LMFDB - Embedded newform 390.6.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [390,6,Mod(1,390)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(390, 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("390.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 390 = 2 \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	390.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:	$62.5496897271$
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	$\chi$	$=$	390.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} +25.0000 q^{5} +36.0000 q^{6} -4.00000 q^{7} -64.0000 q^{8} +81.0000 q^{9} +O(q^{10})$

\(q-4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} +25.0000 q^{5} +36.0000 q^{6} -4.00000 q^{7} -64.0000 q^{8} +81.0000 q^{9} -100.000 q^{10} +96.0000 q^{11} -144.000 q^{12} +169.000 q^{13} +16.0000 q^{14} -225.000 q^{15} +256.000 q^{16} -1182.00 q^{17} -324.000 q^{18} -1168.00 q^{19} +400.000 q^{20} +36.0000 q^{21} -384.000 q^{22} +1800.00 q^{23} +576.000 q^{24} +625.000 q^{25} -676.000 q^{26} -729.000 q^{27} -64.0000 q^{28} +366.000 q^{29} +900.000 q^{30} -4564.00 q^{31} -1024.00 q^{32} -864.000 q^{33} +4728.00 q^{34} -100.000 q^{35} +1296.00 q^{36} +10262.0 q^{37} +4672.00 q^{38} -1521.00 q^{39} -1600.00 q^{40} -14310.0 q^{41} -144.000 q^{42} +12956.0 q^{43} +1536.00 q^{44} +2025.00 q^{45} -7200.00 q^{46} +3252.00 q^{47} -2304.00 q^{48} -16791.0 q^{49} -2500.00 q^{50} +10638.0 q^{51} +2704.00 q^{52} +5670.00 q^{53} +2916.00 q^{54} +2400.00 q^{55} +256.000 q^{56} +10512.0 q^{57} -1464.00 q^{58} +42384.0 q^{59} -3600.00 q^{60} +29678.0 q^{61} +18256.0 q^{62} -324.000 q^{63} +4096.00 q^{64} +4225.00 q^{65} +3456.00 q^{66} -22384.0 q^{67} -18912.0 q^{68} -16200.0 q^{69} +400.000 q^{70} +3732.00 q^{71} -5184.00 q^{72} +37658.0 q^{73} -41048.0 q^{74} -5625.00 q^{75} -18688.0 q^{76} -384.000 q^{77} +6084.00 q^{78} -58792.0 q^{79} +6400.00 q^{80} +6561.00 q^{81} +57240.0 q^{82} -66912.0 q^{83} +576.000 q^{84} -29550.0 q^{85} -51824.0 q^{86} -3294.00 q^{87} -6144.00 q^{88} -132486. q^{89} -8100.00 q^{90} -676.000 q^{91} +28800.0 q^{92} +41076.0 q^{93} -13008.0 q^{94} -29200.0 q^{95} +9216.00 q^{96} -12910.0 q^{97} +67164.0 q^{98} +7776.00 q^{99} +O(q^{100})\)

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
$5$	25.0000	0.447214
$5$	25.0000	0.447214
$6$	36.0000	0.408248
$7$	−4.00000	−0.0308542	−0.0154271	−	0.999881i	$-0.504911\pi$
$7$	−4.00000	−0.0308542	−0.0154271	+	0.999881i	$0.504911\pi$
$8$	−64.0000	−0.353553
$9$	81.0000	0.333333
$10$	−100.000	−0.316228
$11$	96.0000	0.239216	0.119608	−	0.992821i	$-0.461836\pi$
$11$	96.0000	0.239216	0.119608	+	0.992821i	$0.461836\pi$
$12$	−144.000	−0.288675
$13$	169.000	0.277350
$13$	169.000	0.277350
$14$	16.0000	0.0218172
$15$	−225.000	−0.258199
$16$	256.000	0.250000
$17$	−1182.00	−0.991962	−0.495981	−	0.868333i	$-0.665191\pi$
$17$	−1182.00	−0.991962	−0.495981	+	0.868333i	$0.665191\pi$
$18$	−324.000	−0.235702
$19$	−1168.00	−0.742265	−0.371132	−	0.928580i	$-0.621030\pi$
$19$	−1168.00	−0.742265	−0.371132	+	0.928580i	$0.621030\pi$
$20$	400.000	0.223607
$21$	36.0000	0.0178137
$22$	−384.000	−0.169151
$23$	1800.00	0.709501	0.354750	−	0.934961i	$-0.384566\pi$
$23$	1800.00	0.709501	0.354750	+	0.934961i	$0.384566\pi$
$24$	576.000	0.204124
$25$	625.000	0.200000
$26$	−676.000	−0.196116
$27$	−729.000	−0.192450
$28$	−64.0000	−0.0154271
$29$	366.000	0.0808139	0.0404070	−	0.999183i	$-0.487135\pi$
$29$	366.000	0.0808139	0.0404070	+	0.999183i	$0.487135\pi$
$30$	900.000	0.182574
$31$	−4564.00	−0.852985	−0.426493	−	0.904491i	$-0.640251\pi$
$31$	−4564.00	−0.852985	−0.426493	+	0.904491i	$0.640251\pi$
$32$	−1024.00	−0.176777
$33$	−864.000	−0.138111
$34$	4728.00	0.701423
$35$	−100.000	−0.0137984
$36$	1296.00	0.166667
$37$	10262.0	1.23233	0.616166	−	0.787616i	$-0.288685\pi$
$37$	10262.0	1.23233	0.616166	+	0.787616i	$0.288685\pi$
$38$	4672.00	0.524860
$39$	−1521.00	−0.160128
$40$	−1600.00	−0.158114
$41$	−14310.0	−1.32947	−0.664737	−	0.747077i	$-0.731456\pi$
$41$	−14310.0	−1.32947	−0.664737	+	0.747077i	$0.731456\pi$
$42$	−144.000	−0.0125962
$43$	12956.0	1.06856	0.534281	−	0.845307i	$-0.320582\pi$
$43$	12956.0	1.06856	0.534281	+	0.845307i	$0.320582\pi$
$44$	1536.00	0.119608
$45$	2025.00	0.149071
$46$	−7200.00	−0.501693
$47$	3252.00	0.214737	0.107368	−	0.994219i	$-0.465758\pi$
$47$	3252.00	0.214737	0.107368	+	0.994219i	$0.465758\pi$
$48$	−2304.00	−0.144338
$49$	−16791.0	−0.999048
$50$	−2500.00	−0.141421
$51$	10638.0	0.572710
$52$	2704.00	0.138675
$53$	5670.00	0.277264	0.138632	−	0.990344i	$-0.455729\pi$
$53$	5670.00	0.277264	0.138632	+	0.990344i	$0.455729\pi$
$54$	2916.00	0.136083
$55$	2400.00	0.106980
$56$	256.000	0.0109086
$57$	10512.0	0.428547
$58$	−1464.00	−0.0571441
$59$	42384.0	1.58516	0.792578	−	0.609771i	$-0.208739\pi$
$59$	42384.0	1.58516	0.792578	+	0.609771i	$0.208739\pi$
$60$	−3600.00	−0.129099
$61$	29678.0	1.02120	0.510599	−	0.859819i	$-0.329424\pi$
$61$	29678.0	1.02120	0.510599	+	0.859819i	$0.329424\pi$
$62$	18256.0	0.603151
$63$	−324.000	−0.0102847
$64$	4096.00	0.125000
$65$	4225.00	0.124035
$66$	3456.00	0.0976594
$67$	−22384.0	−0.609187	−0.304594	−	0.952482i	$-0.598521\pi$
$67$	−22384.0	−0.609187	−0.304594	+	0.952482i	$0.598521\pi$
$68$	−18912.0	−0.495981
$69$	−16200.0	−0.409631
$70$	400.000	0.00975697
$71$	3732.00	0.0878609	0.0439305	−	0.999035i	$-0.486012\pi$
$71$	3732.00	0.0878609	0.0439305	+	0.999035i	$0.486012\pi$
$72$	−5184.00	−0.117851
$73$	37658.0	0.827085	0.413542	−	0.910485i	$-0.364291\pi$
