LMFDB - Embedded newform 336.6.a.o.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [336,6,Mod(1,336)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(336, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("336.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 336 = 2^{4} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	336.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:	$53.8889634572$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 42)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	336.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+9.00000 q^{3} +44.0000 q^{5} +49.0000 q^{7} +81.0000 q^{9} +O(q^{10})$

$q+9.00000 q^{3} +44.0000 q^{5} +49.0000 q^{7} +81.0000 q^{9} +470.000 q^{11} -1158.00 q^{13} +396.000 q^{15} +1204.00 q^{17} +2644.00 q^{19} +441.000 q^{21} +1190.00 q^{23} -1189.00 q^{25} +729.000 q^{27} +3614.00 q^{29} -5616.00 q^{31} +4230.00 q^{33} +2156.00 q^{35} -6478.00 q^{37} -10422.0 q^{39} +2856.00 q^{41} +13492.0 q^{43} +3564.00 q^{45} +18372.0 q^{47} +2401.00 q^{49} +10836.0 q^{51} -4374.00 q^{53} +20680.0 q^{55} +23796.0 q^{57} -30248.0 q^{59} +19542.0 q^{61} +3969.00 q^{63} -50952.0 q^{65} -54328.0 q^{67} +10710.0 q^{69} +10730.0 q^{71} +35374.0 q^{73} -10701.0 q^{75} +23030.0 q^{77} +49956.0 q^{79} +6561.00 q^{81} +26948.0 q^{83} +52976.0 q^{85} +32526.0 q^{87} +100776. q^{89} -56742.0 q^{91} -50544.0 q^{93} +116336. q^{95} +77134.0 q^{97} +38070.0 q^{99} +O(q^{100})$

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 $

$2$	0	0
$2$	0	0
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	0	0
$5$	44.0000	0.787096	0.393548	−	0.919304i	$-0.371248\pi$
$5$	44.0000	0.787096	0.393548	+	0.919304i	$0.371248\pi$
$6$	0	0
$7$	49.0000	0.377964
$7$	49.0000	0.377964
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	470.000	1.17116	0.585580	−	0.810615i	$-0.300867\pi$
$11$	470.000	1.17116	0.585580	+	0.810615i	$0.300867\pi$
$12$	0	0
$13$	−1158.00	−1.90042	−0.950211	−	0.311606i	$-0.899133\pi$
$13$	−1158.00	−1.90042	−0.950211	+	0.311606i	$0.899133\pi$
$14$	0	0
$15$	396.000	0.454430
$16$	0	0
$17$	1204.00	1.01043	0.505213	−	0.862995i	$-0.331414\pi$
$17$	1204.00	1.01043	0.505213	+	0.862995i	$0.331414\pi$
$18$	0	0
$19$	2644.00	1.68026	0.840132	−	0.542382i	$-0.182478\pi$
$19$	2644.00	1.68026	0.840132	+	0.542382i	$0.182478\pi$
$20$	0	0
$21$	441.000	0.218218
$22$	0	0
$23$	1190.00	0.469059	0.234529	−	0.972109i	$-0.424645\pi$
$23$	1190.00	0.469059	0.234529	+	0.972109i	$0.424645\pi$
$24$	0	0
$25$	−1189.00	−0.380480
$26$	0	0
$27$	729.000	0.192450
$28$	0	0
$29$	3614.00	0.797982	0.398991	−	0.916955i	$-0.369360\pi$
$29$	3614.00	0.797982	0.398991	+	0.916955i	$0.369360\pi$
$30$	0	0
$31$	−5616.00	−1.04960	−0.524799	−	0.851226i	$-0.675859\pi$
$31$	−5616.00	−1.04960	−0.524799	+	0.851226i	$0.675859\pi$
$32$	0	0
$33$	4230.00	0.676169
$34$	0	0
$35$	2156.00	0.297494
$36$	0	0
$37$	−6478.00	−0.777923	−0.388962	−	0.921254i	$-0.627166\pi$
$37$	−6478.00	−0.777923	−0.388962	+	0.921254i	$0.627166\pi$
$38$	0	0
$39$	−10422.0	−1.09721
$40$	0	0
$41$	2856.00	0.265337	0.132669	−	0.991160i	$-0.457645\pi$
$41$	2856.00	0.265337	0.132669	+	0.991160i	$0.457645\pi$
$42$	0	0
$43$	13492.0	1.11277	0.556385	−	0.830925i	$-0.312188\pi$
$43$	13492.0	1.11277	0.556385	+	0.830925i	$0.312188\pi$
$44$	0	0
$45$	3564.00	0.262365
$46$	0	0
$47$	18372.0	1.21314	0.606571	−	0.795029i	$-0.292544\pi$
$47$	18372.0	1.21314	0.606571	+	0.795029i	$0.292544\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	10836.0	0.583369
$52$	0	0
$53$	−4374.00	−0.213889	−0.106945	−	0.994265i	$-0.534107\pi$
$53$	−4374.00	−0.213889	−0.106945	+	0.994265i	$0.534107\pi$
$54$	0	0
$55$	20680.0	0.921815
$56$	0	0
$57$	23796.0	0.970101
$58$	0	0
$59$	−30248.0	−1.13127	−0.565635	−	0.824655i	$-0.691369\pi$
$59$	−30248.0	−1.13127	−0.565635	+	0.824655i	$0.691369\pi$
$60$	0	0
$61$	19542.0	0.672426	0.336213	−	0.941786i	$-0.390854\pi$
$61$	19542.0	0.672426	0.336213	+	0.941786i	$0.390854\pi$
$62$	0	0
$63$	3969.00	0.125988
$64$	0	0
$65$	−50952.0	−1.49581
$66$	0	0
$67$	−54328.0	−1.47855	−0.739276	−	0.673402i	$-0.764832\pi$
$67$	−54328.0	−1.47855	−0.739276	+	0.673402i	$0.764832\pi$
$68$	0	0
$69$	10710.0	0.270811
$70$	0	0
$71$	10730.0	0.252612	0.126306	−	0.991991i	$-0.459688\pi$
$71$	10730.0	0.252612	0.126306	+	0.991991i	$0.459688\pi$
$72$	0	0
$73$	35374.0	0.776921	0.388461	−	0.921465i	$-0.373007\pi$
$73$	35374.0	0.776921	0.388461	+	0.921465i	$0.373007\pi$
$74$	0	0
$75$	−10701.0	−0.219670
$76$	0	0
$77$	23030.0	0.442657
$78$	0	0
$79$	49956.0	0.900575	0.450288	−	0.892884i	$-0.351321\pi$
