LMFDB - Embedded newform 432.9.g.h.271.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [432,9,Mod(271,432)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("432.271"); S:= CuspForms(chi, 9); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(432, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0])) N = Newforms(chi, 9, names="a")

Level:	$ N $	$=$	$ 432 = 2^{4} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	432.g (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,0,0,0,0,0,0,0,0,0,57036] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$175.987559546$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 3658 x^{10} + 10323672 x^{8} + 10806771976 x^{6} + 8657863222624 x^{4} + 575997114675360 x^{2} + 35\!\cdots\!00 $ x^12 + 3658x^10 + 10323672x^8 + 10806771976x^6 + 8657863222624x^4 + 575997114675360*x^2 + 35494966945166400
Coefficient ring:	$\Z[a_1, \ldots, a_{31}]$
Coefficient ring index:	$ 2^{40}\cdot 3^{25} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			271.8
Root			$17.0122 + 29.4661i$ of defining polynomial
Character	$\chi$	$=$	432.271
Dual form			432.9.g.h.271.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+281.444 q^{5} +23.3884i q^{7} +18435.7i q^{11} -44777.0 q^{13} +94354.3 q^{17} -78057.7i q^{19} -501632. i q^{23} -311414. q^{25} +637248. q^{29} -395010. i q^{31} +6582.51i q^{35} +1.01162e6 q^{37} -3.04204e6 q^{41} +2.55366e6i q^{43} +7.20891e6i q^{47} +5.76425e6 q^{49} +1.51025e7 q^{53} +5.18861e6i q^{55} +176235. i q^{59} -1.37032e7 q^{61} -1.26022e7 q^{65} -3.67424e7i q^{67} +1.04049e7i q^{71} -1.27655e7 q^{73} -431181. q^{77} +4.53221e7i q^{79} -6.30178e7i q^{83} +2.65554e7 q^{85} +3.79752e7 q^{89} -1.04726e6i q^{91} -2.19688e7i q^{95} +7.73978e7 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 57036 q^{13} + 306996 q^{25} - 2032140 q^{37} - 6810648 q^{49} + 28116276 q^{61} + 33515148 q^{73} + 95803200 q^{85} + 90106572 q^{97}+O(q^{100}) $ 12 * q + 57036 * q^13 + 306996 * q^25 - 2032140 * q^37 - 6810648 * q^49 + 28116276 * q^61 + 33515148 * q^73 + 95803200 * q^85 + 90106572 * q^97

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	281.444	0.450310	0.225155	−	0.974323i	$-0.427711\pi$
$5$	281.444	0.450310	0.225155	+	0.974323i	$0.427711\pi$
$6$	0	0
$7$	23.3884i	0.00974110i	0.999988	+	0.00487055i	$0.00155035\pi$
$7$	23.3884i	0.00974110i	−0.999988	+	0.00487055i	$0.998450\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	18435.7i	1.25918i	0.776927	+	0.629591i	$0.216778\pi$
$11$	18435.7i	1.25918i	−0.776927	+	0.629591i	$0.783222\pi$
$12$	0	0
$13$	−44777.0	−1.56777	−0.783884	−	0.620907i	$-0.786764\pi$
$13$	−44777.0	−1.56777	−0.783884	+	0.620907i	$0.786764\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	94354.3	1.12971	0.564854	−	0.825191i	$-0.308933\pi$
$17$	94354.3	1.12971	0.564854	+	0.825191i	$0.308933\pi$
$18$	0	0
$19$	− 78057.7i	− 0.598965i	−0.954102	−	0.299482i	$-0.903186\pi$
$19$	− 78057.7i	− 0.598965i	0.954102	−	0.299482i	$-0.0968140\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 501632.i	− 1.79256i	−0.443486	−	0.896281i	$-0.646258\pi$
$23$	− 501632.i	− 1.79256i	0.443486	−	0.896281i	$-0.353742\pi$
$24$	0	0
$25$	−311414.	−0.797221
$26$	0	0
$27$	0	0
$28$	0	0
$29$	637248.	0.900983	0.450492	−	0.892781i	$-0.351249\pi$
$29$	637248.	0.900983	0.450492	+	0.892781i	$0.351249\pi$
$30$	0	0
$31$	− 395010.i	− 0.427721i	−0.976864	−	0.213861i	$-0.931396\pi$
$31$	− 395010.i	− 0.427721i	0.976864	−	0.213861i	$-0.0686039\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	6582.51i	0.00438651i
$36$	0	0
$37$	1.01162e6	0.539770	0.269885	−	0.962893i	$-0.413014\pi$
$37$	1.01162e6	0.539770	0.269885	+	0.962893i	$0.413014\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−3.04204e6	−1.07654	−0.538269	−	0.842773i	$-0.680922\pi$
$41$	−3.04204e6	−1.07654	−0.538269	+	0.842773i	$0.680922\pi$
$42$	0	0
$43$	2.55366e6i	0.746945i	0.927641	+	0.373472i	$0.121833\pi$
$43$	2.55366e6i	0.746945i	−0.927641	+	0.373472i	$0.878167\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	7.20891e6i	1.47733i	0.674072	+	0.738666i	$0.264544\pi$
$47$	7.20891e6i	1.47733i	−0.674072	+	0.738666i	$0.735456\pi$
$48$	0	0
$49$	5.76425e6	0.999905
$50$	0	0
$51$	0	0
$52$	0	0
$53$	1.51025e7	1.91402	0.957008	−	0.290061i	$-0.0936757\pi$
$53$	1.51025e7	1.91402	0.957008	+	0.290061i	$0.0936757\pi$
$54$	0	0
$55$	5.18861e6i	0.567022i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	176235.i	0.0145440i	0.999974	+	0.00727202i	$0.00231478\pi$
$59$	176235.i	0.0145440i	−0.999974	+	0.00727202i	$0.997685\pi$
$60$	0	0
$61$	−1.37032e7	−0.989699	−0.494850	−	0.868979i	$-0.664777\pi$
$61$	−1.37032e7	−0.989699	−0.494850	+	0.868979i	$0.664777\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.26022e7	−0.705982
$66$	0	0
$67$	− 3.67424e7i	− 1.82334i	−0.410919	−	0.911672i	$-0.634792\pi$
