LMFDB - Embedded newform 4032.3.d.o.449.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4032,3,Mod(449,4032)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4032, 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("4032.449");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 4032 = 2^{6} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	4032.d (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:	$109.864042590$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 44x^{10} + 719x^{8} + 5356x^{6} + 17809x^{4} + 20000x^{2} + 144 $ x^12 + 44x^10 + 719x^8 + 5356x^6 + 17809x^4 + 20000*x^2 + 144
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{13} $
Twist minimal:	no (minimal twist has level 2016)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.2
Root			$2.56196i$ of defining polynomial
Character	$\chi$	$=$	4032.449
Dual form			4032.3.d.o.449.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-7.59798i q^{5} +2.64575 q^{7} +O(q^{10})$	$q-7.59798i q^{5} +2.64575 q^{7} +15.9883i q^{11} -13.6731 q^{13} +32.2343i q^{17} +23.4205 q^{19} +7.96534i q^{23} -32.7294 q^{25} -23.0549i q^{29} -29.9819 q^{31} -20.1024i q^{35} -27.4923 q^{37} -19.8136i q^{41} -19.9121 q^{43} -17.4013i q^{47} +7.00000 q^{49} -38.0442i q^{53} +121.479 q^{55} -108.970i q^{59} -36.4222 q^{61} +103.888i q^{65} +88.2321 q^{67} -93.7399i q^{71} +41.3616 q^{73} +42.3010i q^{77} +58.8617 q^{79} -112.269i q^{83} +244.916 q^{85} +47.5136i q^{89} -36.1756 q^{91} -177.949i q^{95} -157.012 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q+O(q^{10}) $ 12 * q	$ 12 q - 16 q^{13} + 64 q^{19} - 124 q^{25} - 160 q^{31} - 56 q^{37} - 64 q^{43} + 84 q^{49} + 160 q^{55} - 104 q^{61} - 64 q^{67} - 64 q^{73} - 32 q^{79} + 184 q^{85} - 96 q^{97}+O(q^{100}) $ 12 * q - 16 * q^13 + 64 * q^19 - 124 * q^25 - 160 * q^31 - 56 * q^37 - 64 * q^43 + 84 * q^49 + 160 * q^55 - 104 * q^61 - 64 * q^67 - 64 * q^73 - 32 * q^79 + 184 * q^85 - 96 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4032\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$1793$	$3781$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$5$	− 7.59798i	− 1.51960i	−0.650159	−	0.759798i	$-0.725298\pi$
$5$	− 7.59798i	− 1.51960i	0.650159	−	0.759798i	$-0.274702\pi$
$6$	0	0
$7$	2.64575	0.377964
$7$	2.64575	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	15.9883i	1.45348i	0.686913	+	0.726740i	$0.258966\pi$
$11$	15.9883i	1.45348i	−0.686913	+	0.726740i	$0.741034\pi$
$12$	0	0
$13$	−13.6731	−1.05178	−0.525888	−	0.850554i	$-0.676267\pi$
$13$	−13.6731	−1.05178	−0.525888	+	0.850554i	$0.676267\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	32.2343i	1.89614i	0.318066	+	0.948069i	$0.396967\pi$
$17$	32.2343i	1.89614i	−0.318066	+	0.948069i	$0.603033\pi$
$18$	0	0
$19$	23.4205	1.23266	0.616330	−	0.787488i	$-0.288619\pi$
$19$	23.4205	1.23266	0.616330	+	0.787488i	$0.288619\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	7.96534i	0.346319i	0.984894	+	0.173160i	$0.0553976\pi$
$23$	7.96534i	0.346319i	−0.984894	+	0.173160i	$0.944602\pi$
$24$	0	0
$25$	−32.7294	−1.30917
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 23.0549i	− 0.794998i	−0.917603	−	0.397499i	$-0.869878\pi$
$29$	− 23.0549i	− 0.794998i	0.917603	−	0.397499i	$-0.130122\pi$
$30$	0	0
$31$	−29.9819	−0.967159	−0.483579	−	0.875300i	$-0.660664\pi$
$31$	−29.9819	−0.967159	−0.483579	+	0.875300i	$0.660664\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 20.1024i	− 0.574354i
$36$	0	0
$37$	−27.4923	−0.743036	−0.371518	−	0.928426i	$-0.621163\pi$
$37$	−27.4923	−0.743036	−0.371518	+	0.928426i	$0.621163\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 19.8136i	− 0.483258i	−0.970369	−	0.241629i	$-0.922318\pi$
$41$	− 19.8136i	− 0.483258i	0.970369	−	0.241629i	$-0.0776817\pi$
$42$	0	0
$43$	−19.9121	−0.463072	−0.231536	−	0.972826i	$-0.574375\pi$
$43$	−19.9121	−0.463072	−0.231536	+	0.972826i	$0.574375\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 17.4013i	− 0.370241i	−0.982716	−	0.185120i	$-0.940733\pi$
$47$	− 17.4013i	− 0.370241i	0.982716	−	0.185120i	$-0.0592675\pi$
$48$	0	0
$49$	7.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 38.0442i	− 0.717815i	−0.933373	−	0.358908i	$-0.883149\pi$
$53$	− 38.0442i	− 0.717815i	0.933373	−	0.358908i	$-0.116851\pi$
$54$	0	0
$55$	121.479	2.20870
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 108.970i	− 1.84694i	−0.383666	−	0.923472i	$-0.625339\pi$
$59$	− 108.970i	− 1.84694i	0.383666	−	0.923472i	$-0.374661\pi$
$60$	0	0
$61$	−36.4222	−0.597085	−0.298543	−	0.954396i	$-0.596500\pi$
$61$	−36.4222	−0.597085	−0.298543	+	0.954396i	$0.596500\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	103.888i	1.59827i
