LMFDB - Embedded newform 80.4.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [80,4,Mod(1,80)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(80, 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("80.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 80 = 2^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	80.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:	$4.72015280046$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 40)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	80.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-4.00000 q^{3} +5.00000 q^{5} -16.0000 q^{7} -11.0000 q^{9} +O(q^{10})$

$q-4.00000 q^{3} +5.00000 q^{5} -16.0000 q^{7} -11.0000 q^{9} -36.0000 q^{11} -42.0000 q^{13} -20.0000 q^{15} -110.000 q^{17} +116.000 q^{19} +64.0000 q^{21} -16.0000 q^{23} +25.0000 q^{25} +152.000 q^{27} +198.000 q^{29} -240.000 q^{31} +144.000 q^{33} -80.0000 q^{35} -258.000 q^{37} +168.000 q^{39} +442.000 q^{41} +292.000 q^{43} -55.0000 q^{45} -392.000 q^{47} -87.0000 q^{49} +440.000 q^{51} +142.000 q^{53} -180.000 q^{55} -464.000 q^{57} +348.000 q^{59} -570.000 q^{61} +176.000 q^{63} -210.000 q^{65} -692.000 q^{67} +64.0000 q^{69} -168.000 q^{71} -134.000 q^{73} -100.000 q^{75} +576.000 q^{77} -784.000 q^{79} -311.000 q^{81} -564.000 q^{83} -550.000 q^{85} -792.000 q^{87} +1034.00 q^{89} +672.000 q^{91} +960.000 q^{93} +580.000 q^{95} -382.000 q^{97} +396.000 q^{99} +O(q^{100})$

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 $

$2$	0	0
$2$	0	0
$3$	−4.00000	−0.769800	−0.384900	−	0.922958i	$-0.625764\pi$
$3$	−4.00000	−0.769800	−0.384900	+	0.922958i	$0.625764\pi$
$4$	0	0
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	−16.0000	−0.863919	−0.431959	−	0.901893i	$-0.642178\pi$
$7$	−16.0000	−0.863919	−0.431959	+	0.901893i	$0.642178\pi$
$8$	0	0
$9$	−11.0000	−0.407407
$10$	0	0
$11$	−36.0000	−0.986764	−0.493382	−	0.869813i	$-0.664240\pi$
$11$	−36.0000	−0.986764	−0.493382	+	0.869813i	$0.664240\pi$
$12$	0	0
$13$	−42.0000	−0.896054	−0.448027	−	0.894020i	$-0.647873\pi$
$13$	−42.0000	−0.896054	−0.448027	+	0.894020i	$0.647873\pi$
$14$	0	0
$15$	−20.0000	−0.344265
$16$	0	0
$17$	−110.000	−1.56935	−0.784674	−	0.619909i	$-0.787170\pi$
$17$	−110.000	−1.56935	−0.784674	+	0.619909i	$0.787170\pi$
$18$	0	0
$19$	116.000	1.40064	0.700322	−	0.713827i	$-0.253040\pi$
$19$	116.000	1.40064	0.700322	+	0.713827i	$0.253040\pi$
$20$	0	0
$21$	64.0000	0.665045
$22$	0	0
$23$	−16.0000	−0.145054	−0.0725268	−	0.997366i	$-0.523106\pi$
$23$	−16.0000	−0.145054	−0.0725268	+	0.997366i	$0.523106\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	152.000	1.08342
$28$	0	0
$29$	198.000	1.26785	0.633925	−	0.773394i	$-0.281443\pi$
$29$	198.000	1.26785	0.633925	+	0.773394i	$0.281443\pi$
$30$	0	0
$31$	−240.000	−1.39049	−0.695246	−	0.718772i	$-0.744705\pi$
$31$	−240.000	−1.39049	−0.695246	+	0.718772i	$0.744705\pi$
$32$	0	0
$33$	144.000	0.759612
$34$	0	0
$35$	−80.0000	−0.386356
$36$	0	0
$37$	−258.000	−1.14635	−0.573175	−	0.819433i	$-0.694288\pi$
$37$	−258.000	−1.14635	−0.573175	+	0.819433i	$0.694288\pi$
$38$	0	0
$39$	168.000	0.689783
$40$	0	0
$41$	442.000	1.68363	0.841815	−	0.539767i	$-0.181488\pi$
$41$	442.000	1.68363	0.841815	+	0.539767i	$0.181488\pi$
$42$	0	0
$43$	292.000	1.03557	0.517786	−	0.855510i	$-0.326756\pi$
$43$	292.000	1.03557	0.517786	+	0.855510i	$0.326756\pi$
$44$	0	0
$45$	−55.0000	−0.182198
$46$	0	0
$47$	−392.000	−1.21658	−0.608288	−	0.793716i	$-0.708143\pi$
$47$	−392.000	−1.21658	−0.608288	+	0.793716i	$0.708143\pi$
$48$	0	0
$49$	−87.0000	−0.253644
$50$	0	0
$51$	440.000	1.20808
$52$	0	0
$53$	142.000	0.368023	0.184011	−	0.982924i	$-0.441092\pi$
$53$	142.000	0.368023	0.184011	+	0.982924i	$0.441092\pi$
$54$	0	0
$55$	−180.000	−0.441294
$56$	0	0
$57$	−464.000	−1.07822
$58$	0	0
$59$	348.000	0.767894	0.383947	−	0.923355i	$-0.374565\pi$
$59$	348.000	0.767894	0.383947	+	0.923355i	$0.374565\pi$
$60$	0	0
$61$	−570.000	−1.19641	−0.598205	−	0.801343i	$-0.704119\pi$
$61$	−570.000	−1.19641	−0.598205	+	0.801343i	$0.704119\pi$
$62$	0	0
$63$	176.000	0.351967
$64$	0	0
$65$	−210.000	−0.400728
$66$	0	0
$67$	−692.000	−1.26181	−0.630905	−	0.775860i	$-0.717316\pi$
$67$	−692.000	−1.26181	−0.630905	+	0.775860i	$0.717316\pi$
$68$	0	0
$69$	64.0000	0.111662
$70$	0	0
$71$	−168.000	−0.280816	−0.140408	−	0.990094i	$-0.544841\pi$
$71$	−168.000	−0.280816	−0.140408	+	0.990094i	$0.544841\pi$
$72$	0	0
$73$	−134.000	−0.214843	−0.107421	−	0.994214i	$-0.534259\pi$
$73$	−134.000	−0.214843	−0.107421	+	0.994214i	$0.534259\pi$
$74$	0	0
