LMFDB - Embedded newform 1440.3.e.a.991.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1440,3,Mod(991,1440)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1440, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1440.991");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1440 = 2^{5} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1440.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:	$39.2371580679$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 160)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			991.2
Root			$0.618034i$ of defining polynomial
Character	$\chi$	$=$	1440.991
Dual form			1440.3.e.a.991.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.23607 q^{5} +1.23607i q^{7} -11.4164i q^{11} +5.41641 q^{13} -6.94427 q^{17} +29.8885i q^{19} +19.1246i q^{23} +5.00000 q^{25} -21.0557 q^{29} -34.4721i q^{31} -2.76393i q^{35} -19.3050 q^{37} +58.1378 q^{41} -62.7639i q^{43} +63.4853i q^{47} +47.4721 q^{49} +98.1378 q^{53} +25.5279i q^{55} -19.2786i q^{59} +1.19350 q^{61} -12.1115 q^{65} -5.01316i q^{67} +84.3607i q^{71} -70.7214 q^{73} +14.1115 q^{77} +124.498i q^{79} +160.652i q^{83} +15.5279 q^{85} -46.2229 q^{89} +6.69505i q^{91} -66.8328i q^{95} +133.331 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 32 q^{13} + 8 q^{17} + 20 q^{25} - 120 q^{29} + 48 q^{37} + 172 q^{49} + 160 q^{53} - 192 q^{61} - 120 q^{65} - 104 q^{73} + 128 q^{77} + 80 q^{85} - 328 q^{89} + 104 q^{97}+O(q^{100}) $ 4 * q - 32 * q^13 + 8 * q^17 + 20 * q^25 - 120 * q^29 + 48 * q^37 + 172 * q^49 + 160 * q^53 - 192 * q^61 - 120 * q^65 - 104 * q^73 + 128 * q^77 + 80 * q^85 - 328 * q^89 + 104 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1440\mathbb{Z}\right)^\times$.

$n$	$577$	$641$	$901$	$991$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$5$	−2.23607	−0.447214
$5$	−2.23607	−0.447214
$6$	0	0
$7$	1.23607i	0.176581i	0.996095	+	0.0882906i	$0.0281404\pi$
$7$	1.23607i	0.176581i	−0.996095	+	0.0882906i	$0.971860\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 11.4164i	− 1.03786i	−0.854818	−	0.518928i	$-0.826331\pi$
$11$	− 11.4164i	− 1.03786i	0.854818	−	0.518928i	$-0.173669\pi$
$12$	0	0
$13$	5.41641	0.416647	0.208323	−	0.978060i	$-0.433199\pi$
$13$	5.41641	0.416647	0.208323	+	0.978060i	$0.433199\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−6.94427	−0.408487	−0.204243	−	0.978920i	$-0.565473\pi$
$17$	−6.94427	−0.408487	−0.204243	+	0.978920i	$0.565473\pi$
$18$	0	0
$19$	29.8885i	1.57308i	0.617539	+	0.786541i	$0.288130\pi$
$19$	29.8885i	1.57308i	−0.617539	+	0.786541i	$0.711870\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	19.1246i	0.831505i	0.909478	+	0.415752i	$0.136482\pi$
$23$	19.1246i	0.831505i	−0.909478	+	0.415752i	$0.863518\pi$
$24$	0	0
$25$	5.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−21.0557	−0.726060	−0.363030	−	0.931778i	$-0.618258\pi$
$29$	−21.0557	−0.726060	−0.363030	+	0.931778i	$0.618258\pi$
$30$	0	0
$31$	− 34.4721i	− 1.11200i	−0.831181	−	0.556002i	$-0.812335\pi$
$31$	− 34.4721i	− 1.11200i	0.831181	−	0.556002i	$-0.187665\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 2.76393i	− 0.0789695i
$36$	0	0
$37$	−19.3050	−0.521755	−0.260878	−	0.965372i	$-0.584012\pi$
$37$	−19.3050	−0.521755	−0.260878	+	0.965372i	$0.584012\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	58.1378	1.41799	0.708997	−	0.705211i	$-0.249148\pi$
$41$	58.1378	1.41799	0.708997	+	0.705211i	$0.249148\pi$
$42$	0	0
$43$	− 62.7639i	− 1.45963i	−0.683647	−	0.729813i	$-0.739607\pi$
$43$	− 62.7639i	− 1.45963i	0.683647	−	0.729813i	$-0.260393\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	63.4853i	1.35075i	0.737474	+	0.675375i	$0.236018\pi$
$47$	63.4853i	1.35075i	−0.737474	+	0.675375i	$0.763982\pi$
$48$	0	0
$49$	47.4721	0.968819
$50$	0	0
$51$	0	0
$52$	0	0
$53$	98.1378	1.85166	0.925828	−	0.377945i	$-0.123369\pi$
$53$	98.1378	1.85166	0.925828	+	0.377945i	$0.123369\pi$
$54$	0	0
$55$	25.5279i	0.464143i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 19.2786i	− 0.326757i	−0.986563	−	0.163378i	$-0.947761\pi$
$59$	− 19.2786i	− 0.326757i	0.986563	−	0.163378i	$-0.0522391\pi$
$60$	0	0
$61$	1.19350	0.0195655	0.00978275	−	0.999952i	$-0.496886\pi$
$61$	1.19350	0.0195655	0.00978275	+	0.999952i	$0.496886\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−12.1115	−0.186330
$66$	0	0
$67$	− 5.01316i	− 0.0748232i	−0.999300	−	0.0374116i	$-0.988089\pi$
$67$	− 5.01316i	− 0.0748232i	0.999300	−	0.0374116i	$-0.0119113\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	84.3607i	1.18818i	0.804399	+	0.594089i	$0.202487\pi$
$71$	84.3607i	1.18818i	−0.804399	+	0.594089i	$0.797513\pi$
$72$	0	0
$73$	−70.7214	−0.968786	−0.484393	−	0.874851i	$-0.660959\pi$
$73$	−70.7214	−0.968786	−0.484393	+	0.874851i	$0.660959\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	14.1115	0.183266
