LMFDB - Embedded newform 882.6.a.n.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [882,6,Mod(1,882)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("882.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(882, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 882 = 2 \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	882.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,4,0,16,-72,0,0,64,0,-288,414] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$141.458529075$
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	$\chi$	$=$	882.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q+4.00000 q^{2} +16.0000 q^{4} -72.0000 q^{5} +64.0000 q^{8} -288.000 q^{10} +414.000 q^{11} +1054.00 q^{13} +256.000 q^{16} -1848.00 q^{17} -236.000 q^{19} -1152.00 q^{20} +1656.00 q^{22} -2898.00 q^{23} +2059.00 q^{25} +4216.00 q^{26} +6522.00 q^{29} -6200.00 q^{31} +1024.00 q^{32} -7392.00 q^{34} +9650.00 q^{37} -944.000 q^{38} -4608.00 q^{40} +8484.00 q^{41} -10804.0 q^{43} +6624.00 q^{44} -11592.0 q^{46} +60.0000 q^{47} +8236.00 q^{50} +16864.0 q^{52} -22506.0 q^{53} -29808.0 q^{55} +26088.0 q^{58} -28176.0 q^{59} +35194.0 q^{61} -24800.0 q^{62} +4096.00 q^{64} -75888.0 q^{65} -28216.0 q^{67} -29568.0 q^{68} +6642.00 q^{71} +52090.0 q^{73} +38600.0 q^{74} -3776.00 q^{76} +43340.0 q^{79} -18432.0 q^{80} +33936.0 q^{82} +25716.0 q^{83} +133056. q^{85} -43216.0 q^{86} +26496.0 q^{88} +98724.0 q^{89} -46368.0 q^{92} +240.000 q^{94} +16992.0 q^{95} +148954. q^{97} +O(q^{100})$

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 $

$2$	4.00000	0.707107
$2$	4.00000	0.707107
$3$	0	0
$3$	0	0
$4$	16.0000	0.500000
$5$	−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
$7$	0	0
$7$	0	0
$8$	64.0000	0.353553
$9$	0	0
$10$	−288.000	−0.910736
$11$	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	$-0.167399\pi$
$13$	1054.00	1.72975	0.864873	+	0.501991i	$0.167399\pi$
$14$	0	0
$15$	0	0
$16$	256.000	0.250000
$17$	−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	$0.523892\pi$
$20$	−1152.00	−0.643988
$21$	0	0
$22$	1656.00	0.729464
$23$	−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
$26$	4216.00	1.22311
$27$	0	0
$28$	0	0
$29$	6522.00	1.44008	0.720039	−	0.693934i	$-0.244124\pi$
$29$	6522.00	1.44008	0.720039	+	0.693934i	$0.244124\pi$
$30$	0	0
$31$	−6200.00	−1.15874	−0.579372	−	0.815063i	$-0.696702\pi$
$31$	−6200.00	−1.15874	−0.579372	+	0.815063i	$0.696702\pi$
$32$	1024.00	0.176777
$33$	0	0
$34$	−7392.00	−1.09664
$35$	0	0
$36$	0	0
$37$	9650.00	1.15884	0.579419	−	0.815030i	$-0.303279\pi$
$37$	9650.00	1.15884	0.579419	+	0.815030i	$0.303279\pi$
$38$	−944.000	−0.106051
$39$	0	0
$40$	−4608.00	−0.455368
$41$	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	$-0.646987\pi$
$43$	−10804.0	−0.891073	−0.445537	+	0.895264i	$0.646987\pi$
$44$	6624.00	0.515809
$45$	0	0
$46$	−11592.0	−0.807725
$47$	60.0000	0.00396193	0.00198096	−	0.999998i	$-0.499369\pi$
$47$	60.0000	0.00396193	0.00198096	+	0.999998i	$0.499369\pi$
$48$	0	0
$49$	0	0
$50$	8236.00	0.465899
$51$	0	0
$52$	16864.0	0.864873
$53$	−22506.0	−1.10055	−0.550274	−	0.834984i	$-0.685477\pi$
$53$	−22506.0	−1.10055	−0.550274	+	0.834984i	$0.685477\pi$
$54$	0	0
$55$	−29808.0	−1.32870
$56$	0	0
$57$	0	0
$58$	26088.0	1.01829
$59$	−28176.0	−1.05378	−0.526889	−	0.849934i	$-0.676642\pi$
$59$	−28176.0	−1.05378	−0.526889	+	0.849934i	$0.676642\pi$
$60$	0	0
$61$	35194.0	1.21100	0.605500	−	0.795845i	$-0.292973\pi$
$61$	35194.0	1.21100	0.605500	+	0.795845i	$0.292973\pi$
$62$	−24800.0	−0.819356
$63$	0	0
$64$	4096.00	0.125000
$65$	−75888.0	−2.22787
$66$	0	0
$67$	−28216.0	−0.767907	−0.383953	−	0.923352i	$-0.625438\pi$
$67$	−28216.0	−0.767907	−0.383953	+	0.923352i	$0.625438\pi$
$68$	−29568.0	−0.775443
$69$	0	0
$70$	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	$-0.306157\pi$
$73$	52090.0	1.14406	0.572028	+	0.820234i	$0.306157\pi$
$74$	38600.0	0.819423
$75$	0	0
