LMFDB - Embedded newform 864.3.e.b.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:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.4
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	864.161
Dual form			864.3.e.b.161.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.82843i q^{5} -11.4853 q^{7} +O(q^{10})$	$q+8.82843i q^{5} -11.4853 q^{7} -14.4853i q^{11} +11.4853 q^{13} +3.17157i q^{17} -31.9706 q^{19} +21.5147i q^{23} -52.9411 q^{25} -6.34315i q^{29} +18.0000 q^{31} -101.397i q^{35} +32.4558 q^{37} -41.6569i q^{41} +27.9411 q^{43} -65.3970i q^{47} +82.9117 q^{49} -98.2254i q^{53} +127.882 q^{55} -29.3970i q^{59} -89.3675 q^{61} +101.397i q^{65} -17.0589 q^{67} -43.8823i q^{71} +31.9706 q^{73} +166.368i q^{77} +55.5442 q^{79} -72.8528i q^{83} -28.0000 q^{85} -4.02439i q^{89} -131.912 q^{91} -282.250i q^{95} +5.91169 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{7}+O(q^{10}) $ 4 * q - 12 * q^7	$ 4 q - 12 q^{7} + 12 q^{13} - 60 q^{19} - 76 q^{25} + 72 q^{31} + 28 q^{37} - 24 q^{43} + 128 q^{49} + 240 q^{55} - 52 q^{61} - 204 q^{67} + 60 q^{73} + 324 q^{79} - 112 q^{85} - 324 q^{91} - 180 q^{97}+O(q^{100}) $ 4 * q - 12 * q^7 + 12 * q^13 - 60 * q^19 - 76 * q^25 + 72 * q^31 + 28 * q^37 - 24 * q^43 + 128 * q^49 + 240 * q^55 - 52 * q^61 - 204 * q^67 + 60 * q^73 + 324 * q^79 - 112 * q^85 - 324 * q^91 - 180 * 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
$5$	8.82843i	1.76569i	0.469669	+	0.882843i	$0.344373\pi$
$5$	8.82843i	1.76569i	−0.469669	+	0.882843i	$0.655627\pi$
$6$	0	0
$7$	−11.4853	−1.64075	−0.820377	−	0.571823i	$-0.806237\pi$
$7$	−11.4853	−1.64075	−0.820377	+	0.571823i	$0.806237\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 14.4853i	− 1.31684i	−0.752649	−	0.658422i	$-0.771224\pi$
$11$	− 14.4853i	− 1.31684i	0.752649	−	0.658422i	$-0.228776\pi$
$12$	0	0
$13$	11.4853	0.883483	0.441742	−	0.897142i	$-0.354361\pi$
$13$	11.4853	0.883483	0.441742	+	0.897142i	$0.354361\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.17157i	0.186563i	0.995640	+	0.0932816i	$0.0297357\pi$
$17$	3.17157i	0.186563i	−0.995640	+	0.0932816i	$0.970264\pi$
$18$	0	0
$19$	−31.9706	−1.68266	−0.841331	−	0.540521i	$-0.818227\pi$
$19$	−31.9706	−1.68266	−0.841331	+	0.540521i	$0.818227\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	21.5147i	0.935423i	0.883881	+	0.467711i	$0.154921\pi$
$23$	21.5147i	0.935423i	−0.883881	+	0.467711i	$0.845079\pi$
$24$	0	0
$25$	−52.9411	−2.11765
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 6.34315i	− 0.218729i	−0.994002	−	0.109365i	$-0.965118\pi$
$29$	− 6.34315i	− 0.218729i	0.994002	−	0.109365i	$-0.0348816\pi$
$30$	0	0
$31$	18.0000	0.580645	0.290323	−	0.956929i	$-0.406237\pi$
$31$	18.0000	0.580645	0.290323	+	0.956929i	$0.406237\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 101.397i	− 2.89706i
$36$	0	0
$37$	32.4558	0.877185	0.438592	−	0.898686i	$-0.355477\pi$
$37$	32.4558	0.877185	0.438592	+	0.898686i	$0.355477\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 41.6569i	− 1.01602i	−0.861351	−	0.508010i	$-0.830381\pi$
$41$	− 41.6569i	− 1.01602i	0.861351	−	0.508010i	$-0.169619\pi$
$42$	0	0
$43$	27.9411	0.649794	0.324897	−	0.945749i	$-0.394670\pi$
$43$	27.9411	0.649794	0.324897	+	0.945749i	$0.394670\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 65.3970i	− 1.39142i	−0.718320	−	0.695712i	$-0.755089\pi$
$47$	− 65.3970i	− 1.39142i	0.718320	−	0.695712i	$-0.244911\pi$
$48$	0	0
$49$	82.9117	1.69208
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 98.2254i	− 1.85331i	−0.375914	−	0.926655i	$-0.622671\pi$
$53$	− 98.2254i	− 1.85331i	0.375914	−	0.926655i	$-0.377329\pi$
$54$	0	0
$55$	127.882	2.32513
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 29.3970i	− 0.498254i	−0.968471	−	0.249127i	$-0.919856\pi$
$59$	− 29.3970i	− 0.498254i	0.968471	−	0.249127i	$-0.0801436\pi$
$60$	0	0
$61$	−89.3675	−1.46504	−0.732521	−	0.680745i	$-0.761656\pi$
$61$	−89.3675	−1.46504	−0.732521	+	0.680745i	$0.761656\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	101.397i	1.55995i
$66$	0	0
$67$	−17.0589	−0.254610	−0.127305	−	0.991864i	$-0.540633\pi$
$67$	−17.0589	−0.254610	−0.127305	+	0.991864i	$0.540633\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 43.8823i	− 0.618060i	−0.951052	−	0.309030i	$-0.899996\pi$
$71$	− 43.8823i	− 0.618060i	0.951052	−	0.309030i	$-0.100004\pi$
$72$	0	0
$73$	31.9706	0.437953	0.218976	−	0.975730i	$-0.429728\pi$
$73$	31.9706	0.437953	0.218976	+	0.975730i	$0.429728\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	166.368i	2.16062i
$78$	0	0
