LMFDB - Embedded newform 8640.2.a.di.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8640,2,Mod(1,8640)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8640, 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("8640.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8640 = 2^{6} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8640.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:	$68.9907473464$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 4320)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	8640.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} -0.561553 q^{7} +3.56155 q^{11} -1.00000 q^{13} -6.68466 q^{17} -7.68466 q^{19} -3.56155 q^{23} +1.00000 q^{25} -4.68466 q^{29} +2.43845 q^{31} -0.561553 q^{35} +5.68466 q^{37} +4.00000 q^{41} +4.68466 q^{43} +3.56155 q^{47} -6.68466 q^{49} +3.56155 q^{55} -1.12311 q^{59} +9.68466 q^{61} -1.00000 q^{65} -5.43845 q^{67} +14.2462 q^{71} +13.6847 q^{73} -2.00000 q^{77} +17.2462 q^{79} +4.87689 q^{83} -6.68466 q^{85} +11.3693 q^{89} +0.561553 q^{91} -7.68466 q^{95} -2.31534 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} + 3 q^{7} + 3 q^{11} - 2 q^{13} - q^{17} - 3 q^{19} - 3 q^{23} + 2 q^{25} + 3 q^{29} + 9 q^{31} + 3 q^{35} - q^{37} + 8 q^{41} - 3 q^{43} + 3 q^{47} - q^{49} + 3 q^{55} + 6 q^{59} + 7 q^{61}+ \cdots - 17 q^{97}+O(q^{100}) $ 2 * q + 2 * q^5 + 3 * q^7 + 3 * q^11 - 2 * q^13 - q^17 - 3 * q^19 - 3 * q^23 + 2 * q^25 + 3 * q^29 + 9 * q^31 + 3 * q^35 - q^37 + 8 * q^41 - 3 * q^43 + 3 * q^47 - q^49 + 3 * q^55 + 6 * q^59 + 7 * q^61 - 2 * q^65 - 15 * q^67 + 12 * q^71 + 15 * q^73 - 4 * q^77 + 18 * q^79 + 18 * q^83 - q^85 - 2 * q^89 - 3 * q^91 - 3 * q^95 - 17 * 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
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−0.561553	−0.212247	−0.106124	−	0.994353i	$-0.533844\pi$
$7$	−0.561553	−0.212247	−0.106124	+	0.994353i	$0.533844\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	3.56155	1.07385	0.536924	−	0.843630i	$-0.319586\pi$
$11$	3.56155	1.07385	0.536924	+	0.843630i	$0.319586\pi$
$12$	0	0
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−6.68466	−1.62127	−0.810634	−	0.585553i	$-0.800877\pi$
$17$	−6.68466	−1.62127	−0.810634	+	0.585553i	$0.800877\pi$
$18$	0	0
$19$	−7.68466	−1.76298	−0.881491	−	0.472201i	$-0.843460\pi$
$19$	−7.68466	−1.76298	−0.881491	+	0.472201i	$0.843460\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3.56155	−0.742635	−0.371318	−	0.928506i	$-0.621094\pi$
$23$	−3.56155	−0.742635	−0.371318	+	0.928506i	$0.621094\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−4.68466	−0.869919	−0.434960	−	0.900450i	$-0.643237\pi$
$29$	−4.68466	−0.869919	−0.434960	+	0.900450i	$0.643237\pi$
$30$	0	0
$31$	2.43845	0.437958	0.218979	−	0.975730i	$-0.429727\pi$
$31$	2.43845	0.437958	0.218979	+	0.975730i	$0.429727\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−0.561553	−0.0949197
$36$	0	0
$37$	5.68466	0.934552	0.467276	−	0.884111i	$-0.345235\pi$
$37$	5.68466	0.934552	0.467276	+	0.884111i	$0.345235\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	4.00000	0.624695	0.312348	−	0.949968i	$-0.398885\pi$
$41$	4.00000	0.624695	0.312348	+	0.949968i	$0.398885\pi$
$42$	0	0
$43$	4.68466	0.714404	0.357202	−	0.934027i	$-0.383731\pi$
$43$	4.68466	0.714404	0.357202	+	0.934027i	$0.383731\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3.56155	0.519506	0.259753	−	0.965675i	$-0.416359\pi$
$47$	3.56155	0.519506	0.259753	+	0.965675i	$0.416359\pi$
$48$	0	0
$49$	−6.68466	−0.954951
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	3.56155	0.480240
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−1.12311	−0.146216	−0.0731079	−	0.997324i	$-0.523292\pi$
$59$	−1.12311	−0.146216	−0.0731079	+	0.997324i	$0.523292\pi$
$60$	0	0
$61$	9.68466	1.23999	0.619997	−	0.784604i	$-0.287134\pi$
$61$	9.68466	1.23999	0.619997	+	0.784604i	$0.287134\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.00000	−0.124035
$66$	0	0
$67$	−5.43845	−0.664412	−0.332206	−	0.943207i	$-0.607793\pi$
$67$	−5.43845	−0.664412	−0.332206	+	0.943207i	$0.607793\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	14.2462	1.69071	0.845357	−	0.534202i	$-0.179388\pi$
$71$	14.2462	1.69071	0.845357	+	0.534202i	$0.179388\pi$
$72$	0	0
$73$	13.6847	1.60167	0.800834	−	0.598886i	$-0.204390\pi$
$73$	13.6847	1.60167	0.800834	+	0.598886i	$0.204390\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.00000	−0.227921
