LMFDB - Embedded newform 462.6.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [462,6,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, 6, names="a")

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

chi := DirichletCharacter("462.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 462 = 2 \cdot 3 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
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:	$74.0973247536$
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)

$f(q)$

$=$

$q+4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} -6.00000 q^{5} -36.0000 q^{6} -49.0000 q^{7} +64.0000 q^{8} +81.0000 q^{9} +O(q^{10})$

\(q+4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} -6.00000 q^{5} -36.0000 q^{6} -49.0000 q^{7} +64.0000 q^{8} +81.0000 q^{9} -24.0000 q^{10} +121.000 q^{11} -144.000 q^{12} +46.0000 q^{13} -196.000 q^{14} +54.0000 q^{15} +256.000 q^{16} -170.000 q^{17} +324.000 q^{18} -1224.00 q^{19} -96.0000 q^{20} +441.000 q^{21} +484.000 q^{22} +2296.00 q^{23} -576.000 q^{24} -3089.00 q^{25} +184.000 q^{26} -729.000 q^{27} -784.000 q^{28} -1154.00 q^{29} +216.000 q^{30} +9636.00 q^{31} +1024.00 q^{32} -1089.00 q^{33} -680.000 q^{34} +294.000 q^{35} +1296.00 q^{36} -8322.00 q^{37} -4896.00 q^{38} -414.000 q^{39} -384.000 q^{40} +3246.00 q^{41} +1764.00 q^{42} -10652.0 q^{43} +1936.00 q^{44} -486.000 q^{45} +9184.00 q^{46} +2860.00 q^{47} -2304.00 q^{48} +2401.00 q^{49} -12356.0 q^{50} +1530.00 q^{51} +736.000 q^{52} -28554.0 q^{53} -2916.00 q^{54} -726.000 q^{55} -3136.00 q^{56} +11016.0 q^{57} -4616.00 q^{58} -1300.00 q^{59} +864.000 q^{60} -30210.0 q^{61} +38544.0 q^{62} -3969.00 q^{63} +4096.00 q^{64} -276.000 q^{65} -4356.00 q^{66} -67228.0 q^{67} -2720.00 q^{68} -20664.0 q^{69} +1176.00 q^{70} -45648.0 q^{71} +5184.00 q^{72} +21390.0 q^{73} -33288.0 q^{74} +27801.0 q^{75} -19584.0 q^{76} -5929.00 q^{77} -1656.00 q^{78} -8184.00 q^{79} -1536.00 q^{80} +6561.00 q^{81} +12984.0 q^{82} -47048.0 q^{83} +7056.00 q^{84} +1020.00 q^{85} -42608.0 q^{86} +10386.0 q^{87} +7744.00 q^{88} +126890. q^{89} -1944.00 q^{90} -2254.00 q^{91} +36736.0 q^{92} -86724.0 q^{93} +11440.0 q^{94} +7344.00 q^{95} -9216.00 q^{96} -46718.0 q^{97} +9604.00 q^{98} +9801.00 q^{99} +O(q^{100})\)

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
$5$	−6.00000	−0.107331	−0.0536656	−	0.998559i	$-0.517091\pi$
$5$	−6.00000	−0.107331	−0.0536656	+	0.998559i	$0.517091\pi$
$6$	−36.0000	−0.408248
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	64.0000	0.353553
$9$	81.0000	0.333333
$10$	−24.0000	−0.0758947
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	−144.000	−0.288675
$13$	46.0000	0.0754917	0.0377459	−	0.999287i	$-0.487982\pi$
$13$	46.0000	0.0754917	0.0377459	+	0.999287i	$0.487982\pi$
$14$	−196.000	−0.267261
$15$	54.0000	0.0619677
$16$	256.000	0.250000
$17$	−170.000	−0.142668	−0.0713340	−	0.997452i	$-0.522726\pi$
$17$	−170.000	−0.142668	−0.0713340	+	0.997452i	$0.522726\pi$
$18$	324.000	0.235702
$19$	−1224.00	−0.777853	−0.388926	−	0.921269i	$-0.627154\pi$
$19$	−1224.00	−0.777853	−0.388926	+	0.921269i	$0.627154\pi$
$20$	−96.0000	−0.0536656
$21$	441.000	0.218218
$22$	484.000	0.213201
$23$	2296.00	0.905008	0.452504	−	0.891762i	$-0.350531\pi$
$23$	2296.00	0.905008	0.452504	+	0.891762i	$0.350531\pi$
$24$	−576.000	−0.204124
$25$	−3089.00	−0.988480
$26$	184.000	0.0533807
$27$	−729.000	−0.192450
$28$	−784.000	−0.188982
$29$	−1154.00	−0.254807	−0.127403	−	0.991851i	$-0.540664\pi$
$29$	−1154.00	−0.254807	−0.127403	+	0.991851i	$0.540664\pi$
$30$	216.000	0.0438178
$31$	9636.00	1.80091	0.900456	−	0.434947i	$-0.143233\pi$
$31$	9636.00	1.80091	0.900456	+	0.434947i	$0.143233\pi$
$32$	1024.00	0.176777
$33$	−1089.00	−0.174078
$34$	−680.000	−0.100882
$35$	294.000	0.0405674
$36$	1296.00	0.166667
$37$	−8322.00	−0.999363	−0.499682	−	0.866209i	$-0.666550\pi$
$37$	−8322.00	−0.999363	−0.499682	+	0.866209i	$0.666550\pi$
$38$	−4896.00	−0.550025
$39$	−414.000	−0.0435852
$40$	−384.000	−0.0379473
$41$	3246.00	0.301571	0.150785	−	0.988567i	$-0.451820\pi$
$41$	3246.00	0.301571	0.150785	+	0.988567i	$0.451820\pi$
$42$	1764.00	0.154303
$43$	−10652.0	−0.878537	−0.439268	−	0.898356i	$-0.644762\pi$
$43$	−10652.0	−0.878537	−0.439268	+	0.898356i	$0.644762\pi$
$44$	1936.00	0.150756
$45$	−486.000	−0.0357771
$46$	9184.00	0.639937
$47$	2860.00	0.188852	0.0944260	−	0.995532i	$-0.469898\pi$
$47$	2860.00	0.188852	0.0944260	+	0.995532i	$0.469898\pi$
$48$	−2304.00	−0.144338
$49$	2401.00	0.142857
$50$	−12356.0	−0.698961
$51$	1530.00	0.0823694
$52$	736.000	0.0377459
$53$	−28554.0	−1.39630	−0.698148	−	0.715954i	$-0.745992\pi$
$53$	−28554.0	−1.39630	−0.698148	+	0.715954i	$0.745992\pi$
$54$	−2916.00	−0.136083
$55$	−726.000	−0.0323616
$56$	−3136.00	−0.133631
$57$	11016.0	0.449094
$58$	−4616.00	−0.180176
$59$	−1300.00	−0.0486198	−0.0243099	−	0.999704i	$-0.507739\pi$
$59$	−1300.00	−0.0486198	−0.0243099	+	0.999704i	$0.507739\pi$
$60$	864.000	0.0309839
$61$	−30210.0	−1.03950	−0.519752	−	0.854317i	$-0.673976\pi$
$61$	−30210.0	−1.03950	−0.519752	+	0.854317i	$0.673976\pi$
$62$	38544.0	1.27344
$63$	−3969.00	−0.125988
$64$	4096.00	0.125000
$65$	−276.000	−0.00810262
$66$	−4356.00	−0.123091
$67$	−67228.0	−1.82963	−0.914815	−	0.403874i	$-0.867664\pi$
$67$	−67228.0	−1.82963	−0.914815	+	0.403874i	$0.867664\pi$
$68$	−2720.00	−0.0713340
$69$	−20664.0	−0.522506
$70$	1176.00	0.0286855
$71$	−45648.0	−1.07467	−0.537336	−	0.843368i	$-0.680569\pi$
$71$	−45648.0	−1.07467	−0.537336	+	0.843368i	$0.680569\pi$
