LMFDB - Embedded newform 462.4.a.f.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)

$f(q)$

$=$

$q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +1.00000 q^{5} -6.00000 q^{6} -7.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$

\(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +1.00000 q^{5} -6.00000 q^{6} -7.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} +2.00000 q^{10} -11.0000 q^{11} -12.0000 q^{12} -43.0000 q^{13} -14.0000 q^{14} -3.00000 q^{15} +16.0000 q^{16} +100.000 q^{17} +18.0000 q^{18} -87.0000 q^{19} +4.00000 q^{20} +21.0000 q^{21} -22.0000 q^{22} -58.0000 q^{23} -24.0000 q^{24} -124.000 q^{25} -86.0000 q^{26} -27.0000 q^{27} -28.0000 q^{28} -223.000 q^{29} -6.00000 q^{30} +88.0000 q^{31} +32.0000 q^{32} +33.0000 q^{33} +200.000 q^{34} -7.00000 q^{35} +36.0000 q^{36} +37.0000 q^{37} -174.000 q^{38} +129.000 q^{39} +8.00000 q^{40} +128.000 q^{41} +42.0000 q^{42} -458.000 q^{43} -44.0000 q^{44} +9.00000 q^{45} -116.000 q^{46} -341.000 q^{47} -48.0000 q^{48} +49.0000 q^{49} -248.000 q^{50} -300.000 q^{51} -172.000 q^{52} -342.000 q^{53} -54.0000 q^{54} -11.0000 q^{55} -56.0000 q^{56} +261.000 q^{57} -446.000 q^{58} -105.000 q^{59} -12.0000 q^{60} +190.000 q^{61} +176.000 q^{62} -63.0000 q^{63} +64.0000 q^{64} -43.0000 q^{65} +66.0000 q^{66} -579.000 q^{67} +400.000 q^{68} +174.000 q^{69} -14.0000 q^{70} +128.000 q^{71} +72.0000 q^{72} -161.000 q^{73} +74.0000 q^{74} +372.000 q^{75} -348.000 q^{76} +77.0000 q^{77} +258.000 q^{78} -396.000 q^{79} +16.0000 q^{80} +81.0000 q^{81} +256.000 q^{82} -420.000 q^{83} +84.0000 q^{84} +100.000 q^{85} -916.000 q^{86} +669.000 q^{87} -88.0000 q^{88} -798.000 q^{89} +18.0000 q^{90} +301.000 q^{91} -232.000 q^{92} -264.000 q^{93} -682.000 q^{94} -87.0000 q^{95} -96.0000 q^{96} +1414.00 q^{97} +98.0000 q^{98} -99.0000 q^{99} +O(q^{100})\)

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
$5$	1.00000	0.0894427	0.0447214	−	0.998999i	$-0.485760\pi$
$5$	1.00000	0.0894427	0.0447214	+	0.998999i	$0.485760\pi$
$6$	−6.00000	−0.408248
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	8.00000	0.353553
$9$	9.00000	0.333333
$10$	2.00000	0.0632456
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	−12.0000	−0.288675
$13$	−43.0000	−0.917389	−0.458694	−	0.888594i	$-0.651683\pi$
$13$	−43.0000	−0.917389	−0.458694	+	0.888594i	$0.651683\pi$
$14$	−14.0000	−0.267261
$15$	−3.00000	−0.0516398
$16$	16.0000	0.250000
$17$	100.000	1.42668	0.713340	−	0.700818i	$-0.247182\pi$
$17$	100.000	1.42668	0.713340	+	0.700818i	$0.247182\pi$
$18$	18.0000	0.235702
$19$	−87.0000	−1.05048	−0.525241	−	0.850953i	$-0.676025\pi$
$19$	−87.0000	−1.05048	−0.525241	+	0.850953i	$0.676025\pi$
$20$	4.00000	0.0447214
$21$	21.0000	0.218218
$22$	−22.0000	−0.213201
$23$	−58.0000	−0.525819	−0.262909	−	0.964821i	$-0.584682\pi$
$23$	−58.0000	−0.525819	−0.262909	+	0.964821i	$0.584682\pi$
$24$	−24.0000	−0.204124
$25$	−124.000	−0.992000
$26$	−86.0000	−0.648692
$27$	−27.0000	−0.192450
$28$	−28.0000	−0.188982
$29$	−223.000	−1.42793	−0.713967	−	0.700180i	$-0.753103\pi$
$29$	−223.000	−1.42793	−0.713967	+	0.700180i	$0.753103\pi$
$30$	−6.00000	−0.0365148
$31$	88.0000	0.509847	0.254924	−	0.966961i	$-0.417950\pi$
$31$	88.0000	0.509847	0.254924	+	0.966961i	$0.417950\pi$
$32$	32.0000	0.176777
$33$	33.0000	0.174078
$34$	200.000	1.00882
$35$	−7.00000	−0.0338062
$36$	36.0000	0.166667
$37$	37.0000	0.164399	0.0821995	−	0.996616i	$-0.473806\pi$
$37$	37.0000	0.164399	0.0821995	+	0.996616i	$0.473806\pi$
$38$	−174.000	−0.742803
$39$	129.000	0.529655
$40$	8.00000	0.0316228
$41$	128.000	0.487567	0.243783	−	0.969830i	$-0.421611\pi$
$41$	128.000	0.487567	0.243783	+	0.969830i	$0.421611\pi$
$42$	42.0000	0.154303
$43$	−458.000	−1.62429	−0.812144	−	0.583458i	$-0.801699\pi$
$43$	−458.000	−1.62429	−0.812144	+	0.583458i	$0.801699\pi$
$44$	−44.0000	−0.150756
$45$	9.00000	0.0298142
$46$	−116.000	−0.371810
$47$	−341.000	−1.05830	−0.529149	−	0.848529i	$-0.677489\pi$
$47$	−341.000	−1.05830	−0.529149	+	0.848529i	$0.677489\pi$
$48$	−48.0000	−0.144338
$49$	49.0000	0.142857
$50$	−248.000	−0.701450
$51$	−300.000	−0.823694
$52$	−172.000	−0.458694
$53$	−342.000	−0.886364	−0.443182	−	0.896432i	$-0.646151\pi$
$53$	−342.000	−0.886364	−0.443182	+	0.896432i	$0.646151\pi$
$54$	−54.0000	−0.136083
$55$	−11.0000	−0.0269680
$56$	−56.0000	−0.133631
$57$	261.000	0.606496
$58$	−446.000	−1.00970
$59$	−105.000	−0.231692	−0.115846	−	0.993267i	$-0.536958\pi$
$59$	−105.000	−0.231692	−0.115846	+	0.993267i	$0.536958\pi$
$60$	−12.0000	−0.0258199
$61$	190.000	0.398803	0.199402	−	0.979918i	$-0.436100\pi$
$61$	190.000	0.398803	0.199402	+	0.979918i	$0.436100\pi$
$62$	176.000	0.360516
$63$	−63.0000	−0.125988
$64$	64.0000	0.125000
$65$	−43.0000	−0.0820537
$66$	66.0000	0.123091
$67$	−579.000	−1.05576	−0.527881	−	0.849318i	$-0.677013\pi$
$67$	−579.000	−1.05576	−0.527881	+	0.849318i	$0.677013\pi$
$68$	400.000	0.713340
$69$	174.000	0.303582
$70$	−14.0000	−0.0239046
$71$	128.000	0.213955	0.106978	−	0.994261i	$-0.465883\pi$
$71$	128.000	0.213955	0.106978	+	0.994261i	$0.465883\pi$
$72$	72.0000	0.117851
$73$	−161.000	−0.258132	−0.129066	−	0.991636i	$-0.541198\pi$
$73$	−161.000	−0.258132	−0.129066	+	0.991636i	$0.541198\pi$
$74$	74.0000	0.116248
$75$	372.000	0.572731
$76$	−348.000	−0.525241