$78$	0	0
$79$	124.498i	1.57593i	0.615720	+	0.787965i	$0.288865\pi$
$79$	124.498i	1.57593i	−0.615720	+	0.787965i	$0.711135\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	160.652i	1.93557i	0.251774	+	0.967786i	$0.418986\pi$
$83$	160.652i	1.93557i	−0.251774	+	0.967786i	$0.581014\pi$
$84$	0	0
$85$	15.5279	0.182681
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−46.2229	−0.519359	−0.259679	−	0.965695i	$-0.583617\pi$
$89$	−46.2229	−0.519359	−0.259679	+	0.965695i	$0.583617\pi$
$90$	0	0
$91$	6.69505i	0.0735720i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 66.8328i	− 0.703503i
$96$	0	0
$97$	133.331	1.37455	0.687275	−	0.726398i	$-0.258807\pi$
$97$	133.331	1.37455	0.687275	+	0.726398i	$0.258807\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	49.7771	0.492842	0.246421	−	0.969163i	$-0.420745\pi$
$101$	49.7771	0.492842	0.246421	+	0.969163i	$0.420745\pi$
$102$	0	0
$103$	196.705i	1.90976i	0.296995	+	0.954879i	$0.404016\pi$
$103$	196.705i	1.90976i	−0.296995	+	0.954879i	$0.595984\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	14.9017i	0.139268i	0.997573	+	0.0696341i	$0.0221832\pi$
$107$	14.9017i	0.139268i	−0.997573	+	0.0696341i	$0.977817\pi$
$108$	0	0
$109$	174.859	1.60421	0.802106	−	0.597182i	$-0.203713\pi$
$109$	174.859	1.60421	0.802106	+	0.597182i	$0.203713\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	33.0557	0.292529	0.146264	−	0.989246i	$-0.453275\pi$
$113$	33.0557	0.292529	0.146264	+	0.989246i	$0.453275\pi$
$114$	0	0
$115$	− 42.7639i	− 0.371860i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 8.58359i	− 0.0721310i
$120$	0	0
$121$	−9.33437	−0.0771435
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−11.1803	−0.0894427
$126$	0	0
$127$	93.7345i	0.738067i	0.929416	+	0.369034i	$0.120311\pi$
$127$	93.7345i	0.738067i	−0.929416	+	0.369034i	$0.879689\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 80.8065i	− 0.616844i	−0.951250	−	0.308422i	$-0.900199\pi$
$131$	− 80.8065i	− 0.616844i	0.951250	−	0.308422i	$-0.0998008\pi$
$132$	0	0
$133$	−36.9443	−0.277776
$134$	0	0
$135$	0	0
$136$	0	0
$137$	192.164	1.40266	0.701329	−	0.712838i	$-0.252591\pi$
$137$	192.164	1.40266	0.701329	+	0.712838i	$0.252591\pi$
$138$	0	0
$139$	− 6.60990i	− 0.0475533i	−0.999717	−	0.0237766i	$-0.992431\pi$
$139$	− 6.60990i	− 0.0475533i	0.999717	−	0.0237766i	$-0.00756905\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 61.8359i	− 0.432419i
$144$	0	0
$145$	47.0820	0.324704
$146$	0	0
$147$	0	0
$148$	0	0
$149$	199.803	1.34096	0.670481	−	0.741927i	$-0.266088\pi$
$149$	199.803	1.34096	0.670481	+	0.741927i	$0.266088\pi$
$150$	0	0
$151$	− 53.2523i	− 0.352664i	−0.984331	−	0.176332i	$-0.943577\pi$
$151$	− 53.2523i	− 0.352664i	0.984331	−	0.176332i	$-0.0564233\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	77.0820i	0.497303i
$156$	0	0
$157$	43.8034	0.279003	0.139501	−	0.990222i	$-0.455450\pi$
$157$	43.8034	0.279003	0.139501	+	0.990222i	$0.455450\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−23.6393	−0.146828
$162$	0	0
$163$	− 188.705i	− 1.15770i	−0.815434	−	0.578850i	$-0.803502\pi$
$163$	− 188.705i	− 1.15770i	0.815434	−	0.578850i	$-0.196498\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	260.705i	1.56111i	0.625088	+	0.780554i	$0.285063\pi$
$167$	260.705i	1.56111i	−0.625088	+	0.780554i	$0.714937\pi$
$168$	0	0
$169$	−139.663	−0.826405
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−28.6950	−0.165867	−0.0829337	−	0.996555i	$-0.526429\pi$
$173$	−28.6950	−0.165867	−0.0829337	+	0.996555i	$0.526429\pi$
$174$	0	0
$175$	6.18034i	0.0353162i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	95.5016i	0.533528i	0.963762	+	0.266764i	$0.0859545\pi$
$179$	95.5016i	0.533528i	−0.963762	+	0.266764i	$0.914046\pi$
$180$	0	0
$181$	−176.885	−0.977268	−0.488634	−	0.872489i	$-0.662505\pi$
$181$	−176.885	−0.977268	−0.488634	+	0.872489i	$0.662505\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	43.1672	0.233336
$186$	0	0
$187$	79.2786i	0.423950i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	231.967i	1.21449i	0.794515	+	0.607245i	$0.207725\pi$
$191$	231.967i	1.21449i	−0.794515	+	0.607245i	$0.792275\pi$
$192$	0	0
$193$	168.387	0.872471	0.436236	−	0.899832i	$-0.356311\pi$
$193$	168.387	0.872471	0.436236	+	0.899832i	$0.356311\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	31.0820	0.157777	0.0788884	−	0.996883i	$-0.474863\pi$
$197$	31.0820	0.157777	0.0788884	+	0.996883i	$0.474863\pi$
$198$	0	0
$199$	− 123.777i	− 0.621995i	−0.950411	−	0.310998i	$-0.899337\pi$
$199$	− 123.777i	− 0.621995i	0.950411	−	0.310998i	$-0.100663\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 26.0263i	− 0.128208i
$204$	0	0
$205$	−130.000	−0.634146
$206$	0	0
$207$	0	0
$208$	0	0
$209$	341.220	1.63263
$210$	0	0
