LMFDB - Embedded newform 270.4.a.f.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:	$0$
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)

$f(q)$

$=$

$q-2.00000 q^{2} +4.00000 q^{4} +5.00000 q^{5} +14.0000 q^{7} -8.00000 q^{8} +O(q^{10})$

\(q-2.00000 q^{2} +4.00000 q^{4} +5.00000 q^{5} +14.0000 q^{7} -8.00000 q^{8} -10.0000 q^{10} +3.00000 q^{11} +47.0000 q^{13} -28.0000 q^{14} +16.0000 q^{16} -39.0000 q^{17} +32.0000 q^{19} +20.0000 q^{20} -6.00000 q^{22} -99.0000 q^{23} +25.0000 q^{25} -94.0000 q^{26} +56.0000 q^{28} +51.0000 q^{29} +83.0000 q^{31} -32.0000 q^{32} +78.0000 q^{34} +70.0000 q^{35} +314.000 q^{37} -64.0000 q^{38} -40.0000 q^{40} -108.000 q^{41} +299.000 q^{43} +12.0000 q^{44} +198.000 q^{46} +531.000 q^{47} -147.000 q^{49} -50.0000 q^{50} +188.000 q^{52} +564.000 q^{53} +15.0000 q^{55} -112.000 q^{56} -102.000 q^{58} +12.0000 q^{59} +230.000 q^{61} -166.000 q^{62} +64.0000 q^{64} +235.000 q^{65} -268.000 q^{67} -156.000 q^{68} -140.000 q^{70} +120.000 q^{71} +1106.00 q^{73} -628.000 q^{74} +128.000 q^{76} +42.0000 q^{77} -739.000 q^{79} +80.0000 q^{80} +216.000 q^{82} +1086.00 q^{83} -195.000 q^{85} -598.000 q^{86} -24.0000 q^{88} -120.000 q^{89} +658.000 q^{91} -396.000 q^{92} -1062.00 q^{94} +160.000 q^{95} -1642.00 q^{97} +294.000 q^{98} +O(q^{100})\)

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
$7$	14.0000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	14.0000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	−8.00000	−0.353553
$9$	0	0
$10$	−10.0000	−0.316228
$11$	3.00000	0.0822304	0.0411152	−	0.999154i	$-0.486909\pi$
$11$	3.00000	0.0822304	0.0411152	+	0.999154i	$0.486909\pi$
$12$	0	0
$13$	47.0000	1.00273	0.501364	−	0.865237i	$-0.332832\pi$
$13$	47.0000	1.00273	0.501364	+	0.865237i	$0.332832\pi$
$14$	−28.0000	−0.534522
$15$	0	0
$16$	16.0000	0.250000
$17$	−39.0000	−0.556405	−0.278203	−	0.960522i	$-0.589739\pi$
$17$	−39.0000	−0.556405	−0.278203	+	0.960522i	$0.589739\pi$
$18$	0	0
$19$	32.0000	0.386384	0.193192	−	0.981161i	$-0.438116\pi$
$19$	32.0000	0.386384	0.193192	+	0.981161i	$0.438116\pi$
$20$	20.0000	0.223607
$21$	0	0
$22$	−6.00000	−0.0581456
$23$	−99.0000	−0.897519	−0.448759	−	0.893653i	$-0.648134\pi$
$23$	−99.0000	−0.897519	−0.448759	+	0.893653i	$0.648134\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	−94.0000	−0.709035
$27$	0	0
$28$	56.0000	0.377964
$29$	51.0000	0.326568	0.163284	−	0.986579i	$-0.447791\pi$
$29$	51.0000	0.326568	0.163284	+	0.986579i	$0.447791\pi$
$30$	0	0
$31$	83.0000	0.480879	0.240439	−	0.970664i	$-0.422708\pi$
$31$	83.0000	0.480879	0.240439	+	0.970664i	$0.422708\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	78.0000	0.393438
$35$	70.0000	0.338062
$36$	0	0
$37$	314.000	1.39517	0.697585	−	0.716502i	$-0.254258\pi$
$37$	314.000	1.39517	0.697585	+	0.716502i	$0.254258\pi$
$38$	−64.0000	−0.273215
$39$	0	0
$40$	−40.0000	−0.158114
$41$	−108.000	−0.411385	−0.205692	−	0.978617i	$-0.565945\pi$
$41$	−108.000	−0.411385	−0.205692	+	0.978617i	$0.565945\pi$
$42$	0	0
$43$	299.000	1.06040	0.530199	−	0.847874i	$-0.322117\pi$
$43$	299.000	1.06040	0.530199	+	0.847874i	$0.322117\pi$
$44$	12.0000	0.0411152
$45$	0	0
$46$	198.000	0.634641
$47$	531.000	1.64796	0.823982	−	0.566616i	$-0.191748\pi$
$47$	531.000	1.64796	0.823982	+	0.566616i	$0.191748\pi$
$48$	0	0
$49$	−147.000	−0.428571
$50$	−50.0000	−0.141421
$51$	0	0
$52$	188.000	0.501364
$53$	564.000	1.46172	0.730862	−	0.682525i	$-0.239118\pi$
$53$	564.000	1.46172	0.730862	+	0.682525i	$0.239118\pi$
$54$	0	0
$55$	15.0000	0.0367745
$56$	−112.000	−0.267261
$57$	0	0
$58$	−102.000	−0.230918
$59$	12.0000	0.0264791	0.0132396	−	0.999912i	$-0.495786\pi$
$59$	12.0000	0.0264791	0.0132396	+	0.999912i	$0.495786\pi$
$60$	0	0
$61$	230.000	0.482762	0.241381	−	0.970430i	$-0.422400\pi$
$61$	230.000	0.482762	0.241381	+	0.970430i	$0.422400\pi$
$62$	−166.000	−0.340033
$63$	0	0
$64$	64.0000	0.125000
$65$	235.000	0.448433
$66$	0	0
$67$	−268.000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−268.000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	−156.000	−0.278203
$69$	0	0
$70$	−140.000	−0.239046
$71$	120.000	0.200583	0.100291	−	0.994958i	$-0.468022\pi$
$71$	120.000	0.200583	0.100291	+	0.994958i	$0.468022\pi$
$72$	0	0
$73$	1106.00	1.77325	0.886627	−	0.462486i	$-0.153042\pi$
$73$	1106.00	1.77325	0.886627	+	0.462486i	$0.153042\pi$
$74$	−628.000	−0.986534
$75$	0	0
$76$	128.000	0.193192
$77$	42.0000	0.0621603
$78$	0	0
$79$	−739.000	−1.05246	−0.526228	−	0.850344i	$-0.676394\pi$
$79$	−739.000	−1.05246	−0.526228	+	0.850344i	$0.676394\pi$
$80$	80.0000	0.111803
$81$	0	0
$82$	216.000	0.290893
$83$	1086.00	1.43619	0.718096	−	0.695944i	$-0.245014\pi$
