LMFDB - Embedded newform 1440.4.a.bk.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1440,4,Mod(1,1440)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1440.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1440, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1440 = 2^{5} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1440.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,0,0,15,0,14,0,0,0,22,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$84.9627504083$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.16773.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 17x + 18 $ x^3 - x^2 - 17*x + 18
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$4.08359$ of defining polynomial
Character	$\chi$	$=$	1440.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+5.00000 q^{5} -10.3344 q^{7} +57.3715 q^{11} -22.3003 q^{13} +106.375 q^{17} +43.7741 q^{19} -16.4428 q^{23} +25.0000 q^{25} +57.5481 q^{29} -175.855 q^{31} -51.6718 q^{35} -280.672 q^{37} -157.622 q^{41} +457.950 q^{43} +18.6412 q^{47} -236.201 q^{49} -370.446 q^{53} +286.858 q^{55} +416.374 q^{59} +867.344 q^{61} -111.501 q^{65} +37.8452 q^{67} -83.6286 q^{71} -12.1984 q^{73} -592.898 q^{77} +175.260 q^{79} +572.402 q^{83} +531.873 q^{85} +622.972 q^{89} +230.459 q^{91} +218.870 q^{95} -1215.34 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 15 q^{5} + 14 q^{7} + 22 q^{11} + 8 q^{13} + 34 q^{17} - 4 q^{19} + 176 q^{23} + 75 q^{25} - 98 q^{29} + 88 q^{31} + 70 q^{35} + 284 q^{37} + 8 q^{41} + 504 q^{43} + 280 q^{47} - 409 q^{49} + 150 q^{53}+ \cdots - 1394 q^{97}+O(q^{100}) $ 3 * q + 15 * q^5 + 14 * q^7 + 22 * q^11 + 8 * q^13 + 34 * q^17 - 4 * q^19 + 176 * q^23 + 75 * q^25 - 98 * q^29 + 88 * q^31 + 70 * q^35 + 284 * q^37 + 8 * q^41 + 504 * q^43 + 280 * q^47 - 409 * q^49 + 150 * q^53 + 110 * q^55 + 350 * q^59 + 350 * q^61 + 40 * q^65 + 804 * q^67 + 500 * q^71 - 486 * q^73 - 788 * q^77 + 1592 * q^79 - 684 * q^83 + 170 * q^85 + 668 * q^89 + 1920 * q^91 - 20 * q^95 - 1394 * q^97

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 $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	−10.3344	−0.558003	−0.279001	−	0.960291i	$-0.590003\pi$
$7$	−10.3344	−0.558003	−0.279001	+	0.960291i	$0.590003\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	57.3715	1.57256	0.786280	−	0.617870i	$-0.212004\pi$
$11$	57.3715	1.57256	0.786280	+	0.617870i	$0.212004\pi$
$12$	0	0
$13$	−22.3003	−0.475768	−0.237884	−	0.971294i	$-0.576454\pi$
$13$	−22.3003	−0.475768	−0.237884	+	0.971294i	$0.576454\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	106.375	1.51763	0.758813	−	0.651309i	$-0.225780\pi$
$17$	106.375	1.51763	0.758813	+	0.651309i	$0.225780\pi$
$18$	0	0
$19$	43.7741	0.528551	0.264275	−	0.964447i	$-0.414867\pi$
$19$	43.7741	0.528551	0.264275	+	0.964447i	$0.414867\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−16.4428	−0.149068	−0.0745339	−	0.997218i	$-0.523747\pi$
$23$	−16.4428	−0.149068	−0.0745339	+	0.997218i	$0.523747\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	57.5481	0.368497	0.184249	−	0.982880i	$-0.441015\pi$
$29$	57.5481	0.368497	0.184249	+	0.982880i	$0.441015\pi$
$30$	0	0
$31$	−175.855	−1.01885	−0.509426	−	0.860514i	$-0.670142\pi$
$31$	−175.855	−1.01885	−0.509426	+	0.860514i	$0.670142\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−51.6718	−0.249546
$36$	0	0
$37$	−280.672	−1.24709	−0.623543	−	0.781789i	$-0.714307\pi$
$37$	−280.672	−1.24709	−0.623543	+	0.781789i	$0.714307\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−157.622	−0.600402	−0.300201	−	0.953876i	$-0.597054\pi$
$41$	−157.622	−0.600402	−0.300201	+	0.953876i	$0.597054\pi$
$42$	0	0
$43$	457.950	1.62411	0.812055	−	0.583580i	$-0.198349\pi$
$43$	457.950	1.62411	0.812055	+	0.583580i	$0.198349\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	18.6412	0.0578532	0.0289266	−	0.999582i	$-0.490791\pi$
$47$	18.6412	0.0578532	0.0289266	+	0.999582i	$0.490791\pi$
$48$	0	0
$49$	−236.201	−0.688633
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−370.446	−0.960088	−0.480044	−	0.877244i	$-0.659379\pi$
$53$	−370.446	−0.960088	−0.480044	+	0.877244i	$0.659379\pi$
$54$	0	0
$55$	286.858	0.703270
$56$	0	0
$57$	0	0
$58$	0	0
$59$	416.374	0.918768	0.459384	−	0.888238i	$-0.348070\pi$
$59$	416.374	0.918768	0.459384	+	0.888238i	$0.348070\pi$
$60$	0	0
$61$	867.344	1.82052	0.910262	−	0.414032i	$-0.135880\pi$
$61$	867.344	1.82052	0.910262	+	0.414032i	$0.135880\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−111.501	−0.212770
$66$	0	0
