LMFDB - Embedded newform 360.3.l.a.161.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [360,3,Mod(161,360)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(360, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("360.161");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 360 = 2^{3} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	360.l (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:	$9.80928951697$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 9 $ x^4 - 4*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.4
Root			$1.58114 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	360.161
Dual form			360.3.l.a.161.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.23607i q^{5} -4.83772 q^{7} +O(q^{10})$	$q+2.23607i q^{5} -4.83772 q^{7} -14.8306i q^{11} -20.1359 q^{13} -6.57484i q^{17} +17.6754 q^{19} -25.1891i q^{23} -5.00000 q^{25} -38.4133i q^{29} -32.9737 q^{31} -10.8175i q^{35} -44.7851 q^{37} +21.2132i q^{41} +16.2719 q^{43} -24.0044i q^{47} -25.5964 q^{49} +42.4264i q^{53} +33.1623 q^{55} -86.3848i q^{59} -38.2719 q^{61} -45.0253i q^{65} +37.2982 q^{67} -19.3400i q^{71} -108.974 q^{73} +71.7464i q^{77} +119.570 q^{79} +123.458i q^{83} +14.7018 q^{85} +63.0317i q^{89} +97.4121 q^{91} +39.5235i q^{95} +109.623 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 32 q^{7}+O(q^{10}) $ 4 * q - 32 * q^7	$ 4 q - 32 q^{7} + 8 q^{13} + 96 q^{19} - 20 q^{25} - 56 q^{31} - 40 q^{37} - 112 q^{43} + 100 q^{49} + 120 q^{55} + 24 q^{61} + 48 q^{67} - 360 q^{73} + 200 q^{79} + 160 q^{85} - 344 q^{91} + 312 q^{97}+O(q^{100}) $ 4 * q - 32 * q^7 + 8 * q^13 + 96 * q^19 - 20 * q^25 - 56 * q^31 - 40 * q^37 - 112 * q^43 + 100 * q^49 + 120 * q^55 + 24 * q^61 + 48 * q^67 - 360 * q^73 + 200 * q^79 + 160 * q^85 - 344 * q^91 + 312 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/360\mathbb{Z}\right)^\times$.

$n$	$181$	$217$	$271$	$281$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$5$	2.23607i	0.447214i
$5$	2.23607i	0.447214i
$6$	0	0
$7$	−4.83772	−0.691103	−0.345552	−	0.938400i	$-0.612308\pi$
$7$	−4.83772	−0.691103	−0.345552	+	0.938400i	$0.612308\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 14.8306i	− 1.34824i	−0.738623	−	0.674119i	$-0.764523\pi$
$11$	− 14.8306i	− 1.34824i	0.738623	−	0.674119i	$-0.235477\pi$
$12$	0	0
$13$	−20.1359	−1.54892	−0.774459	−	0.632624i	$-0.781978\pi$
$13$	−20.1359	−1.54892	−0.774459	+	0.632624i	$0.781978\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 6.57484i	− 0.386755i	−0.981124	−	0.193378i	$-0.938056\pi$
$17$	− 6.57484i	− 0.386755i	0.981124	−	0.193378i	$-0.0619442\pi$
$18$	0	0
$19$	17.6754	0.930287	0.465143	−	0.885235i	$-0.346003\pi$
$19$	17.6754	0.930287	0.465143	+	0.885235i	$0.346003\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 25.1891i	− 1.09518i	−0.836747	−	0.547589i	$-0.815546\pi$
$23$	− 25.1891i	− 1.09518i	0.836747	−	0.547589i	$-0.184454\pi$
$24$	0	0
$25$	−5.00000	−0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 38.4133i	− 1.32460i	−0.749241	−	0.662298i	$-0.769581\pi$
$29$	− 38.4133i	− 1.32460i	0.749241	−	0.662298i	$-0.230419\pi$
$30$	0	0
$31$	−32.9737	−1.06367	−0.531833	−	0.846849i	$-0.678497\pi$
$31$	−32.9737	−1.06367	−0.531833	+	0.846849i	$0.678497\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 10.8175i	− 0.309071i
$36$	0	0
$37$	−44.7851	−1.21041	−0.605203	−	0.796071i	$-0.706908\pi$
$37$	−44.7851	−1.21041	−0.605203	+	0.796071i	$0.706908\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	21.2132i	0.517395i	0.965958	+	0.258698i	$0.0832933\pi$
$41$	21.2132i	0.517395i	−0.965958	+	0.258698i	$0.916707\pi$
$42$	0	0
$43$	16.2719	0.378416	0.189208	−	0.981937i	$-0.439408\pi$
$43$	16.2719	0.378416	0.189208	+	0.981937i	$0.439408\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 24.0044i	− 0.510732i	−0.966845	−	0.255366i	$-0.917804\pi$
$47$	− 24.0044i	− 0.510732i	0.966845	−	0.255366i	$-0.0821959\pi$
$48$	0	0
$49$	−25.5964	−0.522376
$50$	0	0
$51$	0	0
$52$	0	0
$53$	42.4264i	0.800498i	0.916406	+	0.400249i	$0.131076\pi$
$53$	42.4264i	0.800498i	−0.916406	+	0.400249i	$0.868924\pi$
$54$	0	0
$55$	33.1623	0.602951
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 86.3848i	− 1.46415i	−0.681225	−	0.732075i	$-0.738552\pi$
$59$	− 86.3848i	− 1.46415i	0.681225	−	0.732075i	$-0.261448\pi$
$60$	0	0
$61$	−38.2719	−0.627408	−0.313704	−	0.949521i	$-0.601570\pi$
$61$	−38.2719	−0.627408	−0.313704	+	0.949521i	$0.601570\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 45.0253i	− 0.692698i
$66$	0	0
$67$	37.2982	0.556690	0.278345	−	0.960481i	$-0.410214\pi$
$67$	37.2982	0.556690	0.278345	+	0.960481i	$0.410214\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 19.3400i	− 0.272394i	−0.990682	−	0.136197i	$-0.956512\pi$
