LMFDB - Embedded newform 462.6.a.c.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:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	462.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} +106.000 q^{5} +36.0000 q^{6} -49.0000 q^{7} -64.0000 q^{8} +81.0000 q^{9} +O(q^{10})$

\(q-4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} +106.000 q^{5} +36.0000 q^{6} -49.0000 q^{7} -64.0000 q^{8} +81.0000 q^{9} -424.000 q^{10} +121.000 q^{11} -144.000 q^{12} -166.000 q^{13} +196.000 q^{14} -954.000 q^{15} +256.000 q^{16} -842.000 q^{17} -324.000 q^{18} +1200.00 q^{19} +1696.00 q^{20} +441.000 q^{21} -484.000 q^{22} +1724.00 q^{23} +576.000 q^{24} +8111.00 q^{25} +664.000 q^{26} -729.000 q^{27} -784.000 q^{28} +1010.00 q^{29} +3816.00 q^{30} -1328.00 q^{31} -1024.00 q^{32} -1089.00 q^{33} +3368.00 q^{34} -5194.00 q^{35} +1296.00 q^{36} +15718.0 q^{37} -4800.00 q^{38} +1494.00 q^{39} -6784.00 q^{40} -7498.00 q^{41} -1764.00 q^{42} -2456.00 q^{43} +1936.00 q^{44} +8586.00 q^{45} -6896.00 q^{46} -2252.00 q^{47} -2304.00 q^{48} +2401.00 q^{49} -32444.0 q^{50} +7578.00 q^{51} -2656.00 q^{52} +15714.0 q^{53} +2916.00 q^{54} +12826.0 q^{55} +3136.00 q^{56} -10800.0 q^{57} -4040.00 q^{58} -5340.00 q^{59} -15264.0 q^{60} -25718.0 q^{61} +5312.00 q^{62} -3969.00 q^{63} +4096.00 q^{64} -17596.0 q^{65} +4356.00 q^{66} -4572.00 q^{67} -13472.0 q^{68} -15516.0 q^{69} +20776.0 q^{70} -67708.0 q^{71} -5184.00 q^{72} -7406.00 q^{73} -62872.0 q^{74} -72999.0 q^{75} +19200.0 q^{76} -5929.00 q^{77} -5976.00 q^{78} +25560.0 q^{79} +27136.0 q^{80} +6561.00 q^{81} +29992.0 q^{82} +22404.0 q^{83} +7056.00 q^{84} -89252.0 q^{85} +9824.00 q^{86} -9090.00 q^{87} -7744.00 q^{88} +61530.0 q^{89} -34344.0 q^{90} +8134.00 q^{91} +27584.0 q^{92} +11952.0 q^{93} +9008.00 q^{94} +127200. q^{95} +9216.00 q^{96} -72782.0 q^{97} -9604.00 q^{98} +9801.00 q^{99} +O(q^{100})\)

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
$5$	106.000	1.89619	0.948093	−	0.317994i	$-0.103009\pi$
$5$	106.000	1.89619	0.948093	+	0.317994i	$0.103009\pi$
$6$	36.0000	0.408248
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	−64.0000	−0.353553
$9$	81.0000	0.333333
$10$	−424.000	−1.34081
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	−144.000	−0.288675
$13$	−166.000	−0.272427	−0.136213	−	0.990680i	$-0.543493\pi$
$13$	−166.000	−0.272427	−0.136213	+	0.990680i	$0.543493\pi$
$14$	196.000	0.267261
$15$	−954.000	−1.09476
$16$	256.000	0.250000
$17$	−842.000	−0.706626	−0.353313	−	0.935505i	$-0.614945\pi$
$17$	−842.000	−0.706626	−0.353313	+	0.935505i	$0.614945\pi$
$18$	−324.000	−0.235702
$19$	1200.00	0.762601	0.381300	−	0.924451i	$-0.375476\pi$
$19$	1200.00	0.762601	0.381300	+	0.924451i	$0.375476\pi$
$20$	1696.00	0.948093
$21$	441.000	0.218218
$22$	−484.000	−0.213201
$23$	1724.00	0.679544	0.339772	−	0.940508i	$-0.389650\pi$
$23$	1724.00	0.679544	0.339772	+	0.940508i	$0.389650\pi$
$24$	576.000	0.204124
$25$	8111.00	2.59552
$26$	664.000	0.192635
$27$	−729.000	−0.192450
$28$	−784.000	−0.188982
$29$	1010.00	0.223011	0.111506	−	0.993764i	$-0.464433\pi$
$29$	1010.00	0.223011	0.111506	+	0.993764i	$0.464433\pi$
$30$	3816.00	0.774115
$31$	−1328.00	−0.248195	−0.124098	−	0.992270i	$-0.539604\pi$
$31$	−1328.00	−0.248195	−0.124098	+	0.992270i	$0.539604\pi$
$32$	−1024.00	−0.176777
$33$	−1089.00	−0.174078
$34$	3368.00	0.499660
$35$	−5194.00	−0.716691
$36$	1296.00	0.166667
$37$	15718.0	1.88753	0.943763	−	0.330623i	$-0.107259\pi$
$37$	15718.0	1.88753	0.943763	+	0.330623i	$0.107259\pi$
$38$	−4800.00	−0.539240
$39$	1494.00	0.157286
$40$	−6784.00	−0.670403
$41$	−7498.00	−0.696604	−0.348302	−	0.937382i	$-0.613242\pi$
$41$	−7498.00	−0.696604	−0.348302	+	0.937382i	$0.613242\pi$
$42$	−1764.00	−0.154303
$43$	−2456.00	−0.202562	−0.101281	−	0.994858i	$-0.532294\pi$
$43$	−2456.00	−0.202562	−0.101281	+	0.994858i	$0.532294\pi$
$44$	1936.00	0.150756
$45$	8586.00	0.632062
$46$	−6896.00	−0.480510
$47$	−2252.00	−0.148704	−0.0743522	−	0.997232i	$-0.523689\pi$
$47$	−2252.00	−0.148704	−0.0743522	+	0.997232i	$0.523689\pi$
$48$	−2304.00	−0.144338
$49$	2401.00	0.142857
$50$	−32444.0	−1.83531
$51$	7578.00	0.407971
$52$	−2656.00	−0.136213
$53$	15714.0	0.768417	0.384209	−	0.923246i	$-0.374474\pi$
$53$	15714.0	0.768417	0.384209	+	0.923246i	$0.374474\pi$
$54$	2916.00	0.136083
$55$	12826.0	0.571721
$56$	3136.00	0.133631
$57$	−10800.0	−0.440288
$58$	−4040.00	−0.157693
$59$	−5340.00	−0.199715	−0.0998576	−	0.995002i	$-0.531839\pi$
$59$	−5340.00	−0.199715	−0.0998576	+	0.995002i	$0.531839\pi$
$60$	−15264.0	−0.547382
$61$	−25718.0	−0.884938	−0.442469	−	0.896784i	$-0.645897\pi$
$61$	−25718.0	−0.884938	−0.442469	+	0.896784i	$0.645897\pi$
$62$	5312.00	0.175501
$63$	−3969.00	−0.125988
$64$	4096.00	0.125000
$65$	−17596.0	−0.516572
$66$	4356.00	0.123091
$67$	−4572.00	−0.124428	−0.0622142	−	0.998063i	$-0.519816\pi$
$67$	−4572.00	−0.124428	−0.0622142	+	0.998063i	$0.519816\pi$
$68$	−13472.0	−0.353313
$69$	−15516.0	−0.392335
$70$	20776.0	0.506777
$71$	−67708.0	−1.59402	−0.797011	−	0.603965i	$-0.793587\pi$
$71$	−67708.0	−1.59402	−0.797011	+	0.603965i	$0.793587\pi$
$72$	−5184.00	−0.117851
$73$	−7406.00	−0.162658	−0.0813292	−	0.996687i	$-0.525917\pi$
$73$	−7406.00	−0.162658	−0.0813292	+	0.996687i	$0.525917\pi$
$74$	−62872.0	−1.33468
$75$	−72999.0	−1.49852
$76$	19200.0	0.381300
$77$	−5929.00	−0.113961
$78$	−5976.00	−0.111218
$79$	25560.0	0.460779	0.230390	−	0.973098i	$-0.426000\pi$
