LMFDB - Embedded newform 342.4.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [342,4,Mod(1,342)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(342, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("342.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 342 = 2 \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	342.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:	$20.1786532220$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 114)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	342.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+2.00000 q^{2} +4.00000 q^{4} +19.0000 q^{5} +9.00000 q^{7} +8.00000 q^{8} +O(q^{10})$

\(q+2.00000 q^{2} +4.00000 q^{4} +19.0000 q^{5} +9.00000 q^{7} +8.00000 q^{8} +38.0000 q^{10} +13.0000 q^{11} +38.0000 q^{13} +18.0000 q^{14} +16.0000 q^{16} -99.0000 q^{17} -19.0000 q^{19} +76.0000 q^{20} +26.0000 q^{22} -68.0000 q^{23} +236.000 q^{25} +76.0000 q^{26} +36.0000 q^{28} -130.000 q^{29} +262.000 q^{31} +32.0000 q^{32} -198.000 q^{34} +171.000 q^{35} -296.000 q^{37} -38.0000 q^{38} +152.000 q^{40} +8.00000 q^{41} +73.0000 q^{43} +52.0000 q^{44} -136.000 q^{46} +271.000 q^{47} -262.000 q^{49} +472.000 q^{50} +152.000 q^{52} +502.000 q^{53} +247.000 q^{55} +72.0000 q^{56} -260.000 q^{58} -540.000 q^{59} +587.000 q^{61} +524.000 q^{62} +64.0000 q^{64} +722.000 q^{65} +684.000 q^{67} -396.000 q^{68} +342.000 q^{70} -992.000 q^{71} -507.000 q^{73} -592.000 q^{74} -76.0000 q^{76} +117.000 q^{77} +980.000 q^{79} +304.000 q^{80} +16.0000 q^{82} +492.000 q^{83} -1881.00 q^{85} +146.000 q^{86} +104.000 q^{88} -810.000 q^{89} +342.000 q^{91} -272.000 q^{92} +542.000 q^{94} -361.000 q^{95} -1046.00 q^{97} -524.000 q^{98} +O(q^{100})\)

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 $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	19.0000	1.69941	0.849706	−	0.527257i	$-0.176780\pi$
$5$	19.0000	1.69941	0.849706	+	0.527257i	$0.176780\pi$
$6$	0	0
$7$	9.00000	0.485954	0.242977	−	0.970032i	$-0.421876\pi$
$7$	9.00000	0.485954	0.242977	+	0.970032i	$0.421876\pi$
$8$	8.00000	0.353553
$9$	0	0
$10$	38.0000	1.20167
$11$	13.0000	0.356332	0.178166	−	0.984000i	$-0.442984\pi$
$11$	13.0000	0.356332	0.178166	+	0.984000i	$0.442984\pi$
$12$	0	0
$13$	38.0000	0.810716	0.405358	−	0.914158i	$-0.367147\pi$
$13$	38.0000	0.810716	0.405358	+	0.914158i	$0.367147\pi$
$14$	18.0000	0.343622
$15$	0	0
$16$	16.0000	0.250000
$17$	−99.0000	−1.41241	−0.706207	−	0.708006i	$-0.749595\pi$
$17$	−99.0000	−1.41241	−0.706207	+	0.708006i	$0.749595\pi$
$18$	0	0
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	76.0000	0.849706
$21$	0	0
$22$	26.0000	0.251964
$23$	−68.0000	−0.616477	−0.308239	−	0.951309i	$-0.599740\pi$
$23$	−68.0000	−0.616477	−0.308239	+	0.951309i	$0.599740\pi$
$24$	0	0
$25$	236.000	1.88800
$26$	76.0000	0.573263
$27$	0	0
$28$	36.0000	0.242977
$29$	−130.000	−0.832427	−0.416214	−	0.909267i	$-0.636643\pi$
$29$	−130.000	−0.832427	−0.416214	+	0.909267i	$0.636643\pi$
$30$	0	0
$31$	262.000	1.51795	0.758977	−	0.651117i	$-0.225699\pi$
$31$	262.000	1.51795	0.758977	+	0.651117i	$0.225699\pi$
$32$	32.0000	0.176777
$33$	0	0
$34$	−198.000	−0.998727
$35$	171.000	0.825836
$36$	0	0
$37$	−296.000	−1.31519	−0.657596	−	0.753371i	$-0.728427\pi$
$37$	−296.000	−1.31519	−0.657596	+	0.753371i	$0.728427\pi$
$38$	−38.0000	−0.162221
$39$	0	0
$40$	152.000	0.600833
$41$	8.00000	0.0304729	0.0152365	−	0.999884i	$-0.495150\pi$
$41$	8.00000	0.0304729	0.0152365	+	0.999884i	$0.495150\pi$
$42$	0	0
$43$	73.0000	0.258893	0.129446	−	0.991586i	$-0.458680\pi$
$43$	73.0000	0.258893	0.129446	+	0.991586i	$0.458680\pi$
$44$	52.0000	0.178166
$45$	0	0
$46$	−136.000	−0.435915
$47$	271.000	0.841051	0.420526	−	0.907281i	$-0.361846\pi$
$47$	271.000	0.841051	0.420526	+	0.907281i	$0.361846\pi$
$48$	0	0
$49$	−262.000	−0.763848
$50$	472.000	1.33502
$51$	0	0
$52$	152.000	0.405358
$53$	502.000	1.30104	0.650519	−	0.759490i	$-0.274551\pi$
$53$	502.000	1.30104	0.650519	+	0.759490i	$0.274551\pi$
$54$	0	0
$55$	247.000	0.605554
$56$	72.0000	0.171811
$57$	0	0
$58$	−260.000	−0.588615
$59$	−540.000	−1.19156	−0.595780	−	0.803148i	$-0.703157\pi$
$59$	−540.000	−1.19156	−0.595780	+	0.803148i	$0.703157\pi$
$60$	0	0
$61$	587.000	1.23209	0.616046	−	0.787710i	$-0.288733\pi$
$61$	587.000	1.23209	0.616046	+	0.787710i	$0.288733\pi$
$62$	524.000	1.07336
$63$	0	0
$64$	64.0000	0.125000
$65$	722.000	1.37774
$66$	0	0
$67$	684.000	1.24722	0.623611	−	0.781735i	$-0.285665\pi$
$67$	684.000	1.24722	0.623611	+	0.781735i	$0.285665\pi$
$68$	−396.000	−0.706207
$69$	0	0
$70$	342.000	0.583955
$71$	−992.000	−1.65815	−0.829076	−	0.559136i	$-0.811133\pi$
$71$	−992.000	−1.65815	−0.829076	+	0.559136i	$0.811133\pi$