$79$	49956.0	0.900575	0.450288	+	0.892884i	$0.351321\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	26948.0	0.429370	0.214685	−	0.976683i	$-0.431128\pi$
$83$	26948.0	0.429370	0.214685	+	0.976683i	$0.431128\pi$
$84$	0	0
$85$	52976.0	0.795302
$86$	0	0
$87$	32526.0	0.460715
$88$	0	0
$89$	100776.	1.34860	0.674298	−	0.738459i	$-0.264446\pi$
$89$	100776.	1.34860	0.674298	+	0.738459i	$0.264446\pi$
$90$	0	0
$91$	−56742.0	−0.718292
$92$	0	0
$93$	−50544.0	−0.605986
$94$	0	0
$95$	116336.	1.32253
$96$	0	0
$97$	77134.0	0.832370	0.416185	−	0.909280i	$-0.363367\pi$
$97$	77134.0	0.832370	0.416185	+	0.909280i	$0.363367\pi$
$98$	0	0
$99$	38070.0	0.390387
$100$	0	0
$101$	99464.0	0.970203	0.485101	−	0.874458i	$-0.338783\pi$
$101$	99464.0	0.970203	0.485101	+	0.874458i	$0.338783\pi$
$102$	0	0
$103$	−66944.0	−0.621754	−0.310877	−	0.950450i	$-0.600623\pi$
$103$	−66944.0	−0.621754	−0.310877	+	0.950450i	$0.600623\pi$
$104$	0	0
$105$	19404.0	0.171758
$106$	0	0
$107$	228198.	1.92687	0.963435	−	0.267942i	$-0.0863437\pi$
$107$	228198.	1.92687	0.963435	+	0.267942i	$0.0863437\pi$
$108$	0	0
$109$	−95186.0	−0.767374	−0.383687	−	0.923463i	$-0.625346\pi$
$109$	−95186.0	−0.767374	−0.383687	+	0.923463i	$0.625346\pi$
$110$	0	0
$111$	−58302.0	−0.449134
$112$	0	0
$113$	−261142.	−1.92389	−0.961946	−	0.273240i	$-0.911905\pi$
$113$	−261142.	−1.92389	−0.961946	+	0.273240i	$0.911905\pi$
$114$	0	0
$115$	52360.0	0.369194
$116$	0	0
$117$	−93798.0	−0.633474
$118$	0	0
$119$	58996.0	0.381905
$120$	0	0
$121$	59849.0	0.371615
$122$	0	0
$123$	25704.0	0.153193
$124$	0	0
$125$	−189816.	−1.08657
$126$	0	0
$127$	167652.	0.922358	0.461179	−	0.887307i	$-0.347427\pi$
$127$	167652.	0.922358	0.461179	+	0.887307i	$0.347427\pi$
$128$	0	0
$129$	121428.	0.642458
$130$	0	0
$131$	83068.0	0.422917	0.211459	−	0.977387i	$-0.432179\pi$
$131$	83068.0	0.422917	0.211459	+	0.977387i	$0.432179\pi$
$132$	0	0
$133$	129556.	0.635080
$134$	0	0
$135$	32076.0	0.151477
$136$	0	0
$137$	162254.	0.738574	0.369287	−	0.929315i	$-0.379602\pi$
$137$	162254.	0.738574	0.369287	+	0.929315i	$0.379602\pi$
$138$	0	0
$139$	58844.0	0.258324	0.129162	−	0.991623i	$-0.458771\pi$
$139$	58844.0	0.258324	0.129162	+	0.991623i	$0.458771\pi$
$140$	0	0
$141$	165348.	0.700408
$142$	0	0
$143$	−544260.	−2.22570
$144$	0	0
$145$	159016.	0.628088
$146$	0	0
$147$	21609.0	0.0824786
$148$	0	0
$149$	430698.	1.58930	0.794652	−	0.607065i	$-0.207653\pi$
$149$	430698.	1.58930	0.794652	+	0.607065i	$0.207653\pi$
$150$	0	0
$151$	500936.	1.78789	0.893943	−	0.448180i	$-0.147928\pi$
$151$	500936.	1.78789	0.893943	+	0.448180i	$0.147928\pi$
$152$	0	0
$153$	97524.0	0.336808
$154$	0	0
$155$	−247104.	−0.826134
$156$	0	0
$157$	−280258.	−0.907421	−0.453711	−	0.891149i	$-0.649900\pi$
$157$	−280258.	−0.907421	−0.453711	+	0.891149i	$0.649900\pi$
$158$	0	0
$159$	−39366.0	−0.123489
$160$	0	0
$161$	58310.0	0.177288
$162$	0	0
$163$	120016.	0.353810	0.176905	−	0.984228i	$-0.443391\pi$
$163$	120016.	0.353810	0.176905	+	0.984228i	$0.443391\pi$
$164$	0	0
$165$	186120.	0.532210
$166$	0	0
$167$	−546932.	−1.51755	−0.758774	−	0.651355i	$-0.774201\pi$
$167$	−546932.	−1.51755	−0.758774	+	0.651355i	$0.774201\pi$
$168$	0	0
$169$	969671.	2.61161
$170$	0	0
$171$	214164.	0.560088
$172$	0	0
$173$	−367096.	−0.932533	−0.466267	−	0.884644i	$-0.654401\pi$
$173$	−367096.	−0.932533	−0.466267	+	0.884644i	$0.654401\pi$
$174$	0	0
$175$	−58261.0	−0.143808
$176$	0	0
$177$	−272232.	−0.653140
$178$	0	0
$179$	88890.0	0.207358	0.103679	−	0.994611i	$-0.466939\pi$
$179$	88890.0	0.207358	0.103679	+	0.994611i	$0.466939\pi$
$180$	0	0
$181$	−782118.	−1.77450	−0.887250	−	0.461290i	$-0.847387\pi$
$181$	−782118.	−1.77450	−0.887250	+	0.461290i	$0.847387\pi$
$182$	0	0
$183$	175878.	0.388225
$184$	0	0
$185$	−285032.	−0.612300
$186$	0	0
$187$	565880.	1.18337
$188$	0	0
$189$	35721.0	0.0727393
$190$	0	0
$191$	−763350.	−1.51405	−0.757025	−	0.653386i	$-0.773348\pi$
$191$	−763350.	−1.51405	−0.757025	+	0.653386i	$0.773348\pi$
$192$	0	0
$193$	−377002.	−0.728535	−0.364267	−	0.931294i	$-0.618681\pi$
$193$	−377002.	−0.728535	−0.364267	+	0.931294i	$0.618681\pi$
$194$	0	0
$195$	−458568.	−0.863609
$196$	0	0
$197$	−68678.0	−0.126082	−0.0630409	−	0.998011i	$-0.520080\pi$
$197$	−68678.0	−0.126082	−0.0630409	+	0.998011i	$0.520080\pi$
$198$	0	0
$199$	−182576.	−0.326822	−0.163411	−	0.986558i	$-0.552250\pi$
$199$	−182576.	−0.326822	−0.163411	+	0.986558i	$0.552250\pi$
$200$	0	0
$201$	−488952.	−0.853643
$202$	0	0
$203$	177086.	0.301609
$204$	0	0
$205$	125664.	0.208846
$206$	0	0
$207$	96390.0	0.156353
$208$	0	0
