LMFDB - Embedded newform 336.6.a.b.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)

$f(q)$

$=$

$q-9.00000 q^{3} -72.0000 q^{5} -49.0000 q^{7} +81.0000 q^{9} +414.000 q^{11} -1054.00 q^{13} +648.000 q^{15} -1848.00 q^{17} -236.000 q^{19} +441.000 q^{21} -2898.00 q^{23} +2059.00 q^{25} -729.000 q^{27} -6522.00 q^{29} -6200.00 q^{31} -3726.00 q^{33} +3528.00 q^{35} +9650.00 q^{37} +9486.00 q^{39} +8484.00 q^{41} +10804.0 q^{43} -5832.00 q^{45} -60.0000 q^{47} +2401.00 q^{49} +16632.0 q^{51} +22506.0 q^{53} -29808.0 q^{55} +2124.00 q^{57} +28176.0 q^{59} -35194.0 q^{61} -3969.00 q^{63} +75888.0 q^{65} +28216.0 q^{67} +26082.0 q^{69} +6642.00 q^{71} -52090.0 q^{73} -18531.0 q^{75} -20286.0 q^{77} -43340.0 q^{79} +6561.00 q^{81} -25716.0 q^{83} +133056. q^{85} +58698.0 q^{87} +98724.0 q^{89} +51646.0 q^{91} +55800.0 q^{93} +16992.0 q^{95} -148954. q^{97} +33534.0 q^{99} +O(q^{100})$

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
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	0	0
$5$	−72.0000	−1.28798	−0.643988	−	0.765036i	$-0.722721\pi$
$5$	−72.0000	−1.28798	−0.643988	+	0.765036i	$0.722721\pi$
$6$	0	0
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	414.000	1.03162	0.515809	−	0.856704i	$-0.327492\pi$
$11$	414.000	1.03162	0.515809	+	0.856704i	$0.327492\pi$
$12$	0	0
$13$	−1054.00	−1.72975	−0.864873	−	0.501991i	$-0.832601\pi$
$13$	−1054.00	−1.72975	−0.864873	+	0.501991i	$0.832601\pi$
$14$	0	0
$15$	648.000	0.743613
$16$	0	0
$17$	−1848.00	−1.55089	−0.775443	−	0.631418i	$-0.782473\pi$
$17$	−1848.00	−1.55089	−0.775443	+	0.631418i	$0.782473\pi$
$18$	0	0
$19$	−236.000	−0.149978	−0.0749891	−	0.997184i	$-0.523892\pi$
$19$	−236.000	−0.149978	−0.0749891	+	0.997184i	$0.523892\pi$
$20$	0	0
$21$	441.000	0.218218
$22$	0	0
$23$	−2898.00	−1.14230	−0.571148	−	0.820847i	$-0.693502\pi$
$23$	−2898.00	−1.14230	−0.571148	+	0.820847i	$0.693502\pi$
$24$	0	0
$25$	2059.00	0.658880
$26$	0	0
$27$	−729.000	−0.192450
$28$	0	0
$29$	−6522.00	−1.44008	−0.720039	−	0.693934i	$-0.755876\pi$
$29$	−6522.00	−1.44008	−0.720039	+	0.693934i	$0.755876\pi$
$30$	0	0
$31$	−6200.00	−1.15874	−0.579372	−	0.815063i	$-0.696702\pi$
$31$	−6200.00	−1.15874	−0.579372	+	0.815063i	$0.696702\pi$
$32$	0	0
$33$	−3726.00	−0.595605
$34$	0	0
$35$	3528.00	0.486809
$36$	0	0
$37$	9650.00	1.15884	0.579419	−	0.815030i	$-0.303279\pi$
$37$	9650.00	1.15884	0.579419	+	0.815030i	$0.303279\pi$
$38$	0	0
$39$	9486.00	0.998669
$40$	0	0
$41$	8484.00	0.788208	0.394104	−	0.919066i	$-0.371055\pi$
$41$	8484.00	0.788208	0.394104	+	0.919066i	$0.371055\pi$
$42$	0	0
$43$	10804.0	0.891073	0.445537	−	0.895264i	$-0.353013\pi$
$43$	10804.0	0.891073	0.445537	+	0.895264i	$0.353013\pi$
$44$	0	0
$45$	−5832.00	−0.429325
$46$	0	0
$47$	−60.0000	−0.00396193	−0.00198096	−	0.999998i	$-0.500631\pi$
$47$	−60.0000	−0.00396193	−0.00198096	+	0.999998i	$0.500631\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	16632.0	0.895404
$52$	0	0
$53$	22506.0	1.10055	0.550274	−	0.834984i	$-0.314523\pi$
$53$	22506.0	1.10055	0.550274	+	0.834984i	$0.314523\pi$
$54$	0	0
$55$	−29808.0	−1.32870
$56$	0	0
$57$	2124.00	0.0865899
$58$	0	0
$59$	28176.0	1.05378	0.526889	−	0.849934i	$-0.323358\pi$
$59$	28176.0	1.05378	0.526889	+	0.849934i	$0.323358\pi$
$60$	0	0
$61$	−35194.0	−1.21100	−0.605500	−	0.795845i	$-0.707027\pi$
$61$	−35194.0	−1.21100	−0.605500	+	0.795845i	$0.707027\pi$
$62$	0	0
$63$	−3969.00	−0.125988
$64$	0	0
$65$	75888.0	2.22787
$66$	0	0
$67$	28216.0	0.767907	0.383953	−	0.923352i	$-0.374562\pi$
$67$	28216.0	0.767907	0.383953	+	0.923352i	$0.374562\pi$
$68$	0	0
$69$	26082.0	0.659505
$70$	0	0
$71$	6642.00	0.156370	0.0781849	−	0.996939i	$-0.475088\pi$
$71$	6642.00	0.156370	0.0781849	+	0.996939i	$0.475088\pi$
$72$	0	0
$73$	−52090.0	−1.14406	−0.572028	−	0.820234i	$-0.693843\pi$
$73$	−52090.0	−1.14406	−0.572028	+	0.820234i	$0.693843\pi$
$74$	0	0
$75$	−18531.0	−0.380405
$76$	0	0
$77$	−20286.0	−0.389915
$78$	0	0
$79$	−43340.0	−0.781306	−0.390653	−	0.920538i	$-0.627751\pi$
$79$	−43340.0	−0.781306	−0.390653	+	0.920538i	$0.627751\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	−25716.0	−0.409740	−0.204870	−	0.978789i	$-0.565677\pi$
$83$	−25716.0	−0.409740	−0.204870	+	0.978789i	$0.565677\pi$
$84$	0	0
$85$	133056.	1.99750
$86$	0	0
$87$	58698.0	0.831429
$88$	0	0
$89$	98724.0	1.32114	0.660568	−	0.750766i	$-0.270315\pi$
