LMFDB - Embedded newform 441.4.a.i.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 7)
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)

$f(q)$

$=$

$q+1.00000 q^{2} -7.00000 q^{4} +16.0000 q^{5} -15.0000 q^{8} +O(q^{10})$

$q+1.00000 q^{2} -7.00000 q^{4} +16.0000 q^{5} -15.0000 q^{8} +16.0000 q^{10} +8.00000 q^{11} -28.0000 q^{13} +41.0000 q^{16} +54.0000 q^{17} +110.000 q^{19} -112.000 q^{20} +8.00000 q^{22} -48.0000 q^{23} +131.000 q^{25} -28.0000 q^{26} +110.000 q^{29} -12.0000 q^{31} +161.000 q^{32} +54.0000 q^{34} -246.000 q^{37} +110.000 q^{38} -240.000 q^{40} +182.000 q^{41} +128.000 q^{43} -56.0000 q^{44} -48.0000 q^{46} +324.000 q^{47} +131.000 q^{50} +196.000 q^{52} +162.000 q^{53} +128.000 q^{55} +110.000 q^{58} +810.000 q^{59} +488.000 q^{61} -12.0000 q^{62} -167.000 q^{64} -448.000 q^{65} +244.000 q^{67} -378.000 q^{68} +768.000 q^{71} +702.000 q^{73} -246.000 q^{74} -770.000 q^{76} +440.000 q^{79} +656.000 q^{80} +182.000 q^{82} -1302.00 q^{83} +864.000 q^{85} +128.000 q^{86} -120.000 q^{88} +730.000 q^{89} +336.000 q^{92} +324.000 q^{94} +1760.00 q^{95} -294.000 q^{97} +O(q^{100})$

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
$5$	16.0000	1.43108	0.715542	−	0.698570i	$-0.246180\pi$
$5$	16.0000	1.43108	0.715542	+	0.698570i	$0.246180\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−15.0000	−0.662913
$9$	0	0
$10$	16.0000	0.505964
$11$	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
$13$	−28.0000	−0.597369	−0.298685	−	0.954352i	$-0.596548\pi$
$13$	−28.0000	−0.597369	−0.298685	+	0.954352i	$0.596548\pi$
$14$	0	0
$15$	0	0
$16$	41.0000	0.640625
$17$	54.0000	0.770407	0.385204	−	0.922832i	$-0.374131\pi$
$17$	54.0000	0.770407	0.385204	+	0.922832i	$0.374131\pi$
$18$	0	0
$19$	110.000	1.32820	0.664098	−	0.747645i	$-0.268816\pi$
$19$	110.000	1.32820	0.664098	+	0.747645i	$0.268816\pi$
$20$	−112.000	−1.25220
$21$	0	0
$22$	8.00000	0.0775275
$23$	−48.0000	−0.435161	−0.217580	−	0.976042i	$-0.569816\pi$
$23$	−48.0000	−0.435161	−0.217580	+	0.976042i	$0.569816\pi$
$24$	0	0
$25$	131.000	1.04800
$26$	−28.0000	−0.211202
$27$	0	0
$28$	0	0
$29$	110.000	0.704362	0.352181	−	0.935932i	$-0.385440\pi$
$29$	110.000	0.704362	0.352181	+	0.935932i	$0.385440\pi$
$30$	0	0
$31$	−12.0000	−0.0695246	−0.0347623	−	0.999396i	$-0.511067\pi$
$31$	−12.0000	−0.0695246	−0.0347623	+	0.999396i	$0.511067\pi$
$32$	161.000	0.889408
$33$	0	0
$34$	54.0000	0.272380
$35$	0	0
$36$	0	0
$37$	−246.000	−1.09303	−0.546516	−	0.837449i	$-0.684046\pi$
$37$	−246.000	−1.09303	−0.546516	+	0.837449i	$0.684046\pi$
$38$	110.000	0.469588
$39$	0	0
$40$	−240.000	−0.948683
$41$	182.000	0.693259	0.346630	−	0.938002i	$-0.387326\pi$
$41$	182.000	0.693259	0.346630	+	0.938002i	$0.387326\pi$
$42$	0	0
$43$	128.000	0.453949	0.226975	−	0.973901i	$-0.427117\pi$
$43$	128.000	0.453949	0.226975	+	0.973901i	$0.427117\pi$
$44$	−56.0000	−0.191871
$45$	0	0
$46$	−48.0000	−0.153852
$47$	324.000	1.00554	0.502769	−	0.864421i	$-0.332315\pi$
$47$	324.000	1.00554	0.502769	+	0.864421i	$0.332315\pi$
$48$	0	0
$49$	0	0
$50$	131.000	0.370524
$51$	0	0
$52$	196.000	0.522698
$53$	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
$55$	128.000	0.313809
$56$	0	0
$57$	0	0
$58$	110.000	0.249029
$59$	810.000	1.78734	0.893670	−	0.448725i	$-0.148122\pi$
$59$	810.000	1.78734	0.893670	+	0.448725i	$0.148122\pi$
$60$	0	0
$61$	488.000	1.02430	0.512148	−	0.858898i	$-0.328850\pi$
$61$	488.000	1.02430	0.512148	+	0.858898i	$0.328850\pi$
$62$	−12.0000	−0.0245807
$63$	0	0
$64$	−167.000	−0.326172
$65$	−448.000	−0.854886
$66$	0	0
$67$	244.000	0.444916	0.222458	−	0.974942i	$-0.428592\pi$
$67$	244.000	0.444916	0.222458	+	0.974942i	$0.428592\pi$
$68$	−378.000	−0.674106
$69$	0	0
$70$	0	0
$71$	768.000	1.28373	0.641865	−	0.766818i	$-0.278161\pi$
$71$	768.000	1.28373	0.641865	+	0.766818i	$0.278161\pi$
$72$	0	0
$73$	702.000	1.12552	0.562759	−	0.826621i	$-0.309740\pi$
$73$	702.000	1.12552	0.562759	+	0.826621i	$0.309740\pi$
$74$	−246.000	−0.386445
$75$	0	0
$76$	−770.000	−1.16217
$77$	0	0
$78$	0	0
$79$	440.000	0.626631	0.313316	−	0.949649i	$-0.398560\pi$
$79$	440.000	0.626631	0.313316	+	0.949649i	$0.398560\pi$
$80$	656.000	0.916788
$81$	0	0
$82$	182.000	0.245104
$83$	−1302.00	−1.72184	−0.860922	−	0.508737i	$-0.830113\pi$
$83$	−1302.00	−1.72184	−0.860922	+	0.508737i	$0.830113\pi$
$84$	0	0
$85$	864.000	1.10252
$86$	128.000	0.160495
$87$	0	0
$88$	−120.000	−0.145364
$89$	730.000	0.869436	0.434718	−	0.900567i	$-0.356848\pi$
$89$	730.000	0.869436	0.434718	+	0.900567i	$0.356848\pi$
$90$	0	0
$91$	0	0
$92$	336.000	0.380765
$93$	0	0
$94$	324.000	0.355511
$95$	1760.00	1.90076
