LMFDB - Embedded newform 270.4.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [270,4,Mod(1,270)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(270, 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("270.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 270 = 2 \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	270.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:	$15.9305157015$
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	$\chi$	$=$	270.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+2.00000 q^{2} +4.00000 q^{4} -5.00000 q^{5} -22.0000 q^{7} +8.00000 q^{8} +O(q^{10})$

\(q+2.00000 q^{2} +4.00000 q^{4} -5.00000 q^{5} -22.0000 q^{7} +8.00000 q^{8} -10.0000 q^{10} -12.0000 q^{11} +38.0000 q^{13} -44.0000 q^{14} +16.0000 q^{16} -105.000 q^{17} -157.000 q^{19} -20.0000 q^{20} -24.0000 q^{22} -117.000 q^{23} +25.0000 q^{25} +76.0000 q^{26} -88.0000 q^{28} +66.0000 q^{29} -25.0000 q^{31} +32.0000 q^{32} -210.000 q^{34} +110.000 q^{35} +314.000 q^{37} -314.000 q^{38} -40.0000 q^{40} -504.000 q^{41} +380.000 q^{43} -48.0000 q^{44} -234.000 q^{46} -252.000 q^{47} +141.000 q^{49} +50.0000 q^{50} +152.000 q^{52} +3.00000 q^{53} +60.0000 q^{55} -176.000 q^{56} +132.000 q^{58} -318.000 q^{59} +293.000 q^{61} -50.0000 q^{62} +64.0000 q^{64} -190.000 q^{65} -322.000 q^{67} -420.000 q^{68} +220.000 q^{70} -120.000 q^{71} +44.0000 q^{73} +628.000 q^{74} -628.000 q^{76} +264.000 q^{77} +917.000 q^{79} -80.0000 q^{80} -1008.00 q^{82} +309.000 q^{83} +525.000 q^{85} +760.000 q^{86} -96.0000 q^{88} +1272.00 q^{89} -836.000 q^{91} -468.000 q^{92} -504.000 q^{94} +785.000 q^{95} +1328.00 q^{97} +282.000 q^{98} +O(q^{100})\)

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
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−22.0000	−1.18789	−0.593944	−	0.804506i	$-0.702430\pi$
$7$	−22.0000	−1.18789	−0.593944	+	0.804506i	$0.702430\pi$
$8$	8.00000	0.353553
$9$	0	0
$10$	−10.0000	−0.316228
$11$	−12.0000	−0.328921	−0.164461	−	0.986384i	$-0.552588\pi$
$11$	−12.0000	−0.328921	−0.164461	+	0.986384i	$0.552588\pi$
$12$	0	0
$13$	38.0000	0.810716	0.405358	−	0.914158i	$-0.367147\pi$
$13$	38.0000	0.810716	0.405358	+	0.914158i	$0.367147\pi$
$14$	−44.0000	−0.839964
$15$	0	0
$16$	16.0000	0.250000
$17$	−105.000	−1.49801	−0.749007	−	0.662562i	$-0.769469\pi$
$17$	−105.000	−1.49801	−0.749007	+	0.662562i	$0.769469\pi$
$18$	0	0
$19$	−157.000	−1.89570	−0.947849	−	0.318719i	$-0.896747\pi$
$19$	−157.000	−1.89570	−0.947849	+	0.318719i	$0.896747\pi$
$20$	−20.0000	−0.223607
$21$	0	0
$22$	−24.0000	−0.232583
$23$	−117.000	−1.06070	−0.530352	−	0.847778i	$-0.677940\pi$
$23$	−117.000	−1.06070	−0.530352	+	0.847778i	$0.677940\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	76.0000	0.573263
$27$	0	0
$28$	−88.0000	−0.593944
$29$	66.0000	0.422617	0.211308	−	0.977419i	$-0.432228\pi$
$29$	66.0000	0.422617	0.211308	+	0.977419i	$0.432228\pi$
$30$	0	0
$31$	−25.0000	−0.144843	−0.0724215	−	0.997374i	$-0.523073\pi$
$31$	−25.0000	−0.144843	−0.0724215	+	0.997374i	$0.523073\pi$
$32$	32.0000	0.176777
$33$	0	0
$34$	−210.000	−1.05926
$35$	110.000	0.531240
$36$	0	0
$37$	314.000	1.39517	0.697585	−	0.716502i	$-0.254258\pi$
$37$	314.000	1.39517	0.697585	+	0.716502i	$0.254258\pi$
$38$	−314.000	−1.34046
$39$	0	0
$40$	−40.0000	−0.158114
$41$	−504.000	−1.91979	−0.959897	−	0.280352i	$-0.909549\pi$
$41$	−504.000	−1.91979	−0.959897	+	0.280352i	$0.909549\pi$
$42$	0	0
$43$	380.000	1.34766	0.673831	−	0.738886i	$-0.264648\pi$
$43$	380.000	1.34766	0.673831	+	0.738886i	$0.264648\pi$
$44$	−48.0000	−0.164461
$45$	0	0
$46$	−234.000	−0.750031
$47$	−252.000	−0.782085	−0.391042	−	0.920373i	$-0.627885\pi$
$47$	−252.000	−0.782085	−0.391042	+	0.920373i	$0.627885\pi$
$48$	0	0
$49$	141.000	0.411079
$50$	50.0000	0.141421
$51$	0	0
$52$	152.000	0.405358
$53$	3.00000	0.00777513	0.00388756	−	0.999992i	$-0.498763\pi$
$53$	3.00000	0.00777513	0.00388756	+	0.999992i	$0.498763\pi$
$54$	0	0
$55$	60.0000	0.147098
$56$	−176.000	−0.419982
$57$	0	0
$58$	132.000	0.298835
$59$	−318.000	−0.701696	−0.350848	−	0.936432i	$-0.614107\pi$
$59$	−318.000	−0.701696	−0.350848	+	0.936432i	$0.614107\pi$
$60$	0	0
$61$	293.000	0.614997	0.307498	−	0.951549i	$-0.400508\pi$
$61$	293.000	0.614997	0.307498	+	0.951549i	$0.400508\pi$
$62$	−50.0000	−0.102419
$63$	0	0
$64$	64.0000	0.125000
$65$	−190.000	−0.362563
$66$	0	0
$67$	−322.000	−0.587143	−0.293571	−	0.955937i	$-0.594844\pi$
$67$	−322.000	−0.587143	−0.293571	+	0.955937i	$0.594844\pi$
$68$	−420.000	−0.749007
$69$	0	0
$70$	220.000	0.375643
$71$	−120.000	−0.200583	−0.100291	−	0.994958i	$-0.531978\pi$
$71$	−120.000	−0.200583	−0.100291	+	0.994958i	$0.531978\pi$
$72$	0	0
$73$	44.0000	0.0705453	0.0352727	−	0.999378i	$-0.488770\pi$
$73$	44.0000	0.0705453	0.0352727	+	0.999378i	$0.488770\pi$
