LMFDB - Embedded newform 1764.3.c.a.197.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1764,3,Mod(197,1764)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1764, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1764.197");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1764.c (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:	$48.0655186332$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			197.1
Root			$-1.41421i$ of defining polynomial
Character	$\chi$	$=$	1764.197
Dual form			1764.3.c.a.197.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.24264i q^{5} +O(q^{10})$	$q-4.24264i q^{5} +21.2132i q^{11} -23.0000 q^{13} -16.9706i q^{17} +1.00000 q^{19} +16.9706i q^{23} +7.00000 q^{25} -33.9411i q^{29} +49.0000 q^{31} +17.0000 q^{37} -21.2132i q^{41} +47.0000 q^{43} +38.1838i q^{47} -84.8528i q^{53} +90.0000 q^{55} -50.9117i q^{59} +40.0000 q^{61} +97.5807i q^{65} +23.0000 q^{67} +63.6396i q^{71} -17.0000 q^{73} -79.0000 q^{79} -106.066i q^{83} -72.0000 q^{85} -135.765i q^{89} -4.24264i q^{95} +40.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q+O(q^{10}) $ 2 * q	$ 2 q - 46 q^{13} + 2 q^{19} + 14 q^{25} + 98 q^{31} + 34 q^{37} + 94 q^{43} + 180 q^{55} + 80 q^{61} + 46 q^{67} - 34 q^{73} - 158 q^{79} - 144 q^{85} + 80 q^{97}+O(q^{100}) $ 2 * q - 46 * q^13 + 2 * q^19 + 14 * q^25 + 98 * q^31 + 34 * q^37 + 94 * q^43 + 180 * q^55 + 80 * q^61 + 46 * q^67 - 34 * q^73 - 158 * q^79 - 144 * q^85 + 80 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times$.

$n$	$785$	$883$	$1081$
$\chi(n)$	$-1$	$1$	$1$

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
$5$	− 4.24264i	− 0.848528i	−0.905539	−	0.424264i	$-0.860533\pi$
$5$	− 4.24264i	− 0.848528i	0.905539	−	0.424264i	$-0.139467\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	21.2132i	1.92847i	0.265043	+	0.964237i	$0.414614\pi$
$11$	21.2132i	1.92847i	−0.265043	+	0.964237i	$0.585386\pi$
$12$	0	0
$13$	−23.0000	−1.76923	−0.884615	−	0.466321i	$-0.845579\pi$
$13$	−23.0000	−1.76923	−0.884615	+	0.466321i	$0.845579\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 16.9706i	− 0.998268i	−0.866525	−	0.499134i	$-0.833651\pi$
$17$	− 16.9706i	− 0.998268i	0.866525	−	0.499134i	$-0.166349\pi$
$18$	0	0
$19$	1.00000	0.0526316	0.0263158	−	0.999654i	$-0.491622\pi$
$19$	1.00000	0.0526316	0.0263158	+	0.999654i	$0.491622\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	16.9706i	0.737851i	0.929459	+	0.368925i	$0.120274\pi$
$23$	16.9706i	0.737851i	−0.929459	+	0.368925i	$0.879726\pi$
$24$	0	0
$25$	7.00000	0.280000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 33.9411i	− 1.17038i	−0.810895	−	0.585192i	$-0.801019\pi$
$29$	− 33.9411i	− 1.17038i	0.810895	−	0.585192i	$-0.198981\pi$
$30$	0	0
$31$	49.0000	1.58065	0.790323	−	0.612691i	$-0.209913\pi$
$31$	49.0000	1.58065	0.790323	+	0.612691i	$0.209913\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	17.0000	0.459459	0.229730	−	0.973254i	$-0.426216\pi$
$37$	17.0000	0.459459	0.229730	+	0.973254i	$0.426216\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 21.2132i	− 0.517395i	−0.965958	−	0.258698i	$-0.916707\pi$
$41$	− 21.2132i	− 0.517395i	0.965958	−	0.258698i	$-0.0832933\pi$
$42$	0	0
$43$	47.0000	1.09302	0.546512	−	0.837452i	$-0.315955\pi$
$43$	47.0000	1.09302	0.546512	+	0.837452i	$0.315955\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	38.1838i	0.812421i	0.913780	+	0.406210i	$0.133150\pi$
$47$	38.1838i	0.812421i	−0.913780	+	0.406210i	$0.866850\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 84.8528i	− 1.60100i	−0.599335	−	0.800498i	$-0.704568\pi$
$53$	− 84.8528i	− 1.60100i	0.599335	−	0.800498i	$-0.295432\pi$
$54$	0	0
$55$	90.0000	1.63636
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 50.9117i	− 0.862910i	−0.902134	−	0.431455i	$-0.858000\pi$
$59$	− 50.9117i	− 0.862910i	0.902134	−	0.431455i	$-0.142000\pi$
$60$	0	0
$61$	40.0000	0.655738	0.327869	−	0.944723i	$-0.393670\pi$
$61$	40.0000	0.655738	0.327869	+	0.944723i	$0.393670\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	97.5807i	1.50124i
$66$	0	0
$67$	23.0000	0.343284	0.171642	−	0.985159i	$-0.445093\pi$
$67$	23.0000	0.343284	0.171642	+	0.985159i	$0.445093\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	63.6396i	0.896333i	0.893950	+	0.448166i	$0.147923\pi$
$71$	63.6396i	0.896333i	−0.893950	+	0.448166i	$0.852077\pi$
$72$	0	0
$73$	−17.0000	−0.232877	−0.116438	−	0.993198i	$-0.537148\pi$