$71$	− 19.3400i	− 0.272394i	0.990682	−	0.136197i	$-0.0434881\pi$
$72$	0	0
$73$	−108.974	−1.49279	−0.746395	−	0.665503i	$-0.768217\pi$
$73$	−108.974	−1.49279	−0.746395	+	0.665503i	$0.768217\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	71.7464i	0.931772i
$78$	0	0
$79$	119.570	1.51355	0.756773	−	0.653678i	$-0.226775\pi$
$79$	119.570	1.51355	0.756773	+	0.653678i	$0.226775\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	123.458i	1.48745i	0.668486	+	0.743725i	$0.266943\pi$
$83$	123.458i	1.48745i	−0.668486	+	0.743725i	$0.733057\pi$
$84$	0	0
$85$	14.7018	0.172962
$86$	0	0
$87$	0	0
$88$	0	0
$89$	63.0317i	0.708221i	0.935204	+	0.354110i	$0.115216\pi$
$89$	63.0317i	0.708221i	−0.935204	+	0.354110i	$0.884784\pi$
$90$	0	0
$91$	97.4121	1.07046
$92$	0	0
$93$	0	0
$94$	0	0
$95$	39.5235i	0.416037i
$96$	0	0
$97$	109.623	1.13013	0.565066	−	0.825046i	$-0.308851\pi$
$97$	109.623	1.13013	0.565066	+	0.825046i	$0.308851\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	140.696i	1.39303i	0.717544	+	0.696513i	$0.245266\pi$
$101$	140.696i	1.39303i	−0.717544	+	0.696513i	$0.754734\pi$
$102$	0	0
$103$	−143.057	−1.38890	−0.694451	−	0.719540i	$-0.744353\pi$
$103$	−143.057	−1.38890	−0.694451	+	0.719540i	$0.744353\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 88.5992i	− 0.828030i	−0.910270	−	0.414015i	$-0.864126\pi$
$107$	− 88.5992i	− 0.828030i	0.910270	−	0.414015i	$-0.135874\pi$
$108$	0	0
$109$	213.517	1.95888	0.979438	−	0.201746i	$-0.0646617\pi$
$109$	213.517	1.95888	0.979438	+	0.201746i	$0.0646617\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	11.2392i	0.0994622i	0.998763	+	0.0497311i	$0.0158364\pi$
$113$	11.2392i	0.0994622i	−0.998763	+	0.0497311i	$0.984164\pi$
$114$	0	0
$115$	56.3246	0.489779
$116$	0	0
$117$	0	0
$118$	0	0
$119$	31.8072i	0.267288i
$120$	0	0
$121$	−98.9473	−0.817747
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 11.1803i	− 0.0894427i
$126$	0	0
$127$	−36.8377	−0.290061	−0.145030	−	0.989427i	$-0.546328\pi$
$127$	−36.8377	−0.290061	−0.145030	+	0.989427i	$0.546328\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	11.5432i	0.0881161i	0.999029	+	0.0440580i	$0.0140286\pi$
$131$	11.5432i	0.0881161i	−0.999029	+	0.0440580i	$0.985971\pi$
$132$	0	0
$133$	−85.5089	−0.642924
$134$	0	0
$135$	0	0
$136$	0	0
$137$	145.701i	1.06351i	0.846897	+	0.531756i	$0.178468\pi$
$137$	145.701i	1.06351i	−0.846897	+	0.531756i	$0.821532\pi$
$138$	0	0
$139$	182.649	1.31402	0.657011	−	0.753881i	$-0.271820\pi$
$139$	182.649	1.31402	0.657011	+	0.753881i	$0.271820\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	298.629i	2.08831i
$144$	0	0
$145$	85.8947	0.592377
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 142.532i	− 0.956588i	−0.878200	−	0.478294i	$-0.841255\pi$
$149$	− 142.532i	− 0.956588i	0.878200	−	0.478294i	$-0.158745\pi$
$150$	0	0
$151$	−230.438	−1.52608	−0.763041	−	0.646350i	$-0.776295\pi$
$151$	−230.438	−1.52608	−0.763041	+	0.646350i	$0.776295\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 73.7314i	− 0.475686i
$156$	0	0
$157$	−164.136	−1.04545	−0.522726	−	0.852501i	$-0.675085\pi$
$157$	−164.136	−1.04545	−0.522726	+	0.852501i	$0.675085\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	121.858i	0.756881i
$162$	0	0
$163$	222.763	1.36664	0.683322	−	0.730117i	$-0.260535\pi$
$163$	222.763	1.36664	0.683322	+	0.730117i	$0.260535\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	236.180i	1.41425i	0.707089	+	0.707125i	$0.250008\pi$
$167$	236.180i	1.41425i	−0.707089	+	0.707125i	$0.749992\pi$
$168$	0	0
$169$	236.456	1.39915
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 204.106i	− 1.17980i	−0.807476	−	0.589901i	$-0.799167\pi$
$173$	− 204.106i	− 1.17980i	0.807476	−	0.589901i	$-0.200833\pi$
$174$	0	0
$175$	24.1886	0.138221
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 314.110i	− 1.75481i	−0.479753	−	0.877403i	$-0.659274\pi$
$179$	− 314.110i	− 1.75481i	0.479753	−	0.877403i	$-0.340726\pi$
$180$	0	0
$181$	304.763	1.68377	0.841887	−	0.539654i	$-0.181445\pi$
$181$	304.763	1.68377	0.841887	+	0.539654i	$0.181445\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 100.142i	− 0.541310i
$186$	0	0
$187$	−97.5089	−0.521438
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 12.3062i	− 0.0644302i	−0.999481	−	0.0322151i	$-0.989744\pi$
$191$	− 12.3062i	− 0.0644302i	0.999481	−	0.0322151i	$-0.0102562\pi$
$192$	0	0
$193$	−293.895	−1.52277	−0.761385	−	0.648300i	$-0.775480\pi$
$193$	−293.895	−1.52277	−0.761385	+	0.648300i	$0.775480\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 160.346i	− 0.813937i	−0.913442	−	0.406969i	$-0.866586\pi$
$197$	− 160.346i	− 0.813937i	0.913442	−	0.406969i	$-0.133414\pi$
$198$	0	0
$199$	−146.982	−0.738604	−0.369302	−	0.929309i	$-0.620403\pi$
