LMFDB - Embedded newform 112.4.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [112,4,Mod(1,112)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(112, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("112.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 112 = 2^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	112.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:	$6.60821392064$
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	$\chi$	$=$	112.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+2.00000 q^{3} +16.0000 q^{5} +7.00000 q^{7} -23.0000 q^{9} +8.00000 q^{11} +28.0000 q^{13} +32.0000 q^{15} +54.0000 q^{17} +110.000 q^{19} +14.0000 q^{21} -48.0000 q^{23} +131.000 q^{25} -100.000 q^{27} -110.000 q^{29} -12.0000 q^{31} +16.0000 q^{33} +112.000 q^{35} -246.000 q^{37} +56.0000 q^{39} +182.000 q^{41} -128.000 q^{43} -368.000 q^{45} -324.000 q^{47} +49.0000 q^{49} +108.000 q^{51} -162.000 q^{53} +128.000 q^{55} +220.000 q^{57} -810.000 q^{59} -488.000 q^{61} -161.000 q^{63} +448.000 q^{65} -244.000 q^{67} -96.0000 q^{69} +768.000 q^{71} -702.000 q^{73} +262.000 q^{75} +56.0000 q^{77} -440.000 q^{79} +421.000 q^{81} +1302.00 q^{83} +864.000 q^{85} -220.000 q^{87} +730.000 q^{89} +196.000 q^{91} -24.0000 q^{93} +1760.00 q^{95} +294.000 q^{97} -184.000 q^{99} +O(q^{100})$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	2.00000	0.384900	0.192450	−	0.981307i	$-0.438357\pi$
$3$	2.00000	0.384900	0.192450	+	0.981307i	$0.438357\pi$
$4$	0	0
$5$	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
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	0	0
$9$	−23.0000	−0.851852
$10$	0	0
$11$	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	$-0.403452\pi$
$13$	28.0000	0.597369	0.298685	+	0.954352i	$0.403452\pi$
$14$	0	0
$15$	32.0000	0.550824
$16$	0	0
$17$	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	$0.268816\pi$
$20$	0	0
$21$	14.0000	0.145479
$22$	0	0
$23$	−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
$26$	0	0
$27$	−100.000	−0.712778
$28$	0	0
$29$	−110.000	−0.704362	−0.352181	−	0.935932i	$-0.614560\pi$
$29$	−110.000	−0.704362	−0.352181	+	0.935932i	$0.614560\pi$
$30$	0	0
$31$	−12.0000	−0.0695246	−0.0347623	−	0.999396i	$-0.511067\pi$
$31$	−12.0000	−0.0695246	−0.0347623	+	0.999396i	$0.511067\pi$
$32$	0	0
$33$	16.0000	0.0844013
$34$	0	0
$35$	112.000	0.540899
$36$	0	0
$37$	−246.000	−1.09303	−0.546516	−	0.837449i	$-0.684046\pi$
$37$	−246.000	−1.09303	−0.546516	+	0.837449i	$0.684046\pi$
$38$	0	0
$39$	56.0000	0.229928
$40$	0	0
$41$	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	$-0.572883\pi$
$43$	−128.000	−0.453949	−0.226975	+	0.973901i	$0.572883\pi$
$44$	0	0
$45$	−368.000	−1.21907
$46$	0	0
$47$	−324.000	−1.00554	−0.502769	−	0.864421i	$-0.667685\pi$
$47$	−324.000	−1.00554	−0.502769	+	0.864421i	$0.667685\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	108.000	0.296530
$52$	0	0
$53$	−162.000	−0.419857	−0.209928	−	0.977717i	$-0.567323\pi$
$53$	−162.000	−0.419857	−0.209928	+	0.977717i	$0.567323\pi$
$54$	0	0
$55$	128.000	0.313809
$56$	0	0
$57$	220.000	0.511223
$58$	0	0
$59$	−810.000	−1.78734	−0.893670	−	0.448725i	$-0.851878\pi$
$59$	−810.000	−1.78734	−0.893670	+	0.448725i	$0.851878\pi$
$60$	0	0
$61$	−488.000	−1.02430	−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−488.000	−1.02430	−0.512148	+	0.858898i	$0.671150\pi$
$62$	0	0
$63$	−161.000	−0.321970
$64$	0	0
$65$	448.000	0.854886
$66$	0	0
$67$	−244.000	−0.444916	−0.222458	−	0.974942i	$-0.571408\pi$
$67$	−244.000	−0.444916	−0.222458	+	0.974942i	$0.571408\pi$
$68$	0	0
$69$	−96.0000	−0.167493
$70$	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	$-0.690260\pi$
$73$	−702.000	−1.12552	−0.562759	+	0.826621i	$0.690260\pi$
$74$	0	0
$75$	262.000	0.403375
$76$	0	0
