LMFDB - Embedded newform 864.3.e.e.161.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [864,3,Mod(161,864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(864, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("864.161");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 864 = 2^{5} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	864.e (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$23.5422948407$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.2441150464.4
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 14x^{6} + 77x^{4} - 188x^{2} + 196 $ x^8 - 14x^6 + 77x^4 - 188*x^2 + 196
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.4
Root			$2.27249 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	864.161
Dual form			864.3.e.e.161.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.88259i q^{5} +10.9726 q^{7} +O(q^{10})$	$q-1.88259i q^{5} +10.9726 q^{7} -0.171573i q^{11} +6.48528 q^{13} +27.2699i q^{17} +5.32478 q^{19} +3.51472i q^{23} +21.4558 q^{25} -34.8003i q^{29} -26.9469 q^{31} -20.6569i q^{35} +46.4264 q^{37} -23.5047i q^{41} -55.1858 q^{43} +57.5980i q^{47} +71.3970 q^{49} +51.7436i q^{53} -0.323002 q^{55} -82.2843i q^{59} +79.8823 q^{61} -12.2091i q^{65} +44.5362 q^{67} +41.2304i q^{71} +66.3381 q^{73} -1.88259i q^{77} +115.050 q^{79} +36.3381i q^{83} +51.3381 q^{85} -158.027i q^{89} +71.1601 q^{91} -10.0244i q^{95} +62.8823 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 16 q^{13} - 32 q^{25} + 32 q^{37} + 96 q^{49} + 96 q^{61} - 216 q^{73} - 336 q^{85} - 40 q^{97}+O(q^{100}) $ 8 * q - 16 * q^13 - 32 * q^25 + 32 * q^37 + 96 * q^49 + 96 * q^61 - 216 * q^73 - 336 * q^85 - 40 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/864\mathbb{Z}\right)^\times$.

$n$	$325$	$353$	$703$
$\chi(n)$	$1$	$-1$	$1$

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
$5$	− 1.88259i	− 0.376519i	−0.982119	−	0.188259i	$-0.939715\pi$
$5$	− 1.88259i	− 0.376519i	0.982119	−	0.188259i	$-0.0602845\pi$
$6$	0	0
$7$	10.9726	1.56751	0.783754	−	0.621071i	$-0.213302\pi$
$7$	10.9726	1.56751	0.783754	+	0.621071i	$0.213302\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 0.171573i	− 0.0155975i	−0.999970	−	0.00779877i	$-0.997518\pi$
$11$	− 0.171573i	− 0.0155975i	0.999970	−	0.00779877i	$-0.00248245\pi$
$12$	0	0
$13$	6.48528	0.498868	0.249434	−	0.968392i	$-0.419755\pi$
$13$	6.48528	0.498868	0.249434	+	0.968392i	$0.419755\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	27.2699i	1.60411i	0.597249	+	0.802056i	$0.296260\pi$
$17$	27.2699i	1.60411i	−0.597249	+	0.802056i	$0.703740\pi$
$18$	0	0
$19$	5.32478	0.280251	0.140126	−	0.990134i	$-0.455249\pi$
$19$	5.32478	0.280251	0.140126	+	0.990134i	$0.455249\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.51472i	0.152814i	0.997077	+	0.0764069i	$0.0243448\pi$
$23$	3.51472i	0.152814i	−0.997077	+	0.0764069i	$0.975655\pi$
$24$	0	0
$25$	21.4558	0.858234
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 34.8003i	− 1.20001i	−0.799997	−	0.600004i	$-0.795165\pi$
$29$	− 34.8003i	− 1.20001i	0.799997	−	0.600004i	$-0.204835\pi$
$30$	0	0
$31$	−26.9469	−0.869254	−0.434627	−	0.900610i	$-0.643120\pi$
$31$	−26.9469	−0.869254	−0.434627	+	0.900610i	$0.643120\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 20.6569i	− 0.590196i
$36$	0	0
$37$	46.4264	1.25477	0.627384	−	0.778710i	$-0.284126\pi$
$37$	46.4264	1.25477	0.627384	+	0.778710i	$0.284126\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 23.5047i	− 0.573285i	−0.958038	−	0.286643i	$-0.907461\pi$
$41$	− 23.5047i	− 0.573285i	0.958038	−	0.286643i	$-0.0925393\pi$
$42$	0	0
$43$	−55.1858	−1.28339	−0.641695	−	0.766960i	$-0.721769\pi$
$43$	−55.1858	−1.28339	−0.641695	+	0.766960i	$0.721769\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	57.5980i	1.22549i	0.790281	+	0.612744i	$0.209935\pi$
$47$	57.5980i	1.22549i	−0.790281	+	0.612744i	$0.790065\pi$
$48$	0	0
$49$	71.3970	1.45708
$50$	0	0
$51$	0	0
$52$	0	0
$53$	51.7436i	0.976294i	0.872761	+	0.488147i	$0.162327\pi$
$53$	51.7436i	0.976294i	−0.872761	+	0.488147i	$0.837673\pi$
$54$	0	0
$55$	−0.323002	−0.00587276
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 82.2843i	− 1.39465i	−0.716756	−	0.697324i	$-0.754374\pi$
$59$	− 82.2843i	− 1.39465i	0.716756	−	0.697324i	$-0.245626\pi$
$60$	0	0
$61$	79.8823	1.30955	0.654773	−	0.755826i	$-0.272764\pi$
$61$	79.8823	1.30955	0.654773	+	0.755826i	$0.272764\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 12.2091i	− 0.187833i
$66$	0	0
$67$	44.5362	0.664720	0.332360	−	0.943153i	$-0.392155\pi$
$67$	44.5362	0.664720	0.332360	+	0.943153i	$0.392155\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	41.2304i	0.580711i	0.956919	+	0.290355i	$0.0937735\pi$