$73$	37658.0	0.827085	0.413542	+	0.910485i	$0.364291\pi$
$74$	−41048.0	−0.871390
$75$	−5625.00	−0.115470
$76$	−18688.0	−0.371132
$77$	−384.000	−0.00738082
$78$	6084.00	0.113228
$79$	−58792.0	−1.05986	−0.529932	−	0.848040i	$-0.677783\pi$
$79$	−58792.0	−1.05986	−0.529932	+	0.848040i	$0.677783\pi$
$80$	6400.00	0.111803
$81$	6561.00	0.111111
$82$	57240.0	0.940080
$83$	−66912.0	−1.06613	−0.533063	−	0.846075i	$-0.678959\pi$
$83$	−66912.0	−1.06613	−0.533063	+	0.846075i	$0.678959\pi$
$84$	576.000	0.00890685
$85$	−29550.0	−0.443619
$86$	−51824.0	−0.755588
$87$	−3294.00	−0.0466579
$88$	−6144.00	−0.0845755
$89$	−132486.	−1.77294	−0.886472	−	0.462782i	$-0.846851\pi$
$89$	−132486.	−1.77294	−0.886472	+	0.462782i	$0.846851\pi$
$90$	−8100.00	−0.105409
$91$	−676.000	−0.00855743
$92$	28800.0	0.354750
$93$	41076.0	0.492471
$94$	−13008.0	−0.151842
$95$	−29200.0	−0.331951
$96$	9216.00	0.102062
$97$	−12910.0	−0.139315	−0.0696573	−	0.997571i	$-0.522191\pi$
$97$	−12910.0	−0.139315	−0.0696573	+	0.997571i	$0.522191\pi$
$98$	67164.0	0.706434
$99$	7776.00	0.0797385
$100$	10000.0	0.100000
$101$	−70122.0	−0.683992	−0.341996	−	0.939701i	$-0.611103\pi$
$101$	−70122.0	−0.683992	−0.341996	+	0.939701i	$0.611103\pi$
$102$	−42552.0	−0.404967
$103$	−24616.0	−0.228625	−0.114313	−	0.993445i	$-0.536467\pi$
$103$	−24616.0	−0.228625	−0.114313	+	0.993445i	$0.536467\pi$
$104$	−10816.0	−0.0980581
$105$	900.000	0.00796653
$106$	−22680.0	−0.196055
$107$	−66732.0	−0.563475	−0.281738	−	0.959492i	$-0.590911\pi$
$107$	−66732.0	−0.563475	−0.281738	+	0.959492i	$0.590911\pi$
$108$	−11664.0	−0.0962250
$109$	8318.00	0.0670583	0.0335292	−	0.999438i	$-0.489325\pi$
$109$	8318.00	0.0670583	0.0335292	+	0.999438i	$0.489325\pi$
$110$	−9600.00	−0.0756466
$111$	−92358.0	−0.711487
$112$	−1024.00	−0.00771356
$113$	−250494.	−1.84545	−0.922723	−	0.385464i	$-0.874041\pi$
$113$	−250494.	−1.84545	−0.922723	+	0.385464i	$0.874041\pi$
$114$	−42048.0	−0.303028
$115$	45000.0	0.317298
$116$	5856.00	0.0404070
$117$	13689.0	0.0924500
$118$	−169536.	−1.12087
$119$	4728.00	0.0306062
$120$	14400.0	0.0912871
$121$	−151835.	−0.942776
$122$	−118712.	−0.722096
$123$	128790.	0.767572
$124$	−73024.0	−0.426493
$125$	15625.0	0.0894427
$126$	1296.00	0.00727241
$127$	125120.	0.688363	0.344181	−	0.938903i	$-0.388156\pi$
$127$	125120.	0.688363	0.344181	+	0.938903i	$0.388156\pi$
$128$	−16384.0	−0.0883883
$129$	−116604.	−0.616935
$130$	−16900.0	−0.0877058
$131$	−100596.	−0.512156	−0.256078	−	0.966656i	$-0.582430\pi$
$131$	−100596.	−0.512156	−0.256078	+	0.966656i	$0.582430\pi$
$132$	−13824.0	−0.0690556
$133$	4672.00	0.0229020
$134$	89536.0	0.430760
$135$	−18225.0	−0.0860663
$136$	75648.0	0.350712
$137$	−394470.	−1.79561	−0.897806	−	0.440391i	$-0.854840\pi$
$137$	−394470.	−1.79561	−0.897806	+	0.440391i	$0.854840\pi$
$138$	64800.0	0.289653
$139$	−108028.	−0.474241	−0.237121	−	0.971480i	$-0.576204\pi$
$139$	−108028.	−0.474241	−0.237121	+	0.971480i	$0.576204\pi$
$140$	−1600.00	−0.00689922
$141$	−29268.0	−0.123978
$142$	−14928.0	−0.0621271
$143$	16224.0	0.0663465
$144$	20736.0	0.0833333
$145$	9150.00	0.0361411
$146$	−150632.	−0.584837
$147$	151119.	0.576801
$148$	164192.	0.616166
$149$	−201066.	−0.741947	−0.370974	−	0.928643i	$-0.620976\pi$
$149$	−201066.	−0.741947	−0.370974	+	0.928643i	$0.620976\pi$
$150$	22500.0	0.0816497
$151$	−263668.	−0.941055	−0.470528	−	0.882385i	$-0.655936\pi$
$151$	−263668.	−0.941055	−0.470528	+	0.882385i	$0.655936\pi$
$152$	74752.0	0.262430
$153$	−95742.0	−0.330654
$154$	1536.00	0.00521903
$155$	−114100.	−0.381466
$156$	−24336.0	−0.0800641
$157$	210446.	0.681383	0.340692	−	0.940175i	$-0.389339\pi$
$157$	210446.	0.681383	0.340692	+	0.940175i	$0.389339\pi$
$158$	235168.	0.749438
$159$	−51030.0	−0.160078
$160$	−25600.0	−0.0790569
$161$	−7200.00	−0.0218911
$162$	−26244.0	−0.0785674
$163$	−305224.	−0.899808	−0.449904	−	0.893077i	$-0.648542\pi$
$163$	−305224.	−0.899808	−0.449904	+	0.893077i	$0.648542\pi$
$164$	−228960.	−0.664737
$165$	−21600.0	−0.0617652
$166$	267648.	0.753865
$167$	528300.	1.46585	0.732925	−	0.680310i	$-0.238155\pi$
$167$	528300.	1.46585	0.732925	+	0.680310i	$0.238155\pi$
$168$	−2304.00	−0.00629810
$169$	28561.0	0.0769231
$170$	118200.	0.313686
$171$	−94608.0	−0.247422
$172$	207296.	0.534281
$173$	−261090.	−0.663247	−0.331623	−	0.943412i	$-0.607596\pi$
$173$	−261090.	−0.663247	−0.331623	+	0.943412i	$0.607596\pi$
$174$	13176.0	0.0329921
$175$	−2500.00	−0.00617085
$176$	24576.0	0.0598039
$177$	−381456.	−0.915190
$178$	529944.	1.25366
$179$	−405132.	−0.945070	−0.472535	−	0.881312i	$-0.656661\pi$
$179$	−405132.	−0.945070	−0.472535	+	0.881312i	$0.656661\pi$
$180$	32400.0	0.0745356
$181$	−151450.	−0.343616	−0.171808	−	0.985130i	$-0.554961\pi$
$181$	−151450.	−0.343616	−0.171808	+	0.985130i	$0.554961\pi$
$182$	2704.00	0.00605101
$183$	−267102.	−0.589589
$184$	−115200.	−0.250846
$185$	256550.	0.551116
$186$	−164304.	−0.348230
$187$	−113472.	−0.237293
$188$	52032.0	0.107368
$189$	2916.00	0.00593790
$190$	116800.	0.234725
$191$	−553608.	−1.09804	−0.549021	−	0.835809i	$-0.684999\pi$
$191$	−553608.	−1.09804	−0.549021	+	0.835809i	$0.684999\pi$
$192$	−36864.0	−0.0721688
$193$	817874.	1.58050	0.790248	−	0.612788i	$-0.209952\pi$
$193$	817874.	1.58050	0.790248	+	0.612788i	$0.209952\pi$
$194$	51640.0	0.0985104
$195$	−38025.0	−0.0716115
$196$	−268656.	−0.499524
$197$	−797658.	−1.46437	−0.732186	−	0.681105i	$-0.761500\pi$
$197$	−797658.	−1.46437	−0.732186	+	0.681105i	$0.761500\pi$
$198$	−31104.0	−0.0563837