$79$	55.5442	0.703091	0.351545	−	0.936171i	$-0.385656\pi$
$79$	55.5442	0.703091	0.351545	+	0.936171i	$0.385656\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 72.8528i	− 0.877745i	−0.898549	−	0.438872i	$-0.855378\pi$
$83$	− 72.8528i	− 0.877745i	0.898549	−	0.438872i	$-0.144622\pi$
$84$	0	0
$85$	−28.0000	−0.329412
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 4.02439i	− 0.0452178i	−0.999744	−	0.0226089i	$-0.992803\pi$
$89$	− 4.02439i	− 0.0452178i	0.999744	−	0.0226089i	$-0.00719725\pi$
$90$	0	0
$91$	−131.912	−1.44958
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 282.250i	− 2.97105i
$96$	0	0
$97$	5.91169	0.0609452	0.0304726	−	0.999536i	$-0.490299\pi$
$97$	5.91169	0.0609452	0.0304726	+	0.999536i	$0.490299\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	79.1960i	0.784118i	0.919940	+	0.392059i	$0.128237\pi$
$101$	79.1960i	0.784118i	−0.919940	+	0.392059i	$0.871763\pi$
$102$	0	0
$103$	−72.3381	−0.702312	−0.351156	−	0.936317i	$-0.614211\pi$
$103$	−72.3381	−0.702312	−0.351156	+	0.936317i	$0.614211\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	158.485i	1.48117i	0.671962	+	0.740585i	$0.265452\pi$
$107$	158.485i	1.48117i	−0.671962	+	0.740585i	$0.734548\pi$
$108$	0	0
$109$	−109.882	−1.00809	−0.504047	−	0.863676i	$-0.668156\pi$
$109$	−109.882	−1.00809	−0.504047	+	0.863676i	$0.668156\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 128.142i	− 1.13400i	−0.823717	−	0.567001i	$-0.808104\pi$
$113$	− 128.142i	− 1.13400i	0.823717	−	0.567001i	$-0.191896\pi$
$114$	0	0
$115$	−189.941	−1.65166
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 36.4264i	− 0.306104i
$120$	0	0
$121$	−88.8234	−0.734077
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 246.676i	− 1.97341i
$126$	0	0
$127$	34.1177	0.268644	0.134322	−	0.990938i	$-0.457114\pi$
$127$	34.1177	0.268644	0.134322	+	0.990938i	$0.457114\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 158.912i	− 1.21307i	−0.795058	−	0.606533i	$-0.792560\pi$
$131$	− 158.912i	− 1.21307i	0.795058	−	0.606533i	$-0.207440\pi$
$132$	0	0
$133$	367.191	2.76083
$134$	0	0
$135$	0	0
$136$	0	0
$137$	37.9655i	0.277121i	0.990354	+	0.138560i	$0.0442475\pi$
$137$	37.9655i	0.277121i	−0.990354	+	0.138560i	$0.955753\pi$
$138$	0	0
$139$	−154.882	−1.11426	−0.557130	−	0.830425i	$-0.688098\pi$
$139$	−154.882	−1.11426	−0.557130	+	0.830425i	$0.688098\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 166.368i	− 1.16341i
$144$	0	0
$145$	56.0000	0.386207
$146$	0	0
$147$	0	0
$148$	0	0
$149$	114.510i	0.768521i	0.923225	+	0.384261i	$0.125544\pi$
$149$	114.510i	0.768521i	−0.923225	+	0.384261i	$0.874456\pi$
$150$	0	0
$151$	65.4853	0.433677	0.216839	−	0.976207i	$-0.430425\pi$
$151$	65.4853	0.433677	0.216839	+	0.976207i	$0.430425\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	158.912i	1.02524i
$156$	0	0
$157$	−98.0000	−0.624204	−0.312102	−	0.950049i	$-0.601033\pi$
$157$	−98.0000	−0.624204	−0.312102	+	0.950049i	$0.601033\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 247.103i	− 1.53480i
$162$	0	0
$163$	−166.029	−1.01859	−0.509293	−	0.860593i	$-0.670093\pi$
$163$	−166.029	−1.01859	−0.509293	+	0.860593i	$0.670093\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	253.279i	1.51664i	0.651881	+	0.758321i	$0.273980\pi$
$167$	253.279i	1.51664i	−0.651881	+	0.758321i	$0.726020\pi$
$168$	0	0
$169$	−37.0883	−0.219457
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 35.8335i	− 0.207130i	−0.994623	−	0.103565i	$-0.966975\pi$
$173$	− 35.8335i	− 0.207130i	0.994623	−	0.103565i	$-0.0330250\pi$
$174$	0	0
$175$	608.044	3.47454
$176$	0	0
$177$	0	0
$178$	0	0
$179$	43.8823i	0.245152i	0.992459	+	0.122576i	$0.0391156\pi$
$179$	43.8823i	0.245152i	−0.992459	+	0.122576i	$0.960884\pi$
$180$	0	0
$181$	−142.456	−0.787049	−0.393524	−	0.919314i	$-0.628744\pi$
$181$	−142.456	−0.787049	−0.393524	+	0.919314i	$0.628744\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	286.534i	1.54883i
$186$	0	0
$187$	45.9411	0.245674
$188$	0	0
$189$	0	0
$190$	0	0
$191$	167.220i	0.875499i	0.899097	+	0.437750i	$0.144224\pi$
$191$	167.220i	0.875499i	−0.899097	+	0.437750i	$0.855776\pi$
$192$	0	0
$193$	−184.647	−0.956719	−0.478359	−	0.878164i	$-0.658768\pi$
$193$	−184.647	−0.956719	−0.478359	+	0.878164i	$0.658768\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 272.642i	− 1.38397i	−0.721913	−	0.691984i	$-0.756737\pi$
$197$	− 272.642i	− 1.38397i	0.721913	−	0.691984i	$-0.243263\pi$
$198$	0	0
$199$	29.4853	0.148167	0.0740836	−	0.997252i	$-0.476397\pi$
$199$	29.4853	0.148167	0.0740836	+	0.997252i	$0.476397\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	72.8528i	0.358881i
