LMFDB - Embedded newform 270.4.a.b.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)

$f(q)$

$=$

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

\(q-2.00000 q^{2} +4.00000 q^{4} -5.00000 q^{5} +8.00000 q^{7} -8.00000 q^{8} +10.0000 q^{10} -18.0000 q^{11} +8.00000 q^{13} -16.0000 q^{14} +16.0000 q^{16} -15.0000 q^{17} +23.0000 q^{19} -20.0000 q^{20} +36.0000 q^{22} -63.0000 q^{23} +25.0000 q^{25} -16.0000 q^{26} +32.0000 q^{28} -156.000 q^{29} -85.0000 q^{31} -32.0000 q^{32} +30.0000 q^{34} -40.0000 q^{35} +74.0000 q^{37} -46.0000 q^{38} +40.0000 q^{40} -246.000 q^{41} -190.000 q^{43} -72.0000 q^{44} +126.000 q^{46} -288.000 q^{47} -279.000 q^{49} -50.0000 q^{50} +32.0000 q^{52} +177.000 q^{53} +90.0000 q^{55} -64.0000 q^{56} +312.000 q^{58} -792.000 q^{59} -907.000 q^{61} +170.000 q^{62} +64.0000 q^{64} -40.0000 q^{65} -322.000 q^{67} -60.0000 q^{68} +80.0000 q^{70} +270.000 q^{71} +254.000 q^{73} -148.000 q^{74} +92.0000 q^{76} -144.000 q^{77} -1123.00 q^{79} -80.0000 q^{80} +492.000 q^{82} +771.000 q^{83} +75.0000 q^{85} +380.000 q^{86} +144.000 q^{88} +198.000 q^{89} +64.0000 q^{91} -252.000 q^{92} +576.000 q^{94} -115.000 q^{95} -1192.00 q^{97} +558.000 q^{98} +O(q^{100})\)

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
$7$	8.00000	0.431959	0.215980	−	0.976398i	$-0.430705\pi$
$7$	8.00000	0.431959	0.215980	+	0.976398i	$0.430705\pi$
$8$	−8.00000	−0.353553
$9$	0	0
$10$	10.0000	0.316228
$11$	−18.0000	−0.493382	−0.246691	−	0.969094i	$-0.579343\pi$
$11$	−18.0000	−0.493382	−0.246691	+	0.969094i	$0.579343\pi$
$12$	0	0
$13$	8.00000	0.170677	0.0853385	−	0.996352i	$-0.472803\pi$
$13$	8.00000	0.170677	0.0853385	+	0.996352i	$0.472803\pi$
$14$	−16.0000	−0.305441
$15$	0	0
$16$	16.0000	0.250000
$17$	−15.0000	−0.214002	−0.107001	−	0.994259i	$-0.534125\pi$
$17$	−15.0000	−0.214002	−0.107001	+	0.994259i	$0.534125\pi$
$18$	0	0
$19$	23.0000	0.277714	0.138857	−	0.990312i	$-0.455657\pi$
$19$	23.0000	0.277714	0.138857	+	0.990312i	$0.455657\pi$
$20$	−20.0000	−0.223607
$21$	0	0
$22$	36.0000	0.348874
$23$	−63.0000	−0.571148	−0.285574	−	0.958357i	$-0.592184\pi$
$23$	−63.0000	−0.571148	−0.285574	+	0.958357i	$0.592184\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	−16.0000	−0.120687
$27$	0	0
$28$	32.0000	0.215980
$29$	−156.000	−0.998913	−0.499456	−	0.866339i	$-0.666467\pi$
$29$	−156.000	−0.998913	−0.499456	+	0.866339i	$0.666467\pi$
$30$	0	0
$31$	−85.0000	−0.492466	−0.246233	−	0.969211i	$-0.579193\pi$
$31$	−85.0000	−0.492466	−0.246233	+	0.969211i	$0.579193\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	30.0000	0.151322
$35$	−40.0000	−0.193178
$36$	0	0
$37$	74.0000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	74.0000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	−46.0000	−0.196373
$39$	0	0
$40$	40.0000	0.158114
$41$	−246.000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−246.000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	−190.000	−0.673831	−0.336915	−	0.941535i	$-0.609384\pi$
$43$	−190.000	−0.673831	−0.336915	+	0.941535i	$0.609384\pi$
$44$	−72.0000	−0.246691
$45$	0	0
$46$	126.000	0.403863
$47$	−288.000	−0.893811	−0.446906	−	0.894581i	$-0.647474\pi$
$47$	−288.000	−0.893811	−0.446906	+	0.894581i	$0.647474\pi$
$48$	0	0
$49$	−279.000	−0.813411
$50$	−50.0000	−0.141421
$51$	0	0
$52$	32.0000	0.0853385
$53$	177.000	0.458732	0.229366	−	0.973340i	$-0.426335\pi$
$53$	177.000	0.458732	0.229366	+	0.973340i	$0.426335\pi$
$54$	0	0
$55$	90.0000	0.220647
$56$	−64.0000	−0.152721
$57$	0	0
$58$	312.000	0.706338
$59$	−792.000	−1.74762	−0.873810	−	0.486267i	$-0.838358\pi$
$59$	−792.000	−1.74762	−0.873810	+	0.486267i	$0.838358\pi$
$60$	0	0
$61$	−907.000	−1.90376	−0.951881	−	0.306469i	$-0.900853\pi$
$61$	−907.000	−1.90376	−0.951881	+	0.306469i	$0.900853\pi$
$62$	170.000	0.348226
$63$	0	0
$64$	64.0000	0.125000
$65$	−40.0000	−0.0763291
$66$	0	0
$67$	−322.000	−0.587143	−0.293571	−	0.955937i	$-0.594844\pi$
$67$	−322.000	−0.587143	−0.293571	+	0.955937i	$0.594844\pi$
$68$	−60.0000	−0.107001
$69$	0	0
$70$	80.0000	0.136598
$71$	270.000	0.451311	0.225656	−	0.974207i	$-0.427548\pi$
$71$	270.000	0.451311	0.225656	+	0.974207i	$0.427548\pi$
$72$	0	0
$73$	254.000	0.407239	0.203620	−	0.979050i	$-0.434729\pi$
$73$	254.000	0.407239	0.203620	+	0.979050i	$0.434729\pi$
$74$	−148.000	−0.232495
$75$	0	0
$76$	92.0000	0.138857
$77$	−144.000	−0.213121
$78$	0	0
$79$	−1123.00	−1.59933	−0.799667	−	0.600444i	$-0.794991\pi$
$79$	−1123.00	−1.59933	−0.799667	+	0.600444i	$0.794991\pi$
$80$	−80.0000	−0.111803
$81$	0	0
$82$	492.000	0.662589
$83$	771.000	1.01962	0.509809	−	0.860288i	$-0.329716\pi$
