LMFDB - Embedded newform 42.6.a.a.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,-54] 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:	$0$
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)

$f(q)$

$=$

\(q-4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} -54.0000 q^{5} +36.0000 q^{6} +49.0000 q^{7} -64.0000 q^{8} +81.0000 q^{9} +216.000 q^{10} +216.000 q^{11} -144.000 q^{12} +998.000 q^{13} -196.000 q^{14} +486.000 q^{15} +256.000 q^{16} +1302.00 q^{17} -324.000 q^{18} +884.000 q^{19} -864.000 q^{20} -441.000 q^{21} -864.000 q^{22} -2268.00 q^{23} +576.000 q^{24} -209.000 q^{25} -3992.00 q^{26} -729.000 q^{27} +784.000 q^{28} -1482.00 q^{29} -1944.00 q^{30} +8360.00 q^{31} -1024.00 q^{32} -1944.00 q^{33} -5208.00 q^{34} -2646.00 q^{35} +1296.00 q^{36} -4714.00 q^{37} -3536.00 q^{38} -8982.00 q^{39} +3456.00 q^{40} -9786.00 q^{41} +1764.00 q^{42} +19436.0 q^{43} +3456.00 q^{44} -4374.00 q^{45} +9072.00 q^{46} +22200.0 q^{47} -2304.00 q^{48} +2401.00 q^{49} +836.000 q^{50} -11718.0 q^{51} +15968.0 q^{52} +26790.0 q^{53} +2916.00 q^{54} -11664.0 q^{55} -3136.00 q^{56} -7956.00 q^{57} +5928.00 q^{58} +28092.0 q^{59} +7776.00 q^{60} -38866.0 q^{61} -33440.0 q^{62} +3969.00 q^{63} +4096.00 q^{64} -53892.0 q^{65} +7776.00 q^{66} +23948.0 q^{67} +20832.0 q^{68} +20412.0 q^{69} +10584.0 q^{70} -20628.0 q^{71} -5184.00 q^{72} +290.000 q^{73} +18856.0 q^{74} +1881.00 q^{75} +14144.0 q^{76} +10584.0 q^{77} +35928.0 q^{78} -99544.0 q^{79} -13824.0 q^{80} +6561.00 q^{81} +39144.0 q^{82} +19308.0 q^{83} -7056.00 q^{84} -70308.0 q^{85} -77744.0 q^{86} +13338.0 q^{87} -13824.0 q^{88} +36390.0 q^{89} +17496.0 q^{90} +48902.0 q^{91} -36288.0 q^{92} -75240.0 q^{93} -88800.0 q^{94} -47736.0 q^{95} +9216.00 q^{96} -79078.0 q^{97} -9604.00 q^{98} +17496.0 q^{99} +O(q^{100})\)

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
$5$	−54.0000	−0.965981	−0.482991	−	0.875625i	$-0.660450\pi$
$5$	−54.0000	−0.965981	−0.482991	+	0.875625i	$0.660450\pi$
$6$	36.0000	0.408248
$7$	49.0000	0.377964
$7$	49.0000	0.377964
$8$	−64.0000	−0.353553
$9$	81.0000	0.333333
$10$	216.000	0.683052
$11$	216.000	0.538235	0.269118	−	0.963107i	$-0.413268\pi$
$11$	216.000	0.538235	0.269118	+	0.963107i	$0.413268\pi$
$12$	−144.000	−0.288675
$13$	998.000	1.63784	0.818921	−	0.573906i	$-0.194572\pi$
$13$	998.000	1.63784	0.818921	+	0.573906i	$0.194572\pi$
$14$	−196.000	−0.267261
$15$	486.000	0.557710
$16$	256.000	0.250000
$17$	1302.00	1.09267	0.546335	−	0.837567i	$-0.316023\pi$
$17$	1302.00	1.09267	0.546335	+	0.837567i	$0.316023\pi$
$18$	−324.000	−0.235702
$19$	884.000	0.561783	0.280891	−	0.959740i	$-0.409370\pi$
$19$	884.000	0.561783	0.280891	+	0.959740i	$0.409370\pi$
$20$	−864.000	−0.482991
$21$	−441.000	−0.218218
$22$	−864.000	−0.380590
$23$	−2268.00	−0.893971	−0.446986	−	0.894541i	$-0.647502\pi$
$23$	−2268.00	−0.893971	−0.446986	+	0.894541i	$0.647502\pi$
$24$	576.000	0.204124
$25$	−209.000	−0.0668800
$26$	−3992.00	−1.15813
$27$	−729.000	−0.192450
$28$	784.000	0.188982
$29$	−1482.00	−0.327230	−0.163615	−	0.986524i	$-0.552315\pi$
$29$	−1482.00	−0.327230	−0.163615	+	0.986524i	$0.552315\pi$
$30$	−1944.00	−0.394360
$31$	8360.00	1.56244	0.781218	−	0.624259i	$-0.214599\pi$
$31$	8360.00	1.56244	0.781218	+	0.624259i	$0.214599\pi$
$32$	−1024.00	−0.176777
$33$	−1944.00	−0.310750
$34$	−5208.00	−0.772634
$35$	−2646.00	−0.365107
$36$	1296.00	0.166667
$37$	−4714.00	−0.566090	−0.283045	−	0.959107i	$-0.591345\pi$
$37$	−4714.00	−0.566090	−0.283045	+	0.959107i	$0.591345\pi$
$38$	−3536.00	−0.397240
$39$	−8982.00	−0.945609
$40$	3456.00	0.341526
$41$	−9786.00	−0.909171	−0.454585	−	0.890703i	$-0.650213\pi$
$41$	−9786.00	−0.909171	−0.454585	+	0.890703i	$0.650213\pi$
$42$	1764.00	0.154303
$43$	19436.0	1.60301	0.801504	−	0.597989i	$-0.204033\pi$
$43$	19436.0	1.60301	0.801504	+	0.597989i	$0.204033\pi$
$44$	3456.00	0.269118
$45$	−4374.00	−0.321994
$46$	9072.00	0.632133
$47$	22200.0	1.46591	0.732957	−	0.680275i	$-0.238140\pi$
$47$	22200.0	1.46591	0.732957	+	0.680275i	$0.238140\pi$
$48$	−2304.00	−0.144338
$49$	2401.00	0.142857
$50$	836.000	0.0472913
$51$	−11718.0	−0.630853
$52$	15968.0	0.818921
$53$	26790.0	1.31004	0.655018	−	0.755614i	$-0.272661\pi$
$53$	26790.0	1.31004	0.655018	+	0.755614i	$0.272661\pi$
$54$	2916.00	0.136083
$55$	−11664.0	−0.519925
$56$	−3136.00	−0.133631
$57$	−7956.00	−0.324345
$58$	5928.00	0.231387
$59$	28092.0	1.05064	0.525318	−	0.850906i	$-0.323946\pi$
$59$	28092.0	1.05064	0.525318	+	0.850906i	$0.323946\pi$
$60$	7776.00	0.278855
$61$	−38866.0	−1.33735	−0.668675	−	0.743555i	$-0.733138\pi$
$61$	−38866.0	−1.33735	−0.668675	+	0.743555i	$0.733138\pi$
$62$	−33440.0	−1.10481
$63$	3969.00	0.125988
$64$	4096.00	0.125000
$65$	−53892.0	−1.58213
$66$	7776.00	0.219734
$67$	23948.0	0.651752	0.325876	−	0.945413i	$-0.394341\pi$
$67$	23948.0	0.651752	0.325876	+	0.945413i	$0.394341\pi$
$68$	20832.0	0.546335
$69$	20412.0	0.516134
$70$	10584.0	0.258169
$71$	−20628.0	−0.485636	−0.242818	−	0.970072i	$-0.578072\pi$
$71$	−20628.0	−0.485636	−0.242818	+	0.970072i	$0.578072\pi$
$72$	−5184.00	−0.117851
$73$	290.000	0.00636929	0.00318464	−	0.999995i	$-0.498986\pi$
$73$	290.000	0.00636929	0.00318464	+	0.999995i	$0.498986\pi$
$74$	18856.0	0.400286
$75$	1881.00	0.0386132
$76$	14144.0	0.280891
$77$	10584.0	0.203434
$78$	35928.0	0.668646
$79$	−99544.0	−1.79452	−0.897258	−	0.441506i	$-0.854444\pi$
$79$	−99544.0	−1.79452	−0.897258	+	0.441506i	$0.854444\pi$
$80$	−13824.0	−0.241495