$72$	5184.00	0.117851
$73$	21390.0	0.469790	0.234895	−	0.972021i	$-0.424525\pi$
$73$	21390.0	0.469790	0.234895	+	0.972021i	$0.424525\pi$
$74$	−33288.0	−0.706657
$75$	27801.0	0.570699
$76$	−19584.0	−0.388926
$77$	−5929.00	−0.113961
$78$	−1656.00	−0.0308194
$79$	−8184.00	−0.147536	−0.0737680	−	0.997275i	$-0.523502\pi$
$79$	−8184.00	−0.147536	−0.0737680	+	0.997275i	$0.523502\pi$
$80$	−1536.00	−0.0268328
$81$	6561.00	0.111111
$82$	12984.0	0.213243
$83$	−47048.0	−0.749628	−0.374814	−	0.927100i	$-0.622293\pi$
$83$	−47048.0	−0.749628	−0.374814	+	0.927100i	$0.622293\pi$
$84$	7056.00	0.109109
$85$	1020.00	0.0153127
$86$	−42608.0	−0.621219
$87$	10386.0	0.147113
$88$	7744.00	0.106600
$89$	126890.	1.69806	0.849029	−	0.528347i	$-0.177188\pi$
$89$	126890.	1.69806	0.849029	+	0.528347i	$0.177188\pi$
$90$	−1944.00	−0.0252982
$91$	−2254.00	−0.0285332
$92$	36736.0	0.452504
$93$	−86724.0	−1.03976
$94$	11440.0	0.133538
$95$	7344.00	0.0834879
$96$	−9216.00	−0.102062
$97$	−46718.0	−0.504144	−0.252072	−	0.967708i	$-0.581112\pi$
$97$	−46718.0	−0.504144	−0.252072	+	0.967708i	$0.581112\pi$
$98$	9604.00	0.101015
$99$	9801.00	0.100504
$100$	−49424.0	−0.494240
$101$	−70914.0	−0.691717	−0.345859	−	0.938287i	$-0.612412\pi$
$101$	−70914.0	−0.691717	−0.345859	+	0.938287i	$0.612412\pi$
$102$	6120.00	0.0582440
$103$	−137372.	−1.27587	−0.637933	−	0.770092i	$-0.720210\pi$
$103$	−137372.	−1.27587	−0.637933	+	0.770092i	$0.720210\pi$
$104$	2944.00	0.0266904
$105$	−2646.00	−0.0234216
$106$	−114216.	−0.987330
$107$	−141308.	−1.19318	−0.596592	−	0.802545i	$-0.703479\pi$
$107$	−141308.	−1.19318	−0.596592	+	0.802545i	$0.703479\pi$
$108$	−11664.0	−0.0962250
$109$	142206.	1.14644	0.573220	−	0.819401i	$-0.305694\pi$
$109$	142206.	1.14644	0.573220	+	0.819401i	$0.305694\pi$
$110$	−2904.00	−0.0228831
$111$	74898.0	0.576983
$112$	−12544.0	−0.0944911
$113$	−157822.	−1.16271	−0.581355	−	0.813650i	$-0.697477\pi$
$113$	−157822.	−1.16271	−0.581355	+	0.813650i	$0.697477\pi$
$114$	44064.0	0.317557
$115$	−13776.0	−0.0971356
$116$	−18464.0	−0.127403
$117$	3726.00	0.0251639
$118$	−5200.00	−0.0343794
$119$	8330.00	0.0539234
$120$	3456.00	0.0219089
$121$	14641.0	0.0909091
$122$	−120840.	−0.735040
$123$	−29214.0	−0.174112
$124$	154176.	0.900456
$125$	37284.0	0.213426
$126$	−15876.0	−0.0890871
$127$	228008.	1.25441	0.627207	−	0.778853i	$-0.284198\pi$
$127$	228008.	1.25441	0.627207	+	0.778853i	$0.284198\pi$
$128$	16384.0	0.0883883
$129$	95868.0	0.507224
$130$	−1104.00	−0.00572942
$131$	−338512.	−1.72344	−0.861719	−	0.507385i	$-0.830612\pi$
$131$	−338512.	−1.72344	−0.861719	+	0.507385i	$0.830612\pi$
$132$	−17424.0	−0.0870388
$133$	59976.0	0.294001
$134$	−268912.	−1.29374
$135$	4374.00	0.0206559
$136$	−10880.0	−0.0504408
$137$	−266918.	−1.21500	−0.607500	−	0.794319i	$-0.707828\pi$
$137$	−266918.	−1.21500	−0.607500	+	0.794319i	$0.707828\pi$
$138$	−82656.0	−0.369468
$139$	−393032.	−1.72540	−0.862702	−	0.505712i	$-0.831230\pi$
$139$	−393032.	−1.72540	−0.862702	+	0.505712i	$0.831230\pi$
$140$	4704.00	0.0202837
$141$	−25740.0	−0.109034
$142$	−182592.	−0.759908
$143$	5566.00	0.0227616
$144$	20736.0	0.0833333
$145$	6924.00	0.0273487
$146$	85560.0	0.332192
$147$	−21609.0	−0.0824786
$148$	−133152.	−0.499682
$149$	124934.	0.461015	0.230507	−	0.973071i	$-0.425961\pi$
$149$	124934.	0.461015	0.230507	+	0.973071i	$0.425961\pi$
$150$	111204.	0.403545
$151$	−151960.	−0.542359	−0.271180	−	0.962529i	$-0.587414\pi$
$151$	−151960.	−0.542359	−0.271180	+	0.962529i	$0.587414\pi$
$152$	−78336.0	−0.275012
$153$	−13770.0	−0.0475560
$154$	−23716.0	−0.0805823
$155$	−57816.0	−0.193294
$156$	−6624.00	−0.0217926
$157$	−491526.	−1.59147	−0.795733	−	0.605648i	$-0.792914\pi$
$157$	−491526.	−1.59147	−0.795733	+	0.605648i	$0.792914\pi$
$158$	−32736.0	−0.104324
$159$	256986.	0.806151
$160$	−6144.00	−0.0189737
$161$	−112504.	−0.342061
$162$	26244.0	0.0785674
$163$	−60948.0	−0.179676	−0.0898381	−	0.995956i	$-0.528635\pi$
$163$	−60948.0	−0.179676	−0.0898381	+	0.995956i	$0.528635\pi$
$164$	51936.0	0.150785
$165$	6534.00	0.0186840
$166$	−188192.	−0.530067
$167$	557184.	1.54599	0.772996	−	0.634410i	$-0.218757\pi$
$167$	557184.	1.54599	0.772996	+	0.634410i	$0.218757\pi$
$168$	28224.0	0.0771517
$169$	−369177.	−0.994301
$170$	4080.00	0.0108277
$171$	−99144.0	−0.259284
$172$	−170432.	−0.439268
$173$	13134.0	0.0333643	0.0166821	−	0.999861i	$-0.494690\pi$
$173$	13134.0	0.0333643	0.0166821	+	0.999861i	$0.494690\pi$
$174$	41544.0	0.104024
$175$	151361.	0.373610
$176$	30976.0	0.0753778
$177$	11700.0	0.0280707
$178$	507560.	1.20071
$179$	507388.	1.18361	0.591804	−	0.806082i	$-0.298416\pi$
$179$	507388.	1.18361	0.591804	+	0.806082i	$0.298416\pi$
$180$	−7776.00	−0.0178885
$181$	580570.	1.31722	0.658610	−	0.752485i	$-0.271145\pi$
$181$	580570.	1.31722	0.658610	+	0.752485i	$0.271145\pi$
$182$	−9016.00	−0.0201760
$183$	271890.	0.600158
$184$	146944.	0.319969
$185$	49932.0	0.107263
$186$	−346896.	−0.735219
$187$	−20570.0	−0.0430160
$188$	45760.0	0.0944260
$189$	35721.0	0.0727393
$190$	29376.0	0.0590349
$191$	−868560.	−1.72273	−0.861363	−	0.507989i	$-0.830389\pi$
$191$	−868560.	−1.72273	−0.861363	+	0.507989i	$0.830389\pi$
$192$	−36864.0	−0.0721688
$193$	−194782.	−0.376405	−0.188203	−	0.982130i	$-0.560266\pi$
$193$	−194782.	−0.376405	−0.188203	+	0.982130i	$0.560266\pi$
$194$	−186872.	−0.356484
$195$	2484.00	0.00467805
$196$	38416.0	0.0714286
$197$	727926.	1.33635	0.668177	−	0.744002i	$-0.267075\pi$
$197$	727926.	1.33635	0.668177	+	0.744002i	$0.267075\pi$
$198$	39204.0	0.0710669