$71$	41.2304i	0.580711i	−0.956919	+	0.290355i	$0.906227\pi$
$72$	0	0
$73$	66.3381	0.908741	0.454371	−	0.890813i	$-0.349864\pi$
$73$	66.3381	0.908741	0.454371	+	0.890813i	$0.349864\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 1.88259i	− 0.0244493i
$78$	0	0
$79$	115.050	1.45633	0.728167	−	0.685400i	$-0.240373\pi$
$79$	115.050	1.45633	0.728167	+	0.685400i	$0.240373\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	36.3381i	0.437808i	0.975746	+	0.218904i	$0.0702482\pi$
$83$	36.3381i	0.437808i	−0.975746	+	0.218904i	$0.929752\pi$
$84$	0	0
$85$	51.3381	0.603978
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 158.027i	− 1.77558i	−0.460245	−	0.887792i	$-0.652238\pi$
$89$	− 158.027i	− 1.77558i	0.460245	−	0.887792i	$-0.347762\pi$
$90$	0	0
$91$	71.1601	0.781979
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 10.0244i	− 0.105520i
$96$	0	0
$97$	62.8823	0.648271	0.324135	−	0.946011i	$-0.394927\pi$
$97$	62.8823	0.648271	0.324135	+	0.946011i	$0.394927\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	130.702i	1.29408i	0.762458	+	0.647038i	$0.223992\pi$
$101$	130.702i	1.29408i	−0.762458	+	0.647038i	$0.776008\pi$
$102$	0	0
$103$	−72.4521	−0.703419	−0.351709	−	0.936109i	$-0.614399\pi$
$103$	−72.4521	−0.703419	−0.351709	+	0.936109i	$0.614399\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 168.255i	− 1.57248i	−0.617924	−	0.786238i	$-0.712026\pi$
$107$	− 168.255i	− 1.57248i	0.617924	−	0.786238i	$-0.287974\pi$
$108$	0	0
$109$	103.397	0.948596	0.474298	−	0.880364i	$-0.342702\pi$
$109$	103.397	0.948596	0.474298	+	0.880364i	$0.342702\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	140.115i	1.23995i	0.784621	+	0.619976i	$0.212858\pi$
$113$	140.115i	1.23995i	−0.784621	+	0.619976i	$0.787142\pi$
$114$	0	0
$115$	6.61678	0.0575373
$116$	0	0
$117$	0	0
$118$	0	0
$119$	299.220i	2.51446i
$120$	0	0
$121$	120.971	0.999757
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 87.4574i	− 0.699660i
$126$	0	0
$127$	−120.052	−0.945292	−0.472646	−	0.881252i	$-0.656701\pi$
$127$	−120.052	−0.945292	−0.472646	+	0.881252i	$0.656701\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 198.421i	− 1.51467i	−0.653028	−	0.757333i	$-0.726502\pi$
$131$	− 198.421i	− 1.51467i	0.653028	−	0.757333i	$-0.273498\pi$
$132$	0	0
$133$	58.4264	0.439296
$134$	0	0
$135$	0	0
$136$	0	0
$137$	120.375i	0.878650i	0.898328	+	0.439325i	$0.144782\pi$
$137$	120.375i	0.878650i	−0.898328	+	0.439325i	$0.855218\pi$
$138$	0	0
$139$	−11.9416	−0.0859105	−0.0429553	−	0.999077i	$-0.513677\pi$
$139$	−11.9416	−0.0859105	−0.0429553	+	0.999077i	$0.513677\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 1.11270i	− 0.00778111i
$144$	0	0
$145$	−65.5147	−0.451826
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 55.5088i	− 0.372542i	−0.982498	−	0.186271i	$-0.940360\pi$
$149$	− 55.5088i	− 0.372542i	0.982498	−	0.186271i	$-0.0596403\pi$
$150$	0	0
$151$	−203.154	−1.34539	−0.672695	−	0.739920i	$-0.734863\pi$
$151$	−203.154	−1.34539	−0.672695	+	0.739920i	$0.734863\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	50.7300i	0.327290i
$156$	0	0
$157$	−164.971	−1.05077	−0.525384	−	0.850865i	$-0.676078\pi$
$157$	−164.971	−1.05077	−0.525384	+	0.850865i	$0.676078\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	38.5654i	0.239537i
$162$	0	0
$163$	−178.948	−1.09784	−0.548919	−	0.835875i	$-0.684961\pi$
$163$	−178.948	−1.09784	−0.548919	+	0.835875i	$0.684961\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	43.8924i	0.262828i	0.991328	+	0.131414i	$0.0419518\pi$
$167$	43.8924i	0.262828i	−0.991328	+	0.131414i	$0.958048\pi$
$168$	0	0
$169$	−126.941	−0.751131
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 191.858i	− 1.10901i	−0.832181	−	0.554504i	$-0.812908\pi$
$173$	− 191.858i	− 1.10901i	0.832181	−	0.554504i	$-0.187092\pi$
$174$	0	0
$175$	235.425	1.34529
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 144.765i	− 0.808740i	−0.914595	−	0.404370i	$-0.867491\pi$
$179$	− 144.765i	− 0.808740i	0.914595	−	0.404370i	$-0.132509\pi$
$180$	0	0
$181$	−0.735065	−0.00406113	−0.00203057	−	0.999998i	$-0.500646\pi$
$181$	−0.735065	−0.00406113	−0.00203057	+	0.999998i	$0.500646\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 87.4020i	− 0.472443i
$186$	0	0
$187$	4.67877	0.0250202
$188$	0	0
$189$	0	0
$190$	0	0
$191$	290.132i	1.51902i	0.650498	+	0.759508i	$0.274560\pi$
$191$	290.132i	1.51902i	−0.650498	+	0.759508i	$0.725440\pi$
$192$	0	0
$193$	63.6030	0.329549	0.164775	−	0.986331i	$-0.447310\pi$
$193$	63.6030	0.329549	0.164775	+	0.986331i	$0.447310\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 88.3710i	− 0.448584i	−0.974522	−	0.224292i	$-0.927993\pi$