$77$	56.0000	0.0828804
$78$	0	0
$79$	−440.000	−0.626631	−0.313316	−	0.949649i	$-0.601440\pi$
$79$	−440.000	−0.626631	−0.313316	+	0.949649i	$0.601440\pi$
$80$	0	0
$81$	421.000	0.577503
$82$	0	0
$83$	1302.00	1.72184	0.860922	−	0.508737i	$-0.169887\pi$
$83$	1302.00	1.72184	0.860922	+	0.508737i	$0.169887\pi$
$84$	0	0
$85$	864.000	1.10252
$86$	0	0
$87$	−220.000	−0.271109
$88$	0	0
$89$	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
$91$	196.000	0.225784
$92$	0	0
$93$	−24.0000	−0.0267600
$94$	0	0
$95$	1760.00	1.90076
$96$	0	0
$97$	294.000	0.307744	0.153872	−	0.988091i	$-0.450826\pi$
$97$	294.000	0.307744	0.153872	+	0.988091i	$0.450826\pi$
$98$	0	0
$99$	−184.000	−0.186795
$100$	0	0
$101$	−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	$0.731101\pi$
$104$	0	0
$105$	224.000	0.208192
$106$	0	0
$107$	−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	$0.487410\pi$
$110$	0	0
$111$	−492.000	−0.420708
$112$	0	0
$113$	1318.00	1.09723	0.548615	−	0.836075i	$-0.315155\pi$
$113$	1318.00	1.09723	0.548615	+	0.836075i	$0.315155\pi$
$114$	0	0
$115$	−768.000	−0.622751
$116$	0	0
$117$	−644.000	−0.508870
$118$	0	0
$119$	378.000	0.291187
$120$	0	0
$121$	−1267.00	−0.951916
$122$	0	0
$123$	364.000	0.266836
$124$	0	0
$125$	96.0000	0.0686920
$126$	0	0
$127$	1776.00	1.24090	0.620451	−	0.784245i	$-0.286950\pi$
$127$	1776.00	1.24090	0.620451	+	0.784245i	$0.286950\pi$
$128$	0	0
$129$	−256.000	−0.174725
$130$	0	0
$131$	1118.00	0.745650	0.372825	−	0.927902i	$-0.378389\pi$
$131$	1118.00	0.745650	0.372825	+	0.927902i	$0.378389\pi$
$132$	0	0
$133$	770.000	0.502011
$134$	0	0
$135$	−1600.00	−1.02004
$136$	0	0
$137$	2274.00	1.41811	0.709054	−	0.705154i	$-0.249122\pi$
$137$	2274.00	1.41811	0.709054	+	0.705154i	$0.249122\pi$
$138$	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
$141$	−648.000	−0.387032
$142$	0	0
$143$	224.000	0.130992
$144$	0	0
$145$	−1760.00	−1.00800
$146$	0	0
$147$	98.0000	0.0549857
$148$	0	0
$149$	−2010.00	−1.10514	−0.552569	−	0.833467i	$-0.686352\pi$
$149$	−2010.00	−1.10514	−0.552569	+	0.833467i	$0.686352\pi$
$150$	0	0
$151$	−1112.00	−0.599293	−0.299647	−	0.954050i	$-0.596869\pi$
$151$	−1112.00	−0.599293	−0.299647	+	0.954050i	$0.596869\pi$
$152$	0	0
$153$	−1242.00	−0.656273
$154$	0	0
$155$	−192.000	−0.0994956
$156$	0	0
$157$	124.000	0.0630336	0.0315168	−	0.999503i	$-0.489966\pi$
$157$	124.000	0.0630336	0.0315168	+	0.999503i	$0.489966\pi$
$158$	0	0
$159$	−324.000	−0.161603
$160$	0	0
$161$	−336.000	−0.164475
$162$	0	0
$163$	−2008.00	−0.964900	−0.482450	−	0.875924i	$-0.660253\pi$
$163$	−2008.00	−0.964900	−0.482450	+	0.875924i	$0.660253\pi$
$164$	0	0
$165$	256.000	0.120785
$166$	0	0
$167$	−2884.00	−1.33635	−0.668176	−	0.744004i	$-0.732924\pi$
$167$	−2884.00	−1.33635	−0.668176	+	0.744004i	$0.732924\pi$
$168$	0	0
$169$	−1413.00	−0.643150
$170$	0	0
$171$	−2530.00	−1.13143
$172$	0	0
$173$	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
$175$	917.000	0.396107
$176$	0	0
$177$	−1620.00	−0.687947
$178$	0	0
$179$	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	$-0.205287\pi$
$181$	3892.00	1.59829	0.799144	+	0.601140i	$0.205287\pi$
$182$	0	0
$183$	−976.000	−0.394251
$184$	0	0
$185$	−3936.00	−1.56422
$186$	0	0
$187$	432.000	0.168936
$188$	0	0
$189$	−700.000	−0.269405
$190$	0	0
$191$	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	$0.686271\pi$
$194$	0	0
$195$	896.000	0.329046
$196$	0	0
$197$	3334.00	1.20577	0.602887	−	0.797826i	$-0.294017\pi$
$197$	3334.00	1.20577	0.602887	+	0.797826i	$0.294017\pi$
$198$	0	0
$199$	−1860.00	−0.662572	−0.331286	−	0.943530i	$-0.607483\pi$
$199$	−1860.00	−0.662572	−0.331286	+	0.943530i	$0.607483\pi$
$200$	0	0
$201$	−488.000	−0.171248
$202$	0	0
$203$	−770.000	−0.266224
$204$	0	0
$205$	2912.00	0.992112
$206$	0	0
$207$	1104.00	0.370692
$208$	0	0
