LMFDB - Embedded newform 42.6.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [42,6,Mod(1,42)]

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

lf = mfeigenbasis(mf)

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")

//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);

Level:	$ N $	$=$	$ 42 = 2 \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	42.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:	$6.73612043215$
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)

$f(q)$

$=$

\(q-4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} +44.0000 q^{5} +36.0000 q^{6} -49.0000 q^{7} -64.0000 q^{8} +81.0000 q^{9} -176.000 q^{10} -470.000 q^{11} -144.000 q^{12} -1158.00 q^{13} +196.000 q^{14} -396.000 q^{15} +256.000 q^{16} +1204.00 q^{17} -324.000 q^{18} -2644.00 q^{19} +704.000 q^{20} +441.000 q^{21} +1880.00 q^{22} -1190.00 q^{23} +576.000 q^{24} -1189.00 q^{25} +4632.00 q^{26} -729.000 q^{27} -784.000 q^{28} +3614.00 q^{29} +1584.00 q^{30} +5616.00 q^{31} -1024.00 q^{32} +4230.00 q^{33} -4816.00 q^{34} -2156.00 q^{35} +1296.00 q^{36} -6478.00 q^{37} +10576.0 q^{38} +10422.0 q^{39} -2816.00 q^{40} +2856.00 q^{41} -1764.00 q^{42} -13492.0 q^{43} -7520.00 q^{44} +3564.00 q^{45} +4760.00 q^{46} -18372.0 q^{47} -2304.00 q^{48} +2401.00 q^{49} +4756.00 q^{50} -10836.0 q^{51} -18528.0 q^{52} -4374.00 q^{53} +2916.00 q^{54} -20680.0 q^{55} +3136.00 q^{56} +23796.0 q^{57} -14456.0 q^{58} +30248.0 q^{59} -6336.00 q^{60} +19542.0 q^{61} -22464.0 q^{62} -3969.00 q^{63} +4096.00 q^{64} -50952.0 q^{65} -16920.0 q^{66} +54328.0 q^{67} +19264.0 q^{68} +10710.0 q^{69} +8624.00 q^{70} -10730.0 q^{71} -5184.00 q^{72} +35374.0 q^{73} +25912.0 q^{74} +10701.0 q^{75} -42304.0 q^{76} +23030.0 q^{77} -41688.0 q^{78} -49956.0 q^{79} +11264.0 q^{80} +6561.00 q^{81} -11424.0 q^{82} -26948.0 q^{83} +7056.00 q^{84} +52976.0 q^{85} +53968.0 q^{86} -32526.0 q^{87} +30080.0 q^{88} +100776. q^{89} -14256.0 q^{90} +56742.0 q^{91} -19040.0 q^{92} -50544.0 q^{93} +73488.0 q^{94} -116336. q^{95} +9216.00 q^{96} +77134.0 q^{97} -9604.00 q^{98} -38070.0 q^{99} +O(q^{100})\)

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
$5$	44.0000	0.787096	0.393548	−	0.919304i	$-0.371248\pi$
$5$	44.0000	0.787096	0.393548	+	0.919304i	$0.371248\pi$
$6$	36.0000	0.408248
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	−64.0000	−0.353553
$9$	81.0000	0.333333
$10$	−176.000	−0.556561
$11$	−470.000	−1.17116	−0.585580	−	0.810615i	$-0.699133\pi$
$11$	−470.000	−1.17116	−0.585580	+	0.810615i	$0.699133\pi$
$12$	−144.000	−0.288675
$13$	−1158.00	−1.90042	−0.950211	−	0.311606i	$-0.899133\pi$
$13$	−1158.00	−1.90042	−0.950211	+	0.311606i	$0.899133\pi$
$14$	196.000	0.267261
$15$	−396.000	−0.454430
$16$	256.000	0.250000
$17$	1204.00	1.01043	0.505213	−	0.862995i	$-0.331414\pi$
$17$	1204.00	1.01043	0.505213	+	0.862995i	$0.331414\pi$
$18$	−324.000	−0.235702
$19$	−2644.00	−1.68026	−0.840132	−	0.542382i	$-0.817522\pi$
$19$	−2644.00	−1.68026	−0.840132	+	0.542382i	$0.817522\pi$
$20$	704.000	0.393548
$21$	441.000	0.218218
$22$	1880.00	0.828135
$23$	−1190.00	−0.469059	−0.234529	−	0.972109i	$-0.575355\pi$
$23$	−1190.00	−0.469059	−0.234529	+	0.972109i	$0.575355\pi$
$24$	576.000	0.204124
$25$	−1189.00	−0.380480
$26$	4632.00	1.34380
$27$	−729.000	−0.192450
$28$	−784.000	−0.188982
$29$	3614.00	0.797982	0.398991	−	0.916955i	$-0.369360\pi$
$29$	3614.00	0.797982	0.398991	+	0.916955i	$0.369360\pi$
$30$	1584.00	0.321331
$31$	5616.00	1.04960	0.524799	−	0.851226i	$-0.324141\pi$
$31$	5616.00	1.04960	0.524799	+	0.851226i	$0.324141\pi$
$32$	−1024.00	−0.176777
$33$	4230.00	0.676169
$34$	−4816.00	−0.714479
$35$	−2156.00	−0.297494
$36$	1296.00	0.166667
$37$	−6478.00	−0.777923	−0.388962	−	0.921254i	$-0.627166\pi$
$37$	−6478.00	−0.777923	−0.388962	+	0.921254i	$0.627166\pi$
$38$	10576.0	1.18813
$39$	10422.0	1.09721
$40$	−2816.00	−0.278280
$41$	2856.00	0.265337	0.132669	−	0.991160i	$-0.457645\pi$
$41$	2856.00	0.265337	0.132669	+	0.991160i	$0.457645\pi$
$42$	−1764.00	−0.154303
$43$	−13492.0	−1.11277	−0.556385	−	0.830925i	$-0.687812\pi$
$43$	−13492.0	−1.11277	−0.556385	+	0.830925i	$0.687812\pi$
$44$	−7520.00	−0.585580
$45$	3564.00	0.262365
$46$	4760.00	0.331675
$47$	−18372.0	−1.21314	−0.606571	−	0.795029i	$-0.707456\pi$
$47$	−18372.0	−1.21314	−0.606571	+	0.795029i	$0.707456\pi$
$48$	−2304.00	−0.144338
$49$	2401.00	0.142857
$50$	4756.00	0.269040
$51$	−10836.0	−0.583369
$52$	−18528.0	−0.950211
$53$	−4374.00	−0.213889	−0.106945	−	0.994265i	$-0.534107\pi$
$53$	−4374.00	−0.213889	−0.106945	+	0.994265i	$0.534107\pi$
$54$	2916.00	0.136083
$55$	−20680.0	−0.921815
$56$	3136.00	0.133631
$57$	23796.0	0.970101
$58$	−14456.0	−0.564259
$59$	30248.0	1.13127	0.565635	−	0.824655i	$-0.308631\pi$
$59$	30248.0	1.13127	0.565635	+	0.824655i	$0.308631\pi$
$60$	−6336.00	−0.227215
$61$	19542.0	0.672426	0.336213	−	0.941786i	$-0.390854\pi$
$61$	19542.0	0.672426	0.336213	+	0.941786i	$0.390854\pi$
$62$	−22464.0	−0.742178
$63$	−3969.00	−0.125988
$64$	4096.00	0.125000
$65$	−50952.0	−1.49581
$66$	−16920.0	−0.478124
$67$	54328.0	1.47855	0.739276	−	0.673402i	$-0.235168\pi$
$67$	54328.0	1.47855	0.739276	+	0.673402i	$0.235168\pi$
$68$	19264.0	0.505213
$69$	10710.0	0.270811
$70$	8624.00	0.210360
$71$	−10730.0	−0.252612	−0.126306	−	0.991991i	$-0.540312\pi$
$71$	−10730.0	−0.252612	−0.126306	+	0.991991i	$0.540312\pi$
$72$	−5184.00	−0.117851
$73$	35374.0	0.776921	0.388461	−	0.921465i	$-0.373007\pi$
$73$	35374.0	0.776921	0.388461	+	0.921465i	$0.373007\pi$
$74$	25912.0	0.550075
$75$	10701.0	0.219670