$83$	771.000	1.01962	0.509809	+	0.860288i	$0.329716\pi$
$84$	0	0
$85$	75.0000	0.0957046
$86$	380.000	0.476470
$87$	0	0
$88$	144.000	0.174437
$89$	198.000	0.235820	0.117910	−	0.993024i	$-0.462381\pi$
$89$	198.000	0.235820	0.117910	+	0.993024i	$0.462381\pi$
$90$	0	0
$91$	64.0000	0.0737255
$92$	−252.000	−0.285574
$93$	0	0
$94$	576.000	0.632020
$95$	−115.000	−0.124197
$96$	0	0
$97$	−1192.00	−1.24772	−0.623862	−	0.781534i	$-0.714437\pi$
$97$	−1192.00	−1.24772	−0.623862	+	0.781534i	$0.714437\pi$
$98$	558.000	0.575168
$99$	0	0
$100$	100.000	0.100000
$101$	1692.00	1.66693	0.833467	−	0.552570i	$-0.186353\pi$
$101$	1692.00	1.66693	0.833467	+	0.552570i	$0.186353\pi$
$102$	0	0
$103$	1748.00	1.67219	0.836095	−	0.548585i	$-0.184833\pi$
$103$	1748.00	1.67219	0.836095	+	0.548585i	$0.184833\pi$
$104$	−64.0000	−0.0603434
$105$	0	0
$106$	−354.000	−0.324373
$107$	948.000	0.856510	0.428255	−	0.903658i	$-0.359128\pi$
$107$	948.000	0.856510	0.428255	+	0.903658i	$0.359128\pi$
$108$	0	0
$109$	593.000	0.521093	0.260546	−	0.965461i	$-0.416097\pi$
$109$	593.000	0.521093	0.260546	+	0.965461i	$0.416097\pi$
$110$	−180.000	−0.156021
$111$	0	0
$112$	128.000	0.107990
$113$	1062.00	0.884111	0.442056	−	0.896988i	$-0.354249\pi$
$113$	1062.00	0.884111	0.442056	+	0.896988i	$0.354249\pi$
$114$	0	0
$115$	315.000	0.255425
$116$	−624.000	−0.499456
$117$	0	0
$118$	1584.00	1.23575
$119$	−120.000	−0.0924402
$120$	0	0
$121$	−1007.00	−0.756574
$122$	1814.00	1.34616
$123$	0	0
$124$	−340.000	−0.246233
$125$	−125.000	−0.0894427
$126$	0	0
$127$	326.000	0.227778	0.113889	−	0.993493i	$-0.463669\pi$
$127$	326.000	0.227778	0.113889	+	0.993493i	$0.463669\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	80.0000	0.0539728
$131$	990.000	0.660280	0.330140	−	0.943932i	$-0.392904\pi$
$131$	990.000	0.660280	0.330140	+	0.943932i	$0.392904\pi$
$132$	0	0
$133$	184.000	0.119961
$134$	644.000	0.415173
$135$	0	0
$136$	120.000	0.0756611
$137$	147.000	0.0916720	0.0458360	−	0.998949i	$-0.485405\pi$
$137$	147.000	0.0916720	0.0458360	+	0.998949i	$0.485405\pi$
$138$	0	0
$139$	1604.00	0.978773	0.489387	−	0.872067i	$-0.337221\pi$
$139$	1604.00	0.978773	0.489387	+	0.872067i	$0.337221\pi$
$140$	−160.000	−0.0965891
$141$	0	0
$142$	−540.000	−0.319125
$143$	−144.000	−0.0842090
$144$	0	0
$145$	780.000	0.446727
$146$	−508.000	−0.287962
$147$	0	0
$148$	296.000	0.164399
$149$	−1218.00	−0.669681	−0.334840	−	0.942275i	$-0.608682\pi$
$149$	−1218.00	−0.669681	−0.334840	+	0.942275i	$0.608682\pi$
$150$	0	0
$151$	−2248.00	−1.21152	−0.605760	−	0.795647i	$-0.707131\pi$
$151$	−2248.00	−1.21152	−0.605760	+	0.795647i	$0.707131\pi$
$152$	−184.000	−0.0981866
$153$	0	0
$154$	288.000	0.150699
$155$	425.000	0.220238
$156$	0	0
$157$	−2998.00	−1.52399	−0.761995	−	0.647583i	$-0.775780\pi$
$157$	−2998.00	−1.52399	−0.761995	+	0.647583i	$0.775780\pi$
$158$	2246.00	1.13090
$159$	0	0
$160$	160.000	0.0790569
$161$	−504.000	−0.246713
$162$	0	0
$163$	3470.00	1.66743	0.833716	−	0.552194i	$-0.186209\pi$
$163$	3470.00	1.66743	0.833716	+	0.552194i	$0.186209\pi$
$164$	−984.000	−0.468521
$165$	0	0
$166$	−1542.00	−0.720978
$167$	−387.000	−0.179323	−0.0896616	−	0.995972i	$-0.528579\pi$
$167$	−387.000	−0.179323	−0.0896616	+	0.995972i	$0.528579\pi$
$168$	0	0
$169$	−2133.00	−0.970869
$170$	−150.000	−0.0676734
$171$	0	0
$172$	−760.000	−0.336915
$173$	855.000	0.375748	0.187874	−	0.982193i	$-0.439840\pi$
$173$	855.000	0.375748	0.187874	+	0.982193i	$0.439840\pi$
$174$	0	0
$175$	200.000	0.0863919
$176$	−288.000	−0.123346
$177$	0	0
$178$	−396.000	−0.166750
$179$	264.000	0.110236	0.0551181	−	0.998480i	$-0.482446\pi$
$179$	264.000	0.110236	0.0551181	+	0.998480i	$0.482446\pi$
$180$	0	0
$181$	−2551.00	−1.04759	−0.523797	−	0.851843i	$-0.675485\pi$
$181$	−2551.00	−1.04759	−0.523797	+	0.851843i	$0.675485\pi$
$182$	−128.000	−0.0521318
$183$	0	0
$184$	504.000	0.201931
$185$	−370.000	−0.147043
$186$	0	0
$187$	270.000	0.105585
$188$	−1152.00	−0.446906
$189$	0	0
$190$	230.000	0.0878208
$191$	2238.00	0.847832	0.423916	−	0.905701i	$-0.360655\pi$
$191$	2238.00	0.847832	0.423916	+	0.905701i	$0.360655\pi$
$192$	0	0
$193$	2180.00	0.813056	0.406528	−	0.913638i	$-0.366739\pi$
$193$	2180.00	0.813056	0.406528	+	0.913638i	$0.366739\pi$
$194$	2384.00	0.882274
$195$	0	0
$196$	−1116.00	−0.406706
$197$	−2577.00	−0.931998	−0.465999	−	0.884785i	$-0.654305\pi$
$197$	−2577.00	−0.931998	−0.465999	+	0.884785i	$0.654305\pi$
$198$	0	0
$199$	1412.00	0.502985	0.251493	−	0.967859i	$-0.419079\pi$
$199$	1412.00	0.502985	0.251493	+	0.967859i	$0.419079\pi$
$200$	−200.000	−0.0707107
$201$	0	0
$202$	−3384.00	−1.17870
