LMFDB - Embedded newform 42.6.a.c.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [42,6,Mod(1,42)] 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("42.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(42, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 42 = 2 \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	42.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-4,9,16,-72] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$6.73612043215$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	42.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q-4.00000 q^{2} +9.00000 q^{3} +16.0000 q^{4} -72.0000 q^{5} -36.0000 q^{6} +49.0000 q^{7} -64.0000 q^{8} +81.0000 q^{9} +288.000 q^{10} -414.000 q^{11} +144.000 q^{12} -1054.00 q^{13} -196.000 q^{14} -648.000 q^{15} +256.000 q^{16} -1848.00 q^{17} -324.000 q^{18} +236.000 q^{19} -1152.00 q^{20} +441.000 q^{21} +1656.00 q^{22} +2898.00 q^{23} -576.000 q^{24} +2059.00 q^{25} +4216.00 q^{26} +729.000 q^{27} +784.000 q^{28} -6522.00 q^{29} +2592.00 q^{30} +6200.00 q^{31} -1024.00 q^{32} -3726.00 q^{33} +7392.00 q^{34} -3528.00 q^{35} +1296.00 q^{36} +9650.00 q^{37} -944.000 q^{38} -9486.00 q^{39} +4608.00 q^{40} +8484.00 q^{41} -1764.00 q^{42} -10804.0 q^{43} -6624.00 q^{44} -5832.00 q^{45} -11592.0 q^{46} +60.0000 q^{47} +2304.00 q^{48} +2401.00 q^{49} -8236.00 q^{50} -16632.0 q^{51} -16864.0 q^{52} +22506.0 q^{53} -2916.00 q^{54} +29808.0 q^{55} -3136.00 q^{56} +2124.00 q^{57} +26088.0 q^{58} -28176.0 q^{59} -10368.0 q^{60} -35194.0 q^{61} -24800.0 q^{62} +3969.00 q^{63} +4096.00 q^{64} +75888.0 q^{65} +14904.0 q^{66} -28216.0 q^{67} -29568.0 q^{68} +26082.0 q^{69} +14112.0 q^{70} -6642.00 q^{71} -5184.00 q^{72} -52090.0 q^{73} -38600.0 q^{74} +18531.0 q^{75} +3776.00 q^{76} -20286.0 q^{77} +37944.0 q^{78} +43340.0 q^{79} -18432.0 q^{80} +6561.00 q^{81} -33936.0 q^{82} +25716.0 q^{83} +7056.00 q^{84} +133056. q^{85} +43216.0 q^{86} -58698.0 q^{87} +26496.0 q^{88} +98724.0 q^{89} +23328.0 q^{90} -51646.0 q^{91} +46368.0 q^{92} +55800.0 q^{93} -240.000 q^{94} -16992.0 q^{95} -9216.00 q^{96} -148954. q^{97} -9604.00 q^{98} -33534.0 q^{99} +O(q^{100})\)

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 $

$2$	−4.00000	−0.707107
$2$	−4.00000	−0.707107
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	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	$0.722721\pi$
$6$	−36.0000	−0.408248
$7$	49.0000	0.377964
$7$	49.0000	0.377964
$8$	−64.0000	−0.353553
$9$	81.0000	0.333333
$10$	288.000	0.910736
$11$	−414.000	−1.03162	−0.515809	−	0.856704i	$-0.672508\pi$
$11$	−414.000	−1.03162	−0.515809	+	0.856704i	$0.672508\pi$
$12$	144.000	0.288675
$13$	−1054.00	−1.72975	−0.864873	−	0.501991i	$-0.832601\pi$
$13$	−1054.00	−1.72975	−0.864873	+	0.501991i	$0.832601\pi$
$14$	−196.000	−0.267261
$15$	−648.000	−0.743613
$16$	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	$0.782473\pi$
$18$	−324.000	−0.235702
$19$	236.000	0.149978	0.0749891	−	0.997184i	$-0.476108\pi$
$19$	236.000	0.149978	0.0749891	+	0.997184i	$0.476108\pi$
$20$	−1152.00	−0.643988
$21$	441.000	0.218218
$22$	1656.00	0.729464
$23$	2898.00	1.14230	0.571148	−	0.820847i	$-0.306498\pi$
$23$	2898.00	1.14230	0.571148	+	0.820847i	$0.306498\pi$
$24$	−576.000	−0.204124
$25$	2059.00	0.658880
$26$	4216.00	1.22311
$27$	729.000	0.192450
$28$	784.000	0.188982
$29$	−6522.00	−1.44008	−0.720039	−	0.693934i	$-0.755876\pi$
$29$	−6522.00	−1.44008	−0.720039	+	0.693934i	$0.755876\pi$
$30$	2592.00	0.525814
$31$	6200.00	1.15874	0.579372	−	0.815063i	$-0.303298\pi$
$31$	6200.00	1.15874	0.579372	+	0.815063i	$0.303298\pi$
$32$	−1024.00	−0.176777
$33$	−3726.00	−0.595605
$34$	7392.00	1.09664
$35$	−3528.00	−0.486809
$36$	1296.00	0.166667
$37$	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
$39$	−9486.00	−0.998669
$40$	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	$0.371055\pi$
$42$	−1764.00	−0.154303
$43$	−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
$45$	−5832.00	−0.429325
$46$	−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	$0.499369\pi$
$48$	2304.00	0.144338
$49$	2401.00	0.142857
$50$	−8236.00	−0.465899
$51$	−16632.0	−0.895404
$52$	−16864.0	−0.864873
$53$	22506.0	1.10055	0.550274	−	0.834984i	$-0.314523\pi$
$53$	22506.0	1.10055	0.550274	+	0.834984i	$0.314523\pi$
$54$	−2916.00	−0.136083
$55$	29808.0	1.32870
$56$	−3136.00	−0.133631
$57$	2124.00	0.0865899
$58$	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	$0.676642\pi$
$60$	−10368.0	−0.371806
$61$	−35194.0	−1.21100	−0.605500	−	0.795845i	$-0.707027\pi$
$61$	−35194.0	−1.21100	−0.605500	+	0.795845i	$0.707027\pi$
$62$	−24800.0	−0.819356
$63$	3969.00	0.125988
$64$	4096.00	0.125000
$65$	75888.0	2.22787
$66$	14904.0	0.421156
$67$	−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
$69$	26082.0	0.659505
$70$	14112.0	0.344226
$71$	−6642.00	−0.156370	−0.0781849	−	0.996939i	$-0.524912\pi$