$89$	98724.0	1.32114	0.660568	+	0.750766i	$0.270315\pi$
$90$	0	0
$91$	51646.0	0.653782
$92$	0	0
$93$	55800.0	0.669001
$94$	0	0
$95$	16992.0	0.193168
$96$	0	0
$97$	−148954.	−1.60740	−0.803698	−	0.595038i	$-0.797137\pi$
$97$	−148954.	−1.60740	−0.803698	+	0.595038i	$0.797137\pi$
$98$	0	0
$99$	33534.0	0.343872
$100$	0	0
$101$	48348.0	0.471601	0.235801	−	0.971801i	$-0.424229\pi$
$101$	48348.0	0.471601	0.235801	+	0.971801i	$0.424229\pi$
$102$	0	0
$103$	183592.	1.70514	0.852571	−	0.522611i	$-0.175042\pi$
$103$	183592.	1.70514	0.852571	+	0.522611i	$0.175042\pi$
$104$	0	0
$105$	−31752.0	−0.281059
$106$	0	0
$107$	2238.00	0.0188973	0.00944867	−	0.999955i	$-0.496992\pi$
$107$	2238.00	0.0188973	0.00944867	+	0.999955i	$0.496992\pi$
$108$	0	0
$109$	60158.0	0.484984	0.242492	−	0.970153i	$-0.422035\pi$
$109$	60158.0	0.484984	0.242492	+	0.970153i	$0.422035\pi$
$110$	0	0
$111$	−86850.0	−0.669056
$112$	0	0
$113$	−7014.00	−0.0516737	−0.0258369	−	0.999666i	$-0.508225\pi$
$113$	−7014.00	−0.0516737	−0.0258369	+	0.999666i	$0.508225\pi$
$114$	0	0
$115$	208656.	1.47125
$116$	0	0
$117$	−85374.0	−0.576582
$118$	0	0
$119$	90552.0	0.586180
$120$	0	0
$121$	10345.0	0.0642343
$122$	0	0
$123$	−76356.0	−0.455072
$124$	0	0
$125$	76752.0	0.439354
$126$	0	0
$127$	1780.00	0.00979289	0.00489644	−	0.999988i	$-0.498441\pi$
$127$	1780.00	0.00979289	0.00489644	+	0.999988i	$0.498441\pi$
$128$	0	0
$129$	−97236.0	−0.514461
$130$	0	0
$131$	265140.	1.34989	0.674943	−	0.737870i	$-0.264168\pi$
$131$	265140.	1.34989	0.674943	+	0.737870i	$0.264168\pi$
$132$	0	0
$133$	11564.0	0.0566864
$134$	0	0
$135$	52488.0	0.247871
$136$	0	0
$137$	−206730.	−0.941027	−0.470514	−	0.882393i	$-0.655931\pi$
$137$	−206730.	−0.941027	−0.470514	+	0.882393i	$0.655931\pi$
$138$	0	0
$139$	236836.	1.03971	0.519853	−	0.854256i	$-0.325987\pi$
$139$	236836.	1.03971	0.519853	+	0.854256i	$0.325987\pi$
$140$	0	0
$141$	540.000	0.00228742
$142$	0	0
$143$	−436356.	−1.78444
$144$	0	0
$145$	469584.	1.85478
$146$	0	0
$147$	−21609.0	−0.0824786
$148$	0	0
$149$	473706.	1.74801	0.874004	−	0.485919i	$-0.161515\pi$
$149$	473706.	1.74801	0.874004	+	0.485919i	$0.161515\pi$
$150$	0	0
$151$	−394952.	−1.40962	−0.704810	−	0.709396i	$-0.748968\pi$
$151$	−394952.	−1.40962	−0.704810	+	0.709396i	$0.748968\pi$
$152$	0	0
$153$	−149688.	−0.516962
$154$	0	0
$155$	446400.	1.49243
$156$	0	0
$157$	−145090.	−0.469773	−0.234887	−	0.972023i	$-0.575472\pi$
$157$	−145090.	−0.469773	−0.234887	+	0.972023i	$0.575472\pi$
$158$	0	0
$159$	−202554.	−0.635401
$160$	0	0
$161$	142002.	0.431747
$162$	0	0
$163$	−530480.	−1.56387	−0.781934	−	0.623361i	$-0.785767\pi$
$163$	−530480.	−1.56387	−0.781934	+	0.623361i	$0.785767\pi$
$164$	0	0
$165$	268272.	0.767124
$166$	0	0
$167$	312348.	0.866658	0.433329	−	0.901236i	$-0.357339\pi$
$167$	312348.	0.866658	0.433329	+	0.901236i	$0.357339\pi$
$168$	0	0
$169$	739623.	1.99202
$170$	0	0
$171$	−19116.0	−0.0499927
$172$	0	0
$173$	−75108.0	−0.190797	−0.0953984	−	0.995439i	$-0.530412\pi$
$173$	−75108.0	−0.190797	−0.0953984	+	0.995439i	$0.530412\pi$
$174$	0	0
$175$	−100891.	−0.249033
$176$	0	0
$177$	−253584.	−0.608399
$178$	0	0
$179$	−386718.	−0.902115	−0.451057	−	0.892495i	$-0.648953\pi$
$179$	−386718.	−0.902115	−0.451057	+	0.892495i	$0.648953\pi$
$180$	0	0
$181$	−417598.	−0.947462	−0.473731	−	0.880669i	$-0.657093\pi$
$181$	−417598.	−0.947462	−0.473731	+	0.880669i	$0.657093\pi$
$182$	0	0
$183$	316746.	0.699171
$184$	0	0
$185$	−694800.	−1.49256
$186$	0	0
$187$	−765072.	−1.59992
$188$	0	0
$189$	35721.0	0.0727393
$190$	0	0
$191$	988050.	1.95973	0.979863	−	0.199669i	$-0.0639868\pi$
$191$	988050.	1.95973	0.979863	+	0.199669i	$0.0639868\pi$
$192$	0	0
$193$	−409258.	−0.790868	−0.395434	−	0.918494i	$-0.629406\pi$
$193$	−409258.	−0.790868	−0.395434	+	0.918494i	$0.629406\pi$
$194$	0	0
$195$	−682992.	−1.28626
$196$	0	0
$197$	−922230.	−1.69307	−0.846533	−	0.532337i	$-0.821314\pi$
$197$	−922230.	−1.69307	−0.846533	+	0.532337i	$0.821314\pi$
$198$	0	0
$199$	−189488.	−0.339195	−0.169597	−	0.985513i	$-0.554247\pi$
$199$	−189488.	−0.339195	−0.169597	+	0.985513i	$0.554247\pi$
$200$	0	0
$201$	−253944.	−0.443351
$202$	0	0
$203$	319578.	0.544298
$204$	0	0
$205$	−610848.	−1.01519
$206$	0	0
$207$	−234738.	−0.380765
$208$	0	0
$209$	−97704.0	−0.154720
$210$	0	0
$211$	611380.	0.945377	0.472689	−	0.881230i	$-0.343284\pi$
$211$	611380.	0.945377	0.472689	+	0.881230i	$0.343284\pi$
$212$	0	0
