LMFDB - Embedded newform 441.4.a.h.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,4,Mod(1,441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, 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("441.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 441 = 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	441.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:	$26.0198423125$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 147)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	441.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} -7.00000 q^{4} +12.0000 q^{5} -15.0000 q^{8} +O(q^{10})$

$q+1.00000 q^{2} -7.00000 q^{4} +12.0000 q^{5} -15.0000 q^{8} +12.0000 q^{10} -20.0000 q^{11} +84.0000 q^{13} +41.0000 q^{16} -96.0000 q^{17} -12.0000 q^{19} -84.0000 q^{20} -20.0000 q^{22} +176.000 q^{23} +19.0000 q^{25} +84.0000 q^{26} -58.0000 q^{29} +264.000 q^{31} +161.000 q^{32} -96.0000 q^{34} +258.000 q^{37} -12.0000 q^{38} -180.000 q^{40} +156.000 q^{43} +140.000 q^{44} +176.000 q^{46} -408.000 q^{47} +19.0000 q^{50} -588.000 q^{52} +722.000 q^{53} -240.000 q^{55} -58.0000 q^{58} +492.000 q^{59} +492.000 q^{61} +264.000 q^{62} -167.000 q^{64} +1008.00 q^{65} +412.000 q^{67} +672.000 q^{68} -296.000 q^{71} -240.000 q^{73} +258.000 q^{74} +84.0000 q^{76} +776.000 q^{79} +492.000 q^{80} +924.000 q^{83} -1152.00 q^{85} +156.000 q^{86} +300.000 q^{88} -744.000 q^{89} -1232.00 q^{92} -408.000 q^{94} -144.000 q^{95} +168.000 q^{97} +O(q^{100})$

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 $

$2$	1.00000	0.353553	0.176777	−	0.984251i	$-0.443433\pi$
$2$	1.00000	0.353553	0.176777	+	0.984251i	$0.443433\pi$
$3$	0	0
$3$	0	0
$4$	−7.00000	−0.875000
$5$	12.0000	1.07331	0.536656	−	0.843801i	$-0.319687\pi$
$5$	12.0000	1.07331	0.536656	+	0.843801i	$0.319687\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−15.0000	−0.662913
$9$	0	0
$10$	12.0000	0.379473
$11$	−20.0000	−0.548202	−0.274101	−	0.961701i	$-0.588380\pi$
$11$	−20.0000	−0.548202	−0.274101	+	0.961701i	$0.588380\pi$
$12$	0	0
$13$	84.0000	1.79211	0.896054	−	0.443945i	$-0.146421\pi$
$13$	84.0000	1.79211	0.896054	+	0.443945i	$0.146421\pi$
$14$	0	0
$15$	0	0
$16$	41.0000	0.640625
$17$	−96.0000	−1.36961	−0.684806	−	0.728725i	$-0.740113\pi$
$17$	−96.0000	−1.36961	−0.684806	+	0.728725i	$0.740113\pi$
$18$	0	0
$19$	−12.0000	−0.144894	−0.0724471	−	0.997372i	$-0.523081\pi$
$19$	−12.0000	−0.144894	−0.0724471	+	0.997372i	$0.523081\pi$
$20$	−84.0000	−0.939149
$21$	0	0
$22$	−20.0000	−0.193819
$23$	176.000	1.59559	0.797794	−	0.602930i	$-0.206000\pi$
$23$	176.000	1.59559	0.797794	+	0.602930i	$0.206000\pi$
$24$	0	0
$25$	19.0000	0.152000
$26$	84.0000	0.633606
$27$	0	0
$28$	0	0
$29$	−58.0000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−58.0000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	264.000	1.52954	0.764771	−	0.644302i	$-0.222852\pi$
$31$	264.000	1.52954	0.764771	+	0.644302i	$0.222852\pi$
$32$	161.000	0.889408
$33$	0	0
$34$	−96.0000	−0.484231
$35$	0	0
$36$	0	0
$37$	258.000	1.14635	0.573175	−	0.819433i	$-0.305712\pi$
$37$	258.000	1.14635	0.573175	+	0.819433i	$0.305712\pi$
$38$	−12.0000	−0.0512278
$39$	0	0
$40$	−180.000	−0.711512
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	156.000	0.553251	0.276625	−	0.960978i	$-0.410784\pi$
$43$	156.000	0.553251	0.276625	+	0.960978i	$0.410784\pi$
$44$	140.000	0.479677
$45$	0	0
$46$	176.000	0.564126
$47$	−408.000	−1.26623	−0.633116	−	0.774057i	$-0.718224\pi$
$47$	−408.000	−1.26623	−0.633116	+	0.774057i	$0.718224\pi$
$48$	0	0
$49$	0	0
$50$	19.0000	0.0537401
$51$	0	0
$52$	−588.000	−1.56809
$53$	722.000	1.87121	0.935607	−	0.353044i	$-0.114853\pi$
$53$	722.000	1.87121	0.935607	+	0.353044i	$0.114853\pi$
$54$	0	0
$55$	−240.000	−0.588393
$56$	0	0
$57$	0	0
$58$	−58.0000	−0.131306
$59$	492.000	1.08564	0.542822	−	0.839848i	$-0.317356\pi$
$59$	492.000	1.08564	0.542822	+	0.839848i	$0.317356\pi$
$60$	0	0
$61$	492.000	1.03269	0.516345	−	0.856380i	$-0.327292\pi$
$61$	492.000	1.03269	0.516345	+	0.856380i	$0.327292\pi$
$62$	264.000	0.540775
$63$	0	0
$64$	−167.000	−0.326172
$65$	1008.00	1.92349
$66$	0	0
$67$	412.000	0.751251	0.375625	−	0.926772i	$-0.377428\pi$
$67$	412.000	0.751251	0.375625	+	0.926772i	$0.377428\pi$
$68$	672.000	1.19841
$69$	0	0
$70$	0	0
$71$	−296.000	−0.494771	−0.247385	−	0.968917i	$-0.579571\pi$
$71$	−296.000	−0.494771	−0.247385	+	0.968917i	$0.579571\pi$
$72$	0	0
$73$	−240.000	−0.384793	−0.192396	−	0.981317i	$-0.561626\pi$
$73$	−240.000	−0.384793	−0.192396	+	0.981317i	$0.561626\pi$
$74$	258.000	0.405296
$75$	0	0
