LMFDB - Embedded newform 126.6.a.c.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,-10,0,-49,-64,0,40,340] 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 14)
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)

$f(q)$

$=$

\(q-4.00000 q^{2} +16.0000 q^{4} -10.0000 q^{5} -49.0000 q^{7} -64.0000 q^{8} +40.0000 q^{10} +340.000 q^{11} -294.000 q^{13} +196.000 q^{14} +256.000 q^{16} -1226.00 q^{17} +2432.00 q^{19} -160.000 q^{20} -1360.00 q^{22} -2000.00 q^{23} -3025.00 q^{25} +1176.00 q^{26} -784.000 q^{28} +6746.00 q^{29} +8856.00 q^{31} -1024.00 q^{32} +4904.00 q^{34} +490.000 q^{35} +9182.00 q^{37} -9728.00 q^{38} +640.000 q^{40} +14574.0 q^{41} +8108.00 q^{43} +5440.00 q^{44} +8000.00 q^{46} +312.000 q^{47} +2401.00 q^{49} +12100.0 q^{50} -4704.00 q^{52} +14634.0 q^{53} -3400.00 q^{55} +3136.00 q^{56} -26984.0 q^{58} +27656.0 q^{59} +34338.0 q^{61} -35424.0 q^{62} +4096.00 q^{64} +2940.00 q^{65} +12316.0 q^{67} -19616.0 q^{68} -1960.00 q^{70} -36920.0 q^{71} -61718.0 q^{73} -36728.0 q^{74} +38912.0 q^{76} -16660.0 q^{77} -64752.0 q^{79} -2560.00 q^{80} -58296.0 q^{82} +77056.0 q^{83} +12260.0 q^{85} -32432.0 q^{86} -21760.0 q^{88} +8166.00 q^{89} +14406.0 q^{91} -32000.0 q^{92} -1248.00 q^{94} -24320.0 q^{95} +20650.0 q^{97} -9604.00 q^{98} +O(q^{100})\)

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
$5$	−10.0000	−0.178885	−0.0894427	−	0.995992i	$-0.528509\pi$
$5$	−10.0000	−0.178885	−0.0894427	+	0.995992i	$0.528509\pi$
$6$	0	0
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	−64.0000	−0.353553
$9$	0	0
$10$	40.0000	0.126491
$11$	340.000	0.847222	0.423611	−	0.905844i	$-0.360762\pi$
$11$	340.000	0.847222	0.423611	+	0.905844i	$0.360762\pi$
$12$	0	0
$13$	−294.000	−0.482491	−0.241245	−	0.970464i	$-0.577556\pi$
$13$	−294.000	−0.482491	−0.241245	+	0.970464i	$0.577556\pi$
$14$	196.000	0.267261
$15$	0	0
$16$	256.000	0.250000
$17$	−1226.00	−1.02889	−0.514444	−	0.857524i	$-0.672002\pi$
$17$	−1226.00	−1.02889	−0.514444	+	0.857524i	$0.672002\pi$
$18$	0	0
$19$	2432.00	1.54554	0.772769	−	0.634688i	$-0.218871\pi$
$19$	2432.00	1.54554	0.772769	+	0.634688i	$0.218871\pi$
$20$	−160.000	−0.0894427
$21$	0	0
$22$	−1360.00	−0.599076
$23$	−2000.00	−0.788334	−0.394167	−	0.919039i	$-0.628967\pi$
$23$	−2000.00	−0.788334	−0.394167	+	0.919039i	$0.628967\pi$
$24$	0	0
$25$	−3025.00	−0.968000
$26$	1176.00	0.341172
$27$	0	0
$28$	−784.000	−0.188982
$29$	6746.00	1.48954	0.744769	−	0.667323i	$-0.232560\pi$
$29$	6746.00	1.48954	0.744769	+	0.667323i	$0.232560\pi$
$30$	0	0
$31$	8856.00	1.65513	0.827567	−	0.561366i	$-0.189724\pi$
$31$	8856.00	1.65513	0.827567	+	0.561366i	$0.189724\pi$
$32$	−1024.00	−0.176777
$33$	0	0
$34$	4904.00	0.727534
$35$	490.000	0.0676123
$36$	0	0
$37$	9182.00	1.10264	0.551319	−	0.834295i	$-0.314125\pi$
$37$	9182.00	1.10264	0.551319	+	0.834295i	$0.314125\pi$
$38$	−9728.00	−1.09286
$39$	0	0
$40$	640.000	0.0632456
$41$	14574.0	1.35400	0.677001	−	0.735982i	$-0.263279\pi$
$41$	14574.0	1.35400	0.677001	+	0.735982i	$0.263279\pi$
$42$	0	0
$43$	8108.00	0.668717	0.334359	−	0.942446i	$-0.391480\pi$
$43$	8108.00	0.668717	0.334359	+	0.942446i	$0.391480\pi$
$44$	5440.00	0.423611
$45$	0	0
$46$	8000.00	0.557437
$47$	312.000	0.0206020	0.0103010	−	0.999947i	$-0.496721\pi$
$47$	312.000	0.0206020	0.0103010	+	0.999947i	$0.496721\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	12100.0	0.684479
$51$	0	0
$52$	−4704.00	−0.241245
$53$	14634.0	0.715605	0.357803	−	0.933797i	$-0.383526\pi$
$53$	14634.0	0.715605	0.357803	+	0.933797i	$0.383526\pi$
$54$	0	0
$55$	−3400.00	−0.151556
$56$	3136.00	0.133631
$57$	0	0
$58$	−26984.0	−1.05326
$59$	27656.0	1.03433	0.517165	−	0.855886i	$-0.326987\pi$
$59$	27656.0	1.03433	0.517165	+	0.855886i	$0.326987\pi$
$60$	0	0
$61$	34338.0	1.18155	0.590773	−	0.806838i	$-0.298823\pi$
$61$	34338.0	1.18155	0.590773	+	0.806838i	$0.298823\pi$
$62$	−35424.0	−1.17036
$63$	0	0
$64$	4096.00	0.125000
$65$	2940.00	0.0863106
$66$	0	0
$67$	12316.0	0.335184	0.167592	−	0.985856i	$-0.446401\pi$
$67$	12316.0	0.335184	0.167592	+	0.985856i	$0.446401\pi$
$68$	−19616.0	−0.514444
$69$	0	0
$70$	−1960.00	−0.0478091
$71$	−36920.0	−0.869192	−0.434596	−	0.900625i	$-0.643109\pi$
$71$	−36920.0	−0.869192	−0.434596	+	0.900625i	$0.643109\pi$
$72$	0	0
$73$	−61718.0	−1.35552	−0.677758	−	0.735285i	$-0.737048\pi$
$73$	−61718.0	−1.35552	−0.677758	+	0.735285i	$0.737048\pi$
$74$	−36728.0	−0.779683
$75$	0	0
$76$	38912.0	0.772769
$77$	−16660.0	−0.320220
$78$	0	0
$79$	−64752.0	−1.16731	−0.583654	−	0.812002i	$-0.698378\pi$
$79$	−64752.0	−1.16731	−0.583654	+	0.812002i	$0.698378\pi$
$80$	−2560.00	−0.0447214
$81$	0	0
$82$	−58296.0	−0.957424
$83$	77056.0	1.22775	0.613877	−	0.789402i	$-0.289609\pi$
$83$	77056.0	1.22775	0.613877	+	0.789402i	$0.289609\pi$
$84$	0	0
$85$	12260.0	0.184053
$86$	−32432.0	−0.472855
$87$	0	0
$88$	−21760.0	−0.299538