$67$	− 3.67424e7i	− 1.82334i	0.410919	−	0.911672i	$-0.365208\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	1.04049e7i	0.409455i	0.978819	+	0.204728i	$0.0656309\pi$
$71$	1.04049e7i	0.409455i	−0.978819	+	0.204728i	$0.934369\pi$
$72$	0	0
$73$	−1.27655e7	−0.449517	−0.224759	−	0.974414i	$-0.572159\pi$
$73$	−1.27655e7	−0.449517	−0.224759	+	0.974414i	$0.572159\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−431181.	−0.0122658
$78$	0	0
$79$	4.53221e7i	1.16360i	0.813334	+	0.581798i	$0.197650\pi$
$79$	4.53221e7i	1.16360i	−0.813334	+	0.581798i	$0.802350\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 6.30178e7i	− 1.32786i	−0.747797	−	0.663928i	$-0.768888\pi$
$83$	− 6.30178e7i	− 1.32786i	0.747797	−	0.663928i	$-0.231112\pi$
$84$	0	0
$85$	2.65554e7	0.508719
$86$	0	0
$87$	0	0
$88$	0	0
$89$	3.79752e7	0.605257	0.302629	−	0.953109i	$-0.402136\pi$
$89$	3.79752e7	0.605257	0.302629	+	0.953109i	$0.402136\pi$
$90$	0	0
$91$	− 1.04726e6i	− 0.0152718i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 2.19688e7i	− 0.269720i
$96$	0	0
$97$	7.73978e7	0.874262	0.437131	−	0.899398i	$-0.355995\pi$
$97$	7.73978e7	0.874262	0.437131	+	0.899398i	$0.355995\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	6.12667e7	0.588761	0.294380	−	0.955688i	$-0.404887\pi$
$101$	6.12667e7	0.588761	0.294380	+	0.955688i	$0.404887\pi$
$102$	0	0
$103$	1.08143e8i	0.960835i	0.877040	+	0.480417i	$0.159515\pi$
$103$	1.08143e8i	0.960835i	−0.877040	+	0.480417i	$0.840485\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.39322e8i	1.06288i	0.847096	+	0.531440i	$0.178349\pi$
$107$	1.39322e8i	1.06288i	−0.847096	+	0.531440i	$0.821651\pi$
$108$	0	0
$109$	8.33449e7	0.590437	0.295218	−	0.955430i	$-0.404608\pi$
$109$	8.33449e7	0.590437	0.295218	+	0.955430i	$0.404608\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−2.95074e8	−1.80975	−0.904873	−	0.425681i	$-0.860034\pi$
$113$	−2.95074e8	−1.80975	−0.904873	+	0.425681i	$0.860034\pi$
$114$	0	0
$115$	− 1.41181e8i	− 0.807209i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	2.20679e6i	0.0110046i
$120$	0	0
$121$	−1.25516e8	−0.585541
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.97585e8	−0.809307
$126$	0	0
$127$	− 4.93862e7i	− 0.189841i	−0.995485	−	0.0949206i	$-0.969740\pi$
$127$	− 4.93862e7i	− 0.189841i	0.995485	−	0.0949206i	$-0.0302597\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1.85164e8i	0.628740i	0.949300	+	0.314370i	$0.101793\pi$
$131$	1.85164e8i	0.628740i	−0.949300	+	0.314370i	$0.898207\pi$
$132$	0	0
$133$	1.82564e6	0.00583457
$134$	0	0
$135$	0	0
$136$	0	0
$137$	5.56232e7	0.157897	0.0789484	−	0.996879i	$-0.474844\pi$
$137$	5.56232e7	0.157897	0.0789484	+	0.996879i	$0.474844\pi$
$138$	0	0
$139$	3.71559e8i	0.995333i	0.867369	+	0.497666i	$0.165810\pi$
$139$	3.71559e8i	0.995333i	−0.867369	+	0.497666i	$0.834190\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 8.25496e8i	− 1.97411i
$144$	0	0
$145$	1.79350e8	0.405722
$146$	0	0
$147$	0	0
$148$	0	0
$149$	8.18079e8	1.65978	0.829889	−	0.557928i	$-0.188403\pi$
$149$	8.18079e8	1.65978	0.829889	+	0.557928i	$0.188403\pi$
$150$	0	0
$151$	− 1.34848e8i	− 0.259380i	−0.991555	−	0.129690i	$-0.958602\pi$
$151$	− 1.34848e8i	− 0.259380i	0.991555	−	0.129690i	$-0.0413983\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 1.11173e8i	− 0.192607i
$156$	0	0
$157$	5.05470e8	0.831949	0.415974	−	0.909376i	$-0.363441\pi$
$157$	5.05470e8	0.831949	0.415974	+	0.909376i	$0.363441\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1.17324e7	0.0174615
$162$	0	0
$163$	6.96867e8i	0.987187i	0.869693	+	0.493593i	$0.164317\pi$
$163$	6.96867e8i	0.987187i	−0.869693	+	0.493593i	$0.835683\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	9.23757e8i	1.18766i	0.804591	+	0.593829i	$0.202385\pi$
$167$	9.23757e8i	1.18766i	−0.804591	+	0.593829i	$0.797615\pi$
$168$	0	0
$169$	1.18925e9	1.45790
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−1.66993e8	−0.186429	−0.0932144	−	0.995646i	$-0.529714\pi$
$173$	−1.66993e8	−0.186429	−0.0932144	+	0.995646i	$0.529714\pi$
$174$	0	0
$175$	− 7.28348e6i	− 0.00776581i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	5.00340e8i	0.487364i	0.969855	+	0.243682i	$0.0783553\pi$
$179$	5.00340e8i	0.487364i	−0.969855	+	0.243682i	$0.921645\pi$
$180$	0	0
$181$	1.14933e9	1.07086	0.535428	−	0.844581i	$-0.320150\pi$
$181$	1.14933e9	1.07086	0.535428	+	0.844581i	$0.320150\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	2.84713e8	0.243064
$186$	0	0
$187$	1.73949e9i	1.42251i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 3.08471e8i	− 0.231783i	−0.993262	−	0.115891i	$-0.963028\pi$
$191$	− 3.08471e8i	− 0.231783i	0.993262	−	0.115891i	$-0.0369724\pi$
$192$	0	0
$193$	1.29057e9	0.930145	0.465073	−	0.885272i	$-0.346028\pi$
$193$	1.29057e9	0.930145	0.465073	+	0.885272i	$0.346028\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−8.97226e8	−0.595713	−0.297856	−	0.954611i	$-0.596272\pi$