$199$	99316.0	0.177781	0.0888907	−	0.996041i	$-0.471668\pi$
$199$	99316.0	0.177781	0.0888907	+	0.996041i	$0.471668\pi$
$200$	−197696.	−0.349480
$201$	605052.	1.05634
$202$	−283656.	−0.489118
$203$	56546.0	0.0963079
$204$	24480.0	0.0411847
$205$	−19476.0	−0.0323679
$206$	−549488.	−0.902174
$207$	185976.	0.301669
$208$	11776.0	0.0188729
$209$	−148104.	−0.234531
$210$	−10584.0	−0.0165616
$211$	43420.0	0.0671404	0.0335702	−	0.999436i	$-0.489312\pi$
$211$	43420.0	0.0671404	0.0335702	+	0.999436i	$0.489312\pi$
$212$	−456864.	−0.698148
$213$	410832.	0.620462
$214$	−565232.	−0.843708
$215$	63912.0	0.0942945
$216$	−46656.0	−0.0680414
$217$	−472164.	−0.680681
$218$	568824.	0.810656
$219$	−192510.	−0.271233
$220$	−11616.0	−0.0161808
$221$	−7820.00	−0.0107703
$222$	299592.	0.407988
$223$	836164.	1.12598	0.562988	−	0.826465i	$-0.309652\pi$
$223$	836164.	1.12598	0.562988	+	0.826465i	$0.309652\pi$
$224$	−50176.0	−0.0668153
$225$	−250209.	−0.329493
$226$	−631288.	−0.822160
$227$	952352.	1.22668	0.613342	−	0.789817i	$-0.289825\pi$
$227$	952352.	1.22668	0.613342	+	0.789817i	$0.289825\pi$
$228$	176256.	0.224547
$229$	960138.	1.20989	0.604944	−	0.796268i	$-0.293196\pi$
$229$	960138.	1.20989	0.604944	+	0.796268i	$0.293196\pi$
$230$	−55104.0	−0.0686853
$231$	53361.0	0.0657952
$232$	−73856.0	−0.0900878
$233$	1.05519e6	1.27332	0.636662	−	0.771143i	$-0.280314\pi$
$233$	1.05519e6	1.27332	0.636662	+	0.771143i	$0.280314\pi$
$234$	14904.0	0.0177936
$235$	−17160.0	−0.0202697
$236$	−20800.0	−0.0243099
$237$	73656.0	0.0851799
$238$	33320.0	0.0381296
$239$	−1.05556e6	−1.19533	−0.597665	−	0.801746i	$-0.703905\pi$
$239$	−1.05556e6	−1.19533	−0.597665	+	0.801746i	$0.703905\pi$
$240$	13824.0	0.0154919
$241$	87782.0	0.0973560	0.0486780	−	0.998815i	$-0.484499\pi$
$241$	87782.0	0.0973560	0.0486780	+	0.998815i	$0.484499\pi$
$242$	58564.0	0.0642824
$243$	−59049.0	−0.0641500
$244$	−483360.	−0.519752
$245$	−14406.0	−0.0153330
$246$	−116856.	−0.123116
$247$	−56304.0	−0.0587215
$248$	616704.	0.636719
$249$	423432.	0.432798
$250$	149136.	0.150915
$251$	−1.42596e6	−1.42864	−0.714318	−	0.699821i	$-0.753263\pi$
$251$	−1.42596e6	−1.42864	−0.714318	+	0.699821i	$0.753263\pi$
$252$	−63504.0	−0.0629941
$253$	277816.	0.272870
$254$	912032.	0.887004
$255$	−9180.00	−0.00884081
$256$	65536.0	0.0625000
$257$	1.55138e6	1.46516	0.732580	−	0.680681i	$-0.238316\pi$
$257$	1.55138e6	1.46516	0.732580	+	0.680681i	$0.238316\pi$
$258$	383472.	0.358661
$259$	407778.	0.377724
$260$	−4416.00	−0.00405131
$261$	−93474.0	−0.0849356
$262$	−1.35405e6	−1.21866
$263$	−1.21161e6	−1.08012	−0.540061	−	0.841626i	$-0.681599\pi$
$263$	−1.21161e6	−1.08012	−0.540061	+	0.841626i	$0.681599\pi$
$264$	−69696.0	−0.0615457
$265$	171324.	0.149866
$266$	239904.	0.207890
$267$	−1.14201e6	−0.980374
$268$	−1.07565e6	−0.914815
$269$	−538494.	−0.453733	−0.226866	−	0.973926i	$-0.572848\pi$
$269$	−538494.	−0.453733	−0.226866	+	0.973926i	$0.572848\pi$
$270$	17496.0	0.0146059
$271$	572264.	0.473340	0.236670	−	0.971590i	$-0.423944\pi$
$271$	572264.	0.473340	0.236670	+	0.971590i	$0.423944\pi$
$272$	−43520.0	−0.0356670
$273$	20286.0	0.0164736
$274$	−1.06767e6	−0.859135
$275$	−373769.	−0.298038
$276$	−330624.	−0.261253
$277$	757574.	0.593233	0.296617	−	0.954997i	$-0.404142\pi$
$277$	757574.	0.593233	0.296617	+	0.954997i	$0.404142\pi$
$278$	−1.57213e6	−1.22005
$279$	780516.	0.600304
$280$	18816.0	0.0143427
$281$	1.46541e6	1.10712	0.553558	−	0.832810i	$-0.313270\pi$
$281$	1.46541e6	1.10712	0.553558	+	0.832810i	$0.313270\pi$
$282$	−102960.	−0.0770985
$283$	803432.	0.596325	0.298163	−	0.954515i	$-0.403626\pi$
$283$	803432.	0.596325	0.298163	+	0.954515i	$0.403626\pi$
$284$	−730368.	−0.537336
$285$	−66096.0	−0.0482018
$286$	22264.0	0.0160949
$287$	−159054.	−0.113983
$288$	82944.0	0.0589256
$289$	−1.39096e6	−0.979646
$290$	27696.0	0.0193385
$291$	420462.	0.291068
$292$	342240.	0.234895
$293$	−707370.	−0.481368	−0.240684	−	0.970603i	$-0.577372\pi$
$293$	−707370.	−0.481368	−0.240684	+	0.970603i	$0.577372\pi$
$294$	−86436.0	−0.0583212
$295$	7800.00	0.00521843
$296$	−532608.	−0.353328
$297$	−88209.0	−0.0580259
$298$	499736.	0.325987
$299$	105616.	0.0683206
$300$	444816.	0.285350
$301$	521948.	0.332056
$302$	−607840.	−0.383506
$303$	638226.	0.399363
$304$	−313344.	−0.194463
$305$	181260.	0.111571
$306$	−55080.0	−0.0336272
$307$	255712.	0.154848	0.0774239	−	0.996998i	$-0.475331\pi$
$307$	255712.	0.154848	0.0774239	+	0.996998i	$0.475331\pi$
$308$	−94864.0	−0.0569803
$309$	1.23635e6	0.736622
$310$	−231264.	−0.136680
$311$	2.22224e6	1.30283	0.651417	−	0.758720i	$-0.274175\pi$
$311$	2.22224e6	1.30283	0.651417	+	0.758720i	$0.274175\pi$
$312$	−26496.0	−0.0154097
$313$	−3.10395e6	−1.79083	−0.895414	−	0.445235i	$-0.853120\pi$
$313$	−3.10395e6	−1.79083	−0.895414	+	0.445235i	$0.853120\pi$
$314$	−1.96610e6	−1.12534
$315$	23814.0	0.0135225
$316$	−130944.	−0.0737680
$317$	−622914.	−0.348161	−0.174080	−	0.984731i	$-0.555695\pi$
$317$	−622914.	−0.348161	−0.174080	+	0.984731i	$0.555695\pi$
$318$	1.02794e6	0.570035
$319$	−139634.	−0.0768271
$320$	−24576.0	−0.0134164
$321$	1.27177e6	0.688885
$322$	−450016.	−0.241873
$323$	208080.	0.110975
$324$	104976.	0.0555556
$325$	−142094.	−0.0746221
$326$	−243792.	−0.127050
$327$	−1.27985e6	−0.661898
$328$	207744.	0.106621
$329$	−140140.	−0.0713793
$330$	26136.0	0.0132116
$331$	2.37142e6	1.18970	0.594851	−	0.803836i	$-0.297211\pi$
$331$	2.37142e6	1.18970	0.594851	+	0.803836i	$0.297211\pi$
$332$	−752768.	−0.374814
$333$	−674082.	−0.333121
$334$	2.22874e6	1.09318