$89$	8166.00	0.109278	0.0546392	−	0.998506i	$-0.482599\pi$
$89$	8166.00	0.109278	0.0546392	+	0.998506i	$0.482599\pi$
$90$	0	0
$91$	14406.0	0.182364
$92$	−32000.0	−0.394167
$93$	0	0
$94$	−1248.00	−0.0145678
$95$	−24320.0	−0.276474
$96$	0	0
$97$	20650.0	0.222839	0.111419	−	0.993773i	$-0.464460\pi$
$97$	20650.0	0.222839	0.111419	+	0.993773i	$0.464460\pi$
$98$	−9604.00	−0.101015
$99$	0	0
$100$	−48400.0	−0.484000
$101$	−186250.	−1.81674	−0.908370	−	0.418167i	$-0.862673\pi$
$101$	−186250.	−1.81674	−0.908370	+	0.418167i	$0.862673\pi$
$102$	0	0
$103$	−60064.0	−0.557855	−0.278927	−	0.960312i	$-0.589979\pi$
$103$	−60064.0	−0.557855	−0.278927	+	0.960312i	$0.589979\pi$
$104$	18816.0	0.170586
$105$	0	0
$106$	−58536.0	−0.506009
$107$	−47892.0	−0.404393	−0.202196	−	0.979345i	$-0.564808\pi$
$107$	−47892.0	−0.404393	−0.202196	+	0.979345i	$0.564808\pi$
$108$	0	0
$109$	22102.0	0.178183	0.0890913	−	0.996023i	$-0.471604\pi$
$109$	22102.0	0.178183	0.0890913	+	0.996023i	$0.471604\pi$
$110$	13600.0	0.107166
$111$	0	0
$112$	−12544.0	−0.0944911
$113$	245054.	1.80537	0.902684	−	0.430304i	$-0.141594\pi$
$113$	245054.	1.80537	0.902684	+	0.430304i	$0.141594\pi$
$114$	0	0
$115$	20000.0	0.141022
$116$	107936.	0.744769
$117$	0	0
$118$	−110624.	−0.731382
$119$	60074.0	0.388883
$120$	0	0
$121$	−45451.0	−0.282215
$122$	−137352.	−0.835479
$123$	0	0
$124$	141696.	0.827567
$125$	61500.0	0.352047
$126$	0	0
$127$	−96696.0	−0.531985	−0.265992	−	0.963975i	$-0.585700\pi$
$127$	−96696.0	−0.531985	−0.265992	+	0.963975i	$0.585700\pi$
$128$	−16384.0	−0.0883883
$129$	0	0
$130$	−11760.0	−0.0610308
$131$	−134368.	−0.684097	−0.342048	−	0.939682i	$-0.611121\pi$
$131$	−134368.	−0.684097	−0.342048	+	0.939682i	$0.611121\pi$
$132$	0	0
$133$	−119168.	−0.584158
$134$	−49264.0	−0.237011
$135$	0	0
$136$	78464.0	0.363767
$137$	294662.	1.34129	0.670645	−	0.741778i	$-0.266017\pi$
$137$	294662.	1.34129	0.670645	+	0.741778i	$0.266017\pi$
$138$	0	0
$139$	314944.	1.38260	0.691300	−	0.722568i	$-0.257038\pi$
$139$	314944.	1.38260	0.691300	+	0.722568i	$0.257038\pi$
$140$	7840.00	0.0338062
$141$	0	0
$142$	147680.	0.614612
$143$	−99960.0	−0.408777
$144$	0	0
$145$	−67460.0	−0.266457
$146$	246872.	0.958495
$147$	0	0
$148$	146912.	0.551319
$149$	−113622.	−0.419273	−0.209636	−	0.977779i	$-0.567228\pi$
$149$	−113622.	−0.419273	−0.209636	+	0.977779i	$0.567228\pi$
$150$	0	0
$151$	408208.	1.45693	0.728466	−	0.685082i	$-0.240234\pi$
$151$	408208.	1.45693	0.728466	+	0.685082i	$0.240234\pi$
$152$	−155648.	−0.546430
$153$	0	0
$154$	66640.0	0.226430
$155$	−88560.0	−0.296080
$156$	0	0
$157$	293546.	0.950445	0.475223	−	0.879866i	$-0.342368\pi$
$157$	293546.	0.950445	0.475223	+	0.879866i	$0.342368\pi$
$158$	259008.	0.825411
$159$	0	0
$160$	10240.0	0.0316228
$161$	98000.0	0.297962
$162$	0	0
$163$	−317116.	−0.934866	−0.467433	−	0.884029i	$-0.654821\pi$
$163$	−317116.	−0.934866	−0.467433	+	0.884029i	$0.654821\pi$
$164$	233184.	0.677001
$165$	0	0
$166$	−308224.	−0.868153
$167$	−141568.	−0.392802	−0.196401	−	0.980524i	$-0.562925\pi$
$167$	−141568.	−0.392802	−0.196401	+	0.980524i	$0.562925\pi$
$168$	0	0
$169$	−284857.	−0.767203
$170$	−49040.0	−0.130145
$171$	0	0
$172$	129728.	0.334359
$173$	71222.0	0.180925	0.0904626	−	0.995900i	$-0.471165\pi$
$173$	71222.0	0.180925	0.0904626	+	0.995900i	$0.471165\pi$
$174$	0	0
$175$	148225.	0.365870
$176$	87040.0	0.211805
$177$	0	0
$178$	−32664.0	−0.0772715
$179$	−485628.	−1.13285	−0.566423	−	0.824114i	$-0.691673\pi$
$179$	−485628.	−1.13285	−0.566423	+	0.824114i	$0.691673\pi$
$180$	0	0
$181$	657090.	1.49083	0.745416	−	0.666600i	$-0.232251\pi$
$181$	657090.	1.49083	0.745416	+	0.666600i	$0.232251\pi$
$182$	−57624.0	−0.128951
$183$	0	0
$184$	128000.	0.278718
$185$	−91820.0	−0.197246
$186$	0	0
$187$	−416840.	−0.871697
$188$	4992.00	0.0103010
$189$	0	0
$190$	97280.0	0.195497
$191$	−68304.0	−0.135476	−0.0677381	−	0.997703i	$-0.521578\pi$
$191$	−68304.0	−0.135476	−0.0677381	+	0.997703i	$0.521578\pi$
$192$	0	0
$193$	352754.	0.681677	0.340839	−	0.940122i	$-0.389289\pi$
$193$	352754.	0.681677	0.340839	+	0.940122i	$0.389289\pi$
$194$	−82600.0	−0.157571
$195$	0	0
$196$	38416.0	0.0714286
$197$	−196982.	−0.361627	−0.180814	−	0.983517i	$-0.557873\pi$
$197$	−196982.	−0.361627	−0.180814	+	0.983517i	$0.557873\pi$
$198$	0	0
$199$	−1.10392e6	−1.97608	−0.988041	−	0.154192i	$-0.950723\pi$
$199$	−1.10392e6	−1.97608	−0.988041	+	0.154192i	$0.950723\pi$
$200$	193600.	0.342240
$201$	0	0
$202$	745000.	1.28463
$203$	−330554.	−0.562992
$204$	0	0
$205$	−145740.	−0.242211
$206$	240256.	0.394463
$207$	0	0
$208$	−75264.0	−0.120623
$209$	826880.	1.30941
$210$	0	0
$211$	−103444.	−0.159955	−0.0799777	−	0.996797i	$-0.525485\pi$
$211$	−103444.	−0.159955	−0.0799777	+	0.996797i	$0.525485\pi$
$212$	234144.	0.357803
$213$	0	0
$214$	191568.	0.285949
$215$	−81080.0	−0.119624
$216$	0	0
$217$	−433944.	−0.625582
