LMFDB - Embedded newform 432.8.a.q.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [432,8,Mod(1,432)]

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, 0]))

N = Newforms(chi, 8, names="a")

//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: f = N[0] # Warning: the index may be different

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			1.1
Root			$4.53113$ 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)

$f(q)$	$=$	$q+65.8132 q^{5} +738.405 q^{7} -4961.26 q^{11} +5967.24 q^{13} +36651.6 q^{17} -22378.9 q^{19} +51473.6 q^{23} -73793.6 q^{25} -68495.7 q^{29} -150655. q^{31} +48596.8 q^{35} +489027. q^{37} +590635. q^{41} +842643. q^{43} -1.22637e6 q^{47} -278301. q^{49} +958904. q^{53} -326517. q^{55} +316269. q^{59} -29722.9 q^{61} +392723. q^{65} -293025. q^{67} -714537. q^{71} -3.96273e6 q^{73} -3.66342e6 q^{77} -2.53805e6 q^{79} -1.66311e6 q^{83} +2.41216e6 q^{85} +4.64819e6 q^{89} +4.40624e6 q^{91} -1.47283e6 q^{95} +1.47010e7 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	65.8132	0.235461	0.117730	−	0.993046i	$-0.462438\pi$
$5$	65.8132	0.235461	0.117730	+	0.993046i	$0.462438\pi$
$6$	0	0
$7$	738.405	0.813676	0.406838	−	0.913500i	$-0.366631\pi$
$7$	738.405	0.813676	0.406838	+	0.913500i	$0.366631\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−4961.26	−1.12387	−0.561937	−	0.827180i	$-0.689944\pi$
$11$	−4961.26	−1.12387	−0.561937	+	0.827180i	$0.689944\pi$
$12$	0	0
$13$	5967.24	0.753306	0.376653	−	0.926354i	$-0.377075\pi$
$13$	5967.24	0.753306	0.376653	+	0.926354i	$0.377075\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	36651.6	1.80935	0.904674	−	0.426104i	$-0.140114\pi$
$17$	36651.6	1.80935	0.904674	+	0.426104i	$0.140114\pi$
$18$	0	0
$19$	−22378.9	−0.748518	−0.374259	−	0.927324i	$-0.622103\pi$
$19$	−22378.9	−0.748518	−0.374259	+	0.927324i	$0.622103\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	51473.6	0.882139	0.441069	−	0.897473i	$-0.354599\pi$
$23$	51473.6	0.882139	0.441069	+	0.897473i	$0.354599\pi$
$24$	0	0
$25$	−73793.6	−0.944558
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−68495.7	−0.521519	−0.260760	−	0.965404i	$-0.583973\pi$
$29$	−68495.7	−0.521519	−0.260760	+	0.965404i	$0.583973\pi$
$30$	0	0
$31$	−150655.	−0.908275	−0.454137	−	0.890932i	$-0.650052\pi$
$31$	−150655.	−0.908275	−0.454137	+	0.890932i	$0.650052\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	48596.8	0.191589
$36$	0	0
$37$	489027.	1.58718	0.793591	−	0.608451i	$-0.208209\pi$
$37$	489027.	1.58718	0.793591	+	0.608451i	$0.208209\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	590635.	1.33837	0.669184	−	0.743096i	$-0.266644\pi$
$41$	590635.	1.33837	0.669184	+	0.743096i	$0.266644\pi$
$42$	0	0
$43$	842643.	1.61623	0.808117	−	0.589023i	$-0.200487\pi$
$43$	842643.	1.61623	0.808117	+	0.589023i	$0.200487\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−1.22637e6	−1.72297	−0.861486	−	0.507782i	$-0.830466\pi$
$47$	−1.22637e6	−1.72297	−0.861486	+	0.507782i	$0.830466\pi$
$48$	0	0
$49$	−278301.	−0.337932
$50$	0	0
$51$	0	0
$52$	0	0
$53$	958904.	0.884727	0.442364	−	0.896836i	$-0.354140\pi$
$53$	958904.	0.884727	0.442364	+	0.896836i	$0.354140\pi$
$54$	0	0
$55$	−326517.	−0.264628
$56$	0	0
$57$	0	0
$58$	0	0
$59$	316269.	0.200482	0.100241	−	0.994963i	$-0.468039\pi$
$59$	316269.	0.200482	0.100241	+	0.994963i	$0.468039\pi$
$60$	0	0
$61$	−29722.9	−0.0167663	−0.00838315	−	0.999965i	$-0.502668\pi$
$61$	−29722.9	−0.0167663	−0.00838315	+	0.999965i	$0.502668\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	392723.	0.177374
$66$	0	0
$67$	−293025.	−0.119026	−0.0595130	−	0.998228i	$-0.518955\pi$
$67$	−293025.	−0.119026	−0.0595130	+	0.998228i	$0.518955\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−714537.	−0.236930	−0.118465	−	0.992958i	$-0.537797\pi$
$71$	−714537.	−0.236930	−0.118465	+	0.992958i	$0.537797\pi$
$72$	0	0
$73$	−3.96273e6	−1.19224	−0.596121	−	0.802894i	$-0.703292\pi$
$73$	−3.96273e6	−1.19224	−0.596121	+	0.802894i	$0.703292\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−3.66342e6	−0.914470
$78$	0	0
$79$	−2.53805e6	−0.579168	−0.289584	−	0.957153i	$-0.593517\pi$
$79$	−2.53805e6	−0.579168	−0.289584	+	0.957153i	$0.593517\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−1.66311e6	−0.319263	−0.159631	−	0.987177i	$-0.551031\pi$
$83$	−1.66311e6	−0.319263	−0.159631	+	0.987177i	$0.551031\pi$
$84$	0	0
$85$	2.41216e6	0.426030
$86$	0	0
$87$	0	0
$88$	0	0
$89$	4.64819e6	0.698906	0.349453	−	0.936954i	$-0.386368\pi$
