LMFDB - Embedded newform 42.6.a.f.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:	$0$
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)

$f(q)$

$=$

$q+4.00000 q^{2} +9.00000 q^{3} +16.0000 q^{4} +24.0000 q^{5} +36.0000 q^{6} +49.0000 q^{7} +64.0000 q^{8} +81.0000 q^{9} +O(q^{10})$

\(q+4.00000 q^{2} +9.00000 q^{3} +16.0000 q^{4} +24.0000 q^{5} +36.0000 q^{6} +49.0000 q^{7} +64.0000 q^{8} +81.0000 q^{9} +96.0000 q^{10} +66.0000 q^{11} +144.000 q^{12} +98.0000 q^{13} +196.000 q^{14} +216.000 q^{15} +256.000 q^{16} -216.000 q^{17} +324.000 q^{18} -340.000 q^{19} +384.000 q^{20} +441.000 q^{21} +264.000 q^{22} -1038.00 q^{23} +576.000 q^{24} -2549.00 q^{25} +392.000 q^{26} +729.000 q^{27} +784.000 q^{28} -2490.00 q^{29} +864.000 q^{30} -7048.00 q^{31} +1024.00 q^{32} +594.000 q^{33} -864.000 q^{34} +1176.00 q^{35} +1296.00 q^{36} -12238.0 q^{37} -1360.00 q^{38} +882.000 q^{39} +1536.00 q^{40} +6468.00 q^{41} +1764.00 q^{42} -15412.0 q^{43} +1056.00 q^{44} +1944.00 q^{45} -4152.00 q^{46} +20604.0 q^{47} +2304.00 q^{48} +2401.00 q^{49} -10196.0 q^{50} -1944.00 q^{51} +1568.00 q^{52} +32490.0 q^{53} +2916.00 q^{54} +1584.00 q^{55} +3136.00 q^{56} -3060.00 q^{57} -9960.00 q^{58} +34224.0 q^{59} +3456.00 q^{60} +35654.0 q^{61} -28192.0 q^{62} +3969.00 q^{63} +4096.00 q^{64} +2352.00 q^{65} +2376.00 q^{66} +12680.0 q^{67} -3456.00 q^{68} -9342.00 q^{69} +4704.00 q^{70} -42642.0 q^{71} +5184.00 q^{72} +33734.0 q^{73} -48952.0 q^{74} -22941.0 q^{75} -5440.00 q^{76} +3234.00 q^{77} +3528.00 q^{78} -85108.0 q^{79} +6144.00 q^{80} +6561.00 q^{81} +25872.0 q^{82} -106764. q^{83} +7056.00 q^{84} -5184.00 q^{85} -61648.0 q^{86} -22410.0 q^{87} +4224.00 q^{88} +34884.0 q^{89} +7776.00 q^{90} +4802.00 q^{91} -16608.0 q^{92} -63432.0 q^{93} +82416.0 q^{94} -8160.00 q^{95} +9216.00 q^{96} +18662.0 q^{97} +9604.00 q^{98} +5346.00 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	4.00000	0.707107
$2$	4.00000	0.707107
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	16.0000	0.500000
$5$	24.0000	0.429325	0.214663	−	0.976688i	$-0.431135\pi$
$5$	24.0000	0.429325	0.214663	+	0.976688i	$0.431135\pi$
$6$	36.0000	0.408248
$7$	49.0000	0.377964
$7$	49.0000	0.377964
$8$	64.0000	0.353553
$9$	81.0000	0.333333
$10$	96.0000	0.303579
$11$	66.0000	0.164461	0.0822304	−	0.996613i	$-0.473796\pi$
$11$	66.0000	0.164461	0.0822304	+	0.996613i	$0.473796\pi$
$12$	144.000	0.288675
$13$	98.0000	0.160830	0.0804151	−	0.996761i	$-0.474375\pi$
$13$	98.0000	0.160830	0.0804151	+	0.996761i	$0.474375\pi$
$14$	196.000	0.267261
$15$	216.000	0.247871
$16$	256.000	0.250000
$17$	−216.000	−0.181272	−0.0906362	−	0.995884i	$-0.528890\pi$
$17$	−216.000	−0.181272	−0.0906362	+	0.995884i	$0.528890\pi$
$18$	324.000	0.235702
$19$	−340.000	−0.216070	−0.108035	−	0.994147i	$-0.534456\pi$
$19$	−340.000	−0.216070	−0.108035	+	0.994147i	$0.534456\pi$
$20$	384.000	0.214663
$21$	441.000	0.218218
$22$	264.000	0.116291
$23$	−1038.00	−0.409145	−0.204573	−	0.978851i	$-0.565580\pi$
$23$	−1038.00	−0.409145	−0.204573	+	0.978851i	$0.565580\pi$
$24$	576.000	0.204124
$25$	−2549.00	−0.815680
$26$	392.000	0.113724
$27$	729.000	0.192450
$28$	784.000	0.188982
$29$	−2490.00	−0.549800	−0.274900	−	0.961473i	$-0.588645\pi$
$29$	−2490.00	−0.549800	−0.274900	+	0.961473i	$0.588645\pi$
$30$	864.000	0.175271
$31$	−7048.00	−1.31723	−0.658615	−	0.752480i	$-0.728857\pi$
$31$	−7048.00	−1.31723	−0.658615	+	0.752480i	$0.728857\pi$
$32$	1024.00	0.176777
$33$	594.000	0.0949514
$34$	−864.000	−0.128179
$35$	1176.00	0.162270
$36$	1296.00	0.166667
$37$	−12238.0	−1.46962	−0.734812	−	0.678271i	$-0.762730\pi$
$37$	−12238.0	−1.46962	−0.734812	+	0.678271i	$0.762730\pi$
$38$	−1360.00	−0.152785
$39$	882.000	0.0928554
$40$	1536.00	0.151789
$41$	6468.00	0.600911	0.300456	−	0.953796i	$-0.402861\pi$
$41$	6468.00	0.600911	0.300456	+	0.953796i	$0.402861\pi$
$42$	1764.00	0.154303
$43$	−15412.0	−1.27112	−0.635562	−	0.772050i	$-0.719232\pi$
$43$	−15412.0	−1.27112	−0.635562	+	0.772050i	$0.719232\pi$
$44$	1056.00	0.0822304
$45$	1944.00	0.143108
$46$	−4152.00	−0.289310
$47$	20604.0	1.36053	0.680263	−	0.732968i	$-0.261866\pi$
$47$	20604.0	1.36053	0.680263	+	0.732968i	$0.261866\pi$
$48$	2304.00	0.144338
$49$	2401.00	0.142857
$50$	−10196.0	−0.576773
$51$	−1944.00	−0.104658
$52$	1568.00	0.0804151
$53$	32490.0	1.58877	0.794383	−	0.607417i	$-0.207794\pi$
$53$	32490.0	1.58877	0.794383	+	0.607417i	$0.207794\pi$
$54$	2916.00	0.136083
$55$	1584.00	0.0706071
$56$	3136.00	0.133631
$57$	−3060.00	−0.124748
$58$	−9960.00	−0.388767
$59$	34224.0	1.27997	0.639986	−	0.768386i	$-0.278940\pi$
$59$	34224.0	1.27997	0.639986	+	0.768386i	$0.278940\pi$
$60$	3456.00	0.123935
$61$	35654.0	1.22683	0.613414	−	0.789762i	$-0.289796\pi$
$61$	35654.0	1.22683	0.613414	+	0.789762i	$0.289796\pi$
$62$	−28192.0	−0.931422
$63$	3969.00	0.125988
$64$	4096.00	0.125000
$65$	2352.00	0.0690484
$66$	2376.00	0.0671408
$67$	12680.0	0.345090	0.172545	−	0.985002i	$-0.444801\pi$
$67$	12680.0	0.345090	0.172545	+	0.985002i	$0.444801\pi$
$68$	−3456.00	−0.0906362
$69$	−9342.00	−0.236220
$70$	4704.00	0.114742
$71$	−42642.0	−1.00390	−0.501951	−	0.864896i	$-0.667384\pi$
$71$	−42642.0	−1.00390	−0.501951	+	0.864896i	$0.667384\pi$
$72$	5184.00	0.117851
$73$	33734.0	0.740902	0.370451	−	0.928852i	$-0.379203\pi$
$73$	33734.0	0.740902	0.370451	+	0.928852i	$0.379203\pi$
$74$	−48952.0	−1.03918
$75$	−22941.0	−0.470933
$76$	−5440.00	−0.108035