$218$	−88408.0	−0.125994
$219$	0	0
$220$	−54400.0	−0.0757778
$221$	360444.	0.496429
$222$	0	0
$223$	307328.	0.413847	0.206924	−	0.978357i	$-0.433655\pi$
$223$	307328.	0.413847	0.206924	+	0.978357i	$0.433655\pi$
$224$	50176.0	0.0668153
$225$	0	0
$226$	−980216.	−1.27659
$227$	891792.	1.14868	0.574340	−	0.818617i	$-0.305259\pi$
$227$	891792.	1.14868	0.574340	+	0.818617i	$0.305259\pi$
$228$	0	0
$229$	276706.	0.348682	0.174341	−	0.984685i	$-0.444220\pi$
$229$	276706.	0.348682	0.174341	+	0.984685i	$0.444220\pi$
$230$	−80000.0	−0.0997173
$231$	0	0
$232$	−431744.	−0.526631
$233$	−1.47943e6	−1.78528	−0.892639	−	0.450772i	$-0.851149\pi$
$233$	−1.47943e6	−1.78528	−0.892639	+	0.450772i	$0.851149\pi$
$234$	0	0
$235$	−3120.00	−0.00368540
$236$	442496.	0.517165
$237$	0	0
$238$	−240296.	−0.274982
$239$	−1.00034e6	−1.13280	−0.566402	−	0.824129i	$-0.691665\pi$
$239$	−1.00034e6	−1.13280	−0.566402	+	0.824129i	$0.691665\pi$
$240$	0	0
$241$	1.35833e6	1.50648	0.753239	−	0.657747i	$-0.228490\pi$
$241$	1.35833e6	1.50648	0.753239	+	0.657747i	$0.228490\pi$
$242$	181804.	0.199556
$243$	0	0
$244$	549408.	0.590773
$245$	−24010.0	−0.0255551
$246$	0	0
$247$	−715008.	−0.745708
$248$	−566784.	−0.585179
$249$	0	0
$250$	−246000.	−0.248934
$251$	177408.	0.177742	0.0888708	−	0.996043i	$-0.471674\pi$
$251$	177408.	0.177742	0.0888708	+	0.996043i	$0.471674\pi$
$252$	0	0
$253$	−680000.	−0.667894
$254$	386784.	0.376170
$255$	0	0
$256$	65536.0	0.0625000
$257$	−326658.	−0.308504	−0.154252	−	0.988032i	$-0.549297\pi$
$257$	−326658.	−0.308504	−0.154252	+	0.988032i	$0.549297\pi$
$258$	0	0
$259$	−449918.	−0.416758
$260$	47040.0	0.0431553
$261$	0	0
$262$	537472.	0.483730
$263$	34920.0	0.0311304	0.0155652	−	0.999879i	$-0.495045\pi$
$263$	34920.0	0.0311304	0.0155652	+	0.999879i	$0.495045\pi$
$264$	0	0
$265$	−146340.	−0.128011
$266$	476672.	0.413062
$267$	0	0
$268$	197056.	0.167592
$269$	−716458.	−0.603685	−0.301842	−	0.953358i	$-0.597602\pi$
$269$	−716458.	−0.603685	−0.301842	+	0.953358i	$0.597602\pi$
$270$	0	0
$271$	−953376.	−0.788571	−0.394286	−	0.918988i	$-0.629008\pi$
$271$	−953376.	−0.788571	−0.394286	+	0.918988i	$0.629008\pi$
$272$	−313856.	−0.257222
$273$	0	0
$274$	−1.17865e6	−0.948435
$275$	−1.02850e6	−0.820111
$276$	0	0
$277$	−1.84729e6	−1.44656	−0.723279	−	0.690556i	$-0.757366\pi$
$277$	−1.84729e6	−1.44656	−0.723279	+	0.690556i	$0.757366\pi$
$278$	−1.25978e6	−0.977645
$279$	0	0
$280$	−31360.0	−0.0239046
$281$	1.99601e6	1.50798	0.753991	−	0.656885i	$-0.228126\pi$
$281$	1.99601e6	1.50798	0.753991	+	0.656885i	$0.228126\pi$
$282$	0	0
$283$	234088.	0.173745	0.0868726	−	0.996219i	$-0.472313\pi$
$283$	234088.	0.173745	0.0868726	+	0.996219i	$0.472313\pi$
$284$	−590720.	−0.434596
$285$	0	0
$286$	399840.	0.289049
$287$	−714126.	−0.511764
$288$	0	0
$289$	83219.0	0.0586108
$290$	269840.	0.188413
$291$	0	0
$292$	−987488.	−0.677758
$293$	2.50081e6	1.70181	0.850905	−	0.525320i	$-0.176054\pi$
$293$	2.50081e6	1.70181	0.850905	+	0.525320i	$0.176054\pi$
$294$	0	0
$295$	−276560.	−0.185027
$296$	−587648.	−0.389841
$297$	0	0
$298$	454488.	0.296471
$299$	588000.	0.380364
$300$	0	0
$301$	−397292.	−0.252751
$302$	−1.63283e6	−1.03021
$303$	0	0
$304$	622592.	0.386384
$305$	−343380.	−0.211361
$306$	0	0
$307$	2.34203e6	1.41823	0.709115	−	0.705092i	$-0.249095\pi$
$307$	2.34203e6	1.41823	0.709115	+	0.705092i	$0.249095\pi$
$308$	−266560.	−0.160110
$309$	0	0
$310$	354240.	0.209360
$311$	163064.	0.0955998	0.0477999	−	0.998857i	$-0.484779\pi$
$311$	163064.	0.0955998	0.0477999	+	0.998857i	$0.484779\pi$
$312$	0	0
$313$	1.73965e6	1.00369	0.501847	−	0.864957i	$-0.332654\pi$
$313$	1.73965e6	1.00369	0.501847	+	0.864957i	$0.332654\pi$
$314$	−1.17418e6	−0.672066
$315$	0	0
$316$	−1.03603e6	−0.583654
$317$	1.79771e6	1.00478	0.502392	−	0.864640i	$-0.332454\pi$
$317$	1.79771e6	1.00478	0.502392	+	0.864640i	$0.332454\pi$
$318$	0	0
$319$	2.29364e6	1.26197
$320$	−40960.0	−0.0223607
$321$	0	0
$322$	−392000.	−0.210691
$323$	−2.98163e6	−1.59019
$324$	0	0
$325$	889350.	0.467051
$326$	1.26846e6	0.661050
$327$	0	0
$328$	−932736.	−0.478712
$329$	−15288.0	−0.00778683
$330$	0	0
$331$	−2.47541e6	−1.24187	−0.620937	−	0.783861i	$-0.713248\pi$
$331$	−2.47541e6	−1.24187	−0.620937	+	0.783861i	$0.713248\pi$
$332$	1.23290e6	0.613877
$333$	0	0
$334$	566272.	0.277753
$335$	−123160.	−0.0599595
$336$	0	0
$337$	89154.0	0.0427628	0.0213814	−	0.999771i	$-0.493194\pi$
$337$	89154.0	0.0427628	0.0213814	+	0.999771i	$0.493194\pi$
$338$	1.13943e6	0.542494
$339$	0	0
$340$	196160.	0.0920266
$341$	3.01104e6	1.40227
$342$	0	0
$343$	−117649.	−0.0539949
$344$	−518912.	−0.236427
$345$	0	0
$346$	−284888.	−0.127933
$347$	−938556.	−0.418443	−0.209222	−	0.977868i	$-0.567093\pi$
$347$	−938556.	−0.418443	−0.209222	+	0.977868i	$0.567093\pi$
$348$	0	0
$349$	3.34268e6	1.46903	0.734516	−	0.678591i	$-0.237409\pi$
$349$	3.34268e6	1.46903	0.734516	+	0.678591i	$0.237409\pi$
