LMFDB - Embedded newform 475.6.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [475,6,Mod(1,475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(475, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("475.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 475 = 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	475.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:	$76.1823144112$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 19)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	475.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+6.00000 q^{2} -4.00000 q^{3} +4.00000 q^{4} -24.0000 q^{6} -248.000 q^{7} -168.000 q^{8} -227.000 q^{9} +O(q^{10})$

\(q+6.00000 q^{2} -4.00000 q^{3} +4.00000 q^{4} -24.0000 q^{6} -248.000 q^{7} -168.000 q^{8} -227.000 q^{9} +204.000 q^{11} -16.0000 q^{12} +370.000 q^{13} -1488.00 q^{14} -1136.00 q^{16} -1554.00 q^{17} -1362.00 q^{18} +361.000 q^{19} +992.000 q^{21} +1224.00 q^{22} +408.000 q^{23} +672.000 q^{24} +2220.00 q^{26} +1880.00 q^{27} -992.000 q^{28} +6174.00 q^{29} -7840.00 q^{31} -1440.00 q^{32} -816.000 q^{33} -9324.00 q^{34} -908.000 q^{36} +5146.00 q^{37} +2166.00 q^{38} -1480.00 q^{39} -7830.00 q^{41} +5952.00 q^{42} +12532.0 q^{43} +816.000 q^{44} +2448.00 q^{46} -2592.00 q^{47} +4544.00 q^{48} +44697.0 q^{49} +6216.00 q^{51} +1480.00 q^{52} +20778.0 q^{53} +11280.0 q^{54} +41664.0 q^{56} -1444.00 q^{57} +37044.0 q^{58} +18972.0 q^{59} -18418.0 q^{61} -47040.0 q^{62} +56296.0 q^{63} +27712.0 q^{64} -4896.00 q^{66} +11548.0 q^{67} -6216.00 q^{68} -1632.00 q^{69} -72984.0 q^{71} +38136.0 q^{72} -59114.0 q^{73} +30876.0 q^{74} +1444.00 q^{76} -50592.0 q^{77} -8880.00 q^{78} -44752.0 q^{79} +47641.0 q^{81} -46980.0 q^{82} +27660.0 q^{83} +3968.00 q^{84} +75192.0 q^{86} -24696.0 q^{87} -34272.0 q^{88} +20730.0 q^{89} -91760.0 q^{91} +1632.00 q^{92} +31360.0 q^{93} -15552.0 q^{94} +5760.00 q^{96} -14018.0 q^{97} +268182. q^{98} -46308.0 q^{99} +O(q^{100})\)

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 $

$2$	6.00000	1.06066	0.530330	−	0.847791i	$-0.322068\pi$
$2$	6.00000	1.06066	0.530330	+	0.847791i	$0.322068\pi$
$3$	−4.00000	−0.256600	−0.128300	−	0.991735i	$-0.540952\pi$
$3$	−4.00000	−0.256600	−0.128300	+	0.991735i	$0.540952\pi$
$4$	4.00000	0.125000
$5$	0	0
$5$	0	0
$6$	−24.0000	−0.272166
$7$	−248.000	−1.91296	−0.956482	−	0.291793i	$-0.905748\pi$
$7$	−248.000	−1.91296	−0.956482	+	0.291793i	$0.905748\pi$
$8$	−168.000	−0.928078
$9$	−227.000	−0.934156
$10$	0	0
$11$	204.000	0.508333	0.254167	−	0.967160i	$-0.418199\pi$
$11$	204.000	0.508333	0.254167	+	0.967160i	$0.418199\pi$
$12$	−16.0000	−0.0320750
$13$	370.000	0.607216	0.303608	−	0.952797i	$-0.401809\pi$
$13$	370.000	0.607216	0.303608	+	0.952797i	$0.401809\pi$
$14$	−1488.00	−2.02900
$15$	0	0
$16$	−1136.00	−1.10938
$17$	−1554.00	−1.30415	−0.652077	−	0.758153i	$-0.726102\pi$
$17$	−1554.00	−1.30415	−0.652077	+	0.758153i	$0.726102\pi$
$18$	−1362.00	−0.990822
$19$	361.000	0.229416
$19$	361.000	0.229416
$20$	0	0
$21$	992.000	0.490867
$22$	1224.00	0.539169
$23$	408.000	0.160820	0.0804101	−	0.996762i	$-0.474377\pi$
$23$	408.000	0.160820	0.0804101	+	0.996762i	$0.474377\pi$
$24$	672.000	0.238145
$25$	0	0
$26$	2220.00	0.644050
$27$	1880.00	0.496305
$28$	−992.000	−0.239120
$29$	6174.00	1.36324	0.681619	−	0.731707i	$-0.261276\pi$
$29$	6174.00	1.36324	0.681619	+	0.731707i	$0.261276\pi$
$30$	0	0
$31$	−7840.00	−1.46525	−0.732625	−	0.680632i	$-0.761705\pi$
$31$	−7840.00	−1.46525	−0.732625	+	0.680632i	$0.761705\pi$
$32$	−1440.00	−0.248592
$33$	−816.000	−0.130438
$34$	−9324.00	−1.38326
$35$	0	0
$36$	−908.000	−0.116770
$37$	5146.00	0.617967	0.308984	−	0.951067i	$-0.400011\pi$
$37$	5146.00	0.617967	0.308984	+	0.951067i	$0.400011\pi$
$38$	2166.00	0.243332
$39$	−1480.00	−0.155812
$40$	0	0
$41$	−7830.00	−0.727448	−0.363724	−	0.931507i	$-0.618495\pi$
$41$	−7830.00	−0.727448	−0.363724	+	0.931507i	$0.618495\pi$
$42$	5952.00	0.520643
$43$	12532.0	1.03359	0.516796	−	0.856108i	$-0.327125\pi$
$43$	12532.0	1.03359	0.516796	+	0.856108i	$0.327125\pi$
$44$	816.000	0.0635416
$45$	0	0
$46$	2448.00	0.170576
$47$	−2592.00	−0.171155	−0.0855777	−	0.996332i	$-0.527274\pi$
$47$	−2592.00	−0.171155	−0.0855777	+	0.996332i	$0.527274\pi$
$48$	4544.00	0.284666
$49$	44697.0	2.65943
$50$	0	0
$51$	6216.00	0.334646
$52$	1480.00	0.0759020
$53$	20778.0	1.01605	0.508024	−	0.861343i	$-0.330376\pi$
$53$	20778.0	1.01605	0.508024	+	0.861343i	$0.330376\pi$
$54$	11280.0	0.526411
$55$	0	0
$56$	41664.0	1.77538
$57$	−1444.00	−0.0588681
$58$	37044.0	1.44593
$59$	18972.0	0.709550	0.354775	−	0.934952i	$-0.384557\pi$
$59$	18972.0	0.709550	0.354775	+	0.934952i	$0.384557\pi$
$60$	0	0
$61$	−18418.0	−0.633750	−0.316875	−	0.948467i	$-0.602634\pi$
$61$	−18418.0	−0.633750	−0.316875	+	0.948467i	$0.602634\pi$
$62$	−47040.0	−1.55413
$63$	56296.0	1.78701
$64$	27712.0	0.845703
$65$	0	0
$66$	−4896.00	−0.138351
$67$	11548.0	0.314282	0.157141	−	0.987576i	$-0.449772\pi$
$67$	11548.0	0.314282	0.157141	+	0.987576i	$0.449772\pi$
$68$	−6216.00	−0.163019
$69$	−1632.00	−0.0412665
$70$	0	0
$71$	−72984.0	−1.71823	−0.859116	−	0.511781i	$-0.828986\pi$