$77$	3234.00	0.0621603
$78$	3528.00	0.0656587
$79$	−85108.0	−1.53427	−0.767137	−	0.641484i	$-0.778319\pi$
$79$	−85108.0	−1.53427	−0.767137	+	0.641484i	$0.778319\pi$
$80$	6144.00	0.107331
$81$	6561.00	0.111111
$82$	25872.0	0.424908
$83$	−106764.	−1.70110	−0.850550	−	0.525895i	$-0.823730\pi$
$83$	−106764.	−1.70110	−0.850550	+	0.525895i	$0.823730\pi$
$84$	7056.00	0.109109
$85$	−5184.00	−0.0778247
$86$	−61648.0	−0.898820
$87$	−22410.0	−0.317427
$88$	4224.00	0.0581456
$89$	34884.0	0.466822	0.233411	−	0.972378i	$-0.425011\pi$
$89$	34884.0	0.466822	0.233411	+	0.972378i	$0.425011\pi$
$90$	7776.00	0.101193
$91$	4802.00	0.0607881
$92$	−16608.0	−0.204573
$93$	−63432.0	−0.760503
$94$	82416.0	0.962037
$95$	−8160.00	−0.0927644
$96$	9216.00	0.102062
$97$	18662.0	0.201386	0.100693	−	0.994918i	$-0.467894\pi$
$97$	18662.0	0.201386	0.100693	+	0.994918i	$0.467894\pi$
$98$	9604.00	0.101015
$99$	5346.00	0.0548202
$100$	−40784.0	−0.407840
$101$	153084.	1.49323	0.746614	−	0.665257i	$-0.231678\pi$
$101$	153084.	1.49323	0.746614	+	0.665257i	$0.231678\pi$
$102$	−7776.00	−0.0740041
$103$	35864.0	0.333093	0.166547	−	0.986034i	$-0.446738\pi$
$103$	35864.0	0.333093	0.166547	+	0.986034i	$0.446738\pi$
$104$	6272.00	0.0568621
$105$	10584.0	0.0936864
$106$	129960.	1.12343
$107$	−95454.0	−0.805999	−0.403000	−	0.915200i	$-0.632032\pi$
$107$	−95454.0	−0.805999	−0.403000	+	0.915200i	$0.632032\pi$
$108$	11664.0	0.0962250
$109$	212222.	1.71090	0.855449	−	0.517887i	$-0.173281\pi$
$109$	212222.	1.71090	0.855449	+	0.517887i	$0.173281\pi$
$110$	6336.00	0.0499268
$111$	−110142.	−0.848488
$112$	12544.0	0.0944911
$113$	62106.0	0.457549	0.228774	−	0.973479i	$-0.426528\pi$
$113$	62106.0	0.457549	0.228774	+	0.973479i	$0.426528\pi$
$114$	−12240.0	−0.0882103
$115$	−24912.0	−0.175656
$116$	−39840.0	−0.274900
$117$	7938.00	0.0536101
$118$	136896.	0.905077
$119$	−10584.0	−0.0685145
$120$	13824.0	0.0876356
$121$	−156695.	−0.972953
$122$	142616.	0.867498
$123$	58212.0	0.346936
$124$	−112768.	−0.658615
$125$	−136176.	−0.779517
$126$	15876.0	0.0890871
$127$	−53044.0	−0.291828	−0.145914	−	0.989297i	$-0.546612\pi$
$127$	−53044.0	−0.291828	−0.145914	+	0.989297i	$0.546612\pi$
$128$	16384.0	0.0883883
$129$	−138708.	−0.733884
$130$	9408.00	0.0488246
$131$	69324.0	0.352944	0.176472	−	0.984306i	$-0.443532\pi$
$131$	69324.0	0.352944	0.176472	+	0.984306i	$0.443532\pi$
$132$	9504.00	0.0474757
$133$	−16660.0	−0.0816669
$134$	50720.0	0.244015
$135$	17496.0	0.0826236
$136$	−13824.0	−0.0640894
$137$	129846.	0.591054	0.295527	−	0.955334i	$-0.404505\pi$
$137$	129846.	0.591054	0.295527	+	0.955334i	$0.404505\pi$
$138$	−37368.0	−0.167033
$139$	−104356.	−0.458121	−0.229061	−	0.973412i	$-0.573565\pi$
$139$	−104356.	−0.458121	−0.229061	+	0.973412i	$0.573565\pi$
$140$	18816.0	0.0811348
$141$	185436.	0.785500
$142$	−170568.	−0.709867
$143$	6468.00	0.0264503
$144$	20736.0	0.0833333
$145$	−59760.0	−0.236043
$146$	134936.	0.523897
$147$	21609.0	0.0824786
$148$	−195808.	−0.734812
$149$	217194.	0.801461	0.400730	−	0.916196i	$-0.368756\pi$
$149$	217194.	0.801461	0.400730	+	0.916196i	$0.368756\pi$
$150$	−91764.0	−0.333000
$151$	221000.	0.788769	0.394385	−	0.918945i	$-0.370958\pi$
$151$	221000.	0.788769	0.394385	+	0.918945i	$0.370958\pi$
$152$	−21760.0	−0.0763924
$153$	−17496.0	−0.0604241
$154$	12936.0	0.0439540
$155$	−169152.	−0.565520
$156$	14112.0	0.0464277
$157$	−378370.	−1.22509	−0.612544	−	0.790436i	$-0.709854\pi$
$157$	−378370.	−1.22509	−0.612544	+	0.790436i	$0.709854\pi$
$158$	−340432.	−1.08489
$159$	292410.	0.917275
$160$	24576.0	0.0758947
$161$	−50862.0	−0.154642
$162$	26244.0	0.0785674
$163$	104816.	0.309000	0.154500	−	0.987993i	$-0.450623\pi$
$163$	104816.	0.309000	0.154500	+	0.987993i	$0.450623\pi$
$164$	103488.	0.300456
$165$	14256.0	0.0407650
$166$	−427056.	−1.20286
$167$	−426972.	−1.18470	−0.592350	−	0.805681i	$-0.701800\pi$
$167$	−426972.	−1.18470	−0.592350	+	0.805681i	$0.701800\pi$
$168$	28224.0	0.0771517
$169$	−361689.	−0.974134
$170$	−20736.0	−0.0550304
$171$	−27540.0	−0.0720234
$172$	−246592.	−0.635562
$173$	331068.	0.841012	0.420506	−	0.907290i	$-0.361853\pi$
$173$	331068.	0.841012	0.420506	+	0.907290i	$0.361853\pi$
$174$	−89640.0	−0.224455
$175$	−124901.	−0.308298
$176$	16896.0	0.0411152
$177$	308016.	0.738993
$178$	139536.	0.330093
$179$	−400194.	−0.933551	−0.466775	−	0.884376i	$-0.654584\pi$
$179$	−400194.	−0.933551	−0.466775	+	0.884376i	$0.654584\pi$
$180$	31104.0	0.0715542
$181$	588098.	1.33430	0.667150	−	0.744924i	$-0.267514\pi$
$181$	588098.	1.33430	0.667150	+	0.744924i	$0.267514\pi$
$182$	19208.0	0.0429837
$183$	320886.	0.708309
$184$	−66432.0	−0.144655
$185$	−293712.	−0.630946
$186$	−253728.	−0.537757
$187$	−14256.0	−0.0298122
$188$	329664.	0.680263
$189$	35721.0	0.0727393
$190$	−32640.0	−0.0655943
$191$	939342.	1.86312	0.931559	−	0.363590i	$-0.118449\pi$
$191$	939342.	1.86312	0.931559	+	0.363590i	$0.118449\pi$
$192$	36864.0	0.0721688
$193$	338390.	0.653919	0.326960	−	0.945038i	$-0.393976\pi$
$193$	338390.	0.653919	0.326960	+	0.945038i	$0.393976\pi$
$194$	74648.0	0.142401
$195$	21168.0	0.0398651
$196$	38416.0	0.0714286
$197$	−237942.	−0.436823	−0.218412	−	0.975857i	$-0.570088\pi$
$197$	−237942.	−0.436823	−0.218412	+	0.975857i	$0.570088\pi$
$198$	21384.0	0.0387638
$199$	204464.	0.366003	0.183001	−	0.983113i	$-0.441419\pi$
$199$	204464.	0.366003	0.183001	+	0.983113i	$0.441419\pi$
$200$	−163136.	−0.288386
$201$	114120.	0.199238
$202$	612336.	1.05587