$203$	−1248.00	−0.431490
$204$	0	0
$205$	1230.00	0.419058
$206$	−3496.00	−1.18242
$207$	0	0
$208$	128.000	0.0426692
$209$	−414.000	−0.137019
$210$	0	0
$211$	−307.000	−0.100165	−0.0500823	−	0.998745i	$-0.515948\pi$
$211$	−307.000	−0.100165	−0.0500823	+	0.998745i	$0.515948\pi$
$212$	708.000	0.229366
$213$	0	0
$214$	−1896.00	−0.605644
$215$	950.000	0.301346
$216$	0	0
$217$	−680.000	−0.212725
$218$	−1186.00	−0.368468
$219$	0	0
$220$	360.000	0.110324
$221$	−120.000	−0.0365252
$222$	0	0
$223$	5234.00	1.57172	0.785862	−	0.618402i	$-0.212219\pi$
$223$	5234.00	1.57172	0.785862	+	0.618402i	$0.212219\pi$
$224$	−256.000	−0.0763604
$225$	0	0
$226$	−2124.00	−0.625161
$227$	−1509.00	−0.441215	−0.220608	−	0.975363i	$-0.570804\pi$
$227$	−1509.00	−0.441215	−0.220608	+	0.975363i	$0.570804\pi$
$228$	0	0
$229$	1211.00	0.349455	0.174727	−	0.984617i	$-0.444096\pi$
$229$	1211.00	0.349455	0.174727	+	0.984617i	$0.444096\pi$
$230$	−630.000	−0.180613
$231$	0	0
$232$	1248.00	0.353169
$233$	−6246.00	−1.75618	−0.878088	−	0.478499i	$-0.841181\pi$
$233$	−6246.00	−1.75618	−0.878088	+	0.478499i	$0.841181\pi$
$234$	0	0
$235$	1440.00	0.399724
$236$	−3168.00	−0.873810
$237$	0	0
$238$	240.000	0.0653651
$239$	−4650.00	−1.25851	−0.629254	−	0.777200i	$-0.716640\pi$
$239$	−4650.00	−1.25851	−0.629254	+	0.777200i	$0.716640\pi$
$240$	0	0
$241$	−3145.00	−0.840611	−0.420306	−	0.907383i	$-0.638077\pi$
$241$	−3145.00	−0.840611	−0.420306	+	0.907383i	$0.638077\pi$
$242$	2014.00	0.534979
$243$	0	0
$244$	−3628.00	−0.951881
$245$	1395.00	0.363768
$246$	0	0
$247$	184.000	0.0473994
$248$	680.000	0.174113
$249$	0	0
$250$	250.000	0.0632456
$251$	−1020.00	−0.256501	−0.128251	−	0.991742i	$-0.540936\pi$
$251$	−1020.00	−0.256501	−0.128251	+	0.991742i	$0.540936\pi$
$252$	0	0
$253$	1134.00	0.281794
$254$	−652.000	−0.161063
$255$	0	0
$256$	256.000	0.0625000
$257$	−6741.00	−1.63616	−0.818078	−	0.575107i	$-0.804960\pi$
$257$	−6741.00	−1.63616	−0.818078	+	0.575107i	$0.804960\pi$
$258$	0	0
$259$	592.000	0.142027
$260$	−160.000	−0.0381645
$261$	0	0
$262$	−1980.00	−0.466889
$263$	−2340.00	−0.548633	−0.274317	−	0.961639i	$-0.588452\pi$
$263$	−2340.00	−0.548633	−0.274317	+	0.961639i	$0.588452\pi$
$264$	0	0
$265$	−885.000	−0.205151
$266$	−368.000	−0.0848253
$267$	0	0
$268$	−1288.00	−0.293571
$269$	6198.00	1.40483	0.702414	−	0.711769i	$-0.252106\pi$
$269$	6198.00	1.40483	0.702414	+	0.711769i	$0.252106\pi$
$270$	0	0
$271$	875.000	0.196135	0.0980673	−	0.995180i	$-0.468734\pi$
$271$	875.000	0.196135	0.0980673	+	0.995180i	$0.468734\pi$
$272$	−240.000	−0.0535005
$273$	0	0
$274$	−294.000	−0.0648219
$275$	−450.000	−0.0986764
$276$	0	0
$277$	5486.00	1.18997	0.594985	−	0.803737i	$-0.297158\pi$
$277$	5486.00	1.18997	0.594985	+	0.803737i	$0.297158\pi$
$278$	−3208.00	−0.692097
$279$	0	0
$280$	320.000	0.0682988
$281$	3204.00	0.680194	0.340097	−	0.940390i	$-0.389540\pi$
$281$	3204.00	0.680194	0.340097	+	0.940390i	$0.389540\pi$
$282$	0	0
$283$	7322.00	1.53798	0.768989	−	0.639262i	$-0.220760\pi$
$283$	7322.00	1.53798	0.768989	+	0.639262i	$0.220760\pi$
$284$	1080.00	0.225656
$285$	0	0
$286$	288.000	0.0595447
$287$	−1968.00	−0.404764
$288$	0	0
$289$	−4688.00	−0.954203
$290$	−1560.00	−0.315884
$291$	0	0
$292$	1016.00	0.203620
$293$	−1353.00	−0.269772	−0.134886	−	0.990861i	$-0.543067\pi$
$293$	−1353.00	−0.269772	−0.134886	+	0.990861i	$0.543067\pi$
$294$	0	0
$295$	3960.00	0.781560
$296$	−592.000	−0.116248
$297$	0	0
$298$	2436.00	0.473536
$299$	−504.000	−0.0974818
$300$	0	0
$301$	−1520.00	−0.291068
$302$	4496.00	0.856675
$303$	0	0
$304$	368.000	0.0694284
$305$	4535.00	0.851388
$306$	0	0
$307$	1658.00	0.308231	0.154116	−	0.988053i	$-0.450747\pi$
$307$	1658.00	0.308231	0.154116	+	0.988053i	$0.450747\pi$
$308$	−576.000	−0.106561
$309$	0	0
$310$	−850.000	−0.155731
$311$	1044.00	0.190353	0.0951765	−	0.995460i	$-0.469658\pi$
$311$	1044.00	0.190353	0.0951765	+	0.995460i	$0.469658\pi$
$312$	0	0
$313$	2588.00	0.467356	0.233678	−	0.972314i	$-0.424924\pi$
$313$	2588.00	0.467356	0.233678	+	0.972314i	$0.424924\pi$
$314$	5996.00	1.07762
$315$	0	0
$316$	−4492.00	−0.799667
$317$	1449.00	0.256732	0.128366	−	0.991727i	$-0.459027\pi$
$317$	1449.00	0.256732	0.128366	+	0.991727i	$0.459027\pi$
$318$	0	0
$319$	2808.00	0.492846
$320$	−320.000	−0.0559017
$321$	0	0
$322$	1008.00	0.174452
$323$	−345.000	−0.0594313
$324$	0	0
$325$	200.000	0.0341354
$326$	−6940.00	−1.17905
$327$	0	0
$328$	1968.00	0.331295
$329$	−2304.00	−0.386090
$330$	0	0
$331$	4880.00	0.810360	0.405180	−	0.914237i	$-0.367209\pi$
$331$	4880.00	0.810360	0.405180	+	0.914237i	$0.367209\pi$
$332$	3084.00	0.509809
$333$	0	0
