LMFDB - Embedded newform 4320.2.a.bg.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4320,2,Mod(1,4320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4320, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4320.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4320 = 2^{5} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4320.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:	$34.4953736732$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1509.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 7x + 4 $ x^3 - x^2 - 7*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.47735$ of defining polynomial
Character	$\chi$	$=$	4320.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} +3.61463 q^{7} +O(q^{10})$	$q-1.00000 q^{5} +3.61463 q^{7} -2.34008 q^{11} -3.95470 q^{13} -2.34008 q^{17} +1.61463 q^{19} +2.61463 q^{23} +1.00000 q^{25} -4.61463 q^{29} -0.340078 q^{31} -3.61463 q^{35} +7.61463 q^{37} +9.90941 q^{41} +9.29478 q^{43} +1.38537 q^{47} +6.06553 q^{49} -4.00000 q^{53} +2.34008 q^{55} -11.2293 q^{59} +7.61463 q^{61} +3.95470 q^{65} -8.29478 q^{67} +13.9094 q^{71} +14.8439 q^{73} -8.45851 q^{77} +0.0452953 q^{79} +15.1387 q^{83} +2.34008 q^{85} -2.68016 q^{89} -14.2948 q^{91} -1.61463 q^{95} +4.38537 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{5} - q^{7}+O(q^{10}) $ 3 * q - 3 * q^5 - q^7	$ 3 q - 3 q^{5} - q^{7} - 2 q^{11} + 5 q^{13} - 2 q^{17} - 7 q^{19} - 4 q^{23} + 3 q^{25} - 2 q^{29} + 4 q^{31} + q^{35} + 11 q^{37} - 4 q^{41} + 6 q^{43} + 16 q^{47} + 20 q^{49} - 12 q^{53} + 2 q^{55} - 10 q^{59} + 11 q^{61} - 5 q^{65} - 3 q^{67} + 8 q^{71} + 9 q^{73} + 22 q^{77} + 17 q^{79} - 12 q^{83} + 2 q^{85} + 2 q^{89} - 21 q^{91} + 7 q^{95} + 25 q^{97}+O(q^{100}) $ 3 * q - 3 * q^5 - q^7 - 2 * q^11 + 5 * q^13 - 2 * q^17 - 7 * q^19 - 4 * q^23 + 3 * q^25 - 2 * q^29 + 4 * q^31 + q^35 + 11 * q^37 - 4 * q^41 + 6 * q^43 + 16 * q^47 + 20 * q^49 - 12 * q^53 + 2 * q^55 - 10 * q^59 + 11 * q^61 - 5 * q^65 - 3 * q^67 + 8 * q^71 + 9 * q^73 + 22 * q^77 + 17 * q^79 - 12 * q^83 + 2 * q^85 + 2 * q^89 - 21 * q^91 + 7 * q^95 + 25 * q^97

Display 100 coefficients 10 coefficients

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
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	3.61463	1.36620	0.683100	−	0.730325i	$-0.260631\pi$
$7$	3.61463	1.36620	0.683100	+	0.730325i	$0.260631\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−2.34008	−0.705560	−0.352780	−	0.935706i	$-0.614764\pi$
$11$	−2.34008	−0.705560	−0.352780	+	0.935706i	$0.614764\pi$
$12$	0	0
$13$	−3.95470	−1.09684	−0.548419	−	0.836204i	$-0.684770\pi$
$13$	−3.95470	−1.09684	−0.548419	+	0.836204i	$0.684770\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.34008	−0.567552	−0.283776	−	0.958891i	$-0.591587\pi$
$17$	−2.34008	−0.567552	−0.283776	+	0.958891i	$0.591587\pi$
$18$	0	0
$19$	1.61463	0.370421	0.185210	−	0.982699i	$-0.440703\pi$
$19$	1.61463	0.370421	0.185210	+	0.982699i	$0.440703\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.61463	0.545187	0.272594	−	0.962129i	$-0.412119\pi$
$23$	2.61463	0.545187	0.272594	+	0.962129i	$0.412119\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−4.61463	−0.856915	−0.428457	−	0.903562i	$-0.640943\pi$
$29$	−4.61463	−0.856915	−0.428457	+	0.903562i	$0.640943\pi$
$30$	0	0
$31$	−0.340078	−0.0610798	−0.0305399	−	0.999534i	$-0.509723\pi$
$31$	−0.340078	−0.0610798	−0.0305399	+	0.999534i	$0.509723\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.61463	−0.610983
$36$	0	0
$37$	7.61463	1.25184	0.625918	−	0.779888i	$-0.284724\pi$
$37$	7.61463	1.25184	0.625918	+	0.779888i	$0.284724\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	9.90941	1.54759	0.773795	−	0.633436i	$-0.218356\pi$
$41$	9.90941	1.54759	0.773795	+	0.633436i	$0.218356\pi$
$42$	0	0
$43$	9.29478	1.41744	0.708721	−	0.705489i	$-0.249273\pi$
$43$	9.29478	1.41744	0.708721	+	0.705489i	$0.249273\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.38537	0.202077	0.101039	−	0.994882i	$-0.467783\pi$
$47$	1.38537	0.202077	0.101039	+	0.994882i	$0.467783\pi$
$48$	0	0
$49$	6.06553	0.866504
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−4.00000	−0.549442	−0.274721	−	0.961524i	$-0.588586\pi$
$53$	−4.00000	−0.549442	−0.274721	+	0.961524i	$0.588586\pi$
$54$	0	0
$55$	2.34008	0.315536
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−11.2293	−1.46192	−0.730962	−	0.682418i	$-0.760928\pi$
$59$	−11.2293	−1.46192	−0.730962	+	0.682418i	$0.760928\pi$
$60$	0	0
$61$	7.61463	0.974953	0.487477	−	0.873136i	$-0.337917\pi$
$61$	7.61463	0.974953	0.487477	+	0.873136i	$0.337917\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	3.95470	0.490521
