LMFDB - Embedded newform 882.4.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [882,4,Mod(1,882)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(882, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("882.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 882 = 2 \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	882.a (trivial)

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:	yes
Analytic conductor:	$52.0396846251$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 42)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	882.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-2.00000 q^{2} +4.00000 q^{4} +2.00000 q^{5} -8.00000 q^{8} +O(q^{10})$

$q-2.00000 q^{2} +4.00000 q^{4} +2.00000 q^{5} -8.00000 q^{8} -4.00000 q^{10} +8.00000 q^{11} +42.0000 q^{13} +16.0000 q^{16} -2.00000 q^{17} +124.000 q^{19} +8.00000 q^{20} -16.0000 q^{22} -76.0000 q^{23} -121.000 q^{25} -84.0000 q^{26} -254.000 q^{29} +72.0000 q^{31} -32.0000 q^{32} +4.00000 q^{34} +398.000 q^{37} -248.000 q^{38} -16.0000 q^{40} +462.000 q^{41} +212.000 q^{43} +32.0000 q^{44} +152.000 q^{46} -264.000 q^{47} +242.000 q^{50} +168.000 q^{52} +162.000 q^{53} +16.0000 q^{55} +508.000 q^{58} -772.000 q^{59} -30.0000 q^{61} -144.000 q^{62} +64.0000 q^{64} +84.0000 q^{65} -764.000 q^{67} -8.00000 q^{68} +236.000 q^{71} -418.000 q^{73} -796.000 q^{74} +496.000 q^{76} +552.000 q^{79} +32.0000 q^{80} -924.000 q^{82} +1036.00 q^{83} -4.00000 q^{85} -424.000 q^{86} -64.0000 q^{88} +30.0000 q^{89} -304.000 q^{92} +528.000 q^{94} +248.000 q^{95} +1190.00 q^{97} +O(q^{100})$

Display 100 coefficients 10 coefficients

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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	2.00000	0.178885	0.0894427	−	0.995992i	$-0.471491\pi$
$5$	2.00000	0.178885	0.0894427	+	0.995992i	$0.471491\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−8.00000	−0.353553
$9$	0	0
$10$	−4.00000	−0.126491
$11$	8.00000	0.219281	0.109640	−	0.993971i	$-0.465030\pi$
$11$	8.00000	0.219281	0.109640	+	0.993971i	$0.465030\pi$
$12$	0	0
$13$	42.0000	0.896054	0.448027	−	0.894020i	$-0.352127\pi$
$13$	42.0000	0.896054	0.448027	+	0.894020i	$0.352127\pi$
$14$	0	0
$15$	0	0
$16$	16.0000	0.250000
$17$	−2.00000	−0.0285336	−0.0142668	−	0.999898i	$-0.504541\pi$
$17$	−2.00000	−0.0285336	−0.0142668	+	0.999898i	$0.504541\pi$
$18$	0	0
$19$	124.000	1.49724	0.748620	−	0.663000i	$-0.230717\pi$
$19$	124.000	1.49724	0.748620	+	0.663000i	$0.230717\pi$
$20$	8.00000	0.0894427
$21$	0	0
$22$	−16.0000	−0.155055
$23$	−76.0000	−0.689004	−0.344502	−	0.938786i	$-0.611952\pi$
$23$	−76.0000	−0.689004	−0.344502	+	0.938786i	$0.611952\pi$
$24$	0	0
$25$	−121.000	−0.968000
$26$	−84.0000	−0.633606
$27$	0	0
$28$	0	0
$29$	−254.000	−1.62644	−0.813218	−	0.581960i	$-0.802286\pi$
$29$	−254.000	−1.62644	−0.813218	+	0.581960i	$0.802286\pi$
$30$	0	0
$31$	72.0000	0.417148	0.208574	−	0.978007i	$-0.433118\pi$
$31$	72.0000	0.417148	0.208574	+	0.978007i	$0.433118\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	4.00000	0.0201763
$35$	0	0
$36$	0	0
$37$	398.000	1.76840	0.884200	−	0.467109i	$-0.154704\pi$
$37$	398.000	1.76840	0.884200	+	0.467109i	$0.154704\pi$
$38$	−248.000	−1.05871
$39$	0	0
$40$	−16.0000	−0.0632456
$41$	462.000	1.75981	0.879906	−	0.475148i	$-0.157606\pi$
$41$	462.000	1.75981	0.879906	+	0.475148i	$0.157606\pi$
$42$	0	0
$43$	212.000	0.751853	0.375927	−	0.926649i	$-0.377324\pi$
$43$	212.000	0.751853	0.375927	+	0.926649i	$0.377324\pi$
$44$	32.0000	0.109640
$45$	0	0
$46$	152.000	0.487200
$47$	−264.000	−0.819327	−0.409663	−	0.912237i	$-0.634354\pi$
$47$	−264.000	−0.819327	−0.409663	+	0.912237i	$0.634354\pi$
$48$	0	0
$49$	0	0
$50$	242.000	0.684479
$51$	0	0
$52$	168.000	0.448027
$53$	162.000	0.419857	0.209928	−	0.977717i	$-0.432677\pi$
$53$	162.000	0.419857	0.209928	+	0.977717i	$0.432677\pi$
$54$	0	0
$55$	16.0000	0.0392262
$56$	0	0
$57$	0	0
$58$	508.000	1.15006
$59$	−772.000	−1.70349	−0.851744	−	0.523958i	$-0.824455\pi$
$59$	−772.000	−1.70349	−0.851744	+	0.523958i	$0.824455\pi$
$60$	0	0
$61$	−30.0000	−0.0629690	−0.0314845	−	0.999504i	$-0.510023\pi$
$61$	−30.0000	−0.0629690	−0.0314845	+	0.999504i	$0.510023\pi$
$62$	−144.000	−0.294968
$63$	0	0
$64$	64.0000	0.125000
$65$	84.0000	0.160291
$66$	0	0
$67$	−764.000	−1.39310	−0.696548	−	0.717510i	$-0.745282\pi$
$67$	−764.000	−1.39310	−0.696548	+	0.717510i	$0.745282\pi$
$68$	−8.00000	−0.0142668
$69$	0	0
$70$	0	0
$71$	236.000	0.394480	0.197240	−	0.980355i	$-0.436802\pi$
$71$	236.000	0.394480	0.197240	+	0.980355i	$0.436802\pi$
$72$	0	0
$73$	−418.000	−0.670181	−0.335090	−	0.942186i	$-0.608767\pi$
$73$	−418.000	−0.670181	−0.335090	+	0.942186i	$0.608767\pi$