$334$	774.000	0.126801
$335$	1610.00	0.262578
$336$	0	0
$337$	−7744.00	−1.25176	−0.625879	−	0.779920i	$-0.715260\pi$
$337$	−7744.00	−1.25176	−0.625879	+	0.779920i	$0.715260\pi$
$338$	4266.00	0.686508
$339$	0	0
$340$	300.000	0.0478523
$341$	1530.00	0.242974
$342$	0	0
$343$	−4976.00	−0.783320
$344$	1520.00	0.238235
$345$	0	0
$346$	−1710.00	−0.265694
$347$	−804.000	−0.124383	−0.0621916	−	0.998064i	$-0.519809\pi$
$347$	−804.000	−0.124383	−0.0621916	+	0.998064i	$0.519809\pi$
$348$	0	0
$349$	−2815.00	−0.431758	−0.215879	−	0.976420i	$-0.569262\pi$
$349$	−2815.00	−0.431758	−0.215879	+	0.976420i	$0.569262\pi$
$350$	−400.000	−0.0610883
$351$	0	0
$352$	576.000	0.0872185
$353$	3738.00	0.563608	0.281804	−	0.959472i	$-0.409067\pi$
$353$	3738.00	0.563608	0.281804	+	0.959472i	$0.409067\pi$
$354$	0	0
$355$	−1350.00	−0.201833
$356$	792.000	0.117910
$357$	0	0
$358$	−528.000	−0.0779488
$359$	11022.0	1.62039	0.810193	−	0.586163i	$-0.199362\pi$
$359$	11022.0	1.62039	0.810193	+	0.586163i	$0.199362\pi$
$360$	0	0
$361$	−6330.00	−0.922875
$362$	5102.00	0.740760
$363$	0	0
$364$	256.000	0.0368628
$365$	−1270.00	−0.182123
$366$	0	0
$367$	7544.00	1.07301	0.536504	−	0.843898i	$-0.319745\pi$
$367$	7544.00	1.07301	0.536504	+	0.843898i	$0.319745\pi$
$368$	−1008.00	−0.142787
$369$	0	0
$370$	740.000	0.103975
$371$	1416.00	0.198154
$372$	0	0
$373$	−5404.00	−0.750157	−0.375078	−	0.926993i	$-0.622384\pi$
$373$	−5404.00	−0.750157	−0.375078	+	0.926993i	$0.622384\pi$
$374$	−540.000	−0.0746597
$375$	0	0
$376$	2304.00	0.316010
$377$	−1248.00	−0.170491
$378$	0	0
$379$	−2335.00	−0.316467	−0.158233	−	0.987402i	$-0.550580\pi$
$379$	−2335.00	−0.316467	−0.158233	+	0.987402i	$0.550580\pi$
$380$	−460.000	−0.0620987
$381$	0	0
$382$	−4476.00	−0.599508
$383$	6633.00	0.884936	0.442468	−	0.896784i	$-0.354103\pi$
$383$	6633.00	0.884936	0.442468	+	0.896784i	$0.354103\pi$
$384$	0	0
$385$	720.000	0.0953106
$386$	−4360.00	−0.574918
$387$	0	0
$388$	−4768.00	−0.623862
$389$	7566.00	0.986148	0.493074	−	0.869987i	$-0.335873\pi$
$389$	7566.00	0.986148	0.493074	+	0.869987i	$0.335873\pi$
$390$	0	0
$391$	945.000	0.122227
$392$	2232.00	0.287584
$393$	0	0
$394$	5154.00	0.659022
$395$	5615.00	0.715244
$396$	0	0
$397$	−7420.00	−0.938033	−0.469017	−	0.883189i	$-0.655392\pi$
$397$	−7420.00	−0.938033	−0.469017	+	0.883189i	$0.655392\pi$
$398$	−2824.00	−0.355664
$399$	0	0
$400$	400.000	0.0500000
$401$	8502.00	1.05878	0.529389	−	0.848379i	$-0.322421\pi$
$401$	8502.00	1.05878	0.529389	+	0.848379i	$0.322421\pi$
$402$	0	0
$403$	−680.000	−0.0840526
$404$	6768.00	0.833467
$405$	0	0
$406$	2496.00	0.305109
$407$	−1332.00	−0.162223
$408$	0	0
$409$	−1903.00	−0.230067	−0.115033	−	0.993362i	$-0.536697\pi$
$409$	−1903.00	−0.230067	−0.115033	+	0.993362i	$0.536697\pi$
$410$	−2460.00	−0.296319
$411$	0	0
$412$	6992.00	0.836095
$413$	−6336.00	−0.754901
$414$	0	0
$415$	−3855.00	−0.455987
$416$	−256.000	−0.0301717
$417$	0	0
$418$	828.000	0.0968871
$419$	−13482.0	−1.57193	−0.785965	−	0.618271i	$-0.787834\pi$
$419$	−13482.0	−1.57193	−0.785965	+	0.618271i	$0.787834\pi$
$420$	0	0
$421$	−1537.00	−0.177931	−0.0889653	−	0.996035i	$-0.528356\pi$
$421$	−1537.00	−0.177931	−0.0889653	+	0.996035i	$0.528356\pi$
$422$	614.000	0.0708271
$423$	0	0
$424$	−1416.00	−0.162186
$425$	−375.000	−0.0428004
$426$	0	0
$427$	−7256.00	−0.822348
$428$	3792.00	0.428255
$429$	0	0
$430$	−1900.00	−0.213084
$431$	10368.0	1.15872	0.579361	−	0.815071i	$-0.303302\pi$
$431$	10368.0	1.15872	0.579361	+	0.815071i	$0.303302\pi$
$432$	0	0
$433$	−13168.0	−1.46146	−0.730732	−	0.682665i	$-0.760821\pi$
$433$	−13168.0	−1.46146	−0.730732	+	0.682665i	$0.760821\pi$
$434$	1360.00	0.150420
$435$	0	0
$436$	2372.00	0.260546
$437$	−1449.00	−0.158616
$438$	0	0
$439$	7319.00	0.795710	0.397855	−	0.917448i	$-0.369755\pi$
$439$	7319.00	0.795710	0.397855	+	0.917448i	$0.369755\pi$
$440$	−720.000	−0.0780106
$441$	0	0
$442$	240.000	0.0258272
$443$	4119.00	0.441760	0.220880	−	0.975301i	$-0.429107\pi$
$443$	4119.00	0.441760	0.220880	+	0.975301i	$0.429107\pi$
$444$	0	0
$445$	−990.000	−0.105462
$446$	−10468.0	−1.11138
$447$	0	0
$448$	512.000	0.0539949
$449$	5388.00	0.566315	0.283158	−	0.959073i	$-0.408618\pi$
$449$	5388.00	0.566315	0.283158	+	0.959073i	$0.408618\pi$
$450$	0	0
$451$	4428.00	0.462320
$452$	4248.00	0.442056
$453$	0	0
$454$	3018.00	0.311986
$455$	−320.000	−0.0329711
$456$	0	0
$457$	−2752.00	−0.281692	−0.140846	−	0.990032i	$-0.544982\pi$
$457$	−2752.00	−0.281692	−0.140846	+	0.990032i	$0.544982\pi$
$458$	−2422.00	−0.247102
$459$	0	0
$460$	1260.00	0.127713
$461$	4314.00	0.435842	0.217921	−	0.975966i	$-0.430073\pi$