$66$	0	0
$67$	−8.29478	−1.01337	−0.506684	−	0.862132i	$-0.669129\pi$
$67$	−8.29478	−1.01337	−0.506684	+	0.862132i	$0.669129\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	13.9094	1.65074	0.825372	−	0.564590i	$-0.190966\pi$
$71$	13.9094	1.65074	0.825372	+	0.564590i	$0.190966\pi$
$72$	0	0
$73$	14.8439	1.73734	0.868672	−	0.495387i	$-0.164974\pi$
$73$	14.8439	1.73734	0.868672	+	0.495387i	$0.164974\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−8.45851	−0.963936
$78$	0	0
$79$	0.0452953	0.00509612	0.00254806	−	0.999997i	$-0.499189\pi$
$79$	0.0452953	0.00509612	0.00254806	+	0.999997i	$0.499189\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	15.1387	1.66168	0.830842	−	0.556508i	$-0.187859\pi$
$83$	15.1387	1.66168	0.830842	+	0.556508i	$0.187859\pi$
$84$	0	0
$85$	2.34008	0.253817
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−2.68016	−0.284096	−0.142048	−	0.989860i	$-0.545369\pi$
$89$	−2.68016	−0.284096	−0.142048	+	0.989860i	$0.545369\pi$
$90$	0	0
$91$	−14.2948	−1.49850
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.61463	−0.165657
$96$	0	0
$97$	4.38537	0.445267	0.222634	−	0.974902i	$-0.428535\pi$
$97$	4.38537	0.445267	0.222634	+	0.974902i	$0.428535\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	4.61463	0.459173	0.229586	−	0.973288i	$-0.426263\pi$
$101$	4.61463	0.459173	0.229586	+	0.973288i	$0.426263\pi$
$102$	0	0
$103$	−0.934472	−0.0920762	−0.0460381	−	0.998940i	$-0.514660\pi$
$103$	−0.934472	−0.0920762	−0.0460381	+	0.998940i	$0.514660\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−6.00000	−0.580042	−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000	−0.580042	−0.290021	+	0.957020i	$0.593662\pi$
$108$	0	0
$109$	15.2293	1.45870	0.729349	−	0.684142i	$-0.239823\pi$
$109$	15.2293	1.45870	0.729349	+	0.684142i	$0.239823\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−10.3401	−0.972713	−0.486356	−	0.873761i	$-0.661674\pi$
$113$	−10.3401	−0.972713	−0.486356	+	0.873761i	$0.661674\pi$
$114$	0	0
$115$	−2.61463	−0.243815
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−8.45851	−0.775390
$120$	0	0
$121$	−5.52404	−0.502185
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−3.31984	−0.294589	−0.147294	−	0.989093i	$-0.547056\pi$
$127$	−3.31984	−0.294589	−0.147294	+	0.989093i	$0.547056\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1.56933	0.137113	0.0685566	−	0.997647i	$-0.478161\pi$
$131$	1.56933	0.137113	0.0685566	+	0.997647i	$0.478161\pi$
$132$	0	0
$133$	5.83627	0.506069
$134$	0	0
$135$	0	0
$136$	0	0
$137$	10.5491	0.901270	0.450635	−	0.892708i	$-0.351198\pi$
$137$	10.5491	0.901270	0.450635	+	0.892708i	$0.351198\pi$
$138$	0	0
$139$	−0.934472	−0.0792608	−0.0396304	−	0.999214i	$-0.512618\pi$
$139$	−0.934472	−0.0792608	−0.0396304	+	0.999214i	$0.512618\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	9.25432	0.773885
$144$	0	0
$145$	4.61463	0.383224
$146$	0	0
$147$	0	0
$148$	0	0
$149$	17.8439	1.46183	0.730914	−	0.682470i	$-0.239094\pi$
$149$	17.8439	1.46183	0.730914	+	0.682470i	$0.239094\pi$
$150$	0	0
$151$	4.04530	0.329201	0.164601	−	0.986360i	$-0.447366\pi$
$151$	4.04530	0.329201	0.164601	+	0.986360i	$0.447366\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0.340078	0.0273157
$156$	0	0
$157$	0.430668	0.0343711	0.0171855	−	0.999852i	$-0.494529\pi$
$157$	0.430668	0.0343711	0.0171855	+	0.999852i	$0.494529\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	9.45090	0.744835
$162$	0	0
$163$	−14.2293	−1.11452	−0.557261	−	0.830338i	$-0.688148\pi$
$163$	−14.2293	−1.11452	−0.557261	+	0.830338i	$0.688148\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	7.22925	0.559416	0.279708	−	0.960085i	$-0.409762\pi$
$167$	7.22925	0.559416	0.279708	+	0.960085i	$0.409762\pi$
$168$	0	0
$169$	2.63969	0.203053
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−17.2293	−1.30992	−0.654958	−	0.755665i	$-0.727314\pi$
$173$	−17.2293	−1.30992	−0.654958	+	0.755665i	$0.727314\pi$
$174$	0	0
$175$	3.61463	0.273240
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−22.4585	−1.67863	−0.839314	−	0.543647i	$-0.817043\pi$
$179$	−22.4585	−1.67863	−0.839314	+	0.543647i	$0.817043\pi$
$180$	0	0
$181$	17.6146	1.30928	0.654642	−	0.755939i	$-0.272819\pi$
$181$	17.6146	1.30928	0.654642	+	0.755939i	$0.272819\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−7.61463	−0.559839
$186$	0	0
$187$	5.47596	0.400442
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0.0905906	0.00655491	0.00327745	−	0.999995i	$-0.498957\pi$
$191$	0.0905906	0.00655491	0.00327745	+	0.999995i	$0.498957\pi$
$192$	0	0
$193$	16.1637	1.16349	0.581745	−	0.813371i	$-0.302370\pi$