$74$	628.000	0.986534
$75$	0	0
$76$	−628.000	−0.947849
$77$	264.000	0.390722
$78$	0	0
$79$	917.000	1.30596	0.652978	−	0.757377i	$-0.273519\pi$
$79$	917.000	1.30596	0.652978	+	0.757377i	$0.273519\pi$
$80$	−80.0000	−0.111803
$81$	0	0
$82$	−1008.00	−1.35750
$83$	309.000	0.408640	0.204320	−	0.978904i	$-0.434502\pi$
$83$	309.000	0.408640	0.204320	+	0.978904i	$0.434502\pi$
$84$	0	0
$85$	525.000	0.669932
$86$	760.000	0.952941
$87$	0	0
$88$	−96.0000	−0.116291
$89$	1272.00	1.51496	0.757482	−	0.652856i	$-0.226430\pi$
$89$	1272.00	1.51496	0.757482	+	0.652856i	$0.226430\pi$
$90$	0	0
$91$	−836.000	−0.963040
$92$	−468.000	−0.530352
$93$	0	0
$94$	−504.000	−0.553017
$95$	785.000	0.847782
$96$	0	0
$97$	1328.00	1.39008	0.695041	−	0.718970i	$-0.255386\pi$
$97$	1328.00	1.39008	0.695041	+	0.718970i	$0.255386\pi$
$98$	282.000	0.290677
$99$	0	0
$100$	100.000	0.100000
$101$	−492.000	−0.484711	−0.242356	−	0.970187i	$-0.577920\pi$
$101$	−492.000	−0.484711	−0.242356	+	0.970187i	$0.577920\pi$
$102$	0	0
$103$	548.000	0.524233	0.262117	−	0.965036i	$-0.415579\pi$
$103$	548.000	0.524233	0.262117	+	0.965036i	$0.415579\pi$
$104$	304.000	0.286631
$105$	0	0
$106$	6.00000	0.00549784
$107$	732.000	0.661356	0.330678	−	0.943744i	$-0.392723\pi$
$107$	732.000	0.661356	0.330678	+	0.943744i	$0.392723\pi$
$108$	0	0
$109$	−907.000	−0.797017	−0.398508	−	0.917165i	$-0.630472\pi$
$109$	−907.000	−0.797017	−0.398508	+	0.917165i	$0.630472\pi$
$110$	120.000	0.104014
$111$	0	0
$112$	−352.000	−0.296972
$113$	−1542.00	−1.28371	−0.641855	−	0.766826i	$-0.721835\pi$
$113$	−1542.00	−1.28371	−0.641855	+	0.766826i	$0.721835\pi$
$114$	0	0
$115$	585.000	0.474361
$116$	264.000	0.211308
$117$	0	0
$118$	−636.000	−0.496174
$119$	2310.00	1.77947
$120$	0	0
$121$	−1187.00	−0.891811
$122$	586.000	0.434868
$123$	0	0
$124$	−100.000	−0.0724215
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−2554.00	−1.78449	−0.892247	−	0.451547i	$-0.850872\pi$
$127$	−2554.00	−1.78449	−0.892247	+	0.451547i	$0.850872\pi$
$128$	128.000	0.0883883
$129$	0	0
$130$	−380.000	−0.256371
$131$	−150.000	−0.100042	−0.0500212	−	0.998748i	$-0.515929\pi$
$131$	−150.000	−0.100042	−0.0500212	+	0.998748i	$0.515929\pi$
$132$	0	0
$133$	3454.00	2.25188
$134$	−644.000	−0.415173
$135$	0	0
$136$	−840.000	−0.529628
$137$	1653.00	1.03084	0.515421	−	0.856937i	$-0.327636\pi$
$137$	1653.00	1.03084	0.515421	+	0.856937i	$0.327636\pi$
$138$	0	0
$139$	1124.00	0.685874	0.342937	−	0.939358i	$-0.388578\pi$
$139$	1124.00	0.685874	0.342937	+	0.939358i	$0.388578\pi$
$140$	440.000	0.265620
$141$	0	0
$142$	−240.000	−0.141833
$143$	−456.000	−0.266662
$144$	0	0
$145$	−330.000	−0.189000
$146$	88.0000	0.0498831
$147$	0	0
$148$	1256.00	0.697585
$149$	1608.00	0.884111	0.442055	−	0.896988i	$-0.354249\pi$
$149$	1608.00	0.884111	0.442055	+	0.896988i	$0.354249\pi$
$150$	0	0
$151$	−2488.00	−1.34086	−0.670432	−	0.741971i	$-0.733891\pi$
$151$	−2488.00	−1.34086	−0.670432	+	0.741971i	$0.733891\pi$
$152$	−1256.00	−0.670231
$153$	0	0
$154$	528.000	0.276282
$155$	125.000	0.0647758
$156$	0	0
$157$	−2968.00	−1.50874	−0.754370	−	0.656449i	$-0.772058\pi$
$157$	−2968.00	−1.50874	−0.754370	+	0.656449i	$0.772058\pi$
$158$	1834.00	0.923451
$159$	0	0
$160$	−160.000	−0.0790569
$161$	2574.00	1.26000
$162$	0	0
$163$	3170.00	1.52327	0.761637	−	0.648004i	$-0.224396\pi$
$163$	3170.00	1.52327	0.761637	+	0.648004i	$0.224396\pi$
$164$	−2016.00	−0.959897
$165$	0	0
$166$	618.000	0.288952
$167$	327.000	0.151521	0.0757605	−	0.997126i	$-0.475862\pi$
$167$	327.000	0.151521	0.0757605	+	0.997126i	$0.475862\pi$
$168$	0	0
$169$	−753.000	−0.342740
$170$	1050.00	0.473714
$171$	0	0
$172$	1520.00	0.673831
$173$	1305.00	0.573510	0.286755	−	0.958004i	$-0.407423\pi$
$173$	1305.00	0.573510	0.286755	+	0.958004i	$0.407423\pi$
$174$	0	0
$175$	−550.000	−0.237578
$176$	−192.000	−0.0822304
$177$	0	0
$178$	2544.00	1.07124
$179$	−4044.00	−1.68862	−0.844309	−	0.535856i	$-0.819989\pi$
$179$	−4044.00	−1.68862	−0.844309	+	0.535856i	$0.819989\pi$
$180$	0	0
$181$	−1051.00	−0.431603	−0.215802	−	0.976437i	$-0.569236\pi$
$181$	−1051.00	−0.431603	−0.215802	+	0.976437i	$0.569236\pi$
$182$	−1672.00	−0.680972
$183$	0	0
$184$	−936.000	−0.375015
$185$	−1570.00	−0.623939
$186$	0	0
$187$	1260.00	0.492729
$188$	−1008.00	−0.391042
$189$	0	0
$190$	1570.00	0.599472
$191$	−2598.00	−0.984213	−0.492106	−	0.870535i	$-0.663773\pi$
$191$	−2598.00	−0.984213	−0.492106	+	0.870535i	$0.663773\pi$
$192$	0	0
$193$	4370.00	1.62984	0.814921	−	0.579572i	$-0.196780\pi$
$193$	4370.00	1.62984	0.814921	+	0.579572i	$0.196780\pi$
$194$	2656.00	0.982937
$195$	0	0
$196$	564.000	0.205539
$197$	−2943.00	−1.06437	−0.532183	−	0.846629i	$-0.678628\pi$
$197$	−2943.00	−1.06437	−0.532183	+	0.846629i	$0.678628\pi$
$198$	0	0
$199$	−4768.00	−1.69847	−0.849233	−	0.528019i	$-0.822935\pi$
