LMFDB - Embedded newform 1440.2.k.e.721.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1440,2,Mod(721,1440)]

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([0, 1, 0, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1440.721");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1440 = 2^{5} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1440.k (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:	$11.4984578911$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			721.3
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1440.721
Dual form			1440.2.k.e.721.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{5} -0.732051 q^{7} -2.00000i q^{11} -3.46410i q^{13} -3.46410 q^{17} -0.535898i q^{19} -6.19615 q^{23} -1.00000 q^{25} -6.92820i q^{29} +5.46410 q^{31} -0.732051i q^{35} -2.00000i q^{37} -1.46410 q^{41} +5.26795i q^{43} +3.26795 q^{47} -6.46410 q^{49} -11.4641i q^{53} +2.00000 q^{55} -7.46410i q^{59} -8.92820i q^{61} +3.46410 q^{65} -10.7321i q^{67} +5.46410 q^{71} +7.46410 q^{73} +1.46410i q^{77} +1.07180 q^{79} +1.26795i q^{83} -3.46410i q^{85} -8.92820 q^{89} +2.53590i q^{91} +0.535898 q^{95} -14.3923 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{7} - 4 q^{23} - 4 q^{25} + 8 q^{31} + 8 q^{41} + 20 q^{47} - 12 q^{49} + 8 q^{55} + 8 q^{71} + 16 q^{73} + 32 q^{79} - 8 q^{89} + 16 q^{95} - 16 q^{97}+O(q^{100}) $ 4 * q + 4 * q^7 - 4 * q^23 - 4 * q^25 + 8 * q^31 + 8 * q^41 + 20 * q^47 - 12 * q^49 + 8 * q^55 + 8 * q^71 + 16 * q^73 + 32 * q^79 - 8 * q^89 + 16 * q^95 - 16 * 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
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	0	0
$7$	−0.732051	−0.276689	−0.138345	−	0.990384i	$-0.544178\pi$
$7$	−0.732051	−0.276689	−0.138345	+	0.990384i	$0.544178\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 2.00000i	− 0.603023i	−0.953463	−	0.301511i	$-0.902509\pi$
$11$	− 2.00000i	− 0.603023i	0.953463	−	0.301511i	$-0.0974911\pi$
$12$	0	0
$13$	− 3.46410i	− 0.960769i	−0.877058	−	0.480384i	$-0.840497\pi$
$13$	− 3.46410i	− 0.960769i	0.877058	−	0.480384i	$-0.159503\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.46410	−0.840168	−0.420084	−	0.907485i	$-0.637999\pi$
$17$	−3.46410	−0.840168	−0.420084	+	0.907485i	$0.637999\pi$
$18$	0	0
$19$	− 0.535898i	− 0.122944i	−0.998109	−	0.0614718i	$-0.980421\pi$
$19$	− 0.535898i	− 0.122944i	0.998109	−	0.0614718i	$-0.0195794\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−6.19615	−1.29199	−0.645994	−	0.763343i	$-0.723557\pi$
$23$	−6.19615	−1.29199	−0.645994	+	0.763343i	$0.723557\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 6.92820i	− 1.28654i	−0.765641	−	0.643268i	$-0.777578\pi$
$29$	− 6.92820i	− 1.28654i	0.765641	−	0.643268i	$-0.222422\pi$
$30$	0	0
$31$	5.46410	0.981382	0.490691	−	0.871334i	$-0.336744\pi$
$31$	5.46410	0.981382	0.490691	+	0.871334i	$0.336744\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 0.732051i	− 0.123739i
$36$	0	0
$37$	− 2.00000i	− 0.328798i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$	− 2.00000i	− 0.328798i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−1.46410	−0.228654	−0.114327	−	0.993443i	$-0.536471\pi$
$41$	−1.46410	−0.228654	−0.114327	+	0.993443i	$0.536471\pi$
$42$	0	0
$43$	5.26795i	0.803355i	0.915781	+	0.401677i	$0.131573\pi$
$43$	5.26795i	0.803355i	−0.915781	+	0.401677i	$0.868427\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3.26795	0.476679	0.238340	−	0.971182i	$-0.423397\pi$
$47$	3.26795	0.476679	0.238340	+	0.971182i	$0.423397\pi$
$48$	0	0
$49$	−6.46410	−0.923443
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 11.4641i	− 1.57472i	−0.616496	−	0.787358i	$-0.711449\pi$
$53$	− 11.4641i	− 1.57472i	0.616496	−	0.787358i	$-0.288551\pi$
$54$	0	0
$55$	2.00000	0.269680
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 7.46410i	− 0.971743i	−0.874030	−	0.485872i	$-0.838502\pi$
$59$	− 7.46410i	− 0.971743i	0.874030	−	0.485872i	$-0.161498\pi$
$60$	0	0
$61$	− 8.92820i	− 1.14314i	−0.820554	−	0.571570i	$-0.806335\pi$
$61$	− 8.92820i	− 1.14314i	0.820554	−	0.571570i	$-0.193665\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	3.46410	0.429669
$66$	0	0
$67$	− 10.7321i	− 1.31113i	−0.755139	−	0.655564i	$-0.772431\pi$
$67$	− 10.7321i	− 1.31113i	0.755139	−	0.655564i	$-0.227569\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	5.46410	0.648470	0.324235	−	0.945977i	$-0.394893\pi$
$71$	5.46410	0.648470	0.324235	+	0.945977i	$0.394893\pi$
$72$	0	0
$73$	7.46410	0.873607	0.436804	−	0.899557i	$-0.356111\pi$
$73$	7.46410	0.873607	0.436804	+	0.899557i	$0.356111\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.46410i	0.166850i
$78$	0	0
$79$	1.07180	0.120587	0.0602933	−	0.998181i	$-0.480796\pi$
$79$	1.07180	0.120587	0.0602933	+	0.998181i	$0.480796\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1.26795i	0.139176i	0.997576	+	0.0695878i	$0.0221684\pi$