$78$	0	0
$79$	17.2462	1.94035	0.970175	−	0.242405i	$-0.0779362\pi$
$79$	17.2462	1.94035	0.970175	+	0.242405i	$0.0779362\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	4.87689	0.535309	0.267654	−	0.963515i	$-0.413751\pi$
$83$	4.87689	0.535309	0.267654	+	0.963515i	$0.413751\pi$
$84$	0	0
$85$	−6.68466	−0.725053
$86$	0	0
$87$	0	0
$88$	0	0
$89$	11.3693	1.20515	0.602573	−	0.798064i	$-0.294142\pi$
$89$	11.3693	1.20515	0.602573	+	0.798064i	$0.294142\pi$
$90$	0	0
$91$	0.561553	0.0588667
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−7.68466	−0.788429
$96$	0	0
$97$	−2.31534	−0.235087	−0.117544	−	0.993068i	$-0.537502\pi$
$97$	−2.31534	−0.235087	−0.117544	+	0.993068i	$0.537502\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	8.68466	0.864156	0.432078	−	0.901836i	$-0.357781\pi$
$101$	8.68466	0.864156	0.432078	+	0.901836i	$0.357781\pi$
$102$	0	0
$103$	11.4384	1.12706	0.563532	−	0.826094i	$-0.309442\pi$
$103$	11.4384	1.12706	0.563532	+	0.826094i	$0.309442\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	8.24621	0.797191	0.398596	−	0.917127i	$-0.369498\pi$
$107$	8.24621	0.797191	0.398596	+	0.917127i	$0.369498\pi$
$108$	0	0
$109$	−15.3693	−1.47211	−0.736057	−	0.676920i	$-0.763314\pi$
$109$	−15.3693	−1.47211	−0.736057	+	0.676920i	$0.763314\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	5.31534	0.500025	0.250013	−	0.968243i	$-0.419565\pi$
$113$	5.31534	0.500025	0.250013	+	0.968243i	$0.419565\pi$
$114$	0	0
$115$	−3.56155	−0.332117
$116$	0	0
$117$	0	0
$118$	0	0
$119$	3.75379	0.344109
$120$	0	0
$121$	1.68466	0.153151
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	7.12311	0.632073	0.316037	−	0.948747i	$-0.397648\pi$
$127$	7.12311	0.632073	0.316037	+	0.948747i	$0.397648\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	11.8078	1.03165	0.515825	−	0.856694i	$-0.327486\pi$
$131$	11.8078	1.03165	0.515825	+	0.856694i	$0.327486\pi$
$132$	0	0
$133$	4.31534	0.374188
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	0	0
$139$	2.80776	0.238151	0.119076	−	0.992885i	$-0.462007\pi$
$139$	2.80776	0.238151	0.119076	+	0.992885i	$0.462007\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−3.56155	−0.297832
$144$	0	0
$145$	−4.68466	−0.389040
$146$	0	0
$147$	0	0
$148$	0	0
$149$	3.31534	0.271603	0.135802	−	0.990736i	$-0.456639\pi$
$149$	3.31534	0.271603	0.135802	+	0.990736i	$0.456639\pi$
$150$	0	0
$151$	15.0000	1.22068	0.610341	−	0.792139i	$-0.291032\pi$
$151$	15.0000	1.22068	0.610341	+	0.792139i	$0.291032\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2.43845	0.195861
$156$	0	0
$157$	−10.6847	−0.852729	−0.426364	−	0.904552i	$-0.640206\pi$
$157$	−10.6847	−0.852729	−0.426364	+	0.904552i	$0.640206\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	2.00000	0.157622
$162$	0	0
$163$	−16.1231	−1.26286	−0.631430	−	0.775433i	$-0.717532\pi$
$163$	−16.1231	−1.26286	−0.631430	+	0.775433i	$0.717532\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	25.1231	1.94408	0.972042	−	0.234806i	$-0.0754456\pi$
$167$	25.1231	1.94408	0.972042	+	0.234806i	$0.0754456\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1.36932	0.104107	0.0520536	−	0.998644i	$-0.483423\pi$
$173$	1.36932	0.104107	0.0520536	+	0.998644i	$0.483423\pi$
$174$	0	0
$175$	−0.561553	−0.0424494
$176$	0	0
$177$	0	0
$178$	0	0
$179$	14.2462	1.06481	0.532406	−	0.846489i	$-0.321288\pi$
$179$	14.2462	1.06481	0.532406	+	0.846489i	$0.321288\pi$
$180$	0	0
$181$	−0.315342	−0.0234392	−0.0117196	−	0.999931i	$-0.503731\pi$
$181$	−0.315342	−0.0234392	−0.0117196	+	0.999931i	$0.503731\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	5.68466	0.417944
$186$	0	0
$187$	−23.8078	−1.74100
$188$	0	0
$189$	0	0
$190$	0	0
$191$	18.0000	1.30243	0.651217	−	0.758891i	$-0.274259\pi$
$191$	18.0000	1.30243	0.651217	+	0.758891i	$0.274259\pi$
$192$	0	0
$193$	−1.68466	−0.121264	−0.0606322	−	0.998160i	$-0.519312\pi$
$193$	−1.68466	−0.121264	−0.0606322	+	0.998160i	$0.519312\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−15.3693	−1.09502	−0.547509	−	0.836800i	$-0.684424\pi$
$197$	−15.3693	−1.09502	−0.547509	+	0.836800i	$0.684424\pi$
$198$	0	0
$199$	−7.49242	−0.531124	−0.265562	−	0.964094i	$-0.585557\pi$
$199$	−7.49242	−0.531124	−0.265562	+	0.964094i	$0.585557\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	2.63068	0.184638
$204$	0	0
$205$	4.00000	0.279372
