LMFDB - Embedded newform 462.4.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [462,4,Mod(1,462)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(462, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("462.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 462 = 2 \cdot 3 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	462.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:	$27.2588824227$
Analytic rank:	$1$
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	$\chi$	$=$	462.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -14.0000 q^{5} +6.00000 q^{6} -7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$

\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -14.0000 q^{5} +6.00000 q^{6} -7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +28.0000 q^{10} +11.0000 q^{11} -12.0000 q^{12} +38.0000 q^{13} +14.0000 q^{14} +42.0000 q^{15} +16.0000 q^{16} +54.0000 q^{17} -18.0000 q^{18} +40.0000 q^{19} -56.0000 q^{20} +21.0000 q^{21} -22.0000 q^{22} +8.00000 q^{23} +24.0000 q^{24} +71.0000 q^{25} -76.0000 q^{26} -27.0000 q^{27} -28.0000 q^{28} -170.000 q^{29} -84.0000 q^{30} +92.0000 q^{31} -32.0000 q^{32} -33.0000 q^{33} -108.000 q^{34} +98.0000 q^{35} +36.0000 q^{36} +294.000 q^{37} -80.0000 q^{38} -114.000 q^{39} +112.000 q^{40} -258.000 q^{41} -42.0000 q^{42} -52.0000 q^{43} +44.0000 q^{44} -126.000 q^{45} -16.0000 q^{46} -76.0000 q^{47} -48.0000 q^{48} +49.0000 q^{49} -142.000 q^{50} -162.000 q^{51} +152.000 q^{52} -322.000 q^{53} +54.0000 q^{54} -154.000 q^{55} +56.0000 q^{56} -120.000 q^{57} +340.000 q^{58} +260.000 q^{59} +168.000 q^{60} +22.0000 q^{61} -184.000 q^{62} -63.0000 q^{63} +64.0000 q^{64} -532.000 q^{65} +66.0000 q^{66} -436.000 q^{67} +216.000 q^{68} -24.0000 q^{69} -196.000 q^{70} -368.000 q^{71} -72.0000 q^{72} -2.00000 q^{73} -588.000 q^{74} -213.000 q^{75} +160.000 q^{76} -77.0000 q^{77} +228.000 q^{78} -200.000 q^{79} -224.000 q^{80} +81.0000 q^{81} +516.000 q^{82} -952.000 q^{83} +84.0000 q^{84} -756.000 q^{85} +104.000 q^{86} +510.000 q^{87} -88.0000 q^{88} -70.0000 q^{89} +252.000 q^{90} -266.000 q^{91} +32.0000 q^{92} -276.000 q^{93} +152.000 q^{94} -560.000 q^{95} +96.0000 q^{96} -1086.00 q^{97} -98.0000 q^{98} +99.0000 q^{99} +O(q^{100})\)

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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	4.00000	0.500000
$5$	−14.0000	−1.25220	−0.626099	−	0.779744i	$-0.715349\pi$
$5$	−14.0000	−1.25220	−0.626099	+	0.779744i	$0.715349\pi$
$6$	6.00000	0.408248
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	−8.00000	−0.353553
$9$	9.00000	0.333333
$10$	28.0000	0.885438
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	−12.0000	−0.288675
$13$	38.0000	0.810716	0.405358	−	0.914158i	$-0.367147\pi$
$13$	38.0000	0.810716	0.405358	+	0.914158i	$0.367147\pi$
$14$	14.0000	0.267261
$15$	42.0000	0.722957
$16$	16.0000	0.250000
$17$	54.0000	0.770407	0.385204	−	0.922832i	$-0.374131\pi$
$17$	54.0000	0.770407	0.385204	+	0.922832i	$0.374131\pi$
$18$	−18.0000	−0.235702
$19$	40.0000	0.482980	0.241490	−	0.970403i	$-0.422364\pi$
$19$	40.0000	0.482980	0.241490	+	0.970403i	$0.422364\pi$
$20$	−56.0000	−0.626099
$21$	21.0000	0.218218
$22$	−22.0000	−0.213201
$23$	8.00000	0.0725268	0.0362634	−	0.999342i	$-0.488454\pi$
$23$	8.00000	0.0725268	0.0362634	+	0.999342i	$0.488454\pi$
$24$	24.0000	0.204124
$25$	71.0000	0.568000
$26$	−76.0000	−0.573263
$27$	−27.0000	−0.192450
$28$	−28.0000	−0.188982
$29$	−170.000	−1.08856	−0.544279	−	0.838904i	$-0.683197\pi$
$29$	−170.000	−1.08856	−0.544279	+	0.838904i	$0.683197\pi$
$30$	−84.0000	−0.511208
$31$	92.0000	0.533022	0.266511	−	0.963832i	$-0.414129\pi$
$31$	92.0000	0.533022	0.266511	+	0.963832i	$0.414129\pi$
$32$	−32.0000	−0.176777
$33$	−33.0000	−0.174078
$34$	−108.000	−0.544760
$35$	98.0000	0.473286
$36$	36.0000	0.166667
$37$	294.000	1.30631	0.653153	−	0.757226i	$-0.273446\pi$
$37$	294.000	1.30631	0.653153	+	0.757226i	$0.273446\pi$
$38$	−80.0000	−0.341519
$39$	−114.000	−0.468067
$40$	112.000	0.442719
$41$	−258.000	−0.982752	−0.491376	−	0.870948i	$-0.663506\pi$
$41$	−258.000	−0.982752	−0.491376	+	0.870948i	$0.663506\pi$
$42$	−42.0000	−0.154303
$43$	−52.0000	−0.184417	−0.0922084	−	0.995740i	$-0.529393\pi$
$43$	−52.0000	−0.184417	−0.0922084	+	0.995740i	$0.529393\pi$
$44$	44.0000	0.150756
$45$	−126.000	−0.417399
$46$	−16.0000	−0.0512842
$47$	−76.0000	−0.235867	−0.117933	−	0.993022i	$-0.537627\pi$
$47$	−76.0000	−0.235867	−0.117933	+	0.993022i	$0.537627\pi$
$48$	−48.0000	−0.144338
$49$	49.0000	0.142857
$50$	−142.000	−0.401637
$51$	−162.000	−0.444795
$52$	152.000	0.405358
$53$	−322.000	−0.834530	−0.417265	−	0.908785i	$-0.637011\pi$
$53$	−322.000	−0.834530	−0.417265	+	0.908785i	$0.637011\pi$
$54$	54.0000	0.136083
$55$	−154.000	−0.377552
$56$	56.0000	0.133631
$57$	−120.000	−0.278849
$58$	340.000	0.769727
$59$	260.000	0.573714	0.286857	−	0.957973i	$-0.407390\pi$
$59$	260.000	0.573714	0.286857	+	0.957973i	$0.407390\pi$
$60$	168.000	0.361478
$61$	22.0000	0.0461772	0.0230886	−	0.999733i	$-0.492650\pi$
$61$	22.0000	0.0461772	0.0230886	+	0.999733i	$0.492650\pi$
$62$	−184.000	−0.376904
$63$	−63.0000	−0.125988
$64$	64.0000	0.125000
$65$	−532.000	−1.01518
$66$	66.0000	0.123091
$67$	−436.000	−0.795013	−0.397507	−	0.917599i	$-0.630124\pi$
$67$	−436.000	−0.795013	−0.397507	+	0.917599i	$0.630124\pi$
$68$	216.000	0.385204