$83$	1086.00	1.43619	0.718096	+	0.695944i	$0.245014\pi$
$84$	0	0
$85$	−195.000	−0.248832
$86$	−598.000	−0.749814
$87$	0	0
$88$	−24.0000	−0.0290728
$89$	−120.000	−0.142921	−0.0714605	−	0.997443i	$-0.522766\pi$
$89$	−120.000	−0.142921	−0.0714605	+	0.997443i	$0.522766\pi$
$90$	0	0
$91$	658.000	0.757991
$92$	−396.000	−0.448759
$93$	0	0
$94$	−1062.00	−1.16529
$95$	160.000	0.172796
$96$	0	0
$97$	−1642.00	−1.71876	−0.859381	−	0.511336i	$-0.829151\pi$
$97$	−1642.00	−1.71876	−0.859381	+	0.511336i	$0.829151\pi$
$98$	294.000	0.303046
$99$	0	0
$100$	100.000	0.100000
$101$	33.0000	0.0325111	0.0162556	−	0.999868i	$-0.494825\pi$
$101$	33.0000	0.0325111	0.0162556	+	0.999868i	$0.494825\pi$
$102$	0	0
$103$	−1198.00	−1.14604	−0.573022	−	0.819540i	$-0.694229\pi$
$103$	−1198.00	−1.14604	−0.573022	+	0.819540i	$0.694229\pi$
$104$	−376.000	−0.354518
$105$	0	0
$106$	−1128.00	−1.03359
$107$	−1542.00	−1.39318	−0.696592	−	0.717467i	$-0.745301\pi$
$107$	−1542.00	−1.39318	−0.696592	+	0.717467i	$0.745301\pi$
$108$	0	0
$109$	−556.000	−0.488579	−0.244290	−	0.969702i	$-0.578555\pi$
$109$	−556.000	−0.488579	−0.244290	+	0.969702i	$0.578555\pi$
$110$	−30.0000	−0.0260035
$111$	0	0
$112$	224.000	0.188982
$113$	1605.00	1.33616	0.668078	−	0.744091i	$-0.267117\pi$
$113$	1605.00	1.33616	0.668078	+	0.744091i	$0.267117\pi$
$114$	0	0
$115$	−495.000	−0.401383
$116$	204.000	0.163284
$117$	0	0
$118$	−24.0000	−0.0187236
$119$	−546.000	−0.420603
$120$	0	0
$121$	−1322.00	−0.993238
$122$	−460.000	−0.341364
$123$	0	0
$124$	332.000	0.240439
$125$	125.000	0.0894427
$126$	0	0
$127$	1334.00	0.932074	0.466037	−	0.884765i	$-0.345681\pi$
$127$	1334.00	0.932074	0.466037	+	0.884765i	$0.345681\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	−470.000	−0.317090
$131$	−2883.00	−1.92282	−0.961408	−	0.275127i	$-0.911280\pi$
$131$	−2883.00	−1.92282	−0.961408	+	0.275127i	$0.911280\pi$
$132$	0	0
$133$	448.000	0.292079
$134$	536.000	0.345547
$135$	0	0
$136$	312.000	0.196719
$137$	282.000	0.175860	0.0879302	−	0.996127i	$-0.471975\pi$
$137$	282.000	0.175860	0.0879302	+	0.996127i	$0.471975\pi$
$138$	0	0
$139$	−2494.00	−1.52186	−0.760929	−	0.648835i	$-0.775257\pi$
$139$	−2494.00	−1.52186	−0.760929	+	0.648835i	$0.775257\pi$
$140$	280.000	0.169031
$141$	0	0
$142$	−240.000	−0.141833
$143$	141.000	0.0824546
$144$	0	0
$145$	255.000	0.146045
$146$	−2212.00	−1.25388
$147$	0	0
$148$	1256.00	0.697585
$149$	2595.00	1.42678	0.713392	−	0.700766i	$-0.247158\pi$
$149$	2595.00	1.42678	0.713392	+	0.700766i	$0.247158\pi$
$150$	0	0
$151$	1229.00	0.662348	0.331174	−	0.943570i	$-0.392555\pi$
$151$	1229.00	0.662348	0.331174	+	0.943570i	$0.392555\pi$
$152$	−256.000	−0.136608
$153$	0	0
$154$	−84.0000	−0.0439540
$155$	415.000	0.215055
$156$	0	0
$157$	−1591.00	−0.808762	−0.404381	−	0.914591i	$-0.632513\pi$
$157$	−1591.00	−0.808762	−0.404381	+	0.914591i	$0.632513\pi$
$158$	1478.00	0.744199
$159$	0	0
$160$	−160.000	−0.0790569
$161$	−1386.00	−0.678460
$162$	0	0
$163$	−457.000	−0.219601	−0.109801	−	0.993954i	$-0.535021\pi$
$163$	−457.000	−0.219601	−0.109801	+	0.993954i	$0.535021\pi$
$164$	−432.000	−0.205692
$165$	0	0
$166$	−2172.00	−1.01554
$167$	−1164.00	−0.539359	−0.269680	−	0.962950i	$-0.586918\pi$
$167$	−1164.00	−0.539359	−0.269680	+	0.962950i	$0.586918\pi$
$168$	0	0
$169$	12.0000	0.00546199
$170$	390.000	0.175951
$171$	0	0
$172$	1196.00	0.530199
$173$	3942.00	1.73240	0.866199	−	0.499700i	$-0.166556\pi$
$173$	3942.00	1.73240	0.866199	+	0.499700i	$0.166556\pi$
$174$	0	0
$175$	350.000	0.151186
$176$	48.0000	0.0205576
$177$	0	0
$178$	240.000	0.101060
$179$	−1212.00	−0.506085	−0.253042	−	0.967455i	$-0.581431\pi$
$179$	−1212.00	−0.506085	−0.253042	+	0.967455i	$0.581431\pi$
$180$	0	0
$181$	2288.00	0.939590	0.469795	−	0.882776i	$-0.344328\pi$
$181$	2288.00	0.939590	0.469795	+	0.882776i	$0.344328\pi$
$182$	−1316.00	−0.535980
$183$	0	0
$184$	792.000	0.317321
$185$	1570.00	0.623939
$186$	0	0
$187$	−117.000	−0.0457534
$188$	2124.00	0.823982
$189$	0	0
$190$	−320.000	−0.122185
$191$	−1938.00	−0.734182	−0.367091	−	0.930185i	$-0.619646\pi$
$191$	−1938.00	−0.734182	−0.367091	+	0.930185i	$0.619646\pi$
$192$	0	0
$193$	−1498.00	−0.558696	−0.279348	−	0.960190i	$-0.590118\pi$
$193$	−1498.00	−0.558696	−0.279348	+	0.960190i	$0.590118\pi$
$194$	3284.00	1.21535
$195$	0	0
$196$	−588.000	−0.214286
$197$	−2124.00	−0.768166	−0.384083	−	0.923299i	$-0.625482\pi$
$197$	−2124.00	−0.768166	−0.384083	+	0.923299i	$0.625482\pi$
$198$	0	0
$199$	−385.000	−0.137145	−0.0685727	−	0.997646i	$-0.521845\pi$
$199$	−385.000	−0.137145	−0.0685727	+	0.997646i	$0.521845\pi$
$200$	−200.000	−0.0707107
$201$	0	0
$202$	−66.0000	−0.0229888
$203$	714.000	0.246862
$204$	0	0
$205$	−540.000	−0.183977