$83$	1.26795i	0.139176i	−0.997576	+	0.0695878i	$0.977832\pi$
$84$	0	0
$85$	− 3.46410i	− 0.375735i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−8.92820	−0.946388	−0.473194	−	0.880958i	$-0.656899\pi$
$89$	−8.92820	−0.946388	−0.473194	+	0.880958i	$0.656899\pi$
$90$	0	0
$91$	2.53590i	0.265834i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0.535898	0.0549820
$96$	0	0
$97$	−14.3923	−1.46132	−0.730659	−	0.682743i	$-0.760787\pi$
$97$	−14.3923	−1.46132	−0.730659	+	0.682743i	$0.760787\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 2.92820i	− 0.291367i	−0.989331	−	0.145684i	$-0.953462\pi$
$101$	− 2.92820i	− 0.291367i	0.989331	−	0.145684i	$-0.0465381\pi$
$102$	0	0
$103$	−15.6603	−1.54305	−0.771525	−	0.636199i	$-0.780506\pi$
$103$	−15.6603	−1.54305	−0.771525	+	0.636199i	$0.780506\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.73205i	0.264117i	0.991242	+	0.132059i	$0.0421587\pi$
$107$	2.73205i	0.264117i	−0.991242	+	0.132059i	$0.957841\pi$
$108$	0	0
$109$	16.9282i	1.62143i	0.585443	+	0.810714i	$0.300921\pi$
$109$	16.9282i	1.62143i	−0.585443	+	0.810714i	$0.699079\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	12.9282	1.21618	0.608092	−	0.793867i	$-0.291935\pi$
$113$	12.9282	1.21618	0.608092	+	0.793867i	$0.291935\pi$
$114$	0	0
$115$	− 6.19615i	− 0.577794i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	2.53590	0.232465
$120$	0	0
$121$	7.00000	0.636364
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 1.00000i	− 0.0894427i
$126$	0	0
$127$	−16.7321	−1.48473	−0.742365	−	0.669996i	$-0.766296\pi$
$127$	−16.7321	−1.48473	−0.742365	+	0.669996i	$0.766296\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 19.8564i	− 1.73486i	−0.497557	−	0.867431i	$-0.665770\pi$
$131$	− 19.8564i	− 1.73486i	0.497557	−	0.867431i	$-0.334230\pi$
$132$	0	0
$133$	0.392305i	0.0340171i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−4.92820	−0.421045	−0.210522	−	0.977589i	$-0.567516\pi$
$137$	−4.92820	−0.421045	−0.210522	+	0.977589i	$0.567516\pi$
$138$	0	0
$139$	0.535898i	0.0454543i	0.999742	+	0.0227272i	$0.00723490\pi$
$139$	0.535898i	0.0454543i	−0.999742	+	0.0227272i	$0.992765\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−6.92820	−0.579365
$144$	0	0
$145$	6.92820	0.575356
$146$	0	0
$147$	0	0
$148$	0	0
$149$	7.85641i	0.643622i	0.946804	+	0.321811i	$0.104292\pi$
$149$	7.85641i	0.643622i	−0.946804	+	0.321811i	$0.895708\pi$
$150$	0	0
$151$	12.3923	1.00847	0.504236	−	0.863566i	$-0.331774\pi$
$151$	12.3923	1.00847	0.504236	+	0.863566i	$0.331774\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	5.46410i	0.438887i
$156$	0	0
$157$	− 3.07180i	− 0.245156i	−0.992459	−	0.122578i	$-0.960884\pi$
$157$	− 3.07180i	− 0.245156i	0.992459	−	0.122578i	$-0.0391162\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	4.53590	0.357479
$162$	0	0
$163$	0.196152i	0.0153638i	0.999970	+	0.00768192i	$0.00244526\pi$
$163$	0.196152i	0.0153638i	−0.999970	+	0.00768192i	$0.997555\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−9.80385	−0.758645	−0.379322	−	0.925265i	$-0.623843\pi$
$167$	−9.80385	−0.758645	−0.379322	+	0.925265i	$0.623843\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 2.00000i	− 0.152057i	−0.997106	−	0.0760286i	$-0.975776\pi$
$173$	− 2.00000i	− 0.152057i	0.997106	−	0.0760286i	$-0.0242240\pi$
$174$	0	0
$175$	0.732051	0.0553378
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 8.53590i	− 0.638003i	−0.947754	−	0.319002i	$-0.896652\pi$
$179$	− 8.53590i	− 0.638003i	0.947754	−	0.319002i	$-0.103348\pi$
$180$	0	0
$181$	16.0000i	1.18927i	0.803996	+	0.594635i	$0.202704\pi$
$181$	16.0000i	1.18927i	−0.803996	+	0.594635i	$0.797296\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	2.00000	0.147043
$186$	0	0
$187$	6.92820i	0.506640i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	15.3205	1.10855	0.554277	−	0.832333i	$-0.312995\pi$
$191$	15.3205	1.10855	0.554277	+	0.832333i	$0.312995\pi$
$192$	0	0
$193$	0.535898	0.0385748	0.0192874	−	0.999814i	$-0.493860\pi$
$193$	0.535898	0.0385748	0.0192874	+	0.999814i	$0.493860\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	19.4641i	1.38676i	0.720572	+	0.693380i	$0.243879\pi$
$197$	19.4641i	1.38676i	−0.720572	+	0.693380i	$0.756121\pi$
$198$	0	0
$199$	1.85641	0.131597	0.0657986	−	0.997833i	$-0.479041\pi$
$199$	1.85641	0.131597	0.0657986	+	0.997833i	$0.479041\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	5.07180i	0.355970i
$204$	0	0
$205$	− 1.46410i	− 0.102257i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−1.07180	−0.0741377
$210$	0	0
$211$	− 26.7846i	− 1.84393i	−0.387275	−	0.921964i	$-0.626584\pi$
$211$	− 26.7846i	− 1.84393i	0.387275	−	0.921964i	$-0.373416\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−5.26795	−0.359271
$216$	0	0
$217$	−4.00000	−0.271538
$218$	0	0
$219$	0	0
$220$	0	0
$221$	12.0000i	0.807207i
$222$	0	0
$223$	5.80385	0.388654	0.194327	−	0.980937i	$-0.437748\pi$
