LMFDB - Embedded newform 1638.4.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1638,4,Mod(1,1638)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1638, 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("1638.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1638.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:	$96.6451285894$
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	$\chi$	$=$	1638.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+2.00000 q^{2} +4.00000 q^{4} +3.00000 q^{5} -7.00000 q^{7} +8.00000 q^{8} +O(q^{10})$

\(q+2.00000 q^{2} +4.00000 q^{4} +3.00000 q^{5} -7.00000 q^{7} +8.00000 q^{8} +6.00000 q^{10} +23.0000 q^{11} -13.0000 q^{13} -14.0000 q^{14} +16.0000 q^{16} -133.000 q^{17} +103.000 q^{19} +12.0000 q^{20} +46.0000 q^{22} -113.000 q^{23} -116.000 q^{25} -26.0000 q^{26} -28.0000 q^{28} +141.000 q^{29} -90.0000 q^{31} +32.0000 q^{32} -266.000 q^{34} -21.0000 q^{35} -233.000 q^{37} +206.000 q^{38} +24.0000 q^{40} +230.000 q^{41} -337.000 q^{43} +92.0000 q^{44} -226.000 q^{46} -96.0000 q^{47} +49.0000 q^{49} -232.000 q^{50} -52.0000 q^{52} +134.000 q^{53} +69.0000 q^{55} -56.0000 q^{56} +282.000 q^{58} -838.000 q^{59} +295.000 q^{61} -180.000 q^{62} +64.0000 q^{64} -39.0000 q^{65} +468.000 q^{67} -532.000 q^{68} -42.0000 q^{70} -54.0000 q^{71} -553.000 q^{73} -466.000 q^{74} +412.000 q^{76} -161.000 q^{77} -694.000 q^{79} +48.0000 q^{80} +460.000 q^{82} +1010.00 q^{83} -399.000 q^{85} -674.000 q^{86} +184.000 q^{88} -390.000 q^{89} +91.0000 q^{91} -452.000 q^{92} -192.000 q^{94} +309.000 q^{95} -102.000 q^{97} +98.0000 q^{98} +O(q^{100})\)

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
$5$	3.00000	0.268328	0.134164	−	0.990959i	$-0.457165\pi$
$5$	3.00000	0.268328	0.134164	+	0.990959i	$0.457165\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	8.00000	0.353553
$9$	0	0
$10$	6.00000	0.189737
$11$	23.0000	0.630433	0.315216	−	0.949020i	$-0.397923\pi$
$11$	23.0000	0.630433	0.315216	+	0.949020i	$0.397923\pi$
$12$	0	0
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	−14.0000	−0.267261
$15$	0	0
$16$	16.0000	0.250000
$17$	−133.000	−1.89748	−0.948742	−	0.316051i	$-0.897643\pi$
$17$	−133.000	−1.89748	−0.948742	+	0.316051i	$0.897643\pi$
$18$	0	0
$19$	103.000	1.24367	0.621837	−	0.783146i	$-0.286386\pi$
$19$	103.000	1.24367	0.621837	+	0.783146i	$0.286386\pi$
$20$	12.0000	0.134164
$21$	0	0
$22$	46.0000	0.445783
$23$	−113.000	−1.02444	−0.512220	−	0.858854i	$-0.671177\pi$
$23$	−113.000	−1.02444	−0.512220	+	0.858854i	$0.671177\pi$
$24$	0	0
$25$	−116.000	−0.928000
$26$	−26.0000	−0.196116
$27$	0	0
$28$	−28.0000	−0.188982
$29$	141.000	0.902864	0.451432	−	0.892306i	$-0.350913\pi$
$29$	141.000	0.902864	0.451432	+	0.892306i	$0.350913\pi$
$30$	0	0
$31$	−90.0000	−0.521435	−0.260717	−	0.965415i	$-0.583959\pi$
$31$	−90.0000	−0.521435	−0.260717	+	0.965415i	$0.583959\pi$
$32$	32.0000	0.176777
$33$	0	0
$34$	−266.000	−1.34172
$35$	−21.0000	−0.101419
$36$	0	0
$37$	−233.000	−1.03527	−0.517635	−	0.855602i	$-0.673187\pi$
$37$	−233.000	−1.03527	−0.517635	+	0.855602i	$0.673187\pi$
$38$	206.000	0.879411
$39$	0	0
$40$	24.0000	0.0948683
$41$	230.000	0.876097	0.438048	−	0.898951i	$-0.355670\pi$
$41$	230.000	0.876097	0.438048	+	0.898951i	$0.355670\pi$
$42$	0	0
$43$	−337.000	−1.19516	−0.597582	−	0.801808i	$-0.703872\pi$
$43$	−337.000	−1.19516	−0.597582	+	0.801808i	$0.703872\pi$
$44$	92.0000	0.315216
$45$	0	0
$46$	−226.000	−0.724389
$47$	−96.0000	−0.297937	−0.148969	−	0.988842i	$-0.547595\pi$
$47$	−96.0000	−0.297937	−0.148969	+	0.988842i	$0.547595\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	−232.000	−0.656195
$51$	0	0
$52$	−52.0000	−0.138675
$53$	134.000	0.347289	0.173644	−	0.984808i	$-0.444446\pi$
$53$	134.000	0.347289	0.173644	+	0.984808i	$0.444446\pi$
$54$	0	0
$55$	69.0000	0.169163
$56$	−56.0000	−0.133631
$57$	0	0
$58$	282.000	0.638421
$59$	−838.000	−1.84912	−0.924562	−	0.381032i	$-0.875569\pi$
$59$	−838.000	−1.84912	−0.924562	+	0.381032i	$0.875569\pi$
$60$	0	0
$61$	295.000	0.619195	0.309597	−	0.950868i	$-0.399806\pi$
$61$	295.000	0.619195	0.309597	+	0.950868i	$0.399806\pi$
$62$	−180.000	−0.368710
$63$	0	0
$64$	64.0000	0.125000
$65$	−39.0000	−0.0744208
$66$	0	0
$67$	468.000	0.853363	0.426681	−	0.904402i	$-0.359683\pi$
$67$	468.000	0.853363	0.426681	+	0.904402i	$0.359683\pi$
$68$	−532.000	−0.948742
$69$	0	0
$70$	−42.0000	−0.0717137
$71$	−54.0000	−0.0902623	−0.0451311	−	0.998981i	$-0.514371\pi$
$71$	−54.0000	−0.0902623	−0.0451311	+	0.998981i	$0.514371\pi$
$72$	0	0
$73$	−553.000	−0.886627	−0.443313	−	0.896367i	$-0.646197\pi$
$73$	−553.000	−0.886627	−0.443313	+	0.896367i	$0.646197\pi$
$74$	−466.000	−0.732046
$75$	0	0
