LMFDB - Embedded newform 126.6.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [126,6,Mod(1,126)]

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

lf = mfeigenbasis(mf)

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")

//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);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$20.2083612964$
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)

$f(q)$

$=$

$q-4.00000 q^{2} +16.0000 q^{4} -76.0000 q^{5} -49.0000 q^{7} -64.0000 q^{8} +O(q^{10})$

\(q-4.00000 q^{2} +16.0000 q^{4} -76.0000 q^{5} -49.0000 q^{7} -64.0000 q^{8} +304.000 q^{10} -650.000 q^{11} +762.000 q^{13} +196.000 q^{14} +256.000 q^{16} +556.000 q^{17} -2452.00 q^{19} -1216.00 q^{20} +2600.00 q^{22} +2950.00 q^{23} +2651.00 q^{25} -3048.00 q^{26} -784.000 q^{28} +674.000 q^{29} -3024.00 q^{31} -1024.00 q^{32} -2224.00 q^{34} +3724.00 q^{35} +7730.00 q^{37} +9808.00 q^{38} +4864.00 q^{40} +17016.0 q^{41} +21836.0 q^{43} -10400.0 q^{44} -11800.0 q^{46} +23940.0 q^{47} +2401.00 q^{49} -10604.0 q^{50} +12192.0 q^{52} -15594.0 q^{53} +49400.0 q^{55} +3136.00 q^{56} -2696.00 q^{58} -5608.00 q^{59} +150.000 q^{61} +12096.0 q^{62} +4096.00 q^{64} -57912.0 q^{65} -43784.0 q^{67} +8896.00 q^{68} -14896.0 q^{70} +39178.0 q^{71} -23570.0 q^{73} -30920.0 q^{74} -39232.0 q^{76} +31850.0 q^{77} -17892.0 q^{79} -19456.0 q^{80} -68064.0 q^{82} -38972.0 q^{83} -42256.0 q^{85} -87344.0 q^{86} +41600.0 q^{88} -6024.00 q^{89} -37338.0 q^{91} +47200.0 q^{92} -95760.0 q^{94} +186352. q^{95} +108430. q^{97} -9604.00 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−4.00000	−0.707107
$2$	−4.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	16.0000	0.500000
$5$	−76.0000	−1.35953	−0.679765	−	0.733430i	$-0.737918\pi$
$5$	−76.0000	−1.35953	−0.679765	+	0.733430i	$0.737918\pi$
$6$	0	0
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	−64.0000	−0.353553
$9$	0	0
$10$	304.000	0.961332
$11$	−650.000	−1.61969	−0.809845	−	0.586645i	$-0.800449\pi$
$11$	−650.000	−1.61969	−0.809845	+	0.586645i	$0.800449\pi$
$12$	0	0
$13$	762.000	1.25054	0.625269	−	0.780410i	$-0.284989\pi$
$13$	762.000	1.25054	0.625269	+	0.780410i	$0.284989\pi$
$14$	196.000	0.267261
$15$	0	0
$16$	256.000	0.250000
$17$	556.000	0.466608	0.233304	−	0.972404i	$-0.425046\pi$
$17$	556.000	0.466608	0.233304	+	0.972404i	$0.425046\pi$
$18$	0	0
$19$	−2452.00	−1.55825	−0.779124	−	0.626870i	$-0.784336\pi$
$19$	−2452.00	−1.55825	−0.779124	+	0.626870i	$0.784336\pi$
$20$	−1216.00	−0.679765
$21$	0	0
$22$	2600.00	1.14529
$23$	2950.00	1.16279	0.581397	−	0.813620i	$-0.302507\pi$
$23$	2950.00	1.16279	0.581397	+	0.813620i	$0.302507\pi$
$24$	0	0
$25$	2651.00	0.848320
$26$	−3048.00	−0.884263
$27$	0	0
$28$	−784.000	−0.188982
$29$	674.000	0.148821	0.0744106	−	0.997228i	$-0.476292\pi$
$29$	674.000	0.148821	0.0744106	+	0.997228i	$0.476292\pi$
$30$	0	0
$31$	−3024.00	−0.565168	−0.282584	−	0.959243i	$-0.591192\pi$
$31$	−3024.00	−0.565168	−0.282584	+	0.959243i	$0.591192\pi$
$32$	−1024.00	−0.176777
$33$	0	0
$34$	−2224.00	−0.329942
$35$	3724.00	0.513854
$36$	0	0
$37$	7730.00	0.928272	0.464136	−	0.885764i	$-0.346365\pi$
$37$	7730.00	0.928272	0.464136	+	0.885764i	$0.346365\pi$
$38$	9808.00	1.10185
$39$	0	0
$40$	4864.00	0.480666
$41$	17016.0	1.58088	0.790438	−	0.612542i	$-0.209853\pi$
$41$	17016.0	1.58088	0.790438	+	0.612542i	$0.209853\pi$
$42$	0	0
$43$	21836.0	1.80095	0.900476	−	0.434907i	$-0.143219\pi$
$43$	21836.0	1.80095	0.900476	+	0.434907i	$0.143219\pi$
$44$	−10400.0	−0.809845
$45$	0	0
$46$	−11800.0	−0.822219
$47$	23940.0	1.58081	0.790405	−	0.612585i	$-0.209870\pi$
$47$	23940.0	1.58081	0.790405	+	0.612585i	$0.209870\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	−10604.0	−0.599853
$51$	0	0
$52$	12192.0	0.625269
$53$	−15594.0	−0.762549	−0.381275	−	0.924462i	$-0.624515\pi$
$53$	−15594.0	−0.762549	−0.381275	+	0.924462i	$0.624515\pi$
$54$	0	0
$55$	49400.0	2.20201
$56$	3136.00	0.133631
$57$	0	0
$58$	−2696.00	−0.105233
$59$	−5608.00	−0.209738	−0.104869	−	0.994486i	$-0.533442\pi$
$59$	−5608.00	−0.209738	−0.104869	+	0.994486i	$0.533442\pi$
$60$	0	0
$61$	150.000	0.00516139	0.00258069	−	0.999997i	$-0.499179\pi$
$61$	150.000	0.00516139	0.00258069	+	0.999997i	$0.499179\pi$
$62$	12096.0	0.399634
$63$	0	0
$64$	4096.00	0.125000
$65$	−57912.0	−1.70014
$66$	0	0
$67$	−43784.0	−1.19159	−0.595797	−	0.803135i	$-0.703164\pi$
$67$	−43784.0	−1.19159	−0.595797	+	0.803135i	$0.703164\pi$
$68$	8896.00	0.233304
$69$	0	0
$70$	−14896.0	−0.363349
$71$	39178.0	0.922351	0.461176	−	0.887309i	$-0.347428\pi$
$71$	39178.0	0.922351	0.461176	+	0.887309i	$0.347428\pi$
$72$	0	0
$73$	−23570.0	−0.517669	−0.258835	−	0.965922i	$-0.583338\pi$
$73$	−23570.0	−0.517669	−0.258835	+	0.965922i	$0.583338\pi$
$74$	−30920.0	−0.656387
$75$	0	0
$76$	−39232.0	−0.779124
$77$	31850.0	0.612185