$96$	0	0
$97$	−294.000	−0.307744	−0.153872	−	0.988091i	$-0.549174\pi$
$97$	−294.000	−0.307744	−0.153872	+	0.988091i	$0.549174\pi$
$98$	0	0
$99$	0	0
$100$	−917.000	−0.917000
$101$	−688.000	−0.677808	−0.338904	−	0.940821i	$-0.610056\pi$
$101$	−688.000	−0.677808	−0.338904	+	0.940821i	$0.610056\pi$
$102$	0	0
$103$	−1388.00	−1.32780	−0.663901	−	0.747820i	$-0.731101\pi$
$103$	−1388.00	−1.32780	−0.663901	+	0.747820i	$0.731101\pi$
$104$	420.000	0.396004
$105$	0	0
$106$	162.000	0.148442
$107$	−244.000	−0.220452	−0.110226	−	0.993907i	$-0.535157\pi$
$107$	−244.000	−0.220452	−0.110226	+	0.993907i	$0.535157\pi$
$108$	0	0
$109$	90.0000	0.0790866	0.0395433	−	0.999218i	$-0.487410\pi$
$109$	90.0000	0.0790866	0.0395433	+	0.999218i	$0.487410\pi$
$110$	128.000	0.110948
$111$	0	0
$112$	0	0
$113$	−1318.00	−1.09723	−0.548615	−	0.836075i	$-0.684845\pi$
$113$	−1318.00	−1.09723	−0.548615	+	0.836075i	$0.684845\pi$
$114$	0	0
$115$	−768.000	−0.622751
$116$	−770.000	−0.616316
$117$	0	0
$118$	810.000	0.631920
$119$	0	0
$120$	0	0
$121$	−1267.00	−0.951916
$122$	488.000	0.362143
$123$	0	0
$124$	84.0000	0.0608341
$125$	96.0000	0.0686920
$126$	0	0
$127$	−1776.00	−1.24090	−0.620451	−	0.784245i	$-0.713050\pi$
$127$	−1776.00	−1.24090	−0.620451	+	0.784245i	$0.713050\pi$
$128$	−1455.00	−1.00473
$129$	0	0
$130$	−448.000	−0.302248
$131$	−1118.00	−0.745650	−0.372825	−	0.927902i	$-0.621611\pi$
$131$	−1118.00	−0.745650	−0.372825	+	0.927902i	$0.621611\pi$
$132$	0	0
$133$	0	0
$134$	244.000	0.157301
$135$	0	0
$136$	−810.000	−0.510713
$137$	−2274.00	−1.41811	−0.709054	−	0.705154i	$-0.750878\pi$
$137$	−2274.00	−1.41811	−0.709054	+	0.705154i	$0.750878\pi$
$138$	0	0
$139$	210.000	0.128144	0.0640718	−	0.997945i	$-0.479591\pi$
$139$	210.000	0.128144	0.0640718	+	0.997945i	$0.479591\pi$
$140$	0	0
$141$	0	0
$142$	768.000	0.453867
$143$	−224.000	−0.130992
$144$	0	0
$145$	1760.00	1.00800
$146$	702.000	0.397931
$147$	0	0
$148$	1722.00	0.956402
$149$	2010.00	1.10514	0.552569	−	0.833467i	$-0.313648\pi$
$149$	2010.00	1.10514	0.552569	+	0.833467i	$0.313648\pi$
$150$	0	0
$151$	1112.00	0.599293	0.299647	−	0.954050i	$-0.403131\pi$
$151$	1112.00	0.599293	0.299647	+	0.954050i	$0.403131\pi$
$152$	−1650.00	−0.880478
$153$	0	0
$154$	0	0
$155$	−192.000	−0.0994956
$156$	0	0
$157$	−124.000	−0.0630336	−0.0315168	−	0.999503i	$-0.510034\pi$
$157$	−124.000	−0.0630336	−0.0315168	+	0.999503i	$0.510034\pi$
$158$	440.000	0.221548
$159$	0	0
$160$	2576.00	1.27282
$161$	0	0
$162$	0	0
$163$	2008.00	0.964900	0.482450	−	0.875924i	$-0.339747\pi$
$163$	2008.00	0.964900	0.482450	+	0.875924i	$0.339747\pi$
$164$	−1274.00	−0.606602
$165$	0	0
$166$	−1302.00	−0.608764
$167$	2884.00	1.33635	0.668176	−	0.744004i	$-0.267076\pi$
$167$	2884.00	1.33635	0.668176	+	0.744004i	$0.267076\pi$
$168$	0	0
$169$	−1413.00	−0.643150
$170$	864.000	0.389799
$171$	0	0
$172$	−896.000	−0.397206
$173$	2228.00	0.979143	0.489571	−	0.871963i	$-0.337153\pi$
$173$	2228.00	0.979143	0.489571	+	0.871963i	$0.337153\pi$
$174$	0	0
$175$	0	0
$176$	328.000	0.140477
$177$	0	0
$178$	730.000	0.307392
$179$	820.000	0.342400	0.171200	−	0.985236i	$-0.445236\pi$
$179$	820.000	0.342400	0.171200	+	0.985236i	$0.445236\pi$
$180$	0	0
$181$	−3892.00	−1.59829	−0.799144	−	0.601140i	$-0.794713\pi$
$181$	−3892.00	−1.59829	−0.799144	+	0.601140i	$0.794713\pi$
$182$	0	0
$183$	0	0
$184$	720.000	0.288473
$185$	−3936.00	−1.56422
$186$	0	0
$187$	432.000	0.168936
$188$	−2268.00	−0.879845
$189$	0	0
$190$	1760.00	0.672020
$191$	5048.00	1.91236	0.956179	−	0.292782i	$-0.0945810\pi$
$191$	5048.00	1.91236	0.956179	+	0.292782i	$0.0945810\pi$
$192$	0	0
$193$	−2962.00	−1.10471	−0.552356	−	0.833608i	$-0.686271\pi$
$193$	−2962.00	−1.10471	−0.552356	+	0.833608i	$0.686271\pi$
$194$	−294.000	−0.108804
$195$	0	0
$196$	0	0
$197$	−3334.00	−1.20577	−0.602887	−	0.797826i	$-0.705983\pi$
$197$	−3334.00	−1.20577	−0.602887	+	0.797826i	$0.705983\pi$
$198$	0	0
$199$	−1860.00	−0.662572	−0.331286	−	0.943530i	$-0.607483\pi$
$199$	−1860.00	−0.662572	−0.331286	+	0.943530i	$0.607483\pi$
$200$	−1965.00	−0.694732
$201$	0	0
$202$	−688.000	−0.239641
$203$	0	0
$204$	0	0
$205$	2912.00	0.992112
$206$	−1388.00	−0.469449
$207$	0	0
$208$	−1148.00	−0.382690
$209$	880.000	0.291248
$210$	0	0
$211$	−4268.00	−1.39252	−0.696259	−	0.717791i	$-0.745153\pi$
$211$	−4268.00	−1.39252	−0.696259	+	0.717791i	$0.745153\pi$
$212$	−1134.00	−0.367375
$213$	0	0
$214$	−244.000	−0.0779416
$215$	2048.00	0.649639
$216$	0	0
$217$	0	0
$218$	90.0000	0.0279613
$219$	0	0
$220$	−896.000	−0.274583
$221$	−1512.00	−0.460218
$222$	0	0
$223$	5432.00	1.63118	0.815591	−	0.578629i	$-0.196412\pi$
