LMFDB - Embedded newform 720.3.l.a.161.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [720,3,Mod(161,720)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(720, 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("720.161");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 720 = 2^{4} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	720.l (of order $2$, degree $1$, not 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:	$19.6185790339$
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:	no (minimal twist has level 45)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.1
Root			$1.58114 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	720.161
Dual form			720.3.l.a.161.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.23607i q^{5} -7.16228 q^{7} +5.42736i q^{11} +9.81139 q^{13} +12.2317i q^{17} -6.32456 q^{19} -12.0394i q^{23} -5.00000 q^{25} +44.9881i q^{29} +58.2719 q^{31} +16.0153i q^{35} +66.4605 q^{37} -16.4743i q^{41} +43.6228 q^{43} +40.0570i q^{47} +2.29822 q^{49} -13.2242i q^{53} +12.1359 q^{55} +25.1519i q^{59} -35.6754 q^{61} -21.9389i q^{65} -26.7018 q^{67} +92.7301i q^{71} +60.3246 q^{73} -38.8723i q^{77} +96.2192 q^{79} -79.1215i q^{83} +27.3509 q^{85} +107.443i q^{89} -70.2719 q^{91} +14.1421i q^{95} -1.07900 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 16 q^{7} - 24 q^{13} - 20 q^{25} + 56 q^{31} + 152 q^{37} + 48 q^{43} - 92 q^{49} - 40 q^{55} - 168 q^{61} - 208 q^{67} + 216 q^{73} + 56 q^{79} + 160 q^{85} - 104 q^{91} - 232 q^{97}+O(q^{100}) $ 4 * q - 16 * q^7 - 24 * q^13 - 20 * q^25 + 56 * q^31 + 152 * q^37 + 48 * q^43 - 92 * q^49 - 40 * q^55 - 168 * q^61 - 208 * q^67 + 216 * q^73 + 56 * q^79 + 160 * q^85 - 104 * q^91 - 232 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/720\mathbb{Z}\right)^\times$.

$n$	$181$	$271$	$577$	$641$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$7$	−7.16228	−1.02318	−0.511591	−	0.859229i	$-0.670944\pi$
$7$	−7.16228	−1.02318	−0.511591	+	0.859229i	$0.670944\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	5.42736i	0.493396i	0.969092	+	0.246698i	$0.0793456\pi$
$11$	5.42736i	0.493396i	−0.969092	+	0.246698i	$0.920654\pi$
$12$	0	0
$13$	9.81139	0.754722	0.377361	−	0.926066i	$-0.376832\pi$
$13$	9.81139	0.754722	0.377361	+	0.926066i	$0.376832\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	12.2317i	0.719511i	0.933047	+	0.359756i	$0.117140\pi$
$17$	12.2317i	0.719511i	−0.933047	+	0.359756i	$0.882860\pi$
$18$	0	0
$19$	−6.32456	−0.332871	−0.166436	−	0.986052i	$-0.553226\pi$
$19$	−6.32456	−0.332871	−0.166436	+	0.986052i	$0.553226\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 12.0394i	− 0.523454i	−0.965142	−	0.261727i	$-0.915708\pi$
$23$	− 12.0394i	− 0.523454i	0.965142	−	0.261727i	$-0.0842920\pi$
$24$	0	0
$25$	−5.00000	−0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	44.9881i	1.55131i	0.631155	+	0.775657i	$0.282581\pi$
$29$	44.9881i	1.55131i	−0.631155	+	0.775657i	$0.717419\pi$
$30$	0	0
$31$	58.2719	1.87974	0.939869	−	0.341535i	$-0.110947\pi$
$31$	58.2719	1.87974	0.939869	+	0.341535i	$0.110947\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	16.0153i	0.457581i
$36$	0	0
$37$	66.4605	1.79623	0.898115	−	0.439761i	$-0.144937\pi$
$37$	66.4605	1.79623	0.898115	+	0.439761i	$0.144937\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 16.4743i	− 0.401813i	−0.979610	−	0.200906i	$-0.935611\pi$
$41$	− 16.4743i	− 0.401813i	0.979610	−	0.200906i	$-0.0643887\pi$
$42$	0	0
$43$	43.6228	1.01448	0.507242	−	0.861804i	$-0.330665\pi$
$43$	43.6228	1.01448	0.507242	+	0.861804i	$0.330665\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	40.0570i	0.852276i	0.904658	+	0.426138i	$0.140126\pi$
$47$	40.0570i	0.852276i	−0.904658	+	0.426138i	$0.859874\pi$
$48$	0	0
$49$	2.29822	0.0469025
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 13.2242i	− 0.249512i	−0.992187	−	0.124756i	$-0.960185\pi$
$53$	− 13.2242i	− 0.249512i	0.992187	−	0.124756i	$-0.0398149\pi$
$54$	0	0
$55$	12.1359	0.220654
$56$	0	0
$57$	0	0
$58$	0	0
$59$	25.1519i	0.426303i	0.977019	+	0.213151i	$0.0683728\pi$
$59$	25.1519i	0.426303i	−0.977019	+	0.213151i	$0.931627\pi$
$60$	0	0
$61$	−35.6754	−0.584843	−0.292422	−	0.956289i	$-0.594461\pi$
$61$	−35.6754	−0.584843	−0.292422	+	0.956289i	$0.594461\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 21.9389i	− 0.337522i
$66$	0	0
$67$	−26.7018	−0.398534	−0.199267	−	0.979945i	$-0.563856\pi$
$67$	−26.7018	−0.398534	−0.199267	+	0.979945i	$0.563856\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	92.7301i	1.30606i	0.757333	+	0.653029i	$0.226502\pi$
$71$	92.7301i	1.30606i	−0.757333	+	0.653029i	$0.773498\pi$
$72$	0	0
$73$	60.3246	0.826364	0.413182	−	0.910649i	$-0.364417\pi$