$199$	−4768.00	−1.69847	−0.849233	+	0.528019i	$0.822935\pi$
$200$	200.000	0.0707107
$201$	0	0
$202$	−984.000	−0.342743
$203$	−1452.00	−0.502022
$204$	0	0
$205$	2520.00	0.858558
$206$	1096.00	0.370689
$207$	0	0
$208$	608.000	0.202679
$209$	1884.00	0.623536
$210$	0	0
$211$	−1267.00	−0.413383	−0.206692	−	0.978406i	$-0.566270\pi$
$211$	−1267.00	−0.413383	−0.206692	+	0.978406i	$0.566270\pi$
$212$	12.0000	0.00388756
$213$	0	0
$214$	1464.00	0.467649
$215$	−1900.00	−0.602693
$216$	0	0
$217$	550.000	0.172057
$218$	−1814.00	−0.563576
$219$	0	0
$220$	240.000	0.0735491
$221$	−3990.00	−1.21446
$222$	0	0
$223$	−2986.00	−0.896670	−0.448335	−	0.893866i	$-0.647983\pi$
$223$	−2986.00	−0.896670	−0.448335	+	0.893866i	$0.647983\pi$
$224$	−704.000	−0.209991
$225$	0	0
$226$	−3084.00	−0.907720
$227$	5409.00	1.58153	0.790766	−	0.612118i	$-0.209682\pi$
$227$	5409.00	1.58153	0.790766	+	0.612118i	$0.209682\pi$
$228$	0	0
$229$	4331.00	1.24978	0.624892	−	0.780711i	$-0.285143\pi$
$229$	4331.00	1.24978	0.624892	+	0.780711i	$0.285143\pi$
$230$	1170.00	0.335424
$231$	0	0
$232$	528.000	0.149418
$233$	2586.00	0.727101	0.363550	−	0.931575i	$-0.381564\pi$
$233$	2586.00	0.727101	0.363550	+	0.931575i	$0.381564\pi$
$234$	0	0
$235$	1260.00	0.349759
$236$	−1272.00	−0.350848
$237$	0	0
$238$	4620.00	1.25828
$239$	510.000	0.138030	0.0690150	−	0.997616i	$-0.478014\pi$
$239$	510.000	0.138030	0.0690150	+	0.997616i	$0.478014\pi$
$240$	0	0
$241$	−205.000	−0.0547934	−0.0273967	−	0.999625i	$-0.508722\pi$
$241$	−205.000	−0.0547934	−0.0273967	+	0.999625i	$0.508722\pi$
$242$	−2374.00	−0.630605
$243$	0	0
$244$	1172.00	0.307498
$245$	−705.000	−0.183840
$246$	0	0
$247$	−5966.00	−1.53687
$248$	−200.000	−0.0512097
$249$	0	0
$250$	−250.000	−0.0632456
$251$	−4680.00	−1.17689	−0.588444	−	0.808538i	$-0.700259\pi$
$251$	−4680.00	−1.17689	−0.588444	+	0.808538i	$0.700259\pi$
$252$	0	0
$253$	1404.00	0.348888
$254$	−5108.00	−1.26183
$255$	0	0
$256$	256.000	0.0625000
$257$	−6159.00	−1.49489	−0.747447	−	0.664321i	$-0.768721\pi$
$257$	−6159.00	−1.49489	−0.747447	+	0.664321i	$0.768721\pi$
$258$	0	0
$259$	−6908.00	−1.65731
$260$	−760.000	−0.181282
$261$	0	0
$262$	−300.000	−0.0707407
$263$	−6240.00	−1.46302	−0.731511	−	0.681829i	$-0.761185\pi$
$263$	−6240.00	−1.46302	−0.731511	+	0.681829i	$0.761185\pi$
$264$	0	0
$265$	−15.0000	−0.00347714
$266$	6908.00	1.59232
$267$	0	0
$268$	−1288.00	−0.293571
$269$	−7758.00	−1.75841	−0.879207	−	0.476439i	$-0.841927\pi$
$269$	−7758.00	−1.75841	−0.879207	+	0.476439i	$0.841927\pi$
$270$	0	0
$271$	−7345.00	−1.64641	−0.823205	−	0.567745i	$-0.807816\pi$
$271$	−7345.00	−1.64641	−0.823205	+	0.567745i	$0.807816\pi$
$272$	−1680.00	−0.374504
$273$	0	0
$274$	3306.00	0.728915
$275$	−300.000	−0.0657843
$276$	0	0
$277$	−3004.00	−0.651599	−0.325799	−	0.945439i	$-0.605633\pi$
$277$	−3004.00	−0.651599	−0.325799	+	0.945439i	$0.605633\pi$
$278$	2248.00	0.484986
$279$	0	0
$280$	880.000	0.187822
$281$	2046.00	0.434356	0.217178	−	0.976132i	$-0.430315\pi$
$281$	2046.00	0.434356	0.217178	+	0.976132i	$0.430315\pi$
$282$	0	0
$283$	−5488.00	−1.15275	−0.576374	−	0.817186i	$-0.695533\pi$
$283$	−5488.00	−1.15275	−0.576374	+	0.817186i	$0.695533\pi$
$284$	−480.000	−0.100291
$285$	0	0
$286$	−912.000	−0.188558
$287$	11088.0	2.28050
$288$	0	0
$289$	6112.00	1.24405
$290$	−660.000	−0.133643
$291$	0	0
$292$	176.000	0.0352727
$293$	333.000	0.0663961	0.0331981	−	0.999449i	$-0.489431\pi$
$293$	333.000	0.0663961	0.0331981	+	0.999449i	$0.489431\pi$
$294$	0	0
$295$	1590.00	0.313808
$296$	2512.00	0.493267
$297$	0	0
$298$	3216.00	0.625161
$299$	−4446.00	−0.859929
$300$	0	0
$301$	−8360.00	−1.60087
$302$	−4976.00	−0.948135
$303$	0	0
$304$	−2512.00	−0.473925
$305$	−1465.00	−0.275035
$306$	0	0
$307$	2918.00	0.542472	0.271236	−	0.962513i	$-0.412568\pi$
$307$	2918.00	0.542472	0.271236	+	0.962513i	$0.412568\pi$
$308$	1056.00	0.195361
$309$	0	0
$310$	250.000	0.0458034
$311$	−5754.00	−1.04913	−0.524565	−	0.851370i	$-0.675772\pi$
$311$	−5754.00	−1.04913	−0.524565	+	0.851370i	$0.675772\pi$
$312$	0	0
$313$	3368.00	0.608213	0.304106	−	0.952638i	$-0.401642\pi$
$313$	3368.00	0.608213	0.304106	+	0.952638i	$0.401642\pi$
$314$	−5936.00	−1.06684
$315$	0	0
$316$	3668.00	0.652978
$317$	2871.00	0.508680	0.254340	−	0.967115i	$-0.418142\pi$
$317$	2871.00	0.508680	0.254340	+	0.967115i	$0.418142\pi$
$318$	0	0
$319$	−792.000	−0.139008
$320$	−320.000	−0.0559017
$321$	0	0
$322$	5148.00	0.890953
$323$	16485.0	2.83978
$324$	0	0
$325$	950.000	0.162143
$326$	6340.00	1.07712
$327$	0	0
$328$	−4032.00	−0.678750
$329$	5544.00	0.929029
$330$	0	0
$331$	−10540.0	−1.75024	−0.875122	−	0.483902i	$-0.839219\pi$
$331$	−10540.0	−1.75024	−0.875122	+	0.483902i	$0.839219\pi$
$332$	1236.00	0.204320
$333$	0	0
$334$	654.000	0.107142
$335$	1610.00	0.262578
$336$	0	0