$223$	5432.00	1.63118	0.815591	+	0.578629i	$0.196412\pi$
$224$	0	0
$225$	0	0
$226$	−1318.00	−0.387929
$227$	−2046.00	−0.598228	−0.299114	−	0.954217i	$-0.596691\pi$
$227$	−2046.00	−0.598228	−0.299114	+	0.954217i	$0.596691\pi$
$228$	0	0
$229$	2980.00	0.859930	0.429965	−	0.902846i	$-0.358526\pi$
$229$	2980.00	0.859930	0.429965	+	0.902846i	$0.358526\pi$
$230$	−768.000	−0.220176
$231$	0	0
$232$	−1650.00	−0.466930
$233$	−4458.00	−1.25345	−0.626724	−	0.779241i	$-0.715605\pi$
$233$	−4458.00	−1.25345	−0.626724	+	0.779241i	$0.715605\pi$
$234$	0	0
$235$	5184.00	1.43901
$236$	−5670.00	−1.56392
$237$	0	0
$238$	0	0
$239$	−4440.00	−1.20167	−0.600836	−	0.799372i	$-0.705166\pi$
$239$	−4440.00	−1.20167	−0.600836	+	0.799372i	$0.705166\pi$
$240$	0	0
$241$	−3302.00	−0.882575	−0.441287	−	0.897366i	$-0.645478\pi$
$241$	−3302.00	−0.882575	−0.441287	+	0.897366i	$0.645478\pi$
$242$	−1267.00	−0.336553
$243$	0	0
$244$	−3416.00	−0.896258
$245$	0	0
$246$	0	0
$247$	−3080.00	−0.793424
$248$	180.000	0.0460888
$249$	0	0
$250$	96.0000	0.0242863
$251$	1582.00	0.397829	0.198914	−	0.980017i	$-0.436258\pi$
$251$	1582.00	0.397829	0.198914	+	0.980017i	$0.436258\pi$
$252$	0	0
$253$	−384.000	−0.0954224
$254$	−1776.00	−0.438725
$255$	0	0
$256$	−119.000	−0.0290527
$257$	2354.00	0.571356	0.285678	−	0.958326i	$-0.407781\pi$
$257$	2354.00	0.571356	0.285678	+	0.958326i	$0.407781\pi$
$258$	0	0
$259$	0	0
$260$	3136.00	0.748025
$261$	0	0
$262$	−1118.00	−0.263627
$263$	3872.00	0.907824	0.453912	−	0.891046i	$-0.350028\pi$
$263$	3872.00	0.907824	0.453912	+	0.891046i	$0.350028\pi$
$264$	0	0
$265$	2592.00	0.600850
$266$	0	0
$267$	0	0
$268$	−1708.00	−0.389301
$269$	180.000	0.0407985	0.0203992	−	0.999792i	$-0.493506\pi$
$269$	180.000	0.0407985	0.0203992	+	0.999792i	$0.493506\pi$
$270$	0	0
$271$	−2032.00	−0.455480	−0.227740	−	0.973722i	$-0.573134\pi$
$271$	−2032.00	−0.455480	−0.227740	+	0.973722i	$0.573134\pi$
$272$	2214.00	0.493542
$273$	0	0
$274$	−2274.00	−0.501377
$275$	1048.00	0.229806
$276$	0	0
$277$	−5426.00	−1.17696	−0.588478	−	0.808513i	$-0.700273\pi$
$277$	−5426.00	−1.17696	−0.588478	+	0.808513i	$0.700273\pi$
$278$	210.000	0.0453056
$279$	0	0
$280$	0	0
$281$	−842.000	−0.178753	−0.0893764	−	0.995998i	$-0.528487\pi$
$281$	−842.000	−0.178753	−0.0893764	+	0.995998i	$0.528487\pi$
$282$	0	0
$283$	3782.00	0.794405	0.397202	−	0.917731i	$-0.369981\pi$
$283$	3782.00	0.794405	0.397202	+	0.917731i	$0.369981\pi$
$284$	−5376.00	−1.12326
$285$	0	0
$286$	−224.000	−0.0463126
$287$	0	0
$288$	0	0
$289$	−1997.00	−0.406473
$290$	1760.00	0.356382
$291$	0	0
$292$	−4914.00	−0.984829
$293$	−4312.00	−0.859760	−0.429880	−	0.902886i	$-0.641444\pi$
$293$	−4312.00	−0.859760	−0.429880	+	0.902886i	$0.641444\pi$
$294$	0	0
$295$	12960.0	2.55783
$296$	3690.00	0.724584
$297$	0	0
$298$	2010.00	0.390725
$299$	1344.00	0.259952
$300$	0	0
$301$	0	0
$302$	1112.00	0.211882
$303$	0	0
$304$	4510.00	0.850876
$305$	7808.00	1.46585
$306$	0	0
$307$	−2674.00	−0.497112	−0.248556	−	0.968618i	$-0.579956\pi$
$307$	−2674.00	−0.497112	−0.248556	+	0.968618i	$0.579956\pi$
$308$	0	0
$309$	0	0
$310$	−192.000	−0.0351770
$311$	−3768.00	−0.687021	−0.343511	−	0.939149i	$-0.611616\pi$
$311$	−3768.00	−0.687021	−0.343511	+	0.939149i	$0.611616\pi$
$312$	0	0
$313$	−2438.00	−0.440268	−0.220134	−	0.975470i	$-0.570649\pi$
$313$	−2438.00	−0.440268	−0.220134	+	0.975470i	$0.570649\pi$
$314$	−124.000	−0.0222857
$315$	0	0
$316$	−3080.00	−0.548302
$317$	3186.00	0.564491	0.282245	−	0.959342i	$-0.408921\pi$
$317$	3186.00	0.564491	0.282245	+	0.959342i	$0.408921\pi$
$318$	0	0
$319$	880.000	0.154453
$320$	−2672.00	−0.466779
$321$	0	0
$322$	0	0
$323$	5940.00	1.02325
$324$	0	0
$325$	−3668.00	−0.626043
$326$	2008.00	0.341144
$327$	0	0
$328$	−2730.00	−0.459570
$329$	0	0
$330$	0	0
$331$	8672.00	1.44005	0.720025	−	0.693949i	$-0.244131\pi$
$331$	8672.00	1.44005	0.720025	+	0.693949i	$0.244131\pi$
$332$	9114.00	1.50661
$333$	0	0
$334$	2884.00	0.472471
$335$	3904.00	0.636711
$336$	0	0
$337$	814.000	0.131577	0.0657884	−	0.997834i	$-0.479044\pi$
$337$	814.000	0.131577	0.0657884	+	0.997834i	$0.479044\pi$
$338$	−1413.00	−0.227388
$339$	0	0
$340$	−6048.00	−0.964703
$341$	−96.0000	−0.0152454
$342$	0	0
$343$	0	0
$344$	−1920.00	−0.300929
$345$	0	0
$346$	2228.00	0.346179
$347$	−9344.00	−1.44557	−0.722784	−	0.691074i	$-0.757138\pi$
$347$	−9344.00	−1.44557	−0.722784	+	0.691074i	$0.757138\pi$
$348$	0	0
$349$	5180.00	0.794496	0.397248	−	0.917711i	$-0.369965\pi$
$349$	5180.00	0.794496	0.397248	+	0.917711i	$0.369965\pi$
$350$	0	0
$351$	0	0
$352$	1288.00	0.195030
$353$	12178.0	1.83617	0.918087	−	0.396379i	$-0.129733\pi$