$209$	1.24268e6	1.96786
$210$	0	0
$211$	−232652.	−0.359750	−0.179875	−	0.983689i	$-0.557569\pi$
$211$	−232652.	−0.359750	−0.179875	+	0.983689i	$0.557569\pi$
$212$	0	0
$213$	96570.0	0.145846
$214$	0	0
$215$	593648.	0.875856
$216$	0	0
$217$	−275184.	−0.396711
$218$	0	0
$219$	318366.	0.448556
$220$	0	0
$221$	−1.39423e6	−1.92023
$222$	0	0
$223$	−167144.	−0.225076	−0.112538	−	0.993647i	$-0.535898\pi$
$223$	−167144.	−0.225076	−0.112538	+	0.993647i	$0.535898\pi$
$224$	0	0
$225$	−96309.0	−0.126827
$226$	0	0
$227$	−415728.	−0.535482	−0.267741	−	0.963491i	$-0.586277\pi$
$227$	−415728.	−0.535482	−0.267741	+	0.963491i	$0.586277\pi$
$228$	0	0
$229$	473482.	0.596643	0.298322	−	0.954465i	$-0.403573\pi$
$229$	473482.	0.596643	0.298322	+	0.954465i	$0.403573\pi$
$230$	0	0
$231$	207270.	0.255568
$232$	0	0
$233$	−1.55655e6	−1.87833	−0.939166	−	0.343465i	$-0.888399\pi$
$233$	−1.55655e6	−1.87833	−0.939166	+	0.343465i	$0.888399\pi$
$234$	0	0
$235$	808368.	0.954859
$236$	0	0
$237$	449604.	0.519947
$238$	0	0
$239$	−655890.	−0.742739	−0.371370	−	0.928485i	$-0.621112\pi$
$239$	−655890.	−0.742739	−0.371370	+	0.928485i	$0.621112\pi$
$240$	0	0
$241$	−889474.	−0.986485	−0.493243	−	0.869892i	$-0.664189\pi$
$241$	−889474.	−0.986485	−0.493243	+	0.869892i	$0.664189\pi$
$242$	0	0
$243$	59049.0	0.0641500
$244$	0	0
$245$	105644.	0.112442
$246$	0	0
$247$	−3.06175e6	−3.19321
$248$	0	0
$249$	242532.	0.247897
$250$	0	0
$251$	−131832.	−0.132080	−0.0660399	−	0.997817i	$-0.521036\pi$
$251$	−131832.	−0.132080	−0.0660399	+	0.997817i	$0.521036\pi$
$252$	0	0
$253$	559300.	0.549343
$254$	0	0
$255$	476784.	0.459168
$256$	0	0
$257$	−1.46482e6	−1.38341	−0.691704	−	0.722181i	$-0.743140\pi$
$257$	−1.46482e6	−1.38341	−0.691704	+	0.722181i	$0.743140\pi$
$258$	0	0
$259$	−317422.	−0.294027
$260$	0	0
$261$	292734.	0.265994
$262$	0	0
$263$	−1.47969e6	−1.31911	−0.659556	−	0.751656i	$-0.729255\pi$
$263$	−1.47969e6	−1.31911	−0.659556	+	0.751656i	$0.729255\pi$
$264$	0	0
$265$	−192456.	−0.168351
$266$	0	0
$267$	906984.	0.778613
$268$	0	0
$269$	−187852.	−0.158283	−0.0791417	−	0.996863i	$-0.525218\pi$
$269$	−187852.	−0.158283	−0.0791417	+	0.996863i	$0.525218\pi$
$270$	0	0
$271$	−193800.	−0.160299	−0.0801495	−	0.996783i	$-0.525540\pi$
$271$	−193800.	−0.160299	−0.0801495	+	0.996783i	$0.525540\pi$
$272$	0	0
$273$	−510678.	−0.414706
$274$	0	0
$275$	−558830.	−0.445603
$276$	0	0
$277$	−617062.	−0.483203	−0.241601	−	0.970376i	$-0.577673\pi$
$277$	−617062.	−0.483203	−0.241601	+	0.970376i	$0.577673\pi$
$278$	0	0
$279$	−454896.	−0.349866
$280$	0	0
$281$	−1.73129e6	−1.30799	−0.653994	−	0.756499i	$-0.726908\pi$
$281$	−1.73129e6	−1.30799	−0.653994	+	0.756499i	$0.726908\pi$
$282$	0	0
$283$	−356020.	−0.264246	−0.132123	−	0.991233i	$-0.542179\pi$
$283$	−356020.	−0.264246	−0.132123	+	0.991233i	$0.542179\pi$
$284$	0	0
$285$	1.04702e6	0.763562
$286$	0	0
$287$	139944.	0.100288
$288$	0	0
$289$	29759.0	0.0209592
$290$	0	0
$291$	694206.	0.480569
$292$	0	0
$293$	536664.	0.365202	0.182601	−	0.983187i	$-0.441548\pi$
$293$	536664.	0.365202	0.182601	+	0.983187i	$0.441548\pi$
$294$	0	0
$295$	−1.33091e6	−0.890419
$296$	0	0
$297$	342630.	0.225390
$298$	0	0
$299$	−1.37802e6	−0.891410
$300$	0	0
$301$	661108.	0.420587
$302$	0	0
$303$	895176.	0.560147
$304$	0	0
$305$	859848.	0.529264
$306$	0	0
$307$	2.88398e6	1.74641	0.873205	−	0.487353i	$-0.162037\pi$
$307$	2.88398e6	1.74641	0.873205	+	0.487353i	$0.162037\pi$
$308$	0	0
$309$	−602496.	−0.358970
$310$	0	0
$311$	1.02958e6	0.603614	0.301807	−	0.953369i	$-0.402410\pi$
$311$	1.02958e6	0.603614	0.301807	+	0.953369i	$0.402410\pi$
$312$	0	0
$313$	1.39297e6	0.803676	0.401838	−	0.915711i	$-0.368372\pi$
$313$	1.39297e6	0.803676	0.401838	+	0.915711i	$0.368372\pi$
$314$	0	0
$315$	174636.	0.0991648
$316$	0	0
$317$	−780030.	−0.435977	−0.217988	−	0.975951i	$-0.569949\pi$
$317$	−780030.	−0.435977	−0.217988	+	0.975951i	$0.569949\pi$
$318$	0	0
$319$	1.69858e6	0.934565
$320$	0	0
$321$	2.05378e6	1.11248
$322$	0	0
$323$	3.18338e6	1.69778
$324$	0	0
$325$	1.37686e6	0.723073
$326$	0	0
$327$	−856674.	−0.443043
$328$	0	0
$329$	900228.	0.458525
$330$	0	0
$331$	1.41204e6	0.708399	0.354200	−	0.935170i	$-0.384753\pi$
$331$	1.41204e6	0.708399	0.354200	+	0.935170i	$0.384753\pi$
$332$	0	0
$333$	−524718.	−0.259308
$334$	0	0
$335$	−2.39043e6	−1.16376
$336$	0	0
$337$	−634662.	−0.304416	−0.152208	−	0.988348i	$-0.548638\pi$
$337$	−634662.	−0.304416	−0.152208	+	0.988348i	$0.548638\pi$
$338$	0	0
$339$	−2.35028e6	−1.11076
$340$	0	0
$341$	−2.63952e6	−1.22925
$342$	0	0
$343$	117649.	0.0539949
$344$	0	0