$76$	84.0000	0.126782
$77$	0	0
$78$	0	0
$79$	776.000	1.10515	0.552575	−	0.833463i	$-0.313645\pi$
$79$	776.000	1.10515	0.552575	+	0.833463i	$0.313645\pi$
$80$	492.000	0.687591
$81$	0	0
$82$	0	0
$83$	924.000	1.22195	0.610977	−	0.791648i	$-0.290777\pi$
$83$	924.000	1.22195	0.610977	+	0.791648i	$0.290777\pi$
$84$	0	0
$85$	−1152.00	−1.47002
$86$	156.000	0.195604
$87$	0	0
$88$	300.000	0.363410
$89$	−744.000	−0.886111	−0.443055	−	0.896494i	$-0.646105\pi$
$89$	−744.000	−0.886111	−0.443055	+	0.896494i	$0.646105\pi$
$90$	0	0
$91$	0	0
$92$	−1232.00	−1.39614
$93$	0	0
$94$	−408.000	−0.447681
$95$	−144.000	−0.155517
$96$	0	0
$97$	168.000	0.175854	0.0879269	−	0.996127i	$-0.471976\pi$
$97$	168.000	0.175854	0.0879269	+	0.996127i	$0.471976\pi$
$98$	0	0
$99$	0	0
$100$	−133.000	−0.133000
$101$	−1524.00	−1.50142	−0.750711	−	0.660630i	$-0.770289\pi$
$101$	−1524.00	−1.50142	−0.750711	+	0.660630i	$0.770289\pi$
$102$	0	0
$103$	408.000	0.390305	0.195153	−	0.980773i	$-0.437480\pi$
$103$	408.000	0.390305	0.195153	+	0.980773i	$0.437480\pi$
$104$	−1260.00	−1.18801
$105$	0	0
$106$	722.000	0.661574
$107$	820.000	0.740863	0.370432	−	0.928860i	$-0.379210\pi$
$107$	820.000	0.740863	0.370432	+	0.928860i	$0.379210\pi$
$108$	0	0
$109$	−918.000	−0.806683	−0.403342	−	0.915050i	$-0.632151\pi$
$109$	−918.000	−0.806683	−0.403342	+	0.915050i	$0.632151\pi$
$110$	−240.000	−0.208028
$111$	0	0
$112$	0	0
$113$	110.000	0.0915746	0.0457873	−	0.998951i	$-0.485420\pi$
$113$	110.000	0.0915746	0.0457873	+	0.998951i	$0.485420\pi$
$114$	0	0
$115$	2112.00	1.71257
$116$	406.000	0.324967
$117$	0	0
$118$	492.000	0.383833
$119$	0	0
$120$	0	0
$121$	−931.000	−0.699474
$122$	492.000	0.365111
$123$	0	0
$124$	−1848.00	−1.33835
$125$	−1272.00	−0.910169
$126$	0	0
$127$	16.0000	0.0111793	0.00558965	−	0.999984i	$-0.498221\pi$
$127$	16.0000	0.0111793	0.00558965	+	0.999984i	$0.498221\pi$
$128$	−1455.00	−1.00473
$129$	0	0
$130$	1008.00	0.680057
$131$	1692.00	1.12848	0.564239	−	0.825611i	$-0.309169\pi$
$131$	1692.00	1.12848	0.564239	+	0.825611i	$0.309169\pi$
$132$	0	0
$133$	0	0
$134$	412.000	0.265607
$135$	0	0
$136$	1440.00	0.907934
$137$	−1126.00	−0.702195	−0.351097	−	0.936339i	$-0.614191\pi$
$137$	−1126.00	−0.702195	−0.351097	+	0.936339i	$0.614191\pi$
$138$	0	0
$139$	−1092.00	−0.666347	−0.333173	−	0.942866i	$-0.608119\pi$
$139$	−1092.00	−0.666347	−0.333173	+	0.942866i	$0.608119\pi$
$140$	0	0
$141$	0	0
$142$	−296.000	−0.174928
$143$	−1680.00	−0.982438
$144$	0	0
$145$	−696.000	−0.398618
$146$	−240.000	−0.136045
$147$	0	0
$148$	−1806.00	−1.00306
$149$	−1070.00	−0.588307	−0.294154	−	0.955758i	$-0.595038\pi$
$149$	−1070.00	−0.588307	−0.294154	+	0.955758i	$0.595038\pi$
$150$	0	0
$151$	−120.000	−0.0646719	−0.0323360	−	0.999477i	$-0.510295\pi$
$151$	−120.000	−0.0646719	−0.0323360	+	0.999477i	$0.510295\pi$
$152$	180.000	0.0960522
$153$	0	0
$154$	0	0
$155$	3168.00	1.64168
$156$	0	0
$157$	−1836.00	−0.933304	−0.466652	−	0.884441i	$-0.654540\pi$
$157$	−1836.00	−0.933304	−0.466652	+	0.884441i	$0.654540\pi$
$158$	776.000	0.390729
$159$	0	0
$160$	1932.00	0.954613
$161$	0	0
$162$	0	0
$163$	916.000	0.440164	0.220082	−	0.975481i	$-0.429368\pi$
$163$	916.000	0.440164	0.220082	+	0.975481i	$0.429368\pi$
$164$	0	0
$165$	0	0
$166$	924.000	0.432026
$167$	504.000	0.233537	0.116769	−	0.993159i	$-0.462746\pi$
$167$	504.000	0.233537	0.116769	+	0.993159i	$0.462746\pi$
$168$	0	0
$169$	4859.00	2.21165
$170$	−1152.00	−0.519732
$171$	0	0
$172$	−1092.00	−0.484094
$173$	−1836.00	−0.806870	−0.403435	−	0.915008i	$-0.632184\pi$
$173$	−1836.00	−0.806870	−0.403435	+	0.915008i	$0.632184\pi$
$174$	0	0
$175$	0	0
$176$	−820.000	−0.351192
$177$	0	0
$178$	−744.000	−0.313287
$179$	−2372.00	−0.990456	−0.495228	−	0.868763i	$-0.664915\pi$
$179$	−2372.00	−0.990456	−0.495228	+	0.868763i	$0.664915\pi$
$180$	0	0
$181$	1092.00	0.448440	0.224220	−	0.974539i	$-0.428017\pi$
$181$	1092.00	0.448440	0.224220	+	0.974539i	$0.428017\pi$
$182$	0	0
$183$	0	0
$184$	−2640.00	−1.05774
$185$	3096.00	1.23039
$186$	0	0
$187$	1920.00	0.750825
$188$	2856.00	1.10795
$189$	0	0
$190$	−144.000	−0.0549835
$191$	−2512.00	−0.951633	−0.475817	−	0.879545i	$-0.657847\pi$
$191$	−2512.00	−0.951633	−0.475817	+	0.879545i	$0.657847\pi$
$192$	0	0
$193$	−2430.00	−0.906297	−0.453148	−	0.891435i	$-0.649699\pi$
$193$	−2430.00	−0.906297	−0.453148	+	0.891435i	$0.649699\pi$
$194$	168.000	0.0621737
$195$	0	0
$196$	0	0
$197$	1762.00	0.637245	0.318623	−	0.947882i	$-0.396780\pi$
$197$	1762.00	0.637245	0.318623	+	0.947882i	$0.396780\pi$
$198$	0	0
$199$	−3096.00	−1.10286	−0.551431	−	0.834220i	$-0.685918\pi$
