LMFDB - Embedded newform 126.6.a.k.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [126,6,Mod(1,126)] 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("126.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(126, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 126 = 2 \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	126.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,4,0,16,54,0,49,64,0,216,-216] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$20.2083612964$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 42)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	126.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q+4.00000 q^{2} +16.0000 q^{4} +54.0000 q^{5} +49.0000 q^{7} +64.0000 q^{8} +216.000 q^{10} -216.000 q^{11} +998.000 q^{13} +196.000 q^{14} +256.000 q^{16} -1302.00 q^{17} +884.000 q^{19} +864.000 q^{20} -864.000 q^{22} +2268.00 q^{23} -209.000 q^{25} +3992.00 q^{26} +784.000 q^{28} +1482.00 q^{29} +8360.00 q^{31} +1024.00 q^{32} -5208.00 q^{34} +2646.00 q^{35} -4714.00 q^{37} +3536.00 q^{38} +3456.00 q^{40} +9786.00 q^{41} +19436.0 q^{43} -3456.00 q^{44} +9072.00 q^{46} -22200.0 q^{47} +2401.00 q^{49} -836.000 q^{50} +15968.0 q^{52} -26790.0 q^{53} -11664.0 q^{55} +3136.00 q^{56} +5928.00 q^{58} -28092.0 q^{59} -38866.0 q^{61} +33440.0 q^{62} +4096.00 q^{64} +53892.0 q^{65} +23948.0 q^{67} -20832.0 q^{68} +10584.0 q^{70} +20628.0 q^{71} +290.000 q^{73} -18856.0 q^{74} +14144.0 q^{76} -10584.0 q^{77} -99544.0 q^{79} +13824.0 q^{80} +39144.0 q^{82} -19308.0 q^{83} -70308.0 q^{85} +77744.0 q^{86} -13824.0 q^{88} -36390.0 q^{89} +48902.0 q^{91} +36288.0 q^{92} -88800.0 q^{94} +47736.0 q^{95} -79078.0 q^{97} +9604.00 q^{98} +O(q^{100})\)

Coefficient data

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

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

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

$2$	4.00000	0.707107
$2$	4.00000	0.707107
$3$	0	0
$3$	0	0
$4$	16.0000	0.500000
$5$	54.0000	0.965981	0.482991	−	0.875625i	$-0.339550\pi$
$5$	54.0000	0.965981	0.482991	+	0.875625i	$0.339550\pi$
$6$	0	0
$7$	49.0000	0.377964
$7$	49.0000	0.377964
$8$	64.0000	0.353553
$9$	0	0
$10$	216.000	0.683052
$11$	−216.000	−0.538235	−0.269118	−	0.963107i	$-0.586732\pi$
$11$	−216.000	−0.538235	−0.269118	+	0.963107i	$0.586732\pi$
$12$	0	0
$13$	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
$15$	0	0
$16$	256.000	0.250000
$17$	−1302.00	−1.09267	−0.546335	−	0.837567i	$-0.683977\pi$
$17$	−1302.00	−1.09267	−0.546335	+	0.837567i	$0.683977\pi$
$18$	0	0
$19$	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
$21$	0	0
$22$	−864.000	−0.380590
$23$	2268.00	0.893971	0.446986	−	0.894541i	$-0.352498\pi$
$23$	2268.00	0.893971	0.446986	+	0.894541i	$0.352498\pi$
$24$	0	0
$25$	−209.000	−0.0668800
$26$	3992.00	1.15813
$27$	0	0
$28$	784.000	0.188982
$29$	1482.00	0.327230	0.163615	−	0.986524i	$-0.447685\pi$
$29$	1482.00	0.327230	0.163615	+	0.986524i	$0.447685\pi$
$30$	0	0
$31$	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
$33$	0	0
$34$	−5208.00	−0.772634
$35$	2646.00	0.365107
$36$	0	0
$37$	−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
$39$	0	0
$40$	3456.00	0.341526
$41$	9786.00	0.909171	0.454585	−	0.890703i	$-0.349787\pi$
$41$	9786.00	0.909171	0.454585	+	0.890703i	$0.349787\pi$
$42$	0	0
$43$	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
$45$	0	0
$46$	9072.00	0.632133
$47$	−22200.0	−1.46591	−0.732957	−	0.680275i	$-0.761860\pi$
$47$	−22200.0	−1.46591	−0.732957	+	0.680275i	$0.761860\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	−836.000	−0.0472913
$51$	0	0
$52$	15968.0	0.818921
$53$	−26790.0	−1.31004	−0.655018	−	0.755614i	$-0.727339\pi$
$53$	−26790.0	−1.31004	−0.655018	+	0.755614i	$0.727339\pi$
$54$	0	0
$55$	−11664.0	−0.519925
$56$	3136.00	0.133631
$57$	0	0
$58$	5928.00	0.231387
$59$	−28092.0	−1.05064	−0.525318	−	0.850906i	$-0.676054\pi$
$59$	−28092.0	−1.05064	−0.525318	+	0.850906i	$0.676054\pi$
$60$	0	0
$61$	−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
$63$	0	0
$64$	4096.00	0.125000