$74$	−796.000	−1.25045
$75$	0	0
$76$	496.000	0.748620
$77$	0	0
$78$	0	0
$79$	552.000	0.786137	0.393069	−	0.919509i	$-0.371413\pi$
$79$	552.000	0.786137	0.393069	+	0.919509i	$0.371413\pi$
$80$	32.0000	0.0447214
$81$	0	0
$82$	−924.000	−1.24437
$83$	1036.00	1.37007	0.685035	−	0.728510i	$-0.259787\pi$
$83$	1036.00	1.37007	0.685035	+	0.728510i	$0.259787\pi$
$84$	0	0
$85$	−4.00000	−0.00510425
$86$	−424.000	−0.531641
$87$	0	0
$88$	−64.0000	−0.0775275
$89$	30.0000	0.0357303	0.0178651	−	0.999840i	$-0.494313\pi$
$89$	30.0000	0.0357303	0.0178651	+	0.999840i	$0.494313\pi$
$90$	0	0
$91$	0	0
$92$	−304.000	−0.344502
$93$	0	0
$94$	528.000	0.579352
$95$	248.000	0.267834
$96$	0	0
$97$	1190.00	1.24563	0.622815	−	0.782369i	$-0.285989\pi$
$97$	1190.00	1.24563	0.622815	+	0.782369i	$0.285989\pi$
$98$	0	0
$99$	0	0
$100$	−484.000	−0.484000
$101$	1370.00	1.34970	0.674852	−	0.737953i	$-0.264207\pi$
$101$	1370.00	1.34970	0.674852	+	0.737953i	$0.264207\pi$
$102$	0	0
$103$	−464.000	−0.443876	−0.221938	−	0.975061i	$-0.571238\pi$
$103$	−464.000	−0.443876	−0.221938	+	0.975061i	$0.571238\pi$
$104$	−336.000	−0.316803
$105$	0	0
$106$	−324.000	−0.296884
$107$	2136.00	1.92986	0.964930	−	0.262509i	$-0.0845500\pi$
$107$	2136.00	1.92986	0.964930	+	0.262509i	$0.0845500\pi$
$108$	0	0
$109$	−1226.00	−1.07733	−0.538667	−	0.842518i	$-0.681072\pi$
$109$	−1226.00	−1.07733	−0.538667	+	0.842518i	$0.681072\pi$
$110$	−32.0000	−0.0277371
$111$	0	0
$112$	0	0
$113$	−338.000	−0.281384	−0.140692	−	0.990053i	$-0.544933\pi$
$113$	−338.000	−0.281384	−0.140692	+	0.990053i	$0.544933\pi$
$114$	0	0
$115$	−152.000	−0.123253
$116$	−1016.00	−0.813218
$117$	0	0
$118$	1544.00	1.20455
$119$	0	0
$120$	0	0
$121$	−1267.00	−0.951916
$122$	60.0000	0.0445258
$123$	0	0
$124$	288.000	0.208574
$125$	−492.000	−0.352047
$126$	0	0
$127$	2088.00	1.45890	0.729449	−	0.684035i	$-0.239777\pi$
$127$	2088.00	1.45890	0.729449	+	0.684035i	$0.239777\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	−168.000	−0.113343
$131$	−292.000	−0.194749	−0.0973747	−	0.995248i	$-0.531045\pi$
$131$	−292.000	−0.194749	−0.0973747	+	0.995248i	$0.531045\pi$
$132$	0	0
$133$	0	0
$134$	1528.00	0.985068
$135$	0	0
$136$	16.0000	0.0100882
$137$	−818.000	−0.510120	−0.255060	−	0.966925i	$-0.582095\pi$
$137$	−818.000	−0.510120	−0.255060	+	0.966925i	$0.582095\pi$
$138$	0	0
$139$	2156.00	1.31561	0.657804	−	0.753189i	$-0.271485\pi$
$139$	2156.00	1.31561	0.657804	+	0.753189i	$0.271485\pi$
$140$	0	0
$141$	0	0
$142$	−472.000	−0.278939
$143$	336.000	0.196488
$144$	0	0
$145$	−508.000	−0.290946
$146$	836.000	0.473889
$147$	0	0
$148$	1592.00	0.884200
$149$	2850.00	1.56699	0.783494	−	0.621400i	$-0.213436\pi$
$149$	2850.00	1.56699	0.783494	+	0.621400i	$0.213436\pi$
$150$	0	0
$151$	1672.00	0.901096	0.450548	−	0.892752i	$-0.351229\pi$
$151$	1672.00	0.901096	0.450548	+	0.892752i	$0.351229\pi$
$152$	−992.000	−0.529354
$153$	0	0
$154$	0	0
$155$	144.000	0.0746217
$156$	0	0
$157$	−446.000	−0.226718	−0.113359	−	0.993554i	$-0.536161\pi$
$157$	−446.000	−0.226718	−0.113359	+	0.993554i	$0.536161\pi$
$158$	−1104.00	−0.555883
$159$	0	0
$160$	−64.0000	−0.0316228
$161$	0	0
$162$	0	0
$163$	2708.00	1.30127	0.650635	−	0.759391i	$-0.274503\pi$
$163$	2708.00	1.30127	0.650635	+	0.759391i	$0.274503\pi$
$164$	1848.00	0.879906
$165$	0	0
$166$	−2072.00	−0.968785
$167$	896.000	0.415177	0.207589	−	0.978216i	$-0.433439\pi$
$167$	896.000	0.415177	0.207589	+	0.978216i	$0.433439\pi$
$168$	0	0
$169$	−433.000	−0.197087
$170$	8.00000	0.00360925
$171$	0	0
$172$	848.000	0.375927
$173$	4034.00	1.77283	0.886414	−	0.462893i	$-0.153189\pi$
$173$	4034.00	1.77283	0.886414	+	0.462893i	$0.153189\pi$
$174$	0	0
$175$	0	0
$176$	128.000	0.0548202
$177$	0	0
$178$	−60.0000	−0.0252651
$179$	3480.00	1.45311	0.726557	−	0.687106i	$-0.241119\pi$
$179$	3480.00	1.45311	0.726557	+	0.687106i	$0.241119\pi$
$180$	0	0
$181$	2898.00	1.19009	0.595046	−	0.803692i	$-0.297134\pi$
$181$	2898.00	1.19009	0.595046	+	0.803692i	$0.297134\pi$
$182$	0	0
$183$	0	0
$184$	608.000	0.243600
$185$	796.000	0.316341
$186$	0	0
$187$	−16.0000	−0.00625688
$188$	−1056.00	−0.409663
$189$	0	0
$190$	−496.000	−0.189387
$191$	−2652.00	−1.00467	−0.502335	−	0.864673i	$-0.667526\pi$
$191$	−2652.00	−1.00467	−0.502335	+	0.864673i	$0.667526\pi$
$192$	0	0
$193$	146.000	0.0544524	0.0272262	−	0.999629i	$-0.491333\pi$
$193$	146.000	0.0544524	0.0272262	+	0.999629i	$0.491333\pi$
$194$	−2380.00	−0.880794
$195$	0	0
$196$	0	0
$197$	2546.00	0.920787	0.460393	−	0.887715i	$-0.347708\pi$
$197$	2546.00	0.920787	0.460393	+	0.887715i	$0.347708\pi$
$198$	0	0
$199$	2536.00	0.903378	0.451689	−	0.892175i	$-0.350822\pi$
$199$	2536.00	0.903378	0.451689	+	0.892175i	$0.350822\pi$