$213$	−59778.0	−0.0902802
$214$	0	0
$215$	−777888.	−1.14768
$216$	0	0
$217$	303800.	0.437964
$218$	0	0
$219$	468810.	0.660521
$220$	0	0
$221$	1.94779e6	2.68264
$222$	0	0
$223$	783256.	1.05473	0.527365	−	0.849639i	$-0.323180\pi$
$223$	783256.	1.05473	0.527365	+	0.849639i	$0.323180\pi$
$224$	0	0
$225$	166779.	0.219627
$226$	0	0
$227$	−80712.0	−0.103962	−0.0519809	−	0.998648i	$-0.516553\pi$
$227$	−80712.0	−0.103962	−0.0519809	+	0.998648i	$0.516553\pi$
$228$	0	0
$229$	152738.	0.192468	0.0962340	−	0.995359i	$-0.469320\pi$
$229$	152738.	0.192468	0.0962340	+	0.995359i	$0.469320\pi$
$230$	0	0
$231$	182574.	0.225117
$232$	0	0
$233$	−354282.	−0.427523	−0.213761	−	0.976886i	$-0.568572\pi$
$233$	−354282.	−0.427523	−0.213761	+	0.976886i	$0.568572\pi$
$234$	0	0
$235$	4320.00	0.00510287
$236$	0	0
$237$	390060.	0.451087
$238$	0	0
$239$	−275370.	−0.311833	−0.155916	−	0.987770i	$-0.549833\pi$
$239$	−275370.	−0.311833	−0.155916	+	0.987770i	$0.549833\pi$
$240$	0	0
$241$	−584698.	−0.648469	−0.324234	−	0.945977i	$-0.605107\pi$
$241$	−584698.	−0.648469	−0.324234	+	0.945977i	$0.605107\pi$
$242$	0	0
$243$	−59049.0	−0.0641500
$244$	0	0
$245$	−172872.	−0.183996
$246$	0	0
$247$	248744.	0.259424
$248$	0	0
$249$	231444.	0.236563
$250$	0	0
$251$	184752.	0.185099	0.0925497	−	0.995708i	$-0.470498\pi$
$251$	184752.	0.185099	0.0925497	+	0.995708i	$0.470498\pi$
$252$	0	0
$253$	−1.19977e6	−1.17841
$254$	0	0
$255$	−1.19750e6	−1.15326
$256$	0	0
$257$	235980.	0.222865	0.111433	−	0.993772i	$-0.464456\pi$
$257$	235980.	0.222865	0.111433	+	0.993772i	$0.464456\pi$
$258$	0	0
$259$	−472850.	−0.438000
$260$	0	0
$261$	−528282.	−0.480026
$262$	0	0
$263$	244494.	0.217961	0.108981	−	0.994044i	$-0.465241\pi$
$263$	244494.	0.217961	0.108981	+	0.994044i	$0.465241\pi$
$264$	0	0
$265$	−1.62043e6	−1.41748
$266$	0	0
$267$	−888516.	−0.762759
$268$	0	0
$269$	1.52779e6	1.28731	0.643656	−	0.765315i	$-0.277417\pi$
$269$	1.52779e6	1.28731	0.643656	+	0.765315i	$0.277417\pi$
$270$	0	0
$271$	−2.07056e6	−1.71263	−0.856317	−	0.516450i	$-0.827253\pi$
$271$	−2.07056e6	−1.71263	−0.856317	+	0.516450i	$0.827253\pi$
$272$	0	0
$273$	−464814.	−0.377461
$274$	0	0
$275$	852426.	0.679712
$276$	0	0
$277$	−2.40727e6	−1.88506	−0.942530	−	0.334120i	$-0.891561\pi$
$277$	−2.40727e6	−1.88506	−0.942530	+	0.334120i	$0.891561\pi$
$278$	0	0
$279$	−502200.	−0.386248
$280$	0	0
$281$	341886.	0.258295	0.129147	−	0.991625i	$-0.458776\pi$
$281$	341886.	0.258295	0.129147	+	0.991625i	$0.458776\pi$
$282$	0	0
$283$	−578564.	−0.429423	−0.214712	−	0.976678i	$-0.568881\pi$
$283$	−578564.	−0.429423	−0.214712	+	0.976678i	$0.568881\pi$
$284$	0	0
$285$	−152928.	−0.111526
$286$	0	0
$287$	−415716.	−0.297915
$288$	0	0
$289$	1.99525e6	1.40525
$290$	0	0
$291$	1.34059e6	0.928030
$292$	0	0
$293$	780540.	0.531161	0.265580	−	0.964089i	$-0.414436\pi$
$293$	780540.	0.531161	0.265580	+	0.964089i	$0.414436\pi$
$294$	0	0
$295$	−2.02867e6	−1.35724
$296$	0	0
$297$	−301806.	−0.198535
$298$	0	0
$299$	3.05449e6	1.97588
$300$	0	0
$301$	−529396.	−0.336794
$302$	0	0
$303$	−435132.	−0.272279
$304$	0	0
$305$	2.53397e6	1.55974
$306$	0	0
$307$	2.24825e6	1.36144	0.680721	−	0.732543i	$-0.261667\pi$
$307$	2.24825e6	1.36144	0.680721	+	0.732543i	$0.261667\pi$
$308$	0	0
$309$	−1.65233e6	−0.984465
$310$	0	0
$311$	−581412.	−0.340865	−0.170433	−	0.985369i	$-0.554517\pi$
$311$	−581412.	−0.340865	−0.170433	+	0.985369i	$0.554517\pi$
$312$	0	0
$313$	−2.00407e6	−1.15625	−0.578125	−	0.815948i	$-0.696216\pi$
$313$	−2.00407e6	−1.15625	−0.578125	+	0.815948i	$0.696216\pi$
$314$	0	0
$315$	285768.	0.162270
$316$	0	0
$317$	−1.27832e6	−0.714481	−0.357241	−	0.934012i	$-0.616282\pi$
$317$	−1.27832e6	−0.714481	−0.357241	+	0.934012i	$0.616282\pi$
$318$	0	0
$319$	−2.70011e6	−1.48561
$320$	0	0
$321$	−20142.0	−0.0109104
$322$	0	0
$323$	436128.	0.232599
$324$	0	0
$325$	−2.17019e6	−1.13969
$326$	0	0
$327$	−541422.	−0.280005
$328$	0	0
$329$	2940.00	0.00149747
$330$	0	0
$331$	−2.59812e6	−1.30343	−0.651716	−	0.758463i	$-0.725951\pi$
$331$	−2.59812e6	−1.30343	−0.651716	+	0.758463i	$0.725951\pi$
$332$	0	0
$333$	781650.	0.386280
$334$	0	0
$335$	−2.03155e6	−0.989045
$336$	0	0
$337$	3.06190e6	1.46864	0.734321	−	0.678802i	$-0.237501\pi$
$337$	3.06190e6	1.46864	0.734321	+	0.678802i	$0.237501\pi$
$338$	0	0
$339$	63126.0	0.0298338
$340$	0	0
$341$	−2.56680e6	−1.19538
$342$	0	0
$343$	−117649.	−0.0539949
$344$	0	0