$71$	−72984.0	−1.71823	−0.859116	+	0.511781i	$0.828986\pi$
$72$	38136.0	0.866970
$73$	−59114.0	−1.29832	−0.649162	−	0.760650i	$-0.724880\pi$
$73$	−59114.0	−1.29832	−0.649162	+	0.760650i	$0.724880\pi$
$74$	30876.0	0.655453
$75$	0	0
$76$	1444.00	0.0286770
$77$	−50592.0	−0.972423
$78$	−8880.00	−0.165263
$79$	−44752.0	−0.806761	−0.403380	−	0.915032i	$-0.632165\pi$
$79$	−44752.0	−0.806761	−0.403380	+	0.915032i	$0.632165\pi$
$80$	0	0
$81$	47641.0	0.806805
$82$	−46980.0	−0.771575
$83$	27660.0	0.440714	0.220357	−	0.975419i	$-0.429278\pi$
$83$	27660.0	0.440714	0.220357	+	0.975419i	$0.429278\pi$
$84$	3968.00	0.0613583
$85$	0	0
$86$	75192.0	1.09629
$87$	−24696.0	−0.349807
$88$	−34272.0	−0.471773
$89$	20730.0	0.277411	0.138706	−	0.990334i	$-0.455706\pi$
$89$	20730.0	0.277411	0.138706	+	0.990334i	$0.455706\pi$
$90$	0	0
$91$	−91760.0	−1.16158
$92$	1632.00	0.0201025
$93$	31360.0	0.375983
$94$	−15552.0	−0.181538
$95$	0	0
$96$	5760.00	0.0637888
$97$	−14018.0	−0.151271	−0.0756357	−	0.997136i	$-0.524099\pi$
$97$	−14018.0	−0.151271	−0.0756357	+	0.997136i	$0.524099\pi$
$98$	268182.	2.82075
$99$	−46308.0	−0.474863
$100$	0	0
$101$	145782.	1.42200	0.711001	−	0.703191i	$-0.248242\pi$
$101$	145782.	1.42200	0.711001	+	0.703191i	$0.248242\pi$
$102$	37296.0	0.354946
$103$	−90152.0	−0.837302	−0.418651	−	0.908147i	$-0.637497\pi$
$103$	−90152.0	−0.837302	−0.418651	+	0.908147i	$0.637497\pi$
$104$	−62160.0	−0.563544
$105$	0	0
$106$	124668.	1.07768
$107$	−36108.0	−0.304891	−0.152445	−	0.988312i	$-0.548715\pi$
$107$	−36108.0	−0.304891	−0.152445	+	0.988312i	$0.548715\pi$
$108$	7520.00	0.0620381
$109$	32654.0	0.263251	0.131626	−	0.991300i	$-0.457980\pi$
$109$	32654.0	0.263251	0.131626	+	0.991300i	$0.457980\pi$
$110$	0	0
$111$	−20584.0	−0.158570
$112$	281728.	2.12219
$113$	145806.	1.07419	0.537093	−	0.843523i	$-0.319523\pi$
$113$	145806.	1.07419	0.537093	+	0.843523i	$0.319523\pi$
$114$	−8664.00	−0.0624391
$115$	0	0
$116$	24696.0	0.170405
$117$	−83990.0	−0.567235
$118$	113832.	0.752592
$119$	385392.	2.49480
$120$	0	0
$121$	−119435.	−0.741597
$122$	−110508.	−0.672193
$123$	31320.0	0.186663
$124$	−31360.0	−0.183156
$125$	0	0
$126$	337776.	1.89541
$127$	259936.	1.43007	0.715035	−	0.699089i	$-0.246411\pi$
$127$	259936.	1.43007	0.715035	+	0.699089i	$0.246411\pi$
$128$	212352.	1.14560
$129$	−50128.0	−0.265220
$130$	0	0
$131$	−124092.	−0.631780	−0.315890	−	0.948796i	$-0.602303\pi$
$131$	−124092.	−0.631780	−0.315890	+	0.948796i	$0.602303\pi$
$132$	−3264.00	−0.0163048
$133$	−89528.0	−0.438864
$134$	69288.0	0.333347
$135$	0	0
$136$	261072.	1.21036
$137$	−84330.0	−0.383867	−0.191933	−	0.981408i	$-0.561476\pi$
$137$	−84330.0	−0.383867	−0.191933	+	0.981408i	$0.561476\pi$
$138$	−9792.00	−0.0437697
$139$	−155476.	−0.682537	−0.341269	−	0.939966i	$-0.610857\pi$
$139$	−155476.	−0.682537	−0.341269	+	0.939966i	$0.610857\pi$
$140$	0	0
$141$	10368.0	0.0439185
$142$	−437904.	−1.82246
$143$	75480.0	0.308668
$144$	257872.	1.03633
$145$	0	0
$146$	−354684.	−1.37708
$147$	−178788.	−0.682409
$148$	20584.0	0.0772459
$149$	−232794.	−0.859026	−0.429513	−	0.903061i	$-0.641315\pi$
$149$	−232794.	−0.859026	−0.429513	+	0.903061i	$0.641315\pi$
$150$	0	0
$151$	496184.	1.77093	0.885463	−	0.464710i	$-0.153841\pi$
$151$	496184.	1.77093	0.885463	+	0.464710i	$0.153841\pi$
$152$	−60648.0	−0.212916
$153$	352758.	1.21828
$154$	−303552.	−1.03141
$155$	0	0
$156$	−5920.00	−0.0194765
$157$	−128078.	−0.414692	−0.207346	−	0.978268i	$-0.566483\pi$
$157$	−128078.	−0.414692	−0.207346	+	0.978268i	$0.566483\pi$
$158$	−268512.	−0.855699
$159$	−83112.0	−0.260718
$160$	0	0
$161$	−101184.	−0.307643
$162$	285846.	0.855745
$163$	570652.	1.68230	0.841148	−	0.540805i	$-0.181880\pi$
$163$	570652.	1.68230	0.841148	+	0.540805i	$0.181880\pi$
$164$	−31320.0	−0.0909310
$165$	0	0
$166$	165960.	0.467448
$167$	39864.0	0.110609	0.0553044	−	0.998470i	$-0.482387\pi$
$167$	39864.0	0.110609	0.0553044	+	0.998470i	$0.482387\pi$
$168$	−166656.	−0.455562
$169$	−234393.	−0.631288
$170$	0	0
$171$	−81947.0	−0.214310
$172$	50128.0	0.129199
$173$	−223086.	−0.566705	−0.283353	−	0.959016i	$-0.591447\pi$
$173$	−223086.	−0.566705	−0.283353	+	0.959016i	$0.591447\pi$
$174$	−148176.	−0.371026
$175$	0	0
$176$	−231744.	−0.563932
$177$	−75888.0	−0.182071
$178$	124380.	0.294239
$179$	316980.	0.739434	0.369717	−	0.929144i	$-0.379455\pi$
$179$	316980.	0.739434	0.369717	+	0.929144i	$0.379455\pi$
$180$	0	0
$181$	857270.	1.94501	0.972504	−	0.232888i	$-0.0748176\pi$
$181$	857270.	1.94501	0.972504	+	0.232888i	$0.0748176\pi$
$182$	−550560.	−1.23204
$183$	73672.0	0.162620
$184$	−68544.0	−0.149254
$185$	0	0
$186$	188160.	0.398791
$187$	−317016.	−0.662944
$188$	−10368.0	−0.0213944
$189$	−466240.	−0.949413
$190$	0	0
$191$	783600.	1.55421	0.777107	−	0.629368i	$-0.216686\pi$
$191$	783600.	1.55421	0.777107	+	0.629368i	$0.216686\pi$
$192$	−110848.	−0.217008
$193$	−231074.	−0.446537	−0.223269	−	0.974757i	$-0.571673\pi$
$193$	−231074.	−0.446537	−0.223269	+	0.974757i	$0.571673\pi$
$194$	−84108.0	−0.160448
$195$	0	0
$196$	178788.	0.332428
$197$	−438966.	−0.805871	−0.402935	−	0.915228i	$-0.632010\pi$
$197$	−438966.	−0.805871	−0.402935	+	0.915228i	$0.632010\pi$