$337$	5006.00	0.809182	0.404591	−	0.914498i	$-0.367414\pi$
$337$	5006.00	0.809182	0.404591	+	0.914498i	$0.367414\pi$
$338$	−1506.00	−0.242354
$339$	0	0
$340$	2100.00	0.334966
$341$	300.000	0.0476420
$342$	0	0
$343$	4444.00	0.699573
$344$	3040.00	0.476470
$345$	0	0
$346$	2610.00	0.405533
$347$	−36.0000	−0.00556940	−0.00278470	−	0.999996i	$-0.500886\pi$
$347$	−36.0000	−0.00556940	−0.00278470	+	0.999996i	$0.500886\pi$
$348$	0	0
$349$	−6715.00	−1.02993	−0.514965	−	0.857211i	$-0.672195\pi$
$349$	−6715.00	−1.02993	−0.514965	+	0.857211i	$0.672195\pi$
$350$	−1100.00	−0.167993
$351$	0	0
$352$	−384.000	−0.0581456
$353$	12822.0	1.93328	0.966638	−	0.256148i	$-0.0824533\pi$
$353$	12822.0	1.93328	0.966638	+	0.256148i	$0.0824533\pi$
$354$	0	0
$355$	600.000	0.0897034
$356$	5088.00	0.757482
$357$	0	0
$358$	−8088.00	−1.19403
$359$	5478.00	0.805342	0.402671	−	0.915345i	$-0.368082\pi$
$359$	5478.00	0.805342	0.402671	+	0.915345i	$0.368082\pi$
$360$	0	0
$361$	17790.0	2.59367
$362$	−2102.00	−0.305190
$363$	0	0
$364$	−3344.00	−0.481520
$365$	−220.000	−0.0315488
$366$	0	0
$367$	−2446.00	−0.347902	−0.173951	−	0.984754i	$-0.555653\pi$
$367$	−2446.00	−0.347902	−0.173951	+	0.984754i	$0.555653\pi$
$368$	−1872.00	−0.265176
$369$	0	0
$370$	−3140.00	−0.441191
$371$	−66.0000	−0.00923598
$372$	0	0
$373$	11696.0	1.62358	0.811791	−	0.583948i	$-0.198493\pi$
$373$	11696.0	1.62358	0.811791	+	0.583948i	$0.198493\pi$
$374$	2520.00	0.348412
$375$	0	0
$376$	−2016.00	−0.276509
$377$	2508.00	0.342622
$378$	0	0
$379$	−2095.00	−0.283939	−0.141970	−	0.989871i	$-0.545344\pi$
$379$	−2095.00	−0.283939	−0.141970	+	0.989871i	$0.545344\pi$
$380$	3140.00	0.423891
$381$	0	0
$382$	−5196.00	−0.695944
$383$	−11313.0	−1.50931	−0.754657	−	0.656119i	$-0.772197\pi$
$383$	−11313.0	−1.50931	−0.754657	+	0.656119i	$0.772197\pi$
$384$	0	0
$385$	−1320.00	−0.174736
$386$	8740.00	1.15247
$387$	0	0
$388$	5312.00	0.695041
$389$	2124.00	0.276841	0.138420	−	0.990374i	$-0.455797\pi$
$389$	2124.00	0.276841	0.138420	+	0.990374i	$0.455797\pi$
$390$	0	0
$391$	12285.0	1.58895
$392$	1128.00	0.145338
$393$	0	0
$394$	−5886.00	−0.752620
$395$	−4585.00	−0.584041
$396$	0	0
$397$	6410.00	0.810349	0.405175	−	0.914239i	$-0.367211\pi$
$397$	6410.00	0.810349	0.405175	+	0.914239i	$0.367211\pi$
$398$	−9536.00	−1.20100
$399$	0	0
$400$	400.000	0.0500000
$401$	−9882.00	−1.23063	−0.615316	−	0.788280i	$-0.710972\pi$
$401$	−9882.00	−1.23063	−0.615316	+	0.788280i	$0.710972\pi$
$402$	0	0
$403$	−950.000	−0.117426
$404$	−1968.00	−0.242356
$405$	0	0
$406$	−2904.00	−0.354983
$407$	−3768.00	−0.458901
$408$	0	0
$409$	5897.00	0.712929	0.356464	−	0.934309i	$-0.383982\pi$
$409$	5897.00	0.712929	0.356464	+	0.934309i	$0.383982\pi$
$410$	5040.00	0.607092
$411$	0	0
$412$	2192.00	0.262117
$413$	6996.00	0.833537
$414$	0	0
$415$	−1545.00	−0.182750
$416$	1216.00	0.143316
$417$	0	0
$418$	3768.00	0.440906
$419$	6852.00	0.798907	0.399454	−	0.916753i	$-0.369200\pi$
$419$	6852.00	0.798907	0.399454	+	0.916753i	$0.369200\pi$
$420$	0	0
$421$	323.000	0.0373921	0.0186960	−	0.999825i	$-0.494049\pi$
$421$	323.000	0.0373921	0.0186960	+	0.999825i	$0.494049\pi$
$422$	−2534.00	−0.292306
$423$	0	0
$424$	24.0000	0.00274892
$425$	−2625.00	−0.299603
$426$	0	0
$427$	−6446.00	−0.730548
$428$	2928.00	0.330678
$429$	0	0
$430$	−3800.00	−0.426168
$431$	10242.0	1.14464	0.572320	−	0.820030i	$-0.306044\pi$
$431$	10242.0	1.14464	0.572320	+	0.820030i	$0.306044\pi$
$432$	0	0
$433$	−14398.0	−1.59798	−0.798988	−	0.601347i	$-0.794631\pi$
$433$	−14398.0	−1.59798	−0.798988	+	0.601347i	$0.794631\pi$
$434$	1100.00	0.121663
$435$	0	0
$436$	−3628.00	−0.398508
$437$	18369.0	2.01077
$438$	0	0
$439$	4079.00	0.443463	0.221731	−	0.975108i	$-0.428829\pi$
$439$	4079.00	0.443463	0.221731	+	0.975108i	$0.428829\pi$
$440$	480.000	0.0520071
$441$	0	0
$442$	−7980.00	−0.858755
$443$	5781.00	0.620008	0.310004	−	0.950735i	$-0.399670\pi$
$443$	5781.00	0.620008	0.310004	+	0.950735i	$0.399670\pi$
$444$	0	0
$445$	−6360.00	−0.677512
$446$	−5972.00	−0.634041
$447$	0	0
$448$	−1408.00	−0.148486
$449$	−15078.0	−1.58480	−0.792400	−	0.610002i	$-0.791168\pi$
$449$	−15078.0	−1.58480	−0.792400	+	0.610002i	$0.791168\pi$
$450$	0	0
$451$	6048.00	0.631462
$452$	−6168.00	−0.641855
$453$	0	0
$454$	10818.0	1.11831
$455$	4180.00	0.430684
$456$	0	0
$457$	4268.00	0.436868	0.218434	−	0.975852i	$-0.429905\pi$
$457$	4268.00	0.436868	0.218434	+	0.975852i	$0.429905\pi$
$458$	8662.00	0.883731
$459$	0	0
$460$	2340.00	0.237181
$461$	−5634.00	−0.569201	−0.284600	−	0.958646i	$-0.591861\pi$
$461$	−5634.00	−0.569201	−0.284600	+	0.958646i	$0.591861\pi$
$462$	0	0
$463$	4526.00	0.454300	0.227150	−	0.973860i	$-0.427059\pi$
$463$	4526.00	0.454300	0.227150	+	0.973860i	$0.427059\pi$
$464$	1056.00	0.105654
$465$	0	0
$466$	5172.00	0.514138