$199$	−146.982	−0.738604	−0.369302	+	0.929309i	$0.620403\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	185.833i	0.915432i
$204$	0	0
$205$	−47.4342	−0.231386
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 262.138i	− 1.25425i
$210$	0	0
$211$	−398.982	−1.89091	−0.945455	−	0.325751i	$-0.894383\pi$
$211$	−398.982	−1.89091	−0.945455	+	0.325751i	$0.894383\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	36.3850i	0.169233i
$216$	0	0
$217$	159.517	0.735103
$218$	0	0
$219$	0	0
$220$	0	0
$221$	132.391i	0.599052i
$222$	0	0
$223$	−226.092	−1.01386	−0.506932	−	0.861986i	$-0.669221\pi$
$223$	−226.092	−1.01386	−0.506932	+	0.861986i	$0.669221\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 153.504i	− 0.676229i	−0.941105	−	0.338115i	$-0.890211\pi$
$227$	− 153.504i	− 0.676229i	0.941105	−	0.338115i	$-0.109789\pi$
$228$	0	0
$229$	72.8071	0.317935	0.158967	−	0.987284i	$-0.449183\pi$
$229$	72.8071	0.317935	0.158967	+	0.987284i	$0.449183\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 14.9856i	− 0.0643160i	−0.999483	−	0.0321580i	$-0.989762\pi$
$233$	− 14.9856i	− 0.0643160i	0.999483	−	0.0321580i	$-0.0102380\pi$
$234$	0	0
$235$	53.6754	0.228406
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 185.299i	− 0.775311i	−0.921804	−	0.387655i	$-0.873285\pi$
$239$	− 185.299i	− 0.775311i	0.921804	−	0.387655i	$-0.126715\pi$
$240$	0	0
$241$	134.930	0.559874	0.279937	−	0.960018i	$-0.409686\pi$
$241$	134.930	0.559874	0.279937	+	0.960018i	$0.409686\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 57.2354i	− 0.233614i
$246$	0	0
$247$	−355.912	−1.44094
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 388.419i	− 1.54748i	−0.633501	−	0.773742i	$-0.718383\pi$
$251$	− 388.419i	− 1.54748i	0.633501	−	0.773742i	$-0.281617\pi$
$252$	0	0
$253$	−373.570	−1.47656
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 260.227i	− 1.01256i	−0.862370	−	0.506279i	$-0.831021\pi$
$257$	− 260.227i	− 1.01256i	0.862370	−	0.506279i	$-0.168979\pi$
$258$	0	0
$259$	216.658	0.836516
$260$	0	0
$261$	0	0
$262$	0	0
$263$	243.673i	0.926512i	0.886225	+	0.463256i	$0.153319\pi$
$263$	243.673i	0.926512i	−0.886225	+	0.463256i	$0.846681\pi$
$264$	0	0
$265$	−94.8683	−0.357994
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 4.62110i	− 0.0171788i	−0.999963	−	0.00858941i	$-0.997266\pi$
$269$	− 4.62110i	− 0.0171788i	0.999963	−	0.00858941i	$-0.00273413\pi$
$270$	0	0
$271$	−307.947	−1.13634	−0.568169	−	0.822912i	$-0.692348\pi$
$271$	−307.947	−1.13634	−0.568169	+	0.822912i	$0.692348\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	74.1531i	0.269648i
$276$	0	0
$277$	332.680	1.20101	0.600505	−	0.799621i	$-0.294966\pi$
$277$	332.680	1.20101	0.600505	+	0.799621i	$0.294966\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 323.408i	− 1.15092i	−0.817831	−	0.575459i	$-0.804823\pi$
$281$	− 323.408i	− 1.15092i	0.817831	−	0.575459i	$-0.195177\pi$
$282$	0	0
$283$	−181.737	−0.642179	−0.321090	−	0.947049i	$-0.604049\pi$
$283$	−181.737	−0.642179	−0.321090	+	0.947049i	$0.604049\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 102.624i	− 0.357573i
$288$	0	0
$289$	245.772	0.850421
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 378.978i	− 1.29344i	−0.762727	−	0.646720i	$-0.776140\pi$
$293$	− 378.978i	− 1.29344i	0.762727	−	0.646720i	$-0.223860\pi$
$294$	0	0
$295$	193.162	0.654787
$296$	0	0
$297$	0	0
$298$	0	0
$299$	507.206i	1.69634i
$300$	0	0
$301$	−78.7189	−0.261524
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 85.5785i	− 0.280585i
$306$	0	0
$307$	16.5438	0.0538885	0.0269443	−	0.999637i	$-0.491422\pi$
$307$	16.5438	0.0538885	0.0269443	+	0.999637i	$0.491422\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	178.662i	0.574476i	0.957859	+	0.287238i	$0.0927370\pi$
$311$	178.662i	0.574476i	−0.957859	+	0.287238i	$0.907263\pi$
$312$	0	0
$313$	135.088	0.431590	0.215795	−	0.976439i	$-0.430766\pi$
$313$	135.088	0.431590	0.215795	+	0.976439i	$0.430766\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	378.445i	1.19383i	0.802304	+	0.596916i	$0.203607\pi$
$317$	378.445i	1.19383i	−0.802304	+	0.596916i	$0.796393\pi$
$318$	0	0
$319$	−569.693	−1.78587
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 116.213i	− 0.359793i
$324$	0	0
$325$	100.680	0.309784
$326$	0	0
$327$	0	0
$328$	0	0
$329$	116.127i	0.352968i
$330$	0	0
$331$	371.956	1.12373	0.561867	−	0.827228i	$-0.310083\pi$
$331$	371.956	1.12373	0.561867	+	0.827228i	$0.310083\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	83.4014i	0.248959i
$336$	0	0
$337$	−193.956	−0.575537	−0.287768	−	0.957700i	$-0.592913\pi$
$337$	−193.956	−0.575537	−0.287768	+	0.957700i	$0.592913\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	489.020i	1.43408i
$342$	0	0
$343$	360.877	1.05212