$78$	0	0
$79$	−17892.0	−0.322546	−0.161273	−	0.986910i	$-0.551560\pi$
$79$	−17892.0	−0.322546	−0.161273	+	0.986910i	$0.551560\pi$
$80$	−19456.0	−0.339882
$81$	0	0
$82$	−68064.0	−1.11785
$83$	−38972.0	−0.620951	−0.310476	−	0.950581i	$-0.600488\pi$
$83$	−38972.0	−0.620951	−0.310476	+	0.950581i	$0.600488\pi$
$84$	0	0
$85$	−42256.0	−0.634368
$86$	−87344.0	−1.27346
$87$	0	0
$88$	41600.0	0.572647
$89$	−6024.00	−0.0806139	−0.0403070	−	0.999187i	$-0.512834\pi$
$89$	−6024.00	−0.0806139	−0.0403070	+	0.999187i	$0.512834\pi$
$90$	0	0
$91$	−37338.0	−0.472659
$92$	47200.0	0.581397
$93$	0	0
$94$	−95760.0	−1.11780
$95$	186352.	2.11848
$96$	0	0
$97$	108430.	1.17009	0.585046	−	0.811000i	$-0.301076\pi$
$97$	108430.	1.17009	0.585046	+	0.811000i	$0.301076\pi$
$98$	−9604.00	−0.101015
$99$	0	0
$100$	42416.0	0.424160
$101$	70424.0	0.686938	0.343469	−	0.939164i	$-0.388398\pi$
$101$	70424.0	0.686938	0.343469	+	0.939164i	$0.388398\pi$
$102$	0	0
$103$	−31552.0	−0.293045	−0.146522	−	0.989207i	$-0.546808\pi$
$103$	−31552.0	−0.293045	−0.146522	+	0.989207i	$0.546808\pi$
$104$	−48768.0	−0.442132
$105$	0	0
$106$	62376.0	0.539204
$107$	−108282.	−0.914317	−0.457159	−	0.889385i	$-0.651133\pi$
$107$	−108282.	−0.914317	−0.457159	+	0.889385i	$0.651133\pi$
$108$	0	0
$109$	−72146.0	−0.581629	−0.290814	−	0.956779i	$-0.593926\pi$
$109$	−72146.0	−0.581629	−0.290814	+	0.956779i	$0.593926\pi$
$110$	−197600.	−1.55706
$111$	0	0
$112$	−12544.0	−0.0944911
$113$	−220906.	−1.62746	−0.813732	−	0.581240i	$-0.802568\pi$
$113$	−220906.	−1.62746	−0.813732	+	0.581240i	$0.802568\pi$
$114$	0	0
$115$	−224200.	−1.58085
$116$	10784.0	0.0744106
$117$	0	0
$118$	22432.0	0.148307
$119$	−27244.0	−0.176361
$120$	0	0
$121$	261449.	1.62339
$122$	−600.000	−0.00364965
$123$	0	0
$124$	−48384.0	−0.282584
$125$	36024.0	0.206213
$126$	0	0
$127$	−239652.	−1.31847	−0.659237	−	0.751935i	$-0.729121\pi$
$127$	−239652.	−1.31847	−0.659237	+	0.751935i	$0.729121\pi$
$128$	−16384.0	−0.0883883
$129$	0	0
$130$	231648.	1.20218
$131$	274172.	1.39587	0.697935	−	0.716161i	$-0.254103\pi$
$131$	274172.	1.39587	0.697935	+	0.716161i	$0.254103\pi$
$132$	0	0
$133$	120148.	0.588962
$134$	175136.	0.842584
$135$	0	0
$136$	−35584.0	−0.164971
$137$	391154.	1.78052	0.890259	−	0.455455i	$-0.150523\pi$
$137$	391154.	1.78052	0.890259	+	0.455455i	$0.150523\pi$
$138$	0	0
$139$	339364.	1.48980	0.744901	−	0.667175i	$-0.232497\pi$
$139$	339364.	1.48980	0.744901	+	0.667175i	$0.232497\pi$
$140$	59584.0	0.256927
$141$	0	0
$142$	−156712.	−0.652201
$143$	−495300.	−2.02548
$144$	0	0
$145$	−51224.0	−0.202327
$146$	94280.0	0.366047
$147$	0	0
$148$	123680.	0.464136
$149$	29334.0	0.108244	0.0541222	−	0.998534i	$-0.482764\pi$
$149$	29334.0	0.108244	0.0541222	+	0.998534i	$0.482764\pi$
$150$	0	0
$151$	71608.0	0.255575	0.127788	−	0.991802i	$-0.459212\pi$
$151$	71608.0	0.255575	0.127788	+	0.991802i	$0.459212\pi$
$152$	156928.	0.550924
$153$	0	0
$154$	−127400.	−0.432880
$155$	229824.	0.768362
$156$	0	0
$157$	296318.	0.959420	0.479710	−	0.877427i	$-0.340742\pi$
$157$	296318.	0.959420	0.479710	+	0.877427i	$0.340742\pi$
$158$	71568.0	0.228074
$159$	0	0
$160$	77824.0	0.240333
$161$	−144550.	−0.439494
$162$	0	0
$163$	−480400.	−1.41623	−0.708115	−	0.706097i	$-0.750454\pi$
$163$	−480400.	−1.41623	−0.708115	+	0.706097i	$0.750454\pi$
$164$	272256.	0.790438
$165$	0	0
$166$	155888.	0.439079
$167$	−160180.	−0.444444	−0.222222	−	0.974996i	$-0.571331\pi$
$167$	−160180.	−0.444444	−0.222222	+	0.974996i	$0.571331\pi$
$168$	0	0
$169$	209351.	0.563843
$170$	169024.	0.448566
$171$	0	0
$172$	349376.	0.900476
$173$	8984.00	0.0228220	0.0114110	−	0.999935i	$-0.496368\pi$
$173$	8984.00	0.0228220	0.0114110	+	0.999935i	$0.496368\pi$
$174$	0	0
$175$	−129899.	−0.320635
$176$	−166400.	−0.404922
$177$	0	0
$178$	24096.0	0.0570026
$179$	−182886.	−0.426627	−0.213313	−	0.976984i	$-0.568425\pi$
$179$	−182886.	−0.426627	−0.213313	+	0.976984i	$0.568425\pi$
$180$	0	0
$181$	138330.	0.313848	0.156924	−	0.987611i	$-0.449842\pi$
$181$	138330.	0.313848	0.156924	+	0.987611i	$0.449842\pi$
$182$	149352.	0.334220
$183$	0	0
$184$	−188800.	−0.411109
$185$	−587480.	−1.26201
$186$	0	0
$187$	−361400.	−0.755760
$188$	383040.	0.790405
$189$	0	0
$190$	−745408.	−1.49799
$191$	−327222.	−0.649021	−0.324511	−	0.945882i	$-0.605200\pi$
$191$	−327222.	−0.649021	−0.324511	+	0.945882i	$0.605200\pi$
$192$	0	0
$193$	786902.	1.52064	0.760322	−	0.649547i	$-0.225041\pi$
$193$	786902.	1.52064	0.760322	+	0.649547i	$0.225041\pi$
$194$	−433720.	−0.827380
$195$	0	0
$196$	38416.0	0.0714286
$197$	−423098.	−0.776740	−0.388370	−	0.921504i	$-0.626962\pi$
$197$	−423098.	−0.776740	−0.388370	+	0.921504i	$0.626962\pi$
$198$	0	0
$199$	1.02392e6	1.83288	0.916439	−	0.400175i	$-0.131051\pi$
$199$	1.02392e6	1.83288	0.916439	+	0.400175i	$0.131051\pi$
$200$	−169664.	−0.299926
$201$	0	0
$202$	−281696.	−0.485738