$77$	77.0000	0.113961
$78$	258.000	0.374522
$79$	−396.000	−0.563968	−0.281984	−	0.959419i	$-0.590993\pi$
$79$	−396.000	−0.563968	−0.281984	+	0.959419i	$0.590993\pi$
$80$	16.0000	0.0223607
$81$	81.0000	0.111111
$82$	256.000	0.344762
$83$	−420.000	−0.555434	−0.277717	−	0.960663i	$-0.589578\pi$
$83$	−420.000	−0.555434	−0.277717	+	0.960663i	$0.589578\pi$
$84$	84.0000	0.109109
$85$	100.000	0.127606
$86$	−916.000	−1.14854
$87$	669.000	0.824418
$88$	−88.0000	−0.106600
$89$	−798.000	−0.950425	−0.475213	−	0.879871i	$-0.657629\pi$
$89$	−798.000	−0.950425	−0.475213	+	0.879871i	$0.657629\pi$
$90$	18.0000	0.0210819
$91$	301.000	0.346740
$92$	−232.000	−0.262909
$93$	−264.000	−0.294360
$94$	−682.000	−0.748329
$95$	−87.0000	−0.0939580
$96$	−96.0000	−0.102062
$97$	1414.00	1.48010	0.740051	−	0.672550i	$-0.234801\pi$
$97$	1414.00	1.48010	0.740051	+	0.672550i	$0.234801\pi$
$98$	98.0000	0.101015
$99$	−99.0000	−0.100504
$100$	−496.000	−0.496000
$101$	−1764.00	−1.73787	−0.868933	−	0.494929i	$-0.835194\pi$
$101$	−1764.00	−1.73787	−0.868933	+	0.494929i	$0.835194\pi$
$102$	−600.000	−0.582440
$103$	978.000	0.935584	0.467792	−	0.883838i	$-0.345050\pi$
$103$	978.000	0.935584	0.467792	+	0.883838i	$0.345050\pi$
$104$	−344.000	−0.324346
$105$	21.0000	0.0195180
$106$	−684.000	−0.626754
$107$	481.000	0.434580	0.217290	−	0.976107i	$-0.430278\pi$
$107$	481.000	0.434580	0.217290	+	0.976107i	$0.430278\pi$
$108$	−108.000	−0.0962250
$109$	−284.000	−0.249562	−0.124781	−	0.992184i	$-0.539823\pi$
$109$	−284.000	−0.249562	−0.124781	+	0.992184i	$0.539823\pi$
$110$	−22.0000	−0.0190693
$111$	−111.000	−0.0949158
$112$	−112.000	−0.0944911
$113$	1498.00	1.24708	0.623540	−	0.781792i	$-0.285694\pi$
$113$	1498.00	1.24708	0.623540	+	0.781792i	$0.285694\pi$
$114$	522.000	0.428858
$115$	−58.0000	−0.0470307
$116$	−892.000	−0.713967
$117$	−387.000	−0.305796
$118$	−210.000	−0.163831
$119$	−700.000	−0.539234
$120$	−24.0000	−0.0182574
$121$	121.000	0.0909091
$122$	380.000	0.281997
$123$	−384.000	−0.281497
$124$	352.000	0.254924
$125$	−249.000	−0.178170
$126$	−126.000	−0.0890871
$127$	−616.000	−0.430403	−0.215201	−	0.976570i	$-0.569041\pi$
$127$	−616.000	−0.430403	−0.215201	+	0.976570i	$0.569041\pi$
$128$	128.000	0.0883883
$129$	1374.00	0.937783
$130$	−86.0000	−0.0580208
$131$	2748.00	1.83278	0.916389	−	0.400289i	$-0.131090\pi$
$131$	2748.00	1.83278	0.916389	+	0.400289i	$0.131090\pi$
$132$	132.000	0.0870388
$133$	609.000	0.397045
$134$	−1158.00	−0.746537
$135$	−27.0000	−0.0172133
$136$	800.000	0.504408
$137$	2184.00	1.36198	0.680992	−	0.732291i	$-0.261549\pi$
$137$	2184.00	1.36198	0.680992	+	0.732291i	$0.261549\pi$
$138$	348.000	0.214665
$139$	908.000	0.554069	0.277034	−	0.960860i	$-0.410648\pi$
$139$	908.000	0.554069	0.277034	+	0.960860i	$0.410648\pi$
$140$	−28.0000	−0.0169031
$141$	1023.00	0.611008
$142$	256.000	0.151289
$143$	473.000	0.276603
$144$	144.000	0.0833333
$145$	−223.000	−0.127718
$146$	−322.000	−0.182527
$147$	−147.000	−0.0824786
$148$	148.000	0.0821995
$149$	421.000	0.231474	0.115737	−	0.993280i	$-0.463077\pi$
$149$	421.000	0.231474	0.115737	+	0.993280i	$0.463077\pi$
$150$	744.000	0.404982
$151$	948.000	0.510908	0.255454	−	0.966821i	$-0.417775\pi$
$151$	948.000	0.510908	0.255454	+	0.966821i	$0.417775\pi$
$152$	−696.000	−0.371402
$153$	900.000	0.475560
$154$	154.000	0.0805823
$155$	88.0000	0.0456021
$156$	516.000	0.264827
$157$	2368.00	1.20374	0.601869	−	0.798595i	$-0.294423\pi$
$157$	2368.00	1.20374	0.601869	+	0.798595i	$0.294423\pi$
$158$	−792.000	−0.398786
$159$	1026.00	0.511743
$160$	32.0000	0.0158114
$161$	406.000	0.198741
$162$	162.000	0.0785674
$163$	−2603.00	−1.25081	−0.625407	−	0.780299i	$-0.715067\pi$
$163$	−2603.00	−1.25081	−0.625407	+	0.780299i	$0.715067\pi$
$164$	512.000	0.243783
$165$	33.0000	0.0155700
$166$	−840.000	−0.392751
$167$	−1402.00	−0.649641	−0.324820	−	0.945776i	$-0.605304\pi$
$167$	−1402.00	−0.649641	−0.324820	+	0.945776i	$0.605304\pi$
$168$	168.000	0.0771517
$169$	−348.000	−0.158398
$170$	200.000	0.0902312
$171$	−783.000	−0.350161
$172$	−1832.00	−0.812144
$173$	836.000	0.367398	0.183699	−	0.982983i	$-0.441193\pi$
$173$	836.000	0.367398	0.183699	+	0.982983i	$0.441193\pi$
$174$	1338.00	0.582951
$175$	868.000	0.374941
$176$	−176.000	−0.0753778
$177$	315.000	0.133768
$178$	−1596.00	−0.672052
$179$	−3292.00	−1.37461	−0.687306	−	0.726368i	$-0.741207\pi$
$179$	−3292.00	−1.37461	−0.687306	+	0.726368i	$0.741207\pi$
$180$	36.0000	0.0149071
$181$	4838.00	1.98677	0.993386	−	0.114823i	$-0.0366302\pi$
$181$	4838.00	1.98677	0.993386	+	0.114823i	$0.0366302\pi$
$182$	602.000	0.245182
$183$	−570.000	−0.230249
$184$	−464.000	−0.185905
$185$	37.0000	0.0147043
$186$	−528.000	−0.208144
$187$	−1100.00	−0.430160
$188$	−1364.00	−0.529149
$189$	189.000	0.0727393
$190$	−174.000	−0.0664384
$191$	−1232.00	−0.466725	−0.233362	−	0.972390i	$-0.574973\pi$
$191$	−1232.00	−0.466725	−0.233362	+	0.972390i	$0.574973\pi$
$192$	−192.000	−0.0721688
$193$	2366.00	0.882427	0.441213	−	0.897402i	$-0.354548\pi$
$193$	2366.00	0.882427	0.441213	+	0.897402i	$0.354548\pi$
$194$	2828.00	1.04659
$195$	129.000	0.0473738
$196$	196.000	0.0714286
$197$	318.000	0.115008	0.0575040	−	0.998345i	$-0.481686\pi$
$197$	318.000	0.115008	0.0575040	+	0.998345i	$0.481686\pi$
$198$	−198.000	−0.0710669