$76$	412.000	0.621837
$77$	−161.000	−0.238281
$78$	0	0
$79$	−694.000	−0.988368	−0.494184	−	0.869357i	$-0.664533\pi$
$79$	−694.000	−0.988368	−0.494184	+	0.869357i	$0.664533\pi$
$80$	48.0000	0.0670820
$81$	0	0
$82$	460.000	0.619494
$83$	1010.00	1.33569	0.667843	−	0.744302i	$-0.267218\pi$
$83$	1010.00	1.33569	0.667843	+	0.744302i	$0.267218\pi$
$84$	0	0
$85$	−399.000	−0.509149
$86$	−674.000	−0.845108
$87$	0	0
$88$	184.000	0.222892
$89$	−390.000	−0.464493	−0.232247	−	0.972657i	$-0.574608\pi$
$89$	−390.000	−0.464493	−0.232247	+	0.972657i	$0.574608\pi$
$90$	0	0
$91$	91.0000	0.104828
$92$	−452.000	−0.512220
$93$	0	0
$94$	−192.000	−0.210673
$95$	309.000	0.333713
$96$	0	0
$97$	−102.000	−0.106768	−0.0533842	−	0.998574i	$-0.517001\pi$
$97$	−102.000	−0.106768	−0.0533842	+	0.998574i	$0.517001\pi$
$98$	98.0000	0.101015
$99$	0	0
$100$	−464.000	−0.464000
$101$	−1590.00	−1.56644	−0.783222	−	0.621742i	$-0.786425\pi$
$101$	−1590.00	−1.56644	−0.783222	+	0.621742i	$0.786425\pi$
$102$	0	0
$103$	1679.00	1.60618	0.803091	−	0.595856i	$-0.203187\pi$
$103$	1679.00	1.60618	0.803091	+	0.595856i	$0.203187\pi$
$104$	−104.000	−0.0980581
$105$	0	0
$106$	268.000	0.245570
$107$	−30.0000	−0.0271048	−0.0135524	−	0.999908i	$-0.504314\pi$
$107$	−30.0000	−0.0271048	−0.0135524	+	0.999908i	$0.504314\pi$
$108$	0	0
$109$	−1423.00	−1.25045	−0.625223	−	0.780446i	$-0.714992\pi$
$109$	−1423.00	−1.25045	−0.625223	+	0.780446i	$0.714992\pi$
$110$	138.000	0.119616
$111$	0	0
$112$	−112.000	−0.0944911
$113$	−844.000	−0.702627	−0.351313	−	0.936258i	$-0.614265\pi$
$113$	−844.000	−0.702627	−0.351313	+	0.936258i	$0.614265\pi$
$114$	0	0
$115$	−339.000	−0.274886
$116$	564.000	0.451432
$117$	0	0
$118$	−1676.00	−1.30753
$119$	931.000	0.717182
$120$	0	0
$121$	−802.000	−0.602554
$122$	590.000	0.437837
$123$	0	0
$124$	−360.000	−0.260717
$125$	−723.000	−0.517337
$126$	0	0
$127$	328.000	0.229176	0.114588	−	0.993413i	$-0.463445\pi$
$127$	328.000	0.229176	0.114588	+	0.993413i	$0.463445\pi$
$128$	128.000	0.0883883
$129$	0	0
$130$	−78.0000	−0.0526235
$131$	267.000	0.178076	0.0890378	−	0.996028i	$-0.471621\pi$
$131$	267.000	0.178076	0.0890378	+	0.996028i	$0.471621\pi$
$132$	0	0
$133$	−721.000	−0.470065
$134$	936.000	0.603419
$135$	0	0
$136$	−1064.00	−0.670862
$137$	−1503.00	−0.937299	−0.468649	−	0.883384i	$-0.655259\pi$
$137$	−1503.00	−0.937299	−0.468649	+	0.883384i	$0.655259\pi$
$138$	0	0
$139$	958.000	0.584579	0.292290	−	0.956330i	$-0.405583\pi$
$139$	958.000	0.584579	0.292290	+	0.956330i	$0.405583\pi$
$140$	−84.0000	−0.0507093
$141$	0	0
$142$	−108.000	−0.0638251
$143$	−299.000	−0.174851
$144$	0	0
$145$	423.000	0.242264
$146$	−1106.00	−0.626940
$147$	0	0
$148$	−932.000	−0.517635
$149$	−1864.00	−1.02486	−0.512432	−	0.858728i	$-0.671255\pi$
$149$	−1864.00	−1.02486	−0.512432	+	0.858728i	$0.671255\pi$
$150$	0	0
$151$	−929.000	−0.500669	−0.250334	−	0.968159i	$-0.580541\pi$
$151$	−929.000	−0.500669	−0.250334	+	0.968159i	$0.580541\pi$
$152$	824.000	0.439705
$153$	0	0
$154$	−322.000	−0.168490
$155$	−270.000	−0.139916
$156$	0	0
$157$	−85.0000	−0.0432085	−0.0216043	−	0.999767i	$-0.506877\pi$
$157$	−85.0000	−0.0432085	−0.0216043	+	0.999767i	$0.506877\pi$
$158$	−1388.00	−0.698882
$159$	0	0
$160$	96.0000	0.0474342
$161$	791.000	0.387202
$162$	0	0
$163$	−3002.00	−1.44254	−0.721272	−	0.692652i	$-0.756442\pi$
$163$	−3002.00	−1.44254	−0.721272	+	0.692652i	$0.756442\pi$
$164$	920.000	0.438048
$165$	0	0
$166$	2020.00	0.944472
$167$	−1897.00	−0.879008	−0.439504	−	0.898241i	$-0.644846\pi$
$167$	−1897.00	−0.879008	−0.439504	+	0.898241i	$0.644846\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	−798.000	−0.360022
$171$	0	0
$172$	−1348.00	−0.597582
$173$	−2222.00	−0.976506	−0.488253	−	0.872702i	$-0.662366\pi$
$173$	−2222.00	−0.976506	−0.488253	+	0.872702i	$0.662366\pi$
$174$	0	0
$175$	812.000	0.350751
$176$	368.000	0.157608
$177$	0	0
$178$	−780.000	−0.328446
$179$	−20.0000	−0.00835123	−0.00417562	−	0.999991i	$-0.501329\pi$
$179$	−20.0000	−0.00835123	−0.00417562	+	0.999991i	$0.501329\pi$
$180$	0	0
$181$	4750.00	1.95063	0.975317	−	0.220810i	$-0.0708700\pi$
$181$	4750.00	1.95063	0.975317	+	0.220810i	$0.0708700\pi$
$182$	182.000	0.0741249
$183$	0	0
$184$	−904.000	−0.362194
$185$	−699.000	−0.277792
$186$	0	0
$187$	−3059.00	−1.19624
$188$	−384.000	−0.148969
$189$	0	0
$190$	618.000	0.235971
$191$	−909.000	−0.344361	−0.172180	−	0.985065i	$-0.555081\pi$
$191$	−909.000	−0.344361	−0.172180	+	0.985065i	$0.555081\pi$
$192$	0	0
$193$	442.000	0.164849	0.0824245	−	0.996597i	$-0.473734\pi$
$193$	442.000	0.164849	0.0824245	+	0.996597i	$0.473734\pi$
$194$	−204.000	−0.0754966
$195$	0	0
$196$	196.000	0.0714286
$197$	−3286.00	−1.18842	−0.594208	−	0.804312i	$-0.702534\pi$