$75$	−100.000	−0.153960
$76$	0	0
$77$	576.000	0.852484
$78$	0	0
$79$	−784.000	−1.11654	−0.558271	−	0.829658i	$-0.688535\pi$
$79$	−784.000	−1.11654	−0.558271	+	0.829658i	$0.688535\pi$
$80$	0	0
$81$	−311.000	−0.426612
$82$	0	0
$83$	−564.000	−0.745868	−0.372934	−	0.927858i	$-0.621648\pi$
$83$	−564.000	−0.745868	−0.372934	+	0.927858i	$0.621648\pi$
$84$	0	0
$85$	−550.000	−0.701834
$86$	0	0
$87$	−792.000	−0.975992
$88$	0	0
$89$	1034.00	1.23150	0.615752	−	0.787940i	$-0.288852\pi$
$89$	1034.00	1.23150	0.615752	+	0.787940i	$0.288852\pi$
$90$	0	0
$91$	672.000	0.774118
$92$	0	0
$93$	960.000	1.07040
$94$	0	0
$95$	580.000	0.626387
$96$	0	0
$97$	−382.000	−0.399858	−0.199929	−	0.979810i	$-0.564071\pi$
$97$	−382.000	−0.399858	−0.199929	+	0.979810i	$0.564071\pi$
$98$	0	0
$99$	396.000	0.402015
$100$	0	0
$101$	−674.000	−0.664015	−0.332007	−	0.943277i	$-0.607726\pi$
$101$	−674.000	−0.664015	−0.332007	+	0.943277i	$0.607726\pi$
$102$	0	0
$103$	992.000	0.948977	0.474489	−	0.880262i	$-0.342633\pi$
$103$	992.000	0.948977	0.474489	+	0.880262i	$0.342633\pi$
$104$	0	0
$105$	320.000	0.297417
$106$	0	0
$107$	500.000	0.451746	0.225873	−	0.974157i	$-0.427477\pi$
$107$	500.000	0.451746	0.225873	+	0.974157i	$0.427477\pi$
$108$	0	0
$109$	1046.00	0.919162	0.459581	−	0.888136i	$-0.348000\pi$
$109$	1046.00	0.919162	0.459581	+	0.888136i	$0.348000\pi$
$110$	0	0
$111$	1032.00	0.882460
$112$	0	0
$113$	−558.000	−0.464533	−0.232266	−	0.972652i	$-0.574614\pi$
$113$	−558.000	−0.464533	−0.232266	+	0.972652i	$0.574614\pi$
$114$	0	0
$115$	−80.0000	−0.0648699
$116$	0	0
$117$	462.000	0.365059
$118$	0	0
$119$	1760.00	1.35579
$120$	0	0
$121$	−35.0000	−0.0262960
$122$	0	0
$123$	−1768.00	−1.29606
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	328.000	0.229176	0.114588	−	0.993413i	$-0.463445\pi$
$127$	328.000	0.229176	0.114588	+	0.993413i	$0.463445\pi$
$128$	0	0
$129$	−1168.00	−0.797183
$130$	0	0
$131$	212.000	0.141393	0.0706967	−	0.997498i	$-0.477478\pi$
$131$	212.000	0.141393	0.0706967	+	0.997498i	$0.477478\pi$
$132$	0	0
$133$	−1856.00	−1.21004
$134$	0	0
$135$	760.000	0.484521
$136$	0	0
$137$	1434.00	0.894269	0.447135	−	0.894467i	$-0.352444\pi$
$137$	1434.00	0.894269	0.447135	+	0.894467i	$0.352444\pi$
$138$	0	0
$139$	−2196.00	−1.34002	−0.670008	−	0.742354i	$-0.733709\pi$
$139$	−2196.00	−1.34002	−0.670008	+	0.742354i	$0.733709\pi$
$140$	0	0
$141$	1568.00	0.936521
$142$	0	0
$143$	1512.00	0.884194
$144$	0	0
$145$	990.000	0.567000
$146$	0	0
$147$	348.000	0.195255
$148$	0	0
$149$	−2418.00	−1.32946	−0.664732	−	0.747081i	$-0.731454\pi$
$149$	−2418.00	−1.32946	−0.664732	+	0.747081i	$0.731454\pi$
$150$	0	0
$151$	−3672.00	−1.97896	−0.989481	−	0.144666i	$-0.953789\pi$
$151$	−3672.00	−1.97896	−0.989481	+	0.144666i	$0.953789\pi$
$152$	0	0
$153$	1210.00	0.639364
$154$	0	0
$155$	−1200.00	−0.621847
$156$	0	0
$157$	358.000	0.181984	0.0909921	−	0.995852i	$-0.470996\pi$
$157$	358.000	0.181984	0.0909921	+	0.995852i	$0.470996\pi$
$158$	0	0
$159$	−568.000	−0.283304
$160$	0	0
$161$	256.000	0.125314
$162$	0	0
$163$	−2564.00	−1.23207	−0.616037	−	0.787717i	$-0.711263\pi$
$163$	−2564.00	−1.23207	−0.616037	+	0.787717i	$0.711263\pi$
$164$	0	0
$165$	720.000	0.339709
$166$	0	0
$167$	3056.00	1.41605	0.708025	−	0.706187i	$-0.249586\pi$
$167$	3056.00	1.41605	0.708025	+	0.706187i	$0.249586\pi$
$168$	0	0
$169$	−433.000	−0.197087
$170$	0	0
$171$	−1276.00	−0.570633
$172$	0	0
$173$	−234.000	−0.102836	−0.0514182	−	0.998677i	$-0.516374\pi$
$173$	−234.000	−0.102836	−0.0514182	+	0.998677i	$0.516374\pi$
$174$	0	0
$175$	−400.000	−0.172784
$176$	0	0
$177$	−1392.00	−0.591125
$178$	0	0
$179$	−524.000	−0.218802	−0.109401	−	0.993998i	$-0.534893\pi$
$179$	−524.000	−0.218802	−0.109401	+	0.993998i	$0.534893\pi$
$180$	0	0
$181$	−1138.00	−0.467331	−0.233665	−	0.972317i	$-0.575072\pi$
$181$	−1138.00	−0.467331	−0.233665	+	0.972317i	$0.575072\pi$
$182$	0	0
$183$	2280.00	0.920997
$184$	0	0
$185$	−1290.00	−0.512663
$186$	0	0
$187$	3960.00	1.54858
$188$	0	0
$189$	−2432.00	−0.935989
$190$	0	0
$191$	−1520.00	−0.575829	−0.287915	−	0.957656i	$-0.592962\pi$
$191$	−1520.00	−0.575829	−0.287915	+	0.957656i	$0.592962\pi$
$192$	0	0
$193$	−2142.00	−0.798884	−0.399442	−	0.916759i	$-0.630796\pi$
$193$	−2142.00	−0.798884	−0.399442	+	0.916759i	$0.630796\pi$
$194$	0	0
$195$	840.000	0.308480
$196$	0	0
$197$	−2306.00	−0.833988	−0.416994	−	0.908909i	$-0.636916\pi$
$197$	−2306.00	−0.833988	−0.416994	+	0.908909i	$0.636916\pi$
$198$	0	0
$199$	−3288.00	−1.17126	−0.585628	−	0.810580i	$-0.699152\pi$
$199$	−3288.00	−1.17126	−0.585628	+	0.810580i	$0.699152\pi$