$72$	0	0
$73$	−507.000	−0.812875	−0.406437	−	0.913679i	$-0.633229\pi$
$73$	−507.000	−0.812875	−0.406437	+	0.913679i	$0.633229\pi$
$74$	−592.000	−0.929981
$75$	0	0
$76$	−76.0000	−0.114708
$77$	117.000	0.173161
$78$	0	0
$79$	980.000	1.39568	0.697839	−	0.716254i	$-0.254145\pi$
$79$	980.000	1.39568	0.697839	+	0.716254i	$0.254145\pi$
$80$	304.000	0.424853
$81$	0	0
$82$	16.0000	0.0215476
$83$	492.000	0.650651	0.325325	−	0.945602i	$-0.394526\pi$
$83$	492.000	0.650651	0.325325	+	0.945602i	$0.394526\pi$
$84$	0	0
$85$	−1881.00	−2.40027
$86$	146.000	0.183065
$87$	0	0
$88$	104.000	0.125982
$89$	−810.000	−0.964717	−0.482359	−	0.875974i	$-0.660220\pi$
$89$	−810.000	−0.964717	−0.482359	+	0.875974i	$0.660220\pi$
$90$	0	0
$91$	342.000	0.393971
$92$	−272.000	−0.308239
$93$	0	0
$94$	542.000	0.594713
$95$	−361.000	−0.389872
$96$	0	0
$97$	−1046.00	−1.09490	−0.547450	−	0.836839i	$-0.684401\pi$
$97$	−1046.00	−1.09490	−0.547450	+	0.836839i	$0.684401\pi$
$98$	−524.000	−0.540122
$99$	0	0
$100$	944.000	0.944000
$101$	−1682.00	−1.65708	−0.828541	−	0.559929i	$-0.810829\pi$
$101$	−1682.00	−1.65708	−0.828541	+	0.559929i	$0.810829\pi$
$102$	0	0
$103$	978.000	0.935584	0.467792	−	0.883838i	$-0.345050\pi$
$103$	978.000	0.935584	0.467792	+	0.883838i	$0.345050\pi$
$104$	304.000	0.286631
$105$	0	0
$106$	1004.00	0.919973
$107$	−674.000	−0.608954	−0.304477	−	0.952520i	$-0.598482\pi$
$107$	−674.000	−0.608954	−0.304477	+	0.952520i	$0.598482\pi$
$108$	0	0
$109$	−360.000	−0.316346	−0.158173	−	0.987411i	$-0.550560\pi$
$109$	−360.000	−0.316346	−0.158173	+	0.987411i	$0.550560\pi$
$110$	494.000	0.428191
$111$	0	0
$112$	144.000	0.121489
$113$	−1838.00	−1.53013	−0.765064	−	0.643954i	$-0.777293\pi$
$113$	−1838.00	−1.53013	−0.765064	+	0.643954i	$0.777293\pi$
$114$	0	0
$115$	−1292.00	−1.04765
$116$	−520.000	−0.416214
$117$	0	0
$118$	−1080.00	−0.842560
$119$	−891.000	−0.686368
$120$	0	0
$121$	−1162.00	−0.873028
$122$	1174.00	0.871221
$123$	0	0
$124$	1048.00	0.758977
$125$	2109.00	1.50908
$126$	0	0
$127$	−1386.00	−0.968406	−0.484203	−	0.874956i	$-0.660890\pi$
$127$	−1386.00	−0.968406	−0.484203	+	0.874956i	$0.660890\pi$
$128$	128.000	0.0883883
$129$	0	0
$130$	1444.00	0.974209
$131$	1083.00	0.722306	0.361153	−	0.932506i	$-0.382383\pi$
$131$	1083.00	0.722306	0.361153	+	0.932506i	$0.382383\pi$
$132$	0	0
$133$	−171.000	−0.111486
$134$	1368.00	0.881919
$135$	0	0
$136$	−792.000	−0.499364
$137$	−499.000	−0.311186	−0.155593	−	0.987821i	$-0.549729\pi$
$137$	−499.000	−0.311186	−0.155593	+	0.987821i	$0.549729\pi$
$138$	0	0
$139$	−195.000	−0.118991	−0.0594953	−	0.998229i	$-0.518949\pi$
$139$	−195.000	−0.118991	−0.0594953	+	0.998229i	$0.518949\pi$
$140$	684.000	0.412918
$141$	0	0
$142$	−1984.00	−1.17249
$143$	494.000	0.288884
$144$	0	0
$145$	−2470.00	−1.41464
$146$	−1014.00	−0.574789
$147$	0	0
$148$	−1184.00	−0.657596
$149$	1485.00	0.816483	0.408241	−	0.912874i	$-0.366142\pi$
$149$	1485.00	0.816483	0.408241	+	0.912874i	$0.366142\pi$
$150$	0	0
$151$	332.000	0.178926	0.0894628	−	0.995990i	$-0.471485\pi$
$151$	332.000	0.178926	0.0894628	+	0.995990i	$0.471485\pi$
$152$	−152.000	−0.0811107
$153$	0	0
$154$	234.000	0.122443
$155$	4978.00	2.57963
$156$	0	0
$157$	3814.00	1.93879	0.969396	−	0.245502i	$-0.0789529\pi$
$157$	3814.00	1.93879	0.969396	+	0.245502i	$0.0789529\pi$
$158$	1960.00	0.986894
$159$	0	0
$160$	608.000	0.300416
$161$	−612.000	−0.299580
$162$	0	0
$163$	−3352.00	−1.61073	−0.805365	−	0.592780i	$-0.798031\pi$
$163$	−3352.00	−1.61073	−0.805365	+	0.592780i	$0.798031\pi$
$164$	32.0000	0.0152365
$165$	0	0
$166$	984.000	0.460080
$167$	−3714.00	−1.72095	−0.860473	−	0.509496i	$-0.829832\pi$
$167$	−3714.00	−1.72095	−0.860473	+	0.509496i	$0.829832\pi$
$168$	0	0
$169$	−753.000	−0.342740
$170$	−3762.00	−1.69725
$171$	0	0
$172$	292.000	0.129446
$173$	1242.00	0.545824	0.272912	−	0.962039i	$-0.412013\pi$
$173$	1242.00	0.545824	0.272912	+	0.962039i	$0.412013\pi$
$174$	0	0
$175$	2124.00	0.917482
$176$	208.000	0.0890829
$177$	0	0
$178$	−1620.00	−0.682158
$179$	−2310.00	−0.964567	−0.482284	−	0.876015i	$-0.660192\pi$
$179$	−2310.00	−0.964567	−0.482284	+	0.876015i	$0.660192\pi$
$180$	0	0
$181$	82.0000	0.0336741	0.0168370	−	0.999858i	$-0.494640\pi$
$181$	82.0000	0.0336741	0.0168370	+	0.999858i	$0.494640\pi$
$182$	684.000	0.278579
$183$	0	0
$184$	−544.000	−0.217958
$185$	−5624.00	−2.23505
$186$	0	0
$187$	−1287.00	−0.503287
$188$	1084.00	0.420526
$189$	0	0
$190$	−722.000	−0.275681
$191$	1313.00	0.497410	0.248705	−	0.968579i	$-0.419995\pi$
$191$	1313.00	0.497410	0.248705	+	0.968579i	$0.419995\pi$
$192$	0	0
$193$	−2352.00	−0.877206	−0.438603	−	0.898681i	$-0.644526\pi$
$193$	−2352.00	−0.877206	−0.438603	+	0.898681i	$0.644526\pi$