$76$	−3776.00	−0.0749891
$77$	0	0
$78$	0	0
$79$	43340.0	0.781306	0.390653	−	0.920538i	$-0.372249\pi$
$79$	43340.0	0.781306	0.390653	+	0.920538i	$0.372249\pi$
$80$	−18432.0	−0.321994
$81$	0	0
$82$	33936.0	0.557347
$83$	25716.0	0.409740	0.204870	−	0.978789i	$-0.434323\pi$
$83$	25716.0	0.409740	0.204870	+	0.978789i	$0.434323\pi$
$84$	0	0
$85$	133056.	1.99750
$86$	−43216.0	−0.630084
$87$	0	0
$88$	26496.0	0.364732
$89$	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
$91$	0	0
$92$	−46368.0	−0.571148
$93$	0	0
$94$	240.000	0.00280151
$95$	16992.0	0.193168
$96$	0	0
$97$	148954.	1.60740	0.803698	−	0.595038i	$-0.202863\pi$
$97$	148954.	1.60740	0.803698	+	0.595038i	$0.202863\pi$
$98$	0	0
$99$	0	0
$100$	32944.0	0.329440
$101$	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	$0.175042\pi$
$104$	67456.0	0.611557
$105$	0	0
$106$	−90024.0	−0.778204
$107$	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	$0.422035\pi$
$110$	−119232.	−0.939531
$111$	0	0
$112$	0	0
$113$	7014.00	0.0516737	0.0258369	−	0.999666i	$-0.491775\pi$
$113$	7014.00	0.0516737	0.0258369	+	0.999666i	$0.491775\pi$
$114$	0	0
$115$	208656.	1.47125
$116$	104352.	0.720039
$117$	0	0
$118$	−112704.	−0.745134
$119$	0	0
$120$	0	0
$121$	10345.0	0.0642343
$122$	140776.	0.856306
$123$	0	0
$124$	−99200.0	−0.579372
$125$	76752.0	0.439354
$126$	0	0
$127$	−1780.00	−0.00979289	−0.00489644	−	0.999988i	$-0.501559\pi$
$127$	−1780.00	−0.00979289	−0.00489644	+	0.999988i	$0.501559\pi$
$128$	16384.0	0.0883883
$129$	0	0
$130$	−303552.	−1.57534
$131$	−265140.	−1.34989	−0.674943	−	0.737870i	$-0.735832\pi$
$131$	−265140.	−1.34989	−0.674943	+	0.737870i	$0.735832\pi$
$132$	0	0
$133$	0	0
$134$	−112864.	−0.542992
$135$	0	0
$136$	−118272.	−0.548321
$137$	206730.	0.941027	0.470514	−	0.882393i	$-0.344069\pi$
$137$	206730.	0.941027	0.470514	+	0.882393i	$0.344069\pi$
$138$	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
$141$	0	0
$142$	26568.0	0.110570
$143$	436356.	1.78444
$144$	0	0
$145$	−469584.	−1.85478
$146$	208360.	0.808970
$147$	0	0
$148$	154400.	0.579419
$149$	−473706.	−1.74801	−0.874004	−	0.485919i	$-0.838485\pi$
$149$	−473706.	−1.74801	−0.874004	+	0.485919i	$0.838485\pi$
$150$	0	0
$151$	394952.	1.40962	0.704810	−	0.709396i	$-0.251032\pi$
$151$	394952.	1.40962	0.704810	+	0.709396i	$0.251032\pi$
$152$	−15104.0	−0.0530253
$153$	0	0
$154$	0	0
$155$	446400.	1.49243
$156$	0	0
$157$	145090.	0.469773	0.234887	−	0.972023i	$-0.424528\pi$
$157$	145090.	0.469773	0.234887	+	0.972023i	$0.424528\pi$
$158$	173360.	0.552467
$159$	0	0
$160$	−73728.0	−0.227684
$161$	0	0
$162$	0	0
$163$	530480.	1.56387	0.781934	−	0.623361i	$-0.214233\pi$
$163$	530480.	1.56387	0.781934	+	0.623361i	$0.214233\pi$
$164$	135744.	0.394104
$165$	0	0
$166$	102864.	0.289730
$167$	−312348.	−0.866658	−0.433329	−	0.901236i	$-0.642661\pi$
$167$	−312348.	−0.866658	−0.433329	+	0.901236i	$0.642661\pi$
$168$	0	0
$169$	739623.	1.99202
$170$	532224.	1.41245
$171$	0	0
$172$	−172864.	−0.445537
$173$	−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
$175$	0	0
$176$	105984.	0.257904
$177$	0	0
$178$	394896.	0.934185
$179$	−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	$-0.342907\pi$
$181$	417598.	0.947462	0.473731	+	0.880669i	$0.342907\pi$
$182$	0	0
$183$	0	0
$184$	−185472.	−0.403863
$185$	−694800.	−1.49256
$186$	0	0
$187$	−765072.	−1.59992
$188$	960.000	0.00198096
$189$	0	0
$190$	67968.0	0.136590
$191$	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	$0.629406\pi$
$194$	595816.	1.13660
$195$	0	0
$196$	0	0
$197$	922230.	1.69307	0.846533	−	0.532337i	$-0.178686\pi$
$197$	922230.	1.69307	0.846533	+	0.532337i	$0.178686\pi$
$198$	0	0
$199$	−189488.	−0.339195	−0.169597	−	0.985513i	$-0.554247\pi$
$199$	−189488.	−0.339195	−0.169597	+	0.985513i	$0.554247\pi$
$200$	131776.	0.232949
$201$	0	0
$202$	193392.	0.333473
$203$	0	0
$204$	0	0
$205$	−610848.	−1.01519
$206$	734368.	1.20572
$207$	0	0
$208$	269824.	0.432436