$211$	34.2492i	0.162319i	0.996701	+	0.0811593i	$0.0258623\pi$
$211$	34.2492i	0.162319i	−0.996701	+	0.0811593i	$0.974138\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	140.344i	0.652765i
$216$	0	0
$217$	42.6099	0.196359
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−37.6130	−0.170195
$222$	0	0
$223$	202.987i	0.910255i	0.890426	+	0.455127i	$0.150406\pi$
$223$	202.987i	0.910255i	−0.890426	+	0.455127i	$0.849594\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 232.207i	− 1.02294i	−0.859302	−	0.511468i	$-0.829102\pi$
$227$	− 232.207i	− 1.02294i	0.859302	−	0.511468i	$-0.170898\pi$
$228$	0	0
$229$	−163.390	−0.713494	−0.356747	−	0.934201i	$-0.616114\pi$
$229$	−163.390	−0.713494	−0.356747	+	0.934201i	$0.616114\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	11.2198	0.0481537	0.0240768	−	0.999710i	$-0.492335\pi$
$233$	11.2198	0.0481537	0.0240768	+	0.999710i	$0.492335\pi$
$234$	0	0
$235$	− 141.957i	− 0.604074i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	216.721i	0.906784i	0.891311	+	0.453392i	$0.149786\pi$
$239$	216.721i	0.906784i	−0.891311	+	0.453392i	$0.850214\pi$
$240$	0	0
$241$	−49.1409	−0.203904	−0.101952	−	0.994789i	$-0.532509\pi$
$241$	−49.1409	−0.203904	−0.101952	+	0.994789i	$0.532509\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−106.151	−0.433269
$246$	0	0
$247$	161.889i	0.655419i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 282.525i	− 1.12560i	−0.826594	−	0.562798i	$-0.809725\pi$
$251$	− 282.525i	− 1.12560i	0.826594	−	0.562798i	$-0.190275\pi$
$252$	0	0
$253$	218.334	0.862982
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−199.941	−0.777981	−0.388991	−	0.921242i	$-0.627176\pi$
$257$	−199.941	−0.777981	−0.388991	+	0.921242i	$0.627176\pi$
$258$	0	0
$259$	− 23.8622i	− 0.0921322i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 247.761i	− 0.942056i	−0.882118	−	0.471028i	$-0.843883\pi$
$263$	− 247.761i	− 0.942056i	0.882118	−	0.471028i	$-0.156117\pi$
$264$	0	0
$265$	−219.443	−0.828086
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−13.4164	−0.0498751	−0.0249376	−	0.999689i	$-0.507939\pi$
$269$	−13.4164	−0.0498751	−0.0249376	+	0.999689i	$0.507939\pi$
$270$	0	0
$271$	59.4690i	0.219443i	0.993962	+	0.109721i	$0.0349959\pi$
$271$	59.4690i	0.219443i	−0.993962	+	0.109721i	$0.965004\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 57.0820i	− 0.207571i
$276$	0	0
$277$	510.859	1.84426	0.922128	−	0.386884i	$-0.126449\pi$
$277$	510.859	1.84426	0.922128	+	0.386884i	$0.126449\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−489.299	−1.74128	−0.870638	−	0.491924i	$-0.836294\pi$
$281$	−489.299	−1.74128	−0.870638	+	0.491924i	$0.836294\pi$
$282$	0	0
$283$	210.233i	0.742873i	0.928458	+	0.371436i	$0.121135\pi$
$283$	210.233i	0.742873i	−0.928458	+	0.371436i	$0.878865\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	71.8622i	0.250391i
$288$	0	0
$289$	−240.777	−0.833139
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−208.472	−0.711509	−0.355754	−	0.934579i	$-0.615776\pi$
$293$	−208.472	−0.711509	−0.355754	+	0.934579i	$0.615776\pi$
$294$	0	0
$295$	43.1084i	0.146130i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	103.587i	0.346444i
$300$	0	0
$301$	77.5805	0.257742
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−2.66874	−0.00874996
$306$	0	0
$307$	− 229.564i	− 0.747766i	−0.927476	−	0.373883i	$-0.878026\pi$
$307$	− 229.564i	− 0.747766i	0.927476	−	0.373883i	$-0.121974\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 351.692i	− 1.13084i	−0.824802	−	0.565421i	$-0.808714\pi$
$311$	− 351.692i	− 1.13084i	0.824802	−	0.565421i	$-0.191286\pi$
$312$	0	0
$313$	418.551	1.33722	0.668612	−	0.743612i	$-0.266889\pi$
$313$	418.551	1.33722	0.668612	+	0.743612i	$0.266889\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	351.082	1.10751	0.553757	−	0.832678i	$-0.313194\pi$
$317$	351.082	1.10751	0.553757	+	0.832678i	$0.313194\pi$
$318$	0	0
$319$	240.381i	0.753545i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 207.554i	− 0.642583i
$324$	0	0
$325$	27.0820	0.0833294
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−78.4721	−0.238517
$330$	0	0
$331$	− 630.853i	− 1.90590i	−0.303127	−	0.952950i	$-0.598031\pi$
$331$	− 630.853i	− 1.90590i	0.303127	−	0.952950i	$-0.401969\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	11.2098i	0.0334620i
$336$	0	0
$337$	284.780	0.845045	0.422523	−	0.906352i	$-0.361145\pi$
$337$	284.780	0.845045	0.422523	+	0.906352i	$0.361145\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−393.548	−1.15410
$342$	0	0
$343$	119.246i	0.347656i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	118.902i	0.342656i	0.985214	+	0.171328i	$0.0548058\pi$
$347$	118.902i	0.342656i	−0.985214	+	0.171328i	$0.945194\pi$
$348$	0	0
$349$	−19.0495	−0.0545831	−0.0272916	−	0.999628i	$-0.508688\pi$
