LMFDB - Embedded newform 462.6.a.a.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)

$f(q)$

$=$

$q-4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} -78.0000 q^{5} +36.0000 q^{6} +49.0000 q^{7} -64.0000 q^{8} +81.0000 q^{9} +O(q^{10})$

\(q-4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} -78.0000 q^{5} +36.0000 q^{6} +49.0000 q^{7} -64.0000 q^{8} +81.0000 q^{9} +312.000 q^{10} +121.000 q^{11} -144.000 q^{12} -502.000 q^{13} -196.000 q^{14} +702.000 q^{15} +256.000 q^{16} -642.000 q^{17} -324.000 q^{18} -520.000 q^{19} -1248.00 q^{20} -441.000 q^{21} -484.000 q^{22} +1020.00 q^{23} +576.000 q^{24} +2959.00 q^{25} +2008.00 q^{26} -729.000 q^{27} +784.000 q^{28} +4818.00 q^{29} -2808.00 q^{30} +1784.00 q^{31} -1024.00 q^{32} -1089.00 q^{33} +2568.00 q^{34} -3822.00 q^{35} +1296.00 q^{36} +7958.00 q^{37} +2080.00 q^{38} +4518.00 q^{39} +4992.00 q^{40} +2430.00 q^{41} +1764.00 q^{42} +22904.0 q^{43} +1936.00 q^{44} -6318.00 q^{45} -4080.00 q^{46} -11316.0 q^{47} -2304.00 q^{48} +2401.00 q^{49} -11836.0 q^{50} +5778.00 q^{51} -8032.00 q^{52} -12222.0 q^{53} +2916.00 q^{54} -9438.00 q^{55} -3136.00 q^{56} +4680.00 q^{57} -19272.0 q^{58} -15852.0 q^{59} +11232.0 q^{60} +46298.0 q^{61} -7136.00 q^{62} +3969.00 q^{63} +4096.00 q^{64} +39156.0 q^{65} +4356.00 q^{66} +19412.0 q^{67} -10272.0 q^{68} -9180.00 q^{69} +15288.0 q^{70} -17292.0 q^{71} -5184.00 q^{72} -30214.0 q^{73} -31832.0 q^{74} -26631.0 q^{75} -8320.00 q^{76} +5929.00 q^{77} -18072.0 q^{78} +35672.0 q^{79} -19968.0 q^{80} +6561.00 q^{81} -9720.00 q^{82} -43428.0 q^{83} -7056.00 q^{84} +50076.0 q^{85} -91616.0 q^{86} -43362.0 q^{87} -7744.00 q^{88} -14934.0 q^{89} +25272.0 q^{90} -24598.0 q^{91} +16320.0 q^{92} -16056.0 q^{93} +45264.0 q^{94} +40560.0 q^{95} +9216.00 q^{96} +85106.0 q^{97} -9604.00 q^{98} +9801.00 q^{99} +O(q^{100})\)

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
$5$	−78.0000	−1.39531	−0.697653	−	0.716436i	$-0.745772\pi$
$5$	−78.0000	−1.39531	−0.697653	+	0.716436i	$0.745772\pi$
$6$	36.0000	0.408248
$7$	49.0000	0.377964
$7$	49.0000	0.377964
$8$	−64.0000	−0.353553
$9$	81.0000	0.333333
$10$	312.000	0.986631
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	−144.000	−0.288675
$13$	−502.000	−0.823845	−0.411922	−	0.911219i	$-0.635143\pi$
$13$	−502.000	−0.823845	−0.411922	+	0.911219i	$0.635143\pi$
$14$	−196.000	−0.267261
$15$	702.000	0.805581
$16$	256.000	0.250000
$17$	−642.000	−0.538782	−0.269391	−	0.963031i	$-0.586822\pi$
$17$	−642.000	−0.538782	−0.269391	+	0.963031i	$0.586822\pi$
$18$	−324.000	−0.235702
$19$	−520.000	−0.330460	−0.165230	−	0.986255i	$-0.552837\pi$
$19$	−520.000	−0.330460	−0.165230	+	0.986255i	$0.552837\pi$
$20$	−1248.00	−0.697653
$21$	−441.000	−0.218218
$22$	−484.000	−0.213201
$23$	1020.00	0.402050	0.201025	−	0.979586i	$-0.435573\pi$
$23$	1020.00	0.402050	0.201025	+	0.979586i	$0.435573\pi$
$24$	576.000	0.204124
$25$	2959.00	0.946880
$26$	2008.00	0.582546
$27$	−729.000	−0.192450
$28$	784.000	0.188982
$29$	4818.00	1.06383	0.531914	−	0.846798i	$-0.321473\pi$
$29$	4818.00	1.06383	0.531914	+	0.846798i	$0.321473\pi$
$30$	−2808.00	−0.569631
$31$	1784.00	0.333419	0.166710	−	0.986006i	$-0.446686\pi$
$31$	1784.00	0.333419	0.166710	+	0.986006i	$0.446686\pi$
$32$	−1024.00	−0.176777
$33$	−1089.00	−0.174078
$34$	2568.00	0.380976
$35$	−3822.00	−0.527376
$36$	1296.00	0.166667
$37$	7958.00	0.955652	0.477826	−	0.878455i	$-0.341425\pi$
$37$	7958.00	0.955652	0.477826	+	0.878455i	$0.341425\pi$
$38$	2080.00	0.233671
$39$	4518.00	0.475647
$40$	4992.00	0.493315
$41$	2430.00	0.225760	0.112880	−	0.993609i	$-0.463992\pi$
$41$	2430.00	0.225760	0.112880	+	0.993609i	$0.463992\pi$
$42$	1764.00	0.154303
$43$	22904.0	1.88904	0.944518	−	0.328460i	$-0.106530\pi$
$43$	22904.0	1.88904	0.944518	+	0.328460i	$0.106530\pi$
$44$	1936.00	0.150756
$45$	−6318.00	−0.465102
$46$	−4080.00	−0.284293
$47$	−11316.0	−0.747220	−0.373610	−	0.927586i	$-0.621880\pi$
$47$	−11316.0	−0.747220	−0.373610	+	0.927586i	$0.621880\pi$
$48$	−2304.00	−0.144338
$49$	2401.00	0.142857
$50$	−11836.0	−0.669545
$51$	5778.00	0.311066
$52$	−8032.00	−0.411922
$53$	−12222.0	−0.597658	−0.298829	−	0.954307i	$-0.596596\pi$
$53$	−12222.0	−0.597658	−0.298829	+	0.954307i	$0.596596\pi$
$54$	2916.00	0.136083
$55$	−9438.00	−0.420701
$56$	−3136.00	−0.133631
$57$	4680.00	0.190791
$58$	−19272.0	−0.752241
$59$	−15852.0	−0.592863	−0.296431	−	0.955054i	$-0.595797\pi$
$59$	−15852.0	−0.592863	−0.296431	+	0.955054i	$0.595797\pi$
$60$	11232.0	0.402790
$61$	46298.0	1.59308	0.796540	−	0.604586i	$-0.206661\pi$
$61$	46298.0	1.59308	0.796540	+	0.604586i	$0.206661\pi$
$62$	−7136.00	−0.235763
$63$	3969.00	0.125988
$64$	4096.00	0.125000
$65$	39156.0	1.14952
$66$	4356.00	0.123091
$67$	19412.0	0.528303	0.264152	−	0.964481i	$-0.414908\pi$
$67$	19412.0	0.528303	0.264152	+	0.964481i	$0.414908\pi$
$68$	−10272.0	−0.269391
$69$	−9180.00	−0.232124
$70$	15288.0	0.372911
$71$	−17292.0	−0.407098	−0.203549	−	0.979065i	$-0.565248\pi$
$71$	−17292.0	−0.407098	−0.203549	+	0.979065i	$0.565248\pi$
$72$	−5184.00	−0.117851
$73$	−30214.0	−0.663592	−0.331796	−	0.943351i	$-0.607655\pi$
$73$	−30214.0	−0.663592	−0.331796	+	0.943351i	$0.607655\pi$
$74$	−31832.0	−0.675748
$75$	−26631.0	−0.546681
$76$	−8320.00	−0.165230
$77$	5929.00	0.113961
$78$	−18072.0	−0.336333
$79$	35672.0	0.643072	0.321536	−	0.946897i	$-0.395801\pi$
$79$	35672.0	0.643072	0.321536	+	0.946897i	$0.395801\pi$