$350$	−592900.	−0.258709
$351$	0	0
$352$	−348160.	−0.149769
$353$	3.76606e6	1.60861	0.804305	−	0.594217i	$-0.202538\pi$
$353$	3.76606e6	1.60861	0.804305	+	0.594217i	$0.202538\pi$
$354$	0	0
$355$	369200.	0.155486
$356$	130656.	0.0546392
$357$	0	0
$358$	1.94251e6	0.801044
$359$	1.53934e6	0.630376	0.315188	−	0.949029i	$-0.397932\pi$
$359$	1.53934e6	0.630376	0.315188	+	0.949029i	$0.397932\pi$
$360$	0	0
$361$	3.43852e6	1.38869
$362$	−2.62836e6	−1.05418
$363$	0	0
$364$	230496.	0.0911822
$365$	617180.	0.242482
$366$	0	0
$367$	−859312.	−0.333032	−0.166516	−	0.986039i	$-0.553252\pi$
$367$	−859312.	−0.333032	−0.166516	+	0.986039i	$0.553252\pi$
$368$	−512000.	−0.197084
$369$	0	0
$370$	367280.	0.139474
$371$	−717066.	−0.270473
$372$	0	0
$373$	−976586.	−0.363445	−0.181722	−	0.983350i	$-0.558167\pi$
$373$	−976586.	−0.363445	−0.181722	+	0.983350i	$0.558167\pi$
$374$	1.66736e6	0.616383
$375$	0	0
$376$	−19968.0	−0.00728392
$377$	−1.98332e6	−0.718688
$378$	0	0
$379$	106444.	0.0380648	0.0190324	−	0.999819i	$-0.493941\pi$
$379$	106444.	0.0380648	0.0190324	+	0.999819i	$0.493941\pi$
$380$	−389120.	−0.138237
$381$	0	0
$382$	273216.	0.0957961
$383$	2.00634e6	0.698889	0.349445	−	0.936957i	$-0.386370\pi$
$383$	2.00634e6	0.698889	0.349445	+	0.936957i	$0.386370\pi$
$384$	0	0
$385$	166600.	0.0572827
$386$	−1.41102e6	−0.482018
$387$	0	0
$388$	330400.	0.111419
$389$	684002.	0.229184	0.114592	−	0.993413i	$-0.463444\pi$
$389$	684002.	0.229184	0.114592	+	0.993413i	$0.463444\pi$
$390$	0	0
$391$	2.45200e6	0.811108
$392$	−153664.	−0.0505076
$393$	0	0
$394$	787928.	0.255709
$395$	647520.	0.208814
$396$	0	0
$397$	−222870.	−0.0709701	−0.0354850	−	0.999370i	$-0.511298\pi$
$397$	−222870.	−0.0709701	−0.0354850	+	0.999370i	$0.511298\pi$
$398$	4.41568e6	1.39730
$399$	0	0
$400$	−774400.	−0.242000
$401$	−1.90072e6	−0.590279	−0.295140	−	0.955454i	$-0.595366\pi$
$401$	−1.90072e6	−0.590279	−0.295140	+	0.955454i	$0.595366\pi$
$402$	0	0
$403$	−2.60366e6	−0.798587
$404$	−2.98000e6	−0.908370
$405$	0	0
$406$	1.32222e6	0.398096
$407$	3.12188e6	0.934179
$408$	0	0
$409$	1.77715e6	0.525311	0.262656	−	0.964890i	$-0.415402\pi$
$409$	1.77715e6	0.525311	0.262656	+	0.964890i	$0.415402\pi$
$410$	582960.	0.171269
$411$	0	0
$412$	−961024.	−0.278927
$413$	−1.35514e6	−0.390940
$414$	0	0
$415$	−770560.	−0.219627
$416$	301056.	0.0852931
$417$	0	0
$418$	−3.30752e6	−0.925895
$419$	−28056.0	−0.00780712	−0.00390356	−	0.999992i	$-0.501243\pi$
$419$	−28056.0	−0.00780712	−0.00390356	+	0.999992i	$0.501243\pi$
$420$	0	0
$421$	−2.70897e6	−0.744902	−0.372451	−	0.928052i	$-0.621482\pi$
$421$	−2.70897e6	−0.744902	−0.372451	+	0.928052i	$0.621482\pi$
$422$	413776.	0.113106
$423$	0	0
$424$	−936576.	−0.253005
$425$	3.70865e6	0.995964
$426$	0	0
$427$	−1.68256e6	−0.446582
$428$	−766272.	−0.202196
$429$	0	0
$430$	324320.	0.0845868
$431$	−5.53898e6	−1.43627	−0.718136	−	0.695902i	$-0.755005\pi$
$431$	−5.53898e6	−1.43627	−0.718136	+	0.695902i	$0.755005\pi$
$432$	0	0
$433$	−868294.	−0.222560	−0.111280	−	0.993789i	$-0.535495\pi$
$433$	−868294.	−0.222560	−0.111280	+	0.993789i	$0.535495\pi$
$434$	1.73578e6	0.442353
$435$	0	0
$436$	353632.	0.0890913
$437$	−4.86400e6	−1.21840
$438$	0	0
$439$	−1.13767e6	−0.281745	−0.140872	−	0.990028i	$-0.544991\pi$
$439$	−1.13767e6	−0.281745	−0.140872	+	0.990028i	$0.544991\pi$
$440$	217600.	0.0535830
$441$	0	0
$442$	−1.44178e6	−0.351028
$443$	−1.75399e6	−0.424636	−0.212318	−	0.977201i	$-0.568101\pi$
$443$	−1.75399e6	−0.424636	−0.212318	+	0.977201i	$0.568101\pi$
$444$	0	0
$445$	−81660.0	−0.0195483
$446$	−1.22931e6	−0.292634
$447$	0	0
$448$	−200704.	−0.0472456
$449$	−2.41674e6	−0.565736	−0.282868	−	0.959159i	$-0.591286\pi$
$449$	−2.41674e6	−0.565736	−0.282868	+	0.959159i	$0.591286\pi$
$450$	0	0
$451$	4.95516e6	1.14714
$452$	3.92086e6	0.902684
$453$	0	0
$454$	−3.56717e6	−0.812239
$455$	−144060.	−0.0326223
$456$	0	0
$457$	−127430.	−0.0285418	−0.0142709	−	0.999898i	$-0.504543\pi$
$457$	−127430.	−0.0285418	−0.0142709	+	0.999898i	$0.504543\pi$
$458$	−1.10682e6	−0.246556
$459$	0	0
$460$	320000.	0.0705108
$461$	128198.	0.0280950	0.0140475	−	0.999901i	$-0.495528\pi$
$461$	128198.	0.0280950	0.0140475	+	0.999901i	$0.495528\pi$
$462$	0	0
$463$	−4.01653e6	−0.870760	−0.435380	−	0.900247i	$-0.643386\pi$
$463$	−4.01653e6	−0.870760	−0.435380	+	0.900247i	$0.643386\pi$
$464$	1.72698e6	0.372384
$465$	0	0
$466$	5.91774e6	1.26238
$467$	−8.67246e6	−1.84014	−0.920069	−	0.391757i	$-0.871867\pi$
$467$	−8.67246e6	−1.84014	−0.920069	+	0.391757i	$0.871867\pi$
$468$	0	0
$469$	−603484.	−0.126687
$470$	12480.0	0.00260597
$471$	0	0
$472$	−1.76998e6	−0.365691
$473$	2.75672e6	0.566552
$474$	0	0
$475$	−7.35680e6	−1.49608
$476$	961184.	0.194442
$477$	0	0
$478$	4.00138e6	0.801013
$479$	−8.28946e6	−1.65077	−0.825387	−	0.564567i	$-0.809043\pi$
$479$	−8.28946e6	−1.65077	−0.825387	+	0.564567i	$0.809043\pi$