$197$	− 88.3710i	− 0.448584i	0.974522	−	0.224292i	$-0.0720069\pi$
$198$	0	0
$199$	318.204	1.59902	0.799508	−	0.600656i	$-0.205094\pi$
$199$	318.204	1.59902	0.799508	+	0.600656i	$0.205094\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 381.848i	− 1.88102i
$204$	0	0
$205$	−44.2498	−0.215853
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 0.913587i	− 0.00437123i
$210$	0	0
$211$	46.6310	0.221000	0.110500	−	0.993876i	$-0.464755\pi$
$211$	46.6310	0.221000	0.110500	+	0.993876i	$0.464755\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	103.892i	0.483220i
$216$	0	0
$217$	−295.676	−1.36256
$218$	0	0
$219$	0	0
$220$	0	0
$221$	176.853i	0.800239i
$222$	0	0
$223$	−47.2770	−0.212004	−0.106002	−	0.994366i	$-0.533805\pi$
$223$	−47.2770	−0.212004	−0.106002	+	0.994366i	$0.533805\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	342.500i	1.50881i	0.656410	+	0.754404i	$0.272074\pi$
$227$	342.500i	1.50881i	−0.656410	+	0.754404i	$0.727926\pi$
$228$	0	0
$229$	−381.647	−1.66658	−0.833290	−	0.552836i	$-0.813545\pi$
$229$	−381.647	−1.66658	−0.833290	+	0.552836i	$0.813545\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	383.716i	1.64685i	0.567424	+	0.823426i	$0.307940\pi$
$233$	383.716i	1.64685i	−0.567424	+	0.823426i	$0.692060\pi$
$234$	0	0
$235$	108.434	0.461419
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 315.161i	− 1.31867i	−0.751850	−	0.659334i	$-0.770838\pi$
$239$	− 315.161i	− 1.31867i	0.751850	−	0.659334i	$-0.229162\pi$
$240$	0	0
$241$	−256.558	−1.06456	−0.532279	−	0.846569i	$-0.678664\pi$
$241$	−256.558	−1.06456	−0.532279	+	0.846569i	$0.678664\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 134.411i	− 0.548618i
$246$	0	0
$247$	34.5327	0.139808
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 246.000i	− 0.980080i	−0.871700	−	0.490040i	$-0.836982\pi$
$251$	− 246.000i	− 0.980080i	0.871700	−	0.490040i	$-0.163018\pi$
$252$	0	0
$253$	0.603030	0.00238352
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 443.046i	− 1.72391i	−0.506982	−	0.861957i	$-0.669239\pi$
$257$	− 443.046i	− 1.72391i	0.506982	−	0.861957i	$-0.330761\pi$
$258$	0	0
$259$	509.416	1.96686
$260$	0	0
$261$	0	0
$262$	0	0
$263$	479.897i	1.82470i	0.409409	+	0.912351i	$0.365735\pi$
$263$	479.897i	1.82470i	−0.409409	+	0.912351i	$0.634265\pi$
$264$	0	0
$265$	97.4121	0.367593
$266$	0	0
$267$	0	0
$268$	0	0
$269$	418.406i	1.55541i	0.628628	+	0.777706i	$0.283617\pi$
$269$	418.406i	1.55541i	−0.628628	+	0.777706i	$0.716383\pi$
$270$	0	0
$271$	−278.347	−1.02711	−0.513555	−	0.858057i	$-0.671672\pi$
$271$	−278.347	−1.02711	−0.513555	+	0.858057i	$0.671672\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 3.68124i	− 0.0133863i
$276$	0	0
$277$	−192.617	−0.695369	−0.347685	−	0.937612i	$-0.613032\pi$
$277$	−192.617	−0.695369	−0.347685	+	0.937612i	$0.613032\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 435.515i	− 1.54988i	−0.632037	−	0.774939i	$-0.717781\pi$
$281$	− 435.515i	− 1.54988i	0.632037	−	0.774939i	$-0.282219\pi$
$282$	0	0
$283$	−518.774	−1.83312	−0.916562	−	0.399894i	$-0.869047\pi$
$283$	−518.774	−1.83312	−0.916562	+	0.399894i	$0.869047\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 257.907i	− 0.898629i
$288$	0	0
$289$	−454.647	−1.57317
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 117.634i	− 0.401482i	−0.979644	−	0.200741i	$-0.935665\pi$
$293$	− 117.634i	− 0.401482i	0.979644	−	0.200741i	$-0.0643350\pi$
$294$	0	0
$295$	−154.908	−0.525111
$296$	0	0
$297$	0	0
$298$	0	0
$299$	22.7939i	0.0762339i
$300$	0	0
$301$	−605.529	−2.01172
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 150.386i	− 0.493068i
$306$	0	0
$307$	−371.618	−1.21048	−0.605241	−	0.796042i	$-0.706923\pi$
$307$	−371.618	−1.21048	−0.605241	+	0.796042i	$0.706923\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	27.8680i	0.0896076i	0.998996	+	0.0448038i	$0.0142663\pi$
$311$	27.8680i	0.0896076i	−0.998996	+	0.0448038i	$0.985734\pi$
$312$	0	0
$313$	−279.368	−0.892548	−0.446274	−	0.894896i	$-0.647249\pi$
$313$	−279.368	−0.892548	−0.446274	+	0.894896i	$0.647249\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	546.367i	1.72355i	0.507287	+	0.861777i	$0.330648\pi$
$317$	546.367i	1.72355i	−0.507287	+	0.861777i	$0.669352\pi$
$318$	0	0
$319$	−5.97078	−0.0187172
$320$	0	0
$321$	0	0
$322$	0	0
$323$	145.206i	0.449554i
$324$	0	0
$325$	139.147	0.428145
$326$	0	0
$327$	0	0
$328$	0	0
$329$	631.997i	1.92096i
$330$	0	0
$331$	−305.294	−0.922337	−0.461168	−	0.887313i	$-0.652570\pi$
$331$	−305.294	−0.922337	−0.461168	+	0.887313i	$0.652570\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 83.8436i	− 0.250279i