$73$	60.3246	0.826364	0.413182	+	0.910649i	$0.364417\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 38.8723i	− 0.504834i
$78$	0	0
$79$	96.2192	1.21796	0.608982	−	0.793184i	$-0.291578\pi$
$79$	96.2192	1.21796	0.608982	+	0.793184i	$0.291578\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 79.1215i	− 0.953271i	−0.879101	−	0.476635i	$-0.841856\pi$
$83$	− 79.1215i	− 0.953271i	0.879101	−	0.476635i	$-0.158144\pi$
$84$	0	0
$85$	27.3509	0.321775
$86$	0	0
$87$	0	0
$88$	0	0
$89$	107.443i	1.20722i	0.797278	+	0.603612i	$0.206272\pi$
$89$	107.443i	1.20722i	−0.797278	+	0.603612i	$0.793728\pi$
$90$	0	0
$91$	−70.2719	−0.772219
$92$	0	0
$93$	0	0
$94$	0	0
$95$	14.1421i	0.148865i
$96$	0	0
$97$	−1.07900	−0.0111237	−0.00556187	−	0.999985i	$-0.501770\pi$
$97$	−1.07900	−0.0111237	−0.00556187	+	0.999985i	$0.501770\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	170.282i	1.68596i	0.537942	+	0.842982i	$0.319202\pi$
$101$	170.282i	1.68596i	−0.537942	+	0.842982i	$0.680798\pi$
$102$	0	0
$103$	128.460	1.24719	0.623595	−	0.781748i	$-0.285672\pi$
$103$	128.460	1.24719	0.623595	+	0.781748i	$0.285672\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	76.3675i	0.713715i	0.934159	+	0.356858i	$0.116152\pi$
$107$	76.3675i	0.713715i	−0.934159	+	0.356858i	$0.883848\pi$
$108$	0	0
$109$	−13.0790	−0.119991	−0.0599954	−	0.998199i	$-0.519109\pi$
$109$	−13.0790	−0.119991	−0.0599954	+	0.998199i	$0.519109\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	20.7170i	0.183336i	0.995790	+	0.0916680i	$0.0292199\pi$
$113$	20.7170i	0.183336i	−0.995790	+	0.0916680i	$0.970780\pi$
$114$	0	0
$115$	−26.9210	−0.234096
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 87.6068i	− 0.736191i
$120$	0	0
$121$	91.5438	0.756560
$122$	0	0
$123$	0	0
$124$	0	0
$125$	11.1803i	0.0894427i
$126$	0	0
$127$	38.0306	0.299454	0.149727	−	0.988727i	$-0.452161\pi$
$127$	38.0306	0.299454	0.149727	+	0.988727i	$0.452161\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 83.9409i	− 0.640770i	−0.947287	−	0.320385i	$-0.896188\pi$
$131$	− 83.9409i	− 0.640770i	0.947287	−	0.320385i	$-0.103812\pi$
$132$	0	0
$133$	45.2982	0.340588
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 15.5936i	− 0.113822i	−0.998379	−	0.0569109i	$-0.981875\pi$
$137$	− 15.5936i	− 0.113822i	0.998379	−	0.0569109i	$-0.0181251\pi$
$138$	0	0
$139$	−67.8420	−0.488072	−0.244036	−	0.969766i	$-0.578471\pi$
$139$	−67.8420	−0.488072	−0.244036	+	0.969766i	$0.578471\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	53.2499i	0.372377i
$144$	0	0
$145$	100.596	0.693769
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 233.426i	− 1.56662i	−0.621634	−	0.783308i	$-0.713531\pi$
$149$	− 233.426i	− 1.56662i	0.621634	−	0.783308i	$-0.286469\pi$
$150$	0	0
$151$	−185.351	−1.22749	−0.613745	−	0.789505i	$-0.710338\pi$
$151$	−185.351	−1.22749	−0.613745	+	0.789505i	$0.710338\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 130.300i	− 0.840645i
$156$	0	0
$157$	−111.276	−0.708765	−0.354383	−	0.935100i	$-0.615309\pi$
$157$	−111.276	−0.708765	−0.354383	+	0.935100i	$0.615309\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	86.2298i	0.535589i
$162$	0	0
$163$	−118.763	−0.728607	−0.364304	−	0.931280i	$-0.618693\pi$
$163$	−118.763	−0.728607	−0.364304	+	0.931280i	$0.618693\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	221.194i	1.32452i	0.749276	+	0.662258i	$0.230402\pi$
$167$	221.194i	1.32452i	−0.749276	+	0.662258i	$0.769598\pi$
$168$	0	0
$169$	−72.7367	−0.430394
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 190.807i	− 1.10293i	−0.834198	−	0.551466i	$-0.814069\pi$
$173$	− 190.807i	− 1.10293i	0.834198	−	0.551466i	$-0.185931\pi$
$174$	0	0
$175$	35.8114	0.204637
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 58.1005i	− 0.324584i	−0.986743	−	0.162292i	$-0.948111\pi$
$179$	− 58.1005i	− 0.324584i	0.986743	−	0.162292i	$-0.0518886\pi$
$180$	0	0
$181$	−162.921	−0.900116	−0.450058	−	0.892999i	$-0.648597\pi$
$181$	−162.921	−0.900116	−0.450058	+	0.892999i	$0.648597\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 148.610i	− 0.803298i
$186$	0	0
$187$	−66.3858	−0.355004
$188$	0	0
$189$	0	0
$190$	0	0
$191$	100.062i	0.523884i	0.965084	+	0.261942i	$0.0843630\pi$
$191$	100.062i	0.523884i	−0.965084	+	0.261942i	$0.915637\pi$
$192$	0	0
$193$	−61.8947	−0.320698	−0.160349	−	0.987060i	$-0.551262\pi$
$193$	−61.8947	−0.320698	−0.160349	+	0.987060i	$0.551262\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	24.8791i	0.126290i	0.998004	+	0.0631449i	$0.0201130\pi$
$197$	24.8791i	0.126290i	−0.998004	+	0.0631449i	$0.979887\pi$
$198$	0	0
$199$	156.491	0.786387	0.393194	−	0.919456i	$-0.371370\pi$
$199$	156.491	0.786387	0.393194	+	0.919456i	$0.371370\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 322.217i	− 1.58728i