$467$	969.000	0.0960171	0.0480085	−	0.998847i	$-0.484713\pi$
$467$	969.000	0.0960171	0.0480085	+	0.998847i	$0.484713\pi$
$468$	0	0
$469$	7084.00	0.697460
$470$	2520.00	0.247317
$471$	0	0
$472$	−2544.00	−0.248087
$473$	−4560.00	−0.443275
$474$	0	0
$475$	−3925.00	−0.379140
$476$	9240.00	0.889737
$477$	0	0
$478$	1020.00	0.0976019
$479$	9756.00	0.930612	0.465306	−	0.885150i	$-0.345944\pi$
$479$	9756.00	0.930612	0.465306	+	0.885150i	$0.345944\pi$
$480$	0	0
$481$	11932.0	1.13109
$482$	−410.000	−0.0387448
$483$	0	0
$484$	−4748.00	−0.445905
$485$	−6640.00	−0.621664
$486$	0	0
$487$	8768.00	0.815844	0.407922	−	0.913017i	$-0.366254\pi$
$487$	8768.00	0.815844	0.407922	+	0.913017i	$0.366254\pi$
$488$	2344.00	0.217434
$489$	0	0
$490$	−1410.00	−0.129995
$491$	−2274.00	−0.209011	−0.104505	−	0.994524i	$-0.533326\pi$
$491$	−2274.00	−0.209011	−0.104505	+	0.994524i	$0.533326\pi$
$492$	0	0
$493$	−6930.00	−0.633086
$494$	−11932.0	−1.08673
$495$	0	0
$496$	−400.000	−0.0362107
$497$	2640.00	0.238270
$498$	0	0
$499$	−1969.00	−0.176642	−0.0883212	−	0.996092i	$-0.528150\pi$
$499$	−1969.00	−0.176642	−0.0883212	+	0.996092i	$0.528150\pi$
$500$	−500.000	−0.0447214
$501$	0	0
$502$	−9360.00	−0.832186
$503$	−10701.0	−0.948577	−0.474288	−	0.880370i	$-0.657295\pi$
$503$	−10701.0	−0.948577	−0.474288	+	0.880370i	$0.657295\pi$
$504$	0	0
$505$	2460.00	0.216769
$506$	2808.00	0.246701
$507$	0	0
$508$	−10216.0	−0.892247
$509$	12420.0	1.08155	0.540773	−	0.841169i	$-0.318132\pi$
$509$	12420.0	1.08155	0.540773	+	0.841169i	$0.318132\pi$
$510$	0	0
$511$	−968.000	−0.0838000
$512$	512.000	0.0441942
$513$	0	0
$514$	−12318.0	−1.05705
$515$	−2740.00	−0.234444
$516$	0	0
$517$	3024.00	0.257244
$518$	−13816.0	−1.17189
$519$	0	0
$520$	−1520.00	−0.128185
$521$	−18816.0	−1.58223	−0.791117	−	0.611665i	$-0.790500\pi$
$521$	−18816.0	−1.58223	−0.791117	+	0.611665i	$0.790500\pi$
$522$	0	0
$523$	−16798.0	−1.40445	−0.702223	−	0.711957i	$-0.747809\pi$
$523$	−16798.0	−1.40445	−0.702223	+	0.711957i	$0.747809\pi$
$524$	−600.000	−0.0500212
$525$	0	0
$526$	−12480.0	−1.03451
$527$	2625.00	0.216977
$528$	0	0
$529$	1522.00	0.125092
$530$	−30.0000	−0.00245871
$531$	0	0
$532$	13816.0	1.12594
$533$	−19152.0	−1.55641
$534$	0	0
$535$	−3660.00	−0.295767
$536$	−2576.00	−0.207586
$537$	0	0
$538$	−15516.0	−1.24339
$539$	−1692.00	−0.135213
$540$	0	0
$541$	−5890.00	−0.468079	−0.234040	−	0.972227i	$-0.575195\pi$
$541$	−5890.00	−0.468079	−0.234040	+	0.972227i	$0.575195\pi$
$542$	−14690.0	−1.16419
$543$	0	0
$544$	−3360.00	−0.264814
$545$	4535.00	0.356437
$546$	0	0
$547$	5516.00	0.431165	0.215582	−	0.976486i	$-0.430835\pi$
$547$	5516.00	0.431165	0.215582	+	0.976486i	$0.430835\pi$
$548$	6612.00	0.515421
$549$	0	0
$550$	−600.000	−0.0465165
$551$	−10362.0	−0.801154
$552$	0	0
$553$	−20174.0	−1.55133
$554$	−6008.00	−0.460750
$555$	0	0
$556$	4496.00	0.342937
$557$	4146.00	0.315389	0.157694	−	0.987488i	$-0.449594\pi$
$557$	4146.00	0.315389	0.157694	+	0.987488i	$0.449594\pi$
$558$	0	0
$559$	14440.0	1.09257
$560$	1760.00	0.132810
$561$	0	0
$562$	4092.00	0.307136
$563$	−21444.0	−1.60525	−0.802626	−	0.596483i	$-0.796564\pi$
$563$	−21444.0	−1.60525	−0.802626	+	0.596483i	$0.796564\pi$
$564$	0	0
$565$	7710.00	0.574092
$566$	−10976.0	−0.815116
$567$	0	0
$568$	−960.000	−0.0709167
$569$	14778.0	1.08880	0.544399	−	0.838826i	$-0.316758\pi$
$569$	14778.0	1.08880	0.544399	+	0.838826i	$0.316758\pi$
$570$	0	0
$571$	9131.00	0.669213	0.334606	−	0.942358i	$-0.391397\pi$
$571$	9131.00	0.669213	0.334606	+	0.942358i	$0.391397\pi$
$572$	−1824.00	−0.133331
$573$	0	0
$574$	22176.0	1.61256
$575$	−2925.00	−0.212141
$576$	0	0
$577$	−2344.00	−0.169120	−0.0845598	−	0.996418i	$-0.526948\pi$
$577$	−2344.00	−0.169120	−0.0845598	+	0.996418i	$0.526948\pi$
$578$	12224.0	0.879674
$579$	0	0
$580$	−1320.00	−0.0945000
$581$	−6798.00	−0.485419
$582$	0	0
$583$	−36.0000	−0.00255741
$584$	352.000	0.0249415
$585$	0	0
$586$	666.000	0.0469492
$587$	−26829.0	−1.88646	−0.943229	−	0.332142i	$-0.892229\pi$
$587$	−26829.0	−1.88646	−0.943229	+	0.332142i	$0.892229\pi$
$588$	0	0
$589$	3925.00	0.274579
$590$	3180.00	0.221896
$591$	0	0
$592$	5024.00	0.348792
$593$	8181.00	0.566532	0.283266	−	0.959041i	$-0.408582\pi$
$593$	8181.00	0.566532	0.283266	+	0.959041i	$0.408582\pi$
$594$	0	0
$595$	−11550.0	−0.795805
$596$	6432.00	0.442055
$597$	0	0
$598$	−8892.00	−0.608062
$599$	24078.0	1.64240	0.821202	−	0.570637i	$-0.193304\pi$
$599$	24078.0	1.64240	0.821202	+	0.570637i	$0.193304\pi$
$600$	0	0
$601$	22565.0	1.53152	0.765762	−	0.643124i	$-0.222362\pi$
$601$	22565.0	1.53152	0.765762	+	0.643124i	$0.222362\pi$
$602$	−16720.0	−1.13199
$603$	0	0
$604$	−9952.00	−0.670432
$605$	5935.00	0.398830
$606$	0	0
$607$	−14716.0	−0.984026	−0.492013	−	0.870588i	$-0.663739\pi$
$607$	−14716.0	−0.984026	−0.492013	+	0.870588i	$0.663739\pi$
$608$	−5024.00	−0.335115