$66$	0	0
$67$	88.2321	1.31690	0.658449	−	0.752626i	$-0.271213\pi$
$67$	88.2321	1.31690	0.658449	+	0.752626i	$0.271213\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 93.7399i	− 1.32028i	−0.751142	−	0.660140i	$-0.770497\pi$
$71$	− 93.7399i	− 1.32028i	0.751142	−	0.660140i	$-0.229503\pi$
$72$	0	0
$73$	41.3616	0.566598	0.283299	−	0.959032i	$-0.408571\pi$
$73$	41.3616	0.566598	0.283299	+	0.959032i	$0.408571\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	42.3010i	0.549364i
$78$	0	0
$79$	58.8617	0.745084	0.372542	−	0.928015i	$-0.378486\pi$
$79$	58.8617	0.745084	0.372542	+	0.928015i	$0.378486\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 112.269i	− 1.35264i	−0.736606	−	0.676322i	$-0.763573\pi$
$83$	− 112.269i	− 1.35264i	0.736606	−	0.676322i	$-0.236427\pi$
$84$	0	0
$85$	244.916	2.88136
$86$	0	0
$87$	0	0
$88$	0	0
$89$	47.5136i	0.533860i	0.963716	+	0.266930i	$0.0860093\pi$
$89$	47.5136i	0.533860i	−0.963716	+	0.266930i	$0.913991\pi$
$90$	0	0
$91$	−36.1756	−0.397534
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 177.949i	− 1.87315i
$96$	0	0
$97$	−157.012	−1.61868	−0.809338	−	0.587343i	$-0.800174\pi$
$97$	−157.012	−1.61868	−0.809338	+	0.587343i	$0.800174\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 186.370i	− 1.84525i	−0.385702	−	0.922623i	$-0.626041\pi$
$101$	− 186.370i	− 1.84525i	0.385702	−	0.922623i	$-0.373959\pi$
$102$	0	0
$103$	−2.15397	−0.0209123	−0.0104561	−	0.999945i	$-0.503328\pi$
$103$	−2.15397	−0.0209123	−0.0104561	+	0.999945i	$0.503328\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	105.455i	0.985556i	0.870155	+	0.492778i	$0.164019\pi$
$107$	105.455i	0.985556i	−0.870155	+	0.492778i	$0.835981\pi$
$108$	0	0
$109$	132.397	1.21466	0.607328	−	0.794451i	$-0.292241\pi$
$109$	132.397	1.21466	0.607328	+	0.794451i	$0.292241\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 56.5496i	− 0.500439i	−0.968189	−	0.250219i	$-0.919497\pi$
$113$	− 56.5496i	− 0.500439i	0.968189	−	0.250219i	$-0.0805028\pi$
$114$	0	0
$115$	60.5205	0.526265
$116$	0	0
$117$	0	0
$118$	0	0
$119$	85.2840i	0.716672i
$120$	0	0
$121$	−134.625	−1.11260
$122$	0	0
$123$	0	0
$124$	0	0
$125$	58.7277i	0.469821i
$126$	0	0
$127$	−118.496	−0.933036	−0.466518	−	0.884512i	$-0.654492\pi$
$127$	−118.496	−0.933036	−0.466518	+	0.884512i	$0.654492\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 183.469i	− 1.40052i	−0.713885	−	0.700262i	$-0.753066\pi$
$131$	− 183.469i	− 1.40052i	0.713885	−	0.700262i	$-0.246934\pi$
$132$	0	0
$133$	61.9649	0.465901
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 272.567i	− 1.98954i	−0.102134	−	0.994771i	$-0.532567\pi$
$137$	− 272.567i	− 1.98954i	0.102134	−	0.994771i	$-0.467433\pi$
$138$	0	0
$139$	13.9010	0.100007	0.0500037	−	0.998749i	$-0.484077\pi$
$139$	13.9010	0.100007	0.0500037	+	0.998749i	$0.484077\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 218.609i	− 1.52873i
$144$	0	0
$145$	−175.171	−1.20808
$146$	0	0
$147$	0	0
$148$	0	0
$149$	73.1094i	0.490667i	0.969439	+	0.245334i	$0.0788975\pi$
$149$	73.1094i	0.490667i	−0.969439	+	0.245334i	$0.921103\pi$
$150$	0	0
$151$	−225.320	−1.49218	−0.746092	−	0.665843i	$-0.768072\pi$
$151$	−225.320	−1.49218	−0.746092	+	0.665843i	$0.768072\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	227.802i	1.46969i
$156$	0	0
$157$	232.971	1.48389	0.741945	−	0.670461i	$-0.233903\pi$
$157$	232.971	1.48389	0.741945	+	0.670461i	$0.233903\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	21.0743i	0.130896i
$162$	0	0
$163$	−204.651	−1.25552	−0.627762	−	0.778405i	$-0.716029\pi$
$163$	−204.651	−1.25552	−0.627762	+	0.778405i	$0.716029\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	34.9420i	0.209233i	0.994513	+	0.104617i	$0.0333616\pi$
$167$	34.9420i	0.209233i	−0.994513	+	0.104617i	$0.966638\pi$
$168$	0	0
$169$	17.9531	0.106231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 72.6526i	− 0.419957i	−0.977706	−	0.209979i	$-0.932661\pi$
$173$	− 72.6526i	− 0.419957i	0.977706	−	0.209979i	$-0.0673394\pi$
$174$	0	0
$175$	−86.5938	−0.494822
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 280.382i	− 1.56638i	−0.621784	−	0.783189i	$-0.713592\pi$
$179$	− 280.382i	− 1.56638i	0.621784	−	0.783189i	$-0.286408\pi$
$180$	0	0
$181$	67.1393	0.370936	0.185468	−	0.982650i	$-0.440620\pi$
$181$	67.1393	0.370936	0.185468	+	0.982650i	$0.440620\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	208.886i	1.12911i
$186$	0	0
$187$	−515.372	−2.75600
$188$	0	0
$189$	0	0
$190$	0	0
$191$	237.891i	1.24550i	0.782420	+	0.622751i	$0.213985\pi$
$191$	237.891i	1.24550i	−0.782420	+	0.622751i	$0.786015\pi$
$192$	0	0
