LMFDB - Embedded newform 42.6.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [42,6,Mod(1,42)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(42, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("42.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 42 = 2 \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	42.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$6.73612043215$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	42.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} +76.0000 q^{5} -36.0000 q^{6} -49.0000 q^{7} +64.0000 q^{8} +81.0000 q^{9} +O(q^{10})$

\(q+4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} +76.0000 q^{5} -36.0000 q^{6} -49.0000 q^{7} +64.0000 q^{8} +81.0000 q^{9} +304.000 q^{10} +650.000 q^{11} -144.000 q^{12} +762.000 q^{13} -196.000 q^{14} -684.000 q^{15} +256.000 q^{16} -556.000 q^{17} +324.000 q^{18} -2452.00 q^{19} +1216.00 q^{20} +441.000 q^{21} +2600.00 q^{22} -2950.00 q^{23} -576.000 q^{24} +2651.00 q^{25} +3048.00 q^{26} -729.000 q^{27} -784.000 q^{28} -674.000 q^{29} -2736.00 q^{30} -3024.00 q^{31} +1024.00 q^{32} -5850.00 q^{33} -2224.00 q^{34} -3724.00 q^{35} +1296.00 q^{36} +7730.00 q^{37} -9808.00 q^{38} -6858.00 q^{39} +4864.00 q^{40} -17016.0 q^{41} +1764.00 q^{42} +21836.0 q^{43} +10400.0 q^{44} +6156.00 q^{45} -11800.0 q^{46} -23940.0 q^{47} -2304.00 q^{48} +2401.00 q^{49} +10604.0 q^{50} +5004.00 q^{51} +12192.0 q^{52} +15594.0 q^{53} -2916.00 q^{54} +49400.0 q^{55} -3136.00 q^{56} +22068.0 q^{57} -2696.00 q^{58} +5608.00 q^{59} -10944.0 q^{60} +150.000 q^{61} -12096.0 q^{62} -3969.00 q^{63} +4096.00 q^{64} +57912.0 q^{65} -23400.0 q^{66} -43784.0 q^{67} -8896.00 q^{68} +26550.0 q^{69} -14896.0 q^{70} -39178.0 q^{71} +5184.00 q^{72} -23570.0 q^{73} +30920.0 q^{74} -23859.0 q^{75} -39232.0 q^{76} -31850.0 q^{77} -27432.0 q^{78} -17892.0 q^{79} +19456.0 q^{80} +6561.00 q^{81} -68064.0 q^{82} +38972.0 q^{83} +7056.00 q^{84} -42256.0 q^{85} +87344.0 q^{86} +6066.00 q^{87} +41600.0 q^{88} +6024.00 q^{89} +24624.0 q^{90} -37338.0 q^{91} -47200.0 q^{92} +27216.0 q^{93} -95760.0 q^{94} -186352. q^{95} -9216.00 q^{96} +108430. q^{97} +9604.00 q^{98} +52650.0 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	4.00000	0.707107
$2$	4.00000	0.707107
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	16.0000	0.500000
$5$	76.0000	1.35953	0.679765	−	0.733430i	$-0.262082\pi$
$5$	76.0000	1.35953	0.679765	+	0.733430i	$0.262082\pi$
$6$	−36.0000	−0.408248
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	64.0000	0.353553
$9$	81.0000	0.333333
$10$	304.000	0.961332
$11$	650.000	1.61969	0.809845	−	0.586645i	$-0.199551\pi$
$11$	650.000	1.61969	0.809845	+	0.586645i	$0.199551\pi$
$12$	−144.000	−0.288675
$13$	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
$15$	−684.000	−0.784925
$16$	256.000	0.250000
$17$	−556.000	−0.466608	−0.233304	−	0.972404i	$-0.574954\pi$
$17$	−556.000	−0.466608	−0.233304	+	0.972404i	$0.574954\pi$
$18$	324.000	0.235702
$19$	−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
$21$	441.000	0.218218
$22$	2600.00	1.14529
$23$	−2950.00	−1.16279	−0.581397	−	0.813620i	$-0.697493\pi$
$23$	−2950.00	−1.16279	−0.581397	+	0.813620i	$0.697493\pi$
$24$	−576.000	−0.204124
$25$	2651.00	0.848320
$26$	3048.00	0.884263
$27$	−729.000	−0.192450
$28$	−784.000	−0.188982
$29$	−674.000	−0.148821	−0.0744106	−	0.997228i	$-0.523708\pi$
$29$	−674.000	−0.148821	−0.0744106	+	0.997228i	$0.523708\pi$
$30$	−2736.00	−0.555026
$31$	−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
$33$	−5850.00	−0.935128
$34$	−2224.00	−0.329942
$35$	−3724.00	−0.513854
$36$	1296.00	0.166667
$37$	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
$39$	−6858.00	−0.721998
$40$	4864.00	0.480666
$41$	−17016.0	−1.58088	−0.790438	−	0.612542i	$-0.790147\pi$
$41$	−17016.0	−1.58088	−0.790438	+	0.612542i	$0.790147\pi$
$42$	1764.00	0.154303
$43$	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
$45$	6156.00	0.453176
$46$	−11800.0	−0.822219
$47$	−23940.0	−1.58081	−0.790405	−	0.612585i	$-0.790130\pi$
$47$	−23940.0	−1.58081	−0.790405	+	0.612585i	$0.790130\pi$
$48$	−2304.00	−0.144338
$49$	2401.00	0.142857
$50$	10604.0	0.599853
$51$	5004.00	0.269396
$52$	12192.0	0.625269
$53$	15594.0	0.762549	0.381275	−	0.924462i	$-0.375485\pi$
$53$	15594.0	0.762549	0.381275	+	0.924462i	$0.375485\pi$
$54$	−2916.00	−0.136083
$55$	49400.0	2.20201
$56$	−3136.00	−0.133631
$57$	22068.0	0.899655
$58$	−2696.00	−0.105233
$59$	5608.00	0.209738	0.104869	−	0.994486i	$-0.466558\pi$
$59$	5608.00	0.209738	0.104869	+	0.994486i	$0.466558\pi$
$60$	−10944.0	−0.392462
$61$	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