$65$	53892.0	1.58213
$66$	0	0
$67$	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
$69$	0	0
$70$	10584.0	0.258169
$71$	20628.0	0.485636	0.242818	−	0.970072i	$-0.421928\pi$
$71$	20628.0	0.485636	0.242818	+	0.970072i	$0.421928\pi$
$72$	0	0
$73$	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
$75$	0	0
$76$	14144.0	0.280891
$77$	−10584.0	−0.203434
$78$	0	0
$79$	−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
$81$	0	0
$82$	39144.0	0.642881
$83$	−19308.0	−0.307639	−0.153820	−	0.988099i	$-0.549158\pi$
$83$	−19308.0	−0.307639	−0.153820	+	0.988099i	$0.549158\pi$
$84$	0	0
$85$	−70308.0	−1.05550
$86$	77744.0	1.13350
$87$	0	0
$88$	−13824.0	−0.190295
$89$	−36390.0	−0.486975	−0.243488	−	0.969904i	$-0.578292\pi$
$89$	−36390.0	−0.486975	−0.243488	+	0.969904i	$0.578292\pi$
$90$	0	0
$91$	48902.0	0.619046
$92$	36288.0	0.446986
$93$	0	0
$94$	−88800.0	−1.03656
$95$	47736.0	0.542671
$96$	0	0
$97$	−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
$99$	0	0
$100$	−3344.00	−0.0334400
$101$	−184626.	−1.80090	−0.900450	−	0.434960i	$-0.856762\pi$
$101$	−184626.	−1.80090	−0.900450	+	0.434960i	$0.856762\pi$
$102$	0	0
$103$	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
$105$	0	0
$106$	−107160.	−0.926335
$107$	−149592.	−1.26313	−0.631566	−	0.775322i	$-0.717588\pi$
$107$	−149592.	−1.26313	−0.631566	+	0.775322i	$0.717588\pi$
$108$	0	0
$109$	−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
$111$	0	0
$112$	12544.0	0.0944911
$113$	71022.0	0.523235	0.261618	−	0.965172i	$-0.415744\pi$
$113$	71022.0	0.523235	0.261618	+	0.965172i	$0.415744\pi$
$114$	0	0
$115$	122472.	0.863559
$116$	23712.0	0.163615
$117$	0	0
$118$	−112368.	−0.742912
$119$	−63798.0	−0.412990
$120$	0	0
$121$	−114395.	−0.710303
$122$	−155464.	−0.945650
$123$	0	0
$124$	133760.	0.781218
$125$	−180036.	−1.03059
$126$	0	0
$127$	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
$129$	0	0
$130$	215568.	1.11873
$131$	−81180.0	−0.413305	−0.206653	−	0.978414i	$-0.566257\pi$
$131$	−81180.0	−0.413305	−0.206653	+	0.978414i	$0.566257\pi$
$132$	0	0
$133$	43316.0	0.212334
$134$	95792.0	0.460858
$135$	0	0
$136$	−83328.0	−0.386317
$137$	260910.	1.18765	0.593826	−	0.804593i	$-0.297617\pi$
$137$	260910.	1.18765	0.593826	+	0.804593i	$0.297617\pi$
$138$	0	0
$139$	−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
$141$	0	0
$142$	82512.0	0.343397
$143$	−215568.	−0.881544
$144$	0	0
$145$	80028.0	0.316098
$146$	1160.00	0.00450377
$147$	0	0
$148$	−75424.0	−0.283045
$149$	398970.	1.47223	0.736113	−	0.676859i	$-0.236659\pi$
$149$	398970.	1.47223	0.736113	+	0.676859i	$0.236659\pi$
$150$	0	0
$151$	−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
$153$	0	0
$154$	−42336.0	−0.143849
$155$	451440.	1.50928
$156$	0	0
$157$	−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
$159$	0	0
$160$	55296.0	0.170763
$161$	111132.	0.337889
$162$	0	0
$163$	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
$165$	0	0
$166$	−77232.0	−0.217534
$167$	100848.	0.279818	0.139909	−	0.990164i	$-0.455319\pi$
$167$	100848.	0.279818	0.139909	+	0.990164i	$0.455319\pi$
$168$	0	0
$169$	624711.	1.68253
$170$	−281232.	−0.746350
$171$	0	0
$172$	310976.	0.801504
$173$	668838.	1.69905	0.849524	−	0.527550i	$-0.176889\pi$
$173$	668838.	1.69905	0.849524	+	0.527550i	$0.176889\pi$
$174$	0	0
$175$	−10241.0	−0.0252783
$176$	−55296.0	−0.134559
$177$	0	0
$178$	−145560.	−0.344344
$179$	614856.	1.43430	0.717151	−	0.696917i	$-0.245446\pi$
$179$	614856.	1.43430	0.717151	+	0.696917i	$0.245446\pi$
$180$	0	0
$181$	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
$183$	0	0
$184$	145152.	0.316066
$185$	−254556.	−0.546832
$186$	0	0
$187$	281232.	0.588113
$188$	−355200.	−0.732957
$189$	0	0
$190$	190944.	0.383727
$191$	41916.0	0.0831374	0.0415687	−	0.999136i	$-0.486764\pi$
$191$	41916.0	0.0831374	0.0415687	+	0.999136i	$0.486764\pi$
$192$	0	0
$193$	−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
$195$	0	0
$196$	38416.0	0.0714286
$197$	−824886.	−1.51436	−0.757179	−	0.653208i	$-0.773423\pi$
$197$	−824886.	−1.51436	−0.757179	+	0.653208i	$0.773423\pi$