$193$	−133.526	−0.691847	−0.345923	−	0.938263i	$-0.612434\pi$
$193$	−133.526	−0.691847	−0.345923	+	0.938263i	$0.612434\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	45.5663i	0.231301i	0.993290	+	0.115650i	$0.0368953\pi$
$197$	45.5663i	0.231301i	−0.993290	+	0.115650i	$0.963105\pi$
$198$	0	0
$199$	−168.351	−0.845986	−0.422993	−	0.906133i	$-0.639021\pi$
$199$	−168.351	−0.845986	−0.422993	+	0.906133i	$0.639021\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 60.9977i	− 0.300481i
$204$	0	0
$205$	−150.543	−0.734358
$206$	0	0
$207$	0	0
$208$	0	0
$209$	374.454i	1.79165i
$210$	0	0
$211$	188.623	0.893949	0.446974	−	0.894547i	$-0.352502\pi$
$211$	188.623	0.893949	0.446974	+	0.894547i	$0.352502\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	151.292i	0.703683i
$216$	0	0
$217$	−79.3247	−0.365552
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 440.743i	− 1.99431i
$222$	0	0
$223$	219.326	0.983523	0.491761	−	0.870730i	$-0.336353\pi$
$223$	219.326	0.983523	0.491761	+	0.870730i	$0.336353\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	318.370i	1.40251i	0.712911	+	0.701255i	$0.247376\pi$
$227$	318.370i	1.40251i	−0.712911	+	0.701255i	$0.752624\pi$
$228$	0	0
$229$	236.519	1.03283	0.516417	−	0.856337i	$-0.327266\pi$
$229$	236.519	1.03283	0.516417	+	0.856337i	$0.327266\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	378.533i	1.62460i	0.583237	+	0.812302i	$0.301786\pi$
$233$	378.533i	1.62460i	−0.583237	+	0.812302i	$0.698214\pi$
$234$	0	0
$235$	−132.215	−0.562617
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 409.885i	− 1.71500i	−0.514486	−	0.857499i	$-0.672017\pi$
$239$	− 409.885i	− 1.71500i	0.514486	−	0.857499i	$-0.327983\pi$
$240$	0	0
$241$	405.153	1.68113	0.840565	−	0.541710i	$-0.182223\pi$
$241$	405.153	1.68113	0.840565	+	0.541710i	$0.182223\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 53.1859i	− 0.217085i
$246$	0	0
$247$	−320.231	−1.29648
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 324.644i	− 1.29340i	−0.762744	−	0.646701i	$-0.776148\pi$
$251$	− 324.644i	− 1.29340i	0.762744	−	0.646701i	$-0.223852\pi$
$252$	0	0
$253$	−127.352	−0.503368
$254$	0	0
$255$	0	0
$256$	0	0
$257$	125.364i	0.487796i	0.969801	+	0.243898i	$0.0784263\pi$
$257$	125.364i	0.487796i	−0.969801	+	0.243898i	$0.921574\pi$
$258$	0	0
$259$	−72.7378	−0.280841
$260$	0	0
$261$	0	0
$262$	0	0
$263$	139.300i	0.529659i	0.964295	+	0.264830i	$0.0853157\pi$
$263$	139.300i	0.529659i	−0.964295	+	0.264830i	$0.914684\pi$
$264$	0	0
$265$	−289.059	−1.09079
$266$	0	0
$267$	0	0
$268$	0	0
$269$	137.020i	0.509370i	0.967024	+	0.254685i	$0.0819718\pi$
$269$	137.020i	0.509370i	−0.967024	+	0.254685i	$0.918028\pi$
$270$	0	0
$271$	133.499	0.492615	0.246308	−	0.969192i	$-0.420783\pi$
$271$	133.499	0.492615	0.246308	+	0.969192i	$0.420783\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 523.286i	− 1.90286i
$276$	0	0
$277$	−134.839	−0.486783	−0.243392	−	0.969928i	$-0.578260\pi$
$277$	−134.839	−0.486783	−0.243392	+	0.969928i	$0.578260\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	117.970i	0.419824i	0.977720	+	0.209912i	$0.0673177\pi$
$281$	117.970i	0.419824i	−0.977720	+	0.209912i	$0.932682\pi$
$282$	0	0
$283$	19.1183	0.0675560	0.0337780	−	0.999429i	$-0.489246\pi$
$283$	19.1183	0.0675560	0.0337780	+	0.999429i	$0.489246\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 52.4218i	− 0.182654i
$288$	0	0
$289$	−750.052	−2.59534
$290$	0	0
$291$	0	0
$292$	0	0
$293$	102.725i	0.350597i	0.984515	+	0.175298i	$0.0560890\pi$
$293$	102.725i	0.350597i	−0.984515	+	0.175298i	$0.943911\pi$
$294$	0	0
$295$	−827.950	−2.80661
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 108.911i	− 0.364250i
$300$	0	0
$301$	−52.6825	−0.175025
$302$	0	0
$303$	0	0
$304$	0	0
$305$	276.735i	0.907329i
$306$	0	0
$307$	35.2385	0.114783	0.0573917	−	0.998352i	$-0.481722\pi$
$307$	35.2385	0.114783	0.0573917	+	0.998352i	$0.481722\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 328.518i	− 1.05633i	−0.849143	−	0.528164i	$-0.822881\pi$
$311$	− 328.518i	− 1.05633i	0.849143	−	0.528164i	$-0.177119\pi$
$312$	0	0
$313$	290.295	0.927459	0.463729	−	0.885977i	$-0.346511\pi$
$313$	290.295	0.927459	0.463729	+	0.885977i	$0.346511\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 37.4331i	− 0.118086i	−0.998255	−	0.0590428i	$-0.981195\pi$
$317$	− 37.4331i	− 0.118086i	0.998255	−	0.0590428i	$-0.0188048\pi$
$318$	0	0
$319$	368.609	1.15551
$320$	0	0
$321$	0	0
$322$	0	0
$323$	754.945i	2.33729i
$324$	0	0
$325$	447.511	1.37696
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 46.0395i	− 0.139938i
$330$	0	0
$331$	−169.871	−0.513207	−0.256603	−	0.966517i	$-0.582603\pi$