$73$	−17.0000	−0.232877	−0.116438	+	0.993198i	$0.537148\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−79.0000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$79$	−79.0000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 106.066i	− 1.27790i	−0.769247	−	0.638952i	$-0.779368\pi$
$83$	− 106.066i	− 1.27790i	0.769247	−	0.638952i	$-0.220632\pi$
$84$	0	0
$85$	−72.0000	−0.847059
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 135.765i	− 1.52544i	−0.646727	−	0.762722i	$-0.723863\pi$
$89$	− 135.765i	− 1.52544i	0.646727	−	0.762722i	$-0.276137\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 4.24264i	− 0.0446594i
$96$	0	0
$97$	40.0000	0.412371	0.206186	−	0.978513i	$-0.433895\pi$
$97$	40.0000	0.412371	0.206186	+	0.978513i	$0.433895\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 21.2132i	− 0.210032i	−0.994471	−	0.105016i	$-0.966511\pi$
$101$	− 21.2132i	− 0.210032i	0.994471	−	0.105016i	$-0.0334893\pi$
$102$	0	0
$103$	−23.0000	−0.223301	−0.111650	−	0.993748i	$-0.535614\pi$
$103$	−23.0000	−0.223301	−0.111650	+	0.993748i	$0.535614\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 16.9706i	− 0.158603i	−0.996851	−	0.0793017i	$-0.974731\pi$
$107$	− 16.9706i	− 0.158603i	0.996851	−	0.0793017i	$-0.0252690\pi$
$108$	0	0
$109$	71.0000	0.651376	0.325688	−	0.945477i	$-0.394404\pi$
$109$	71.0000	0.651376	0.325688	+	0.945477i	$0.394404\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 131.522i	− 1.16391i	−0.813221	−	0.581955i	$-0.802288\pi$
$113$	− 131.522i	− 1.16391i	0.813221	−	0.581955i	$-0.197712\pi$
$114$	0	0
$115$	72.0000	0.626087
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−329.000	−2.71901
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 135.765i	− 1.08612i
$126$	0	0
$127$	17.0000	0.133858	0.0669291	−	0.997758i	$-0.478680\pi$
$127$	17.0000	0.133858	0.0669291	+	0.997758i	$0.478680\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 55.1543i	− 0.421025i	−0.977591	−	0.210513i	$-0.932487\pi$
$131$	− 55.1543i	− 0.421025i	0.977591	−	0.210513i	$-0.0675133\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 67.8823i	− 0.495491i	−0.968825	−	0.247745i	$-0.920310\pi$
$137$	− 67.8823i	− 0.495491i	0.968825	−	0.247745i	$-0.0796897\pi$
$138$	0	0
$139$	31.0000	0.223022	0.111511	−	0.993763i	$-0.464431\pi$
$139$	31.0000	0.223022	0.111511	+	0.993763i	$0.464431\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 487.904i	− 3.41191i
$144$	0	0
$145$	−144.000	−0.993103
$146$	0	0
$147$	0	0
$148$	0	0
$149$	169.706i	1.13896i	0.822004	+	0.569482i	$0.192856\pi$
$149$	169.706i	1.13896i	−0.822004	+	0.569482i	$0.807144\pi$
$150$	0	0
$151$	−40.0000	−0.264901	−0.132450	−	0.991190i	$-0.542285\pi$
$151$	−40.0000	−0.264901	−0.132450	+	0.991190i	$0.542285\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 207.889i	− 1.34122i
$156$	0	0
$157$	280.000	1.78344	0.891720	−	0.452588i	$-0.149499\pi$
$157$	280.000	1.78344	0.891720	+	0.452588i	$0.149499\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	200.000	1.22699	0.613497	−	0.789697i	$-0.289762\pi$
$163$	200.000	1.22699	0.613497	+	0.789697i	$0.289762\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	106.066i	0.635126i	0.948237	+	0.317563i	$0.102864\pi$
$167$	106.066i	0.635126i	−0.948237	+	0.317563i	$0.897136\pi$
$168$	0	0
$169$	360.000	2.13018
$170$	0	0
$171$	0	0
$172$	0	0
$173$	254.558i	1.47144i	0.677288	+	0.735718i	$0.263155\pi$
$173$	254.558i	1.47144i	−0.677288	+	0.735718i	$0.736845\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 199.404i	− 1.11399i	−0.830516	−	0.556995i	$-0.811954\pi$
$179$	− 199.404i	− 1.11399i	0.830516	−	0.556995i	$-0.188046\pi$
$180$	0	0
$181$	271.000	1.49724	0.748619	−	0.663001i	$-0.230717\pi$
$181$	271.000	1.49724	0.748619	+	0.663001i	$0.230717\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 72.1249i	− 0.389864i
$186$	0	0
$187$	360.000	1.92513
$188$	0	0
$189$	0	0
$190$	0	0
$191$	241.831i	1.26613i	0.774099	+	0.633064i	$0.218203\pi$
$191$	241.831i	1.26613i	−0.774099	+	0.633064i	$0.781797\pi$
$192$	0	0
$193$	−103.000	−0.533679	−0.266839	−	0.963741i	$-0.585979\pi$
$193$	−103.000	−0.533679	−0.266839	+	0.963741i	$0.585979\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 152.735i	− 0.775305i	−0.921806	−	0.387652i	$-0.873286\pi$
$197$	− 152.735i	− 0.775305i	0.921806	−	0.387652i	$-0.126714\pi$
$198$	0	0
$199$	160.000	0.804020	0.402010	−	0.915635i	$-0.368312\pi$
$199$	160.000	0.804020	0.402010	+	0.915635i	$0.368312\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−90.0000	−0.439024