$203$	−122010.	−0.207805
$204$	−31104.0	−0.0523288
$205$	155232.	0.257986
$206$	143456.	0.235532
$207$	−84078.0	−0.136382
$208$	25088.0	0.0402076
$209$	−22440.0	−0.0355351
$210$	42336.0	0.0662463
$211$	−348724.	−0.539232	−0.269616	−	0.962968i	$-0.586897\pi$
$211$	−348724.	−0.539232	−0.269616	+	0.962968i	$0.586897\pi$
$212$	519840.	0.794383
$213$	−383778.	−0.579604
$214$	−381816.	−0.569928
$215$	−369888.	−0.545725
$216$	46656.0	0.0680414
$217$	−345352.	−0.497866
$218$	848888.	1.20979
$219$	303606.	0.427760
$220$	25344.0	0.0353036
$221$	−21168.0	−0.0291541
$222$	−440568.	−0.599971
$223$	1.47006e6	1.97957	0.989787	−	0.142554i	$-0.0455316\pi$
$223$	1.47006e6	1.97957	0.989787	+	0.142554i	$0.0455316\pi$
$224$	50176.0	0.0668153
$225$	−206469.	−0.271893
$226$	248424.	0.323536
$227$	−589560.	−0.759387	−0.379694	−	0.925112i	$-0.623971\pi$
$227$	−589560.	−0.759387	−0.379694	+	0.925112i	$0.623971\pi$
$228$	−48960.0	−0.0623741
$229$	−1.04534e6	−1.31725	−0.658627	−	0.752469i	$-0.728863\pi$
$229$	−1.04534e6	−1.31725	−0.658627	+	0.752469i	$0.728863\pi$
$230$	−99648.0	−0.124208
$231$	29106.0	0.0358883
$232$	−159360.	−0.194383
$233$	651222.	0.785849	0.392925	−	0.919571i	$-0.371463\pi$
$233$	651222.	0.785849	0.392925	+	0.919571i	$0.371463\pi$
$234$	31752.0	0.0379080
$235$	494496.	0.584108
$236$	547584.	0.639986
$237$	−765972.	−0.885813
$238$	−42336.0	−0.0484471
$239$	−513462.	−0.581452	−0.290726	−	0.956806i	$-0.593897\pi$
$239$	−513462.	−0.581452	−0.290726	+	0.956806i	$0.593897\pi$
$240$	55296.0	0.0619677
$241$	−694714.	−0.770484	−0.385242	−	0.922816i	$-0.625882\pi$
$241$	−694714.	−0.770484	−0.385242	+	0.922816i	$0.625882\pi$
$242$	−626780.	−0.687981
$243$	59049.0	0.0641500
$244$	570464.	0.613414
$245$	57624.0	0.0613322
$246$	232848.	0.245321
$247$	−33320.0	−0.0347506
$248$	−451072.	−0.465711
$249$	−960876.	−0.982130
$250$	−544704.	−0.551202
$251$	−1.39608e6	−1.39870	−0.699352	−	0.714777i	$-0.746528\pi$
$251$	−1.39608e6	−1.39870	−0.699352	+	0.714777i	$0.746528\pi$
$252$	63504.0	0.0629941
$253$	−68508.0	−0.0672884
$254$	−212176.	−0.206354
$255$	−46656.0	−0.0449321
$256$	65536.0	0.0625000
$257$	−1.00520e6	−0.949339	−0.474670	−	0.880164i	$-0.657432\pi$
$257$	−1.00520e6	−0.949339	−0.474670	+	0.880164i	$0.657432\pi$
$258$	−554832.	−0.518934
$259$	−599662.	−0.555466
$260$	37632.0	0.0345242
$261$	−201690.	−0.183267
$262$	277296.	0.249569
$263$	1.25301e6	1.11703	0.558515	−	0.829494i	$-0.311371\pi$
$263$	1.25301e6	1.11703	0.558515	+	0.829494i	$0.311371\pi$
$264$	38016.0	0.0335704
$265$	779760.	0.682097
$266$	−66640.0	−0.0577472
$267$	313956.	0.269520
$268$	202880.	0.172545
$269$	−1.76069e6	−1.48355	−0.741774	−	0.670650i	$-0.766015\pi$
$269$	−1.76069e6	−1.48355	−0.741774	+	0.670650i	$0.766015\pi$
$270$	69984.0	0.0584237
$271$	770528.	0.637331	0.318666	−	0.947867i	$-0.396765\pi$
$271$	770528.	0.637331	0.318666	+	0.947867i	$0.396765\pi$
$272$	−55296.0	−0.0453181
$273$	43218.0	0.0350960
$274$	519384.	0.417938
$275$	−168234.	−0.134147
$276$	−149472.	−0.118110
$277$	707738.	0.554208	0.277104	−	0.960840i	$-0.410625\pi$
$277$	707738.	0.554208	0.277104	+	0.960840i	$0.410625\pi$
$278$	−417424.	−0.323941
$279$	−570888.	−0.439077
$280$	75264.0	0.0573710
$281$	2.30432e6	1.74091	0.870456	−	0.492247i	$-0.163824\pi$
$281$	2.30432e6	1.74091	0.870456	+	0.492247i	$0.163824\pi$
$282$	741744.	0.555432
$283$	1.60903e6	1.19426	0.597128	−	0.802146i	$-0.296308\pi$
$283$	1.60903e6	1.19426	0.597128	+	0.802146i	$0.296308\pi$
$284$	−682272.	−0.501951
$285$	−73440.0	−0.0535575
$286$	25872.0	0.0187032
$287$	316932.	0.227123
$288$	82944.0	0.0589256
$289$	−1.37320e6	−0.967140
$290$	−239040.	−0.166907
$291$	167958.	0.116270
$292$	539744.	0.370451
$293$	517020.	0.351834	0.175917	−	0.984405i	$-0.443711\pi$
$293$	517020.	0.351834	0.175917	+	0.984405i	$0.443711\pi$
$294$	86436.0	0.0583212
$295$	821376.	0.549524
$296$	−783232.	−0.519590
$297$	48114.0	0.0316505
$298$	868776.	0.566718
$299$	−101724.	−0.0658030
$300$	−367056.	−0.235467
$301$	−755188.	−0.480440
$302$	884000.	0.557744
$303$	1.37776e6	0.862116
$304$	−87040.0	−0.0540176
$305$	855696.	0.526708
$306$	−69984.0	−0.0427263
$307$	1.35002e6	0.817512	0.408756	−	0.912644i	$-0.365963\pi$
$307$	1.35002e6	0.817512	0.408756	+	0.912644i	$0.365963\pi$
$308$	51744.0	0.0310802
$309$	322776.	0.192311
$310$	−676608.	−0.399883
$311$	1.34538e6	0.788758	0.394379	−	0.918948i	$-0.370960\pi$
$311$	1.34538e6	0.788758	0.394379	+	0.918948i	$0.370960\pi$
$312$	56448.0	0.0328293
$313$	256154.	0.147788	0.0738942	−	0.997266i	$-0.476457\pi$
$313$	256154.	0.147788	0.0738942	+	0.997266i	$0.476457\pi$
$314$	−1.51348e6	−0.866269
$315$	95256.0	0.0540899
$316$	−1.36173e6	−0.767137
$317$	1.84629e6	1.03193	0.515967	−	0.856609i	$-0.327433\pi$
$317$	1.84629e6	1.03193	0.515967	+	0.856609i	$0.327433\pi$
$318$	1.16964e6	0.648611
$319$	−164340.	−0.0904204
$320$	98304.0	0.0536656
$321$	−859086.	−0.465344
$322$	−203448.	−0.109349
$323$	73440.0	0.0391675
$324$	104976.	0.0555556
$325$	−249802.	−0.131186
$326$	419264.	0.218496
$327$	1.91000e6	0.987788
$328$	413952.	0.212454
$329$	1.00960e6	0.514231
$330$	57024.0	0.0288252
$331$	−3.33238e6	−1.67180	−0.835900	−	0.548881i	$-0.815054\pi$
$331$	−3.33238e6	−1.67180	−0.835900	+	0.548881i	$0.815054\pi$
$332$	−1.70822e6	−0.850550
$333$	−991278.	−0.489875
$334$	−1.70789e6	−0.837709
$335$	304320.	0.148156
$336$	112896.	0.0545545
$337$	−1.63481e6	−0.784136	−0.392068	−	0.919936i	$-0.628240\pi$