$80$	−19968.0	−0.348827
$81$	6561.00	0.111111
$82$	−9720.00	−0.159636
$83$	−43428.0	−0.691950	−0.345975	−	0.938244i	$-0.612452\pi$
$83$	−43428.0	−0.691950	−0.345975	+	0.938244i	$0.612452\pi$
$84$	−7056.00	−0.109109
$85$	50076.0	0.751765
$86$	−91616.0	−1.33575
$87$	−43362.0	−0.614202
$88$	−7744.00	−0.106600
$89$	−14934.0	−0.199849	−0.0999243	−	0.994995i	$-0.531860\pi$
$89$	−14934.0	−0.199849	−0.0999243	+	0.994995i	$0.531860\pi$
$90$	25272.0	0.328877
$91$	−24598.0	−0.311384
$92$	16320.0	0.201025
$93$	−16056.0	−0.192500
$94$	45264.0	0.528364
$95$	40560.0	0.461093
$96$	9216.00	0.102062
$97$	85106.0	0.918398	0.459199	−	0.888333i	$-0.348137\pi$
$97$	85106.0	0.918398	0.459199	+	0.888333i	$0.348137\pi$
$98$	−9604.00	−0.101015
$99$	9801.00	0.100504
$100$	47344.0	0.473440
$101$	−72150.0	−0.703773	−0.351887	−	0.936043i	$-0.614460\pi$
$101$	−72150.0	−0.703773	−0.351887	+	0.936043i	$0.614460\pi$
$102$	−23112.0	−0.219957
$103$	11744.0	0.109074	0.0545372	−	0.998512i	$-0.482632\pi$
$103$	11744.0	0.109074	0.0545372	+	0.998512i	$0.482632\pi$
$104$	32128.0	0.291273
$105$	34398.0	0.304481
$106$	48888.0	0.422608
$107$	−217932.	−1.84019	−0.920093	−	0.391701i	$-0.871887\pi$
$107$	−217932.	−1.84019	−0.920093	+	0.391701i	$0.871887\pi$
$108$	−11664.0	−0.0962250
$109$	87674.0	0.706813	0.353407	−	0.935470i	$-0.385023\pi$
$109$	87674.0	0.706813	0.353407	+	0.935470i	$0.385023\pi$
$110$	37752.0	0.297480
$111$	−71622.0	−0.551746
$112$	12544.0	0.0944911
$113$	145818.	1.07427	0.537137	−	0.843495i	$-0.319506\pi$
$113$	145818.	1.07427	0.537137	+	0.843495i	$0.319506\pi$
$114$	−18720.0	−0.134910
$115$	−79560.0	−0.560984
$116$	77088.0	0.531914
$117$	−40662.0	−0.274615
$118$	63408.0	0.419217
$119$	−31458.0	−0.203640
$120$	−44928.0	−0.284816
$121$	14641.0	0.0909091
$122$	−185192.	−1.12648
$123$	−21870.0	−0.130342
$124$	28544.0	0.166710
$125$	12948.0	0.0741187
$126$	−15876.0	−0.0890871
$127$	−341752.	−1.88019	−0.940095	−	0.340912i	$-0.889264\pi$
$127$	−341752.	−1.88019	−0.940095	+	0.340912i	$0.889264\pi$
$128$	−16384.0	−0.0883883
$129$	−206136.	−1.09064
$130$	−156624.	−0.812830
$131$	−110460.	−0.562376	−0.281188	−	0.959653i	$-0.590728\pi$
$131$	−110460.	−0.562376	−0.281188	+	0.959653i	$0.590728\pi$
$132$	−17424.0	−0.0870388
$133$	−25480.0	−0.124902
$134$	−77648.0	−0.373567
$135$	56862.0	0.268527
$136$	41088.0	0.190488
$137$	−123510.	−0.562213	−0.281106	−	0.959677i	$-0.590701\pi$
$137$	−123510.	−0.562213	−0.281106	+	0.959677i	$0.590701\pi$
$138$	36720.0	0.164136
$139$	134720.	0.591419	0.295709	−	0.955278i	$-0.404444\pi$
$139$	134720.	0.591419	0.295709	+	0.955278i	$0.404444\pi$
$140$	−61152.0	−0.263688
$141$	101844.	0.431407
$142$	69168.0	0.287862
$143$	−60742.0	−0.248399
$144$	20736.0	0.0833333
$145$	−375804.	−1.48437
$146$	120856.	0.469230
$147$	−21609.0	−0.0824786
$148$	127328.	0.477826
$149$	−387270.	−1.42905	−0.714526	−	0.699609i	$-0.753358\pi$
$149$	−387270.	−1.42905	−0.714526	+	0.699609i	$0.753358\pi$
$150$	106524.	0.386562
$151$	−463792.	−1.65532	−0.827658	−	0.561233i	$-0.810327\pi$
$151$	−463792.	−1.65532	−0.827658	+	0.561233i	$0.810327\pi$
$152$	33280.0	0.116835
$153$	−52002.0	−0.179594
$154$	−23716.0	−0.0805823
$155$	−139152.	−0.465222
$156$	72288.0	0.237823
$157$	−371122.	−1.20162	−0.600811	−	0.799391i	$-0.705155\pi$
$157$	−371122.	−1.20162	−0.600811	+	0.799391i	$0.705155\pi$
$158$	−142688.	−0.454721
$159$	109998.	0.345058
$160$	79872.0	0.246658
$161$	49980.0	0.151961
$162$	−26244.0	−0.0785674
$163$	−76156.0	−0.224510	−0.112255	−	0.993679i	$-0.535807\pi$
$163$	−76156.0	−0.224510	−0.112255	+	0.993679i	$0.535807\pi$
$164$	38880.0	0.112880
$165$	84942.0	0.242892
$166$	173712.	0.489282
$167$	172200.	0.477795	0.238898	−	0.971045i	$-0.423214\pi$
$167$	172200.	0.477795	0.238898	+	0.971045i	$0.423214\pi$
$168$	28224.0	0.0771517
$169$	−119289.	−0.321280
$170$	−200304.	−0.531578
$171$	−42120.0	−0.110153
$172$	366464.	0.944518
$173$	−601134.	−1.52706	−0.763530	−	0.645772i	$-0.776535\pi$
$173$	−601134.	−1.52706	−0.763530	+	0.645772i	$0.776535\pi$
$174$	173448.	0.434306
$175$	144991.	0.357887
$176$	30976.0	0.0753778
$177$	142668.	0.342289
$178$	59736.0	0.141314
$179$	223572.	0.521537	0.260768	−	0.965401i	$-0.416024\pi$
$179$	223572.	0.521537	0.260768	+	0.965401i	$0.416024\pi$
$180$	−101088.	−0.232551
$181$	181622.	0.412071	0.206035	−	0.978545i	$-0.433944\pi$
$181$	181622.	0.412071	0.206035	+	0.978545i	$0.433944\pi$
$182$	98392.0	0.220182
$183$	−416682.	−0.919765
$184$	−65280.0	−0.142146
$185$	−620724.	−1.33343
$186$	64224.0	0.136118
$187$	−77682.0	−0.162449
$188$	−181056.	−0.373610
$189$	−35721.0	−0.0727393
$190$	−162240.	−0.326042
$191$	117252.	0.232561	0.116280	−	0.993216i	$-0.462903\pi$
$191$	117252.	0.232561	0.116280	+	0.993216i	$0.462903\pi$
$192$	−36864.0	−0.0721688
$193$	−410758.	−0.793766	−0.396883	−	0.917869i	$-0.629908\pi$
$193$	−410758.	−0.793766	−0.396883	+	0.917869i	$0.629908\pi$
$194$	−340424.	−0.649405
$195$	−352404.	−0.663673
$196$	38416.0	0.0714286
$197$	190506.	0.349738	0.174869	−	0.984592i	$-0.444050\pi$
$197$	190506.	0.349738	0.174869	+	0.984592i	$0.444050\pi$
$198$	−39204.0	−0.0710669
$199$	−893560.	−1.59952	−0.799762	−	0.600317i	$-0.795041\pi$
$199$	−893560.	−1.59952	−0.799762	+	0.600317i	$0.795041\pi$
$200$	−189376.	−0.334773
$201$	−174708.	−0.305016
$202$	288600.	0.497643
$203$	236082.	0.402090
$204$	92448.0	0.155533
$205$	−189540.	−0.315004
$206$	−46976.0	−0.0771273
$207$	82620.0	0.134017