$193$	16.1637	1.16349	0.581745	+	0.813371i	$0.302370\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	21.1387	1.50607	0.753034	−	0.657982i	$-0.228590\pi$
$197$	21.1387	1.50607	0.753034	+	0.657982i	$0.228590\pi$
$198$	0	0
$199$	−27.8641	−1.97523	−0.987617	−	0.156882i	$-0.949856\pi$
$199$	−27.8641	−1.97523	−0.987617	+	0.156882i	$0.949856\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−16.6802	−1.17072
$204$	0	0
$205$	−9.90941	−0.692103
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−3.77835	−0.261354
$210$	0	0
$211$	5.52404	0.380290	0.190145	−	0.981756i	$-0.439104\pi$
$211$	5.52404	0.380290	0.190145	+	0.981756i	$0.439104\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−9.29478	−0.633899
$216$	0	0
$217$	−1.22925	−0.0834472
$218$	0	0
$219$	0	0
$220$	0	0
$221$	9.25432	0.622513
$222$	0	0
$223$	6.09059	0.407856	0.203928	−	0.978986i	$-0.434629\pi$
$223$	6.09059	0.407856	0.203928	+	0.978986i	$0.434629\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−3.45090	−0.229044	−0.114522	−	0.993421i	$-0.536534\pi$
$227$	−3.45090	−0.229044	−0.114522	+	0.993421i	$0.536534\pi$
$228$	0	0
$229$	10.6802	0.705765	0.352882	−	0.935668i	$-0.385202\pi$
$229$	10.6802	0.705765	0.352882	+	0.935668i	$0.385202\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−7.31984	−0.479539	−0.239769	−	0.970830i	$-0.577072\pi$
$233$	−7.31984	−0.479539	−0.239769	+	0.970830i	$0.577072\pi$
$234$	0	0
$235$	−1.38537	−0.0903718
$236$	0	0
$237$	0	0
$238$	0	0
$239$	10.7707	0.696702	0.348351	−	0.937364i	$-0.386742\pi$
$239$	10.7707	0.696702	0.348351	+	0.937364i	$0.386742\pi$
$240$	0	0
$241$	14.3603	0.925029	0.462514	−	0.886612i	$-0.346947\pi$
$241$	14.3603	0.925029	0.462514	+	0.886612i	$0.346947\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−6.06553	−0.387512
$246$	0	0
$247$	−6.38537	−0.406292
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−7.11082	−0.448831	−0.224416	−	0.974494i	$-0.572047\pi$
$251$	−7.11082	−0.448831	−0.224416	+	0.974494i	$0.572047\pi$
$252$	0	0
$253$	−6.11843	−0.384662
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−24.0278	−1.49882	−0.749408	−	0.662109i	$-0.769662\pi$
$257$	−24.0278	−1.49882	−0.749408	+	0.662109i	$0.769662\pi$
$258$	0	0
$259$	27.5240	1.71026
$260$	0	0
$261$	0	0
$262$	0	0
$263$	23.8188	1.46873	0.734366	−	0.678754i	$-0.237480\pi$
$263$	23.8188	1.46873	0.734366	+	0.678754i	$0.237480\pi$
$264$	0	0
$265$	4.00000	0.245718
$266$	0	0
$267$	0	0
$268$	0	0
$269$	2.74568	0.167407	0.0837036	−	0.996491i	$-0.473325\pi$
$269$	2.74568	0.167407	0.0837036	+	0.996491i	$0.473325\pi$
$270$	0	0
$271$	11.5240	0.700035	0.350018	−	0.936743i	$-0.386176\pi$
$271$	11.5240	0.700035	0.350018	+	0.936743i	$0.386176\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−2.34008	−0.141112
$276$	0	0
$277$	26.3679	1.58429	0.792147	−	0.610330i	$-0.208963\pi$
$277$	26.3679	1.58429	0.792147	+	0.610330i	$0.208963\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	14.7707	0.881149	0.440574	−	0.897716i	$-0.354775\pi$
$281$	14.7707	0.881149	0.440574	+	0.897716i	$0.354775\pi$
$282$	0	0
$283$	14.4585	0.859469	0.429735	−	0.902955i	$-0.358607\pi$
$283$	14.4585	0.859469	0.429735	+	0.902955i	$0.358607\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	35.8188	2.11432
$288$	0	0
$289$	−11.5240	−0.677884
$290$	0	0
$291$	0	0
$292$	0	0
$293$	18.3679	1.07307	0.536533	−	0.843880i	$-0.319734\pi$
$293$	18.3679	1.07307	0.536533	+	0.843880i	$0.319734\pi$
$294$	0	0
$295$	11.2293	0.653792
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−10.3401	−0.597982
$300$	0	0
$301$	33.5972	1.93651
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−7.61463	−0.436012
$306$	0	0
$307$	1.29478	0.0738971	0.0369486	−	0.999317i	$-0.488236\pi$
$307$	1.29478	0.0738971	0.0369486	+	0.999317i	$0.488236\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	1.31984	0.0748415	0.0374208	−	0.999300i	$-0.488086\pi$
$311$	1.31984	0.0748415	0.0374208	+	0.999300i	$0.488086\pi$
$312$	0	0
$313$	22.2042	1.25506	0.627528	−	0.778594i	$-0.284067\pi$
$313$	22.2042	1.25506	0.627528	+	0.778594i	$0.284067\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	34.2773	1.92521	0.962603	−	0.270915i	$-0.0873263\pi$
$317$	34.2773	1.92521	0.962603	+	0.270915i	$0.0873263\pi$
$318$	0	0
$319$	10.7986	0.604605
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−3.77835	−0.210233
$324$	0	0
$325$	−3.95470	−0.219368
$326$	0	0
$327$	0	0
$328$	0	0
$329$	5.00761	0.276078
$330$	0	0
$331$	15.6146	0.858258	0.429129	−	0.903243i	$-0.358821\pi$
$331$	15.6146	0.858258	0.429129	+	0.903243i	$0.358821\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	8.29478	0.453192
$336$	0	0