$331$	−169.871	−0.513207	−0.256603	+	0.966517i	$0.582603\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 670.386i	− 2.00115i
$336$	0	0
$337$	23.1974	0.0688349	0.0344175	−	0.999408i	$-0.489042\pi$
$337$	23.1974	0.0688349	0.0344175	+	0.999408i	$0.489042\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 479.359i	− 1.40575i
$342$	0	0
$343$	18.5203	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 65.8953i	− 0.189900i	−0.995482	−	0.0949500i	$-0.969731\pi$
$347$	− 65.8953i	− 0.189900i	0.995482	−	0.0949500i	$-0.0302691\pi$
$348$	0	0
$349$	−356.672	−1.02198	−0.510992	−	0.859586i	$-0.670722\pi$
$349$	−356.672	−1.02198	−0.510992	+	0.859586i	$0.670722\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 660.912i	− 1.87227i	−0.351638	−	0.936136i	$-0.614375\pi$
$353$	− 660.912i	− 1.87227i	0.351638	−	0.936136i	$-0.385625\pi$
$354$	0	0
$355$	−712.235	−2.00629
$356$	0	0
$357$	0	0
$358$	0	0
$359$	265.729i	0.740192i	0.928993	+	0.370096i	$0.120675\pi$
$359$	265.729i	0.740192i	−0.928993	+	0.370096i	$0.879325\pi$
$360$	0	0
$361$	187.521	0.519449
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 314.265i	− 0.861000i
$366$	0	0
$367$	−504.859	−1.37564	−0.687819	−	0.725882i	$-0.741432\pi$
$367$	−504.859	−1.37564	−0.687819	+	0.725882i	$0.741432\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 100.656i	− 0.271309i
$372$	0	0
$373$	−376.433	−1.00920	−0.504602	−	0.863352i	$-0.668361\pi$
$373$	−376.433	−1.00920	−0.504602	+	0.863352i	$0.668361\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	315.232i	0.836159i
$378$	0	0
$379$	−730.419	−1.92723	−0.963614	−	0.267298i	$-0.913869\pi$
$379$	−730.419	−1.92723	−0.963614	+	0.267298i	$0.913869\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 329.591i	− 0.860551i	−0.902698	−	0.430275i	$-0.858416\pi$
$383$	− 329.591i	− 0.860551i	0.902698	−	0.430275i	$-0.141584\pi$
$384$	0	0
$385$	321.402	0.834812
$386$	0	0
$387$	0	0
$388$	0	0
$389$	212.863i	0.547205i	0.961843	+	0.273602i	$0.0882152\pi$
$389$	212.863i	0.547205i	−0.961843	+	0.273602i	$0.911785\pi$
$390$	0	0
$391$	−256.757	−0.656668
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 447.230i	− 1.13223i
$396$	0	0
$397$	525.086	1.32264	0.661318	−	0.750106i	$-0.269997\pi$
$397$	525.086	1.32264	0.661318	+	0.750106i	$0.269997\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 449.796i	− 1.12169i	−0.827923	−	0.560843i	$-0.810477\pi$
$401$	− 449.796i	− 1.12169i	0.827923	−	0.560843i	$-0.189523\pi$
$402$	0	0
$403$	409.945	1.01723
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 439.555i	− 1.07999i
$408$	0	0
$409$	−241.664	−0.590866	−0.295433	−	0.955363i	$-0.595464\pi$
$409$	−241.664	−0.590866	−0.295433	+	0.955363i	$0.595464\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 288.307i	− 0.698079i
$414$	0	0
$415$	−853.021	−2.05547
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 597.571i	− 1.42618i	−0.701070	−	0.713092i	$-0.747294\pi$
$419$	− 597.571i	− 1.42618i	0.701070	−	0.713092i	$-0.252706\pi$
$420$	0	0
$421$	−773.357	−1.83695	−0.918476	−	0.395476i	$-0.870580\pi$
$421$	−773.357	−1.83695	−0.918476	+	0.395476i	$0.870580\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 1055.01i	− 2.48238i
$426$	0	0
$427$	−96.3641	−0.225677
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 133.936i	− 0.310757i	−0.987855	−	0.155379i	$-0.950340\pi$
$431$	− 133.936i	− 0.310757i	0.987855	−	0.155379i	$-0.0496598\pi$
$432$	0	0
$433$	522.822	1.20744	0.603720	−	0.797196i	$-0.293684\pi$
$433$	522.822	1.20744	0.603720	+	0.797196i	$0.293684\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	186.552i	0.426893i
$438$	0	0
$439$	664.678	1.51407	0.757037	−	0.653372i	$-0.226646\pi$
$439$	664.678	1.51407	0.757037	+	0.653372i	$0.226646\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 747.553i	− 1.68748i	−0.536753	−	0.843739i	$-0.680349\pi$
$443$	− 747.553i	− 1.68748i	0.536753	−	0.843739i	$-0.319651\pi$
$444$	0	0
$445$	361.007	0.811253
$446$	0	0
$447$	0	0
$448$	0	0
$449$	256.828i	0.572000i	0.958230	+	0.286000i	$0.0923257\pi$
$449$	256.828i	0.572000i	−0.958230	+	0.286000i	$0.907674\pi$
$450$	0	0
$451$	316.785	0.702406
$452$	0	0
$453$	0	0
$454$	0	0
$455$	274.861i	0.604091i
$456$	0	0
$457$	335.849	0.734899	0.367450	−	0.930043i	$-0.380231\pi$
$457$	335.849	0.734899	0.367450	+	0.930043i	$0.380231\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	348.270i	0.755466i	0.925915	+	0.377733i	$0.123296\pi$
$461$	348.270i	0.755466i	−0.925915	+	0.377733i	$0.876704\pi$
$462$	0	0
$463$	694.584	1.50018	0.750090	−	0.661335i	$-0.230010\pi$
$463$	694.584	1.50018	0.750090	+	0.661335i	$0.230010\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	351.357i	0.752371i	0.926544	+	0.376185i	$0.122764\pi$