$200$	0	0
$201$	2768.00	0.971342
$202$	0	0
$203$	−3168.00	−1.09532
$204$	0	0
$205$	2210.00	0.752942
$206$	0	0
$207$	176.000	0.0590959
$208$	0	0
$209$	−4176.00	−1.38211
$210$	0	0
$211$	3876.00	1.26462	0.632310	−	0.774715i	$-0.282107\pi$
$211$	3876.00	1.26462	0.632310	+	0.774715i	$0.282107\pi$
$212$	0	0
$213$	672.000	0.216172
$214$	0	0
$215$	1460.00	0.463122
$216$	0	0
$217$	3840.00	1.20127
$218$	0	0
$219$	536.000	0.165386
$220$	0	0
$221$	4620.00	1.40622
$222$	0	0
$223$	5688.00	1.70806	0.854028	−	0.520226i	$-0.174152\pi$
$223$	5688.00	1.70806	0.854028	+	0.520226i	$0.174152\pi$
$224$	0	0
$225$	−275.000	−0.0814815
$226$	0	0
$227$	2796.00	0.817520	0.408760	−	0.912642i	$-0.365961\pi$
$227$	2796.00	0.817520	0.408760	+	0.912642i	$0.365961\pi$
$228$	0	0
$229$	4446.00	1.28297	0.641485	−	0.767136i	$-0.278319\pi$
$229$	4446.00	1.28297	0.641485	+	0.767136i	$0.278319\pi$
$230$	0	0
$231$	−2304.00	−0.656243
$232$	0	0
$233$	2522.00	0.709106	0.354553	−	0.935036i	$-0.384633\pi$
$233$	2522.00	0.709106	0.354553	+	0.935036i	$0.384633\pi$
$234$	0	0
$235$	−1960.00	−0.544069
$236$	0	0
$237$	3136.00	0.859515
$238$	0	0
$239$	−816.000	−0.220848	−0.110424	−	0.993885i	$-0.535221\pi$
$239$	−816.000	−0.220848	−0.110424	+	0.993885i	$0.535221\pi$
$240$	0	0
$241$	−5422.00	−1.44922	−0.724609	−	0.689160i	$-0.757980\pi$
$241$	−5422.00	−1.44922	−0.724609	+	0.689160i	$0.757980\pi$
$242$	0	0
$243$	−2860.00	−0.755017
$244$	0	0
$245$	−435.000	−0.113433
$246$	0	0
$247$	−4872.00	−1.25505
$248$	0	0
$249$	2256.00	0.574169
$250$	0	0
$251$	5900.00	1.48368	0.741842	−	0.670575i	$-0.233952\pi$
$251$	5900.00	1.48368	0.741842	+	0.670575i	$0.233952\pi$
$252$	0	0
$253$	576.000	0.143134
$254$	0	0
$255$	2200.00	0.540272
$256$	0	0
$257$	5250.00	1.27426	0.637132	−	0.770754i	$-0.280120\pi$
$257$	5250.00	1.27426	0.637132	+	0.770754i	$0.280120\pi$
$258$	0	0
$259$	4128.00	0.990353
$260$	0	0
$261$	−2178.00	−0.516532
$262$	0	0
$263$	−6240.00	−1.46302	−0.731511	−	0.681829i	$-0.761185\pi$
$263$	−6240.00	−1.46302	−0.731511	+	0.681829i	$0.761185\pi$
$264$	0	0
$265$	710.000	0.164585
$266$	0	0
$267$	−4136.00	−0.948012
$268$	0	0
$269$	−714.000	−0.161834	−0.0809170	−	0.996721i	$-0.525785\pi$
$269$	−714.000	−0.161834	−0.0809170	+	0.996721i	$0.525785\pi$
$270$	0	0
$271$	−2144.00	−0.480586	−0.240293	−	0.970700i	$-0.577243\pi$
$271$	−2144.00	−0.480586	−0.240293	+	0.970700i	$0.577243\pi$
$272$	0	0
$273$	−2688.00	−0.595916
$274$	0	0
$275$	−900.000	−0.197353
$276$	0	0
$277$	−4466.00	−0.968722	−0.484361	−	0.874868i	$-0.660948\pi$
$277$	−4466.00	−0.968722	−0.484361	+	0.874868i	$0.660948\pi$
$278$	0	0
$279$	2640.00	0.566497
$280$	0	0
$281$	−5302.00	−1.12559	−0.562795	−	0.826596i	$-0.690274\pi$
$281$	−5302.00	−1.12559	−0.562795	+	0.826596i	$0.690274\pi$
$282$	0	0
$283$	6932.00	1.45606	0.728029	−	0.685546i	$-0.240436\pi$
$283$	6932.00	1.45606	0.728029	+	0.685546i	$0.240436\pi$
$284$	0	0
$285$	−2320.00	−0.482193
$286$	0	0
$287$	−7072.00	−1.45452
$288$	0	0
$289$	7187.00	1.46285
$290$	0	0
$291$	1528.00	0.307811
$292$	0	0
$293$	−4034.00	−0.804330	−0.402165	−	0.915567i	$-0.631742\pi$
$293$	−4034.00	−0.804330	−0.402165	+	0.915567i	$0.631742\pi$
$294$	0	0
$295$	1740.00	0.343413
$296$	0	0
$297$	−5472.00	−1.06908
$298$	0	0
$299$	672.000	0.129976
$300$	0	0
$301$	−4672.00	−0.894650
$302$	0	0
$303$	2696.00	0.511159
$304$	0	0
$305$	−2850.00	−0.535051
$306$	0	0
$307$	3836.00	0.713134	0.356567	−	0.934270i	$-0.383947\pi$
$307$	3836.00	0.713134	0.356567	+	0.934270i	$0.383947\pi$
$308$	0	0
$309$	−3968.00	−0.730523
$310$	0	0
$311$	−664.000	−0.121067	−0.0605337	−	0.998166i	$-0.519280\pi$
$311$	−664.000	−0.121067	−0.0605337	+	0.998166i	$0.519280\pi$
$312$	0	0
$313$	2986.00	0.539229	0.269615	−	0.962968i	$-0.413104\pi$
$313$	2986.00	0.539229	0.269615	+	0.962968i	$0.413104\pi$
$314$	0	0
$315$	880.000	0.157404
$316$	0	0
$317$	2726.00	0.482989	0.241494	−	0.970402i	$-0.422362\pi$
$317$	2726.00	0.482989	0.241494	+	0.970402i	$0.422362\pi$
$318$	0	0
$319$	−7128.00	−1.25107
$320$	0	0
$321$	−2000.00	−0.347754
$322$	0	0
$323$	−12760.0	−2.19810
$324$	0	0
$325$	−1050.00	−0.179211
$326$	0	0
$327$	−4184.00	−0.707571
$328$	0	0
$329$	6272.00	1.05102
$330$	0	0
$331$	9212.00	1.52972	0.764860	−	0.644197i	$-0.222808\pi$
$331$	9212.00	1.52972	0.764860	+	0.644197i	$0.222808\pi$
$332$	0	0
$333$	2838.00	0.467031
$334$	0	0
$335$	−3460.00	−0.564298
$336$	0	0
$337$	−3278.00	−0.529864	−0.264932	−	0.964267i	$-0.585349\pi$
$337$	−3278.00	−0.529864	−0.264932	+	0.964267i	$0.585349\pi$
$338$	0	0
$339$	2232.00	0.357598