$345$	−1.87790e6	−0.849426
$346$	0	0
$347$	1.42550e6	0.635540	0.317770	−	0.948168i	$-0.397066\pi$
$347$	1.42550e6	0.635540	0.317770	+	0.948168i	$0.397066\pi$
$348$	0	0
$349$	2.93322e6	1.28908	0.644542	−	0.764569i	$-0.277048\pi$
$349$	2.93322e6	1.28908	0.644542	+	0.764569i	$0.277048\pi$
$350$	0	0
$351$	768366.	0.332890
$352$	0	0
$353$	−2.01276e6	−0.859716	−0.429858	−	0.902896i	$-0.641436\pi$
$353$	−2.01276e6	−0.859716	−0.429858	+	0.902896i	$0.641436\pi$
$354$	0	0
$355$	−478224.	−0.201400
$356$	0	0
$357$	−814968.	−0.338431
$358$	0	0
$359$	−4.07710e6	−1.66961	−0.834806	−	0.550545i	$-0.814420\pi$
$359$	−4.07710e6	−1.66961	−0.834806	+	0.550545i	$0.814420\pi$
$360$	0	0
$361$	−2.42040e6	−0.977507
$362$	0	0
$363$	−93105.0	−0.0370857
$364$	0	0
$365$	3.75048e6	1.47352
$366$	0	0
$367$	−594752.	−0.230500	−0.115250	−	0.993337i	$-0.536767\pi$
$367$	−594752.	−0.230500	−0.115250	+	0.993337i	$0.536767\pi$
$368$	0	0
$369$	687204.	0.262736
$370$	0	0
$371$	−1.10279e6	−0.415968
$372$	0	0
$373$	2.04522e6	0.761147	0.380573	−	0.924751i	$-0.375727\pi$
$373$	2.04522e6	0.761147	0.380573	+	0.924751i	$0.375727\pi$
$374$	0	0
$375$	−690768.	−0.253661
$376$	0	0
$377$	6.87419e6	2.49097
$378$	0	0
$379$	3.22198e6	1.15219	0.576096	−	0.817382i	$-0.304575\pi$
$379$	3.22198e6	1.15219	0.576096	+	0.817382i	$0.304575\pi$
$380$	0	0
$381$	−16020.0	−0.00565393
$382$	0	0
$383$	1.72966e6	0.602508	0.301254	−	0.953544i	$-0.402595\pi$
$383$	1.72966e6	0.602508	0.301254	+	0.953544i	$0.402595\pi$
$384$	0	0
$385$	1.46059e6	0.502200
$386$	0	0
$387$	875124.	0.297024
$388$	0	0
$389$	−1.74919e6	−0.586087	−0.293043	−	0.956099i	$-0.594668\pi$
$389$	−1.74919e6	−0.586087	−0.293043	+	0.956099i	$0.594668\pi$
$390$	0	0
$391$	5.35550e6	1.77157
$392$	0	0
$393$	−2.38626e6	−0.779357
$394$	0	0
$395$	3.12048e6	1.00630
$396$	0	0
$397$	1.88205e6	0.599313	0.299657	−	0.954047i	$-0.403128\pi$
$397$	1.88205e6	0.599313	0.299657	+	0.954047i	$0.403128\pi$
$398$	0	0
$399$	−104076.	−0.0327279
$400$	0	0
$401$	4.18124e6	1.29851	0.649253	−	0.760573i	$-0.275082\pi$
$401$	4.18124e6	1.29851	0.649253	+	0.760573i	$0.275082\pi$
$402$	0	0
$403$	6.53480e6	2.00433
$404$	0	0
$405$	−472392.	−0.143108
$406$	0	0
$407$	3.99510e6	1.19548
$408$	0	0
$409$	−471682.	−0.139425	−0.0697126	−	0.997567i	$-0.522208\pi$
$409$	−471682.	−0.139425	−0.0697126	+	0.997567i	$0.522208\pi$
$410$	0	0
$411$	1.86057e6	0.543302
$412$	0	0
$413$	−1.38062e6	−0.398291
$414$	0	0
$415$	1.85155e6	0.527735
$416$	0	0
$417$	−2.13152e6	−0.600275
$418$	0	0
$419$	3.54094e6	0.985333	0.492666	−	0.870218i	$-0.336022\pi$
$419$	3.54094e6	0.985333	0.492666	+	0.870218i	$0.336022\pi$
$420$	0	0
$421$	2.72763e6	0.750032	0.375016	−	0.927018i	$-0.377637\pi$
$421$	2.72763e6	0.750032	0.375016	+	0.927018i	$0.377637\pi$
$422$	0	0
$423$	−4860.00	−0.00132064
$424$	0	0
$425$	−3.80503e6	−1.02185
$426$	0	0
$427$	1.72451e6	0.457715
$428$	0	0
$429$	3.92720e6	1.03024
$430$	0	0
$431$	4.76517e6	1.23562	0.617810	−	0.786327i	$-0.288020\pi$
$431$	4.76517e6	1.23562	0.617810	+	0.786327i	$0.288020\pi$
$432$	0	0
$433$	6.90300e6	1.76937	0.884684	−	0.466191i	$-0.154374\pi$
$433$	6.90300e6	1.76937	0.884684	+	0.466191i	$0.154374\pi$
$434$	0	0
$435$	−4.22626e6	−1.07086
$436$	0	0
$437$	683928.	0.171319
$438$	0	0
$439$	−5.40126e6	−1.33762	−0.668811	−	0.743432i	$-0.733197\pi$
$439$	−5.40126e6	−1.33762	−0.668811	+	0.743432i	$0.733197\pi$
$440$	0	0
$441$	194481.	0.0476190
$442$	0	0
$443$	4.05863e6	0.982586	0.491293	−	0.870994i	$-0.336524\pi$
$443$	4.05863e6	0.982586	0.491293	+	0.870994i	$0.336524\pi$
$444$	0	0
$445$	−7.10813e6	−1.70159
$446$	0	0
$447$	−4.26335e6	−1.00921
$448$	0	0
$449$	212994.	0.0498599	0.0249300	−	0.999689i	$-0.492064\pi$
$449$	212994.	0.0498599	0.0249300	+	0.999689i	$0.492064\pi$
$450$	0	0
$451$	3.51238e6	0.813129
$452$	0	0
$453$	3.55457e6	0.813844
$454$	0	0
$455$	−3.71851e6	−0.842055
$456$	0	0
$457$	−916150.	−0.205199	−0.102600	−	0.994723i	$-0.532716\pi$
$457$	−916150.	−0.205199	−0.102600	+	0.994723i	$0.532716\pi$
$458$	0	0
$459$	1.34719e6	0.298468
$460$	0	0
$461$	−4.15835e6	−0.911315	−0.455657	−	0.890155i	$-0.650596\pi$
$461$	−4.15835e6	−0.911315	−0.455657	+	0.890155i	$0.650596\pi$
$462$	0	0
$463$	−8.40799e6	−1.82280	−0.911401	−	0.411519i	$-0.864998\pi$
$463$	−8.40799e6	−1.82280	−0.911401	+	0.411519i	$0.864998\pi$
$464$	0	0
$465$	−4.01760e6	−0.861657
$466$	0	0
$467$	−72048.0	−0.0152873	−0.00764363	−	0.999971i	$-0.502433\pi$