$197$	−3286.00	−1.18842	−0.594208	+	0.804312i	$0.702534\pi$
$198$	0	0
$199$	3993.00	1.42239	0.711197	−	0.702993i	$-0.248154\pi$
$199$	3993.00	1.42239	0.711197	+	0.702993i	$0.248154\pi$
$200$	−928.000	−0.328098
$201$	0	0
$202$	−3180.00	−1.10764
$203$	−987.000	−0.341250
$204$	0	0
$205$	690.000	0.235081
$206$	3358.00	1.13574
$207$	0	0
$208$	−208.000	−0.0693375
$209$	2369.00	0.784053
$210$	0	0
$211$	3445.00	1.12400	0.561999	−	0.827138i	$-0.310032\pi$
$211$	3445.00	1.12400	0.561999	+	0.827138i	$0.310032\pi$
$212$	536.000	0.173644
$213$	0	0
$214$	−60.0000	−0.0191660
$215$	−1011.00	−0.320696
$216$	0	0
$217$	630.000	0.197084
$218$	−2846.00	−0.884199
$219$	0	0
$220$	276.000	0.0845814
$221$	1729.00	0.526268
$222$	0	0
$223$	−1420.00	−0.426414	−0.213207	−	0.977007i	$-0.568391\pi$
$223$	−1420.00	−0.426414	−0.213207	+	0.977007i	$0.568391\pi$
$224$	−224.000	−0.0668153
$225$	0	0
$226$	−1688.00	−0.496832
$227$	510.000	0.149118	0.0745592	−	0.997217i	$-0.476245\pi$
$227$	510.000	0.149118	0.0745592	+	0.997217i	$0.476245\pi$
$228$	0	0
$229$	−1746.00	−0.503838	−0.251919	−	0.967748i	$-0.581062\pi$
$229$	−1746.00	−0.503838	−0.251919	+	0.967748i	$0.581062\pi$
$230$	−678.000	−0.194374
$231$	0	0
$232$	1128.00	0.319210
$233$	−1020.00	−0.286792	−0.143396	−	0.989665i	$-0.545802\pi$
$233$	−1020.00	−0.286792	−0.143396	+	0.989665i	$0.545802\pi$
$234$	0	0
$235$	−288.000	−0.0799449
$236$	−3352.00	−0.924562
$237$	0	0
$238$	1862.00	0.507124
$239$	2824.00	0.764307	0.382154	−	0.924099i	$-0.375183\pi$
$239$	2824.00	0.764307	0.382154	+	0.924099i	$0.375183\pi$
$240$	0	0
$241$	2538.00	0.678369	0.339185	−	0.940720i	$-0.389849\pi$
$241$	2538.00	0.678369	0.339185	+	0.940720i	$0.389849\pi$
$242$	−1604.00	−0.426070
$243$	0	0
$244$	1180.00	0.309597
$245$	147.000	0.0383326
$246$	0	0
$247$	−1339.00	−0.344933
$248$	−720.000	−0.184355
$249$	0	0
$250$	−1446.00	−0.365812
$251$	−4725.00	−1.18820	−0.594102	−	0.804389i	$-0.702493\pi$
$251$	−4725.00	−1.18820	−0.594102	+	0.804389i	$0.702493\pi$
$252$	0	0
$253$	−2599.00	−0.645841
$254$	656.000	0.162052
$255$	0	0
$256$	256.000	0.0625000
$257$	2642.00	0.641258	0.320629	−	0.947205i	$-0.396106\pi$
$257$	2642.00	0.641258	0.320629	+	0.947205i	$0.396106\pi$
$258$	0	0
$259$	1631.00	0.391295
$260$	−156.000	−0.0372104
$261$	0	0
$262$	534.000	0.125918
$263$	5152.00	1.20793	0.603966	−	0.797010i	$-0.293586\pi$
$263$	5152.00	1.20793	0.603966	+	0.797010i	$0.293586\pi$
$264$	0	0
$265$	402.000	0.0931874
$266$	−1442.00	−0.332386
$267$	0	0
$268$	1872.00	0.426681
$269$	−518.000	−0.117409	−0.0587045	−	0.998275i	$-0.518697\pi$
$269$	−518.000	−0.117409	−0.0587045	+	0.998275i	$0.518697\pi$
$270$	0	0
$271$	−3642.00	−0.816368	−0.408184	−	0.912900i	$-0.633838\pi$
$271$	−3642.00	−0.816368	−0.408184	+	0.912900i	$0.633838\pi$
$272$	−2128.00	−0.474371
$273$	0	0
$274$	−3006.00	−0.662770
$275$	−2668.00	−0.585042
$276$	0	0
$277$	4876.00	1.05766	0.528828	−	0.848729i	$-0.322632\pi$
$277$	4876.00	1.05766	0.528828	+	0.848729i	$0.322632\pi$
$278$	1916.00	0.413360
$279$	0	0
$280$	−168.000	−0.0358569
$281$	−2202.00	−0.467474	−0.233737	−	0.972300i	$-0.575096\pi$
$281$	−2202.00	−0.467474	−0.233737	+	0.972300i	$0.575096\pi$
$282$	0	0
$283$	−4036.00	−0.847757	−0.423879	−	0.905719i	$-0.639332\pi$
$283$	−4036.00	−0.847757	−0.423879	+	0.905719i	$0.639332\pi$
$284$	−216.000	−0.0451311
$285$	0	0
$286$	−598.000	−0.123638
$287$	−1610.00	−0.331133
$288$	0	0
$289$	12776.0	2.60045
$290$	846.000	0.171306
$291$	0	0
$292$	−2212.00	−0.443313
$293$	−5346.00	−1.06593	−0.532964	−	0.846138i	$-0.678922\pi$
$293$	−5346.00	−1.06593	−0.532964	+	0.846138i	$0.678922\pi$
$294$	0	0
$295$	−2514.00	−0.496172
$296$	−1864.00	−0.366023
$297$	0	0
$298$	−3728.00	−0.724689
$299$	1469.00	0.284129
$300$	0	0
$301$	2359.00	0.451729
$302$	−1858.00	−0.354026
$303$	0	0
$304$	1648.00	0.310919
$305$	885.000	0.166147
$306$	0	0
$307$	−1556.00	−0.289269	−0.144635	−	0.989485i	$-0.546201\pi$
$307$	−1556.00	−0.289269	−0.144635	+	0.989485i	$0.546201\pi$
$308$	−644.000	−0.119141
$309$	0	0
$310$	−540.000	−0.0989353
$311$	−3884.00	−0.708172	−0.354086	−	0.935213i	$-0.615208\pi$
$311$	−3884.00	−0.708172	−0.354086	+	0.935213i	$0.615208\pi$
$312$	0	0
$313$	−5022.00	−0.906902	−0.453451	−	0.891281i	$-0.649807\pi$
$313$	−5022.00	−0.906902	−0.453451	+	0.891281i	$0.649807\pi$
$314$	−170.000	−0.0305530
$315$	0	0
$316$	−2776.00	−0.494184
$317$	−5654.00	−1.00177	−0.500884	−	0.865515i	$-0.666992\pi$
$317$	−5654.00	−1.00177	−0.500884	+	0.865515i	$0.666992\pi$
$318$	0	0
$319$	3243.00	0.569195
$320$	192.000	0.0335410
$321$	0	0
$322$	1582.00	0.273793
$323$	−13699.0	−2.35985
$324$	0	0
$325$	1508.00	0.257381
$326$	−6004.00	−1.02003
$327$	0	0
$328$	1840.00	0.309747