$81$	6561.00	0.111111
$82$	39144.0	0.642881
$83$	19308.0	0.307639	0.153820	−	0.988099i	$-0.450842\pi$
$83$	19308.0	0.307639	0.153820	+	0.988099i	$0.450842\pi$
$84$	−7056.00	−0.109109
$85$	−70308.0	−1.05550
$86$	−77744.0	−1.13350
$87$	13338.0	0.188926
$88$	−13824.0	−0.190295
$89$	36390.0	0.486975	0.243488	−	0.969904i	$-0.421708\pi$
$89$	36390.0	0.486975	0.243488	+	0.969904i	$0.421708\pi$
$90$	17496.0	0.227684
$91$	48902.0	0.619046
$92$	−36288.0	−0.446986
$93$	−75240.0	−0.902072
$94$	−88800.0	−1.03656
$95$	−47736.0	−0.542671
$96$	9216.00	0.102062
$97$	−79078.0	−0.853348	−0.426674	−	0.904405i	$-0.640315\pi$
$97$	−79078.0	−0.853348	−0.426674	+	0.904405i	$0.640315\pi$
$98$	−9604.00	−0.101015
$99$	17496.0	0.179412
$100$	−3344.00	−0.0334400
$101$	184626.	1.80090	0.900450	−	0.434960i	$-0.143238\pi$
$101$	184626.	1.80090	0.900450	+	0.434960i	$0.143238\pi$
$102$	46872.0	0.446080
$103$	64592.0	0.599909	0.299955	−	0.953953i	$-0.403028\pi$
$103$	64592.0	0.599909	0.299955	+	0.953953i	$0.403028\pi$
$104$	−63872.0	−0.579065
$105$	23814.0	0.210794
$106$	−107160.	−0.926335
$107$	149592.	1.26313	0.631566	−	0.775322i	$-0.282412\pi$
$107$	149592.	1.26313	0.631566	+	0.775322i	$0.282412\pi$
$108$	−11664.0	−0.0962250
$109$	−63826.0	−0.514555	−0.257277	−	0.966338i	$-0.582825\pi$
$109$	−63826.0	−0.514555	−0.257277	+	0.966338i	$0.582825\pi$
$110$	46656.0	0.367643
$111$	42426.0	0.326832
$112$	12544.0	0.0944911
$113$	−71022.0	−0.523235	−0.261618	−	0.965172i	$-0.584256\pi$
$113$	−71022.0	−0.523235	−0.261618	+	0.965172i	$0.584256\pi$
$114$	31824.0	0.229347
$115$	122472.	0.863559
$116$	−23712.0	−0.163615
$117$	80838.0	0.545948
$118$	−112368.	−0.742912
$119$	63798.0	0.412990
$120$	−31104.0	−0.197180
$121$	−114395.	−0.710303
$122$	155464.	0.945650
$123$	88074.0	0.524910
$124$	133760.	0.781218
$125$	180036.	1.03059
$126$	−15876.0	−0.0890871
$127$	269624.	1.48337	0.741685	−	0.670749i	$-0.234027\pi$
$127$	269624.	1.48337	0.741685	+	0.670749i	$0.234027\pi$
$128$	−16384.0	−0.0883883
$129$	−174924.	−0.925497
$130$	215568.	1.11873
$131$	81180.0	0.413305	0.206653	−	0.978414i	$-0.433743\pi$
$131$	81180.0	0.413305	0.206653	+	0.978414i	$0.433743\pi$
$132$	−31104.0	−0.155375
$133$	43316.0	0.212334
$134$	−95792.0	−0.460858
$135$	39366.0	0.185903
$136$	−83328.0	−0.386317
$137$	−260910.	−1.18765	−0.593826	−	0.804593i	$-0.702383\pi$
$137$	−260910.	−1.18765	−0.593826	+	0.804593i	$0.702383\pi$
$138$	−81648.0	−0.364962
$139$	−297964.	−1.30806	−0.654029	−	0.756470i	$-0.726922\pi$
$139$	−297964.	−1.30806	−0.654029	+	0.756470i	$0.726922\pi$
$140$	−42336.0	−0.182553
$141$	−199800.	−0.846346
$142$	82512.0	0.343397
$143$	215568.	0.881544
$144$	20736.0	0.0833333
$145$	80028.0	0.316098
$146$	−1160.00	−0.00450377
$147$	−21609.0	−0.0824786
$148$	−75424.0	−0.283045
$149$	−398970.	−1.47223	−0.736113	−	0.676859i	$-0.763341\pi$
$149$	−398970.	−1.47223	−0.736113	+	0.676859i	$0.763341\pi$
$150$	−7524.00	−0.0273036
$151$	−224968.	−0.802931	−0.401466	−	0.915874i	$-0.631499\pi$
$151$	−224968.	−0.802931	−0.401466	+	0.915874i	$0.631499\pi$
$152$	−56576.0	−0.198620
$153$	105462.	0.364223
$154$	−42336.0	−0.143849
$155$	−451440.	−1.50928
$156$	−143712.	−0.472804
$157$	−233218.	−0.755115	−0.377557	−	0.925986i	$-0.623236\pi$
$157$	−233218.	−0.755115	−0.377557	+	0.925986i	$0.623236\pi$
$158$	398176.	1.26891
$159$	−241110.	−0.756349
$160$	55296.0	0.170763
$161$	−111132.	−0.337889
$162$	−26244.0	−0.0785674
$163$	466220.	1.37443	0.687214	−	0.726455i	$-0.258834\pi$
$163$	466220.	1.37443	0.687214	+	0.726455i	$0.258834\pi$
$164$	−156576.	−0.454585
$165$	104976.	0.300179
$166$	−77232.0	−0.217534
$167$	−100848.	−0.279818	−0.139909	−	0.990164i	$-0.544681\pi$
$167$	−100848.	−0.279818	−0.139909	+	0.990164i	$0.544681\pi$
$168$	28224.0	0.0771517
$169$	624711.	1.68253
$170$	281232.	0.746350
$171$	71604.0	0.187261
$172$	310976.	0.801504
$173$	−668838.	−1.69905	−0.849524	−	0.527550i	$-0.823111\pi$
$173$	−668838.	−1.69905	−0.849524	+	0.527550i	$0.823111\pi$
$174$	−53352.0	−0.133591
$175$	−10241.0	−0.0252783
$176$	55296.0	0.134559
$177$	−252828.	−0.606585
$178$	−145560.	−0.344344
$179$	−614856.	−1.43430	−0.717151	−	0.696917i	$-0.754554\pi$
$179$	−614856.	−1.43430	−0.717151	+	0.696917i	$0.754554\pi$
$180$	−69984.0	−0.160997
$181$	540686.	1.22673	0.613365	−	0.789800i	$-0.289816\pi$
$181$	540686.	1.22673	0.613365	+	0.789800i	$0.289816\pi$
$182$	−195608.	−0.437732
$183$	349794.	0.772120
$184$	145152.	0.316066
$185$	254556.	0.546832
$186$	300960.	0.637862
$187$	281232.	0.588113
$188$	355200.	0.732957
$189$	−35721.0	−0.0727393
$190$	190944.	0.383727
$191$	−41916.0	−0.0831374	−0.0415687	−	0.999136i	$-0.513236\pi$
$191$	−41916.0	−0.0831374	−0.0415687	+	0.999136i	$0.513236\pi$
$192$	−36864.0	−0.0721688
$193$	−533998.	−1.03192	−0.515960	−	0.856612i	$-0.672565\pi$
$193$	−533998.	−1.03192	−0.515960	+	0.856612i	$0.672565\pi$
$194$	316312.	0.603408
$195$	485028.	0.913441
$196$	38416.0	0.0714286
$197$	824886.	1.51436	0.757179	−	0.653208i	$-0.226577\pi$
$197$	824886.	1.51436	0.757179	+	0.653208i	$0.226577\pi$
$198$	−69984.0	−0.126863
$199$	−399544.	−0.715207	−0.357604	−	0.933873i	$-0.616406\pi$
$199$	−399544.	−0.715207	−0.357604	+	0.933873i	$0.616406\pi$
$200$	13376.0	0.0236457
$201$	−215532.	−0.376289
$202$	−738504.	−1.27343
$203$	−72618.0	−0.123681
$204$	−187488.	−0.315426
$205$	528444.	0.878242
$206$	−258368.	−0.424200
$207$	−183708.	−0.297990
$208$	255488.	0.409461
$209$	190944.	0.302371
$210$	−95256.0	−0.149054