$203$	−33026.0	−0.0562491
$204$	0	0
$205$	−1.29322e6	−2.14925
$206$	126208.	0.207214
$207$	0	0
$208$	195072.	0.312634
$209$	1.59380e6	2.52388
$210$	0	0
$211$	461516.	0.713642	0.356821	−	0.934173i	$-0.383861\pi$
$211$	461516.	0.713642	0.356821	+	0.934173i	$0.383861\pi$
$212$	−249504.	−0.381275
$213$	0	0
$214$	433128.	0.646520
$215$	−1.65954e6	−2.44845
$216$	0	0
$217$	148176.	0.213613
$218$	288584.	0.411274
$219$	0	0
$220$	790400.	1.10101
$221$	423672.	0.583511
$222$	0	0
$223$	995048.	1.33993	0.669965	−	0.742393i	$-0.266309\pi$
$223$	995048.	1.33993	0.669965	+	0.742393i	$0.266309\pi$
$224$	50176.0	0.0668153
$225$	0	0
$226$	883624.	1.15079
$227$	95568.0	0.123097	0.0615486	−	0.998104i	$-0.480396\pi$
$227$	95568.0	0.123097	0.0615486	+	0.998104i	$0.480396\pi$
$228$	0	0
$229$	−1.04409e6	−1.31567	−0.657836	−	0.753161i	$-0.728528\pi$
$229$	−1.04409e6	−1.31567	−0.657836	+	0.753161i	$0.728528\pi$
$230$	896800.	1.11783
$231$	0	0
$232$	−43136.0	−0.0526163
$233$	1.16941e6	1.41116	0.705581	−	0.708629i	$-0.250686\pi$
$233$	1.16941e6	1.41116	0.705581	+	0.708629i	$0.250686\pi$
$234$	0	0
$235$	−1.81944e6	−2.14916
$236$	−89728.0	−0.104869
$237$	0	0
$238$	108976.	0.124706
$239$	27342.0	0.0309625	0.0154812	−	0.999880i	$-0.495072\pi$
$239$	27342.0	0.0309625	0.0154812	+	0.999880i	$0.495072\pi$
$240$	0	0
$241$	−907714.	−1.00671	−0.503357	−	0.864078i	$-0.667902\pi$
$241$	−907714.	−1.00671	−0.503357	+	0.864078i	$0.667902\pi$
$242$	−1.04580e6	−1.14791
$243$	0	0
$244$	2400.00	0.00258069
$245$	−182476.	−0.194218
$246$	0	0
$247$	−1.86842e6	−1.94865
$248$	193536.	0.199817
$249$	0	0
$250$	−144096.	−0.145815
$251$	−44088.0	−0.0441709	−0.0220854	−	0.999756i	$-0.507031\pi$
$251$	−44088.0	−0.0441709	−0.0220854	+	0.999756i	$0.507031\pi$
$252$	0	0
$253$	−1.91750e6	−1.88336
$254$	958608.	0.932302
$255$	0	0
$256$	65536.0	0.0625000
$257$	829200.	0.783117	0.391558	−	0.920153i	$-0.371936\pi$
$257$	829200.	0.783117	0.391558	+	0.920153i	$0.371936\pi$
$258$	0	0
$259$	−378770.	−0.350854
$260$	−926592.	−0.850071
$261$	0	0
$262$	−1.09669e6	−0.987029
$263$	−1.31947e6	−1.17627	−0.588137	−	0.808761i	$-0.700139\pi$
$263$	−1.31947e6	−1.17627	−0.588137	+	0.808761i	$0.700139\pi$
$264$	0	0
$265$	1.18514e6	1.03671
$266$	−480592.	−0.416459
$267$	0	0
$268$	−700544.	−0.595797
$269$	783788.	0.660416	0.330208	−	0.943908i	$-0.392881\pi$
$269$	783788.	0.660416	0.330208	+	0.943908i	$0.392881\pi$
$270$	0	0
$271$	955080.	0.789981	0.394990	−	0.918685i	$-0.370748\pi$
$271$	955080.	0.789981	0.394990	+	0.918685i	$0.370748\pi$
$272$	142336.	0.116652
$273$	0	0
$274$	−1.56462e6	−1.25902
$275$	−1.72315e6	−1.37401
$276$	0	0
$277$	1.91273e6	1.49780	0.748901	−	0.662682i	$-0.230582\pi$
$277$	1.91273e6	1.49780	0.748901	+	0.662682i	$0.230582\pi$
$278$	−1.35746e6	−1.05345
$279$	0	0
$280$	−238336.	−0.181675
$281$	1.02620e6	0.775295	0.387648	−	0.921808i	$-0.373288\pi$
$281$	1.02620e6	0.775295	0.387648	+	0.921808i	$0.373288\pi$
$282$	0	0
$283$	1.74668e6	1.29642	0.648211	−	0.761461i	$-0.275518\pi$
$283$	1.74668e6	1.29642	0.648211	+	0.761461i	$0.275518\pi$
$284$	626848.	0.461176
$285$	0	0
$286$	1.98120e6	1.43223
$287$	−833784.	−0.597515
$288$	0	0
$289$	−1.11072e6	−0.782277
$290$	204896.	0.143067
$291$	0	0
$292$	−377120.	−0.258835
$293$	−2.23212e6	−1.51897	−0.759484	−	0.650526i	$-0.774548\pi$
$293$	−2.23212e6	−1.51897	−0.759484	+	0.650526i	$0.774548\pi$
$294$	0	0
$295$	426208.	0.285146
$296$	−494720.	−0.328194
$297$	0	0
$298$	−117336.	−0.0765404
$299$	2.24790e6	1.45412
$300$	0	0
$301$	−1.06996e6	−0.680696
$302$	−286432.	−0.180719
$303$	0	0
$304$	−627712.	−0.389562
$305$	−11400.0	−0.00701706
$306$	0	0
$307$	1.85324e6	1.12224	0.561119	−	0.827735i	$-0.310371\pi$
$307$	1.85324e6	1.12224	0.561119	+	0.827735i	$0.310371\pi$
$308$	509600.	0.306092
$309$	0	0
$310$	−919296.	−0.543314
$311$	450956.	0.264383	0.132191	−	0.991224i	$-0.457799\pi$
$311$	450956.	0.264383	0.132191	+	0.991224i	$0.457799\pi$
$312$	0	0
$313$	1.60263e6	0.924642	0.462321	−	0.886713i	$-0.347017\pi$
$313$	1.60263e6	0.924642	0.462321	+	0.886713i	$0.347017\pi$
$314$	−1.18527e6	−0.678413
$315$	0	0
$316$	−286272.	−0.161273
$317$	20862.0	0.0116602	0.00583012	−	0.999983i	$-0.498144\pi$
$317$	20862.0	0.0116602	0.00583012	+	0.999983i	$0.498144\pi$
$318$	0	0
$319$	−438100.	−0.241044
$320$	−311296.	−0.169941
$321$	0	0
$322$	578200.	0.310770
$323$	−1.36331e6	−0.727091
$324$	0	0
$325$	2.02006e6	1.06086
$326$	1.92160e6	1.00143
$327$	0	0
$328$	−1.08902e6	−0.558924
$329$	−1.17306e6	−0.597490
$330$	0	0
$331$	2.07621e6	1.04160	0.520801	−	0.853678i	$-0.325633\pi$
$331$	2.07621e6	1.04160	0.520801	+	0.853678i	$0.325633\pi$
$332$	−623552.	−0.310476
$333$	0	0
$334$	640720.	0.314269
$335$	3.32758e6	1.62001
$336$	0	0
$337$	1.20508e6	0.578019	0.289009	−	0.957326i	$-0.406674\pi$
$337$	1.20508e6	0.578019	0.289009	+	0.957326i	$0.406674\pi$
