LMFDB - Embedded newform 1200.4.a.bb.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1200,4,Mod(1,1200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1200, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1200.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$1200 = 2^{4} \cdot 3 \cdot 5^{2}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	1200.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:	$70.8022920069$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 150)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1200.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

f(q)

=

q+3.00000 q^{3} +1.00000 q^{7} +9.00000 q^{9} -42.0000 q^{11} -67.0000 q^{13} +54.0000 q^{17} +115.000 q^{19} +3.00000 q^{21} +162.000 q^{23} +27.0000 q^{27} -210.000 q^{29} +193.000 q^{31} -126.000 q^{33} -286.000 q^{37} -201.000 q^{39} +12.0000 q^{41} -263.000 q^{43} -414.000 q^{47} -342.000 q^{49} +162.000 q^{51} -192.000 q^{53} +345.000 q^{57} -690.000 q^{59} -733.000 q^{61} +9.00000 q^{63} -299.000 q^{67} +486.000 q^{69} +228.000 q^{71} +938.000 q^{73} -42.0000 q^{77} +160.000 q^{79} +81.0000 q^{81} +462.000 q^{83} -630.000 q^{87} -240.000 q^{89} -67.0000 q^{91} +579.000 q^{93} -511.000 q^{97} -378.000 q^{99} +O(q^{100})

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
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.00000	0.0539949	0.0269975	−	0.999636i	$-0.491405\pi$
$7$	1.00000	0.0539949	0.0269975	+	0.999636i	$0.491405\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−42.0000	−1.15123	−0.575613	−	0.817723i	$-0.695236\pi$
$11$	−42.0000	−1.15123	−0.575613	+	0.817723i	$0.695236\pi$
$12$	0	0
$13$	−67.0000	−1.42942	−0.714710	−	0.699421i	$-0.753441\pi$
$13$	−67.0000	−1.42942	−0.714710	+	0.699421i	$0.753441\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	54.0000	0.770407	0.385204	−	0.922832i	$-0.374131\pi$
$17$	54.0000	0.770407	0.385204	+	0.922832i	$0.374131\pi$
$18$	0	0
$19$	115.000	1.38857	0.694284	−	0.719701i	$-0.255721\pi$
$19$	115.000	1.38857	0.694284	+	0.719701i	$0.255721\pi$
$20$	0	0
$21$	3.00000	0.0311740
$22$	0	0
$23$	162.000	1.46867	0.734333	−	0.678789i	$-0.237495\pi$
$23$	162.000	1.46867	0.734333	+	0.678789i	$0.237495\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−210.000	−1.34469	−0.672345	−	0.740238i	$-0.734713\pi$
$29$	−210.000	−1.34469	−0.672345	+	0.740238i	$0.734713\pi$
$30$	0	0
$31$	193.000	1.11819	0.559094	−	0.829104i	$-0.311149\pi$
$31$	193.000	1.11819	0.559094	+	0.829104i	$0.311149\pi$
$32$	0	0
$33$	−126.000	−0.664660
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−286.000	−1.27076	−0.635380	−	0.772200i	$-0.719156\pi$
$37$	−286.000	−1.27076	−0.635380	+	0.772200i	$0.719156\pi$
$38$	0	0
$39$	−201.000	−0.825276
$40$	0	0
$41$	12.0000	0.0457094	0.0228547	−	0.999739i	$-0.492724\pi$
$41$	12.0000	0.0457094	0.0228547	+	0.999739i	$0.492724\pi$
$42$	0	0
$43$	−263.000	−0.932724	−0.466362	−	0.884594i	$-0.654436\pi$
$43$	−263.000	−0.932724	−0.466362	+	0.884594i	$0.654436\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−414.000	−1.28485	−0.642427	−	0.766347i	$-0.722072\pi$
$47$	−414.000	−1.28485	−0.642427	+	0.766347i	$0.722072\pi$
$48$	0	0
$49$	−342.000	−0.997085
$50$	0	0
$51$	162.000	0.444795
$52$	0	0
$53$	−192.000	−0.497608	−0.248804	−	0.968554i	$-0.580038\pi$
$53$	−192.000	−0.497608	−0.248804	+	0.968554i	$0.580038\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	345.000	0.801691
$58$	0	0
$59$	−690.000	−1.52255	−0.761274	−	0.648430i	$-0.775426\pi$
$59$	−690.000	−1.52255	−0.761274	+	0.648430i	$0.775426\pi$
$60$	0	0
$61$	−733.000	−1.53854	−0.769271	−	0.638923i	$-0.779380\pi$
$61$	−733.000	−1.53854	−0.769271	+	0.638923i	$0.779380\pi$
$62$	0	0
$63$	9.00000	0.0179983
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−299.000	−0.545204	−0.272602	−	0.962127i	$-0.587884\pi$
$67$	−299.000	−0.545204	−0.272602	+	0.962127i	$0.587884\pi$
$68$	0	0
$69$	486.000	0.847935
$70$	0	0
$71$	228.000	0.381107	0.190554	−	0.981677i	$-0.438972\pi$
$71$	228.000	0.381107	0.190554	+	0.981677i	$0.438972\pi$
$72$	0	0
$73$	938.000	1.50390	0.751949	−	0.659221i	$-0.229114\pi$
$73$	938.000	1.50390	0.751949	+	0.659221i	$0.229114\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−42.0000	−0.0621603
$78$	0	0
$79$	160.000	0.227866	0.113933	−	0.993488i	$-0.463655\pi$
$79$	160.000	0.227866	0.113933	+	0.993488i	$0.463655\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	462.000	0.610977	0.305488	−	0.952196i	$-0.401180\pi$