$204$	0	0
$205$	−36.8377	−0.179696
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 34.3256i	− 0.164237i
$210$	0	0
$211$	−237.789	−1.12696	−0.563482	−	0.826128i	$-0.690539\pi$
$211$	−237.789	−1.12696	−0.563482	+	0.826128i	$0.690539\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 97.5435i	− 0.453691i
$216$	0	0
$217$	−417.359	−1.92332
$218$	0	0
$219$	0	0
$220$	0	0
$221$	120.010i	0.543031i
$222$	0	0
$223$	182.302	0.817500	0.408750	−	0.912646i	$-0.365965\pi$
$223$	182.302	0.817500	0.408750	+	0.912646i	$0.365965\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 406.078i	− 1.78889i	−0.447180	−	0.894444i	$-0.647572\pi$
$227$	− 406.078i	− 1.78889i	0.447180	−	0.894444i	$-0.352428\pi$
$228$	0	0
$229$	−27.2982	−0.119206	−0.0596031	−	0.998222i	$-0.518984\pi$
$229$	−27.2982	−0.119206	−0.0596031	+	0.998222i	$0.518984\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 356.382i	− 1.52954i	−0.644306	−	0.764768i	$-0.722854\pi$
$233$	− 356.382i	− 1.52954i	0.644306	−	0.764768i	$-0.277146\pi$
$234$	0	0
$235$	89.5701	0.381149
$236$	0	0
$237$	0	0
$238$	0	0
$239$	271.690i	1.13678i	0.822760	+	0.568389i	$0.192433\pi$
$239$	271.690i	1.13678i	−0.822760	+	0.568389i	$0.807567\pi$
$240$	0	0
$241$	−224.438	−0.931280	−0.465640	−	0.884974i	$-0.654176\pi$
$241$	−224.438	−0.931280	−0.465640	+	0.884974i	$0.654176\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 5.13898i	− 0.0209754i
$246$	0	0
$247$	−62.0527	−0.251225
$248$	0	0
$249$	0	0
$250$	0	0
$251$	318.775i	1.27002i	0.772504	+	0.635010i	$0.219004\pi$
$251$	318.775i	1.27002i	−0.772504	+	0.635010i	$0.780996\pi$
$252$	0	0
$253$	65.3423	0.258270
$254$	0	0
$255$	0	0
$256$	0	0
$257$	371.975i	1.44738i	0.690128	+	0.723688i	$0.257554\pi$
$257$	371.975i	1.44738i	−0.690128	+	0.723688i	$0.742446\pi$
$258$	0	0
$259$	−476.009	−1.83787
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 238.549i	− 0.907031i	−0.891248	−	0.453515i	$-0.850170\pi$
$263$	− 238.549i	− 0.907031i	0.891248	−	0.453515i	$-0.149830\pi$
$264$	0	0
$265$	−29.5701	−0.111585
$266$	0	0
$267$	0	0
$268$	0	0
$269$	125.871i	0.467922i	0.972246	+	0.233961i	$0.0751688\pi$
$269$	125.871i	0.467922i	−0.972246	+	0.233961i	$0.924831\pi$
$270$	0	0
$271$	258.649	0.954425	0.477212	−	0.878788i	$-0.341647\pi$
$271$	258.649	0.954425	0.477212	+	0.878788i	$0.341647\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 27.1368i	− 0.0986793i
$276$	0	0
$277$	227.715	0.822074	0.411037	−	0.911619i	$-0.365167\pi$
$277$	227.715	0.822074	0.411037	+	0.911619i	$0.365167\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 241.384i	− 0.859016i	−0.903063	−	0.429508i	$-0.858687\pi$
$281$	− 241.384i	− 0.859016i	0.903063	−	0.429508i	$-0.141313\pi$
$282$	0	0
$283$	−208.333	−0.736159	−0.368080	−	0.929794i	$-0.619985\pi$
$283$	−208.333	−0.736159	−0.368080	+	0.929794i	$0.619985\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	117.994i	0.411128i
$288$	0	0
$289$	139.386	0.482304
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 201.693i	− 0.688372i	−0.938901	−	0.344186i	$-0.888155\pi$
$293$	− 201.693i	− 0.688372i	0.938901	−	0.344186i	$-0.111845\pi$
$294$	0	0
$295$	56.2413	0.190648
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 118.124i	− 0.395062i
$300$	0	0
$301$	−312.438	−1.03800
$302$	0	0
$303$	0	0
$304$	0	0
$305$	79.7727i	0.261550i
$306$	0	0
$307$	−342.824	−1.11669	−0.558346	−	0.829608i	$-0.688564\pi$
$307$	−342.824	−1.11669	−0.558346	+	0.829608i	$0.688564\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	217.640i	0.699807i	0.936786	+	0.349903i	$0.113786\pi$
$311$	217.640i	0.699807i	−0.936786	+	0.349903i	$0.886214\pi$
$312$	0	0
$313$	281.895	0.900622	0.450311	−	0.892872i	$-0.351313\pi$
$313$	281.895	0.900622	0.450311	+	0.892872i	$0.351313\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	15.4013i	0.0485847i	0.999705	+	0.0242923i	$0.00773325\pi$
$317$	15.4013i	0.0485847i	−0.999705	+	0.0242923i	$0.992267\pi$
$318$	0	0
$319$	−244.167	−0.765412
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 77.3600i	− 0.239505i
$324$	0	0
$325$	−49.0569	−0.150944
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 286.899i	− 0.872034i
$330$	0	0
$331$	−375.517	−1.13449	−0.567247	−	0.823548i	$-0.691991\pi$
$331$	−375.517	−1.13449	−0.567247	+	0.823548i	$0.691991\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	59.7070i	0.178230i
$336$	0	0
$337$	188.114	0.558201	0.279101	−	0.960262i	$-0.409964\pi$
$337$	188.114	0.558201	0.279101	+	0.960262i	$0.409964\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	316.262i	0.927456i
$342$	0	0
$343$	334.491	0.975193
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 513.793i	− 1.48067i	−0.672237	−	0.740336i	$-0.734666\pi$