$329$	672.000	0.112610
$330$	0	0
$331$	3214.00	0.533708	0.266854	−	0.963737i	$-0.414016\pi$
$331$	3214.00	0.533708	0.266854	+	0.963737i	$0.414016\pi$
$332$	4040.00	0.667843
$333$	0	0
$334$	−3794.00	−0.621552
$335$	1404.00	0.228981
$336$	0	0
$337$	−657.000	−0.106199	−0.0530995	−	0.998589i	$-0.516910\pi$
$337$	−657.000	−0.106199	−0.0530995	+	0.998589i	$0.516910\pi$
$338$	338.000	0.0543928
$339$	0	0
$340$	−1596.00	−0.254574
$341$	−2070.00	−0.328730
$342$	0	0
$343$	−343.000	−0.0539949
$344$	−2696.00	−0.422554
$345$	0	0
$346$	−4444.00	−0.690494
$347$	−5336.00	−0.825509	−0.412754	−	0.910842i	$-0.635433\pi$
$347$	−5336.00	−0.825509	−0.412754	+	0.910842i	$0.635433\pi$
$348$	0	0
$349$	5816.00	0.892044	0.446022	−	0.895022i	$-0.352840\pi$
$349$	5816.00	0.892044	0.446022	+	0.895022i	$0.352840\pi$
$350$	1624.00	0.248018
$351$	0	0
$352$	736.000	0.111446
$353$	910.000	0.137208	0.0686040	−	0.997644i	$-0.478146\pi$
$353$	910.000	0.137208	0.0686040	+	0.997644i	$0.478146\pi$
$354$	0	0
$355$	−162.000	−0.0242199
$356$	−1560.00	−0.232247
$357$	0	0
$358$	−40.0000	−0.00590521
$359$	−8274.00	−1.21639	−0.608196	−	0.793787i	$-0.708107\pi$
$359$	−8274.00	−1.21639	−0.608196	+	0.793787i	$0.708107\pi$
$360$	0	0
$361$	3750.00	0.546727
$362$	9500.00	1.37931
$363$	0	0
$364$	364.000	0.0524142
$365$	−1659.00	−0.237907
$366$	0	0
$367$	5624.00	0.799919	0.399960	−	0.916533i	$-0.369024\pi$
$367$	5624.00	0.799919	0.399960	+	0.916533i	$0.369024\pi$
$368$	−1808.00	−0.256110
$369$	0	0
$370$	−1398.00	−0.196429
$371$	−938.000	−0.131263
$372$	0	0
$373$	246.000	0.0341485	0.0170743	−	0.999854i	$-0.494565\pi$
$373$	246.000	0.0341485	0.0170743	+	0.999854i	$0.494565\pi$
$374$	−6118.00	−0.845867
$375$	0	0
$376$	−768.000	−0.105337
$377$	−1833.00	−0.250409
$378$	0	0
$379$	−802.000	−0.108696	−0.0543482	−	0.998522i	$-0.517308\pi$
$379$	−802.000	−0.108696	−0.0543482	+	0.998522i	$0.517308\pi$
$380$	1236.00	0.166856
$381$	0	0
$382$	−1818.00	−0.243500
$383$	−2597.00	−0.346477	−0.173238	−	0.984880i	$-0.555423\pi$
$383$	−2597.00	−0.346477	−0.173238	+	0.984880i	$0.555423\pi$
$384$	0	0
$385$	−483.000	−0.0639376
$386$	884.000	0.116566
$387$	0	0
$388$	−408.000	−0.0533842
$389$	−5986.00	−0.780211	−0.390106	−	0.920770i	$-0.627562\pi$
$389$	−5986.00	−0.780211	−0.390106	+	0.920770i	$0.627562\pi$
$390$	0	0
$391$	15029.0	1.94386
$392$	392.000	0.0505076
$393$	0	0
$394$	−6572.00	−0.840336
$395$	−2082.00	−0.265207
$396$	0	0
$397$	14524.0	1.83612	0.918059	−	0.396444i	$-0.129756\pi$
$397$	14524.0	1.83612	0.918059	+	0.396444i	$0.129756\pi$
$398$	7986.00	1.00578
$399$	0	0
$400$	−1856.00	−0.232000
$401$	7334.00	0.913323	0.456661	−	0.889641i	$-0.349045\pi$
$401$	7334.00	0.913323	0.456661	+	0.889641i	$0.349045\pi$
$402$	0	0
$403$	1170.00	0.144620
$404$	−6360.00	−0.783222
$405$	0	0
$406$	−1974.00	−0.241300
$407$	−5359.00	−0.652668
$408$	0	0
$409$	2981.00	0.360394	0.180197	−	0.983631i	$-0.442327\pi$
$409$	2981.00	0.360394	0.180197	+	0.983631i	$0.442327\pi$
$410$	1380.00	0.166228
$411$	0	0
$412$	6716.00	0.803091
$413$	5866.00	0.698903
$414$	0	0
$415$	3030.00	0.358402
$416$	−416.000	−0.0490290
$417$	0	0
$418$	4738.00	0.554409
$419$	169.000	0.0197045	0.00985226	−	0.999951i	$-0.496864\pi$
$419$	169.000	0.0197045	0.00985226	+	0.999951i	$0.496864\pi$
$420$	0	0
$421$	9950.00	1.15186	0.575930	−	0.817499i	$-0.304640\pi$
$421$	9950.00	1.15186	0.575930	+	0.817499i	$0.304640\pi$
$422$	6890.00	0.794787
$423$	0	0
$424$	1072.00	0.122785
$425$	15428.0	1.76087
$426$	0	0
$427$	−2065.00	−0.234034
$428$	−120.000	−0.0135524
$429$	0	0
$430$	−2022.00	−0.226766
$431$	7462.00	0.833949	0.416974	−	0.908918i	$-0.363090\pi$
$431$	7462.00	0.833949	0.416974	+	0.908918i	$0.363090\pi$
$432$	0	0
$433$	−782.000	−0.0867910	−0.0433955	−	0.999058i	$-0.513818\pi$
$433$	−782.000	−0.0867910	−0.0433955	+	0.999058i	$0.513818\pi$
$434$	1260.00	0.139359
$435$	0	0
$436$	−5692.00	−0.625223
$437$	−11639.0	−1.27407
$438$	0	0
$439$	1079.00	0.117307	0.0586536	−	0.998278i	$-0.481319\pi$
$439$	1079.00	0.117307	0.0586536	+	0.998278i	$0.481319\pi$
$440$	552.000	0.0598081
$441$	0	0
$442$	3458.00	0.372127
$443$	−4538.00	−0.486697	−0.243349	−	0.969939i	$-0.578246\pi$
$443$	−4538.00	−0.486697	−0.243349	+	0.969939i	$0.578246\pi$
$444$	0	0
$445$	−1170.00	−0.124637
$446$	−2840.00	−0.301520
$447$	0	0
$448$	−448.000	−0.0472456
$449$	15773.0	1.65785	0.828924	−	0.559361i	$-0.188953\pi$
$449$	15773.0	1.65785	0.828924	+	0.559361i	$0.188953\pi$
$450$	0	0
$451$	5290.00	0.552320
$452$	−3376.00	−0.351313
$453$	0	0
$454$	1020.00	0.105443
$455$	273.000	0.0281284
$456$	0	0
$457$	−3112.00	−0.318541	−0.159270	−	0.987235i	$-0.550914\pi$
$457$	−3112.00	−0.318541	−0.159270	+	0.987235i	$0.550914\pi$