$83$	462.000	0.610977	0.305488	+	0.952196i	$0.401180\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−630.000	−0.776357
$88$	0	0
$89$	−240.000	−0.285842	−0.142921	−	0.989734i	$-0.545650\pi$
$89$	−240.000	−0.285842	−0.142921	+	0.989734i	$0.545650\pi$
$90$	0	0
$91$	−67.0000	−0.0771814
$92$	0	0
$93$	579.000	0.645586
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−511.000	−0.534889	−0.267444	−	0.963573i	$-0.586179\pi$
$97$	−511.000	−0.534889	−0.267444	+	0.963573i	$0.586179\pi$
$98$	0	0
$99$	−378.000	−0.383742
$100$	0	0
$101$	912.000	0.898489	0.449245	−	0.893409i	$-0.351693\pi$
$101$	912.000	0.898489	0.449245	+	0.893409i	$0.351693\pi$
$102$	0	0
$103$	−668.000	−0.639029	−0.319515	−	0.947581i	$-0.603520\pi$
$103$	−668.000	−0.639029	−0.319515	+	0.947581i	$0.603520\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1296.00	1.17093	0.585463	−	0.810699i	$-0.300913\pi$
$107$	1296.00	1.17093	0.585463	+	0.810699i	$0.300913\pi$
$108$	0	0
$109$	−1735.00	−1.52461	−0.762307	−	0.647216i	$-0.775933\pi$
$109$	−1735.00	−1.52461	−0.762307	+	0.647216i	$0.775933\pi$
$110$	0	0
$111$	−858.000	−0.733673
$112$	0	0
$113$	−1092.00	−0.909086	−0.454543	−	0.890725i	$-0.650197\pi$
$113$	−1092.00	−0.909086	−0.454543	+	0.890725i	$0.650197\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−603.000	−0.476473
$118$	0	0
$119$	54.0000	0.0415981
$120$	0	0
$121$	433.000	0.325319
$122$	0	0
$123$	36.0000	0.0263903
$124$	0	0
$125$	0	0
$126$	0	0
$127$	16.0000	0.0111793	0.00558965	−	0.999984i	$-0.498221\pi$
$127$	16.0000	0.0111793	0.00558965	+	0.999984i	$0.498221\pi$
$128$	0	0
$129$	−789.000	−0.538508
$130$	0	0
$131$	−1992.00	−1.32856	−0.664282	−	0.747482i	$-0.731263\pi$
$131$	−1992.00	−1.32856	−0.664282	+	0.747482i	$0.731263\pi$
$132$	0	0
$133$	115.000	0.0749757
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2346.00	−1.46301	−0.731505	−	0.681836i	$-0.761182\pi$
$137$	−2346.00	−1.46301	−0.731505	+	0.681836i	$0.761182\pi$
$138$	0	0
$139$	−2900.00	−1.76960	−0.884801	−	0.465968i	$-0.845706\pi$
$139$	−2900.00	−1.76960	−0.884801	+	0.465968i	$0.845706\pi$
$140$	0	0
$141$	−1242.00	−0.741810
$142$	0	0
$143$	2814.00	1.64558
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−1026.00	−0.575667
$148$	0	0
$149$	−2070.00	−1.13813	−0.569064	−	0.822293i	$-0.692694\pi$
$149$	−2070.00	−1.13813	−0.569064	+	0.822293i	$0.692694\pi$
$150$	0	0
$151$	−2237.00	−1.20559	−0.602796	−	0.797895i	$-0.705947\pi$
$151$	−2237.00	−1.20559	−0.602796	+	0.797895i	$0.705947\pi$
$152$	0	0
$153$	486.000	0.256802
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−241.000	−0.122509	−0.0612544	−	0.998122i	$-0.519510\pi$
$157$	−241.000	−0.122509	−0.0612544	+	0.998122i	$0.519510\pi$
$158$	0	0
$159$	−576.000	−0.287294
$160$	0	0
$161$	162.000	0.0793006
$162$	0	0
$163$	3547.00	1.70443	0.852216	−	0.523190i	$-0.175258\pi$
$163$	3547.00	1.70443	0.852216	+	0.523190i	$0.175258\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−984.000	−0.455953	−0.227977	−	0.973667i	$-0.573211\pi$
$167$	−984.000	−0.455953	−0.227977	+	0.973667i	$0.573211\pi$
$168$	0	0
$169$	2292.00	1.04324
$170$	0	0
$171$	1035.00	0.462856
$172$	0	0
$173$	3618.00	1.59001	0.795004	−	0.606604i	$-0.207469\pi$
$173$	3618.00	1.59001	0.795004	+	0.606604i	$0.207469\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−2070.00	−0.879044
$178$	0	0
$179$	150.000	0.0626342	0.0313171	−	0.999509i	$-0.490030\pi$
$179$	150.000	0.0626342	0.0313171	+	0.999509i	$0.490030\pi$
$180$	0	0
$181$	197.000	0.0809000	0.0404500	−	0.999182i	$-0.487121\pi$
$181$	197.000	0.0809000	0.0404500	+	0.999182i	$0.487121\pi$
$182$	0	0
$183$	−2199.00	−0.888277
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−2268.00	−0.886912
$188$	0	0
$189$	27.0000	0.0103913
$190$	0	0
$191$	−1302.00	−0.493243	−0.246622	−	0.969112i	$-0.579320\pi$
$191$	−1302.00	−0.493243	−0.246622	+	0.969112i	$0.579320\pi$
$192$	0	0
$193$	4163.00	1.55264	0.776319	−	0.630340i	$-0.217084\pi$
$193$	4163.00	1.55264	0.776319	+	0.630340i	$0.217084\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3054.00	1.10451	0.552255	−	0.833675i	$-0.313767\pi$
$197$	3054.00	1.10451	0.552255	+	0.833675i	$0.313767\pi$
$198$	0	0
$199$	−3425.00	−1.22006	−0.610030	−	0.792379i	$-0.708842\pi$
$199$	−3425.00	−1.22006	−0.610030	+	0.792379i	$0.708842\pi$
$200$	0	0
$201$	−897.000	−0.314774
$202$	0	0
