LMFDB - Embedded newform 40.4.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [40,4,Mod(1,40)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(40, 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("40.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 40 = 2^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	40.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:	$2.36007640023$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	40.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-6.00000 q^{3} -5.00000 q^{5} -34.0000 q^{7} +9.00000 q^{9} +O(q^{10})$

$q-6.00000 q^{3} -5.00000 q^{5} -34.0000 q^{7} +9.00000 q^{9} +16.0000 q^{11} +58.0000 q^{13} +30.0000 q^{15} -70.0000 q^{17} +4.00000 q^{19} +204.000 q^{21} -134.000 q^{23} +25.0000 q^{25} +108.000 q^{27} -242.000 q^{29} +100.000 q^{31} -96.0000 q^{33} +170.000 q^{35} -438.000 q^{37} -348.000 q^{39} -138.000 q^{41} +178.000 q^{43} -45.0000 q^{45} +22.0000 q^{47} +813.000 q^{49} +420.000 q^{51} +162.000 q^{53} -80.0000 q^{55} -24.0000 q^{57} -268.000 q^{59} +250.000 q^{61} -306.000 q^{63} -290.000 q^{65} +422.000 q^{67} +804.000 q^{69} -852.000 q^{71} +306.000 q^{73} -150.000 q^{75} -544.000 q^{77} -456.000 q^{79} -891.000 q^{81} +434.000 q^{83} +350.000 q^{85} +1452.00 q^{87} -726.000 q^{89} -1972.00 q^{91} -600.000 q^{93} -20.0000 q^{95} +1378.00 q^{97} +144.000 q^{99} +O(q^{100})$

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
$3$	−6.00000	−1.15470	−0.577350	−	0.816497i	$-0.695913\pi$
$3$	−6.00000	−1.15470	−0.577350	+	0.816497i	$0.695913\pi$
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−34.0000	−1.83583	−0.917914	−	0.396780i	$-0.870128\pi$
$7$	−34.0000	−1.83583	−0.917914	+	0.396780i	$0.870128\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	16.0000	0.438562	0.219281	−	0.975662i	$-0.429629\pi$
$11$	16.0000	0.438562	0.219281	+	0.975662i	$0.429629\pi$
$12$	0	0
$13$	58.0000	1.23741	0.618704	−	0.785624i	$-0.287658\pi$
$13$	58.0000	1.23741	0.618704	+	0.785624i	$0.287658\pi$
$14$	0	0
$15$	30.0000	0.516398
$16$	0	0
$17$	−70.0000	−0.998676	−0.499338	−	0.866407i	$-0.666423\pi$
$17$	−70.0000	−0.998676	−0.499338	+	0.866407i	$0.666423\pi$
$18$	0	0
$19$	4.00000	0.0482980	0.0241490	−	0.999708i	$-0.492312\pi$
$19$	4.00000	0.0482980	0.0241490	+	0.999708i	$0.492312\pi$
$20$	0	0
$21$	204.000	2.11983
$22$	0	0
$23$	−134.000	−1.21482	−0.607412	−	0.794387i	$-0.707792\pi$
$23$	−134.000	−1.21482	−0.607412	+	0.794387i	$0.707792\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	108.000	0.769800
$28$	0	0
$29$	−242.000	−1.54960	−0.774798	−	0.632209i	$-0.782148\pi$
$29$	−242.000	−1.54960	−0.774798	+	0.632209i	$0.782148\pi$
$30$	0	0
$31$	100.000	0.579372	0.289686	−	0.957122i	$-0.406449\pi$
$31$	100.000	0.579372	0.289686	+	0.957122i	$0.406449\pi$
$32$	0	0
$33$	−96.0000	−0.506408
$34$	0	0
$35$	170.000	0.821007
$36$	0	0
$37$	−438.000	−1.94613	−0.973064	−	0.230534i	$-0.925953\pi$
$37$	−438.000	−1.94613	−0.973064	+	0.230534i	$0.925953\pi$
$38$	0	0
$39$	−348.000	−1.42884
$40$	0	0
$41$	−138.000	−0.525658	−0.262829	−	0.964842i	$-0.584656\pi$
$41$	−138.000	−0.525658	−0.262829	+	0.964842i	$0.584656\pi$
$42$	0	0
$43$	178.000	0.631273	0.315637	−	0.948880i	$-0.397782\pi$
$43$	178.000	0.631273	0.315637	+	0.948880i	$0.397782\pi$
$44$	0	0
$45$	−45.0000	−0.149071
$46$	0	0
$47$	22.0000	0.0682772	0.0341386	−	0.999417i	$-0.489131\pi$
$47$	22.0000	0.0682772	0.0341386	+	0.999417i	$0.489131\pi$
$48$	0	0
$49$	813.000	2.37026
$50$	0	0
$51$	420.000	1.15317
$52$	0	0
$53$	162.000	0.419857	0.209928	−	0.977717i	$-0.432677\pi$
$53$	162.000	0.419857	0.209928	+	0.977717i	$0.432677\pi$
$54$	0	0
$55$	−80.0000	−0.196131
$56$	0	0
$57$	−24.0000	−0.0557698
$58$	0	0
$59$	−268.000	−0.591367	−0.295683	−	0.955286i	$-0.595547\pi$
$59$	−268.000	−0.591367	−0.295683	+	0.955286i	$0.595547\pi$
$60$	0	0
$61$	250.000	0.524741	0.262371	−	0.964967i	$-0.415496\pi$
$61$	250.000	0.524741	0.262371	+	0.964967i	$0.415496\pi$
$62$	0	0
$63$	−306.000	−0.611942
$64$	0	0
$65$	−290.000	−0.553386
$66$	0	0
$67$	422.000	0.769485	0.384743	−	0.923024i	$-0.374290\pi$
$67$	422.000	0.769485	0.384743	+	0.923024i	$0.374290\pi$
$68$	0	0
$69$	804.000	1.40276
$70$	0	0
$71$	−852.000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−852.000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	0	0
$73$	306.000	0.490611	0.245305	−	0.969446i	$-0.421112\pi$
$73$	306.000	0.490611	0.245305	+	0.969446i	$0.421112\pi$
$74$	0	0
$75$	−150.000	−0.230940
$76$	0	0
$77$	−544.000	−0.805124
$78$	0	0