$335$	403368.	0.196376
$336$	112896.	0.0545545
$337$	3.26295e6	1.56508	0.782539	−	0.622601i	$-0.213924\pi$
$337$	3.26295e6	1.56508	0.782539	+	0.622601i	$0.213924\pi$
$338$	−1.47671e6	−0.703077
$339$	1.42040e6	0.671291
$340$	16320.0	0.00765637
$341$	1.16596e6	0.542995
$342$	−396576.	−0.183342
$343$	−117649.	−0.0539949
$344$	−681728.	−0.310610
$345$	123984.	0.0560813
$346$	52536.0	0.0235921
$347$	−2.72691e6	−1.21576	−0.607879	−	0.794030i	$-0.707979\pi$
$347$	−2.72691e6	−1.21576	−0.607879	+	0.794030i	$0.707979\pi$
$348$	166176.	0.0735564
$349$	2.50415e6	1.10052	0.550259	−	0.834994i	$-0.314529\pi$
$349$	2.50415e6	1.10052	0.550259	+	0.834994i	$0.314529\pi$
$350$	605444.	0.264182
$351$	−33534.0	−0.0145284
$352$	123904.	0.0533002
$353$	−2.28557e6	−0.976241	−0.488120	−	0.872776i	$-0.662317\pi$
$353$	−2.28557e6	−0.976241	−0.488120	+	0.872776i	$0.662317\pi$
$354$	46800.0	0.0198490
$355$	273888.	0.115346
$356$	2.03024e6	0.849029
$357$	−74970.0	−0.0311327
$358$	2.02955e6	0.836937
$359$	−2.06593e6	−0.846017	−0.423008	−	0.906126i	$-0.639026\pi$
$359$	−2.06593e6	−0.846017	−0.423008	+	0.906126i	$0.639026\pi$
$360$	−31104.0	−0.0126491
$361$	−977923.	−0.394945
$362$	2.32228e6	0.931415
$363$	−131769.	−0.0524864
$364$	−36064.0	−0.0142666
$365$	−128340.	−0.0504231
$366$	1.08756e6	0.424376
$367$	−618172.	−0.239576	−0.119788	−	0.992799i	$-0.538222\pi$
$367$	−618172.	−0.239576	−0.119788	+	0.992799i	$0.538222\pi$
$368$	587776.	0.226252
$369$	262926.	0.100524
$370$	199728.	0.0758463
$371$	1.39915e6	0.527750
$372$	−1.38758e6	−0.519879
$373$	−13178.0	−0.00490430	−0.00245215	−	0.999997i	$-0.500781\pi$
$373$	−13178.0	−0.00490430	−0.00245215	+	0.999997i	$0.500781\pi$
$374$	−82280.0	−0.0304169
$375$	−335556.	−0.123222
$376$	183040.	0.0667692
$377$	−53084.0	−0.0192358
$378$	142884.	0.0514344
$379$	2.87877e6	1.02946	0.514730	−	0.857353i	$-0.327892\pi$
$379$	2.87877e6	1.02946	0.514730	+	0.857353i	$0.327892\pi$
$380$	117504.	0.0417440
$381$	−2.05207e6	−0.724236
$382$	−3.47424e6	−1.21815
$383$	2.34668e6	0.817441	0.408720	−	0.912660i	$-0.365975\pi$
$383$	2.34668e6	0.817441	0.408720	+	0.912660i	$0.365975\pi$
$384$	−147456.	−0.0510310
$385$	35574.0	0.0122315
$386$	−779128.	−0.266159
$387$	−862812.	−0.292846
$388$	−747488.	−0.252072
$389$	−1.70539e6	−0.571414	−0.285707	−	0.958317i	$-0.592228\pi$
$389$	−1.70539e6	−0.571414	−0.285707	+	0.958317i	$0.592228\pi$
$390$	9936.00	0.00330788
$391$	−390320.	−0.129116
$392$	153664.	0.0505076
$393$	3.04661e6	0.995028
$394$	2.91170e6	0.944945
$395$	49104.0	0.0158352
$396$	156816.	0.0502519
$397$	5.55012e6	1.76737	0.883683	−	0.468087i	$-0.155057\pi$
$397$	5.55012e6	1.76737	0.883683	+	0.468087i	$0.155057\pi$
$398$	397264.	0.125710
$399$	−539784.	−0.169741
$400$	−790784.	−0.247120
$401$	−725726.	−0.225378	−0.112689	−	0.993630i	$-0.535946\pi$
$401$	−725726.	−0.225378	−0.112689	+	0.993630i	$0.535946\pi$
$402$	2.42021e6	0.746943
$403$	443256.	0.135954
$404$	−1.13462e6	−0.345859
$405$	−39366.0	−0.0119257
$406$	226184.	0.0681000
$407$	−1.00696e6	−0.301319
$408$	97920.0	0.0291220
$409$	5.40903e6	1.59886	0.799431	−	0.600758i	$-0.205134\pi$
$409$	5.40903e6	1.59886	0.799431	+	0.600758i	$0.205134\pi$
$410$	−77904.0	−0.0228876
$411$	2.40226e6	0.701481
$412$	−2.19795e6	−0.637933
$413$	63700.0	0.0183766
$414$	743904.	0.213312
$415$	282288.	0.0804585
$416$	47104.0	0.0133452
$417$	3.53729e6	0.996163
$418$	−592416.	−0.165839
$419$	4.36353e6	1.21424	0.607118	−	0.794612i	$-0.292326\pi$
$419$	4.36353e6	1.21424	0.607118	+	0.794612i	$0.292326\pi$
$420$	−42336.0	−0.0117108
$421$	23814.0	0.00654828	0.00327414	−	0.999995i	$-0.498958\pi$
$421$	23814.0	0.00654828	0.00327414	+	0.999995i	$0.498958\pi$
$422$	173680.	0.0474754
$423$	231660.	0.0629506
$424$	−1.82746e6	−0.493665
$425$	525130.	0.141024
$426$	1.64333e6	0.438733
$427$	1.48029e6	0.392896
$428$	−2.26093e6	−0.596592
$429$	−50094.0	−0.0131414
$430$	255648.	0.0666763
$431$	−4.25480e6	−1.10328	−0.551640	−	0.834082i	$-0.685998\pi$
$431$	−4.25480e6	−1.10328	−0.551640	+	0.834082i	$0.685998\pi$
$432$	−186624.	−0.0481125
$433$	5.15496e6	1.32131	0.660657	−	0.750688i	$-0.270278\pi$
$433$	5.15496e6	1.32131	0.660657	+	0.750688i	$0.270278\pi$
$434$	−1.88866e6	−0.481314
$435$	−62316.0	−0.0157898
$436$	2.27530e6	0.573220
$437$	−2.81030e6	−0.703963
$438$	−770040.	−0.191791
$439$	1.34646e6	0.333452	0.166726	−	0.986003i	$-0.446680\pi$
$439$	1.34646e6	0.333452	0.166726	+	0.986003i	$0.446680\pi$
$440$	−46464.0	−0.0114416
$441$	194481.	0.0476190
$442$	−31280.0	−0.00761572
$443$	−255852.	−0.0619412	−0.0309706	−	0.999520i	$-0.509860\pi$
$443$	−255852.	−0.0619412	−0.0309706	+	0.999520i	$0.509860\pi$
$444$	1.19837e6	0.288491
$445$	−761340.	−0.182255
$446$	3.34466e6	0.796186
$447$	−1.12441e6	−0.266167
$448$	−200704.	−0.0472456
$449$	−2.38363e6	−0.557986	−0.278993	−	0.960293i	$-0.590001\pi$
$449$	−2.38363e6	−0.557986	−0.278993	+	0.960293i	$0.590001\pi$
$450$	−1.00084e6	−0.232987
$451$	392766.	0.0909269
$452$	−2.52515e6	−0.581355
$453$	1.36764e6	0.313131
$454$	3.80941e6	0.867397
$455$	13524.0	0.00306250
$456$	705024.	0.158779
$457$	4.10510e6	0.919460	0.459730	−	0.888059i	$-0.347946\pi$
$457$	4.10510e6	0.919460	0.459730	+	0.888059i	$0.347946\pi$
$458$	3.84055e6	0.855520
$459$	123930.	0.0274565
$460$	−220416.	−0.0485678
$461$	−1.10068e6	−0.241218	−0.120609	−	0.992700i	$-0.538485\pi$
$461$	−1.10068e6	−0.241218	−0.120609	+	0.992700i	$0.538485\pi$
$462$	213444.	0.0465242
$463$	2.25743e6	0.489398	0.244699	−	0.969599i	$-0.421311\pi$