$345$	471240.	0.213154
$346$	0	0
$347$	−3.07942e6	−1.37292	−0.686460	−	0.727167i	$-0.740837\pi$
$347$	−3.07942e6	−1.37292	−0.686460	+	0.727167i	$0.740837\pi$
$348$	0	0
$349$	−2.60671e6	−1.14559	−0.572796	−	0.819698i	$-0.694141\pi$
$349$	−2.60671e6	−1.14559	−0.572796	+	0.819698i	$0.694141\pi$
$350$	0	0
$351$	−844182.	−0.365736
$352$	0	0
$353$	−63132.0	−0.0269658	−0.0134829	−	0.999909i	$-0.504292\pi$
$353$	−63132.0	−0.0269658	−0.0134829	+	0.999909i	$0.504292\pi$
$354$	0	0
$355$	472120.	0.198830
$356$	0	0
$357$	530964.	0.220493
$358$	0	0
$359$	−479270.	−0.196266	−0.0981328	−	0.995173i	$-0.531287\pi$
$359$	−479270.	−0.196266	−0.0981328	+	0.995173i	$0.531287\pi$
$360$	0	0
$361$	4.51464e6	1.82329
$362$	0	0
$363$	538641.	0.214552
$364$	0	0
$365$	1.55646e6	0.611512
$366$	0	0
$367$	1.33451e6	0.517199	0.258599	−	0.965985i	$-0.416739\pi$
$367$	1.33451e6	0.517199	0.258599	+	0.965985i	$0.416739\pi$
$368$	0	0
$369$	231336.	0.0884458
$370$	0	0
$371$	−214326.	−0.0808426
$372$	0	0
$373$	−1.69759e6	−0.631774	−0.315887	−	0.948797i	$-0.602302\pi$
$373$	−1.69759e6	−0.631774	−0.315887	+	0.948797i	$0.602302\pi$
$374$	0	0
$375$	−1.70834e6	−0.627332
$376$	0	0
$377$	−4.18501e6	−1.51650
$378$	0	0
$379$	−2.51074e6	−0.897850	−0.448925	−	0.893569i	$-0.648193\pi$
$379$	−2.51074e6	−0.897850	−0.448925	+	0.893569i	$0.648193\pi$
$380$	0	0
$381$	1.50887e6	0.532524
$382$	0	0
$383$	−559144.	−0.194772	−0.0973860	−	0.995247i	$-0.531048\pi$
$383$	−559144.	−0.194772	−0.0973860	+	0.995247i	$0.531048\pi$
$384$	0	0
$385$	1.01332e6	0.348413
$386$	0	0
$387$	1.09285e6	0.370923
$388$	0	0
$389$	4.51055e6	1.51132	0.755658	−	0.654966i	$-0.227317\pi$
$389$	4.51055e6	1.51132	0.755658	+	0.654966i	$0.227317\pi$
$390$	0	0
$391$	1.43276e6	0.473949
$392$	0	0
$393$	747612.	0.244171
$394$	0	0
$395$	2.19806e6	0.708839
$396$	0	0
$397$	5.19862e6	1.65543	0.827717	−	0.561145i	$-0.189639\pi$
$397$	5.19862e6	1.65543	0.827717	+	0.561145i	$0.189639\pi$
$398$	0	0
$399$	1.16600e6	0.366664
$400$	0	0
$401$	−6.34816e6	−1.97146	−0.985728	−	0.168346i	$-0.946157\pi$
$401$	−6.34816e6	−1.97146	−0.985728	+	0.168346i	$0.946157\pi$
$402$	0	0
$403$	6.50333e6	1.99468
$404$	0	0
$405$	288684.	0.0874551
$406$	0	0
$407$	−3.04466e6	−0.911072
$408$	0	0
$409$	−181642.	−0.0536918	−0.0268459	−	0.999640i	$-0.508546\pi$
$409$	−181642.	−0.0536918	−0.0268459	+	0.999640i	$0.508546\pi$
$410$	0	0
$411$	1.46029e6	0.426416
$412$	0	0
$413$	−1.48215e6	−0.427580
$414$	0	0
$415$	1.18571e6	0.337955
$416$	0	0
$417$	529596.	0.149144
$418$	0	0
$419$	5.62699e6	1.56582	0.782909	−	0.622136i	$-0.213735\pi$
$419$	5.62699e6	1.56582	0.782909	+	0.622136i	$0.213735\pi$
$420$	0	0
$421$	−4.42671e6	−1.21724	−0.608619	−	0.793462i	$-0.708276\pi$
$421$	−4.42671e6	−1.21724	−0.608619	+	0.793462i	$0.708276\pi$
$422$	0	0
$423$	1.48813e6	0.404381
$424$	0	0
$425$	−1.43156e6	−0.384447
$426$	0	0
$427$	957558.	0.254153
$428$	0	0
$429$	−4.89834e6	−1.28501
$430$	0	0
$431$	1.44163e6	0.373817	0.186909	−	0.982377i	$-0.440153\pi$
$431$	1.44163e6	0.373817	0.186909	+	0.982377i	$0.440153\pi$
$432$	0	0
$433$	−3.89661e6	−0.998775	−0.499387	−	0.866379i	$-0.666442\pi$
$433$	−3.89661e6	−0.998775	−0.499387	+	0.866379i	$0.666442\pi$
$434$	0	0
$435$	1.43114e6	0.362627
$436$	0	0
$437$	3.14636e6	0.788143
$438$	0	0
$439$	−5.11207e6	−1.26601	−0.633003	−	0.774149i	$-0.718178\pi$
$439$	−5.11207e6	−1.26601	−0.633003	+	0.774149i	$0.718178\pi$
$440$	0	0
$441$	194481.	0.0476190
$442$	0	0
$443$	−5.44070e6	−1.31718	−0.658591	−	0.752501i	$-0.728847\pi$
$443$	−5.44070e6	−1.31718	−0.658591	+	0.752501i	$0.728847\pi$
$444$	0	0
$445$	4.43414e6	1.06147
$446$	0	0
$447$	3.87628e6	0.917586
$448$	0	0
$449$	−1.31525e6	−0.307887	−0.153943	−	0.988080i	$-0.549197\pi$
$449$	−1.31525e6	−0.307887	−0.153943	+	0.988080i	$0.549197\pi$
$450$	0	0
$451$	1.34232e6	0.310753
$452$	0	0
$453$	4.50842e6	1.03224
$454$	0	0
$455$	−2.49665e6	−0.565365
$456$	0	0
$457$	2.77604e6	0.621778	0.310889	−	0.950446i	$-0.399373\pi$
$457$	2.77604e6	0.621778	0.310889	+	0.950446i	$0.399373\pi$
$458$	0	0
$459$	877716.	0.194456
$460$	0	0
$461$	138080.	0.0302607	0.0151303	−	0.999886i	$-0.495184\pi$
$461$	138080.	0.0302607	0.0151303	+	0.999886i	$0.495184\pi$
$462$	0	0
$463$	364076.	0.0789295	0.0394648	−	0.999221i	$-0.487435\pi$
$463$	364076.	0.0789295	0.0394648	+	0.999221i	$0.487435\pi$
$464$	0	0
$465$	−2.22394e6	−0.476969
$466$	0	0
$467$	5.73897e6	1.21770	0.608852	−	0.793284i	$-0.291630\pi$
$467$	5.73897e6	1.21770	0.608852	+	0.793284i	$0.291630\pi$
$468$	0	0
$469$	−2.66207e6	−0.558840
$470$	0	0
$471$	−2.52232e6	−0.523900
$472$	0	0