$203$	−210.000	−0.0726065
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1458.00	0.489556
$208$	0	0
$209$	−4830.00	−1.59856
$210$	0	0
$211$	2443.00	0.797076	0.398538	−	0.917152i	$-0.369518\pi$
$211$	2443.00	0.797076	0.398538	+	0.917152i	$0.369518\pi$
$212$	0	0
$213$	684.000	0.220032
$214$	0	0
$215$	0	0
$216$	0	0
$217$	193.000	0.0603765
$218$	0	0
$219$	2814.00	0.868276
$220$	0	0
$221$	−3618.00	−1.10124
$222$	0	0
$223$	−23.0000	−0.00690670	−0.00345335	−	0.999994i	$-0.501099\pi$
$223$	−23.0000	−0.00690670	−0.00345335	+	0.999994i	$0.501099\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1956.00	0.571913	0.285957	−	0.958243i	$-0.407689\pi$
$227$	1956.00	0.571913	0.285957	+	0.958243i	$0.407689\pi$
$228$	0	0
$229$	1805.00	0.520864	0.260432	−	0.965492i	$-0.416135\pi$
$229$	1805.00	0.520864	0.260432	+	0.965492i	$0.416135\pi$
$230$	0	0
$231$	−126.000	−0.0358883
$232$	0	0
$233$	3468.00	0.975091	0.487546	−	0.873098i	$-0.337892\pi$
$233$	3468.00	0.975091	0.487546	+	0.873098i	$0.337892\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	480.000	0.131558
$238$	0	0
$239$	−2640.00	−0.714508	−0.357254	−	0.934007i	$-0.616287\pi$
$239$	−2640.00	−0.714508	−0.357254	+	0.934007i	$0.616287\pi$
$240$	0	0
$241$	−5383.00	−1.43879	−0.719397	−	0.694599i	$-0.755582\pi$
$241$	−5383.00	−1.43879	−0.719397	+	0.694599i	$0.755582\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−7705.00	−1.98485
$248$	0	0
$249$	1386.00	0.352748
$250$	0	0
$251$	5028.00	1.26440	0.632200	−	0.774805i	$-0.282152\pi$
$251$	5028.00	1.26440	0.632200	+	0.774805i	$0.282152\pi$
$252$	0	0
$253$	−6804.00	−1.69077
$254$	0	0
$255$	0	0
$256$	0	0
$257$	564.000	0.136892	0.0684462	−	0.997655i	$-0.478196\pi$
$257$	564.000	0.136892	0.0684462	+	0.997655i	$0.478196\pi$
$258$	0	0
$259$	−286.000	−0.0686146
$260$	0	0
$261$	−1890.00	−0.448230
$262$	0	0
$263$	1812.00	0.424839	0.212420	−	0.977179i	$-0.431866\pi$
$263$	1812.00	0.424839	0.212420	+	0.977179i	$0.431866\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−720.000	−0.165031
$268$	0	0
$269$	−5190.00	−1.17636	−0.588178	−	0.808731i	$-0.700155\pi$
$269$	−5190.00	−1.17636	−0.588178	+	0.808731i	$0.700155\pi$
$270$	0	0
$271$	−4592.00	−1.02931	−0.514657	−	0.857396i	$-0.672081\pi$
$271$	−4592.00	−1.02931	−0.514657	+	0.857396i	$0.672081\pi$
$272$	0	0
$273$	−201.000	−0.0445607
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−2191.00	−0.475251	−0.237625	−	0.971357i	$-0.576369\pi$
$277$	−2191.00	−0.475251	−0.237625	+	0.971357i	$0.576369\pi$
$278$	0	0
$279$	1737.00	0.372729
$280$	0	0
$281$	7842.00	1.66482	0.832410	−	0.554160i	$-0.186960\pi$
$281$	7842.00	1.66482	0.832410	+	0.554160i	$0.186960\pi$
$282$	0	0
$283$	247.000	0.0518821	0.0259410	−	0.999663i	$-0.491742\pi$
$283$	247.000	0.0518821	0.0259410	+	0.999663i	$0.491742\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	12.0000	0.00246808
$288$	0	0
$289$	−1997.00	−0.406473
$290$	0	0
$291$	−1533.00	−0.308818
$292$	0	0
$293$	−5442.00	−1.08507	−0.542534	−	0.840034i	$-0.682535\pi$
$293$	−5442.00	−1.08507	−0.542534	+	0.840034i	$0.682535\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−1134.00	−0.221553
$298$	0	0
$299$	−10854.0	−2.09934
$300$	0	0
$301$	−263.000	−0.0503624
$302$	0	0
$303$	2736.00	0.518743
$304$	0	0
$305$	0	0
$306$	0	0
$307$	3871.00	0.719641	0.359820	−	0.933022i	$-0.382838\pi$
$307$	3871.00	0.719641	0.359820	+	0.933022i	$0.382838\pi$
$308$	0	0
$309$	−2004.00	−0.368944
$310$	0	0
$311$	5718.00	1.04257	0.521283	−	0.853384i	$-0.325454\pi$
$311$	5718.00	1.04257	0.521283	+	0.853384i	$0.325454\pi$
$312$	0	0
$313$	−3637.00	−0.656790	−0.328395	−	0.944540i	$-0.606508\pi$
$313$	−3637.00	−0.656790	−0.328395	+	0.944540i	$0.606508\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−1296.00	−0.229623	−0.114812	−	0.993387i	$-0.536626\pi$
$317$	−1296.00	−0.229623	−0.114812	+	0.993387i	$0.536626\pi$
$318$	0	0
$319$	8820.00	1.54804
$320$	0	0
$321$	3888.00	0.676034
$322$	0	0
$323$	6210.00	1.06976
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−5205.00	−0.880236
$328$	0	0
$329$	−414.000	−0.0693756
$330$	0	0
$331$	−5132.00	−0.852206	−0.426103	−	0.904675i	$-0.640114\pi$
$331$	−5132.00	−0.852206	−0.426103	+	0.904675i	$0.640114\pi$
$332$	0	0
$333$	−2574.00	−0.423587
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−6751.00	−1.09125	−0.545624	−	0.838030i	$-0.683707\pi$
$337$	−6751.00	−1.09125	−0.545624	+	0.838030i	$0.683707\pi$