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−27.3693	−1.89318
$210$	0	0
$211$	−13.6847	−0.942091	−0.471045	−	0.882109i	$-0.656123\pi$
$211$	−13.6847	−0.942091	−0.471045	+	0.882109i	$0.656123\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	4.68466	0.319491
$216$	0	0
$217$	−1.36932	−0.0929553
$218$	0	0
$219$	0	0
$220$	0	0
$221$	6.68466	0.449659
$222$	0	0
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−4.49242	−0.298173	−0.149086	−	0.988824i	$-0.547633\pi$
$227$	−4.49242	−0.298173	−0.149086	+	0.988824i	$0.547633\pi$
$228$	0	0
$229$	7.36932	0.486978	0.243489	−	0.969904i	$-0.421708\pi$
$229$	7.36932	0.486978	0.243489	+	0.969904i	$0.421708\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−25.3693	−1.66200	−0.831000	−	0.556273i	$-0.812231\pi$
$233$	−25.3693	−1.66200	−0.831000	+	0.556273i	$0.812231\pi$
$234$	0	0
$235$	3.56155	0.232330
$236$	0	0
$237$	0	0
$238$	0	0
$239$	23.6155	1.52756	0.763781	−	0.645476i	$-0.223341\pi$
$239$	23.6155	1.52756	0.763781	+	0.645476i	$0.223341\pi$
$240$	0	0
$241$	−19.0000	−1.22390	−0.611949	−	0.790897i	$-0.709614\pi$
$241$	−19.0000	−1.22390	−0.611949	+	0.790897i	$0.709614\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−6.68466	−0.427067
$246$	0	0
$247$	7.68466	0.488963
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−23.8078	−1.50273	−0.751366	−	0.659885i	$-0.770605\pi$
$251$	−23.8078	−1.50273	−0.751366	+	0.659885i	$0.770605\pi$
$252$	0	0
$253$	−12.6847	−0.797478
$254$	0	0
$255$	0	0
$256$	0	0
$257$	12.6847	0.791247	0.395624	−	0.918413i	$-0.370528\pi$
$257$	12.6847	0.791247	0.395624	+	0.918413i	$0.370528\pi$
$258$	0	0
$259$	−3.19224	−0.198356
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−16.4924	−1.01697	−0.508483	−	0.861072i	$-0.669794\pi$
$263$	−16.4924	−1.01697	−0.508483	+	0.861072i	$0.669794\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	8.05398	0.491060	0.245530	−	0.969389i	$-0.421038\pi$
$269$	8.05398	0.491060	0.245530	+	0.969389i	$0.421038\pi$
$270$	0	0
$271$	19.6847	1.19576	0.597879	−	0.801586i	$-0.296010\pi$
$271$	19.6847	1.19576	0.597879	+	0.801586i	$0.296010\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	3.56155	0.214770
$276$	0	0
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	9.36932	0.558927	0.279463	−	0.960156i	$-0.409843\pi$
$281$	9.36932	0.558927	0.279463	+	0.960156i	$0.409843\pi$
$282$	0	0
$283$	−30.7386	−1.82722	−0.913611	−	0.406589i	$-0.866718\pi$
$283$	−30.7386	−1.82722	−0.913611	+	0.406589i	$0.866718\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2.24621	−0.132590
$288$	0	0
$289$	27.6847	1.62851
$290$	0	0
$291$	0	0
$292$	0	0
$293$	2.00000	0.116841	0.0584206	−	0.998292i	$-0.481394\pi$
$293$	2.00000	0.116841	0.0584206	+	0.998292i	$0.481394\pi$
$294$	0	0
$295$	−1.12311	−0.0653897
$296$	0	0
$297$	0	0
$298$	0	0
$299$	3.56155	0.205970
$300$	0	0
$301$	−2.63068	−0.151630
$302$	0	0
$303$	0	0
$304$	0	0
$305$	9.68466	0.554542
$306$	0	0
$307$	−5.06913	−0.289311	−0.144655	−	0.989482i	$-0.546207\pi$
$307$	−5.06913	−0.289311	−0.144655	+	0.989482i	$0.546207\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	22.8769	1.29723	0.648615	−	0.761117i	$-0.275349\pi$
$311$	22.8769	1.29723	0.648615	+	0.761117i	$0.275349\pi$
$312$	0	0
$313$	−23.0540	−1.30309	−0.651544	−	0.758611i	$-0.725878\pi$
$313$	−23.0540	−1.30309	−0.651544	+	0.758611i	$0.725878\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−12.0000	−0.673987	−0.336994	−	0.941507i	$-0.609410\pi$
$317$	−12.0000	−0.673987	−0.336994	+	0.941507i	$0.609410\pi$
$318$	0	0
$319$	−16.6847	−0.934162
$320$	0	0
$321$	0	0
$322$	0	0
$323$	51.3693	2.85827
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−2.00000	−0.110264
$330$	0	0
$331$	12.5616	0.690445	0.345223	−	0.938521i	$-0.387803\pi$
$331$	12.5616	0.690445	0.345223	+	0.938521i	$0.387803\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−5.43845	−0.297134
$336$	0	0
$337$	23.0540	1.25583	0.627915	−	0.778282i	$-0.283909\pi$
$337$	23.0540	1.25583	0.627915	+	0.778282i	$0.283909\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	8.68466	0.470301
$342$	0	0
$343$	7.68466	0.414933
$344$	0	0
$345$	0	0
$346$	0	0
$347$	18.7386	1.00594	0.502971	−	0.864303i	$-0.332240\pi$
$347$	18.7386	1.00594	0.502971	+	0.864303i	$0.332240\pi$
$348$	0	0
$349$	25.6847	1.37487	0.687434	−	0.726247i	$-0.258737\pi$
$349$	25.6847	1.37487	0.687434	+	0.726247i	$0.258737\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	14.0540	0.748018	0.374009	−	0.927425i	$-0.377983\pi$