$199$	467192.	0.836301	0.418151	−	0.908378i	$-0.362678\pi$
$199$	467192.	0.836301	0.418151	+	0.908378i	$0.362678\pi$
$200$	−40000.0	−0.0707107
$201$	201456.	0.351714
$202$	280488.	0.483655
$203$	−1464.00	−0.00249345
$204$	170208.	0.286355
$205$	−357750.	−0.594559
$206$	98464.0	0.161663
$207$	145800.	0.236500
$208$	43264.0	0.0693375
$209$	−112128.	−0.177561
$210$	−3600.00	−0.00563319
$211$	180332.	0.278847	0.139424	−	0.990233i	$-0.455475\pi$
$211$	180332.	0.278847	0.139424	+	0.990233i	$0.455475\pi$
$212$	90720.0	0.138632
$213$	−33588.0	−0.0507265
$214$	266928.	0.398437
$215$	323900.	0.477876
$216$	46656.0	0.0680414
$217$	18256.0	0.0263182
$218$	−33272.0	−0.0474174
$219$	−338922.	−0.477518
$220$	38400.0	0.0534902
$221$	−199758.	−0.275121
$222$	369432.	0.503097
$223$	1.26676e6	1.70582	0.852911	−	0.522057i	$-0.174835\pi$
$223$	1.26676e6	1.70582	0.852911	+	0.522057i	$0.174835\pi$
$224$	4096.00	0.00545431
$225$	50625.0	0.0666667
$226$	1.00198e6	1.30493
$227$	−380880.	−0.490595	−0.245298	−	0.969448i	$-0.578886\pi$
$227$	−380880.	−0.490595	−0.245298	+	0.969448i	$0.578886\pi$
$228$	168192.	0.214273
$229$	115622.	0.145697	0.0728487	−	0.997343i	$-0.476791\pi$
$229$	115622.	0.145697	0.0728487	+	0.997343i	$0.476791\pi$
$230$	−180000.	−0.224364
$231$	3456.00	0.00426132
$232$	−23424.0	−0.0285720
$233$	−863478.	−1.04199	−0.520993	−	0.853561i	$-0.674438\pi$
$233$	−863478.	−1.04199	−0.520993	+	0.853561i	$0.674438\pi$
$234$	−54756.0	−0.0653720
$235$	81300.0	0.0960331
$236$	678144.	0.792578
$237$	529128.	0.611913
$238$	−18912.0	−0.0216419
$239$	−1.07634e6	−1.21886	−0.609431	−	0.792839i	$-0.708602\pi$
$239$	−1.07634e6	−1.21886	−0.609431	+	0.792839i	$0.708602\pi$
$240$	−57600.0	−0.0645497
$241$	−1.22467e6	−1.35824	−0.679120	−	0.734027i	$-0.737638\pi$
$241$	−1.22467e6	−1.35824	−0.679120	+	0.734027i	$0.737638\pi$
$242$	607340.	0.666643
$243$	−59049.0	−0.0641500
$244$	474848.	0.510599
$245$	−419775.	−0.446788
$246$	−515160.	−0.542756
$247$	−197392.	−0.205867
$248$	292096.	0.301576
$249$	602208.	0.615528
$250$	−62500.0	−0.0632456
$251$	662604.	0.663850	0.331925	−	0.943306i	$-0.392302\pi$
$251$	662604.	0.663850	0.331925	+	0.943306i	$0.392302\pi$
$252$	−5184.00	−0.00514237
$253$	172800.	0.169724
$254$	−500480.	−0.486746
$255$	265950.	0.256124
$256$	65536.0	0.0625000
$257$	1.11703e6	1.05495	0.527473	−	0.849572i	$-0.323139\pi$
$257$	1.11703e6	1.05495	0.527473	+	0.849572i	$0.323139\pi$
$258$	466416.	0.436239
$259$	−41048.0	−0.0380227
$260$	67600.0	0.0620174
$261$	29646.0	0.0269380
$262$	402384.	0.362149
$263$	1.39891e6	1.24710	0.623550	−	0.781784i	$-0.285690\pi$
$263$	1.39891e6	1.24710	0.623550	+	0.781784i	$0.285690\pi$
$264$	55296.0	0.0488297
$265$	141750.	0.123996
$266$	−18688.0	−0.0161942
$267$	1.19237e6	1.02361
$268$	−358144.	−0.304594
$269$	953214.	0.803174	0.401587	−	0.915821i	$-0.368459\pi$
$269$	953214.	0.803174	0.401587	+	0.915821i	$0.368459\pi$
$270$	72900.0	0.0608581
$271$	326036.	0.269676	0.134838	−	0.990868i	$-0.456949\pi$
$271$	326036.	0.269676	0.134838	+	0.990868i	$0.456949\pi$
$272$	−302592.	−0.247991
$273$	6084.00	0.00494063
$274$	1.57788e6	1.26969
$275$	60000.0	0.0478431
$276$	−259200.	−0.204815
$277$	−1.43948e6	−1.12722	−0.563608	−	0.826043i	$-0.690587\pi$
$277$	−1.43948e6	−1.12722	−0.563608	+	0.826043i	$0.690587\pi$
$278$	432112.	0.335339
$279$	−369684.	−0.284328
$280$	6400.00	0.00487848
$281$	−2.21041e6	−1.66996	−0.834980	−	0.550280i	$-0.814521\pi$
$281$	−2.21041e6	−1.66996	−0.834980	+	0.550280i	$0.814521\pi$
$282$	117072.	0.0876658
$283$	−601876.	−0.446726	−0.223363	−	0.974735i	$-0.571703\pi$
$283$	−601876.	−0.446726	−0.223363	+	0.974735i	$0.571703\pi$
$284$	59712.0	0.0439305
$285$	262800.	0.191652
$286$	−64896.0	−0.0469140
$287$	57240.0	0.0410199
$288$	−82944.0	−0.0589256
$289$	−22733.0	−0.0160108
$290$	−36600.0	−0.0255556
$291$	116190.	0.0804334
$292$	602528.	0.413542
$293$	−776346.	−0.528307	−0.264153	−	0.964481i	$-0.585093\pi$
$293$	−776346.	−0.528307	−0.264153	+	0.964481i	$0.585093\pi$
$294$	−604476.	−0.407860
$295$	1.05960e6	0.708903
$296$	−656768.	−0.435695
$297$	−69984.0	−0.0460371
$298$	804264.	0.524636
$299$	304200.	0.196780
$300$	−90000.0	−0.0577350
$301$	−51824.0	−0.0329697
$302$	1.05467e6	0.665426
$303$	631098.	0.394903
$304$	−299008.	−0.185566
$305$	741950.	0.456694
$306$	382968.	0.233808
$307$	2.35561e6	1.42645	0.713226	−	0.700934i	$-0.247233\pi$
$307$	2.35561e6	1.42645	0.713226	+	0.700934i	$0.247233\pi$
$308$	−6144.00	−0.00369041
$309$	221544.	0.131997
$310$	456400.	0.269738
$311$	−1.79148e6	−1.05029	−0.525147	−	0.851011i	$-0.675990\pi$
$311$	−1.79148e6	−1.05029	−0.525147	+	0.851011i	$0.675990\pi$
$312$	97344.0	0.0566139
$313$	710186.	0.409743	0.204871	−	0.978789i	$-0.434322\pi$
$313$	710186.	0.409743	0.204871	+	0.978789i	$0.434322\pi$
$314$	−841784.	−0.481811
$315$	−8100.00	−0.00459948
$316$	−940672.	−0.529932
$317$	1.30741e6	0.730739	0.365370	−	0.930863i	$-0.380943\pi$
$317$	1.30741e6	0.730739	0.365370	+	0.930863i	$0.380943\pi$
$318$	204120.	0.113193
$319$	35136.0	0.0193319
$320$	102400.	0.0559017
$321$	600588.	0.325322
$322$	28800.0	0.0154794
$323$	1.38058e6	0.736299
$324$	104976.	0.0555556
$325$	105625.	0.0554700
$326$	1.22090e6	0.636260
$327$	−74862.0	−0.0387161
$328$	915840.	0.470040
$329$	−13008.0	−0.00662553
$330$	86400.0	0.0436746
$331$	−798232.	−0.400460	−0.200230	−	0.979749i	$-0.564169\pi$
$331$	−798232.	−0.400460	−0.200230	+	0.979749i	$0.564169\pi$
$332$	−1.07059e6	−0.533063
$333$	831222.	0.410777
$334$	−2.11320e6	−1.03651
$335$	−559600.	−0.272437