$76$	−42304.0	−0.840132
$77$	23030.0	0.442657
$78$	−41688.0	−0.775844
$79$	−49956.0	−0.900575	−0.450288	−	0.892884i	$-0.648679\pi$
$79$	−49956.0	−0.900575	−0.450288	+	0.892884i	$0.648679\pi$
$80$	11264.0	0.196774
$81$	6561.00	0.111111
$82$	−11424.0	−0.187622
$83$	−26948.0	−0.429370	−0.214685	−	0.976683i	$-0.568872\pi$
$83$	−26948.0	−0.429370	−0.214685	+	0.976683i	$0.568872\pi$
$84$	7056.00	0.109109
$85$	52976.0	0.795302
$86$	53968.0	0.786847
$87$	−32526.0	−0.460715
$88$	30080.0	0.414068
$89$	100776.	1.34860	0.674298	−	0.738459i	$-0.264446\pi$
$89$	100776.	1.34860	0.674298	+	0.738459i	$0.264446\pi$
$90$	−14256.0	−0.185520
$91$	56742.0	0.718292
$92$	−19040.0	−0.234529
$93$	−50544.0	−0.605986
$94$	73488.0	0.857821
$95$	−116336.	−1.32253
$96$	9216.00	0.102062
$97$	77134.0	0.832370	0.416185	−	0.909280i	$-0.363367\pi$
$97$	77134.0	0.832370	0.416185	+	0.909280i	$0.363367\pi$
$98$	−9604.00	−0.101015
$99$	−38070.0	−0.390387
$100$	−19024.0	−0.190240
$101$	99464.0	0.970203	0.485101	−	0.874458i	$-0.338783\pi$
$101$	99464.0	0.970203	0.485101	+	0.874458i	$0.338783\pi$
$102$	43344.0	0.412504
$103$	66944.0	0.621754	0.310877	−	0.950450i	$-0.399377\pi$
$103$	66944.0	0.621754	0.310877	+	0.950450i	$0.399377\pi$
$104$	74112.0	0.671901
$105$	19404.0	0.171758
$106$	17496.0	0.151243
$107$	−228198.	−1.92687	−0.963435	−	0.267942i	$-0.913656\pi$
$107$	−228198.	−1.92687	−0.963435	+	0.267942i	$0.913656\pi$
$108$	−11664.0	−0.0962250
$109$	−95186.0	−0.767374	−0.383687	−	0.923463i	$-0.625346\pi$
$109$	−95186.0	−0.767374	−0.383687	+	0.923463i	$0.625346\pi$
$110$	82720.0	0.651822
$111$	58302.0	0.449134
$112$	−12544.0	−0.0944911
$113$	−261142.	−1.92389	−0.961946	−	0.273240i	$-0.911905\pi$
$113$	−261142.	−1.92389	−0.961946	+	0.273240i	$0.911905\pi$
$114$	−95184.0	−0.685965
$115$	−52360.0	−0.369194
$116$	57824.0	0.398991
$117$	−93798.0	−0.633474
$118$	−120992.	−0.799929
$119$	−58996.0	−0.381905
$120$	25344.0	0.160665
$121$	59849.0	0.371615
$122$	−78168.0	−0.475477
$123$	−25704.0	−0.153193
$124$	89856.0	0.524799
$125$	−189816.	−1.08657
$126$	15876.0	0.0890871
$127$	−167652.	−0.922358	−0.461179	−	0.887307i	$-0.652573\pi$
$127$	−167652.	−0.922358	−0.461179	+	0.887307i	$0.652573\pi$
$128$	−16384.0	−0.0883883
$129$	121428.	0.642458
$130$	203808.	1.05770
$131$	−83068.0	−0.422917	−0.211459	−	0.977387i	$-0.567821\pi$
$131$	−83068.0	−0.422917	−0.211459	+	0.977387i	$0.567821\pi$
$132$	67680.0	0.338085
$133$	129556.	0.635080
$134$	−217312.	−1.04549
$135$	−32076.0	−0.151477
$136$	−77056.0	−0.357239
$137$	162254.	0.738574	0.369287	−	0.929315i	$-0.379602\pi$
$137$	162254.	0.738574	0.369287	+	0.929315i	$0.379602\pi$
$138$	−42840.0	−0.191492
$139$	−58844.0	−0.258324	−0.129162	−	0.991623i	$-0.541229\pi$
$139$	−58844.0	−0.258324	−0.129162	+	0.991623i	$0.541229\pi$
$140$	−34496.0	−0.148747
$141$	165348.	0.700408
$142$	42920.0	0.178624
$143$	544260.	2.22570
$144$	20736.0	0.0833333
$145$	159016.	0.628088
$146$	−141496.	−0.549366
$147$	−21609.0	−0.0824786
$148$	−103648.	−0.388962
$149$	430698.	1.58930	0.794652	−	0.607065i	$-0.207653\pi$
$149$	430698.	1.58930	0.794652	+	0.607065i	$0.207653\pi$
$150$	−42804.0	−0.155330
$151$	−500936.	−1.78789	−0.893943	−	0.448180i	$-0.852072\pi$
$151$	−500936.	−1.78789	−0.893943	+	0.448180i	$0.852072\pi$
$152$	169216.	0.594063
$153$	97524.0	0.336808
$154$	−92120.0	−0.313006
$155$	247104.	0.826134
$156$	166752.	0.548605
$157$	−280258.	−0.907421	−0.453711	−	0.891149i	$-0.649900\pi$
$157$	−280258.	−0.907421	−0.453711	+	0.891149i	$0.649900\pi$
$158$	199824.	0.636803
$159$	39366.0	0.123489
$160$	−45056.0	−0.139140
$161$	58310.0	0.177288
$162$	−26244.0	−0.0785674
$163$	−120016.	−0.353810	−0.176905	−	0.984228i	$-0.556609\pi$
$163$	−120016.	−0.353810	−0.176905	+	0.984228i	$0.556609\pi$
$164$	45696.0	0.132669
$165$	186120.	0.532210
$166$	107792.	0.303610
$167$	546932.	1.51755	0.758774	−	0.651355i	$-0.225799\pi$
$167$	546932.	1.51755	0.758774	+	0.651355i	$0.225799\pi$
$168$	−28224.0	−0.0771517
$169$	969671.	2.61161
$170$	−211904.	−0.562363
$171$	−214164.	−0.560088
$172$	−215872.	−0.556385
$173$	−367096.	−0.932533	−0.466267	−	0.884644i	$-0.654401\pi$
$173$	−367096.	−0.932533	−0.466267	+	0.884644i	$0.654401\pi$
$174$	130104.	0.325775
$175$	58261.0	0.143808
$176$	−120320.	−0.292790
$177$	−272232.	−0.653140
$178$	−403104.	−0.953602
$179$	−88890.0	−0.207358	−0.103679	−	0.994611i	$-0.533061\pi$
$179$	−88890.0	−0.207358	−0.103679	+	0.994611i	$0.533061\pi$
$180$	57024.0	0.131183
$181$	−782118.	−1.77450	−0.887250	−	0.461290i	$-0.847387\pi$
$181$	−782118.	−1.77450	−0.887250	+	0.461290i	$0.847387\pi$
$182$	−226968.	−0.507909
$183$	−175878.	−0.388225
$184$	76160.0	0.165837
$185$	−285032.	−0.612300
$186$	202176.	0.428496
$187$	−565880.	−1.18337
$188$	−293952.	−0.606571
$189$	35721.0	0.0727393
$190$	465344.	0.935169
$191$	763350.	1.51405	0.757025	−	0.653386i	$-0.226652\pi$
$191$	763350.	1.51405	0.757025	+	0.653386i	$0.226652\pi$
$192$	−36864.0	−0.0721688
$193$	−377002.	−0.728535	−0.364267	−	0.931294i	$-0.618681\pi$
$193$	−377002.	−0.728535	−0.364267	+	0.931294i	$0.618681\pi$
$194$	−308536.	−0.588575
$195$	458568.	0.863609
$196$	38416.0	0.0714286
$197$	−68678.0	−0.126082	−0.0630409	−	0.998011i	$-0.520080\pi$
$197$	−68678.0	−0.126082	−0.0630409	+	0.998011i	$0.520080\pi$
$198$	152280.	0.276045
$199$	182576.	0.326822	0.163411	−	0.986558i	$-0.447750\pi$
$199$	182576.	0.326822	0.163411	+	0.986558i	$0.447750\pi$
$200$	76096.0	0.134520
$201$	−488952.	−0.853643
$202$	−397856.	−0.686037
$203$	−177086.	−0.301609
$204$	−173376.	−0.291685
$205$	125664.	0.208846
$206$	−267776.	−0.439646