$198$	−277848.	−0.503668
$199$	−385000.	−0.689173	−0.344586	−	0.938755i	$-0.611981\pi$
$199$	−385000.	−0.689173	−0.344586	+	0.938755i	$0.611981\pi$
$200$	0	0
$201$	−46192.0	−0.0806448
$202$	874692.	1.50826
$203$	−1.53115e6	−2.60782
$204$	24864.0	0.0418307
$205$	0	0
$206$	−540912.	−0.888093
$207$	−92616.0	−0.150231
$208$	−420320.	−0.673630
$209$	73644.0	0.116620
$210$	0	0
$211$	438740.	0.678424	0.339212	−	0.940710i	$-0.389840\pi$
$211$	438740.	0.678424	0.339212	+	0.940710i	$0.389840\pi$
$212$	83112.0	0.127006
$213$	291936.	0.440899
$214$	−216648.	−0.323385
$215$	0	0
$216$	−315840.	−0.460609
$217$	1.94432e6	2.80297
$218$	195924.	0.279220
$219$	236456.	0.333150
$220$	0	0
$221$	−574980.	−0.791903
$222$	−123504.	−0.168189
$223$	372352.	0.501408	0.250704	−	0.968064i	$-0.419338\pi$
$223$	372352.	0.501408	0.250704	+	0.968064i	$0.419338\pi$
$224$	357120.	0.475548
$225$	0	0
$226$	874836.	1.13935
$227$	−1.01101e6	−1.30224	−0.651121	−	0.758974i	$-0.725701\pi$
$227$	−1.01101e6	−1.30224	−0.651121	+	0.758974i	$0.725701\pi$
$228$	−5776.00	−0.00735851
$229$	571382.	0.720009	0.360004	−	0.932951i	$-0.382775\pi$
$229$	571382.	0.720009	0.360004	+	0.932951i	$0.382775\pi$
$230$	0	0
$231$	202368.	0.249524
$232$	−1.03723e6	−1.26519
$233$	594678.	0.717616	0.358808	−	0.933411i	$-0.383183\pi$
$233$	594678.	0.717616	0.358808	+	0.933411i	$0.383183\pi$
$234$	−503940.	−0.601643
$235$	0	0
$236$	75888.0	0.0886938
$237$	179008.	0.207015
$238$	2.31235e6	2.64613
$239$	−380544.	−0.430933	−0.215467	−	0.976511i	$-0.569127\pi$
$239$	−380544.	−0.430933	−0.215467	+	0.976511i	$0.569127\pi$
$240$	0	0
$241$	−1.07678e6	−1.19422	−0.597111	−	0.802159i	$-0.703685\pi$
$241$	−1.07678e6	−1.19422	−0.597111	+	0.802159i	$0.703685\pi$
$242$	−716610.	−0.786583
$243$	−647404.	−0.703331
$244$	−73672.0	−0.0792187
$245$	0	0
$246$	187920.	0.197986
$247$	133570.	0.139305
$248$	1.31712e6	1.35987
$249$	−110640.	−0.113087
$250$	0	0
$251$	−113316.	−0.113529	−0.0567645	−	0.998388i	$-0.518078\pi$
$251$	−113316.	−0.113529	−0.0567645	+	0.998388i	$0.518078\pi$
$252$	225184.	0.223376
$253$	83232.0	0.0817502
$254$	1.55962e6	1.51682
$255$	0	0
$256$	387328.	0.369385
$257$	−553218.	−0.522473	−0.261236	−	0.965275i	$-0.584130\pi$
$257$	−553218.	−0.522473	−0.261236	+	0.965275i	$0.584130\pi$
$258$	−300768.	−0.281308
$259$	−1.27621e6	−1.18215
$260$	0	0
$261$	−1.40150e6	−1.27348
$262$	−744552.	−0.670103
$263$	−824088.	−0.734656	−0.367328	−	0.930091i	$-0.619727\pi$
$263$	−824088.	−0.734656	−0.367328	+	0.930091i	$0.619727\pi$
$264$	137088.	0.121057
$265$	0	0
$266$	−537168.	−0.465485
$267$	−82920.0	−0.0711838
$268$	46192.0	0.0392853
$269$	1.52158e6	1.28208	0.641039	−	0.767508i	$-0.278504\pi$
$269$	1.52158e6	1.28208	0.641039	+	0.767508i	$0.278504\pi$
$270$	0	0
$271$	−1.08304e6	−0.895821	−0.447911	−	0.894078i	$-0.647832\pi$
$271$	−1.08304e6	−0.895821	−0.447911	+	0.894078i	$0.647832\pi$
$272$	1.76534e6	1.44680
$273$	367040.	0.298062
$274$	−505980.	−0.407152
$275$	0	0
$276$	−6528.00	−0.00515831
$277$	−658598.	−0.515728	−0.257864	−	0.966181i	$-0.583019\pi$
$277$	−658598.	−0.515728	−0.257864	+	0.966181i	$0.583019\pi$
$278$	−932856.	−0.723940
$279$	1.77968e6	1.36877
$280$	0	0
$281$	356346.	0.269219	0.134610	−	0.990899i	$-0.457022\pi$
$281$	356346.	0.269219	0.134610	+	0.990899i	$0.457022\pi$
$282$	62208.0	0.0465826
$283$	−405116.	−0.300686	−0.150343	−	0.988634i	$-0.548038\pi$
$283$	−405116.	−0.300686	−0.150343	+	0.988634i	$0.548038\pi$
$284$	−291936.	−0.214779
$285$	0	0
$286$	452880.	0.327392
$287$	1.94184e6	1.39158
$288$	326880.	0.232224
$289$	995059.	0.700816
$290$	0	0
$291$	56072.0	0.0388162
$292$	−236456.	−0.162291
$293$	948570.	0.645506	0.322753	−	0.946483i	$-0.395392\pi$
$293$	948570.	0.645506	0.322753	+	0.946483i	$0.395392\pi$
$294$	−1.07273e6	−0.723805
$295$	0	0
$296$	−864528.	−0.573522
$297$	383520.	0.252288
$298$	−1.39676e6	−0.911134
$299$	150960.	0.0976526
$300$	0	0
$301$	−3.10794e6	−1.97722
$302$	2.97710e6	1.87835
$303$	−583128.	−0.364886
$304$	−410096.	−0.254508
$305$	0	0
$306$	2.11655e6	1.29218
$307$	1.16235e6	0.703866	0.351933	−	0.936025i	$-0.385524\pi$
$307$	1.16235e6	0.703866	0.351933	+	0.936025i	$0.385524\pi$
$308$	−202368.	−0.121553
$309$	360608.	0.214852
$310$	0	0
$311$	−2.55718e6	−1.49920	−0.749600	−	0.661891i	$-0.769754\pi$
$311$	−2.55718e6	−1.49920	−0.749600	+	0.661891i	$0.769754\pi$
$312$	248640.	0.144605
$313$	1.05943e6	0.611240	0.305620	−	0.952154i	$-0.401136\pi$
$313$	1.05943e6	0.611240	0.305620	+	0.952154i	$0.401136\pi$
$314$	−768468.	−0.439847
$315$	0	0
$316$	−179008.	−0.100845
$317$	−2.64425e6	−1.47793	−0.738967	−	0.673742i	$-0.764686\pi$
$317$	−2.64425e6	−1.47793	−0.738967	+	0.673742i	$0.764686\pi$
$318$	−498672.	−0.276533
$319$	1.25950e6	0.692979
$320$	0	0
$321$	144432.	0.0782350
$322$	−607104.	−0.326305
$323$	−560994.	−0.299193
$324$	190564.	0.100851
$325$	0	0
$326$	3.42391e6	1.78434
$327$	−130616.	−0.0675503
$328$	1.31544e6	0.675128
$329$	642816.	0.327414
$330$	0	0
$331$	2.39083e6	1.19944	0.599720	−	0.800210i	$-0.295279\pi$
$331$	2.39083e6	1.19944	0.599720	+	0.800210i	$0.295279\pi$
$332$	110640.	0.0550893
$333$	−1.16814e6	−0.577278
$334$	239184.	0.117318