$336$	0	0
$337$	159.206	0.472422	0.236211	−	0.971702i	$-0.424094\pi$
$337$	159.206	0.472422	0.236211	+	0.971702i	$0.424094\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	4.62335i	0.0135582i
$342$	0	0
$343$	245.752	0.716478
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 296.387i	− 0.854141i	−0.904218	−	0.427070i	$-0.859546\pi$
$347$	− 296.387i	− 0.854141i	0.904218	−	0.427070i	$-0.140454\pi$
$348$	0	0
$349$	108.839	0.311858	0.155929	−	0.987768i	$-0.450163\pi$
$349$	108.839	0.311858	0.155929	+	0.987768i	$0.450163\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 12.3200i	− 0.0349008i	−0.999848	−	0.0174504i	$-0.994445\pi$
$353$	− 12.3200i	− 0.0349008i	0.999848	−	0.0174504i	$-0.00555492\pi$
$354$	0	0
$355$	77.6201	0.218648
$356$	0	0
$357$	0	0
$358$	0	0
$359$	92.2254i	0.256895i	0.991716	+	0.128448i	$0.0409994\pi$
$359$	92.2254i	0.256895i	−0.991716	+	0.128448i	$0.959001\pi$
$360$	0	0
$361$	−332.647	−0.921459
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 124.888i	− 0.342158i
$366$	0	0
$367$	13.0673	0.0356058	0.0178029	−	0.999842i	$-0.494333\pi$
$367$	13.0673	0.0356058	0.0178029	+	0.999842i	$0.494333\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	567.759i	1.53035i
$372$	0	0
$373$	−233.632	−0.626361	−0.313180	−	0.949694i	$-0.601394\pi$
$373$	−233.632	−0.626361	−0.313180	+	0.949694i	$0.601394\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 225.689i	− 0.598646i
$378$	0	0
$379$	683.528	1.80351	0.901753	−	0.432253i	$-0.142281\pi$
$379$	683.528	1.80351	0.901753	+	0.432253i	$0.142281\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 279.858i	− 0.730699i	−0.930870	−	0.365350i	$-0.880949\pi$
$383$	− 279.858i	− 0.730699i	0.930870	−	0.365350i	$-0.119051\pi$
$384$	0	0
$385$	−3.54416	−0.00920560
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 260.434i	− 0.669497i	−0.942308	−	0.334748i	$-0.891349\pi$
$389$	− 260.434i	− 0.669497i	0.942308	−	0.334748i	$-0.108651\pi$
$390$	0	0
$391$	−95.8460	−0.245130
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 216.593i	− 0.548337i
$396$	0	0
$397$	167.809	0.422693	0.211346	−	0.977411i	$-0.432215\pi$
$397$	167.809	0.422693	0.211346	+	0.977411i	$0.432215\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 60.2430i	− 0.150232i	−0.997175	−	0.0751159i	$-0.976067\pi$
$401$	− 60.2430i	− 0.150232i	0.997175	−	0.0751159i	$-0.0239327\pi$
$402$	0	0
$403$	−174.758	−0.433643
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 7.96551i	− 0.0195713i
$408$	0	0
$409$	532.765	1.30260	0.651301	−	0.758819i	$-0.274223\pi$
$409$	532.765	1.30260	0.651301	+	0.758819i	$0.274223\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 902.869i	− 2.18612i
$414$	0	0
$415$	68.4098	0.164843
$416$	0	0
$417$	0	0
$418$	0	0
$419$	483.255i	1.15335i	0.816973	+	0.576676i	$0.195651\pi$
$419$	483.255i	1.15335i	−0.816973	+	0.576676i	$0.804349\pi$
$420$	0	0
$421$	−29.2061	−0.0693731	−0.0346865	−	0.999398i	$-0.511043\pi$
$421$	−29.2061	−0.0693731	−0.0346865	+	0.999398i	$0.511043\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	585.098i	1.37670i
$426$	0	0
$427$	876.512	2.05272
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 634.971i	− 1.47325i	−0.676302	−	0.736625i	$-0.736418\pi$
$431$	− 634.971i	− 1.47325i	0.676302	−	0.736625i	$-0.263582\pi$
$432$	0	0
$433$	−28.5004	−0.0658209	−0.0329104	−	0.999458i	$-0.510478\pi$
$433$	−28.5004	−0.0658209	−0.0329104	+	0.999458i	$0.510478\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	18.7151i	0.0428263i
$438$	0	0
$439$	287.547	0.655006	0.327503	−	0.944850i	$-0.393793\pi$
$439$	287.547	0.655006	0.327503	+	0.944850i	$0.393793\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 0.176624i	0 0.000398699i	−1.00000		0.000199349i	$-0.999937\pi$
$443$	− 0.176624i	0 0.000398699i	1.00000		0.000199349i	$-6.34549e-5\pi$
$444$	0	0
$445$	−297.500	−0.668540
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 36.5166i	− 0.0813287i	−0.999173	−	0.0406644i	$-0.987053\pi$
$449$	− 36.5166i	− 0.0813287i	0.999173	−	0.0406644i	$-0.0129474\pi$
$450$	0	0
$451$	−4.03277	−0.00894184
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 133.966i	− 0.294430i
$456$	0	0
$457$	−243.235	−0.532244	−0.266122	−	0.963939i	$-0.585742\pi$
$457$	−243.235	−0.532244	−0.266122	+	0.963939i	$0.585742\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	155.231i	0.336726i	0.985725	+	0.168363i	$0.0538481\pi$
$461$	155.231i	0.336726i	−0.985725	+	0.168363i	$0.946152\pi$
$462$	0	0
$463$	−444.550	−0.960151	−0.480076	−	0.877227i	$-0.659391\pi$
$463$	−444.550	−0.960151	−0.480076	+	0.877227i	$0.659391\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	853.357i	1.82732i	0.406482	+	0.913659i	$0.366755\pi$