$67$	37.8452	0.0690078	0.0345039	−	0.999405i	$-0.489015\pi$
$67$	37.8452	0.0690078	0.0345039	+	0.999405i	$0.489015\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−83.6286	−0.139787	−0.0698936	−	0.997554i	$-0.522266\pi$
$71$	−83.6286	−0.139787	−0.0698936	+	0.997554i	$0.522266\pi$
$72$	0	0
$73$	−12.1984	−0.0195578	−0.00977889	−	0.999952i	$-0.503113\pi$
$73$	−12.1984	−0.0195578	−0.00977889	+	0.999952i	$0.503113\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−592.898	−0.877493
$78$	0	0
$79$	175.260	0.249598	0.124799	−	0.992182i	$-0.460171\pi$
$79$	175.260	0.249598	0.124799	+	0.992182i	$0.460171\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	572.402	0.756980	0.378490	−	0.925605i	$-0.376443\pi$
$83$	572.402	0.756980	0.378490	+	0.925605i	$0.376443\pi$
$84$	0	0
$85$	531.873	0.678703
$86$	0	0
$87$	0	0
$88$	0	0
$89$	622.972	0.741965	0.370983	−	0.928640i	$-0.379021\pi$
$89$	622.972	0.741965	0.370983	+	0.928640i	$0.379021\pi$
$90$	0	0
$91$	230.459	0.265480
$92$	0	0
$93$	0	0
$94$	0	0
$95$	218.870	0.236375
$96$	0	0
$97$	−1215.34	−1.27216	−0.636080	−	0.771623i	$-0.719445\pi$
$97$	−1215.34	−1.27216	−0.636080	+	0.771623i	$0.719445\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1557.64	1.53456	0.767282	−	0.641309i	$-0.221608\pi$
$101$	1557.64	1.53456	0.767282	+	0.641309i	$0.221608\pi$
$102$	0	0
$103$	1692.90	1.61948	0.809742	−	0.586786i	$-0.199607\pi$
$103$	1692.90	1.61948	0.809742	+	0.586786i	$0.199607\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1746.09	−1.57758	−0.788790	−	0.614663i	$-0.789292\pi$
$107$	−1746.09	−1.57758	−0.788790	+	0.614663i	$0.789292\pi$
$108$	0	0
$109$	−213.939	−0.187997	−0.0939984	−	0.995572i	$-0.529965\pi$
$109$	−213.939	−0.187997	−0.0939984	+	0.995572i	$0.529965\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−245.278	−0.204193	−0.102096	−	0.994775i	$-0.532555\pi$
$113$	−245.278	−0.204193	−0.102096	+	0.994775i	$0.532555\pi$
$114$	0	0
$115$	−82.2139	−0.0666651
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1099.31	−0.846839
$120$	0	0
$121$	1960.49	1.47295
$122$	0	0
$123$	0	0
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	1234.24	0.862372	0.431186	−	0.902263i	$-0.358095\pi$
$127$	1234.24	0.862372	0.431186	+	0.902263i	$0.358095\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	358.201	0.238902	0.119451	−	0.992840i	$-0.461887\pi$
$131$	358.201	0.238902	0.119451	+	0.992840i	$0.461887\pi$
$132$	0	0
$133$	−452.377	−0.294933
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2331.07	1.45370	0.726851	−	0.686796i	$-0.240983\pi$
$137$	2331.07	1.45370	0.726851	+	0.686796i	$0.240983\pi$
$138$	0	0
$139$	2213.32	1.35058	0.675291	−	0.737551i	$-0.264018\pi$
$139$	2213.32	1.35058	0.675291	+	0.737551i	$0.264018\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−1279.40	−0.748174
$144$	0	0
$145$	287.741	0.164797
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2533.98	−1.39323	−0.696617	−	0.717443i	$-0.745312\pi$
$149$	−2533.98	−1.39323	−0.696617	+	0.717443i	$0.745312\pi$
$150$	0	0
$151$	3508.98	1.89110	0.945552	−	0.325471i	$-0.105523\pi$
$151$	3508.98	1.89110	0.945552	+	0.325471i	$0.105523\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−879.273	−0.455644
$156$	0	0
$157$	−520.132	−0.264402	−0.132201	−	0.991223i	$-0.542204\pi$
$157$	−520.132	−0.264402	−0.132201	+	0.991223i	$0.542204\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	169.926	0.0831802
$162$	0	0
$163$	−727.388	−0.349530	−0.174765	−	0.984610i	$-0.555917\pi$
$163$	−727.388	−0.349530	−0.174765	+	0.984610i	$0.555917\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2812.14	1.30305	0.651527	−	0.758626i	$-0.274129\pi$
$167$	2812.14	1.30305	0.651527	+	0.758626i	$0.274129\pi$
$168$	0	0
$169$	−1699.70	−0.773645
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−2375.56	−1.04399	−0.521996	−	0.852948i	$-0.674813\pi$
$173$	−2375.56	−1.04399	−0.521996	+	0.852948i	$0.674813\pi$
$174$	0	0
$175$	−258.359	−0.111601
$176$	0	0
$177$	0	0
$178$	0	0
$179$	1677.63	0.700512	0.350256	−	0.936654i	$-0.386095\pi$
$179$	1677.63	0.700512	0.350256	+	0.936654i	$0.386095\pi$
$180$	0	0
$181$	−1671.89	−0.686580	−0.343290	−	0.939230i	$-0.611541\pi$
$181$	−1671.89	−0.686580	−0.343290	+	0.939230i	$0.611541\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−1403.36	−0.557714
$186$	0	0
$187$	6102.87	2.38656
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1270.29	0.481228	0.240614	−	0.970621i	$-0.422651\pi$
$191$	1270.29	0.481228	0.240614	+	0.970621i	$0.422651\pi$
$192$	0	0
$193$	482.865	0.180090	0.0900450	−	0.995938i	$-0.471299\pi$