$207$	−96390.0	−0.156353
$208$	−296448.	−0.475106
$209$	1.24268e6	1.96786
$210$	−77616.0	−0.121452
$211$	232652.	0.359750	0.179875	−	0.983689i	$-0.442431\pi$
$211$	232652.	0.359750	0.179875	+	0.983689i	$0.442431\pi$
$212$	−69984.0	−0.106945
$213$	96570.0	0.145846
$214$	912792.	1.36250
$215$	−593648.	−0.875856
$216$	46656.0	0.0680414
$217$	−275184.	−0.396711
$218$	380744.	0.542615
$219$	−318366.	−0.448556
$220$	−330880.	−0.460908
$221$	−1.39423e6	−1.92023
$222$	−233208.	−0.317586
$223$	167144.	0.225076	0.112538	−	0.993647i	$-0.464102\pi$
$223$	167144.	0.225076	0.112538	+	0.993647i	$0.464102\pi$
$224$	50176.0	0.0668153
$225$	−96309.0	−0.126827
$226$	1.04457e6	1.36040
$227$	415728.	0.535482	0.267741	−	0.963491i	$-0.413723\pi$
$227$	415728.	0.535482	0.267741	+	0.963491i	$0.413723\pi$
$228$	380736.	0.485050
$229$	473482.	0.596643	0.298322	−	0.954465i	$-0.403573\pi$
$229$	473482.	0.596643	0.298322	+	0.954465i	$0.403573\pi$
$230$	209440.	0.261060
$231$	−207270.	−0.255568
$232$	−231296.	−0.282129
$233$	−1.55655e6	−1.87833	−0.939166	−	0.343465i	$-0.888399\pi$
$233$	−1.55655e6	−1.87833	−0.939166	+	0.343465i	$0.888399\pi$
$234$	375192.	0.447934
$235$	−808368.	−0.954859
$236$	483968.	0.565635
$237$	449604.	0.519947
$238$	235984.	0.270048
$239$	655890.	0.742739	0.371370	−	0.928485i	$-0.378888\pi$
$239$	655890.	0.742739	0.371370	+	0.928485i	$0.378888\pi$
$240$	−101376.	−0.113608
$241$	−889474.	−0.986485	−0.493243	−	0.869892i	$-0.664189\pi$
$241$	−889474.	−0.986485	−0.493243	+	0.869892i	$0.664189\pi$
$242$	−239396.	−0.262772
$243$	−59049.0	−0.0641500
$244$	312672.	0.336213
$245$	105644.	0.112442
$246$	102816.	0.108324
$247$	3.06175e6	3.19321
$248$	−359424.	−0.371089
$249$	242532.	0.247897
$250$	759264.	0.768321
$251$	131832.	0.132080	0.0660399	−	0.997817i	$-0.478964\pi$
$251$	131832.	0.132080	0.0660399	+	0.997817i	$0.478964\pi$
$252$	−63504.0	−0.0629941
$253$	559300.	0.549343
$254$	670608.	0.652205
$255$	−476784.	−0.459168
$256$	65536.0	0.0625000
$257$	−1.46482e6	−1.38341	−0.691704	−	0.722181i	$-0.743140\pi$
$257$	−1.46482e6	−1.38341	−0.691704	+	0.722181i	$0.743140\pi$
$258$	−485712.	−0.454286
$259$	317422.	0.294027
$260$	−815232.	−0.747907
$261$	292734.	0.265994
$262$	332272.	0.299048
$263$	1.47969e6	1.31911	0.659556	−	0.751656i	$-0.270745\pi$
$263$	1.47969e6	1.31911	0.659556	+	0.751656i	$0.270745\pi$
$264$	−270720.	−0.239062
$265$	−192456.	−0.168351
$266$	−518224.	−0.449069
$267$	−906984.	−0.778613
$268$	869248.	0.739276
$269$	−187852.	−0.158283	−0.0791417	−	0.996863i	$-0.525218\pi$
$269$	−187852.	−0.158283	−0.0791417	+	0.996863i	$0.525218\pi$
$270$	128304.	0.107110
$271$	193800.	0.160299	0.0801495	−	0.996783i	$-0.474460\pi$
$271$	193800.	0.160299	0.0801495	+	0.996783i	$0.474460\pi$
$272$	308224.	0.252606
$273$	−510678.	−0.414706
$274$	−649016.	−0.522251
$275$	558830.	0.445603
$276$	171360.	0.135406
$277$	−617062.	−0.483203	−0.241601	−	0.970376i	$-0.577673\pi$
$277$	−617062.	−0.483203	−0.241601	+	0.970376i	$0.577673\pi$
$278$	235376.	0.182663
$279$	454896.	0.349866
$280$	137984.	0.105180
$281$	−1.73129e6	−1.30799	−0.653994	−	0.756499i	$-0.726908\pi$
$281$	−1.73129e6	−1.30799	−0.653994	+	0.756499i	$0.726908\pi$
$282$	−661392.	−0.495263
$283$	356020.	0.264246	0.132123	−	0.991233i	$-0.457821\pi$
$283$	356020.	0.264246	0.132123	+	0.991233i	$0.457821\pi$
$284$	−171680.	−0.126306
$285$	1.04702e6	0.763562
$286$	−2.17704e6	−1.57381
$287$	−139944.	−0.100288
$288$	−82944.0	−0.0589256
$289$	29759.0	0.0209592
$290$	−636064.	−0.444126
$291$	−694206.	−0.480569
$292$	565984.	0.388461
$293$	536664.	0.365202	0.182601	−	0.983187i	$-0.441548\pi$
$293$	536664.	0.365202	0.182601	+	0.983187i	$0.441548\pi$
$294$	86436.0	0.0583212
$295$	1.33091e6	0.890419
$296$	414592.	0.275037
$297$	342630.	0.225390
$298$	−1.72279e6	−1.12381
$299$	1.37802e6	0.891410
$300$	171216.	0.109835
$301$	661108.	0.420587
$302$	2.00374e6	1.26423
$303$	−895176.	−0.560147
$304$	−676864.	−0.420066
$305$	859848.	0.529264
$306$	−390096.	−0.238160
$307$	−2.88398e6	−1.74641	−0.873205	−	0.487353i	$-0.837963\pi$
$307$	−2.88398e6	−1.74641	−0.873205	+	0.487353i	$0.837963\pi$
$308$	368480.	0.221328
$309$	−602496.	−0.358970
$310$	−988416.	−0.584165
$311$	−1.02958e6	−0.603614	−0.301807	−	0.953369i	$-0.597590\pi$
$311$	−1.02958e6	−0.603614	−0.301807	+	0.953369i	$0.597590\pi$
$312$	−667008.	−0.387922
$313$	1.39297e6	0.803676	0.401838	−	0.915711i	$-0.368372\pi$
$313$	1.39297e6	0.803676	0.401838	+	0.915711i	$0.368372\pi$
$314$	1.12103e6	0.641644
$315$	−174636.	−0.0991648
$316$	−799296.	−0.450288
$317$	−780030.	−0.435977	−0.217988	−	0.975951i	$-0.569949\pi$
$317$	−780030.	−0.435977	−0.217988	+	0.975951i	$0.569949\pi$
$318$	−157464.	−0.0873200
$319$	−1.69858e6	−0.934565
$320$	180224.	0.0983870
$321$	2.05378e6	1.11248
$322$	−233240.	−0.125361
$323$	−3.18338e6	−1.69778
$324$	104976.	0.0555556
$325$	1.37686e6	0.723073
$326$	480064.	0.250181
$327$	856674.	0.443043
$328$	−182784.	−0.0938110
$329$	900228.	0.458525
$330$	−744480.	−0.376329
$331$	−1.41204e6	−0.708399	−0.354200	−	0.935170i	$-0.615247\pi$
$331$	−1.41204e6	−0.708399	−0.354200	+	0.935170i	$0.615247\pi$
$332$	−431168.	−0.214685
$333$	−524718.	−0.259308
$334$	−2.18773e6	−1.07307
$335$	2.39043e6	1.16376
$336$	112896.	0.0545545
$337$	−634662.	−0.304416	−0.152208	−	0.988348i	$-0.548638\pi$
$337$	−634662.	−0.304416	−0.152208	+	0.988348i	$0.548638\pi$
$338$	−3.87868e6	−1.84668
$339$	2.35028e6	1.11076
$340$	847616.	0.397651
$341$	−2.63952e6	−1.22925
$342$	856656.	0.396042
$343$	−117649.	−0.0539949
$344$	863488.	0.393423
$345$	471240.	0.213154