$199$	−3096.00	−1.10286	−0.551431	+	0.834220i	$0.685918\pi$
$200$	−285.000	−0.100763
$201$	0	0
$202$	−1524.00	−0.530833
$203$	0	0
$204$	0	0
$205$	0	0
$206$	408.000	0.137994
$207$	0	0
$208$	3444.00	1.14807
$209$	240.000	0.0794313
$210$	0	0
$211$	156.000	0.0508980	0.0254490	−	0.999676i	$-0.491898\pi$
$211$	156.000	0.0508980	0.0254490	+	0.999676i	$0.491898\pi$
$212$	−5054.00	−1.63731
$213$	0	0
$214$	820.000	0.261935
$215$	1872.00	0.593811
$216$	0	0
$217$	0	0
$218$	−918.000	−0.285206
$219$	0	0
$220$	1680.00	0.514844
$221$	−8064.00	−2.45449
$222$	0	0
$223$	−5040.00	−1.51347	−0.756734	−	0.653723i	$-0.773206\pi$
$223$	−5040.00	−1.51347	−0.756734	+	0.653723i	$0.773206\pi$
$224$	0	0
$225$	0	0
$226$	110.000	0.0323765
$227$	2172.00	0.635069	0.317535	−	0.948247i	$-0.397145\pi$
$227$	2172.00	0.635069	0.317535	+	0.948247i	$0.397145\pi$
$228$	0	0
$229$	−2700.00	−0.779131	−0.389566	−	0.920999i	$-0.627375\pi$
$229$	−2700.00	−0.779131	−0.389566	+	0.920999i	$0.627375\pi$
$230$	2112.00	0.605483
$231$	0	0
$232$	870.000	0.246200
$233$	3802.00	1.06900	0.534501	−	0.845168i	$-0.320500\pi$
$233$	3802.00	1.06900	0.534501	+	0.845168i	$0.320500\pi$
$234$	0	0
$235$	−4896.00	−1.35906
$236$	−3444.00	−0.949938
$237$	0	0
$238$	0	0
$239$	4408.00	1.19301	0.596506	−	0.802609i	$-0.296555\pi$
$239$	4408.00	1.19301	0.596506	+	0.802609i	$0.296555\pi$
$240$	0	0
$241$	−3096.00	−0.827514	−0.413757	−	0.910387i	$-0.635784\pi$
$241$	−3096.00	−0.827514	−0.413757	+	0.910387i	$0.635784\pi$
$242$	−931.000	−0.247301
$243$	0	0
$244$	−3444.00	−0.903605
$245$	0	0
$246$	0	0
$247$	−1008.00	−0.259666
$248$	−3960.00	−1.01395
$249$	0	0
$250$	−1272.00	−0.321793
$251$	−924.000	−0.232360	−0.116180	−	0.993228i	$-0.537065\pi$
$251$	−924.000	−0.232360	−0.116180	+	0.993228i	$0.537065\pi$
$252$	0	0
$253$	−3520.00	−0.874706
$254$	16.0000	0.00395248
$255$	0	0
$256$	−119.000	−0.0290527
$257$	−2760.00	−0.669899	−0.334950	−	0.942236i	$-0.608719\pi$
$257$	−2760.00	−0.669899	−0.334950	+	0.942236i	$0.608719\pi$
$258$	0	0
$259$	0	0
$260$	−7056.00	−1.68306
$261$	0	0
$262$	1692.00	0.398978
$263$	2360.00	0.553323	0.276661	−	0.960967i	$-0.410772\pi$
$263$	2360.00	0.553323	0.276661	+	0.960967i	$0.410772\pi$
$264$	0	0
$265$	8664.00	2.00840
$266$	0	0
$267$	0	0
$268$	−2884.00	−0.657345
$269$	4020.00	0.911166	0.455583	−	0.890193i	$-0.349431\pi$
$269$	4020.00	0.911166	0.455583	+	0.890193i	$0.349431\pi$
$270$	0	0
$271$	−4800.00	−1.07594	−0.537969	−	0.842965i	$-0.680808\pi$
$271$	−4800.00	−1.07594	−0.537969	+	0.842965i	$0.680808\pi$
$272$	−3936.00	−0.877408
$273$	0	0
$274$	−1126.00	−0.248263
$275$	−380.000	−0.0833268
$276$	0	0
$277$	6446.00	1.39820	0.699102	−	0.715022i	$-0.253583\pi$
$277$	6446.00	1.39820	0.699102	+	0.715022i	$0.253583\pi$
$278$	−1092.00	−0.235589
$279$	0	0
$280$	0	0
$281$	2602.00	0.552393	0.276196	−	0.961101i	$-0.410926\pi$
$281$	2602.00	0.552393	0.276196	+	0.961101i	$0.410926\pi$
$282$	0	0
$283$	6900.00	1.44934	0.724669	−	0.689098i	$-0.241993\pi$
$283$	6900.00	1.44934	0.724669	+	0.689098i	$0.241993\pi$
$284$	2072.00	0.432925
$285$	0	0
$286$	−1680.00	−0.347344
$287$	0	0
$288$	0	0
$289$	4303.00	0.875840
$290$	−696.000	−0.140933
$291$	0	0
$292$	1680.00	0.336694
$293$	−4452.00	−0.887674	−0.443837	−	0.896107i	$-0.646383\pi$
$293$	−4452.00	−0.887674	−0.443837	+	0.896107i	$0.646383\pi$
$294$	0	0
$295$	5904.00	1.16523
$296$	−3870.00	−0.759930
$297$	0	0
$298$	−1070.00	−0.207998
$299$	14784.0	2.85947
$300$	0	0
$301$	0	0
$302$	−120.000	−0.0228650
$303$	0	0
$304$	−492.000	−0.0928228
$305$	5904.00	1.10840
$306$	0	0
$307$	2436.00	0.452866	0.226433	−	0.974027i	$-0.427294\pi$
$307$	2436.00	0.452866	0.226433	+	0.974027i	$0.427294\pi$
$308$	0	0
$309$	0	0
$310$	3168.00	0.580420
$311$	−7488.00	−1.36529	−0.682646	−	0.730750i	$-0.739171\pi$
$311$	−7488.00	−1.36529	−0.682646	+	0.730750i	$0.739171\pi$
$312$	0	0
$313$	1752.00	0.316386	0.158193	−	0.987408i	$-0.449433\pi$
$313$	1752.00	0.316386	0.158193	+	0.987408i	$0.449433\pi$
$314$	−1836.00	−0.329973
$315$	0	0
$316$	−5432.00	−0.967006
$317$	1562.00	0.276753	0.138376	−	0.990380i	$-0.455812\pi$
$317$	1562.00	0.276753	0.138376	+	0.990380i	$0.455812\pi$
$318$	0	0
$319$	1160.00	0.203597
$320$	−2004.00	−0.350084
$321$	0	0
$322$	0	0
$323$	1152.00	0.198449
$324$	0	0
$325$	1596.00	0.272400
$326$	916.000	0.155621
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−7092.00	−1.17768	−0.588839	−	0.808250i	$-0.700415\pi$
$331$	−7092.00	−1.17768	−0.588839	+	0.808250i	$0.700415\pi$
$332$	−6468.00	−1.06921
$333$	0	0
$334$	504.000	0.0825678
$335$	4944.00	0.806327
$336$	0	0