$79$	−456.000	−0.649418	−0.324709	−	0.945814i	$-0.605266\pi$
$79$	−456.000	−0.649418	−0.324709	+	0.945814i	$0.605266\pi$
$80$	0	0
$81$	−891.000	−1.22222
$82$	0	0
$83$	434.000	0.573948	0.286974	−	0.957938i	$-0.407351\pi$
$83$	434.000	0.573948	0.286974	+	0.957938i	$0.407351\pi$
$84$	0	0
$85$	350.000	0.446622
$86$	0	0
$87$	1452.00	1.78932
$88$	0	0
$89$	−726.000	−0.864672	−0.432336	−	0.901712i	$-0.642311\pi$
$89$	−726.000	−0.864672	−0.432336	+	0.901712i	$0.642311\pi$
$90$	0	0
$91$	−1972.00	−2.27167
$92$	0	0
$93$	−600.000	−0.669001
$94$	0	0
$95$	−20.0000	−0.0215995
$96$	0	0
$97$	1378.00	1.44242	0.721210	−	0.692717i	$-0.243586\pi$
$97$	1378.00	1.44242	0.721210	+	0.692717i	$0.243586\pi$
$98$	0	0
$99$	144.000	0.146187
$100$	0	0
$101$	126.000	0.124133	0.0620667	−	0.998072i	$-0.480231\pi$
$101$	126.000	0.124133	0.0620667	+	0.998072i	$0.480231\pi$
$102$	0	0
$103$	−1262.00	−1.20727	−0.603634	−	0.797262i	$-0.706281\pi$
$103$	−1262.00	−1.20727	−0.603634	+	0.797262i	$0.706281\pi$
$104$	0	0
$105$	−1020.00	−0.948017
$106$	0	0
$107$	510.000	0.460781	0.230390	−	0.973098i	$-0.426000\pi$
$107$	510.000	0.460781	0.230390	+	0.973098i	$0.426000\pi$
$108$	0	0
$109$	26.0000	0.0228472	0.0114236	−	0.999935i	$-0.496364\pi$
$109$	26.0000	0.0228472	0.0114236	+	0.999935i	$0.496364\pi$
$110$	0	0
$111$	2628.00	2.24720
$112$	0	0
$113$	1242.00	1.03396	0.516980	−	0.855997i	$-0.327056\pi$
$113$	1242.00	1.03396	0.516980	+	0.855997i	$0.327056\pi$
$114$	0	0
$115$	670.000	0.543285
$116$	0	0
$117$	522.000	0.412469
$118$	0	0
$119$	2380.00	1.83340
$120$	0	0
$121$	−1075.00	−0.807663
$122$	0	0
$123$	828.000	0.606978
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−978.000	−0.683334	−0.341667	−	0.939821i	$-0.610992\pi$
$127$	−978.000	−0.683334	−0.341667	+	0.939821i	$0.610992\pi$
$128$	0	0
$129$	−1068.00	−0.728931
$130$	0	0
$131$	−912.000	−0.608258	−0.304129	−	0.952631i	$-0.598365\pi$
$131$	−912.000	−0.608258	−0.304129	+	0.952631i	$0.598365\pi$
$132$	0	0
$133$	−136.000	−0.0886669
$134$	0	0
$135$	−540.000	−0.344265
$136$	0	0
$137$	−926.000	−0.577471	−0.288735	−	0.957409i	$-0.593235\pi$
$137$	−926.000	−0.577471	−0.288735	+	0.957409i	$0.593235\pi$
$138$	0	0
$139$	516.000	0.314867	0.157434	−	0.987530i	$-0.449678\pi$
$139$	516.000	0.314867	0.157434	+	0.987530i	$0.449678\pi$
$140$	0	0
$141$	−132.000	−0.0788398
$142$	0	0
$143$	928.000	0.542680
$144$	0	0
$145$	1210.00	0.693000
$146$	0	0
$147$	−4878.00	−2.73694
$148$	0	0
$149$	−958.000	−0.526728	−0.263364	−	0.964697i	$-0.584832\pi$
$149$	−958.000	−0.526728	−0.263364	+	0.964697i	$0.584832\pi$
$150$	0	0
$151$	332.000	0.178926	0.0894628	−	0.995990i	$-0.471485\pi$
$151$	332.000	0.178926	0.0894628	+	0.995990i	$0.471485\pi$
$152$	0	0
$153$	−630.000	−0.332892
$154$	0	0
$155$	−500.000	−0.259103
$156$	0	0
$157$	−1022.00	−0.519519	−0.259759	−	0.965673i	$-0.583643\pi$
$157$	−1022.00	−0.519519	−0.259759	+	0.965673i	$0.583643\pi$
$158$	0	0
$159$	−972.000	−0.484809
$160$	0	0
$161$	4556.00	2.23021
$162$	0	0
$163$	−926.000	−0.444969	−0.222484	−	0.974936i	$-0.571417\pi$
$163$	−926.000	−0.444969	−0.222484	+	0.974936i	$0.571417\pi$
$164$	0	0
$165$	480.000	0.226472
$166$	0	0
$167$	654.000	0.303042	0.151521	−	0.988454i	$-0.451583\pi$
$167$	654.000	0.303042	0.151521	+	0.988454i	$0.451583\pi$
$168$	0	0
$169$	1167.00	0.531179
$170$	0	0
$171$	36.0000	0.0160993
$172$	0	0
$173$	−1294.00	−0.568676	−0.284338	−	0.958724i	$-0.591774\pi$
$173$	−1294.00	−0.568676	−0.284338	+	0.958724i	$0.591774\pi$
$174$	0	0
$175$	−850.000	−0.367165
$176$	0	0
$177$	1608.00	0.682851
$178$	0	0
$179$	−2836.00	−1.18420	−0.592102	−	0.805863i	$-0.701702\pi$
$179$	−2836.00	−1.18420	−0.592102	+	0.805863i	$0.701702\pi$
$180$	0	0
$181$	1742.00	0.715369	0.357685	−	0.933842i	$-0.383566\pi$
$181$	1742.00	0.715369	0.357685	+	0.933842i	$0.383566\pi$
$182$	0	0
$183$	−1500.00	−0.605919
$184$	0	0
$185$	2190.00	0.870335
$186$	0	0
$187$	−1120.00	−0.437981
$188$	0	0
$189$	−3672.00	−1.41322
$190$	0	0
$191$	4460.00	1.68960	0.844802	−	0.535079i	$-0.179718\pi$
$191$	4460.00	1.68960	0.844802	+	0.535079i	$0.179718\pi$
$192$	0	0
$193$	−3782.00	−1.41054	−0.705270	−	0.708939i	$-0.749174\pi$
$193$	−3782.00	−1.41054	−0.705270	+	0.708939i	$0.749174\pi$
$194$	0	0
$195$	1740.00	0.638995
$196$	0	0
$197$	4474.00	1.61807	0.809034	−	0.587762i	$-0.199991\pi$
$197$	4474.00	1.61807	0.809034	+	0.587762i	$0.199991\pi$
$198$	0	0
$199$	3608.00	1.28525	0.642624	−	0.766182i	$-0.277846\pi$
$199$	3608.00	1.28525	0.642624	+	0.766182i	$0.277846\pi$
$200$	0	0
$201$	−2532.00	−0.888525
$202$	0	0
$203$	8228.00	2.84479