$353$	12178.0	1.83617	0.918087	+	0.396379i	$0.129733\pi$
$354$	0	0
$355$	12288.0	1.83712
$356$	−5110.00	−0.760757
$357$	0	0
$358$	820.000	0.121057
$359$	−440.000	−0.0646861	−0.0323431	−	0.999477i	$-0.510297\pi$
$359$	−440.000	−0.0646861	−0.0323431	+	0.999477i	$0.510297\pi$
$360$	0	0
$361$	5241.00	0.764106
$362$	−3892.00	−0.565080
$363$	0	0
$364$	0	0
$365$	11232.0	1.61071
$366$	0	0
$367$	9816.00	1.39616	0.698080	−	0.716019i	$-0.254038\pi$
$367$	9816.00	1.39616	0.698080	+	0.716019i	$0.254038\pi$
$368$	−1968.00	−0.278775
$369$	0	0
$370$	−3936.00	−0.553035
$371$	0	0
$372$	0	0
$373$	−442.000	−0.0613563	−0.0306781	−	0.999529i	$-0.509767\pi$
$373$	−442.000	−0.0613563	−0.0306781	+	0.999529i	$0.509767\pi$
$374$	432.000	0.0597278
$375$	0	0
$376$	−4860.00	−0.666583
$377$	−3080.00	−0.420764
$378$	0	0
$379$	−3960.00	−0.536706	−0.268353	−	0.963321i	$-0.586479\pi$
$379$	−3960.00	−0.536706	−0.268353	+	0.963321i	$0.586479\pi$
$380$	−12320.0	−1.66316
$381$	0	0
$382$	5048.00	0.676121
$383$	6708.00	0.894942	0.447471	−	0.894298i	$-0.352325\pi$
$383$	6708.00	0.894942	0.447471	+	0.894298i	$0.352325\pi$
$384$	0	0
$385$	0	0
$386$	−2962.00	−0.390575
$387$	0	0
$388$	2058.00	0.269276
$389$	13350.0	1.74003	0.870015	−	0.493025i	$-0.164109\pi$
$389$	13350.0	1.74003	0.870015	+	0.493025i	$0.164109\pi$
$390$	0	0
$391$	−2592.00	−0.335251
$392$	0	0
$393$	0	0
$394$	−3334.00	−0.426306
$395$	7040.00	0.896762
$396$	0	0
$397$	1356.00	0.171425	0.0857125	−	0.996320i	$-0.472683\pi$
$397$	1356.00	0.171425	0.0857125	+	0.996320i	$0.472683\pi$
$398$	−1860.00	−0.234255
$399$	0	0
$400$	5371.00	0.671375
$401$	−6222.00	−0.774843	−0.387421	−	0.921903i	$-0.626634\pi$
$401$	−6222.00	−0.774843	−0.387421	+	0.921903i	$0.626634\pi$
$402$	0	0
$403$	336.000	0.0415319
$404$	4816.00	0.593082
$405$	0	0
$406$	0	0
$407$	−1968.00	−0.239681
$408$	0	0
$409$	−5150.00	−0.622619	−0.311309	−	0.950309i	$-0.600768\pi$
$409$	−5150.00	−0.622619	−0.311309	+	0.950309i	$0.600768\pi$
$410$	2912.00	0.350764
$411$	0	0
$412$	9716.00	1.16183
$413$	0	0
$414$	0	0
$415$	−20832.0	−2.46410
$416$	−4508.00	−0.531305
$417$	0	0
$418$	880.000	0.102972
$419$	2310.00	0.269334	0.134667	−	0.990891i	$-0.457004\pi$
$419$	2310.00	0.269334	0.134667	+	0.990891i	$0.457004\pi$
$420$	0	0
$421$	1262.00	0.146095	0.0730476	−	0.997328i	$-0.476727\pi$
$421$	1262.00	0.146095	0.0730476	+	0.997328i	$0.476727\pi$
$422$	−4268.00	−0.492329
$423$	0	0
$424$	−2430.00	−0.278328
$425$	7074.00	0.807387
$426$	0	0
$427$	0	0
$428$	1708.00	0.192896
$429$	0	0
$430$	2048.00	0.229682
$431$	4488.00	0.501576	0.250788	−	0.968042i	$-0.419310\pi$
$431$	4488.00	0.501576	0.250788	+	0.968042i	$0.419310\pi$
$432$	0	0
$433$	−17038.0	−1.89098	−0.945490	−	0.325652i	$-0.894416\pi$
$433$	−17038.0	−1.89098	−0.945490	+	0.325652i	$0.894416\pi$
$434$	0	0
$435$	0	0
$436$	−630.000	−0.0692008
$437$	−5280.00	−0.577979
$438$	0	0
$439$	−16200.0	−1.76124	−0.880619	−	0.473824i	$-0.842873\pi$
$439$	−16200.0	−1.76124	−0.880619	+	0.473824i	$0.842873\pi$
$440$	−1920.00	−0.208028
$441$	0	0
$442$	−1512.00	−0.162712
$443$	8772.00	0.940791	0.470395	−	0.882456i	$-0.344111\pi$
$443$	8772.00	0.940791	0.470395	+	0.882456i	$0.344111\pi$
$444$	0	0
$445$	11680.0	1.24424
$446$	5432.00	0.576710
$447$	0	0
$448$	0	0
$449$	−2130.00	−0.223877	−0.111939	−	0.993715i	$-0.535706\pi$
$449$	−2130.00	−0.223877	−0.111939	+	0.993715i	$0.535706\pi$
$450$	0	0
$451$	1456.00	0.152019
$452$	9226.00	0.960076
$453$	0	0
$454$	−2046.00	−0.211506
$455$	0	0
$456$	0	0
$457$	10534.0	1.07825	0.539124	−	0.842226i	$-0.318755\pi$
$457$	10534.0	1.07825	0.539124	+	0.842226i	$0.318755\pi$
$458$	2980.00	0.304031
$459$	0	0
$460$	5376.00	0.544907
$461$	−9268.00	−0.936342	−0.468171	−	0.883638i	$-0.655087\pi$
$461$	−9268.00	−0.936342	−0.468171	+	0.883638i	$0.655087\pi$
$462$	0	0
$463$	−9392.00	−0.942728	−0.471364	−	0.881939i	$-0.656238\pi$
$463$	−9392.00	−0.942728	−0.471364	+	0.881939i	$0.656238\pi$
$464$	4510.00	0.451232
$465$	0	0
$466$	−4458.00	−0.443161
$467$	−10806.0	−1.07075	−0.535377	−	0.844613i	$-0.679830\pi$
$467$	−10806.0	−1.07075	−0.535377	+	0.844613i	$0.679830\pi$
$468$	0	0
$469$	0	0
$470$	5184.00	0.508766
$471$	0	0
$472$	−12150.0	−1.18485
$473$	1024.00	0.0995424
$474$	0	0
$475$	14410.0	1.39195
$476$	0	0
$477$	0	0
$478$	−4440.00	−0.424855
$479$	4940.00	0.471220	0.235610	−	0.971848i	$-0.424291\pi$
$479$	4940.00	0.471220	0.235610	+	0.971848i	$0.424291\pi$
$480$	0	0
$481$	6888.00	0.652943
$482$	−3302.00	−0.312037
$483$	0	0
$484$	8869.00	0.832926
$485$	−4704.00	−0.440407
$486$	0	0
$487$	−5216.00	−0.485338	−0.242669	−	0.970109i	$-0.578023\pi$
$487$	−5216.00	−0.485338	−0.242669	+	0.970109i	$0.578023\pi$