$200$	968.000	0.342240
$201$	0	0
$202$	−2740.00	−0.954385
$203$	0	0
$204$	0	0
$205$	924.000	0.314805
$206$	928.000	0.313868
$207$	0	0
$208$	672.000	0.224014
$209$	992.000	0.328316
$210$	0	0
$211$	−1300.00	−0.424150	−0.212075	−	0.977253i	$-0.568022\pi$
$211$	−1300.00	−0.424150	−0.212075	+	0.977253i	$0.568022\pi$
$212$	648.000	0.209928
$213$	0	0
$214$	−4272.00	−1.36462
$215$	424.000	0.134496
$216$	0	0
$217$	0	0
$218$	2452.00	0.761791
$219$	0	0
$220$	64.0000	0.0196131
$221$	−84.0000	−0.0255677
$222$	0	0
$223$	−2576.00	−0.773550	−0.386775	−	0.922174i	$-0.626411\pi$
$223$	−2576.00	−0.773550	−0.386775	+	0.922174i	$0.626411\pi$
$224$	0	0
$225$	0	0
$226$	676.000	0.198968
$227$	−1836.00	−0.536826	−0.268413	−	0.963304i	$-0.586499\pi$
$227$	−1836.00	−0.536826	−0.268413	+	0.963304i	$0.586499\pi$
$228$	0	0
$229$	1874.00	0.540775	0.270387	−	0.962752i	$-0.412848\pi$
$229$	1874.00	0.540775	0.270387	+	0.962752i	$0.412848\pi$
$230$	304.000	0.0871529
$231$	0	0
$232$	2032.00	0.575032
$233$	−3730.00	−1.04876	−0.524379	−	0.851485i	$-0.675702\pi$
$233$	−3730.00	−1.04876	−0.524379	+	0.851485i	$0.675702\pi$
$234$	0	0
$235$	−528.000	−0.146566
$236$	−3088.00	−0.851744
$237$	0	0
$238$	0	0
$239$	−2004.00	−0.542377	−0.271188	−	0.962526i	$-0.587417\pi$
$239$	−2004.00	−0.542377	−0.271188	+	0.962526i	$0.587417\pi$
$240$	0	0
$241$	646.000	0.172666	0.0863330	−	0.996266i	$-0.472485\pi$
$241$	646.000	0.172666	0.0863330	+	0.996266i	$0.472485\pi$
$242$	2534.00	0.673106
$243$	0	0
$244$	−120.000	−0.0314845
$245$	0	0
$246$	0	0
$247$	5208.00	1.34161
$248$	−576.000	−0.147484
$249$	0	0
$250$	984.000	0.248934
$251$	1260.00	0.316855	0.158427	−	0.987371i	$-0.449358\pi$
$251$	1260.00	0.316855	0.158427	+	0.987371i	$0.449358\pi$
$252$	0	0
$253$	−608.000	−0.151086
$254$	−4176.00	−1.03160
$255$	0	0
$256$	256.000	0.0625000
$257$	5910.00	1.43446	0.717229	−	0.696838i	$-0.245410\pi$
$257$	5910.00	1.43446	0.717229	+	0.696838i	$0.245410\pi$
$258$	0	0
$259$	0	0
$260$	336.000	0.0801455
$261$	0	0
$262$	584.000	0.137709
$263$	−2988.00	−0.700563	−0.350281	−	0.936645i	$-0.613914\pi$
$263$	−2988.00	−0.700563	−0.350281	+	0.936645i	$0.613914\pi$
$264$	0	0
$265$	324.000	0.0751063
$266$	0	0
$267$	0	0
$268$	−3056.00	−0.696548
$269$	−1318.00	−0.298736	−0.149368	−	0.988782i	$-0.547724\pi$
$269$	−1318.00	−0.298736	−0.149368	+	0.988782i	$0.547724\pi$
$270$	0	0
$271$	5640.00	1.26423	0.632114	−	0.774876i	$-0.282188\pi$
$271$	5640.00	1.26423	0.632114	+	0.774876i	$0.282188\pi$
$272$	−32.0000	−0.00713340
$273$	0	0
$274$	1636.00	0.360709
$275$	−968.000	−0.212264
$276$	0	0
$277$	6446.00	1.39820	0.699102	−	0.715022i	$-0.253583\pi$
$277$	6446.00	1.39820	0.699102	+	0.715022i	$0.253583\pi$
$278$	−4312.00	−0.930275
$279$	0	0
$280$	0	0
$281$	−4930.00	−1.04662	−0.523308	−	0.852144i	$-0.675302\pi$
$281$	−4930.00	−1.04662	−0.523308	+	0.852144i	$0.675302\pi$
$282$	0	0
$283$	6260.00	1.31491	0.657453	−	0.753496i	$-0.271634\pi$
$283$	6260.00	1.31491	0.657453	+	0.753496i	$0.271634\pi$
$284$	944.000	0.197240
$285$	0	0
$286$	−672.000	−0.138938
$287$	0	0
$288$	0	0
$289$	−4909.00	−0.999186
$290$	1016.00	0.205730
$291$	0	0
$292$	−1672.00	−0.335090
$293$	−2310.00	−0.460586	−0.230293	−	0.973121i	$-0.573968\pi$
$293$	−2310.00	−0.460586	−0.230293	+	0.973121i	$0.573968\pi$
$294$	0	0
$295$	−1544.00	−0.304729
$296$	−3184.00	−0.625224
$297$	0	0
$298$	−5700.00	−1.10803
$299$	−3192.00	−0.617385
$300$	0	0
$301$	0	0
$302$	−3344.00	−0.637171
$303$	0	0
$304$	1984.00	0.374310
$305$	−60.0000	−0.0112642
$306$	0	0
$307$	−196.000	−0.0364375	−0.0182187	−	0.999834i	$-0.505800\pi$
$307$	−196.000	−0.0364375	−0.0182187	+	0.999834i	$0.505800\pi$
$308$	0	0
$309$	0	0
$310$	−288.000	−0.0527655
$311$	−6736.00	−1.22818	−0.614089	−	0.789237i	$-0.710477\pi$
$311$	−6736.00	−1.22818	−0.614089	+	0.789237i	$0.710477\pi$
$312$	0	0
$313$	−394.000	−0.0711508	−0.0355754	−	0.999367i	$-0.511326\pi$
$313$	−394.000	−0.0711508	−0.0355754	+	0.999367i	$0.511326\pi$
$314$	892.000	0.160314
$315$	0	0
$316$	2208.00	0.393069
$317$	6714.00	1.18958	0.594788	−	0.803882i	$-0.297236\pi$
$317$	6714.00	1.18958	0.594788	+	0.803882i	$0.297236\pi$
$318$	0	0
$319$	−2032.00	−0.356646
$320$	128.000	0.0223607
$321$	0	0
$322$	0	0
$323$	−248.000	−0.0427216
$324$	0	0
$325$	−5082.00	−0.867380
$326$	−5416.00	−0.920136
$327$	0	0
$328$	−3696.00	−0.622187
$329$	0	0
$330$	0	0
$331$	692.000	0.114912	0.0574558	−	0.998348i	$-0.481701\pi$
$331$	692.000	0.114912	0.0574558	+	0.998348i	$0.481701\pi$
$332$	4144.00	0.685035
$333$	0	0
$334$	−1792.00	−0.293574
$335$	−1528.00	−0.249205
$336$	0	0
$337$	−1566.00	−0.253132	−0.126566	−	0.991958i	$-0.540396\pi$
$337$	−1566.00	−0.253132	−0.126566	+	0.991958i	$0.540396\pi$