$71$	−6642.00	−0.156370	−0.0781849	+	0.996939i	$0.524912\pi$
$72$	−5184.00	−0.117851
$73$	−52090.0	−1.14406	−0.572028	−	0.820234i	$-0.693843\pi$
$73$	−52090.0	−1.14406	−0.572028	+	0.820234i	$0.693843\pi$
$74$	−38600.0	−0.819423
$75$	18531.0	0.380405
$76$	3776.00	0.0749891
$77$	−20286.0	−0.389915
$78$	37944.0	0.706166
$79$	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
$81$	6561.00	0.111111
$82$	−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	$0.434323\pi$
$84$	7056.00	0.109109
$85$	133056.	1.99750
$86$	43216.0	0.630084
$87$	−58698.0	−0.831429
$88$	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	$0.270315\pi$
$90$	23328.0	0.303579
$91$	−51646.0	−0.653782
$92$	46368.0	0.571148
$93$	55800.0	0.669001
$94$	−240.000	−0.00280151
$95$	−16992.0	−0.193168
$96$	−9216.00	−0.102062
$97$	−148954.	−1.60740	−0.803698	−	0.595038i	$-0.797137\pi$
$97$	−148954.	−1.60740	−0.803698	+	0.595038i	$0.797137\pi$
$98$	−9604.00	−0.101015
$99$	−33534.0	−0.343872
$100$	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	$0.424229\pi$
$102$	66528.0	0.633146
$103$	−183592.	−1.70514	−0.852571	−	0.522611i	$-0.824958\pi$
$103$	−183592.	−1.70514	−0.852571	+	0.522611i	$0.824958\pi$
$104$	67456.0	0.611557
$105$	−31752.0	−0.281059
$106$	−90024.0	−0.778204
$107$	−2238.00	−0.0188973	−0.00944867	−	0.999955i	$-0.503008\pi$
$107$	−2238.00	−0.0188973	−0.00944867	+	0.999955i	$0.503008\pi$
$108$	11664.0	0.0962250
$109$	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
$111$	86850.0	0.669056
$112$	12544.0	0.0944911
$113$	−7014.00	−0.0516737	−0.0258369	−	0.999666i	$-0.508225\pi$
$113$	−7014.00	−0.0516737	−0.0258369	+	0.999666i	$0.508225\pi$
$114$	−8496.00	−0.0612283
$115$	−208656.	−1.47125
$116$	−104352.	−0.720039
$117$	−85374.0	−0.576582
$118$	112704.	0.745134
$119$	−90552.0	−0.586180
$120$	41472.0	0.262907
$121$	10345.0	0.0642343
$122$	140776.	0.856306
$123$	76356.0	0.455072
$124$	99200.0	0.579372
$125$	76752.0	0.439354
$126$	−15876.0	−0.0890871
$127$	−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
$129$	−97236.0	−0.514461
$130$	−303552.	−1.57534
$131$	−265140.	−1.34989	−0.674943	−	0.737870i	$-0.735832\pi$
$131$	−265140.	−1.34989	−0.674943	+	0.737870i	$0.735832\pi$
$132$	−59616.0	−0.297802
$133$	11564.0	0.0566864
$134$	112864.	0.542992
$135$	−52488.0	−0.247871
$136$	118272.	0.548321
$137$	−206730.	−0.941027	−0.470514	−	0.882393i	$-0.655931\pi$
$137$	−206730.	−0.941027	−0.470514	+	0.882393i	$0.655931\pi$
$138$	−104328.	−0.466341
$139$	−236836.	−1.03971	−0.519853	−	0.854256i	$-0.674013\pi$
$139$	−236836.	−1.03971	−0.519853	+	0.854256i	$0.674013\pi$
$140$	−56448.0	−0.243404
$141$	540.000	0.00228742
$142$	26568.0	0.110570
$143$	436356.	1.78444
$144$	20736.0	0.0833333
$145$	469584.	1.85478
$146$	208360.	0.808970
$147$	21609.0	0.0824786
$148$	154400.	0.579419
$149$	473706.	1.74801	0.874004	−	0.485919i	$-0.161515\pi$
$149$	473706.	1.74801	0.874004	+	0.485919i	$0.161515\pi$
$150$	−74124.0	−0.268987
$151$	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
$153$	−149688.	−0.516962
$154$	81144.0	0.275711
$155$	−446400.	−1.49243
$156$	−151776.	−0.499335
$157$	−145090.	−0.469773	−0.234887	−	0.972023i	$-0.575472\pi$
$157$	−145090.	−0.469773	−0.234887	+	0.972023i	$0.575472\pi$
$158$	−173360.	−0.552467
$159$	202554.	0.635401
$160$	73728.0	0.227684
$161$	142002.	0.431747
$162$	−26244.0	−0.0785674
$163$	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
$165$	268272.	0.767124
$166$	−102864.	−0.289730
$167$	−312348.	−0.866658	−0.433329	−	0.901236i	$-0.642661\pi$
$167$	−312348.	−0.866658	−0.433329	+	0.901236i	$0.642661\pi$
$168$	−28224.0	−0.0771517
$169$	739623.	1.99202
$170$	−532224.	−1.41245
$171$	19116.0	0.0499927
$172$	−172864.	−0.445537
$173$	−75108.0	−0.190797	−0.0953984	−	0.995439i	$-0.530412\pi$
$173$	−75108.0	−0.190797	−0.0953984	+	0.995439i	$0.530412\pi$
$174$	234792.	0.587909
$175$	100891.	0.249033
$176$	−105984.	−0.257904
$177$	−253584.	−0.608399
$178$	−394896.	−0.934185
$179$	386718.	0.902115	0.451057	−	0.892495i	$-0.351047\pi$
$179$	386718.	0.902115	0.451057	+	0.892495i	$0.351047\pi$
$180$	−93312.0	−0.214663
$181$	−417598.	−0.947462	−0.473731	−	0.880669i	$-0.657093\pi$
$181$	−417598.	−0.947462	−0.473731	+	0.880669i	$0.657093\pi$
$182$	206584.	0.462294
$183$	−316746.	−0.699171
$184$	−185472.	−0.403863
$185$	−694800.	−1.49256
$186$	−223200.	−0.473055
$187$	765072.	1.59992
$188$	960.000	0.00198096
$189$	35721.0	0.0727393
$190$	67968.0	0.136590
$191$	−988050.	−1.95973	−0.979863	−	0.199669i	$-0.936013\pi$
$191$	−988050.	−1.95973	−0.979863	+	0.199669i	$0.936013\pi$
$192$	36864.0	0.0721688
$193$	−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
$195$	682992.	1.28626
$196$	38416.0	0.0714286
$197$	−922230.	−1.69307	−0.846533	−	0.532337i	$-0.821314\pi$
$197$	−922230.	−1.69307	−0.846533	+	0.532337i	$0.821314\pi$
$198$	134136.	0.243155
$199$	189488.	0.339195	0.169597	−	0.985513i	$-0.445753\pi$
