LMFDB - Embedded newform 80.6.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [80,6,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, 6, names="a")

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

chi := DirichletCharacter("80.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 80 = 2^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
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:	$12.8307055850$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 10)
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)

$f(q)$

$=$

$q-24.0000 q^{3} +25.0000 q^{5} +172.000 q^{7} +333.000 q^{9} +O(q^{10})$

$q-24.0000 q^{3} +25.0000 q^{5} +172.000 q^{7} +333.000 q^{9} -132.000 q^{11} -946.000 q^{13} -600.000 q^{15} -222.000 q^{17} -500.000 q^{19} -4128.00 q^{21} -3564.00 q^{23} +625.000 q^{25} -2160.00 q^{27} +2190.00 q^{29} -2312.00 q^{31} +3168.00 q^{33} +4300.00 q^{35} -11242.0 q^{37} +22704.0 q^{39} +1242.00 q^{41} -20624.0 q^{43} +8325.00 q^{45} -6588.00 q^{47} +12777.0 q^{49} +5328.00 q^{51} -21066.0 q^{53} -3300.00 q^{55} +12000.0 q^{57} -7980.00 q^{59} +16622.0 q^{61} +57276.0 q^{63} -23650.0 q^{65} -1808.00 q^{67} +85536.0 q^{69} +24528.0 q^{71} +20474.0 q^{73} -15000.0 q^{75} -22704.0 q^{77} +46240.0 q^{79} -29079.0 q^{81} +51576.0 q^{83} -5550.00 q^{85} -52560.0 q^{87} -110310. q^{89} -162712. q^{91} +55488.0 q^{93} -12500.0 q^{95} -78382.0 q^{97} -43956.0 q^{99} +O(q^{100})$

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
$3$	−24.0000	−1.53960	−0.769800	−	0.638285i	$-0.779644\pi$
$3$	−24.0000	−1.53960	−0.769800	+	0.638285i	$0.779644\pi$
$4$	0	0
$5$	25.0000	0.447214
$5$	25.0000	0.447214
$6$	0	0
$7$	172.000	1.32673	0.663366	−	0.748295i	$-0.269127\pi$
$7$	172.000	1.32673	0.663366	+	0.748295i	$0.269127\pi$
$8$	0	0
$9$	333.000	1.37037
$10$	0	0
$11$	−132.000	−0.328921	−0.164461	−	0.986384i	$-0.552588\pi$
$11$	−132.000	−0.328921	−0.164461	+	0.986384i	$0.552588\pi$
$12$	0	0
$13$	−946.000	−1.55250	−0.776252	−	0.630423i	$-0.782882\pi$
$13$	−946.000	−1.55250	−0.776252	+	0.630423i	$0.782882\pi$
$14$	0	0
$15$	−600.000	−0.688530
$16$	0	0
$17$	−222.000	−0.186308	−0.0931538	−	0.995652i	$-0.529695\pi$
$17$	−222.000	−0.186308	−0.0931538	+	0.995652i	$0.529695\pi$
$18$	0	0
$19$	−500.000	−0.317750	−0.158875	−	0.987299i	$-0.550787\pi$
$19$	−500.000	−0.317750	−0.158875	+	0.987299i	$0.550787\pi$
$20$	0	0
$21$	−4128.00	−2.04264
$22$	0	0
$23$	−3564.00	−1.40481	−0.702406	−	0.711777i	$-0.747891\pi$
$23$	−3564.00	−1.40481	−0.702406	+	0.711777i	$0.747891\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	−2160.00	−0.570222
$28$	0	0
$29$	2190.00	0.483559	0.241779	−	0.970331i	$-0.422269\pi$
$29$	2190.00	0.483559	0.241779	+	0.970331i	$0.422269\pi$
$30$	0	0
$31$	−2312.00	−0.432099	−0.216050	−	0.976382i	$-0.569317\pi$
$31$	−2312.00	−0.432099	−0.216050	+	0.976382i	$0.569317\pi$
$32$	0	0
$33$	3168.00	0.506408
$34$	0	0
$35$	4300.00	0.593333
$36$	0	0
$37$	−11242.0	−1.35002	−0.675009	−	0.737810i	$-0.735860\pi$
$37$	−11242.0	−1.35002	−0.675009	+	0.737810i	$0.735860\pi$
$38$	0	0
$39$	22704.0	2.39024
$40$	0	0
$41$	1242.00	0.115388	0.0576942	−	0.998334i	$-0.481625\pi$
$41$	1242.00	0.115388	0.0576942	+	0.998334i	$0.481625\pi$
$42$	0	0
$43$	−20624.0	−1.70099	−0.850495	−	0.525983i	$-0.823697\pi$
$43$	−20624.0	−1.70099	−0.850495	+	0.525983i	$0.823697\pi$
$44$	0	0
$45$	8325.00	0.612848
$46$	0	0
$47$	−6588.00	−0.435020	−0.217510	−	0.976058i	$-0.569793\pi$
$47$	−6588.00	−0.435020	−0.217510	+	0.976058i	$0.569793\pi$
$48$	0	0
$49$	12777.0	0.760219
$50$	0	0
$51$	5328.00	0.286839
$52$	0	0
$53$	−21066.0	−1.03013	−0.515065	−	0.857151i	$-0.672232\pi$
$53$	−21066.0	−1.03013	−0.515065	+	0.857151i	$0.672232\pi$
$54$	0	0
$55$	−3300.00	−0.147098
$56$	0	0
$57$	12000.0	0.489209
$58$	0	0
$59$	−7980.00	−0.298451	−0.149225	−	0.988803i	$-0.547678\pi$
$59$	−7980.00	−0.298451	−0.149225	+	0.988803i	$0.547678\pi$
$60$	0	0
$61$	16622.0	0.571951	0.285975	−	0.958237i	$-0.407682\pi$
$61$	16622.0	0.571951	0.285975	+	0.958237i	$0.407682\pi$
$62$	0	0
$63$	57276.0	1.81811
$64$	0	0
$65$	−23650.0	−0.694301
$66$	0	0
$67$	−1808.00	−0.0492052	−0.0246026	−	0.999697i	$-0.507832\pi$
$67$	−1808.00	−0.0492052	−0.0246026	+	0.999697i	$0.507832\pi$
$68$	0	0
$69$	85536.0	2.16285
$70$	0	0
$71$	24528.0	0.577452	0.288726	−	0.957412i	$-0.406768\pi$
$71$	24528.0	0.577452	0.288726	+	0.957412i	$0.406768\pi$
$72$	0	0
$73$	20474.0	0.449672	0.224836	−	0.974397i	$-0.427815\pi$
$73$	20474.0	0.449672	0.224836	+	0.974397i	$0.427815\pi$
$74$	0	0
$75$	−15000.0	−0.307920
$76$	0	0
$77$	−22704.0	−0.436391
$78$	0	0