$69$	−24.0000	−0.0418733
$70$	−196.000	−0.334664
$71$	−368.000	−0.615121	−0.307560	−	0.951529i	$-0.599513\pi$
$71$	−368.000	−0.615121	−0.307560	+	0.951529i	$0.599513\pi$
$72$	−72.0000	−0.117851
$73$	−2.00000	−0.00320661	−0.00160330	−	0.999999i	$-0.500510\pi$
$73$	−2.00000	−0.00320661	−0.00160330	+	0.999999i	$0.500510\pi$
$74$	−588.000	−0.923697
$75$	−213.000	−0.327935
$76$	160.000	0.241490
$77$	−77.0000	−0.113961
$78$	228.000	0.330973
$79$	−200.000	−0.284832	−0.142416	−	0.989807i	$-0.545487\pi$
$79$	−200.000	−0.284832	−0.142416	+	0.989807i	$0.545487\pi$
$80$	−224.000	−0.313050
$81$	81.0000	0.111111
$82$	516.000	0.694911
$83$	−952.000	−1.25898	−0.629491	−	0.777007i	$-0.716737\pi$
$83$	−952.000	−1.25898	−0.629491	+	0.777007i	$0.716737\pi$
$84$	84.0000	0.109109
$85$	−756.000	−0.964703
$86$	104.000	0.130402
$87$	510.000	0.628480
$88$	−88.0000	−0.106600
$89$	−70.0000	−0.0833706	−0.0416853	−	0.999131i	$-0.513273\pi$
$89$	−70.0000	−0.0833706	−0.0416853	+	0.999131i	$0.513273\pi$
$90$	252.000	0.295146
$91$	−266.000	−0.306422
$92$	32.0000	0.0362634
$93$	−276.000	−0.307741
$94$	152.000	0.166783
$95$	−560.000	−0.604787
$96$	96.0000	0.102062
$97$	−1086.00	−1.13677	−0.568385	−	0.822763i	$-0.692431\pi$
$97$	−1086.00	−1.13677	−0.568385	+	0.822763i	$0.692431\pi$
$98$	−98.0000	−0.101015
$99$	99.0000	0.100504
$100$	284.000	0.284000
$101$	102.000	0.100489	0.0502445	−	0.998737i	$-0.484000\pi$
$101$	102.000	0.100489	0.0502445	+	0.998737i	$0.484000\pi$
$102$	324.000	0.314517
$103$	188.000	0.179847	0.0899233	−	0.995949i	$-0.471338\pi$
$103$	188.000	0.179847	0.0899233	+	0.995949i	$0.471338\pi$
$104$	−304.000	−0.286631
$105$	−294.000	−0.273252
$106$	644.000	0.590102
$107$	−1716.00	−1.55039	−0.775196	−	0.631721i	$-0.782349\pi$
$107$	−1716.00	−1.55039	−0.775196	+	0.631721i	$0.782349\pi$
$108$	−108.000	−0.0962250
$109$	310.000	0.272409	0.136205	−	0.990681i	$-0.456510\pi$
$109$	310.000	0.272409	0.136205	+	0.990681i	$0.456510\pi$
$110$	308.000	0.266970
$111$	−882.000	−0.754196
$112$	−112.000	−0.0944911
$113$	−1822.00	−1.51681	−0.758404	−	0.651785i	$-0.774021\pi$
$113$	−1822.00	−1.51681	−0.758404	+	0.651785i	$0.774021\pi$
$114$	240.000	0.197176
$115$	−112.000	−0.0908179
$116$	−680.000	−0.544279
$117$	342.000	0.270239
$118$	−520.000	−0.405677
$119$	−378.000	−0.291187
$120$	−336.000	−0.255604
$121$	121.000	0.0909091
$122$	−44.0000	−0.0326522
$123$	774.000	0.567392
$124$	368.000	0.266511
$125$	756.000	0.540950
$126$	126.000	0.0890871
$127$	−1576.00	−1.10116	−0.550580	−	0.834782i	$-0.685594\pi$
$127$	−1576.00	−1.10116	−0.550580	+	0.834782i	$0.685594\pi$
$128$	−128.000	−0.0883883
$129$	156.000	0.106473
$130$	1064.00	0.717838
$131$	1072.00	0.714970	0.357485	−	0.933919i	$-0.383634\pi$
$131$	1072.00	0.714970	0.357485	+	0.933919i	$0.383634\pi$
$132$	−132.000	−0.0870388
$133$	−280.000	−0.182549
$134$	872.000	0.562159
$135$	378.000	0.240986
$136$	−432.000	−0.272380
$137$	−246.000	−0.153410	−0.0767051	−	0.997054i	$-0.524440\pi$
$137$	−246.000	−0.153410	−0.0767051	+	0.997054i	$0.524440\pi$
$138$	48.0000	0.0296089
$139$	1640.00	1.00074	0.500370	−	0.865811i	$-0.333197\pi$
$139$	1640.00	1.00074	0.500370	+	0.865811i	$0.333197\pi$
$140$	392.000	0.236643
$141$	228.000	0.136178
$142$	736.000	0.434956
$143$	418.000	0.244440
$144$	144.000	0.0833333
$145$	2380.00	1.36309
$146$	4.00000	0.00226741
$147$	−147.000	−0.0824786
$148$	1176.00	0.653153
$149$	−850.000	−0.467347	−0.233674	−	0.972315i	$-0.575075\pi$
$149$	−850.000	−0.467347	−0.233674	+	0.972315i	$0.575075\pi$
$150$	426.000	0.231885
$151$	−1448.00	−0.780375	−0.390187	−	0.920735i	$-0.627590\pi$
$151$	−1448.00	−0.780375	−0.390187	+	0.920735i	$0.627590\pi$
$152$	−320.000	−0.170759
$153$	486.000	0.256802
$154$	154.000	0.0805823
$155$	−1288.00	−0.667449
$156$	−456.000	−0.234033
$157$	−1886.00	−0.958721	−0.479360	−	0.877618i	$-0.659131\pi$
$157$	−1886.00	−0.958721	−0.479360	+	0.877618i	$0.659131\pi$
$158$	400.000	0.201407
$159$	966.000	0.481816
$160$	448.000	0.221359
$161$	−56.0000	−0.0274125
$162$	−162.000	−0.0785674
$163$	228.000	0.109560	0.0547802	−	0.998498i	$-0.482554\pi$
$163$	228.000	0.109560	0.0547802	+	0.998498i	$0.482554\pi$
$164$	−1032.00	−0.491376
$165$	462.000	0.217980
$166$	1904.00	0.890235
$167$	1664.00	0.771043	0.385522	−	0.922699i	$-0.374022\pi$
$167$	1664.00	0.771043	0.385522	+	0.922699i	$0.374022\pi$
$168$	−168.000	−0.0771517
$169$	−753.000	−0.342740
$170$	1512.00	0.682148
$171$	360.000	0.160993
$172$	−208.000	−0.0922084
$173$	2438.00	1.07143	0.535716	−	0.844398i	$-0.320042\pi$
$173$	2438.00	1.07143	0.535716	+	0.844398i	$0.320042\pi$
$174$	−1020.00	−0.444402
$175$	−497.000	−0.214684
$176$	176.000	0.0753778
$177$	−780.000	−0.331234
$178$	140.000	0.0589519
$179$	1620.00	0.676450	0.338225	−	0.941065i	$-0.390174\pi$
$179$	1620.00	0.676450	0.338225	+	0.941065i	$0.390174\pi$
$180$	−504.000	−0.208700
$181$	3602.00	1.47920	0.739598	−	0.673049i	$-0.235016\pi$
$181$	3602.00	1.47920	0.739598	+	0.673049i	$0.235016\pi$
$182$	532.000	0.216673
$183$	−66.0000	−0.0266604
$184$	−64.0000	−0.0256421
$185$	−4116.00	−1.63575
$186$	552.000	0.217605
$187$	594.000	0.232287
$188$	−304.000	−0.117933
$189$	189.000	0.0727393
$190$	1120.00	0.427649
$191$	3472.00	1.31531	0.657657	−	0.753317i	$-0.271547\pi$
$191$	3472.00	1.31531	0.657657	+	0.753317i	$0.271547\pi$
$192$	−192.000	−0.0721688
$193$	−222.000	−0.0827975	−0.0413987	−	0.999143i	$-0.513181\pi$