$461$	4314.00	0.435842	0.217921	+	0.975966i	$0.430073\pi$
$462$	0	0
$463$	−5794.00	−0.581577	−0.290788	−	0.956787i	$-0.593918\pi$
$463$	−5794.00	−0.581577	−0.290788	+	0.956787i	$0.593918\pi$
$464$	−2496.00	−0.249728
$465$	0	0
$466$	12492.0	1.24180
$467$	−6309.00	−0.625151	−0.312576	−	0.949893i	$-0.601192\pi$
$467$	−6309.00	−0.625151	−0.312576	+	0.949893i	$0.601192\pi$
$468$	0	0
$469$	−2576.00	−0.253622
$470$	−2880.00	−0.282648
$471$	0	0
$472$	6336.00	0.617877
$473$	3420.00	0.332456
$474$	0	0
$475$	575.000	0.0555428
$476$	−480.000	−0.0462201
$477$	0	0
$478$	9300.00	0.889900
$479$	−14826.0	−1.41423	−0.707116	−	0.707097i	$-0.750004\pi$
$479$	−14826.0	−1.41423	−0.707116	+	0.707097i	$0.750004\pi$
$480$	0	0
$481$	592.000	0.0561182
$482$	6290.00	0.594402
$483$	0	0
$484$	−4028.00	−0.378287
$485$	5960.00	0.557999
$486$	0	0
$487$	6758.00	0.628818	0.314409	−	0.949288i	$-0.398194\pi$
$487$	6758.00	0.628818	0.314409	+	0.949288i	$0.398194\pi$
$488$	7256.00	0.673081
$489$	0	0
$490$	−2790.00	−0.257223
$491$	14574.0	1.33954	0.669771	−	0.742567i	$-0.266392\pi$
$491$	14574.0	1.33954	0.669771	+	0.742567i	$0.266392\pi$
$492$	0	0
$493$	2340.00	0.213769
$494$	−368.000	−0.0335164
$495$	0	0
$496$	−1360.00	−0.123117
$497$	2160.00	0.194948
$498$	0	0
$499$	12611.0	1.13135	0.565677	−	0.824627i	$-0.308615\pi$
$499$	12611.0	1.13135	0.565677	+	0.824627i	$0.308615\pi$
$500$	−500.000	−0.0447214
$501$	0	0
$502$	2040.00	0.181374
$503$	−15639.0	−1.38630	−0.693150	−	0.720794i	$-0.743778\pi$
$503$	−15639.0	−1.38630	−0.693150	+	0.720794i	$0.743778\pi$
$504$	0	0
$505$	−8460.00	−0.745475
$506$	−2268.00	−0.199259
$507$	0	0
$508$	1304.00	0.113889
$509$	−15420.0	−1.34279	−0.671394	−	0.741100i	$-0.734304\pi$
$509$	−15420.0	−1.34279	−0.671394	+	0.741100i	$0.734304\pi$
$510$	0	0
$511$	2032.00	0.175911
$512$	−512.000	−0.0441942
$513$	0	0
$514$	13482.0	1.15694
$515$	−8740.00	−0.747826
$516$	0	0
$517$	5184.00	0.440990
$518$	−1184.00	−0.100429
$519$	0	0
$520$	320.000	0.0269864
$521$	−10494.0	−0.882439	−0.441219	−	0.897399i	$-0.645454\pi$
$521$	−10494.0	−0.882439	−0.441219	+	0.897399i	$0.645454\pi$
$522$	0	0
$523$	−10708.0	−0.895274	−0.447637	−	0.894215i	$-0.647734\pi$
$523$	−10708.0	−0.895274	−0.447637	+	0.894215i	$0.647734\pi$
$524$	3960.00	0.330140
$525$	0	0
$526$	4680.00	0.387942
$527$	1275.00	0.105389
$528$	0	0
$529$	−8198.00	−0.673790
$530$	1770.00	0.145064
$531$	0	0
$532$	736.000	0.0599805
$533$	−1968.00	−0.159932
$534$	0	0
$535$	−4740.00	−0.383043
$536$	2576.00	0.207586
$537$	0	0
$538$	−12396.0	−0.993363
$539$	5022.00	0.401323
$540$	0	0
$541$	23030.0	1.83020	0.915099	−	0.403229i	$-0.132112\pi$
$541$	23030.0	1.83020	0.915099	+	0.403229i	$0.132112\pi$
$542$	−1750.00	−0.138688
$543$	0	0
$544$	480.000	0.0378306
$545$	−2965.00	−0.233040
$546$	0	0
$547$	−3814.00	−0.298126	−0.149063	−	0.988828i	$-0.547626\pi$
$547$	−3814.00	−0.298126	−0.149063	+	0.988828i	$0.547626\pi$
$548$	588.000	0.0458360
$549$	0	0
$550$	900.000	0.0697748
$551$	−3588.00	−0.277412
$552$	0	0
$553$	−8984.00	−0.690847
$554$	−10972.0	−0.841436
$555$	0	0
$556$	6416.00	0.489387
$557$	−22266.0	−1.69379	−0.846895	−	0.531761i	$-0.821531\pi$
$557$	−22266.0	−1.69379	−0.846895	+	0.531761i	$0.821531\pi$
$558$	0	0
$559$	−1520.00	−0.115007
$560$	−640.000	−0.0482945
$561$	0	0
$562$	−6408.00	−0.480970
$563$	23844.0	1.78491	0.892455	−	0.451136i	$-0.148981\pi$
$563$	23844.0	1.78491	0.892455	+	0.451136i	$0.148981\pi$
$564$	0	0
$565$	−5310.00	−0.395387
$566$	−14644.0	−1.08751
$567$	0	0
$568$	−2160.00	−0.159563
$569$	−7488.00	−0.551693	−0.275846	−	0.961202i	$-0.588958\pi$
$569$	−7488.00	−0.551693	−0.275846	+	0.961202i	$0.588958\pi$
$570$	0	0
$571$	5111.00	0.374586	0.187293	−	0.982304i	$-0.440029\pi$
$571$	5111.00	0.374586	0.187293	+	0.982304i	$0.440029\pi$
$572$	−576.000	−0.0421045
$573$	0	0
$574$	3936.00	0.286212
$575$	−1575.00	−0.114230
$576$	0	0
$577$	6986.00	0.504040	0.252020	−	0.967722i	$-0.418905\pi$
$577$	6986.00	0.504040	0.252020	+	0.967722i	$0.418905\pi$
$578$	9376.00	0.674724
$579$	0	0
$580$	3120.00	0.223364
$581$	6168.00	0.440433
$582$	0	0
$583$	−3186.00	−0.226330
$584$	−2032.00	−0.143981
$585$	0	0
$586$	2706.00	0.190757
$587$	−20571.0	−1.44643	−0.723216	−	0.690622i	$-0.757337\pi$
$587$	−20571.0	−1.44643	−0.723216	+	0.690622i	$0.757337\pi$
$588$	0	0
$589$	−1955.00	−0.136765
$590$	−7920.00	−0.552646
$591$	0	0
$592$	1184.00	0.0821995
$593$	−23241.0	−1.60943	−0.804716	−	0.593660i	$-0.797683\pi$
$593$	−23241.0	−1.60943	−0.804716	+	0.593660i	$0.797683\pi$
$594$	0	0
$595$	600.000	0.0413405
$596$	−4872.00	−0.334840
$597$	0	0
$598$	1008.00	0.0689301