$338$	−837404.	−0.398697
$339$	0	0
$340$	−676096.	−0.317184
$341$	1.96560e6	0.915396
$342$	0	0
$343$	−117649.	−0.0539949
$344$	−1.39750e6	−0.636732
$345$	0	0
$346$	−35936.0	−0.0161376
$347$	876642.	0.390840	0.195420	−	0.980720i	$-0.437393\pi$
$347$	876642.	0.390840	0.195420	+	0.980720i	$0.437393\pi$
$348$	0	0
$349$	−1.29593e6	−0.569532	−0.284766	−	0.958597i	$-0.591916\pi$
$349$	−1.29593e6	−0.569532	−0.284766	+	0.958597i	$0.591916\pi$
$350$	519596.	0.226723
$351$	0	0
$352$	665600.	0.286323
$353$	3.99040e6	1.70443	0.852215	−	0.523192i	$-0.175259\pi$
$353$	3.99040e6	1.70443	0.852215	+	0.523192i	$0.175259\pi$
$354$	0	0
$355$	−2.97753e6	−1.25396
$356$	−96384.0	−0.0403070
$357$	0	0
$358$	731544.	0.301671
$359$	−4.06452e6	−1.66446	−0.832229	−	0.554432i	$-0.812936\pi$
$359$	−4.06452e6	−1.66446	−0.832229	+	0.554432i	$0.812936\pi$
$360$	0	0
$361$	3.53620e6	1.42814
$362$	−553320.	−0.221924
$363$	0	0
$364$	−597408.	−0.236329
$365$	1.79132e6	0.703787
$366$	0	0
$367$	−1.67243e6	−0.648162	−0.324081	−	0.946029i	$-0.605055\pi$
$367$	−1.67243e6	−0.648162	−0.324081	+	0.946029i	$0.605055\pi$
$368$	755200.	0.290698
$369$	0	0
$370$	2.34992e6	0.892378
$371$	764106.	0.288216
$372$	0	0
$373$	3.16769e6	1.17888	0.589441	−	0.807812i	$-0.299348\pi$
$373$	3.16769e6	1.17888	0.589441	+	0.807812i	$0.299348\pi$
$374$	1.44560e6	0.534403
$375$	0	0
$376$	−1.53216e6	−0.558901
$377$	513588.	0.186106
$378$	0	0
$379$	−4.20388e6	−1.50332	−0.751662	−	0.659548i	$-0.770748\pi$
$379$	−4.20388e6	−1.50332	−0.751662	+	0.659548i	$0.770748\pi$
$380$	2.98163e6	1.05924
$381$	0	0
$382$	1.30889e6	0.458927
$383$	342616.	0.119347	0.0596734	−	0.998218i	$-0.480994\pi$
$383$	342616.	0.119347	0.0596734	+	0.998218i	$0.480994\pi$
$384$	0	0
$385$	−2.42060e6	−0.832283
$386$	−3.14761e6	−1.07526
$387$	0	0
$388$	1.73488e6	0.585046
$389$	3.83959e6	1.28650	0.643252	−	0.765654i	$-0.277585\pi$
$389$	3.83959e6	1.28650	0.643252	+	0.765654i	$0.277585\pi$
$390$	0	0
$391$	1.64020e6	0.542569
$392$	−153664.	−0.0505076
$393$	0	0
$394$	1.69239e6	0.549238
$395$	1.35979e6	0.438510
$396$	0	0
$397$	3.43894e6	1.09509	0.547543	−	0.836777i	$-0.315563\pi$
$397$	3.43894e6	1.09509	0.547543	+	0.836777i	$0.315563\pi$
$398$	−4.09568e6	−1.29604
$399$	0	0
$400$	678656.	0.212080
$401$	3.89421e6	1.20937	0.604684	−	0.796466i	$-0.293299\pi$
$401$	3.89421e6	1.20937	0.604684	+	0.796466i	$0.293299\pi$
$402$	0	0
$403$	−2.30429e6	−0.706764
$404$	1.12678e6	0.343469
$405$	0	0
$406$	132104.	0.0397741
$407$	−5.02450e6	−1.50351
$408$	0	0
$409$	−1.64679e6	−0.486778	−0.243389	−	0.969929i	$-0.578259\pi$
$409$	−1.64679e6	−0.486778	−0.243389	+	0.969929i	$0.578259\pi$
$410$	5.17286e6	1.51975
$411$	0	0
$412$	−504832.	−0.146522
$413$	274792.	0.0792737
$414$	0	0
$415$	2.96187e6	0.844201
$416$	−780288.	−0.221066
$417$	0	0
$418$	−6.37520e6	−1.78465
$419$	1.67659e6	0.466544	0.233272	−	0.972412i	$-0.425057\pi$
$419$	1.67659e6	0.466544	0.233272	+	0.972412i	$0.425057\pi$
$420$	0	0
$421$	−566742.	−0.155840	−0.0779202	−	0.996960i	$-0.524828\pi$
$421$	−566742.	−0.155840	−0.0779202	+	0.996960i	$0.524828\pi$
$422$	−1.84606e6	−0.504621
$423$	0	0
$424$	998016.	0.269602
$425$	1.47396e6	0.395833
$426$	0	0
$427$	−7350.00	−0.00195082
$428$	−1.73251e6	−0.457159
$429$	0	0
$430$	6.63814e6	1.73131
$431$	−6.68468e6	−1.73335	−0.866677	−	0.498870i	$-0.833749\pi$
$431$	−6.68468e6	−1.73335	−0.866677	+	0.498870i	$0.833749\pi$
$432$	0	0
$433$	6.91337e6	1.77203	0.886013	−	0.463661i	$-0.153464\pi$
$433$	6.91337e6	1.77203	0.886013	+	0.463661i	$0.153464\pi$
$434$	−592704.	−0.151047
$435$	0	0
$436$	−1.15434e6	−0.290814
$437$	−7.23340e6	−1.81192
$438$	0	0
$439$	−4.56281e6	−1.12998	−0.564990	−	0.825098i	$-0.691120\pi$
$439$	−4.56281e6	−1.12998	−0.564990	+	0.825098i	$0.691120\pi$
$440$	−3.16160e6	−0.778530
$441$	0	0
$442$	−1.69469e6	−0.412605
$443$	−4.59760e6	−1.11307	−0.556534	−	0.830825i	$-0.687869\pi$
$443$	−4.59760e6	−1.11307	−0.556534	+	0.830825i	$0.687869\pi$
$444$	0	0
$445$	457824.	0.109597
$446$	−3.98019e6	−0.947473
$447$	0	0
$448$	−200704.	−0.0472456
$449$	−1.70658e6	−0.399494	−0.199747	−	0.979848i	$-0.564012\pi$
$449$	−1.70658e6	−0.399494	−0.199747	+	0.979848i	$0.564012\pi$
$450$	0	0
$451$	−1.10604e7	−2.56053
$452$	−3.53450e6	−0.813732
$453$	0	0
$454$	−382272.	−0.0870428
$455$	2.83769e6	0.642593
$456$	0	0
$457$	−6.93916e6	−1.55423	−0.777117	−	0.629356i	$-0.783319\pi$
$457$	−6.93916e6	−1.55423	−0.777117	+	0.629356i	$0.783319\pi$
$458$	4.17634e6	0.930320
$459$	0	0
$460$	−3.58720e6	−0.790426
$461$	2.61805e6	0.573753	0.286877	−	0.957968i	$-0.407383\pi$
$461$	2.61805e6	0.573753	0.286877	+	0.957968i	$0.407383\pi$
$462$	0	0
$463$	7.13602e6	1.54705	0.773524	−	0.633767i	$-0.218492\pi$
$463$	7.13602e6	1.54705	0.773524	+	0.633767i	$0.218492\pi$
$464$	172544.	0.0372053
$465$	0	0
$466$	−4.67764e6	−0.997843