$79$	25560.0	0.460779	0.230390	+	0.973098i	$0.426000\pi$
$80$	27136.0	0.474046
$81$	6561.00	0.111111
$82$	29992.0	0.492573
$83$	22404.0	0.356969	0.178484	−	0.983943i	$-0.442881\pi$
$83$	22404.0	0.356969	0.178484	+	0.983943i	$0.442881\pi$
$84$	7056.00	0.109109
$85$	−89252.0	−1.33989
$86$	9824.00	0.143233
$87$	−9090.00	−0.128755
$88$	−7744.00	−0.106600
$89$	61530.0	0.823402	0.411701	−	0.911319i	$-0.364935\pi$
$89$	61530.0	0.823402	0.411701	+	0.911319i	$0.364935\pi$
$90$	−34344.0	−0.446935
$91$	8134.00	0.102968
$92$	27584.0	0.339772
$93$	11952.0	0.143296
$94$	9008.00	0.105150
$95$	127200.	1.44603
$96$	9216.00	0.102062
$97$	−72782.0	−0.785407	−0.392703	−	0.919665i	$-0.628460\pi$
$97$	−72782.0	−0.785407	−0.392703	+	0.919665i	$0.628460\pi$
$98$	−9604.00	−0.101015
$99$	9801.00	0.100504
$100$	129776.	1.29776
$101$	99402.0	0.969598	0.484799	−	0.874626i	$-0.338893\pi$
$101$	99402.0	0.969598	0.484799	+	0.874626i	$0.338893\pi$
$102$	−30312.0	−0.288479
$103$	105384.	0.978772	0.489386	−	0.872067i	$-0.337221\pi$
$103$	105384.	0.978772	0.489386	+	0.872067i	$0.337221\pi$
$104$	10624.0	0.0963174
$105$	46746.0	0.413782
$106$	−62856.0	−0.543353
$107$	52228.0	0.441005	0.220503	−	0.975386i	$-0.429230\pi$
$107$	52228.0	0.441005	0.220503	+	0.975386i	$0.429230\pi$
$108$	−11664.0	−0.0962250
$109$	144010.	1.16098	0.580492	−	0.814266i	$-0.302860\pi$
$109$	144010.	1.16098	0.580492	+	0.814266i	$0.302860\pi$
$110$	−51304.0	−0.404268
$111$	−141462.	−1.08976
$112$	−12544.0	−0.0944911
$113$	56794.0	0.418414	0.209207	−	0.977871i	$-0.432912\pi$
$113$	56794.0	0.418414	0.209207	+	0.977871i	$0.432912\pi$
$114$	43200.0	0.311330
$115$	182744.	1.28854
$116$	16160.0	0.111506
$117$	−13446.0	−0.0908089
$118$	21360.0	0.141220
$119$	41258.0	0.267080
$120$	61056.0	0.387057
$121$	14641.0	0.0909091
$122$	102872.	0.625745
$123$	67482.0	0.402184
$124$	−21248.0	−0.124098
$125$	528516.	3.02540
$126$	15876.0	0.0890871
$127$	100808.	0.554607	0.277304	−	0.960782i	$-0.410559\pi$
$127$	100808.	0.554607	0.277304	+	0.960782i	$0.410559\pi$
$128$	−16384.0	−0.0883883
$129$	22104.0	0.116949
$130$	70384.0	0.365271
$131$	−52068.0	−0.265090	−0.132545	−	0.991177i	$-0.542315\pi$
$131$	−52068.0	−0.265090	−0.132545	+	0.991177i	$0.542315\pi$
$132$	−17424.0	−0.0870388
$133$	−58800.0	−0.288236
$134$	18288.0	0.0879841
$135$	−77274.0	−0.364921
$136$	53888.0	0.249830
$137$	296298.	1.34874	0.674369	−	0.738395i	$-0.264416\pi$
$137$	296298.	1.34874	0.674369	+	0.738395i	$0.264416\pi$
$138$	62064.0	0.277423
$139$	−307960.	−1.35194	−0.675970	−	0.736929i	$-0.736275\pi$
$139$	−307960.	−1.35194	−0.675970	+	0.736929i	$0.736275\pi$
$140$	−83104.0	−0.358345
$141$	20268.0	0.0858545
$142$	270832.	1.12714
$143$	−20086.0	−0.0821397
$144$	20736.0	0.0833333
$145$	107060.	0.422870
$146$	29624.0	0.115017
$147$	−21609.0	−0.0824786
$148$	251488.	0.943763
$149$	359770.	1.32758	0.663788	−	0.747921i	$-0.268948\pi$
$149$	359770.	1.32758	0.663788	+	0.747921i	$0.268948\pi$
$150$	291996.	1.05962
$151$	−418848.	−1.49491	−0.747453	−	0.664314i	$-0.768724\pi$
$151$	−418848.	−1.49491	−0.747453	+	0.664314i	$0.768724\pi$
$152$	−76800.0	−0.269620
$153$	−68202.0	−0.235542
$154$	23716.0	0.0805823
$155$	−140768.	−0.470625
$156$	23904.0	0.0786428
$157$	331558.	1.07352	0.536760	−	0.843735i	$-0.319648\pi$
$157$	331558.	1.07352	0.536760	+	0.843735i	$0.319648\pi$
$158$	−102240.	−0.325820
$159$	−141426.	−0.443646
$160$	−108544.	−0.335201
$161$	−84476.0	−0.256844
$162$	−26244.0	−0.0785674
$163$	509284.	1.50138	0.750691	−	0.660654i	$-0.229721\pi$
$163$	509284.	1.50138	0.750691	+	0.660654i	$0.229721\pi$
$164$	−119968.	−0.348302
$165$	−115434.	−0.330084
$166$	−89616.0	−0.252415
$167$	507128.	1.40710	0.703552	−	0.710643i	$-0.251596\pi$
$167$	507128.	1.40710	0.703552	+	0.710643i	$0.251596\pi$
$168$	−28224.0	−0.0771517
$169$	−343737.	−0.925784
$170$	357008.	0.947449
$171$	97200.0	0.254200
$172$	−39296.0	−0.101281
$173$	−102046.	−0.259227	−0.129614	−	0.991565i	$-0.541374\pi$
$173$	−102046.	−0.259227	−0.129614	+	0.991565i	$0.541374\pi$
$174$	36360.0	0.0910439
$175$	−397439.	−0.981014
$176$	30976.0	0.0753778
$177$	48060.0	0.115306
$178$	−246120.	−0.582233
$179$	−541180.	−1.26244	−0.631218	−	0.775606i	$-0.717445\pi$
$179$	−541180.	−1.26244	−0.631218	+	0.775606i	$0.717445\pi$
$180$	137376.	0.316031
$181$	539102.	1.22314	0.611568	−	0.791192i	$-0.290539\pi$
$181$	539102.	1.22314	0.611568	+	0.791192i	$0.290539\pi$
$182$	−32536.0	−0.0728091
$183$	231462.	0.510919
$184$	−110336.	−0.240255
$185$	1.66611e6	3.57910
$186$	−47808.0	−0.101325
$187$	−101882.	−0.213056
$188$	−36032.0	−0.0743522
$189$	35721.0	0.0727393
$190$	−508800.	−1.02250
$191$	111172.	0.220502	0.110251	−	0.993904i	$-0.464835\pi$
$191$	111172.	0.220502	0.110251	+	0.993904i	$0.464835\pi$
$192$	−36864.0	−0.0721688
$193$	−166.000	−0.000320786 0	−0.000160393	−	1.00000i	$-0.500051\pi$
$193$	−166.000	−0.000320786 0	−0.000160393		1.00000i	$0.500051\pi$
$194$	291128.	0.555366
$195$	158364.	0.298243
$196$	38416.0	0.0714286
$197$	−420342.	−0.771680	−0.385840	−	0.922566i	$-0.626088\pi$
$197$	−420342.	−0.771680	−0.385840	+	0.922566i	$0.626088\pi$
$198$	−39204.0	−0.0710669
$199$	−718160.	−1.28555	−0.642774	−	0.766056i	$-0.722217\pi$
$199$	−718160.	−1.28555	−0.642774	+	0.766056i	$0.722217\pi$
$200$	−519104.	−0.917655
$201$	41148.0	0.0718387
$202$	−397608.	−0.685609
$203$	−49490.0	−0.0842903
$204$	121248.	0.203985
$205$	−794788.	−1.32089
$206$	−421536.	−0.692096