$199$	−806.000	−0.287115	−0.143557	−	0.989642i	$-0.545854\pi$
$199$	−806.000	−0.287115	−0.143557	+	0.989642i	$0.545854\pi$
$200$	−992.000	−0.350725
$201$	1737.00	0.609545
$202$	−3528.00	−1.22886
$203$	1561.00	0.539708
$204$	−1200.00	−0.411847
$205$	128.000	0.0436093
$206$	1956.00	0.661558
$207$	−522.000	−0.175273
$208$	−688.000	−0.229347
$209$	957.000	0.316732
$210$	42.0000	0.0138013
$211$	4462.00	1.45581	0.727907	−	0.685676i	$-0.240493\pi$
$211$	4462.00	1.45581	0.727907	+	0.685676i	$0.240493\pi$
$212$	−1368.00	−0.443182
$213$	−384.000	−0.123527
$214$	962.000	0.307294
$215$	−458.000	−0.145281
$216$	−216.000	−0.0680414
$217$	−616.000	−0.192704
$218$	−568.000	−0.176467
$219$	483.000	0.149032
$220$	−44.0000	−0.0134840
$221$	−4300.00	−1.30882
$222$	−222.000	−0.0671156
$223$	3802.00	1.14171	0.570854	−	0.821052i	$-0.306612\pi$
$223$	3802.00	1.14171	0.570854	+	0.821052i	$0.306612\pi$
$224$	−224.000	−0.0668153
$225$	−1116.00	−0.330667
$226$	2996.00	0.881818
$227$	−3370.00	−0.985351	−0.492676	−	0.870213i	$-0.663981\pi$
$227$	−3370.00	−0.985351	−0.492676	+	0.870213i	$0.663981\pi$
$228$	1044.00	0.303248
$229$	−898.000	−0.259133	−0.129567	−	0.991571i	$-0.541359\pi$
$229$	−898.000	−0.259133	−0.129567	+	0.991571i	$0.541359\pi$
$230$	−116.000	−0.0332557
$231$	−231.000	−0.0657952
$232$	−1784.00	−0.504851
$233$	−462.000	−0.129900	−0.0649498	−	0.997889i	$-0.520689\pi$
$233$	−462.000	−0.129900	−0.0649498	+	0.997889i	$0.520689\pi$
$234$	−774.000	−0.216231
$235$	−341.000	−0.0946570
$236$	−420.000	−0.115846
$237$	1188.00	0.325607
$238$	−1400.00	−0.381296
$239$	−5665.00	−1.53322	−0.766608	−	0.642116i	$-0.778057\pi$
$239$	−5665.00	−1.53322	−0.766608	+	0.642116i	$0.778057\pi$
$240$	−48.0000	−0.0129099
$241$	1593.00	0.425785	0.212892	−	0.977076i	$-0.431712\pi$
$241$	1593.00	0.425785	0.212892	+	0.977076i	$0.431712\pi$
$242$	242.000	0.0642824
$243$	−243.000	−0.0641500
$244$	760.000	0.199402
$245$	49.0000	0.0127775
$246$	−768.000	−0.199048
$247$	3741.00	0.963701
$248$	704.000	0.180258
$249$	1260.00	0.320680
$250$	−498.000	−0.125985
$251$	4545.00	1.14294	0.571470	−	0.820623i	$-0.306373\pi$
$251$	4545.00	1.14294	0.571470	+	0.820623i	$0.306373\pi$
$252$	−252.000	−0.0629941
$253$	638.000	0.158540
$254$	−1232.00	−0.304341
$255$	−300.000	−0.0736734
$256$	256.000	0.0625000
$257$	−2859.00	−0.693928	−0.346964	−	0.937878i	$-0.612787\pi$
$257$	−2859.00	−0.693928	−0.346964	+	0.937878i	$0.612787\pi$
$258$	2748.00	0.663112
$259$	−259.000	−0.0621370
$260$	−172.000	−0.0410269
$261$	−2007.00	−0.475978
$262$	5496.00	1.29597
$263$	−6279.00	−1.47217	−0.736083	−	0.676891i	$-0.763327\pi$
$263$	−6279.00	−1.47217	−0.736083	+	0.676891i	$0.763327\pi$
$264$	264.000	0.0615457
$265$	−342.000	−0.0792788
$266$	1218.00	0.280753
$267$	2394.00	0.548728
$268$	−2316.00	−0.527881
$269$	−6270.00	−1.42115	−0.710574	−	0.703623i	$-0.751565\pi$
$269$	−6270.00	−1.42115	−0.710574	+	0.703623i	$0.751565\pi$
$270$	−54.0000	−0.0121716
$271$	6721.00	1.50654	0.753269	−	0.657713i	$-0.228476\pi$
$271$	6721.00	1.50654	0.753269	+	0.657713i	$0.228476\pi$
$272$	1600.00	0.356670
$273$	−903.000	−0.200191
$274$	4368.00	0.963068
$275$	1364.00	0.299099
$276$	696.000	0.151791
$277$	−5388.00	−1.16871	−0.584357	−	0.811497i	$-0.698653\pi$
$277$	−5388.00	−1.16871	−0.584357	+	0.811497i	$0.698653\pi$
$278$	1816.00	0.391786
$279$	792.000	0.169949
$280$	−56.0000	−0.0119523
$281$	4765.00	1.01159	0.505794	−	0.862654i	$-0.331200\pi$
$281$	4765.00	1.01159	0.505794	+	0.862654i	$0.331200\pi$
$282$	2046.00	0.432048
$283$	−903.000	−0.189674	−0.0948371	−	0.995493i	$-0.530233\pi$
$283$	−903.000	−0.189674	−0.0948371	+	0.995493i	$0.530233\pi$
$284$	512.000	0.106978
$285$	261.000	0.0542467
$286$	946.000	0.195588
$287$	−896.000	−0.184283
$288$	288.000	0.0589256
$289$	5087.00	1.03542
$290$	−446.000	−0.0903104
$291$	−4242.00	−0.854538
$292$	−644.000	−0.129066
$293$	672.000	0.133989	0.0669943	−	0.997753i	$-0.478659\pi$
$293$	672.000	0.133989	0.0669943	+	0.997753i	$0.478659\pi$
$294$	−294.000	−0.0583212
$295$	−105.000	−0.0207232
$296$	296.000	0.0581238
$297$	297.000	0.0580259
$298$	842.000	0.163677
$299$	2494.00	0.482380
$300$	1488.00	0.286366
$301$	3206.00	0.613923
$302$	1896.00	0.361267
$303$	5292.00	1.00336
$304$	−1392.00	−0.262621
$305$	190.000	0.0356701
$306$	1800.00	0.336272
$307$	−1876.00	−0.348759	−0.174379	−	0.984679i	$-0.555792\pi$
$307$	−1876.00	−0.348759	−0.174379	+	0.984679i	$0.555792\pi$
$308$	308.000	0.0569803
$309$	−2934.00	−0.540160
$310$	176.000	0.0322456
$311$	−4688.00	−0.854766	−0.427383	−	0.904071i	$-0.640564\pi$
$311$	−4688.00	−0.854766	−0.427383	+	0.904071i	$0.640564\pi$
$312$	1032.00	0.187261
$313$	4912.00	0.887037	0.443519	−	0.896265i	$-0.353730\pi$
$313$	4912.00	0.887037	0.443519	+	0.896265i	$0.353730\pi$
$314$	4736.00	0.851172
$315$	−63.0000	−0.0112687
$316$	−1584.00	−0.281984
$317$	−558.000	−0.0988656	−0.0494328	−	0.998777i	$-0.515741\pi$
$317$	−558.000	−0.0988656	−0.0494328	+	0.998777i	$0.515741\pi$
$318$	2052.00	0.361857
$319$	2453.00	0.430538
$320$	64.0000	0.0111803
$321$	−1443.00	−0.250905
$322$	812.000	0.140531
$323$	−8700.00	−1.49870
$324$	324.000	0.0555556
$325$	5332.00	0.910050
$326$	−5206.00	−0.884459
$327$	852.000	0.144085
$328$	1024.00	0.172381
$329$	2387.00	0.399999
$330$	66.0000	0.0110096
$331$	4660.00	0.773827	0.386914	−	0.922116i	$-0.373541\pi$