$206$	2396.00	0.810375
$207$	0	0
$208$	752.000	0.250682
$209$	96.0000	0.0317725
$210$	0	0
$211$	3170.00	1.03427	0.517137	−	0.855903i	$-0.326998\pi$
$211$	3170.00	1.03427	0.517137	+	0.855903i	$0.326998\pi$
$212$	2256.00	0.730862
$213$	0	0
$214$	3084.00	0.985130
$215$	1495.00	0.474224
$216$	0	0
$217$	1162.00	0.363510
$218$	1112.00	0.345478
$219$	0	0
$220$	60.0000	0.0183873
$221$	−1833.00	−0.557923
$222$	0	0
$223$	1388.00	0.416804	0.208402	−	0.978043i	$-0.433174\pi$
$223$	1388.00	0.416804	0.208402	+	0.978043i	$0.433174\pi$
$224$	−448.000	−0.133631
$225$	0	0
$226$	−3210.00	−0.944805
$227$	−4644.00	−1.35786	−0.678928	−	0.734205i	$-0.737555\pi$
$227$	−4644.00	−1.35786	−0.678928	+	0.734205i	$0.737555\pi$
$228$	0	0
$229$	4736.00	1.36665	0.683327	−	0.730113i	$-0.260532\pi$
$229$	4736.00	1.36665	0.683327	+	0.730113i	$0.260532\pi$
$230$	990.000	0.283820
$231$	0	0
$232$	−408.000	−0.115459
$233$	2814.00	0.791207	0.395604	−	0.918421i	$-0.370535\pi$
$233$	2814.00	0.791207	0.395604	+	0.918421i	$0.370535\pi$
$234$	0	0
$235$	2655.00	0.736992
$236$	48.0000	0.0132396
$237$	0	0
$238$	1092.00	0.297411
$239$	−2202.00	−0.595965	−0.297982	−	0.954571i	$-0.596314\pi$
$239$	−2202.00	−0.595965	−0.297982	+	0.954571i	$0.596314\pi$
$240$	0	0
$241$	3485.00	0.931488	0.465744	−	0.884920i	$-0.345787\pi$
$241$	3485.00	0.931488	0.465744	+	0.884920i	$0.345787\pi$
$242$	2644.00	0.702325
$243$	0	0
$244$	920.000	0.241381
$245$	−735.000	−0.191663
$246$	0	0
$247$	1504.00	0.387438
$248$	−664.000	−0.170016
$249$	0	0
$250$	−250.000	−0.0632456
$251$	−6345.00	−1.59559	−0.797795	−	0.602929i	$-0.794000\pi$
$251$	−6345.00	−1.59559	−0.797795	+	0.602929i	$0.794000\pi$
$252$	0	0
$253$	−297.000	−0.0738033
$254$	−2668.00	−0.659076
$255$	0	0
$256$	256.000	0.0625000
$257$	525.000	0.127426	0.0637132	−	0.997968i	$-0.479706\pi$
$257$	525.000	0.127426	0.0637132	+	0.997968i	$0.479706\pi$
$258$	0	0
$259$	4396.00	1.05465
$260$	940.000	0.224217
$261$	0	0
$262$	5766.00	1.35964
$263$	5196.00	1.21825	0.609124	−	0.793075i	$-0.291521\pi$
$263$	5196.00	1.21825	0.609124	+	0.793075i	$0.291521\pi$
$264$	0	0
$265$	2820.00	0.653703
$266$	−896.000	−0.206531
$267$	0	0
$268$	−1072.00	−0.244339
$269$	−7479.00	−1.69518	−0.847589	−	0.530654i	$-0.821946\pi$
$269$	−7479.00	−1.69518	−0.847589	+	0.530654i	$0.821946\pi$
$270$	0	0
$271$	−856.000	−0.191876	−0.0959378	−	0.995387i	$-0.530585\pi$
$271$	−856.000	−0.191876	−0.0959378	+	0.995387i	$0.530585\pi$
$272$	−624.000	−0.139101
$273$	0	0
$274$	−564.000	−0.124352
$275$	75.0000	0.0164461
$276$	0	0
$277$	−7054.00	−1.53009	−0.765043	−	0.643979i	$-0.777282\pi$
$277$	−7054.00	−1.53009	−0.765043	+	0.643979i	$0.777282\pi$
$278$	4988.00	1.07612
$279$	0	0
$280$	−560.000	−0.119523
$281$	1014.00	0.215268	0.107634	−	0.994191i	$-0.465673\pi$
$281$	1014.00	0.215268	0.107634	+	0.994191i	$0.465673\pi$
$282$	0	0
$283$	992.000	0.208368	0.104184	−	0.994558i	$-0.466777\pi$
$283$	992.000	0.208368	0.104184	+	0.994558i	$0.466777\pi$
$284$	480.000	0.100291
$285$	0	0
$286$	−282.000	−0.0583042
$287$	−1512.00	−0.310977
$288$	0	0
$289$	−3392.00	−0.690413
$290$	−510.000	−0.103270
$291$	0	0
$292$	4424.00	0.886627
$293$	−4950.00	−0.986970	−0.493485	−	0.869754i	$-0.664277\pi$
$293$	−4950.00	−0.986970	−0.493485	+	0.869754i	$0.664277\pi$
$294$	0	0
$295$	60.0000	0.0118418
$296$	−2512.00	−0.493267
$297$	0	0
$298$	−5190.00	−1.00889
$299$	−4653.00	−0.899966
$300$	0	0
$301$	4186.00	0.801585
$302$	−2458.00	−0.468351
$303$	0	0
$304$	512.000	0.0965961
$305$	1150.00	0.215898
$306$	0	0
$307$	−4777.00	−0.888071	−0.444035	−	0.896009i	$-0.646454\pi$
$307$	−4777.00	−0.888071	−0.444035	+	0.896009i	$0.646454\pi$
$308$	168.000	0.0310802
$309$	0	0
$310$	−830.000	−0.152067
$311$	−7692.00	−1.40249	−0.701243	−	0.712922i	$-0.747371\pi$
$311$	−7692.00	−1.40249	−0.701243	+	0.712922i	$0.747371\pi$
$312$	0	0
$313$	−2932.00	−0.529477	−0.264739	−	0.964320i	$-0.585286\pi$
$313$	−2932.00	−0.529477	−0.264739	+	0.964320i	$0.585286\pi$
$314$	3182.00	0.571881
$315$	0	0
$316$	−2956.00	−0.526228
$317$	8352.00	1.47980	0.739898	−	0.672720i	$-0.234874\pi$
$317$	8352.00	1.47980	0.739898	+	0.672720i	$0.234874\pi$
$318$	0	0
$319$	153.000	0.0268538
$320$	320.000	0.0559017
$321$	0	0
$322$	2772.00	0.479744
$323$	−1248.00	−0.214986
$324$	0	0
$325$	1175.00	0.200545
$326$	914.000	0.155282
$327$	0	0
$328$	864.000	0.145446
$329$	7434.00	1.24574
$330$	0	0
$331$	−3070.00	−0.509796	−0.254898	−	0.966968i	$-0.582042\pi$
$331$	−3070.00	−0.509796	−0.254898	+	0.966968i	$0.582042\pi$
$332$	4344.00	0.718096
$333$	0	0
$334$	2328.00	0.381385
$335$	−1340.00	−0.218543
$336$	0	0
$337$	−1672.00	−0.270266	−0.135133	−	0.990827i	$-0.543146\pi$
$337$	−1672.00	−0.270266	−0.135133	+	0.990827i	$0.543146\pi$