$197$	−8.97226e8	−0.595713	−0.297856	+	0.954611i	$0.596272\pi$
$198$	0	0
$199$	− 2.80181e8i	− 0.178660i	−0.996002	−	0.0893299i	$-0.971527\pi$
$199$	− 2.80181e8i	− 0.178660i	0.996002	−	0.0893299i	$-0.0284725\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1.49042e7i	0.00877657i
$204$	0	0
$205$	−8.56163e8	−0.484776
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1.43905e9	0.754206
$210$	0	0
$211$	− 9.73562e8i	− 0.491172i	−0.969375	−	0.245586i	$-0.921020\pi$
$211$	− 9.73562e8i	− 0.491172i	0.969375	−	0.245586i	$-0.0789804\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	7.18710e8i	0.336357i
$216$	0	0
$217$	9.23864e6	0.00416648
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−4.22491e9	−1.77112
$222$	0	0
$223$	3.12348e9i	1.26305i	0.775358	+	0.631523i	$0.217570\pi$
$223$	3.12348e9i	1.26305i	−0.775358	+	0.631523i	$0.782430\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 8.76571e7i	− 0.0330129i	−0.999864	−	0.0165065i	$-0.994746\pi$
$227$	− 8.76571e7i	− 0.0330129i	0.999864	−	0.0165065i	$-0.00525441\pi$
$228$	0	0
$229$	4.06777e9	1.47916	0.739579	−	0.673070i	$-0.235025\pi$
$229$	4.06777e9	1.47916	0.739579	+	0.673070i	$0.235025\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	4.50672e9	1.52910	0.764551	−	0.644563i	$-0.222961\pi$
$233$	4.50672e9	1.52910	0.764551	+	0.644563i	$0.222961\pi$
$234$	0	0
$235$	2.02890e9i	0.665257i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 8.37062e8i	− 0.256546i	−0.991739	−	0.128273i	$-0.959057\pi$
$239$	− 8.37062e8i	− 0.256546i	0.991739	−	0.128273i	$-0.0409434\pi$
$240$	0	0
$241$	6.15014e9	1.82313	0.911563	−	0.411160i	$-0.134876\pi$
$241$	6.15014e9	1.82313	0.911563	+	0.411160i	$0.134876\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.62231e9	0.450267
$246$	0	0
$247$	3.49519e9i	0.939038i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1.21090e9i	0.305079i	0.988297	+	0.152540i	$0.0487451\pi$
$251$	1.21090e9i	0.305079i	−0.988297	+	0.152540i	$0.951255\pi$
$252$	0	0
$253$	9.24794e9	2.25716
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−2.03803e9	−0.467172	−0.233586	−	0.972336i	$-0.575046\pi$
$257$	−2.03803e9	−0.467172	−0.233586	+	0.972336i	$0.575046\pi$
$258$	0	0
$259$	2.36601e7i	0.00525796i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 7.74386e9i	− 1.61858i	−0.587408	−	0.809291i	$-0.699852\pi$
$263$	− 7.74386e9i	− 1.61858i	0.587408	−	0.809291i	$-0.300148\pi$
$264$	0	0
$265$	4.25051e9	0.861901
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−2.01126e9	−0.384113	−0.192056	−	0.981384i	$-0.561516\pi$
$269$	−2.01126e9	−0.384113	−0.192056	+	0.981384i	$0.561516\pi$
$270$	0	0
$271$	6.63609e9i	1.23037i	0.788383	+	0.615184i	$0.210918\pi$
$271$	6.63609e9i	1.23037i	−0.788383	+	0.615184i	$0.789082\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 5.74114e9i	− 1.00385i
$276$	0	0
$277$	3.03332e9	0.515228	0.257614	−	0.966248i	$-0.417064\pi$
$277$	3.03332e9	0.515228	0.257614	+	0.966248i	$0.417064\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−1.15594e10	−1.85400	−0.926998	−	0.375065i	$-0.877620\pi$
$281$	−1.15594e10	−1.85400	−0.926998	+	0.375065i	$0.877620\pi$
$282$	0	0
$283$	6.64711e9i	1.03630i	0.855288	+	0.518152i	$0.173380\pi$
$283$	6.64711e9i	1.03630i	−0.855288	+	0.518152i	$0.826620\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 7.11484e7i	− 0.0104867i
$288$	0	0
$289$	1.92698e9	0.276239
$290$	0	0
$291$	0	0
$292$	0	0
$293$	1.19517e10	1.62166	0.810830	−	0.585282i	$-0.199016\pi$
$293$	1.19517e10	1.62166	0.810830	+	0.585282i	$0.199016\pi$
$294$	0	0
$295$	4.96004e7i	0.00654933i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2.24616e10i	2.81032i
$300$	0	0
$301$	−5.97259e7	−0.00727606
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−3.85669e9	−0.445672
$306$	0	0
$307$	− 7.03462e9i	− 0.791930i	−0.918266	−	0.395965i	$-0.870410\pi$
$307$	− 7.03462e9i	− 0.791930i	0.918266	−	0.395965i	$-0.129590\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	1.19961e10i	1.28233i	0.767402	+	0.641166i	$0.221549\pi$
$311$	1.19961e10i	1.28233i	−0.767402	+	0.641166i	$0.778451\pi$
$312$	0	0
$313$	1.09732e10	1.14329	0.571645	−	0.820501i	$-0.306305\pi$
$313$	1.09732e10	1.14329	0.571645	+	0.820501i	$0.306305\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2.40572e9	0.238237	0.119118	−	0.992880i	$-0.461993\pi$
$317$	2.40572e9	0.238237	0.119118	+	0.992880i	$0.461993\pi$
$318$	0	0
$319$	1.17481e10i	1.13450i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 7.36508e9i	− 0.676655i
$324$	0	0
$325$	1.39442e10	1.24986
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−1.68605e8	−0.0143908
$330$	0	0
$331$	8.04237e9i	0.669996i	0.942219	+	0.334998i	$0.108736\pi$
$331$	8.04237e9i	0.669996i	−0.942219	+	0.334998i	$0.891264\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 1.03409e10i	− 0.821070i