$211$	868868.	1.34353	0.671765	−	0.740764i	$-0.265536\pi$
$211$	868868.	1.34353	0.671765	+	0.740764i	$0.265536\pi$
$212$	428640.	0.655018
$213$	185652.	0.280382
$214$	−598368.	−0.893170
$215$	−1.04954e6	−1.54848
$216$	46656.0	0.0680414
$217$	409640.	0.590545
$218$	255304.	0.363845
$219$	−2610.00	−0.00367731
$220$	−186624.	−0.259963
$221$	1.29940e6	1.78962
$222$	−169704.	−0.231105
$223$	−626656.	−0.843853	−0.421927	−	0.906630i	$-0.638646\pi$
$223$	−626656.	−0.843853	−0.421927	+	0.906630i	$0.638646\pi$
$224$	−50176.0	−0.0668153
$225$	−16929.0	−0.0222933
$226$	284088.	0.369983
$227$	−450396.	−0.580136	−0.290068	−	0.957006i	$-0.593678\pi$
$227$	−450396.	−0.580136	−0.290068	+	0.957006i	$0.593678\pi$
$228$	−127296.	−0.162173
$229$	−1.06453e6	−1.34143	−0.670717	−	0.741714i	$-0.734013\pi$
$229$	−1.06453e6	−1.34143	−0.670717	+	0.741714i	$0.734013\pi$
$230$	−489888.	−0.610629
$231$	−95256.0	−0.117453
$232$	94848.0	0.115693
$233$	1.43618e6	1.73308	0.866540	−	0.499108i	$-0.166339\pi$
$233$	1.43618e6	1.73308	0.866540	+	0.499108i	$0.166339\pi$
$234$	−323352.	−0.386043
$235$	−1.19880e6	−1.41605
$236$	449472.	0.525318
$237$	895896.	1.03606
$238$	−255192.	−0.292028
$239$	−997860.	−1.12999	−0.564995	−	0.825094i	$-0.691122\pi$
$239$	−997860.	−1.12999	−0.564995	+	0.825094i	$0.691122\pi$
$240$	124416.	0.139427
$241$	−227974.	−0.252838	−0.126419	−	0.991977i	$-0.540348\pi$
$241$	−227974.	−0.252838	−0.126419	+	0.991977i	$0.540348\pi$
$242$	457580.	0.502260
$243$	−59049.0	−0.0641500
$244$	−621856.	−0.668675
$245$	−129654.	−0.137997
$246$	−352296.	−0.371168
$247$	882232.	0.920111
$248$	−535040.	−0.552404
$249$	−173772.	−0.177616
$250$	−720144.	−0.728734
$251$	1.51657e6	1.51942	0.759712	−	0.650260i	$-0.225340\pi$
$251$	1.51657e6	1.51942	0.759712	+	0.650260i	$0.225340\pi$
$252$	63504.0	0.0629941
$253$	−489888.	−0.481167
$254$	−1.07850e6	−1.04890
$255$	632772.	0.609392
$256$	65536.0	0.0625000
$257$	455886.	0.430550	0.215275	−	0.976553i	$-0.430935\pi$
$257$	455886.	0.430550	0.215275	+	0.976553i	$0.430935\pi$
$258$	699696.	0.654425
$259$	−230986.	−0.213962
$260$	−862272.	−0.791063
$261$	−120042.	−0.109077
$262$	−324720.	−0.292251
$263$	−752652.	−0.670973	−0.335486	−	0.942045i	$-0.608901\pi$
$263$	−752652.	−0.670973	−0.335486	+	0.942045i	$0.608901\pi$
$264$	124416.	0.109867
$265$	−1.44666e6	−1.26547
$266$	−173264.	−0.150143
$267$	−327510.	−0.281155
$268$	383168.	0.325876
$269$	143682.	0.121066	0.0605329	−	0.998166i	$-0.480720\pi$
$269$	143682.	0.121066	0.0605329	+	0.998166i	$0.480720\pi$
$270$	−157464.	−0.131453
$271$	757496.	0.626552	0.313276	−	0.949662i	$-0.398574\pi$
$271$	757496.	0.626552	0.313276	+	0.949662i	$0.398574\pi$
$272$	333312.	0.273167
$273$	−440118.	−0.357407
$274$	1.04364e6	0.839797
$275$	−45144.0	−0.0359972
$276$	326592.	0.258067
$277$	−1.16214e6	−0.910035	−0.455018	−	0.890482i	$-0.650367\pi$
$277$	−1.16214e6	−0.910035	−0.455018	+	0.890482i	$0.650367\pi$
$278$	1.19186e6	0.924936
$279$	677160.	0.520812
$280$	169344.	0.129085
$281$	−414366.	−0.313053	−0.156527	−	0.987674i	$-0.550030\pi$
$281$	−414366.	−0.313053	−0.156527	+	0.987674i	$0.550030\pi$
$282$	799200.	0.598457
$283$	120428.	0.0893843	0.0446922	−	0.999001i	$-0.485769\pi$
$283$	120428.	0.0893843	0.0446922	+	0.999001i	$0.485769\pi$
$284$	−330048.	−0.242818
$285$	429624.	0.313312
$286$	−862272.	−0.623346
$287$	−479514.	−0.343634
$288$	−82944.0	−0.0589256
$289$	275347.	0.193926
$290$	−320112.	−0.223515
$291$	711702.	0.492681
$292$	4640.00	0.00318464
$293$	2.20159e6	1.49819	0.749094	−	0.662463i	$-0.230489\pi$
$293$	2.20159e6	1.49819	0.749094	+	0.662463i	$0.230489\pi$
$294$	86436.0	0.0583212
$295$	−1.51697e6	−1.01490
$296$	301696.	0.200143
$297$	−157464.	−0.103583
$298$	1.59588e6	1.04102
$299$	−2.26346e6	−1.46418
$300$	30096.0	0.0193066
$301$	952364.	0.605880
$302$	899872.	0.567758
$303$	−1.66163e6	−1.03975
$304$	226304.	0.140446
$305$	2.09876e6	1.29186
$306$	−421848.	−0.257545
$307$	110900.	0.0671561	0.0335781	−	0.999436i	$-0.489310\pi$
$307$	110900.	0.0671561	0.0335781	+	0.999436i	$0.489310\pi$
$308$	169344.	0.101717
$309$	−581328.	−0.346358
$310$	1.80576e6	1.06722
$311$	−910608.	−0.533864	−0.266932	−	0.963715i	$-0.586010\pi$
$311$	−910608.	−0.533864	−0.266932	+	0.963715i	$0.586010\pi$
$312$	574848.	0.334323
$313$	3.12247e6	1.80152	0.900758	−	0.434322i	$-0.143012\pi$
$313$	3.12247e6	1.80152	0.900758	+	0.434322i	$0.143012\pi$
$314$	932872.	0.533947
$315$	−214326.	−0.121702
$316$	−1.59270e6	−0.897258
$317$	−2.76688e6	−1.54647	−0.773237	−	0.634117i	$-0.781364\pi$
$317$	−2.76688e6	−1.54647	−0.773237	+	0.634117i	$0.781364\pi$
$318$	964440.	0.534820
$319$	−320112.	−0.176127
$320$	−221184.	−0.120748
$321$	−1.34633e6	−0.729270
$322$	444528.	0.238924
$323$	1.15097e6	0.613842
$324$	104976.	0.0555556
$325$	−208582.	−0.109539
$326$	−1.86488e6	−0.971867
$327$	574434.	0.297078
$328$	626304.	0.321440
$329$	1.08780e6	0.554063
$330$	−419904.	−0.212259
$331$	3.22257e6	1.61671	0.808356	−	0.588694i	$-0.200358\pi$
$331$	3.22257e6	1.61671	0.808356	+	0.588694i	$0.200358\pi$
$332$	308928.	0.153820
$333$	−381834.	−0.188697
$334$	403392.	0.197861
$335$	−1.29319e6	−0.629580
$336$	−112896.	−0.0545545
$337$	1.63306e6	0.783298	0.391649	−	0.920115i	$-0.371905\pi$
$337$	1.63306e6	0.783298	0.391649	+	0.920115i	$0.371905\pi$
$338$	−2.49884e6	−1.18973
$339$	639198.	0.302090
$340$	−1.12493e6	−0.527749
$341$	1.80576e6	0.840958
$342$	−286416.	−0.132413
$343$	117649.	0.0539949
$344$	−1.24390e6	−0.566749
$345$	−1.10225e6	−0.498576