$199$	189488.	0.339195	0.169597	+	0.985513i	$0.445753\pi$
$200$	−131776.	−0.232949
$201$	−253944.	−0.443351
$202$	−193392.	−0.333473
$203$	−319578.	−0.544298
$204$	−266112.	−0.447702
$205$	−610848.	−1.01519
$206$	734368.	1.20572
$207$	234738.	0.380765
$208$	−269824.	−0.432436
$209$	−97704.0	−0.154720
$210$	127008.	0.198739
$211$	−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
$213$	−59778.0	−0.0902802
$214$	8952.00	0.0133624
$215$	777888.	1.14768
$216$	−46656.0	−0.0680414
$217$	303800.	0.437964
$218$	−240632.	−0.342935
$219$	−468810.	−0.660521
$220$	476928.	0.664349
$221$	1.94779e6	2.68264
$222$	−347400.	−0.473094
$223$	−783256.	−1.05473	−0.527365	−	0.849639i	$-0.676820\pi$
$223$	−783256.	−1.05473	−0.527365	+	0.849639i	$0.676820\pi$
$224$	−50176.0	−0.0668153
$225$	166779.	0.219627
$226$	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	$0.483447\pi$
$228$	33984.0	0.0432950
$229$	152738.	0.192468	0.0962340	−	0.995359i	$-0.469320\pi$
$229$	152738.	0.192468	0.0962340	+	0.995359i	$0.469320\pi$
$230$	834624.	1.04033
$231$	−182574.	−0.225117
$232$	417408.	0.509144
$233$	−354282.	−0.427523	−0.213761	−	0.976886i	$-0.568572\pi$
$233$	−354282.	−0.427523	−0.213761	+	0.976886i	$0.568572\pi$
$234$	341496.	0.407705
$235$	−4320.00	−0.00510287
$236$	−450816.	−0.526889
$237$	390060.	0.451087
$238$	362208.	0.414492
$239$	275370.	0.311833	0.155916	−	0.987770i	$-0.450167\pi$
$239$	275370.	0.311833	0.155916	+	0.987770i	$0.450167\pi$
$240$	−165888.	−0.185903
$241$	−584698.	−0.648469	−0.324234	−	0.945977i	$-0.605107\pi$
$241$	−584698.	−0.648469	−0.324234	+	0.945977i	$0.605107\pi$
$242$	−41380.0	−0.0454205
$243$	59049.0	0.0641500
$244$	−563104.	−0.605500
$245$	−172872.	−0.183996
$246$	−305424.	−0.321785
$247$	−248744.	−0.259424
$248$	−396800.	−0.409678
$249$	231444.	0.236563
$250$	−307008.	−0.310670
$251$	−184752.	−0.185099	−0.0925497	−	0.995708i	$-0.529502\pi$
$251$	−184752.	−0.185099	−0.0925497	+	0.995708i	$0.529502\pi$
$252$	63504.0	0.0629941
$253$	−1.19977e6	−1.17841
$254$	7120.00	0.00692462
$255$	1.19750e6	1.15326
$256$	65536.0	0.0625000
$257$	235980.	0.222865	0.111433	−	0.993772i	$-0.464456\pi$
$257$	235980.	0.222865	0.111433	+	0.993772i	$0.464456\pi$
$258$	388944.	0.363779
$259$	472850.	0.438000
$260$	1.21421e6	1.11393
$261$	−528282.	−0.480026
$262$	1.06056e6	0.954513
$263$	−244494.	−0.217961	−0.108981	−	0.994044i	$-0.534759\pi$
$263$	−244494.	−0.217961	−0.108981	+	0.994044i	$0.534759\pi$
$264$	238464.	0.210578
$265$	−1.62043e6	−1.41748
$266$	−46256.0	−0.0400833
$267$	888516.	0.762759
$268$	−451456.	−0.383953
$269$	1.52779e6	1.28731	0.643656	−	0.765315i	$-0.277417\pi$
$269$	1.52779e6	1.28731	0.643656	+	0.765315i	$0.277417\pi$
$270$	209952.	0.175271
$271$	2.07056e6	1.71263	0.856317	−	0.516450i	$-0.172747\pi$
$271$	2.07056e6	1.71263	0.856317	+	0.516450i	$0.172747\pi$
$272$	−473088.	−0.387721
$273$	−464814.	−0.377461
$274$	826920.	0.665407
$275$	−852426.	−0.679712
$276$	417312.	0.329753
$277$	−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
$279$	502200.	0.386248
$280$	225792.	0.172113
$281$	341886.	0.258295	0.129147	−	0.991625i	$-0.458776\pi$
$281$	341886.	0.258295	0.129147	+	0.991625i	$0.458776\pi$
$282$	−2160.00	−0.00161745
$283$	578564.	0.429423	0.214712	−	0.976678i	$-0.431119\pi$
$283$	578564.	0.429423	0.214712	+	0.976678i	$0.431119\pi$
$284$	−106272.	−0.0781849
$285$	−152928.	−0.111526
$286$	−1.74542e6	−1.26179
$287$	415716.	0.297915
$288$	−82944.0	−0.0589256
$289$	1.99525e6	1.40525
$290$	−1.87834e6	−1.31153
$291$	−1.34059e6	−0.928030
$292$	−833440.	−0.572028
$293$	780540.	0.531161	0.265580	−	0.964089i	$-0.414436\pi$
$293$	780540.	0.531161	0.265580	+	0.964089i	$0.414436\pi$
$294$	−86436.0	−0.0583212
$295$	2.02867e6	1.35724
$296$	−617600.	−0.409711
$297$	−301806.	−0.198535
$298$	−1.89482e6	−1.23603
$299$	−3.05449e6	−1.97588
$300$	296496.	0.190202
$301$	−529396.	−0.336794
$302$	−1.57981e6	−0.996752
$303$	435132.	0.272279
$304$	60416.0	0.0374945
$305$	2.53397e6	1.55974
$306$	598752.	0.365547
$307$	−2.24825e6	−1.36144	−0.680721	−	0.732543i	$-0.738333\pi$
$307$	−2.24825e6	−1.36144	−0.680721	+	0.732543i	$0.738333\pi$
$308$	−324576.	−0.194957
$309$	−1.65233e6	−0.984465
$310$	1.78560e6	1.05531
$311$	581412.	0.340865	0.170433	−	0.985369i	$-0.445483\pi$
$311$	581412.	0.340865	0.170433	+	0.985369i	$0.445483\pi$
$312$	607104.	0.353083
$313$	−2.00407e6	−1.15625	−0.578125	−	0.815948i	$-0.696216\pi$
$313$	−2.00407e6	−1.15625	−0.578125	+	0.815948i	$0.696216\pi$
$314$	580360.	0.332180
$315$	−285768.	−0.162270
$316$	693440.	0.390653
$317$	−1.27832e6	−0.714481	−0.357241	−	0.934012i	$-0.616282\pi$
$317$	−1.27832e6	−0.714481	−0.357241	+	0.934012i	$0.616282\pi$
$318$	−810216.	−0.449296
$319$	2.70011e6	1.48561
$320$	−294912.	−0.160997
$321$	−20142.0	−0.0109104
$322$	−568008.	−0.305292
$323$	−436128.	−0.232599
$324$	104976.	0.0555556
$325$	−2.17019e6	−1.13969
$326$	−2.12192e6	−1.10582
$327$	541422.	0.280005