$467$	351.357i	0.752371i	−0.926544	+	0.376185i	$0.877236\pi$
$468$	0	0
$469$	233.440	0.497740
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 318.360i	− 0.673066i
$474$	0	0
$475$	−766.539	−1.61377
$476$	0	0
$477$	0	0
$478$	0	0
$479$	204.236i	0.426380i	0.977011	+	0.213190i	$0.0683853\pi$
$479$	204.236i	0.426380i	−0.977011	+	0.213190i	$0.931615\pi$
$480$	0	0
$481$	375.905	0.781506
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1192.97i	2.45973i
$486$	0	0
$487$	−783.629	−1.60910	−0.804548	−	0.593888i	$-0.797592\pi$
$487$	−783.629	−1.60910	−0.804548	+	0.593888i	$0.797592\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 767.058i	− 1.56224i	−0.624383	−	0.781119i	$-0.714649\pi$
$491$	− 767.058i	− 1.56224i	0.624383	−	0.781119i	$-0.285351\pi$
$492$	0	0
$493$	743.161	1.50743
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 248.013i	− 0.499019i
$498$	0	0
$499$	614.441	1.23135	0.615673	−	0.788002i	$-0.288884\pi$
$499$	614.441	1.23135	0.615673	+	0.788002i	$0.288884\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	659.785i	1.31170i	0.754891	+	0.655850i	$0.227690\pi$
$503$	659.785i	1.31170i	−0.754891	+	0.655850i	$0.772310\pi$
$504$	0	0
$505$	−1416.04	−2.80403
$506$	0	0
$507$	0	0
$508$	0	0
$509$	525.085i	1.03160i	0.856709	+	0.515800i	$0.172505\pi$
$509$	525.085i	1.03160i	−0.856709	+	0.515800i	$0.827495\pi$
$510$	0	0
$511$	109.433	0.214154
$512$	0	0
$513$	0	0
$514$	0	0
$515$	16.3658i	0.0317783i
$516$	0	0
$517$	278.217	0.538137
$518$	0	0
$519$	0	0
$520$	0	0
$521$	101.327i	0.194486i	0.995261	+	0.0972431i	$0.0310024\pi$
$521$	101.327i	0.194486i	−0.995261	+	0.0972431i	$0.968998\pi$
$522$	0	0
$523$	145.620	0.278432	0.139216	−	0.990262i	$-0.455542\pi$
$523$	145.620	0.278432	0.139216	+	0.990262i	$0.455542\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 966.447i	− 1.83387i
$528$	0	0
$529$	465.553	0.880063
$530$	0	0
$531$	0	0
$532$	0	0
$533$	270.913i	0.508279i
$534$	0	0
$535$	801.242	1.49765
$536$	0	0
$537$	0	0
$538$	0	0
$539$	111.918i	0.207640i
$540$	0	0
$541$	−324.233	−0.599322	−0.299661	−	0.954046i	$-0.596874\pi$
$541$	−324.233	−0.599322	−0.299661	+	0.954046i	$0.596874\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 1005.95i	− 1.84579i
$546$	0	0
$547$	692.428	1.26587	0.632933	−	0.774207i	$-0.281851\pi$
$547$	692.428	1.26587	0.632933	+	0.774207i	$0.281851\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 539.959i	− 0.979962i
$552$	0	0
$553$	155.733	0.281615
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 335.715i	− 0.602719i	−0.953511	−	0.301360i	$-0.902560\pi$
$557$	− 335.715i	− 0.602719i	0.953511	−	0.301360i	$-0.0974405\pi$
$558$	0	0
$559$	272.260	0.487048
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 306.666i	− 0.544700i	−0.962198	−	0.272350i	$-0.912199\pi$
$563$	− 306.666i	− 0.544700i	0.962198	−	0.272350i	$-0.0878008\pi$
$564$	0	0
$565$	−429.663	−0.760465
$566$	0	0
$567$	0	0
$568$	0	0
$569$	736.974i	1.29521i	0.761977	+	0.647605i	$0.224229\pi$
$569$	736.974i	1.29521i	−0.761977	+	0.647605i	$0.775771\pi$
$570$	0	0
$571$	501.899	0.878982	0.439491	−	0.898247i	$-0.355159\pi$
$571$	501.899	0.878982	0.439491	+	0.898247i	$0.355159\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 260.701i	− 0.453392i
$576$	0	0
$577$	644.392	1.11680	0.558399	−	0.829573i	$-0.311416\pi$
$577$	644.392	1.11680	0.558399	+	0.829573i	$0.311416\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 297.037i	− 0.511251i
$582$	0	0
$583$	608.262	1.04333
$584$	0	0
$585$	0	0
$586$	0	0
$587$	272.221i	0.463750i	0.972746	+	0.231875i	$0.0744860\pi$
$587$	272.221i	0.463750i	−0.972746	+	0.231875i	$0.925514\pi$
$588$	0	0
$589$	−702.192	−1.19218
$590$	0	0
$591$	0	0
$592$	0	0
$593$	90.7452i	0.153027i	0.997069	+	0.0765137i	$0.0243789\pi$
$593$	90.7452i	0.153027i	−0.997069	+	0.0765137i	$0.975621\pi$
$594$	0	0
$595$	647.987	1.08905
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 763.189i	− 1.27411i	−0.770820	−	0.637053i	$-0.780153\pi$
$599$	− 763.189i	− 1.27411i	0.770820	−	0.637053i	$-0.219847\pi$
$600$	0	0
$601$	−243.581	−0.405292	−0.202646	−	0.979252i	$-0.564954\pi$
$601$	−243.581	−0.405292	−0.202646	+	0.979252i	$0.564954\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	1022.88i	1.69071i
$606$	0	0
$607$	234.356	0.386089	0.193045	−	0.981190i	$-0.438164\pi$
$607$	234.356	0.386089	0.193045	+	0.981190i	$0.438164\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	237.929i	0.389410i
$612$	0	0
$613$	−1032.32	−1.68405	−0.842024	−	0.539440i	$-0.818636\pi$
$613$	−1032.32	−1.68405	−0.842024	+	0.539440i	$0.818636\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	290.151i	0.470262i	0.971964	+	0.235131i	$0.0755519\pi$