$206$	0	0
$207$	0	0
$208$	0	0
$209$	21.2132i	0.101499i
$210$	0	0
$211$	80.0000	0.379147	0.189573	−	0.981867i	$-0.439289\pi$
$211$	80.0000	0.379147	0.189573	+	0.981867i	$0.439289\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 199.404i	− 0.927461i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	390.323i	1.76617i
$222$	0	0
$223$	−104.000	−0.466368	−0.233184	−	0.972433i	$-0.574914\pi$
$223$	−104.000	−0.466368	−0.233184	+	0.972433i	$0.574914\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 148.492i	− 0.654152i	−0.944998	−	0.327076i	$-0.893937\pi$
$227$	− 148.492i	− 0.654152i	0.944998	−	0.327076i	$-0.106063\pi$
$228$	0	0
$229$	31.0000	0.135371	0.0676856	−	0.997707i	$-0.478439\pi$
$229$	31.0000	0.135371	0.0676856	+	0.997707i	$0.478439\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	173.948i	0.746559i	0.927719	+	0.373280i	$0.121767\pi$
$233$	173.948i	0.746559i	−0.927719	+	0.373280i	$0.878233\pi$
$234$	0	0
$235$	162.000	0.689362
$236$	0	0
$237$	0	0
$238$	0	0
$239$	326.683i	1.36688i	0.730009	+	0.683438i	$0.239516\pi$
$239$	326.683i	1.36688i	−0.730009	+	0.683438i	$0.760484\pi$
$240$	0	0
$241$	130.000	0.539419	0.269710	−	0.962942i	$-0.413072\pi$
$241$	130.000	0.539419	0.269710	+	0.962942i	$0.413072\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−23.0000	−0.0931174
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 458.205i	− 1.82552i	−0.408498	−	0.912759i	$-0.633947\pi$
$251$	− 458.205i	− 1.82552i	0.408498	−	0.912759i	$-0.366053\pi$
$252$	0	0
$253$	−360.000	−1.42292
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 106.066i	− 0.412708i	−0.978477	−	0.206354i	$-0.933840\pi$
$257$	− 106.066i	− 0.412708i	0.978477	−	0.206354i	$-0.0661599\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 169.706i	− 0.645269i	−0.946524	−	0.322634i	$-0.895432\pi$
$263$	− 169.706i	− 0.645269i	0.946524	−	0.322634i	$-0.104568\pi$
$264$	0	0
$265$	−360.000	−1.35849
$266$	0	0
$267$	0	0
$268$	0	0
$269$	224.860i	0.835911i	0.908468	+	0.417955i	$0.137253\pi$
$269$	224.860i	0.835911i	−0.908468	+	0.417955i	$0.862747\pi$
$270$	0	0
$271$	−200.000	−0.738007	−0.369004	−	0.929428i	$-0.620301\pi$
$271$	−200.000	−0.738007	−0.369004	+	0.929428i	$0.620301\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	148.492i	0.539972i
$276$	0	0
$277$	137.000	0.494585	0.247292	−	0.968941i	$-0.420459\pi$
$277$	137.000	0.494585	0.247292	+	0.968941i	$0.420459\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 84.8528i	− 0.301967i	−0.988536	−	0.150984i	$-0.951756\pi$
$281$	− 84.8528i	− 0.301967i	0.988536	−	0.150984i	$-0.0482441\pi$
$282$	0	0
$283$	−257.000	−0.908127	−0.454064	−	0.890969i	$-0.650026\pi$
$283$	−257.000	−0.908127	−0.454064	+	0.890969i	$0.650026\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.00000	0.00346021
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 339.411i	− 1.15840i	−0.815185	−	0.579200i	$-0.803365\pi$
$293$	− 339.411i	− 1.15840i	0.815185	−	0.579200i	$-0.196635\pi$
$294$	0	0
$295$	−216.000	−0.732203
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 390.323i	− 1.30543i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 169.706i	− 0.556412i
$306$	0	0
$307$	127.000	0.413681	0.206840	−	0.978375i	$-0.433682\pi$
$307$	127.000	0.413681	0.206840	+	0.978375i	$0.433682\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	318.198i	1.02314i	0.859240	+	0.511572i	$0.170937\pi$
$311$	318.198i	1.02314i	−0.859240	+	0.511572i	$0.829063\pi$
$312$	0	0
$313$	−137.000	−0.437700	−0.218850	−	0.975759i	$-0.570230\pi$
$313$	−137.000	−0.437700	−0.218850	+	0.975759i	$0.570230\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 407.294i	− 1.28484i	−0.766354	−	0.642419i	$-0.777931\pi$
$317$	− 407.294i	− 1.28484i	0.766354	−	0.642419i	$-0.222069\pi$
$318$	0	0
$319$	720.000	2.25705
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 16.9706i	− 0.0525404i
$324$	0	0
$325$	−161.000	−0.495385
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−241.000	−0.728097	−0.364048	−	0.931380i	$-0.618606\pi$
$331$	−241.000	−0.728097	−0.364048	+	0.931380i	$0.618606\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 97.5807i	− 0.291286i
$336$	0	0
$337$	527.000	1.56380	0.781899	−	0.623405i	$-0.214251\pi$
$337$	527.000	1.56380	0.781899	+	0.623405i	$0.214251\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	1039.45i	3.04823i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 254.558i	− 0.733598i	−0.930300	−	0.366799i	$-0.880454\pi$