$331$	4660.00	0.773827	0.386914	+	0.922116i	$0.373541\pi$
$332$	−1680.00	−0.277717
$333$	333.000	0.0547997
$334$	−2804.00	−0.459365
$335$	−579.000	−0.0944303
$336$	336.000	0.0545545
$337$	−9906.00	−1.60123	−0.800615	−	0.599180i	$-0.795493\pi$
$337$	−9906.00	−1.60123	−0.800615	+	0.599180i	$0.795493\pi$
$338$	−696.000	−0.112004
$339$	−4494.00	−0.720002
$340$	400.000	0.0638031
$341$	−968.000	−0.153725
$342$	−1566.00	−0.247601
$343$	−343.000	−0.0539949
$344$	−3664.00	−0.574272
$345$	174.000	0.0271532
$346$	1672.00	0.259790
$347$	−10664.0	−1.64978	−0.824890	−	0.565294i	$-0.808763\pi$
$347$	−10664.0	−1.64978	−0.824890	+	0.565294i	$0.808763\pi$
$348$	2676.00	0.412209
$349$	5489.00	0.841889	0.420945	−	0.907086i	$-0.361699\pi$
$349$	5489.00	0.841889	0.420945	+	0.907086i	$0.361699\pi$
$350$	1736.00	0.265123
$351$	1161.00	0.176552
$352$	−352.000	−0.0533002
$353$	−1845.00	−0.278185	−0.139093	−	0.990279i	$-0.544419\pi$
$353$	−1845.00	−0.278185	−0.139093	+	0.990279i	$0.544419\pi$
$354$	630.000	0.0945879
$355$	128.000	0.0191367
$356$	−3192.00	−0.475213
$357$	2100.00	0.311327
$358$	−6584.00	−0.971998
$359$	−848.000	−0.124668	−0.0623339	−	0.998055i	$-0.519854\pi$
$359$	−848.000	−0.124668	−0.0623339	+	0.998055i	$0.519854\pi$
$360$	72.0000	0.0105409
$361$	710.000	0.103514
$362$	9676.00	1.40486
$363$	−363.000	−0.0524864
$364$	1204.00	0.173370
$365$	−161.000	−0.0230880
$366$	−1140.00	−0.162811
$367$	−8474.00	−1.20528	−0.602642	−	0.798012i	$-0.705885\pi$
$367$	−8474.00	−1.20528	−0.602642	+	0.798012i	$0.705885\pi$
$368$	−928.000	−0.131455
$369$	1152.00	0.162522
$370$	74.0000	0.0103975
$371$	2394.00	0.335014
$372$	−1056.00	−0.147180
$373$	11396.0	1.58194	0.790969	−	0.611857i	$-0.209577\pi$
$373$	11396.0	1.58194	0.790969	+	0.611857i	$0.209577\pi$
$374$	−2200.00	−0.304169
$375$	747.000	0.102866
$376$	−2728.00	−0.374165
$377$	9589.00	1.30997
$378$	378.000	0.0514344
$379$	2155.00	0.292071	0.146036	−	0.989279i	$-0.453349\pi$
$379$	2155.00	0.292071	0.146036	+	0.989279i	$0.453349\pi$
$380$	−348.000	−0.0469790
$381$	1848.00	0.248493
$382$	−2464.00	−0.330024
$383$	−5864.00	−0.782340	−0.391170	−	0.920318i	$-0.627930\pi$
$383$	−5864.00	−0.782340	−0.391170	+	0.920318i	$0.627930\pi$
$384$	−384.000	−0.0510310
$385$	77.0000	0.0101929
$386$	4732.00	0.623970
$387$	−4122.00	−0.541429
$388$	5656.00	0.740051
$389$	10104.0	1.31695	0.658474	−	0.752603i	$-0.271202\pi$
$389$	10104.0	1.31695	0.658474	+	0.752603i	$0.271202\pi$
$390$	258.000	0.0334983
$391$	−5800.00	−0.750175
$392$	392.000	0.0505076
$393$	−8244.00	−1.05815
$394$	636.000	0.0813229
$395$	−396.000	−0.0504428
$396$	−396.000	−0.0502519
$397$	−5868.00	−0.741830	−0.370915	−	0.928667i	$-0.620956\pi$
$397$	−5868.00	−0.741830	−0.370915	+	0.928667i	$0.620956\pi$
$398$	−1612.00	−0.203021
$399$	−1827.00	−0.229234
$400$	−1984.00	−0.248000
$401$	−2418.00	−0.301120	−0.150560	−	0.988601i	$-0.548108\pi$
$401$	−2418.00	−0.301120	−0.150560	+	0.988601i	$0.548108\pi$
$402$	3474.00	0.431013
$403$	−3784.00	−0.467728
$404$	−7056.00	−0.868933
$405$	81.0000	0.00993808
$406$	3122.00	0.381631
$407$	−407.000	−0.0495682
$408$	−2400.00	−0.291220
$409$	−2354.00	−0.284591	−0.142296	−	0.989824i	$-0.545448\pi$
$409$	−2354.00	−0.284591	−0.142296	+	0.989824i	$0.545448\pi$
$410$	256.000	0.0308364
$411$	−6552.00	−0.786341
$412$	3912.00	0.467792
$413$	735.000	0.0875714
$414$	−1044.00	−0.123937
$415$	−420.000	−0.0496795
$416$	−1376.00	−0.162173
$417$	−2724.00	−0.319892
$418$	1914.00	0.223964
$419$	−11443.0	−1.33419	−0.667097	−	0.744971i	$-0.732463\pi$
$419$	−11443.0	−1.33419	−0.667097	+	0.744971i	$0.732463\pi$
$420$	84.0000	0.00975900
$421$	−13325.0	−1.54257	−0.771284	−	0.636492i	$-0.780385\pi$
$421$	−13325.0	−1.54257	−0.771284	+	0.636492i	$0.780385\pi$
$422$	8924.00	1.02942
$423$	−3069.00	−0.352766
$424$	−2736.00	−0.313377
$425$	−12400.0	−1.41527
$426$	−768.000	−0.0873468
$427$	−1330.00	−0.150734
$428$	1924.00	0.217290
$429$	−1419.00	−0.159697
$430$	−916.000	−0.102729
$431$	−8503.00	−0.950290	−0.475145	−	0.879907i	$-0.657604\pi$
$431$	−8503.00	−0.950290	−0.475145	+	0.879907i	$0.657604\pi$
$432$	−432.000	−0.0481125
$433$	4496.00	0.498993	0.249497	−	0.968376i	$-0.419735\pi$
$433$	4496.00	0.498993	0.249497	+	0.968376i	$0.419735\pi$
$434$	−1232.00	−0.136262
$435$	669.000	0.0737381
$436$	−1136.00	−0.124781
$437$	5046.00	0.552364
$438$	966.000	0.105382
$439$	−7247.00	−0.787883	−0.393941	−	0.919136i	$-0.628889\pi$
$439$	−7247.00	−0.787883	−0.393941	+	0.919136i	$0.628889\pi$
$440$	−88.0000	−0.00953463
$441$	441.000	0.0476190
$442$	−8600.00	−0.925476
$443$	8568.00	0.918912	0.459456	−	0.888201i	$-0.348044\pi$
$443$	8568.00	0.918912	0.459456	+	0.888201i	$0.348044\pi$
$444$	−444.000	−0.0474579
$445$	−798.000	−0.0850086
$446$	7604.00	0.807309
$447$	−1263.00	−0.133642
$448$	−448.000	−0.0472456
$449$	−7812.00	−0.821094	−0.410547	−	0.911840i	$-0.634662\pi$
$449$	−7812.00	−0.821094	−0.410547	+	0.911840i	$0.634662\pi$
$450$	−2232.00	−0.233817
$451$	−1408.00	−0.147007
$452$	5992.00	0.623540
$453$	−2844.00	−0.294973
$454$	−6740.00	−0.696749
$455$	301.000	0.0310134
$456$	2088.00	0.214429
$457$	4262.00	0.436254	0.218127	−	0.975920i	$-0.430005\pi$
$457$	4262.00	0.436254	0.218127	+	0.975920i	$0.430005\pi$
$458$	−1796.00	−0.183235
$459$	−2700.00	−0.274565
$460$	−232.000	−0.0235153