$198$	0	0
$199$	−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
$201$	0	0
$202$	−738504.	−1.27343
$203$	72618.0	0.123681
$204$	0	0
$205$	528444.	0.878242
$206$	258368.	0.424200
$207$	0	0
$208$	255488.	0.409461
$209$	−190944.	−0.302371
$210$	0	0
$211$	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
$213$	0	0
$214$	−598368.	−0.893170
$215$	1.04954e6	1.54848
$216$	0	0
$217$	409640.	0.590545
$218$	−255304.	−0.363845
$219$	0	0
$220$	−186624.	−0.259963
$221$	−1.29940e6	−1.78962
$222$	0	0
$223$	−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
$225$	0	0
$226$	284088.	0.369983
$227$	450396.	0.580136	0.290068	−	0.957006i	$-0.406322\pi$
$227$	450396.	0.580136	0.290068	+	0.957006i	$0.406322\pi$
$228$	0	0
$229$	−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
$231$	0	0
$232$	94848.0	0.115693
$233$	−1.43618e6	−1.73308	−0.866540	−	0.499108i	$-0.833661\pi$
$233$	−1.43618e6	−1.73308	−0.866540	+	0.499108i	$0.833661\pi$
$234$	0	0
$235$	−1.19880e6	−1.41605
$236$	−449472.	−0.525318
$237$	0	0
$238$	−255192.	−0.292028
$239$	997860.	1.12999	0.564995	−	0.825094i	$-0.308878\pi$
$239$	997860.	1.12999	0.564995	+	0.825094i	$0.308878\pi$
$240$	0	0
$241$	−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
$243$	0	0
$244$	−621856.	−0.668675
$245$	129654.	0.137997
$246$	0	0
$247$	882232.	0.920111
$248$	535040.	0.552404
$249$	0	0
$250$	−720144.	−0.728734
$251$	−1.51657e6	−1.51942	−0.759712	−	0.650260i	$-0.774660\pi$
$251$	−1.51657e6	−1.51942	−0.759712	+	0.650260i	$0.774660\pi$
$252$	0	0
$253$	−489888.	−0.481167
$254$	1.07850e6	1.04890
$255$	0	0
$256$	65536.0	0.0625000
$257$	−455886.	−0.430550	−0.215275	−	0.976553i	$-0.569065\pi$
$257$	−455886.	−0.430550	−0.215275	+	0.976553i	$0.569065\pi$
$258$	0	0
$259$	−230986.	−0.213962
$260$	862272.	0.791063
$261$	0	0
$262$	−324720.	−0.292251
$263$	752652.	0.670973	0.335486	−	0.942045i	$-0.391099\pi$
$263$	752652.	0.670973	0.335486	+	0.942045i	$0.391099\pi$
$264$	0	0
$265$	−1.44666e6	−1.26547
$266$	173264.	0.150143
$267$	0	0
$268$	383168.	0.325876
$269$	−143682.	−0.121066	−0.0605329	−	0.998166i	$-0.519280\pi$
$269$	−143682.	−0.121066	−0.0605329	+	0.998166i	$0.519280\pi$
$270$	0	0
$271$	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
$273$	0	0
$274$	1.04364e6	0.839797
$275$	45144.0	0.0359972
$276$	0	0
$277$	−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
$279$	0	0
$280$	169344.	0.129085
$281$	414366.	0.313053	0.156527	−	0.987674i	$-0.449970\pi$
$281$	414366.	0.313053	0.156527	+	0.987674i	$0.449970\pi$
$282$	0	0
$283$	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
$285$	0	0
$286$	−862272.	−0.623346
$287$	479514.	0.343634
$288$	0	0
$289$	275347.	0.193926
$290$	320112.	0.223515
$291$	0	0
$292$	4640.00	0.00318464
$293$	−2.20159e6	−1.49819	−0.749094	−	0.662463i	$-0.769511\pi$
$293$	−2.20159e6	−1.49819	−0.749094	+	0.662463i	$0.769511\pi$
$294$	0	0
$295$	−1.51697e6	−1.01490
$296$	−301696.	−0.200143
$297$	0	0
$298$	1.59588e6	1.04102
$299$	2.26346e6	1.46418
$300$	0	0
$301$	952364.	0.605880
$302$	−899872.	−0.567758
$303$	0	0
$304$	226304.	0.140446
$305$	−2.09876e6	−1.29186
$306$	0	0
$307$	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
$309$	0	0
$310$	1.80576e6	1.06722
$311$	910608.	0.533864	0.266932	−	0.963715i	$-0.413990\pi$
$311$	910608.	0.533864	0.266932	+	0.963715i	$0.413990\pi$
$312$	0	0
$313$	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
$315$	0	0
$316$	−1.59270e6	−0.897258
$317$	2.76688e6	1.54647	0.773237	−	0.634117i	$-0.218636\pi$
$317$	2.76688e6	1.54647	0.773237	+	0.634117i	$0.218636\pi$
$318$	0	0
$319$	−320112.	−0.176127
$320$	221184.	0.120748
$321$	0	0
$322$	444528.	0.238924
$323$	−1.15097e6	−0.613842
$324$	0	0
$325$	−208582.	−0.109539
$326$	1.86488e6	0.971867
$327$	0	0
$328$	626304.	0.321440
$329$	−1.08780e6	−0.554063
$330$	0	0
$331$	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
$333$	0	0
$334$	403392.	0.197861
$335$	1.29319e6	0.629580
$336$	0	0
$337$	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
$339$	0	0
$340$	−1.12493e6	−0.527749
$341$	−1.80576e6	−0.840958