$480$	0	0
$481$	−2.69951e6	−0.532013
$482$	−5.43332e6	−1.06524
$483$	0	0
$484$	−727216.	−0.141107
$485$	−206500.	−0.0398626
$486$	0	0
$487$	−8.91770e6	−1.70385	−0.851923	−	0.523667i	$-0.824563\pi$
$487$	−8.91770e6	−1.70385	−0.851923	+	0.523667i	$0.824563\pi$
$488$	−2.19763e6	−0.417739
$489$	0	0
$490$	96040.0	0.0180702
$491$	5.71537e6	1.06989	0.534947	−	0.844886i	$-0.320332\pi$
$491$	5.71537e6	1.06989	0.534947	+	0.844886i	$0.320332\pi$
$492$	0	0
$493$	−8.27060e6	−1.53257
$494$	2.86003e6	0.527295
$495$	0	0
$496$	2.26714e6	0.413784
$497$	1.80908e6	0.328524
$498$	0	0
$499$	125116.	0.0224937	0.0112469	−	0.999937i	$-0.496420\pi$
$499$	125116.	0.0224937	0.0112469	+	0.999937i	$0.496420\pi$
$500$	984000.	0.176023
$501$	0	0
$502$	−709632.	−0.125682
$503$	2.77116e6	0.488362	0.244181	−	0.969730i	$-0.421481\pi$
$503$	2.77116e6	0.488362	0.244181	+	0.969730i	$0.421481\pi$
$504$	0	0
$505$	1.86250e6	0.324988
$506$	2.72000e6	0.472272
$507$	0	0
$508$	−1.54714e6	−0.265992
$509$	138534.	0.0237007	0.0118504	−	0.999930i	$-0.496228\pi$
$509$	138534.	0.0237007	0.0118504	+	0.999930i	$0.496228\pi$
$510$	0	0
$511$	3.02418e6	0.512337
$512$	−262144.	−0.0441942
$513$	0	0
$514$	1.30663e6	0.218145
$515$	600640.	0.0997921
$516$	0	0
$517$	106080.	0.0174545
$518$	1.79967e6	0.294692
$519$	0	0
$520$	−188160.	−0.0305154
$521$	1.80281e6	0.290976	0.145488	−	0.989360i	$-0.453525\pi$
$521$	1.80281e6	0.290976	0.145488	+	0.989360i	$0.453525\pi$
$522$	0	0
$523$	9.77247e6	1.56225	0.781124	−	0.624375i	$-0.214646\pi$
$523$	9.77247e6	1.56225	0.781124	+	0.624375i	$0.214646\pi$
$524$	−2.14989e6	−0.342048
$525$	0	0
$526$	−139680.	−0.0220125
$527$	−1.08575e7	−1.70295
$528$	0	0
$529$	−2.43634e6	−0.378529
$530$	585360.	0.0905177
$531$	0	0
$532$	−1.90669e6	−0.292079
$533$	−4.28476e6	−0.653293
$534$	0	0
$535$	478920.	0.0723400
$536$	−788224.	−0.118505
$537$	0	0
$538$	2.86583e6	0.426869
$539$	816340.	0.121032
$540$	0	0
$541$	2.45504e6	0.360633	0.180316	−	0.983609i	$-0.442288\pi$
$541$	2.45504e6	0.360633	0.180316	+	0.983609i	$0.442288\pi$
$542$	3.81350e6	0.557604
$543$	0	0
$544$	1.25542e6	0.181883
$545$	−221020.	−0.0318743
$546$	0	0
$547$	1.32081e7	1.88744	0.943721	−	0.330743i	$-0.107299\pi$
$547$	1.32081e7	1.88744	0.943721	+	0.330743i	$0.107299\pi$
$548$	4.71459e6	0.670645
$549$	0	0
$550$	4.11400e6	0.579906
$551$	1.64063e7	2.30214
$552$	0	0
$553$	3.17285e6	0.441201
$554$	7.38916e6	1.02287
$555$	0	0
$556$	5.03910e6	0.691300
$557$	−7.83293e6	−1.06976	−0.534880	−	0.844928i	$-0.679643\pi$
$557$	−7.83293e6	−1.06976	−0.534880	+	0.844928i	$0.679643\pi$
$558$	0	0
$559$	−2.38375e6	−0.322650
$560$	125440.	0.0169031
$561$	0	0
$562$	−7.98402e6	−1.06630
$563$	−3.57908e6	−0.475883	−0.237942	−	0.971279i	$-0.576473\pi$
$563$	−3.57908e6	−0.475883	−0.237942	+	0.971279i	$0.576473\pi$
$564$	0	0
$565$	−2.45054e6	−0.322954
$566$	−936352.	−0.122856
$567$	0	0
$568$	2.36288e6	0.307306
$569$	3.39581e6	0.439707	0.219853	−	0.975533i	$-0.429442\pi$
$569$	3.39581e6	0.439707	0.219853	+	0.975533i	$0.429442\pi$
$570$	0	0
$571$	−1.47695e6	−0.189572	−0.0947862	−	0.995498i	$-0.530217\pi$
$571$	−1.47695e6	−0.189572	−0.0947862	+	0.995498i	$0.530217\pi$
$572$	−1.59936e6	−0.204388
$573$	0	0
$574$	2.85650e6	0.361872
$575$	6.05000e6	0.763108
$576$	0	0
$577$	−1.49961e7	−1.87516	−0.937580	−	0.347771i	$-0.886939\pi$
$577$	−1.49961e7	−1.87516	−0.937580	+	0.347771i	$0.886939\pi$
$578$	−332876.	−0.0414441
$579$	0	0
$580$	−1.07936e6	−0.133228
$581$	−3.77574e6	−0.464047
$582$	0	0
$583$	4.97556e6	0.606276
$584$	3.94995e6	0.479247
$585$	0	0
$586$	−1.00032e7	−1.20336
$587$	3.29291e6	0.394444	0.197222	−	0.980359i	$-0.436808\pi$
$587$	3.29291e6	0.394444	0.197222	+	0.980359i	$0.436808\pi$
$588$	0	0
$589$	2.15378e7	2.55807
$590$	1.10624e6	0.130834
$591$	0	0
$592$	2.35059e6	0.275660
$593$	1.17908e7	1.37692	0.688459	−	0.725275i	$-0.258287\pi$
$593$	1.17908e7	1.37692	0.688459	+	0.725275i	$0.258287\pi$
$594$	0	0
$595$	−600740.	−0.0695655
$596$	−1.81795e6	−0.209636
$597$	0	0
$598$	−2.35200e6	−0.268958
$599$	1.52642e6	0.173823	0.0869117	−	0.996216i	$-0.472300\pi$
$599$	1.52642e6	0.173823	0.0869117	+	0.996216i	$0.472300\pi$
$600$	0	0
$601$	−1.00142e7	−1.13092	−0.565458	−	0.824777i	$-0.691301\pi$
$601$	−1.00142e7	−1.13092	−0.565458	+	0.824777i	$0.691301\pi$
$602$	1.58917e6	0.178722
$603$	0	0
$604$	6.53133e6	0.728466
$605$	454510.	0.0504841
$606$	0	0
$607$	1.20660e7	1.32920	0.664599	−	0.747200i	$-0.268602\pi$
$607$	1.20660e7	1.32920	0.664599	+	0.747200i	$0.268602\pi$
$608$	−2.49037e6	−0.273215
$609$	0	0
$610$	1.37352e6	0.149455
$611$	−91728.0	−0.00994029
$612$	0	0
$613$	5.81950e6	0.625511	0.312755	−	0.949834i	$-0.398748\pi$
$613$	5.81950e6	0.625511	0.312755	+	0.949834i	$0.398748\pi$
$614$	−9.36813e6	−1.00284
$615$	0	0
$616$	1.06624e6	0.113215
$617$	4.16589e6	0.440550	0.220275	−	0.975438i	$-0.429305\pi$