$347$	− 254.558i	− 0.733598i	0.930300	−	0.366799i	$-0.119546\pi$
$348$	0	0
$349$	400.000	1.14613	0.573066	−	0.819509i	$-0.305754\pi$
$349$	400.000	1.14613	0.573066	+	0.819509i	$0.305754\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 403.051i	− 1.14179i	−0.821024	−	0.570894i	$-0.806597\pi$
$353$	− 403.051i	− 1.14179i	0.821024	−	0.570894i	$-0.193403\pi$
$354$	0	0
$355$	270.000	0.760563
$356$	0	0
$357$	0	0
$358$	0	0
$359$	33.9411i	0.0945435i	0.998882	+	0.0472718i	$0.0150527\pi$
$359$	33.9411i	0.0945435i	−0.998882	+	0.0472718i	$0.984947\pi$
$360$	0	0
$361$	−360.000	−0.997230
$362$	0	0
$363$	0	0
$364$	0	0
$365$	72.1249i	0.197602i
$366$	0	0
$367$	223.000	0.607629	0.303815	−	0.952731i	$-0.401740\pi$
$367$	223.000	0.607629	0.303815	+	0.952731i	$0.401740\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	617.000	1.65416	0.827078	−	0.562088i	$-0.190002\pi$
$373$	617.000	1.65416	0.827078	+	0.562088i	$0.190002\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	780.646i	2.07068i
$378$	0	0
$379$	−361.000	−0.952507	−0.476253	−	0.879308i	$-0.658005\pi$
$379$	−361.000	−0.952507	−0.476253	+	0.879308i	$0.658005\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 424.264i	− 1.10774i	−0.832603	−	0.553870i	$-0.813151\pi$
$383$	− 424.264i	− 1.10774i	0.832603	−	0.553870i	$-0.186849\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 106.066i	− 0.272663i	−0.990663	−	0.136332i	$-0.956469\pi$
$389$	− 106.066i	− 0.272663i	0.990663	−	0.136332i	$-0.0435313\pi$
$390$	0	0
$391$	288.000	0.736573
$392$	0	0
$393$	0	0
$394$	0	0
$395$	335.169i	0.848528i
$396$	0	0
$397$	−377.000	−0.949622	−0.474811	−	0.880088i	$-0.657484\pi$
$397$	−377.000	−0.949622	−0.474811	+	0.880088i	$0.657484\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 84.8528i	− 0.211603i	−0.994387	−	0.105802i	$-0.966259\pi$
$401$	− 84.8528i	− 0.211603i	0.994387	−	0.105802i	$-0.0337408\pi$
$402$	0	0
$403$	−1127.00	−2.79653
$404$	0	0
$405$	0	0
$406$	0	0
$407$	360.624i	0.886055i
$408$	0	0
$409$	−689.000	−1.68460	−0.842298	−	0.539012i	$-0.818798\pi$
$409$	−689.000	−1.68460	−0.842298	+	0.539012i	$0.818798\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−450.000	−1.08434
$416$	0	0
$417$	0	0
$418$	0	0
$419$	691.550i	1.65048i	0.564783	+	0.825239i	$0.308960\pi$
$419$	691.550i	1.65048i	−0.564783	+	0.825239i	$0.691040\pi$
$420$	0	0
$421$	281.000	0.667458	0.333729	−	0.942669i	$-0.391693\pi$
$421$	281.000	0.667458	0.333729	+	0.942669i	$0.391693\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 118.794i	− 0.279515i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	436.992i	1.01390i	0.861975	+	0.506951i	$0.169228\pi$
$431$	436.992i	1.01390i	−0.861975	+	0.506951i	$0.830772\pi$
$432$	0	0
$433$	−593.000	−1.36952	−0.684758	−	0.728771i	$-0.740092\pi$
$433$	−593.000	−1.36952	−0.684758	+	0.728771i	$0.740092\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	16.9706i	0.0388342i
$438$	0	0
$439$	472.000	1.07517	0.537585	−	0.843209i	$-0.319337\pi$
$439$	472.000	1.07517	0.537585	+	0.843209i	$0.319337\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 763.675i	− 1.72387i	−0.507017	−	0.861936i	$-0.669252\pi$
$443$	− 763.675i	− 1.72387i	0.507017	−	0.861936i	$-0.330748\pi$
$444$	0	0
$445$	−576.000	−1.29438
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 649.124i	− 1.44571i	−0.690999	−	0.722855i	$-0.742829\pi$
$449$	− 649.124i	− 1.44571i	0.690999	−	0.722855i	$-0.257171\pi$
$450$	0	0
$451$	450.000	0.997783
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	353.000	0.772429	0.386214	−	0.922409i	$-0.373782\pi$
$457$	353.000	0.772429	0.386214	+	0.922409i	$0.373782\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	543.058i	1.17800i	0.808133	+	0.589000i	$0.200478\pi$
$461$	543.058i	1.17800i	−0.808133	+	0.589000i	$0.799522\pi$
$462$	0	0
$463$	−337.000	−0.727862	−0.363931	−	0.931426i	$-0.618566\pi$
$463$	−337.000	−0.727862	−0.363931	+	0.931426i	$0.618566\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	4.24264i	0.00908488i	0.999990	+	0.00454244i	$0.00144591\pi$
$467$	4.24264i	0.00908488i	−0.999990	+	0.00454244i	$0.998554\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	997.021i	2.10787i
$474$	0	0
$475$	7.00000	0.0147368
$476$	0	0
$477$	0	0
$478$	0	0
$479$	424.264i	0.885729i	0.896589	+	0.442864i	$0.146038\pi$
$479$	424.264i	0.885729i	−0.896589	+	0.442864i	$0.853962\pi$
$480$	0	0
$481$	−391.000	−0.812890
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 169.706i	− 0.349909i