$337$	−1.63481e6	−0.784136	−0.392068	+	0.919936i	$0.628240\pi$
$338$	−1.44676e6	−0.688816
$339$	558954.	0.264166
$340$	−82944.0	−0.0389124
$341$	−465168.	−0.216633
$342$	−110160.	−0.0509282
$343$	117649.	0.0539949
$344$	−986368.	−0.449410
$345$	−224208.	−0.101415
$346$	1.32427e6	0.594685
$347$	−841530.	−0.375185	−0.187593	−	0.982247i	$-0.560068\pi$
$347$	−841530.	−0.375185	−0.187593	+	0.982247i	$0.560068\pi$
$348$	−358560.	−0.158713
$349$	−977242.	−0.429476	−0.214738	−	0.976672i	$-0.568890\pi$
$349$	−977242.	−0.429476	−0.214738	+	0.976672i	$0.568890\pi$
$350$	−499604.	−0.218000
$351$	71442.0	0.0309518
$352$	67584.0	0.0290728
$353$	3.45857e6	1.47727	0.738634	−	0.674106i	$-0.235471\pi$
$353$	3.45857e6	1.47727	0.738634	+	0.674106i	$0.235471\pi$
$354$	1.23206e6	0.522547
$355$	−1.02341e6	−0.431001
$356$	558144.	0.233411
$357$	−95256.0	−0.0395569
$358$	−1.60078e6	−0.660120
$359$	−3.47301e6	−1.42223	−0.711115	−	0.703076i	$-0.751810\pi$
$359$	−3.47301e6	−1.42223	−0.711115	+	0.703076i	$0.751810\pi$
$360$	124416.	0.0505964
$361$	−2.36050e6	−0.953314
$362$	2.35239e6	0.943492
$363$	−1.41026e6	−0.561734
$364$	76832.0	0.0303941
$365$	809616.	0.318088
$366$	1.28354e6	0.500850
$367$	3.11994e6	1.20915	0.604575	−	0.796548i	$-0.293343\pi$
$367$	3.11994e6	1.20915	0.604575	+	0.796548i	$0.293343\pi$
$368$	−265728.	−0.102286
$369$	523908.	0.200304
$370$	−1.17485e6	−0.446146
$371$	1.59201e6	0.600497
$372$	−1.01491e6	−0.380252
$373$	−2.01673e6	−0.750543	−0.375272	−	0.926915i	$-0.622451\pi$
$373$	−2.01673e6	−0.750543	−0.375272	+	0.926915i	$0.622451\pi$
$374$	−57024.0	−0.0210804
$375$	−1.22558e6	−0.450054
$376$	1.31866e6	0.481019
$377$	−244020.	−0.0884244
$378$	142884.	0.0514344
$379$	−5.38083e6	−1.92420	−0.962102	−	0.272690i	$-0.912087\pi$
$379$	−5.38083e6	−1.92420	−0.962102	+	0.272690i	$0.912087\pi$
$380$	−130560.	−0.0463822
$381$	−477396.	−0.168487
$382$	3.75737e6	1.31742
$383$	807432.	0.281261	0.140630	−	0.990062i	$-0.455087\pi$
$383$	807432.	0.281261	0.140630	+	0.990062i	$0.455087\pi$
$384$	147456.	0.0510310
$385$	77616.0	0.0266870
$386$	1.35356e6	0.462391
$387$	−1.24837e6	−0.423708
$388$	298592.	0.100693
$389$	891390.	0.298671	0.149336	−	0.988787i	$-0.452286\pi$
$389$	891390.	0.298671	0.149336	+	0.988787i	$0.452286\pi$
$390$	84672.0	0.0281889
$391$	224208.	0.0741667
$392$	153664.	0.0505076
$393$	623916.	0.203772
$394$	−951768.	−0.308881
$395$	−2.04259e6	−0.658702
$396$	85536.0	0.0274101
$397$	1.12345e6	0.357749	0.178875	−	0.983872i	$-0.442754\pi$
$397$	1.12345e6	0.357749	0.178875	+	0.983872i	$0.442754\pi$
$398$	817856.	0.258803
$399$	−149940.	−0.0471504
$400$	−652544.	−0.203920
$401$	1.72037e6	0.534271	0.267136	−	0.963659i	$-0.413923\pi$
$401$	1.72037e6	0.534271	0.267136	+	0.963659i	$0.413923\pi$
$402$	456480.	0.140882
$403$	−690704.	−0.211850
$404$	2.44934e6	0.746614
$405$	157464.	0.0477028
$406$	−488040.	−0.146940
$407$	−807708.	−0.241695
$408$	−124416.	−0.0370021
$409$	77246.0	0.0228332	0.0114166	−	0.999935i	$-0.496366\pi$
$409$	77246.0	0.0228332	0.0114166	+	0.999935i	$0.496366\pi$
$410$	620928.	0.182424
$411$	1.16861e6	0.341245
$412$	573824.	0.166547
$413$	1.67698e6	0.483784
$414$	−336312.	−0.0964365
$415$	−2.56234e6	−0.730324
$416$	100352.	0.0284310
$417$	−939204.	−0.264496
$418$	−89760.0	−0.0251271
$419$	−5.20615e6	−1.44871	−0.724356	−	0.689427i	$-0.757863\pi$
$419$	−5.20615e6	−1.44871	−0.724356	+	0.689427i	$0.757863\pi$
$420$	169344.	0.0468432
$421$	1.71847e6	0.472539	0.236270	−	0.971688i	$-0.424075\pi$
$421$	1.71847e6	0.472539	0.236270	+	0.971688i	$0.424075\pi$
$422$	−1.39490e6	−0.381295
$423$	1.66892e6	0.453509
$424$	2.07936e6	0.561714
$425$	550584.	0.147860
$426$	−1.53511e6	−0.409842
$427$	1.74705e6	0.463697
$428$	−1.52726e6	−0.403000
$429$	58212.0	0.0152711
$430$	−1.47955e6	−0.385886
$431$	−580626.	−0.150558	−0.0752789	−	0.997163i	$-0.523985\pi$
$431$	−580626.	−0.150558	−0.0752789	+	0.997163i	$0.523985\pi$
$432$	186624.	0.0481125
$433$	4.15087e6	1.06395	0.531973	−	0.846761i	$-0.321451\pi$
$433$	4.15087e6	1.06395	0.531973	+	0.846761i	$0.321451\pi$
$434$	−1.38141e6	−0.352045
$435$	−537840.	−0.136279
$436$	3.39555e6	0.855449
$437$	352920.	0.0884042
$438$	1.21442e6	0.302472
$439$	3.88407e6	0.961891	0.480946	−	0.876750i	$-0.340293\pi$
$439$	3.88407e6	0.961891	0.480946	+	0.876750i	$0.340293\pi$
$440$	101376.	0.0249634
$441$	194481.	0.0476190
$442$	−84672.0	−0.0206150
$443$	−2.31499e6	−0.560453	−0.280226	−	0.959934i	$-0.590410\pi$
$443$	−2.31499e6	−0.560453	−0.280226	+	0.959934i	$0.590410\pi$
$444$	−1.76227e6	−0.424244
$445$	837216.	0.200418
$446$	5.88022e6	1.39977
$447$	1.95475e6	0.462723
$448$	200704.	0.0472456
$449$	−1.92281e6	−0.450113	−0.225056	−	0.974346i	$-0.572257\pi$
$449$	−1.92281e6	−0.450113	−0.225056	+	0.974346i	$0.572257\pi$
$450$	−825876.	−0.192258
$451$	426888.	0.0988263
$452$	993696.	0.228774
$453$	1.98900e6	0.455396
$454$	−2.35824e6	−0.536968
$455$	115248.	0.0260979
$456$	−195840.	−0.0441051
$457$	6.86215e6	1.53699	0.768493	−	0.639858i	$-0.221007\pi$
$457$	6.86215e6	1.53699	0.768493	+	0.639858i	$0.221007\pi$
$458$	−4.18137e6	−0.931440
$459$	−157464.	−0.0348859
$460$	−398592.	−0.0878282
$461$	2.97167e6	0.651250	0.325625	−	0.945499i	$-0.394425\pi$
$461$	2.97167e6	0.651250	0.325625	+	0.945499i	$0.394425\pi$
$462$	116424.	0.0253768
$463$	4.87423e6	1.05670	0.528352	−	0.849025i	$-0.322810\pi$
$463$	4.87423e6	1.05670	0.528352	+	0.849025i	$0.322810\pi$
$464$	−637440.	−0.137450