$89$	4.64819e6	0.698906	0.349453	+	0.936954i	$0.386368\pi$
$90$	0	0
$91$	4.40624e6	0.612947
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.47283e6	−0.176246
$96$	0	0
$97$	1.47010e7	1.63548	0.817738	−	0.575590i	$-0.195228\pi$
$97$	1.47010e7	1.63548	0.817738	+	0.575590i	$0.195228\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	9.28380e6	0.896604	0.448302	−	0.893882i	$-0.352029\pi$
$101$	9.28380e6	0.896604	0.448302	+	0.893882i	$0.352029\pi$
$102$	0	0
$103$	−4.11049e6	−0.370650	−0.185325	−	0.982677i	$-0.559334\pi$
$103$	−4.11049e6	−0.370650	−0.185325	+	0.982677i	$0.559334\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.77931e7	1.40414	0.702068	−	0.712110i	$-0.252260\pi$
$107$	1.77931e7	1.40414	0.702068	+	0.712110i	$0.252260\pi$
$108$	0	0
$109$	−1.72244e7	−1.27394	−0.636972	−	0.770887i	$-0.719813\pi$
$109$	−1.72244e7	−1.27394	−0.636972	+	0.770887i	$0.719813\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−2.16751e6	−0.141315	−0.0706573	−	0.997501i	$-0.522510\pi$
$113$	−2.16751e6	−0.141315	−0.0706573	+	0.997501i	$0.522510\pi$
$114$	0	0
$115$	3.38764e6	0.207709
$116$	0	0
$117$	0	0
$118$	0	0
$119$	2.70638e7	1.47222
$120$	0	0
$121$	5.12697e6	0.263095
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−9.99825e6	−0.457867
$126$	0	0
$127$	−9.66827e6	−0.418828	−0.209414	−	0.977827i	$-0.567156\pi$
$127$	−9.66827e6	−0.418828	−0.209414	+	0.977827i	$0.567156\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	4.22431e7	1.64175	0.820873	−	0.571111i	$-0.193487\pi$
$131$	4.22431e7	1.64175	0.820873	+	0.571111i	$0.193487\pi$
$132$	0	0
$133$	−1.65247e7	−0.609051
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−8.01451e6	−0.266290	−0.133145	−	0.991097i	$-0.542508\pi$
$137$	−8.01451e6	−0.266290	−0.133145	+	0.991097i	$0.542508\pi$
$138$	0	0
$139$	−1.46010e7	−0.461138	−0.230569	−	0.973056i	$-0.574059\pi$
$139$	−1.46010e7	−0.461138	−0.230569	+	0.973056i	$0.574059\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−2.96050e7	−0.846622
$144$	0	0
$145$	−4.50792e6	−0.122797
$146$	0	0
$147$	0	0
$148$	0	0
$149$	4.07087e7	1.00817	0.504086	−	0.863653i	$-0.331829\pi$
$149$	4.07087e7	1.00817	0.504086	+	0.863653i	$0.331829\pi$
$150$	0	0
$151$	5.29940e7	1.25259	0.626293	−	0.779588i	$-0.284572\pi$
$151$	5.29940e7	1.25259	0.626293	+	0.779588i	$0.284572\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−9.91508e6	−0.213863
$156$	0	0
$157$	4.10669e7	0.846922	0.423461	−	0.905914i	$-0.360815\pi$
$157$	4.10669e7	0.846922	0.423461	+	0.905914i	$0.360815\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	3.80083e7	0.717775
$162$	0	0
$163$	4.33772e7	0.784522	0.392261	−	0.919854i	$-0.371693\pi$
$163$	4.33772e7	0.784522	0.392261	+	0.919854i	$0.371693\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3.86845e7	−0.642731	−0.321365	−	0.946955i	$-0.604142\pi$
$167$	−3.86845e7	−0.642731	−0.321365	+	0.946955i	$0.604142\pi$
$168$	0	0
$169$	−2.71406e7	−0.432529
$170$	0	0
$171$	0	0
$172$	0	0
$173$	5.30547e7	0.779045	0.389523	−	0.921017i	$-0.372640\pi$
$173$	5.30547e7	0.779045	0.389523	+	0.921017i	$0.372640\pi$
$174$	0	0
$175$	−5.44896e7	−0.768564
$176$	0	0
$177$	0	0
$178$	0	0
$179$	1.34132e8	1.74802	0.874012	−	0.485904i	$-0.161509\pi$
$179$	1.34132e8	1.74802	0.874012	+	0.485904i	$0.161509\pi$
$180$	0	0
$181$	6.35105e6	0.0796105	0.0398053	−	0.999207i	$-0.487326\pi$
$181$	6.35105e6	0.0796105	0.0398053	+	0.999207i	$0.487326\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	3.21845e7	0.373719
$186$	0	0
$187$	−1.81839e8	−2.03348
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1.15843e8	1.20297	0.601483	−	0.798885i	$-0.294577\pi$
$191$	1.15843e8	1.20297	0.601483	+	0.798885i	$0.294577\pi$
$192$	0	0
$193$	5.73856e7	0.574582	0.287291	−	0.957843i	$-0.407245\pi$
$193$	5.73856e7	0.574582	0.287291	+	0.957843i	$0.407245\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1.04846e8	0.977059	0.488530	−	0.872547i	$-0.337533\pi$
$197$	1.04846e8	0.977059	0.488530	+	0.872547i	$0.337533\pi$
$198$	0	0
$199$	2.10623e8	1.89461	0.947304	−	0.320335i	$-0.103795\pi$
$199$	2.10623e8	1.89461	0.947304	+	0.320335i	$0.103795\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−5.05776e7	−0.424348
$204$	0	0
$205$	3.88716e7	0.315133
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1.11028e8	0.841240
$210$	0	0
$211$	6.49193e7	0.475757	0.237878	−	0.971295i	$-0.423548\pi$
$211$	6.49193e7	0.475757	0.237878	+	0.971295i	$0.423548\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	5.54570e7	0.380559
$216$	0	0
$217$	−1.11244e8	−0.739041