$349$	−19.0495	−0.0545831	−0.0272916	+	0.999628i	$0.508688\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	96.3344	0.272902	0.136451	−	0.990647i	$-0.456430\pi$
$353$	96.3344	0.272902	0.136451	+	0.990647i	$0.456430\pi$
$354$	0	0
$355$	− 188.636i	− 0.531370i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 15.0031i	− 0.0417914i	−0.999782	−	0.0208957i	$-0.993348\pi$
$359$	− 15.0031i	− 0.0417914i	0.999782	−	0.0208957i	$-0.00665179\pi$
$360$	0	0
$361$	−532.325	−1.47458
$362$	0	0
$363$	0	0
$364$	0	0
$365$	158.138	0.433254
$366$	0	0
$367$	439.348i	1.19713i	0.801073	+	0.598566i	$0.204263\pi$
$367$	439.348i	1.19713i	−0.801073	+	0.598566i	$0.795737\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	121.305i	0.326968i
$372$	0	0
$373$	−346.689	−0.929461	−0.464730	−	0.885452i	$-0.653849\pi$
$373$	−346.689	−0.929461	−0.464730	+	0.885452i	$0.653849\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−114.046	−0.302510
$378$	0	0
$379$	− 338.768i	− 0.893846i	−0.894572	−	0.446923i	$-0.852520\pi$
$379$	− 338.768i	− 0.893846i	0.894572	−	0.446923i	$-0.147480\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	142.351i	0.371673i	0.982581	+	0.185836i	$0.0594994\pi$
$383$	142.351i	0.371673i	−0.982581	+	0.185836i	$0.940501\pi$
$384$	0	0
$385$	−31.5542	−0.0819589
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−91.6331	−0.235561	−0.117780	−	0.993040i	$-0.537578\pi$
$389$	−91.6331	−0.235561	−0.117780	+	0.993040i	$0.537578\pi$
$390$	0	0
$391$	− 132.807i	− 0.339659i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 278.387i	− 0.704777i
$396$	0	0
$397$	−384.354	−0.968147	−0.484074	−	0.875027i	$-0.660843\pi$
$397$	−384.354	−0.968147	−0.484074	+	0.875027i	$0.660843\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	416.545	1.03877	0.519383	−	0.854542i	$-0.326162\pi$
$401$	416.545	1.03877	0.519383	+	0.854542i	$0.326162\pi$
$402$	0	0
$403$	− 186.715i	− 0.463313i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	220.393i	0.541507i
$408$	0	0
$409$	764.354	1.86884	0.934419	−	0.356177i	$-0.115920\pi$
$409$	764.354	1.86884	0.934419	+	0.356177i	$0.115920\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	23.8297	0.0576991
$414$	0	0
$415$	− 359.230i	− 0.865614i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	117.613i	0.280699i	0.990102	+	0.140350i	$0.0448227\pi$
$419$	117.613i	0.280699i	−0.990102	+	0.140350i	$0.955177\pi$
$420$	0	0
$421$	−429.915	−1.02118	−0.510588	−	0.859826i	$-0.670572\pi$
$421$	−429.915	−1.02118	−0.510588	+	0.859826i	$0.670572\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−34.7214	−0.0816973
$426$	0	0
$427$	1.47524i	0.00345490i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 596.466i	− 1.38391i	−0.721940	−	0.691956i	$-0.756749\pi$
$431$	− 596.466i	− 1.38391i	0.721940	−	0.691956i	$-0.243251\pi$
$432$	0	0
$433$	−678.551	−1.56709	−0.783546	−	0.621333i	$-0.786591\pi$
$433$	−678.551	−1.56709	−0.783546	+	0.621333i	$0.786591\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−571.607	−1.30802
$438$	0	0
$439$	324.774i	0.739804i	0.929071	+	0.369902i	$0.120609\pi$
$439$	324.774i	0.739804i	−0.929071	+	0.369902i	$0.879391\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 49.4064i	− 0.111527i	−0.998444	−	0.0557634i	$-0.982241\pi$
$443$	− 49.4064i	− 0.111527i	0.998444	−	0.0557634i	$-0.0177592\pi$
$444$	0	0
$445$	103.358	0.232264
$446$	0	0
$447$	0	0
$448$	0	0
$449$	186.859	0.416167	0.208084	−	0.978111i	$-0.433277\pi$
$449$	186.859	0.416167	0.208084	+	0.978111i	$0.433277\pi$
$450$	0	0
$451$	− 663.724i	− 1.47167i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 14.9706i	− 0.0329024i
$456$	0	0
$457$	−645.214	−1.41185	−0.705923	−	0.708289i	$-0.749468\pi$
$457$	−645.214	−1.41185	−0.705923	+	0.708289i	$0.749468\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−10.7864	−0.0233978	−0.0116989	−	0.999932i	$-0.503724\pi$
$461$	−10.7864	−0.0233978	−0.0116989	+	0.999932i	$0.503724\pi$
$462$	0	0
$463$	− 594.233i	− 1.28344i	−0.766939	−	0.641720i	$-0.778221\pi$
$463$	− 594.233i	− 1.28344i	0.766939	−	0.641720i	$-0.221779\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 246.148i	− 0.527083i	−0.964648	−	0.263542i	$-0.915109\pi$
$467$	− 246.148i	− 0.527083i	0.964648	−	0.263542i	$-0.0848906\pi$
$468$	0	0
$469$	6.19660	0.0132124
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−716.539	−1.51488
$474$	0	0
$475$	149.443i	0.314616i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 722.610i	− 1.50858i	−0.656541	−	0.754290i	$-0.727981\pi$
$479$	− 722.610i	− 1.50858i	0.656541	−	0.754290i	$-0.272019\pi$
$480$	0	0
$481$	−104.563	−0.217388
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−298.138	−0.614717
$486$	0	0
$487$	482.403i	0.990561i	0.868733	+	0.495281i	$0.164935\pi$