$223$	5.80385	0.388654	0.194327	+	0.980937i	$0.437748\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	10.0526i	0.667212i	0.942713	+	0.333606i	$0.108265\pi$
$227$	10.0526i	0.667212i	−0.942713	+	0.333606i	$0.891735\pi$
$228$	0	0
$229$	− 4.00000i	− 0.264327i	−0.991228	−	0.132164i	$-0.957808\pi$
$229$	− 4.00000i	− 0.264327i	0.991228	−	0.132164i	$-0.0421925\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	5.32051	0.348558	0.174279	−	0.984696i	$-0.444241\pi$
$233$	5.32051	0.348558	0.174279	+	0.984696i	$0.444241\pi$
$234$	0	0
$235$	3.26795i	0.213177i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−20.0000	−1.29369	−0.646846	−	0.762620i	$-0.723912\pi$
$239$	−20.0000	−1.29369	−0.646846	+	0.762620i	$0.723912\pi$
$240$	0	0
$241$	16.3923	1.05592	0.527961	−	0.849269i	$-0.322957\pi$
$241$	16.3923	1.05592	0.527961	+	0.849269i	$0.322957\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 6.46410i	− 0.412976i
$246$	0	0
$247$	−1.85641	−0.118120
$248$	0	0
$249$	0	0
$250$	0	0
$251$	24.9282i	1.57345i	0.617301	+	0.786727i	$0.288226\pi$
$251$	24.9282i	1.57345i	−0.617301	+	0.786727i	$0.711774\pi$
$252$	0	0
$253$	12.3923i	0.779098i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2.00000	0.124757	0.0623783	−	0.998053i	$-0.480131\pi$
$257$	2.00000	0.124757	0.0623783	+	0.998053i	$0.480131\pi$
$258$	0	0
$259$	1.46410i	0.0909748i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−11.6603	−0.719002	−0.359501	−	0.933145i	$-0.617053\pi$
$263$	−11.6603	−0.719002	−0.359501	+	0.933145i	$0.617053\pi$
$264$	0	0
$265$	11.4641	0.704234
$266$	0	0
$267$	0	0
$268$	0	0
$269$	8.92820i	0.544362i	0.962246	+	0.272181i	$0.0877450\pi$
$269$	8.92820i	0.544362i	−0.962246	+	0.272181i	$0.912255\pi$
$270$	0	0
$271$	19.3205	1.17364	0.586819	−	0.809718i	$-0.300380\pi$
$271$	19.3205	1.17364	0.586819	+	0.809718i	$0.300380\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2.00000i	0.120605i
$276$	0	0
$277$	2.00000i	0.120168i	0.998193	+	0.0600842i	$0.0191369\pi$
$277$	2.00000i	0.120168i	−0.998193	+	0.0600842i	$0.980863\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−10.5359	−0.628519	−0.314260	−	0.949337i	$-0.601756\pi$
$281$	−10.5359	−0.628519	−0.314260	+	0.949337i	$0.601756\pi$
$282$	0	0
$283$	9.66025i	0.574242i	0.957894	+	0.287121i	$0.0926983\pi$
$283$	9.66025i	0.574242i	−0.957894	+	0.287121i	$0.907302\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.07180	0.0632662
$288$	0	0
$289$	−5.00000	−0.294118
$290$	0	0
$291$	0	0
$292$	0	0
$293$	15.8564i	0.926341i	0.886269	+	0.463171i	$0.153288\pi$
$293$	15.8564i	0.926341i	−0.886269	+	0.463171i	$0.846712\pi$
$294$	0	0
$295$	7.46410	0.434577
$296$	0	0
$297$	0	0
$298$	0	0
$299$	21.4641i	1.24130i
$300$	0	0
$301$	− 3.85641i	− 0.222280i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	8.92820	0.511227
$306$	0	0
$307$	24.9808i	1.42573i	0.701303	+	0.712864i	$0.252602\pi$
$307$	24.9808i	1.42573i	−0.701303	+	0.712864i	$0.747398\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	31.3205	1.77602	0.888012	−	0.459821i	$-0.152086\pi$
$311$	31.3205	1.77602	0.888012	+	0.459821i	$0.152086\pi$
$312$	0	0
$313$	−4.14359	−0.234210	−0.117105	−	0.993120i	$-0.537361\pi$
$313$	−4.14359	−0.234210	−0.117105	+	0.993120i	$0.537361\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 8.53590i	− 0.479424i	−0.970844	−	0.239712i	$-0.922947\pi$
$317$	− 8.53590i	− 0.479424i	0.970844	−	0.239712i	$-0.0770530\pi$
$318$	0	0
$319$	−13.8564	−0.775810
$320$	0	0
$321$	0	0
$322$	0	0
$323$	1.85641i	0.103293i
$324$	0	0
$325$	3.46410i	0.192154i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−2.39230	−0.131892
$330$	0	0
$331$	14.0000i	0.769510i	0.923019	+	0.384755i	$0.125714\pi$
$331$	14.0000i	0.769510i	−0.923019	+	0.384755i	$0.874286\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	10.7321	0.586355
$336$	0	0
$337$	−19.8564	−1.08165	−0.540824	−	0.841136i	$-0.681887\pi$
$337$	−19.8564	−1.08165	−0.540824	+	0.841136i	$0.681887\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 10.9282i	− 0.591795i
$342$	0	0
$343$	9.85641	0.532196
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 1.66025i	− 0.0891271i	−0.999007	−	0.0445636i	$-0.985810\pi$
$347$	− 1.66025i	− 0.0891271i	0.999007	−	0.0445636i	$-0.0141897\pi$
$348$	0	0
$349$	− 28.0000i	− 1.49881i	−0.662114	−	0.749403i	$-0.730341\pi$
$349$	− 28.0000i	− 1.49881i	0.662114	−	0.749403i	$-0.269659\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−12.9282	−0.688099	−0.344049	−	0.938952i	$-0.611799\pi$
$353$	−12.9282	−0.688099	−0.344049	+	0.938952i	$0.611799\pi$
$354$	0	0
$355$	5.46410i	0.290004i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	18.9282	0.998992	0.499496	−	0.866316i	$-0.333518\pi$
$359$	18.9282	0.998992	0.499496	+	0.866316i	$0.333518\pi$
$360$	0	0
$361$	18.7128	0.984885
$362$	0	0
$363$	0	0
$364$	0	0
$365$	7.46410i	0.390689i
$366$	0	0
$367$	−2.87564	−0.150107	−0.0750537	−	0.997179i	$-0.523913\pi$