$207$	139644.	0.226515
$208$	−42496.0	−0.0681067
$209$	145200.	0.229933
$210$	−186984.	−0.292588
$211$	−369368.	−0.571154	−0.285577	−	0.958356i	$-0.592185\pi$
$211$	−369368.	−0.571154	−0.285577	+	0.958356i	$0.592185\pi$
$212$	251424.	0.384209
$213$	609372.	0.920309
$214$	−208912.	−0.311838
$215$	−260336.	−0.384094
$216$	46656.0	0.0680414
$217$	65072.0	0.0938091
$218$	−576040.	−0.820940
$219$	66654.0	0.0939109
$220$	205216.	0.285861
$221$	139772.	0.192504
$222$	565848.	0.770579
$223$	−470216.	−0.633192	−0.316596	−	0.948561i	$-0.602540\pi$
$223$	−470216.	−0.633192	−0.316596	+	0.948561i	$0.602540\pi$
$224$	50176.0	0.0668153
$225$	656991.	0.865173
$226$	−227176.	−0.295863
$227$	1.27195e6	1.63834	0.819171	−	0.573549i	$-0.194434\pi$
$227$	1.27195e6	1.63834	0.819171	+	0.573549i	$0.194434\pi$
$228$	−172800.	−0.220144
$229$	526430.	0.663364	0.331682	−	0.943391i	$-0.392384\pi$
$229$	526430.	0.663364	0.331682	+	0.943391i	$0.392384\pi$
$230$	−730976.	−0.911137
$231$	53361.0	0.0657952
$232$	−64640.0	−0.0788463
$233$	1.01645e6	1.22659	0.613293	−	0.789856i	$-0.289845\pi$
$233$	1.01645e6	1.22659	0.613293	+	0.789856i	$0.289845\pi$
$234$	53784.0	0.0642116
$235$	−238712.	−0.281971
$236$	−85440.0	−0.0998576
$237$	−230040.	−0.266031
$238$	−165032.	−0.188854
$239$	456560.	0.517015	0.258507	−	0.966009i	$-0.416769\pi$
$239$	456560.	0.517015	0.258507	+	0.966009i	$0.416769\pi$
$240$	−244224.	−0.273691
$241$	898482.	0.996476	0.498238	−	0.867040i	$-0.333981\pi$
$241$	898482.	0.996476	0.498238	+	0.867040i	$0.333981\pi$
$242$	−58564.0	−0.0642824
$243$	−59049.0	−0.0641500
$244$	−411488.	−0.442469
$245$	254506.	0.270884
$246$	−269928.	−0.284387
$247$	−199200.	−0.207753
$248$	84992.0	0.0877503
$249$	−201636.	−0.206096
$250$	−2.11406e6	−2.13928
$251$	952092.	0.953882	0.476941	−	0.878935i	$-0.341746\pi$
$251$	952092.	0.953882	0.476941	+	0.878935i	$0.341746\pi$
$252$	−63504.0	−0.0629941
$253$	208604.	0.204890
$254$	−403232.	−0.392167
$255$	803268.	0.773589
$256$	65536.0	0.0625000
$257$	1.78442e6	1.68525	0.842624	−	0.538502i	$-0.181010\pi$
$257$	1.78442e6	1.68525	0.842624	+	0.538502i	$0.181010\pi$
$258$	−88416.0	−0.0826954
$259$	−770182.	−0.713418
$260$	−281536.	−0.258286
$261$	81810.0	0.0743370
$262$	208272.	0.187447
$263$	1.00334e6	0.894459	0.447230	−	0.894419i	$-0.352411\pi$
$263$	1.00334e6	0.894459	0.447230	+	0.894419i	$0.352411\pi$
$264$	69696.0	0.0615457
$265$	1.66568e6	1.45706
$266$	235200.	0.203814
$267$	−553770.	−0.475391
$268$	−73152.0	−0.0622142
$269$	1.74537e6	1.47064	0.735321	−	0.677719i	$-0.237032\pi$
$269$	1.74537e6	1.47064	0.735321	+	0.677719i	$0.237032\pi$
$270$	309096.	0.258038
$271$	−1.98265e6	−1.63992	−0.819959	−	0.572422i	$-0.806004\pi$
$271$	−1.98265e6	−1.63992	−0.819959	+	0.572422i	$0.806004\pi$
$272$	−215552.	−0.176657
$273$	−73206.0	−0.0594484
$274$	−1.18519e6	−0.953701
$275$	981431.	0.782579
$276$	−248256.	−0.196167
$277$	−426102.	−0.333668	−0.166834	−	0.985985i	$-0.553354\pi$
$277$	−426102.	−0.333668	−0.166834	+	0.985985i	$0.553354\pi$
$278$	1.23184e6	0.955966
$279$	−107568.	−0.0827318
$280$	332416.	0.253388
$281$	−274538.	−0.207413	−0.103707	−	0.994608i	$-0.533070\pi$
$281$	−274538.	−0.207413	−0.103707	+	0.994608i	$0.533070\pi$
$282$	−81072.0	−0.0607083
$283$	−14616.0	−0.0108483	−0.00542416	−	0.999985i	$-0.501727\pi$
$283$	−14616.0	−0.0108483	−0.00542416	+	0.999985i	$0.501727\pi$
$284$	−1.08333e6	−0.797011
$285$	−1.14480e6	−0.834867
$286$	80344.0	0.0580816
$287$	367402.	0.263291
$288$	−82944.0	−0.0589256
$289$	−710893.	−0.500679
$290$	−428240.	−0.299014
$291$	655038.	0.453455
$292$	−118496.	−0.0813292
$293$	−1.64977e6	−1.12267	−0.561337	−	0.827588i	$-0.689713\pi$
$293$	−1.64977e6	−1.12267	−0.561337	+	0.827588i	$0.689713\pi$
$294$	86436.0	0.0583212
$295$	−566040.	−0.378697
$296$	−1.00595e6	−0.667341
$297$	−88209.0	−0.0580259
$298$	−1.43908e6	−0.938738
$299$	−286184.	−0.185126
$300$	−1.16798e6	−0.749262
$301$	120344.	0.0765611
$302$	1.67539e6	1.05706
$303$	−894618.	−0.559798
$304$	307200.	0.190650
$305$	−2.72611e6	−1.67801
$306$	272808.	0.166553
$307$	1.15309e6	0.698259	0.349129	−	0.937074i	$-0.386477\pi$
$307$	1.15309e6	0.698259	0.349129	+	0.937074i	$0.386477\pi$
$308$	−94864.0	−0.0569803
$309$	−948456.	−0.565094
$310$	563072.	0.332782
$311$	−3.16759e6	−1.85707	−0.928534	−	0.371248i	$-0.878930\pi$
$311$	−3.16759e6	−1.85707	−0.928534	+	0.371248i	$0.878930\pi$
$312$	−95616.0	−0.0556089
$313$	556714.	0.321197	0.160598	−	0.987020i	$-0.448658\pi$
$313$	556714.	0.321197	0.160598	+	0.987020i	$0.448658\pi$
$314$	−1.32623e6	−0.759094
$315$	−420714.	−0.238897
$316$	408960.	0.230390
$317$	1.24886e6	0.698015	0.349008	−	0.937120i	$-0.386519\pi$
$317$	1.24886e6	0.698015	0.349008	+	0.937120i	$0.386519\pi$
$318$	565704.	0.313705
$319$	122210.	0.0672404
$320$	434176.	0.237023
$321$	−470052.	−0.254615
$322$	337904.	0.181616
$323$	−1.01040e6	−0.538874
$324$	104976.	0.0555556
$325$	−1.34643e6	−0.707089
$326$	−2.03714e6	−1.06164
$327$	−1.29609e6	−0.670295
$328$	479872.	0.246287
$329$	110348.	0.0562050
$330$	461736.	0.233404
$331$	3.28433e6	1.64770	0.823848	−	0.566811i	$-0.191823\pi$
$331$	3.28433e6	1.64770	0.823848	+	0.566811i	$0.191823\pi$
$332$	358464.	0.178484
$333$	1.27316e6	0.629175
$334$	−2.02851e6	−0.994973
$335$	−484632.	−0.235939
$336$	112896.	0.0545545
$337$	2.54966e6	1.22295	0.611473	−	0.791265i	$-0.290577\pi$
$337$	2.54966e6	1.22295	0.611473	+	0.791265i	$0.290577\pi$
$338$	1.37495e6	0.654628