$338$	866.000	0.139362
$339$	0	0
$340$	−16.0000	−0.00255212
$341$	576.000	0.0914726
$342$	0	0
$343$	0	0
$344$	−1696.00	−0.265820
$345$	0	0
$346$	−8068.00	−1.25358
$347$	5328.00	0.824271	0.412135	−	0.911123i	$-0.364783\pi$
$347$	5328.00	0.824271	0.412135	+	0.911123i	$0.364783\pi$
$348$	0	0
$349$	−11326.0	−1.73715	−0.868577	−	0.495554i	$-0.834965\pi$
$349$	−11326.0	−1.73715	−0.868577	+	0.495554i	$0.834965\pi$
$350$	0	0
$351$	0	0
$352$	−256.000	−0.0387638
$353$	−2130.00	−0.321157	−0.160579	−	0.987023i	$-0.551336\pi$
$353$	−2130.00	−0.321157	−0.160579	+	0.987023i	$0.551336\pi$
$354$	0	0
$355$	472.000	0.0705666
$356$	120.000	0.0178651
$357$	0	0
$358$	−6960.00	−1.02751
$359$	−3044.00	−0.447510	−0.223755	−	0.974645i	$-0.571832\pi$
$359$	−3044.00	−0.447510	−0.223755	+	0.974645i	$0.571832\pi$
$360$	0	0
$361$	8517.00	1.24173
$362$	−5796.00	−0.841522
$363$	0	0
$364$	0	0
$365$	−836.000	−0.119886
$366$	0	0
$367$	−12416.0	−1.76597	−0.882984	−	0.469404i	$-0.844469\pi$
$367$	−12416.0	−1.76597	−0.882984	+	0.469404i	$0.844469\pi$
$368$	−1216.00	−0.172251
$369$	0	0
$370$	−1592.00	−0.223687
$371$	0	0
$372$	0	0
$373$	−7442.00	−1.03306	−0.516531	−	0.856268i	$-0.672777\pi$
$373$	−7442.00	−1.03306	−0.516531	+	0.856268i	$0.672777\pi$
$374$	32.0000	0.00442428
$375$	0	0
$376$	2112.00	0.289676
$377$	−10668.0	−1.45737
$378$	0	0
$379$	100.000	0.0135532	0.00677659	−	0.999977i	$-0.497843\pi$
$379$	100.000	0.0135532	0.00677659	+	0.999977i	$0.497843\pi$
$380$	992.000	0.133917
$381$	0	0
$382$	5304.00	0.710409
$383$	8080.00	1.07799	0.538993	−	0.842310i	$-0.318805\pi$
$383$	8080.00	1.07799	0.538993	+	0.842310i	$0.318805\pi$
$384$	0	0
$385$	0	0
$386$	−292.000	−0.0385036
$387$	0	0
$388$	4760.00	0.622815
$389$	5482.00	0.714520	0.357260	−	0.934005i	$-0.383711\pi$
$389$	5482.00	0.714520	0.357260	+	0.934005i	$0.383711\pi$
$390$	0	0
$391$	152.000	0.0196598
$392$	0	0
$393$	0	0
$394$	−5092.00	−0.651095
$395$	1104.00	0.140629
$396$	0	0
$397$	−10446.0	−1.32058	−0.660289	−	0.751011i	$-0.729566\pi$
$397$	−10446.0	−1.32058	−0.660289	+	0.751011i	$0.729566\pi$
$398$	−5072.00	−0.638785
$399$	0	0
$400$	−1936.00	−0.242000
$401$	11334.0	1.41145	0.705727	−	0.708484i	$-0.250621\pi$
$401$	11334.0	1.41145	0.705727	+	0.708484i	$0.250621\pi$
$402$	0	0
$403$	3024.00	0.373787
$404$	5480.00	0.674852
$405$	0	0
$406$	0	0
$407$	3184.00	0.387776
$408$	0	0
$409$	−8594.00	−1.03899	−0.519494	−	0.854474i	$-0.673879\pi$
$409$	−8594.00	−1.03899	−0.519494	+	0.854474i	$0.673879\pi$
$410$	−1848.00	−0.222601
$411$	0	0
$412$	−1856.00	−0.221938
$413$	0	0
$414$	0	0
$415$	2072.00	0.245085
$416$	−1344.00	−0.158401
$417$	0	0
$418$	−1984.00	−0.232155
$419$	10500.0	1.22424	0.612122	−	0.790763i	$-0.290316\pi$
$419$	10500.0	1.22424	0.612122	+	0.790763i	$0.290316\pi$
$420$	0	0
$421$	−12066.0	−1.39682	−0.698410	−	0.715698i	$-0.746109\pi$
$421$	−12066.0	−1.39682	−0.698410	+	0.715698i	$0.746109\pi$
$422$	2600.00	0.299919
$423$	0	0
$424$	−1296.00	−0.148442
$425$	242.000	0.0276205
$426$	0	0
$427$	0	0
$428$	8544.00	0.964930
$429$	0	0
$430$	−848.000	−0.0951028
$431$	−4332.00	−0.484142	−0.242071	−	0.970259i	$-0.577827\pi$
$431$	−4332.00	−0.484142	−0.242071	+	0.970259i	$0.577827\pi$
$432$	0	0
$433$	1918.00	0.212871	0.106436	−	0.994320i	$-0.466056\pi$
$433$	1918.00	0.212871	0.106436	+	0.994320i	$0.466056\pi$
$434$	0	0
$435$	0	0
$436$	−4904.00	−0.538667
$437$	−9424.00	−1.03160
$438$	0	0
$439$	7992.00	0.868878	0.434439	−	0.900701i	$-0.356947\pi$
$439$	7992.00	0.868878	0.434439	+	0.900701i	$0.356947\pi$
$440$	−128.000	−0.0138685
$441$	0	0
$442$	168.000	0.0180791
$443$	−3184.00	−0.341482	−0.170741	−	0.985316i	$-0.554616\pi$
$443$	−3184.00	−0.341482	−0.170741	+	0.985316i	$0.554616\pi$
$444$	0	0
$445$	60.0000	0.00639162
$446$	5152.00	0.546983
$447$	0	0
$448$	0	0
$449$	−11426.0	−1.20095	−0.600475	−	0.799644i	$-0.705022\pi$
$449$	−11426.0	−1.20095	−0.600475	+	0.799644i	$0.705022\pi$
$450$	0	0
$451$	3696.00	0.385893
$452$	−1352.00	−0.140692
$453$	0	0
$454$	3672.00	0.379594
$455$	0	0
$456$	0	0
$457$	−16934.0	−1.73335	−0.866673	−	0.498877i	$-0.833746\pi$
$457$	−16934.0	−1.73335	−0.866673	+	0.498877i	$0.833746\pi$
$458$	−3748.00	−0.382385
$459$	0	0
$460$	−608.000	−0.0616264
$461$	−17038.0	−1.72134	−0.860671	−	0.509161i	$-0.829956\pi$
$461$	−17038.0	−1.72134	−0.860671	+	0.509161i	$0.829956\pi$
$462$	0	0
$463$	−13592.0	−1.36431	−0.682153	−	0.731209i	$-0.738956\pi$
$463$	−13592.0	−1.36431	−0.682153	+	0.731209i	$0.738956\pi$
$464$	−4064.00	−0.406609
$465$	0	0
$466$	7460.00	0.741583
$467$	8612.00	0.853353	0.426676	−	0.904404i	$-0.359684\pi$
$467$	8612.00	0.853353	0.426676	+	0.904404i	$0.359684\pi$
$468$	0	0
$469$	0	0