$346$	1.46838e6	0.659401
$347$	3.07942e6	1.37292	0.686460	−	0.727167i	$-0.259163\pi$
$347$	3.07942e6	1.37292	0.686460	+	0.727167i	$0.259163\pi$
$348$	−520416.	−0.230358
$349$	−2.60671e6	−1.14559	−0.572796	−	0.819698i	$-0.694141\pi$
$349$	−2.60671e6	−1.14559	−0.572796	+	0.819698i	$0.694141\pi$
$350$	−233044.	−0.101688
$351$	844182.	0.365736
$352$	481280.	0.207034
$353$	−63132.0	−0.0269658	−0.0134829	−	0.999909i	$-0.504292\pi$
$353$	−63132.0	−0.0269658	−0.0134829	+	0.999909i	$0.504292\pi$
$354$	1.08893e6	0.461839
$355$	−472120.	−0.198830
$356$	1.61242e6	0.674298
$357$	530964.	0.220493
$358$	355560.	0.146624
$359$	479270.	0.196266	0.0981328	−	0.995173i	$-0.468713\pi$
$359$	479270.	0.196266	0.0981328	+	0.995173i	$0.468713\pi$
$360$	−228096.	−0.0927601
$361$	4.51464e6	1.82329
$362$	3.12847e6	1.25476
$363$	−538641.	−0.214552
$364$	907872.	0.359146
$365$	1.55646e6	0.611512
$366$	703512.	0.274517
$367$	−1.33451e6	−0.517199	−0.258599	−	0.965985i	$-0.583261\pi$
$367$	−1.33451e6	−0.517199	−0.258599	+	0.965985i	$0.583261\pi$
$368$	−304640.	−0.117265
$369$	231336.	0.0884458
$370$	1.14013e6	0.432962
$371$	214326.	0.0808426
$372$	−808704.	−0.302993
$373$	−1.69759e6	−0.631774	−0.315887	−	0.948797i	$-0.602302\pi$
$373$	−1.69759e6	−0.631774	−0.315887	+	0.948797i	$0.602302\pi$
$374$	2.26352e6	0.836769
$375$	1.70834e6	0.627332
$376$	1.17581e6	0.428911
$377$	−4.18501e6	−1.51650
$378$	−142884.	−0.0514344
$379$	2.51074e6	0.897850	0.448925	−	0.893569i	$-0.351807\pi$
$379$	2.51074e6	0.897850	0.448925	+	0.893569i	$0.351807\pi$
$380$	−1.86138e6	−0.661264
$381$	1.50887e6	0.532524
$382$	−3.05340e6	−1.07060
$383$	559144.	0.194772	0.0973860	−	0.995247i	$-0.468952\pi$
$383$	559144.	0.194772	0.0973860	+	0.995247i	$0.468952\pi$
$384$	147456.	0.0510310
$385$	1.01332e6	0.348413
$386$	1.50801e6	0.515152
$387$	−1.09285e6	−0.370923
$388$	1.23414e6	0.416185
$389$	4.51055e6	1.51132	0.755658	−	0.654966i	$-0.227317\pi$
$389$	4.51055e6	1.51132	0.755658	+	0.654966i	$0.227317\pi$
$390$	−1.83427e6	−0.610664
$391$	−1.43276e6	−0.473949
$392$	−153664.	−0.0505076
$393$	747612.	0.244171
$394$	274712.	0.0891532
$395$	−2.19806e6	−0.708839
$396$	−609120.	−0.195193
$397$	5.19862e6	1.65543	0.827717	−	0.561145i	$-0.189639\pi$
$397$	5.19862e6	1.65543	0.827717	+	0.561145i	$0.189639\pi$
$398$	−730304.	−0.231098
$399$	−1.16600e6	−0.366664
$400$	−304384.	−0.0951200
$401$	−6.34816e6	−1.97146	−0.985728	−	0.168346i	$-0.946157\pi$
$401$	−6.34816e6	−1.97146	−0.985728	+	0.168346i	$0.946157\pi$
$402$	1.95581e6	0.603616
$403$	−6.50333e6	−1.99468
$404$	1.59142e6	0.485101
$405$	288684.	0.0874551
$406$	708344.	0.213270
$407$	3.04466e6	0.911072
$408$	693504.	0.206252
$409$	−181642.	−0.0536918	−0.0268459	−	0.999640i	$-0.508546\pi$
$409$	−181642.	−0.0536918	−0.0268459	+	0.999640i	$0.508546\pi$
$410$	−502656.	−0.147676
$411$	−1.46029e6	−0.426416
$412$	1.07110e6	0.310877
$413$	−1.48215e6	−0.427580
$414$	385560.	0.110558
$415$	−1.18571e6	−0.337955
$416$	1.18579e6	0.335950
$417$	529596.	0.149144
$418$	−4.97072e6	−1.39149
$419$	−5.62699e6	−1.56582	−0.782909	−	0.622136i	$-0.786265\pi$
$419$	−5.62699e6	−1.56582	−0.782909	+	0.622136i	$0.786265\pi$
$420$	310464.	0.0858792
$421$	−4.42671e6	−1.21724	−0.608619	−	0.793462i	$-0.708276\pi$
$421$	−4.42671e6	−1.21724	−0.608619	+	0.793462i	$0.708276\pi$
$422$	−930608.	−0.254382
$423$	−1.48813e6	−0.404381
$424$	279936.	0.0756213
$425$	−1.43156e6	−0.384447
$426$	−386280.	−0.103128
$427$	−957558.	−0.254153
$428$	−3.65117e6	−0.963435
$429$	−4.89834e6	−1.28501
$430$	2.37459e6	0.619324
$431$	−1.44163e6	−0.373817	−0.186909	−	0.982377i	$-0.559847\pi$
$431$	−1.44163e6	−0.373817	−0.186909	+	0.982377i	$0.559847\pi$
$432$	−186624.	−0.0481125
$433$	−3.89661e6	−0.998775	−0.499387	−	0.866379i	$-0.666442\pi$
$433$	−3.89661e6	−0.998775	−0.499387	+	0.866379i	$0.666442\pi$
$434$	1.10074e6	0.280517
$435$	−1.43114e6	−0.362627
$436$	−1.52298e6	−0.383687
$437$	3.14636e6	0.788143
$438$	1.27346e6	0.317177
$439$	5.11207e6	1.26601	0.633003	−	0.774149i	$-0.281822\pi$
$439$	5.11207e6	1.26601	0.633003	+	0.774149i	$0.281822\pi$
$440$	1.32352e6	0.325911
$441$	194481.	0.0476190
$442$	5.57693e6	1.35781
$443$	5.44070e6	1.31718	0.658591	−	0.752501i	$-0.271153\pi$
$443$	5.44070e6	1.31718	0.658591	+	0.752501i	$0.271153\pi$
$444$	932832.	0.224567
$445$	4.43414e6	1.06147
$446$	−668576.	−0.159153
$447$	−3.87628e6	−0.917586
$448$	−200704.	−0.0472456
$449$	−1.31525e6	−0.307887	−0.153943	−	0.988080i	$-0.549197\pi$
$449$	−1.31525e6	−0.307887	−0.153943	+	0.988080i	$0.549197\pi$
$450$	385236.	0.0896800
$451$	−1.34232e6	−0.310753
$452$	−4.17827e6	−0.961946
$453$	4.50842e6	1.03224
$454$	−1.66291e6	−0.378643
$455$	2.49665e6	0.565365
$456$	−1.52294e6	−0.342982
$457$	2.77604e6	0.621778	0.310889	−	0.950446i	$-0.399373\pi$
$457$	2.77604e6	0.621778	0.310889	+	0.950446i	$0.399373\pi$
$458$	−1.89393e6	−0.421891
$459$	−877716.	−0.194456
$460$	−837760.	−0.184597
$461$	138080.	0.0302607	0.0151303	−	0.999886i	$-0.495184\pi$
$461$	138080.	0.0302607	0.0151303	+	0.999886i	$0.495184\pi$
$462$	829080.	0.180714
$463$	−364076.	−0.0789295	−0.0394648	−	0.999221i	$-0.512565\pi$
$463$	−364076.	−0.0789295	−0.0394648	+	0.999221i	$0.512565\pi$
$464$	925184.	0.199496
$465$	−2.22394e6	−0.476969
$466$	6.22618e6	1.32818
$467$	−5.73897e6	−1.21770	−0.608852	−	0.793284i	$-0.708370\pi$
$467$	−5.73897e6	−1.21770	−0.608852	+	0.793284i	$0.708370\pi$
$468$	−1.50077e6	−0.316737
$469$	−2.66207e6	−0.558840
$470$	3.23347e6	0.675188
$471$	2.52232e6	0.523900
$472$	−1.93587e6	−0.399965
$473$	6.34124e6	1.30323
$474$	−1.79842e6	−0.367658