$209$	−97704.0	−0.154720
$210$	0	0
$211$	−611380.	−0.945377	−0.472689	−	0.881230i	$-0.656716\pi$
$211$	−611380.	−0.945377	−0.472689	+	0.881230i	$0.656716\pi$
$212$	−360096.	−0.550274
$213$	0	0
$214$	8952.00	0.0133624
$215$	777888.	1.14768
$216$	0	0
$217$	0	0
$218$	240632.	0.342935
$219$	0	0
$220$	−476928.	−0.664349
$221$	−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
$225$	0	0
$226$	28056.0	0.0365388
$227$	80712.0	0.103962	0.0519809	−	0.998648i	$-0.483447\pi$
$227$	80712.0	0.103962	0.0519809	+	0.998648i	$0.483447\pi$
$228$	0	0
$229$	−152738.	−0.192468	−0.0962340	−	0.995359i	$-0.530680\pi$
$229$	−152738.	−0.192468	−0.0962340	+	0.995359i	$0.530680\pi$
$230$	834624.	1.04033
$231$	0	0
$232$	417408.	0.509144
$233$	354282.	0.427523	0.213761	−	0.976886i	$-0.431428\pi$
$233$	354282.	0.427523	0.213761	+	0.976886i	$0.431428\pi$
$234$	0	0
$235$	−4320.00	−0.00510287
$236$	−450816.	−0.526889
$237$	0	0
$238$	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	$-0.394893\pi$
$241$	584698.	0.648469	0.324234	+	0.945977i	$0.394893\pi$
$242$	41380.0	0.0454205
$243$	0	0
$244$	563104.	0.605500
$245$	0	0
$246$	0	0
$247$	−248744.	−0.259424
$248$	−396800.	−0.409678
$249$	0	0
$250$	307008.	0.310670
$251$	−184752.	−0.185099	−0.0925497	−	0.995708i	$-0.529502\pi$
$251$	−184752.	−0.185099	−0.0925497	+	0.995708i	$0.529502\pi$
$252$	0	0
$253$	−1.19977e6	−1.17841
$254$	−7120.00	−0.00692462
$255$	0	0
$256$	65536.0	0.0625000
$257$	235980.	0.222865	0.111433	−	0.993772i	$-0.464456\pi$
$257$	235980.	0.222865	0.111433	+	0.993772i	$0.464456\pi$
$258$	0	0
$259$	0	0
$260$	−1.21421e6	−1.11393
$261$	0	0
$262$	−1.06056e6	−0.954513
$263$	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
$267$	0	0
$268$	−451456.	−0.383953
$269$	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	$0.827253\pi$
$272$	−473088.	−0.387721
$273$	0	0
$274$	826920.	0.665407
$275$	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	$0.891561\pi$
$278$	947344.	0.735183
$279$	0	0
$280$	0	0
$281$	−341886.	−0.258295	−0.129147	−	0.991625i	$-0.541224\pi$
$281$	−341886.	−0.258295	−0.129147	+	0.991625i	$0.541224\pi$
$282$	0	0
$283$	−578564.	−0.429423	−0.214712	−	0.976678i	$-0.568881\pi$
$283$	−578564.	−0.429423	−0.214712	+	0.976678i	$0.568881\pi$
$284$	106272.	0.0781849
$285$	0	0
$286$	1.74542e6	1.26179
$287$	0	0
$288$	0	0
$289$	1.99525e6	1.40525
$290$	−1.87834e6	−1.31153
$291$	0	0
$292$	833440.	0.572028
$293$	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
$296$	617600.	0.409711
$297$	0	0
$298$	−1.89482e6	−1.23603
$299$	−3.05449e6	−1.97588
$300$	0	0
$301$	0	0
$302$	1.57981e6	0.996752
$303$	0	0
$304$	−60416.0	−0.0374945
$305$	−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
$309$	0	0
$310$	1.78560e6	1.05531
$311$	581412.	0.340865	0.170433	−	0.985369i	$-0.445483\pi$
$311$	581412.	0.340865	0.170433	+	0.985369i	$0.445483\pi$
$312$	0	0
$313$	2.00407e6	1.15625	0.578125	−	0.815948i	$-0.303784\pi$
$313$	2.00407e6	1.15625	0.578125	+	0.815948i	$0.303784\pi$
$314$	580360.	0.332180
$315$	0	0
$316$	693440.	0.390653
$317$	1.27832e6	0.714481	0.357241	−	0.934012i	$-0.383718\pi$
$317$	1.27832e6	0.714481	0.357241	+	0.934012i	$0.383718\pi$
$318$	0	0
$319$	2.70011e6	1.48561
$320$	−294912.	−0.160997
$321$	0	0
$322$	0	0
$323$	436128.	0.232599
$324$	0	0
$325$	2.17019e6	1.13969
$326$	2.12192e6	1.10582
$327$	0	0
$328$	542976.	0.278674
$329$	0	0
$330$	0	0
$331$	2.59812e6	1.30343	0.651716	−	0.758463i	$-0.274049\pi$
$331$	2.59812e6	1.30343	0.651716	+	0.758463i	$0.274049\pi$
$332$	411456.	0.204870
$333$	0	0
$334$	−1.24939e6	−0.612819
$335$	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	$0.237501\pi$
$338$	2.95849e6	1.40857
$339$	0	0
$340$	2.12890e6	0.998751
$341$	−2.56680e6	−1.19538
$342$	0	0
$343$	0	0
$344$	−691456.	−0.315042
$345$	0	0
$346$	−300432.	−0.134914
$347$	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	$-0.722952\pi$