$193$	−222.000	−0.0827975	−0.0413987	+	0.999143i	$0.513181\pi$
$194$	2172.00	0.803817
$195$	1596.00	0.586112
$196$	196.000	0.0714286
$197$	−3906.00	−1.41264	−0.706322	−	0.707890i	$-0.749647\pi$
$197$	−3906.00	−1.41264	−0.706322	+	0.707890i	$0.749647\pi$
$198$	−198.000	−0.0710669
$199$	1900.00	0.676821	0.338411	−	0.940999i	$-0.390111\pi$
$199$	1900.00	0.676821	0.338411	+	0.940999i	$0.390111\pi$
$200$	−568.000	−0.200818
$201$	1308.00	0.459001
$202$	−204.000	−0.0710564
$203$	1190.00	0.411437
$204$	−648.000	−0.222397
$205$	3612.00	1.23060
$206$	−376.000	−0.127171
$207$	72.0000	0.0241756
$208$	608.000	0.202679
$209$	440.000	0.145624
$210$	588.000	0.193218
$211$	1172.00	0.382388	0.191194	−	0.981552i	$-0.438764\pi$
$211$	1172.00	0.382388	0.191194	+	0.981552i	$0.438764\pi$
$212$	−1288.00	−0.417265
$213$	1104.00	0.355140
$214$	3432.00	1.09629
$215$	728.000	0.230926
$216$	216.000	0.0680414
$217$	−644.000	−0.201463
$218$	−620.000	−0.192622
$219$	6.00000	0.00185134
$220$	−616.000	−0.188776
$221$	2052.00	0.624581
$222$	1764.00	0.533297
$223$	−2372.00	−0.712291	−0.356145	−	0.934431i	$-0.615909\pi$
$223$	−2372.00	−0.712291	−0.356145	+	0.934431i	$0.615909\pi$
$224$	224.000	0.0668153
$225$	639.000	0.189333
$226$	3644.00	1.07255
$227$	1024.00	0.299406	0.149703	−	0.988731i	$-0.452168\pi$
$227$	1024.00	0.299406	0.149703	+	0.988731i	$0.452168\pi$
$228$	−480.000	−0.139424
$229$	290.000	0.0836845	0.0418422	−	0.999124i	$-0.486677\pi$
$229$	290.000	0.0836845	0.0418422	+	0.999124i	$0.486677\pi$
$230$	224.000	0.0642179
$231$	231.000	0.0657952
$232$	1360.00	0.384864
$233$	−5022.00	−1.41203	−0.706013	−	0.708199i	$-0.749508\pi$
$233$	−5022.00	−1.41203	−0.706013	+	0.708199i	$0.749508\pi$
$234$	−684.000	−0.191088
$235$	1064.00	0.295352
$236$	1040.00	0.286857
$237$	600.000	0.164448
$238$	756.000	0.205900
$239$	200.000	0.0541294	0.0270647	−	0.999634i	$-0.491384\pi$
$239$	200.000	0.0541294	0.0270647	+	0.999634i	$0.491384\pi$
$240$	672.000	0.180739
$241$	−1498.00	−0.400393	−0.200196	−	0.979756i	$-0.564158\pi$
$241$	−1498.00	−0.400393	−0.200196	+	0.979756i	$0.564158\pi$
$242$	−242.000	−0.0642824
$243$	−243.000	−0.0641500
$244$	88.0000	0.0230886
$245$	−686.000	−0.178885
$246$	−1548.00	−0.401207
$247$	1520.00	0.391560
$248$	−736.000	−0.188452
$249$	2856.00	0.726874
$250$	−1512.00	−0.382509
$251$	−7468.00	−1.87799	−0.938996	−	0.343928i	$-0.888242\pi$
$251$	−7468.00	−1.87799	−0.938996	+	0.343928i	$0.888242\pi$
$252$	−252.000	−0.0629941
$253$	88.0000	0.0218676
$254$	3152.00	0.778638
$255$	2268.00	0.556971
$256$	256.000	0.0625000
$257$	2194.00	0.532521	0.266261	−	0.963901i	$-0.414212\pi$
$257$	2194.00	0.532521	0.266261	+	0.963901i	$0.414212\pi$
$258$	−312.000	−0.0752879
$259$	−2058.00	−0.493737
$260$	−2128.00	−0.507588
$261$	−1530.00	−0.362853
$262$	−2144.00	−0.505560
$263$	3288.00	0.770900	0.385450	−	0.922729i	$-0.374046\pi$
$263$	3288.00	0.770900	0.385450	+	0.922729i	$0.374046\pi$
$264$	264.000	0.0615457
$265$	4508.00	1.04500
$266$	560.000	0.129082
$267$	210.000	0.0481340
$268$	−1744.00	−0.397507
$269$	−6710.00	−1.52088	−0.760439	−	0.649410i	$-0.775016\pi$
$269$	−6710.00	−1.52088	−0.760439	+	0.649410i	$0.775016\pi$
$270$	−756.000	−0.170403
$271$	4952.00	1.11001	0.555005	−	0.831847i	$-0.312716\pi$
$271$	4952.00	1.11001	0.555005	+	0.831847i	$0.312716\pi$
$272$	864.000	0.192602
$273$	798.000	0.176913
$274$	492.000	0.108477
$275$	781.000	0.171258
$276$	−96.0000	−0.0209367
$277$	−3666.00	−0.795193	−0.397597	−	0.917560i	$-0.630156\pi$
$277$	−3666.00	−0.795193	−0.397597	+	0.917560i	$0.630156\pi$
$278$	−3280.00	−0.707631
$279$	828.000	0.177674
$280$	−784.000	−0.167332
$281$	−6798.00	−1.44318	−0.721592	−	0.692319i	$-0.756589\pi$
$281$	−6798.00	−1.44318	−0.721592	+	0.692319i	$0.756589\pi$
$282$	−456.000	−0.0962922
$283$	−1992.00	−0.418417	−0.209209	−	0.977871i	$-0.567089\pi$
$283$	−1992.00	−0.418417	−0.209209	+	0.977871i	$0.567089\pi$
$284$	−1472.00	−0.307560
$285$	1680.00	0.349174
$286$	−836.000	−0.172845
$287$	1806.00	0.371445
$288$	−288.000	−0.0589256
$289$	−1997.00	−0.406473
$290$	−4760.00	−0.963851
$291$	3258.00	0.656314
$292$	−8.00000	−0.00160330
$293$	−4002.00	−0.797950	−0.398975	−	0.916962i	$-0.630634\pi$
$293$	−4002.00	−0.797950	−0.398975	+	0.916962i	$0.630634\pi$
$294$	294.000	0.0583212
$295$	−3640.00	−0.718403
$296$	−2352.00	−0.461849
$297$	−297.000	−0.0580259
$298$	1700.00	0.330464
$299$	304.000	0.0587986
$300$	−852.000	−0.163967
$301$	364.000	0.0697030
$302$	2896.00	0.551808
$303$	−306.000	−0.0580173
$304$	640.000	0.120745
$305$	−308.000	−0.0578230
$306$	−972.000	−0.181587
$307$	5344.00	0.993479	0.496740	−	0.867900i	$-0.334530\pi$
$307$	5344.00	0.993479	0.496740	+	0.867900i	$0.334530\pi$
$308$	−308.000	−0.0569803
$309$	−564.000	−0.103834
$310$	2576.00	0.471958
$311$	−2268.00	−0.413526	−0.206763	−	0.978391i	$-0.566293\pi$
$311$	−2268.00	−0.413526	−0.206763	+	0.978391i	$0.566293\pi$
$312$	912.000	0.165487
$313$	−7422.00	−1.34031	−0.670154	−	0.742222i	$-0.733772\pi$
$313$	−7422.00	−1.34031	−0.670154	+	0.742222i	$0.733772\pi$
$314$	3772.00	0.677918
$315$	882.000	0.157762
$316$	−800.000	−0.142416
$317$	−7626.00	−1.35116	−0.675582	−	0.737285i	$-0.736107\pi$
$317$	−7626.00	−1.35116	−0.675582	+	0.737285i	$0.736107\pi$
$318$	−1932.00	−0.340696
$319$	−1870.00	−0.328213
$320$	−896.000	−0.156525
$321$	5148.00	0.895119
$322$	112.000	0.0193836