$328$	−542976.	−0.278674
$329$	2940.00	0.00149747
$330$	−1.07309e6	−0.542438
$331$	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
$333$	781650.	0.386280
$334$	1.24939e6	0.612819
$335$	2.03155e6	0.989045
$336$	112896.	0.0545545
$337$	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
$339$	−63126.0	−0.0298338
$340$	2.12890e6	0.998751
$341$	−2.56680e6	−1.19538
$342$	−76464.0	−0.0353502
$343$	117649.	0.0539949
$344$	691456.	0.315042
$345$	−1.87790e6	−0.849426
$346$	300432.	0.134914
$347$	−1.42550e6	−0.635540	−0.317770	−	0.948168i	$-0.602934\pi$
$347$	−1.42550e6	−0.635540	−0.317770	+	0.948168i	$0.602934\pi$
$348$	−939168.	−0.415715
$349$	2.93322e6	1.28908	0.644542	−	0.764569i	$-0.277048\pi$
$349$	2.93322e6	1.28908	0.644542	+	0.764569i	$0.277048\pi$
$350$	−403564.	−0.176093
$351$	−768366.	−0.332890
$352$	423936.	0.182366
$353$	−2.01276e6	−0.859716	−0.429858	−	0.902896i	$-0.641436\pi$
$353$	−2.01276e6	−0.859716	−0.429858	+	0.902896i	$0.641436\pi$
$354$	1.01434e6	0.430203
$355$	478224.	0.201400
$356$	1.57958e6	0.660568
$357$	−814968.	−0.338431
$358$	−1.54687e6	−0.637891
$359$	4.07710e6	1.66961	0.834806	−	0.550545i	$-0.185580\pi$
$359$	4.07710e6	1.66961	0.834806	+	0.550545i	$0.185580\pi$
$360$	373248.	0.151789
$361$	−2.42040e6	−0.977507
$362$	1.67039e6	0.669957
$363$	93105.0	0.0370857
$364$	−826336.	−0.326891
$365$	3.75048e6	1.47352
$366$	1.26698e6	0.494389
$367$	594752.	0.230500	0.115250	−	0.993337i	$-0.463233\pi$
$367$	594752.	0.230500	0.115250	+	0.993337i	$0.463233\pi$
$368$	741888.	0.285574
$369$	687204.	0.262736
$370$	2.77920e6	1.05540
$371$	1.10279e6	0.415968
$372$	892800.	0.334501
$373$	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
$375$	690768.	0.253661
$376$	−3840.00	−0.00140075
$377$	6.87419e6	2.49097
$378$	−142884.	−0.0514344
$379$	−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
$381$	−16020.0	−0.00565393
$382$	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	$0.597405\pi$
$384$	−147456.	−0.0510310
$385$	1.46059e6	0.502200
$386$	1.63703e6	0.559228
$387$	−875124.	−0.297024
$388$	−2.38326e6	−0.803698
$389$	−1.74919e6	−0.586087	−0.293043	−	0.956099i	$-0.594668\pi$
$389$	−1.74919e6	−0.586087	−0.293043	+	0.956099i	$0.594668\pi$
$390$	−2.73197e6	−0.909524
$391$	−5.35550e6	−1.77157
$392$	−153664.	−0.0505076
$393$	−2.38626e6	−0.779357
$394$	3.68892e6	1.19718
$395$	−3.12048e6	−1.00630
$396$	−536544.	−0.171936
$397$	1.88205e6	0.599313	0.299657	−	0.954047i	$-0.403128\pi$
$397$	1.88205e6	0.599313	0.299657	+	0.954047i	$0.403128\pi$
$398$	−757952.	−0.239847
$399$	104076.	0.0327279
$400$	527104.	0.164720
$401$	4.18124e6	1.29851	0.649253	−	0.760573i	$-0.275082\pi$
$401$	4.18124e6	1.29851	0.649253	+	0.760573i	$0.275082\pi$
$402$	1.01578e6	0.313497
$403$	−6.53480e6	−2.00433
$404$	773568.	0.235801
$405$	−472392.	−0.143108
$406$	1.27831e6	0.384877
$407$	−3.99510e6	−1.19548
$408$	1.06445e6	0.316573
$409$	−471682.	−0.139425	−0.0697126	−	0.997567i	$-0.522208\pi$
$409$	−471682.	−0.139425	−0.0697126	+	0.997567i	$0.522208\pi$
$410$	2.44339e6	0.717850
$411$	−1.86057e6	−0.543302
$412$	−2.93747e6	−0.852571
$413$	−1.38062e6	−0.398291
$414$	−938952.	−0.269242
$415$	−1.85155e6	−0.527735
$416$	1.07930e6	0.305779
$417$	−2.13152e6	−0.600275
$418$	390816.	0.109404
$419$	−3.54094e6	−0.985333	−0.492666	−	0.870218i	$-0.663978\pi$
$419$	−3.54094e6	−0.985333	−0.492666	+	0.870218i	$0.663978\pi$
$420$	−508032.	−0.140530
$421$	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
$423$	4860.00	0.00132064
$424$	−1.44038e6	−0.389102
$425$	−3.80503e6	−1.02185
$426$	239112.	0.0638377
$427$	−1.72451e6	−0.457715
$428$	−35808.0	−0.00944867
$429$	3.92720e6	1.03024
$430$	−3.11155e6	−0.811533
$431$	−4.76517e6	−1.23562	−0.617810	−	0.786327i	$-0.711980\pi$
$431$	−4.76517e6	−1.23562	−0.617810	+	0.786327i	$0.711980\pi$
$432$	186624.	0.0481125
$433$	6.90300e6	1.76937	0.884684	−	0.466191i	$-0.154374\pi$
$433$	6.90300e6	1.76937	0.884684	+	0.466191i	$0.154374\pi$
$434$	−1.21520e6	−0.309687
$435$	4.22626e6	1.07086
$436$	962528.	0.242492
$437$	683928.	0.171319
$438$	1.87524e6	0.467059
$439$	5.40126e6	1.33762	0.668811	−	0.743432i	$-0.266803\pi$
$439$	5.40126e6	1.33762	0.668811	+	0.743432i	$0.266803\pi$
$440$	−1.90771e6	−0.469766
$441$	194481.	0.0476190
$442$	−7.79117e6	−1.89691
$443$	−4.05863e6	−0.982586	−0.491293	−	0.870994i	$-0.663476\pi$
$443$	−4.05863e6	−0.982586	−0.491293	+	0.870994i	$0.663476\pi$
$444$	1.38960e6	0.334528
$445$	−7.10813e6	−1.70159
$446$	3.13302e6	0.745807
$447$	4.26335e6	1.00921
$448$	200704.	0.0472456
$449$	212994.	0.0498599	0.0249300	−	0.999689i	$-0.492064\pi$
$449$	212994.	0.0498599	0.0249300	+	0.999689i	$0.492064\pi$
$450$	−667116.	−0.155300
$451$	−3.51238e6	−0.813129
$452$	−112224.	−0.0258369
$453$	3.55457e6	0.813844
$454$	−322848.	−0.0735120
$455$	3.71851e6	0.842055
$456$	−135936.	−0.0306142
$457$	−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
$459$	−1.34719e6	−0.298468