$336$	0	0
$337$	1.23899e10	0.960609	0.480304	−	0.877102i	$-0.340526\pi$
$337$	1.23899e10	0.960609	0.480304	+	0.877102i	$0.340526\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	7.28228e9	0.538579
$342$	0	0
$343$	2.69646e8i	0.0194813i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 1.72806e10i	− 1.19190i	−0.803021	−	0.595951i	$-0.796775\pi$
$347$	− 1.72806e10i	− 1.19190i	0.803021	−	0.595951i	$-0.203225\pi$
$348$	0	0
$349$	6.66046e9	0.448954	0.224477	−	0.974479i	$-0.427933\pi$
$349$	6.66046e9	0.448954	0.224477	+	0.974479i	$0.427933\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	2.30231e10	1.48274	0.741371	−	0.671096i	$-0.234176\pi$
$353$	2.30231e10	1.48274	0.741371	+	0.671096i	$0.234176\pi$
$354$	0	0
$355$	2.92841e9i	0.184382i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 1.26084e9i	− 0.0759072i	−0.999280	−	0.0379536i	$-0.987916\pi$
$359$	− 1.26084e9i	− 0.0759072i	0.999280	−	0.0379536i	$-0.0120839\pi$
$360$	0	0
$361$	1.08906e10	0.641241
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−3.59277e9	−0.202422
$366$	0	0
$367$	2.95471e10i	1.62873i	0.580350	+	0.814367i	$0.302916\pi$
$367$	2.95471e10i	1.62873i	−0.580350	+	0.814367i	$0.697084\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	3.53223e8i	0.0186446i
$372$	0	0
$373$	2.20480e10	1.13903	0.569513	−	0.821982i	$-0.307132\pi$
$373$	2.20480e10	1.13903	0.569513	+	0.821982i	$0.307132\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2.85341e10	−1.41253
$378$	0	0
$379$	1.84796e10i	0.895643i	0.894123	+	0.447821i	$0.147800\pi$
$379$	1.84796e10i	0.895643i	−0.894123	+	0.447821i	$0.852200\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	2.23271e9i	0.103762i	0.998653	+	0.0518809i	$0.0165216\pi$
$383$	2.23271e9i	0.103762i	−0.998653	+	0.0518809i	$0.983478\pi$
$384$	0	0
$385$	−1.21353e8	−0.00552342
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−1.14835e10	−0.501508	−0.250754	−	0.968051i	$-0.580678\pi$
$389$	−1.14835e10	−0.501508	−0.250754	+	0.968051i	$0.580678\pi$
$390$	0	0
$391$	− 4.73312e10i	− 2.02507i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	1.27556e10i	0.523979i
$396$	0	0
$397$	−6.65121e9	−0.267756	−0.133878	−	0.990998i	$-0.542743\pi$
$397$	−6.65121e9	−0.267756	−0.133878	+	0.990998i	$0.542743\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	3.44492e10	1.33230	0.666149	−	0.745819i	$-0.267942\pi$
$401$	3.44492e10	1.33230	0.666149	+	0.745819i	$0.267942\pi$
$402$	0	0
$403$	1.76874e10i	0.670568i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.86499e10i	0.679669i
$408$	0	0
$409$	−7.81290e9	−0.279202	−0.139601	−	0.990208i	$-0.544582\pi$
$409$	−7.81290e9	−0.279202	−0.139601	+	0.990208i	$0.544582\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−4.12186e6	−0.000141675 0
$414$	0	0
$415$	− 1.77360e10i	− 0.597946i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	4.63977e9i	0.150536i	0.997163	+	0.0752680i	$0.0239812\pi$
$419$	4.63977e9i	0.150536i	−0.997163	+	0.0752680i	$0.976019\pi$
$420$	0	0
$421$	−4.57018e10	−1.45481	−0.727403	−	0.686210i	$-0.759273\pi$
$421$	−4.57018e10	−1.45481	−0.727403	+	0.686210i	$0.759273\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−2.93833e10	−0.900627
$426$	0	0
$427$	− 3.20496e8i	− 0.00964076i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	1.90498e10i	0.552054i	0.961150	+	0.276027i	$0.0890179\pi$
$431$	1.90498e10i	0.552054i	−0.961150	+	0.276027i	$0.910982\pi$
$432$	0	0
$433$	1.32123e10	0.375862	0.187931	−	0.982182i	$-0.439822\pi$
$433$	1.32123e10	0.375862	0.187931	+	0.982182i	$0.439822\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−3.91563e10	−1.07368
$438$	0	0
$439$	− 6.29790e9i	− 0.169565i	−0.996399	−	0.0847827i	$-0.972980\pi$
$439$	− 6.29790e9i	− 0.169565i	0.996399	−	0.0847827i	$-0.0270196\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 6.36986e10i	− 1.65392i	−0.562259	−	0.826961i	$-0.690068\pi$
$443$	− 6.36986e10i	− 1.65392i	0.562259	−	0.826961i	$-0.309932\pi$
$444$	0	0
$445$	1.06879e10	0.272553
$446$	0	0
$447$	0	0
$448$	0	0
$449$	4.30959e10	1.06035	0.530177	−	0.847887i	$-0.322126\pi$
$449$	4.30959e10	1.06035	0.530177	+	0.847887i	$0.322126\pi$
$450$	0	0
$451$	− 5.60821e10i	− 1.35556i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 2.94745e8i	− 0.00687704i
$456$	0	0
$457$	−3.84852e10	−0.882325	−0.441162	−	0.897427i	$-0.645434\pi$
$457$	−3.84852e10	−0.882325	−0.441162	+	0.897427i	$0.645434\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−8.45292e9	−0.187156	−0.0935779	−	0.995612i	$-0.529830\pi$
$461$	−8.45292e9	−0.187156	−0.0935779	+	0.995612i	$0.529830\pi$
$462$	0	0
$463$	5.80849e10i	1.26398i	0.774978	+	0.631989i	$0.217761\pi$
$463$	5.80849e10i	1.26398i	−0.774978	+	0.631989i	$0.782239\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 1.24229e10i	− 0.261189i	−0.991436	−	0.130595i	$-0.958311\pi$