$208$	−128512.	−0.205961
$209$	−62920.0	−0.0996375
$210$	−137592.	−0.215300
$211$	294392.	0.455218	0.227609	−	0.973753i	$-0.426909\pi$
$211$	294392.	0.455218	0.227609	+	0.973753i	$0.426909\pi$
$212$	−195552.	−0.298829
$213$	155628.	0.235038
$214$	871728.	1.30121
$215$	−1.78651e6	−2.63578
$216$	46656.0	0.0680414
$217$	87416.0	0.126021
$218$	−350696.	−0.499792
$219$	271926.	0.383125
$220$	−151008.	−0.210350
$221$	322284.	0.443872
$222$	286488.	0.390143
$223$	−399040.	−0.537346	−0.268673	−	0.963231i	$-0.586585\pi$
$223$	−399040.	−0.537346	−0.268673	+	0.963231i	$0.586585\pi$
$224$	−50176.0	−0.0668153
$225$	239679.	0.315627
$226$	−583272.	−0.759626
$227$	86100.0	0.110902	0.0554509	−	0.998461i	$-0.482340\pi$
$227$	86100.0	0.110902	0.0554509	+	0.998461i	$0.482340\pi$
$228$	74880.0	0.0953957
$229$	371126.	0.467663	0.233831	−	0.972277i	$-0.424874\pi$
$229$	371126.	0.467663	0.233831	+	0.972277i	$0.424874\pi$
$230$	318240.	0.396675
$231$	−53361.0	−0.0657952
$232$	−308352.	−0.376120
$233$	−57786.0	−0.0697321	−0.0348661	−	0.999392i	$-0.511100\pi$
$233$	−57786.0	−0.0697321	−0.0348661	+	0.999392i	$0.511100\pi$
$234$	162648.	0.194182
$235$	882648.	1.04260
$236$	−253632.	−0.296431
$237$	−321048.	−0.371278
$238$	125832.	0.143995
$239$	725616.	0.821698	0.410849	−	0.911703i	$-0.365232\pi$
$239$	725616.	0.821698	0.410849	+	0.911703i	$0.365232\pi$
$240$	179712.	0.201395
$241$	625370.	0.693577	0.346788	−	0.937943i	$-0.387272\pi$
$241$	625370.	0.693577	0.346788	+	0.937943i	$0.387272\pi$
$242$	−58564.0	−0.0642824
$243$	−59049.0	−0.0641500
$244$	740768.	0.796540
$245$	−187278.	−0.199329
$246$	87480.0	0.0921661
$247$	261040.	0.272248
$248$	−114176.	−0.117881
$249$	390852.	0.399497
$250$	−51792.0	−0.0524098
$251$	267084.	0.267586	0.133793	−	0.991009i	$-0.457284\pi$
$251$	267084.	0.267586	0.133793	+	0.991009i	$0.457284\pi$
$252$	63504.0	0.0629941
$253$	123420.	0.121223
$254$	1.36701e6	1.32950
$255$	−450684.	−0.434032
$256$	65536.0	0.0625000
$257$	−63726.0	−0.0601844	−0.0300922	−	0.999547i	$-0.509580\pi$
$257$	−63726.0	−0.0601844	−0.0300922	+	0.999547i	$0.509580\pi$
$258$	824544.	0.771196
$259$	389942.	0.361202
$260$	626496.	0.574758
$261$	390258.	0.354610
$262$	441840.	0.397660
$263$	−153888.	−0.137188	−0.0685939	−	0.997645i	$-0.521851\pi$
$263$	−153888.	−0.137188	−0.0685939	+	0.997645i	$0.521851\pi$
$264$	69696.0	0.0615457
$265$	953316.	0.833916
$266$	101920.	0.0883192
$267$	134406.	0.115383
$268$	310592.	0.264152
$269$	1.79997e6	1.51665	0.758324	−	0.651878i	$-0.226019\pi$
$269$	1.79997e6	1.51665	0.758324	+	0.651878i	$0.226019\pi$
$270$	−227448.	−0.189877
$271$	−1.05129e6	−0.869558	−0.434779	−	0.900537i	$-0.643173\pi$
$271$	−1.05129e6	−0.869558	−0.434779	+	0.900537i	$0.643173\pi$
$272$	−164352.	−0.134695
$273$	221382.	0.179778
$274$	494040.	0.397544
$275$	358039.	0.285495
$276$	−146880.	−0.116062
$277$	2.14313e6	1.67822	0.839110	−	0.543961i	$-0.183076\pi$
$277$	2.14313e6	1.67822	0.839110	+	0.543961i	$0.183076\pi$
$278$	−538880.	−0.418196
$279$	144504.	0.111140
$280$	244608.	0.186456
$281$	893910.	0.675349	0.337674	−	0.941263i	$-0.390360\pi$
$281$	893910.	0.675349	0.337674	+	0.941263i	$0.390360\pi$
$282$	−407376.	−0.305051
$283$	665216.	0.493738	0.246869	−	0.969049i	$-0.420598\pi$
$283$	665216.	0.493738	0.246869	+	0.969049i	$0.420598\pi$
$284$	−276672.	−0.203549
$285$	−365040.	−0.266212
$286$	242968.	0.175644
$287$	119070.	0.0853292
$288$	−82944.0	−0.0589256
$289$	−1.00769e6	−0.709714
$290$	1.50322e6	1.04961
$291$	−765954.	−0.530237
$292$	−483424.	−0.331796
$293$	−2.20009e6	−1.49717	−0.748584	−	0.663040i	$-0.769266\pi$
$293$	−2.20009e6	−1.49717	−0.748584	+	0.663040i	$0.769266\pi$
$294$	86436.0	0.0583212
$295$	1.23646e6	0.827225
$296$	−509312.	−0.337874
$297$	−88209.0	−0.0580259
$298$	1.54908e6	1.01049
$299$	−512040.	−0.331227
$300$	−426096.	−0.273341
$301$	1.12230e6	0.713988
$302$	1.85517e6	1.17049
$303$	649350.	0.406324
$304$	−133120.	−0.0826151
$305$	−3.61124e6	−2.22284
$306$	208008.	0.126992
$307$	−491272.	−0.297493	−0.148746	−	0.988875i	$-0.547524\pi$
$307$	−491272.	−0.297493	−0.148746	+	0.988875i	$0.547524\pi$
$308$	94864.0	0.0569803
$309$	−105696.	−0.0629742
$310$	556608.	0.328962
$311$	−105180.	−0.0616641	−0.0308320	−	0.999525i	$-0.509816\pi$
$311$	−105180.	−0.0616641	−0.0308320	+	0.999525i	$0.509816\pi$
$312$	−289152.	−0.168167
$313$	−1.89175e6	−1.09145	−0.545724	−	0.837965i	$-0.683745\pi$
$313$	−1.89175e6	−1.09145	−0.545724	+	0.837965i	$0.683745\pi$
$314$	1.48449e6	0.849674
$315$	−309582.	−0.175792
$316$	570752.	0.321536
$317$	−672198.	−0.375707	−0.187853	−	0.982197i	$-0.560153\pi$
$317$	−672198.	−0.375707	−0.187853	+	0.982197i	$0.560153\pi$
$318$	−439992.	−0.243993
$319$	582978.	0.320756
$320$	−319488.	−0.174413
$321$	1.96139e6	1.06243
$322$	−199920.	−0.107453
$323$	333840.	0.178046
$324$	104976.	0.0555556
$325$	−1.48542e6	−0.780082
$326$	304624.	0.158752
$327$	−789066.	−0.408079
$328$	−155520.	−0.0798181
$329$	−554484.	−0.282422
$330$	−339768.	−0.171750
$331$	−1.07290e6	−0.538256	−0.269128	−	0.963104i	$-0.586736\pi$
$331$	−1.07290e6	−0.538256	−0.269128	+	0.963104i	$0.586736\pi$
$332$	−694848.	−0.345975
$333$	644598.	0.318551
$334$	−688800.	−0.337852
$335$	−1.51414e6	−0.737145
$336$	−112896.	−0.0545545
$337$	1.03582e6	0.496831	0.248416	−	0.968654i	$-0.420090\pi$
$337$	1.03582e6	0.496831	0.248416	+	0.968654i	$0.420090\pi$
$338$	477156.	0.227179
$339$	−1.31236e6	−0.620232
$340$	801216.	0.375883