$342$	0	0
$343$	117649.	0.0539949
$344$	1.24390e6	0.566749
$345$	0	0
$346$	2.67535e6	1.20141
$347$	−1.03642e6	−0.462073	−0.231036	−	0.972945i	$-0.574212\pi$
$347$	−1.03642e6	−0.462073	−0.231036	+	0.972945i	$0.574212\pi$
$348$	0	0
$349$	−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
$351$	0	0
$352$	−221184.	−0.0951474
$353$	−238806.	−0.102002	−0.0510010	−	0.998699i	$-0.516241\pi$
$353$	−238806.	−0.102002	−0.0510010	+	0.998699i	$0.516241\pi$
$354$	0	0
$355$	1.11391e6	0.469116
$356$	−582240.	−0.243488
$357$	0	0
$358$	2.45942e6	1.01421
$359$	2.66428e6	1.09105	0.545523	−	0.838096i	$-0.316331\pi$
$359$	2.66428e6	1.09105	0.545523	+	0.838096i	$0.316331\pi$
$360$	0	0
$361$	−1.69464e6	−0.684400
$362$	2.16274e6	0.867429
$363$	0	0
$364$	782432.	0.309523
$365$	15660.0	0.00615261
$366$	0	0
$367$	−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
$369$	0	0
$370$	−1.01822e6	−0.386669
$371$	−1.31271e6	−0.495147
$372$	0	0
$373$	−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
$375$	0	0
$376$	−1.42080e6	−0.518279
$377$	1.47904e6	0.535951
$378$	0	0
$379$	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
$381$	0	0
$382$	167664.	0.0587870
$383$	2.55686e6	0.890657	0.445329	−	0.895367i	$-0.353087\pi$
$383$	2.55686e6	0.890657	0.445329	+	0.895367i	$0.353087\pi$
$384$	0	0
$385$	−571536.	−0.196513
$386$	−2.13599e6	−0.729678
$387$	0	0
$388$	−1.26525e6	−0.426674
$389$	−2.91785e6	−0.977664	−0.488832	−	0.872378i	$-0.662577\pi$
$389$	−2.91785e6	−0.977664	−0.488832	+	0.872378i	$0.662577\pi$
$390$	0	0
$391$	−2.95294e6	−0.976815
$392$	153664.	0.0505076
$393$	0	0
$394$	−3.29954e6	−1.07081
$395$	−5.37538e6	−1.73347
$396$	0	0
$397$	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
$399$	0	0
$400$	−53504.0	−0.0167200
$401$	−990666.	−0.307657	−0.153828	−	0.988098i	$-0.549160\pi$
$401$	−990666.	−0.307657	−0.153828	+	0.988098i	$0.549160\pi$
$402$	0	0
$403$	8.34328e6	2.55902
$404$	−2.95402e6	−0.900450
$405$	0	0
$406$	290472.	0.0874559
$407$	1.01822e6	0.304689
$408$	0	0
$409$	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
$411$	0	0
$412$	1.03347e6	0.299955
$413$	−1.37651e6	−0.397103
$414$	0	0
$415$	−1.04263e6	−0.297174
$416$	1.02195e6	0.289532
$417$	0	0
$418$	−763776.	−0.213809
$419$	−605220.	−0.168414	−0.0842070	−	0.996448i	$-0.526836\pi$
$419$	−605220.	−0.168414	−0.0842070	+	0.996448i	$0.526836\pi$
$420$	0	0
$421$	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
$423$	0	0
$424$	−1.71456e6	−0.463167
$425$	272118.	0.0730777
$426$	0	0
$427$	−1.90443e6	−0.505471
$428$	−2.39347e6	−0.631566
$429$	0	0
$430$	4.19818e6	1.09494
$431$	−5.37594e6	−1.39400	−0.696998	−	0.717074i	$-0.745481\pi$
$431$	−5.37594e6	−1.39400	−0.696998	+	0.717074i	$0.745481\pi$
$432$	0	0
$433$	−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
$435$	0	0
$436$	−1.02122e6	−0.257277
$437$	2.00491e6	0.502217
$438$	0	0
$439$	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
$441$	0	0
$442$	−5.19758e6	−1.26545
$443$	−2.14094e6	−0.518318	−0.259159	−	0.965835i	$-0.583445\pi$
$443$	−2.14094e6	−0.518318	−0.259159	+	0.965835i	$0.583445\pi$
$444$	0	0
$445$	−1.96506e6	−0.470409
$446$	−2.50662e6	−0.596695
$447$	0	0
$448$	200704.	0.0472456
$449$	6.97808e6	1.63350	0.816752	−	0.576990i	$-0.195773\pi$
$449$	6.97808e6	1.63350	0.816752	+	0.576990i	$0.195773\pi$
$450$	0	0
$451$	−2.11378e6	−0.489348
$452$	1.13635e6	0.261618
$453$	0	0
$454$	1.80158e6	0.410218
$455$	2.64071e6	0.597987
$456$	0	0
$457$	−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
$459$	0	0
$460$	1.95955e6	0.431780
$461$	7.83001e6	1.71597	0.857985	−	0.513674i	$-0.171716\pi$
$461$	7.83001e6	1.71597	0.857985	+	0.513674i	$0.171716\pi$
$462$	0	0
$463$	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
$465$	0	0
$466$	−5.74471e6	−1.22547
$467$	1.79329e6	0.380504	0.190252	−	0.981735i	$-0.439070\pi$
$467$	1.79329e6	0.380504	0.190252	+	0.981735i	$0.439070\pi$
$468$	0	0
$469$	1.17345e6	0.246339
$470$	−4.79520e6	−1.00130
$471$	0	0
$472$	−1.79789e6	−0.371456
$473$	−4.19818e6	−0.862795
$474$	0	0
$475$	−184756.	−0.0375720