$194$	−2092.00	−0.774211
$195$	0	0
$196$	−1048.00	−0.381924
$197$	3826.00	1.38371	0.691856	−	0.722036i	$-0.256793\pi$
$197$	3826.00	1.38371	0.691856	+	0.722036i	$0.256793\pi$
$198$	0	0
$199$	2225.00	0.792593	0.396297	−	0.918123i	$-0.370295\pi$
$199$	2225.00	0.792593	0.396297	+	0.918123i	$0.370295\pi$
$200$	1888.00	0.667509
$201$	0	0
$202$	−3364.00	−1.17173
$203$	−1170.00	−0.404522
$204$	0	0
$205$	152.000	0.0517861
$206$	1956.00	0.661558
$207$	0	0
$208$	608.000	0.202679
$209$	−247.000	−0.0817481
$210$	0	0
$211$	1632.00	0.532472	0.266236	−	0.963908i	$-0.414220\pi$
$211$	1632.00	0.532472	0.266236	+	0.963908i	$0.414220\pi$
$212$	2008.00	0.650519
$213$	0	0
$214$	−1348.00	−0.430595
$215$	1387.00	0.439966
$216$	0	0
$217$	2358.00	0.737657
$218$	−720.000	−0.223691
$219$	0	0
$220$	988.000	0.302777
$221$	−3762.00	−1.14507
$222$	0	0
$223$	1148.00	0.344734	0.172367	−	0.985033i	$-0.444858\pi$
$223$	1148.00	0.344734	0.172367	+	0.985033i	$0.444858\pi$
$224$	288.000	0.0859054
$225$	0	0
$226$	−3676.00	−1.08196
$227$	−5794.00	−1.69410	−0.847051	−	0.531511i	$-0.821624\pi$
$227$	−5794.00	−1.69410	−0.847051	+	0.531511i	$0.821624\pi$
$228$	0	0
$229$	2525.00	0.728632	0.364316	−	0.931275i	$-0.381303\pi$
$229$	2525.00	0.728632	0.364316	+	0.931275i	$0.381303\pi$
$230$	−2584.00	−0.740800
$231$	0	0
$232$	−1040.00	−0.294308
$233$	1227.00	0.344993	0.172497	−	0.985010i	$-0.444817\pi$
$233$	1227.00	0.344993	0.172497	+	0.985010i	$0.444817\pi$
$234$	0	0
$235$	5149.00	1.42929
$236$	−2160.00	−0.595780
$237$	0	0
$238$	−1782.00	−0.485336
$239$	−675.000	−0.182687	−0.0913433	−	0.995819i	$-0.529116\pi$
$239$	−675.000	−0.182687	−0.0913433	+	0.995819i	$0.529116\pi$
$240$	0	0
$241$	−6068.00	−1.62188	−0.810942	−	0.585126i	$-0.801045\pi$
$241$	−6068.00	−1.62188	−0.810942	+	0.585126i	$0.801045\pi$
$242$	−2324.00	−0.617324
$243$	0	0
$244$	2348.00	0.616046
$245$	−4978.00	−1.29809
$246$	0	0
$247$	−722.000	−0.185991
$248$	2096.00	0.536678
$249$	0	0
$250$	4218.00	1.06708
$251$	3763.00	0.946289	0.473144	−	0.880985i	$-0.343119\pi$
$251$	3763.00	0.946289	0.473144	+	0.880985i	$0.343119\pi$
$252$	0	0
$253$	−884.000	−0.219670
$254$	−2772.00	−0.684767
$255$	0	0
$256$	256.000	0.0625000
$257$	−4304.00	−1.04465	−0.522327	−	0.852745i	$-0.674936\pi$
$257$	−4304.00	−1.04465	−0.522327	+	0.852745i	$0.674936\pi$
$258$	0	0
$259$	−2664.00	−0.639123
$260$	2888.00	0.688870
$261$	0	0
$262$	2166.00	0.510748
$263$	8247.00	1.93358	0.966791	−	0.255570i	$-0.0822632\pi$
$263$	8247.00	1.93358	0.966791	+	0.255570i	$0.0822632\pi$
$264$	0	0
$265$	9538.00	2.21100
$266$	−342.000	−0.0788322
$267$	0	0
$268$	2736.00	0.623611
$269$	5430.00	1.23075	0.615377	−	0.788233i	$-0.289004\pi$
$269$	5430.00	1.23075	0.615377	+	0.788233i	$0.289004\pi$
$270$	0	0
$271$	5452.00	1.22209	0.611043	−	0.791597i	$-0.290750\pi$
$271$	5452.00	1.22209	0.611043	+	0.791597i	$0.290750\pi$
$272$	−1584.00	−0.353103
$273$	0	0
$274$	−998.000	−0.220042
$275$	3068.00	0.672754
$276$	0	0
$277$	−6311.00	−1.36892	−0.684461	−	0.729050i	$-0.739962\pi$
$277$	−6311.00	−1.36892	−0.684461	+	0.729050i	$0.739962\pi$
$278$	−390.000	−0.0841390
$279$	0	0
$280$	1368.00	0.291977
$281$	6158.00	1.30731	0.653657	−	0.756791i	$-0.273234\pi$
$281$	6158.00	1.30731	0.653657	+	0.756791i	$0.273234\pi$
$282$	0	0
$283$	5853.00	1.22942	0.614708	−	0.788755i	$-0.289274\pi$
$283$	5853.00	1.22942	0.614708	+	0.788755i	$0.289274\pi$
$284$	−3968.00	−0.829076
$285$	0	0
$286$	988.000	0.204272
$287$	72.0000	0.0148085
$288$	0	0
$289$	4888.00	0.994911
$290$	−4940.00	−1.00030
$291$	0	0
$292$	−2028.00	−0.406437
$293$	5892.00	1.17479	0.587396	−	0.809299i	$-0.300153\pi$
$293$	5892.00	1.17479	0.587396	+	0.809299i	$0.300153\pi$
$294$	0	0
$295$	−10260.0	−2.02495
$296$	−2368.00	−0.464991
$297$	0	0
$298$	2970.00	0.577341
$299$	−2584.00	−0.499788
$300$	0	0
$301$	657.000	0.125810
$302$	664.000	0.126520
$303$	0	0
$304$	−304.000	−0.0573539
$305$	11153.0	2.09383
$306$	0	0
$307$	1824.00	0.339092	0.169546	−	0.985522i	$-0.445770\pi$
$307$	1824.00	0.339092	0.169546	+	0.985522i	$0.445770\pi$
$308$	468.000	0.0865804
$309$	0	0
$310$	9956.00	1.82407
$311$	−9757.00	−1.77900	−0.889500	−	0.456936i	$-0.848947\pi$
$311$	−9757.00	−1.77900	−0.889500	+	0.456936i	$0.848947\pi$
$312$	0	0
$313$	−5722.00	−1.03331	−0.516656	−	0.856193i	$-0.672823\pi$
$313$	−5722.00	−1.03331	−0.516656	+	0.856193i	$0.672823\pi$
$314$	7628.00	1.37093
$315$	0	0
$316$	3920.00	0.697839
$317$	−10224.0	−1.81147	−0.905737	−	0.423841i	$-0.860682\pi$
$317$	−10224.0	−1.81147	−0.905737	+	0.423841i	$0.860682\pi$
$318$	0	0
$319$	−1690.00	−0.296620
$320$	1216.00	0.212426
$321$	0	0
$322$	−1224.00	−0.211835
$323$	1881.00	0.324030
$324$	0	0
$325$	8968.00	1.53063