$204$	0	0
$205$	690.000	0.235081
$206$	0	0
$207$	−1206.00	−0.404941
$208$	0	0
$209$	64.0000	0.0211817
$210$	0	0
$211$	−256.000	−0.0835250	−0.0417625	−	0.999128i	$-0.513297\pi$
$211$	−256.000	−0.0835250	−0.0417625	+	0.999128i	$0.513297\pi$
$212$	0	0
$213$	5112.00	1.64445
$214$	0	0
$215$	−890.000	−0.282314
$216$	0	0
$217$	−3400.00	−1.06363
$218$	0	0
$219$	−1836.00	−0.566509
$220$	0	0
$221$	−4060.00	−1.23577
$222$	0	0
$223$	−5158.00	−1.54890	−0.774451	−	0.632634i	$-0.781974\pi$
$223$	−5158.00	−1.54890	−0.774451	+	0.632634i	$0.781974\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	0	0
$227$	−2226.00	−0.650858	−0.325429	−	0.945566i	$-0.605509\pi$
$227$	−2226.00	−0.650858	−0.325429	+	0.945566i	$0.605509\pi$
$228$	0	0
$229$	2086.00	0.601951	0.300975	−	0.953632i	$-0.402688\pi$
$229$	2086.00	0.601951	0.300975	+	0.953632i	$0.402688\pi$
$230$	0	0
$231$	3264.00	0.929677
$232$	0	0
$233$	−5718.00	−1.60772	−0.803860	−	0.594819i	$-0.797224\pi$
$233$	−5718.00	−1.60772	−0.803860	+	0.594819i	$0.797224\pi$
$234$	0	0
$235$	−110.000	−0.0305345
$236$	0	0
$237$	2736.00	0.749883
$238$	0	0
$239$	−3624.00	−0.980825	−0.490412	−	0.871491i	$-0.663154\pi$
$239$	−3624.00	−0.980825	−0.490412	+	0.871491i	$0.663154\pi$
$240$	0	0
$241$	−82.0000	−0.0219174	−0.0109587	−	0.999940i	$-0.503488\pi$
$241$	−82.0000	−0.0219174	−0.0109587	+	0.999940i	$0.503488\pi$
$242$	0	0
$243$	2430.00	0.641500
$244$	0	0
$245$	−4065.00	−1.06001
$246$	0	0
$247$	232.000	0.0597644
$248$	0	0
$249$	−2604.00	−0.662738
$250$	0	0
$251$	−5040.00	−1.26742	−0.633709	−	0.773571i	$-0.718468\pi$
$251$	−5040.00	−1.26742	−0.633709	+	0.773571i	$0.718468\pi$
$252$	0	0
$253$	−2144.00	−0.532775
$254$	0	0
$255$	−2100.00	−0.515714
$256$	0	0
$257$	−2310.00	−0.560676	−0.280338	−	0.959901i	$-0.590447\pi$
$257$	−2310.00	−0.560676	−0.280338	+	0.959901i	$0.590447\pi$
$258$	0	0
$259$	14892.0	3.57276
$260$	0	0
$261$	−2178.00	−0.516532
$262$	0	0
$263$	−4110.00	−0.963625	−0.481813	−	0.876274i	$-0.660021\pi$
$263$	−4110.00	−0.963625	−0.481813	+	0.876274i	$0.660021\pi$
$264$	0	0
$265$	−810.000	−0.187766
$266$	0	0
$267$	4356.00	0.998438
$268$	0	0
$269$	746.000	0.169087	0.0845435	−	0.996420i	$-0.473057\pi$
$269$	746.000	0.169087	0.0845435	+	0.996420i	$0.473057\pi$
$270$	0	0
$271$	−4596.00	−1.03021	−0.515105	−	0.857127i	$-0.672247\pi$
$271$	−4596.00	−1.03021	−0.515105	+	0.857127i	$0.672247\pi$
$272$	0	0
$273$	11832.0	2.62310
$274$	0	0
$275$	400.000	0.0877124
$276$	0	0
$277$	−2206.00	−0.478504	−0.239252	−	0.970957i	$-0.576902\pi$
$277$	−2206.00	−0.478504	−0.239252	+	0.970957i	$0.576902\pi$
$278$	0	0
$279$	900.000	0.193124
$280$	0	0
$281$	8278.00	1.75738	0.878691	−	0.477392i	$-0.158418\pi$
$281$	8278.00	1.75738	0.878691	+	0.477392i	$0.158418\pi$
$282$	0	0
$283$	1178.00	0.247438	0.123719	−	0.992317i	$-0.460518\pi$
$283$	1178.00	0.247438	0.123719	+	0.992317i	$0.460518\pi$
$284$	0	0
$285$	120.000	0.0249410
$286$	0	0
$287$	4692.00	0.965017
$288$	0	0
$289$	−13.0000	−0.00264604
$290$	0	0
$291$	−8268.00	−1.66556
$292$	0	0
$293$	106.000	0.0211351	0.0105676	−	0.999944i	$-0.496636\pi$
$293$	106.000	0.0211351	0.0105676	+	0.999944i	$0.496636\pi$
$294$	0	0
$295$	1340.00	0.264467
$296$	0	0
$297$	1728.00	0.337605
$298$	0	0
$299$	−7772.00	−1.50323
$300$	0	0
$301$	−6052.00	−1.15891
$302$	0	0
$303$	−756.000	−0.143337
$304$	0	0
$305$	−1250.00	−0.234671
$306$	0	0
$307$	8134.00	1.51216	0.756078	−	0.654482i	$-0.227113\pi$
$307$	8134.00	1.51216	0.756078	+	0.654482i	$0.227113\pi$
$308$	0	0
$309$	7572.00	1.39403
$310$	0	0
$311$	−4396.00	−0.801525	−0.400763	−	0.916182i	$-0.631255\pi$
$311$	−4396.00	−0.801525	−0.400763	+	0.916182i	$0.631255\pi$
$312$	0	0
$313$	4826.00	0.871507	0.435753	−	0.900066i	$-0.356482\pi$
$313$	4826.00	0.871507	0.435753	+	0.900066i	$0.356482\pi$
$314$	0	0
$315$	1530.00	0.273669
$316$	0	0
$317$	7026.00	1.24486	0.622428	−	0.782677i	$-0.286146\pi$
$317$	7026.00	1.24486	0.622428	+	0.782677i	$0.286146\pi$
$318$	0	0
$319$	−3872.00	−0.679594
$320$	0	0
$321$	−3060.00	−0.532064
$322$	0	0
$323$	−280.000	−0.0482341
$324$	0	0
$325$	1450.00	0.247482
$326$	0	0
$327$	−156.000	−0.0263817
$328$	0	0
$329$	−748.000	−0.125345
$330$	0	0
$331$	8808.00	1.46263	0.731316	−	0.682038i	$-0.238906\pi$
$331$	8808.00	1.46263	0.731316	+	0.682038i	$0.238906\pi$
$332$	0	0
$333$	−3942.00	−0.648710
$334$	0	0
$335$	−2110.00	−0.344124
$336$	0	0
$337$	5602.00	0.905520	0.452760	−	0.891632i	$-0.350439\pi$
$337$	5602.00	0.905520	0.452760	+	0.891632i	$0.350439\pi$
$338$	0	0
$339$	−7452.00	−1.19391
$340$	0	0
$341$	1600.00	0.254090