$367$	−2.87564	−0.150107	−0.0750537	+	0.997179i	$0.523913\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	8.39230i	0.435707i
$372$	0	0
$373$	− 25.7128i	− 1.33136i	−0.746238	−	0.665679i	$-0.768142\pi$
$373$	− 25.7128i	− 1.33136i	0.746238	−	0.665679i	$-0.231858\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−24.0000	−1.23606
$378$	0	0
$379$	36.2487i	1.86197i	0.365056	+	0.930986i	$0.381050\pi$
$379$	36.2487i	1.86197i	−0.365056	+	0.930986i	$0.618950\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	21.1244	1.07940	0.539702	−	0.841856i	$-0.318537\pi$
$383$	21.1244	1.07940	0.539702	+	0.841856i	$0.318537\pi$
$384$	0	0
$385$	−1.46410	−0.0746175
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 6.78461i	− 0.343993i	−0.985098	−	0.171997i	$-0.944978\pi$
$389$	− 6.78461i	− 0.343993i	0.985098	−	0.171997i	$-0.0550218\pi$
$390$	0	0
$391$	21.4641	1.08549
$392$	0	0
$393$	0	0
$394$	0	0
$395$	1.07180i	0.0539279i
$396$	0	0
$397$	32.2487i	1.61852i	0.587453	+	0.809258i	$0.300131\pi$
$397$	32.2487i	1.61852i	−0.587453	+	0.809258i	$0.699869\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	7.85641	0.392330	0.196165	−	0.980571i	$-0.437151\pi$
$401$	7.85641	0.392330	0.196165	+	0.980571i	$0.437151\pi$
$402$	0	0
$403$	− 18.9282i	− 0.942881i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−4.00000	−0.198273
$408$	0	0
$409$	−11.3205	−0.559763	−0.279882	−	0.960035i	$-0.590295\pi$
$409$	−11.3205	−0.559763	−0.279882	+	0.960035i	$0.590295\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	5.46410i	0.268871i
$414$	0	0
$415$	−1.26795	−0.0622412
$416$	0	0
$417$	0	0
$418$	0	0
$419$	18.3923i	0.898523i	0.893400	+	0.449261i	$0.148313\pi$
$419$	18.3923i	0.898523i	−0.893400	+	0.449261i	$0.851687\pi$
$420$	0	0
$421$	0.143594i	0.00699832i	0.999994	+	0.00349916i	$0.00111382\pi$
$421$	0.143594i	0.00699832i	−0.999994	+	0.00349916i	$0.998886\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	3.46410	0.168034
$426$	0	0
$427$	6.53590i	0.316294i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−21.4641	−1.03389	−0.516945	−	0.856019i	$-0.672931\pi$
$431$	−21.4641	−1.03389	−0.516945	+	0.856019i	$0.672931\pi$
$432$	0	0
$433$	−19.4641	−0.935385	−0.467693	−	0.883891i	$-0.654915\pi$
$433$	−19.4641	−0.935385	−0.467693	+	0.883891i	$0.654915\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3.32051i	0.158841i
$438$	0	0
$439$	−40.7846	−1.94654	−0.973272	−	0.229657i	$-0.926240\pi$
$439$	−40.7846	−1.94654	−0.973272	+	0.229657i	$0.926240\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	20.9808i	0.996826i	0.866940	+	0.498413i	$0.166084\pi$
$443$	20.9808i	0.996826i	−0.866940	+	0.498413i	$0.833916\pi$
$444$	0	0
$445$	− 8.92820i	− 0.423237i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	23.3205	1.10056	0.550281	−	0.834979i	$-0.314520\pi$
$449$	23.3205	1.10056	0.550281	+	0.834979i	$0.314520\pi$
$450$	0	0
$451$	2.92820i	0.137884i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−2.53590	−0.118885
$456$	0	0
$457$	−26.7846	−1.25293	−0.626466	−	0.779449i	$-0.715499\pi$
$457$	−26.7846	−1.25293	−0.626466	+	0.779449i	$0.715499\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	10.9282i	0.508977i	0.967076	+	0.254489i	$0.0819071\pi$
$461$	10.9282i	0.508977i	−0.967076	+	0.254489i	$0.918093\pi$
$462$	0	0
$463$	11.2679	0.523666	0.261833	−	0.965113i	$-0.415673\pi$
$463$	11.2679	0.523666	0.261833	+	0.965113i	$0.415673\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 25.6603i	− 1.18741i	−0.804681	−	0.593707i	$-0.797664\pi$
$467$	− 25.6603i	− 1.18741i	0.804681	−	0.593707i	$-0.202336\pi$
$468$	0	0
$469$	7.85641i	0.362775i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	10.5359	0.484441
$474$	0	0
$475$	0.535898i	0.0245887i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	5.85641	0.267586	0.133793	−	0.991009i	$-0.457284\pi$
$479$	5.85641	0.267586	0.133793	+	0.991009i	$0.457284\pi$
$480$	0	0
$481$	−6.92820	−0.315899
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 14.3923i	− 0.653521i
$486$	0	0
$487$	−6.58846	−0.298551	−0.149276	−	0.988796i	$-0.547694\pi$
$487$	−6.58846	−0.298551	−0.149276	+	0.988796i	$0.547694\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 16.9282i	− 0.763959i	−0.924171	−	0.381980i	$-0.875242\pi$
$491$	− 16.9282i	− 0.763959i	0.924171	−	0.381980i	$-0.124758\pi$
$492$	0	0
$493$	24.0000i	1.08091i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−4.00000	−0.179425
$498$	0	0
$499$	− 31.4641i	− 1.40853i	−0.709939	−	0.704263i	$-0.751277\pi$
$499$	− 31.4641i	− 1.40853i	0.709939	−	0.704263i	$-0.248723\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−0.339746	−0.0151485	−0.00757426	−	0.999971i	$-0.502411\pi$
$503$	−0.339746	−0.0151485	−0.00757426	+	0.999971i	$0.502411\pi$
$504$	0	0
$505$	2.92820	0.130303
$506$	0	0
$507$	0	0
$508$	0	0
$509$	1.85641i	0.0822838i	0.999153	+	0.0411419i	$0.0130996\pi$
$509$	1.85641i	0.0822838i	−0.999153	+	0.0411419i	$0.986900\pi$