$338$	0	0
$339$	−3276.00	−0.524861
$340$	0	0
$341$	−8106.00	−1.28729
$342$	0	0
$343$	−685.000	−0.107832
$344$	0	0
$345$	0	0
$346$	0	0
$347$	5226.00	0.808491	0.404246	−	0.914651i	$-0.367534\pi$
$347$	5226.00	0.808491	0.404246	+	0.914651i	$0.367534\pi$
$348$	0	0
$349$	−6190.00	−0.949407	−0.474704	−	0.880146i	$-0.657445\pi$
$349$	−6190.00	−0.949407	−0.474704	+	0.880146i	$0.657445\pi$
$350$	0	0
$351$	−1809.00	−0.275092
$352$	0	0
$353$	6618.00	0.997849	0.498924	−	0.866646i	$-0.333729\pi$
$353$	6618.00	0.997849	0.498924	+	0.866646i	$0.333729\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	162.000	0.0240167
$358$	0	0
$359$	3420.00	0.502787	0.251394	−	0.967885i	$-0.419111\pi$
$359$	3420.00	0.502787	0.251394	+	0.967885i	$0.419111\pi$
$360$	0	0
$361$	6366.00	0.928124
$362$	0	0
$363$	1299.00	0.187823
$364$	0	0
$365$	0	0
$366$	0	0
$367$	871.000	0.123885	0.0619425	−	0.998080i	$-0.480270\pi$
$367$	871.000	0.123885	0.0619425	+	0.998080i	$0.480270\pi$
$368$	0	0
$369$	108.000	0.0152365
$370$	0	0
$371$	−192.000	−0.0268683
$372$	0	0
$373$	6383.00	0.886057	0.443028	−	0.896508i	$-0.353904\pi$
$373$	6383.00	0.886057	0.443028	+	0.896508i	$0.353904\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	14070.0	1.92213
$378$	0	0
$379$	9865.00	1.33702	0.668511	−	0.743703i	$-0.266932\pi$
$379$	9865.00	1.33702	0.668511	+	0.743703i	$0.266932\pi$
$380$	0	0
$381$	48.0000	0.00645437
$382$	0	0
$383$	−9828.00	−1.31119	−0.655597	−	0.755111i	$-0.727583\pi$
$383$	−9828.00	−1.31119	−0.655597	+	0.755111i	$0.727583\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−2367.00	−0.310908
$388$	0	0
$389$	12540.0	1.63446	0.817228	−	0.576315i	$-0.195510\pi$
$389$	12540.0	1.63446	0.817228	+	0.576315i	$0.195510\pi$
$390$	0	0
$391$	8748.00	1.13147
$392$	0	0
$393$	−5976.00	−0.767047
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−1381.00	−0.174585	−0.0872927	−	0.996183i	$-0.527822\pi$
$397$	−1381.00	−0.174585	−0.0872927	+	0.996183i	$0.527822\pi$
$398$	0	0
$399$	345.000	0.0432872
$400$	0	0
$401$	14232.0	1.77235	0.886175	−	0.463351i	$-0.153353\pi$
$401$	14232.0	1.77235	0.886175	+	0.463351i	$0.153353\pi$
$402$	0	0
$403$	−12931.0	−1.59836
$404$	0	0
$405$	0	0
$406$	0	0
$407$	12012.0	1.46293
$408$	0	0
$409$	2645.00	0.319772	0.159886	−	0.987135i	$-0.448887\pi$
$409$	2645.00	0.319772	0.159886	+	0.987135i	$0.448887\pi$
$410$	0	0
$411$	−7038.00	−0.844669
$412$	0	0
$413$	−690.000	−0.0822099
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−8700.00	−1.02168
$418$	0	0
$419$	−3000.00	−0.349784	−0.174892	−	0.984588i	$-0.555958\pi$
$419$	−3000.00	−0.349784	−0.174892	+	0.984588i	$0.555958\pi$
$420$	0	0
$421$	−11338.0	−1.31254	−0.656271	−	0.754525i	$-0.727867\pi$
$421$	−11338.0	−1.31254	−0.656271	+	0.754525i	$0.727867\pi$
$422$	0	0
$423$	−3726.00	−0.428284
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−733.000	−0.0830734
$428$	0	0
$429$	8442.00	0.950078
$430$	0	0
$431$	3258.00	0.364112	0.182056	−	0.983288i	$-0.441725\pi$
$431$	3258.00	0.364112	0.182056	+	0.983288i	$0.441725\pi$
$432$	0	0
$433$	1163.00	0.129077	0.0645384	−	0.997915i	$-0.479443\pi$
$433$	1163.00	0.129077	0.0645384	+	0.997915i	$0.479443\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	18630.0	2.03934
$438$	0	0
$439$	−6695.00	−0.727870	−0.363935	−	0.931424i	$-0.618567\pi$
$439$	−6695.00	−0.727870	−0.363935	+	0.931424i	$0.618567\pi$
$440$	0	0
$441$	−3078.00	−0.332362
$442$	0	0
$443$	−16368.0	−1.75546	−0.877728	−	0.479159i	$-0.840942\pi$
$443$	−16368.0	−1.75546	−0.877728	+	0.479159i	$0.840942\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−6210.00	−0.657098
$448$	0	0
$449$	16380.0	1.72165	0.860824	−	0.508903i	$-0.169949\pi$
$449$	16380.0	1.72165	0.860824	+	0.508903i	$0.169949\pi$
$450$	0	0
$451$	−504.000	−0.0526218
$452$	0	0
$453$	−6711.00	−0.696049
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−13786.0	−1.41112	−0.705560	−	0.708650i	$-0.749304\pi$
$457$	−13786.0	−1.41112	−0.705560	+	0.708650i	$0.749304\pi$
$458$	0	0
$459$	1458.00	0.148265
$460$	0	0
$461$	11832.0	1.19538	0.597691	−	0.801726i	$-0.296085\pi$
$461$	11832.0	1.19538	0.597691	+	0.801726i	$0.296085\pi$
$462$	0	0
$463$	−3008.00	−0.301930	−0.150965	−	0.988539i	$-0.548238\pi$
$463$	−3008.00	−0.301930	−0.150965	+	0.988539i	$0.548238\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−4434.00	−0.439360	−0.219680	−	0.975572i	$-0.570501\pi$
$467$	−4434.00	−0.439360	−0.219680	+	0.975572i	$0.570501\pi$