$353$	14.0540	0.748018	0.374009	+	0.927425i	$0.377983\pi$
$354$	0	0
$355$	14.2462	0.756110
$356$	0	0
$357$	0	0
$358$	0	0
$359$	10.8769	0.574061	0.287030	−	0.957922i	$-0.407332\pi$
$359$	10.8769	0.574061	0.287030	+	0.957922i	$0.407332\pi$
$360$	0	0
$361$	40.0540	2.10810
$362$	0	0
$363$	0	0
$364$	0	0
$365$	13.6847	0.716288
$366$	0	0
$367$	−29.0540	−1.51661	−0.758303	−	0.651903i	$-0.773971\pi$
$367$	−29.0540	−1.51661	−0.758303	+	0.651903i	$0.773971\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	5.00000	0.258890	0.129445	−	0.991587i	$-0.458680\pi$
$373$	5.00000	0.258890	0.129445	+	0.991587i	$0.458680\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4.68466	0.241272
$378$	0	0
$379$	−21.9309	−1.12651	−0.563257	−	0.826282i	$-0.690452\pi$
$379$	−21.9309	−1.12651	−0.563257	+	0.826282i	$0.690452\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−12.1922	−0.622994	−0.311497	−	0.950247i	$-0.600830\pi$
$383$	−12.1922	−0.622994	−0.311497	+	0.950247i	$0.600830\pi$
$384$	0	0
$385$	−2.00000	−0.101929
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−32.0540	−1.62520	−0.812601	−	0.582821i	$-0.801949\pi$
$389$	−32.0540	−1.62520	−0.812601	+	0.582821i	$0.801949\pi$
$390$	0	0
$391$	23.8078	1.20401
$392$	0	0
$393$	0	0
$394$	0	0
$395$	17.2462	0.867751
$396$	0	0
$397$	−9.31534	−0.467524	−0.233762	−	0.972294i	$-0.575104\pi$
$397$	−9.31534	−0.467524	−0.233762	+	0.972294i	$0.575104\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	16.0000	0.799002	0.399501	−	0.916733i	$-0.369183\pi$
$401$	16.0000	0.799002	0.399501	+	0.916733i	$0.369183\pi$
$402$	0	0
$403$	−2.43845	−0.121468
$404$	0	0
$405$	0	0
$406$	0	0
$407$	20.2462	1.00357
$408$	0	0
$409$	14.3693	0.710517	0.355258	−	0.934768i	$-0.384393\pi$
$409$	14.3693	0.710517	0.355258	+	0.934768i	$0.384393\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0.630683	0.0310339
$414$	0	0
$415$	4.87689	0.239397
$416$	0	0
$417$	0	0
$418$	0	0
$419$	20.4384	0.998483	0.499242	−	0.866463i	$-0.333612\pi$
$419$	20.4384	0.998483	0.499242	+	0.866463i	$0.333612\pi$
$420$	0	0
$421$	−19.0540	−0.928634	−0.464317	−	0.885669i	$-0.653700\pi$
$421$	−19.0540	−0.928634	−0.464317	+	0.885669i	$0.653700\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−6.68466	−0.324254
$426$	0	0
$427$	−5.43845	−0.263185
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−5.61553	−0.270490	−0.135245	−	0.990812i	$-0.543182\pi$
$431$	−5.61553	−0.270490	−0.135245	+	0.990812i	$0.543182\pi$
$432$	0	0
$433$	−20.7386	−0.996635	−0.498318	−	0.866995i	$-0.666049\pi$
$433$	−20.7386	−0.996635	−0.498318	+	0.866995i	$0.666049\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	27.3693	1.30925
$438$	0	0
$439$	4.49242	0.214412	0.107206	−	0.994237i	$-0.465810\pi$
$439$	4.49242	0.214412	0.107206	+	0.994237i	$0.465810\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	26.2462	1.24700	0.623498	−	0.781825i	$-0.285711\pi$
$443$	26.2462	1.24700	0.623498	+	0.781825i	$0.285711\pi$
$444$	0	0
$445$	11.3693	0.538957
$446$	0	0
$447$	0	0
$448$	0	0
$449$	22.0000	1.03824	0.519122	−	0.854700i	$-0.326259\pi$
$449$	22.0000	1.03824	0.519122	+	0.854700i	$0.326259\pi$
$450$	0	0
$451$	14.2462	0.670828
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0.561553	0.0263260
$456$	0	0
$457$	−31.3693	−1.46739	−0.733697	−	0.679476i	$-0.762207\pi$
$457$	−31.3693	−1.46739	−0.733697	+	0.679476i	$0.762207\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−18.0000	−0.838344	−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000	−0.838344	−0.419172	+	0.907907i	$0.637680\pi$
$462$	0	0
$463$	1.68466	0.0782927	0.0391464	−	0.999233i	$-0.487536\pi$
$463$	1.68466	0.0782927	0.0391464	+	0.999233i	$0.487536\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−19.8617	−0.919092	−0.459546	−	0.888154i	$-0.651988\pi$
$467$	−19.8617	−0.919092	−0.459546	+	0.888154i	$0.651988\pi$
$468$	0	0
$469$	3.05398	0.141019
$470$	0	0
$471$	0	0
$472$	0	0
$473$	16.6847	0.767161
$474$	0	0
$475$	−7.68466	−0.352596
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	−5.68466	−0.259198
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−2.31534	−0.105134
$486$	0	0
$487$	−3.19224	−0.144654	−0.0723270	−	0.997381i	$-0.523043\pi$
$487$	−3.19224	−0.144654	−0.0723270	+	0.997381i	$0.523043\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−29.6155	−1.33653	−0.668265	−	0.743923i	$-0.732963\pi$
