LMFDB - Embedded newform 138.6.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [138,6,Mod(1,138)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(138, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("138.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 138 = 2 \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	138.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:	$22.1329671342$
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	$\chi$	$=$	138.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} -44.0000 q^{5} +36.0000 q^{6} -70.0000 q^{7} -64.0000 q^{8} +81.0000 q^{9} +O(q^{10})$

\(q-4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} -44.0000 q^{5} +36.0000 q^{6} -70.0000 q^{7} -64.0000 q^{8} +81.0000 q^{9} +176.000 q^{10} -136.000 q^{11} -144.000 q^{12} -1022.00 q^{13} +280.000 q^{14} +396.000 q^{15} +256.000 q^{16} +484.000 q^{17} -324.000 q^{18} -1046.00 q^{19} -704.000 q^{20} +630.000 q^{21} +544.000 q^{22} +529.000 q^{23} +576.000 q^{24} -1189.00 q^{25} +4088.00 q^{26} -729.000 q^{27} -1120.00 q^{28} +2618.00 q^{29} -1584.00 q^{30} -4860.00 q^{31} -1024.00 q^{32} +1224.00 q^{33} -1936.00 q^{34} +3080.00 q^{35} +1296.00 q^{36} +14918.0 q^{37} +4184.00 q^{38} +9198.00 q^{39} +2816.00 q^{40} +7530.00 q^{41} -2520.00 q^{42} +16186.0 q^{43} -2176.00 q^{44} -3564.00 q^{45} -2116.00 q^{46} +29160.0 q^{47} -2304.00 q^{48} -11907.0 q^{49} +4756.00 q^{50} -4356.00 q^{51} -16352.0 q^{52} +9896.00 q^{53} +2916.00 q^{54} +5984.00 q^{55} +4480.00 q^{56} +9414.00 q^{57} -10472.0 q^{58} -2004.00 q^{59} +6336.00 q^{60} -2570.00 q^{61} +19440.0 q^{62} -5670.00 q^{63} +4096.00 q^{64} +44968.0 q^{65} -4896.00 q^{66} +46118.0 q^{67} +7744.00 q^{68} -4761.00 q^{69} -12320.0 q^{70} -32688.0 q^{71} -5184.00 q^{72} -46830.0 q^{73} -59672.0 q^{74} +10701.0 q^{75} -16736.0 q^{76} +9520.00 q^{77} -36792.0 q^{78} -34338.0 q^{79} -11264.0 q^{80} +6561.00 q^{81} -30120.0 q^{82} +31736.0 q^{83} +10080.0 q^{84} -21296.0 q^{85} -64744.0 q^{86} -23562.0 q^{87} +8704.00 q^{88} -60792.0 q^{89} +14256.0 q^{90} +71540.0 q^{91} +8464.00 q^{92} +43740.0 q^{93} -116640. q^{94} +46024.0 q^{95} +9216.00 q^{96} -19218.0 q^{97} +47628.0 q^{98} -11016.0 q^{99} +O(q^{100})\)

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
$5$	−44.0000	−0.787096	−0.393548	−	0.919304i	$-0.628752\pi$
$5$	−44.0000	−0.787096	−0.393548	+	0.919304i	$0.628752\pi$
$6$	36.0000	0.408248
$7$	−70.0000	−0.539949	−0.269975	−	0.962867i	$-0.587015\pi$
$7$	−70.0000	−0.539949	−0.269975	+	0.962867i	$0.587015\pi$
$8$	−64.0000	−0.353553
$9$	81.0000	0.333333
$10$	176.000	0.556561
$11$	−136.000	−0.338889	−0.169444	−	0.985540i	$-0.554197\pi$
$11$	−136.000	−0.338889	−0.169444	+	0.985540i	$0.554197\pi$
$12$	−144.000	−0.288675
$13$	−1022.00	−1.67723	−0.838615	−	0.544725i	$-0.816634\pi$
$13$	−1022.00	−1.67723	−0.838615	+	0.544725i	$0.816634\pi$
$14$	280.000	0.381802
$15$	396.000	0.454430
$16$	256.000	0.250000
$17$	484.000	0.406184	0.203092	−	0.979160i	$-0.434901\pi$
$17$	484.000	0.406184	0.203092	+	0.979160i	$0.434901\pi$
$18$	−324.000	−0.235702
$19$	−1046.00	−0.664734	−0.332367	−	0.943150i	$-0.607847\pi$
$19$	−1046.00	−0.664734	−0.332367	+	0.943150i	$0.607847\pi$
$20$	−704.000	−0.393548
$21$	630.000	0.311740
$22$	544.000	0.239631
$23$	529.000	0.208514
$23$	529.000	0.208514
$24$	576.000	0.204124
$25$	−1189.00	−0.380480
$26$	4088.00	1.18598
$27$	−729.000	−0.192450
$28$	−1120.00	−0.269975
$29$	2618.00	0.578062	0.289031	−	0.957320i	$-0.406667\pi$
$29$	2618.00	0.578062	0.289031	+	0.957320i	$0.406667\pi$
$30$	−1584.00	−0.321331
$31$	−4860.00	−0.908306	−0.454153	−	0.890924i	$-0.650058\pi$
$31$	−4860.00	−0.908306	−0.454153	+	0.890924i	$0.650058\pi$
$32$	−1024.00	−0.176777
$33$	1224.00	0.195658
$34$	−1936.00	−0.287216
$35$	3080.00	0.424992
$36$	1296.00	0.166667
$37$	14918.0	1.79146	0.895728	−	0.444602i	$-0.146655\pi$
$37$	14918.0	1.79146	0.895728	+	0.444602i	$0.146655\pi$
$38$	4184.00	0.470038
$39$	9198.00	0.968349
$40$	2816.00	0.278280
$41$	7530.00	0.699577	0.349788	−	0.936829i	$-0.386253\pi$
$41$	7530.00	0.699577	0.349788	+	0.936829i	$0.386253\pi$
$42$	−2520.00	−0.220433
$43$	16186.0	1.33496	0.667480	−	0.744628i	$-0.267373\pi$
$43$	16186.0	1.33496	0.667480	+	0.744628i	$0.267373\pi$
$44$	−2176.00	−0.169444
$45$	−3564.00	−0.262365
$46$	−2116.00	−0.147442
$47$	29160.0	1.92550	0.962749	−	0.270398i	$-0.0871554\pi$
$47$	29160.0	1.92550	0.962749	+	0.270398i	$0.0871554\pi$
$48$	−2304.00	−0.144338
$49$	−11907.0	−0.708455
$50$	4756.00	0.269040
$51$	−4356.00	−0.234511
$52$	−16352.0	−0.838615
$53$	9896.00	0.483916	0.241958	−	0.970287i	$-0.422210\pi$
$53$	9896.00	0.483916	0.241958	+	0.970287i	$0.422210\pi$
$54$	2916.00	0.136083
$55$	5984.00	0.266738
$56$	4480.00	0.190901
$57$	9414.00	0.383784
$58$	−10472.0	−0.408752
$59$	−2004.00	−0.0749493	−0.0374747	−	0.999298i	$-0.511931\pi$
$59$	−2004.00	−0.0749493	−0.0374747	+	0.999298i	$0.511931\pi$
$60$	6336.00	0.227215
$61$	−2570.00	−0.0884318	−0.0442159	−	0.999022i	$-0.514079\pi$
$61$	−2570.00	−0.0884318	−0.0442159	+	0.999022i	$0.514079\pi$
$62$	19440.0	0.642269
$63$	−5670.00	−0.179983
$64$	4096.00	0.125000
$65$	44968.0	1.32014
$66$	−4896.00	−0.138351
$67$	46118.0	1.25511	0.627557	−	0.778570i	$-0.284055\pi$
$67$	46118.0	1.25511	0.627557	+	0.778570i	$0.284055\pi$
$68$	7744.00	0.203092
$69$	−4761.00	−0.120386
$70$	−12320.0	−0.300515
$71$	−32688.0	−0.769560	−0.384780	−	0.923008i	$-0.625723\pi$
$71$	−32688.0	−0.769560	−0.384780	+	0.923008i	$0.625723\pi$
$72$	−5184.00	−0.117851