$79$	46240.0	0.833585	0.416793	−	0.909002i	$-0.363154\pi$
$79$	46240.0	0.833585	0.416793	+	0.909002i	$0.363154\pi$
$80$	0	0
$81$	−29079.0	−0.492455
$82$	0	0
$83$	51576.0	0.821774	0.410887	−	0.911686i	$-0.365219\pi$
$83$	51576.0	0.821774	0.410887	+	0.911686i	$0.365219\pi$
$84$	0	0
$85$	−5550.00	−0.0833193
$86$	0	0
$87$	−52560.0	−0.744487
$88$	0	0
$89$	−110310.	−1.47618	−0.738091	−	0.674701i	$-0.764272\pi$
$89$	−110310.	−1.47618	−0.738091	+	0.674701i	$0.764272\pi$
$90$	0	0
$91$	−162712.	−2.05976
$92$	0	0
$93$	55488.0	0.665260
$94$	0	0
$95$	−12500.0	−0.142102
$96$	0	0
$97$	−78382.0	−0.845838	−0.422919	−	0.906168i	$-0.638994\pi$
$97$	−78382.0	−0.845838	−0.422919	+	0.906168i	$0.638994\pi$
$98$	0	0
$99$	−43956.0	−0.450744
$100$	0	0
$101$	141942.	1.38455	0.692273	−	0.721636i	$-0.256609\pi$
$101$	141942.	1.38455	0.692273	+	0.721636i	$0.256609\pi$
$102$	0	0
$103$	436.000	0.00404943	0.00202471	−	0.999998i	$-0.499356\pi$
$103$	436.000	0.00404943	0.00202471	+	0.999998i	$0.499356\pi$
$104$	0	0
$105$	−103200.	−0.913496
$106$	0	0
$107$	−232968.	−1.96715	−0.983574	−	0.180508i	$-0.942226\pi$
$107$	−232968.	−1.96715	−0.983574	+	0.180508i	$0.942226\pi$
$108$	0	0
$109$	−174850.	−1.40961	−0.704806	−	0.709400i	$-0.748966\pi$
$109$	−174850.	−1.40961	−0.704806	+	0.709400i	$0.748966\pi$
$110$	0	0
$111$	269808.	2.07849
$112$	0	0
$113$	182994.	1.34816	0.674079	−	0.738659i	$-0.264541\pi$
$113$	182994.	1.34816	0.674079	+	0.738659i	$0.264541\pi$
$114$	0	0
$115$	−89100.0	−0.628251
$116$	0	0
$117$	−315018.	−2.12751
$118$	0	0
$119$	−38184.0	−0.247180
$120$	0	0
$121$	−143627.	−0.891811
$122$	0	0
$123$	−29808.0	−0.177652
$124$	0	0
$125$	15625.0	0.0894427
$126$	0	0
$127$	122452.	0.673685	0.336842	−	0.941561i	$-0.390641\pi$
$127$	122452.	0.673685	0.336842	+	0.941561i	$0.390641\pi$
$128$	0	0
$129$	494976.	2.61885
$130$	0	0
$131$	241908.	1.23161	0.615803	−	0.787900i	$-0.288832\pi$
$131$	241908.	1.23161	0.615803	+	0.787900i	$0.288832\pi$
$132$	0	0
$133$	−86000.0	−0.421570
$134$	0	0
$135$	−54000.0	−0.255011
$136$	0	0
$137$	277098.	1.26134	0.630670	−	0.776051i	$-0.282780\pi$
$137$	277098.	1.26134	0.630670	+	0.776051i	$0.282780\pi$
$138$	0	0
$139$	193540.	0.849638	0.424819	−	0.905278i	$-0.360338\pi$
$139$	193540.	0.849638	0.424819	+	0.905278i	$0.360338\pi$
$140$	0	0
$141$	158112.	0.669757
$142$	0	0
$143$	124872.	0.510652
$144$	0	0
$145$	54750.0	0.216254
$146$	0	0
$147$	−306648.	−1.17043
$148$	0	0
$149$	140550.	0.518639	0.259320	−	0.965792i	$-0.416502\pi$
$149$	140550.	0.518639	0.259320	+	0.965792i	$0.416502\pi$
$150$	0	0
$151$	−433952.	−1.54881	−0.774407	−	0.632688i	$-0.781952\pi$
$151$	−433952.	−1.54881	−0.774407	+	0.632688i	$0.781952\pi$
$152$	0	0
$153$	−73926.0	−0.255310
$154$	0	0
$155$	−57800.0	−0.193241
$156$	0	0
$157$	−555922.	−1.79997	−0.899984	−	0.435923i	$-0.856422\pi$
$157$	−555922.	−1.79997	−0.899984	+	0.435923i	$0.856422\pi$
$158$	0	0
$159$	505584.	1.58599
$160$	0	0
$161$	−613008.	−1.86381
$162$	0	0
$163$	66616.0	0.196386	0.0981928	−	0.995167i	$-0.468694\pi$
$163$	66616.0	0.196386	0.0981928	+	0.995167i	$0.468694\pi$
$164$	0	0
$165$	79200.0	0.226472
$166$	0	0
$167$	205692.	0.570724	0.285362	−	0.958420i	$-0.407886\pi$
$167$	205692.	0.570724	0.285362	+	0.958420i	$0.407886\pi$
$168$	0	0
$169$	523623.	1.41027
$170$	0	0
$171$	−166500.	−0.435436
$172$	0	0
$173$	433854.	1.10212	0.551059	−	0.834466i	$-0.314224\pi$
$173$	433854.	1.10212	0.551059	+	0.834466i	$0.314224\pi$
$174$	0	0
$175$	107500.	0.265346
$176$	0	0
$177$	191520.	0.459495
$178$	0	0
$179$	489180.	1.14113	0.570566	−	0.821252i	$-0.306724\pi$
$179$	489180.	1.14113	0.570566	+	0.821252i	$0.306724\pi$
$180$	0	0
$181$	719462.	1.63234	0.816172	−	0.577810i	$-0.196092\pi$
$181$	719462.	1.63234	0.816172	+	0.577810i	$0.196092\pi$
$182$	0	0
$183$	−398928.	−0.880576
$184$	0	0
$185$	−281050.	−0.603746
$186$	0	0
$187$	29304.0	0.0612806
$188$	0	0
$189$	−371520.	−0.756533
$190$	0	0
$191$	185928.	0.368775	0.184387	−	0.982854i	$-0.440970\pi$
$191$	185928.	0.368775	0.184387	+	0.982854i	$0.440970\pi$
$192$	0	0
$193$	−591406.	−1.14286	−0.571429	−	0.820651i	$-0.693611\pi$
$193$	−591406.	−1.14286	−0.571429	+	0.820651i	$0.693611\pi$
$194$	0	0
$195$	567600.	1.06895
$196$	0	0
$197$	449478.	0.825169	0.412584	−	0.910919i	$-0.364626\pi$
$197$	449478.	0.825169	0.412584	+	0.910919i	$0.364626\pi$
$198$	0	0
$199$	−157160.	−0.281326	−0.140663	−	0.990058i	$-0.544923\pi$
$199$	−157160.	−0.281326	−0.140663	+	0.990058i	$0.544923\pi$
$200$	0	0
$201$	43392.0	0.0757564
$202$	0	0
$203$	376680.	0.641553
$204$	0	0
$205$	31050.0	0.0516032
$206$	0	0
$207$	−1.18681e6	−1.92511
$208$	0	0