$338$	−24.0000	−0.00386221
$339$	0	0
$340$	−780.000	−0.124416
$341$	249.000	0.0395428
$342$	0	0
$343$	−6860.00	−1.07990
$344$	−2392.00	−0.374907
$345$	0	0
$346$	−7884.00	−1.22499
$347$	−5076.00	−0.785285	−0.392643	−	0.919691i	$-0.628439\pi$
$347$	−5076.00	−0.785285	−0.392643	+	0.919691i	$0.628439\pi$
$348$	0	0
$349$	8594.00	1.31813	0.659063	−	0.752087i	$-0.270953\pi$
$349$	8594.00	1.31813	0.659063	+	0.752087i	$0.270953\pi$
$350$	−700.000	−0.106904
$351$	0	0
$352$	−96.0000	−0.0145364
$353$	12711.0	1.91654	0.958269	−	0.285866i	$-0.0922813\pi$
$353$	12711.0	1.91654	0.958269	+	0.285866i	$0.0922813\pi$
$354$	0	0
$355$	600.000	0.0897034
$356$	−480.000	−0.0714605
$357$	0	0
$358$	2424.00	0.357856
$359$	−1464.00	−0.215228	−0.107614	−	0.994193i	$-0.534321\pi$
$359$	−1464.00	−0.215228	−0.107614	+	0.994193i	$0.534321\pi$
$360$	0	0
$361$	−5835.00	−0.850707
$362$	−4576.00	−0.664390
$363$	0	0
$364$	2632.00	0.378995
$365$	5530.00	0.793023
$366$	0	0
$367$	−7630.00	−1.08524	−0.542620	−	0.839979i	$-0.682567\pi$
$367$	−7630.00	−1.08524	−0.542620	+	0.839979i	$0.682567\pi$
$368$	−1584.00	−0.224380
$369$	0	0
$370$	−3140.00	−0.441191
$371$	7896.00	1.10496
$372$	0	0
$373$	−3883.00	−0.539019	−0.269510	−	0.962998i	$-0.586862\pi$
$373$	−3883.00	−0.539019	−0.269510	+	0.962998i	$0.586862\pi$
$374$	234.000	0.0323525
$375$	0	0
$376$	−4248.00	−0.582643
$377$	2397.00	0.327458
$378$	0	0
$379$	−13768.0	−1.86600	−0.933001	−	0.359874i	$-0.882820\pi$
$379$	−13768.0	−1.86600	−0.933001	+	0.359874i	$0.882820\pi$
$380$	640.000	0.0863982
$381$	0	0
$382$	3876.00	0.519145
$383$	−14139.0	−1.88634	−0.943171	−	0.332307i	$-0.892173\pi$
$383$	−14139.0	−1.88634	−0.943171	+	0.332307i	$0.892173\pi$
$384$	0	0
$385$	210.000	0.0277989
$386$	2996.00	0.395058
$387$	0	0
$388$	−6568.00	−0.859381
$389$	567.000	0.0739024	0.0369512	−	0.999317i	$-0.488235\pi$
$389$	567.000	0.0739024	0.0369512	+	0.999317i	$0.488235\pi$
$390$	0	0
$391$	3861.00	0.499384
$392$	1176.00	0.151523
$393$	0	0
$394$	4248.00	0.543176
$395$	−3695.00	−0.470672
$396$	0	0
$397$	−6685.00	−0.845115	−0.422557	−	0.906336i	$-0.638867\pi$
$397$	−6685.00	−0.845115	−0.422557	+	0.906336i	$0.638867\pi$
$398$	770.000	0.0969764
$399$	0	0
$400$	400.000	0.0500000
$401$	4572.00	0.569364	0.284682	−	0.958622i	$-0.408112\pi$
$401$	4572.00	0.569364	0.284682	+	0.958622i	$0.408112\pi$
$402$	0	0
$403$	3901.00	0.482190
$404$	132.000	0.0162556
$405$	0	0
$406$	−1428.00	−0.174558
$407$	942.000	0.114725
$408$	0	0
$409$	−25.0000	−0.00302242	−0.00151121	−	0.999999i	$-0.500481\pi$
$409$	−25.0000	−0.00302242	−0.00151121	+	0.999999i	$0.500481\pi$
$410$	1080.00	0.130091
$411$	0	0
$412$	−4792.00	−0.573022
$413$	168.000	0.0200163
$414$	0	0
$415$	5430.00	0.642285
$416$	−1504.00	−0.177259
$417$	0	0
$418$	−192.000	−0.0224666
$419$	12453.0	1.45195	0.725977	−	0.687719i	$-0.241388\pi$
$419$	12453.0	1.45195	0.725977	+	0.687719i	$0.241388\pi$
$420$	0	0
$421$	5048.00	0.584381	0.292191	−	0.956360i	$-0.405616\pi$
$421$	5048.00	0.584381	0.292191	+	0.956360i	$0.405616\pi$
$422$	−6340.00	−0.731342
$423$	0	0
$424$	−4512.00	−0.516797
$425$	−975.000	−0.111281
$426$	0	0
$427$	3220.00	0.364934
$428$	−6168.00	−0.696592
$429$	0	0
$430$	−2990.00	−0.335327
$431$	5400.00	0.603501	0.301750	−	0.953387i	$-0.402429\pi$
$431$	5400.00	0.603501	0.301750	+	0.953387i	$0.402429\pi$
$432$	0	0
$433$	−6298.00	−0.698990	−0.349495	−	0.936938i	$-0.613647\pi$
$433$	−6298.00	−0.698990	−0.349495	+	0.936938i	$0.613647\pi$
$434$	−2324.00	−0.257040
$435$	0	0
$436$	−2224.00	−0.244290
$437$	−3168.00	−0.346787
$438$	0	0
$439$	−6208.00	−0.674924	−0.337462	−	0.941339i	$-0.609568\pi$
$439$	−6208.00	−0.674924	−0.337462	+	0.941339i	$0.609568\pi$
$440$	−120.000	−0.0130018
$441$	0	0
$442$	3666.00	0.394511
$443$	−3360.00	−0.360358	−0.180179	−	0.983634i	$-0.557668\pi$
$443$	−3360.00	−0.360358	−0.180179	+	0.983634i	$0.557668\pi$
$444$	0	0
$445$	−600.000	−0.0639162
$446$	−2776.00	−0.294725
$447$	0	0
$448$	896.000	0.0944911
$449$	14394.0	1.51291	0.756453	−	0.654048i	$-0.226931\pi$
$449$	14394.0	1.51291	0.756453	+	0.654048i	$0.226931\pi$
$450$	0	0
$451$	−324.000	−0.0338283
$452$	6420.00	0.668078
$453$	0	0
$454$	9288.00	0.960149
$455$	3290.00	0.338984
$456$	0	0
$457$	−916.000	−0.0937608	−0.0468804	−	0.998901i	$-0.514928\pi$
$457$	−916.000	−0.0937608	−0.0468804	+	0.998901i	$0.514928\pi$
$458$	−9472.00	−0.966370
$459$	0	0
$460$	−1980.00	−0.200691
$461$	8550.00	0.863803	0.431902	−	0.901921i	$-0.357843\pi$
$461$	8550.00	0.863803	0.431902	+	0.901921i	$0.357843\pi$
$462$	0	0
$463$	3734.00	0.374803	0.187401	−	0.982283i	$-0.439993\pi$
$463$	3734.00	0.374803	0.187401	+	0.982283i	$0.439993\pi$
$464$	816.000	0.0816419
$465$	0	0