$73$	−46830.0	−1.02853	−0.514265	−	0.857631i	$-0.671935\pi$
$73$	−46830.0	−1.02853	−0.514265	+	0.857631i	$0.671935\pi$
$74$	−59672.0	−1.26675
$75$	10701.0	0.219670
$76$	−16736.0	−0.332367
$77$	9520.00	0.182983
$78$	−36792.0	−0.684726
$79$	−34338.0	−0.619024	−0.309512	−	0.950896i	$-0.600166\pi$
$79$	−34338.0	−0.619024	−0.309512	+	0.950896i	$0.600166\pi$
$80$	−11264.0	−0.196774
$81$	6561.00	0.111111
$82$	−30120.0	−0.494675
$83$	31736.0	0.505658	0.252829	−	0.967511i	$-0.418639\pi$
$83$	31736.0	0.505658	0.252829	+	0.967511i	$0.418639\pi$
$84$	10080.0	0.155870
$85$	−21296.0	−0.319706
$86$	−64744.0	−0.943960
$87$	−23562.0	−0.333744
$88$	8704.00	0.119815
$89$	−60792.0	−0.813526	−0.406763	−	0.913534i	$-0.633343\pi$
$89$	−60792.0	−0.813526	−0.406763	+	0.913534i	$0.633343\pi$
$90$	14256.0	0.185520
$91$	71540.0	0.905619
$92$	8464.00	0.104257
$93$	43740.0	0.524411
$94$	−116640.	−1.36153
$95$	46024.0	0.523209
$96$	9216.00	0.102062
$97$	−19218.0	−0.207386	−0.103693	−	0.994609i	$-0.533066\pi$
$97$	−19218.0	−0.207386	−0.103693	+	0.994609i	$0.533066\pi$
$98$	47628.0	0.500953
$99$	−11016.0	−0.112963
$100$	−19024.0	−0.190240
$101$	12326.0	0.120232	0.0601158	−	0.998191i	$-0.480853\pi$
$101$	12326.0	0.120232	0.0601158	+	0.998191i	$0.480853\pi$
$102$	17424.0	0.165824
$103$	−179210.	−1.66444	−0.832222	−	0.554443i	$-0.812932\pi$
$103$	−179210.	−1.66444	−0.832222	+	0.554443i	$0.812932\pi$
$104$	65408.0	0.592990
$105$	−27720.0	−0.245369
$106$	−39584.0	−0.342180
$107$	75608.0	0.638423	0.319211	−	0.947684i	$-0.396582\pi$
$107$	75608.0	0.638423	0.319211	+	0.947684i	$0.396582\pi$
$108$	−11664.0	−0.0962250
$109$	81450.0	0.656636	0.328318	−	0.944567i	$-0.393518\pi$
$109$	81450.0	0.656636	0.328318	+	0.944567i	$0.393518\pi$
$110$	−23936.0	−0.188612
$111$	−134262.	−1.03430
$112$	−17920.0	−0.134987
$113$	123544.	0.910176	0.455088	−	0.890446i	$-0.349608\pi$
$113$	123544.	0.910176	0.455088	+	0.890446i	$0.349608\pi$
$114$	−37656.0	−0.271376
$115$	−23276.0	−0.164121
$116$	41888.0	0.289031
$117$	−82782.0	−0.559077
$118$	8016.00	0.0529972
$119$	−33880.0	−0.219319
$120$	−25344.0	−0.160665
$121$	−142555.	−0.885154
$122$	10280.0	0.0625307
$123$	−67770.0	−0.403901
$124$	−77760.0	−0.454153
$125$	189816.	1.08657
$126$	22680.0	0.127267
$127$	−144236.	−0.793532	−0.396766	−	0.917920i	$-0.629868\pi$
$127$	−144236.	−0.793532	−0.396766	+	0.917920i	$0.629868\pi$
$128$	−16384.0	−0.0883883
$129$	−145674.	−0.770740
$130$	−179872.	−0.933480
$131$	211260.	1.07557	0.537785	−	0.843082i	$-0.319261\pi$
$131$	211260.	1.07557	0.537785	+	0.843082i	$0.319261\pi$
$132$	19584.0	0.0978288
$133$	73220.0	0.358922
$134$	−184472.	−0.887500
$135$	32076.0	0.151477
$136$	−30976.0	−0.143608
$137$	−47928.0	−0.218166	−0.109083	−	0.994033i	$-0.534791\pi$
$137$	−47928.0	−0.218166	−0.109083	+	0.994033i	$0.534791\pi$
$138$	19044.0	0.0851257
$139$	94220.0	0.413624	0.206812	−	0.978381i	$-0.433691\pi$
$139$	94220.0	0.413624	0.206812	+	0.978381i	$0.433691\pi$
$140$	49280.0	0.212496
$141$	−262440.	−1.11169
$142$	130752.	0.544161
$143$	138992.	0.568394
$144$	20736.0	0.0833333
$145$	−115192.	−0.454990
$146$	187320.	0.727281
$147$	107163.	0.409027
$148$	238688.	0.895728
$149$	−497980.	−1.83758	−0.918790	−	0.394747i	$-0.870832\pi$
$149$	−497980.	−1.83758	−0.918790	+	0.394747i	$0.870832\pi$
$150$	−42804.0	−0.155330
$151$	362128.	1.29247	0.646234	−	0.763139i	$-0.276343\pi$
$151$	362128.	1.29247	0.646234	+	0.763139i	$0.276343\pi$
$152$	66944.0	0.235019
$153$	39204.0	0.135395
$154$	−38080.0	−0.129388
$155$	213840.	0.714924
$156$	147168.	0.484174
$157$	154110.	0.498978	0.249489	−	0.968378i	$-0.419737\pi$
$157$	154110.	0.498978	0.249489	+	0.968378i	$0.419737\pi$
$158$	137352.	0.437716
$159$	−89064.0	−0.279389
$160$	45056.0	0.139140
$161$	−37030.0	−0.112587
$162$	−26244.0	−0.0785674
$163$	388720.	1.14596	0.572978	−	0.819571i	$-0.305788\pi$
$163$	388720.	1.14596	0.572978	+	0.819571i	$0.305788\pi$
$164$	120480.	0.349788
$165$	−53856.0	−0.154001
$166$	−126944.	−0.357554
$167$	368248.	1.02176	0.510880	−	0.859652i	$-0.329320\pi$
$167$	368248.	1.02176	0.510880	+	0.859652i	$0.329320\pi$
$168$	−40320.0	−0.110217
$169$	673191.	1.81310
$170$	85184.0	0.226066
$171$	−84726.0	−0.221578
$172$	258976.	0.667480
$173$	−42486.0	−0.107927	−0.0539636	−	0.998543i	$-0.517185\pi$
$173$	−42486.0	−0.107927	−0.0539636	+	0.998543i	$0.517185\pi$
$174$	94248.0	0.235993
$175$	83230.0	0.205440
$176$	−34816.0	−0.0847222
$177$	18036.0	0.0432720
$178$	243168.	0.575250
$179$	697484.	1.62705	0.813526	−	0.581528i	$-0.197545\pi$
$179$	697484.	1.62705	0.813526	+	0.581528i	$0.197545\pi$
$180$	−57024.0	−0.131183
$181$	−833622.	−1.89135	−0.945677	−	0.325108i	$-0.894599\pi$
$181$	−833622.	−1.89135	−0.945677	+	0.325108i	$0.894599\pi$
$182$	−286160.	−0.640369
$183$	23130.0	0.0510561
$184$	−33856.0	−0.0737210
$185$	−656392.	−1.41005
$186$	−174960.	−0.370814
$187$	−65824.0	−0.137651
$188$	466560.	0.962749
$189$	51030.0	0.103913
$190$	−184096.	−0.369965
$191$	225620.	0.447501	0.223751	−	0.974646i	$-0.428170\pi$
$191$	225620.	0.447501	0.223751	+	0.974646i	$0.428170\pi$
$192$	−36864.0	−0.0721688
$193$	476462.	0.920736	0.460368	−	0.887728i	$-0.347718\pi$
$193$	476462.	0.920736	0.460368	+	0.887728i	$0.347718\pi$
$194$	76872.0	0.146644
$195$	−404712.	−0.762184
$196$	−190512.	−0.354227
$197$	−366894.	−0.673558	−0.336779	−	0.941584i	$-0.609338\pi$
$197$	−366894.	−0.673558	−0.336779	+	0.941584i	$0.609338\pi$
$198$	44064.0	0.0798769
$199$	−449986.	−0.805501	−0.402751	−	0.915310i	$-0.631946\pi$