$467$	− 1.24229e10i	− 0.261189i	0.991436	−	0.130595i	$-0.0416886\pi$
$468$	0	0
$469$	8.59346e8	0.0177614
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−4.70784e10	−0.940540
$474$	0	0
$475$	2.43083e10i	0.477507i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	5.43435e10i	1.03230i	0.856499	+	0.516149i	$0.172635\pi$
$479$	5.43435e10i	1.03230i	−0.856499	+	0.516149i	$0.827365\pi$
$480$	0	0
$481$	−4.52972e10	−0.846235
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2.17831e10	0.393689
$486$	0	0
$487$	− 1.23281e9i	− 0.0219169i	−0.999940	−	0.0109585i	$-0.996512\pi$
$487$	− 1.23281e9i	− 0.0219169i	0.999940	−	0.0109585i	$-0.00348826\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 1.69739e10i	− 0.292050i	−0.989281	−	0.146025i	$-0.953352\pi$
$491$	− 1.69739e10i	− 0.292050i	0.989281	−	0.146025i	$-0.0466479\pi$
$492$	0	0
$493$	6.01271e10	1.01785
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−2.43355e8	−0.00398854
$498$	0	0
$499$	− 6.22259e10i	− 1.00362i	−0.864978	−	0.501809i	$-0.832668\pi$
$499$	− 6.22259e10i	− 1.00362i	0.864978	−	0.501809i	$-0.167332\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 9.30365e10i	− 1.45339i	−0.686962	−	0.726693i	$-0.741056\pi$
$503$	− 9.30365e10i	− 1.45339i	0.686962	−	0.726693i	$-0.258944\pi$
$504$	0	0
$505$	1.72431e10	0.265125
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−7.16721e9	−0.106777	−0.0533887	−	0.998574i	$-0.517002\pi$
$509$	−7.16721e9	−0.106777	−0.0533887	+	0.998574i	$0.517002\pi$
$510$	0	0
$511$	− 2.98564e8i	− 0.00437879i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	3.04361e10i	0.432674i
$516$	0	0
$517$	−1.32901e11	−1.86023
$518$	0	0
$519$	0	0
$520$	0	0
$521$	8.79857e10	1.19416	0.597078	−	0.802183i	$-0.296328\pi$
$521$	8.79857e10	1.19416	0.597078	+	0.802183i	$0.296328\pi$
$522$	0	0
$523$	− 1.48988e11i	− 1.99134i	−0.0929812	−	0.995668i	$-0.529640\pi$
$523$	− 1.48988e11i	− 1.99134i	0.0929812	−	0.995668i	$-0.470360\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 3.72709e10i	− 0.483200i
$528$	0	0
$529$	−1.73324e11	−2.21328
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1.36214e11	1.68776
$534$	0	0
$535$	3.92112e10i	0.478625i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1.06268e11i	1.25906i
$540$	0	0
$541$	−1.63747e11	−1.91154	−0.955769	−	0.294117i	$-0.904974\pi$
$541$	−1.63747e11	−1.91154	−0.955769	+	0.294117i	$0.904974\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	2.34569e10	0.265880
$546$	0	0
$547$	− 8.91292e10i	− 0.995567i	−0.867301	−	0.497784i	$-0.834147\pi$
$547$	− 8.91292e10i	− 0.995567i	0.867301	−	0.497784i	$-0.165853\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 4.97421e10i	− 0.539657i
$552$	0	0
$553$	−1.06001e9	−0.0113347
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−1.64327e11	−1.70722	−0.853608	−	0.520916i	$-0.825591\pi$
$557$	−1.64327e11	−1.70722	−0.853608	+	0.520916i	$0.825591\pi$
$558$	0	0
$559$	− 1.14345e11i	− 1.17104i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	1.70133e11i	1.69339i	0.532082	+	0.846693i	$0.321410\pi$
$563$	1.70133e11i	1.69339i	−0.532082	+	0.846693i	$0.678590\pi$
$564$	0	0
$565$	−8.30468e10	−0.814947
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−4.60379e10	−0.439205	−0.219602	−	0.975589i	$-0.570476\pi$
$569$	−4.60379e10	−0.439205	−0.219602	+	0.975589i	$0.570476\pi$
$570$	0	0
$571$	− 1.38854e11i	− 1.30621i	−0.757266	−	0.653107i	$-0.773465\pi$
$571$	− 1.38854e11i	− 1.30621i	0.757266	−	0.653107i	$-0.226535\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1.56216e11i	1.42907i
$576$	0	0
$577$	2.21757e10	0.200066	0.100033	−	0.994984i	$-0.468105\pi$
$577$	2.21757e10	0.200066	0.100033	+	0.994984i	$0.468105\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	1.47388e9	0.0129348
$582$	0	0
$583$	2.78425e11i	2.41010i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 1.45439e11i	− 1.22498i	−0.790478	−	0.612490i	$-0.790168\pi$
$587$	− 1.45439e11i	− 1.22498i	0.790478	−	0.612490i	$-0.209832\pi$
$588$	0	0
$589$	−3.08335e10	−0.256190
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−1.38982e11	−1.12393	−0.561964	−	0.827162i	$-0.689954\pi$
$593$	−1.38982e11	−1.12393	−0.561964	+	0.827162i	$0.689954\pi$
$594$	0	0
$595$	6.21088e8i	0.00495548i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	9.36649e10i	0.727561i	0.931485	+	0.363780i	$0.118514\pi$
$599$	9.36649e10i	0.727561i	−0.931485	+	0.363780i	$0.881486\pi$
$600$	0	0
$601$	−2.08604e11	−1.59892	−0.799458	−	0.600722i	$-0.794880\pi$
$601$	−2.08604e11	−1.59892	−0.799458	+	0.600722i	$0.794880\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−3.53256e10	−0.263675
$606$	0	0
$607$	− 2.06930e11i	− 1.52429i	−0.647405	−	0.762146i	$-0.724146\pi$
$607$	− 2.06930e11i	− 1.52429i	0.647405	−	0.762146i	$-0.275854\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 3.22794e11i	− 2.31611i