$209$	66000.0	0.104515
$210$	0	0
$211$	−253052.	−0.391294	−0.195647	−	0.980674i	$-0.562681\pi$
$211$	−253052.	−0.391294	−0.195647	+	0.980674i	$0.562681\pi$
$212$	0	0
$213$	−588672.	−0.889046
$214$	0	0
$215$	−515600.	−0.760706
$216$	0	0
$217$	−397664.	−0.573280
$218$	0	0
$219$	−491376.	−0.692315
$220$	0	0
$221$	210012.	0.289243
$222$	0	0
$223$	−1.07344e6	−1.44550	−0.722749	−	0.691111i	$-0.757122\pi$
$223$	−1.07344e6	−1.44550	−0.722749	+	0.691111i	$0.757122\pi$
$224$	0	0
$225$	208125.	0.274074
$226$	0	0
$227$	626832.	0.807396	0.403698	−	0.914892i	$-0.367725\pi$
$227$	626832.	0.807396	0.403698	+	0.914892i	$0.367725\pi$
$228$	0	0
$229$	−116650.	−0.146993	−0.0734964	−	0.997295i	$-0.523416\pi$
$229$	−116650.	−0.146993	−0.0734964	+	0.997295i	$0.523416\pi$
$230$	0	0
$231$	544896.	0.671868
$232$	0	0
$233$	−743046.	−0.896656	−0.448328	−	0.893869i	$-0.647980\pi$
$233$	−743046.	−0.896656	−0.448328	+	0.893869i	$0.647980\pi$
$234$	0	0
$235$	−164700.	−0.194547
$236$	0	0
$237$	−1.10976e6	−1.28339
$238$	0	0
$239$	−978720.	−1.10832	−0.554158	−	0.832411i	$-0.686960\pi$
$239$	−978720.	−1.10832	−0.554158	+	0.832411i	$0.686960\pi$
$240$	0	0
$241$	−1.13280e6	−1.25635	−0.628174	−	0.778073i	$-0.716197\pi$
$241$	−1.13280e6	−1.25635	−0.628174	+	0.778073i	$0.716197\pi$
$242$	0	0
$243$	1.22278e6	1.32841
$244$	0	0
$245$	319425.	0.339980
$246$	0	0
$247$	473000.	0.493309
$248$	0	0
$249$	−1.23782e6	−1.26520
$250$	0	0
$251$	−905652.	−0.907355	−0.453677	−	0.891166i	$-0.649888\pi$
$251$	−905652.	−0.907355	−0.453677	+	0.891166i	$0.649888\pi$
$252$	0	0
$253$	470448.	0.462073
$254$	0	0
$255$	133200.	0.128278
$256$	0	0
$257$	1.93994e6	1.83212	0.916062	−	0.401036i	$-0.131350\pi$
$257$	1.93994e6	1.83212	0.916062	+	0.401036i	$0.131350\pi$
$258$	0	0
$259$	−1.93362e6	−1.79111
$260$	0	0
$261$	729270.	0.662654
$262$	0	0
$263$	805476.	0.718064	0.359032	−	0.933325i	$-0.383107\pi$
$263$	805476.	0.718064	0.359032	+	0.933325i	$0.383107\pi$
$264$	0	0
$265$	−526650.	−0.460689
$266$	0	0
$267$	2.64744e6	2.27273
$268$	0	0
$269$	−858690.	−0.723529	−0.361764	−	0.932270i	$-0.617825\pi$
$269$	−858690.	−0.723529	−0.361764	+	0.932270i	$0.617825\pi$
$270$	0	0
$271$	383608.	0.317296	0.158648	−	0.987335i	$-0.449287\pi$
$271$	383608.	0.317296	0.158648	+	0.987335i	$0.449287\pi$
$272$	0	0
$273$	3.90509e6	3.17120
$274$	0	0
$275$	−82500.0	−0.0657843
$276$	0	0
$277$	2.01076e6	1.57456	0.787282	−	0.616593i	$-0.211488\pi$
$277$	2.01076e6	1.57456	0.787282	+	0.616593i	$0.211488\pi$
$278$	0	0
$279$	−769896.	−0.592136
$280$	0	0
$281$	202602.	0.153066	0.0765329	−	0.997067i	$-0.475615\pi$
$281$	202602.	0.153066	0.0765329	+	0.997067i	$0.475615\pi$
$282$	0	0
$283$	221536.	0.164429	0.0822145	−	0.996615i	$-0.473801\pi$
$283$	221536.	0.164429	0.0822145	+	0.996615i	$0.473801\pi$
$284$	0	0
$285$	300000.	0.218781
$286$	0	0
$287$	213624.	0.153089
$288$	0	0
$289$	−1.37057e6	−0.965289
$290$	0	0
$291$	1.88117e6	1.30225
$292$	0	0
$293$	−322506.	−0.219467	−0.109733	−	0.993961i	$-0.535000\pi$
$293$	−322506.	−0.219467	−0.109733	+	0.993961i	$0.535000\pi$
$294$	0	0
$295$	−199500.	−0.133471
$296$	0	0
$297$	285120.	0.187558
$298$	0	0
$299$	3.37154e6	2.18098
$300$	0	0
$301$	−3.54733e6	−2.25676
$302$	0	0
$303$	−3.40661e6	−2.13165
$304$	0	0
$305$	415550.	0.255784
$306$	0	0
$307$	−1.44301e6	−0.873822	−0.436911	−	0.899505i	$-0.643927\pi$
$307$	−1.44301e6	−0.873822	−0.436911	+	0.899505i	$0.643927\pi$
$308$	0	0
$309$	−10464.0	−0.00623450
$310$	0	0
$311$	−171312.	−0.100435	−0.0502177	−	0.998738i	$-0.515992\pi$
$311$	−171312.	−0.100435	−0.0502177	+	0.998738i	$0.515992\pi$
$312$	0	0
$313$	−1.02689e6	−0.592463	−0.296232	−	0.955116i	$-0.595730\pi$
$313$	−1.02689e6	−0.592463	−0.296232	+	0.955116i	$0.595730\pi$
$314$	0	0
$315$	1.43190e6	0.813086
$316$	0	0
$317$	752958.	0.420845	0.210423	−	0.977610i	$-0.432516\pi$
$317$	752958.	0.420845	0.210423	+	0.977610i	$0.432516\pi$
$318$	0	0
$319$	−289080.	−0.159053
$320$	0	0
$321$	5.59123e6	3.02862
$322$	0	0
$323$	111000.	0.0591993
$324$	0	0
$325$	−591250.	−0.310501
$326$	0	0
$327$	4.19640e6	2.17024
$328$	0	0
$329$	−1.13314e6	−0.577155
$330$	0	0
$331$	−1.99413e6	−1.00042	−0.500212	−	0.865903i	$-0.666745\pi$
$331$	−1.99413e6	−1.00042	−0.500212	+	0.865903i	$0.666745\pi$
$332$	0	0
$333$	−3.74359e6	−1.85002
$334$	0	0
$335$	−45200.0	−0.0220053
$336$	0	0
$337$	−987022.	−0.473426	−0.236713	−	0.971580i	$-0.576070\pi$
$337$	−987022.	−0.473426	−0.236713	+	0.971580i	$0.576070\pi$
$338$	0	0
$339$	−4.39186e6	−2.07562
$340$	0	0
$341$	305184.	0.142127
$342$	0	0
$343$	−693160.	−0.318125
$344$	0	0
$345$	2.13840e6	0.967256
$346$	0	0