$323$	2160.00	0.372092
$324$	324.000	0.0555556
$325$	2698.00	0.460487
$326$	−456.000	−0.0774709
$327$	−930.000	−0.157276
$328$	2064.00	0.347455
$329$	532.000	0.0891493
$330$	−924.000	−0.154135
$331$	1492.00	0.247758	0.123879	−	0.992297i	$-0.460467\pi$
$331$	1492.00	0.247758	0.123879	+	0.992297i	$0.460467\pi$
$332$	−3808.00	−0.629491
$333$	2646.00	0.435435
$334$	−3328.00	−0.545210
$335$	6104.00	0.995514
$336$	336.000	0.0545545
$337$	74.0000	0.0119615	0.00598077	−	0.999982i	$-0.498096\pi$
$337$	74.0000	0.0119615	0.00598077	+	0.999982i	$0.498096\pi$
$338$	1506.00	0.242354
$339$	5466.00	0.875730
$340$	−3024.00	−0.482351
$341$	1012.00	0.160712
$342$	−720.000	−0.113840
$343$	−343.000	−0.0539949
$344$	416.000	0.0652012
$345$	336.000	0.0524337
$346$	−4876.00	−0.757617
$347$	11724.0	1.81377	0.906884	−	0.421381i	$-0.138454\pi$
$347$	11724.0	1.81377	0.906884	+	0.421381i	$0.138454\pi$
$348$	2040.00	0.314240
$349$	6350.00	0.973948	0.486974	−	0.873417i	$-0.338101\pi$
$349$	6350.00	0.973948	0.486974	+	0.873417i	$0.338101\pi$
$350$	994.000	0.151804
$351$	−1026.00	−0.156022
$352$	−352.000	−0.0533002
$353$	−5822.00	−0.877829	−0.438915	−	0.898529i	$-0.644637\pi$
$353$	−5822.00	−0.877829	−0.438915	+	0.898529i	$0.644637\pi$
$354$	1560.00	0.234218
$355$	5152.00	0.770253
$356$	−280.000	−0.0416853
$357$	1134.00	0.168117
$358$	−3240.00	−0.478322
$359$	13320.0	1.95822	0.979112	−	0.203320i	$-0.0651731\pi$
$359$	13320.0	1.95822	0.979112	+	0.203320i	$0.0651731\pi$
$360$	1008.00	0.147573
$361$	−5259.00	−0.766730
$362$	−7204.00	−1.04595
$363$	−363.000	−0.0524864
$364$	−1064.00	−0.153211
$365$	28.0000	0.00401531
$366$	132.000	0.0188518
$367$	−11076.0	−1.57537	−0.787687	−	0.616075i	$-0.788722\pi$
$367$	−11076.0	−1.57537	−0.787687	+	0.616075i	$0.788722\pi$
$368$	128.000	0.0181317
$369$	−2322.00	−0.327584
$370$	8232.00	1.15665
$371$	2254.00	0.315423
$372$	−1104.00	−0.153870
$373$	−3922.00	−0.544433	−0.272216	−	0.962236i	$-0.587757\pi$
$373$	−3922.00	−0.544433	−0.272216	+	0.962236i	$0.587757\pi$
$374$	−1188.00	−0.164251
$375$	−2268.00	−0.312317
$376$	608.000	0.0833915
$377$	−6460.00	−0.882512
$378$	−378.000	−0.0514344
$379$	−11620.0	−1.57488	−0.787440	−	0.616392i	$-0.788594\pi$
$379$	−11620.0	−1.57488	−0.787440	+	0.616392i	$0.788594\pi$
$380$	−2240.00	−0.302394
$381$	4728.00	0.635755
$382$	−6944.00	−0.930068
$383$	3628.00	0.484026	0.242013	−	0.970273i	$-0.422192\pi$
$383$	3628.00	0.484026	0.242013	+	0.970273i	$0.422192\pi$
$384$	384.000	0.0510310
$385$	1078.00	0.142701
$386$	444.000	0.0585466
$387$	−468.000	−0.0614723
$388$	−4344.00	−0.568385
$389$	9750.00	1.27081	0.635404	−	0.772180i	$-0.280833\pi$
$389$	9750.00	1.27081	0.635404	+	0.772180i	$0.280833\pi$
$390$	−3192.00	−0.414444
$391$	432.000	0.0558751
$392$	−392.000	−0.0505076
$393$	−3216.00	−0.412788
$394$	7812.00	0.998891
$395$	2800.00	0.356667
$396$	396.000	0.0502519
$397$	−6606.00	−0.835128	−0.417564	−	0.908648i	$-0.637116\pi$
$397$	−6606.00	−0.835128	−0.417564	+	0.908648i	$0.637116\pi$
$398$	−3800.00	−0.478585
$399$	840.000	0.105395
$400$	1136.00	0.142000
$401$	2.00000	0.000249065 0	0.000124533	−	1.00000i	$-0.499960\pi$
$401$	2.00000	0.000249065 0	0.000124533		1.00000i	$0.499960\pi$
$402$	−2616.00	−0.324563
$403$	3496.00	0.432129
$404$	408.000	0.0502445
$405$	−1134.00	−0.139133
$406$	−2380.00	−0.290930
$407$	3234.00	0.393866
$408$	1296.00	0.157259
$409$	−13930.0	−1.68409	−0.842047	−	0.539405i	$-0.818649\pi$
$409$	−13930.0	−1.68409	−0.842047	+	0.539405i	$0.818649\pi$
$410$	−7224.00	−0.870166
$411$	738.000	0.0885714
$412$	752.000	0.0899233
$413$	−1820.00	−0.216843
$414$	−144.000	−0.0170947
$415$	13328.0	1.57650
$416$	−1216.00	−0.143316
$417$	−4920.00	−0.577778
$418$	−880.000	−0.102972
$419$	−7740.00	−0.902443	−0.451222	−	0.892412i	$-0.649012\pi$
$419$	−7740.00	−0.902443	−0.451222	+	0.892412i	$0.649012\pi$
$420$	−1176.00	−0.136626
$421$	−5618.00	−0.650367	−0.325184	−	0.945651i	$-0.605426\pi$
$421$	−5618.00	−0.650367	−0.325184	+	0.945651i	$0.605426\pi$
$422$	−2344.00	−0.270389
$423$	−684.000	−0.0786223
$424$	2576.00	0.295051
$425$	3834.00	0.437591
$426$	−2208.00	−0.251122
$427$	−154.000	−0.0174534
$428$	−6864.00	−0.775196
$429$	−1254.00	−0.141127
$430$	−1456.00	−0.163290
$431$	−9008.00	−1.00673	−0.503364	−	0.864074i	$-0.667905\pi$
$431$	−9008.00	−1.00673	−0.503364	+	0.864074i	$0.667905\pi$
$432$	−432.000	−0.0481125
$433$	−10702.0	−1.18777	−0.593886	−	0.804549i	$-0.702407\pi$
$433$	−10702.0	−1.18777	−0.593886	+	0.804549i	$0.702407\pi$
$434$	1288.00	0.142456
$435$	−7140.00	−0.786981
$436$	1240.00	0.136205
$437$	320.000	0.0350290
$438$	−12.0000	−0.00130909
$439$	−4960.00	−0.539243	−0.269622	−	0.962966i	$-0.586899\pi$
$439$	−4960.00	−0.539243	−0.269622	+	0.962966i	$0.586899\pi$
$440$	1232.00	0.133485
$441$	441.000	0.0476190
$442$	−4104.00	−0.441646
$443$	188.000	0.0201629	0.0100814	−	0.999949i	$-0.496791\pi$
$443$	188.000	0.0201629	0.0100814	+	0.999949i	$0.496791\pi$
$444$	−3528.00	−0.377098
$445$	980.000	0.104397
$446$	4744.00	0.503666
$447$	2550.00	0.269823
$448$	−448.000	−0.0472456
$449$	−3150.00	−0.331086	−0.165543	−	0.986203i	$-0.552938\pi$
$449$	−3150.00	−0.331086	−0.165543	+	0.986203i	$0.552938\pi$
$450$	−1278.00	−0.133879
$451$	−2838.00	−0.296311
$452$	−7288.00	−0.758404
$453$	4344.00	0.450550
$454$	−2048.00	−0.211712
$455$	3724.00	0.383701
$456$	960.000	0.0985880