$510$	0	0
$511$	−5.46410	−0.241718
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 15.6603i	− 0.690073i
$516$	0	0
$517$	− 6.53590i	− 0.287448i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	43.8564	1.92138	0.960692	−	0.277616i	$-0.0895444\pi$
$521$	43.8564	1.92138	0.960692	+	0.277616i	$0.0895444\pi$
$522$	0	0
$523$	11.8038i	0.516146i	0.966125	+	0.258073i	$0.0830875\pi$
$523$	11.8038i	0.516146i	−0.966125	+	0.258073i	$0.916912\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−18.9282	−0.824525
$528$	0	0
$529$	15.3923	0.669231
$530$	0	0
$531$	0	0
$532$	0	0
$533$	5.07180i	0.219684i
$534$	0	0
$535$	−2.73205	−0.118117
$536$	0	0
$537$	0	0
$538$	0	0
$539$	12.9282i	0.556857i
$540$	0	0
$541$	− 26.9282i	− 1.15773i	−0.815422	−	0.578867i	$-0.803495\pi$
$541$	− 26.9282i	− 1.15773i	0.815422	−	0.578867i	$-0.196505\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−16.9282	−0.725125
$546$	0	0
$547$	− 33.2679i	− 1.42243i	−0.702972	−	0.711217i	$-0.748144\pi$
$547$	− 33.2679i	− 1.42243i	0.702972	−	0.711217i	$-0.251856\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−3.71281	−0.158171
$552$	0	0
$553$	−0.784610	−0.0333650
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 14.7846i	− 0.626444i	−0.949680	−	0.313222i	$-0.898592\pi$
$557$	− 14.7846i	− 0.626444i	0.949680	−	0.313222i	$-0.101408\pi$
$558$	0	0
$559$	18.2487	0.771838
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 22.0526i	− 0.929405i	−0.885467	−	0.464702i	$-0.846161\pi$
$563$	− 22.0526i	− 0.929405i	0.885467	−	0.464702i	$-0.153839\pi$
$564$	0	0
$565$	12.9282i	0.543894i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	13.4641	0.564445	0.282222	−	0.959349i	$-0.408928\pi$
$569$	13.4641	0.564445	0.282222	+	0.959349i	$0.408928\pi$
$570$	0	0
$571$	− 6.78461i	− 0.283927i	−0.989872	−	0.141964i	$-0.954658\pi$
$571$	− 6.78461i	− 0.283927i	0.989872	−	0.141964i	$-0.0453416\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	6.19615	0.258397
$576$	0	0
$577$	39.5692	1.64729	0.823644	−	0.567107i	$-0.191937\pi$
$577$	39.5692	1.64729	0.823644	+	0.567107i	$0.191937\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 0.928203i	− 0.0385084i
$582$	0	0
$583$	−22.9282	−0.949589
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 3.80385i	− 0.157002i	−0.996914	−	0.0785008i	$-0.974987\pi$
$587$	− 3.80385i	− 0.157002i	0.996914	−	0.0785008i	$-0.0250133\pi$
$588$	0	0
$589$	− 2.92820i	− 0.120655i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−32.6410	−1.34041	−0.670203	−	0.742178i	$-0.733793\pi$
$593$	−32.6410	−1.34041	−0.670203	+	0.742178i	$0.733793\pi$
$594$	0	0
$595$	2.53590i	0.103962i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−34.6410	−1.41539	−0.707697	−	0.706516i	$-0.750266\pi$
$599$	−34.6410	−1.41539	−0.707697	+	0.706516i	$0.750266\pi$
$600$	0	0
$601$	18.5359	0.756095	0.378048	−	0.925786i	$-0.376596\pi$
$601$	18.5359	0.756095	0.378048	+	0.925786i	$0.376596\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	7.00000i	0.284590i
$606$	0	0
$607$	−30.9808	−1.25747	−0.628735	−	0.777619i	$-0.716427\pi$
$607$	−30.9808	−1.25747	−0.628735	+	0.777619i	$0.716427\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 11.3205i	− 0.457979i
$612$	0	0
$613$	26.3923i	1.06598i	0.846123	+	0.532988i	$0.178931\pi$
$613$	26.3923i	1.06598i	−0.846123	+	0.532988i	$0.821069\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	20.5359	0.826744	0.413372	−	0.910562i	$-0.364351\pi$
$617$	20.5359	0.826744	0.413372	+	0.910562i	$0.364351\pi$
$618$	0	0
$619$	− 1.32051i	− 0.0530757i	−0.999648	−	0.0265379i	$-0.991552\pi$
$619$	− 1.32051i	− 0.0530757i	0.999648	−	0.0265379i	$-0.00844825\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	6.53590	0.261855
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	6.92820i	0.276246i
$630$	0	0
$631$	23.3205	0.928375	0.464187	−	0.885737i	$-0.346346\pi$
$631$	23.3205	0.928375	0.464187	+	0.885737i	$0.346346\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 16.7321i	− 0.663991i
$636$	0	0
$637$	22.3923i	0.887215i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−0.392305	−0.0154951	−0.00774755	−	0.999970i	$-0.502466\pi$
$641$	−0.392305	−0.0154951	−0.00774755	+	0.999970i	$0.502466\pi$
$642$	0	0
$643$	− 39.1244i	− 1.54291i	−0.636281	−	0.771457i	$-0.719528\pi$
$643$	− 39.1244i	− 1.54291i	0.636281	−	0.771457i	$-0.280472\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	16.7321	0.657805	0.328902	−	0.944364i	$-0.393321\pi$
$647$	16.7321	0.657805	0.328902	+	0.944364i	$0.393321\pi$
$648$	0	0
$649$	−14.9282	−0.585983
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 12.2487i	− 0.479329i	−0.970856	−	0.239665i	$-0.922963\pi$
$653$	− 12.2487i	− 0.479329i	0.970856	−	0.239665i	$-0.0770375\pi$
$654$	0	0
$655$	19.8564	0.775854
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 17.3205i	− 0.674711i	−0.941377	−	0.337356i	$-0.890468\pi$