$476$	−1.02077e6	−0.206495
$477$	0	0
$478$	3.99144e6	0.799024
$479$	6.59657e6	1.31365	0.656824	−	0.754044i	$-0.271899\pi$
$479$	6.59657e6	1.31365	0.656824	+	0.754044i	$0.271899\pi$
$480$	0	0
$481$	−4.70457e6	−0.927166
$482$	−911896.	−0.178784
$483$	0	0
$484$	−1.83032e6	−0.355151
$485$	−4.27021e6	−0.824319
$486$	0	0
$487$	−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
$489$	0	0
$490$	518616.	0.0975789
$491$	−381264.	−0.0713710	−0.0356855	−	0.999363i	$-0.511361\pi$
$491$	−381264.	−0.0713710	−0.0356855	+	0.999363i	$0.511361\pi$
$492$	0	0
$493$	−1.92956e6	−0.357554
$494$	3.52893e6	0.650617
$495$	0	0
$496$	2.14016e6	0.390609
$497$	1.01077e6	0.183553
$498$	0	0
$499$	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
$501$	0	0
$502$	−6.06629e6	−1.07439
$503$	4.02300e6	0.708974	0.354487	−	0.935061i	$-0.384656\pi$
$503$	4.02300e6	0.708974	0.354487	+	0.935061i	$0.384656\pi$
$504$	0	0
$505$	−9.96980e6	−1.73964
$506$	−1.95955e6	−0.340236
$507$	0	0
$508$	4.31398e6	0.741685
$509$	1.94715e6	0.333123	0.166562	−	0.986031i	$-0.446734\pi$
$509$	1.94715e6	0.333123	0.166562	+	0.986031i	$0.446734\pi$
$510$	0	0
$511$	14210.0	0.00240736
$512$	262144.	0.0441942
$513$	0	0
$514$	−1.82354e6	−0.304445
$515$	3.48797e6	0.579501
$516$	0	0
$517$	4.79520e6	0.789006
$518$	−923944.	−0.151294
$519$	0	0
$520$	3.44909e6	0.559366
$521$	−7.38569e6	−1.19206	−0.596028	−	0.802963i	$-0.703255\pi$
$521$	−7.38569e6	−1.19206	−0.596028	+	0.802963i	$0.703255\pi$
$522$	0	0
$523$	−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
$525$	0	0
$526$	3.01061e6	0.474449
$527$	−1.08847e7	−1.70722
$528$	0	0
$529$	−1.29252e6	−0.200816
$530$	−5.78664e6	−0.894822
$531$	0	0
$532$	693056.	0.106167
$533$	9.76643e6	1.48908
$534$	0	0
$535$	−8.07797e6	−1.22016
$536$	1.53267e6	0.230429
$537$	0	0
$538$	−574728.	−0.0856065
$539$	−518616.	−0.0768907
$540$	0	0
$541$	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
$543$	0	0
$544$	−1.33325e6	−0.193158
$545$	−3.44660e6	−0.497050
$546$	0	0
$547$	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
$549$	0	0
$550$	180576.	0.0254538
$551$	1.31009e6	0.183832
$552$	0	0
$553$	−4.87766e6	−0.678263
$554$	−4.64855e6	−0.643492
$555$	0	0
$556$	−4.76742e6	−0.654029
$557$	1.52258e6	0.207942	0.103971	−	0.994580i	$-0.466845\pi$
$557$	1.52258e6	0.207942	0.103971	+	0.994580i	$0.466845\pi$
$558$	0	0
$559$	1.93971e7	2.62548
$560$	677376.	0.0912767
$561$	0	0
$562$	1.65746e6	0.221362
$563$	7.86462e6	1.04570	0.522850	−	0.852425i	$-0.324869\pi$
$563$	7.86462e6	1.04570	0.522850	+	0.852425i	$0.324869\pi$
$564$	0	0
$565$	3.83519e6	0.505435
$566$	481712.	0.0632043
$567$	0	0
$568$	1.32019e6	0.171698
$569$	1.46321e6	0.189464	0.0947321	−	0.995503i	$-0.469801\pi$
$569$	1.46321e6	0.189464	0.0947321	+	0.995503i	$0.469801\pi$
$570$	0	0
$571$	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
$573$	0	0
$574$	1.91806e6	0.242986
$575$	−474012.	−0.0597888
$576$	0	0
$577$	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
$579$	0	0
$580$	1.28045e6	0.158049
$581$	−946092.	−0.116277
$582$	0	0
$583$	5.78664e6	0.705107
$584$	18560.0	0.00225188
$585$	0	0
$586$	−8.80634e6	−1.05938
$587$	−5.12929e6	−0.614416	−0.307208	−	0.951642i	$-0.599395\pi$
$587$	−5.12929e6	−0.614416	−0.307208	+	0.951642i	$0.599395\pi$
$588$	0	0
$589$	7.39024e6	0.877749
$590$	−6.06787e6	−0.717640
$591$	0	0
$592$	−1.20678e6	−0.141522
$593$	2.75433e6	0.321647	0.160823	−	0.986983i	$-0.448585\pi$
$593$	2.75433e6	0.321647	0.160823	+	0.986983i	$0.448585\pi$
$594$	0	0
$595$	−3.44509e6	−0.398941
$596$	6.38352e6	0.736113
$597$	0	0
$598$	9.05386e6	1.03533
$599$	9.88616e6	1.12580	0.562899	−	0.826525i	$-0.309686\pi$
$599$	9.88616e6	1.12580	0.562899	+	0.826525i	$0.309686\pi$
$600$	0	0
$601$	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
$603$	0	0
$604$	−3.59949e6	−0.401466
$605$	−6.17733e6	−0.686139
$606$	0	0
$607$	−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
$609$	0	0
$610$	−8.39506e6	−0.913480
$611$	−2.21556e7	−2.40094
$612$	0	0
$613$	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
$615$	0	0
$616$	−677376.	−0.0719247