$340$	0	0
$341$	8640.00	1.37209
$342$	0	0
$343$	6880.00	1.08305
$344$	0	0
$345$	320.000	0.0499369
$346$	0	0
$347$	−4956.00	−0.766721	−0.383360	−	0.923599i	$-0.625233\pi$
$347$	−4956.00	−0.766721	−0.383360	+	0.923599i	$0.625233\pi$
$348$	0	0
$349$	4678.00	0.717500	0.358750	−	0.933434i	$-0.383203\pi$
$349$	4678.00	0.717500	0.358750	+	0.933434i	$0.383203\pi$
$350$	0	0
$351$	−6384.00	−0.970805
$352$	0	0
$353$	1890.00	0.284970	0.142485	−	0.989797i	$-0.454491\pi$
$353$	1890.00	0.284970	0.142485	+	0.989797i	$0.454491\pi$
$354$	0	0
$355$	−840.000	−0.125585
$356$	0	0
$357$	−7040.00	−1.04369
$358$	0	0
$359$	6472.00	0.951474	0.475737	−	0.879588i	$-0.342181\pi$
$359$	6472.00	0.951474	0.475737	+	0.879588i	$0.342181\pi$
$360$	0	0
$361$	6597.00	0.961802
$362$	0	0
$363$	140.000	0.0202427
$364$	0	0
$365$	−670.000	−0.0960806
$366$	0	0
$367$	−1960.00	−0.278777	−0.139389	−	0.990238i	$-0.544514\pi$
$367$	−1960.00	−0.278777	−0.139389	+	0.990238i	$0.544514\pi$
$368$	0	0
$369$	−4862.00	−0.685923
$370$	0	0
$371$	−2272.00	−0.317942
$372$	0	0
$373$	8750.00	1.21463	0.607316	−	0.794460i	$-0.292246\pi$
$373$	8750.00	1.21463	0.607316	+	0.794460i	$0.292246\pi$
$374$	0	0
$375$	−500.000	−0.0688530
$376$	0	0
$377$	−8316.00	−1.13606
$378$	0	0
$379$	380.000	0.0515021	0.0257510	−	0.999668i	$-0.491802\pi$
$379$	380.000	0.0515021	0.0257510	+	0.999668i	$0.491802\pi$
$380$	0	0
$381$	−1312.00	−0.176419
$382$	0	0
$383$	9688.00	1.29252	0.646258	−	0.763119i	$-0.276333\pi$
$383$	9688.00	1.29252	0.646258	+	0.763119i	$0.276333\pi$
$384$	0	0
$385$	2880.00	0.381243
$386$	0	0
$387$	−3212.00	−0.421900
$388$	0	0
$389$	3870.00	0.504413	0.252207	−	0.967673i	$-0.418844\pi$
$389$	3870.00	0.504413	0.252207	+	0.967673i	$0.418844\pi$
$390$	0	0
$391$	1760.00	0.227639
$392$	0	0
$393$	−848.000	−0.108845
$394$	0	0
$395$	−3920.00	−0.499333
$396$	0	0
$397$	1622.00	0.205053	0.102526	−	0.994730i	$-0.467307\pi$
$397$	1622.00	0.205053	0.102526	+	0.994730i	$0.467307\pi$
$398$	0	0
$399$	7424.00	0.931491
$400$	0	0
$401$	9906.00	1.23362	0.616811	−	0.787112i	$-0.288424\pi$
$401$	9906.00	1.23362	0.616811	+	0.787112i	$0.288424\pi$
$402$	0	0
$403$	10080.0	1.24596
$404$	0	0
$405$	−1555.00	−0.190787
$406$	0	0
$407$	9288.00	1.13118
$408$	0	0
$409$	−4214.00	−0.509459	−0.254730	−	0.967012i	$-0.581986\pi$
$409$	−4214.00	−0.509459	−0.254730	+	0.967012i	$0.581986\pi$
$410$	0	0
$411$	−5736.00	−0.688409
$412$	0	0
$413$	−5568.00	−0.663398
$414$	0	0
$415$	−2820.00	−0.333562
$416$	0	0
$417$	8784.00	1.03155
$418$	0	0
$419$	7012.00	0.817562	0.408781	−	0.912632i	$-0.365954\pi$
$419$	7012.00	0.817562	0.408781	+	0.912632i	$0.365954\pi$
$420$	0	0
$421$	−1602.00	−0.185455	−0.0927277	−	0.995692i	$-0.529559\pi$
$421$	−1602.00	−0.185455	−0.0927277	+	0.995692i	$0.529559\pi$
$422$	0	0
$423$	4312.00	0.495642
$424$	0	0
$425$	−2750.00	−0.313870
$426$	0	0
$427$	9120.00	1.03360
$428$	0	0
$429$	−6048.00	−0.680653
$430$	0	0
$431$	3584.00	0.400546	0.200273	−	0.979740i	$-0.435817\pi$
$431$	3584.00	0.400546	0.200273	+	0.979740i	$0.435817\pi$
$432$	0	0
$433$	−3470.00	−0.385121	−0.192561	−	0.981285i	$-0.561679\pi$
$433$	−3470.00	−0.385121	−0.192561	+	0.981285i	$0.561679\pi$
$434$	0	0
$435$	−3960.00	−0.436477
$436$	0	0
$437$	−1856.00	−0.203168
$438$	0	0
$439$	3416.00	0.371382	0.185691	−	0.982608i	$-0.440548\pi$
$439$	3416.00	0.371382	0.185691	+	0.982608i	$0.440548\pi$
$440$	0	0
$441$	957.000	0.103337
$442$	0	0
$443$	−9708.00	−1.04118	−0.520588	−	0.853808i	$-0.674287\pi$
$443$	−9708.00	−1.04118	−0.520588	+	0.853808i	$0.674287\pi$
$444$	0	0
$445$	5170.00	0.550745
$446$	0	0
$447$	9672.00	1.02342
$448$	0	0
$449$	−10366.0	−1.08954	−0.544768	−	0.838587i	$-0.683382\pi$
$449$	−10366.0	−1.08954	−0.544768	+	0.838587i	$0.683382\pi$
$450$	0	0
$451$	−15912.0	−1.66135
$452$	0	0
$453$	14688.0	1.52340
$454$	0	0
$455$	3360.00	0.346196
$456$	0	0
$457$	−16742.0	−1.71369	−0.856847	−	0.515572i	$-0.827580\pi$
$457$	−16742.0	−1.71369	−0.856847	+	0.515572i	$0.827580\pi$
$458$	0	0
$459$	−16720.0	−1.70027
$460$	0	0
$461$	−1258.00	−0.127095	−0.0635476	−	0.997979i	$-0.520241\pi$
$461$	−1258.00	−0.127095	−0.0635476	+	0.997979i	$0.520241\pi$
$462$	0	0
$463$	−13528.0	−1.35788	−0.678941	−	0.734193i	$-0.737561\pi$
$463$	−13528.0	−1.35788	−0.678941	+	0.734193i	$0.737561\pi$
$464$	0	0
$465$	4800.00	0.478698
$466$	0	0
$467$	−6916.00	−0.685298	−0.342649	−	0.939463i	$-0.611324\pi$
$467$	−6916.00	−0.685298	−0.342649	+	0.939463i	$0.611324\pi$
$468$	0	0
$469$	11072.0	1.09010
$470$	0	0
$471$	−1432.00	−0.140091
$472$	0	0
$473$	−10512.0	−1.02187
$474$	0	0