$336$	9216.00	0.00445343
$337$	212642.	0.101994	0.0509970	−	0.998699i	$-0.483760\pi$
$337$	212642.	0.101994	0.0509970	+	0.998699i	$0.483760\pi$
$338$	−114244.	−0.0543928
$339$	2.25445e6	1.06547
$340$	−472800.	−0.221810
$341$	−438144.	−0.204047
$342$	378432.	0.174953
$343$	134392.	0.0616791
$344$	−829184.	−0.377794
$345$	−405000.	−0.183192
$346$	1.04436e6	0.468986
$347$	−2.76731e6	−1.23377	−0.616884	−	0.787054i	$-0.711605\pi$
$347$	−2.76731e6	−1.23377	−0.616884	+	0.787054i	$0.711605\pi$
$348$	−52704.0	−0.0233290
$349$	3.35853e6	1.47600	0.737998	−	0.674803i	$-0.235771\pi$
$349$	3.35853e6	1.47600	0.737998	+	0.674803i	$0.235771\pi$
$350$	10000.0	0.00436345
$351$	−123201.	−0.0533761
$352$	−98304.0	−0.0422877
$353$	−1.59281e6	−0.680344	−0.340172	−	0.940363i	$-0.610485\pi$
$353$	−1.59281e6	−0.680344	−0.340172	+	0.940363i	$0.610485\pi$
$354$	1.52582e6	0.647137
$355$	93300.0	0.0392926
$356$	−2.11978e6	−0.886472
$357$	−42552.0	−0.0176705
$358$	1.62053e6	0.668265
$359$	408060.	0.167104	0.0835522	−	0.996503i	$-0.473373\pi$
$359$	408060.	0.167104	0.0835522	+	0.996503i	$0.473373\pi$
$360$	−129600.	−0.0527046
$361$	−1.11188e6	−0.449043
$362$	605800.	0.242973
$363$	1.36652e6	0.544312
$364$	−10816.0	−0.00427871
$365$	941450.	0.369884
$366$	1.06841e6	0.416902
$367$	1.21402e6	0.470503	0.235251	−	0.971935i	$-0.424409\pi$
$367$	1.21402e6	0.470503	0.235251	+	0.971935i	$0.424409\pi$
$368$	460800.	0.177375
$369$	−1.15911e6	−0.443158
$370$	−1.02620e6	−0.389698
$371$	−22680.0	−0.00855477
$372$	657216.	0.246236
$373$	−416026.	−0.154828	−0.0774138	−	0.996999i	$-0.524666\pi$
$373$	−416026.	−0.154828	−0.0774138	+	0.996999i	$0.524666\pi$
$374$	453888.	0.167791
$375$	−140625.	−0.0516398
$376$	−208128.	−0.0759208
$377$	61854.0	0.0224137
$378$	−11664.0	−0.00419873
$379$	931160.	0.332986	0.166493	−	0.986043i	$-0.446756\pi$
$379$	931160.	0.332986	0.166493	+	0.986043i	$0.446756\pi$
$380$	−467200.	−0.165975
$381$	−1.12608e6	−0.397426
$382$	2.21443e6	0.776433
$383$	549804.	0.191519	0.0957593	−	0.995405i	$-0.469472\pi$
$383$	549804.	0.191519	0.0957593	+	0.995405i	$0.469472\pi$
$384$	147456.	0.0510310
$385$	−9600.00	−0.00330080
$386$	−3.27150e6	−1.11758
$387$	1.04944e6	0.356187
$388$	−206560.	−0.0696573
$389$	2.83877e6	0.951167	0.475584	−	0.879671i	$-0.342237\pi$
$389$	2.83877e6	0.951167	0.475584	+	0.879671i	$0.342237\pi$
$390$	152100.	0.0506370
$391$	−2.12760e6	−0.703798
$392$	1.07462e6	0.353217
$393$	905364.	0.295694
$394$	3.19063e6	1.03547
$395$	−1.46980e6	−0.473986
$396$	124416.	0.0398693
$397$	−1.65647e6	−0.527480	−0.263740	−	0.964594i	$-0.584956\pi$
$397$	−1.65647e6	−0.527480	−0.263740	+	0.964594i	$0.584956\pi$
$398$	−1.86877e6	−0.591354
$399$	−42048.0	−0.0132225
$400$	160000.	0.0500000
$401$	−5.22113e6	−1.62145	−0.810726	−	0.585426i	$-0.800927\pi$
$401$	−5.22113e6	−1.62145	−0.810726	+	0.585426i	$0.800927\pi$
$402$	−805824.	−0.248700
$403$	−771316.	−0.236575
$404$	−1.12195e6	−0.341996
$405$	164025.	0.0496904
$406$	5856.00	0.00176314
$407$	985152.	0.294793
$408$	−680832.	−0.202483
$409$	−1.44372e6	−0.426751	−0.213375	−	0.976970i	$-0.568446\pi$
$409$	−1.44372e6	−0.426751	−0.213375	+	0.976970i	$0.568446\pi$
$410$	1.43100e6	0.420417
$411$	3.55023e6	1.03670
$412$	−393856.	−0.114313
$413$	−169536.	−0.0489088
$414$	−583200.	−0.167231
$415$	−1.67280e6	−0.476786
$416$	−173056.	−0.0490290
$417$	972252.	0.273803
$418$	448512.	0.125555
$419$	1.93964e6	0.539743	0.269871	−	0.962896i	$-0.413019\pi$
$419$	1.93964e6	0.539743	0.269871	+	0.962896i	$0.413019\pi$
$420$	14400.0	0.00398327
$421$	−2.46822e6	−0.678701	−0.339350	−	0.940660i	$-0.610207\pi$
$421$	−2.46822e6	−0.678701	−0.339350	+	0.940660i	$0.610207\pi$
$422$	−721328.	−0.197175
$423$	263412.	0.0715788
$424$	−362880.	−0.0980276
$425$	−738750.	−0.198392
$426$	134352.	0.0358691
$427$	−118712.	−0.0315083
$428$	−1.06771e6	−0.281738
$429$	−146016.	−0.0383052
$430$	−1.29560e6	−0.337909
$431$	−1.64375e6	−0.426228	−0.213114	−	0.977027i	$-0.568361\pi$
$431$	−1.64375e6	−0.426228	−0.213114	+	0.977027i	$0.568361\pi$
$432$	−186624.	−0.0481125
$433$	−495790.	−0.127080	−0.0635401	−	0.997979i	$-0.520239\pi$
$433$	−495790.	−0.127080	−0.0635401	+	0.997979i	$0.520239\pi$
$434$	−73024.0	−0.0186098
$435$	−82350.0	−0.0208661
$436$	133088.	0.0335292
$437$	−2.10240e6	−0.526637
$438$	1.35569e6	0.337656
$439$	1.03518e6	0.256362	0.128181	−	0.991751i	$-0.459086\pi$
$439$	1.03518e6	0.256362	0.128181	+	0.991751i	$0.459086\pi$
$440$	−153600.	−0.0378233
$441$	−1.36007e6	−0.333016
$442$	799032.	0.194540
$443$	−1.41400e6	−0.342325	−0.171163	−	0.985243i	$-0.554752\pi$
$443$	−1.41400e6	−0.342325	−0.171163	+	0.985243i	$0.554752\pi$
$444$	−1.47773e6	−0.355744
$445$	−3.31215e6	−0.792885
$446$	−5.06706e6	−1.20620
$447$	1.80959e6	0.428363
$448$	−16384.0	−0.00385678
$449$	−267294.	−0.0625710	−0.0312855	−	0.999510i	$-0.509960\pi$
$449$	−267294.	−0.0625710	−0.0312855	+	0.999510i	$0.509960\pi$
$450$	−202500.	−0.0471405
$451$	−1.37376e6	−0.318031
$452$	−4.00790e6	−0.922723
$453$	2.37301e6	0.543318
$454$	1.52352e6	0.346903
$455$	−16900.0	−0.00382700
$456$	−672768.	−0.151514
$457$	−3.14239e6	−0.703833	−0.351916	−	0.936031i	$-0.614470\pi$
$457$	−3.14239e6	−0.703833	−0.351916	+	0.936031i	$0.614470\pi$
$458$	−462488.	−0.103024
$459$	861678.	0.190903
$460$	720000.	0.158649
$461$	−6.95839e6	−1.52495	−0.762476	−	0.647016i	$-0.776017\pi$
$461$	−6.95839e6	−1.52495	−0.762476	+	0.647016i	$0.776017\pi$
$462$	−13824.0	−0.00301321
$463$	−3.99790e6	−0.866721	−0.433361	−	0.901221i	$-0.642672\pi$