$617$	−6.28370e6	−0.664511	−0.332256	−	0.943189i	$-0.607810\pi$
$617$	−6.28370e6	−0.664511	−0.332256	+	0.943189i	$0.607810\pi$
$618$	0	0
$619$	−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
$621$	0	0
$622$	3.64243e6	0.377499
$623$	−1.78311e6	−0.184059
$624$	0	0
$625$	−9.06882e6	−0.928647
$626$	1.24899e7	1.27386
$627$	0	0
$628$	−3.73149e6	−0.377557
$629$	6.13763e6	0.618549
$630$	0	0
$631$	−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
$633$	0	0
$634$	1.10675e7	1.09352
$635$	1.45597e7	1.43291
$636$	0	0
$637$	2.39620e6	0.233978
$638$	−1.28045e6	−0.124540
$639$	0	0
$640$	884736.	0.0853815
$641$	−5.93231e6	−0.570268	−0.285134	−	0.958488i	$-0.592038\pi$
$641$	−5.93231e6	−0.570268	−0.285134	+	0.958488i	$0.592038\pi$
$642$	0	0
$643$	−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
$645$	0	0
$646$	−4.60387e6	−0.434052
$647$	4.97050e6	0.466809	0.233404	−	0.972380i	$-0.425013\pi$
$647$	4.97050e6	0.466809	0.233404	+	0.972380i	$0.425013\pi$
$648$	0	0
$649$	6.06787e6	0.565490
$650$	−834328.	−0.0774557
$651$	0	0
$652$	7.45952e6	0.687214
$653$	1.83355e7	1.68271	0.841354	−	0.540484i	$-0.181759\pi$
$653$	1.83355e7	1.68271	0.841354	+	0.540484i	$0.181759\pi$
$654$	0	0
$655$	−4.38372e6	−0.399245
$656$	2.50522e6	0.227293
$657$	0	0
$658$	−4.35120e6	−0.391782
$659$	−9.01402e6	−0.808546	−0.404273	−	0.914638i	$-0.632475\pi$
$659$	−9.01402e6	−0.808546	−0.404273	+	0.914638i	$0.632475\pi$
$660$	0	0
$661$	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
$663$	0	0
$664$	−1.23571e6	−0.108767
$665$	2.33906e6	0.205111
$666$	0	0
$667$	3.36118e6	0.292534
$668$	1.61357e6	0.139909
$669$	0	0
$670$	5.17277e6	0.445180
$671$	8.39506e6	0.719809
$672$	0	0
$673$	−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
$675$	0	0
$676$	9.99538e6	0.841264
$677$	−985074.	−0.0826033	−0.0413016	−	0.999147i	$-0.513150\pi$
$677$	−985074.	−0.0826033	−0.0413016	+	0.999147i	$0.513150\pi$
$678$	0	0
$679$	−3.87482e6	−0.322535
$680$	−4.49971e6	−0.373175
$681$	0	0
$682$	−7.22304e6	−0.594647
$683$	1.88208e7	1.54379	0.771894	−	0.635752i	$-0.219310\pi$
$683$	1.88208e7	1.54379	0.771894	+	0.635752i	$0.219310\pi$
$684$	0	0
$685$	1.40891e7	1.14725
$686$	470596.	0.0381802
$687$	0	0
$688$	4.97562e6	0.400752
$689$	−2.67364e7	−2.14563
$690$	0	0
$691$	−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
$693$	0	0
$694$	−4.14566e6	−0.326735
$695$	−1.60901e7	−1.26356
$696$	0	0
$697$	−1.27414e7	−0.993423
$698$	−1.69199e7	−1.31450
$699$	0	0
$700$	−163856.	−0.0126391
$701$	1.41489e6	0.108750	0.0543748	−	0.998521i	$-0.482683\pi$
$701$	1.41489e6	0.108750	0.0543748	+	0.998521i	$0.482683\pi$
$702$	0	0
$703$	−4.16718e6	−0.318019
$704$	−884736.	−0.0672794
$705$	0	0
$706$	−955224.	−0.0721263
$707$	−9.04667e6	−0.680676
$708$	0	0
$709$	−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
$711$	0	0
$712$	−2.32896e6	−0.172172
$713$	1.89605e7	1.39677
$714$	0	0
$715$	−1.16407e7	−0.851555
$716$	9.83770e6	0.717151
$717$	0	0
$718$	1.06571e7	0.771486
$719$	−1.08854e6	−0.0785279	−0.0392639	−	0.999229i	$-0.512501\pi$
$719$	−1.08854e6	−0.0785279	−0.0392639	+	0.999229i	$0.512501\pi$
$720$	0	0
$721$	3.16501e6	0.226744
$722$	−6.77857e6	−0.483944
$723$	0	0
$724$	8.65098e6	0.613365
$725$	−309738.	−0.0218851
$726$	0	0
$727$	−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
$729$	0	0
$730$	62640.0	0.00435055
$731$	−2.53057e7	−1.75156
$732$	0	0
$733$	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
$735$	0	0
$736$	2.32243e6	0.158033
$737$	−5.17277e6	−0.350796
$738$	0	0
$739$	−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
$741$	0	0
$742$	−5.25084e6	−0.350122
$743$	−1.73678e7	−1.15418	−0.577088	−	0.816682i	$-0.695811\pi$
$743$	−1.73678e7	−1.15418	−0.577088	+	0.816682i	$0.695811\pi$
$744$	0	0
$745$	2.15444e7	1.42214
$746$	−1.58660e7	−1.04380
$747$	0	0
$748$	4.49971e6	0.294056
$749$	−7.33001e6	−0.477419
$750$	0	0
$751$	−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
$753$	0	0
$754$	5.91614e6	0.378975
$755$	−1.21483e7	−0.775617
$756$	0	0
$757$	−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