$491$	−29.6155	−1.33653	−0.668265	+	0.743923i	$0.732963\pi$
$492$	0	0
$493$	31.3153	1.41037
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−8.00000	−0.358849
$498$	0	0
$499$	0	0		−	1.00000i	$-0.5\pi$
$499$	0	0			1.00000i	$0.5\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	29.4233	1.31192	0.655960	−	0.754796i	$-0.272264\pi$
$503$	29.4233	1.31192	0.655960	+	0.754796i	$0.272264\pi$
$504$	0	0
$505$	8.68466	0.386462
$506$	0	0
$507$	0	0
$508$	0	0
$509$	38.6847	1.71467	0.857334	−	0.514761i	$-0.172119\pi$
$509$	38.6847	1.71467	0.857334	+	0.514761i	$0.172119\pi$
$510$	0	0
$511$	−7.68466	−0.339949
$512$	0	0
$513$	0	0
$514$	0	0
$515$	11.4384	0.504038
$516$	0	0
$517$	12.6847	0.557871
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−31.3693	−1.37431	−0.687157	−	0.726509i	$-0.741142\pi$
$521$	−31.3693	−1.37431	−0.687157	+	0.726509i	$0.741142\pi$
$522$	0	0
$523$	37.1080	1.62262	0.811309	−	0.584618i	$-0.198756\pi$
$523$	37.1080	1.62262	0.811309	+	0.584618i	$0.198756\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−16.3002	−0.710047
$528$	0	0
$529$	−10.3153	−0.448493
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−4.00000	−0.173259
$534$	0	0
$535$	8.24621	0.356515
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−23.8078	−1.02547
$540$	0	0
$541$	−9.68466	−0.416376	−0.208188	−	0.978089i	$-0.566757\pi$
$541$	−9.68466	−0.416376	−0.208188	+	0.978089i	$0.566757\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−15.3693	−0.658349
$546$	0	0
$547$	−18.3693	−0.785415	−0.392708	−	0.919663i	$-0.628462\pi$
$547$	−18.3693	−0.785415	−0.392708	+	0.919663i	$0.628462\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	36.0000	1.53365
$552$	0	0
$553$	−9.68466	−0.411834
$554$	0	0
$555$	0	0
$556$	0	0
$557$	23.3693	0.990190	0.495095	−	0.868839i	$-0.335133\pi$
$557$	23.3693	0.990190	0.495095	+	0.868839i	$0.335133\pi$
$558$	0	0
$559$	−4.68466	−0.198140
$560$	0	0
$561$	0	0
$562$	0	0
$563$	43.1231	1.81742	0.908711	−	0.417426i	$-0.137068\pi$
$563$	43.1231	1.81742	0.908711	+	0.417426i	$0.137068\pi$
$564$	0	0
$565$	5.31534	0.223618
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−7.36932	−0.308938	−0.154469	−	0.987998i	$-0.549367\pi$
$569$	−7.36932	−0.308938	−0.154469	+	0.987998i	$0.549367\pi$
$570$	0	0
$571$	35.4384	1.48305	0.741526	−	0.670924i	$-0.234102\pi$
$571$	35.4384	1.48305	0.741526	+	0.670924i	$0.234102\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−3.56155	−0.148527
$576$	0	0
$577$	42.4233	1.76610	0.883052	−	0.469274i	$-0.155484\pi$
$577$	42.4233	1.76610	0.883052	+	0.469274i	$0.155484\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−2.73863	−0.113618
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−6.00000	−0.247647	−0.123823	−	0.992304i	$-0.539516\pi$
$587$	−6.00000	−0.247647	−0.123823	+	0.992304i	$0.539516\pi$
$588$	0	0
$589$	−18.7386	−0.772112
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−7.31534	−0.300405	−0.150203	−	0.988655i	$-0.547993\pi$
$593$	−7.31534	−0.300405	−0.150203	+	0.988655i	$0.547993\pi$
$594$	0	0
$595$	3.75379	0.153890
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−11.6155	−0.474598	−0.237299	−	0.971437i	$-0.576262\pi$
$599$	−11.6155	−0.474598	−0.237299	+	0.971437i	$0.576262\pi$
$600$	0	0
$601$	−7.94602	−0.324125	−0.162063	−	0.986780i	$-0.551815\pi$
$601$	−7.94602	−0.324125	−0.162063	+	0.986780i	$0.551815\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	1.68466	0.0684911
$606$	0	0
$607$	−24.5616	−0.996923	−0.498461	−	0.866912i	$-0.666101\pi$
$607$	−24.5616	−0.996923	−0.498461	+	0.866912i	$0.666101\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−3.56155	−0.144085
$612$	0	0
$613$	−40.3693	−1.63050	−0.815251	−	0.579108i	$-0.803401\pi$
$613$	−40.3693	−1.63050	−0.815251	+	0.579108i	$0.803401\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−0.0539753	−0.00217296	−0.00108648	−	0.999999i	$-0.500346\pi$
$617$	−0.0539753	−0.00217296	−0.00108648	+	0.999999i	$0.500346\pi$
$618$	0	0
$619$	−14.8078	−0.595174	−0.297587	−	0.954695i	$-0.596182\pi$
$619$	−14.8078	−0.595174	−0.297587	+	0.954695i	$0.596182\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−6.38447	−0.255788
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−38.0000	−1.51516
$630$	0	0
$631$	−11.0540	−0.440052	−0.220026	−	0.975494i	$-0.570614\pi$