$218$	0	0
$219$	0	0
$220$	0	0
$221$	2.18709e8	1.36299
$222$	0	0
$223$	−1.40879e8	−0.850705	−0.425352	−	0.905028i	$-0.639850\pi$
$223$	−1.40879e8	−0.850705	−0.425352	+	0.905028i	$0.639850\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3.40163e6	0.0193017	0.00965086	−	0.999953i	$-0.496928\pi$
$227$	3.40163e6	0.0193017	0.00965086	+	0.999953i	$0.496928\pi$
$228$	0	0
$229$	−2.76169e7	−0.151967	−0.0759836	−	0.997109i	$-0.524210\pi$
$229$	−2.76169e7	−0.151967	−0.0759836	+	0.997109i	$0.524210\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	3.51458e8	1.82023	0.910117	−	0.414351i	$-0.135992\pi$
$233$	3.51458e8	1.82023	0.910117	+	0.414351i	$0.135992\pi$
$234$	0	0
$235$	−8.07112e7	−0.405692
$236$	0	0
$237$	0	0
$238$	0	0
$239$	3.79195e8	1.79668	0.898339	−	0.439302i	$-0.144774\pi$
$239$	3.79195e8	1.79668	0.898339	+	0.439302i	$0.144774\pi$
$240$	0	0
$241$	2.99276e7	0.137725	0.0688624	−	0.997626i	$-0.478063\pi$
$241$	2.99276e7	0.137725	0.0688624	+	0.997626i	$0.478063\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1.83159e7	−0.0795696
$246$	0	0
$247$	−1.33540e8	−0.563863
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1.71456e7	0.0684374	0.0342187	−	0.999414i	$-0.489106\pi$
$251$	1.71456e7	0.0684374	0.0342187	+	0.999414i	$0.489106\pi$
$252$	0	0
$253$	−2.55374e8	−0.991414
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−9.79449e7	−0.359928	−0.179964	−	0.983673i	$-0.557598\pi$
$257$	−9.79449e7	−0.359928	−0.179964	+	0.983673i	$0.557598\pi$
$258$	0	0
$259$	3.61100e8	1.29145
$260$	0	0
$261$	0	0
$262$	0	0
$263$	8.92540e7	0.302540	0.151270	−	0.988492i	$-0.451664\pi$
$263$	8.92540e7	0.302540	0.151270	+	0.988492i	$0.451664\pi$
$264$	0	0
$265$	6.31085e7	0.208318
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−5.34653e8	−1.67471	−0.837354	−	0.546662i	$-0.815898\pi$
$269$	−5.34653e8	−1.67471	−0.837354	+	0.546662i	$0.815898\pi$
$270$	0	0
$271$	−2.11501e8	−0.645535	−0.322768	−	0.946478i	$-0.604613\pi$
$271$	−2.11501e8	−0.645535	−0.322768	+	0.946478i	$0.604613\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	3.66110e8	1.06157
$276$	0	0
$277$	−3.24017e8	−0.915986	−0.457993	−	0.888956i	$-0.651432\pi$
$277$	−3.24017e8	−0.915986	−0.457993	+	0.888956i	$0.651432\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	4.91210e7	0.132067	0.0660336	−	0.997817i	$-0.478966\pi$
$281$	4.91210e7	0.132067	0.0660336	+	0.997817i	$0.478966\pi$
$282$	0	0
$283$	4.03017e8	1.05699	0.528495	−	0.848936i	$-0.322756\pi$
$283$	4.03017e8	1.05699	0.528495	+	0.848936i	$0.322756\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4.36128e8	1.08900
$288$	0	0
$289$	9.33004e8	2.27374
$290$	0	0
$291$	0	0
$292$	0	0
$293$	4.85906e8	1.12854	0.564268	−	0.825592i	$-0.309159\pi$
$293$	4.85906e8	1.12854	0.564268	+	0.825592i	$0.309159\pi$
$294$	0	0
$295$	2.08147e7	0.0472056
$296$	0	0
$297$	0	0
$298$	0	0
$299$	3.07155e8	0.664521
$300$	0	0
$301$	6.22212e8	1.31509
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−1.95616e6	−0.00394780
$306$	0	0
$307$	−2.82215e8	−0.556668	−0.278334	−	0.960484i	$-0.589782\pi$
$307$	−2.82215e8	−0.556668	−0.278334	+	0.960484i	$0.589782\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−7.04440e7	−0.132795	−0.0663977	−	0.997793i	$-0.521151\pi$
$311$	−7.04440e7	−0.132795	−0.0663977	+	0.997793i	$0.521151\pi$
$312$	0	0
$313$	−4.78096e8	−0.881273	−0.440636	−	0.897686i	$-0.645247\pi$
$313$	−4.78096e8	−0.881273	−0.440636	+	0.897686i	$0.645247\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6.89470e8	1.21565	0.607824	−	0.794072i	$-0.292043\pi$
$317$	6.89470e8	1.21565	0.607824	+	0.794072i	$0.292043\pi$
$318$	0	0
$319$	3.39825e8	0.586123
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−8.20225e8	−1.35433
$324$	0	0
$325$	−4.40344e8	−0.711542
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−9.05555e8	−1.40194
$330$	0	0
$331$	−5.36833e8	−0.813657	−0.406829	−	0.913504i	$-0.633365\pi$
$331$	−5.36833e8	−0.813657	−0.406829	+	0.913504i	$0.633365\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−1.92849e7	−0.0280259
$336$	0	0
$337$	−1.24011e9	−1.76505	−0.882524	−	0.470268i	$-0.844157\pi$
$337$	−1.24011e9	−1.76505	−0.882524	+	0.470268i	$0.844157\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	7.47438e8	1.02079
$342$	0	0
$343$	−8.13607e8	−1.08864
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−4.86766e7	−0.0625413	−0.0312706	−	0.999511i	$-0.509955\pi$
$347$	−4.86766e7	−0.0625413	−0.0312706	+	0.999511i	$0.509955\pi$
$348$	0	0
$349$	1.23665e9	1.55725	0.778626	−	0.627488i	$-0.215917\pi$