$467$	2.17398e6	0.461278	0.230639	−	0.973039i	$-0.425918\pi$
$467$	2.17398e6	0.461278	0.230639	+	0.973039i	$0.425918\pi$
$468$	0	0
$469$	2.14542e6	0.450380
$470$	7.27776e6	1.51968
$471$	0	0
$472$	358912.	0.0741537
$473$	−1.41934e7	−2.91698
$474$	0	0
$475$	−6.50025e6	−1.32189
$476$	−435904.	−0.0881807
$477$	0	0
$478$	−109368.	−0.0218938
$479$	4.63294e6	0.922609	0.461305	−	0.887242i	$-0.347381\pi$
$479$	4.63294e6	0.922609	0.461305	+	0.887242i	$0.347381\pi$
$480$	0	0
$481$	5.89026e6	1.16084
$482$	3.63086e6	0.711855
$483$	0	0
$484$	4.18318e6	0.811696
$485$	−8.24068e6	−1.59077
$486$	0	0
$487$	−4.56645e6	−0.872481	−0.436241	−	0.899830i	$-0.643690\pi$
$487$	−4.56645e6	−0.872481	−0.436241	+	0.899830i	$0.643690\pi$
$488$	−9600.00	−0.00182483
$489$	0	0
$490$	729904.	0.137333
$491$	5.31429e6	0.994813	0.497407	−	0.867518i	$-0.334286\pi$
$491$	5.31429e6	0.994813	0.497407	+	0.867518i	$0.334286\pi$
$492$	0	0
$493$	374744.	0.0694412
$494$	7.47370e6	1.37790
$495$	0	0
$496$	−774144.	−0.141292
$497$	−1.91972e6	−0.348616
$498$	0	0
$499$	−2.46314e6	−0.442831	−0.221415	−	0.975180i	$-0.571068\pi$
$499$	−2.46314e6	−0.442831	−0.221415	+	0.975180i	$0.571068\pi$
$500$	576384.	0.103107
$501$	0	0
$502$	176352.	0.0312335
$503$	−2.79924e6	−0.493310	−0.246655	−	0.969103i	$-0.579331\pi$
$503$	−2.79924e6	−0.493310	−0.246655	+	0.969103i	$0.579331\pi$
$504$	0	0
$505$	−5.35222e6	−0.933912
$506$	7.67000e6	1.33174
$507$	0	0
$508$	−3.83443e6	−0.659237
$509$	−1.99914e6	−0.342018	−0.171009	−	0.985269i	$-0.554703\pi$
$509$	−1.99914e6	−0.342018	−0.171009	+	0.985269i	$0.554703\pi$
$510$	0	0
$511$	1.15493e6	0.195661
$512$	−262144.	−0.0441942
$513$	0	0
$514$	−3.31680e6	−0.553747
$515$	2.39795e6	0.398403
$516$	0	0
$517$	−1.55610e7	−2.56042
$518$	1.51508e6	0.248091
$519$	0	0
$520$	3.70637e6	0.601091
$521$	−3.52160e6	−0.568390	−0.284195	−	0.958767i	$-0.591726\pi$
$521$	−3.52160e6	−0.568390	−0.284195	+	0.958767i	$0.591726\pi$
$522$	0	0
$523$	2.60685e6	0.416737	0.208369	−	0.978050i	$-0.433185\pi$
$523$	2.60685e6	0.416737	0.208369	+	0.978050i	$0.433185\pi$
$524$	4.38675e6	0.697935
$525$	0	0
$526$	5.27786e6	0.831752
$527$	−1.68134e6	−0.263712
$528$	0	0
$529$	2.26616e6	0.352088
$530$	−4.74058e6	−0.733063
$531$	0	0
$532$	1.92237e6	0.294481
$533$	1.29662e7	1.97694
$534$	0	0
$535$	8.22943e6	1.24304
$536$	2.80218e6	0.421292
$537$	0	0
$538$	−3.13515e6	−0.466985
$539$	−1.56065e6	−0.231384
$540$	0	0
$541$	−1.37441e6	−0.201894	−0.100947	−	0.994892i	$-0.532187\pi$
$541$	−1.37441e6	−0.201894	−0.100947	+	0.994892i	$0.532187\pi$
$542$	−3.82032e6	−0.558601
$543$	0	0
$544$	−569344.	−0.0824855
$545$	5.48310e6	0.790742
$546$	0	0
$547$	−8.78398e6	−1.25523	−0.627614	−	0.778524i	$-0.715968\pi$
$547$	−8.78398e6	−1.25523	−0.627614	+	0.778524i	$0.715968\pi$
$548$	6.25846e6	0.890259
$549$	0	0
$550$	6.89260e6	0.971575
$551$	−1.65265e6	−0.231900
$552$	0	0
$553$	876708.	0.121911
$554$	−7.65092e6	−1.05911
$555$	0	0
$556$	5.42982e6	0.744901
$557$	−6.29262e6	−0.859396	−0.429698	−	0.902973i	$-0.641380\pi$
$557$	−6.29262e6	−0.859396	−0.429698	+	0.902973i	$0.641380\pi$
$558$	0	0
$559$	1.66390e7	2.25216
$560$	953344.	0.128463
$561$	0	0
$562$	−4.10481e6	−0.548216
$563$	−4.86582e6	−0.646971	−0.323485	−	0.946233i	$-0.604855\pi$
$563$	−4.86582e6	−0.646971	−0.323485	+	0.946233i	$0.604855\pi$
$564$	0	0
$565$	1.67889e7	2.21259
$566$	−6.98670e6	−0.916709
$567$	0	0
$568$	−2.50739e6	−0.326100
$569$	4.46383e6	0.577998	0.288999	−	0.957329i	$-0.406678\pi$
$569$	4.46383e6	0.577998	0.288999	+	0.957329i	$0.406678\pi$
$570$	0	0
$571$	8.17054e6	1.04872	0.524361	−	0.851496i	$-0.324304\pi$
$571$	8.17054e6	1.04872	0.524361	+	0.851496i	$0.324304\pi$
$572$	−7.92480e6	−1.01274
$573$	0	0
$574$	3.33514e6	0.422507
$575$	7.82045e6	0.986421
$576$	0	0
$577$	−5.50343e6	−0.688167	−0.344084	−	0.938939i	$-0.611810\pi$
$577$	−5.50343e6	−0.688167	−0.344084	+	0.938939i	$0.611810\pi$
$578$	4.44288e6	0.553153
$579$	0	0
$580$	−819584.	−0.101163
$581$	1.90963e6	0.234697
$582$	0	0
$583$	1.01361e7	1.23509
$584$	1.50848e6	0.183024
$585$	0	0
$586$	8.92848e6	1.07407
$587$	−8.14251e6	−0.975356	−0.487678	−	0.873024i	$-0.662156\pi$
$587$	−8.14251e6	−0.975356	−0.487678	+	0.873024i	$0.662156\pi$
$588$	0	0
$589$	7.41485e6	0.880672
$590$	−1.70483e6	−0.201628
$591$	0	0
$592$	1.97888e6	0.232068
$593$	2.73136e6	0.318964	0.159482	−	0.987201i	$-0.449018\pi$
$593$	2.73136e6	0.318964	0.159482	+	0.987201i	$0.449018\pi$
$594$	0	0
$595$	2.07054e6	0.239768
$596$	469344.	0.0541222
$597$	0	0
$598$	−8.99160e6	−1.02822
$599$	−1.23733e6	−0.140902	−0.0704510	−	0.997515i	$-0.522444\pi$
$599$	−1.23733e6	−0.140902	−0.0704510	+	0.997515i	$0.522444\pi$
$600$	0	0
$601$	−1.59756e7	−1.80414	−0.902071	−	0.431587i	$-0.857954\pi$
$601$	−1.59756e7	−1.80414	−0.902071	+	0.431587i	$0.857954\pi$
$602$	4.27986e6	0.481324
$603$	0	0
$604$	1.14573e6	0.127788
$605$	−1.98701e7	−2.20705