$473$	6.34124e6	1.30323
$474$	0	0
$475$	−3.14372e6	−0.639307
$476$	0	0
$477$	−354294.	−0.0712964
$478$	0	0
$479$	−5.51996e6	−1.09925	−0.549625	−	0.835411i	$-0.685230\pi$
$479$	−5.51996e6	−1.09925	−0.549625	+	0.835411i	$0.685230\pi$
$480$	0	0
$481$	7.50152e6	1.47838
$482$	0	0
$483$	524790.	0.102357
$484$	0	0
$485$	3.39390e6	0.655155
$486$	0	0
$487$	−4.23022e6	−0.808241	−0.404121	−	0.914706i	$-0.632422\pi$
$487$	−4.23022e6	−0.808241	−0.404121	+	0.914706i	$0.632422\pi$
$488$	0	0
$489$	1.08014e6	0.204272
$490$	0	0
$491$	7.21423e6	1.35047	0.675237	−	0.737601i	$-0.264041\pi$
$491$	7.21423e6	1.35047	0.675237	+	0.737601i	$0.264041\pi$
$492$	0	0
$493$	4.35126e6	0.806301
$494$	0	0
$495$	1.67508e6	0.307272
$496$	0	0
$497$	525770.	0.0954783
$498$	0	0
$499$	224804.	0.0404159	0.0202080	−	0.999796i	$-0.493567\pi$
$499$	224804.	0.0404159	0.0202080	+	0.999796i	$0.493567\pi$
$500$	0	0
$501$	−4.92239e6	−0.876156
$502$	0	0
$503$	−5.06983e6	−0.893457	−0.446728	−	0.894670i	$-0.647411\pi$
$503$	−5.06983e6	−0.893457	−0.446728	+	0.894670i	$0.647411\pi$
$504$	0	0
$505$	4.37642e6	0.763643
$506$	0	0
$507$	8.72704e6	1.50781
$508$	0	0
$509$	5.48135e6	0.937763	0.468881	−	0.883261i	$-0.344657\pi$
$509$	5.48135e6	0.937763	0.468881	+	0.883261i	$0.344657\pi$
$510$	0	0
$511$	1.73333e6	0.293649
$512$	0	0
$513$	1.92748e6	0.323367
$514$	0	0
$515$	−2.94554e6	−0.489380
$516$	0	0
$517$	8.63484e6	1.42078
$518$	0	0
$519$	−3.30386e6	−0.538398
$520$	0	0
$521$	7.88932e6	1.27334	0.636671	−	0.771136i	$-0.280311\pi$
$521$	7.88932e6	1.27334	0.636671	+	0.771136i	$0.280311\pi$
$522$	0	0
$523$	9.74950e6	1.55858	0.779288	−	0.626666i	$-0.215581\pi$
$523$	9.74950e6	1.55858	0.779288	+	0.626666i	$0.215581\pi$
$524$	0	0
$525$	−524349.	−0.0830275
$526$	0	0
$527$	−6.76166e6	−1.06054
$528$	0	0
$529$	−5.02024e6	−0.779984
$530$	0	0
$531$	−2.45009e6	−0.377090
$532$	0	0
$533$	−3.30725e6	−0.504253
$534$	0	0
$535$	1.00407e7	1.51663
$536$	0	0
$537$	800010.	0.119718
$538$	0	0
$539$	1.12847e6	0.167309
$540$	0	0
$541$	−4.17454e6	−0.613219	−0.306610	−	0.951835i	$-0.599195\pi$
$541$	−4.17454e6	−0.613219	−0.306610	+	0.951835i	$0.599195\pi$
$542$	0	0
$543$	−7.03906e6	−1.02451
$544$	0	0
$545$	−4.18818e6	−0.603997
$546$	0	0
$547$	2.72887e6	0.389955	0.194978	−	0.980808i	$-0.437537\pi$
$547$	2.72887e6	0.389955	0.194978	+	0.980808i	$0.437537\pi$
$548$	0	0
$549$	1.58290e6	0.224142
$550$	0	0
$551$	9.55542e6	1.34082
$552$	0	0
$553$	2.44784e6	0.340385
$554$	0	0
$555$	−2.56529e6	−0.353512
$556$	0	0
$557$	−7.35103e6	−1.00395	−0.501973	−	0.864883i	$-0.667392\pi$
$557$	−7.35103e6	−1.00395	−0.501973	+	0.864883i	$0.667392\pi$
$558$	0	0
$559$	−1.56237e7	−2.11473
$560$	0	0
$561$	5.09292e6	0.683219
$562$	0	0
$563$	1.37821e7	1.83250	0.916248	−	0.400612i	$-0.131202\pi$
$563$	1.37821e7	1.83250	0.916248	+	0.400612i	$0.131202\pi$
$564$	0	0
$565$	−1.14902e7	−1.51429
$566$	0	0
$567$	321489.	0.0419961
$568$	0	0
$569$	2.71217e6	0.351186	0.175593	−	0.984463i	$-0.443816\pi$
$569$	2.71217e6	0.351186	0.175593	+	0.984463i	$0.443816\pi$
$570$	0	0
$571$	−4.94398e6	−0.634580	−0.317290	−	0.948329i	$-0.602773\pi$
$571$	−4.94398e6	−0.634580	−0.317290	+	0.948329i	$0.602773\pi$
$572$	0	0
$573$	−6.87015e6	−0.874137
$574$	0	0
$575$	−1.41491e6	−0.178468
$576$	0	0
$577$	−1.32683e7	−1.65911	−0.829556	−	0.558424i	$-0.811406\pi$
$577$	−1.32683e7	−1.65911	−0.829556	+	0.558424i	$0.811406\pi$
$578$	0	0
$579$	−3.39302e6	−0.420620
$580$	0	0
$581$	1.32045e6	0.162286
$582$	0	0
$583$	−2.05578e6	−0.250499
$584$	0	0
$585$	−4.12711e6	−0.498605
$586$	0	0
$587$	−1.67205e6	−0.200287	−0.100144	−	0.994973i	$-0.531930\pi$
$587$	−1.67205e6	−0.200287	−0.100144	+	0.994973i	$0.531930\pi$
$588$	0	0
$589$	−1.48487e7	−1.76360
$590$	0	0
$591$	−618102.	−0.0727933
$592$	0	0
$593$	5.11693e6	0.597548	0.298774	−	0.954324i	$-0.403422\pi$
$593$	5.11693e6	0.597548	0.298774	+	0.954324i	$0.403422\pi$
$594$	0	0
$595$	2.59582e6	0.300596
$596$	0	0
$597$	−1.64318e6	−0.188691
$598$	0	0
$599$	7.15931e6	0.815275	0.407638	−	0.913144i	$-0.366353\pi$
$599$	7.15931e6	0.815275	0.407638	+	0.913144i	$0.366353\pi$
$600$	0	0
$601$	−9.63384e6	−1.08796	−0.543980	−	0.839098i	$-0.683083\pi$
$601$	−9.63384e6	−1.08796	−0.543980	+	0.839098i	$0.683083\pi$
$602$	0	0
$603$	−4.40057e6	−0.492851
$604$	0	0
$605$	2.63336e6	0.292497
$606$	0	0
$607$	−5.52115e6	−0.608216	−0.304108	−	0.952638i	$-0.598358\pi$
$607$	−5.52115e6	−0.608216	−0.304108	+	0.952638i	$0.598358\pi$
$608$	0	0
$609$	1.59377e6	0.174134
$610$	0	0
$611$	−2.12748e7	−2.30548
$612$	0	0
$613$	4.79890e6	0.515811	0.257906	−	0.966170i	$-0.416968\pi$