$326$	−6704.00	−1.13896
$327$	0	0
$328$	64.0000	0.0107738
$329$	2439.00	0.408713
$330$	0	0
$331$	1712.00	0.284290	0.142145	−	0.989846i	$-0.454600\pi$
$331$	1712.00	0.284290	0.142145	+	0.989846i	$0.454600\pi$
$332$	1968.00	0.325325
$333$	0	0
$334$	−7428.00	−1.21689
$335$	12996.0	2.11954
$336$	0	0
$337$	934.000	0.150974	0.0754870	−	0.997147i	$-0.475949\pi$
$337$	934.000	0.150974	0.0754870	+	0.997147i	$0.475949\pi$
$338$	−1506.00	−0.242354
$339$	0	0
$340$	−7524.00	−1.20014
$341$	3406.00	0.540895
$342$	0	0
$343$	−5445.00	−0.857150
$344$	584.000	0.0915325
$345$	0	0
$346$	2484.00	0.385956
$347$	6491.00	1.00419	0.502097	−	0.864811i	$-0.332562\pi$
$347$	6491.00	1.00419	0.502097	+	0.864811i	$0.332562\pi$
$348$	0	0
$349$	3205.00	0.491575	0.245788	−	0.969324i	$-0.420953\pi$
$349$	3205.00	0.491575	0.245788	+	0.969324i	$0.420953\pi$
$350$	4248.00	0.648758
$351$	0	0
$352$	416.000	0.0629911
$353$	3522.00	0.531040	0.265520	−	0.964105i	$-0.414456\pi$
$353$	3522.00	0.531040	0.265520	+	0.964105i	$0.414456\pi$
$354$	0	0
$355$	−18848.0	−2.81788
$356$	−3240.00	−0.482359
$357$	0	0
$358$	−4620.00	−0.682052
$359$	7365.00	1.08276	0.541379	−	0.840779i	$-0.317903\pi$
$359$	7365.00	1.08276	0.541379	+	0.840779i	$0.317903\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	164.000	0.0238112
$363$	0	0
$364$	1368.00	0.196985
$365$	−9633.00	−1.38141
$366$	0	0
$367$	10624.0	1.51109	0.755543	−	0.655099i	$-0.227373\pi$
$367$	10624.0	1.51109	0.755543	+	0.655099i	$0.227373\pi$
$368$	−1088.00	−0.154119
$369$	0	0
$370$	−11248.0	−1.58042
$371$	4518.00	0.632245
$372$	0	0
$373$	2528.00	0.350925	0.175462	−	0.984486i	$-0.443858\pi$
$373$	2528.00	0.350925	0.175462	+	0.984486i	$0.443858\pi$
$374$	−2574.00	−0.355878
$375$	0	0
$376$	2168.00	0.297357
$377$	−4940.00	−0.674862
$378$	0	0
$379$	−6810.00	−0.922972	−0.461486	−	0.887148i	$-0.652683\pi$
$379$	−6810.00	−0.922972	−0.461486	+	0.887148i	$0.652683\pi$
$380$	−1444.00	−0.194936
$381$	0	0
$382$	2626.00	0.351722
$383$	8222.00	1.09693	0.548466	−	0.836173i	$-0.315212\pi$
$383$	8222.00	1.09693	0.548466	+	0.836173i	$0.315212\pi$
$384$	0	0
$385$	2223.00	0.294272
$386$	−4704.00	−0.620278
$387$	0	0
$388$	−4184.00	−0.547450
$389$	−1725.00	−0.224835	−0.112418	−	0.993661i	$-0.535859\pi$
$389$	−1725.00	−0.224835	−0.112418	+	0.993661i	$0.535859\pi$
$390$	0	0
$391$	6732.00	0.870721
$392$	−2096.00	−0.270061
$393$	0	0
$394$	7652.00	0.978432
$395$	18620.0	2.37183
$396$	0	0
$397$	11669.0	1.47519	0.737595	−	0.675244i	$-0.235961\pi$
$397$	11669.0	1.47519	0.737595	+	0.675244i	$0.235961\pi$
$398$	4450.00	0.560448
$399$	0	0
$400$	3776.00	0.472000
$401$	−9592.00	−1.19452	−0.597259	−	0.802049i	$-0.703744\pi$
$401$	−9592.00	−1.19452	−0.597259	+	0.802049i	$0.703744\pi$
$402$	0	0
$403$	9956.00	1.23063
$404$	−6728.00	−0.828541
$405$	0	0
$406$	−2340.00	−0.286040
$407$	−3848.00	−0.468644
$408$	0	0
$409$	9130.00	1.10379	0.551894	−	0.833914i	$-0.313905\pi$
$409$	9130.00	1.10379	0.551894	+	0.833914i	$0.313905\pi$
$410$	304.000	0.0366183
$411$	0	0
$412$	3912.00	0.467792
$413$	−4860.00	−0.579044
$414$	0	0
$415$	9348.00	1.10572
$416$	1216.00	0.143316
$417$	0	0
$418$	−494.000	−0.0578046
$419$	3660.00	0.426737	0.213368	−	0.976972i	$-0.431557\pi$
$419$	3660.00	0.426737	0.213368	+	0.976972i	$0.431557\pi$
$420$	0	0
$421$	−13438.0	−1.55565	−0.777824	−	0.628482i	$-0.783677\pi$
$421$	−13438.0	−1.55565	−0.777824	+	0.628482i	$0.783677\pi$
$422$	3264.00	0.376514
$423$	0	0
$424$	4016.00	0.459986
$425$	−23364.0	−2.66664
$426$	0	0
$427$	5283.00	0.598741
$428$	−2696.00	−0.304477
$429$	0	0
$430$	2774.00	0.311103
$431$	1458.00	0.162945	0.0814726	−	0.996676i	$-0.474038\pi$
$431$	1458.00	0.162945	0.0814726	+	0.996676i	$0.474038\pi$
$432$	0	0
$433$	−13342.0	−1.48078	−0.740388	−	0.672180i	$-0.765358\pi$
$433$	−13342.0	−1.48078	−0.740388	+	0.672180i	$0.765358\pi$
$434$	4716.00	0.521602
$435$	0	0
$436$	−1440.00	−0.158173
$437$	1292.00	0.141430
$438$	0	0
$439$	4490.00	0.488146	0.244073	−	0.969757i	$-0.421516\pi$
$439$	4490.00	0.488146	0.244073	+	0.969757i	$0.421516\pi$
$440$	1976.00	0.214096
$441$	0	0
$442$	−7524.00	−0.809684
$443$	−2713.00	−0.290967	−0.145484	−	0.989361i	$-0.546474\pi$
$443$	−2713.00	−0.290967	−0.145484	+	0.989361i	$0.546474\pi$
$444$	0	0
$445$	−15390.0	−1.63945
$446$	2296.00	0.243764
$447$	0	0
$448$	576.000	0.0607443
$449$	−2140.00	−0.224928	−0.112464	−	0.993656i	$-0.535874\pi$
$449$	−2140.00	−0.224928	−0.112464	+	0.993656i	$0.535874\pi$
$450$	0	0
$451$	104.000	0.0108585
$452$	−7352.00	−0.765064
$453$	0	0
$454$	−11588.0	−1.19791
$455$	6498.00	0.669519
$456$	0	0
$457$	−2221.00	−0.227339	−0.113670	−	0.993519i	$-0.536261\pi$
$457$	−2221.00	−0.227339	−0.113670	+	0.993519i	$0.536261\pi$
$458$	5050.00	0.515221