$463$	2.25743e6	0.489398	0.244699	+	0.969599i	$0.421311\pi$
$464$	−295424.	−0.0637017
$465$	520344.	0.111598
$466$	4.22074e6	0.900377
$467$	−5.58149e6	−1.18429	−0.592145	−	0.805831i	$-0.701719\pi$
$467$	−5.58149e6	−1.18429	−0.592145	+	0.805831i	$0.701719\pi$
$468$	59616.0	0.0125820
$469$	3.29417e6	0.691535
$470$	−68640.0	−0.0143329
$471$	4.42373e6	0.918833
$472$	−83200.0	−0.0171897
$473$	−1.28889e6	−0.264889
$474$	294624.	0.0602313
$475$	3.78094e6	0.768892
$476$	133280.	0.0269617
$477$	−2.31287e6	−0.465432
$478$	−4.22224e6	−0.845227
$479$	−8.22136e6	−1.63721	−0.818606	−	0.574356i	$-0.805253\pi$
$479$	−8.22136e6	−1.63721	−0.818606	+	0.574356i	$0.805253\pi$
$480$	55296.0	0.0109545
$481$	−382812.	−0.0754437
$482$	351128.	0.0688411
$483$	1.01254e6	0.197489
$484$	234256.	0.0454545
$485$	280308.	0.0541105
$486$	−236196.	−0.0453609
$487$	−125416.	−0.0239624	−0.0119812	−	0.999928i	$-0.503814\pi$
$487$	−125416.	−0.0239624	−0.0119812	+	0.999928i	$0.503814\pi$
$488$	−1.93344e6	−0.367520
$489$	548532.	0.103736
$490$	−57624.0	−0.0108421
$491$	−1.32364e6	−0.247779	−0.123890	−	0.992296i	$-0.539537\pi$
$491$	−1.32364e6	−0.247779	−0.123890	+	0.992296i	$0.539537\pi$
$492$	−467424.	−0.0870559
$493$	196180.	0.0363528
$494$	−225216.	−0.0415223
$495$	−58806.0	−0.0107872
$496$	2.46682e6	0.450228
$497$	2.23675e6	0.406188
$498$	1.69373e6	0.306034
$499$	−3.80640e6	−0.684325	−0.342163	−	0.939641i	$-0.611159\pi$
$499$	−3.80640e6	−0.684325	−0.342163	+	0.939641i	$0.611159\pi$
$500$	596544.	0.106713
$501$	−5.01466e6	−0.892579
$502$	−5.70382e6	−1.01020
$503$	−1.80757e6	−0.318548	−0.159274	−	0.987234i	$-0.550915\pi$
$503$	−1.80757e6	−0.318548	−0.159274	+	0.987234i	$0.550915\pi$
$504$	−254016.	−0.0445435
$505$	425484.	0.0742429
$506$	1.11126e6	0.192948
$507$	3.32259e6	0.574060
$508$	3.64813e6	0.627207
$509$	−1.44768e6	−0.247672	−0.123836	−	0.992303i	$-0.539520\pi$
$509$	−1.44768e6	−0.247672	−0.123836	+	0.992303i	$0.539520\pi$
$510$	−36720.0	−0.00625140
$511$	−1.04811e6	−0.177564
$512$	262144.	0.0441942
$513$	892296.	0.149698
$514$	6.20551e6	1.03602
$515$	824232.	0.136940
$516$	1.53389e6	0.253612
$517$	346060.	0.0569410
$518$	1.63111e6	0.267091
$519$	−118206.	−0.0192629
$520$	−17664.0	−0.00286471
$521$	2.43537e6	0.393071	0.196535	−	0.980497i	$-0.437031\pi$
$521$	2.43537e6	0.393071	0.196535	+	0.980497i	$0.437031\pi$
$522$	−373896.	−0.0600585
$523$	3.07518e6	0.491604	0.245802	−	0.969320i	$-0.420949\pi$
$523$	3.07518e6	0.491604	0.245802	+	0.969320i	$0.420949\pi$
$524$	−5.41619e6	−0.861719
$525$	−1.36225e6	−0.215704
$526$	−4.84643e6	−0.763762
$527$	−1.63812e6	−0.256933
$528$	−278784.	−0.0435194
$529$	−1.16473e6	−0.180961
$530$	685296.	0.105971
$531$	−105300.	−0.0162066
$532$	959616.	0.147000
$533$	149316.	0.0227661
$534$	−4.56804e6	−0.693229
$535$	847848.	0.128066
$536$	−4.30259e6	−0.646872
$537$	−4.56649e6	−0.683356
$538$	−2.15398e6	−0.320838
$539$	290521.	0.0430730
$540$	69984.0	0.0103280
$541$	5.94155e6	0.872784	0.436392	−	0.899757i	$-0.356256\pi$
$541$	5.94155e6	0.872784	0.436392	+	0.899757i	$0.356256\pi$
$542$	2.28906e6	0.334702
$543$	−5.22513e6	−0.760497
$544$	−174080.	−0.0252204
$545$	−853236.	−0.123049
$546$	81144.0	0.0116486
$547$	−9.96259e6	−1.42365	−0.711826	−	0.702356i	$-0.752132\pi$
$547$	−9.96259e6	−1.42365	−0.711826	+	0.702356i	$0.752132\pi$
$548$	−4.27069e6	−0.607500
$549$	−2.44701e6	−0.346501
$550$	−1.49508e6	−0.210745
$551$	1.41250e6	0.198202
$552$	−1.32250e6	−0.184734
$553$	401016.	0.0557634
$554$	3.03030e6	0.419479
$555$	−449388.	−0.0619283
$556$	−6.28851e6	−0.862702
$557$	−5.05552e6	−0.690443	−0.345222	−	0.938521i	$-0.612196\pi$
$557$	−5.05552e6	−0.690443	−0.345222	+	0.938521i	$0.612196\pi$
$558$	3.12206e6	0.424479
$559$	−489992.	−0.0663223
$560$	75264.0	0.0101419
$561$	185130.	0.0248353
$562$	5.86164e6	0.782850
$563$	−1.31910e7	−1.75391	−0.876955	−	0.480572i	$-0.840429\pi$
$563$	−1.31910e7	−1.75391	−0.876955	+	0.480572i	$0.840429\pi$
$564$	−411840.	−0.0545169
$565$	946932.	0.124795
$566$	3.21373e6	0.421665
$567$	−321489.	−0.0419961
$568$	−2.92147e6	−0.379954
$569$	−1.66246e6	−0.215264	−0.107632	−	0.994191i	$-0.534327\pi$
$569$	−1.66246e6	−0.215264	−0.107632	+	0.994191i	$0.534327\pi$
$570$	−264384.	−0.0340838
$571$	−8.11341e6	−1.04139	−0.520695	−	0.853743i	$-0.674327\pi$
$571$	−8.11341e6	−1.04139	−0.520695	+	0.853743i	$0.674327\pi$
$572$	89056.0	0.0113808
$573$	7.81704e6	0.994617
$574$	−636216.	−0.0805981
$575$	−7.09234e6	−0.894582
$576$	331776.	0.0416667
$577$	−6.48554e6	−0.810974	−0.405487	−	0.914101i	$-0.632898\pi$
$577$	−6.48554e6	−0.810974	−0.405487	+	0.914101i	$0.632898\pi$
$578$	−5.56383e6	−0.692714
$579$	1.75304e6	0.217318
$580$	110784.	0.0136744
$581$	2.30535e6	0.283333
$582$	1.68185e6	0.205816
$583$	−3.45503e6	−0.420999
$584$	1.36896e6	0.166096
$585$	−22356.0	−0.00270087
$586$	−2.82948e6	−0.340379
$587$	−284644.	−0.0340963	−0.0170481	−	0.999855i	$-0.505427\pi$
$587$	−284644.	−0.0340963	−0.0170481	+	0.999855i	$0.505427\pi$
$588$	−345744.	−0.0412393
$589$	−1.17945e7	−1.40084
$590$	31200.0	0.00368998
$591$	−6.55133e6	−0.771545
$592$	−2.13043e6	−0.249841
$593$	−1.06577e7	−1.24459	−0.622296	−	0.782782i	$-0.713800\pi$
$593$	−1.06577e7	−1.24459	−0.622296	+	0.782782i	$0.713800\pi$
$594$	−352836.	−0.0410305
$595$	−49980.0	−0.00578767
$596$	1.99894e6	0.230507
$597$	−893844.	−0.102642
$598$	422464.	0.0483100
$599$	−1.08883e7	−1.23991	−0.619957	−	0.784636i	$-0.712850\pi$
$599$	−1.08883e7	−1.23991	−0.619957	+	0.784636i	$0.712850\pi$
$600$	1.77926e6	0.201773
$601$	1.09700e7	1.23885	0.619427	−	0.785054i	$-0.287365\pi$