$612$	0	0
$613$	9.61960e10	0.681263	0.340632	−	0.940197i	$-0.389359\pi$
$613$	9.61960e10	0.681263	0.340632	+	0.940197i	$0.389359\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1.50368e11	1.03756	0.518782	−	0.854907i	$-0.326386\pi$
$617$	1.50368e11	1.03756	0.518782	+	0.854907i	$0.326386\pi$
$618$	0	0
$619$	1.32770e11i	0.904350i	0.891929	+	0.452175i	$0.149352\pi$
$619$	1.32770e11i	0.904350i	−0.891929	+	0.452175i	$0.850648\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	8.88178e8i	0.00589587i
$624$	0	0
$625$	6.60373e10	0.432782
$626$	0	0
$627$	0	0
$628$	0	0
$629$	9.54504e10	0.609783
$630$	0	0
$631$	− 2.13375e11i	− 1.34594i	−0.739670	−	0.672970i	$-0.765018\pi$
$631$	− 2.13375e11i	− 1.34594i	0.739670	−	0.672970i	$-0.234982\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 1.38994e10i	− 0.0854874i
$636$	0	0
$637$	−2.58106e11	−1.56762
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−2.20634e10	−0.130689	−0.0653446	−	0.997863i	$-0.520815\pi$
$641$	−2.20634e10	−0.130689	−0.0653446	+	0.997863i	$0.520815\pi$
$642$	0	0
$643$	− 4.70805e10i	− 0.275421i	−0.990473	−	0.137711i	$-0.956026\pi$
$643$	− 4.70805e10i	− 0.275421i	0.990473	−	0.137711i	$-0.0439744\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 9.60972e10i	− 0.548395i	−0.961673	−	0.274198i	$-0.911588\pi$
$647$	− 9.60972e10i	− 0.548395i	0.961673	−	0.274198i	$-0.0884123\pi$
$648$	0	0
$649$	−3.24902e9	−0.0183136
$650$	0	0
$651$	0	0
$652$	0	0
$653$	2.52293e11	1.38756	0.693781	−	0.720187i	$-0.255944\pi$
$653$	2.52293e11	1.38756	0.693781	+	0.720187i	$0.255944\pi$
$654$	0	0
$655$	5.21132e10i	0.283128i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	1.11758e11i	0.592569i	0.955100	+	0.296284i	$0.0957476\pi$
$659$	1.11758e11i	0.592569i	−0.955100	+	0.296284i	$0.904252\pi$
$660$	0	0
$661$	−3.57895e10	−0.187478	−0.0937388	−	0.995597i	$-0.529882\pi$
$661$	−3.57895e10	−0.187478	−0.0937388	+	0.995597i	$0.529882\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	5.13816e8	0.00262737
$666$	0	0
$667$	− 3.19664e11i	− 1.61507i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 2.52628e11i	− 1.24621i
$672$	0	0
$673$	−1.87846e11	−0.915675	−0.457837	−	0.889036i	$-0.651376\pi$
$673$	−1.87846e11	−0.915675	−0.457837	+	0.889036i	$0.651376\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	6.74445e10	0.321064	0.160532	−	0.987031i	$-0.448679\pi$
$677$	6.74445e10	0.321064	0.160532	+	0.987031i	$0.448679\pi$
$678$	0	0
$679$	1.81021e9i	0.00851627i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	1.74004e11i	0.799605i	0.916601	+	0.399802i	$0.130921\pi$
$683$	1.74004e11i	0.799605i	−0.916601	+	0.399802i	$0.869079\pi$
$684$	0	0
$685$	1.56548e10	0.0711025
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−6.76246e11	−3.00073
$690$	0	0
$691$	3.19656e11i	1.40207i	0.713125	+	0.701036i	$0.247279\pi$
$691$	3.19656e11i	1.40207i	−0.713125	+	0.701036i	$0.752721\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1.04573e11i	0.448208i
$696$	0	0
$697$	−2.87030e11	−1.21617
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−6.11911e10	−0.253406	−0.126703	−	0.991941i	$-0.540439\pi$
$701$	−6.11911e10	−0.253406	−0.126703	+	0.991941i	$0.540439\pi$
$702$	0	0
$703$	− 7.89644e10i	− 0.323303i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.43293e9i	0.00573518i
$708$	0	0
$709$	−9.28241e10	−0.367347	−0.183673	−	0.982987i	$-0.558799\pi$
$709$	−9.28241e10	−0.367347	−0.183673	+	0.982987i	$0.558799\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−1.98150e11	−0.766717
$714$	0	0
$715$	− 2.32331e11i	− 0.888960i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	2.12759e11i	0.796108i	0.917362	+	0.398054i	$0.130314\pi$
$719$	2.12759e11i	0.796108i	−0.917362	+	0.398054i	$0.869686\pi$
$720$	0	0
$721$	−2.52929e9	−0.00935959
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−1.98448e11	−0.718283
$726$	0	0
$727$	3.79275e11i	1.35774i	0.734259	+	0.678870i	$0.237530\pi$
$727$	3.79275e11i	1.35774i	−0.734259	+	0.678870i	$0.762470\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	2.40948e11i	0.843829i
$732$	0	0
$733$	2.28187e11	0.790452	0.395226	−	0.918584i	$-0.370666\pi$
$733$	2.28187e11	0.790452	0.395226	+	0.918584i	$0.370666\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	6.77372e11	2.29592
$738$	0	0
$739$	− 4.39440e11i	− 1.47340i	−0.676218	−	0.736702i	$-0.736382\pi$
$739$	− 4.39440e11i	− 1.47340i	0.676218	−	0.736702i	$-0.263618\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 3.99231e11i	− 1.30999i	−0.755632	−	0.654996i	$-0.772670\pi$
$743$	− 3.99231e11i	− 1.30999i	0.755632	−	0.654996i	$-0.227330\pi$
$744$	0	0
$745$	2.30243e11	0.747415
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−3.25851e9	−0.0103536
$750$	0	0
$751$	3.65748e11i	1.14980i	0.818224	+	0.574900i	$0.194959\pi$