$617$	4.16589e6	0.440550	0.220275	+	0.975438i	$0.429305\pi$
$618$	0	0
$619$	−8.08090e6	−0.847683	−0.423841	−	0.905736i	$-0.639319\pi$
$619$	−8.08090e6	−0.847683	−0.423841	+	0.905736i	$0.639319\pi$
$620$	−1.41696e6	−0.148040
$621$	0	0
$622$	−652256.	−0.0675993
$623$	−400134.	−0.0413034
$624$	0	0
$625$	8.83812e6	0.905024
$626$	−6.95860e6	−0.709718
$627$	0	0
$628$	4.69674e6	0.475223
$629$	−1.12571e7	−1.13449
$630$	0	0
$631$	−8.40878e6	−0.840735	−0.420368	−	0.907354i	$-0.638099\pi$
$631$	−8.40878e6	−0.840735	−0.420368	+	0.907354i	$0.638099\pi$
$632$	4.14413e6	0.412706
$633$	0	0
$634$	−7.19086e6	−0.710489
$635$	966960.	0.0951643
$636$	0	0
$637$	−705894.	−0.0689272
$638$	−9.17456e6	−0.892347
$639$	0	0
$640$	163840.	0.0158114
$641$	−6.29760e6	−0.605383	−0.302691	−	0.953089i	$-0.597885\pi$
$641$	−6.29760e6	−0.605383	−0.302691	+	0.953089i	$0.597885\pi$
$642$	0	0
$643$	4.39762e6	0.419460	0.209730	−	0.977759i	$-0.432741\pi$
$643$	4.39762e6	0.419460	0.209730	+	0.977759i	$0.432741\pi$
$644$	1.56800e6	0.148981
$645$	0	0
$646$	1.19265e7	1.12443
$647$	−6.55397e6	−0.615522	−0.307761	−	0.951464i	$-0.599580\pi$
$647$	−6.55397e6	−0.615522	−0.307761	+	0.951464i	$0.599580\pi$
$648$	0	0
$649$	9.40304e6	0.876308
$650$	−3.55740e6	−0.330255
$651$	0	0
$652$	−5.07386e6	−0.467433
$653$	−3.79652e6	−0.348420	−0.174210	−	0.984709i	$-0.555737\pi$
$653$	−3.79652e6	−0.348420	−0.174210	+	0.984709i	$0.555737\pi$
$654$	0	0
$655$	1.34368e6	0.122375
$656$	3.73094e6	0.338500
$657$	0	0
$658$	61152.0	0.00550612
$659$	8.82684e6	0.791757	0.395879	−	0.918303i	$-0.370440\pi$
$659$	8.82684e6	0.791757	0.395879	+	0.918303i	$0.370440\pi$
$660$	0	0
$661$	−341270.	−0.0303805	−0.0151902	−	0.999885i	$-0.504835\pi$
$661$	−341270.	−0.0303805	−0.0151902	+	0.999885i	$0.504835\pi$
$662$	9.90165e6	0.878137
$663$	0	0
$664$	−4.93158e6	−0.434076
$665$	1.19168e6	0.104497
$666$	0	0
$667$	−1.34920e7	−1.17425
$668$	−2.26509e6	−0.196401
$669$	0	0
$670$	492640.	0.0423977
$671$	1.16749e7	1.00103
$672$	0	0
$673$	4.41807e6	0.376006	0.188003	−	0.982168i	$-0.439799\pi$
$673$	4.41807e6	0.376006	0.188003	+	0.982168i	$0.439799\pi$
$674$	−356616.	−0.0302379
$675$	0	0
$676$	−4.55771e6	−0.383601
$677$	−1.63858e7	−1.37403	−0.687014	−	0.726644i	$-0.741079\pi$
$677$	−1.63858e7	−1.37403	−0.687014	+	0.726644i	$0.741079\pi$
$678$	0	0
$679$	−1.01185e6	−0.0842251
$680$	−784640.	−0.0650726
$681$	0	0
$682$	−1.20442e7	−0.991552
$683$	1.75399e7	1.43872	0.719360	−	0.694638i	$-0.244435\pi$
$683$	1.75399e7	1.43872	0.719360	+	0.694638i	$0.244435\pi$
$684$	0	0
$685$	−2.94662e6	−0.239937
$686$	470596.	0.0381802
$687$	0	0
$688$	2.07565e6	0.167179
$689$	−4.30240e6	−0.345273
$690$	0	0
$691$	3.14638e6	0.250678	0.125339	−	0.992114i	$-0.459998\pi$
$691$	3.14638e6	0.250678	0.125339	+	0.992114i	$0.459998\pi$
$692$	1.13955e6	0.0904626
$693$	0	0
$694$	3.75422e6	0.295884
$695$	−3.14944e6	−0.247327
$696$	0	0
$697$	−1.78677e7	−1.39312
$698$	−1.33707e7	−1.03876
$699$	0	0
$700$	2.37160e6	0.182935
$701$	1.90919e7	1.46742	0.733709	−	0.679464i	$-0.237788\pi$
$701$	1.90919e7	1.46742	0.733709	+	0.679464i	$0.237788\pi$
$702$	0	0
$703$	2.23306e7	1.70417
$704$	1.39264e6	0.105903
$705$	0	0
$706$	−1.50642e7	−1.13746
$707$	9.12625e6	0.686663
$708$	0	0
$709$	990974.	0.0740366	0.0370183	−	0.999315i	$-0.488214\pi$
$709$	990974.	0.0740366	0.0370183	+	0.999315i	$0.488214\pi$
$710$	−1.47680e6	−0.109945
$711$	0	0
$712$	−522624.	−0.0386358
$713$	−1.77120e7	−1.30480
$714$	0	0
$715$	999600.	0.0731242
$716$	−7.77005e6	−0.566423
$717$	0	0
$718$	−6.15738e6	−0.445743
$719$	1.69014e7	1.21928	0.609638	−	0.792680i	$-0.291315\pi$
$719$	1.69014e7	1.21928	0.609638	+	0.792680i	$0.291315\pi$
$720$	0	0
$721$	2.94314e6	0.210849
$722$	−1.37541e7	−0.981950
$723$	0	0
$724$	1.05134e7	0.745416
$725$	−2.04066e7	−1.44187
$726$	0	0
$727$	−2.34302e7	−1.64414	−0.822071	−	0.569384i	$-0.807182\pi$
$727$	−2.34302e7	−1.64414	−0.822071	+	0.569384i	$0.807182\pi$
$728$	−921984.	−0.0644755
$729$	0	0
$730$	−2.46872e6	−0.171461
$731$	−9.94041e6	−0.688035
$732$	0	0
$733$	975810.	0.0670819	0.0335409	−	0.999437i	$-0.489322\pi$
$733$	975810.	0.0670819	0.0335409	+	0.999437i	$0.489322\pi$
$734$	3.43725e6	0.235489
$735$	0	0
$736$	2.04800e6	0.139359
$737$	4.18744e6	0.283975
$738$	0	0
$739$	−6.30208e6	−0.424495	−0.212247	−	0.977216i	$-0.568078\pi$
$739$	−6.30208e6	−0.424495	−0.212247	+	0.977216i	$0.568078\pi$
$740$	−1.46912e6	−0.0986229
$741$	0	0
$742$	2.86826e6	0.191253
$743$	6.95698e6	0.462326	0.231163	−	0.972915i	$-0.425747\pi$
$743$	6.95698e6	0.462326	0.231163	+	0.972915i	$0.425747\pi$
$744$	0	0
$745$	1.13622e6	0.0750018
$746$	3.90634e6	0.256994
$747$	0	0
$748$	−6.66944e6	−0.435848
$749$	2.34671e6	0.152846
$750$	0	0
$751$	2.74535e7	1.77622	0.888112	−	0.459628i	$-0.152017\pi$
$751$	2.74535e7	1.77622	0.888112	+	0.459628i	$0.152017\pi$
$752$	79872.0	0.00515051
$753$	0	0
$754$	7.93330e6	0.508189