$470$	1056.00	0.103638
$471$	0	0
$472$	6176.00	0.602274
$473$	1696.00	0.164867
$474$	0	0
$475$	−15004.0	−1.44933
$476$	0	0
$477$	0	0
$478$	4008.00	0.383518
$479$	7432.00	0.708928	0.354464	−	0.935070i	$-0.384663\pi$
$479$	7432.00	0.708928	0.354464	+	0.935070i	$0.384663\pi$
$480$	0	0
$481$	16716.0	1.58458
$482$	−1292.00	−0.122093
$483$	0	0
$484$	−5068.00	−0.475958
$485$	2380.00	0.222825
$486$	0	0
$487$	−6616.00	−0.615605	−0.307802	−	0.951450i	$-0.599594\pi$
$487$	−6616.00	−0.615605	−0.307802	+	0.951450i	$0.599594\pi$
$488$	240.000	0.0222629
$489$	0	0
$490$	0	0
$491$	−17040.0	−1.56620	−0.783100	−	0.621896i	$-0.786363\pi$
$491$	−17040.0	−1.56620	−0.783100	+	0.621896i	$0.786363\pi$
$492$	0	0
$493$	508.000	0.0464081
$494$	−10416.0	−0.948660
$495$	0	0
$496$	1152.00	0.104287
$497$	0	0
$498$	0	0
$499$	−2948.00	−0.264470	−0.132235	−	0.991218i	$-0.542215\pi$
$499$	−2948.00	−0.264470	−0.132235	+	0.991218i	$0.542215\pi$
$500$	−1968.00	−0.176023
$501$	0	0
$502$	−2520.00	−0.224050
$503$	17304.0	1.53389	0.766946	−	0.641712i	$-0.221776\pi$
$503$	17304.0	1.53389	0.766946	+	0.641712i	$0.221776\pi$
$504$	0	0
$505$	2740.00	0.241442
$506$	1216.00	0.106834
$507$	0	0
$508$	8352.00	0.729449
$509$	4650.00	0.404927	0.202463	−	0.979290i	$-0.435105\pi$
$509$	4650.00	0.404927	0.202463	+	0.979290i	$0.435105\pi$
$510$	0	0
$511$	0	0
$512$	−512.000	−0.0441942
$513$	0	0
$514$	−11820.0	−1.01431
$515$	−928.000	−0.0794030
$516$	0	0
$517$	−2112.00	−0.179663
$518$	0	0
$519$	0	0
$520$	−672.000	−0.0566714
$521$	16854.0	1.41725	0.708625	−	0.705585i	$-0.249316\pi$
$521$	16854.0	1.41725	0.708625	+	0.705585i	$0.249316\pi$
$522$	0	0
$523$	124.000	0.0103674	0.00518369	−	0.999987i	$-0.498350\pi$
$523$	124.000	0.0103674	0.00518369	+	0.999987i	$0.498350\pi$
$524$	−1168.00	−0.0973747
$525$	0	0
$526$	5976.00	0.495373
$527$	−144.000	−0.0119027
$528$	0	0
$529$	−6391.00	−0.525273
$530$	−648.000	−0.0531082
$531$	0	0
$532$	0	0
$533$	19404.0	1.57689
$534$	0	0
$535$	4272.00	0.345224
$536$	6112.00	0.492534
$537$	0	0
$538$	2636.00	0.211238
$539$	0	0
$540$	0	0
$541$	5382.00	0.427708	0.213854	−	0.976866i	$-0.431398\pi$
$541$	5382.00	0.427708	0.213854	+	0.976866i	$0.431398\pi$
$542$	−11280.0	−0.893944
$543$	0	0
$544$	64.0000	0.00504408
$545$	−2452.00	−0.192720
$546$	0	0
$547$	17460.0	1.36478	0.682391	−	0.730987i	$-0.260940\pi$
$547$	17460.0	1.36478	0.682391	+	0.730987i	$0.260940\pi$
$548$	−3272.00	−0.255060
$549$	0	0
$550$	1936.00	0.150093
$551$	−31496.0	−2.43516
$552$	0	0
$553$	0	0
$554$	−12892.0	−0.988680
$555$	0	0
$556$	8624.00	0.657804
$557$	9514.00	0.723736	0.361868	−	0.932229i	$-0.382139\pi$
$557$	9514.00	0.723736	0.361868	+	0.932229i	$0.382139\pi$
$558$	0	0
$559$	8904.00	0.673701
$560$	0	0
$561$	0	0
$562$	9860.00	0.740069
$563$	3988.00	0.298533	0.149267	−	0.988797i	$-0.452309\pi$
$563$	3988.00	0.298533	0.149267	+	0.988797i	$0.452309\pi$
$564$	0	0
$565$	−676.000	−0.0503355
$566$	−12520.0	−0.929779
$567$	0	0
$568$	−1888.00	−0.139470
$569$	−11346.0	−0.835939	−0.417969	−	0.908461i	$-0.637258\pi$
$569$	−11346.0	−0.835939	−0.417969	+	0.908461i	$0.637258\pi$
$570$	0	0
$571$	−8436.00	−0.618276	−0.309138	−	0.951017i	$-0.600041\pi$
$571$	−8436.00	−0.618276	−0.309138	+	0.951017i	$0.600041\pi$
$572$	1344.00	0.0982438
$573$	0	0
$574$	0	0
$575$	9196.00	0.666956
$576$	0	0
$577$	−2098.00	−0.151371	−0.0756853	−	0.997132i	$-0.524114\pi$
$577$	−2098.00	−0.151371	−0.0756853	+	0.997132i	$0.524114\pi$
$578$	9818.00	0.706531
$579$	0	0
$580$	−2032.00	−0.145473
$581$	0	0
$582$	0	0
$583$	1296.00	0.0920666
$584$	3344.00	0.236945
$585$	0	0
$586$	4620.00	0.325683
$587$	9436.00	0.663484	0.331742	−	0.943370i	$-0.392364\pi$
$587$	9436.00	0.663484	0.331742	+	0.943370i	$0.392364\pi$
$588$	0	0
$589$	8928.00	0.624570
$590$	3088.00	0.215476
$591$	0	0
$592$	6368.00	0.442100
$593$	−1314.00	−0.0909941	−0.0454971	−	0.998964i	$-0.514487\pi$
$593$	−1314.00	−0.0909941	−0.0454971	+	0.998964i	$0.514487\pi$
$594$	0	0
$595$	0	0
$596$	11400.0	0.783494
$597$	0	0
$598$	6384.00	0.436557
$599$	8940.00	0.609814	0.304907	−	0.952382i	$-0.401375\pi$
$599$	8940.00	0.609814	0.304907	+	0.952382i	$0.401375\pi$
$600$	0	0
$601$	−16058.0	−1.08988	−0.544941	−	0.838474i	$-0.683448\pi$
$601$	−16058.0	−1.08988	−0.544941	+	0.838474i	$0.683448\pi$
$602$	0	0
$603$	0	0
$604$	6688.00	0.450548
$605$	−2534.00	−0.170284
$606$	0	0
$607$	−3936.00	−0.263192	−0.131596	−	0.991303i	$-0.542010\pi$
$607$	−3936.00	−0.263192	−0.131596	+	0.991303i	$0.542010\pi$
$608$	−3968.00	−0.264677
$609$	0	0
$610$	120.000	0.00796501
$611$	−11088.0	−0.734161
$612$	0	0
$613$	174.000	0.0114646	0.00573230	−	0.999984i	$-0.498175\pi$
$613$	174.000	0.0114646	0.00573230	+	0.999984i	$0.498175\pi$