$193$	482.865	0.180090	0.0900450	+	0.995938i	$0.471299\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1767.21	−0.639128	−0.319564	−	0.947565i	$-0.603536\pi$
$197$	−1767.21	−0.639128	−0.319564	+	0.947565i	$0.603536\pi$
$198$	0	0
$199$	2817.00	1.00348	0.501738	−	0.865020i	$-0.332694\pi$
$199$	2817.00	1.00348	0.501738	+	0.865020i	$0.332694\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−594.723	−0.205623
$204$	0	0
$205$	−788.112	−0.268508
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2511.39	0.831178
$210$	0	0
$211$	4281.22	1.39683	0.698415	−	0.715693i	$-0.253889\pi$
$211$	4281.22	1.39683	0.698415	+	0.715693i	$0.253889\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	2289.75	0.726324
$216$	0	0
$217$	1817.34	0.568522
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−2372.18	−0.722037
$222$	0	0
$223$	−908.793	−0.272903	−0.136451	−	0.990647i	$-0.543570\pi$
$223$	−908.793	−0.272903	−0.136451	+	0.990647i	$0.543570\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−4177.76	−1.22153	−0.610766	−	0.791811i	$-0.709138\pi$
$227$	−4177.76	−1.22153	−0.610766	+	0.791811i	$0.709138\pi$
$228$	0	0
$229$	2714.79	0.783398	0.391699	−	0.920093i	$-0.371888\pi$
$229$	2714.79	0.783398	0.391699	+	0.920093i	$0.371888\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	2783.28	0.782571	0.391285	−	0.920269i	$-0.372031\pi$
$233$	2783.28	0.782571	0.391285	+	0.920269i	$0.372031\pi$
$234$	0	0
$235$	93.2061	0.0258727
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−4573.36	−1.23777	−0.618883	−	0.785483i	$-0.712415\pi$
$239$	−4573.36	−1.23777	−0.618883	+	0.785483i	$0.712415\pi$
$240$	0	0
$241$	−3901.38	−1.04278	−0.521390	−	0.853318i	$-0.674586\pi$
$241$	−3901.38	−1.04278	−0.521390	+	0.853318i	$0.674586\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1181.01	−0.307966
$246$	0	0
$247$	−976.173	−0.251467
$248$	0	0
$249$	0	0
$250$	0	0
$251$	3014.39	0.758035	0.379018	−	0.925389i	$-0.376262\pi$
$251$	3014.39	0.758035	0.379018	+	0.925389i	$0.376262\pi$
$252$	0	0
$253$	−943.348	−0.234418
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1980.46	0.480691	0.240345	−	0.970687i	$-0.422739\pi$
$257$	1980.46	0.480691	0.240345	+	0.970687i	$0.422739\pi$
$258$	0	0
$259$	2900.56	0.695878
$260$	0	0
$261$	0	0
$262$	0	0
$263$	1212.21	0.284213	0.142107	−	0.989851i	$-0.454612\pi$
$263$	1212.21	0.284213	0.142107	+	0.989851i	$0.454612\pi$
$264$	0	0
$265$	−1852.23	−0.429364
$266$	0	0
$267$	0	0
$268$	0	0
$269$	1852.56	0.419898	0.209949	−	0.977712i	$-0.432670\pi$
$269$	1852.56	0.419898	0.209949	+	0.977712i	$0.432670\pi$
$270$	0	0
$271$	4855.55	1.08839	0.544194	−	0.838959i	$-0.316835\pi$
$271$	4855.55	1.08839	0.544194	+	0.838959i	$0.316835\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1434.29	0.314512
$276$	0	0
$277$	5676.17	1.23122	0.615611	−	0.788051i	$-0.288909\pi$
$277$	5676.17	1.23122	0.615611	+	0.788051i	$0.288909\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	8152.07	1.73065	0.865323	−	0.501214i	$-0.167113\pi$
$281$	8152.07	1.73065	0.865323	+	0.501214i	$0.167113\pi$
$282$	0	0
$283$	8396.78	1.76373	0.881867	−	0.471498i	$-0.156287\pi$
$283$	8396.78	1.76373	0.881867	+	0.471498i	$0.156287\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1628.93	0.335026
$288$	0	0
$289$	6402.56	1.30319
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−2103.84	−0.419480	−0.209740	−	0.977757i	$-0.567262\pi$
$293$	−2103.84	−0.419480	−0.209740	+	0.977757i	$0.567262\pi$
$294$	0	0
$295$	2081.87	0.410885
$296$	0	0
$297$	0	0
$298$	0	0
$299$	366.679	0.0709216
$300$	0	0
$301$	−4732.62	−0.906258
$302$	0	0
$303$	0	0
$304$	0	0
$305$	4336.72	0.814163
$306$	0	0
$307$	1118.19	0.207879	0.103939	−	0.994584i	$-0.466855\pi$
$307$	1118.19	0.207879	0.103939	+	0.994584i	$0.466855\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	1525.00	0.278054	0.139027	−	0.990289i	$-0.455602\pi$
$311$	1525.00	0.278054	0.139027	+	0.990289i	$0.455602\pi$
$312$	0	0
$313$	4812.52	0.869072	0.434536	−	0.900655i	$-0.356912\pi$
$313$	4812.52	0.869072	0.434536	+	0.900655i	$0.356912\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−2600.96	−0.460834	−0.230417	−	0.973092i	$-0.574009\pi$
$317$	−2600.96	−0.460834	−0.230417	+	0.973092i	$0.574009\pi$
$318$	0	0
$319$	3301.63	0.579484
$320$	0	0
$321$	0	0
$322$	0	0
$323$	4656.45	0.802142
$324$	0	0
$325$	−557.507	−0.0951535
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−192.645	−0.0322823
$330$	0	0
$331$	−7973.83	−1.32411	−0.662056	−	0.749454i	$-0.730316\pi$