$465$	−1.52237e6	−0.326503
$466$	2.60489e6	0.555679
$467$	−8.17301e6	−1.73416	−0.867081	−	0.498167i	$-0.834007\pi$
$467$	−8.17301e6	−1.73416	−0.867081	+	0.498167i	$0.834007\pi$
$468$	127008.	0.0268050
$469$	621320.	0.130432
$470$	1.97798e6	0.413027
$471$	−3.40533e6	−0.707305
$472$	2.19034e6	0.452539
$473$	−1.01719e6	−0.209050
$474$	−3.06389e6	−0.626364
$475$	866660.	0.176244
$476$	−169344.	−0.0342572
$477$	2.63169e6	0.529589
$478$	−2.05385e6	−0.411148
$479$	2.34397e6	0.466782	0.233391	−	0.972383i	$-0.425018\pi$
$479$	2.34397e6	0.466782	0.233391	+	0.972383i	$0.425018\pi$
$480$	221184.	0.0438178
$481$	−1.19932e6	−0.236360
$482$	−2.77886e6	−0.544814
$483$	−457758.	−0.0892829
$484$	−2.50712e6	−0.486476
$485$	447888.	0.0864600
$486$	236196.	0.0453609
$487$	316928.	0.0605534	0.0302767	−	0.999542i	$-0.490361\pi$
$487$	316928.	0.0605534	0.0302767	+	0.999542i	$0.490361\pi$
$488$	2.28186e6	0.433749
$489$	943344.	0.178401
$490$	230496.	0.0433684
$491$	−5.20041e6	−0.973495	−0.486748	−	0.873543i	$-0.661817\pi$
$491$	−5.20041e6	−0.973495	−0.486748	+	0.873543i	$0.661817\pi$
$492$	931392.	0.173468
$493$	537840.	0.0996634
$494$	−133280.	−0.0245724
$495$	128304.	0.0235357
$496$	−1.80429e6	−0.329308
$497$	−2.08946e6	−0.379440
$498$	−3.84350e6	−0.694471
$499$	−4.86773e6	−0.875135	−0.437568	−	0.899185i	$-0.644160\pi$
$499$	−4.86773e6	−0.875135	−0.437568	+	0.899185i	$0.644160\pi$
$500$	−2.17882e6	−0.389758
$501$	−3.84275e6	−0.683987
$502$	−5.58432e6	−0.989034
$503$	426888.	0.0752305	0.0376153	−	0.999292i	$-0.488024\pi$
$503$	426888.	0.0752305	0.0376153	+	0.999292i	$0.488024\pi$
$504$	254016.	0.0445435
$505$	3.67402e6	0.641081
$506$	−274032.	−0.0475801
$507$	−3.25520e6	−0.562416
$508$	−848704.	−0.145914
$509$	−9.41621e6	−1.61095	−0.805474	−	0.592631i	$-0.798089\pi$
$509$	−9.41621e6	−1.61095	−0.805474	+	0.592631i	$0.798089\pi$
$510$	−186624.	−0.0317718
$511$	1.65297e6	0.280035
$512$	262144.	0.0441942
$513$	−247860.	−0.0415827
$514$	−4.02082e6	−0.671284
$515$	860736.	0.143005
$516$	−2.21933e6	−0.366942
$517$	1.35986e6	0.223753
$518$	−2.39865e6	−0.392773
$519$	2.97961e6	0.485558
$520$	150528.	0.0244123
$521$	1.84039e6	0.297041	0.148520	−	0.988909i	$-0.452549\pi$
$521$	1.84039e6	0.297041	0.148520	+	0.988909i	$0.452549\pi$
$522$	−806760.	−0.129589
$523$	−979108.	−0.156522	−0.0782612	−	0.996933i	$-0.524937\pi$
$523$	−979108.	−0.156522	−0.0782612	+	0.996933i	$0.524937\pi$
$524$	1.10918e6	0.176472
$525$	−1.12411e6	−0.177996
$526$	5.01204e6	0.789860
$527$	1.52237e6	0.238777
$528$	152064.	0.0237379
$529$	−5.35890e6	−0.832600
$530$	3.11904e6	0.482316
$531$	2.77214e6	0.426658
$532$	−266560.	−0.0408334
$533$	633864.	0.0966447
$534$	1.25582e6	0.190579
$535$	−2.29090e6	−0.346036
$536$	811520.	0.122008
$537$	−3.60175e6	−0.538986
$538$	−7.04275e6	−1.04903
$539$	158466.	0.0234944
$540$	279936.	0.0413118
$541$	5.96117e6	0.875666	0.437833	−	0.899056i	$-0.355746\pi$
$541$	5.96117e6	0.875666	0.437833	+	0.899056i	$0.355746\pi$
$542$	3.08211e6	0.450661
$543$	5.29288e6	0.770358
$544$	−221184.	−0.0320447
$545$	5.09333e6	0.734531
$546$	172872.	0.0248166
$547$	8.73025e6	1.24755	0.623775	−	0.781604i	$-0.285598\pi$
$547$	8.73025e6	1.24755	0.623775	+	0.781604i	$0.285598\pi$
$548$	2.07754e6	0.295527
$549$	2.88797e6	0.408943
$550$	−672936.	−0.0948565
$551$	846600.	0.118795
$552$	−597888.	−0.0835165
$553$	−4.17029e6	−0.579901
$554$	2.83095e6	0.391885
$555$	−2.64341e6	−0.364277
$556$	−1.66970e6	−0.229061
$557$	−3.01066e6	−0.411172	−0.205586	−	0.978639i	$-0.565910\pi$
$557$	−3.01066e6	−0.411172	−0.205586	+	0.978639i	$0.565910\pi$
$558$	−2.28355e6	−0.310474
$559$	−1.51038e6	−0.204435
$560$	301056.	0.0405674
$561$	−128304.	−0.0172121
$562$	9.21727e6	1.23101
$563$	1.17573e7	1.56327	0.781637	−	0.623733i	$-0.214385\pi$
$563$	1.17573e7	1.56327	0.781637	+	0.623733i	$0.214385\pi$
$564$	2.96698e6	0.392750
$565$	1.49054e6	0.196437
$566$	6.43611e6	0.844467
$567$	321489.	0.0419961
$568$	−2.72909e6	−0.354933
$569$	1.31578e7	1.70374	0.851870	−	0.523754i	$-0.175469\pi$
$569$	1.31578e7	1.70374	0.851870	+	0.523754i	$0.175469\pi$
$570$	−293760.	−0.0378709
$571$	−1.03344e7	−1.32647	−0.663234	−	0.748412i	$-0.730817\pi$
$571$	−1.03344e7	−1.32647	−0.663234	+	0.748412i	$0.730817\pi$
$572$	103488.	0.0132251
$573$	8.45408e6	1.07567
$574$	1.26773e6	0.160600
$575$	2.64586e6	0.333732
$576$	331776.	0.0416667
$577$	−7.88133e6	−0.985508	−0.492754	−	0.870169i	$-0.664010\pi$
$577$	−7.88133e6	−0.985508	−0.492754	+	0.870169i	$0.664010\pi$
$578$	−5.49280e6	−0.683872
$579$	3.04551e6	0.377541
$580$	−956160.	−0.118021
$581$	−5.23144e6	−0.642955
$582$	671832.	0.0822154
$583$	2.14434e6	0.261290
$584$	2.15898e6	0.261948
$585$	190512.	0.0230161
$586$	2.06808e6	0.248784
$587$	−554568.	−0.0664293	−0.0332146	−	0.999448i	$-0.510574\pi$
$587$	−554568.	−0.0664293	−0.0332146	+	0.999448i	$0.510574\pi$
$588$	345744.	0.0412393
$589$	2.39632e6	0.284614
$590$	3.28550e6	0.388572
$591$	−2.14148e6	−0.252200
$592$	−3.13293e6	−0.367406
$593$	−9.20369e6	−1.07479	−0.537397	−	0.843329i	$-0.680592\pi$
$593$	−9.20369e6	−1.07479	−0.537397	+	0.843329i	$0.680592\pi$
$594$	192456.	0.0223803
$595$	−254016.	−0.0294150
$596$	3.47510e6	0.400730
$597$	1.84018e6	0.211312
$598$	−406896.	−0.0465297
$599$	8.54295e6	0.972839	0.486419	−	0.873725i	$-0.338303\pi$
$599$	8.54295e6	0.972839	0.486419	+	0.873725i	$0.338303\pi$
$600$	−1.46822e6	−0.166500
$601$	−9.61555e6	−1.08590	−0.542948	−	0.839767i	$-0.682692\pi$