$759$	0	0
$760$	3.05510e6	0.191863
$761$	−2.57535e6	−0.161204	−0.0806018	−	0.996746i	$-0.525684\pi$
$761$	−2.57535e6	−0.161204	−0.0806018	+	0.996746i	$0.525684\pi$
$762$	0	0
$763$	−3.12747e6	−0.194483
$764$	670656.	0.0415687
$765$	0	0
$766$	1.02275e7	0.629790
$767$	−2.80358e7	−1.72078
$768$	0	0
$769$	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
$771$	0	0
$772$	−8.54397e6	−0.515960
$773$	1.72921e7	1.04088	0.520439	−	0.853899i	$-0.325768\pi$
$773$	1.72921e7	1.04088	0.520439	+	0.853899i	$0.325768\pi$
$774$	0	0
$775$	−1.74724e6	−0.104496
$776$	−5.06099e6	−0.301704
$777$	0	0
$778$	−1.16714e7	−0.691313
$779$	8.65082e6	0.510756
$780$	0	0
$781$	−4.45565e6	−0.261387
$782$	−1.18117e7	−0.690712
$783$	0	0
$784$	614656.	0.0357143
$785$	−1.25938e7	−0.729427
$786$	0	0
$787$	−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
$789$	0	0
$790$	−2.15015e7	−1.22575
$791$	3.48008e6	0.197764
$792$	0	0
$793$	−3.87883e7	−2.19037
$794$	1.00286e7	0.564533
$795$	0	0
$796$	−6.39270e6	−0.357604
$797$	−2.91057e6	−0.162305	−0.0811526	−	0.996702i	$-0.525860\pi$
$797$	−2.91057e6	−0.162305	−0.0811526	+	0.996702i	$0.525860\pi$
$798$	0	0
$799$	2.89044e7	1.60176
$800$	−214016.	−0.0118228
$801$	0	0
$802$	−3.96266e6	−0.217546
$803$	−62640.0	−0.00342817
$804$	0	0
$805$	6.00113e6	0.326395
$806$	3.33731e7	1.80950
$807$	0	0
$808$	−1.18161e7	−0.636714
$809$	1.16252e7	0.624496	0.312248	−	0.950001i	$-0.398918\pi$
$809$	1.16252e7	0.624496	0.312248	+	0.950001i	$0.398918\pi$
$810$	0	0
$811$	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
$813$	0	0
$814$	4.07290e6	0.215448
$815$	2.51759e7	1.32767
$816$	0	0
$817$	1.71814e7	0.900542
$818$	1.80730e7	0.944379
$819$	0	0
$820$	8.45510e6	0.439121
$821$	2.22870e7	1.15397	0.576984	−	0.816755i	$-0.304229\pi$
$821$	2.22870e7	1.15397	0.576984	+	0.816755i	$0.304229\pi$
$822$	0	0
$823$	−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
$825$	0	0
$826$	−5.50603e6	−0.280795
$827$	2.37457e7	1.20732	0.603658	−	0.797244i	$-0.293709\pi$
$827$	2.37457e7	1.20732	0.603658	+	0.797244i	$0.293709\pi$
$828$	0	0
$829$	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
$831$	0	0
$832$	4.08781e6	0.204730
$833$	−3.12610e6	−0.156096
$834$	0	0
$835$	5.44579e6	0.270299
$836$	−3.05510e6	−0.151186
$837$	0	0
$838$	−2.42088e6	−0.119087
$839$	−1.00872e7	−0.494729	−0.247365	−	0.968922i	$-0.579565\pi$
$839$	−1.00872e7	−0.494729	−0.247365	+	0.968922i	$0.579565\pi$
$840$	0	0
$841$	−1.83148e7	−0.892920
$842$	1.79957e7	0.874761
$843$	0	0
$844$	1.39019e7	0.671765
$845$	3.37344e7	1.62529
$846$	0	0
$847$	−5.60536e6	−0.268469
$848$	−6.85824e6	−0.327509
$849$	0	0
$850$	1.08847e6	0.0516737
$851$	−1.06914e7	−0.506068
$852$	0	0
$853$	−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
$855$	0	0
$856$	−9.57389e6	−0.446585
$857$	−2.45612e6	−0.114234	−0.0571172	−	0.998367i	$-0.518191\pi$
$857$	−2.45612e6	−0.114234	−0.0571172	+	0.998367i	$0.518191\pi$
$858$	0	0
$859$	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
$861$	0	0
$862$	−2.15038e7	−0.985703
$863$	−1.05199e7	−0.480824	−0.240412	−	0.970671i	$-0.577283\pi$
$863$	−1.05199e7	−0.480824	−0.240412	+	0.970671i	$0.577283\pi$
$864$	0	0
$865$	3.61173e7	1.64125
$866$	−7.94246e6	−0.359882
$867$	0	0
$868$	6.55424e6	0.295273
$869$	2.15015e7	0.965872
$870$	0	0
$871$	2.39001e7	1.06747
$872$	−4.08486e6	−0.181922
$873$	0	0
$874$	8.01965e6	0.355121
$875$	−8.82176e6	−0.389525
$876$	0	0
$877$	−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
$879$	0	0
$880$	−2.98598e6	−0.129981
$881$	1.18134e7	0.512786	0.256393	−	0.966573i	$-0.417466\pi$
$881$	1.18134e7	0.512786	0.256393	+	0.966573i	$0.417466\pi$
$882$	0	0
$883$	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
$885$	0	0
$886$	−8.56378e6	−0.366506
$887$	−4.47728e7	−1.91075	−0.955377	−	0.295388i	$-0.904551\pi$
$887$	−4.47728e7	−1.91075	−0.955377	+	0.295388i	$0.904551\pi$
$888$	0	0
$889$	1.32116e7	0.560661
$890$	−7.86024e6	−0.332630
$891$	0	0
$892$	−1.00265e7	−0.421927
$893$	−1.96248e7	−0.823525
$894$	0	0
$895$	3.32022e7	1.38551
$896$	802816.	0.0334077
$897$	0	0