$457$	−12806.0	−1.31081	−0.655404	−	0.755278i	$-0.727502\pi$
$457$	−12806.0	−1.31081	−0.655404	+	0.755278i	$0.727502\pi$
$458$	−580.000	−0.0591738
$459$	−1458.00	−0.148265
$460$	−448.000	−0.0454089
$461$	−1778.00	−0.179631	−0.0898153	−	0.995958i	$-0.528628\pi$
$461$	−1778.00	−0.179631	−0.0898153	+	0.995958i	$0.528628\pi$
$462$	−462.000	−0.0465242
$463$	−15672.0	−1.57309	−0.786544	−	0.617534i	$-0.788132\pi$
$463$	−15672.0	−1.57309	−0.786544	+	0.617534i	$0.788132\pi$
$464$	−2720.00	−0.272140
$465$	3864.00	0.385352
$466$	10044.0	0.998453
$467$	4.00000	0.000396355 0	0.000198178	−	1.00000i	$-0.499937\pi$
$467$	4.00000	0.000396355 0	0.000198178		1.00000i	$0.499937\pi$
$468$	1368.00	0.135119
$469$	3052.00	0.300487
$470$	−2128.00	−0.208845
$471$	5658.00	0.553518
$472$	−2080.00	−0.202838
$473$	−572.000	−0.0556038
$474$	−1200.00	−0.116282
$475$	2840.00	0.274333
$476$	−1512.00	−0.145593
$477$	−2898.00	−0.278177
$478$	−400.000	−0.0382753
$479$	14640.0	1.39649	0.698245	−	0.715859i	$-0.253965\pi$
$479$	14640.0	1.39649	0.698245	+	0.715859i	$0.253965\pi$
$480$	−1344.00	−0.127802
$481$	11172.0	1.05904
$482$	2996.00	0.283120
$483$	168.000	0.0158266
$484$	484.000	0.0454545
$485$	15204.0	1.42346
$486$	486.000	0.0453609
$487$	−19896.0	−1.85128	−0.925640	−	0.378404i	$-0.876473\pi$
$487$	−19896.0	−1.85128	−0.925640	+	0.378404i	$0.876473\pi$
$488$	−176.000	−0.0163261
$489$	−684.000	−0.0632547
$490$	1372.00	0.126491
$491$	7972.00	0.732732	0.366366	−	0.930471i	$-0.380602\pi$
$491$	7972.00	0.732732	0.366366	+	0.930471i	$0.380602\pi$
$492$	3096.00	0.283696
$493$	−9180.00	−0.838634
$494$	−3040.00	−0.276875
$495$	−1386.00	−0.125851
$496$	1472.00	0.133256
$497$	2576.00	0.232494
$498$	−5712.00	−0.513978
$499$	−2260.00	−0.202748	−0.101374	−	0.994848i	$-0.532324\pi$
$499$	−2260.00	−0.202748	−0.101374	+	0.994848i	$0.532324\pi$
$500$	3024.00	0.270475
$501$	−4992.00	−0.445162
$502$	14936.0	1.32794
$503$	6288.00	0.557392	0.278696	−	0.960379i	$-0.410098\pi$
$503$	6288.00	0.557392	0.278696	+	0.960379i	$0.410098\pi$
$504$	504.000	0.0445435
$505$	−1428.00	−0.125832
$506$	−176.000	−0.0154628
$507$	2259.00	0.197881
$508$	−6304.00	−0.550580
$509$	12010.0	1.04584	0.522921	−	0.852381i	$-0.324842\pi$
$509$	12010.0	1.04584	0.522921	+	0.852381i	$0.324842\pi$
$510$	−4536.00	−0.393838
$511$	14.0000	0.00121198
$512$	−512.000	−0.0441942
$513$	−1080.00	−0.0929496
$514$	−4388.00	−0.376549
$515$	−2632.00	−0.225203
$516$	624.000	0.0532366
$517$	−836.000	−0.0711165
$518$	4116.00	0.349125
$519$	−7314.00	−0.618591
$520$	4256.00	0.358919
$521$	10842.0	0.911702	0.455851	−	0.890056i	$-0.349335\pi$
$521$	10842.0	0.911702	0.455851	+	0.890056i	$0.349335\pi$
$522$	3060.00	0.256576
$523$	18808.0	1.57250	0.786249	−	0.617910i	$-0.212020\pi$
$523$	18808.0	1.57250	0.786249	+	0.617910i	$0.212020\pi$
$524$	4288.00	0.357485
$525$	1491.00	0.123948
$526$	−6576.00	−0.545109
$527$	4968.00	0.410644
$528$	−528.000	−0.0435194
$529$	−12103.0	−0.994740
$530$	−9016.00	−0.738925
$531$	2340.00	0.191238
$532$	−1120.00	−0.0912747
$533$	−9804.00	−0.796732
$534$	−420.000	−0.0340359
$535$	24024.0	1.94140
$536$	3488.00	0.281080
$537$	−4860.00	−0.390548
$538$	13420.0	1.07542
$539$	539.000	0.0430730
$540$	1512.00	0.120493
$541$	15622.0	1.24148	0.620741	−	0.784015i	$-0.286832\pi$
$541$	15622.0	1.24148	0.620741	+	0.784015i	$0.286832\pi$
$542$	−9904.00	−0.784895
$543$	−10806.0	−0.854014
$544$	−1728.00	−0.136190
$545$	−4340.00	−0.341110
$546$	−1596.00	−0.125096
$547$	2284.00	0.178532	0.0892658	−	0.996008i	$-0.471548\pi$
$547$	2284.00	0.178532	0.0892658	+	0.996008i	$0.471548\pi$
$548$	−984.000	−0.0767051
$549$	198.000	0.0153924
$550$	−1562.00	−0.121098
$551$	−6800.00	−0.525753
$552$	192.000	0.0148045
$553$	1400.00	0.107657
$554$	7332.00	0.562287
$555$	12348.0	0.944403
$556$	6560.00	0.500370
$557$	16854.0	1.28209	0.641047	−	0.767501i	$-0.278500\pi$
$557$	16854.0	1.28209	0.641047	+	0.767501i	$0.278500\pi$
$558$	−1656.00	−0.125635
$559$	−1976.00	−0.149510
$560$	1568.00	0.118322
$561$	−1782.00	−0.134111
$562$	13596.0	1.02049
$563$	528.000	0.0395250	0.0197625	−	0.999805i	$-0.493709\pi$
$563$	528.000	0.0395250	0.0197625	+	0.999805i	$0.493709\pi$
$564$	912.000	0.0680889
$565$	25508.0	1.89934
$566$	3984.00	0.295866
$567$	−567.000	−0.0419961
$568$	2944.00	0.217478
$569$	8050.00	0.593099	0.296550	−	0.955017i	$-0.404164\pi$
$569$	8050.00	0.593099	0.296550	+	0.955017i	$0.404164\pi$
$570$	−3360.00	−0.246903
$571$	−11308.0	−0.828765	−0.414383	−	0.910103i	$-0.636002\pi$
$571$	−11308.0	−0.828765	−0.414383	+	0.910103i	$0.636002\pi$
$572$	1672.00	0.122220
$573$	−10416.0	−0.759397
$574$	−3612.00	−0.262652
$575$	568.000	0.0411952
$576$	576.000	0.0416667
$577$	5274.00	0.380519	0.190260	−	0.981734i	$-0.439067\pi$
$577$	5274.00	0.380519	0.190260	+	0.981734i	$0.439067\pi$
$578$	3994.00	0.287420
$579$	666.000	0.0478031
$580$	9520.00	0.681546
$581$	6664.00	0.475851
$582$	−6516.00	−0.464084
$583$	−3542.00	−0.251620
$584$	16.0000	0.00113371
$585$	−4788.00	−0.338392
$586$	8004.00	0.564236
$587$	11604.0	0.815926	0.407963	−	0.912999i	$-0.366239\pi$
$587$	11604.0	0.815926	0.407963	+	0.912999i	$0.366239\pi$
$588$	−588.000	−0.0412393
$589$	3680.00	0.257439
$590$	7280.00	0.507988
$591$	11718.0	0.815591
$592$	4704.00	0.326576
$593$	−13002.0	−0.900385	−0.450192	−	0.892932i	$-0.648645\pi$
$593$	−13002.0	−0.900385	−0.450192	+	0.892932i	$0.648645\pi$
$594$	594.000	0.0410305