$461$	8316.00	0.840162	0.420081	−	0.907487i	$-0.362002\pi$
$461$	8316.00	0.840162	0.420081	+	0.907487i	$0.362002\pi$
$462$	−462.000	−0.0465242
$463$	−16103.0	−1.61635	−0.808175	−	0.588943i	$-0.799544\pi$
$463$	−16103.0	−1.61635	−0.808175	+	0.588943i	$0.799544\pi$
$464$	−3568.00	−0.356983
$465$	−264.000	−0.0263284
$466$	−924.000	−0.0918529
$467$	−8913.00	−0.883179	−0.441589	−	0.897217i	$-0.645585\pi$
$467$	−8913.00	−0.883179	−0.441589	+	0.897217i	$0.645585\pi$
$468$	−1548.00	−0.152898
$469$	4053.00	0.399041
$470$	−682.000	−0.0669326
$471$	−7104.00	−0.694979
$472$	−840.000	−0.0819155
$473$	5038.00	0.489741
$474$	2376.00	0.230239
$475$	10788.0	1.04208
$476$	−2800.00	−0.269617
$477$	−3078.00	−0.295455
$478$	−11330.0	−1.08415
$479$	−4100.00	−0.391093	−0.195547	−	0.980694i	$-0.562648\pi$
$479$	−4100.00	−0.391093	−0.195547	+	0.980694i	$0.562648\pi$
$480$	−96.0000	−0.00912871
$481$	−1591.00	−0.150818
$482$	3186.00	0.301075
$483$	−1218.00	−0.114743
$484$	484.000	0.0454545
$485$	1414.00	0.132384
$486$	−486.000	−0.0453609
$487$	10496.0	0.976631	0.488315	−	0.872667i	$-0.337612\pi$
$487$	10496.0	0.976631	0.488315	+	0.872667i	$0.337612\pi$
$488$	1520.00	0.140998
$489$	7809.00	0.722158
$490$	98.0000	0.00903508
$491$	2625.00	0.241272	0.120636	−	0.992697i	$-0.461507\pi$
$491$	2625.00	0.241272	0.120636	+	0.992697i	$0.461507\pi$
$492$	−1536.00	−0.140748
$493$	−22300.0	−2.03720
$494$	7482.00	0.681439
$495$	−99.0000	−0.00898933
$496$	1408.00	0.127462
$497$	−896.000	−0.0808674
$498$	2520.00	0.226755
$499$	−12397.0	−1.11216	−0.556078	−	0.831130i	$-0.687694\pi$
$499$	−12397.0	−1.11216	−0.556078	+	0.831130i	$0.687694\pi$
$500$	−996.000	−0.0890849
$501$	4206.00	0.375070
$502$	9090.00	0.808180
$503$	−1854.00	−0.164345	−0.0821727	−	0.996618i	$-0.526186\pi$
$503$	−1854.00	−0.164345	−0.0821727	+	0.996618i	$0.526186\pi$
$504$	−504.000	−0.0445435
$505$	−1764.00	−0.155440
$506$	1276.00	0.112105
$507$	1044.00	0.0914510
$508$	−2464.00	−0.215201
$509$	17530.0	1.52653	0.763265	−	0.646086i	$-0.223595\pi$
$509$	17530.0	1.52653	0.763265	+	0.646086i	$0.223595\pi$
$510$	−600.000	−0.0520950
$511$	1127.00	0.0975647
$512$	512.000	0.0441942
$513$	2349.00	0.202165
$514$	−5718.00	−0.490681
$515$	978.000	0.0836812
$516$	5496.00	0.468891
$517$	3751.00	0.319089
$518$	−518.000	−0.0439375
$519$	−2508.00	−0.212117
$520$	−344.000	−0.0290104
$521$	−22255.0	−1.87142	−0.935709	−	0.352772i	$-0.885239\pi$
$521$	−22255.0	−1.87142	−0.935709	+	0.352772i	$0.885239\pi$
$522$	−4014.00	−0.336567
$523$	−7789.00	−0.651222	−0.325611	−	0.945504i	$-0.605570\pi$
$523$	−7789.00	−0.651222	−0.325611	+	0.945504i	$0.605570\pi$
$524$	10992.0	0.916389
$525$	−2604.00	−0.216472
$526$	−12558.0	−1.04098
$527$	8800.00	0.727389
$528$	528.000	0.0435194
$529$	−8803.00	−0.723514
$530$	−684.000	−0.0560586
$531$	−945.000	−0.0772307
$532$	2436.00	0.198523
$533$	−5504.00	−0.447288
$534$	4788.00	0.388009
$535$	481.000	0.0388700
$536$	−4632.00	−0.373269
$537$	9876.00	0.793633
$538$	−12540.0	−1.00490
$539$	−539.000	−0.0430730
$540$	−108.000	−0.00860663
$541$	8288.00	0.658649	0.329324	−	0.944217i	$-0.393179\pi$
$541$	8288.00	0.658649	0.329324	+	0.944217i	$0.393179\pi$
$542$	13442.0	1.06528
$543$	−14514.0	−1.14706
$544$	3200.00	0.252204
$545$	−284.000	−0.0223215
$546$	−1806.00	−0.141556
$547$	−6432.00	−0.502765	−0.251383	−	0.967888i	$-0.580885\pi$
$547$	−6432.00	−0.502765	−0.251383	+	0.967888i	$0.580885\pi$
$548$	8736.00	0.680992
$549$	1710.00	0.132934
$550$	2728.00	0.211495
$551$	19401.0	1.50002
$552$	1392.00	0.107332
$553$	2772.00	0.213160
$554$	−10776.0	−0.826405
$555$	−111.000	−0.00848953
$556$	3632.00	0.277034
$557$	−9815.00	−0.746634	−0.373317	−	0.927704i	$-0.621780\pi$
$557$	−9815.00	−0.746634	−0.373317	+	0.927704i	$0.621780\pi$
$558$	1584.00	0.120172
$559$	19694.0	1.49010
$560$	−112.000	−0.00845154
$561$	3300.00	0.248353
$562$	9530.00	0.715300
$563$	−20042.0	−1.50030	−0.750151	−	0.661267i	$-0.770019\pi$
$563$	−20042.0	−1.50030	−0.750151	+	0.661267i	$0.770019\pi$
$564$	4092.00	0.305504
$565$	1498.00	0.111542
$566$	−1806.00	−0.134120
$567$	−567.000	−0.0419961
$568$	1024.00	0.0756445
$569$	−8562.00	−0.630822	−0.315411	−	0.948955i	$-0.602142\pi$
$569$	−8562.00	−0.630822	−0.315411	+	0.948955i	$0.602142\pi$
$570$	522.000	0.0383582
$571$	27004.0	1.97913	0.989564	−	0.144093i	$-0.0460265\pi$
$571$	27004.0	1.97913	0.989564	+	0.144093i	$0.0460265\pi$
$572$	1892.00	0.138302
$573$	3696.00	0.269464
$574$	−1792.00	−0.130308
$575$	7192.00	0.521612
$576$	576.000	0.0416667
$577$	14360.0	1.03607	0.518037	−	0.855358i	$-0.326663\pi$
$577$	14360.0	1.03607	0.518037	+	0.855358i	$0.326663\pi$
$578$	10174.0	0.732150
$579$	−7098.00	−0.509469
$580$	−892.000	−0.0638591
$581$	2940.00	0.209934
$582$	−8484.00	−0.604249
$583$	3762.00	0.267249
$584$	−1288.00	−0.0912634
$585$	−387.000	−0.0273512
$586$	1344.00	0.0947442
$587$	15453.0	1.08656	0.543282	−	0.839550i	$-0.317181\pi$
$587$	15453.0	1.08656	0.543282	+	0.839550i	$0.317181\pi$
$588$	−588.000	−0.0412393
$589$	−7656.00	−0.535586
$590$	−210.000	−0.0146535
$591$	−954.000	−0.0663999
$592$	592.000	0.0410997
$593$	18000.0	1.24649	0.623247	−	0.782025i	$-0.285813\pi$
$593$	18000.0	1.24649	0.623247	+	0.782025i	$0.285813\pi$
$594$	594.000	0.0410305
$595$	−700.000	−0.0482306
$596$	1684.00	0.115737
$597$	2418.00	0.165766