$335$	0	0
$336$	−1.12691e6	−0.544555
$337$	1.12382e6	0.539042	0.269521	−	0.962994i	$-0.413135\pi$
$337$	1.12382e6	0.539042	0.269521	+	0.962994i	$0.413135\pi$
$338$	−1.40636e6	−0.669583
$339$	−583224.	−0.275636
$340$	0	0
$341$	−1.59936e6	−0.744835
$342$	−491682.	−0.227310
$343$	−6.91672e6	−3.17442
$344$	−2.10538e6	−0.959254
$345$	0	0
$346$	−1.33852e6	−0.601081
$347$	−2.00387e6	−0.893399	−0.446699	−	0.894684i	$-0.647401\pi$
$347$	−2.00387e6	−0.893399	−0.446699	+	0.894684i	$0.647401\pi$
$348$	−98784.0	−0.0437259
$349$	3.60808e6	1.58567	0.792834	−	0.609437i	$-0.208605\pi$
$349$	3.60808e6	1.58567	0.792834	+	0.609437i	$0.208605\pi$
$350$	0	0
$351$	695600.	0.301364
$352$	−293760.	−0.126368
$353$	1.82825e6	0.780908	0.390454	−	0.920622i	$-0.372318\pi$
$353$	1.82825e6	0.780908	0.390454	+	0.920622i	$0.372318\pi$
$354$	−455328.	−0.193115
$355$	0	0
$356$	82920.0	0.0346764
$357$	−1.54157e6	−0.640165
$358$	1.90188e6	0.784288
$359$	−2.61012e6	−1.06887	−0.534434	−	0.845210i	$-0.679475\pi$
$359$	−2.61012e6	−1.06887	−0.534434	+	0.845210i	$0.679475\pi$
$360$	0	0
$361$	130321.	0.0526316
$362$	5.14362e6	2.06299
$363$	477740.	0.190294
$364$	−367040.	−0.145198
$365$	0	0
$366$	442032.	0.172485
$367$	−391136.	−0.151587	−0.0757936	−	0.997124i	$-0.524149\pi$
$367$	−391136.	−0.151587	−0.0757936	+	0.997124i	$0.524149\pi$
$368$	−463488.	−0.178410
$369$	1.77741e6	0.679550
$370$	0	0
$371$	−5.15294e6	−1.94366
$372$	125440.	0.0469979
$373$	−262070.	−0.0975316	−0.0487658	−	0.998810i	$-0.515529\pi$
$373$	−262070.	−0.0975316	−0.0487658	+	0.998810i	$0.515529\pi$
$374$	−1.90210e6	−0.703159
$375$	0	0
$376$	435456.	0.158845
$377$	2.28438e6	0.827780
$378$	−2.79744e6	−1.00700
$379$	824060.	0.294687	0.147343	−	0.989085i	$-0.452928\pi$
$379$	824060.	0.294687	0.147343	+	0.989085i	$0.452928\pi$
$380$	0	0
$381$	−1.03974e6	−0.366956
$382$	4.70160e6	1.64849
$383$	3.89779e6	1.35776	0.678878	−	0.734251i	$-0.262467\pi$
$383$	3.89779e6	1.35776	0.678878	+	0.734251i	$0.262467\pi$
$384$	−849408.	−0.293960
$385$	0	0
$386$	−1.38644e6	−0.473624
$387$	−2.84476e6	−0.965537
$388$	−56072.0	−0.0189089
$389$	1.97401e6	0.661416	0.330708	−	0.943733i	$-0.392713\pi$
$389$	1.97401e6	0.661416	0.330708	+	0.943733i	$0.392713\pi$
$390$	0	0
$391$	−634032.	−0.209734
$392$	−7.50910e6	−2.46816
$393$	496368.	0.162115
$394$	−2.63380e6	−0.854755
$395$	0	0
$396$	−185232.	−0.0593578
$397$	2.84403e6	0.905646	0.452823	−	0.891600i	$-0.350417\pi$
$397$	2.84403e6	0.905646	0.452823	+	0.891600i	$0.350417\pi$
$398$	−2.31000e6	−0.730978
$399$	358112.	0.112613
$400$	0	0
$401$	5.80235e6	1.80195	0.900976	−	0.433869i	$-0.142852\pi$
$401$	5.80235e6	1.80195	0.900976	+	0.433869i	$0.142852\pi$
$402$	−277152.	−0.0855368
$403$	−2.90080e6	−0.889724
$404$	583128.	0.177750
$405$	0	0
$406$	−9.18691e6	−2.76601
$407$	1.04978e6	0.314133
$408$	−1.04429e6	−0.310577
$409$	5.35337e6	1.58241	0.791205	−	0.611551i	$-0.209454\pi$
$409$	5.35337e6	1.58241	0.791205	+	0.611551i	$0.209454\pi$
$410$	0	0
$411$	337320.	0.0985003
$412$	−360608.	−0.104663
$413$	−4.70506e6	−1.35734
$414$	−555696.	−0.159344
$415$	0	0
$416$	−532800.	−0.150949
$417$	621904.	0.175139
$418$	441864.	0.123694
$419$	−1.14198e6	−0.317778	−0.158889	−	0.987296i	$-0.550791\pi$
$419$	−1.14198e6	−0.317778	−0.158889	+	0.987296i	$0.550791\pi$
$420$	0	0
$421$	−740794.	−0.203701	−0.101850	−	0.994800i	$-0.532476\pi$
$421$	−740794.	−0.203701	−0.101850	+	0.994800i	$0.532476\pi$
$422$	2.63244e6	0.719577
$423$	588384.	0.159886
$424$	−3.49070e6	−0.942971
$425$	0	0
$426$	1.75162e6	0.467644
$427$	4.56766e6	1.21234
$428$	−144432.	−0.0381113
$429$	−301920.	−0.0792043
$430$	0	0
$431$	1.30354e6	0.338010	0.169005	−	0.985615i	$-0.445945\pi$
$431$	1.30354e6	0.338010	0.169005	+	0.985615i	$0.445945\pi$
$432$	−2.13568e6	−0.550588
$433$	7.23557e6	1.85461	0.927305	−	0.374306i	$-0.122119\pi$
$433$	7.23557e6	1.85461	0.927305	+	0.374306i	$0.122119\pi$
$434$	1.16659e7	2.97300
$435$	0	0
$436$	130616.	0.0329064
$437$	147288.	0.0368947
$438$	1.41874e6	0.353359
$439$	−415336.	−0.102858	−0.0514290	−	0.998677i	$-0.516378\pi$
$439$	−415336.	−0.102858	−0.0514290	+	0.998677i	$0.516378\pi$
$440$	0	0
$441$	−1.01462e7	−2.48432
$442$	−3.44988e6	−0.839940
$443$	−3.93203e6	−0.951935	−0.475967	−	0.879463i	$-0.657902\pi$
$443$	−3.93203e6	−0.951935	−0.475967	+	0.879463i	$0.657902\pi$
$444$	−82336.0	−0.0198213
$445$	0	0
$446$	2.23411e6	0.531824
$447$	931176.	0.220426
$448$	−6.87258e6	−1.61780
$449$	−4.76624e6	−1.11573	−0.557866	−	0.829931i	$-0.688380\pi$
$449$	−4.76624e6	−1.11573	−0.557866	+	0.829931i	$0.688380\pi$
$450$	0	0
$451$	−1.59732e6	−0.369786
$452$	583224.	0.134273
$453$	−1.98474e6	−0.454420
$454$	−6.06607e6	−1.38124
$455$	0	0
$456$	242592.	0.0546342
$457$	−2.72215e6	−0.609708	−0.304854	−	0.952399i	$-0.598608\pi$
$457$	−2.72215e6	−0.609708	−0.304854	+	0.952399i	$0.598608\pi$
$458$	3.42829e6	0.763685
$459$	−2.92152e6	−0.647258
$460$	0	0
$461$	2.46675e6	0.540596	0.270298	−	0.962777i	$-0.412878\pi$
$461$	2.46675e6	0.540596	0.270298	+	0.962777i	$0.412878\pi$
$462$	1.21421e6	0.264660
$463$	1.63734e6	0.354967	0.177483	−	0.984124i	$-0.443204\pi$
$463$	1.63734e6	0.354967	0.177483	+	0.984124i	$0.443204\pi$