$488$	−7320.00	−0.679018
$489$	0	0
$490$	0	0
$491$	−4412.00	−0.405521	−0.202760	−	0.979228i	$-0.564991\pi$
$491$	−4412.00	−0.405521	−0.202760	+	0.979228i	$0.564991\pi$
$492$	0	0
$493$	5940.00	0.542645
$494$	−3080.00	−0.280518
$495$	0	0
$496$	−492.000	−0.0445392
$497$	0	0
$498$	0	0
$499$	19060.0	1.70991	0.854953	−	0.518706i	$-0.173586\pi$
$499$	19060.0	1.70991	0.854953	+	0.518706i	$0.173586\pi$
$500$	−672.000	−0.0601055
$501$	0	0
$502$	1582.00	0.140654
$503$	12768.0	1.13180	0.565902	−	0.824473i	$-0.308528\pi$
$503$	12768.0	1.13180	0.565902	+	0.824473i	$0.308528\pi$
$504$	0	0
$505$	−11008.0	−0.969999
$506$	−384.000	−0.0337369
$507$	0	0
$508$	12432.0	1.08579
$509$	−5500.00	−0.478945	−0.239473	−	0.970903i	$-0.576975\pi$
$509$	−5500.00	−0.478945	−0.239473	+	0.970903i	$0.576975\pi$
$510$	0	0
$511$	0	0
$512$	11521.0	0.994455
$513$	0	0
$514$	2354.00	0.202005
$515$	−22208.0	−1.90020
$516$	0	0
$517$	2592.00	0.220495
$518$	0	0
$519$	0	0
$520$	6720.00	0.566714
$521$	−7338.00	−0.617051	−0.308526	−	0.951216i	$-0.599836\pi$
$521$	−7338.00	−0.617051	−0.308526	+	0.951216i	$0.599836\pi$
$522$	0	0
$523$	17582.0	1.46999	0.734997	−	0.678070i	$-0.237183\pi$
$523$	17582.0	1.46999	0.734997	+	0.678070i	$0.237183\pi$
$524$	7826.00	0.652444
$525$	0	0
$526$	3872.00	0.320964
$527$	−648.000	−0.0535623
$528$	0	0
$529$	−9863.00	−0.810635
$530$	2592.00	0.212433
$531$	0	0
$532$	0	0
$533$	−5096.00	−0.414132
$534$	0	0
$535$	−3904.00	−0.315485
$536$	−3660.00	−0.294940
$537$	0	0
$538$	180.000	0.0144244
$539$	0	0
$540$	0	0
$541$	−1618.00	−0.128583	−0.0642914	−	0.997931i	$-0.520479\pi$
$541$	−1618.00	−0.128583	−0.0642914	+	0.997931i	$0.520479\pi$
$542$	−2032.00	−0.161037
$543$	0	0
$544$	8694.00	0.685206
$545$	1440.00	0.113179
$546$	0	0
$547$	16144.0	1.26192	0.630958	−	0.775817i	$-0.282662\pi$
$547$	16144.0	1.26192	0.630958	+	0.775817i	$0.282662\pi$
$548$	15918.0	1.24085
$549$	0	0
$550$	1048.00	0.0812489
$551$	12100.0	0.935531
$552$	0	0
$553$	0	0
$554$	−5426.00	−0.416117
$555$	0	0
$556$	−1470.00	−0.112126
$557$	−4654.00	−0.354033	−0.177016	−	0.984208i	$-0.556645\pi$
$557$	−4654.00	−0.354033	−0.177016	+	0.984208i	$0.556645\pi$
$558$	0	0
$559$	−3584.00	−0.271175
$560$	0	0
$561$	0	0
$562$	−842.000	−0.0631986
$563$	10078.0	0.754418	0.377209	−	0.926128i	$-0.376884\pi$
$563$	10078.0	0.754418	0.377209	+	0.926128i	$0.376884\pi$
$564$	0	0
$565$	−21088.0	−1.57023
$566$	3782.00	0.280865
$567$	0	0
$568$	−11520.0	−0.851001
$569$	5930.00	0.436904	0.218452	−	0.975848i	$-0.429899\pi$
$569$	5930.00	0.436904	0.218452	+	0.975848i	$0.429899\pi$
$570$	0	0
$571$	−19048.0	−1.39603	−0.698016	−	0.716082i	$-0.745933\pi$
$571$	−19048.0	−1.39603	−0.698016	+	0.716082i	$0.745933\pi$
$572$	1568.00	0.114618
$573$	0	0
$574$	0	0
$575$	−6288.00	−0.456048
$576$	0	0
$577$	14366.0	1.03651	0.518253	−	0.855227i	$-0.326582\pi$
$577$	14366.0	1.03651	0.518253	+	0.855227i	$0.326582\pi$
$578$	−1997.00	−0.143710
$579$	0	0
$580$	−12320.0	−0.882000
$581$	0	0
$582$	0	0
$583$	1296.00	0.0920666
$584$	−10530.0	−0.746121
$585$	0	0
$586$	−4312.00	−0.303971
$587$	−3626.00	−0.254959	−0.127480	−	0.991841i	$-0.540689\pi$
$587$	−3626.00	−0.254959	−0.127480	+	0.991841i	$0.540689\pi$
$588$	0	0
$589$	−1320.00	−0.0923424
$590$	12960.0	0.904330
$591$	0	0
$592$	−10086.0	−0.700223
$593$	−1062.00	−0.0735432	−0.0367716	−	0.999324i	$-0.511707\pi$
$593$	−1062.00	−0.0735432	−0.0367716	+	0.999324i	$0.511707\pi$
$594$	0	0
$595$	0	0
$596$	−14070.0	−0.966996
$597$	0	0
$598$	1344.00	0.0919068
$599$	10200.0	0.695761	0.347880	−	0.937539i	$-0.386902\pi$
$599$	10200.0	0.695761	0.347880	+	0.937539i	$0.386902\pi$
$600$	0	0
$601$	25158.0	1.70751	0.853757	−	0.520671i	$-0.174318\pi$
$601$	25158.0	1.70751	0.853757	+	0.520671i	$0.174318\pi$
$602$	0	0
$603$	0	0
$604$	−7784.00	−0.524382
$605$	−20272.0	−1.36227
$606$	0	0
$607$	−25664.0	−1.71609	−0.858047	−	0.513570i	$-0.828323\pi$
$607$	−25664.0	−1.71609	−0.858047	+	0.513570i	$0.828323\pi$
$608$	17710.0	1.18131
$609$	0	0
$610$	7808.00	0.518257
$611$	−9072.00	−0.600677
$612$	0	0
$613$	19018.0	1.25307	0.626533	−	0.779395i	$-0.284473\pi$
$613$	19018.0	1.25307	0.626533	+	0.779395i	$0.284473\pi$
$614$	−2674.00	−0.175755
$615$	0	0
$616$	0	0
$617$	−17334.0	−1.13102	−0.565511	−	0.824741i	$-0.691321\pi$
$617$	−17334.0	−1.13102	−0.565511	+	0.824741i	$0.691321\pi$
$618$	0	0
$619$	−18730.0	−1.21619	−0.608096	−	0.793864i	$-0.708066\pi$
$619$	−18730.0	−1.21619	−0.608096	+	0.793864i	$0.708066\pi$
$620$	1344.00	0.0870586
$621$	0	0
$622$	−3768.00	−0.242899
$623$	0	0
$624$	0	0
$625$	−14839.0	−0.949696
$626$	−2438.00	−0.155658
$627$	0	0