$341$	215864.	0.100530
$342$	168480.	0.0778902
$343$	117649.	0.0539949
$344$	−1.46586e6	−0.667875
$345$	716040.	0.323884
$346$	2.40454e6	1.07979
$347$	−1.52821e6	−0.681334	−0.340667	−	0.940184i	$-0.610653\pi$
$347$	−1.52821e6	−0.681334	−0.340667	+	0.940184i	$0.610653\pi$
$348$	−693792.	−0.307101
$349$	−1.45205e6	−0.638141	−0.319071	−	0.947731i	$-0.603371\pi$
$349$	−1.45205e6	−0.638141	−0.319071	+	0.947731i	$0.603371\pi$
$350$	−579964.	−0.253064
$351$	365958.	0.158549
$352$	−123904.	−0.0533002
$353$	−184926.	−0.0789880	−0.0394940	−	0.999220i	$-0.512575\pi$
$353$	−184926.	−0.0789880	−0.0394940	+	0.999220i	$0.512575\pi$
$354$	−570672.	−0.242035
$355$	1.34878e6	0.568027
$356$	−238944.	−0.0999243
$357$	283122.	0.117572
$358$	−894288.	−0.368782
$359$	1.09661e6	0.449071	0.224536	−	0.974466i	$-0.427913\pi$
$359$	1.09661e6	0.449071	0.224536	+	0.974466i	$0.427913\pi$
$360$	404352.	0.164438
$361$	−2.20570e6	−0.890796
$362$	−726488.	−0.291378
$363$	−131769.	−0.0524864
$364$	−393568.	−0.155692
$365$	2.35669e6	0.925914
$366$	1.66673e6	0.650372
$367$	3.73587e6	1.44786	0.723930	−	0.689873i	$-0.242334\pi$
$367$	3.73587e6	1.44786	0.723930	+	0.689873i	$0.242334\pi$
$368$	261120.	0.100513
$369$	196830.	0.0752533
$370$	2.48290e6	0.942875
$371$	−598878.	−0.225893
$372$	−256896.	−0.0962498
$373$	4.10724e6	1.52854	0.764272	−	0.644893i	$-0.223098\pi$
$373$	4.10724e6	1.52854	0.764272	+	0.644893i	$0.223098\pi$
$374$	310728.	0.114869
$375$	−116532.	−0.0427924
$376$	724224.	0.264182
$377$	−2.41864e6	−0.876430
$378$	142884.	0.0514344
$379$	3.18638e6	1.13946	0.569731	−	0.821832i	$-0.307048\pi$
$379$	3.18638e6	1.13946	0.569731	+	0.821832i	$0.307048\pi$
$380$	648960.	0.230547
$381$	3.07577e6	1.08553
$382$	−469008.	−0.164445
$383$	−4.20383e6	−1.46436	−0.732180	−	0.681111i	$-0.761497\pi$
$383$	−4.20383e6	−1.46436	−0.732180	+	0.681111i	$0.761497\pi$
$384$	147456.	0.0510310
$385$	−462462.	−0.159010
$386$	1.64303e6	0.561278
$387$	1.85522e6	0.629679
$388$	1.36170e6	0.459199
$389$	−3.32540e6	−1.11422	−0.557108	−	0.830440i	$-0.688089\pi$
$389$	−3.32540e6	−1.11422	−0.557108	+	0.830440i	$0.688089\pi$
$390$	1.40962e6	0.469288
$391$	−654840.	−0.216617
$392$	−153664.	−0.0505076
$393$	994140.	0.324688
$394$	−762024.	−0.247302
$395$	−2.78242e6	−0.897283
$396$	156816.	0.0502519
$397$	5.34467e6	1.70194	0.850971	−	0.525213i	$-0.176014\pi$
$397$	5.34467e6	1.70194	0.850971	+	0.525213i	$0.176014\pi$
$398$	3.57424e6	1.13103
$399$	229320.	0.0721124
$400$	757504.	0.236720
$401$	−17262.0	−0.00536081	−0.00268040	−	0.999996i	$-0.500853\pi$
$401$	−17262.0	−0.00536081	−0.00268040	+	0.999996i	$0.500853\pi$
$402$	698832.	0.215679
$403$	−895568.	−0.274686
$404$	−1.15440e6	−0.351887
$405$	−511758.	−0.155034
$406$	−944328.	−0.284320
$407$	962918.	0.288140
$408$	−369792.	−0.109978
$409$	−591886.	−0.174956	−0.0874782	−	0.996166i	$-0.527881\pi$
$409$	−591886.	−0.174956	−0.0874782	+	0.996166i	$0.527881\pi$
$410$	758160.	0.222742
$411$	1.11159e6	0.324594
$412$	187904.	0.0545372
$413$	−776748.	−0.224081
$414$	−330480.	−0.0947642
$415$	3.38738e6	0.965482
$416$	514048.	0.145637
$417$	−1.21248e6	−0.341456
$418$	251680.	0.0704544
$419$	1.64303e6	0.457204	0.228602	−	0.973520i	$-0.426585\pi$
$419$	1.64303e6	0.457204	0.228602	+	0.973520i	$0.426585\pi$
$420$	550368.	0.152240
$421$	−2.62575e6	−0.722019	−0.361010	−	0.932562i	$-0.617568\pi$
$421$	−2.62575e6	−0.722019	−0.361010	+	0.932562i	$0.617568\pi$
$422$	−1.17757e6	−0.321888
$423$	−916596.	−0.249073
$424$	782208.	0.211304
$425$	−1.89968e6	−0.510161
$426$	−622512.	−0.166197
$427$	2.26860e6	0.602128
$428$	−3.48691e6	−0.920093
$429$	546678.	0.143413
$430$	7.14605e6	1.86378
$431$	−1.79868e6	−0.466402	−0.233201	−	0.972429i	$-0.574920\pi$
$431$	−1.79868e6	−0.466402	−0.233201	+	0.972429i	$0.574920\pi$
$432$	−186624.	−0.0481125
$433$	−2.84174e6	−0.728392	−0.364196	−	0.931322i	$-0.618656\pi$
$433$	−2.84174e6	−0.728392	−0.364196	+	0.931322i	$0.618656\pi$
$434$	−349664.	−0.0891100
$435$	3.38224e6	0.857000
$436$	1.40278e6	0.353407
$437$	−530400.	−0.132862
$438$	−1.08770e6	−0.270910
$439$	6.94278e6	1.71938	0.859690	−	0.510816i	$-0.170657\pi$
$439$	6.94278e6	1.71938	0.859690	+	0.510816i	$0.170657\pi$
$440$	604032.	0.148740
$441$	194481.	0.0476190
$442$	−1.28914e6	−0.313865
$443$	887508.	0.214864	0.107432	−	0.994212i	$-0.465737\pi$
$443$	887508.	0.214864	0.107432	+	0.994212i	$0.465737\pi$
$444$	−1.14595e6	−0.275873
$445$	1.16485e6	0.278850
$446$	1.59616e6	0.379961
$447$	3.48543e6	0.825064
$448$	200704.	0.0472456
$449$	−6.17391e6	−1.44525	−0.722627	−	0.691238i	$-0.757066\pi$
$449$	−6.17391e6	−1.44525	−0.722627	+	0.691238i	$0.757066\pi$
$450$	−958716.	−0.223182
$451$	294030.	0.0680691
$452$	2.33309e6	0.537137
$453$	4.17413e6	0.955697
$454$	−344400.	−0.0784194
$455$	1.91864e6	0.434476
$456$	−299520.	−0.0674549
$457$	−6.54564e6	−1.46609	−0.733046	−	0.680179i	$-0.761902\pi$
$457$	−6.54564e6	−1.46609	−0.733046	+	0.680179i	$0.761902\pi$
$458$	−1.48450e6	−0.330687
$459$	468018.	0.103689
$460$	−1.27296e6	−0.280492
$461$	3.59248e6	0.787304	0.393652	−	0.919260i	$-0.371212\pi$
$461$	3.59248e6	0.787304	0.393652	+	0.919260i	$0.371212\pi$
$462$	213444.	0.0465242
$463$	−1.51974e6	−0.329472	−0.164736	−	0.986338i	$-0.552677\pi$
$463$	−1.51974e6	−0.329472	−0.164736	+	0.986338i	$0.552677\pi$
$464$	1.23341e6	0.265957
$465$	1.25237e6	0.268596
$466$	231144.	0.0493081
$467$	1.33352e6	0.282949	0.141475	−	0.989942i	$-0.454816\pi$