$204$	0	0
$205$	367.765	1.79397
$206$	0	0
$207$	0	0
$208$	0	0
$209$	463.103i	2.21580i
$210$	0	0
$211$	−23.2355	−0.110121	−0.0550604	−	0.998483i	$-0.517535\pi$
$211$	−23.2355	−0.110121	−0.0550604	+	0.998483i	$0.517535\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	246.676i	1.14733i
$216$	0	0
$217$	−206.735	−0.952696
$218$	0	0
$219$	0	0
$220$	0	0
$221$	36.4264i	0.164825i
$222$	0	0
$223$	253.882	1.13849	0.569243	−	0.822170i	$-0.307236\pi$
$223$	253.882	1.13849	0.569243	+	0.822170i	$0.307236\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 302.059i	− 1.33066i	−0.746551	−	0.665328i	$-0.768292\pi$
$227$	− 302.059i	− 1.33066i	0.746551	−	0.665328i	$-0.231708\pi$
$228$	0	0
$229$	−309.765	−1.35268	−0.676342	−	0.736588i	$-0.736436\pi$
$229$	−309.765	−1.35268	−0.676342	+	0.736588i	$0.736436\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 272.902i	− 1.17125i	−0.810582	−	0.585626i	$-0.800849\pi$
$233$	− 272.902i	− 1.17125i	0.810582	−	0.585626i	$-0.199151\pi$
$234$	0	0
$235$	577.352	2.45682
$236$	0	0
$237$	0	0
$238$	0	0
$239$	405.588i	1.69702i	0.529179	+	0.848510i	$0.322500\pi$
$239$	405.588i	1.69702i	−0.529179	+	0.848510i	$0.677500\pi$
$240$	0	0
$241$	−77.2355	−0.320479	−0.160240	−	0.987078i	$-0.551227\pi$
$241$	−77.2355	−0.320479	−0.160240	+	0.987078i	$0.551227\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	731.980i	2.98767i
$246$	0	0
$247$	−367.191	−1.48660
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 171.265i	− 0.682330i	−0.940003	−	0.341165i	$-0.889178\pi$
$251$	− 171.265i	− 0.682330i	0.940003	−	0.341165i	$-0.110822\pi$
$252$	0	0
$253$	311.647	1.23181
$254$	0	0
$255$	0	0
$256$	0	0
$257$	13.5391i	0.0526813i	0.999653	+	0.0263407i	$0.00838547\pi$
$257$	13.5391i	0.0526813i	−0.999653	+	0.0263407i	$0.991615\pi$
$258$	0	0
$259$	−372.765	−1.43925
$260$	0	0
$261$	0	0
$262$	0	0
$263$	289.706i	1.10154i	0.834656	+	0.550771i	$0.185666\pi$
$263$	289.706i	1.10154i	−0.834656	+	0.550771i	$0.814334\pi$
$264$	0	0
$265$	867.176	3.27236
$266$	0	0
$267$	0	0
$268$	0	0
$269$	89.0639i	0.331093i	0.986202	+	0.165546i	$0.0529387\pi$
$269$	89.0639i	0.331093i	−0.986202	+	0.165546i	$0.947061\pi$
$270$	0	0
$271$	−172.955	−0.638212	−0.319106	−	0.947719i	$-0.603383\pi$
$271$	−172.955	−0.638212	−0.319106	+	0.947719i	$0.603383\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	766.867i	2.78861i
$276$	0	0
$277$	−281.294	−1.01550	−0.507750	−	0.861504i	$-0.669523\pi$
$277$	−281.294	−1.01550	−0.507750	+	0.861504i	$0.669523\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 353.657i	− 1.25857i	−0.777177	−	0.629283i	$-0.783349\pi$
$281$	− 353.657i	− 1.25857i	0.777177	−	0.629283i	$-0.216651\pi$
$282$	0	0
$283$	−471.235	−1.66514	−0.832570	−	0.553920i	$-0.813131\pi$
$283$	−471.235	−1.66514	−0.832570	+	0.553920i	$0.813131\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	478.441i	1.66704i
$288$	0	0
$289$	278.941	0.965194
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 527.740i	− 1.80116i	−0.434689	−	0.900580i	$-0.643142\pi$
$293$	− 527.740i	− 1.80116i	0.434689	−	0.900580i	$-0.356858\pi$
$294$	0	0
$295$	259.529	0.879759
$296$	0	0
$297$	0	0
$298$	0	0
$299$	247.103i	0.826430i
$300$	0	0
$301$	−320.912	−1.06615
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 788.975i	− 2.58680i
$306$	0	0
$307$	−483.588	−1.57520	−0.787602	−	0.616184i	$-0.788678\pi$
$307$	−483.588	−1.57520	−0.787602	+	0.616184i	$0.788678\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	52.1909i	0.167816i	0.996473	+	0.0839082i	$0.0267403\pi$
$311$	52.1909i	0.167816i	−0.996473	+	0.0839082i	$0.973260\pi$
$312$	0	0
$313$	−576.647	−1.84232	−0.921161	−	0.389182i	$-0.872758\pi$
$313$	−576.647	−1.84232	−0.921161	+	0.389182i	$0.872758\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 339.598i	− 1.07129i	−0.844444	−	0.535644i	$-0.820069\pi$
$317$	− 339.598i	− 1.07129i	0.844444	−	0.535644i	$-0.179931\pi$
$318$	0	0
$319$	−91.8823	−0.288032
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 101.397i	− 0.313923i
$324$	0	0
$325$	−608.044	−1.87090
$326$	0	0
$327$	0	0
$328$	0	0
$329$	751.103i	2.28299i
$330$	0	0
$331$	−132.588	−0.400568	−0.200284	−	0.979738i	$-0.564186\pi$
$331$	−132.588	−0.400568	−0.200284	+	0.979738i	$0.564186\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 150.603i	− 0.449561i
$336$	0	0
$337$	221.912	0.658492	0.329246	−	0.944244i	$-0.393206\pi$
$337$	221.912	0.658492	0.329246	+	0.944244i	$0.393206\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 260.735i	− 0.764619i
$342$	0	0
$343$	−389.485	−1.13553