$468$	0	0
$469$	−299.000	−0.0294382
$470$	0	0
$471$	−723.000	−0.0707305
$472$	0	0
$473$	11046.0	1.07378
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−1728.00	−0.165869
$478$	0	0
$479$	−7410.00	−0.706830	−0.353415	−	0.935467i	$-0.614980\pi$
$479$	−7410.00	−0.706830	−0.353415	+	0.935467i	$0.614980\pi$
$480$	0	0
$481$	19162.0	1.81645
$482$	0	0
$483$	486.000	0.0457842
$484$	0	0
$485$	0	0
$486$	0	0
$487$	8671.00	0.806818	0.403409	−	0.915020i	$-0.367825\pi$
$487$	8671.00	0.806818	0.403409	+	0.915020i	$0.367825\pi$
$488$	0	0
$489$	10641.0	0.984055
$490$	0	0
$491$	19368.0	1.78017	0.890087	−	0.455790i	$-0.150643\pi$
$491$	19368.0	1.78017	0.890087	+	0.455790i	$0.150643\pi$
$492$	0	0
$493$	−11340.0	−1.03596
$494$	0	0
$495$	0	0
$496$	0	0
$497$	228.000	0.0205779
$498$	0	0
$499$	8875.00	0.796192	0.398096	−	0.917344i	$-0.369671\pi$
$499$	8875.00	0.796192	0.398096	+	0.917344i	$0.369671\pi$
$500$	0	0
$501$	−2952.00	−0.263245
$502$	0	0
$503$	10452.0	0.926504	0.463252	−	0.886227i	$-0.346682\pi$
$503$	10452.0	0.926504	0.463252	+	0.886227i	$0.346682\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	6876.00	0.602315
$508$	0	0
$509$	−19770.0	−1.72159	−0.860796	−	0.508951i	$-0.830033\pi$
$509$	−19770.0	−1.72159	−0.860796	+	0.508951i	$0.830033\pi$
$510$	0	0
$511$	938.000	0.0812029
$512$	0	0
$513$	3105.00	0.267230
$514$	0	0
$515$	0	0
$516$	0	0
$517$	17388.0	1.47916
$518$	0	0
$519$	10854.0	0.917992
$520$	0	0
$521$	−11238.0	−0.945001	−0.472501	−	0.881330i	$-0.656649\pi$
$521$	−11238.0	−0.945001	−0.472501	+	0.881330i	$0.656649\pi$
$522$	0	0
$523$	7447.00	0.622628	0.311314	−	0.950307i	$-0.399231\pi$
$523$	7447.00	0.622628	0.311314	+	0.950307i	$0.399231\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	10422.0	0.861460
$528$	0	0
$529$	14077.0	1.15698
$530$	0	0
$531$	−6210.00	−0.507516
$532$	0	0
$533$	−804.000	−0.0653379
$534$	0	0
$535$	0	0
$536$	0	0
$537$	450.000	0.0361619
$538$	0	0
$539$	14364.0	1.14787
$540$	0	0
$541$	−17623.0	−1.40050	−0.700251	−	0.713896i	$-0.746929\pi$
$541$	−17623.0	−1.40050	−0.700251	+	0.713896i	$0.746929\pi$
$542$	0	0
$543$	591.000	0.0467076
$544$	0	0
$545$	0	0
$546$	0	0
$547$	10096.0	0.789166	0.394583	−	0.918860i	$-0.370889\pi$
$547$	10096.0	0.789166	0.394583	+	0.918860i	$0.370889\pi$
$548$	0	0
$549$	−6597.00	−0.512847
$550$	0	0
$551$	−24150.0	−1.86720
$552$	0	0
$553$	160.000	0.0123036
$554$	0	0
$555$	0	0
$556$	0	0
$557$	14514.0	1.10409	0.552045	−	0.833814i	$-0.313848\pi$
$557$	14514.0	1.10409	0.552045	+	0.833814i	$0.313848\pi$
$558$	0	0
$559$	17621.0	1.33325
$560$	0	0
$561$	−6804.00	−0.512059
$562$	0	0
$563$	10242.0	0.766694	0.383347	−	0.923604i	$-0.374771\pi$
$563$	10242.0	0.766694	0.383347	+	0.923604i	$0.374771\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	81.0000	0.00599944
$568$	0	0
$569$	−6750.00	−0.497319	−0.248660	−	0.968591i	$-0.579990\pi$
$569$	−6750.00	−0.497319	−0.248660	+	0.968591i	$0.579990\pi$
$570$	0	0
$571$	−17117.0	−1.25451	−0.627254	−	0.778815i	$-0.715821\pi$
$571$	−17117.0	−1.25451	−0.627254	+	0.778815i	$0.715821\pi$
$572$	0	0
$573$	−3906.00	−0.284774
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−301.000	−0.0217171	−0.0108586	−	0.999941i	$-0.503456\pi$
$577$	−301.000	−0.0217171	−0.0108586	+	0.999941i	$0.503456\pi$
$578$	0	0
$579$	12489.0	0.896416
$580$	0	0
$581$	462.000	0.0329897
$582$	0	0
$583$	8064.00	0.572859
$584$	0	0
$585$	0	0
$586$	0	0
$587$	15456.0	1.08678	0.543388	−	0.839482i	$-0.317141\pi$
$587$	15456.0	1.08678	0.543388	+	0.839482i	$0.317141\pi$
$588$	0	0
$589$	22195.0	1.55268
$590$	0	0
$591$	9162.00	0.637689
$592$	0	0
$593$	−9492.00	−0.657318	−0.328659	−	0.944449i	$-0.606597\pi$
$593$	−9492.00	−0.657318	−0.328659	+	0.944449i	$0.606597\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−10275.0	−0.704402
$598$	0	0
$599$	−1500.00	−0.102318	−0.0511589	−	0.998691i	$-0.516291\pi$
$599$	−1500.00	−0.102318	−0.0511589	+	0.998691i	$0.516291\pi$
$600$	0	0
$601$	14627.0	0.992758	0.496379	−	0.868106i	$-0.334663\pi$
$601$	14627.0	0.992758	0.496379	+	0.868106i	$0.334663\pi$
$602$	0	0
$603$	−2691.00	−0.181735
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−16184.0	−1.08219	−0.541094	−	0.840962i	$-0.681990\pi$
$607$	−16184.0	−1.08219	−0.541094	+	0.840962i	$0.681990\pi$
$608$	0	0
$609$	−630.000	−0.0419194
$610$	0	0
$611$	27738.0	1.83659
$612$	0	0
$613$	−18502.0	−1.21907	−0.609534	−	0.792760i	$-0.708643\pi$