$613$	4.79890e6	0.515811	0.257906	+	0.966170i	$0.416968\pi$
$614$	0	0
$615$	1.13098e6	0.120577
$616$	0	0
$617$	2.71826e6	0.287461	0.143730	−	0.989617i	$-0.454090\pi$
$617$	2.71826e6	0.287461	0.143730	+	0.989617i	$0.454090\pi$
$618$	0	0
$619$	4.34335e6	0.455615	0.227807	−	0.973706i	$-0.426844\pi$
$619$	4.34335e6	0.455615	0.227807	+	0.973706i	$0.426844\pi$
$620$	0	0
$621$	867510.	0.0902704
$622$	0	0
$623$	4.93802e6	0.509722
$624$	0	0
$625$	−4.63628e6	−0.474755
$626$	0	0
$627$	1.11841e7	1.13614
$628$	0	0
$629$	−7.79951e6	−0.786033
$630$	0	0
$631$	−1.11952e6	−0.111933	−0.0559667	−	0.998433i	$-0.517824\pi$
$631$	−1.11952e6	−0.111933	−0.0559667	+	0.998433i	$0.517824\pi$
$632$	0	0
$633$	−2.09387e6	−0.207702
$634$	0	0
$635$	7.37669e6	0.725984
$636$	0	0
$637$	−2.78036e6	−0.271489
$638$	0	0
$639$	869130.	0.0842040
$640$	0	0
$641$	−3.96588e6	−0.381237	−0.190618	−	0.981664i	$-0.561049\pi$
$641$	−3.96588e6	−0.381237	−0.190618	+	0.981664i	$0.561049\pi$
$642$	0	0
$643$	−1.92086e7	−1.83218	−0.916092	−	0.400968i	$-0.868674\pi$
$643$	−1.92086e7	−1.83218	−0.916092	+	0.400968i	$0.868674\pi$
$644$	0	0
$645$	5.34283e6	0.505676
$646$	0	0
$647$	−4.72739e6	−0.443977	−0.221989	−	0.975049i	$-0.571255\pi$
$647$	−4.72739e6	−0.443977	−0.221989	+	0.975049i	$0.571255\pi$
$648$	0	0
$649$	−1.42166e7	−1.32490
$650$	0	0
$651$	−2.47666e6	−0.229041
$652$	0	0
$653$	1.21159e7	1.11192	0.555958	−	0.831210i	$-0.312352\pi$
$653$	1.21159e7	1.11192	0.555958	+	0.831210i	$0.312352\pi$
$654$	0	0
$655$	3.65499e6	0.332877
$656$	0	0
$657$	2.86529e6	0.258974
$658$	0	0
$659$	−5.91457e6	−0.530530	−0.265265	−	0.964176i	$-0.585459\pi$
$659$	−5.91457e6	−0.530530	−0.265265	+	0.964176i	$0.585459\pi$
$660$	0	0
$661$	1.41779e7	1.26214	0.631072	−	0.775724i	$-0.282615\pi$
$661$	1.41779e7	1.26214	0.631072	+	0.775724i	$0.282615\pi$
$662$	0	0
$663$	−1.25481e7	−1.10865
$664$	0	0
$665$	5.70046e6	0.499869
$666$	0	0
$667$	4.30066e6	0.374301
$668$	0	0
$669$	−1.50430e6	−0.129948
$670$	0	0
$671$	9.18474e6	0.787518
$672$	0	0
$673$	1.34245e7	1.14251	0.571256	−	0.820772i	$-0.306456\pi$
$673$	1.34245e7	1.14251	0.571256	+	0.820772i	$0.306456\pi$
$674$	0	0
$675$	−866781.	−0.0732234
$676$	0	0
$677$	−8.67312e6	−0.727283	−0.363642	−	0.931539i	$-0.618467\pi$
$677$	−8.67312e6	−0.727283	−0.363642	+	0.931539i	$0.618467\pi$
$678$	0	0
$679$	3.77957e6	0.314606
$680$	0	0
$681$	−3.74155e6	−0.309161
$682$	0	0
$683$	−1.49777e7	−1.22855	−0.614275	−	0.789092i	$-0.710552\pi$
$683$	−1.49777e7	−1.22855	−0.614275	+	0.789092i	$0.710552\pi$
$684$	0	0
$685$	7.13918e6	0.581329
$686$	0	0
$687$	4.26134e6	0.344472
$688$	0	0
$689$	5.06509e6	0.406480
$690$	0	0
$691$	8.76742e6	0.698517	0.349258	−	0.937026i	$-0.386434\pi$
$691$	8.76742e6	0.698517	0.349258	+	0.937026i	$0.386434\pi$
$692$	0	0
$693$	1.86543e6	0.147552
$694$	0	0
$695$	2.58914e6	0.203326
$696$	0	0
$697$	3.43862e6	0.268104
$698$	0	0
$699$	−1.40089e7	−1.08445
$700$	0	0
$701$	−1.99459e7	−1.53306	−0.766529	−	0.642209i	$-0.778018\pi$
$701$	−1.99459e7	−1.53306	−0.766529	+	0.642209i	$0.778018\pi$
$702$	0	0
$703$	−1.71278e7	−1.30712
$704$	0	0
$705$	7.27531e6	0.551288
$706$	0	0
$707$	4.87374e6	0.366702
$708$	0	0
$709$	−2.66387e7	−1.99020	−0.995100	−	0.0988699i	$-0.968477\pi$
$709$	−2.66387e7	−1.99020	−0.995100	+	0.0988699i	$0.968477\pi$
$710$	0	0
$711$	4.04644e6	0.300192
$712$	0	0
$713$	−6.68304e6	−0.492323
$714$	0	0
$715$	−2.39474e7	−1.75184
$716$	0	0
$717$	−5.90301e6	−0.428821
$718$	0	0
$719$	1.17408e6	0.0846985	0.0423492	−	0.999103i	$-0.486516\pi$
$719$	1.17408e6	0.0846985	0.0423492	+	0.999103i	$0.486516\pi$
$720$	0	0
$721$	−3.28026e6	−0.235001
$722$	0	0
$723$	−8.00527e6	−0.569548
$724$	0	0
$725$	−4.29705e6	−0.303616
$726$	0	0
$727$	−1.05734e7	−0.741957	−0.370979	−	0.928641i	$-0.620978\pi$
$727$	−1.05734e7	−0.741957	−0.370979	+	0.928641i	$0.620978\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	1.62444e7	1.12437
$732$	0	0
$733$	2.31670e7	1.59261	0.796306	−	0.604894i	$-0.206785\pi$
$733$	2.31670e7	1.59261	0.796306	+	0.604894i	$0.206785\pi$
$734$	0	0
$735$	950796.	0.0649186
$736$	0	0
$737$	−2.55342e7	−1.73162
$738$	0	0
$739$	1.55334e7	1.04630	0.523148	−	0.852242i	$-0.324757\pi$
$739$	1.55334e7	1.04630	0.523148	+	0.852242i	$0.324757\pi$
$740$	0	0
$741$	−2.75558e7	−1.84360
$742$	0	0
$743$	−2.54630e7	−1.69215	−0.846074	−	0.533066i	$-0.821040\pi$
$743$	−2.54630e7	−1.69215	−0.846074	+	0.533066i	$0.821040\pi$
$744$	0	0
$745$	1.89507e7	1.25094
$746$	0	0
$747$	2.18279e6	0.143123
$748$	0	0
$749$	1.11817e7	0.728288
$750$	0	0
$751$	−9.88512e6	−0.639561	−0.319781	−	0.947492i	$-0.603609\pi$