$467$	−72048.0	−0.0152873	−0.00764363	+	0.999971i	$0.502433\pi$
$468$	0	0
$469$	−1.38258e6	−0.290241
$470$	0	0
$471$	1.30581e6	0.271224
$472$	0	0
$473$	4.47286e6	0.919247
$474$	0	0
$475$	−485924.	−0.0988176
$476$	0	0
$477$	1.82299e6	0.366849
$478$	0	0
$479$	−4.80560e6	−0.956994	−0.478497	−	0.878089i	$-0.658818\pi$
$479$	−4.80560e6	−0.956994	−0.478497	+	0.878089i	$0.658818\pi$
$480$	0	0
$481$	−1.01711e7	−2.00450
$482$	0	0
$483$	−1.27802e6	−0.249270
$484$	0	0
$485$	1.07247e7	2.07029
$486$	0	0
$487$	−4.41805e6	−0.844127	−0.422064	−	0.906566i	$-0.638694\pi$
$487$	−4.41805e6	−0.844127	−0.422064	+	0.906566i	$0.638694\pi$
$488$	0	0
$489$	4.77432e6	0.902899
$490$	0	0
$491$	5.64998e6	1.05765	0.528826	−	0.848730i	$-0.322632\pi$
$491$	5.64998e6	1.05765	0.528826	+	0.848730i	$0.322632\pi$
$492$	0	0
$493$	1.20527e7	2.23339
$494$	0	0
$495$	−2.41445e6	−0.442899
$496$	0	0
$497$	−325458.	−0.0591022
$498$	0	0
$499$	9.22344e6	1.65822	0.829109	−	0.559087i	$-0.188848\pi$
$499$	9.22344e6	1.65822	0.829109	+	0.559087i	$0.188848\pi$
$500$	0	0
$501$	−2.81113e6	−0.500365
$502$	0	0
$503$	−1.45562e6	−0.256525	−0.128262	−	0.991740i	$-0.540940\pi$
$503$	−1.45562e6	−0.256525	−0.128262	+	0.991740i	$0.540940\pi$
$504$	0	0
$505$	−3.48106e6	−0.607411
$506$	0	0
$507$	−6.65661e6	−1.15009
$508$	0	0
$509$	−367344.	−0.0628461	−0.0314231	−	0.999506i	$-0.510004\pi$
$509$	−367344.	−0.0628461	−0.0314231	+	0.999506i	$0.510004\pi$
$510$	0	0
$511$	2.55241e6	0.432412
$512$	0	0
$513$	172044.	0.0288633
$514$	0	0
$515$	−1.32186e7	−2.19618
$516$	0	0
$517$	−24840.0	−0.00408719
$518$	0	0
$519$	675972.	0.110157
$520$	0	0
$521$	5.76362e6	0.930254	0.465127	−	0.885244i	$-0.346009\pi$
$521$	5.76362e6	0.930254	0.465127	+	0.885244i	$0.346009\pi$
$522$	0	0
$523$	−235100.	−0.0375836	−0.0187918	−	0.999823i	$-0.505982\pi$
$523$	−235100.	−0.0375836	−0.0187918	+	0.999823i	$0.505982\pi$
$524$	0	0
$525$	908019.	0.143779
$526$	0	0
$527$	1.14576e7	1.79708
$528$	0	0
$529$	1.96206e6	0.304841
$530$	0	0
$531$	2.28226e6	0.351259
$532$	0	0
$533$	−8.94214e6	−1.36340
$534$	0	0
$535$	−161136.	−0.0243393
$536$	0	0
$537$	3.48046e6	0.520836
$538$	0	0
$539$	994014.	0.147374
$540$	0	0
$541$	109010.	0.0160130	0.00800651	−	0.999968i	$-0.497451\pi$
$541$	109010.	0.0160130	0.00800651	+	0.999968i	$0.497451\pi$
$542$	0	0
$543$	3.75838e6	0.547018
$544$	0	0
$545$	−4.33138e6	−0.624647
$546$	0	0
$547$	−1.61953e6	−0.231430	−0.115715	−	0.993282i	$-0.536916\pi$
$547$	−1.61953e6	−0.231430	−0.115715	+	0.993282i	$0.536916\pi$
$548$	0	0
$549$	−2.85071e6	−0.403667
$550$	0	0
$551$	1.53919e6	0.215980
$552$	0	0
$553$	2.12366e6	0.295306
$554$	0	0
$555$	6.25320e6	0.861727
$556$	0	0
$557$	−6.62986e6	−0.905454	−0.452727	−	0.891649i	$-0.649549\pi$
$557$	−6.62986e6	−0.905454	−0.452727	+	0.891649i	$0.649549\pi$
$558$	0	0
$559$	−1.13874e7	−1.54133
$560$	0	0
$561$	6.88565e6	0.923714
$562$	0	0
$563$	−7.85294e6	−1.04415	−0.522073	−	0.852901i	$-0.674841\pi$
$563$	−7.85294e6	−1.04415	−0.522073	+	0.852901i	$0.674841\pi$
$564$	0	0
$565$	505008.	0.0665545
$566$	0	0
$567$	−321489.	−0.0419961
$568$	0	0
$569$	−2.48155e6	−0.321323	−0.160661	−	0.987010i	$-0.551363\pi$
$569$	−2.48155e6	−0.321323	−0.160661	+	0.987010i	$0.551363\pi$
$570$	0	0
$571$	−1.13675e7	−1.45907	−0.729533	−	0.683945i	$-0.760263\pi$
$571$	−1.13675e7	−1.45907	−0.729533	+	0.683945i	$0.760263\pi$
$572$	0	0
$573$	−8.89245e6	−1.13145
$574$	0	0
$575$	−5.96698e6	−0.752636
$576$	0	0
$577$	−8.20505e6	−1.02599	−0.512993	−	0.858393i	$-0.671463\pi$
$577$	−8.20505e6	−1.02599	−0.512993	+	0.858393i	$0.671463\pi$
$578$	0	0
$579$	3.68332e6	0.456608
$580$	0	0
$581$	1.26008e6	0.154867
$582$	0	0
$583$	9.31748e6	1.13534
$584$	0	0
$585$	6.14693e6	0.742623
$586$	0	0
$587$	1.38400e7	1.65783	0.828917	−	0.559371i	$-0.188957\pi$
$587$	1.38400e7	1.65783	0.828917	+	0.559371i	$0.188957\pi$
$588$	0	0
$589$	1.46320e6	0.173786
$590$	0	0
$591$	8.30007e6	0.977492
$592$	0	0
$593$	−5.38951e6	−0.629380	−0.314690	−	0.949195i	$-0.601901\pi$
$593$	−5.38951e6	−0.629380	−0.314690	+	0.949195i	$0.601901\pi$
$594$	0	0
$595$	−6.51974e6	−0.754985
$596$	0	0
$597$	1.70539e6	0.195834
$598$	0	0
$599$	1.14885e7	1.30827	0.654134	−	0.756379i	$-0.273033\pi$
$599$	1.14885e7	1.30827	0.654134	+	0.756379i	$0.273033\pi$
$600$	0	0
$601$	−2.79225e6	−0.315333	−0.157666	−	0.987492i	$-0.550397\pi$
$601$	−2.79225e6	−0.315333	−0.157666	+	0.987492i	$0.550397\pi$