$331$	−7973.83	−1.32411	−0.662056	+	0.749454i	$0.730316\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	189.226	0.0308612
$336$	0	0
$337$	4760.22	0.769453	0.384727	−	0.923031i	$-0.374296\pi$
$337$	4760.22	0.769453	0.384727	+	0.923031i	$0.374296\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−10089.0	−1.60221
$342$	0	0
$343$	5985.67	0.942262
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−1446.88	−0.223840	−0.111920	−	0.993717i	$-0.535700\pi$
$347$	−1446.88	−0.223840	−0.111920	+	0.993717i	$0.535700\pi$
$348$	0	0
$349$	−6332.39	−0.971247	−0.485624	−	0.874168i	$-0.661407\pi$
$349$	−6332.39	−0.971247	−0.485624	+	0.874168i	$0.661407\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−322.985	−0.0486990	−0.0243495	−	0.999704i	$-0.507751\pi$
$353$	−322.985	−0.0486990	−0.0243495	+	0.999704i	$0.507751\pi$
$354$	0	0
$355$	−418.143	−0.0625147
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−1921.57	−0.282497	−0.141249	−	0.989974i	$-0.545112\pi$
$359$	−1921.57	−0.282497	−0.141249	+	0.989974i	$0.545112\pi$
$360$	0	0
$361$	−4942.83	−0.720634
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−60.9921	−0.00874650
$366$	0	0
$367$	4855.18	0.690568	0.345284	−	0.938498i	$-0.387783\pi$
$367$	4855.18	0.690568	0.345284	+	0.938498i	$0.387783\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	3828.32	0.535732
$372$	0	0
$373$	−206.627	−0.0286830	−0.0143415	−	0.999897i	$-0.504565\pi$
$373$	−206.627	−0.0286830	−0.0143415	+	0.999897i	$0.504565\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−1283.34	−0.175319
$378$	0	0
$379$	−4021.99	−0.545107	−0.272554	−	0.962141i	$-0.587868\pi$
$379$	−4021.99	−0.545107	−0.272554	+	0.962141i	$0.587868\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−9435.15	−1.25878	−0.629391	−	0.777088i	$-0.716696\pi$
$383$	−9435.15	−1.25878	−0.629391	+	0.777088i	$0.716696\pi$
$384$	0	0
$385$	−2964.49	−0.392427
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−11462.1	−1.49397	−0.746984	−	0.664842i	$-0.768499\pi$
$389$	−11462.1	−1.49397	−0.746984	+	0.664842i	$0.768499\pi$
$390$	0	0
$391$	−1749.10	−0.226229
$392$	0	0
$393$	0	0
$394$	0	0
$395$	876.299	0.111624
$396$	0	0
$397$	−620.723	−0.0784716	−0.0392358	−	0.999230i	$-0.512492\pi$
$397$	−620.723	−0.0784716	−0.0392358	+	0.999230i	$0.512492\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2466.48	0.307157	0.153579	−	0.988136i	$-0.450920\pi$
$401$	2466.48	0.307157	0.153579	+	0.988136i	$0.450920\pi$
$402$	0	0
$403$	3921.60	0.484737
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−16102.6	−1.96112
$408$	0	0
$409$	3155.44	0.381483	0.190742	−	0.981640i	$-0.438911\pi$
$409$	3155.44	0.381483	0.190742	+	0.981640i	$0.438911\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−4302.96	−0.512675
$414$	0	0
$415$	2862.01	0.338532
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−5651.57	−0.658944	−0.329472	−	0.944165i	$-0.606871\pi$
$419$	−5651.57	−0.658944	−0.329472	+	0.944165i	$0.606871\pi$
$420$	0	0
$421$	3403.15	0.393965	0.196983	−	0.980407i	$-0.436886\pi$
$421$	3403.15	0.393965	0.196983	+	0.980407i	$0.436886\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	2659.36	0.303525
$426$	0	0
$427$	−8963.44	−1.01586
$428$	0	0
$429$	0	0
$430$	0	0
$431$	13949.4	1.55898	0.779488	−	0.626418i	$-0.215479\pi$
$431$	13949.4	1.55898	0.779488	+	0.626418i	$0.215479\pi$
$432$	0	0
$433$	−9477.04	−1.05182	−0.525909	−	0.850541i	$-0.676275\pi$
$433$	−9477.04	−1.05182	−0.525909	+	0.850541i	$0.676275\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−719.768	−0.0787899
$438$	0	0
$439$	−11944.7	−1.29860	−0.649302	−	0.760530i	$-0.724939\pi$
$439$	−11944.7	−1.29860	−0.649302	+	0.760530i	$0.724939\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−13457.6	−1.44332	−0.721660	−	0.692248i	$-0.756620\pi$
$443$	−13457.6	−1.44332	−0.721660	+	0.692248i	$0.756620\pi$
$444$	0	0
$445$	3114.86	0.331817
$446$	0	0
$447$	0	0
$448$	0	0
$449$	13366.9	1.40496	0.702478	−	0.711706i	$-0.252077\pi$
$449$	13366.9	1.40496	0.702478	+	0.711706i	$0.252077\pi$
$450$	0	0
$451$	−9043.04	−0.944169
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1152.29	0.118726
$456$	0	0
$457$	−11634.1	−1.19086	−0.595429	−	0.803408i	$-0.703018\pi$
$457$	−11634.1	−1.19086	−0.595429	+	0.803408i	$0.703018\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−13499.4	−1.36384	−0.681921	−	0.731426i	$-0.738855\pi$
$461$	−13499.4	−1.36384	−0.681921	+	0.731426i	$0.738855\pi$
$462$	0	0
$463$	−3591.21	−0.360470	−0.180235	−	0.983624i	$-0.557686\pi$