$342$	0	0
$343$	−15980.0	−2.51557
$344$	0	0
$345$	−4020.00	−0.627332
$346$	0	0
$347$	−6634.00	−1.02632	−0.513158	−	0.858294i	$-0.671525\pi$
$347$	−6634.00	−1.02632	−0.513158	+	0.858294i	$0.671525\pi$
$348$	0	0
$349$	3198.00	0.490501	0.245251	−	0.969460i	$-0.421130\pi$
$349$	3198.00	0.490501	0.245251	+	0.969460i	$0.421130\pi$
$350$	0	0
$351$	6264.00	0.952557
$352$	0	0
$353$	−5230.00	−0.788569	−0.394284	−	0.918988i	$-0.629008\pi$
$353$	−5230.00	−0.788569	−0.394284	+	0.918988i	$0.629008\pi$
$354$	0	0
$355$	4260.00	0.636894
$356$	0	0
$357$	−14280.0	−2.11702
$358$	0	0
$359$	−312.000	−0.0458683	−0.0229342	−	0.999737i	$-0.507301\pi$
$359$	−312.000	−0.0458683	−0.0229342	+	0.999737i	$0.507301\pi$
$360$	0	0
$361$	−6843.00	−0.997667
$362$	0	0
$363$	6450.00	0.932609
$364$	0	0
$365$	−1530.00	−0.219408
$366$	0	0
$367$	10790.0	1.53470	0.767348	−	0.641231i	$-0.221576\pi$
$367$	10790.0	1.53470	0.767348	+	0.641231i	$0.221576\pi$
$368$	0	0
$369$	−1242.00	−0.175219
$370$	0	0
$371$	−5508.00	−0.770785
$372$	0	0
$373$	−4190.00	−0.581635	−0.290818	−	0.956778i	$-0.593927\pi$
$373$	−4190.00	−0.581635	−0.290818	+	0.956778i	$0.593927\pi$
$374$	0	0
$375$	750.000	0.103280
$376$	0	0
$377$	−14036.0	−1.91748
$378$	0	0
$379$	−6980.00	−0.946012	−0.473006	−	0.881059i	$-0.656831\pi$
$379$	−6980.00	−0.946012	−0.473006	+	0.881059i	$0.656831\pi$
$380$	0	0
$381$	5868.00	0.789047
$382$	0	0
$383$	13962.0	1.86273	0.931364	−	0.364089i	$-0.118620\pi$
$383$	13962.0	1.86273	0.931364	+	0.364089i	$0.118620\pi$
$384$	0	0
$385$	2720.00	0.360062
$386$	0	0
$387$	1602.00	0.210424
$388$	0	0
$389$	3810.00	0.496593	0.248296	−	0.968684i	$-0.420129\pi$
$389$	3810.00	0.496593	0.248296	+	0.968684i	$0.420129\pi$
$390$	0	0
$391$	9380.00	1.21321
$392$	0	0
$393$	5472.00	0.702356
$394$	0	0
$395$	2280.00	0.290428
$396$	0	0
$397$	−9158.00	−1.15775	−0.578875	−	0.815416i	$-0.696508\pi$
$397$	−9158.00	−1.15775	−0.578875	+	0.815416i	$0.696508\pi$
$398$	0	0
$399$	816.000	0.102384
$400$	0	0
$401$	4866.00	0.605976	0.302988	−	0.952994i	$-0.402016\pi$
$401$	4866.00	0.605976	0.302988	+	0.952994i	$0.402016\pi$
$402$	0	0
$403$	5800.00	0.716920
$404$	0	0
$405$	4455.00	0.546594
$406$	0	0
$407$	−7008.00	−0.853498
$408$	0	0
$409$	13486.0	1.63042	0.815208	−	0.579169i	$-0.196623\pi$
$409$	13486.0	1.63042	0.815208	+	0.579169i	$0.196623\pi$
$410$	0	0
$411$	5556.00	0.666806
$412$	0	0
$413$	9112.00	1.08565
$414$	0	0
$415$	−2170.00	−0.256677
$416$	0	0
$417$	−3096.00	−0.363577
$418$	0	0
$419$	5628.00	0.656195	0.328098	−	0.944644i	$-0.393593\pi$
$419$	5628.00	0.656195	0.328098	+	0.944644i	$0.393593\pi$
$420$	0	0
$421$	7938.00	0.918942	0.459471	−	0.888193i	$-0.348039\pi$
$421$	7938.00	0.918942	0.459471	+	0.888193i	$0.348039\pi$
$422$	0	0
$423$	198.000	0.0227591
$424$	0	0
$425$	−1750.00	−0.199735
$426$	0	0
$427$	−8500.00	−0.963334
$428$	0	0
$429$	−5568.00	−0.626633
$430$	0	0
$431$	1916.00	0.214131	0.107066	−	0.994252i	$-0.465855\pi$
$431$	1916.00	0.214131	0.107066	+	0.994252i	$0.465855\pi$
$432$	0	0
$433$	−16510.0	−1.83238	−0.916189	−	0.400746i	$-0.868751\pi$
$433$	−16510.0	−1.83238	−0.916189	+	0.400746i	$0.868751\pi$
$434$	0	0
$435$	−7260.00	−0.800208
$436$	0	0
$437$	−536.000	−0.0586736
$438$	0	0
$439$	−1256.00	−0.136550	−0.0682752	−	0.997667i	$-0.521750\pi$
$439$	−1256.00	−0.136550	−0.0682752	+	0.997667i	$0.521750\pi$
$440$	0	0
$441$	7317.00	0.790087
$442$	0	0
$443$	−12222.0	−1.31080	−0.655400	−	0.755282i	$-0.727500\pi$
$443$	−12222.0	−1.31080	−0.655400	+	0.755282i	$0.727500\pi$
$444$	0	0
$445$	3630.00	0.386693
$446$	0	0
$447$	5748.00	0.608213
$448$	0	0
$449$	−5946.00	−0.624965	−0.312482	−	0.949924i	$-0.601160\pi$
$449$	−5946.00	−0.624965	−0.312482	+	0.949924i	$0.601160\pi$
$450$	0	0
$451$	−2208.00	−0.230534
$452$	0	0
$453$	−1992.00	−0.206606
$454$	0	0
$455$	9860.00	1.01592
$456$	0	0
$457$	1258.00	0.128768	0.0643838	−	0.997925i	$-0.479492\pi$
$457$	1258.00	0.128768	0.0643838	+	0.997925i	$0.479492\pi$
$458$	0	0
$459$	−7560.00	−0.768781
$460$	0	0
$461$	16422.0	1.65911	0.829554	−	0.558426i	$-0.188595\pi$
$461$	16422.0	1.65911	0.829554	+	0.558426i	$0.188595\pi$
$462$	0	0
$463$	2658.00	0.266799	0.133399	−	0.991062i	$-0.457411\pi$
$463$	2658.00	0.266799	0.133399	+	0.991062i	$0.457411\pi$
$464$	0	0
$465$	3000.00	0.299186
$466$	0	0
$467$	3686.00	0.365241	0.182621	−	0.983183i	$-0.441542\pi$
$467$	3686.00	0.365241	0.182621	+	0.983183i	$0.441542\pi$
$468$	0	0
$469$	−14348.0	−1.41264
$470$	0	0
$471$	6132.00	0.599889
$472$	0	0
$473$	2848.00	0.276852
$474$	0	0
$475$	100.000	0.00965961