$602$	0	0
$603$	2.28550e6	0.255969
$604$	0	0
$605$	−744840.	−0.0827322
$606$	0	0
$607$	−713888.	−0.0786427	−0.0393213	−	0.999227i	$-0.512520\pi$
$607$	−713888.	−0.0786427	−0.0393213	+	0.999227i	$0.512520\pi$
$608$	0	0
$609$	−2.87620e6	−0.314251
$610$	0	0
$611$	63240.0	0.00685313
$612$	0	0
$613$	3.10972e6	0.334249	0.167124	−	0.985936i	$-0.446552\pi$
$613$	3.10972e6	0.334249	0.167124	+	0.985936i	$0.446552\pi$
$614$	0	0
$615$	5.49763e6	0.586122
$616$	0	0
$617$	3.62384e6	0.383227	0.191613	−	0.981470i	$-0.438628\pi$
$617$	3.62384e6	0.383227	0.191613	+	0.981470i	$0.438628\pi$
$618$	0	0
$619$	−4.25196e6	−0.446028	−0.223014	−	0.974815i	$-0.571590\pi$
$619$	−4.25196e6	−0.446028	−0.223014	+	0.974815i	$0.571590\pi$
$620$	0	0
$621$	2.11264e6	0.219835
$622$	0	0
$623$	−4.83748e6	−0.499343
$624$	0	0
$625$	−1.19605e7	−1.22476
$626$	0	0
$627$	879336.	0.0893277
$628$	0	0
$629$	−1.78332e7	−1.79723
$630$	0	0
$631$	1.70299e7	1.70270	0.851349	−	0.524600i	$-0.175785\pi$
$631$	1.70299e7	1.70270	0.851349	+	0.524600i	$0.175785\pi$
$632$	0	0
$633$	−5.50242e6	−0.545814
$634$	0	0
$635$	−128160.	−0.0126130
$636$	0	0
$637$	−2.53065e6	−0.247107
$638$	0	0
$639$	538002.	0.0521233
$640$	0	0
$641$	−1.80938e6	−0.173934	−0.0869669	−	0.996211i	$-0.527717\pi$
$641$	−1.80938e6	−0.173934	−0.0869669	+	0.996211i	$0.527717\pi$
$642$	0	0
$643$	−1.53012e7	−1.45948	−0.729740	−	0.683725i	$-0.760359\pi$
$643$	−1.53012e7	−1.45948	−0.729740	+	0.683725i	$0.760359\pi$
$644$	0	0
$645$	7.00099e6	0.662614
$646$	0	0
$647$	−1.67546e7	−1.57352	−0.786762	−	0.617256i	$-0.788244\pi$
$647$	−1.67546e7	−1.57352	−0.786762	+	0.617256i	$0.788244\pi$
$648$	0	0
$649$	1.16649e7	1.08710
$650$	0	0
$651$	−2.73420e6	−0.252859
$652$	0	0
$653$	1.47859e7	1.35695	0.678477	−	0.734622i	$-0.262640\pi$
$653$	1.47859e7	1.35695	0.678477	+	0.734622i	$0.262640\pi$
$654$	0	0
$655$	−1.90901e7	−1.73862
$656$	0	0
$657$	−4.21929e6	−0.381352
$658$	0	0
$659$	−933762.	−0.0837573	−0.0418786	−	0.999123i	$-0.513334\pi$
$659$	−933762.	−0.0837573	−0.0418786	+	0.999123i	$0.513334\pi$
$660$	0	0
$661$	6.09724e6	0.542787	0.271394	−	0.962468i	$-0.412516\pi$
$661$	6.09724e6	0.542787	0.271394	+	0.962468i	$0.412516\pi$
$662$	0	0
$663$	−1.75301e7	−1.54882
$664$	0	0
$665$	−832608.	−0.0730107
$666$	0	0
$667$	1.89008e7	1.64500
$668$	0	0
$669$	−7.04930e6	−0.608949
$670$	0	0
$671$	−1.45703e7	−1.24929
$672$	0	0
$673$	2.09190e6	0.178034	0.0890171	−	0.996030i	$-0.471627\pi$
$673$	2.09190e6	0.178034	0.0890171	+	0.996030i	$0.471627\pi$
$674$	0	0
$675$	−1.50101e6	−0.126802
$676$	0	0
$677$	1.36453e6	0.114423	0.0572113	−	0.998362i	$-0.481779\pi$
$677$	1.36453e6	0.114423	0.0572113	+	0.998362i	$0.481779\pi$
$678$	0	0
$679$	7.29875e6	0.607539
$680$	0	0
$681$	726408.	0.0600223
$682$	0	0
$683$	−1.15483e7	−0.947254	−0.473627	−	0.880726i	$-0.657055\pi$
$683$	−1.15483e7	−0.947254	−0.473627	+	0.880726i	$0.657055\pi$
$684$	0	0
$685$	1.48846e7	1.21202
$686$	0	0
$687$	−1.37464e6	−0.111121
$688$	0	0
$689$	−2.37213e7	−1.90367
$690$	0	0
$691$	−4.17744e6	−0.332824	−0.166412	−	0.986056i	$-0.553218\pi$
$691$	−4.17744e6	−0.332824	−0.166412	+	0.986056i	$0.553218\pi$
$692$	0	0
$693$	−1.64317e6	−0.129972
$694$	0	0
$695$	−1.70522e7	−1.33912
$696$	0	0
$697$	−1.56784e7	−1.22242
$698$	0	0
$699$	3.18854e6	0.246830
$700$	0	0
$701$	−1.70278e7	−1.30877	−0.654385	−	0.756161i	$-0.727073\pi$
$701$	−1.70278e7	−1.30877	−0.654385	+	0.756161i	$0.727073\pi$
$702$	0	0
$703$	−2.27740e6	−0.173800
$704$	0	0
$705$	−38880.0	−0.00294614
$706$	0	0
$707$	−2.36905e6	−0.178249
$708$	0	0
$709$	2.26932e7	1.69543	0.847715	−	0.530452i	$-0.177978\pi$
$709$	2.26932e7	1.69543	0.847715	+	0.530452i	$0.177978\pi$
$710$	0	0
$711$	−3.51054e6	−0.260435
$712$	0	0
$713$	1.79676e7	1.32363
$714$	0	0
$715$	3.14176e7	2.29831
$716$	0	0
$717$	2.47833e6	0.180037
$718$	0	0
$719$	−5.12544e6	−0.369751	−0.184875	−	0.982762i	$-0.559188\pi$
$719$	−5.12544e6	−0.369751	−0.184875	+	0.982762i	$0.559188\pi$
$720$	0	0
$721$	−8.99601e6	−0.644483
$722$	0	0
$723$	5.26228e6	0.374394
$724$	0	0
$725$	−1.34288e7	−0.948838
$726$	0	0
$727$	1.54328e7	1.08295	0.541476	−	0.840716i	$-0.317866\pi$
$727$	1.54328e7	1.08295	0.541476	+	0.840716i	$0.317866\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	−1.99658e7	−1.38195
$732$	0	0
$733$	−6.84465e6	−0.470535	−0.235267	−	0.971931i	$-0.575597\pi$