$463$	−3.99790e6	−0.866721	−0.433361	+	0.901221i	$0.642672\pi$
$464$	93696.0	0.0202035
$465$	1.02690e6	0.220240
$466$	3.45391e6	0.736795
$467$	−4.22824e6	−0.897154	−0.448577	−	0.893744i	$-0.648069\pi$
$467$	−4.22824e6	−0.897154	−0.448577	+	0.893744i	$0.648069\pi$
$468$	219024.	0.0462250
$469$	89536.0	0.0187960
$470$	−325200.	−0.0679056
$471$	−1.89401e6	−0.393397
$472$	−2.71258e6	−0.560437
$473$	1.24378e6	0.255617
$474$	−2.11651e6	−0.432688
$475$	−730000.	−0.148453
$476$	75648.0	0.0153031
$477$	459270.	0.0924213
$478$	4.30536e6	0.861866
$479$	2.85536e6	0.568621	0.284310	−	0.958732i	$-0.408235\pi$
$479$	2.85536e6	0.568621	0.284310	+	0.958732i	$0.408235\pi$
$480$	230400.	0.0456435
$481$	1.73428e6	0.341787
$482$	4.89868e6	0.960421
$483$	64800.0	0.0126388
$484$	−2.42936e6	−0.471388
$485$	−322750.	−0.0623034
$486$	236196.	0.0453609
$487$	23636.0	0.00451598	0.00225799	−	0.999997i	$-0.499281\pi$
$487$	23636.0	0.00451598	0.00225799	+	0.999997i	$0.499281\pi$
$488$	−1.89939e6	−0.361048
$489$	2.74702e6	0.519504
$490$	1.67910e6	0.315927
$491$	−4.09856e6	−0.767234	−0.383617	−	0.923492i	$-0.625322\pi$
$491$	−4.09856e6	−0.767234	−0.383617	+	0.923492i	$0.625322\pi$
$492$	2.06064e6	0.383786
$493$	−432612.	−0.0801643
$494$	789568.	0.145570
$495$	194400.	0.0356602
$496$	−1.16838e6	−0.213246
$497$	−14928.0	−0.00271088
$498$	−2.40883e6	−0.435244
$499$	4.98176e6	0.895636	0.447818	−	0.894125i	$-0.352201\pi$
$499$	4.98176e6	0.895636	0.447818	+	0.894125i	$0.352201\pi$
$500$	250000.	0.0447214
$501$	−4.75470e6	−0.846309
$502$	−2.65042e6	−0.469413
$503$	−7.95216e6	−1.40141	−0.700705	−	0.713451i	$-0.747131\pi$
$503$	−7.95216e6	−1.40141	−0.700705	+	0.713451i	$0.747131\pi$
$504$	20736.0	0.00363621
$505$	−1.75305e6	−0.305890
$506$	−691200.	−0.120013
$507$	−257049.	−0.0444116
$508$	2.00192e6	0.344181
$509$	6.58078e6	1.12586	0.562928	−	0.826506i	$-0.309675\pi$
$509$	6.58078e6	1.12586	0.562928	+	0.826506i	$0.309675\pi$
$510$	−1.06380e6	−0.181107
$511$	−150632.	−0.0255191
$512$	−262144.	−0.0441942
$513$	851472.	0.142849
$514$	−4.46810e6	−0.745960
$515$	−615400.	−0.102244
$516$	−1.86566e6	−0.308467
$517$	312192.	0.0513683
$518$	164192.	0.0268861
$519$	2.34981e6	0.382926
$520$	−270400.	−0.0438529
$521$	−200934.	−0.0324309	−0.0162155	−	0.999869i	$-0.505162\pi$
$521$	−200934.	−0.0324309	−0.0162155	+	0.999869i	$0.505162\pi$
$522$	−118584.	−0.0190480
$523$	6.61466e6	1.05743	0.528717	−	0.848798i	$-0.322673\pi$
$523$	6.61466e6	1.05743	0.528717	+	0.848798i	$0.322673\pi$
$524$	−1.60954e6	−0.256078
$525$	22500.0	0.00356274
$526$	−5.59565e6	−0.881832
$527$	5.39465e6	0.846129
$528$	−221184.	−0.0345278
$529$	−3.19634e6	−0.496609
$530$	−567000.	−0.0876786
$531$	3.43310e6	0.528385
$532$	74752.0	0.0114510
$533$	−2.41839e6	−0.368730
$534$	−4.76950e6	−0.723801
$535$	−1.66830e6	−0.251994
$536$	1.43258e6	0.215380
$537$	3.64619e6	0.545636
$538$	−3.81286e6	−0.567930
$539$	−1.61194e6	−0.238988
$540$	−291600.	−0.0430331
$541$	−1.34964e7	−1.98256	−0.991279	−	0.131780i	$-0.957931\pi$
$541$	−1.34964e7	−1.98256	−0.991279	+	0.131780i	$0.957931\pi$
$542$	−1.30414e6	−0.190690
$543$	1.36305e6	0.198387
$544$	1.21037e6	0.175356
$545$	207950.	0.0299894
$546$	−24336.0	−0.00349356
$547$	4.30963e6	0.615845	0.307923	−	0.951411i	$-0.400366\pi$
$547$	4.30963e6	0.615845	0.307923	+	0.951411i	$0.400366\pi$
$548$	−6.31152e6	−0.897806
$549$	2.40392e6	0.340399
$550$	−240000.	−0.0338302
$551$	−427488.	−0.0599853
$552$	1.03680e6	0.144826
$553$	235168.	0.0327013
$554$	5.75793e6	0.797061
$555$	−2.30895e6	−0.318187
$556$	−1.72845e6	−0.237121
$557$	9.46846e6	1.29313	0.646564	−	0.762860i	$-0.276205\pi$
$557$	9.46846e6	1.29313	0.646564	+	0.762860i	$0.276205\pi$
$558$	1.47874e6	0.201050
$559$	2.18956e6	0.296366
$560$	−25600.0	−0.00344961
$561$	1.02125e6	0.137001
$562$	8.84162e6	1.18084
$563$	7.23220e6	0.961611	0.480805	−	0.876827i	$-0.340344\pi$
$563$	7.23220e6	0.961611	0.480805	+	0.876827i	$0.340344\pi$
$564$	−468288.	−0.0619891
$565$	−6.26235e6	−0.825308
$566$	2.40750e6	0.315883
$567$	−26244.0	−0.00342825
$568$	−238848.	−0.0310635
$569$	3.18201e6	0.412022	0.206011	−	0.978550i	$-0.433952\pi$
$569$	3.18201e6	0.412022	0.206011	+	0.978550i	$0.433952\pi$
$570$	−1.05120e6	−0.135518
$571$	−4.72042e6	−0.605885	−0.302943	−	0.953009i	$-0.597969\pi$
$571$	−4.72042e6	−0.605885	−0.302943	+	0.953009i	$0.597969\pi$
$572$	259584.	0.0331732
$573$	4.98247e6	0.633955
$574$	−228960.	−0.0290055
$575$	1.12500e6	0.141900
$576$	331776.	0.0416667
$577$	−3.26592e6	−0.408381	−0.204191	−	0.978931i	$-0.565456\pi$
$577$	−3.26592e6	−0.408381	−0.204191	+	0.978931i	$0.565456\pi$
$578$	90932.0	0.0113213
$579$	−7.36087e6	−0.912499
$580$	146400.	0.0180705
$581$	267648.	0.0328945
$582$	−464760.	−0.0568750
$583$	544320.	0.0663259
$584$	−2.41011e6	−0.292419
$585$	342225.	0.0413449
$586$	3.10538e6	0.373569
$587$	465576.	0.0557693	0.0278847	−	0.999611i	$-0.491123\pi$
$587$	465576.	0.0557693	0.0278847	+	0.999611i	$0.491123\pi$
$588$	2.41790e6	0.288400
$589$	5.33075e6	0.633141
$590$	−4.23840e6	−0.501270
$591$	7.17892e6	0.845455
$592$	2.62707e6	0.308083
$593$	6.52048e6	0.761453	0.380726	−	0.924688i	$-0.375674\pi$
$593$	6.52048e6	0.761453	0.380726	+	0.924688i	$0.375674\pi$
$594$	279936.	0.0325531
$595$	118200.	0.0136875
$596$	−3.21706e6	−0.370974
$597$	−4.20473e6	−0.482839
$598$	−1.21680e6	−0.139145
$599$	4.14010e6	0.471458	0.235729	−	0.971819i	$-0.424252\pi$
$599$	4.14010e6	0.471458	0.235729	+	0.971819i	$0.424252\pi$
$600$	360000.	0.0408248
$601$	9.11115e6	1.02893	0.514466	−	0.857511i	$-0.327990\pi$