$347$	−2.20601e6	−0.983520	−0.491760	−	0.870731i	$-0.663646\pi$
$347$	−2.20601e6	−0.983520	−0.491760	+	0.870731i	$0.663646\pi$
$348$	0	0
$349$	2.74187e6	1.20499	0.602495	−	0.798123i	$-0.294173\pi$
$349$	2.74187e6	1.20499	0.602495	+	0.798123i	$0.294173\pi$
$350$	0	0
$351$	2.04336e6	0.885273
$352$	0	0
$353$	−2.38957e6	−1.02066	−0.510331	−	0.859978i	$-0.670477\pi$
$353$	−2.38957e6	−1.02066	−0.510331	+	0.859978i	$0.670477\pi$
$354$	0	0
$355$	613200.	0.258245
$356$	0	0
$357$	916416.	0.380559
$358$	0	0
$359$	279480.	0.114450	0.0572248	−	0.998361i	$-0.481775\pi$
$359$	279480.	0.114450	0.0572248	+	0.998361i	$0.481775\pi$
$360$	0	0
$361$	−2.22610e6	−0.899035
$362$	0	0
$363$	3.44705e6	1.37303
$364$	0	0
$365$	511850.	0.201099
$366$	0	0
$367$	2.47637e6	0.959734	0.479867	−	0.877341i	$-0.340685\pi$
$367$	2.47637e6	0.959734	0.479867	+	0.877341i	$0.340685\pi$
$368$	0	0
$369$	413586.	0.158125
$370$	0	0
$371$	−3.62335e6	−1.36671
$372$	0	0
$373$	2.74525e6	1.02167	0.510835	−	0.859679i	$-0.329336\pi$
$373$	2.74525e6	1.02167	0.510835	+	0.859679i	$0.329336\pi$
$374$	0	0
$375$	−375000.	−0.137706
$376$	0	0
$377$	−2.07174e6	−0.750727
$378$	0	0
$379$	1.18906e6	0.425212	0.212606	−	0.977138i	$-0.431805\pi$
$379$	1.18906e6	0.425212	0.212606	+	0.977138i	$0.431805\pi$
$380$	0	0
$381$	−2.93885e6	−1.03721
$382$	0	0
$383$	−3.25760e6	−1.13475	−0.567377	−	0.823458i	$-0.692042\pi$
$383$	−3.25760e6	−1.13475	−0.567377	+	0.823458i	$0.692042\pi$
$384$	0	0
$385$	−567600.	−0.195160
$386$	0	0
$387$	−6.86779e6	−2.33099
$388$	0	0
$389$	1.98351e6	0.664600	0.332300	−	0.943174i	$-0.392175\pi$
$389$	1.98351e6	0.664600	0.332300	+	0.943174i	$0.392175\pi$
$390$	0	0
$391$	791208.	0.261727
$392$	0	0
$393$	−5.80579e6	−1.89618
$394$	0	0
$395$	1.15600e6	0.372791
$396$	0	0
$397$	4.97416e6	1.58396	0.791978	−	0.610549i	$-0.209051\pi$
$397$	4.97416e6	1.58396	0.791978	+	0.610549i	$0.209051\pi$
$398$	0	0
$399$	2.06400e6	0.649049
$400$	0	0
$401$	−1.34264e6	−0.416963	−0.208482	−	0.978026i	$-0.566852\pi$
$401$	−1.34264e6	−0.416963	−0.208482	+	0.978026i	$0.566852\pi$
$402$	0	0
$403$	2.18715e6	0.670836
$404$	0	0
$405$	−726975.	−0.220233
$406$	0	0
$407$	1.48394e6	0.444050
$408$	0	0
$409$	−1.09423e6	−0.323445	−0.161722	−	0.986836i	$-0.551705\pi$
$409$	−1.09423e6	−0.323445	−0.161722	+	0.986836i	$0.551705\pi$
$410$	0	0
$411$	−6.65035e6	−1.94196
$412$	0	0
$413$	−1.37256e6	−0.395964
$414$	0	0
$415$	1.28940e6	0.367509
$416$	0	0
$417$	−4.64496e6	−1.30810
$418$	0	0
$419$	954060.	0.265485	0.132743	−	0.991151i	$-0.457622\pi$
$419$	954060.	0.265485	0.132743	+	0.991151i	$0.457622\pi$
$420$	0	0
$421$	−1.59390e6	−0.438284	−0.219142	−	0.975693i	$-0.570326\pi$
$421$	−1.59390e6	−0.438284	−0.219142	+	0.975693i	$0.570326\pi$
$422$	0	0
$423$	−2.19380e6	−0.596138
$424$	0	0
$425$	−138750.	−0.0372615
$426$	0	0
$427$	2.85898e6	0.758826
$428$	0	0
$429$	−2.99693e6	−0.786200
$430$	0	0
$431$	2.64665e6	0.686283	0.343141	−	0.939284i	$-0.388509\pi$
$431$	2.64665e6	0.686283	0.343141	+	0.939284i	$0.388509\pi$
$432$	0	0
$433$	3.72355e6	0.954416	0.477208	−	0.878790i	$-0.341649\pi$
$433$	3.72355e6	0.954416	0.477208	+	0.878790i	$0.341649\pi$
$434$	0	0
$435$	−1.31400e6	−0.332945
$436$	0	0
$437$	1.78200e6	0.446379
$438$	0	0
$439$	2.58340e6	0.639780	0.319890	−	0.947455i	$-0.396354\pi$
$439$	2.58340e6	0.639780	0.319890	+	0.947455i	$0.396354\pi$
$440$	0	0
$441$	4.25474e6	1.04178
$442$	0	0
$443$	−7.56206e6	−1.83076	−0.915379	−	0.402593i	$-0.868109\pi$
$443$	−7.56206e6	−1.83076	−0.915379	+	0.402593i	$0.868109\pi$
$444$	0	0
$445$	−2.75775e6	−0.660169
$446$	0	0
$447$	−3.37320e6	−0.798497
$448$	0	0
$449$	4.30773e6	1.00840	0.504200	−	0.863587i	$-0.331788\pi$
$449$	4.30773e6	1.00840	0.504200	+	0.863587i	$0.331788\pi$
$450$	0	0
$451$	−163944.	−0.0379537
$452$	0	0
$453$	1.04148e7	2.38456
$454$	0	0
$455$	−4.06780e6	−0.921152
$456$	0	0
$457$	−2.24354e6	−0.502509	−0.251254	−	0.967921i	$-0.580843\pi$
$457$	−2.24354e6	−0.502509	−0.251254	+	0.967921i	$0.580843\pi$
$458$	0	0
$459$	479520.	0.106237
$460$	0	0
$461$	1.65670e6	0.363071	0.181536	−	0.983384i	$-0.441893\pi$
$461$	1.65670e6	0.363071	0.181536	+	0.983384i	$0.441893\pi$
$462$	0	0
$463$	2.89160e6	0.626881	0.313441	−	0.949608i	$-0.398518\pi$
$463$	2.89160e6	0.626881	0.313441	+	0.949608i	$0.398518\pi$
$464$	0	0
$465$	1.38720e6	0.297514
$466$	0	0
$467$	6.52699e6	1.38491	0.692454	−	0.721462i	$-0.256530\pi$
$467$	6.52699e6	1.38491	0.692454	+	0.721462i	$0.256530\pi$
$468$	0	0
$469$	−310976.	−0.0652822
$470$	0	0
$471$	1.33421e7	2.77123
$472$	0	0
$473$	2.72237e6	0.559492
$474$	0	0
$475$	−312500.	−0.0635501
$476$	0	0
$477$	−7.01498e6	−1.41166