$487$	482.403i	0.990561i	−0.868733	+	0.495281i	$0.835065\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 235.967i	− 0.480585i	−0.970700	−	0.240293i	$-0.922757\pi$
$491$	− 235.967i	− 0.480585i	0.970700	−	0.240293i	$-0.0772434\pi$
$492$	0	0
$493$	146.217	0.296586
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−104.276	−0.209810
$498$	0	0
$499$	591.331i	1.18503i	0.805558	+	0.592516i	$0.201865\pi$
$499$	591.331i	1.18503i	−0.805558	+	0.592516i	$0.798135\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	708.843i	1.40923i	0.709590	+	0.704615i	$0.248880\pi$
$503$	708.843i	1.40923i	−0.709590	+	0.704615i	$0.751120\pi$
$504$	0	0
$505$	−111.305	−0.220406
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−801.489	−1.57463	−0.787317	−	0.616548i	$-0.788531\pi$
$509$	−801.489	−1.57463	−0.787317	+	0.616548i	$0.788531\pi$
$510$	0	0
$511$	− 87.4164i	− 0.171069i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 439.846i	− 0.854070i
$516$	0	0
$517$	724.774	1.40188
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−285.882	−0.548718	−0.274359	−	0.961627i	$-0.588466\pi$
$521$	−285.882	−0.548718	−0.274359	+	0.961627i	$0.588466\pi$
$522$	0	0
$523$	493.013i	0.942664i	0.881956	+	0.471332i	$0.156227\pi$
$523$	493.013i	0.942664i	−0.881956	+	0.471332i	$0.843773\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	239.384i	0.454239i
$528$	0	0
$529$	163.249	0.308600
$530$	0	0
$531$	0	0
$532$	0	0
$533$	314.898	0.590803
$534$	0	0
$535$	− 33.3212i	− 0.0622826i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 541.961i	− 1.00549i
$540$	0	0
$541$	−906.774	−1.67611	−0.838054	−	0.545588i	$-0.816306\pi$
$541$	−906.774	−1.67611	−0.838054	+	0.545588i	$0.816306\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−390.997	−0.717425
$546$	0	0
$547$	111.420i	0.203693i	0.994800	+	0.101847i	$0.0324751\pi$
$547$	111.420i	0.203693i	−0.994800	+	0.101847i	$0.967525\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 629.325i	− 1.14215i
$552$	0	0
$553$	−153.889	−0.278279
$554$	0	0
$555$	0	0
$556$	0	0
$557$	462.255	0.829902	0.414951	−	0.909844i	$-0.363799\pi$
$557$	462.255	0.829902	0.414951	+	0.909844i	$0.363799\pi$
$558$	0	0
$559$	− 339.955i	− 0.608149i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 640.555i	− 1.13775i	−0.822423	−	0.568876i	$-0.807378\pi$
$563$	− 640.555i	− 1.13775i	0.822423	−	0.568876i	$-0.192622\pi$
$564$	0	0
$565$	−73.9149	−0.130823
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−817.915	−1.43746	−0.718730	−	0.695289i	$-0.755276\pi$
$569$	−817.915	−1.43746	−0.718730	+	0.695289i	$0.755276\pi$
$570$	0	0
$571$	− 470.642i	− 0.824242i	−0.911129	−	0.412121i	$-0.864788\pi$
$571$	− 470.642i	− 0.824242i	0.911129	−	0.412121i	$-0.135212\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	95.6231i	0.166301i
$576$	0	0
$577$	1051.17	1.82178	0.910890	−	0.412649i	$-0.135396\pi$
$577$	1051.17	1.82178	0.910890	+	0.412649i	$0.135396\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−198.577	−0.341786
$582$	0	0
$583$	− 1120.38i	− 1.92175i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 267.708i	− 0.456062i	−0.973654	−	0.228031i	$-0.926771\pi$
$587$	− 267.708i	− 0.456062i	0.973654	−	0.228031i	$-0.0732287\pi$
$588$	0	0
$589$	1030.32	1.74927
$590$	0	0
$591$	0	0
$592$	0	0
$593$	168.440	0.284047	0.142023	−	0.989863i	$-0.454639\pi$
$593$	168.440	0.284047	0.142023	+	0.989863i	$0.454639\pi$
$594$	0	0
$595$	19.1935i	0.0322580i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 626.715i	− 1.04627i	−0.852250	−	0.523135i	$-0.824763\pi$
$599$	− 626.715i	− 1.04627i	0.852250	−	0.523135i	$-0.175237\pi$
$600$	0	0
$601$	−359.252	−0.597758	−0.298879	−	0.954291i	$-0.596613\pi$
$601$	−359.252	−0.597758	−0.298879	+	0.954291i	$0.596613\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	20.8723	0.0344996
$606$	0	0
$607$	679.728i	1.11982i	0.828555	+	0.559908i	$0.189164\pi$
$607$	679.728i	1.11982i	−0.828555	+	0.559908i	$0.810836\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	343.862i	0.562786i
$612$	0	0
$613$	−748.958	−1.22179	−0.610896	−	0.791711i	$-0.709191\pi$
$613$	−748.958	−1.22179	−0.610896	+	0.791711i	$0.709191\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	947.876	1.53627	0.768133	−	0.640290i	$-0.221186\pi$
$617$	947.876	1.53627	0.768133	+	0.640290i	$0.221186\pi$
$618$	0	0
$619$	1074.32i	1.73558i	0.496934	+	0.867788i	$0.334459\pi$
$619$	1074.32i	1.73558i	−0.496934	+	0.867788i	$0.665541\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 57.1347i	− 0.0917089i
$624$	0	0
$625$	25.0000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	134.059	0.213130
$630$	0	0
$631$	− 1061.67i	− 1.68253i	−0.540627	−	0.841263i	$-0.681813\pi$
$631$	− 1061.67i	− 1.68253i	0.540627	−	0.841263i	$-0.318187\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 209.597i	− 0.330074i
$636$	0	0
$637$	257.128	0.403655