$349$	−2.93322e6	−1.28908	−0.644542	+	0.764569i	$0.722952\pi$
$350$	0	0
$351$	0	0
$352$	423936.	0.182366
$353$	−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
$356$	1.57958e6	0.660568
$357$	0	0
$358$	−1.54687e6	−0.637891
$359$	−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
$362$	1.67039e6	0.669957
$363$	0	0
$364$	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	$0.536767\pi$
$368$	−741888.	−0.285574
$369$	0	0
$370$	−2.77920e6	−1.05540
$371$	0	0
$372$	0	0
$373$	2.04522e6	0.761147	0.380573	−	0.924751i	$-0.375727\pi$
$373$	2.04522e6	0.761147	0.380573	+	0.924751i	$0.375727\pi$
$374$	−3.06029e6	−1.13131
$375$	0	0
$376$	3840.00	0.00140075
$377$	6.87419e6	2.49097
$378$	0	0
$379$	−3.22198e6	−1.15219	−0.576096	−	0.817382i	$-0.695425\pi$
$379$	−3.22198e6	−1.15219	−0.576096	+	0.817382i	$0.695425\pi$
$380$	271872.	0.0965841
$381$	0	0
$382$	3.95220e6	1.38574
$383$	−1.72966e6	−0.602508	−0.301254	−	0.953544i	$-0.597405\pi$
$383$	−1.72966e6	−0.602508	−0.301254	+	0.953544i	$0.597405\pi$
$384$	0	0
$385$	0	0
$386$	−1.63703e6	−0.559228
$387$	0	0
$388$	2.38326e6	0.803698
$389$	1.74919e6	0.586087	0.293043	−	0.956099i	$-0.405332\pi$
$389$	1.74919e6	0.586087	0.293043	+	0.956099i	$0.405332\pi$
$390$	0	0
$391$	5.35550e6	1.77157
$392$	0	0
$393$	0	0
$394$	3.68892e6	1.19718
$395$	−3.12048e6	−1.00630
$396$	0	0
$397$	−1.88205e6	−0.599313	−0.299657	−	0.954047i	$-0.596872\pi$
$397$	−1.88205e6	−0.599313	−0.299657	+	0.954047i	$0.596872\pi$
$398$	−757952.	−0.239847
$399$	0	0
$400$	527104.	0.164720
$401$	−4.18124e6	−1.29851	−0.649253	−	0.760573i	$-0.724918\pi$
$401$	−4.18124e6	−1.29851	−0.649253	+	0.760573i	$0.724918\pi$
$402$	0	0
$403$	−6.53480e6	−2.00433
$404$	773568.	0.235801
$405$	0	0
$406$	0	0
$407$	3.99510e6	1.19548
$408$	0	0
$409$	471682.	0.139425	0.0697126	−	0.997567i	$-0.477792\pi$
$409$	471682.	0.139425	0.0697126	+	0.997567i	$0.477792\pi$
$410$	−2.44339e6	−0.717850
$411$	0	0
$412$	2.93747e6	0.852571
$413$	0	0
$414$	0	0
$415$	−1.85155e6	−0.527735
$416$	1.07930e6	0.305779
$417$	0	0
$418$	−390816.	−0.109404
$419$	−3.54094e6	−0.985333	−0.492666	−	0.870218i	$-0.663978\pi$
$419$	−3.54094e6	−0.985333	−0.492666	+	0.870218i	$0.663978\pi$
$420$	0	0
$421$	2.72763e6	0.750032	0.375016	−	0.927018i	$-0.377637\pi$
$421$	2.72763e6	0.750032	0.375016	+	0.927018i	$0.377637\pi$
$422$	−2.44552e6	−0.668483
$423$	0	0
$424$	−1.44038e6	−0.389102
$425$	−3.80503e6	−1.02185
$426$	0	0
$427$	0	0
$428$	35808.0	0.00944867
$429$	0	0
$430$	3.11155e6	0.811533
$431$	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	$-0.845626\pi$
$433$	−6.90300e6	−1.76937	−0.884684	+	0.466191i	$0.845626\pi$
$434$	0	0
$435$	0	0
$436$	962528.	0.242492
$437$	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	$0.733197\pi$
$440$	−1.90771e6	−0.469766
$441$	0	0
$442$	−7.79117e6	−1.89691
$443$	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
$446$	3.13302e6	0.745807
$447$	0	0
$448$	0	0
$449$	−212994.	−0.0498599	−0.0249300	−	0.999689i	$-0.507936\pi$
$449$	−212994.	−0.0498599	−0.0249300	+	0.999689i	$0.507936\pi$
$450$	0	0
$451$	3.51238e6	0.813129
$452$	112224.	0.0258369
$453$	0	0
$454$	322848.	0.0735120
$455$	0	0
$456$	0	0
$457$	−916150.	−0.205199	−0.102600	−	0.994723i	$-0.532716\pi$
$457$	−916150.	−0.205199	−0.102600	+	0.994723i	$0.532716\pi$
$458$	−610952.	−0.136095
$459$	0	0
$460$	3.33850e6	0.735625
$461$	−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	$-0.135002\pi$
$463$	8.40799e6	1.82280	0.911401	+	0.411519i	$0.135002\pi$
$464$	1.66963e6	0.360019
$465$	0	0
$466$	1.41713e6	0.302304
$467$	72048.0	0.0152873	0.00764363	−	0.999971i	$-0.497567\pi$
$467$	72048.0	0.0152873	0.00764363	+	0.999971i	$0.497567\pi$
$468$	0	0
$469$	0	0
$470$	−17280.0	−0.00360827
$471$	0	0
$472$	−1.80326e6	−0.372567
$473$	−4.47286e6	−0.919247
$474$	0	0
$475$	−485924.	−0.0988176
$476$	0	0
$477$	0	0
$478$	−1.10148e6	−0.220499
$479$	4.80560e6	0.956994	0.478497	−	0.878089i	$-0.341182\pi$