$486$	0	0
$487$	287.000	0.589322	0.294661	−	0.955602i	$-0.404793\pi$
$487$	287.000	0.589322	0.294661	+	0.955602i	$0.404793\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	339.411i	0.691265i	0.938370	+	0.345633i	$0.112336\pi$
$491$	339.411i	0.691265i	−0.938370	+	0.345633i	$0.887664\pi$
$492$	0	0
$493$	−576.000	−1.16836
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−121.000	−0.242485	−0.121242	−	0.992623i	$-0.538688\pi$
$499$	−121.000	−0.242485	−0.121242	+	0.992623i	$0.538688\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	292.742i	0.581992i	0.956724	+	0.290996i	$0.0939867\pi$
$503$	292.742i	0.581992i	−0.956724	+	0.290996i	$0.906013\pi$
$504$	0	0
$505$	−90.0000	−0.178218
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 29.6985i	− 0.0583467i	−0.999574	−	0.0291734i	$-0.990713\pi$
$509$	− 29.6985i	− 0.0583467i	0.999574	−	0.0291734i	$-0.00928749\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	97.5807i	0.189477i
$516$	0	0
$517$	−810.000	−1.56673
$518$	0	0
$519$	0	0
$520$	0	0
$521$	848.528i	1.62865i	0.580407	+	0.814326i	$0.302893\pi$
$521$	848.528i	1.62865i	−0.580407	+	0.814326i	$0.697107\pi$
$522$	0	0
$523$	127.000	0.242830	0.121415	−	0.992602i	$-0.461257\pi$
$523$	127.000	0.242830	0.121415	+	0.992602i	$0.461257\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 831.558i	− 1.57791i
$528$	0	0
$529$	241.000	0.455577
$530$	0	0
$531$	0	0
$532$	0	0
$533$	487.904i	0.915392i
$534$	0	0
$535$	−72.0000	−0.134579
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−1.00000	−0.00184843	−0.000924214	−	1.00000i	$-0.500294\pi$
$541$	−1.00000	−0.00184843	−0.000924214		1.00000i	$0.500294\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 301.227i	− 0.552711i
$546$	0	0
$547$	−544.000	−0.994516	−0.497258	−	0.867603i	$-0.665660\pi$
$547$	−544.000	−0.994516	−0.497258	+	0.867603i	$0.665660\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 33.9411i	− 0.0615991i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	258.801i	0.464634i	0.972640	+	0.232317i	$0.0746307\pi$
$557$	258.801i	0.464634i	−0.972640	+	0.232317i	$0.925369\pi$
$558$	0	0
$559$	−1081.00	−1.93381
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 547.301i	− 0.972115i	−0.873927	−	0.486057i	$-0.838435\pi$
$563$	− 547.301i	− 0.972115i	0.873927	−	0.486057i	$-0.161565\pi$
$564$	0	0
$565$	−558.000	−0.987611
$566$	0	0
$567$	0	0
$568$	0	0
$569$	700.036i	1.23029i	0.788414	+	0.615146i	$0.210903\pi$
$569$	700.036i	1.23029i	−0.788414	+	0.615146i	$0.789097\pi$
$570$	0	0
$571$	689.000	1.20665	0.603327	−	0.797494i	$-0.293841\pi$
$571$	689.000	1.20665	0.603327	+	0.797494i	$0.293841\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	118.794i	0.206598i
$576$	0	0
$577$	−953.000	−1.65165	−0.825823	−	0.563929i	$-0.809289\pi$
$577$	−953.000	−1.65165	−0.825823	+	0.563929i	$0.809289\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	1800.00	3.08748
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 780.646i	− 1.32989i	−0.746892	−	0.664945i	$-0.768455\pi$
$587$	− 780.646i	− 1.32989i	0.746892	−	0.664945i	$-0.231545\pi$
$588$	0	0
$589$	49.0000	0.0831919
$590$	0	0
$591$	0	0
$592$	0	0
$593$	869.741i	1.46668i	0.679862	+	0.733340i	$0.262040\pi$
$593$	869.741i	1.46668i	−0.679862	+	0.733340i	$0.737960\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 1018.23i	− 1.69989i	−0.526872	−	0.849945i	$-0.676635\pi$
$599$	− 1018.23i	− 1.69989i	0.526872	−	0.849945i	$-0.323365\pi$
$600$	0	0
$601$	961.000	1.59900	0.799501	−	0.600665i	$-0.205097\pi$
$601$	961.000	1.59900	0.799501	+	0.600665i	$0.205097\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	1395.83i	2.30716i
$606$	0	0
$607$	−743.000	−1.22405	−0.612026	−	0.790837i	$-0.709645\pi$
$607$	−743.000	−1.22405	−0.612026	+	0.790837i	$0.709645\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 878.227i	− 1.43736i
$612$	0	0
$613$	560.000	0.913540	0.456770	−	0.889585i	$-0.349006\pi$
$613$	560.000	0.913540	0.456770	+	0.889585i	$0.349006\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	46.6690i	0.0756387i	0.999285	+	0.0378193i	$0.0120411\pi$
$617$	46.6690i	0.0756387i	−0.999285	+	0.0378193i	$0.987959\pi$
$618$	0	0
$619$	−209.000	−0.337641	−0.168821	−	0.985647i	$-0.553996\pi$
$619$	−209.000	−0.337641	−0.168821	+	0.985647i	$0.553996\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−401.000	−0.641600
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 288.500i	− 0.458664i
$630$	0	0