$595$	5292.00	0.364623
$596$	−3400.00	−0.233674
$597$	−5700.00	−0.390763
$598$	−608.000	−0.0415769
$599$	16320.0	1.11322	0.556609	−	0.830775i	$-0.312102\pi$
$599$	16320.0	1.11322	0.556609	+	0.830775i	$0.312102\pi$
$600$	1704.00	0.115943
$601$	−9258.00	−0.628356	−0.314178	−	0.949364i	$-0.601729\pi$
$601$	−9258.00	−0.628356	−0.314178	+	0.949364i	$0.601729\pi$
$602$	−728.000	−0.0492875
$603$	−3924.00	−0.265004
$604$	−5792.00	−0.390187
$605$	−1694.00	−0.113836
$606$	612.000	0.0410244
$607$	25824.0	1.72679	0.863397	−	0.504525i	$-0.168332\pi$
$607$	25824.0	1.72679	0.863397	+	0.504525i	$0.168332\pi$
$608$	−1280.00	−0.0853797
$609$	−3570.00	−0.237543
$610$	616.000	0.0408871
$611$	−2888.00	−0.191221
$612$	1944.00	0.128401
$613$	15518.0	1.02246	0.511228	−	0.859445i	$-0.329191\pi$
$613$	15518.0	1.02246	0.511228	+	0.859445i	$0.329191\pi$
$614$	−10688.0	−0.702496
$615$	−10836.0	−0.710487
$616$	616.000	0.0402911
$617$	−14486.0	−0.945194	−0.472597	−	0.881279i	$-0.656683\pi$
$617$	−14486.0	−0.945194	−0.472597	+	0.881279i	$0.656683\pi$
$618$	1128.00	0.0734220
$619$	12460.0	0.809062	0.404531	−	0.914524i	$-0.367435\pi$
$619$	12460.0	0.809062	0.404531	+	0.914524i	$0.367435\pi$
$620$	−5152.00	−0.333725
$621$	−216.000	−0.0139578
$622$	4536.00	0.292407
$623$	490.000	0.0315111
$624$	−1824.00	−0.117017
$625$	−19459.0	−1.24538
$626$	14844.0	0.947741
$627$	−1320.00	−0.0840761
$628$	−7544.00	−0.479360
$629$	15876.0	1.00639
$630$	−1764.00	−0.111555
$631$	−16648.0	−1.05031	−0.525156	−	0.851006i	$-0.675993\pi$
$631$	−16648.0	−1.05031	−0.525156	+	0.851006i	$0.675993\pi$
$632$	1600.00	0.100703
$633$	−3516.00	−0.220772
$634$	15252.0	0.955417
$635$	22064.0	1.37887
$636$	3864.00	0.240908
$637$	1862.00	0.115817
$638$	3740.00	0.232082
$639$	−3312.00	−0.205040
$640$	1792.00	0.110680
$641$	−10638.0	−0.655500	−0.327750	−	0.944764i	$-0.606290\pi$
$641$	−10638.0	−0.655500	−0.327750	+	0.944764i	$0.606290\pi$
$642$	−10296.0	−0.632945
$643$	4588.00	0.281389	0.140694	−	0.990053i	$-0.455066\pi$
$643$	4588.00	0.281389	0.140694	+	0.990053i	$0.455066\pi$
$644$	−224.000	−0.0137063
$645$	−2184.00	−0.133325
$646$	−4320.00	−0.263109
$647$	−8796.00	−0.534477	−0.267238	−	0.963630i	$-0.586111\pi$
$647$	−8796.00	−0.534477	−0.267238	+	0.963630i	$0.586111\pi$
$648$	−648.000	−0.0392837
$649$	2860.00	0.172981
$650$	−5396.00	−0.325613
$651$	1932.00	0.116315
$652$	912.000	0.0547802
$653$	18878.0	1.13132	0.565661	−	0.824638i	$-0.308621\pi$
$653$	18878.0	1.13132	0.565661	+	0.824638i	$0.308621\pi$
$654$	1860.00	0.111211
$655$	−15008.0	−0.895284
$656$	−4128.00	−0.245688
$657$	−18.0000	−0.00106887
$658$	−1064.00	−0.0630381
$659$	−20780.0	−1.22834	−0.614168	−	0.789175i	$-0.710508\pi$
$659$	−20780.0	−1.22834	−0.614168	+	0.789175i	$0.710508\pi$
$660$	1848.00	0.108990
$661$	10402.0	0.612089	0.306045	−	0.952017i	$-0.400994\pi$
$661$	10402.0	0.612089	0.306045	+	0.952017i	$0.400994\pi$
$662$	−2984.00	−0.175191
$663$	−6156.00	−0.360602
$664$	7616.00	0.445118
$665$	3920.00	0.228588
$666$	−5292.00	−0.307899
$667$	−1360.00	−0.0789496
$668$	6656.00	0.385522
$669$	7116.00	0.411241
$670$	−12208.0	−0.703935
$671$	242.000	0.0139230
$672$	−672.000	−0.0385758
$673$	23018.0	1.31839	0.659197	−	0.751971i	$-0.270896\pi$
$673$	23018.0	1.31839	0.659197	+	0.751971i	$0.270896\pi$
$674$	−148.000	−0.00845808
$675$	−1917.00	−0.109312
$676$	−3012.00	−0.171370
$677$	−16866.0	−0.957479	−0.478739	−	0.877957i	$-0.658906\pi$
$677$	−16866.0	−0.957479	−0.478739	+	0.877957i	$0.658906\pi$
$678$	−10932.0	−0.619234
$679$	7602.00	0.429658
$680$	6048.00	0.341074
$681$	−3072.00	−0.172862
$682$	−2024.00	−0.113641
$683$	4668.00	0.261517	0.130758	−	0.991414i	$-0.458259\pi$
$683$	4668.00	0.261517	0.130758	+	0.991414i	$0.458259\pi$
$684$	1440.00	0.0804967
$685$	3444.00	0.192100
$686$	686.000	0.0381802
$687$	−870.000	−0.0483152
$688$	−832.000	−0.0461042
$689$	−12236.0	−0.676567
$690$	−672.000	−0.0370762
$691$	−33108.0	−1.82270	−0.911351	−	0.411629i	$-0.864960\pi$
$691$	−33108.0	−1.82270	−0.911351	+	0.411629i	$0.864960\pi$
$692$	9752.00	0.535716
$693$	−693.000	−0.0379869
$694$	−23448.0	−1.28253
$695$	−22960.0	−1.25313
$696$	−4080.00	−0.222201
$697$	−13932.0	−0.757119
$698$	−12700.0	−0.688685
$699$	15066.0	0.815234
$700$	−1988.00	−0.107342
$701$	7302.00	0.393428	0.196714	−	0.980461i	$-0.436973\pi$
$701$	7302.00	0.393428	0.196714	+	0.980461i	$0.436973\pi$
$702$	2052.00	0.110324
$703$	11760.0	0.630920
$704$	704.000	0.0376889
$705$	−3192.00	−0.170522
$706$	11644.0	0.620719
$707$	−714.000	−0.0379812
$708$	−3120.00	−0.165617
$709$	−10570.0	−0.559894	−0.279947	−	0.960015i	$-0.590317\pi$
$709$	−10570.0	−0.559894	−0.279947	+	0.960015i	$0.590317\pi$
$710$	−10304.0	−0.544651
$711$	−1800.00	−0.0949441
$712$	560.000	0.0294760
$713$	736.000	0.0386584
$714$	−2268.00	−0.118876
$715$	−5852.00	−0.306087
$716$	6480.00	0.338225
$717$	−600.000	−0.0312516
$718$	−26640.0	−1.38467
$719$	20220.0	1.04879	0.524394	−	0.851476i	$-0.324292\pi$
$719$	20220.0	1.04879	0.524394	+	0.851476i	$0.324292\pi$
$720$	−2016.00	−0.104350
$721$	−1316.00	−0.0679756
$722$	10518.0	0.542160
$723$	4494.00	0.231167
$724$	14408.0	0.739598
$725$	−12070.0	−0.618301
$726$	726.000	0.0371135
$727$	−29996.0	−1.53025	−0.765124	−	0.643883i	$-0.777322\pi$
$727$	−29996.0	−1.53025	−0.765124	+	0.643883i	$0.777322\pi$
$728$	2128.00	0.108336
$729$	729.000	0.0370370
$730$	−56.0000	−0.00283925