$733$	−6.84465e6	−0.470535	−0.235267	+	0.971931i	$0.575597\pi$
$734$	0	0
$735$	1.55585e6	0.106230
$736$	0	0
$737$	1.16814e7	0.792186
$738$	0	0
$739$	−2.99389e6	−0.201662	−0.100831	−	0.994904i	$-0.532150\pi$
$739$	−2.99389e6	−0.201662	−0.100831	+	0.994904i	$0.532150\pi$
$740$	0	0
$741$	−2.23870e6	−0.149779
$742$	0	0
$743$	−2.23250e7	−1.48361	−0.741804	−	0.670617i	$-0.766029\pi$
$743$	−2.23250e7	−1.48361	−0.741804	+	0.670617i	$0.766029\pi$
$744$	0	0
$745$	−3.41068e7	−2.25139
$746$	0	0
$747$	−2.08300e6	−0.136580
$748$	0	0
$749$	−109662.	−0.00714252
$750$	0	0
$751$	1.41440e7	0.915110	0.457555	−	0.889181i	$-0.348725\pi$
$751$	1.41440e7	0.915110	0.457555	+	0.889181i	$0.348725\pi$
$752$	0	0
$753$	−1.66277e6	−0.106867
$754$	0	0
$755$	2.84365e7	1.81556
$756$	0	0
$757$	−8.15367e6	−0.517147	−0.258573	−	0.965992i	$-0.583252\pi$
$757$	−8.15367e6	−0.517147	−0.258573	+	0.965992i	$0.583252\pi$
$758$	0	0
$759$	1.07979e7	0.680357
$760$	0	0
$761$	−2.25745e6	−0.141305	−0.0706524	−	0.997501i	$-0.522508\pi$
$761$	−2.25745e6	−0.141305	−0.0706524	+	0.997501i	$0.522508\pi$
$762$	0	0
$763$	−2.94774e6	−0.183307
$764$	0	0
$765$	1.07775e7	0.665834
$766$	0	0
$767$	−2.96975e7	−1.82277
$768$	0	0
$769$	−748774.	−0.0456599	−0.0228299	−	0.999739i	$-0.507268\pi$
$769$	−748774.	−0.0456599	−0.0228299	+	0.999739i	$0.507268\pi$
$770$	0	0
$771$	−2.12382e6	−0.128671
$772$	0	0
$773$	−9.46225e6	−0.569568	−0.284784	−	0.958592i	$-0.591922\pi$
$773$	−9.46225e6	−0.569568	−0.284784	+	0.958592i	$0.591922\pi$
$774$	0	0
$775$	−1.27658e7	−0.763473
$776$	0	0
$777$	4.25565e6	0.252879
$778$	0	0
$779$	−2.00222e6	−0.118214
$780$	0	0
$781$	2.74979e6	0.161314
$782$	0	0
$783$	4.75454e6	0.277143
$784$	0	0
$785$	1.04465e7	0.605056
$786$	0	0
$787$	1.99634e7	1.14894	0.574470	−	0.818525i	$-0.305208\pi$
$787$	1.99634e7	1.14894	0.574470	+	0.818525i	$0.305208\pi$
$788$	0	0
$789$	−2.20045e6	−0.125840
$790$	0	0
$791$	343686.	0.0195308
$792$	0	0
$793$	3.70945e7	2.09472
$794$	0	0
$795$	1.45839e7	0.818381
$796$	0	0
$797$	−4.05368e6	−0.226050	−0.113025	−	0.993592i	$-0.536054\pi$
$797$	−4.05368e6	−0.226050	−0.113025	+	0.993592i	$0.536054\pi$
$798$	0	0
$799$	110880.	0.00614450
$800$	0	0
$801$	7.99664e6	0.440379
$802$	0	0
$803$	−2.15653e7	−1.18023
$804$	0	0
$805$	−1.02241e7	−0.556080
$806$	0	0
$807$	−1.37501e7	−0.743229
$808$	0	0
$809$	1.85432e7	0.996124	0.498062	−	0.867141i	$-0.334045\pi$
$809$	1.85432e7	0.996124	0.498062	+	0.867141i	$0.334045\pi$
$810$	0	0
$811$	−1.63648e7	−0.873690	−0.436845	−	0.899537i	$-0.643904\pi$
$811$	−1.63648e7	−0.873690	−0.436845	+	0.899537i	$0.643904\pi$
$812$	0	0
$813$	1.86350e7	0.988790
$814$	0	0
$815$	3.81946e7	2.01422
$816$	0	0
$817$	−2.54974e6	−0.133642
$818$	0	0
$819$	4.18333e6	0.217927
$820$	0	0
$821$	−5.45014e6	−0.282195	−0.141098	−	0.989996i	$-0.545063\pi$
$821$	−5.45014e6	−0.282195	−0.141098	+	0.989996i	$0.545063\pi$
$822$	0	0
$823$	2.19153e7	1.12784	0.563921	−	0.825829i	$-0.309292\pi$
$823$	2.19153e7	1.12784	0.563921	+	0.825829i	$0.309292\pi$
$824$	0	0
$825$	−7.67183e6	−0.392432
$826$	0	0
$827$	−8.11859e6	−0.412778	−0.206389	−	0.978470i	$-0.566171\pi$
$827$	−8.11859e6	−0.412778	−0.206389	+	0.978470i	$0.566171\pi$
$828$	0	0
$829$	1.60662e7	0.811943	0.405972	−	0.913886i	$-0.366933\pi$
$829$	1.60662e7	0.811943	0.405972	+	0.913886i	$0.366933\pi$
$830$	0	0
$831$	2.16654e7	1.08834
$832$	0	0
$833$	−4.43705e6	−0.221555
$834$	0	0
$835$	−2.24891e7	−1.11623
$836$	0	0
$837$	4.51980e6	0.223000
$838$	0	0
$839$	2.63504e7	1.29236	0.646178	−	0.763186i	$-0.276366\pi$
$839$	2.63504e7	1.29236	0.646178	+	0.763186i	$0.276366\pi$
$840$	0	0
$841$	2.20253e7	1.07382
$842$	0	0
$843$	−3.07697e6	−0.149127
$844$	0	0
$845$	−5.32529e7	−2.56567
$846$	0	0
$847$	−506905.	−0.0242783
$848$	0	0
$849$	5.20708e6	0.247927
$850$	0	0
$851$	−2.79657e7	−1.32374
$852$	0	0
$853$	2.78129e7	1.30880	0.654400	−	0.756148i	$-0.272921\pi$
$853$	2.78129e7	1.30880	0.654400	+	0.756148i	$0.272921\pi$
$854$	0	0
$855$	1.37635e6	0.0643894
$856$	0	0
$857$	−2.69363e6	−0.125281	−0.0626406	−	0.998036i	$-0.519952\pi$
$857$	−2.69363e6	−0.125281	−0.0626406	+	0.998036i	$0.519952\pi$
$858$	0	0
$859$	1.88389e7	0.871109	0.435555	−	0.900162i	$-0.356552\pi$
$859$	1.88389e7	0.871109	0.435555	+	0.900162i	$0.356552\pi$
$860$	0	0
$861$	3.74144e6	0.172001
$862$	0	0
$863$	1.28630e7	0.587917	0.293959	−	0.955818i	$-0.405027\pi$
$863$	1.28630e7	0.587917	0.293959	+	0.955818i	$0.405027\pi$