$631$	−11.0540	−0.440052	−0.220026	+	0.975494i	$0.570614\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	7.12311	0.282672
$636$	0	0
$637$	6.68466	0.264856
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−37.3693	−1.47600	−0.738000	−	0.674801i	$-0.764229\pi$
$641$	−37.3693	−1.47600	−0.738000	+	0.674801i	$0.764229\pi$
$642$	0	0
$643$	33.1771	1.30838	0.654188	−	0.756332i	$-0.273010\pi$
$643$	33.1771	1.30838	0.654188	+	0.756332i	$0.273010\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	40.4924	1.59192	0.795961	−	0.605348i	$-0.206966\pi$
$647$	40.4924	1.59192	0.795961	+	0.605348i	$0.206966\pi$
$648$	0	0
$649$	−4.00000	−0.157014
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−22.7386	−0.889832	−0.444916	−	0.895572i	$-0.646766\pi$
$653$	−22.7386	−0.889832	−0.444916	+	0.895572i	$0.646766\pi$
$654$	0	0
$655$	11.8078	0.461368
$656$	0	0
$657$	0	0
$658$	0	0
$659$	30.7386	1.19741	0.598704	−	0.800971i	$-0.295683\pi$
$659$	30.7386	1.19741	0.598704	+	0.800971i	$0.295683\pi$
$660$	0	0
$661$	−21.0540	−0.818905	−0.409452	−	0.912331i	$-0.634280\pi$
$661$	−21.0540	−0.818905	−0.409452	+	0.912331i	$0.634280\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	4.31534	0.167342
$666$	0	0
$667$	16.6847	0.646033
$668$	0	0
$669$	0	0
$670$	0	0
$671$	34.4924	1.33157
$672$	0	0
$673$	13.0540	0.503194	0.251597	−	0.967832i	$-0.419044\pi$
$673$	13.0540	0.503194	0.251597	+	0.967832i	$0.419044\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	11.3693	0.436958	0.218479	−	0.975842i	$-0.429890\pi$
$677$	11.3693	0.436958	0.218479	+	0.975842i	$0.429890\pi$
$678$	0	0
$679$	1.30019	0.0498966
$680$	0	0
$681$	0	0
$682$	0	0
$683$	40.4924	1.54940	0.774700	−	0.632329i	$-0.217901\pi$
$683$	40.4924	1.54940	0.774700	+	0.632329i	$0.217901\pi$
$684$	0	0
$685$	−6.00000	−0.229248
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	16.8769	0.642027	0.321014	−	0.947075i	$-0.395976\pi$
$691$	16.8769	0.642027	0.321014	+	0.947075i	$0.395976\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	2.80776	0.106505
$696$	0	0
$697$	−26.7386	−1.01280
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−34.7926	−1.31410	−0.657049	−	0.753848i	$-0.728196\pi$
$701$	−34.7926	−1.31410	−0.657049	+	0.753848i	$0.728196\pi$
$702$	0	0
$703$	−43.6847	−1.64760
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−4.87689	−0.183414
$708$	0	0
$709$	11.6847	0.438827	0.219413	−	0.975632i	$-0.429586\pi$
$709$	11.6847	0.438827	0.219413	+	0.975632i	$0.429586\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−8.68466	−0.325243
$714$	0	0
$715$	−3.56155	−0.133195
$716$	0	0
$717$	0	0
$718$	0	0
$719$	4.87689	0.181877	0.0909387	−	0.995856i	$-0.471013\pi$
$719$	4.87689	0.181877	0.0909387	+	0.995856i	$0.471013\pi$
$720$	0	0
$721$	−6.42329	−0.239216
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−4.68466	−0.173984
$726$	0	0
$727$	−24.0000	−0.890111	−0.445055	−	0.895503i	$-0.646816\pi$
$727$	−24.0000	−0.890111	−0.445055	+	0.895503i	$0.646816\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−31.3153	−1.15824
$732$	0	0
$733$	−40.7386	−1.50472	−0.752358	−	0.658755i	$-0.771084\pi$
$733$	−40.7386	−1.50472	−0.752358	+	0.658755i	$0.771084\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−19.3693	−0.713478
$738$	0	0
$739$	7.50758	0.276171	0.138085	−	0.990420i	$-0.455905\pi$
$739$	7.50758	0.276171	0.138085	+	0.990420i	$0.455905\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−39.1771	−1.43727	−0.718634	−	0.695389i	$-0.755232\pi$
$743$	−39.1771	−1.43727	−0.718634	+	0.695389i	$0.755232\pi$
$744$	0	0
$745$	3.31534	0.121465
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−4.63068	−0.169201
$750$	0	0
$751$	−3.73863	−0.136425	−0.0682123	−	0.997671i	$-0.521730\pi$
$751$	−3.73863	−0.136425	−0.0682123	+	0.997671i	$0.521730\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	15.0000	0.545906
$756$	0	0
$757$	−45.1080	−1.63948	−0.819738	−	0.572738i	$-0.805881\pi$
$757$	−45.1080	−1.63948	−0.819738	+	0.572738i	$0.805881\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	22.0000	0.797499	0.398750	−	0.917060i	$-0.369444\pi$
$761$	22.0000	0.797499	0.398750	+	0.917060i	$0.369444\pi$
$762$	0	0
$763$	8.63068	0.312452
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.12311	0.0405530
$768$	0	0
$769$	15.0000	0.540914	0.270457	−	0.962732i	$-0.412825\pi$
$769$	15.0000	0.540914	0.270457	+	0.962732i	$0.412825\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	14.6307	0.526229	0.263115	−	0.964765i	$-0.415250\pi$