$478$	0	0
$479$	5.96232e6	1.18734	0.593672	−	0.804707i	$-0.297678\pi$
$479$	5.96232e6	1.18734	0.593672	+	0.804707i	$0.297678\pi$
$480$	0	0
$481$	1.06349e7	2.09591
$482$	0	0
$483$	1.47122e7	2.86952
$484$	0	0
$485$	−1.95955e6	−0.378270
$486$	0	0
$487$	−2.99191e6	−0.571644	−0.285822	−	0.958283i	$-0.592267\pi$
$487$	−2.99191e6	−0.571644	−0.285822	+	0.958283i	$0.592267\pi$
$488$	0	0
$489$	−1.59878e6	−0.302355
$490$	0	0
$491$	1.20419e6	0.225419	0.112710	−	0.993628i	$-0.464047\pi$
$491$	1.20419e6	0.225419	0.112710	+	0.993628i	$0.464047\pi$
$492$	0	0
$493$	−486180.	−0.0900907
$494$	0	0
$495$	−1.09890e6	−0.201579
$496$	0	0
$497$	4.21882e6	0.766125
$498$	0	0
$499$	−9.20546e6	−1.65499	−0.827493	−	0.561477i	$-0.810233\pi$
$499$	−9.20546e6	−1.65499	−0.827493	+	0.561477i	$0.810233\pi$
$500$	0	0
$501$	−4.93661e6	−0.878687
$502$	0	0
$503$	3.35956e6	0.592055	0.296027	−	0.955179i	$-0.404338\pi$
$503$	3.35956e6	0.592055	0.296027	+	0.955179i	$0.404338\pi$
$504$	0	0
$505$	3.54855e6	0.619188
$506$	0	0
$507$	−1.25670e7	−2.17125
$508$	0	0
$509$	−2.53701e6	−0.434038	−0.217019	−	0.976167i	$-0.569633\pi$
$509$	−2.53701e6	−0.434038	−0.217019	+	0.976167i	$0.569633\pi$
$510$	0	0
$511$	3.52153e6	0.596594
$512$	0	0
$513$	1.08000e6	0.181188
$514$	0	0
$515$	10900.0	0.00181096
$516$	0	0
$517$	869616.	0.143087
$518$	0	0
$519$	−1.04125e7	−1.69682
$520$	0	0
$521$	−9.31580e6	−1.50358	−0.751789	−	0.659404i	$-0.770809\pi$
$521$	−9.31580e6	−1.50358	−0.751789	+	0.659404i	$0.770809\pi$
$522$	0	0
$523$	5.02802e6	0.803790	0.401895	−	0.915686i	$-0.368352\pi$
$523$	5.02802e6	0.803790	0.401895	+	0.915686i	$0.368352\pi$
$524$	0	0
$525$	−2.58000e6	−0.408528
$526$	0	0
$527$	513264.	0.0805034
$528$	0	0
$529$	6.26575e6	0.973496
$530$	0	0
$531$	−2.65734e6	−0.408988
$532$	0	0
$533$	−1.17493e6	−0.179141
$534$	0	0
$535$	−5.82420e6	−0.879735
$536$	0	0
$537$	−1.17403e7	−1.75689
$538$	0	0
$539$	−1.68656e6	−0.250052
$540$	0	0
$541$	134222.	0.0197165	0.00985827	−	0.999951i	$-0.496862\pi$
$541$	134222.	0.0197165	0.00985827	+	0.999951i	$0.496862\pi$
$542$	0	0
$543$	−1.72671e7	−2.51316
$544$	0	0
$545$	−4.37125e6	−0.630397
$546$	0	0
$547$	−605648.	−0.0865470	−0.0432735	−	0.999063i	$-0.513779\pi$
$547$	−605648.	−0.0865470	−0.0432735	+	0.999063i	$0.513779\pi$
$548$	0	0
$549$	5.53513e6	0.783784
$550$	0	0
$551$	−1.09500e6	−0.153651
$552$	0	0
$553$	7.95328e6	1.10594
$554$	0	0
$555$	6.74520e6	0.929528
$556$	0	0
$557$	−7.06240e6	−0.964527	−0.482264	−	0.876026i	$-0.660185\pi$
$557$	−7.06240e6	−0.964527	−0.482264	+	0.876026i	$0.660185\pi$
$558$	0	0
$559$	1.95103e7	2.64079
$560$	0	0
$561$	−703296.	−0.0943476
$562$	0	0
$563$	1.03029e7	1.36990	0.684952	−	0.728588i	$-0.259823\pi$
$563$	1.03029e7	1.36990	0.684952	+	0.728588i	$0.259823\pi$
$564$	0	0
$565$	4.57485e6	0.602915
$566$	0	0
$567$	−5.00159e6	−0.653357
$568$	0	0
$569$	1.04769e6	0.135660	0.0678300	−	0.997697i	$-0.478392\pi$
$569$	1.04769e6	0.135660	0.0678300	+	0.997697i	$0.478392\pi$
$570$	0	0
$571$	−1.40765e7	−1.80677	−0.903385	−	0.428830i	$-0.858926\pi$
$571$	−1.40765e7	−1.80677	−0.903385	+	0.428830i	$0.858926\pi$
$572$	0	0
$573$	−4.46227e6	−0.567766
$574$	0	0
$575$	−2.22750e6	−0.280962
$576$	0	0
$577$	1.62682e6	0.203423	0.101711	−	0.994814i	$-0.467568\pi$
$577$	1.62682e6	0.203423	0.101711	+	0.994814i	$0.467568\pi$
$578$	0	0
$579$	1.41937e7	1.75955
$580$	0	0
$581$	8.87107e6	1.09027
$582$	0	0
$583$	2.78071e6	0.338832
$584$	0	0
$585$	−7.87545e6	−0.951449
$586$	0	0
$587$	−6.96089e6	−0.833814	−0.416907	−	0.908949i	$-0.636886\pi$
$587$	−6.96089e6	−0.833814	−0.416907	+	0.908949i	$0.636886\pi$
$588$	0	0
$589$	1.15600e6	0.137300
$590$	0	0
$591$	−1.07875e7	−1.27043
$592$	0	0
$593$	−1.13639e7	−1.32706	−0.663529	−	0.748150i	$-0.730942\pi$
$593$	−1.13639e7	−1.32706	−0.663529	+	0.748150i	$0.730942\pi$
$594$	0	0
$595$	−954600.	−0.110542
$596$	0	0
$597$	3.77184e6	0.433129
$598$	0	0
$599$	−1.48688e7	−1.69321	−0.846603	−	0.532224i	$-0.821356\pi$
$599$	−1.48688e7	−1.69321	−0.846603	+	0.532224i	$0.821356\pi$
$600$	0	0
$601$	−1.23612e6	−0.139596	−0.0697981	−	0.997561i	$-0.522236\pi$
$601$	−1.23612e6	−0.139596	−0.0697981	+	0.997561i	$0.522236\pi$
$602$	0	0
$603$	−602064.	−0.0674294
$604$	0	0
$605$	−3.59067e6	−0.398830
$606$	0	0
$607$	1.24498e7	1.37149	0.685743	−	0.727844i	$-0.259478\pi$
$607$	1.24498e7	1.37149	0.685743	+	0.727844i	$0.259478\pi$
$608$	0	0
$609$	−9.04032e6	−0.987735
$610$	0	0
$611$	6.23225e6	0.675370
$612$	0	0
$613$	−8.73491e6	−0.938873	−0.469437	−	0.882966i	$-0.655543\pi$
$613$	−8.73491e6	−0.938873	−0.469437	+	0.882966i	$0.655543\pi$
$614$	0	0
$615$	−745200.	−0.0794484