$209$	880.000	0.291248
$210$	0	0
$211$	4268.00	1.39252	0.696259	−	0.717791i	$-0.254847\pi$
$211$	4268.00	1.39252	0.696259	+	0.717791i	$0.254847\pi$
$212$	0	0
$213$	1536.00	0.494108
$214$	0	0
$215$	−2048.00	−0.649639
$216$	0	0
$217$	−84.0000	−0.0262778
$218$	0	0
$219$	−1404.00	−0.433212
$220$	0	0
$221$	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
$225$	−3013.00	−0.892741
$226$	0	0
$227$	2046.00	0.598228	0.299114	−	0.954217i	$-0.403309\pi$
$227$	2046.00	0.598228	0.299114	+	0.954217i	$0.403309\pi$
$228$	0	0
$229$	−2980.00	−0.859930	−0.429965	−	0.902846i	$-0.641474\pi$
$229$	−2980.00	−0.859930	−0.429965	+	0.902846i	$0.641474\pi$
$230$	0	0
$231$	112.000	0.0319007
$232$	0	0
$233$	4458.00	1.25345	0.626724	−	0.779241i	$-0.284395\pi$
$233$	4458.00	1.25345	0.626724	+	0.779241i	$0.284395\pi$
$234$	0	0
$235$	−5184.00	−1.43901
$236$	0	0
$237$	−880.000	−0.241190
$238$	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	$-0.354522\pi$
$241$	3302.00	0.882575	0.441287	+	0.897366i	$0.354522\pi$
$242$	0	0
$243$	3542.00	0.935059
$244$	0	0
$245$	784.000	0.204441
$246$	0	0
$247$	3080.00	0.793424
$248$	0	0
$249$	2604.00	0.662738
$250$	0	0
$251$	−1582.00	−0.397829	−0.198914	−	0.980017i	$-0.563742\pi$
$251$	−1582.00	−0.397829	−0.198914	+	0.980017i	$0.563742\pi$
$252$	0	0
$253$	−384.000	−0.0954224
$254$	0	0
$255$	1728.00	0.424359
$256$	0	0
$257$	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
$259$	−1722.00	−0.413127
$260$	0	0
$261$	2530.00	0.600012
$262$	0	0
$263$	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
$267$	1460.00	0.334646
$268$	0	0
$269$	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	$0.573134\pi$
$272$	0	0
$273$	392.000	0.0869045
$274$	0	0
$275$	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	$0.700273\pi$
$278$	0	0
$279$	276.000	0.0592247
$280$	0	0
$281$	842.000	0.178753	0.0893764	−	0.995998i	$-0.471513\pi$
$281$	842.000	0.178753	0.0893764	+	0.995998i	$0.471513\pi$
$282$	0	0
$283$	3782.00	0.794405	0.397202	−	0.917731i	$-0.369981\pi$
$283$	3782.00	0.794405	0.397202	+	0.917731i	$0.369981\pi$
$284$	0	0
$285$	3520.00	0.731603
$286$	0	0
$287$	1274.00	0.262027
$288$	0	0
$289$	−1997.00	−0.406473
$290$	0	0
$291$	588.000	0.118451
$292$	0	0
$293$	−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
$296$	0	0
$297$	−800.000	−0.156299
$298$	0	0
$299$	−1344.00	−0.259952
$300$	0	0
$301$	−896.000	−0.171577
$302$	0	0
$303$	−1376.00	−0.260888
$304$	0	0
$305$	−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
$309$	−2776.00	−0.511072
$310$	0	0
$311$	3768.00	0.687021	0.343511	−	0.939149i	$-0.388384\pi$
$311$	3768.00	0.687021	0.343511	+	0.939149i	$0.388384\pi$
$312$	0	0
$313$	2438.00	0.440268	0.220134	−	0.975470i	$-0.429351\pi$
$313$	2438.00	0.440268	0.220134	+	0.975470i	$0.429351\pi$
$314$	0	0
$315$	−2576.00	−0.460766
$316$	0	0
$317$	−3186.00	−0.564491	−0.282245	−	0.959342i	$-0.591079\pi$
$317$	−3186.00	−0.564491	−0.282245	+	0.959342i	$0.591079\pi$
$318$	0	0
$319$	−880.000	−0.154453
$320$	0	0
$321$	−488.000	−0.0848520
$322$	0	0
$323$	5940.00	1.02325
$324$	0	0
$325$	3668.00	0.626043
$326$	0	0
$327$	180.000	0.0304404
$328$	0	0
$329$	−2268.00	−0.380057
$330$	0	0
$331$	−8672.00	−1.44005	−0.720025	−	0.693949i	$-0.755869\pi$
$331$	−8672.00	−1.44005	−0.720025	+	0.693949i	$0.755869\pi$
$332$	0	0
$333$	5658.00	0.931101
$334$	0	0
$335$	−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	$0.479044\pi$
$338$	0	0
$339$	2636.00	0.422324
$340$	0	0
$341$	−96.0000	−0.0152454
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	−1536.00	−0.239697
$346$	0	0
$347$	−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	$-0.630035\pi$
$349$	−5180.00	−0.794496	−0.397248	+	0.917711i	$0.630035\pi$
$350$	0	0
$351$	−2800.00	−0.425792
$352$	0	0
$353$	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