$337$	23.5240	1.28144	0.640718	−	0.767776i	$-0.278637\pi$
$337$	23.5240	1.28144	0.640718	+	0.767776i	$0.278637\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0.795808	0.0430954
$342$	0	0
$343$	−3.37777	−0.182382
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−20.5491	−1.10313	−0.551567	−	0.834131i	$-0.685970\pi$
$347$	−20.5491	−1.10313	−0.551567	+	0.834131i	$0.685970\pi$
$348$	0	0
$349$	−1.74568	−0.0934443	−0.0467222	−	0.998908i	$-0.514878\pi$
$349$	−1.74568	−0.0934443	−0.0467222	+	0.998908i	$0.514878\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−30.2495	−1.61002	−0.805009	−	0.593263i	$-0.797840\pi$
$353$	−30.2495	−1.61002	−0.805009	+	0.593263i	$0.797840\pi$
$354$	0	0
$355$	−13.9094	−0.738235
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−5.31984	−0.280771	−0.140385	−	0.990097i	$-0.544834\pi$
$359$	−5.31984	−0.280771	−0.140385	+	0.990097i	$0.544834\pi$
$360$	0	0
$361$	−16.3930	−0.862788
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−14.8439	−0.776964
$366$	0	0
$367$	13.5240	0.705949	0.352974	−	0.935633i	$-0.385170\pi$
$367$	13.5240	0.705949	0.352974	+	0.935633i	$0.385170\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−14.4585	−0.750648
$372$	0	0
$373$	−19.8641	−1.02852	−0.514262	−	0.857633i	$-0.671934\pi$
$373$	−19.8641	−1.02852	−0.514262	+	0.857633i	$0.671934\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	18.2495	0.939896
$378$	0	0
$379$	2.16373	0.111143	0.0555716	−	0.998455i	$-0.482302\pi$
$379$	2.16373	0.111143	0.0555716	+	0.998455i	$0.482302\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	12.7457	0.651274	0.325637	−	0.945495i	$-0.394421\pi$
$383$	12.7457	0.651274	0.325637	+	0.945495i	$0.394421\pi$
$384$	0	0
$385$	8.45851	0.431085
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−29.2042	−1.48071	−0.740356	−	0.672215i	$-0.765343\pi$
$389$	−29.2042	−1.48071	−0.740356	+	0.672215i	$0.765343\pi$
$390$	0	0
$391$	−6.11843	−0.309422
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−0.0452953	−0.00227905
$396$	0	0
$397$	1.11082	0.0557506	0.0278753	−	0.999611i	$-0.491126\pi$
$397$	1.11082	0.0557506	0.0278753	+	0.999611i	$0.491126\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−15.8188	−0.789954	−0.394977	−	0.918691i	$-0.629247\pi$
$401$	−15.8188	−0.789954	−0.394977	+	0.918691i	$0.629247\pi$
$402$	0	0
$403$	1.34491	0.0669946
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−17.8188	−0.883246
$408$	0	0
$409$	26.0481	1.28799	0.643997	−	0.765028i	$-0.277275\pi$
$409$	26.0481	1.28799	0.643997	+	0.765028i	$0.277275\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−40.5896	−1.99728
$414$	0	0
$415$	−15.1387	−0.743128
$416$	0	0
$417$	0	0
$418$	0	0
$419$	32.7986	1.60232	0.801158	−	0.598453i	$-0.204218\pi$
$419$	32.7986	1.60232	0.801158	+	0.598453i	$0.204218\pi$
$420$	0	0
$421$	−18.2042	−0.887218	−0.443609	−	0.896220i	$-0.646302\pi$
$421$	−18.2042	−0.887218	−0.443609	+	0.896220i	$0.646302\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−2.34008	−0.113510
$426$	0	0
$427$	27.5240	1.33198
$428$	0	0
$429$	0	0
$430$	0	0
$431$	32.5896	1.56978	0.784892	−	0.619633i	$-0.212718\pi$
$431$	32.5896	1.56978	0.784892	+	0.619633i	$0.212718\pi$
$432$	0	0
$433$	5.45090	0.261954	0.130977	−	0.991385i	$-0.458189\pi$
$433$	5.45090	0.261954	0.130977	+	0.991385i	$0.458189\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4.22165	0.201949
$438$	0	0
$439$	25.3603	1.21038	0.605191	−	0.796080i	$-0.293097\pi$
$439$	25.3603	1.21038	0.605191	+	0.796080i	$0.293097\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−37.9094	−1.80113	−0.900565	−	0.434721i	$-0.856847\pi$
$443$	−37.9094	−1.80113	−0.900565	+	0.434721i	$0.856847\pi$
$444$	0	0
$445$	2.68016	0.127052
$446$	0	0
$447$	0	0
$448$	0	0
$449$	11.9094	0.562040	0.281020	−	0.959702i	$-0.409327\pi$
$449$	11.9094	0.562040	0.281020	+	0.959702i	$0.409327\pi$
$450$	0	0
$451$	−23.1888	−1.09192
$452$	0	0
$453$	0	0
$454$	0	0
$455$	14.2948	0.670150
$456$	0	0
$457$	−27.0481	−1.26526	−0.632628	−	0.774456i	$-0.718024\pi$
$457$	−27.0481	−1.26526	−0.632628	+	0.774456i	$0.718024\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−4.45851	−0.207653	−0.103827	−	0.994595i	$-0.533109\pi$
$461$	−4.45851	−0.207653	−0.103827	+	0.994595i	$0.533109\pi$
$462$	0	0
$463$	−38.1136	−1.77129	−0.885645	−	0.464364i	$-0.846283\pi$
$463$	−38.1136	−1.77129	−0.885645	+	0.464364i	$0.846283\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−2.68016	−0.124023	−0.0620114	−	0.998075i	$-0.519752\pi$
$467$	−2.68016	−0.124023	−0.0620114	+	0.998075i	$0.519752\pi$
$468$	0	0
$469$	−29.9825	−1.38447
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−21.7505	−1.00009