$467$	1.33352e6	0.282949	0.141475	+	0.989942i	$0.454816\pi$
$468$	−650592.	−0.137307
$469$	951188.	0.199680
$470$	−3.53059e6	−0.737230
$471$	3.34010e6	0.693756
$472$	1.01453e6	0.209609
$473$	2.77138e6	0.569566
$474$	1.28419e6	0.262533
$475$	−1.53868e6	−0.312906
$476$	−503328.	−0.101820
$477$	−989982.	−0.199219
$478$	−2.90246e6	−0.581028
$479$	5.15705e6	1.02698	0.513490	−	0.858095i	$-0.328352\pi$
$479$	5.15705e6	1.02698	0.513490	+	0.858095i	$0.328352\pi$
$480$	−718848.	−0.142408
$481$	−3.99492e6	−0.787309
$482$	−2.50148e6	−0.490433
$483$	−449820.	−0.0877346
$484$	234256.	0.0454545
$485$	−6.63827e6	−1.28145
$486$	236196.	0.0453609
$487$	−3.42899e6	−0.655155	−0.327578	−	0.944824i	$-0.606232\pi$
$487$	−3.42899e6	−0.655155	−0.327578	+	0.944824i	$0.606232\pi$
$488$	−2.96307e6	−0.563239
$489$	685404.	0.129621
$490$	749112.	0.140947
$491$	9.44840e6	1.76870	0.884351	−	0.466822i	$-0.154601\pi$
$491$	9.44840e6	1.76870	0.884351	+	0.466822i	$0.154601\pi$
$492$	−349920.	−0.0651712
$493$	−3.09316e6	−0.573171
$494$	−1.04416e6	−0.192508
$495$	−764478.	−0.140234
$496$	456704.	0.0833548
$497$	−847308.	−0.153869
$498$	−1.56341e6	−0.282487
$499$	−4.07717e6	−0.733006	−0.366503	−	0.930417i	$-0.619445\pi$
$499$	−4.07717e6	−0.733006	−0.366503	+	0.930417i	$0.619445\pi$
$500$	207168.	0.0370593
$501$	−1.54980e6	−0.275855
$502$	−1.06834e6	−0.189212
$503$	−4.32338e6	−0.761910	−0.380955	−	0.924594i	$-0.624405\pi$
$503$	−4.32338e6	−0.761910	−0.380955	+	0.924594i	$0.624405\pi$
$504$	−254016.	−0.0445435
$505$	5.62770e6	0.981980
$506$	−493680.	−0.0857174
$507$	1.07360e6	0.185491
$508$	−5.46803e6	−0.940095
$509$	−6.35043e6	−1.08645	−0.543224	−	0.839588i	$-0.682796\pi$
$509$	−6.35043e6	−1.08645	−0.543224	+	0.839588i	$0.682796\pi$
$510$	1.80274e6	0.306907
$511$	−1.48049e6	−0.250814
$512$	−262144.	−0.0441942
$513$	379080.	0.0635971
$514$	254904.	0.0425568
$515$	−916032.	−0.152192
$516$	−3.29818e6	−0.545318
$517$	−1.36924e6	−0.225295
$518$	−1.55977e6	−0.255409
$519$	5.41021e6	0.881648
$520$	−2.50598e6	−0.406415
$521$	3.12777e6	0.504825	0.252412	−	0.967620i	$-0.418776\pi$
$521$	3.12777e6	0.504825	0.252412	+	0.967620i	$0.418776\pi$
$522$	−1.56103e6	−0.250747
$523$	−6.76898e6	−1.08210	−0.541052	−	0.840989i	$-0.681974\pi$
$523$	−6.76898e6	−1.08210	−0.541052	+	0.840989i	$0.681974\pi$
$524$	−1.76736e6	−0.281188
$525$	−1.30492e6	−0.206626
$526$	615552.	0.0970064
$527$	−1.14533e6	−0.179640
$528$	−278784.	−0.0435194
$529$	−5.39594e6	−0.838355
$530$	−3.81326e6	−0.589668
$531$	−1.28401e6	−0.197621
$532$	−407680.	−0.0624511
$533$	−1.21986e6	−0.185991
$534$	−537624.	−0.0815879
$535$	1.69987e7	2.56762
$536$	−1.24237e6	−0.186783
$537$	−2.01215e6	−0.301109
$538$	−7.19988e6	−1.07243
$539$	290521.	0.0430730
$540$	909792.	0.134263
$541$	−7.42759e6	−1.09108	−0.545538	−	0.838086i	$-0.683675\pi$
$541$	−7.42759e6	−1.09108	−0.545538	+	0.838086i	$0.683675\pi$
$542$	4.20515e6	0.614870
$543$	−1.63460e6	−0.237909
$544$	657408.	0.0952440
$545$	−6.83857e6	−0.986221
$546$	−885528.	−0.127122
$547$	−1.14050e7	−1.62977	−0.814885	−	0.579623i	$-0.803200\pi$
$547$	−1.14050e7	−1.62977	−0.814885	+	0.579623i	$0.803200\pi$
$548$	−1.97616e6	−0.281106
$549$	3.75014e6	0.531027
$550$	−1.43216e6	−0.201875
$551$	−2.50536e6	−0.351553
$552$	587520.	0.0820682
$553$	1.74793e6	0.243058
$554$	−8.57252e6	−1.18668
$555$	5.58652e6	0.769854
$556$	2.15552e6	0.295709
$557$	−7.50733e6	−1.02529	−0.512646	−	0.858600i	$-0.671335\pi$
$557$	−7.50733e6	−1.02529	−0.512646	+	0.858600i	$0.671335\pi$
$558$	−578016.	−0.0785877
$559$	−1.14978e7	−1.55627
$560$	−978432.	−0.131844
$561$	699138.	0.0937898
$562$	−3.57564e6	−0.477544
$563$	3.67782e6	0.489012	0.244506	−	0.969648i	$-0.421374\pi$
$563$	3.67782e6	0.489012	0.244506	+	0.969648i	$0.421374\pi$
$564$	1.62950e6	0.215704
$565$	−1.13738e7	−1.49894
$566$	−2.66086e6	−0.349126
$567$	321489.	0.0419961
$568$	1.10669e6	0.143931
$569$	330558.	0.0428023	0.0214011	−	0.999771i	$-0.493187\pi$
$569$	330558.	0.0428023	0.0214011	+	0.999771i	$0.493187\pi$
$570$	1.46016e6	0.188241
$571$	−3.91658e6	−0.502709	−0.251354	−	0.967895i	$-0.580876\pi$
$571$	−3.91658e6	−0.502709	−0.251354	+	0.967895i	$0.580876\pi$
$572$	−971872.	−0.124199
$573$	−1.05527e6	−0.134269
$574$	−476280.	−0.0603368
$575$	3.01818e6	0.380694
$576$	331776.	0.0416667
$577$	−1.64371e6	−0.205535	−0.102767	−	0.994705i	$-0.532770\pi$
$577$	−1.64371e6	−0.205535	−0.102767	+	0.994705i	$0.532770\pi$
$578$	4.03077e6	0.501844
$579$	3.69682e6	0.458281
$580$	−6.01286e6	−0.742184
$581$	−2.12797e6	−0.261532
$582$	3.06382e6	0.374934
$583$	−1.47886e6	−0.180201
$584$	1.93370e6	0.234615
$585$	3.17164e6	0.383172
$586$	8.80034e6	1.05866
$587$	1.43860e7	1.72323	0.861616	−	0.507560i	$-0.169452\pi$
$587$	1.43860e7	1.72323	0.861616	+	0.507560i	$0.169452\pi$
$588$	−345744.	−0.0412393
$589$	−927680.	−0.110182
$590$	−4.94582e6	−0.584936
$591$	−1.71455e6	−0.201921
$592$	2.03725e6	0.238913
$593$	−1.49626e7	−1.74731	−0.873653	−	0.486549i	$-0.838255\pi$
$593$	−1.49626e7	−1.74731	−0.873653	+	0.486549i	$0.838255\pi$
$594$	352836.	0.0410305
$595$	2.45372e6	0.284141
$596$	−6.19632e6	−0.714526
$597$	8.04204e6	0.923486
$598$	2.04816e6	0.234213
$599$	904956.	0.103053	0.0515265	−	0.998672i	$-0.483591\pi$
$599$	904956.	0.103053	0.0515265	+	0.998672i	$0.483591\pi$
$600$	1.70438e6	0.193281
$601$	6.11184e6	0.690217	0.345109	−	0.938563i	$-0.387842\pi$
$601$	6.11184e6	0.690217	0.345109	+	0.938563i	$0.387842\pi$
$602$	−4.48918e6	−0.504866