$628$	868.000	0.0551544
$629$	−13284.0	−0.842079
$630$	0	0
$631$	−6928.00	−0.437083	−0.218541	−	0.975828i	$-0.570130\pi$
$631$	−6928.00	−0.437083	−0.218541	+	0.975828i	$0.570130\pi$
$632$	−6600.00	−0.415402
$633$	0	0
$634$	3186.00	0.199578
$635$	−28416.0	−1.77583
$636$	0	0
$637$	0	0
$638$	880.000	0.0546074
$639$	0	0
$640$	−23280.0	−1.43785
$641$	−16302.0	−1.00451	−0.502255	−	0.864720i	$-0.667496\pi$
$641$	−16302.0	−1.00451	−0.502255	+	0.864720i	$0.667496\pi$
$642$	0	0
$643$	−4718.00	−0.289362	−0.144681	−	0.989478i	$-0.546216\pi$
$643$	−4718.00	−0.289362	−0.144681	+	0.989478i	$0.546216\pi$
$644$	0	0
$645$	0	0
$646$	5940.00	0.361774
$647$	−21436.0	−1.30253	−0.651264	−	0.758851i	$-0.725761\pi$
$647$	−21436.0	−1.30253	−0.651264	+	0.758851i	$0.725761\pi$
$648$	0	0
$649$	6480.00	0.391930
$650$	−3668.00	−0.221340
$651$	0	0
$652$	−14056.0	−0.844287
$653$	−4458.00	−0.267159	−0.133580	−	0.991038i	$-0.542647\pi$
$653$	−4458.00	−0.267159	−0.133580	+	0.991038i	$0.542647\pi$
$654$	0	0
$655$	−17888.0	−1.06709
$656$	7462.00	0.444119
$657$	0	0
$658$	0	0
$659$	26640.0	1.57473	0.787365	−	0.616487i	$-0.211445\pi$
$659$	26640.0	1.57473	0.787365	+	0.616487i	$0.211445\pi$
$660$	0	0
$661$	−7432.00	−0.437324	−0.218662	−	0.975801i	$-0.570169\pi$
$661$	−7432.00	−0.437324	−0.218662	+	0.975801i	$0.570169\pi$
$662$	8672.00	0.509134
$663$	0	0
$664$	19530.0	1.14143
$665$	0	0
$666$	0	0
$667$	−5280.00	−0.306510
$668$	−20188.0	−1.16931
$669$	0	0
$670$	3904.00	0.225111
$671$	3904.00	0.224608
$672$	0	0
$673$	58.0000	0.00332204	0.00166102	−	0.999999i	$-0.499471\pi$
$673$	58.0000	0.00332204	0.00166102	+	0.999999i	$0.499471\pi$
$674$	814.000	0.0465194
$675$	0	0
$676$	9891.00	0.562756
$677$	−21516.0	−1.22146	−0.610729	−	0.791840i	$-0.709124\pi$
$677$	−21516.0	−1.22146	−0.610729	+	0.791840i	$0.709124\pi$
$678$	0	0
$679$	0	0
$680$	−12960.0	−0.730873
$681$	0	0
$682$	−96.0000	−0.00539007
$683$	−18108.0	−1.01447	−0.507235	−	0.861808i	$-0.669332\pi$
$683$	−18108.0	−1.01447	−0.507235	+	0.861808i	$0.669332\pi$
$684$	0	0
$685$	−36384.0	−2.02943
$686$	0	0
$687$	0	0
$688$	5248.00	0.290811
$689$	−4536.00	−0.250810
$690$	0	0
$691$	10078.0	0.554827	0.277413	−	0.960751i	$-0.410523\pi$
$691$	10078.0	0.554827	0.277413	+	0.960751i	$0.410523\pi$
$692$	−15596.0	−0.856750
$693$	0	0
$694$	−9344.00	−0.511086
$695$	3360.00	0.183384
$696$	0	0
$697$	9828.00	0.534092
$698$	5180.00	0.280897
$699$	0	0
$700$	0	0
$701$	−18762.0	−1.01089	−0.505443	−	0.862860i	$-0.668671\pi$
$701$	−18762.0	−1.01089	−0.505443	+	0.862860i	$0.668671\pi$
$702$	0	0
$703$	−27060.0	−1.45176
$704$	−1336.00	−0.0715233
$705$	0	0
$706$	12178.0	0.649186
$707$	0	0
$708$	0	0
$709$	6810.00	0.360726	0.180363	−	0.983600i	$-0.442273\pi$
$709$	6810.00	0.360726	0.180363	+	0.983600i	$0.442273\pi$
$710$	12288.0	0.649522
$711$	0	0
$712$	−10950.0	−0.576360
$713$	576.000	0.0302544
$714$	0	0
$715$	−3584.00	−0.187460
$716$	−5740.00	−0.299600
$717$	0	0
$718$	−440.000	−0.0228700
$719$	4860.00	0.252083	0.126041	−	0.992025i	$-0.459773\pi$
$719$	4860.00	0.252083	0.126041	+	0.992025i	$0.459773\pi$
$720$	0	0
$721$	0	0
$722$	5241.00	0.270152
$723$	0	0
$724$	27244.0	1.39850
$725$	14410.0	0.738171
$726$	0	0
$727$	13636.0	0.695641	0.347821	−	0.937561i	$-0.386922\pi$
$727$	13636.0	0.695641	0.347821	+	0.937561i	$0.386922\pi$
$728$	0	0
$729$	0	0
$730$	11232.0	0.569473
$731$	6912.00	0.349726
$732$	0	0
$733$	−2088.00	−0.105214	−0.0526071	−	0.998615i	$-0.516753\pi$
$733$	−2088.00	−0.105214	−0.0526071	+	0.998615i	$0.516753\pi$
$734$	9816.00	0.493617
$735$	0	0
$736$	−7728.00	−0.387035
$737$	1952.00	0.0975615
$738$	0	0
$739$	−5160.00	−0.256852	−0.128426	−	0.991719i	$-0.540992\pi$
$739$	−5160.00	−0.256852	−0.128426	+	0.991719i	$0.540992\pi$
$740$	27552.0	1.36869
$741$	0	0
$742$	0	0
$743$	28152.0	1.39004	0.695018	−	0.718992i	$-0.255396\pi$
$743$	28152.0	1.39004	0.695018	+	0.718992i	$0.255396\pi$
$744$	0	0
$745$	32160.0	1.58155
$746$	−442.000	−0.0216927
$747$	0	0
$748$	−3024.00	−0.147819
$749$	0	0
$750$	0	0
$751$	−16808.0	−0.816688	−0.408344	−	0.912828i	$-0.633894\pi$
$751$	−16808.0	−0.816688	−0.408344	+	0.912828i	$0.633894\pi$
$752$	13284.0	0.644172
$753$	0	0
$754$	−3080.00	−0.148763
$755$	17792.0	0.857639
$756$	0	0
$757$	21674.0	1.04063	0.520314	−	0.853975i	$-0.325815\pi$
$757$	21674.0	1.04063	0.520314	+	0.853975i	$0.325815\pi$
$758$	−3960.00	−0.189754
$759$	0	0
$760$	−26400.0	−1.26004
$761$	7422.00	0.353544	0.176772	−	0.984252i	$-0.443434\pi$
$761$	7422.00	0.353544	0.176772	+	0.984252i	$0.443434\pi$
$762$	0	0
$763$	0	0
$764$	−35336.0	−1.67331
$765$	0	0
$766$	6708.00	0.316410
$767$	−22680.0	−1.06770
$768$	0	0