$463$	−3591.21	−0.360470	−0.180235	+	0.983624i	$0.557686\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−7570.42	−0.750144	−0.375072	−	0.926996i	$-0.622382\pi$
$467$	−7570.42	−0.750144	−0.375072	+	0.926996i	$0.622382\pi$
$468$	0	0
$469$	−391.106	−0.0385066
$470$	0	0
$471$	0	0
$472$	0	0
$473$	26273.3	2.55401
$474$	0	0
$475$	1094.35	0.105710
$476$	0	0
$477$	0	0
$478$	0	0
$479$	6459.91	0.616202	0.308101	−	0.951354i	$-0.400307\pi$
$479$	6459.91	0.616202	0.308101	+	0.951354i	$0.400307\pi$
$480$	0	0
$481$	6259.06	0.593323
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−6076.72	−0.568927
$486$	0	0
$487$	−11097.4	−1.03259	−0.516294	−	0.856411i	$-0.672689\pi$
$487$	−11097.4	−1.03259	−0.516294	+	0.856411i	$0.672689\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	17047.6	1.56690	0.783450	−	0.621455i	$-0.213458\pi$
$491$	17047.6	1.56690	0.783450	+	0.621455i	$0.213458\pi$
$492$	0	0
$493$	6121.66	0.559241
$494$	0	0
$495$	0	0
$496$	0	0
$497$	864.248	0.0780017
$498$	0	0
$499$	13993.7	1.25540	0.627699	−	0.778456i	$-0.283997\pi$
$499$	13993.7	1.25540	0.627699	+	0.778456i	$0.283997\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	19540.5	1.73214	0.866071	−	0.499920i	$-0.166637\pi$
$503$	19540.5	1.73214	0.866071	+	0.499920i	$0.166637\pi$
$504$	0	0
$505$	7788.20	0.686278
$506$	0	0
$507$	0	0
$508$	0	0
$509$	6787.91	0.591098	0.295549	−	0.955328i	$-0.404497\pi$
$509$	6787.91	0.591098	0.295549	+	0.955328i	$0.404497\pi$
$510$	0	0
$511$	126.063	0.0109133
$512$	0	0
$513$	0	0
$514$	0	0
$515$	8464.52	0.724255
$516$	0	0
$517$	1069.47	0.0909777
$518$	0	0
$519$	0	0
$520$	0	0
$521$	10006.7	0.841465	0.420732	−	0.907185i	$-0.361773\pi$
$521$	10006.7	0.841465	0.420732	+	0.907185i	$0.361773\pi$
$522$	0	0
$523$	−21487.6	−1.79653	−0.898265	−	0.439454i	$-0.855172\pi$
$523$	−21487.6	−1.79653	−0.898265	+	0.439454i	$0.855172\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−18706.5	−1.54624
$528$	0	0
$529$	−11896.6	−0.977779
$530$	0	0
$531$	0	0
$532$	0	0
$533$	3515.02	0.285652
$534$	0	0
$535$	−8730.46	−0.705515
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−13551.2	−1.08292
$540$	0	0
$541$	13111.1	1.04194	0.520971	−	0.853574i	$-0.325570\pi$
$541$	13111.1	1.04194	0.520971	+	0.853574i	$0.325570\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−1069.70	−0.0840747
$546$	0	0
$547$	4242.34	0.331608	0.165804	−	0.986159i	$-0.446978\pi$
$547$	4242.34	0.331608	0.165804	+	0.986159i	$0.446978\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	2519.12	0.194769
$552$	0	0
$553$	−1811.20	−0.139277
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−6390.85	−0.486156	−0.243078	−	0.970007i	$-0.578157\pi$
$557$	−6390.85	−0.486156	−0.243078	+	0.970007i	$0.578157\pi$
$558$	0	0
$559$	−10212.4	−0.772699
$560$	0	0
$561$	0	0
$562$	0	0
$563$	10303.6	0.771306	0.385653	−	0.922644i	$-0.373976\pi$
$563$	10303.6	0.771306	0.385653	+	0.922644i	$0.373976\pi$
$564$	0	0
$565$	−1226.39	−0.0913178
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−9526.59	−0.701890	−0.350945	−	0.936396i	$-0.614140\pi$
$569$	−9526.59	−0.701890	−0.350945	+	0.936396i	$0.614140\pi$
$570$	0	0
$571$	13652.4	1.00058	0.500292	−	0.865857i	$-0.333226\pi$
$571$	13652.4	1.00058	0.500292	+	0.865857i	$0.333226\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−411.070	−0.0298136
$576$	0	0
$577$	−20645.6	−1.48958	−0.744790	−	0.667299i	$-0.767450\pi$
$577$	−20645.6	−1.48958	−0.744790	+	0.667299i	$0.767450\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−5915.41	−0.422397
$582$	0	0
$583$	−21253.1	−1.50980
$584$	0	0
$585$	0	0
$586$	0	0
$587$	5652.57	0.397456	0.198728	−	0.980055i	$-0.436319\pi$
$587$	5652.57	0.397456	0.198728	+	0.980055i	$0.436319\pi$
$588$	0	0
$589$	−7697.87	−0.538515
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−7818.01	−0.541395	−0.270697	−	0.962664i	$-0.587254\pi$
$593$	−7818.01	−0.541395	−0.270697	+	0.962664i	$0.587254\pi$
$594$	0	0
$595$	−5496.57	−0.378718
$596$	0	0
$597$	0	0
$598$	0	0
$599$	7086.03	0.483351	0.241676	−	0.970357i	$-0.422303\pi$
$599$	7086.03	0.483351	0.241676	+	0.970357i	$0.422303\pi$
$600$	0	0
$601$	26401.7	1.79193	0.895963	−	0.444128i	$-0.146487\pi$
$601$	26401.7	1.79193	0.895963	+	0.444128i	$0.146487\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	9802.46	0.658722
$606$	0	0
$607$	−13149.4	−0.879268	−0.439634	−	0.898177i	$-0.644892\pi$
$607$	−13149.4	−0.879268	−0.439634	+	0.898177i	$0.644892\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−415.704	−0.0275247
$612$	0	0
$613$	−4922.18	−0.324315	−0.162157	−	0.986765i	$-0.551845\pi$