$476$	0	0
$477$	1458.00	0.139952
$478$	0	0
$479$	88.0000	0.00839420	0.00419710	−	0.999991i	$-0.498664\pi$
$479$	88.0000	0.00839420	0.00419710	+	0.999991i	$0.498664\pi$
$480$	0	0
$481$	−25404.0	−2.40816
$482$	0	0
$483$	−27336.0	−2.57522
$484$	0	0
$485$	−6890.00	−0.645070
$486$	0	0
$487$	−14714.0	−1.36911	−0.684553	−	0.728963i	$-0.740003\pi$
$487$	−14714.0	−1.36911	−0.684553	+	0.728963i	$0.740003\pi$
$488$	0	0
$489$	5556.00	0.513806
$490$	0	0
$491$	−7344.00	−0.675010	−0.337505	−	0.941324i	$-0.609583\pi$
$491$	−7344.00	−0.675010	−0.337505	+	0.941324i	$0.609583\pi$
$492$	0	0
$493$	16940.0	1.54754
$494$	0	0
$495$	−720.000	−0.0653770
$496$	0	0
$497$	28968.0	2.61447
$498$	0	0
$499$	1604.00	0.143898	0.0719488	−	0.997408i	$-0.477078\pi$
$499$	1604.00	0.143898	0.0719488	+	0.997408i	$0.477078\pi$
$500$	0	0
$501$	−3924.00	−0.349923
$502$	0	0
$503$	14802.0	1.31210	0.656052	−	0.754715i	$-0.272225\pi$
$503$	14802.0	1.31210	0.656052	+	0.754715i	$0.272225\pi$
$504$	0	0
$505$	−630.000	−0.0555141
$506$	0	0
$507$	−7002.00	−0.613353
$508$	0	0
$509$	−22514.0	−1.96054	−0.980271	−	0.197660i	$-0.936666\pi$
$509$	−22514.0	−1.96054	−0.980271	+	0.197660i	$0.936666\pi$
$510$	0	0
$511$	−10404.0	−0.900677
$512$	0	0
$513$	432.000	0.0371799
$514$	0	0
$515$	6310.00	0.539906
$516$	0	0
$517$	352.000	0.0299438
$518$	0	0
$519$	7764.00	0.656651
$520$	0	0
$521$	−6710.00	−0.564243	−0.282121	−	0.959379i	$-0.591038\pi$
$521$	−6710.00	−0.564243	−0.282121	+	0.959379i	$0.591038\pi$
$522$	0	0
$523$	7930.00	0.663011	0.331505	−	0.943453i	$-0.392443\pi$
$523$	7930.00	0.663011	0.331505	+	0.943453i	$0.392443\pi$
$524$	0	0
$525$	5100.00	0.423966
$526$	0	0
$527$	−7000.00	−0.578605
$528$	0	0
$529$	5789.00	0.475795
$530$	0	0
$531$	−2412.00	−0.197122
$532$	0	0
$533$	−8004.00	−0.650454
$534$	0	0
$535$	−2550.00	−0.206068
$536$	0	0
$537$	17016.0	1.36740
$538$	0	0
$539$	13008.0	1.03951
$540$	0	0
$541$	4918.00	0.390834	0.195417	−	0.980720i	$-0.437394\pi$
$541$	4918.00	0.390834	0.195417	+	0.980720i	$0.437394\pi$
$542$	0	0
$543$	−10452.0	−0.826037
$544$	0	0
$545$	−130.000	−0.0102176
$546$	0	0
$547$	−3922.00	−0.306568	−0.153284	−	0.988182i	$-0.548985\pi$
$547$	−3922.00	−0.306568	−0.153284	+	0.988182i	$0.548985\pi$
$548$	0	0
$549$	2250.00	0.174914
$550$	0	0
$551$	−968.000	−0.0748424
$552$	0	0
$553$	15504.0	1.19222
$554$	0	0
$555$	−13140.0	−1.00498
$556$	0	0
$557$	17786.0	1.35299	0.676496	−	0.736446i	$-0.263497\pi$
$557$	17786.0	1.35299	0.676496	+	0.736446i	$0.263497\pi$
$558$	0	0
$559$	10324.0	0.781143
$560$	0	0
$561$	6720.00	0.505737
$562$	0	0
$563$	20266.0	1.51707	0.758535	−	0.651633i	$-0.225916\pi$
$563$	20266.0	1.51707	0.758535	+	0.651633i	$0.225916\pi$
$564$	0	0
$565$	−6210.00	−0.462401
$566$	0	0
$567$	30294.0	2.24379
$568$	0	0
$569$	13358.0	0.984177	0.492088	−	0.870545i	$-0.336234\pi$
$569$	13358.0	0.984177	0.492088	+	0.870545i	$0.336234\pi$
$570$	0	0
$571$	16360.0	1.19903	0.599514	−	0.800364i	$-0.295361\pi$
$571$	16360.0	1.19903	0.599514	+	0.800364i	$0.295361\pi$
$572$	0	0
$573$	−26760.0	−1.95099
$574$	0	0
$575$	−3350.00	−0.242965
$576$	0	0
$577$	−15574.0	−1.12366	−0.561832	−	0.827251i	$-0.689903\pi$
$577$	−15574.0	−1.12366	−0.561832	+	0.827251i	$0.689903\pi$
$578$	0	0
$579$	22692.0	1.62875
$580$	0	0
$581$	−14756.0	−1.05367
$582$	0	0
$583$	2592.00	0.184133
$584$	0	0
$585$	−2610.00	−0.184462
$586$	0	0
$587$	6654.00	0.467870	0.233935	−	0.972252i	$-0.424840\pi$
$587$	6654.00	0.467870	0.233935	+	0.972252i	$0.424840\pi$
$588$	0	0
$589$	400.000	0.0279825
$590$	0	0
$591$	−26844.0	−1.86838
$592$	0	0
$593$	−17742.0	−1.22863	−0.614314	−	0.789062i	$-0.710567\pi$
$593$	−17742.0	−1.22863	−0.614314	+	0.789062i	$0.710567\pi$
$594$	0	0
$595$	−11900.0	−0.819920
$596$	0	0
$597$	−21648.0	−1.48408
$598$	0	0
$599$	15840.0	1.08048	0.540238	−	0.841512i	$-0.318334\pi$
$599$	15840.0	1.08048	0.540238	+	0.841512i	$0.318334\pi$
$600$	0	0
$601$	−3002.00	−0.203751	−0.101875	−	0.994797i	$-0.532484\pi$
$601$	−3002.00	−0.203751	−0.101875	+	0.994797i	$0.532484\pi$
$602$	0	0
$603$	3798.00	0.256495
$604$	0	0
$605$	5375.00	0.361198
$606$	0	0
$607$	−23610.0	−1.57875	−0.789374	−	0.613912i	$-0.789595\pi$
$607$	−23610.0	−1.57875	−0.789374	+	0.613912i	$0.789595\pi$
$608$	0	0
$609$	−49368.0	−3.28488
$610$	0	0
$611$	1276.00	0.0844868
$612$	0	0
$613$	23850.0	1.57144	0.785720	−	0.618583i	$-0.212293\pi$
$613$	23850.0	1.57144	0.785720	+	0.618583i	$0.212293\pi$
$614$	0	0
$615$	−4140.00	−0.271449
$616$	0	0
$617$	−5334.00	−0.348037	−0.174018	−	0.984742i	$-0.555675\pi$
$617$	−5334.00	−0.348037	−0.174018	+	0.984742i	$0.555675\pi$