$349$	1.23665e9	1.55725	0.778626	+	0.627488i	$0.215917\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	6.06858e7	0.0734304	0.0367152	−	0.999326i	$-0.488311\pi$
$353$	6.06858e7	0.0734304	0.0367152	+	0.999326i	$0.488311\pi$
$354$	0	0
$355$	−4.70260e7	−0.0557878
$356$	0	0
$357$	0	0
$358$	0	0
$359$	5.36061e8	0.611482	0.305741	−	0.952115i	$-0.401096\pi$
$359$	5.36061e8	0.611482	0.305741	+	0.952115i	$0.401096\pi$
$360$	0	0
$361$	−3.93055e8	−0.439722
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−2.60800e8	−0.280726
$366$	0	0
$367$	4.31327e8	0.455487	0.227743	−	0.973721i	$-0.426865\pi$
$367$	4.31327e8	0.455487	0.227743	+	0.973721i	$0.426865\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	7.08059e8	0.719881
$372$	0	0
$373$	2.69689e8	0.269080	0.134540	−	0.990908i	$-0.457044\pi$
$373$	2.69689e8	0.269080	0.134540	+	0.990908i	$0.457044\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−4.08730e8	−0.392864
$378$	0	0
$379$	8.03807e8	0.758429	0.379214	−	0.925309i	$-0.376194\pi$
$379$	8.03807e8	0.758429	0.379214	+	0.925309i	$0.376194\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−9.83923e8	−0.894882	−0.447441	−	0.894314i	$-0.647664\pi$
$383$	−9.83923e8	−0.894882	−0.447441	+	0.894314i	$0.647664\pi$
$384$	0	0
$385$	−2.41102e8	−0.215322
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−2.16250e8	−0.186265	−0.0931326	−	0.995654i	$-0.529688\pi$
$389$	−2.16250e8	−0.186265	−0.0931326	+	0.995654i	$0.529688\pi$
$390$	0	0
$391$	1.88659e9	1.59610
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−1.67037e8	−0.136371
$396$	0	0
$397$	4.33533e8	0.347741	0.173870	−	0.984769i	$-0.444373\pi$
$397$	4.33533e8	0.347741	0.173870	+	0.984769i	$0.444373\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2.42713e9	1.87970	0.939849	−	0.341591i	$-0.110966\pi$
$401$	2.42713e9	1.87970	0.939849	+	0.341591i	$0.110966\pi$
$402$	0	0
$403$	−8.98993e8	−0.684209
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−2.42619e9	−1.78379
$408$	0	0
$409$	9.18976e8	0.664160	0.332080	−	0.943251i	$-0.392250\pi$
$409$	9.18976e8	0.664160	0.332080	+	0.943251i	$0.392250\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	2.33535e8	0.163127
$414$	0	0
$415$	−1.09455e8	−0.0751738
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−1.10024e9	−0.730699	−0.365349	−	0.930870i	$-0.619050\pi$
$419$	−1.10024e9	−0.730699	−0.365349	+	0.930870i	$0.619050\pi$
$420$	0	0
$421$	−2.14909e9	−1.40367	−0.701837	−	0.712337i	$-0.747637\pi$
$421$	−2.14909e9	−1.40367	−0.701837	+	0.712337i	$0.747637\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−2.70466e9	−1.70904
$426$	0	0
$427$	−2.19476e7	−0.0136423
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−1.61850e9	−0.973737	−0.486869	−	0.873475i	$-0.661861\pi$
$431$	−1.61850e9	−0.973737	−0.486869	+	0.873475i	$0.661861\pi$
$432$	0	0
$433$	1.16527e9	0.689794	0.344897	−	0.938641i	$-0.387914\pi$
$433$	1.16527e9	0.689794	0.344897	+	0.938641i	$0.387914\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1.15192e9	−0.660296
$438$	0	0
$439$	−2.08096e9	−1.17392	−0.586958	−	0.809617i	$-0.699675\pi$
$439$	−2.08096e9	−1.17392	−0.586958	+	0.809617i	$0.699675\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−2.23100e9	−1.21923	−0.609616	−	0.792697i	$-0.708676\pi$
$443$	−2.23100e9	−1.21923	−0.609616	+	0.792697i	$0.708676\pi$
$444$	0	0
$445$	3.05912e8	0.164565
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−2.52631e9	−1.31712	−0.658559	−	0.752529i	$-0.728834\pi$
$449$	−2.52631e9	−1.31712	−0.658559	+	0.752529i	$0.728834\pi$
$450$	0	0
$451$	−2.93030e9	−1.50416
$452$	0	0
$453$	0	0
$454$	0	0
$455$	2.89989e8	0.144325
$456$	0	0
$457$	−1.87627e9	−0.919579	−0.459790	−	0.888028i	$-0.652075\pi$
$457$	−1.87627e9	−0.919579	−0.459790	+	0.888028i	$0.652075\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−1.97321e9	−0.938036	−0.469018	−	0.883189i	$-0.655392\pi$
$461$	−1.97321e9	−0.938036	−0.469018	+	0.883189i	$0.655392\pi$
$462$	0	0
$463$	1.60344e9	0.750790	0.375395	−	0.926865i	$-0.377507\pi$
$463$	1.60344e9	0.750790	0.375395	+	0.926865i	$0.377507\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	1.32390e9	0.601515	0.300757	−	0.953701i	$-0.402761\pi$
$467$	1.32390e9	0.601515	0.300757	+	0.953701i	$0.402761\pi$
$468$	0	0
$469$	−2.16371e8	−0.0968486
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−4.18057e9	−1.81644
$474$	0	0
$475$	1.65142e9	0.707018
$476$	0	0
$477$	0	0
$478$	0	0
$479$	2.44225e9	1.01535	0.507675	−	0.861548i	$-0.330505\pi$