$659$	− 17.3205i	− 0.674711i	0.941377	−	0.337356i	$-0.109532\pi$
$660$	0	0
$661$	− 8.14359i	− 0.316749i	−0.987379	−	0.158375i	$-0.949375\pi$
$661$	− 8.14359i	− 0.316749i	0.987379	−	0.158375i	$-0.0506253\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−0.392305	−0.0152129
$666$	0	0
$667$	42.9282i	1.66219i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−17.8564	−0.689339
$672$	0	0
$673$	12.5359	0.483223	0.241612	−	0.970373i	$-0.422324\pi$
$673$	12.5359	0.483223	0.241612	+	0.970373i	$0.422324\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 17.6077i	− 0.676719i	−0.941017	−	0.338359i	$-0.890128\pi$
$677$	− 17.6077i	− 0.676719i	0.941017	−	0.338359i	$-0.109872\pi$
$678$	0	0
$679$	10.5359	0.404331
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 16.9808i	− 0.649751i	−0.945757	−	0.324875i	$-0.894678\pi$
$683$	− 16.9808i	− 0.649751i	0.945757	−	0.324875i	$-0.105322\pi$
$684$	0	0
$685$	− 4.92820i	− 0.188297i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−39.7128	−1.51294
$690$	0	0
$691$	− 18.0000i	− 0.684752i	−0.939563	−	0.342376i	$-0.888768\pi$
$691$	− 18.0000i	− 0.684752i	0.939563	−	0.342376i	$-0.111232\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−0.535898	−0.0203278
$696$	0	0
$697$	5.07180	0.192108
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 19.0718i	− 0.720332i	−0.932888	−	0.360166i	$-0.882720\pi$
$701$	− 19.0718i	− 0.720332i	0.932888	−	0.360166i	$-0.117280\pi$
$702$	0	0
$703$	−1.07180	−0.0404236
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2.14359i	0.0806181i
$708$	0	0
$709$	− 12.7846i	− 0.480136i	−0.970756	−	0.240068i	$-0.922830\pi$
$709$	− 12.7846i	− 0.480136i	0.970756	−	0.240068i	$-0.0771698\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−33.8564	−1.26793
$714$	0	0
$715$	− 6.92820i	− 0.259100i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−1.85641	−0.0692323	−0.0346161	−	0.999401i	$-0.511021\pi$
$719$	−1.85641	−0.0692323	−0.0346161	+	0.999401i	$0.511021\pi$
$720$	0	0
$721$	11.4641	0.426945
$722$	0	0
$723$	0	0
$724$	0	0
$725$	6.92820i	0.257307i
$726$	0	0
$727$	24.0526	0.892060	0.446030	−	0.895018i	$-0.352837\pi$
$727$	24.0526	0.892060	0.446030	+	0.895018i	$0.352837\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 18.2487i	− 0.674953i
$732$	0	0
$733$	− 35.0718i	− 1.29541i	−0.761893	−	0.647703i	$-0.775730\pi$
$733$	− 35.0718i	− 1.29541i	0.761893	−	0.647703i	$-0.224270\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−21.4641	−0.790640
$738$	0	0
$739$	− 29.3205i	− 1.07857i	−0.842123	−	0.539286i	$-0.818694\pi$
$739$	− 29.3205i	− 1.07857i	0.842123	−	0.539286i	$-0.181306\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	10.9808	0.402845	0.201423	−	0.979504i	$-0.435444\pi$
$743$	10.9808	0.402845	0.201423	+	0.979504i	$0.435444\pi$
$744$	0	0
$745$	−7.85641	−0.287836
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 2.00000i	− 0.0730784i
$750$	0	0
$751$	−26.2487	−0.957829	−0.478915	−	0.877862i	$-0.658970\pi$
$751$	−26.2487	−0.957829	−0.478915	+	0.877862i	$0.658970\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	12.3923i	0.451002i
$756$	0	0
$757$	− 19.0718i	− 0.693176i	−0.938017	−	0.346588i	$-0.887340\pi$
$757$	− 19.0718i	− 0.693176i	0.938017	−	0.346588i	$-0.112660\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	5.71281	0.207089	0.103545	−	0.994625i	$-0.466982\pi$
$761$	5.71281	0.207089	0.103545	+	0.994625i	$0.466982\pi$
$762$	0	0
$763$	− 12.3923i	− 0.448632i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−25.8564	−0.933621
$768$	0	0
$769$	12.9282	0.466203	0.233101	−	0.972452i	$-0.425113\pi$
$769$	12.9282	0.466203	0.233101	+	0.972452i	$0.425113\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	22.3923i	0.805395i	0.915333	+	0.402698i	$0.131927\pi$
$773$	22.3923i	0.805395i	−0.915333	+	0.402698i	$0.868073\pi$
$774$	0	0
$775$	−5.46410	−0.196276
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0.784610i	0.0281116i
$780$	0	0
$781$	− 10.9282i	− 0.391042i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	3.07180	0.109637
$786$	0	0
$787$	− 16.5885i	− 0.591315i	−0.955294	−	0.295657i	$-0.904461\pi$
$787$	− 16.5885i	− 0.591315i	0.955294	−	0.295657i	$-0.0955387\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−9.46410	−0.336505
$792$	0	0
$793$	−30.9282	−1.09829
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 50.1051i	− 1.77481i	−0.460986	−	0.887407i	$-0.652504\pi$
$797$	− 50.1051i	− 1.77481i	0.460986	−	0.887407i	$-0.347496\pi$
$798$	0	0
$799$	−11.3205	−0.400491
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 14.9282i	− 0.526805i
$804$	0	0
$805$	4.53590i	0.159869i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−23.8564	−0.838747	−0.419373	−	0.907814i	$-0.637750\pi$
$809$	−23.8564	−0.838747	−0.419373	+	0.907814i	$0.637750\pi$
$810$	0	0
$811$	28.9282i	1.01581i	0.861414	+	0.507903i	$0.169579\pi$