$479$	4.80560e6	0.956994	0.478497	+	0.878089i	$0.341182\pi$
$480$	0	0
$481$	1.01711e7	2.00450
$482$	2.33879e6	0.458537
$483$	0	0
$484$	165520.	0.0321172
$485$	−1.07247e7	−2.07029
$486$	0	0
$487$	4.41805e6	0.844127	0.422064	−	0.906566i	$-0.361306\pi$
$487$	4.41805e6	0.844127	0.422064	+	0.906566i	$0.361306\pi$
$488$	2.25242e6	0.428153
$489$	0	0
$490$	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
$494$	−994976.	−0.183441
$495$	0	0
$496$	−1.58720e6	−0.289686
$497$	0	0
$498$	0	0
$499$	−9.22344e6	−1.65822	−0.829109	−	0.559087i	$-0.811152\pi$
$499$	−9.22344e6	−1.65822	−0.829109	+	0.559087i	$0.811152\pi$
$500$	1.22803e6	0.219677
$501$	0	0
$502$	−739008.	−0.130885
$503$	1.45562e6	0.256525	0.128262	−	0.991740i	$-0.459060\pi$
$503$	1.45562e6	0.256525	0.128262	+	0.991740i	$0.459060\pi$
$504$	0	0
$505$	−3.48106e6	−0.607411
$506$	−4.79909e6	−0.833264
$507$	0	0
$508$	−28480.0	−0.00489644
$509$	−367344.	−0.0628461	−0.0314231	−	0.999506i	$-0.510004\pi$
$509$	−367344.	−0.0628461	−0.0314231	+	0.999506i	$0.510004\pi$
$510$	0	0
$511$	0	0
$512$	262144.	0.0441942
$513$	0	0
$514$	943920.	0.157590
$515$	−1.32186e7	−2.19618
$516$	0	0
$517$	24840.0	0.00408719
$518$	0	0
$519$	0	0
$520$	−4.85683e6	−0.787671
$521$	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	$0.505982\pi$
$524$	−4.24224e6	−0.674943
$525$	0	0
$526$	977976.	0.154122
$527$	1.14576e7	1.79708
$528$	0	0
$529$	1.96206e6	0.304841
$530$	6.48173e6	1.00231
$531$	0	0
$532$	0	0
$533$	8.94214e6	1.36340
$534$	0	0
$535$	−161136.	−0.0243393
$536$	−1.80582e6	−0.271496
$537$	0	0
$538$	6.11117e6	0.910266
$539$	0	0
$540$	0	0
$541$	109010.	0.0160130	0.00800651	−	0.999968i	$-0.497451\pi$
$541$	109010.	0.0160130	0.00800651	+	0.999968i	$0.497451\pi$
$542$	−8.28224e6	−1.21102
$543$	0	0
$544$	−1.89235e6	−0.274160
$545$	−4.33138e6	−0.624647
$546$	0	0
$547$	1.61953e6	0.231430	0.115715	−	0.993282i	$-0.463084\pi$
$547$	1.61953e6	0.231430	0.115715	+	0.993282i	$0.463084\pi$
$548$	3.30768e6	0.470514
$549$	0	0
$550$	3.40970e6	0.480629
$551$	−1.53919e6	−0.215980
$552$	0	0
$553$	0	0
$554$	−9.62908e6	−1.33294
$555$	0	0
$556$	3.78938e6	0.519853
$557$	6.62986e6	0.905454	0.452727	−	0.891649i	$-0.350451\pi$
$557$	6.62986e6	0.905454	0.452727	+	0.891649i	$0.350451\pi$
$558$	0	0
$559$	−1.13874e7	−1.54133
$560$	0	0
$561$	0	0
$562$	−1.36754e6	−0.182642
$563$	7.85294e6	1.04415	0.522073	−	0.852901i	$-0.325159\pi$
$563$	7.85294e6	1.04415	0.522073	+	0.852901i	$0.325159\pi$
$564$	0	0
$565$	−505008.	−0.0665545
$566$	−2.31426e6	−0.303648
$567$	0	0
$568$	425088.	0.0552851
$569$	2.48155e6	0.321323	0.160661	−	0.987010i	$-0.448637\pi$
$569$	2.48155e6	0.321323	0.160661	+	0.987010i	$0.448637\pi$
$570$	0	0
$571$	1.13675e7	1.45907	0.729533	−	0.683945i	$-0.239737\pi$
$571$	1.13675e7	1.45907	0.729533	+	0.683945i	$0.239737\pi$
$572$	6.98170e6	0.892218
$573$	0	0
$574$	0	0
$575$	−5.96698e6	−0.752636
$576$	0	0
$577$	8.20505e6	1.02599	0.512993	−	0.858393i	$-0.328537\pi$
$577$	8.20505e6	1.02599	0.512993	+	0.858393i	$0.328537\pi$
$578$	7.98099e6	0.993658
$579$	0	0
$580$	−7.51334e6	−0.927392
$581$	0	0
$582$	0	0
$583$	−9.31748e6	−1.13534
$584$	3.33376e6	0.404485
$585$	0	0
$586$	3.12216e6	0.375587
$587$	−1.38400e7	−1.65783	−0.828917	−	0.559371i	$-0.811043\pi$
$587$	−1.38400e7	−1.65783	−0.828917	+	0.559371i	$0.811043\pi$
$588$	0	0
$589$	1.46320e6	0.173786
$590$	8.11469e6	0.959714
$591$	0	0
$592$	2.47040e6	0.289710
$593$	−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
$595$	0	0
$596$	−7.57930e6	−0.874004
$597$	0	0
$598$	−1.22180e7	−1.39716
$599$	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	$-0.449603\pi$
$601$	2.79225e6	0.315333	0.157666	+	0.987492i	$0.449603\pi$
$602$	0	0
$603$	0	0
$604$	6.31923e6	0.704810
$605$	−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	$0.512520\pi$
$608$	−241664.	−0.0265126
$609$	0	0
$610$	−1.01359e7	−1.10290
$611$	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	$0.446552\pi$