$606$	0	0
$607$	−1.88275e6	−0.207406	−0.103703	−	0.994608i	$-0.533069\pi$
$607$	−1.88275e6	−0.207406	−0.103703	+	0.994608i	$0.533069\pi$
$608$	2.51085e6	0.275462
$609$	0	0
$610$	45600.0	0.00496181
$611$	1.82423e7	1.97686
$612$	0	0
$613$	−9.82804e6	−1.05637	−0.528185	−	0.849130i	$-0.677127\pi$
$613$	−9.82804e6	−1.05637	−0.528185	+	0.849130i	$0.677127\pi$
$614$	−7.41294e6	−0.793542
$615$	0	0
$616$	−2.03840e6	−0.216440
$617$	8.21262e6	0.868498	0.434249	−	0.900793i	$-0.357014\pi$
$617$	8.21262e6	0.868498	0.434249	+	0.900793i	$0.357014\pi$
$618$	0	0
$619$	6.98465e6	0.732686	0.366343	−	0.930480i	$-0.380610\pi$
$619$	6.98465e6	0.732686	0.366343	+	0.930480i	$0.380610\pi$
$620$	3.67718e6	0.384181
$621$	0	0
$622$	−1.80382e6	−0.186947
$623$	295176.	0.0304692
$624$	0	0
$625$	−1.10222e7	−1.12867
$626$	−6.41054e6	−0.653820
$627$	0	0
$628$	4.74109e6	0.479710
$629$	4.29788e6	0.433139
$630$	0	0
$631$	1.26789e7	1.26767	0.633837	−	0.773467i	$-0.281479\pi$
$631$	1.26789e7	1.26767	0.633837	+	0.773467i	$0.281479\pi$
$632$	1.14509e6	0.114037
$633$	0	0
$634$	−83448.0	−0.00824504
$635$	1.82136e7	1.79250
$636$	0	0
$637$	1.82956e6	0.178648
$638$	1.75240e6	0.170444
$639$	0	0
$640$	1.24518e6	0.120167
$641$	1.40324e7	1.34892	0.674460	−	0.738311i	$-0.264376\pi$
$641$	1.40324e7	1.34892	0.674460	+	0.738311i	$0.264376\pi$
$642$	0	0
$643$	1.30368e6	0.124349	0.0621745	−	0.998065i	$-0.480196\pi$
$643$	1.30368e6	0.124349	0.0621745	+	0.998065i	$0.480196\pi$
$644$	−2.31280e6	−0.219747
$645$	0	0
$646$	5.45325e6	0.514131
$647$	−1.57110e6	−0.147551	−0.0737757	−	0.997275i	$-0.523505\pi$
$647$	−1.57110e6	−0.147551	−0.0737757	+	0.997275i	$0.523505\pi$
$648$	0	0
$649$	3.64520e6	0.339711
$650$	−8.08025e6	−0.750138
$651$	0	0
$652$	−7.68640e6	−0.708115
$653$	8.34115e6	0.765496	0.382748	−	0.923853i	$-0.374978\pi$
$653$	8.34115e6	0.765496	0.382748	+	0.923853i	$0.374978\pi$
$654$	0	0
$655$	−2.08371e7	−1.89773
$656$	4.35610e6	0.395219
$657$	0	0
$658$	4.69224e6	0.422489
$659$	−6.18334e6	−0.554638	−0.277319	−	0.960778i	$-0.589446\pi$
$659$	−6.18334e6	−0.554638	−0.277319	+	0.960778i	$0.589446\pi$
$660$	0	0
$661$	928966.	0.0826982	0.0413491	−	0.999145i	$-0.486834\pi$
$661$	928966.	0.0826982	0.0413491	+	0.999145i	$0.486834\pi$
$662$	−8.30485e6	−0.736524
$663$	0	0
$664$	2.49421e6	0.219539
$665$	−9.13125e6	−0.800711
$666$	0	0
$667$	1.98830e6	0.173048
$668$	−2.56288e6	−0.222222
$669$	0	0
$670$	−1.33103e7	−1.14552
$671$	−97500.0	−0.00835985
$672$	0	0
$673$	1.79131e7	1.52452	0.762259	−	0.647272i	$-0.224090\pi$
$673$	1.79131e7	1.52452	0.762259	+	0.647272i	$0.224090\pi$
$674$	−4.82033e6	−0.408721
$675$	0	0
$676$	3.34962e6	0.281922
$677$	4.96397e6	0.416253	0.208126	−	0.978102i	$-0.433263\pi$
$677$	4.96397e6	0.416253	0.208126	+	0.978102i	$0.433263\pi$
$678$	0	0
$679$	−5.31307e6	−0.442253
$680$	2.70438e6	0.224283
$681$	0	0
$682$	−7.86240e6	−0.647283
$683$	−89526.0	−0.00734340	−0.00367170	−	0.999993i	$-0.501169\pi$
$683$	−89526.0	−0.00734340	−0.00367170	+	0.999993i	$0.501169\pi$
$684$	0	0
$685$	−2.97277e7	−2.42067
$686$	470596.	0.0381802
$687$	0	0
$688$	5.59002e6	0.450238
$689$	−1.18826e7	−0.953596
$690$	0	0
$691$	−142396.	−0.0113450	−0.00567248	−	0.999984i	$-0.501806\pi$
$691$	−142396.	−0.0113450	−0.00567248	+	0.999984i	$0.501806\pi$
$692$	143744.	0.0114110
$693$	0	0
$694$	−3.50657e6	−0.276365
$695$	−2.57917e7	−2.02543
$696$	0	0
$697$	9.46090e6	0.737650
$698$	5.18372e6	0.402720
$699$	0	0
$700$	−2.07838e6	−0.160317
$701$	−1.03935e7	−0.798852	−0.399426	−	0.916765i	$-0.630791\pi$
$701$	−1.03935e7	−0.798852	−0.399426	+	0.916765i	$0.630791\pi$
$702$	0	0
$703$	−1.89540e7	−1.44648
$704$	−2.66240e6	−0.202461
$705$	0	0
$706$	−1.59616e7	−1.20521
$707$	−3.45078e6	−0.259638
$708$	0	0
$709$	4.65503e6	0.347782	0.173891	−	0.984765i	$-0.444366\pi$
$709$	4.65503e6	0.347782	0.173891	+	0.984765i	$0.444366\pi$
$710$	1.19101e7	0.886686
$711$	0	0
$712$	385536.	0.0285013
$713$	−8.92080e6	−0.657173
$714$	0	0
$715$	3.76428e7	2.75370
$716$	−2.92618e6	−0.213313
$717$	0	0
$718$	1.62581e7	1.17695
$719$	6.72134e6	0.484880	0.242440	−	0.970166i	$-0.422052\pi$
$719$	6.72134e6	0.484880	0.242440	+	0.970166i	$0.422052\pi$
$720$	0	0
$721$	1.54605e6	0.110760
$722$	−1.41448e7	−1.00984
$723$	0	0
$724$	2.21328e6	0.156924
$725$	1.78677e6	0.126248
$726$	0	0
$727$	−1.24076e7	−0.870670	−0.435335	−	0.900269i	$-0.643370\pi$
$727$	−1.24076e7	−0.870670	−0.435335	+	0.900269i	$0.643370\pi$
$728$	2.38963e6	0.167110
$729$	0	0
$730$	−7.16528e6	−0.497652
$731$	1.21408e7	0.840339
$732$	0	0
$733$	1.35958e7	0.934641	0.467321	−	0.884088i	$-0.345219\pi$
$733$	1.35958e7	0.934641	0.467321	+	0.884088i	$0.345219\pi$
$734$	6.68973e6	0.458319
$735$	0	0
$736$	−3.02080e6	−0.205555
$737$	2.84596e7	1.93001
$738$	0	0
$739$	2.56819e6	0.172988	0.0864941	−	0.996252i	$-0.472434\pi$
$739$	2.56819e6	0.172988	0.0864941	+	0.996252i	$0.472434\pi$