$339$	−511146.	−0.241572
$340$	−1.42803e6	−0.669947
$341$	−160688.	−0.0748337
$342$	−388800.	−0.179747
$343$	−117649.	−0.0539949
$344$	157184.	0.0716164
$345$	−1.64470e6	−0.743940
$346$	408184.	0.183301
$347$	−1.36269e6	−0.607539	−0.303769	−	0.952746i	$-0.598245\pi$
$347$	−1.36269e6	−0.607539	−0.303769	+	0.952746i	$0.598245\pi$
$348$	−145440.	−0.0643777
$349$	2.60885e6	1.14653	0.573265	−	0.819370i	$-0.305676\pi$
$349$	2.60885e6	1.14653	0.573265	+	0.819370i	$0.305676\pi$
$350$	1.58976e6	0.693682
$351$	121014.	0.0524285
$352$	−123904.	−0.0533002
$353$	−1.77973e6	−0.760180	−0.380090	−	0.924950i	$-0.624107\pi$
$353$	−1.77973e6	−0.760180	−0.380090	+	0.924950i	$0.624107\pi$
$354$	−192240.	−0.0815334
$355$	−7.17705e6	−3.02256
$356$	984480.	0.411701
$357$	−371322.	−0.154198
$358$	2.16472e6	0.892677
$359$	−4.46120e6	−1.82690	−0.913452	−	0.406947i	$-0.866593\pi$
$359$	−4.46120e6	−1.82690	−0.913452	+	0.406947i	$0.866593\pi$
$360$	−549504.	−0.223468
$361$	−1.03610e6	−0.418440
$362$	−2.15641e6	−0.864887
$363$	−131769.	−0.0524864
$364$	130144.	0.0514838
$365$	−785036.	−0.308431
$366$	−925848.	−0.361274
$367$	−2.88327e6	−1.11743	−0.558715	−	0.829360i	$-0.688705\pi$
$367$	−2.88327e6	−1.11743	−0.558715	+	0.829360i	$0.688705\pi$
$368$	441344.	0.169886
$369$	−607338.	−0.232201
$370$	−6.66443e6	−2.53081
$371$	−769986.	−0.290434
$372$	191232.	0.0716479
$373$	2.89375e6	1.07694	0.538468	−	0.842646i	$-0.319003\pi$
$373$	2.89375e6	1.07694	0.538468	+	0.842646i	$0.319003\pi$
$374$	407528.	0.150653
$375$	−4.75664e6	−1.74672
$376$	144128.	0.0525749
$377$	−167660.	−0.0607542
$378$	−142884.	−0.0514344
$379$	4.94734e6	1.76919	0.884593	−	0.466363i	$-0.154436\pi$
$379$	4.94734e6	1.76919	0.884593	+	0.466363i	$0.154436\pi$
$380$	2.03520e6	0.723016
$381$	−907272.	−0.320203
$382$	−444688.	−0.155918
$383$	3.05712e6	1.06492	0.532459	−	0.846456i	$-0.321268\pi$
$383$	3.05712e6	1.06492	0.532459	+	0.846456i	$0.321268\pi$
$384$	147456.	0.0510310
$385$	−628474.	−0.216090
$386$	664.000	0.000226830 0
$387$	−198936.	−0.0675205
$388$	−1.16451e6	−0.392703
$389$	5.20185e6	1.74295	0.871473	−	0.490444i	$-0.163165\pi$
$389$	5.20185e6	1.74295	0.871473	+	0.490444i	$0.163165\pi$
$390$	−633456.	−0.210889
$391$	−1.45161e6	−0.480184
$392$	−153664.	−0.0505076
$393$	468612.	0.153050
$394$	1.68137e6	0.545660
$395$	2.70936e6	0.873723
$396$	156816.	0.0502519
$397$	−4.68244e6	−1.49106	−0.745532	−	0.666470i	$-0.767804\pi$
$397$	−4.68244e6	−1.49106	−0.745532	+	0.666470i	$0.767804\pi$
$398$	2.87264e6	0.909020
$399$	529200.	0.166413
$400$	2.07642e6	0.648880
$401$	−2.50280e6	−0.777257	−0.388629	−	0.921394i	$-0.627051\pi$
$401$	−2.50280e6	−0.777257	−0.388629	+	0.921394i	$0.627051\pi$
$402$	−164592.	−0.0507976
$403$	220448.	0.0676151
$404$	1.59043e6	0.484799
$405$	695466.	0.210687
$406$	197960.	0.0596022
$407$	1.90188e6	0.569111
$408$	−484992.	−0.144239
$409$	−446630.	−0.132020	−0.0660100	−	0.997819i	$-0.521027\pi$
$409$	−446630.	−0.132020	−0.0660100	+	0.997819i	$0.521027\pi$
$410$	3.17915e6	0.934010
$411$	−2.66668e6	−0.778694
$412$	1.68614e6	0.489386
$413$	261660.	0.0754853
$414$	−558576.	−0.160170
$415$	2.37482e6	0.676879
$416$	169984.	0.0481587
$417$	2.77164e6	0.780543
$418$	−580800.	−0.162587
$419$	2.35154e6	0.654361	0.327180	−	0.944962i	$-0.393902\pi$
$419$	2.35154e6	0.654361	0.327180	+	0.944962i	$0.393902\pi$
$420$	747936.	0.206891
$421$	−57818.0	−0.0158986	−0.00794928	−	0.999968i	$-0.502530\pi$
$421$	−57818.0	−0.0158986	−0.00794928	+	0.999968i	$0.502530\pi$
$422$	1.47747e6	0.403867
$423$	−182412.	−0.0495681
$424$	−1.00570e6	−0.271677
$425$	−6.82946e6	−1.83406
$426$	−2.43749e6	−0.650756
$427$	1.26018e6	0.334475
$428$	835648.	0.220503
$429$	180774.	0.0474234
$430$	1.04134e6	0.271596
$431$	4.65663e6	1.20748	0.603738	−	0.797183i	$-0.293677\pi$
$431$	4.65663e6	1.20748	0.603738	+	0.797183i	$0.293677\pi$
$432$	−186624.	−0.0481125
$433$	4.26283e6	1.09264	0.546322	−	0.837575i	$-0.316027\pi$
$433$	4.26283e6	1.09264	0.546322	+	0.837575i	$0.316027\pi$
$434$	−260288.	−0.0663330
$435$	−963540.	−0.244144
$436$	2.30416e6	0.580492
$437$	2.06880e6	0.518221
$438$	−266616.	−0.0664050
$439$	−5.29748e6	−1.31192	−0.655961	−	0.754795i	$-0.727736\pi$
$439$	−5.29748e6	−1.31192	−0.655961	+	0.754795i	$0.727736\pi$
$440$	−820864.	−0.202134
$441$	194481.	0.0476190
$442$	−559088.	−0.136121
$443$	−4.49536e6	−1.08832	−0.544158	−	0.838983i	$-0.683151\pi$
$443$	−4.49536e6	−1.08832	−0.544158	+	0.838983i	$0.683151\pi$
$444$	−2.26339e6	−0.544882
$445$	6.52218e6	1.56132
$446$	1.88086e6	0.447734
$447$	−3.23793e6	−0.766476
$448$	−200704.	−0.0472456
$449$	−5.74527e6	−1.34491	−0.672457	−	0.740136i	$-0.734761\pi$
$449$	−5.74527e6	−1.34491	−0.672457	+	0.740136i	$0.734761\pi$
$450$	−2.62796e6	−0.611770
$451$	−907258.	−0.210034
$452$	908704.	0.209207
$453$	3.76963e6	0.863085
$454$	−5.08779e6	−1.15848
$455$	862204.	0.195246
$456$	691200.	0.155665
$457$	−7.60970e6	−1.70442	−0.852211	−	0.523198i	$-0.824739\pi$
$457$	−7.60970e6	−1.70442	−0.852211	+	0.523198i	$0.824739\pi$
$458$	−2.10572e6	−0.469069
$459$	613818.	0.135990
$460$	2.92390e6	0.644271
$461$	5.31328e6	1.16442	0.582211	−	0.813038i	$-0.302188\pi$
$461$	5.31328e6	1.16442	0.582211	+	0.813038i	$0.302188\pi$
$462$	−213444.	−0.0465242
$463$	1.39734e6	0.302936	0.151468	−	0.988462i	$-0.451600\pi$
$463$	1.39734e6	0.302936	0.151468	+	0.988462i	$0.451600\pi$
$464$	258560.	0.0557528
$465$	1.26691e6	0.271715
$466$	−4.06582e6	−0.867327