$618$	0	0
$619$	−2164.00	−0.140515	−0.0702573	−	0.997529i	$-0.522382\pi$
$619$	−2164.00	−0.140515	−0.0702573	+	0.997529i	$0.522382\pi$
$620$	0	0
$621$	−14472.0	−0.935171
$622$	0	0
$623$	24684.0	1.58739
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−384.000	−0.0244585
$628$	0	0
$629$	30660.0	1.94355
$630$	0	0
$631$	−25220.0	−1.59111	−0.795557	−	0.605879i	$-0.792821\pi$
$631$	−25220.0	−1.59111	−0.795557	+	0.605879i	$0.792821\pi$
$632$	0	0
$633$	1536.00	0.0964463
$634$	0	0
$635$	4890.00	0.305596
$636$	0	0
$637$	47154.0	2.93298
$638$	0	0
$639$	−7668.00	−0.474713
$640$	0	0
$641$	−12306.0	−0.758280	−0.379140	−	0.925339i	$-0.623780\pi$
$641$	−12306.0	−0.758280	−0.379140	+	0.925339i	$0.623780\pi$
$642$	0	0
$643$	−27414.0	−1.68134	−0.840671	−	0.541547i	$-0.817839\pi$
$643$	−27414.0	−1.68134	−0.840671	+	0.541547i	$0.817839\pi$
$644$	0	0
$645$	5340.00	0.325988
$646$	0	0
$647$	−21834.0	−1.32671	−0.663356	−	0.748304i	$-0.730869\pi$
$647$	−21834.0	−1.32671	−0.663356	+	0.748304i	$0.730869\pi$
$648$	0	0
$649$	−4288.00	−0.259351
$650$	0	0
$651$	20400.0	1.22817
$652$	0	0
$653$	−23998.0	−1.43815	−0.719077	−	0.694931i	$-0.755435\pi$
$653$	−23998.0	−1.43815	−0.719077	+	0.694931i	$0.755435\pi$
$654$	0	0
$655$	4560.00	0.272021
$656$	0	0
$657$	2754.00	0.163537
$658$	0	0
$659$	−32004.0	−1.89180	−0.945902	−	0.324452i	$-0.894820\pi$
$659$	−32004.0	−1.89180	−0.945902	+	0.324452i	$0.894820\pi$
$660$	0	0
$661$	−8526.00	−0.501699	−0.250849	−	0.968026i	$-0.580710\pi$
$661$	−8526.00	−0.501699	−0.250849	+	0.968026i	$0.580710\pi$
$662$	0	0
$663$	24360.0	1.42694
$664$	0	0
$665$	680.000	0.0396530
$666$	0	0
$667$	32428.0	1.88248
$668$	0	0
$669$	30948.0	1.78852
$670$	0	0
$671$	4000.00	0.230132
$672$	0	0
$673$	8178.00	0.468408	0.234204	−	0.972187i	$-0.424752\pi$
$673$	8178.00	0.468408	0.234204	+	0.972187i	$0.424752\pi$
$674$	0	0
$675$	2700.00	0.153960
$676$	0	0
$677$	−16646.0	−0.944989	−0.472495	−	0.881334i	$-0.656646\pi$
$677$	−16646.0	−0.944989	−0.472495	+	0.881334i	$0.656646\pi$
$678$	0	0
$679$	−46852.0	−2.64803
$680$	0	0
$681$	13356.0	0.751546
$682$	0	0
$683$	−22446.0	−1.25750	−0.628750	−	0.777608i	$-0.716433\pi$
$683$	−22446.0	−1.25750	−0.628750	+	0.777608i	$0.716433\pi$
$684$	0	0
$685$	4630.00	0.258253
$686$	0	0
$687$	−12516.0	−0.695073
$688$	0	0
$689$	9396.00	0.519534
$690$	0	0
$691$	35336.0	1.94536	0.972681	−	0.232147i	$-0.0745750\pi$
$691$	35336.0	1.94536	0.972681	+	0.232147i	$0.0745750\pi$
$692$	0	0
$693$	−4896.00	−0.268375
$694$	0	0
$695$	−2580.00	−0.140813
$696$	0	0
$697$	9660.00	0.524962
$698$	0	0
$699$	34308.0	1.85643
$700$	0	0
$701$	3482.00	0.187608	0.0938041	−	0.995591i	$-0.470097\pi$
$701$	3482.00	0.187608	0.0938041	+	0.995591i	$0.470097\pi$
$702$	0	0
$703$	−1752.00	−0.0939942
$704$	0	0
$705$	660.000	0.0352582
$706$	0	0
$707$	−4284.00	−0.227887
$708$	0	0
$709$	−19402.0	−1.02773	−0.513863	−	0.857872i	$-0.671786\pi$
$709$	−19402.0	−1.02773	−0.513863	+	0.857872i	$0.671786\pi$
$710$	0	0
$711$	−4104.00	−0.216473
$712$	0	0
$713$	−13400.0	−0.703834
$714$	0	0
$715$	−4640.00	−0.242694
$716$	0	0
$717$	21744.0	1.13256
$718$	0	0
$719$	−9896.00	−0.513294	−0.256647	−	0.966505i	$-0.582618\pi$
$719$	−9896.00	−0.513294	−0.256647	+	0.966505i	$0.582618\pi$
$720$	0	0
$721$	42908.0	2.21633
$722$	0	0
$723$	492.000	0.0253080
$724$	0	0
$725$	−6050.00	−0.309919
$726$	0	0
$727$	494.000	0.0252014	0.0126007	−	0.999921i	$-0.495989\pi$
$727$	494.000	0.0252014	0.0126007	+	0.999921i	$0.495989\pi$
$728$	0	0
$729$	9477.00	0.481481
$730$	0	0
$731$	−12460.0	−0.630437
$732$	0	0
$733$	9282.00	0.467720	0.233860	−	0.972270i	$-0.424864\pi$
$733$	9282.00	0.467720	0.233860	+	0.972270i	$0.424864\pi$
$734$	0	0
$735$	24390.0	1.22400
$736$	0	0
$737$	6752.00	0.337467
$738$	0	0
$739$	−3252.00	−0.161877	−0.0809383	−	0.996719i	$-0.525792\pi$
$739$	−3252.00	−0.161877	−0.0809383	+	0.996719i	$0.525792\pi$
$740$	0	0
$741$	−1392.00	−0.0690100
$742$	0	0
$743$	−4710.00	−0.232561	−0.116281	−	0.993216i	$-0.537097\pi$
$743$	−4710.00	−0.232561	−0.116281	+	0.993216i	$0.537097\pi$
$744$	0	0
$745$	4790.00	0.235560
$746$	0	0
$747$	3906.00	0.191316
$748$	0	0
$749$	−17340.0	−0.845914
$750$	0	0
$751$	25764.0	1.25185	0.625927	−	0.779882i	$-0.284721\pi$
$751$	25764.0	1.25185	0.625927	+	0.779882i	$0.284721\pi$
$752$	0	0
$753$	30240.0	1.46349
$754$	0	0
$755$	−1660.00	−0.0800180
$756$	0	0
$757$	−30094.0	−1.44489	−0.722447	−	0.691426i	$-0.756983\pi$
$757$	−30094.0	−1.44489	−0.722447	+	0.691426i	$0.756983\pi$
$758$	0	0
$759$	12864.0	0.615196
$760$	0	0