$598$	4988.00	0.341094
$599$	27180.0	1.85400	0.926999	−	0.375064i	$-0.122379\pi$
$599$	27180.0	1.85400	0.926999	+	0.375064i	$0.122379\pi$
$600$	2976.00	0.202491
$601$	−16153.0	−1.09633	−0.548165	−	0.836370i	$-0.684674\pi$
$601$	−16153.0	−1.09633	−0.548165	+	0.836370i	$0.684674\pi$
$602$	6412.00	0.434109
$603$	−5211.00	−0.351921
$604$	3792.00	0.255454
$605$	121.000	0.00813116
$606$	10584.0	0.709481
$607$	−11443.0	−0.765168	−0.382584	−	0.923921i	$-0.624966\pi$
$607$	−11443.0	−0.765168	−0.382584	+	0.923921i	$0.624966\pi$
$608$	−2784.00	−0.185701
$609$	−4683.00	−0.311601
$610$	380.000	0.0252225
$611$	14663.0	0.970870
$612$	3600.00	0.237780
$613$	−9488.00	−0.625150	−0.312575	−	0.949893i	$-0.601191\pi$
$613$	−9488.00	−0.625150	−0.312575	+	0.949893i	$0.601191\pi$
$614$	−3752.00	−0.246610
$615$	−384.000	−0.0251778
$616$	616.000	0.0402911
$617$	22902.0	1.49433	0.747164	−	0.664640i	$-0.231415\pi$
$617$	22902.0	1.49433	0.747164	+	0.664640i	$0.231415\pi$
$618$	−5868.00	−0.381951
$619$	19448.0	1.26281	0.631406	−	0.775452i	$-0.282478\pi$
$619$	19448.0	1.26281	0.631406	+	0.775452i	$0.282478\pi$
$620$	352.000	0.0228011
$621$	1566.00	0.101194
$622$	−9376.00	−0.604411
$623$	5586.00	0.359227
$624$	2064.00	0.132414
$625$	15251.0	0.976064
$626$	9824.00	0.627230
$627$	−2871.00	−0.182866
$628$	9472.00	0.601869
$629$	3700.00	0.234545
$630$	−126.000	−0.00796819
$631$	−13608.0	−0.858520	−0.429260	−	0.903181i	$-0.641226\pi$
$631$	−13608.0	−0.858520	−0.429260	+	0.903181i	$0.641226\pi$
$632$	−3168.00	−0.199393
$633$	−13386.0	−0.840515
$634$	−1116.00	−0.0699086
$635$	−616.000	−0.0384964
$636$	4104.00	0.255871
$637$	−2107.00	−0.131056
$638$	4906.00	0.304436
$639$	1152.00	0.0713183
$640$	128.000	0.00790569
$641$	17628.0	1.08622	0.543108	−	0.839663i	$-0.317248\pi$
$641$	17628.0	1.08622	0.543108	+	0.839663i	$0.317248\pi$
$642$	−2886.00	−0.177416
$643$	−21074.0	−1.29250	−0.646250	−	0.763126i	$-0.723664\pi$
$643$	−21074.0	−1.29250	−0.646250	+	0.763126i	$0.723664\pi$
$644$	1624.00	0.0993704
$645$	1374.00	0.0838778
$646$	−17400.0	−1.05974
$647$	−4899.00	−0.297681	−0.148840	−	0.988861i	$-0.547554\pi$
$647$	−4899.00	−0.297681	−0.148840	+	0.988861i	$0.547554\pi$
$648$	648.000	0.0392837
$649$	1155.00	0.0698578
$650$	10664.0	0.643502
$651$	1848.00	0.111258
$652$	−10412.0	−0.625407
$653$	7760.00	0.465042	0.232521	−	0.972591i	$-0.425303\pi$
$653$	7760.00	0.465042	0.232521	+	0.972591i	$0.425303\pi$
$654$	1704.00	0.101883
$655$	2748.00	0.163929
$656$	2048.00	0.121892
$657$	−1449.00	−0.0860439
$658$	4774.00	0.282842
$659$	18365.0	1.08558	0.542791	−	0.839868i	$-0.317367\pi$
$659$	18365.0	1.08558	0.542791	+	0.839868i	$0.317367\pi$
$660$	132.000	0.00778499
$661$	13286.0	0.781794	0.390897	−	0.920435i	$-0.372165\pi$
$661$	13286.0	0.781794	0.390897	+	0.920435i	$0.372165\pi$
$662$	9320.00	0.547178
$663$	12900.0	0.755648
$664$	−3360.00	−0.196375
$665$	609.000	0.0355128
$666$	666.000	0.0387492
$667$	12934.0	0.750834
$668$	−5608.00	−0.324820
$669$	−11406.0	−0.659165
$670$	−1158.00	−0.0667723
$671$	−2090.00	−0.120244
$672$	672.000	0.0385758
$673$	−15242.0	−0.873010	−0.436505	−	0.899702i	$-0.643784\pi$
$673$	−15242.0	−0.873010	−0.436505	+	0.899702i	$0.643784\pi$
$674$	−19812.0	−1.13224
$675$	3348.00	0.190910
$676$	−1392.00	−0.0791989
$677$	14170.0	0.804427	0.402214	−	0.915546i	$-0.368241\pi$
$677$	14170.0	0.804427	0.402214	+	0.915546i	$0.368241\pi$
$678$	−8988.00	−0.509118
$679$	−9898.00	−0.559426
$680$	800.000	0.0451156
$681$	10110.0	0.568893
$682$	−1936.00	−0.108700
$683$	−7116.00	−0.398662	−0.199331	−	0.979932i	$-0.563877\pi$
$683$	−7116.00	−0.398662	−0.199331	+	0.979932i	$0.563877\pi$
$684$	−3132.00	−0.175080
$685$	2184.00	0.121819
$686$	−686.000	−0.0381802
$687$	2694.00	0.149611
$688$	−7328.00	−0.406072
$689$	14706.0	0.813141
$690$	348.000	0.0192002
$691$	2010.00	0.110657	0.0553285	−	0.998468i	$-0.482379\pi$
$691$	2010.00	0.110657	0.0553285	+	0.998468i	$0.482379\pi$
$692$	3344.00	0.183699
$693$	693.000	0.0379869
$694$	−21328.0	−1.16657
$695$	908.000	0.0495574
$696$	5352.00	0.291476
$697$	12800.0	0.695602
$698$	10978.0	0.595306
$699$	1386.00	0.0749976
$700$	3472.00	0.187470
$701$	7794.00	0.419936	0.209968	−	0.977708i	$-0.432664\pi$
$701$	7794.00	0.419936	0.209968	+	0.977708i	$0.432664\pi$
$702$	2322.00	0.124841
$703$	−3219.00	−0.172698
$704$	−704.000	−0.0376889
$705$	1023.00	0.0546502
$706$	−3690.00	−0.196707
$707$	12348.0	0.656852
$708$	1260.00	0.0668838
$709$	29991.0	1.58863	0.794313	−	0.607509i	$-0.207831\pi$
$709$	29991.0	1.58863	0.794313	+	0.607509i	$0.207831\pi$
$710$	256.000	0.0135317
$711$	−3564.00	−0.187989
$712$	−6384.00	−0.336026
$713$	−5104.00	−0.268087
$714$	4200.00	0.220142
$715$	473.000	0.0247401
$716$	−13168.0	−0.687306
$717$	16995.0	0.885202
$718$	−1696.00	−0.0881534
$719$	19613.0	1.01730	0.508652	−	0.860972i	$-0.330144\pi$
$719$	19613.0	1.01730	0.508652	+	0.860972i	$0.330144\pi$
$720$	144.000	0.00745356
$721$	−6846.00	−0.353618
$722$	1420.00	0.0731952
$723$	−4779.00	−0.245827
$724$	19352.0	0.993386
$725$	27652.0	1.41651
$726$	−726.000	−0.0371135
$727$	16646.0	0.849197	0.424598	−	0.905382i	$-0.360415\pi$
$727$	16646.0	0.849197	0.424598	+	0.905382i	$0.360415\pi$
$728$	2408.00	0.122591
$729$	729.000	0.0370370
$730$	−322.000	−0.0163257
$731$	−45800.0	−2.31734
$732$	−2280.00	−0.115125