$614$	392.000	0.0257652
$615$	0	0
$616$	0	0
$617$	−16018.0	−1.04515	−0.522577	−	0.852592i	$-0.675029\pi$
$617$	−16018.0	−1.04515	−0.522577	+	0.852592i	$0.675029\pi$
$618$	0	0
$619$	3068.00	0.199214	0.0996069	−	0.995027i	$-0.468241\pi$
$619$	3068.00	0.199214	0.0996069	+	0.995027i	$0.468241\pi$
$620$	576.000	0.0373108
$621$	0	0
$622$	13472.0	0.868453
$623$	0	0
$624$	0	0
$625$	14141.0	0.905024
$626$	788.000	0.0503112
$627$	0	0
$628$	−1784.00	−0.113359
$629$	−796.000	−0.0504588
$630$	0	0
$631$	24656.0	1.55553	0.777765	−	0.628555i	$-0.216353\pi$
$631$	24656.0	1.55553	0.777765	+	0.628555i	$0.216353\pi$
$632$	−4416.00	−0.277942
$633$	0	0
$634$	−13428.0	−0.841158
$635$	4176.00	0.260976
$636$	0	0
$637$	0	0
$638$	4064.00	0.252187
$639$	0	0
$640$	−256.000	−0.0158114
$641$	−7594.00	−0.467933	−0.233966	−	0.972245i	$-0.575171\pi$
$641$	−7594.00	−0.467933	−0.233966	+	0.972245i	$0.575171\pi$
$642$	0	0
$643$	3724.00	0.228398	0.114199	−	0.993458i	$-0.463570\pi$
$643$	3724.00	0.228398	0.114199	+	0.993458i	$0.463570\pi$
$644$	0	0
$645$	0	0
$646$	496.000	0.0302088
$647$	3792.00	0.230416	0.115208	−	0.993341i	$-0.463247\pi$
$647$	3792.00	0.230416	0.115208	+	0.993341i	$0.463247\pi$
$648$	0	0
$649$	−6176.00	−0.373543
$650$	10164.0	0.613331
$651$	0	0
$652$	10832.0	0.650635
$653$	−24702.0	−1.48034	−0.740171	−	0.672418i	$-0.765256\pi$
$653$	−24702.0	−1.48034	−0.740171	+	0.672418i	$0.765256\pi$
$654$	0	0
$655$	−584.000	−0.0348378
$656$	7392.00	0.439953
$657$	0	0
$658$	0	0
$659$	20144.0	1.19074	0.595371	−	0.803451i	$-0.297005\pi$
$659$	20144.0	1.19074	0.595371	+	0.803451i	$0.297005\pi$
$660$	0	0
$661$	2522.00	0.148403	0.0742015	−	0.997243i	$-0.476359\pi$
$661$	2522.00	0.148403	0.0742015	+	0.997243i	$0.476359\pi$
$662$	−1384.00	−0.0812548
$663$	0	0
$664$	−8288.00	−0.484393
$665$	0	0
$666$	0	0
$667$	19304.0	1.12062
$668$	3584.00	0.207589
$669$	0	0
$670$	3056.00	0.176214
$671$	−240.000	−0.0138079
$672$	0	0
$673$	−10414.0	−0.596479	−0.298239	−	0.954491i	$-0.596399\pi$
$673$	−10414.0	−0.596479	−0.298239	+	0.954491i	$0.596399\pi$
$674$	3132.00	0.178991
$675$	0	0
$676$	−1732.00	−0.0985435
$677$	−22230.0	−1.26199	−0.630996	−	0.775786i	$-0.717353\pi$
$677$	−22230.0	−1.26199	−0.630996	+	0.775786i	$0.717353\pi$
$678$	0	0
$679$	0	0
$680$	32.0000	0.00180462
$681$	0	0
$682$	−1152.00	−0.0646809
$683$	−18192.0	−1.01918	−0.509588	−	0.860418i	$-0.670202\pi$
$683$	−18192.0	−1.01918	−0.509588	+	0.860418i	$0.670202\pi$
$684$	0	0
$685$	−1636.00	−0.0912531
$686$	0	0
$687$	0	0
$688$	3392.00	0.187963
$689$	6804.00	0.376214
$690$	0	0
$691$	−8108.00	−0.446372	−0.223186	−	0.974776i	$-0.571646\pi$
$691$	−8108.00	−0.446372	−0.223186	+	0.974776i	$0.571646\pi$
$692$	16136.0	0.886414
$693$	0	0
$694$	−10656.0	−0.582848
$695$	4312.00	0.235343
$696$	0	0
$697$	−924.000	−0.0502138
$698$	22652.0	1.22835
$699$	0	0
$700$	0	0
$701$	5794.00	0.312177	0.156089	−	0.987743i	$-0.450111\pi$
$701$	5794.00	0.312177	0.156089	+	0.987743i	$0.450111\pi$
$702$	0	0
$703$	49352.0	2.64772
$704$	512.000	0.0274101
$705$	0	0
$706$	4260.00	0.227092
$707$	0	0
$708$	0	0
$709$	−1954.00	−0.103504	−0.0517518	−	0.998660i	$-0.516480\pi$
$709$	−1954.00	−0.103504	−0.0517518	+	0.998660i	$0.516480\pi$
$710$	−944.000	−0.0498982
$711$	0	0
$712$	−240.000	−0.0126326
$713$	−5472.00	−0.287417
$714$	0	0
$715$	672.000	0.0351488
$716$	13920.0	0.726557
$717$	0	0
$718$	6088.00	0.316438
$719$	−32016.0	−1.66063	−0.830317	−	0.557292i	$-0.811840\pi$
$719$	−32016.0	−1.66063	−0.830317	+	0.557292i	$0.811840\pi$
$720$	0	0
$721$	0	0
$722$	−17034.0	−0.878033
$723$	0	0
$724$	11592.0	0.595046
$725$	30734.0	1.57439
$726$	0	0
$727$	23072.0	1.17702	0.588510	−	0.808490i	$-0.299715\pi$
$727$	23072.0	1.17702	0.588510	+	0.808490i	$0.299715\pi$
$728$	0	0
$729$	0	0
$730$	1672.00	0.0847719
$731$	−424.000	−0.0214531
$732$	0	0
$733$	−31782.0	−1.60149	−0.800747	−	0.599003i	$-0.795564\pi$
$733$	−31782.0	−1.60149	−0.800747	+	0.599003i	$0.795564\pi$
$734$	24832.0	1.24873
$735$	0	0
$736$	2432.00	0.121800
$737$	−6112.00	−0.305480
$738$	0	0
$739$	−24396.0	−1.21437	−0.607186	−	0.794559i	$-0.707702\pi$
$739$	−24396.0	−1.21437	−0.607186	+	0.794559i	$0.707702\pi$
$740$	3184.00	0.158170
$741$	0	0
$742$	0	0
$743$	32604.0	1.60986	0.804929	−	0.593371i	$-0.202203\pi$
$743$	32604.0	1.60986	0.804929	+	0.593371i	$0.202203\pi$
$744$	0	0
$745$	5700.00	0.280311
$746$	14884.0	0.730485
$747$	0	0
$748$	−64.0000	−0.00312844
$749$	0	0
$750$	0	0
$751$	−7680.00	−0.373165	−0.186583	−	0.982439i	$-0.559741\pi$
$751$	−7680.00	−0.373165	−0.186583	+	0.982439i	$0.559741\pi$
$752$	−4224.00	−0.204832
$753$	0	0
$754$	21336.0	1.03052
$755$	3344.00	0.161193
$756$	0	0
$757$	366.000	0.0175727	0.00878633	−	0.999961i	$-0.497203\pi$