$459$	0	0
$460$	−5168.00	−0.523824
$461$	−16907.0	−1.70811	−0.854054	−	0.520185i	$-0.825863\pi$
$461$	−16907.0	−1.70811	−0.854054	+	0.520185i	$0.825863\pi$
$462$	0	0
$463$	−2237.00	−0.224540	−0.112270	−	0.993678i	$-0.535812\pi$
$463$	−2237.00	−0.224540	−0.112270	+	0.993678i	$0.535812\pi$
$464$	−2080.00	−0.208107
$465$	0	0
$466$	2454.00	0.243947
$467$	311.000	0.0308166	0.0154083	−	0.999881i	$-0.495095\pi$
$467$	311.000	0.0308166	0.0154083	+	0.999881i	$0.495095\pi$
$468$	0	0
$469$	6156.00	0.606093
$470$	10298.0	1.01066
$471$	0	0
$472$	−4320.00	−0.421280
$473$	949.000	0.0922517
$474$	0	0
$475$	−4484.00	−0.433137
$476$	−3564.00	−0.343184
$477$	0	0
$478$	−1350.00	−0.129179
$479$	−12640.0	−1.20571	−0.602856	−	0.797850i	$-0.705971\pi$
$479$	−12640.0	−1.20571	−0.602856	+	0.797850i	$0.705971\pi$
$480$	0	0
$481$	−11248.0	−1.06625
$482$	−12136.0	−1.14685
$483$	0	0
$484$	−4648.00	−0.436514
$485$	−19874.0	−1.86068
$486$	0	0
$487$	13324.0	1.23977	0.619885	−	0.784693i	$-0.287179\pi$
$487$	13324.0	1.23977	0.619885	+	0.784693i	$0.287179\pi$
$488$	4696.00	0.435611
$489$	0	0
$490$	−9956.00	−0.917890
$491$	−5552.00	−0.510302	−0.255151	−	0.966901i	$-0.582125\pi$
$491$	−5552.00	−0.510302	−0.255151	+	0.966901i	$0.582125\pi$
$492$	0	0
$493$	12870.0	1.17573
$494$	−1444.00	−0.131515
$495$	0	0
$496$	4192.00	0.379489
$497$	−8928.00	−0.805786
$498$	0	0
$499$	−15125.0	−1.35689	−0.678445	−	0.734651i	$-0.737346\pi$
$499$	−15125.0	−1.35689	−0.678445	+	0.734651i	$0.737346\pi$
$500$	8436.00	0.754539
$501$	0	0
$502$	7526.00	0.669127
$503$	1632.00	0.144667	0.0723333	−	0.997381i	$-0.476955\pi$
$503$	1632.00	0.144667	0.0723333	+	0.997381i	$0.476955\pi$
$504$	0	0
$505$	−31958.0	−2.81606
$506$	−1768.00	−0.155330
$507$	0	0
$508$	−5544.00	−0.484203
$509$	−4410.00	−0.384027	−0.192014	−	0.981392i	$-0.561502\pi$
$509$	−4410.00	−0.384027	−0.192014	+	0.981392i	$0.561502\pi$
$510$	0	0
$511$	−4563.00	−0.395020
$512$	512.000	0.0441942
$513$	0	0
$514$	−8608.00	−0.738682
$515$	18582.0	1.58994
$516$	0	0
$517$	3523.00	0.299693
$518$	−5328.00	−0.451928
$519$	0	0
$520$	5776.00	0.487105
$521$	2488.00	0.209215	0.104608	−	0.994514i	$-0.466641\pi$
$521$	2488.00	0.209215	0.104608	+	0.994514i	$0.466641\pi$
$522$	0	0
$523$	−17662.0	−1.47668	−0.738342	−	0.674427i	$-0.764391\pi$
$523$	−17662.0	−1.47668	−0.738342	+	0.674427i	$0.764391\pi$
$524$	4332.00	0.361153
$525$	0	0
$526$	16494.0	1.36725
$527$	−25938.0	−2.14398
$528$	0	0
$529$	−7543.00	−0.619956
$530$	19076.0	1.56341
$531$	0	0
$532$	−684.000	−0.0557428
$533$	304.000	0.0247049
$534$	0	0
$535$	−12806.0	−1.03486
$536$	5472.00	0.440960
$537$	0	0
$538$	10860.0	0.870275
$539$	−3406.00	−0.272183
$540$	0	0
$541$	10487.0	0.833404	0.416702	−	0.909043i	$-0.363186\pi$
$541$	10487.0	0.833404	0.416702	+	0.909043i	$0.363186\pi$
$542$	10904.0	0.864146
$543$	0	0
$544$	−3168.00	−0.249682
$545$	−6840.00	−0.537603
$546$	0	0
$547$	13174.0	1.02976	0.514881	−	0.857262i	$-0.327836\pi$
$547$	13174.0	1.02976	0.514881	+	0.857262i	$0.327836\pi$
$548$	−1996.00	−0.155593
$549$	0	0
$550$	6136.00	0.475709
$551$	2470.00	0.190972
$552$	0	0
$553$	8820.00	0.678236
$554$	−12622.0	−0.967974
$555$	0	0
$556$	−780.000	−0.0594953
$557$	14321.0	1.08941	0.544704	−	0.838628i	$-0.316642\pi$
$557$	14321.0	1.08941	0.544704	+	0.838628i	$0.316642\pi$
$558$	0	0
$559$	2774.00	0.209889
$560$	2736.00	0.206459
$561$	0	0
$562$	12316.0	0.924411
$563$	7212.00	0.539875	0.269937	−	0.962878i	$-0.412997\pi$
$563$	7212.00	0.539875	0.269937	+	0.962878i	$0.412997\pi$
$564$	0	0
$565$	−34922.0	−2.60032
$566$	11706.0	0.869328
$567$	0	0
$568$	−7936.00	−0.586245
$569$	−3490.00	−0.257133	−0.128566	−	0.991701i	$-0.541038\pi$
$569$	−3490.00	−0.257133	−0.128566	+	0.991701i	$0.541038\pi$
$570$	0	0
$571$	6152.00	0.450881	0.225441	−	0.974257i	$-0.427618\pi$
$571$	6152.00	0.450881	0.225441	+	0.974257i	$0.427618\pi$
$572$	1976.00	0.144442
$573$	0	0
$574$	144.000	0.0104712
$575$	−16048.0	−1.16391
$576$	0	0
$577$	−20941.0	−1.51089	−0.755446	−	0.655210i	$-0.772580\pi$
$577$	−20941.0	−1.51089	−0.755446	+	0.655210i	$0.772580\pi$
$578$	9776.00	0.703509
$579$	0	0
$580$	−9880.00	−0.707318
$581$	4428.00	0.316187
$582$	0	0
$583$	6526.00	0.463601
$584$	−4056.00	−0.287395
$585$	0	0
$586$	11784.0	0.830704
$587$	23331.0	1.64050	0.820250	−	0.572005i	$-0.193834\pi$
$587$	23331.0	1.64050	0.820250	+	0.572005i	$0.193834\pi$
$588$	0	0
$589$	−4978.00	−0.348243
$590$	−20520.0	−1.43186
$591$	0	0
$592$	−4736.00	−0.328798
$593$	18542.0	1.28403	0.642014	−	0.766693i	$-0.278099\pi$
$593$	18542.0	1.28403	0.642014	+	0.766693i	$0.278099\pi$
$594$	0	0
$595$	−16929.0	−1.16642
$596$	5940.00	0.408241
$597$	0	0
$598$	−5168.00	−0.353403