$337$	366.000	0.0591611	0.0295805	−	0.999562i	$-0.490583\pi$
$337$	366.000	0.0591611	0.0295805	+	0.999562i	$0.490583\pi$
$338$	4859.00	0.781937
$339$	0	0
$340$	8064.00	1.28627
$341$	−5280.00	−0.838499
$342$	0	0
$343$	0	0
$344$	−2340.00	−0.366757
$345$	0	0
$346$	−1836.00	−0.285272
$347$	6364.00	0.984546	0.492273	−	0.870441i	$-0.336166\pi$
$347$	6364.00	0.984546	0.492273	+	0.870441i	$0.336166\pi$
$348$	0	0
$349$	10500.0	1.61046	0.805232	−	0.592960i	$-0.202041\pi$
$349$	10500.0	1.61046	0.805232	+	0.592960i	$0.202041\pi$
$350$	0	0
$351$	0	0
$352$	−3220.00	−0.487576
$353$	408.000	0.0615174	0.0307587	−	0.999527i	$-0.490208\pi$
$353$	408.000	0.0615174	0.0307587	+	0.999527i	$0.490208\pi$
$354$	0	0
$355$	−3552.00	−0.531044
$356$	5208.00	0.775347
$357$	0	0
$358$	−2372.00	−0.350179
$359$	11936.0	1.75476	0.877379	−	0.479798i	$-0.159290\pi$
$359$	11936.0	1.75476	0.877379	+	0.479798i	$0.159290\pi$
$360$	0	0
$361$	−6715.00	−0.979006
$362$	1092.00	0.158548
$363$	0	0
$364$	0	0
$365$	−2880.00	−0.413003
$366$	0	0
$367$	2448.00	0.348187	0.174093	−	0.984729i	$-0.444301\pi$
$367$	2448.00	0.348187	0.174093	+	0.984729i	$0.444301\pi$
$368$	7216.00	1.02217
$369$	0	0
$370$	3096.00	0.435009
$371$	0	0
$372$	0	0
$373$	11374.0	1.57888	0.789442	−	0.613826i	$-0.210370\pi$
$373$	11374.0	1.57888	0.789442	+	0.613826i	$0.210370\pi$
$374$	1920.00	0.265457
$375$	0	0
$376$	6120.00	0.839401
$377$	−4872.00	−0.665572
$378$	0	0
$379$	−5892.00	−0.798553	−0.399277	−	0.916830i	$-0.630739\pi$
$379$	−5892.00	−0.798553	−0.399277	+	0.916830i	$0.630739\pi$
$380$	1008.00	0.136077
$381$	0	0
$382$	−2512.00	−0.336453
$383$	−10488.0	−1.39925	−0.699624	−	0.714511i	$-0.746649\pi$
$383$	−10488.0	−1.39925	−0.699624	+	0.714511i	$0.746649\pi$
$384$	0	0
$385$	0	0
$386$	−2430.00	−0.320424
$387$	0	0
$388$	−1176.00	−0.153872
$389$	−4514.00	−0.588352	−0.294176	−	0.955751i	$-0.595045\pi$
$389$	−4514.00	−0.588352	−0.294176	+	0.955751i	$0.595045\pi$
$390$	0	0
$391$	−16896.0	−2.18534
$392$	0	0
$393$	0	0
$394$	1762.00	0.225300
$395$	9312.00	1.18617
$396$	0	0
$397$	6036.00	0.763068	0.381534	−	0.924355i	$-0.375396\pi$
$397$	6036.00	0.763068	0.381534	+	0.924355i	$0.375396\pi$
$398$	−3096.00	−0.389921
$399$	0	0
$400$	779.000	0.0973750
$401$	6770.00	0.843086	0.421543	−	0.906808i	$-0.361489\pi$
$401$	6770.00	0.843086	0.421543	+	0.906808i	$0.361489\pi$
$402$	0	0
$403$	22176.0	2.74110
$404$	10668.0	1.31374
$405$	0	0
$406$	0	0
$407$	−5160.00	−0.628432
$408$	0	0
$409$	−12504.0	−1.51169	−0.755847	−	0.654748i	$-0.772775\pi$
$409$	−12504.0	−1.51169	−0.755847	+	0.654748i	$0.772775\pi$
$410$	0	0
$411$	0	0
$412$	−2856.00	−0.341517
$413$	0	0
$414$	0	0
$415$	11088.0	1.31154
$416$	13524.0	1.59392
$417$	0	0
$418$	240.000	0.0280832
$419$	9492.00	1.10672	0.553359	−	0.832943i	$-0.313346\pi$
$419$	9492.00	1.10672	0.553359	+	0.832943i	$0.313346\pi$
$420$	0	0
$421$	5182.00	0.599894	0.299947	−	0.953956i	$-0.403031\pi$
$421$	5182.00	0.599894	0.299947	+	0.953956i	$0.403031\pi$
$422$	156.000	0.0179952
$423$	0	0
$424$	−10830.0	−1.24045
$425$	−1824.00	−0.208181
$426$	0	0
$427$	0	0
$428$	−5740.00	−0.648256
$429$	0	0
$430$	1872.00	0.209944
$431$	5720.00	0.639264	0.319632	−	0.947542i	$-0.396441\pi$
$431$	5720.00	0.639264	0.319632	+	0.947542i	$0.396441\pi$
$432$	0	0
$433$	−13608.0	−1.51030	−0.755149	−	0.655554i	$-0.772435\pi$
$433$	−13608.0	−1.51030	−0.755149	+	0.655554i	$0.772435\pi$
$434$	0	0
$435$	0	0
$436$	6426.00	0.705848
$437$	−2112.00	−0.231191
$438$	0	0
$439$	−12864.0	−1.39855	−0.699277	−	0.714851i	$-0.746495\pi$
$439$	−12864.0	−1.39855	−0.699277	+	0.714851i	$0.746495\pi$
$440$	3600.00	0.390053
$441$	0	0
$442$	−8064.00	−0.867795
$443$	13252.0	1.42127	0.710634	−	0.703562i	$-0.248408\pi$
$443$	13252.0	1.42127	0.710634	+	0.703562i	$0.248408\pi$
$444$	0	0
$445$	−8928.00	−0.951074
$446$	−5040.00	−0.535092
$447$	0	0
$448$	0	0
$449$	−226.000	−0.0237541	−0.0118771	−	0.999929i	$-0.503781\pi$
$449$	−226.000	−0.0237541	−0.0118771	+	0.999929i	$0.503781\pi$
$450$	0	0
$451$	0	0
$452$	−770.000	−0.0801278
$453$	0	0
$454$	2172.00	0.224531
$455$	0	0
$456$	0	0
$457$	−11334.0	−1.16014	−0.580068	−	0.814568i	$-0.696974\pi$
$457$	−11334.0	−1.16014	−0.580068	+	0.814568i	$0.696974\pi$
$458$	−2700.00	−0.275464
$459$	0	0
$460$	−14784.0	−1.49849
$461$	−1596.00	−0.161243	−0.0806216	−	0.996745i	$-0.525691\pi$
$461$	−1596.00	−0.161243	−0.0806216	+	0.996745i	$0.525691\pi$
$462$	0	0
$463$	12728.0	1.27758	0.638791	−	0.769380i	$-0.279435\pi$
$463$	12728.0	1.27758	0.638791	+	0.769380i	$0.279435\pi$
$464$	−2378.00	−0.237922
$465$	0	0
$466$	3802.00	0.377949
$467$	−3012.00	−0.298456	−0.149228	−	0.988803i	$-0.547679\pi$