$751$	3.65748e11i	1.14980i	−0.818224	+	0.574900i	$0.805041\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 3.79522e10i	− 0.116802i
$756$	0	0
$757$	4.43728e11	1.35124	0.675621	−	0.737249i	$-0.263875\pi$
$757$	4.43728e11	1.35124	0.675621	+	0.737249i	$0.263875\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	5.25382e11	1.56652	0.783261	−	0.621694i	$-0.213555\pi$
$761$	5.25382e11	1.56652	0.783261	+	0.621694i	$0.213555\pi$
$762$	0	0
$763$	1.94930e9i	0.00575150i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 7.89130e9i	− 0.0228017i
$768$	0	0
$769$	3.30392e11	0.944767	0.472383	−	0.881393i	$-0.343394\pi$
$769$	3.30392e11	0.944767	0.472383	+	0.881393i	$0.343394\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−4.39513e11	−1.23099	−0.615494	−	0.788142i	$-0.711043\pi$
$773$	−4.39513e11	−1.23099	−0.615494	+	0.788142i	$0.711043\pi$
$774$	0	0
$775$	1.23012e11i	0.340988i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	2.37455e11i	0.644808i
$780$	0	0
$781$	−1.91822e11	−0.515579
$782$	0	0
$783$	0	0
$784$	0	0
$785$	1.42261e11	0.374635
$786$	0	0
$787$	2.54826e11i	0.664269i	0.943232	+	0.332135i	$0.107769\pi$
$787$	2.54826e11i	0.664269i	−0.943232	+	0.332135i	$0.892231\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 6.90131e9i	− 0.0176289i
$792$	0	0
$793$	6.13590e11	1.55162
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.50959e11	0.374133	0.187066	−	0.982347i	$-0.440102\pi$
$797$	1.50959e11	0.374133	0.187066	+	0.982347i	$0.440102\pi$
$798$	0	0
$799$	6.80192e11i	1.66895i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 2.35341e11i	− 0.566024i
$804$	0	0
$805$	3.30200e9	0.00786310
$806$	0	0
$807$	0	0
$808$	0	0
$809$	5.70448e10	0.133175	0.0665874	−	0.997781i	$-0.478789\pi$
$809$	5.70448e10	0.133175	0.0665874	+	0.997781i	$0.478789\pi$
$810$	0	0
$811$	− 6.34392e11i	− 1.46647i	−0.679973	−	0.733237i	$-0.738009\pi$
$811$	− 6.34392e11i	− 1.46647i	0.679973	−	0.733237i	$-0.261991\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1.96129e11i	0.444540i
$816$	0	0
$817$	1.99332e11	0.447393
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1.90988e11	−0.420372	−0.210186	−	0.977661i	$-0.567407\pi$
$821$	−1.90988e11	−0.420372	−0.210186	+	0.977661i	$0.567407\pi$
$822$	0	0
$823$	3.46201e11i	0.754622i	0.926087	+	0.377311i	$0.123151\pi$
$823$	3.46201e11i	0.754622i	−0.926087	+	0.377311i	$0.876849\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	5.59159e11i	1.19540i	0.801720	+	0.597700i	$0.203919\pi$
$827$	5.59159e11i	1.19540i	−0.801720	+	0.597700i	$0.796081\pi$
$828$	0	0
$829$	5.07979e11	1.07554	0.537771	−	0.843091i	$-0.319266\pi$
$829$	5.07979e11	1.07554	0.537771	+	0.843091i	$0.319266\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	5.43882e11	1.12960
$834$	0	0
$835$	2.59986e11i	0.534815i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 4.60725e11i	− 0.929809i	−0.885361	−	0.464905i	$-0.846089\pi$
$839$	− 4.60725e11i	− 0.929809i	0.885361	−	0.464905i	$-0.153911\pi$
$840$	0	0
$841$	−9.41611e10	−0.188229
$842$	0	0
$843$	0	0
$844$	0	0
$845$	3.34708e11	0.656506
$846$	0	0
$847$	− 2.93561e9i	− 0.00570381i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 5.07460e11i	− 0.967572i
$852$	0	0
$853$	−6.53220e11	−1.23385	−0.616926	−	0.787021i	$-0.711622\pi$
$853$	−6.53220e11	−1.23385	−0.616926	+	0.787021i	$0.711622\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−6.24697e10	−0.115810	−0.0579050	−	0.998322i	$-0.518442\pi$
$857$	−6.24697e10	−0.115810	−0.0579050	+	0.998322i	$0.518442\pi$
$858$	0	0
$859$	− 8.00732e11i	− 1.47067i	−0.677706	−	0.735333i	$-0.737026\pi$
$859$	− 8.00732e11i	− 1.47067i	0.677706	−	0.735333i	$-0.262974\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 8.00083e11i	− 1.44242i	−0.692716	−	0.721210i	$-0.743586\pi$
$863$	− 8.00083e11i	− 1.44242i	0.692716	−	0.721210i	$-0.256414\pi$
$864$	0	0
$865$	−4.69991e10	−0.0839508
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−8.35545e11	−1.46518
$870$	0	0
$871$	1.64522e12i	2.85858i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 4.62118e9i	− 0.00788354i
$876$	0	0
$877$	9.95688e11	1.68316	0.841579	−	0.540133i	$-0.181626\pi$
$877$	9.95688e11	1.68316	0.841579	+	0.540133i	$0.181626\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−1.11519e12	−1.85117	−0.925586	−	0.378536i	$-0.876428\pi$
$881$	−1.11519e12	−1.85117	−0.925586	+	0.378536i	$0.876428\pi$
$882$	0	0
$883$	− 8.38877e11i	− 1.37992i	−0.723845	−	0.689962i	$-0.757627\pi$
$883$	− 8.38877e11i	− 1.37992i	0.723845	−	0.689962i	$-0.242373\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	9.54412e11i	1.54185i	0.636928	+	0.770923i	$0.280205\pi$
$887$	9.54412e11i	1.54185i	−0.636928	+	0.770923i	$0.719795\pi$
$888$	0	0
$889$	1.15506e9	0.00184926
$890$	0	0
$891$	0	0
$892$	0	0
$893$	5.62711e11	0.884869
$894$	0	0
$895$	1.40818e11i	0.219465i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 2.51719e11i	− 0.385370i