$599$	2700.00	0.184172	0.0920860	−	0.995751i	$-0.470647\pi$
$599$	2700.00	0.184172	0.0920860	+	0.995751i	$0.470647\pi$
$600$	0	0
$601$	17452.0	1.18450	0.592248	−	0.805756i	$-0.298241\pi$
$601$	17452.0	1.18450	0.592248	+	0.805756i	$0.298241\pi$
$602$	1314.00	0.0889612
$603$	0	0
$604$	1328.00	0.0894628
$605$	−22078.0	−1.48363
$606$	0	0
$607$	5114.00	0.341962	0.170981	−	0.985274i	$-0.445306\pi$
$607$	5114.00	0.341962	0.170981	+	0.985274i	$0.445306\pi$
$608$	−608.000	−0.0405554
$609$	0	0
$610$	22306.0	1.48056
$611$	10298.0	0.681853
$612$	0	0
$613$	19333.0	1.27382	0.636911	−	0.770938i	$-0.280212\pi$
$613$	19333.0	1.27382	0.636911	+	0.770938i	$0.280212\pi$
$614$	3648.00	0.239774
$615$	0	0
$616$	936.000	0.0612216
$617$	−1599.00	−0.104333	−0.0521664	−	0.998638i	$-0.516613\pi$
$617$	−1599.00	−0.104333	−0.0521664	+	0.998638i	$0.516613\pi$
$618$	0	0
$619$	3580.00	0.232459	0.116230	−	0.993222i	$-0.462919\pi$
$619$	3580.00	0.232459	0.116230	+	0.993222i	$0.462919\pi$
$620$	19912.0	1.28981
$621$	0	0
$622$	−19514.0	−1.25794
$623$	−7290.00	−0.468808
$624$	0	0
$625$	10571.0	0.676544
$626$	−11444.0	−0.730662
$627$	0	0
$628$	15256.0	0.969396
$629$	29304.0	1.85759
$630$	0	0
$631$	18917.0	1.19346	0.596730	−	0.802442i	$-0.296466\pi$
$631$	18917.0	1.19346	0.596730	+	0.802442i	$0.296466\pi$
$632$	7840.00	0.493447
$633$	0	0
$634$	−20448.0	−1.28091
$635$	−26334.0	−1.64572
$636$	0	0
$637$	−9956.00	−0.619264
$638$	−3380.00	−0.209742
$639$	0	0
$640$	2432.00	0.150208
$641$	8058.00	0.496524	0.248262	−	0.968693i	$-0.420141\pi$
$641$	8058.00	0.496524	0.248262	+	0.968693i	$0.420141\pi$
$642$	0	0
$643$	4153.00	0.254710	0.127355	−	0.991857i	$-0.459351\pi$
$643$	4153.00	0.254710	0.127355	+	0.991857i	$0.459351\pi$
$644$	−2448.00	−0.149790
$645$	0	0
$646$	3762.00	0.229124
$647$	−17519.0	−1.06452	−0.532259	−	0.846582i	$-0.678657\pi$
$647$	−17519.0	−1.06452	−0.532259	+	0.846582i	$0.678657\pi$
$648$	0	0
$649$	−7020.00	−0.424590
$650$	17936.0	1.08232
$651$	0	0
$652$	−13408.0	−0.805365
$653$	3957.00	0.237135	0.118568	−	0.992946i	$-0.462170\pi$
$653$	3957.00	0.237135	0.118568	+	0.992946i	$0.462170\pi$
$654$	0	0
$655$	20577.0	1.22750
$656$	128.000	0.00761823
$657$	0	0
$658$	4878.00	0.289003
$659$	14890.0	0.880170	0.440085	−	0.897956i	$-0.354948\pi$
$659$	14890.0	0.880170	0.440085	+	0.897956i	$0.354948\pi$
$660$	0	0
$661$	11812.0	0.695058	0.347529	−	0.937669i	$-0.387021\pi$
$661$	11812.0	0.695058	0.347529	+	0.937669i	$0.387021\pi$
$662$	3424.00	0.201023
$663$	0	0
$664$	3936.00	0.230040
$665$	−3249.00	−0.189460
$666$	0	0
$667$	8840.00	0.513173
$668$	−14856.0	−0.860473
$669$	0	0
$670$	25992.0	1.49874
$671$	7631.00	0.439034
$672$	0	0
$673$	9308.00	0.533131	0.266565	−	0.963817i	$-0.414111\pi$
$673$	9308.00	0.533131	0.266565	+	0.963817i	$0.414111\pi$
$674$	1868.00	0.106755
$675$	0	0
$676$	−3012.00	−0.171370
$677$	−9814.00	−0.557138	−0.278569	−	0.960416i	$-0.589860\pi$
$677$	−9814.00	−0.557138	−0.278569	+	0.960416i	$0.589860\pi$
$678$	0	0
$679$	−9414.00	−0.532071
$680$	−15048.0	−0.848624
$681$	0	0
$682$	6812.00	0.382471
$683$	29942.0	1.67745	0.838725	−	0.544555i	$-0.183301\pi$
$683$	29942.0	1.67745	0.838725	+	0.544555i	$0.183301\pi$
$684$	0	0
$685$	−9481.00	−0.528833
$686$	−10890.0	−0.606096
$687$	0	0
$688$	1168.00	0.0647232
$689$	19076.0	1.05477
$690$	0	0
$691$	−25353.0	−1.39576	−0.697882	−	0.716212i	$-0.745874\pi$
$691$	−25353.0	−1.39576	−0.697882	+	0.716212i	$0.745874\pi$
$692$	4968.00	0.272912
$693$	0	0
$694$	12982.0	0.710072
$695$	−3705.00	−0.202214
$696$	0	0
$697$	−792.000	−0.0430404
$698$	6410.00	0.347596
$699$	0	0
$700$	8496.00	0.458741
$701$	5038.00	0.271445	0.135722	−	0.990747i	$-0.456665\pi$
$701$	5038.00	0.271445	0.135722	+	0.990747i	$0.456665\pi$
$702$	0	0
$703$	5624.00	0.301726
$704$	832.000	0.0445414
$705$	0	0
$706$	7044.00	0.375502
$707$	−15138.0	−0.805266
$708$	0	0
$709$	30150.0	1.59705	0.798524	−	0.601963i	$-0.205615\pi$
$709$	30150.0	1.59705	0.798524	+	0.601963i	$0.205615\pi$
$710$	−37696.0	−1.99254
$711$	0	0
$712$	−6480.00	−0.341079
$713$	−17816.0	−0.935785
$714$	0	0
$715$	9386.00	0.490932
$716$	−9240.00	−0.482284
$717$	0	0
$718$	14730.0	0.765625
$719$	425.000	0.0220443	0.0110221	−	0.999939i	$-0.496491\pi$
$719$	425.000	0.0220443	0.0110221	+	0.999939i	$0.496491\pi$
$720$	0	0
$721$	8802.00	0.454651
$722$	722.000	0.0372161
$723$	0	0
$724$	328.000	0.0168370
$725$	−30680.0	−1.57162
$726$	0	0
$727$	−18941.0	−0.966276	−0.483138	−	0.875544i	$-0.660503\pi$
$727$	−18941.0	−0.966276	−0.483138	+	0.875544i	$0.660503\pi$
$728$	2736.00	0.139290
$729$	0	0
$730$	−19266.0	−0.976804
$731$	−7227.00	−0.365664
$732$	0	0
$733$	−27702.0	−1.39590	−0.697951	−	0.716145i	$-0.745905\pi$
$733$	−27702.0	−1.39590	−0.697951	+	0.716145i	$0.745905\pi$