$773$	14.6307	0.526229	0.263115	+	0.964765i	$0.415250\pi$
$774$	0	0
$775$	2.43845	0.0875916
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−30.7386	−1.10133
$780$	0	0
$781$	50.7386	1.81557
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−10.6847	−0.381352
$786$	0	0
$787$	−21.7386	−0.774899	−0.387449	−	0.921891i	$-0.626644\pi$
$787$	−21.7386	−0.774899	−0.387449	+	0.921891i	$0.626644\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−2.98485	−0.106129
$792$	0	0
$793$	−9.68466	−0.343912
$794$	0	0
$795$	0	0
$796$	0	0
$797$	5.26137	0.186367	0.0931836	−	0.995649i	$-0.470296\pi$
$797$	5.26137	0.186367	0.0931836	+	0.995649i	$0.470296\pi$
$798$	0	0
$799$	−23.8078	−0.842258
$800$	0	0
$801$	0	0
$802$	0	0
$803$	48.7386	1.71995
$804$	0	0
$805$	2.00000	0.0704907
$806$	0	0
$807$	0	0
$808$	0	0
$809$	36.0000	1.26569	0.632846	−	0.774277i	$-0.281886\pi$
$809$	36.0000	1.26569	0.632846	+	0.774277i	$0.281886\pi$
$810$	0	0
$811$	12.3845	0.434878	0.217439	−	0.976074i	$-0.430230\pi$
$811$	12.3845	0.434878	0.217439	+	0.976074i	$0.430230\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−16.1231	−0.564768
$816$	0	0
$817$	−36.0000	−1.25948
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−22.0000	−0.767805	−0.383903	−	0.923374i	$-0.625420\pi$
$821$	−22.0000	−0.767805	−0.383903	+	0.923374i	$0.625420\pi$
$822$	0	0
$823$	−53.7926	−1.87509	−0.937546	−	0.347861i	$-0.886908\pi$
$823$	−53.7926	−1.87509	−0.937546	+	0.347861i	$0.886908\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−44.2462	−1.53859	−0.769296	−	0.638893i	$-0.779393\pi$
$827$	−44.2462	−1.53859	−0.769296	+	0.638893i	$0.779393\pi$
$828$	0	0
$829$	22.4233	0.778793	0.389397	−	0.921070i	$-0.372684\pi$
$829$	22.4233	0.778793	0.389397	+	0.921070i	$0.372684\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	44.6847	1.54823
$834$	0	0
$835$	25.1231	0.869421
$836$	0	0
$837$	0	0
$838$	0	0
$839$	31.1231	1.07449	0.537244	−	0.843427i	$-0.319465\pi$
$839$	31.1231	1.07449	0.537244	+	0.843427i	$0.319465\pi$
$840$	0	0
$841$	−7.05398	−0.243241
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−12.0000	−0.412813
$846$	0	0
$847$	−0.946025	−0.0325058
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−20.2462	−0.694031
$852$	0	0
$853$	45.1080	1.54447	0.772233	−	0.635339i	$-0.219140\pi$
$853$	45.1080	1.54447	0.772233	+	0.635339i	$0.219140\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	9.36932	0.320050	0.160025	−	0.987113i	$-0.448843\pi$
$857$	9.36932	0.320050	0.160025	+	0.987113i	$0.448843\pi$
$858$	0	0
$859$	−6.94602	−0.236995	−0.118498	−	0.992954i	$-0.537808\pi$
$859$	−6.94602	−0.236995	−0.118498	+	0.992954i	$0.537808\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−34.6847	−1.18068	−0.590340	−	0.807155i	$-0.701006\pi$
$863$	−34.6847	−1.18068	−0.590340	+	0.807155i	$0.701006\pi$
$864$	0	0
$865$	1.36932	0.0465582
$866$	0	0
$867$	0	0
$868$	0	0
$869$	61.4233	2.08364
$870$	0	0
$871$	5.43845	0.184275
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−0.561553	−0.0189839
$876$	0	0
$877$	11.7386	0.396385	0.198193	−	0.980163i	$-0.436493\pi$
$877$	11.7386	0.396385	0.198193	+	0.980163i	$0.436493\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	12.7386	0.429175	0.214588	−	0.976705i	$-0.431159\pi$
$881$	12.7386	0.429175	0.214588	+	0.976705i	$0.431159\pi$
$882$	0	0
$883$	−25.3002	−0.851419	−0.425710	−	0.904860i	$-0.639975\pi$
$883$	−25.3002	−0.851419	−0.425710	+	0.904860i	$0.639975\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	24.1922	0.812296	0.406148	−	0.913807i	$-0.366872\pi$
$887$	24.1922	0.812296	0.406148	+	0.913807i	$0.366872\pi$
$888$	0	0
$889$	−4.00000	−0.134156
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−27.3693	−0.915879
$894$	0	0
$895$	14.2462	0.476198
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−11.4233	−0.380988
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−0.315342	−0.0104823
$906$	0	0
$907$	27.7386	0.921046	0.460523	−	0.887648i	$-0.347662\pi$
$907$	27.7386	0.921046	0.460523	+	0.887648i	$0.347662\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−9.75379	−0.323157	−0.161579	−	0.986860i	$-0.551659\pi$
$911$	−9.75379	−0.323157	−0.161579	+	0.986860i	$0.551659\pi$
$912$	0	0
$913$	17.3693	0.574840
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−6.63068	−0.218964
$918$	0	0
$919$	−40.6847	−1.34206	−0.671032	−	0.741429i	$-0.734148\pi$