$356$	0	0
$357$	756.000	0.112078
$358$	0	0
$359$	−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
$362$	0	0
$363$	−2534.00	−0.366393
$364$	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	$0.254038\pi$
$368$	0	0
$369$	−4186.00	−0.590554
$370$	0	0
$371$	−1134.00	−0.158691
$372$	0	0
$373$	−442.000	−0.0613563	−0.0306781	−	0.999529i	$-0.509767\pi$
$373$	−442.000	−0.0613563	−0.0306781	+	0.999529i	$0.509767\pi$
$374$	0	0
$375$	192.000	0.0264396
$376$	0	0
$377$	−3080.00	−0.420764
$378$	0	0
$379$	3960.00	0.536706	0.268353	−	0.963321i	$-0.413521\pi$
$379$	3960.00	0.536706	0.268353	+	0.963321i	$0.413521\pi$
$380$	0	0
$381$	3552.00	0.477623
$382$	0	0
$383$	−6708.00	−0.894942	−0.447471	−	0.894298i	$-0.647675\pi$
$383$	−6708.00	−0.894942	−0.447471	+	0.894298i	$0.647675\pi$
$384$	0	0
$385$	896.000	0.118609
$386$	0	0
$387$	2944.00	0.386697
$388$	0	0
$389$	−13350.0	−1.74003	−0.870015	−	0.493025i	$-0.835891\pi$
$389$	−13350.0	−1.74003	−0.870015	+	0.493025i	$0.835891\pi$
$390$	0	0
$391$	−2592.00	−0.335251
$392$	0	0
$393$	2236.00	0.287001
$394$	0	0
$395$	−7040.00	−0.896762
$396$	0	0
$397$	−1356.00	−0.171425	−0.0857125	−	0.996320i	$-0.527317\pi$
$397$	−1356.00	−0.171425	−0.0857125	+	0.996320i	$0.527317\pi$
$398$	0	0
$399$	1540.00	0.193224
$400$	0	0
$401$	6222.00	0.774843	0.387421	−	0.921903i	$-0.373366\pi$
$401$	6222.00	0.774843	0.387421	+	0.921903i	$0.373366\pi$
$402$	0	0
$403$	−336.000	−0.0415319
$404$	0	0
$405$	6736.00	0.826456
$406$	0	0
$407$	−1968.00	−0.239681
$408$	0	0
$409$	5150.00	0.622619	0.311309	−	0.950309i	$-0.399232\pi$
$409$	5150.00	0.622619	0.311309	+	0.950309i	$0.399232\pi$
$410$	0	0
$411$	4548.00	0.545830
$412$	0	0
$413$	−5670.00	−0.675551
$414$	0	0
$415$	20832.0	2.46410
$416$	0	0
$417$	420.000	0.0493225
$418$	0	0
$419$	−2310.00	−0.269334	−0.134667	−	0.990891i	$-0.542996\pi$
$419$	−2310.00	−0.269334	−0.134667	+	0.990891i	$0.542996\pi$
$420$	0	0
$421$	1262.00	0.146095	0.0730476	−	0.997328i	$-0.476727\pi$
$421$	1262.00	0.146095	0.0730476	+	0.997328i	$0.476727\pi$
$422$	0	0
$423$	7452.00	0.856569
$424$	0	0
$425$	7074.00	0.807387
$426$	0	0
$427$	−3416.00	−0.387147
$428$	0	0
$429$	448.000	0.0504188
$430$	0	0
$431$	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	$-0.105584\pi$
$433$	17038.0	1.89098	0.945490	+	0.325652i	$0.105584\pi$
$434$	0	0
$435$	−3520.00	−0.387979
$436$	0	0
$437$	−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	$0.842873\pi$
$440$	0	0
$441$	−1127.00	−0.121693
$442$	0	0
$443$	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
$446$	0	0
$447$	−4020.00	−0.425368
$448$	0	0
$449$	2130.00	0.223877	0.111939	−	0.993715i	$-0.464294\pi$
$449$	2130.00	0.223877	0.111939	+	0.993715i	$0.464294\pi$
$450$	0	0
$451$	1456.00	0.152019
$452$	0	0
$453$	−2224.00	−0.230668
$454$	0	0
$455$	3136.00	0.323116
$456$	0	0
$457$	10534.0	1.07825	0.539124	−	0.842226i	$-0.318755\pi$
$457$	10534.0	1.07825	0.539124	+	0.842226i	$0.318755\pi$
$458$	0	0
$459$	−5400.00	−0.549129
$460$	0	0
$461$	−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	$-0.343762\pi$
$463$	9392.00	0.942728	0.471364	+	0.881939i	$0.343762\pi$
$464$	0	0
$465$	−384.000	−0.0382959
$466$	0	0
$467$	10806.0	1.07075	0.535377	−	0.844613i	$-0.320170\pi$
$467$	10806.0	1.07075	0.535377	+	0.844613i	$0.320170\pi$
$468$	0	0
$469$	−1708.00	−0.168162
$470$	0	0
$471$	248.000	0.0242616
$472$	0	0
$473$	−1024.00	−0.0995424
$474$	0	0
$475$	14410.0	1.39195
$476$	0	0
$477$	3726.00	0.357656
$478$	0	0
$479$	−4940.00	−0.471220	−0.235610	−	0.971848i	$-0.575709\pi$
$479$	−4940.00	−0.471220	−0.235610	+	0.971848i	$0.575709\pi$
$480$	0	0
$481$	−6888.00	−0.652943
$482$	0	0
$483$	−672.000	−0.0633065
$484$	0	0
$485$	4704.00	0.440407
$486$	0	0
$487$	5216.00	0.485338	0.242669	−	0.970109i	$-0.421977\pi$