$347$	− 513.793i	− 1.48067i	0.672237	−	0.740336i	$-0.265334\pi$
$348$	0	0
$349$	112.535	0.322451	0.161225	−	0.986918i	$-0.448455\pi$
$349$	112.535	0.322451	0.161225	+	0.986918i	$0.448455\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 428.172i	− 1.21295i	−0.795102	−	0.606475i	$-0.792583\pi$
$353$	− 428.172i	− 1.21295i	0.795102	−	0.606475i	$-0.207417\pi$
$354$	0	0
$355$	207.351	0.584087
$356$	0	0
$357$	0	0
$358$	0	0
$359$	56.1961i	0.156535i	0.996932	+	0.0782676i	$0.0249389\pi$
$359$	56.1961i	0.156535i	−0.996932	+	0.0782676i	$0.975061\pi$
$360$	0	0
$361$	−321.000	−0.889197
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 134.890i	− 0.369561i
$366$	0	0
$367$	154.364	0.420610	0.210305	−	0.977636i	$-0.432554\pi$
$367$	154.364	0.420610	0.210305	+	0.977636i	$0.432554\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	94.7151i	0.255297i
$372$	0	0
$373$	557.285	1.49406	0.747030	−	0.664790i	$-0.231479\pi$
$373$	557.285	1.49406	0.747030	+	0.664790i	$0.231479\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	441.396i	1.17081i
$378$	0	0
$379$	−147.404	−0.388928	−0.194464	−	0.980910i	$-0.562297\pi$
$379$	−147.404	−0.388928	−0.194464	+	0.980910i	$0.562297\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 736.619i	− 1.92329i	−0.274302	−	0.961644i	$-0.588447\pi$
$383$	− 736.619i	− 1.92329i	0.274302	−	0.961644i	$-0.411553\pi$
$384$	0	0
$385$	−86.9210	−0.225769
$386$	0	0
$387$	0	0
$388$	0	0
$389$	296.408i	0.761975i	0.924580	+	0.380987i	$0.124416\pi$
$389$	296.408i	0.761975i	−0.924580	+	0.380987i	$0.875584\pi$
$390$	0	0
$391$	147.263	0.376631
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 215.153i	− 0.544690i
$396$	0	0
$397$	−457.057	−1.15128	−0.575638	−	0.817704i	$-0.695246\pi$
$397$	−457.057	−1.15128	−0.575638	+	0.817704i	$0.695246\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	391.141i	0.975415i	0.873007	+	0.487707i	$0.162167\pi$
$401$	391.141i	0.975415i	−0.873007	+	0.487707i	$0.837833\pi$
$402$	0	0
$403$	571.728	1.41868
$404$	0	0
$405$	0	0
$406$	0	0
$407$	360.705i	0.886253i
$408$	0	0
$409$	−411.842	−1.00695	−0.503474	−	0.864010i	$-0.667945\pi$
$409$	−411.842	−1.00695	−0.503474	+	0.864010i	$0.667945\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 180.145i	− 0.436186i
$414$	0	0
$415$	−176.921	−0.426316
$416$	0	0
$417$	0	0
$418$	0	0
$419$	653.447i	1.55954i	0.626066	+	0.779770i	$0.284664\pi$
$419$	653.447i	1.55954i	−0.626066	+	0.779770i	$0.715336\pi$
$420$	0	0
$421$	125.035	0.296995	0.148497	−	0.988913i	$-0.452556\pi$
$421$	125.035	0.296995	0.148497	+	0.988913i	$0.452556\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 61.1584i	− 0.143902i
$426$	0	0
$427$	255.517	0.598401
$428$	0	0
$429$	0	0
$430$	0	0
$431$	397.208i	0.921596i	0.887505	+	0.460798i	$0.152437\pi$
$431$	397.208i	0.921596i	−0.887505	+	0.460798i	$0.847563\pi$
$432$	0	0
$433$	560.114	1.29357	0.646783	−	0.762674i	$-0.276114\pi$
$433$	560.114	1.29357	0.646783	+	0.762674i	$0.276114\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	76.1441i	0.174243i
$438$	0	0
$439$	664.386	1.51341	0.756704	−	0.653758i	$-0.226809\pi$
$439$	664.386	1.51341	0.756704	+	0.653758i	$0.226809\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	371.305i	0.838160i	0.907949	+	0.419080i	$0.137647\pi$
$443$	371.305i	0.838160i	−0.907949	+	0.419080i	$0.862353\pi$
$444$	0	0
$445$	240.250	0.539887
$446$	0	0
$447$	0	0
$448$	0	0
$449$	585.471i	1.30395i	0.758243	+	0.651973i	$0.226058\pi$
$449$	585.471i	1.30395i	−0.758243	+	0.651973i	$0.773942\pi$
$450$	0	0
$451$	89.4121	0.198253
$452$	0	0
$453$	0	0
$454$	0	0
$455$	157.133i	0.345347i
$456$	0	0
$457$	168.641	0.369017	0.184508	−	0.982831i	$-0.440931\pi$
$457$	168.641	0.369017	0.184508	+	0.982831i	$0.440931\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 298.492i	− 0.647487i	−0.946145	−	0.323744i	$-0.895058\pi$
$461$	− 298.492i	− 0.647487i	0.946145	−	0.323744i	$-0.104942\pi$
$462$	0	0
$463$	595.285	1.28571	0.642856	−	0.765987i	$-0.277749\pi$
$463$	595.285	1.28571	0.642856	+	0.765987i	$0.277749\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	623.655i	1.33545i	0.744408	+	0.667725i	$0.232732\pi$
$467$	623.655i	1.33545i	−0.744408	+	0.667725i	$0.767268\pi$
$468$	0	0
$469$	191.246	0.407773
$470$	0	0
$471$	0	0
$472$	0	0
$473$	236.756i	0.500542i
$474$	0	0
$475$	31.6228	0.0665743
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 131.857i	− 0.275276i	−0.990483	−	0.137638i	$-0.956049\pi$
$479$	− 131.857i	− 0.275276i	0.990483	−	0.137638i	$-0.0439510\pi$
$480$	0	0
$481$	652.070	1.35565
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2.41272i	0.00497468i
$486$	0	0
$487$	−41.8028	−0.0858375	−0.0429187	−	0.999079i	$-0.513666\pi$