$474$	0	0
$475$	1.61463	0.0740842
$476$	0	0
$477$	0	0
$478$	0	0
$479$	31.0076	1.41677	0.708387	−	0.705824i	$-0.249423\pi$
$479$	31.0076	1.41677	0.708387	+	0.705824i	$0.249423\pi$
$480$	0	0
$481$	−30.1136	−1.37306
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−4.38537	−0.199130
$486$	0	0
$487$	−35.5240	−1.60975	−0.804874	−	0.593446i	$-0.797767\pi$
$487$	−35.5240	−1.60975	−0.804874	+	0.593446i	$0.797767\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−13.8689	−0.625897	−0.312948	−	0.949770i	$-0.601317\pi$
$491$	−13.8689	−0.625897	−0.312948	+	0.949770i	$0.601317\pi$
$492$	0	0
$493$	10.7986	0.486344
$494$	0	0
$495$	0	0
$496$	0	0
$497$	50.2773	2.25525
$498$	0	0
$499$	−30.2773	−1.35540	−0.677700	−	0.735339i	$-0.737023\pi$
$499$	−30.2773	−1.35540	−0.677700	+	0.735339i	$0.737023\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−19.6627	−0.876716	−0.438358	−	0.898800i	$-0.644440\pi$
$503$	−19.6627	−0.876716	−0.438358	+	0.898800i	$0.644440\pi$
$504$	0	0
$505$	−4.61463	−0.205348
$506$	0	0
$507$	0	0
$508$	0	0
$509$	42.4334	1.88083	0.940415	−	0.340030i	$-0.110437\pi$
$509$	42.4334	1.88083	0.940415	+	0.340030i	$0.110437\pi$
$510$	0	0
$511$	53.6551	2.37356
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0.934472	0.0411777
$516$	0	0
$517$	−3.24188	−0.142578
$518$	0	0
$519$	0	0
$520$	0	0
$521$	19.7784	0.866505	0.433253	−	0.901272i	$-0.357366\pi$
$521$	19.7784	0.866505	0.433253	+	0.901272i	$0.357366\pi$
$522$	0	0
$523$	−26.6878	−1.16697	−0.583487	−	0.812122i	$-0.698312\pi$
$523$	−26.6878	−1.16697	−0.583487	+	0.812122i	$0.698312\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0.795808	0.0346660
$528$	0	0
$529$	−16.1637	−0.702771
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−39.1888	−1.69745
$534$	0	0
$535$	6.00000	0.259403
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−14.1938	−0.611371
$540$	0	0
$541$	−7.06553	−0.303771	−0.151885	−	0.988398i	$-0.548534\pi$
$541$	−7.06553	−0.303771	−0.151885	+	0.988398i	$0.548534\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−15.2293	−0.652350
$546$	0	0
$547$	41.5896	1.77824	0.889121	−	0.457673i	$-0.151317\pi$
$547$	41.5896	1.77824	0.889121	+	0.457673i	$0.151317\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−7.45090	−0.317419
$552$	0	0
$553$	0.163726	0.00696232
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−22.4990	−0.953312	−0.476656	−	0.879090i	$-0.658151\pi$
$557$	−22.4990	−0.953312	−0.476656	+	0.879090i	$0.658151\pi$
$558$	0	0
$559$	−36.7581	−1.55470
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−26.0405	−1.09747	−0.548737	−	0.835995i	$-0.684891\pi$
$563$	−26.0405	−1.09747	−0.548737	+	0.835995i	$0.684891\pi$
$564$	0	0
$565$	10.3401	0.435010
$566$	0	0
$567$	0	0
$568$	0	0
$569$	12.7707	0.535378	0.267689	−	0.963505i	$-0.413740\pi$
$569$	12.7707	0.535378	0.267689	+	0.963505i	$0.413740\pi$
$570$	0	0
$571$	−43.9825	−1.84061	−0.920306	−	0.391199i	$-0.872060\pi$
$571$	−43.9825	−1.84061	−0.920306	+	0.391199i	$0.872060\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	2.61463	0.109037
$576$	0	0
$577$	−17.0655	−0.710447	−0.355224	−	0.934781i	$-0.615595\pi$
$577$	−17.0655	−0.710447	−0.355224	+	0.934781i	$0.615595\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	54.7206	2.27019
$582$	0	0
$583$	9.36031	0.387664
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−5.18879	−0.214164	−0.107082	−	0.994250i	$-0.534151\pi$
$587$	−5.18879	−0.214164	−0.107082	+	0.994250i	$0.534151\pi$
$588$	0	0
$589$	−0.549099	−0.0226252
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−11.1108	−0.456267	−0.228133	−	0.973630i	$-0.573262\pi$
$593$	−11.1108	−0.456267	−0.228133	+	0.973630i	$0.573262\pi$
$594$	0	0
$595$	8.45851	0.346765
$596$	0	0
$597$	0	0
$598$	0	0
$599$	36.2369	1.48060	0.740299	−	0.672278i	$-0.234684\pi$
$599$	36.2369	1.48060	0.740299	+	0.672278i	$0.234684\pi$
$600$	0	0
$601$	20.5240	0.837193	0.418596	−	0.908172i	$-0.362522\pi$
$601$	20.5240	0.837193	0.418596	+	0.908172i	$0.362522\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	5.52404	0.224584
$606$	0	0
$607$	−32.7533	−1.32942	−0.664708	−	0.747104i	$-0.731444\pi$
$607$	−32.7533	−1.32942	−0.664708	+	0.747104i	$0.731444\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−5.47874	−0.221646
$612$	0	0
$613$	−20.7255	−0.837093	−0.418547	−	0.908195i	$-0.637460\pi$
$613$	−20.7255	−0.837093	−0.418547	+	0.908195i	$0.637460\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−24.1184	−0.970971	−0.485486	−	0.874245i	$-0.661357\pi$
$617$	−24.1184	−0.970971	−0.485486	+	0.874245i	$0.661357\pi$
$618$	0	0
$619$	−7.43345	−0.298775	−0.149388	−	0.988779i	$-0.547730\pi$