$467$	−6.51409e6	−1.38217	−0.691085	−	0.722773i	$-0.742867\pi$
$467$	−6.51409e6	−1.38217	−0.691085	+	0.722773i	$0.742867\pi$
$468$	−215136.	−0.0454045
$469$	224028.	0.0470295
$470$	954848.	0.199384
$471$	−2.98402e6	−0.619797
$472$	341760.	0.0706100
$473$	−297176.	−0.0610746
$474$	920160.	0.188112
$475$	9.73320e6	1.97935
$476$	660128.	0.133540
$477$	1.27283e6	0.256139
$478$	−1.82624e6	−0.365585
$479$	8.47996e6	1.68871	0.844355	−	0.535784i	$-0.179984\pi$
$479$	8.47996e6	1.68871	0.844355	+	0.535784i	$0.179984\pi$
$480$	976896.	0.193529
$481$	−2.60919e6	−0.514213
$482$	−3.59393e6	−0.704615
$483$	760284.	0.148289
$484$	234256.	0.0454545
$485$	−7.71489e6	−1.48928
$486$	236196.	0.0453609
$487$	−385552.	−0.0736649	−0.0368324	−	0.999321i	$-0.511727\pi$
$487$	−385552.	−0.0736649	−0.0368324	+	0.999321i	$0.511727\pi$
$488$	1.64595e6	0.312873
$489$	−4.58356e6	−0.866823
$490$	−1.01802e6	−0.191544
$491$	1.93381e6	0.362002	0.181001	−	0.983483i	$-0.442066\pi$
$491$	1.93381e6	0.362002	0.181001	+	0.983483i	$0.442066\pi$
$492$	1.07971e6	0.201092
$493$	−850420.	−0.157585
$494$	796800.	0.146903
$495$	1.03891e6	0.190574
$496$	−339968.	−0.0620489
$497$	3.31769e6	0.602483
$498$	806544.	0.145732
$499$	4.49414e6	0.807970	0.403985	−	0.914766i	$-0.367625\pi$
$499$	4.49414e6	0.807970	0.403985	+	0.914766i	$0.367625\pi$
$500$	8.45626e6	1.51270
$501$	−4.56415e6	−0.812392
$502$	−3.80837e6	−0.674496
$503$	2.91074e6	0.512961	0.256480	−	0.966549i	$-0.417437\pi$
$503$	2.91074e6	0.512961	0.256480	+	0.966549i	$0.417437\pi$
$504$	254016.	0.0445435
$505$	1.05366e7	1.83854
$506$	−834416.	−0.144879
$507$	3.09363e6	0.534501
$508$	1.61293e6	0.277304
$509$	8.88025e6	1.51926	0.759628	−	0.650358i	$-0.225381\pi$
$509$	8.88025e6	1.51926	0.759628	+	0.650358i	$0.225381\pi$
$510$	−3.21307e6	−0.547010
$511$	362894.	0.0614791
$512$	−262144.	−0.0441942
$513$	−874800.	−0.146763
$514$	−7.13767e6	−1.19165
$515$	1.11707e7	1.85593
$516$	353664.	0.0584745
$517$	−272492.	−0.0448361
$518$	3.08073e6	0.504463
$519$	918414.	0.149665
$520$	1.12614e6	0.182636
$521$	−1.75212e6	−0.282793	−0.141397	−	0.989953i	$-0.545159\pi$
$521$	−1.75212e6	−0.282793	−0.141397	+	0.989953i	$0.545159\pi$
$522$	−327240.	−0.0525642
$523$	−8.88014e6	−1.41960	−0.709799	−	0.704404i	$-0.751214\pi$
$523$	−8.88014e6	−1.41960	−0.709799	+	0.704404i	$0.751214\pi$
$524$	−833088.	−0.132545
$525$	3.57695e6	0.566389
$526$	−4.01338e6	−0.632478
$527$	1.11818e6	0.175381
$528$	−278784.	−0.0435194
$529$	−3.46417e6	−0.538220
$530$	−6.66274e6	−1.03030
$531$	−432540.	−0.0665717
$532$	−940800.	−0.144118
$533$	1.24467e6	0.189773
$534$	2.21508e6	0.336152
$535$	5.53617e6	0.836228
$536$	292608.	0.0439921
$537$	4.87062e6	0.728867
$538$	−6.98148e6	−1.03990
$539$	290521.	0.0430730
$540$	−1.23638e6	−0.182461
$541$	−1.61196e6	−0.236788	−0.118394	−	0.992967i	$-0.537775\pi$
$541$	−1.61196e6	−0.236788	−0.118394	+	0.992967i	$0.537775\pi$
$542$	7.93059e6	1.15960
$543$	−4.85192e6	−0.706178
$544$	862208.	0.124915
$545$	1.52651e7	2.20144
$546$	292824.	0.0420364
$547$	1.02093e7	1.45891	0.729453	−	0.684031i	$-0.239775\pi$
$547$	1.02093e7	1.45891	0.729453	+	0.684031i	$0.239775\pi$
$548$	4.74077e6	0.674369
$549$	−2.08316e6	−0.294979
$550$	−3.92572e6	−0.553367
$551$	1.21200e6	0.170068
$552$	993024.	0.138711
$553$	−1.25244e6	−0.174158
$554$	1.70441e6	0.235939
$555$	−1.49950e7	−2.06639
$556$	−4.92736e6	−0.675970
$557$	3.33306e6	0.455203	0.227601	−	0.973754i	$-0.426912\pi$
$557$	3.33306e6	0.455203	0.227601	+	0.973754i	$0.426912\pi$
$558$	430272.	0.0585002
$559$	407696.	0.0551832
$560$	−1.32966e6	−0.179173
$561$	916938.	0.123008
$562$	1.09815e6	0.146663
$563$	114004.	0.0151583	0.00757913	−	0.999971i	$-0.497587\pi$
$563$	114004.	0.0151583	0.00757913	+	0.999971i	$0.497587\pi$
$564$	324288.	0.0429273
$565$	6.02016e6	0.793391
$566$	58464.0	0.00767092
$567$	−321489.	−0.0419961
$568$	4.33331e6	0.563572
$569$	−1.43745e7	−1.86129	−0.930643	−	0.365929i	$-0.880751\pi$
$569$	−1.43745e7	−1.86129	−0.930643	+	0.365929i	$0.880751\pi$
$570$	4.57920e6	0.590340
$571$	127072.	0.0163102	0.00815511	−	0.999967i	$-0.497404\pi$
$571$	127072.	0.0163102	0.00815511	+	0.999967i	$0.497404\pi$
$572$	−321376.	−0.0410699
$573$	−1.00055e6	−0.127307
$574$	−1.46961e6	−0.186175
$575$	1.39834e7	1.76377
$576$	331776.	0.0416667
$577$	−1.47342e7	−1.84242	−0.921208	−	0.389071i	$-0.872796\pi$
$577$	−1.47342e7	−1.84242	−0.921208	+	0.389071i	$0.872796\pi$
$578$	2.84357e6	0.354034
$579$	1494.00	0.000185206 0
$580$	1.71296e6	0.211435
$581$	−1.09780e6	−0.134922
$582$	−2.62015e6	−0.320641
$583$	1.90139e6	0.231687
$584$	473984.	0.0575084
$585$	−1.42528e6	−0.172191
$586$	6.59906e6	0.793850
$587$	5.61959e6	0.673146	0.336573	−	0.941657i	$-0.390732\pi$
$587$	5.61959e6	0.673146	0.336573	+	0.941657i	$0.390732\pi$
$588$	−345744.	−0.0412393
$589$	−1.59360e6	−0.189274
$590$	2.26416e6	0.267779
$591$	3.78308e6	0.445530
$592$	4.02381e6	0.471882
$593$	−1.00426e7	−1.17276	−0.586382	−	0.810035i	$-0.699448\pi$
$593$	−1.00426e7	−1.17276	−0.586382	+	0.810035i	$0.699448\pi$
$594$	352836.	0.0410305
$595$	4.37335e6	0.506433
$596$	5.75632e6	0.663788
$597$	6.46344e6	0.742212
$598$	1.14474e6	0.130904
$599$	4.62174e6	0.526306	0.263153	−	0.964754i	$-0.415238\pi$
$599$	4.62174e6	0.526306	0.263153	+	0.964754i	$0.415238\pi$
$600$	4.67194e6	0.529808
$601$	−8.28348e6	−0.935463	−0.467731	−	0.883871i	$-0.654929\pi$
$601$	−8.28348e6	−0.935463	−0.467731	+	0.883871i	$0.654929\pi$