$199$	−449986.	−0.805501	−0.402751	+	0.915310i	$0.631946\pi$
$200$	76096.0	0.134520
$201$	−415062.	−0.724641
$202$	−49304.0	−0.0850166
$203$	−183260.	−0.312124
$204$	−69696.0	−0.117255
$205$	−331320.	−0.550634
$206$	716840.	1.17694
$207$	42849.0	0.0695048
$208$	−261632.	−0.419307
$209$	142256.	0.225271
$210$	110880.	0.173502
$211$	328144.	0.507409	0.253705	−	0.967282i	$-0.418351\pi$
$211$	328144.	0.507409	0.253705	+	0.967282i	$0.418351\pi$
$212$	158336.	0.241958
$213$	294192.	0.444306
$214$	−302432.	−0.451433
$215$	−712184.	−1.05074
$216$	46656.0	0.0680414
$217$	340200.	0.490439
$218$	−325800.	−0.464312
$219$	421470.	0.593822
$220$	95744.0	0.133369
$221$	−494648.	−0.681264
$222$	537048.	0.731359
$223$	201896.	0.271873	0.135936	−	0.990718i	$-0.456596\pi$
$223$	201896.	0.271873	0.135936	+	0.990718i	$0.456596\pi$
$224$	71680.0	0.0954504
$225$	−96309.0	−0.126827
$226$	−494176.	−0.643592
$227$	−547980.	−0.705830	−0.352915	−	0.935655i	$-0.614810\pi$
$227$	−547980.	−0.705830	−0.352915	+	0.935655i	$0.614810\pi$
$228$	150624.	0.191892
$229$	891794.	1.12377	0.561883	−	0.827217i	$-0.310077\pi$
$229$	891794.	1.12377	0.561883	+	0.827217i	$0.310077\pi$
$230$	93104.0	0.116051
$231$	−85680.0	−0.105645
$232$	−167552.	−0.204376
$233$	−1.15849e6	−1.39798	−0.698990	−	0.715131i	$-0.746367\pi$
$233$	−1.15849e6	−1.39798	−0.698990	+	0.715131i	$0.746367\pi$
$234$	331128.	0.395327
$235$	−1.28304e6	−1.51555
$236$	−32064.0	−0.0374747
$237$	309042.	0.357393
$238$	135520.	0.155082
$239$	79176.0	0.0896600	0.0448300	−	0.998995i	$-0.485725\pi$
$239$	79176.0	0.0896600	0.0448300	+	0.998995i	$0.485725\pi$
$240$	101376.	0.113608
$241$	744118.	0.825276	0.412638	−	0.910895i	$-0.364607\pi$
$241$	744118.	0.825276	0.412638	+	0.910895i	$0.364607\pi$
$242$	570220.	0.625899
$243$	−59049.0	−0.0641500
$244$	−41120.0	−0.0442159
$245$	523908.	0.557622
$246$	271080.	0.285601
$247$	1.06901e6	1.11491
$248$	311040.	0.321135
$249$	−285624.	−0.291942
$250$	−759264.	−0.768321
$251$	381020.	0.381736	0.190868	−	0.981616i	$-0.438870\pi$
$251$	381020.	0.381736	0.190868	+	0.981616i	$0.438870\pi$
$252$	−90720.0	−0.0899915
$253$	−71944.0	−0.0706632
$254$	576944.	0.561112
$255$	191664.	0.184582
$256$	65536.0	0.0625000
$257$	−293174.	−0.276881	−0.138440	−	0.990371i	$-0.544209\pi$
$257$	−293174.	−0.276881	−0.138440	+	0.990371i	$0.544209\pi$
$258$	582696.	0.544995
$259$	−1.04426e6	−0.967296
$260$	719488.	0.660070
$261$	212058.	0.192687
$262$	−845040.	−0.760543
$263$	−512164.	−0.456583	−0.228291	−	0.973593i	$-0.573314\pi$
$263$	−512164.	−0.456583	−0.228291	+	0.973593i	$0.573314\pi$
$264$	−78336.0	−0.0691754
$265$	−435424.	−0.380888
$266$	−292880.	−0.253796
$267$	547128.	0.469689
$268$	737888.	0.627557
$269$	−604166.	−0.509068	−0.254534	−	0.967064i	$-0.581922\pi$
$269$	−604166.	−0.509068	−0.254534	+	0.967064i	$0.581922\pi$
$270$	−128304.	−0.107110
$271$	−340660.	−0.281772	−0.140886	−	0.990026i	$-0.544995\pi$
$271$	−340660.	−0.281772	−0.140886	+	0.990026i	$0.544995\pi$
$272$	123904.	0.101546
$273$	−643860.	−0.522859
$274$	191712.	0.154267
$275$	161704.	0.128940
$276$	−76176.0	−0.0601929
$277$	−489890.	−0.383618	−0.191809	−	0.981432i	$-0.561435\pi$
$277$	−489890.	−0.383618	−0.191809	+	0.981432i	$0.561435\pi$
$278$	−376880.	−0.292477
$279$	−393660.	−0.302769
$280$	−197120.	−0.150257
$281$	−897072.	−0.677737	−0.338869	−	0.940834i	$-0.610044\pi$
$281$	−897072.	−0.677737	−0.338869	+	0.940834i	$0.610044\pi$
$282$	1.04976e6	0.786081
$283$	199486.	0.148063	0.0740315	−	0.997256i	$-0.476413\pi$
$283$	199486.	0.148063	0.0740315	+	0.997256i	$0.476413\pi$
$284$	−523008.	−0.384780
$285$	−414216.	−0.302075
$286$	−555968.	−0.401915
$287$	−527100.	−0.377736
$288$	−82944.0	−0.0589256
$289$	−1.18560e6	−0.835014
$290$	460768.	0.321727
$291$	172962.	0.119734
$292$	−749280.	−0.514265
$293$	−2.26369e6	−1.54045	−0.770225	−	0.637772i	$-0.779856\pi$
$293$	−2.26369e6	−1.54045	−0.770225	+	0.637772i	$0.779856\pi$
$294$	−428652.	−0.289225
$295$	88176.0	0.0589923
$296$	−954752.	−0.633376
$297$	99144.0	0.0652192
$298$	1.99192e6	1.29937
$299$	−540638.	−0.349727
$300$	171216.	0.109835
$301$	−1.13302e6	−0.720811
$302$	−1.44851e6	−0.913913
$303$	−110934.	−0.0694158
$304$	−267776.	−0.166183
$305$	113080.	0.0696043
$306$	−156816.	−0.0957385
$307$	2.03506e6	1.23234	0.616172	−	0.787611i	$-0.288683\pi$
$307$	2.03506e6	1.23234	0.616172	+	0.787611i	$0.288683\pi$
$308$	152320.	0.0914914
$309$	1.61289e6	0.960967
$310$	−855360.	−0.505527
$311$	1.62901e6	0.955042	0.477521	−	0.878620i	$-0.341536\pi$
$311$	1.62901e6	0.955042	0.477521	+	0.878620i	$0.341536\pi$
$312$	−588672.	−0.342363
$313$	1.68721e6	0.973438	0.486719	−	0.873559i	$-0.338194\pi$
$313$	1.68721e6	0.973438	0.486719	+	0.873559i	$0.338194\pi$
$314$	−616440.	−0.352831
$315$	249480.	0.141664
$316$	−549408.	−0.309512
$317$	1.32353e6	0.739751	0.369876	−	0.929081i	$-0.379400\pi$
$317$	1.32353e6	0.739751	0.369876	+	0.929081i	$0.379400\pi$
$318$	356256.	0.197558
$319$	−356048.	−0.195899
$320$	−180224.	−0.0983870
$321$	−680472.	−0.368594
$322$	148120.	0.0796112
$323$	−506264.	−0.270004
$324$	104976.	0.0555556
$325$	1.21516e6	0.638152
$326$	−1.55488e6	−0.810313
$327$	−733050.	−0.379109
$328$	−481920.	−0.247338
$329$	−2.04120e6	−1.03967
$330$	215424.	0.108895
$331$	3.14113e6	1.57585	0.787926	−	0.615769i	$-0.211155\pi$
$331$	3.14113e6	1.57585	0.787926	+	0.615769i	$0.211155\pi$
$332$	507776.	0.252829
$333$	1.20836e6	0.597152
$334$	−1.47299e6	−0.722494
$335$	−2.02919e6	−0.987896
$336$	161280.	0.0779350