$613$	−4922.18	−0.324315	−0.162157	+	0.986765i	$0.551845\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	18351.1	1.19739	0.598694	−	0.800978i	$-0.295686\pi$
$617$	18351.1	1.19739	0.598694	+	0.800978i	$0.295686\pi$
$618$	0	0
$619$	−30123.7	−1.95601	−0.978007	−	0.208574i	$-0.933118\pi$
$619$	−30123.7	−1.95601	−0.978007	+	0.208574i	$0.933118\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−6438.02	−0.414019
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−29856.4	−1.89261
$630$	0	0
$631$	−16825.6	−1.06152	−0.530758	−	0.847523i	$-0.678093\pi$
$631$	−16825.6	−1.06152	−0.530758	+	0.847523i	$0.678093\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	6171.21	0.385664
$636$	0	0
$637$	5267.35	0.327629
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−11519.6	−0.709821	−0.354910	−	0.934900i	$-0.615489\pi$
$641$	−11519.6	−0.709821	−0.354910	+	0.934900i	$0.615489\pi$
$642$	0	0
$643$	−17883.5	−1.09682	−0.548411	−	0.836209i	$-0.684767\pi$
$643$	−17883.5	−1.09682	−0.548411	+	0.836209i	$0.684767\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−28208.5	−1.71405	−0.857025	−	0.515275i	$-0.827690\pi$
$647$	−28208.5	−1.71405	−0.857025	+	0.515275i	$0.827690\pi$
$648$	0	0
$649$	23888.0	1.44482
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−15006.4	−0.899305	−0.449652	−	0.893204i	$-0.648452\pi$
$653$	−15006.4	−0.899305	−0.449652	+	0.893204i	$0.648452\pi$
$654$	0	0
$655$	1791.01	0.106840
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−23076.6	−1.36409	−0.682047	−	0.731308i	$-0.738910\pi$
$659$	−23076.6	−1.36409	−0.682047	+	0.731308i	$0.738910\pi$
$660$	0	0
$661$	1476.85	0.0869030	0.0434515	−	0.999056i	$-0.486165\pi$
$661$	1476.85	0.0869030	0.0434515	+	0.999056i	$0.486165\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−2261.88	−0.131898
$666$	0	0
$667$	−946.252	−0.0549311
$668$	0	0
$669$	0	0
$670$	0	0
$671$	49760.8	2.86289
$672$	0	0
$673$	23118.1	1.32413	0.662064	−	0.749447i	$-0.269681\pi$
$673$	23118.1	1.32413	0.662064	+	0.749447i	$0.269681\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−22723.4	−1.29000	−0.645000	−	0.764182i	$-0.723143\pi$
$677$	−22723.4	−1.29000	−0.645000	+	0.764182i	$0.723143\pi$
$678$	0	0
$679$	12559.8	0.709869
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−21993.0	−1.23212	−0.616060	−	0.787699i	$-0.711272\pi$
$683$	−21993.0	−1.23212	−0.616060	+	0.787699i	$0.711272\pi$
$684$	0	0
$685$	11655.4	0.650115
$686$	0	0
$687$	0	0
$688$	0	0
$689$	8261.04	0.456779
$690$	0	0
$691$	−12542.9	−0.690527	−0.345263	−	0.938506i	$-0.612210\pi$
$691$	−12542.9	−0.690527	−0.345263	+	0.938506i	$0.612210\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	11066.6	0.603999
$696$	0	0
$697$	−16767.0	−0.911186
$698$	0	0
$699$	0	0
$700$	0	0
$701$	29128.1	1.56940	0.784702	−	0.619873i	$-0.212816\pi$
$701$	29128.1	1.56940	0.784702	+	0.619873i	$0.212816\pi$
$702$	0	0
$703$	−12286.2	−0.659148
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−16097.2	−0.856292
$708$	0	0
$709$	−29067.0	−1.53968	−0.769842	−	0.638235i	$-0.779665\pi$
$709$	−29067.0	−1.53968	−0.769842	+	0.638235i	$0.779665\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	2891.54	0.151878
$714$	0	0
$715$	−6397.00	−0.334593
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−1699.49	−0.0881505	−0.0440753	−	0.999028i	$-0.514034\pi$
$719$	−1699.49	−0.0881505	−0.0440753	+	0.999028i	$0.514034\pi$
$720$	0	0
$721$	−17495.1	−0.903677
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1438.70	0.0736995
$726$	0	0
$727$	6029.01	0.307570	0.153785	−	0.988104i	$-0.450854\pi$
$727$	6029.01	0.307570	0.153785	+	0.988104i	$0.450854\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	48714.3	2.46479
$732$	0	0
$733$	−24004.9	−1.20960	−0.604802	−	0.796376i	$-0.706748\pi$
$733$	−24004.9	−1.20960	−0.604802	+	0.796376i	$0.706748\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	2171.24	0.108519
$738$	0	0
$739$	−31261.8	−1.55614	−0.778068	−	0.628180i	$-0.783800\pi$
$739$	−31261.8	−1.55614	−0.778068	+	0.628180i	$0.783800\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−7595.49	−0.375036	−0.187518	−	0.982261i	$-0.560044\pi$
$743$	−7595.49	−0.375036	−0.187518	+	0.982261i	$0.560044\pi$
$744$	0	0
$745$	−12669.9	−0.623073
$746$	0	0
$747$	0	0
$748$	0	0
$749$	18044.7	0.880294
$750$	0	0
$751$	−15968.8	−0.775914	−0.387957	−	0.921677i	$-0.626819\pi$
$751$	−15968.8	−0.775914	−0.387957	+	0.921677i	$0.626819\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	17544.9	0.845727
$756$	0	0
$757$	10683.8	0.512956	0.256478	−	0.966550i	$-0.417438\pi$