$811$	28.9282i	1.01581i	−0.861414	+	0.507903i	$0.830421\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−0.196152	−0.00687092
$816$	0	0
$817$	2.82309	0.0987673
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 34.7846i	− 1.21399i	−0.794705	−	0.606996i	$-0.792375\pi$
$821$	− 34.7846i	− 1.21399i	0.794705	−	0.606996i	$-0.207625\pi$
$822$	0	0
$823$	9.12436	0.318055	0.159028	−	0.987274i	$-0.449164\pi$
$823$	9.12436	0.318055	0.159028	+	0.987274i	$0.449164\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	23.1244i	0.804113i	0.915615	+	0.402056i	$0.131704\pi$
$827$	23.1244i	0.804113i	−0.915615	+	0.402056i	$0.868296\pi$
$828$	0	0
$829$	28.9282i	1.00472i	0.864659	+	0.502359i	$0.167534\pi$
$829$	28.9282i	1.00472i	−0.864659	+	0.502359i	$0.832466\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	22.3923	0.775847
$834$	0	0
$835$	− 9.80385i	− 0.339276i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	24.7846	0.855660	0.427830	−	0.903859i	$-0.359278\pi$
$839$	24.7846	0.855660	0.427830	+	0.903859i	$0.359278\pi$
$840$	0	0
$841$	−19.0000	−0.655172
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1.00000i	0.0344010i
$846$	0	0
$847$	−5.12436	−0.176075
$848$	0	0
$849$	0	0
$850$	0	0
$851$	12.3923i	0.424803i
$852$	0	0
$853$	− 21.6077i	− 0.739833i	−0.929065	−	0.369917i	$-0.879386\pi$
$853$	− 21.6077i	− 0.739833i	0.929065	−	0.369917i	$-0.120614\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	19.8564	0.678282	0.339141	−	0.940736i	$-0.389864\pi$
$857$	19.8564	0.678282	0.339141	+	0.940736i	$0.389864\pi$
$858$	0	0
$859$	− 28.2487i	− 0.963834i	−0.876217	−	0.481917i	$-0.839941\pi$
$859$	− 28.2487i	− 0.963834i	0.876217	−	0.481917i	$-0.160059\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	47.6603	1.62237	0.811187	−	0.584787i	$-0.198822\pi$
$863$	47.6603	1.62237	0.811187	+	0.584787i	$0.198822\pi$
$864$	0	0
$865$	2.00000	0.0680020
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 2.14359i	− 0.0727164i
$870$	0	0
$871$	−37.1769	−1.25969
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0.732051i	0.0247478i
$876$	0	0
$877$	− 1.71281i	− 0.0578376i	−0.999582	−	0.0289188i	$-0.990794\pi$
$877$	− 1.71281i	− 0.0578376i	0.999582	−	0.0289188i	$-0.00920642\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−9.46410	−0.318854	−0.159427	−	0.987210i	$-0.550965\pi$
$881$	−9.46410	−0.318854	−0.159427	+	0.987210i	$0.550965\pi$
$882$	0	0
$883$	27.9090i	0.939211i	0.882876	+	0.469606i	$0.155604\pi$
$883$	27.9090i	0.939211i	−0.882876	+	0.469606i	$0.844396\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−13.9090	−0.467017	−0.233509	−	0.972355i	$-0.575021\pi$
$887$	−13.9090	−0.467017	−0.233509	+	0.972355i	$0.575021\pi$
$888$	0	0
$889$	12.2487	0.410809
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 1.75129i	− 0.0586046i
$894$	0	0
$895$	8.53590	0.285324
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 37.8564i	− 1.26258i
$900$	0	0
$901$	39.7128i	1.32303i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−16.0000	−0.531858
$906$	0	0
$907$	− 4.87564i	− 0.161893i	−0.996718	−	0.0809466i	$-0.974206\pi$
$907$	− 4.87564i	− 0.161893i	0.996718	−	0.0809466i	$-0.0257943\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−49.1769	−1.62930	−0.814652	−	0.579950i	$-0.803072\pi$
$911$	−49.1769	−1.62930	−0.814652	+	0.579950i	$0.803072\pi$
$912$	0	0
$913$	2.53590	0.0839260
$914$	0	0
$915$	0	0
$916$	0	0
$917$	14.5359i	0.480018i
$918$	0	0
$919$	38.9282	1.28412	0.642061	−	0.766653i	$-0.278079\pi$
$919$	38.9282	1.28412	0.642061	+	0.766653i	$0.278079\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 18.9282i	− 0.623029i
$924$	0	0
$925$	2.00000i	0.0657596i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	17.4641	0.572979	0.286489	−	0.958083i	$-0.407512\pi$
$929$	17.4641	0.572979	0.286489	+	0.958083i	$0.407512\pi$
$930$	0	0
$931$	3.46410i	0.113531i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−6.92820	−0.226576
$936$	0	0
$937$	4.24871	0.138799	0.0693997	−	0.997589i	$-0.477892\pi$
$937$	4.24871	0.138799	0.0693997	+	0.997589i	$0.477892\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	32.0000i	1.04317i	0.853199	+	0.521585i	$0.174659\pi$
$941$	32.0000i	1.04317i	−0.853199	+	0.521585i	$0.825341\pi$
$942$	0	0
$943$	9.07180	0.295418
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 3.12436i	− 0.101528i	−0.998711	−	0.0507640i	$-0.983834\pi$
$947$	− 3.12436i	− 0.101528i	0.998711	−	0.0507640i	$-0.0161656\pi$
$948$	0	0
$949$	− 25.8564i	− 0.839334i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	17.2154	0.557661	0.278831	−	0.960340i	$-0.410053\pi$
$953$	17.2154	0.557661	0.278831	+	0.960340i	$0.410053\pi$
$954$	0	0
$955$	15.3205i	0.495760i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	3.60770	0.116499
$960$	0	0
$961$	−1.14359	−0.0368901
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0.535898i	0.0172512i
$966$	0	0
$967$	−16.3397	−0.525451	−0.262725	−	0.964871i	$-0.584621\pi$