$769$	−13790.0	−0.646658	−0.323329	−	0.946287i	$-0.604802\pi$
$769$	−13790.0	−0.646658	−0.323329	+	0.946287i	$0.604802\pi$
$770$	0	0
$771$	0	0
$772$	20734.0	0.966623
$773$	−6232.00	−0.289973	−0.144987	−	0.989434i	$-0.546314\pi$
$773$	−6232.00	−0.289973	−0.144987	+	0.989434i	$0.546314\pi$
$774$	0	0
$775$	−1572.00	−0.0728618
$776$	4410.00	0.204007
$777$	0	0
$778$	13350.0	0.615194
$779$	20020.0	0.920784
$780$	0	0
$781$	6144.00	0.281498
$782$	−2592.00	−0.118529
$783$	0	0
$784$	0	0
$785$	−1984.00	−0.0902064
$786$	0	0
$787$	1766.00	0.0799887	0.0399943	−	0.999200i	$-0.487266\pi$
$787$	1766.00	0.0799887	0.0399943	+	0.999200i	$0.487266\pi$
$788$	23338.0	1.05505
$789$	0	0
$790$	7040.00	0.317053
$791$	0	0
$792$	0	0
$793$	−13664.0	−0.611883
$794$	1356.00	0.0606079
$795$	0	0
$796$	13020.0	0.579751
$797$	1204.00	0.0535105	0.0267552	−	0.999642i	$-0.491483\pi$
$797$	1204.00	0.0535105	0.0267552	+	0.999642i	$0.491483\pi$
$798$	0	0
$799$	17496.0	0.774673
$800$	21091.0	0.932099
$801$	0	0
$802$	−6222.00	−0.273948
$803$	5616.00	0.246805
$804$	0	0
$805$	0	0
$806$	336.000	0.0146837
$807$	0	0
$808$	10320.0	0.449327
$809$	7050.00	0.306384	0.153192	−	0.988196i	$-0.451045\pi$
$809$	7050.00	0.306384	0.153192	+	0.988196i	$0.451045\pi$
$810$	0	0
$811$	−23282.0	−1.00807	−0.504033	−	0.863684i	$-0.668151\pi$
$811$	−23282.0	−1.00807	−0.504033	+	0.863684i	$0.668151\pi$
$812$	0	0
$813$	0	0
$814$	−1968.00	−0.0847400
$815$	32128.0	1.38085
$816$	0	0
$817$	14080.0	0.602934
$818$	−5150.00	−0.220129
$819$	0	0
$820$	−20384.0	−0.868098
$821$	−10142.0	−0.431131	−0.215565	−	0.976489i	$-0.569159\pi$
$821$	−10142.0	−0.431131	−0.215565	+	0.976489i	$0.569159\pi$
$822$	0	0
$823$	−9192.00	−0.389323	−0.194662	−	0.980870i	$-0.562361\pi$
$823$	−9192.00	−0.389323	−0.194662	+	0.980870i	$0.562361\pi$
$824$	20820.0	0.880217
$825$	0	0
$826$	0	0
$827$	46716.0	1.96430	0.982149	−	0.188104i	$-0.0602344\pi$
$827$	46716.0	1.96430	0.982149	+	0.188104i	$0.0602344\pi$
$828$	0	0
$829$	−11240.0	−0.470906	−0.235453	−	0.971886i	$-0.575657\pi$
$829$	−11240.0	−0.470906	−0.235453	+	0.971886i	$0.575657\pi$
$830$	−20832.0	−0.871192
$831$	0	0
$832$	4676.00	0.194845
$833$	0	0
$834$	0	0
$835$	46144.0	1.91243
$836$	−6160.00	−0.254842
$837$	0	0
$838$	2310.00	0.0952239
$839$	700.000	0.0288042	0.0144021	−	0.999896i	$-0.495416\pi$
$839$	700.000	0.0288042	0.0144021	+	0.999896i	$0.495416\pi$
$840$	0	0
$841$	−12289.0	−0.503875
$842$	1262.00	0.0516525
$843$	0	0
$844$	29876.0	1.21845
$845$	−22608.0	−0.920401
$846$	0	0
$847$	0	0
$848$	6642.00	0.268971
$849$	0	0
$850$	7074.00	0.285454
$851$	11808.0	0.475644
$852$	0	0
$853$	37492.0	1.50493	0.752463	−	0.658635i	$-0.228866\pi$
$853$	37492.0	1.50493	0.752463	+	0.658635i	$0.228866\pi$
$854$	0	0
$855$	0	0
$856$	3660.00	0.146140
$857$	28894.0	1.15169	0.575846	−	0.817558i	$-0.304673\pi$
$857$	28894.0	1.15169	0.575846	+	0.817558i	$0.304673\pi$
$858$	0	0
$859$	2770.00	0.110025	0.0550123	−	0.998486i	$-0.482480\pi$
$859$	2770.00	0.110025	0.0550123	+	0.998486i	$0.482480\pi$
$860$	−14336.0	−0.568434
$861$	0	0
$862$	4488.00	0.177334
$863$	−17688.0	−0.697690	−0.348845	−	0.937180i	$-0.613426\pi$
$863$	−17688.0	−0.697690	−0.348845	+	0.937180i	$0.613426\pi$
$864$	0	0
$865$	35648.0	1.40124
$866$	−17038.0	−0.668562
$867$	0	0
$868$	0	0
$869$	3520.00	0.137408
$870$	0	0
$871$	−6832.00	−0.265779
$872$	−1350.00	−0.0524275
$873$	0	0
$874$	−5280.00	−0.204346
$875$	0	0
$876$	0	0
$877$	−33566.0	−1.29241	−0.646205	−	0.763164i	$-0.723645\pi$
$877$	−33566.0	−1.29241	−0.646205	+	0.763164i	$0.723645\pi$
$878$	−16200.0	−0.622692
$879$	0	0
$880$	5248.00	0.201034
$881$	−16758.0	−0.640853	−0.320426	−	0.947273i	$-0.603826\pi$
$881$	−16758.0	−0.640853	−0.320426	+	0.947273i	$0.603826\pi$
$882$	0	0
$883$	11468.0	0.437066	0.218533	−	0.975830i	$-0.429873\pi$
$883$	11468.0	0.437066	0.218533	+	0.975830i	$0.429873\pi$
$884$	10584.0	0.402691
$885$	0	0
$886$	8772.00	0.332620
$887$	−50356.0	−1.90619	−0.953094	−	0.302674i	$-0.902121\pi$
$887$	−50356.0	−1.90619	−0.953094	+	0.302674i	$0.902121\pi$
$888$	0	0
$889$	0	0
$890$	11680.0	0.439904
$891$	0	0
$892$	−38024.0	−1.42728
$893$	35640.0	1.33555
$894$	0	0
$895$	13120.0	0.490004
$896$	0	0
$897$	0	0
$898$	−2130.00	−0.0791526
$899$	−1320.00	−0.0489705
$900$	0	0
$901$	8748.00	0.323461
$902$	1456.00	0.0537467
$903$	0	0
$904$	19770.0	0.727368
$905$	−62272.0	−2.28728
$906$	0	0
$907$	−8716.00	−0.319085	−0.159542	−	0.987191i	$-0.551002\pi$
$907$	−8716.00	−0.319085	−0.159542	+	0.987191i	$0.551002\pi$
$908$	14322.0	0.523450
$909$	0	0
$910$	0	0
$911$	−7632.00	−0.277563	−0.138781	−	0.990323i	$-0.544318\pi$