$602$	−481376.	−0.0541369
$603$	−370332.	−0.0414761
$604$	−6.70157e6	−0.747453
$605$	1.55195e6	0.172381
$606$	3.57847e6	0.395837
$607$	−6.70103e6	−0.738193	−0.369097	−	0.929391i	$-0.620333\pi$
$607$	−6.70103e6	−0.738193	−0.369097	+	0.929391i	$0.620333\pi$
$608$	−1.22880e6	−0.134810
$609$	445410.	0.0486650
$610$	1.09044e7	1.18653
$611$	373832.	0.0405110
$612$	−1.09123e6	−0.117771
$613$	−7.11837e6	−0.765119	−0.382560	−	0.923931i	$-0.624957\pi$
$613$	−7.11837e6	−0.765119	−0.382560	+	0.923931i	$0.624957\pi$
$614$	−4.61235e6	−0.493744
$615$	7.15309e6	0.762616
$616$	379456.	0.0402911
$617$	869298.	0.0919297	0.0459648	−	0.998943i	$-0.485364\pi$
$617$	869298.	0.0919297	0.0459648	+	0.998943i	$0.485364\pi$
$618$	3.79382e6	0.399582
$619$	−1.43251e7	−1.50270	−0.751348	−	0.659906i	$-0.770596\pi$
$619$	−1.43251e7	−1.50270	−0.751348	+	0.659906i	$0.770596\pi$
$620$	−2.25229e6	−0.235312
$621$	−1.25680e6	−0.130778
$622$	1.26704e7	1.31315
$623$	−3.01497e6	−0.311217
$624$	382464.	0.0393214
$625$	3.06758e7	3.14120
$626$	−2.22686e6	−0.227120
$627$	−1.30680e6	−0.132752
$628$	5.30493e6	0.536760
$629$	−1.32346e7	−1.33378
$630$	1.68286e6	0.168926
$631$	−5.58305e6	−0.558210	−0.279105	−	0.960261i	$-0.590038\pi$
$631$	−5.58305e6	−0.558210	−0.279105	+	0.960261i	$0.590038\pi$
$632$	−1.63584e6	−0.162910
$633$	3.32431e6	0.329756
$634$	−4.99543e6	−0.493571
$635$	1.06856e7	1.05164
$636$	−2.26282e6	−0.221823
$637$	−398566.	−0.0389181
$638$	−488840.	−0.0475461
$639$	−5.48435e6	−0.531340
$640$	−1.73670e6	−0.167601
$641$	−3.44372e6	−0.331042	−0.165521	−	0.986206i	$-0.552930\pi$
$641$	−3.44372e6	−0.331042	−0.165521	+	0.986206i	$0.552930\pi$
$642$	1.88021e6	0.180040
$643$	−1.06140e6	−0.101240	−0.0506198	−	0.998718i	$-0.516120\pi$
$643$	−1.06140e6	−0.101240	−0.0506198	+	0.998718i	$0.516120\pi$
$644$	−1.35162e6	−0.128422
$645$	2.34302e6	0.221757
$646$	4.04160e6	0.381041
$647$	−7.37005e6	−0.692165	−0.346083	−	0.938204i	$-0.612488\pi$
$647$	−7.37005e6	−0.692165	−0.346083	+	0.938204i	$0.612488\pi$
$648$	−419904.	−0.0392837
$649$	−646140.	−0.0602164
$650$	5.38570e6	0.499987
$651$	−585648.	−0.0541607
$652$	8.14854e6	0.750691
$653$	1.89691e6	0.174086	0.0870432	−	0.996205i	$-0.472258\pi$
$653$	1.89691e6	0.174086	0.0870432	+	0.996205i	$0.472258\pi$
$654$	5.18436e6	0.473970
$655$	−5.51921e6	−0.502659
$656$	−1.91949e6	−0.174151
$657$	−599886.	−0.0542195
$658$	−441392.	−0.0397429
$659$	426020.	0.0382135	0.0191067	−	0.999817i	$-0.493918\pi$
$659$	426020.	0.0382135	0.0191067	+	0.999817i	$0.493918\pi$
$660$	−1.84694e6	−0.165042
$661$	−1.04010e6	−0.0925914	−0.0462957	−	0.998928i	$-0.514742\pi$
$661$	−1.04010e6	−0.0925914	−0.0462957	+	0.998928i	$0.514742\pi$
$662$	−1.31373e7	−1.16510
$663$	−1.25795e6	−0.111142
$664$	−1.43386e6	−0.126208
$665$	−6.23280e6	−0.546549
$666$	−5.09263e6	−0.444894
$667$	1.74124e6	0.151546
$668$	8.11405e6	0.703552
$669$	4.23194e6	0.365573
$670$	1.93853e6	0.166834
$671$	−3.11188e6	−0.266819
$672$	−451584.	−0.0385758
$673$	−1.48761e6	−0.126605	−0.0633024	−	0.997994i	$-0.520163\pi$
$673$	−1.48761e6	−0.126605	−0.0633024	+	0.997994i	$0.520163\pi$
$674$	−1.01986e7	−0.864753
$675$	−5.91292e6	−0.499508
$676$	−5.49979e6	−0.462892
$677$	1.21030e7	1.01490	0.507449	−	0.861682i	$-0.330589\pi$
$677$	1.21030e7	1.01490	0.507449	+	0.861682i	$0.330589\pi$
$678$	2.04458e6	0.170817
$679$	3.56632e6	0.296856
$680$	5.71213e6	0.473724
$681$	−1.14475e7	−0.945898
$682$	642752.	0.0529155
$683$	−1.53881e7	−1.26222	−0.631108	−	0.775695i	$-0.717400\pi$
$683$	−1.53881e7	−1.26222	−0.631108	+	0.775695i	$0.717400\pi$
$684$	1.55520e6	0.127100
$685$	3.14076e7	2.55746
$686$	470596.	0.0381802
$687$	−4.73787e6	−0.382993
$688$	−628736.	−0.0506404
$689$	−2.60852e6	−0.209337
$690$	6.57878e6	0.526045
$691$	1.07177e7	0.853898	0.426949	−	0.904276i	$-0.359588\pi$
$691$	1.07177e7	0.853898	0.426949	+	0.904276i	$0.359588\pi$
$692$	−1.63274e6	−0.129614
$693$	−480249.	−0.0379869
$694$	5.45077e6	0.429595
$695$	−3.26438e7	−2.56353
$696$	581760.	0.0455219
$697$	6.31332e6	0.492239
$698$	−1.04354e7	−0.810719
$699$	−9.14809e6	−0.708170
$700$	−6.35902e6	−0.490507
$701$	1.15187e7	0.885338	0.442669	−	0.896685i	$-0.354032\pi$
$701$	1.15187e7	0.885338	0.442669	+	0.896685i	$0.354032\pi$
$702$	−484056.	−0.0370726
$703$	1.88616e7	1.43943
$704$	495616.	0.0376889
$705$	2.14841e6	0.162796
$706$	7.11890e6	0.537528
$707$	−4.87070e6	−0.366474
$708$	768960.	0.0576528
$709$	−1.27812e7	−0.954894	−0.477447	−	0.878661i	$-0.658438\pi$
$709$	−1.27812e7	−0.954894	−0.477447	+	0.878661i	$0.658438\pi$
$710$	2.87082e7	2.13727
$711$	2.07036e6	0.153593
$712$	−3.93792e6	−0.291117
$713$	−2.28947e6	−0.168660
$714$	1.48529e6	0.109035
$715$	−2.12912e6	−0.155752
$716$	−8.65888e6	−0.631218
$717$	−4.10904e6	−0.298499
$718$	1.78448e7	1.29182
$719$	9.03422e6	0.651731	0.325866	−	0.945416i	$-0.394344\pi$
$719$	9.03422e6	0.651731	0.325866	+	0.945416i	$0.394344\pi$
$720$	2.19802e6	0.158015
$721$	−5.16382e6	−0.369941
$722$	4.14440e6	0.295882
$723$	−8.08634e6	−0.575316
$724$	8.62563e6	0.611568
$725$	8.19211e6	0.578830
$726$	527076.	0.0371135
$727$	−2.72466e7	−1.91195	−0.955974	−	0.293451i	$-0.905196\pi$
$727$	−2.72466e7	−1.91195	−0.955974	+	0.293451i	$0.905196\pi$
$728$	−520576.	−0.0364046
$729$	531441.	0.0370370
$730$	3.14014e6	0.218093
$731$	2.06795e6	0.143135
$732$	3.70339e6	0.255459
$733$	−2.15464e7	−1.48120	−0.740602	−	0.671944i	$-0.765460\pi$
$733$	−2.15464e7	−1.48120	−0.740602	+	0.671944i	$0.765460\pi$