$919$	−40.6847	−1.34206	−0.671032	+	0.741429i	$0.734148\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−14.2462	−0.468920
$924$	0	0
$925$	5.68466	0.186910
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−32.0000	−1.04989	−0.524943	−	0.851137i	$-0.675913\pi$
$929$	−32.0000	−1.04989	−0.524943	+	0.851137i	$0.675913\pi$
$930$	0	0
$931$	51.3693	1.68356
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−23.8078	−0.778597
$936$	0	0
$937$	26.3153	0.859685	0.429842	−	0.902904i	$-0.358569\pi$
$937$	26.3153	0.859685	0.429842	+	0.902904i	$0.358569\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	37.4233	1.21996	0.609982	−	0.792415i	$-0.291177\pi$
$941$	37.4233	1.21996	0.609982	+	0.792415i	$0.291177\pi$
$942$	0	0
$943$	−14.2462	−0.463920
$944$	0	0
$945$	0	0
$946$	0	0
$947$	24.7386	0.803898	0.401949	−	0.915662i	$-0.368333\pi$
$947$	24.7386	0.803898	0.401949	+	0.915662i	$0.368333\pi$
$948$	0	0
$949$	−13.6847	−0.444223
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−37.3153	−1.20876	−0.604381	−	0.796695i	$-0.706580\pi$
$953$	−37.3153	−1.20876	−0.604381	+	0.796695i	$0.706580\pi$
$954$	0	0
$955$	18.0000	0.582466
$956$	0	0
$957$	0	0
$958$	0	0
$959$	3.36932	0.108801
$960$	0	0
$961$	−25.0540	−0.808193
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−1.68466	−0.0542311
$966$	0	0
$967$	−44.4233	−1.42856	−0.714278	−	0.699862i	$-0.753245\pi$
$967$	−44.4233	−1.42856	−0.714278	+	0.699862i	$0.753245\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−1.69981	−0.0545496	−0.0272748	−	0.999628i	$-0.508683\pi$
$971$	−1.69981	−0.0545496	−0.0272748	+	0.999628i	$0.508683\pi$
$972$	0	0
$973$	−1.57671	−0.0505469
$974$	0	0
$975$	0	0
$976$	0	0
$977$	59.4233	1.90112	0.950560	−	0.310542i	$-0.100510\pi$
$977$	59.4233	1.90112	0.950560	+	0.310542i	$0.100510\pi$
$978$	0	0
$979$	40.4924	1.29414
$980$	0	0
$981$	0	0
$982$	0	0
$983$	3.17708	0.101333	0.0506666	−	0.998716i	$-0.483865\pi$
$983$	3.17708	0.101333	0.0506666	+	0.998716i	$0.483865\pi$
$984$	0	0
$985$	−15.3693	−0.489707
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−16.6847	−0.530541
$990$	0	0
$991$	46.4773	1.47640	0.738200	−	0.674582i	$-0.235676\pi$
$991$	46.4773	1.47640	0.738200	+	0.674582i	$0.235676\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−7.49242	−0.237526
$996$	0	0
$997$	5.31534	0.168339	0.0841693	−	0.996451i	$-0.473176\pi$
$997$	5.31534	0.168339	0.0841693	+	0.996451i	$0.473176\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8640.2.a.di.1.1		2
3.2	odd	2		8640.2.a.cu.1.1		2
4.3	odd	2		8640.2.a.cx.1.2		2
8.3	odd	2		4320.2.a.n.1.2	✓	2
8.5	even	2		4320.2.a.u.1.1	yes	2
12.11	even	2		8640.2.a.cj.1.2		2
24.5	odd	2		4320.2.a.be.1.1	yes	2
24.11	even	2		4320.2.a.x.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4320.2.a.n.1.2	✓	2	8.3	odd	2
4320.2.a.u.1.1	yes	2	8.5	even	2
4320.2.a.x.1.2	yes	2	24.11	even	2
4320.2.a.be.1.1	yes	2	24.5	odd	2
8640.2.a.cj.1.2		2	12.11	even	2
8640.2.a.cu.1.1		2	3.2	odd	2
8640.2.a.cx.1.2		2	4.3	odd	2
8640.2.a.di.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:	\( N \)	\(=\)	\( 8640 = 2^{6} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8640.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} -0.561553 q^{7} +3.56155 q^{11} -1.00000 q^{13} -6.68466 q^{17} -7.68466 q^{19} -3.56155 q^{23} +1.00000 q^{25} -4.68466 q^{29} +2.43845 q^{31} -0.561553 q^{35} +5.68466 q^{37} +4.00000 q^{41} +4.68466 q^{43} +3.56155 q^{47} -6.68466 q^{49} +3.56155 q^{55} -1.12311 q^{59} +9.68466 q^{61} -1.00000 q^{65} -5.43845 q^{67} +14.2462 q^{71} +13.6847 q^{73} -2.00000 q^{77} +17.2462 q^{79} +4.87689 q^{83} -6.68466 q^{85} +11.3693 q^{89} +0.561553 q^{91} -7.68466 q^{95} -2.31534 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} + 3 q^{7} + 3 q^{11} - 2 q^{13} - q^{17} - 3 q^{19} - 3 q^{23} + 2 q^{25} + 3 q^{29} + 9 q^{31} + 3 q^{35} - q^{37} + 8 q^{41} - 3 q^{43} + 3 q^{47} - q^{49} + 3 q^{55} + 6 q^{59} + 7 q^{61}+ \cdots - 17 q^{97}+O(q^{100}) \) 2 * q + 2 * q^5 + 3 * q^7 + 3 * q^11 - 2 * q^13 - q^17 - 3 * q^19 - 3 * q^23 + 2 * q^25 + 3 * q^29 + 9 * q^31 + 3 * q^35 - q^37 + 8 * q^41 - 3 * q^43 + 3 * q^47 - q^49 + 3 * q^55 + 6 * q^59 + 7 * q^61 - 2 * q^65 - 15 * q^67 + 12 * q^71 + 15 * q^73 - 4 * q^77 + 18 * q^79 + 18 * q^83 - q^85 - 2 * q^89 - 3 * q^91 - 3 * q^95 - 17 * q^97

Introduction

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