$337$	1.16582e6	0.559188	0.279594	−	0.960118i	$-0.409800\pi$
$337$	1.16582e6	0.559188	0.279594	+	0.960118i	$0.409800\pi$
$338$	−2.69276e6	−1.28205
$339$	−1.11190e6	−0.525491
$340$	−340736.	−0.159853
$341$	660960.	0.307815
$342$	338904.	0.156679
$343$	2.00998e6	0.922479
$344$	−1.03590e6	−0.471980
$345$	209484.	0.0947552
$346$	169944.	0.0763160
$347$	−403828.	−0.180042	−0.0900208	−	0.995940i	$-0.528693\pi$
$347$	−403828.	−0.180042	−0.0900208	+	0.995940i	$0.528693\pi$
$348$	−376992.	−0.166872
$349$	−2.87189e6	−1.26213	−0.631064	−	0.775731i	$-0.717382\pi$
$349$	−2.87189e6	−1.26213	−0.631064	+	0.775731i	$0.717382\pi$
$350$	−332920.	−0.145268
$351$	745038.	0.322783
$352$	139264.	0.0599076
$353$	3.09901e6	1.32369	0.661846	−	0.749640i	$-0.269773\pi$
$353$	3.09901e6	1.32369	0.661846	+	0.749640i	$0.269773\pi$
$354$	−72144.0	−0.0305979
$355$	1.43827e6	0.605718
$356$	−972672.	−0.406763
$357$	304920.	0.126624
$358$	−2.78994e6	−1.15050
$359$	116628.	0.0477603	0.0238801	−	0.999715i	$-0.492398\pi$
$359$	116628.	0.0477603	0.0238801	+	0.999715i	$0.492398\pi$
$360$	228096.	0.0927601
$361$	−1.38198e6	−0.558129
$362$	3.33449e6	1.33739
$363$	1.28300e6	0.511044
$364$	1.14464e6	0.452809
$365$	2.06052e6	0.809552
$366$	−92520.0	−0.0361021
$367$	2.44342e6	0.946964	0.473482	−	0.880804i	$-0.342997\pi$
$367$	2.44342e6	0.946964	0.473482	+	0.880804i	$0.342997\pi$
$368$	135424.	0.0521286
$369$	609930.	0.233192
$370$	2.62557e6	0.997055
$371$	−692720.	−0.261290
$372$	699840.	0.262205
$373$	−1.62318e6	−0.604079	−0.302040	−	0.953295i	$-0.597668\pi$
$373$	−1.62318e6	−0.604079	−0.302040	+	0.953295i	$0.597668\pi$
$374$	263296.	0.0973342
$375$	−1.70834e6	−0.627332
$376$	−1.86624e6	−0.680766
$377$	−2.67560e6	−0.969543
$378$	−204120.	−0.0734778
$379$	3.70856e6	1.32620	0.663098	−	0.748533i	$-0.269241\pi$
$379$	3.70856e6	1.32620	0.663098	+	0.748533i	$0.269241\pi$
$380$	736384.	0.261605
$381$	1.29812e6	0.458146
$382$	−902480.	−0.316431
$383$	3.93251e6	1.36985	0.684924	−	0.728614i	$-0.259835\pi$
$383$	3.93251e6	1.36985	0.684924	+	0.728614i	$0.259835\pi$
$384$	147456.	0.0510310
$385$	−418880.	−0.144025
$386$	−1.90585e6	−0.651058
$387$	1.31107e6	0.444987
$388$	−307488.	−0.103693
$389$	−386712.	−0.129573	−0.0647864	−	0.997899i	$-0.520637\pi$
$389$	−386712.	−0.129573	−0.0647864	+	0.997899i	$0.520637\pi$
$390$	1.61885e6	0.538945
$391$	256036.	0.0846953
$392$	762048.	0.250477
$393$	−1.90134e6	−0.620981
$394$	1.46758e6	0.476277
$395$	1.51087e6	0.487231
$396$	−176256.	−0.0564815
$397$	4.61248e6	1.46878	0.734392	−	0.678725i	$-0.237467\pi$
$397$	4.61248e6	1.46878	0.734392	+	0.678725i	$0.237467\pi$
$398$	1.79994e6	0.569576
$399$	−658980.	−0.207224
$400$	−304384.	−0.0951200
$401$	−890100.	−0.276425	−0.138213	−	0.990403i	$-0.544136\pi$
$401$	−890100.	−0.276425	−0.138213	+	0.990403i	$0.544136\pi$
$402$	1.66025e6	0.512398
$403$	4.96692e6	1.52344
$404$	197216.	0.0601158
$405$	−288684.	−0.0874551
$406$	733040.	0.220705
$407$	−2.02885e6	−0.607105
$408$	278784.	0.0829120
$409$	1.41281e6	0.417613	0.208807	−	0.977957i	$-0.433042\pi$
$409$	1.41281e6	0.417613	0.208807	+	0.977957i	$0.433042\pi$
$410$	1.32528e6	0.389357
$411$	431352.	0.125958
$412$	−2.86736e6	−0.832222
$413$	140280.	0.0404688
$414$	−171396.	−0.0491473
$415$	−1.39638e6	−0.398001
$416$	1.04653e6	0.296495
$417$	−847980.	−0.238806
$418$	−569024.	−0.159291
$419$	−1.34849e6	−0.375243	−0.187621	−	0.982241i	$-0.560078\pi$
$419$	−1.34849e6	−0.375243	−0.187621	+	0.982241i	$0.560078\pi$
$420$	−443520.	−0.122685
$421$	1.42710e6	0.392418	0.196209	−	0.980562i	$-0.437137\pi$
$421$	1.42710e6	0.392418	0.196209	+	0.980562i	$0.437137\pi$
$422$	−1.31258e6	−0.358792
$423$	2.36196e6	0.641832
$424$	−633344.	−0.171090
$425$	−575476.	−0.154545
$426$	−1.17677e6	−0.314172
$427$	179900.	0.0477487
$428$	1.20973e6	0.319211
$429$	−1.25093e6	−0.328163
$430$	2.84874e6	0.742987
$431$	3.69521e6	0.958177	0.479088	−	0.877767i	$-0.340967\pi$
$431$	3.69521e6	0.958177	0.479088	+	0.877767i	$0.340967\pi$
$432$	−186624.	−0.0481125
$433$	1.11543e6	0.285907	0.142953	−	0.989729i	$-0.454340\pi$
$433$	1.11543e6	0.285907	0.142953	+	0.989729i	$0.454340\pi$
$434$	−1.36080e6	−0.346793
$435$	1.03673e6	0.262689
$436$	1.30320e6	0.328318
$437$	−553334.	−0.138607
$438$	−1.68588e6	−0.419896
$439$	−799888.	−0.198092	−0.0990462	−	0.995083i	$-0.531579\pi$
$439$	−799888.	−0.198092	−0.0990462	+	0.995083i	$0.531579\pi$
$440$	−382976.	−0.0943061
$441$	−964467.	−0.236152
$442$	1.97859e6	0.481727
$443$	8.16768e6	1.97738	0.988688	−	0.149988i	$-0.0479235\pi$
$443$	8.16768e6	1.97738	0.988688	+	0.149988i	$0.0479235\pi$
$444$	−2.14819e6	−0.517149
$445$	2.67485e6	0.640323
$446$	−807584.	−0.192243
$447$	4.48182e6	1.06093
$448$	−286720.	−0.0674937
$449$	−2.28235e6	−0.534277	−0.267138	−	0.963658i	$-0.586078\pi$
$449$	−2.28235e6	−0.534277	−0.267138	+	0.963658i	$0.586078\pi$
$450$	385236.	0.0896800
$451$	−1.02408e6	−0.237079
$452$	1.97670e6	0.455088
$453$	−3.25915e6	−0.746207
$454$	2.19192e6	0.499097
$455$	−3.14776e6	−0.712809
$456$	−602496.	−0.135688
$457$	10310.0	0.00230924	0.00115462	−	0.999999i	$-0.499632\pi$
$457$	10310.0	0.00230924	0.00115462	+	0.999999i	$0.499632\pi$
$458$	−3.56718e6	−0.794622
$459$	−352836.	−0.0781702
$460$	−372416.	−0.0820604
$461$	−8.70968e6	−1.90875	−0.954377	−	0.298606i	$-0.903479\pi$
$461$	−8.70968e6	−1.90875	−0.954377	+	0.298606i	$0.903479\pi$
$462$	342720.	0.0747024
$463$	−7.59838e6	−1.64729	−0.823643	−	0.567109i	$-0.808062\pi$
$463$	−7.59838e6	−1.64729	−0.823643	+	0.567109i	$0.808062\pi$
$464$	670208.	0.144516
$465$	−1.92456e6	−0.412761