$63$	−3969.00	−0.125988
$64$	4096.00	0.125000
$65$	57912.0	1.70014
$66$	−23400.0	−0.661235
$67$	−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
$69$	26550.0	0.671339
$70$	−14896.0	−0.363349
$71$	−39178.0	−0.922351	−0.461176	−	0.887309i	$-0.652572\pi$
$71$	−39178.0	−0.922351	−0.461176	+	0.887309i	$0.652572\pi$
$72$	5184.00	0.117851
$73$	−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
$75$	−23859.0	−0.489778
$76$	−39232.0	−0.779124
$77$	−31850.0	−0.612185
$78$	−27432.0	−0.510530
$79$	−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
$81$	6561.00	0.111111
$82$	−68064.0	−1.11785
$83$	38972.0	0.620951	0.310476	−	0.950581i	$-0.399512\pi$
$83$	38972.0	0.620951	0.310476	+	0.950581i	$0.399512\pi$
$84$	7056.00	0.109109
$85$	−42256.0	−0.634368
$86$	87344.0	1.27346
$87$	6066.00	0.0859220
$88$	41600.0	0.572647
$89$	6024.00	0.0806139	0.0403070	−	0.999187i	$-0.487166\pi$
$89$	6024.00	0.0806139	0.0403070	+	0.999187i	$0.487166\pi$
$90$	24624.0	0.320444
$91$	−37338.0	−0.472659
$92$	−47200.0	−0.581397
$93$	27216.0	0.326300
$94$	−95760.0	−1.11780
$95$	−186352.	−2.11848
$96$	−9216.00	−0.102062
$97$	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
$99$	52650.0	0.539896
$100$	42416.0	0.424160
$101$	−70424.0	−0.686938	−0.343469	−	0.939164i	$-0.611602\pi$
$101$	−70424.0	−0.686938	−0.343469	+	0.939164i	$0.611602\pi$
$102$	20016.0	0.190492
$103$	−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
$105$	33516.0	0.296674
$106$	62376.0	0.539204
$107$	108282.	0.914317	0.457159	−	0.889385i	$-0.348867\pi$
$107$	108282.	0.914317	0.457159	+	0.889385i	$0.348867\pi$
$108$	−11664.0	−0.0962250
$109$	−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
$111$	−69570.0	−0.535938
$112$	−12544.0	−0.0944911
$113$	220906.	1.62746	0.813732	−	0.581240i	$-0.197432\pi$
$113$	220906.	1.62746	0.813732	+	0.581240i	$0.197432\pi$
$114$	88272.0	0.636152
$115$	−224200.	−1.58085
$116$	−10784.0	−0.0744106
$117$	61722.0	0.416846
$118$	22432.0	0.148307
$119$	27244.0	0.176361
$120$	−43776.0	−0.277513
$121$	261449.	1.62339
$122$	600.000	0.00364965
$123$	153144.	0.912719
$124$	−48384.0	−0.282584
$125$	−36024.0	−0.206213
$126$	−15876.0	−0.0890871
$127$	−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
$129$	−196524.	−1.03978
$130$	231648.	1.20218
$131$	−274172.	−1.39587	−0.697935	−	0.716161i	$-0.745897\pi$
$131$	−274172.	−1.39587	−0.697935	+	0.716161i	$0.745897\pi$
$132$	−93600.0	−0.467564
$133$	120148.	0.588962
$134$	−175136.	−0.842584
$135$	−55404.0	−0.261642
$136$	−35584.0	−0.164971
$137$	−391154.	−1.78052	−0.890259	−	0.455455i	$-0.849477\pi$
$137$	−391154.	−1.78052	−0.890259	+	0.455455i	$0.849477\pi$
$138$	106200.	0.474708
$139$	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
$141$	215460.	0.912681
$142$	−156712.	−0.652201
$143$	495300.	2.02548
$144$	20736.0	0.0833333
$145$	−51224.0	−0.202327
$146$	−94280.0	−0.366047
$147$	−21609.0	−0.0824786
$148$	123680.	0.464136
$149$	−29334.0	−0.108244	−0.0541222	−	0.998534i	$-0.517236\pi$
$149$	−29334.0	−0.108244	−0.0541222	+	0.998534i	$0.517236\pi$
$150$	−95436.0	−0.346325
$151$	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
$153$	−45036.0	−0.155536
$154$	−127400.	−0.432880
$155$	−229824.	−0.768362
$156$	−109728.	−0.360999
$157$	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
$159$	−140346.	−0.440258
$160$	77824.0	0.240333
$161$	144550.	0.439494
$162$	26244.0	0.0785674
$163$	−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
$165$	−444600.	−1.27133
$166$	155888.	0.439079
$167$	160180.	0.444444	0.222222	−	0.974996i	$-0.428669\pi$
$167$	160180.	0.444444	0.222222	+	0.974996i	$0.428669\pi$
$168$	28224.0	0.0771517
$169$	209351.	0.563843
$170$	−169024.	−0.448566
$171$	−198612.	−0.519416
$172$	349376.	0.900476
$173$	−8984.00	−0.0228220	−0.0114110	−	0.999935i	$-0.503632\pi$
$173$	−8984.00	−0.0228220	−0.0114110	+	0.999935i	$0.503632\pi$
$174$	24264.0	0.0607560
$175$	−129899.	−0.320635
$176$	166400.	0.404922
$177$	−50472.0	−0.121093
$178$	24096.0	0.0570026
$179$	182886.	0.426627	0.213313	−	0.976984i	$-0.431575\pi$
$179$	182886.	0.426627	0.213313	+	0.976984i	$0.431575\pi$
$180$	98496.0	0.226588
$181$	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
$183$	−1350.00	−0.00297993
$184$	−188800.	−0.411109
$185$	587480.	1.26201
$186$	108864.	0.230729
$187$	−361400.	−0.755760
$188$	−383040.	−0.790405
$189$	35721.0	0.0727393
$190$	−745408.	−1.49799
$191$	327222.	0.649021	0.324511	−	0.945882i	$-0.394800\pi$
$191$	327222.	0.649021	0.324511	+	0.945882i	$0.394800\pi$
$192$	−36864.0	−0.0721688
$193$	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
$195$	−521208.	−0.981577