$601$	1.09700e7	1.23885	0.619427	+	0.785054i	$0.287365\pi$
$602$	2.08779e6	0.234799
$603$	−5.44547e6	−0.609877
$604$	−2.43136e6	−0.271180
$605$	−87846.0	−0.00975739
$606$	2.55290e6	0.282392
$607$	−6.27562e6	−0.691329	−0.345664	−	0.938358i	$-0.612346\pi$
$607$	−6.27562e6	−0.691329	−0.345664	+	0.938358i	$0.612346\pi$
$608$	−1.25338e6	−0.137506
$609$	−508914.	−0.0556034
$610$	725040.	0.0788928
$611$	131560.	0.0142568
$612$	−220320.	−0.0237780
$613$	5.05980e6	0.543854	0.271927	−	0.962318i	$-0.412339\pi$
$613$	5.05980e6	0.543854	0.271927	+	0.962318i	$0.412339\pi$
$614$	1.02285e6	0.109494
$615$	175284.	0.0186876
$616$	−379456.	−0.0402911
$617$	6.26540e6	0.662577	0.331288	−	0.943530i	$-0.392517\pi$
$617$	6.26540e6	0.662577	0.331288	+	0.943530i	$0.392517\pi$
$618$	4.94539e6	0.520870
$619$	−1.16275e7	−1.21972	−0.609859	−	0.792510i	$-0.708774\pi$
$619$	−1.16275e7	−1.21972	−0.609859	+	0.792510i	$0.708774\pi$
$620$	−925056.	−0.0966471
$621$	−1.67378e6	−0.174169
$622$	8.88894e6	0.921243
$623$	−6.21761e6	−0.641805
$624$	−105984.	−0.0108963
$625$	9.42942e6	0.965573
$626$	−1.24158e7	−1.26631
$627$	1.33294e6	0.135407
$628$	−7.86442e6	−0.795733
$629$	1.41474e6	0.142577
$630$	95256.0	0.00956183
$631$	1.39076e7	1.39052	0.695260	−	0.718758i	$-0.255289\pi$
$631$	1.39076e7	1.39052	0.695260	+	0.718758i	$0.255289\pi$
$632$	−523776.	−0.0521618
$633$	−390780.	−0.0387635
$634$	−2.49166e6	−0.246187
$635$	−1.36805e6	−0.134638
$636$	4.11178e6	0.403076
$637$	110446.	0.0107845
$638$	−558536.	−0.0543250
$639$	−3.69749e6	−0.358224
$640$	−98304.0	−0.00948683
$641$	−9.66485e6	−0.929073	−0.464537	−	0.885554i	$-0.653779\pi$
$641$	−9.66485e6	−0.929073	−0.464537	+	0.885554i	$0.653779\pi$
$642$	5.08709e6	0.487115
$643$	7.94445e6	0.757768	0.378884	−	0.925444i	$-0.376308\pi$
$643$	7.94445e6	0.757768	0.378884	+	0.925444i	$0.376308\pi$
$644$	−1.80006e6	−0.171030
$645$	−575208.	−0.0544409
$646$	832320.	0.0784710
$647$	−1.19378e7	−1.12115	−0.560573	−	0.828105i	$-0.689419\pi$
$647$	−1.19378e7	−1.12115	−0.560573	+	0.828105i	$0.689419\pi$
$648$	419904.	0.0392837
$649$	−157300.	−0.0146594
$650$	−568376.	−0.0527658
$651$	4.24948e6	0.392991
$652$	−975168.	−0.0898381
$653$	4.15738e6	0.381537	0.190769	−	0.981635i	$-0.438902\pi$
$653$	4.15738e6	0.381537	0.190769	+	0.981635i	$0.438902\pi$
$654$	−5.11942e6	−0.468033
$655$	2.03107e6	0.184979
$656$	830976.	0.0753926
$657$	1.73259e6	0.156597
$658$	−560560.	−0.0504728
$659$	−1.15343e7	−1.03461	−0.517305	−	0.855801i	$-0.673065\pi$
$659$	−1.15343e7	−1.03461	−0.517305	+	0.855801i	$0.673065\pi$
$660$	104544.	0.00934199
$661$	−3.54671e6	−0.315734	−0.157867	−	0.987460i	$-0.550462\pi$
$661$	−3.54671e6	−0.315734	−0.157867	+	0.987460i	$0.550462\pi$
$662$	9.48568e6	0.841247
$663$	70380.0	0.00621821
$664$	−3.01107e6	−0.265034
$665$	−359856.	−0.0315555
$666$	−2.69633e6	−0.235552
$667$	−2.64958e6	−0.230602
$668$	8.91494e6	0.772996
$669$	−7.52548e6	−0.650083
$670$	1.61347e6	0.138859
$671$	−3.65541e6	−0.313422
$672$	451584.	0.0385758
$673$	−1.76805e7	−1.50473	−0.752364	−	0.658748i	$-0.771086\pi$
$673$	−1.76805e7	−1.50473	−0.752364	+	0.658748i	$0.771086\pi$
$674$	1.30518e7	1.10668
$675$	2.25188e6	0.190233
$676$	−5.90683e6	−0.497150
$677$	−1.20909e7	−1.01388	−0.506939	−	0.861982i	$-0.669223\pi$
$677$	−1.20909e7	−1.01388	−0.506939	+	0.861982i	$0.669223\pi$
$678$	5.68159e6	0.474674
$679$	2.28918e6	0.190549
$680$	65280.0	0.00541387
$681$	−8.57117e6	−0.708227
$682$	4.66382e6	0.383956
$683$	−6.89752e6	−0.565771	−0.282886	−	0.959154i	$-0.591292\pi$
$683$	−6.89752e6	−0.565771	−0.282886	+	0.959154i	$0.591292\pi$
$684$	−1.58630e6	−0.129642
$685$	1.60151e6	0.130408
$686$	−470596.	−0.0381802
$687$	−8.64124e6	−0.698529
$688$	−2.72691e6	−0.219634
$689$	−1.31348e6	−0.105409
$690$	495936.	0.0396555
$691$	1.93101e6	0.153847	0.0769236	−	0.997037i	$-0.475490\pi$
$691$	1.93101e6	0.153847	0.0769236	+	0.997037i	$0.475490\pi$
$692$	210144.	0.0166821
$693$	−480249.	−0.0379869
$694$	−1.09076e7	−0.859670
$695$	2.35819e6	0.185190
$696$	664704.	0.0520122
$697$	−551820.	−0.0430245
$698$	1.00166e7	0.778183
$699$	−9.49667e6	−0.735154
$700$	2.42178e6	0.186805
$701$	4.49995e6	0.345870	0.172935	−	0.984933i	$-0.444675\pi$
$701$	4.49995e6	0.345870	0.172935	+	0.984933i	$0.444675\pi$
$702$	−134136.	−0.0102731
$703$	1.01861e7	0.777358
$704$	495616.	0.0376889
$705$	154440.	0.0117027
$706$	−9.14226e6	−0.690307
$707$	3.47479e6	0.261445
$708$	187200.	0.0140353
$709$	6.33531e6	0.473317	0.236659	−	0.971593i	$-0.423948\pi$
$709$	6.33531e6	0.473317	0.236659	+	0.971593i	$0.423948\pi$
$710$	1.09555e6	0.0815619
$711$	−662904.	−0.0491787
$712$	8.12096e6	0.600354
$713$	2.21243e7	1.62984
$714$	−299880.	−0.0220142
$715$	−33396.0	−0.00244303
$716$	8.11821e6	0.591804
$717$	9.50004e6	0.690125
$718$	−8.26371e6	−0.598224
$719$	1.24254e7	0.896370	0.448185	−	0.893941i	$-0.352071\pi$
$719$	1.24254e7	0.896370	0.448185	+	0.893941i	$0.352071\pi$
$720$	−124416.	−0.00894427
$721$	6.73123e6	0.482232
$722$	−3.91169e6	−0.279268
$723$	−790038.	−0.0562085
$724$	9.28912e6	0.658610
$725$	3.56471e6	0.251871
$726$	−527076.	−0.0371135
$727$	6.74779e6	0.473506	0.236753	−	0.971570i	$-0.423917\pi$
$727$	6.74779e6	0.473506	0.236753	+	0.971570i	$0.423917\pi$
$728$	−144256.	−0.0100880
$729$	531441.	0.0370370
$730$	−513360.	−0.0356545
$731$	1.81084e6	0.125339
$732$	4.35024e6	0.300079
$733$	−1.56038e7	−1.07268	−0.536340	−	0.844002i	$-0.680193\pi$
$733$	−1.56038e7	−1.07268	−0.536340	+	0.844002i	$0.680193\pi$
$734$	−2.47269e6	−0.169406