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 230.059i	− 0.662994i	−0.943456	−	0.331497i	$-0.892446\pi$
$347$	− 230.059i	− 0.662994i	0.943456	−	0.331497i	$-0.107554\pi$
$348$	0	0
$349$	213.544	0.611874	0.305937	−	0.952052i	$-0.401030\pi$
$349$	213.544	0.611874	0.305937	+	0.952052i	$0.401030\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	210.843i	0.597288i	0.954365	+	0.298644i	$0.0965343\pi$
$353$	210.843i	0.597288i	−0.954365	+	0.298644i	$0.903466\pi$
$354$	0	0
$355$	387.411	1.09130
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 367.456i	− 1.02355i	−0.859118	−	0.511777i	$-0.828987\pi$
$359$	− 367.456i	− 1.02355i	0.859118	−	0.511777i	$-0.171013\pi$
$360$	0	0
$361$	661.117	1.83135
$362$	0	0
$363$	0	0
$364$	0	0
$365$	282.250i	0.773287i
$366$	0	0
$367$	−280.955	−0.765546	−0.382773	−	0.923842i	$-0.625031\pi$
$367$	−280.955	−0.765546	−0.382773	+	0.923842i	$0.625031\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	1128.15i	3.04083i
$372$	0	0
$373$	503.749	1.35053	0.675267	−	0.737573i	$-0.264028\pi$
$373$	503.749	1.35053	0.675267	+	0.737573i	$0.264028\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 72.8528i	− 0.193244i
$378$	0	0
$379$	629.735	1.66157	0.830785	−	0.556593i	$-0.187892\pi$
$379$	629.735	1.66157	0.830785	+	0.556593i	$0.187892\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 101.823i	− 0.265857i	−0.991126	−	0.132929i	$-0.957562\pi$
$383$	− 101.823i	− 0.265857i	0.991126	−	0.132929i	$-0.0424381\pi$
$384$	0	0
$385$	−1468.76	−3.81497
$386$	0	0
$387$	0	0
$388$	0	0
$389$	476.995i	1.22621i	0.790002	+	0.613104i	$0.210079\pi$
$389$	476.995i	1.22621i	−0.790002	+	0.613104i	$0.789921\pi$
$390$	0	0
$391$	−68.2355	−0.174515
$392$	0	0
$393$	0	0
$394$	0	0
$395$	490.368i	1.24144i
$396$	0	0
$397$	241.647	0.608682	0.304341	−	0.952563i	$-0.401564\pi$
$397$	241.647	0.608682	0.304341	+	0.952563i	$0.401564\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	418.274i	1.04308i	0.853228	+	0.521539i	$0.174642\pi$
$401$	418.274i	1.04308i	−0.853228	+	0.521539i	$0.825358\pi$
$402$	0	0
$403$	206.735	0.512990
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 470.132i	− 1.15512i
$408$	0	0
$409$	−21.4996	−0.0525662	−0.0262831	−	0.999655i	$-0.508367\pi$
$409$	−21.4996	−0.0525662	−0.0262831	+	0.999655i	$0.508367\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	337.632i	0.817512i
$414$	0	0
$415$	643.176	1.54982
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 375.338i	− 0.895795i	−0.894085	−	0.447897i	$-0.852173\pi$
$419$	− 375.338i	− 0.895795i	0.894085	−	0.447897i	$-0.147827\pi$
$420$	0	0
$421$	582.515	1.38365	0.691823	−	0.722067i	$-0.256808\pi$
$421$	582.515	1.38365	0.691823	+	0.722067i	$0.256808\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 167.907i	− 0.395074i
$426$	0	0
$427$	1026.41	2.40377
$428$	0	0
$429$	0	0
$430$	0	0
$431$	440.309i	1.02160i	0.859700	+	0.510799i	$0.170650\pi$
$431$	440.309i	1.02160i	−0.859700	+	0.510799i	$0.829350\pi$
$432$	0	0
$433$	−91.4701	−0.211247	−0.105624	−	0.994406i	$-0.533684\pi$
$433$	−91.4701	−0.211247	−0.105624	+	0.994406i	$0.533684\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 687.838i	− 1.57400i
$438$	0	0
$439$	−630.000	−1.43508	−0.717540	−	0.696517i	$-0.754732\pi$
$439$	−630.000	−1.43508	−0.717540	+	0.696517i	$0.754732\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 696.146i	− 1.57144i	−0.618585	−	0.785718i	$-0.712294\pi$
$443$	− 696.146i	− 1.57144i	0.618585	−	0.785718i	$-0.287706\pi$
$444$	0	0
$445$	35.5290	0.0798405
$446$	0	0
$447$	0	0
$448$	0	0
$449$	489.328i	1.08982i	0.838495	+	0.544909i	$0.183436\pi$
$449$	489.328i	1.08982i	−0.838495	+	0.544909i	$0.816564\pi$
$450$	0	0
$451$	−603.411	−1.33794
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 1164.57i	− 2.55950i
$456$	0	0
$457$	399.235	0.873599	0.436799	−	0.899559i	$-0.356112\pi$
$457$	399.235	0.873599	0.436799	+	0.899559i	$0.356112\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	146.818i	0.318478i	0.987240	+	0.159239i	$0.0509040\pi$
$461$	146.818i	0.318478i	−0.987240	+	0.159239i	$0.949096\pi$
$462$	0	0
$463$	557.132	1.20331	0.601654	−	0.798756i	$-0.294508\pi$
$463$	557.132	1.20331	0.601654	+	0.798756i	$0.294508\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	852.073i	1.82457i	0.409558	+	0.912284i	$0.365683\pi$
$467$	852.073i	1.82457i	−0.409558	+	0.912284i	$0.634317\pi$
$468$	0	0
$469$	195.926	0.417753
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 404.735i	− 0.855677i
$474$	0	0
$475$	1692.56	3.56328
$476$	0	0
$477$	0	0
$478$	0	0
$479$	301.206i	0.628823i	0.949287	+	0.314411i	$0.101807\pi$