$196$	38416.0	0.0714286
$197$	423098.	0.776740	0.388370	−	0.921504i	$-0.373038\pi$
$197$	423098.	0.776740	0.388370	+	0.921504i	$0.373038\pi$
$198$	210600.	0.381764
$199$	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
$201$	394056.	0.687967
$202$	−281696.	−0.485738
$203$	33026.0	0.0562491
$204$	80064.0	0.134698
$205$	−1.29322e6	−2.14925
$206$	−126208.	−0.207214
$207$	−238950.	−0.387598
$208$	195072.	0.312634
$209$	−1.59380e6	−2.52388
$210$	134064.	0.209780
$211$	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
$213$	352602.	0.532520
$214$	433128.	0.646520
$215$	1.65954e6	2.44845
$216$	−46656.0	−0.0680414
$217$	148176.	0.213613
$218$	−288584.	−0.411274
$219$	212130.	0.298877
$220$	790400.	1.10101
$221$	−423672.	−0.583511
$222$	−278280.	−0.378965
$223$	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
$225$	214731.	0.282773
$226$	883624.	1.15079
$227$	−95568.0	−0.123097	−0.0615486	−	0.998104i	$-0.519604\pi$
$227$	−95568.0	−0.123097	−0.0615486	+	0.998104i	$0.519604\pi$
$228$	353088.	0.449827
$229$	−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
$231$	286650.	0.353445
$232$	−43136.0	−0.0526163
$233$	−1.16941e6	−1.41116	−0.705581	−	0.708629i	$-0.749314\pi$
$233$	−1.16941e6	−1.41116	−0.705581	+	0.708629i	$0.749314\pi$
$234$	246888.	0.294754
$235$	−1.81944e6	−2.14916
$236$	89728.0	0.104869
$237$	161028.	0.186222
$238$	108976.	0.124706
$239$	−27342.0	−0.0309625	−0.0154812	−	0.999880i	$-0.504928\pi$
$239$	−27342.0	−0.0309625	−0.0154812	+	0.999880i	$0.504928\pi$
$240$	−175104.	−0.196231
$241$	−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
$243$	−59049.0	−0.0641500
$244$	2400.00	0.00258069
$245$	182476.	0.194218
$246$	612576.	0.645390
$247$	−1.86842e6	−1.94865
$248$	−193536.	−0.199817
$249$	−350748.	−0.358506
$250$	−144096.	−0.145815
$251$	44088.0	0.0441709	0.0220854	−	0.999756i	$-0.492969\pi$
$251$	44088.0	0.0441709	0.0220854	+	0.999756i	$0.492969\pi$
$252$	−63504.0	−0.0629941
$253$	−1.91750e6	−1.88336
$254$	−958608.	−0.932302
$255$	380304.	0.366252
$256$	65536.0	0.0625000
$257$	−829200.	−0.783117	−0.391558	−	0.920153i	$-0.628064\pi$
$257$	−829200.	−0.783117	−0.391558	+	0.920153i	$0.628064\pi$
$258$	−786096.	−0.735235
$259$	−378770.	−0.350854
$260$	926592.	0.850071
$261$	−54594.0	−0.0496071
$262$	−1.09669e6	−0.987029
$263$	1.31947e6	1.17627	0.588137	−	0.808761i	$-0.299861\pi$
$263$	1.31947e6	1.17627	0.588137	+	0.808761i	$0.299861\pi$
$264$	−374400.	−0.330618
$265$	1.18514e6	1.03671
$266$	480592.	0.416459
$267$	−54216.0	−0.0465425
$268$	−700544.	−0.595797
$269$	−783788.	−0.660416	−0.330208	−	0.943908i	$-0.607119\pi$
$269$	−783788.	−0.660416	−0.330208	+	0.943908i	$0.607119\pi$
$270$	−221616.	−0.185009
$271$	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
$273$	336042.	0.272890
$274$	−1.56462e6	−1.25902
$275$	1.72315e6	1.37401
$276$	424800.	0.335669
$277$	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
$279$	−244944.	−0.188389
$280$	−238336.	−0.181675
$281$	−1.02620e6	−0.775295	−0.387648	−	0.921808i	$-0.626712\pi$
$281$	−1.02620e6	−0.775295	−0.387648	+	0.921808i	$0.626712\pi$
$282$	861840.	0.645363
$283$	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
$285$	1.67717e6	1.22311
$286$	1.98120e6	1.43223
$287$	833784.	0.597515
$288$	82944.0	0.0589256
$289$	−1.11072e6	−0.782277
$290$	−204896.	−0.143067
$291$	−975870.	−0.675553
$292$	−377120.	−0.258835
$293$	2.23212e6	1.51897	0.759484	−	0.650526i	$-0.225452\pi$
$293$	2.23212e6	1.51897	0.759484	+	0.650526i	$0.225452\pi$
$294$	−86436.0	−0.0583212
$295$	426208.	0.285146
$296$	494720.	0.328194
$297$	−473850.	−0.311709
$298$	−117336.	−0.0765404
$299$	−2.24790e6	−1.45412
$300$	−381744.	−0.244889
$301$	−1.06996e6	−0.680696
$302$	286432.	0.180719
$303$	633816.	0.396604
$304$	−627712.	−0.389562
$305$	11400.0	0.00701706
$306$	−180144.	−0.109981
$307$	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
$309$	283968.	0.169189
$310$	−919296.	−0.543314
$311$	−450956.	−0.264383	−0.132191	−	0.991224i	$-0.542201\pi$
$311$	−450956.	−0.264383	−0.132191	+	0.991224i	$0.542201\pi$
$312$	−438912.	−0.255265
$313$	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
$315$	−301644.	−0.171285
$316$	−286272.	−0.161273
$317$	−20862.0	−0.0116602	−0.00583012	−	0.999983i	$-0.501856\pi$
$317$	−20862.0	−0.0116602	−0.00583012	+	0.999983i	$0.501856\pi$
$318$	−561384.	−0.311309
$319$	−438100.	−0.241044
$320$	311296.	0.169941
$321$	−974538.	−0.527881
$322$	578200.	0.310770
$323$	1.36331e6	0.727091
$324$	104976.	0.0555556
$325$	2.02006e6	1.06086
$326$	−1.92160e6	−1.00143
$327$	649314.	0.335804
$328$	−1.08902e6	−0.558924
$329$	1.17306e6	0.597490