$467$	−3012.00	−0.298456	−0.149228	+	0.988803i	$0.547679\pi$
$468$	0	0
$469$	0	0
$470$	−4896.00	−0.480501
$471$	0	0
$472$	−7380.00	−0.719687
$473$	−3120.00	−0.303293
$474$	0	0
$475$	−228.000	−0.0220239
$476$	0	0
$477$	0	0
$478$	4408.00	0.421793
$479$	−4296.00	−0.409790	−0.204895	−	0.978784i	$-0.565685\pi$
$479$	−4296.00	−0.409790	−0.204895	+	0.978784i	$0.565685\pi$
$480$	0	0
$481$	21672.0	2.05438
$482$	−3096.00	−0.292570
$483$	0	0
$484$	6517.00	0.612040
$485$	2016.00	0.188746
$486$	0	0
$487$	−8184.00	−0.761504	−0.380752	−	0.924677i	$-0.624335\pi$
$487$	−8184.00	−0.761504	−0.380752	+	0.924677i	$0.624335\pi$
$488$	−7380.00	−0.684584
$489$	0	0
$490$	0	0
$491$	12164.0	1.11803	0.559016	−	0.829157i	$-0.311179\pi$
$491$	12164.0	1.11803	0.559016	+	0.829157i	$0.311179\pi$
$492$	0	0
$493$	5568.00	0.508661
$494$	−1008.00	−0.0918058
$495$	0	0
$496$	10824.0	0.979863
$497$	0	0
$498$	0	0
$499$	972.000	0.0871998	0.0435999	−	0.999049i	$-0.486117\pi$
$499$	972.000	0.0871998	0.0435999	+	0.999049i	$0.486117\pi$
$500$	8904.00	0.796398
$501$	0	0
$502$	−924.000	−0.0821517
$503$	7728.00	0.685039	0.342519	−	0.939511i	$-0.388720\pi$
$503$	7728.00	0.685039	0.342519	+	0.939511i	$0.388720\pi$
$504$	0	0
$505$	−18288.0	−1.61150
$506$	−3520.00	−0.309255
$507$	0	0
$508$	−112.000	−0.00978188
$509$	11604.0	1.01049	0.505244	−	0.862977i	$-0.331403\pi$
$509$	11604.0	1.01049	0.505244	+	0.862977i	$0.331403\pi$
$510$	0	0
$511$	0	0
$512$	11521.0	0.994455
$513$	0	0
$514$	−2760.00	−0.236845
$515$	4896.00	0.418919
$516$	0	0
$517$	8160.00	0.694152
$518$	0	0
$519$	0	0
$520$	−15120.0	−1.27511
$521$	−10848.0	−0.912206	−0.456103	−	0.889927i	$-0.650755\pi$
$521$	−10848.0	−0.912206	−0.456103	+	0.889927i	$0.650755\pi$
$522$	0	0
$523$	18132.0	1.51598	0.757989	−	0.652267i	$-0.226182\pi$
$523$	18132.0	1.51598	0.757989	+	0.652267i	$0.226182\pi$
$524$	−11844.0	−0.987419
$525$	0	0
$526$	2360.00	0.195629
$527$	−25344.0	−2.09488
$528$	0	0
$529$	18809.0	1.54590
$530$	8664.00	0.710076
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	9840.00	0.795178
$536$	−6180.00	−0.498014
$537$	0	0
$538$	4020.00	0.322146
$539$	0	0
$540$	0	0
$541$	6950.00	0.552318	0.276159	−	0.961112i	$-0.410938\pi$
$541$	6950.00	0.552318	0.276159	+	0.961112i	$0.410938\pi$
$542$	−4800.00	−0.380402
$543$	0	0
$544$	−15456.0	−1.21814
$545$	−11016.0	−0.865823
$546$	0	0
$547$	17012.0	1.32976	0.664882	−	0.746949i	$-0.268482\pi$
$547$	17012.0	1.32976	0.664882	+	0.746949i	$0.268482\pi$
$548$	7882.00	0.614420
$549$	0	0
$550$	−380.000	−0.0294605
$551$	696.000	0.0538123
$552$	0	0
$553$	0	0
$554$	6446.00	0.494340
$555$	0	0
$556$	7644.00	0.583054
$557$	−3926.00	−0.298653	−0.149327	−	0.988788i	$-0.547711\pi$
$557$	−3926.00	−0.298653	−0.149327	+	0.988788i	$0.547711\pi$
$558$	0	0
$559$	13104.0	0.991485
$560$	0	0
$561$	0	0
$562$	2602.00	0.195300
$563$	−18828.0	−1.40942	−0.704712	−	0.709494i	$-0.748924\pi$
$563$	−18828.0	−1.40942	−0.704712	+	0.709494i	$0.748924\pi$
$564$	0	0
$565$	1320.00	0.0982882
$566$	6900.00	0.512418
$567$	0	0
$568$	4440.00	0.327990
$569$	−11990.0	−0.883387	−0.441693	−	0.897166i	$-0.645622\pi$
$569$	−11990.0	−0.883387	−0.441693	+	0.897166i	$0.645622\pi$
$570$	0	0
$571$	−15716.0	−1.15183	−0.575914	−	0.817510i	$-0.695354\pi$
$571$	−15716.0	−1.15183	−0.575914	+	0.817510i	$0.695354\pi$
$572$	11760.0	0.859633
$573$	0	0
$574$	0	0
$575$	3344.00	0.242529
$576$	0	0
$577$	13872.0	1.00086	0.500432	−	0.865776i	$-0.333174\pi$
$577$	13872.0	1.00086	0.500432	+	0.865776i	$0.333174\pi$
$578$	4303.00	0.309656
$579$	0	0
$580$	4872.00	0.348791
$581$	0	0
$582$	0	0
$583$	−14440.0	−1.02580
$584$	3600.00	0.255084
$585$	0	0
$586$	−4452.00	−0.313840
$587$	8820.00	0.620171	0.310085	−	0.950709i	$-0.399642\pi$
$587$	8820.00	0.620171	0.310085	+	0.950709i	$0.399642\pi$
$588$	0	0
$589$	−3168.00	−0.221622
$590$	5904.00	0.411973
$591$	0	0
$592$	10578.0	0.734380
$593$	−16872.0	−1.16838	−0.584191	−	0.811617i	$-0.698588\pi$
$593$	−16872.0	−1.16838	−0.584191	+	0.811617i	$0.698588\pi$
$594$	0	0
$595$	0	0
$596$	7490.00	0.514769
$597$	0	0
$598$	14784.0	1.01097
$599$	6056.00	0.413091	0.206545	−	0.978437i	$-0.433778\pi$
$599$	6056.00	0.413091	0.206545	+	0.978437i	$0.433778\pi$
$600$	0	0
$601$	−10752.0	−0.729756	−0.364878	−	0.931055i	$-0.618889\pi$
$601$	−10752.0	−0.729756	−0.364878	+	0.931055i	$0.618889\pi$
$602$	0	0
$603$	0	0
$604$	840.000	0.0565879
$605$	−11172.0	−0.750754
$606$	0	0
$607$	−20256.0	−1.35447	−0.677237	−	0.735765i	$-0.736823\pi$
$607$	−20256.0	−1.35447	−0.677237	+	0.735765i	$0.736823\pi$
$608$	−1932.00	−0.128870
$609$	0	0