$599$	−20208.0	−1.37842	−0.689212	−	0.724559i	$-0.742043\pi$
$599$	−20208.0	−1.37842	−0.689212	+	0.724559i	$0.742043\pi$
$600$	0	0
$601$	−9055.00	−0.614578	−0.307289	−	0.951616i	$-0.599422\pi$
$601$	−9055.00	−0.614578	−0.307289	+	0.951616i	$0.599422\pi$
$602$	3040.00	0.205816
$603$	0	0
$604$	−8992.00	−0.605760
$605$	5035.00	0.338350
$606$	0	0
$607$	15554.0	1.04006	0.520031	−	0.854148i	$-0.325920\pi$
$607$	15554.0	1.04006	0.520031	+	0.854148i	$0.325920\pi$
$608$	−736.000	−0.0490933
$609$	0	0
$610$	−9070.00	−0.602022
$611$	−2304.00	−0.152553
$612$	0	0
$613$	−5632.00	−0.371084	−0.185542	−	0.982636i	$-0.559404\pi$
$613$	−5632.00	−0.371084	−0.185542	+	0.982636i	$0.559404\pi$
$614$	−3316.00	−0.217953
$615$	0	0
$616$	1152.00	0.0753497
$617$	9141.00	0.596439	0.298219	−	0.954497i	$-0.403607\pi$
$617$	9141.00	0.596439	0.298219	+	0.954497i	$0.403607\pi$
$618$	0	0
$619$	−13372.0	−0.868281	−0.434141	−	0.900845i	$-0.642948\pi$
$619$	−13372.0	−0.868281	−0.434141	+	0.900845i	$0.642948\pi$
$620$	1700.00	0.110119
$621$	0	0
$622$	−2088.00	−0.134600
$623$	1584.00	0.101865
$624$	0	0
$625$	625.000	0.0400000
$626$	−5176.00	−0.330471
$627$	0	0
$628$	−11992.0	−0.761995
$629$	−1110.00	−0.0703634
$630$	0	0
$631$	11165.0	0.704392	0.352196	−	0.935926i	$-0.385435\pi$
$631$	11165.0	0.704392	0.352196	+	0.935926i	$0.385435\pi$
$632$	8984.00	0.565450
$633$	0	0
$634$	−2898.00	−0.181537
$635$	−1630.00	−0.101865
$636$	0	0
$637$	−2232.00	−0.138831
$638$	−5616.00	−0.348495
$639$	0	0
$640$	640.000	0.0395285
$641$	912.000	0.0561963	0.0280982	−	0.999605i	$-0.491055\pi$
$641$	912.000	0.0561963	0.0280982	+	0.999605i	$0.491055\pi$
$642$	0	0
$643$	−27952.0	−1.71434	−0.857169	−	0.515035i	$-0.827779\pi$
$643$	−27952.0	−1.71434	−0.857169	+	0.515035i	$0.827779\pi$
$644$	−2016.00	−0.123356
$645$	0	0
$646$	690.000	0.0420243
$647$	6285.00	0.381899	0.190950	−	0.981600i	$-0.438843\pi$
$647$	6285.00	0.381899	0.190950	+	0.981600i	$0.438843\pi$
$648$	0	0
$649$	14256.0	0.862245
$650$	−400.000	−0.0241374
$651$	0	0
$652$	13880.0	0.833716
$653$	16497.0	0.988633	0.494317	−	0.869282i	$-0.335418\pi$
$653$	16497.0	0.988633	0.494317	+	0.869282i	$0.335418\pi$
$654$	0	0
$655$	−4950.00	−0.295286
$656$	−3936.00	−0.234261
$657$	0	0
$658$	4608.00	0.273007
$659$	−14844.0	−0.877451	−0.438725	−	0.898621i	$-0.644570\pi$
$659$	−14844.0	−0.877451	−0.438725	+	0.898621i	$0.644570\pi$
$660$	0	0
$661$	31934.0	1.87911	0.939553	−	0.342404i	$-0.111241\pi$
$661$	31934.0	1.87911	0.939553	+	0.342404i	$0.111241\pi$
$662$	−9760.00	−0.573011
$663$	0	0
$664$	−6168.00	−0.360489
$665$	−920.000	−0.0536482
$666$	0	0
$667$	9828.00	0.570527
$668$	−1548.00	−0.0896616
$669$	0	0
$670$	−3220.00	−0.185671
$671$	16326.0	0.939282
$672$	0	0
$673$	−24352.0	−1.39480	−0.697400	−	0.716682i	$-0.745660\pi$
$673$	−24352.0	−1.39480	−0.697400	+	0.716682i	$0.745660\pi$
$674$	15488.0	0.885127
$675$	0	0
$676$	−8532.00	−0.485435
$677$	10374.0	0.588929	0.294465	−	0.955662i	$-0.404859\pi$
$677$	10374.0	0.588929	0.294465	+	0.955662i	$0.404859\pi$
$678$	0	0
$679$	−9536.00	−0.538966
$680$	−600.000	−0.0338367
$681$	0	0
$682$	−3060.00	−0.171809
$683$	7347.00	0.411603	0.205802	−	0.978594i	$-0.434020\pi$
$683$	7347.00	0.411603	0.205802	+	0.978594i	$0.434020\pi$
$684$	0	0
$685$	−735.000	−0.0409969
$686$	9952.00	0.553891
$687$	0	0
$688$	−3040.00	−0.168458
$689$	1416.00	0.0782951
$690$	0	0
$691$	−5371.00	−0.295691	−0.147845	−	0.989010i	$-0.547234\pi$
$691$	−5371.00	−0.295691	−0.147845	+	0.989010i	$0.547234\pi$
$692$	3420.00	0.187874
$693$	0	0
$694$	1608.00	0.0879522
$695$	−8020.00	−0.437721
$696$	0	0
$697$	3690.00	0.200529
$698$	5630.00	0.305299
$699$	0	0
$700$	800.000	0.0431959
$701$	−7086.00	−0.381790	−0.190895	−	0.981610i	$-0.561139\pi$
$701$	−7086.00	−0.381790	−0.190895	+	0.981610i	$0.561139\pi$
$702$	0	0
$703$	1702.00	0.0913117
$704$	−1152.00	−0.0616728
$705$	0	0
$706$	−7476.00	−0.398531
$707$	13536.0	0.720048
$708$	0	0
$709$	17186.0	0.910344	0.455172	−	0.890404i	$-0.349578\pi$
$709$	17186.0	0.910344	0.455172	+	0.890404i	$0.349578\pi$
$710$	2700.00	0.142717
$711$	0	0
$712$	−1584.00	−0.0833749
$713$	5355.00	0.281271
$714$	0	0
$715$	720.000	0.0376594
$716$	1056.00	0.0551181
$717$	0	0
$718$	−22044.0	−1.14579
$719$	23814.0	1.23520	0.617602	−	0.786490i	$-0.288104\pi$
$719$	23814.0	1.23520	0.617602	+	0.786490i	$0.288104\pi$
$720$	0	0
$721$	13984.0	0.722318
$722$	12660.0	0.652571
$723$	0	0
$724$	−10204.0	−0.523797
$725$	−3900.00	−0.199783
$726$	0	0
$727$	−22732.0	−1.15967	−0.579837	−	0.814732i	$-0.696884\pi$
$727$	−22732.0	−1.15967	−0.579837	+	0.814732i	$0.696884\pi$
$728$	−512.000	−0.0260659
$729$	0	0
$730$	2540.00	0.128780