$330$	−1.77840e6	−0.898969
$331$	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
$333$	626130.	0.309424
$334$	640720.	0.314269
$335$	−3.32758e6	−1.62001
$336$	112896.	0.0545545
$337$	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
$339$	−1.98815e6	−0.939617
$340$	−676096.	−0.317184
$341$	−1.96560e6	−0.915396
$342$	−794448.	−0.367282
$343$	−117649.	−0.0539949
$344$	1.39750e6	0.636732
$345$	2.01780e6	0.912705
$346$	−35936.0	−0.0161376
$347$	−876642.	−0.390840	−0.195420	−	0.980720i	$-0.562607\pi$
$347$	−876642.	−0.390840	−0.195420	+	0.980720i	$0.562607\pi$
$348$	97056.0	0.0429610
$349$	−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
$351$	−555498.	−0.240666
$352$	665600.	0.286323
$353$	−3.99040e6	−1.70443	−0.852215	−	0.523192i	$-0.824741\pi$
$353$	−3.99040e6	−1.70443	−0.852215	+	0.523192i	$0.824741\pi$
$354$	−201888.	−0.0856253
$355$	−2.97753e6	−1.25396
$356$	96384.0	0.0403070
$357$	−245196.	−0.101822
$358$	731544.	0.301671
$359$	4.06452e6	1.66446	0.832229	−	0.554432i	$-0.187064\pi$
$359$	4.06452e6	1.66446	0.832229	+	0.554432i	$0.187064\pi$
$360$	393984.	0.160222
$361$	3.53620e6	1.42814
$362$	553320.	0.221924
$363$	−2.35304e6	−0.937266
$364$	−597408.	−0.236329
$365$	−1.79132e6	−0.703787
$366$	−5400.00	−0.00210713
$367$	−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
$369$	−1.37830e6	−0.526959
$370$	2.34992e6	0.892378
$371$	−764106.	−0.288216
$372$	435456.	0.163150
$373$	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
$375$	324216.	0.119057
$376$	−1.53216e6	−0.558901
$377$	−513588.	−0.186106
$378$	142884.	0.0514344
$379$	−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
$381$	2.15687e6	0.761222
$382$	1.30889e6	0.458927
$383$	−342616.	−0.119347	−0.0596734	−	0.998218i	$-0.519006\pi$
$383$	−342616.	−0.119347	−0.0596734	+	0.998218i	$0.519006\pi$
$384$	−147456.	−0.0510310
$385$	−2.42060e6	−0.832283
$386$	3.14761e6	1.07526
$387$	1.76872e6	0.600317
$388$	1.73488e6	0.585046
$389$	−3.83959e6	−1.28650	−0.643252	−	0.765654i	$-0.722415\pi$
$389$	−3.83959e6	−1.28650	−0.643252	+	0.765654i	$0.722415\pi$
$390$	−2.08483e6	−0.694080
$391$	1.64020e6	0.542569
$392$	153664.	0.0505076
$393$	2.46755e6	0.805906
$394$	1.69239e6	0.549238
$395$	−1.35979e6	−0.438510
$396$	842400.	0.269948
$397$	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
$399$	−1.08133e6	−0.340038
$400$	678656.	0.212080
$401$	−3.89421e6	−1.20937	−0.604684	−	0.796466i	$-0.706701\pi$
$401$	−3.89421e6	−1.20937	−0.604684	+	0.796466i	$0.706701\pi$
$402$	1.57622e6	0.486466
$403$	−2.30429e6	−0.706764
$404$	−1.12678e6	−0.343469
$405$	498636.	0.151059
$406$	132104.	0.0397741
$407$	5.02450e6	1.50351
$408$	320256.	0.0952460
$409$	−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
$411$	3.52039e6	1.02798
$412$	−504832.	−0.146522
$413$	−274792.	−0.0792737
$414$	−955800.	−0.274073
$415$	2.96187e6	0.844201
$416$	780288.	0.221066
$417$	−3.05428e6	−0.860138
$418$	−6.37520e6	−1.78465
$419$	−1.67659e6	−0.466544	−0.233272	−	0.972412i	$-0.574943\pi$
$419$	−1.67659e6	−0.466544	−0.233272	+	0.972412i	$0.574943\pi$
$420$	536256.	0.148337
$421$	−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
$423$	−1.93914e6	−0.526936
$424$	998016.	0.269602
$425$	−1.47396e6	−0.395833
$426$	1.41041e6	0.376548
$427$	−7350.00	−0.00195082
$428$	1.73251e6	0.457159
$429$	−4.45770e6	−1.16941
$430$	6.63814e6	1.73131
$431$	6.68468e6	1.73335	0.866677	−	0.498870i	$-0.166251\pi$
$431$	6.68468e6	1.73335	0.866677	+	0.498870i	$0.166251\pi$
$432$	−186624.	−0.0481125
$433$	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
$435$	461016.	0.116813
$436$	−1.15434e6	−0.290814
$437$	7.23340e6	1.81192
$438$	848520.	0.211338
$439$	−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
$441$	194481.	0.0476190
$442$	−1.69469e6	−0.412605
$443$	4.59760e6	1.11307	0.556534	−	0.830825i	$-0.312131\pi$
$443$	4.59760e6	1.11307	0.556534	+	0.830825i	$0.312131\pi$
$444$	−1.11312e6	−0.267969
$445$	457824.	0.109597
$446$	3.98019e6	0.947473
$447$	264006.	0.0624950
$448$	−200704.	−0.0472456
$449$	1.70658e6	0.399494	0.199747	−	0.979848i	$-0.435988\pi$
$449$	1.70658e6	0.399494	0.199747	+	0.979848i	$0.435988\pi$
$450$	858924.	0.199951
$451$	−1.10604e7	−2.56053
$452$	3.53450e6	0.813732
$453$	−644472.	−0.147557
$454$	−382272.	−0.0870428
$455$	−2.83769e6	−0.642593
$456$	1.41235e6	0.318076
$457$	−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
$459$	405324.	0.0897988
$460$	−3.58720e6	−0.790426
$461$	−2.61805e6	−0.573753	−0.286877	−	0.957968i	$-0.592617\pi$
$461$	−2.61805e6	−0.573753	−0.286877	+	0.957968i	$0.592617\pi$
$462$	1.14660e6	0.249923
$463$	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