$466$	−5628.00	−0.559468
$467$	9840.00	0.975034	0.487517	−	0.873113i	$-0.337903\pi$
$467$	9840.00	0.975034	0.487517	+	0.873113i	$0.337903\pi$
$468$	0	0
$469$	−3752.00	−0.369406
$470$	−5310.00	−0.521132
$471$	0	0
$472$	−96.0000	−0.00936178
$473$	897.000	0.0871968
$474$	0	0
$475$	800.000	0.0772769
$476$	−2184.00	−0.210301
$477$	0	0
$478$	4404.00	0.421411
$479$	−17280.0	−1.64832	−0.824158	−	0.566360i	$-0.808351\pi$
$479$	−17280.0	−1.64832	−0.824158	+	0.566360i	$0.808351\pi$
$480$	0	0
$481$	14758.0	1.39897
$482$	−6970.00	−0.658661
$483$	0	0
$484$	−5288.00	−0.496619
$485$	−8210.00	−0.768653
$486$	0	0
$487$	−4588.00	−0.426904	−0.213452	−	0.976954i	$-0.568471\pi$
$487$	−4588.00	−0.426904	−0.213452	+	0.976954i	$0.568471\pi$
$488$	−1840.00	−0.170682
$489$	0	0
$490$	1470.00	0.135526
$491$	636.000	0.0584568	0.0292284	−	0.999573i	$-0.490695\pi$
$491$	636.000	0.0584568	0.0292284	+	0.999573i	$0.490695\pi$
$492$	0	0
$493$	−1989.00	−0.181704
$494$	−3008.00	−0.273960
$495$	0	0
$496$	1328.00	0.120220
$497$	1680.00	0.151626
$498$	0	0
$499$	−11716.0	−1.05106	−0.525531	−	0.850774i	$-0.676133\pi$
$499$	−11716.0	−1.05106	−0.525531	+	0.850774i	$0.676133\pi$
$500$	500.000	0.0447214
$501$	0	0
$502$	12690.0	1.12825
$503$	4653.00	0.412459	0.206230	−	0.978504i	$-0.433881\pi$
$503$	4653.00	0.412459	0.206230	+	0.978504i	$0.433881\pi$
$504$	0	0
$505$	165.000	0.0145394
$506$	594.000	0.0521868
$507$	0	0
$508$	5336.00	0.466037
$509$	16479.0	1.43501	0.717504	−	0.696555i	$-0.245285\pi$
$509$	16479.0	1.43501	0.717504	+	0.696555i	$0.245285\pi$
$510$	0	0
$511$	15484.0	1.34045
$512$	−512.000	−0.0441942
$513$	0	0
$514$	−1050.00	−0.0901041
$515$	−5990.00	−0.512526
$516$	0	0
$517$	1593.00	0.135513
$518$	−8792.00	−0.745750
$519$	0	0
$520$	−1880.00	−0.158545
$521$	3120.00	0.262360	0.131180	−	0.991359i	$-0.458123\pi$
$521$	3120.00	0.262360	0.131180	+	0.991359i	$0.458123\pi$
$522$	0	0
$523$	17645.0	1.47526	0.737631	−	0.675204i	$-0.235944\pi$
$523$	17645.0	1.47526	0.737631	+	0.675204i	$0.235944\pi$
$524$	−11532.0	−0.961408
$525$	0	0
$526$	−10392.0	−0.861431
$527$	−3237.00	−0.267563
$528$	0	0
$529$	−2366.00	−0.194460
$530$	−5640.00	−0.462238
$531$	0	0
$532$	1792.00	0.146040
$533$	−5076.00	−0.412507
$534$	0	0
$535$	−7710.00	−0.623051
$536$	2144.00	0.172774
$537$	0	0
$538$	14958.0	1.19867
$539$	−441.000	−0.0352416
$540$	0	0
$541$	−2182.00	−0.173404	−0.0867019	−	0.996234i	$-0.527633\pi$
$541$	−2182.00	−0.173404	−0.0867019	+	0.996234i	$0.527633\pi$
$542$	1712.00	0.135677
$543$	0	0
$544$	1248.00	0.0983595
$545$	−2780.00	−0.218499
$546$	0	0
$547$	−4033.00	−0.315244	−0.157622	−	0.987499i	$-0.550383\pi$
$547$	−4033.00	−0.315244	−0.157622	+	0.987499i	$0.550383\pi$
$548$	1128.00	0.0879302
$549$	0	0
$550$	−150.000	−0.0116291
$551$	1632.00	0.126181
$552$	0	0
$553$	−10346.0	−0.795582
$554$	14108.0	1.08193
$555$	0	0
$556$	−9976.00	−0.760929
$557$	−960.000	−0.0730278	−0.0365139	−	0.999333i	$-0.511625\pi$
$557$	−960.000	−0.0730278	−0.0365139	+	0.999333i	$0.511625\pi$
$558$	0	0
$559$	14053.0	1.06329
$560$	1120.00	0.0845154
$561$	0	0
$562$	−2028.00	−0.152217
$563$	−23754.0	−1.77817	−0.889087	−	0.457739i	$-0.848660\pi$
$563$	−23754.0	−1.77817	−0.889087	+	0.457739i	$0.848660\pi$
$564$	0	0
$565$	8025.00	0.597547
$566$	−1984.00	−0.147339
$567$	0	0
$568$	−960.000	−0.0709167
$569$	−22536.0	−1.66038	−0.830192	−	0.557478i	$-0.811769\pi$
$569$	−22536.0	−1.66038	−0.830192	+	0.557478i	$0.811769\pi$
$570$	0	0
$571$	17726.0	1.29914	0.649571	−	0.760301i	$-0.274949\pi$
$571$	17726.0	1.29914	0.649571	+	0.760301i	$0.274949\pi$
$572$	564.000	0.0412273
$573$	0	0
$574$	3024.00	0.219894
$575$	−2475.00	−0.179504
$576$	0	0
$577$	17168.0	1.23867	0.619336	−	0.785126i	$-0.287402\pi$
$577$	17168.0	1.23867	0.619336	+	0.785126i	$0.287402\pi$
$578$	6784.00	0.488196
$579$	0	0
$580$	1020.00	0.0730227
$581$	15204.0	1.08566
$582$	0	0
$583$	1692.00	0.120198
$584$	−8848.00	−0.626940
$585$	0	0
$586$	9900.00	0.697893
$587$	7542.00	0.530309	0.265155	−	0.964206i	$-0.414577\pi$
$587$	7542.00	0.530309	0.265155	+	0.964206i	$0.414577\pi$
$588$	0	0
$589$	2656.00	0.185804
$590$	−120.000	−0.00837343
$591$	0	0
$592$	5024.00	0.348792
$593$	−15543.0	−1.07635	−0.538174	−	0.842834i	$-0.680886\pi$
$593$	−15543.0	−1.07635	−0.538174	+	0.842834i	$0.680886\pi$
$594$	0	0
$595$	−2730.00	−0.188099
$596$	10380.0	0.713392
$597$	0	0
$598$	9306.00	0.636372
$599$	16026.0	1.09316	0.546581	−	0.837406i	$-0.315929\pi$
$599$	16026.0	1.09316	0.546581	+	0.837406i	$0.315929\pi$
$600$	0	0
$601$	10469.0	0.710548	0.355274	−	0.934762i	$-0.384388\pi$
$601$	10469.0	0.710548	0.355274	+	0.934762i	$0.384388\pi$
$602$	−8372.00	−0.566806
$603$	0	0
$604$	4916.00	0.331174
$605$	−6610.00	−0.444190