$346$	2.67535e6	1.20141
$347$	1.03642e6	0.462073	0.231036	−	0.972945i	$-0.425788\pi$
$347$	1.03642e6	0.462073	0.231036	+	0.972945i	$0.425788\pi$
$348$	213408.	0.0944632
$349$	−4.22999e6	−1.85898	−0.929491	−	0.368844i	$-0.879754\pi$
$349$	−4.22999e6	−1.85898	−0.929491	+	0.368844i	$0.879754\pi$
$350$	40964.0	0.0178744
$351$	−727542.	−0.315203
$352$	−221184.	−0.0951474
$353$	238806.	0.102002	0.0510010	−	0.998699i	$-0.483759\pi$
$353$	238806.	0.102002	0.0510010	+	0.998699i	$0.483759\pi$
$354$	1.01131e6	0.428921
$355$	1.11391e6	0.469116
$356$	582240.	0.243488
$357$	−574182.	−0.238440
$358$	2.45942e6	1.01421
$359$	−2.66428e6	−1.09105	−0.545523	−	0.838096i	$-0.683669\pi$
$359$	−2.66428e6	−1.09105	−0.545523	+	0.838096i	$0.683669\pi$
$360$	279936.	0.113842
$361$	−1.69464e6	−0.684400
$362$	−2.16274e6	−0.867429
$363$	1.02956e6	0.410094
$364$	782432.	0.309523
$365$	−15660.0	−0.00615261
$366$	−1.39918e6	−0.545971
$367$	−1.71083e6	−0.663044	−0.331522	−	0.943448i	$-0.607562\pi$
$367$	−1.71083e6	−0.663044	−0.331522	+	0.943448i	$0.607562\pi$
$368$	−580608.	−0.223493
$369$	−792666.	−0.303057
$370$	−1.01822e6	−0.386669
$371$	1.31271e6	0.495147
$372$	−1.20384e6	−0.451036
$373$	−3.96649e6	−1.47616	−0.738081	−	0.674712i	$-0.764268\pi$
$373$	−3.96649e6	−1.47616	−0.738081	+	0.674712i	$0.764268\pi$
$374$	−1.12493e6	−0.415859
$375$	−1.62032e6	−0.595009
$376$	−1.42080e6	−0.518279
$377$	−1.47904e6	−0.535951
$378$	142884.	0.0514344
$379$	828668.	0.296335	0.148167	−	0.988962i	$-0.452663\pi$
$379$	828668.	0.296335	0.148167	+	0.988962i	$0.452663\pi$
$380$	−763776.	−0.271336
$381$	−2.42662e6	−0.856424
$382$	167664.	0.0587870
$383$	−2.55686e6	−0.890657	−0.445329	−	0.895367i	$-0.646913\pi$
$383$	−2.55686e6	−0.890657	−0.445329	+	0.895367i	$0.646913\pi$
$384$	147456.	0.0510310
$385$	−571536.	−0.196513
$386$	2.13599e6	0.729678
$387$	1.57432e6	0.534336
$388$	−1.26525e6	−0.426674
$389$	2.91785e6	0.977664	0.488832	−	0.872378i	$-0.337423\pi$
$389$	2.91785e6	0.977664	0.488832	+	0.872378i	$0.337423\pi$
$390$	−1.94011e6	−0.645900
$391$	−2.95294e6	−0.976815
$392$	−153664.	−0.0505076
$393$	−730620.	−0.238622
$394$	−3.29954e6	−1.07081
$395$	5.37538e6	1.73347
$396$	279936.	0.0897059
$397$	2.50715e6	0.798370	0.399185	−	0.916870i	$-0.369293\pi$
$397$	2.50715e6	0.798370	0.399185	+	0.916870i	$0.369293\pi$
$398$	1.59818e6	0.505728
$399$	−389844.	−0.122591
$400$	−53504.0	−0.0167200
$401$	990666.	0.307657	0.153828	−	0.988098i	$-0.450840\pi$
$401$	990666.	0.307657	0.153828	+	0.988098i	$0.450840\pi$
$402$	862128.	0.266077
$403$	8.34328e6	2.55902
$404$	2.95402e6	0.900450
$405$	−354294.	−0.107331
$406$	290472.	0.0874559
$407$	−1.01822e6	−0.304689
$408$	749952.	0.223040
$409$	4.51824e6	1.33555	0.667777	−	0.744362i	$-0.267246\pi$
$409$	4.51824e6	1.33555	0.667777	+	0.744362i	$0.267246\pi$
$410$	−2.11378e6	−0.621011
$411$	2.34819e6	0.685691
$412$	1.03347e6	0.299955
$413$	1.37651e6	0.397103
$414$	734832.	0.210711
$415$	−1.04263e6	−0.297174
$416$	−1.02195e6	−0.289532
$417$	2.68168e6	0.755207
$418$	−763776.	−0.213809
$419$	605220.	0.168414	0.0842070	−	0.996448i	$-0.473164\pi$
$419$	605220.	0.168414	0.0842070	+	0.996448i	$0.473164\pi$
$420$	381024.	0.105397
$421$	4.49893e6	1.23710	0.618549	−	0.785746i	$-0.287721\pi$
$421$	4.49893e6	1.23710	0.618549	+	0.785746i	$0.287721\pi$
$422$	−3.47547e6	−0.950020
$423$	1.79820e6	0.488638
$424$	−1.71456e6	−0.463167
$425$	−272118.	−0.0730777
$426$	−742608.	−0.198260
$427$	−1.90443e6	−0.505471
$428$	2.39347e6	0.631566
$429$	−1.94011e6	−0.508960
$430$	4.19818e6	1.09494
$431$	5.37594e6	1.39400	0.696998	−	0.717074i	$-0.254519\pi$
$431$	5.37594e6	1.39400	0.696998	+	0.717074i	$0.254519\pi$
$432$	−186624.	−0.0481125
$433$	−1.98561e6	−0.508950	−0.254475	−	0.967079i	$-0.581903\pi$
$433$	−1.98561e6	−0.508950	−0.254475	+	0.967079i	$0.581903\pi$
$434$	−1.63856e6	−0.417578
$435$	−720252.	−0.182499
$436$	−1.02122e6	−0.257277
$437$	−2.00491e6	−0.502217
$438$	10440.0	0.00260025
$439$	3.38727e6	0.838859	0.419429	−	0.907788i	$-0.362230\pi$
$439$	3.38727e6	0.838859	0.419429	+	0.907788i	$0.362230\pi$
$440$	746496.	0.183821
$441$	194481.	0.0476190
$442$	−5.19758e6	−1.26545
$443$	2.14094e6	0.518318	0.259159	−	0.965835i	$-0.416555\pi$
$443$	2.14094e6	0.518318	0.259159	+	0.965835i	$0.416555\pi$
$444$	678816.	0.163416
$445$	−1.96506e6	−0.470409
$446$	2.50662e6	0.596695
$447$	3.59073e6	0.849990
$448$	200704.	0.0472456
$449$	−6.97808e6	−1.63350	−0.816752	−	0.576990i	$-0.804227\pi$
$449$	−6.97808e6	−1.63350	−0.816752	+	0.576990i	$0.804227\pi$
$450$	67716.0	0.0157638
$451$	−2.11378e6	−0.489348
$452$	−1.13635e6	−0.261618
$453$	2.02471e6	0.463573
$454$	1.80158e6	0.410218
$455$	−2.64071e6	−0.597987
$456$	509184.	0.114673
$457$	−5.17999e6	−1.16021	−0.580107	−	0.814540i	$-0.696989\pi$
$457$	−5.17999e6	−1.16021	−0.580107	+	0.814540i	$0.696989\pi$
$458$	4.25812e6	0.948537
$459$	−949158.	−0.210284
$460$	1.95955e6	0.431780
$461$	−7.83001e6	−1.71597	−0.857985	−	0.513674i	$-0.828284\pi$
$461$	−7.83001e6	−1.71597	−0.857985	+	0.513674i	$0.828284\pi$
$462$	381024.	0.0830515
$463$	165320.	0.0358404	0.0179202	−	0.999839i	$-0.494296\pi$
$463$	165320.	0.0358404	0.0179202	+	0.999839i	$0.494296\pi$
$464$	−379392.	−0.0818075
$465$	4.06296e6	0.871385
$466$	−5.74471e6	−1.22547
$467$	−1.79329e6	−0.380504	−0.190252	−	0.981735i	$-0.560930\pi$
$467$	−1.79329e6	−0.380504	−0.190252	+	0.981735i	$0.560930\pi$
$468$	1.29341e6	0.272974
$469$	1.17345e6	0.246339
$470$	4.79520e6	1.00130
$471$	2.09896e6	0.435966