$487$	5216.00	0.485338	0.242669	+	0.970109i	$0.421977\pi$
$488$	0	0
$489$	−4016.00	−0.371390
$490$	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
$494$	0	0
$495$	−2944.00	−0.267319
$496$	0	0
$497$	5376.00	0.485204
$498$	0	0
$499$	−19060.0	−1.70991	−0.854953	−	0.518706i	$-0.826414\pi$
$499$	−19060.0	−1.70991	−0.854953	+	0.518706i	$0.826414\pi$
$500$	0	0
$501$	−5768.00	−0.514362
$502$	0	0
$503$	−12768.0	−1.13180	−0.565902	−	0.824473i	$-0.691472\pi$
$503$	−12768.0	−1.13180	−0.565902	+	0.824473i	$0.691472\pi$
$504$	0	0
$505$	−11008.0	−0.969999
$506$	0	0
$507$	−2826.00	−0.247548
$508$	0	0
$509$	−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
$511$	−4914.00	−0.425406
$512$	0	0
$513$	−11000.0	−0.946709
$514$	0	0
$515$	−22208.0	−1.90020
$516$	0	0
$517$	−2592.00	−0.220495
$518$	0	0
$519$	4456.00	0.376872
$520$	0	0
$521$	−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	$0.237183\pi$
$524$	0	0
$525$	1834.00	0.152462
$526$	0	0
$527$	−648.000	−0.0535623
$528$	0	0
$529$	−9863.00	−0.810635
$530$	0	0
$531$	18630.0	1.52255
$532$	0	0
$533$	5096.00	0.414132
$534$	0	0
$535$	−3904.00	−0.315485
$536$	0	0
$537$	1640.00	0.131790
$538$	0	0
$539$	392.000	0.0313259
$540$	0	0
$541$	−1618.00	−0.128583	−0.0642914	−	0.997931i	$-0.520479\pi$
$541$	−1618.00	−0.128583	−0.0642914	+	0.997931i	$0.520479\pi$
$542$	0	0
$543$	7784.00	0.615181
$544$	0	0
$545$	1440.00	0.113179
$546$	0	0
$547$	−16144.0	−1.26192	−0.630958	−	0.775817i	$-0.717338\pi$
$547$	−16144.0	−1.26192	−0.630958	+	0.775817i	$0.717338\pi$
$548$	0	0
$549$	11224.0	0.872548
$550$	0	0
$551$	−12100.0	−0.935531
$552$	0	0
$553$	−3080.00	−0.236844
$554$	0	0
$555$	−7872.00	−0.602068
$556$	0	0
$557$	4654.00	0.354033	0.177016	−	0.984208i	$-0.443355\pi$
$557$	4654.00	0.354033	0.177016	+	0.984208i	$0.443355\pi$
$558$	0	0
$559$	−3584.00	−0.271175
$560$	0	0
$561$	864.000	0.0650234
$562$	0	0
$563$	−10078.0	−0.754418	−0.377209	−	0.926128i	$-0.623116\pi$
$563$	−10078.0	−0.754418	−0.377209	+	0.926128i	$0.623116\pi$
$564$	0	0
$565$	21088.0	1.57023
$566$	0	0
$567$	2947.00	0.218276
$568$	0	0
$569$	−5930.00	−0.436904	−0.218452	−	0.975848i	$-0.570101\pi$
$569$	−5930.00	−0.436904	−0.218452	+	0.975848i	$0.570101\pi$
$570$	0	0
$571$	19048.0	1.39603	0.698016	−	0.716082i	$-0.254067\pi$
$571$	19048.0	1.39603	0.698016	+	0.716082i	$0.254067\pi$
$572$	0	0
$573$	10096.0	0.736067
$574$	0	0
$575$	−6288.00	−0.456048
$576$	0	0
$577$	−14366.0	−1.03651	−0.518253	−	0.855227i	$-0.673418\pi$
$577$	−14366.0	−1.03651	−0.518253	+	0.855227i	$0.673418\pi$
$578$	0	0
$579$	−5924.00	−0.425204
$580$	0	0
$581$	9114.00	0.650796
$582$	0	0
$583$	−1296.00	−0.0920666
$584$	0	0
$585$	−10304.0	−0.728236
$586$	0	0
$587$	3626.00	0.254959	0.127480	−	0.991841i	$-0.459311\pi$
$587$	3626.00	0.254959	0.127480	+	0.991841i	$0.459311\pi$
$588$	0	0
$589$	−1320.00	−0.0923424
$590$	0	0
$591$	6668.00	0.464103
$592$	0	0
$593$	−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
$595$	6048.00	0.416712
$596$	0	0
$597$	−3720.00	−0.255024
$598$	0	0
$599$	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	$-0.825682\pi$
$601$	−25158.0	−1.70751	−0.853757	+	0.520671i	$0.825682\pi$
$602$	0	0
$603$	5612.00	0.379002
$604$	0	0
$605$	−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	$0.828323\pi$
$608$	0	0
$609$	−1540.00	−0.102470
$610$	0	0
$611$	−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	$0.284473\pi$
$614$	0	0
$615$	5824.00	0.381864
$616$	0	0
$617$	17334.0	1.13102	0.565511	−	0.824741i	$-0.308679\pi$
$617$	17334.0	1.13102	0.565511	+	0.824741i	$0.308679\pi$
$618$	0	0
$619$	−18730.0	−1.21619	−0.608096	−	0.793864i	$-0.708066\pi$
$619$	−18730.0	−1.21619	−0.608096	+	0.793864i	$0.708066\pi$
$620$	0	0
$621$	4800.00	0.310173
$622$	0	0
$623$	5110.00	0.328616
$624$	0	0
$625$	−14839.0	−0.949696
$626$	0	0