$610$	5904.00	0.391879
$611$	−34272.0	−2.26923
$612$	0	0
$613$	−28190.0	−1.85740	−0.928698	−	0.370838i	$-0.879071\pi$
$613$	−28190.0	−1.85740	−0.928698	+	0.370838i	$0.879071\pi$
$614$	2436.00	0.160112
$615$	0	0
$616$	0	0
$617$	−29318.0	−1.91296	−0.956482	−	0.291793i	$-0.905748\pi$
$617$	−29318.0	−1.91296	−0.956482	+	0.291793i	$0.905748\pi$
$618$	0	0
$619$	−24348.0	−1.58098	−0.790492	−	0.612473i	$-0.790175\pi$
$619$	−24348.0	−1.58098	−0.790492	+	0.612473i	$0.790175\pi$
$620$	−22176.0	−1.43647
$621$	0	0
$622$	−7488.00	−0.482703
$623$	0	0
$624$	0	0
$625$	−17639.0	−1.12890
$626$	1752.00	0.111859
$627$	0	0
$628$	12852.0	0.816641
$629$	−24768.0	−1.57006
$630$	0	0
$631$	−25184.0	−1.58884	−0.794421	−	0.607368i	$-0.792226\pi$
$631$	−25184.0	−1.58884	−0.794421	+	0.607368i	$0.792226\pi$
$632$	−11640.0	−0.732618
$633$	0	0
$634$	1562.00	0.0978469
$635$	192.000	0.0119989
$636$	0	0
$637$	0	0
$638$	1160.00	0.0719825
$639$	0	0
$640$	−17460.0	−1.07839
$641$	−32318.0	−1.99140	−0.995698	−	0.0926628i	$-0.970462\pi$
$641$	−32318.0	−1.99140	−0.995698	+	0.0926628i	$0.970462\pi$
$642$	0	0
$643$	3948.00	0.242137	0.121068	−	0.992644i	$-0.461368\pi$
$643$	3948.00	0.242137	0.121068	+	0.992644i	$0.461368\pi$
$644$	0	0
$645$	0	0
$646$	1152.00	0.0701623
$647$	13848.0	0.841454	0.420727	−	0.907187i	$-0.361775\pi$
$647$	13848.0	0.841454	0.420727	+	0.907187i	$0.361775\pi$
$648$	0	0
$649$	−9840.00	−0.595152
$650$	1596.00	0.0963081
$651$	0	0
$652$	−6412.00	−0.385143
$653$	3158.00	0.189253	0.0946264	−	0.995513i	$-0.469834\pi$
$653$	3158.00	0.189253	0.0946264	+	0.995513i	$0.469834\pi$
$654$	0	0
$655$	20304.0	1.21121
$656$	0	0
$657$	0	0
$658$	0	0
$659$	24596.0	1.45391	0.726953	−	0.686687i	$-0.240936\pi$
$659$	24596.0	1.45391	0.726953	+	0.686687i	$0.240936\pi$
$660$	0	0
$661$	15468.0	0.910190	0.455095	−	0.890443i	$-0.349605\pi$
$661$	15468.0	0.910190	0.455095	+	0.890443i	$0.349605\pi$
$662$	−7092.00	−0.416372
$663$	0	0
$664$	−13860.0	−0.810049
$665$	0	0
$666$	0	0
$667$	−10208.0	−0.592587
$668$	−3528.00	−0.204345
$669$	0	0
$670$	4944.00	0.285080
$671$	−9840.00	−0.566124
$672$	0	0
$673$	13470.0	0.771516	0.385758	−	0.922600i	$-0.373940\pi$
$673$	13470.0	0.771516	0.385758	+	0.922600i	$0.373940\pi$
$674$	366.000	0.0209166
$675$	0	0
$676$	−34013.0	−1.93520
$677$	−9564.00	−0.542946	−0.271473	−	0.962446i	$-0.587511\pi$
$677$	−9564.00	−0.542946	−0.271473	+	0.962446i	$0.587511\pi$
$678$	0	0
$679$	0	0
$680$	17280.0	0.974497
$681$	0	0
$682$	−5280.00	−0.296454
$683$	−13852.0	−0.776035	−0.388018	−	0.921652i	$-0.626840\pi$
$683$	−13852.0	−0.776035	−0.388018	+	0.921652i	$0.626840\pi$
$684$	0	0
$685$	−13512.0	−0.753674
$686$	0	0
$687$	0	0
$688$	6396.00	0.354426
$689$	60648.0	3.35342
$690$	0	0
$691$	324.000	0.0178373	0.00891863	−	0.999960i	$-0.497161\pi$
$691$	324.000	0.0178373	0.00891863	+	0.999960i	$0.497161\pi$
$692$	12852.0	0.706011
$693$	0	0
$694$	6364.00	0.348090
$695$	−13104.0	−0.715199
$696$	0	0
$697$	0	0
$698$	10500.0	0.569385
$699$	0	0
$700$	0	0
$701$	−24922.0	−1.34278	−0.671392	−	0.741103i	$-0.734303\pi$
$701$	−24922.0	−1.34278	−0.671392	+	0.741103i	$0.734303\pi$
$702$	0	0
$703$	−3096.00	−0.166099
$704$	3340.00	0.178808
$705$	0	0
$706$	408.000	0.0217497
$707$	0	0
$708$	0	0
$709$	−17886.0	−0.947423	−0.473711	−	0.880680i	$-0.657086\pi$
$709$	−17886.0	−0.947423	−0.473711	+	0.880680i	$0.657086\pi$
$710$	−3552.00	−0.187752
$711$	0	0
$712$	11160.0	0.587414
$713$	46464.0	2.44052
$714$	0	0
$715$	−20160.0	−1.05446
$716$	16604.0	0.866649
$717$	0	0
$718$	11936.0	0.620401
$719$	−6792.00	−0.352293	−0.176147	−	0.984364i	$-0.556363\pi$
$719$	−6792.00	−0.352293	−0.176147	+	0.984364i	$0.556363\pi$
$720$	0	0
$721$	0	0
$722$	−6715.00	−0.346131
$723$	0	0
$724$	−7644.00	−0.392385
$725$	−1102.00	−0.0564514
$726$	0	0
$727$	−1512.00	−0.0771348	−0.0385674	−	0.999256i	$-0.512279\pi$
$727$	−1512.00	−0.0771348	−0.0385674	+	0.999256i	$0.512279\pi$
$728$	0	0
$729$	0	0
$730$	−2880.00	−0.146019
$731$	−14976.0	−0.757739
$732$	0	0
$733$	11244.0	0.566585	0.283292	−	0.959034i	$-0.408573\pi$
$733$	11244.0	0.566585	0.283292	+	0.959034i	$0.408573\pi$
$734$	2448.00	0.123103
$735$	0	0
$736$	28336.0	1.41913
$737$	−8240.00	−0.411838
$738$	0	0
$739$	−1996.00	−0.0993559	−0.0496780	−	0.998765i	$-0.515820\pi$
$739$	−1996.00	−0.0993559	−0.0496780	+	0.998765i	$0.515820\pi$
$740$	−21672.0	−1.07659
$741$	0	0
$742$	0	0
$743$	656.000	0.0323907	0.0161954	−	0.999869i	$-0.494845\pi$
$743$	656.000	0.0323907	0.0161954	+	0.999869i	$0.494845\pi$
$744$	0	0
$745$	−12840.0	−0.631438
$746$	11374.0	0.558219
$747$	0	0
$748$	−13440.0	−0.656972