$464$	−7.01366e6	−1.51234
$465$	0	0
$466$	3.56807e6	0.761147
$467$	−5.94565e6	−1.26156	−0.630779	−	0.775962i	$-0.717265\pi$
$467$	−5.94565e6	−1.26156	−0.630779	+	0.775962i	$0.717265\pi$
$468$	−335960.	−0.0709044
$469$	−2.86390e6	−0.601210
$470$	0	0
$471$	512312.	0.106410
$472$	−3.18730e6	−0.658518
$473$	2.55653e6	0.525409
$474$	1.07405e6	0.219572
$475$	0	0
$476$	1.54157e6	0.311850
$477$	−4.71661e6	−0.949147
$478$	−2.28326e6	−0.457074
$479$	−3.85531e6	−0.767752	−0.383876	−	0.923385i	$-0.625411\pi$
$479$	−3.85531e6	−0.767752	−0.383876	+	0.923385i	$0.625411\pi$
$480$	0	0
$481$	1.90402e6	0.375240
$482$	−6.46069e6	−1.26666
$483$	404736.	0.0789413
$484$	−477740.	−0.0926997
$485$	0	0
$486$	−3.88442e6	−0.745995
$487$	4.54847e6	0.869047	0.434523	−	0.900661i	$-0.356917\pi$
$487$	4.54847e6	0.869047	0.434523	+	0.900661i	$0.356917\pi$
$488$	3.09422e6	0.588169
$489$	−2.28261e6	−0.431677
$490$	0	0
$491$	1.66820e6	0.312281	0.156140	−	0.987735i	$-0.450095\pi$
$491$	1.66820e6	0.312281	0.156140	+	0.987735i	$0.450095\pi$
$492$	125280.	0.0233329
$493$	−9.59440e6	−1.77787
$494$	801420.	0.147755
$495$	0	0
$496$	8.90624e6	1.62551
$497$	1.81000e7	3.28691
$498$	−663840.	−0.119947
$499$	−956044.	−0.171880	−0.0859402	−	0.996300i	$-0.527389\pi$
$499$	−956044.	−0.171880	−0.0859402	+	0.996300i	$0.527389\pi$
$500$	0	0
$501$	−159456.	−0.0283822
$502$	−679896.	−0.120416
$503$	−5.70410e6	−1.00523	−0.502617	−	0.864509i	$-0.667630\pi$
$503$	−5.70410e6	−1.00523	−0.502617	+	0.864509i	$0.667630\pi$
$504$	−9.45773e6	−1.65848
$505$	0	0
$506$	499392.	0.0867092
$507$	937572.	0.161989
$508$	1.03974e6	0.178759
$509$	6.66109e6	1.13960	0.569798	−	0.821785i	$-0.307022\pi$
$509$	6.66109e6	1.13960	0.569798	+	0.821785i	$0.307022\pi$
$510$	0	0
$511$	1.46603e7	2.48365
$512$	−4.47130e6	−0.753804
$513$	678680.	0.113860
$514$	−3.31931e6	−0.554166
$515$	0	0
$516$	−200512.	−0.0331525
$517$	−528768.	−0.0870039
$518$	−7.65725e6	−1.25386
$519$	892344.	0.145417
$520$	0	0
$521$	5.91547e6	0.954761	0.477380	−	0.878697i	$-0.341586\pi$
$521$	5.91547e6	0.954761	0.477380	+	0.878697i	$0.341586\pi$
$522$	−8.40899e6	−1.35073
$523$	−5.56872e6	−0.890227	−0.445114	−	0.895474i	$-0.646837\pi$
$523$	−5.56872e6	−0.890227	−0.445114	+	0.895474i	$0.646837\pi$
$524$	−496368.	−0.0789724
$525$	0	0
$526$	−4.94453e6	−0.779221
$527$	1.21834e7	1.91091
$528$	926976.	0.144705
$529$	−6.26988e6	−0.974137
$530$	0	0
$531$	−4.30664e6	−0.662831
$532$	−358112.	−0.0548580
$533$	−2.89710e6	−0.441718
$534$	−497520.	−0.0755018
$535$	0	0
$536$	−1.94006e6	−0.291678
$537$	−1.26792e6	−0.189739
$538$	9.12949e6	1.35985
$539$	9.11819e6	1.35188
$540$	0	0
$541$	744878.	0.109419	0.0547094	−	0.998502i	$-0.482577\pi$
$541$	744878.	0.109419	0.0547094	+	0.998502i	$0.482577\pi$
$542$	−6.49824e6	−0.950162
$543$	−3.42908e6	−0.499089
$544$	2.23776e6	0.324202
$545$	0	0
$546$	2.20224e6	0.316143
$547$	6.02403e6	0.860833	0.430416	−	0.902631i	$-0.358367\pi$
$547$	6.02403e6	0.860833	0.430416	+	0.902631i	$0.358367\pi$
$548$	−337320.	−0.0479834
$549$	4.18089e6	0.592021
$550$	0	0
$551$	2.22881e6	0.312748
$552$	274176.	0.0382985
$553$	1.10985e7	1.54330
$554$	−3.95159e6	−0.547012
$555$	0	0
$556$	−621904.	−0.0853172
$557$	5.29744e6	0.723483	0.361741	−	0.932278i	$-0.382182\pi$
$557$	5.29744e6	0.723483	0.361741	+	0.932278i	$0.382182\pi$
$558$	1.06781e7	1.45180
$559$	4.63684e6	0.627614
$560$	0	0
$561$	1.26806e6	0.170112
$562$	2.13808e6	0.285550
$563$	6.33332e6	0.842094	0.421047	−	0.907039i	$-0.361663\pi$
$563$	6.33332e6	0.842094	0.421047	+	0.907039i	$0.361663\pi$
$564$	41472.0	0.00548981
$565$	0	0
$566$	−2.43070e6	−0.318926
$567$	−1.18150e7	−1.54339
$568$	1.22613e7	1.59465
$569$	−3.07508e6	−0.398176	−0.199088	−	0.979982i	$-0.563798\pi$
$569$	−3.07508e6	−0.398176	−0.199088	+	0.979982i	$0.563798\pi$
$570$	0	0
$571$	359036.	0.0460837	0.0230419	−	0.999735i	$-0.492665\pi$
$571$	359036.	0.0460837	0.0230419	+	0.999735i	$0.492665\pi$
$572$	301920.	0.0385835
$573$	−3.13440e6	−0.398812
$574$	1.16510e7	1.47600
$575$	0	0
$576$	−6.29062e6	−0.790019
$577$	−6.43104e6	−0.804159	−0.402079	−	0.915605i	$-0.631712\pi$
$577$	−6.43104e6	−0.804159	−0.402079	+	0.915605i	$0.631712\pi$
$578$	5.97035e6	0.743328
$579$	924296.	0.114582
$580$	0	0
$581$	−6.85968e6	−0.843070
$582$	336432.	0.0411709
$583$	4.23871e6	0.516491
$584$	9.93115e6	1.20495
$585$	0	0
$586$	5.69142e6	0.684663
$587$	7.47330e6	0.895194	0.447597	−	0.894235i	$-0.352280\pi$
$587$	7.47330e6	0.895194	0.447597	+	0.894235i	$0.352280\pi$
$588$	−715152.	−0.0853012
$589$	−2.83024e6	−0.336151
$590$	0	0
$591$	1.75586e6	0.206786
$592$	−5.84586e6	−0.685557
$593$	9.41451e6	1.09941	0.549707	−	0.835358i	$-0.314739\pi$
$593$	9.41451e6	1.09941	0.549707	+	0.835358i	$0.314739\pi$
$594$	2.30112e6	0.267592
$595$	0	0
$596$	−931176.	−0.107378
$597$	1.54000e6	0.176842
$598$	905760.	0.103576
$599$	−1.22340e6	−0.139316	−0.0696581	−	0.997571i	$-0.522191\pi$
$599$	−1.22340e6	−0.139316	−0.0696581	+	0.997571i	$0.522191\pi$
$600$	0	0
$601$	4.64268e6	0.524304	0.262152	−	0.965027i	$-0.415568\pi$
$601$	4.64268e6	0.524304	0.262152	+	0.965027i	$0.415568\pi$
$602$	−1.86476e7	−2.09716
$603$	−2.62140e6	−0.293589