$603$	1.57237e6	0.176101
$604$	−7.42067e6	−0.827658
$605$	−1.14200e6	−0.126846
$606$	−2.59740e6	−0.287314
$607$	−1.03513e7	−1.14031	−0.570155	−	0.821537i	$-0.693117\pi$
$607$	−1.03513e7	−1.14031	−0.570155	+	0.821537i	$0.693117\pi$
$608$	532480.	0.0584177
$609$	−2.12474e6	−0.232147
$610$	1.44450e7	1.57178
$611$	5.68063e6	0.615593
$612$	−832032.	−0.0897969
$613$	4.54354e6	0.488363	0.244182	−	0.969730i	$-0.421481\pi$
$613$	4.54354e6	0.488363	0.244182	+	0.969730i	$0.421481\pi$
$614$	1.96509e6	0.210359
$615$	1.70586e6	0.181868
$616$	−379456.	−0.0402911
$617$	−8.34929e6	−0.882951	−0.441476	−	0.897273i	$-0.645545\pi$
$617$	−8.34929e6	−0.882951	−0.441476	+	0.897273i	$0.645545\pi$
$618$	422784.	0.0445295
$619$	5.15954e6	0.541233	0.270617	−	0.962687i	$-0.412772\pi$
$619$	5.15954e6	0.541233	0.270617	+	0.962687i	$0.412772\pi$
$620$	−2.22643e6	−0.232611
$621$	−743580.	−0.0773747
$622$	420720.	0.0436031
$623$	−731766.	−0.0755357
$624$	1.15661e6	0.118912
$625$	−1.02568e7	−1.05030
$626$	7.56700e6	0.771770
$627$	566280.	0.0575258
$628$	−5.93795e6	−0.600811
$629$	−5.10904e6	−0.514888
$630$	1.23833e6	0.124304
$631$	3.33812e6	0.333756	0.166878	−	0.985978i	$-0.446631\pi$
$631$	3.33812e6	0.333756	0.166878	+	0.985978i	$0.446631\pi$
$632$	−2.28301e6	−0.227360
$633$	−2.64953e6	−0.262821
$634$	2.68879e6	0.265665
$635$	2.66567e7	2.62344
$636$	1.75997e6	0.172529
$637$	−1.20530e6	−0.117692
$638$	−2.33191e6	−0.226809
$639$	−1.40065e6	−0.135699
$640$	1.27795e6	0.123329
$641$	−1.51004e7	−1.45159	−0.725795	−	0.687911i	$-0.758528\pi$
$641$	−1.51004e7	−1.45159	−0.725795	+	0.687911i	$0.758528\pi$
$642$	−7.84555e6	−0.751253
$643$	−1.71050e7	−1.63153	−0.815766	−	0.578382i	$-0.803684\pi$
$643$	−1.71050e7	−1.63153	−0.815766	+	0.578382i	$0.803684\pi$
$644$	799680.	0.0759804
$645$	1.60786e7	1.52177
$646$	−1.33536e6	−0.125897
$647$	−8.36243e6	−0.785365	−0.392683	−	0.919674i	$-0.628453\pi$
$647$	−8.36243e6	−0.785365	−0.392683	+	0.919674i	$0.628453\pi$
$648$	−419904.	−0.0392837
$649$	−1.91809e6	−0.178755
$650$	5.94167e6	0.551601
$651$	−786744.	−0.0727580
$652$	−1.21850e6	−0.112255
$653$	1.29485e7	1.18832	0.594162	−	0.804345i	$-0.297484\pi$
$653$	1.29485e7	1.18832	0.594162	+	0.804345i	$0.297484\pi$
$654$	3.15626e6	0.288555
$655$	8.61588e6	0.784687
$656$	622080.	0.0564400
$657$	−2.44733e6	−0.221197
$658$	2.21794e6	0.199703
$659$	−5.39982e6	−0.484357	−0.242179	−	0.970232i	$-0.577862\pi$
$659$	−5.39982e6	−0.484357	−0.242179	+	0.970232i	$0.577862\pi$
$660$	1.35907e6	0.121446
$661$	−3.63908e6	−0.323958	−0.161979	−	0.986794i	$-0.551788\pi$
$661$	−3.63908e6	−0.323958	−0.161979	+	0.986794i	$0.551788\pi$
$662$	4.29160e6	0.380605
$663$	−2.90056e6	−0.256270
$664$	2.77939e6	0.244641
$665$	1.98744e6	0.174277
$666$	−2.57839e6	−0.225249
$667$	4.91436e6	0.427713
$668$	2.75520e6	0.238898
$669$	3.59136e6	0.310237
$670$	6.05654e6	0.521240
$671$	5.60206e6	0.480332
$672$	451584.	0.0385758
$673$	5.82483e6	0.495730	0.247865	−	0.968795i	$-0.420271\pi$
$673$	5.82483e6	0.495730	0.247865	+	0.968795i	$0.420271\pi$
$674$	−4.14327e6	−0.351313
$675$	−2.15711e6	−0.182227
$676$	−1.90862e6	−0.160640
$677$	−1.36657e6	−0.114593	−0.0572966	−	0.998357i	$-0.518248\pi$
$677$	−1.36657e6	−0.114593	−0.0572966	+	0.998357i	$0.518248\pi$
$678$	5.24945e6	0.438570
$679$	4.17019e6	0.347122
$680$	−3.20486e6	−0.265789
$681$	−774900.	−0.0640292
$682$	−863456.	−0.0710852
$683$	5.67865e6	0.465794	0.232897	−	0.972501i	$-0.425180\pi$
$683$	5.67865e6	0.465794	0.232897	+	0.972501i	$0.425180\pi$
$684$	−673920.	−0.0550767
$685$	9.63378e6	0.784459
$686$	−470596.	−0.0381802
$687$	−3.34013e6	−0.270005
$688$	5.86342e6	0.472259
$689$	6.13544e6	0.492377
$690$	−2.86416e6	−0.229021
$691$	2.89483e6	0.230636	0.115318	−	0.993329i	$-0.463211\pi$
$691$	2.89483e6	0.230636	0.115318	+	0.993329i	$0.463211\pi$
$692$	−9.61814e6	−0.763530
$693$	480249.	0.0379869
$694$	6.11285e6	0.481776
$695$	−1.05082e7	−0.825210
$696$	2.77517e6	0.217153
$697$	−1.56006e6	−0.121635
$698$	5.80818e6	0.451234
$699$	520074.	0.0402599
$700$	2.31986e6	0.178944
$701$	1.75699e7	1.35044	0.675218	−	0.737618i	$-0.264050\pi$
$701$	1.75699e7	1.35044	0.675218	+	0.737618i	$0.264050\pi$
$702$	−1.46383e6	−0.112111
$703$	−4.13816e6	−0.315805
$704$	495616.	0.0376889
$705$	−7.94383e6	−0.601946
$706$	739704.	0.0558530
$707$	−3.53535e6	−0.266001
$708$	2.28269e6	0.171145
$709$	−1.92613e7	−1.43903	−0.719515	−	0.694477i	$-0.755636\pi$
$709$	−1.92613e7	−1.43903	−0.719515	+	0.694477i	$0.755636\pi$
$710$	−5.39510e6	−0.401656
$711$	2.88943e6	0.214357
$712$	955776.	0.0706572
$713$	1.81968e6	0.134051
$714$	−1.13249e6	−0.0831358
$715$	4.73788e6	0.346592
$716$	3.57715e6	0.260768
$717$	−6.53054e6	−0.474407
$718$	−4.38643e6	−0.317541
$719$	−2.88337e6	−0.208007	−0.104004	−	0.994577i	$-0.533165\pi$
$719$	−2.88337e6	−0.208007	−0.104004	+	0.994577i	$0.533165\pi$
$720$	−1.61741e6	−0.116276
$721$	575456.	0.0412263
$722$	8.82280e6	0.629888
$723$	−5.62833e6	−0.400437
$724$	2.90595e6	0.206035
$725$	1.42565e7	1.00732
$726$	527076.	0.0371135
$727$	5.01217e6	0.351714	0.175857	−	0.984416i	$-0.443730\pi$
$727$	5.01217e6	0.351714	0.175857	+	0.984416i	$0.443730\pi$
$728$	1.57427e6	0.110091
$729$	531441.	0.0370370
$730$	−9.42677e6	−0.654720
$731$	−1.47044e7	−1.01778
$732$	−6.66691e6	−0.459883
$733$	416426.	0.0286271	0.0143136	−	0.999898i	$-0.495444\pi$
$733$	416426.	0.0286271	0.0143136	+	0.999898i	$0.495444\pi$
$734$	−1.49435e7	−1.02379
$735$	1.68550e6	0.115083