$731$	2850.00	0.144201
$732$	0	0
$733$	4664.00	0.235019	0.117509	−	0.993072i	$-0.462509\pi$
$733$	4664.00	0.235019	0.117509	+	0.993072i	$0.462509\pi$
$734$	−15088.0	−0.758731
$735$	0	0
$736$	2016.00	0.100966
$737$	5796.00	0.289686
$738$	0	0
$739$	5501.00	0.273826	0.136913	−	0.990583i	$-0.456282\pi$
$739$	5501.00	0.273826	0.136913	+	0.990583i	$0.456282\pi$
$740$	−1480.00	−0.0735215
$741$	0	0
$742$	−2832.00	−0.140116
$743$	27096.0	1.33789	0.668947	−	0.743310i	$-0.266745\pi$
$743$	27096.0	1.33789	0.668947	+	0.743310i	$0.266745\pi$
$744$	0	0
$745$	6090.00	0.299490
$746$	10808.0	0.530441
$747$	0	0
$748$	1080.00	0.0527924
$749$	7584.00	0.369978
$750$	0	0
$751$	−5659.00	−0.274967	−0.137483	−	0.990504i	$-0.543901\pi$
$751$	−5659.00	−0.274967	−0.137483	+	0.990504i	$0.543901\pi$
$752$	−4608.00	−0.223453
$753$	0	0
$754$	2496.00	0.120556
$755$	11240.0	0.541809
$756$	0	0
$757$	37694.0	1.80979	0.904895	−	0.425634i	$-0.139949\pi$
$757$	37694.0	1.80979	0.904895	+	0.425634i	$0.139949\pi$
$758$	4670.00	0.223776
$759$	0	0
$760$	920.000	0.0439104
$761$	6588.00	0.313817	0.156909	−	0.987613i	$-0.449847\pi$
$761$	6588.00	0.313817	0.156909	+	0.987613i	$0.449847\pi$
$762$	0	0
$763$	4744.00	0.225091
$764$	8952.00	0.423916
$765$	0	0
$766$	−13266.0	−0.625744
$767$	−6336.00	−0.298279
$768$	0	0
$769$	−19.0000	−0.000890972 0	−0.000445486	−	1.00000i	$-0.500142\pi$
$769$	−19.0000	−0.000890972 0	−0.000445486		1.00000i	$0.500142\pi$
$770$	−1440.00	−0.0673948
$771$	0	0
$772$	8720.00	0.406528
$773$	−33639.0	−1.56521	−0.782607	−	0.622516i	$-0.786111\pi$
$773$	−33639.0	−1.56521	−0.782607	+	0.622516i	$0.786111\pi$
$774$	0	0
$775$	−2125.00	−0.0984932
$776$	9536.00	0.441137
$777$	0	0
$778$	−15132.0	−0.697312
$779$	−5658.00	−0.260230
$780$	0	0
$781$	−4860.00	−0.222669
$782$	−1890.00	−0.0864274
$783$	0	0
$784$	−4464.00	−0.203353
$785$	14990.0	0.681549
$786$	0	0
$787$	23474.0	1.06322	0.531612	−	0.846988i	$-0.321586\pi$
$787$	23474.0	1.06322	0.531612	+	0.846988i	$0.321586\pi$
$788$	−10308.0	−0.465999
$789$	0	0
$790$	−11230.0	−0.505754
$791$	8496.00	0.381900
$792$	0	0
$793$	−7256.00	−0.324928
$794$	14840.0	0.663290
$795$	0	0
$796$	5648.00	0.251493
$797$	−7917.00	−0.351863	−0.175931	−	0.984402i	$-0.556294\pi$
$797$	−7917.00	−0.351863	−0.175931	+	0.984402i	$0.556294\pi$
$798$	0	0
$799$	4320.00	0.191277
$800$	−800.000	−0.0353553
$801$	0	0
$802$	−17004.0	−0.748668
$803$	−4572.00	−0.200925
$804$	0	0
$805$	2520.00	0.110333
$806$	1360.00	0.0594342
$807$	0	0
$808$	−13536.0	−0.589350
$809$	41202.0	1.79059	0.895294	−	0.445476i	$-0.146966\pi$
$809$	41202.0	1.79059	0.895294	+	0.445476i	$0.146966\pi$
$810$	0	0
$811$	35492.0	1.53674	0.768368	−	0.640008i	$-0.221069\pi$
$811$	35492.0	1.53674	0.768368	+	0.640008i	$0.221069\pi$
$812$	−4992.00	−0.215745
$813$	0	0
$814$	2664.00	0.114709
$815$	−17350.0	−0.745698
$816$	0	0
$817$	−4370.00	−0.187132
$818$	3806.00	0.162682
$819$	0	0
$820$	4920.00	0.209529
$821$	7146.00	0.303772	0.151886	−	0.988398i	$-0.451465\pi$
$821$	7146.00	0.303772	0.151886	+	0.988398i	$0.451465\pi$
$822$	0	0
$823$	8882.00	0.376193	0.188097	−	0.982151i	$-0.439768\pi$
$823$	8882.00	0.376193	0.188097	+	0.982151i	$0.439768\pi$
$824$	−13984.0	−0.591208
$825$	0	0
$826$	12672.0	0.533796
$827$	21705.0	0.912644	0.456322	−	0.889815i	$-0.349166\pi$
$827$	21705.0	0.912644	0.456322	+	0.889815i	$0.349166\pi$
$828$	0	0
$829$	29018.0	1.21573	0.607863	−	0.794042i	$-0.292027\pi$
$829$	29018.0	1.21573	0.607863	+	0.794042i	$0.292027\pi$
$830$	7710.00	0.322431
$831$	0	0
$832$	512.000	0.0213346
$833$	4185.00	0.174072
$834$	0	0
$835$	1935.00	0.0801957
$836$	−1656.00	−0.0685095
$837$	0	0
$838$	26964.0	1.11152
$839$	−31164.0	−1.28236	−0.641180	−	0.767390i	$-0.721555\pi$
$839$	−31164.0	−1.28236	−0.641180	+	0.767390i	$0.721555\pi$
$840$	0	0
$841$	−53.0000	−0.00217311
$842$	3074.00	0.125816
$843$	0	0
$844$	−1228.00	−0.0500823
$845$	10665.0	0.434186
$846$	0	0
$847$	−8056.00	−0.326809
$848$	2832.00	0.114683
$849$	0	0
$850$	750.000	0.0302645
$851$	−4662.00	−0.187792
$852$	0	0
$853$	49160.0	1.97328	0.986639	−	0.162921i	$-0.0520916\pi$
$853$	49160.0	1.97328	0.986639	+	0.162921i	$0.0520916\pi$
$854$	14512.0	0.581488
$855$	0	0
$856$	−7584.00	−0.302822
$857$	2349.00	0.0936293	0.0468147	−	0.998904i	$-0.485093\pi$
$857$	2349.00	0.0936293	0.0468147	+	0.998904i	$0.485093\pi$
$858$	0	0
$859$	−28195.0	−1.11991	−0.559954	−	0.828524i	$-0.689181\pi$
$859$	−28195.0	−1.11991	−0.559954	+	0.828524i	$0.689181\pi$
$860$	3800.00	0.150673
$861$	0	0
$862$	−20736.0	−0.819340
$863$	23997.0	0.946544	0.473272	−	0.880916i	$-0.343073\pi$
$863$	23997.0	0.946544	0.473272	+	0.880916i	$0.343073\pi$