$761$	22362.0	1.06521	0.532603	−	0.846365i	$-0.321214\pi$
$761$	22362.0	1.06521	0.532603	+	0.846365i	$0.321214\pi$
$762$	0	0
$763$	−884.000	−0.0419436
$764$	0	0
$765$	3150.00	0.148874
$766$	0	0
$767$	−15544.0	−0.731762
$768$	0	0
$769$	−30398.0	−1.42546	−0.712731	−	0.701438i	$-0.752542\pi$
$769$	−30398.0	−1.42546	−0.712731	+	0.701438i	$0.752542\pi$
$770$	0	0
$771$	13860.0	0.647413
$772$	0	0
$773$	1290.00	0.0600234	0.0300117	−	0.999550i	$-0.490446\pi$
$773$	1290.00	0.0600234	0.0300117	+	0.999550i	$0.490446\pi$
$774$	0	0
$775$	2500.00	0.115874
$776$	0	0
$777$	−89352.0	−4.12546
$778$	0	0
$779$	−552.000	−0.0253883
$780$	0	0
$781$	−13632.0	−0.624573
$782$	0	0
$783$	−26136.0	−1.19288
$784$	0	0
$785$	5110.00	0.232336
$786$	0	0
$787$	14.0000	0.000634112 0	0.000317056	−	1.00000i	$-0.499899\pi$
$787$	14.0000	0.000634112 0	0.000317056		1.00000i	$0.499899\pi$
$788$	0	0
$789$	24660.0	1.11270
$790$	0	0
$791$	−42228.0	−1.89817
$792$	0	0
$793$	14500.0	0.649319
$794$	0	0
$795$	4860.00	0.216813
$796$	0	0
$797$	−38814.0	−1.72505	−0.862523	−	0.506017i	$-0.831117\pi$
$797$	−38814.0	−1.72505	−0.862523	+	0.506017i	$0.831117\pi$
$798$	0	0
$799$	−1540.00	−0.0681868
$800$	0	0
$801$	−6534.00	−0.288224
$802$	0	0
$803$	4896.00	0.215163
$804$	0	0
$805$	−22780.0	−0.997378
$806$	0	0
$807$	−4476.00	−0.195245
$808$	0	0
$809$	27402.0	1.19086	0.595428	−	0.803408i	$-0.296982\pi$
$809$	27402.0	1.19086	0.595428	+	0.803408i	$0.296982\pi$
$810$	0	0
$811$	−28576.0	−1.23729	−0.618643	−	0.785672i	$-0.712317\pi$
$811$	−28576.0	−1.23729	−0.618643	+	0.785672i	$0.712317\pi$
$812$	0	0
$813$	27576.0	1.18958
$814$	0	0
$815$	4630.00	0.198996
$816$	0	0
$817$	712.000	0.0304893
$818$	0	0
$819$	−17748.0	−0.757223
$820$	0	0
$821$	31762.0	1.35018	0.675092	−	0.737733i	$-0.264104\pi$
$821$	31762.0	1.35018	0.675092	+	0.737733i	$0.264104\pi$
$822$	0	0
$823$	20506.0	0.868523	0.434261	−	0.900787i	$-0.357009\pi$
$823$	20506.0	0.868523	0.434261	+	0.900787i	$0.357009\pi$
$824$	0	0
$825$	−2400.00	−0.101282
$826$	0	0
$827$	13014.0	0.547208	0.273604	−	0.961842i	$-0.411784\pi$
$827$	13014.0	0.547208	0.273604	+	0.961842i	$0.411784\pi$
$828$	0	0
$829$	−22790.0	−0.954800	−0.477400	−	0.878686i	$-0.658421\pi$
$829$	−22790.0	−0.954800	−0.477400	+	0.878686i	$0.658421\pi$
$830$	0	0
$831$	13236.0	0.552529
$832$	0	0
$833$	−56910.0	−2.36712
$834$	0	0
$835$	−3270.00	−0.135525
$836$	0	0
$837$	10800.0	0.446001
$838$	0	0
$839$	23696.0	0.975062	0.487531	−	0.873106i	$-0.337898\pi$
$839$	23696.0	0.975062	0.487531	+	0.873106i	$0.337898\pi$
$840$	0	0
$841$	34175.0	1.40125
$842$	0	0
$843$	−49668.0	−2.02925
$844$	0	0
$845$	−5835.00	−0.237550
$846$	0	0
$847$	36550.0	1.48273
$848$	0	0
$849$	−7068.00	−0.285716
$850$	0	0
$851$	58692.0	2.36420
$852$	0	0
$853$	5306.00	0.212982	0.106491	−	0.994314i	$-0.466038\pi$
$853$	5306.00	0.212982	0.106491	+	0.994314i	$0.466038\pi$
$854$	0	0
$855$	−180.000	−0.00719985
$856$	0	0
$857$	−21054.0	−0.839196	−0.419598	−	0.907710i	$-0.637829\pi$
$857$	−21054.0	−0.839196	−0.419598	+	0.907710i	$0.637829\pi$
$858$	0	0
$859$	7364.00	0.292499	0.146249	−	0.989248i	$-0.453280\pi$
$859$	7364.00	0.292499	0.146249	+	0.989248i	$0.453280\pi$
$860$	0	0
$861$	−28152.0	−1.11431
$862$	0	0
$863$	17226.0	0.679467	0.339733	−	0.940522i	$-0.389663\pi$
$863$	17226.0	0.679467	0.339733	+	0.940522i	$0.389663\pi$
$864$	0	0
$865$	6470.00	0.254320
$866$	0	0
$867$	78.0000	0.00305539
$868$	0	0
$869$	−7296.00	−0.284810
$870$	0	0
$871$	24476.0	0.952167
$872$	0	0
$873$	12402.0	0.480807
$874$	0	0
$875$	4250.00	0.164201
$876$	0	0
$877$	21202.0	0.816352	0.408176	−	0.912903i	$-0.366165\pi$
$877$	21202.0	0.816352	0.408176	+	0.912903i	$0.366165\pi$
$878$	0	0
$879$	−636.000	−0.0244047
$880$	0	0
$881$	−29490.0	−1.12774	−0.563872	−	0.825862i	$-0.690689\pi$
$881$	−29490.0	−1.12774	−0.563872	+	0.825862i	$0.690689\pi$
$882$	0	0
$883$	2570.00	0.0979472	0.0489736	−	0.998800i	$-0.484405\pi$
$883$	2570.00	0.0979472	0.0489736	+	0.998800i	$0.484405\pi$
$884$	0	0
$885$	−8040.00	−0.305380
$886$	0	0
$887$	36334.0	1.37540	0.687698	−	0.725997i	$-0.258621\pi$
$887$	36334.0	1.37540	0.687698	+	0.725997i	$0.258621\pi$
$888$	0	0
$889$	33252.0	1.25448
$890$	0	0
$891$	−14256.0	−0.536020
$892$	0	0
$893$	88.0000	0.00329766
$894$	0	0
$895$	14180.0	0.529592
$896$	0	0
$897$	46632.0	1.73578
$898$	0	0
$899$	−24200.0	−0.897792
$900$	0	0
$901$	−11340.0	−0.419301
$902$	0	0
$903$	36312.0	1.33819
$904$	0	0
$905$	−8710.00	−0.319923
$906$	0	0
$907$	−12474.0	−0.456662	−0.228331	−	0.973584i	$-0.573327\pi$
$907$	−12474.0	−0.456662	−0.228331	+	0.973584i	$0.573327\pi$