$460$	−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	$0.650596\pi$
$462$	730296.	0.159182
$463$	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
$465$	−4.01760e6	−0.861657
$466$	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	$0.497567\pi$
$468$	−1.36598e6	−0.288291
$469$	−1.38258e6	−0.290241
$470$	17280.0	0.00360827
$471$	−1.30581e6	−0.271224
$472$	1.80326e6	0.372567
$473$	4.47286e6	0.919247
$474$	−1.56024e6	−0.318967
$475$	485924.	0.0988176
$476$	−1.44883e6	−0.293090
$477$	1.82299e6	0.366849
$478$	−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	$0.341182\pi$
$480$	663552.	0.131453
$481$	−1.01711e7	−2.00450
$482$	2.33879e6	0.458537
$483$	1.27802e6	0.249270
$484$	165520.	0.0321172
$485$	1.07247e7	2.07029
$486$	−236196.	−0.0453609
$487$	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
$489$	4.77432e6	0.902899
$490$	691488.	0.130105
$491$	−5.64998e6	−1.05765	−0.528826	−	0.848730i	$-0.677368\pi$
$491$	−5.64998e6	−1.05765	−0.528826	+	0.848730i	$0.677368\pi$
$492$	1.22170e6	0.227536
$493$	1.20527e7	2.23339
$494$	994976.	0.183441
$495$	2.41445e6	0.442899
$496$	1.58720e6	0.289686
$497$	−325458.	−0.0591022
$498$	−925776.	−0.167276
$499$	−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
$501$	−2.81113e6	−0.500365
$502$	739008.	0.130885
$503$	1.45562e6	0.256525	0.128262	−	0.991740i	$-0.459060\pi$
$503$	1.45562e6	0.256525	0.128262	+	0.991740i	$0.459060\pi$
$504$	−254016.	−0.0445435
$505$	−3.48106e6	−0.607411
$506$	4.79909e6	0.833264
$507$	6.65661e6	1.15009
$508$	−28480.0	−0.00489644
$509$	−367344.	−0.0628461	−0.0314231	−	0.999506i	$-0.510004\pi$
$509$	−367344.	−0.0628461	−0.0314231	+	0.999506i	$0.510004\pi$
$510$	−4.79002e6	−0.815477
$511$	−2.55241e6	−0.432412
$512$	−262144.	−0.0441942
$513$	172044.	0.0288633
$514$	−943920.	−0.157590
$515$	1.32186e7	2.19618
$516$	−1.55578e6	−0.257231
$517$	−24840.0	−0.00408719
$518$	−1.89140e6	−0.309713
$519$	−675972.	−0.110157
$520$	−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	$0.346009\pi$
$522$	2.11313e6	0.339429
$523$	235100.	0.0375836	0.0187918	−	0.999823i	$-0.494018\pi$
$523$	235100.	0.0375836	0.0187918	+	0.999823i	$0.494018\pi$
$524$	−4.24224e6	−0.674943
$525$	908019.	0.143779
$526$	977976.	0.154122
$527$	−1.14576e7	−1.79708
$528$	−953856.	−0.148901
$529$	1.96206e6	0.304841
$530$	6.48173e6	1.00231
$531$	−2.28226e6	−0.351259
$532$	185024.	0.0283432
$533$	−8.94214e6	−1.36340
$534$	−3.55406e6	−0.539352
$535$	161136.	0.0243393
$536$	1.80582e6	0.271496
$537$	3.48046e6	0.520836
$538$	−6.11117e6	−0.910266
$539$	−994014.	−0.147374
$540$	−839808.	−0.123935
$541$	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
$543$	−3.75838e6	−0.547018
$544$	1.89235e6	0.274160
$545$	−4.33138e6	−0.624647
$546$	1.85926e6	0.266906
$547$	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
$549$	−2.85071e6	−0.403667
$550$	3.40970e6	0.480629
$551$	−1.53919e6	−0.215980
$552$	−1.66925e6	−0.233170
$553$	2.12366e6	0.295306
$554$	9.62908e6	1.33294
$555$	−6.25320e6	−0.861727
$556$	−3.78938e6	−0.519853
$557$	−6.62986e6	−0.905454	−0.452727	−	0.891649i	$-0.649549\pi$
$557$	−6.62986e6	−0.905454	−0.452727	+	0.891649i	$0.649549\pi$
$558$	−2.00880e6	−0.273119
$559$	1.13874e7	1.54133
$560$	−903168.	−0.121702
$561$	6.88565e6	0.923714
$562$	−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	$0.325159\pi$
$564$	8640.00	0.00114371
$565$	505008.	0.0665545
$566$	−2.31426e6	−0.303648
$567$	321489.	0.0419961
$568$	425088.	0.0552851
$569$	−2.48155e6	−0.321323	−0.160661	−	0.987010i	$-0.551363\pi$
$569$	−2.48155e6	−0.321323	−0.160661	+	0.987010i	$0.551363\pi$
$570$	611712.	0.0788606
$571$	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
$573$	−8.89245e6	−1.13145
$574$	−1.66286e6	−0.210658
$575$	5.96698e6	0.752636
$576$	331776.	0.0416667
$577$	−8.20505e6	−1.02599	−0.512993	−	0.858393i	$-0.671463\pi$
$577$	−8.20505e6	−1.02599	−0.512993	+	0.858393i	$0.671463\pi$
$578$	−7.98099e6	−0.993658
$579$	−3.68332e6	−0.456608
$580$	7.51334e6	0.927392
$581$	1.26008e6	0.154867
$582$	5.36234e6	0.656217
$583$	−9.31748e6	−1.13534
$584$	3.33376e6	0.404485
$585$	6.14693e6	0.742623
$586$	−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	$0.811043\pi$
$588$	345744.	0.0412393
$589$	1.46320e6	0.173786
$590$	−8.11469e6	−0.959714
$591$	−8.30007e6	−0.977492
$592$	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	$0.601901\pi$
$594$	1.20722e6	0.140385
$595$	6.51974e6	0.754985
$596$	7.57930e6	0.874004
$597$	1.70539e6	0.195834
$598$	1.22180e7	1.39716
$599$	−1.14885e7	−1.30827	−0.654134	−	0.756379i	$-0.726967\pi$
$599$	−1.14885e7	−1.30827	−0.654134	+	0.756379i	$0.726967\pi$
$600$	−1.18598e6	−0.134493
$601$	−2.79225e6	−0.315333	−0.157666	−	0.987492i	$-0.550397\pi$
$601$	−2.79225e6	−0.315333	−0.157666	+	0.987492i	$0.550397\pi$
$602$	2.11758e6	0.238149
$603$	−2.28550e6	−0.255969
$604$	6.31923e6	0.704810
$605$	−744840.	−0.0827322
$606$	−1.74053e6	−0.192530