$755$	−4.08208e6	−0.260624
$756$	0	0
$757$	−1.96889e7	−1.24877	−0.624384	−	0.781118i	$-0.714650\pi$
$757$	−1.96889e7	−1.24877	−0.624384	+	0.781118i	$0.714650\pi$
$758$	−425776.	−0.0269159
$759$	0	0
$760$	1.55648e6	0.0977484
$761$	2.82079e7	1.76567	0.882835	−	0.469684i	$-0.155632\pi$
$761$	2.82079e7	1.76567	0.882835	+	0.469684i	$0.155632\pi$
$762$	0	0
$763$	−1.08300e6	−0.0673467
$764$	−1.09286e6	−0.0677381
$765$	0	0
$766$	−8.02538e6	−0.494189
$767$	−8.13086e6	−0.499055
$768$	0	0
$769$	−1.38081e6	−0.0842009	−0.0421005	−	0.999113i	$-0.513405\pi$
$769$	−1.38081e6	−0.0842009	−0.0421005	+	0.999113i	$0.513405\pi$
$770$	−666400.	−0.0405050
$771$	0	0
$772$	5.64406e6	0.340839
$773$	1.54347e7	0.929074	0.464537	−	0.885554i	$-0.346221\pi$
$773$	1.54347e7	0.929074	0.464537	+	0.885554i	$0.346221\pi$
$774$	0	0
$775$	−2.67894e7	−1.60217
$776$	−1.32160e6	−0.0787854
$777$	0	0
$778$	−2.73601e6	−0.162057
$779$	3.54440e7	2.09266
$780$	0	0
$781$	−1.25528e7	−0.736399
$782$	−9.80800e6	−0.573540
$783$	0	0
$784$	614656.	0.0357143
$785$	−2.93546e6	−0.170021
$786$	0	0
$787$	−7.10107e6	−0.408683	−0.204342	−	0.978900i	$-0.565505\pi$
$787$	−7.10107e6	−0.408683	−0.204342	+	0.978900i	$0.565505\pi$
$788$	−3.15171e6	−0.180814
$789$	0	0
$790$	−2.59008e6	−0.147654
$791$	−1.20076e7	−0.682365
$792$	0	0
$793$	−1.00954e7	−0.570085
$794$	891480.	0.0501834
$795$	0	0
$796$	−1.76627e7	−0.988041
$797$	−6.48182e6	−0.361452	−0.180726	−	0.983533i	$-0.557845\pi$
$797$	−6.48182e6	−0.361452	−0.180726	+	0.983533i	$0.557845\pi$
$798$	0	0
$799$	−382512.	−0.0211972
$800$	3.09760e6	0.171120
$801$	0	0
$802$	7.60289e6	0.417391
$803$	−2.09841e7	−1.14842
$804$	0	0
$805$	−980000.	−0.0533011
$806$	1.04147e7	0.564686
$807$	0	0
$808$	1.19200e7	0.642315
$809$	−1.60578e7	−0.862610	−0.431305	−	0.902206i	$-0.641947\pi$
$809$	−1.60578e7	−0.862610	−0.431305	+	0.902206i	$0.641947\pi$
$810$	0	0
$811$	4.84775e6	0.258814	0.129407	−	0.991592i	$-0.458693\pi$
$811$	4.84775e6	0.258814	0.129407	+	0.991592i	$0.458693\pi$
$812$	−5.28886e6	−0.281496
$813$	0	0
$814$	−1.24875e7	−0.660564
$815$	3.17116e6	0.167234
$816$	0	0
$817$	1.97187e7	1.03353
$818$	−7.10862e6	−0.371451
$819$	0	0
$820$	−2.33184e6	−0.121106
$821$	−2.17976e7	−1.12863	−0.564314	−	0.825560i	$-0.690859\pi$
$821$	−2.17976e7	−1.12863	−0.564314	+	0.825560i	$0.690859\pi$
$822$	0	0
$823$	3.20206e7	1.64790	0.823948	−	0.566665i	$-0.191767\pi$
$823$	3.20206e7	1.64790	0.823948	+	0.566665i	$0.191767\pi$
$824$	3.84410e6	0.197231
$825$	0	0
$826$	5.42058e6	0.276436
$827$	−2.19008e7	−1.11352	−0.556758	−	0.830675i	$-0.687955\pi$
$827$	−2.19008e7	−1.11352	−0.556758	+	0.830675i	$0.687955\pi$
$828$	0	0
$829$	−1.45999e7	−0.737844	−0.368922	−	0.929460i	$-0.620273\pi$
$829$	−1.45999e7	−0.737844	−0.368922	+	0.929460i	$0.620273\pi$
$830$	3.08224e6	0.155300
$831$	0	0
$832$	−1.20422e6	−0.0603113
$833$	−2.94363e6	−0.146984
$834$	0	0
$835$	1.41568e6	0.0702666
$836$	1.32301e7	0.654707
$837$	0	0
$838$	112224.	0.00552047
$839$	−4.60947e6	−0.226072	−0.113036	−	0.993591i	$-0.536058\pi$
$839$	−4.60947e6	−0.226072	−0.113036	+	0.993591i	$0.536058\pi$
$840$	0	0
$841$	2.49974e7	1.21872
$842$	1.08359e7	0.526725
$843$	0	0
$844$	−1.65510e6	−0.0799777
$845$	2.84857e6	0.137241
$846$	0	0
$847$	2.22710e6	0.106667
$848$	3.74630e6	0.178901
$849$	0	0
$850$	−1.48346e7	−0.704253
$851$	−1.83640e7	−0.869247
$852$	0	0
$853$	−1.98437e7	−0.933793	−0.466897	−	0.884312i	$-0.654628\pi$
$853$	−1.98437e7	−0.933793	−0.466897	+	0.884312i	$0.654628\pi$
$854$	6.73025e6	0.315781
$855$	0	0
$856$	3.06509e6	0.142974
$857$	1.22960e6	0.0571888	0.0285944	−	0.999591i	$-0.490897\pi$
$857$	1.22960e6	0.0571888	0.0285944	+	0.999591i	$0.490897\pi$
$858$	0	0
$859$	3.33041e7	1.53998	0.769989	−	0.638058i	$-0.220262\pi$
$859$	3.33041e7	1.53998	0.769989	+	0.638058i	$0.220262\pi$
$860$	−1.29728e6	−0.0598119
$861$	0	0
$862$	2.21559e7	1.01560
$863$	2.36616e7	1.08148	0.540738	−	0.841191i	$-0.318145\pi$
$863$	2.36616e7	1.08148	0.540738	+	0.841191i	$0.318145\pi$
$864$	0	0
$865$	−712220.	−0.0323649
$866$	3.47318e6	0.157374
$867$	0	0
$868$	−6.94310e6	−0.312791
$869$	−2.20157e7	−0.988969
$870$	0	0
$871$	−3.62090e6	−0.161723
$872$	−1.41453e6	−0.0629971
$873$	0	0
$874$	1.94560e7	0.861539
$875$	−3.01350e6	−0.133061
$876$	0	0
$877$	−2.37812e7	−1.04408	−0.522042	−	0.852920i	$-0.674830\pi$
$877$	−2.37812e7	−1.04408	−0.522042	+	0.852920i	$0.674830\pi$
$878$	4.55069e6	0.199224
$879$	0	0
$880$	−870400.	−0.0378889
$881$	1.41871e7	0.615818	0.307909	−	0.951416i	$-0.400371\pi$
$881$	1.41871e7	0.615818	0.307909	+	0.951416i	$0.400371\pi$
$882$	0	0
$883$	2.09281e7	0.903293	0.451647	−	0.892197i	$-0.350837\pi$
$883$	2.09281e7	0.903293	0.451647	+	0.892197i	$0.350837\pi$
$884$	5.76710e6	0.248214
$885$	0	0
$886$	7.01595e6	0.300263
$887$	7.98586e6	0.340810	0.170405	−	0.985374i	$-0.445492\pi$
$887$	7.98586e6	0.340810	0.170405	+	0.985374i	$0.445492\pi$