$735$	129654.	0.00885253
$736$	2.35110e6	0.159984
$737$	−8.13459e6	−0.551654
$738$	1.05170e6	0.0710809
$739$	1.34127e7	0.903452	0.451726	−	0.892157i	$-0.350809\pi$
$739$	1.34127e7	0.903452	0.451726	+	0.892157i	$0.350809\pi$
$740$	798912.	0.0536315
$741$	506736.	0.0339029
$742$	5.59658e6	0.373176
$743$	−1.36970e7	−0.910237	−0.455119	−	0.890431i	$-0.650403\pi$
$743$	−1.36970e7	−0.910237	−0.455119	+	0.890431i	$0.650403\pi$
$744$	−5.55034e6	−0.367610
$745$	−749604.	−0.0494813
$746$	−52712.0	−0.00346787
$747$	−3.81089e6	−0.249876
$748$	−329120.	−0.0215080
$749$	6.92409e6	0.450981
$750$	−1.34222e6	−0.0871308
$751$	6.80019e6	0.439968	0.219984	−	0.975503i	$-0.429399\pi$
$751$	6.80019e6	0.439968	0.219984	+	0.975503i	$0.429399\pi$
$752$	732160.	0.0472130
$753$	1.28336e7	0.824824
$754$	−212336.	−0.0136018
$755$	911760.	0.0582121
$756$	571536.	0.0363696
$757$	−5.39840e6	−0.342394	−0.171197	−	0.985237i	$-0.554763\pi$
$757$	−5.39840e6	−0.342394	−0.171197	+	0.985237i	$0.554763\pi$
$758$	1.15151e7	0.727938
$759$	−2.50034e6	−0.157542
$760$	470016.	0.0295174
$761$	9.75289e6	0.610480	0.305240	−	0.952275i	$-0.401263\pi$
$761$	9.75289e6	0.610480	0.305240	+	0.952275i	$0.401263\pi$
$762$	−8.20829e6	−0.512112
$763$	−6.96809e6	−0.433314
$764$	−1.38970e7	−0.861363
$765$	82620.0	0.00510425
$766$	9.38670e6	0.578018
$767$	−59800.0	−0.00367039
$768$	−589824.	−0.0360844
$769$	−6.52007e6	−0.397591	−0.198795	−	0.980041i	$-0.563703\pi$
$769$	−6.52007e6	−0.397591	−0.198795	+	0.980041i	$0.563703\pi$
$770$	142296.	0.00864900
$771$	−1.39624e7	−0.845910
$772$	−3.11651e6	−0.188203
$773$	−1.39519e7	−0.839819	−0.419909	−	0.907566i	$-0.637938\pi$
$773$	−1.39519e7	−0.839819	−0.419909	+	0.907566i	$0.637938\pi$
$774$	−3.45125e6	−0.207073
$775$	−2.97656e7	−1.78017
$776$	−2.98995e6	−0.178242
$777$	−3.67000e6	−0.218079
$778$	−6.82158e6	−0.404051
$779$	−3.97310e6	−0.234577
$780$	39744.0	0.00233903
$781$	−5.52341e6	−0.324026
$782$	−1.56128e6	−0.0912986
$783$	841266.	0.0490376
$784$	614656.	0.0357143
$785$	2.94916e6	0.170814
$786$	1.21864e7	0.703591
$787$	6.56130e6	0.377619	0.188809	−	0.982014i	$-0.439537\pi$
$787$	6.56130e6	0.377619	0.188809	+	0.982014i	$0.439537\pi$
$788$	1.16468e7	0.668177
$789$	1.09045e7	0.623609
$790$	196416.	0.0111972
$791$	7.73328e6	0.439463
$792$	627264.	0.0355335
$793$	−1.38966e6	−0.0784740
$794$	2.22005e7	1.24972
$795$	−1.54192e6	−0.0865253
$796$	1.58906e6	0.0888907
$797$	−1.53202e7	−0.854314	−0.427157	−	0.904177i	$-0.640485\pi$
$797$	−1.53202e7	−0.854314	−0.427157	+	0.904177i	$0.640485\pi$
$798$	−2.15914e6	−0.120025
$799$	−486200.	−0.0269431
$800$	−3.16314e6	−0.174740
$801$	1.02781e7	0.566019
$802$	−2.90290e6	−0.159366
$803$	2.58819e6	0.141647
$804$	9.68083e6	0.528169
$805$	675024.	0.0367138
$806$	1.77302e6	0.0961340
$807$	4.84645e6	0.261963
$808$	−4.53850e6	−0.244559
$809$	1.37633e7	0.739351	0.369675	−	0.929161i	$-0.379469\pi$
$809$	1.37633e7	0.739351	0.369675	+	0.929161i	$0.379469\pi$
$810$	−157464.	−0.00843274
$811$	4.42244e6	0.236108	0.118054	−	0.993007i	$-0.462334\pi$
$811$	4.42244e6	0.236108	0.118054	+	0.993007i	$0.462334\pi$
$812$	904736.	0.0481539
$813$	−5.15038e6	−0.273283
$814$	−4.02785e6	−0.213065
$815$	365688.	0.0192849
$816$	391680.	0.0205924
$817$	1.30380e7	0.683372
$818$	2.16361e7	1.13057
$819$	−182574.	−0.00951107
$820$	−311616.	−0.0161840
$821$	2.79067e7	1.44494	0.722470	−	0.691402i	$-0.243007\pi$
$821$	2.79067e7	1.44494	0.722470	+	0.691402i	$0.243007\pi$
$822$	9.60905e6	0.496022
$823$	3.01935e7	1.55387	0.776935	−	0.629581i	$-0.216774\pi$
$823$	3.01935e7	1.55387	0.776935	+	0.629581i	$0.216774\pi$
$824$	−8.79181e6	−0.451087
$825$	3.36392e6	0.172072
$826$	254800.	0.0129942
$827$	2.17929e7	1.10803	0.554015	−	0.832507i	$-0.313095\pi$
$827$	2.17929e7	1.10803	0.554015	+	0.832507i	$0.313095\pi$
$828$	2.97562e6	0.150835
$829$	−8.10216e6	−0.409463	−0.204731	−	0.978818i	$-0.565632\pi$
$829$	−8.10216e6	−0.409463	−0.204731	+	0.978818i	$0.565632\pi$
$830$	1.12915e6	0.0568928
$831$	−6.81817e6	−0.342503
$832$	188416.	0.00943647
$833$	−408170.	−0.0203811
$834$	1.41492e7	0.704393
$835$	−3.34310e6	−0.165933
$836$	−2.36966e6	−0.117266
$837$	−7.02464e6	−0.346586
$838$	1.74541e7	0.858595
$839$	−6.65445e6	−0.326368	−0.163184	−	0.986596i	$-0.552176\pi$
$839$	−6.65445e6	−0.326368	−0.163184	+	0.986596i	$0.552176\pi$
$840$	−169344.	−0.00828079
$841$	−1.91794e7	−0.935074
$842$	95256.0	0.00463033
$843$	−1.31887e7	−0.639194
$844$	694720.	0.0335702
$845$	2.21506e6	0.106720
$846$	926640.	0.0445128
$847$	−717409.	−0.0343604
$848$	−7.30982e6	−0.349074
$849$	−7.23089e6	−0.344288
$850$	2.10052e6	0.0997194
$851$	−1.91073e7	−0.904432
$852$	6.57331e6	0.310231
$853$	1.70163e7	0.800742	0.400371	−	0.916353i	$-0.368881\pi$
$853$	1.70163e7	0.800742	0.400371	+	0.916353i	$0.368881\pi$
$854$	5.92116e6	0.277819
$855$	594864.	0.0278293
$856$	−9.04371e6	−0.421854
$857$	−2.51134e7	−1.16803	−0.584013	−	0.811744i	$-0.698518\pi$
$857$	−2.51134e7	−1.16803	−0.584013	+	0.811744i	$0.698518\pi$
$858$	−200376.	−0.00929239
$859$	1.30076e7	0.601469	0.300734	−	0.953708i	$-0.402768\pi$
$859$	1.30076e7	0.601469	0.300734	+	0.953708i	$0.402768\pi$
$860$	1.02259e6	0.0471472
$861$	1.43149e6	0.0658081
$862$	−1.70192e7	−0.780137
$863$	3.03499e7	1.38717	0.693585	−	0.720375i	$-0.256030\pi$
$863$	3.03499e7	1.38717	0.693585	+	0.720375i	$0.256030\pi$
$864$	−746496.	−0.0340207
$865$	−78804.0	−0.00358103
$866$	2.06198e7	0.934309
$867$	1.25186e7	0.565599
$868$	−7.55462e6	−0.340340
$869$	−990264.	−0.0444838
$870$	−249264.	−0.0111651