$479$	301.206i	0.628823i	−0.949287	+	0.314411i	$0.898193\pi$
$480$	0	0
$481$	372.765	0.774978
$482$	0	0
$483$	0	0
$484$	0	0
$485$	52.1909i	0.107610i
$486$	0	0
$487$	−357.780	−0.734660	−0.367330	−	0.930091i	$-0.619728\pi$
$487$	−357.780	−0.734660	−0.367330	+	0.930091i	$0.619728\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 218.132i	− 0.444261i	−0.975017	−	0.222130i	$-0.928699\pi$
$491$	− 218.132i	− 0.444261i	0.975017	−	0.222130i	$-0.0713011\pi$
$492$	0	0
$493$	20.1177	0.0408068
$494$	0	0
$495$	0	0
$496$	0	0
$497$	504.000i	1.01408i
$498$	0	0
$499$	−457.529	−0.916892	−0.458446	−	0.888722i	$-0.651594\pi$
$499$	−457.529	−0.916892	−0.458446	+	0.888722i	$0.651594\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 530.632i	− 1.05493i	−0.849576	−	0.527467i	$-0.823142\pi$
$503$	− 530.632i	− 1.05493i	0.849576	−	0.527467i	$-0.176858\pi$
$504$	0	0
$505$	−699.176	−1.38451
$506$	0	0
$507$	0	0
$508$	0	0
$509$	180.759i	0.355127i	0.984109	+	0.177563i	$0.0568215\pi$
$509$	180.759i	0.355127i	−0.984109	+	0.177563i	$0.943179\pi$
$510$	0	0
$511$	−367.191	−0.718573
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 638.632i	− 1.24006i
$516$	0	0
$517$	−947.294	−1.83229
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 603.505i	− 1.15836i	−0.815200	−	0.579179i	$-0.803373\pi$
$521$	− 603.505i	− 1.15836i	0.815200	−	0.579179i	$-0.196627\pi$
$522$	0	0
$523$	−119.029	−0.227588	−0.113794	−	0.993504i	$-0.536300\pi$
$523$	−119.029	−0.227588	−0.113794	+	0.993504i	$0.536300\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	57.0883i	0.108327i
$528$	0	0
$529$	66.1169	0.124985
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 478.441i	− 0.897637i
$534$	0	0
$535$	−1399.18	−2.61528
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 1201.00i	− 2.22820i
$540$	0	0
$541$	−727.337	−1.34443	−0.672216	−	0.740355i	$-0.734657\pi$
$541$	−727.337	−1.34443	−0.672216	+	0.740355i	$0.734657\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 970.087i	− 1.77998i
$546$	0	0
$547$	−5.91169	−0.0108075	−0.00540374	−	0.999985i	$-0.501720\pi$
$547$	−5.91169	−0.0108075	−0.00540374	+	0.999985i	$0.501720\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	202.794i	0.368047i
$552$	0	0
$553$	−637.940	−1.15360
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 72.4466i	− 0.130066i	−0.997883	−	0.0650329i	$-0.979285\pi$
$557$	− 72.4466i	− 0.130066i	0.997883	−	0.0650329i	$-0.0207152\pi$
$558$	0	0
$559$	320.912	0.574082
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 525.734i	− 0.933809i	−0.884308	−	0.466904i	$-0.845369\pi$
$563$	− 525.734i	− 0.933809i	0.884308	−	0.466904i	$-0.154631\pi$
$564$	0	0
$565$	1131.29	2.00229
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 931.269i	− 1.63668i	−0.574737	−	0.818338i	$-0.694896\pi$
$569$	− 931.269i	− 1.63668i	0.574737	−	0.818338i	$-0.305104\pi$
$570$	0	0
$571$	145.471	0.254765	0.127383	−	0.991854i	$-0.459342\pi$
$571$	145.471	0.254765	0.127383	+	0.991854i	$0.459342\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 1139.01i	− 1.98089i
$576$	0	0
$577$	937.205	1.62427	0.812136	−	0.583468i	$-0.198305\pi$
$577$	937.205	1.62427	0.812136	+	0.583468i	$0.198305\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	836.735i	1.44016i
$582$	0	0
$583$	−1422.82	−2.44052
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 939.838i	− 1.60109i	−0.599275	−	0.800543i	$-0.704544\pi$
$587$	− 939.838i	− 1.60109i	0.599275	−	0.800543i	$-0.295456\pi$
$588$	0	0
$589$	−575.470	−0.977029
$590$	0	0
$591$	0	0
$592$	0	0
$593$	280.950i	0.473778i	0.971537	+	0.236889i	$0.0761278\pi$
$593$	280.950i	0.473778i	−0.971537	+	0.236889i	$0.923872\pi$
$594$	0	0
$595$	321.588	0.540484
$596$	0	0
$597$	0	0
$598$	0	0
$599$	575.147i	0.960179i	0.877220	+	0.480089i	$0.159396\pi$
$599$	575.147i	0.960179i	−0.877220	+	0.480089i	$0.840604\pi$
$600$	0	0
$601$	399.470	0.664676	0.332338	−	0.943160i	$-0.392163\pi$
$601$	399.470	0.664676	0.332338	+	0.943160i	$0.392163\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 784.171i	− 1.29615i
$606$	0	0
$607$	−842.044	−1.38722	−0.693611	−	0.720350i	$-0.743981\pi$
$607$	−842.044	−1.38722	−0.693611	+	0.720350i	$0.743981\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 751.103i	− 1.22930i
$612$	0	0
$613$	−350.809	−0.572282	−0.286141	−	0.958187i	$-0.592373\pi$
$613$	−350.809	−0.572282	−0.286141	+	0.958187i	$0.592373\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	407.387i	0.660270i	0.943934	+	0.330135i	$0.107094\pi$
$617$	407.387i	0.660270i	−0.943934	+	0.330135i	$0.892906\pi$
$618$	0	0
$619$	−829.234	−1.33963	−0.669817	−	0.742526i	$-0.733628\pi$