$614$	8.99301e6	0.962685
$615$	0	0
$616$	0	0
$617$	−3.62384e6	−0.383227	−0.191613	−	0.981470i	$-0.561372\pi$
$617$	−3.62384e6	−0.383227	−0.191613	+	0.981470i	$0.561372\pi$
$618$	0	0
$619$	−4.25196e6	−0.446028	−0.223014	−	0.974815i	$-0.571590\pi$
$619$	−4.25196e6	−0.446028	−0.223014	+	0.974815i	$0.571590\pi$
$620$	7.14240e6	0.746217
$621$	0	0
$622$	2.32565e6	0.241028
$623$	0	0
$624$	0	0
$625$	−1.19605e7	−1.22476
$626$	8.01628e6	0.817593
$627$	0	0
$628$	2.32144e6	0.234887
$629$	−1.78332e7	−1.79723
$630$	0	0
$631$	−1.70299e7	−1.70270	−0.851349	−	0.524600i	$-0.824215\pi$
$631$	−1.70299e7	−1.70270	−0.851349	+	0.524600i	$0.824215\pi$
$632$	2.77376e6	0.276233
$633$	0	0
$634$	5.11327e6	0.505214
$635$	128160.	0.0126130
$636$	0	0
$637$	0	0
$638$	1.08004e7	1.05048
$639$	0	0
$640$	−1.17965e6	−0.113842
$641$	1.80938e6	0.173934	0.0869669	−	0.996211i	$-0.472283\pi$
$641$	1.80938e6	0.173934	0.0869669	+	0.996211i	$0.472283\pi$
$642$	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
$645$	0	0
$646$	1.74451e6	0.164472
$647$	1.67546e7	1.57352	0.786762	−	0.617256i	$-0.211756\pi$
$647$	1.67546e7	1.57352	0.786762	+	0.617256i	$0.211756\pi$
$648$	0	0
$649$	−1.16649e7	−1.08710
$650$	8.68074e6	0.805886
$651$	0	0
$652$	8.48768e6	0.781934
$653$	−1.47859e7	−1.35695	−0.678477	−	0.734622i	$-0.737360\pi$
$653$	−1.47859e7	−1.35695	−0.678477	+	0.734622i	$0.737360\pi$
$654$	0	0
$655$	1.90901e7	1.73862
$656$	2.17190e6	0.197052
$657$	0	0
$658$	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	$-0.587484\pi$
$661$	−6.09724e6	−0.542787	−0.271394	+	0.962468i	$0.587484\pi$
$662$	1.03925e7	0.921666
$663$	0	0
$664$	1.64582e6	0.144865
$665$	0	0
$666$	0	0
$667$	−1.89008e7	−1.64500
$668$	−4.99757e6	−0.433329
$669$	0	0
$670$	8.12621e6	0.699360
$671$	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	$0.471627\pi$
$674$	1.22476e7	1.03849
$675$	0	0
$676$	1.18340e7	0.996010
$677$	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
$679$	0	0
$680$	8.51558e6	0.706223
$681$	0	0
$682$	−1.02672e7	−0.845261
$683$	−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
$687$	0	0
$688$	−2.76582e6	−0.222768
$689$	−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	$0.553218\pi$
$692$	−1.20173e6	−0.0953984
$693$	0	0
$694$	5.70199e6	0.449395
$695$	−1.70522e7	−1.33912
$696$	0	0
$697$	−1.56784e7	−1.22242
$698$	−1.17329e7	−0.911520
$699$	0	0
$700$	0	0
$701$	1.70278e7	1.30877	0.654385	−	0.756161i	$-0.272927\pi$
$701$	1.70278e7	1.30877	0.654385	+	0.756161i	$0.272927\pi$
$702$	0	0
$703$	−2.27740e6	−0.173800
$704$	1.69574e6	0.128952
$705$	0	0
$706$	−8.05104e6	−0.607911
$707$	0	0
$708$	0	0
$709$	2.26932e7	1.69543	0.847715	−	0.530452i	$-0.177978\pi$
$709$	2.26932e7	1.69543	0.847715	+	0.530452i	$0.177978\pi$
$710$	−1.91290e6	−0.142412
$711$	0	0
$712$	6.31834e6	0.467092
$713$	1.79676e7	1.32363
$714$	0	0
$715$	−3.14176e7	−2.29831
$716$	−6.18749e6	−0.451057
$717$	0	0
$718$	−1.63084e7	−1.18059
$719$	5.12544e6	0.369751	0.184875	−	0.982762i	$-0.440812\pi$
$719$	5.12544e6	0.369751	0.184875	+	0.982762i	$0.440812\pi$
$720$	0	0
$721$	0	0
$722$	−9.68161e6	−0.691202
$723$	0	0
$724$	6.68157e6	0.473731
$725$	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
$729$	0	0
$730$	−1.50019e7	−1.04193
$731$	1.99658e7	1.38195
$732$	0	0
$733$	6.84465e6	0.470535	0.235267	−	0.971931i	$-0.424403\pi$
$733$	6.84465e6	0.470535	0.235267	+	0.971931i	$0.424403\pi$
$734$	−2.37901e6	−0.162988
$735$	0	0
$736$	−2.96755e6	−0.201931
$737$	−1.16814e7	−0.792186
$738$	0	0
$739$	2.99389e6	0.201662	0.100831	−	0.994904i	$-0.467850\pi$
$739$	2.99389e6	0.201662	0.100831	+	0.994904i	$0.467850\pi$
$740$	−1.11168e7	−0.746278
$741$	0	0
$742$	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
$746$	8.18089e6	0.538212
$747$	0	0
$748$	−1.22412e7	−0.799960
$749$	0	0
$750$	0	0
$751$	−1.41440e7	−0.915110	−0.457555	−	0.889181i	$-0.651275\pi$
$751$	−1.41440e7	−0.915110	−0.457555	+	0.889181i	$0.651275\pi$