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 649.856i	− 1.87278i	−0.350957	−	0.936392i	$-0.614144\pi$
$347$	− 649.856i	− 1.87278i	0.350957	−	0.936392i	$-0.385856\pi$
$348$	0	0
$349$	−343.675	−0.984743	−0.492372	−	0.870385i	$-0.663870\pi$
$349$	−343.675	−0.984743	−0.492372	+	0.870385i	$0.663870\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 461.195i	− 1.30650i	−0.757142	−	0.653250i	$-0.773405\pi$
$353$	− 461.195i	− 1.30650i	0.757142	−	0.653250i	$-0.226595\pi$
$354$	0	0
$355$	43.2456	0.121818
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 248.380i	− 0.691867i	−0.938259	−	0.345933i	$-0.887562\pi$
$359$	− 248.380i	− 0.691867i	0.938259	−	0.345933i	$-0.112438\pi$
$360$	0	0
$361$	−48.5787	−0.134567
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 243.673i	− 0.667596i
$366$	0	0
$367$	59.6007	0.162400	0.0811999	−	0.996698i	$-0.474125\pi$
$367$	59.6007	0.162400	0.0811999	+	0.996698i	$0.474125\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 205.247i	− 0.553227i
$372$	0	0
$373$	110.460	0.296141	0.148070	−	0.988977i	$-0.452694\pi$
$373$	110.460	0.296141	0.148070	+	0.988977i	$0.452694\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	773.487i	2.05169i
$378$	0	0
$379$	637.579	1.68227	0.841133	−	0.540829i	$-0.181889\pi$
$379$	637.579	1.68227	0.841133	+	0.540829i	$0.181889\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	574.728i	1.50060i	0.661100	+	0.750298i	$0.270090\pi$
$383$	574.728i	1.50060i	−0.661100	+	0.750298i	$0.729910\pi$
$384$	0	0
$385$	−160.430	−0.416701
$386$	0	0
$387$	0	0
$388$	0	0
$389$	71.4364i	0.183641i	0.995776	+	0.0918206i	$0.0292686\pi$
$389$	71.4364i	0.183641i	−0.995776	+	0.0918206i	$0.970731\pi$
$390$	0	0
$391$	−165.614	−0.423566
$392$	0	0
$393$	0	0
$394$	0	0
$395$	267.367i	0.676878i
$396$	0	0
$397$	365.767	0.921328	0.460664	−	0.887575i	$-0.347611\pi$
$397$	365.767	0.921328	0.460664	+	0.887575i	$0.347611\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	728.791i	1.81743i	0.417413	+	0.908717i	$0.362937\pi$
$401$	728.791i	1.81743i	−0.417413	+	0.908717i	$0.637063\pi$
$402$	0	0
$403$	663.956	1.64753
$404$	0	0
$405$	0	0
$406$	0	0
$407$	664.190i	1.63192i
$408$	0	0
$409$	−401.035	−0.980525	−0.490263	−	0.871575i	$-0.663099\pi$
$409$	−401.035	−0.980525	−0.490263	+	0.871575i	$0.663099\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	417.906i	1.01188i
$414$	0	0
$415$	−276.061	−0.665208
$416$	0	0
$417$	0	0
$418$	0	0
$419$	356.623i	0.851130i	0.904928	+	0.425565i	$0.139925\pi$
$419$	356.623i	0.851130i	−0.904928	+	0.425565i	$0.860075\pi$
$420$	0	0
$421$	272.719	0.647788	0.323894	−	0.946093i	$-0.395008\pi$
$421$	272.719	0.647788	0.323894	+	0.946093i	$0.395008\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	32.8742i	0.0773510i
$426$	0	0
$427$	185.149	0.433604
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 469.680i	− 1.08974i	−0.838519	−	0.544872i	$-0.816578\pi$
$431$	− 469.680i	− 1.08974i	0.838519	−	0.544872i	$-0.183422\pi$
$432$	0	0
$433$	361.412	0.834670	0.417335	−	0.908753i	$-0.362964\pi$
$433$	361.412	0.834670	0.417335	+	0.908753i	$0.362964\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 445.229i	− 1.01883i
$438$	0	0
$439$	−250.105	−0.569716	−0.284858	−	0.958570i	$-0.591946\pi$
$439$	−250.105	−0.569716	−0.284858	+	0.958570i	$0.591946\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	356.319i	0.804333i	0.915567	+	0.402166i	$0.131743\pi$
$443$	356.319i	0.804333i	−0.915567	+	0.402166i	$0.868257\pi$
$444$	0	0
$445$	−140.943	−0.316726
$446$	0	0
$447$	0	0
$448$	0	0
$449$	282.657i	0.629525i	0.949171	+	0.314762i	$0.101925\pi$
$449$	282.657i	0.629525i	−0.949171	+	0.314762i	$0.898075\pi$
$450$	0	0
$451$	314.605	0.697572
$452$	0	0
$453$	0	0
$454$	0	0
$455$	217.820i	0.478725i
$456$	0	0
$457$	−413.851	−0.905581	−0.452791	−	0.891617i	$-0.649571\pi$
$457$	−413.851	−0.905581	−0.452791	+	0.891617i	$0.649571\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 177.453i	− 0.384931i	−0.981304	−	0.192465i	$-0.938352\pi$
$461$	− 177.453i	− 0.384931i	0.981304	−	0.192465i	$-0.0616483\pi$
$462$	0	0
$463$	267.101	0.576892	0.288446	−	0.957496i	$-0.406861\pi$
$463$	267.101	0.576892	0.288446	+	0.957496i	$0.406861\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 260.525i	− 0.557870i	−0.960310	−	0.278935i	$-0.910019\pi$
$467$	− 260.525i	− 0.557870i	0.960310	−	0.278935i	$-0.0899814\pi$
$468$	0	0
$469$	−180.438	−0.384730
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 241.322i	− 0.510195i
$474$	0	0
$475$	−88.3772	−0.186057
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 149.746i	− 0.312621i	−0.987708	−	0.156311i	$-0.950040\pi$
$479$	− 149.746i	− 0.312621i	0.987708	−	0.156311i	$-0.0499601\pi$
$480$	0	0
$481$	901.789	1.87482
$482$	0	0
$483$	0	0
$484$	0	0
$485$	245.124i	0.505410i
$486$	0	0