$475$	3.14372e6	0.639307
$476$	−943936.	−0.190952
$477$	−354294.	−0.0712964
$478$	−2.62356e6	−0.525196
$479$	5.51996e6	1.09925	0.549625	−	0.835411i	$-0.314770\pi$
$479$	5.51996e6	1.09925	0.549625	+	0.835411i	$0.314770\pi$
$480$	405504.	0.0803326
$481$	7.50152e6	1.47838
$482$	3.55790e6	0.697550
$483$	−524790.	−0.102357
$484$	957584.	0.185808
$485$	3.39390e6	0.655155
$486$	236196.	0.0453609
$487$	4.23022e6	0.808241	0.404121	−	0.914706i	$-0.367578\pi$
$487$	4.23022e6	0.808241	0.404121	+	0.914706i	$0.367578\pi$
$488$	−1.25069e6	−0.237738
$489$	1.08014e6	0.204272
$490$	−422576.	−0.0795087
$491$	−7.21423e6	−1.35047	−0.675237	−	0.737601i	$-0.735959\pi$
$491$	−7.21423e6	−1.35047	−0.675237	+	0.737601i	$0.735959\pi$
$492$	−411264.	−0.0765963
$493$	4.35126e6	0.806301
$494$	−1.22470e7	−2.25794
$495$	−1.67508e6	−0.307272
$496$	1.43770e6	0.262399
$497$	525770.	0.0954783
$498$	−970128.	−0.175289
$499$	−224804.	−0.0404159	−0.0202080	−	0.999796i	$-0.506433\pi$
$499$	−224804.	−0.0404159	−0.0202080	+	0.999796i	$0.506433\pi$
$500$	−3.03706e6	−0.543285
$501$	−4.92239e6	−0.876156
$502$	−527328.	−0.0933946
$503$	5.06983e6	0.893457	0.446728	−	0.894670i	$-0.352589\pi$
$503$	5.06983e6	0.893457	0.446728	+	0.894670i	$0.352589\pi$
$504$	254016.	0.0445435
$505$	4.37642e6	0.763643
$506$	−2.23720e6	−0.388444
$507$	−8.72704e6	−1.50781
$508$	−2.68243e6	−0.461179
$509$	5.48135e6	0.937763	0.468881	−	0.883261i	$-0.344657\pi$
$509$	5.48135e6	0.937763	0.468881	+	0.883261i	$0.344657\pi$
$510$	1.90714e6	0.324681
$511$	−1.73333e6	−0.293649
$512$	−262144.	−0.0441942
$513$	1.92748e6	0.323367
$514$	5.85926e6	0.978217
$515$	2.94554e6	0.489380
$516$	1.94285e6	0.321229
$517$	8.63484e6	1.42078
$518$	−1.26969e6	−0.207909
$519$	3.30386e6	0.538398
$520$	3.26093e6	0.528850
$521$	7.88932e6	1.27334	0.636671	−	0.771136i	$-0.280311\pi$
$521$	7.88932e6	1.27334	0.636671	+	0.771136i	$0.280311\pi$
$522$	−1.17094e6	−0.188086
$523$	−9.74950e6	−1.55858	−0.779288	−	0.626666i	$-0.784419\pi$
$523$	−9.74950e6	−1.55858	−0.779288	+	0.626666i	$0.784419\pi$
$524$	−1.32909e6	−0.211459
$525$	−524349.	−0.0830275
$526$	−5.91876e6	−0.932752
$527$	6.76166e6	1.06054
$528$	1.08288e6	0.169042
$529$	−5.02024e6	−0.779984
$530$	769824.	0.119042
$531$	2.45009e6	0.377090
$532$	2.07290e6	0.317540
$533$	−3.30725e6	−0.504253
$534$	3.62794e6	0.550562
$535$	−1.00407e7	−1.51663
$536$	−3.47699e6	−0.522747
$537$	800010.	0.119718
$538$	751408.	0.111923
$539$	−1.12847e6	−0.167309
$540$	−513216.	−0.0757383
$541$	−4.17454e6	−0.613219	−0.306610	−	0.951835i	$-0.599195\pi$
$541$	−4.17454e6	−0.613219	−0.306610	+	0.951835i	$0.599195\pi$
$542$	−775200.	−0.113348
$543$	7.03906e6	1.02451
$544$	−1.23290e6	−0.178620
$545$	−4.18818e6	−0.603997
$546$	2.04271e6	0.293242
$547$	−2.72887e6	−0.389955	−0.194978	−	0.980808i	$-0.562463\pi$
$547$	−2.72887e6	−0.389955	−0.194978	+	0.980808i	$0.562463\pi$
$548$	2.59606e6	0.369287
$549$	1.58290e6	0.224142
$550$	−2.23532e6	−0.315089
$551$	−9.55542e6	−1.34082
$552$	−685440.	−0.0957462
$553$	2.44784e6	0.340385
$554$	2.46825e6	0.341676
$555$	2.56529e6	0.353512
$556$	−941504.	−0.129162
$557$	−7.35103e6	−1.00395	−0.501973	−	0.864883i	$-0.667392\pi$
$557$	−7.35103e6	−1.00395	−0.501973	+	0.864883i	$0.667392\pi$
$558$	−1.81958e6	−0.247393
$559$	1.56237e7	2.11473
$560$	−551936.	−0.0743736
$561$	5.09292e6	0.683219
$562$	6.92516e6	0.924888
$563$	−1.37821e7	−1.83250	−0.916248	−	0.400612i	$-0.868798\pi$
$563$	−1.37821e7	−1.83250	−0.916248	+	0.400612i	$0.868798\pi$
$564$	2.64557e6	0.350204
$565$	−1.14902e7	−1.51429
$566$	−1.42408e6	−0.186850
$567$	−321489.	−0.0419961
$568$	686720.	0.0893118
$569$	2.71217e6	0.351186	0.175593	−	0.984463i	$-0.443816\pi$
$569$	2.71217e6	0.351186	0.175593	+	0.984463i	$0.443816\pi$
$570$	−4.18810e6	−0.539920
$571$	4.94398e6	0.634580	0.317290	−	0.948329i	$-0.397227\pi$
$571$	4.94398e6	0.634580	0.317290	+	0.948329i	$0.397227\pi$
$572$	8.70816e6	1.11285
$573$	−6.87015e6	−0.874137
$574$	559776.	0.0709144
$575$	1.41491e6	0.178468
$576$	331776.	0.0416667
$577$	−1.32683e7	−1.65911	−0.829556	−	0.558424i	$-0.811406\pi$
$577$	−1.32683e7	−1.65911	−0.829556	+	0.558424i	$0.811406\pi$
$578$	−119036.	−0.0148204
$579$	3.39302e6	0.420620
$580$	2.54426e6	0.314044
$581$	1.32045e6	0.162286
$582$	2.77682e6	0.339814
$583$	2.05578e6	0.250499
$584$	−2.26394e6	−0.274683
$585$	−4.12711e6	−0.498605
$586$	−2.14666e6	−0.258237
$587$	1.67205e6	0.200287	0.100144	−	0.994973i	$-0.468070\pi$
$587$	1.67205e6	0.200287	0.100144	+	0.994973i	$0.468070\pi$
$588$	−345744.	−0.0412393
$589$	−1.48487e7	−1.76360
$590$	−5.32365e6	−0.629621
$591$	618102.	0.0727933
$592$	−1.65837e6	−0.194481
$593$	5.11693e6	0.597548	0.298774	−	0.954324i	$-0.403422\pi$
$593$	5.11693e6	0.597548	0.298774	+	0.954324i	$0.403422\pi$
$594$	−1.37052e6	−0.159375
$595$	−2.59582e6	−0.300596
$596$	6.89117e6	0.794652
$597$	−1.64318e6	−0.188691
$598$	−5.51208e6	−0.630322
$599$	−7.15931e6	−0.815275	−0.407638	−	0.913144i	$-0.633647\pi$
$599$	−7.15931e6	−0.815275	−0.407638	+	0.913144i	$0.633647\pi$
$600$	−684864.	−0.0776652
$601$	−9.63384e6	−1.08796	−0.543980	−	0.839098i	$-0.683083\pi$
$601$	−9.63384e6	−1.08796	−0.543980	+	0.839098i	$0.683083\pi$
$602$	−2.64443e6	−0.297400
$603$	4.40057e6	0.492851
$604$	−8.01498e6	−0.893943
$605$	2.63336e6	0.292497
$606$	3.58070e6	0.396084
$607$	5.52115e6	0.608216	0.304108	−	0.952638i	$-0.401642\pi$
$607$	5.52115e6	0.608216	0.304108	+	0.952638i	$0.401642\pi$
$608$	2.70746e6	0.297031
$609$	1.59377e6	0.174134
$610$	−3.43939e6	−0.374246
$611$	2.12748e7	2.30548
$612$	1.56038e6	0.168404
$613$	4.79890e6	0.515811	0.257906	−	0.966170i	$-0.416968\pi$