$740$	−9.39968e6	−0.631006
$741$	0	0
$742$	−3.05642e6	−0.203800
$743$	2.02133e7	1.34327	0.671637	−	0.740880i	$-0.265591\pi$
$743$	2.02133e7	1.34327	0.671637	+	0.740880i	$0.265591\pi$
$744$	0	0
$745$	−2.22938e6	−0.147161
$746$	−1.26707e7	−0.833595
$747$	0	0
$748$	−5.78240e6	−0.377880
$749$	5.30582e6	0.345579
$750$	0	0
$751$	7.04813e6	0.456010	0.228005	−	0.973660i	$-0.426780\pi$
$751$	7.04813e6	0.456010	0.228005	+	0.973660i	$0.426780\pi$
$752$	6.12864e6	0.395202
$753$	0	0
$754$	−2.05435e6	−0.131597
$755$	−5.44221e6	−0.347462
$756$	0	0
$757$	−2.04120e7	−1.29463	−0.647315	−	0.762223i	$-0.724108\pi$
$757$	−2.04120e7	−1.29463	−0.647315	+	0.762223i	$0.724108\pi$
$758$	1.68155e7	1.06301
$759$	0	0
$760$	−1.19265e7	−0.748997
$761$	5.07974e6	0.317965	0.158983	−	0.987281i	$-0.449179\pi$
$761$	5.07974e6	0.317965	0.158983	+	0.987281i	$0.449179\pi$
$762$	0	0
$763$	3.53515e6	0.219835
$764$	−5.23555e6	−0.324511
$765$	0	0
$766$	−1.37046e6	−0.0843909
$767$	−4.27330e6	−0.262286
$768$	0	0
$769$	2.33898e7	1.42630	0.713149	−	0.701012i	$-0.247268\pi$
$769$	2.33898e7	1.42630	0.713149	+	0.701012i	$0.247268\pi$
$770$	9.68240e6	0.588513
$771$	0	0
$772$	1.25904e7	0.760322
$773$	1.11253e6	0.0669672	0.0334836	−	0.999439i	$-0.489340\pi$
$773$	1.11253e6	0.0669672	0.0334836	+	0.999439i	$0.489340\pi$
$774$	0	0
$775$	−8.01662e6	−0.479443
$776$	−6.93952e6	−0.413690
$777$	0	0
$778$	−1.53584e7	−0.909696
$779$	−4.17232e7	−2.46340
$780$	0	0
$781$	−2.54657e7	−1.49392
$782$	−6.56080e6	−0.383654
$783$	0	0
$784$	614656.	0.0357143
$785$	−2.25202e7	−1.30436
$786$	0	0
$787$	2.00812e6	0.115572	0.0577859	−	0.998329i	$-0.481596\pi$
$787$	2.00812e6	0.115572	0.0577859	+	0.998329i	$0.481596\pi$
$788$	−6.76957e6	−0.388370
$789$	0	0
$790$	−5.43917e6	−0.310074
$791$	1.08244e7	0.615124
$792$	0	0
$793$	114300.	0.00645451
$794$	−1.37558e7	−0.774343
$795$	0	0
$796$	1.63827e7	0.916439
$797$	−3.00897e7	−1.67792	−0.838961	−	0.544191i	$-0.816837\pi$
$797$	−3.00897e7	−1.67792	−0.838961	+	0.544191i	$0.816837\pi$
$798$	0	0
$799$	1.33106e7	0.737619
$800$	−2.71462e6	−0.149963
$801$	0	0
$802$	−1.55768e7	−0.855152
$803$	1.53205e7	0.838463
$804$	0	0
$805$	1.09858e7	0.597506
$806$	9.21715e6	0.499757
$807$	0	0
$808$	−4.50714e6	−0.242869
$809$	1.88207e6	0.101103	0.0505515	−	0.998721i	$-0.483902\pi$
$809$	1.88207e6	0.101103	0.0505515	+	0.998721i	$0.483902\pi$
$810$	0	0
$811$	4.88220e6	0.260654	0.130327	−	0.991471i	$-0.458397\pi$
$811$	4.88220e6	0.260654	0.130327	+	0.991471i	$0.458397\pi$
$812$	−528416.	−0.0281246
$813$	0	0
$814$	2.00980e7	1.06314
$815$	3.65104e7	1.92541
$816$	0	0
$817$	−5.35419e7	−2.80633
$818$	6.58718e6	0.344204
$819$	0	0
$820$	−2.06915e7	−1.07462
$821$	−8.37096e6	−0.433429	−0.216714	−	0.976235i	$-0.569534\pi$
$821$	−8.37096e6	−0.433429	−0.216714	+	0.976235i	$0.569534\pi$
$822$	0	0
$823$	−2.02090e7	−1.04003	−0.520015	−	0.854157i	$-0.674074\pi$
$823$	−2.02090e7	−1.04003	−0.520015	+	0.854157i	$0.674074\pi$
$824$	2.01933e6	0.103607
$825$	0	0
$826$	−1.09917e6	−0.0560549
$827$	1.31059e7	0.666352	0.333176	−	0.942865i	$-0.391880\pi$
$827$	1.31059e7	0.666352	0.333176	+	0.942865i	$0.391880\pi$
$828$	0	0
$829$	3.18667e7	1.61046	0.805232	−	0.592960i	$-0.202041\pi$
$829$	3.18667e7	1.61046	0.805232	+	0.592960i	$0.202041\pi$
$830$	−1.18475e7	−0.596941
$831$	0	0
$832$	3.12115e6	0.156317
$833$	1.33496e6	0.0666583
$834$	0	0
$835$	1.21737e7	0.604235
$836$	2.55008e7	1.26194
$837$	0	0
$838$	−6.70637e6	−0.329896
$839$	−9.94742e6	−0.487872	−0.243936	−	0.969791i	$-0.578439\pi$
$839$	−9.94742e6	−0.487872	−0.243936	+	0.969791i	$0.578439\pi$
$840$	0	0
$841$	−2.00569e7	−0.977852
$842$	2.26697e6	0.110196
$843$	0	0
$844$	7.38426e6	0.356821
$845$	−1.59107e7	−0.766561
$846$	0	0
$847$	−1.28110e7	−0.613585
$848$	−3.99206e6	−0.190637
$849$	0	0
$850$	−5.89582e6	−0.279896
$851$	2.28035e7	1.07939
$852$	0	0
$853$	−6.52611e6	−0.307102	−0.153551	−	0.988141i	$-0.549071\pi$
$853$	−6.52611e6	−0.307102	−0.153551	+	0.988141i	$0.549071\pi$
$854$	29400.0	0.00137944
$855$	0	0
$856$	6.93005e6	0.323260
$857$	8.76238e6	0.407540	0.203770	−	0.979019i	$-0.434681\pi$
$857$	8.76238e6	0.407540	0.203770	+	0.979019i	$0.434681\pi$
$858$	0	0
$859$	6.47942e6	0.299608	0.149804	−	0.988716i	$-0.452136\pi$
$859$	6.47942e6	0.299608	0.149804	+	0.988716i	$0.452136\pi$
$860$	−2.65526e7	−1.22422
$861$	0	0
$862$	2.67387e7	1.22567
$863$	1.83417e7	0.838323	0.419162	−	0.907912i	$-0.362324\pi$
$863$	1.83417e7	0.838323	0.419162	+	0.907912i	$0.362324\pi$
$864$	0	0
$865$	−682784.	−0.0310272
$866$	−2.76535e7	−1.25301
$867$	0	0
$868$	2.37082e6	0.106807
$869$	1.16298e7	0.522424
$870$	0	0
$871$	−3.33634e7	−1.49013
$872$	4.61734e6	0.205637
$873$	0	0
$874$	2.89336e7	1.28122
$875$	−1.76518e6	−0.0779413
$876$	0	0
$877$	2.69065e7	1.18129	0.590647	−	0.806930i	$-0.298873\pi$
$877$	2.69065e7	1.18129	0.590647	+	0.806930i	$0.298873\pi$