$479$	2.44225e9	1.01535	0.507675	+	0.861548i	$0.330505\pi$
$480$	0	0
$481$	2.91814e9	1.19563
$482$	0	0
$483$	0	0
$484$	0	0
$485$	9.67517e8	0.385090
$486$	0	0
$487$	1.37309e7	0.00538701	0.00269350	−	0.999996i	$-0.499143\pi$
$487$	1.37309e7	0.00538701	0.00269350	+	0.999996i	$0.499143\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	6.32242e8	0.241045	0.120522	−	0.992711i	$-0.461543\pi$
$491$	6.32242e8	0.241045	0.120522	+	0.992711i	$0.461543\pi$
$492$	0	0
$493$	−2.51048e9	−0.943610
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−5.27618e8	−0.192784
$498$	0	0
$499$	2.43365e9	0.876812	0.438406	−	0.898777i	$-0.355543\pi$
$499$	2.43365e9	0.876812	0.438406	+	0.898777i	$0.355543\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−2.00281e9	−0.701700	−0.350850	−	0.936432i	$-0.614107\pi$
$503$	−2.00281e9	−0.701700	−0.350850	+	0.936432i	$0.614107\pi$
$504$	0	0
$505$	6.10997e8	0.211115
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−5.28309e8	−0.177573	−0.0887863	−	0.996051i	$-0.528299\pi$
$509$	−5.28309e8	−0.177573	−0.0887863	+	0.996051i	$0.528299\pi$
$510$	0	0
$511$	−2.92610e9	−0.970099
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−2.70525e8	−0.0872733
$516$	0	0
$517$	6.08433e9	1.93640
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−1.76283e9	−0.546109	−0.273055	−	0.961999i	$-0.588034\pi$
$521$	−1.76283e9	−0.546109	−0.273055	+	0.961999i	$0.588034\pi$
$522$	0	0
$523$	3.39318e9	1.03717	0.518586	−	0.855026i	$-0.326459\pi$
$523$	3.39318e9	1.03717	0.518586	+	0.855026i	$0.326459\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−5.52175e9	−1.64339
$528$	0	0
$529$	−7.55295e8	−0.221831
$530$	0	0
$531$	0	0
$532$	0	0
$533$	3.52446e9	1.00820
$534$	0	0
$535$	1.17102e9	0.330619
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1.38073e9	0.379793
$540$	0	0
$541$	−3.83907e9	−1.04240	−0.521202	−	0.853433i	$-0.674516\pi$
$541$	−3.83907e9	−1.04240	−0.521202	+	0.853433i	$0.674516\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−1.13359e9	−0.299964
$546$	0	0
$547$	−4.31235e9	−1.12657	−0.563285	−	0.826263i	$-0.690463\pi$
$547$	−4.31235e9	−1.12657	−0.563285	+	0.826263i	$0.690463\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1.53286e9	0.390366
$552$	0	0
$553$	−1.87410e9	−0.471255
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−1.84109e9	−0.451421	−0.225710	−	0.974194i	$-0.572470\pi$
$557$	−1.84109e9	−0.451421	−0.225710	+	0.974194i	$0.572470\pi$
$558$	0	0
$559$	5.02825e9	1.21752
$560$	0	0
$561$	0	0
$562$	0	0
$563$	1.16055e9	0.274085	0.137042	−	0.990565i	$-0.456240\pi$
$563$	1.16055e9	0.274085	0.137042	+	0.990565i	$0.456240\pi$
$564$	0	0
$565$	−1.42651e8	−0.0332740
$566$	0	0
$567$	0	0
$568$	0	0
$569$	7.02920e8	0.159960	0.0799802	−	0.996796i	$-0.474514\pi$
$569$	7.02920e8	0.159960	0.0799802	+	0.996796i	$0.474514\pi$
$570$	0	0
$571$	7.95929e8	0.178915	0.0894577	−	0.995991i	$-0.471487\pi$
$571$	7.95929e8	0.178915	0.0894577	+	0.995991i	$0.471487\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−3.79842e9	−0.833232
$576$	0	0
$577$	−4.44712e9	−0.963749	−0.481874	−	0.876240i	$-0.660044\pi$
$577$	−4.44712e9	−0.963749	−0.481874	+	0.876240i	$0.660044\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−1.22805e9	−0.259776
$582$	0	0
$583$	−4.75737e9	−0.994323
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−5.50846e9	−1.12408	−0.562039	−	0.827111i	$-0.689983\pi$
$587$	−5.50846e9	−1.12408	−0.562039	+	0.827111i	$0.689983\pi$
$588$	0	0
$589$	3.37150e9	0.679859
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−8.05850e9	−1.58695	−0.793475	−	0.608603i	$-0.791730\pi$
$593$	−8.05850e9	−1.58695	−0.793475	+	0.608603i	$0.791730\pi$
$594$	0	0
$595$	1.78115e9	0.346650
$596$	0	0
$597$	0	0
$598$	0	0
$599$	3.39566e9	0.645551	0.322775	−	0.946476i	$-0.395384\pi$
$599$	3.39566e9	0.645551	0.322775	+	0.946476i	$0.395384\pi$
$600$	0	0
$601$	1.35177e9	0.254005	0.127002	−	0.991902i	$-0.459464\pi$
$601$	1.35177e9	0.254005	0.127002	+	0.991902i	$0.459464\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	3.37423e8	0.0619485
$606$	0	0
$607$	−4.91297e9	−0.891627	−0.445814	−	0.895126i	$-0.647086\pi$
$607$	−4.91297e9	−0.891627	−0.445814	+	0.895126i	$0.647086\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−7.31802e9	−1.29793
$612$	0	0
$613$	−2.06110e9	−0.361399	−0.180700	−	0.983538i	$-0.557836\pi$
$613$	−2.06110e9	−0.361399	−0.180700	+	0.983538i	$0.557836\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−3.72277e9	−0.638071	−0.319035	−	0.947743i	$-0.603359\pi$