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−497.404	−0.775981	−0.387991	−	0.921663i	$-0.626831\pi$
$641$	−497.404	−0.775981	−0.387991	+	0.921663i	$0.626831\pi$
$642$	0	0
$643$	604.397i	0.939964i	0.882676	+	0.469982i	$0.155740\pi$
$643$	604.397i	0.939964i	−0.882676	+	0.469982i	$0.844260\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	693.151i	1.07133i	0.844430	+	0.535665i	$0.179939\pi$
$647$	693.151i	1.07133i	−0.844430	+	0.535665i	$0.820061\pi$
$648$	0	0
$649$	−220.093	−0.339126
$650$	0	0
$651$	0	0
$652$	0	0
$653$	948.853	1.45307	0.726534	−	0.687131i	$-0.241130\pi$
$653$	948.853	1.45307	0.726534	+	0.687131i	$0.241130\pi$
$654$	0	0
$655$	180.689i	0.275861i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	889.260i	1.34941i	0.738088	+	0.674704i	$0.235729\pi$
$659$	889.260i	1.34941i	−0.738088	+	0.674704i	$0.764271\pi$
$660$	0	0
$661$	−664.354	−1.00507	−0.502537	−	0.864555i	$-0.667600\pi$
$661$	−664.354	−1.00507	−0.502537	+	0.864555i	$0.667600\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	82.6099	0.124225
$666$	0	0
$667$	− 402.683i	− 0.603722i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 13.6254i	− 0.0203062i
$672$	0	0
$673$	429.437	0.638093	0.319046	−	0.947739i	$-0.396637\pi$
$673$	429.437	0.638093	0.319046	+	0.947739i	$0.396637\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−336.918	−0.497663	−0.248832	−	0.968547i	$-0.580047\pi$
$677$	−336.918	−0.497663	−0.248832	+	0.968547i	$0.580047\pi$
$678$	0	0
$679$	164.807i	0.242719i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	818.200i	1.19795i	0.800767	+	0.598975i	$0.204425\pi$
$683$	818.200i	1.19795i	−0.800767	+	0.598975i	$0.795575\pi$
$684$	0	0
$685$	−429.692	−0.627288
$686$	0	0
$687$	0	0
$688$	0	0
$689$	531.554	0.771486
$690$	0	0
$691$	− 356.689i	− 0.516192i	−0.966119	−	0.258096i	$-0.916905\pi$
$691$	− 356.689i	− 0.516192i	0.966119	−	0.258096i	$-0.0830951\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	14.7802i	0.0212665i
$696$	0	0
$697$	−403.724	−0.579232
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−50.0851	−0.0714481	−0.0357241	−	0.999362i	$-0.511374\pi$
$701$	−50.0851	−0.0714481	−0.0357241	+	0.999362i	$0.511374\pi$
$702$	0	0
$703$	− 576.997i	− 0.820764i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	61.5279i	0.0870267i
$708$	0	0
$709$	−698.604	−0.985337	−0.492668	−	0.870217i	$-0.663978\pi$
$709$	−698.604	−0.985337	−0.492668	+	0.870217i	$0.663978\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	659.266	0.924637
$714$	0	0
$715$	138.269i	0.193384i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 246.387i	− 0.342680i	−0.985212	−	0.171340i	$-0.945190\pi$
$719$	− 246.387i	− 0.342680i	0.985212	−	0.171340i	$-0.0548097\pi$
$720$	0	0
$721$	−243.141	−0.337227
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−105.279	−0.145212
$726$	0	0
$727$	256.134i	0.352316i	0.984362	+	0.176158i	$0.0563670\pi$
$727$	256.134i	0.352316i	−0.984362	+	0.176158i	$0.943633\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	435.850i	0.596238i
$732$	0	0
$733$	897.574	1.22452	0.612261	−	0.790656i	$-0.290260\pi$
$733$	897.574	1.22452	0.612261	+	0.790656i	$0.290260\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−57.2322	−0.0776557
$738$	0	0
$739$	− 268.486i	− 0.363310i	−0.983362	−	0.181655i	$-0.941855\pi$
$739$	− 268.486i	− 0.363310i	0.983362	−	0.181655i	$-0.0581454\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 612.049i	− 0.823753i	−0.911240	−	0.411877i	$-0.864873\pi$
$743$	− 612.049i	− 0.823753i	0.911240	−	0.411877i	$-0.135127\pi$
$744$	0	0
$745$	−446.774	−0.599697
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−18.4195	−0.0245921
$750$	0	0
$751$	− 142.485i	− 0.189726i	−0.995490	−	0.0948632i	$-0.969759\pi$
$751$	− 142.485i	− 0.189726i	0.995490	−	0.0948632i	$-0.0302414\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	119.076i	0.157716i
$756$	0	0
$757$	−636.342	−0.840610	−0.420305	−	0.907383i	$-0.638077\pi$
$757$	−636.342	−0.840610	−0.420305	+	0.907383i	$0.638077\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−591.666	−0.777484	−0.388742	−	0.921347i	$-0.627090\pi$
$761$	−591.666	−0.777484	−0.388742	+	0.921347i	$0.627090\pi$
$762$	0	0
$763$	216.138i	0.283274i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 104.421i	− 0.136142i
$768$	0	0
$769$	0.216701	0.000281796 0	0.000140898	−	1.00000i	$-0.499955\pi$
$769$	0.216701	0.000281796 0	0.000140898		1.00000i	$0.499955\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−762.807	−0.986813	−0.493407	−	0.869799i	$-0.664249\pi$
$773$	−762.807	−0.986813	−0.493407	+	0.869799i	$0.664249\pi$
$774$	0	0
$775$	− 172.361i	− 0.222401i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1737.65i	2.23062i
$780$	0	0
$781$	963.096	1.23316
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−97.9474	−0.124774