$487$	−41.8028	−0.0858375	−0.0429187	+	0.999079i	$0.513666\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 178.817i	− 0.364189i	−0.983281	−	0.182095i	$-0.941712\pi$
$491$	− 178.817i	− 0.364189i	0.983281	−	0.182095i	$-0.0582877\pi$
$492$	0	0
$493$	−550.280	−1.11619
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 664.159i	− 1.33634i
$498$	0	0
$499$	39.0961	0.0783489	0.0391744	−	0.999232i	$-0.487527\pi$
$499$	39.0961	0.0783489	0.0391744	+	0.999232i	$0.487527\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	578.698i	1.15049i	0.817980	+	0.575247i	$0.195094\pi$
$503$	578.698i	1.15049i	−0.817980	+	0.575247i	$0.804906\pi$
$504$	0	0
$505$	380.763	0.753986
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 355.743i	− 0.698905i	−0.936954	−	0.349453i	$-0.886368\pi$
$509$	− 355.743i	− 0.698905i	0.936954	−	0.349453i	$-0.113632\pi$
$510$	0	0
$511$	−432.061	−0.845521
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 287.246i	− 0.557760i
$516$	0	0
$517$	−217.404	−0.420510
$518$	0	0
$519$	0	0
$520$	0	0
$521$	810.952i	1.55653i	0.627936	+	0.778265i	$0.283900\pi$
$521$	810.952i	1.55653i	−0.627936	+	0.778265i	$0.716100\pi$
$522$	0	0
$523$	−720.483	−1.37760	−0.688798	−	0.724953i	$-0.741861\pi$
$523$	−720.483	−1.37760	−0.688798	+	0.724953i	$0.741861\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	712.764i	1.35249i
$528$	0	0
$529$	384.052	0.725996
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 161.636i	− 0.303257i
$534$	0	0
$535$	170.763	0.319183
$536$	0	0
$537$	0	0
$538$	0	0
$539$	12.4733i	0.0231415i
$540$	0	0
$541$	347.149	0.641680	0.320840	−	0.947133i	$-0.396035\pi$
$541$	347.149	0.641680	0.320840	+	0.947133i	$0.396035\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	29.2455i	0.0536615i
$546$	0	0
$547$	720.833	1.31779	0.658896	−	0.752234i	$-0.271024\pi$
$547$	720.833	1.31779	0.658896	+	0.752234i	$0.271024\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 284.530i	− 0.516388i
$552$	0	0
$553$	−689.149	−1.24620
$554$	0	0
$555$	0	0
$556$	0	0
$557$	429.102i	0.770380i	0.922837	+	0.385190i	$0.125864\pi$
$557$	429.102i	0.770380i	−0.922837	+	0.385190i	$0.874136\pi$
$558$	0	0
$559$	428.000	0.765653
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 670.820i	− 1.19151i	−0.803166	−	0.595755i	$-0.796853\pi$
$563$	− 670.820i	− 1.19151i	0.803166	−	0.595755i	$-0.203147\pi$
$564$	0	0
$565$	46.3246	0.0819904
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 368.663i	− 0.647914i	−0.946072	−	0.323957i	$-0.894987\pi$
$569$	− 368.663i	− 0.647914i	0.946072	−	0.323957i	$-0.105013\pi$
$570$	0	0
$571$	−124.289	−0.217669	−0.108834	−	0.994060i	$-0.534712\pi$
$571$	−124.289	−0.217669	−0.108834	+	0.994060i	$0.534712\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	60.1972i	0.104691i
$576$	0	0
$577$	504.236	0.873893	0.436947	−	0.899487i	$-0.356060\pi$
$577$	504.236	0.873893	0.436947	+	0.899487i	$0.356060\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	566.690i	0.975370i
$582$	0	0
$583$	71.7722	0.123108
$584$	0	0
$585$	0	0
$586$	0	0
$587$	39.2256i	0.0668238i	0.999442	+	0.0334119i	$0.0106373\pi$
$587$	39.2256i	0.0668238i	−0.999442	+	0.0334119i	$0.989363\pi$
$588$	0	0
$589$	−368.544	−0.625711
$590$	0	0
$591$	0	0
$592$	0	0
$593$	621.670i	1.04835i	0.851611	+	0.524174i	$0.175626\pi$
$593$	621.670i	1.04835i	−0.851611	+	0.524174i	$0.824374\pi$
$594$	0	0
$595$	−195.895	−0.329235
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 1119.77i	− 1.86940i	−0.355436	−	0.934701i	$-0.615668\pi$
$599$	− 1119.77i	− 1.86940i	0.355436	−	0.934701i	$-0.384332\pi$
$600$	0	0
$601$	−323.789	−0.538751	−0.269375	−	0.963035i	$-0.586817\pi$
$601$	−323.789	−0.538751	−0.269375	+	0.963035i	$0.586817\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 204.698i	− 0.338344i
$606$	0	0
$607$	−1025.63	−1.68966	−0.844832	−	0.535031i	$-0.820300\pi$
$607$	−1025.63	−1.68966	−0.844832	+	0.535031i	$0.820300\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	393.014i	0.643232i
$612$	0	0
$613$	904.153	1.47496	0.737482	−	0.675367i	$-0.236015\pi$
$613$	904.153	1.47496	0.737482	+	0.675367i	$0.236015\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 710.716i	− 1.15189i	−0.817488	−	0.575945i	$-0.804634\pi$
$617$	− 710.716i	− 1.15189i	0.817488	−	0.575945i	$-0.195366\pi$
$618$	0	0
$619$	583.737	0.943032	0.471516	−	0.881858i	$-0.343707\pi$
$619$	583.737	0.943032	0.471516	+	0.881858i	$0.343707\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 769.537i	− 1.23521i
$624$	0	0
$625$	25.0000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	812.924i	1.29241i
$630$	0	0
$631$	20.0968	0.0318491	0.0159246	−	0.999873i	$-0.494931\pi$
$631$	20.0968	0.0318491	0.0159246	+	0.999873i	$0.494931\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 85.0390i	− 0.133920i
$636$	0	0