$601$	−9.61555e6	−1.08590	−0.542948	+	0.839767i	$0.682692\pi$
$602$	−3.02075e6	−0.339722
$603$	1.02708e6	0.115030
$604$	3.53600e6	0.394385
$605$	−3.76068e6	−0.417713
$606$	5.51102e6	0.609608
$607$	2.21264e6	0.243747	0.121873	−	0.992546i	$-0.461110\pi$
$607$	2.21264e6	0.243747	0.121873	+	0.992546i	$0.461110\pi$
$608$	−348160.	−0.0381962
$609$	−1.09809e6	−0.119976
$610$	3.42278e6	0.372439
$611$	2.01919e6	0.218814
$612$	−279936.	−0.0302121
$613$	−7.96215e6	−0.855814	−0.427907	−	0.903823i	$-0.640749\pi$
$613$	−7.96215e6	−0.855814	−0.427907	+	0.903823i	$0.640749\pi$
$614$	5.40008e6	0.578068
$615$	1.39709e6	0.148948
$616$	206976.	0.0219770
$617$	−1.37397e7	−1.45299	−0.726497	−	0.687170i	$-0.758853\pi$
$617$	−1.37397e7	−1.45299	−0.726497	+	0.687170i	$0.758853\pi$
$618$	1.29110e6	0.135985
$619$	−8.70113e6	−0.912744	−0.456372	−	0.889789i	$-0.650851\pi$
$619$	−8.70113e6	−0.912744	−0.456372	+	0.889789i	$0.650851\pi$
$620$	−2.70643e6	−0.282760
$621$	−756702.	−0.0787401
$622$	5.38152e6	0.557736
$623$	1.70932e6	0.176442
$624$	225792.	0.0232138
$625$	4.69740e6	0.481014
$626$	1.02462e6	0.104502
$627$	−201960.	−0.0205162
$628$	−6.05392e6	−0.612544
$629$	2.64341e6	0.266402
$630$	381024.	0.0382473
$631$	445412.	0.0445337	0.0222668	−	0.999752i	$-0.492912\pi$
$631$	445412.	0.0445337	0.0222668	+	0.999752i	$0.492912\pi$
$632$	−5.44691e6	−0.542447
$633$	−3.13852e6	−0.311326
$634$	7.38516e6	0.729687
$635$	−1.27306e6	−0.125289
$636$	4.67856e6	0.458637
$637$	235298.	0.0229757
$638$	−657360.	−0.0639369
$639$	−3.45400e6	−0.334634
$640$	393216.	0.0379473
$641$	−8.00119e6	−0.769147	−0.384573	−	0.923094i	$-0.625651\pi$
$641$	−8.00119e6	−0.769147	−0.384573	+	0.923094i	$0.625651\pi$
$642$	−3.43634e6	−0.329048
$643$	−1.58402e7	−1.51090	−0.755448	−	0.655209i	$-0.772581\pi$
$643$	−1.58402e7	−1.51090	−0.755448	+	0.655209i	$0.772581\pi$
$644$	−813792.	−0.0773212
$645$	−3.32899e6	−0.315075
$646$	293760.	0.0276956
$647$	1.30187e6	0.122266	0.0611331	−	0.998130i	$-0.480529\pi$
$647$	1.30187e6	0.122266	0.0611331	+	0.998130i	$0.480529\pi$
$648$	419904.	0.0392837
$649$	2.25878e6	0.210505
$650$	−999208.	−0.0927625
$651$	−3.10817e6	−0.287443
$652$	1.67706e6	0.154500
$653$	7.34149e6	0.673753	0.336877	−	0.941549i	$-0.390629\pi$
$653$	7.34149e6	0.673753	0.336877	+	0.941549i	$0.390629\pi$
$654$	7.63999e6	0.698471
$655$	1.66378e6	0.151528
$656$	1.65581e6	0.150228
$657$	2.73245e6	0.246967
$658$	4.03838e6	0.363616
$659$	−6.18934e6	−0.555176	−0.277588	−	0.960700i	$-0.589535\pi$
$659$	−6.18934e6	−0.555176	−0.277588	+	0.960700i	$0.589535\pi$
$660$	228096.	0.0203825
$661$	−1.96690e7	−1.75097	−0.875484	−	0.483248i	$-0.839457\pi$
$661$	−1.96690e7	−1.75097	−0.875484	+	0.483248i	$0.839457\pi$
$662$	−1.33295e7	−1.18214
$663$	−190512.	−0.0168321
$664$	−6.83290e6	−0.601429
$665$	−399840.	−0.0350616
$666$	−3.96511e6	−0.346394
$667$	2.58462e6	0.224948
$668$	−6.83155e6	−0.592350
$669$	1.32305e7	1.14291
$670$	1.21728e6	0.104762
$671$	2.35316e6	0.201765
$672$	451584.	0.0385758
$673$	7.18259e6	0.611285	0.305642	−	0.952146i	$-0.401129\pi$
$673$	7.18259e6	0.611285	0.305642	+	0.952146i	$0.401129\pi$
$674$	−6.53922e6	−0.554468
$675$	−1.85822e6	−0.156978
$676$	−5.78702e6	−0.487067
$677$	−1.89192e7	−1.58647	−0.793234	−	0.608917i	$-0.791604\pi$
$677$	−1.89192e7	−1.58647	−0.793234	+	0.608917i	$0.791604\pi$
$678$	2.23582e6	0.186794
$679$	914438.	0.0761167
$680$	−331776.	−0.0275152
$681$	−5.30604e6	−0.438432
$682$	−1.86067e6	−0.153182
$683$	2.12204e7	1.74061	0.870306	−	0.492512i	$-0.163921\pi$
$683$	2.12204e7	1.74061	0.870306	+	0.492512i	$0.163921\pi$
$684$	−440640.	−0.0360117
$685$	3.11630e6	0.253754
$686$	470596.	0.0381802
$687$	−9.40808e6	−0.760517
$688$	−3.94547e6	−0.317781
$689$	3.18402e6	0.255522
$690$	−896832.	−0.0717114
$691$	1.63276e7	1.30085	0.650424	−	0.759571i	$-0.274591\pi$
$691$	1.63276e7	1.30085	0.650424	+	0.759571i	$0.274591\pi$
$692$	5.29709e6	0.420506
$693$	261954.	0.0207201
$694$	−3.36612e6	−0.265296
$695$	−2.50454e6	−0.196683
$696$	−1.43424e6	−0.112227
$697$	−1.39709e6	−0.108929
$698$	−3.90897e6	−0.303685
$699$	5.86100e6	0.453710
$700$	−1.99842e6	−0.154149
$701$	−5.40470e6	−0.415409	−0.207705	−	0.978192i	$-0.566599\pi$
$701$	−5.40470e6	−0.415409	−0.207705	+	0.978192i	$0.566599\pi$
$702$	285768.	0.0218862
$703$	4.16092e6	0.317542
$704$	270336.	0.0205576
$705$	4.45046e6	0.337235
$706$	1.38343e7	1.04459
$707$	7.50112e6	0.564387
$708$	4.92826e6	0.369496
$709$	2.21195e7	1.65257	0.826284	−	0.563253i	$-0.190450\pi$
$709$	2.21195e7	1.65257	0.826284	+	0.563253i	$0.190450\pi$
$710$	−4.09363e6	−0.304763
$711$	−6.89375e6	−0.511424
$712$	2.23258e6	0.165046
$713$	7.31582e6	0.538939
$714$	−381024.	−0.0279709
$715$	155232.	0.0113558
$716$	−6.40310e6	−0.466775
$717$	−4.62116e6	−0.335701
$718$	−1.38920e7	−1.00567
$719$	2.55819e7	1.84548	0.922742	−	0.385418i	$-0.125943\pi$
$719$	2.55819e7	1.84548	0.922742	+	0.385418i	$0.125943\pi$
$720$	497664.	0.0357771
$721$	1.75734e6	0.125897
$722$	−9.44200e6	−0.674095
$723$	−6.25243e6	−0.444839
$724$	9.40957e6	0.667150
$725$	6.34701e6	0.448460
$726$	−5.64102e6	−0.397206
$727$	−9.29438e6	−0.652205	−0.326103	−	0.945334i	$-0.605735\pi$
$727$	−9.29438e6	−0.652205	−0.326103	+	0.945334i	$0.605735\pi$
$728$	307328.	0.0214918
$729$	531441.	0.0370370
$730$	3.23846e6	0.224922
$731$	3.32899e6	0.230420
$732$	5.13418e6	0.354155
$733$	3.40699e6	0.234213	0.117107	−	0.993119i	$-0.462638\pi$