$458$	−3492.00	−0.356267
$459$	0	0
$460$	−1356.00	−0.137443
$461$	9305.00	0.940080	0.470040	−	0.882645i	$-0.344239\pi$
$461$	9305.00	0.940080	0.470040	+	0.882645i	$0.344239\pi$
$462$	0	0
$463$	13763.0	1.38147	0.690735	−	0.723108i	$-0.257287\pi$
$463$	13763.0	1.38147	0.690735	+	0.723108i	$0.257287\pi$
$464$	2256.00	0.225716
$465$	0	0
$466$	−2040.00	−0.202792
$467$	−14625.0	−1.44917	−0.724587	−	0.689183i	$-0.757969\pi$
$467$	−14625.0	−1.44917	−0.724587	+	0.689183i	$0.757969\pi$
$468$	0	0
$469$	−3276.00	−0.322541
$470$	−576.000	−0.0565296
$471$	0	0
$472$	−6704.00	−0.653764
$473$	−7751.00	−0.753470
$474$	0	0
$475$	−11948.0	−1.15413
$476$	3724.00	0.358591
$477$	0	0
$478$	5648.00	0.540447
$479$	6797.00	0.648357	0.324178	−	0.945996i	$-0.394912\pi$
$479$	6797.00	0.648357	0.324178	+	0.945996i	$0.394912\pi$
$480$	0	0
$481$	3029.00	0.287132
$482$	5076.00	0.479679
$483$	0	0
$484$	−3208.00	−0.301277
$485$	−306.000	−0.0286490
$486$	0	0
$487$	−13400.0	−1.24684	−0.623421	−	0.781886i	$-0.714258\pi$
$487$	−13400.0	−1.24684	−0.623421	+	0.781886i	$0.714258\pi$
$488$	2360.00	0.218918
$489$	0	0
$490$	294.000	0.0271052
$491$	−3176.00	−0.291916	−0.145958	−	0.989291i	$-0.546626\pi$
$491$	−3176.00	−0.291916	−0.145958	+	0.989291i	$0.546626\pi$
$492$	0	0
$493$	−18753.0	−1.71317
$494$	−2678.00	−0.243905
$495$	0	0
$496$	−1440.00	−0.130359
$497$	378.000	0.0341159
$498$	0	0
$499$	−12164.0	−1.09125	−0.545627	−	0.838028i	$-0.683708\pi$
$499$	−12164.0	−1.09125	−0.545627	+	0.838028i	$0.683708\pi$
$500$	−2892.00	−0.258668
$501$	0	0
$502$	−9450.00	−0.840188
$503$	−14808.0	−1.31264	−0.656318	−	0.754484i	$-0.727887\pi$
$503$	−14808.0	−1.31264	−0.656318	+	0.754484i	$0.727887\pi$
$504$	0	0
$505$	−4770.00	−0.420321
$506$	−5198.00	−0.456678
$507$	0	0
$508$	1312.00	0.114588
$509$	10977.0	0.955888	0.477944	−	0.878390i	$-0.341382\pi$
$509$	10977.0	0.955888	0.477944	+	0.878390i	$0.341382\pi$
$510$	0	0
$511$	3871.00	0.335113
$512$	512.000	0.0441942
$513$	0	0
$514$	5284.00	0.453438
$515$	5037.00	0.430984
$516$	0	0
$517$	−2208.00	−0.187829
$518$	3262.00	0.276687
$519$	0	0
$520$	−312.000	−0.0263117
$521$	14885.0	1.25168	0.625838	−	0.779953i	$-0.284757\pi$
$521$	14885.0	1.25168	0.625838	+	0.779953i	$0.284757\pi$
$522$	0	0
$523$	−1054.00	−0.0881228	−0.0440614	−	0.999029i	$-0.514030\pi$
$523$	−1054.00	−0.0881228	−0.0440614	+	0.999029i	$0.514030\pi$
$524$	1068.00	0.0890378
$525$	0	0
$526$	10304.0	0.854136
$527$	11970.0	0.989414
$528$	0	0
$529$	602.000	0.0494781
$530$	804.000	0.0658934
$531$	0	0
$532$	−2884.00	−0.235032
$533$	−2990.00	−0.242986
$534$	0	0
$535$	−90.0000	−0.00727297
$536$	3744.00	0.301709
$537$	0	0
$538$	−1036.00	−0.0830207
$539$	1127.00	0.0900618
$540$	0	0
$541$	21697.0	1.72426	0.862132	−	0.506684i	$-0.169129\pi$
$541$	21697.0	1.72426	0.862132	+	0.506684i	$0.169129\pi$
$542$	−7284.00	−0.577259
$543$	0	0
$544$	−4256.00	−0.335431
$545$	−4269.00	−0.335530
$546$	0	0
$547$	23836.0	1.86317	0.931585	−	0.363524i	$-0.118427\pi$
$547$	23836.0	1.86317	0.931585	+	0.363524i	$0.118427\pi$
$548$	−6012.00	−0.468649
$549$	0	0
$550$	−5336.00	−0.413687
$551$	14523.0	1.12287
$552$	0	0
$553$	4858.00	0.373568
$554$	9752.00	0.747875
$555$	0	0
$556$	3832.00	0.292290
$557$	−4902.00	−0.372898	−0.186449	−	0.982465i	$-0.559698\pi$
$557$	−4902.00	−0.372898	−0.186449	+	0.982465i	$0.559698\pi$
$558$	0	0
$559$	4381.00	0.331479
$560$	−336.000	−0.0253546
$561$	0	0
$562$	−4404.00	−0.330554
$563$	3147.00	0.235578	0.117789	−	0.993039i	$-0.462419\pi$
$563$	3147.00	0.235578	0.117789	+	0.993039i	$0.462419\pi$
$564$	0	0
$565$	−2532.00	−0.188535
$566$	−8072.00	−0.599455
$567$	0	0
$568$	−432.000	−0.0319125
$569$	15394.0	1.13418	0.567091	−	0.823655i	$-0.308069\pi$
$569$	15394.0	1.13418	0.567091	+	0.823655i	$0.308069\pi$
$570$	0	0
$571$	−11588.0	−0.849287	−0.424643	−	0.905361i	$-0.639601\pi$
$571$	−11588.0	−0.849287	−0.424643	+	0.905361i	$0.639601\pi$
$572$	−1196.00	−0.0874253
$573$	0	0
$574$	−3220.00	−0.234147
$575$	13108.0	0.950681
$576$	0	0
$577$	25474.0	1.83795	0.918974	−	0.394317i	$-0.129019\pi$
$577$	25474.0	1.83795	0.918974	+	0.394317i	$0.129019\pi$
$578$	25552.0	1.83879
$579$	0	0
$580$	1692.00	0.121132
$581$	−7070.00	−0.504842
$582$	0	0
$583$	3082.00	0.218942
$584$	−4424.00	−0.313470
$585$	0	0
$586$	−10692.0	−0.753724
$587$	10122.0	0.711720	0.355860	−	0.934539i	$-0.384188\pi$
$587$	10122.0	0.711720	0.355860	+	0.934539i	$0.384188\pi$
$588$	0	0
$589$	−9270.00	−0.648495
$590$	−5028.00	−0.350847
$591$	0	0
$592$	−3728.00	−0.258817
$593$	19524.0	1.35203	0.676016	−	0.736887i	$-0.263705\pi$
$593$	19524.0	1.35203	0.676016	+	0.736887i	$0.263705\pi$
$594$	0	0
$595$	2793.00	0.192440
$596$	−7456.00	−0.512432
$597$	0	0
$598$	2938.00	0.200909