$731$	−2808.00	−0.142076
$732$	−264.000	−0.0133302
$733$	−2282.00	−0.114990	−0.0574949	−	0.998346i	$-0.518311\pi$
$733$	−2282.00	−0.114990	−0.0574949	+	0.998346i	$0.518311\pi$
$734$	22152.0	1.11396
$735$	2058.00	0.103280
$736$	−256.000	−0.0128210
$737$	−4796.00	−0.239705
$738$	4644.00	0.231637
$739$	3260.00	0.162275	0.0811374	−	0.996703i	$-0.474145\pi$
$739$	3260.00	0.162275	0.0811374	+	0.996703i	$0.474145\pi$
$740$	−16464.0	−0.817877
$741$	−4560.00	−0.226067
$742$	−4508.00	−0.223038
$743$	−14512.0	−0.716546	−0.358273	−	0.933617i	$-0.616634\pi$
$743$	−14512.0	−0.716546	−0.358273	+	0.933617i	$0.616634\pi$
$744$	2208.00	0.108803
$745$	11900.0	0.585211
$746$	7844.00	0.384972
$747$	−8568.00	−0.419661
$748$	2376.00	0.116143
$749$	12012.0	0.585993
$750$	4536.00	0.220842
$751$	−20128.0	−0.978004	−0.489002	−	0.872283i	$-0.662639\pi$
$751$	−20128.0	−0.978004	−0.489002	+	0.872283i	$0.662639\pi$
$752$	−1216.00	−0.0589667
$753$	22404.0	1.08426
$754$	12920.0	0.624030
$755$	20272.0	0.977184
$756$	756.000	0.0363696
$757$	−3066.00	−0.147207	−0.0736035	−	0.997288i	$-0.523450\pi$
$757$	−3066.00	−0.147207	−0.0736035	+	0.997288i	$0.523450\pi$
$758$	23240.0	1.11361
$759$	−264.000	−0.0126253
$760$	4480.00	0.213825
$761$	26982.0	1.28528	0.642639	−	0.766169i	$-0.277839\pi$
$761$	26982.0	1.28528	0.642639	+	0.766169i	$0.277839\pi$
$762$	−9456.00	−0.449547
$763$	−2170.00	−0.102961
$764$	13888.0	0.657657
$765$	−6804.00	−0.321568
$766$	−7256.00	−0.342258
$767$	9880.00	0.465119
$768$	−768.000	−0.0360844
$769$	3550.00	0.166471	0.0832355	−	0.996530i	$-0.473475\pi$
$769$	3550.00	0.166471	0.0832355	+	0.996530i	$0.473475\pi$
$770$	−2156.00	−0.100905
$771$	−6582.00	−0.307451
$772$	−888.000	−0.0413987
$773$	−33062.0	−1.53837	−0.769183	−	0.639028i	$-0.779337\pi$
$773$	−33062.0	−1.53837	−0.769183	+	0.639028i	$0.779337\pi$
$774$	936.000	0.0434675
$775$	6532.00	0.302757
$776$	8688.00	0.401909
$777$	6174.00	0.285059
$778$	−19500.0	−0.898598
$779$	−10320.0	−0.474650
$780$	6384.00	0.293056
$781$	−4048.00	−0.185466
$782$	−864.000	−0.0395097
$783$	4590.00	0.209493
$784$	784.000	0.0357143
$785$	26404.0	1.20051
$786$	6432.00	0.291885
$787$	18824.0	0.852609	0.426304	−	0.904580i	$-0.359815\pi$
$787$	18824.0	0.852609	0.426304	+	0.904580i	$0.359815\pi$
$788$	−15624.0	−0.706322
$789$	−9864.00	−0.445079
$790$	−5600.00	−0.252201
$791$	12754.0	0.573300
$792$	−792.000	−0.0355335
$793$	836.000	0.0374366
$794$	13212.0	0.590524
$795$	−13524.0	−0.603329
$796$	7600.00	0.338411
$797$	41314.0	1.83616	0.918078	−	0.396399i	$-0.129740\pi$
$797$	41314.0	1.83616	0.918078	+	0.396399i	$0.129740\pi$
$798$	−1680.00	−0.0745255
$799$	−4104.00	−0.181713
$800$	−2272.00	−0.100409
$801$	−630.000	−0.0277902
$802$	−4.00000	−0.000176116 0
$803$	−22.0000	−0.000966828 0
$804$	5232.00	0.229501
$805$	784.000	0.0343259
$806$	−6992.00	−0.305562
$807$	20130.0	0.878079
$808$	−816.000	−0.0355282
$809$	16090.0	0.699251	0.349626	−	0.936889i	$-0.386309\pi$
$809$	16090.0	0.699251	0.349626	+	0.936889i	$0.386309\pi$
$810$	2268.00	0.0983820
$811$	4472.00	0.193629	0.0968145	−	0.995302i	$-0.469135\pi$
$811$	4472.00	0.193629	0.0968145	+	0.995302i	$0.469135\pi$
$812$	4760.00	0.205718
$813$	−14856.0	−0.640864
$814$	−6468.00	−0.278505
$815$	−3192.00	−0.137191
$816$	−2592.00	−0.111199
$817$	−2080.00	−0.0890698
$818$	27860.0	1.19083
$819$	−2394.00	−0.102141
$820$	14448.0	0.615300
$821$	−44338.0	−1.88478	−0.942392	−	0.334512i	$-0.891429\pi$
$821$	−44338.0	−1.88478	−0.942392	+	0.334512i	$0.891429\pi$
$822$	−1476.00	−0.0626295
$823$	−19432.0	−0.823034	−0.411517	−	0.911402i	$-0.635001\pi$
$823$	−19432.0	−0.823034	−0.411517	+	0.911402i	$0.635001\pi$
$824$	−1504.00	−0.0635853
$825$	−2343.00	−0.0988761
$826$	3640.00	0.153331
$827$	15764.0	0.662839	0.331420	−	0.943483i	$-0.392472\pi$
$827$	15764.0	0.662839	0.331420	+	0.943483i	$0.392472\pi$
$828$	288.000	0.0120878
$829$	16570.0	0.694210	0.347105	−	0.937826i	$-0.387165\pi$
$829$	16570.0	0.694210	0.347105	+	0.937826i	$0.387165\pi$
$830$	−26656.0	−1.11475
$831$	10998.0	0.459105
$832$	2432.00	0.101339
$833$	2646.00	0.110058
$834$	9840.00	0.408551
$835$	−23296.0	−0.965499
$836$	1760.00	0.0728120
$837$	−2484.00	−0.102580
$838$	15480.0	0.638124
$839$	−5580.00	−0.229610	−0.114805	−	0.993388i	$-0.536624\pi$
$839$	−5580.00	−0.229610	−0.114805	+	0.993388i	$0.536624\pi$
$840$	2352.00	0.0966092
$841$	4511.00	0.184960
$842$	11236.0	0.459879
$843$	20394.0	0.833223
$844$	4688.00	0.191194
$845$	10542.0	0.429178
$846$	1368.00	0.0555943
$847$	−847.000	−0.0343604
$848$	−5152.00	−0.208633
$849$	5976.00	0.241573
$850$	−7668.00	−0.309424
$851$	2352.00	0.0947421
$852$	4416.00	0.177570
$853$	−17522.0	−0.703332	−0.351666	−	0.936126i	$-0.614385\pi$
$853$	−17522.0	−0.703332	−0.351666	+	0.936126i	$0.614385\pi$
$854$	308.000	0.0123414
$855$	−5040.00	−0.201596
$856$	13728.0	0.548146
$857$	−37146.0	−1.48061	−0.740305	−	0.672271i	$-0.765319\pi$
$857$	−37146.0	−1.48061	−0.740305	+	0.672271i	$0.765319\pi$
$858$	2508.00	0.0997922
$859$	46820.0	1.85969	0.929847	−	0.367945i	$-0.119939\pi$
$859$	46820.0	1.85969	0.929847	+	0.367945i	$0.119939\pi$
$860$	2912.00	0.115463
$861$	−5418.00	−0.214454
$862$	18016.0	0.711865
$863$	25808.0	1.01798	0.508989	−	0.860773i	$-0.330019\pi$
$863$	25808.0	1.01798	0.508989	+	0.860773i	$0.330019\pi$
$864$	864.000	0.0340207
$865$	−34132.0	−1.34164
$866$	21404.0	0.839882
$867$	5991.00	0.234677