$898$	2.79123e7	1.15506
$899$	1.23895e7	0.511276
$900$	0	0
$901$	3.48806e7	1.43144
$902$	−8.45510e6	−0.346021
$903$	0	0
$904$	4.54541e6	0.184992
$905$	2.91970e7	1.18500
$906$	0	0
$907$	−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
$909$	0	0
$910$	1.05628e7	0.422841
$911$	−5.27869e6	−0.210732	−0.105366	−	0.994434i	$-0.533601\pi$
$911$	−5.27869e6	−0.210732	−0.105366	+	0.994434i	$0.533601\pi$
$912$	0	0
$913$	4.17053e6	0.165582
$914$	−2.07200e7	−0.820396
$915$	0	0
$916$	−1.70325e7	−0.670717
$917$	−3.97782e6	−0.156215
$918$	0	0
$919$	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
$921$	0	0
$922$	3.13200e7	1.21337
$923$	2.05867e7	0.795396
$924$	0	0
$925$	985226.	0.0378601
$926$	661280.	0.0253430
$927$	0	0
$928$	1.51757e6	0.0578467
$929$	−1.38042e7	−0.524774	−0.262387	−	0.964963i	$-0.584510\pi$
$929$	−1.38042e7	−0.524774	−0.262387	+	0.964963i	$0.584510\pi$
$930$	0	0
$931$	2.12248e6	0.0802547
$932$	−2.29788e7	−0.866540
$933$	0	0
$934$	7.17317e6	0.269057
$935$	1.51865e7	0.568106
$936$	0	0
$937$	−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
$939$	0	0
$940$	−1.91808e7	−0.708023
$941$	3.25570e7	1.19859	0.599295	−	0.800528i	$-0.295448\pi$
$941$	3.25570e7	1.19859	0.599295	+	0.800528i	$0.295448\pi$
$942$	0	0
$943$	2.21946e7	0.812773
$944$	−7.19155e6	−0.262659
$945$	0	0
$946$	−1.67927e7	−0.610088
$947$	5.27117e6	0.190999	0.0954997	−	0.995429i	$-0.469555\pi$
$947$	5.27117e6	0.190999	0.0954997	+	0.995429i	$0.469555\pi$
$948$	0	0
$949$	289420.	0.0104319
$950$	−739024.	−0.0265674
$951$	0	0
$952$	−4.08307e6	−0.146014
$953$	−8.20579e6	−0.292677	−0.146338	−	0.989235i	$-0.546749\pi$
$953$	−8.20579e6	−0.292677	−0.146338	+	0.989235i	$0.546749\pi$
$954$	0	0
$955$	2.26346e6	0.0803092
$956$	1.59658e7	0.564995
$957$	0	0
$958$	2.63863e7	0.928890
$959$	1.27846e7	0.448890
$960$	0	0
$961$	4.12604e7	1.44120
$962$	−1.88183e7	−0.655605
$963$	0	0
$964$	−3.64758e6	−0.126419
$965$	−2.88359e7	−0.996816
$966$	0	0
$967$	−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
$969$	0	0
$970$	−1.70808e7	−0.582881
$971$	3.67702e7	1.25155	0.625774	−	0.780004i	$-0.284783\pi$
$971$	3.67702e7	1.25155	0.625774	+	0.780004i	$0.284783\pi$
$972$	0	0
$973$	−1.46002e7	−0.494399
$974$	−2.38957e7	−0.807091
$975$	0	0
$976$	−9.94970e6	−0.334338
$977$	1.85183e7	0.620674	0.310337	−	0.950627i	$-0.399558\pi$
$977$	1.85183e7	0.620674	0.310337	+	0.950627i	$0.399558\pi$
$978$	0	0
$979$	7.86024e6	0.262107
$980$	2.07446e6	0.0689987
$981$	0	0
$982$	−1.52506e6	−0.0504670
$983$	−2.72169e7	−0.898370	−0.449185	−	0.893439i	$-0.648286\pi$
$983$	−2.72169e7	−0.898370	−0.449185	+	0.893439i	$0.648286\pi$
$984$	0	0
$985$	−4.45438e7	−1.46284
$986$	−7.71826e6	−0.252829
$987$	0	0
$988$	1.41157e7	0.460056
$989$	4.40808e7	1.43304
$990$	0	0
$991$	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
$993$	0	0
$994$	4.04309e6	0.129792
$995$	−2.15754e7	−0.690877
$996$	0	0
$997$	−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
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	126.6.a.k.1.1		1
3.2	odd	2		42.6.a.a.1.1	✓	1
4.3	odd	2		1008.6.a.x.1.1		1
7.6	odd	2		882.6.a.o.1.1		1
12.11	even	2		336.6.a.j.1.1		1
15.2	even	4		1050.6.g.o.799.1		2
15.8	even	4		1050.6.g.o.799.2		2
15.14	odd	2		1050.6.a.n.1.1		1
21.2	odd	6		294.6.e.r.67.1		2
21.5	even	6		294.6.e.h.67.1		2
21.11	odd	6		294.6.e.r.79.1		2
21.17	even	6		294.6.e.h.79.1		2
21.20	even	2		294.6.a.h.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.6.a.a.1.1	✓	1	3.2	odd	2
126.6.a.k.1.1		1	1.1	even	1	trivial
294.6.a.h.1.1		1	21.20	even	2
294.6.e.h.67.1		2	21.5	even	6
294.6.e.h.79.1		2	21.17	even	6
294.6.e.r.67.1		2	21.2	odd	6
294.6.e.r.79.1		2	21.11	odd	6
336.6.a.j.1.1		1	12.11	even	2
882.6.a.o.1.1		1	7.6	odd	2
1008.6.a.x.1.1		1	4.3	odd	2
1050.6.a.n.1.1		1	15.14	odd	2
1050.6.g.o.799.1		2	15.2	even	4
1050.6.g.o.799.2		2	15.8	even	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:	\( N \)	\(=\)	\( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	126.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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