$466$	4.63394e6	0.988521
$467$	−6.11311e6	−1.29709	−0.648545	−	0.761177i	$-0.724622\pi$
$467$	−6.11311e6	−1.29709	−0.648545	+	0.761177i	$0.724622\pi$
$468$	−1.32451e6	−0.279538
$469$	−3.22826e6	−0.677698
$470$	5.13216e6	1.07166
$471$	−1.38699e6	−0.288085
$472$	128256.	0.0264986
$473$	−2.20130e6	−0.452403
$474$	−1.23617e6	−0.252715
$475$	1.24369e6	0.252918
$476$	−542080.	−0.109659
$477$	801576.	0.161305
$478$	−316704.	−0.0633992
$479$	8.56955e6	1.70655	0.853276	−	0.521460i	$-0.174612\pi$
$479$	8.56955e6	1.70655	0.853276	+	0.521460i	$0.174612\pi$
$480$	−405504.	−0.0803326
$481$	−1.52462e7	−3.00468
$482$	−2.97647e6	−0.583558
$483$	333270.	0.0650023
$484$	−2.28088e6	−0.442577
$485$	845592.	0.163232
$486$	236196.	0.0453609
$487$	−6.01064e6	−1.14841	−0.574207	−	0.818710i	$-0.694689\pi$
$487$	−6.01064e6	−1.14841	−0.574207	+	0.818710i	$0.694689\pi$
$488$	164480.	0.0312654
$489$	−3.49848e6	−0.661618
$490$	−2.09563e6	−0.394298
$491$	5.60296e6	1.04885	0.524425	−	0.851457i	$-0.324280\pi$
$491$	5.60296e6	1.04885	0.524425	+	0.851457i	$0.324280\pi$
$492$	−1.08432e6	−0.201950
$493$	1.26711e6	0.234800
$494$	−4.27605e6	−0.788361
$495$	484704.	0.0889127
$496$	−1.24416e6	−0.227076
$497$	2.28816e6	0.415523
$498$	1.14250e6	0.206434
$499$	3.62344e6	0.651433	0.325716	−	0.945467i	$-0.394395\pi$
$499$	3.62344e6	0.651433	0.325716	+	0.945467i	$0.394395\pi$
$500$	3.03706e6	0.543285
$501$	−3.31423e6	−0.589914
$502$	−1.52408e6	−0.269928
$503$	−6.20844e6	−1.09411	−0.547057	−	0.837095i	$-0.684252\pi$
$503$	−6.20844e6	−1.09411	−0.547057	+	0.837095i	$0.684252\pi$
$504$	362880.	0.0636336
$505$	−542344.	−0.0946338
$506$	287776.	0.0499664
$507$	−6.05872e6	−1.04679
$508$	−2.30778e6	−0.396766
$509$	7.87063e6	1.34653	0.673263	−	0.739403i	$-0.264892\pi$
$509$	7.87063e6	1.34653	0.673263	+	0.739403i	$0.264892\pi$
$510$	−766656.	−0.130519
$511$	3.27810e6	0.555354
$512$	−262144.	−0.0441942
$513$	762534.	0.127928
$514$	1.17270e6	0.195784
$515$	7.88524e6	1.31008
$516$	−2.33078e6	−0.385370
$517$	−3.96576e6	−0.652529
$518$	4.17704e6	0.683981
$519$	382374.	0.0623118
$520$	−2.87795e6	−0.466740
$521$	−4.26558e6	−0.688469	−0.344234	−	0.938884i	$-0.611861\pi$
$521$	−4.26558e6	−0.688469	−0.344234	+	0.938884i	$0.611861\pi$
$522$	−848232.	−0.136251
$523$	−7.30523e6	−1.16783	−0.583915	−	0.811815i	$-0.698480\pi$
$523$	−7.30523e6	−1.16783	−0.583915	+	0.811815i	$0.698480\pi$
$524$	3.38016e6	0.537785
$525$	−749070.	−0.118611
$526$	2.04866e6	0.322853
$527$	−2.35224e6	−0.368939
$528$	313344.	0.0489144
$529$	279841.	0.0434783
$530$	1.74170e6	0.269329
$531$	−162324.	−0.0249831
$532$	1.17152e6	0.179461
$533$	−7.69566e6	−1.17335
$534$	−2.18851e6	−0.332121
$535$	−3.32675e6	−0.502500
$536$	−2.95155e6	−0.443750
$537$	−6.27736e6	−0.939379
$538$	2.41666e6	0.359965
$539$	1.61935e6	0.240087
$540$	513216.	0.0757383
$541$	2.66795e6	0.391909	0.195954	−	0.980613i	$-0.437220\pi$
$541$	2.66795e6	0.391909	0.195954	+	0.980613i	$0.437220\pi$
$542$	1.36264e6	0.199243
$543$	7.50260e6	1.09197
$544$	−495616.	−0.0718039
$545$	−3.58380e6	−0.516836
$546$	2.57544e6	0.369717
$547$	−4.62874e6	−0.661446	−0.330723	−	0.943728i	$-0.607293\pi$
$547$	−4.62874e6	−0.661446	−0.330723	+	0.943728i	$0.607293\pi$
$548$	−766848.	−0.109083
$549$	−208170.	−0.0294773
$550$	−646816.	−0.0911746
$551$	−2.73843e6	−0.384257
$552$	304704.	0.0425628
$553$	2.40366e6	0.334241
$554$	1.95956e6	0.271259
$555$	5.90753e6	0.814092
$556$	1.50752e6	0.206812
$557$	1.12247e7	1.53299	0.766493	−	0.642252i	$-0.222000\pi$
$557$	1.12247e7	1.53299	0.766493	+	0.642252i	$0.222000\pi$
$558$	1.57464e6	0.214090
$559$	−1.65421e7	−2.23904
$560$	788480.	0.106248
$561$	592416.	0.0794730
$562$	3.58829e6	0.479233
$563$	3.47856e6	0.462518	0.231259	−	0.972892i	$-0.425715\pi$
$563$	3.47856e6	0.462518	0.231259	+	0.972892i	$0.425715\pi$
$564$	−4.19904e6	−0.555843
$565$	−5.43594e6	−0.716396
$566$	−797944.	−0.104696
$567$	−459270.	−0.0599944
$568$	2.09203e6	0.272081
$569$	4.70263e6	0.608920	0.304460	−	0.952525i	$-0.401524\pi$
$569$	4.70263e6	0.608920	0.304460	+	0.952525i	$0.401524\pi$
$570$	1.65686e6	0.213599
$571$	−1.12116e7	−1.43906	−0.719530	−	0.694462i	$-0.755643\pi$
$571$	−1.12116e7	−1.43906	−0.719530	+	0.694462i	$0.755643\pi$
$572$	2.22387e6	0.284197
$573$	−2.03058e6	−0.258365
$574$	2.10840e6	0.267100
$575$	−628981.	−0.0793356
$576$	331776.	0.0416667
$577$	−5.83226e6	−0.729285	−0.364642	−	0.931148i	$-0.618809\pi$
$577$	−5.83226e6	−0.729285	−0.364642	+	0.931148i	$0.618809\pi$
$578$	4.74240e6	0.590444
$579$	−4.28816e6	−0.531587
$580$	−1.84307e6	−0.227495
$581$	−2.22152e6	−0.273030
$582$	−691848.	−0.0846649
$583$	−1.34586e6	−0.163994
$584$	2.99712e6	0.363640
$585$	3.64241e6	0.440047
$586$	9.05475e6	1.08926
$587$	−1.49393e6	−0.178952	−0.0894758	−	0.995989i	$-0.528519\pi$
$587$	−1.49393e6	−0.178952	−0.0894758	+	0.995989i	$0.528519\pi$
$588$	1.71461e6	0.204513
$589$	5.08356e6	0.603781
$590$	−352704.	−0.0417139
$591$	3.30205e6	0.388879
$592$	3.81901e6	0.447864
$593$	1.16103e7	1.35584	0.677919	−	0.735137i	$-0.262882\pi$
$593$	1.16103e7	1.35584	0.677919	+	0.735137i	$0.262882\pi$
$594$	−396576.	−0.0461169
$595$	1.49072e6	0.172625
$596$	−7.96768e6	−0.918790
$597$	4.04987e6	0.465056
$598$	2.16255e6	0.247294
$599$	1.74222e7	1.98398	0.991989	−	0.126323i	$-0.0403175\pi$
$599$	1.74222e7	1.98398	0.991989	+	0.126323i	$0.0403175\pi$
$600$	−684864.	−0.0776652
$601$	−4.96150e6	−0.560308	−0.280154	−	0.959955i	$-0.590385\pi$
$601$	−4.96150e6	−0.560308	−0.280154	+	0.959955i	$0.590385\pi$
$602$	4.53208e6	0.509690