$911$	−7632.00	−0.277563	−0.138781	+	0.990323i	$0.544318\pi$
$912$	0	0
$913$	−10416.0	−0.377568
$914$	10534.0	0.381219
$915$	0	0
$916$	−20860.0	−0.752439
$917$	0	0
$918$	0	0
$919$	−23080.0	−0.828443	−0.414221	−	0.910176i	$-0.635946\pi$
$919$	−23080.0	−0.828443	−0.414221	+	0.910176i	$0.635946\pi$
$920$	11520.0	0.412830
$921$	0	0
$922$	−9268.00	−0.331047
$923$	−21504.0	−0.766861
$924$	0	0
$925$	−32226.0	−1.14550
$926$	−9392.00	−0.333305
$927$	0	0
$928$	17710.0	0.626465
$929$	45110.0	1.59312	0.796561	−	0.604558i	$-0.206650\pi$
$929$	45110.0	1.59312	0.796561	+	0.604558i	$0.206650\pi$
$930$	0	0
$931$	0	0
$932$	31206.0	1.09677
$933$	0	0
$934$	−10806.0	−0.378569
$935$	6912.00	0.241761
$936$	0	0
$937$	−16674.0	−0.581340	−0.290670	−	0.956823i	$-0.593878\pi$
$937$	−16674.0	−0.581340	−0.290670	+	0.956823i	$0.593878\pi$
$938$	0	0
$939$	0	0
$940$	−36288.0	−1.25913
$941$	43832.0	1.51847	0.759236	−	0.650815i	$-0.225573\pi$
$941$	43832.0	1.51847	0.759236	+	0.650815i	$0.225573\pi$
$942$	0	0
$943$	−8736.00	−0.301679
$944$	33210.0	1.14501
$945$	0	0
$946$	1024.00	0.0351936
$947$	736.000	0.0252553	0.0126277	−	0.999920i	$-0.495980\pi$
$947$	736.000	0.0252553	0.0126277	+	0.999920i	$0.495980\pi$
$948$	0	0
$949$	−19656.0	−0.672351
$950$	14410.0	0.492129
$951$	0	0
$952$	0	0
$953$	−38138.0	−1.29634	−0.648169	−	0.761496i	$-0.724465\pi$
$953$	−38138.0	−1.29634	−0.648169	+	0.761496i	$0.724465\pi$
$954$	0	0
$955$	80768.0	2.73674
$956$	31080.0	1.05146
$957$	0	0
$958$	4940.00	0.166601
$959$	0	0
$960$	0	0
$961$	−29647.0	−0.995166
$962$	6888.00	0.230850
$963$	0	0
$964$	23114.0	0.772253
$965$	−47392.0	−1.58094
$966$	0	0
$967$	26224.0	0.872086	0.436043	−	0.899926i	$-0.356380\pi$
$967$	26224.0	0.872086	0.436043	+	0.899926i	$0.356380\pi$
$968$	19005.0	0.631037
$969$	0	0
$970$	−4704.00	−0.155708
$971$	18762.0	0.620084	0.310042	−	0.950723i	$-0.399657\pi$
$971$	18762.0	0.620084	0.310042	+	0.950723i	$0.399657\pi$
$972$	0	0
$973$	0	0
$974$	−5216.00	−0.171593
$975$	0	0
$976$	20008.0	0.656189
$977$	−38394.0	−1.25725	−0.628625	−	0.777709i	$-0.716382\pi$
$977$	−38394.0	−1.25725	−0.628625	+	0.777709i	$0.716382\pi$
$978$	0	0
$979$	5840.00	0.190651
$980$	0	0
$981$	0	0
$982$	−4412.00	−0.143373
$983$	5388.00	0.174822	0.0874112	−	0.996172i	$-0.472141\pi$
$983$	5388.00	0.174822	0.0874112	+	0.996172i	$0.472141\pi$
$984$	0	0
$985$	−53344.0	−1.72556
$986$	5940.00	0.191854
$987$	0	0
$988$	21560.0	0.694246
$989$	−6144.00	−0.197541
$990$	0	0
$991$	25472.0	0.816493	0.408247	−	0.912872i	$-0.366140\pi$
$991$	25472.0	0.816493	0.408247	+	0.912872i	$0.366140\pi$
$992$	−1932.00	−0.0618357
$993$	0	0
$994$	0	0
$995$	−29760.0	−0.948196
$996$	0	0
$997$	17096.0	0.543065	0.271532	−	0.962429i	$-0.412470\pi$
$997$	17096.0	0.543065	0.271532	+	0.962429i	$0.412470\pi$
$998$	19060.0	0.604543
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.4.a.i.1.1		1
3.2	odd	2		49.4.a.b.1.1		1
7.2	even	3		441.4.e.e.361.1		2
7.3	odd	6		441.4.e.h.226.1		2
7.4	even	3		441.4.e.e.226.1		2
7.5	odd	6		441.4.e.h.361.1		2
7.6	odd	2		63.4.a.b.1.1		1
12.11	even	2		784.4.a.g.1.1		1
15.14	odd	2		1225.4.a.j.1.1		1
21.2	odd	6		49.4.c.b.18.1		2
21.5	even	6		49.4.c.c.18.1		2
21.11	odd	6		49.4.c.b.30.1		2
21.17	even	6		49.4.c.c.30.1		2
21.20	even	2		7.4.a.a.1.1	✓	1
28.27	even	2		1008.4.a.c.1.1		1
35.34	odd	2		1575.4.a.e.1.1		1
84.83	odd	2		112.4.a.f.1.1		1
105.62	odd	4		175.4.b.b.99.1		2
105.83	odd	4		175.4.b.b.99.2		2
105.104	even	2		175.4.a.b.1.1		1
168.83	odd	2		448.4.a.e.1.1		1
168.125	even	2		448.4.a.i.1.1		1
231.230	odd	2		847.4.a.b.1.1		1
273.272	even	2		1183.4.a.b.1.1		1
357.356	even	2		2023.4.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.4.a.a.1.1	✓	1	21.20	even	2
49.4.a.b.1.1		1	3.2	odd	2
49.4.c.b.18.1		2	21.2	odd	6
49.4.c.b.30.1		2	21.11	odd	6
49.4.c.c.18.1		2	21.5	even	6
49.4.c.c.30.1		2	21.17	even	6
63.4.a.b.1.1		1	7.6	odd	2
112.4.a.f.1.1		1	84.83	odd	2
175.4.a.b.1.1		1	105.104	even	2
175.4.b.b.99.1		2	105.62	odd	4
175.4.b.b.99.2		2	105.83	odd	4
441.4.a.i.1.1		1	1.1	even	1	trivial
441.4.e.e.226.1		2	7.4	even	3
441.4.e.e.361.1		2	7.2	even	3
441.4.e.h.226.1		2	7.3	odd	6
441.4.e.h.361.1		2	7.5	odd	6
448.4.a.e.1.1		1	168.83	odd	2
448.4.a.i.1.1		1	168.125	even	2
784.4.a.g.1.1		1	12.11	even	2
847.4.a.b.1.1		1	231.230	odd	2
1008.4.a.c.1.1		1	28.27	even	2
1183.4.a.b.1.1		1	273.272	even	2
1225.4.a.j.1.1		1	15.14	odd	2
1575.4.a.e.1.1		1	35.34	odd	2
2023.4.a.a.1.1		1	357.356	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:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	441.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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