$616$	0	0
$617$	1.25495e7	1.32713	0.663565	−	0.748119i	$-0.269043\pi$
$617$	1.25495e7	1.32713	0.663565	+	0.748119i	$0.269043\pi$
$618$	0	0
$619$	1.46658e7	1.53843	0.769216	−	0.638988i	$-0.220647\pi$
$619$	1.46658e7	1.53843	0.769216	+	0.638988i	$0.220647\pi$
$620$	0	0
$621$	7.69824e6	0.801055
$622$	0	0
$623$	−1.89733e7	−1.95850
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	−1.58400e6	−0.160911
$628$	0	0
$629$	2.49572e6	0.251519
$630$	0	0
$631$	196288.	0.0196255	0.00981274	−	0.999952i	$-0.496876\pi$
$631$	196288.	0.0196255	0.00981274	+	0.999952i	$0.496876\pi$
$632$	0	0
$633$	6.07325e6	0.602437
$634$	0	0
$635$	3.06130e6	0.301281
$636$	0	0
$637$	−1.20870e7	−1.18024
$638$	0	0
$639$	8.16782e6	0.791324
$640$	0	0
$641$	−1.11596e7	−1.07276	−0.536381	−	0.843976i	$-0.680209\pi$
$641$	−1.11596e7	−1.07276	−0.536381	+	0.843976i	$0.680209\pi$
$642$	0	0
$643$	2.25158e6	0.214763	0.107381	−	0.994218i	$-0.465753\pi$
$643$	2.25158e6	0.214763	0.107381	+	0.994218i	$0.465753\pi$
$644$	0	0
$645$	1.23744e7	1.17118
$646$	0	0
$647$	−8.05319e6	−0.756323	−0.378161	−	0.925740i	$-0.623444\pi$
$647$	−8.05319e6	−0.756323	−0.378161	+	0.925740i	$0.623444\pi$
$648$	0	0
$649$	1.05336e6	0.0981669
$650$	0	0
$651$	9.54394e6	0.882623
$652$	0	0
$653$	−416466.	−0.0382205	−0.0191103	−	0.999817i	$-0.506083\pi$
$653$	−416466.	−0.0382205	−0.0191103	+	0.999817i	$0.506083\pi$
$654$	0	0
$655$	6.04770e6	0.550791
$656$	0	0
$657$	6.81784e6	0.616217
$658$	0	0
$659$	−1.31721e7	−1.18152	−0.590761	−	0.806847i	$-0.701172\pi$
$659$	−1.31721e7	−1.18152	−0.590761	+	0.806847i	$0.701172\pi$
$660$	0	0
$661$	−1.69494e6	−0.150886	−0.0754432	−	0.997150i	$-0.524037\pi$
$661$	−1.69494e6	−0.150886	−0.0754432	+	0.997150i	$0.524037\pi$
$662$	0	0
$663$	−5.04029e6	−0.445319
$664$	0	0
$665$	−2.15000e6	−0.188532
$666$	0	0
$667$	−7.80516e6	−0.679309
$668$	0	0
$669$	2.57627e7	2.22549
$670$	0	0
$671$	−2.19410e6	−0.188127
$672$	0	0
$673$	−8.91605e6	−0.758813	−0.379406	−	0.925230i	$-0.623872\pi$
$673$	−8.91605e6	−0.758813	−0.379406	+	0.925230i	$0.623872\pi$
$674$	0	0
$675$	−1.35000e6	−0.114044
$676$	0	0
$677$	−1.42894e7	−1.19824	−0.599118	−	0.800661i	$-0.704482\pi$
$677$	−1.42894e7	−1.19824	−0.599118	+	0.800661i	$0.704482\pi$
$678$	0	0
$679$	−1.34817e7	−1.12220
$680$	0	0
$681$	−1.50440e7	−1.24307
$682$	0	0
$683$	5.33314e6	0.437452	0.218726	−	0.975786i	$-0.429810\pi$
$683$	5.33314e6	0.437452	0.218726	+	0.975786i	$0.429810\pi$
$684$	0	0
$685$	6.92745e6	0.564088
$686$	0	0
$687$	2.79960e6	0.226310
$688$	0	0
$689$	1.99284e7	1.59928
$690$	0	0
$691$	−698252.	−0.0556310	−0.0278155	−	0.999613i	$-0.508855\pi$
$691$	−698252.	−0.0556310	−0.0278155	+	0.999613i	$0.508855\pi$
$692$	0	0
$693$	−7.56043e6	−0.598017
$694$	0	0
$695$	4.83850e6	0.379969
$696$	0	0
$697$	−275724.	−0.0214977
$698$	0	0
$699$	1.78331e7	1.38049
$700$	0	0
$701$	1.79880e7	1.38257	0.691285	−	0.722582i	$-0.257045\pi$
$701$	1.79880e7	1.38257	0.691285	+	0.722582i	$0.257045\pi$
$702$	0	0
$703$	5.62100e6	0.428968
$704$	0	0
$705$	3.95280e6	0.299524
$706$	0	0
$707$	2.44140e7	1.83692
$708$	0	0
$709$	−1.39464e7	−1.04195	−0.520975	−	0.853572i	$-0.674432\pi$
$709$	−1.39464e7	−1.04195	−0.520975	+	0.853572i	$0.674432\pi$
$710$	0	0
$711$	1.53979e7	1.14232
$712$	0	0
$713$	8.23997e6	0.607018
$714$	0	0
$715$	3.12180e6	0.228370
$716$	0	0
$717$	2.34893e7	1.70636
$718$	0	0
$719$	−6.22272e6	−0.448909	−0.224454	−	0.974485i	$-0.572060\pi$
$719$	−6.22272e6	−0.448909	−0.224454	+	0.974485i	$0.572060\pi$
$720$	0	0
$721$	74992.0	0.00537250
$722$	0	0
$723$	2.71872e7	1.93427
$724$	0	0
$725$	1.36875e6	0.0967117
$726$	0	0
$727$	7.76729e6	0.545047	0.272523	−	0.962149i	$-0.412142\pi$
$727$	7.76729e6	0.545047	0.272523	+	0.962149i	$0.412142\pi$
$728$	0	0
$729$	−2.22804e7	−1.55276
$730$	0	0
$731$	4.57853e6	0.316907
$732$	0	0
$733$	2.42083e7	1.66420	0.832099	−	0.554627i	$-0.187139\pi$
$733$	2.42083e7	1.66420	0.832099	+	0.554627i	$0.187139\pi$
$734$	0	0
$735$	−7.66620e6	−0.523434
$736$	0	0
$737$	238656.	0.0161847
$738$	0	0
$739$	−1.26850e7	−0.854434	−0.427217	−	0.904149i	$-0.640506\pi$
$739$	−1.26850e7	−0.854434	−0.427217	+	0.904149i	$0.640506\pi$
$740$	0	0
$741$	−1.13520e7	−0.759498
$742$	0	0
$743$	−1.97632e7	−1.31337	−0.656684	−	0.754166i	$-0.728041\pi$
$743$	−1.97632e7	−1.31337	−0.656684	+	0.754166i	$0.728041\pi$
$744$	0	0
$745$	3.51375e6	0.231942
$746$	0	0
$747$	1.71748e7	1.12613
$748$	0	0
$749$	−4.00705e7	−2.60988
$750$	0	0
$751$	9.01761e6	0.583434	0.291717	−	0.956505i	$-0.405774\pi$
$751$	9.01761e6	0.583434	0.291717	+	0.956505i	$0.405774\pi$
$752$	0	0
$753$	2.17356e7	1.39696
$754$	0	0
$755$	−1.08488e7	−0.692651