$900$	0	0
$901$	1.42499e12	2.16228
$902$	0	0
$903$	0	0
$904$	0	0
$905$	3.23472e11	0.482217
$906$	0	0
$907$	9.64332e11i	1.42494i	0.701701	+	0.712472i	$0.252424\pi$
$907$	9.64332e11i	1.42494i	−0.701701	+	0.712472i	$0.747576\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1.24511e12i	1.80774i	0.427808	+	0.903870i	$0.359286\pi$
$911$	1.24511e12i	1.80774i	−0.427808	+	0.903870i	$0.640714\pi$
$912$	0	0
$913$	1.16178e12	1.67201
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−4.33068e9	−0.00612462
$918$	0	0
$919$	3.83262e11i	0.537321i	0.963235	+	0.268660i	$0.0865809\pi$
$919$	3.83262e11i	0.537321i	−0.963235	+	0.268660i	$0.913419\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 4.65903e11i	− 0.641931i
$924$	0	0
$925$	−3.15032e11	−0.430316
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−4.00117e11	−0.537185	−0.268592	−	0.963254i	$-0.586558\pi$
$929$	−4.00117e11	−0.537185	−0.268592	+	0.963254i	$0.586558\pi$
$930$	0	0
$931$	− 4.49944e11i	− 0.598908i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	4.89568e11i	0.640570i
$936$	0	0
$937$	1.10143e12	1.42889	0.714446	−	0.699691i	$-0.246679\pi$
$937$	1.10143e12	1.42889	0.714446	+	0.699691i	$0.246679\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1.39883e11	0.178404	0.0892022	−	0.996014i	$-0.471568\pi$
$941$	1.39883e11	0.178404	0.0892022	+	0.996014i	$0.471568\pi$
$942$	0	0
$943$	1.52599e12i	1.92976i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	2.60667e11i	0.324105i	0.986782	+	0.162053i	$0.0518115\pi$
$947$	2.60667e11i	0.324105i	−0.986782	+	0.162053i	$0.948189\pi$
$948$	0	0
$949$	5.71601e11	0.704739
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−7.51963e11	−0.911643	−0.455822	−	0.890071i	$-0.650655\pi$
$953$	−7.51963e11	−0.911643	−0.455822	+	0.890071i	$0.650655\pi$
$954$	0	0
$955$	− 8.68173e10i	− 0.104374i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1.30094e9i	0.00153809i
$960$	0	0
$961$	6.96858e11	0.817054
$962$	0	0
$963$	0	0
$964$	0	0
$965$	3.63222e11	0.418854
$966$	0	0
$967$	− 4.17530e10i	− 0.0477509i	−0.999715	−	0.0238754i	$-0.992399\pi$
$967$	− 4.17530e10i	− 0.0477509i	0.999715	−	0.0238754i	$-0.00760051\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 3.96128e11i	− 0.445614i	−0.974863	−	0.222807i	$-0.928478\pi$
$971$	− 3.96128e11i	− 0.445614i	0.974863	−	0.222807i	$-0.0715220\pi$
$972$	0	0
$973$	−8.69016e9	−0.00969563
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−6.41869e11	−0.704479	−0.352239	−	0.935910i	$-0.614580\pi$
$977$	−6.41869e11	−0.704479	−0.352239	+	0.935910i	$0.614580\pi$
$978$	0	0
$979$	7.00099e11i	0.762129i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 1.01844e12i	− 1.09074i	−0.838195	−	0.545371i	$-0.816389\pi$
$983$	− 1.01844e12i	− 1.09074i	0.838195	−	0.545371i	$-0.183611\pi$
$984$	0	0
$985$	−2.52519e11	−0.268255
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1.28100e12	1.33895
$990$	0	0
$991$	− 1.56768e11i	− 0.162540i	−0.996692	−	0.0812702i	$-0.974102\pi$
$991$	− 1.56768e11i	− 0.162540i	0.996692	−	0.0812702i	$-0.0258977\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 7.88552e10i	− 0.0804523i
$996$	0	0
$997$	2.78658e11	0.282027	0.141014	−	0.990008i	$-0.454964\pi$
$997$	2.78658e11	0.282027	0.141014	+	0.990008i	$0.454964\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	432.9.g.h.271.8	yes	12
3.2	odd	2	inner	432.9.g.h.271.6	yes	12
4.3	odd	2	inner	432.9.g.h.271.7	yes	12
12.11	even	2	inner	432.9.g.h.271.5	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
432.9.g.h.271.5	✓	12	12.11	even	2	inner
432.9.g.h.271.6	yes	12	3.2	odd	2	inner
432.9.g.h.271.7	yes	12	4.3	odd	2	inner
432.9.g.h.271.8	yes	12	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 432 = 2^{4} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	432.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+281.444 q^{5} +23.3884i q^{7} +18435.7i q^{11} -44777.0 q^{13} +94354.3 q^{17} -78057.7i q^{19} -501632. i q^{23} -311414. q^{25} +637248. q^{29} -395010. i q^{31} +6582.51i q^{35} +1.01162e6 q^{37} -3.04204e6 q^{41} +2.55366e6i q^{43} +7.20891e6i q^{47} +5.76425e6 q^{49} +1.51025e7 q^{53} +5.18861e6i q^{55} +176235. i q^{59} -1.37032e7 q^{61} -1.26022e7 q^{65} -3.67424e7i q^{67} +1.04049e7i q^{71} -1.27655e7 q^{73} -431181. q^{77} +4.53221e7i q^{79} -6.30178e7i q^{83} +2.65554e7 q^{85} +3.79752e7 q^{89} -1.04726e6i q^{91} -2.19688e7i q^{95} +7.73978e7 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 57036 q^{13} + 306996 q^{25} - 2032140 q^{37} - 6810648 q^{49} + 28116276 q^{61} + 33515148 q^{73} + 95803200 q^{85} + 90106572 q^{97}+O(q^{100}) \) 12 * q + 57036 * q^13 + 306996 * q^25 - 2032140 * q^37 - 6810648 * q^49 + 28116276 * q^61 + 33515148 * q^73 + 95803200 * q^85 + 90106572 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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