$619$	−7.43345	−0.298775	−0.149388	+	0.988779i	$0.547730\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−9.68776	−0.388132
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−17.8188	−0.710483
$630$	0	0
$631$	−13.6551	−0.543601	−0.271800	−	0.962354i	$-0.587619\pi$
$631$	−13.6551	−0.543601	−0.271800	+	0.962354i	$0.587619\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	3.31984	0.131744
$636$	0	0
$637$	−23.9874	−0.950414
$638$	0	0
$639$	0	0
$640$	0	0
$641$	37.7784	1.49216	0.746078	−	0.665859i	$-0.231935\pi$
$641$	37.7784	1.49216	0.746078	+	0.665859i	$0.231935\pi$
$642$	0	0
$643$	−0.0655284	−0.00258419	−0.00129209	−	0.999999i	$-0.500411\pi$
$643$	−0.0655284	−0.00258419	−0.00129209	+	0.999999i	$0.500411\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−38.2773	−1.50484	−0.752418	−	0.658685i	$-0.771113\pi$
$647$	−38.2773	−1.50484	−0.752418	+	0.658685i	$0.771113\pi$
$648$	0	0
$649$	26.2773	1.03148
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−41.7282	−1.63295	−0.816476	−	0.577380i	$-0.804075\pi$
$653$	−41.7282	−1.63295	−0.816476	+	0.577380i	$0.804075\pi$
$654$	0	0
$655$	−1.56933	−0.0613189
$656$	0	0
$657$	0	0
$658$	0	0
$659$	23.8188	0.927849	0.463925	−	0.885875i	$-0.346441\pi$
$659$	23.8188	0.927849	0.463925	+	0.885875i	$0.346441\pi$
$660$	0	0
$661$	−3.02506	−0.117661	−0.0588306	−	0.998268i	$-0.518737\pi$
$661$	−3.02506	−0.117661	−0.0588306	+	0.998268i	$0.518737\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−5.83627	−0.226321
$666$	0	0
$667$	−12.0655	−0.467179
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−17.8188	−0.687888
$672$	0	0
$673$	−39.6551	−1.52859	−0.764296	−	0.644866i	$-0.776913\pi$
$673$	−39.6551	−1.52859	−0.764296	+	0.644866i	$0.776913\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−25.9499	−0.997335	−0.498667	−	0.866793i	$-0.666177\pi$
$677$	−25.9499	−0.997335	−0.498667	+	0.866793i	$0.666177\pi$
$678$	0	0
$679$	15.8515	0.608324
$680$	0	0
$681$	0	0
$682$	0	0
$683$	35.8188	1.37057	0.685285	−	0.728275i	$-0.259678\pi$
$683$	35.8188	1.37057	0.685285	+	0.728275i	$0.259678\pi$
$684$	0	0
$685$	−10.5491	−0.403060
$686$	0	0
$687$	0	0
$688$	0	0
$689$	15.8188	0.602649
$690$	0	0
$691$	−20.6802	−0.786710	−0.393355	−	0.919387i	$-0.628686\pi$
$691$	−20.6802	−0.786710	−0.393355	+	0.919387i	$0.628686\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0.934472	0.0354465
$696$	0	0
$697$	−23.1888	−0.878338
$698$	0	0
$699$	0	0
$700$	0	0
$701$	30.4334	1.14946	0.574728	−	0.818345i	$-0.305108\pi$
$701$	30.4334	1.14946	0.574728	+	0.818345i	$0.305108\pi$
$702$	0	0
$703$	12.2948	0.463707
$704$	0	0
$705$	0	0
$706$	0	0
$707$	16.6802	0.627322
$708$	0	0
$709$	8.88435	0.333659	0.166829	−	0.985986i	$-0.446647\pi$
$709$	8.88435	0.333659	0.166829	+	0.985986i	$0.446647\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−0.889176	−0.0332999
$714$	0	0
$715$	−9.25432	−0.346092
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−4.13106	−0.154062	−0.0770312	−	0.997029i	$-0.524544\pi$
$719$	−4.13106	−0.154062	−0.0770312	+	0.997029i	$0.524544\pi$
$720$	0	0
$721$	−3.37777	−0.125795
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−4.61463	−0.171383
$726$	0	0
$727$	−22.2773	−0.826220	−0.413110	−	0.910681i	$-0.635558\pi$
$727$	−22.2773	−0.826220	−0.413110	+	0.910681i	$0.635558\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−21.7505	−0.804472
$732$	0	0
$733$	7.90941	0.292141	0.146070	−	0.989274i	$-0.453337\pi$
$733$	7.90941	0.292141	0.146070	+	0.989274i	$0.453337\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	19.4104	0.714993
$738$	0	0
$739$	33.7282	1.24071	0.620356	−	0.784320i	$-0.286988\pi$
$739$	33.7282	1.24071	0.620356	+	0.784320i	$0.286988\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−31.7128	−1.16343	−0.581715	−	0.813392i	$-0.697618\pi$
$743$	−31.7128	−1.16343	−0.581715	+	0.813392i	$0.697618\pi$
$744$	0	0
$745$	−17.8439	−0.653749
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−21.6878	−0.792454
$750$	0	0
$751$	39.8641	1.45466	0.727331	−	0.686287i	$-0.240760\pi$
$751$	39.8641	1.45466	0.727331	+	0.686287i	$0.240760\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−4.04530	−0.147223
$756$	0	0
$757$	17.6425	0.641226	0.320613	−	0.947210i	$-0.396111\pi$
$757$	17.6425	0.641226	0.320613	+	0.947210i	$0.396111\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	23.9094	0.866715	0.433358	−	0.901222i	$-0.357329\pi$
$761$	23.9094	0.866715	0.433358	+	0.901222i	$0.357329\pi$
$762$	0	0
$763$	55.0481	1.99287
$764$	0	0
$765$	0	0
$766$	0	0
$767$	44.4084	1.60349
$768$	0	0
$769$	−36.8188	−1.32772	−0.663860	−	0.747857i	$-0.731083\pi$