$472$	−1.79789e6	−0.371456
$473$	4.19818e6	0.862795
$474$	−3.58358e6	−0.732608
$475$	−184756.	−0.0375720
$476$	1.02077e6	0.206495
$477$	2.16999e6	0.436678
$478$	3.99144e6	0.799024
$479$	−6.59657e6	−1.31365	−0.656824	−	0.754044i	$-0.728101\pi$
$479$	−6.59657e6	−1.31365	−0.656824	+	0.754044i	$0.728101\pi$
$480$	−497664.	−0.0985901
$481$	−4.70457e6	−0.927166
$482$	911896.	0.178784
$483$	1.00019e6	0.195080
$484$	−1.83032e6	−0.355151
$485$	4.27021e6	0.824319
$486$	236196.	0.0453609
$487$	−5.97393e6	−1.14140	−0.570700	−	0.821159i	$-0.693328\pi$
$487$	−5.97393e6	−1.14140	−0.570700	+	0.821159i	$0.693328\pi$
$488$	2.48742e6	0.472825
$489$	−4.19598e6	−0.793526
$490$	518616.	0.0975789
$491$	381264.	0.0713710	0.0356855	−	0.999363i	$-0.488639\pi$
$491$	381264.	0.0713710	0.0356855	+	0.999363i	$0.488639\pi$
$492$	1.40918e6	0.262455
$493$	−1.92956e6	−0.357554
$494$	−3.52893e6	−0.650617
$495$	−944784.	−0.173308
$496$	2.14016e6	0.390609
$497$	−1.01077e6	−0.183553
$498$	695088.	0.125593
$499$	1.54351e6	0.277497	0.138748	−	0.990328i	$-0.455692\pi$
$499$	1.54351e6	0.277497	0.138748	+	0.990328i	$0.455692\pi$
$500$	2.88058e6	0.515293
$501$	907632.	0.161553
$502$	−6.06629e6	−1.07439
$503$	−4.02300e6	−0.708974	−0.354487	−	0.935061i	$-0.615344\pi$
$503$	−4.02300e6	−0.708974	−0.354487	+	0.935061i	$0.615344\pi$
$504$	−254016.	−0.0445435
$505$	−9.96980e6	−1.73964
$506$	1.95955e6	0.340236
$507$	−5.62240e6	−0.971408
$508$	4.31398e6	0.741685
$509$	−1.94715e6	−0.333123	−0.166562	−	0.986031i	$-0.553266\pi$
$509$	−1.94715e6	−0.333123	−0.166562	+	0.986031i	$0.553266\pi$
$510$	−2.53109e6	−0.430905
$511$	14210.0	0.00240736
$512$	−262144.	−0.0441942
$513$	−644436.	−0.108115
$514$	−1.82354e6	−0.304445
$515$	−3.48797e6	−0.579501
$516$	−2.79878e6	−0.462749
$517$	4.79520e6	0.789006
$518$	923944.	0.151294
$519$	6.01954e6	0.980946
$520$	3.44909e6	0.559366
$521$	7.38569e6	1.19206	0.596028	−	0.802963i	$-0.296745\pi$
$521$	7.38569e6	1.19206	0.596028	+	0.802963i	$0.296745\pi$
$522$	480168.	0.0771289
$523$	−329740.	−0.0527130	−0.0263565	−	0.999653i	$-0.508390\pi$
$523$	−329740.	−0.0527130	−0.0263565	+	0.999653i	$0.508390\pi$
$524$	1.29888e6	0.206653
$525$	92169.0	0.0145944
$526$	3.01061e6	0.474449
$527$	1.08847e7	1.70722
$528$	−497664.	−0.0776875
$529$	−1.29252e6	−0.200816
$530$	5.78664e6	0.894822
$531$	2.27545e6	0.350212
$532$	693056.	0.106167
$533$	−9.76643e6	−1.48908
$534$	1.31004e6	0.198807
$535$	−8.07797e6	−1.22016
$536$	−1.53267e6	−0.230429
$537$	5.53370e6	0.828095
$538$	−574728.	−0.0856065
$539$	518616.	0.0768907
$540$	629856.	0.0929516
$541$	87086.0	0.0127925	0.00639625	−	0.999980i	$-0.497964\pi$
$541$	87086.0	0.0127925	0.00639625	+	0.999980i	$0.497964\pi$
$542$	−3.02998e6	−0.443039
$543$	−4.86617e6	−0.708252
$544$	−1.33325e6	−0.193158
$545$	3.44660e6	0.497050
$546$	1.76047e6	0.252725
$547$	6.91531e6	0.988196	0.494098	−	0.869406i	$-0.335498\pi$
$547$	6.91531e6	0.988196	0.494098	+	0.869406i	$0.335498\pi$
$548$	−4.17456e6	−0.593826
$549$	−3.14815e6	−0.445784
$550$	180576.	0.0254538
$551$	−1.31009e6	−0.183832
$552$	−1.30637e6	−0.182481
$553$	−4.87766e6	−0.678263
$554$	4.64855e6	0.643492
$555$	−2.29100e6	−0.315714
$556$	−4.76742e6	−0.654029
$557$	−1.52258e6	−0.207942	−0.103971	−	0.994580i	$-0.533155\pi$
$557$	−1.52258e6	−0.207942	−0.103971	+	0.994580i	$0.533155\pi$
$558$	−2.70864e6	−0.368270
$559$	1.93971e7	2.62548
$560$	−677376.	−0.0912767
$561$	−2.53109e6	−0.339547
$562$	1.65746e6	0.221362
$563$	−7.86462e6	−1.04570	−0.522850	−	0.852425i	$-0.675131\pi$
$563$	−7.86462e6	−1.04570	−0.522850	+	0.852425i	$0.675131\pi$
$564$	−3.19680e6	−0.423173
$565$	3.83519e6	0.505435
$566$	−481712.	−0.0632043
$567$	321489.	0.0419961
$568$	1.32019e6	0.171698
$569$	−1.46321e6	−0.189464	−0.0947321	−	0.995503i	$-0.530199\pi$
$569$	−1.46321e6	−0.189464	−0.0947321	+	0.995503i	$0.530199\pi$
$570$	−1.71850e6	−0.221545
$571$	9.19855e6	1.18067	0.590336	−	0.807158i	$-0.298995\pi$
$571$	9.19855e6	1.18067	0.590336	+	0.807158i	$0.298995\pi$
$572$	3.44909e6	0.440772
$573$	377244.	0.0479994
$574$	1.91806e6	0.242986
$575$	474012.	0.0597888
$576$	331776.	0.0416667
$577$	3.28939e6	0.411317	0.205658	−	0.978624i	$-0.434066\pi$
$577$	3.28939e6	0.411317	0.205658	+	0.978624i	$0.434066\pi$
$578$	−1.10139e6	−0.137126
$579$	4.80598e6	0.595780
$580$	1.28045e6	0.158049
$581$	946092.	0.116277
$582$	−2.84681e6	−0.348378
$583$	5.78664e6	0.705107
$584$	−18560.0	−0.00225188
$585$	−4.36525e6	−0.527375
$586$	−8.80634e6	−1.05938
$587$	5.12929e6	0.614416	0.307208	−	0.951642i	$-0.400605\pi$
$587$	5.12929e6	0.614416	0.307208	+	0.951642i	$0.400605\pi$
$588$	−345744.	−0.0412393
$589$	7.39024e6	0.877749
$590$	6.06787e6	0.717640
$591$	−7.42397e6	−0.874315
$592$	−1.20678e6	−0.141522
$593$	−2.75433e6	−0.321647	−0.160823	−	0.986983i	$-0.551415\pi$
$593$	−2.75433e6	−0.321647	−0.160823	+	0.986983i	$0.551415\pi$
$594$	629856.	0.0732445
$595$	−3.44509e6	−0.398941
$596$	−6.38352e6	−0.736113
$597$	3.59590e6	0.412925
$598$	9.05386e6	1.03533
$599$	−9.88616e6	−1.12580	−0.562899	−	0.826525i	$-0.690314\pi$
$599$	−9.88616e6	−1.12580	−0.562899	+	0.826525i	$0.690314\pi$
$600$	−120384.	−0.0136518
$601$	1.37039e7	1.54760	0.773798	−	0.633433i	$-0.218355\pi$
$601$	1.37039e7	1.54760	0.773798	+	0.633433i	$0.218355\pi$
$602$	−3.80946e6	−0.428422
$603$	1.93979e6	0.217251
$604$	−3.59949e6	−0.401466
$605$	6.17733e6	0.686139
$606$	6.64654e6	0.735214
$607$	−7.85310e6	−0.865107	−0.432553	−	0.901608i	$-0.642387\pi$
$607$	−7.85310e6	−0.865107	−0.432553	+	0.901608i	$0.642387\pi$