$733$	−36346.0	−1.83147	−0.915737	−	0.401779i	$-0.868392\pi$
$733$	−36346.0	−1.83147	−0.915737	+	0.401779i	$0.868392\pi$
$734$	−16948.0	−0.852264
$735$	−147.000	−0.00737711
$736$	−1856.00	−0.0929525
$737$	6369.00	0.318324
$738$	2304.00	0.114921
$739$	−25698.0	−1.27918	−0.639591	−	0.768715i	$-0.720896\pi$
$739$	−25698.0	−1.27918	−0.639591	+	0.768715i	$0.720896\pi$
$740$	148.000	0.00735215
$741$	−11223.0	−0.556393
$742$	4788.00	0.236891
$743$	−27543.0	−1.35997	−0.679983	−	0.733228i	$-0.738013\pi$
$743$	−27543.0	−1.35997	−0.679983	+	0.733228i	$0.738013\pi$
$744$	−2112.00	−0.104072
$745$	421.000	0.0207037
$746$	22792.0	1.11860
$747$	−3780.00	−0.185145
$748$	−4400.00	−0.215080
$749$	−3367.00	−0.164256
$750$	1494.00	0.0727376
$751$	26005.0	1.26356	0.631782	−	0.775146i	$-0.282324\pi$
$751$	26005.0	1.26356	0.631782	+	0.775146i	$0.282324\pi$
$752$	−5456.00	−0.264574
$753$	−13635.0	−0.659877
$754$	19178.0	0.926289
$755$	948.000	0.0456970
$756$	756.000	0.0363696
$757$	17621.0	0.846032	0.423016	−	0.906122i	$-0.360971\pi$
$757$	17621.0	0.846032	0.423016	+	0.906122i	$0.360971\pi$
$758$	4310.00	0.206525
$759$	−1914.00	−0.0915333
$760$	−696.000	−0.0332192
$761$	22968.0	1.09407	0.547036	−	0.837109i	$-0.315756\pi$
$761$	22968.0	1.09407	0.547036	+	0.837109i	$0.315756\pi$
$762$	3696.00	0.175711
$763$	1988.00	0.0943256
$764$	−4928.00	−0.233362
$765$	900.000	0.0425354
$766$	−11728.0	−0.553198
$767$	4515.00	0.212552
$768$	−768.000	−0.0360844
$769$	28273.0	1.32581	0.662907	−	0.748702i	$-0.269323\pi$
$769$	28273.0	1.32581	0.662907	+	0.748702i	$0.269323\pi$
$770$	154.000	0.00720750
$771$	8577.00	0.400640
$772$	9464.00	0.441213
$773$	−13239.0	−0.616007	−0.308004	−	0.951385i	$-0.599661\pi$
$773$	−13239.0	−0.616007	−0.308004	+	0.951385i	$0.599661\pi$
$774$	−8244.00	−0.382848
$775$	−10912.0	−0.505769
$776$	11312.0	0.523295
$777$	777.000	0.0358748
$778$	20208.0	0.931224
$779$	−11136.0	−0.512180
$780$	516.000	0.0236869
$781$	−1408.00	−0.0645099
$782$	−11600.0	−0.530454
$783$	6021.00	0.274806
$784$	784.000	0.0357143
$785$	2368.00	0.107666
$786$	−16488.0	−0.748228
$787$	1775.00	0.0803963	0.0401982	−	0.999192i	$-0.487201\pi$
$787$	1775.00	0.0803963	0.0401982	+	0.999192i	$0.487201\pi$
$788$	1272.00	0.0575040
$789$	18837.0	0.849956
$790$	−792.000	−0.0356685
$791$	−10486.0	−0.471352
$792$	−792.000	−0.0355335
$793$	−8170.00	−0.365858
$794$	−11736.0	−0.524553
$795$	1026.00	0.0457717
$796$	−3224.00	−0.143557
$797$	−16481.0	−0.732481	−0.366240	−	0.930520i	$-0.619355\pi$
$797$	−16481.0	−0.732481	−0.366240	+	0.930520i	$0.619355\pi$
$798$	−3654.00	−0.162093
$799$	−34100.0	−1.50985
$800$	−3968.00	−0.175362
$801$	−7182.00	−0.316808
$802$	−4836.00	−0.212924
$803$	1771.00	0.0778297
$804$	6948.00	0.304772
$805$	406.000	0.0177759
$806$	−7568.00	−0.330734
$807$	18810.0	0.820500
$808$	−14112.0	−0.614429
$809$	−29079.0	−1.26374	−0.631868	−	0.775076i	$-0.717712\pi$
$809$	−29079.0	−1.26374	−0.631868	+	0.775076i	$0.717712\pi$
$810$	162.000	0.00702728
$811$	−40381.0	−1.74842	−0.874210	−	0.485548i	$-0.838620\pi$
$811$	−40381.0	−1.74842	−0.874210	+	0.485548i	$0.838620\pi$
$812$	6244.00	0.269854
$813$	−20163.0	−0.869800
$814$	−814.000	−0.0350500
$815$	−2603.00	−0.111876
$816$	−4800.00	−0.205924
$817$	39846.0	1.70629
$818$	−4708.00	−0.201236
$819$	2709.00	0.115580
$820$	512.000	0.0218047
$821$	−6345.00	−0.269722	−0.134861	−	0.990865i	$-0.543059\pi$
$821$	−6345.00	−0.269722	−0.134861	+	0.990865i	$0.543059\pi$
$822$	−13104.0	−0.556027
$823$	7757.00	0.328544	0.164272	−	0.986415i	$-0.447472\pi$
$823$	7757.00	0.328544	0.164272	+	0.986415i	$0.447472\pi$
$824$	7824.00	0.330779
$825$	−4092.00	−0.172685
$826$	1470.00	0.0619223
$827$	−15659.0	−0.658424	−0.329212	−	0.944256i	$-0.606783\pi$
$827$	−15659.0	−0.658424	−0.329212	+	0.944256i	$0.606783\pi$
$828$	−2088.00	−0.0876365
$829$	−5384.00	−0.225566	−0.112783	−	0.993620i	$-0.535976\pi$
$829$	−5384.00	−0.225566	−0.112783	+	0.993620i	$0.535976\pi$
$830$	−840.000	−0.0351287
$831$	16164.0	0.674757
$832$	−2752.00	−0.114674
$833$	4900.00	0.203811
$834$	−5448.00	−0.226198
$835$	−1402.00	−0.0581056
$836$	3828.00	0.158366
$837$	−2376.00	−0.0981202
$838$	−22886.0	−0.943417
$839$	40395.0	1.66221	0.831103	−	0.556119i	$-0.187710\pi$
$839$	40395.0	1.66221	0.831103	+	0.556119i	$0.187710\pi$
$840$	168.000	0.00690066
$841$	25340.0	1.03899
$842$	−26650.0	−1.09076
$843$	−14295.0	−0.584040
$844$	17848.0	0.727907
$845$	−348.000	−0.0141675
$846$	−6138.00	−0.249443
$847$	−847.000	−0.0343604
$848$	−5472.00	−0.221591
$849$	2709.00	0.109508
$850$	−24800.0	−1.00074
$851$	−2146.00	−0.0864441
$852$	−1536.00	−0.0617635
$853$	−33202.0	−1.33273	−0.666363	−	0.745628i	$-0.732150\pi$
$853$	−33202.0	−1.33273	−0.666363	+	0.745628i	$0.732150\pi$
$854$	−2660.00	−0.106585
$855$	−783.000	−0.0313193
$856$	3848.00	0.153647
$857$	21034.0	0.838399	0.419199	−	0.907894i	$-0.362311\pi$
$857$	21034.0	0.838399	0.419199	+	0.907894i	$0.362311\pi$
$858$	−2838.00	−0.112923
$859$	20920.0	0.830944	0.415472	−	0.909606i	$-0.363616\pi$
$859$	20920.0	0.830944	0.415472	+	0.909606i	$0.363616\pi$
$860$	−1832.00	−0.0726403
$861$	2688.00	0.106396
$862$	−17006.0	−0.671957
$863$	8354.00	0.329517	0.164759	−	0.986334i	$-0.447315\pi$
$863$	8354.00	0.329517	0.164759	+	0.986334i	$0.447315\pi$
$864$	−864.000	−0.0340207