$617$	−3.72277e9	−0.638071	−0.319035	+	0.947743i	$0.603359\pi$
$618$	0	0
$619$	−3.30066e9	−0.559349	−0.279675	−	0.960095i	$-0.590227\pi$
$619$	−3.30066e9	−0.559349	−0.279675	+	0.960095i	$0.590227\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	3.43224e9	0.568683
$624$	0	0
$625$	5.10711e9	0.836749
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1.79236e10	2.87177
$630$	0	0
$631$	−5.12709e9	−0.812397	−0.406199	−	0.913785i	$-0.633146\pi$
$631$	−5.12709e9	−0.812397	−0.406199	+	0.913785i	$0.633146\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−6.36300e8	−0.0986174
$636$	0	0
$637$	−1.66069e9	−0.254566
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−4.51842e9	−0.677615	−0.338808	−	0.940856i	$-0.610024\pi$
$641$	−4.51842e9	−0.677615	−0.338808	+	0.940856i	$0.610024\pi$
$642$	0	0
$643$	9.70600e9	1.43980	0.719900	−	0.694078i	$-0.244188\pi$
$643$	9.70600e9	1.43980	0.719900	+	0.694078i	$0.244188\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	6.12536e9	0.889132	0.444566	−	0.895746i	$-0.353358\pi$
$647$	6.12536e9	0.889132	0.444566	+	0.895746i	$0.353358\pi$
$648$	0	0
$649$	−1.56910e9	−0.225317
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1.30103e10	1.82849	0.914243	−	0.405166i	$-0.132786\pi$
$653$	1.30103e10	1.82849	0.914243	+	0.405166i	$0.132786\pi$
$654$	0	0
$655$	2.78015e9	0.386566
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−1.23137e10	−1.67606	−0.838032	−	0.545621i	$-0.816294\pi$
$659$	−1.23137e10	−1.67606	−0.838032	+	0.545621i	$0.816294\pi$
$660$	0	0
$661$	6.97776e9	0.939746	0.469873	−	0.882734i	$-0.344300\pi$
$661$	6.97776e9	0.939746	0.469873	+	0.882734i	$0.344300\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1.08755e9	−0.143407
$666$	0	0
$667$	−3.52572e9	−0.460053
$668$	0	0
$669$	0	0
$670$	0	0
$671$	1.47463e8	0.0188432
$672$	0	0
$673$	−1.06462e10	−1.34630	−0.673152	−	0.739504i	$-0.735060\pi$
$673$	−1.06462e10	−1.34630	−0.673152	+	0.739504i	$0.735060\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	4.33348e9	0.536756	0.268378	−	0.963314i	$-0.413512\pi$
$677$	4.33348e9	0.536756	0.268378	+	0.963314i	$0.413512\pi$
$678$	0	0
$679$	1.08553e10	1.33075
$680$	0	0
$681$	0	0
$682$	0	0
$683$	6.16980e9	0.740966	0.370483	−	0.928839i	$-0.379192\pi$
$683$	6.16980e9	0.740966	0.370483	+	0.928839i	$0.379192\pi$
$684$	0	0
$685$	−5.27461e8	−0.0627008
$686$	0	0
$687$	0	0
$688$	0	0
$689$	5.72201e9	0.666471
$690$	0	0
$691$	−7.57273e9	−0.873131	−0.436565	−	0.899672i	$-0.643805\pi$
$691$	−7.57273e9	−0.873131	−0.436565	+	0.899672i	$0.643805\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−9.60939e8	−0.108580
$696$	0	0
$697$	2.16477e10	2.42158
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−4.42696e9	−0.485391	−0.242696	−	0.970102i	$-0.578032\pi$
$701$	−4.42696e9	−0.485391	−0.242696	+	0.970102i	$0.578032\pi$
$702$	0	0
$703$	−1.09439e10	−1.18803
$704$	0	0
$705$	0	0
$706$	0	0
$707$	6.85520e9	0.729545
$708$	0	0
$709$	−1.13232e10	−1.19319	−0.596594	−	0.802544i	$-0.703480\pi$
$709$	−1.13232e10	−1.19319	−0.596594	+	0.802544i	$0.703480\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−7.75474e9	−0.801224
$714$	0	0
$715$	−1.94840e9	−0.199346
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1.02156e10	1.02497	0.512487	−	0.858695i	$-0.328724\pi$
$719$	1.02156e10	1.02497	0.512487	+	0.858695i	$0.328724\pi$
$720$	0	0
$721$	−3.03521e9	−0.301589
$722$	0	0
$723$	0	0
$724$	0	0
$725$	5.05455e9	0.492605
$726$	0	0
$727$	1.95355e10	1.88562	0.942811	−	0.333329i	$-0.108172\pi$
$727$	1.95355e10	1.88562	0.942811	+	0.333329i	$0.108172\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	3.08842e10	2.92433
$732$	0	0
$733$	−1.32745e10	−1.24496	−0.622478	−	0.782637i	$-0.713874\pi$
$733$	−1.32745e10	−1.24496	−0.622478	+	0.782637i	$0.713874\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.45377e9	0.133770
$738$	0	0
$739$	−1.69495e9	−0.154490	−0.0772452	−	0.997012i	$-0.524612\pi$
$739$	−1.69495e9	−0.154490	−0.0772452	+	0.997012i	$0.524612\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	9.34493e9	0.835824	0.417912	−	0.908487i	$-0.362762\pi$
$743$	9.34493e9	0.835824	0.417912	+	0.908487i	$0.362762\pi$
$744$	0	0
$745$	2.67917e9	0.237385
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1.31385e10	1.14251
$750$	0	0
$751$	1.31194e10	1.13025	0.565124	−	0.825006i	$-0.308828\pi$
$751$	1.31194e10	1.13025	0.565124	+	0.825006i	$0.308828\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	3.48770e9	0.294934
$756$	0	0
$757$	5.72593e9	0.479745	0.239873	−	0.970804i	$-0.422894\pi$