$769$	−36.8188	−1.32772	−0.663860	+	0.747857i	$0.731083\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−11.3198	−0.407147	−0.203573	−	0.979060i	$-0.565256\pi$
$773$	−11.3198	−0.407147	−0.203573	+	0.979060i	$0.565256\pi$
$774$	0	0
$775$	−0.340078	−0.0122160
$776$	0	0
$777$	0	0
$778$	0	0
$779$	16.0000	0.573259
$780$	0	0
$781$	−32.5491	−1.16470
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−0.430668	−0.0153712
$786$	0	0
$787$	34.1791	1.21835	0.609177	−	0.793034i	$-0.291500\pi$
$787$	34.1791	1.21835	0.609177	+	0.793034i	$0.291500\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−37.3755	−1.32892
$792$	0	0
$793$	−30.1136	−1.06937
$794$	0	0
$795$	0	0
$796$	0	0
$797$	32.5491	1.15295	0.576474	−	0.817115i	$-0.304428\pi$
$797$	32.5491	1.15295	0.576474	+	0.817115i	$0.304428\pi$
$798$	0	0
$799$	−3.24188	−0.114689
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−34.7358	−1.22580
$804$	0	0
$805$	−9.45090	−0.333101
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−26.2773	−0.923862	−0.461931	−	0.886916i	$-0.652843\pi$
$809$	−26.2773	−0.923862	−0.461931	+	0.886916i	$0.652843\pi$
$810$	0	0
$811$	−28.6802	−1.00710	−0.503548	−	0.863967i	$-0.667972\pi$
$811$	−28.6802	−1.00710	−0.503548	+	0.863967i	$0.667972\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	14.2293	0.498429
$816$	0	0
$817$	15.0076	0.525050
$818$	0	0
$819$	0	0
$820$	0	0
$821$	3.36031	0.117276	0.0586378	−	0.998279i	$-0.481324\pi$
$821$	3.36031	0.117276	0.0586378	+	0.998279i	$0.481324\pi$
$822$	0	0
$823$	2.42584	0.0845594	0.0422797	−	0.999106i	$-0.486538\pi$
$823$	2.42584	0.0845594	0.0422797	+	0.999106i	$0.486538\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0.458508	0.0159439	0.00797194	−	0.999968i	$-0.497462\pi$
$827$	0.458508	0.0159439	0.00797194	+	0.999968i	$0.497462\pi$
$828$	0	0
$829$	29.8014	1.03504	0.517522	−	0.855670i	$-0.326855\pi$
$829$	29.8014	1.03504	0.517522	+	0.855670i	$0.326855\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−14.1938	−0.491786
$834$	0	0
$835$	−7.22925	−0.250179
$836$	0	0
$837$	0	0
$838$	0	0
$839$	34.9575	1.20687	0.603433	−	0.797414i	$-0.293799\pi$
$839$	34.9575	1.20687	0.603433	+	0.797414i	$0.293799\pi$
$840$	0	0
$841$	−7.70522	−0.265697
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−2.63969	−0.0908081
$846$	0	0
$847$	−19.9673	−0.686086
$848$	0	0
$849$	0	0
$850$	0	0
$851$	19.9094	0.682486
$852$	0	0
$853$	−16.5443	−0.566465	−0.283232	−	0.959051i	$-0.591407\pi$
$853$	−16.5443	−0.566465	−0.283232	+	0.959051i	$0.591407\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	16.8613	0.575972	0.287986	−	0.957635i	$-0.407014\pi$
$857$	16.8613	0.575972	0.287986	+	0.957635i	$0.407014\pi$
$858$	0	0
$859$	−2.84388	−0.0970320	−0.0485160	−	0.998822i	$-0.515449\pi$
$859$	−2.84388	−0.0970320	−0.0485160	+	0.998822i	$0.515449\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	55.6627	1.89478	0.947390	−	0.320081i	$-0.103710\pi$
$863$	55.6627	1.89478	0.947390	+	0.320081i	$0.103710\pi$
$864$	0	0
$865$	17.2293	0.585812
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−0.105995	−0.00359562
$870$	0	0
$871$	32.8034	1.11150
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−3.61463	−0.122197
$876$	0	0
$877$	−52.7811	−1.78229	−0.891146	−	0.453717i	$-0.850098\pi$
$877$	−52.7811	−1.78229	−0.891146	+	0.453717i	$0.850098\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	36.4585	1.22832	0.614159	−	0.789182i	$-0.289495\pi$
$881$	36.4585	1.22832	0.614159	+	0.789182i	$0.289495\pi$
$882$	0	0
$883$	−49.9825	−1.68205	−0.841023	−	0.540999i	$-0.818046\pi$
$883$	−49.9825	−1.68205	−0.841023	+	0.540999i	$0.818046\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	4.43345	0.148861	0.0744303	−	0.997226i	$-0.476286\pi$
$887$	4.43345	0.148861	0.0744303	+	0.997226i	$0.476286\pi$
$888$	0	0
$889$	−12.0000	−0.402467
$890$	0	0
$891$	0	0
$892$	0	0
$893$	2.23686	0.0748537
$894$	0	0
$895$	22.4585	0.750705
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1.56933	0.0523401
$900$	0	0
$901$	9.36031	0.311837
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−17.6146	−0.585530
$906$	0	0
$907$	17.0000	0.564476	0.282238	−	0.959344i	$-0.408923\pi$
$907$	17.0000	0.564476	0.282238	+	0.959344i	$0.408923\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	35.8188	1.18673	0.593365	−	0.804933i	$-0.297799\pi$
$911$	35.8188	1.18673	0.593365	+	0.804933i	$0.297799\pi$
$912$	0	0
$913$	−35.4256	−1.17242
$914$	0	0
$915$	0	0
$916$	0	0
$917$	5.67255	0.187324
$918$	0	0
$919$	7.24188	0.238888	0.119444	−	0.992841i	$-0.461889\pi$
$919$	7.24188	0.238888	0.119444	+	0.992841i	$0.461889\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−55.0076	−1.81060