$786$	0	0
$787$	1311.80i	1.66684i	0.552642	+	0.833419i	$0.313620\pi$
$787$	1311.80i	1.66684i	−0.552642	+	0.833419i	$0.686380\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	40.8591i	0.0516550i
$792$	0	0
$793$	6.46446	0.00815190
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1341.85	1.68363	0.841813	−	0.539769i	$-0.181489\pi$
$797$	1341.85	1.68363	0.841813	+	0.539769i	$0.181489\pi$
$798$	0	0
$799$	− 440.859i	− 0.551764i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	807.384i	1.00546i
$804$	0	0
$805$	52.8591	0.0656635
$806$	0	0
$807$	0	0
$808$	0	0
$809$	53.8948	0.0666190	0.0333095	−	0.999445i	$-0.489395\pi$
$809$	53.8948	0.0666190	0.0333095	+	0.999445i	$0.489395\pi$
$810$	0	0
$811$	20.9644i	0.0258500i	0.999916	+	0.0129250i	$0.00411427\pi$
$811$	20.9644i	0.0258500i	−0.999916	+	0.0129250i	$0.995886\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	421.957i	0.517739i
$816$	0	0
$817$	1875.92	2.29611
$818$	0	0
$819$	0	0
$820$	0	0
$821$	363.240	0.442436	0.221218	−	0.975224i	$-0.428997\pi$
$821$	363.240	0.442436	0.221218	+	0.975224i	$0.428997\pi$
$822$	0	0
$823$	1487.42i	1.80732i	0.428256	+	0.903658i	$0.359128\pi$
$823$	1487.42i	1.80732i	−0.428256	+	0.903658i	$0.640872\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 948.981i	− 1.14750i	−0.819031	−	0.573749i	$-0.805489\pi$
$827$	− 948.981i	− 1.14750i	0.819031	−	0.573749i	$-0.194511\pi$
$828$	0	0
$829$	1114.06	1.34386	0.671930	−	0.740614i	$-0.265465\pi$
$829$	1114.06	1.34386	0.671930	+	0.740614i	$0.265465\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−329.659	−0.395750
$834$	0	0
$835$	− 582.954i	− 0.698149i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 145.272i	− 0.173149i	−0.996245	−	0.0865747i	$-0.972408\pi$
$839$	− 145.272i	− 0.173149i	0.996245	−	0.0865747i	$-0.0275921\pi$
$840$	0	0
$841$	−397.656	−0.472837
$842$	0	0
$843$	0	0
$844$	0	0
$845$	312.295	0.369580
$846$	0	0
$847$	− 11.5379i	− 0.0136221i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 369.200i	− 0.433842i
$852$	0	0
$853$	−127.423	−0.149382	−0.0746909	−	0.997207i	$-0.523797\pi$
$853$	−127.423	−0.149382	−0.0746909	+	0.997207i	$0.523797\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	576.715	0.672946	0.336473	−	0.941693i	$-0.390766\pi$
$857$	576.715	0.672946	0.336473	+	0.941693i	$0.390766\pi$
$858$	0	0
$859$	513.155i	0.597386i	0.954349	+	0.298693i	$0.0965507\pi$
$859$	513.155i	0.597386i	−0.954349	+	0.298693i	$0.903449\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 774.488i	− 0.897437i	−0.893673	−	0.448719i	$-0.851881\pi$
$863$	− 774.488i	− 0.897437i	0.893673	−	0.448719i	$-0.148119\pi$
$864$	0	0
$865$	64.1641	0.0741781
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1421.33	1.63559
$870$	0	0
$871$	− 27.1533i	− 0.0311749i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 13.8197i	− 0.0157939i
$876$	0	0
$877$	1335.13	1.52239	0.761194	−	0.648524i	$-0.224613\pi$
$877$	1335.13	1.52239	0.761194	+	0.648524i	$0.224613\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−29.7570	−0.0337764	−0.0168882	−	0.999857i	$-0.505376\pi$
$881$	−29.7570	−0.0337764	−0.0168882	+	0.999857i	$0.505376\pi$
$882$	0	0
$883$	− 105.544i	− 0.119529i	−0.998213	−	0.0597645i	$-0.980965\pi$
$883$	− 105.544i	− 0.119529i	0.998213	−	0.0597645i	$-0.0190350\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 60.2268i	− 0.0678994i	−0.999424	−	0.0339497i	$-0.989191\pi$
$887$	− 60.2268i	− 0.0678994i	0.999424	−	0.0339497i	$-0.0108086\pi$
$888$	0	0
$889$	−115.862	−0.130329
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−1897.48	−2.12484
$894$	0	0
$895$	− 213.548i	− 0.238601i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	725.836i	0.807381i
$900$	0	0
$901$	−681.495	−0.756377
$902$	0	0
$903$	0	0
$904$	0	0
$905$	395.528	0.437047
$906$	0	0
$907$	155.473i	0.171414i	0.996320	+	0.0857072i	$0.0273150\pi$
$907$	155.473i	0.171414i	−0.996320	+	0.0857072i	$0.972685\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 686.630i	− 0.753710i	−0.926272	−	0.376855i	$-0.877005\pi$
$911$	− 686.630i	− 0.753710i	0.926272	−	0.376855i	$-0.122995\pi$
$912$	0	0
$913$	1834.07	2.00884
$914$	0	0
$915$	0	0
$916$	0	0
$917$	99.8823	0.108923
$918$	0	0
$919$	− 1339.16i	− 1.45719i	−0.684943	−	0.728597i	$-0.740173\pi$
$919$	− 1339.16i	− 1.45719i	0.684943	−	0.728597i	$-0.259827\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	456.932i	0.495051i
$924$	0	0
$925$	−96.5248	−0.104351
$926$	0	0
$927$	0	0
$928$	0	0
$929$	636.302	0.684932	0.342466	−	0.939530i	$-0.388738\pi$
$929$	636.302	0.684932	0.342466	+	0.939530i	$0.388738\pi$
$930$	0	0
$931$	1418.87i	1.52403i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 177.272i	− 0.189596i
$936$	0	0
$937$	−310.210	−0.331068	−0.165534	−	0.986204i	$-0.552935\pi$