$757$	5.72593e9	0.479745	0.239873	+	0.970804i	$0.422894\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−1.47621e10	−1.21423	−0.607115	−	0.794614i	$-0.707673\pi$
$761$	−1.47621e10	−1.21423	−0.607115	+	0.794614i	$0.707673\pi$
$762$	0	0
$763$	−1.27186e10	−1.03658
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.88726e9	0.151024
$768$	0	0
$769$	5.63551e9	0.446880	0.223440	−	0.974718i	$-0.428271\pi$
$769$	5.63551e9	0.446880	0.223440	+	0.974718i	$0.428271\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	4.65338e9	0.362360	0.181180	−	0.983450i	$-0.442008\pi$
$773$	4.65338e9	0.362360	0.181180	+	0.983450i	$0.442008\pi$
$774$	0	0
$775$	1.11174e10	0.857918
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−1.32178e10	−1.00179
$780$	0	0
$781$	3.54501e9	0.266280
$782$	0	0
$783$	0	0
$784$	0	0
$785$	2.70275e9	0.199417
$786$	0	0
$787$	1.23587e10	0.903780	0.451890	−	0.892074i	$-0.350750\pi$
$787$	1.23587e10	0.903780	0.451890	+	0.892074i	$0.350750\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−1.60050e9	−0.114984
$792$	0	0
$793$	−1.77364e8	−0.0126302
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.70444e10	1.19256	0.596278	−	0.802778i	$-0.296646\pi$
$797$	1.70444e10	1.19256	0.596278	+	0.802778i	$0.296646\pi$
$798$	0	0
$799$	−4.49484e10	−3.11746
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1.96602e10	1.33993
$804$	0	0
$805$	2.50145e9	0.169008
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1.00497e10	0.667315	0.333658	−	0.942694i	$-0.391717\pi$
$809$	1.00497e10	0.667315	0.333658	+	0.942694i	$0.391717\pi$
$810$	0	0
$811$	−5.26821e9	−0.346809	−0.173404	−	0.984851i	$-0.555477\pi$
$811$	−5.26821e9	−0.346809	−0.173404	+	0.984851i	$0.555477\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	2.85480e9	0.184724
$816$	0	0
$817$	−1.88575e10	−1.20978
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1.82202e10	1.14909	0.574543	−	0.818474i	$-0.305180\pi$
$821$	1.82202e10	1.14909	0.574543	+	0.818474i	$0.305180\pi$
$822$	0	0
$823$	1.35868e9	0.0849603	0.0424802	−	0.999097i	$-0.486474\pi$
$823$	1.35868e9	0.0849603	0.0424802	+	0.999097i	$0.486474\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−8.90171e9	−0.547273	−0.273637	−	0.961833i	$-0.588227\pi$
$827$	−8.90171e9	−0.547273	−0.273637	+	0.961833i	$0.588227\pi$
$828$	0	0
$829$	6.14205e9	0.374432	0.187216	−	0.982319i	$-0.440054\pi$
$829$	6.14205e9	0.374432	0.187216	+	0.982319i	$0.440054\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−1.02002e10	−0.611436
$834$	0	0
$835$	−2.54595e9	−0.151338
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−1.05464e10	−0.616504	−0.308252	−	0.951305i	$-0.599744\pi$
$839$	−1.05464e10	−0.616504	−0.308252	+	0.951305i	$0.599744\pi$
$840$	0	0
$841$	−1.25582e10	−0.728018
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−1.78621e9	−0.101844
$846$	0	0
$847$	3.78578e9	0.214074
$848$	0	0
$849$	0	0
$850$	0	0
$851$	2.51720e10	1.40012
$852$	0	0
$853$	3.41878e9	0.188604	0.0943018	−	0.995544i	$-0.469938\pi$
$853$	3.41878e9	0.188604	0.0943018	+	0.995544i	$0.469938\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−2.17037e10	−1.17788	−0.588941	−	0.808176i	$-0.700455\pi$
$857$	−2.17037e10	−1.17788	−0.588941	+	0.808176i	$0.700455\pi$
$858$	0	0
$859$	2.19211e10	1.18001	0.590005	−	0.807400i	$-0.299126\pi$
$859$	2.19211e10	1.18001	0.590005	+	0.807400i	$0.299126\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1.35347e10	0.716822	0.358411	−	0.933564i	$-0.383319\pi$
$863$	1.35347e10	0.716822	0.358411	+	0.933564i	$0.383319\pi$
$864$	0	0
$865$	3.49170e9	0.183434
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1.25919e10	0.650912
$870$	0	0
$871$	−1.74855e9	−0.0896631
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−7.38276e9	−0.372555
$876$	0	0
$877$	2.45197e10	1.22749	0.613743	−	0.789506i	$-0.289663\pi$
$877$	2.45197e10	1.22749	0.613743	+	0.789506i	$0.289663\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−2.13536e10	−1.05210	−0.526048	−	0.850455i	$-0.676327\pi$
$881$	−2.13536e10	−1.05210	−0.526048	+	0.850455i	$0.676327\pi$
$882$	0	0
$883$	1.64504e9	0.0804109	0.0402054	−	0.999191i	$-0.487199\pi$
$883$	1.64504e9	0.0804109	0.0402054	+	0.999191i	$0.487199\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−9.65301e9	−0.464441	−0.232220	−	0.972663i	$-0.574599\pi$
$887$	−9.65301e9	−0.464441	−0.232220	+	0.972663i	$0.574599\pi$
$888$	0	0
$889$	−7.13909e9	−0.340790
$890$	0	0
$891$	0	0
$892$	0	0
$893$	2.74448e10	1.28967
$894$	0	0
$895$	8.82767e9	0.411591