$619$	−829.234	−1.33963	−0.669817	+	0.742526i	$0.733628\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	46.2212i	0.0741914i
$624$	0	0
$625$	854.235	1.36678
$626$	0	0
$627$	0	0
$628$	0	0
$629$	102.936i	0.163650i
$630$	0	0
$631$	30.8377	0.0488711	0.0244355	−	0.999701i	$-0.492221\pi$
$631$	30.8377	0.0488711	0.0244355	+	0.999701i	$0.492221\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	301.206i	0.474340i
$636$	0	0
$637$	952.264	1.49492
$638$	0	0
$639$	0	0
$640$	0	0
$641$	91.2162i	0.142303i	0.997466	+	0.0711515i	$0.0226674\pi$
$641$	91.2162i	0.142303i	−0.997466	+	0.0711515i	$0.977333\pi$
$642$	0	0
$643$	−24.1766	−0.0375997	−0.0187999	−	0.999823i	$-0.505985\pi$
$643$	−24.1766	−0.0375997	−0.0187999	+	0.999823i	$0.505985\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	388.971i	0.601191i	0.953752	+	0.300595i	$0.0971854\pi$
$647$	388.971i	0.601191i	−0.953752	+	0.300595i	$0.902815\pi$
$648$	0	0
$649$	−425.823	−0.656122
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 46.7737i	− 0.0716290i	−0.999358	−	0.0358145i	$-0.988597\pi$
$653$	− 46.7737i	− 0.0716290i	0.999358	−	0.0358145i	$-0.0114025\pi$
$654$	0	0
$655$	1402.94	2.14189
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 60.4996i	− 0.0918051i	−0.998946	−	0.0459026i	$-0.985384\pi$
$659$	− 60.4996i	− 0.0918051i	0.998946	−	0.0459026i	$-0.0146164\pi$
$660$	0	0
$661$	−695.573	−1.05230	−0.526152	−	0.850391i	$-0.676366\pi$
$661$	−695.573	−1.05230	−0.526152	+	0.850391i	$0.676366\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	3241.72i	4.87476i
$666$	0	0
$667$	136.471	0.204604
$668$	0	0
$669$	0	0
$670$	0	0
$671$	1294.51i	1.92923i
$672$	0	0
$673$	55.3818	0.0822910	0.0411455	−	0.999153i	$-0.486899\pi$
$673$	55.3818	0.0822910	0.0411455	+	0.999153i	$0.486899\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	317.563i	0.469075i	0.972107	+	0.234537i	$0.0753575\pi$
$677$	317.563i	0.469075i	−0.972107	+	0.234537i	$0.924643\pi$
$678$	0	0
$679$	−67.8974	−0.0999962
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 942.823i	− 1.38041i	−0.723612	−	0.690207i	$-0.757520\pi$
$683$	− 942.823i	− 1.38041i	0.723612	−	0.690207i	$-0.242480\pi$
$684$	0	0
$685$	−335.176	−0.489308
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 1128.15i	− 1.63737i
$690$	0	0
$691$	402.999	0.583211	0.291606	−	0.956539i	$-0.405810\pi$
$691$	402.999	0.583211	0.291606	+	0.956539i	$0.405810\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 1367.37i	− 1.96743i
$696$	0	0
$697$	132.118	0.189552
$698$	0	0
$699$	0	0
$700$	0	0
$701$	650.965i	0.928623i	0.885672	+	0.464311i	$0.153698\pi$
$701$	650.965i	0.928623i	−0.885672	+	0.464311i	$0.846302\pi$
$702$	0	0
$703$	−1037.63	−1.47601
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 909.588i	− 1.28655i
$708$	0	0
$709$	576.338	0.812889	0.406444	−	0.913676i	$-0.366769\pi$
$709$	576.338	0.812889	0.406444	+	0.913676i	$0.366769\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	387.265i	0.543149i
$714$	0	0
$715$	1468.76	2.05421
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 662.912i	− 0.921991i	−0.887402	−	0.460996i	$-0.847492\pi$
$719$	− 662.912i	− 0.921991i	0.887402	−	0.460996i	$-0.152508\pi$
$720$	0	0
$721$	830.823	1.15232
$722$	0	0
$723$	0	0
$724$	0	0
$725$	335.813i	0.463191i
$726$	0	0
$727$	105.058	0.144509	0.0722545	−	0.997386i	$-0.476981\pi$
$727$	105.058	0.144509	0.0722545	+	0.997386i	$0.476981\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	88.6173i	0.121228i
$732$	0	0
$733$	−1217.29	−1.66070	−0.830350	−	0.557242i	$-0.811860\pi$
$733$	−1217.29	−1.66070	−0.830350	+	0.557242i	$0.811860\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	247.103i	0.335282i
$738$	0	0
$739$	1202.53	1.62724	0.813618	−	0.581399i	$-0.197494\pi$
$739$	1202.53	1.62724	0.813618	+	0.581399i	$0.197494\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 735.984i	− 0.990557i	−0.868734	−	0.495279i	$-0.835066\pi$
$743$	− 735.984i	− 0.990557i	0.868734	−	0.495279i	$-0.164934\pi$
$744$	0	0
$745$	−1010.94	−1.35697
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 1820.25i	− 2.43024i
$750$	0	0
$751$	925.721	1.23265	0.616325	−	0.787492i	$-0.288621\pi$
$751$	925.721	1.23265	0.616325	+	0.787492i	$0.288621\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	578.132i	0.765738i
$756$	0	0
$757$	−319.897	−0.422586	−0.211293	−	0.977423i	$-0.567767\pi$
$757$	−319.897	−0.422586	−0.211293	+	0.977423i	$0.567767\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 1246.49i	− 1.63797i	−0.573816	−	0.818984i	$-0.694538\pi$
$761$	− 1246.49i	− 1.63797i	0.573816	−	0.818984i	$-0.305462\pi$