$627$	1760.00	0.112101
$628$	0	0
$629$	−13284.0	−0.842079
$630$	0	0
$631$	6928.00	0.437083	0.218541	−	0.975828i	$-0.429870\pi$
$631$	6928.00	0.437083	0.218541	+	0.975828i	$0.429870\pi$
$632$	0	0
$633$	8536.00	0.535980
$634$	0	0
$635$	28416.0	1.77583
$636$	0	0
$637$	1372.00	0.0853385
$638$	0	0
$639$	−17664.0	−1.09355
$640$	0	0
$641$	16302.0	1.00451	0.502255	−	0.864720i	$-0.332504\pi$
$641$	16302.0	1.00451	0.502255	+	0.864720i	$0.332504\pi$
$642$	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
$645$	−4096.00	−0.250046
$646$	0	0
$647$	21436.0	1.30253	0.651264	−	0.758851i	$-0.274239\pi$
$647$	21436.0	1.30253	0.651264	+	0.758851i	$0.274239\pi$
$648$	0	0
$649$	−6480.00	−0.391930
$650$	0	0
$651$	−168.000	−0.0101143
$652$	0	0
$653$	4458.00	0.267159	0.133580	−	0.991038i	$-0.457353\pi$
$653$	4458.00	0.267159	0.133580	+	0.991038i	$0.457353\pi$
$654$	0	0
$655$	17888.0	1.06709
$656$	0	0
$657$	16146.0	0.958775
$658$	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	$-0.429831\pi$
$661$	7432.00	0.437324	0.218662	+	0.975801i	$0.429831\pi$
$662$	0	0
$663$	3024.00	0.177138
$664$	0	0
$665$	12320.0	0.718420
$666$	0	0
$667$	5280.00	0.306510
$668$	0	0
$669$	10864.0	0.627842
$670$	0	0
$671$	−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	$0.499471\pi$
$674$	0	0
$675$	−13100.0	−0.746991
$676$	0	0
$677$	−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
$679$	2058.00	0.116316
$680$	0	0
$681$	4092.00	0.230258
$682$	0	0
$683$	−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
$687$	−5960.00	−0.330987
$688$	0	0
$689$	−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	$0.410523\pi$
$692$	0	0
$693$	−1288.00	−0.0706018
$694$	0	0
$695$	3360.00	0.183384
$696$	0	0
$697$	9828.00	0.534092
$698$	0	0
$699$	8916.00	0.482452
$700$	0	0
$701$	18762.0	1.01089	0.505443	−	0.862860i	$-0.331329\pi$
$701$	18762.0	1.01089	0.505443	+	0.862860i	$0.331329\pi$
$702$	0	0
$703$	−27060.0	−1.45176
$704$	0	0
$705$	−10368.0	−0.553874
$706$	0	0
$707$	−4816.00	−0.256187
$708$	0	0
$709$	6810.00	0.360726	0.180363	−	0.983600i	$-0.442273\pi$
$709$	6810.00	0.360726	0.180363	+	0.983600i	$0.442273\pi$
$710$	0	0
$711$	10120.0	0.533797
$712$	0	0
$713$	576.000	0.0302544
$714$	0	0
$715$	3584.00	0.187460
$716$	0	0
$717$	−8880.00	−0.462524
$718$	0	0
$719$	−4860.00	−0.252083	−0.126041	−	0.992025i	$-0.540227\pi$
$719$	−4860.00	−0.252083	−0.126041	+	0.992025i	$0.540227\pi$
$720$	0	0
$721$	−9716.00	−0.501862
$722$	0	0
$723$	6604.00	0.339703
$724$	0	0
$725$	−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
$729$	−4283.00	−0.217599
$730$	0	0
$731$	−6912.00	−0.349726
$732$	0	0
$733$	2088.00	0.105214	0.0526071	−	0.998615i	$-0.483247\pi$
$733$	2088.00	0.105214	0.0526071	+	0.998615i	$0.483247\pi$
$734$	0	0
$735$	1568.00	0.0786892
$736$	0	0
$737$	−1952.00	−0.0975615
$738$	0	0
$739$	5160.00	0.256852	0.128426	−	0.991719i	$-0.459008\pi$
$739$	5160.00	0.256852	0.128426	+	0.991719i	$0.459008\pi$
$740$	0	0
$741$	6160.00	0.305389
$742$	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
$746$	0	0
$747$	−29946.0	−1.46676
$748$	0	0
$749$	−1708.00	−0.0833230
$750$	0	0
$751$	16808.0	0.816688	0.408344	−	0.912828i	$-0.366106\pi$
$751$	16808.0	0.816688	0.408344	+	0.912828i	$0.366106\pi$
$752$	0	0
$753$	−3164.00	−0.153124
$754$	0	0
$755$	−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	$0.325815\pi$
$758$	0	0
$759$	−768.000	−0.0367281
$760$	0	0
$761$	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
$763$	630.000	0.0298919
$764$	0	0
$765$	−19872.0	−0.939181
$766$	0	0
$767$	−22680.0	−1.06770
$768$	0	0
$769$	13790.0	0.646658	0.323329	−	0.946287i	$-0.395198\pi$
$769$	13790.0	0.646658	0.323329	+	0.946287i	$0.395198\pi$
$770$	0	0
$771$	4708.00	0.219915
$772$	0	0
$773$	−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