$608$	−905216.	−0.0993101
$609$	653562.	0.0714075
$610$	−8.39506e6	−0.913480
$611$	2.21556e7	2.40094
$612$	1.68739e6	0.182112
$613$	1.46977e7	1.57978	0.789892	−	0.613246i	$-0.210136\pi$
$613$	1.46977e7	1.57978	0.789892	+	0.613246i	$0.210136\pi$
$614$	−443600.	−0.0474865
$615$	−4.75600e6	−0.507053
$616$	−677376.	−0.0719247
$617$	6.28370e6	0.664511	0.332256	−	0.943189i	$-0.392190\pi$
$617$	6.28370e6	0.664511	0.332256	+	0.943189i	$0.392190\pi$
$618$	2.32531e6	0.244912
$619$	−2.26692e6	−0.237799	−0.118900	−	0.992906i	$-0.537937\pi$
$619$	−2.26692e6	−0.237799	−0.118900	+	0.992906i	$0.537937\pi$
$620$	−7.22304e6	−0.754642
$621$	1.65337e6	0.172045
$622$	3.64243e6	0.377499
$623$	1.78311e6	0.184059
$624$	−2.29939e6	−0.236402
$625$	−9.06882e6	−0.928647
$626$	−1.24899e7	−1.27386
$627$	−1.71850e6	−0.174574
$628$	−3.73149e6	−0.377557
$629$	−6.13763e6	−0.618549
$630$	857304.	0.0860565
$631$	−1.17477e7	−1.17457	−0.587285	−	0.809380i	$-0.699803\pi$
$631$	−1.17477e7	−1.17457	−0.587285	+	0.809380i	$0.699803\pi$
$632$	6.37082e6	0.634457
$633$	−7.81981e6	−0.775688
$634$	1.10675e7	1.09352
$635$	−1.45597e7	−1.43291
$636$	−3.85776e6	−0.378175
$637$	2.39620e6	0.233978
$638$	1.28045e6	0.124540
$639$	−1.67087e6	−0.161879
$640$	884736.	0.0853815
$641$	5.93231e6	0.570268	0.285134	−	0.958488i	$-0.407962\pi$
$641$	5.93231e6	0.570268	0.285134	+	0.958488i	$0.407962\pi$
$642$	5.38531e6	0.515672
$643$	−6.94443e6	−0.662383	−0.331191	−	0.943564i	$-0.607451\pi$
$643$	−6.94443e6	−0.662383	−0.331191	+	0.943564i	$0.607451\pi$
$644$	−1.77811e6	−0.168945
$645$	9.44590e6	0.894013
$646$	−4.60387e6	−0.434052
$647$	−4.97050e6	−0.466809	−0.233404	−	0.972380i	$-0.574987\pi$
$647$	−4.97050e6	−0.466809	−0.233404	+	0.972380i	$0.574987\pi$
$648$	−419904.	−0.0392837
$649$	6.06787e6	0.565490
$650$	834328.	0.0774557
$651$	−3.68676e6	−0.340951
$652$	7.45952e6	0.687214
$653$	−1.83355e7	−1.68271	−0.841354	−	0.540484i	$-0.818241\pi$
$653$	−1.83355e7	−1.68271	−0.841354	+	0.540484i	$0.818241\pi$
$654$	−2.29774e6	−0.210066
$655$	−4.38372e6	−0.399245
$656$	−2.50522e6	−0.227293
$657$	23490.0	0.00212310
$658$	−4.35120e6	−0.391782
$659$	9.01402e6	0.808546	0.404273	−	0.914638i	$-0.367525\pi$
$659$	9.01402e6	0.808546	0.404273	+	0.914638i	$0.367525\pi$
$660$	1.67962e6	0.150089
$661$	699398.	0.0622617	0.0311308	−	0.999515i	$-0.490089\pi$
$661$	699398.	0.0622617	0.0311308	+	0.999515i	$0.490089\pi$
$662$	−1.28903e7	−1.14319
$663$	−1.16946e7	−1.03324
$664$	−1.23571e6	−0.108767
$665$	−2.33906e6	−0.205111
$666$	1.52734e6	0.133429
$667$	3.36118e6	0.292534
$668$	−1.61357e6	−0.139909
$669$	5.63990e6	0.487199
$670$	5.17277e6	0.445180
$671$	−8.39506e6	−0.719809
$672$	451584.	0.0385758
$673$	−5.80603e6	−0.494130	−0.247065	−	0.968999i	$-0.579466\pi$
$673$	−5.80603e6	−0.494130	−0.247065	+	0.968999i	$0.579466\pi$
$674$	−6.53223e6	−0.553875
$675$	152361.	0.0128711
$676$	9.99538e6	0.841264
$677$	985074.	0.0826033	0.0413016	−	0.999147i	$-0.486850\pi$
$677$	985074.	0.0826033	0.0413016	+	0.999147i	$0.486850\pi$
$678$	−2.55679e6	−0.213610
$679$	−3.87482e6	−0.322535
$680$	4.49971e6	0.373175
$681$	4.05356e6	0.334942
$682$	−7.22304e6	−0.594647
$683$	−1.88208e7	−1.54379	−0.771894	−	0.635752i	$-0.780690\pi$
$683$	−1.88208e7	−1.54379	−0.771894	+	0.635752i	$0.780690\pi$
$684$	1.14566e6	0.0936304
$685$	1.40891e7	1.14725
$686$	−470596.	−0.0381802
$687$	9.58077e6	0.774477
$688$	4.97562e6	0.400752
$689$	2.67364e7	2.14563
$690$	4.40899e6	0.352547
$691$	−1.93385e7	−1.54073	−0.770366	−	0.637601i	$-0.779927\pi$
$691$	−1.93385e7	−1.54073	−0.770366	+	0.637601i	$0.779927\pi$
$692$	−1.07014e7	−0.849524
$693$	857304.	0.0678113
$694$	−4.14566e6	−0.326735
$695$	1.60901e7	1.26356
$696$	−853632.	−0.0667956
$697$	−1.27414e7	−0.993423
$698$	1.69199e7	1.31450
$699$	−1.29256e7	−1.00059
$700$	−163856.	−0.0126391
$701$	−1.41489e6	−0.108750	−0.0543748	−	0.998521i	$-0.517317\pi$
$701$	−1.41489e6	−0.108750	−0.0543748	+	0.998521i	$0.517317\pi$
$702$	2.91017e6	0.222882
$703$	−4.16718e6	−0.318019
$704$	884736.	0.0672794
$705$	1.07892e7	0.817554
$706$	−955224.	−0.0721263
$707$	9.04667e6	0.680676
$708$	−4.04525e6	−0.303293
$709$	−754906.	−0.0563998	−0.0281999	−	0.999602i	$-0.508977\pi$
$709$	−754906.	−0.0563998	−0.0281999	+	0.999602i	$0.508977\pi$
$710$	−4.45565e6	−0.331715
$711$	−8.06306e6	−0.598172
$712$	−2.32896e6	−0.172172
$713$	−1.89605e7	−1.39677
$714$	2.29673e6	0.168603
$715$	−1.16407e7	−0.851555
$716$	−9.83770e6	−0.717151
$717$	8.98074e6	0.652400
$718$	1.06571e7	0.771486
$719$	1.08854e6	0.0785279	0.0392639	−	0.999229i	$-0.487499\pi$
$719$	1.08854e6	0.0785279	0.0392639	+	0.999229i	$0.487499\pi$
$720$	−1.11974e6	−0.0804984
$721$	3.16501e6	0.226744
$722$	6.77857e6	0.483944
$723$	2.05177e6	0.145976
$724$	8.65098e6	0.613365
$725$	309738.	0.0218851
$726$	−4.11822e6	−0.289980
$727$	−755392.	−0.0530074	−0.0265037	−	0.999649i	$-0.508437\pi$
$727$	−755392.	−0.0530074	−0.0265037	+	0.999649i	$0.508437\pi$
$728$	−3.12973e6	−0.218866
$729$	531441.	0.0370370
$730$	62640.0	0.00435055
$731$	2.53057e7	1.75156
$732$	5.59670e6	0.386060
$733$	1.56369e6	0.107495	0.0537477	−	0.998555i	$-0.482883\pi$
$733$	1.56369e6	0.107495	0.0537477	+	0.998555i	$0.482883\pi$
$734$	6.84333e6	0.468843
$735$	1.16689e6	0.0796728
$736$	2.32243e6	0.158033
$737$	5.17277e6	0.350796
$738$	3.17066e6	0.214294
$739$	−1.05544e7	−0.710922	−0.355461	−	0.934691i	$-0.615676\pi$
$739$	−1.05544e7	−0.710922	−0.355461	+	0.934691i	$0.615676\pi$
$740$	4.07290e6	0.273416