$603$	3.73556e6	0.418372
$604$	5.79405e6	0.646234
$605$	6.27242e6	0.696701
$606$	443736.	0.0490844
$607$	1.09248e7	1.20349	0.601743	−	0.798690i	$-0.294473\pi$
$607$	1.09248e7	1.20349	0.601743	+	0.798690i	$0.294473\pi$
$608$	1.07110e6	0.117509
$609$	1.64934e6	0.180205
$610$	−452320.	−0.0492177
$611$	−2.98015e7	−3.22950
$612$	627264.	0.0676974
$613$	1.24209e7	1.33506	0.667532	−	0.744581i	$-0.267351\pi$
$613$	1.24209e7	1.33506	0.667532	+	0.744581i	$0.267351\pi$
$614$	−8.14026e6	−0.871399
$615$	2.98188e6	0.317909
$616$	−609280.	−0.0646942
$617$	1.21179e7	1.28149	0.640743	−	0.767755i	$-0.278626\pi$
$617$	1.21179e7	1.28149	0.640743	+	0.767755i	$0.278626\pi$
$618$	−6.45156e6	−0.679506
$619$	−7.56403e6	−0.793463	−0.396732	−	0.917935i	$-0.629856\pi$
$619$	−7.56403e6	−0.793463	−0.396732	+	0.917935i	$0.629856\pi$
$620$	3.42144e6	0.357462
$621$	−385641.	−0.0401286
$622$	−6.51603e6	−0.675316
$623$	4.25544e6	0.439263
$624$	2.35469e6	0.242087
$625$	−4.63628e6	−0.474755
$626$	−6.74884e6	−0.688325
$627$	−1.28030e6	−0.130060
$628$	2.46576e6	0.249489
$629$	7.22031e6	0.727661
$630$	−997920.	−0.100172
$631$	−8.80623e6	−0.880475	−0.440237	−	0.897881i	$-0.645106\pi$
$631$	−8.80623e6	−0.880475	−0.440237	+	0.897881i	$0.645106\pi$
$632$	2.19763e6	0.218858
$633$	−2.95330e6	−0.292953
$634$	−5.29412e6	−0.523083
$635$	6.34638e6	0.624586
$636$	−1.42502e6	−0.139695
$637$	1.21690e7	1.18824
$638$	1.42419e6	0.138521
$639$	−2.64773e6	−0.256520
$640$	720896.	0.0695701
$641$	7.08252e6	0.680836	0.340418	−	0.940274i	$-0.389431\pi$
$641$	7.08252e6	0.680836	0.340418	+	0.940274i	$0.389431\pi$
$642$	2.72189e6	0.260635
$643$	−6.46139e6	−0.616309	−0.308155	−	0.951336i	$-0.599711\pi$
$643$	−6.46139e6	−0.616309	−0.308155	+	0.951336i	$0.599711\pi$
$644$	−592480.	−0.0562936
$645$	6.40966e6	0.606646
$646$	2.02506e6	0.190922
$647$	−385672.	−0.0362207	−0.0181104	−	0.999836i	$-0.505765\pi$
$647$	−385672.	−0.0362207	−0.0181104	+	0.999836i	$0.505765\pi$
$648$	−419904.	−0.0392837
$649$	272544.	0.0253995
$650$	−4.86063e6	−0.451242
$651$	−3.06180e6	−0.283155
$652$	6.21952e6	0.572978
$653$	−1.05644e7	−0.969530	−0.484765	−	0.874644i	$-0.661095\pi$
$653$	−1.05644e7	−0.969530	−0.484765	+	0.874644i	$0.661095\pi$
$654$	2.93220e6	0.268071
$655$	−9.29544e6	−0.846577
$656$	1.92768e6	0.174894
$657$	−3.79323e6	−0.342843
$658$	8.16480e6	0.735158
$659$	2.08997e7	1.87468	0.937340	−	0.348415i	$-0.113280\pi$
$659$	2.08997e7	1.87468	0.937340	+	0.348415i	$0.113280\pi$
$660$	−861696.	−0.0770006
$661$	−1.74326e7	−1.55188	−0.775941	−	0.630805i	$-0.782725\pi$
$661$	−1.74326e7	−1.55188	−0.775941	+	0.630805i	$0.782725\pi$
$662$	−1.25645e7	−1.11430
$663$	4.45183e6	0.393328
$664$	−2.03110e6	−0.178777
$665$	−3.22168e6	−0.282506
$666$	−4.83343e6	−0.422250
$667$	1.38492e6	0.120534
$668$	5.89197e6	0.510880
$669$	−1.81706e6	−0.156966
$670$	8.11677e6	0.698548
$671$	349520.	0.0299686
$672$	−645120.	−0.0551083
$673$	1.45788e7	1.24075	0.620376	−	0.784305i	$-0.286980\pi$
$673$	1.45788e7	1.24075	0.620376	+	0.784305i	$0.286980\pi$
$674$	−4.66329e6	−0.395405
$675$	866781.	0.0732234
$676$	1.07711e7	0.906550
$677$	1.80302e7	1.51192	0.755961	−	0.654616i	$-0.227170\pi$
$677$	1.80302e7	1.51192	0.755961	+	0.654616i	$0.227170\pi$
$678$	4.44758e6	0.371578
$679$	1.34526e6	0.111978
$680$	1.36294e6	0.113033
$681$	4.93182e6	0.407511
$682$	−2.64384e6	−0.217658
$683$	−2.74348e6	−0.225035	−0.112517	−	0.993650i	$-0.535891\pi$
$683$	−2.74348e6	−0.225035	−0.112517	+	0.993650i	$0.535891\pi$
$684$	−1.35562e6	−0.110789
$685$	2.10883e6	0.171718
$686$	−8.03992e6	−0.652291
$687$	−8.02615e6	−0.648807
$688$	4.14362e6	0.333740
$689$	−1.01137e7	−0.811638
$690$	−837936.	−0.0670021
$691$	6.23295e6	0.496591	0.248295	−	0.968684i	$-0.420130\pi$
$691$	6.23295e6	0.496591	0.248295	+	0.968684i	$0.420130\pi$
$692$	−679776.	−0.0539636
$693$	771120.	0.0609942
$694$	1.61531e6	0.127309
$695$	−4.14568e6	−0.325562
$696$	1.50797e6	0.117996
$697$	3.64452e6	0.284157
$698$	1.14875e7	0.892460
$699$	1.04264e7	0.807124
$700$	1.33168e6	0.102720
$701$	−1.84340e7	−1.41685	−0.708425	−	0.705786i	$-0.750594\pi$
$701$	−1.84340e7	−1.41685	−0.708425	+	0.705786i	$0.750594\pi$
$702$	−2.98015e6	−0.228242
$703$	−1.56042e7	−1.19084
$704$	−557056.	−0.0423611
$705$	1.15474e7	0.875004
$706$	−1.23961e7	−0.935991
$707$	−862820.	−0.0649190
$708$	288576.	0.0216360
$709$	2.27746e7	1.70151	0.850756	−	0.525561i	$-0.176145\pi$
$709$	2.27746e7	1.70151	0.850756	+	0.525561i	$0.176145\pi$
$710$	−5.75309e6	−0.428307
$711$	−2.78138e6	−0.206341
$712$	3.89069e6	0.287625
$713$	−2.57094e6	−0.189395
$714$	−1.21968e6	−0.0895366
$715$	−6.11565e6	−0.447381
$716$	1.11597e7	0.813526
$717$	−712584.	−0.0517652
$718$	−466512.	−0.0337716
$719$	−5.28691e6	−0.381399	−0.190700	−	0.981648i	$-0.561076\pi$
$719$	−5.28691e6	−0.381399	−0.190700	+	0.981648i	$0.561076\pi$
$720$	−912384.	−0.0655913
$721$	1.25447e7	0.898715
$722$	5.52793e6	0.394657
$723$	−6.69706e6	−0.476473
$724$	−1.33380e7	−0.945677
$725$	−3.11280e6	−0.219941
$726$	−5.13198e6	−0.361363
$727$	2.12360e7	1.49017	0.745086	−	0.666968i	$-0.232408\pi$
$727$	2.12360e7	1.49017	0.745086	+	0.666968i	$0.232408\pi$
$728$	−4.57856e6	−0.320185
$729$	531441.	0.0370370
$730$	−8.24208e6	−0.572440
$731$	7.83402e6	0.542240
$732$	370080.	0.0255281
$733$	−6.33805e6	−0.435708	−0.217854	−	0.975981i	$-0.569906\pi$
$733$	−6.33805e6	−0.435708	−0.217854	+	0.975981i	$0.569906\pi$
$734$	−9.77369e6	−0.669604
$735$	−4.71517e6	−0.321943
$736$	−541696.	−0.0368605
$737$	−6.27205e6	−0.425344
$738$	−2.43972e6	−0.164892