$733$	3.40699e6	0.234213	0.117107	+	0.993119i	$0.462638\pi$
$734$	1.24797e7	0.854999
$735$	518616.	0.0354101
$736$	−1.06291e6	−0.0723274
$737$	836880.	0.0567537
$738$	2.09563e6	0.141636
$739$	2.18135e7	1.46932	0.734658	−	0.678438i	$-0.237343\pi$
$739$	2.18135e7	1.46932	0.734658	+	0.678438i	$0.237343\pi$
$740$	−4.69939e6	−0.315473
$741$	−299880.	−0.0200633
$742$	6.36804e6	0.424616
$743$	3.79246e6	0.252028	0.126014	−	0.992028i	$-0.459782\pi$
$743$	3.79246e6	0.252028	0.126014	+	0.992028i	$0.459782\pi$
$744$	−4.05965e6	−0.268878
$745$	5.21266e6	0.344087
$746$	−8.06692e6	−0.530714
$747$	−8.64788e6	−0.567033
$748$	−228096.	−0.0149061
$749$	−4.67725e6	−0.304639
$750$	−4.90234e6	−0.318236
$751$	−2.01483e7	−1.30358	−0.651790	−	0.758400i	$-0.725982\pi$
$751$	−2.01483e7	−1.30358	−0.651790	+	0.758400i	$0.725982\pi$
$752$	5.27462e6	0.340132
$753$	−1.25647e7	−0.807542
$754$	−976080.	−0.0625255
$755$	5.30400e6	0.338638
$756$	571536.	0.0363696
$757$	1.18427e7	0.751126	0.375563	−	0.926797i	$-0.377449\pi$
$757$	1.18427e7	0.751126	0.375563	+	0.926797i	$0.377449\pi$
$758$	−2.15233e7	−1.36062
$759$	−616572.	−0.0388490
$760$	−522240.	−0.0327972
$761$	2.97791e6	0.186402	0.0932008	−	0.995647i	$-0.470290\pi$
$761$	2.97791e6	0.186402	0.0932008	+	0.995647i	$0.470290\pi$
$762$	−1.90958e6	−0.119138
$763$	1.03989e7	0.646659
$764$	1.50295e7	0.931559
$765$	−419904.	−0.0259416
$766$	3.22973e6	0.198881
$767$	3.35395e6	0.205858
$768$	589824.	0.0360844
$769$	−2.02441e7	−1.23447	−0.617237	−	0.786777i	$-0.711748\pi$
$769$	−2.02441e7	−1.23447	−0.617237	+	0.786777i	$0.711748\pi$
$770$	310464.	0.0188705
$771$	−9.04684e6	−0.548101
$772$	5.41424e6	0.326960
$773$	−7.37953e6	−0.444202	−0.222101	−	0.975024i	$-0.571291\pi$
$773$	−7.37953e6	−0.444202	−0.222101	+	0.975024i	$0.571291\pi$
$774$	−4.99349e6	−0.299607
$775$	1.79654e7	1.07444
$776$	1.19437e6	0.0712006
$777$	−5.39696e6	−0.320698
$778$	3.56556e6	0.211193
$779$	−2.19912e6	−0.129839
$780$	338688.	0.0199326
$781$	−2.81437e6	−0.165103
$782$	896832.	0.0524438
$783$	−1.81521e6	−0.105809
$784$	614656.	0.0357143
$785$	−9.08088e6	−0.525961
$786$	2.49566e6	0.144089
$787$	1.36289e7	0.784377	0.392188	−	0.919885i	$-0.371718\pi$
$787$	1.36289e7	0.784377	0.392188	+	0.919885i	$0.371718\pi$
$788$	−3.80707e6	−0.218412
$789$	1.12771e7	0.644918
$790$	−8.17037e6	−0.465773
$791$	3.04319e6	0.172937
$792$	342144.	0.0193819
$793$	3.49409e6	0.197311
$794$	4.49382e6	0.252967
$795$	7.01784e6	0.393809
$796$	3.27142e6	0.183001
$797$	−1.49548e7	−0.833938	−0.416969	−	0.908921i	$-0.636908\pi$
$797$	−1.49548e7	−0.833938	−0.416969	+	0.908921i	$0.636908\pi$
$798$	−599760.	−0.0333404
$799$	−4.45046e6	−0.246626
$800$	−2.61018e6	−0.144193
$801$	2.82560e6	0.155607
$802$	6.88150e6	0.377787
$803$	2.22644e6	0.121849
$804$	1.82592e6	0.0996189
$805$	−1.22069e6	−0.0663919
$806$	−2.76282e6	−0.149801
$807$	−1.58462e7	−0.856527
$808$	9.79738e6	0.527936
$809$	2.87242e7	1.54304	0.771519	−	0.636206i	$-0.219497\pi$
$809$	2.87242e7	1.54304	0.771519	+	0.636206i	$0.219497\pi$
$810$	629856.	0.0337310
$811$	−1.52265e7	−0.812922	−0.406461	−	0.913668i	$-0.633237\pi$
$811$	−1.52265e7	−0.812922	−0.406461	+	0.913668i	$0.633237\pi$
$812$	−1.95216e6	−0.103902
$813$	6.93475e6	0.367963
$814$	−3.23083e6	−0.170904
$815$	2.51558e6	0.132661
$816$	−497664.	−0.0261644
$817$	5.24008e6	0.274652
$818$	308984.	0.0161455
$819$	388962.	0.0202627
$820$	2.48371e6	0.128993
$821$	−3.31001e7	−1.71384	−0.856921	−	0.515447i	$-0.827626\pi$
$821$	−3.31001e7	−1.71384	−0.856921	+	0.515447i	$0.827626\pi$
$822$	4.67446e6	0.241297
$823$	−1.35915e7	−0.699470	−0.349735	−	0.936849i	$-0.613728\pi$
$823$	−1.35915e7	−0.699470	−0.349735	+	0.936849i	$0.613728\pi$
$824$	2.29530e6	0.117766
$825$	−1.51411e6	−0.0774500
$826$	6.70790e6	0.342087
$827$	3.13936e6	0.159616	0.0798082	−	0.996810i	$-0.474569\pi$
$827$	3.13936e6	0.159616	0.0798082	+	0.996810i	$0.474569\pi$
$828$	−1.34525e6	−0.0681909
$829$	1.27081e7	0.642234	0.321117	−	0.947040i	$-0.395942\pi$
$829$	1.27081e7	0.642234	0.321117	+	0.947040i	$0.395942\pi$
$830$	−1.02493e7	−0.516417
$831$	6.36964e6	0.319972
$832$	401408.	0.0201038
$833$	−518616.	−0.0258960
$834$	−3.75682e6	−0.187027
$835$	−1.02473e7	−0.508621
$836$	−359040.	−0.0177675
$837$	−5.13799e6	−0.253501
$838$	−2.08246e7	−1.02439
$839$	−2.98312e7	−1.46307	−0.731536	−	0.681803i	$-0.761196\pi$
$839$	−2.98312e7	−1.46307	−0.731536	+	0.681803i	$0.761196\pi$
$840$	677376.	0.0331231
$841$	−1.43110e7	−0.697720
$842$	6.87390e6	0.334136
$843$	2.07389e7	1.00512
$844$	−5.57958e6	−0.269616
$845$	−8.68054e6	−0.418220
$846$	6.67570e6	0.320679
$847$	−7.67806e6	−0.367742
$848$	8.31744e6	0.397192
$849$	1.44813e7	0.689504
$850$	2.20234e6	0.104553
$851$	1.27030e7	0.601290
$852$	−6.14045e6	−0.289802
$853$	−1.92215e7	−0.904515	−0.452257	−	0.891888i	$-0.649381\pi$
$853$	−1.92215e7	−0.904515	−0.452257	+	0.891888i	$0.649381\pi$
$854$	6.98818e6	0.327884
$855$	−660960.	−0.0309215
$856$	−6.10906e6	−0.284964
$857$	−2.65655e7	−1.23556	−0.617782	−	0.786349i	$-0.711969\pi$
$857$	−2.65655e7	−1.23556	−0.617782	+	0.786349i	$0.711969\pi$
$858$	232848.	0.0107983
$859$	−9.16844e6	−0.423948	−0.211974	−	0.977275i	$-0.567989\pi$
$859$	−9.16844e6	−0.423948	−0.211974	+	0.977275i	$0.567989\pi$
$860$	−5.91821e6	−0.272863
$861$	2.85239e6	0.131130
$862$	−2.32250e6	−0.106460
$863$	−2.92196e7	−1.33551	−0.667755	−	0.744381i	$-0.732745\pi$
$863$	−2.92196e7	−1.33551	−0.667755	+	0.744381i	$0.732745\pi$
$864$	746496.	0.0340207