$741$	−7.94009e6	−0.531227
$742$	−5.25084e6	−0.350122
$743$	1.73678e7	1.15418	0.577088	−	0.816682i	$-0.304189\pi$
$743$	1.73678e7	1.15418	0.577088	+	0.816682i	$0.304189\pi$
$744$	4.81536e6	0.318931
$745$	2.15444e7	1.42214
$746$	1.58660e7	1.04380
$747$	1.56395e6	0.102546
$748$	4.49971e6	0.294056
$749$	7.33001e6	0.477419
$750$	6.48130e6	0.420735
$751$	−2.80181e7	−1.81276	−0.906378	−	0.422467i	$-0.861164\pi$
$751$	−2.80181e7	−1.81276	−0.906378	+	0.422467i	$0.861164\pi$
$752$	5.68320e6	0.366478
$753$	−1.36491e7	−0.877239
$754$	5.91614e6	0.378975
$755$	1.21483e7	0.775617
$756$	−571536.	−0.0363696
$757$	−1.01979e7	−0.646801	−0.323401	−	0.946262i	$-0.604826\pi$
$757$	−1.01979e7	−0.646801	−0.323401	+	0.946262i	$0.604826\pi$
$758$	−3.31467e6	−0.209540
$759$	4.40899e6	0.277802
$760$	3.05510e6	0.191863
$761$	2.57535e6	0.161204	0.0806018	−	0.996746i	$-0.474316\pi$
$761$	2.57535e6	0.161204	0.0806018	+	0.996746i	$0.474316\pi$
$762$	9.70646e6	0.605583
$763$	−3.12747e6	−0.194483
$764$	−670656.	−0.0415687
$765$	−5.69495e6	−0.351833
$766$	1.02275e7	0.629790
$767$	2.80358e7	1.72078
$768$	−589824.	−0.0360844
$769$	971234.	0.0592254	0.0296127	−	0.999561i	$-0.490573\pi$
$769$	971234.	0.0592254	0.0296127	+	0.999561i	$0.490573\pi$
$770$	2.28614e6	0.138956
$771$	−4.10297e6	−0.248578
$772$	−8.54397e6	−0.515960
$773$	−1.72921e7	−1.04088	−0.520439	−	0.853899i	$-0.674232\pi$
$773$	−1.72921e7	−1.04088	−0.520439	+	0.853899i	$0.674232\pi$
$774$	−6.29726e6	−0.377833
$775$	−1.74724e6	−0.104496
$776$	5.06099e6	0.301704
$777$	2.07887e6	0.123531
$778$	−1.16714e7	−0.691313
$779$	−8.65082e6	−0.510756
$780$	7.76045e6	0.456720
$781$	−4.45565e6	−0.261387
$782$	1.18117e7	0.690712
$783$	1.08038e6	0.0629755
$784$	614656.	0.0357143
$785$	1.25938e7	0.729427
$786$	2.92248e6	0.168731
$787$	−1.65515e7	−0.952576	−0.476288	−	0.879289i	$-0.658018\pi$
$787$	−1.65515e7	−0.952576	−0.476288	+	0.879289i	$0.658018\pi$
$788$	1.31982e7	0.757179
$789$	6.77387e6	0.387386
$790$	−2.15015e7	−1.22575
$791$	−3.48008e6	−0.197764
$792$	−1.11974e6	−0.0634316
$793$	−3.87883e7	−2.19037
$794$	−1.00286e7	−0.564533
$795$	1.30199e7	0.730619
$796$	−6.39270e6	−0.357604
$797$	2.91057e6	0.162305	0.0811526	−	0.996702i	$-0.474140\pi$
$797$	2.91057e6	0.162305	0.0811526	+	0.996702i	$0.474140\pi$
$798$	1.55938e6	0.0866849
$799$	2.89044e7	1.60176
$800$	214016.	0.0118228
$801$	2.94759e6	0.162325
$802$	−3.96266e6	−0.217546
$803$	62640.0	0.00342817
$804$	−3.44851e6	−0.188145
$805$	6.00113e6	0.326395
$806$	−3.33731e7	−1.80950
$807$	−1.29314e6	−0.0698974
$808$	−1.18161e7	−0.636714
$809$	−1.16252e7	−0.624496	−0.312248	−	0.950001i	$-0.601082\pi$
$809$	−1.16252e7	−0.624496	−0.312248	+	0.950001i	$0.601082\pi$
$810$	1.41718e6	0.0758947
$811$	3.09020e7	1.64981	0.824906	−	0.565270i	$-0.191228\pi$
$811$	3.09020e7	1.64981	0.824906	+	0.565270i	$0.191228\pi$
$812$	−1.16189e6	−0.0618407
$813$	−6.81746e6	−0.361740
$814$	4.07290e6	0.215448
$815$	−2.51759e7	−1.32767
$816$	−2.99981e6	−0.157713
$817$	1.71814e7	0.900542
$818$	−1.80730e7	−0.944379
$819$	3.96106e6	0.206349
$820$	8.45510e6	0.439121
$821$	−2.22870e7	−1.15397	−0.576984	−	0.816755i	$-0.695771\pi$
$821$	−2.22870e7	−1.15397	−0.576984	+	0.816755i	$0.695771\pi$
$822$	−9.39276e6	−0.484857
$823$	−1.64895e7	−0.848610	−0.424305	−	0.905519i	$-0.639482\pi$
$823$	−1.64895e7	−0.848610	−0.424305	+	0.905519i	$0.639482\pi$
$824$	−4.13389e6	−0.212100
$825$	406296.	0.0207830
$826$	−5.50603e6	−0.280795
$827$	−2.37457e7	−1.20732	−0.603658	−	0.797244i	$-0.706291\pi$
$827$	−2.37457e7	−1.20732	−0.603658	+	0.797244i	$0.706291\pi$
$828$	−2.93933e6	−0.148995
$829$	2.60865e7	1.31835	0.659173	−	0.751991i	$-0.270906\pi$
$829$	2.60865e7	1.31835	0.659173	+	0.751991i	$0.270906\pi$
$830$	4.17053e6	0.210134
$831$	1.04592e7	0.525409
$832$	4.08781e6	0.204730
$833$	3.12610e6	0.156096
$834$	−1.07267e7	−0.534012
$835$	5.44579e6	0.270299
$836$	3.05510e6	0.151186
$837$	−6.09444e6	−0.300691
$838$	−2.42088e6	−0.119087
$839$	1.00872e7	0.494729	0.247365	−	0.968922i	$-0.420435\pi$
$839$	1.00872e7	0.494729	0.247365	+	0.968922i	$0.420435\pi$
$840$	−1.52410e6	−0.0745271
$841$	−1.83148e7	−0.892920
$842$	−1.79957e7	−0.874761
$843$	3.72929e6	0.180741
$844$	1.39019e7	0.671765
$845$	−3.37344e7	−1.62529
$846$	−7.19280e6	−0.345519
$847$	−5.60536e6	−0.268469
$848$	6.85824e6	0.327509
$849$	−1.08385e6	−0.0516061
$850$	1.08847e6	0.0516737
$851$	1.06914e7	0.506068
$852$	2.97043e6	0.140191
$853$	−2.43630e7	−1.14646	−0.573229	−	0.819395i	$-0.694309\pi$
$853$	−2.43630e7	−1.14646	−0.573229	+	0.819395i	$0.694309\pi$
$854$	7.61774e6	0.357422
$855$	−3.86662e6	−0.180890
$856$	−9.57389e6	−0.446585
$857$	2.45612e6	0.114234	0.0571172	−	0.998367i	$-0.481809\pi$
$857$	2.45612e6	0.114234	0.0571172	+	0.998367i	$0.481809\pi$
$858$	7.76045e6	0.359889
$859$	8.62982e6	0.399042	0.199521	−	0.979894i	$-0.436061\pi$
$859$	8.62982e6	0.399042	0.199521	+	0.979894i	$0.436061\pi$
$860$	−1.67927e7	−0.774238
$861$	4.31563e6	0.198397
$862$	−2.15038e7	−0.985703
$863$	1.05199e7	0.480824	0.240412	−	0.970671i	$-0.422717\pi$
$863$	1.05199e7	0.480824	0.240412	+	0.970671i	$0.422717\pi$
$864$	746496.	0.0340207
$865$	3.61173e7	1.64125
$866$	7.94246e6	0.359882
$867$	−2.47812e6	−0.111963
$868$	6.55424e6	0.295273
$869$	−2.15015e7	−0.965872
$870$	2.88101e6	0.129047
$871$	2.39001e7	1.06747
$872$	4.08486e6	0.181922
$873$	−6.40532e6	−0.284449
$874$	8.01965e6	0.355121
$875$	8.82176e6	0.389525
$876$	−41760.0	−0.00183865
$877$	−1.14540e7	−0.502872	−0.251436	−	0.967874i	$-0.580903\pi$