$617$	290.151i	0.470262i	−0.971964	+	0.235131i	$0.924448\pi$
$618$	0	0
$619$	272.201	0.439743	0.219871	−	0.975529i	$-0.429436\pi$
$619$	272.201	0.439743	0.219871	+	0.975529i	$0.429436\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	125.709i	0.201780i
$624$	0	0
$625$	−372.022	−0.595236
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 886.196i	− 1.40890i
$630$	0	0
$631$	986.769	1.56382	0.781909	−	0.623393i	$-0.214246\pi$
$631$	986.769	1.56382	0.781909	+	0.623393i	$0.214246\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	900.328i	1.41784i
$636$	0	0
$637$	−95.7115	−0.150254
$638$	0	0
$639$	0	0
$640$	0	0
$641$	43.7459i	0.0682464i	0.999418	+	0.0341232i	$0.0108639\pi$
$641$	43.7459i	0.0682464i	−0.999418	+	0.0341232i	$0.989136\pi$
$642$	0	0
$643$	45.0634	0.0700830	0.0350415	−	0.999386i	$-0.488844\pi$
$643$	45.0634	0.0700830	0.0350415	+	0.999386i	$0.488844\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	374.803i	0.579294i	0.957134	+	0.289647i	$0.0935379\pi$
$647$	374.803i	0.579294i	−0.957134	+	0.289647i	$0.906462\pi$
$648$	0	0
$649$	1742.24	2.68450
$650$	0	0
$651$	0	0
$652$	0	0
$653$	582.059i	0.891361i	0.895192	+	0.445681i	$0.147038\pi$
$653$	582.059i	0.891361i	−0.895192	+	0.445681i	$0.852962\pi$
$654$	0	0
$655$	−1393.99	−2.12823
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 1067.83i	− 1.62038i	−0.586168	−	0.810190i	$-0.699364\pi$
$659$	− 1067.83i	− 1.62038i	0.586168	−	0.810190i	$-0.300636\pi$
$660$	0	0
$661$	−867.759	−1.31280	−0.656399	−	0.754414i	$-0.727921\pi$
$661$	−867.759	−1.31280	−0.656399	+	0.754414i	$0.727921\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 470.808i	− 0.707982i
$666$	0	0
$667$	183.640	0.275323
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 582.328i	− 0.867851i
$672$	0	0
$673$	619.945	0.921166	0.460583	−	0.887617i	$-0.347640\pi$
$673$	619.945	0.921166	0.460583	+	0.887617i	$0.347640\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 98.7785i	− 0.145906i	−0.997335	−	0.0729531i	$-0.976758\pi$
$677$	− 98.7785i	− 0.145906i	0.997335	−	0.0729531i	$-0.0232424\pi$
$678$	0	0
$679$	−415.413	−0.611802
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 768.220i	− 1.12477i	−0.826875	−	0.562386i	$-0.809883\pi$
$683$	− 768.220i	− 1.12477i	0.826875	−	0.562386i	$-0.190117\pi$
$684$	0	0
$685$	−2070.96	−3.02330
$686$	0	0
$687$	0	0
$688$	0	0
$689$	520.181i	0.754980i
$690$	0	0
$691$	−576.287	−0.833991	−0.416995	−	0.908909i	$-0.636917\pi$
$691$	−576.287	−0.833991	−0.416995	+	0.908909i	$0.636917\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 105.620i	− 0.151971i
$696$	0	0
$697$	638.678	0.916324
$698$	0	0
$699$	0	0
$700$	0	0
$701$	156.372i	0.223070i	0.993761	+	0.111535i	$0.0355767\pi$
$701$	156.372i	0.223070i	−0.993761	+	0.111535i	$0.964423\pi$
$702$	0	0
$703$	−643.885	−0.915910
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 493.088i	− 0.697438i
$708$	0	0
$709$	−509.397	−0.718472	−0.359236	−	0.933247i	$-0.616963\pi$
$709$	−509.397	−0.718472	−0.359236	+	0.933247i	$0.616963\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 238.816i	− 0.334946i
$714$	0	0
$715$	−1660.99	−2.32306
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 1239.18i	− 1.72348i	−0.507354	−	0.861738i	$-0.669376\pi$
$719$	− 1239.18i	− 1.72348i	0.507354	−	0.861738i	$-0.330624\pi$
$720$	0	0
$721$	−5.69886	−0.00790410
$722$	0	0
$723$	0	0
$724$	0	0
$725$	754.574i	1.04079i
$726$	0	0
$727$	274.267	0.377258	0.188629	−	0.982048i	$-0.439596\pi$
$727$	274.267	0.377258	0.188629	+	0.982048i	$0.439596\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 641.853i	− 0.878048i
$732$	0	0
$733$	831.350	1.13417	0.567087	−	0.823658i	$-0.308070\pi$
$733$	831.350	1.13417	0.567087	+	0.823658i	$0.308070\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1410.68i	1.91408i
$738$	0	0
$739$	558.704	0.756027	0.378014	−	0.925800i	$-0.376607\pi$
$739$	558.704	0.756027	0.378014	+	0.925800i	$0.376607\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	464.348i	0.624964i	0.949924	+	0.312482i	$0.101160\pi$
$743$	464.348i	0.624964i	−0.949924	+	0.312482i	$0.898840\pi$
$744$	0	0
$745$	555.484	0.745616
$746$	0	0
$747$	0	0
$748$	0	0
$749$	279.006i	0.372505i
$750$	0	0
$751$	−884.392	−1.17762	−0.588810	−	0.808272i	$-0.700403\pi$
$751$	−884.392	−1.17762	−0.588810	+	0.808272i	$0.700403\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1711.98i	2.26752i
$756$	0	0
$757$	−616.656	−0.814605	−0.407302	−	0.913293i	$-0.633531\pi$
$757$	−616.656	−0.814605	−0.407302	+	0.913293i	$0.633531\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	432.071i	0.567767i	0.958859	+	0.283884i	$0.0916229\pi$