$734$	1.15331e7	0.790143
$735$	−2.29055e6	−0.156395
$736$	−1.76538e6	−0.120128
$737$	−553212.	−0.0375165
$738$	2.42935e6	0.164191
$739$	−9.48912e6	−0.639168	−0.319584	−	0.947558i	$-0.603543\pi$
$739$	−9.48912e6	−0.639168	−0.319584	+	0.947558i	$0.603543\pi$
$740$	2.66577e7	1.78955
$741$	1.79280e6	0.119946
$742$	3.07994e6	0.205368
$743$	2.16227e7	1.43694	0.718469	−	0.695559i	$-0.244843\pi$
$743$	2.16227e7	1.43694	0.718469	+	0.695559i	$0.244843\pi$
$744$	−764928.	−0.0506627
$745$	3.81356e7	2.51733
$746$	−1.15750e7	−0.761508
$747$	1.81472e6	0.118990
$748$	−1.63011e6	−0.106528
$749$	−2.55917e6	−0.166684
$750$	1.90266e7	1.23512
$751$	5.95283e6	0.385145	0.192572	−	0.981283i	$-0.438317\pi$
$751$	5.95283e6	0.385145	0.192572	+	0.981283i	$0.438317\pi$
$752$	−576512.	−0.0371761
$753$	−8.56883e6	−0.550724
$754$	670640.	0.0429597
$755$	−4.43979e7	−2.83462
$756$	571536.	0.0363696
$757$	1.10220e7	0.699072	0.349536	−	0.936923i	$-0.386339\pi$
$757$	1.10220e7	0.699072	0.349536	+	0.936923i	$0.386339\pi$
$758$	−1.97894e7	−1.25100
$759$	−1.87744e6	−0.118293
$760$	−8.14080e6	−0.511250
$761$	−1.99788e7	−1.25057	−0.625285	−	0.780396i	$-0.715017\pi$
$761$	−1.99788e7	−1.25057	−0.625285	+	0.780396i	$0.715017\pi$
$762$	3.62909e6	0.226418
$763$	−7.05649e6	−0.438811
$764$	1.77875e6	0.110251
$765$	−7.22941e6	−0.446632
$766$	−1.22285e7	−0.753011
$767$	886440.	0.0544078
$768$	−589824.	−0.0360844
$769$	2.01005e7	1.22572	0.612859	−	0.790192i	$-0.290019\pi$
$769$	2.01005e7	1.22572	0.612859	+	0.790192i	$0.290019\pi$
$770$	2.51390e6	0.152799
$771$	−1.60598e7	−0.972978
$772$	−2656.00	−0.000160393 0
$773$	−1.23534e7	−0.743598	−0.371799	−	0.928313i	$-0.621259\pi$
$773$	−1.23534e7	−0.743598	−0.371799	+	0.928313i	$0.621259\pi$
$774$	795744.	0.0477442
$775$	−1.07714e7	−0.644196
$776$	4.65805e6	0.277683
$777$	6.93164e6	0.411892
$778$	−2.08074e7	−1.23245
$779$	−8.99760e6	−0.531231
$780$	2.53382e6	0.149121
$781$	−8.19267e6	−0.480615
$782$	5.80643e6	0.339541
$783$	−736290.	−0.0429185
$784$	614656.	0.0357143
$785$	3.51451e7	2.03559
$786$	−1.87445e6	−0.108222
$787$	5.80285e6	0.333968	0.166984	−	0.985960i	$-0.446597\pi$
$787$	5.80285e6	0.333968	0.166984	+	0.985960i	$0.446597\pi$
$788$	−6.72547e6	−0.385840
$789$	−9.03010e6	−0.516416
$790$	−1.08374e7	−0.617816
$791$	−2.78291e6	−0.158146
$792$	−627264.	−0.0355335
$793$	4.26919e6	0.241081
$794$	1.87298e7	1.05434
$795$	−1.49912e7	−0.841235
$796$	−1.14906e7	−0.642774
$797$	−1.80032e7	−1.00393	−0.501965	−	0.864888i	$-0.667389\pi$
$797$	−1.80032e7	−1.00393	−0.501965	+	0.864888i	$0.667389\pi$
$798$	−2.11680e6	−0.117672
$799$	1.89618e6	0.105078
$800$	−8.30566e6	−0.458827
$801$	4.98393e6	0.274467
$802$	1.00112e7	0.549604
$803$	−896126.	−0.0490434
$804$	658368.	0.0359194
$805$	−8.95446e6	−0.487023
$806$	−881792.	−0.0478111
$807$	−1.57083e7	−0.849075
$808$	−6.36173e6	−0.342805
$809$	−7.34925e6	−0.394795	−0.197398	−	0.980324i	$-0.563249\pi$
$809$	−7.34925e6	−0.394795	−0.197398	+	0.980324i	$0.563249\pi$
$810$	−2.78186e6	−0.148978
$811$	−2.12776e7	−1.13598	−0.567991	−	0.823035i	$-0.692279\pi$
$811$	−2.12776e7	−1.13598	−0.567991	+	0.823035i	$0.692279\pi$
$812$	−791840.	−0.0421451
$813$	1.78438e7	0.946808
$814$	−7.60751e6	−0.402422
$815$	5.39841e7	2.84690
$816$	1.93997e6	0.101993
$817$	−2.94720e6	−0.154474
$818$	1.78652e6	0.0933522
$819$	658854.	0.0343225
$820$	−1.27166e7	−0.660445
$821$	−1.13924e6	−0.0589870	−0.0294935	−	0.999565i	$-0.509389\pi$
$821$	−1.13924e6	−0.0589870	−0.0294935	+	0.999565i	$0.509389\pi$
$822$	1.06667e7	0.550620
$823$	543104.	0.0279501	0.0139751	−	0.999902i	$-0.495551\pi$
$823$	543104.	0.0279501	0.0139751	+	0.999902i	$0.495551\pi$
$824$	−6.74458e6	−0.346048
$825$	−8.83288e6	−0.451822
$826$	−1.04664e6	−0.0533761
$827$	−2.12436e7	−1.08010	−0.540050	−	0.841633i	$-0.681595\pi$
$827$	−2.12436e7	−1.08010	−0.540050	+	0.841633i	$0.681595\pi$
$828$	2.23430e6	0.113257
$829$	−1.50221e6	−0.0759179	−0.0379590	−	0.999279i	$-0.512086\pi$
$829$	−1.50221e6	−0.0759179	−0.0379590	+	0.999279i	$0.512086\pi$
$830$	−9.49930e6	−0.478626
$831$	3.83492e6	0.192643
$832$	−679936.	−0.0340533
$833$	−2.02164e6	−0.100947
$834$	−1.10866e7	−0.551927
$835$	5.37556e7	2.66813
$836$	2.32320e6	0.114966
$837$	968112.	0.0477652
$838$	−9.40616e6	−0.462703
$839$	−3.05629e7	−1.49896	−0.749478	−	0.662029i	$-0.769696\pi$
$839$	−3.05629e7	−1.49896	−0.749478	+	0.662029i	$0.769696\pi$
$840$	−2.99174e6	−0.146294
$841$	−1.94910e7	−0.950266
$842$	231272.	0.0112420
$843$	2.47084e6	0.119750
$844$	−5.90989e6	−0.285577
$845$	−3.64361e7	−1.75546
$846$	729648.	0.0350500
$847$	−717409.	−0.0343604
$848$	4.02278e6	0.192104
$849$	131544.	0.00626328
$850$	2.73178e7	1.29688
$851$	2.70978e7	1.28266
$852$	9.74995e6	0.460154
$853$	2.55225e7	1.20102	0.600511	−	0.799617i	$-0.294964\pi$
$853$	2.55225e7	1.20102	0.600511	+	0.799617i	$0.294964\pi$
$854$	−5.04073e6	−0.236509
$855$	1.03032e7	0.482011
$856$	−3.34259e6	−0.155919
$857$	−2.11822e7	−0.985186	−0.492593	−	0.870260i	$-0.663951\pi$
$857$	−2.11822e7	−0.985186	−0.492593	+	0.870260i	$0.663951\pi$
$858$	−723096.	−0.0335334
$859$	−1.50271e7	−0.694854	−0.347427	−	0.937707i	$-0.612945\pi$
$859$	−1.50271e7	−0.694854	−0.347427	+	0.937707i	$0.612945\pi$
$860$	−4.16538e6	−0.192047
$861$	−3.30662e6	−0.152011
$862$	−1.86265e7	−0.853815
$863$	4.32316e6	0.197594	0.0987972	−	0.995108i	$-0.468500\pi$
$863$	4.32316e6	0.197594	0.0987972	+	0.995108i	$0.468500\pi$
$864$	746496.	0.0340207
$865$	−1.08169e7	−0.491543