$864$	0	0
$865$	−4275.00	−0.168040
$866$	26336.0	1.03341
$867$	0	0
$868$	−2720.00	−0.106363
$869$	20214.0	0.789083
$870$	0	0
$871$	−2576.00	−0.100212
$872$	−4744.00	−0.184234
$873$	0	0
$874$	2898.00	0.112158
$875$	−1000.00	−0.0386356
$876$	0	0
$877$	46286.0	1.78217	0.891087	−	0.453832i	$-0.149943\pi$
$877$	46286.0	1.78217	0.891087	+	0.453832i	$0.149943\pi$
$878$	−14638.0	−0.562652
$879$	0	0
$880$	1440.00	0.0551618
$881$	−39636.0	−1.51574	−0.757872	−	0.652403i	$-0.773761\pi$
$881$	−39636.0	−1.51574	−0.757872	+	0.652403i	$0.773761\pi$
$882$	0	0
$883$	−16744.0	−0.638143	−0.319072	−	0.947731i	$-0.603371\pi$
$883$	−16744.0	−0.638143	−0.319072	+	0.947731i	$0.603371\pi$
$884$	−480.000	−0.0182626
$885$	0	0
$886$	−8238.00	−0.312371
$887$	−1251.00	−0.0473557	−0.0236778	−	0.999720i	$-0.507538\pi$
$887$	−1251.00	−0.0473557	−0.0236778	+	0.999720i	$0.507538\pi$
$888$	0	0
$889$	2608.00	0.0983909
$890$	1980.00	0.0745728
$891$	0	0
$892$	20936.0	0.785862
$893$	−6624.00	−0.248224
$894$	0	0
$895$	−1320.00	−0.0492991
$896$	−1024.00	−0.0381802
$897$	0	0
$898$	−10776.0	−0.400445
$899$	13260.0	0.491931
$900$	0	0
$901$	−2655.00	−0.0981697
$902$	−8856.00	−0.326910
$903$	0	0
$904$	−8496.00	−0.312580
$905$	12755.0	0.468498
$906$	0	0
$907$	−36988.0	−1.35410	−0.677049	−	0.735938i	$-0.736741\pi$
$907$	−36988.0	−1.35410	−0.677049	+	0.735938i	$0.736741\pi$
$908$	−6036.00	−0.220608
$909$	0	0
$910$	640.000	0.0233141
$911$	−16404.0	−0.596585	−0.298292	−	0.954475i	$-0.596417\pi$
$911$	−16404.0	−0.596585	−0.298292	+	0.954475i	$0.596417\pi$
$912$	0	0
$913$	−13878.0	−0.503061
$914$	5504.00	0.199186
$915$	0	0
$916$	4844.00	0.174727
$917$	7920.00	0.285214
$918$	0	0
$919$	−664.000	−0.0238339	−0.0119169	−	0.999929i	$-0.503793\pi$
$919$	−664.000	−0.0238339	−0.0119169	+	0.999929i	$0.503793\pi$
$920$	−2520.00	−0.0903065
$921$	0	0
$922$	−8628.00	−0.308187
$923$	2160.00	0.0770285
$924$	0	0
$925$	1850.00	0.0657596
$926$	11588.0	0.411237
$927$	0	0
$928$	4992.00	0.176585
$929$	39642.0	1.40001	0.700006	−	0.714137i	$-0.253180\pi$
$929$	39642.0	1.40001	0.700006	+	0.714137i	$0.253180\pi$
$930$	0	0
$931$	−6417.00	−0.225895
$932$	−24984.0	−0.878088
$933$	0	0
$934$	12618.0	0.442049
$935$	−1350.00	−0.0472190
$936$	0	0
$937$	−36028.0	−1.25612	−0.628059	−	0.778165i	$-0.716151\pi$
$937$	−36028.0	−1.25612	−0.628059	+	0.778165i	$0.716151\pi$
$938$	5152.00	0.179338
$939$	0	0
$940$	5760.00	0.199862
$941$	−23058.0	−0.798798	−0.399399	−	0.916777i	$-0.630781\pi$
$941$	−23058.0	−0.798798	−0.399399	+	0.916777i	$0.630781\pi$
$942$	0	0
$943$	15498.0	0.535190
$944$	−12672.0	−0.436905
$945$	0	0
$946$	−6840.00	−0.235082
$947$	−19953.0	−0.684673	−0.342337	−	0.939577i	$-0.611218\pi$
$947$	−19953.0	−0.684673	−0.342337	+	0.939577i	$0.611218\pi$
$948$	0	0
$949$	2032.00	0.0695063
$950$	−1150.00	−0.0392747
$951$	0	0
$952$	960.000	0.0326825
$953$	−25638.0	−0.871455	−0.435727	−	0.900079i	$-0.643509\pi$
$953$	−25638.0	−0.871455	−0.435727	+	0.900079i	$0.643509\pi$
$954$	0	0
$955$	−11190.0	−0.379162
$956$	−18600.0	−0.629254
$957$	0	0
$958$	29652.0	1.00001
$959$	1176.00	0.0395986
$960$	0	0
$961$	−22566.0	−0.757477
$962$	−1184.00	−0.0396816
$963$	0	0
$964$	−12580.0	−0.420306
$965$	−10900.0	−0.363610
$966$	0	0
$967$	−27034.0	−0.899023	−0.449511	−	0.893275i	$-0.648402\pi$
$967$	−27034.0	−0.899023	−0.449511	+	0.893275i	$0.648402\pi$
$968$	8056.00	0.267489
$969$	0	0
$970$	−11920.0	−0.394565
$971$	−14802.0	−0.489206	−0.244603	−	0.969623i	$-0.578658\pi$
$971$	−14802.0	−0.489206	−0.244603	+	0.969623i	$0.578658\pi$
$972$	0	0
$973$	12832.0	0.422790
$974$	−13516.0	−0.444641
$975$	0	0
$976$	−14512.0	−0.475940
$977$	9186.00	0.300805	0.150402	−	0.988625i	$-0.451943\pi$
$977$	9186.00	0.300805	0.150402	+	0.988625i	$0.451943\pi$
$978$	0	0
$979$	−3564.00	−0.116349
$980$	5580.00	0.181884
$981$	0	0
$982$	−29148.0	−0.947200
$983$	−31647.0	−1.02684	−0.513419	−	0.858138i	$-0.671621\pi$
$983$	−31647.0	−1.02684	−0.513419	+	0.858138i	$0.671621\pi$
$984$	0	0
$985$	12885.0	0.416802
$986$	−4680.00	−0.151158
$987$	0	0
$988$	736.000	0.0236997
$989$	11970.0	0.384857
$990$	0	0
$991$	−48823.0	−1.56500	−0.782499	−	0.622651i	$-0.786055\pi$
$991$	−48823.0	−1.56500	−0.782499	+	0.622651i	$0.786055\pi$
$992$	2720.00	0.0870565
$993$	0	0
$994$	−4320.00	−0.137849
$995$	−7060.00	−0.224942
$996$	0	0
$997$	−13066.0	−0.415050	−0.207525	−	0.978230i	$-0.566541\pi$
$997$	−13066.0	−0.415050	−0.207525	+	0.978230i	$0.566541\pi$
$998$	−25222.0	−0.799988
$999$	0	0

Twists

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

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