$752$	15360.0	0.000990482 0
$753$	0	0
$754$	2.74968e7	1.76138
$755$	−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	$0.583252\pi$
$758$	−1.28879e7	−0.814723
$759$	0	0
$760$	1.08749e6	0.0682952
$761$	−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
$763$	0	0
$764$	1.58088e7	0.979863
$765$	0	0
$766$	−6.91862e6	−0.426037
$767$	−2.96975e7	−1.82277
$768$	0	0
$769$	748774.	0.0456599	0.0228299	−	0.999739i	$-0.492732\pi$
$769$	748774.	0.0456599	0.0228299	+	0.999739i	$0.492732\pi$
$770$	0	0
$771$	0	0
$772$	−6.54813e6	−0.395434
$773$	−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
$776$	9.53306e6	0.568300
$777$	0	0
$778$	6.99674e6	0.414426
$779$	−2.00222e6	−0.118214
$780$	0	0
$781$	2.74979e6	0.161314
$782$	2.14220e7	1.25269
$783$	0	0
$784$	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	$0.305208\pi$
$788$	1.47557e7	0.846533
$789$	0	0
$790$	−1.24819e7	−0.711564
$791$	0	0
$792$	0	0
$793$	3.70945e7	2.09472
$794$	−7.52818e6	−0.423779
$795$	0	0
$796$	−3.03181e6	−0.169597
$797$	−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
$800$	2.10842e6	0.116475
$801$	0	0
$802$	−1.67250e7	−0.918182
$803$	2.15653e7	1.18023
$804$	0	0
$805$	0	0
$806$	−2.61392e7	−1.41728
$807$	0	0
$808$	3.09427e6	0.166736
$809$	−1.85432e7	−0.996124	−0.498062	−	0.867141i	$-0.665955\pi$
$809$	−1.85432e7	−0.996124	−0.498062	+	0.867141i	$0.665955\pi$
$810$	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
$813$	0	0
$814$	1.59804e7	0.845331
$815$	−3.81946e7	−2.01422
$816$	0	0
$817$	2.54974e6	0.133642
$818$	1.88673e6	0.0985884
$819$	0	0
$820$	−9.77357e6	−0.507596
$821$	5.45014e6	0.282195	0.141098	−	0.989996i	$-0.454937\pi$
$821$	5.45014e6	0.282195	0.141098	+	0.989996i	$0.454937\pi$
$822$	0	0
$823$	−2.19153e7	−1.12784	−0.563921	−	0.825829i	$-0.690708\pi$
$823$	−2.19153e7	−1.12784	−0.563921	+	0.825829i	$0.690708\pi$
$824$	1.17499e7	0.602859
$825$	0	0
$826$	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	$-0.633067\pi$
$829$	−1.60662e7	−0.811943	−0.405972	+	0.913886i	$0.633067\pi$
$830$	−7.40621e6	−0.373165
$831$	0	0
$832$	4.31718e6	0.216218
$833$	0	0
$834$	0	0
$835$	2.24891e7	1.11623
$836$	−1.56326e6	−0.0773600
$837$	0	0
$838$	−1.41637e7	−0.696736
$839$	−2.63504e7	−1.29236	−0.646178	−	0.763186i	$-0.723634\pi$
$839$	−2.63504e7	−1.29236	−0.646178	+	0.763186i	$0.723634\pi$
$840$	0	0
$841$	2.20253e7	1.07382
$842$	1.09105e7	0.530352
$843$	0	0
$844$	−9.78208e6	−0.472689
$845$	−5.32529e7	−2.56567
$846$	0	0
$847$	0	0
$848$	−5.76154e6	−0.275137
$849$	0	0
$850$	−1.52201e7	−0.722555
$851$	−2.79657e7	−1.32374
$852$	0	0
$853$	−2.78129e7	−1.30880	−0.654400	−	0.756148i	$-0.727079\pi$
$853$	−2.78129e7	−1.30880	−0.654400	+	0.756148i	$0.727079\pi$
$854$	0	0
$855$	0	0
$856$	143232.	0.00668122
$857$	−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	$0.356552\pi$
$860$	1.24462e7	0.573840
$861$	0	0
$862$	1.90607e7	0.873716
$863$	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
$866$	−2.76120e7	−1.25113
$867$	0	0
$868$	0	0
$869$	1.79428e7	0.806009
$870$	0	0
$871$	−2.97397e7	−1.32828
$872$	3.85011e6	0.171468
$873$	0	0
$874$	2.73571e6	0.121141
$875$	0	0
$876$	0	0
$877$	−1.58811e7	−0.697240	−0.348620	−	0.937264i	$-0.613350\pi$
$877$	−1.58811e7	−0.697240	−0.348620	+	0.937264i	$0.613350\pi$
$878$	−2.16050e7	−0.945842
$879$	0	0
$880$	−7.63085e6	−0.332174
$881$	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	$-0.339780\pi$
$883$	2.23513e7	0.964721	0.482361	+	0.875973i	$0.339780\pi$
$884$	−3.11647e7	−1.34132
$885$	0	0
$886$	1.62345e7	0.694793
$887$	−8.91140e6	−0.380309	−0.190155	−	0.981754i	$-0.560899\pi$
$887$	−8.91140e6	−0.380309	−0.190155	+	0.981754i	$0.560899\pi$
$888$	0	0
$889$	0	0
$890$	−2.84325e7	−1.20321
$891$	0	0
$892$	1.25321e7	0.527365
$893$	−14160.0	−0.000594203 0
$894$	0	0
$895$	2.78437e7	1.16190
$896$	0	0
$897$	0	0
$898$	−851976.	−0.0352563
$899$	−4.04364e7	−1.66868
$900$	0	0
$901$	4.15911e7	1.70682
$902$	1.40495e7	0.574969