$613$	−18502.0	−1.21907	−0.609534	+	0.792760i	$0.708643\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−13896.0	−0.906697	−0.453348	−	0.891333i	$-0.649771\pi$
$617$	−13896.0	−0.906697	−0.453348	+	0.891333i	$0.649771\pi$
$618$	0	0
$619$	9895.00	0.642510	0.321255	−	0.946993i	$-0.395895\pi$
$619$	9895.00	0.642510	0.321255	+	0.946993i	$0.395895\pi$
$620$	0	0
$621$	4374.00	0.282645
$622$	0	0
$623$	−240.000	−0.0154340
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−14490.0	−0.922926
$628$	0	0
$629$	−15444.0	−0.979003
$630$	0	0
$631$	−467.000	−0.0294627	−0.0147314	−	0.999891i	$-0.504689\pi$
$631$	−467.000	−0.0294627	−0.0147314	+	0.999891i	$0.504689\pi$
$632$	0	0
$633$	7329.00	0.460192
$634$	0	0
$635$	0	0
$636$	0	0
$637$	22914.0	1.42525
$638$	0	0
$639$	2052.00	0.127036
$640$	0	0
$641$	30612.0	1.88627	0.943137	−	0.332405i	$-0.107860\pi$
$641$	30612.0	1.88627	0.943137	+	0.332405i	$0.107860\pi$
$642$	0	0
$643$	1852.00	0.113586	0.0567930	−	0.998386i	$-0.481913\pi$
$643$	1852.00	0.113586	0.0567930	+	0.998386i	$0.481913\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	21156.0	1.28551	0.642757	−	0.766070i	$-0.277790\pi$
$647$	21156.0	1.28551	0.642757	+	0.766070i	$0.277790\pi$
$648$	0	0
$649$	28980.0	1.75280
$650$	0	0
$651$	579.000	0.0348584
$652$	0	0
$653$	−9702.00	−0.581422	−0.290711	−	0.956811i	$-0.593892\pi$
$653$	−9702.00	−0.581422	−0.290711	+	0.956811i	$0.593892\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	8442.00	0.501300
$658$	0	0
$659$	−1980.00	−0.117041	−0.0585204	−	0.998286i	$-0.518638\pi$
$659$	−1980.00	−0.117041	−0.0585204	+	0.998286i	$0.518638\pi$
$660$	0	0
$661$	−20158.0	−1.18617	−0.593083	−	0.805142i	$-0.702089\pi$
$661$	−20158.0	−1.18617	−0.593083	+	0.805142i	$0.702089\pi$
$662$	0	0
$663$	−10854.0	−0.635799
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−34020.0	−1.97490
$668$	0	0
$669$	−69.0000	−0.00398758
$670$	0	0
$671$	30786.0	1.77121
$672$	0	0
$673$	−16882.0	−0.966944	−0.483472	−	0.875360i	$-0.660624\pi$
$673$	−16882.0	−0.966944	−0.483472	+	0.875360i	$0.660624\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	20934.0	1.18842	0.594209	−	0.804311i	$-0.297465\pi$
$677$	20934.0	1.18842	0.594209	+	0.804311i	$0.297465\pi$
$678$	0	0
$679$	−511.000	−0.0288813
$680$	0	0
$681$	5868.00	0.330194
$682$	0	0
$683$	8712.00	0.488075	0.244038	−	0.969766i	$-0.421528\pi$
$683$	8712.00	0.488075	0.244038	+	0.969766i	$0.421528\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	5415.00	0.300721
$688$	0	0
$689$	12864.0	0.711291
$690$	0	0
$691$	14128.0	0.777792	0.388896	−	0.921282i	$-0.372856\pi$
$691$	14128.0	0.777792	0.388896	+	0.921282i	$0.372856\pi$
$692$	0	0
$693$	−378.000	−0.0207201
$694$	0	0
$695$	0	0
$696$	0	0
$697$	648.000	0.0352148
$698$	0	0
$699$	10404.0	0.562969
$700$	0	0
$701$	−28278.0	−1.52360	−0.761801	−	0.647811i	$-0.775685\pi$
$701$	−28278.0	−1.52360	−0.761801	+	0.647811i	$0.775685\pi$
$702$	0	0
$703$	−32890.0	−1.76454
$704$	0	0
$705$	0	0
$706$	0	0
$707$	912.000	0.0485138
$708$	0	0
$709$	8885.00	0.470639	0.235320	−	0.971918i	$-0.424386\pi$
$709$	8885.00	0.470639	0.235320	+	0.971918i	$0.424386\pi$
$710$	0	0
$711$	1440.00	0.0759553
$712$	0	0
$713$	31266.0	1.64225
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−7920.00	−0.412521
$718$	0	0
$719$	7530.00	0.390572	0.195286	−	0.980746i	$-0.437436\pi$
$719$	7530.00	0.390572	0.195286	+	0.980746i	$0.437436\pi$
$720$	0	0
$721$	−668.000	−0.0345043
$722$	0	0
$723$	−16149.0	−0.830688
$724$	0	0
$725$	0	0
$726$	0	0
$727$	1801.00	0.0918781	0.0459391	−	0.998944i	$-0.485372\pi$
$727$	1801.00	0.0918781	0.0459391	+	0.998944i	$0.485372\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−14202.0	−0.718577
$732$	0	0
$733$	−7882.00	−0.397174	−0.198587	−	0.980083i	$-0.563635\pi$
$733$	−7882.00	−0.397174	−0.198587	+	0.980083i	$0.563635\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	12558.0	0.627652
$738$	0	0
$739$	−33860.0	−1.68547	−0.842734	−	0.538331i	$-0.819055\pi$
$739$	−33860.0	−1.68547	−0.842734	+	0.538331i	$0.819055\pi$
$740$	0	0
$741$	−23115.0	−1.14595
$742$	0	0
$743$	20652.0	1.01972	0.509858	−	0.860259i	$-0.329698\pi$
$743$	20652.0	1.01972	0.509858	+	0.860259i	$0.329698\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	4158.00	0.203659
$748$	0	0
$749$	1296.00	0.0632240
$750$	0	0
$751$	−7472.00	−0.363059	−0.181529	−	0.983386i	$-0.558105\pi$
$751$	−7472.00	−0.363059	−0.181529	+	0.983386i	$0.558105\pi$