$475$	2900.00	0.280129
$476$	0	0
$477$	−1562.00	−0.149935
$478$	0	0
$479$	−1728.00	−0.164832	−0.0824158	−	0.996598i	$-0.526264\pi$
$479$	−1728.00	−0.164832	−0.0824158	+	0.996598i	$0.526264\pi$
$480$	0	0
$481$	10836.0	1.02719
$482$	0	0
$483$	−1024.00	−0.0964671
$484$	0	0
$485$	−1910.00	−0.178822
$486$	0	0
$487$	−16656.0	−1.54981	−0.774903	−	0.632080i	$-0.782201\pi$
$487$	−16656.0	−1.54981	−0.774903	+	0.632080i	$0.782201\pi$
$488$	0	0
$489$	10256.0	0.948451
$490$	0	0
$491$	1084.00	0.0996339	0.0498169	−	0.998758i	$-0.484136\pi$
$491$	1084.00	0.0996339	0.0498169	+	0.998758i	$0.484136\pi$
$492$	0	0
$493$	−21780.0	−1.98970
$494$	0	0
$495$	1980.00	0.179787
$496$	0	0
$497$	2688.00	0.242602
$498$	0	0
$499$	−5804.00	−0.520687	−0.260343	−	0.965516i	$-0.583836\pi$
$499$	−5804.00	−0.520687	−0.260343	+	0.965516i	$0.583836\pi$
$500$	0	0
$501$	−12224.0	−1.09008
$502$	0	0
$503$	−10512.0	−0.931823	−0.465911	−	0.884831i	$-0.654273\pi$
$503$	−10512.0	−0.931823	−0.465911	+	0.884831i	$0.654273\pi$
$504$	0	0
$505$	−3370.00	−0.296956
$506$	0	0
$507$	1732.00	0.151718
$508$	0	0
$509$	−4314.00	−0.375667	−0.187834	−	0.982201i	$-0.560147\pi$
$509$	−4314.00	−0.375667	−0.187834	+	0.982201i	$0.560147\pi$
$510$	0	0
$511$	2144.00	0.185607
$512$	0	0
$513$	17632.0	1.51749
$514$	0	0
$515$	4960.00	0.424396
$516$	0	0
$517$	14112.0	1.20047
$518$	0	0
$519$	936.000	0.0791635
$520$	0	0
$521$	−1190.00	−0.100067	−0.0500334	−	0.998748i	$-0.515933\pi$
$521$	−1190.00	−0.100067	−0.0500334	+	0.998748i	$0.515933\pi$
$522$	0	0
$523$	3780.00	0.316038	0.158019	−	0.987436i	$-0.449489\pi$
$523$	3780.00	0.316038	0.158019	+	0.987436i	$0.449489\pi$
$524$	0	0
$525$	1600.00	0.133009
$526$	0	0
$527$	26400.0	2.18217
$528$	0	0
$529$	−11911.0	−0.978959
$530$	0	0
$531$	−3828.00	−0.312846
$532$	0	0
$533$	−18564.0	−1.50862
$534$	0	0
$535$	2500.00	0.202027
$536$	0	0
$537$	2096.00	0.168434
$538$	0	0
$539$	3132.00	0.250287
$540$	0	0
$541$	−11002.0	−0.874331	−0.437165	−	0.899381i	$-0.644018\pi$
$541$	−11002.0	−0.874331	−0.437165	+	0.899381i	$0.644018\pi$
$542$	0	0
$543$	4552.00	0.359751
$544$	0	0
$545$	5230.00	0.411062
$546$	0	0
$547$	−5908.00	−0.461806	−0.230903	−	0.972977i	$-0.574168\pi$
$547$	−5908.00	−0.461806	−0.230903	+	0.972977i	$0.574168\pi$
$548$	0	0
$549$	6270.00	0.487426
$550$	0	0
$551$	22968.0	1.77581
$552$	0	0
$553$	12544.0	0.964602
$554$	0	0
$555$	5160.00	0.394648
$556$	0	0
$557$	14806.0	1.12630	0.563151	−	0.826354i	$-0.309589\pi$
$557$	14806.0	1.12630	0.563151	+	0.826354i	$0.309589\pi$
$558$	0	0
$559$	−12264.0	−0.927928
$560$	0	0
$561$	−15840.0	−1.19210
$562$	0	0
$563$	684.000	0.0512028	0.0256014	−	0.999672i	$-0.491850\pi$
$563$	684.000	0.0512028	0.0256014	+	0.999672i	$0.491850\pi$
$564$	0	0
$565$	−2790.00	−0.207745
$566$	0	0
$567$	4976.00	0.368558
$568$	0	0
$569$	−2582.00	−0.190234	−0.0951169	−	0.995466i	$-0.530323\pi$
$569$	−2582.00	−0.190234	−0.0951169	+	0.995466i	$0.530323\pi$
$570$	0	0
$571$	2540.00	0.186157	0.0930785	−	0.995659i	$-0.470329\pi$
$571$	2540.00	0.186157	0.0930785	+	0.995659i	$0.470329\pi$
$572$	0	0
$573$	6080.00	0.443273
$574$	0	0
$575$	−400.000	−0.0290107
$576$	0	0
$577$	22786.0	1.64401	0.822005	−	0.569480i	$-0.192856\pi$
$577$	22786.0	1.64401	0.822005	+	0.569480i	$0.192856\pi$
$578$	0	0
$579$	8568.00	0.614981
$580$	0	0
$581$	9024.00	0.644369
$582$	0	0
$583$	−5112.00	−0.363152
$584$	0	0
$585$	2310.00	0.163259
$586$	0	0
$587$	−7884.00	−0.554357	−0.277178	−	0.960818i	$-0.589399\pi$
$587$	−7884.00	−0.554357	−0.277178	+	0.960818i	$0.589399\pi$
$588$	0	0
$589$	−27840.0	−1.94758
$590$	0	0
$591$	9224.00	0.642005
$592$	0	0
$593$	−21902.0	−1.51671	−0.758354	−	0.651843i	$-0.773996\pi$
$593$	−21902.0	−1.51671	−0.758354	+	0.651843i	$0.773996\pi$
$594$	0	0
$595$	8800.00	0.606327
$596$	0	0
$597$	13152.0	0.901634
$598$	0	0
$599$	−15080.0	−1.02863	−0.514317	−	0.857600i	$-0.671955\pi$
$599$	−15080.0	−1.02863	−0.514317	+	0.857600i	$0.671955\pi$
$600$	0	0
$601$	−19702.0	−1.33721	−0.668603	−	0.743619i	$-0.733108\pi$
$601$	−19702.0	−1.33721	−0.668603	+	0.743619i	$0.733108\pi$
$602$	0	0
$603$	7612.00	0.514071
$604$	0	0
$605$	−175.000	−0.0117599
$606$	0	0
$607$	−7320.00	−0.489472	−0.244736	−	0.969590i	$-0.578701\pi$
$607$	−7320.00	−0.489472	−0.244736	+	0.969590i	$0.578701\pi$
$608$	0	0
$609$	12672.0	0.843178
$610$	0	0
$611$	16464.0	1.09012
$612$	0	0
$613$	24350.0	1.60438	0.802192	−	0.597066i	$-0.203667\pi$
$613$	24350.0	1.60438	0.802192	+	0.597066i	$0.203667\pi$
$614$	0	0
$615$	−8840.00	−0.579615
$616$	0	0
$617$	19546.0	1.27535	0.637676	−	0.770305i	$-0.279896\pi$