$609$	0	0
$610$	−2930.00	−0.194479
$611$	−9576.00	−0.634048
$612$	0	0
$613$	−7552.00	−0.497590	−0.248795	−	0.968556i	$-0.580034\pi$
$613$	−7552.00	−0.497590	−0.248795	+	0.968556i	$0.580034\pi$
$614$	5836.00	0.383586
$615$	0	0
$616$	2112.00	0.138141
$617$	−3981.00	−0.259755	−0.129878	−	0.991530i	$-0.541458\pi$
$617$	−3981.00	−0.259755	−0.129878	+	0.991530i	$0.541458\pi$
$618$	0	0
$619$	13928.0	0.904384	0.452192	−	0.891921i	$-0.350642\pi$
$619$	13928.0	0.904384	0.452192	+	0.891921i	$0.350642\pi$
$620$	500.000	0.0323879
$621$	0	0
$622$	−11508.0	−0.741847
$623$	−27984.0	−1.79961
$624$	0	0
$625$	625.000	0.0400000
$626$	6736.00	0.430071
$627$	0	0
$628$	−11872.0	−0.754370
$629$	−32970.0	−2.08998
$630$	0	0
$631$	18605.0	1.17378	0.586889	−	0.809668i	$-0.300353\pi$
$631$	18605.0	1.17378	0.586889	+	0.809668i	$0.300353\pi$
$632$	7336.00	0.461725
$633$	0	0
$634$	5742.00	0.359691
$635$	12770.0	0.798050
$636$	0	0
$637$	5358.00	0.333268
$638$	−1584.00	−0.0982934
$639$	0	0
$640$	−640.000	−0.0395285
$641$	14928.0	0.919845	0.459922	−	0.887959i	$-0.347877\pi$
$641$	14928.0	0.919845	0.459922	+	0.887959i	$0.347877\pi$
$642$	0	0
$643$	−6082.00	−0.373018	−0.186509	−	0.982453i	$-0.559717\pi$
$643$	−6082.00	−0.373018	−0.186509	+	0.982453i	$0.559717\pi$
$644$	10296.0	0.629999
$645$	0	0
$646$	32970.0	2.00803
$647$	4875.00	0.296223	0.148111	−	0.988971i	$-0.452681\pi$
$647$	4875.00	0.296223	0.148111	+	0.988971i	$0.452681\pi$
$648$	0	0
$649$	3816.00	0.230803
$650$	1900.00	0.114653
$651$	0	0
$652$	12680.0	0.761637
$653$	−5157.00	−0.309049	−0.154525	−	0.987989i	$-0.549385\pi$
$653$	−5157.00	−0.309049	−0.154525	+	0.987989i	$0.549385\pi$
$654$	0	0
$655$	750.000	0.0447403
$656$	−8064.00	−0.479949
$657$	0	0
$658$	11088.0	0.656923
$659$	−1506.00	−0.0890219	−0.0445109	−	0.999009i	$-0.514173\pi$
$659$	−1506.00	−0.0890219	−0.0445109	+	0.999009i	$0.514173\pi$
$660$	0	0
$661$	−2386.00	−0.140400	−0.0702002	−	0.997533i	$-0.522364\pi$
$661$	−2386.00	−0.140400	−0.0702002	+	0.997533i	$0.522364\pi$
$662$	−21080.0	−1.23761
$663$	0	0
$664$	2472.00	0.144476
$665$	−17270.0	−1.00707
$666$	0	0
$667$	−7722.00	−0.448271
$668$	1308.00	0.0757605
$669$	0	0
$670$	3220.00	0.185671
$671$	−3516.00	−0.202286
$672$	0	0
$673$	21158.0	1.21186	0.605929	−	0.795518i	$-0.292801\pi$
$673$	21158.0	1.21186	0.605929	+	0.795518i	$0.292801\pi$
$674$	10012.0	0.572178
$675$	0	0
$676$	−3012.00	−0.171370
$677$	26826.0	1.52291	0.761453	−	0.648220i	$-0.224486\pi$
$677$	26826.0	1.52291	0.761453	+	0.648220i	$0.224486\pi$
$678$	0	0
$679$	−29216.0	−1.65126
$680$	4200.00	0.236857
$681$	0	0
$682$	600.000	0.0336880
$683$	32493.0	1.82037	0.910183	−	0.414206i	$-0.135941\pi$
$683$	32493.0	1.82037	0.910183	+	0.414206i	$0.135941\pi$
$684$	0	0
$685$	−8265.00	−0.461006
$686$	8888.00	0.494673
$687$	0	0
$688$	6080.00	0.336915
$689$	114.000	0.00630342
$690$	0	0
$691$	−7531.00	−0.414606	−0.207303	−	0.978277i	$-0.566469\pi$
$691$	−7531.00	−0.414606	−0.207303	+	0.978277i	$0.566469\pi$
$692$	5220.00	0.286755
$693$	0	0
$694$	−72.0000	−0.00393816
$695$	−5620.00	−0.306732
$696$	0	0
$697$	52920.0	2.87588
$698$	−13430.0	−0.728271
$699$	0	0
$700$	−2200.00	−0.118789
$701$	24306.0	1.30959	0.654797	−	0.755805i	$-0.272754\pi$
$701$	24306.0	1.30959	0.654797	+	0.755805i	$0.272754\pi$
$702$	0	0
$703$	−49298.0	−2.64482
$704$	−768.000	−0.0411152
$705$	0	0
$706$	25644.0	1.36703
$707$	10824.0	0.575783
$708$	0	0
$709$	−27454.0	−1.45424	−0.727120	−	0.686510i	$-0.759142\pi$
$709$	−27454.0	−1.45424	−0.727120	+	0.686510i	$0.759142\pi$
$710$	1200.00	0.0634299
$711$	0	0
$712$	10176.0	0.535620
$713$	2925.00	0.153635
$714$	0	0
$715$	2280.00	0.119255
$716$	−16176.0	−0.844309
$717$	0	0
$718$	10956.0	0.569463
$719$	−5334.00	−0.276668	−0.138334	−	0.990386i	$-0.544175\pi$
$719$	−5334.00	−0.276668	−0.138334	+	0.990386i	$0.544175\pi$
$720$	0	0
$721$	−12056.0	−0.622731
$722$	35580.0	1.83400
$723$	0	0
$724$	−4204.00	−0.215802
$725$	1650.00	0.0845234
$726$	0	0
$727$	26048.0	1.32884	0.664420	−	0.747359i	$-0.268679\pi$
$727$	26048.0	1.32884	0.664420	+	0.747359i	$0.268679\pi$
$728$	−6688.00	−0.340486
$729$	0	0
$730$	−440.000	−0.0223084
$731$	−39900.0	−2.01882
$732$	0	0
$733$	−33136.0	−1.66972	−0.834861	−	0.550461i	$-0.814452\pi$
$733$	−33136.0	−1.66972	−0.834861	+	0.550461i	$0.814452\pi$
$734$	−4892.00	−0.246004
$735$	0	0
$736$	−3744.00	−0.187508
$737$	3864.00	0.193124
$738$	0	0
$739$	−11599.0	−0.577370	−0.288685	−	0.957424i	$-0.593218\pi$
$739$	−11599.0	−0.577370	−0.288685	+	0.957424i	$0.593218\pi$
$740$	−6280.00	−0.311969
$741$	0	0
$742$	−132.000	−0.00653083
$743$	26424.0	1.30471	0.652357	−	0.757912i	$-0.273780\pi$
$743$	26424.0	1.30471	0.652357	+	0.757912i	$0.273780\pi$
$744$	0	0
$745$	−8040.00	−0.395386
$746$	23392.0	1.14805
$747$	0	0
$748$	5040.00	0.246365
$749$	−16104.0	−0.785617