$757$	10683.8	0.512956	0.256478	+	0.966550i	$0.417438\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	592.024	0.0282009	0.0141004	−	0.999901i	$-0.495512\pi$
$761$	592.024	0.0282009	0.0141004	+	0.999901i	$0.495512\pi$
$762$	0	0
$763$	2210.92	0.104903
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−9285.25	−0.437120
$768$	0	0
$769$	14173.1	0.664623	0.332312	−	0.943170i	$-0.392171\pi$
$769$	14173.1	0.664623	0.332312	+	0.943170i	$0.392171\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	5085.16	0.236611	0.118306	−	0.992977i	$-0.462254\pi$
$773$	5085.16	0.236611	0.118306	+	0.992977i	$0.462254\pi$
$774$	0	0
$775$	−4396.36	−0.203770
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−6899.78	−0.317343
$780$	0	0
$781$	−4797.90	−0.219824
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−2600.66	−0.118244
$786$	0	0
$787$	−33276.2	−1.50720	−0.753602	−	0.657331i	$-0.771685\pi$
$787$	−33276.2	−1.50720	−0.753602	+	0.657331i	$0.771685\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	2534.79	0.113940
$792$	0	0
$793$	−19342.0	−0.866147
$794$	0	0
$795$	0	0
$796$	0	0
$797$	2060.07	0.0915575	0.0457788	−	0.998952i	$-0.485423\pi$
$797$	2060.07	0.0915575	0.0457788	+	0.998952i	$0.485423\pi$
$798$	0	0
$799$	1982.95	0.0877995
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−699.842	−0.0307558
$804$	0	0
$805$	849.628	0.0371993
$806$	0	0
$807$	0	0
$808$	0	0
$809$	36723.3	1.59595	0.797973	−	0.602693i	$-0.205906\pi$
$809$	36723.3	1.59595	0.797973	+	0.602693i	$0.205906\pi$
$810$	0	0
$811$	−2568.35	−0.111205	−0.0556023	−	0.998453i	$-0.517708\pi$
$811$	−2568.35	−0.111205	−0.0556023	+	0.998453i	$0.517708\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−3636.94	−0.156315
$816$	0	0
$817$	20046.3	0.858425
$818$	0	0
$819$	0	0
$820$	0	0
$821$	10030.5	0.426393	0.213196	−	0.977009i	$-0.431613\pi$
$821$	10030.5	0.426393	0.213196	+	0.977009i	$0.431613\pi$
$822$	0	0
$823$	34953.3	1.48043	0.740216	−	0.672370i	$-0.234723\pi$
$823$	34953.3	1.48043	0.740216	+	0.672370i	$0.234723\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−14173.8	−0.595976	−0.297988	−	0.954570i	$-0.596316\pi$
$827$	−14173.8	−0.595976	−0.297988	+	0.954570i	$0.596316\pi$
$828$	0	0
$829$	15693.6	0.657490	0.328745	−	0.944419i	$-0.393374\pi$
$829$	15693.6	0.657490	0.328745	+	0.944419i	$0.393374\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−25125.8	−1.04509
$834$	0	0
$835$	14060.7	0.582743
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−42935.1	−1.76673	−0.883363	−	0.468689i	$-0.844726\pi$
$839$	−42935.1	−1.76673	−0.883363	+	0.468689i	$0.844726\pi$
$840$	0	0
$841$	−21077.2	−0.864210
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−8498.49	−0.345985
$846$	0	0
$847$	−20260.4	−0.821908
$848$	0	0
$849$	0	0
$850$	0	0
$851$	4615.03	0.185900
$852$	0	0
$853$	22098.1	0.887017	0.443509	−	0.896270i	$-0.353734\pi$
$853$	22098.1	0.887017	0.443509	+	0.896270i	$0.353734\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	27366.0	1.09079	0.545393	−	0.838180i	$-0.316380\pi$
$857$	27366.0	1.09079	0.545393	+	0.838180i	$0.316380\pi$
$858$	0	0
$859$	21030.1	0.835316	0.417658	−	0.908604i	$-0.362851\pi$
$859$	21030.1	0.835316	0.417658	+	0.908604i	$0.362851\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−17303.3	−0.682516	−0.341258	−	0.939970i	$-0.610853\pi$
$863$	−17303.3	−0.682516	−0.341258	+	0.939970i	$0.610853\pi$
$864$	0	0
$865$	−11877.8	−0.466888
$866$	0	0
$867$	0	0
$868$	0	0
$869$	10054.9	0.392509
$870$	0	0
$871$	−843.957	−0.0328317
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−1291.79	−0.0499093
$876$	0	0
$877$	13678.2	0.526659	0.263330	−	0.964706i	$-0.415179\pi$
$877$	13678.2	0.526659	0.263330	+	0.964706i	$0.415179\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−21656.6	−0.828183	−0.414092	−	0.910235i	$-0.635901\pi$
$881$	−21656.6	−0.828183	−0.414092	+	0.910235i	$0.635901\pi$
$882$	0	0
$883$	2409.29	0.0918222	0.0459111	−	0.998946i	$-0.485381\pi$
$883$	2409.29	0.0918222	0.0459111	+	0.998946i	$0.485381\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−30291.4	−1.14666	−0.573330	−	0.819325i	$-0.694349\pi$
$887$	−30291.4	−1.14666	−0.573330	+	0.819325i	$0.694349\pi$
$888$	0	0
$889$	−12755.1	−0.481206
$890$	0	0
$891$	0	0
$892$	0	0
$893$	816.002	0.0305783
$894$	0	0
$895$	8388.13	0.313279
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−10120.1	−0.375444
$900$	0	0
$901$	−39406.0	−1.45705
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−8359.47	−0.307048
$906$	0	0
$907$	−8343.45	−0.305446	−0.152723	−	0.988269i	$-0.548804\pi$