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1.03192e10	0.473683
$900$	0	0
$901$	3.51454e10	1.60078
$902$	0	0
$903$	0	0
$904$	0	0
$905$	4.17983e8	0.0187451
$906$	0	0
$907$	8.05417e9	0.358423	0.179211	−	0.983811i	$-0.442645\pi$
$907$	8.05417e9	0.358423	0.179211	+	0.983811i	$0.442645\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−5.73799e9	−0.251446	−0.125723	−	0.992065i	$-0.540125\pi$
$911$	−5.73799e9	−0.251446	−0.125723	+	0.992065i	$0.540125\pi$
$912$	0	0
$913$	8.25114e9	0.358811
$914$	0	0
$915$	0	0
$916$	0	0
$917$	3.11925e10	1.33585
$918$	0	0
$919$	−3.67326e10	−1.56116	−0.780579	−	0.625057i	$-0.785076\pi$
$919$	−3.67326e10	−1.56116	−0.780579	+	0.625057i	$0.785076\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−4.26381e9	−0.178481
$924$	0	0
$925$	−3.60871e10	−1.49919
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−3.50768e9	−0.143537	−0.0717687	−	0.997421i	$-0.522864\pi$
$929$	−3.50768e9	−0.143537	−0.0717687	+	0.997421i	$0.522864\pi$
$930$	0	0
$931$	6.22809e9	0.252948
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−1.19674e10	−0.478805
$936$	0	0
$937$	−1.68124e10	−0.667638	−0.333819	−	0.942637i	$-0.608337\pi$
$937$	−1.68124e10	−0.667638	−0.333819	+	0.942637i	$0.608337\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	3.51454e10	1.37501	0.687503	−	0.726182i	$-0.258707\pi$
$941$	3.51454e10	1.37501	0.687503	+	0.726182i	$0.258707\pi$
$942$	0	0
$943$	3.04021e10	1.18063
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−3.30590e10	−1.26493	−0.632463	−	0.774591i	$-0.717956\pi$
$947$	−3.30590e10	−1.26493	−0.632463	+	0.774591i	$0.717956\pi$
$948$	0	0
$949$	−2.36466e10	−0.898124
$950$	0	0
$951$	0	0
$952$	0	0
$953$	6.24805e9	0.233840	0.116920	−	0.993141i	$-0.462698\pi$
$953$	6.24805e9	0.233840	0.116920	+	0.993141i	$0.462698\pi$
$954$	0	0
$955$	7.62401e9	0.283251
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−5.91795e9	−0.216674
$960$	0	0
$961$	−4.81574e9	−0.175037
$962$	0	0
$963$	0	0
$964$	0	0
$965$	3.77673e9	0.135291
$966$	0	0
$967$	−2.21264e10	−0.786899	−0.393449	−	0.919346i	$-0.628718\pi$
$967$	−2.21264e10	−0.786899	−0.393449	+	0.919346i	$0.628718\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−5.45446e9	−0.191199	−0.0955993	−	0.995420i	$-0.530477\pi$
$971$	−5.45446e9	−0.191199	−0.0955993	+	0.995420i	$0.530477\pi$
$972$	0	0
$973$	−1.07815e10	−0.375217
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−4.95373e10	−1.69942	−0.849711	−	0.527248i	$-0.823224\pi$
$977$	−4.95373e10	−1.69942	−0.849711	+	0.527248i	$0.823224\pi$
$978$	0	0
$979$	−2.30609e10	−0.785482
$980$	0	0
$981$	0	0
$982$	0	0
$983$	8.72042e9	0.292820	0.146410	−	0.989224i	$-0.453228\pi$
$983$	8.72042e9	0.292820	0.146410	+	0.989224i	$0.453228\pi$
$984$	0	0
$985$	6.90027e9	0.230059
$986$	0	0
$987$	0	0
$988$	0	0
$989$	4.33739e10	1.42574
$990$	0	0
$991$	−1.64359e10	−0.536458	−0.268229	−	0.963355i	$-0.586438\pi$
$991$	−1.64359e10	−0.536458	−0.268229	+	0.963355i	$0.586438\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	1.38618e10	0.446106
$996$	0	0
$997$	−3.18717e10	−1.01853	−0.509264	−	0.860611i	$-0.670082\pi$
$997$	−3.18717e10	−1.01853	−0.509264	+	0.860611i	$0.670082\pi$
$998$	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.1.1		2
3.2	odd	2		432.8.a.j.1.2		2
4.3	odd	2		27.8.a.e.1.1	yes	2
12.11	even	2		27.8.a.b.1.2	✓	2
36.7	odd	6		81.8.c.d.28.2		4
36.11	even	6		81.8.c.h.28.1		4
36.23	even	6		81.8.c.h.55.1		4
36.31	odd	6		81.8.c.d.55.2		4

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

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

\(f(q)\)	\(=\)	\(q+65.8132 q^{5} +738.405 q^{7} -4961.26 q^{11} +5967.24 q^{13} +36651.6 q^{17} -22378.9 q^{19} +51473.6 q^{23} -73793.6 q^{25} -68495.7 q^{29} -150655. q^{31} +48596.8 q^{35} +489027. q^{37} +590635. q^{41} +842643. q^{43} -1.22637e6 q^{47} -278301. q^{49} +958904. q^{53} -326517. q^{55} +316269. q^{59} -29722.9 q^{61} +392723. q^{65} -293025. q^{67} -714537. q^{71} -3.96273e6 q^{73} -3.66342e6 q^{77} -2.53805e6 q^{79} -1.66311e6 q^{83} +2.41216e6 q^{85} +4.64819e6 q^{89} +4.40624e6 q^{91} -1.47283e6 q^{95} +1.47010e7 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 180 q^{5} - 700 q^{7} - 10890 q^{11} - 5480 q^{13} + 16416 q^{17} - 16024 q^{19} - 24372 q^{23} - 138880 q^{25} - 143280 q^{29} + 38708 q^{31} - 115650 q^{35} + 455620 q^{37} + 731880 q^{41} + 1088840 q^{43}+ \cdots + 4098670 q^{97}+O(q^{100}) \) 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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