$734$	21248.0	1.06850
$735$	0	0
$736$	−2176.00	−0.108979
$737$	8892.00	0.444425
$738$	0	0
$739$	4685.00	0.233208	0.116604	−	0.993179i	$-0.462799\pi$
$739$	4685.00	0.233208	0.116604	+	0.993179i	$0.462799\pi$
$740$	−22496.0	−1.11753
$741$	0	0
$742$	9036.00	0.447065
$743$	36172.0	1.78603	0.893016	−	0.450025i	$-0.148585\pi$
$743$	36172.0	1.78603	0.893016	+	0.450025i	$0.148585\pi$
$744$	0	0
$745$	28215.0	1.38754
$746$	5056.00	0.248141
$747$	0	0
$748$	−5148.00	−0.251644
$749$	−6066.00	−0.295924
$750$	0	0
$751$	2912.00	0.141492	0.0707459	−	0.997494i	$-0.477462\pi$
$751$	2912.00	0.141492	0.0707459	+	0.997494i	$0.477462\pi$
$752$	4336.00	0.210263
$753$	0	0
$754$	−9880.00	−0.477199
$755$	6308.00	0.304068
$756$	0	0
$757$	27259.0	1.30878	0.654389	−	0.756158i	$-0.272926\pi$
$757$	27259.0	1.30878	0.654389	+	0.756158i	$0.272926\pi$
$758$	−13620.0	−0.652639
$759$	0	0
$760$	−2888.00	−0.137840
$761$	−147.000	−0.00700229	−0.00350115	−	0.999994i	$-0.501114\pi$
$761$	−147.000	−0.00700229	−0.00350115	+	0.999994i	$0.501114\pi$
$762$	0	0
$763$	−3240.00	−0.153730
$764$	5252.00	0.248705
$765$	0	0
$766$	16444.0	0.775647
$767$	−20520.0	−0.966016
$768$	0	0
$769$	−14645.0	−0.686752	−0.343376	−	0.939198i	$-0.611570\pi$
$769$	−14645.0	−0.686752	−0.343376	+	0.939198i	$0.611570\pi$
$770$	4446.00	0.208081
$771$	0	0
$772$	−9408.00	−0.438603
$773$	1632.00	0.0759366	0.0379683	−	0.999279i	$-0.487911\pi$
$773$	1632.00	0.0759366	0.0379683	+	0.999279i	$0.487911\pi$
$774$	0	0
$775$	61832.0	2.86590
$776$	−8368.00	−0.387105
$777$	0	0
$778$	−3450.00	−0.158983
$779$	−152.000	−0.00699097
$780$	0	0
$781$	−12896.0	−0.590852
$782$	13464.0	0.615693
$783$	0	0
$784$	−4192.00	−0.190962
$785$	72466.0	3.29481
$786$	0	0
$787$	−15776.0	−0.714554	−0.357277	−	0.933999i	$-0.616295\pi$
$787$	−15776.0	−0.714554	−0.357277	+	0.933999i	$0.616295\pi$
$788$	15304.0	0.691856
$789$	0	0
$790$	37240.0	1.67714
$791$	−16542.0	−0.743572
$792$	0	0
$793$	22306.0	0.998877
$794$	23338.0	1.04312
$795$	0	0
$796$	8900.00	0.396297
$797$	17256.0	0.766925	0.383462	−	0.923557i	$-0.374732\pi$
$797$	17256.0	0.766925	0.383462	+	0.923557i	$0.374732\pi$
$798$	0	0
$799$	−26829.0	−1.18791
$800$	7552.00	0.333754
$801$	0	0
$802$	−19184.0	−0.844652
$803$	−6591.00	−0.289653
$804$	0	0
$805$	−11628.0	−0.509110
$806$	19912.0	0.870186
$807$	0	0
$808$	−13456.0	−0.585867
$809$	−25845.0	−1.12319	−0.561596	−	0.827412i	$-0.689812\pi$
$809$	−25845.0	−1.12319	−0.561596	+	0.827412i	$0.689812\pi$
$810$	0	0
$811$	10962.0	0.474634	0.237317	−	0.971432i	$-0.423732\pi$
$811$	10962.0	0.474634	0.237317	+	0.971432i	$0.423732\pi$
$812$	−4680.00	−0.202261
$813$	0	0
$814$	−7696.00	−0.331382
$815$	−63688.0	−2.73729
$816$	0	0
$817$	−1387.00	−0.0593941
$818$	18260.0	0.780496
$819$	0	0
$820$	608.000	0.0258930
$821$	−627.000	−0.0266534	−0.0133267	−	0.999911i	$-0.504242\pi$
$821$	−627.000	−0.0266534	−0.0133267	+	0.999911i	$0.504242\pi$
$822$	0	0
$823$	38153.0	1.61595	0.807977	−	0.589214i	$-0.200563\pi$
$823$	38153.0	1.61595	0.807977	+	0.589214i	$0.200563\pi$
$824$	7824.00	0.330779
$825$	0	0
$826$	−9720.00	−0.409446
$827$	−23744.0	−0.998379	−0.499190	−	0.866493i	$-0.666369\pi$
$827$	−23744.0	−0.998379	−0.499190	+	0.866493i	$0.666369\pi$
$828$	0	0
$829$	−18060.0	−0.756634	−0.378317	−	0.925676i	$-0.623497\pi$
$829$	−18060.0	−0.756634	−0.378317	+	0.925676i	$0.623497\pi$
$830$	18696.0	0.781865
$831$	0	0
$832$	2432.00	0.101339
$833$	25938.0	1.07887
$834$	0	0
$835$	−70566.0	−2.92460
$836$	−988.000	−0.0408740
$837$	0	0
$838$	7320.00	0.301748
$839$	8590.00	0.353468	0.176734	−	0.984259i	$-0.443447\pi$
$839$	8590.00	0.353468	0.176734	+	0.984259i	$0.443447\pi$
$840$	0	0
$841$	−7489.00	−0.307065
$842$	−26876.0	−1.10001
$843$	0	0
$844$	6528.00	0.266236
$845$	−14307.0	−0.582457
$846$	0	0
$847$	−10458.0	−0.424252
$848$	8032.00	0.325259
$849$	0	0
$850$	−46728.0	−1.88560
$851$	20128.0	0.810786
$852$	0	0
$853$	−222.000	−0.00891106	−0.00445553	−	0.999990i	$-0.501418\pi$
$853$	−222.000	−0.00891106	−0.00445553	+	0.999990i	$0.501418\pi$
$854$	10566.0	0.423374
$855$	0	0
$856$	−5392.00	−0.215298
$857$	16956.0	0.675853	0.337926	−	0.941173i	$-0.390274\pi$
$857$	16956.0	0.675853	0.337926	+	0.941173i	$0.390274\pi$
$858$	0	0
$859$	11245.0	0.446652	0.223326	−	0.974744i	$-0.428308\pi$
$859$	11245.0	0.446652	0.223326	+	0.974744i	$0.428308\pi$
$860$	5548.00	0.219983
$861$	0	0
$862$	2916.00	0.115220
$863$	−12468.0	−0.491791	−0.245896	−	0.969296i	$-0.579082\pi$
$863$	−12468.0	−0.491791	−0.245896	+	0.969296i	$0.579082\pi$
$864$	0	0
$865$	23598.0	0.927579
$866$	−26684.0	−1.04707
$867$	0	0
$868$	9432.00	0.368828
$869$	12740.0	0.497324
$870$	0	0
$871$	25992.0	1.01114
$872$	−2880.00	−0.111845