$752$	0	0
$753$	15084.0	0.730002
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−32251.0	−1.54846	−0.774229	−	0.632906i	$-0.781862\pi$
$757$	−32251.0	−1.54846	−0.774229	+	0.632906i	$0.781862\pi$
$758$	0	0
$759$	−20412.0	−0.976164
$760$	0	0
$761$	16812.0	0.800834	0.400417	−	0.916333i	$-0.368865\pi$
$761$	16812.0	0.800834	0.400417	+	0.916333i	$0.368865\pi$
$762$	0	0
$763$	−1735.00	−0.0823214
$764$	0	0
$765$	0	0
$766$	0	0
$767$	46230.0	2.17636
$768$	0	0
$769$	−34645.0	−1.62462	−0.812309	−	0.583228i	$-0.801789\pi$
$769$	−34645.0	−1.62462	−0.812309	+	0.583228i	$0.801789\pi$
$770$	0	0
$771$	1692.00	0.0790349
$772$	0	0
$773$	−8412.00	−0.391408	−0.195704	−	0.980663i	$-0.562699\pi$
$773$	−8412.00	−0.391408	−0.195704	+	0.980663i	$0.562699\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−858.000	−0.0396146
$778$	0	0
$779$	1380.00	0.0634706
$780$	0	0
$781$	−9576.00	−0.438740
$782$	0	0
$783$	−5670.00	−0.258786
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−18329.0	−0.830188	−0.415094	−	0.909778i	$-0.636251\pi$
$787$	−18329.0	−0.830188	−0.415094	+	0.909778i	$0.636251\pi$
$788$	0	0
$789$	5436.00	0.245281
$790$	0	0
$791$	−1092.00	−0.0490860
$792$	0	0
$793$	49111.0	2.19922
$794$	0	0
$795$	0	0
$796$	0	0
$797$	16044.0	0.713059	0.356529	−	0.934284i	$-0.383960\pi$
$797$	16044.0	0.713059	0.356529	+	0.934284i	$0.383960\pi$
$798$	0	0
$799$	−22356.0	−0.989860
$800$	0	0
$801$	−2160.00	−0.0952807
$802$	0	0
$803$	−39396.0	−1.73133
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−15570.0	−0.679170
$808$	0	0
$809$	−24000.0	−1.04301	−0.521505	−	0.853248i	$-0.674629\pi$
$809$	−24000.0	−1.04301	−0.521505	+	0.853248i	$0.674629\pi$
$810$	0	0
$811$	−5117.00	−0.221556	−0.110778	−	0.993845i	$-0.535334\pi$
$811$	−5117.00	−0.221556	−0.110778	+	0.993845i	$0.535334\pi$
$812$	0	0
$813$	−13776.0	−0.594275
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−30245.0	−1.29515
$818$	0	0
$819$	−603.000	−0.0257271
$820$	0	0
$821$	13542.0	0.575663	0.287831	−	0.957681i	$-0.407066\pi$
$821$	13542.0	0.575663	0.287831	+	0.957681i	$0.407066\pi$
$822$	0	0
$823$	−1283.00	−0.0543409	−0.0271705	−	0.999631i	$-0.508650\pi$
$823$	−1283.00	−0.0543409	−0.0271705	+	0.999631i	$0.508650\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−16344.0	−0.687227	−0.343613	−	0.939111i	$-0.611651\pi$
$827$	−16344.0	−0.687227	−0.343613	+	0.939111i	$0.611651\pi$
$828$	0	0
$829$	−790.000	−0.0330975	−0.0165488	−	0.999863i	$-0.505268\pi$
$829$	−790.000	−0.0330975	−0.0165488	+	0.999863i	$0.505268\pi$
$830$	0	0
$831$	−6573.00	−0.274386
$832$	0	0
$833$	−18468.0	−0.768161
$834$	0	0
$835$	0	0
$836$	0	0
$837$	5211.00	0.215195
$838$	0	0
$839$	9990.00	0.411076	0.205538	−	0.978649i	$-0.434106\pi$
$839$	9990.00	0.411076	0.205538	+	0.978649i	$0.434106\pi$
$840$	0	0
$841$	19711.0	0.808192
$842$	0	0
$843$	23526.0	0.961184
$844$	0	0
$845$	0	0
$846$	0	0
$847$	433.000	0.0175656
$848$	0	0
$849$	741.000	0.0299541
$850$	0	0
$851$	−46332.0	−1.86632
$852$	0	0
$853$	24743.0	0.993182	0.496591	−	0.867985i	$-0.334585\pi$
$853$	24743.0	0.993182	0.496591	+	0.867985i	$0.334585\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−23556.0	−0.938924	−0.469462	−	0.882953i	$-0.655552\pi$
$857$	−23556.0	−0.938924	−0.469462	+	0.882953i	$0.655552\pi$
$858$	0	0
$859$	34000.0	1.35048	0.675242	−	0.737597i	$-0.264039\pi$
$859$	34000.0	1.35048	0.675242	+	0.737597i	$0.264039\pi$
$860$	0	0
$861$	36.0000	0.00142494
$862$	0	0
$863$	37032.0	1.46070	0.730350	−	0.683073i	$-0.239357\pi$
$863$	37032.0	1.46070	0.730350	+	0.683073i	$0.239357\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−5991.00	−0.234677
$868$	0	0
$869$	−6720.00	−0.262325
$870$	0	0
$871$	20033.0	0.779325
$872$	0	0
$873$	−4599.00	−0.178296
$874$	0	0
$875$	0	0
$876$	0	0
$877$	2519.00	0.0969904	0.0484952	−	0.998823i	$-0.484557\pi$
$877$	2519.00	0.0969904	0.0484952	+	0.998823i	$0.484557\pi$
$878$	0	0
$879$	−16326.0	−0.626465
$880$	0	0
$881$	43992.0	1.68232	0.841162	−	0.540783i	$-0.181872\pi$
$881$	43992.0	1.68232	0.841162	+	0.540783i	$0.181872\pi$
$882$	0	0
$883$	19177.0	0.730869	0.365435	−	0.930837i	$-0.380920\pi$
$883$	19177.0	0.730869	0.365435	+	0.930837i	$0.380920\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−44994.0	−1.70321	−0.851607	−	0.524181i	$-0.824372\pi$
$887$	−44994.0	−1.70321	−0.851607	+	0.524181i	$0.824372\pi$
$888$	0	0
$889$	16.0000	0.000603625 0
$890$	0	0
$891$	−3402.00	−0.127914