$967$	−16.3397	−0.525451	−0.262725	+	0.964871i	$0.584621\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	36.9282i	1.18508i	0.805540	+	0.592541i	$0.201875\pi$
$971$	36.9282i	1.18508i	−0.805540	+	0.592541i	$0.798125\pi$
$972$	0	0
$973$	− 0.392305i	− 0.0125767i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	24.5359	0.784973	0.392486	−	0.919758i	$-0.371615\pi$
$977$	24.5359	0.784973	0.392486	+	0.919758i	$0.371615\pi$
$978$	0	0
$979$	17.8564i	0.570693i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−48.7321	−1.55431	−0.777156	−	0.629309i	$-0.783338\pi$
$983$	−48.7321	−1.55431	−0.777156	+	0.629309i	$0.783338\pi$
$984$	0	0
$985$	−19.4641	−0.620178
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 32.6410i	− 1.03792i
$990$	0	0
$991$	−41.4641	−1.31715	−0.658575	−	0.752515i	$-0.728841\pi$
$991$	−41.4641	−1.31715	−0.658575	+	0.752515i	$0.728841\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	1.85641i	0.0588520i
$996$	0	0
$997$	− 11.1769i	− 0.353976i	−0.984213	−	0.176988i	$-0.943365\pi$
$997$	− 11.1769i	− 0.353976i	0.984213	−	0.176988i	$-0.0566354\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1440.2.k.e.721.3		4
3.2	odd	2		160.2.d.a.81.1		4
4.3	odd	2		360.2.k.e.181.4		4
5.2	odd	4		7200.2.d.o.2449.2		4
5.3	odd	4		7200.2.d.n.2449.3		4
5.4	even	2		7200.2.k.j.3601.3		4
8.3	odd	2		360.2.k.e.181.3		4
8.5	even	2	inner	1440.2.k.e.721.1		4
12.11	even	2		40.2.d.a.21.1	✓	4
15.2	even	4		800.2.f.c.49.1		4
15.8	even	4		800.2.f.e.49.4		4
15.14	odd	2		800.2.d.e.401.4		4
20.3	even	4		1800.2.d.l.1549.3		4
20.7	even	4		1800.2.d.p.1549.2		4
20.19	odd	2		1800.2.k.j.901.1		4
24.5	odd	2		160.2.d.a.81.4		4
24.11	even	2		40.2.d.a.21.2	yes	4
40.3	even	4		1800.2.d.p.1549.1		4
40.13	odd	4		7200.2.d.o.2449.3		4
40.19	odd	2		1800.2.k.j.901.2		4
40.27	even	4		1800.2.d.l.1549.4		4
40.29	even	2		7200.2.k.j.3601.4		4
40.37	odd	4		7200.2.d.n.2449.2		4
48.5	odd	4		1280.2.a.d.1.1		2
48.11	even	4		1280.2.a.o.1.2		2
48.29	odd	4		1280.2.a.n.1.2		2
48.35	even	4		1280.2.a.a.1.1		2
60.23	odd	4		200.2.f.e.149.2		4
60.47	odd	4		200.2.f.c.149.3		4
60.59	even	2		200.2.d.f.101.4		4
120.29	odd	2		800.2.d.e.401.1		4
120.53	even	4		800.2.f.c.49.2		4
120.59	even	2		200.2.d.f.101.3		4
120.77	even	4		800.2.f.e.49.3		4
120.83	odd	4		200.2.f.c.149.4		4
120.107	odd	4		200.2.f.e.149.1		4
240.29	odd	4		6400.2.a.be.1.1		2
240.59	even	4		6400.2.a.z.1.1		2
240.149	odd	4		6400.2.a.cj.1.2		2
240.179	even	4		6400.2.a.ce.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.2.d.a.21.1	✓	4	12.11	even	2
40.2.d.a.21.2	yes	4	24.11	even	2
160.2.d.a.81.1		4	3.2	odd	2
160.2.d.a.81.4		4	24.5	odd	2
200.2.d.f.101.3		4	120.59	even	2
200.2.d.f.101.4		4	60.59	even	2
200.2.f.c.149.3		4	60.47	odd	4
200.2.f.c.149.4		4	120.83	odd	4
200.2.f.e.149.1		4	120.107	odd	4
200.2.f.e.149.2		4	60.23	odd	4
360.2.k.e.181.3		4	8.3	odd	2
360.2.k.e.181.4		4	4.3	odd	2
800.2.d.e.401.1		4	120.29	odd	2
800.2.d.e.401.4		4	15.14	odd	2
800.2.f.c.49.1		4	15.2	even	4
800.2.f.c.49.2		4	120.53	even	4
800.2.f.e.49.3		4	120.77	even	4
800.2.f.e.49.4		4	15.8	even	4
1280.2.a.a.1.1		2	48.35	even	4
1280.2.a.d.1.1		2	48.5	odd	4
1280.2.a.n.1.2		2	48.29	odd	4
1280.2.a.o.1.2		2	48.11	even	4
1440.2.k.e.721.1		4	8.5	even	2	inner
1440.2.k.e.721.3		4	1.1	even	1	trivial
1800.2.d.l.1549.3		4	20.3	even	4
1800.2.d.l.1549.4		4	40.27	even	4
1800.2.d.p.1549.1		4	40.3	even	4
1800.2.d.p.1549.2		4	20.7	even	4
1800.2.k.j.901.1		4	20.19	odd	2
1800.2.k.j.901.2		4	40.19	odd	2
6400.2.a.z.1.1		2	240.59	even	4
6400.2.a.be.1.1		2	240.29	odd	4
6400.2.a.ce.1.2		2	240.179	even	4
6400.2.a.cj.1.2		2	240.149	odd	4
7200.2.d.n.2449.2		4	40.37	odd	4
7200.2.d.n.2449.3		4	5.3	odd	4
7200.2.d.o.2449.2		4	5.2	odd	4
7200.2.d.o.2449.3		4	40.13	odd	4
7200.2.k.j.3601.3		4	5.4	even	2
7200.2.k.j.3601.4		4	40.29	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:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1440.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{5} -0.732051 q^{7} -2.00000i q^{11} -3.46410i q^{13} -3.46410 q^{17} -0.535898i q^{19} -6.19615 q^{23} -1.00000 q^{25} -6.92820i q^{29} +5.46410 q^{31} -0.732051i q^{35} -2.00000i q^{37} -1.46410 q^{41} +5.26795i q^{43} +3.26795 q^{47} -6.46410 q^{49} -11.4641i q^{53} +2.00000 q^{55} -7.46410i q^{59} -8.92820i q^{61} +3.46410 q^{65} -10.7321i q^{67} +5.46410 q^{71} +7.46410 q^{73} +1.46410i q^{77} +1.07180 q^{79} +1.26795i q^{83} -3.46410i q^{85} -8.92820 q^{89} +2.53590i q^{91} +0.535898 q^{95} -14.3923 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{7} - 4 q^{23} - 4 q^{25} + 8 q^{31} + 8 q^{41} + 20 q^{47} - 12 q^{49} + 8 q^{55} + 8 q^{71} + 16 q^{73} + 32 q^{79} - 8 q^{89} + 16 q^{95} - 16 q^{97}+O(q^{100}) \) 4 * q + 4 * q^7 - 4 * q^23 - 4 * q^25 + 8 * q^31 + 8 * q^41 + 20 * q^47 - 12 * q^49 + 8 * q^55 + 8 * q^71 + 16 * q^73 + 32 * q^79 - 8 * q^89 + 16 * q^95 - 16 * q^97

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