$871$	−3.09249e6	−0.138122
$872$	9.10118e6	0.405328
$873$	−3.78416e6	−0.168048
$874$	−1.12412e7	−0.497777
$875$	−1.82692e6	−0.0806675
$876$	−3.08016e6	−0.135617
$877$	4.05025e7	1.77821	0.889105	−	0.457704i	$-0.151328\pi$
$877$	4.05025e7	1.77821	0.889105	+	0.457704i	$0.151328\pi$
$878$	5.38586e6	0.235786
$879$	6.36633e6	0.277918
$880$	−185856.	−0.00809040
$881$	1.06585e7	0.462654	0.231327	−	0.972876i	$-0.425693\pi$
$881$	1.06585e7	0.462654	0.231327	+	0.972876i	$0.425693\pi$
$882$	777924.	0.0336718
$883$	−2.90352e7	−1.25321	−0.626603	−	0.779339i	$-0.715555\pi$
$883$	−2.90352e7	−1.25321	−0.626603	+	0.779339i	$0.715555\pi$
$884$	−125120.	−0.00538513
$885$	−70200.0	−0.00301286
$886$	−1.02341e6	−0.0437990
$887$	2.50206e7	1.06780	0.533898	−	0.845549i	$-0.320727\pi$
$887$	2.50206e7	1.06780	0.533898	+	0.845549i	$0.320727\pi$
$888$	4.79347e6	0.203994
$889$	−1.11724e7	−0.474124
$890$	−3.04536e6	−0.128873
$891$	793881.	0.0335013
$892$	1.33786e7	0.562988
$893$	−3.50064e6	−0.146899
$894$	−4.49762e6	−0.188209
$895$	−3.04433e6	−0.127038
$896$	−802816.	−0.0334077
$897$	−950544.	−0.0394449
$898$	−9.53452e6	−0.394555
$899$	−1.11199e7	−0.458884
$900$	−4.00334e6	−0.164747
$901$	4.85418e6	0.199207
$902$	1.57106e6	0.0642950
$903$	−4.69753e6	−0.191712
$904$	−1.01006e7	−0.411080
$905$	−3.48342e6	−0.141379
$906$	5.47056e6	0.221417
$907$	−2.03759e7	−0.822430	−0.411215	−	0.911538i	$-0.634896\pi$
$907$	−2.03759e7	−0.822430	−0.411215	+	0.911538i	$0.634896\pi$
$908$	1.52376e7	0.613342
$909$	−5.74403e6	−0.230572
$910$	54096.0	0.00216552
$911$	−4.32590e6	−0.172695	−0.0863477	−	0.996265i	$-0.527520\pi$
$911$	−4.32590e6	−0.172695	−0.0863477	+	0.996265i	$0.527520\pi$
$912$	2.82010e6	0.112273
$913$	−5.69281e6	−0.226021
$914$	1.64204e7	0.650157
$915$	−1.63134e6	−0.0644157
$916$	1.53622e7	0.604944
$917$	1.65871e7	0.651399
$918$	495720.	0.0194147
$919$	−3.05674e7	−1.19390	−0.596952	−	0.802277i	$-0.703622\pi$
$919$	−3.05674e7	−1.19390	−0.596952	+	0.802277i	$0.703622\pi$
$920$	−881664.	−0.0343426
$921$	−2.30141e6	−0.0894014
$922$	−4.40273e6	−0.170567
$923$	−2.09981e6	−0.0811289
$924$	853776.	0.0328976
$925$	2.57067e7	0.987851
$926$	9.02973e6	0.346057
$927$	−1.11271e7	−0.425289
$928$	−1.18170e6	−0.0450439
$929$	4.21306e7	1.60162	0.800808	−	0.598921i	$-0.204403\pi$
$929$	4.21306e7	1.60162	0.800808	+	0.598921i	$0.204403\pi$
$930$	2.08138e6	0.0789120
$931$	−2.93882e6	−0.111122
$932$	1.68830e7	0.636662
$933$	−2.00001e7	−0.752192
$934$	−2.23260e7	−0.837420
$935$	123420.	0.00461696
$936$	238464.	0.00889679
$937$	2.47469e7	0.920815	0.460408	−	0.887708i	$-0.347703\pi$
$937$	2.47469e7	0.920815	0.460408	+	0.887708i	$0.347703\pi$
$938$	1.31767e7	0.488989
$939$	2.79356e7	1.03394
$940$	−274560.	−0.0101349
$941$	3.28483e7	1.20931	0.604656	−	0.796487i	$-0.293311\pi$
$941$	3.28483e7	1.20931	0.604656	+	0.796487i	$0.293311\pi$
$942$	1.76949e7	0.649713
$943$	7.45282e6	0.272924
$944$	−332800.	−0.0121550
$945$	−214326.	−0.00780720
$946$	−5.15557e6	−0.187305
$947$	−2.36632e6	−0.0857428	−0.0428714	−	0.999081i	$-0.513651\pi$
$947$	−2.36632e6	−0.0857428	−0.0428714	+	0.999081i	$0.513651\pi$
$948$	1.17850e6	0.0425900
$949$	983940.	0.0354653
$950$	1.51237e7	0.543689
$951$	5.60623e6	0.201011
$952$	533120.	0.0190648
$953$	−4.27359e7	−1.52427	−0.762133	−	0.647420i	$-0.775848\pi$
$953$	−4.27359e7	−1.52427	−0.762133	+	0.647420i	$0.775848\pi$
$954$	−9.25150e6	−0.329110
$955$	5.21136e6	0.184902
$956$	−1.68890e7	−0.597665
$957$	1.25671e6	0.0443562
$958$	−3.28854e7	−1.15768
$959$	1.30790e7	0.459227
$960$	221184.	0.00774597
$961$	6.42233e7	2.24328
$962$	−1.53125e6	−0.0533467
$963$	−1.14459e7	−0.397728
$964$	1.40451e6	0.0486780
$965$	1.16869e6	0.0404000
$966$	4.05014e6	0.139646
$967$	−5.37312e7	−1.84782	−0.923912	−	0.382606i	$-0.875027\pi$
$967$	−5.37312e7	−1.84782	−0.923912	+	0.382606i	$0.875027\pi$
$968$	937024.	0.0321412
$969$	−1.87272e6	−0.0640713
$970$	1.12123e6	0.0382619
$971$	−1.33627e7	−0.454828	−0.227414	−	0.973798i	$-0.573027\pi$
$971$	−1.33627e7	−0.454828	−0.227414	+	0.973798i	$0.573027\pi$
$972$	−944784.	−0.0320750
$973$	1.92586e7	0.652142
$974$	−501664.	−0.0169440
$975$	1.27885e6	0.0430831
$976$	−7.73376e6	−0.259876
$977$	3.50902e7	1.17611	0.588057	−	0.808819i	$-0.299893\pi$
$977$	3.50902e7	1.17611	0.588057	+	0.808819i	$0.299893\pi$
$978$	2.19413e6	0.0733525
$979$	1.53537e7	0.511984
$980$	−230496.	−0.00766652
$981$	1.15187e7	0.382147
$982$	−5.29454e6	−0.175206
$983$	−1.03230e7	−0.340739	−0.170370	−	0.985380i	$-0.554496\pi$
$983$	−1.03230e7	−0.340739	−0.170370	+	0.985380i	$0.554496\pi$
$984$	−1.86970e6	−0.0615578
$985$	−4.36756e6	−0.143433
$986$	784720.	0.0257053
$987$	1.26126e6	0.0412109
$988$	−900864.	−0.0293607
$989$	−2.44570e7	−0.795083
$990$	−235224.	−0.00762770
$991$	2.65205e7	0.857821	0.428911	−	0.903347i	$-0.358898\pi$
$991$	2.65205e7	0.857821	0.428911	+	0.903347i	$0.358898\pi$
$992$	9.86726e6	0.318359
$993$	−2.13428e7	−0.686875
$994$	8.94701e6	0.287218
$995$	−595896.	−0.0190815
$996$	6.77491e6	0.216399
$997$	3.07548e7	0.979883	0.489942	−	0.871755i	$-0.337018\pi$
$997$	3.07548e7	0.979883	0.489942	+	0.871755i	$0.337018\pi$
$998$	−1.52256e7	−0.483891
$999$	6.06674e6	0.192328

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	462.6.a.d.1.1	✓	1

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

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:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	462.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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