$465$	2.06842e6	0.443614
$466$	−4.67764e6	−0.997843
$467$	−2.17398e6	−0.461278	−0.230639	−	0.973039i	$-0.574082\pi$
$467$	−2.17398e6	−0.461278	−0.230639	+	0.973039i	$0.574082\pi$
$468$	987552.	0.208423
$469$	2.14542e6	0.450380
$470$	−7.27776e6	−1.51968
$471$	−2.66686e6	−0.553922
$472$	358912.	0.0741537
$473$	1.41934e7	2.91698
$474$	644112.	0.131679
$475$	−6.50025e6	−1.32189
$476$	435904.	0.0881807
$477$	1.26311e6	0.254183
$478$	−109368.	−0.0218938
$479$	−4.63294e6	−0.922609	−0.461305	−	0.887242i	$-0.652619\pi$
$479$	−4.63294e6	−0.922609	−0.461305	+	0.887242i	$0.652619\pi$
$480$	−700416.	−0.138756
$481$	5.89026e6	1.16084
$482$	−3.63086e6	−0.711855
$483$	−1.30095e6	−0.253742
$484$	4.18318e6	0.811696
$485$	8.24068e6	1.59077
$486$	−236196.	−0.0453609
$487$	−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
$489$	4.32360e6	0.817661
$490$	729904.	0.137333
$491$	−5.31429e6	−0.994813	−0.497407	−	0.867518i	$-0.665714\pi$
$491$	−5.31429e6	−0.994813	−0.497407	+	0.867518i	$0.665714\pi$
$492$	2.45030e6	0.456360
$493$	374744.	0.0694412
$494$	−7.47370e6	−1.37790
$495$	4.00140e6	0.734005
$496$	−774144.	−0.141292
$497$	1.91972e6	0.348616
$498$	−1.40299e6	−0.253502
$499$	−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
$501$	−1.44162e6	−0.256600
$502$	176352.	0.0312335
$503$	2.79924e6	0.493310	0.246655	−	0.969103i	$-0.420669\pi$
$503$	2.79924e6	0.493310	0.246655	+	0.969103i	$0.420669\pi$
$504$	−254016.	−0.0445435
$505$	−5.35222e6	−0.933912
$506$	−7.67000e6	−1.33174
$507$	−1.88416e6	−0.325535
$508$	−3.83443e6	−0.659237
$509$	1.99914e6	0.342018	0.171009	−	0.985269i	$-0.445297\pi$
$509$	1.99914e6	0.342018	0.171009	+	0.985269i	$0.445297\pi$
$510$	1.52122e6	0.258980
$511$	1.15493e6	0.195661
$512$	262144.	0.0441942
$513$	1.78751e6	0.299885
$514$	−3.31680e6	−0.553747
$515$	−2.39795e6	−0.398403
$516$	−3.14438e6	−0.519890
$517$	−1.55610e7	−2.56042
$518$	−1.51508e6	−0.248091
$519$	80856.0	0.0131763
$520$	3.70637e6	0.601091
$521$	3.52160e6	0.568390	0.284195	−	0.958767i	$-0.408274\pi$
$521$	3.52160e6	0.568390	0.284195	+	0.958767i	$0.408274\pi$
$522$	−218376.	−0.0350775
$523$	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
$525$	1.16909e6	0.185119
$526$	5.27786e6	0.831752
$527$	1.68134e6	0.263712
$528$	−1.49760e6	−0.233782
$529$	2.26616e6	0.352088
$530$	4.74058e6	0.733063
$531$	454248.	0.0699128
$532$	1.92237e6	0.294481
$533$	−1.29662e7	−1.97694
$534$	−216864.	−0.0329105
$535$	8.22943e6	1.24304
$536$	−2.80218e6	−0.421292
$537$	−1.64597e6	−0.246313
$538$	−3.13515e6	−0.466985
$539$	1.56065e6	0.231384
$540$	−886464.	−0.130821
$541$	−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
$543$	−1.24497e6	−0.181200
$544$	−569344.	−0.0824855
$545$	−5.48310e6	−0.790742
$546$	1.34417e6	0.192962
$547$	−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
$549$	12150.0	0.00172046
$550$	6.89260e6	0.971575
$551$	1.65265e6	0.231900
$552$	1.69920e6	0.237354
$553$	876708.	0.121911
$554$	7.65092e6	1.05911
$555$	−5.28732e6	−0.728623
$556$	5.42982e6	0.744901
$557$	6.29262e6	0.859396	0.429698	−	0.902973i	$-0.358620\pi$
$557$	6.29262e6	0.859396	0.429698	+	0.902973i	$0.358620\pi$
$558$	−979776.	−0.133211
$559$	1.66390e7	2.25216
$560$	−953344.	−0.128463
$561$	3.25260e6	0.436338
$562$	−4.10481e6	−0.548216
$563$	4.86582e6	0.646971	0.323485	−	0.946233i	$-0.395145\pi$
$563$	4.86582e6	0.646971	0.323485	+	0.946233i	$0.395145\pi$
$564$	3.44736e6	0.456340
$565$	1.67889e7	2.21259
$566$	6.98670e6	0.916709
$567$	−321489.	−0.0419961
$568$	−2.50739e6	−0.326100
$569$	−4.46383e6	−0.577998	−0.288999	−	0.957329i	$-0.593322\pi$
$569$	−4.46383e6	−0.577998	−0.288999	+	0.957329i	$0.593322\pi$
$570$	6.70867e6	0.864867
$571$	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
$573$	−2.94500e6	−0.374713
$574$	3.33514e6	0.422507
$575$	−7.82045e6	−0.986421
$576$	331776.	0.0416667
$577$	−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
$579$	−7.08212e6	−0.877944
$580$	−819584.	−0.101163
$581$	−1.90963e6	−0.234697
$582$	−3.90348e6	−0.477688
$583$	1.01361e7	1.23509
$584$	−1.50848e6	−0.183024
$585$	4.69087e6	0.566714
$586$	8.92848e6	1.07407
$587$	8.14251e6	0.975356	0.487678	−	0.873024i	$-0.337844\pi$
$587$	8.14251e6	0.975356	0.487678	+	0.873024i	$0.337844\pi$
$588$	−345744.	−0.0412393
$589$	7.41485e6	0.880672
$590$	1.70483e6	0.201628
$591$	−3.80788e6	−0.448451
$592$	1.97888e6	0.232068
$593$	−2.73136e6	−0.318964	−0.159482	−	0.987201i	$-0.550982\pi$
$593$	−2.73136e6	−0.318964	−0.159482	+	0.987201i	$0.550982\pi$
$594$	−1.89540e6	−0.220412
$595$	2.07054e6	0.239768
$596$	−469344.	−0.0541222
$597$	−9.21528e6	−1.05821
$598$	−8.99160e6	−1.02822