$607$	713888.	0.0786427	0.0393213	−	0.999227i	$-0.487480\pi$
$607$	713888.	0.0786427	0.0393213	+	0.999227i	$0.487480\pi$
$608$	−241664.	−0.0265126
$609$	−2.87620e6	−0.314251
$610$	−1.01359e7	−1.10290
$611$	−63240.0	−0.00685313
$612$	−2.39501e6	−0.258481
$613$	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
$615$	−5.49763e6	−0.586122
$616$	1.29830e6	0.137856
$617$	3.62384e6	0.383227	0.191613	−	0.981470i	$-0.438628\pi$
$617$	3.62384e6	0.383227	0.191613	+	0.981470i	$0.438628\pi$
$618$	6.60931e6	0.696122
$619$	4.25196e6	0.446028	0.223014	−	0.974815i	$-0.428410\pi$
$619$	4.25196e6	0.446028	0.223014	+	0.974815i	$0.428410\pi$
$620$	−7.14240e6	−0.746217
$621$	2.11264e6	0.219835
$622$	−2.32565e6	−0.241028
$623$	4.83748e6	0.499343
$624$	−2.42842e6	−0.249667
$625$	−1.19605e7	−1.22476
$626$	8.01628e6	0.817593
$627$	−879336.	−0.0893277
$628$	−2.32144e6	−0.234887
$629$	−1.78332e7	−1.79723
$630$	1.14307e6	0.114742
$631$	−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
$633$	−5.50242e6	−0.545814
$634$	5.11327e6	0.505214
$635$	128160.	0.0126130
$636$	3.24086e6	0.317701
$637$	−2.53065e6	−0.247107
$638$	−1.08004e7	−1.05048
$639$	−538002.	−0.0521233
$640$	1.17965e6	0.113842
$641$	−1.80938e6	−0.173934	−0.0869669	−	0.996211i	$-0.527717\pi$
$641$	−1.80938e6	−0.173934	−0.0869669	+	0.996211i	$0.527717\pi$
$642$	80568.0	0.00771481
$643$	1.53012e7	1.45948	0.729740	−	0.683725i	$-0.239641\pi$
$643$	1.53012e7	1.45948	0.729740	+	0.683725i	$0.239641\pi$
$644$	2.27203e6	0.215874
$645$	7.00099e6	0.662614
$646$	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	$0.211756\pi$
$648$	−419904.	−0.0392837
$649$	1.16649e7	1.08710
$650$	8.68074e6	0.805886
$651$	2.73420e6	0.252859
$652$	8.48768e6	0.781934
$653$	1.47859e7	1.35695	0.678477	−	0.734622i	$-0.262640\pi$
$653$	1.47859e7	1.35695	0.678477	+	0.734622i	$0.262640\pi$
$654$	−2.16569e6	−0.197994
$655$	1.90901e7	1.73862
$656$	2.17190e6	0.197052
$657$	−4.21929e6	−0.381352
$658$	−11760.0	−0.00105887
$659$	933762.	0.0837573	0.0418786	−	0.999123i	$-0.486666\pi$
$659$	933762.	0.0837573	0.0418786	+	0.999123i	$0.486666\pi$
$660$	4.29235e6	0.383562
$661$	6.09724e6	0.542787	0.271394	−	0.962468i	$-0.412516\pi$
$661$	6.09724e6	0.542787	0.271394	+	0.962468i	$0.412516\pi$
$662$	−1.03925e7	−0.921666
$663$	1.75301e7	1.54882
$664$	−1.64582e6	−0.144865
$665$	−832608.	−0.0730107
$666$	−3.12660e6	−0.273141
$667$	−1.89008e7	−1.64500
$668$	−4.99757e6	−0.433329
$669$	−7.04930e6	−0.608949
$670$	−8.12621e6	−0.699360
$671$	1.45703e7	1.24929
$672$	−451584.	−0.0385758
$673$	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
$675$	1.50101e6	0.126802
$676$	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	$0.481779\pi$
$678$	252504.	0.0210957
$679$	−7.29875e6	−0.607539
$680$	−8.51558e6	−0.706223
$681$	726408.	0.0600223
$682$	1.02672e7	0.845261
$683$	1.15483e7	0.947254	0.473627	−	0.880726i	$-0.342945\pi$
$683$	1.15483e7	0.947254	0.473627	+	0.880726i	$0.342945\pi$
$684$	305856.	0.0249964
$685$	1.48846e7	1.21202
$686$	−470596.	−0.0381802
$687$	1.37464e6	0.111121
$688$	−2.76582e6	−0.222768
$689$	−2.37213e7	−1.90367
$690$	7.51162e6	0.600635
$691$	4.17744e6	0.332824	0.166412	−	0.986056i	$-0.446782\pi$
$691$	4.17744e6	0.332824	0.166412	+	0.986056i	$0.446782\pi$
$692$	−1.20173e6	−0.0953984
$693$	−1.64317e6	−0.129972
$694$	5.70199e6	0.449395
$695$	1.70522e7	1.33912
$696$	3.75667e6	0.293955
$697$	−1.56784e7	−1.22242
$698$	−1.17329e7	−0.911520
$699$	−3.18854e6	−0.246830
$700$	1.61426e6	0.124517
$701$	−1.70278e7	−1.30877	−0.654385	−	0.756161i	$-0.727073\pi$
$701$	−1.70278e7	−1.30877	−0.654385	+	0.756161i	$0.727073\pi$
$702$	3.07346e6	0.235389
$703$	2.27740e6	0.173800
$704$	−1.69574e6	−0.128952
$705$	−38880.0	−0.00294614
$706$	8.05104e6	0.607911
$707$	2.36905e6	0.178249
$708$	−4.05734e6	−0.304200
$709$	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
$711$	3.51054e6	0.260435
$712$	−6.31834e6	−0.467092
$713$	1.79676e7	1.32363
$714$	3.25987e6	0.239307
$715$	−3.14176e7	−2.29831
$716$	6.18749e6	0.451057
$717$	2.47833e6	0.180037
$718$	−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	$0.440812\pi$
$720$	−1.49299e6	−0.107331
$721$	−8.99601e6	−0.644483
$722$	9.68161e6	0.691202
$723$	−5.26228e6	−0.374394
$724$	−6.68157e6	−0.473731
$725$	−1.34288e7	−0.948838
$726$	−372420.	−0.0262235
$727$	−1.54328e7	−1.08295	−0.541476	−	0.840716i	$-0.682134\pi$
$727$	−1.54328e7	−1.08295	−0.541476	+	0.840716i	$0.682134\pi$
$728$	3.30534e6	0.231147
$729$	531441.	0.0370370
$730$	−1.50019e7	−1.04193
$731$	1.99658e7	1.38195
$732$	−5.06794e6	−0.349586
$733$	−6.84465e6	−0.470535	−0.235267	−	0.971931i	$-0.575597\pi$
$733$	−6.84465e6	−0.470535	−0.235267	+	0.971931i	$0.575597\pi$
$734$	−2.37901e6	−0.162988
$735$	−1.55585e6	−0.106230
$736$	−2.96755e6	−0.201931
$737$	1.16814e7	0.792186
$738$	−2.74882e6	−0.185782
$739$	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
$741$	−2.23870e6	−0.149779
$742$	−4.41118e6	−0.294134