$907$	−8343.45	−0.305446	−0.152723	+	0.988269i	$0.548804\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	25369.4	0.922641	0.461320	−	0.887234i	$-0.347376\pi$
$911$	25369.4	0.922641	0.461320	+	0.887234i	$0.347376\pi$
$912$	0	0
$913$	32839.6	1.19040
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−3701.78	−0.133308
$918$	0	0
$919$	−37432.5	−1.34362	−0.671808	−	0.740726i	$-0.734482\pi$
$919$	−37432.5	−1.34362	−0.671808	+	0.740726i	$0.734482\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	1864.94	0.0665062
$924$	0	0
$925$	−7016.80	−0.249417
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−41578.0	−1.46839	−0.734193	−	0.678941i	$-0.762439\pi$
$929$	−41578.0	−1.46839	−0.734193	+	0.678941i	$0.762439\pi$
$930$	0	0
$931$	−10339.5	−0.363977
$932$	0	0
$933$	0	0
$934$	0	0
$935$	30514.4	1.06730
$936$	0	0
$937$	16023.2	0.558649	0.279325	−	0.960197i	$-0.409889\pi$
$937$	16023.2	0.558649	0.279325	+	0.960197i	$0.409889\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−3075.17	−0.106533	−0.0532665	−	0.998580i	$-0.516963\pi$
$941$	−3075.17	−0.106533	−0.0532665	+	0.998580i	$0.516963\pi$
$942$	0	0
$943$	2591.75	0.0895006
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−46724.7	−1.60332	−0.801662	−	0.597778i	$-0.796050\pi$
$947$	−46724.7	−1.60332	−0.801662	+	0.597778i	$0.796050\pi$
$948$	0	0
$949$	272.028	0.00930496
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−5521.52	−0.187681	−0.0938403	−	0.995587i	$-0.529914\pi$
$953$	−5521.52	−0.187681	−0.0938403	+	0.995587i	$0.529914\pi$
$954$	0	0
$955$	6351.43	0.215212
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−24090.1	−0.811169
$960$	0	0
$961$	1133.82	0.0380593
$962$	0	0
$963$	0	0
$964$	0	0
$965$	2414.33	0.0805387
$966$	0	0
$967$	41158.9	1.36875	0.684374	−	0.729131i	$-0.260075\pi$
$967$	41158.9	1.36875	0.684374	+	0.729131i	$0.260075\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	54672.7	1.80693	0.903466	−	0.428660i	$-0.141014\pi$
$971$	54672.7	1.80693	0.903466	+	0.428660i	$0.141014\pi$
$972$	0	0
$973$	−22873.2	−0.753629
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−44517.0	−1.45775	−0.728876	−	0.684646i	$-0.759957\pi$
$977$	−44517.0	−1.45775	−0.728876	+	0.684646i	$0.759957\pi$
$978$	0	0
$979$	35740.9	1.16679
$980$	0	0
$981$	0	0
$982$	0	0
$983$	33117.6	1.07456	0.537278	−	0.843405i	$-0.319453\pi$
$983$	33117.6	1.07456	0.537278	+	0.843405i	$0.319453\pi$
$984$	0	0
$985$	−8836.03	−0.285827
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−7529.98	−0.242103
$990$	0	0
$991$	−54402.1	−1.74383	−0.871917	−	0.489653i	$-0.837123\pi$
$991$	−54402.1	−1.74383	−0.871917	+	0.489653i	$0.837123\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	14085.0	0.448768
$996$	0	0
$997$	55607.7	1.76641	0.883207	−	0.468984i	$-0.155380\pi$
$997$	55607.7	1.76641	0.883207	+	0.468984i	$0.155380\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1440.4.a.bk.1.1	yes	3
3.2	odd	2		1440.4.a.bi.1.1	yes	3
4.3	odd	2		1440.4.a.bj.1.3	yes	3
12.11	even	2		1440.4.a.bh.1.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1440.4.a.bh.1.3	✓	3	12.11	even	2
1440.4.a.bi.1.1	yes	3	3.2	odd	2
1440.4.a.bj.1.3	yes	3	4.3	odd	2
1440.4.a.bk.1.1	yes	3	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1440.a (trivial)

\(f(q)\)	\(=\)	\(q+5.00000 q^{5} -10.3344 q^{7} +57.3715 q^{11} -22.3003 q^{13} +106.375 q^{17} +43.7741 q^{19} -16.4428 q^{23} +25.0000 q^{25} +57.5481 q^{29} -175.855 q^{31} -51.6718 q^{35} -280.672 q^{37} -157.622 q^{41} +457.950 q^{43} +18.6412 q^{47} -236.201 q^{49} -370.446 q^{53} +286.858 q^{55} +416.374 q^{59} +867.344 q^{61} -111.501 q^{65} +37.8452 q^{67} -83.6286 q^{71} -12.1984 q^{73} -592.898 q^{77} +175.260 q^{79} +572.402 q^{83} +531.873 q^{85} +622.972 q^{89} +230.459 q^{91} +218.870 q^{95} -1215.34 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 15 q^{5} + 14 q^{7} + 22 q^{11} + 8 q^{13} + 34 q^{17} - 4 q^{19} + 176 q^{23} + 75 q^{25} - 98 q^{29} + 88 q^{31} + 70 q^{35} + 284 q^{37} + 8 q^{41} + 504 q^{43} + 280 q^{47} - 409 q^{49} + 150 q^{53}+ \cdots - 1394 q^{97}+O(q^{100}) \) 3 * q + 15 * q^5 + 14 * q^7 + 22 * q^11 + 8 * q^13 + 34 * q^17 - 4 * q^19 + 176 * q^23 + 75 * q^25 - 98 * q^29 + 88 * q^31 + 70 * q^35 + 284 * q^37 + 8 * q^41 + 504 * q^43 + 280 * q^47 - 409 * q^49 + 150 * q^53 + 110 * q^55 + 350 * q^59 + 350 * q^61 + 40 * q^65 + 804 * q^67 + 500 * q^71 - 486 * q^73 - 788 * q^77 + 1592 * q^79 - 684 * q^83 + 170 * q^85 + 668 * q^89 + 1920 * q^91 - 20 * q^95 - 1394 * q^97

Introduction

L-functions

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