$864$	0	0
$865$	5.40778e6	0.245741
$866$	0	0
$867$	−1.79572e7	−0.811319
$868$	0	0
$869$	−1.79428e7	−0.806009
$870$	0	0
$871$	−2.97397e7	−1.32828
$872$	0	0
$873$	−1.20653e7	−0.535799
$874$	0	0
$875$	−3.76085e6	−0.166060
$876$	0	0
$877$	−1.58811e7	−0.697240	−0.348620	−	0.937264i	$-0.613350\pi$
$877$	−1.58811e7	−0.697240	−0.348620	+	0.937264i	$0.613350\pi$
$878$	0	0
$879$	−7.02486e6	−0.306666
$880$	0	0
$881$	1.73681e7	0.753899	0.376950	−	0.926234i	$-0.376973\pi$
$881$	1.73681e7	0.753899	0.376950	+	0.926234i	$0.376973\pi$
$882$	0	0
$883$	−2.23513e7	−0.964721	−0.482361	−	0.875973i	$-0.660220\pi$
$883$	−2.23513e7	−0.964721	−0.482361	+	0.875973i	$0.660220\pi$
$884$	0	0
$885$	1.82580e7	0.783603
$886$	0	0
$887$	8.91140e6	0.380309	0.190155	−	0.981754i	$-0.439101\pi$
$887$	8.91140e6	0.380309	0.190155	+	0.981754i	$0.439101\pi$
$888$	0	0
$889$	−87220.0	−0.00370136
$890$	0	0
$891$	2.71625e6	0.114624
$892$	0	0
$893$	14160.0	0.000594203 0
$894$	0	0
$895$	2.78437e7	1.16190
$896$	0	0
$897$	−2.74904e7	−1.14078
$898$	0	0
$899$	4.04364e7	1.66868
$900$	0	0
$901$	−4.15911e7	−1.70682
$902$	0	0
$903$	4.76456e6	0.194448
$904$	0	0
$905$	3.00671e7	1.22031
$906$	0	0
$907$	4.53327e6	0.182976	0.0914878	−	0.995806i	$-0.470838\pi$
$907$	4.53327e6	0.182976	0.0914878	+	0.995806i	$0.470838\pi$
$908$	0	0
$909$	3.91619e6	0.157200
$910$	0	0
$911$	9.83085e6	0.392460	0.196230	−	0.980558i	$-0.437130\pi$
$911$	9.83085e6	0.392460	0.196230	+	0.980558i	$0.437130\pi$
$912$	0	0
$913$	−1.06464e7	−0.422695
$914$	0	0
$915$	−2.28057e7	−0.900515
$916$	0	0
$917$	−1.29919e7	−0.510209
$918$	0	0
$919$	4.82049e7	1.88279	0.941396	−	0.337304i	$-0.109515\pi$
$919$	4.82049e7	1.88279	0.941396	+	0.337304i	$0.109515\pi$
$920$	0	0
$921$	−2.02343e7	−0.786029
$922$	0	0
$923$	−7.00067e6	−0.270480
$924$	0	0
$925$	1.98694e7	0.763536
$926$	0	0
$927$	1.48710e7	0.568381
$928$	0	0
$929$	−6.77017e6	−0.257371	−0.128686	−	0.991685i	$-0.541076\pi$
$929$	−6.77017e6	−0.257371	−0.128686	+	0.991685i	$0.541076\pi$
$930$	0	0
$931$	−566636.	−0.0214255
$932$	0	0
$933$	5.23271e6	0.196799
$934$	0	0
$935$	5.50852e7	2.06066
$936$	0	0
$937$	1.25127e7	0.465590	0.232795	−	0.972526i	$-0.425213\pi$
$937$	1.25127e7	0.465590	0.232795	+	0.972526i	$0.425213\pi$
$938$	0	0
$939$	1.80366e7	0.667562
$940$	0	0
$941$	−1.38659e7	−0.510473	−0.255236	−	0.966879i	$-0.582153\pi$
$941$	−1.38659e7	−0.510473	−0.255236	+	0.966879i	$0.582153\pi$
$942$	0	0
$943$	−2.45866e7	−0.900367
$944$	0	0
$945$	−2.57191e6	−0.0936864
$946$	0	0
$947$	−4.89287e6	−0.177292	−0.0886460	−	0.996063i	$-0.528254\pi$
$947$	−4.89287e6	−0.177292	−0.0886460	+	0.996063i	$0.528254\pi$
$948$	0	0
$949$	5.49029e7	1.97893
$950$	0	0
$951$	1.15049e7	0.412506
$952$	0	0
$953$	−1.40055e7	−0.499535	−0.249768	−	0.968306i	$-0.580354\pi$
$953$	−1.40055e7	−0.499535	−0.249768	+	0.968306i	$0.580354\pi$
$954$	0	0
$955$	−7.11396e7	−2.52408
$956$	0	0
$957$	2.43010e7	0.857717
$958$	0	0
$959$	1.01298e7	0.355675
$960$	0	0
$961$	9.81085e6	0.342687
$962$	0	0
$963$	181278.	0.00629911
$964$	0	0
$965$	2.94666e7	1.01862
$966$	0	0
$967$	1.02386e7	0.352106	0.176053	−	0.984381i	$-0.443667\pi$
$967$	1.02386e7	0.352106	0.176053	+	0.984381i	$0.443667\pi$
$968$	0	0
$969$	−3.92515e6	−0.134291
$970$	0	0
$971$	−1.17452e7	−0.399773	−0.199886	−	0.979819i	$-0.564057\pi$
$971$	−1.17452e7	−0.399773	−0.199886	+	0.979819i	$0.564057\pi$
$972$	0	0
$973$	−1.16050e7	−0.392972
$974$	0	0
$975$	1.95317e7	0.658003
$976$	0	0
$977$	2.26195e7	0.758134	0.379067	−	0.925369i	$-0.376245\pi$
$977$	2.26195e7	0.758134	0.379067	+	0.925369i	$0.376245\pi$
$978$	0	0
$979$	4.08717e7	1.36291
$980$	0	0
$981$	4.87280e6	0.161661
$982$	0	0
$983$	−2.79575e7	−0.922813	−0.461407	−	0.887189i	$-0.652655\pi$
$983$	−2.79575e7	−0.922813	−0.461407	+	0.887189i	$0.652655\pi$
$984$	0	0
$985$	6.64006e7	2.18063
$986$	0	0
$987$	−26460.0	−0.000864564 0
$988$	0	0
$989$	−3.13100e7	−1.01787
$990$	0	0
$991$	1.66475e7	0.538474	0.269237	−	0.963074i	$-0.413228\pi$
$991$	1.66475e7	0.538474	0.269237	+	0.963074i	$0.413228\pi$
$992$	0	0
$993$	2.33830e7	0.752537
$994$	0	0
$995$	1.36431e7	0.436874
$996$	0	0
$997$	−5.17280e7	−1.64812	−0.824058	−	0.566505i	$-0.808295\pi$
$997$	−5.17280e7	−1.64812	−0.824058	+	0.566505i	$0.808295\pi$
$998$	0	0
$999$	−7.03485e6	−0.223019

Twists

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

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