$599$	18157.0	1.23852	0.619261	−	0.785185i	$-0.287432\pi$
$599$	18157.0	1.23852	0.619261	+	0.785185i	$0.287432\pi$
$600$	0	0
$601$	7438.00	0.504829	0.252415	−	0.967619i	$-0.418775\pi$
$601$	7438.00	0.504829	0.252415	+	0.967619i	$0.418775\pi$
$602$	4718.00	0.319421
$603$	0	0
$604$	−3716.00	−0.250334
$605$	−2406.00	−0.161682
$606$	0	0
$607$	9451.00	0.631967	0.315984	−	0.948765i	$-0.397665\pi$
$607$	9451.00	0.631967	0.315984	+	0.948765i	$0.397665\pi$
$608$	3296.00	0.219853
$609$	0	0
$610$	1770.00	0.117484
$611$	1248.00	0.0826329
$612$	0	0
$613$	−12631.0	−0.832237	−0.416119	−	0.909310i	$-0.636610\pi$
$613$	−12631.0	−0.832237	−0.416119	+	0.909310i	$0.636610\pi$
$614$	−3112.00	−0.204544
$615$	0	0
$616$	−1288.00	−0.0842451
$617$	−16065.0	−1.04822	−0.524111	−	0.851650i	$-0.675602\pi$
$617$	−16065.0	−1.04822	−0.524111	+	0.851650i	$0.675602\pi$
$618$	0	0
$619$	−21773.0	−1.41378	−0.706891	−	0.707323i	$-0.749903\pi$
$619$	−21773.0	−1.41378	−0.706891	+	0.707323i	$0.749903\pi$
$620$	−1080.00	−0.0699578
$621$	0	0
$622$	−7768.00	−0.500753
$623$	2730.00	0.175562
$624$	0	0
$625$	12331.0	0.789184
$626$	−10044.0	−0.641276
$627$	0	0
$628$	−340.000	−0.0216043
$629$	30989.0	1.96441
$630$	0	0
$631$	15575.0	0.982616	0.491308	−	0.870986i	$-0.336519\pi$
$631$	15575.0	0.982616	0.491308	+	0.870986i	$0.336519\pi$
$632$	−5552.00	−0.349441
$633$	0	0
$634$	−11308.0	−0.708357
$635$	984.000	0.0614943
$636$	0	0
$637$	−637.000	−0.0396214
$638$	6486.00	0.402482
$639$	0	0
$640$	384.000	0.0237171
$641$	−8814.00	−0.543108	−0.271554	−	0.962423i	$-0.587538\pi$
$641$	−8814.00	−0.543108	−0.271554	+	0.962423i	$0.587538\pi$
$642$	0	0
$643$	10629.0	0.651892	0.325946	−	0.945388i	$-0.394317\pi$
$643$	10629.0	0.651892	0.325946	+	0.945388i	$0.394317\pi$
$644$	3164.00	0.193601
$645$	0	0
$646$	−27398.0	−1.66867
$647$	18624.0	1.13166	0.565831	−	0.824521i	$-0.308556\pi$
$647$	18624.0	1.13166	0.565831	+	0.824521i	$0.308556\pi$
$648$	0	0
$649$	−19274.0	−1.16575
$650$	3016.00	0.181996
$651$	0	0
$652$	−12008.0	−0.721272
$653$	−29561.0	−1.77153	−0.885767	−	0.464131i	$-0.846367\pi$
$653$	−29561.0	−1.77153	−0.885767	+	0.464131i	$0.846367\pi$
$654$	0	0
$655$	801.000	0.0477827
$656$	3680.00	0.219024
$657$	0	0
$658$	1344.00	0.0796270
$659$	−18432.0	−1.08954	−0.544771	−	0.838585i	$-0.683384\pi$
$659$	−18432.0	−1.08954	−0.544771	+	0.838585i	$0.683384\pi$
$660$	0	0
$661$	18956.0	1.11544	0.557718	−	0.830031i	$-0.311677\pi$
$661$	18956.0	1.11544	0.557718	+	0.830031i	$0.311677\pi$
$662$	6428.00	0.377389
$663$	0	0
$664$	8080.00	0.472236
$665$	−2163.00	−0.126132
$666$	0	0
$667$	−15933.0	−0.924930
$668$	−7588.00	−0.439504
$669$	0	0
$670$	2808.00	0.161914
$671$	6785.00	0.390361
$672$	0	0
$673$	32299.0	1.84998	0.924989	−	0.379994i	$-0.124074\pi$
$673$	32299.0	1.84998	0.924989	+	0.379994i	$0.124074\pi$
$674$	−1314.00	−0.0750940
$675$	0	0
$676$	676.000	0.0384615
$677$	5786.00	0.328470	0.164235	−	0.986421i	$-0.447484\pi$
$677$	5786.00	0.328470	0.164235	+	0.986421i	$0.447484\pi$
$678$	0	0
$679$	714.000	0.0403546
$680$	−3192.00	−0.180011
$681$	0	0
$682$	−4140.00	−0.232447
$683$	17761.0	0.995030	0.497515	−	0.867455i	$-0.334246\pi$
$683$	17761.0	0.995030	0.497515	+	0.867455i	$0.334246\pi$
$684$	0	0
$685$	−4509.00	−0.251504
$686$	−686.000	−0.0381802
$687$	0	0
$688$	−5392.00	−0.298791
$689$	−1742.00	−0.0963206
$690$	0	0
$691$	−592.000	−0.0325915	−0.0162958	−	0.999867i	$-0.505187\pi$
$691$	−592.000	−0.0325915	−0.0162958	+	0.999867i	$0.505187\pi$
$692$	−8888.00	−0.488253
$693$	0	0
$694$	−10672.0	−0.583723
$695$	2874.00	0.156859
$696$	0	0
$697$	−30590.0	−1.66238
$698$	11632.0	0.630770
$699$	0	0
$700$	3248.00	0.175376
$701$	3498.00	0.188470	0.0942351	−	0.995550i	$-0.469959\pi$
$701$	3498.00	0.188470	0.0942351	+	0.995550i	$0.469959\pi$
$702$	0	0
$703$	−23999.0	−1.28754
$704$	1472.00	0.0788041
$705$	0	0
$706$	1820.00	0.0970207
$707$	11130.0	0.592060
$708$	0	0
$709$	−19378.0	−1.02645	−0.513227	−	0.858253i	$-0.671550\pi$
$709$	−19378.0	−1.02645	−0.513227	+	0.858253i	$0.671550\pi$
$710$	−324.000	−0.0171261
$711$	0	0
$712$	−3120.00	−0.164223
$713$	10170.0	0.534179
$714$	0	0
$715$	−897.000	−0.0469173
$716$	−80.0000	−0.00417562
$717$	0	0
$718$	−16548.0	−0.860120
$719$	−2856.00	−0.148137	−0.0740687	−	0.997253i	$-0.523598\pi$
$719$	−2856.00	−0.148137	−0.0740687	+	0.997253i	$0.523598\pi$
$720$	0	0
$721$	−11753.0	−0.607080
$722$	7500.00	0.386594
$723$	0	0
$724$	19000.0	0.975317
$725$	−16356.0	−0.837857
$726$	0	0
$727$	19699.0	1.00495	0.502473	−	0.864593i	$-0.332424\pi$
$727$	19699.0	1.00495	0.502473	+	0.864593i	$0.332424\pi$
$728$	728.000	0.0370625
$729$	0	0
$730$	−3318.00	−0.168226
$731$	44821.0	2.26780
$732$	0	0