$903$	0	0
$904$	448896.	0.0182694
$905$	−3.00671e7	−1.22031
$906$	0	0
$907$	−4.53327e6	−0.182976	−0.0914878	−	0.995806i	$-0.529162\pi$
$907$	−4.53327e6	−0.182976	−0.0914878	+	0.995806i	$0.529162\pi$
$908$	1.29139e6	0.0519809
$909$	0	0
$910$	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
$914$	−3.66460e6	−0.145098
$915$	0	0
$916$	−2.44381e6	−0.0962340
$917$	0	0
$918$	0	0
$919$	−4.82049e7	−1.88279	−0.941396	−	0.337304i	$-0.890485\pi$
$919$	−4.82049e7	−1.88279	−0.941396	+	0.337304i	$0.890485\pi$
$920$	1.33540e7	0.520165
$921$	0	0
$922$	−1.66334e7	−0.644397
$923$	7.00067e6	0.270480
$924$	0	0
$925$	1.98694e7	0.763536
$926$	3.36320e7	1.28892
$927$	0	0
$928$	6.67853e6	0.254572
$929$	−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
$931$	0	0
$932$	5.66851e6	0.213761
$933$	0	0
$934$	288192.	0.0108097
$935$	5.50852e7	2.06066
$936$	0	0
$937$	−1.25127e7	−0.465590	−0.232795	−	0.972526i	$-0.574787\pi$
$937$	−1.25127e7	−0.465590	−0.232795	+	0.972526i	$0.574787\pi$
$938$	0	0
$939$	0	0
$940$	−69120.0	−0.00255143
$941$	−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
$944$	−7.21306e6	−0.263445
$945$	0	0
$946$	−1.78914e7	−0.650006
$947$	−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
$950$	−1.94370e6	−0.0698746
$951$	0	0
$952$	0	0
$953$	1.40055e7	0.499535	0.249768	−	0.968306i	$-0.419646\pi$
$953$	1.40055e7	0.499535	0.249768	+	0.968306i	$0.419646\pi$
$954$	0	0
$955$	−7.11396e7	−2.52408
$956$	−4.40592e6	−0.155916
$957$	0	0
$958$	1.92224e7	0.676697
$959$	0	0
$960$	0	0
$961$	9.81085e6	0.342687
$962$	4.06844e7	1.41739
$963$	0	0
$964$	9.35517e6	0.324234
$965$	2.94666e7	1.01862
$966$	0	0
$967$	−1.02386e7	−0.352106	−0.176053	−	0.984381i	$-0.556333\pi$
$967$	−1.02386e7	−0.352106	−0.176053	+	0.984381i	$0.556333\pi$
$968$	662080.	0.0227103
$969$	0	0
$970$	−4.28988e7	−1.46391
$971$	1.17452e7	0.399773	0.199886	−	0.979819i	$-0.435943\pi$
$971$	1.17452e7	0.399773	0.199886	+	0.979819i	$0.435943\pi$
$972$	0	0
$973$	0	0
$974$	1.76722e7	0.596888
$975$	0	0
$976$	9.00966e6	0.302750
$977$	−2.26195e7	−0.758134	−0.379067	−	0.925369i	$-0.623755\pi$
$977$	−2.26195e7	−0.758134	−0.379067	+	0.925369i	$0.623755\pi$
$978$	0	0
$979$	4.08717e7	1.36291
$980$	0	0
$981$	0	0
$982$	2.25999e7	0.747873
$983$	2.79575e7	0.922813	0.461407	−	0.887189i	$-0.347345\pi$
$983$	2.79575e7	0.922813	0.461407	+	0.887189i	$0.347345\pi$
$984$	0	0
$985$	−6.64006e7	−2.18063
$986$	−4.82106e7	−1.57925
$987$	0	0
$988$	−3.97990e6	−0.129712
$989$	3.13100e7	1.01787
$990$	0	0
$991$	−1.66475e7	−0.538474	−0.269237	−	0.963074i	$-0.586772\pi$
$991$	−1.66475e7	−0.538474	−0.269237	+	0.963074i	$0.586772\pi$
$992$	−6.34880e6	−0.204839
$993$	0	0
$994$	0	0
$995$	1.36431e7	0.436874
$996$	0	0
$997$	5.17280e7	1.64812	0.824058	−	0.566505i	$-0.191705\pi$
$997$	5.17280e7	1.64812	0.824058	+	0.566505i	$0.191705\pi$
$998$	−3.68938e7	−1.17254
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.6.a.n.1.1		1
3.2	odd	2		294.6.a.c.1.1		1
7.6	odd	2		126.6.a.l.1.1		1
21.2	odd	6		294.6.e.n.67.1		2
21.5	even	6		294.6.e.l.67.1		2
21.11	odd	6		294.6.e.n.79.1		2
21.17	even	6		294.6.e.l.79.1		2
21.20	even	2		42.6.a.c.1.1	✓	1
28.27	even	2		1008.6.a.ba.1.1		1
84.83	odd	2		336.6.a.b.1.1		1
105.62	odd	4		1050.6.g.b.799.1		2
105.83	odd	4		1050.6.g.b.799.2		2
105.104	even	2		1050.6.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.6.a.c.1.1	✓	1	21.20	even	2
126.6.a.l.1.1		1	7.6	odd	2
294.6.a.c.1.1		1	3.2	odd	2
294.6.e.l.67.1		2	21.5	even	6
294.6.e.l.79.1		2	21.17	even	6
294.6.e.n.67.1		2	21.2	odd	6
294.6.e.n.79.1		2	21.11	odd	6
336.6.a.b.1.1		1	84.83	odd	2
882.6.a.n.1.1		1	1.1	even	1	trivial
1008.6.a.ba.1.1		1	28.27	even	2
1050.6.a.g.1.1		1	105.104	even	2
1050.6.g.b.799.1		2	105.62	odd	4
1050.6.g.b.799.2		2	105.83	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	882.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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