$487$	628.749	1.29107	0.645533	−	0.763732i	$-0.276635\pi$
$487$	628.749	1.29107	0.645533	+	0.763732i	$0.276635\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 390.341i	− 0.794992i	−0.917604	−	0.397496i	$-0.869879\pi$
$491$	− 390.341i	− 0.794992i	0.917604	−	0.397496i	$-0.130121\pi$
$492$	0	0
$493$	−252.561	−0.512294
$494$	0	0
$495$	0	0
$496$	0	0
$497$	93.5615i	0.188253i
$498$	0	0
$499$	−254.097	−0.509212	−0.254606	−	0.967045i	$-0.581946\pi$
$499$	−254.097	−0.509212	−0.254606	+	0.967045i	$0.581946\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 830.503i	− 1.65110i	−0.564330	−	0.825549i	$-0.690865\pi$
$503$	− 830.503i	− 1.65110i	0.564330	−	0.825549i	$-0.309135\pi$
$504$	0	0
$505$	−314.605	−0.622980
$506$	0	0
$507$	0	0
$508$	0	0
$509$	808.291i	1.58800i	0.607919	+	0.793999i	$0.292005\pi$
$509$	808.291i	1.58800i	−0.607919	+	0.793999i	$0.707995\pi$
$510$	0	0
$511$	527.184	1.03167
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 319.885i	− 0.621136i
$516$	0	0
$517$	−356.000	−0.688588
$518$	0	0
$519$	0	0
$520$	0	0
$521$	591.948i	1.13618i	0.822968	+	0.568088i	$0.192317\pi$
$521$	591.948i	1.13618i	−0.822968	+	0.568088i	$0.807683\pi$
$522$	0	0
$523$	529.781	1.01297	0.506483	−	0.862250i	$-0.330945\pi$
$523$	529.781	1.01297	0.506483	+	0.862250i	$0.330945\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	216.796i	0.411378i
$528$	0	0
$529$	−105.491	−0.199416
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 427.148i	− 0.801403i
$534$	0	0
$535$	198.114	0.370306
$536$	0	0
$537$	0	0
$538$	0	0
$539$	379.611i	0.704288i
$540$	0	0
$541$	−413.307	−0.763968	−0.381984	−	0.924169i	$-0.624759\pi$
$541$	−413.307	−0.763968	−0.381984	+	0.924169i	$0.624759\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	477.440i	0.876036i
$546$	0	0
$547$	−691.851	−1.26481	−0.632405	−	0.774638i	$-0.717932\pi$
$547$	−691.851	−1.26481	−0.632405	+	0.774638i	$0.717932\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 678.971i	− 1.23225i
$552$	0	0
$553$	−578.447	−1.04602
$554$	0	0
$555$	0	0
$556$	0	0
$557$	111.996i	0.201069i	0.994934	+	0.100535i	$0.0320553\pi$
$557$	111.996i	0.201069i	−0.994934	+	0.100535i	$0.967945\pi$
$558$	0	0
$559$	−327.650	−0.586136
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 397.455i	− 0.705960i	−0.935631	−	0.352980i	$-0.885168\pi$
$563$	− 397.455i	− 0.705960i	0.935631	−	0.352980i	$-0.114832\pi$
$564$	0	0
$565$	−25.1317	−0.0444808
$566$	0	0
$567$	0	0
$568$	0	0
$569$	340.056i	0.597639i	0.954310	+	0.298819i	$0.0965928\pi$
$569$	340.056i	0.597639i	−0.954310	+	0.298819i	$0.903407\pi$
$570$	0	0
$571$	−547.412	−0.958690	−0.479345	−	0.877627i	$-0.659126\pi$
$571$	−547.412	−0.958690	−0.479345	+	0.877627i	$0.659126\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	125.946i	0.219036i
$576$	0	0
$577$	140.974	0.244322	0.122161	−	0.992510i	$-0.461018\pi$
$577$	140.974	0.244322	0.122161	+	0.992510i	$0.461018\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 597.257i	− 1.02798i
$582$	0	0
$583$	629.210	1.07926
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 222.057i	− 0.378291i	−0.981949	−	0.189145i	$-0.939428\pi$
$587$	− 222.057i	− 0.378291i	0.981949	−	0.189145i	$-0.0605717\pi$
$588$	0	0
$589$	−582.824	−0.989515
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 948.975i	− 1.60029i	−0.599804	−	0.800147i	$-0.704755\pi$
$593$	− 948.975i	− 1.60029i	0.599804	−	0.800147i	$-0.295245\pi$
$594$	0	0
$595$	−71.1231	−0.119535
$596$	0	0
$597$	0	0
$598$	0	0
$599$	31.7951i	0.0530804i	0.999648	+	0.0265402i	$0.00844900\pi$
$599$	31.7951i	0.0530804i	−0.999648	+	0.0265402i	$0.991551\pi$
$600$	0	0
$601$	−520.561	−0.866158	−0.433079	−	0.901356i	$-0.642573\pi$
$601$	−520.561	−0.866158	−0.433079	+	0.901356i	$0.642573\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 221.253i	− 0.365707i
$606$	0	0
$607$	676.505	1.11451	0.557253	−	0.830343i	$-0.311856\pi$
$607$	676.505	1.11451	0.557253	+	0.830343i	$0.311856\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	483.351i	0.791082i
$612$	0	0
$613$	479.820	0.782741	0.391370	−	0.920233i	$-0.372001\pi$
$613$	479.820	0.782741	0.391370	+	0.920233i	$0.372001\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 514.687i	− 0.834177i	−0.908866	−	0.417088i	$-0.863051\pi$
$617$	− 514.687i	− 0.834177i	0.908866	−	0.417088i	$-0.136949\pi$
$618$	0	0
$619$	−830.894	−1.34232	−0.671158	−	0.741314i	$-0.734203\pi$
$619$	−830.894	−1.34232	−0.671158	+	0.741314i	$0.734203\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 304.930i	− 0.489454i
$624$	0	0
$625$	25.0000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	294.454i	0.468131i
$630$	0	0
$631$	−481.324	−0.762795	−0.381398	−	0.924411i	$-0.624557\pi$
$631$	−481.324	−0.762795	−0.381398	+	0.924411i	$0.624557\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 82.3717i	− 0.129719i