$888$	0	0
$889$	4.73810e6	0.201071
$890$	326640.	0.0138227
$891$	0	0
$892$	4.91725e6	0.206924
$893$	758784.	0.0318412
$894$	0	0
$895$	4.85628e6	0.202650
$896$	802816.	0.0334077
$897$	0	0
$898$	9.66695e6	0.400036
$899$	5.97426e7	2.46538
$900$	0	0
$901$	−1.79413e7	−0.736278
$902$	−1.98206e7	−0.811150
$903$	0	0
$904$	−1.56835e7	−0.638294
$905$	−6.57090e6	−0.266688
$906$	0	0
$907$	−2.31861e7	−0.935856	−0.467928	−	0.883767i	$-0.654999\pi$
$907$	−2.31861e7	−0.935856	−0.467928	+	0.883767i	$0.654999\pi$
$908$	1.42687e7	0.574340
$909$	0	0
$910$	576240.	0.0230675
$911$	−1.65299e7	−0.659895	−0.329948	−	0.943999i	$-0.607031\pi$
$911$	−1.65299e7	−0.659895	−0.329948	+	0.943999i	$0.607031\pi$
$912$	0	0
$913$	2.61990e7	1.04018
$914$	509720.	0.0201821
$915$	0	0
$916$	4.42730e6	0.174341
$917$	6.58403e6	0.258564
$918$	0	0
$919$	1.28087e7	0.500283	0.250142	−	0.968209i	$-0.419523\pi$
$919$	1.28087e7	0.500283	0.250142	+	0.968209i	$0.419523\pi$
$920$	−1.28000e6	−0.0498586
$921$	0	0
$922$	−512792.	−0.0198662
$923$	1.08545e7	0.419377
$924$	0	0
$925$	−2.77756e7	−1.06735
$926$	1.60661e7	0.615720
$927$	0	0
$928$	−6.90790e6	−0.263315
$929$	−2.97319e7	−1.13027	−0.565136	−	0.824998i	$-0.691176\pi$
$929$	−2.97319e7	−1.13027	−0.565136	+	0.824998i	$0.691176\pi$
$930$	0	0
$931$	5.83923e6	0.220791
$932$	−2.36709e7	−0.892639
$933$	0	0
$934$	3.46899e7	1.30117
$935$	4.16840e6	0.155934
$936$	0	0
$937$	1.10970e7	0.412911	0.206456	−	0.978456i	$-0.433807\pi$
$937$	1.10970e7	0.412911	0.206456	+	0.978456i	$0.433807\pi$
$938$	2.41394e6	0.0895816
$939$	0	0
$940$	−49920.0	−0.00184270
$941$	−3.74313e7	−1.37804	−0.689019	−	0.724743i	$-0.741958\pi$
$941$	−3.74313e7	−1.37804	−0.689019	+	0.724743i	$0.741958\pi$
$942$	0	0
$943$	−2.91480e7	−1.06741
$944$	7.07994e6	0.258583
$945$	0	0
$946$	−1.10269e7	−0.400613
$947$	−1.50907e7	−0.546808	−0.273404	−	0.961899i	$-0.588150\pi$
$947$	−1.50907e7	−0.546808	−0.273404	+	0.961899i	$0.588150\pi$
$948$	0	0
$949$	1.81451e7	0.654024
$950$	2.94272e7	1.05789
$951$	0	0
$952$	−3.84474e6	−0.137491
$953$	2.15741e7	0.769484	0.384742	−	0.923024i	$-0.374290\pi$
$953$	2.15741e7	0.769484	0.384742	+	0.923024i	$0.374290\pi$
$954$	0	0
$955$	683040.	0.0242347
$956$	−1.60055e7	−0.566402
$957$	0	0
$958$	3.31579e7	1.16727
$959$	−1.44384e7	−0.506960
$960$	0	0
$961$	4.97996e7	1.73947
$962$	1.07980e7	0.376190
$963$	0	0
$964$	2.17333e7	0.753239
$965$	−3.52754e6	−0.121942
$966$	0	0
$967$	−3.29467e7	−1.13304	−0.566520	−	0.824048i	$-0.691711\pi$
$967$	−3.29467e7	−1.13304	−0.566520	+	0.824048i	$0.691711\pi$
$968$	2.90886e6	0.0997781
$969$	0	0
$970$	826000.	0.0281871
$971$	−2.24599e7	−0.764470	−0.382235	−	0.924065i	$-0.624846\pi$
$971$	−2.24599e7	−0.764470	−0.382235	+	0.924065i	$0.624846\pi$
$972$	0	0
$973$	−1.54323e7	−0.522573
$974$	3.56708e7	1.20480
$975$	0	0
$976$	8.79053e6	0.295386
$977$	5.16236e7	1.73026	0.865132	−	0.501545i	$-0.167235\pi$
$977$	5.16236e7	1.73026	0.865132	+	0.501545i	$0.167235\pi$
$978$	0	0
$979$	2.77644e6	0.0925831
$980$	−384160.	−0.0127775
$981$	0	0
$982$	−2.28615e7	−0.756529
$983$	1.10202e7	0.363751	0.181876	−	0.983322i	$-0.441783\pi$
$983$	1.10202e7	0.363751	0.181876	+	0.983322i	$0.441783\pi$
$984$	0	0
$985$	1.96982e6	0.0646898
$986$	3.30824e7	1.08369
$987$	0	0
$988$	−1.14401e7	−0.372854
$989$	−1.62160e7	−0.527173
$990$	0	0
$991$	3.21029e7	1.03839	0.519194	−	0.854656i	$-0.326232\pi$
$991$	3.21029e7	1.03839	0.519194	+	0.854656i	$0.326232\pi$
$992$	−9.06854e6	−0.292589
$993$	0	0
$994$	−7.23632e6	−0.232301
$995$	1.10392e7	0.353492
$996$	0	0
$997$	2.81772e7	0.897759	0.448879	−	0.893592i	$-0.351823\pi$
$997$	2.81772e7	0.897759	0.448879	+	0.893592i	$0.351823\pi$
$998$	−500464.	−0.0159055
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	126.6.a.c.1.1		1
3.2	odd	2		14.6.a.b.1.1	✓	1
4.3	odd	2		1008.6.a.n.1.1		1
7.6	odd	2		882.6.a.g.1.1		1
12.11	even	2		112.6.a.d.1.1		1
15.2	even	4		350.6.c.f.99.2		2
15.8	even	4		350.6.c.f.99.1		2
15.14	odd	2		350.6.a.b.1.1		1
21.2	odd	6		98.6.c.a.67.1		2
21.5	even	6		98.6.c.b.67.1		2
21.11	odd	6		98.6.c.a.79.1		2
21.17	even	6		98.6.c.b.79.1		2
21.20	even	2		98.6.a.b.1.1		1
24.5	odd	2		448.6.a.f.1.1		1
24.11	even	2		448.6.a.k.1.1		1
84.83	odd	2		784.6.a.h.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.6.a.b.1.1	✓	1	3.2	odd	2
98.6.a.b.1.1		1	21.20	even	2
98.6.c.a.67.1		2	21.2	odd	6
98.6.c.a.79.1		2	21.11	odd	6
98.6.c.b.67.1		2	21.5	even	6
98.6.c.b.79.1		2	21.17	even	6
112.6.a.d.1.1		1	12.11	even	2
126.6.a.c.1.1		1	1.1	even	1	trivial
350.6.a.b.1.1		1	15.14	odd	2
350.6.c.f.99.1		2	15.8	even	4
350.6.c.f.99.2		2	15.2	even	4
448.6.a.f.1.1		1	24.5	odd	2
448.6.a.k.1.1		1	24.11	even	2
784.6.a.h.1.1		1	84.83	odd	2
882.6.a.g.1.1		1	7.6	odd	2
1008.6.a.n.1.1		1	4.3	odd	2

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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