$751$	−9.88512e6	−0.639561	−0.319781	+	0.947492i	$0.603609\pi$
$752$	0	0
$753$	−1.18649e6	−0.0762563
$754$	0	0
$755$	2.20412e7	1.40724
$756$	0	0
$757$	−6.41980e6	−0.407176	−0.203588	−	0.979057i	$-0.565260\pi$
$757$	−6.41980e6	−0.407176	−0.203588	+	0.979057i	$0.565260\pi$
$758$	0	0
$759$	5.03370e6	0.317163
$760$	0	0
$761$	−5.05052e6	−0.316136	−0.158068	−	0.987428i	$-0.550527\pi$
$761$	−5.05052e6	−0.316136	−0.158068	+	0.987428i	$0.550527\pi$
$762$	0	0
$763$	−4.66411e6	−0.290040
$764$	0	0
$765$	4.29106e6	0.265101
$766$	0	0
$767$	3.50272e7	2.14989
$768$	0	0
$769$	2.28169e6	0.139136	0.0695682	−	0.997577i	$-0.477838\pi$
$769$	2.28169e6	0.139136	0.0695682	+	0.997577i	$0.477838\pi$
$770$	0	0
$771$	−1.31833e7	−0.798711
$772$	0	0
$773$	2.73777e7	1.64797	0.823984	−	0.566613i	$-0.191746\pi$
$773$	2.73777e7	1.64797	0.823984	+	0.566613i	$0.191746\pi$
$774$	0	0
$775$	6.67742e6	0.399351
$776$	0	0
$777$	−2.85680e6	−0.169757
$778$	0	0
$779$	7.55126e6	0.445837
$780$	0	0
$781$	5.04310e6	0.295849
$782$	0	0
$783$	2.63461e6	0.153572
$784$	0	0
$785$	−1.23314e7	−0.714227
$786$	0	0
$787$	−2.04263e7	−1.17558	−0.587791	−	0.809013i	$-0.700002\pi$
$787$	−2.04263e7	−1.17558	−0.587791	+	0.809013i	$0.700002\pi$
$788$	0	0
$789$	−1.33172e7	−0.761589
$790$	0	0
$791$	−1.27960e7	−0.727163
$792$	0	0
$793$	−2.26296e7	−1.27789
$794$	0	0
$795$	−1.73210e6	−0.0971977
$796$	0	0
$797$	2.31557e7	1.29126	0.645628	−	0.763652i	$-0.276596\pi$
$797$	2.31557e7	1.29126	0.645628	+	0.763652i	$0.276596\pi$
$798$	0	0
$799$	2.21199e7	1.22579
$800$	0	0
$801$	8.16286e6	0.449532
$802$	0	0
$803$	1.66258e7	0.909899
$804$	0	0
$805$	2.56564e6	0.139542
$806$	0	0
$807$	−1.69067e6	−0.0913849
$808$	0	0
$809$	1.14894e7	0.617203	0.308601	−	0.951191i	$-0.400139\pi$
$809$	1.14894e7	0.617203	0.308601	+	0.951191i	$0.400139\pi$
$810$	0	0
$811$	−1.72443e7	−0.920648	−0.460324	−	0.887751i	$-0.652267\pi$
$811$	−1.72443e7	−0.920648	−0.460324	+	0.887751i	$0.652267\pi$
$812$	0	0
$813$	−1.74420e6	−0.0925486
$814$	0	0
$815$	5.28070e6	0.278482
$816$	0	0
$817$	3.56728e7	1.86975
$818$	0	0
$819$	−4.59610e6	−0.239431
$820$	0	0
$821$	1.37077e6	0.0709752	0.0354876	−	0.999370i	$-0.488702\pi$
$821$	1.37077e6	0.0709752	0.0354876	+	0.999370i	$0.488702\pi$
$822$	0	0
$823$	−8.56851e6	−0.440967	−0.220483	−	0.975391i	$-0.570763\pi$
$823$	−8.56851e6	−0.440967	−0.220483	+	0.975391i	$0.570763\pi$
$824$	0	0
$825$	−5.02947e6	−0.257269
$826$	0	0
$827$	1.33258e6	0.0677533	0.0338766	−	0.999426i	$-0.489215\pi$
$827$	1.33258e6	0.0677533	0.0338766	+	0.999426i	$0.489215\pi$
$828$	0	0
$829$	6.25659e6	0.316193	0.158096	−	0.987424i	$-0.449464\pi$
$829$	6.25659e6	0.316193	0.158096	+	0.987424i	$0.449464\pi$
$830$	0	0
$831$	−5.55356e6	−0.278977
$832$	0	0
$833$	2.89080e6	0.144346
$834$	0	0
$835$	−2.40650e7	−1.19446
$836$	0	0
$837$	−4.09406e6	−0.201995
$838$	0	0
$839$	1.61258e6	0.0790891	0.0395445	−	0.999218i	$-0.487409\pi$
$839$	1.61258e6	0.0790891	0.0395445	+	0.999218i	$0.487409\pi$
$840$	0	0
$841$	−7.45015e6	−0.363225
$842$	0	0
$843$	−1.55816e7	−0.755168
$844$	0	0
$845$	4.26655e7	2.05558
$846$	0	0
$847$	2.93260e6	0.140457
$848$	0	0
$849$	−3.20418e6	−0.152562
$850$	0	0
$851$	−7.70882e6	−0.364892
$852$	0	0
$853$	−3.44919e7	−1.62310	−0.811548	−	0.584286i	$-0.801375\pi$
$853$	−3.44919e7	−1.62310	−0.811548	+	0.584286i	$0.801375\pi$
$854$	0	0
$855$	9.42322e6	0.440843
$856$	0	0
$857$	−1.85487e7	−0.862704	−0.431352	−	0.902184i	$-0.641963\pi$
$857$	−1.85487e7	−0.862704	−0.431352	+	0.902184i	$0.641963\pi$
$858$	0	0
$859$	2.56435e7	1.18575	0.592877	−	0.805293i	$-0.297992\pi$
$859$	2.56435e7	1.18575	0.592877	+	0.805293i	$0.297992\pi$
$860$	0	0
$861$	1.25950e6	0.0579014
$862$	0	0
$863$	2.49899e7	1.14219	0.571093	−	0.820885i	$-0.306519\pi$
$863$	2.49899e7	1.14219	0.571093	+	0.820885i	$0.306519\pi$
$864$	0	0
$865$	−1.61522e7	−0.733993
$866$	0	0
$867$	267831.	0.0121008
$868$	0	0
$869$	2.34793e7	1.05472
$870$	0	0
$871$	6.29118e7	2.80987
$872$	0	0
$873$	6.24785e6	0.277457
$874$	0	0
$875$	−9.30098e6	−0.410685
$876$	0	0
$877$	−3.46337e7	−1.52055	−0.760274	−	0.649602i	$-0.774935\pi$
$877$	−3.46337e7	−1.52055	−0.760274	+	0.649602i	$0.774935\pi$
$878$	0	0
$879$	4.82998e6	0.210850
$880$	0	0
$881$	2.92434e7	1.26937	0.634685	−	0.772771i	$-0.281130\pi$
$881$	2.92434e7	1.26937	0.634685	+	0.772771i	$0.281130\pi$
$882$	0	0
$883$	3.76532e7	1.62518	0.812588	−	0.582839i	$-0.198058\pi$
$883$	3.76532e7	1.62518	0.812588	+	0.582839i	$0.198058\pi$
$884$	0	0
$885$	−1.19782e7	−0.514084
$886$	0	0
$887$	2.08201e6	0.0888534	0.0444267	−	0.999013i	$-0.485854\pi$