$750$	0	0
$751$	13661.0	0.663778	0.331889	−	0.943319i	$-0.392314\pi$
$751$	13661.0	0.663778	0.331889	+	0.943319i	$0.392314\pi$
$752$	−4032.00	−0.195521
$753$	0	0
$754$	5016.00	0.242270
$755$	12440.0	0.599653
$756$	0	0
$757$	−22846.0	−1.09690	−0.548449	−	0.836184i	$-0.684782\pi$
$757$	−22846.0	−1.09690	−0.548449	+	0.836184i	$0.684782\pi$
$758$	−4190.00	−0.200775
$759$	0	0
$760$	6280.00	0.299736
$761$	20862.0	0.993754	0.496877	−	0.867821i	$-0.334480\pi$
$761$	20862.0	0.993754	0.496877	+	0.867821i	$0.334480\pi$
$762$	0	0
$763$	19954.0	0.946767
$764$	−10392.0	−0.492106
$765$	0	0
$766$	−22626.0	−1.06725
$767$	−12084.0	−0.568876
$768$	0	0
$769$	−22219.0	−1.04192	−0.520961	−	0.853581i	$-0.674426\pi$
$769$	−22219.0	−1.04192	−0.520961	+	0.853581i	$0.674426\pi$
$770$	−2640.00	−0.123557
$771$	0	0
$772$	17480.0	0.814921
$773$	17619.0	0.819808	0.409904	−	0.912129i	$-0.365562\pi$
$773$	17619.0	0.819808	0.409904	+	0.912129i	$0.365562\pi$
$774$	0	0
$775$	−625.000	−0.0289686
$776$	10624.0	0.491468
$777$	0	0
$778$	4248.00	0.195756
$779$	79128.0	3.63935
$780$	0	0
$781$	1440.00	0.0659760
$782$	24570.0	1.12356
$783$	0	0
$784$	2256.00	0.102770
$785$	14840.0	0.674729
$786$	0	0
$787$	15584.0	0.705857	0.352929	−	0.935650i	$-0.385186\pi$
$787$	15584.0	0.705857	0.352929	+	0.935650i	$0.385186\pi$
$788$	−11772.0	−0.532183
$789$	0	0
$790$	−9170.00	−0.412980
$791$	33924.0	1.52490
$792$	0	0
$793$	11134.0	0.498588
$794$	12820.0	0.573003
$795$	0	0
$796$	−19072.0	−0.849233
$797$	−13023.0	−0.578793	−0.289397	−	0.957209i	$-0.593455\pi$
$797$	−13023.0	−0.578793	−0.289397	+	0.957209i	$0.593455\pi$
$798$	0	0
$799$	26460.0	1.17157
$800$	800.000	0.0353553
$801$	0	0
$802$	−19764.0	−0.870188
$803$	−528.000	−0.0232039
$804$	0	0
$805$	−12870.0	−0.563488
$806$	−1900.00	−0.0830331
$807$	0	0
$808$	−3936.00	−0.171371
$809$	−25872.0	−1.12436	−0.562182	−	0.827013i	$-0.690038\pi$
$809$	−25872.0	−1.12436	−0.562182	+	0.827013i	$0.690038\pi$
$810$	0	0
$811$	22052.0	0.954809	0.477405	−	0.878684i	$-0.341578\pi$
$811$	22052.0	0.954809	0.477405	+	0.878684i	$0.341578\pi$
$812$	−5808.00	−0.251011
$813$	0	0
$814$	−7536.00	−0.324492
$815$	−15850.0	−0.681229
$816$	0	0
$817$	−59660.0	−2.55476
$818$	11794.0	0.504117
$819$	0	0
$820$	10080.0	0.429279
$821$	1914.00	0.0813630	0.0406815	−	0.999172i	$-0.487047\pi$
$821$	1914.00	0.0813630	0.0406815	+	0.999172i	$0.487047\pi$
$822$	0	0
$823$	−11068.0	−0.468780	−0.234390	−	0.972143i	$-0.575309\pi$
$823$	−11068.0	−0.468780	−0.234390	+	0.972143i	$0.575309\pi$
$824$	4384.00	0.185345
$825$	0	0
$826$	13992.0	0.589399
$827$	−21405.0	−0.900030	−0.450015	−	0.893021i	$-0.648581\pi$
$827$	−21405.0	−0.900030	−0.450015	+	0.893021i	$0.648581\pi$
$828$	0	0
$829$	−40042.0	−1.67758	−0.838791	−	0.544453i	$-0.816737\pi$
$829$	−40042.0	−1.67758	−0.838791	+	0.544453i	$0.816737\pi$
$830$	−3090.00	−0.129223
$831$	0	0
$832$	2432.00	0.101339
$833$	−14805.0	−0.615802
$834$	0	0
$835$	−1635.00	−0.0677623
$836$	7536.00	0.311768
$837$	0	0
$838$	13704.0	0.564913
$839$	17904.0	0.736728	0.368364	−	0.929682i	$-0.379918\pi$
$839$	17904.0	0.736728	0.368364	+	0.929682i	$0.379918\pi$
$840$	0	0
$841$	−20033.0	−0.821395
$842$	646.000	0.0264402
$843$	0	0
$844$	−5068.00	−0.206692
$845$	3765.00	0.153278
$846$	0	0
$847$	26114.0	1.05937
$848$	48.0000	0.00194378
$849$	0	0
$850$	−5250.00	−0.211851
$851$	−36738.0	−1.47986
$852$	0	0
$853$	−21580.0	−0.866219	−0.433110	−	0.901341i	$-0.642584\pi$
$853$	−21580.0	−0.866219	−0.433110	+	0.901341i	$0.642584\pi$
$854$	−12892.0	−0.516575
$855$	0	0
$856$	5856.00	0.233825
$857$	17151.0	0.683625	0.341813	−	0.939768i	$-0.388959\pi$
$857$	17151.0	0.683625	0.341813	+	0.939768i	$0.388959\pi$
$858$	0	0
$859$	33425.0	1.32764	0.663822	−	0.747891i	$-0.268933\pi$
$859$	33425.0	1.32764	0.663822	+	0.747891i	$0.268933\pi$
$860$	−7600.00	−0.301346
$861$	0	0
$862$	20484.0	0.809383
$863$	6603.00	0.260450	0.130225	−	0.991484i	$-0.458430\pi$
$863$	6603.00	0.260450	0.130225	+	0.991484i	$0.458430\pi$
$864$	0	0
$865$	−6525.00	−0.256482
$866$	−28796.0	−1.12994
$867$	0	0
$868$	2200.00	0.0860286
$869$	−11004.0	−0.429557
$870$	0	0
$871$	−12236.0	−0.476006
$872$	−7256.00	−0.281788
$873$	0	0
$874$	36738.0	1.42183
$875$	2750.00	0.106248
$876$	0	0
$877$	−43384.0	−1.67044	−0.835219	−	0.549918i	$-0.814659\pi$
$877$	−43384.0	−1.67044	−0.835219	+	0.549918i	$0.814659\pi$
$878$	8158.00	0.313575
$879$	0	0
$880$	960.000	0.0367745
$881$	12726.0	0.486663	0.243331	−	0.969943i	$-0.421760\pi$
$881$	12726.0	0.486663	0.243331	+	0.969943i	$0.421760\pi$
$882$	0	0
$883$	2786.00	0.106179	0.0530897	−	0.998590i	$-0.483093\pi$
$883$	2786.00	0.106179	0.0530897	+	0.998590i	$0.483093\pi$
$884$	−15960.0	−0.607232
$885$	0	0
$886$	11562.0	0.438412
$887$	−4389.00	−0.166142	−0.0830711	−	0.996544i	$-0.526473\pi$