$762$	0	0
$763$	1262.03	1.65403
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 337.632i	− 0.440199i
$768$	0	0
$769$	−217.000	−0.282185	−0.141092	−	0.989996i	$-0.545061\pi$
$769$	−217.000	−0.282185	−0.141092	+	0.989996i	$0.545061\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	270.303i	0.349680i	0.984597	+	0.174840i	$0.0559408\pi$
$773$	270.303i	0.349680i	−0.984597	+	0.174840i	$0.944059\pi$
$774$	0	0
$775$	−952.940	−1.22960
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1331.79i	1.70962i
$780$	0	0
$781$	−635.647	−0.813888
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 865.186i	− 1.10215i
$786$	0	0
$787$	−593.205	−0.753755	−0.376878	−	0.926263i	$-0.623002\pi$
$787$	−593.205	−0.753755	−0.376878	+	0.926263i	$0.623002\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	1471.75i	1.86062i
$792$	0	0
$793$	−1026.41	−1.29434
$794$	0	0
$795$	0	0
$796$	0	0
$797$	665.137i	0.834551i	0.908780	+	0.417275i	$0.137015\pi$
$797$	665.137i	0.834551i	−0.908780	+	0.417275i	$0.862985\pi$
$798$	0	0
$799$	207.411	0.259589
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 463.103i	− 0.576716i
$804$	0	0
$805$	2181.53	2.70997
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 675.452i	− 0.834922i	−0.908695	−	0.417461i	$-0.862920\pi$
$809$	− 675.452i	− 0.834922i	0.908695	−	0.417461i	$-0.137080\pi$
$810$	0	0
$811$	527.117	0.649959	0.324980	−	0.945721i	$-0.394643\pi$
$811$	527.117	0.649959	0.324980	+	0.945721i	$0.394643\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 1465.78i	− 1.79850i
$816$	0	0
$817$	−893.294	−1.09338
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 885.994i	− 1.07916i	−0.841933	−	0.539582i	$-0.818582\pi$
$821$	− 885.994i	− 1.07916i	0.841933	−	0.539582i	$-0.181418\pi$
$822$	0	0
$823$	1169.42	1.42093	0.710465	−	0.703733i	$-0.248485\pi$
$823$	1169.42	1.42093	0.710465	+	0.703733i	$0.248485\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1169.90i	1.41463i	0.706900	+	0.707314i	$0.250093\pi$
$827$	1169.90i	1.41463i	−0.706900	+	0.707314i	$0.749907\pi$
$828$	0	0
$829$	1419.40	1.71218	0.856089	−	0.516828i	$-0.172887\pi$
$829$	1419.40	1.71218	0.856089	+	0.516828i	$0.172887\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	262.960i	0.315679i
$834$	0	0
$835$	−2236.06	−2.67791
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1549.50i	1.84684i	0.383791	+	0.923420i	$0.374618\pi$
$839$	1549.50i	1.84684i	−0.383791	+	0.923420i	$0.625382\pi$
$840$	0	0
$841$	800.765	0.952158
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 327.431i	− 0.387493i
$846$	0	0
$847$	1020.16	1.20444
$848$	0	0
$849$	0	0
$850$	0	0
$851$	698.278i	0.820539i
$852$	0	0
$853$	248.809	0.291687	0.145844	−	0.989308i	$-0.453410\pi$
$853$	248.809	0.291687	0.145844	+	0.989308i	$0.453410\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 359.627i	− 0.419634i	−0.977741	−	0.209817i	$-0.932713\pi$
$857$	− 359.627i	− 0.419634i	0.977741	−	0.209817i	$-0.0672869\pi$
$858$	0	0
$859$	341.735	0.397829	0.198914	−	0.980017i	$-0.436258\pi$
$859$	341.735	0.397829	0.198914	+	0.980017i	$0.436258\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 758.985i	− 0.879473i	−0.898127	−	0.439736i	$-0.855072\pi$
$863$	− 758.985i	− 0.879473i	0.898127	−	0.439736i	$-0.144928\pi$
$864$	0	0
$865$	316.353	0.365726
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 804.573i	− 0.925860i
$870$	0	0
$871$	−195.926	−0.224944
$872$	0	0
$873$	0	0
$874$	0	0
$875$	2833.15i	3.23788i
$876$	0	0
$877$	603.014	0.687588	0.343794	−	0.939045i	$-0.388288\pi$
$877$	603.014	0.687588	0.343794	+	0.939045i	$0.388288\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1296.20i	1.47128i	0.677372	+	0.735641i	$0.263119\pi$
$881$	1296.20i	1.47128i	−0.677372	+	0.735641i	$0.736881\pi$
$882$	0	0
$883$	−348.442	−0.394611	−0.197306	−	0.980342i	$-0.563219\pi$
$883$	−348.442	−0.394611	−0.197306	+	0.980342i	$0.563219\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	151.675i	0.170998i	0.996338	+	0.0854991i	$0.0272485\pi$
$887$	151.675i	0.170998i	−0.996338	+	0.0854991i	$0.972752\pi$
$888$	0	0
$889$	−391.852	−0.440778
$890$	0	0
$891$	0	0
$892$	0	0
$893$	2090.78i	2.34130i
$894$	0	0
$895$	−387.411	−0.432862
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 114.177i	− 0.127004i
$900$	0	0
$901$	311.529	0.345759
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 1257.66i	− 1.38968i
$906$	0	0
$907$	−600.734	−0.662331	−0.331165	−	0.943573i	$-0.607442\pi$
$907$	−600.734	−0.662331	−0.331165	+	0.943573i	$0.607442\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 935.147i	− 1.02651i	−0.858237	−	0.513253i	$-0.828440\pi$