$761$	432.071i	0.567767i	−0.958859	+	0.283884i	$0.908377\pi$
$762$	0	0
$763$	350.291	0.459097
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1489.95i	1.94257i
$768$	0	0
$769$	−676.958	−0.880310	−0.440155	−	0.897922i	$-0.645076\pi$
$769$	−676.958	−0.880310	−0.440155	+	0.897922i	$0.645076\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 1505.43i	− 1.94752i	−0.227572	−	0.973761i	$-0.573079\pi$
$773$	− 1505.43i	− 1.94752i	0.227572	−	0.973761i	$-0.426921\pi$
$774$	0	0
$775$	981.290	1.26618
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 464.045i	− 0.595693i
$780$	0	0
$781$	1498.74	1.91900
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 1770.11i	− 2.25492i
$786$	0	0
$787$	−1421.31	−1.80599	−0.902995	−	0.429652i	$-0.858636\pi$
$787$	−1421.31	−1.80599	−0.902995	+	0.429652i	$0.858636\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 149.616i	− 0.189148i
$792$	0	0
$793$	498.003	0.627999
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 311.357i	− 0.390661i	−0.980737	−	0.195331i	$-0.937422\pi$
$797$	− 311.357i	− 0.390661i	0.980737	−	0.195331i	$-0.0625780\pi$
$798$	0	0
$799$	560.920	0.702027
$800$	0	0
$801$	0	0
$802$	0	0
$803$	661.301i	0.823538i
$804$	0	0
$805$	160.122	0.198910
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 757.935i	− 0.936879i	−0.883496	−	0.468440i	$-0.844816\pi$
$809$	− 757.935i	− 0.936879i	0.883496	−	0.468440i	$-0.155184\pi$
$810$	0	0
$811$	−468.771	−0.578016	−0.289008	−	0.957327i	$-0.593325\pi$
$811$	−468.771	−0.578016	−0.289008	+	0.957327i	$0.593325\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1554.93i	1.90789i
$816$	0	0
$817$	−466.352	−0.570810
$818$	0	0
$819$	0	0
$820$	0	0
$821$	344.371i	0.419453i	0.977760	+	0.209727i	$0.0672574\pi$
$821$	344.371i	0.419453i	−0.977760	+	0.209727i	$0.932743\pi$
$822$	0	0
$823$	339.392	0.412384	0.206192	−	0.978512i	$-0.433893\pi$
$823$	339.392	0.412384	0.206192	+	0.978512i	$0.433893\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	20.4474i	0.0247248i	0.999924	+	0.0123624i	$0.00393518\pi$
$827$	20.4474i	0.0247248i	−0.999924	+	0.0123624i	$0.996065\pi$
$828$	0	0
$829$	−1005.94	−1.21343	−0.606717	−	0.794918i	$-0.707514\pi$
$829$	−1005.94	−1.21343	−0.606717	+	0.794918i	$0.707514\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	225.640i	0.270877i
$834$	0	0
$835$	265.489	0.317950
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 34.1308i	− 0.0406803i	−0.999793	−	0.0203401i	$-0.993525\pi$
$839$	− 34.1308i	− 0.0406803i	0.999793	−	0.0203401i	$-0.00647492\pi$
$840$	0	0
$841$	309.470	0.367978
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 136.407i	− 0.161428i
$846$	0	0
$847$	−356.185	−0.420525
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 218.986i	− 0.257327i
$852$	0	0
$853$	722.627	0.847159	0.423580	−	0.905859i	$-0.360773\pi$
$853$	722.627	0.847159	0.423580	+	0.905859i	$0.360773\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	212.693i	0.248183i	0.992271	+	0.124091i	$0.0396016\pi$
$857$	212.693i	0.248183i	−0.992271	+	0.124091i	$0.960398\pi$
$858$	0	0
$859$	241.343	0.280958	0.140479	−	0.990084i	$-0.455136\pi$
$859$	241.343	0.280958	0.140479	+	0.990084i	$0.455136\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	466.768i	0.540867i	0.962739	+	0.270434i	$0.0871670\pi$
$863$	466.768i	0.540867i	−0.962739	+	0.270434i	$0.912833\pi$
$864$	0	0
$865$	−552.013	−0.638166
$866$	0	0
$867$	0	0
$868$	0	0
$869$	941.097i	1.08297i
$870$	0	0
$871$	−1206.40	−1.38508
$872$	0	0
$873$	0	0
$874$	0	0
$875$	155.379i	0.177576i
$876$	0	0
$877$	−1211.14	−1.38101	−0.690504	−	0.723329i	$-0.742611\pi$
$877$	−1211.14	−1.38101	−0.690504	+	0.723329i	$0.742611\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 705.274i	− 0.800538i	−0.916398	−	0.400269i	$-0.868917\pi$
$881$	− 705.274i	− 0.800538i	0.916398	−	0.400269i	$-0.131083\pi$
$882$	0	0
$883$	−752.778	−0.852523	−0.426262	−	0.904600i	$-0.640170\pi$
$883$	−752.778	−0.852523	−0.426262	+	0.904600i	$0.640170\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 991.843i	− 1.11820i	−0.829100	−	0.559100i	$-0.811147\pi$
$887$	− 991.843i	− 1.11820i	0.829100	−	0.559100i	$-0.188853\pi$
$888$	0	0
$889$	−313.510	−0.352654
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 407.548i	− 0.456381i
$894$	0	0
$895$	−2130.33	−2.38026
$896$	0	0
$897$	0	0
$898$	0	0
$899$	691.232i	0.768889i
$900$	0	0
$901$	1226.33	1.36108
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 510.124i	− 0.563673i
$906$	0	0
$907$	−1339.55	−1.47690	−0.738451	−	0.674307i	$-0.764443\pi$