$601$	9.11115e6	1.02893	0.514466	+	0.857511i	$0.327990\pi$
$602$	207296.	0.0233131
$603$	−1.81310e6	−0.203062
$604$	−4.21869e6	−0.470528
$605$	−3.79588e6	−0.421622
$606$	−2.52439e6	−0.279238
$607$	−6.99090e6	−0.770126	−0.385063	−	0.922890i	$-0.625820\pi$
$607$	−6.99090e6	−0.770126	−0.385063	+	0.922890i	$0.625820\pi$
$608$	1.19603e6	0.131215
$609$	13176.0	0.00143960
$610$	−2.96780e6	−0.322931
$611$	549588.	0.0595572
$612$	−1.53187e6	−0.165327
$613$	2.18322e6	0.234664	0.117332	−	0.993093i	$-0.462566\pi$
$613$	2.18322e6	0.234664	0.117332	+	0.993093i	$0.462566\pi$
$614$	−9.42243e6	−1.00865
$615$	3.21975e6	0.343269
$616$	24576.0	0.00260951
$617$	−3.32713e6	−0.351849	−0.175925	−	0.984404i	$-0.556291\pi$
$617$	−3.32713e6	−0.351849	−0.175925	+	0.984404i	$0.556291\pi$
$618$	−886176.	−0.0933359
$619$	4.82619e6	0.506265	0.253133	−	0.967432i	$-0.418539\pi$
$619$	4.82619e6	0.506265	0.253133	+	0.967432i	$0.418539\pi$
$620$	−1.82560e6	−0.190733
$621$	−1.31220e6	−0.136544
$622$	7.16592e6	0.742670
$623$	529944.	0.0547028
$624$	−389376.	−0.0400320
$625$	390625.	0.0400000
$626$	−2.84074e6	−0.289732
$627$	1.00915e6	0.102515
$628$	3.36714e6	0.340692
$629$	−1.21297e7	−1.22243
$630$	32400.0	0.00325232
$631$	1.69252e6	0.169224	0.0846119	−	0.996414i	$-0.473035\pi$
$631$	1.69252e6	0.169224	0.0846119	+	0.996414i	$0.473035\pi$
$632$	3.76269e6	0.374719
$633$	−1.62299e6	−0.160993
$634$	−5.22962e6	−0.516711
$635$	3.12800e6	0.307845
$636$	−816480.	−0.0800392
$637$	−2.83768e6	−0.277086
$638$	−140544.	−0.0136698
$639$	302292.	0.0292870
$640$	−409600.	−0.0395285
$641$	5.58088e6	0.536485	0.268243	−	0.963351i	$-0.413557\pi$
$641$	5.58088e6	0.536485	0.268243	+	0.963351i	$0.413557\pi$
$642$	−2.40235e6	−0.230038
$643$	−294544.	−0.0280946	−0.0140473	−	0.999901i	$-0.504472\pi$
$643$	−294544.	−0.0280946	−0.0140473	+	0.999901i	$0.504472\pi$
$644$	−115200.	−0.0109456
$645$	−2.91510e6	−0.275902
$646$	−5.52230e6	−0.520642
$647$	1.91246e7	1.79611	0.898053	−	0.439887i	$-0.144982\pi$
$647$	1.91246e7	1.79611	0.898053	+	0.439887i	$0.144982\pi$
$648$	−419904.	−0.0392837
$649$	4.06886e6	0.379194
$650$	−422500.	−0.0392232
$651$	−164304.	−0.0151948
$652$	−4.88358e6	−0.449904
$653$	8.63955e6	0.792881	0.396441	−	0.918060i	$-0.370245\pi$
$653$	8.63955e6	0.792881	0.396441	+	0.918060i	$0.370245\pi$
$654$	299448.	0.0273764
$655$	−2.51490e6	−0.229043
$656$	−3.66336e6	−0.332369
$657$	3.05030e6	0.275695
$658$	52032.0	0.00468496
$659$	1.64908e7	1.47921	0.739604	−	0.673042i	$-0.235013\pi$
$659$	1.64908e7	1.47921	0.739604	+	0.673042i	$0.235013\pi$
$660$	−345600.	−0.0308826
$661$	1.22734e7	1.09260	0.546300	−	0.837590i	$-0.316036\pi$
$661$	1.22734e7	1.09260	0.546300	+	0.837590i	$0.316036\pi$
$662$	3.19293e6	0.283168
$663$	1.79782e6	0.158841
$664$	4.28237e6	0.376933
$665$	116800.	0.0102421
$666$	−3.32489e6	−0.290463
$667$	658800.	0.0573375
$668$	8.45280e6	0.732925
$669$	−1.14009e7	−0.984856
$670$	2.23840e6	0.192642
$671$	2.84909e6	0.244287
$672$	−36864.0	−0.00314905
$673$	−2.19461e7	−1.86776	−0.933878	−	0.357591i	$-0.883598\pi$
$673$	−2.19461e7	−1.86776	−0.933878	+	0.357591i	$0.883598\pi$
$674$	−850568.	−0.0721206
$675$	−455625.	−0.0384900
$676$	456976.	0.0384615
$677$	−1.15550e6	−0.0968941	−0.0484471	−	0.998826i	$-0.515427\pi$
$677$	−1.15550e6	−0.0968941	−0.0484471	+	0.998826i	$0.515427\pi$
$678$	−9.01778e6	−0.753400
$679$	51640.0	0.00429845
$680$	1.89120e6	0.156843
$681$	3.42792e6	0.283245
$682$	1.75258e6	0.144283
$683$	−8.59166e6	−0.704734	−0.352367	−	0.935862i	$-0.614623\pi$
$683$	−8.59166e6	−0.704734	−0.352367	+	0.935862i	$0.614623\pi$
$684$	−1.51373e6	−0.123711
$685$	−9.86175e6	−0.803022
$686$	−537568.	−0.0436137
$687$	−1.04060e6	−0.0841184
$688$	3.31674e6	0.267141
$689$	958230.	0.0768992
$690$	1.62000e6	0.129537
$691$	−1.80227e6	−0.143590	−0.0717952	−	0.997419i	$-0.522873\pi$
$691$	−1.80227e6	−0.143590	−0.0717952	+	0.997419i	$0.522873\pi$
$692$	−4.17744e6	−0.331623
$693$	−31104.0	−0.00246027
$694$	1.10692e7	0.872406
$695$	−2.70070e6	−0.212087
$696$	210816.	0.0164961
$697$	1.69144e7	1.31879
$698$	−1.34341e7	−1.04369
$699$	7.77130e6	0.601590
$700$	−40000.0	−0.00308542
$701$	626670.	0.0481664	0.0240832	−	0.999710i	$-0.492333\pi$
$701$	626670.	0.0481664	0.0240832	+	0.999710i	$0.492333\pi$
$702$	492804.	0.0377426
$703$	−1.19860e7	−0.914717
$704$	393216.	0.0299020
$705$	−731700.	−0.0554447
$706$	6.37126e6	0.481076
$707$	280488.	0.0211040
$708$	−6.10330e6	−0.457595
$709$	4.22865e6	0.315926	0.157963	−	0.987445i	$-0.449507\pi$
$709$	4.22865e6	0.315926	0.157963	+	0.987445i	$0.449507\pi$
$710$	−373200.	−0.0277841
$711$	−4.76215e6	−0.353288
$712$	8.47910e6	0.626830
$713$	−8.21520e6	−0.605194
$714$	170208.	0.0124949
$715$	405600.	0.0296710
$716$	−6.48211e6	−0.472535
$717$	9.68706e6	0.703711
$718$	−1.63224e6	−0.118161
$719$	1.50827e7	1.08807	0.544033	−	0.839064i	$-0.316896\pi$
$719$	1.50827e7	1.08807	0.544033	+	0.839064i	$0.316896\pi$
$720$	518400.	0.0372678
$721$	98464.0	0.00705406
$722$	4.44750e6	0.317521
$723$	1.10220e7	0.784180
$724$	−2.42320e6	−0.171808
$725$	228750.	0.0161628
$726$	−5.46606e6	−0.384887
$727$	1.54296e7	1.08272	0.541362	−	0.840790i	$-0.317909\pi$
$727$	1.54296e7	1.08272	0.541362	+	0.840790i	$0.317909\pi$
$728$	43264.0	0.00302551
$729$	531441.	0.0370370
$730$	−3.76580e6	−0.261547
$731$	−1.53140e7	−1.05997
$732$	−4.27363e6	−0.294795
$733$	7.58022e6	0.521101	0.260551	−	0.965460i	$-0.416096\pi$
$733$	7.58022e6	0.521101	0.260551	+	0.965460i	$0.416096\pi$
$734$	−4.85610e6	−0.332696
$735$	3.77797e6	0.257953