$617$	19546.0	1.27535	0.637676	+	0.770305i	$0.279896\pi$
$618$	0	0
$619$	−3476.00	−0.225706	−0.112853	−	0.993612i	$-0.535999\pi$
$619$	−3476.00	−0.225706	−0.112853	+	0.993612i	$0.535999\pi$
$620$	0	0
$621$	−2432.00	−0.157154
$622$	0	0
$623$	−16544.0	−1.06392
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	16704.0	1.06394
$628$	0	0
$629$	28380.0	1.79902
$630$	0	0
$631$	−21880.0	−1.38039	−0.690197	−	0.723621i	$-0.742476\pi$
$631$	−21880.0	−1.38039	−0.690197	+	0.723621i	$0.742476\pi$
$632$	0	0
$633$	−15504.0	−0.973505
$634$	0	0
$635$	1640.00	0.102490
$636$	0	0
$637$	3654.00	0.227279
$638$	0	0
$639$	1848.00	0.114406
$640$	0	0
$641$	20994.0	1.29362	0.646812	−	0.762649i	$-0.276102\pi$
$641$	20994.0	1.29362	0.646812	+	0.762649i	$0.276102\pi$
$642$	0	0
$643$	18204.0	1.11648	0.558239	−	0.829680i	$-0.311477\pi$
$643$	18204.0	1.11648	0.558239	+	0.829680i	$0.311477\pi$
$644$	0	0
$645$	−5840.00	−0.356511
$646$	0	0
$647$	2064.00	0.125416	0.0627080	−	0.998032i	$-0.480026\pi$
$647$	2064.00	0.125416	0.0627080	+	0.998032i	$0.480026\pi$
$648$	0	0
$649$	−12528.0	−0.757730
$650$	0	0
$651$	−15360.0	−0.924740
$652$	0	0
$653$	9942.00	0.595805	0.297902	−	0.954596i	$-0.403713\pi$
$653$	9942.00	0.595805	0.297902	+	0.954596i	$0.403713\pi$
$654$	0	0
$655$	1060.00	0.0632330
$656$	0	0
$657$	1474.00	0.0875285
$658$	0	0
$659$	−24236.0	−1.43263	−0.716313	−	0.697779i	$-0.754172\pi$
$659$	−24236.0	−1.43263	−0.716313	+	0.697779i	$0.754172\pi$
$660$	0	0
$661$	17614.0	1.03647	0.518234	−	0.855239i	$-0.326590\pi$
$661$	17614.0	1.03647	0.518234	+	0.855239i	$0.326590\pi$
$662$	0	0
$663$	−18480.0	−1.08251
$664$	0	0
$665$	−9280.00	−0.541147
$666$	0	0
$667$	−3168.00	−0.183906
$668$	0	0
$669$	−22752.0	−1.31486
$670$	0	0
$671$	20520.0	1.18057
$672$	0	0
$673$	13058.0	0.747918	0.373959	−	0.927445i	$-0.378000\pi$
$673$	13058.0	0.747918	0.373959	+	0.927445i	$0.378000\pi$
$674$	0	0
$675$	3800.00	0.216685
$676$	0	0
$677$	−33186.0	−1.88396	−0.941980	−	0.335668i	$-0.891038\pi$
$677$	−33186.0	−1.88396	−0.941980	+	0.335668i	$0.891038\pi$
$678$	0	0
$679$	6112.00	0.345445
$680$	0	0
$681$	−11184.0	−0.629327
$682$	0	0
$683$	31716.0	1.77684	0.888418	−	0.459035i	$-0.151805\pi$
$683$	31716.0	1.77684	0.888418	+	0.459035i	$0.151805\pi$
$684$	0	0
$685$	7170.00	0.399929
$686$	0	0
$687$	−17784.0	−0.987630
$688$	0	0
$689$	−5964.00	−0.329768
$690$	0	0
$691$	2084.00	0.114731	0.0573655	−	0.998353i	$-0.481730\pi$
$691$	2084.00	0.114731	0.0573655	+	0.998353i	$0.481730\pi$
$692$	0	0
$693$	−6336.00	−0.347308
$694$	0	0
$695$	−10980.0	−0.599274
$696$	0	0
$697$	−48620.0	−2.64220
$698$	0	0
$699$	−10088.0	−0.545870
$700$	0	0
$701$	−7418.00	−0.399678	−0.199839	−	0.979829i	$-0.564042\pi$
$701$	−7418.00	−0.399678	−0.199839	+	0.979829i	$0.564042\pi$
$702$	0	0
$703$	−29928.0	−1.60563
$704$	0	0
$705$	7840.00	0.418825
$706$	0	0
$707$	10784.0	0.573655
$708$	0	0
$709$	−18242.0	−0.966280	−0.483140	−	0.875543i	$-0.660504\pi$
$709$	−18242.0	−0.966280	−0.483140	+	0.875543i	$0.660504\pi$
$710$	0	0
$711$	8624.00	0.454888
$712$	0	0
$713$	3840.00	0.201696
$714$	0	0
$715$	7560.00	0.395424
$716$	0	0
$717$	3264.00	0.170009
$718$	0	0
$719$	−3024.00	−0.156851	−0.0784257	−	0.996920i	$-0.524989\pi$
$719$	−3024.00	−0.156851	−0.0784257	+	0.996920i	$0.524989\pi$
$720$	0	0
$721$	−15872.0	−0.819839
$722$	0	0
$723$	21688.0	1.11561
$724$	0	0
$725$	4950.00	0.253570
$726$	0	0
$727$	26176.0	1.33537	0.667685	−	0.744444i	$-0.267285\pi$
$727$	26176.0	1.33537	0.667685	+	0.744444i	$0.267285\pi$
$728$	0	0
$729$	19837.0	1.00782
$730$	0	0
$731$	−32120.0	−1.62517
$732$	0	0
$733$	−17818.0	−0.897848	−0.448924	−	0.893570i	$-0.648193\pi$
$733$	−17818.0	−0.897848	−0.448924	+	0.893570i	$0.648193\pi$
$734$	0	0
$735$	1740.00	0.0873209
$736$	0	0
$737$	24912.0	1.24511
$738$	0	0
$739$	22052.0	1.09769	0.548847	−	0.835923i	$-0.315067\pi$
$739$	22052.0	1.09769	0.548847	+	0.835923i	$0.315067\pi$
$740$	0	0
$741$	19488.0	0.966140
$742$	0	0
$743$	−15840.0	−0.782117	−0.391059	−	0.920366i	$-0.627891\pi$
$743$	−15840.0	−0.782117	−0.391059	+	0.920366i	$0.627891\pi$
$744$	0	0
$745$	−12090.0	−0.594555
$746$	0	0
$747$	6204.00	0.303872
$748$	0	0
$749$	−8000.00	−0.390272
$750$	0	0
$751$	−21024.0	−1.02154	−0.510770	−	0.859717i	$-0.670640\pi$
$751$	−21024.0	−1.02154	−0.510770	+	0.859717i	$0.670640\pi$
$752$	0	0
$753$	−23600.0	−1.14214
$754$	0	0
$755$	−18360.0	−0.885018
$756$	0	0
$757$	−38034.0	−1.82612	−0.913058	−	0.407831i	$-0.866285\pi$
$757$	−38034.0	−1.82612	−0.913058	+	0.407831i	$0.866285\pi$
$758$	0	0
$759$	−2304.00	−0.110184
$760$	0	0