$467$	853.357i	1.82732i	−0.406482	+	0.913659i	$0.633245\pi$
$468$	0	0
$469$	488.676	1.04195
$470$	0	0
$471$	0	0
$472$	0	0
$473$	9.46838i	0.0200177i
$474$	0	0
$475$	114.248	0.240521
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 445.882i	− 0.930861i	−0.885085	−	0.465430i	$-0.845900\pi$
$479$	− 445.882i	− 0.930861i	0.885085	−	0.465430i	$-0.154100\pi$
$480$	0	0
$481$	301.088	0.625963
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 118.382i	− 0.244086i
$486$	0	0
$487$	−438.413	−0.900232	−0.450116	−	0.892970i	$-0.648617\pi$
$487$	−438.413	−0.900232	−0.450116	+	0.892970i	$0.648617\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	186.754i	0.380355i	0.981750	+	0.190178i	$0.0609064\pi$
$491$	186.754i	0.380355i	−0.981750	+	0.190178i	$0.939094\pi$
$492$	0	0
$493$	948.999	1.92495
$494$	0	0
$495$	0	0
$496$	0	0
$497$	452.403i	0.910268i
$498$	0	0
$499$	−432.775	−0.867284	−0.433642	−	0.901085i	$-0.642772\pi$
$499$	−432.775	−0.867284	−0.433642	+	0.901085i	$0.642772\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	45.6913i	0.0908377i	0.998968	+	0.0454188i	$0.0144622\pi$
$503$	45.6913i	0.0908377i	−0.998968	+	0.0454188i	$0.985538\pi$
$504$	0	0
$505$	246.058	0.487244
$506$	0	0
$507$	0	0
$508$	0	0
$509$	267.051i	0.524658i	0.964978	+	0.262329i	$0.0844906\pi$
$509$	267.051i	0.524658i	−0.964978	+	0.262329i	$0.915509\pi$
$510$	0	0
$511$	727.898	1.42446
$512$	0	0
$513$	0	0
$514$	0	0
$515$	136.398i	0.264850i
$516$	0	0
$517$	9.88225	0.0191146
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 144.101i	− 0.276586i	−0.990391	−	0.138293i	$-0.955838\pi$
$521$	− 144.101i	− 0.276586i	0.990391	−	0.138293i	$-0.0441616\pi$
$522$	0	0
$523$	−631.083	−1.20666	−0.603330	−	0.797491i	$-0.706160\pi$
$523$	−631.083	−1.20666	−0.603330	+	0.797491i	$0.706160\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 734.839i	− 1.39438i
$528$	0	0
$529$	516.647	0.976648
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 152.435i	− 0.285994i
$534$	0	0
$535$	−316.755	−0.592066
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 12.2498i	− 0.0227269i
$540$	0	0
$541$	235.765	0.435794	0.217897	−	0.975972i	$-0.430080\pi$
$541$	235.765	0.435794	0.217897	+	0.975972i	$0.430080\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 194.654i	− 0.357164i
$546$	0	0
$547$	−813.418	−1.48705	−0.743526	−	0.668707i	$-0.766848\pi$
$547$	−813.418	−1.48705	−0.743526	+	0.668707i	$0.766848\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 185.304i	− 0.336304i
$552$	0	0
$553$	1262.40	2.28281
$554$	0	0
$555$	0	0
$556$	0	0
$557$	175.773i	0.315571i	0.987473	+	0.157786i	$0.0504355\pi$
$557$	175.773i	0.315571i	−0.987473	+	0.157786i	$0.949565\pi$
$558$	0	0
$559$	−357.895	−0.640242
$560$	0	0
$561$	0	0
$562$	0	0
$563$	572.927i	1.01763i	0.860875	+	0.508816i	$0.169917\pi$
$563$	572.927i	1.01763i	−0.860875	+	0.508816i	$0.830083\pi$
$564$	0	0
$565$	263.779	0.466865
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 814.664i	− 1.43175i	−0.698230	−	0.715873i	$-0.746029\pi$
$569$	− 814.664i	− 1.43175i	0.698230	−	0.715873i	$-0.253971\pi$
$570$	0	0
$571$	−128.284	−0.224665	−0.112333	−	0.993671i	$-0.535832\pi$
$571$	−128.284	−0.224665	−0.112333	+	0.993671i	$0.535832\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	75.4113i	0.131150i
$576$	0	0
$577$	30.4996	0.0528589	0.0264294	−	0.999651i	$-0.491586\pi$
$577$	30.4996	0.0528589	0.0264294	+	0.999651i	$0.491586\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	398.722i	0.686268i
$582$	0	0
$583$	8.87780	0.0152278
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 193.794i	− 0.330143i	−0.986282	−	0.165071i	$-0.947215\pi$
$587$	− 193.794i	− 0.330143i	0.986282	−	0.165071i	$-0.0527855\pi$
$588$	0	0
$589$	−143.486	−0.243610
$590$	0	0
$591$	0	0
$592$	0	0
$593$	684.488i	1.15428i	0.816645	+	0.577140i	$0.195831\pi$
$593$	684.488i	1.15428i	−0.816645	+	0.577140i	$0.804169\pi$
$594$	0	0
$595$	563.310	0.946740
$596$	0	0
$597$	0	0
$598$	0	0
$599$	814.244i	1.35934i	0.733519	+	0.679669i	$0.237877\pi$
$599$	814.244i	1.35934i	−0.733519	+	0.679669i	$0.762123\pi$
$600$	0	0
$601$	−531.368	−0.884139	−0.442069	−	0.896981i	$-0.645756\pi$
$601$	−531.368	−0.884139	−0.442069	+	0.896981i	$0.645756\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 227.738i	− 0.376427i
$606$	0	0
$607$	464.880	0.765865	0.382933	−	0.923776i	$-0.374914\pi$
$607$	464.880	0.765865	0.382933	+	0.923776i	$0.374914\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	373.539i	0.611357i
$612$	0	0
$613$	1120.59	1.82804	0.914019	−	0.405672i	$-0.132962\pi$