$637$	22.5487	0.0353983
$638$	0	0
$639$	0	0
$640$	0	0
$641$	341.607i	0.532928i	0.963845	+	0.266464i	$0.0858553\pi$
$641$	341.607i	0.532928i	−0.963845	+	0.266464i	$0.914145\pi$
$642$	0	0
$643$	469.693	0.730471	0.365235	−	0.930915i	$-0.380989\pi$
$643$	469.693	0.730471	0.365235	+	0.930915i	$0.380989\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	572.099i	0.884234i	0.896957	+	0.442117i	$0.145772\pi$
$647$	572.099i	0.884234i	−0.896957	+	0.442117i	$0.854228\pi$
$648$	0	0
$649$	−136.508	−0.210336
$650$	0	0
$651$	0	0
$652$	0	0
$653$	213.540i	0.327014i	0.986542	+	0.163507i	$0.0522806\pi$
$653$	213.540i	0.327014i	−0.986542	+	0.163507i	$0.947719\pi$
$654$	0	0
$655$	−187.698	−0.286561
$656$	0	0
$657$	0	0
$658$	0	0
$659$	420.983i	0.638820i	0.947617	+	0.319410i	$0.103485\pi$
$659$	420.983i	0.638820i	−0.947617	+	0.319410i	$0.896515\pi$
$660$	0	0
$661$	434.272	0.656992	0.328496	−	0.944505i	$-0.393458\pi$
$661$	434.272	0.656992	0.328496	+	0.944505i	$0.393458\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 101.290i	− 0.152316i
$666$	0	0
$667$	541.631	0.812041
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 193.623i	− 0.288560i
$672$	0	0
$673$	72.7801	0.108143	0.0540714	−	0.998537i	$-0.482780\pi$
$673$	72.7801	0.108143	0.0540714	+	0.998537i	$0.482780\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	172.106i	0.254219i	0.991889	+	0.127109i	$0.0405700\pi$
$677$	172.106i	0.254219i	−0.991889	+	0.127109i	$0.959430\pi$
$678$	0	0
$679$	7.72811	0.0113816
$680$	0	0
$681$	0	0
$682$	0	0
$683$	792.592i	1.16046i	0.814454	+	0.580228i	$0.197037\pi$
$683$	792.592i	1.16046i	−0.814454	+	0.580228i	$0.802963\pi$
$684$	0	0
$685$	−34.8683	−0.0509027
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 129.747i	− 0.188313i
$690$	0	0
$691$	−154.851	−0.224097	−0.112049	−	0.993703i	$-0.535741\pi$
$691$	−154.851	−0.224097	−0.112049	+	0.993703i	$0.535741\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	151.699i	0.218272i
$696$	0	0
$697$	201.509	0.289109
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 950.544i	− 1.35598i	−0.735070	−	0.677991i	$-0.762851\pi$
$701$	− 950.544i	− 1.35598i	0.735070	−	0.677991i	$-0.237149\pi$
$702$	0	0
$703$	−420.333	−0.597913
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 1219.61i	− 1.72505i
$708$	0	0
$709$	390.350	0.550564	0.275282	−	0.961363i	$-0.411229\pi$
$709$	390.350	0.550564	0.275282	+	0.961363i	$0.411229\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 701.561i	− 0.983956i
$714$	0	0
$715$	119.070	0.166532
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 655.227i	− 0.911303i	−0.890158	−	0.455651i	$-0.849406\pi$
$719$	− 655.227i	− 0.911303i	0.890158	−	0.455651i	$-0.150594\pi$
$720$	0	0
$721$	−920.070	−1.27610
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 224.940i	− 0.310263i
$726$	0	0
$727$	−1424.25	−1.95908	−0.979539	−	0.201254i	$-0.935498\pi$
$727$	−1424.25	−1.95908	−0.979539	+	0.201254i	$0.935498\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	533.580i	0.729932i
$732$	0	0
$733$	−946.749	−1.29161	−0.645805	−	0.763503i	$-0.723478\pi$
$733$	−946.749	−1.29161	−0.645805	+	0.763503i	$0.723478\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 144.920i	− 0.196635i
$738$	0	0
$739$	−591.429	−0.800310	−0.400155	−	0.916447i	$-0.631044\pi$
$739$	−591.429	−0.800310	−0.400155	+	0.916447i	$0.631044\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	732.202i	0.985467i	0.870180	+	0.492734i	$0.164002\pi$
$743$	732.202i	0.985467i	−0.870180	+	0.492734i	$0.835998\pi$
$744$	0	0
$745$	−521.956	−0.700612
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 546.965i	− 0.730261i
$750$	0	0
$751$	215.359	0.286764	0.143382	−	0.989667i	$-0.454202\pi$
$751$	215.359	0.286764	0.143382	+	0.989667i	$0.454202\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	414.457i	0.548950i
$756$	0	0
$757$	276.258	0.364938	0.182469	−	0.983212i	$-0.441591\pi$
$757$	276.258	0.364938	0.182469	+	0.983212i	$0.441591\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 893.373i	− 1.17395i	−0.809606	−	0.586973i	$-0.800319\pi$
$761$	− 893.373i	− 1.17395i	0.809606	−	0.586973i	$-0.199681\pi$
$762$	0	0
$763$	93.6754	0.122773
$764$	0	0
$765$	0	0
$766$	0	0
$767$	246.775i	0.321740i
$768$	0	0
$769$	284.316	0.369722	0.184861	−	0.982765i	$-0.440817\pi$
$769$	284.316	0.369722	0.184861	+	0.982765i	$0.440817\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	1059.64i	1.37081i	0.728162	+	0.685405i	$0.240375\pi$
$773$	1059.64i	1.37081i	−0.728162	+	0.685405i	$0.759625\pi$
$774$	0	0
$775$	−291.359	−0.375948
$776$	0	0
$777$	0	0
$778$	0	0
$779$	104.193i	0.133752i
$780$	0	0
$781$	−503.280	−0.644404
$782$	0	0
$783$	0	0
$784$	0	0
$785$	248.821i	0.316970i
$786$	0	0