$739$	1.92112e7	1.29403	0.647013	−	0.762479i	$-0.276018\pi$
$739$	1.92112e7	1.29403	0.647013	+	0.762479i	$0.276018\pi$
$740$	−1.05023e7	−0.705024
$741$	−9.62111e6	−0.643694
$742$	2.77088e6	0.184760
$743$	7.63612e6	0.507458	0.253729	−	0.967275i	$-0.418343\pi$
$743$	7.63612e6	0.507458	0.253729	+	0.967275i	$0.418343\pi$
$744$	−2.79936e6	−0.185407
$745$	2.19111e7	1.44635
$746$	6.49271e6	0.427149
$747$	2.57062e6	0.168553
$748$	−1.05318e6	−0.0688256
$749$	−5.29256e6	−0.344716
$750$	6.83338e6	0.443590
$751$	−2.79372e7	−1.80752	−0.903761	−	0.428038i	$-0.859205\pi$
$751$	−2.79372e7	−1.80752	−0.903761	+	0.428038i	$0.859205\pi$
$752$	7.46496e6	0.481374
$753$	−3.42918e6	−0.220396
$754$	1.07024e7	0.685571
$755$	−1.59336e7	−1.01730
$756$	816480.	0.0519566
$757$	−1.04036e6	−0.0659847	−0.0329923	−	0.999456i	$-0.510504\pi$
$757$	−1.04036e6	−0.0659847	−0.0329923	+	0.999456i	$0.510504\pi$
$758$	−1.48342e7	−0.937762
$759$	647496.	0.0407974
$760$	−2.94554e6	−0.184982
$761$	−3.84256e6	−0.240524	−0.120262	−	0.992742i	$-0.538374\pi$
$761$	−3.84256e6	−0.240524	−0.120262	+	0.992742i	$0.538374\pi$
$762$	−5.19250e6	−0.323958
$763$	−5.70150e6	−0.354550
$764$	3.60992e6	0.223751
$765$	−1.72498e6	−0.106569
$766$	−1.57300e7	−0.968629
$767$	2.04809e6	0.125707
$768$	−589824.	−0.0360844
$769$	2.75351e6	0.167908	0.0839538	−	0.996470i	$-0.473245\pi$
$769$	2.75351e6	0.167908	0.0839538	+	0.996470i	$0.473245\pi$
$770$	1.67552e6	0.101841
$771$	2.63857e6	0.159857
$772$	7.62339e6	0.460368
$773$	2.02611e7	1.21959	0.609795	−	0.792559i	$-0.291252\pi$
$773$	2.02611e7	1.21959	0.609795	+	0.792559i	$0.291252\pi$
$774$	−5.24426e6	−0.314653
$775$	5.77854e6	0.345592
$776$	1.22995e6	0.0733219
$777$	9.39834e6	0.558468
$778$	1.54685e6	0.0916218
$779$	−7.87638e6	−0.465032
$780$	−6.47539e6	−0.381092
$781$	4.44557e6	0.260795
$782$	−1.02414e6	−0.0598886
$783$	−1.90852e6	−0.111248
$784$	−3.04819e6	−0.177114
$785$	−6.78084e6	−0.392744
$786$	7.60536e6	0.439100
$787$	−1.58192e7	−0.910435	−0.455218	−	0.890380i	$-0.650438\pi$
$787$	−1.58192e7	−0.910435	−0.455218	+	0.890380i	$0.650438\pi$
$788$	−5.87030e6	−0.336779
$789$	4.60948e6	0.263608
$790$	−6.04349e6	−0.344524
$791$	−8.64808e6	−0.491449
$792$	705024.	0.0399384
$793$	2.62654e6	0.148320
$794$	−1.84499e7	−1.03859
$795$	3.91882e6	0.219906
$796$	−7.19978e6	−0.402751
$797$	−1.84010e7	−1.02611	−0.513057	−	0.858355i	$-0.671487\pi$
$797$	−1.84010e7	−1.02611	−0.513057	+	0.858355i	$0.671487\pi$
$798$	2.63592e6	0.146529
$799$	1.41134e7	0.782107
$800$	1.21754e6	0.0672600
$801$	−4.92415e6	−0.271175
$802$	3.56040e6	0.195462
$803$	6.36888e6	0.348557
$804$	−6.64099e6	−0.362320
$805$	1.62932e6	0.0886169
$806$	−1.98677e7	−1.07723
$807$	5.43749e6	0.293910
$808$	−788864.	−0.0425083
$809$	2.17890e7	1.17049	0.585243	−	0.810858i	$-0.300999\pi$
$809$	2.17890e7	1.17049	0.585243	+	0.810858i	$0.300999\pi$
$810$	1.15474e6	0.0618401
$811$	2.82392e7	1.50765	0.753826	−	0.657074i	$-0.228206\pi$
$811$	2.82392e7	1.50765	0.753826	+	0.657074i	$0.228206\pi$
$812$	−2.93216e6	−0.156062
$813$	3.06594e6	0.162681
$814$	8.11539e6	0.429288
$815$	−1.71037e7	−0.901977
$816$	−1.11514e6	−0.0586276
$817$	−1.69306e7	−0.887393
$818$	−5.65122e6	−0.295297
$819$	5.79474e6	0.301873
$820$	−5.30112e6	−0.275317
$821$	−2.25177e7	−1.16591	−0.582956	−	0.812504i	$-0.698104\pi$
$821$	−2.25177e7	−1.16591	−0.582956	+	0.812504i	$0.698104\pi$
$822$	−1.72541e6	−0.0890661
$823$	−5.53926e6	−0.285070	−0.142535	−	0.989790i	$-0.545525\pi$
$823$	−5.53926e6	−0.285070	−0.142535	+	0.989790i	$0.545525\pi$
$824$	1.14694e7	0.588470
$825$	−1.45534e6	−0.0744438
$826$	−561120.	−0.0286158
$827$	−1.40451e7	−0.714101	−0.357050	−	0.934085i	$-0.616218\pi$
$827$	−1.40451e7	−0.714101	−0.357050	+	0.934085i	$0.616218\pi$
$828$	685584.	0.0347524
$829$	3.64779e7	1.84350	0.921751	−	0.387782i	$-0.126759\pi$
$829$	3.64779e7	1.84350	0.921751	+	0.387782i	$0.126759\pi$
$830$	5.58554e6	0.281430
$831$	4.40901e6	0.221482
$832$	−4.18611e6	−0.209654
$833$	−5.76299e6	−0.287763
$834$	3.39192e6	0.168861
$835$	−1.62029e7	−0.804224
$836$	2.27610e6	0.112635
$837$	3.54294e6	0.174804
$838$	5.39395e6	0.265337
$839$	−1.34296e7	−0.658655	−0.329328	−	0.944216i	$-0.606822\pi$
$839$	−1.34296e7	−0.658655	−0.329328	+	0.944216i	$0.606822\pi$
$840$	1.77408e6	0.0867511
$841$	−1.36572e7	−0.665844
$842$	−5.70839e6	−0.277481
$843$	8.07365e6	0.391292
$844$	5.25030e6	0.253705
$845$	−2.96204e7	−1.42708
$846$	−9.44784e6	−0.453844
$847$	9.97885e6	0.477938
$848$	2.53338e6	0.120979
$849$	−1.79537e6	−0.0854842
$850$	2.30190e6	0.109280
$851$	7.89162e6	0.373545
$852$	4.70707e6	0.222153
$853$	3.74589e7	1.76272	0.881358	−	0.472449i	$-0.156630\pi$
$853$	3.74589e7	1.76272	0.881358	+	0.472449i	$0.156630\pi$
$854$	−719600.	−0.0337634
$855$	3.72794e6	0.174403
$856$	−4.83891e6	−0.225717
$857$	3.08502e7	1.43485	0.717425	−	0.696636i	$-0.245321\pi$
$857$	3.08502e7	1.43485	0.717425	+	0.696636i	$0.245321\pi$
$858$	5.00371e6	0.232046
$859$	−1.72110e7	−0.795835	−0.397918	−	0.917421i	$-0.630267\pi$
$859$	−1.72110e7	−0.795835	−0.397918	+	0.917421i	$0.630267\pi$
$860$	−1.13949e7	−0.525371
$861$	4.74390e6	0.218086
$862$	−1.47808e7	−0.677533
$863$	−2.53707e6	−0.115959	−0.0579797	−	0.998318i	$-0.518466\pi$
$863$	−2.53707e6	−0.115959	−0.0579797	+	0.998318i	$0.518466\pi$
$864$	746496.	0.0340207
$865$	1.86938e6	0.0849490
$866$	−4.46174e6	−0.202166
$867$	1.06704e7	0.482096
$868$	5.44320e6	0.245219
$869$	4.66997e6	0.209780
$870$	−4.14691e6	−0.185749
$871$	−4.71326e7	−2.10512
$872$	−5.21280e6	−0.232156
$873$	−1.55666e6	−0.0691286