$599$	1.23733e6	0.140902	0.0704510	−	0.997515i	$-0.477556\pi$
$599$	1.23733e6	0.140902	0.0704510	+	0.997515i	$0.477556\pi$
$600$	−1.52698e6	−0.173163
$601$	−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
$603$	−3.54650e6	−0.397198
$604$	1.14573e6	0.127788
$605$	1.98701e7	2.20705
$606$	2.53526e6	0.280441
$607$	−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
$609$	−297234.	−0.0324755
$610$	45600.0	0.00496181
$611$	−1.82423e7	−1.97686
$612$	−720576.	−0.0777681
$613$	−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
$615$	1.16389e7	1.24087
$616$	−2.03840e6	−0.216440
$617$	−8.21262e6	−0.868498	−0.434249	−	0.900793i	$-0.642986\pi$
$617$	−8.21262e6	−0.868498	−0.434249	+	0.900793i	$0.642986\pi$
$618$	1.13587e6	0.119635
$619$	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
$621$	2.15055e6	0.223780
$622$	−1.80382e6	−0.186947
$623$	−295176.	−0.0304692
$624$	−1.75565e6	−0.180499
$625$	−1.10222e7	−1.12867
$626$	6.41054e6	0.653820
$627$	1.43442e7	1.45716
$628$	4.74109e6	0.479710
$629$	−4.29788e6	−0.433139
$630$	−1.20658e6	−0.121116
$631$	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
$633$	−4.15364e6	−0.412022
$634$	−83448.0	−0.00824504
$635$	−1.82136e7	−1.79250
$636$	−2.24554e6	−0.220129
$637$	1.82956e6	0.178648
$638$	−1.75240e6	−0.170444
$639$	−3.17342e6	−0.307450
$640$	1.24518e6	0.120167
$641$	−1.40324e7	−1.34892	−0.674460	−	0.738311i	$-0.735624\pi$
$641$	−1.40324e7	−1.34892	−0.674460	+	0.738311i	$0.735624\pi$
$642$	−3.89815e6	−0.373268
$643$	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
$645$	−1.49358e7	−1.41361
$646$	5.45325e6	0.514131
$647$	1.57110e6	0.147551	0.0737757	−	0.997275i	$-0.476495\pi$
$647$	1.57110e6	0.147551	0.0737757	+	0.997275i	$0.476495\pi$
$648$	419904.	0.0392837
$649$	3.64520e6	0.339711
$650$	8.08025e6	0.750138
$651$	−1.33358e6	−0.123330
$652$	−7.68640e6	−0.708115
$653$	−8.34115e6	−0.765496	−0.382748	−	0.923853i	$-0.625022\pi$
$653$	−8.34115e6	−0.765496	−0.382748	+	0.923853i	$0.625022\pi$
$654$	2.59726e6	0.237449
$655$	−2.08371e7	−1.89773
$656$	−4.35610e6	−0.395219
$657$	−1.90917e6	−0.172556
$658$	4.69224e6	0.422489
$659$	6.18334e6	0.554638	0.277319	−	0.960778i	$-0.410554\pi$
$659$	6.18334e6	0.554638	0.277319	+	0.960778i	$0.410554\pi$
$660$	−7.11360e6	−0.635667
$661$	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
$663$	3.81305e6	0.336890
$664$	2.49421e6	0.219539
$665$	9.13125e6	0.800711
$666$	2.50452e6	0.218796
$667$	1.98830e6	0.173048
$668$	2.56288e6	0.222222
$669$	−8.95543e6	−0.773609
$670$	−1.33103e7	−1.14552
$671$	97500.0	0.00835985
$672$	451584.	0.0385758
$673$	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
$675$	−1.93258e6	−0.163259
$676$	3.34962e6	0.281922
$677$	−4.96397e6	−0.416253	−0.208126	−	0.978102i	$-0.566737\pi$
$677$	−4.96397e6	−0.416253	−0.208126	+	0.978102i	$0.566737\pi$
$678$	−7.95262e6	−0.664409
$679$	−5.31307e6	−0.442253
$680$	−2.70438e6	−0.224283
$681$	860112.	0.0710701
$682$	−7.86240e6	−0.647283
$683$	89526.0	0.00734340	0.00367170	−	0.999993i	$-0.498831\pi$
$683$	89526.0	0.00734340	0.00367170	+	0.999993i	$0.498831\pi$
$684$	−3.17779e6	−0.259708
$685$	−2.97277e7	−2.42067
$686$	−470596.	−0.0381802
$687$	9.39677e6	0.759603
$688$	5.59002e6	0.450238
$689$	1.18826e7	0.953596
$690$	8.07120e6	0.645380
$691$	−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
$693$	−2.57985e6	−0.204062
$694$	−3.50657e6	−0.276365
$695$	2.57917e7	2.02543
$696$	388224.	0.0303780
$697$	9.46090e6	0.737650
$698$	−5.18372e6	−0.402720
$699$	1.05247e7	0.814735
$700$	−2.07838e6	−0.160317
$701$	1.03935e7	0.798852	0.399426	−	0.916765i	$-0.369209\pi$
$701$	1.03935e7	0.798852	0.399426	+	0.916765i	$0.369209\pi$
$702$	−2.22199e6	−0.170177
$703$	−1.89540e7	−1.44648
$704$	2.66240e6	0.202461
$705$	1.63750e7	1.24082
$706$	−1.59616e7	−1.20521
$707$	3.45078e6	0.259638
$708$	−807552.	−0.0605463
$709$	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
$711$	−1.44925e6	−0.107515
$712$	385536.	0.0285013
$713$	8.92080e6	0.657173
$714$	−980784.	−0.0719992
$715$	3.76428e7	2.75370
$716$	2.92618e6	0.213313
$717$	246078.	0.0178762
$718$	1.62581e7	1.17695
$719$	−6.72134e6	−0.484880	−0.242440	−	0.970166i	$-0.577948\pi$
$719$	−6.72134e6	−0.484880	−0.242440	+	0.970166i	$0.577948\pi$
$720$	1.57594e6	0.113294
$721$	1.54605e6	0.110760
$722$	1.41448e7	1.00984
$723$	8.16943e6	0.581227
$724$	2.21328e6	0.156924
$725$	−1.78677e6	−0.126248
$726$	−9.41216e6	−0.662747
$727$	−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
$729$	531441.	0.0370370
$730$	−7.16528e6	−0.497652
$731$	−1.21408e7	−0.840339
$732$	−21600.0	−0.00148996
$733$	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
$735$	−1.64228e6	−0.112132
