53} +1.99632e7 q^{55} +1.01924e7i q^{59} +1.22909e7 q^{61} +1.08524e7i q^{65} -3.31500e7 q^{67} -2.04368e7i q^{71} +1.03423e7 q^{73} +1.01744e8i q^{77} +2.57943e7 q^{79} -2.24588e7i q^{83} -8.07199e7 q^{85} +7.34429e6i q^{89} -5.53105e7 q^{91} +1.47623e7i q^{95} -1.55238e7 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 1698 q^{7}+O(q^{10}) \) 6 * q + 1698 * q^7	\( 6 q + 1698 q^{7} + 41844 q^{13} + 36384 q^{19} - 1646688 q^{25} - 2058474 q^{31} - 9395880 q^{37} + 2737284 q^{43} + 28900656 q^{49} + 26674542 q^{55} - 40180776 q^{61} - 111355284 q^{67} + 12821718 q^{73} - 21820404 q^{79} - 83844396 q^{85} - 156632964 q^{91} + 53735106 q^{97}+O(q^{100}) \) 6 * q + 1698 * q^7 + 41844 * q^13 + 36384 * q^19 - 1646688 * q^25 - 2058474 * q^31 - 9395880 * q^37 + 2737284 * q^43 + 28900656 * q^49 + 26674542 * q^55 - 40180776 * q^61 - 111355284 * q^67 + 12821718 * q^73 - 21820404 * q^79 - 83844396 * q^85 - 156632964 * q^91 + 53735106 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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