$613$	1120.59	1.82804	0.914019	+	0.405672i	$0.132962\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 561.372i	− 0.909841i	−0.890532	−	0.454921i	$-0.849668\pi$
$617$	− 561.372i	− 0.909841i	0.890532	−	0.454921i	$-0.150332\pi$
$618$	0	0
$619$	−471.340	−0.761454	−0.380727	−	0.924687i	$-0.624326\pi$
$619$	−471.340	−0.761454	−0.380727	+	0.924687i	$0.624326\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 1733.96i	− 2.78324i
$624$	0	0
$625$	371.749	0.594799
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1266.04i	2.01279i
$630$	0	0
$631$	431.160	0.683296	0.341648	−	0.939828i	$-0.389015\pi$
$631$	431.160	0.683296	0.341648	+	0.939828i	$0.389015\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	226.009i	0.355920i
$636$	0	0
$637$	463.029	0.726891
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 399.580i	− 0.623370i	−0.950186	−	0.311685i	$-0.899107\pi$
$641$	− 399.580i	− 0.623370i	0.950186	−	0.311685i	$-0.100893\pi$
$642$	0	0
$643$	1013.66	1.57646	0.788231	−	0.615380i	$-0.210997\pi$
$643$	1013.66	1.57646	0.788231	+	0.615380i	$0.210997\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 28.4710i	− 0.0440046i	−0.999758	−	0.0220023i	$-0.992996\pi$
$647$	− 28.4710i	− 0.0440046i	0.999758	−	0.0220023i	$-0.00700412\pi$
$648$	0	0
$649$	−14.1177	−0.0217531
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 274.692i	− 0.420662i	−0.977630	−	0.210331i	$-0.932546\pi$
$653$	− 274.692i	− 0.420662i	0.977630	−	0.210331i	$-0.0674542\pi$
$654$	0	0
$655$	−373.547	−0.570300
$656$	0	0
$657$	0	0
$658$	0	0
$659$	87.7065i	0.133090i	0.997783	+	0.0665451i	$0.0211976\pi$
$659$	87.7065i	0.133090i	−0.997783	+	0.0665451i	$0.978802\pi$
$660$	0	0
$661$	−212.368	−0.321282	−0.160641	−	0.987013i	$-0.551356\pi$
$661$	−212.368	−0.321282	−0.160641	+	0.987013i	$0.551356\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 109.993i	− 0.165403i
$666$	0	0
$667$	122.313	0.183378
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 13.7056i	− 0.0204257i
$672$	0	0
$673$	1099.17	1.63325	0.816623	−	0.577171i	$-0.195843\pi$
$673$	1099.17	1.63325	0.816623	+	0.577171i	$0.195843\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	715.745i	1.05723i	0.848862	+	0.528615i	$0.177288\pi$
$677$	715.745i	1.05723i	−0.848862	+	0.528615i	$0.822712\pi$
$678$	0	0
$679$	689.979	1.01617
$680$	0	0
$681$	0	0
$682$	0	0
$683$	588.823i	0.862112i	0.902325	+	0.431056i	$0.141859\pi$
$683$	588.823i	0.862112i	−0.902325	+	0.431056i	$0.858141\pi$
$684$	0	0
$685$	226.617	0.330828
$686$	0	0
$687$	0	0
$688$	0	0
$689$	335.572i	0.487042i
$690$	0	0
$691$	466.975	0.675796	0.337898	−	0.941183i	$-0.390284\pi$
$691$	466.975	0.675796	0.337898	+	0.941183i	$0.390284\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	22.4811i	0.0323469i
$696$	0	0
$697$	640.971	0.919613
$698$	0	0
$699$	0	0
$700$	0	0
$701$	374.525i	0.534273i	0.963659	+	0.267136i	$0.0860774\pi$
$701$	374.525i	0.534273i	−0.963659	+	0.267136i	$0.913923\pi$
$702$	0	0
$703$	247.210	0.351650
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1434.13i	2.02847i
$708$	0	0
$709$	−259.647	−0.366215	−0.183108	−	0.983093i	$-0.558616\pi$
$709$	−259.647	−0.366215	−0.183108	+	0.983093i	$0.558616\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 94.7107i	− 0.132834i
$714$	0	0
$715$	−2.09476	−0.00292973
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 1124.90i	− 1.56454i	−0.622942	−	0.782268i	$-0.714063\pi$
$719$	− 1124.90i	− 1.56454i	0.622942	−	0.782268i	$-0.285937\pi$
$720$	0	0
$721$	−794.985	−1.10261
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 746.669i	− 1.02989i
$726$	0	0
$727$	185.241	0.254803	0.127401	−	0.991851i	$-0.459336\pi$
$727$	185.241	0.254803	0.127401	+	0.991851i	$0.459336\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 1504.91i	− 2.05870i
$732$	0	0
$733$	62.0732	0.0846837	0.0423419	−	0.999103i	$-0.486518\pi$
$733$	62.0732	0.0846837	0.0423419	+	0.999103i	$0.486518\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 7.64121i	− 0.0103680i
$738$	0	0
$739$	−156.357	−0.211579	−0.105789	−	0.994389i	$-0.533737\pi$
$739$	−156.357	−0.211579	−0.105789	+	0.994389i	$0.533737\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	371.220i	0.499624i	0.968294	+	0.249812i	$0.0803688\pi$
$743$	371.220i	0.499624i	−0.968294	+	0.249812i	$0.919631\pi$
$744$	0	0
$745$	−104.500	−0.140269
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 1846.19i	− 2.46487i
$750$	0	0
$751$	−380.653	−0.506861	−0.253431	−	0.967354i	$-0.581559\pi$
$751$	−380.653	−0.506861	−0.253431	+	0.967354i	$0.581559\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	382.456i	0.506564i
$756$	0	0
$757$	351.088	0.463789	0.231895	−	0.972741i	$-0.425508\pi$