$887$	2.08201e6	0.0888534	0.0444267	+	0.999013i	$0.485854\pi$
$888$	0	0
$889$	8.21495e6	0.348618
$890$	0	0
$891$	3.08367e6	0.130129
$892$	0	0
$893$	4.85756e7	2.03840
$894$	0	0
$895$	3.91116e6	0.163210
$896$	0	0
$897$	−1.24022e7	−0.514656
$898$	0	0
$899$	−2.02962e7	−0.837560
$900$	0	0
$901$	−5.26630e6	−0.216119
$902$	0	0
$903$	5.94997e6	0.242826
$904$	0	0
$905$	−3.44132e7	−1.39670
$906$	0	0
$907$	−1.62350e7	−0.655291	−0.327645	−	0.944801i	$-0.606255\pi$
$907$	−1.62350e7	−0.655291	−0.327645	+	0.944801i	$0.606255\pi$
$908$	0	0
$909$	8.05658e6	0.323401
$910$	0	0
$911$	2.58656e7	1.03259	0.516294	−	0.856412i	$-0.327311\pi$
$911$	2.58656e7	1.03259	0.516294	+	0.856412i	$0.327311\pi$
$912$	0	0
$913$	1.26656e7	0.502860
$914$	0	0
$915$	7.73863e6	0.305571
$916$	0	0
$917$	4.07033e6	0.159848
$918$	0	0
$919$	1.23266e7	0.481453	0.240726	−	0.970593i	$-0.422614\pi$
$919$	1.23266e7	0.481453	0.240726	+	0.970593i	$0.422614\pi$
$920$	0	0
$921$	2.59558e7	1.00829
$922$	0	0
$923$	−1.24253e7	−0.480069
$924$	0	0
$925$	7.70234e6	0.295984
$926$	0	0
$927$	−5.42246e6	−0.207251
$928$	0	0
$929$	−4.15128e7	−1.57813	−0.789064	−	0.614310i	$-0.789434\pi$
$929$	−4.15128e7	−1.57813	−0.789064	+	0.614310i	$0.789434\pi$
$930$	0	0
$931$	6.34824e6	0.240038
$932$	0	0
$933$	9.26622e6	0.348497
$934$	0	0
$935$	2.48987e7	0.931425
$936$	0	0
$937$	2.26895e7	0.844260	0.422130	−	0.906535i	$-0.361283\pi$
$937$	2.26895e7	0.844260	0.422130	+	0.906535i	$0.361283\pi$
$938$	0	0
$939$	1.25367e7	0.464002
$940$	0	0
$941$	1.58213e7	0.582464	0.291232	−	0.956652i	$-0.405935\pi$
$941$	1.58213e7	0.582464	0.291232	+	0.956652i	$0.405935\pi$
$942$	0	0
$943$	3.39864e6	0.124459
$944$	0	0
$945$	1.57172e6	0.0572528
$946$	0	0
$947$	−5.17579e7	−1.87543	−0.937716	−	0.347402i	$-0.887064\pi$
$947$	−5.17579e7	−1.87543	−0.937716	+	0.347402i	$0.887064\pi$
$948$	0	0
$949$	−4.09631e7	−1.47648
$950$	0	0
$951$	−7.02027e6	−0.251711
$952$	0	0
$953$	2.29818e7	0.819695	0.409848	−	0.912154i	$-0.365582\pi$
$953$	2.29818e7	0.819695	0.409848	+	0.912154i	$0.365582\pi$
$954$	0	0
$955$	−3.35874e7	−1.19170
$956$	0	0
$957$	1.52872e7	0.539571
$958$	0	0
$959$	7.95045e6	0.279155
$960$	0	0
$961$	2.91030e6	0.101655
$962$	0	0
$963$	1.84840e7	0.642290
$964$	0	0
$965$	−1.65881e7	−0.573427
$966$	0	0
$967$	−3.20783e7	−1.10318	−0.551588	−	0.834117i	$-0.685978\pi$
$967$	−3.20783e7	−1.10318	−0.551588	+	0.834117i	$0.685978\pi$
$968$	0	0
$969$	2.86504e7	0.980214
$970$	0	0
$971$	−7.31101e6	−0.248845	−0.124423	−	0.992229i	$-0.539708\pi$
$971$	−7.31101e6	−0.248845	−0.124423	+	0.992229i	$0.539708\pi$
$972$	0	0
$973$	2.88336e6	0.0976374
$974$	0	0
$975$	1.23918e7	0.417466
$976$	0	0
$977$	7.58600e6	0.254259	0.127129	−	0.991886i	$-0.459424\pi$
$977$	7.58600e6	0.254259	0.127129	+	0.991886i	$0.459424\pi$
$978$	0	0
$979$	4.73647e7	1.57942
$980$	0	0
$981$	−7.71007e6	−0.255791
$982$	0	0
$983$	2.47823e7	0.818007	0.409004	−	0.912533i	$-0.365876\pi$
$983$	2.47823e7	0.818007	0.409004	+	0.912533i	$0.365876\pi$
$984$	0	0
$985$	−3.02183e6	−0.0992384
$986$	0	0
$987$	8.10205e6	0.264729
$988$	0	0
$989$	1.60555e7	0.521954
$990$	0	0
$991$	7.63530e6	0.246969	0.123484	−	0.992347i	$-0.460593\pi$
$991$	7.63530e6	0.246969	0.123484	+	0.992347i	$0.460593\pi$
$992$	0	0
$993$	1.27084e7	0.408995
$994$	0	0
$995$	−8.03334e6	−0.257240
$996$	0	0
$997$	−2.89785e7	−0.923289	−0.461644	−	0.887065i	$-0.652740\pi$
$997$	−2.89785e7	−0.923289	−0.461644	+	0.887065i	$0.652740\pi$
$998$	0	0
$999$	−4.72246e6	−0.149711

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.6.a.o.1.1		1
3.2	odd	2		1008.6.a.g.1.1		1
4.3	odd	2		42.6.a.b.1.1	✓	1
12.11	even	2		126.6.a.h.1.1		1
20.3	even	4		1050.6.g.l.799.2		2
20.7	even	4		1050.6.g.l.799.1		2
20.19	odd	2		1050.6.a.o.1.1		1
28.3	even	6		294.6.e.k.79.1		2
28.11	odd	6		294.6.e.o.79.1		2
28.19	even	6		294.6.e.k.67.1		2
28.23	odd	6		294.6.e.o.67.1		2
28.27	even	2		294.6.a.f.1.1		1
84.83	odd	2		882.6.a.v.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.6.a.b.1.1	✓	1	4.3	odd	2
126.6.a.h.1.1		1	12.11	even	2
294.6.a.f.1.1		1	28.27	even	2
294.6.e.k.67.1		2	28.19	even	6
294.6.e.k.79.1		2	28.3	even	6
294.6.e.o.67.1		2	28.23	odd	6
294.6.e.o.79.1		2	28.11	odd	6
336.6.a.o.1.1		1	1.1	even	1	trivial
882.6.a.v.1.1		1	84.83	odd	2
1008.6.a.g.1.1		1	3.2	odd	2
1050.6.a.o.1.1		1	20.19	odd	2
1050.6.g.l.799.1		2	20.7	even	4
1050.6.g.l.799.2		2	20.3	even	4

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:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	336.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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