$868$	−2576.00	−0.100732
$869$	−2200.00	−0.0858802
$870$	14280.0	0.556480
$871$	−16568.0	−0.644530
$872$	−2480.00	−0.0963112
$873$	−9774.00	−0.378923
$874$	−640.000	−0.0247692
$875$	−5292.00	−0.204460
$876$	24.0000	0.000925668 0
$877$	−43546.0	−1.67667	−0.838337	−	0.545152i	$-0.816472\pi$
$877$	−43546.0	−1.67667	−0.838337	+	0.545152i	$0.816472\pi$
$878$	9920.00	0.381303
$879$	12006.0	0.460697
$880$	−2464.00	−0.0943880
$881$	−10278.0	−0.393047	−0.196524	−	0.980499i	$-0.562965\pi$
$881$	−10278.0	−0.393047	−0.196524	+	0.980499i	$0.562965\pi$
$882$	−882.000	−0.0336718
$883$	20708.0	0.789218	0.394609	−	0.918849i	$-0.370880\pi$
$883$	20708.0	0.789218	0.394609	+	0.918849i	$0.370880\pi$
$884$	8208.00	0.312291
$885$	10920.0	0.414770
$886$	−376.000	−0.0142573
$887$	−30296.0	−1.14683	−0.573416	−	0.819264i	$-0.694382\pi$
$887$	−30296.0	−1.14683	−0.573416	+	0.819264i	$0.694382\pi$
$888$	7056.00	0.266648
$889$	11032.0	0.416200
$890$	−1960.00	−0.0738195
$891$	891.000	0.0335013
$892$	−9488.00	−0.356145
$893$	−3040.00	−0.113919
$894$	−5100.00	−0.190794
$895$	−22680.0	−0.847049
$896$	896.000	0.0334077
$897$	−912.000	−0.0339474
$898$	6300.00	0.234113
$899$	−15640.0	−0.580226
$900$	2556.00	0.0946667
$901$	−17388.0	−0.642928
$902$	5676.00	0.209523
$903$	−1092.00	−0.0402431
$904$	14576.0	0.536273
$905$	−50428.0	−1.85225
$906$	−8688.00	−0.318587
$907$	29604.0	1.08378	0.541888	−	0.840451i	$-0.317710\pi$
$907$	29604.0	1.08378	0.541888	+	0.840451i	$0.317710\pi$
$908$	4096.00	0.149703
$909$	918.000	0.0334963
$910$	−7448.00	−0.271317
$911$	12112.0	0.440492	0.220246	−	0.975444i	$-0.429314\pi$
$911$	12112.0	0.440492	0.220246	+	0.975444i	$0.429314\pi$
$912$	−1920.00	−0.0697122
$913$	−10472.0	−0.379598
$914$	25612.0	0.926881
$915$	924.000	0.0333842
$916$	1160.00	0.0418422
$917$	−7504.00	−0.270233
$918$	2916.00	0.104839
$919$	−33320.0	−1.19600	−0.598001	−	0.801496i	$-0.704038\pi$
$919$	−33320.0	−1.19600	−0.598001	+	0.801496i	$0.704038\pi$
$920$	896.000	0.0321090
$921$	−16032.0	−0.573586
$922$	3556.00	0.127018
$923$	−13984.0	−0.498688
$924$	924.000	0.0328976
$925$	20874.0	0.741982
$926$	31344.0	1.11234
$927$	1692.00	0.0599488
$928$	5440.00	0.192432
$929$	−45950.0	−1.62279	−0.811394	−	0.584499i	$-0.801291\pi$
$929$	−45950.0	−1.62279	−0.811394	+	0.584499i	$0.801291\pi$
$930$	−7728.00	−0.272485
$931$	1960.00	0.0689972
$932$	−20088.0	−0.706013
$933$	6804.00	0.238749
$934$	−8.00000	−0.000280266 0
$935$	−8316.00	−0.290869
$936$	−2736.00	−0.0955438
$937$	11054.0	0.385399	0.192699	−	0.981258i	$-0.438276\pi$
$937$	11054.0	0.385399	0.192699	+	0.981258i	$0.438276\pi$
$938$	−6104.00	−0.212476
$939$	22266.0	0.773827
$940$	4256.00	0.147676
$941$	−55818.0	−1.93370	−0.966852	−	0.255339i	$-0.917813\pi$
$941$	−55818.0	−1.93370	−0.966852	+	0.255339i	$0.917813\pi$
$942$	−11316.0	−0.391396
$943$	−2064.00	−0.0712758
$944$	4160.00	0.143428
$945$	−2646.00	−0.0910840
$946$	1144.00	0.0393178
$947$	−33636.0	−1.15420	−0.577098	−	0.816675i	$-0.695815\pi$
$947$	−33636.0	−1.15420	−0.577098	+	0.816675i	$0.695815\pi$
$948$	2400.00	0.0822240
$949$	−76.0000	−0.00259965
$950$	−5680.00	−0.193983
$951$	22878.0	0.780095
$952$	3024.00	0.102950
$953$	31418.0	1.06792	0.533961	−	0.845509i	$-0.320703\pi$
$953$	31418.0	1.06792	0.533961	+	0.845509i	$0.320703\pi$
$954$	5796.00	0.196701
$955$	−48608.0	−1.64703
$956$	800.000	0.0270647
$957$	5610.00	0.189494
$958$	−29280.0	−0.987467
$959$	1722.00	0.0579836
$960$	2688.00	0.0903696
$961$	−21327.0	−0.715887
$962$	−22344.0	−0.748856
$963$	−15444.0	−0.516797
$964$	−5992.00	−0.200196
$965$	3108.00	0.103679
$966$	−336.000	−0.0111911
$967$	−13176.0	−0.438171	−0.219086	−	0.975706i	$-0.570307\pi$
$967$	−13176.0	−0.438171	−0.219086	+	0.975706i	$0.570307\pi$
$968$	−968.000	−0.0321412
$969$	−6480.00	−0.214827
$970$	−30408.0	−1.00654
$971$	27132.0	0.896712	0.448356	−	0.893855i	$-0.352010\pi$
$971$	27132.0	0.896712	0.448356	+	0.893855i	$0.352010\pi$
$972$	−972.000	−0.0320750
$973$	−11480.0	−0.378245
$974$	39792.0	1.30905
$975$	−8094.00	−0.265862
$976$	352.000	0.0115443
$977$	27154.0	0.889185	0.444592	−	0.895733i	$-0.353349\pi$
$977$	27154.0	0.889185	0.444592	+	0.895733i	$0.353349\pi$
$978$	1368.00	0.0447278
$979$	−770.000	−0.0251372
$980$	−2744.00	−0.0894427
$981$	2790.00	0.0908031
$982$	−15944.0	−0.518120
$983$	35068.0	1.13784	0.568919	−	0.822393i	$-0.307362\pi$
$983$	35068.0	1.13784	0.568919	+	0.822393i	$0.307362\pi$
$984$	−6192.00	−0.200603
$985$	54684.0	1.76891
$986$	18360.0	0.593004
$987$	−1596.00	−0.0514704
$988$	6080.00	0.195780
$989$	−416.000	−0.0133752
$990$	2772.00	0.0889898
$991$	26072.0	0.835726	0.417863	−	0.908510i	$-0.362779\pi$
$991$	26072.0	0.835726	0.417863	+	0.908510i	$0.362779\pi$
$992$	−2944.00	−0.0942259
$993$	−4476.00	−0.143043
$994$	−5152.00	−0.164398
$995$	−26600.0	−0.847514
$996$	11424.0	0.363437
$997$	−19866.0	−0.631056	−0.315528	−	0.948916i	$-0.602182\pi$
$997$	−19866.0	−0.631056	−0.315528	+	0.948916i	$0.602182\pi$
$998$	4520.00	0.143365
$999$	−7938.00	−0.251399

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	462.4.a.b.1.1	✓	1
3.2	odd	2		1386.4.a.m.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
462.4.a.b.1.1	✓	1	1.1	even	1	trivial
1386.4.a.m.1.1		1	3.2	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	462.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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