$873$	0	0
$874$	2584.00	0.100006
$875$	18981.0	0.733343
$876$	0	0
$877$	8194.00	0.315498	0.157749	−	0.987479i	$-0.449576\pi$
$877$	8194.00	0.315498	0.157749	+	0.987479i	$0.449576\pi$
$878$	8980.00	0.345171
$879$	0	0
$880$	3952.00	0.151389
$881$	28833.0	1.10262	0.551310	−	0.834300i	$-0.314128\pi$
$881$	28833.0	1.10262	0.551310	+	0.834300i	$0.314128\pi$
$882$	0	0
$883$	29413.0	1.12098	0.560491	−	0.828161i	$-0.310613\pi$
$883$	29413.0	1.12098	0.560491	+	0.828161i	$0.310613\pi$
$884$	−15048.0	−0.572533
$885$	0	0
$886$	−5426.00	−0.205745
$887$	−24544.0	−0.929095	−0.464547	−	0.885548i	$-0.653783\pi$
$887$	−24544.0	−0.929095	−0.464547	+	0.885548i	$0.653783\pi$
$888$	0	0
$889$	−12474.0	−0.470601
$890$	−30780.0	−1.15927
$891$	0	0
$892$	4592.00	0.172367
$893$	−5149.00	−0.192950
$894$	0	0
$895$	−43890.0	−1.63920
$896$	1152.00	0.0429527
$897$	0	0
$898$	−4280.00	−0.159048
$899$	−34060.0	−1.26359
$900$	0	0
$901$	−49698.0	−1.83760
$902$	208.000	0.00767810
$903$	0	0
$904$	−14704.0	−0.540982
$905$	1558.00	0.0572262
$906$	0	0
$907$	−26326.0	−0.963771	−0.481886	−	0.876234i	$-0.660048\pi$
$907$	−26326.0	−0.963771	−0.481886	+	0.876234i	$0.660048\pi$
$908$	−23176.0	−0.847051
$909$	0	0
$910$	12996.0	0.473421
$911$	42738.0	1.55431	0.777153	−	0.629311i	$-0.216663\pi$
$911$	42738.0	1.55431	0.777153	+	0.629311i	$0.216663\pi$
$912$	0	0
$913$	6396.00	0.231847
$914$	−4442.00	−0.160753
$915$	0	0
$916$	10100.0	0.364316
$917$	9747.00	0.351008
$918$	0	0
$919$	−38680.0	−1.38840	−0.694198	−	0.719784i	$-0.744241\pi$
$919$	−38680.0	−1.38840	−0.694198	+	0.719784i	$0.744241\pi$
$920$	−10336.0	−0.370400
$921$	0	0
$922$	−33814.0	−1.20781
$923$	−37696.0	−1.34429
$924$	0	0
$925$	−69856.0	−2.48308
$926$	−4474.00	−0.158774
$927$	0	0
$928$	−4160.00	−0.147154
$929$	−32270.0	−1.13966	−0.569830	−	0.821763i	$-0.692991\pi$
$929$	−32270.0	−1.13966	−0.569830	+	0.821763i	$0.692991\pi$
$930$	0	0
$931$	4978.00	0.175239
$932$	4908.00	0.172497
$933$	0	0
$934$	622.000	0.0217906
$935$	−24453.0	−0.855293
$936$	0	0
$937$	30989.0	1.08043	0.540217	−	0.841526i	$-0.318342\pi$
$937$	30989.0	1.08043	0.540217	+	0.841526i	$0.318342\pi$
$938$	12312.0	0.428573
$939$	0	0
$940$	20596.0	0.714646
$941$	8318.00	0.288161	0.144080	−	0.989566i	$-0.453978\pi$
$941$	8318.00	0.288161	0.144080	+	0.989566i	$0.453978\pi$
$942$	0	0
$943$	−544.000	−0.0187859
$944$	−8640.00	−0.297890
$945$	0	0
$946$	1898.00	0.0652318
$947$	46716.0	1.60303	0.801513	−	0.597977i	$-0.204029\pi$
$947$	46716.0	1.60303	0.801513	+	0.597977i	$0.204029\pi$
$948$	0	0
$949$	−19266.0	−0.659010
$950$	−8968.00	−0.306274
$951$	0	0
$952$	−7128.00	−0.242668
$953$	50212.0	1.70674	0.853372	−	0.521303i	$-0.174554\pi$
$953$	50212.0	1.70674	0.853372	+	0.521303i	$0.174554\pi$
$954$	0	0
$955$	24947.0	0.845305
$956$	−2700.00	−0.0913433
$957$	0	0
$958$	−25280.0	−0.852568
$959$	−4491.00	−0.151222
$960$	0	0
$961$	38853.0	1.30419
$962$	−22496.0	−0.753950
$963$	0	0
$964$	−24272.0	−0.810942
$965$	−44688.0	−1.49073
$966$	0	0
$967$	−12336.0	−0.410237	−0.205118	−	0.978737i	$-0.565758\pi$
$967$	−12336.0	−0.410237	−0.205118	+	0.978737i	$0.565758\pi$
$968$	−9296.00	−0.308662
$969$	0	0
$970$	−39748.0	−1.31570
$971$	−40532.0	−1.33958	−0.669791	−	0.742550i	$-0.733616\pi$
$971$	−40532.0	−1.33958	−0.669791	+	0.742550i	$0.733616\pi$
$972$	0	0
$973$	−1755.00	−0.0578240
$974$	26648.0	0.876650
$975$	0	0
$976$	9392.00	0.308023
$977$	19426.0	0.636124	0.318062	−	0.948070i	$-0.396968\pi$
$977$	19426.0	0.636124	0.318062	+	0.948070i	$0.396968\pi$
$978$	0	0
$979$	−10530.0	−0.343759
$980$	−19912.0	−0.649046
$981$	0	0
$982$	−11104.0	−0.360838
$983$	−34428.0	−1.11707	−0.558536	−	0.829480i	$-0.688637\pi$
$983$	−34428.0	−1.11707	−0.558536	+	0.829480i	$0.688637\pi$
$984$	0	0
$985$	72694.0	2.35150
$986$	25740.0	0.831368
$987$	0	0
$988$	−2888.00	−0.0929955
$989$	−4964.00	−0.159602
$990$	0	0
$991$	10072.0	0.322853	0.161427	−	0.986885i	$-0.448390\pi$
$991$	10072.0	0.322853	0.161427	+	0.986885i	$0.448390\pi$
$992$	8384.00	0.268339
$993$	0	0
$994$	−17856.0	−0.569777
$995$	42275.0	1.34694
$996$	0	0
$997$	12589.0	0.399897	0.199949	−	0.979806i	$-0.435922\pi$
$997$	12589.0	0.399897	0.199949	+	0.979806i	$0.435922\pi$
$998$	−30250.0	−0.959466
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	342.4.a.e.1.1		1
3.2	odd	2		114.4.a.a.1.1	✓	1
12.11	even	2		912.4.a.e.1.1		1
57.56	even	2		2166.4.a.h.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.4.a.a.1.1	✓	1	3.2	odd	2
342.4.a.e.1.1		1	1.1	even	1	trivial
912.4.a.e.1.1		1	12.11	even	2
2166.4.a.h.1.1		1	57.56	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	342.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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