$757$	366.000	0.0175727	0.00878633	+	0.999961i	$0.497203\pi$
$758$	−200.000	−0.00958355
$759$	0	0
$760$	−1984.00	−0.0946937
$761$	29374.0	1.39922	0.699610	−	0.714525i	$-0.253357\pi$
$761$	29374.0	1.39922	0.699610	+	0.714525i	$0.253357\pi$
$762$	0	0
$763$	0	0
$764$	−10608.0	−0.502335
$765$	0	0
$766$	−16160.0	−0.762251
$767$	−32424.0	−1.52642
$768$	0	0
$769$	38990.0	1.82837	0.914184	−	0.405299i	$-0.132833\pi$
$769$	38990.0	1.82837	0.914184	+	0.405299i	$0.132833\pi$
$770$	0	0
$771$	0	0
$772$	584.000	0.0272262
$773$	−20470.0	−0.952464	−0.476232	−	0.879320i	$-0.657998\pi$
$773$	−20470.0	−0.952464	−0.476232	+	0.879320i	$0.657998\pi$
$774$	0	0
$775$	−8712.00	−0.403799
$776$	−9520.00	−0.440397
$777$	0	0
$778$	−10964.0	−0.505242
$779$	57288.0	2.63486
$780$	0	0
$781$	1888.00	0.0865019
$782$	−304.000	−0.0139016
$783$	0	0
$784$	0	0
$785$	−892.000	−0.0405565
$786$	0	0
$787$	−29916.0	−1.35501	−0.677503	−	0.735520i	$-0.736938\pi$
$787$	−29916.0	−1.35501	−0.677503	+	0.735520i	$0.736938\pi$
$788$	10184.0	0.460393
$789$	0	0
$790$	−2208.00	−0.0994394
$791$	0	0
$792$	0	0
$793$	−1260.00	−0.0564236
$794$	20892.0	0.933790
$795$	0	0
$796$	10144.0	0.451689
$797$	4914.00	0.218398	0.109199	−	0.994020i	$-0.465172\pi$
$797$	4914.00	0.218398	0.109199	+	0.994020i	$0.465172\pi$
$798$	0	0
$799$	528.000	0.0233783
$800$	3872.00	0.171120
$801$	0	0
$802$	−22668.0	−0.998049
$803$	−3344.00	−0.146958
$804$	0	0
$805$	0	0
$806$	−6048.00	−0.264307
$807$	0	0
$808$	−10960.0	−0.477192
$809$	−34250.0	−1.48846	−0.744231	−	0.667922i	$-0.767184\pi$
$809$	−34250.0	−1.48846	−0.744231	+	0.667922i	$0.767184\pi$
$810$	0	0
$811$	41804.0	1.81003	0.905017	−	0.425376i	$-0.139858\pi$
$811$	41804.0	1.81003	0.905017	+	0.425376i	$0.139858\pi$
$812$	0	0
$813$	0	0
$814$	−6368.00	−0.274199
$815$	5416.00	0.232778
$816$	0	0
$817$	26288.0	1.12570
$818$	17188.0	0.734675
$819$	0	0
$820$	3696.00	0.157402
$821$	−30862.0	−1.31193	−0.655963	−	0.754793i	$-0.727737\pi$
$821$	−30862.0	−1.31193	−0.655963	+	0.754793i	$0.727737\pi$
$822$	0	0
$823$	10576.0	0.447942	0.223971	−	0.974596i	$-0.428098\pi$
$823$	10576.0	0.447942	0.223971	+	0.974596i	$0.428098\pi$
$824$	3712.00	0.156934
$825$	0	0
$826$	0	0
$827$	10680.0	0.449069	0.224534	−	0.974466i	$-0.427914\pi$
$827$	10680.0	0.449069	0.224534	+	0.974466i	$0.427914\pi$
$828$	0	0
$829$	1178.00	0.0493530	0.0246765	−	0.999695i	$-0.492144\pi$
$829$	1178.00	0.0493530	0.0246765	+	0.999695i	$0.492144\pi$
$830$	−4144.00	−0.173302
$831$	0	0
$832$	2688.00	0.112007
$833$	0	0
$834$	0	0
$835$	1792.00	0.0742691
$836$	3968.00	0.164158
$837$	0	0
$838$	−21000.0	−0.865672
$839$	−5600.00	−0.230433	−0.115217	−	0.993340i	$-0.536756\pi$
$839$	−5600.00	−0.230433	−0.115217	+	0.993340i	$0.536756\pi$
$840$	0	0
$841$	40127.0	1.64529
$842$	24132.0	0.987700
$843$	0	0
$844$	−5200.00	−0.212075
$845$	−866.000	−0.0352560
$846$	0	0
$847$	0	0
$848$	2592.00	0.104964
$849$	0	0
$850$	−484.000	−0.0195307
$851$	−30248.0	−1.21843
$852$	0	0
$853$	826.000	0.0331556	0.0165778	−	0.999863i	$-0.494723\pi$
$853$	826.000	0.0331556	0.0165778	+	0.999863i	$0.494723\pi$
$854$	0	0
$855$	0	0
$856$	−17088.0	−0.682308
$857$	45918.0	1.83026	0.915128	−	0.403164i	$-0.132090\pi$
$857$	45918.0	1.83026	0.915128	+	0.403164i	$0.132090\pi$
$858$	0	0
$859$	−42380.0	−1.68334	−0.841669	−	0.539994i	$-0.818426\pi$
$859$	−42380.0	−1.68334	−0.841669	+	0.539994i	$0.818426\pi$
$860$	1696.00	0.0672478
$861$	0	0
$862$	8664.00	0.342340
$863$	26524.0	1.04622	0.523110	−	0.852265i	$-0.324772\pi$
$863$	26524.0	1.04622	0.523110	+	0.852265i	$0.324772\pi$
$864$	0	0
$865$	8068.00	0.317133
$866$	−3836.00	−0.150523
$867$	0	0
$868$	0	0
$869$	4416.00	0.172385
$870$	0	0
$871$	−32088.0	−1.24829
$872$	9808.00	0.380895
$873$	0	0
$874$	18848.0	0.729454
$875$	0	0
$876$	0	0
$877$	20614.0	0.793712	0.396856	−	0.917881i	$-0.370101\pi$
$877$	20614.0	0.793712	0.396856	+	0.917881i	$0.370101\pi$
$878$	−15984.0	−0.614389
$879$	0	0
$880$	256.000	0.00980654
$881$	−23730.0	−0.907473	−0.453737	−	0.891136i	$-0.649909\pi$
$881$	−23730.0	−0.907473	−0.453737	+	0.891136i	$0.649909\pi$
$882$	0	0
$883$	−9028.00	−0.344073	−0.172036	−	0.985091i	$-0.555035\pi$
$883$	−9028.00	−0.344073	−0.172036	+	0.985091i	$0.555035\pi$
$884$	−336.000	−0.0127838
$885$	0	0
$886$	6368.00	0.241464
$887$	37200.0	1.40818	0.704089	−	0.710112i	$-0.251356\pi$
$887$	37200.0	1.40818	0.704089	+	0.710112i	$0.251356\pi$
$888$	0	0
$889$	0	0
$890$	−120.000	−0.00451956
$891$	0	0
$892$	−10304.0	−0.386775
$893$	−32736.0	−1.22673
$894$	0	0
$895$	6960.00	0.259941
$896$	0	0
$897$	0	0
$898$	22852.0	0.849199
$899$	−18288.0	−0.678464
$900$	0	0
$901$	−324.000	−0.0119800
$902$	−7392.00	−0.272868