$631$	560.000	0.887480	0.443740	−	0.896156i	$-0.353651\pi$
$631$	560.000	0.887480	0.443740	+	0.896156i	$0.353651\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 72.1249i	− 0.113583i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 118.794i	− 0.185326i	−0.995698	−	0.0926630i	$-0.970462\pi$
$641$	− 118.794i	− 0.185326i	0.995698	−	0.0926630i	$-0.0295379\pi$
$642$	0	0
$643$	−113.000	−0.175739	−0.0878694	−	0.996132i	$-0.528006\pi$
$643$	−113.000	−0.175739	−0.0878694	+	0.996132i	$0.528006\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	530.330i	0.819676i	0.912158	+	0.409838i	$0.134415\pi$
$647$	530.330i	0.819676i	−0.912158	+	0.409838i	$0.865585\pi$
$648$	0	0
$649$	1080.00	1.66410
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 21.2132i	− 0.0324858i	−0.999868	−	0.0162429i	$-0.994830\pi$
$653$	− 21.2132i	− 0.0324858i	0.999868	−	0.0162429i	$-0.00517050\pi$
$654$	0	0
$655$	−234.000	−0.357252
$656$	0	0
$657$	0	0
$658$	0	0
$659$	933.381i	1.41636i	0.706032	+	0.708180i	$0.250483\pi$
$659$	933.381i	1.41636i	−0.706032	+	0.708180i	$0.749517\pi$
$660$	0	0
$661$	−311.000	−0.470499	−0.235250	−	0.971935i	$-0.575591\pi$
$661$	−311.000	−0.470499	−0.235250	+	0.971935i	$0.575591\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	576.000	0.863568
$668$	0	0
$669$	0	0
$670$	0	0
$671$	848.528i	1.26457i
$672$	0	0
$673$	−223.000	−0.331352	−0.165676	−	0.986180i	$-0.552981\pi$
$673$	−223.000	−0.331352	−0.165676	+	0.986180i	$0.552981\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	407.294i	0.601615i	0.953685	+	0.300808i	$0.0972562\pi$
$677$	407.294i	0.601615i	−0.953685	+	0.300808i	$0.902744\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	322.441i	0.472095i	0.971742	+	0.236047i	$0.0758521\pi$
$683$	322.441i	0.472095i	−0.971742	+	0.236047i	$0.924148\pi$
$684$	0	0
$685$	−288.000	−0.420438
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1951.61i	2.83253i
$690$	0	0
$691$	799.000	1.15630	0.578148	−	0.815932i	$-0.303776\pi$
$691$	799.000	1.15630	0.578148	+	0.815932i	$0.303776\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 131.522i	− 0.189240i
$696$	0	0
$697$	−360.000	−0.516499
$698$	0	0
$699$	0	0
$700$	0	0
$701$	458.205i	0.653645i	0.945086	+	0.326823i	$0.105978\pi$
$701$	458.205i	0.653645i	−0.945086	+	0.326823i	$0.894022\pi$
$702$	0	0
$703$	17.0000	0.0241821
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−40.0000	−0.0564175	−0.0282087	−	0.999602i	$-0.508980\pi$
$709$	−40.0000	−0.0564175	−0.0282087	+	0.999602i	$0.508980\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	831.558i	1.16628i
$714$	0	0
$715$	−2070.00	−2.89510
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 835.800i	− 1.16245i	−0.813744	−	0.581224i	$-0.802574\pi$
$719$	− 835.800i	− 1.16245i	0.813744	−	0.581224i	$-0.197426\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 237.588i	− 0.327707i
$726$	0	0
$727$	943.000	1.29711	0.648556	−	0.761167i	$-0.275373\pi$
$727$	943.000	1.29711	0.648556	+	0.761167i	$0.275373\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 797.616i	− 1.09113i
$732$	0	0
$733$	−527.000	−0.718963	−0.359482	−	0.933152i	$-0.617046\pi$
$733$	−527.000	−0.718963	−0.359482	+	0.933152i	$0.617046\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	487.904i	0.662013i
$738$	0	0
$739$	−991.000	−1.34100	−0.670501	−	0.741909i	$-0.733921\pi$
$739$	−991.000	−1.34100	−0.670501	+	0.741909i	$0.733921\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	530.330i	0.713769i	0.934149	+	0.356884i	$0.116161\pi$
$743$	530.330i	0.713769i	−0.934149	+	0.356884i	$0.883839\pi$
$744$	0	0
$745$	720.000	0.966443
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	761.000	1.01332	0.506658	−	0.862147i	$-0.330881\pi$
$751$	761.000	1.01332	0.506658	+	0.862147i	$0.330881\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	169.706i	0.224776i
$756$	0	0
$757$	1190.00	1.57199	0.785997	−	0.618230i	$-0.212150\pi$
$757$	1190.00	1.57199	0.785997	+	0.618230i	$0.212150\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	797.616i	1.04812i	0.851683	+	0.524058i	$0.175583\pi$
$761$	797.616i	1.04812i	−0.851683	+	0.524058i	$0.824417\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1170.97i	1.52669i
$768$	0	0
$769$	−599.000	−0.778934	−0.389467	−	0.921040i	$-0.627341\pi$
$769$	−599.000	−0.778934	−0.389467	+	0.921040i	$0.627341\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 360.624i	− 0.466526i	−0.972414	−	0.233263i	$-0.925060\pi$