$749$	0	0
$750$	0	0
$751$	1056.00	0.0513102	0.0256551	−	0.999671i	$-0.491833\pi$
$751$	1056.00	0.0513102	0.0256551	+	0.999671i	$0.491833\pi$
$752$	−16728.0	−0.811180
$753$	0	0
$754$	−4872.00	−0.235315
$755$	−1440.00	−0.0694132
$756$	0	0
$757$	−18702.0	−0.897934	−0.448967	−	0.893548i	$-0.648208\pi$
$757$	−18702.0	−0.897934	−0.448967	+	0.893548i	$0.648208\pi$
$758$	−5892.00	−0.282331
$759$	0	0
$760$	2160.00	0.103094
$761$	17904.0	0.852851	0.426425	−	0.904523i	$-0.359773\pi$
$761$	17904.0	0.852851	0.426425	+	0.904523i	$0.359773\pi$
$762$	0	0
$763$	0	0
$764$	17584.0	0.832679
$765$	0	0
$766$	−10488.0	−0.494709
$767$	41328.0	1.94559
$768$	0	0
$769$	7560.00	0.354513	0.177257	−	0.984165i	$-0.443278\pi$
$769$	7560.00	0.354513	0.177257	+	0.984165i	$0.443278\pi$
$770$	0	0
$771$	0	0
$772$	17010.0	0.793009
$773$	−14292.0	−0.665003	−0.332502	−	0.943103i	$-0.607893\pi$
$773$	−14292.0	−0.665003	−0.332502	+	0.943103i	$0.607893\pi$
$774$	0	0
$775$	5016.00	0.232490
$776$	−2520.00	−0.116576
$777$	0	0
$778$	−4514.00	−0.208014
$779$	0	0
$780$	0	0
$781$	5920.00	0.271235
$782$	−16896.0	−0.772634
$783$	0	0
$784$	0	0
$785$	−22032.0	−1.00173
$786$	0	0
$787$	−26364.0	−1.19412	−0.597062	−	0.802195i	$-0.703665\pi$
$787$	−26364.0	−1.19412	−0.597062	+	0.802195i	$0.703665\pi$
$788$	−12334.0	−0.557590
$789$	0	0
$790$	9312.00	0.419375
$791$	0	0
$792$	0	0
$793$	41328.0	1.85069
$794$	6036.00	0.269785
$795$	0	0
$796$	21672.0	0.965005
$797$	17220.0	0.765325	0.382662	−	0.923888i	$-0.375007\pi$
$797$	17220.0	0.765325	0.382662	+	0.923888i	$0.375007\pi$
$798$	0	0
$799$	39168.0	1.73425
$800$	3059.00	0.135190
$801$	0	0
$802$	6770.00	0.298076
$803$	4800.00	0.210944
$804$	0	0
$805$	0	0
$806$	22176.0	0.969127
$807$	0	0
$808$	22860.0	0.995312
$809$	−16442.0	−0.714549	−0.357274	−	0.933999i	$-0.616294\pi$
$809$	−16442.0	−0.714549	−0.357274	+	0.933999i	$0.616294\pi$
$810$	0	0
$811$	31332.0	1.35662	0.678308	−	0.734778i	$-0.262714\pi$
$811$	31332.0	1.35662	0.678308	+	0.734778i	$0.262714\pi$
$812$	0	0
$813$	0	0
$814$	−5160.00	−0.222184
$815$	10992.0	0.472433
$816$	0	0
$817$	−1872.00	−0.0801628
$818$	−12504.0	−0.534465
$819$	0	0
$820$	0	0
$821$	25810.0	1.09717	0.548584	−	0.836095i	$-0.315167\pi$
$821$	25810.0	1.09717	0.548584	+	0.836095i	$0.315167\pi$
$822$	0	0
$823$	12368.0	0.523841	0.261921	−	0.965089i	$-0.415644\pi$
$823$	12368.0	0.523841	0.261921	+	0.965089i	$0.415644\pi$
$824$	−6120.00	−0.258738
$825$	0	0
$826$	0	0
$827$	−6316.00	−0.265573	−0.132786	−	0.991145i	$-0.542392\pi$
$827$	−6316.00	−0.265573	−0.132786	+	0.991145i	$0.542392\pi$
$828$	0	0
$829$	23868.0	0.999964	0.499982	−	0.866036i	$-0.333340\pi$
$829$	23868.0	0.999964	0.499982	+	0.866036i	$0.333340\pi$
$830$	11088.0	0.463699
$831$	0	0
$832$	−14028.0	−0.584535
$833$	0	0
$834$	0	0
$835$	6048.00	0.250658
$836$	−1680.00	−0.0695024
$837$	0	0
$838$	9492.00	0.391284
$839$	−48216.0	−1.98403	−0.992015	−	0.126120i	$-0.959748\pi$
$839$	−48216.0	−1.98403	−0.992015	+	0.126120i	$0.959748\pi$
$840$	0	0
$841$	−21025.0	−0.862069
$842$	5182.00	0.212094
$843$	0	0
$844$	−1092.00	−0.0445358
$845$	58308.0	2.37379
$846$	0	0
$847$	0	0
$848$	29602.0	1.19875
$849$	0	0
$850$	−1824.00	−0.0736032
$851$	45408.0	1.82910
$852$	0	0
$853$	27300.0	1.09582	0.547910	−	0.836537i	$-0.315424\pi$
$853$	27300.0	1.09582	0.547910	+	0.836537i	$0.315424\pi$
$854$	0	0
$855$	0	0
$856$	−12300.0	−0.491128
$857$	8640.00	0.344384	0.172192	−	0.985063i	$-0.444915\pi$
$857$	8640.00	0.344384	0.172192	+	0.985063i	$0.444915\pi$
$858$	0	0
$859$	−24372.0	−0.968058	−0.484029	−	0.875052i	$-0.660827\pi$
$859$	−24372.0	−0.968058	−0.484029	+	0.875052i	$0.660827\pi$
$860$	−13104.0	−0.519585
$861$	0	0
$862$	5720.00	0.226014
$863$	−2176.00	−0.0858307	−0.0429154	−	0.999079i	$-0.513665\pi$
$863$	−2176.00	−0.0858307	−0.0429154	+	0.999079i	$0.513665\pi$
$864$	0	0
$865$	−22032.0	−0.866024
$866$	−13608.0	−0.533971
$867$	0	0
$868$	0	0
$869$	−15520.0	−0.605846
$870$	0	0
$871$	34608.0	1.34632
$872$	13770.0	0.534760
$873$	0	0
$874$	−2112.00	−0.0817385
$875$	0	0
$876$	0	0
$877$	−27574.0	−1.06170	−0.530848	−	0.847467i	$-0.678127\pi$
$877$	−27574.0	−1.06170	−0.530848	+	0.847467i	$0.678127\pi$
$878$	−12864.0	−0.494464
$879$	0	0
$880$	−9840.00	−0.376939
$881$	16968.0	0.648884	0.324442	−	0.945906i	$-0.394824\pi$
$881$	16968.0	0.648884	0.324442	+	0.945906i	$0.394824\pi$
$882$	0	0
$883$	−1860.00	−0.0708879	−0.0354439	−	0.999372i	$-0.511285\pi$
$883$	−1860.00	−0.0708879	−0.0354439	+	0.999372i	$0.511285\pi$
$884$	56448.0	2.14768
$885$	0	0
$886$	13252.0	0.502494
$887$	2280.00	0.0863077	0.0431538	−	0.999068i	$-0.486259\pi$