$756$	0	0
$757$	−1.12556e6	−0.0713887	−0.0356944	−	0.999363i	$-0.511364\pi$
$757$	−1.12556e6	−0.0713887	−0.0356944	+	0.999363i	$0.511364\pi$
$758$	0	0
$759$	−1.12908e7	−0.711407
$760$	0	0
$761$	2.25747e7	1.41306	0.706529	−	0.707684i	$-0.250260\pi$
$761$	2.25747e7	1.41306	0.706529	+	0.707684i	$0.250260\pi$
$762$	0	0
$763$	−3.00742e7	−1.87018
$764$	0	0
$765$	−1.84815e6	−0.114178
$766$	0	0
$767$	7.54908e6	0.463346
$768$	0	0
$769$	−632350.	−0.0385604	−0.0192802	−	0.999814i	$-0.506137\pi$
$769$	−632350.	−0.0385604	−0.0192802	+	0.999814i	$0.506137\pi$
$770$	0	0
$771$	−4.65585e7	−2.82074
$772$	0	0
$773$	−1.25867e7	−0.757643	−0.378822	−	0.925470i	$-0.623671\pi$
$773$	−1.25867e7	−0.757643	−0.378822	+	0.925470i	$0.623671\pi$
$774$	0	0
$775$	−1.44500e6	−0.0864199
$776$	0	0
$777$	4.64070e7	2.75760
$778$	0	0
$779$	−621000.	−0.0366647
$780$	0	0
$781$	−3.23770e6	−0.189937
$782$	0	0
$783$	−4.73040e6	−0.275736
$784$	0	0
$785$	−1.38980e7	−0.804970
$786$	0	0
$787$	−2.15792e7	−1.24194	−0.620968	−	0.783836i	$-0.713260\pi$
$787$	−2.15792e7	−1.24194	−0.620968	+	0.783836i	$0.713260\pi$
$788$	0	0
$789$	−1.93314e7	−1.10553
$790$	0	0
$791$	3.14750e7	1.78864
$792$	0	0
$793$	−1.57244e7	−0.887956
$794$	0	0
$795$	1.26396e7	0.709276
$796$	0	0
$797$	−3.09760e7	−1.72735	−0.863673	−	0.504052i	$-0.831842\pi$
$797$	−3.09760e7	−1.72735	−0.863673	+	0.504052i	$0.831842\pi$
$798$	0	0
$799$	1.46254e6	0.0810475
$800$	0	0
$801$	−3.67332e7	−2.02292
$802$	0	0
$803$	−2.70257e6	−0.147907
$804$	0	0
$805$	−1.53252e7	−0.833521
$806$	0	0
$807$	2.06086e7	1.11395
$808$	0	0
$809$	4.24929e6	0.228268	0.114134	−	0.993465i	$-0.463591\pi$
$809$	4.24929e6	0.228268	0.114134	+	0.993465i	$0.463591\pi$
$810$	0	0
$811$	−3.42333e6	−0.182767	−0.0913833	−	0.995816i	$-0.529129\pi$
$811$	−3.42333e6	−0.182767	−0.0913833	+	0.995816i	$0.529129\pi$
$812$	0	0
$813$	−9.20659e6	−0.488509
$814$	0	0
$815$	1.66540e6	0.0878263
$816$	0	0
$817$	1.03120e7	0.540490
$818$	0	0
$819$	−5.41831e7	−2.82263
$820$	0	0
$821$	3.10571e7	1.60806	0.804030	−	0.594588i	$-0.202685\pi$
$821$	3.10571e7	1.60806	0.804030	+	0.594588i	$0.202685\pi$
$822$	0	0
$823$	3.11904e7	1.60517	0.802584	−	0.596538i	$-0.203458\pi$
$823$	3.11904e7	1.60517	0.802584	+	0.596538i	$0.203458\pi$
$824$	0	0
$825$	1.98000e6	0.101282
$826$	0	0
$827$	8.28487e6	0.421233	0.210616	−	0.977569i	$-0.432453\pi$
$827$	8.28487e6	0.421233	0.210616	+	0.977569i	$0.432453\pi$
$828$	0	0
$829$	−1.81688e7	−0.918208	−0.459104	−	0.888383i	$-0.651829\pi$
$829$	−1.81688e7	−0.918208	−0.459104	+	0.888383i	$0.651829\pi$
$830$	0	0
$831$	−4.82582e7	−2.42420
$832$	0	0
$833$	−2.83649e6	−0.141635
$834$	0	0
$835$	5.14230e6	0.255236
$836$	0	0
$837$	4.99392e6	0.246393
$838$	0	0
$839$	1.02743e7	0.503902	0.251951	−	0.967740i	$-0.418928\pi$
$839$	1.02743e7	0.503902	0.251951	+	0.967740i	$0.418928\pi$
$840$	0	0
$841$	−1.57150e7	−0.766171
$842$	0	0
$843$	−4.86245e6	−0.235660
$844$	0	0
$845$	1.30906e7	0.630691
$846$	0	0
$847$	−2.47038e7	−1.18319
$848$	0	0
$849$	−5.31686e6	−0.253155
$850$	0	0
$851$	4.00665e7	1.89652
$852$	0	0
$853$	6.28597e6	0.295801	0.147901	−	0.989002i	$-0.452748\pi$
$853$	6.28597e6	0.295801	0.147901	+	0.989002i	$0.452748\pi$
$854$	0	0
$855$	−4.16250e6	−0.194733
$856$	0	0
$857$	1.54050e7	0.716490	0.358245	−	0.933628i	$-0.383375\pi$
$857$	1.54050e7	0.716490	0.358245	+	0.933628i	$0.383375\pi$
$858$	0	0
$859$	−1.43526e7	−0.663664	−0.331832	−	0.943338i	$-0.607667\pi$
$859$	−1.43526e7	−0.663664	−0.331832	+	0.943338i	$0.607667\pi$
$860$	0	0
$861$	−5.12698e6	−0.235697
$862$	0	0
$863$	−1.33278e7	−0.609158	−0.304579	−	0.952487i	$-0.598516\pi$
$863$	−1.33278e7	−0.609158	−0.304579	+	0.952487i	$0.598516\pi$
$864$	0	0
$865$	1.08464e7	0.492882
$866$	0	0
$867$	3.28938e7	1.48616
$868$	0	0
$869$	−6.10368e6	−0.274184
$870$	0	0
$871$	1.71037e6	0.0763913
$872$	0	0
$873$	−2.61012e7	−1.15911
$874$	0	0
$875$	2.68750e6	0.118667
$876$	0	0
$877$	3.24846e7	1.42620	0.713098	−	0.701065i	$-0.247292\pi$
$877$	3.24846e7	1.42620	0.713098	+	0.701065i	$0.247292\pi$
$878$	0	0
$879$	7.74014e6	0.337891
$880$	0	0
$881$	1.54600e7	0.671073	0.335537	−	0.942027i	$-0.391082\pi$
$881$	1.54600e7	0.671073	0.335537	+	0.942027i	$0.391082\pi$
$882$	0	0
$883$	1.69478e6	0.0731494	0.0365747	−	0.999331i	$-0.488355\pi$
$883$	1.69478e6	0.0731494	0.0365747	+	0.999331i	$0.488355\pi$
$884$	0	0
$885$	4.78800e6	0.205492
$886$	0	0
$887$	2.87257e6	0.122592	0.0612960	−	0.998120i	$-0.480477\pi$
$887$	2.87257e6	0.122592	0.0612960	+	0.998120i	$0.480477\pi$
$888$	0	0
$889$	2.10617e7	0.893799
$890$	0	0
$891$	3.83843e6	0.161979
$892$	0	0
$893$	3.29400e6	0.138228