$878$	1.82512e7	0.799017
$879$	0	0
$880$	1.26464e7	0.550504
$881$	1.52174e7	0.660542	0.330271	−	0.943886i	$-0.392860\pi$
$881$	1.52174e7	0.660542	0.330271	+	0.943886i	$0.392860\pi$
$882$	0	0
$883$	−2.61520e7	−1.12877	−0.564383	−	0.825513i	$-0.690886\pi$
$883$	−2.61520e7	−1.12877	−0.564383	+	0.825513i	$0.690886\pi$
$884$	6.77875e6	0.291756
$885$	0	0
$886$	1.83904e7	0.787058
$887$	1.08021e7	0.460997	0.230499	−	0.973073i	$-0.425964\pi$
$887$	1.08021e7	0.460997	0.230499	+	0.973073i	$0.425964\pi$
$888$	0	0
$889$	1.17429e7	0.498337
$890$	−1.83130e6	−0.0774968
$891$	0	0
$892$	1.59208e7	0.669965
$893$	−5.87009e7	−2.46329
$894$	0	0
$895$	1.38993e7	0.580011
$896$	802816.	0.0334077
$897$	0	0
$898$	6.82631e6	0.282485
$899$	−2.03818e6	−0.0841090
$900$	0	0
$901$	−8.67026e6	−0.355812
$902$	4.42416e7	1.81057
$903$	0	0
$904$	1.41380e7	0.575395
$905$	−1.05131e7	−0.426686
$906$	0	0
$907$	−9.84167e6	−0.397238	−0.198619	−	0.980077i	$-0.563646\pi$
$907$	−9.84167e6	−0.397238	−0.198619	+	0.980077i	$0.563646\pi$
$908$	1.52909e6	0.0615486
$909$	0	0
$910$	−1.13508e7	−0.454382
$911$	−2.72509e7	−1.08789	−0.543945	−	0.839121i	$-0.683070\pi$
$911$	−2.72509e7	−1.08789	−0.543945	+	0.839121i	$0.683070\pi$
$912$	0	0
$913$	2.53318e7	1.00575
$914$	2.77566e7	1.09901
$915$	0	0
$916$	−1.67054e7	−0.657836
$917$	−1.34344e7	−0.527589
$918$	0	0
$919$	2.86432e7	1.11875	0.559374	−	0.828916i	$-0.311042\pi$
$919$	2.86432e7	1.11875	0.559374	+	0.828916i	$0.311042\pi$
$920$	1.43488e7	0.558915
$921$	0	0
$922$	−1.04722e7	−0.405705
$923$	2.98536e7	1.15343
$924$	0	0
$925$	2.04922e7	0.787472
$926$	−2.85441e7	−1.09393
$927$	0	0
$928$	−690176.	−0.0263081
$929$	−6.78492e6	−0.257932	−0.128966	−	0.991649i	$-0.541166\pi$
$929$	−6.78492e6	−0.257932	−0.128966	+	0.991649i	$0.541166\pi$
$930$	0	0
$931$	−5.88725e6	−0.222607
$932$	1.87106e7	0.705581
$933$	0	0
$934$	−8.69590e6	−0.326173
$935$	2.74664e7	1.02748
$936$	0	0
$937$	3.00308e7	1.11742	0.558712	−	0.829362i	$-0.311296\pi$
$937$	3.00308e7	1.11742	0.558712	+	0.829362i	$0.311296\pi$
$938$	−8.58166e6	−0.318467
$939$	0	0
$940$	−2.91110e7	−1.07458
$941$	−2.30725e7	−0.849415	−0.424707	−	0.905331i	$-0.639623\pi$
$941$	−2.30725e7	−0.849415	−0.424707	+	0.905331i	$0.639623\pi$
$942$	0	0
$943$	5.01972e7	1.83823
$944$	−1.43565e6	−0.0524346
$945$	0	0
$946$	5.67736e7	2.06262
$947$	−2.71433e7	−0.983531	−0.491765	−	0.870728i	$-0.663648\pi$
$947$	−2.71433e7	−0.983531	−0.491765	+	0.870728i	$0.663648\pi$
$948$	0	0
$949$	−1.79603e7	−0.647365
$950$	2.60010e7	0.934719
$951$	0	0
$952$	1.74362e6	0.0623532
$953$	1.61552e7	0.576209	0.288104	−	0.957599i	$-0.406975\pi$
$953$	1.61552e7	0.576209	0.288104	+	0.957599i	$0.406975\pi$
$954$	0	0
$955$	2.48689e7	0.882364
$956$	437472.	0.0154812
$957$	0	0
$958$	−1.85318e7	−0.652383
$959$	−1.91665e7	−0.672973
$960$	0	0
$961$	−1.94846e7	−0.680585
$962$	−2.35610e7	−0.820837
$963$	0	0
$964$	−1.45234e7	−0.503357
$965$	−5.98046e7	−2.06736
$966$	0	0
$967$	−3.80323e7	−1.30793	−0.653967	−	0.756523i	$-0.726897\pi$
$967$	−3.80323e7	−1.30793	−0.653967	+	0.756523i	$0.726897\pi$
$968$	−1.67327e7	−0.573956
$969$	0	0
$970$	3.29627e7	1.12485
$971$	2.23104e7	0.759379	0.379689	−	0.925114i	$-0.376031\pi$
$971$	2.23104e7	0.759379	0.379689	+	0.925114i	$0.376031\pi$
$972$	0	0
$973$	−1.66288e7	−0.563093
$974$	1.82658e7	0.616937
$975$	0	0
$976$	38400.0	0.00129035
$977$	−3.06930e7	−1.02873	−0.514367	−	0.857570i	$-0.671973\pi$
$977$	−3.06930e7	−1.02873	−0.514367	+	0.857570i	$0.671973\pi$
$978$	0	0
$979$	3.91560e6	0.130569
$980$	−2.91962e6	−0.0971092
$981$	0	0
$982$	−2.12572e7	−0.703439
$983$	1.52706e7	0.504048	0.252024	−	0.967721i	$-0.418904\pi$
$983$	1.52706e7	0.504048	0.252024	+	0.967721i	$0.418904\pi$
$984$	0	0
$985$	3.21554e7	1.05600
$986$	−1.49898e6	−0.0491024
$987$	0	0
$988$	−2.98948e7	−0.974323
$989$	6.44162e7	2.09413
$990$	0	0
$991$	3.16279e7	1.02303	0.511513	−	0.859276i	$-0.329085\pi$
$991$	3.16279e7	1.02303	0.511513	+	0.859276i	$0.329085\pi$
$992$	3.09658e6	0.0999085
$993$	0	0
$994$	7.67889e6	0.246509
$995$	−7.78179e7	−2.49185
$996$	0	0
$997$	−3.55842e7	−1.13376	−0.566878	−	0.823802i	$-0.691849\pi$
$997$	−3.55842e7	−1.13376	−0.566878	+	0.823802i	$0.691849\pi$
$998$	9.85256e6	0.313129
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.6.a.e.1.1	✓	1	3.2	odd	2
126.6.a.a.1.1		1	1.1	even	1	trivial
294.6.a.k.1.1		1	21.20	even	2
294.6.e.c.67.1		2	21.5	even	6
294.6.e.c.79.1		2	21.17	even	6
294.6.e.d.67.1		2	21.2	odd	6
294.6.e.d.79.1		2	21.11	odd	6
336.6.a.q.1.1		1	12.11	even	2
882.6.a.j.1.1		1	7.6	odd	2
1008.6.a.d.1.1		1	4.3	odd	2
1050.6.a.f.1.1		1	15.14	odd	2
1050.6.g.h.799.1		2	15.8	even	4
1050.6.g.h.799.2		2	15.2	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 $\F_{q}$

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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