$874$	2.21334e6	0.0980096
$875$	−1.32871e7	−0.586693
$876$	6.74352e6	0.296911
$877$	1.26269e7	0.554368	0.277184	−	0.960817i	$-0.410599\pi$
$877$	1.26269e7	0.554368	0.277184	+	0.960817i	$0.410599\pi$
$878$	3.19955e6	0.140073
$879$	2.03732e7	0.889379
$880$	1.53190e6	0.0666845
$881$	−6.28380e6	−0.272761	−0.136381	−	0.990657i	$-0.543547\pi$
$881$	−6.28380e6	−0.272761	−0.136381	+	0.990657i	$0.543547\pi$
$882$	3.85787e6	0.166984
$883$	−8.46036e6	−0.365163	−0.182582	−	0.983191i	$-0.558445\pi$
$883$	−8.46036e6	−0.365163	−0.182582	+	0.983191i	$0.558445\pi$
$884$	−7.91437e6	−0.340632
$885$	−793584.	−0.0340592
$886$	−3.26707e7	−1.39822
$887$	−2.76151e7	−1.17852	−0.589262	−	0.807942i	$-0.700581\pi$
$887$	−2.76151e7	−1.17852	−0.589262	+	0.807942i	$0.700581\pi$
$888$	8.59277e6	0.365680
$889$	1.00965e7	0.428467
$890$	−1.06994e7	−0.452777
$891$	−892296.	−0.0376543
$892$	3.23034e6	0.135936
$893$	−3.05014e7	−1.27994
$894$	−1.79273e7	−0.750189
$895$	−3.06893e7	−1.28065
$896$	1.14688e6	0.0477252
$897$	4.86574e6	0.201915
$898$	9.12940e6	0.377791
$899$	−1.27235e7	−0.525057
$900$	−1.54094e6	−0.0634133
$901$	4.78966e6	0.196559
$902$	4.09632e6	0.167640
$903$	1.01972e7	0.416160
$904$	−7.90682e6	−0.321796
$905$	3.66794e7	1.48868
$906$	1.30366e7	0.527648
$907$	−6.84459e6	−0.276267	−0.138133	−	0.990414i	$-0.544110\pi$
$907$	−6.84459e6	−0.276267	−0.138133	+	0.990414i	$0.544110\pi$
$908$	−8.76768e6	−0.352915
$909$	998406.	0.0400772
$910$	1.25910e7	0.504032
$911$	2.48852e7	0.993446	0.496723	−	0.867909i	$-0.334536\pi$
$911$	2.48852e7	0.993446	0.496723	+	0.867909i	$0.334536\pi$
$912$	2.40998e6	0.0959460
$913$	−4.31610e6	−0.171362
$914$	−41240.0	−0.00163288
$915$	−1.01772e6	−0.0401861
$916$	1.42687e7	0.561883
$917$	−1.47882e7	−0.580754
$918$	1.41134e6	0.0552747
$919$	−1.91853e7	−0.749342	−0.374671	−	0.927158i	$-0.622244\pi$
$919$	−1.91853e7	−0.749342	−0.374671	+	0.927158i	$0.622244\pi$
$920$	1.48966e6	0.0580255
$921$	−1.83156e7	−0.711494
$922$	3.48387e7	1.34969
$923$	3.34071e7	1.29073
$924$	−1.37088e6	−0.0528226
$925$	−1.77375e7	−0.681613
$926$	3.03935e7	1.16481
$927$	−1.45160e7	−0.554815
$928$	−2.68083e6	−0.102188
$929$	3.58672e7	1.36351	0.681756	−	0.731580i	$-0.261217\pi$
$929$	3.58672e7	1.36351	0.681756	+	0.731580i	$0.261217\pi$
$930$	7.69824e6	0.291866
$931$	1.24547e7	0.470934
$932$	−1.85358e7	−0.698990
$933$	−1.46611e7	−0.551393
$934$	2.44524e7	0.917181
$935$	2.89626e6	0.108345
$936$	5.29805e6	0.197663
$937$	2.27726e7	0.847351	0.423675	−	0.905814i	$-0.360740\pi$
$937$	2.27726e7	0.847351	0.423675	+	0.905814i	$0.360740\pi$
$938$	1.29130e7	0.479205
$939$	−1.51849e7	−0.562015
$940$	−2.05286e7	−0.757775
$941$	−3.02768e7	−1.11464	−0.557321	−	0.830297i	$-0.688171\pi$
$941$	−3.02768e7	−1.11464	−0.557321	+	0.830297i	$0.688171\pi$
$942$	5.54796e6	0.203707
$943$	3.98337e6	0.145872
$944$	−513024.	−0.0187373
$945$	−2.24532e6	−0.0817897
$946$	8.80518e6	0.319897
$947$	2.27834e7	0.825551	0.412775	−	0.910833i	$-0.364559\pi$
$947$	2.27834e7	0.825551	0.412775	+	0.910833i	$0.364559\pi$
$948$	4.94467e6	0.178697
$949$	4.78603e7	1.72508
$950$	−4.97478e6	−0.178840
$951$	−1.19118e7	−0.427096
$952$	2.16832e6	0.0775409
$953$	3.61745e6	0.129024	0.0645120	−	0.997917i	$-0.479451\pi$
$953$	3.61745e6	0.129024	0.0645120	+	0.997917i	$0.479451\pi$
$954$	−3.20630e6	−0.114060
$955$	−9.92728e6	−0.352226
$956$	1.26682e6	0.0448300
$957$	3.20443e6	0.113102
$958$	−3.42782e7	−1.20671
$959$	3.35496e6	0.117799
$960$	1.62202e6	0.0568038
$961$	−5.00955e6	−0.174981
$962$	6.09848e7	2.12463
$963$	6.12425e6	0.212808
$964$	1.19059e7	0.412638
$965$	−2.09643e7	−0.724707
$966$	−1.33308e6	−0.0459635
$967$	−1.05880e7	−0.364121	−0.182061	−	0.983287i	$-0.558277\pi$
$967$	−1.05880e7	−0.364121	−0.182061	+	0.983287i	$0.558277\pi$
$968$	9.12352e6	0.312949
$969$	4.55638e6	0.155887
$970$	−3.38237e6	−0.115423
$971$	4.05879e7	1.38149	0.690746	−	0.723097i	$-0.257282\pi$
$971$	4.05879e7	1.38149	0.690746	+	0.723097i	$0.257282\pi$
$972$	−944784.	−0.0320750
$973$	−6.59540e6	−0.223336
$974$	2.40426e7	0.812051
$975$	−1.09364e7	−0.368437
$976$	−657920.	−0.0221080
$977$	−5.70586e7	−1.91243	−0.956213	−	0.292671i	$-0.905456\pi$
$977$	−5.70586e7	−1.91243	−0.956213	+	0.292671i	$0.905456\pi$
$978$	1.39939e7	0.467835
$979$	8.26771e6	0.275695
$980$	8.38253e6	0.278811
$981$	6.59745e6	0.218879
$982$	−2.24118e7	−0.741649
$983$	−4.75361e7	−1.56906	−0.784530	−	0.620091i	$-0.787096\pi$
$983$	−4.75361e7	−1.56906	−0.784530	+	0.620091i	$0.787096\pi$
$984$	4.33728e6	0.142800
$985$	1.61433e7	0.530155
$986$	−5.06845e6	−0.166029
$987$	1.83708e7	0.600254
$988$	1.71042e7	0.557455
$989$	8.56239e6	0.278358
$990$	−1.93882e6	−0.0628707
$991$	−3.57955e7	−1.15783	−0.578915	−	0.815388i	$-0.696524\pi$
$991$	−3.57955e7	−1.15783	−0.578915	+	0.815388i	$0.696524\pi$
$992$	4.97664e6	0.160567
$993$	−2.82702e7	−0.909819
$994$	−9.15264e6	−0.293819
$995$	1.97994e7	0.634007
$996$	−4.56998e6	−0.145971
$997$	4.18371e7	1.33298	0.666490	−	0.745514i	$-0.267796\pi$
$997$	4.18371e7	1.33298	0.666490	+	0.745514i	$0.267796\pi$
$998$	−1.44938e7	−0.460633
$999$	−1.08752e7	−0.344766

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.6.a.a.1.1	✓	1
3.2	odd	2		414.6.a.d.1.1		1
4.3	odd	2		1104.6.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.6.a.a.1.1	✓	1	1.1	even	1	trivial
414.6.a.d.1.1		1	3.2	odd	2
1104.6.a.c.1.1		1	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 138 = 2 \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	138.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more