$866$	−1.70513e7	−0.772616
$867$	6.39804e6	0.289067
$868$	1.04115e6	0.0469045
$869$	3.09276e6	0.138930
$870$	3.85416e6	0.172636
$871$	758952.	0.0338976
$872$	−9.21664e6	−0.410470
$873$	−5.89534e6	−0.261802
$874$	−8.27520e6	−0.366438
$875$	−2.58973e7	−1.14349
$876$	1.06646e6	0.0469554
$877$	1.42430e7	0.625319	0.312660	−	0.949865i	$-0.398780\pi$
$877$	1.42430e7	0.625319	0.312660	+	0.949865i	$0.398780\pi$
$878$	2.11899e7	0.927669
$879$	1.48479e7	0.648176
$880$	3.28346e6	0.142930
$881$	−1.98050e7	−0.859676	−0.429838	−	0.902906i	$-0.641429\pi$
$881$	−1.98050e7	−0.859676	−0.429838	+	0.902906i	$0.641429\pi$
$882$	−777924.	−0.0336718
$883$	1.78078e7	0.768613	0.384306	−	0.923206i	$-0.374441\pi$
$883$	1.78078e7	0.768613	0.384306	+	0.923206i	$0.374441\pi$
$884$	2.23635e6	0.0962519
$885$	5.09436e6	0.218641
$886$	1.79814e7	0.769555
$887$	1.64591e7	0.702420	0.351210	−	0.936297i	$-0.385770\pi$
$887$	1.64591e7	0.702420	0.351210	+	0.936297i	$0.385770\pi$
$888$	9.05357e6	0.385290
$889$	−4.93959e6	−0.209622
$890$	−2.60887e7	−1.10402
$891$	793881.	0.0335013
$892$	−7.52346e6	−0.316596
$893$	−2.70240e6	−0.113402
$894$	1.29517e7	0.541980
$895$	−5.73651e7	−2.39381
$896$	802816.	0.0334077
$897$	2.57566e6	0.106883
$898$	2.29811e7	0.950998
$899$	−1.34128e6	−0.0553503
$900$	1.05119e7	0.432587
$901$	−1.32312e7	−0.542984
$902$	3.62903e6	0.148516
$903$	−1.08310e6	−0.0442026
$904$	−3.63482e6	−0.147932
$905$	5.71448e7	2.31929
$906$	−1.50785e7	−0.610293
$907$	−2.58722e7	−1.04427	−0.522137	−	0.852861i	$-0.674865\pi$
$907$	−2.58722e7	−1.04427	−0.522137	+	0.852861i	$0.674865\pi$
$908$	2.03512e7	0.819171
$909$	8.05156e6	0.323199
$910$	−3.44882e6	−0.138060
$911$	3.31944e7	1.32516	0.662581	−	0.748990i	$-0.269461\pi$
$911$	3.31944e7	1.32516	0.662581	+	0.748990i	$0.269461\pi$
$912$	−2.76480e6	−0.110072
$913$	2.71088e6	0.107630
$914$	3.04388e7	1.20521
$915$	2.45350e7	0.968797
$916$	8.42288e6	0.331682
$917$	2.55133e6	0.100194
$918$	−2.45527e6	−0.0961597
$919$	−1.56096e7	−0.609681	−0.304841	−	0.952403i	$-0.598603\pi$
$919$	−1.56096e7	−0.609681	−0.304841	+	0.952403i	$0.598603\pi$
$920$	−1.16956e7	−0.455568
$921$	−1.03778e7	−0.403140
$922$	−2.12531e7	−0.823371
$923$	1.12395e7	0.434254
$924$	853776.	0.0328976
$925$	1.27489e8	4.89911
$926$	−5.58938e6	−0.214208
$927$	8.53610e6	0.326257
$928$	−1.03424e6	−0.0394232
$929$	−1.22660e7	−0.466296	−0.233148	−	0.972441i	$-0.574903\pi$
$929$	−1.22660e7	−0.466296	−0.233148	+	0.972441i	$0.574903\pi$
$930$	−5.06765e6	−0.192132
$931$	2.88120e6	0.108943
$932$	1.62633e7	0.613293
$933$	2.85083e7	1.07218
$934$	2.60564e7	0.977342
$935$	−1.07995e7	−0.403993
$936$	860544.	0.0321058
$937$	2.00704e7	0.746804	0.373402	−	0.927670i	$-0.378191\pi$
$937$	2.00704e7	0.746804	0.373402	+	0.927670i	$0.378191\pi$
$938$	−896112.	−0.0332549
$939$	−5.01043e6	−0.185443
$940$	−3.81939e6	−0.140986
$941$	−4.83600e6	−0.178038	−0.0890189	−	0.996030i	$-0.528373\pi$
$941$	−4.83600e6	−0.178038	−0.0890189	+	0.996030i	$0.528373\pi$
$942$	1.19361e7	0.438263
$943$	−1.29266e7	−0.473373
$944$	−1.36704e6	−0.0499288
$945$	3.78643e6	0.137927
$946$	1.18870e6	0.0431863
$947$	2.57220e7	0.932031	0.466015	−	0.884777i	$-0.345689\pi$
$947$	2.57220e7	0.932031	0.466015	+	0.884777i	$0.345689\pi$
$948$	−3.68064e6	−0.133016
$949$	1.22940e6	0.0443125
$950$	−3.89328e7	−1.39961
$951$	−1.12397e7	−0.402999
$952$	−2.64051e6	−0.0944269
$953$	−3.91626e7	−1.39682	−0.698409	−	0.715699i	$-0.746108\pi$
$953$	−3.91626e7	−1.39682	−0.698409	+	0.715699i	$0.746108\pi$
$954$	−5.09134e6	−0.181118
$955$	1.17842e7	0.418112
$956$	7.30496e6	0.258507
$957$	−1.09989e6	−0.0388212
$958$	−3.39198e7	−1.19410
$959$	−1.45186e7	−0.509775
$960$	−3.90758e6	−0.136845
$961$	−2.68656e7	−0.938399
$962$	1.04368e7	0.363603
$963$	4.23047e6	0.147002
$964$	1.43757e7	0.498238
$965$	−17596.0	−0.000608269 0
$966$	−3.04114e6	−0.104856
$967$	−2.38367e7	−0.819746	−0.409873	−	0.912143i	$-0.634427\pi$
$967$	−2.38367e7	−0.819746	−0.409873	+	0.912143i	$0.634427\pi$
$968$	−937024.	−0.0321412
$969$	9.09360e6	0.311119
$970$	3.08596e7	1.05308
$971$	999412.	0.0340170	0.0170085	−	0.999855i	$-0.494586\pi$
$971$	999412.	0.0340170	0.0170085	+	0.999855i	$0.494586\pi$
$972$	−944784.	−0.0320750
$973$	1.50900e7	0.510985
$974$	1.54221e6	0.0520889
$975$	1.21178e7	0.408238
$976$	−6.58381e6	−0.221234
$977$	2.36994e6	0.0794329	0.0397165	−	0.999211i	$-0.487355\pi$
$977$	2.36994e6	0.0794329	0.0397165	+	0.999211i	$0.487355\pi$
$978$	1.83342e7	0.612936
$979$	7.44513e6	0.248265
$980$	4.07210e6	0.135442
$981$	1.16648e7	0.386995
$982$	−7.73525e6	−0.255974
$983$	2.51680e7	0.830741	0.415371	−	0.909652i	$-0.363652\pi$
$983$	2.51680e7	0.830741	0.415371	+	0.909652i	$0.363652\pi$
$984$	−4.31885e6	−0.142194
$985$	−4.45563e7	−1.46325
$986$	3.40168e6	0.111430
$987$	−993132.	−0.0324500
$988$	−3.18720e6	−0.103876
$989$	−4.23414e6	−0.137650
$990$	−4.15562e6	−0.134756
$991$	2.24675e6	0.0726727	0.0363363	−	0.999340i	$-0.488431\pi$
$991$	2.24675e6	0.0726727	0.0363363	+	0.999340i	$0.488431\pi$
$992$	1.35987e6	0.0438752
$993$	−2.95590e7	−0.951298
$994$	−1.32708e7	−0.426020
$995$	−7.61250e7	−2.43764
$996$	−3.22618e6	−0.103048
$997$	−3.29015e7	−1.04828	−0.524140	−	0.851632i	$-0.675613\pi$
$997$	−3.29015e7	−1.04828	−0.524140	+	0.851632i	$0.675613\pi$
$998$	−1.79766e7	−0.571321
$999$	−1.14584e7	−0.363255

Twists

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

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