$865$	7.94563e6	0.361067
$866$	1.66035e7	0.752324
$867$	−1.23588e7	−0.558379
$868$	−5.52563e6	−0.248933
$869$	−5.61713e6	−0.252328
$870$	−2.15136e6	−0.0963640
$871$	1.24264e6	0.0555009
$872$	1.35822e7	0.604894
$873$	1.51162e6	0.0671286
$874$	1.41168e6	0.0625112
$875$	−6.67262e6	−0.294630
$876$	4.85770e6	0.213880
$877$	9.71286e6	0.426430	0.213215	−	0.977005i	$-0.431606\pi$
$877$	9.71286e6	0.426430	0.213215	+	0.977005i	$0.431606\pi$
$878$	1.55363e7	0.680160
$879$	4.65318e6	0.203132
$880$	405504.	0.0176518
$881$	1.65372e7	0.717833	0.358917	−	0.933370i	$-0.383146\pi$
$881$	1.65372e7	0.717833	0.358917	+	0.933370i	$0.383146\pi$
$882$	777924.	0.0336718
$883$	−2.39487e7	−1.03367	−0.516833	−	0.856086i	$-0.672889\pi$
$883$	−2.39487e7	−1.03367	−0.516833	+	0.856086i	$0.672889\pi$
$884$	−338688.	−0.0145770
$885$	7.39238e6	0.317268
$886$	−9.25994e6	−0.396300
$887$	−4.62846e6	−0.197527	−0.0987637	−	0.995111i	$-0.531489\pi$
$887$	−4.62846e6	−0.197527	−0.0987637	+	0.995111i	$0.531489\pi$
$888$	−7.04909e6	−0.299986
$889$	−2.59916e6	−0.110301
$890$	3.34886e6	0.141717
$891$	433026.	0.0182734
$892$	2.35209e7	0.989787
$893$	−7.00536e6	−0.293969
$894$	7.81898e6	0.327195
$895$	−9.60466e6	−0.400797
$896$	802816.	0.0334077
$897$	−915516.	−0.0379914
$898$	−7.69126e6	−0.318278
$899$	1.75495e7	0.724212
$900$	−3.30350e6	−0.135947
$901$	−7.01784e6	−0.287999
$902$	1.70755e6	0.0698808
$903$	−6.79669e6	−0.277382
$904$	3.97478e6	0.161768
$905$	1.41144e7	0.572848
$906$	7.95600e6	0.322014
$907$	2.06126e7	0.831983	0.415991	−	0.909369i	$-0.363435\pi$
$907$	2.06126e7	0.831983	0.415991	+	0.909369i	$0.363435\pi$
$908$	−9.43296e6	−0.379694
$909$	1.23998e7	0.497743
$910$	460992.	0.0184540
$911$	−3.46749e6	−0.138427	−0.0692133	−	0.997602i	$-0.522049\pi$
$911$	−3.46749e6	−0.138427	−0.0692133	+	0.997602i	$0.522049\pi$
$912$	−783360.	−0.0311870
$913$	−7.04642e6	−0.279764
$914$	2.74486e7	1.08681
$915$	7.70126e6	0.304095
$916$	−1.67255e7	−0.658627
$917$	3.39688e6	0.133400
$918$	−629856.	−0.0246680
$919$	−3.61227e7	−1.41088	−0.705442	−	0.708767i	$-0.749252\pi$
$919$	−3.61227e7	−1.41088	−0.705442	+	0.708767i	$0.749252\pi$
$920$	−1.59437e6	−0.0621039
$921$	1.21502e7	0.471991
$922$	1.18867e7	0.460504
$923$	−4.17892e6	−0.161458
$924$	465696.	0.0179441
$925$	3.11947e7	1.19874
$926$	1.94969e7	0.747203
$927$	2.90498e6	0.111031
$928$	−2.54976e6	−0.0971917
$929$	1.29366e7	0.491792	0.245896	−	0.969296i	$-0.420918\pi$
$929$	1.29366e7	0.491792	0.245896	+	0.969296i	$0.420918\pi$
$930$	−6.08947e6	−0.230873
$931$	−816340.	−0.0308672
$932$	1.04196e7	0.392925
$933$	1.21084e7	0.455390
$934$	−3.26920e7	−1.22624
$935$	−342144.	−0.0127991
$936$	508032.	0.0189540
$937$	5.01394e7	1.86565	0.932824	−	0.360332i	$-0.117336\pi$
$937$	5.01394e7	1.86565	0.932824	+	0.360332i	$0.117336\pi$
$938$	2.48528e6	0.0922292
$939$	2.30539e6	0.0853257
$940$	7.91194e6	0.292054
$941$	−1.05568e7	−0.388651	−0.194325	−	0.980937i	$-0.562252\pi$
$941$	−1.05568e7	−0.388651	−0.194325	+	0.980937i	$0.562252\pi$
$942$	−1.36213e7	−0.500140
$943$	−6.71378e6	−0.245860
$944$	8.76134e6	0.319993
$945$	857304.	0.0312288
$946$	−4.06877e6	−0.147821
$947$	−3.14684e6	−0.114025	−0.0570124	−	0.998373i	$-0.518157\pi$
$947$	−3.14684e6	−0.114025	−0.0570124	+	0.998373i	$0.518157\pi$
$948$	−1.22556e7	−0.442906
$949$	3.30593e6	0.119159
$950$	3.46664e6	0.124623
$951$	1.66166e7	0.595787
$952$	−677376.	−0.0242235
$953$	5.22829e7	1.86478	0.932389	−	0.361455i	$-0.117720\pi$
$953$	5.22829e7	1.86478	0.932389	+	0.361455i	$0.117720\pi$
$954$	1.05268e7	0.374476
$955$	2.25442e7	0.799883
$956$	−8.21539e6	−0.290726
$957$	−1.47906e6	−0.0522043
$958$	9.37589e6	0.330064
$959$	6.36245e6	0.223397
$960$	884736.	0.0309839
$961$	2.10452e7	0.735095
$962$	−4.79730e6	−0.167132
$963$	−7.73177e6	−0.268666
$964$	−1.11154e7	−0.385242
$965$	8.12136e6	0.280744
$966$	−1.83103e6	−0.0631325
$967$	−2.48235e7	−0.853682	−0.426841	−	0.904327i	$-0.640374\pi$
$967$	−2.48235e7	−0.853682	−0.426841	+	0.904327i	$0.640374\pi$
$968$	−1.00285e7	−0.343991
$969$	660960.	0.0226134
$970$	1.79155e6	0.0611364
$971$	1.33077e7	0.452956	0.226478	−	0.974016i	$-0.427279\pi$
$971$	1.33077e7	0.452956	0.226478	+	0.974016i	$0.427279\pi$
$972$	944784.	0.0320750
$973$	−5.11344e6	−0.173154
$974$	1.26771e6	0.0428177
$975$	−2.24822e6	−0.0757403
$976$	9.12742e6	0.306707
$977$	8.17705e6	0.274069	0.137035	−	0.990566i	$-0.456243\pi$
$977$	8.17705e6	0.274069	0.137035	+	0.990566i	$0.456243\pi$
$978$	3.77338e6	0.126149
$979$	2.30234e6	0.0767739
$980$	921984.	0.0306661
$981$	1.71900e7	0.570299
$982$	−2.08016e7	−0.688365
$983$	−1.32465e7	−0.437238	−0.218619	−	0.975810i	$-0.570155\pi$
$983$	−1.32465e7	−0.437238	−0.218619	+	0.975810i	$0.570155\pi$
$984$	3.72557e6	0.122661
$985$	−5.71061e6	−0.187539
$986$	2.15136e6	0.0704727
$987$	9.08636e6	0.296891
$988$	−533120.	−0.0173753
$989$	1.59977e7	0.520075
$990$	513216.	0.0166423
$991$	−1.48550e7	−0.480494	−0.240247	−	0.970712i	$-0.577228\pi$
$991$	−1.48550e7	−0.480494	−0.240247	+	0.970712i	$0.577228\pi$
$992$	−7.21715e6	−0.232856
$993$	−2.99914e7	−0.965215
$994$	−8.35783e6	−0.268304
$995$	4.90714e6	0.157134
$996$	−1.53740e7	−0.491065
$997$	−3.33769e6	−0.106343	−0.0531714	−	0.998585i	$-0.516933\pi$
$997$	−3.33769e6	−0.106343	−0.0531714	+	0.998585i	$0.516933\pi$
$998$	−1.94709e7	−0.618814
$999$	−8.92150e6	−0.282829

Twists

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

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