$606$	0	0
$607$	−8074.00	−0.539891	−0.269945	−	0.962876i	$-0.587006\pi$
$607$	−8074.00	−0.539891	−0.269945	+	0.962876i	$0.587006\pi$
$608$	−1024.00	−0.0683038
$609$	0	0
$610$	−2300.00	−0.152663
$611$	24957.0	1.65246
$612$	0	0
$613$	26855.0	1.76943	0.884717	−	0.466128i	$-0.154351\pi$
$613$	26855.0	1.76943	0.884717	+	0.466128i	$0.154351\pi$
$614$	9554.00	0.627961
$615$	0	0
$616$	−336.000	−0.0219770
$617$	24447.0	1.59514	0.797568	−	0.603229i	$-0.206119\pi$
$617$	24447.0	1.59514	0.797568	+	0.603229i	$0.206119\pi$
$618$	0	0
$619$	1850.00	0.120126	0.0600628	−	0.998195i	$-0.480870\pi$
$619$	1850.00	0.120126	0.0600628	+	0.998195i	$0.480870\pi$
$620$	1660.00	0.107528
$621$	0	0
$622$	15384.0	0.991708
$623$	−1680.00	−0.108038
$624$	0	0
$625$	625.000	0.0400000
$626$	5864.00	0.374397
$627$	0	0
$628$	−6364.00	−0.404381
$629$	−12246.0	−0.776280
$630$	0	0
$631$	21728.0	1.37081	0.685403	−	0.728164i	$-0.259626\pi$
$631$	21728.0	1.37081	0.685403	+	0.728164i	$0.259626\pi$
$632$	5912.00	0.372099
$633$	0	0
$634$	−16704.0	−1.04637
$635$	6670.00	0.416836
$636$	0	0
$637$	−6909.00	−0.429740
$638$	−306.000	−0.0189885
$639$	0	0
$640$	−640.000	−0.0395285
$641$	23862.0	1.47035	0.735173	−	0.677879i	$-0.237101\pi$
$641$	23862.0	1.47035	0.735173	+	0.677879i	$0.237101\pi$
$642$	0	0
$643$	10523.0	0.645391	0.322696	−	0.946503i	$-0.395411\pi$
$643$	10523.0	0.645391	0.322696	+	0.946503i	$0.395411\pi$
$644$	−5544.00	−0.339230
$645$	0	0
$646$	2496.00	0.152018
$647$	5484.00	0.333228	0.166614	−	0.986022i	$-0.446717\pi$
$647$	5484.00	0.333228	0.166614	+	0.986022i	$0.446717\pi$
$648$	0	0
$649$	36.0000	0.00217739
$650$	−2350.00	−0.141807
$651$	0	0
$652$	−1828.00	−0.109801
$653$	−26784.0	−1.60511	−0.802557	−	0.596576i	$-0.796527\pi$
$653$	−26784.0	−1.60511	−0.802557	+	0.596576i	$0.796527\pi$
$654$	0	0
$655$	−14415.0	−0.859909
$656$	−1728.00	−0.102846
$657$	0	0
$658$	−14868.0	−0.880874
$659$	−12120.0	−0.716431	−0.358216	−	0.933639i	$-0.616615\pi$
$659$	−12120.0	−0.716431	−0.358216	+	0.933639i	$0.616615\pi$
$660$	0	0
$661$	−18226.0	−1.07248	−0.536240	−	0.844066i	$-0.680156\pi$
$661$	−18226.0	−1.07248	−0.536240	+	0.844066i	$0.680156\pi$
$662$	6140.00	0.360480
$663$	0	0
$664$	−8688.00	−0.507771
$665$	2240.00	0.130622
$666$	0	0
$667$	−5049.00	−0.293101
$668$	−4656.00	−0.269680
$669$	0	0
$670$	2680.00	0.154533
$671$	690.000	0.0396977
$672$	0	0
$673$	−11062.0	−0.633594	−0.316797	−	0.948493i	$-0.602607\pi$
$673$	−11062.0	−0.633594	−0.316797	+	0.948493i	$0.602607\pi$
$674$	3344.00	0.191107
$675$	0	0
$676$	48.0000	0.00273100
$677$	−9348.00	−0.530684	−0.265342	−	0.964154i	$-0.585485\pi$
$677$	−9348.00	−0.530684	−0.265342	+	0.964154i	$0.585485\pi$
$678$	0	0
$679$	−22988.0	−1.29926
$680$	1560.00	0.0879754
$681$	0	0
$682$	−498.000	−0.0279610
$683$	19248.0	1.07834	0.539169	−	0.842198i	$-0.318739\pi$
$683$	19248.0	1.07834	0.539169	+	0.842198i	$0.318739\pi$
$684$	0	0
$685$	1410.00	0.0786472
$686$	13720.0	0.763604
$687$	0	0
$688$	4784.00	0.265099
$689$	26508.0	1.46571
$690$	0	0
$691$	−17710.0	−0.974993	−0.487496	−	0.873125i	$-0.662090\pi$
$691$	−17710.0	−0.974993	−0.487496	+	0.873125i	$0.662090\pi$
$692$	15768.0	0.866199
$693$	0	0
$694$	10152.0	0.555280
$695$	−12470.0	−0.680596
$696$	0	0
$697$	4212.00	0.228897
$698$	−17188.0	−0.932056
$699$	0	0
$700$	1400.00	0.0755929
$701$	−19437.0	−1.04725	−0.523627	−	0.851947i	$-0.675422\pi$
$701$	−19437.0	−1.04725	−0.523627	+	0.851947i	$0.675422\pi$
$702$	0	0
$703$	10048.0	0.539072
$704$	192.000	0.0102788
$705$	0	0
$706$	−25422.0	−1.35520
$707$	462.000	0.0245761
$708$	0	0
$709$	−19516.0	−1.03376	−0.516882	−	0.856057i	$-0.672907\pi$
$709$	−19516.0	−1.03376	−0.516882	+	0.856057i	$0.672907\pi$
$710$	−1200.00	−0.0634299
$711$	0	0
$712$	960.000	0.0505302
$713$	−8217.00	−0.431598
$714$	0	0
$715$	705.000	0.0368748
$716$	−4848.00	−0.253042
$717$	0	0
$718$	2928.00	0.152189
$719$	17358.0	0.900340	0.450170	−	0.892943i	$-0.351363\pi$
$719$	17358.0	0.900340	0.450170	+	0.892943i	$0.351363\pi$
$720$	0	0
$721$	−16772.0	−0.866327
$722$	11670.0	0.601541
$723$	0	0
$724$	9152.00	0.469795
$725$	1275.00	0.0653135
$726$	0	0
$727$	24428.0	1.24620	0.623098	−	0.782144i	$-0.285874\pi$
$727$	24428.0	1.24620	0.623098	+	0.782144i	$0.285874\pi$
$728$	−5264.00	−0.267990
$729$	0	0
$730$	−11060.0	−0.560752
$731$	−11661.0	−0.590010
$732$	0	0
$733$	−21418.0	−1.07925	−0.539626	−	0.841905i	$-0.681434\pi$
$733$	−21418.0	−1.07925	−0.539626	+	0.841905i	$0.681434\pi$
$734$	15260.0	0.767380
$735$	0	0
$736$	3168.00	0.158660
$737$	−804.000	−0.0401842
$738$	0	0
$739$	−664.000	−0.0330523	−0.0165261	−	0.999863i	$-0.505261\pi$
$739$	−664.000	−0.0330523	−0.0165261	+	0.999863i	$0.505261\pi$
$740$	6280.00	0.311969