$907$	−1339.55	−1.47690	−0.738451	+	0.674307i	$0.764443\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	206.322i	0.226479i	0.993568	+	0.113240i	$0.0361228\pi$
$911$	206.322i	0.226479i	−0.993568	+	0.113240i	$0.963877\pi$
$912$	0	0
$913$	1795.00	1.96604
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 485.413i	− 0.529349i
$918$	0	0
$919$	1293.69	1.40772	0.703859	−	0.710339i	$-0.251458\pi$
$919$	1293.69	1.40772	0.703859	+	0.710339i	$0.251458\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	1281.71i	1.38864i
$924$	0	0
$925$	899.806	0.972764
$926$	0	0
$927$	0	0
$928$	0	0
$929$	721.666i	0.776821i	0.921487	+	0.388410i	$0.126976\pi$
$929$	721.666i	0.776821i	−0.921487	+	0.388410i	$0.873024\pi$
$930$	0	0
$931$	163.944	0.176094
$932$	0	0
$933$	0	0
$934$	0	0
$935$	3915.79i	4.18801i
$936$	0	0
$937$	30.7529	0.0328206	0.0164103	−	0.999865i	$-0.494776\pi$
$937$	30.7529	0.0328206	0.0164103	+	0.999865i	$0.494776\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	341.703i	0.363128i	0.983379	+	0.181564i	$0.0581159\pi$
$941$	341.703i	0.363128i	−0.983379	+	0.181564i	$0.941884\pi$
$942$	0	0
$943$	157.822	0.167362
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1882.00i	1.98733i	0.112394	+	0.993664i	$0.464148\pi$
$947$	1882.00i	1.98733i	−0.112394	+	0.993664i	$0.535852\pi$
$948$	0	0
$949$	−565.541	−0.595933
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 155.432i	− 0.163098i	−0.996669	−	0.0815488i	$-0.974013\pi$
$953$	− 155.432i	− 0.163098i	0.996669	−	0.0815488i	$-0.0259867\pi$
$954$	0	0
$955$	1807.49	1.89266
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 721.145i	− 0.751976i
$960$	0	0
$961$	−62.0843	−0.0646038
$962$	0	0
$963$	0	0
$964$	0	0
$965$	1014.53i	1.05133i
$966$	0	0
$967$	1488.38	1.53917	0.769585	−	0.638544i	$-0.220463\pi$
$967$	1488.38	1.53917	0.769585	+	0.638544i	$0.220463\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	506.143i	0.521260i	0.965439	+	0.260630i	$0.0839302\pi$
$971$	506.143i	0.521260i	−0.965439	+	0.260630i	$0.916070\pi$
$972$	0	0
$973$	36.7787	0.0377993
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 797.673i	− 0.816451i	−0.912881	−	0.408226i	$-0.866148\pi$
$977$	− 797.673i	− 0.816451i	0.912881	−	0.408226i	$-0.133852\pi$
$978$	0	0
$979$	−759.660	−0.775955
$980$	0	0
$981$	0	0
$982$	0	0
$983$	65.9444i	0.0670848i	0.999437	+	0.0335424i	$0.0106789\pi$
$983$	65.9444i	0.0670848i	−0.999437	+	0.0335424i	$0.989321\pi$
$984$	0	0
$985$	346.212	0.351484
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 158.607i	− 0.160371i
$990$	0	0
$991$	−222.739	−0.224762	−0.112381	−	0.993665i	$-0.535848\pi$
$991$	−222.739	−0.224762	−0.112381	+	0.993665i	$0.535848\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	1279.13i	1.28556i
$996$	0	0
$997$	1648.77	1.65373	0.826866	−	0.562398i	$-0.190121\pi$
$997$	1648.77	1.65373	0.826866	+	0.562398i	$0.190121\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4032.3.d.o.449.2		12
3.2	odd	2	inner	4032.3.d.o.449.11		12
4.3	odd	2		4032.3.d.n.449.2		12
8.3	odd	2		2016.3.d.f.449.11	yes	12
8.5	even	2		2016.3.d.e.449.11	yes	12
12.11	even	2		4032.3.d.n.449.11		12
24.5	odd	2		2016.3.d.e.449.2	✓	12
24.11	even	2		2016.3.d.f.449.2	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2016.3.d.e.449.2	✓	12	24.5	odd	2
2016.3.d.e.449.11	yes	12	8.5	even	2
2016.3.d.f.449.2	yes	12	24.11	even	2
2016.3.d.f.449.11	yes	12	8.3	odd	2
4032.3.d.n.449.2		12	4.3	odd	2
4032.3.d.n.449.11		12	12.11	even	2
4032.3.d.o.449.2		12	1.1	even	1	trivial
4032.3.d.o.449.11		12	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	4032.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-7.59798i q^{5} +2.64575 q^{7} +O(q^{10})\)	\(q-7.59798i q^{5} +2.64575 q^{7} +15.9883i q^{11} -13.6731 q^{13} +32.2343i q^{17} +23.4205 q^{19} +7.96534i q^{23} -32.7294 q^{25} -23.0549i q^{29} -29.9819 q^{31} -20.1024i q^{35} -27.4923 q^{37} -19.8136i q^{41} -19.9121 q^{43} -17.4013i q^{47} +7.00000 q^{49} -38.0442i q^{53} +121.479 q^{55} -108.970i q^{59} -36.4222 q^{61} +103.888i q^{65} +88.2321 q^{67} -93.7399i q^{71} +41.3616 q^{73} +42.3010i q^{77} +58.8617 q^{79} -112.269i q^{83} +244.916 q^{85} +47.5136i q^{89} -36.1756 q^{91} -177.949i q^{95} -157.012 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q+O(q^{10}) \) 12 * q	\( 12 q - 16 q^{13} + 64 q^{19} - 124 q^{25} - 160 q^{31} - 56 q^{37} - 64 q^{43} + 84 q^{49} + 160 q^{55} - 104 q^{61} - 64 q^{67} - 64 q^{73} - 32 q^{79} + 184 q^{85} - 96 q^{97}+O(q^{100}) \) 12 * q - 16 * q^13 + 64 * q^19 - 124 * q^25 - 160 * q^31 - 56 * q^37 - 64 * q^43 + 84 * q^49 + 160 * q^55 - 104 * q^61 - 64 * q^67 - 64 * q^73 - 32 * q^79 + 184 * q^85 - 96 * q^97

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