$887$	−4389.00	−0.166142	−0.0830711	+	0.996544i	$0.526473\pi$
$888$	0	0
$889$	56188.0	2.11978
$890$	−12720.0	−0.479073
$891$	0	0
$892$	−11944.0	−0.448335
$893$	39564.0	1.48260
$894$	0	0
$895$	20220.0	0.755173
$896$	−2816.00	−0.104995
$897$	0	0
$898$	−30156.0	−1.12062
$899$	−1650.00	−0.0612131
$900$	0	0
$901$	−315.000	−0.0116472
$902$	12096.0	0.446511
$903$	0	0
$904$	−12336.0	−0.453860
$905$	5255.00	0.193019
$906$	0	0
$907$	−24868.0	−0.910395	−0.455198	−	0.890390i	$-0.650431\pi$
$907$	−24868.0	−0.910395	−0.455198	+	0.890390i	$0.650431\pi$
$908$	21636.0	0.790766
$909$	0	0
$910$	8360.00	0.304540
$911$	−126.000	−0.00458240	−0.00229120	−	0.999997i	$-0.500729\pi$
$911$	−126.000	−0.00458240	−0.00229120	+	0.999997i	$0.500729\pi$
$912$	0	0
$913$	−3708.00	−0.134411
$914$	8536.00	0.308912
$915$	0	0
$916$	17324.0	0.624892
$917$	3300.00	0.118839
$918$	0	0
$919$	−16144.0	−0.579479	−0.289740	−	0.957106i	$-0.593569\pi$
$919$	−16144.0	−0.579479	−0.289740	+	0.957106i	$0.593569\pi$
$920$	4680.00	0.167712
$921$	0	0
$922$	−11268.0	−0.402486
$923$	−4560.00	−0.162616
$924$	0	0
$925$	7850.00	0.279034
$926$	9052.00	0.321239
$927$	0	0
$928$	2112.00	0.0747088
$929$	−42192.0	−1.49007	−0.745035	−	0.667026i	$-0.767567\pi$
$929$	−42192.0	−1.49007	−0.745035	+	0.667026i	$0.767567\pi$
$930$	0	0
$931$	−22137.0	−0.779281
$932$	10344.0	0.363550
$933$	0	0
$934$	1938.00	0.0678943
$935$	−6300.00	−0.220355
$936$	0	0
$937$	3272.00	0.114079	0.0570393	−	0.998372i	$-0.481834\pi$
$937$	3272.00	0.114079	0.0570393	+	0.998372i	$0.481834\pi$
$938$	14168.0	0.493179
$939$	0	0
$940$	5040.00	0.174879
$941$	20838.0	0.721891	0.360945	−	0.932587i	$-0.382454\pi$
$941$	20838.0	0.721891	0.360945	+	0.932587i	$0.382454\pi$
$942$	0	0
$943$	58968.0	2.03633
$944$	−5088.00	−0.175424
$945$	0	0
$946$	−9120.00	−0.313443
$947$	13353.0	0.458199	0.229099	−	0.973403i	$-0.426422\pi$
$947$	13353.0	0.458199	0.229099	+	0.973403i	$0.426422\pi$
$948$	0	0
$949$	1672.00	0.0571922
$950$	−7850.00	−0.268092
$951$	0	0
$952$	18480.0	0.629139
$953$	13098.0	0.445211	0.222605	−	0.974909i	$-0.428544\pi$
$953$	13098.0	0.445211	0.222605	+	0.974909i	$0.428544\pi$
$954$	0	0
$955$	12990.0	0.440153
$956$	2040.00	0.0690150
$957$	0	0
$958$	19512.0	0.658042
$959$	−36366.0	−1.22452
$960$	0	0
$961$	−29166.0	−0.979021
$962$	23864.0	0.799799
$963$	0	0
$964$	−820.000	−0.0273967
$965$	−21850.0	−0.728887
$966$	0	0
$967$	19826.0	0.659319	0.329659	−	0.944100i	$-0.393066\pi$
$967$	19826.0	0.659319	0.329659	+	0.944100i	$0.393066\pi$
$968$	−9496.00	−0.315303
$969$	0	0
$970$	−13280.0	−0.439583
$971$	23322.0	0.770792	0.385396	−	0.922751i	$-0.374065\pi$
$971$	23322.0	0.770792	0.385396	+	0.922751i	$0.374065\pi$
$972$	0	0
$973$	−24728.0	−0.814741
$974$	17536.0	0.576889
$975$	0	0
$976$	4688.00	0.153749
$977$	−47346.0	−1.55039	−0.775196	−	0.631721i	$-0.782349\pi$
$977$	−47346.0	−1.55039	−0.775196	+	0.631721i	$0.782349\pi$
$978$	0	0
$979$	−15264.0	−0.498304
$980$	−2820.00	−0.0919200
$981$	0	0
$982$	−4548.00	−0.147793
$983$	−33033.0	−1.07181	−0.535905	−	0.844278i	$-0.680029\pi$
$983$	−33033.0	−1.07181	−0.535905	+	0.844278i	$0.680029\pi$
$984$	0	0
$985$	14715.0	0.475999
$986$	−13860.0	−0.447660
$987$	0	0
$988$	−23864.0	−0.768436
$989$	−44460.0	−1.42947
$990$	0	0
$991$	6017.00	0.192872	0.0964361	−	0.995339i	$-0.469256\pi$
$991$	6017.00	0.192872	0.0964361	+	0.995339i	$0.469256\pi$
$992$	−800.000	−0.0256049
$993$	0	0
$994$	5280.00	0.168482
$995$	23840.0	0.759577
$996$	0	0
$997$	−34216.0	−1.08689	−0.543446	−	0.839444i	$-0.682881\pi$
$997$	−34216.0	−1.08689	−0.543446	+	0.839444i	$0.682881\pi$
$998$	−3938.00	−0.124905
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	270.4.a.g.1.1	yes	1
3.2	odd	2		270.4.a.c.1.1	✓	1
4.3	odd	2		2160.4.a.i.1.1		1
5.2	odd	4		1350.4.c.i.649.2		2
5.3	odd	4		1350.4.c.i.649.1		2
5.4	even	2		1350.4.a.l.1.1		1
9.2	odd	6		810.4.e.s.271.1		2
9.4	even	3		810.4.e.k.541.1		2
9.5	odd	6		810.4.e.s.541.1		2
9.7	even	3		810.4.e.k.271.1		2
12.11	even	2		2160.4.a.r.1.1		1
15.2	even	4		1350.4.c.l.649.1		2
15.8	even	4		1350.4.c.l.649.2		2
15.14	odd	2		1350.4.a.z.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
270.4.a.c.1.1	✓	1	3.2	odd	2
270.4.a.g.1.1	yes	1	1.1	even	1	trivial
810.4.e.k.271.1		2	9.7	even	3
810.4.e.k.541.1		2	9.4	even	3
810.4.e.s.271.1		2	9.2	odd	6
810.4.e.s.541.1		2	9.5	odd	6
1350.4.a.l.1.1		1	5.4	even	2
1350.4.a.z.1.1		1	15.14	odd	2
1350.4.c.i.649.1		2	5.3	odd	4
1350.4.c.i.649.2		2	5.2	odd	4
1350.4.c.l.649.1		2	15.2	even	4
1350.4.c.l.649.2		2	15.8	even	4
2160.4.a.i.1.1		1	4.3	odd	2
2160.4.a.r.1.1		1	12.11	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:	\( N \)	\(=\)	\( 270 = 2 \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	270.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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