$736$	−1.84320e6	−0.125423
$737$	−2.14886e6	−0.145727
$738$	4.63644e6	0.313360
$739$	−1.01597e7	−0.684338	−0.342169	−	0.939638i	$-0.611162\pi$
$739$	−1.01597e7	−0.684338	−0.342169	+	0.939638i	$0.611162\pi$
$740$	4.10480e6	0.275558
$741$	1.77653e6	0.118857
$742$	90720.0	0.00604914
$743$	1.69873e7	1.12889	0.564445	−	0.825470i	$-0.309090\pi$
$743$	1.69873e7	1.12889	0.564445	+	0.825470i	$0.309090\pi$
$744$	−2.62886e6	−0.174115
$745$	−5.02665e6	−0.331809
$746$	1.66410e6	0.109480
$747$	−5.41987e6	−0.355376
$748$	−1.81555e6	−0.118646
$749$	266928.	0.0173856
$750$	562500.	0.0365148
$751$	2.05374e7	1.32875	0.664377	−	0.747397i	$-0.268697\pi$
$751$	2.05374e7	1.32875	0.664377	+	0.747397i	$0.268697\pi$
$752$	832512.	0.0536841
$753$	−5.96344e6	−0.383274
$754$	−247416.	−0.0158489
$755$	−6.59170e6	−0.420853
$756$	46656.0	0.00296895
$757$	1.32570e7	0.840823	0.420411	−	0.907334i	$-0.361886\pi$
$757$	1.32570e7	0.840823	0.420411	+	0.907334i	$0.361886\pi$
$758$	−3.72464e6	−0.235457
$759$	−1.55520e6	−0.0979900
$760$	1.86880e6	0.117362
$761$	−1.06308e7	−0.665433	−0.332717	−	0.943027i	$-0.607965\pi$
$761$	−1.06308e7	−0.665433	−0.332717	+	0.943027i	$0.607965\pi$
$762$	4.50432e6	0.281023
$763$	−33272.0	−0.00206903
$764$	−8.85773e6	−0.549021
$765$	−2.39355e6	−0.147873
$766$	−2.19922e6	−0.135424
$767$	7.16290e6	0.439643
$768$	−589824.	−0.0360844
$769$	−2.23975e7	−1.36579	−0.682895	−	0.730516i	$-0.739279\pi$
$769$	−2.23975e7	−1.36579	−0.682895	+	0.730516i	$0.739279\pi$
$770$	38400.0	0.00233402
$771$	−1.00532e7	−0.609074
$772$	1.30860e7	0.790248
$773$	−1.73985e7	−1.04728	−0.523640	−	0.851939i	$-0.675426\pi$
$773$	−1.73985e7	−1.04728	−0.523640	+	0.851939i	$0.675426\pi$
$774$	−4.19774e6	−0.251863
$775$	−2.85250e6	−0.170597
$776$	826240.	0.0492552
$777$	369432.	0.0219524
$778$	−1.13551e7	−0.672577
$779$	1.67141e7	0.986822
$780$	−608400.	−0.0358057
$781$	358272.	0.0210177
$782$	8.51040e6	0.497660
$783$	−266814.	−0.0155526
$784$	−4.29850e6	−0.249762
$785$	5.26115e6	0.304724
$786$	−3.62146e6	−0.209087
$787$	−2.58584e7	−1.48821	−0.744105	−	0.668063i	$-0.767124\pi$
$787$	−2.58584e7	−1.48821	−0.744105	+	0.668063i	$0.767124\pi$
$788$	−1.27625e7	−0.732186
$789$	−1.25902e7	−0.720013
$790$	5.87920e6	0.335159
$791$	1.00198e6	0.0569398
$792$	−497664.	−0.0281918
$793$	5.01558e6	0.283229
$794$	6.62586e6	0.372985
$795$	−1.27575e6	−0.0715892
$796$	7.47507e6	0.418151
$797$	2.56181e7	1.42857	0.714283	−	0.699857i	$-0.246753\pi$
$797$	2.56181e7	1.42857	0.714283	+	0.699857i	$0.246753\pi$
$798$	168192.	0.00934971
$799$	−3.84386e6	−0.213011
$800$	−640000.	−0.0353553
$801$	−1.07314e7	−0.590981
$802$	2.08845e7	1.14654
$803$	3.61517e6	0.197852
$804$	3.22330e6	0.175857
$805$	−180000.	−0.00979000
$806$	3.08526e6	0.167284
$807$	−8.57893e6	−0.463713
$808$	4.48781e6	0.241828
$809$	3.29115e7	1.76798	0.883988	−	0.467510i	$-0.154849\pi$
$809$	3.29115e7	1.76798	0.883988	+	0.467510i	$0.154849\pi$
$810$	−656100.	−0.0351364
$811$	−1.16343e7	−0.621137	−0.310568	−	0.950551i	$-0.600519\pi$
$811$	−1.16343e7	−0.621137	−0.310568	+	0.950551i	$0.600519\pi$
$812$	−23424.0	−0.00124673
$813$	−2.93432e6	−0.155698
$814$	−3.94061e6	−0.208450
$815$	−7.63060e6	−0.402406
$816$	2.72333e6	0.143177
$817$	−1.51326e7	−0.793156
$818$	5.77487e6	0.301758
$819$	−54756.0	−0.00285248
$820$	−5.72400e6	−0.297280
$821$	−2.40157e7	−1.24348	−0.621739	−	0.783225i	$-0.713574\pi$
$821$	−2.40157e7	−1.24348	−0.621739	+	0.783225i	$0.713574\pi$
$822$	−1.42009e7	−0.733056
$823$	3.32629e7	1.71183	0.855914	−	0.517119i	$-0.172995\pi$
$823$	3.32629e7	1.71183	0.855914	+	0.517119i	$0.172995\pi$
$824$	1.57542e6	0.0808313
$825$	−540000.	−0.0276222
$826$	678144.	0.0345837
$827$	5.66755e6	0.288159	0.144079	−	0.989566i	$-0.453978\pi$
$827$	5.66755e6	0.288159	0.144079	+	0.989566i	$0.453978\pi$
$828$	2.33280e6	0.118250
$829$	1.94822e7	0.984579	0.492290	−	0.870431i	$-0.336160\pi$
$829$	1.94822e7	0.984579	0.492290	+	0.870431i	$0.336160\pi$
$830$	6.69120e6	0.337139
$831$	1.29553e7	0.650798
$832$	692224.	0.0346688
$833$	1.98470e7	0.991018
$834$	−3.88901e6	−0.193608
$835$	1.32075e7	0.655548
$836$	−1.79405e6	−0.0887807
$837$	3.32716e6	0.164157
$838$	−7.75858e6	−0.381656
$839$	2.01195e7	0.986762	0.493381	−	0.869813i	$-0.335761\pi$
$839$	2.01195e7	0.986762	0.493381	+	0.869813i	$0.335761\pi$
$840$	−57600.0	−0.00281659
$841$	−2.03772e7	−0.993469
$842$	9.87287e6	0.479914
$843$	1.98937e7	0.964152
$844$	2.88531e6	0.139424
$845$	714025.	0.0344010
$846$	−1.05365e6	−0.0506139
$847$	607340.	0.0290886
$848$	1.45152e6	0.0693160
$849$	5.41688e6	0.257917
$850$	2.95500e6	0.140285
$851$	1.84716e7	0.874341
$852$	−537408.	−0.0253633
$853$	−1.32934e7	−0.625553	−0.312776	−	0.949827i	$-0.601259\pi$
$853$	−1.32934e7	−0.625553	−0.312776	+	0.949827i	$0.601259\pi$
$854$	474848.	0.0222797
$855$	−2.36520e6	−0.110650
$856$	4.27085e6	0.199219
$857$	−4.55012e6	−0.211627	−0.105813	−	0.994386i	$-0.533745\pi$
$857$	−4.55012e6	−0.211627	−0.105813	+	0.994386i	$0.533745\pi$
$858$	584064.	0.0270858
$859$	−1.91436e6	−0.0885200	−0.0442600	−	0.999020i	$-0.514093\pi$
$859$	−1.91436e6	−0.0885200	−0.0442600	+	0.999020i	$0.514093\pi$
$860$	5.18240e6	0.238938
$861$	−515160.	−0.0236829
$862$	6.57499e6	0.301389
$863$	−1.84847e7	−0.844863	−0.422432	−	0.906395i	$-0.638823\pi$
$863$	−1.84847e7	−0.844863	−0.422432	+	0.906395i	$0.638823\pi$
$864$	746496.	0.0340207
$865$	−6.52725e6	−0.296613
$866$	1.98316e6	0.0898593
$867$	204597.	0.00924382
$868$	292096.	0.0131591
$869$	−5.64403e6	−0.253536
$870$	329400.	0.0147545