$604$	1.98474e6	0.221366
$605$	0	0
$606$	−3.49877e6	−0.387020
$607$	2.06646e6	0.227644	0.113822	−	0.993501i	$-0.463691\pi$
$607$	2.06646e6	0.227644	0.113822	+	0.993501i	$0.463691\pi$
$608$	−519840.	−0.0570310
$609$	6.12461e6	0.669168
$610$	0	0
$611$	−959040.	−0.103928
$612$	1.41103e6	0.152285
$613$	1.60408e7	1.72415	0.862077	−	0.506778i	$-0.169163\pi$
$613$	1.60408e7	1.72415	0.862077	+	0.506778i	$0.169163\pi$
$614$	6.97409e6	0.746563
$615$	0	0
$616$	8.49946e6	0.902484
$617$	−8.10983e6	−0.857628	−0.428814	−	0.903393i	$-0.641068\pi$
$617$	−8.10983e6	−0.857628	−0.428814	+	0.903393i	$0.641068\pi$
$618$	2.16365e6	0.227885
$619$	6.43147e6	0.674658	0.337329	−	0.941387i	$-0.390477\pi$
$619$	6.43147e6	0.674658	0.337329	+	0.941387i	$0.390477\pi$
$620$	0	0
$621$	767040.	0.0798158
$622$	−1.53431e7	−1.59014
$623$	−5.14104e6	−0.530678
$624$	1.68128e6	0.172854
$625$	0	0
$626$	6.35658e6	0.648317
$627$	−294576.	−0.0299246
$628$	−512312.	−0.0518365
$629$	−7.99688e6	−0.805924
$630$	0	0
$631$	5.74292e6	0.574195	0.287097	−	0.957901i	$-0.407310\pi$
$631$	5.74292e6	0.574195	0.287097	+	0.957901i	$0.407310\pi$
$632$	7.51834e6	0.748737
$633$	−1.75496e6	−0.174084
$634$	−1.58655e7	−1.56759
$635$	0	0
$636$	−332448.	−0.0325897
$637$	1.65379e7	1.61485
$638$	7.55698e6	0.735015
$639$	1.65674e7	1.60510
$640$	0	0
$641$	6.62781e6	0.637125	0.318563	−	0.947902i	$-0.396800\pi$
$641$	6.62781e6	0.637125	0.318563	+	0.947902i	$0.396800\pi$
$642$	866592.	0.0829807
$643$	1.67112e7	1.59397	0.796986	−	0.603997i	$-0.206426\pi$
$643$	1.67112e7	1.59397	0.796986	+	0.603997i	$0.206426\pi$
$644$	−404736.	−0.0384554
$645$	0	0
$646$	−3.36596e6	−0.317342
$647$	599400.	0.0562932	0.0281466	−	0.999604i	$-0.491039\pi$
$647$	599400.	0.0562932	0.0281466	+	0.999604i	$0.491039\pi$
$648$	−8.00369e6	−0.748777
$649$	3.87029e6	0.360688
$650$	0	0
$651$	−7.77728e6	−0.719242
$652$	2.28261e6	0.210287
$653$	−1.12433e7	−1.03183	−0.515916	−	0.856639i	$-0.672549\pi$
$653$	−1.12433e7	−1.03183	−0.515916	+	0.856639i	$0.672549\pi$
$654$	−783696.	−0.0716479
$655$	0	0
$656$	8.89488e6	0.807013
$657$	1.34189e7	1.21284
$658$	3.85690e6	0.347275
$659$	1.00891e7	0.904981	0.452490	−	0.891769i	$-0.350536\pi$
$659$	1.00891e7	0.904981	0.452490	+	0.891769i	$0.350536\pi$
$660$	0	0
$661$	2.21936e7	1.97571	0.987855	−	0.155377i	$-0.0496593\pi$
$661$	2.21936e7	1.97571	0.987855	+	0.155377i	$0.0496593\pi$
$662$	1.43450e7	1.27220
$663$	2.29992e6	0.203202
$664$	−4.64688e6	−0.409017
$665$	0	0
$666$	−7.00885e6	−0.612296
$667$	2.51899e6	0.219236
$668$	159456.	0.0138261
$669$	−1.48941e6	−0.128661
$670$	0	0
$671$	−3.75727e6	−0.322156
$672$	−1.42848e6	−0.122026
$673$	−1.00785e7	−0.857742	−0.428871	−	0.903366i	$-0.641089\pi$
$673$	−1.00785e7	−0.857742	−0.428871	+	0.903366i	$0.641089\pi$
$674$	6.74293e6	0.571741
$675$	0	0
$676$	−937572.	−0.0789111
$677$	6.39388e6	0.536158	0.268079	−	0.963397i	$-0.413611\pi$
$677$	6.39388e6	0.536158	0.268079	+	0.963397i	$0.413611\pi$
$678$	−3.49934e6	−0.292356
$679$	3.47646e6	0.289377
$680$	0	0
$681$	4.04405e6	0.334155
$682$	−9.59616e6	−0.790017
$683$	2.25396e7	1.84882	0.924411	−	0.381398i	$-0.124557\pi$
$683$	2.25396e7	1.84882	0.924411	+	0.381398i	$0.124557\pi$
$684$	−327788.	−0.0267888
$685$	0	0
$686$	−4.15003e7	−3.36698
$687$	−2.28553e6	−0.184754
$688$	−1.42364e7	−1.14664
$689$	7.68786e6	0.616961
$690$	0	0
$691$	−2.24043e6	−0.178499	−0.0892495	−	0.996009i	$-0.528447\pi$
$691$	−2.24043e6	−0.178499	−0.0892495	+	0.996009i	$0.528447\pi$
$692$	−892344.	−0.0708381
$693$	1.14844e7	0.908395
$694$	−1.20232e7	−0.947593
$695$	0	0
$696$	4.14893e6	0.324648
$697$	1.21678e7	0.948704
$698$	2.16485e7	1.68186
$699$	−2.37871e6	−0.184140
$700$	0	0
$701$	7.24184e6	0.556614	0.278307	−	0.960492i	$-0.410227\pi$
$701$	7.24184e6	0.556614	0.278307	+	0.960492i	$0.410227\pi$
$702$	4.17360e6	0.319645
$703$	1.85771e6	0.141771
$704$	5.65325e6	0.429899
$705$	0	0
$706$	1.09695e7	0.828278
$707$	−3.61539e7	−2.72024
$708$	−303552.	−0.0227588
$709$	−1.10476e7	−0.825377	−0.412688	−	0.910872i	$-0.635410\pi$
$709$	−1.10476e7	−0.825377	−0.412688	+	0.910872i	$0.635410\pi$
$710$	0	0
$711$	1.01587e7	0.753641
$712$	−3.48264e6	−0.257459
$713$	−3.19872e6	−0.235642
$714$	−9.24941e6	−0.678998
$715$	0	0
$716$	1.26792e6	0.0924292
$717$	1.52218e6	0.110578
$718$	−1.56607e7	−1.13371
$719$	−1.76408e7	−1.27261	−0.636305	−	0.771438i	$-0.719538\pi$
$719$	−1.76408e7	−1.27261	−0.636305	+	0.771438i	$0.719538\pi$
$720$	0	0
$721$	2.23577e7	1.60173
$722$	781926.	0.0558242
$723$	4.30713e6	0.306438
$724$	3.42908e6	0.243126
$725$	0	0
$726$	2.86644e6	0.201837
$727$	−1.02224e7	−0.717325	−0.358662	−	0.933467i	$-0.616767\pi$
$727$	−1.02224e7	−0.717325	−0.358662	+	0.933467i	$0.616767\pi$
$728$	1.54157e7	1.07804
$729$	−8.98715e6	−0.626330
$730$	0	0
$731$	−1.94747e7	−1.34796
$732$	294688.	0.0203275
$733$	2.27060e7	1.56092	0.780460	−	0.625206i	$-0.214985\pi$
$733$	2.27060e7	1.56092	0.780460	+	0.625206i	$0.214985\pi$
$734$	−2.34682e6	−0.160783
$735$	0	0
$736$	−587520.	−0.0399786
$737$	2.35579e6	0.159760
$738$	1.06645e7	0.720772
$739$	1.39817e7	0.941779	0.470889	−	0.882192i	$-0.343933\pi$
$739$	1.39817e7	0.941779	0.470889	+	0.882192i	$0.343933\pi$
$740$	0	0