$924$	0	0
$925$	7.61463	0.250367
$926$	0	0
$927$	0	0
$928$	0	0
$929$	13.9094	0.456353	0.228176	−	0.973620i	$-0.426724\pi$
$929$	13.9094	0.456353	0.228176	+	0.973620i	$0.426724\pi$
$930$	0	0
$931$	9.79357	0.320971
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−5.47596	−0.179083
$936$	0	0
$937$	−4.93447	−0.161202	−0.0806011	−	0.996746i	$-0.525684\pi$
$937$	−4.93447	−0.161202	−0.0806011	+	0.996746i	$0.525684\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	51.9749	1.69433	0.847167	−	0.531326i	$-0.178306\pi$
$941$	51.9749	1.69433	0.847167	+	0.531326i	$0.178306\pi$
$942$	0	0
$943$	25.9094	0.843726
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−19.7282	−0.641081	−0.320541	−	0.947235i	$-0.603865\pi$
$947$	−19.7282	−0.641081	−0.320541	+	0.947235i	$0.603865\pi$
$948$	0	0
$949$	−58.7032	−1.90559
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−54.5769	−1.76792	−0.883960	−	0.467562i	$-0.845132\pi$
$953$	−54.5769	−1.76792	−0.883960	+	0.467562i	$0.845132\pi$
$954$	0	0
$955$	−0.0905906	−0.00293144
$956$	0	0
$957$	0	0
$958$	0	0
$959$	38.1311	1.23132
$960$	0	0
$961$	−30.8843	−0.996269
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−16.1637	−0.520329
$966$	0	0
$967$	−5.70522	−0.183467	−0.0917337	−	0.995784i	$-0.529241\pi$
$967$	−5.70522	−0.183467	−0.0917337	+	0.995784i	$0.529241\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	8.29961	0.266347	0.133174	−	0.991093i	$-0.457483\pi$
$971$	8.29961	0.266347	0.133174	+	0.991093i	$0.457483\pi$
$972$	0	0
$973$	−3.37777	−0.108286
$974$	0	0
$975$	0	0
$976$	0	0
$977$	6.24949	0.199939	0.0999694	−	0.994991i	$-0.468126\pi$
$977$	6.24949	0.199939	0.0999694	+	0.994991i	$0.468126\pi$
$978$	0	0
$979$	6.27177	0.200447
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−46.5645	−1.48518	−0.742588	−	0.669748i	$-0.766402\pi$
$983$	−46.5645	−1.48518	−0.742588	+	0.669748i	$0.766402\pi$
$984$	0	0
$985$	−21.1387	−0.673534
$986$	0	0
$987$	0	0
$988$	0	0
$989$	24.3024	0.772771
$990$	0	0
$991$	30.4132	0.966108	0.483054	−	0.875591i	$-0.339528\pi$
$991$	30.4132	0.966108	0.483054	+	0.875591i	$0.339528\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	27.8641	0.883352
$996$	0	0
$997$	−22.4711	−0.711668	−0.355834	−	0.934549i	$-0.615803\pi$
$997$	−22.4711	−0.711668	−0.355834	+	0.934549i	$0.615803\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4320.2.a.bg.1.3	✓	3
3.2	odd	2		4320.2.a.bi.1.3	yes	3
4.3	odd	2		4320.2.a.bh.1.1	yes	3
8.3	odd	2		8640.2.a.dn.1.1		3
8.5	even	2		8640.2.a.dm.1.3		3
12.11	even	2		4320.2.a.bj.1.1	yes	3
24.5	odd	2		8640.2.a.dk.1.3		3
24.11	even	2		8640.2.a.dl.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4320.2.a.bg.1.3	✓	3	1.1	even	1	trivial
4320.2.a.bh.1.1	yes	3	4.3	odd	2
4320.2.a.bi.1.3	yes	3	3.2	odd	2
4320.2.a.bj.1.1	yes	3	12.11	even	2
8640.2.a.dk.1.3		3	24.5	odd	2
8640.2.a.dl.1.1		3	24.11	even	2
8640.2.a.dm.1.3		3	8.5	even	2
8640.2.a.dn.1.1		3	8.3	odd	2

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Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4320 = 2^{5} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4320.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} +3.61463 q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} +3.61463 q^{7} -2.34008 q^{11} -3.95470 q^{13} -2.34008 q^{17} +1.61463 q^{19} +2.61463 q^{23} +1.00000 q^{25} -4.61463 q^{29} -0.340078 q^{31} -3.61463 q^{35} +7.61463 q^{37} +9.90941 q^{41} +9.29478 q^{43} +1.38537 q^{47} +6.06553 q^{49} -4.00000 q^{53} +2.34008 q^{55} -11.2293 q^{59} +7.61463 q^{61} +3.95470 q^{65} -8.29478 q^{67} +13.9094 q^{71} +14.8439 q^{73} -8.45851 q^{77} +0.0452953 q^{79} +15.1387 q^{83} +2.34008 q^{85} -2.68016 q^{89} -14.2948 q^{91} -1.61463 q^{95} +4.38537 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{5} - q^{7}+O(q^{10}) \) 3 * q - 3 * q^5 - q^7	\( 3 q - 3 q^{5} - q^{7} - 2 q^{11} + 5 q^{13} - 2 q^{17} - 7 q^{19} - 4 q^{23} + 3 q^{25} - 2 q^{29} + 4 q^{31} + q^{35} + 11 q^{37} - 4 q^{41} + 6 q^{43} + 16 q^{47} + 20 q^{49} - 12 q^{53} + 2 q^{55} - 10 q^{59} + 11 q^{61} - 5 q^{65} - 3 q^{67} + 8 q^{71} + 9 q^{73} + 22 q^{77} + 17 q^{79} - 12 q^{83} + 2 q^{85} + 2 q^{89} - 21 q^{91} + 7 q^{95} + 25 q^{97}+O(q^{100}) \) 3 * q - 3 * q^5 - q^7 - 2 * q^11 + 5 * q^13 - 2 * q^17 - 7 * q^19 - 4 * q^23 + 3 * q^25 - 2 * q^29 + 4 * q^31 + q^35 + 11 * q^37 - 4 * q^41 + 6 * q^43 + 16 * q^47 + 20 * q^49 - 12 * q^53 + 2 * q^55 - 10 * q^59 + 11 * q^61 - 5 * q^65 - 3 * q^67 + 8 * q^71 + 9 * q^73 + 22 * q^77 + 17 * q^79 - 12 * q^83 + 2 * q^85 + 2 * q^89 - 21 * q^91 + 7 * q^95 + 25 * q^97

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