$892$	0	0
$893$	−47610.0	−1.78411
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−32562.0	−1.21206
$898$	0	0
$899$	−40530.0	−1.50362
$900$	0	0
$901$	−10368.0	−0.383361
$902$	0	0
$903$	−789.000	−0.0290767
$904$	0	0
$905$	0	0
$906$	0	0
$907$	52396.0	1.91817	0.959085	−	0.283117i	$-0.0913686\pi$
$907$	52396.0	1.91817	0.959085	+	0.283117i	$0.0913686\pi$
$908$	0	0
$909$	8208.00	0.299496
$910$	0	0
$911$	−7242.00	−0.263379	−0.131689	−	0.991291i	$-0.542040\pi$
$911$	−7242.00	−0.263379	−0.131689	+	0.991291i	$0.542040\pi$
$912$	0	0
$913$	−19404.0	−0.703372
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1992.00	−0.0717357
$918$	0	0
$919$	−4085.00	−0.146629	−0.0733143	−	0.997309i	$-0.523358\pi$
$919$	−4085.00	−0.146629	−0.0733143	+	0.997309i	$0.523358\pi$
$920$	0	0
$921$	11613.0	0.415485
$922$	0	0
$923$	−15276.0	−0.544762
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−6012.00	−0.213010
$928$	0	0
$929$	−3030.00	−0.107009	−0.0535043	−	0.998568i	$-0.517039\pi$
$929$	−3030.00	−0.107009	−0.0535043	+	0.998568i	$0.517039\pi$
$930$	0	0
$931$	−39330.0	−1.38452
$932$	0	0
$933$	17154.0	0.601926
$934$	0	0
$935$	0	0
$936$	0	0
$937$	5759.00	0.200788	0.100394	−	0.994948i	$-0.467990\pi$
$937$	5759.00	0.200788	0.100394	+	0.994948i	$0.467990\pi$
$938$	0	0
$939$	−10911.0	−0.379198
$940$	0	0
$941$	−258.000	−0.00893790	−0.00446895	−	0.999990i	$-0.501423\pi$
$941$	−258.000	−0.00893790	−0.00446895	+	0.999990i	$0.501423\pi$
$942$	0	0
$943$	1944.00	0.0671319
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1374.00	−0.0471478	−0.0235739	−	0.999722i	$-0.507505\pi$
$947$	−1374.00	−0.0471478	−0.0235739	+	0.999722i	$0.507505\pi$
$948$	0	0
$949$	−62846.0	−2.14970
$950$	0	0
$951$	−3888.00	−0.132573
$952$	0	0
$953$	9288.00	0.315706	0.157853	−	0.987463i	$-0.449543\pi$
$953$	9288.00	0.315706	0.157853	+	0.987463i	$0.449543\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	26460.0	0.893762
$958$	0	0
$959$	−2346.00	−0.0789951
$960$	0	0
$961$	7458.00	0.250344
$962$	0	0
$963$	11664.0	0.390309
$964$	0	0
$965$	0	0
$966$	0	0
$967$	21616.0	0.718846	0.359423	−	0.933175i	$-0.382974\pi$
$967$	21616.0	0.718846	0.359423	+	0.933175i	$0.382974\pi$
$968$	0	0
$969$	18630.0	0.617628
$970$	0	0
$971$	19098.0	0.631188	0.315594	−	0.948894i	$-0.397796\pi$
$971$	19098.0	0.631188	0.315594	+	0.948894i	$0.397796\pi$
$972$	0	0
$973$	−2900.00	−0.0955496
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−18246.0	−0.597483	−0.298742	−	0.954334i	$-0.596567\pi$
$977$	−18246.0	−0.597483	−0.298742	+	0.954334i	$0.596567\pi$
$978$	0	0
$979$	10080.0	0.329069
$980$	0	0
$981$	−15615.0	−0.508204
$982$	0	0
$983$	38772.0	1.25802	0.629011	−	0.777397i	$-0.283460\pi$
$983$	38772.0	1.25802	0.629011	+	0.777397i	$0.283460\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−1242.00	−0.0400540
$988$	0	0
$989$	−42606.0	−1.36986
$990$	0	0
$991$	23053.0	0.738953	0.369477	−	0.929240i	$-0.379537\pi$
$991$	23053.0	0.738953	0.369477	+	0.929240i	$0.379537\pi$
$992$	0	0
$993$	−15396.0	−0.492021
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−10366.0	−0.329282	−0.164641	−	0.986354i	$-0.552647\pi$
$997$	−10366.0	−0.329282	−0.164641	+	0.986354i	$0.552647\pi$
$998$	0	0
$999$	−7722.00	−0.244558

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.4.a.bb.1.1		1
4.3	odd	2		150.4.a.a.1.1	✓	1
5.2	odd	4		1200.4.f.c.49.1		2
5.3	odd	4		1200.4.f.c.49.2		2
5.4	even	2		1200.4.a.i.1.1		1
12.11	even	2		450.4.a.o.1.1		1
20.3	even	4		150.4.c.e.49.2		2
20.7	even	4		150.4.c.e.49.1		2
20.19	odd	2		150.4.a.h.1.1	yes	1
60.23	odd	4		450.4.c.a.199.1		2
60.47	odd	4		450.4.c.a.199.2		2
60.59	even	2		450.4.a.f.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.4.a.a.1.1	✓	1	4.3	odd	2
150.4.a.h.1.1	yes	1	20.19	odd	2
150.4.c.e.49.1		2	20.7	even	4
150.4.c.e.49.2		2	20.3	even	4
450.4.a.f.1.1		1	60.59	even	2
450.4.a.o.1.1		1	12.11	even	2
450.4.c.a.199.1		2	60.23	odd	4
450.4.c.a.199.2		2	60.47	odd	4
1200.4.a.i.1.1		1	5.4	even	2
1200.4.a.bb.1.1		1	1.1	even	1	trivial
1200.4.f.c.49.1		2	5.2	odd	4
1200.4.f.c.49.2		2	5.3	odd	4

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Classical	Maass
Hilbert	Bianchi

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

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1200.4.a.bb.1.1

Level	$1200$
Weight	$4$
Character	1200.1
Self dual	yes
Analytic conductor	$70.802$
Analytic rank	$1$
Dimension	$1$
CM	no
Inner twists	$1$