$636$	0	0
$637$	515.409	0.809119
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 1076.40i	− 1.67925i	−0.543163	−	0.839627i	$-0.682774\pi$
$641$	− 1076.40i	− 1.67925i	0.543163	−	0.839627i	$-0.317226\pi$
$642$	0	0
$643$	1021.36	1.58843	0.794214	−	0.607638i	$-0.207883\pi$
$643$	1021.36	1.58843	0.794214	+	0.607638i	$0.207883\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 972.866i	− 1.50366i	−0.659359	−	0.751829i	$-0.729172\pi$
$647$	− 972.866i	− 1.50366i	0.659359	−	0.751829i	$-0.270828\pi$
$648$	0	0
$649$	−1281.14	−1.97402
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 309.260i	− 0.473599i	−0.971559	−	0.236799i	$-0.923902\pi$
$653$	− 309.260i	− 0.473599i	0.971559	−	0.236799i	$-0.0760984\pi$
$654$	0	0
$655$	−25.8114	−0.0394067
$656$	0	0
$657$	0	0
$658$	0	0
$659$	294.758i	0.447281i	0.974672	+	0.223641i	$0.0717942\pi$
$659$	294.758i	0.447281i	−0.974672	+	0.223641i	$0.928206\pi$
$660$	0	0
$661$	232.552	0.351819	0.175909	−	0.984406i	$-0.443713\pi$
$661$	232.552	0.351819	0.175909	+	0.984406i	$0.443713\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 191.204i	− 0.287524i
$666$	0	0
$667$	−967.596	−1.45067
$668$	0	0
$669$	0	0
$670$	0	0
$671$	567.596i	0.845895i
$672$	0	0
$673$	37.5174	0.0557466	0.0278733	−	0.999611i	$-0.491127\pi$
$673$	37.5174	0.0557466	0.0278733	+	0.999611i	$0.491127\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	111.630i	0.164890i	0.996596	+	0.0824448i	$0.0262728\pi$
$677$	111.630i	0.164890i	−0.996596	+	0.0824448i	$0.973727\pi$
$678$	0	0
$679$	−530.325	−0.781038
$680$	0	0
$681$	0	0
$682$	0	0
$683$	80.5971i	0.118005i	0.998258	+	0.0590023i	$0.0187919\pi$
$683$	80.5971i	0.118005i	−0.998258	+	0.0590023i	$0.981208\pi$
$684$	0	0
$685$	−325.798	−0.475617
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 854.296i	− 1.23991i
$690$	0	0
$691$	280.377	0.405756	0.202878	−	0.979204i	$-0.434971\pi$
$691$	280.377	0.405756	0.202878	+	0.979204i	$0.434971\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	408.416i	0.587649i
$696$	0	0
$697$	139.473	0.200105
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 182.365i	− 0.260150i	−0.991504	−	0.130075i	$-0.958478\pi$
$701$	− 182.365i	− 0.260150i	0.991504	−	0.130075i	$-0.0415218\pi$
$702$	0	0
$703$	−791.596	−1.12603
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 680.646i	− 0.962725i
$708$	0	0
$709$	−42.7374	−0.0602784	−0.0301392	−	0.999546i	$-0.509595\pi$
$709$	−42.7374	−0.0602784	−0.0301392	+	0.999546i	$0.509595\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	830.577i	1.16490i
$714$	0	0
$715$	−667.754	−0.933921
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 765.586i	− 1.06479i	−0.846495	−	0.532396i	$-0.821292\pi$
$719$	− 765.586i	− 1.06479i	0.846495	−	0.532396i	$-0.178708\pi$
$720$	0	0
$721$	692.070	0.959875
$722$	0	0
$723$	0	0
$724$	0	0
$725$	192.066i	0.264919i
$726$	0	0
$727$	201.653	0.277377	0.138689	−	0.990336i	$-0.455711\pi$
$727$	201.653	0.277377	0.138689	+	0.990336i	$0.455711\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 106.985i	− 0.146354i
$732$	0	0
$733$	489.057	0.667199	0.333600	−	0.942715i	$-0.391737\pi$
$733$	489.057	0.667199	0.333600	+	0.942715i	$0.391737\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 553.156i	− 0.750551i
$738$	0	0
$739$	−760.306	−1.02883	−0.514415	−	0.857541i	$-0.671991\pi$
$739$	−760.306	−1.02883	−0.514415	+	0.857541i	$0.671991\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 771.776i	− 1.03873i	−0.854553	−	0.519365i	$-0.826169\pi$
$743$	− 771.776i	− 1.03873i	0.854553	−	0.519365i	$-0.173831\pi$
$744$	0	0
$745$	318.710	0.427799
$746$	0	0
$747$	0	0
$748$	0	0
$749$	428.618i	0.572254i
$750$	0	0
$751$	1366.53	1.81962	0.909809	−	0.415026i	$-0.136228\pi$
$751$	1366.53	1.81962	0.909809	+	0.415026i	$0.136228\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 515.276i	− 0.682485i
$756$	0	0
$757$	10.6271	0.0140384	0.00701919	−	0.999975i	$-0.497766\pi$
$757$	10.6271	0.0140384	0.00701919	+	0.999975i	$0.497766\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 20.6052i	− 0.0270765i	−0.999908	−	0.0135383i	$-0.995691\pi$
$761$	− 20.6052i	− 0.0270765i	0.999908	−	0.0135383i	$-0.00430950\pi$
$762$	0	0
$763$	−1032.94	−1.35379
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1739.44i	2.26785i
$768$	0	0
$769$	1235.93	1.60719	0.803595	−	0.595177i	$-0.202918\pi$
$769$	1235.93	1.60719	0.803595	+	0.595177i	$0.202918\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	54.0189i	0.0698822i	0.999389	+	0.0349411i	$0.0111244\pi$
$773$	54.0189i	0.0698822i	−0.999389	+	0.0349411i	$0.988876\pi$
$774$	0	0
$775$	164.868	0.212733
$776$	0	0
$777$	0	0
$778$	0	0
$779$	374.953i	0.481326i
$780$	0	0
$781$	−286.824	−0.367253