$761$	37802.0	1.80069	0.900343	−	0.435182i	$-0.143316\pi$
$761$	37802.0	1.80069	0.900343	+	0.435182i	$0.143316\pi$
$762$	0	0
$763$	−16736.0	−0.794081
$764$	0	0
$765$	6050.00	0.285932
$766$	0	0
$767$	−14616.0	−0.688075
$768$	0	0
$769$	15042.0	0.705369	0.352684	−	0.935742i	$-0.385269\pi$
$769$	15042.0	0.705369	0.352684	+	0.935742i	$0.385269\pi$
$770$	0	0
$771$	−21000.0	−0.980929
$772$	0	0
$773$	5950.00	0.276852	0.138426	−	0.990373i	$-0.455796\pi$
$773$	5950.00	0.276852	0.138426	+	0.990373i	$0.455796\pi$
$774$	0	0
$775$	−6000.00	−0.278099
$776$	0	0
$777$	−16512.0	−0.762374
$778$	0	0
$779$	51272.0	2.35816
$780$	0	0
$781$	6048.00	0.277099
$782$	0	0
$783$	30096.0	1.37362
$784$	0	0
$785$	1790.00	0.0813858
$786$	0	0
$787$	−23364.0	−1.05824	−0.529121	−	0.848546i	$-0.677478\pi$
$787$	−23364.0	−1.05824	−0.529121	+	0.848546i	$0.677478\pi$
$788$	0	0
$789$	24960.0	1.12624
$790$	0	0
$791$	8928.00	0.401319
$792$	0	0
$793$	23940.0	1.07205
$794$	0	0
$795$	−2840.00	−0.126697
$796$	0	0
$797$	19846.0	0.882034	0.441017	−	0.897499i	$-0.354618\pi$
$797$	19846.0	0.882034	0.441017	+	0.897499i	$0.354618\pi$
$798$	0	0
$799$	43120.0	1.90923
$800$	0	0
$801$	−11374.0	−0.501724
$802$	0	0
$803$	4824.00	0.211999
$804$	0	0
$805$	1280.00	0.0560423
$806$	0	0
$807$	2856.00	0.124580
$808$	0	0
$809$	24762.0	1.07613	0.538063	−	0.842905i	$-0.319156\pi$
$809$	24762.0	1.07613	0.538063	+	0.842905i	$0.319156\pi$
$810$	0	0
$811$	−16644.0	−0.720653	−0.360327	−	0.932826i	$-0.617335\pi$
$811$	−16644.0	−0.720653	−0.360327	+	0.932826i	$0.617335\pi$
$812$	0	0
$813$	8576.00	0.369955
$814$	0	0
$815$	−12820.0	−0.551000
$816$	0	0
$817$	33872.0	1.45047
$818$	0	0
$819$	−7392.00	−0.315381
$820$	0	0
$821$	3182.00	0.135265	0.0676325	−	0.997710i	$-0.478455\pi$
$821$	3182.00	0.135265	0.0676325	+	0.997710i	$0.478455\pi$
$822$	0	0
$823$	7504.00	0.317829	0.158914	−	0.987292i	$-0.449201\pi$
$823$	7504.00	0.317829	0.158914	+	0.987292i	$0.449201\pi$
$824$	0	0
$825$	3600.00	0.151922
$826$	0	0
$827$	−12604.0	−0.529969	−0.264984	−	0.964253i	$-0.585367\pi$
$827$	−12604.0	−0.529969	−0.264984	+	0.964253i	$0.585367\pi$
$828$	0	0
$829$	12230.0	0.512383	0.256191	−	0.966626i	$-0.417532\pi$
$829$	12230.0	0.512383	0.256191	+	0.966626i	$0.417532\pi$
$830$	0	0
$831$	17864.0	0.745722
$832$	0	0
$833$	9570.00	0.398056
$834$	0	0
$835$	15280.0	0.633277
$836$	0	0
$837$	−36480.0	−1.50649
$838$	0	0
$839$	−9656.00	−0.397333	−0.198666	−	0.980067i	$-0.563661\pi$
$839$	−9656.00	−0.397333	−0.198666	+	0.980067i	$0.563661\pi$
$840$	0	0
$841$	14815.0	0.607446
$842$	0	0
$843$	21208.0	0.866480
$844$	0	0
$845$	−2165.00	−0.0881400
$846$	0	0
$847$	560.000	0.0227176
$848$	0	0
$849$	−27728.0	−1.12087
$850$	0	0
$851$	4128.00	0.166282
$852$	0	0
$853$	5806.00	0.233052	0.116526	−	0.993188i	$-0.462824\pi$
$853$	5806.00	0.233052	0.116526	+	0.993188i	$0.462824\pi$
$854$	0	0
$855$	−6380.00	−0.255195
$856$	0	0
$857$	−39094.0	−1.55826	−0.779128	−	0.626865i	$-0.784338\pi$
$857$	−39094.0	−1.55826	−0.779128	+	0.626865i	$0.784338\pi$
$858$	0	0
$859$	18876.0	0.749756	0.374878	−	0.927074i	$-0.377685\pi$
$859$	18876.0	0.749756	0.374878	+	0.927074i	$0.377685\pi$
$860$	0	0
$861$	28288.0	1.11969
$862$	0	0
$863$	−32296.0	−1.27389	−0.636946	−	0.770909i	$-0.719803\pi$
$863$	−32296.0	−1.27389	−0.636946	+	0.770909i	$0.719803\pi$
$864$	0	0
$865$	−1170.00	−0.0459898
$866$	0	0
$867$	−28748.0	−1.12611
$868$	0	0
$869$	28224.0	1.10176
$870$	0	0
$871$	29064.0	1.13065
$872$	0	0
$873$	4202.00	0.162905
$874$	0	0
$875$	−2000.00	−0.0772712
$876$	0	0
$877$	−9578.00	−0.368787	−0.184393	−	0.982853i	$-0.559032\pi$
$877$	−9578.00	−0.368787	−0.184393	+	0.982853i	$0.559032\pi$
$878$	0	0
$879$	16136.0	0.619174
$880$	0	0
$881$	−41710.0	−1.59506	−0.797529	−	0.603281i	$-0.793860\pi$
$881$	−41710.0	−1.59506	−0.797529	+	0.603281i	$0.793860\pi$
$882$	0	0
$883$	−2260.00	−0.0861326	−0.0430663	−	0.999072i	$-0.513713\pi$
$883$	−2260.00	−0.0861326	−0.0430663	+	0.999072i	$0.513713\pi$
$884$	0	0
$885$	−6960.00	−0.264359
$886$	0	0
$887$	33696.0	1.27554	0.637768	−	0.770228i	$-0.279858\pi$
$887$	33696.0	1.27554	0.637768	+	0.770228i	$0.279858\pi$
$888$	0	0
$889$	−5248.00	−0.197989
$890$	0	0
$891$	11196.0	0.420965
$892$	0	0
$893$	−45472.0	−1.70399
$894$	0	0
$895$	−2620.00	−0.0978513
$896$	0	0
$897$	−2688.00	−0.100055
$898$	0	0
$899$	−47520.0	−1.76294
$900$	0	0
$901$	−15620.0	−0.577556
$902$	0	0
$903$	18688.0	0.688702
$904$	0	0
$905$	−5690.00	−0.208997
$906$	0	0
$907$	−7756.00	−0.283940	−0.141970	−	0.989871i	$-0.545344\pi$
$907$	−7756.00	−0.283940	−0.141970	+	0.989871i	$0.545344\pi$