$736$	−1.04448e6	−0.0710732
$737$	2.34885e6	0.159289
$738$	−787320.	−0.0532121
$739$	2.26842e7	1.52796	0.763979	−	0.645241i	$-0.223243\pi$
$739$	2.26842e7	1.52796	0.763979	+	0.645241i	$0.223243\pi$
$740$	−9.93158e6	−0.666713
$741$	−2.34936e6	−0.157182
$742$	2.39551e6	0.159731
$743$	2.74750e7	1.82585	0.912925	−	0.408128i	$-0.133818\pi$
$743$	2.74750e7	1.82585	0.912925	+	0.408128i	$0.133818\pi$
$744$	1.02758e6	0.0680589
$745$	3.02071e7	1.99397
$746$	−1.64290e7	−1.08084
$747$	−3.51767e6	−0.230650
$748$	−1.24291e6	−0.0812244
$749$	−1.06787e7	−0.695525
$750$	466128.	0.0302588
$751$	−2.96805e6	−0.192031	−0.0960154	−	0.995380i	$-0.530610\pi$
$751$	−2.96805e6	−0.192031	−0.0960154	+	0.995380i	$0.530610\pi$
$752$	−2.89690e6	−0.186805
$753$	−2.40376e6	−0.154491
$754$	9.67454e6	0.619729
$755$	3.61758e7	2.30967
$756$	−571536.	−0.0363696
$757$	1.75593e7	1.11370	0.556849	−	0.830614i	$-0.312010\pi$
$757$	1.75593e7	1.11370	0.556849	+	0.830614i	$0.312010\pi$
$758$	−1.27455e7	−0.805721
$759$	−1.11078e6	−0.0699880
$760$	−2.59584e6	−0.163021
$761$	−2.23450e7	−1.39868	−0.699341	−	0.714789i	$-0.746523\pi$
$761$	−2.23450e7	−1.39868	−0.699341	+	0.714789i	$0.746523\pi$
$762$	−1.23031e7	−0.767584
$763$	4.29603e6	0.267150
$764$	1.87603e6	0.116280
$765$	4.05616e6	0.250588
$766$	1.68153e7	1.03546
$767$	7.95770e6	0.488427
$768$	−589824.	−0.0360844
$769$	−1.11300e7	−0.678705	−0.339353	−	0.940659i	$-0.610208\pi$
$769$	−1.11300e7	−0.678705	−0.339353	+	0.940659i	$0.610208\pi$
$770$	1.84985e6	0.112437
$771$	573534.	0.0347475
$772$	−6.57213e6	−0.396883
$773$	1.37300e7	0.826461	0.413230	−	0.910627i	$-0.364400\pi$
$773$	1.37300e7	0.826461	0.413230	+	0.910627i	$0.364400\pi$
$774$	−7.42090e6	−0.445250
$775$	5.27886e6	0.315708
$776$	−5.44678e6	−0.324703
$777$	−3.50948e6	−0.208540
$778$	1.33016e7	0.787870
$779$	−1.26360e6	−0.0746047
$780$	−5.63846e6	−0.331837
$781$	−2.09233e6	−0.122745
$782$	2.61936e6	0.153172
$783$	−3.51232e6	−0.204734
$784$	614656.	0.0357143
$785$	2.89475e7	1.67663
$786$	−3.97656e6	−0.229589
$787$	−1.86313e7	−1.07227	−0.536137	−	0.844131i	$-0.680117\pi$
$787$	−1.86313e7	−1.07227	−0.536137	+	0.844131i	$0.680117\pi$
$788$	3.04810e6	0.174869
$789$	1.38499e6	0.0792054
$790$	1.11297e7	0.634475
$791$	7.14508e6	0.406037
$792$	−627264.	−0.0355335
$793$	−2.32416e7	−1.31245
$794$	−2.13787e7	−1.20345
$795$	−8.57984e6	−0.481462
$796$	−1.42970e7	−0.799762
$797$	5.06039e6	0.282188	0.141094	−	0.989996i	$-0.454938\pi$
$797$	5.06039e6	0.282188	0.141094	+	0.989996i	$0.454938\pi$
$798$	−917280.	−0.0509911
$799$	7.26487e6	0.402588
$800$	−3.03002e6	−0.167386
$801$	−1.20965e6	−0.0666162
$802$	69048.0	0.00379066
$803$	−3.65589e6	−0.200080
$804$	−2.79533e6	−0.152508
$805$	−3.89844e6	−0.212032
$806$	3.58227e6	0.194232
$807$	−1.61997e7	−0.875637
$808$	4.61760e6	0.248822
$809$	−1.19684e7	−0.642933	−0.321467	−	0.946921i	$-0.604176\pi$
$809$	−1.19684e7	−0.642933	−0.321467	+	0.946921i	$0.604176\pi$
$810$	2.04703e6	0.109626
$811$	−2.69460e7	−1.43861	−0.719305	−	0.694695i	$-0.755539\pi$
$811$	−2.69460e7	−1.43861	−0.719305	+	0.694695i	$0.755539\pi$
$812$	3.77731e6	0.201045
$813$	9.46159e6	0.502039
$814$	−3.85167e6	−0.203746
$815$	5.94017e6	0.313260
$816$	1.47917e6	0.0777664
$817$	−1.19101e7	−0.624251
$818$	2.36754e6	0.123713
$819$	−1.99244e6	−0.103795
$820$	−3.03264e6	−0.157502
$821$	−9.46724e6	−0.490191	−0.245096	−	0.969499i	$-0.578819\pi$
$821$	−9.46724e6	−0.490191	−0.245096	+	0.969499i	$0.578819\pi$
$822$	−4.44636e6	−0.229522
$823$	−2.20562e7	−1.13509	−0.567546	−	0.823342i	$-0.692107\pi$
$823$	−2.20562e7	−1.13509	−0.567546	+	0.823342i	$0.692107\pi$
$824$	−751616.	−0.0385636
$825$	−3.22235e6	−0.164831
$826$	3.10699e6	0.158449
$827$	1.22724e7	0.623971	0.311986	−	0.950087i	$-0.399006\pi$
$827$	1.22724e7	0.623971	0.311986	+	0.950087i	$0.399006\pi$
$828$	1.32192e6	0.0670084
$829$	6.34453e6	0.320637	0.160318	−	0.987065i	$-0.448748\pi$
$829$	6.34453e6	0.320637	0.160318	+	0.987065i	$0.448748\pi$
$830$	−1.35495e7	−0.682699
$831$	−1.92882e7	−0.968921
$832$	−2.05619e6	−0.102981
$833$	−1.54144e6	−0.0769688
$834$	4.84992e6	0.241446
$835$	−1.34316e7	−0.666671
$836$	−1.00672e6	−0.0498188
$837$	−1.30054e6	−0.0641666
$838$	−6.57211e6	−0.323292
$839$	1.82048e7	0.892857	0.446428	−	0.894819i	$-0.352696\pi$
$839$	1.82048e7	0.892857	0.446428	+	0.894819i	$0.352696\pi$
$840$	−2.20147e6	−0.107650
$841$	2.70198e6	0.131732
$842$	1.05030e7	0.510545
$843$	−8.04519e6	−0.389913
$844$	4.71027e6	0.227609
$845$	9.30454e6	0.448284
$846$	3.66638e6	0.176121
$847$	717409.	0.0343604
$848$	−3.12883e6	−0.149414
$849$	−5.98694e6	−0.285060
$850$	7.59871e6	0.360739
$851$	8.11716e6	0.384220
$852$	2.49005e6	0.117519
$853$	−595294.	−0.0280130	−0.0140065	−	0.999902i	$-0.504459\pi$
$853$	−595294.	−0.0280130	−0.0140065	+	0.999902i	$0.504459\pi$
$854$	−9.07441e6	−0.425769
$855$	3.28536e6	0.153698
$856$	1.39476e7	0.650604
$857$	−1.98720e7	−0.924250	−0.462125	−	0.886815i	$-0.652913\pi$
$857$	−1.98720e7	−0.924250	−0.462125	+	0.886815i	$0.652913\pi$
$858$	−2.18671e6	−0.101408
$859$	−4.44092e6	−0.205348	−0.102674	−	0.994715i	$-0.532740\pi$
$859$	−4.44092e6	−0.205348	−0.102674	+	0.994715i	$0.532740\pi$
$860$	−2.85842e7	−1.31789
$861$	−1.07163e6	−0.0492648
$862$	7.19472e6	0.329796
$863$	−3.04580e7	−1.39211	−0.696055	−	0.717988i	$-0.745063\pi$
$863$	−3.04580e7	−1.39211	−0.696055	+	0.717988i	$0.745063\pi$
$864$	746496.	0.0340207
$865$	4.68885e7	2.13072
$866$	1.13670e7	0.515051