$865$	836.000	0.0328611
$866$	8992.00	0.352841
$867$	−15261.0	−0.597798
$868$	−2464.00	−0.0963521
$869$	4356.00	0.170043
$870$	1338.00	0.0521407
$871$	24897.0	0.968545
$872$	−2272.00	−0.0882335
$873$	12726.0	0.493368
$874$	10092.0	0.390580
$875$	1743.00	0.0673419
$876$	1932.00	0.0745162
$877$	19564.0	0.753283	0.376642	−	0.926359i	$-0.377079\pi$
$877$	19564.0	0.753283	0.376642	+	0.926359i	$0.377079\pi$
$878$	−14494.0	−0.557117
$879$	−2016.00	−0.0773584
$880$	−176.000	−0.00674200
$881$	32061.0	1.22606	0.613032	−	0.790058i	$-0.289950\pi$
$881$	32061.0	1.22606	0.613032	+	0.790058i	$0.289950\pi$
$882$	882.000	0.0336718
$883$	46711.0	1.78024	0.890119	−	0.455728i	$-0.150621\pi$
$883$	46711.0	1.78024	0.890119	+	0.455728i	$0.150621\pi$
$884$	−17200.0	−0.654410
$885$	315.000	0.0119645
$886$	17136.0	0.649769
$887$	−7120.00	−0.269522	−0.134761	−	0.990878i	$-0.543027\pi$
$887$	−7120.00	−0.269522	−0.134761	+	0.990878i	$0.543027\pi$
$888$	−888.000	−0.0335578
$889$	4312.00	0.162677
$890$	−1596.00	−0.0601102
$891$	−891.000	−0.0335013
$892$	15208.0	0.570854
$893$	29667.0	1.11172
$894$	−2526.00	−0.0944990
$895$	−3292.00	−0.122949
$896$	−896.000	−0.0334077
$897$	−7482.00	−0.278502
$898$	−15624.0	−0.580601
$899$	−19624.0	−0.728028
$900$	−4464.00	−0.165333
$901$	−34200.0	−1.26456
$902$	−2816.00	−0.103950
$903$	−9618.00	−0.354449
$904$	11984.0	0.440909
$905$	4838.00	0.177702
$906$	−5688.00	−0.208577
$907$	−21940.0	−0.803204	−0.401602	−	0.915814i	$-0.631546\pi$
$907$	−21940.0	−0.803204	−0.401602	+	0.915814i	$0.631546\pi$
$908$	−13480.0	−0.492676
$909$	−15876.0	−0.579289
$910$	602.000	0.0219298
$911$	66.0000	0.00240030	0.00120015	−	0.999999i	$-0.499618\pi$
$911$	66.0000	0.00240030	0.00120015	+	0.999999i	$0.499618\pi$
$912$	4176.00	0.151624
$913$	4620.00	0.167470
$914$	8524.00	0.308478
$915$	−570.000	−0.0205941
$916$	−3592.00	−0.129567
$917$	−19236.0	−0.692725
$918$	−5400.00	−0.194147
$919$	−42594.0	−1.52889	−0.764443	−	0.644691i	$-0.776986\pi$
$919$	−42594.0	−1.52889	−0.764443	+	0.644691i	$0.776986\pi$
$920$	−464.000	−0.0166279
$921$	5628.00	0.201356
$922$	16632.0	0.594084
$923$	−5504.00	−0.196280
$924$	−924.000	−0.0328976
$925$	−4588.00	−0.163084
$926$	−32206.0	−1.14293
$927$	8802.00	0.311861
$928$	−7136.00	−0.252425
$929$	−6609.00	−0.233406	−0.116703	−	0.993167i	$-0.537233\pi$
$929$	−6609.00	−0.233406	−0.116703	+	0.993167i	$0.537233\pi$
$930$	−528.000	−0.0186170
$931$	−4263.00	−0.150069
$932$	−1848.00	−0.0649498
$933$	14064.0	0.493499
$934$	−17826.0	−0.624502
$935$	−1100.00	−0.0384747
$936$	−3096.00	−0.108115
$937$	10098.0	0.352068	0.176034	−	0.984384i	$-0.443673\pi$
$937$	10098.0	0.352068	0.176034	+	0.984384i	$0.443673\pi$
$938$	8106.00	0.282164
$939$	−14736.0	−0.512131
$940$	−1364.00	−0.0473285
$941$	−41824.0	−1.44891	−0.724455	−	0.689323i	$-0.757908\pi$
$941$	−41824.0	−1.44891	−0.724455	+	0.689323i	$0.757908\pi$
$942$	−14208.0	−0.491424
$943$	−7424.00	−0.256372
$944$	−1680.00	−0.0579230
$945$	189.000	0.00650600
$946$	10076.0	0.346299
$947$	−28838.0	−0.989556	−0.494778	−	0.869020i	$-0.664751\pi$
$947$	−28838.0	−0.989556	−0.494778	+	0.869020i	$0.664751\pi$
$948$	4752.00	0.162804
$949$	6923.00	0.236807
$950$	21576.0	0.736861
$951$	1674.00	0.0570801
$952$	−5600.00	−0.190648
$953$	16165.0	0.549460	0.274730	−	0.961521i	$-0.411411\pi$
$953$	16165.0	0.549460	0.274730	+	0.961521i	$0.411411\pi$
$954$	−6156.00	−0.208918
$955$	−1232.00	−0.0417451
$956$	−22660.0	−0.766608
$957$	−7359.00	−0.248571
$958$	−8200.00	−0.276545
$959$	−15288.0	−0.514781
$960$	−192.000	−0.00645497
$961$	−22047.0	−0.740056
$962$	−3182.00	−0.106644
$963$	4329.00	0.144860
$964$	6372.00	0.212892
$965$	2366.00	0.0789267
$966$	−2436.00	−0.0811356
$967$	−40114.0	−1.33400	−0.667001	−	0.745057i	$-0.732422\pi$
$967$	−40114.0	−1.33400	−0.667001	+	0.745057i	$0.732422\pi$
$968$	968.000	0.0321412
$969$	26100.0	0.865276
$970$	2828.00	0.0936099
$971$	−6515.00	−0.215321	−0.107660	−	0.994188i	$-0.534336\pi$
$971$	−6515.00	−0.215321	−0.107660	+	0.994188i	$0.534336\pi$
$972$	−972.000	−0.0320750
$973$	−6356.00	−0.209418
$974$	20992.0	0.690582
$975$	−15996.0	−0.525417
$976$	3040.00	0.0997008
$977$	12790.0	0.418821	0.209411	−	0.977828i	$-0.432845\pi$
$977$	12790.0	0.418821	0.209411	+	0.977828i	$0.432845\pi$
$978$	15618.0	0.510643
$979$	8778.00	0.286564
$980$	196.000	0.00638877
$981$	−2556.00	−0.0831874
$982$	5250.00	0.170605
$983$	4788.00	0.155355	0.0776773	−	0.996979i	$-0.475250\pi$
$983$	4788.00	0.155355	0.0776773	+	0.996979i	$0.475250\pi$
$984$	−3072.00	−0.0995242
$985$	318.000	0.0102866
$986$	−44600.0	−1.44052
$987$	−7161.00	−0.230939
$988$	14964.0	0.481850
$989$	26564.0	0.854081
$990$	−198.000	−0.00635642
$991$	26989.0	0.865120	0.432560	−	0.901605i	$-0.357610\pi$
$991$	26989.0	0.865120	0.432560	+	0.901605i	$0.357610\pi$
$992$	2816.00	0.0901291
$993$	−13980.0	−0.446769
$994$	−1792.00	−0.0571819
$995$	−806.000	−0.0256803
$996$	5040.00	0.160340
$997$	−55518.0	−1.76356	−0.881782	−	0.471658i	$-0.843656\pi$
$997$	−55518.0	−1.76356	−0.881782	+	0.471658i	$0.843656\pi$
$998$	−24794.0	−0.786413
$999$	−999.000	−0.0316386

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
462.4.a.f.1.1	✓	1	1.1	even	1	trivial
1386.4.a.d.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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