$887$	2280.00	0.0863077	0.0431538	+	0.999068i	$0.486259\pi$
$888$	0	0
$889$	0	0
$890$	−8928.00	−0.336255
$891$	0	0
$892$	35280.0	1.32428
$893$	4896.00	0.183470
$894$	0	0
$895$	−28464.0	−1.06307
$896$	0	0
$897$	0	0
$898$	−226.000	−0.00839835
$899$	−15312.0	−0.568058
$900$	0	0
$901$	−69312.0	−2.56284
$902$	0	0
$903$	0	0
$904$	−1650.00	−0.0607060
$905$	13104.0	0.481317
$906$	0	0
$907$	36084.0	1.32100	0.660501	−	0.750825i	$-0.270344\pi$
$907$	36084.0	1.32100	0.660501	+	0.750825i	$0.270344\pi$
$908$	−15204.0	−0.555686
$909$	0	0
$910$	0	0
$911$	−24152.0	−0.878366	−0.439183	−	0.898398i	$-0.644732\pi$
$911$	−24152.0	−0.878366	−0.439183	+	0.898398i	$0.644732\pi$
$912$	0	0
$913$	−18480.0	−0.669878
$914$	−11334.0	−0.410170
$915$	0	0
$916$	18900.0	0.681740
$917$	0	0
$918$	0	0
$919$	36336.0	1.30426	0.652130	−	0.758108i	$-0.273876\pi$
$919$	36336.0	1.30426	0.652130	+	0.758108i	$0.273876\pi$
$920$	−31680.0	−1.13528
$921$	0	0
$922$	−1596.00	−0.0570081
$923$	−24864.0	−0.886683
$924$	0	0
$925$	4902.00	0.174245
$926$	12728.0	0.451693
$927$	0	0
$928$	−9338.00	−0.330318
$929$	432.000	0.0152567	0.00762834	−	0.999971i	$-0.497572\pi$
$929$	432.000	0.0152567	0.00762834	+	0.999971i	$0.497572\pi$
$930$	0	0
$931$	0	0
$932$	−26614.0	−0.935376
$933$	0	0
$934$	−3012.00	−0.105520
$935$	23040.0	0.805870
$936$	0	0
$937$	−22176.0	−0.773168	−0.386584	−	0.922254i	$-0.626345\pi$
$937$	−22176.0	−0.773168	−0.386584	+	0.922254i	$0.626345\pi$
$938$	0	0
$939$	0	0
$940$	34272.0	1.18918
$941$	−43524.0	−1.50780	−0.753901	−	0.656988i	$-0.771830\pi$
$941$	−43524.0	−1.50780	−0.753901	+	0.656988i	$0.771830\pi$
$942$	0	0
$943$	0	0
$944$	20172.0	0.695490
$945$	0	0
$946$	−3120.00	−0.107230
$947$	−1868.00	−0.0640991	−0.0320495	−	0.999486i	$-0.510203\pi$
$947$	−1868.00	−0.0640991	−0.0320495	+	0.999486i	$0.510203\pi$
$948$	0	0
$949$	−20160.0	−0.689590
$950$	−228.000	−0.00778663
$951$	0	0
$952$	0	0
$953$	9238.00	0.314006	0.157003	−	0.987598i	$-0.449817\pi$
$953$	9238.00	0.314006	0.157003	+	0.987598i	$0.449817\pi$
$954$	0	0
$955$	−30144.0	−1.02140
$956$	−30856.0	−1.04389
$957$	0	0
$958$	−4296.00	−0.144883
$959$	0	0
$960$	0	0
$961$	39905.0	1.33950
$962$	21672.0	0.726334
$963$	0	0
$964$	21672.0	0.724075
$965$	−29160.0	−0.972739
$966$	0	0
$967$	−30616.0	−1.01814	−0.509071	−	0.860724i	$-0.670011\pi$
$967$	−30616.0	−1.01814	−0.509071	+	0.860724i	$0.670011\pi$
$968$	13965.0	0.463690
$969$	0	0
$970$	2016.00	0.0667318
$971$	−27540.0	−0.910196	−0.455098	−	0.890441i	$-0.650396\pi$
$971$	−27540.0	−0.910196	−0.455098	+	0.890441i	$0.650396\pi$
$972$	0	0
$973$	0	0
$974$	−8184.00	−0.269232
$975$	0	0
$976$	20172.0	0.661568
$977$	16402.0	0.537100	0.268550	−	0.963266i	$-0.413456\pi$
$977$	16402.0	0.537100	0.268550	+	0.963266i	$0.413456\pi$
$978$	0	0
$979$	14880.0	0.485768
$980$	0	0
$981$	0	0
$982$	12164.0	0.395284
$983$	55176.0	1.79028	0.895138	−	0.445789i	$-0.147077\pi$
$983$	55176.0	1.79028	0.895138	+	0.445789i	$0.147077\pi$
$984$	0	0
$985$	21144.0	0.683963
$986$	5568.00	0.179839
$987$	0	0
$988$	7056.00	0.227208
$989$	27456.0	0.882760
$990$	0	0
$991$	27096.0	0.868550	0.434275	−	0.900780i	$-0.357005\pi$
$991$	27096.0	0.868550	0.434275	+	0.900780i	$0.357005\pi$
$992$	42504.0	1.36039
$993$	0	0
$994$	0	0
$995$	−37152.0	−1.18372
$996$	0	0
$997$	16812.0	0.534044	0.267022	−	0.963691i	$-0.413960\pi$
$997$	16812.0	0.534044	0.267022	+	0.963691i	$0.413960\pi$
$998$	972.000	0.0308298
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.4.a.h.1.1		1
3.2	odd	2		147.4.a.e.1.1	yes	1
7.2	even	3		441.4.e.f.361.1		2
7.3	odd	6		441.4.e.g.226.1		2
7.4	even	3		441.4.e.f.226.1		2
7.5	odd	6		441.4.e.g.361.1		2
7.6	odd	2		441.4.a.g.1.1		1
12.11	even	2		2352.4.a.b.1.1		1
21.2	odd	6		147.4.e.e.67.1		2
21.5	even	6		147.4.e.f.67.1		2
21.11	odd	6		147.4.e.e.79.1		2
21.17	even	6		147.4.e.f.79.1		2
21.20	even	2		147.4.a.d.1.1	✓	1
84.83	odd	2		2352.4.a.bi.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.4.a.d.1.1	✓	1	21.20	even	2
147.4.a.e.1.1	yes	1	3.2	odd	2
147.4.e.e.67.1		2	21.2	odd	6
147.4.e.e.79.1		2	21.11	odd	6
147.4.e.f.67.1		2	21.5	even	6
147.4.e.f.79.1		2	21.17	even	6
441.4.a.g.1.1		1	7.6	odd	2
441.4.a.h.1.1		1	1.1	even	1	trivial
441.4.e.f.226.1		2	7.4	even	3
441.4.e.f.361.1		2	7.2	even	3
441.4.e.g.226.1		2	7.3	odd	6
441.4.e.g.361.1		2	7.5	odd	6
2352.4.a.b.1.1		1	12.11	even	2
2352.4.a.bi.1.1		1	84.83	odd	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:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	441.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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