$871$	−3.78290e6	−0.168958
$872$	−532352.	−0.0237087
$873$	−1.04571e6	−0.0464382
$874$	8.40960e6	0.372389
$875$	−62500.0	−0.00275969
$876$	−5.42275e6	−0.238759
$877$	1.04774e7	0.459995	0.229997	−	0.973191i	$-0.426128\pi$
$877$	1.04774e7	0.459995	0.229997	+	0.973191i	$0.426128\pi$
$878$	−4.14070e6	−0.181275
$879$	6.98711e6	0.305018
$880$	614400.	0.0267451
$881$	2.75883e7	1.19753	0.598763	−	0.800926i	$-0.295659\pi$
$881$	2.75883e7	1.19753	0.598763	+	0.800926i	$0.295659\pi$
$882$	5.44028e6	0.235478
$883$	3.09300e6	0.133499	0.0667494	−	0.997770i	$-0.478737\pi$
$883$	3.09300e6	0.133499	0.0667494	+	0.997770i	$0.478737\pi$
$884$	−3.19613e6	−0.137560
$885$	−9.53640e6	−0.409285
$886$	5.65598e6	0.242060
$887$	3.59071e7	1.53240	0.766199	−	0.642603i	$-0.222146\pi$
$887$	3.59071e7	1.53240	0.766199	+	0.642603i	$0.222146\pi$
$888$	5.91091e6	0.251549
$889$	−500480.	−0.0212389
$890$	1.32486e7	0.560654
$891$	629856.	0.0265795
$892$	2.02682e7	0.852911
$893$	−3.79834e6	−0.159391
$894$	−7.23838e6	−0.302899
$895$	−1.01283e7	−0.422648
$896$	65536.0	0.00272716
$897$	−2.73780e6	−0.113611
$898$	1.06918e6	0.0442444
$899$	−1.67042e6	−0.0689330
$900$	810000.	0.0333333
$901$	−6.70194e6	−0.275035
$902$	5.49504e6	0.224882
$903$	466416.	0.0190351
$904$	1.60316e7	0.652464
$905$	−3.78625e6	−0.153670
$906$	−9.49205e6	−0.384184
$907$	2.74801e7	1.10917	0.554587	−	0.832126i	$-0.312876\pi$
$907$	2.74801e7	1.10917	0.554587	+	0.832126i	$0.312876\pi$
$908$	−6.09408e6	−0.245298
$909$	−5.67988e6	−0.227997
$910$	67600.0	0.00270610
$911$	9.21756e6	0.367976	0.183988	−	0.982928i	$-0.441099\pi$
$911$	9.21756e6	0.367976	0.183988	+	0.982928i	$0.441099\pi$
$912$	2.69107e6	0.107137
$913$	−6.42355e6	−0.255034
$914$	1.25696e7	0.497685
$915$	−6.67755e6	−0.263672
$916$	1.84995e6	0.0728487
$917$	402384.	0.0158022
$918$	−3.44671e6	−0.134989
$919$	−1.39774e7	−0.545930	−0.272965	−	0.962024i	$-0.588004\pi$
$919$	−1.39774e7	−0.545930	−0.272965	+	0.962024i	$0.588004\pi$
$920$	−2.88000e6	−0.112182
$921$	−2.12005e7	−0.823562
$922$	2.78335e7	1.07830
$923$	630708.	0.0243682
$924$	55296.0	0.00213066
$925$	6.41375e6	0.246466
$926$	1.59916e7	0.612864
$927$	−1.99390e6	−0.0762085
$928$	−374784.	−0.0142860
$929$	1.43621e7	0.545984	0.272992	−	0.962016i	$-0.411987\pi$
$929$	1.43621e7	0.545984	0.272992	+	0.962016i	$0.411987\pi$
$930$	−4.10760e6	−0.155733
$931$	1.96119e7	0.741558
$932$	−1.38156e7	−0.520993
$933$	1.61233e7	0.606388
$934$	1.69129e7	0.634384
$935$	−2.83680e6	−0.106121
$936$	−876096.	−0.0326860
$937$	−3.10118e7	−1.15392	−0.576962	−	0.816771i	$-0.695762\pi$
$937$	−3.10118e7	−1.15392	−0.576962	+	0.816771i	$0.695762\pi$
$938$	−358144.	−0.0132908
$939$	−6.39167e6	−0.236565
$940$	1.30080e6	0.0480165
$941$	1.78804e7	0.658269	0.329135	−	0.944283i	$-0.393243\pi$
$941$	1.78804e7	0.658269	0.329135	+	0.944283i	$0.393243\pi$
$942$	7.57606e6	0.278174
$943$	−2.57580e7	−0.943263
$944$	1.08503e7	0.396289
$945$	72900.0	0.00265551
$946$	−4.97510e6	−0.180748
$947$	−1.83420e7	−0.664617	−0.332308	−	0.943171i	$-0.607827\pi$
$947$	−1.83420e7	−0.664617	−0.332308	+	0.943171i	$0.607827\pi$
$948$	8.46605e6	0.305957
$949$	6.36420e6	0.229392
$950$	2.92000e6	0.104972
$951$	−1.17667e7	−0.421892
$952$	−302592.	−0.0108209
$953$	−8.87079e6	−0.316395	−0.158198	−	0.987407i	$-0.550568\pi$
$953$	−8.87079e6	−0.316395	−0.158198	+	0.987407i	$0.550568\pi$
$954$	−1.83708e6	−0.0653517
$955$	−1.38402e7	−0.491059
$956$	−1.72214e7	−0.609431
$957$	−316224.	−0.0111613
$958$	−1.14215e7	−0.402076
$959$	1.57788e6	0.0554023
$960$	−921600.	−0.0322749
$961$	−7.79906e6	−0.272417
$962$	−6.93711e6	−0.241680
$963$	−5.40529e6	−0.187825
$964$	−1.95947e7	−0.679120
$965$	2.04468e7	0.706819
$966$	−259200.	−0.00893701
$967$	4.18026e7	1.43760	0.718798	−	0.695219i	$-0.244692\pi$
$967$	4.18026e7	1.43760	0.718798	+	0.695219i	$0.244692\pi$
$968$	9.71744e6	0.333322
$969$	−1.24252e7	−0.425102
$970$	1.29100e6	0.0440552
$971$	−5.56803e7	−1.89519	−0.947597	−	0.319467i	$-0.896496\pi$
$971$	−5.56803e7	−1.89519	−0.947597	+	0.319467i	$0.896496\pi$
$972$	−944784.	−0.0320750
$973$	432112.	0.0146324
$974$	−94544.0	−0.00319328
$975$	−950625.	−0.0320256
$976$	7.59757e6	0.255300
$977$	−4.82682e7	−1.61780	−0.808900	−	0.587946i	$-0.799937\pi$
$977$	−4.82682e7	−1.61780	−0.808900	+	0.587946i	$0.799937\pi$
$978$	−1.09881e7	−0.367345
$979$	−1.27187e7	−0.424116
$980$	−6.71640e6	−0.223394
$981$	673758.	0.0223528
$982$	1.63943e7	0.542517
$983$	1.23503e7	0.407656	0.203828	−	0.979007i	$-0.434662\pi$
$983$	1.23503e7	0.407656	0.203828	+	0.979007i	$0.434662\pi$
$984$	−8.24256e6	−0.271378
$985$	−1.99414e7	−0.654887
$986$	1.73045e6	0.0566848
$987$	117072.	0.00382525
$988$	−3.15827e6	−0.102934
$989$	2.33208e7	0.758146
$990$	−777600.	−0.0252155
$991$	1.56614e7	0.506577	0.253288	−	0.967391i	$-0.418488\pi$
$991$	1.56614e7	0.506577	0.253288	+	0.967391i	$0.418488\pi$
$992$	4.67354e6	0.150788
$993$	7.18409e6	0.231206
$994$	59712.0	0.00191688
$995$	1.16798e7	0.374005
$996$	9.63533e6	0.307764
$997$	−4.30263e6	−0.137087	−0.0685435	−	0.997648i	$-0.521835\pi$
$997$	−4.30263e6	−0.137087	−0.0685435	+	0.997648i	$0.521835\pi$
$998$	−1.99270e7	−0.633310
$999$	−7.48100e6	−0.237162

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	390.6.a.a.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.6.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:	\( N \)	\(=\)	\( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	390.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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