$787$	875.517	1.11247	0.556237	−	0.831024i	$-0.312245\pi$
$787$	875.517	1.11247	0.556237	+	0.831024i	$0.312245\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 148.381i	− 0.187586i
$792$	0	0
$793$	−350.026	−0.441394
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 742.449i	− 0.931555i	−0.884902	−	0.465777i	$-0.845775\pi$
$797$	− 742.449i	− 0.931555i	0.884902	−	0.465777i	$-0.154225\pi$
$798$	0	0
$799$	−489.964	−0.613222
$800$	0	0
$801$	0	0
$802$	0	0
$803$	327.403i	0.407725i
$804$	0	0
$805$	192.816	0.239523
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 113.720i	− 0.140568i	−0.997527	−	0.0702842i	$-0.977609\pi$
$809$	− 113.720i	− 0.140568i	0.997527	−	0.0702842i	$-0.0223906\pi$
$810$	0	0
$811$	1466.03	1.80769	0.903844	−	0.427863i	$-0.140733\pi$
$811$	1466.03	1.80769	0.903844	+	0.427863i	$0.140733\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	265.562i	0.325843i
$816$	0	0
$817$	−275.895	−0.337692
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 550.073i	− 0.670003i	−0.942218	−	0.335002i	$-0.891263\pi$
$821$	− 550.073i	− 0.670003i	0.942218	−	0.335002i	$-0.108737\pi$
$822$	0	0
$823$	−1392.51	−1.69199	−0.845997	−	0.533187i	$-0.820994\pi$
$823$	−1392.51	−1.69199	−0.845997	+	0.533187i	$0.820994\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 955.922i	− 1.15589i	−0.816075	−	0.577945i	$-0.803855\pi$
$827$	− 955.922i	− 1.15589i	0.816075	−	0.577945i	$-0.196145\pi$
$828$	0	0
$829$	−1652.69	−1.99360	−0.996798	−	0.0799552i	$-0.974522\pi$
$829$	−1652.69	−1.99360	−0.996798	+	0.0799552i	$0.974522\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	28.1111i	0.0337469i
$834$	0	0
$835$	494.605	0.592341
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 1568.11i	− 1.86903i	−0.355927	−	0.934514i	$-0.615835\pi$
$839$	− 1568.11i	− 1.86903i	0.355927	−	0.934514i	$-0.384165\pi$
$840$	0	0
$841$	−1182.93	−1.40657
$842$	0	0
$843$	0	0
$844$	0	0
$845$	162.644i	0.192478i
$846$	0	0
$847$	−655.662	−0.774099
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 800.147i	− 0.940243i
$852$	0	0
$853$	−651.232	−0.763461	−0.381730	−	0.924274i	$-0.624672\pi$
$853$	−651.232	−0.763461	−0.381730	+	0.924274i	$0.624672\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	299.131i	0.349044i	0.984653	+	0.174522i	$0.0558380\pi$
$857$	299.131i	0.349044i	−0.984653	+	0.174522i	$0.944162\pi$
$858$	0	0
$859$	−1095.72	−1.27558	−0.637788	−	0.770212i	$-0.720150\pi$
$859$	−1095.72	−1.27558	−0.637788	+	0.770212i	$0.720150\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 1221.95i	− 1.41593i	−0.706247	−	0.707965i	$-0.749613\pi$
$863$	− 1221.95i	− 1.41593i	0.706247	−	0.707965i	$-0.250387\pi$
$864$	0	0
$865$	−426.658	−0.493246
$866$	0	0
$867$	0	0
$868$	0	0
$869$	522.216i	0.600939i
$870$	0	0
$871$	−261.982	−0.300782
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 80.0767i	− 0.0915162i
$876$	0	0
$877$	766.399	0.873887	0.436944	−	0.899489i	$-0.356061\pi$
$877$	766.399	0.873887	0.436944	+	0.899489i	$0.356061\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 310.097i	− 0.351983i	−0.984392	−	0.175992i	$-0.943687\pi$
$881$	− 310.097i	− 0.351983i	0.984392	−	0.175992i	$-0.0563132\pi$
$882$	0	0
$883$	122.236	0.138433	0.0692165	−	0.997602i	$-0.477950\pi$
$883$	122.236	0.138433	0.0692165	+	0.997602i	$0.477950\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 265.444i	− 0.299261i	−0.988742	−	0.149630i	$-0.952192\pi$
$887$	− 265.444i	− 0.299261i	0.988742	−	0.149630i	$-0.0478084\pi$
$888$	0	0
$889$	−272.386	−0.306396
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 253.343i	− 0.283698i
$894$	0	0
$895$	−129.917	−0.145158
$896$	0	0
$897$	0	0
$898$	0	0
$899$	2621.54i	2.91606i
$900$	0	0
$901$	161.754	0.179527
$902$	0	0
$903$	0	0
$904$	0	0
$905$	364.302i	0.402544i
$906$	0	0
$907$	−672.622	−0.741590	−0.370795	−	0.928715i	$-0.620915\pi$
$907$	−672.622	−0.741590	−0.370795	+	0.928715i	$0.620915\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 1402.48i	− 1.53949i	−0.638350	−	0.769746i	$-0.720383\pi$
$911$	− 1402.48i	− 1.53949i	0.638350	−	0.769746i	$-0.279617\pi$
$912$	0	0
$913$	429.421	0.470340
$914$	0	0
$915$	0	0
$916$	0	0
$917$	601.208i	0.655625i
$918$	0	0
$919$	338.255	0.368068	0.184034	−	0.982920i	$-0.441084\pi$
$919$	338.255	0.368068	0.184034	+	0.982920i	$0.441084\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	909.811i	0.985711i
$924$	0	0
$925$	−332.302	−0.359246
$926$	0	0
$927$	0	0
$928$	0	0
$929$	148.207i	0.159533i	0.996814	+	0.0797667i	$0.0254175\pi$
$929$	148.207i	0.159533i	−0.996814	+	0.0797667i	$0.974582\pi$
$930$	0	0
$931$	−14.5352	−0.0156125
$932$	0	0
$933$	0	0
$934$	0	0
$935$	148.443i	0.158763i
$936$	0	0
$937$	−1416.72	−1.51197	−0.755987	−	0.654587i	$-0.772843\pi$