$908$	0	0
$909$	7414.00	0.270525
$910$	0	0
$911$	5312.00	0.193188	0.0965941	−	0.995324i	$-0.469205\pi$
$911$	5312.00	0.193188	0.0965941	+	0.995324i	$0.469205\pi$
$912$	0	0
$913$	20304.0	0.735996
$914$	0	0
$915$	11400.0	0.411882
$916$	0	0
$917$	−3392.00	−0.122152
$918$	0	0
$919$	23576.0	0.846246	0.423123	−	0.906072i	$-0.360934\pi$
$919$	23576.0	0.846246	0.423123	+	0.906072i	$0.360934\pi$
$920$	0	0
$921$	−15344.0	−0.548971
$922$	0	0
$923$	7056.00	0.251626
$924$	0	0
$925$	−6450.00	−0.229270
$926$	0	0
$927$	−10912.0	−0.386620
$928$	0	0
$929$	−19038.0	−0.672354	−0.336177	−	0.941799i	$-0.609134\pi$
$929$	−19038.0	−0.672354	−0.336177	+	0.941799i	$0.609134\pi$
$930$	0	0
$931$	−10092.0	−0.355265
$932$	0	0
$933$	2656.00	0.0931978
$934$	0	0
$935$	19800.0	0.692545
$936$	0	0
$937$	20570.0	0.717175	0.358587	−	0.933496i	$-0.383259\pi$
$937$	20570.0	0.717175	0.358587	+	0.933496i	$0.383259\pi$
$938$	0	0
$939$	−11944.0	−0.415099
$940$	0	0
$941$	−21386.0	−0.740875	−0.370438	−	0.928857i	$-0.620792\pi$
$941$	−21386.0	−0.740875	−0.370438	+	0.928857i	$0.620792\pi$
$942$	0	0
$943$	−7072.00	−0.244216
$944$	0	0
$945$	−12160.0	−0.418587
$946$	0	0
$947$	−38020.0	−1.30463	−0.652315	−	0.757948i	$-0.726202\pi$
$947$	−38020.0	−1.30463	−0.652315	+	0.757948i	$0.726202\pi$
$948$	0	0
$949$	5628.00	0.192511
$950$	0	0
$951$	−10904.0	−0.371805
$952$	0	0
$953$	20202.0	0.686681	0.343340	−	0.939211i	$-0.388442\pi$
$953$	20202.0	0.686681	0.343340	+	0.939211i	$0.388442\pi$
$954$	0	0
$955$	−7600.00	−0.257519
$956$	0	0
$957$	28512.0	0.963074
$958$	0	0
$959$	−22944.0	−0.772576
$960$	0	0
$961$	27809.0	0.933470
$962$	0	0
$963$	−5500.00	−0.184045
$964$	0	0
$965$	−10710.0	−0.357272
$966$	0	0
$967$	−29840.0	−0.992337	−0.496168	−	0.868226i	$-0.665260\pi$
$967$	−29840.0	−0.992337	−0.496168	+	0.868226i	$0.665260\pi$
$968$	0	0
$969$	51040.0	1.69210
$970$	0	0
$971$	12476.0	0.412332	0.206166	−	0.978517i	$-0.433901\pi$
$971$	12476.0	0.412332	0.206166	+	0.978517i	$0.433901\pi$
$972$	0	0
$973$	35136.0	1.15767
$974$	0	0
$975$	4200.00	0.137957
$976$	0	0
$977$	−36974.0	−1.21075	−0.605375	−	0.795940i	$-0.706977\pi$
$977$	−36974.0	−1.21075	−0.605375	+	0.795940i	$0.706977\pi$
$978$	0	0
$979$	−37224.0	−1.21520
$980$	0	0
$981$	−11506.0	−0.374473
$982$	0	0
$983$	16368.0	0.531087	0.265543	−	0.964099i	$-0.414449\pi$
$983$	16368.0	0.531087	0.265543	+	0.964099i	$0.414449\pi$
$984$	0	0
$985$	−11530.0	−0.372971
$986$	0	0
$987$	−25088.0	−0.809078
$988$	0	0
$989$	−4672.00	−0.150213
$990$	0	0
$991$	49552.0	1.58837	0.794183	−	0.607678i	$-0.207899\pi$
$991$	49552.0	1.58837	0.794183	+	0.607678i	$0.207899\pi$
$992$	0	0
$993$	−36848.0	−1.17758
$994$	0	0
$995$	−16440.0	−0.523802
$996$	0	0
$997$	24414.0	0.775526	0.387763	−	0.921759i	$-0.373248\pi$
$997$	24414.0	0.775526	0.387763	+	0.921759i	$0.373248\pi$
$998$	0	0
$999$	−39216.0	−1.24198

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	80.4.a.b.1.1		1
3.2	odd	2		720.4.a.d.1.1		1
4.3	odd	2		40.4.a.b.1.1	✓	1
5.2	odd	4		400.4.c.h.49.2		2
5.3	odd	4		400.4.c.h.49.1		2
5.4	even	2		400.4.a.p.1.1		1
8.3	odd	2		320.4.a.e.1.1		1
8.5	even	2		320.4.a.j.1.1		1
12.11	even	2		360.4.a.f.1.1		1
16.3	odd	4		1280.4.d.d.641.2		2
16.5	even	4		1280.4.d.m.641.2		2
16.11	odd	4		1280.4.d.d.641.1		2
16.13	even	4		1280.4.d.m.641.1		2
20.3	even	4		200.4.c.f.49.2		2
20.7	even	4		200.4.c.f.49.1		2
20.19	odd	2		200.4.a.d.1.1		1
28.27	even	2		1960.4.a.e.1.1		1
40.19	odd	2		1600.4.a.bk.1.1		1
40.29	even	2		1600.4.a.q.1.1		1
60.23	odd	4		1800.4.f.d.649.1		2
60.47	odd	4		1800.4.f.d.649.2		2
60.59	even	2		1800.4.a.h.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.4.a.b.1.1	✓	1	4.3	odd	2
80.4.a.b.1.1		1	1.1	even	1	trivial
200.4.a.d.1.1		1	20.19	odd	2
200.4.c.f.49.1		2	20.7	even	4
200.4.c.f.49.2		2	20.3	even	4
320.4.a.e.1.1		1	8.3	odd	2
320.4.a.j.1.1		1	8.5	even	2
360.4.a.f.1.1		1	12.11	even	2
400.4.a.p.1.1		1	5.4	even	2
400.4.c.h.49.1		2	5.3	odd	4
400.4.c.h.49.2		2	5.2	odd	4
720.4.a.d.1.1		1	3.2	odd	2
1280.4.d.d.641.1		2	16.11	odd	4
1280.4.d.d.641.2		2	16.3	odd	4
1280.4.d.m.641.1		2	16.13	even	4
1280.4.d.m.641.2		2	16.5	even	4
1600.4.a.q.1.1		1	40.29	even	2
1600.4.a.bk.1.1		1	40.19	odd	2
1800.4.a.h.1.1		1	60.59	even	2
1800.4.f.d.649.1		2	60.23	odd	4
1800.4.f.d.649.2		2	60.47	odd	4
1960.4.a.e.1.1		1	28.27	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:	\( N \)	\(=\)	\( 80 = 2^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	80.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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