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 367.019i	− 0.467540i
$786$	0	0
$787$	28.0612	0.0356559	0.0178280	−	0.999841i	$-0.494325\pi$
$787$	28.0612	0.0356559	0.0178280	+	0.999841i	$0.494325\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 54.3722i	− 0.0687386i
$792$	0	0
$793$	770.641	0.971804
$794$	0	0
$795$	0	0
$796$	0	0
$797$	835.564i	1.04839i	0.851599	+	0.524193i	$0.175633\pi$
$797$	835.564i	1.04839i	−0.851599	+	0.524193i	$0.824367\pi$
$798$	0	0
$799$	−157.825	−0.197528
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1616.15i	2.01264i
$804$	0	0
$805$	−272.483	−0.338488
$806$	0	0
$807$	0	0
$808$	0	0
$809$	793.187i	0.980453i	0.871595	+	0.490227i	$0.163086\pi$
$809$	793.187i	0.980453i	−0.871595	+	0.490227i	$0.836914\pi$
$810$	0	0
$811$	298.105	0.367577	0.183789	−	0.982966i	$-0.441164\pi$
$811$	298.105	0.367577	0.183789	+	0.982966i	$0.441164\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	498.113i	0.611182i
$816$	0	0
$817$	287.613	0.352035
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 1143.48i	− 1.39280i	−0.717656	−	0.696398i	$-0.754785\pi$
$821$	− 1143.48i	− 1.39280i	0.717656	−	0.696398i	$-0.245215\pi$
$822$	0	0
$823$	−460.399	−0.559416	−0.279708	−	0.960085i	$-0.590238\pi$
$823$	−460.399	−0.559416	−0.279708	+	0.960085i	$0.590238\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 250.204i	− 0.302544i	−0.988492	−	0.151272i	$-0.951663\pi$
$827$	− 250.204i	− 0.302544i	0.988492	−	0.151272i	$-0.0483370\pi$
$828$	0	0
$829$	−171.815	−0.207256	−0.103628	−	0.994616i	$-0.533045\pi$
$829$	−171.815	−0.207256	−0.103628	+	0.994616i	$0.533045\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	168.292i	0.202032i
$834$	0	0
$835$	−528.114	−0.632472
$836$	0	0
$837$	0	0
$838$	0	0
$839$	915.442i	1.09111i	0.838075	+	0.545555i	$0.183681\pi$
$839$	915.442i	1.09111i	−0.838075	+	0.545555i	$0.816319\pi$
$840$	0	0
$841$	−634.579	−0.754553
$842$	0	0
$843$	0	0
$844$	0	0
$845$	528.732i	0.625719i
$846$	0	0
$847$	478.680	0.565147
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1128.10i	1.32561i
$852$	0	0
$853$	1242.33	1.45642	0.728211	−	0.685353i	$-0.240352\pi$
$853$	1242.33	1.45642	0.728211	+	0.685353i	$0.240352\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 716.993i	− 0.836632i	−0.908302	−	0.418316i	$-0.862621\pi$
$857$	− 716.993i	− 0.836632i	0.908302	−	0.418316i	$-0.137379\pi$
$858$	0	0
$859$	551.509	0.642036	0.321018	−	0.947073i	$-0.395975\pi$
$859$	551.509	0.642036	0.321018	+	0.947073i	$0.395975\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 1513.33i	− 1.75356i	−0.480887	−	0.876782i	$-0.659685\pi$
$863$	− 1513.33i	− 1.75356i	0.480887	−	0.876782i	$-0.340315\pi$
$864$	0	0
$865$	456.394	0.527623
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 1773.30i	− 2.04062i
$870$	0	0
$871$	−751.035	−0.862267
$872$	0	0
$873$	0	0
$874$	0	0
$875$	54.0874i	0.0618141i
$876$	0	0
$877$	−946.355	−1.07908	−0.539541	−	0.841959i	$-0.681402\pi$
$877$	−946.355	−1.07908	−0.539541	+	0.841959i	$0.681402\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1239.83i	1.40730i	0.710547	+	0.703650i	$0.248447\pi$
$881$	1239.83i	1.40730i	−0.710547	+	0.703650i	$0.751553\pi$
$882$	0	0
$883$	−1156.41	−1.30964	−0.654820	−	0.755785i	$-0.727255\pi$
$883$	−1156.41	−1.30964	−0.654820	+	0.755785i	$0.727255\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1100.19i	1.24035i	0.784464	+	0.620174i	$0.212938\pi$
$887$	1100.19i	1.24035i	−0.784464	+	0.620174i	$0.787062\pi$
$888$	0	0
$889$	178.211	0.200462
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 424.288i	− 0.475127i
$894$	0	0
$895$	702.372	0.784773
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1266.63i	1.40893i
$900$	0	0
$901$	278.947	0.309597
$902$	0	0
$903$	0	0
$904$	0	0
$905$	681.471i	0.753006i
$906$	0	0
$907$	−570.430	−0.628919	−0.314460	−	0.949271i	$-0.601823\pi$
$907$	−570.430	−0.628919	−0.314460	+	0.949271i	$0.601823\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 1321.74i	− 1.45087i	−0.688290	−	0.725435i	$-0.741638\pi$
$911$	− 1321.74i	− 1.45087i	0.688290	−	0.725435i	$-0.258362\pi$
$912$	0	0
$913$	1830.96	2.00544
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 55.8428i	− 0.0608973i
$918$	0	0
$919$	1748.03	1.90210	0.951048	−	0.309044i	$-0.100009\pi$
$919$	1748.03	1.90210	0.951048	+	0.309044i	$0.100009\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	389.429i	0.421917i
$924$	0	0
$925$	223.925	0.242081
$926$	0	0
$927$	0	0
$928$	0	0
$929$	556.666i	0.599210i	0.954063	+	0.299605i	$0.0968548\pi$
$929$	556.666i	0.599210i	−0.954063	+	0.299605i	$0.903145\pi$
$930$	0	0
$931$	−452.429	−0.485960
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 218.037i	− 0.233194i
$936$	0	0
$937$	−117.701	−0.125615	−0.0628074	−	0.998026i	$-0.520005\pi$