$736$	−3.02080e6	−0.205555
$737$	−2.84596e7	−1.93001
$738$	−5.51318e6	−0.372616
$739$	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
$741$	1.68158e7	1.12505
$742$	−3.05642e6	−0.203800
$743$	−2.02133e7	−1.34327	−0.671637	−	0.740880i	$-0.734409\pi$
$743$	−2.02133e7	−1.34327	−0.671637	+	0.740880i	$0.734409\pi$
$744$	1.74182e6	0.115364
$745$	−2.22938e6	−0.147161
$746$	1.26707e7	0.833595
$747$	3.15673e6	0.206984
$748$	−5.78240e6	−0.377880
$749$	−5.30582e6	−0.345579
$750$	1.29686e6	0.0841863
$751$	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
$753$	−396792.	−0.0255021
$754$	−2.05435e6	−0.131597
$755$	5.44221e6	0.347462
$756$	571536.	0.0363696
$757$	−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
$759$	1.72575e7	1.08736
$760$	−1.19265e7	−0.748997
$761$	−5.07974e6	−0.317965	−0.158983	−	0.987281i	$-0.550821\pi$
$761$	−5.07974e6	−0.317965	−0.158983	+	0.987281i	$0.550821\pi$
$762$	8.62747e6	0.538265
$763$	3.53515e6	0.219835
$764$	5.23555e6	0.324511
$765$	−3.42274e6	−0.211456
$766$	−1.37046e6	−0.0843909
$767$	4.27330e6	0.262286
$768$	−589824.	−0.0360844
$769$	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
$771$	7.46280e6	0.452133
$772$	1.25904e7	0.760322
$773$	−1.11253e6	−0.0669672	−0.0334836	−	0.999439i	$-0.510660\pi$
$773$	−1.11253e6	−0.0669672	−0.0334836	+	0.999439i	$0.510660\pi$
$774$	7.07486e6	0.424488
$775$	−8.01662e6	−0.479443
$776$	6.93952e6	0.413690
$777$	3.40893e6	0.202566
$778$	−1.53584e7	−0.909696
$779$	4.17232e7	2.46340
$780$	−8.33933e6	−0.490789
$781$	−2.54657e7	−1.49392
$782$	6.56080e6	0.383654
$783$	491346.	0.0286407
$784$	614656.	0.0357143
$785$	2.25202e7	1.30436
$786$	9.87019e6	0.569861
$787$	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
$789$	−1.18752e7	−0.679123
$790$	−5.43917e6	−0.310074
$791$	−1.08244e7	−0.615124
$792$	3.36960e6	0.190882
$793$	114300.	0.00645451
$794$	1.37558e7	0.774343
$795$	−1.06663e7	−0.598544
$796$	1.63827e7	0.916439
$797$	3.00897e7	1.67792	0.838961	−	0.544191i	$-0.183163\pi$
$797$	3.00897e7	1.67792	0.838961	+	0.544191i	$0.183163\pi$
$798$	−4.32533e6	−0.240443
$799$	1.33106e7	0.737619
$800$	2.71462e6	0.149963
$801$	487944.	0.0268713
$802$	−1.55768e7	−0.855152
$803$	−1.53205e7	−0.838463
$804$	6.30490e6	0.343984
$805$	1.09858e7	0.597506
$806$	−9.21715e6	−0.499757
$807$	7.05409e6	0.381292
$808$	−4.50714e6	−0.242869
$809$	−1.88207e6	−0.101103	−0.0505515	−	0.998721i	$-0.516098\pi$
$809$	−1.88207e6	−0.101103	−0.0505515	+	0.998721i	$0.516098\pi$
$810$	1.99454e6	0.106815
$811$	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
$813$	−8.59572e6	−0.456096
$814$	2.00980e7	1.06314
$815$	−3.65104e7	−1.92541
$816$	1.28102e6	0.0673491
$817$	−5.35419e7	−2.80633
$818$	−6.58718e6	−0.344204
$819$	−3.02438e6	−0.157553
$820$	−2.06915e7	−1.07462
$821$	8.37096e6	0.433429	0.216714	−	0.976235i	$-0.430466\pi$
$821$	8.37096e6	0.433429	0.216714	+	0.976235i	$0.430466\pi$
$822$	1.40815e7	0.726893
$823$	−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
$825$	−1.55084e7	−0.793288
$826$	−1.09917e6	−0.0560549
$827$	−1.31059e7	−0.666352	−0.333176	−	0.942865i	$-0.608120\pi$
$827$	−1.31059e7	−0.666352	−0.333176	+	0.942865i	$0.608120\pi$
$828$	−3.82320e6	−0.193799
$829$	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
$831$	−1.72146e7	−0.864756
$832$	3.12115e6	0.156317
$833$	−1.33496e6	−0.0666583
$834$	−1.22171e7	−0.608209
$835$	1.21737e7	0.604235
$836$	−2.55008e7	−1.26194
$837$	2.20450e6	0.108767
$838$	−6.70637e6	−0.329896
$839$	9.94742e6	0.487872	0.243936	−	0.969791i	$-0.421561\pi$
$839$	9.94742e6	0.487872	0.243936	+	0.969791i	$0.421561\pi$
$840$	2.14502e6	0.104890
$841$	−2.00569e7	−0.977852
$842$	−2.26697e6	−0.110196
$843$	9.23582e6	0.447617
$844$	7.38426e6	0.356821
$845$	1.59107e7	0.766561
$846$	−7.75656e6	−0.372600
$847$	−1.28110e7	−0.613585
$848$	3.99206e6	0.190637
$849$	−1.57201e7	−0.748489
$850$	−5.89582e6	−0.279896
$851$	−2.28035e7	−1.07939
$852$	5.64163e6	0.266260
$853$	−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
$855$	−1.50945e7	−0.706161
$856$	6.93005e6	0.323260
$857$	−8.76238e6	−0.407540	−0.203770	−	0.979019i	$-0.565319\pi$
$857$	−8.76238e6	−0.407540	−0.203770	+	0.979019i	$0.565319\pi$
$858$	−1.78308e7	−0.826899
$859$	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
$861$	−7.50406e6	−0.344975
$862$	2.67387e7	1.22567
$863$	−1.83417e7	−0.838323	−0.419162	−	0.907912i	$-0.637676\pi$
$863$	−1.83417e7	−0.838323	−0.419162	+	0.907912i	$0.637676\pi$
$864$	−746496.	−0.0340207
$865$	−682784.	−0.0310272
$866$	2.76535e7	1.25301
$867$	9.99649e6	0.451648
$868$	2.37082e6	0.106807
$869$	−1.16298e7	−0.522424
$870$	1.84406e6	0.0825996
$871$	−3.33634e7	−1.49013