$867$	9.06924e6	0.409754
$868$	1.39866e6	0.0630103
$869$	4.31631e6	0.193894
$870$	−1.35289e7	−0.605990
$871$	−9.74482e6	−0.435240
$872$	−5.61114e6	−0.249896
$873$	6.89359e6	0.306133
$874$	2.12160e6	0.0939474
$875$	634452.	0.0280142
$876$	4.35082e6	0.191562
$877$	4.11013e7	1.80450	0.902249	−	0.431215i	$-0.141915\pi$
$877$	4.11013e7	1.80450	0.902249	+	0.431215i	$0.141915\pi$
$878$	−2.77711e7	−1.21579
$879$	1.98008e7	0.864390
$880$	−2.41613e6	−0.105175
$881$	2.45200e7	1.06434	0.532171	−	0.846637i	$-0.321376\pi$
$881$	2.45200e7	1.06434	0.532171	+	0.846637i	$0.321376\pi$
$882$	−777924.	−0.0336718
$883$	−3.08283e6	−0.133060	−0.0665300	−	0.997784i	$-0.521193\pi$
$883$	−3.08283e6	−0.133060	−0.0665300	+	0.997784i	$0.521193\pi$
$884$	5.15654e6	0.221936
$885$	−1.11281e7	−0.477599
$886$	−3.55003e6	−0.151932
$887$	2.37525e7	1.01368	0.506839	−	0.862041i	$-0.330814\pi$
$887$	2.37525e7	1.01368	0.506839	+	0.862041i	$0.330814\pi$
$888$	4.58381e6	0.195072
$889$	−1.67458e7	−0.710645
$890$	−4.65941e6	−0.197177
$891$	793881.	0.0335013
$892$	−6.38464e6	−0.268673
$893$	5.88432e6	0.246926
$894$	−1.39417e7	−0.583408
$895$	−1.74386e7	−0.727703
$896$	−802816.	−0.0334077
$897$	4.60836e6	0.191234
$898$	2.46956e7	1.02195
$899$	8.59531e6	0.354701
$900$	3.83486e6	0.157813
$901$	7.84652e6	0.322007
$902$	−1.17612e6	−0.0481322
$903$	−1.01007e7	−0.412221
$904$	−9.33235e6	−0.379813
$905$	−1.41665e7	−0.574965
$906$	−1.66965e7	−0.675780
$907$	2.50973e7	1.01300	0.506500	−	0.862240i	$-0.330939\pi$
$907$	2.50973e7	1.01300	0.506500	+	0.862240i	$0.330939\pi$
$908$	1.37760e6	0.0554509
$909$	−5.84415e6	−0.234591
$910$	−7.67458e6	−0.307221
$911$	2.39629e7	0.956627	0.478314	−	0.878189i	$-0.341248\pi$
$911$	2.39629e7	0.956627	0.478314	+	0.878189i	$0.341248\pi$
$912$	1.19808e6	0.0476978
$913$	−5.25479e6	−0.208631
$914$	2.61826e7	1.03668
$915$	3.25012e7	1.28335
$916$	5.93802e6	0.233831
$917$	−5.41254e6	−0.212558
$918$	−1.87207e6	−0.0733189
$919$	9.36056e6	0.365606	0.182803	−	0.983150i	$-0.441483\pi$
$919$	9.36056e6	0.365606	0.182803	+	0.983150i	$0.441483\pi$
$920$	5.09184e6	0.198338
$921$	4.42145e6	0.171757
$922$	−1.43699e7	−0.556708
$923$	8.68058e6	0.335386
$924$	−853776.	−0.0328976
$925$	2.35477e7	0.904887
$926$	6.07898e6	0.232972
$927$	951264.	0.0363581
$928$	−4.93363e6	−0.188060
$929$	2.56159e6	0.0973800	0.0486900	−	0.998814i	$-0.484495\pi$
$929$	2.56159e6	0.0973800	0.0486900	+	0.998814i	$0.484495\pi$
$930$	−5.00947e6	−0.189926
$931$	−1.24852e6	−0.0472086
$932$	−924576.	−0.0348661
$933$	946620.	0.0356018
$934$	−5.33410e6	−0.200075
$935$	6.05920e6	0.226666
$936$	2.60237e6	0.0970910
$937$	1.22738e7	0.456701	0.228350	−	0.973579i	$-0.426667\pi$
$937$	1.22738e7	0.456701	0.228350	+	0.973579i	$0.426667\pi$
$938$	−3.80475e6	−0.141195
$939$	1.70258e7	0.630148
$940$	1.41224e7	0.521300
$941$	3.76486e7	1.38604	0.693019	−	0.720919i	$-0.256280\pi$
$941$	3.76486e7	1.38604	0.693019	+	0.720919i	$0.256280\pi$
$942$	−1.33604e7	−0.490560
$943$	2.47860e6	0.0907668
$944$	−4.05811e6	−0.148216
$945$	2.78624e6	0.101494
$946$	−1.10855e7	−0.402744
$947$	−2.06302e7	−0.747530	−0.373765	−	0.927524i	$-0.621933\pi$
$947$	−2.06302e7	−0.747530	−0.373765	+	0.927524i	$0.621933\pi$
$948$	−5.13677e6	−0.185639
$949$	1.51674e7	0.546697
$950$	6.15472e6	0.221258
$951$	6.04978e6	0.216914
$952$	2.01331e6	0.0719977
$953$	−2.79681e7	−0.997540	−0.498770	−	0.866734i	$-0.666215\pi$
$953$	−2.79681e7	−0.997540	−0.498770	+	0.866734i	$0.666215\pi$
$954$	3.95993e6	0.140869
$955$	−9.14566e6	−0.324494
$956$	1.16099e7	0.410849
$957$	−5.24680e6	−0.185189
$958$	−2.06282e7	−0.726185
$959$	−6.05199e6	−0.212496
$960$	2.87539e6	0.100698
$961$	−2.54465e7	−0.888832
$962$	1.59797e7	0.556711
$963$	−1.76525e7	−0.613395
$964$	1.00059e7	0.346788
$965$	3.20391e7	1.10755
$966$	1.79928e6	0.0620377
$967$	2.74786e7	0.944994	0.472497	−	0.881332i	$-0.343353\pi$
$967$	2.74786e7	0.944994	0.472497	+	0.881332i	$0.343353\pi$
$968$	−937024.	−0.0321412
$969$	−3.00456e6	−0.102795
$970$	2.65531e7	0.906119
$971$	−3.26904e7	−1.11269	−0.556343	−	0.830953i	$-0.687796\pi$
$971$	−3.26904e7	−1.11269	−0.556343	+	0.830953i	$0.687796\pi$
$972$	−944784.	−0.0320750
$973$	6.60128e6	0.223535
$974$	1.37160e7	0.463265
$975$	1.33688e7	0.450381
$976$	1.18523e7	0.398270
$977$	−4.94406e7	−1.65709	−0.828547	−	0.559920i	$-0.810832\pi$
$977$	−4.94406e7	−1.65709	−0.828547	+	0.559920i	$0.810832\pi$
$978$	−2.74162e6	−0.0916557
$979$	−1.80701e6	−0.0602566
$980$	−2.99645e6	−0.0996647
$981$	7.10159e6	0.235604
$982$	−3.77936e7	−1.25066
$983$	7.23892e6	0.238940	0.119470	−	0.992838i	$-0.461880\pi$
$983$	7.23892e6	0.238940	0.119470	+	0.992838i	$0.461880\pi$
$984$	1.39968e6	0.0460830
$985$	−1.48595e7	−0.487992
$986$	1.23726e7	0.405293
$987$	4.99036e6	0.163057
$988$	4.17664e6	0.136124
$989$	2.33621e7	0.759488
$990$	3.05791e6	0.0991601
$991$	3.59078e7	1.16146	0.580730	−	0.814096i	$-0.302767\pi$
$991$	3.59078e7	1.16146	0.580730	+	0.814096i	$0.302767\pi$
$992$	−1.82682e6	−0.0589407
$993$	9.65610e6	0.310763
$994$	3.38923e6	0.108802
$995$	6.96977e7	2.23183
$996$	6.25363e6	0.199749
$997$	351914.	0.0112124	0.00560620	−	0.999984i	$-0.498215\pi$
$997$	351914.	0.0112124	0.00560620	+	0.999984i	$0.498215\pi$
$998$	1.63087e7	0.518314
$999$	−5.80138e6	−0.183915

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
462.6.a.a.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

Related objects

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