$937$	−310.210	−0.331068	−0.165534	+	0.986204i	$0.552935\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−1322.66	−1.40559	−0.702793	−	0.711394i	$-0.748064\pi$
$941$	−1322.66	−1.40559	−0.702793	+	0.711394i	$0.748064\pi$
$942$	0	0
$943$	1111.86i	1.17907i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	724.883i	0.765452i	0.923862	+	0.382726i	$0.125015\pi$
$947$	724.883i	0.765452i	−0.923862	+	0.382726i	$0.874985\pi$
$948$	0	0
$949$	−383.056	−0.403641
$950$	0	0
$951$	0	0
$952$	0	0
$953$	298.223	0.312931	0.156465	−	0.987683i	$-0.449990\pi$
$953$	298.223	0.312931	0.156465	+	0.987683i	$0.449990\pi$
$954$	0	0
$955$	− 518.695i	− 0.543136i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	237.528i	0.247683i
$960$	0	0
$961$	−227.328	−0.236554
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−376.525	−0.390181
$966$	0	0
$967$	249.236i	0.257742i	0.991661	+	0.128871i	$0.0411352\pi$
$967$	249.236i	0.257742i	−0.991661	+	0.128871i	$0.958865\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 339.246i	− 0.349378i	−0.984624	−	0.174689i	$-0.944108\pi$
$971$	− 339.246i	− 0.349378i	0.984624	−	0.174689i	$-0.0558920\pi$
$972$	0	0
$973$	8.17029	0.00839701
$974$	0	0
$975$	0	0
$976$	0	0
$977$	40.8854	0.0418479	0.0209240	−	0.999781i	$-0.493339\pi$
$977$	40.8854	0.0418479	0.0209240	+	0.999781i	$0.493339\pi$
$978$	0	0
$979$	527.700i	0.539019i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	5.56423i	0.00566045i	0.999996	+	0.00283023i	$0.000900890\pi$
$983$	5.56423i	0.00566045i	−0.999996	+	0.00283023i	$0.999099\pi$
$984$	0	0
$985$	−69.5016	−0.0705600
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1200.34	1.21369
$990$	0	0
$991$	− 1562.45i	− 1.57664i	−0.615267	−	0.788319i	$-0.710952\pi$
$991$	− 1562.45i	− 1.57664i	0.615267	−	0.788319i	$-0.289048\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	276.774i	0.278165i
$996$	0	0
$997$	−704.224	−0.706343	−0.353172	−	0.935559i	$-0.614897\pi$
$997$	−704.224	−0.706343	−0.353172	+	0.935559i	$0.614897\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1440.3.e.a.991.2		4
3.2	odd	2		160.3.b.b.31.3	yes	4
4.3	odd	2	inner	1440.3.e.a.991.1		4
8.3	odd	2		2880.3.e.h.2431.3		4
8.5	even	2		2880.3.e.h.2431.4		4
12.11	even	2		160.3.b.b.31.2	✓	4
15.2	even	4		800.3.h.e.799.4		4
15.8	even	4		800.3.h.h.799.2		4
15.14	odd	2		800.3.b.g.351.2		4
24.5	odd	2		320.3.b.d.191.2		4
24.11	even	2		320.3.b.d.191.3		4
48.5	odd	4		1280.3.g.b.1151.4		4
48.11	even	4		1280.3.g.c.1151.2		4
48.29	odd	4		1280.3.g.c.1151.1		4
48.35	even	4		1280.3.g.b.1151.3		4
60.23	odd	4		800.3.h.e.799.3		4
60.47	odd	4		800.3.h.h.799.1		4
60.59	even	2		800.3.b.g.351.3		4
120.29	odd	2		1600.3.b.u.1151.3		4
120.53	even	4		1600.3.h.f.1599.3		4
120.59	even	2		1600.3.b.u.1151.2		4
120.77	even	4		1600.3.h.k.1599.1		4
120.83	odd	4		1600.3.h.k.1599.2		4
120.107	odd	4		1600.3.h.f.1599.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.3.b.b.31.2	✓	4	12.11	even	2
160.3.b.b.31.3	yes	4	3.2	odd	2
320.3.b.d.191.2		4	24.5	odd	2
320.3.b.d.191.3		4	24.11	even	2
800.3.b.g.351.2		4	15.14	odd	2
800.3.b.g.351.3		4	60.59	even	2
800.3.h.e.799.3		4	60.23	odd	4
800.3.h.e.799.4		4	15.2	even	4
800.3.h.h.799.1		4	60.47	odd	4
800.3.h.h.799.2		4	15.8	even	4
1280.3.g.b.1151.3		4	48.35	even	4
1280.3.g.b.1151.4		4	48.5	odd	4
1280.3.g.c.1151.1		4	48.29	odd	4
1280.3.g.c.1151.2		4	48.11	even	4
1440.3.e.a.991.1		4	4.3	odd	2	inner
1440.3.e.a.991.2		4	1.1	even	1	trivial
1600.3.b.u.1151.2		4	120.59	even	2
1600.3.b.u.1151.3		4	120.29	odd	2
1600.3.h.f.1599.3		4	120.53	even	4
1600.3.h.f.1599.4		4	120.107	odd	4
1600.3.h.k.1599.1		4	120.77	even	4
1600.3.h.k.1599.2		4	120.83	odd	4
2880.3.e.h.2431.3		4	8.3	odd	2
2880.3.e.h.2431.4		4	8.5	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1440.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.23607 q^{5} +1.23607i q^{7} -11.4164i q^{11} +5.41641 q^{13} -6.94427 q^{17} +29.8885i q^{19} +19.1246i q^{23} +5.00000 q^{25} -21.0557 q^{29} -34.4721i q^{31} -2.76393i q^{35} -19.3050 q^{37} +58.1378 q^{41} -62.7639i q^{43} +63.4853i q^{47} +47.4721 q^{49} +98.1378 q^{53} +25.5279i q^{55} -19.2786i q^{59} +1.19350 q^{61} -12.1115 q^{65} -5.01316i q^{67} +84.3607i q^{71} -70.7214 q^{73} +14.1115 q^{77} +124.498i q^{79} +160.652i q^{83} +15.5279 q^{85} -46.2229 q^{89} +6.69505i q^{91} -66.8328i q^{95} +133.331 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 32 q^{13} + 8 q^{17} + 20 q^{25} - 120 q^{29} + 48 q^{37} + 172 q^{49} + 160 q^{53} - 192 q^{61} - 120 q^{65} - 104 q^{73} + 128 q^{77} + 80 q^{85} - 328 q^{89} + 104 q^{97}+O(q^{100}) \) 4 * q - 32 * q^13 + 8 * q^17 + 20 * q^25 - 120 * q^29 + 48 * q^37 + 172 * q^49 + 160 * q^53 - 192 * q^61 - 120 * q^65 - 104 * q^73 + 128 * q^77 + 80 * q^85 - 328 * q^89 + 104 * q^97

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