$776$	0	0
$777$	−3444.00	−0.159013
$778$	0	0
$779$	20020.0	0.920784
$780$	0	0
$781$	6144.00	0.281498
$782$	0	0
$783$	11000.0	0.502054
$784$	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	$0.487266\pi$
$788$	0	0
$789$	7744.00	0.349422
$790$	0	0
$791$	9226.00	0.414714
$792$	0	0
$793$	−13664.0	−0.611883
$794$	0	0
$795$	−5184.00	−0.231267
$796$	0	0
$797$	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
$800$	0	0
$801$	−16790.0	−0.740631
$802$	0	0
$803$	−5616.00	−0.246805
$804$	0	0
$805$	−5376.00	−0.235378
$806$	0	0
$807$	360.000	0.0157033
$808$	0	0
$809$	−7050.00	−0.306384	−0.153192	−	0.988196i	$-0.548955\pi$
$809$	−7050.00	−0.306384	−0.153192	+	0.988196i	$0.548955\pi$
$810$	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
$813$	−4064.00	−0.175315
$814$	0	0
$815$	−32128.0	−1.38085
$816$	0	0
$817$	−14080.0	−0.602934
$818$	0	0
$819$	−4508.00	−0.192335
$820$	0	0
$821$	10142.0	0.431131	0.215565	−	0.976489i	$-0.430841\pi$
$821$	10142.0	0.431131	0.215565	+	0.976489i	$0.430841\pi$
$822$	0	0
$823$	9192.00	0.389323	0.194662	−	0.980870i	$-0.437639\pi$
$823$	9192.00	0.389323	0.194662	+	0.980870i	$0.437639\pi$
$824$	0	0
$825$	2096.00	0.0884525
$826$	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	$-0.424343\pi$
$829$	11240.0	0.470906	0.235453	+	0.971886i	$0.424343\pi$
$830$	0	0
$831$	−10852.0	−0.453010
$832$	0	0
$833$	2646.00	0.110058
$834$	0	0
$835$	−46144.0	−1.91243
$836$	0	0
$837$	1200.00	0.0495556
$838$	0	0
$839$	−700.000	−0.0288042	−0.0144021	−	0.999896i	$-0.504584\pi$
$839$	−700.000	−0.0288042	−0.0144021	+	0.999896i	$0.504584\pi$
$840$	0	0
$841$	−12289.0	−0.503875
$842$	0	0
$843$	1684.00	0.0688019
$844$	0	0
$845$	−22608.0	−0.920401
$846$	0	0
$847$	−8869.00	−0.359790
$848$	0	0
$849$	7564.00	0.305767
$850$	0	0
$851$	11808.0	0.475644
$852$	0	0
$853$	−37492.0	−1.50493	−0.752463	−	0.658635i	$-0.771134\pi$
$853$	−37492.0	−1.50493	−0.752463	+	0.658635i	$0.771134\pi$
$854$	0	0
$855$	−40480.0	−1.61917
$856$	0	0
$857$	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	$0.482480\pi$
$860$	0	0
$861$	2548.00	0.100854
$862$	0	0
$863$	−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
$866$	0	0
$867$	−3994.00	−0.156451
$868$	0	0
$869$	−3520.00	−0.137408
$870$	0	0
$871$	−6832.00	−0.265779
$872$	0	0
$873$	−6762.00	−0.262152
$874$	0	0
$875$	672.000	0.0259631
$876$	0	0
$877$	−33566.0	−1.29241	−0.646205	−	0.763164i	$-0.723645\pi$
$877$	−33566.0	−1.29241	−0.646205	+	0.763164i	$0.723645\pi$
$878$	0	0
$879$	−8624.00	−0.330922
$880$	0	0
$881$	−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	$-0.570127\pi$
$883$	−11468.0	−0.437066	−0.218533	+	0.975830i	$0.570127\pi$
$884$	0	0
$885$	−25920.0	−0.984510
$886$	0	0
$887$	50356.0	1.90619	0.953094	−	0.302674i	$-0.0978793\pi$
$887$	50356.0	1.90619	0.953094	+	0.302674i	$0.0978793\pi$
$888$	0	0
$889$	12432.0	0.469017
$890$	0	0
$891$	3368.00	0.126636
$892$	0	0
$893$	−35640.0	−1.33555
$894$	0	0
$895$	13120.0	0.490004
$896$	0	0
$897$	−2688.00	−0.100055
$898$	0	0
$899$	1320.00	0.0489705
$900$	0	0
$901$	−8748.00	−0.323461
$902$	0	0
$903$	−1792.00	−0.0660399
$904$	0	0
$905$	62272.0	2.28728
$906$	0	0
$907$	8716.00	0.319085	0.159542	−	0.987191i	$-0.448998\pi$
$907$	8716.00	0.319085	0.159542	+	0.987191i	$0.448998\pi$
$908$	0	0
$909$	15824.0	0.577392
$910$	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
$914$	0	0
$915$	−15616.0	−0.564207
$916$	0	0
$917$	7826.00	0.281829
$918$	0	0
$919$	23080.0	0.828443	0.414221	−	0.910176i	$-0.364054\pi$
$919$	23080.0	0.828443	0.414221	+	0.910176i	$0.364054\pi$
$920$	0	0
$921$	−5348.00	−0.191338
$922$	0	0
$923$	21504.0	0.766861
$924$	0	0
$925$	−32226.0	−1.14550
$926$	0	0
$927$	31924.0	1.13109
$928$	0	0
$929$	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
$931$	5390.00	0.189742
$932$	0	0