$903$	0	0
$904$	2704.00	0.0994842
$905$	5796.00	0.212890
$906$	0	0
$907$	23988.0	0.878179	0.439090	−	0.898443i	$-0.355301\pi$
$907$	23988.0	0.878179	0.439090	+	0.898443i	$0.355301\pi$
$908$	−7344.00	−0.268413
$909$	0	0
$910$	0	0
$911$	−15276.0	−0.555561	−0.277781	−	0.960645i	$-0.589599\pi$
$911$	−15276.0	−0.555561	−0.277781	+	0.960645i	$0.589599\pi$
$912$	0	0
$913$	8288.00	0.300430
$914$	33868.0	1.22566
$915$	0	0
$916$	7496.00	0.270387
$917$	0	0
$918$	0	0
$919$	−10760.0	−0.386224	−0.193112	−	0.981177i	$-0.561858\pi$
$919$	−10760.0	−0.386224	−0.193112	+	0.981177i	$0.561858\pi$
$920$	1216.00	0.0435764
$921$	0	0
$922$	34076.0	1.21717
$923$	9912.00	0.353475
$924$	0	0
$925$	−48158.0	−1.71181
$926$	27184.0	0.964710
$927$	0	0
$928$	8128.00	0.287516
$929$	−52890.0	−1.86788	−0.933942	−	0.357424i	$-0.883655\pi$
$929$	−52890.0	−1.86788	−0.933942	+	0.357424i	$0.883655\pi$
$930$	0	0
$931$	0	0
$932$	−14920.0	−0.524379
$933$	0	0
$934$	−17224.0	−0.603412
$935$	−32.0000	−0.00111926
$936$	0	0
$937$	6118.00	0.213305	0.106652	−	0.994296i	$-0.465987\pi$
$937$	6118.00	0.213305	0.106652	+	0.994296i	$0.465987\pi$
$938$	0	0
$939$	0	0
$940$	−2112.00	−0.0732828
$941$	−32230.0	−1.11654	−0.558272	−	0.829658i	$-0.688535\pi$
$941$	−32230.0	−1.11654	−0.558272	+	0.829658i	$0.688535\pi$
$942$	0	0
$943$	−35112.0	−1.21252
$944$	−12352.0	−0.425872
$945$	0	0
$946$	−3392.00	−0.116579
$947$	18544.0	0.636324	0.318162	−	0.948036i	$-0.396934\pi$
$947$	18544.0	0.636324	0.318162	+	0.948036i	$0.396934\pi$
$948$	0	0
$949$	−17556.0	−0.600518
$950$	30008.0	1.02483
$951$	0	0
$952$	0	0
$953$	−25930.0	−0.881380	−0.440690	−	0.897659i	$-0.645266\pi$
$953$	−25930.0	−0.881380	−0.440690	+	0.897659i	$0.645266\pi$
$954$	0	0
$955$	−5304.00	−0.179721
$956$	−8016.00	−0.271188
$957$	0	0
$958$	−14864.0	−0.501288
$959$	0	0
$960$	0	0
$961$	−24607.0	−0.825988
$962$	−33432.0	−1.12047
$963$	0	0
$964$	2584.00	0.0863330
$965$	292.000	0.00974074
$966$	0	0
$967$	8192.00	0.272427	0.136214	−	0.990680i	$-0.456507\pi$
$967$	8192.00	0.272427	0.136214	+	0.990680i	$0.456507\pi$
$968$	10136.0	0.336553
$969$	0	0
$970$	−4760.00	−0.157561
$971$	−54444.0	−1.79937	−0.899686	−	0.436537i	$-0.856205\pi$
$971$	−54444.0	−1.79937	−0.899686	+	0.436537i	$0.856205\pi$
$972$	0	0
$973$	0	0
$974$	13232.0	0.435298
$975$	0	0
$976$	−480.000	−0.0157422
$977$	25446.0	0.833255	0.416627	−	0.909077i	$-0.363212\pi$
$977$	25446.0	0.833255	0.416627	+	0.909077i	$0.363212\pi$
$978$	0	0
$979$	240.000	0.00783497
$980$	0	0
$981$	0	0
$982$	34080.0	1.10747
$983$	33192.0	1.07697	0.538484	−	0.842635i	$-0.318997\pi$
$983$	33192.0	1.07697	0.538484	+	0.842635i	$0.318997\pi$
$984$	0	0
$985$	5092.00	0.164715
$986$	−1016.00	−0.0328154
$987$	0	0
$988$	20832.0	0.670804
$989$	−16112.0	−0.518030
$990$	0	0
$991$	11024.0	0.353369	0.176685	−	0.984268i	$-0.443463\pi$
$991$	11024.0	0.353369	0.176685	+	0.984268i	$0.443463\pi$
$992$	−2304.00	−0.0737420
$993$	0	0
$994$	0	0
$995$	5072.00	0.161601
$996$	0	0
$997$	40714.0	1.29331	0.646653	−	0.762785i	$-0.276168\pi$
$997$	40714.0	1.29331	0.646653	+	0.762785i	$0.276168\pi$
$998$	5896.00	0.187009
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.4.a.d.1.1		1
3.2	odd	2		294.4.a.h.1.1		1
7.2	even	3		882.4.g.r.361.1		2
7.3	odd	6		882.4.g.s.667.1		2
7.4	even	3		882.4.g.r.667.1		2
7.5	odd	6		882.4.g.s.361.1		2
7.6	odd	2		126.4.a.c.1.1		1
12.11	even	2		2352.4.a.ba.1.1		1
21.2	odd	6		294.4.e.d.67.1		2
21.5	even	6		294.4.e.a.67.1		2
21.11	odd	6		294.4.e.d.79.1		2
21.17	even	6		294.4.e.a.79.1		2
21.20	even	2		42.4.a.b.1.1	✓	1
28.27	even	2		1008.4.a.j.1.1		1
84.83	odd	2		336.4.a.d.1.1		1
105.62	odd	4		1050.4.g.n.799.2		2
105.83	odd	4		1050.4.g.n.799.1		2
105.104	even	2		1050.4.a.d.1.1		1
168.83	odd	2		1344.4.a.t.1.1		1
168.125	even	2		1344.4.a.f.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.4.a.b.1.1	✓	1	21.20	even	2
126.4.a.c.1.1		1	7.6	odd	2
294.4.a.h.1.1		1	3.2	odd	2
294.4.e.a.67.1		2	21.5	even	6
294.4.e.a.79.1		2	21.17	even	6
294.4.e.d.67.1		2	21.2	odd	6
294.4.e.d.79.1		2	21.11	odd	6
336.4.a.d.1.1		1	84.83	odd	2
882.4.a.d.1.1		1	1.1	even	1	trivial
882.4.g.r.361.1		2	7.2	even	3
882.4.g.r.667.1		2	7.4	even	3
882.4.g.s.361.1		2	7.5	odd	6
882.4.g.s.667.1		2	7.3	odd	6
1008.4.a.j.1.1		1	28.27	even	2
1050.4.a.d.1.1		1	105.104	even	2
1050.4.g.n.799.1		2	105.83	odd	4
1050.4.g.n.799.2		2	105.62	odd	4
1344.4.a.f.1.1		1	168.125	even	2
1344.4.a.t.1.1		1	168.83	odd	2
2352.4.a.ba.1.1		1	12.11	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:	\( N \)	\(=\)	\( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	882.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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