$937$	−1416.72	−1.51197	−0.755987	+	0.654587i	$0.772843\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 1398.92i	− 1.48663i	−0.668939	−	0.743317i	$-0.733251\pi$
$941$	− 1398.92i	− 1.48663i	0.668939	−	0.743317i	$-0.266749\pi$
$942$	0	0
$943$	−198.342	−0.210330
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 1050.57i	− 1.10937i	−0.832060	−	0.554686i	$-0.812839\pi$
$947$	− 1050.57i	− 1.10937i	0.832060	−	0.554686i	$-0.187161\pi$
$948$	0	0
$949$	591.868	0.623675
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 551.928i	− 0.579148i	−0.957156	−	0.289574i	$-0.906486\pi$
$953$	− 551.928i	− 0.579148i	0.957156	−	0.289574i	$-0.0935136\pi$
$954$	0	0
$955$	223.745	0.234288
$956$	0	0
$957$	0	0
$958$	0	0
$959$	111.686i	0.116461i
$960$	0	0
$961$	2434.61	2.53342
$962$	0	0
$963$	0	0
$964$	0	0
$965$	138.401i	0.143420i
$966$	0	0
$967$	357.093	0.369279	0.184639	−	0.982806i	$-0.440888\pi$
$967$	357.093	0.369279	0.184639	+	0.982806i	$0.440888\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 308.206i	− 0.317411i	−0.987326	−	0.158705i	$-0.949268\pi$
$971$	− 308.206i	− 0.317411i	0.987326	−	0.158705i	$-0.0507320\pi$
$972$	0	0
$973$	485.903	0.499387
$974$	0	0
$975$	0	0
$976$	0	0
$977$	253.280i	0.259243i	0.991564	+	0.129621i	$0.0413762\pi$
$977$	253.280i	0.259243i	−0.991564	+	0.129621i	$0.958624\pi$
$978$	0	0
$979$	−583.132	−0.595640
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 1068.73i	− 1.08722i	−0.839339	−	0.543609i	$-0.817058\pi$
$983$	− 1068.73i	− 1.08722i	0.839339	−	0.543609i	$-0.182942\pi$
$984$	0	0
$985$	55.6313	0.0564785
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 525.194i	− 0.531035i
$990$	0	0
$991$	−280.631	−0.283179	−0.141590	−	0.989925i	$-0.545221\pi$
$991$	−280.631	−0.283179	−0.141590	+	0.989925i	$0.545221\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 349.925i	− 0.351683i
$996$	0	0
$997$	−356.574	−0.357647	−0.178824	−	0.983881i	$-0.557229\pi$
$997$	−356.574	−0.357647	−0.178824	+	0.983881i	$0.557229\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	720.3.l.a.161.1		4
3.2	odd	2	inner	720.3.l.a.161.3		4
4.3	odd	2		45.3.c.a.26.1	✓	4
5.2	odd	4		3600.3.c.i.449.2		8
5.3	odd	4		3600.3.c.i.449.8		8
5.4	even	2		3600.3.l.v.1601.4		4
8.3	odd	2		2880.3.l.g.1601.4		4
8.5	even	2		2880.3.l.c.1601.3		4
12.11	even	2		45.3.c.a.26.4	yes	4
15.2	even	4		3600.3.c.i.449.1		8
15.8	even	4		3600.3.c.i.449.7		8
15.14	odd	2		3600.3.l.v.1601.3		4
20.3	even	4		225.3.d.b.224.1		8
20.7	even	4		225.3.d.b.224.8		8
20.19	odd	2		225.3.c.c.26.4		4
24.5	odd	2		2880.3.l.c.1601.1		4
24.11	even	2		2880.3.l.g.1601.2		4
36.7	odd	6		405.3.i.d.296.1		8
36.11	even	6		405.3.i.d.296.4		8
36.23	even	6		405.3.i.d.26.1		8
36.31	odd	6		405.3.i.d.26.4		8
60.23	odd	4		225.3.d.b.224.7		8
60.47	odd	4		225.3.d.b.224.2		8
60.59	even	2		225.3.c.c.26.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.3.c.a.26.1	✓	4	4.3	odd	2
45.3.c.a.26.4	yes	4	12.11	even	2
225.3.c.c.26.1		4	60.59	even	2
225.3.c.c.26.4		4	20.19	odd	2
225.3.d.b.224.1		8	20.3	even	4
225.3.d.b.224.2		8	60.47	odd	4
225.3.d.b.224.7		8	60.23	odd	4
225.3.d.b.224.8		8	20.7	even	4
405.3.i.d.26.1		8	36.23	even	6
405.3.i.d.26.4		8	36.31	odd	6
405.3.i.d.296.1		8	36.7	odd	6
405.3.i.d.296.4		8	36.11	even	6
720.3.l.a.161.1		4	1.1	even	1	trivial
720.3.l.a.161.3		4	3.2	odd	2	inner
2880.3.l.c.1601.1		4	24.5	odd	2
2880.3.l.c.1601.3		4	8.5	even	2
2880.3.l.g.1601.2		4	24.11	even	2
2880.3.l.g.1601.4		4	8.3	odd	2
3600.3.c.i.449.1		8	15.2	even	4
3600.3.c.i.449.2		8	5.2	odd	4
3600.3.c.i.449.7		8	15.8	even	4
3600.3.c.i.449.8		8	5.3	odd	4
3600.3.l.v.1601.3		4	15.14	odd	2
3600.3.l.v.1601.4		4	5.4	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	720.l (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.23607i q^{5} -7.16228 q^{7} +5.42736i q^{11} +9.81139 q^{13} +12.2317i q^{17} -6.32456 q^{19} -12.0394i q^{23} -5.00000 q^{25} +44.9881i q^{29} +58.2719 q^{31} +16.0153i q^{35} +66.4605 q^{37} -16.4743i q^{41} +43.6228 q^{43} +40.0570i q^{47} +2.29822 q^{49} -13.2242i q^{53} +12.1359 q^{55} +25.1519i q^{59} -35.6754 q^{61} -21.9389i q^{65} -26.7018 q^{67} +92.7301i q^{71} +60.3246 q^{73} -38.8723i q^{77} +96.2192 q^{79} -79.1215i q^{83} +27.3509 q^{85} +107.443i q^{89} -70.2719 q^{91} +14.1421i q^{95} -1.07900 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 16 q^{7} - 24 q^{13} - 20 q^{25} + 56 q^{31} + 152 q^{37} + 48 q^{43} - 92 q^{49} - 40 q^{55} - 168 q^{61} - 208 q^{67} + 216 q^{73} + 56 q^{79} + 160 q^{85} - 104 q^{91} - 232 q^{97}+O(q^{100}) \) 4 * q - 16 * q^7 - 24 * q^13 - 20 * q^25 + 56 * q^31 + 152 * q^37 + 48 * q^43 - 92 * q^49 - 40 * q^55 - 168 * q^61 - 208 * q^67 + 216 * q^73 + 56 * q^79 + 160 * q^85 - 104 * q^91 - 232 * q^97

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