$773$	− 360.624i	− 0.466526i	0.972414	−	0.233263i	$-0.0749403\pi$
$774$	0	0
$775$	343.000	0.442581
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 21.2132i	− 0.0272313i
$780$	0	0
$781$	−1350.00	−1.72855
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 1187.94i	− 1.51330i
$786$	0	0
$787$	−170.000	−0.216010	−0.108005	−	0.994150i	$-0.534446\pi$
$787$	−170.000	−0.216010	−0.108005	+	0.994150i	$0.534446\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−920.000	−1.16015
$794$	0	0
$795$	0	0
$796$	0	0
$797$	322.441i	0.404568i	0.979327	+	0.202284i	$0.0648364\pi$
$797$	322.441i	0.404568i	−0.979327	+	0.202284i	$0.935164\pi$
$798$	0	0
$799$	648.000	0.811014
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 360.624i	− 0.449096i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	309.713i	0.382834i	0.981509	+	0.191417i	$0.0613083\pi$
$809$	309.713i	0.382834i	−0.981509	+	0.191417i	$0.938692\pi$
$810$	0	0
$811$	−1190.00	−1.46732	−0.733662	−	0.679514i	$-0.762190\pi$
$811$	−1190.00	−1.46732	−0.733662	+	0.679514i	$0.762190\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 848.528i	− 1.04114i
$816$	0	0
$817$	47.0000	0.0575275
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 572.756i	− 0.697633i	−0.937191	−	0.348816i	$-0.886584\pi$
$821$	− 572.756i	− 0.697633i	0.937191	−	0.348816i	$-0.113416\pi$
$822$	0	0
$823$	1160.00	1.40948	0.704739	−	0.709467i	$-0.251064\pi$
$823$	1160.00	1.40948	0.704739	+	0.709467i	$0.251064\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	343.654i	0.415543i	0.978177	+	0.207771i	$0.0666210\pi$
$827$	343.654i	0.415543i	−0.978177	+	0.207771i	$0.933379\pi$
$828$	0	0
$829$	−1079.00	−1.30157	−0.650784	−	0.759263i	$-0.725560\pi$
$829$	−1079.00	−1.30157	−0.650784	+	0.759263i	$0.725560\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	450.000	0.538922
$836$	0	0
$837$	0	0
$838$	0	0
$839$	33.9411i	0.0404543i	0.999795	+	0.0202271i	$0.00643893\pi$
$839$	33.9411i	0.0404543i	−0.999795	+	0.0202271i	$0.993561\pi$
$840$	0	0
$841$	−311.000	−0.369798
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 1527.35i	− 1.80752i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	288.500i	0.339012i
$852$	0	0
$853$	1033.00	1.21102	0.605510	−	0.795838i	$-0.292969\pi$
$853$	1033.00	1.21102	0.605510	+	0.795838i	$0.292969\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 237.588i	− 0.277232i	−0.990346	−	0.138616i	$-0.955735\pi$
$857$	− 237.588i	− 0.277232i	0.990346	−	0.138616i	$-0.0442654\pi$
$858$	0	0
$859$	−890.000	−1.03609	−0.518044	−	0.855354i	$-0.673340\pi$
$859$	−890.000	−1.03609	−0.518044	+	0.855354i	$0.673340\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 462.448i	− 0.535861i	−0.963438	−	0.267930i	$-0.913660\pi$
$863$	− 462.448i	− 0.535861i	0.963438	−	0.267930i	$-0.0863397\pi$
$864$	0	0
$865$	1080.00	1.24855
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 1675.84i	− 1.92847i
$870$	0	0
$871$	−529.000	−0.607348
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−1504.00	−1.71494	−0.857469	−	0.514536i	$-0.827964\pi$
$877$	−1504.00	−1.71494	−0.857469	+	0.514536i	$0.827964\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	254.558i	0.288943i	0.989509	+	0.144471i	$0.0461481\pi$
$881$	254.558i	0.288943i	−0.989509	+	0.144471i	$0.953852\pi$
$882$	0	0
$883$	1313.00	1.48698	0.743488	−	0.668749i	$-0.233170\pi$
$883$	1313.00	1.48698	0.743488	+	0.668749i	$0.233170\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1166.73i	1.31536i	0.753297	+	0.657681i	$0.228463\pi$
$887$	1166.73i	1.31536i	−0.753297	+	0.657681i	$0.771537\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	38.1838i	0.0427590i
$894$	0	0
$895$	−846.000	−0.945251
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 1663.12i	− 1.84996i
$900$	0	0
$901$	−1440.00	−1.59822
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 1149.76i	− 1.27045i
$906$	0	0
$907$	−817.000	−0.900772	−0.450386	−	0.892834i	$-0.648714\pi$
$907$	−817.000	−0.900772	−0.450386	+	0.892834i	$0.648714\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 899.440i	− 0.987310i	−0.869658	−	0.493655i	$-0.835660\pi$
$911$	− 899.440i	− 0.987310i	0.869658	−	0.493655i	$-0.164340\pi$
$912$	0	0
$913$	2250.00	2.46440
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−1729.00	−1.88139	−0.940696	−	0.339250i	$-0.889827\pi$
$919$	−1729.00	−1.88139	−0.940696	+	0.339250i	$0.889827\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 1463.71i	− 1.58582i
$924$	0	0
$925$	119.000	0.128649
$926$	0	0