$872$	−4.61734e6	−0.205637
$873$	8.78283e6	0.390031
$874$	2.89336e7	1.28122
$875$	1.76518e6	0.0779413
$876$	3.39408e6	0.149438
$877$	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
$879$	−2.00891e7	−0.876976
$880$	1.26464e7	0.550504
$881$	−1.52174e7	−0.660542	−0.330271	−	0.943886i	$-0.607140\pi$
$881$	−1.52174e7	−0.660542	−0.330271	+	0.943886i	$0.607140\pi$
$882$	777924.	0.0336718
$883$	−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
$885$	−3.83587e6	−0.164629
$886$	1.83904e7	0.787058
$887$	−1.08021e7	−0.460997	−0.230499	−	0.973073i	$-0.574036\pi$
$887$	−1.08021e7	−0.460997	−0.230499	+	0.973073i	$0.574036\pi$
$888$	−4.45248e6	−0.189483
$889$	1.17429e7	0.498337
$890$	1.83130e6	0.0774968
$891$	4.26465e6	0.179965
$892$	1.59208e7	0.669965
$893$	5.87009e7	2.46329
$894$	1.05602e6	0.0441906
$895$	1.38993e7	0.580011
$896$	−802816.	−0.0334077
$897$	2.02311e7	0.839534
$898$	6.82631e6	0.282485
$899$	2.03818e6	0.0841090
$900$	3.43570e6	0.141387
$901$	−8.67026e6	−0.355812
$902$	−4.42416e7	−1.81057
$903$	9.62968e6	0.393000
$904$	1.41380e7	0.575395
$905$	1.05131e7	0.426686
$906$	−2.57789e6	−0.104338
$907$	−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
$909$	−5.70434e6	−0.228979
$910$	−1.13508e7	−0.454382
$911$	2.72509e7	1.08789	0.543945	−	0.839121i	$-0.316930\pi$
$911$	2.72509e7	1.08789	0.543945	+	0.839121i	$0.316930\pi$
$912$	5.64941e6	0.224914
$913$	2.53318e7	1.00575
$914$	−2.77566e7	−1.09901
$915$	−102600.	−0.00405130
$916$	−1.67054e7	−0.657836
$917$	1.34344e7	0.527589
$918$	1.62130e6	0.0634974
$919$	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
$921$	−1.66791e7	−0.647924
$922$	−1.04722e7	−0.405705
$923$	−2.98536e7	−1.15343
$924$	4.58640e6	0.176723
$925$	2.04922e7	0.787472
$926$	2.85441e7	1.09393
$927$	−2.55571e6	−0.0976816
$928$	−690176.	−0.0263081
$929$	6.78492e6	0.257932	0.128966	−	0.991649i	$-0.458834\pi$
$929$	6.78492e6	0.257932	0.128966	+	0.991649i	$0.458834\pi$
$930$	8.27366e6	0.313683
$931$	−5.88725e6	−0.222607
$932$	−1.87106e7	−0.705581
$933$	4.05860e6	0.152641
$934$	−8.69590e6	−0.326173
$935$	−2.74664e7	−1.02748
$936$	3.95021e6	0.147377
$937$	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
$939$	−1.44237e7	−0.533842
$940$	−2.91110e7	−1.07458
$941$	2.30725e7	0.849415	0.424707	−	0.905331i	$-0.360377\pi$
$941$	2.30725e7	0.849415	0.424707	+	0.905331i	$0.360377\pi$
$942$	−1.06674e7	−0.391682
$943$	5.01972e7	1.83823
$944$	1.43565e6	0.0524346
$945$	2.71480e6	0.0988912
$946$	5.67736e7	2.06262
$947$	2.71433e7	0.983531	0.491765	−	0.870728i	$-0.336352\pi$
$947$	2.71433e7	0.983531	0.491765	+	0.870728i	$0.336352\pi$
$948$	2.57645e6	0.0931109
$949$	−1.79603e7	−0.647365
$950$	−2.60010e7	−0.934719
$951$	187758.	0.00673205
$952$	1.74362e6	0.0623532
$953$	−1.61552e7	−0.576209	−0.288104	−	0.957599i	$-0.593025\pi$
$953$	−1.61552e7	−0.576209	−0.288104	+	0.957599i	$0.593025\pi$
$954$	5.05246e6	0.179735
$955$	2.48689e7	0.882364
$956$	−437472.	−0.0154812
$957$	3.94290e6	0.139167
$958$	−1.85318e7	−0.652383
$959$	1.91665e7	0.672973
$960$	−2.80166e6	−0.0981156
$961$	−1.94846e7	−0.680585
$962$	2.35610e7	0.820837
$963$	8.77084e6	0.304772
$964$	−1.45234e7	−0.503357
$965$	5.98046e7	2.06736
$966$	−5.20380e6	−0.179423
$967$	−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
$969$	−1.22698e7	−0.419786
$970$	3.29627e7	1.12485
$971$	−2.23104e7	−0.759379	−0.379689	−	0.925114i	$-0.623969\pi$
$971$	−2.23104e7	−0.759379	−0.379689	+	0.925114i	$0.623969\pi$
$972$	−944784.	−0.0320750
$973$	−1.66288e7	−0.563093
$974$	−1.82658e7	−0.616937
$975$	−1.81806e7	−0.612485
$976$	38400.0	0.00129035
$977$	3.06930e7	1.02873	0.514367	−	0.857570i	$-0.328027\pi$
$977$	3.06930e7	1.02873	0.514367	+	0.857570i	$0.328027\pi$
$978$	1.72944e7	0.578174
$979$	3.91560e6	0.130569
$980$	2.91962e6	0.0971092
$981$	−5.84383e6	−0.193876
$982$	−2.12572e7	−0.703439
$983$	−1.52706e7	−0.504048	−0.252024	−	0.967721i	$-0.581096\pi$
$983$	−1.52706e7	−0.504048	−0.252024	+	0.967721i	$0.581096\pi$
$984$	9.80122e6	0.322695
$985$	3.21554e7	1.05600
$986$	1.49898e6	0.0491024
$987$	−1.05575e7	−0.344961
$988$	−2.98948e7	−0.974323
$989$	−6.44162e7	−2.09413
$990$	1.60056e7	0.519020
$991$	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
$993$	−1.86859e7	−0.601369
$994$	7.67889e6	0.246509
$995$	7.78179e7	2.49185
$996$	−5.61197e6	−0.179253
$997$	−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
$999$	−5.63517e6	−0.178646

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 42 = 2 \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	42.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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