$933$	7536.00	0.264435
$934$	0	0
$935$	6912.00	0.241761
$936$	0	0
$937$	16674.0	0.581340	0.290670	−	0.956823i	$-0.406122\pi$
$937$	16674.0	0.581340	0.290670	+	0.956823i	$0.406122\pi$
$938$	0	0
$939$	4876.00	0.169459
$940$	0	0
$941$	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
$944$	0	0
$945$	−11200.0	−0.385541
$946$	0	0
$947$	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
$950$	0	0
$951$	−6372.00	−0.217273
$952$	0	0
$953$	38138.0	1.29634	0.648169	−	0.761496i	$-0.275535\pi$
$953$	38138.0	1.29634	0.648169	+	0.761496i	$0.275535\pi$
$954$	0	0
$955$	80768.0	2.73674
$956$	0	0
$957$	−1760.00	−0.0594490
$958$	0	0
$959$	15918.0	0.535995
$960$	0	0
$961$	−29647.0	−0.995166
$962$	0	0
$963$	5612.00	0.187792
$964$	0	0
$965$	−47392.0	−1.58094
$966$	0	0
$967$	−26224.0	−0.872086	−0.436043	−	0.899926i	$-0.643620\pi$
$967$	−26224.0	−0.872086	−0.436043	+	0.899926i	$0.643620\pi$
$968$	0	0
$969$	11880.0	0.393850
$970$	0	0
$971$	−18762.0	−0.620084	−0.310042	−	0.950723i	$-0.600343\pi$
$971$	−18762.0	−0.620084	−0.310042	+	0.950723i	$0.600343\pi$
$972$	0	0
$973$	1470.00	0.0484337
$974$	0	0
$975$	7336.00	0.240964
$976$	0	0
$977$	38394.0	1.25725	0.628625	−	0.777709i	$-0.283618\pi$
$977$	38394.0	1.25725	0.628625	+	0.777709i	$0.283618\pi$
$978$	0	0
$979$	5840.00	0.190651
$980$	0	0
$981$	−2070.00	−0.0673700
$982$	0	0
$983$	−5388.00	−0.174822	−0.0874112	−	0.996172i	$-0.527859\pi$
$983$	−5388.00	−0.174822	−0.0874112	+	0.996172i	$0.527859\pi$
$984$	0	0
$985$	53344.0	1.72556
$986$	0	0
$987$	−4536.00	−0.146284
$988$	0	0
$989$	6144.00	0.197541
$990$	0	0
$991$	−25472.0	−0.816493	−0.408247	−	0.912872i	$-0.633860\pi$
$991$	−25472.0	−0.816493	−0.408247	+	0.912872i	$0.633860\pi$
$992$	0	0
$993$	−17344.0	−0.554275
$994$	0	0
$995$	−29760.0	−0.948196
$996$	0	0
$997$	−17096.0	−0.543065	−0.271532	−	0.962429i	$-0.587530\pi$
$997$	−17096.0	−0.543065	−0.271532	+	0.962429i	$0.587530\pi$
$998$	0	0
$999$	24600.0	0.779089

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.4.a.f.1.1		1
3.2	odd	2		1008.4.a.c.1.1		1
4.3	odd	2		7.4.a.a.1.1	✓	1
7.6	odd	2		784.4.a.g.1.1		1
8.3	odd	2		448.4.a.i.1.1		1
8.5	even	2		448.4.a.e.1.1		1
12.11	even	2		63.4.a.b.1.1		1
20.3	even	4		175.4.b.b.99.2		2
20.7	even	4		175.4.b.b.99.1		2
20.19	odd	2		175.4.a.b.1.1		1
28.3	even	6		49.4.c.b.30.1		2
28.11	odd	6		49.4.c.c.30.1		2
28.19	even	6		49.4.c.b.18.1		2
28.23	odd	6		49.4.c.c.18.1		2
28.27	even	2		49.4.a.b.1.1		1
44.43	even	2		847.4.a.b.1.1		1
52.51	odd	2		1183.4.a.b.1.1		1
60.59	even	2		1575.4.a.e.1.1		1
68.67	odd	2		2023.4.a.a.1.1		1
84.11	even	6		441.4.e.h.226.1		2
84.23	even	6		441.4.e.h.361.1		2
84.47	odd	6		441.4.e.e.361.1		2
84.59	odd	6		441.4.e.e.226.1		2
84.83	odd	2		441.4.a.i.1.1		1
140.139	even	2		1225.4.a.j.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.4.a.a.1.1	✓	1	4.3	odd	2
49.4.a.b.1.1		1	28.27	even	2
49.4.c.b.18.1		2	28.19	even	6
49.4.c.b.30.1		2	28.3	even	6
49.4.c.c.18.1		2	28.23	odd	6
49.4.c.c.30.1		2	28.11	odd	6
63.4.a.b.1.1		1	12.11	even	2
112.4.a.f.1.1		1	1.1	even	1	trivial
175.4.a.b.1.1		1	20.19	odd	2
175.4.b.b.99.1		2	20.7	even	4
175.4.b.b.99.2		2	20.3	even	4
441.4.a.i.1.1		1	84.83	odd	2
441.4.e.e.226.1		2	84.59	odd	6
441.4.e.e.361.1		2	84.47	odd	6
441.4.e.h.226.1		2	84.11	even	6
441.4.e.h.361.1		2	84.23	even	6
448.4.a.e.1.1		1	8.5	even	2
448.4.a.i.1.1		1	8.3	odd	2
784.4.a.g.1.1		1	7.6	odd	2
847.4.a.b.1.1		1	44.43	even	2
1008.4.a.c.1.1		1	3.2	odd	2
1183.4.a.b.1.1		1	52.51	odd	2
1225.4.a.j.1.1		1	140.139	even	2
1575.4.a.e.1.1		1	60.59	even	2
2023.4.a.a.1.1		1	68.67	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:	\( N \)	\(=\)	\( 112 = 2^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	112.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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