$757$	351.088	0.463789	0.231895	+	0.972741i	$0.425508\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 300.772i	− 0.395232i	−0.980280	−	0.197616i	$-0.936680\pi$
$761$	− 300.772i	− 0.395232i	0.980280	−	0.197616i	$-0.0633199\pi$
$762$	0	0
$763$	1134.53	1.48693
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 533.637i	− 0.695745i
$768$	0	0
$769$	−378.955	−0.492790	−0.246395	−	0.969170i	$-0.579246\pi$
$769$	−378.955	−0.492790	−0.246395	+	0.969170i	$0.579246\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 249.305i	− 0.322516i	−0.986912	−	0.161258i	$-0.948445\pi$
$773$	− 249.305i	− 0.322516i	0.986912	−	0.161258i	$-0.0515552\pi$
$774$	0	0
$775$	−578.168	−0.746023
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 125.157i	− 0.160664i
$780$	0	0
$781$	7.07403	0.00905765
$782$	0	0
$783$	0	0
$784$	0	0
$785$	310.572i	0.395634i
$786$	0	0
$787$	−1126.62	−1.43154	−0.715769	−	0.698337i	$-0.753924\pi$
$787$	−1126.62	−1.43154	−0.715769	+	0.698337i	$0.753924\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	1537.42i	1.94364i
$792$	0	0
$793$	518.059	0.653290
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 451.324i	− 0.566278i	−0.959079	−	0.283139i	$-0.908624\pi$
$797$	− 451.324i	− 0.566278i	0.959079	−	0.283139i	$-0.0913758\pi$
$798$	0	0
$799$	−1570.69	−1.96582
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 11.3818i	− 0.0141741i
$804$	0	0
$805$	72.6030	0.0901901
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 684.710i	− 0.846365i	−0.906044	−	0.423183i	$-0.860913\pi$
$809$	− 684.710i	− 0.846365i	0.906044	−	0.423183i	$-0.139087\pi$
$810$	0	0
$811$	−1324.62	−1.63331	−0.816656	−	0.577125i	$-0.804174\pi$
$811$	−1324.62	−1.63331	−0.816656	+	0.577125i	$0.804174\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	336.886i	0.413357i
$816$	0	0
$817$	−293.852	−0.359672
$818$	0	0
$819$	0	0
$820$	0	0
$821$	837.836i	1.02051i	0.860024	+	0.510253i	$0.170448\pi$
$821$	837.836i	1.02051i	−0.860024	+	0.510253i	$0.829552\pi$
$822$	0	0
$823$	217.033	0.263710	0.131855	−	0.991269i	$-0.457907\pi$
$823$	217.033	0.263710	0.131855	+	0.991269i	$0.457907\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 458.215i	− 0.554069i	−0.960860	−	0.277035i	$-0.910648\pi$
$827$	− 458.215i	− 0.554069i	0.960860	−	0.277035i	$-0.0893517\pi$
$828$	0	0
$829$	−335.706	−0.404953	−0.202476	−	0.979287i	$-0.564899\pi$
$829$	−335.706	−0.404953	−0.202476	+	0.979287i	$0.564899\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1946.99i	2.33732i
$834$	0	0
$835$	82.6314	0.0989598
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 1202.59i	− 1.43336i	−0.697401	−	0.716681i	$-0.745660\pi$
$839$	− 1202.59i	− 1.43336i	0.697401	−	0.716681i	$-0.254340\pi$
$840$	0	0
$841$	−370.058	−0.440021
$842$	0	0
$843$	0	0
$844$	0	0
$845$	238.978i	0.282815i
$846$	0	0
$847$	1327.36	1.56713
$848$	0	0
$849$	0	0
$850$	0	0
$851$	163.176i	0.191746i
$852$	0	0
$853$	−730.471	−0.856355	−0.428178	−	0.903695i	$-0.640844\pi$
$853$	−730.471	−0.856355	−0.428178	+	0.903695i	$0.640844\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1244.03i	1.45161i	0.687899	+	0.725807i	$0.258533\pi$
$857$	1244.03i	1.45161i	−0.687899	+	0.725807i	$0.741467\pi$
$858$	0	0
$859$	−1584.24	−1.84428	−0.922141	−	0.386855i	$-0.873561\pi$
$859$	−1584.24	−1.84428	−0.922141	+	0.386855i	$0.873561\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	746.392i	0.864881i	0.901663	+	0.432440i	$0.142347\pi$
$863$	746.392i	0.864881i	−0.901663	+	0.432440i	$0.857653\pi$
$864$	0	0
$865$	−361.191	−0.417562
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 19.7395i	− 0.0227152i
$870$	0	0
$871$	288.830	0.331607
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 959.632i	− 1.09672i
$876$	0	0
$877$	−1351.97	−1.54158	−0.770792	−	0.637087i	$-0.780139\pi$
$877$	−1351.97	−1.54158	−0.770792	+	0.637087i	$0.780139\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 649.799i	− 0.737569i	−0.929515	−	0.368785i	$-0.879774\pi$
$881$	− 649.799i	− 0.737569i	0.929515	−	0.368785i	$-0.120226\pi$
$882$	0	0
$883$	829.881	0.939843	0.469922	−	0.882708i	$-0.344282\pi$
$883$	829.881	0.939843	0.469922	+	0.882708i	$0.344282\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	296.662i	0.334455i	0.985918	+	0.167228i	$0.0534815\pi$
$887$	296.662i	0.334455i	−0.985918	+	0.167228i	$0.946519\pi$
$888$	0	0
$889$	−1317.28	−1.48175
$890$	0	0
$891$	0	0
$892$	0	0
$893$	306.696i	0.343445i
$894$	0	0
$895$	−272.533	−0.304506
$896$	0	0
$897$	0	0
$898$	0	0
$899$	937.759i	1.04311i
$900$	0	0
$901$	−1411.04	−1.56608
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1.38383i	0.00152909i
$906$	0	0
$907$	964.450	1.06334	0.531670	−	0.846952i	$-0.321565\pi$