$613$	4.79890e6	0.515811	0.257906	+	0.966170i	$0.416968\pi$
$614$	1.15359e7	1.23490
$615$	−1.13098e6	−0.120577
$616$	−1.47392e6	−0.156503
$617$	2.71826e6	0.287461	0.143730	−	0.989617i	$-0.454090\pi$
$617$	2.71826e6	0.287461	0.143730	+	0.989617i	$0.454090\pi$
$618$	2.40998e6	0.253830
$619$	−4.34335e6	−0.455615	−0.227807	−	0.973706i	$-0.573156\pi$
$619$	−4.34335e6	−0.455615	−0.227807	+	0.973706i	$0.573156\pi$
$620$	3.95366e6	0.413067
$621$	867510.	0.0902704
$622$	4.11832e6	0.426819
$623$	−4.93802e6	−0.509722
$624$	2.66803e6	0.274302
$625$	−4.63628e6	−0.474755
$626$	−5.57188e6	−0.568285
$627$	−1.11841e7	−1.13614
$628$	−4.48413e6	−0.453711
$629$	−7.79951e6	−0.786033
$630$	698544.	0.0701201
$631$	1.11952e6	0.111933	0.0559667	−	0.998433i	$-0.482176\pi$
$631$	1.11952e6	0.111933	0.0559667	+	0.998433i	$0.482176\pi$
$632$	3.19718e6	0.318401
$633$	−2.09387e6	−0.207702
$634$	3.12012e6	0.308282
$635$	−7.37669e6	−0.725984
$636$	629856.	0.0617445
$637$	−2.78036e6	−0.271489
$638$	6.79432e6	0.660837
$639$	−869130.	−0.0842040
$640$	−720896.	−0.0695701
$641$	−3.96588e6	−0.381237	−0.190618	−	0.981664i	$-0.561049\pi$
$641$	−3.96588e6	−0.381237	−0.190618	+	0.981664i	$0.561049\pi$
$642$	−8.21513e6	−0.786641
$643$	1.92086e7	1.83218	0.916092	−	0.400968i	$-0.131326\pi$
$643$	1.92086e7	1.83218	0.916092	+	0.400968i	$0.131326\pi$
$644$	932960.	0.0886438
$645$	5.34283e6	0.505676
$646$	1.27335e7	1.20051
$647$	4.72739e6	0.443977	0.221989	−	0.975049i	$-0.428745\pi$
$647$	4.72739e6	0.443977	0.221989	+	0.975049i	$0.428745\pi$
$648$	−419904.	−0.0392837
$649$	−1.42166e7	−1.32490
$650$	−5.50745e6	−0.511290
$651$	2.47666e6	0.229041
$652$	−1.92026e6	−0.176905
$653$	1.21159e7	1.11192	0.555958	−	0.831210i	$-0.312352\pi$
$653$	1.21159e7	1.11192	0.555958	+	0.831210i	$0.312352\pi$
$654$	−3.42670e6	−0.313279
$655$	−3.65499e6	−0.332877
$656$	731136.	0.0663344
$657$	2.86529e6	0.258974
$658$	−3.60091e6	−0.324226
$659$	5.91457e6	0.530530	0.265265	−	0.964176i	$-0.414541\pi$
$659$	5.91457e6	0.530530	0.265265	+	0.964176i	$0.414541\pi$
$660$	2.97792e6	0.266105
$661$	1.41779e7	1.26214	0.631072	−	0.775724i	$-0.282615\pi$
$661$	1.41779e7	1.26214	0.631072	+	0.775724i	$0.282615\pi$
$662$	5.64818e6	0.500914
$663$	1.25481e7	1.10865
$664$	1.72467e6	0.151805
$665$	5.70046e6	0.499869
$666$	2.09887e6	0.183358
$667$	−4.30066e6	−0.374301
$668$	8.75091e6	0.758774
$669$	−1.50430e6	−0.129948
$670$	−9.56173e6	−0.822904
$671$	−9.18474e6	−0.787518
$672$	−451584.	−0.0385758
$673$	1.34245e7	1.14251	0.571256	−	0.820772i	$-0.306456\pi$
$673$	1.34245e7	1.14251	0.571256	+	0.820772i	$0.306456\pi$
$674$	2.53865e6	0.215255
$675$	866781.	0.0732234
$676$	1.55147e7	1.30580
$677$	−8.67312e6	−0.727283	−0.363642	−	0.931539i	$-0.618467\pi$
$677$	−8.67312e6	−0.727283	−0.363642	+	0.931539i	$0.618467\pi$
$678$	−9.40111e6	−0.785425
$679$	−3.77957e6	−0.314606
$680$	−3.39046e6	−0.281182
$681$	−3.74155e6	−0.309161
$682$	1.05581e7	0.869209
$683$	1.49777e7	1.22855	0.614275	−	0.789092i	$-0.289448\pi$
$683$	1.49777e7	1.22855	0.614275	+	0.789092i	$0.289448\pi$
$684$	−3.42662e6	−0.280044
$685$	7.13918e6	0.581329
$686$	470596.	0.0381802
$687$	−4.26134e6	−0.344472
$688$	−3.45395e6	−0.278192
$689$	5.06509e6	0.406480
$690$	−1.88496e6	−0.150723
$691$	−8.76742e6	−0.698517	−0.349258	−	0.937026i	$-0.613566\pi$
$691$	−8.76742e6	−0.698517	−0.349258	+	0.937026i	$0.613566\pi$
$692$	−5.87354e6	−0.466267
$693$	1.86543e6	0.147552
$694$	−1.23177e7	−0.970802
$695$	−2.58914e6	−0.203326
$696$	2.08166e6	0.162887
$697$	3.43862e6	0.268104
$698$	1.04269e7	0.810056
$699$	1.40089e7	1.08445
$700$	932176.	0.0719040
$701$	−1.99459e7	−1.53306	−0.766529	−	0.642209i	$-0.778018\pi$
$701$	−1.99459e7	−1.53306	−0.766529	+	0.642209i	$0.778018\pi$
$702$	−3.37673e6	−0.258615
$703$	1.71278e7	1.30712
$704$	−1.92512e6	−0.146395
$705$	7.27531e6	0.551288
$706$	252528.	0.0190677
$707$	−4.87374e6	−0.366702
$708$	−4.35571e6	−0.326570
$709$	−2.66387e7	−1.99020	−0.995100	−	0.0988699i	$-0.968477\pi$
$709$	−2.66387e7	−1.99020	−0.995100	+	0.0988699i	$0.968477\pi$
$710$	1.88848e6	0.140594
$711$	−4.04644e6	−0.300192
$712$	−6.44966e6	−0.476801
$713$	−6.68304e6	−0.492323
$714$	−2.12386e6	−0.155912
$715$	2.39474e7	1.75184
$716$	−1.42224e6	−0.103679
$717$	−5.90301e6	−0.428821
$718$	−1.91708e6	−0.138781
$719$	−1.17408e6	−0.0846985	−0.0423492	−	0.999103i	$-0.513484\pi$
$719$	−1.17408e6	−0.0846985	−0.0423492	+	0.999103i	$0.513484\pi$
$720$	912384.	0.0655913
$721$	−3.28026e6	−0.235001
$722$	−1.80585e7	−1.28926
$723$	8.00527e6	0.569548
$724$	−1.25139e7	−0.887250
$725$	−4.29705e6	−0.303616
$726$	2.15456e6	0.151711
$727$	1.05734e7	0.741957	0.370979	−	0.928641i	$-0.379022\pi$
$727$	1.05734e7	0.741957	0.370979	+	0.928641i	$0.379022\pi$
$728$	−3.63149e6	−0.253955
$729$	531441.	0.0370370
$730$	−6.22582e6	−0.432404
$731$	−1.62444e7	−1.12437
$732$	−2.81405e6	−0.194113
$733$	2.31670e7	1.59261	0.796306	−	0.604894i	$-0.206785\pi$
$733$	2.31670e7	1.59261	0.796306	+	0.604894i	$0.206785\pi$
$734$	5.33805e6	0.365715
$735$	−950796.	−0.0649186
$736$	1.21856e6	0.0829187
$737$	−2.55342e7	−1.73162
$738$	−925344.	−0.0625406
$739$	−1.55334e7	−1.04630	−0.523148	−	0.852242i	$-0.675243\pi$
$739$	−1.55334e7	−1.04630	−0.523148	+	0.852242i	$0.675243\pi$
$740$	−4.56051e6	−0.306150
$741$	−2.75558e7	−1.84360
$742$	−857304.	−0.0571643
$743$	2.54630e7	1.69215	0.846074	−	0.533066i	$-0.178960\pi$
$743$	2.54630e7	1.69215	0.846074	+	0.533066i	$0.178960\pi$
$744$	3.23482e6	0.214248
$745$	1.89507e7	1.25094
$746$	6.79038e6	0.446732
$747$	−2.18279e6	−0.143123
$748$	−9.05408e6	−0.591685