$733$	18776.0	0.946122	0.473061	−	0.881030i	$-0.343149\pi$
$733$	18776.0	0.946122	0.473061	+	0.881030i	$0.343149\pi$
$734$	11248.0	0.565628
$735$	0	0
$736$	−3616.00	−0.181097
$737$	10764.0	0.537988
$738$	0	0
$739$	−26158.0	−1.30208	−0.651040	−	0.759043i	$-0.725667\pi$
$739$	−26158.0	−1.30208	−0.651040	+	0.759043i	$0.725667\pi$
$740$	−2796.00	−0.138896
$741$	0	0
$742$	−1876.00	−0.0928169
$743$	−33700.0	−1.66397	−0.831987	−	0.554795i	$-0.812797\pi$
$743$	−33700.0	−1.66397	−0.831987	+	0.554795i	$0.812797\pi$
$744$	0	0
$745$	−5592.00	−0.275000
$746$	492.000	0.0241466
$747$	0	0
$748$	−12236.0	−0.598118
$749$	210.000	0.0102446
$750$	0	0
$751$	−14416.0	−0.700462	−0.350231	−	0.936663i	$-0.613897\pi$
$751$	−14416.0	−0.700462	−0.350231	+	0.936663i	$0.613897\pi$
$752$	−1536.00	−0.0744843
$753$	0	0
$754$	−3666.00	−0.177066
$755$	−2787.00	−0.134343
$756$	0	0
$757$	−27724.0	−1.33110	−0.665552	−	0.746351i	$-0.731804\pi$
$757$	−27724.0	−1.33110	−0.665552	+	0.746351i	$0.731804\pi$
$758$	−1604.00	−0.0768600
$759$	0	0
$760$	2472.00	0.117985
$761$	24476.0	1.16591	0.582953	−	0.812506i	$-0.301897\pi$
$761$	24476.0	1.16591	0.582953	+	0.812506i	$0.301897\pi$
$762$	0	0
$763$	9961.00	0.472624
$764$	−3636.00	−0.172180
$765$	0	0
$766$	−5194.00	−0.244996
$767$	10894.0	0.512855
$768$	0	0
$769$	27869.0	1.30687	0.653434	−	0.756983i	$-0.273328\pi$
$769$	27869.0	1.30687	0.653434	+	0.756983i	$0.273328\pi$
$770$	−966.000	−0.0452107
$771$	0	0
$772$	1768.00	0.0824245
$773$	−15407.0	−0.716884	−0.358442	−	0.933552i	$-0.616692\pi$
$773$	−15407.0	−0.716884	−0.358442	+	0.933552i	$0.616692\pi$
$774$	0	0
$775$	10440.0	0.483891
$776$	−816.000	−0.0377483
$777$	0	0
$778$	−11972.0	−0.551693
$779$	23690.0	1.08958
$780$	0	0
$781$	−1242.00	−0.0569043
$782$	30058.0	1.37452
$783$	0	0
$784$	784.000	0.0357143
$785$	−255.000	−0.0115941
$786$	0	0
$787$	20675.0	0.936447	0.468224	−	0.883610i	$-0.344894\pi$
$787$	20675.0	0.936447	0.468224	+	0.883610i	$0.344894\pi$
$788$	−13144.0	−0.594208
$789$	0	0
$790$	−4164.00	−0.187530
$791$	5908.00	0.265568
$792$	0	0
$793$	−3835.00	−0.171734
$794$	29048.0	1.29833
$795$	0	0
$796$	15972.0	0.711197
$797$	37704.0	1.67571	0.837857	−	0.545890i	$-0.183808\pi$
$797$	37704.0	1.67571	0.837857	+	0.545890i	$0.183808\pi$
$798$	0	0
$799$	12768.0	0.565331
$800$	−3712.00	−0.164049
$801$	0	0
$802$	14668.0	0.645817
$803$	−12719.0	−0.558959
$804$	0	0
$805$	2373.00	0.103897
$806$	2340.00	0.102262
$807$	0	0
$808$	−12720.0	−0.553822
$809$	−31714.0	−1.37825	−0.689125	−	0.724642i	$-0.742005\pi$
$809$	−31714.0	−1.37825	−0.689125	+	0.724642i	$0.742005\pi$
$810$	0	0
$811$	−2493.00	−0.107942	−0.0539711	−	0.998542i	$-0.517188\pi$
$811$	−2493.00	−0.107942	−0.0539711	+	0.998542i	$0.517188\pi$
$812$	−3948.00	−0.170625
$813$	0	0
$814$	−10718.0	−0.461506
$815$	−9006.00	−0.387075
$816$	0	0
$817$	−34711.0	−1.48639
$818$	5962.00	0.254837
$819$	0	0
$820$	2760.00	0.117541
$821$	−17932.0	−0.762279	−0.381140	−	0.924518i	$-0.624468\pi$
$821$	−17932.0	−0.762279	−0.381140	+	0.924518i	$0.624468\pi$
$822$	0	0
$823$	22.0000	0.000931800 0	0.000465900	−	1.00000i	$-0.499852\pi$
$823$	22.0000	0.000931800 0	0.000465900		1.00000i	$0.499852\pi$
$824$	13432.0	0.567871
$825$	0	0
$826$	11732.0	0.494199
$827$	−27225.0	−1.14475	−0.572374	−	0.819993i	$-0.693977\pi$
$827$	−27225.0	−1.14475	−0.572374	+	0.819993i	$0.693977\pi$
$828$	0	0
$829$	40759.0	1.70762	0.853811	−	0.520583i	$-0.174285\pi$
$829$	40759.0	1.70762	0.853811	+	0.520583i	$0.174285\pi$
$830$	6060.00	0.253429
$831$	0	0
$832$	−832.000	−0.0346688
$833$	−6517.00	−0.271069
$834$	0	0
$835$	−5691.00	−0.235862
$836$	9476.00	0.392027
$837$	0	0
$838$	338.000	0.0139332
$839$	−3680.00	−0.151428	−0.0757138	−	0.997130i	$-0.524124\pi$
$839$	−3680.00	−0.151428	−0.0757138	+	0.997130i	$0.524124\pi$
$840$	0	0
$841$	−4508.00	−0.184837
$842$	19900.0	0.814488
$843$	0	0
$844$	13780.0	0.561999
$845$	507.000	0.0206406
$846$	0	0
$847$	5614.00	0.227744
$848$	2144.00	0.0868222
$849$	0	0
$850$	30856.0	1.24512
$851$	26329.0	1.06057
$852$	0	0
$853$	−12112.0	−0.486175	−0.243087	−	0.970004i	$-0.578160\pi$
$853$	−12112.0	−0.486175	−0.243087	+	0.970004i	$0.578160\pi$
$854$	−4130.00	−0.165487
$855$	0	0
$856$	−240.000	−0.00958298
$857$	−21322.0	−0.849878	−0.424939	−	0.905222i	$-0.639704\pi$
$857$	−21322.0	−0.849878	−0.424939	+	0.905222i	$0.639704\pi$
$858$	0	0
$859$	−43598.0	−1.73172	−0.865858	−	0.500289i	$-0.833227\pi$
$859$	−43598.0	−1.73172	−0.865858	+	0.500289i	$0.833227\pi$
$860$	−4044.00	−0.160348
$861$	0	0
$862$	14924.0	0.589691
$863$	37560.0	1.48153	0.740763	−	0.671766i	$-0.234464\pi$
$863$	37560.0	1.48153	0.740763	+	0.671766i	$0.234464\pi$
$864$	0	0
$865$	−6666.00	−0.262024
$866$	−1564.00	−0.0613705
$867$	0	0