$908$	0	0
$909$	1134.00	0.0413778
$910$	0	0
$911$	−41132.0	−1.49590	−0.747949	−	0.663756i	$-0.768961\pi$
$911$	−41132.0	−1.49590	−0.747949	+	0.663756i	$0.768961\pi$
$912$	0	0
$913$	6944.00	0.251712
$914$	0	0
$915$	7500.00	0.270975
$916$	0	0
$917$	31008.0	1.11666
$918$	0	0
$919$	−38416.0	−1.37892	−0.689460	−	0.724324i	$-0.742152\pi$
$919$	−38416.0	−1.37892	−0.689460	+	0.724324i	$0.742152\pi$
$920$	0	0
$921$	−48804.0	−1.74609
$922$	0	0
$923$	−49416.0	−1.76224
$924$	0	0
$925$	−10950.0	−0.389226
$926$	0	0
$927$	−11358.0	−0.402423
$928$	0	0
$929$	41302.0	1.45864	0.729319	−	0.684174i	$-0.239837\pi$
$929$	41302.0	1.45864	0.729319	+	0.684174i	$0.239837\pi$
$930$	0	0
$931$	3252.00	0.114479
$932$	0	0
$933$	26376.0	0.925521
$934$	0	0
$935$	5600.00	0.195871
$936$	0	0
$937$	−26150.0	−0.911722	−0.455861	−	0.890051i	$-0.650669\pi$
$937$	−26150.0	−0.911722	−0.455861	+	0.890051i	$0.650669\pi$
$938$	0	0
$939$	−28956.0	−1.00633
$940$	0	0
$941$	35254.0	1.22130	0.610652	−	0.791899i	$-0.290907\pi$
$941$	35254.0	1.22130	0.610652	+	0.791899i	$0.290907\pi$
$942$	0	0
$943$	18492.0	0.638582
$944$	0	0
$945$	18360.0	0.632011
$946$	0	0
$947$	18550.0	0.636530	0.318265	−	0.948002i	$-0.396900\pi$
$947$	18550.0	0.636530	0.318265	+	0.948002i	$0.396900\pi$
$948$	0	0
$949$	17748.0	0.607086
$950$	0	0
$951$	−42156.0	−1.43744
$952$	0	0
$953$	17322.0	0.588788	0.294394	−	0.955684i	$-0.404882\pi$
$953$	17322.0	0.588788	0.294394	+	0.955684i	$0.404882\pi$
$954$	0	0
$955$	−22300.0	−0.755614
$956$	0	0
$957$	23232.0	0.784727
$958$	0	0
$959$	31484.0	1.06014
$960$	0	0
$961$	−19791.0	−0.664328
$962$	0	0
$963$	4590.00	0.153594
$964$	0	0
$965$	18910.0	0.630813
$966$	0	0
$967$	35190.0	1.17025	0.585126	−	0.810942i	$-0.301045\pi$
$967$	35190.0	1.17025	0.585126	+	0.810942i	$0.301045\pi$
$968$	0	0
$969$	1680.00	0.0556960
$970$	0	0
$971$	−40696.0	−1.34500	−0.672501	−	0.740096i	$-0.734780\pi$
$971$	−40696.0	−1.34500	−0.672501	+	0.740096i	$0.734780\pi$
$972$	0	0
$973$	−17544.0	−0.578042
$974$	0	0
$975$	−8700.00	−0.285767
$976$	0	0
$977$	44306.0	1.45084	0.725422	−	0.688304i	$-0.241645\pi$
$977$	44306.0	1.45084	0.725422	+	0.688304i	$0.241645\pi$
$978$	0	0
$979$	−11616.0	−0.379212
$980$	0	0
$981$	234.000	0.00761574
$982$	0	0
$983$	−18798.0	−0.609932	−0.304966	−	0.952363i	$-0.598645\pi$
$983$	−18798.0	−0.609932	−0.304966	+	0.952363i	$0.598645\pi$
$984$	0	0
$985$	−22370.0	−0.723622
$986$	0	0
$987$	4488.00	0.144736
$988$	0	0
$989$	−23852.0	−0.766885
$990$	0	0
$991$	2468.00	0.0791106	0.0395553	−	0.999217i	$-0.487406\pi$
$991$	2468.00	0.0791106	0.0395553	+	0.999217i	$0.487406\pi$
$992$	0	0
$993$	−52848.0	−1.68890
$994$	0	0
$995$	−18040.0	−0.574780
$996$	0	0
$997$	−61086.0	−1.94043	−0.970217	−	0.242237i	$-0.922119\pi$
$997$	−61086.0	−1.94043	−0.970217	+	0.242237i	$0.922119\pi$
$998$	0	0
$999$	−47304.0	−1.49813

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	40.4.a.a.1.1	✓	1
3.2	odd	2		360.4.a.h.1.1		1
4.3	odd	2		80.4.a.e.1.1		1
5.2	odd	4		200.4.c.c.49.2		2
5.3	odd	4		200.4.c.c.49.1		2
5.4	even	2		200.4.a.i.1.1		1
7.6	odd	2		1960.4.a.h.1.1		1
8.3	odd	2		320.4.a.c.1.1		1
8.5	even	2		320.4.a.l.1.1		1
12.11	even	2		720.4.a.bd.1.1		1
15.2	even	4		1800.4.f.j.649.1		2
15.8	even	4		1800.4.f.j.649.2		2
15.14	odd	2		1800.4.a.bi.1.1		1
16.3	odd	4		1280.4.d.a.641.2		2
16.5	even	4		1280.4.d.p.641.2		2
16.11	odd	4		1280.4.d.a.641.1		2
16.13	even	4		1280.4.d.p.641.1		2
20.3	even	4		400.4.c.f.49.2		2
20.7	even	4		400.4.c.f.49.1		2
20.19	odd	2		400.4.a.e.1.1		1
40.19	odd	2		1600.4.a.br.1.1		1
40.29	even	2		1600.4.a.j.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.4.a.a.1.1	✓	1	1.1	even	1	trivial
80.4.a.e.1.1		1	4.3	odd	2
200.4.a.i.1.1		1	5.4	even	2
200.4.c.c.49.1		2	5.3	odd	4
200.4.c.c.49.2		2	5.2	odd	4
320.4.a.c.1.1		1	8.3	odd	2
320.4.a.l.1.1		1	8.5	even	2
360.4.a.h.1.1		1	3.2	odd	2
400.4.a.e.1.1		1	20.19	odd	2
400.4.c.f.49.1		2	20.7	even	4
400.4.c.f.49.2		2	20.3	even	4
720.4.a.bd.1.1		1	12.11	even	2
1280.4.d.a.641.1		2	16.11	odd	4
1280.4.d.a.641.2		2	16.3	odd	4
1280.4.d.p.641.1		2	16.13	even	4
1280.4.d.p.641.2		2	16.5	even	4
1600.4.a.j.1.1		1	40.29	even	2
1600.4.a.br.1.1		1	40.19	odd	2
1800.4.a.bi.1.1		1	15.14	odd	2
1800.4.f.j.649.1		2	15.2	even	4
1800.4.f.j.649.2		2	15.8	even	4
1960.4.a.h.1.1		1	7.6	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:	\( N \)	\(=\)	\( 40 = 2^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	40.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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