$743$	2.23250e7	1.48361	0.741804	−	0.670617i	$-0.233971\pi$
$743$	2.23250e7	1.48361	0.741804	+	0.670617i	$0.233971\pi$
$744$	−3.57120e6	−0.236528
$745$	−3.41068e7	−2.25139
$746$	−8.18089e6	−0.538212
$747$	2.08300e6	0.136580
$748$	1.22412e7	0.799960
$749$	−109662.	−0.00714252
$750$	−2.76307e6	−0.179366
$751$	−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
$753$	−1.66277e6	−0.106867
$754$	−2.74968e7	−1.76138
$755$	−2.84365e7	−1.81556
$756$	571536.	0.0363696
$757$	−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
$759$	−1.07979e7	−0.680357
$760$	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	$0.522508\pi$
$762$	64080.0	0.00399793
$763$	2.94774e6	0.183307
$764$	−1.58088e7	−0.979863
$765$	1.07775e7	0.665834
$766$	6.91862e6	0.426037
$767$	2.96975e7	1.82277
$768$	589824.	0.0360844
$769$	−748774.	−0.0456599	−0.0228299	−	0.999739i	$-0.507268\pi$
$769$	−748774.	−0.0456599	−0.0228299	+	0.999739i	$0.507268\pi$
$770$	−5.84237e6	−0.355109
$771$	2.12382e6	0.128671
$772$	−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	$0.591922\pi$
$774$	3.50050e6	0.210028
$775$	1.27658e7	0.763473
$776$	9.53306e6	0.568300
$777$	4.25565e6	0.252879
$778$	6.99674e6	0.414426
$779$	2.00222e6	0.118214
$780$	1.09279e7	0.643130
$781$	2.74979e6	0.161314
$782$	2.14220e7	1.25269
$783$	−4.75454e6	−0.277143
$784$	614656.	0.0357143
$785$	1.04465e7	0.605056
$786$	9.54504e6	0.551089
$787$	−1.99634e7	−1.14894	−0.574470	−	0.818525i	$-0.694792\pi$
$787$	−1.99634e7	−1.14894	−0.574470	+	0.818525i	$0.694792\pi$
$788$	−1.47557e7	−0.846533
$789$	−2.20045e6	−0.125840
$790$	1.24819e7	0.711564
$791$	−343686.	−0.0195308
$792$	2.14618e6	0.121577
$793$	3.70945e7	2.09472
$794$	−7.52818e6	−0.423779
$795$	−1.45839e7	−0.818381
$796$	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	$0.536054\pi$
$798$	−416304.	−0.0231421
$799$	−110880.	−0.00614450
$800$	−2.10842e6	−0.116475
$801$	7.99664e6	0.440379
$802$	−1.67250e7	−0.918182
$803$	2.15653e7	1.18023
$804$	−4.06310e6	−0.221676
$805$	−1.02241e7	−0.556080
$806$	2.61392e7	1.41728
$807$	1.37501e7	0.743229
$808$	−3.09427e6	−0.166736
$809$	1.85432e7	0.996124	0.498062	−	0.867141i	$-0.334045\pi$
$809$	1.85432e7	0.996124	0.498062	+	0.867141i	$0.334045\pi$
$810$	1.88957e6	0.101193
$811$	1.63648e7	0.873690	0.436845	−	0.899537i	$-0.356096\pi$
$811$	1.63648e7	0.873690	0.436845	+	0.899537i	$0.356096\pi$
$812$	−5.11325e6	−0.272149
$813$	1.86350e7	0.988790
$814$	1.59804e7	0.845331
$815$	−3.81946e7	−2.01422
$816$	−4.25779e6	−0.223851
$817$	−2.54974e6	−0.133642
$818$	1.88673e6	0.0985884
$819$	−4.18333e6	−0.217927
$820$	−9.77357e6	−0.507596
$821$	−5.45014e6	−0.282195	−0.141098	−	0.989996i	$-0.545063\pi$
$821$	−5.45014e6	−0.282195	−0.141098	+	0.989996i	$0.545063\pi$
$822$	7.44228e6	0.384173
$823$	−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
$825$	−7.67183e6	−0.392432
$826$	5.52250e6	0.281634
$827$	8.11859e6	0.412778	0.206389	−	0.978470i	$-0.433829\pi$
$827$	8.11859e6	0.412778	0.206389	+	0.978470i	$0.433829\pi$
$828$	3.75581e6	0.190383
$829$	1.60662e7	0.811943	0.405972	−	0.913886i	$-0.366933\pi$
$829$	1.60662e7	0.811943	0.405972	+	0.913886i	$0.366933\pi$
$830$	7.40621e6	0.373165
$831$	−2.16654e7	−1.08834
$832$	−4.31718e6	−0.216218
$833$	−4.43705e6	−0.221555
$834$	8.52610e6	0.424458
$835$	2.24891e7	1.11623
$836$	−1.56326e6	−0.0773600
$837$	4.51980e6	0.223000
$838$	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	$0.723634\pi$
$840$	2.03213e6	0.0993694
$841$	2.20253e7	1.07382
$842$	−1.09105e7	−0.530352
$843$	3.07697e6	0.149127
$844$	−9.78208e6	−0.472689
$845$	−5.32529e7	−2.56567
$846$	−19440.0	−0.000933835 0
$847$	506905.	0.0242783
$848$	5.76154e6	0.275137
$849$	5.20708e6	0.247927
$850$	1.52201e7	0.722555
$851$	2.79657e7	1.32374
$852$	−956448.	−0.0451401
$853$	2.78129e7	1.30880	0.654400	−	0.756148i	$-0.272921\pi$
$853$	2.78129e7	1.30880	0.654400	+	0.756148i	$0.272921\pi$
$854$	6.89802e6	0.323653
$855$	−1.37635e6	−0.0643894
$856$	143232.	0.00668122
$857$	−2.69363e6	−0.125281	−0.0626406	−	0.998036i	$-0.519952\pi$
$857$	−2.69363e6	−0.125281	−0.0626406	+	0.998036i	$0.519952\pi$
$858$	−1.57088e7	−0.728493
$859$	−1.88389e7	−0.871109	−0.435555	−	0.900162i	$-0.643448\pi$
$859$	−1.88389e7	−0.871109	−0.435555	+	0.900162i	$0.643448\pi$
$860$	1.24462e7	0.573840
$861$	3.74144e6	0.172001
$862$	1.90607e7	0.873716
$863$	−1.28630e7	−0.587917	−0.293959	−	0.955818i	$-0.594973\pi$
$863$	−1.28630e7	−0.587917	−0.293959	+	0.955818i	$0.594973\pi$
$864$	−746496.	−0.0340207
$865$	5.40778e6	0.245741
$866$	−2.76120e7	−1.25113
$867$	1.79572e7	0.811319
$868$	4.86080e6	0.218982
$869$	−1.79428e7	−0.806009
$870$	−1.69050e7	−0.757212
$871$	2.97397e7	1.32828
$872$	−3.85011e6	−0.171468
$873$	−1.20653e7	−0.535799
$874$	−2.73571e6	−0.121141
$875$	3.76085e6	0.166060
$876$	−7.50096e6	−0.330260
$877$	−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
$879$	7.02486e6	0.306666