$937$	−117.701	−0.125615	−0.0628074	+	0.998026i	$0.520005\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1714.17i	1.82165i	0.412797	+	0.910823i	$0.364552\pi$
$941$	1714.17i	1.82165i	−0.412797	+	0.910823i	$0.635448\pi$
$942$	0	0
$943$	534.342	0.566640
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1151.68i	1.21613i	0.793886	+	0.608066i	$0.208054\pi$
$947$	1151.68i	1.21613i	−0.793886	+	0.608066i	$0.791946\pi$
$948$	0	0
$949$	2194.29	2.31221
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1272.18i	1.33493i	0.744643	+	0.667463i	$0.232620\pi$
$953$	1272.18i	1.33493i	−0.744643	+	0.667463i	$0.767380\pi$
$954$	0	0
$955$	27.5174	0.0288141
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 704.862i	− 0.734997i
$960$	0	0
$961$	126.263	0.131387
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 657.168i	− 0.681004i
$966$	0	0
$967$	1358.70	1.40506	0.702532	−	0.711652i	$-0.252053\pi$
$967$	1358.70	1.40506	0.702532	+	0.711652i	$0.252053\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1818.71i	1.87303i	0.350632	+	0.936513i	$0.385967\pi$
$971$	1818.71i	1.87303i	−0.350632	+	0.936513i	$0.614033\pi$
$972$	0	0
$973$	−883.606	−0.908125
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 594.974i	− 0.608981i	−0.952515	−	0.304490i	$-0.901514\pi$
$977$	− 594.974i	− 0.608981i	0.952515	−	0.304490i	$-0.0984862\pi$
$978$	0	0
$979$	934.799	0.954850
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 1209.81i	− 1.23073i	−0.788241	−	0.615366i	$-0.789008\pi$
$983$	− 1209.81i	− 1.23073i	0.788241	−	0.615366i	$-0.210992\pi$
$984$	0	0
$985$	358.544	0.364004
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 409.874i	− 0.414433i
$990$	0	0
$991$	−236.596	−0.238745	−0.119373	−	0.992850i	$-0.538088\pi$
$991$	−236.596	−0.238745	−0.119373	+	0.992850i	$0.538088\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 328.662i	− 0.330314i
$996$	0	0
$997$	614.355	0.616204	0.308102	−	0.951353i	$-0.400306\pi$
$997$	614.355	0.616204	0.308102	+	0.951353i	$0.400306\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	360.3.l.a.161.4	yes	4
3.2	odd	2	inner	360.3.l.a.161.2	✓	4
4.3	odd	2		720.3.l.d.161.3		4
5.2	odd	4		1800.3.c.b.449.3		8
5.3	odd	4		1800.3.c.b.449.5		8
5.4	even	2		1800.3.l.g.1601.1		4
8.3	odd	2		2880.3.l.h.1601.1		4
8.5	even	2		2880.3.l.a.1601.2		4
12.11	even	2		720.3.l.d.161.1		4
15.2	even	4		1800.3.c.b.449.4		8
15.8	even	4		1800.3.c.b.449.6		8
15.14	odd	2		1800.3.l.g.1601.2		4
20.3	even	4		3600.3.c.l.449.4		8
20.7	even	4		3600.3.c.l.449.6		8
20.19	odd	2		3600.3.l.m.1601.4		4
24.5	odd	2		2880.3.l.a.1601.4		4
24.11	even	2		2880.3.l.h.1601.3		4
60.23	odd	4		3600.3.c.l.449.3		8
60.47	odd	4		3600.3.c.l.449.5		8
60.59	even	2		3600.3.l.m.1601.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.3.l.a.161.2	✓	4	3.2	odd	2	inner
360.3.l.a.161.4	yes	4	1.1	even	1	trivial
720.3.l.d.161.1		4	12.11	even	2
720.3.l.d.161.3		4	4.3	odd	2
1800.3.c.b.449.3		8	5.2	odd	4
1800.3.c.b.449.4		8	15.2	even	4
1800.3.c.b.449.5		8	5.3	odd	4
1800.3.c.b.449.6		8	15.8	even	4
1800.3.l.g.1601.1		4	5.4	even	2
1800.3.l.g.1601.2		4	15.14	odd	2
2880.3.l.a.1601.2		4	8.5	even	2
2880.3.l.a.1601.4		4	24.5	odd	2
2880.3.l.h.1601.1		4	8.3	odd	2
2880.3.l.h.1601.3		4	24.11	even	2
3600.3.c.l.449.3		8	60.23	odd	4
3600.3.c.l.449.4		8	20.3	even	4
3600.3.c.l.449.5		8	60.47	odd	4
3600.3.c.l.449.6		8	20.7	even	4
3600.3.l.m.1601.3		4	60.59	even	2
3600.3.l.m.1601.4		4	20.19	odd	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:	\( N \)	\(=\)	\( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	360.l (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.23607i q^{5} -4.83772 q^{7} +O(q^{10})\)	\(q+2.23607i q^{5} -4.83772 q^{7} -14.8306i q^{11} -20.1359 q^{13} -6.57484i q^{17} +17.6754 q^{19} -25.1891i q^{23} -5.00000 q^{25} -38.4133i q^{29} -32.9737 q^{31} -10.8175i q^{35} -44.7851 q^{37} +21.2132i q^{41} +16.2719 q^{43} -24.0044i q^{47} -25.5964 q^{49} +42.4264i q^{53} +33.1623 q^{55} -86.3848i q^{59} -38.2719 q^{61} -45.0253i q^{65} +37.2982 q^{67} -19.3400i q^{71} -108.974 q^{73} +71.7464i q^{77} +119.570 q^{79} +123.458i q^{83} +14.7018 q^{85} +63.0317i q^{89} +97.4121 q^{91} +39.5235i q^{95} +109.623 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 32 q^{7}+O(q^{10}) \) 4 * q - 32 * q^7	\( 4 q - 32 q^{7} + 8 q^{13} + 96 q^{19} - 20 q^{25} - 56 q^{31} - 40 q^{37} - 112 q^{43} + 100 q^{49} + 120 q^{55} + 24 q^{61} + 48 q^{67} - 360 q^{73} + 200 q^{79} + 160 q^{85} - 344 q^{91} + 312 q^{97}+O(q^{100}) \) 4 * q - 32 * q^7 + 8 * q^13 + 96 * q^19 - 20 * q^25 - 56 * q^31 - 40 * q^37 - 112 * q^43 + 100 * q^49 + 120 * q^55 + 24 * q^61 + 48 * q^67 - 360 * q^73 + 200 * q^79 + 160 * q^85 - 344 * q^91 + 312 * q^97

Introduction

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