$749$	1.11817e7	0.728288
$750$	−6.83338e6	−0.443590
$751$	9.88512e6	0.639561	0.319781	−	0.947492i	$-0.396391\pi$
$751$	9.88512e6	0.639561	0.319781	+	0.947492i	$0.396391\pi$
$752$	−4.70323e6	−0.303286
$753$	−1.18649e6	−0.0762563
$754$	1.67400e7	1.07233
$755$	−2.20412e7	−1.40724
$756$	571536.	0.0363696
$757$	−6.41980e6	−0.407176	−0.203588	−	0.979057i	$-0.565260\pi$
$757$	−6.41980e6	−0.407176	−0.203588	+	0.979057i	$0.565260\pi$
$758$	−1.00430e7	−0.634876
$759$	−5.03370e6	−0.317163
$760$	7.44550e6	0.467585
$761$	−5.05052e6	−0.316136	−0.158068	−	0.987428i	$-0.550527\pi$
$761$	−5.05052e6	−0.316136	−0.158068	+	0.987428i	$0.550527\pi$
$762$	−6.03547e6	−0.376551
$763$	4.66411e6	0.290040
$764$	1.22136e7	0.757025
$765$	4.29106e6	0.265101
$766$	−2.23658e6	−0.137725
$767$	−3.50272e7	−2.14989
$768$	−589824.	−0.0360844
$769$	2.28169e6	0.139136	0.0695682	−	0.997577i	$-0.477838\pi$
$769$	2.28169e6	0.139136	0.0695682	+	0.997577i	$0.477838\pi$
$770$	−4.05328e6	−0.246365
$771$	1.31833e7	0.798711
$772$	−6.03203e6	−0.364267
$773$	2.73777e7	1.64797	0.823984	−	0.566613i	$-0.191746\pi$
$773$	2.73777e7	1.64797	0.823984	+	0.566613i	$0.191746\pi$
$774$	4.37141e6	0.262282
$775$	−6.67742e6	−0.399351
$776$	−4.93658e6	−0.294287
$777$	−2.85680e6	−0.169757
$778$	−1.80422e7	−1.06866
$779$	−7.55126e6	−0.445837
$780$	7.33709e6	0.431805
$781$	5.04310e6	0.295849
$782$	5.73104e6	0.335133
$783$	−2.63461e6	−0.153572
$784$	614656.	0.0357143
$785$	−1.23314e7	−0.714227
$786$	−2.99045e6	−0.172655
$787$	2.04263e7	1.17558	0.587791	−	0.809013i	$-0.299998\pi$
$787$	2.04263e7	1.17558	0.587791	+	0.809013i	$0.299998\pi$
$788$	−1.09885e6	−0.0630409
$789$	−1.33172e7	−0.761589
$790$	8.79226e6	0.501225
$791$	1.27960e7	0.727163
$792$	2.43648e6	0.138023
$793$	−2.26296e7	−1.27789
$794$	−2.07945e7	−1.17057
$795$	1.73210e6	0.0971977
$796$	2.92122e6	0.163411
$797$	2.31557e7	1.29126	0.645628	−	0.763652i	$-0.276596\pi$
$797$	2.31557e7	1.29126	0.645628	+	0.763652i	$0.276596\pi$
$798$	4.66402e6	0.259270
$799$	−2.21199e7	−1.22579
$800$	1.21754e6	0.0672600
$801$	8.16286e6	0.449532
$802$	2.53926e7	1.39403
$803$	−1.66258e7	−0.909899
$804$	−7.82323e6	−0.426821
$805$	2.56564e6	0.139542
$806$	2.60133e7	1.41045
$807$	1.69067e6	0.0913849
$808$	−6.36570e6	−0.343018
$809$	1.14894e7	0.617203	0.308601	−	0.951191i	$-0.400139\pi$
$809$	1.14894e7	0.617203	0.308601	+	0.951191i	$0.400139\pi$
$810$	−1.15474e6	−0.0618401
$811$	1.72443e7	0.920648	0.460324	−	0.887751i	$-0.347733\pi$
$811$	1.72443e7	0.920648	0.460324	+	0.887751i	$0.347733\pi$
$812$	−2.83338e6	−0.150804
$813$	−1.74420e6	−0.0925486
$814$	−1.21786e7	−0.644225
$815$	−5.28070e6	−0.278482
$816$	−2.77402e6	−0.145842
$817$	3.56728e7	1.86975
$818$	726568.	0.0379658
$819$	4.59610e6	0.239431
$820$	2.01062e6	0.104423
$821$	1.37077e6	0.0709752	0.0354876	−	0.999370i	$-0.488702\pi$
$821$	1.37077e6	0.0709752	0.0354876	+	0.999370i	$0.488702\pi$
$822$	5.84114e6	0.301522
$823$	8.56851e6	0.440967	0.220483	−	0.975391i	$-0.429237\pi$
$823$	8.56851e6	0.440967	0.220483	+	0.975391i	$0.429237\pi$
$824$	−4.28442e6	−0.219823
$825$	−5.02947e6	−0.257269
$826$	5.92861e6	0.302345
$827$	−1.33258e6	−0.0677533	−0.0338766	−	0.999426i	$-0.510785\pi$
$827$	−1.33258e6	−0.0677533	−0.0338766	+	0.999426i	$0.510785\pi$
$828$	−1.54224e6	−0.0781765
$829$	6.25659e6	0.316193	0.158096	−	0.987424i	$-0.449464\pi$
$829$	6.25659e6	0.316193	0.158096	+	0.987424i	$0.449464\pi$
$830$	4.74285e6	0.238970
$831$	5.55356e6	0.278977
$832$	−4.74317e6	−0.237553
$833$	2.89080e6	0.144346
$834$	−2.11838e6	−0.105460
$835$	2.40650e7	1.19446
$836$	1.98829e7	0.983929
$837$	−4.09406e6	−0.201995
$838$	2.25080e7	1.10720
$839$	−1.61258e6	−0.0790891	−0.0395445	−	0.999218i	$-0.512591\pi$
$839$	−1.61258e6	−0.0790891	−0.0395445	+	0.999218i	$0.512591\pi$
$840$	−1.24186e6	−0.0607258
$841$	−7.45015e6	−0.363225
$842$	1.77068e7	0.860718
$843$	1.55816e7	0.755168
$844$	3.72243e6	0.179875
$845$	4.26655e7	2.05558
$846$	5.95253e6	0.285940
$847$	−2.93260e6	−0.140457
$848$	−1.11974e6	−0.0534723
$849$	−3.20418e6	−0.152562
$850$	5.72622e6	0.271845
$851$	7.70882e6	0.364892
$852$	1.54512e6	0.0729228
$853$	−3.44919e7	−1.62310	−0.811548	−	0.584286i	$-0.801375\pi$
$853$	−3.44919e7	−1.62310	−0.811548	+	0.584286i	$0.801375\pi$
$854$	3.83023e6	0.179713
$855$	−9.42322e6	−0.440843
$856$	1.46047e7	0.681251
$857$	−1.85487e7	−0.862704	−0.431352	−	0.902184i	$-0.641963\pi$
$857$	−1.85487e7	−0.862704	−0.431352	+	0.902184i	$0.641963\pi$
$858$	1.95934e7	0.908638
$859$	−2.56435e7	−1.18575	−0.592877	−	0.805293i	$-0.702008\pi$
$859$	−2.56435e7	−1.18575	−0.592877	+	0.805293i	$0.702008\pi$
$860$	−9.49837e6	−0.437928
$861$	1.25950e6	0.0579014
$862$	5.76650e6	0.264329
$863$	−2.49899e7	−1.14219	−0.571093	−	0.820885i	$-0.693481\pi$
$863$	−2.49899e7	−1.14219	−0.571093	+	0.820885i	$0.693481\pi$
$864$	746496.	0.0340207
$865$	−1.61522e7	−0.733993
$866$	1.55865e7	0.706241
$867$	−267831.	−0.0121008
$868$	−4.40294e6	−0.198355
$869$	2.34793e7	1.05472
$870$	5.72458e6	0.256416
$871$	−6.29118e7	−2.80987
$872$	6.09190e6	0.271308
$873$	6.24785e6	0.277457
$874$	−1.25854e7	−0.557301
$875$	9.30098e6	0.410685
$876$	−5.09386e6	−0.224278
$877$	−3.46337e7	−1.52055	−0.760274	−	0.649602i	$-0.774935\pi$
$877$	−3.46337e7	−1.52055	−0.760274	+	0.649602i	$0.774935\pi$
$878$	−2.04483e7	−0.895201
$879$	−4.82998e6	−0.210850
$880$	−5.29408e6	−0.230454
$881$	2.92434e7	1.26937	0.634685	−	0.772771i	$-0.281130\pi$
$881$	2.92434e7	1.26937	0.634685	+	0.772771i	$0.281130\pi$
$882$	−777924.	−0.0336718
$883$	−3.76532e7	−1.62518	−0.812588	−	0.582839i	$-0.801942\pi$