$877$	−1.14540e7	−0.502872	−0.251436	+	0.967874i	$0.580903\pi$
$878$	−1.35491e7	−0.593163
$879$	−1.98143e7	−0.864980
$880$	−2.98598e6	−0.129981
$881$	−1.18134e7	−0.512786	−0.256393	−	0.966573i	$-0.582534\pi$
$881$	−1.18134e7	−0.512786	−0.256393	+	0.966573i	$0.582534\pi$
$882$	−777924.	−0.0336718
$883$	4.63221e6	0.199934	0.0999670	−	0.994991i	$-0.468126\pi$
$883$	4.63221e6	0.199934	0.0999670	+	0.994991i	$0.468126\pi$
$884$	2.07903e7	0.894810
$885$	1.36527e7	0.585950
$886$	−8.56378e6	−0.366506
$887$	4.47728e7	1.91075	0.955377	−	0.295388i	$-0.0954490\pi$
$887$	4.47728e7	1.91075	0.955377	+	0.295388i	$0.0954490\pi$
$888$	−2.71526e6	−0.115553
$889$	1.32116e7	0.560661
$890$	7.86024e6	0.332630
$891$	1.41718e6	0.0598039
$892$	−1.00265e7	−0.421927
$893$	1.96248e7	0.823525
$894$	−1.43629e7	−0.601034
$895$	3.32022e7	1.38551
$896$	−802816.	−0.0334077
$897$	2.03712e7	0.845347
$898$	2.79123e7	1.15506
$899$	−1.23895e7	−0.511276
$900$	−270864.	−0.0111467
$901$	3.48806e7	1.43144
$902$	8.45510e6	0.346021
$903$	−8.57128e6	−0.349805
$904$	4.54541e6	0.184992
$905$	−2.91970e7	−1.18500
$906$	−8.09885e6	−0.327795
$907$	−2.08357e7	−0.840986	−0.420493	−	0.907296i	$-0.638143\pi$
$907$	−2.08357e7	−0.840986	−0.420493	+	0.907296i	$0.638143\pi$
$908$	−7.20634e6	−0.290068
$909$	1.49547e7	0.600300
$910$	1.05628e7	0.422841
$911$	5.27869e6	0.210732	0.105366	−	0.994434i	$-0.466399\pi$
$911$	5.27869e6	0.210732	0.105366	+	0.994434i	$0.466399\pi$
$912$	−2.03674e6	−0.0810863
$913$	4.17053e6	0.165582
$914$	2.07200e7	0.820396
$915$	−1.88889e7	−0.745853
$916$	−1.70325e7	−0.670717
$917$	3.97782e6	0.156215
$918$	3.79663e6	0.148693
$919$	2.51286e7	0.981477	0.490738	−	0.871307i	$-0.336727\pi$
$919$	2.51286e7	0.981477	0.490738	+	0.871307i	$0.336727\pi$
$920$	−7.83821e6	−0.305314
$921$	−998100.	−0.0387726
$922$	3.13200e7	1.21337
$923$	−2.05867e7	−0.795396
$924$	−1.52410e6	−0.0587263
$925$	985226.	0.0378601
$926$	−661280.	−0.0253430
$927$	5.23195e6	0.199970
$928$	1.51757e6	0.0578467
$929$	1.38042e7	0.524774	0.262387	−	0.964963i	$-0.415490\pi$
$929$	1.38042e7	0.524774	0.262387	+	0.964963i	$0.415490\pi$
$930$	−1.62518e7	−0.616162
$931$	2.12248e6	0.0802547
$932$	2.29788e7	0.866540
$933$	8.19547e6	0.308226
$934$	7.17317e6	0.269057
$935$	−1.51865e7	−0.568106
$936$	−5.17363e6	−0.193022
$937$	−4.73307e7	−1.76114	−0.880570	−	0.473915i	$-0.842840\pi$
$937$	−4.73307e7	−1.76114	−0.880570	+	0.473915i	$0.842840\pi$
$938$	−4.69381e6	−0.174188
$939$	−2.81023e7	−1.04011
$940$	−1.91808e7	−0.708023
$941$	−3.25570e7	−1.19859	−0.599295	−	0.800528i	$-0.704552\pi$
$941$	−3.25570e7	−1.19859	−0.599295	+	0.800528i	$0.704552\pi$
$942$	−8.39585e6	−0.308274
$943$	2.21946e7	0.812773
$944$	7.19155e6	0.262659
$945$	1.92893e6	0.0702648
$946$	−1.67927e7	−0.610088
$947$	−5.27117e6	−0.190999	−0.0954997	−	0.995429i	$-0.530445\pi$
$947$	−5.27117e6	−0.190999	−0.0954997	+	0.995429i	$0.530445\pi$
$948$	1.43343e7	0.518032
$949$	289420.	0.0104319
$950$	739024.	0.0265674
$951$	2.49019e7	0.892857
$952$	−4.08307e6	−0.146014
$953$	8.20579e6	0.292677	0.146338	−	0.989235i	$-0.453251\pi$
$953$	8.20579e6	0.292677	0.146338	+	0.989235i	$0.453251\pi$
$954$	−8.67996e6	−0.308778
$955$	2.26346e6	0.0803092
$956$	−1.59658e7	−0.564995
$957$	2.88101e6	0.101687
$958$	2.63863e7	0.928890
$959$	−1.27846e7	−0.448890
$960$	1.99066e6	0.0697137
$961$	4.12604e7	1.44120
$962$	1.88183e7	0.655605
$963$	1.21170e7	0.421044
$964$	−3.64758e6	−0.126419
$965$	2.88359e7	0.996816
$966$	−4.00075e6	−0.137943
$967$	−1.18118e7	−0.406210	−0.203105	−	0.979157i	$-0.565103\pi$
$967$	−1.18118e7	−0.406210	−0.203105	+	0.979157i	$0.565103\pi$
$968$	7.32128e6	0.251130
$969$	−1.03587e7	−0.354402
$970$	−1.70808e7	−0.582881
$971$	−3.67702e7	−1.25155	−0.625774	−	0.780004i	$-0.715217\pi$
$971$	−3.67702e7	−1.25155	−0.625774	+	0.780004i	$0.715217\pi$
$972$	−944784.	−0.0320750
$973$	−1.46002e7	−0.494399
$974$	2.38957e7	0.807091
$975$	1.87724e6	0.0632423
$976$	−9.94970e6	−0.334338
$977$	−1.85183e7	−0.620674	−0.310337	−	0.950627i	$-0.600442\pi$
$977$	−1.85183e7	−0.620674	−0.310337	+	0.950627i	$0.600442\pi$
$978$	1.67839e7	0.561108
$979$	7.86024e6	0.262107
$980$	−2.07446e6	−0.0689987
$981$	−5.16991e6	−0.171518
$982$	−1.52506e6	−0.0504670
$983$	2.72169e7	0.898370	0.449185	−	0.893439i	$-0.351714\pi$
$983$	2.72169e7	0.898370	0.449185	+	0.893439i	$0.351714\pi$
$984$	−5.63674e6	−0.185584
$985$	−4.45438e7	−1.46284
$986$	7.71826e6	0.252829
$987$	−9.79020e6	−0.319889
$988$	1.41157e7	0.460056
$989$	−4.40808e7	−1.43304
$990$	3.77914e6	0.122548
$991$	1.63398e7	0.528522	0.264261	−	0.964451i	$-0.414872\pi$
$991$	1.63398e7	0.528522	0.264261	+	0.964451i	$0.414872\pi$
$992$	−8.56064e6	−0.276202
$993$	−2.90031e7	−0.933409
$994$	4.04309e6	0.129792
$995$	2.15754e7	0.690877
$996$	−2.78035e6	−0.0888079
$997$	−3.02062e7	−0.962406	−0.481203	−	0.876609i	$-0.659800\pi$
$997$	−3.02062e7	−0.962406	−0.481203	+	0.876609i	$0.659800\pi$
$998$	−6.17403e6	−0.196220
$999$	3.43651e6	0.108944

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.6.a.a.1.1	✓	1	1.1	even	1	trivial
126.6.a.k.1.1		1	3.2	odd	2
294.6.a.h.1.1		1	7.6	odd	2
294.6.e.h.67.1		2	7.5	odd	6
294.6.e.h.79.1		2	7.3	odd	6
294.6.e.r.67.1		2	7.2	even	3
294.6.e.r.79.1		2	7.4	even	3
336.6.a.j.1.1		1	4.3	odd	2
882.6.a.o.1.1		1	21.20	even	2
1008.6.a.x.1.1		1	12.11	even	2
1050.6.a.n.1.1		1	5.4	even	2
1050.6.g.o.799.1		2	5.2	odd	4
1050.6.g.o.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

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