$741$	0	0
$742$	−15792.0	−0.781324
$743$	−34209.0	−1.68911	−0.844553	−	0.535471i	$-0.820134\pi$
$743$	−34209.0	−1.68911	−0.844553	+	0.535471i	$0.820134\pi$
$744$	0	0
$745$	12975.0	0.638077
$746$	7766.00	0.381144
$747$	0	0
$748$	−468.000	−0.0228767
$749$	−21588.0	−1.05315
$750$	0	0
$751$	6857.00	0.333176	0.166588	−	0.986027i	$-0.446725\pi$
$751$	6857.00	0.333176	0.166588	+	0.986027i	$0.446725\pi$
$752$	8496.00	0.411991
$753$	0	0
$754$	−4794.00	−0.231548
$755$	6145.00	0.296211
$756$	0	0
$757$	−23719.0	−1.13881	−0.569407	−	0.822056i	$-0.692827\pi$
$757$	−23719.0	−1.13881	−0.569407	+	0.822056i	$0.692827\pi$
$758$	27536.0	1.31946
$759$	0	0
$760$	−1280.00	−0.0610927
$761$	−14418.0	−0.686796	−0.343398	−	0.939190i	$-0.611578\pi$
$761$	−14418.0	−0.686796	−0.343398	+	0.939190i	$0.611578\pi$
$762$	0	0
$763$	−7784.00	−0.369331
$764$	−7752.00	−0.367091
$765$	0	0
$766$	28278.0	1.33385
$767$	564.000	0.0265513
$768$	0	0
$769$	−4849.00	−0.227385	−0.113693	−	0.993516i	$-0.536268\pi$
$769$	−4849.00	−0.227385	−0.113693	+	0.993516i	$0.536268\pi$
$770$	−420.000	−0.0196568
$771$	0	0
$772$	−5992.00	−0.279348
$773$	−36258.0	−1.68708	−0.843538	−	0.537070i	$-0.819531\pi$
$773$	−36258.0	−1.68708	−0.843538	+	0.537070i	$0.819531\pi$
$774$	0	0
$775$	2075.00	0.0961757
$776$	13136.0	0.607674
$777$	0	0
$778$	−1134.00	−0.0522569
$779$	−3456.00	−0.158953
$780$	0	0
$781$	360.000	0.0164940
$782$	−7722.00	−0.353118
$783$	0	0
$784$	−2352.00	−0.107143
$785$	−7955.00	−0.361689
$786$	0	0
$787$	−18877.0	−0.855009	−0.427505	−	0.904013i	$-0.640607\pi$
$787$	−18877.0	−0.855009	−0.427505	+	0.904013i	$0.640607\pi$
$788$	−8496.00	−0.384083
$789$	0	0
$790$	7390.00	0.332816
$791$	22470.0	1.01004
$792$	0	0
$793$	10810.0	0.484079
$794$	13370.0	0.597586
$795$	0	0
$796$	−1540.00	−0.0685727
$797$	−16200.0	−0.719992	−0.359996	−	0.932954i	$-0.617222\pi$
$797$	−16200.0	−0.719992	−0.359996	+	0.932954i	$0.617222\pi$
$798$	0	0
$799$	−20709.0	−0.916936
$800$	−800.000	−0.0353553
$801$	0	0
$802$	−9144.00	−0.402601
$803$	3318.00	0.145815
$804$	0	0
$805$	−6930.00	−0.303417
$806$	−7802.00	−0.340960
$807$	0	0
$808$	−264.000	−0.0114944
$809$	−26760.0	−1.16296	−0.581478	−	0.813562i	$-0.697525\pi$
$809$	−26760.0	−1.16296	−0.581478	+	0.813562i	$0.697525\pi$
$810$	0	0
$811$	−10510.0	−0.455063	−0.227531	−	0.973771i	$-0.573065\pi$
$811$	−10510.0	−0.455063	−0.227531	+	0.973771i	$0.573065\pi$
$812$	2856.00	0.123431
$813$	0	0
$814$	−1884.00	−0.0811231
$815$	−2285.00	−0.0982087
$816$	0	0
$817$	9568.00	0.409721
$818$	50.0000	0.00213717
$819$	0	0
$820$	−2160.00	−0.0919884
$821$	−28230.0	−1.20004	−0.600021	−	0.799985i	$-0.704841\pi$
$821$	−28230.0	−1.20004	−0.600021	+	0.799985i	$0.704841\pi$
$822$	0	0
$823$	−39868.0	−1.68859	−0.844296	−	0.535877i	$-0.819981\pi$
$823$	−39868.0	−1.68859	−0.844296	+	0.535877i	$0.819981\pi$
$824$	9584.00	0.405187
$825$	0	0
$826$	−336.000	−0.0141537
$827$	32394.0	1.36209	0.681046	−	0.732241i	$-0.261525\pi$
$827$	32394.0	1.36209	0.681046	+	0.732241i	$0.261525\pi$
$828$	0	0
$829$	34820.0	1.45880	0.729402	−	0.684085i	$-0.239798\pi$
$829$	34820.0	1.45880	0.729402	+	0.684085i	$0.239798\pi$
$830$	−10860.0	−0.454164
$831$	0	0
$832$	3008.00	0.125341
$833$	5733.00	0.238459
$834$	0	0
$835$	−5820.00	−0.241209
$836$	384.000	0.0158863
$837$	0	0
$838$	−24906.0	−1.02669
$839$	−1146.00	−0.0471565	−0.0235783	−	0.999722i	$-0.507506\pi$
$839$	−1146.00	−0.0471565	−0.0235783	+	0.999722i	$0.507506\pi$
$840$	0	0
$841$	−21788.0	−0.893354
$842$	−10096.0	−0.413220
$843$	0	0
$844$	12680.0	0.517137
$845$	60.0000	0.00244268
$846$	0	0
$847$	−18508.0	−0.750817
$848$	9024.00	0.365431
$849$	0	0
$850$	1950.00	0.0786876
$851$	−31086.0	−1.25219
$852$	0	0
$853$	−19393.0	−0.778433	−0.389217	−	0.921146i	$-0.627254\pi$
$853$	−19393.0	−0.778433	−0.389217	+	0.921146i	$0.627254\pi$
$854$	−6440.00	−0.258047
$855$	0	0
$856$	12336.0	0.492565
$857$	−8430.00	−0.336013	−0.168007	−	0.985786i	$-0.553733\pi$
$857$	−8430.00	−0.336013	−0.168007	+	0.985786i	$0.553733\pi$
$858$	0	0
$859$	15470.0	0.614470	0.307235	−	0.951634i	$-0.400596\pi$
$859$	15470.0	0.614470	0.307235	+	0.951634i	$0.400596\pi$
$860$	5980.00	0.237112
$861$	0	0
$862$	−10800.0	−0.426740
$863$	5871.00	0.231577	0.115789	−	0.993274i	$-0.463060\pi$
$863$	5871.00	0.231577	0.115789	+	0.993274i	$0.463060\pi$
$864$	0	0
$865$	19710.0	0.774752
$866$	12596.0	0.494260
$867$	0	0
$868$	4648.00	0.181755
$869$	−2217.00	−0.0865438
$870$	0	0
$871$	−12596.0	−0.490011
$872$	4448.00	0.172739
$873$	0	0
$874$	6336.00	0.245216
$875$	1750.00	0.0676123
$876$	0	0
$877$	−11299.0	−0.435051	−0.217526	−	0.976055i	$-0.569799\pi$
$877$	−11299.0	−0.435051	−0.217526	+	0.976055i	$0.569799\pi$
$878$	12416.0	0.477243