$927$	0	0
$928$	0	0
$929$	954.594i	1.02755i	0.857925	+	0.513775i	$0.171753\pi$
$929$	954.594i	1.02755i	−0.857925	+	0.513775i	$0.828247\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 1527.35i	− 1.63353i
$936$	0	0
$937$	−377.000	−0.402348	−0.201174	−	0.979556i	$-0.564476\pi$
$937$	−377.000	−0.402348	−0.201174	+	0.979556i	$0.564476\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 967.322i	− 1.02797i	−0.857798	−	0.513986i	$-0.828168\pi$
$941$	− 967.322i	− 1.02797i	0.857798	−	0.513986i	$-0.171832\pi$
$942$	0	0
$943$	360.000	0.381760
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1336.43i	1.41123i	0.708597	+	0.705613i	$0.249328\pi$
$947$	1336.43i	1.41123i	−0.708597	+	0.705613i	$0.750672\pi$
$948$	0	0
$949$	391.000	0.412013
$950$	0	0
$951$	0	0
$952$	0	0
$953$	763.675i	0.801338i	0.916223	+	0.400669i	$0.131222\pi$
$953$	763.675i	0.801338i	−0.916223	+	0.400669i	$0.868778\pi$
$954$	0	0
$955$	1026.00	1.07435
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	1440.00	1.49844
$962$	0	0
$963$	0	0
$964$	0	0
$965$	436.992i	0.452841i
$966$	0	0
$967$	−1057.00	−1.09307	−0.546536	−	0.837436i	$-0.684054\pi$
$967$	−1057.00	−1.09307	−0.546536	+	0.837436i	$0.684054\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 1103.09i	− 1.13603i	−0.823018	−	0.568016i	$-0.807711\pi$
$971$	− 1103.09i	− 1.13603i	0.823018	−	0.568016i	$-0.192289\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 318.198i	− 0.325689i	−0.986652	−	0.162844i	$-0.947933\pi$
$977$	− 318.198i	− 0.325689i	0.986652	−	0.162844i	$-0.0520669\pi$
$978$	0	0
$979$	2880.00	2.94178
$980$	0	0
$981$	0	0
$982$	0	0
$983$	526.087i	0.535186i	0.963532	+	0.267593i	$0.0862282\pi$
$983$	526.087i	0.535186i	−0.963532	+	0.267593i	$0.913772\pi$
$984$	0	0
$985$	−648.000	−0.657868
$986$	0	0
$987$	0	0
$988$	0	0
$989$	797.616i	0.806488i
$990$	0	0
$991$	431.000	0.434914	0.217457	−	0.976070i	$-0.430224\pi$
$991$	431.000	0.434914	0.217457	+	0.976070i	$0.430224\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 678.823i	− 0.682234i
$996$	0	0
$997$	−137.000	−0.137412	−0.0687061	−	0.997637i	$-0.521887\pi$
$997$	−137.000	−0.137412	−0.0687061	+	0.997637i	$0.521887\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.3.c.a.197.1		2
3.2	odd	2	inner	1764.3.c.a.197.2		2
7.2	even	3		1764.3.bk.a.557.1		4
7.3	odd	6		252.3.bk.a.233.1	yes	4
7.4	even	3		1764.3.bk.a.1745.2		4
7.5	odd	6		252.3.bk.a.53.2	yes	4
7.6	odd	2		1764.3.c.d.197.2		2
21.2	odd	6		1764.3.bk.a.557.2		4
21.5	even	6		252.3.bk.a.53.1	✓	4
21.11	odd	6		1764.3.bk.a.1745.1		4
21.17	even	6		252.3.bk.a.233.2	yes	4
21.20	even	2		1764.3.c.d.197.1		2
28.3	even	6		1008.3.dc.c.737.1		4
28.19	even	6		1008.3.dc.c.305.2		4
84.47	odd	6		1008.3.dc.c.305.1		4
84.59	odd	6		1008.3.dc.c.737.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.3.bk.a.53.1	✓	4	21.5	even	6
252.3.bk.a.53.2	yes	4	7.5	odd	6
252.3.bk.a.233.1	yes	4	7.3	odd	6
252.3.bk.a.233.2	yes	4	21.17	even	6
1008.3.dc.c.305.1		4	84.47	odd	6
1008.3.dc.c.305.2		4	28.19	even	6
1008.3.dc.c.737.1		4	28.3	even	6
1008.3.dc.c.737.2		4	84.59	odd	6
1764.3.c.a.197.1		2	1.1	even	1	trivial
1764.3.c.a.197.2		2	3.2	odd	2	inner
1764.3.c.d.197.1		2	21.20	even	2
1764.3.c.d.197.2		2	7.6	odd	2
1764.3.bk.a.557.1		4	7.2	even	3
1764.3.bk.a.557.2		4	21.2	odd	6
1764.3.bk.a.1745.1		4	21.11	odd	6
1764.3.bk.a.1745.2		4	7.4	even	3

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:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1764.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-4.24264i q^{5} +O(q^{10})\)	\(q-4.24264i q^{5} +21.2132i q^{11} -23.0000 q^{13} -16.9706i q^{17} +1.00000 q^{19} +16.9706i q^{23} +7.00000 q^{25} -33.9411i q^{29} +49.0000 q^{31} +17.0000 q^{37} -21.2132i q^{41} +47.0000 q^{43} +38.1838i q^{47} -84.8528i q^{53} +90.0000 q^{55} -50.9117i q^{59} +40.0000 q^{61} +97.5807i q^{65} +23.0000 q^{67} +63.6396i q^{71} -17.0000 q^{73} -79.0000 q^{79} -106.066i q^{83} -72.0000 q^{85} -135.765i q^{89} -4.24264i q^{95} +40.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q+O(q^{10}) \) 2 * q	\( 2 q - 46 q^{13} + 2 q^{19} + 14 q^{25} + 98 q^{31} + 34 q^{37} + 94 q^{43} + 180 q^{55} + 80 q^{61} + 46 q^{67} - 34 q^{73} - 158 q^{79} - 144 q^{85} + 80 q^{97}+O(q^{100}) \) 2 * q - 46 * q^13 + 2 * q^19 + 14 * q^25 + 98 * q^31 + 34 * q^37 + 94 * q^43 + 180 * q^55 + 80 * q^61 + 46 * q^67 - 34 * q^73 - 158 * q^79 - 144 * q^85 + 80 * q^97

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