$894$	0	0
$895$	1.22295e7	0.510330
$896$	0	0
$897$	−8.09171e7	−3.35783
$898$	0	0
$899$	−5.06328e6	−0.208945
$900$	0	0
$901$	4.67665e6	0.191921
$902$	0	0
$903$	8.51359e7	3.47451
$904$	0	0
$905$	1.79866e7	0.730006
$906$	0	0
$907$	−3.95422e7	−1.59603	−0.798017	−	0.602635i	$-0.794118\pi$
$907$	−3.95422e7	−1.59603	−0.798017	+	0.602635i	$0.794118\pi$
$908$	0	0
$909$	4.72667e7	1.89734
$910$	0	0
$911$	−1.13178e7	−0.451819	−0.225909	−	0.974148i	$-0.572535\pi$
$911$	−1.13178e7	−0.451819	−0.225909	+	0.974148i	$0.572535\pi$
$912$	0	0
$913$	−6.80803e6	−0.270299
$914$	0	0
$915$	−9.97320e6	−0.393806
$916$	0	0
$917$	4.16082e7	1.63401
$918$	0	0
$919$	−8.51348e6	−0.332520	−0.166260	−	0.986082i	$-0.553169\pi$
$919$	−8.51348e6	−0.332520	−0.166260	+	0.986082i	$0.553169\pi$
$920$	0	0
$921$	3.46322e7	1.34534
$922$	0	0
$923$	−2.32035e7	−0.896497
$924$	0	0
$925$	−7.02625e6	−0.270003
$926$	0	0
$927$	145188.	0.00554921
$928$	0	0
$929$	−7.54587e6	−0.286860	−0.143430	−	0.989660i	$-0.545813\pi$
$929$	−7.54587e6	−0.286860	−0.143430	+	0.989660i	$0.545813\pi$
$930$	0	0
$931$	−6.38850e6	−0.241560
$932$	0	0
$933$	4.11149e6	0.154630
$934$	0	0
$935$	732600.	0.0274055
$936$	0	0
$937$	−1.84500e7	−0.686512	−0.343256	−	0.939242i	$-0.611530\pi$
$937$	−1.84500e7	−0.686512	−0.343256	+	0.939242i	$0.611530\pi$
$938$	0	0
$939$	2.46453e7	0.912157
$940$	0	0
$941$	6.75046e6	0.248519	0.124259	−	0.992250i	$-0.460344\pi$
$941$	6.75046e6	0.248519	0.124259	+	0.992250i	$0.460344\pi$
$942$	0	0
$943$	−4.42649e6	−0.162099
$944$	0	0
$945$	−9.28800e6	−0.338332
$946$	0	0
$947$	−6.45677e6	−0.233959	−0.116980	−	0.993134i	$-0.537321\pi$
$947$	−6.45677e6	−0.233959	−0.116980	+	0.993134i	$0.537321\pi$
$948$	0	0
$949$	−1.93684e7	−0.698117
$950$	0	0
$951$	−1.80710e7	−0.647934
$952$	0	0
$953$	−3.96648e7	−1.41473	−0.707364	−	0.706849i	$-0.750116\pi$
$953$	−3.96648e7	−1.41473	−0.707364	+	0.706849i	$0.750116\pi$
$954$	0	0
$955$	4.64820e6	0.164921
$956$	0	0
$957$	6.93792e6	0.244878
$958$	0	0
$959$	4.76609e7	1.67346
$960$	0	0
$961$	−2.32838e7	−0.813290
$962$	0	0
$963$	−7.75783e7	−2.69572
$964$	0	0
$965$	−1.47851e7	−0.511102
$966$	0	0
$967$	3.43015e7	1.17963	0.589816	−	0.807538i	$-0.299200\pi$
$967$	3.43015e7	1.17963	0.589816	+	0.807538i	$0.299200\pi$
$968$	0	0
$969$	−2.66400e6	−0.0911433
$970$	0	0
$971$	5.77115e6	0.196433	0.0982164	−	0.995165i	$-0.468686\pi$
$971$	5.77115e6	0.196433	0.0982164	+	0.995165i	$0.468686\pi$
$972$	0	0
$973$	3.32889e7	1.12724
$974$	0	0
$975$	1.41900e7	0.478047
$976$	0	0
$977$	7.08746e6	0.237549	0.118775	−	0.992921i	$-0.462103\pi$
$977$	7.08746e6	0.237549	0.118775	+	0.992921i	$0.462103\pi$
$978$	0	0
$979$	1.45609e7	0.485548
$980$	0	0
$981$	−5.82250e7	−1.93169
$982$	0	0
$983$	−4.59362e7	−1.51625	−0.758126	−	0.652108i	$-0.773885\pi$
$983$	−4.59362e7	−1.51625	−0.758126	+	0.652108i	$0.773885\pi$
$984$	0	0
$985$	1.12370e7	0.369027
$986$	0	0
$987$	2.71953e7	0.888588
$988$	0	0
$989$	7.35039e7	2.38957
$990$	0	0
$991$	4.50298e7	1.45652	0.728260	−	0.685301i	$-0.240329\pi$
$991$	4.50298e7	1.45652	0.728260	+	0.685301i	$0.240329\pi$
$992$	0	0
$993$	4.78592e7	1.54025
$994$	0	0
$995$	−3.92900e6	−0.125813
$996$	0	0
$997$	−2.37364e7	−0.756271	−0.378136	−	0.925750i	$-0.623435\pi$
$997$	−2.37364e7	−0.756271	−0.378136	+	0.925750i	$0.623435\pi$
$998$	0	0
$999$	2.42827e7	0.769810

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	80.6.a.a.1.1		1
3.2	odd	2		720.6.a.j.1.1		1
4.3	odd	2		10.6.a.b.1.1	✓	1
5.2	odd	4		400.6.c.b.49.2		2
5.3	odd	4		400.6.c.b.49.1		2
5.4	even	2		400.6.a.n.1.1		1
8.3	odd	2		320.6.a.b.1.1		1
8.5	even	2		320.6.a.o.1.1		1
12.11	even	2		90.6.a.d.1.1		1
20.3	even	4		50.6.b.a.49.2		2
20.7	even	4		50.6.b.a.49.1		2
20.19	odd	2		50.6.a.d.1.1		1
28.27	even	2		490.6.a.a.1.1		1
60.23	odd	4		450.6.c.h.199.1		2
60.47	odd	4		450.6.c.h.199.2		2
60.59	even	2		450.6.a.l.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.6.a.b.1.1	✓	1	4.3	odd	2
50.6.a.d.1.1		1	20.19	odd	2
50.6.b.a.49.1		2	20.7	even	4
50.6.b.a.49.2		2	20.3	even	4
80.6.a.a.1.1		1	1.1	even	1	trivial
90.6.a.d.1.1		1	12.11	even	2
320.6.a.b.1.1		1	8.3	odd	2
320.6.a.o.1.1		1	8.5	even	2
400.6.a.n.1.1		1	5.4	even	2
400.6.c.b.49.1		2	5.3	odd	4
400.6.c.b.49.2		2	5.2	odd	4
450.6.a.l.1.1		1	60.59	even	2
450.6.c.h.199.1		2	60.23	odd	4
450.6.c.h.199.2		2	60.47	odd	4
490.6.a.a.1.1		1	28.27	even	2
720.6.a.j.1.1		1	3.2	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:	\( N \)	\(=\)	\( 80 = 2^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	80.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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