$907$	964.450	1.06334	0.531670	+	0.846952i	$0.321565\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 399.941i	− 0.439013i	−0.975611	−	0.219507i	$-0.929555\pi$
$911$	− 399.941i	− 0.439013i	0.975611	−	0.219507i	$-0.0704448\pi$
$912$	0	0
$913$	6.23463	0.00682873
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 2177.19i	− 2.37425i
$918$	0	0
$919$	−1363.99	−1.48421	−0.742107	−	0.670281i	$-0.766174\pi$
$919$	−1363.99	−1.48421	−0.742107	+	0.670281i	$0.766174\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	267.391i	0.289698i
$924$	0	0
$925$	996.118	1.07688
$926$	0	0
$927$	0	0
$928$	0	0
$929$	78.7364i	0.0847539i	0.999102	+	0.0423770i	$0.0134930\pi$
$929$	78.7364i	0.0847539i	−0.999102	+	0.0423770i	$0.986507\pi$
$930$	0	0
$931$	380.173	0.408349
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 8.80822i	− 0.00942056i
$936$	0	0
$937$	−861.705	−0.919642	−0.459821	−	0.888012i	$-0.652086\pi$
$937$	−861.705	−0.919642	−0.459821	+	0.888012i	$0.652086\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	641.991i	0.682243i	0.940019	+	0.341122i	$0.110807\pi$
$941$	641.991i	0.682243i	−0.940019	+	0.341122i	$0.889193\pi$
$942$	0	0
$943$	82.6124	0.0876060
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 1259.50i	− 1.32999i	−0.746849	−	0.664994i	$-0.768434\pi$
$947$	− 1259.50i	− 1.32999i	0.746849	−	0.664994i	$-0.231566\pi$
$948$	0	0
$949$	430.221	0.453342
$950$	0	0
$951$	0	0
$952$	0	0
$953$	867.155i	0.909921i	0.890512	+	0.454961i	$0.150347\pi$
$953$	867.155i	0.909921i	−0.890512	+	0.454961i	$0.849653\pi$
$954$	0	0
$955$	546.200	0.571938
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1320.82i	1.37729i
$960$	0	0
$961$	−234.865	−0.244397
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 119.739i	− 0.124081i
$966$	0	0
$967$	−157.972	−0.163363	−0.0816813	−	0.996659i	$-0.526029\pi$
$967$	−157.972	−0.163363	−0.0816813	+	0.996659i	$0.526029\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 826.466i	− 0.851149i	−0.904923	−	0.425575i	$-0.860072\pi$
$971$	− 826.466i	− 0.851149i	0.904923	−	0.425575i	$-0.139928\pi$
$972$	0	0
$973$	−131.029	−0.134665
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 1777.33i	− 1.81917i	−0.415514	−	0.909587i	$-0.636398\pi$
$977$	− 1777.33i	− 1.81917i	0.415514	−	0.909587i	$-0.363602\pi$
$978$	0	0
$979$	−27.1131	−0.0276947
$980$	0	0
$981$	0	0
$982$	0	0
$983$	587.324i	0.597481i	0.954334	+	0.298740i	$0.0965665\pi$
$983$	587.324i	0.597481i	−0.954334	+	0.298740i	$0.903434\pi$
$984$	0	0
$985$	−166.367	−0.168900
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 193.962i	− 0.196120i
$990$	0	0
$991$	1132.76	1.14304	0.571522	−	0.820587i	$-0.306353\pi$
$991$	1132.76	1.14304	0.571522	+	0.820587i	$0.306353\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 599.049i	− 0.602059i
$996$	0	0
$997$	−1296.22	−1.30012	−0.650060	−	0.759883i	$-0.725256\pi$
$997$	−1296.22	−1.30012	−0.650060	+	0.759883i	$0.725256\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	864.3.e.e.161.4	yes	8
3.2	odd	2	inner	864.3.e.e.161.6	yes	8
4.3	odd	2	inner	864.3.e.e.161.3	✓	8
8.3	odd	2		1728.3.e.v.1025.5		8
8.5	even	2		1728.3.e.v.1025.6		8
12.11	even	2	inner	864.3.e.e.161.5	yes	8
24.5	odd	2		1728.3.e.v.1025.4		8
24.11	even	2		1728.3.e.v.1025.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
864.3.e.e.161.3	✓	8	4.3	odd	2	inner
864.3.e.e.161.4	yes	8	1.1	even	1	trivial
864.3.e.e.161.5	yes	8	12.11	even	2	inner
864.3.e.e.161.6	yes	8	3.2	odd	2	inner
1728.3.e.v.1025.3		8	24.11	even	2
1728.3.e.v.1025.4		8	24.5	odd	2
1728.3.e.v.1025.5		8	8.3	odd	2
1728.3.e.v.1025.6		8	8.5	even	2

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:	\( N \)	\(=\)	\( 864 = 2^{5} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	864.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.88259i q^{5} +10.9726 q^{7} +O(q^{10})\)	\(q-1.88259i q^{5} +10.9726 q^{7} -0.171573i q^{11} +6.48528 q^{13} +27.2699i q^{17} +5.32478 q^{19} +3.51472i q^{23} +21.4558 q^{25} -34.8003i q^{29} -26.9469 q^{31} -20.6569i q^{35} +46.4264 q^{37} -23.5047i q^{41} -55.1858 q^{43} +57.5980i q^{47} +71.3970 q^{49} +51.7436i q^{53} -0.323002 q^{55} -82.2843i q^{59} +79.8823 q^{61} -12.2091i q^{65} +44.5362 q^{67} +41.2304i q^{71} +66.3381 q^{73} -1.88259i q^{77} +115.050 q^{79} +36.3381i q^{83} +51.3381 q^{85} -158.027i q^{89} +71.1601 q^{91} -10.0244i q^{95} +62.8823 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 16 q^{13} - 32 q^{25} + 32 q^{37} + 96 q^{49} + 96 q^{61} - 216 q^{73} - 336 q^{85} - 40 q^{97}+O(q^{100}) \) 8 * q - 16 * q^13 - 32 * q^25 + 32 * q^37 + 96 * q^49 + 96 * q^61 - 216 * q^73 - 336 * q^85 - 40 * q^97

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