$868$	2520.00	0.0985419
$869$	−15962.0	−0.623100
$870$	0	0
$871$	−6084.00	−0.236680
$872$	−11384.0	−0.442100
$873$	0	0
$874$	−23278.0	−0.900904
$875$	5061.00	0.195535
$876$	0	0
$877$	43230.0	1.66451	0.832254	−	0.554395i	$-0.187050\pi$
$877$	43230.0	1.66451	0.832254	+	0.554395i	$0.187050\pi$
$878$	2158.00	0.0829487
$879$	0	0
$880$	1104.00	0.0422907
$881$	25691.0	0.982465	0.491233	−	0.871028i	$-0.336547\pi$
$881$	25691.0	0.982465	0.491233	+	0.871028i	$0.336547\pi$
$882$	0	0
$883$	−13789.0	−0.525523	−0.262761	−	0.964861i	$-0.584633\pi$
$883$	−13789.0	−0.525523	−0.262761	+	0.964861i	$0.584633\pi$
$884$	6916.00	0.263134
$885$	0	0
$886$	−9076.00	−0.344147
$887$	−10482.0	−0.396788	−0.198394	−	0.980122i	$-0.563573\pi$
$887$	−10482.0	−0.396788	−0.198394	+	0.980122i	$0.563573\pi$
$888$	0	0
$889$	−2296.00	−0.0866202
$890$	−2340.00	−0.0881314
$891$	0	0
$892$	−5680.00	−0.213207
$893$	−9888.00	−0.370537
$894$	0	0
$895$	−60.0000	−0.00224087
$896$	−896.000	−0.0334077
$897$	0	0
$898$	31546.0	1.17228
$899$	−12690.0	−0.470784
$900$	0	0
$901$	−17822.0	−0.658975
$902$	10580.0	0.390549
$903$	0	0
$904$	−6752.00	−0.248416
$905$	14250.0	0.523410
$906$	0	0
$907$	5340.00	0.195493	0.0977463	−	0.995211i	$-0.468837\pi$
$907$	5340.00	0.195493	0.0977463	+	0.995211i	$0.468837\pi$
$908$	2040.00	0.0745592
$909$	0	0
$910$	546.000	0.0198898
$911$	−21495.0	−0.781736	−0.390868	−	0.920447i	$-0.627825\pi$
$911$	−21495.0	−0.781736	−0.390868	+	0.920447i	$0.627825\pi$
$912$	0	0
$913$	23230.0	0.842060
$914$	−6224.00	−0.225242
$915$	0	0
$916$	−6984.00	−0.251919
$917$	−1869.00	−0.0673062
$918$	0	0
$919$	−52070.0	−1.86902	−0.934511	−	0.355935i	$-0.884163\pi$
$919$	−52070.0	−1.86902	−0.934511	+	0.355935i	$0.884163\pi$
$920$	−2712.00	−0.0971869
$921$	0	0
$922$	18610.0	0.664737
$923$	702.000	0.0250342
$924$	0	0
$925$	27028.0	0.960730
$926$	27526.0	0.976847
$927$	0	0
$928$	4512.00	0.159605
$929$	4386.00	0.154898	0.0774489	−	0.996996i	$-0.475323\pi$
$929$	4386.00	0.154898	0.0774489	+	0.996996i	$0.475323\pi$
$930$	0	0
$931$	5047.00	0.177668
$932$	−4080.00	−0.143396
$933$	0	0
$934$	−29250.0	−1.02472
$935$	−9177.00	−0.320984
$936$	0	0
$937$	33374.0	1.16359	0.581794	−	0.813337i	$-0.302351\pi$
$937$	33374.0	1.16359	0.581794	+	0.813337i	$0.302351\pi$
$938$	−6552.00	−0.228071
$939$	0	0
$940$	−1152.00	−0.0399724
$941$	22742.0	0.787851	0.393926	−	0.919142i	$-0.371117\pi$
$941$	22742.0	0.787851	0.393926	+	0.919142i	$0.371117\pi$
$942$	0	0
$943$	−25990.0	−0.897509
$944$	−13408.0	−0.462281
$945$	0	0
$946$	−15502.0	−0.532784
$947$	45239.0	1.55234	0.776172	−	0.630521i	$-0.217159\pi$
$947$	45239.0	1.55234	0.776172	+	0.630521i	$0.217159\pi$
$948$	0	0
$949$	7189.00	0.245906
$950$	−23896.0	−0.816093
$951$	0	0
$952$	7448.00	0.253562
$953$	46536.0	1.58179	0.790897	−	0.611950i	$-0.209615\pi$
$953$	46536.0	1.58179	0.790897	+	0.611950i	$0.209615\pi$
$954$	0	0
$955$	−2727.00	−0.0924017
$956$	11296.0	0.382154
$957$	0	0
$958$	13594.0	0.458457
$959$	10521.0	0.354266
$960$	0	0
$961$	−21691.0	−0.728106
$962$	6058.00	0.203033
$963$	0	0
$964$	10152.0	0.339185
$965$	1326.00	0.0442336
$966$	0	0
$967$	−56219.0	−1.86958	−0.934789	−	0.355205i	$-0.884411\pi$
$967$	−56219.0	−1.86958	−0.934789	+	0.355205i	$0.884411\pi$
$968$	−6416.00	−0.213035
$969$	0	0
$970$	−612.000	−0.0202579
$971$	35376.0	1.16918	0.584588	−	0.811330i	$-0.301256\pi$
$971$	35376.0	1.16918	0.584588	+	0.811330i	$0.301256\pi$
$972$	0	0
$973$	−6706.00	−0.220950
$974$	−26800.0	−0.881650
$975$	0	0
$976$	4720.00	0.154799
$977$	−37025.0	−1.21242	−0.606210	−	0.795304i	$-0.707311\pi$
$977$	−37025.0	−1.21242	−0.606210	+	0.795304i	$0.707311\pi$
$978$	0	0
$979$	−8970.00	−0.292832
$980$	588.000	0.0191663
$981$	0	0
$982$	−6352.00	−0.206416
$983$	−47063.0	−1.52704	−0.763518	−	0.645786i	$-0.776530\pi$
$983$	−47063.0	−1.52704	−0.763518	+	0.645786i	$0.776530\pi$
$984$	0	0
$985$	−9858.00	−0.318885
$986$	−37506.0	−1.21139
$987$	0	0
$988$	−5356.00	−0.172467
$989$	38081.0	1.22437
$990$	0	0
$991$	−2138.00	−0.0685326	−0.0342663	−	0.999413i	$-0.510909\pi$
$991$	−2138.00	−0.0685326	−0.0342663	+	0.999413i	$0.510909\pi$
$992$	−2880.00	−0.0921775
$993$	0	0
$994$	756.000	0.0241236
$995$	11979.0	0.381668
$996$	0	0
$997$	42282.0	1.34311	0.671557	−	0.740953i	$-0.265626\pi$
$997$	42282.0	1.34311	0.671557	+	0.740953i	$0.265626\pi$
$998$	−24328.0	−0.771633
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1638.4.a.k.1.1	yes	1
3.2	odd	2		1638.4.a.c.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1638.4.a.c.1.1	✓	1	3.2	odd	2
1638.4.a.k.1.1	yes	1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1638.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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