$741$	−534280.	−0.0357457
$742$	−3.09177e7	−2.06156
$743$	−2.12383e7	−1.41140	−0.705698	−	0.708513i	$-0.749366\pi$
$743$	−2.12383e7	−1.41140	−0.705698	+	0.708513i	$0.749366\pi$
$744$	−5.26848e6	−0.348942
$745$	0	0
$746$	−1.57242e6	−0.103448
$747$	−6.27882e6	−0.411696
$748$	−1.26806e6	−0.0828681
$749$	8.95478e6	0.583244
$750$	0	0
$751$	1.08008e6	0.0698805	0.0349403	−	0.999389i	$-0.488876\pi$
$751$	1.08008e6	0.0698805	0.0349403	+	0.999389i	$0.488876\pi$
$752$	2.94451e6	0.189875
$753$	453264.	0.0291316
$754$	1.37063e7	0.877993
$755$	0	0
$756$	−1.86496e6	−0.118677
$757$	6.16086e6	0.390752	0.195376	−	0.980728i	$-0.437407\pi$
$757$	6.16086e6	0.390752	0.195376	+	0.980728i	$0.437407\pi$
$758$	4.94436e6	0.312563
$759$	−332928.	−0.0209771
$760$	0	0
$761$	−6.96471e6	−0.435955	−0.217977	−	0.975954i	$-0.569946\pi$
$761$	−6.96471e6	−0.435955	−0.217977	+	0.975954i	$0.569946\pi$
$762$	−6.23846e6	−0.389216
$763$	−8.09819e6	−0.503590
$764$	3.13440e6	0.194277
$765$	0	0
$766$	2.33868e7	1.44012
$767$	7.01964e6	0.430850
$768$	−1.54931e6	−0.0947842
$769$	1.08349e7	0.660710	0.330355	−	0.943857i	$-0.392832\pi$
$769$	1.08349e7	0.660710	0.330355	+	0.943857i	$0.392832\pi$
$770$	0	0
$771$	2.21287e6	0.134067
$772$	−924296.	−0.0558172
$773$	−2.62798e7	−1.58188	−0.790940	−	0.611894i	$-0.790408\pi$
$773$	−2.62798e7	−1.58188	−0.790940	+	0.611894i	$0.790408\pi$
$774$	−1.70686e7	−1.02411
$775$	0	0
$776$	2.35502e6	0.140392
$777$	5.10483e6	0.303339
$778$	1.18440e7	0.701537
$779$	−2.82663e6	−0.166888
$780$	0	0
$781$	−1.48887e7	−0.873434
$782$	−3.80419e6	−0.222457
$783$	1.16071e7	0.676581
$784$	−5.07758e7	−2.95030
$785$	0	0
$786$	2.97821e6	0.171949
$787$	1.54390e7	0.888550	0.444275	−	0.895890i	$-0.353461\pi$
$787$	1.54390e7	0.888550	0.444275	+	0.895890i	$0.353461\pi$
$788$	−1.75586e6	−0.100734
$789$	3.29635e6	0.188513
$790$	0	0
$791$	−3.61599e7	−2.05488
$792$	7.77974e6	0.440709
$793$	−6.81466e6	−0.384823
$794$	1.70642e7	0.960583
$795$	0	0
$796$	−1.54000e6	−0.0861466
$797$	−1.19255e7	−0.665016	−0.332508	−	0.943100i	$-0.607895\pi$
$797$	−1.19255e7	−0.665016	−0.332508	+	0.943100i	$0.607895\pi$
$798$	2.14867e6	0.119444
$799$	4.02797e6	0.223213
$800$	0	0
$801$	−4.70571e6	−0.259146
$802$	3.48141e7	1.91126
$803$	−1.20593e7	−0.659981
$804$	−184768.	−0.0100806
$805$	0	0
$806$	−1.74048e7	−0.943695
$807$	−6.08633e6	−0.328982
$808$	−2.44914e7	−1.31973
$809$	3.49378e7	1.87683	0.938414	−	0.345514i	$-0.112295\pi$
$809$	3.49378e7	1.87683	0.938414	+	0.345514i	$0.112295\pi$
$810$	0	0
$811$	−5.93602e6	−0.316915	−0.158458	−	0.987366i	$-0.550652\pi$
$811$	−5.93602e6	−0.316915	−0.158458	+	0.987366i	$0.550652\pi$
$812$	−6.12461e6	−0.325978
$813$	4.33216e6	0.229868
$814$	6.29870e6	0.333189
$815$	0	0
$816$	−7.06138e6	−0.371248
$817$	4.52405e6	0.237122
$818$	3.21202e7	1.67840
$819$	2.08295e7	1.08510
$820$	0	0
$821$	2.08358e7	1.07883	0.539415	−	0.842040i	$-0.318645\pi$
$821$	2.08358e7	1.07883	0.539415	+	0.842040i	$0.318645\pi$
$822$	2.02392e6	0.104475
$823$	2.37266e7	1.22106	0.610528	−	0.791995i	$-0.290957\pi$
$823$	2.37266e7	1.22106	0.610528	+	0.791995i	$0.290957\pi$
$824$	1.51455e7	0.777082
$825$	0	0
$826$	−2.82303e7	−1.43968
$827$	1.11613e7	0.567480	0.283740	−	0.958901i	$-0.408425\pi$
$827$	1.11613e7	0.567480	0.283740	+	0.958901i	$0.408425\pi$
$828$	−370464.	−0.0187789
$829$	−8.58071e6	−0.433647	−0.216824	−	0.976211i	$-0.569570\pi$
$829$	−8.58071e6	−0.433647	−0.216824	+	0.976211i	$0.569570\pi$
$830$	0	0
$831$	2.63439e6	0.132336
$832$	1.02534e7	0.513525
$833$	−6.94591e7	−3.46830
$834$	3.73142e6	0.185763
$835$	0	0
$836$	294576.	0.0145775
$837$	−1.47392e7	−0.727211
$838$	−6.85188e6	−0.337054
$839$	1.09293e7	0.536027	0.268014	−	0.963415i	$-0.413633\pi$
$839$	1.09293e7	0.536027	0.268014	+	0.963415i	$0.413633\pi$
$840$	0	0
$841$	1.76071e7	0.858417
$842$	−4.44476e6	−0.216057
$843$	−1.42538e6	−0.0690817
$844$	1.75496e6	0.0848030
$845$	0	0
$846$	3.53030e6	0.169585
$847$	2.96199e7	1.41865
$848$	−2.36038e7	−1.12718
$849$	1.62046e6	0.0771561
$850$	0	0
$851$	2.09957e6	0.0993816
$852$	1.16774e6	0.0551123
$853$	−3.84963e7	−1.81153	−0.905767	−	0.423775i	$-0.860705\pi$
$853$	−3.84963e7	−1.81153	−0.905767	+	0.423775i	$0.860705\pi$
$854$	2.74060e7	1.28588
$855$	0	0
$856$	6.06614e6	0.282962
$857$	6.86045e6	0.319081	0.159540	−	0.987191i	$-0.448999\pi$
$857$	6.86045e6	0.319081	0.159540	+	0.987191i	$0.448999\pi$
$858$	−1.81152e6	−0.0840088
$859$	−6.58868e6	−0.304660	−0.152330	−	0.988330i	$-0.548678\pi$
$859$	−6.58868e6	−0.304660	−0.152330	+	0.988330i	$0.548678\pi$
$860$	0	0
$861$	−7.76736e6	−0.357080
$862$	7.82122e6	0.358514
$863$	1.66861e7	0.762657	0.381328	−	0.924440i	$-0.375467\pi$
$863$	1.66861e7	0.762657	0.381328	+	0.924440i	$0.375467\pi$
$864$	−2.70720e6	−0.123378
$865$	0	0
$866$	4.34134e7	1.96711
$867$	−3.98024e6	−0.179830
$868$	7.77728e6	0.350371
$869$	−9.12941e6	−0.410103
$870$	0	0
$871$	4.27276e6	0.190837
$872$	−5.48587e6	−0.244317
$873$	3.18209e6	0.141311
$874$	883728.	0.0391327
$875$	0	0
$876$	945824.	0.0416438
$877$	−3.34255e7	−1.46750	−0.733751	−	0.679418i	$-0.762232\pi$
$877$	−3.34255e7	−1.46750	−0.733751	+	0.679418i	$0.762232\pi$
$878$	−2.49202e6	−0.109097