$883$	−3.76532e7	−1.62518	−0.812588	+	0.582839i	$0.801942\pi$
$884$	−2.23077e7	−0.960117
$885$	−1.19782e7	−0.514084
$886$	−2.17628e7	−0.931388
$887$	−2.08201e6	−0.0888534	−0.0444267	−	0.999013i	$-0.514146\pi$
$887$	−2.08201e6	−0.0888534	−0.0444267	+	0.999013i	$0.514146\pi$
$888$	−3.73133e6	−0.158793
$889$	8.21495e6	0.348618
$890$	−1.77366e7	−0.750576
$891$	−3.08367e6	−0.130129
$892$	2.67430e6	0.112538
$893$	4.85756e7	2.03840
$894$	1.55051e7	0.648831
$895$	−3.91116e6	−0.163210
$896$	802816.	0.0334077
$897$	−1.24022e7	−0.514656
$898$	5.26098e6	0.217709
$899$	2.02962e7	0.837560
$900$	−1.54094e6	−0.0634133
$901$	−5.26630e6	−0.216119
$902$	5.36928e6	0.219735
$903$	−5.94997e6	−0.242826
$904$	1.67131e7	0.680198
$905$	−3.44132e7	−1.39670
$906$	−1.80337e7	−0.729901
$907$	1.62350e7	0.655291	0.327645	−	0.944801i	$-0.393745\pi$
$907$	1.62350e7	0.655291	0.327645	+	0.944801i	$0.393745\pi$
$908$	6.65165e6	0.267741
$909$	8.05658e6	0.323401
$910$	−9.98659e6	−0.399773
$911$	−2.58656e7	−1.03259	−0.516294	−	0.856412i	$-0.672689\pi$
$911$	−2.58656e7	−1.03259	−0.516294	+	0.856412i	$0.672689\pi$
$912$	6.09178e6	0.242525
$913$	1.26656e7	0.502860
$914$	−1.11042e7	−0.439664
$915$	−7.73863e6	−0.305571
$916$	7.57571e6	0.298322
$917$	4.07033e6	0.159848
$918$	3.51086e6	0.137501
$919$	−1.23266e7	−0.481453	−0.240726	−	0.970593i	$-0.577386\pi$
$919$	−1.23266e7	−0.481453	−0.240726	+	0.970593i	$0.577386\pi$
$920$	3.35104e6	0.130530
$921$	2.59558e7	1.00829
$922$	−552320.	−0.0213975
$923$	1.24253e7	0.480069
$924$	−3.31632e6	−0.127784
$925$	7.70234e6	0.295984
$926$	1.45630e6	0.0558116
$927$	5.42246e6	0.207251
$928$	−3.70074e6	−0.141065
$929$	−4.15128e7	−1.57813	−0.789064	−	0.614310i	$-0.789434\pi$
$929$	−4.15128e7	−1.57813	−0.789064	+	0.614310i	$0.789434\pi$
$930$	8.89574e6	0.337268
$931$	−6.34824e6	−0.240038
$932$	−2.49047e7	−0.939166
$933$	9.26622e6	0.348497
$934$	2.29559e7	0.861046
$935$	−2.48987e7	−0.931425
$936$	6.00307e6	0.223967
$937$	2.26895e7	0.844260	0.422130	−	0.906535i	$-0.361283\pi$
$937$	2.26895e7	0.844260	0.422130	+	0.906535i	$0.361283\pi$
$938$	1.06483e7	0.395160
$939$	−1.25367e7	−0.464002
$940$	−1.29339e7	−0.477430
$941$	1.58213e7	0.582464	0.291232	−	0.956652i	$-0.405935\pi$
$941$	1.58213e7	0.582464	0.291232	+	0.956652i	$0.405935\pi$
$942$	−1.00893e7	−0.370453
$943$	−3.39864e6	−0.124459
$944$	7.74349e6	0.282818
$945$	1.57172e6	0.0572528
$946$	−2.53650e7	−0.921523
$947$	5.17579e7	1.87543	0.937716	−	0.347402i	$-0.112936\pi$
$947$	5.17579e7	1.87543	0.937716	+	0.347402i	$0.112936\pi$
$948$	7.19366e6	0.259974
$949$	−4.09631e7	−1.47648
$950$	−1.25749e7	−0.452058
$951$	7.02027e6	0.251711
$952$	3.77574e6	0.135024
$953$	2.29818e7	0.819695	0.409848	−	0.912154i	$-0.365582\pi$
$953$	2.29818e7	0.819695	0.409848	+	0.912154i	$0.365582\pi$
$954$	1.41718e6	0.0504142
$955$	3.35874e7	1.19170
$956$	1.04942e7	0.371370
$957$	1.52872e7	0.539571
$958$	−2.20798e7	−0.777288
$959$	−7.95045e6	−0.279155
$960$	−1.62202e6	−0.0568038
$961$	2.91030e6	0.101655
$962$	−3.00061e7	−1.04537
$963$	−1.84840e7	−0.642290
$964$	−1.42316e7	−0.493243
$965$	−1.65881e7	−0.573427
$966$	2.09916e6	0.0723774
$967$	3.20783e7	1.10318	0.551588	−	0.834117i	$-0.314022\pi$
$967$	3.20783e7	1.10318	0.551588	+	0.834117i	$0.314022\pi$
$968$	−3.83034e6	−0.131386
$969$	2.86504e7	0.980214
$970$	−1.35756e7	−0.463265
$971$	7.31101e6	0.248845	0.124423	−	0.992229i	$-0.460292\pi$
$971$	7.31101e6	0.248845	0.124423	+	0.992229i	$0.460292\pi$
$972$	−944784.	−0.0320750
$973$	2.88336e6	0.0976374
$974$	−1.69209e7	−0.571513
$975$	−1.23918e7	−0.417466
$976$	5.00275e6	0.168106
$977$	7.58600e6	0.254259	0.127129	−	0.991886i	$-0.459424\pi$
$977$	7.58600e6	0.254259	0.127129	+	0.991886i	$0.459424\pi$
$978$	−4.32058e6	−0.144442
$979$	−4.73647e7	−1.57942
$980$	1.69030e6	0.0562211
$981$	−7.71007e6	−0.255791
$982$	2.88569e7	0.954929
$983$	−2.47823e7	−0.818007	−0.409004	−	0.912533i	$-0.634124\pi$
$983$	−2.47823e7	−0.818007	−0.409004	+	0.912533i	$0.634124\pi$
$984$	1.64506e6	0.0541618
$985$	−3.02183e6	−0.0992384
$986$	−1.74050e7	−0.570141
$987$	−8.10205e6	−0.264729
$988$	4.89880e7	1.59661
$989$	1.60555e7	0.521954
$990$	6.70032e6	0.217274
$991$	−7.63530e6	−0.246969	−0.123484	−	0.992347i	$-0.539407\pi$
$991$	−7.63530e6	−0.246969	−0.123484	+	0.992347i	$0.539407\pi$
$992$	−5.75078e6	−0.185544
$993$	1.27084e7	0.408995
$994$	−2.10308e6	−0.0675134
$995$	8.03334e6	0.257240
$996$	3.88051e6	0.123948
$997$	−2.89785e7	−0.923289	−0.461644	−	0.887065i	$-0.652740\pi$
$997$	−2.89785e7	−0.923289	−0.461644	+	0.887065i	$0.652740\pi$
$998$	899216.	0.0285784
$999$	4.72246e6	0.149711

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	42.6.a.b.1.1	✓	1
3.2	odd	2		126.6.a.h.1.1		1
4.3	odd	2		336.6.a.o.1.1		1
5.2	odd	4		1050.6.g.l.799.1		2
5.3	odd	4		1050.6.g.l.799.2		2
5.4	even	2		1050.6.a.o.1.1		1
7.2	even	3		294.6.e.o.67.1		2
7.3	odd	6		294.6.e.k.79.1		2
7.4	even	3		294.6.e.o.79.1		2
7.5	odd	6		294.6.e.k.67.1		2
7.6	odd	2		294.6.a.f.1.1		1
12.11	even	2		1008.6.a.g.1.1		1
21.20	even	2		882.6.a.v.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.6.a.b.1.1	✓	1	1.1	even	1	trivial
126.6.a.h.1.1		1	3.2	odd	2
294.6.a.f.1.1		1	7.6	odd	2
294.6.e.k.67.1		2	7.5	odd	6
294.6.e.k.79.1		2	7.3	odd	6
294.6.e.o.67.1		2	7.2	even	3
294.6.e.o.79.1		2	7.4	even	3
336.6.a.o.1.1		1	4.3	odd	2
882.6.a.v.1.1		1	21.20	even	2
1008.6.a.g.1.1		1	12.11	even	2
1050.6.a.o.1.1		1	5.4	even	2
1050.6.g.l.799.1		2	5.2	odd	4
1050.6.g.l.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 $\F_{q}$

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

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