$911$	− 935.147i	− 1.02651i	0.858237	−	0.513253i	$-0.171560\pi$
$912$	0	0
$913$	−1055.29	−1.15585
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1825.15i	1.99034i
$918$	0	0
$919$	−1596.12	−1.73680	−0.868398	−	0.495867i	$-0.834850\pi$
$919$	−1596.12	−1.73680	−0.868398	+	0.495867i	$0.834850\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 504.000i	− 0.546046i
$924$	0	0
$925$	−1718.25	−1.85757
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 1603.14i	− 1.72566i	−0.505497	−	0.862828i	$-0.668691\pi$
$929$	− 1603.14i	− 1.72566i	0.505497	−	0.862828i	$-0.331309\pi$
$930$	0	0
$931$	−2650.73	−2.84719
$932$	0	0
$933$	0	0
$934$	0	0
$935$	405.588i	0.433784i
$936$	0	0
$937$	−83.5299	−0.0891461	−0.0445730	−	0.999006i	$-0.514193\pi$
$937$	−83.5299	−0.0891461	−0.0445730	+	0.999006i	$0.514193\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 846.416i	− 0.899486i	−0.893158	−	0.449743i	$-0.851516\pi$
$941$	− 846.416i	− 0.899486i	0.893158	−	0.449743i	$-0.148484\pi$
$942$	0	0
$943$	896.235	0.950409
$944$	0	0
$945$	0	0
$946$	0	0
$947$	860.601i	0.908766i	0.890806	+	0.454383i	$0.150140\pi$
$947$	860.601i	0.908766i	−0.890806	+	0.454383i	$0.849860\pi$
$948$	0	0
$949$	367.191	0.386924
$950$	0	0
$951$	0	0
$952$	0	0
$953$	361.466i	0.379293i	0.981852	+	0.189646i	$0.0607341\pi$
$953$	361.466i	0.379293i	−0.981852	+	0.189646i	$0.939266\pi$
$954$	0	0
$955$	−1476.29	−1.54586
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 436.045i	− 0.454687i
$960$	0	0
$961$	−637.000	−0.662851
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 1630.14i	− 1.68926i
$966$	0	0
$967$	362.220	0.374582	0.187291	−	0.982305i	$-0.440029\pi$
$967$	362.220	0.374582	0.187291	+	0.982305i	$0.440029\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1526.92i	1.57253i	0.617891	+	0.786264i	$0.287987\pi$
$971$	1526.92i	1.57253i	−0.617891	+	0.786264i	$0.712013\pi$
$972$	0	0
$973$	1778.87	1.82823
$974$	0	0
$975$	0	0
$976$	0	0
$977$	631.050i	0.645905i	0.946415	+	0.322953i	$0.104675\pi$
$977$	631.050i	0.645905i	−0.946415	+	0.322953i	$0.895325\pi$
$978$	0	0
$979$	−58.2944	−0.0595448
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1043.57i	1.06162i	0.847490	+	0.530811i	$0.178112\pi$
$983$	1043.57i	1.06162i	−0.847490	+	0.530811i	$0.821888\pi$
$984$	0	0
$985$	2407.00	2.44365
$986$	0	0
$987$	0	0
$988$	0	0
$989$	601.145i	0.607832i
$990$	0	0
$991$	608.044	0.613566	0.306783	−	0.951780i	$-0.400747\pi$
$991$	608.044	0.613566	0.306783	+	0.951780i	$0.400747\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	260.309i	0.261617i
$996$	0	0
$997$	−1292.23	−1.29612	−0.648061	−	0.761588i	$-0.724420\pi$
$997$	−1292.23	−1.29612	−0.648061	+	0.761588i	$0.724420\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	864.3.e.b.161.4	yes	4
3.2	odd	2	inner	864.3.e.b.161.1	✓	4
4.3	odd	2		864.3.e.d.161.4	yes	4
8.3	odd	2		1728.3.e.r.1025.1		4
8.5	even	2		1728.3.e.o.1025.1		4
12.11	even	2		864.3.e.d.161.1	yes	4
24.5	odd	2		1728.3.e.o.1025.4		4
24.11	even	2		1728.3.e.r.1025.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
864.3.e.b.161.1	✓	4	3.2	odd	2	inner
864.3.e.b.161.4	yes	4	1.1	even	1	trivial
864.3.e.d.161.1	yes	4	12.11	even	2
864.3.e.d.161.4	yes	4	4.3	odd	2
1728.3.e.o.1025.1		4	8.5	even	2
1728.3.e.o.1025.4		4	24.5	odd	2
1728.3.e.r.1025.1		4	8.3	odd	2
1728.3.e.r.1025.4		4	24.11	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)

\(f(q)\)	\(=\)	\(q+8.82843i q^{5} -11.4853 q^{7} +O(q^{10})\)	\(q+8.82843i q^{5} -11.4853 q^{7} -14.4853i q^{11} +11.4853 q^{13} +3.17157i q^{17} -31.9706 q^{19} +21.5147i q^{23} -52.9411 q^{25} -6.34315i q^{29} +18.0000 q^{31} -101.397i q^{35} +32.4558 q^{37} -41.6569i q^{41} +27.9411 q^{43} -65.3970i q^{47} +82.9117 q^{49} -98.2254i q^{53} +127.882 q^{55} -29.3970i q^{59} -89.3675 q^{61} +101.397i q^{65} -17.0589 q^{67} -43.8823i q^{71} +31.9706 q^{73} +166.368i q^{77} +55.5442 q^{79} -72.8528i q^{83} -28.0000 q^{85} -4.02439i q^{89} -131.912 q^{91} -282.250i q^{95} +5.91169 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{7}+O(q^{10}) \) 4 * q - 12 * q^7	\( 4 q - 12 q^{7} + 12 q^{13} - 60 q^{19} - 76 q^{25} + 72 q^{31} + 28 q^{37} - 24 q^{43} + 128 q^{49} + 240 q^{55} - 52 q^{61} - 204 q^{67} + 60 q^{73} + 324 q^{79} - 112 q^{85} - 324 q^{91} - 180 q^{97}+O(q^{100}) \) 4 * q - 12 * q^7 + 12 * q^13 - 60 * q^19 - 76 * q^25 + 72 * q^31 + 28 * q^37 - 24 * q^43 + 128 * q^49 + 240 * q^55 - 52 * q^61 - 204 * q^67 + 60 * q^73 + 324 * q^79 - 112 * q^85 - 324 * q^91 - 180 * q^97

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