$879$	−3.79428e6	−0.165637
$880$	0	0
$881$	−2.04334e6	−0.0886954	−0.0443477	−	0.999016i	$-0.514121\pi$
$881$	−2.04334e6	−0.0886954	−0.0443477	+	0.999016i	$0.514121\pi$
$882$	−6.08773e7	−2.63502
$883$	−4.25705e7	−1.83741	−0.918706	−	0.394942i	$-0.870765\pi$
$883$	−4.25705e7	−1.83741	−0.918706	+	0.394942i	$0.870765\pi$
$884$	−2.29992e6	−0.0989879
$885$	0	0
$886$	−2.35922e7	−1.00968
$887$	1.47058e7	0.627597	0.313798	−	0.949490i	$-0.398398\pi$
$887$	1.47058e7	0.627597	0.313798	+	0.949490i	$0.398398\pi$
$888$	3.45811e6	0.147166
$889$	−6.44641e7	−2.73567
$890$	0	0
$891$	9.71876e6	0.410126
$892$	1.48941e6	0.0626760
$893$	−935712.	−0.0392657
$894$	5.58706e6	0.233797
$895$	0	0
$896$	−5.26633e7	−2.19148
$897$	−603840.	−0.0250577
$898$	−2.85974e7	−1.18341
$899$	−4.84042e7	−1.99748
$900$	0	0
$901$	−3.22890e7	−1.32508
$902$	−9.58392e6	−0.392217
$903$	1.24317e7	0.507356
$904$	−2.44954e7	−0.996928
$905$	0	0
$906$	−1.19084e7	−0.481985
$907$	3.89732e7	1.57307	0.786534	−	0.617547i	$-0.211873\pi$
$907$	3.89732e7	1.57307	0.786534	+	0.617547i	$0.211873\pi$
$908$	−4.04405e6	−0.162780
$909$	−3.30925e7	−1.32837
$910$	0	0
$911$	3.47532e7	1.38739	0.693695	−	0.720268i	$-0.255981\pi$
$911$	3.47532e7	1.38739	0.693695	+	0.720268i	$0.255981\pi$
$912$	1.64038e6	0.0653068
$913$	5.64264e6	0.224030
$914$	−1.63329e7	−0.646693
$915$	0	0
$916$	2.28553e6	0.0900011
$917$	3.07748e7	1.20857
$918$	−1.75291e7	−0.686520
$919$	−1.81251e7	−0.707931	−0.353966	−	0.935258i	$-0.615167\pi$
$919$	−1.81251e7	−0.707931	−0.353966	+	0.935258i	$0.615167\pi$
$920$	0	0
$921$	−4.64939e6	−0.180612
$922$	1.48005e7	0.573389
$923$	−2.70041e7	−1.04334
$924$	809472.	0.0311905
$925$	0	0
$926$	9.82406e6	0.376499
$927$	2.04645e7	0.782171
$928$	−8.89056e6	−0.338890
$929$	−4.55062e6	−0.172994	−0.0864971	−	0.996252i	$-0.527567\pi$
$929$	−4.55062e6	−0.172994	−0.0864971	+	0.996252i	$0.527567\pi$
$930$	0	0
$931$	1.61356e7	0.610115
$932$	2.37871e6	0.0897020
$933$	1.02287e7	0.384695
$934$	−3.56739e7	−1.33808
$935$	0	0
$936$	1.41103e7	0.526438
$937$	−5.07809e6	−0.188952	−0.0944760	−	0.995527i	$-0.530118\pi$
$937$	−5.07809e6	−0.188952	−0.0944760	+	0.995527i	$0.530118\pi$
$938$	−1.71834e7	−0.637680
$939$	−4.23772e6	−0.156844
$940$	0	0
$941$	1.40795e7	0.518339	0.259170	−	0.965832i	$-0.416551\pi$
$941$	1.40795e7	0.518339	0.259170	+	0.965832i	$0.416551\pi$
$942$	3.07387e6	0.112865
$943$	−3.19464e6	−0.116988
$944$	−2.15522e7	−0.787157
$945$	0	0
$946$	1.53392e7	0.557281
$947$	−2.09673e7	−0.759746	−0.379873	−	0.925039i	$-0.624032\pi$
$947$	−2.09673e7	−0.759746	−0.379873	+	0.925039i	$0.624032\pi$
$948$	716032.	0.0258769
$949$	−2.18722e7	−0.788364
$950$	0	0
$951$	1.05770e7	0.379238
$952$	−6.47459e7	−2.31537
$953$	1.79751e6	0.0641120	0.0320560	−	0.999486i	$-0.489795\pi$
$953$	1.79751e6	0.0641120	0.0320560	+	0.999486i	$0.489795\pi$
$954$	−2.82996e7	−1.00672
$955$	0	0
$956$	−1.52218e6	−0.0538667
$957$	−5.03798e6	−0.177818
$958$	−2.31319e7	−0.814324
$959$	2.09138e7	0.734323
$960$	0	0
$961$	3.28364e7	1.14696
$962$	1.14241e7	0.398002
$963$	8.19652e6	0.284815
$964$	−4.30713e6	−0.149278
$965$	0	0
$966$	2.42842e6	0.0837298
$967$	2.30807e7	0.793750	0.396875	−	0.917873i	$-0.370095\pi$
$967$	2.30807e7	0.793750	0.396875	+	0.917873i	$0.370095\pi$
$968$	2.00651e7	0.688260
$969$	2.24398e6	0.0767730
$970$	0	0
$971$	−5.52696e7	−1.88121	−0.940607	−	0.339498i	$-0.889743\pi$
$971$	−5.52696e7	−1.88121	−0.940607	+	0.339498i	$0.889743\pi$
$972$	−2.58962e6	−0.0879164
$973$	3.85580e7	1.30567
$974$	2.72908e7	0.921763
$975$	0	0
$976$	2.09228e7	0.703066
$977$	3.06110e7	1.02599	0.512993	−	0.858393i	$-0.328537\pi$
$977$	3.06110e7	1.02599	0.512993	+	0.858393i	$0.328537\pi$
$978$	−1.36956e7	−0.457863
$979$	4.22892e6	0.141017
$980$	0	0
$981$	−7.41246e6	−0.245918
$982$	1.00092e7	0.331224
$983$	−1.55466e7	−0.513160	−0.256580	−	0.966523i	$-0.582596\pi$
$983$	−1.55466e7	−0.513160	−0.256580	+	0.966523i	$0.582596\pi$
$984$	−5.26176e6	−0.173238
$985$	0	0
$986$	−5.75664e7	−1.88572
$987$	−2.57126e6	−0.0840144
$988$	534280.	0.0174131
$989$	5.11306e6	0.166223
$990$	0	0
$991$	−7.44554e6	−0.240831	−0.120415	−	0.992724i	$-0.538423\pi$
$991$	−7.44554e6	−0.240831	−0.120415	+	0.992724i	$0.538423\pi$
$992$	1.12896e7	0.364250
$993$	−9.56331e6	−0.307776
$994$	1.08600e8	3.48630
$995$	0	0
$996$	−442560.	−0.0141359
$997$	2.71926e7	0.866388	0.433194	−	0.901301i	$-0.357386\pi$
$997$	2.71926e7	0.866388	0.433194	+	0.901301i	$0.357386\pi$
$998$	−5.73626e6	−0.182307
$999$	9.67448e6	0.306700

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.6.a.b.1.1		1
5.4	even	2		19.6.a.a.1.1	✓	1
15.14	odd	2		171.6.a.d.1.1		1
20.19	odd	2		304.6.a.a.1.1		1
35.34	odd	2		931.6.a.a.1.1		1
95.94	odd	2		361.6.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.6.a.a.1.1	✓	1	5.4	even	2
171.6.a.d.1.1		1	15.14	odd	2
304.6.a.a.1.1		1	20.19	odd	2
361.6.a.c.1.1		1	95.94	odd	2
475.6.a.b.1.1		1	1.1	even	1	trivial
931.6.a.a.1.1		1	35.34	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:	\( N \)	\(=\)	\( 475 = 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	475.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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