$879$	0	0
$880$	240.000	0.00919363
$881$	29682.0	1.13509	0.567544	−	0.823343i	$-0.307894\pi$
$881$	29682.0	1.13509	0.567544	+	0.823343i	$0.307894\pi$
$882$	0	0
$883$	40316.0	1.53651	0.768257	−	0.640142i	$-0.221124\pi$
$883$	40316.0	1.53651	0.768257	+	0.640142i	$0.221124\pi$
$884$	−7332.00	−0.278961
$885$	0	0
$886$	6720.00	0.254811
$887$	−21945.0	−0.830711	−0.415356	−	0.909659i	$-0.636343\pi$
$887$	−21945.0	−0.830711	−0.415356	+	0.909659i	$0.636343\pi$
$888$	0	0
$889$	18676.0	0.704581
$890$	1200.00	0.0451956
$891$	0	0
$892$	5552.00	0.208402
$893$	16992.0	0.636748
$894$	0	0
$895$	−6060.00	−0.226328
$896$	−1792.00	−0.0668153
$897$	0	0
$898$	−28788.0	−1.06979
$899$	4233.00	0.157039
$900$	0	0
$901$	−21996.0	−0.813311
$902$	648.000	0.0239202
$903$	0	0
$904$	−12840.0	−0.472403
$905$	11440.0	0.420197
$906$	0	0
$907$	24911.0	0.911969	0.455985	−	0.889988i	$-0.349287\pi$
$907$	24911.0	0.911969	0.455985	+	0.889988i	$0.349287\pi$
$908$	−18576.0	−0.678928
$909$	0	0
$910$	−6580.00	−0.239698
$911$	33264.0	1.20975	0.604877	−	0.796319i	$-0.293222\pi$
$911$	33264.0	1.20975	0.604877	+	0.796319i	$0.293222\pi$
$912$	0	0
$913$	3258.00	0.118099
$914$	1832.00	0.0662989
$915$	0	0
$916$	18944.0	0.683327
$917$	−40362.0	−1.45351
$918$	0	0
$919$	−23191.0	−0.832427	−0.416214	−	0.909267i	$-0.636643\pi$
$919$	−23191.0	−0.832427	−0.416214	+	0.909267i	$0.636643\pi$
$920$	3960.00	0.141910
$921$	0	0
$922$	−17100.0	−0.610801
$923$	5640.00	0.201130
$924$	0	0
$925$	7850.00	0.279034
$926$	−7468.00	−0.265026
$927$	0	0
$928$	−1632.00	−0.0577296
$929$	2160.00	0.0762834	0.0381417	−	0.999272i	$-0.487856\pi$
$929$	2160.00	0.0762834	0.0381417	+	0.999272i	$0.487856\pi$
$930$	0	0
$931$	−4704.00	−0.165593
$932$	11256.0	0.395604
$933$	0	0
$934$	−19680.0	−0.689453
$935$	−585.000	−0.0204615
$936$	0	0
$937$	2066.00	0.0720312	0.0360156	−	0.999351i	$-0.488533\pi$
$937$	2066.00	0.0720312	0.0360156	+	0.999351i	$0.488533\pi$
$938$	7504.00	0.261209
$939$	0	0
$940$	10620.0	0.368496
$941$	−22233.0	−0.770218	−0.385109	−	0.922871i	$-0.625836\pi$
$941$	−22233.0	−0.770218	−0.385109	+	0.922871i	$0.625836\pi$
$942$	0	0
$943$	10692.0	0.369225
$944$	192.000	0.00661978
$945$	0	0
$946$	−1794.00	−0.0616575
$947$	−17754.0	−0.609216	−0.304608	−	0.952478i	$-0.598525\pi$
$947$	−17754.0	−0.609216	−0.304608	+	0.952478i	$0.598525\pi$
$948$	0	0
$949$	51982.0	1.77809
$950$	−1600.00	−0.0546430
$951$	0	0
$952$	4368.00	0.148706
$953$	33891.0	1.15198	0.575990	−	0.817457i	$-0.304617\pi$
$953$	33891.0	1.15198	0.575990	+	0.817457i	$0.304617\pi$
$954$	0	0
$955$	−9690.00	−0.328336
$956$	−8808.00	−0.297982
$957$	0	0
$958$	34560.0	1.16554
$959$	3948.00	0.132938
$960$	0	0
$961$	−22902.0	−0.768756
$962$	−29516.0	−0.989225
$963$	0	0
$964$	13940.0	0.465744
$965$	−7490.00	−0.249857
$966$	0	0
$967$	51074.0	1.69848	0.849239	−	0.528008i	$-0.177061\pi$
$967$	51074.0	1.69848	0.849239	+	0.528008i	$0.177061\pi$
$968$	10576.0	0.351163
$969$	0	0
$970$	16420.0	0.543520
$971$	20967.0	0.692959	0.346479	−	0.938058i	$-0.387377\pi$
$971$	20967.0	0.692959	0.346479	+	0.938058i	$0.387377\pi$
$972$	0	0
$973$	−34916.0	−1.15042
$974$	9176.00	0.301867
$975$	0	0
$976$	3680.00	0.120691
$977$	31749.0	1.03965	0.519826	−	0.854272i	$-0.325997\pi$
$977$	31749.0	1.03965	0.519826	+	0.854272i	$0.325997\pi$
$978$	0	0
$979$	−360.000	−0.0117525
$980$	−2940.00	−0.0958315
$981$	0	0
$982$	−1272.00	−0.0413352
$983$	47325.0	1.53554	0.767769	−	0.640727i	$-0.221367\pi$
$983$	47325.0	1.53554	0.767769	+	0.640727i	$0.221367\pi$
$984$	0	0
$985$	−10620.0	−0.343534
$986$	3978.00	0.128484
$987$	0	0
$988$	6016.00	0.193719
$989$	−29601.0	−0.951726
$990$	0	0
$991$	2363.00	0.0757449	0.0378724	−	0.999283i	$-0.487942\pi$
$991$	2363.00	0.0757449	0.0378724	+	0.999283i	$0.487942\pi$
$992$	−2656.00	−0.0850081
$993$	0	0
$994$	−3360.00	−0.107216
$995$	−1925.00	−0.0613333
$996$	0	0
$997$	45569.0	1.44753	0.723764	−	0.690048i	$-0.242411\pi$
$997$	45569.0	1.44753	0.723764	+	0.690048i	$0.242411\pi$
$998$	23432.0	0.743213
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
270.4.a.f.1.1	✓	1	1.1	even	1	trivial
270.4.a.j.1.1	yes	1	3.2	odd	2
810.4.e.f.271.1		2	9.2	odd	6
810.4.e.f.541.1		2	9.5	odd	6
810.4.e.n.271.1		2	9.7	even	3
810.4.e.n.541.1		2	9.4	even	3
1350.4.a.e.1.1		1	15.14	odd	2
1350.4.a.r.1.1		1	5.4	even	2
1350.4.c.j.649.1		2	15.8	even	4
1350.4.c.j.649.2		2	15.2	even	4
1350.4.c.k.649.1		2	5.2	odd	4
1350.4.c.k.649.2		2	5.3	odd	4
2160.4.a.b.1.1		1	12.11	even	2
2160.4.a.l.1.1		1	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 270 = 2 \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	270.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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