$880$	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	$0.376973\pi$
$882$	−777924.	−0.0336718
$883$	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
$885$	1.82580e7	0.783603
$886$	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	$0.560899\pi$
$888$	−5.55840e6	−0.236547
$889$	−87220.0	−0.00370136
$890$	2.84325e7	1.20321
$891$	−2.71625e6	−0.114624
$892$	−1.25321e7	−0.527365
$893$	14160.0	0.000594203 0
$894$	−1.70534e7	−0.713621
$895$	−2.78437e7	−1.16190
$896$	−802816.	−0.0334077
$897$	−2.74904e7	−1.14078
$898$	−851976.	−0.0352563
$899$	−4.04364e7	−1.66868
$900$	2.66846e6	0.109813
$901$	−4.15911e7	−1.70682
$902$	1.40495e7	0.574969
$903$	−4.76456e6	−0.194448
$904$	448896.	0.0182694
$905$	3.00671e7	1.22031
$906$	−1.42183e7	−0.575475
$907$	−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
$909$	3.91619e6	0.157200
$910$	−1.48740e7	−0.595423
$911$	−9.83085e6	−0.392460	−0.196230	−	0.980558i	$-0.562870\pi$
$911$	−9.83085e6	−0.392460	−0.196230	+	0.980558i	$0.562870\pi$
$912$	543744.	0.0216475
$913$	−1.06464e7	−0.422695
$914$	3.66460e6	0.145098
$915$	2.28057e7	0.900515
$916$	2.44381e6	0.0962340
$917$	−1.29919e7	−0.510209
$918$	5.38877e6	0.211049
$919$	−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
$921$	−2.02343e7	−0.786029
$922$	1.66334e7	0.644397
$923$	7.00067e6	0.270480
$924$	−2.92118e6	−0.112559
$925$	1.98694e7	0.763536
$926$	−3.36320e7	−1.28892
$927$	−1.48710e7	−0.568381
$928$	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	$0.541076\pi$
$930$	1.60704e7	0.609283
$931$	566636.	0.0214255
$932$	−5.66851e6	−0.213761
$933$	5.23271e6	0.196799
$934$	−288192.	−0.0108097
$935$	−5.50852e7	−2.06066
$936$	5.46394e6	0.203852
$937$	1.25127e7	0.465590	0.232795	−	0.972526i	$-0.425213\pi$
$937$	1.25127e7	0.465590	0.232795	+	0.972526i	$0.425213\pi$
$938$	5.53034e6	0.205232
$939$	−1.80366e7	−0.667562
$940$	−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	$0.582153\pi$
$942$	5.22324e6	0.191784
$943$	2.45866e7	0.900367
$944$	−7.21306e6	−0.263445
$945$	−2.57191e6	−0.0936864
$946$	−1.78914e7	−0.650006
$947$	4.89287e6	0.177292	0.0886460	−	0.996063i	$-0.471746\pi$
$947$	4.89287e6	0.177292	0.0886460	+	0.996063i	$0.471746\pi$
$948$	6.24096e6	0.225544
$949$	5.49029e7	1.97893
$950$	−1.94370e6	−0.0698746
$951$	−1.15049e7	−0.412506
$952$	5.79533e6	0.207246
$953$	−1.40055e7	−0.499535	−0.249768	−	0.968306i	$-0.580354\pi$
$953$	−1.40055e7	−0.499535	−0.249768	+	0.968306i	$0.580354\pi$
$954$	−7.29194e6	−0.259401
$955$	7.11396e7	2.52408
$956$	4.40592e6	0.155916
$957$	2.43010e7	0.857717
$958$	−1.92224e7	−0.676697
$959$	−1.01298e7	−0.355675
$960$	−2.65421e6	−0.0929516
$961$	9.81085e6	0.342687
$962$	4.06844e7	1.41739
$963$	−181278.	−0.00629911
$964$	−9.35517e6	−0.324234
$965$	2.94666e7	1.01862
$966$	−5.11207e6	−0.176260
$967$	−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
$969$	−3.92515e6	−0.134291
$970$	−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	$0.435943\pi$
$972$	944784.	0.0320750
$973$	−1.16050e7	−0.392972
$974$	−1.76722e7	−0.596888
$975$	−1.95317e7	−0.658003
$976$	−9.00966e6	−0.302750
$977$	2.26195e7	0.758134	0.379067	−	0.925369i	$-0.376245\pi$
$977$	2.26195e7	0.758134	0.379067	+	0.925369i	$0.376245\pi$
$978$	−1.90973e7	−0.638446
$979$	−4.08717e7	−1.36291
$980$	−2.76595e6	−0.0919982
$981$	4.87280e6	0.161661
$982$	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	$0.347345\pi$
$984$	−4.88678e6	−0.160892
$985$	6.64006e7	2.18063
$986$	−4.82106e7	−1.57925
$987$	26460.0	0.000864564 0
$988$	−3.97990e6	−0.129712
$989$	−3.13100e7	−1.01787
$990$	−9.65779e6	−0.313177
$991$	−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
$993$	2.33830e7	0.752537
$994$	1.30183e6	0.0417916
$995$	−1.36431e7	−0.436874
$996$	3.70310e6	0.118282
$997$	−5.17280e7	−1.64812	−0.824058	−	0.566505i	$-0.808295\pi$
$997$	−5.17280e7	−1.64812	−0.824058	+	0.566505i	$0.808295\pi$
$998$	3.68938e7	1.17254
$999$	7.03485e6	0.223019

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	42.6.a.c.1.1	✓	1
3.2	odd	2		126.6.a.l.1.1		1
4.3	odd	2		336.6.a.b.1.1		1
5.2	odd	4		1050.6.g.b.799.1		2
5.3	odd	4		1050.6.g.b.799.2		2
5.4	even	2		1050.6.a.g.1.1		1
7.2	even	3		294.6.e.l.67.1		2
7.3	odd	6		294.6.e.n.79.1		2
7.4	even	3		294.6.e.l.79.1		2
7.5	odd	6		294.6.e.n.67.1		2
7.6	odd	2		294.6.a.c.1.1		1
12.11	even	2		1008.6.a.ba.1.1		1
21.20	even	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	1.1	even	1	trivial
126.6.a.l.1.1		1	3.2	odd	2
294.6.a.c.1.1		1	7.6	odd	2
294.6.e.l.67.1		2	7.2	even	3
294.6.e.l.79.1		2	7.4	even	3
294.6.e.n.67.1		2	7.5	odd	6
294.6.e.n.79.1		2	7.3	odd	6
336.6.a.b.1.1		1	4.3	odd	2
882.6.a.n.1.1		1	21.20	even	2
1008.6.a.ba.1.1		1	12.11	even	2
1050.6.a.g.1.1		1	5.4	even	2
1050.6.g.b.799.1		2	5.2	odd	4
1050.6.g.b.799.2		2	5.3	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:	\( N \)	\(=\)	\( 42 = 2 \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	42.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more